{"text": "\\input{../../style/preamble} \n\\input{../../latex-math/basic-math}\n\\input{../../latex-math/basic-ml}\n\\input{../../latex-math/ml-bagging.tex}\n\\input{../../latex-math/ml-boosting.tex}\n\\input{../../latex-math/ml-trees.tex}\n\n\\newcommand{\\titlefigure}{figure/fig-gb-concept-2.png}\n\\newcommand{\\learninggoals}{\n \\item Understand idea of forward stagewise modelling\n \\item Understand fitting process of gradient boosting for regression problems\n}\n\n\\title{Introduction to Machine Learning}\n\\date{}\n\n\\begin{document}\n\n\\lecturechapter{Gradient Boosting}\n\\lecture{Introduction to Machine Learning}\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% \\section{Gradient Boosting}\n\n% \\begin{vbframe}{Gradient descent}\n% % \\begin{vbframe}{Forward stagewise additive modeling}\n% \n% % Let's recall gradient descent from numerical optimization.\n% Let $\\risk(\\theta)$ be (just for the next few slides) an arbitrary, differentiable, unconstrained objective function, which we want to minimize. The gradient $\\nabla \\risk(\\theta)$ is the direction of the steepest ascent, $-\\nabla \\risk(\\theta)$ of \\textbf{steepest descent}.\n% \n% \\lz\n% \n% For an intermediate solution $\\theta^{[k]}$ during minimization, we can iteratively improve by updating\n% $$\n% \\theta^{[k+1]} = \\theta^{[k]} - \\beta \\nabla \\risk(\\theta^{[k]}) \\qquad \\text{for } 0 < \\beta \\leq c\\left(\\theta^{[k]}\\right)\n% $$\n% % \\enquote{Walking down the hill, towards the valley.}\n% \n% % $f(x_1, x_2) = -\\sin(0.8 x_1) \\cdot \\frac{1}{2\\pi} \\exp\\left( (x_2 x_1 + \\pi / 2)^2 \\right)$\n% <>=\n% #modified from ../../../cim1/2017/11-Optimierung/functions.\n% \n% sd_plot = function(col = terrain_hcl, theta = 40, phi = 40, xlab = \"x\", ylab = \"y\") {\n% if (is.function(col)) col = col(nrow(z) * ncol(z))\n% \n% par(mfrow = c(1, 2))\n% \n% par(mar = rep(0.5, 4))\n% require(\"colorspace\")\n% pmat = persp2(x, y, z, theta = theta, phi = phi, ticktype = \"detailed\",\n% xlab = xlab, ylab = ylab, zlab = \"\", col = col, lwd = 0.3, border = NA)\n% \n% for (j in seq_along(p)) {\n% t3d = trans3d(p[[j]][[1]], p[[j]][[2]], do.call(foo, p[[j]]), pmat)\n% if (j > 1) {\n% t3d2 = trans3d(p[[j - 1]][[1]], p[[j - 1]][[2]], do.call(foo, p[[j- 1]]), pmat)\n% lines(c(t3d$x, t3d2$x), c(t3d$y, t3d2$y))\n% points(x = t3d2$x, y = t3d2$y, pch = 16, col = heat_hcl(1))\n% }\n% points(x = t3d$x, y = t3d$y, pch = 16, col = heat_hcl(1))\n% }\n% par(mar = c(4.1, 4.1, 1.1, 1.1))\n% image(x, y, z, col = col, xlab = xlab, ylab = ylab, useRaster = TRUE)\n% contour(x, y, z, add = TRUE, nlevels = 15)\n% for (j in seq_along(p)) {\n% if (j > 1) {\n% lines(c(p[[j]][1], p[[j - 1]][1]), c(p[[j]][2], p[[j - 1]][2]))\n% points(p[[j - 1]][1], p[[j - 1]][2], pch = 16, col = heat_hcl(1))\n% }\n% points(p[[j]][1], p[[j]][2], pch = 16, col = heat_hcl(1))\n% }\n% invisible(NULL)\n% }\n% \n% @\n% <>=\n% foo = function(x, y) {\n% -1 * sin(.8 * pi*x) * dnorm(-y * x, mean = pi / 2, sd = 0.8)\n% }\n% \n% x = seq(0, 2.5, length = 50)\n% y = seq(-3, 1, length = 50)\n% z = outer(x, y, foo)\n% p = c(list(list(1.8, -.5)), optim0(1.8, -.5, FUN = foo, maximum = FALSE, maxit = 19))\n% \n% sd_plot(phi = 35, theta = -20, xlab = \"x_1\", ylab = \"x_2\", col = viridis::viridis)\n% @\n% \n% % {\\scriptsize step size $\\beta = 1$}\n% \n% % \\framebreak\n% \n% % $\\beta$ is called \\textbf{step size}, and can be set by\n% % \\begin{itemize}\n% % \\item fixing it to a (smallish) constant\n% % \\item adapting it according to previous gradient values, the local Hessian, etc.\n% % \\item line search methods, which solve $\\beta^{[k]} = \\argmin_{\\beta} f\\left(x^{[k]} - \\beta \\nabla f\\left(x^{[k]}\\right)\\right)$. Only one real parameter $\\beta$, i.e, \\enquote{easy} to solve\\dots\n% % \\end{itemize}\n% \n% \n% \n% \n% \\end{vbframe}\n\n\n\\begin{vbframe}{Forward stagewise additive modeling}\n\nAssume a regression problem for now (as this is simpler to explain);\nand assume a space of base learners $\\mathcal{B}$.\n\n\\lz\n\n% A weak learner should have the property that it delivers better predictions than by random chance (e.g. for a balanced training set a misclassification error less than 1/2).\n\n% \\lz\n\nWe want to learn an additive model:\n\n$$\n\\fx = \\sum_{m=1}^M \\betam \\bmmxth.\n$$\n\nHence, we minimize the empirical risk:\n\n$$\n\\riskef = \\sum_{i=1}^n L\\left(\\yi,\\fxi \\right) =\n\\sum_{i=1}^n L\\left(\\yi, \\sum_{m=1}^M \\betam \\bmmxth\\right)\n$$\n\n\n% \\framebreak\n\n% A common \\textbf{loss} for \\textbf{regression} is the \\textbf{squared error} with\n% $\\Lxy = (y-\\fx)^2$.\n\n% \\lz\n\n% Apparently, $\\risk$ depends on the \\textbf{base learners} $b(x, \\thetam)$,\n% or rather their parameters $\\thetam$ and weights $\\betam$. Hence, we have to optimize these.\n\n% \\lz\n\n\\framebreak\n\\textbf{Why is gradient boosting a good choice for this problem?}\n\\begin{itemize}\n\\item Because of the additive structure it is difficult to jointly minimize $\\riskef$ w.r.t. $\\left(\\left(\\beta^{[1]}, \\bm{\\theta}^{[1]}\\right), \\ldots, \\left(\\beta^{[M]}, \\bm{\\theta}^{[M]}\\right)\\right)$, which is a very high-dimensional parameter space (though this is less of a problem nowadays, especially in the\ncase of numeric parameter spaces).\n% - however, this is nowadays especially in the case of numeric parameter spaces not a real problem anymore.\n\\item Considering trees as base learners is worse as we would have to grow $M$ trees in parallel so they\n work optimally together as an ensemble.\n\\item Stagewise additive modeling has nice properties, which we want to make use of, e.g. for regularization, early stopping, \\dots\n\\end{itemize}\n\n\\framebreak\n\nHence, we add additive components in a greedy fashion by sequentially minimizing the risk only w.r.t. the next additive component:\n\n$$ \\min \\limits_{\\beta, \\bm{\\theta}} \\sum_{i=1}^n L\\left(\\yi, \\fmdh\\left(\\xi\\right) + \\beta b\\left(\\xi, \\bm{\\theta}\\right)\\right) $$\n\n\\lz\n\nDoing this iteratively is called \\textbf{forward stagewise additive modeling}.\n\n\\input{algorithms/forward_stagewise_additive_modeling.tex}\n\n\\end{vbframe}\n\n\n% \\section{Gradient Boosting}\n\n\\begin{vbframe}{Gradient boosting}\n\n% \\begin{footnotesize}\nThis is not really an algorithm, but an abstract principle.\nTo find $b\\left(\\xv, \\thetam\\right)$ and $\\betam$, we use gradient descent, but in function space!\n\n\\lz\n\\begin{columns}\n\\column{5cm}\n% \\textbf{Thought experiment:}\nConsider a model $f$ whose predictions we can arbitrarily define for each training $\\xi$, i.e.,\n$f$ is a finite vector\n $$\\left(f\\left(\\xv^{(1)}\\right), \\ldots, f\\left(\\xv^{(n)}\\right)\\right)^\\top $$\n\nThis implies $n$ parameters $\\fxi$ (and the model would provide no generalization...).\n\nAlso, we let's assume $L$ to be differentiable.\n\n\\column{5cm}\n\\begin{center}\n \\vspace{-1cm}\n \\includegraphics[width=\\textwidth]{figure/fig-gb-concept-2.png}\n\\end{center}\n\n\\end{columns}\n% \\end{footnotesize}\n\\end{vbframe}\n\n\\begin{vbframe}{Gradient boosting}\n\n\\textbf{Aim:} Define a movement in function space so we can push our current function towards the data points.\n\n\\vspace*{0.1cm}\n\\textbf{Given:} Regression problem with one feature $x$ and target variable $y$.\n\n\\vspace*{0.1cm}\n\\textbf{Initialization:} Set all parameters to the optimal constant value (e.g., the mean of $y$ for $L2$).\n\n\\begin{figure}\n \\includegraphics[width=\\textwidth]{figure/fig-gb-concept-1.png}\n\\end{figure}\n\n\n\n\\end{vbframe}\n\n\\begin{frame}{Pseudo Residuals}\n% \\begin{footnotesize}\nHow do we distort our $f$ to move it towards the labels and reduce risk?\n\\vspace*{0.2cm}\nLet's minimize risk with GD.\n\nSo, we calculate the (negative) gradient of the risk for each parameter, which (weirdly) here are the outputs $\\fxi$ \n(0 if $i \\neq j$):\n% \\footnote{The gradient of all terms with $i \\neq j$ are 0.}\n\n$$\n\\tilde{r}^{(i)} = - \\fp{\\riske}{\\fxi} = - \\fp{\\sum_j L(y^{(j)}, f(\\xv^{(j)}))}{\\fxi} = - \\fp{\\Lxyi}{\\fxi}.\n$$\n\n\\vspace*{0.1cm}\n\nAt each point, we would like to change the output via:\n$ \\tilde{r}(f) = - \\fp{L(y, f)}{f} .$\n\n\\begin{columns}\n\\begin{column}{0.45\\textwidth}\n\\begin{center}\n \\textbf{L2 Example:} The PRs \n match the usual residuals:\n $$ \\tilde{r}(f) = - \\fp{0.5(y - f)^2}{f} = y - f $$\n\\end{center}\n\\end{column}\n\\begin{column}{0.55\\textwidth}\n\\begin{center}\n \\includegraphics[width=0.7\\textwidth]{figure/pseudo_residual_1.png}\n\\end{center}\n\\end{column}\n\\end{columns}\n\n% \\end{footnotesize}\n\n\\end{frame}\n\n\\begin{vbframe}{Boosting as Gradient Descent}\n\n%\\begin{footnotesize}\n\\vspace*{0.2cm}\nCombining this with \\enquote{forward stagewise modeling}, we are at $\\fmd$ during minimization.\nHere, we calculate the direction of the negative gradient or vector of PRs:\n\n$$ \\rmi = -\\left[\\fp{\\Lxyi}{f(\\xi)}\\right]_{f=\\fmd} $$\n\n\\lz\n\nThe gradient descent update for each vector component of $f$ is:\n\n$$\n \\fm (\\xi) = \\fmd (\\xi) + \\beta \\rmi.\n$$\n\nLike this, we should \\enquote{nudge} $f$ in the direction best risk reduction.\n\n%\\end{footnotesize}\n\\end{vbframe}\n\n\n\\begin{vbframe}{Gradient boosting}\n\n\\textbf{Iteration 1:}\n\nLet's move our function $\\fxi$ a fraction towards the pseudo-residuals with a learning rate of $\\beta = 0.6$.\n\n\n\\begin{figure}\n \\includegraphics[width=\\textwidth]{figure/fig-gb-concept-pseudo-resi-1.png}\n\\end{figure}\n\n\\framebreak\n\n\\textbf{Iteration 2:}\n\nLet's move our function $\\fxi$ a fraction towards the pseudo-residuals with a learning rate of $\\beta = 0.6$.\n\n\n\\begin{figure}\n \\includegraphics[width=\\textwidth]{figure/fig-gb-concept-pseudo-resi-2.png}\n\\end{figure}\n\n\\framebreak\n% \\begin{footnotesize}\n% We find our $\\betam$ by minimizing with line search:\n\n% $$\n % \\betam = \\argmin_{\\beta} \\sumin L(\\yi, \\fmdh(x) + \\beta b(x, \\thetamh)),\n% $$\n\n% where $h(x, \\thetam) = \\rmm$.\n\n% \\lz\n\n%What is the point of doing all this? \nAs said, such a model parameterization is pointless.\n\n\\vspace*{0.3cm}\n\n\nSo, we restrict our additive components to $b\\left(\\xv, \\thetam\\right) \\in \\mathcal{B}$.\n\n% \\framebreak\n\nThe pseudo-residuals are calculated exactly as stated above,\nthen we fit a simple model $b(\\xv, \\thetam)$ to them:\n$$ \\thetamh = \\argmin_{\\bm{\\theta}} \\sum_{i=1}^n \\left(\\rmi - b(\\xi, \\bm{\\theta})\\right)^2. $$\n\n\\lz\n\n\\begin{columns}\n\\column{5cm}\nSo, evaluated on the training data,\n$b(\\xv, \\thetam)$ corresponds as closely as possible to the negative risk gradient and generalizes over $\\Xspace$.\n\n\n\\column{5cm}\n\\vspace*{-1cm}\n\\begin{figure}[th]\n \\includegraphics[width=\\textwidth]{figure/fig-gb-concept-idea.png}\n\\end{figure}\n\n\n\\end{columns}\n% \\end{footnotesize}\n\n\\framebreak\n% \\begin{footnotesize}\n\\textbf{In a nutshell}: One boosting iteration is exactly one approximated gradient descent step in function space,\nwhich minimizes the empirical risk as much as possible.\n\n\n\\vspace*{0.1cm}\n\\textbf{Iteration 1:}\n\\begin{figure}\n \\includegraphics[width=0.9\\textwidth]{figure/fig-gb-concept-idea-1.png}\n\\end{figure}\n% \\end{footnotesize}\n\\framebreak\n% \\begin{footnotesize}\nInstead of moving the function values for each observation by a fraction closer to the observed data, we fit a regression base learner to the pseudo-residuals (right plot). \n\n\n\\vspace*{0.1cm}\n\\textbf{Iteration 2:}\n\\begin{figure}\n \\includegraphics[width=0.9\\textwidth]{figure/fig-gb-concept-idea-2.png}\n\\end{figure}\n% \\end{footnotesize}\n\\framebreak\n% \\begin{footnotesize}\nThis BL is added to the of the ensemble, weighted by a learning rate (here: $\\beta = 0.4$).\nThen we iterate.\n\n\n\\vspace*{0.1cm}\n\\textbf{Iteration 3:}\n\\begin{figure}\n \\includegraphics[width=\\textwidth]{figure/fig-gb-concept-idea-3.png}\n\\end{figure}\n\n% \\begin{footnotesize}\n% This procedure is continued stepwise until the boosting algorithm terminates.\n% \\end{footnotesize}\n% \\end{footnotesize}\n\\end{vbframe}\n%% Combining this with the iterative additive procedure\n%% of \\enquote{forward stagewise modelling}, we are at the spot $\\fmd$ during minimization.\n%% At this point, we now calculate the direction of the negative gradient:\n%%\n%% $$ \\rmi = -\\left[\\fp{\\Lxyi}{f(\\xi)}\\right]_{f=\\fmd} $$\n%%\n%% We will call these $\\rmi$ \\textbf{pseudo residuals}. For squared loss they match the usual residuals\n%%\n%%\n%% $$\n%% - \\fp{\\Lxy}{\\fx} = - \\fp{0.5(y - \\fx)^2}{\\fx} = y - \\fx\n%% $$\n%%\n%%\n%% \\framebreak\n%%\n%% % We find our $\\betam$ by minimizing with line search:\n%%\n%% % $$\n%% % \\betam = \\argmin_{\\beta} \\sumin L(\\yi, \\fmdh(x) + \\beta b(x, \\thetamh)),\n%% % $$\n%%\n%% % where $h(x, \\thetam) = \\rmm$.\n%%\n%% % \\lz\n%%\n%% What is the point in doing all this? A model parameterized in this way is senseless,\n%% as it is just memorizing the instances of the training data...?\n%%\n%% \\lz\n%%\n%% So, we restrict our additive components to $b\\left(x, \\thetam\\right) \\in \\mathcal{B}$.\n%%\n%% % \\framebreak\n%%\n%% The pseudo-residuals are calculated exactly as stated above,\n%% then we fit a regression model $b(\\bm{x}, \\thetam)$ to them:\n%% $$ \\thetamh = \\argmin_{\\thetab} \\sum_{i=1}^n (\\rmi - b(\\xi, \\thetab))^2 $$\n%% So, evaluated on the training data,\n%% our $b(x, \\thetam)$ corresponds as closely as possible to the negative\n%% loss function gradient and generalizes to the whole space.\n%%\n%% \\lz\n%%\n%% \\textbf{In a nutshell}: One boosting iteration is exactly one approximated gradient step in function space,\n%% which minimizes the empirical risk as much as possible.\n%%\n%% \\end{vbframe}\n\n\\begin{vbframe}{Gradient boosting algorithm}\n\n\\input{algorithms/gradient_boosting_general.tex}\n\nNote that we also initialize the model in a loss-optimal manner. %Also, the constant learning rate can be replaced by a line-search, however a small constant learning rate is commonly used.\n\n\\end{vbframe}\n\n\\begin{vbframe}{Line Search}\nThe learning rate, as always, influences how we converge. \nAlthough a small constant LR is commonly used, we can also run line search,\n% to solve this one-dimensional optimization problem:\n$$\\betamh = \\argmin_{\\beta} \\sumin L(\\yi, \\fmd(\\xv) + \\beta b(\\xv, \\thetamh))$$\n\nAlternatively, an (inexact) backtracking line search can be used to find the \n$\\betam$ that minimizes the above equation.\n\n\\end{vbframe}\n\n\\endlecture\n\\end{document}\n\n", "meta": {"hexsha": "8d435864226dd68e124870dbc2fc128c35c41ef8", "size": 14022, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "slides/boosting/slides-boosting-gradient-boosting-concept.tex", "max_stars_repo_name": "jukaje/lecture_i2ml", "max_stars_repo_head_hexsha": "cd4900f5190e9d319867b4c0eb9d8e19f659fb62", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "slides/boosting/slides-boosting-gradient-boosting-concept.tex", "max_issues_repo_name": "jukaje/lecture_i2ml", "max_issues_repo_head_hexsha": "cd4900f5190e9d319867b4c0eb9d8e19f659fb62", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "slides/boosting/slides-boosting-gradient-boosting-concept.tex", "max_forks_repo_name": "jukaje/lecture_i2ml", "max_forks_repo_head_hexsha": "cd4900f5190e9d319867b4c0eb9d8e19f659fb62", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.6827133479, "max_line_length": 316, "alphanum_fraction": 0.6676650977, "num_tokens": 4614, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765155565327, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.3499320178907083}} {"text": "\\section{Introduction}\n\tIn plasticity model, plastic deformation depends on the internal state variables which are measurable variables like strain rate, temperature, stress and material variables. The relation between all the internal state variables is expressed using flow rule.\n\t\nIn general, flow rule is the resistance to the plasticity and expressed as the ratio of the plastic driving force to the flow resistance. During plastic deformation, the driving forces direct the internal slip mechanisms which causes shearing. A general power law flow rule representation equ.(\\ref{power_law_equ})$^{\\cite{chowdhury2017}}$ \\par\n\n\\begin{equation}\\label{power_law_equ}\n\\dot{\\gamma}^{(\\alpha)}=\\dot{\\gamma}_{0}\\left|\\frac{\\tau^{\\alpha}}{g^{\\alpha}}\\right|^{n} \\operatorname{sgn}\\left(\\tau^{\\alpha}\\right)\n\\end{equation}\n\nIn equ.(\\ref{power_law_equ}) $\\dot{\\gamma}^{(\\alpha)}$ is slip rate of the slip system $\\alpha$, $\\tau^{\\alpha}$ is the shear stress, $g^{\\alpha}$ is the internal state variable which depends on temperature.\n\nPlasticity is mediated through dislocation. The basic types of dislocations are edge, screw and mixed dislocation. During plastic deformation, dislocation motion will occur in multiple slip system simultaneously. This lead to the interaction of dislocation with dislocations of other slip systems. This collective dislocation dynamics plays an important role in understanding the material response like fatigue, creep, yield.$^{\\cite{Messerschmidt2010}}$ \n\nThe entanglement of different dislocation generate a complex stress field. These complex stress field creates a low energy barriers which act as obstacles for dislocation movement having long range interactions. This complex system of interaction leads to the formation of dislocation avalanches. Dislocation avalanches are the basic mechanism of plastic flow in solids at the nano-scale. Dislocation avalanches are also associated with strain bursts which is a sudden increase in strain rate.\nThey follow power law distribution.\n$^{\\cite{Papanikolaou_2017}}$\n\nA comprehensive overview on dislocation avalanche physics and the phenomena leading to strain bursts in crystal plasticity is discussed further.\n\n\n\n\n", "meta": {"hexsha": "8dc42bbccbc717cbbeb80b059360384261f84cc0", "size": 2195, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Document/Introduction.tex", "max_stars_repo_name": "viswambhar-yasa/IGTO", "max_stars_repo_head_hexsha": "03001db03b17d1d2120740b9d3d7f0efe1f93faf", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2020-11-23T15:55:08.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-15T17:47:19.000Z", "max_issues_repo_path": "Document/Introduction.tex", "max_issues_repo_name": "bhan5426/IGTO", "max_issues_repo_head_hexsha": "03001db03b17d1d2120740b9d3d7f0efe1f93faf", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Document/Introduction.tex", "max_forks_repo_name": "bhan5426/IGTO", "max_forks_repo_head_hexsha": "03001db03b17d1d2120740b9d3d7f0efe1f93faf", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-12-27T01:21:33.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-24T10:50:31.000Z", "avg_line_length": 95.4347826087, "max_line_length": 493, "alphanum_fraction": 0.8, "num_tokens": 476, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3499320008841906}} {"text": " \n%% \\listfiles\n\\documentclass[apj]{emulateapj}\n%\\documentclass[preprint2,12pt]{emulateapj}\n%% \\usepackage{natbib}\n\\usepackage{graphicx}\n\\usepackage{epsfig}\n\\usepackage{amssymb,amsmath}\n\\usepackage{array}\n\\usepackage{threeparttable}\n\\usepackage{hyperref,graphicx} \n\n\\doublespace\n\n%definitions\n\\newcommand{\\Msol}{${\\rm M_{\\sun}}$}\n\n\n%% Editing markup...\n\\usepackage{color}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% WARNING: This LaTeX block was generated automatically by authors.py\n% Do not change by hand: your changes will be lost.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n% --------------------- Ancillary information ---------------------\n\\shortauthors{SURP et al.}\n\\shorttitle{CITA Final Project}\n\n\n\n\n\\begin{document}\n\n\\title{CITA Final Project: Introduction to Galpy}\n %% ---------\n \n\\author{Yumna Arshad}\n%\\altaffiltext{1}{CITA, University of Toronto}\n \n\\section{Introduction}\nMy final project aims to introduce the methods of galpy, a module in python used for modelling the orbital dynamics of objects in the galaxy. \nThe project consists of 2 parts. \nThe first part is an introductory approach to galpy and focuses on specifically learning how to use both potential and orbit objects from the module as well as the various methods associated with them.\nThe second part consists of modelling the accretion of a globular cluster, that is initially bound to a satellite galaxy of the Milky Way, onto the Milky Way galaxy itself. \n\n\n\\section{Part 1: Intro to Galpy Methods}\n\nFor this part of the project there are 3 separate tasks: 1) plotting rotation curve of MW galaxy, 2) integrating orbit of Sun and 3) integrating orbit of a specific tidal stream named GD -1.\n\n\n\\subsection{Rotation Curve of MW Galaxy}\nTo plot the rotation curve of the MW galaxy I created 3 different potential instances corresponding to the potential of the disk, bulge and halo separately and plotted them on one plot using the plotRotcurve method. Then I summed up all the potentials to create the overall galactic rotation curve which peaks at around 225 km/s and slowly declines afterward, this is the sum of the bulge, disk and halo potentials.\nA plot of these rotation curves can be seen in Fig.\\ref{fig:Q1_rot_curve}.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q1a.pdf}\n \\caption{Plot of the rotation curve of the Milky Way Galaxy including its labelled components: bulge, halo and disk.}\n \\label{fig:Q1_rot_curve}\n\\end{figure}\n\n\\subsection{Orbit of Sun in MW Galaxy}\nTo plot the orbit of the Sun in the MW Galaxy I initialized an Orbit instance and integrated it for 10 Gyr using the MW potential in part 1 to properly model the orbit.\nI used the plot method of the sun orbit object to display the plot. \nA figure of this plot is shown in Fig.\\ref{fig:Q1_Sun}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q1b.pdf}\n \\caption{Plot of the orbit of the Sun in Milky Way potential for 10 Gyr (integrated forward). The orbit is plotted as: vertical height above plane of galaxy in kpc as a function of galactocentric radius in kpc.}\n \\label{fig:Q1_Sun}\n\\end{figure}\n\n\n\n\\subsection{GD-1 Tidal Stream}\nThe path of GD-1 tidal stream in the sky is plotted by: initializing an orbit object with the given coordinates for the stream as specified in paper by Bovy and Webb \\cite{GD-1}.\nFigures showing the GD-1 stream in declination vs right ascension are shown in Fig.\\ref{fig:Q1_radec_f} (integrated forwards) and in Fig.\\ref{fig:Q1_radec_b} (integrated backwards).\nFigures showing the GD-1 stream in distance (from the Sun) vs right ascension are shown in Fig.\\ref{fig:Q1_radist_f} (integrated forwards) and in Fig.\\ref{fig:Q1_radist_b} (integrated backwards).\nFor all 4 orbits, the integration period is 50 Myr either forwards or backwards in time.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q1c_1.pdf}\n \\caption{Tidal stream GD-1's declination vs right ascension integrated forwards in time.}\n \\label{fig:Q1_radec_f}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q1c_3.pdf}\n \\caption{Tidal stream GD-1's declination vs right ascension integrated backwards in time.}\n \\label{fig:Q1_radec_b}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q1c_2.pdf}\n \\caption{Tidal stream GD-1's distance (from Sun) vs right ascension integrated forwards in time.}\n \\label{fig:Q1_radist_f}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q1c_4.pdf}\n \\caption{Tidal stream GD-1's distance (from Sun) vs right ascension integrated backwards in time.}\n \\label{fig:Q1_radist_b}\n\\end{figure}\n\n\n\\section{Part 2: Stimulating Accretion of Globular Cluster onto Milky Way}\n\n\\subsection{Method}\nTo stimulate the accretion of a globular cluster onto the Milky Way Galaxy: the first thing I did was import and integrate the orbits of all the satellite galaxies of the Milky Way and plotted their apocenters and pericenters vs their current galactocentric radii. \nUsing these plots, the satellite with the smallest pericenter is identified to be TucanaIII and the time at which it 1st reaches its smallest galactocentric radius is -1.77 Gyr (i.e 1.77 Gyr in the past). This satellite will be used to model the accretion of the globular cluster. \nTo begin with, the orbit of TucanaIII integrated backwards for 10 Gyr is plotted with the effects of dynamical friction included. Dynamical friction is implemented using an instance of the ChandrasekharDynamicalFrictionForce with the density of the MW potential (imported using MWPotential2014 which is a list of potential objects making up the Milky Way) and given satellite parameters such as mass and size. Then, the orbit of the satellite is integrated back in time using the combined potential of the MW and the dynamical friction. This orbit is shown in Fig.\\ref{fig:Q2_cdf}. The radius of the satellite at 10 Gyr is determined to be 240.6 kpc.\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q2c.pdf}\n \\caption{Plot showing the orbit of TucanaIII satellite galaxy, the satellite with the smallest pericenter radius, integrated backwards in time for 10 Gyr. The orbit is shown in radius vs time.}\n \\label{fig:Q2_cdf}\n\\end{figure}\n\nA star-cluster is then initialized to be on a circular orbit of radius 4 kpc within the satellite galaxy, the orbital velocity can be determined using Eq.\\ref{equation: v_circ} and this gives a velocity of $\\approx$ 328 km/s.\n\n\\begin{equation}\n v = \\sqrt{\\frac{GM}{r}}\n \\label{equation: v_circ}\n\\end{equation}\n\nAs well, the satellite + cluster system are moved to the radius determined above (i.e. 240.6 kpc). To model the orbit of the satellite characerized by a Hernquist potential, a MovingObjectPotential object instance is created with the associated orbit being that of the satellite as previously determined, i.e. orbit of TucanaIII. \nFinally, the orbits of both the satellite and the star-cluster are integrated and plotted. The results are shown in Fig.\\ref{fig:Q2_satellite} and Fig.\\ref{fig:Q2_star_cluster}, respectively. \n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q2_sat.pdf}\n \\caption{Plot showing the orbit of the satellite galaxy integrated forwards in time for 12 Gyr. The orbit is shown in radius vs time and includes the effects of dynamical friction.}\n \\label{fig:Q2_satellite}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=1.0\\columnwidth]{Q2_sc.pdf}\n \\caption{Plot showing the orbit of the star-cluster system within the satellite galaxy for 12 Gyr. Note that r is the galactocentric distance, plotted as a function of time.}\n \\label{fig:Q2_star_cluster}\n\\end{figure}\n\n\n\n\n\\subsection{Comment on Orbits of Star Cluster and Satellite Galaxy}\nThe orbit of the satellite galaxy as shown in Fig.\\ref{fig:Q2_satellite} is the same as the orbit in Fig.\\ref{fig:Q2_cdf} since it initially starts at its position at -10 Gyr and follows the same path forward in time as it did to go back.\nThe star cluster's orbit within the satellite follows a bound orbit such that it becomes accreted onto the Milky Way galaxy as the satellite spirals in and remains bound to the MW galaxy.\n\n\n\\begin{thebibliography}{99}\n\n\\bibitem{GD-1}\nJ. J. Webb and J. Bovy, “Searching for the GD-1 stream progenitor inGaiaDR2 with directN-body simulations,” Monthly Notices of the Royal Astronomical Society, vol. 485, no. 4, pp. 5929–5938, 2019.\n\n\n\n\\end{thebibliography}\n\n\n\\end{document}\n\n", "meta": {"hexsha": "116e3535f65e6c86fdc474a7382c45341a570e3f", "size": 8652, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Final Project/main.tex", "max_stars_repo_name": "yumnaarshad/CTA200", "max_stars_repo_head_hexsha": "abc1ed35e39365d9344342abfe201586428c1cac", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Final Project/main.tex", "max_issues_repo_name": "yumnaarshad/CTA200", "max_issues_repo_head_hexsha": "abc1ed35e39365d9344342abfe201586428c1cac", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Final Project/main.tex", "max_forks_repo_name": "yumnaarshad/CTA200", "max_forks_repo_head_hexsha": "abc1ed35e39365d9344342abfe201586428c1cac", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.1590909091, "max_line_length": 650, "alphanum_fraction": 0.753814147, "num_tokens": 2253, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621764862150634, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.349931999626837}} {"text": "\\documentclass[12pt,a4paper]{article}\n\n\\usepackage[utf8]{inputenc}\n\\usepackage[english]{babel}\n%\\usepackage[english]{isodate}\n%\\usepackage[parfill]{parskip}\n\n\\usepackage{graphicx}\n\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{newtxtext,newtxmath}\n\\usepackage{bm}\n\\usepackage{listings}\n\\usepackage{color}\n\\usepackage{url}\n\\usepackage{intmacros}\n\\usepackage{syntax}\n\\usepackage{manual}\n\n\\lstset{%\n% backgroundcolor=\\color{yellow!20},%\n basicstyle=\\ttfamily,%\n %numbers=left, numberstyle=\\tiny, stepnumber=2, numbersep=5pt,%\n numbers=left, numberstyle=\\tiny, numbersep=5pt,%\n\tframe=single,%\n }%\n\n\\lstset{emph={% \n let, var, init, at, wait, once, when, goto, then, end, prop, param%\n },emphstyle={\\color{blue}\\bfseries}%\n}\n\n\\newcommand{\\LNCS}[1]{\\textit{LNCS}~#1}\n\\newcommand{\\ENTCS}[1]{\\textit{ENTCS}~#1}\n\n\\begin{document}\n\n\\title{HySIA Manual (version~0.1.1)}\n\\author{Daisuke Ishii \\quad \\texttt{dsksh@acm.org}}\n\n\\maketitle\n\n\\setlength{\\grammarparsep}{8pt plus 1pt minus 1pt} % increase separation between rules\n\\setlength{\\grammarindent}{8em} % increase separation between LHS/RHS \n\n\\emph{HySIA} is a reliable simulator and verifier for hybrid systems.\n%\nHySIA supports nonlinear \\emph{hybrid automata} (HAs) whose ODEs, guards, and reset functions are specified with nonlinear expressions. It assumes a deterministic class of HA; a transition to another location happens whenever a guard condition holds.\n%\nMain functionalities of HySIA are as follows:\n\n\\textbf{Simulation}.\n%\nHySIA simulates an HA based on interval analysis;\nit computes an overapproximation of a bounded trajectory that is composed of \\emph{boxes} (i.e., closed interval vectors) and \\emph{parallelotopes} (linear transformation of boxes).\n%\nIntensive use of interval analysis techniques distinguishes HySIA from other reachability analysis tools.\n%\nFirst, the simulation process carefully reduces \\emph{wrapping effect} that can expand an enclosure interval. As a result, HySIA is able to simulate HA for more number of steps than other overapproximation-based tools; e.g., it can simulate a periodic bouncing ball for more than a thousand steps.\n%\nSecond, HySIA assures the soundness of each interval computation, so that the resulting overapproximation is verified to contain a theoretical trajectory.\n%This verification often fails, e.g., when an ODE is \\emph{stiff} or when a trajectory and a guard are close to tangent.\n%Thanks to this \\emph{quasi-complete} manner, the simulation process of HySIA is decidable and performs efficiently.\n%\nSee Reference~\\cite{GI2016} for details of the underlying method.\n\n\\textbf{Monitoring}.\n%\nHySIA takes a temporal property as an input and monitors whether a simulated trajectory satisfies the property; otherwise, HySIA computes (an interval overapproximation of) a robustness signal for the property.\nThe verification process is based on a monitoring procedure of the \\emph{signal temporal logic} (STL) formulas~\\cite{Maler2004}, which is extended to handle overapproximation of trajectories.\n%The soundness of interval computation is again utilized here to evaluate the logical negation against an approximated trajectory.\n%\nSee Reference~\\cite{Ishii2016IEICE} for more details.\n\n\n\\section{Short Tutorial}\n\nHySIA provides a specification language for hybrid automata and temporal logic properties.\nAn HA is simulated and analyzed about a property using the \\verb|hysia| command.\nThis section exemplifies the modeling, simulation, and verification of an HA using HySIA.\n\n\\subsection{Modeling Hybrid Automata}\n\nBelow is an example specification that describes a simple bouncing ball model (\\texttt{bb.ha}):\n\\begin{lstlisting}\n(* first part *)\nlet g = 1\nlet c = 0.9\nlet pertb = [-1e-5, 1e-5]\n\n(* second part *)\nvar y, vy\n\ninit Loc, 1+pertb, 0+pertb\n\nat Loc wait vy, -g\n once (y, -vy) goto Loc then y, -c*vy\nend\n\n(* third part *)\nparam order = 20\nparam t_max = 100\nparam dump_interval = 0.1\n\\end{lstlisting}\n%\nThe specification consists of three parts.\nThe first part (Lines~1--4) defines three constants \\verb|g|, \\verb|c|, and \\verb|pertb| to be used in the second part.\nThe values of constants can be either real values or intervals.\n\nThe second part (Lines~6--13) describes an HA.\nLine~7 declared the state variables \\verb|y| and \\verb|vy| of the HA, which are evaluated over timeline.\nLine~9 describes the initial state as a comma-separated list of a location and values for each state variable.\nLines~11--13 defines a location named \\verb|Loc|.\nAfter the keyward \\verb|wait|, the derivatives of the state variable are specified as $\\frac{d}{dt}\\mathtt{y} = \\mathtt{vy}$ and $\\frac{d}{dt}\\mathtt{vy} = -\\mathtt{g}$.\nAt Line~12, an inter-location transition is specified.\nThe tuple after \\verb|once| tells that the guard condition for this transition is $\\mathtt{y} = 0 ~\\land~ \\mathtt{vy} > 0$. (Only the left-hand side of the implicit form is described.)\nWhenever the guard is satisfied, an execution will transit to the same location \\verb|Loc| with a reset of the state variables as $\\mathtt{y} := \\mathtt{y}$ and $\\mathtt{vy} := -\\mathtt{c}\\,\\mathtt{vy}$.\n\nThe last part (Lines~15--18) configures some parameters of the simulator implementation, i.e., the order \\verb|order| of Taylor coefficient expansion, the time horizon \\verb|t_max| of a continuous state evolution, and \\verb|dump_interval| that bounds the step size when computing a dumped data.\n\nIn general, an HA consists of multiple locations. The bouncing ball system can be modeled with two locations by modelling the second part as follows: %(\\texttt{bb1.ha}):\n\\begin{lstlisting}\ninit Fall, 1+pertb, -0+pertb\n\nat Fall wait vy, -g\n once (y, vy) goto Rise then y, -c*vy\nend\n\nat Rise wait vy, -g\n once (vy, -y) goto Fall then y, vy\nend\n\\end{lstlisting}\n\n\\subsection{Simulation of a Model}\n\nOnce a specification is prepared, a user can simulate the model for 10 transitions with the following command:\n\\begin{verbatim}\n$ hysia bb.ha -n 10 -dump\n\\end{verbatim}\nThe option \\verb|-dump| let HySIA to output the result of the simulation to the file \\verb|pped.dat| in a JSON format.\nThe output data consists of a set of \\emph{boxes} (interval vectors) that encloses the trajectories of the HA.\n(In this example, the interval value \\verb|pertb| allows perturbation of trajectories; these trajectories are enclosed within the boxes.)\nThe result can be visualized as shown in Figure~\\ref{f:bb}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.8\\linewidth]{bb.eps}\n\\caption{Dumped trajectory enclosure of the bouncing ball example}\n\\label{f:bb}\n\\end{figure}\n\nSimulation and verification of HySIA are all computed with validated interval analysis.\nThus, a resulting interval enclosure of a system's state expands as longer the simulation length and more the \\emph{wrapping effect} occurs.\nHowever, thanks to the underlying \\emph{parallelotope}-based simulation method for wrapping effect reduction, HySIA is able to simulate a large number of jumps for various HA.\nIndeed, for the bouncing ball model, when we modify the parameter values as \\verb|c = 1| and \\verb|pertb = 0|, HySIA can simulate for more than a thousand steps.\n\nAs we have seen so far, HySIA allows models to involve some uncertainties derived by interval values.\nHySIA computes (the overapproximation of) the reachable region with respect to the uncertainties.\nHowever, here we may encounter a drawback of HySIA; a simulation is not always successful but may result in an error.\n%\nWhen we modify the value of \\verb|pertb| to a slightly large interval \\verb|[-1e-4, 1e-4]| and run a simulation, \\verb|hysia| will output as follows:\n\\begin{verbatim}\n$ hysia bb.ha -n 10 -dump\nstep 0 (0.000000 < inf) at Loc\nstep 1 (1.414043 < inf) at Loc\nstep 2 (3.959155 < inf) at Loc\nstep 3 (6.249714 < inf) at Loc\nstep 4 (8.310768 < inf) at Loc\nstep 5 (10.163552 < inf) at Loc\nstep 6 (11.809965 < inf) at Loc\nlibc++abi.dylib: terminating with uncaught exception of type std::\nruntime_error: zero in the derivative\nAbort trap: 6\n\\end{verbatim}\nThe simulation fails after the sixth jump. The output implies that HySIA fails in the detection of a discrete change because a state enclosure becomes too large.\nIn fact, in each detection, HySIA checks whether the orientation between the trajectory and the guard is regular enough so that it is sure that each trajectory within an enclosure satisfies the guard.\n\n\n\\subsection{Verification of STL Properties}\n\nHySIA provides a function for verifying properties described in the \\emph{signal temporal logic} (STL).\nFor example, we can add an STL property \n\\begin{lstlisting}\nprop G[0,10] F[0,1] y-0.3\n\\end{lstlisting}\nin the above mentioned model.\n(It should be added in between the second and third parts.)\nThe property intuitively claims that, for the duration of $10$ time units, the height of the ball (\\verb|y|) goes beyond $0.3$ (\\verb|y-0.3| is interpreted as $\\texttt{y} - 0.3 > 0$) within every duration of $1$ time unit.\n\nThe verification can be done with the following command:\n\\begin{verbatim}\n$ hysia bb.ha -a\n\\end{verbatim}\nThe \\verb|hysia| command calculates the \\emph{necessary} simulation length for the verification (in this case, 11 time units), performs a simulation, and evaluates the satisfiability of this property.\nAccordingly, the verification succeeds with the following output:\n\\begin{verbatim}\n$ hysia bb.ha -a\nstep 0 (0.000000 < 11.000000) at Loc\nstep 1 (1.414196 < 11.000000) at Loc\nstep 2 (3.959734 < 11.000000) at Loc\nstep 3 (6.250722 < 11.000000) at Loc\nstep 4 (8.312608 < 11.000000) at Loc\nstep 5 (10.168295 < 11.000000) at Loc\ntrue, 0.036239\n\\end{verbatim}\n\n\n\\section{Getting the Tool}\n\nThe HySIA tool can be obtained and used in three ways.\n\n\\subsection{Web Demonstration}\n\nA web demonstration site is available at:\n\\begin{quote}\n\\url{http://bit.ly/hysia}\n\\end{quote}\n%\nThrough a web browser, you can access a GUI, load the basic examples, modify the HA/STL specification, simulate, and verify the specification.\n\n\n\\subsection{Docker Image}\n\nFor those who are familier with \\emph{Docker},\\footnote{\\url{https://www.docker.com/}} Docker containers are available at: \n\\begin{quote} \n\\url{https://hub.docker.com/r/dsksh/}\n\\end{quote} \n\nThe image \\verb|dsksh/hysia-web| contains the server program for the web demonstration. A container can be launched by:\n\\begin{verbatim}\n$ docker run -p 8080:80 dsksh/hysia-web\n\\end{verbatim}\nThen, the server will be available at:\n\\begin{quote}\n\\url{http://localhost:8080}\n\\end{quote}\n\n\n\\subsection{Source Distribution}\n\nThe HySIA source code is distributed via GitHub:\n\\begin{quote}\n\\url{https://github.com/dsksh/hysia}.\n\\end{quote}\n\n\\subsubsection*{Requirements}\n\nIn addition to a standard UNIX-like environment, the following softwares are required to compile HySIA:\n\\begin{itemize}\n \\item \\emph{C/C++ compiler}.\n We have compiled with both \\verb|gcc| (versions~4.7.4–4.8.4) and \\verb|clang| (Apple LLVM version~8).\n %\n \\item \\emph{OCaml compiler}.\n We have tested with versions~3.12–4.02.\n %\n \\item \\emph{CAPD library}.\\footnote{http://capd.ii.uj.edu.pl/}\n HySIA is built on an old release of the CAPD-DynSys 3.0 library, which was distributed around 2014, and is not available on the official site.\n A (slightly modified) source package is available at:\n \\begin{quote}\\small\n \\hspace{-4em}\n \\url{https://www.dropbox.com/s/3uf7t2nsizfebno/capdDynSys-201406.zip}\n \\end{quote}\n %\n \\item \\emph{Boost library}.\\footnote{http://www.boost.org/}\n HySIA uses \\verb|shared_ptr|.\n %\n \\item \\emph{OUnit}.\\footnote{http://ounit.forge.ocamlcore.org/} Optional for test cases compilation.\n \\item \\emph{Eliom}.\\footnote{http://ocsigen.org/eliom/} Optional for building the web application.\n\\end{itemize}\n\n\n\\subsubsection*{Build}\n\nIn the root directory of the source code, HySIA can be compiled with:\n\\begin{verbatim}\n$ ./configure; make\n\\end{verbatim}\n\nWhen compilation succeeds, the program file\n\\begin{verbatim}\nsrc_ocaml/hss.opt\n\\end{verbatim}\nis generated.\n\n\n\\section{Examples}\n\n\\subsection{Simple Rotation System}\n\nTBD.\n\n\\subsection{Bouncing Planet}\n\nTBD.\n\n\n\\section{Reference Manual}\n\n\\subsection{Command-Line Tool}\n\nThe basic syntax for the command-line execution is as follows:\n\\begin{grammar}\n ::= `hysia' \n\\end{grammar}\n%\nThe \\texttt{hysia} command accepts the following options:\n\\begin{description}\n\\item[\\texttt{-h}, \\texttt{-help}, or \\texttt{--help}] Displays a summary of the options accepted by the command.\n\\item[\\texttt{-n}] Specifies the number steps to simulate (default is $\\infty$).\n\\item[\\texttt{-t}] Specifies the max simulation time (default is $\\infty$).\n\\item[\\texttt{-a}] Decides the simulation length automatically from the STL property.\n\\item[\\texttt{-g}] Sets the debug flag.\n\\item[\\texttt{-dump}] Activates dumping plot to the file ``pped.dat.''\n\\item[\\texttt{-cm_thres}] Sets the threshold for character matrix selection.\n\\end{description}\n\nThe option \\texttt{-cm_thres} specifies the character matrix $B$ to be used in the parallelotope method (see the corresponding publication for the detail). It is selected as follows:\n\\begin{itemize}\n\t\\item $-1$: $B := (\\mathrm{mid}\\J) A$.\n\t\\item $0$: $B := I$ (i.e., identity matrix).\n\t\\item $1$: $B := \\mathrm{orthogonalize}((\\mathrm{mid}\\J) A)$.\n\t\\item $n > 1$ (default): \n\t\t\\[\n\t\t\tB := \\begin{cases}\n\t\t\t\t(\\mathrm{mid}\\J) A & \\text{if $\\kappa((\\mathrm{mid}\\J) A) < n$} \\\\\n\t\t\t\t\\mathrm{orthogonalize}((\\mathrm{mid}\\J) A) & \\text{otherwise.}\n\t\t\t\\end{cases}\n\t\t\\]\n\\end{itemize}\n\n\n\\subsection{Solving Parameters}\n\n\\begin{description}\n\\item[\\texttt{order}] Order of Taylor expansion.\n\\item[\\texttt{t_max}] Max time horizon assumed in the simulation of each step.\n\\item[\\texttt{h_min}] Min time CAPD integration can take.\n\\item[\\texttt{epsilon}] Specifies the precision of the event detection.\n\\item[\\texttt{dump_interval}] Sets the precision of the dumped flowpipe data.\n\\item[\\texttt{delta}] Parameter for the box inflation process.\n\\item[\\texttt{tau}] Parameter for the box inflation process.\n\\item[\\texttt{cm_thres}] Parameter for the character matrix selection.\n\\end{description}\n\n\n\\subsection{Specification Language}\n\nThis section describes the grammar of the specification language of HySIA.\nA specification consists of the definition of a hybrid automaton, an STL formula, and solving parameter configurations.\n\n\\subsubsection{Lexical conventions}\n\nThe lexical class of digits, letters, and identifiers is the following:\n\\begin{grammar}\n ::= [`0'--`9']\n\n ::= [`a'--`z' `A'--`Z']\n\n ::= ( | | `_')*\n\\end{grammar}\n\nThe syntax of various numeral expressions is as follows:\n\\begin{grammar}\n ::= +\n\n ::= \n%+ ~~|~~ + `.' *\n+ ~ (`.' *)? ~ ( (`e'|`E') (`+'|`-')? + )?\n%\\alt + `.' * (`e'|`E') +\n%\\alt + (`e'|`E') +\n%\\alt + (`e'|`E') (`+'|`-') +\n\n ::= ~~|~~ `-' \n\n ::= ~|~ \n\n ::= `(' `,' `)'\n\n ::= \n\\alt `(' `)'\n\n ::= `,' ~~|~~ \n\\end{grammar}\n\n\n\\noindent\n\\textbf{Comments.}\nComments are either enclosed between \\texttt{(*} and \\texttt{*)} (can be nested) or prefixed with \\texttt{\\#}.\n\n\n\\subsubsection{Toplevel syntax}\n\nThe syntax for the toplevel of specifications is the following:\n\\begin{grammar}\n ::= + ? *\n\n ::= \n`let' `=' \n\\alt `let' `=' `R' \n\\alt `var' \n\\alt `init' \n\\alt `at' ? * `end'\n\n ::= `prop' \n\n ::= `param' `=' \n\n\n ::= `wait' \n\n ::= `inv' \n\n ::= (`when' | `once') `(' `,' `)' `goto' `then' \n\\end{grammar}\n\n\n\\subsubsection{Expressions}\n\nThe operators and function application in expressions have the priorities and associativities as shown in the table below (from lowest to greatest priority):\n\\begin{table}[ht]\n\t\\centering\n\t%\\caption{\\label{t:operators} Terms} \n\t%\\small\n \\begin{tabular}{|l|l|} \\hline\n\t\tconstruct & associativity \\\\\n\t\t\\hline\n\t\t`\\texttt{+}', `\\texttt{-}' & left \\\\\n\t\t`\\texttt{*}', `\\texttt{/}' & left \\\\\n\t\tfunction application & left \\\\\n\t\t`\\texttt{\\^}', `\\texttt{-}' (unary) & --- \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nThe syntax for expressions is the following:\n\\begin{grammar}\n ::= `+' ~~|~~ `-' \n\\alt `*' ~~|~~ `/' \n\\alt `-' ~~|~~ `^' \n\\alt \n\\alt ~~|~~ \n\\alt `(' `)' \n\n ::= `sqrt' | `exp' | `log' | `sin' | `cos' | `atan' | `asin' | `acos'\n\n ::= ~~|~~ `(' `)'\n\n ::= `,' ~|~ \n\\end{grammar}\n\n\n\\subsubsection{STL formulae}\n\nThe operators in formulae have the following priorities and associativities:\n\\begin{table}[ht]\n\t\\centering\n\t%\\caption{\\label{t:operators} Terms} \n\t%\\small\n \\begin{tabular}{|l|l|} \\hline\n\t\tconstruct & associativity \\\\\n\t\t\\hline\n\t\t`\\texttt{->}' & right \\\\\n\t\t`\\texttt{|}' & right \\\\\n\t\t`\\texttt{\\&}' & right \\\\\n\t\t`\\texttt{U} [$\\cdot$,$\\cdot$]' & right \\\\\n\t\t`\\texttt{G} [$\\cdot$,$\\cdot$]', `\\texttt{F} [$\\cdot$,$\\cdot$]' & --- \\\\\n\t\t`\\texttt{!}' & --- \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\nThe syntax for formulae is the following:\n\\begin{grammar}\n ::= `true' ~~|~~ `false'\n\\alt ~~|~~ \n\\alt `!' \n\\alt (`&'|`|'|`->') \n\\alt (`F'|`G') \n\\alt `U' \n\\alt `(' `)'\n\\end{grammar}\n\n\n\\begin{thebibliography}{99}\n\\bibitem{Donze2013}\n A. Donz\\'{e}, T. Ferr\\`{e}re, O. Maler: \n {Efficient Robust Monitoring for STL},\n Proc. of CAV, pp.~264--279, \\LNCS{8044}, 2013.\n\n\\bibitem{GI2016}\n A. Goldsztejn, D. Ishii: \n {A Parallelotope Method for Hybrid System Simulation},\n Reliable Computing, 23:163--185, 2016.\n\n\\bibitem{Ishii2016IEICE}\n D. Ishii, N. Yonezaki, A. Goldsztejn: \n {Monitoring Temporal Properties using Interval Analysis}. \n IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences,\n E99-A, 2016.\n\n\\bibitem{Ishii2015-NSV}\n D. Ishii, N. Yonezaki, A. Goldsztejn: \n {Monitoring Bounded LTL Properties Using Interval Analysis}. \n Proc. of 8th International Workshop on Numerical Software Verification (NSV), \n \\ENTCS{317}, pp.~85--100, 2015.\n\n\\bibitem{Maler2004}\n O. Maler and D. Nickovic:\n Monitoring Temporal Properties of Continuous Signals,\n Proc. of FORMATS, pp.~152--166, \\LNCS{3253}, 2004.\n\\end{thebibliography}\n\n\\end{document}\n", "meta": {"hexsha": "85cfe79256af17609d4e2ffc1b108b5b57e5e59e", "size": 19017, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/manual.tex", "max_stars_repo_name": "dsksh/hysia", "max_stars_repo_head_hexsha": "5c3dd4d1b0e3946ad7a7a81e0b7c83154e69a71c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/manual.tex", "max_issues_repo_name": "dsksh/hysia", "max_issues_repo_head_hexsha": "5c3dd4d1b0e3946ad7a7a81e0b7c83154e69a71c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/manual.tex", "max_forks_repo_name": "dsksh/hysia", "max_forks_repo_head_hexsha": "5c3dd4d1b0e3946ad7a7a81e0b7c83154e69a71c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-07-31T10:21:01.000Z", "max_forks_repo_forks_event_max_datetime": "2019-07-31T10:21:01.000Z", "avg_line_length": 36.9262135922, "max_line_length": 297, "alphanum_fraction": 0.7136246516, "num_tokens": 5609, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6584175139669997, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3497575550127163}} {"text": "\n\\documentclass{article}\n\n\\usepackage{verbatim}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{array}\n\\usepackage{fullpage}\n\n\\author{B.~Barras}\n\\title{An introduction to syntax of Coq V8}\n\n%% Le _ est un caractère normal\n\\catcode`\\_=13\n\\let\\subscr=_\n\\def_{\\ifmmode\\sb\\else\\subscr\\fi}\n\n\\def\\NT#1{\\langle\\textit{#1}\\rangle}\n\\def\\NTL#1#2{\\langle\\textit{#1}\\rangle_{#2}}\n\\def\\TERM#1{\\textsf{\\bf #1}}\n\n\\newenvironment{transbox}\n {\\begin{center}\\tt\\begin{tabular}{l|ll} \\hfil\\textrm{V7} & \\hfil\\textrm{V8} \\\\ \\hline}\n {\\end{tabular}\\end{center}}\n\\def\\TRANS#1#2\n {\\begin{tabular}[t]{@{}l@{}}#1\\end{tabular} & \n \\begin{tabular}[t]{@{}l@{}}#2\\end{tabular} \\\\}\n\\def\\TRANSCOM#1#2#3\n {\\begin{tabular}[t]{@{}l@{}}#1\\end{tabular} & \n \\begin{tabular}[t]{@{}l@{}}#2\\end{tabular} & #3 \\\\}\n\n\\begin{document}\n\n\\maketitle\n\nThe goal of this document is to introduce by example to the new syntax of\nCoq. It is strongly recommended to read first the definition of the new\nsyntax, but this document should also be useful for the eager user who wants\nto start with the new syntax quickly.\n\n\n\\section{Changes in lexical conventions w.r.t. V7}\n\n\\subsection{Identifiers}\n\nThe lexical conventions changed: \\TERM{_} is not a regular identifier\nanymore. It is used in terms as a placeholder for subterms to be inferred\nat type-checking, and in patterns as a non-binding variable.\n\nFurthermore, only letters (unicode letters), digits, single quotes and\n_ are allowed after the first character.\n\n\\subsection{Quoted string}\n\nQuoted strings are used typically to give a filename (which may not\nbe a regular identifier). As before they are written between double\nquotes (\"). Unlike for V7, there is no escape character: characters\nare written normaly but the double quote which is doubled.\n\n\\section{Main changes in terms w.r.t. V7}\n\n\n\\subsection{Precedence of application}\n\nIn the new syntax, parentheses are not really part of the syntax of\napplication. The precedence of application (10) is tighter than all\nprefix and infix notations. It makes it possible to remove parentheses\nin many contexts.\n\n\\begin{transbox}\n\\TRANS{(A x)->(f x)=(g y)}{A x -> f x = g y}\n\\TRANS{(f [x]x)}{f (fun x => x)}\n\\end{transbox}\n\n\n\\subsection{Arithmetics and scopes}\n\nThe specialized notation for \\TERM{Z} and \\TERM{R} (introduced by\nsymbols \\TERM{`} and \\TERM{``}) have disappeared. They have been\nreplaced by the general notion of scope.\n\n\\begin{center}\n\\begin{tabular}{l|l|l}\ntype & scope name & delimiter \\\\\n\\hline\ntypes & type_scope & \\TERM{T} \\\\\n\\TERM{bool} & bool_scope & \\\\\n\\TERM{nat} & nat_scope & \\TERM{nat} \\\\\n\\TERM{Z} & Z_scope & \\TERM{Z} \\\\\n\\TERM{R} & R_scope & \\TERM{R} \\\\\n\\TERM{positive} & positive_scope & \\TERM{P}\n\\end{tabular}\n\\end{center}\n\nIn order to use notations of arithmetics on \\TERM{Z}, its scope must be opened with command \\verb+Open Scope Z_scope.+ Another possibility is using the scope change notation (\\TERM{\\%}). The latter notation is to be used when notations of several scopes appear in the same expression.\n\nIn examples below, scope changes are not needed if the appropriate scope\nhas been opened. Scope nat_scope is opened in the initial state of Coq.\n\\begin{transbox}\n\\TRANSCOM{`0+x=x+0`}{0+x=x+0}{\\textrm{Z_scope}}\n\\TRANSCOM{``0 + [if b then ``1`` else ``2``]``}{0 + if b then 1 else 2}{\\textrm{R_scope}}\n\\TRANSCOM{(0)}{0}{\\textrm{nat_scope}}\n\\end{transbox}\n\nBelow is a table that tells which notation is available in which\nscope. The relative precedences and associativity of operators is the\nsame as in usual mathematics. See the reference manual for more\ndetails. However, it is important to remember that unlike V7, the type\noperators for product and sum are left associative, in order not to\nclash with arithmetic operators.\n\n\\begin{center}\n\\begin{tabular}{l|l}\nscope & notations \\\\\n\\hline\nnat_scope & $+ ~- ~* ~< ~\\leq ~> ~\\geq$ \\\\\nZ_scope & $+ ~- ~* ~/ ~\\TERM{mod} ~< ~\\leq ~> ~\\geq ~?=$ \\\\\nR_scope & $+ ~- ~* ~/ ~< ~\\leq ~> ~\\geq$ \\\\\ntype_scope & $* ~+$ \\\\\nbool_scope & $\\TERM{\\&\\&} ~\\TERM{$||$} ~\\TERM{-}$ \\\\\nlist_scope & $\\TERM{::} ~\\TERM{++}$\n\\end{tabular}\n\\end{center}\n(Note: $\\leq$ is written \\TERM{$<=$})\n\n\n\n\\subsection{Notation for implicit arguments}\n\nThe explicitation of arguments is closer to the \\emph{bindings} notation in\ntactics. Argument positions follow the argument names of the head constant.\n\n\\begin{transbox}\n\\TRANS{f 1!t1 2!t2}{f (x:=t1) (y:=t2)}\n\\TRANS{!f t1 t2}{@f t1 t2}\n\\end{transbox}\n\n\n\\subsection{Universal quantification}\n\nThe universal quantification and dependent product types are now\nmaterialized with the \\TERM{forall} keyword before the binders and a\ncomma after the binders.\n\nThe syntax of binders also changed significantly. A binder can simply be\na name when its type can be inferred. In other cases, the name and the type\nof the variable are put between parentheses. When several consecutive\nvariables have the same type, they can be grouped. Finally, if all variables\nhave the same type parentheses can be omitted.\n\n\\begin{transbox}\n\\TRANS{(x:A)B}{forall (x:~A), B ~~\\textrm{or}~~ forall x:~A, B}\n\\TRANS{(x,y:nat)P}{forall (x y :~nat), P ~~\\textrm{or}~~ forall x y :~nat, P}\n\\TRANS{(x,y:nat;z:A)P}{forall (x y :~nat) (z:A), P}\n\\TRANS{(x,y,z,t:?)P}{forall x y z t, P}\n\\TRANS{(x,y:nat;z:?)P}{forall (x y :~nat) z, P}\n\\end{transbox}\n\n\\subsection{Abstraction}\n\nThe notation for $\\lambda$-abstraction follows that of universal\nquantification. The binders are surrounded by keyword \\TERM{fun}\nand $\\Rightarrow$ (\\verb+=>+ in ascii).\n\n\\begin{transbox}\n\\TRANS{[x,y:nat; z](f a b c)}{fun (x y:nat) z => f a b c}\n\\end{transbox}\n\n\n\\subsection{Pattern-matching}\n\nBeside the usage of the keyword pair \\TERM{match}/\\TERM{with} instead of\n\\TERM{Cases}/\\TERM{of}, the main change is the notation for the type of\nbranches and return type. It is no longer written between \\TERM{$<$ $>$} before\nthe \\TERM{Cases} keyword, but interleaved with the destructured objects.\n\nThe idea is that for each destructured object, one may specify a variable\nname to tell how the branches types depend on this destructured objects (case\nof a dependent elimination), and also how they depend on the value of the\narguments of the inductive type of the destructured objects. The type of\nbranches is then given after the keyword \\TERM{return}, unless it can be\ninferred.\n\nMoreover, when the destructured object is a variable, one may use this\nvariable in the return type.\n\n\\begin{transbox}\n\\TRANS{Cases n of\\\\~~ O => O \\\\| (S k) => (1) end}{match n with\\\\~~ 0 => 0 \\\\| (S k) => 1 end}\n\\TRANS{Cases m n of \\\\~~0 0 => t \\\\| ... end}{match m, n with \\\\~~0, 0 => t \\\\| .. end}\n\\TRANS{<[n:nat](P n)>Cases T of ... end}{match T as n return P n with ... end}\n\\TRANS{<[n:nat][p:(even n)]\\~{}(odd n)>Cases p of\\\\~~ ... \\\\end}{match p in even n return \\~{} odd n with\\\\~~ ...\\\\end}\n\\end{transbox}\n\n\n\\subsection{Fixpoints and cofixpoints}\n\nAn easier syntax for non-mutual fixpoints is provided, making it very close\nto the usual notation for non-recursive functions. The decreasing argument\nis now indicated by an annotation between curly braces, regardless of the\nbinders grouping. The annotation can be omitted if the binders introduce only\none variable. The type of the result can be omitted if inferable.\n\n\\begin{transbox}\n\\TRANS{Fix plus\\{plus [n:nat] : nat -> nat :=\\\\~~ [m]...\\}}{fix plus (n m:nat) \\{struct n\\}: nat := ...}\n\\TRANS{Fix fact\\{fact [n:nat]: nat :=\\\\\n~~Cases n of\\\\~~~~ O => (1) \\\\~~| (S k) => (mult n (fact k)) end\\}}{fix fact\n (n:nat) :=\\\\\n~~match n with \\\\~~~~0 => 1 \\\\~~| (S k) => n * fact k end}\n\\end{transbox}\n\nThere is a syntactic sugar for mutual fixpoints associated to a local\ndefinition:\n\n\\begin{transbox}\n\\TRANS{let f := Fix f \\{f [x:A] : T := M\\} in\\\\(g (f y))}{let fix f (x:A) : T := M in\\\\g (f x)}\n\\end{transbox}\n\nThe same applies to cofixpoints, annotations are not allowed in that case.\n\n\\subsection{Notation for type cast}\n\n\\begin{transbox}\n\\TRANS{O :: nat}{0 : nat}\n\\end{transbox}\n\n\\section{Main changes in tactics w.r.t. V7}\n\nThe main change is that all tactic names are lowercase. This also holds for\nLtac keywords.\n\n\\subsection{Ltac}\n\nDefinitions of macros are introduced by \\TERM{Ltac} instead of\n\\TERM{Tactic Definition}, \\TERM{Meta Definition} or \\TERM{Recursive\nDefinition}.\n\nRules of a match command are not between square brackets anymore.\n\nContext (understand a term with a placeholder) instantiation \\TERM{inst}\nbecame \\TERM{context}. Syntax is unified with subterm matching.\n\n\\begin{transbox}\n\\TRANS{match t with [C[x=y]] => inst C[y=x]}{match t with context C[x=y] => context C[y=x]}\n\\end{transbox}\n\n\\subsection{Named arguments of theorems}\n\n\\begin{transbox}\n\\TRANS{Apply thm with x:=t 1:=u}{apply thm with (x:=t) (1:=u)}\n\\end{transbox}\n\n\n\\subsection{Occurrences}\n\nTo avoid ambiguity between a numeric literal and the optionnal\noccurrence numbers of this term, the occurrence numbers are put after\nthe term itself. This applies to tactic \\TERM{pattern} and also\n\\TERM{unfold}\n\\begin{transbox}\n\\TRANS{Pattern 1 2 (f x) 3 4 d y z}{pattern (f x at 1 2) (d at 3 4) y z}\n\\end{transbox}\n\n\\section{Main changes in vernacular commands w.r.t. V7}\n\n\n\\subsection{Binders}\n\nThe binders of vernacular commands changed in the same way as those of\nfixpoints. This also holds for parameters of inductive definitions.\n\n\n\\begin{transbox}\n\\TRANS{Definition x [a:A] : T := M}{Definition x (a:A) : T := M}\n\\TRANS{Inductive and [A,B:Prop]: Prop := \\\\~~conj : A->B->(and A B)}%\n {Inductive and (A B:Prop): Prop := \\\\~~conj : A -> B -> and A B}\n\\end{transbox}\n\n\\subsection{Hints}\n\nThe syntax of \\emph{extern} hints changed: the pattern and the tactic\nto be applied are separated by a \\TERM{$\\Rightarrow$}.\n\\begin{transbox}\n\\TRANS{Hint Extern 4 (toto ?) Apply lemma}{Hint Extern 4 (toto _) => apply lemma}\n\\end{transbox}\n\n\\end{document}\n", "meta": {"hexsha": "ae4b569b36ae2f1c717cb09046ae98a5a8999c90", "size": 9770, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "coq/dev/v8-syntax/memo-v8.tex", "max_stars_repo_name": "reichel3/TacTok", "max_stars_repo_head_hexsha": "c344e76263de04311af8a0030c07aec95d87f71c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2020-11-23T02:45:36.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-18T03:03:33.000Z", "max_issues_repo_path": "coq/dev/v8-syntax/memo-v8.tex", "max_issues_repo_name": "reichel3/TacTok", "max_issues_repo_head_hexsha": "c344e76263de04311af8a0030c07aec95d87f71c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2021-02-23T03:03:51.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-13T00:07:38.000Z", "max_forks_repo_path": "coq/dev/v8-syntax/memo-v8.tex", "max_forks_repo_name": "reichel3/TacTok", "max_forks_repo_head_hexsha": "c344e76263de04311af8a0030c07aec95d87f71c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-01-19T17:56:28.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-28T04:39:41.000Z", "avg_line_length": 34.0418118467, "max_line_length": 284, "alphanum_fraction": 0.7022517912, "num_tokens": 3005, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.34975754789168645}} {"text": "\\documentclass[11pt, oneside]{article} \t% use \"amsart\" instead of \"article\" for AMSLaTeX format\n\\usepackage{geometry} \t\t% See geometry.pdf to learn the layout options. There are lots.\n\\geometry{letterpaper} \t\t% ... or a4paper or a5paper or ... \n%\\geometry{landscape} \t\t% Activate for rotated page geometry\n%\\usepackage[parfill]{parskip} \t\t% Activate to begin paragraphs with an empty line rather than an indent\n\\usepackage{graphicx}\t\t\t\t% Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode\n\t\t\t\t\t\t\t\t% TeX will automatically convert eps --> pdf in pdflatex\t\t\n\\usepackage{amssymb}\n\\usepackage{amsmath}\n\\PassOptionsToPackage{hyphens}{url} % url is loaded by hyperref\n\\usepackage[unicode=true]{hyperref}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\\usepackage{lineno}\n\\linenumbers\n% math typesetting \n\\let\\vec\\mathbf\n\\newcommand{\\matrx}[1]{{\\left[ \\stackrel{}{#1}\\right]}}\n\n%SetFonts\n\n%SetFonts\n\n\\pagestyle{headings}\n\\markright{Yoav Ram, \\today \\; v2\\hfill}\n\n\n\n\\begin{document}\nHere I describe how I analyzed the stability of a modifier allele $m$ with vertical transmission rate $\\rho$ to invasion by a modifier $M$ with a vertical transmission rate $P$, as described in the main text, eqs. 34-35, in environmental regime $AkBl$.\n\\\\\nThe analysis is similar to that used in the main text to analyze stability in $A1B1$, but it uses computation because the cases where $k>1$ or $l>1$ cannot be analyzed using closed form expressions.\n\\\\\nThe analysis includes the following steps for a fixed $W, w, k,$ and $l$:\n\\begin{enumerate}\n\n\\item Find the stable frequency vector $\\vec{x^*}$ without invader $M$, i.e. $x^*_3=x^*_4=0$. This is done by minimizing the expression $|x_{k+l} - x_{0}|$ where $x_{t}$ is defined in eq. 9 of the main text. The minimization is done by iterating the recurrence until it converges, i.e. until the difference is smaller than machine precision (roughly $10^-8$ when subtracting similar small numbers).\n\n\\item Define $F_A(x)$ by eq. 35 of main text with $w_A=W$ and $w_B=w$ ($W>w$), and similarly $F_B(x)$ with $w_B=W$ and $w_A=w$.\n\n\\item Define, similar to eq. 37 of main text,\n\\begin{equation}\n\\vec{F} = \\underbrace{F_A \\circ \\ldots \\circ F_A}_{k \\text{ times}} \\circ\n\t\\underbrace{F_B \\circ \\ldots \\circ F_B}_{l \\text{ times}}.\n\\label{eq:F}\\end{equation}\n\n\\item Calculate the linear approximation $\\vec{J}$ of $\\vec{F}$ near $\\vec{x^*}$, that is, the Jacobian matrix of $\\vec{F}$ at $\\vec{x^*}$:\n\\begin{equation}\n\\vec{J}_{ij} = \\frac{\\partial F_i}{\\partial x_j}.\n\\label{eq:jacobian}\\end{equation}\n\n\\item Define $\\vec{L}=\\vec{L_{ex}}$ as in eqs. 40-45 in the main text such that\n\\begin{equation}\n\\vec{J} = \\begin{pmatrix}\n\\vec{L_{in}} & * \\\\\n* & \\vec{L_{ex}}\n\\end{pmatrix}.\n\\end{equation}\n\n\\item Calculate the eigenvalues $\\lambda_1 > \\lambda_2$ of $\\vec{L}$; by the Perron-Frobenius theorem, the leading eigenvalue $\\lambda_1$ is real and positive. Denote $\\lambda_1(\\rho, P)$ the resulting leading eigenvalue with resident rate $\\rho$ and invader rate $P$. Note that \n\\begin{equation}\n\\lambda_1(\\rho, \\rho) = 1,\n\\end{equation}\nfor any $\\rho \\in (0,1)$.\n\n\\item The evolutionary stable rate $\\rho^*$ is defined as stable to invasion, that is, \n\\begin{equation}\n\\lambda_1(\\rho^*,\\rho^* \\pm dP) < 1 = \\lambda_1(\\rho^*,\\rho^*)\n\\end{equation}\nfor a small enough $dP>0$.\nTherefore, \n\\begin{equation}\n\\frac{\\partial \\lambda_1}{\\partial P}\\big(\\rho^*,\\rho^*\\big) = 0. \n\\label{eq:dλdP=0} \\end{equation}\n\n\\item We use Brent's root-finding method (Brent, 1971) to find $\\rho^*$ that satisfies \\autoref{eq:dλdP=0}. If, due to numerical instability of the process, we have\n\\begin{equation}\n\\frac{\\partial \\lambda_1}{\\partial P}\\big(0,0\\big) \\cdot \\frac{\\partial \\lambda_1}{\\partial P}\\big(1,1) > 0,\n\\end{equation}\ni.e., the partial derivative sign is identical at $\\rho=P=0$ and $\\rho=P=1$, then we cannot use Brent's method. In these cases we assume that the partial derivative doesn't have a root in $(0,1)$ and we determine the stable rate $\\rho^*$ by the rule\n\\begin{equation}\n\\rho^* = \\begin{cases}\n0, \\quad \\text{if } \\frac{\\partial \\lambda_1}{\\partial P}\\big(0,0\\big) \\le 0, \\\\\n1, \\quad \\text{if } \\frac{\\partial \\lambda_1}{\\partial P}\\big(0,0\\big) > 0.\n\\end{cases}\n\\end{equation}\n\n\\end{enumerate}\n\n\\paragraph{Results}\n\\autoref{fig:AkBk_stable_modifier_w_0.1}, \\autoref{fig:AkBk_stable_modifier_w_0.5} and \\autoref{fig:AkBk_stable_modifier_w_0.9}, show the leading eigenvalue $\\lambda_1$ of the external stability matrix $\\vec{L}$ for different choices of environmental cycles $AkBk$, resident rate $\\rho$ and invader rate $P$. The arrows show a series of invading modifiers generated using Algorithm~\\autoref{algorithm} (this algorithm cannot be efficiently used to determine the stable rate as it requires fine tuning of the meta-parameter $\\eta$).\n\nThe leading eigenvalue calculation is good for small $k$ but for large $k$ and especially for $w=0.1$ the calculation is unstable.\nThis is probably because when the environment is constant for a long period of time, the frequencies are close to the boundaries (i.e. 0 and 1) and the gradient just after the environment changes is very sharp.\nTherefore, the calculation of $\\vec{J}$ is unstable.\nAlso, because most of the time the frequencies don't change by much, the elements of $\\vec{L}$ are small, leading to unstable calculation of the leading eigenvalue.\n\\\\\n\n\\begin{algorithm} \n% https://en.wikibooks.org/wiki/LaTeX/Algorithms\n\\caption{Modifier gradient ascent algorithm}\\label{algorithm}\n\\begin{algorithmic}[1]\n\\State $\\eta > 0$ is a step size\n\\State $\\rho_{0} = 0.5$ is a starting condition\n\\State $t = 0$ is an iteration counter\n\\While {$\\frac{|\\rho_{t} - \\rho_{t-1}|}{\\rho_{t-1}} < \\epsilon \\;\\text{and}\\; \\eta > \\eta_0$}\n\t\\State $P_{t} \\gets \\rho_{t} + \\eta \\frac{\\partial \\lambda_1(\\rho_{t}, \\rho_{t})}{\\partial P}$\n\t\\If {$\\lambda_1(\\rho_{t}, P_{t})$} \n\t\\State {$\\rho_{t+1} \\gets P_{t}$}\n\t\\Else\n\t\\State {$\\eta \\gets \\eta / 2$}\n\t\\EndIf\n\t\\State $t \\gets t+1$\n\\EndWhile\n\\State $\\rho^* \\gets \\rho_t$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\linewidth]{../figures/AkBk_stable_optimal_rate.pdf}\n \\caption{Fitness \"optimal\" and evolutionary stable vertical transmission rate in $AkBk$ selection regime. \\textbf{(A)} The vertical transmission rate $\\hat{rho}$ that maximized the geometric average of the population mean fitness is zero, i.e. complete oblique transmission, when selection cycles quickly between favoring phenotype $A$ and $B$, and then abruptly transitions to ~0.2, followed by a slow decrease (see Fig. SX for details on the abrupt transition). \n \\textbf{(B)} The evolutionary stable rate $\\rho^*$ which cannot be invaded by modifiers with either higher or lower vertical transmission rate $P$, rapidly increases from zero when selection cycles are short ($k=1$ or $2$) to roughly 1 when the environment cycles are longer. The stable rate was found by finding the root of $\\frac{\\partial \\lambda_1}{\\partial P}$, the leading eigenvalue of the external stability matrix w.r.t to the rate of the invading modifier (see Supplementary Information for details). The dashed lines shows $1-\\frac{1}{k-1}$, which seems to fit the values for $w=0.5$ (Carja et al. 2011). The values for $w=0.1$ (blue) could not be calculated for $k>21$ due to numerical instability when selection is strong and the duration between selection fluctuations is long.\n In all cases, $W=1$}\n \\label{fig:AkBk_stable_modifier_w_0.1}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\linewidth]{{../figures/AkBk_stable_modifier_w_0.1}.pdf}\n \\caption{The leading eigenvalue of the external stability matrix in $AkBk$ with selection. The colors show $\\lambda_1(W, w, \\rho, P, k, l)$, the leading eigenvalue of the external stability matrix for combinations of $\\rho$ (y-axis) and $P$ (x-axis), the vertical transmission rates of resident and invader modifiers, respectively. Purple shows values lower than 1, leading to stability of $\\rho$; white equals to 1; green larger than 1, leading to invasion of $P$.\n The arrows show an \"evolutionary\" path $\\rho_{0} \\to \\rho_{1} \\to \\ldots \\to \\rho^*$: at each step, a modifier with rate $P_t$ invades the resident modifier with rate $\\rho_t$ and becomes the resident modifier $\\rho_{t+1}$, until consecutive invaders have similar rates (see Algorithm~\\autoref{algorithm}).\n Here, $W=1$ and $w=0.1$.}\n \\label{fig:AkBk_stable_modifier_w_0.1}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\linewidth]{{../figures/AkBk_stable_modifier_w_0.5}.pdf}\n \\caption{The leading eigenvalue of the external stability matrix in $AkBk$ with intermediate selection.\n Same as~\\autoref{fig:AkBk_stable_modifier_w_0.1} except here, $W=1$ and $w=0.5$.}\n \\label{fig:AkBk_stable_modifier_w_0.5}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width=\\linewidth]{{../figures/AkBk_stable_modifier_w_0.9}.pdf}\n \\caption{The leading eigenvalue of the external stability matrix in $AkBk$ with intermediate selection.\n Same as~\\autoref{fig:AkBk_stable_modifier_w_0.1} except here, $W=1$ and $w=0.9$.}\n \\label{fig:AkBk_stable_modifier_w_0.9}\n\\end{figure}\n\n\\paragraph{Note}\nCrucially, the Jacobian $\\vec{J}$ in~\\autoref{eq:jacobian} can be calculated using \\textit{automatic differentiation} with good precision from a function that iteratively calculates $\\vec{F}$ according to~\\autoref{eq:F}. Similarly, the partial derivative $\\frac{\\partial \\lambda_1}{\\partial P}$ in Algorithm~\\autoref{algorithm} can be calculated from a function that calculates $\\lambda_1$ using simple arithmetic operations, as the eigenvalues can be found from a quadratic formula.\n\\\\\nNote that \\textit{automatic differentiation} does not mean \\textit{symbolic} or \\textit{numerical differentiation}, which can lead to inefficient or inaccurate estimation of $\\vec{J}$ when $k$ is not very small. Rather, from Bartholomew-Biggs et al. (2000): \n\\begin{quotation}\\textit{\nAutomatic differentiation is a set of techniques for transforming a program that calculates numerical values of a function, into a program which calculates numerical values for derivatives of that function with about the same accuracy and efficiency as the function values themselves.\n}\\end{quotation}\n\n\\paragraph{References} ~\\\\\n\\begin{itemize}\n\n\\item Bartholomew-Biggs, M., S. Brown, B. Christianson, and L. Dixon. 2000. Automatic differentiation of algorithms. J. Comput. Appl. Math. 124:171–190.\n\n\\item Brent, R. P. 1971. An algorithm with guaranteed convergence for finding a zero of a function. Comput. J. 14:422–425.\n\n\\end{itemize}\n\n\\end{document} ", "meta": {"hexsha": "62ac2f5c5999e8e415be5df3caa6cd4cfff17a33", "size": 10688, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ms/stability_analysis.tex", "max_stars_repo_name": "yoavram/Milpitas", "max_stars_repo_head_hexsha": "9dc3090cc5bca1a933ef21df75e7d907a89a37b0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ms/stability_analysis.tex", "max_issues_repo_name": "yoavram/Milpitas", "max_issues_repo_head_hexsha": "9dc3090cc5bca1a933ef21df75e7d907a89a37b0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ms/stability_analysis.tex", "max_forks_repo_name": "yoavram/Milpitas", "max_forks_repo_head_hexsha": "9dc3090cc5bca1a933ef21df75e7d907a89a37b0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 62.8705882353, "max_line_length": 792, "alphanum_fraction": 0.7308196108, "num_tokens": 3192, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093585306514, "lm_q2_score": 0.658417500561683, "lm_q1q2_score": 0.34975753811872645}} {"text": "\\section{Model Functions}\n\nThis module is intended to be used an approximation of fuel slosh attached to the spacecraft. It is intended to have multiple fuel slosh particles attached to the spacecraft. Also it is recommended to attach at least 3 fuel slosh particles with different sloshing directions to get motion across all axes. Below is a list of functions that this model performs:\n\n\\begin{itemize}\n\t\\item Compute it's contributions to the mass properties of the spacecraft\n\t\\item Provides matrix contributions for the back substitution method\n\t\\item Compute it's derivatives for $\\rho$, $\\dot{\\rho}$, and $\\dot{m}$\n\t\\item Adds energy and momentum contributions to the spacecraft\n\\end{itemize}\n\n\\section{Model Assumptions and Limitations}\nBelow is a summary of the assumptions/limitations:\n\n\\begin{itemize}\n\t\\item Is an approximation to sloshing fuel\n\t\\item Is derived in such a manner that does not require constraints to be met\n\t\\item A single fuel slosh particle can only move along one direction, $\\hat{\\bm p}_j$, as seen in Figure~\\ref{fig:Flex_Slosh_Figure}\n\t\\item Multiple fuel slosh particles are not interconnected, each fuel slosh particle is attached the rigid body hub independently\n\t\\item Only constant linear spring and damping terms\n\t\\item The mass, spring and damping coefficients can be attenuated to approximate frequencies expected by the fuel slosh\n\t\\item There are no travel limits for the fuel slosh therefore the particles could travel past the limits of the fuel tank boundary (this can be avoided by varying the mass and/or spring constant while considering the expected accelerations of the s/c to stay within the tank)\n\t\\item The mass of the fuel slosh particles add to the total mass of the fuel tank\n\t\\item This model could be used with other fuel slosh models like a pendulum based fuel slosh model\n\\end{itemize}", "meta": {"hexsha": "1621d0e67496c8d82cb309b2810c300f5e4a0e6c", "size": 1848, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/simulation/dynamics/LinearSpringMassDamper/_Documentation/secModelFunctions.tex", "max_stars_repo_name": "ian-cooke/basilisk_mag", "max_stars_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_stars_repo_licenses": ["0BSD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/simulation/dynamics/LinearSpringMassDamper/_Documentation/secModelFunctions.tex", "max_issues_repo_name": "ian-cooke/basilisk_mag", "max_issues_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_issues_repo_licenses": ["0BSD"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-03-13T20:52:22.000Z", "max_issues_repo_issues_event_max_datetime": "2019-03-13T20:52:22.000Z", "max_forks_repo_path": "src/simulation/dynamics/LinearSpringMassDamper/_Documentation/secModelFunctions.tex", "max_forks_repo_name": "ian-cooke/basilisk_mag", "max_forks_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_forks_repo_licenses": ["0BSD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 73.92, "max_line_length": 360, "alphanum_fraction": 0.7981601732, "num_tokens": 416, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3497261564093474}} {"text": "\n%\\section{Calibration and Validation}\n%\\section{Validation}\n\\label{sec:validation}\n\n% VALIDATION TODO\n%\n% show major effort\n% table of results\n% show hierarchy\n% precisely define cases \n%\n%\n\n%\n% meshed vanes are 24x more expensive\n%\n\nThe previous chapters briefly outlined the physical phenomenon under\nconsideration, the mathematical models proposed to simulate it,\nand the numerical solution of these models for a variety of system \nconfigurations and scenarios. Before these simulations can be used \nas a tool to evaluate proposed system designs, it is necessary to\nvalidate that the physical models in use accurately represent\nreality. As defined by Moser {\\em et al.}\\cite{Moser2012Validating},\n``validation is the process of determining whether a mathematical model\nis a sufficient representation of reality for the purposes for which the\nmodel will be used--that is, for predicting specified QoIs (Quantities\nof Interest) to inform a specific decision.'' \n\nThis chapter contains a discussion of the validation of the\ncomputational models against existing experimental data and high\nfidelity simulations. This chapter does not exhaustively detail the\nvalidation studies performed in the course of this study. Rather, this\nchapter discusses four representative cases and the overall validation\napproach pursued. \n%A more detailed set of validation data is provided in \n%Appendix~\\ref{app-validation}. \n% The limited validation that was performed\n% for the 2016 field tests in discussed in Chapter~\\ref{sec:field}. \n\nA challenge in this project is the scarcity of experimental data. Only\ntwo or three cases of experimental measurements are available. These\nmeasurements, for reasons detailed in the next section, are not\nsufficient to provide confidence in the output of simulations across a\nwide variety of scenarios. Therefore, a high fidelity model using \nmeshed vanes with enforced no-slip velocity boundary conditions along\nthe surface of the turning vanes was developed. These ``gridded'' runs have\nbeen validated against the experimental data, which they match quite\nclosely. However, as detailed in Section~\\ref{subsec:vane}, \nexplicitly meshing the vanes would be far too \nexpensive to permit a rapid exploration of a variety of\nsystem configurations. Instead, this high fidelity model is used \nto generate additional reliable data to permit validation of lower\nfidelity models, such as the virtual vanes. Likewise, the results of the\nunsteady virtual vane simulations can be used as validation data for a\nfurther reduced, steady Navier-Stokes model. This hierarchy of\nvalidation is shown in Figure~\\ref{fig:val_hier}, with data sources that\ngenerate more reliable data at the top, and models that are less\nreliable, but also less computationally expensive at the bottom. In\nterms of expense, the steady virtual vane model generates a solution in\napproximately two minutes, versus twelve hours for the unsteady virtual\nvane model. The gridded vanes require another factor of ten in\ncomputational time, and many more man-hours hours of work to generate\nthe mesh (which were generated using gridgen). An example of one of the\ngridded meshes complexity is shown in Figure~\\ref{fig:gridded_mesh}. \nTherefore, it is unrealistic to perform parameter sweeps or\nsystem configuration investigations with the gridded vanes and these\nresults are used only for validation studies. Instead, the\nsteady model is used, with promising results re-evaluated with unsteady\nvirtual vane models. %and so on up the hierarchy if more confidence is necessary. \n\n%\n% https://www.draw.io/\n%\n \\begin{figure}[!htb]\n \\begin{center}\n \\includegraphics[width = 8 cm]{figs/validation_hierarchy}\n \\caption{This figure depicts the validation hierarchy. The\n experimental measurements are at the top, where the data is expected\n to be the most reliable, but simultaneously the most\n limited. Moving down the table leads to simulated data sources that\n are less reliable but increasingly cheaper in time to generate. At\n the bottom are the steady virtual vane solutions.} \n \\label{fig:val_hier}\n \\end{center}\n \\end{figure}\n\n \\begin{figure}[!htb]\n \\begin{center}\n \\includegraphics[width = 8 cm]{figs/gridded_mesh_example}\n \\caption{An example of the gridded mesh, where the turning vanes\n are explicitly represented and a no-slip boundary condition is\n imposed on the surface. This mesh was generated using gridgen.} \n \\label{fig:gridded_mesh}\n \\end{center}\n \\end{figure}\n\nThree kinds of experimental validation data are available. These are\ndata generated in the laboratory using a heated plate, data from\nexperiments in the wind tunnel (``Wind-only''), and measurements from field\ntests (``Field'') conducted in Arizona. The available data from\nthese cases and the gridded vanes created to mimic them are are\nsummarized in Table~\\ref{tab:val_data}. Every case shown has been\nsimulated using the virtual vanes. \n\n%\\large\n\\begin{table}[h]\n\\centering\n\\begin{tabular}{l|l|l|l|}\n & Wind-Only & Thermal-Only & Field \\\\\n \\hline \nExperiment & Straight Vanes $60^{\\circ}$ & Straight Vanes $60^{\\circ}$ & June 2014 \\\\\n & & Straight Vanes $30^{\\circ}$ & August 2014 \\\\\n & & Hybrid (Two tier) & August 2015 \\\\\n \\hline \nGridded & Straight Vanes $60^{\\circ}$ & Straight Vanes $60^{\\circ}$ & \\\\\n & Straight Vanes $30^{\\circ}$ & Straight Vanes $30^{\\circ}$ & \\\\\n \\hline \n\\end{tabular}\n \\caption{Available truth data from the laboratory experiments \n (cold wind and thermal-only), the field tests, and the gridded\n vanes.} \n \\label{tab:val_data}\n\\end{table}\n%\n%\n%\n% http://www.tablesgenerator.com/\n%\n%\n\n\n%\n% experimental challenges\n%\n%\\normalsize\n\\section{Thermal-Only Validation}\nThis section provides examples of the validation performed with the\nrichest experimental data set, the measurements in the laboratory. All of the\nthermal-only the data was generated in a laboratory setting at Georgia\nTech. The general system configuration is depicted in Figure\n\\ref{fig:lab_image}. These data were taken using stereo particle image\nvelocimetry (PIV) at Georgia Tech by Mark Simpson and Ari Glezer, and\nthe errors in in measurement and sampling are \nnot quoted. The particles are seeded outside of the array vanes and\npermitted to naturally convect into the turning vane enclosure. The\nparticles were from a glycol-water theatrical fog (Rosco Fog Fluid). \nOnly velocity measurements are available. Several\npotentially important quantities, such as the pressure and\ntemperature, have not been measured. \n\n%\n% http://convertonlinefree.com/ConvertImageEN.aspx\n%\n \\begin{figure}[!htb]\n \\begin{center}\n \\includegraphics[width = 12 cm]{figs/Optimized-lab_setup}\n \\caption{An example of the single tier straight vane laboratory\n configuration. The apparatus is shown with a turbine, but that was\n removed for data gathering. The particles for PIV were seeded\n outside of the turning vanes and entrained into the central region.}\n \\label{fig:lab_image}\n \\end{center}\n \\end{figure}\n\nWhile no sensitivity analysis has been performed, it is likely that the\nlargest uncertainty in the laboratory simulation is a result of the\nventilation of the laboratory. The heated plate at the bottom of the\napparatus generated enough heat to cause an increase in room\ntemperature (30+ Kelvin), which greatly impacts the SoV\nperformance, as the ground to air thermal gradient drives the\nvortex. The laboratory is cooled to maintain\ntemperature by two inlet HVAC ducts in the room. \n%While efforts have been made to characterize the level of ventilation being\n%used, these numbers come with non-trivial uncertainties attached. \nOne vent continuously provides air at 288 Kelvin with a flow rate estimated \nto be 1 $\\text{m}^3$/s.\n%(4-6 m/s with an approximate area of 0.2 $m^2$)\nThe other vent is active only if the room temperature exceeds 301 Kelvin, \nwith a flow rate also estimated at 1 $\\text{m}^3$/s\\cite{mark_comm}.\nFinally, the air leaves through the cracks around the laboratory doors and \nexhaust vents. Preliminary results indicated that an inflow rate of 1\n$\\text{m}^3$/s, the lower bound of the possible inflow rates results in\nexcessive heating of the room, while inflow conditions at the maximum\ninflow rate of 2 $\\text{m}^3$/s result in a simulated room that is too cold,\ncompared to the laboratory. \n\nOur simulated vortices are sensitive to ambient room temperature and thus \nthe inflow rate. It is likely that the laboratory is run where one of\nthe vents is operating intermittently. \nTo mimic these conditions in our simulations, Dirichlet boundary conditions \non parts of the sides of the computational domain are used to\nestablish a constant inflow of cool air at the rates \nproscribed by our collaborators. Over the remainder of the side walls, \nadiabatic thermal boundary conditions are are used. \n\nThe most significant boundary condition disparity is that flow leaves the\ndomain through the top boundary, instead of out of the sides of the\nroom. Preliminary results suggested that the SoV phenomenon was not\nsensitive to these boundary condition details. The important element is\nthe global energy balance in the room. The flow rate into the room is\nadjusted to 1.3 $\\text{m}^3$/s for the validation results discussed here. \n\n% \\begin{figure}[!htb]\n% \\begin{center}\n% \\includegraphics[width = 12 cm]{figs/hybrid_profile}\n% \\caption{Azimuthal and vertical velocity profiles as a function of\n% radius. The simulation and experimental data broadly agree, with\n% the simulation also exhibiting the characteristic ``twin-peak''\n% structure of the hybrid vanes in the azimuthal velocity. }\n% \\label{fig:lab}\n% \\end{center}\n% \\end{figure}\n\n\\begin{figure}[htp!]\n\n \\centering\n \\includegraphics[width =0.47\\textwidth]{figs/sim_vs_exp_30_vt}\n% \\caption{Azimuthal velocity}\n \\hfill\n \\includegraphics[width =0.47\\textwidth]{figs/sim_vs_exp_30_vz}%\n %\\caption{Vertical velocity} \n \\caption{Azimuthal (left figure) and vertical (right figure) velocity \n as a function of radius for the thermal-only cases. Shown are single\n tier straight $30^{\\circ}$ vanes. $V_{\\theta}^V$ (gold line) is\n the virtual vane simulation, $V_{\\theta}^E$ (blue line) the experiment,\n and $V_{\\theta}^G$ (red line) the gridded vane. These results were all\n generated by unsteady simulations and then temporally averaged. \n The lack of smoothness in the data is believed to be\n attributable to finite-time averaging, particularly in the case of the\n gridded vanes, which were expensive calculations. } \n \\label{fig:val_lab} \n\\end{figure}\n\nFigure~\\ref{fig:val_lab} is a direct comparison\nbetween laboratory measurements for a simple single tier vane\nconfiguration ($30^{\\circ}$ straight vanes) and nominally identical\nsimulations with the gridded and \nvirtual vanes. The simulations and experiment broadly agree. The\nsimulation correctly reproduce the peak structure in the azimuthal\nvelocity observed for this configuration in the experiment. The gridded\nvanes closely represent the peak radial location, while the\nvirtual vanes over-predict the radial location, likely due to the\nincreased eddy diffusivity that exists in the virtual vanes. The radial\nlocation and magnitude of peak vertical velocity also closely agrees\nwith experiment. \n\nSimilar validation comparisons have been made between several other\nconfigurations with similar levels of agreement, notably the\n$60^{\\circ}$ single tier straight vane case, and the two-tier hybrid\nvanes. \n%Some of these cases are detailed in Appendix~\\ref{app-validation}.\nThese validation studies have provided a level of confidence that our\nsimulations accurately reproduce the phenomena observed in laboratory.\n\n%\\todo{slices and images of thermal only?} \n\n\\section{Wind Cases}\n\nThe laboratory thermal vortex experiments described in the previous\nsection did not include the effects of the wind, but experience in\nthe field indicated that ambient winds were both pervasive and intense\n(see Section~\\ref{subsec:field_predict} for more details). To ensure that\nthe virtual vanes accurately represent the impact of ambient winds, a\nvalidation study was performed using the data obtained in the wind\ntunnel. \n\nA numerical experiment was performed in which the 60 degree single tier\nstraight vanes were placed in a isothermal wind. The boundary conditions\nare as detailed in Section~\\ref{sec:bc}, but in isothermal\nconditions. These results were compared to an identical experimental\nconfiguration placed in a wind tunnel. However, no measurements (of\nvelocity or any quantity) were made for the vanes in these\nconditions. Qualitative comparisons, based on descriptions of observed\nstructures and videos of smoke visualization were made between the\nsimulations and the wind tunnel experiments. The initial validation was\nfound to have significant qualitative differences between the virtual\nvanes and the experimental images, with the flow visibly exiting out the\nback of the vanes instead of being contained within. As a result of\nthis inconsistency, the virtual vane model was refined to include a\nseparation model, which is detailed in Section~\\ref{sec:separation}. The\nimages did not identify any inconsistencies between the refined\nsimulation and experiment. \n\n\n\\begin{figure}\n \\begin{subfigure}{0.49\\textwidth}\n \\centering\n \\includegraphics[width=.95\\linewidth]{figs/gridded_wind}\n \\caption{Streamwise Velocity: Gridded Vanes}\n \\end{subfigure}\n %\\hfill\n \\begin{subfigure}{0.49\\textwidth} \n \\centering\n \\includegraphics[width=.95\\linewidth]{figs/virtual_wind}\n \\caption{Streamwise Velocity: Virtual Vanes}\n \\end{subfigure} \\\\\n \\begin{subfigure}{0.49\\textwidth}\n \\centering\n \\includegraphics[width=.95\\linewidth]{figs/gridded_wind_span}\n \\caption{Spanwise Velocity: Gridded Vanes}\n \\end{subfigure}\n %\\hfill\n \\begin{subfigure}{0.49\\textwidth}\n \\centering\n \\includegraphics[width=.95\\linewidth]{figs/virtual_wind_span}\n \\caption{Spanwise Velocity: Virtual Vanes}\n \\end{subfigure}\n \\caption{Horizontal slices through the top of the vanes for the\n wind validation cases. On the left are the explicitly gridded vanes,\n and on the right the virtual vanes. The streamwise velocity (top\n images), which moves left to right, shows penetration through the\n region where the vanes are aligned with the flow in both the gridded\n and virtual vanes. The second row shows the spanwise velocity (red for\n flow moving up the page, blue for flow moving down the page) where it\n can be seen that the virtual vane case correctly reproduces the\n direction and magnitude of velocity inside the vanes.} \n \\label{fig:wind_val}\n\\end{figure}\n\n%\n% Attached in the plate from which we inject the fog from. The holes used\n% for seeding are the center one (0.5\u001b$B!I\u001b(B diameter) and the other 4 ones\n% which are 0.25\u001b$B!I\u001b(B diameter. The smallest holes are for alignment pins in\n% the tunnel so disregard those ones. Please note that this file is in\n% inches and the previous one in millimeters. \n% Hidalgo Ardana Pablo\n%\n% 11/26/14\n%\nHowever, these results are limited, and are\nonly for the cold wind, as the wind tunnel did not include a heated plate.\nTo provide a more quantitative validation study, the virtual vanes were\ncompared to gridded vanes for a cold wind case. Both cases had identical\nboundary conditions, as detailed in Section~\\ref{sec:bc}. A two meter\nper second inlet velocity was selected. Figure~\\ref{fig:wind_val}\ncontains images of the simulated averaged \nstreamwise and spanwise velocity in a horizontal plane at approximately\nthe height of the vanes obtained from simulations with gridded and\nvirtual vanes. \nThe streamwise velocity penetrates through the region where the vanes\nare aligned with the flow in both the gridded and virtual vanes. This\nindicates the virtual vane region as defined in\nSection~\\ref{subsec:vane} is not turning the flow in locations where\nthe flow velocity is aligned with the vane direction, as intended. \nThe second row contains images of the spanwise velocity, where it can be\nseen that the virtual vane case correctly reproduces the direction and\nmagnitude of velocity inside the vanes. While the wake has similar\nstructure between the two cases for the streamwise velocity, the\nspanwise velocity in the wake is not as closely represented between the\ncases. While there are some differences in the\ndetails of these simulations, the overall character of the flow\ninside the vanes is quite similar. This demonstrates that the virtual\nvane formulation can indeed accurately represent the interaction with the wind, \nat least in the case without thermal differences. \n\n\\section{Comparisons between Steady and Unsteady Virtual Vanes}\n\\label{sec:steady_val}\n\nAt the bottom of the validation hierarchy are the steady virtual\nvanes. This is expected to be the most inaccurate model. \nSimultaneously, this is by far the cheapest computationally, and \nreduces the run-time from approximately 12 hours (in the case of the\nunsteady) to less than two minutes. This reduced run-time permits\nrapid exploration of the SoV configuration space, making the steady\nvirtual vanes an invaluable design tool. For this reason, the steady\nmodel is used extensively in Chapters~\\ref{sec:results} and\n\\ref{sec:field} to explore new SoV vane design concepts. \n\nHowever, the results of the steady cases must first be\ncompared to the transient case to ensure the output is consistent. \nFigure~\\ref{fig:rom_compare} depicts such a case, where the streamwise\nvelocity in a wind case was used as a direct comparison between the\nsteady and unsteady virtual vane cases. This was a hot-wind case, with\nan ambient freestream velocity of 3 m/s and a temperature difference of 60\nKelvin. The boundary conditions are precisely as described in\nSection~\\ref{sec:bc}. \n%The vanes are an axi-symmetric two-tier structure,\n%with inner vane angles of and degrees, respectively. \n\nThese two solutions are similar, with comparable velocity magnitudes and \nconsistent signs. A comparison between the azimuthal velocities are\nshown in Figure~\\ref{fig:rom_az}, which makes clear the more diffuse and\nweaker magnitude peak in the steady solution. Ultimately, the steady\nsolution's principle use as a design tool is driven by the response in\nkinetic energy flux to sensitivity to small perturbations in the SoV\ndesign (such as vane or cone geometry). To measure this, a comparison\nwas made between steady and unsteady solutions to changes from a base state \nin response to small ($1^{\\circ}-5^{\\circ}$) perturbations in the vane\nangles. The results of this are shown in Figure~\\ref{fig:rom_it}, where\nthe steady solution typically underestimates, but broadly agrees with,\nthe change in kinetic energy flux attributable to a perturbation in the\nsystem design parameters. It is for this reason that the steady\nsolution is believed to be a useful tool to explore the system\nconfiguration space, as it accurately represents favorable design\nadjustments, and so can be used in an optimization effort to rapidly\nprobe various configurations and drive the system towards peak kinetic\nenergy flux generation. \n\n\\begin{figure}[htp!]\n \\begin{subfigure}{0.49\\textwidth}\n \\centering\n \\includegraphics[width =0.95\\textwidth]{figs/rom_steady}\n \\caption{Steady Virtual Vanes}\n \\end{subfigure}\n \\begin{subfigure}{0.49\\textwidth}\n %\\hfill\n \\includegraphics[width =0.95\\textwidth]{figs/rom_unsteady}%\n \\caption{Unsteady Virtual Vanes}\n \\end{subfigure}\n \\caption{A comparison between the streamwise velocity in the averaged\n transient virtual vane solution (right image) and the steady virtual\n vane solution (left image). These horizontal slices were taken at the\n height of the second tier of vanes. The black lines indicate the\n annular vane forcing region. While the steady solution is more diffuse,\n it possesses a similar qualitative structure as the higher fidelity\n solution. The unsteady solution has a larger peak velocity inside the\n apparatus, while simultaneously possessing a larger and more intense\n wake region.}\n \\label{fig:rom_compare} \n\\end{figure}\n\n% Flow features smeared\n% what type of vanes are these?\n% calibration\n\n\\begin{figure}[htp!]\n \\centering\n \\includegraphics[width =0.7\\textwidth]{figs/rom_radial}\n \\caption{The azimuthal velocity profile as a function of radius for the\n steady and unsteady cases. The profile is extracted from $\\theta =\n 90^{\\circ}$. The unsteady vanes were temporally averaged. The steady\n solution has a similar radial peak location, but a lower velocity\n magnitude and a more diffuse structure.} \n \\label{fig:rom_az}\n\\end{figure}\n\n\n% The results between these simulations are broadly consistent. \n% Across a range of previously tested configurations the kinetic energy\n% flux typically agrees to within 20\\% and the sensitivities of the energy\n% flux to small perturbations in the system configuration correlate well\n% with the results of the higher fidelity transient model. \n\n\\begin{figure}[htp!]\n \\centering\n \\includegraphics[width =0.7\\textwidth]{figs/rom_iterate}\n \\caption{A comparison between the change in kinetic energy flux due to\n a perturbation in system parameters (vane angle) between the steady and\n unsteady virtual vanes. For each iteration, a design parameter was\n changed, and the \\% change in kinetic energy flux was recorded.} \n \\label{fig:rom_it} \n\\end{figure}\n\n\\section{Field Configurations}\n\\label{sec:field_val}\n\nSeveral field tests have been performed by the experimental team. After\neach field test, qualitative observations, measurements and lessons\nlearned are provided by the field team. Actual measurements\nare limited. Due to the complexity of the configuration\n(two vane tiers and a cone) gridded vanes cases have not been developed \nfor the field configurations. This section provides a discussion of some of\nthe results from the latest field test, as an example of typical\nvalidations performed. \n\n \\begin{figure}[!htb]\n \\begin{center}\n \\includegraphics[width = 12 cm]{figs/validate_field}\n \\caption{A comparison between simulated and experimental data for the\n August 2015 field test. Azimuthal velocity data from the actual field\n test is shown in blue. Two virtual vane simulations with different\n scenario parameters are shown in red and gold. The velocity field was\n temporally averaged but not averaged in space, to reproduce\n the measurements from the field.}\n \\label{fig:field_val}\n \\end{center}\n \\end{figure}\n%\n% provide an example of these validations below\n%\n\nFigure~\\ref{fig:field_val} shows velocity data from the \nAugust 2015 field test in blue. These results were \nobtained using an anemometer at fixed\nazimuthal location (believed to be at a ninety degree angle, where the zero\nis defined to be aligned with the streamwise flow direction) to measure the\nazimuthal velocity. A time\nseries from approximately an hour was gathered. This data included large\nscenario uncertainties, with estimated 3 m/s variations in wind, 20\ndegree wind heading changes, and ten degree Celsius shifts in\ntemperature. Other than the velocities, the only other measurement\ntaken was the temperature. Due to a DAQ malfunction, the\ntemperature was only measured at one location at one meter in height. \nA solidworks CAD file provided by the experimental team\ndefined the vane and cone geometry, which were then represented in the\nsimulations as virtual vanes and a solid surface, as described in \nSections~\\ref{subsec:vane} and \\ref{subsec:solid_surface}. Hence, the\nscenario uncertainty was significant, while the uncertainty in the\nsystem apparatus was small. \n\n\\begin{table}[!htb]\n\\centering\n \\caption{The range of conditions represented in\n Figure~\\ref{fig:field_val}. The heading is with respect to the presumed\n wind direction. The heading may at first appear not to be significant,\n as the vanes are axisymmetric. However, the rake holding\n the anemometer was placed at a $90^{\\circ}$ relative to the presumed\n freestream velocity, and the SoV solution structure is not symmetric.}\n\\begin{tabular}{l|l|l|l}\nCase Name & Wind Velocity (m/s) & Heading & $T_\\text{Surface}$\n (Celsius) \\\\ \n \\hline\n Peak (Red Line) & 5.0 & $0^{\\circ}$ & $60^{\\circ}$ \\\\\n Match (Black Line) & 3.0 & $0^{\\circ}$ & $60^{\\circ}$ \\\\\n Minimum (Gold Line) & 2.0 & $20^{\\circ}$ & $20^{\\circ}$ \\\\\n\\end{tabular}\n \\label{tab:val_cond}\n\\end{table}\n\n\nTo span the range of scenario conditions, several simulations were conducted\nwith different parameters. These simulations were performed with\nunsteady virtual vanes on a wind blown domain as detailed in\nSection~\\ref{sec:bc}. The azimuthal velocity from two such\nsimulations (red and gold lines) are plotted against the experimental\ndata (blue line) in Figure~\\ref{fig:field_val}. The red line represents\nan upper bound, with the strongest wind speeds and highest thermal\ngradient. The gold line is a lower bound, with a more modest ambient\nfreestream velocity, a lower temperature gradient, and an indirect wind\nheading. \n\nThese simulations accurately bound the experimental data. Furthermore, a\n``Matching'' case was identified that is broadly consistent with the\nfield results. This corresponded to a 3 m/s wind velocity and a 60\ndegree Celsius temperature difference. These range of conditions are\nsummarized in Table~\\ref{tab:val_cond}. \nAs the rake was at fixed azimuthal location, the kinetic energy fluxes were\ncompared by assuming azimuthal symmetry and integrating in a horizontal\nplane at the top of the vanes (where a turbine to extract this energy\nwould likely be placed). By this metric, the ``Matching'' simulation\nkinetic energy flux agrees with the experimental estimate within\n10\\%. This is likely an optimistic measure, as the simulations indicate\nthat the velocity field is highly azimuthally asymmetric. Nevertheless,\nno significant inconsistencies have been identified between the\nexperimental results and the simulations. \n\n\n%\n% validation story is incomplete\n% \n% you have done: \n%\n% 1) comparions between laboratory + gridded + virtual \n% \n% 2) comparisons between gridded + virtual in laboratory\n% + field configurations, thermal only and wind\n%\n% 3) comparisons of virtual vanes to field observations \n% quantitative + qualitative\n%\n%\n% You need to discuss what needs to be validated -- the\n% comparison to gridded vanes is a useful validation tool \n%\n", "meta": {"hexsha": "286bcb3c6a0ca8a2b3635e5f859c04bb49ecd454", "size": 26440, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "disputatio/dissertation/chapter-validation.tex", "max_stars_repo_name": "nicholasmalaya/paleologos", "max_stars_repo_head_hexsha": "11959056caa80d3c910759b714a0f8e42f986f0f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-04T17:49:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-04T17:49:42.000Z", "max_issues_repo_path": "disputatio/dissertation/chapter-validation.tex", "max_issues_repo_name": "nicholasmalaya/paleologos", "max_issues_repo_head_hexsha": "11959056caa80d3c910759b714a0f8e42f986f0f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "disputatio/dissertation/chapter-validation.tex", "max_forks_repo_name": "nicholasmalaya/paleologos", "max_forks_repo_head_hexsha": "11959056caa80d3c910759b714a0f8e42f986f0f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-01-04T16:08:18.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-16T19:34:24.000Z", "avg_line_length": 46.6313932981, "max_line_length": 87, "alphanum_fraction": 0.7808245083, "num_tokens": 6295, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6442250996557036, "lm_q1q2_score": 0.34972615270122964}} {"text": "%==============================================================================\n\\chapter{Quantification of the Difference}\n\\label{chap:quantifing}\n%==============================================================================\n\nHaving shown explicitly the discrepancy between the pure and the thermal \ndescriptions as well as where it is most significant, the author will then\nquantify the difference. Related definitions of distances in quantum \nmechanics are introduced in \\cref{chap:distappd}.\n%In this chapter, it will be explained why the difference between the two cases \n%is expected to be negatively correlated to the mass of the black hole, then \n%the difference will be evaluated. \n\n%------------------------------------------------------------------------------\n\\section{The Canonical State} %Popescu--Winter Theorem\n\\label{sec:arbitrary-canonical}\n%------------------------------------------------------------------------------\n\nIt has been shown in \\cite{Popescu2006} that a small subsystem in a large, \nisolated system which is subject to a \\emph{constraint}, is expected to be very \nclose to a \\emph{canonical state} of it, which reduces to the usual \nthermodynamic \\emph{canonical ensemble} when the total system is under an \n\\emph{energy constraint}.\n\nDenoted by $U$, the large system has all its possible pure states in Hilbert \nspace $\\mscrH_U$ which has a finite dimension $d_R = \\dim \\mscrH_U$. A global \nconstraint $R$ is also imposed, which restricts the physical Hilbert space of \n$U$ to $\\mscrH_R \\subseteq \\mscrH_U$. The \\emph{equiprobable state} of $U$ is\n\\begin{equation}\n\\mscrE_R \\coloneqq d_R^{-1} \\Bbbone_R,\n\\end{equation}\nwhere $\\Bbbone_R$ is the identity operator on $\\mscrH_R$. An energy constraint, \nfor instance, reads\n\\begin{equation}\n\\abr{H_U} = E_R,\n\\label{eq:ene-constraint}\n\\end{equation}\nso that a state $\\Ket{\\alpha}$ in $\\mscrH_R^{\\rbr{\\text{E}}}$ satisfies\n\\begin{equation}\nE_R = \\Braket{\\alpha | H_U | \\alpha} = \\sum_E \\vbr{\\Braket{E | \\alpha}}^2 E,\n\\end{equation}\nwhere $\\Ket{E}$ is an energy eigenstate of $U$ with eigenvalue $E$.\n\nThe subsystem in concern is named $S$, with \\emph{all possible} states in the\nHilbert space $\\mscrH_S$; the rest of $U$ is called the \\emph{environment} and \ndenoted by $E$, the pure states of which lie in $\\mscrH_E$. One has\n\\begin{equation}\n\\mscrH_S \\otimes \\mscrH_E = \\mscrH_U \\supseteq \\mscrH_R,\\qquad\nH_U = H_S + H_E + H_\\text{int},\n\\end{equation}\nwhere $H_S$, $H_E$ and $H_\\text{int}$ are the system, environment and \ninteraction Hamiltonian, respectively. \n\nThe \\emph{canonical state} of $S$ is defined as\n\\begin{equation}\n{\\Omega}_S \\coloneqq \\tr_E {\\mscrE}_R,\n\\end{equation}\nwhere the trace is taken over all the degrees of freedom in the environment. \nWhen the energy constraint \\cref{eq:ene-constraint} is used, it can be shown \nthat $\\Omega_S^{(E)}$ is a canonical ensemble\\footnote{A derivation of \n\\cref{eq:canonical-state-def} in classical statistical physics can be found in \n\\cite[sec.~28]{Landau1980}.}\n\\begin{equation}\n\\Omega_S^{(E)} \\propto \\ee^{- H_S/T_\\text{m}}\n= \\sum_{E_S} \\ee^{-E_S/T_\\text{m}} \\Ket{E_S}\\Bra{E_S},\n\\label{eq:canonical-state-def}\n\\end{equation}\nwhere $T_\\text{m}$ can be identified with a temperature, and $\\Ket{E_S}$ is an\neigenstate of the system with eigenvalue $E$. In this case it reduces to \nthe traditional thermodynamic statistical physics.\n\n%To state the theorems, one additional definition is needed: \nThe environment also has an \\emph{effective} dimension\n\\begin{equation}\nd_E^\\text{eff} \\coloneqq \\rbr{\\tr \\Omega_E^2}^{-1} \\geq d_R/d_S,\n\\label{eq:def-effective-dim}\n\\end{equation}\nwhere ${\\Omega}_E = \\tr_S {\\mscrE}_R$. When no constraint is enforced, so that\n$\\mscrH_S \\otimes \\mscrH_E \\equiv \\mscrH_U = \\mscrH_R$, \n\\cref{eq:def-effective-dim} reduces to\n\\begin{equation}\nd_E^\\text{eff} = d_R/d_S = d_E.\n\\end{equation}\nDetailed discussions about $d_E^\\text{eff}$ can be found in \\cite{Popescu2006}.\n\nEquipped with all the definitions above, one picks an arbitrary pure state \n$\\Ket{\\phi}\\in\\mscrH_R$ and denote the \\emph{reduced state} of $S$ by\n\\begin{equation}\n\\rfun{\\rho_S}{\\phi} = \\tr_E \\Ket{\\phi}\\Bra{\\phi}\n\\end{equation}\nThen a lemma states that the average \\emph{trace distance}\\footnote{See \n\\cref{sec:trace-dist}.} between $\\rho_S$ and $\\Omega_S$ is very small in terms \nof the ratio between $d_S$ and $d_S/d_E^\\text{eff}$, i.e.\\\n\\begin{equation}\n\\abr{\\rfun{T}{\\rfun{\\rho_S}{\\phi}, {\\Omega}_S}} \\le \\frac{1}{2}\n\\sqrt{\\frac{d_S}{d_E^\\text{eff}}}.\n\\end{equation}\nIn a typical division where $d_S/d_E$ is small, this average distance will also \nbe tiny.\n\nMore over, the main theorem asserts that those ${\\rho}_S$'s which are close to \n${\\Omega}_S$ dominate; the probability of a large deviation is exponentially \nsmall with respect to the deviation. For an arbitrary $\\epsilon > 0$, the \ntheorem states that\n\\begin{equation}\n\\frac{\\sfun{V}{\\cbr{\\Ket{\\phi} \\in \\mscrH_R |\n\\rfun{T}{\\rfun{\\rho_S}{\\phi}, \\Omega_S} \\geq \\eta}}}\n{\\sfun{V}{\\cbr{\\Ket{\\phi} \\in \\mscrH_R}}} \\leq \\eta',\n\\label{eq:main-theorem-Popescu}\n\\end{equation}\nwhere\n\\begin{equation}\n\\eta = \\epsilon + \\frac{1}{2}\\sqrt{\\frac{d_S}{d_E^\\text{eff}}}; \\qquad\n\\eta' = 4\\rfun{\\exp}{-C d_R \\epsilon^2},\\quad C = \\frac{2}{9\\pp^3}.\n\\label{eq:main-theorem-supp}\n\\end{equation}\nTo understand the theorem, first note that the left-hand side of \n\\cref{eq:main-theorem-Popescu} is a probability measure. More over, one can \nchoose $\\epsilon = d_R^{-1/3}$ as well, so that\n\\begin{equation}\n\\eta = \\epsilon + \\frac{1}{2}\\sqrt{\\frac{d_S}{d_E^\\text{eff}}} \\gtrsim \nd_R^{-1/3};\\qquad\n\\eta' = 4\\rfun{\\exp}{-C d_R \\epsilon^2} = 4\\rfun{\\exp}{-C d_R^{+1/3}},\n\\end{equation}\nwhere $d_E^\\text{eff} \\gg d_S$ is also assumed. In this case, the probability \nof the deviation greater than $d_R^{-1/3}$ is smaller than an \n\\emph{exponential} of $d_R^{+1/3}$.\n\nFurther explanations and proofs of the lemma and the theorem can \nbe found in \\cite{Popescu2006,Popescu2007}.\n\n%------------------------------------------------------------------------------\n\\section{Distance between the Pure and Mixed States}\n\\label{sec:dist-radiation}\n%------------------------------------------------------------------------------\n\nTo apply the aforementioned formalism to the Hawking radiation, one recognises \nthe universe in the CGHS model as the total isolated system $U$, whereas the \ngravitational degrees of freedom as $E$, and the system in concern $S$ is the \nradiation field. It has been shown in \\cite{Demers1996,Kiefer2001} that the \nreduced state of the radiation field can indeed be \\emph{thermal} in certain \n\\emph{decoherence} schemes, while in the collapsing case discussed in \n\\cref{sec:hawrad-1+1dila,sec:1+1ddlt}, the radiation field remains pure.\n\nIn this section, the trace distance between the pure and thermal descriptions \nwill be evaluated. Due to technical difficulties, the distance has not been able \nto be derived exactly. Instead, a lower and an upper bound have been set to the \ndistance by \\emph{Fuchs-van de Graaf inequality} in \\cref{eq:ineq-fvdg}, where \nonly the calculable \\emph{Fidelity} (see \\cref{sec:fidelity}) is needed.\n\nNote that the wave functional of the Hawking radiation field \n\\cref{eq:squeezed-wave-functional} is, roughly speaking, the superposition of \nquantum-mechanical wave functions per Fourier mode,\n\\begin{equation}\n\\sfun{\\chi_b}{g} \\sim \\sum_p \\rfun{\\chi_b^\\rbr{p}}{g_p}\n= \\sum_p \\Braket{g_p | \\chi_b^\\rbr{p}},\n\\label{eq:wave-function-decomposed}\n\\end{equation}\nwhere $g_p \\coloneqq \\rfun{g}{p}$ is the Fourier transform of the field $g$ \nevaluated at $p$. On the other hand, the thermal state of the free field can \nalso be sloppily written as the product of the quantum-mechanical density \noperators per mode,\n\\begin{equation}\n{\\rho}_\\text{th} \\sim \\bigotimes_p {\\rho}_\\text{th}^\\rbr{p}.\n\\end{equation}\nBy \\cref{eq:fidelity-pure-mixed}, the fidelity of $\\sfun{\\chi_b}{g}$ and \n$\\rho_\\text{th}$ can be reduced to the product of the fidelity per mode, because\n\\begin{equation}\n\\Braket{\\chi_b | {\\rho}_\\text{th} | \\chi_b} \\sim \\prod_p \n\\Braket{\\chi_b^\\rbr{p} | {\\rho}_\\text{th}^\\rbr{p} | \\chi_b^\\rbr{p}}.\n\\label{eq:expectation-decomposed}\n\\end{equation}\n\nSince $\\Ket{\\chi_b^\\rbr{p}}$ is just a quantum-mechanical general Gaussian \nstate (see \\cref{sec:single-harosc}), the result in \\cref{eq:fidelity-gg-th} \ncan be adapted by substituting\n\\begin{equation}\n\\Omega = \\vbr{p},\\qquad \\omega = p\\,\\coth\\frac{\\pp p}{2\\lambda}\\quad\n\\text{and}\\quad T = T_\\text{HD} \\equiv \\frac{\\lambda}{2\\pp},\n\\end{equation}\nyielding the \\emph{fidelity per mode}\n\\begin{equation}\nF^\\rbr{p} = \\frac{\\sqrt{u-1}}{\\sqrt[4]{u^2+u+1}},\\qquad\nu \\coloneqq \\ee^q \\equiv \\ee^{\\vbr{p}/T_\\text{HD}},\n\\label{eq:fidelity-pt}\n\\end{equation}\nso that the trace distance per Fourier mode can be evaluated, see \n\\cref{fig:trace-dist-per-mode}.\n\n\\begin{figure}\n\\begin{center}\n\\input{./graphics/graph_trace_dist_mode_bounds}\n\\end{center}\n\\caption[Possible value of the trace distance per Fourier mode]{Possible value \nof the trace distance between mode wave functions in \n\\cref{eq:wave-function-decomposed} and the corresponding thermal density \noperators. One sees that the difference becomes exponentially small with respect \nto $p/T_\\text{HD}$, which suggests it would be difficult to distinguish the pure \nand thermal descriptions by detecting the high-energy modes in the Hawking \nradiation, confirming the results in \\cref{sec:corr_Fourier}.\n\\label{fig:trace-dist-per-mode}}\n%, evaluated in terms of lower and upper bound by fidelity\n\\end{figure}\n\nTo approach the trace distance for the wave functional and the total thermal \nstate, one needs to deal with the product with continuous index in \n\\cref{eq:expectation-decomposed}. A popular way to go to the `continuous limit' \nin \\emph{summation} is\n\\begin{equation}\n\\sum_p \\rfun{g}{p} \\to \\frac{1}{2\\pp\\Lambda} \\int \\dd p\\,\\rfun{g}{p},\n\\end{equation}\nwhere $\\Lambda$ has the same dimension as $p$ in order to fix the dimension; in \nthe box-normalisation scheme, for example, the corresponding $\\Lambda$ would be \nproportional to the volume of the box $V$. A similar method may be used to \nnormalise the product, namely\n\\begin{equation}\n\\prod_p \\rfun{f}{p} = \\cfun{\\exp}{\\sum_p \\ln\\rfun{f}{p}} \\to \n\\cfun{\\exp}{\\frac{1}{2\\pp\\Lambda}\\int \\dd p\\,\\ln\\rfun{f}{p}}.\n\\end{equation}\nNote that the wave functional \\cref{eq:squeezed-wave-functional}, which has \nalways been dealt with, can also be seen as being normalised by the method.\nOne thus derives\n\\begin{equation}\nF = \\cfun{\\exp}{\\frac{2}{2\\pp\\Lambda}\\int_0^{+\\infty}\\dd p\\,\\ln F^\\rbr{p}}\n= \\rfun{\\exp}{-\\frac{\\pp}{9} \\frac{T_\\text{HD}}{\\Lambda}},\n\\end{equation}\nwhich is shown in \\cref{fig:trace-dist-total}.\n\n\\begin{figure}\n\\begin{center}\n\\input{./graphics/graph_trace_dist_total_bounds}\n\\end{center}\n\\caption[Possible value of the trace distance between pure and thermal \ndescriptions]{Possible value of the trace distance between the pure and thermal \ndescriptions of the Hawking radiation field. One sees that the difference \nbetween the two cases is positively (negatively) correlated with the Hawking \ntemperature temperature (mass of black hole). \n\\label{fig:trace-dist-total}}\n\\end{figure}\n\n%------------------------------------------------------------------------------\n\\section{Discussion}\n\\label{sec:dist-discussion}\n%------------------------------------------------------------------------------\nStrictly speaking, the formalism in \\cref{sec:arbitrary-canonical} is \\emph{not}\napplicable to the radiation field considered here, because they were proved for \n\\emph{finite dimensional systems}, while the cases in this work are all \n\\emph{infinite-dimensional}. However, since a regularised result can be \nobtained, it is believable that a mathematically rigorous approach exists, which \nwill justify the result, similar to the renormalisation procedure in quantum \nfield theory.\n\nThough the asymptotic behaviour of the trace distance with respect to Hawking \ntemperature has clearly been revealed by setting bounds to it, it is still \nappealing to calculate its \\emph{exact value}. One possible approach is by \nusing the results in \\cite{Joos1985}, where the eigenfunctions and eigenvalues \nof a general Gaussian density operator have been explicitly solved.\n\nThe evaluation of the trace distance was motivated by \\citeauthor{Hsu2009} in \n\\cite{Hsu2009}, where operational meaning of the distance was also discussed in \nterms of toy models, which is unfortunately not applicable to the field system. \nIn future works, it is expected that the distance between the pure and thermal \nstates will be understood in terms of \\emph{experiments} or \\emph{observations}, \nwhere the essence of physical science lies.\n\n\n%%% Local Variables: \n%%% mode: latex\n%%% TeX-master: \"../mythesis\"\n%%% End: \n", "meta": {"hexsha": "eaaa35262fd7c611f73d5dd9390beff24031777d", "size": 12770, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ubonn-thesis-current/mythesis/sections/thesis_dist.tex", "max_stars_repo_name": "cmp0xff/Masterarbeit", "max_stars_repo_head_hexsha": "b29c84f9a29e4a7c9a3499658a1dfa7f87d64c9c", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ubonn-thesis-current/mythesis/sections/thesis_dist.tex", "max_issues_repo_name": "cmp0xff/Masterarbeit", "max_issues_repo_head_hexsha": "b29c84f9a29e4a7c9a3499658a1dfa7f87d64c9c", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ubonn-thesis-current/mythesis/sections/thesis_dist.tex", "max_forks_repo_name": "cmp0xff/Masterarbeit", "max_forks_repo_head_hexsha": "b29c84f9a29e4a7c9a3499658a1dfa7f87d64c9c", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.1217712177, "max_line_length": 81, "alphanum_fraction": 0.7041503524, "num_tokens": 3764, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442250928250375, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3497261489931118}} {"text": "\\hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block}{}\\section{numpp\\+:\\+:matrix\\+:\\+:sparse\\+:\\+:block$<$ T, Rows, Columns $>$ Class Template Reference}\n\\label{classnumpp_1_1matrix_1_1sparse_1_1block}\\index{numpp\\+::matrix\\+::sparse\\+::block$<$ T, Rows, Columns $>$@{numpp\\+::matrix\\+::sparse\\+::block$<$ T, Rows, Columns $>$}}\n\n\nBlock version of sparse matrix. \n\n\n\n\n{\\ttfamily \\#include $<$sparse\\+\\_\\+block.\\+hpp$>$}\n\n\\subsection*{Public Types}\n\\begin{DoxyCompactItemize}\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_ae63cfc45208606404488f5fe326bb1d3}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_ae63cfc45208606404488f5fe326bb1d3}} \nusing {\\bfseries value\\+\\_\\+type} = T\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a9cc87ccc78269d5061ede24a826095b7}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a9cc87ccc78269d5061ede24a826095b7}} \nusing {\\bfseries size\\+\\_\\+type} = std\\+::size\\+\\_\\+t\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a2e487b67c00d269e7e64f59548e9de61}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a2e487b67c00d269e7e64f59548e9de61}} \nusing {\\bfseries reference} = T \\&\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a469a9838360107bb0ad15b5f8b22910f}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a469a9838360107bb0ad15b5f8b22910f}} \nusing {\\bfseries const\\+\\_\\+reference} = const T \\&\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a51437641f0fb291201fa88ade1ecc187}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a51437641f0fb291201fa88ade1ecc187}} \nusing {\\bfseries pointer} = T $\\ast$\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a810d06f6c2e6b0e92f73bbb971eeff7c}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a810d06f6c2e6b0e92f73bbb971eeff7c}} \nusing {\\bfseries const\\+\\_\\+pointer} = const T $\\ast$\n\\end{DoxyCompactItemize}\n\\subsection*{Public Member Functions}\n\\begin{DoxyCompactItemize}\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a83470439e34e65f8abc0d59c81bf09c4}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a83470439e34e65f8abc0d59c81bf09c4}} \n{\\bfseries block} (const \\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{block} \\&)=default\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a48e8a9c7c0610bb7e8b84b67db6e5a6c}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a48e8a9c7c0610bb7e8b84b67db6e5a6c}} \n{\\bfseries block} (\\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{block} \\&\\&)=default\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_aecf67898fb5cd432a940c4cb22b6790b}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_aecf67898fb5cd432a940c4cb22b6790b}} \n\\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{block} \\& {\\bfseries operator=} (\\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{block} \\&\\&)=default\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a8ba9f28a47fd22af6fc63d6ba2a467ce}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a8ba9f28a47fd22af6fc63d6ba2a467ce}} \n\\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{block} \\& {\\bfseries operator=} (const \\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{block} \\&)=default\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_ab613bba3b224199917249af76563a7c3}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_ab613bba3b224199917249af76563a7c3}} \n{\\bfseries block} (std\\+::initializer\\+\\_\\+list$<$ std\\+::initializer\\+\\_\\+list$<$ T $>$$>$ elements)\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a0cafdc2b3674aa15cfe3575ed8d797c4}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a0cafdc2b3674aa15cfe3575ed8d797c4}} \nsize\\+\\_\\+type {\\bfseries size} () const noexcept\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a36977c382b70c3865ab252138d4617f4}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a36977c382b70c3865ab252138d4617f4}} \nconst std\\+::array$<$ std\\+::vector$<$ T $>$, Rows $>$ \\& {\\bfseries data} () const noexcept\n\\item \n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_acd407b6661d5051136ec040a009651d8}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_acd407b6661d5051136ec040a009651d8}} \nstd\\+::array$<$ std\\+::vector$<$ T $>$, Rows $>$ \\& {\\bfseries data} () noexcept\n\\item \n{\\footnotesize template$<$typename U $>$ }\\\\auto \\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block_a9ee2ec26bf97fe00dfc11957de26156c}{operator$\\ast$} (const \\hyperlink{classnumpp_1_1vector}{numpp\\+::vector}$<$ U, Columns, false $>$ \\&vec)\n\\end{DoxyCompactItemize}\n\n\n\\subsection{Detailed Description}\n\\subsubsection*{template$<$typename T, std\\+::size\\+\\_\\+t Rows, std\\+::size\\+\\_\\+t Columns$>$\\newline\nclass numpp\\+::matrix\\+::sparse\\+::block$<$ T, Rows, Columns $>$}\n\nBlock version of sparse matrix. \n\n\n\\begin{DoxyTemplParams}{Template Parameters}\n{\\em T} & arithmetic type contained in matrix class \\\\\n\\hline\n{\\em Rows} & number of rows in matrix \\\\\n\\hline\n{\\em Columns} & number of columns in matrix\\\\\n\\hline\n\\end{DoxyTemplParams}\n{\\bfseries This implementation is an experiment in cache hit-\\/rate.~\\newline\n } Whole sparse matrix is contained in one array, divided based on blocks of contiguous non-\\/zero numbers.~\\newline\nAfter each block operation calculating row is performed.\n\n\\begin{DoxyWarning}{Warning}\n{\\bfseries For parallelization of the sparse matrix dense vector multiplication you need compiler with Open\\+MP support} \n\nOnly sparse matrix dense vector multiplication implemented \n\n{\\bfseries Each row has to be seperate std\\+::initializer\\+\\_\\+list}\n\\end{DoxyWarning}\n{\\bfseries Pros\\+:}\n\\begin{DoxyItemize}\n\\item Obviously lower memory overhead, for 50\\% sparse matrix with no blocks it contains only 150\\% storage of dense matrix.\n\\item Higher cache locality (at least in theory)\n\\end{DoxyItemize}\n\n{\\bfseries Cons\\+:}\n\\begin{DoxyItemize}\n\\item Impossible S\\+I\\+MD optimization\n\\item Additional operations of read for every block\n\\end{DoxyItemize}\n\n{\\bfseries Example\\+:} \n\\begin{DoxyCode}\n\\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1nested}{numpp::matrix::sparse::nested} mat\\{\\{0, 0, 1\\},\n \\{0, 2.3, 0\\},\n \\{0, 0, 13.7\\}\n \\};\nconstexpr numpp::matrix::vector vec\\{1,2,3\\};\n\\textcolor{keyword}{auto} result = mat*vec;\n\\end{DoxyCode}\n\n\n\n\\begin{DoxyCode}\n\\textcolor{preprocessor}{#include\"numpp/structures/matrices/sparse.hpp\"}\n\\end{DoxyCode}\n \n\n\\subsection{Member Function Documentation}\n\\mbox{\\Hypertarget{classnumpp_1_1matrix_1_1sparse_1_1block_a9ee2ec26bf97fe00dfc11957de26156c}\\label{classnumpp_1_1matrix_1_1sparse_1_1block_a9ee2ec26bf97fe00dfc11957de26156c}} \n\\index{numpp\\+::matrix\\+::sparse\\+::block@{numpp\\+::matrix\\+::sparse\\+::block}!operator$\\ast$@{operator$\\ast$}}\n\\index{operator$\\ast$@{operator$\\ast$}!numpp\\+::matrix\\+::sparse\\+::block@{numpp\\+::matrix\\+::sparse\\+::block}}\n\\subsubsection{\\texorpdfstring{operator$\\ast$()}{operator*()}}\n{\\footnotesize\\ttfamily template$<$typename T , std\\+::size\\+\\_\\+t Rows, std\\+::size\\+\\_\\+t Columns$>$ \\\\\ntemplate$<$typename U $>$ \\\\\nauto \\hyperlink{classnumpp_1_1matrix_1_1sparse_1_1block}{numpp\\+::matrix\\+::sparse\\+::block}$<$ T, Rows, Columns $>$\\+::operator$\\ast$ (\\begin{DoxyParamCaption}\\item[{const \\hyperlink{classnumpp_1_1vector}{numpp\\+::vector}$<$ U, Columns, false $>$ \\&}]{vec }\\end{DoxyParamCaption})\\hspace{0.3cm}{\\ttfamily [inline]}}\n\nFunction multiplying sparse matrix and dense vector\n\n\n\\begin{DoxyParams}{Parameters}\n{\\em vec} & Vector which will be multiplied \\\\\n\\hline\n\\end{DoxyParams}\n\\begin{DoxyReturn}{Returns}\nvector$<$\\+U, Rows, false$>$ \n\\end{DoxyReturn}\n\n\nThe documentation for this class was generated from the following file\\+:\\begin{DoxyCompactItemize}\n\\item \nstructures/matrices/sparse/sparse\\+\\_\\+block.\\+hpp\\end{DoxyCompactItemize}\n", "meta": {"hexsha": "efc3c28b1b7817e435bfed9091c928d3a8cbab7d", "size": 7867, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/classnumpp_1_1matrix_1_1sparse_1_1block.tex", "max_stars_repo_name": "szymonmaszke/numpp", "max_stars_repo_head_hexsha": "9149c9d81f70a6ce833fdd1d2f0f2b584e2ac4d9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 10, "max_stars_repo_stars_event_min_datetime": "2018-06-06T01:51:17.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-02T15:17:00.000Z", "max_issues_repo_path": "docs/classnumpp_1_1matrix_1_1sparse_1_1block.tex", "max_issues_repo_name": "vyzyv/numpp", "max_issues_repo_head_hexsha": "9149c9d81f70a6ce833fdd1d2f0f2b584e2ac4d9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2018-11-28T12:15:46.000Z", "max_issues_repo_issues_event_max_datetime": "2018-12-16T00:03:38.000Z", "max_forks_repo_path": "docs/classnumpp_1_1matrix_1_1sparse_1_1block.tex", "max_forks_repo_name": "szymonmaszke/numpp", "max_forks_repo_head_hexsha": "9149c9d81f70a6ce833fdd1d2f0f2b584e2ac4d9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2017-08-06T13:58:27.000Z", "max_forks_repo_forks_event_max_datetime": "2018-04-06T06:45:22.000Z", "avg_line_length": 55.013986014, "max_line_length": 316, "alphanum_fraction": 0.7799669506, "num_tokens": 3042, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.626124191181315, "lm_q1q2_score": 0.3495820469459447}} {"text": "\\documentclass[main.tex]{subfiles}\n\\begin{document}\n\n\\marginpar{Tuesday\\\\ 2020-11-3, \\\\ compiled \\\\ \\today}\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{figures/roche-equipotential}\n\\caption{Equipotential lines for the Roche potential \\(\\phi _R\\). \\(M_{1, 2}\\) mark the two masses, while the Lagrange points\\(L\\) are the ones in which the potential is stationary: \\(\\nabla \\phi _R = 0\\).}\n\\label{fig:roche-equipotential}\n\\end{figure}\n\n% The equation of motion of a particle in the corotating frame is \n% %\n% \\begin{align}\n% \\pdv{\\vec{v}}{t} \n% + \\qty(\\vec{v} \\cdot \\vec{\\nabla}) \\vec{v}\n% = - \\frac{1}{\\rho } \\vec{\\nabla} P \n% - \\vec{\\omega} \\wedge \\vec{v}\n% - \\vec{\\nabla} \\phi _R\n% \\,,\n% \\end{align}\n% %\n% where \\(\\phi _R\\) is the Roche potential, which can be written in terms of the parameters \\(q = M_2 / M_1 \\) and \\(a\\);\n%\n% \\todo[inline]{there might be a wrong sign in the last lecture!} \n%\n% \\begin{align}\n% \\phi _R = - \\frac{GM_1}{\\abs{\\vec{r} - \\vec{r}_1}}\n% - \\frac{GM_2}{\\abs{\\vec{r} - \\vec{r}_2}}\n% - \\frac{1}{2} \\qty(\\vec{\\omega} \\wedge \\vec{r})^2\n% \\,.\n% \\end{align}\n% \\todo[inline]{insert diagram for the quantities in the potential}\n\nAlso, if \\(\\theta \\) is the angle between \\(\\vec{\\omega}\\) and \\(\\vec{r}\\) we can write \n%\n\\begin{align}\n\\qty(\\vec{\\omega} \\wedge \\vec{r})^2 = \\omega^2 r^2 \\sin^2 \\theta \n\\,.\n\\end{align}\n\nLet us come back to the Roche potential, whose equipotential contours we sketch in figure \\ref{fig:roche-equipotential}. \nIn terms of the aforementioned variables \\(q\\) and \\(\\theta \\) it can be written as\n%\n\\begin{align}\n\\phi _R &= \n- G (M_1 + M_2 ) \\qty[ + \\frac{1}{2} \\frac{\\omega^2 r^2 \\sin^2\\theta }{G (M_1 + M_2 )}\n+ \\frac{GM_1 }{G (M_1 + M_2 ) \\abs{\\vec{r} - \\vec{r}_1 }}\n+ \\frac{GM_2 }{G (M_1 + M_2 ) \\abs{\\vec{r} - \\vec{r}_2 }}\n ] \\\\\n &= - G (M_1 + M_2 )\n \\qty[\n \\frac{1}{2} \\frac{\\omega^2r^2 \\sin^2\\theta }{G(M_1 + M_2 )} \n + \\frac{1}{(1 + q) \\abs{\\vec{r} - \\vec{r}_1}}\n + \\frac{q}{(1 + q) \\abs{\\vec{r} - \\vec{r}_2}}\n ]\n\\,.\n\\end{align}\n\nWe can then use Cartesian coordinates centered in \\(M_1 \\): \nthen, \n%\n\\begin{align}\n\\phi _R = - \\frac{G (M_1 + M_2 )}{2 a^3}\n\\qty[ \\frac{2 a^3}{1 + q}\n\\frac{1}{\\sqrt{x^2 +y^2 +z^2}}\n+\n\\frac{2 q a^3}{1 + q} \n\\frac{1}{\\sqrt{(x-a)^2 + y^2 +z^2}}\n+ \\qty[(x - x _{\\text{CM}})^2 + y^2]\n]\n\\,,\n\\end{align}\n%\nwhere \n%\n\\begin{align}\nx _{\\text{CM}} = \\frac{qa}{1 + q}\n\\,,\n\\end{align}\n%\nso if we rescale all the spatial coordinates by the major semi axis \\(a\\) we get \n%\n\\begin{align}\n\\phi _R = - \\frac{G (M_1 + M_2 )}{2a}\n\\qty[\n\\frac{2}{1 +q}\n\\frac{1}{\\sqrt{x^2 +y^2 +z^2}} \n+\n\\frac{2 q}{1 + q} \n\\frac{1}{\\sqrt{(x-1)^2 + y^2 + z^2}}\n+ \\qty[\\qty(x - \\frac{q}{1 + q})^2 + y^2]\n]\n\\,.\n\\end{align}\n\nIf the stars are stationary, or in slow evolution. Then, their surfaces will be (at least in first approximation) at rest. Therefore, the derivative terms vanish, and we get \n%\n\\begin{align}\n0 = - \\frac{1}{\\rho } \\vec{\\nabla} P - \\vec{\\nabla} \\phi _R\n\\,,\n\\end{align}\n%\nbut the surface is defined by \\(\\vec{\\nabla} P = 0\\), which by this equation corresponds to \\(\\vec{\\nabla} \\phi _R = 0\\). \n\nThe star will then take the shape of an equi-Roche potential surface. \nWhat is this shape? See \\ref{fig:roche-equipotential}.\n\n% \\todo[inline]{Make 2D plots, cuts of the surface}\n\n\\subsection{The Roche lobe}\n\nWe have roughly spherical contours near the stars, and a figure-eight contour eventually.\nThe center of this contour is called \\(L_1 \\), the first or ``inner'' Lagrange point. \n\nAnalogy with a dog food container for Roche Lobe overflow.\n\nIs the overflow stable? It depends on whether the volume of the Roche lobe increases or decreases as the donor star loses mass: in the latter case. \n\n% Tight worn jeans? \n\nWe can introduce a characteristic \\textbf{radius} of the Roche Lobe, calculated by \n%\n\\begin{align}\nV _{\\text{lobe}} = \\frac{4}{3} \\pi R _{\\text{lobe}}^3\n\\,,\n\\end{align}\n%\nalthough the lobe is not a sphere this allows us to give a characteristic number. \nWe can calculate \\cite[]{eggletonApproximationsRadiiRoche1983}:\n%\n\\begin{align}\n\\frac{R_{\\text{lobe}}}{a} = f(q) \\approx\n\\begin{cases}\n \\num{.38} + \\num{.2} \\log_{10} q & \\num{.5} < q < \\num{20}\\\\\n \\num{.46} \\qty( \\frac{q}{1 + q})^{1/3}\n & 0 < q < \\num{.5}\n\\end{cases} \n\\,,\n\\end{align}\n%\nand we can see that the dependence on \\(q\\) is quite weak. A plot of this is shown in figure \\ref{fig:roche-lobe-radius}:\n%\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{figures/roche-lobe-radius}\n\\caption{Radius of the Roche lobe as a function of \\(q\\).}\n\\label{fig:roche-lobe-radius}\n\\end{figure}\n\nHowever, the orbital separation \\(a\\) is itself a function of \\(q\\), and the dependence \\(a(q)\\) is much more relevant than the dependence of \\(R _{\\text{lobe}} / a\\). \n\nIn order to calculate \\(a(q)\\), let us make some assumptions. \n\\begin{enumerate}\n \\item \\(M_1 + M_2 = \\const\\). This is realistic: it is hard for the binary star system as a whole to lose or gain mass.\n \\item The total angular momentum \\(L _{\\text{tot}}\\) is a constant. This is a bit tricky: even a small amount of mass loss can result in high angular momentum loss. \n \\item \\(L _{\\text{tot}} = L _{\\text{orb}}\\): all the angular momentum is orbital. We are neglecting the spins of the stars, and the angular momentum of the gas. This is realistic, since tidal forces move the configuration towards a tidally locked state. \n\\end{enumerate}\n\nThe conservation of mass tells us that \\(M_1 (1 + q)\\) is constant. \nThe orbital angular momentum is written in terms of the distances of the stars from the center of mass:\n%\n\\begin{align}\nL _{\\text{orb}} &= M_1 v_1 a_1 + M_2 v_2 a_2 \\\\\n&= M_1 a_1^2 \\omega + M_2 a_2^2 \\omega \n\\,,\n\\end{align}\n%\nso if we place the origin of our coordinates in the center of mass we can write everything in terms of the orbital separation \\(a = a_1 + a_2 \\):\n% \\todo[inline]{So \\(a\\) is not the semimajor axis}\n%\n\\begin{align}\n-M_1 a_1 + M_2 a_2 &= 0 \\\\\n- (M_1 + M_2 ) a_1 + M_2 a &= 0 \\\\\na_1 &= \\frac{qa}{1+q} \\\\\na_2 &= \\frac{a}{1 +q}\n\\,.\n\\end{align}\n\nIn terms of angular momentum we get \n%\n\\begin{align}\n\\qty[\n M_1 \\frac{q^2 a^2}{(1 + q)^2}\n + M_2 \\frac{a^2}{(1+q)^2}\n]\\omega &= \\const \\\\\nM_1 a^2\n\\qty[\n \\frac{q^2}{(1 + q)^2}\n + \\frac{q}{(1+q)^2}\n]\\omega &= \\const \n\\,,\n\\end{align}\n%\nwhich means \n%\n\\begin{align}\n\\frac{a^2 M_1 \\omega q}{1 + q} =\n\\frac{a^2 M_1 2 \\pi q}{P(1 + q)} \n= \\const\n\\,,\n\\end{align}\n%\nsince \\(\\omega = 2\\pi / P\\), where \\(P\\) is the orbital period. \nWe also know that (because of mass conservation) \\(M_1 \\propto (1 + q)^{-1}\\), therefore \n%\n\\begin{align}\n\\frac{a^2 q}{(1+q)^2} \\frac{1}{P}\n\\,,\n\\end{align}\n%\nbut also Kepler's third law tells us that \\(a^3 \\propto P^2\\): so, \\(P \\propto a^{3/2}\\). This yields \n%\n\\begin{align}\n\\frac{a^{2 - 3/2} q}{(1+q)^2} = \\sqrt{a} \\frac{q}{(1+q)^2} = \\const\n\\,,\n\\end{align}\n%\ntherefore \n%\n\\begin{align}\na \\propto \\frac{(1 + q)^{4}}{q^2}\n\\,.\n\\end{align}\n\nThe Roche lobe \\(R_2\\) for \\(M_2\\) is given by \\(R_2 = a f\\), therefore\n%\n\\begin{align}\n\\log R_2 = \\log a + \\log f\n\\,,\n\\end{align}\n%\nand if we consider a variation of \\(q\\), the variation of \\(\\log R_2 \\) will be given by the sum of the variations of the two logarithms. \n\nBecause of the expression we derived, we have \n%\n\\begin{align}\n\\log a &= 4 \\log (1 +q) - 2 \\log q + \\const \\\\\n\\Delta \\log a &= 4 \\frac{\\Delta q}{1 + q} - 2 \\frac{\\Delta q}{q } \\\\\n&= \\frac{4q -2 -2 q}{(1 + q) q} \\Delta q = 2 \\frac{\\Delta q}{q} \\frac{q -1}{q +1}\n\\,.\n\\end{align}\n\nThen, the fractional variation is approximately \n%\n\\begin{align}\n\\frac{\\Delta R_2}{R_2} = \\Delta \\log R_2 \\approx \n2\\frac{\\Delta q}{q} \\frac{q -1}{q +1}\n\\,,\n\\end{align}\n%\nsince \\(\\Delta \\log f\\) is negligible (see figure \\ref{fig:roche-lobe-relative-corrections}). \nAs the star \\(M_2 \\) donates mass, \\(M_2 \\) decreases and \\(M_1 \\) increases. Then, \\(q = M_2 / M_1 \\) decreases, therefore \\(\\Delta q < 0\\). \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\textwidth]{figures/roche-lobe-relative-corrections.pdf}\n\\caption{Relative size of the corrections --- \\(\\Delta \\) means derivative with respect to \\(q\\). One can notice that for typical \\(q \\gtrsim 2\\) the magnitude of \\(\\Delta \\log a\\) is about one order of magnitude larger than that of \\(\\Delta \\log f\\). Also, we can see that \\(\\Delta \\log a + \\Delta \\log f\\) is indeed positive for \\(q > 1\\) --- the curve is not shown in the region in which it is negative.}\n\\label{fig:roche-lobe-relative-corrections}\n\\end{figure}\n\nIf we want the Roche lobe to shrink, we want \\(\\Delta R_2 < 0\\): so, we want \\(q - 1 > 0\\). \nThis means that \\textbf{Roche lobe accretion is self-sustaining, or stable iff the donor star is larger than the receiver}. \n% Including \\(\\Delta f\\), things don't change by much. \nIf we include \\(\\Delta f\\), as one can see in figure \\ref{fig:roche-lobe-relative-corrections}, the picture is quite similar --- the noticeable correction we have is that systems with \\(\\num{.8} < q < 1\\) might still have Roche Lobe Overflow. \n\nIs this the case in real systems? Usually the mass of the BH is of at least a few solar masses, so we need an unusually large companion. \nFor example, the companion to the BH in Cygnus X-1 is a very large O-star. \n\n\\section{Accretion disks}\n\nNow we will discuss what happens when this is indeed the case. What is the fate of the matter passing through the inner Lagrange point? \n\n\\subsection{Circularization}\n\nLet us change the perspective: we are not comoving with respect to the orbiting stars anymore. \nLet us call \\(b_1 \\) the separation of the \\(L_1 \\) point from the center of star 1. This is slightly larger than \\(R_1 \\). \n\nIf we are stationary at the center the companion compact star will appear to rotate at a large velocity compared to us. \n\nThe components of this velocity can be decomposed into the parallel and perpendicular to the separation vector between the star, however the component \\(v _ \\parallel\\) will be of the order of the speed of sound in the gas: roughly, \n%\n\\begin{align}\nv_{\\parallel} \\approx c_s = \\sqrt{ \\pdv{P}{\\rho }} = \\sqrt{\\frac{k_B T}{\\mu m_p}} \\approx \\num{10} \\sqrt{ \\frac{T}{\\SI{e4}{K}}} \\SI{}{km /s}\n\\,;\n\\end{align}\n%\nwhile the component \\(v_\\perp\\) will be large: of the order of \n%\n\\begin{align}\nv_\\perp \\sim b_1 \\omega \n\\,,\n\\end{align}\n%\nand the distance \\(b_1\\) will be roughly given by \\cite[]{plavecTablesRocheModel1964}\n%\n\\begin{align}\nb_1 \\approx a (\\num{.5} - \\num{.227} \\log q)\n\\,,\n\\end{align}\n%\nwhile by Kepler's third law \n%\n\\begin{align}\n4 \\pi^2 a^3 = G (M_1 + M_2 ) P^2\n\\,,\n\\end{align}\n%\nso \n%\n\\begin{align}\na \\approx \\num{3e11} \\qty(\\frac{M_1 }{M_{\\odot}})^{1/3}\n\\qty(1 + q)^{1/3}\n\\qty(\\frac{P}{\\SI{1}{d}})^{2/3} \n\\SI{}{cm}\n\\,,\n\\end{align}\n%\nwhile the angular velocity is \n%\n\\begin{align}\n\\omega = \\frac{2 \\pi }{P} \\approx \\num{7e-5} \\qty(\\frac{P}{\\SI{1}{d}})^{-1} \\SI{}{rad/s}\n\\,.\n\\end{align}\n\nThis means that the perpendicular velocity is approximately \n%\n\\begin{align}\nv_{\\perp} \\approx \\num{100} \\qty(\\frac{M_1 }{M_{\\odot}})^{1/3}\n\\qty(1 + q)^{1/3}\n\\qty(\\frac{P}{\\SI{1}{d}})^{-1/3}\n\\SI{}{km/s}\n\\,,\n\\end{align}\n%\nan order of magnitude more than the speed of sound. \n\n\\end{document}\n", "meta": {"hexsha": "47b46f48d1f612033bea2e875cd48861e29bb638", "size": 11128, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ap_third_semester/compact_objects/nov03.tex", "max_stars_repo_name": "jacopok/notes", "max_stars_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-10-10T13:10:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T14:52:50.000Z", "max_issues_repo_path": "ap_third_semester/compact_objects/nov03.tex", "max_issues_repo_name": "jacopok/notes", "max_issues_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ap_third_semester/compact_objects/nov03.tex", "max_forks_repo_name": "jacopok/notes", "max_forks_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-03T16:20:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-06T16:11:07.000Z", "avg_line_length": 32.7294117647, "max_line_length": 407, "alphanum_fraction": 0.6454888569, "num_tokens": 4032, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34958202995258797}} {"text": "\\documentclass[../notes.tex]{subfiles}\n\n\\pagestyle{main}\n\\renewcommand{\\chaptermark}[1]{\\markboth{\\chaptername\\ \\thechapter\\ (#1)}{}}\n\\setcounter{chapter}{19}\n\n\\begin{document}\n\n\n\n\n\\chapter{The Second Law of Thermodynamics}\n\\section{Thermodynamic Cycles}\n\\begin{itemize}\n \\item \\marginnote{8/23:}Examples: Engines and pumps.\n \\item In a thermodynamic cycle, the energy doesn't change, so $W_\\text{on gas}=-Q$, i.e., $W_\\text{by gas}=Q_\\text{net}$.\n \\item Thus, $W_\\text{by engine}=Q_\\text{in}-Q_\\text{out}$ per cycle.\n \\item \\textbf{Otto cycle}: The thermodynamic cycle used in car engines. \\emph{Also known as} \\textbf{auto cycle}.\n \\begin{figure}[h!]\n \\centering\n \\begin{tikzpicture}[\n scale=1.8,\n every node/.append style={black},\n pics/point/.style args={#1:#2}{code={\n \\node [circle,fill=pix,inner sep=1.5pt,label={#1:$#2$}] {};\n }}\n ]\n \\footnotesize\n \\draw [stealth-stealth] (0,2) node[above]{$p$} -- (0,0) -- (3,0) node[right]{$V$};\n \n \\draw [blx,thick,dashed] plot[domain=1:2.5,samples=500] (\\x,{1.67/\\x});\n \\draw [blx,thick,dashed] plot[domain=0.6:2.5,samples=500] (\\x,{1/\\x});\n \\draw [blx,thick,dashed] plot[domain=0.3:2.5,samples=500] (\\x,{0.5/\\x});\n \\draw [blx,thick,dashed] plot[domain=0.12:2.5,samples=500] (\\x,{0.2/\\x});\n \n \\draw [orx,very thick,postaction={decorate},decoration={\n markings,\n mark=at position 0.25 with {\\arrow{>} \\node[above right,fill=white,inner sep=1.5pt]{$Q=0$};},\n mark=at position 0.465 with {\\arrow{>} \\node[right=3pt,fill=white,inner sep=1.5pt]{$Q_\\text{out}$};},\n mark=at position 0.65 with {\\arrow{>} \\node[below left,fill=white,inner sep=1.5pt]{$Q=0$};},\n mark=at position 0.91 with {\\arrow{>} \\node[left =3pt,fill=white,inner sep=1.5pt]{$Q_\\text{in}$};}\n }] plot[domain=0.8:2] (\\x,{0.845/\\x^(1.756)}) -- plot[domain=2:0.8] (\\x,{0.4/\\x^2}) -- cycle;\n \n \\pic at (0.8,{1/0.8}) {point=:};\n \\end{tikzpicture}\n \\caption{Otto cycle.}\n \\label{fig:ottoCycle}\n \\end{figure}\n \\begin{itemize}\n \\item Two adiabatic and two isochoric processes.\n \\end{itemize}\n \\item \\textbf{Stirling engine cycle}: The thermodynamic cycle used in steam engines.\n \\begin{figure}[h!]\n \\centering\n \\begin{tikzpicture}[\n scale=1.8,\n every node/.append style={black},\n pics/point/.style args={#1:#2}{code={\n \\node [circle,fill=pix,inner sep=1.5pt,label={#1:$#2$}] {};\n }}\n ]\n \\footnotesize\n \\draw [stealth-stealth] (0,2) node[above]{$p$} -- (0,0) -- (3,0) node[right]{$V$};\n \\draw [very thin,dashed] (0.8,{0.5/0.8}) -- (0.8,0) node[below]{$V_1$};\n \\draw [very thin,dashed] (2,{0.5/2}) -- (2,0) node[below]{$V_2$};\n \n \\draw [blx,thick,dashed] plot[domain=1:2.5,samples=500] (\\x,{1.67/\\x});\n \\draw [blx,thick,dashed] plot[domain=0.6:2.5,samples=500] (\\x,{1/\\x}) node[right]{$T_1$};\n \\draw [blx,thick,dashed] plot[domain=0.3:2.5,samples=500] (\\x,{0.5/\\x}) node[right]{$T_2$};\n \\draw [blx,thick,dashed] plot[domain=0.12:2.5,samples=500] (\\x,{0.2/\\x});\n \n \\draw [orx,very thick,postaction={decorate},decoration={\n markings,\n mark=at position 0.25 with {\\arrow{>} \\node[above=4pt,fill=white,inner sep=1.5pt]{$+Q_2$};},\n mark=at position 0.45 with {\\arrow{>} \\node[right=3pt,fill=white,inner sep=1.5pt]{$-Q_3$};},\n mark=at position 0.65 with {\\arrow{>} \\node[below=3pt,fill=white,inner sep=1.5pt]{$-Q_4$};},\n mark=at position 0.91 with {\\arrow{>} \\node[left =3pt,fill=white,inner sep=1.5pt]{$+Q_1$};}\n }] plot[domain=0.8:2] (\\x,{1/\\x}) -- plot[domain=2:0.8] (\\x,{0.5/\\x}) -- cycle;\n \\draw [orx,very thick] plot[domain=0.8:2] (\\x,{1/\\x}) -- plot[domain=2:0.8] (\\x,{0.5/\\x}) -- cycle;\n \n \\pic at (0.8,{1/0.8}) {point=:};\n \\end{tikzpicture}\n \\caption{Stirling engine cycle.}\n \\label{fig:stirlingCycle}\n \\end{figure}\n \\begin{itemize}\n \\item Two isothermal and two isochoric processes.\n \\item For the isothermal processes, $\\Delta E_\\text{int}=0$, so $Q=W_\\text{by gas}=nRT\\ln(V_f/V_i)$.\n \\item For the isochoric process, $Q=nC_V\\Delta T$.\n \\item $Q_1+Q_3=nC_V(T_1-T_2)+nC_V(T_2-T_1)=0$, so we only have to worry about $Q_2$ and $Q_4$ to determine the work done by the gas:\n \\begin{align*}\n W_\\text{by gas} &= Q_2+Q_4\\\\\n &= nR\\ln\\left( \\frac{V_2}{V_1} \\right)\\cdot(T_1-T_2)\n \\end{align*}\n \\item To get more work, you want the \\textbf{compression factor} to be as big as possible.\n \\item You also want the difference between the two temperatures to be as big as possible.\n \\item The engine is powered until $T_1=T_2$, because at that point you can't exchange heat.\n \\end{itemize}\n \\item \\textbf{Compression factor}: The ratio of the final volume to the initial volume, i.e., $V_2/V_1$.\n \\item \\textbf{Efficiency}: The quotient of the work an engine does and the heat you put in per cycle. \\emph{Denoted by} $\\bm{e}$. \\emph{Given by}\n \\begin{align*}\n e &= \\frac{W_\\text{engine}}{Q_\\text{in}}\\\\\n &= \\frac{Q_\\text{in}-Q_\\text{out}}{Q_\\text{in}}\\\\\n &= 1-\\frac{Q_\\text{out}}{Q_\\text{in}}\n \\end{align*}\n \\item \\textbf{Carnot cycle}.\n \\begin{figure}[h!]\n \\centering\n \\begin{tikzpicture}[\n scale=1.8,\n every node/.append style={black},\n pics/point/.style args={#1:#2}{code={\n \\node [circle,fill=pix,inner sep=1.5pt,label={#1:$#2$}] {};\n }}\n ]\n \\footnotesize\n \\draw [stealth-stealth] (0,2) node[above]{$p$} -- (0,0) -- (3,0) node[right]{$V$};\n \n \\draw [blx,thick,dashed] plot[domain=1:2.5,samples=500] (\\x,{1.67/\\x}) node[right]{$T_H$};\n \\draw [blx,thick,dashed] plot[domain=0.6:2.5,samples=500] (\\x,{1/\\x});\n \\draw [blx,thick,dashed] plot[domain=0.3:2.5,samples=500] (\\x,{0.5/\\x}) node[right]{$T_L$};\n \\draw [blx,thick,dashed] plot[domain=0.12:2.5,samples=500] (\\x,{0.2/\\x});\n \n \\draw [orx,very thick,postaction={decorate},decoration={\n markings,\n mark=at position 0.13 with {\\arrow{>} \\node[above=6pt,xshift=2pt,fill=white,inner sep=1.5pt]{$Q_\\text{in}$};},\n mark=at position 0.33 with {\\arrow{>} \\node[right=3pt,fill=white,inner sep=1.5pt]{$Q=0$};},\n mark=at position 0.58 with {\\arrow{>} \\node[below=3pt,fill=white,inner sep=1.5pt]{$Q_\\text{out}$};},\n mark=at position 0.81 with {\\arrow{>} \\node[left =3pt,fill=white,inner sep=1.5pt]{$Q=0$};}\n }] plot[domain=1.1:1.6] (\\x,{1.67/\\x}) plot[domain=1.6:2] (\\x,{21.178/\\x^(6.404)}) plot[domain=2:1.4] (\\x,{0.5/\\x}) plot[domain=1.4:1.1] (\\x,{2.69/\\x^(6.001)});\n \n \\pic at (1.1,{1.67/1.1}) {point=:};\n \\end{tikzpicture}\n \\caption{Carnot cycle.}\n \\label{fig:carnotCycle}\n \\end{figure}\n \\begin{itemize}\n \\item Two isothermal and two adiabatic processes.\n \\item For this engine, $e=1-Q_\\text{out}/Q_\\text{in}=1-T_L/T_H$.\n \\begin{itemize}\n \\item Thus, as the temperatures approach each other, efficiency approaches 1.\n \\item But, you cannot run an engine when the heat bath temperatures are equal to each other.\n \\end{itemize}\n \\item Therefore, the Carnot cycle is the most efficient engine you can build, but it's just not physically viable.\n \\begin{itemize}\n \\item A car run on the Carnot cycle would have great mileage but wouldn't get you very far.\n \\end{itemize}\n \\item Conclusion: The Carnot cycle is interesting, but it does not yield a particularly useful engine.\n \\end{itemize}\n\\end{itemize}\n\n\n\n\\section{The Second Law}\n\\begin{itemize}\n \\item \\textbf{Second Law of Thermodynamics}: It is not possible to remove heat at a high temperature and convert it entirely to work done by the engine. In other words, some heat is always exhausted to the low temperature.\n \\item Maximizing engine efficiency:\n \\begin{itemize}\n \\item You want to maximize the temperature difference.\n \\item You could technically run an engine between air temperature and ice, but it's hard to lug around a bunch of ice.\n \\item Additionally, you can't have $T_H$ be too high because hotter engines emit nastier exhaust (specifically, exhaust that contributes more to acid rain).\n \\item Recall that the area on a $pV$-graph encompassed by the cycle is equal to the work done by the gas.\n \\end{itemize}\n \\item If you run a heat engine backwards, you get a \\textbf{heat pump}.\n \\item \\textbf{Heat pump}: A device that removes heat from low temperature sinks and exhausts it into high temperature sinks.\n \\begin{itemize}\n \\item This is an air conditioner.\n \\item The second law of thermodynamics asserts that you can't remove heat at a low temperature and move it to a high temperature without doing work on the gas. Thus, you have to plug in the air conditioner and provide power --- heat won't magically flow from cold to hot.\n \\end{itemize}\n\\end{itemize}\n\n\n\n\\section{Statistical Mechanics}\n\\begin{itemize}\n \\item Free expansion.\n \\begin{itemize}\n \\item Imagine you have gas on one side of a thermally insulated container separated from the rest of the container by a partition.\n \\item If you remove the partition, the gas will expand out to fill the entire container.\n \\item However, the gas is not working against a force ($p=0$) and heat is not flowing in ($Q=0$).\n \\item Therefore, $\\dd{E_\\text{int}}=\\dd{Q}-p\\dd{V}=0-0\\dd{V}=0$.\n \\item From this, we can conclude that $\\Delta T=0$.\n \\item Note that this is an \\textbf{irreversible process}.\n \\end{itemize}\n \\item \\textbf{Irreversible process}: A process that looks like it could occur on a microscopic level but could not on a macroscopic level.\n \\begin{itemize}\n \\item Imagine a video of the milk molecules moving in a cup of coffee (which would look normal forwards and backwards) vs. a video of milk spreading out in a cup of coffee (which most certainly would not look normal backwards).\n \\end{itemize}\n \\item \\textbf{Macrostate}: A set of thermal variables ($n,p,V,T$) defining a system at an instant in time.\n \\item \\textbf{Microstate}: A way of arranging the molecules in a system that produces a macrostate.\n \\begin{itemize}\n \\item Multiple microstates can correspond to the same macrostate.\n \\end{itemize}\n \\item Consider a box with 10 molecules inside.\n \\begin{itemize}\n \\item Each molecules has a 50/50 chance of being on the left \\emph{or} right side of the box.\n \\item We want to achieve the macrostate defined by 10 molecules on the left side and 0 on the right.\n \\begin{itemize}\n \\item There is only 1 corresponding microstate (namely that just described).\n \\end{itemize}\n \\item We want to achieve the macrostate defined by 3 molecules on the left side and 7 on the right.\n \\begin{itemize}\n \\item There are $\\frac{10!}{7!3!}=120$ corresponding microstates.\n \\end{itemize}\n \\item For the even 5/5 split, there are $\\binom{10}{5}=252$ corresponding microstates.\n \\end{itemize}\n \\item If there are $N$ molecules and we want $m$ on the left and $N-m$ on the right, the number of microstates that will realize this macrostate is $\\frac{N!}{m!(N-m)!}$.\n \\begin{itemize}\n \\item This corresponds to a binomial distribution.\n \\end{itemize}\n \\item \\textbf{Fundamental Assumption of Statistical Physics}: All microstates of a system are equally probable.\n \\item \\textbf{Number of microstates of a macrostate}: \\emph{Denoted by} $\\bm{\\Omega_\\textbf{macro}}$.\n \\item It follows from the fundamental assumption that\n \\begin{equation*}\n P_\\text{macro} \\propto \\Omega_\\text{macro}\n \\end{equation*}\n \\item Relating this back to our example, since $P_{m,N-m}\\propto\\Omega{m,N-m}$ we have that $(5,5)$ is approximately twice as likely as $(3,7)$.\n \\item If we have $N_A$ of molecules, it is going to be much more likely that the molecules are evenly (or almost evenly) distributed on both sides of the box than anything else.\n \\begin{itemize}\n \\item $\\Omega_{N_A,0}=1$ still, but $\\Omega_{N_A/2,N_A/2}=\\frac{N_A!}{(N_A!)^2}\\approx 10^{N_A/3}\\approx 10^{\\num{2e23}}$, which is a huge number.\n \\end{itemize}\n \\item If you make $N$ and $m$ continuous rather than integer quantities, the binomial distribution becomes a Gaussian distribution.\n \\begin{itemize}\n \\item Thus, Gaussian distributions describe the probability distribution corresponding to the number of molecules on the left and the right.\n \\item The standard deviation of the Gaussian distribution of $N_A$ macrostates is $\\sigma\\approx\\sqrt{N_A}$.\n \\item Thus, the fraction of the width of the central peak to all states is $\\sigma/N_A\\approx 10^{-12}$.\n \\item Therefore, the width of the spike is approximately one-trillionth the width of the whole distribution, meaning that although you won't typically see a perfect 50/50 split, you won't see very large fluctuations (even 49/51 is very unlikely).\n \\end{itemize}\n \\item It's not that you \\emph{can't} have all of the air molecules move to one side of the room; it's that you \\emph{won't} have this happen.\n\\end{itemize}\n\n\n\n\\section{Office Hours (Gazes)}\n\\begin{itemize}\n \\item Preference for sine vs. cosine when describing harmonic waves?\n \\begin{itemize}\n \\item No preference.\n \\item Gazes tends to start with sines, but it really doesn't matter; there is no canonical preference.\n \\end{itemize}\n \\item We say that the open end of a pipe is a node and we often say that closed ends are antinodes. However, on the midterm, there was a question with a tuning fork at one end of a pipe with water at the bottom, and we were supposed to infer that the open end with the tuning fork was an antinode and the end with water was a node. As such, I'm wondering if there's any unambiguous way we can identify open and closed ends of pipes.\n \\begin{itemize}\n \\item We do assume the end with the tuning fork to be a pressure node and the water to be a pressure antinode.\n \\item It's a \\emph{pressure} node at the end of the tube, but a displacement antinode.\n \\item Some transmission does occur at the end of a tube.\n \\item Flaring the ends of instruments cuts down on reflection and increases transmission.\n \\end{itemize}\n \\item How do we know that even rays with not so well understood geometry converge on the image?\n \\begin{itemize}\n \\item A computer can just brute force the problem and come to this conclusion.\n \\end{itemize}\n \\item When refracting an object through multiple lenses, the side the \\emph{original} object is on is always the $+s$ and $-s'$ side, even when the first image may be on the other side of the next lens?\n \\begin{itemize}\n \\item Yes.\n \\item Because the rays just continue on through the second lens. Those \\emph{rays} act as the object and effectively determine the sign.\n \\end{itemize}\n \\item Plano-convex lens on top of a mirror?\n \\begin{itemize}\n \\item You can make the contraption image the same way as a concave mirror.\n \\item Allows analogy of concave mirrors to regular lenses.\n \\item Also helps define our sign convention: flips what defines positive image distance.\n \\end{itemize}\n \\item When do we make use of vibration? In Problem 19.43, shouldn't we have needed to use $\\frac{7}{2}nR\\Delta T$ for every degree Kelvin above $\\SI{1000}{\\kelvin}$?\n \\begin{itemize}\n \\item Didn't quite cross over the threshold.\n \\item Won't have to account for this on the final.\n \\item Everything is monoatomic on the final!\n \\end{itemize}\n\\end{itemize}\n\n\n\n\\section{Entropy}\n\\begin{itemize}\n \\item \\marginnote{8/24:}\\textbf{Entropy} (of a macrostate): The following quantity, where $k$ is the Boltzmann constant. \\emph{Denoted by} $\\bm{S_\\textbf{macro}}$.\n \\begin{equation*}\n S_\\text{macro} = k\\ln\\Omega_\\text{macro}\n \\end{equation*}\n \\item $\\Delta S>0$ (i.e., entropy increases) for an irreversible process.\n \\item Entropies of multiple systems:\n \\begin{itemize}\n \\item Consider 2 systems in macrostates $A$ and $B$, respectively.\n \\item The number of microstates pertaining to each macrostate is $\\Omega_A$ and $\\Omega_B$, respectively.\n \\item We know that $P_A\\propto\\Omega_A$ and $P_B\\propto\\Omega_B$.\n \\item Moreover, the probability $P_{AB}$ that the combination of the two systems in in microstate $AB$ is equal to $P_AP_B$ by fundamental probability laws, and the total number of microstates for the combination is equal to $\\Omega_A\\Omega_B$.\n \\item It follows that\n \\begin{align*}\n S_{AB} &= k\\ln(\\Omega_A\\Omega_B)\\\\\n &= k\\ln\\Omega_A+k\\ln\\Omega_B\\\\\n &= S_A+S_B\n \\end{align*}\n i.e., that entropies are additive.\n \\end{itemize}\n \\item Primary consequences of the second law of thermodynamics:\n \\begin{itemize}\n \\item Wrt heat engines: You will never see gas taking in $Q_\\text{in}$ and converting entirely to $W_\\text{by gas}$ (i.e., there will always be some $Q_\\text{out}$).\n \\item Wrt heat pumps: You will never see $Q$ extracted from $T_L$ and exhausted to $T_H$ without some $W_\\text{on gas}$.\n \\item Wrt entropy: You will never see a decrease in entropy.\n \\end{itemize}\n \\item Change in entropy for a reversible process:\n \\begin{itemize}\n \\item Consider the Carnot cycle (Figure \\ref{fig:carnotCycle}).\n \\item For the cycle, $\\Delta S_\\text{gas}=0$.\n \\item If we add or subtract $Q$ of heat at a constant $T$,\n \\begin{equation*}\n \\Delta S = \\frac{Q}{T}\n \\end{equation*}\n \\item Thus,\n \\begin{align*}\n \\Delta S_\\text{univ} &= \\Delta S_{Q_\\text{in}}+\\Delta S_{Q_\\text{out}}\\\\\n &= \\frac{Q_\\text{in}}{T_H}-\\frac{Q_\\text{out}}{T_L}\\\\\n \\intertext{The above is true for any thermodynamic cycle. But since $Q_\\text{in}/Q_\\text{out}=T_H/T_L$ for a Carnot cycle\\dots}\n &= \\frac{Q_\\text{out}T_H/T_L}{T_H}-\\frac{Q_\\text{out}}{T_L}\\\\\n &= 0\n \\end{align*}\n for a Carnot cycle.\n \\item Thus, overall, $\\Delta S=0$. This is actually true of any reversible process.\n \\end{itemize}\n \\item Thus, since every process is either reversible or irreversible, $\\Delta S_\\text{univ}\\geq 0$ for every process in this universe.\n\\end{itemize}\n\n\n\n\n\\end{document}", "meta": {"hexsha": "c7605f50c44445f7013f753d963acb547bfa5459", "size": 19075, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Notes/Chapter20/chapter20.tex", "max_stars_repo_name": "shadypuck/PHYS13300Notes", "max_stars_repo_head_hexsha": "61c7dcb457b6ce79feba5d9a46e991c88cdcde68", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Notes/Chapter20/chapter20.tex", "max_issues_repo_name": "shadypuck/PHYS13300Notes", "max_issues_repo_head_hexsha": "61c7dcb457b6ce79feba5d9a46e991c88cdcde68", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Notes/Chapter20/chapter20.tex", "max_forks_repo_name": "shadypuck/PHYS13300Notes", "max_forks_repo_head_hexsha": "61c7dcb457b6ce79feba5d9a46e991c88cdcde68", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 58.6923076923, "max_line_length": 436, "alphanum_fraction": 0.6379030144, "num_tokens": 5744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34958202995258797}} {"text": "% 06Holonomic.tex\n\n\\subsection{6.1. The Robenius Integrability Condition}\n\nCan one always find a surface orthogonal to a family of curves in $\\mathbb{R}^3$?\n\n\\subsection{6.2. Integrability and Constraints}\n\n\\subsection{6.3. Heuristic Thermodynamics via Caratheodory}\n\nCan one go adiabatically from some state to any nearby state?\n\n\\subsubsection{6.3a. Introduction}\n\n\n\n\\subsubsection{6.3b. The First Law of Thermodynamics}\n\nConsider system of regions of fluids separated by ``diathermous'' membranes \\\\\n\\phantom{\\quad \\, } allow only passage of heat, not fluids \\\\\n\nassume system connected \\\\\nassume each state in thermal equilibrium\n\nLet $p_i,v_i$ (uniform) pressure and volume of $i$th region \\\\\n\\phantom{\\quad \\, } at thermal equilibrium, ``equations of state'' $p_iv_i = n_iRT_i$ \\\\\n\\phantom{\\quad \\quad \\, } $p_i v_i = n_iRT_i$ eliminate all but 1 pressure\n\\[\n\\Longrightarrow p_1, v_1,v_2, \\dots v_n\n\\]\nassume globally defined energy function $U$\n\npath in $M^{n+1}$ represents sequence of states each in equilibrium, i.e. assume very slow changes in time, quasi-static irreversible processes, e.g. ``stirring'' \\\\\non $M$, $\\text{dim}{M} = n+1$, assume $\\exists \\, $ work 1-form $W$, work done by system \n\\[\nW = p_i dv_i = p_i(U,v_1\\dots v_n)dv_i \\quad \\quad \\, i=1 \\dots n \n\\]\nheat 1-form, heat added or removed from system, assume $Q \\neq 0$\n\\[\nQ = \\sum_{i=0}^n Q_i(U,v_1 \\dots v_n) dv_i \\quad \\, (v_0 = U)\n\\]\n\n1st. law of thermodynamics \n\\[\ndU = Q-W\n\\]\nenergy conservation \n\n\\subsubsection*{6.3c. Some Elementary Changes of State}\n\n1. Heating at constant volume. \n\n\\phantom{\\quad \\, } path $\\gamma_I \\in M$, $\\text{dim}M = n+1$ s.t. $dv_1 = \\dots = dv_n =0$. $W=0$\n\\phantom{\\quad \\, } $dU = Q_0 dU$. $\\dot{\\gamma}_I = c_0 \\frac{\\partial}{ \\partial U}$\n\n2. Quasi-static adiabatic process. No heat exchanged,\n\\[\nQ(\\dot{\\gamma}_{\\text{II}}) = 0 \\quad \\, \\text{ so } dU = -W\n\\]\n\n3. Stirring at constant volume adiabatic, but not quasistatic \\\\\n\\phantom{\\quad \\, } $Q,W$ makes no sense but \\\\\n\\phantom{\\quad \\, } work is being done by (or on) system, $U(y') - U(x)$, difference of internal energy \\\\\nassume connected mechanical manifold $V$, $\\text{dim}V =n$ \\\\\n\\phantom{assume } diff. $\\pi:M \\to V$ \\\\\n\\phantom{assume diff} $\\pi$ onto \\\\\n\\phantom{assume diff} $\\pi_*$ onto \\\\\n\\phantom{assume diff} $\\pi$ submersion\n\nBy main thm. on submanifolds of Sec. 1.3d, \\\\\n\\phantom{\\quad \\, } if $v\\in V$, then $\\pi^{-1}(v)$ 1-dim. embedded submanifold of $M$ \\\\\n\\phantom{\\quad \\quad \\, } assume $\\forall \\, \\pi^{-1}(v)$ connected, we're assuming given any pair of states \\\\\n\\phantom{\\quad \\quad \\quad \\, } lying on $\\pi^{-1}(v)$, 1 of them can be obtained by other by ``heating at constant volume''\n\nassume $W$ on $M$ is $0$ when $\\left. W \\right|_{\\pi^{-1}(v)} =0$ \\\\\non the other hand, $Q \\neq 0$ on $\\pi^{-1}(v)$; $dU = Q \\neq 0$ (first law) \\\\\n\\phantom{\\quad \\, } $(U,v^1 \\dots v^n)$ local coordinate system for $M$ ($U$ global coordinate)\n\n\\subsubsection{6.3d. The Second Law of Thermodynamics} \n\ncyclic process starts and ends at the same state\n\nKelvin 2nd. law of thermodynamics \\\\\n\\phantom{\\quad \\, } $\\nexists $ quasistatic cyclic process can $Q$ converted entirely into $W$ \\\\\n\nCaratheodory (1909) 2nd. law of thermodynamics \\\\\n\\phantom{\\quad \\, } $\\forall \\, $ neighborhood $N \\ni $ state $x$, $\\exists \\, y$ not accessible from $x$ via quasistatic adiabatic paths, i.e. paths s.t. $Q=0$\n\n\n\n", "meta": {"hexsha": "67ad927db88602dede75bb21e775f08be54e2cf8", "size": 3400, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "LaTeX_and_pdfs/the geometry of physics problems/06Holonomic.tex", "max_stars_repo_name": "wacfeldwang333/mathphysics", "max_stars_repo_head_hexsha": "59eb794dfa46e2b80e43df0440bb8ec3c472d973", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 50, "max_stars_repo_stars_event_min_datetime": "2017-01-10T14:24:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T11:19:23.000Z", "max_issues_repo_path": "LaTeX_and_pdfs/the geometry of physics problems/06Holonomic.tex", "max_issues_repo_name": "wacfeldwang333/mathphysics", "max_issues_repo_head_hexsha": "59eb794dfa46e2b80e43df0440bb8ec3c472d973", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2017-09-29T09:29:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-12T03:12:29.000Z", "max_forks_repo_path": "LaTeX_and_pdfs/the geometry of physics problems/06Holonomic.tex", "max_forks_repo_name": "wacfeldwang333/mathphysics", "max_forks_repo_head_hexsha": "59eb794dfa46e2b80e43df0440bb8ec3c472d973", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 25, "max_forks_repo_forks_event_min_datetime": "2018-01-21T05:33:31.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T20:15:13.000Z", "avg_line_length": 37.3626373626, "max_line_length": 165, "alphanum_fraction": 0.6694117647, "num_tokens": 1142, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34958202995258797}} {"text": "\\clearpage\n\\makeatletter\n\\efloat@restorefloats\n\\makeatother\n\n\n\\begin{appendix}\n\\hypertarget{parallel-set-of-non-bayesian-analyses}{%\n\\section{Parallel set of non-Bayesian\nanalyses}\\label{parallel-set-of-non-bayesian-analyses}}\n\nIn this appendix, we report the output of a parallel set of non-Bayesian\nlinear mixed-effects models for the primary analyses (Reaction Time,\nAccuracy, EWMA) in Experiments 1 and 2. The \\(p\\) values for each\ncoefficient were computed via the Satterthwaite approximation using the\n\\texttt{lmerTest} R package (Kuznetsova, Brockhoff, \\& Christensen,\n2017).\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:mle rt trio}Output of the MLE model predicting log reaction time as a function of center stimulus type in Experiment 1.}\n\\begin{tabular}{llllll}\n\\toprule\nterm & \\multicolumn{1}{c}{estimate} & \\multicolumn{1}{c}{std.error} & \\multicolumn{1}{c}{statistic} & \\multicolumn{1}{c}{df} & \\multicolumn{1}{c}{p.value}\\\\\n\\midrule\nIntercept (ASL) & 6.96 & 0.08 & 91.84 & 62.58 & < .001\\\\\nFace & -0.37 & 0.11 & -3.32 & 84.45 & < .01\\\\\nObject & -1.12 & 0.10 & -11.13 & 79.76 & < .001\\\\\nBullseye & -1.30 & 0.13 & -10.04 & 97.77 & < .001\\\\\nscale(Months) & -0.01 & 0.04 & -0.18 & 75.55 & 0.86\\\\\n\\bottomrule\n\\addlinespace\n\\end{tabular}\n\\begin{tablenotes}[para]\n\\normalsize{\\textit{Note.} Model specification: log(RT) \\textasciitilde{} stimuli + scale(Months) + (1 | subid) + (1 | target\\_image)}\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:mle acc trio}Output of the MLE model predicting accuracy as a function of center stimulus type in Experiment 1.}\n\\begin{tabular}{lllll}\n\\toprule\nterm & \\multicolumn{1}{c}{estimate} & \\multicolumn{1}{c}{std.error} & \\multicolumn{1}{c}{statistic} & \\multicolumn{1}{c}{p.value}\\\\\n\\midrule\nIntercept (ASL) & 1.07 & 0.14 & 7.66 & < .001\\\\\nFace & -0.64 & 0.20 & -3.19 & < .01\\\\\nObject & -1.14 & 0.19 & -6.16 & < .001\\\\\nBullseye & -1.15 & 0.22 & -5.26 & < .001\\\\\n\\bottomrule\n\\addlinespace\n\\end{tabular}\n\\begin{tablenotes}[para]\n\\normalsize{\\textit{Note.} Model specification: correct \\textasciitilde{} stimuli + (1 | subid) + (1 | target\\_image)}\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:mle ewma trio prop lang}Output of the MLE model estimating the difference in random responding in the EWMA as a function of center stimulus type in Experiment 1.}\n\\begin{tabular}{lllll}\n\\toprule\nterm & \\multicolumn{1}{c}{estimate} & \\multicolumn{1}{c}{std.error} & \\multicolumn{1}{c}{statistic} & \\multicolumn{1}{c}{p.value}\\\\\n\\midrule\nIntercept (ASL) & 1.73 & 0.21 & 8.31 & < .001\\\\\nFace & -1.80 & 0.29 & -6.26 & < .001\\\\\n\\bottomrule\n\\addlinespace\n\\end{tabular}\n\\begin{tablenotes}[para]\n\\normalsize{\\textit{Note.} Model specification: guess \\textasciitilde{} stimuli + (1 | subid) + (1 | target\\_image)}\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:mle rt noise}Output of the MLE model predicting log reaction time as a function of noise condition and age category (adults vs. children) in Experiment 2.}\n\\begin{tabular}{llllll}\n\\toprule\nterm & \\multicolumn{1}{c}{estimate} & \\multicolumn{1}{c}{std.error} & \\multicolumn{1}{c}{statistic} & \\multicolumn{1}{c}{df} & \\multicolumn{1}{c}{p.value}\\\\\n\\midrule\nIntercept (Clear) & -0.76 & 0.05 & -13.92 & 49.62 & < .001\\\\\nnoise & 0.09 & 0.04 & 2.31 & 54.35 & 0.02\\\\\nadults & 0.16 & 0.06 & 2.88 & 56.01 & < .01\\\\\n\\bottomrule\n\\addlinespace\n\\end{tabular}\n\\begin{tablenotes}[para]\n\\normalsize{\\textit{Note.} Model specification: log(rt) \\textasciitilde{} noise\\_condition + age\\_category + (noise\\_condition | subid) + (1 | target\\_image)}\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:mle acc noise}Output of the MLE model predicting accuracy as a function of noise condition and age category (adults vs. children) in Experiment 2.}\n\\begin{tabular}{lllll}\n\\toprule\nterm & \\multicolumn{1}{c}{estimate} & \\multicolumn{1}{c}{std.error} & \\multicolumn{1}{c}{statistic} & \\multicolumn{1}{c}{p.value}\\\\\n\\midrule\nIntercept (Clear) & 0.48 & 0.13 & 3.78 & < .001\\\\\nnoise & 0.22 & 0.14 & 1.52 & 0.13\\\\\nadults & 1.75 & 0.15 & 11.54 & < .001\\\\\n\\bottomrule\n\\addlinespace\n\\end{tabular}\n\\begin{tablenotes}[para]\n\\normalsize{\\textit{Note.} Model specification: correct \\textasciitilde{} noise\\_condition + age\\_category + (noise\\_condition | subid) + (1 | target\\_image)}\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:mle ewma noise}Output of the MLE model estimating the difference in random responding (EWMA model) as a function of noise condition and age category (adults vs. children) in Experiment 2.}\n\\begin{tabular}{lllll}\n\\toprule\nterm & \\multicolumn{1}{c}{estimate} & \\multicolumn{1}{c}{std.error} & \\multicolumn{1}{c}{statistic} & \\multicolumn{1}{c}{p.value}\\\\\n\\midrule\nIntercept (Clear) & 1.81 & 0.15 & 12.23 & < .001\\\\\nnoise & 0.73 & 0.17 & 4.18 & < .001\\\\\nchildren & -2.56 & 0.10 & -25.75 & < .001\\\\\n\\bottomrule\n\\addlinespace\n\\end{tabular}\n\\begin{tablenotes}[para]\n\\normalsize{\\textit{Note.} Model specification: guess\\_num \\textasciitilde{} noise\\_condition + age\\_category + (noise\\_condition | subid) + (noise\\_condition | target\\_image)}\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\hypertarget{bayesian-data-analysis-models-experiment-1}{%\n\\section{Bayesian data analysis models: Experiment\n1}\\label{bayesian-data-analysis-models-experiment-1}}\n\nIn this appendix, we report the full output of the Bayesian linear mixed\neffects models in Experiment 1.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-rt-model}Output of the regression predicting reaction time (milliseconds) as a function of center stimulus type in Experiment 1.}\n\\begin{tabular}{lll}\n\\toprule\nCenter Stimulus Type & \\multicolumn{1}{c}{Mean RT} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nBullseye & 288.71 & [229.8, 355.17]\\\\\nObject & 344.51 & [295.71, 397.31]\\\\\nFace & 724.78 & [603.2, 862.81]\\\\\nASL & 1,047.89 & [897.29, 1213.52]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-acc-model}Output of the logistic regression predicting accuracy as a function of center stimulus type in Experiment 1.}\n\\begin{tabular}{lll}\n\\toprule\nCenter Stimulus Type & \\multicolumn{1}{c}{Mean Accuracy} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nObject & 0.46 & [0.37, 0.53]\\\\\nBullseye & 0.48 & [0.38, 0.58]\\\\\nFace & 0.63 & [0.54, 0.7]\\\\\nASL & 0.76 & [0.7, 0.81]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-acc-contrasts}Output of the model estimating differences in Accuracy for specific contrasts of interest in Experiment 1.}\n\\begin{tabular}{lll}\n\\toprule\nContrast & \\multicolumn{1}{c}{Mean Difference Accuracy} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nObject vs. Chance & -0.04 & [-0.13, 0.03]\\\\\nBullseye vs. Chance & -0.02 & [-0.12, 0.08]\\\\\nASL vs. Face & 0.13 & [0.04, 0.23]\\\\\nFace vs. Object/Bullseye & 0.16 & [0.07, 0.24]\\\\\nASL vs. English & 0.23 & [0.17, 0.29]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-rt-contrasts}Output of the model estimating differences in RT for specific contrasts of interest in Experiment 1.}\n\\begin{tabular}{lll}\n\\toprule\nContrast & \\multicolumn{1}{c}{Mean Difference RT} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nASL vs. Face & 323.10 & [132.3, 522.6]\\\\\nFace vs. Object/Bullseye & 408.20 & [286.6, 546.2]\\\\\nASL vs. English & 595.20 & [444.6, 760.8]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-ewma-cuts}Output of the model estimating the point in the Reaction Time distribution when children's Exponentially Weighted Moving Average statistic crossed the pre-defined guessing threshold for the ASL and Face center stimulus types in Experiment 1.}\n\\begin{tabular}{lll}\n\\toprule\nCenter Stimulus Type & \\multicolumn{1}{c}{Mean EWMA Cut Point} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nASL & 0.68 & [0.59, 0.78]\\\\\nFace & 0.90 & [0.77, 1.03]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-guess-cuts}Output of the model estimating the mean proportion of shifts categorized as language-driven by the Exponentially Weighted Moving Average model for the ASL and Face center stimulus types in Experiment 1.}\n\\begin{tabular}{lll}\n\\toprule\nCenter Stimulus Type & \\multicolumn{1}{c}{Mean Language-driven} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nFace & 0.34 & [0.23, 0.46]\\\\\nAsl & 0.75 & [0.65, 0.84]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:trio-hddm}Summary of the Drift Diffusion Model output for the drift rate and boundary separation parameters for both all four center stimulus types in Experiment 1.}\n\\begin{tabular}{llll}\n\\toprule\nParameter & \\multicolumn{1}{c}{Center Stim Type} & \\multicolumn{1}{c}{Mean Param Estimate} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nBoundary & Face & 1.34 & [1.21, 1.47]\\\\\nBoundary & ASL & 1.76 & [1.65, 1.88]\\\\\nDrift & Face & 0.55 & [0.3, 0.8]\\\\\nDrift & ASL & 0.63 & [0.44, 0.82]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\hypertarget{bayesian-data-analysis-models-experiment-2}{%\n\\section{Bayesian data analysis models: Experiment\n2}\\label{bayesian-data-analysis-models-experiment-2}}\n\nIn this appendix, we report the full output of the Bayesian linear mixed\neffects models in Experiment 2.\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:noise-acc-model}Output of the logistic regression predicting accuracy as a function of noise condition and age group in Experiment 2.}\n\\begin{tabular}{llll}\n\\toprule\nNoise Condition & \\multicolumn{1}{c}{Age Group} & \\multicolumn{1}{c}{Mean Accuracy} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nClear & children & 0.61 & [0.54, 0.68]\\\\\nNoise & children & 0.67 & [0.6, 0.74]\\\\\nClear & adults & 0.90 & [0.87, 0.93]\\\\\nNoise & adults & 0.92 & [0.89, 0.95]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:noise-rt-model}Output of the regression estimating reaction times (milliseconds) as a function of noise condition and age group in Experiment 2.}\n\\begin{tabular}{llll}\n\\toprule\nNoise Condition & \\multicolumn{1}{c}{Age Group} & \\multicolumn{1}{c}{Mean RT} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nClear & children & 455.70 & [407, 503.6]\\\\\nNoise & children & 500.20 & [446.6, 555.6]\\\\\nClear & adults & 542.40 & [486.2, 602.4]\\\\\nNoise & adults & 595.20 & [532.4, 665.3]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:noise-guess-cuts}Output of the model estimating the mean proportion of shifts categorized as language-driven by the Exponentially Weighted Moving Average model for the each noise condition and age group in Experiment 2.}\n\\begin{tabular}{llll}\n\\toprule\nNoise Condition & \\multicolumn{1}{c}{Age Group} & \\multicolumn{1}{c}{Mean Language-driven} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nClear & children & 0.36 & [0.32, 0.4]\\\\\nNoise & children & 0.47 & [0.43, 0.51]\\\\\nClear & adults & 0.81 & [0.77, 0.86]\\\\\nNoise & adults & 0.93 & [0.89, 0.97]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{threeparttable}\n\\caption{\\label{tab:noise-hddm}Summary of the Drift Diffusion Model output for the drift rate and boundary separation parameters for both processing contexts and age groups in Experiment 2.}\n\\begin{tabular}{llll}\n\\toprule\nParameter & \\multicolumn{1}{c}{Age Group} & \\multicolumn{1}{c}{Mean Parameter Estimate} & \\multicolumn{1}{c}{95\\% HDI}\\\\\n\\midrule\nDrift & Children & 0.59 & [0.3, 0.89]\\\\\nDrift & Adults & 1.86 & [1.48, 2.25]\\\\\nBoundary & Children & 1.16 & [0.94, 1.39]\\\\\nBoundary & Adults & 1.78 & [1.53, 2.05]\\\\\n\\bottomrule\n\\end{tabular}\n\\end{threeparttable}\n\\end{center}\n\\end{table}\n\\end{appendix}\n", "meta": {"hexsha": "c7ae22dbb29504399418034067fde657a0d1a577", "size": 12698, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "writing/journal_submission/paper/speed-acc-appendix.tex", "max_stars_repo_name": "kemacdonald/speed-acc", "max_stars_repo_head_hexsha": "012087b4b604a882c813a2edc7df2af8b9c8e3e5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "writing/journal_submission/paper/speed-acc-appendix.tex", "max_issues_repo_name": "kemacdonald/speed-acc", "max_issues_repo_head_hexsha": "012087b4b604a882c813a2edc7df2af8b9c8e3e5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2016-07-07T21:15:31.000Z", "max_issues_repo_issues_event_max_datetime": "2016-07-07T21:16:27.000Z", "max_forks_repo_path": "writing/journal_submission/paper/speed-acc-appendix.tex", "max_forks_repo_name": "kemacdonald/speed-acc", "max_forks_repo_head_hexsha": "012087b4b604a882c813a2edc7df2af8b9c8e3e5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2016-06-23T22:48:28.000Z", "max_forks_repo_forks_event_max_datetime": "2016-06-23T22:48:28.000Z", "avg_line_length": 35.7690140845, "max_line_length": 277, "alphanum_fraction": 0.7072767365, "num_tokens": 4653, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283033, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34958202995258786}} {"text": "\\documentclass{article}\n\\title{Compsci 590T Homework 8}\n\\usepackage[margin=1in]{geometry}\n\\usepackage[loose,nice]{units}\n\\usepackage{enumitem}\n\\usepackage{amsmath}\n\\usepackage{multirow,array}\n\\newcommand{\\pvec}[1]{\\vec{#1}\\mkern2mu\\vphantom{#1}}\n\\usepackage{amssymb}\n\\usepackage{cancel}\n\\usepackage{mathabx}\n\\usepackage{hyperref}\n\\newcommand\\tab[1][1cm]{\\hspace*{#1}}\n\\usepackage{tikz}\n\\newcommand*\\circled[1]{\\tikz[baseline=(char.base)]{\n \\node[shape=circle,draw,inner sep=2pt] (char) {#1};}}\n\n\\usetikzlibrary{positioning}\n\\tikzset{main node/.style={circle,fill=blue!20,draw,minimum size=4mm,inner sep=0pt},}\n\\author{Student: Brian Sun, Collaborator: Thai On}\n\\date{}\n\n\\begin{document}\n\\maketitle\n\n\n\\section{Problem 1}\nAssume player $i$ (players who values all rooms equally) is not always pay the least amount of rent in any envy-free outcome. If this is true, there will exist an envy-free outcome that at least one player $i'$ who don't value all room the same and pay less than $i$.\\\\\\\\\nTherefore the room $i'$ get is cost less than the room $i$ get.\\\\\nHowever this violate the evny-free property: $v_{i\\sigma(i)} - p_{\\sigma(i)} \\geq v_{ij} - p_j,\\ \\forall j$.\\\\\nSince $i$ values all rooms equally and the room $i'$ get is cost less than the room $i$ get. \\\\\nSo $p_{\\sigma(i)} > p_k \\tab$(define room $k$ is the room player $i'$ get)\\\\\nwhich leads to $v_{i\\sigma(i)} - p_{\\sigma(i)} \\not\\geq v_{ik} - p_k \\tab$\\\\\\\\\nSo the statement \"player $i$ (players who values all rooms equally) is not always pay the least amount of rent in any envy-free outcome\" is false. Therefore player $i$ pays the least amount of rent in any envy-free outcome, since if there exist an player who pay less than $i$, than it's not envy-free anymore.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 2}\n\nBase on the envy-free property we have: $\\overbrace{v_{i\\sigma(i)}- p_{\\sigma(i)}}^{A} \\geq \\overbrace{v_{ij} - p_j}^{B},\\ \\forall j$\\\\\ndefine $A = v_{i\\sigma(i)}- p_{\\sigma(i)}$ and $B = v_{ij} - p_j$\\\\\nTo satisfy envy-free there will have two different outcome:\\\\\n1: when $A>B$\\\\\n2: when $A=B$\\\\\\\\\nFor case 1:\\\\\nif $A > B$ that means there exist an room $j^*$ such that $v_{i\\sigma(i)}- p_{\\sigma(i)} > v_{ij^*} - p_{j^*}$\\\\\nsince room $j^*$ will assign to a player $i'$ and because everyone have the same valuation for all the rooms so $i' = i$.\\\\\nTherefore we have $v_{i'j^*} - p_{j^*} < v_{i\\sigma(i)} - p_{\\sigma(i)}$,\nsince $v_{i'j^*} - p_{j^*} \\not \\geq v_{i\\sigma(i)} - p_{\\sigma(i)}$ which break the envy-free property.\nTherefore case 1 cannot happen, since when $A > B$, then the outcome is not envy-free.\\\\\\\\\nFor case 2:\\\\\nsince base on case 1 we know $A > B$ cannot satisfy envy-free, so we have to assume $A = B$ for all players, so the outcome satisfy envy-free.\\\\\\\\\nwhen $A = B$ there will have three cases:\\\\\n$a$: $A = B > 0$\\\\\n$b$: $A = B < 0$\\\\\n$c$: $A = B = 0$\\\\\\\\\nFor case $(a)$:\\\\\nif $A = B > 0$ happen, that means $v_{i\\sigma(i)} > p_{\\sigma(i)},\\ \\forall i$\\\\\nTherefore $\\sum_{i = 1}^n v_{i\\sigma(i)} > \\sum_{i = 1}^n p_{\\sigma(i)}$\\\\\\\\\n$\\sum_{i}^n v_{i\\sigma(i)}$ means the sum of all the player value their assign room. Since there are $n$ players and $n$ rooms and everyone cannot get the same room, therefore $\\sum_{i = 1}^n v_{i\\sigma(i)} = \\sum_{j = 1}^n v_{ij}$\\\\\n$\\sum_{i = 1}^n p_{\\sigma(i)}$ means the sum of all the room price that assign to players. Since there are $n$ players and $n$ rooms and everyone cannot get the same room, therefore $\\sum_{i = 1}^n p_{\\sigma(i)} = \\sum_{j = 1}^n p_{j}$\\\\\\\\\nSince $\\sum_{i = 1}^n v_{i\\sigma(i)} = \\sum_{j = 1}^n v_{ij}$ and $\\sum_{i = 1}^n p_{\\sigma(i)} = \\sum_{j = 1}^n p_{j}$ and $\\sum_{i = 1}^n v_{i\\sigma(i)} > \\sum_{i = 1}^n p_{\\sigma(i)}$\\\\\nso we got $\\sum_{j = 1}^n v_{ij} > \\sum_{j = 1}^n p_{j}$\\\\\nWhich this break the requirement from the question: $\\sum_{j = 1}^n v_{ij} = \\sum_{j = 1}^n p_{j}$\\\\\nSince case $a$ break the requirement from the question, therefore case $a$ cannot happen too.\\\\\\\\\nFor case $(b)$:\\\\\nif $A = B > 0$ happen, that means $v_{i\\sigma(i)} < p_{\\sigma(i)},\\ \\forall i$\\\\\nTherefore $\\sum_{i = 1}^n v_{i\\sigma(i)} < \\sum_{i = 1}^n p_{\\sigma(i)}$\\\\\\\\\nSince $\\sum_{i = 1}^n v_{i\\sigma(i)} = \\sum_{j = 1}^n v_{ij}$ and $\\sum_{i = 1}^n p_{\\sigma(i)} = \\sum_{j = 1}^n p_{j}$ and $\\sum_{i = 1}^n v_{i\\sigma(i)} < \\sum_{i = 1}^n p_{\\sigma(i)}$\\\\\nso we got $\\sum_{j = 1}^n v_{ij} < \\sum_{j = 1}^n p_{j}$\\\\\nWhich this break the requirement from the question: $\\sum_{j = 1}^n v_{ij} = \\sum_{j = 1}^n p_{j}$\\\\\nSince case $b$ break the requirement from the question, therefore case $b$ cannot happen too.\\\\\\\\\nFor case $(c)$:\\\\\nif $A=B=0$ happen, that means $v_{i\\sigma(i)} = p_{\\sigma(i)},\\ \\forall i$\\\\\nTherefore $\\sum_{i = 1}^n v_{i\\sigma(i)} = \\sum_{i = 1}^n p_{\\sigma(i)}$\\\\\\\\\nSince $\\sum_{i = 1}^n v_{i\\sigma(i)} = \\sum_{j = 1}^n v_{ij}$ and $\\sum_{i = 1}^n p_{\\sigma(i)} = \\sum_{j = 1}^n p_{j}$ and $\\sum_{i = 1}^n v_{i\\sigma(i)} = \\sum_{i = 1}^n p_{\\sigma(i)}$\\\\\nso we got $\\sum_{j = 1}^n v_{ij} = \\sum_{j = 1}^n p_{j}$.\\\\\nSince case $(c)$ is the only case that satisfy envy-free and $\\sum_{j = 1}^n v_{ij} = \\sum_{j = 1}^n p_{j}$. Therefore there exist only one unique EF price vector. That is $p_{\\sigma(i)} = v_{i\\sigma(i)},\\ \\forall i$ which is set all the room price equal to player's room valuation. \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 3}\n\nIf this is not true that means there exist an room $j^*$ such that $p_{j} > p_{j^*}$ and the outcome is envy-free.\\\\\nBase on the given information we know every player value room $j$ the least, therefore $v_{ij} < v_{ij^*}$\\\\\\\\\nAssume player $i$ got assign to room $j$\n\\\\therefore base on the envy-free property we know $v_{ij} - p_j \\geq v_{ij^*} - p_{j^*} \\longrightarrow (\\star)$\\\\\\\\\nHowever since $p_{j} > p_{j^*} = -p_{j} < -p_{j^*}$ and $v_{ij} < v_{ij^*}$ we know $v_{ij} - p_j < v_{ij^*} - p_{j^*}$ which contradict with $(\\star)$.\\\\\\\\\nTherefore the statement \"there exist an room $j^*$ such that $p_{j} > p_{j^*}$ and the outcome is envy-free\" is false. So the price of room $j$ should be the lowest, since if there are room lower that $j$ than it's not envy-free anymore.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 1.a}\nBase on the VCG mechanism determine the payment for each player we have:\n$$p_i = \\sum_{j \\not = i} v_j(o_{-i}^*) - \\sum_{j \\not = i}v_j(o^*) = \\sum_{j \\not = i}[v_j(o_{-i}^*) - v_j(o^*)] \\longrightarrow (\\star)$$\nDefine $S^*$ as the set contain all the successful bidders from $o^*$. Another way to say this is $S^*$ has all the bidders form $o^*$ who's utility is greater than $0$.\\\\\\\\\nSince there are $k$ items and each bidder only can have one item, therefore both $S^*$ will have the size of $k$ bidders.\\\\\\\\\nBase on VCG mechanism, $o^*_{-i}$ is the socially optimal outcome without $i$. Therefore given any $i$ was in $S^*$ that means we have to make a new set $S^*_{-i}$ to find the substitute bidder $i'$ for $i$.\\\\\nSince we are making the socially optimal outcome, therefore $i'$ should be the highest bidder amount all the bidders who are not in $S^*$. Because if $i'$ is not the highest bidder amount all the bidders who are not in $S^*$, then it's not socially optimal outcome anymore.\\\\\\\\\nBase on $(\\star)$ we know how we assign the value for these successful bidders is base on the sum of $v(j) \\in S^*_{-i}$ minus the sum of $v(j) \\in S^* \\setminus i$ $\\tab \\bullet$($j$ is a player)\\\\\nSince the only different between $S^*$ and $S^*_{-i}$ is $S^*$ have $i$ but don't have $i'$ and $S^*_{-i}$ have $i'$ but don't have $i$. Therefore the only different between $S^*\\setminus i$ and $S^*_{-i}$ is $S^*_{-i}$ have $i'$ but $S^*$ don't have.\\\\\\\\\nSince the only different between $S^*\\setminus i$ and $S^*_{-i}$ is $S^*_{-i}$ have $i'$ but $S^*$ don't have. Therefore the sum of $v(j) \\in S^*_{-i}$ minus the sum of $v(j) \\in S^* \\setminus i$ is always equal to $v(i')$\\\\\\\\\nSince no matter which bidder we take away from $S^*$ VCG will always find the same bidder $i'$ to substitute $i$. Therefore all the successful bidder will have the same pay, since base on $(\\star)$ the formula will always return $v(i')$.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 1.b}\n It is possible for example: \\\\\n Assume the auction have 4 bidders and 3 items A,B and C.\\\\\n set A and B are identical\\\\\n then we can make a table of each player bids the items.\n \\begin{center}\n\\begin{tabular}{ | m{4em} | m{4em}| m{4em} | m{4em} |} \n \\hline\n & A,B & C \\\\ \n \\hline\n $p_1$ & 100 & 100 \\\\ \n \\hline\n $p_2$ & 100 & 200 \\\\ \n \\hline\n $p_3$ & 200 & 100 \\\\ \n \\hline\n $p_4$ & 50 & 100 \\\\ \n \\hline\n\\end{tabular}\n\\end{center}\nBase on the VCG mechanism we know $o^* = 500$ with $p_1$ get item A, $p_2$ get item C and $p_3$ get item B.\\\\\nset $W^+_i = \\sum_{j \\not = i} v_j(o^*)$\\\\\nset $W^-_i = \\sum_{j \\not = i} v_j(o_{-i}^*)$\\\\\\\\\nBase on VCG mechanism we know they way it determine the payment for bidder $i$ will be:\n\\begin{align*}\n&W^+_1 = 400\\\\\n&W^-_1 = 450\\\\\n&p_1 = 450 - 400 = 50\\\\\\\\\n&W^+_2 = 300\\\\\n&W^-_2 = 400\\\\\n&p_2 = 400 - 300 = 100\\\\\\\\\n&W^+_3 = 300\\\\\n&W^-_3 = 350\\\\\n&p_3 = 350 - 300 = 50\\\\\n\\end{align*}\nSince bidder $2$ and $3$ receive items they value the same ($p_2$ get item C and $p_3$ get item B). But $p_3$ pay $50$ and $p_2$ pay $100$. Therefore it is possible that two bidders who receive items they value the same will pay different prices.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 2}\n\nVCG cannot extract truthful valuations from the bidders.\\\\ \nFor example there exist cases such as:\\\\\n$v_1 > v_2 > v_3 > ... v_n$\\\\\nIn this case if $p_1$ works with $p_2$, they can maximize their joint utility by let $p_2$ say $v_2 = 0$ therefore base on the valuation, we will have $v_1 > v_3 > ... v_n$\\\\\nBase on this valuation, VCG will let $p_1$ be the winner of the auction and let $p_1$ pay the second highest price, which is $v_3$. Since this is the single item auction, therefore everyone except $p_1$ utility will be $0$ and $p_1$ utility will be $u_1 = v_1 - v_3$. Therefore $p_1$ and $p_2$ joint utility will be $u_1 + u_2 = v_1 - v_3 + 0$\\\\\\\\\nWhich compare to if they don't work together, $p_1$ will be the winner of the auction. $p_1$ will still be the winner, since $p_1$ is the highest and base on VCG, $p_1$ will pay the second price which is $v_2$. Since this is the single item auction therefore everyone's utility will be $0$ except $p_1$'s utility will be: $u_1 = v_1 - v_2$. $p_1$ and $p_2$ joint utility will be $v_1 - v_2 + 0$\\\\\\\\\nSince $v_3 < v_2$ therefore $v_1 - v_2 + 0 < v_1 - v_3 + 0$ Which means $p_1$ and $p_2$ can have a higher joint utility when they work together. therefore VCG cannot extract truthful valuations from the bidders in this case.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 3.a}\n\nThird price auction is not truthful.\nIf the third price auction is truthful, that means all bidders best response is always to bid truthfully. however this is not truth base on the case where the second highest bidder $i$ bidding honestly or dishonestly will win or lose the auction.\\\\\\\\\nIf $i$ bidding honestly, and $i$ is the second highest bidder, then $i$ cannot win the auction since he is not the highest bidder, therefore $i$'s utility is $0$.\\\\\nIf $i$ bidding untruthfully by increase his value higher the the highest bidder, than $i$ become the highest bidder and win the auction. Since $i$ win the auction so he will pay the third highest prise. Since $i$'s true value for the item is higher than the third highest bidder, therefore it is good for $i$ since he win the auction and pay less than what he actually think the item is.\\\\\\\\\nTherefore third price auction is not truthful, since the second highest bidder can lie his bid to be highest and win the auction without paying more than what he actually values it.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 3.b}\n\nThis auction is truthful since players lie to increase or decrease their value will not increase their utility.\\\\\\\\\nTo prove this is true, we are going to look at:\\\\\n1. when player $i$ win by telling the truth\\\\\n2. when player $i$ lose by telling the truth\\\\\\\\\nFor the first case:\\\\\nwhen player $i$ win, that means $i$ is going to pay the second highest bid, or the reserve price (whichever is higher).\\\\\nIf $i$ lie by increase his bid, this action will not effect the second highest bid or the reserve price, so he is still going to pay $max(b_2,r)$. Which means his utility still the same $u_i = v_i - max(b_2,r)$.\\\\\nIf $i$ lie by decrease his bid, if $b_i$ still greater than $max(b_2,r)$, then this action will still not effect the second highest bid or the reserve price. So he is still going to pay $max(b_2,r)$ which means his utility still the same $u_i = v_i - max(b_2,r)$.\\\\\nIf $i$ lie by decrease his bid, if $b_i$ less or equal to $max(b_2,r)$, then this action will change the fact that $i$ is the winner, since $b_1 \\leq max(b_2,r)$. Since $i$ is not the winner anymore therefore his utility will from $u_i = v_i - max(b_2,r)$ to $u_i = 0$ which is worse than before.\\\\\nIn the first case, since no matter how $i$ lie by increase or decrease the value his utility will either be the same or become worse, therefore there is no reason for bidder $i$ to lie.\\\\\\\\\nFor the second case:\\\\\nwhen player $i$ lose the auction, that means it's either: \\\\\na. $b_i$ is the highest bid but less than $r$\\\\\nb. $b_i$ is not the highest bid which $b_i < max(b_1,r)$\\\\\\\\\nFor case a: \\\\\nIf $i$ increase his bid higher but still not greater than $r$, than this action will not change the fact that he lose the auction. Since $i$ still lose the auction, so his utility will still be $0$ like before.\\\\ \nIf $i$ increase his bid higher than $r$, than this action will make him become the winner, but this will decrease his utility since his true valuation is $v_i < r$ and now $i$ has to pay $r$ therefore his utility will be $u_i = v_i - r < 0$ which is worse than before.\\\\\nIf $i$ decrease his bid, than this action ill not change the fact that he lose the auction. Since $i$ still lose the auction, so his utility will still be $0$ like before.\\\\\\\\\nFor case b:\\\\\nIf $i$ increase his bid higher but will not greater than $max(b_1,r)$ than this action will not change the fact that he lose the auction. Since $i$ still lose the auction, so his utility will still be $0$ like before.\\\\\nIf $i$ increase his bid higher than $max(b_1,r)$, than this action will make him become the winner, but this will decrease his utility since his true valuation is $v_i < r$ and now $i$ has to pay $r$ therefore his utility will be $u_i = v_i - max(b_1,r) < 0$ which is worse than before.\\\\\nIf $i$ decrease his bid, than this action ill not change the fact that he lose the auction. Since $i$ still lose the auction, so his utility will still be $0$ like before.\\\\\\\\\nIn the second case, since no matter how $i$ lie by increase or decrease the value, his utility will either be the same or become worse, therefore there is no reason for bidder $i$ to lie.\\\\\\\\\nSince no matter the player win or lose they cannot increase their utility by give a wrong valuation, therefore this auction is truthful.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Problem 4}\nAssume bidder $i'$ bid $\\frac{n-1}{n}v_{i'}$ and bidder $i$ bids $s_i$\\\\\n$i$ will win when $v_{i'} < \\frac{n-1}{n}s_i$ and will have the utility of $v_i - s_i$, but will lose when $v_{i'} > \\frac{n-1}{n}s_i$ and will have the utility $0$.\nBase on these information we can have expected probability for player $i$: \n\\begin{align*}\nE[u_i] &= \\int_0^{\\frac{n-1}{n}s_i} (v_r - s_r) dv_{i'} + \\int_{\\frac{n-1}{n}s_i}^1 (0) dv_{i'}\\\\\n&= \\frac{n}{n-1}s_iv_i - \\frac{n}{n-1}s_i^2\n\\end{align*}\nFind player $i$ best bid by derivative the result:\n\\begin{align*}\n\\frac{d}{ds_i}(\\frac{n}{n-1}s_iv_i - \\frac{n}{n-1}s_i^2) &= 0\\\\\n\\frac{n}{n-1}v_i - 2\\frac{n}{n-1}s_i &= 0\\\\\ns_i = \\frac{1}{2}v_i\n\\end{align*}\nThis apply to when $i'$ bit $s_{i'}$\\\\\nSince this is only for two player game, so $n = 2$\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\end{document}", "meta": {"hexsha": "a1675742a731fc3c0d0644523ae4e4eec4c26d5f", "size": 16127, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "590T hw8.tex", "max_stars_repo_name": "briansun1210/Game-Theory-Prov", "max_stars_repo_head_hexsha": "ce14c776b8719bac1cd0c790959b05bc3987e447", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2022-01-25T20:24:09.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-25T20:29:11.000Z", "max_issues_repo_path": "590T hw8.tex", "max_issues_repo_name": "briansun1210/Game-Theory-Prov", "max_issues_repo_head_hexsha": "ce14c776b8719bac1cd0c790959b05bc3987e447", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "590T hw8.tex", "max_forks_repo_name": "briansun1210/Game-Theory-Prov", "max_forks_repo_head_hexsha": "ce14c776b8719bac1cd0c790959b05bc3987e447", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 75.3598130841, "max_line_length": 398, "alphanum_fraction": 0.6546784895, "num_tokens": 5399, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34958202995258786}} {"text": "%!TEX root = thesis-kdyoung.tex\n\n% \\begin{savequote}%[45mm]\n% If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.\n% \\qauthor{John von Neumann}\n% \\end{savequote}\n\\begin{savequote}%[45mm]\n---Everything should be made as simple as possible, but not simpler.\n\\qauthor{Albert Einstein}\n\\end{savequote}\n\n\\chapter{Introduction}\n\\label{chap:intro}\nAssembly lines are flow-based production systems widely used by\nthe manufacturing industry.\nDesigning these systems to meet the requirements set\nby the industry gives rise to the Assembly Line\nBalancing Problem (\\albp{}).\nAn assembly line problem consists of a sequence\nof \\emph{stations} and a set of \\emph{tasks}.\nStations are fixed points along the assembly\nline which are operated by either human personnel\nor autonomous machinery.\nA workpiece is launched down the line and \na subset of the tasks are performed on it at each station.\nOnce all required tasks are finished, the completed product\nexits the assembly line.\n% The construction of a single product is completed\n% once all tasks have performed on it \n% To completely construct a single product\n% the all tasks are performed along the line.\nThe workload of tasks must be divided among the stations\nto optimize some performance measure,\nwhile also respecting physical restrictions.\nA standard example of such a restriction would be a\nprecedence relation, which can require\none task to be completed before another can begin\nprocessing on the product.\nThe line itself is a mechanism used to\ntransport the products between stations;\nmost commonly the line is a conveyor belt.\n\nThere are two common objectives when designing an assembly\nline, which each lead a distinct problem.\nThe first is called the type-1 problem and is concerned with\nthe design of a new assembly line where the production rate\nof the product is predefined and the aim is to construct the line\nwith as few stations as possible, in-order to meet this demand.\nThe second problem, called type-2, considers the redesign of an existing\nassembly line where the number of stations is fixed and the rate\nof production is optimized.\nThe second case can arise when a manufacturing company alters their\nproduction process or changes the products they offer.\nOur thesis focuses on solving the type-2 problem.\n\nThe \\emph{cycle time} of an assembly line defines the time each station\nis given to perform its assigned tasks.\nAs such, finding the minimal possible cycle time \nis the primary aim when optimizing the production\nrate of an assembly line.\nSince all stations are connected by a single line,\nthe station with the largest workload will define the\noverall cycle time of the line.\nTo achieve the minimal cycle time value,\nwe aim to balance the workload of the tasks\nas evenly as possible among the stations.\n\nThe structure of the paper is as follows.\nThe remainder of Chapter \\ref{chap:intro}\nprovides illustrative examples of the problem\nand closes with some motivation.\nIn Chapter \\ref{chap:theory} we detail a selection\nof the relevant theory for our problem, spanning\nthe fields of Operations Research and Artificial Intelligence.\nChapter \\ref{chap:mip} presents the mixed-integer\nprograms which were used to compare our\nsolution methodology against.\nThe construction of our logic-based Benders\ndecomposition is presented in Chapter \\ref{chap:benders},\ntogether with many possible modelling choices.\nThe results of our computational experiments are \ngiven in Chapter \\ref{chap:exp}.\nWe conclude in Chapter \\ref{chap:conc}\nwith some remarks on the performance of our\nmethod and finally note a range of possible directions\nfor future research on this topic.\n\n\\section{Preliminaries and Examples}\n\\label{sec:intro:prelim}\nThe fundamental problem in this area is called the\nSimple \\albp{}, or \\sab{}; we denote\nthe type-2 version by \\sab{2}.\nWe now detail some straightforward examples\nto illustrate how the workload\nof an assembly line can be balanced.\nIn Example \\ref{ex:intro:simple} we provide an instance\nof the \\sab{2} with a possible feasible solution and the optimal solution given\nin Figures \\ref{fig:intro:exSchedFeas} and \\ref{fig:intro:exSchedOpt}\nrespectively. Note, that $t_i$ denotes the processing time of task $i$ and\n$c$ denotes the cycle time of the assembly line.\n\n\\begin{example}\\label{ex:intro:simple}\n\tConsider an instance of the problem with four tasks $T_1,\\ldots,T_4$,\n\tand three stations $S_1,\\ldots,S_3$,\n\talong the assembly line.\n\tThe aim is to find an assignment of the tasks to the stations\n\twhich minimizes the cycle time.\n\tThis assignment must respect the precedence relations listed in Figure \\ref{fig:intro:exPrec}, \n\t\\eg task $T_1$ must be completed before task $T_2$.\n\tThe processing time of each task is included in Figure \\ref{fig:intro:exPrec}\n\tnext to each task.\n\n\tA feasible solution to this problem is given in Figure \\ref{fig:intro:exSchedFeas},\n\twhere the stations are spaced along the horizontal axis.\n\tEach station's workload begins at the horizontal axis and proceeds\n\tupward through time until all assigned tasks are completed.\n\tIn this solution, the workload of $S_3$ is the largest\n\tand as such defines the cycle time as $t_3+t_4=2+9=11$.\n\n\tThis feasible solution can be improved by moving task $T_3$ from station\n\t$S_3$ to $S_2$.\n\tFor this assignment to remain feasible, $T_3$ must be processed\n\tafter $T_2$, due to the precedence relations.\n\tThis new assignment leads to the optimal solution listed in\n\tFigure \\ref{fig:intro:exSchedOpt} with a cycle time of nine.\t\\qed\n\\end{example}\n\n\\begin{figure}[tpb]\n\t\\centering\n\t\\caption{Precedence graph}\n\t\\vspace{2mm}\n\t\\includegraphics[width=0.45\\textwidth]{images/precgraphSimple.eps}\n\t\\label{fig:intro:exPrec}\n\\end{figure}\n\n\\begin{figure}[tpb]\n\t\\centering\n\t\\begin{minipage}{0.45\\textwidth}\n\t\t\\centering\n\t\t\\caption{Feasible solution}\n\t\t\\vspace{2mm}\n\t\t\\includegraphics[width=\\linewidth]{images/exSimpleFeas.eps}\n\t\t\\label{fig:intro:exSchedFeas}\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}{0.45\\textwidth}\n\t\t\\centering\n\t\t\\vfill\n\t\t\\caption{Optimal solution}\n\t\t\\vspace{2mm}\n\t\t\\includegraphics[width=\\linewidth]{images/exSimpleOpt.eps}\n\t\t\\label{fig:intro:exSchedOpt}\n\t\\end{minipage}\n\\end{figure}\n\nThe variant of the \\albp{} which we are concerned with adds the consideration\nof sequence-dependent setup times between consecutive tasks\nwithin a station's workload.\nTheoretical approaches to the \\albp{} usually assume that the assembly\nline workers can decide on an arbitrary precedence-feasible sequence\nto execute their assigned tasks, which will not affect the station's total\nprocessing time.\nHowever, in practice there can be non-trivial setup costs, due\nto walking times or tool changes, which can account for a considerable amount\nof a station's processing time.\nAdding this consideration to the problem\nleads to a scheduling problem arising within each station.\nThis variant of the problem is called the SetUp Assembly Line Balancing and\nScheduling Problem (\\sua{}).\n\nTo realistically model the setup costs that occur\nin assembly lines, two types of setups are introduced;\n\\emph{forward} and \\emph{backward} setups.\nA forward setup time, denoted by $\\phi$, occurs between two consecutive\ntasks within a station's task sequence if both tasks\nare performed on the same product.\nWhereas a backward setup time, denoted by $\\beta$, occurs between the last task\nperformed on a product and the first task performed on\nthe next product along the line.\n\n\\begin{figure}[tpb]\n\t\\centering\n\t\\caption{Cyclic task sequence of a station (adapted from \\cite{Scholl2013})}\n\t\\vspace{2mm}\n\t\\includegraphics[width=0.8\\textwidth]{images/IntroForwBackSetupEx.eps}\n\t\\label{fig:intro:forwBackDifference}\n\\end{figure}\n\nTo see how these setups are differentiated, \nFigure \\ref{fig:intro:forwBackDifference} depicts the sequence of tasks \n$T_i,\\:T_j,\\:T_k$ which are performed in a cyclic manner on consecutive\nproducts.\nThe execution of tasks proceeds chronologically from right to left,\n\\ie $T_1$ is the first task performed in cycle $p$.\nForward setups arise between consecutive tasks operating\non the same product.\nA backward setup cost is incurred between the last\ntask of cycle $p$ and the first task of the next cycle, $p+1$.\nEach cycle begins at time zero with the execution of task $T_i$\ntaking $t_i$ time units to process.\nThen after performing the forward setups $\\phi_{ij}$ and $\\phi_{jk}$\ntogether with processing times $t_j$ and $t_k$,\nthe station operator must perform the necessary backward\nsetup operation before moving to the next\nworkpiece along the line. \nThus, when including this backward setup time $\\beta_{ki}$,\nthe following condition must be satisfied to have a \nfeasible cycle time\n$t_i+\\phi_{ij}+t_j+\\phi_{jk}+t_k+\\beta_{ki} \\leq c$.\nIn Figure \\ref{fig:intro:forwBackDifference}, the gray arrow from $T_k$ to $T_i$ implies\nthe cyclic nature of the sequence.\n\n% Example \\ref{ex:intro:simpleSetup} alters the previous\n% example into an instance of the \\sua{2}.\n\n\\begin{example}\\label{ex:intro:simpleSetup}\n\tAgain consider the instance from Example \\ref{ex:intro:simple},\n\tbut now with sequence-dependent setup times\n\tbetween tasks.\n\tThe arrays of forward and backward setup costs\n\tcan be found in Tables \\ref{tab:intro:forwSetupTimes}\n\tand \\ref{tab:intro:backSetupTimes} respectively.\n\tNote, that some of the entries in these arrays are\n\tomitted as the corresponding sequence of tasks is not \n\tpossible due to the precedence relations or\n\tlogical restrictions.\n\n\tThe previous optimal solution is amended\n\tto include the required setup costs and is given in Figure \n\t\\ref{fig:intro:exSchedSetupFeas}.\n\tNote that the setup cost of any task to itself is \n\tdefined as zero.\n\t% This results in a large forward setup, $\\phi_{23}$,\n\t% occurring on station $S_2$.\n\tTo reach the optimal solution we must now consider\n\tthe sequence of tasks within each station.\n\n\tBy moving task $T_3$ to station $S_3$ and considering\n\tthe sequencing of $T_3$ and $T_4$, we find the optimal\n\tsolution to this problem, given in Figure \\ref{fig:intro:exSchedSetupOpt}.\n\tThe optimal cycle time is 13 in this case.\t\\qed\n\\end{example}\n\n\\begin{table}[tpb]\n\t\\centering\n\t\\begin{minipage}{0.45\\textwidth}\n\t\t\\def\\arraystretch{1.1}\n\t\t\\centering\n\t\t\\caption{Forward setup times}\n\t\t\\vspace{2mm}\n\t\t\\begin{tabular}{lllll}\n\t\t\t\\toprule\n\t\t\t$\\phi$ & $T_1$ & $T_2$ & $T_3$ & $T_4$ \\\\\\midrule\\midrule\n\t\t\t$T_1$ & -- & 3 & 3 & 3 \\\\\n\t\t\t$T_2$ & -- & -- & 2 & 3 \\\\\n\t\t\t$T_3$ & -- & -- & -- & 1 \\\\\n\t\t\t$T_4$ & -- & -- & 2 & -- \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{tab:intro:forwSetupTimes}\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}{0.45\\textwidth}\n\t\t\\def\\arraystretch{1.1}\n\t\t\\centering\n\t\t\\caption{Backward setup times}\n\t\t\\vspace{2mm}\n\t\t\\begin{tabular}{lllll}\n\t\t\t\\toprule\n\t\t\t$\\beta$ & $T_1$ & $T_2$ & $T_3$ & $T_4$ \\\\\\midrule\\midrule\n\t\t\t$T_1$ & 0 & -- & -- & -- \\\\\n\t\t\t$T_2$ & 3 & 0 & -- & -- \\\\\n\t\t\t$T_3$ & 3 & 5 & 0 & 3 \\\\\n\t\t\t$T_4$ & 3 & 3 & 1 & 0 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\t\\label{tab:intro:backSetupTimes}\n\t\\end{minipage}\n\\end{table}\n\n\\begin{figure}[tpb]\n\t\\centering\n\t\\begin{minipage}{0.47\\textwidth}\n\t\t\\centering\n\t\t\\caption{Feasible solution with setup}\n\t\t\\vspace{2mm}\n\t\t\\includegraphics[width=\\linewidth]{images/exSimpleSetupFeas.eps}\n\t\t\\label{fig:intro:exSchedSetupFeas}\n\t\\end{minipage}\n\t\\hfill\n\t\\begin{minipage}{0.47\\textwidth}\n\t\t\\centering\n\t\t\\caption{Optimal solution with setup}\n\t\t\\vspace{2mm}\n\t\t\\includegraphics[width=\\linewidth]{images/exSimpleSetupOpt.eps}\n\t\t\\label{fig:intro:exSchedSetupOpt}\n\t\\end{minipage}\n\\end{figure}\n\n\\section{Motivation}\n\\label{sec:intro:motiv}\nIn this section we provide motivation for why the \\albp{}\nwith setup costs\nis an important problem to be solved and why the approach we\nchose to utilize is well-suited to the problem.\n\n\\subsection{Why the SUALBSP?}\nAssembly lines were originally designed for the production\nof a single type of product in high volumes.\nAlthough, assembly lines able to produce only a single\nproduct are not suitable for consumer-centric markets, \nwhere there is a need to tailor the products\nmore closely to the user's needs.\nSequence-dependent setup times are commonly considered\nin job shop scheduling problems and a range of other\nproduction problems (\\cf \\cite{Allahverdi2008a,Allahverdi2008b,Allahverdi2015}).\nIn those problems, the setup times arise because the products\nare typically assumed to be highly diverse,\nleading to additional configuration effort being required\nbetween any two tasks.\nIn the context of operational planning of assembly lines,\nsequence-dependent setup times have been sparsely\nconsidered by the literature.\n\\authciteb{Scholl2013} list a number of possible\npractical settings where setup times between tasks should not\nbe taken as negligible. Some include the following:\n\\begin{itemize}\n\t\\item In automotive assembly lines where large workpieces (vehicles)\n\tneed to be constructed, work can be performed at numerous\n\tmounting positions on the item.\n\tThe size of the car body can result in walking distances\n\tbetween mounting positions being non-trivial.\n\tThus, the preparation time (setup) can \n\tbe a critical aspect when sequencing the required tasks.\n\t\\item Fixed material containers can be placed along the moving \n\tconveyor system to allow workers to collect parts or tools needed.\n\tFurther to walking times between these containers and the workpiece,\n\tthe time taken to retrieve what is required from the container\n\tmust also be accounted for.\n\tAt a major German car manufacturer, these times contributed $10-15\\%$\n\tof the total cycle time \\cite{Scholl2013}.\n\t\\item Specific tools are often required to perform a task.\n\tIf consecutive tasks need different tools for their\n\texecution a tool-change between tasks will be necessary.\n\tRobotic assemblers used to perform the tasks of a station\n\tcan be built to be highly flexible in the types of tasks\n\tthey can perform.\n\tHowever, this flexibility can mean that numerous tool changes\n\tare required to switch from one task to another\n\tin a sequence.\n\tThus, these sequence-dependent setup costs from tool-changes\n\tare typical in robotic assembly lines.\n\\end{itemize}\nConsequently, accurately modelling the setup costs incurred\ncan be an important aspect when balancing an assembly line.\n\nIn practice, setup times are accounted for in simpler ways, such as\nusing an approximation or incorporating the setup time\ninto the task's processing time.\n\\citeauthor{Scholl2013} note that by using such approximations\n``the planning team need to strike a fragile balance between an underestimation\nof setups, which lead to infeasible line balances, and an overestimation\nof setups, which leads to an allocation of excessive resources.''\nThe procedure used to estimate the cost of setups\ncan be time consuming and is prone to getting stuck in sub-optimal solutions.\n\nFor these reasons we feel that further research into the \neffect of setup times on the \\albp{} is of interest to the\nacademic community and the manufacturing industry.\n\n\\subsection{Why Benders Decomposition?}\nBenders decomposition is a well-known approach\nto optimization problems.\nIt is naturally suited\nto problems where the decisions can be separated into\ntwo distinct sets.\nAdvances in the past few decades to the theory of\nthis method has broadened its applicability to a \nwider array of problems.\nIn our case, a solution to the \\sua{2} can naturally\nbe divided into the assignment portion, which\nassigns each task to a station, and the scheduling\nportion, which decides on the time that each task\nbegins execution within a station.\n\nWe can view this separation of concerns in the following\nway.\nThe manager of an assembly line needs a new item\nto be put into production and so assigns the required\nset of tasks to the stations along the line.\nWhen calculating the line's cycle time the manager only estimates the setup times\nwithin each station.\nEach station operator must then decide if a possible sequencing\nof their assigned tasks exists which can respect the manager's cycle time;\nor if a precedence-feasible sequence exists at all.\n% The worker at each station must then decide if there exists\n% a possible sequencing of their assigned tasks which can\n% respect the manager's cycle time; or if there a\n% precedence-feasible sequence exists at all.\nIf at least one worker cannot find a feasible task-sequence\nthat respects the manager's cycle time\nestimate, then a revision\nmay need to be made to the assignment of tasks.\nThe give-and-take between the assignment and scheduling portions\ncontinues until a mutually satisfactory solution is found.\nThis informal process of communicating information between\nthe station workers and the assembly line manager could be time\nconsuming and not guarantee optimality.\n\nThe informal feedback loop between the two halves of the\ndecision problem suggests that Benders decomposition could be\nemployed to mimic this process.\nThe decision making process can naturally be divided\ninto a master problem, which assigns the tasks\nto the stations along the line,\nand several independent scheduling problems.\nIn total, there is a sub-problem\nfor each of the $m$ stations, which needs to\ndecide the exact execution time of all tasks\nthat station has been assigned.\nEach sub-problem is similar to the asymmetric\nTravelling Salesperson Problem (TSP)\nwith some forbidden paths due to the precedence\nrelations.\nWith this interpretation, the tasks are viewed as the \ncities and distances between them are the setup times.\nTo complete the iterative loop, the sub-problems\nrelay their information back to the master as\nBenders cuts and the assignment problem is re-optimized.\nIn the words of J. N. Hooker,\n``the Benders cuts added to the master problem are\nthe mathematical equivalent of telephone calls''\nfrom the station workers to the line manager \\cite{Hooker2007}.\n\nThe reader may be familiar with the classical version\nof the Benders decomposition method due to the \nwork of \\authciteb{Benders1962} and \\authciteb{Geoffrion1972},\nhowever for the \\sua{}, this approach is inappropriate.\nDue to the sub-problems being highly-combinatorial discrete scheduling\nproblems, formulating them as a linear or non-linear program\nwill not be practical.\nWe instead explore how the more general method of \\emph{logic-based}\nBenders decomposition can be used to solve the \\sua{2}.\n\nBy employing this approach we are able to exploit the comparative\nadvantages of multiple solving technologies to tackle each\nportion of the problem.\nMixed-Integer Programs (MIPs) are well-suited to the assignment\nproblem of the master and Constraint Programming (CP), from \nthe Computer Science discipline, is an effective solving technology\nfor scheduling problems.\n\n\\section{Problem Definition}\n\\label{sec:intro:probDef}\nHere we present the core notation that will be used for the\nremainder of the thesis.\nAlong the conveyor belt of the assembly line\nthere are \\emph{work stations} $K=\\{1,2,\\ldots,m\\}$.\nWorkpieces (or products/items) move down the \nconveyor belt from station to station.\nA workpiece remains at each station for \none cycle, which has a duration given by the \\emph{cycle time}.\nThe objective of the \\albp{} is to optimally partition\nthe total work required among the stations with respect to\na performance measure.\n\nThe work required to complete a single piece is separated into\na set of \\emph{non-preemptive tasks} $V=\\{1,2,\\ldots,n\\}$,\neach with a discrete processing time $t_i$.\nPhysical and technical conditions impose a set of \n\\emph{precedence relations} $E\\subseteq V\\times V$,\nwhich prevent some non-allowed task orderings from occurring.\nIf we consider the graph $G=(V,E)$ with tasks as the vertex set\nand directed edges defined by the precedence relations,\nthen $G$ is a directed acyclic graph (DAG).\t\nWithout loss of generality, we may assume that the vertices\nare numbered topologically so that the following holds:\n$(i,j) \\notin E$ if $i>j$.\n\nBetween each pair of tasks, discrete\n\\emph{forward} and \\emph{backward setup} times\nare predefined.\nWe denote the forward setup time between $i$ and $j$\nby $\\phi_{ij}$ and the backward setup time\nby $\\beta_{ij}$.\nWhen $i=j$, the forward setup time is undefined\nas a task can never follow itself in a station's\nforward work load.\nThe backward setup $\\beta_{ii}$, \\ie the setup cost\nof a task to itself, is defined as zero.\n\nTo fully specify a feasible solution to the \\sua{2},\none must give the following:\nan assignment of tasks to the stations\nand the execution time window of each task within\nits assigned station.\nFrom this specification, the cycle time is calculated by\nthe total processing time of the station with the\nlargest workload.\n\n", "meta": {"hexsha": "f26e43553bfb1c92e8890bd0e0ec55a39cab9192", "size": 20592, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Assembly Task Scheduling/sualbsp-2-master/thesis/chap_intro.tex", "max_stars_repo_name": "BillChan226/ODA-Multi-Manipulator", "max_stars_repo_head_hexsha": "6863558a9fdb946fb16c67a7660172154274cfe0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Assembly Task Scheduling/sualbsp-2-master/thesis/chap_intro.tex", "max_issues_repo_name": "BillChan226/ODA-Multi-Manipulator", "max_issues_repo_head_hexsha": "6863558a9fdb946fb16c67a7660172154274cfe0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Assembly Task Scheduling/sualbsp-2-master/thesis/chap_intro.tex", "max_forks_repo_name": "BillChan226/ODA-Multi-Manipulator", "max_forks_repo_head_hexsha": "6863558a9fdb946fb16c67a7660172154274cfe0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.1017964072, "max_line_length": 118, "alphanum_fraction": 0.7816142191, "num_tokens": 5174, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269796369904, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34958202074957434}} {"text": "\\section{SEDflow} \\label{sec:sedflow}\nIn this section, we present \\sedflow, which applies ANPE to galaxy SED modeling\nfor a scalable and accelerated approach.\nFor our SED model, we use the state-of-the-art PROVABGS model from\n\\cite{hahn2022}. \nAlthough many SED models have been recently used in the\nliterature~(\\emph{e.g.} {\\sc Bagpipes}, \\citealt{carnall2018}; \n{\\sc Prospector}, \\citealt{leja2017, johnson2021}), we choose PROVABGS because\nit will be used to analyze >10 million galaxy spectrophotometry measured by the\nDESI Bright Galaxy Survey~(Hahn~\\etal~in prep.).\nBelow, we describe the PROVABGS model, the construction of the\n\\sedflow~training data using PROVABGS, and the training procedure for \\sedflow.\n\n% section explaining specific SED set up \n\\subsection{SED Modeling: PROVABGS} \\label{sec:provabgs}\nWe use the state-of-the-art SPS model of the\nPROVABGS~\\citep{hahn2022}. \nThe SED of a galaxy is modeled as a composite of stellar populations defined by\nstellar evolution theory (in the form of isochrones, stellar spectral\nlibraries, and an initial mass function) and its star\nformation and chemical enrichment histories (SFH and ZH), attenuated by\ndust~\\citep[see][for a review]{walcher2011, conroy2013}. \nThe PROVABGS model, in particular, utilizes a non-parametric SFH with a\nstarburst, a non-parametric ZH that varies with time, and a flexible dust\nattenuation prescription.\n\n% highlight advantages of provabgs \nThe SFH has two components: one based on non-negative matrix factorization\n(NMF) bases and the other, a starburst component.\nThe SFH contribution from the NMF component is a linear combination of four NMF\nSFH basis functions, derived from performing NMF~\\citep{lee1999, cichocki2009,\nfevotte2011} on SFHs of galaxies in the Illustris cosmological hydrodynamical\nsimulation~\\citep{vogelsberger2014, genel2014, nelson2015}.\nThe NMF SFH prescription provides a compact and flexible representation of the\nSFH.\nThe second starburst component consists of a single stellar population (SSP)\nand adds stochasticity to the SFH. \n\nThe ZH is similar defined using two NMF bases dervied from Illustris. \nThis ZH prescription enables us to flexibly model a wide range of ZHs and,\nunlike most SED models, it does not assume constant metallicity over time,\nwhich can significantly bias inferred galaxy properties~\\citep{thorne2021}. \nThe stellar evolution theory is based on Flexible Stellar Population\nSynthesis~\\citep[FSPS;][]{conroy2009, conroy2010c} with the MIST\nisochrones~\\citep{paxton2011, paxton2013, paxton2015, choi2016, dotter2016}, \nthe \\cite{chabrier2003} initial mass function (IMF), and a combination of the\nMILES~\\citep{sanchez-blazquez2006} and BaSeL~\\citep{lejeune1997, lejeune1998,\nwestera2002} libraries.\nThe SFH and ZH are binned into 43 logarithmically-space time and SSPs are\nevalulated at each time bin using FSPS. \nThe SSPs are summed up to get the unattenuated rest-frame galaxy SED. \n\nLastly, PROVABGS attenuates the light from the composite stellar population\nusing the two component \\cite{charlot2000} dust attenuation model with\ndiffuse-dust (ISM) and birth cloud (BC) components. \nAll SSPs are attenuated by the diffuse dust using the \\cite{kriek2013}\nattenuation curve.\nThen, the BC component provides extra dust attenuation on SSPs younger than 100\nMyr with young stars that are embedded in modecular clouds and HII regions. \nIn total the PROVABGS SED model has 12 parameters: stellar mass ($M_*$),\nsix SFH parameters ($\\beta_1, \\beta_2, \\beta_3, \\beta_4, t_{\\rm burst}, f_{\\rm\nburst}$), two ZH parameters ($\\gamma_1, \\gamma2$), and three dust attenuation\nparameters ($\\tau_{\\rm BC}, \\tau_{\\rm ISM}, n_{\\rm dust}$). \nEach PROVABGS model evaluation takes ${\\sim}340$ ms. \n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.85\\textwidth]{figs/training.pdf}\n \\caption{\\label{fig:data}\n The distribution of SED model parameters, redshift, photometric\n magnitudes, and uncertainties of the data used to train \\sedflow.\n We plot a subset of the parameters ($\\log M_*$, $\\beta_1$) and photometric\n bands for clarity.\n The training data was constructed by sampling SED model parameters from the\n prior and forward modeling optical photometry using the PROVABGS and noise\n models (Section~\\ref{sec:sedflow}). \n For comparison, we present the distribution of redshift, magnitudes, and\n uncertainties for galaxies in the NSA catalog (blue). \n \\emph{The training set encompasses the observations; thus, {\\sc SEDflow}\n can be used to infer the posterior for the NSA galaxies}.\n }\n\\end{center}\n\\end{figure}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% training data \n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Training Data} \\label{sec:training}\nIn this section, we describe how we construct the training data for\n\\sedflow~using the PROVABGS SED model.\nFirst, we sample $N_{\\rm train}$ model parameters from a prior: $\\theta'\\sim p(\\theta)$. \nWe use the same priors as \\cite{hahn2022}: uniform priors over $M_*,\nt_{\\rm burst}, f_{\\rm burst}, \\gamma_1, \\gamma_2, \\tau_{\\rm BC}, \\tau_{\\rm ISM},\nn_{\\rm dust}$ with broad conservative ranges and Dirichlet prior over $\\beta_1,\n\\beta_2, \\beta_3, \\beta_4$, chosen to normalize the NMF SFH.\nFor each $\\theta'$, we also uniformly sample a redshift within the range of the\nNSA: $z' \\sim \\mathcal{U}(0., 0.2)$. \nNext, we forward model mock observables. \nWe calculate the rest-frame galaxy SED from PROVABGS and redshift it: \n$F(\\lambda;\\theta', z)$. \nAfterwards, we convolve $F$ with optical broadband filters, $R_X$, to generate\nnoiseless photometric fluxes:\n\\begin{equation}\n f_X(\\theta', z') = \\int F(\\lambda;\\theta', z') \\, R_X(\\lambda) \\, {\\rm d}\\lambda\n\\end{equation}\nThe next step of the forward model is to apply noise. \nWe assign photometric uncertainties, $\\sigma'_X$, by sampling an estimate of\nthe observed $p(\\sigma_X | f_X)$ of NSA galaxies. \nThen, we apply Gaussian noise\n\\begin{equation} \\label{eq:noise} \n \\hat{f}_X(\\theta', z', \\sigma'_x) = f_X(\\theta', z') + n_X \\quad {\\rm where}~n_X \\sim \\mathcal{N}(0, \\sigma'_X)\n\\end{equation}\nto derive the forward modeled photometric flux.\n\nFor our estimate of $p(\\sigma_X | f_X)$, we use an empirical estimate based on\nNSA photometry and measured uncertainties. \nFor each of the five optical bands, we separately estimate \n\\begin{equation}\n \\hat{p}(\\sigma_X | f_X) = \\mathcal{N} \\big( \\mu_{\\sigma_X}(f_X),\n \\sigma_{\\sigma_X}(f_X) \\big)\n\\end{equation}\nas a Gaussian in magnitude-space. \n$\\mu_{\\sigma_X}$ and $\\sigma_{\\sigma_X}$ are the median and standard deviation\nof $\\sigma_X$ as a function of $f_X$ that we estimate by evaluating them in\n$f_X$ bins and interpolating over the bins. \nAny $\\theta'$ that is assigned a negative $\\sigma'_X$ is removed from our\ntraining data. \nWe also remove any training data with $f_X(\\theta')$ outside the range of NSA\nphotometry. \n\nAs SBI requires an accurate noise model, one might be concerned about the\nsimplicity of our model.\nIf the noise model is incorrect, estimates of the parameters $\\theta$ would be\nbiased by an amount that is impossible to predict.\nWe therefore add the noise variances $\\sigma_X$ to the conditioning variables\nof the posterior model, \\emph{i.e.} we train and evaluate \n$p_\\phi(\\btheta\\given\\bfi{x},\\{\\sigma_X\\})$.\nThis means we merely have to ensure that $\\sigma'_X$ spans the observed\n$\\sigma_X$ values in order to have a posterior that is robust to our choice of\nnoise model.\nWe discuss this further in Section~\\ref{sec:discuss}.\n\nIn total, we constuct $N_{\\rm train} = 1,131,561$ sets of SED parameters,\nredshift, photometric uncertainties, and mock NSA photometry. \nIn Figure~\\ref{fig:data}, we present the distribution of the training data\n$\\{(\\theta', z, \\sigma'_X, \\hat{f}_X) \\}$ (black).\nWe include select SED model parameters ($\\log M_*$, $\\beta_1$), redshift,\nphotometry in the $g$ and $r$ bands, and photometric uncertainty in the $r$\nband.\nWe also include the $(z, \\sigma_X, f_X)$ distribution of NSA galaxies (blue),\nfor comparison.\nThe photometry and uncertainties are in magnitude-space. \nThe distribution of the training data spans the distribution of NSA galaxies.\n\n\\subsection{Training \\sedflow} \\label{sec:anpe_train}\nFor \\sedflow, we use a MAF normalizing flow model (Section~\\ref{sec:flow}) with \n15 MADE blocks, each with 2 hidden layers and 500 hidden units.\nIn total, the model has 7,890,330 parameters, $\\bphi$. \nWe determine this architecture through experimentation. \nOur goal is to determine $\\bphi$ of the MAF model, \n$p_\\phi(\\btheta\\given\\bfi{x})$, so that it accurately estimates the\nposterior probability distribution $p(\\btheta\\given\\bfi{x})$. \n$\\btheta$ represent the SED parameters and $\\bfi{x} = (f_X, \\sigma_X, z)$.\nWe do this by minimizing the KL divergence between \n$p_\\phi(\\btheta\\given\\bfi{x})$ and $p(\\btheta\\given\\bfi{x})$: \n$D_{\\rm KL} (p\\,||\\,p_\\phi)$.\n\nIn practice, we split the training data into a training and validation set with\na 90/10 split. \nAfterwards, we maximize the total log likelihood \n$\\sum_i \\log p_\\phi(\\btheta_i\\given \\bfi{x}_i)$ over training set, which is\nequivalent to minimizing $D_{\\rm KL} (p\\,||\\,p_\\phi)$.\nWe use the {\\sc Adam} optimizer~\\citep{kingma2017} with a learning rate of $5\\times10^{-4}$. \nTo prevent overfitting, we evaluate the total log likelihood on the validation\ndata at every training epoch and stop the training when the validation log\nlikelihood fails to increase after 20 epochs. \nTraining our model with a batch size of 50 takes roughly a day on a single 2.6\nGHz Intel Skylake CPU. \nGiven our small batch size, we find similar training times when using CPUs or\nGPUs. \n", "meta": {"hexsha": "31f04de37ae15633d90f5b63a3905652edb90c2c", "size": 9699, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/paper/sedflow.tex", "max_stars_repo_name": "changhoonhahn/SEDflow", "max_stars_repo_head_hexsha": "4561ecfe3a38cc4c25df263d971a87e8a83f88ce", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 18, "max_stars_repo_stars_event_min_datetime": "2022-03-16T03:11:04.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T16:01:42.000Z", "max_issues_repo_path": "docs/paper/sedflow.tex", "max_issues_repo_name": "changhoonhahn/SEDflow", "max_issues_repo_head_hexsha": "4561ecfe3a38cc4c25df263d971a87e8a83f88ce", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/paper/sedflow.tex", "max_forks_repo_name": "changhoonhahn/SEDflow", "max_forks_repo_head_hexsha": "4561ecfe3a38cc4c25df263d971a87e8a83f88ce", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 53.5856353591, "max_line_length": 116, "alphanum_fraction": 0.7456438808, "num_tokens": 2735, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.34932965773726726}} {"text": "% Filename: p2_set_and_functions@tikz_for_teachers.tex\n% This code is part of LaTeX with Vim.\n% \n% Description: TikZ for teachers is free book about TikZ and Sage.\n% \n% Created: 30.03.12 08:25:52 PM\n% Last Change: 30.03.12 08:26:12 PM\n% \n% Author: Raniere Gaia Costa da Silva, r.gaia.cs@gmail.com\n% Organization: \n% \n% Copyright (c) 2010, 2011, 2012, Raniere Gaia Costa da Silva. All rights \n% reserved.\n% \n% This file is license under the terms of a Creative Commons Attribution \n% 3.0 Unported License, or (at your option) any later version. More details\n% at .\n\\chapter{Conjuntos e Fun\\c{c}\\~{o}es}\n\n\\section{No\\c{c}\\~{o}es de L\\'{o}gica}\n\n\\section{Conjuntos}\n\n\\example{codes/sets01@tikz_for_teachers}\n\n\\example{codes/sets02@tikz_for_teachers}\n\n\\example{codes/sets03@tikz_for_teachers}\n\n\\example{codes/sets04@tikz_for_teachers}\n\n\\section{Conjuntos Num\\'{e}ricos}\n\n\\section{Rela\\c{c}\\~{o}es}\n\n\\section{Fun\\c{c}\\~{o}es}\n\n\n", "meta": {"hexsha": "911122b78a77a401b5f241ee1680132b647d65fb", "size": 969, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "pt/tikz_for_teachers/p2_set_and_functions@tikz_for_teachers.tex", "max_stars_repo_name": "r-gaia-cs/latex_with_vim", "max_stars_repo_head_hexsha": "1ede258a90230247bf302918ded851c2856a7356", "max_stars_repo_licenses": ["Vim"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "pt/tikz_for_teachers/p2_set_and_functions@tikz_for_teachers.tex", "max_issues_repo_name": "r-gaia-cs/latex_with_vim", "max_issues_repo_head_hexsha": "1ede258a90230247bf302918ded851c2856a7356", "max_issues_repo_licenses": ["Vim"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "pt/tikz_for_teachers/p2_set_and_functions@tikz_for_teachers.tex", "max_forks_repo_name": "r-gaia-cs/latex_with_vim", "max_forks_repo_head_hexsha": "1ede258a90230247bf302918ded851c2856a7356", "max_forks_repo_licenses": ["Vim"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.8461538462, "max_line_length": 75, "alphanum_fraction": 0.7244582043, "num_tokens": 352, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213219520929, "lm_q2_score": 0.6723317123102956, "lm_q1q2_score": 0.349290659969759}} {"text": "\\documentclass[bigger]{beamer}\n\n\\input{header-beam} % change to header-handout for handouts\n\n% ====================\n\\title[Lecture 19]{Logic I F13 Lecture 19}\n\\date{November 21, 2013}\n% ====================\n\n\\include{header}\n\n\\setlength{\\fitchprfwidth}{14em}\n\n\\section{Numerical Quantification}\n\n\\subsec{Numerical Quantification}{\n\n\n\\bit\n\\item Cardinal numbers can be determiners:\n\\bit\n\\item \\emph{Three cubes} are left of \\emph{two dodecahedra}\n\\eit\n\\item Not always clear if ``three cubes'' means \\emph{exactly} or \\emph{at least} three\n\\item We'll assume the latter---the ``exactly'' is implicated, not implied\n\\bit\n\\item Do you have two dollars? Yes, I have two dollars. (Uncontroversially true even if you have more than 2\\$)\n\\item How much money do you have? I have two dollars. (True but misleading if you have more.)\n\\eit\n\\item FOL can express all of:\n\\bit\n\\item \\emph{At least $n$} cubes are \\dots \n\\item \\emph{Exactly $n$} cubes are \\dots\n\\item \\emph{At most $n$} cubes are \\dots\n\\eit\n\\eit\n\n}\n\n\\subsec{At least $n$}{\n\n\\bit\n\\item At least 1 cube is small:\n\\[\n\\exists x(Cube(x) \\land Small(x))\n\\]\n\\item At least 2 cubes are small:\n\\[\n\\exists x\\exists y(x \\neq y \\land Cube(x) \\land Small(x) \\land Cube(y) \\land Small(y))\n\\]\n\\eit\n}\n\n\n\\subsec{$\\exists^{\\ge n}x\\, A(x)$}{\n\n\\bit\n\\item $\\exists^{\\ge n}x\\, A(x)$:\n\\begin{align*}\n\\exists x_1\\dots\\exists x_n(x_1 \\neq x_2 \\land x_1 \\neq x_3 \\land \\dots \\land x_1 \\neq x_n & {} \\land {}\\\\\nx_2 \\neq x_3 \\land \\dots \\land x_2 \\neq x_n & {} \\land {}\\\\\n\\ddots\\qquad &\\\\\nx_{n-1} \\neq x_n & {} \\land {}\\\\\n A(x_1) \\land \\dots \\land A(x_n)) \n\\end{align*}\n\\eit\n}\n\n\\subsec{$\\exists^{\\ge n}x\\, A(x)$}{\n\n\n\\bit\n\\item NB: must state that \\emph{every pair} of variables is different, e.g.,\n\\[\\sf\\exists x_1\\exists x_2\\exists x_3(x_1 \\neq x_2 \\land x_2 \\neq x_3 \\land Cube(x_1) \\land Cube(x_2) \\land Cube(x_3))\\]\nonly says ``There are at least two cubes''! \n\\item At least $n$ As are Bs:\\[\n\\exists^{\\ge n} x(A(x) \\land B(x))\n\\]\n\\eit\n}\n\n\\subsec{Exactly One}{\n\n\\bit\n\\item There is exactly one cube:\n\\[\n\\exists x(Cube(x) \\land \\lnot \\exists y(Cube(y) \\land x \\neq y))\n\\]\n\\item This is equivalent to:\n\\[\n\\exists x(Cube(x) \\land \\forall y(Cube(y) \\to x = y))\n\\]\n\\item In general: ``x has property A \\emph{uniquely}'':\n\\begin{align*}\nA(x) \\land & \\forall y(A(y) \\to x = y)\\\\\n\\text{or just:\\qquad} & \\forall y(A(y) \\liff x = y)\n\\end{align*}\n\\eit\n}\n\n\\subsec{$\\exists^{=n} x\\, A(x)$}{\n\n\\bit\n\\item There are exactly $n$ things that satisfy A:\n\\begin{align*}\n\\exists x_1\\dots\\exists x_n(x_1 \\neq x_2 \\land x_1 \\neq x_3 \\land \\dots \\land x_1 \\neq x_n & {} \\land {}\\\\\nx_2 \\neq x_3 \\land \\dots \\land x_2 \\neq x_n & {} \\land {}\\\\\n\\ddots\\qquad &\\\\\nx_{n-1} \\neq x_n & {} \\land {}\\\\\n\\only<1>{ A(x_1) \\land \\dots \\land A(x_n) &{} \\land {}}\\\\\n\\forall y(A(y) \\only<1>{\\to}\\only<2>{\\leftrightarrow} (y = x_1 \\lor \\dots \\lor y=x_n)))\n\\end{align*}\n\\item Exactly $n$ As are Bs:\n\\[\n\\exists^{=n}x (A(x) \\land B(x))\n\\]\n\\eit\n\n}\n\n\\subsec{At most $n$}{\n\n\\bit\n\\item There are \\emph{at most $n$} As $\\IFF$ There aren't \\emph{at least $n+1$} As\n\\[\n\\exists^{\\emph{\\le n}}x\\, A(x) \\IFF \\emph{\\lnot}\\exists^{\\emph{\\ge(n+1)}}x\\, A(x)\n\\]\n\\item $\\lnot\\exists^{\\ge(n+1)}x\\, A(x)$ is equivalent to:\n\\begin{align*}\n\\forall x_1\\dots\\forall x_{n+1}((A(x_1) \\land \\dots \\land A(x_{n+1})) & {} \\to {}\\\\\n(x_1 = x_2 \\lor x_1 = x_3 \\lor \\dots \\lor x_1 = x_{n+1} & {} \\lor {}\\\\\nx_2 = x_3 \\lor \\dots \\lor x_2 = x_{n+1} & {} \\lor {}\\\\\n\\ddots\\qquad &\\\\\nx_n = x_{n+1} & ))\n\\end{align*}\n\\eit\n}\n\n\\subsec{Proofs With Numerical Claims}{\n\n\\fitchprf{\n\\pline{\\exists x\\, P(x)}\\\\\n\\pline{\\forall x\\forall y ((P(x) \\land P(y)) \\to x = y)}}{\n\\pline{\\exists x(P(x) \\land \\forall y(P(y) \\to x = y))}\n}\n}\n\n\\section{``The'', ``Both'', ``Neither''}\n\n\\subsec{Definite Descriptions}{\n\n\\bit\n\\item Definite description: \\emph{the so-and-so}\n\\item Russell's analysis of definite description: to say\\\\\n\\bit\n\\item ``The A is B''\\\\\n\\eit\nis to say:\n\\bit\n\\item There is a unique A\n\\item It is B\n\\eit\n\\item In FOL:\n\\[\n\\exists x(A(x) \\land \\forall y(A(y) \\to x = y) \\land B(x))\n\\]\n\\item or more succinctly:\n\\[\n\\exists x(\\forall y(A(y) \\liff x = y) \\land B(x))\n\\]\n\\eit\n\n}\n\n\\subsec{The vs. Exactly One}{\n\n\\bit \\item Compare:\n\\ben\n\\item The cube is small\n\\[\\exists x(Cube(x) \\land \\forall y(Cube(y) \\to x = y) \\land Small(x))\\]\n\\item There is exactly one small cube\n\\[\\exists x(Cube(x) \\land \\forall y((Cube(y) \\emph{{}\\land Small(y)}) \\to x = y) \\land Small(x))\\]\n\\een\n\\item (2) can be true without (1), but not vice versa\n\\item So (2) is a FO consequence of (1), but not vice versa\n\\eit\n}\n\n\\subsec{Strawson's Analysis}{\n\n\\bit\n\\item According to Russell, ``The cube is small'' is false if there is no cube, or if there is more than one\n\\item Strawson disagrees\n\\item ``There is a unique cube'' is not part of what is \\emph{said}, but is only \\emph{presupposed}\n\\eit\n\n}\n\n\\subsec{``Both'' and ``Neither''}{\n\n\\bit\n\\item ``Both cubes are small'': There are exactly 2 cubes, and each of them is small:\n\\begin{align*}\n\\exists x\\exists y(x \\neq y \\land Cube(x) \\land Cube(y) &{} \\land {}\\\\\n\\forall z(Cube(z) \\to (z = x \\lor z = y)) & {}\\land {}\\\\\n\\forall z(Cube(z) \\to Small(z))&)\n\\end{align*}\n\\item NB: ``Both cubes are small'' implies ``There are exactly two small cubes'', but not vice versa!\n\\item ``Neither cube is small'': There are exactly 2 cubes, and none of them are small:\n\\begin{align*}\n\\exists x\\exists y(x \\neq y \\land Cube(x) \\land Cube(y) & {}\\land {}\\\\\n\\forall z(Cube(z) \\to (z = x \\lor z = y)) &{} \\land {}\\\\\n\\forall z(Cube(z) \\to \\lnot Small(z))&)\n\\end{align*} \n\\eit\n\n}\n\n\\section{Truth-Functional Completeness}\n\n\\subsec{Truth Functions}{\n\n\\begin{block}{Definition}\n\nAn ($n$-place) \\emph{truth function} $t$ is a mapping of $n$-tuples of \\T{} and \\F\\ to either \\T\\ or \\F.\n\\medskip\n\n$n$-place truth functions correspond to truth tables of sentence with $n$ atomic sentences.\n\\medskip\n\nA sentence $S$ containing the atomic sentences $A_1$, \\dots, $A_n$\n\\emph{expresses} the truth function $t$ iff the truth value of $S$ on\nthe truth-value assignment which assigns $v_i$ to $A_i$ is $t(v_1,\n\\dots, v_n)$.\n\\medskip\n\nAn $n$-place truth function is \\emph{expressible} if there is a\nsentence containing atomic sentences $A_1$, \\dots, $A_n$ that\nexpresses it.\n\n\\end{block}\n}\n\n\\subsec{Examples}{\n\n\\begin{tabular}{cc@{\\qquad}cc}\n$\\begin{array}{cc|c}\n&& t_\\land\\\\ \\hline \n\\T & \\T & \\T \\\\\n\\T & \\F & \\F \\\\\n\\F & \\T & \\F \\\\\n\\F & \\F & \\F\n\\end{array}$\n&\n$A_1 \\land A_2$ &\n$\\begin{array}{cc|c}\n&& t_\\lor \\\\ \\hline\n\\T & \\T & \\T \\\\\n\\T & \\F & \\T \\\\\n\\F & \\T & \\T \\\\\n\\F & \\F & \\F\n\\end{array}$\n& $A_1 \\lor A_2$\n\\end{tabular}\n}\n\n\\subsec{Examples}{\n\n\\begin{tabular}{cc@{\\qquad}cc}\n$\\begin{array}{cc|c}\n& & t_1\\\\ \\hline\n\\T & \\T & \\T \\\\\n\\T & \\F & \\T \\\\\n\\F & \\T & \\F \\\\\n\\F & \\F & \\F\n\\end{array}$\n& $A_1$ &\n$\\begin{array}{cc|c}\n& & t_{XOR}\\\\ \\hline\n\\T & \\T & \\F \\\\\n\\T & \\F & \\T \\\\\n\\F & \\T & \\T \\\\\n\\F & \\F & \\F\n\\end{array}$\n&\n$(A_1 \\lor A_2) \\land \\lnot(A_1 \\land A_2)$ \\\\ \\ \\\\\n$\\begin{array}{cc|c}\n& & t_2\\\\ \\hline\n\\T & \\T & \\T \\\\\n\\T & \\F & \\F \\\\\n\\F & \\T & \\T \\\\\n\\F & \\F & \\F\n\\end{array}$\n& $A_2$ \n\\end{tabular}\n}\n\n\n\n\\subsec{Truth-Functional Completeness}{\n\n\\begin{block}{Definition}\n\nA set of connectives is \\emph{truth-functionally complete} if every\ntruth function is expressible by a sentence containing only those\nconnectives.\n\n\\end{block}\n\n}\n\n\\subsec{$\\{\\land, \\lor\\}$ Not Truth-Functionally Complete}{\n\n\\bit\n\\item $\\{\\land, \\lor\\}$ is not truth-functionally complete\n\\item Remember: To be truth functionally complete, \\emph{every} truth\n function would have to be expressible using only $\\land$ and $\\lor$\n\\item Which 2-place truth-functions can be expressed using $\\land$ and\n $\\lor$?\n\\item Not this one:\n\\[\n\\begin{array}{cc|c}\n\\T & \\T & \\F \\\\\n\\T & \\F & \\T \\\\\n\\F & \\T & \\T \\\\\n\\F & \\F & \\F\n\\end{array}\n\\]\n \\eit\n\n}\n\n\n\\subsec{Course Evaluation}{\n\n\\bit\n\\item Please fill out TWO TA evaluations, one for Jared and one for Dan.\n\\item More TopHat instant feedback questions? Or is TopHat not worth the trouble?\n\\item Three tests good, or would you prefer a midterm \nand a final after classes end?\n\\item Timing of homeworks---prefer Friday or Monday?\n\\item PASS sessions: did you go? Was it useful?\n\\eit\n}\n\\end{document}\n\n\\subsec{$\\{\\land, \\lor, \\lnot\\}$ is Truth-functionally Complete}{\n\\[\n\\begin{array}{lll|l@{\\qquad}l}\nA_1 & A_2 & A_3 & t_\\mathrm{ODD} & S \\\\\n\\hline\n\\T & \\T & \\T & \\T & \\uncover<2->{(A_1 \\land A_2 \\land A_3)} \\uncover<6->{{}\\lor}\\\\\n\\T & \\T & \\F & \\F \\\\\n\\T & \\F & \\T & \\F \\\\\n\\T & \\F & \\F & \\T & \\uncover<3->{(A_1 \\land \\lnot A_2 \\land \\lnot A_3)} \\uncover<6->{{}\\lor}\\\\\n\\F & \\T & \\T & \\F \\\\\n\\F & \\T & \\F & \\T & \\uncover<4->{(\\lnot A_1 \\land A_2 \\land \\lnot A_3)} \\uncover<6->{{}\\lor}\\\\\n\\F & \\F & \\T & \\T & \\uncover<5->{(\\lnot A_1 \\land \\lnot A_2 \\land A_3)} \\uncover<6->{{}\\lor}\\\\\n\\F & \\F & \\F & \\F\n\\end{array}\\]\n}\n\n\\subsec{$\\{\\land, \\lor, \\lnot\\}$ is Truth-Functionally Complete}{\n\n\\bit\n\\item $\\lnot A_1 \\land A_2 \\land \\lnot A_3$ is true in, and only in line \\F\\ \\T\\ \\F\n\\item Combine using $\\lor$: make $S$ true in all (and only) the lines where it is supposed to be true\n\\item NB: Procedure results in DNF!\n\\eit\n\n}\n\n\\subsec{The ``neither\\dots nor \\dots'' connective: $\\downarrow$}{\n\n\\[\n\\begin{array}{cc|c}\nP & Q & (P \\downarrow Q)\\\\\n\\hline\n\\T & \\T & \\F\\\\\n\\T & \\F & \\F\\\\\n\\F & \\T & \\F\\\\\n\\F & \\F & \\T\n\\end{array}\n\\]\n\n}\n\n\\subsec{$\\{\\downarrow\\}$ is Truth-Functionally Complete}{\n\n\\bit\n\\item Already know that $\\{\\lnot, \\land, \\lor\\}$ is truth-functionally\n complete, i.e.,\n\\item Every truth-function can be expressed using only $\\lor$, $\\land$, $\\lnot$\n\\item To show $\\downarrow$ is truth-functionally complete, suffices to\n show that \\emph{every sentence containing only $\\lnot$, $\\lor$, $\\land$ is\n tautologically equivalent to one containing only $\\downarrow$}\n\\item For that, it suffices to show that any negated sentence,\n conjunction, disjunction, can be expressed using only $\\downarrow$\n\\eit\n\n}\n\n\\subsec{Expressing $\\lnot$ Using $\\downarrow$}{\n\n\\begin{columns}\n\\begin{column}{3cm}\n\\[\\begin{array}{cc|c}\nP & Q & (P \\downarrow Q)\\\\\n\\hline\n\\T & \\T & \\F\\\\\n\\T & \\F & \\F\\\\\n\\F & \\T & \\F\\\\\n\\F & \\F & \\T\n\\end{array}\\]\n\\end{column}\n\\begin{column}{7cm}\n\\bits\n\\item Note how $P \\downarrow Q$ is \\F{} in the first line and \\T{} in the last (when $P$ and $Q$ have same truth value)\n\\item So $P \\downarrow P$ is \\F{} if $P$ is \\T, and \\T{} if $P$ is \\F, i.e., \\[\n\\lnot P \\Leftrightarrow (P \\downarrow P)\n\\]\n\\eit\n\\end{column}\n\\end{columns}\n\n}\n\n\\subsec{Expressing $\\lor$ Using $\\downarrow$}{\n\n\\begin{columns}\n\\begin{column}{3cm}\n\\[\\begin{array}{cc|c}\nP & Q & (P \\downarrow Q)\\\\\n\\hline\n\\T & \\T & \\F\\\\\n\\T & \\F & \\F\\\\\n\\F & \\T & \\F\\\\\n\\F & \\F & \\T\n\\end{array}\\]\n\\end{column}\n\\begin{column}{7cm}\n\\bits\n\\item $P \\downarrow Q$ is the ``neither \\dots nor'' connective, which can also\nbe expressed as $\\lnot(P \\lor Q)$, i.e.,\n\\[ \\lnot(P \\lor Q) \\Leftrightarrow P \\downarrow Q \\] \n\\item Negate both sides:\n\\[\nP \\lor Q \\Leftrightarrow \\lnot(P \\downarrow Q)\n\\]\n\\item Apply what we figured out in last slide:\n\\[\nP \\lor Q \\Leftrightarrow (P \\downarrow Q)\\downarrow(P \\downarrow Q)\n\\]\n\\eit\n\\end{column}\n\\end{columns}\n\n}\n\n\\subsec{Expressing $\\land$ Using $\\downarrow$}{\n\n\\begin{columns}\n\\begin{column}{3cm}\n\\[\\begin{array}{cc|c}\nP & Q & (P \\downarrow Q)\\\\\n\\hline\n\\T & \\T & \\F\\\\\n\\T & \\F & \\F\\\\\n\\F & \\T & \\F\\\\\n\\F & \\F & \\T\n\\end{array}\\]\n\\end{column}\n\\begin{column}{7cm}\n\\bits\n\\item $P \\downarrow Q$ is the ``neither \\dots nor'' connective, which can also\nbe expressed as $\\lnot P \\land \\lnot Q$, i.e.,\n\\[ (\\lnot P \\land \\lnot Q) \\Leftrightarrow P \\downarrow Q \\] \n\\item Equivalence holds for \\emph{all sentences} $P$, $Q$, so also if we replace $P$ by $\\lnot R$ and $Q$ by $\\lnot S$:\n\\[\n\\lnot\\lnot R \\land \\lnot\\lnot S \\Leftrightarrow (\\lnot R \\downarrow \\lnot S)\n\\]\n\\item Delete $\\lnot\\lnot$'s, and express $\\lnot$ using $\\downarrow$:\n\\[\nR \\land S \\Leftrightarrow (R \\downarrow R)\\downarrow(S \\downarrow S)\n\\]\n\\eit\n\\end{column}\n\\end{columns}\n\n}\n\n\\subsec{Truth-Functionally Complete Sets of Connectives}{\n\n\\bit\n\\item De Morgan's Law: $\\land$ can be expressed by $\\lor$ and $\\lnot$\n\\item Similarly: $\\lor$ can be expressed by $\\land$, $\\lnot$\n\\item So $\\{\\lor, \\lnot\\}$ and $\\{\\land, \\lnot\\}$ are\n truth-functionally complete\n\\item $\\{\\to, \\bot\\}$ is truth-functionally complete (HW)\n\\item $\\{\\to, \\lnot\\}$ is truth-functionally complete \n\\item No other sets of connectives that don't contain one of these sets are complete \n\\item ``Neither \\dots nor'' is truth-functionally complete by itself\n\\item ``Not both'' connective is truth-functionally complete by itself (HW)\n\\item No other 2-place connectives are complete by themselves\n\\eit\n\n}\n\n\\end{document}\n\n\n\n\n", "meta": {"hexsha": "b0481cee5f4cf65484fc0241599d8a49f6e145ba", "size": 12586, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "279-lec19.tex", "max_stars_repo_name": "rzach/phil279", "max_stars_repo_head_hexsha": "722ec82ae7a4593d40c72083d830c4e3e4864dc0", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2015-09-23T13:42:54.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-21T10:48:55.000Z", "max_issues_repo_path": "279-lec19.tex", "max_issues_repo_name": "rzach/phil279", "max_issues_repo_head_hexsha": "722ec82ae7a4593d40c72083d830c4e3e4864dc0", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "279-lec19.tex", "max_forks_repo_name": "rzach/phil279", "max_forks_repo_head_hexsha": "722ec82ae7a4593d40c72083d830c4e3e4864dc0", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 24.486381323, "max_line_length": 121, "alphanum_fraction": 0.6242650564, "num_tokens": 4777, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213219520929, "lm_q2_score": 0.6723317057447908, "lm_q1q2_score": 0.3492906565588393}} {"text": "\\documentclass{article}\n\n\\usepackage[utf8]{inputenc}\n\\usepackage[english]{babel}\n\\usepackage{a4}\n\\usepackage[T1]{fontenc}\n\\usepackage[cyr]{aeguill}\n\\usepackage{graphicx}\n\\usepackage{amsmath}\n\\usepackage{authblk}\n\\usepackage{listings}\n\\usepackage{subfigure}\n\\usepackage{float}\n\\usepackage{geometry}\n\\usepackage[a4paper]{geometry} % Marges plus larges\n\\geometry{hmargin=2.5cm,vmargin=1.65cm}\n\\usepackage[sectionbib]{chapterbib}\n\n\n\n\\title{TS114 Signal processing \\\\ MICA Project}\n\n\n\n\\author{ABIED Imad \\\\ Email: imad.abied@bordeaux-inp.fr\n\n\\and AHALLI Mohamed \\\\ Email: mohamed.ahalli@bordeaux-inp.fr\n\n\\and BIGI Mohamed \\\\ Email: mohamed.bigi@bordeaux-inp.fr }\n\n\\date{11/05/2021}\n\n \n\\sloppy \n\n\\begin{document}\n\\maketitle\n\\begin{center}\n ENSEIRB-MATMECA\n\\end{center}\n\n\\tableofcontents\n\n\\newpage\n\n\\section{Introduction}\n\nThe goal of this project was to analyse signals obtained from measuring heart electrical activity. Such a signal is called electrocardiogram, ECG for short.\nECG respects the pattern described in figure 1. Detecting this pattern consists of identifying the four characteristic points P, Q, R,S and T in each R-R interval from a real ECG signal, as shown in figure 2. The algorithm for this task is presented in the technical part below.\n\n\\begin{figure}[htbp]\n\n\\centerline{\\includegraphics[scale = 0.5]{figure_1.png}}\n\\caption{ECG pattern}\n\n\\centerline{\\includegraphics[scale = 0.3]{figure_2.png}}\n\\caption{Real ECG signal}\n\n\\end{figure}\n\nFurthermore, cardiac pathologies are defined from the PQRST mathematical properties. Therefore, the identification of cardiac pathologies can be automatic. Algorithms for automatic identification are discussed in the technical part of this report.\n\nPQRST mathematical properties can change over time for different reasons. For example, the heart rate increases after doing a physical activity. This is why the spectrogram was used during the project. The spectrogram was employed to understand how ECG characteristics change in the course of time, enabling us to choose the right time interval to apply desired algorithms. For that reason the first section of the technical part is dedicated to spectrograms.\n\n\\section{Data visualization (spectrogram) }\nSpectrograms are used to locally represent the spectrum of a signal. As the spectrum is a statistical measurement, it’s necessary to define the term “locally” by a window of a certain length noted N. The larger the window, the greater frequency precision is obtained at the price of losing time accuracy. Conversely, The shorter window, the greater time accuracy is obtained at the cost of losing frequency precision. This is because there is more data to process in order to establish the spectre, which is a statistical measure. Nevertheless, the spectrum computed by a large window is not “very local\".\n\n\\begin{figure}[H]\n\n\\begin{subfigure}\n\n\\centerline{\\includegraphics[scale = 0.6]{figure_3_large_window.png}}\n\\caption{Spectrogram with a large window.}\n\\end{subfigure}\n\n\\begin{subfigure}\n\n\\centerline{\\includegraphics[scale = 0.2]{figure_4_short_window.png}}\n\\caption{Spectrogram with a short window.}\n\\end{subfigure}\n\n\\end{figure}\n\nThis duality between frequency precision and time accuracy can be shown by the example ./src/duality\\_time\\_frequence.m. In this script the spectrogram of the same signal is plotted using a large window in figure 3 and shorter one in figure 4.\n\nIn figure 3, the yellow line is thin compared to the one in figure 3, facilitating to us the reading of frequency value. On the other hand, it’s very difficult to read the transition at 0.25s in comparison with figure 4.\n\n\\section{QRS complex Detection}\n\nThe QRS complex is going to be detected using the Pan and Tompkins algorithm.The first step of this algorithm eliminates the interference of T and P waves with the QRS complex. A well designed filter is proposed by Pan and Tompkins for this task. Its transfer function is given by :\n\n\\begin{equation}\n H(z) = \\frac{(1-z^{-6})^{2}}{(1-z^{-1})^{2}}\\times\\frac{(-1 + 32z^{-16} -32z^{-17} +z^{-32})}{(1-z^{-1})}\n\\end{equation}\n\nObserving the fact that the R wave is very sharp,the derivative of the ECG signal is going to take big absolute values at the QRS complex.Pan and Tompkins suggests this time a filter with a transfer function equal to :\n\n\\begin{equation}\n H(z) = \\frac{1}{8T_s}\\times(-z^{-2} - 2z^{-1} +2z +z^{2})\n\\end{equation}\n\nAs a consequence, if a well chosen window is used for moving window integration,the result will be a signal which takes big values for every QRS complex.\n\nIn order to enhance the QRS complex domain, a thresholding operation is applied with a threshold equal to the mean of ECG signal after integration.\n\nAt this point, it’s certain that the computed maximum for every QRS domain corresponds to the unique pick called R.\n\nOnce the R pick is detected, Q and S are detected by searching consecutively the first minimum on the left and the first minimum on the right.\n\nAll those steps are applied to an ECG signal. The result at every step are plotted in figure 5.\n\n\n\\begin{figure}[htbp]\n\n\\centerline{\\includegraphics[scale = 0.4]{figure_5.png}}\n\\caption{ECG signal in each step.}\n\n\\end{figure}\n\n\\begin{tabular}{ |p{5cm}|p{5cm}|p{5cm}| }\n\\hline\n\\multicolumn{3}{|c|}{Used filters analyse} \\\\\n\\hline\nband-pass filter & high-pass filter & five-point differentiation filter\\\\\n\\hline\nNature : band-pass & Nature : high-pass & Nature : differentiation\\\\\nType : Infinite Impulse Response & Type : Infinite Impulse Response & Type : Finite Impulse Response \\\\\nCausal : Yes & Causal : Yes & Causal : No \\\\\nGroup delay : 5 samples &Group delay : 16.49 samples & Group delay : 0 \\\\\nLinear phase : Yes & Linear phase : Yes & Linear phase : Yes\\\\\n\\hline\n\\end{tabular}\n\n\\newpage\n\n\\section{P and T wave detection}\n\\subsection{About P and T waves}\n\nGenerally, P and T waves in an ECG (electrocardiogram) signal are lower in amplitude compared to a QRS complex, and, they are contaminated with noise from various sources. These factors make the detection of P and T waves within an ECG a challenging task. Unlike a P wave, T waves are slightly asymmetrical, the peak of the wave is a little closer to its end than to its beginning.\n\\subsection{P and T waves detection method}\nT waves are considered to be the highest peak between the first R peak and 0.7 times the R-R interval. While P waves have the highest peak in the remaining of the interval, as shown in the figures below.\n\n\\begin{figure}[htbp]\n\n\\centerline{\\includegraphics[scale=0.5]{T_70.png}}\n\\caption{Highest peak on 70\\% of R-R interval.}\n\n\\end{figure}\n\n\\begin{figure}[htbp]\n\n\\centerline{\\includegraphics[scale=0.5]{P_30_30.png}}\n\\caption{Highest peak on the remaining 30\\% of R-R interval.}\n\n\\end{figure}\n\n\nIn order to detect P and T waves, two filters are used:\n \\begin{equation}\n G_1(z) = 1 -z^{-6}\n \\end{equation}\n \n and\n \n \\begin{equation}\n G_2(z) = \\frac{1-z^{-8}}{1-z^{-1}}\n \\end{equation}\n \n The first filter \\textit{$G_1(z)$} is a differentiator, it allows the detection of maximum, minimum, as well as null values. This is achieved by determining where the signal, after applying the differentiator \\textit{$G_1(z)$}, crosses the level 0.\n \n The Algorithm functions as follows: The R-R interval is divided into two parts, a part containing 70\\% of the R-R interval, in which T waves are located; and another part containing the remaining 30\\% of the R-R interval, in which P waves are located. In each part, the locations in which the signal (after crossing both filers (\\textit{$G_1(z)$} and (\\textit{$G_2(z)$}) crossed the level 0 were determined. As mentioned before, these locations correspond to either a maximum, minimum, or null values in the original ECG signal. Therefore, the locations that provide a maximum value, are where P and T waves are located.\n \n The figures below illustrate the locations of P and T waves in the original signal,as well as their location after each filter.\n \n\n\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=0.5]{ori_sig.png}}\n\\caption{Original ECG with locations of P and T waves}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=0.5]{sig_G1.png}}\n\\caption{ECG after undergoing first filter}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centerline{\\includegraphics[scale=0.5]{sig_G2.png}}\n\\caption{ECG after undergoing second filter}\n\\end{figure}\n \n\n\\section{Tachycardia/Bradycardia}\n\n\\subsection{Bradycardia}\n\nBradycardia is defined as a heart rate (HR) of less than 50 or 60 bpm (beats per minute), compared to a normal heart rate of 60 to 100 bpm. A slow heart rate is in general a sign of good health and fitness. However, a heart rate that is too slow (\\textit{ie} Bradycardia ), is a sign of a problem with the heart's electrical system. It means that the heart's natural pacemaker isn't working right or that the electrical pathways of the heart are disrupted. The heart beat can be so slow that it can't pump enough blood for the body. Which could be life-threatening if left untreated.\n\n\\subsection{Tachycardia}\n \nTachycardia is defined as a heart rate (HR) of over 100 bpm (beats per minute). With atrial or supraventricular tachycardia, electrical signals in the heart’s upper chambers fire abnormally. This interferes with the heart’s natural pacemaker, and causes abnormal heart rates. This rapid heartbeat keeps the heart’s chambers from filling completely between contractions, which compromises blood flow to the rest of the body. \n\n\\subsection{Detecting cardiac rhythm anomalies: Tachycardia and Bradycardia}\n\nAn algorithm that calculates the heart rate of a patient, would make it possible to detect cardiac rhythm anomalies. The R waves mark the moment in which the heart beats. Therefore, the duration between consecutive R peaks would help determine the heart rate.\nThe value calculated is:\n\n\\begin{equation}\n \\overline{\\Delta} = \\frac{1}{N}\\times\\sum_{n=0}^{N-1}\\Delta_n\n\\end{equation}\n\n$\\overline{\\Delta}$ is the mean of R-R intervals duration. It's value was converted to bpm (beats per minute) by multiplying with \\textit{$F_s$} and by 60.\n\n\n\\section{Other pathologies}\n\n\\subsection{Ectopic beats}\nAn ectopic heartbeat is when the heart either skips a beat, or adds an extra beat. They are also called premature heartbeats. Ectopic heartbeats are usually not a cause for concern, and they may occur for no known reason. Despite the skipped or added beat, the heart functions normally.\n\n\\subsection{Detecting ectopic beats}\n\nDetecting premature heartbeats, comes down to detecting irregularities in R-R intervals, since, as mentioned earlier (\\textit{section 5.3}), R peaks characterize the moment in which the heart beats.\n\nThe algorithm devised, calculated the length of different R-R intervals,and chose their maximum value, since an ectopic beat is placed either to close or too far from a regular beat, either a maximum or a minimum value would have been appropriate.\nBy calculating the same value for normal patients, a threshold $\\epsilon$ was determined as $\\epsilon = 1$.\nThis value (\\textit{\\textbf{max}}) was calculated for different patients, and compared with $\\epsilon$.\nThe results were consistent, \\textit{\\textbf{max}} was Superior to $\\epsilon$ in patients with ectopic beats, and inferior or equal on other cases.\n\n\\section{Fibrillation}\n\n\\subsection{Atrial fibrillation}\n\n\nAtrial fibrillation is characterized by a strong heart rhythm irregularity until the point of considering it as white noise. The main property of a white noise is that its autocorrelation function is null except in zero. That is why autocorrelation function of the studied ECG is estimated with the formula :\n\n\\begin{equation}\n \\hat{\\gamma_k} = \\frac{1}{N-k-1}\\times\\sum_{n=0}^{N-k-1} (\\Delta_{n+k} - \\overline{\\Delta})(\\Delta_{n} - \\overline{\\Delta})\n\\end{equation}\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.4]{figure_6_edge_effect.png}}\n\\caption{Edge effect.}\n\\end{figure}\n\nFigure 11 is an example of an estimated autocorrelation function. At the right of the figure, big random values are remarked because, in this area,the autocorrelation is estimated using a small number of samples so the mean is not representative enough. For that reason, the studied domain of the autocorrelation function is not going to include the values at the right edge.\n\nThe heart rhythm is going to be considered as a white noise if 40\\% of its autocorrelation function at zero is still a maximum of the autocorrelation function.\n\n\\subsection{Ventricular fibrillation}\n\nVentricular fibrillation, or V-fib, is considered the most serious cardiac rhythm disturbance.\nDisordered electrical activity causes the heart’s lower chambers (ventricles) to quiver, or fibrillate, instead of contracting (or beating) normally. This prohibits the heart from pumping blood. Thus leading to collapse and cardiac arrest\n\n\\subsection{Detecting ventricular fibrillation}\n\nTwo main properties are used to detect Ventricular fibrillation: The similarity between the ECG (electrocardiogram) of the patient and the pure sine; as well as a rapid heart rate between 240 and 600 bpm (beats per minute).\nThe similarity between Ventricular fibrillation ECG and the pure sine function was detected by resorting to the Fourier transform. Plotting the two Fourier transforms, illustrates the similarity between them, as shown below.\n\\textbf{N.B: FT = Fourier transform}\n\n\\newpage\n\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.6]{Fourrier_VF.png}}\n\\caption{TF of ECG}\n\\end{figure}\n\\begin{figure}\n\\centerline{\\includegraphics[scale=0.6]{sin_fou.png}}\n\\caption{TF of sine}\n\\end{figure}\n\n\\newpage\n\nThe FT are similar, with two important peaks at symmetric values of frequency. In order to detect the two peaks, a threshold was applied to the FT,in order to calculate the number of values superior to $0.8\\times\\textit{n}$; where \\textit{n} is the maximum value of the FT. An ECG with a ventricular fibrillation generally has a low number of values above $0.8\\times\\textit{n}$, compared to other ECGs, combining this with the condition of a heart rate between 240 and 600 bpm, a case of ventricular fibrillation can be determined\n\n\n\n\n\\section{Conclusion}\nThis report has discussed the implementation of algorithms used to detect PQRST waves and automatic heart pathology identification. Besides that, it has introduced the utility of spectrograms as well as their limitations. The results obtained were satisfying. However, we really wished to develop a graphic user interface to make our work valuable but we could not because we did not have enough time.\n\n\\newpgae\n\n\\begin{thebibliography}{2} \n \\bibitem[1]{cle} P and T waves https://pubmed.ncbi.nlm.nih.gov/29484531/\n \\bibitem[2]{cle} Bradycardia https://www.uofmhealth.org/health-library/aa107571\n \\bibitem[3]{cle} tachy https://pubmed.ncbi.nlm.nih.gov/29484531/\n \\bibitem[4]{cle} ectopic https://www.medicalnewstoday.com/articles/323202#what-is-an-ectopic-heartbeat\n \\bibitem[5]{cle} Ventr fibr https://www.heart.org/en/health-topics/arrhythmia/about-arrhythmia/ventricular-fibrillation\n \n\\end{thebibliography} \n\n\\end{document}\n\n\n", "meta": {"hexsha": "7017a890a53923ad8b0b4d7f843a8bca1267d6fa", "size": 15119, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "documents/Report_LaTeX/main.tex", "max_stars_repo_name": "ImadABID/MICA_Project", "max_stars_repo_head_hexsha": "892f46ac33c5b2abb495c49768003642d6bf982d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "documents/Report_LaTeX/main.tex", "max_issues_repo_name": "ImadABID/MICA_Project", "max_issues_repo_head_hexsha": "892f46ac33c5b2abb495c49768003642d6bf982d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "documents/Report_LaTeX/main.tex", "max_forks_repo_name": "ImadABID/MICA_Project", "max_forks_repo_head_hexsha": "892f46ac33c5b2abb495c49768003642d6bf982d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 51.2508474576, "max_line_length": 621, "alphanum_fraction": 0.7722732985, "num_tokens": 3836, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.3492906531479197}} {"text": "% Template file for TeXShop by Michael Sharpe, LPPL\n\\documentclass[11pt]{article} \n\\usepackage[margin=0.9in]{geometry} \n\\usepackage[parfill]{parskip}% Begin paragraphs with an empty line rather than an indent\n\\usepackage{graphicx}\n\\usepackage{amsmath,amsthm}\n\\usepackage{natbib}\n\n\\title{MIT 6.862 Applied Machine Learning \\\\ Project Progress Report}\n\\author{Matthew West, Fall 2019}\n\\begin{document}\n\\date{}\n\\maketitle\n\n\\section{Introduction}\nThis is the progress report for my MIT course 6.862 project, which aims to explore classification in the context of medical diagnosis, and specifically in datasets derived from images of breast cancer. The focus thus far has been on the establishment of a data pipeline and exploratory data analysis, before comparing multiple candidate machine learning models on the basis of classification accuracy.\n\nThis report focuses on the Wisconsin Breast Cancer dataset, \\cite{Dua:2019} an analysis of which first appeared in Street et al., which used a decision tree-based algorithm to classify the data with approximately 97\\% accuracy. \\cite{street1993nuclear, mangasarian1993mathematical} As outlined in the proposal, it was decided that comparison between traditional machine learning on structured data derived from images and the raw images themselves was a desirable direction to take the project in. To this end, I have contacted an original author of this paper in an attempt to acquire the corresponding raw images, though this has not proved fruitful as of yet. There remains the option to extend the analysis to further datasets outlined in the proposal, though such a comparison still won't be straightforward and will introduce substantial roadblocks in the form of data extraction and image segmentation. \n\nThe report is structured as follows: Following this introduction, an account of the progress so far will be given regarding the data, methods, results, and discussion, followed by a brief outline of what will be done with the time remaining, and outlining a specific plot that will appear in the final report.\n\n\\section{Progress}\n\\subsection{Data}\nThe data used in the initial portion of the project thus far is the Wisconsin Breast Cancer dataset, collected by a team at the University of Wisconsin-Madison in the early 1990's. The dataset uses features derived from a set of fine-needle aspirate (FNA) images of breast tissue in patients with either malignant or benign tumours. It is a structured dataset with 569 instances and 30 features, an ID number, and a label, with 212 malignant examples and 357 benign examples. There are 10 base features (radius, texture, perimeter, area, smoothness, compactness, concavity, concave points, symmetry, fractal dimension), and for each of these the mean, standard error, and largest/worst example of these features are included over each image, resulting in 30 features in total. \n\nThe dataset was checked for missing values, before dropping a non-explanatory ID column, an unexplained column of all NaN values, and separating the labels from the feature vectors.\n\n\n\\subsection{Method}\n\nThe machine learning problem at hand is one of classification. If we denote our hypothesis by $\\text{h}(x; \\lambda)$ where the feature vector for each tumour instance is denoted as $x^{(i)}$ and $\\lambda$ denotes a specific vector of parameters that determine a hypothesis, our problem is to identify the optimal $\\lambda$ subject to minimising\n\n$$\\frac{1}{n}\\sum_{i=1}^{n} L(\\text{h}(x; \\lambda), y^{(i)}),$$\n\nwhere $L(g, a)$ is the loss function chosen for a given algorithm, providing a measure of how far away the guess from the hypothesis $g$ is from the actual instance label, $a$. Specifically, this will be evaluated using 5-fold cross validation, as minimising training error will not be a relevant or interesting metric. Selected hypotheses from many such algorithms will be compared to see which of these minimises validation error and is therefore the preferred model in terms of classification accuracy. Multiple algorithms have been investigated so far, with promising results from xgboost, naive Bayes and logistic regression, as seen in table \\ref{table:1}. These were mainly implemented using scikit-learn in Python 3 on my local machine.\n\nA principle component decomposition was done on the dataset, allowing 2 or 3 principle components to represent a substantial portion of the variation across all 30 features. This allows for an intuitive understanding of how separable the dataset is, as two or three principle components can be plotted and understood visually, something that isn't readily doable for higher dimensional data. It is also possible to directly train and test models on the principle component representation of the data, though one loses the interpretability for which features are most important in predicting the outcome variable.\n\n\\subsection{Results}\nThe first result was, having used PCA to represent the data in both 2 and 3 dimensions, it was confirmed that the dataset would be somewhat separable, if not perfectly linearly separable. This was achieved by confirming that the PCA plots showed natural clusters of malignant and benign instances in the reduced latent space.\n\nFurthermore, it was determined preliminarily that tree-based methods such as xgboost are performing better than other algorithms, with xgboost consisently obtaining 96\\% classification accuracy without too much parameter tuning or feature selection. An additional bonus of using tree-based methods is that they allow some quantitative insight into which features are most important in classification, which appear to be those derived from texture and concave points.\n\n\\begin{table}[h!]\n\t\\centering\n\t\\begin{tabular}{ |c|c|c|c| } \n\t\t\\hline\n\t\tAlgorithm & 5-fold accuracy set & 5-fold accuracy ($\\%$) \\\\\n\t\t\\hline\n\t\t\\textbf{xgboost}\t& [0.930, 0.957, 0.992, 0.965, 0.973] & $96\\pm2$ \\\\ \n\t\t\\textbf{Naive Bayes} & [0.922, 0.922, 0.956 0.947, 0.956] & $94\\pm2$ \\\\ \n\t\t\\textbf{Logistic regression} & [0.930, 0.939, 0.973, 0.947, 0.965] & $95\\pm2$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\caption{Table of cross-validated results from selected classification algorithms}\n\\label{table:1}\n\\end{table}\n\n\n\\subsection{Discussion}\nGetting rather good results using default hyperameters can be straightforward for some datasets. Marginal performance increases then come from the selection of algorithms together with the set of hyperparameters tuned for that algorithm. Without exploring this vast solution space comprehensively, it has still been possible to obtain classification accuracies close to that initial benchmark described in the orginal paper. One insight that has helped to overcome the `noise' associated with the accuracy metric across this solution space was the decision to use 5-fold cross validation when investigating accuracy.\n\nA key lesson learnt thus far was how to think about exploratory data analysis in a systematic way, such as by using PCA to represent the data. This paralleled nicely with the content covered in class on autoencoders, but simplifying it to a linear combination of original features. This has allowed me to abstract conceptually from thinking about raw features to latent spaces, which is an invaluable tool in applied machine learning. \n\n\n\\section{Looking Ahead}\nAn outline for the next month leading up to the conclusion of the project is given in this section.\n\\begin{itemize}\n\t\\item An investigation into precision and recall. In the context of medical diagnostics, the use of machine learning algorithms is controversial due to the direct impact of such diagnostics on human survival. Therefore, perhaps unlike many other domains in which machine learning is applied, special consideration must be given to precision and recall, as false positives are in general much less serious than false negatives. This presents a different set of potential metrics, and some of these will be investigated to see if a different algorithm is preferred when we take the domain-specific selection of metric into account. Deadline: 11/14\n\t\\item It will hopefully be possible to acquire a set or perhaps representative subset of the original Wisconsin Breast Cancer images from the researchers that first collected this data. If this is the case, then the focus of this milestone will be to investigate the use of convolutional neural networks (CNN's) on these raw images, and compare performance on any relevant metrics. It is of particular interest to me to see if there are advantages to using CNN's in the small dataset regime, despite not expecting a substantial increase in resultant accuracy (e.g. interpretability, automated feature extraction). I identified this as ``time-allowing\" in the project proposal because of the potential time commitment of pre-processing these larger datasets for use in CNN's and also deriving structured datasets from these similar in form to the Wisconsin dataset, for the purposes of making this comparison. Deadline: 12/01\n\\end{itemize}\n\n\\section{Projected Results Section}\nThe key result of this initial stage of the project will be a comparison of accuracies on the Wisconsin dataset using one model for each of the algorithms selected for comparison, by means of a violinplot or boxplot. The horizontal axis will be discrete in the different algorithms, and the vertical axis will be classification accuracy.\n\nEach of these models will have been tuned to select the optimal set of any relevant hyperparameters, using 5 or 10-fold cross-validation. This plot allows the reader to assess visually which algorithm performs better on this dataset, as well as how much variance there is in each estimate of accuracy across the set of folds used for validation. I envision including a set of similar figures that concern performance not just in terms of accuracy, but also on other metrics that take into account sensitivity and specificity.\n\\nocite{*}\n\\bibliographystyle{plain}\n\\bibliography{MIT}\n\n\\end{document}", "meta": {"hexsha": "d845928bf8f4b680245cc4c77093753a99f7ff1b", "size": 9965, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Project/Progress Report/Progress_Report.tex", "max_stars_repo_name": "mwestt/MIT-6.862", "max_stars_repo_head_hexsha": "76e0c15898667ff5c05bfae38f7431cc6e0cff7c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Project/Progress Report/Progress_Report.tex", "max_issues_repo_name": "mwestt/MIT-6.862", "max_issues_repo_head_hexsha": "76e0c15898667ff5c05bfae38f7431cc6e0cff7c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Project/Progress Report/Progress_Report.tex", "max_forks_repo_name": "mwestt/MIT-6.862", "max_forks_repo_head_hexsha": "76e0c15898667ff5c05bfae38f7431cc6e0cff7c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-11-18T02:22:00.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-18T02:22:00.000Z", "avg_line_length": 123.024691358, "max_line_length": 925, "alphanum_fraction": 0.8031108881, "num_tokens": 2141, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.3492906531479197}} {"text": "%2multibyte Version: 5.50.0.2960 CodePage: 65001\r\n\r\n\\documentclass{article}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\usepackage{amsmath}\r\n\r\n\\setcounter{MaxMatrixCols}{10}\r\n%TCIDATA{OutputFilter=LATEX.DLL}\r\n%TCIDATA{Version=5.50.0.2960}\r\n%TCIDATA{Codepage=65001}\r\n%TCIDATA{}\r\n%TCIDATA{BibliographyScheme=Manual}\r\n%TCIDATA{Created=Monday, November 30, 2015 16:55:25}\r\n%TCIDATA{LastRevised=Wednesday, December 09, 2015 17:26:48}\r\n%TCIDATA{}\r\n%TCIDATA{}\r\n%TCIDATA{CSTFile=40 LaTeX article.cst}\r\n\r\n\\newtheorem{theorem}{Theorem}\r\n\\newtheorem{acknowledgement}[theorem]{Acknowledgement}\r\n\\newtheorem{algorithm}[theorem]{Algorithm}\r\n\\newtheorem{axiom}[theorem]{Axiom}\r\n\\newtheorem{case}[theorem]{Case}\r\n\\newtheorem{claim}[theorem]{Claim}\r\n\\newtheorem{conclusion}[theorem]{Conclusion}\r\n\\newtheorem{condition}[theorem]{Condition}\r\n\\newtheorem{conjecture}[theorem]{Conjecture}\r\n\\newtheorem{corollary}[theorem]{Corollary}\r\n\\newtheorem{criterion}[theorem]{Criterion}\r\n\\newtheorem{definition}[theorem]{Definition}\r\n\\newtheorem{example}[theorem]{Example}\r\n\\newtheorem{exercise}[theorem]{Exercise}\r\n\\newtheorem{lemma}[theorem]{Lemma}\r\n\\newtheorem{notation}[theorem]{Notation}\r\n\\newtheorem{problem}[theorem]{Problem}\r\n\\newtheorem{proposition}[theorem]{Proposition}\r\n\\newtheorem{remark}[theorem]{Remark}\r\n\\newtheorem{solution}[theorem]{Solution}\r\n\\newtheorem{summary}[theorem]{Summary}\r\n\\newenvironment{proof}[1][Proof]{\\noindent\\textbf{#1.} }{\\ \\rule{0.5em}{0.5em}}\r\n\\input{tcilatex}\r\n\\begin{document}\r\n\r\n\r\n\\subsection{Model Assumptions}\r\n\r\nThe world economy is composed of $N$ countries. At a given point in time $t$%\r\n, each country $n$ is endowed with $L_{nt}$ units of a primary (non\r\nproduced) input, which we interpret as equipped labour. There are $J$\r\nsectors (or broad classes of goods) in the economy, whose output is combined\r\ninto a final good through a Cobb-Douglas aggregate. In formulas, aggregate\r\ngross output in the economy is given by: \r\n\\begin{equation}\r\nQ_{nt}=\\prod_{j=1}^{J}\\left( Q_{nt}^{j}\\right) ^{\\alpha ^{j}}\r\n\\label{aggregate}\r\n\\end{equation}%\r\nwhere $Q_{t}^{j}$ is the gross output in sector $j$ and $\\sum_{j=1}^{J}%\r\n\\alpha ^{j}=1$. Competitive firms in each sector $j$ produce a composite\r\ngood according to the following constant-elasticity-of-substitution (CES)\r\ntechnology: \r\n\\begin{equation}\r\nQ_{nt}^{j}=\\left[ \\int_{0}^{1}q_{nt}(\\omega ^{j})^{\\frac{\\eta -1}{\\eta }%\r\n}d\\omega ^{j}\\right] ^{\\frac{\\eta }{\\eta -1}} \\label{sectoroutput}\r\n\\end{equation}%\r\nwhere $q_{nt}(\\omega ^{j})$ is the quantity of good $\\omega ^{j}$ used by\r\ncountry $n$ in sector $j$ at time $t$, and $\\eta >0$ is the elasticity of\r\nsubstitution across goods within a given sector. The intermediate goods $%\r\n\\omega ^{j}$ can be produced locally or imported from other countries.\r\nDelivering a good from country $n$ to country $m$ in sector $j$ and time\r\nperiod $t$ results in $0<\\kappa _{mnt}^{j}\\leq 1$ goods arriving at $m$; we\r\nassume that $\\kappa _{mnt}^{j}\\geq \\kappa _{mkt}^{j}\\kappa _{knt}^{j}\\quad\r\n\\forall m,n,k,j,t$ and $\\kappa _{nnt}^{j}=1$. All costs incurred are net\r\nlosses.\\footnote{%\r\nIn the calibration, the $\\kappa $s will reflect all trading costs, including\r\ntariffs; so implicitly we adopt the extreme assumption that tariff revenues\r\nare wasted---or at least not rebated back to agents in a way that would\r\ninteract with the allocation of resources in the economy.} Under the\r\nassumption of perfect competition, goods are sourced from the lowest-cost\r\nproducer, after adjusting for transport costs. The technology for producing $%\r\nq_{nt}(\\omega ^{j})$ is given accordingly by the country of origin ($m$)\r\nwith the lowest cost (with $m=n$ when the good is produced locally): \r\n\\begin{equation}\r\nx_{mt}(\\omega ^{j})=A_{mt}^{j}z_{m}(\\omega ^{j})l_{mt}(\\omega ^{j})^{\\beta\r\n^{j}}M_{mt}(\\omega ^{j})^{1-\\beta ^{j}}\r\n\\end{equation}%\r\nwhere $x_{mt}(\\omega ^{j})$ is the production of good $\\omega ^{j}$ by\r\ncountry $m$ at time $t$, $M_{mt}(\\omega ^{j})$ is the amount of the\r\naggregate composite good used by country $m$ to produce $x_{mt}(\\omega ^{j})$\r\nunits of good $\\omega ^{j}$ and $l_{mt}(\\omega ^{j})$ is the corresponding\r\namount of equipped labour. Total factor productivity (TFP) levels vary\r\nacross countries, sectors, and goods. Specifically, each intermediate good $%\r\n\\omega ^{j}$ in sector $j$ of country $n$ has a time-invariant idiosyncratic\r\nproductivity factor $z_{n}(\\omega ^{j})$ and a time-varying factor $%\r\nA_{nt}^{j}$ common to all the goods $\\omega ^{j}$ in sector $j$. Building on\r\nthe literature, we assume the productivities $z_{n}(\\omega ^{j})$ follow a\r\nsector-specific, time-invariant Fr\\'{e}chet distribution $F_{n}^{j}(z)=\\exp\r\n(-T_{n}^{j}z^{-\\theta })$. A higher $T_{n}^{j}$ shifts the distribution of\r\nproductivities to the right, that is leading to probabilistically higher\r\nproductivities. A higher $\\theta $ decreases the dispersion of the\r\nproductivity distribution, and hence reduces the scope for comparative\r\nadvantage. Shocks to $A_{nt}^{j}$ over time are interpreted as standard\r\nsectoral total factor productivity (TFP) shocks.\r\n\r\nThe single final good can be used both as input in the production of\r\nintermediaries $\\omega^{j}$ or for final consumption, $C_{nt}$. Hence,\r\nmarket clearing in the good markets implies: \r\n\\begin{equation*}\r\nQ_{nt} = C_{nt} + \\sum_{j=1}^{J}\\int_{0}^{1}M_{nt}(\\omega^{j})d\\omega ^{j}, \r\n\\end{equation*}\r\nwhere the integral aggregates over the unit-size continuum of goods $%\r\n\\omega^{j}$ entering in the production of each sector's $j$ aggregate good.\r\n\r\nClearing in the input market within a sector implies: \r\n\\begin{equation*}\r\nL_{nt}^{j}=\\int_{0}^{1}l_{nt}(\\omega ^{j})d\\omega ^{j}, \r\n\\end{equation*}%\r\nwhere $l_{nt}(\\omega ^{j})$ denotes the amount of equipped labour used in\r\nthe production of good $\\omega ^{j}$ by country $n$. We assume there is\r\nperfect risk-sharing within a country, but no risk-sharing across countries.%\r\n\\footnote{%\r\nTo motivate the lack of risk-sharing across countries, see our discussion of\r\nFigures 1 and 2.} The (equipped) labour shares allocated to each sector, $%\r\nL_{nt}^{j}/L_{nt},$ with $\\sum\\nolimits_{k=1}^{J}L_{nt}^{k}=L_{nt}$, are\r\ndetermined ex ante (before the realization of the shocks), but can be\r\nadjusted ex-post, at an adjustment cost. At the beginning of each period, a\r\nrepresentative consumer decides on the ex-ante optimal allocation of the\r\nprimary input $L_{nt}$ into different sectors in order to maximize the\r\nexpected value of utility. Then (stochastic) shocks to productivity $%\r\nA_{nt}^{j}$ are realized; equipped labour is reallocated freely within a\r\nsector, while reallocation across sectors implies a disutility cost $%\r\n\\Upsilon .$ Production and consumption take place. The disutility cost of\r\nex-post reallocation across sectors aims at capturing the idea that in the\r\nshort run, it is costly to reallocate productive factors across sectors.\r\n\r\nThe representative consumer's budget constraint in each period is: \r\n\\begin{equation*}\r\nP_{nt}C_{nt}=\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}, \r\n\\end{equation*}%\r\nwhere $P_{nt}$ is the price of the aggregate good (\\ref{aggregate}), $%\r\nw_{nt}^{j}L_{nt}^{j}$ is the nominal value-added generated in sector $j$.\r\nLifetime utility is given by \r\n\\begin{equation*}\r\nU_{n}=\\sum\\limits_{t=0}^{\\infty }\\delta ^{t}u(C_{nt})+\\Upsilon , \r\n\\end{equation*}%\r\nwhere $u^{\\prime }>0$, $u^{\\prime \\prime }\\leq 0$ and $\\delta $ is the\r\ndiscount factor and $\\Upsilon $ is the sum of quadratic deviations of\r\nsectoral shares from the original sectoral allocation. The cost $\\Upsilon $\r\nis relevant ex-post, when consumers need to reallocate inputs across\r\nsectors. Ex ante, however, the optimal allocation is such that the expected\r\nadjustment costs will be zero. Because there is no intertemporal trade and\r\nno capital in the economy, and given that input shares ex ante will equal\r\nthe expected optimal allocation, each period consumers maximize the expected\r\nstatic utility flow $E\\left[ u\\left( C_{nt}\\right) \\right] $ and the\r\nequilibrium is simply a sequence of static equilibria. In making his labor\r\nallocation decisions the representative consumer takes into account the\r\njoint probability distribution function of sectoral productivities, $%\r\nA_{nt}^{j}$s.\r\n\r\nIn the analysis, we assume log utility and therefore the consumer solves: \r\n\\begin{equation}\r\nL_{nt}^{k\\ast }=\\arg \\max E_{t-1}\\left[ \\ln \\left( \\frac{%\r\n\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}}{P_{nt}}\\right) \\right]\r\n,s.t.:\\sum\\nolimits_{j=1}^{J}L_{nt}^{j}=L_{nt}, \\label{eq:log:utility}\r\n\\end{equation}%\r\nwhere $E_{t-1}$ indicates that the expectation is taken before the\r\nrealization of period $t$ shocks. This\\ maximization problem leads to the\r\nfollowing first-order conditions for the ex-ante allocation of inputs to\r\nsectors: \r\n\\begin{equation}\r\n\\psi _{nt}^{k\\ast }=\\frac{L_{nt}^{k\\ast }}{L_{nt}}=E_{t-1}\\left[ \\frac{%\r\nw_{nt}^{k}L_{nt}^{k}}{\\sum_{k}w_{nt}^{k}L_{nt}^{k}}\\right] ,\\qquad \\forall\r\nk,t. \\label{eq:FOC}\r\n\\end{equation}%\r\nIn words, the share of resources allocated ex ante to a given sector equals\r\nits expected share in value added. To gain intuition on this expression note\r\nthat $1/\\sum_{k}w_{nt}^{k}L_{nt}^{k}$ is the marginal utility of consumption\r\nin period $t$; thus, more resources are allocated to higher value-added\r\nsectors, after appropriately weighting by marginal utility. Consider, for\r\nfurther intuition, a (small) sector whose productivity is negatively\r\ncorrelated with the rest of the economy (that is, it has high value added\r\nwhen the rest of the economy has low value added); in states of the world in\r\nwhich overall income is low, the marginal utility of consumption $%\r\n1/\\sum_{k}w_{nt}^{k}L_{nt}^{k}$ will be high and hence the optimal\r\nallocation entails allocating more resources to this sector. In the closed\r\neconomy, the value-added share is pinned down by the Cobb--Douglas\r\ncoefficients $\\alpha ^{j}\\beta ^{j}$, as with Cobb-Douglas technology there\r\nis no variation on expenditures (and sales) shares---and log-utility implies\r\nthe shares determine the sectoral allocation of resources. In the open\r\neconomy this result no longer holds as a country's sectoral shares depend on\r\nits absolute and comparative advantage as well as trading costs vis-\\`{a}%\r\n-vis other countries.\r\n\r\nEx post, as said, labour can potentially be reallocated after incurring the\r\najustment costs. The ex-post sectoral input allocation solves:\r\n\r\n\\begin{equation}\r\nL_{nt}^{k}=\\arg \\max \\left[ \\ln \\left( \\frac{%\r\n\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}}{P_{nt}}\\right) +\\frac{\\varrho }{2}%\r\n\\sum_{j=1}^{J}\\left[ \\psi _{nt}^{j}-\\psi _{nt}^{j\\ast }\\right] ^{2}\\right] ,%\r\n\\text{ }s.t.:\\sum\\nolimits_{j=1}^{J}L_{nt}^{j}=L_{nt}\r\n\\end{equation}%\r\nwith $\\psi _{nt}^{j}\\equiv \\frac{L_{nt}^{j}}{L_{nt}}$ and $%\r\n\\sum_{j=1}^{J}\\psi _{nt}^{j}=\\sum_{j=1}^{J}\\psi _{nt}^{j\\ast }=1.$ The first\r\norder conditions satisfy%\r\n\\begin{equation}\r\n\\frac{w_{nt}^{k}}{\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}}+\\varrho \\left[ \\psi\r\n_{nt}^{k}-\\psi _{nt}^{k\\ast }\\right]/L_{nt} =\\lambda\r\n\\end{equation}%\r\nwhere $\\lambda $ is given by $\\lambda =\\frac{\\frac{1}{J}%\r\n\\sum_{j=1}^{J}w_{nt}^{j}}{\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}}$ and hence\r\n\r\n\\begin{equation*}\r\n\\psi _{nt}^{k}=\\psi _{nt}^{k\\ast }+\\frac{1}{\\varrho }\\left[ \\frac{\\frac{1}{J}%\r\n\\sum_{j=1}^{J}w_{nt}^{j}-w_{nt}^{k}}{\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}/L_{nt}}%\r\n\\right] \r\n\\end{equation*}%\r\nIn words, the ex post input shares $\\psi _{nt}^{k}$ equal the ex-ante\r\noptimal shares $\\psi _{nt}^{k\\ast }$ plus a fraction of the percentage differential\r\nbetween the average equipped labour cost in the economy $\\frac{1}{J}%\r\n\\sum_{j=1}^{J}w_{nt}^{j}$ and the sectoral input cost $w_{nt}^{k}$. \r\n(Note that the denominator is the average input cost in the economy.)\r\nThe\r\nadjustment cost parameter $\\varrho $ determines the elasticity of sectoral\r\nadjustment to the cost differential. In the extreme, when adjustment costs\r\nare infinite, $\\varrho \\rightarrow \\infty $, the economy simply stays at the\r\ninitial sectoral input shares $\\psi _{nt}^{k}=\\psi _{nt}^{k\\ast }$. As\r\nadjustment costs decrease, the sectoral shares adjust in response to the\r\ncost differential.\r\n\r\n\\bigskip\r\n\r\nPAUSE\r\n\r\n\\subsection{Model Solution}\r\n\r\nWe first discuss the solution under autarky, and then turn to the solution\r\nunder free trade.\r\n\r\n\\paragraph{Solution under Autarky}\r\n\r\nWe solve the model backwards in two stages. First, we solve the model taking\r\nthe sectoral allocation of nonproduced inputs $L_{t}^{j}$ as fixed. We then\r\nsolve for the ex-ante optimal $L_{t}^{j\\prime }s$ before the shocks are\r\nrealized. In the analysis of the autarky case, we omit the country-specific\r\nsubscripts $n$ for convenience.\r\n\r\nThe demand for each sector's composite good is given by \r\n\\begin{equation}\r\nQ_{t}^{j}=\\alpha _{t}^{j}\\left( \\frac{P_{t}^{j}}{P_{t}}\\right) ^{-1}Q_{t},\r\n\\label{QJ}\r\n\\end{equation}%\r\nand the demand for each intermediate good $\\omega ^{j}$ is \r\n\\begin{equation*}\r\nq_{t}(\\omega ^{j})=\\left[ \\frac{p_{t}(\\omega ^{j})}{P_{t}^{j}}\\right]\r\n^{-\\eta }Q_{t}^{j}, \r\n\\end{equation*}%\r\nwhere \r\n\\begin{equation}\r\nP_{t}^{j}=\\left[ \\int_{0}^{1}p_{t}(\\omega ^{j})^{1-\\eta }d\\omega ^{j}\\right]\r\n^{\\frac{1}{1-\\eta }} \\label{PJ}\r\n\\end{equation}%\r\nis the aggregate price index in sector $j$, and the economy-wide price index\r\nis given by: \r\n\\begin{equation}\r\nP_{t}=\\prod_{j=1}^{J}\\alpha _{t}^{j^{-\\alpha _{t}^{j}}}\\left(\r\nP_{t}^{j}\\right) ^{\\alpha _{t}^{j}}. \\label{Pt}\r\n\\end{equation}%\r\nThe demand for non-produced inputs $l_{t}(\\omega ^{j})$ and produced inputs $%\r\nM_{t}(\\omega ^{j})$ are given, respectively, by $l_{t}^{j}(\\omega\r\n^{j})=\\beta ^{j}\\frac{p_{t}(\\omega ^{j})q_{t}(\\omega ^{j})}{w_{t}^{j}}$ and $%\r\nM_{t}(\\omega ^{j})=(1-\\beta ^{j})\\frac{p_{t}(\\omega ^{j})q_{t}(\\omega ^{j})}{%\r\nP_{t}}.$ Aggregating over all goods $\\omega ^{j}$ in a given sector, we\r\nobtain \r\n\\begin{equation}\r\nw_{t}^{j}L_{t}^{j}=\\beta ^{j}P_{t}^{j}Q_{t}^{j} \\label{Labor share}\r\n\\end{equation}%\r\nand, correspondingly, $P_{t}M_{t}^{j}=(1-\\beta ^{j})P_{t}^{j}Q_{t}^{j}.$\r\nLabour shares are given by:\r\n\r\n\\begin{equation*}\r\n\\frac{L_{t}^{j}}{L_{t}}=\\psi _{nt}^{k}=\\psi _{nt}^{k\\ast }+\\frac{1}{\\varrho }%\r\n\\left[ \\frac{\\frac{1}{J}\\sum_{j=1}^{J}w_{nt}^{j}-w_{nt}^{k}}{%\r\n\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}}\\right] \r\n\\end{equation*}%\r\nand the initial labour allocation, $\\psi _{nt}^{k\\ast }$, is taken as given.\r\n\r\nUsing the input demand functions and the zero profit condition the autarky\r\nprices of intermediate goods are given by: \r\n\\begin{equation}\r\np_{t}(\\omega ^{j})=B^{j}\\left[ A_{t}^{j}\\cdot z(\\omega ^{j})\\right]\r\n^{-1}\\left( w_{t}^{j}\\right) ^{\\beta ^{j}}P_{t}^{1-\\beta ^{j}},\r\n\\label{Frechet}\r\n\\end{equation}%\r\nwhere $B^{j}=$ $\\left( \\beta ^{j}\\right) ^{-\\beta ^{j}}(1-\\beta\r\n^{j})^{-(1-\\beta ^{j})}$. Using (\\ref{Frechet}) and the properties of the Fr%\r\n\\'{e}chet distribution, we can express the sectoral price index as: \r\n\\begin{equation}\r\nP_{t}^{j}=\\xi B^{j}\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{\\frac{1}{%\r\n\\theta }}\\right] ^{-1}\\left( w_{t}^{j}\\right) ^{\\beta ^{j}}P_{t}^{1-\\beta\r\n^{j}}\r\n\\end{equation}%\r\nwhere $\\xi =\\left[ \\Gamma \\left( \\frac{\\theta +1-\\eta }{\\theta }\\right) %\r\n\\right] ,$ and $\\Gamma $ is the gamma function.\r\n\r\nUsing (\\ref{QJ}), (\\ref{Labor share}), and (\\ref{Pt}) we obtain real GDP:\r\n\r\n\\begin{eqnarray*}\r\nY_{t} &=&\\sum_{j=1}^{J}\\frac{w_{t}^{j}L_{t}^{j}}{P_{t}}=\\sum_{j=1}^{J}\\beta\r\n^{j}\\frac{P_{t}^{j}Q_{t}^{j}}{P_{t}}=\\sum_{j=1}^{J}\\beta ^{j}\\alpha\r\n_{t}^{j}Q_{t} \\\\\r\n\\text{since }w_{t}^{j}L_{t}^{j} &=&\\beta ^{j}P_{t}^{j}Q_{t}^{j}=\\beta\r\n^{j}\\alpha _{t}^{j}P_{t}Q_{t} \\\\\r\nQ_{t} &=&\\frac{1}{\\alpha _{t}^{j}\\beta ^{j}}\\frac{w_{t}^{j}}{P_{t}}L_{t}^{j}\r\n\\\\\r\n\\text{and since }\\frac{w_{t}^{j}}{P_{t}} &=&\\left( \\xi B^{j}\\right)\r\n^{-1/\\beta ^{j}}\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{\\frac{1}{\\theta }%\r\n}\\right] ^{1/\\beta ^{j}}\\left( \\frac{P_{t}^{j}}{P_{t}}\\right) ^{\\frac{1}{%\r\n\\beta ^{j}}}\r\n\\end{eqnarray*}%\r\nWe hence obtain%\r\n\\begin{eqnarray*}\r\nQ_{t} &=&\\left( \\xi B^{j}\\right) ^{-1/\\beta ^{j}}\\left[ A_{t}^{j}\\cdot\r\n\\left( T^{j}\\right) ^{\\frac{1}{\\theta }}\\right] ^{1/\\beta ^{j}}\\left( \\frac{%\r\nP_{t}^{j}}{P_{t}}\\right) ^{\\frac{1}{\\beta ^{j}}}\\frac{1}{\\alpha\r\n_{t}^{j}\\beta ^{j}}L_{t}^{j} \\\\\r\n\\frac{P_{t}^{j}}{P_{t}} &=&\\left( \\xi B^{j}\\right) \\left[ A_{t}^{j}\\cdot\r\n\\left( T^{j}\\right) ^{\\frac{1}{\\theta }}\\right] ^{-1}\\left( \\frac{1}{\\alpha\r\n_{t}^{j}\\beta ^{j}}L_{t}^{j}\\right) ^{-\\beta ^{j}}\\left( Q_{t}\\right)\r\n^{\\beta ^{j}} \\\\\r\n\\prod_{j=1}^{J}\\alpha _{t}^{j^{-\\alpha _{t}^{j}}}\\left( P_{t}^{j}\\right)\r\n^{\\alpha _{t}^{j}} &=&\\prod_{j=1}^{J}\\alpha _{t}^{j^{-\\alpha\r\n_{t}^{j}}}\\left( \\xi B^{j}\\right) ^{\\alpha _{t}^{j}}\\left[ A_{t}^{j}\\cdot\r\n\\left( T^{j}\\right) ^{\\frac{1}{\\theta }}\\right] ^{-\\alpha _{t}^{j}}\\left( \r\n\\frac{1}{\\alpha _{t}^{j}\\beta ^{j}}L_{t}^{j}\\right) ^{-\\beta ^{j}\\alpha\r\n_{t}^{j}}\\left( Q_{t}\\right) ^{\\beta ^{j}\\alpha _{t}^{j}} \\\\\r\n1 &=&\\prod_{j=1}^{J}\\alpha _{t}^{j^{-\\alpha _{t}^{j}}}\\left( \\xi\r\nB^{j}\\right) ^{\\alpha _{t}^{j}}\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{%\r\n\\frac{1}{\\theta }}\\right] ^{-\\alpha _{t}^{j}}\\left( \\frac{1}{\\alpha\r\n_{t}^{j}\\beta ^{j}}L_{t}^{j}\\right) ^{-\\beta ^{j}\\alpha _{t}^{j}}Q_{t}^{\\bar{%\r\n\\beta}} \\\\\r\nQ_{t} &=&\\prod_{j=1}^{J}\\left( \\xi B^{j}\\right) ^{-\\alpha ^{j}/\\bar{\\beta}}%\r\n\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{\\frac{1}{\\theta }}\\right]\r\n^{\\alpha ^{j}/\\bar{\\beta}}\\alpha _{t}^{j^{-\\alpha _{t}^{j}/\\bar{\\beta}%\r\n}}\\left( \\alpha _{t}^{j}\\beta ^{j}\\right) ^{-\\frac{\\beta ^{j}\\alpha _{t}^{j}%\r\n}{\\bar{\\beta}}}\\left( L_{t}^{j}\\right) ^{\\frac{\\beta ^{j}\\alpha _{t}^{j}}{%\r\n\\bar{\\beta}}} \\\\\r\nY_{t} &=&\\sum_{j=1}^{J}\\frac{w_{t}^{j}L_{t}^{j}}{P_{t}}=\\sum_{k=1}^{J}\\beta\r\n^{k}\\alpha _{t}^{k}Q_{t}=\\bar{\\beta}\\prod_{j=1}^{J}\\left( \\xi B^{j}\\right)\r\n^{-\\alpha ^{j}/\\bar{\\beta}}\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{\\frac{%\r\n1}{\\theta }}\\right] ^{\\alpha ^{j}/\\bar{\\beta}}\\alpha _{t}^{j^{-\\alpha\r\n_{t}^{j}/\\bar{\\beta}}}\\left( \\alpha _{t}^{j}\\beta ^{j}\\right) ^{-\\frac{\\beta\r\n^{j}\\alpha _{t}^{j}}{\\bar{\\beta}}}\\left( L_{t}^{j}\\right) ^{\\frac{\\beta\r\n^{j}\\alpha _{t}^{j}}{\\bar{\\beta}}} \\\\\r\nY_{t} &=&\\prod_{j=1}^{J}R^{j}\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{%\r\n\\frac{1}{\\theta }}\\right] ^{\\alpha ^{j}/\\bar{\\beta}}\\left( L_{t}^{j}\\right)\r\n^{\\frac{\\beta ^{j}\\alpha _{t}^{j}}{\\bar{\\beta}}}\r\n\\end{eqnarray*}\r\n\r\n\\bigskip where $\\bar{\\beta}=\\sum_{j=1}^{J}\\alpha ^{j}\\beta ^{j}$ and $%\r\nR_{j}\\propto \\prod_{j=1}^{J}\\left( \\beta ^{j}\\alpha ^{j}\\right) ^{-\\frac{%\r\n\\alpha ^{j}\\beta ^{j}}{\\bar{\\beta}}}(B^{j})^{-\\frac{\\alpha ^{j}}{\\bar{\\beta}}%\r\n}\\left( T^{j}\\right) ^{\\frac{\\alpha ^{j}}{\\bar{\\beta}\\cdot \\theta }}$ is a\r\ntime-invariant product. We now need to determine $L_{nt}^{k}$\r\n\r\n\\begin{equation}\r\n\\psi _{nt}^{k\\ast }=\\frac{L_{nt}^{k\\ast }}{L_{nt}}=E_{t-1}\\left[ \\frac{%\r\nw_{nt}^{k}L_{nt}^{k}}{\\sum_{k}w_{nt}^{k}L_{nt}^{k}}\\right] ,\\qquad \\forall\r\nk,t.\r\n\\end{equation}%\r\n\\begin{equation*}\r\n\\psi _{nt}^{k}=\\psi _{nt}^{k\\ast }+\\frac{1}{\\varrho }\\left[ \\frac{\\frac{1}{J}%\r\n\\sum_{j=1}^{J}w_{nt}^{j}-w_{nt}^{k}}{\\sum_{j=1}^{J}w_{nt}^{j}L_{nt}^{j}}%\r\n\\right] \r\n\\end{equation*}\r\n\r\nWe can now move one step backward and solve for the allocation of the\r\nprimary input across sectors, $L_{t}^{j},$ $j=1,...,J$. \r\n\r\n$\\psi _{nt}^{k\\ast }=\\frac{\\alpha ^{j}\\beta ^{j}}{\\bar{\\beta}}$.$.$Hence:\r\n\r\n\\begin{equation*}\r\n\\psi _{t}^{k}=\\frac{\\alpha ^{j}\\beta ^{j}}{\\bar{\\beta}}+\\frac{1}{\\varrho }%\r\n\\left[ \\frac{\\bar{w}_{t}-w_{t}^{k}}{\\sum_{j=1}^{J}w_{t}^{j}L_{t}^{j}}\\right] \r\n\\end{equation*}\r\n\r\n\\begin{eqnarray}\r\nY_{t} &=&\\sum_{j=1}^{J}\\frac{w_{t}^{j}L_{t}^{j}}{P_{t}}=\\prod_{j=1}^{J}R_{j}%\r\n\\left[ A_{t}^{j}\\right] ^{\\frac{\\alpha ^{j}}{\\bar{\\beta}}}\\left( \\frac{%\r\n\\alpha ^{j}\\beta ^{j}}{\\bar{\\beta}}+\\frac{1}{\\varrho }\\left[ \\frac{\\bar{w}%\r\n_{t}-w_{t}^{k}}{\\sum_{j=1}^{J}w_{t}^{j}L_{t}^{j}}\\right] \\right) ^{\\frac{%\r\n\\alpha ^{j}\\beta ^{j}}{\\bar{\\beta}}}L_{t} \\label{expost} \\\\\r\nY_{t} &=&\\sum_{j=1}^{J}\\frac{w_{t}^{j}L_{t}^{j}}{P_{t}}=\\prod_{j=1}^{J}R_{j}%\r\n\\left[ A_{t}^{j}\\right] ^{\\frac{\\alpha ^{j}}{\\bar{\\beta}}}\\left( \\frac{%\r\n\\alpha ^{j}\\beta ^{j}}{\\bar{\\beta}}+\\frac{1}{\\varrho }\\left[ \\frac{\\frac{%\r\n\\bar{w}_{t}-w_{t}^{k}}{P_{t}}}{Y_{t}}\\right] \\right) ^{\\frac{\\alpha\r\n^{j}\\beta ^{j}}{\\bar{\\beta}}}L_{t}\r\n\\end{eqnarray}\r\n\r\n\\begin{equation*}\r\n\\frac{\\bar{w}_{t}-w_{t}^{k}}{P_{t}}=\\sum_{j\\neq k}^{J}\\left( \\xi\r\nB^{j}\\right) ^{-1/\\beta ^{j}}\\left[ A_{t}^{j}\\cdot \\left( T^{j}\\right) ^{%\r\n\\frac{1}{\\theta }}\\right] ^{1/\\beta ^{j}}\\left( \\frac{P_{t}^{j}}{P_{t}}%\r\n\\right) ^{\\frac{1}{\\beta ^{j}}}\r\n\\end{equation*}\r\n\r\n\\bigskip \r\n\r\n\\bigskip \r\n\r\n\\bigskip \r\n\r\nSTOP HERE\r\n\r\n\\bigskip\r\n\r\n\\paragraph{Solution with International Trade}\r\n\r\nThe key difference in the internationally open economy is that inputs can\r\npotentially be sourced from different countries. Delivering a unit of good $%\r\n\\omega ^{j}$ produced in country $m$ to country $n$ costs: \r\n\\begin{equation*}\r\np_{nmt}^{j}(\\omega ^{j})=\\frac{B^{j}\\left( w_{mt}^{j}\\right) ^{\\beta\r\n^{j}}P_{mt}^{1-\\beta ^{j}}}{A_{mt}^{j}\\kappa _{nmt}^{j}z_{m}(\\omega ^{j})} \r\n\\end{equation*}%\r\nwhere $B^{j}\\left( w_{mt}^{j}\\right) ^{\\beta ^{j}}P_{mt}^{1-\\beta ^{j}}$ is\r\nthe cost of the input bundle in country of origin $m$, sector $j$, at time $%\r\nt $. Because of perfect competition, the price paid in country $n$, denoted $%\r\np_{nt}(\\omega ^{j})$, will be the minimum price across all $N$ potential\r\ntrading partners: $p_{nt}^{j}(\\omega ^{j})=\\min \\left\\{ p_{nmt}^{j}(\\omega\r\n^{j});\\text{ }m=1,...,N\\right\\} .$ Producers of the aggregate good in (\\ref%\r\n{aggregate}) minimize production costs taking prices as given. We assume the\r\ndistribution of efficiencies for any good $\\omega ^{j}$ in sector $j$ and\r\ncountry $n$ are independent across countries and sectors and follow a\r\ntime-invariant Fr\\'{e}chet distribution: $F_{n}^{j}(z)=\\exp\r\n(-T_{n}^{j}z^{-\\theta }).$ Under this assumption, the distribution of prices\r\nin sector $j$ of country $n$, conditional on $\\left\\{ A_{mt}^{j}\\right\\}\r\n_{m=1,...N}$ is given by $G_{nt}^{j}(p)|_{\\left\\{ A_{t}^{j}\\right\\} }=\\Pr\r\n(P_{nt}^{j},>=stealth]\n\t\\begin{tikzpicture}[node distance=1cm]\n\t\t\\node (compute) [generalRect] {Compute Descriptors};\n\t\t\\node (descriptor) [generalRect, right = of compute] {Descriptor Function};\n\t\t\\node (pca) [generalRect, fill=white, above = 0.5cm of descriptor, dashed] {PCA Function};\n\t\t\\node (compare) [generalRect, below= of compute] {Compare Descriptors};\n\t\t\\node (query) [generalRect, fill=white, left = of compare, dashed] {Query Image};\n\t\t\\node (dist) [generalRect, right = of compare] {Distance Measure};\n\t\t\\node (display) [generalRect, below= of compare, fill=white] {Display Results};\n\t\t\\node (plotPR) [generalRect, below= of display, fill=white] {\\shortstack{Plot Precision-Recall Graph}};\n\t\t\n\t\t\\draw [arrow] (compute) -- (compare);\n\t\t\\draw [arrow, <->] (compute) -- (descriptor);\n\t\t\\draw [arrow, <->, dashed] (pca.west) -- (compute.east);\n\t\t\\draw [arrow] (compare) -- (display);\n\t\t\\draw [arrow, <->] (compare) -- (dist);\n\t\t\\draw [arrow, dashed] (query) -- (compare);\n\t\t\\draw [arrow] (display) -- (plotPR);\n\t\\end{tikzpicture}\n\t\\caption{System design}\n\t\\label{fig:sys-operation}\n\\end{figure}\n\n\\chapter{Description Techniques} \\label{sec:techniques-implemented}\nThis Chapter discusses the different descriptor generation techniques implemented, along with the theory of their operation and the difficulties which were encountered whilst implementing them. Details of how each technique was tested is given in Appendix \\ref{app:test-methodology}.\n\n\\section{Global Colour Histogram} \\label{sec:color-histogram}\nA global colour histogram quantises the RGB space into a series of bins. There are $Q$ bins for each dimension, and therefore $Q^3$ bins for the total image. Each pixel in the image is then sorted into the appropriate bin, and the sum of the bin totals is normalised to 1. This vector histogram, of length $Q^3$, can therefore be used to describe the image. A diagrammatic representation of this technique is shown in Figure \\ref{fig:rgb-hist-diag}. This Figure shows the RGB space quantised with a quantisation level of $Q=2$, yielding $2^2=4$ bins. $P_1$, $P_2$, and $P_3$ represent points in the RGB space which will be quantised to their containing boxes.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\tdplotsetmaincoords{65}{135}\n\t\\newcommand{\\histbox}[3]\n\t{\n\t\t\\draw[dashed, color=#3] (#1) -- (#2x);\n\t\t\\draw[dashed, color=#3] (#1) -- (#2y);\n\t\t\\draw[dashed, color=#3] (#1) -- (#2z);\n\t\t\\draw[dashed, color=#3] (#2x) -- (#2xy);\n\t\t\\draw[dashed, color=#3] (#2y) -- (#2xy);\n\t\t\\draw[dashed, color=#3] (#2x) -- (#2xz);\n\t\t\\draw[dashed, color=#3] (#2z) -- (#2xz);\n\t\t\\draw[dashed, color=#3] (#2y) -- (#2yz);\n\t\t\\draw[dashed, color=#3] (#2z) -- (#2yz);\n\t\t\\draw[dashed, color=#3] (#2xy) -- (#2);\n\t\t\\draw[dashed, color=#3] (#2xz) -- (#2);\n\t\t\\draw[dashed, color=#3] (#2yz) -- (#2);\n\t}\n\t\\begin{tikzpicture}[scale=2,tdplot_main_coords]\n\t\\coordinate (O) at (0,0,0);\n\t\\tdplotgetpolarcoords{1}{1}{1}\n\t\\tdplotsetcoord{A}{1.73205}{\\tdplotrestheta}{\\tdplotresphi}\n\t\\tdplotgetpolarcoords{2}{1}{1}\n\t\\tdplotsetcoord{B}{2.44949}{\\tdplotrestheta}{\\tdplotresphi}\n\t\\tdplotgetpolarcoords{1}{2}{1}\n\t\\tdplotsetcoord{C}{2.44949}{\\tdplotrestheta}{\\tdplotresphi}\n\t\\tdplotgetpolarcoords{1}{1}{2}\n\t\\tdplotsetcoord{D}{2.44949}{\\tdplotrestheta}{\\tdplotresphi}\n\t\n\t\\tdplotgetpolarcoords{2}{2}{1}\n\t\\tdplotsetcoord{E}{3}{\\tdplotrestheta}{\\tdplotresphi}\n\t\\tdplotgetpolarcoords{2}{1}{2}\n\t\\tdplotsetcoord{F}{3}{\\tdplotrestheta}{\\tdplotresphi}\n\t\\tdplotgetpolarcoords{1}{2}{2}\n\t\\tdplotsetcoord{G}{3}{\\tdplotrestheta}{\\tdplotresphi}\n\t\n\t\\tdplotgetpolarcoords{2}{2}{2}\n\t\\tdplotsetcoord{H}{3.464102}{\\tdplotrestheta}{\\tdplotresphi}\n\t\n\t\\draw[thick,->] (0,0,0) -- (2.25,0,0) node[anchor=north east]{R};\n\t\\draw[thick,->] (0,0,0) -- (0,2.25,0) node[anchor=north west]{G};\n\t\\draw[thick,->] (0,0,0) -- (0,0,2.25) node[anchor=south]{B};\n\t\n\t\n\t\\draw plot [mark=*, mark size=2, mark options={color={rgb:red,1.5;green,0.5;blue,0.5}}] coordinates{(1.5,0.5,0.5)};\n\t\\draw plot [mark=*, mark size=2, mark options={color={rgb:red,0.3;green,1.6;blue,1.7}}] coordinates{(0.3,1.6,1.7)};\n\t\\draw plot [mark=*, mark size=2, mark options={color={rgb:red,1.8;green,1.8;blue,0.2}}] coordinates{(1.8,1.8,0.2)};\n\t\\histbox{O}{A}{blue}\n\t\\histbox{O}{B}{blue}\n\t\\histbox{O}{C}{blue}\n\t\\histbox{O}{D}{blue}\n\t\\histbox{O}{E}{blue}\n\t\\histbox{O}{F}{blue}\n\t\\histbox{O}{G}{blue}\n\t\\histbox{O}{H}{blue}\n\t\n\t\\node[left] at (1.5,0.5,0.5) {$P_1$};\n\t\\node[below] at(0.3,1.6,1.7) {$P_2$};\n\t\\node[right] at(1.8,1.8,0.2) {$P_3$};\n\t\n\t\\end{tikzpicture}\n\t\\caption{Visualisation of RGB histogram}\n\t\\label{fig:rgb-hist-diag}\n\\end{figure}\n\n\nThe implementation of this function was relatively straightforward, since example code for the function had been provided \\cite{collomosse2016}. The only modifications made to the code were to explicitly state the histogram bin edges. This was necessary to provide a robust descriptor since the provided code normalised the histogram to the range of the input. Therefore an image containing only values in one area of the RGB space, would appear similar to an image with uniform distribution. The function is implemented as \\texttt{H = vs\\_compute\\_rgb\\_histogram(img, Q)} where \\texttt{H} is the descriptor returned, \\texttt{img} is the image input, and \\texttt{Q} is the quantisation level for each image dimension. The only variable parameter is therefore $Q$.\n\n\\section{Gridding} \\label{sec:gridding-concatenation}\nGridding of features allows descriptors to apply locally to parts of an image. The implementation of this is fairly simple in \\gls{matlab}. The image can be split using the native array indexing of \\gls{matlab} to select a subset of the image array, and the descriptor function can then be applied to each of these sub images. The resultant descriptors are then concatenated into a matrix.\n\nGridding is implemented as \\texttt{F = vs\\_grid(img, h\\_level, v\\_level, compute\\_function)}, where \\texttt{img} is the image input, \\texttt{h\\_level} is the horizontal quantisation level, \\texttt{v\\_level} is the vertical quantisation level and \\texttt{compute\\_function} is the descriptor generation function. This configuration therefore gives two independent variables, \\texttt{h\\_level} and \\texttt{v\\_level}.\n\nHowever, since most of the images in the dataset have an aspect ratio of approximately 1.5:1, it makes sense to set the vertical quantisation level to 1.5 times the horizontal quantisation level. Therefore the overall quantisation level, $Q$ can be defined, where \\texttt{h\\_level = ceil(1.5*Q)}, and \\texttt{v\\_level = Q}.\n\n\\section{Edge Orientation Histogram} \\label{sec:texture-histogram}\nTexture is an important concept in computer vision, but is difficult to define. Texture loosely corresponds to a repeating pattern of edge characteristics, examples of which would include grass pebbles, and hair \\cite[p. 194]{forsyth2012}.\n\nOne method of creating a texture based descriptor is to use an edge orientation histogram. This works as follows:\n\\begin{enumerate}\n\t\\item Edges are detected in the image.\n\t\\item An estimation of the orientation of each edge is calculated.\n\t\\item A histogram is created of the edge orientations.\n\\end{enumerate}\n\nThis process is implemented using two functions: an edge orientation function, and a histogram generation function. The edge orientation function returns a two dimensional matrix where each element contains a value corresponding to an estimate of the edge orientation at that location. This estimate is in the range $-\\pi$ to $+\\pi$, but is normalised to $0$ to $1$. Weak edges are set to a normalised value of 0.5, corresponding to \\SI{0}{\\radian}.\n\nIn order to perform the edge detection, and edge angle estimation, the Sobel operator is used \\cite{sobel2015}. The Sobel operator is formed of two kernels which are convolved with input image in order to perform edge detection in the x and y directions respectively. The kernels are defined in Equation \\ref{eq:sobel-kernels}. \n\n\\begin{equation}\n\tK_x = \n\t\\begin{bmatrix}\n\t1 & 0 & -1 \\\\\n\t2 & 0 & -2 \\\\\n\t1 & 0 & -1 \\\\\n\t\\end{bmatrix}\n\tK_y = \n\t\\begin{bmatrix}\n\t1 & 2 & 1 \\\\\n\t0 & 0 & 0 \\\\\n\t-1 & -2 & -1 \\\\\n\t\\end{bmatrix}\n\t\\label{eq:sobel-kernels}\n\\end{equation}\n\nThe edge magnitude and angle are calculated as described in Equations \\ref{eq:sobel-mag} and \\ref{eq:sobel-ang} respectively. It should be noted that the operators in these equations apply on an element-by-element basis, not to the matrices as a whole. In addition, the $K_y$ value is inverted in order to allow the performance to match the \\gls{matlab} library function \\texttt{imgradient}, this allows for easier testing. The pixels with edge magnitudes above a certain threshold are passed to the histogram generation function.\n\\begin{equation}\n\\text{Mag} = \\sqrt{{K_x}^2 + {K_y}^2}\n\\label{eq:sobel-mag}\n\\end{equation}\n\\begin{equation}\n\\theta = \\atantwo(-K_y, K_x)\n\\label{eq:sobel-ang}\n\\end{equation}\n\nThe edge orientation function is implemented as \\texttt{F = vs\\_edge\\_detect(img, compute\\_function, strength)} where \\texttt{F} is the descriptor returned, \\texttt{img} is the image input, \\texttt{compute\\_function} is the histogram generation function, and \\texttt{strength} is the minimum edge strength to consider (range 0--1).\n\nThe histogram generation function is \\texttt{H = vs\\_compute\\_histogram(img, Q)} where \\texttt{H} is the histogram returned, \\texttt{img} is the image input and \\texttt{Q} is the quantisation factor. When \\texttt{vs\\_compute\\_histogram} is used as the \\texttt{compute\\_function} in \\texttt{vs\\_edge\\_detect}, the system can produce an edge orientation histogram, with two independent parameters. These parameters determine the minimum strength edge to consider, as well as the quantisation level of the histogram. These parameters will be referred to as $E$ and $Q$ respectively.\n\nAdditionally, since texture is an inherently local feature, the image will be gridded. The descriptor function handle is therefore : \\texttt{@(x)vs\\_grid(x, v\\_level, h\\_level, @(x)vs\\_edge\\_detect(x, @(x)vs\\_compute\\_histogram(x,Q),E))}.\n\n\\section{Concatenation of Descriptors} \\label{sec:conc-desc}\nDescriptors focusing upon an individual aspect of an image, such as colour or texture have already been discussed, as well as how the can be applied to multiple sections of an image rather than globally.\n\nIt stands to reason therefore that multiple descriptors could be combined to produce a descriptor of even greater effectiveness, such as a combination of an RGB histogram and a texture histogram. This is possible in \\gls{matlab} using the native array indexing interface, but it is clearer to use the \\texttt{horzcat} function.\n\nA combined descriptor can therefore be invoked using the function \\texttt{@(x)horzcat( func\\_1(x) .* M, func\\_2(x))} where \\texttt{texture\\_func(x)} and \\texttt{color\\_func(x)} are function handles for two descriptor functions with a single parameter, \\texttt{x} for the input image. The only other parameter in this function is \\texttt{M}. \\texttt{M} defines a weighting factor between the two descriptors, and can be used to give a greater weighting to one over the other. This works by using a multiplier value to either spread all of the values in the descriptor further apart, or bring them closer together.\n\n\\chapter{Experimental Results} \\label{sec:results}\n\nIn order to evaluate the performance of each descriptor, two test images have been chosen from the data set: 9\\_23\\_s and 13\\_1\\_s. These images have been chosen because they represent two contrasting examples of what could be presented to the algorithm. Image 9\\_23\\_s shows a sheep in a field. This image has two strongly dominant colours: the green of the grass, and the cream of the sheep's wool, as well as two regions of largely uniform texture: the sheep's body, and the grass. Image 13\\_1\\_s strongly contrasts this. This image shows a collection of differently coloured books on a shelf. The image therefore contains a large variety of colours distributed around the image. In addition to this the transitions between the books, as well as the text on the spine of the books means that a large amount of varied texture is distributed across the image.\n\nThe fact that these images contrast in many different ways means that they should be ideal to evaluate the performance of each descriptor in a fair way, as well as determine the optimum parameters for each descriptor function. The performance of the descriptors across the wider images in the dataset will then be presented in Section \\ref{sec:results}.\n\nIn order to evaluate performance, a precision-recall characteristic is plotted for each test. This characteristic is a plot showing the proportion of the returned results which have been relevant (precision), vs the number of relevant results returned (recall). The ideal precision-recall characteristic would be a horizontal line at precision = 1, a system which produced this characteristic would return all the relevant images, and none of the irrelevant images. Most practical systems produce a characteristic which slopes from the top left to the bottom right, this occurs because systems first return the results they are most sure about. These results are likely to be correct, and so precision will be high, but since not many results have yet been returned, recall will be low. As the system returns an increasing number of results, an increasing number of irrelevant results will be returned also, causing precision to decrease as recall increases.\n\n\\section{Global Colour Histogram} \\label{sec:global-colour-histogram-results}\nThe global colour descriptor has one parameter the quantisation level, $Q$, which produces a descriptor of length $Q^3$, as discussed in Section \\ref{sec:color-histogram}.\n\nIn order to analyse the effect of quantisation level on performance, results will be generated for various values of $Q$, and the performance of each image analysed for each test value. The expected result is that higher values of $Q$ will produce better results, up to a certain point, after which performance will decrease. This should happen because at very high quantisation levels values, there are too many possible bins, and as such nominally similar colours will be sorted into different bins. Since each bin is a different dimension in the descriptor, they will not therefore be considered similar by the distance measure function.\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/9_23_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/global_colour_hist/9_23_s/pr_2.txt}{Q=2}\n\t\t\\prplotadd{data/global_colour_hist/9_23_s/pr_3.txt}{Q=3}\n\t\t\\prplotadd{data/global_colour_hist/9_23_s/pr_4.txt}{Q=4}\n\t\t%\\prplotadd{data/global_colour_hist/9_23_s/pr_5.txt}{Q=5}\n\t\t\\prplotadd{data/global_colour_hist/9_23_s/pr_6.txt}{Q=6}\n\t\t%\\prplotadd{data/global_colour_hist/9_23_s/pr_7.txt}{Q=7}\n\t\t\\prplotadd{data/global_colour_hist/9_23_s/pr_8.txt}{Q=8}\n\t\t%\\prplotadd{data/global_colour_hist/9_23_s/pr_9.txt}{Q=9}\n\t\t\\prplotadd{data/global_colour_hist/9_23_s/pr_10.txt}{Q=10}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Global colour histogram results for query image 9\\_23\\_s}\n\t\\label{fig:colour-hist-sheep}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/13_1_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/global_colour_hist/13_1_s/pr_2.txt}{Q=2}\n\t\t\\prplotadd{data/global_colour_hist/13_1_s/pr_3.txt}{Q=3}\n\t\t\\prplotadd{data/global_colour_hist/13_1_s/pr_4.txt}{Q=4}\n\t\t%\\prplotadd{data/global_colour_hist/13_1_s/pr_5.txt}{Q=5}\n\t\t\\prplotadd{data/global_colour_hist/13_1_s/pr_6.txt}{Q=6}\n\t\t%\\prplotadd{data/global_colour_hist/13_1_s/pr_7.txt}{Q=$}\n\t\t\\prplotadd{data/global_colour_hist/13_1_s/pr_8.txt}{Q=8}\n\t\t%\\prplotadd{data/global_colour_hist/13_1_s/pr_9.txt}{Q=9}\n\t\t\\prplotadd{data/global_colour_hist/13_1_s/pr_10.txt}{Q=10}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Global colour histogram results for query image 13\\_1\\_s}\n\t\\label{fig:colour-hist-books}\n\\end{figure}\n\nThe results shown in Figures \\ref{fig:colour-hist-sheep} and \\ref{fig:colour-hist-books} match expectations suggesting that a very low quantisation value does not deliver good results, yet performance also drops off at higher quantisation levels. Both test images suggest that a quantisation level of $Q=4$, yielding $64$ bins, gives the best result, since the line joining these data points is closest to the top right hand corner.\n\nThe results also show that query image 13\\_1\\_s performs better than 9\\_23\\_s. The reason for this is likely due to the fact that 13\\_1\\_s has a much more varied colour distribution, allowing it to have a much more unique histogram signature. This means that similar images can be better matched. In contrast to this, the green field will dominate the histogram of image 9\\_23\\_s meaning the system may confuse it with other images containing a green background, such as those of other farm animals in a field.\n\n\\FloatBarrier\n\\section{Gridding} \\label{sec:gridding-results}\n\nAs discussed in Section \\ref{sec:gridding-concatenation}, the independent parameters for the gridding function can be reduced to a single quantisation factor $Q$. In a similar manner to that of Section \\ref{sec:global-colour-histogram-results} the optimum quantisation factor will be determined by computing descriptors for various values of $Q$ and then, using the two test images, determining which has best results. \n\nThe same effect of increasing Quantisation factor as experienced with the global colour histogram is expected. Increasing the value should increase performance up to a maximum, at this point the performance should decrease with increasing quantisation factor. One reason for this is that tighter grids mean that small movements in the object could potentially result in the image being quantised in a very different way, meaning visually similar images would be distant from each other in the quantised space.\n\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/9_23_s}\n\t\t\\subcaption{Query image}\n\t\t\\label{fig:grid-color-sheep-img}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/gridding/color_hist/9_23_s/pr_1.txt}{Q=1}\n\t\t\\prplotadd{data/gridding/color_hist/9_23_s/pr_2.txt}{Q=2}\n\t\t\\prplotadd{data/gridding/color_hist/9_23_s/pr_3.txt}{Q=3}\n\t\t\\prplotadd{data/gridding/color_hist/9_23_s/pr_4.txt}{Q=4}\n\t\t\\prplotadd{data/gridding/color_hist/9_23_s/pr_8.txt}{Q=8}\n\t\t\\prplotadd{data/gridding/color_hist/9_23_s/pr_15.txt}{Q=15}\n\t\t\\prplotadd{data/global_colour_hist/9_23_s/pr_4.txt}{No grid}\n\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\t\\label{fig:grid-color-sheep-graph}\n\t\\end{minipage}\n\t\\caption{Gridded global colour histogram results for query image 9\\_23\\_s}\n\t\\label{fig:grid-color-sheep}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/13_1_s}\n\t\t\\subcaption{Query image}\n\t\t\\label{fig:grid-color-books-img}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/gridding/color_hist/13_1_s/pr_1.txt}{Q=1}\n\t\t\\prplotadd{data/gridding/color_hist/13_1_s/pr_2.txt}{Q=2}\n\t\t\\prplotadd{data/gridding/color_hist/13_1_s/pr_3.txt}{Q=3}\n\t\t\\prplotadd{data/gridding/color_hist/13_1_s/pr_4.txt}{Q=4}\n\t\t\\prplotadd{data/gridding/color_hist/13_1_s/pr_8.txt}{Q=8}\n\t\t\\prplotadd{data/gridding/color_hist/13_1_s/pr_15.txt}{Q=15}\n\t\t\\prplotadd{data/global_colour_hist/13_1_s/pr_4.txt}{No grid}\n\t\t\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\t\\label{fig:grid-color-books-graph}\n\t\\end{minipage}\n\t\\caption{Gridded global colour histogram results for query image 13\\_1\\_s}\n\t\\label{fig:grid-color-books}\n\\end{figure}\n\nFigures \\ref{fig:grid-color-sheep} and \\ref{fig:grid-color-books} show the results of the gridding process.\n\nFigure \\ref{fig:grid-color-sheep} shows little variation across gridding quantisation levels, this is likely due to the fact that the image has only two dominant colours, so all grids have similar colour histograms. Gridding does, in the general case, show slight performance improvement with this image however. Notably increasing precision values for higher recall levels.\n\nFigure \\ref{fig:grid-color-books} on the other hand shows great variation in the result with grid size. Increasing the grid quantisation level to 2 greatly increases the performance up to a recall of around 0.4, and a grid quantisation level of 1 also increases the performance by a large amount also. In addition to this, much higher quantisation steps, such as 8 and 15 show greatly decreased performance.\n\nThe optimum quantisation factor is therefore $Q=2$, yielding \\texttt{h\\_level = 3}, and \\texttt{v\\_level = 2} using the formulae defined in Section \\ref{sec:gridding-concatenation}.\n\n\n\\FloatBarrier\n\\section{Edge Orientation Histogram} \\label{sec:texture-histogram-results}\nAs discussed in Section \\ref{sec:texture-histogram}, the gridded edge orientation histogram implementation has two variable parameters, $E$ and $Q$ (setting the grid quantisation factors to the optimised values from Section \\ref{sec:gridding-results}). Evaluating the effect of changing both together is computationally expensive, and the results are hard to visualise, since a 3-axis plot would be required. Therefore the two variables will be evaluated separately.\n\nInitially the effect of changing the histogram quantisation level, $Q$, will be investigated using a constant $E$ of $0$. This will leave all edges in the image for the consideration of the histogram. The expected result is similar to increasing the quantisation level of the RGB histogram -- increasing $Q$ will increase performance up to a point, but increasing past that point will result in poorer results\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/9_23_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/grid_text_hist/E_0/9_23_s/pr_4.txt}{Q=4}\n\t\t\\prplotadd{data/grid_text_hist/E_0/9_23_s/pr_8.txt}{Q=8}\n\t\t\\prplotadd{data/grid_text_hist/E_0/9_23_s/pr_10.txt}{Q=10}\n\t\t\\prplotadd{data/grid_text_hist/E_0/9_23_s/pr_12.txt}{Q=12}\n\t\t\\prplotadd{data/grid_text_hist/E_0/9_23_s/pr_15.txt}{Q=16}\n\t\t\\prplotadd{data/grid_text_hist/E_0/9_23_s/pr_18.txt}{Q=18}\n\t\t\\prplotadd{data/grid_text_hist/E_0/9_23_s/pr_20.txt}{Q=20}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Gridded texture histogram results for query image 9\\_23\\_s, with $E=0$}\n\t\\label{fig:grid-text-sheep}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/13_1_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/grid_text_hist/E_0/13_1_s/pr_4.txt}{Q=4}\n\t\t\\prplotadd{data/grid_text_hist/E_0/13_1_s/pr_8.txt}{Q=8}\n\t\t\\prplotadd{data/grid_text_hist/E_0/13_1_s/pr_10.txt}{Q=10}\n\t\t\\prplotadd{data/grid_text_hist/E_0/13_1_s/pr_12.txt}{Q=12}\n\t\t\\prplotadd{data/grid_text_hist/E_0/13_1_s/pr_15.txt}{Q=16}\n\t\t\\prplotadd{data/grid_text_hist/E_0/13_1_s/pr_18.txt}{Q=18}\n\t\t\\prplotadd{data/grid_text_hist/E_0/13_1_s/pr_20.txt}{Q=20}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Gridded texture histogram results for query image 13\\_1\\_s, with $E=0$}\n\t\\label{fig:grid-text-books}\n\\end{figure}\n\nFigure \\ref{fig:grid-text-sheep} shows that for the sheep image, there is very little difference in performance with quantisation levels greater than approximately 4. Figure \\ref{fig:grid-text-books} shows that for the books, on the other hand, performance is strongly dependant on quantisation level. Higher quantisation levels continue to improve performance, up to approximately $Q=16$, after this point performance decreases.\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/9_23_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/grid_text_hist/Q_16/9_23_s/pr_0.txt}{E=0}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_1.txt}{E=0.1}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_2.txt}{E=0.2}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_3.txt}{E=0.3}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_4.txt}{E=0.4}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_5.txt}{E=0.5}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_6.txt}{E=0.6}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_7.txt}{E=0.7}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_8.txt}{E=0.8}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_9.txt}{E=0.9}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/9_23_s/pr_10.txt}{E=1.0}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Gridded texture histogram results for query image 9\\_23\\_s, with $Q=16$}\n\t\\label{fig:grid-text-sheep-q16}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/13_1_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/grid_text_hist/Q_16/13_1_s/pr_0.txt}{E=0}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_1.txt}{E=0.1}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_2.txt}{E=0.2}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_3.txt}{E=0.3}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_4.txt}{E=0.4}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_5.txt}{E=0.5}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_6.txt}{E=0.6}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_7.txt}{E=0.7}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_8.txt}{E=0.8}\n\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_9.txt}{E=0.9}\n%\t\t\\prplotadd{data/grid_text_hist/Q_16/13_1_s/pr_10.txt}{E=1.0}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Gridded texture histogram results for query image 13\\_1\\_s, with $Q=16$}\n\t\\label{fig:grid-text-books-q16}\n\\end{figure}\n\nFigures \\ref{fig:grid-text-sheep-q16} and \\ref{fig:grid-text-books-q16} show the result of varying $E$, whilst keeping $Q$ at a constant value of $16$. The results are interesting. For the bookshelves, increasing $E$ above $0$ improves performance up to approximately $E=0.3$, however, for the sheep increasing $E$ above $0$ greatly hinders performance. This is likely due to the fact that whilst the books have lots of strong edges (at the edges of the books), which will remain with increasing values of $E$, the sheep image likely lacks these strong edges, and so increasing $E$ removes almost all of the relevant edges from the generated descriptor.\n\n\\FloatBarrier\n\\section{Concatenation of Descriptors} \\label{sec:conc-desc-results}\nSections \\ref{sec:gridding-results} and \\ref{sec:texture-histogram-results} demonstrate the effectiveness of gridded colour histogram and texture descriptors. It stands to reason therefore that these descriptors could be combined to produce a descriptor of even greater effectiveness. This is possible using the methodology described in Section \\ref{sec:conc-desc}.\n\nIn order to combine the gridded texture and gridded global colour histogram results, the parameters of these functions which were found to be optimal in Sections \\ref{sec:gridding-results} and \\ref{sec:texture-histogram-results} will be used. The only free parameter of the concatenation function is therefore $M$. $M$ is a multiplier that is applied to the texture function in order to adjust the weighting between the texture and colour descriptors. When $M$ is $1$ the two descriptors are equally weighted, however since the colour descriptor is 4 times as long as the texture descriptor (a 64 bin histogram, vs and 8 bin histogram), it is effectively weighted more highly since there are many more dimensions in which values can be distant. Whilst weighting of $M=4$ should therefore make up for this difference in descriptor length, a different value may provide better results.\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/9_23_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/grid_text_hist/Q_16/9_23_s/pr_0.txt}{Texture only}\n\t\t\\prplotadd{data/gridding/color_hist/9_23_s/pr_2.txt}{Colour only}\n\t\t\\prplotadd{data/concat/9_23_s/pr_1.txt}{W = 1}\n\t\t\\prplotadd{data/concat/9_23_s/pr_4.txt}{W = 4}\n\t\t\\prplotadd{data/concat/9_23_s/pr_10.txt}{W = 10}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Concatenated descriptor results for query image 9\\_23\\_s}\n\t\\label{fig:concat-sheep}\n\\end{figure}\n\n\\begin{figure}[ht]\n\t\\begin{minipage}[]{0.3\\linewidth}\n\t\t\\centering\n\t\t\\includegraphics[width = 0.9\\linewidth]{figs/msrc/13_1_s}\n\t\t\\subcaption{Query image}\n\t\\end{minipage}\n\t\\begin{minipage}[]{0.7\\linewidth}\n\t\t\\centering\n\t\t\\begin{tikzpicture}\n\t\t\\prplot{data/grid_text_hist/Q_16/13_1_s/pr_0.txt}{Texture only}\n\t\t\\prplotadd{data/gridding/color_hist/13_1_s/pr_2.txt}{Colour only}\n\t\t\\prplotadd{data/concat/13_1_s/pr_1.txt}{W = 1}\n\t\t\\prplotadd{data/concat/13_1_s/pr_4.txt}{W = 4}\n\t\t\\prplotadd{data/concat/13_1_s/pr_10.txt}{W = 10}\n\t\t\\prplotclose\n\t\t\\end{tikzpicture}\n\t\t\\subcaption{Resultant graph}\n\t\\end{minipage}\n\t\\caption{Concatenated descriptor results for query image 13\\_1\\_s}\n\t\\label{fig:concat-books}\n\\end{figure}\n\nFigure \\ref{fig:concat-sheep} shows that the concatenated descriptor remains very close to the plot of only the RGB histogram for all values of $W$ plotted. This implies that all of the descriptors are very close in the texture plot, leading to the colour plot dominating the results.\n\nFigure \\ref{fig:concat-books} on the other hand shows that the concatenated descriptor does vary between the two plots with differing values of $W$. This implies that both descriptors are discriminative. The results show that at a value of $W=1$, the results are close to the RGB histogram, but at $W=10$, the results are closer to the edge orientation histogram. A value of $W=4$ provides a compromise between the two as expected. The results are poorer than the RGB histogram for this particular query image, but this is unlikely to hold true for all query images since some images will likely be described by colour well, and others by texture well.\n\n\\section{Overall Results} \\label{sec:overall-results}\n\nFollowing the characterisation each algorithm, discussed in the previous parts of this section, it has been possible to test the performance of each descriptor against the entire database of images. This is achieved by calculating \\gls{map}. \\gls{map} is a single numeric measure for determining the performance of a descriptor across a dataset. Is is the mean of \\gls{ap}, which is defined in Equation \\ref{eq:map}. In this equation P$(i)$ represents the precision at that value of $i$, and Rel$(i)$ is $1$ if the result is relevant, or $0$ if it is not.\n\n\\begin{equation}\n\\text{\\glsentryshort{ap}} = \\frac{\\sum_{i=1}^{n} \\left( \\text{P}(i) \\times \\text{Rel}(i) \\right) }{\\sum_{i=1}^{n} \\text{Rel}(i) } \\label{eq:map}\n\\end{equation}\n\nTable \\ref{tbl:map-stats} tabulates the \\gls{map} statistics for all images in the dataset, both within each category, as well as globally. These results show \\gls{map} across the entire dataset, so a lot of the numbers are low. The reason for this is that whilst the images have similar Level 2 content, some of the images are very different in terms of colour and texture content. In order to account for this, Table \\ref{tbl:map-stats-at-10} tabulates \\gls{map} statistics for the first 10 relevant recalled images. This data removes the effect of the aforementioned `outlier' images in each category, and better demonstrates how well each type of image content responds to each descriptor.\n\nThe results in Table \\ref{tbl:map-stats-at-10} show that the image sets respond well to some descriptors, and some respond poorly. One example of this is category 4, where the \\gls{map} is 0.71 for the edge orientation histogram, but 0.25 and 0.44 for the global colour histogram and gridded global colour histogram. This can be explained by looking at the type of image category 4 is. Category 4 contains images of aeroplanes. Almost all of these images have a smooth aeroplane body in the middle of the image, with a clear sky at the top, and plain grass at the bottom. This distinctive texture signature is what allows this high \\gls{ap} figure. Other image sets which respond particularly well to the edge orientation histogram are categories 7 and 13 respectively. Category 7 contains cars, and category 13 books on shelves, both of these have distinct texture signatures.\n\nIn general, most of the categories responded better to the gridded colour histogram than the global colour histogram. This is expected since encoding the spatial aspect of the colour data should allow for a more discriminative descriptor. A notable example which does not is category 18. Category 18 is a series of images which all contain water, however the water is in a different location in almost every image, this kind of spatial variance cannot be encoded in the gridded descriptor, so it performs poorly.\n\nTwo categories which responded particularly well to the colour descriptor are categories 2 and 13. These are images of trees, mostly on a background of a washed-out sky, and bookshelves of colourful books respectively. Both images likely work well with this descriptor because they contain a variety of colours which are both common across most of the images, and distinctive in the image set itself. \n\nIn general, concatenation of colour and texture descriptors improved performance over the individual descriptors. The cases where they did not are cases where one descriptor had a much greater performance over the other, such as category 4, which responded particularly well to the texture descriptor and category 2 which responded particularly well to the colour descriptors. One example of a category which showed great improvement with descriptor concatenation are categories 7 and 10. These categories contain differently coloured cars in different orientations and differently coloured flowers, with varying shapes. Concatenation helps with the cars because the variance of colour across the set means that the similarly coloured cars strongly match each other in a colour histogram, but all others are distant, whereas all of the cars are likely moderate matches with the edge orientation histogram. The flowers have a similar effect.\n\nOne category which performed particularly poorly in all cases is category 16. This category contains photographs of dogs, of various sizes and colours, all taken from different camera angles and on different backgrounds. This is an example of Level 2 concepts which do not translate well at all to Level 1 attempts at interpretation.\n\n\n\\begin{table}[ht]\n\t\\rowcolors{2}{gray!25}{white}\n\t\\caption{\\glsentryshort{map} statistics for all methods and categories}\n\t\\label{tbl:map-stats}\n\t\\centering\n\t\\pgfplotstableset{ % Makes bold a whole row of a table\n\t\thighlightrow/.style={\n\t\t\tpostproc cell content/.append code={\n\t\t\t\t\\count0=\\pgfplotstablerow\n\t\t\t\t\\advance\\count0 by1\n\t\t\t\t\\ifnum\\count0=#1\n\t\t\t\t\\pgfkeysalso{@cell content/.add={$\\bf}{$}}\n\t\t\t\t\\fi\n\t\t\t},\n\t\t},\n\t}\n\t\\pgfplotstabletypeset[\n\tcol sep=comma,\n\tstring type,\n\tcolumns/set/.style={column name={\\textbf{Image category}}, column type={r}},\n\tcolumns/gch/.style={column name={\\textbf{\\shortstack{Global colour\\\\histogram}}}, column type={r}},\n\tcolumns/gridch/.style={column name={\\textbf{\\shortstack{Gridded\\\\ colour\\\\histogram}}}, column type={r}},\n\tcolumns/eoh/.style={column name={\\textbf{\\shortstack{Gridded\\\\edge orienatation\\\\histogram}}}, column type={r}},\n\tcolumns/concat/.style={column name={\\textbf{\\shortstack{Concatenated\\\\descriptors}}}, column type={r}},\n\tevery head row/.style={before row=\\toprule, after row=\\midrule},\n\tevery last row/.style={after row=\\bottomrule},\n\thighlightrow={21},\n\t]{data/global_stats/map.txt}\n\\end{table}\n\n\\begin{table}[ht]\n\t\\rowcolors{2}{gray!25}{white}\n\t\\caption{\\glsentryshort{map} statistics for first 10 images in each set, over all methods and categories}\n\t\\label{tbl:map-stats-at-10}\n\t\\centering\n\t\\pgfplotstableset{ % Makes bold a whole row of a table\n\t\thighlightrow/.style={\n\t\t\tpostproc cell content/.append code={\n\t\t\t\t\\count0=\\pgfplotstablerow\n\t\t\t\t\\advance\\count0 by1\n\t\t\t\t\\ifnum\\count0=#1\n\t\t\t\t\\pgfkeysalso{@cell content/.add={$\\bf}{$}}\n\t\t\t\t\\fi\n\t\t\t},\n\t\t},\n\t}\n\t\\pgfplotstabletypeset[\n\tcol sep=comma,\n\tstring type,\n\tcolumns/set/.style={column name={\\textbf{Image category}}, column type={r}},\n\tcolumns/gch/.style={column name={\\textbf{\\shortstack{Global colour\\\\histogram}}}, column type={r}},\n\tcolumns/gridch/.style={column name={\\textbf{\\shortstack{Gridded\\\\ colour\\\\histogram}}}, column type={r}},\n\tcolumns/eoh/.style={column name={\\textbf{\\shortstack{Gridded\\\\edge orienatation\\\\histogram}}}, column type={r}},\n\tcolumns/concat/.style={column name={\\textbf{\\shortstack{Concatenated\\\\descriptors}}}, column type={r}},\n\tevery head row/.style={before row=\\toprule, after row=\\midrule},\n\tevery last row/.style={after row=\\bottomrule},\n\thighlightrow={21},\n\t]{data/global_stats/map_at_10.txt}\n\\end{table}\n\n\\chapter{Conclusion} \\label{sec:conclusion}\nThis report presents several methods of generating descriptors for images, and uses test data to set their adjustable parameters, the results of which is presented in Section \\ref{sec:results}. The overall results show that the \\gls{matlab} system is capable of performing with a reasonable \\gls{map} over the first 10 images in each set. However performance does suffer when considered over the whole dataset. Overall the concatenated descriptors perform best over the entire dataset with a \\gls{map} of 0.21, or 0.36 if only the first 10 images are considered. A global colour histogram performs most poorly overall with a \\gls{map} of 0.16 over the entire dataset, or 0.26 if only the first 10 images are considered.\n\nThe Appendices present further results which extend the investigation in areas other than investigation of different descriptors. Appendix \\ref{app:pca} investigates \\gls{pca} and shows how descriptors can be made much more compact without impact on performance. The investigated descriptor, a gridded colour histogram, was able to be reduced to 26\\% of it's original size without noticeable performance impact.\n\nAppendix \\ref{app:dist-measures} investigates various distance measures which can be used to compare descriptors and tests their relative performance. The $L_2$ norm, which has been used throughout the remainder of this investigation was found to be the strongest overall performer.\n\nIn addition Appendix \\ref{app:test-methodology} presents the test methodology for verification of the functionality of the descriptor generation functions.\n\nTo conclude the system presented by the report is a modular and extensible visual search system, with characterised performance, and scope to be extended beyond its current functionality. ", "meta": {"hexsha": "f275e3df13df30f69c79fc874f5fc60452413810", "size": 42246, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/docs/body.tex", "max_stars_repo_name": "joshtyler/visual-search", "max_stars_repo_head_hexsha": "63fd7512f18596d5bd9475c7624b631a2205bcb8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2016-12-05T13:05:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-13T22:42:39.000Z", "max_issues_repo_path": "docs/docs/body.tex", "max_issues_repo_name": "joshtyler/visual-search", "max_issues_repo_head_hexsha": "63fd7512f18596d5bd9475c7624b631a2205bcb8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/docs/body.tex", "max_forks_repo_name": "joshtyler/visual-search", "max_forks_repo_head_hexsha": "63fd7512f18596d5bd9475c7624b631a2205bcb8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-02-26T21:33:35.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-26T21:33:35.000Z", "avg_line_length": 73.2166377816, "max_line_length": 958, "alphanum_fraction": 0.7682384131, "num_tokens": 12132, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041652, "lm_q2_score": 0.6825737473266735, "lm_q1q2_score": 0.34928432044220414}} {"text": "\\section*{Representing Time}\r\n\r\n\r\n\\subsection*{\\\\\r\nThe underspecification of time representations in OWL axioms}\r\n\r\nBoth an axiom in the OBO representational syntax \\todo{more on OBO} \r\n%\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mclass{CellMembrane}\\;\\mirel{continuant\\_part\\_of}\\;\\mclass{Cell}\r\n\\end{split}\r\n\\label{eq:cellobo}\r\n\\end{equation}\r\n%\r\nand the OWL axiom (often considered equivalent)\r\n%\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mclass{CellMembrane}\\;\\mathtt{subClassOf}\\;\\mclass{continuant\\_part\\_of}\\;\\mathtt{some}\\;\\mclass{Cell}\r\n\\end{split}\r\n\\label{eq:celldl}\r\n\\end{equation}\r\n%\r\ndo not exhibit any explicit commitment with regard to time. This is not a side issue, because it makes a\r\ndifference whether, for example, a cell membrane is \\emph{always} part of some cell or\r\nonly \\emph{at some time}. \r\nThe lack of temporal definition of OWL statements is especially unsatisfactory\r\nwhen it comes to transitive properties like $\\mirel{continuant\\_part\\_of}$ or $\\mirel{located\\_in}$,\r\nwhere the suppression of the temporal factor can produce plainly wrong entailments,\r\nespecially at the level of individuals, such as in the following example:\r\n%\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mirel{located\\_in} (\\mirel{Thrombus\\#398}, \\mirel{Heart\\#431}) \\\\\r\n\\mirel{located\\_in} (\\mirel{Heart\\#431}, \\mirel{Patient\\#900})\r\n\\end{split}\r\n\\label{eq:trans}\r\n\\end{equation}\r\n%\r\nIf we now assume that \\mirel{Heart\\#431} is later transplanted to \\mirel{Patient\\#115}, we also have\r\n\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mirel{located\\_in} (\\mirel{Heart\\#431}, \\mirel{Patient\\#115})\r\n\\end{split}\r\n\\label{eq:trans}\r\n\\end{equation}\r\n\r\nThis would entail that one of the patients is located within the other, which is plainly wrong. \r\nAlso, if we assume that the thrombus was no longer in the heart when it was transplanted to Patient\\#115, then the inference to\r\n%\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mirel{located\\_in} (\\mirel{Thrombus\\#398}, \\mirel{Patient\\#115})\r\n\\end{split}\r\n\\label{eq:transEntailment}\r\n\\end{equation}\r\n%\r\nwhich follows from the transitivity of the relation $\\mirel{located\\_in}$, is obviously invalid.\r\n\r\nThis has a far-reaching consequence: by default, OWL statements are temporally ambiguous in a way that may entail unintended consequences when domain ontologies are used for reasoning with real data. (A lack of an explicit treatment of temporal context has previously been implicated in an assessment of the quality of existential restrictions in OBO Foundry ontologies \\cite{boeker2011}.)\r\n\r\n\\subsection*{Strengths of Relatedness}\r\n\r\nWe can now propose a distinction between different temporal \\emph{strengths} of relatedness. These are to be satisfied by temporally qualified such as parthood defined in the OBO Relation Ontology and apply to relations between continuants only. \r\nWe begin by characterising the ontological status of regions of time , henceforth referred to by the symbol $t$. \r\nTimes are either time intervals or time points. %, assuming that the latter can be approximated by infinitesimally small intervals.\r\nWe restrict ourselves to those ternary relations for which it can be assumed that whenever they hold for a time interval, \r\nthey also hold for any of its subintervals, including time points. For example, if a car is located in a garage during a day, then it is located therein at any time interval during that day. If a patient's cholesterol level is elevated for a whole year, it is elevated over any duration of time during this year. \r\nIn the following, ternary relations are marked by the superscript $^t$, like in $\\mrelt{rel}$, from which binary relations are distinguished by\r\n$^b$, like in $\\mrelb{rel}$. \r\n\r\nFormally:\r\n\\begin{equation}\r\n\\begin{split}\r\n\\forall a, b, t, t^\\prime:\\; \\mrelt{rel}(a, b, t) \\wedge \\mirel{within}(t^\\prime,t) \\rightarrow\r\n\\mrelt{rel}(a,b,t^\\prime)\r\n\\end{split}\r\n\\label{eq:temporarily:temp}\r\n\\end{equation}\r\n\r\nThese are called time-distributive relations.\r\n\r\n\\subsubsection*{Temporary Generic Relatedness (TGR)}\r\n\r\nInformally: for all instances $a$ of \\mclass{A} there is some time $t$ and some instance $b$ of\r\n\\mclass{B} such that $a$ is related to $b$ at $t$. Examples:\r\n\\begin{enumerate}[(a)]\r\n\\item for all apple seeds there is\r\nsome apple such that the seed is part of the apple at some time;\r\n\\item for all\r\ntrees there is some leaf such that the leaf is part of the tree at some time.\r\n\\end{enumerate}\r\n\r\nFormally:\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mclass{TemporarilyRelated}(\\mclass{A},\\mclass{B}) =_{def}&\\;\r\n\\forall a, t:\\; \\mrelt{inst}(\\mclass{A}, a, t) \\\\\r\n&\\ \\rightarrow\r\n\\exists b, t^\\prime:\\;(\\mrelt{inst}(\\mclass{B},b,t^\\prime) \\wedge\r\n\\mrelt{rel}(a,b,t^\\prime) \\wedge \\mirel{within}(t^\\prime,t))\r\n\\end{split}\r\n\\label{eq:temporarily:cls}\r\n\\end{equation}\r\n\r\n\\subsubsection*{Permanent Generic Relatedness (PGR)}\r\n\r\nInformally: for all instances $a$ of \\mclass{A} there is, at all times $t$ for which\r\n$a$ exists,\r\nsome instance $b$ of \\mclass{B} such that $a$ is related to $b$ at $t$, but not necessarily\r\nalways the same $b$ at all times $t$. Examples:\r\n\\begin{enumerate}[(a)]\r\n\\item all cells have a water molecule as\r\npart at all times, but not always the same water molecule;\r\n\\item every bacterial colony has some bacteria as parts at all times, but not\r\nalways the same bacteria.\r\n\\end{enumerate}\r\n\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mclass{PermanentlyGenericallyRelated}(\\mclass{A},\\mclass{B}) =_{def}&\\;\r\n\\forall a, t:\\; \\mrelt{inst}(\\mclass{A}, a, t) \\\\\r\n&\\ \\rightarrow\r\n\\exists b:\\;(\\mrelt{inst}(\\mclass{B},b,t) \\wedge\r\n\\mrelt{rel}(a,b,t))\r\n\\end{split}\r\n\\label{eq:generically:cls}\r\n\\end{equation}\r\n\r\nIt is as PGR relations that the standard interpretation of OBO binary class-level relations are understood \r\nsuch as in formula (\\ref{eq:cellobo}) above, according to \\cite{OBO:RO}.\r\n\r\n\\subsubsection*{Permanent Specific Relatedness (PSR)}\r\n\r\nInformally, for all instances $a$ of \\mclass{A} there is, at all times $t$ that $a$ exists, an\r\ninstance $b$ of \\mclass{B} such that $a$ is related to $b$ at $t$; in this case it is the\r\nsame $b$ at all times $t$. Examples:\r\n\\begin{enumerate}[(a)]\r\n\\item a human being has a brain as part at all times, and it is necessarily the same brain;\r\n\\item a radioactively marked molecule of DNA has the radioactive isotope as part\r\nat all times, and it is necessarily the same radioactive isotope.\r\n\\end{enumerate}\r\n\r\n\\begin{equation}\r\n\\begin{split}\r\n\\mclass{Permanently}&\\mclass{SpecificallyRelated}(\\mclass{A},\\mclass{B}) =_{def}\\;\r\n\\forall a, t:\\; \\mrelt{inst}(\\mclass{A}, a, t) \\\\\r\n&\\ \\rightarrow\r\n\\exists b:\\;\\big(\\mrelt{inst}(\\mclass{B},b,t) \\wedge\r\n\\mrelt{rel}(a,b,t))\r\n\\\\\r\n&\\quad\\quad \\wedge \\forall t^\\prime: (\\mrelt{inst}(\\mclass{A},a,t^\\prime)\r\n\\rightarrow (\\mrelt{rel}(a,b,t^\\prime) \\wedge\r\n\\mrelt{inst}(\\mclass{B},b,t^\\prime))\\big)\r\n\\end{split}\r\n\\label{eq:specifically:cls}\r\n\\end{equation}\r\n", "meta": {"hexsha": "28e5b7b3a916dbf5edb2293413e0533863aa69dc", "size": 6840, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/OWL-TIME/bfo-owl-time-problem.tex", "max_stars_repo_name": "zhengj2007/bfo", "max_stars_repo_head_hexsha": "bf988c5747ff1ac517eeb55a534cab2f62ee52e2", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/OWL-TIME/bfo-owl-time-problem.tex", "max_issues_repo_name": "zhengj2007/bfo", "max_issues_repo_head_hexsha": "bf988c5747ff1ac517eeb55a534cab2f62ee52e2", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 193, "max_issues_repo_issues_event_min_datetime": "2015-07-23T20:52:43.000Z", "max_issues_repo_issues_event_max_datetime": "2015-07-24T02:41:37.000Z", "max_forks_repo_path": "docs/OWL-TIME/bfo-owl-time-problem.tex", "max_forks_repo_name": "zhengj2007/bfo", "max_forks_repo_head_hexsha": "bf988c5747ff1ac517eeb55a534cab2f62ee52e2", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.75, "max_line_length": 390, "alphanum_fraction": 0.7257309942, "num_tokens": 2047, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3492843138337172}} {"text": "\\section{Results and discussions}\r\n\\label{section:results_and_discussions}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nIn this section, five DL models of semantic segmentation approach including Res-UNet, VGG16 encoder-decoder, FCN-DenseNet, PSPNet, and GCN were evaluated on exemplary three damage scenarios of an RMS of the numerically calculated full wavefield interpolated at the bottom surface of the plate in order to identify the delamination.\r\nAdditionally, an experimental scenario was also used to evaluate the performance of the models to show the DL capabilities of generalization.\r\nFor each model, the mean and the max \\(IoU\\) are calculated and presented as a metric of comparison.\r\nMoreover, the Accuracy of classification, Precision, Recall, and F1 score depicted in Eqs~(\\ref{accuracy}-\\ref{f1_score}) respectively, were calculated for each model.\r\n\\begin{equation}\r\n\t\\rm Accuracy\\ = \\frac{TP+TN}{(Total\\ number\\ of\\ tested\\ samples)}\r\n\t\\label{accuracy}\r\n\\end{equation}\r\n\r\n\\begin{equation}\r\n\t\\rm Pricision\\ =\\frac{TP}{TP+FP}\r\n\t\\label{pricisoin}\r\n\\end{equation}\r\n\\begin{equation}\r\n\t\\rm Recall\\ = \\frac{TP}{TP+ FN}\r\n\t\\label{recall}\r\n\\end{equation}\r\n\\begin{equation}\r\n\t\\rm F1 score =\\frac{2 \\times (Pricision\\times Recall)}{(Pricision + Recall)} \r\n\t\\label{f1_score}\r\n\\end{equation}\r\nwhere the Positive/Negative refers to the predicted output as (damage or non-damage) respectively, True Positive (TP) and True Negative (TN) represent the correct classification, and the False Positive (FP) and False Negative (FN) represent the incorrect classification.\r\n\r\nIn this work, all semantic segmentation models were implemented and trained with Keras API~\\cite{chollet2015keras} running on top of TensorFlow.\r\nFurther, we have replaced the previous NVIDIA RTX2080 GPU utilised in our work~\\cite{Ijjeh2021} with a superior NVIDIA Tesla V100 GPU with \\(32\\)GB memory that enables us to increase the total number of learnable parameters of the DL models that will allow them to learn and extract more features. \r\nFor training purposes, we have used the K-fold cross-validation technique with five folds. \r\nAccordingly, each model has trained for \\(5\\) iterations. \r\nFor each iteration, we calculated the mean accuracy of all folds.\r\nFurther, the number of epochs equals \\(20\\) per iteration.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\subsection{Numerical scenarios}\r\nIn the following, three scenarios of numerical data regarding delamination of different locations, shapes, and angles are presented. \r\nThe first exemplary delamination scenario is shown in Fig.~\\ref{fig:softmax_448}. \r\nThe delamination is located at the left edge of the plate, and it is surrounded by a line to represent its shape and location as shown in Fig.~\\ref{fig:RMS_flat_shell_Vz_448}.\r\nFigures~\\ref{fig:unet_pred_448}-~\\ref{fig:gcn_pred_448} show the predicted output of the Res-UNet, VGG16 encoder-decoder, PSPNet, FCN-DenseNet and GCN models, respectively. \r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nAll models properly indicate the location of the delamination. \r\nMoreover, pixels related to delamination location are clustered into a single spot without any additional noise. \r\nVGG16 encoder-decoder and GCN models better visually resemble the actual shape of the delamination than other models.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\begin{figure}[!h]\r\n\t\\centering\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{RMS_flat_shell_Vz_448_with_square_label.png}\r\n\t\t\\caption{RMS bottom with label}\r\n\t\t\\label{fig:RMS_flat_shell_Vz_448}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[scale=1.0]{m1_rand_single_delam_448.png}\r\n%\t\t\\caption{Ground truth}\r\n%\t\t\\label{fig:m1_rand_single_delam_448}\r\n%\t\\end{subfigure}\r\n%\t\\hfil\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{residual_unet_num_269.png}\r\n\t\t\\caption{Res-UNet}\r\n\t\t\\label{fig:unet_pred_448}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{VGG16_ecoder_decoder_num_269.png}\r\n\t\t\\caption{VGG16 encoder-decoder}\r\n\t\t\\label{fig:vgg16_pred_448}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{PSPNet_num_269.png}\r\n\t\t\\caption{PSPNet}\r\n\t\t\\label{fig:pspnet_pred_448}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{FCN_DenseNet_num_269.png}\r\n\t\t\\caption{FCN-DenseNet}\r\n\t\t\\label{fig:fcn_densenet_pred_448}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{GCN_num_269.png}\r\n\t\t\\caption{GCN}\r\n\t\t\\label{fig:gcn_pred_448}\r\n\t\\end{subfigure}\r\n\t\\caption{First delamination scenario based on numerical data}\r\n\t\\label{fig:softmax_448}\r\n\\end{figure} \r\n\\clearpage\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nIn the second delamination scenario, shown in Fig.~\\ref{fig:385_softmax}, the delamination is located at the upper left corner of the plate, and it is surrounded by an ellipse to represent its shape and location as shown in Fig.~\\ref{fig:RMS_flat_shell_Vz_385}.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nThis is the most challenging damage scenario because of reflections coming from plate edges overshadows reflections from damage.\r\nAs a result, changes in RMS patterns are barely visible.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nFigures~\\ref{fig:Unet_Pred__softmax_385} -~\\ref{fig:gcn_pred_385} show the predicted output of the Res-UNet, VGG16 encoder-decoder, PSPNet, FCN-DenseNet and GCN models, respectively. \r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nAll models perform reasonably well, considering the difficult damage scenario.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\begin{figure}[!h]\r\n\t\\centering\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{RMS_flat_shell_Vz_27_385_with_label.png}\r\n\t\t\\caption{RMS bottom with label}\r\n\t\t\\label{fig:RMS_flat_shell_Vz_385}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[scale=1.0]{m1_rand_single_delam_385.png}\r\n%\t\t\\caption{Ground truth}\r\n%\t\t\\label{fig:m1_rand_single_delam_385}\r\n%\t\\end{subfigure}\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{residual_unet_num_17.png}\r\n\t\t\\caption{Res-UNet}\r\n\t\t\\label{fig:Unet_Pred__softmax_385}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{VGG16_ecoder_decoder_num_17.png}\r\n\t\t\\caption{VGG16 encoder-decoder}\t\t\t\\label{fig:vgg16_pred__softmax_385}\t\t\t\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{PSPNet_num_17.png}\r\n\t\t\\caption{PSPNet}\r\n\t\t\\label{fig:pspnet_pred__softmax_385}\r\n\t\\end{subfigure}\t\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{FCN_DenseNet_num_17.png}\r\n\t\t\\caption{FCN-DenseNet}\r\n\t\t\\label{fig:fcn_densenet_pred__softmax_385}\r\n\t\\end{subfigure}\t\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{GCN_num_17.png}\r\n\t\t\\caption{GCN}\r\n\t\t\\label{fig:gcn_pred_385}\r\n\t\\end{subfigure}\r\n\t\\caption{Second delamination scenario based on numerical data}\r\n\t\\label{fig:385_softmax}\r\n\\end{figure}\r\n\\clearpage\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nThe third delamination scenario is shown in Figure~\\ref{fig:475_softmax}. \r\nThe delamination is located at the upper middle of the plate and it is surrounded by an ellipse to represent its shape and location as shown in Fig.~\\ref{fig:RMS_flat_shell_Vz_475}.\r\nFigures~\\ref{fig:Unet_Pred__softmax_475} -~\\ref{fig:gcn_pred_475} show the predicted output of the Res-UNet, VGG16 encoder-decoder, PSPNet, FCN-DenseNet and GCN models, respectively. \r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nIn this case, the elliptical shape of delamination is best preserved by Res-UNet and FCN-DenseNet. \r\nHowever, the highest value of IoU is obtained by GCN (see Table~\\ref{tab:table_numerical_scenarios}).\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\begin{figure}[!h]\r\n\t\\centering\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{RMS_flat_shell_Vz_27_475_with_label.png}\r\n\t\t\\caption{RMS bottom with label}\r\n\t\t\\label{fig:RMS_flat_shell_Vz_475}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[scale=1.0]{m1_rand_single_delam_475.png}\r\n%\t\t\\caption{Ground truth}\r\n%\t\t\\label{fig:m1_rand_single_delam_475}\r\n%\t\\end{subfigure}\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{residual_unet_num_377.png}\r\n\t\t\\caption{Res-UNet}\r\n\t\t\\label{fig:Unet_Pred__softmax_475}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{VGG16_ecoder_decoder_num_377.png}\r\n\t\t\\caption{VGG16 encoder-decoder}\t\t\t\\label{fig:vgg16_pred__softmax_475}\t\t\t\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{PSPNet_num_377.png}\r\n\t\t\\caption{PSPNet}\r\n\t\t\\label{fig:pspnet_pred__softmax_475}\r\n\t\\end{subfigure}\t\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{FCN_DenseNet_num_377.png}\r\n\t\t\\caption{FCN-DenseNet}\r\n\t\t\\label{fig:fcn_densenet_pred__softmax_475}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1.0]{GCN_num_377.png}\r\n\t\t\\caption{GCN}\r\n\t\t\\label{fig:gcn_pred_475}\r\n\t\\end{subfigure}\t\r\n\t\\caption{Third delamination scenario based on numerical data}\r\n\t\\label{fig:475_softmax}\r\n\\end{figure}\r\n\\clearpage\r\nThe \\(IoU\\) values for all models regarding the predicted delamination are presented in Table.~\\ref{tab:table_numerical_scenarios}.\r\nFor the first and third scenarios, the GCN model has the highest \\(IoU\\) compared to the other models, and for the second scenario, the VGG16 encoder-decoder model has the highest \\(IoU\\) compared to the other models.\r\nFurther, in all models, the predicted outputs have no noise regarding delamination identification.\r\n\\begin{table}[]\r\n\t\\centering\r\n\t\\caption{\\(IoU\\) of Numerical scenarios}\r\n\t\\label{tab:table_numerical_scenarios}\r\n\t\\resizebox{\\textwidth}{!}\r\n\t{\r\n\t\t\\begin{tabular}{cccc}\\hline\r\n\t\t\tModel & 1st scenario & 2nd scenario & 3rd scenario \\\\ \\hline\r\n\t\t\tRes-UNet & \\(0.498\\) & \\(0.782\\) & \\(0.816\\) \\\\ \r\n\t\t\tVGG16 encoder-decoder & \\(0.512\\) & \\(0.787\\) & \\(0.662\\) \\\\\r\n\t\t\tFCN-DenseNet & \\(0.734\\) & \\(0.612\\) & \\(0.866\\) \\\\ \r\n\t\t\tPSPNet & \\(0.389\\) & \\(0.496\\) & \\(0.646\\) \\\\ \r\n\t\t\tGCN & \\(0.791\\) & \\(0.696\\) & \\(0.875\\) \\\\ \\hline\r\n\t\t\\end{tabular}\r\n\t}\r\n\\end{table}\r\nTable~\\ref{tab:table_iou} presents the mean and maximum values of \\(IoU\\) calculated for the previously unseen numerical test set (380 cases) for all models.\r\nTable~\\ref{tab:table_iou} shows that all models have a relatively high \\(IoU\\), indicating their ability to detect and localise the delamination, which is higher compared to the traditional signal processing techniques such as the adaptive wavenumber filtering presented in our previous work~\\cite{Ijjeh2021}. \r\nThe mean \\(IoU\\) for the adaptive wavenumber filtering technique regarding the whole testing samples was \\(0.373\\) compared to the previous FCN-DenseNet model, which has a mean \\(IoU\\) of \\(0.623\\).\r\n\\begin{table}[]\r\n\t\\centering\r\n\t\\caption{Analysis of numerical data}\r\n\t\\label{tab:table_iou}\r\n\t\\begin{tabular}{ccc}\\hline\r\n\t\tModel & mean \\(IoU\\) & max \\(IoU\\) \\\\ \\hline\r\n\t\tRes-UNet & \\(0.664\\) & \\(0.888\\) \\\\ \r\n\t\tVGG16 encoder-decoder & \\(0.572\\) & \\(0.841\\) \\\\ \r\n\t\tFCN-DenseNet & \\(0.680\\) & \\(0.920\\) \\\\ \r\n\t\tPSPNet & \\(0.549\\) & \\(0.914\\) \\\\ \r\n\t\tGCN & \\(0.763\\) & \\(0.931\\) \\\\ \\hline\r\n\t\\end{tabular}\r\n\\end{table}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nFurther, in Table~\\ref{tab:table_performance} the TP, TN, FP, and FN are presented for all models regarding the test set. \r\n\\begin{table}[]\r\n\t\\centering\r\n\t\\caption{Model classification performance}\r\n\t\\label{tab:table_performance}\r\n\t\\resizebox{\\textwidth}{!}\r\n\t{\r\n\t\t\\begin{tabular}{ccccc} \\hline\r\n\t\t\tModel& True Positive & True Negative & False Positive & False Negative \\\\ \\hline\r\n\t\t\tRes-UNet & 376 & 376 & 4 & 0 \\\\ \r\n\t\t\tVGG16 encoder-decoder & 373 & 373 & 7 & 0 \\\\ \r\n\t\t\tFCN-DenseNet & 378 & 378 & 2 & 0 \\\\ \r\n\t\t\tPSPNet & 368 & 368 & 12 & 0 \\\\ \r\n\t\t\tGCN & 380 & 380 & 0 & 0 \\\\ \\hline\r\n\t\t\\end{tabular}\r\n\t}\r\n\\end{table}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nMoreover, Table~\\ref{tab:evaluation_metric} presents the classification accuracy, precision, recall, and the F1-score values for all models as additional evaluation metrics.\r\nAs shown in Table~\\ref{tab:evaluation_metric}, all models have high classification accuracy, which indicates that all the presented models are capable of predicting the presence of the delamination in all the numerically generated cases. \r\nHowever, the best performance was achieved by GCN.\r\n\\begin{table}[]\r\n\t\\centering\r\n\t\\caption{Evaluation metric}\r\n\t\\label{tab:evaluation_metric}\r\n\t\\resizebox{\\textwidth}{!}\r\n\t{\r\n\t\t\\begin{tabular}{ccccc} \\hline\r\n\t\t\tModel& Accuracy & Precision & Recall & F1-Score \\\\ \\hline\r\n\t\t\tRes-UNet & \\(0.994\\) & \\(0.989\\) & \\(1.00\\) & \\(0.994\\) \\\\ \r\n\t\t\tVGG16 encoder-decoder & \\(0.991\\) & \\(0.981\\) & \\(1.00\\) & \\(0.991\\)\\\\ \r\n\t\t\tFCN-DenseNet & \\(0.997\\) & \\(0.994\\) & \\(1.00\\) & \\(0.994\\) \\\\ \r\n\t\t\tPSPNet & \\(0.984\\) & \\(0.968\\) & \\(1.00\\) & \\(0.984\\) \\\\ \r\n\t\t\tGCN & \\(1.00\\) & \\(1.00\\) & \\(1.00\\) & \\(1.00\\) \\\\ \\hline\r\n\t\t\\end{tabular}\r\n\t}\r\n\\end{table}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%Figures~\\ref{fig:res_unet_iou_loss}-\\ref{fig:GCN_iou_loss} show the accuracy and the loss graphs of the training and validation phases during epochs for Res-UNet, VGG16 encoder-decoder, FCN-DenseNet, PSPNet and GCN models, respectively.\r\n%\\begin{figure} [!h]\r\n%\t\\centering\r\n%\t%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t \\centering\t\t\\includegraphics[width=\\textwidth]{Unet_kfold_iou_per_epochs_softmax.png}\t\\caption{}\r\n%\t \\label{fig:unet_accuracy_metric}\r\n%\t\\end{subfigure}\r\n%\t\\hfill\t\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t \\centering\r\n%\t \\includegraphics[width=\\textwidth]{Unet_kfold_loss_per_epochs_softmax.png}\r\n%\t \\caption{}\r\n%\t \\label{fig:unet_loss_metric}\r\n%\t\\end{subfigure}\r\n%\t\\caption{Accuracy and loss of the Res-UNet model}\r\n%\t\\label{fig:res_unet_iou_loss}\r\n%\\end{figure}\r\n%\t%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%\\begin{figure}[!h]\r\n%\t\\centering\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[width=\\textwidth]{FCN_VGG16_iou_per_epochs_softmax.png}\r\n%\t\t\\caption{}\r\n%\t\t\\label{fig:vgg16_accuracy_metric}\r\n%\t\\end{subfigure}\t\t\r\n%\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[width=\\textwidth]{FCN_VGG16_loss_per_epochs_softmax.png}\r\n%\t\t\\caption{}\r\n%\t\t\\label{fig:vgg16_loss_metric}\r\n%\t\\end{subfigure}\r\n%\t\\caption{Accuracy and loss of the VGG16 encoder-decoder model}\r\n%\t\\label{fig:Vgg16_iou_loss}\r\n%\\end{figure}\r\n%\t%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%\\begin{figure}[!h]\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\\centering\r\n%\t\\includegraphics[width=\\textwidth]{FCN_DenseNet_iou_per_epochs_softmax.png}\r\n%\t\\caption{}\r\n%\t\\label{fig:fcn_densenet_accuracy_metric}\r\n%\t\\end{subfigure}\r\n%\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\\centering\r\n%\t\\includegraphics[width=\\textwidth]{FCN_DenseNet_loss_per_epochs_softmax.png}\r\n%\t\\caption{}\r\n%\t\\label{fig:fcn_densenet_loss_metric}\r\n%\t\\end{subfigure}\t\r\n%\\caption{Accuracy and loss of the FCN-DenseNet model}\r\n%\\label{fig:FCN_DenseNet_iou_loss}\r\n%\\end{figure}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%\\begin{figure} [!h]\r\n%\t\\centering\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[width=\\textwidth]{PSPNet_kfold_iou_per_epochs_softmax.png}\r\n%\t\t\\caption{}\r\n%\t\t\\label{fig:psp_accuracy_metric}\r\n%\t\\end{subfigure}\r\n%\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[width=\\textwidth]{PSPNet_kfold_loss_per_epochs_softmax.png}\r\n%\t\\caption{}\r\n%\t\\label{fig:psp_loss_metric}\r\n%\t\\end{subfigure}\r\n%\t\\caption{Accuracy and loss of the PSPNet model}\r\n%\t\\label{fig:PSPNet_iou_loss}\r\n%\\end{figure}\r\n%\t%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%\\begin{figure} [!h]\r\n%\t\\centering\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[width=\\textwidth]{GCN_kfold_iou_per_epochs_softmax.png}\r\n%\t\t\\caption{}\r\n%\t\t\\label{fig:gcn_accuracy_metric}\r\n%\t\\end{subfigure}\r\n%\t\\hfill\r\n%\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n%\t\t\\centering\r\n%\t\t\\includegraphics[width=\\textwidth]{GCN_kfold_loss_per_epochs_softmax.png}\t\t\t\r\n%\t\t\\caption{}\r\n%\t\t\\label{fig:gcn_loss_metric}\r\n%\t\\end{subfigure}\r\n%\t\\caption{Accuracy and loss of the GCN model}\r\n%\t\\label{fig:GCN_iou_loss}\t\r\n%\\end{figure}\r\n%\\clearpage\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\r\nMoreover, the total number of parameters in any DL model is a sum of the trainable parameters (weights of convolution filters) and non-trainable parameters (biases and pooling filters).\r\nTrainable parameters are continuously updated until we reach the minimum loss value, while the non-trainable parameters are not changed during the whole training process.\r\nTable~\\ref{tab:table_parameters} shows the total number of parameters for all implemented models.\r\nFurther, the total number of parameters can reflect the computation complexity of the model.\r\nIt can be noted that as the number of total parameters increases, the required time for training increases.\r\n\\begin{table}[]\r\n\t\\centering\r\n\t\\caption{Model parameters}\r\n\t\\label{tab:table_parameters}\r\n\t\t\\begin{tabular}{cc}\\hline\r\n\t\t\tModel & Total parameters (\\(\\approx\\)) \\\\ \\hline\r\n\t\t\tRes-UNet & \\(52\\times 10^6\\) \\\\ \r\n\t\t\tVGG16 encoder-decoder & \\(37.3\\times 10^6\\) \\\\\r\n\t\t\tFCN-DenseNet & \\(2.5\\times 10^6\\) \\\\ \r\n\t\t\tPSPNet & \\(6.6\\times 10^6\\) \\\\ \r\n\t\t\tGCN & \\(36\\times 10^6\\) \\\\ \\hline\r\n\t\t\\end{tabular}\r\n\\end{table}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\subsection{Experimental scenario}\r\nIn this scenario, an experimental case of CFRP with Teflon insert as artificial delamination is investigated presented in Fig.~\\ref{fig:Exp_ERMS_teflon}. \r\nSimilar to the synthetic data set, we applied a frequency of \\(50\\) kHz to excite a signal in a transducer placed at the centre of the plate. \r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nA0 mode wavelength for this particular CFRP material at such frequency is about 20 mm. \r\nThe measurements were performed by Polytec PSV-400 SLDV on a bottom surface of the plate of dimensions 500 by 500 mm. \r\nThe measurement grid spacing was 1 mm and sampling frequency 512 kHz. \r\nThe measured full wavefield was further processed by an energy compensated RMS takes into account wave attenuation. \r\nThe results of such operation are shown in Fig.~\\ref{fig:Delamination}.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nThe delamination is surrounded by a square frame representing its shape and location. \r\nFigures~(\\ref{fig:unet_exp_7_} - \\ref{fig:gcn_exp}) shows delamination prediction maps for Res-UNet, VGG16 encoder-decoder, PSPNet, FCN-DenseNet and GCN models, receptively.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nThe performance of the models were evaluated by using IoU metric which takes into account not only damage location but also its shape and size. \r\nThe following results were obtained: the Res-UNet IoU = 0.577, the VGG16 encoder-decoder \\(IoU = 0.624\\), the\r\nPSPNet \\(IoU = 0.488\\), the FCN-DenseNet \\(IoU = 0.537\\) and the GCN \\(IoU =\r\n0.723\\). \r\nSimilarly to the numerical dataset, the best accuracy was achieved by using GCN.\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\nAs shown, the models are capable of precise detection and localisation of the delamination. \r\nWe can see that the models can identify the delamination with almost free noise, which indicates the models are capable of generalising and detecting the delamination on previously unseen data. \r\nConsidering the presented models were trained only on the numerically generated dataset, the models show high generalisation capability.\r\nThe performance of the models can be further improved when they are trained on the experimental and the numerical data, as new features will be learned. \r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\begin{figure} [!h]\r\n\t\\centering\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1]{ERMS_with_label.png}\r\n\t\t\\caption{ERMS CFRP Teflon inserted \\& Label}\r\n\t\t\\label{fig:Delamination}\t\r\n\t\\end{subfigure}\t\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1]{residual_unet_decoder_exp_7.png}\r\n\t\t\\caption{Res-UNet} \r\n\t\t\\label{fig:unet_exp_7_}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1]{VGG16_ecoder_decoder_exp_7.png}\r\n\t\t\\caption{VGG16 encoder-decoder} \r\n\t\t\\label{fig:vgg16_exp_7_}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1]{pspnet_exp_7.png}\r\n\t\t\\caption{PSPNet} \r\n\t\t\\label{fig:pspnet_exp_7_}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1]{FCN_DenseNet_exp_7.png}\r\n\t\t\\caption{FCN-DenseNet} \r\n\t\t\\label{fig:fcn_densenet_exp}\r\n\t\\end{subfigure}\r\n\t\\hfill\r\n\t\\begin{subfigure}[b]{0.47\\textwidth}\r\n\t\t\\centering\r\n\t\t\\includegraphics[scale=1]{GCN_exp_7.png}\r\n\t\t\\caption{GCN} \r\n\t\t\\label{fig:gcn_exp}\r\n\t\\end{subfigure}\r\n\t\\caption{Experimental results}\r\n\t\\label{fig:Exp_ERMS_teflon}\r\n\\end{figure}\r\n\\clearpage\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n", "meta": {"hexsha": "e9b5045a54249ce0fc0dbf27fb70d54bb4d22c7e", "size": 22737, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "reports/journal_papers/MSSP_2/results_and_discussions.tex", "max_stars_repo_name": "IFFM-PAS-MISD/aidd", "max_stars_repo_head_hexsha": "9fb0ad6d5e6d94531c34778a66127e5913a3830c", "max_stars_repo_licenses": ["RSA-MD"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-03T05:36:07.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-03T05:36:07.000Z", "max_issues_repo_path": "reports/journal_papers/MSSP_2/results_and_discussions.tex", "max_issues_repo_name": "IFFM-PAS-MISD/aidd", "max_issues_repo_head_hexsha": "9fb0ad6d5e6d94531c34778a66127e5913a3830c", "max_issues_repo_licenses": ["RSA-MD"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "reports/journal_papers/MSSP_2/results_and_discussions.tex", "max_forks_repo_name": "IFFM-PAS-MISD/aidd", "max_forks_repo_head_hexsha": "9fb0ad6d5e6d94531c34778a66127e5913a3830c", "max_forks_repo_licenses": ["RSA-MD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.1721991701, "max_line_length": 333, "alphanum_fraction": 0.6543519374, "num_tokens": 6657, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3491914270934561}} {"text": "%======================================================================\r\n\\chapter{Conclusions and Future Considerations}\r\n%======================================================================\r\n\r\nA summary of the main conclusions from the development of the up-scaling framework as well as the implementation and testing of the framework are presented here. These conclusions represent a successful completion of the research objectives for the thesis. That being said, there are substantial limitations to this research. A series of recommendations are provided to address some of these limitations and to provide guidance on how to extend this research.\r\n\r\n%----------------------------------------------------------------------\r\n\\section{Conclusions}\r\n%----------------------------------------------------------------------\r\nA multi-scale framework for up-scaling \\acrshort{dem} simulations has been developed to address the computational demands of simulating microscale phenomenon in a macroscale domain in the context of NFR. Up-scaling is achieved by matching homogenized stress-strain curves from REV-scale \\acrshort{dem} simulations to single element continuum models using \\acrshort{pso} and LMA optimization algorithms. A Drucker-Prager plasticity model with ductile damage is implemented in the \\acrshort{cdm} model to empirically capture the effect of the degradation (damage) of the NFR as deformation takes place.\r\n\r\n\\subsection*{1. Deformable DEM Homogenization}\r\n\r\nHomogenization algorithms were developed for homogenizing \\acrshort{dem} simulations with deformable bodies to assess the spatially averaged stress-strain behaviour of the REV from the microscale displacements, strains, and stresses. In this homogenization process, the resultant inter-block contact forces and block displacement from the \\acrshort{dem} simulations are converted to average stresses and strains. To apply the homogenization algorithms, a method of automatically assessing a suitable REV given a sufficiently large domain was developed. These algorithms were implemented in Python\\textsuperscript{TM} as the HODS software, which was used as a module for MOUSE.\r\n\r\n\\subsection*{2. Parameterization Methodology}\r\n\r\nTwo examples of the parameterization methodology are presented. Here, the key parameters required to capture the salient features of the model are isolated in order to be able to run the parameter estimation algorithms effectively. the main aspect of this parameterization methodology is the functional assumptions of the hardening/softening and damage evolution functions. In addition, the parameters are rewritten in term of physically meaningful parameters to provide more insight into the mechanics. The Drucker-Prager model with ductile damage is shown to be a reasonable \\acrshort{cdm} model approach to represent NFR in a continuum context, including effects of pressure dependent yield and the triaxiality based damage initiation criterion. Compared to a full \\acrshort{dem} simulation, the \\acrshort{cdm} model shows a good fit pre-damage, but is unable to emulate the subtle post-yield oscillations arising from non-continuous yielding in the NFR.\r\n\r\n\\subsection*{3. Up-Scaling Framework}\r\n\r\nMOUSE software was created and written in Python\\textsuperscript{TM} to provide an implementation of the up-scaling framework presented in this thesis using in house and third-party software modules. The software itself provides a platform through which the four software elements of the up-scaling framework (\\acrshort{dem} module, homogenization module, parameter estimation module, and macroscale module) can communicate with each other. The communication is facilitated by MOUSE through modules which wrap the third party software in such a way that the I/O routines to and from the modules are performed in a consistent capacity regardless of the third party software being used. A consistent set of data protocols were developed for the modules to effectively transfer data between them.\r\n\r\n\\subsection*{4. Framework Verification}\r\n\r\nThe parameter estimation module was tested and yielded an appropriate parameter set that both matched the \\acrshort{dem} data and provided realistic parameters. Additionally, the Drucker-Prager model with ductile damage was found to provide a far superior fit than the damage plasticity model for quasi-brittle materials. Most importantly, the DNS of the slope stability analysis showed that with this up-scaling framework, very significant computational gains can be had with an acceptable error. Very comparable results ($<5\\%$ error) to full \\acrshort{dem} solutions were obtained with the \\acrshort{cdm} method but required two orders of magnitude less computational time. The computational demands were again able to be reduced by another order of magnitude by using a selectively refined continuum mesh at the locations in the domain with high stress gradients.\r\n\r\n%----------------------------------------------------------------------\r\n\\section{Recommendations}\r\n%----------------------------------------------------------------------\r\n\r\nThe main limitation of the presented up-scaling implementation is the macroscale constitutive model module. In the current ABAQUS\\textsuperscript{TM} module, the constitutive models consider only isotropic behaviour. In addition, the model does not consider the effects of pore pressure in the rock mass or fluid flow in any capacity. Though the isotropic assumptions for the elasto-plastic constitutive relationships are likely sufficiently accurate, future implementations of the macroscale constitutive model should consider anisotropic damage behaviour, as anisotropic implementations are found to be completely insufficient. In the case of the Drucker-Prager model with ductile damage, the exponential Johnson-Cook triaxiality based damage initiation criterion provides an excellent fit for monotonic loading, but does not model cyclic loading well. Here, it would be ideal for the cyclic loading capacity of the damage plasticity model for quasi-brittle materials to be incorporated as well. Ultimately, the available damage material subroutines in ABAQUS\\textsuperscript{TM} are insufficient for the key physical characteristics in the system to be captured. As such, a custom anisotropic damage implementation is recommended for the macroscale constitutive model.\r\n\r\nRetrospectively, the functional form of the hardening curve is overly complex. Though it is often necessary to model the softening of the material in the plasticity model, with \\acrshort{cdm} the damage can implicitly model the softening behaviour. Here, it is observed that for the Barcelona model used for the hardening/softening curve, only the hardening portion of the curve is ever used. As such, for future implementations of the plasticity hardening functions, a simpler exponential function could be applied which would also have the benefit of decreasing the number of parameters that need to be estimated, leading to more consistent solutions and faster convergence of the optimization algorithms. \r\n\r\nFurthermore, it is speculated by the author that portions of the parameterization methodology could be modified to yield more consistent solutions. In the parameterization formulations presented in this thesis, too much emphasis was placed on creating physically meaningful parameters rather than numerically consistent parameters. This inconsistency is the case with certain paired parameters if one of the parameters is highly sensitive.\r\n\r\nAdditionally, effectively searching a 11+ dimensional parameter space is computationally expensive. By dividing the problem and exploiting features of the curves and constitutive models, it may be possible to increase the convergence rate and effectively get a better, faster solution. Instead of searching the entire parameter space, it is possible to split the parameters into groups (e.g. pre-damage and post damage). Here, the damage parameters don't actually affect the plasticity calculations until damage is initiated. As such, the plasticity parameters can be estimated using the pre-damage curve and the damage parameters can subsequently be estimated using the post-damage curve.\r\n\r\nA more rigorous examination of other available optimization routines and associated optimization parameters would be another way to potentially reduce the computational cost and increase the accuracy of the parameter estimation process. \\acrshort{pso} and LMA were used in this research, but Dynamically Dimensioned Search (DDS), \r\nReal-Coded Genetic Algorithm (RGA) and Simulated Annealing (SA) were also briefly investigated. As previously stated, rigorously assessing the most effective algorithm was not a priority of this research, but the \\acrshort{pso} + LMA combination was chosen for its simplicity and effectiveness. The other algorithms, if properly applied may provide a faster and more accurate solution.\r\n\t\r\nFor this up-scaling methodology to be more accurate, 3D \\acrshort{dem} simulations are required to capture accurate physical responses of these complex systems. In addition to modifying the \\acrshort{dem} simulations, the associated homogenization algorithms would have to be modified to provide the 3D stress and strain tensors. \r\n\r\nHydro-mechanically coupled \\acrshort{dem} simulations are also an important consideration for more accurate simulations. Here, the homogenization module should be modified to assess the homogenized fluid properties such as pore pressure and flow velocity vectors and the macroscale model needs to be modified to simulate poroelastic physics.\r\n\r\nUltimately, rigorous validation studies should be done with real-world applications to show the viability of this approach. Incorporating some of the above reccomendations will allow this up-scaling framework to validly be applied to complex geomechanical problems in order to avoid the computational demands of \\acrshort{dem} modelling. ", "meta": {"hexsha": "5d84a1cc17026a1ef8b6f681d8a8c019414836d9", "size": 9962, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Chapter6.tex", "max_stars_repo_name": "yetisir/masters-thesis", "max_stars_repo_head_hexsha": "026fef44fdfcf1175a14b5dd131ef65009653d67", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-03-07T16:17:25.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-07T16:17:25.000Z", "max_issues_repo_path": "Chapter6.tex", "max_issues_repo_name": "yetisir/masters-thesis", "max_issues_repo_head_hexsha": "026fef44fdfcf1175a14b5dd131ef65009653d67", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter6.tex", "max_forks_repo_name": "yetisir/masters-thesis", "max_forks_repo_head_hexsha": "026fef44fdfcf1175a14b5dd131ef65009653d67", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 211.9574468085, "max_line_length": 1272, "alphanum_fraction": 0.7909054407, "num_tokens": 1854, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34919141938779563}} {"text": "\\documentclass[%handout,\n\tsans,\n\t12pt,\n\t%slidescentered,% center text on slide\n\t%draft,\t\t\t% compile as draft version\n\t%notes,\t\t\t% include notes in slides\n\t%compress\t\t% compress navigation bar\n]{beamer}\n\n\\beamertemplatenavigationsymbolsempty\n\n\\usetheme{default}\n\\usecolortheme{orchid}\n\\setbeamertemplate{frametitle}\n{\n \\vspace*{1.5em}\\insertframetitle\\vspace*{-1.5em}\n}\n\\setbeamertemplate{footline}[frame number]\n\n\\usepackage[T1]{fontenc}\n\\usepackage[utf8x]{inputenc}\n\n\\usepackage{mathpazo}\n\\usepackage[british]{babel}\n\\usepackage{csquotes}\n\n\\newcommand{\\high}[1]{{\\usebeamercolor[fg]{structure} #1}}\n\\newcommand{\\bad}[1]{\\textcolor{red}{#1}}\n\\newcommand{\\gray}[1]{\\textcolor{darkgray}{#1}}\n\\newcommand{\\black}[1]{\\textcolor{black}{#1}}\n\n\\usepackage{amsmath,amssymb}\n\\usepackage{upgreek}\n\\usepackage{booktabs}\n\\usepackage{hyperref}\n\\usepackage{graphicx}\n\\usepackage{colortbl}\n\\usepackage{url}\n\\usepackage{setspace}\n\\usepackage{wrapfig}\n\\usepackage{tabularx}\n\\usepackage{xspace}\n\\usepackage{mathpartir}\n\n\\usepackage{tikz}\n\\usetikzlibrary{trees, positioning}\n\\usetikzlibrary{shapes.geometric}\n\n\\usepackage{isabelle,isabellesym}\n\\isabellestyle{it}\n\\def\\isacartoucheopen{}%\n\\def\\isacartoucheclose{}%\n\n\\newcommand{\\NN}{\\mathbb{N}}\n\\newcommand{\\QQ}{\\mathbb{Q}}\n\\newcommand{\\RR}{\\mathbb{R}}\n\\newcommand{\\CC}{\\mathbb{C}}\n\\renewcommand{\\epsilon}{\\varepsilon}\n\\renewcommand{\\phi}{\\varphi}\n\\def\\braces#1{[#1]}\n\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\n\\newcommand{\\eg}{e.\\,g.\\xspace}\n\\newcommand{\\ie}{i.\\,e.\\xspace}\n\\DeclareMathOperator\\caret{\\char`\\^}\n\n\\newcommand{\\hastype}{\\,:\\,}\n\\newcommand{\\cons}{::}\n\\newcommand{\\corrto}{\\overset{\\scriptscriptstyle\\wedge}{=}}\n\\newcommand{\\listapp}{\\mathbin{@}}\n\\newcommand{\\listnil}{[\\hskip0.3mm]}\n\\newcommand{\\listnth}{\\mathbin{!}}\n\\newcommand{\\expectation}{\\text{\\upshape E}}\n\n\\usepackage{manfnt}\n\\newenvironment{danger}{\\medbreak\\noindent\\hangindent=2pc\\hangafter=-2%\n \\clubpenalty=10000%\n \\hbox to0pt{\\hskip-\\hangindent\\hskip0.25em\\raisebox{-0.25em}[0pt][0pt]{\\dbend}\\hfill}\\small\\ignorespaces}%\n {\\medbreak\\par}\n %\\raisebox{-1.05em}[0pt][0pt]{\\Huge\\hskip.15em \\stixdanger}\n\n\\newcommand{\\etAl}{\\textit{et al.}\\xspace}\n\n%\\definecolor{mybg}{rgb}{0.9,0.9,0.9}\n\\definecolor{mybg}{rgb}{1,1,1}\n\\setbeamercolor{background canvas}{bg=mybg}\n\n\\title{Verification of an Approximation Algorithm for the Metric Travelling Salesperson Problem}\n\\author{\\normalsize Fabian Hellauer}\n\\institute[]{\\footnotesize Technische Universität München}\n\\date{\\footnotesize16 October 2019}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{frame}\n\\begin{center}\n\\includegraphics[width=5cm]{isabelle.pdf}\n\\end{center}\n\\end{frame}\n\n\\newcommand{\\pivot}[1]{{\\color{red}#1}}\n\\newcommand{\\ltpiv}[1]{{\\color{blue}#1}}\n\\newcommand{\\gtpiv}[1]{{\\color{olive}#1}}\n\n\\section{Verification of Algorithms}\n\\begin{frame}{Verification of Algorithms}%\nReasons:\n\\begin{itemize}\n\t\\item a basis for reliable software\\pause\n\t\\item find mistakes in algorithm explanations\\pause\n\t\\item prove that verification of executable programs\n\tis feasible\\ % when using a modern tool like\n\t\\pause when using \\includegraphics[height=1em]{isabelle.pdf}Isabelle\\pause\n\\end{itemize}\nMathematical results might be a requirement.\n%In this case here, from graph theory.\n\\end{frame}\n\n\\section{Metric TSP}\n\\begin{frame}\n\tTSP is the problem of finding a tour of minimal cost within a graph.\\\\\\pause\n\tWe consider the special case where the nodes form a \\textit{metric space} with their weights.\\\\\\pause\n\tIn particular,\\pause\n\t\\begin{itemize}\n\t\t\\item \\textit{weight} is total\n\t\t% All TSP cities are connected by an edge, in other words we require the graph to be complete.\n\t\t\\item $\\textit{weight}\\ v_1\\ v_3\\ \\leq\\ \\textit{weight}\\ v_1\\ v_2\\ +\\ \\textit{weight}\\ v_2\\ v_3$\n\t\t\\item $0\\ \\leq\\ \\textit{weight}\\ v_1\\ v_2$\\pause\n\t\\end{itemize}\n\tIn such a graph, the cost of a \\textit{minimum spanning tree} is a lower bound on the optimal tour cost.\n\\end{frame}\n\n\\section{Approximation Algorithms}\n\\begin{frame}{Approximation Algorithms}\n\tWhat are Approximation Algorithms?\\pause%based on (metric) TSP\n\t\\vspace{1em}\n\t\\includegraphics[width=\\linewidth]{OPT_line}\n\t%OPT is the performance of an *actual* algorithm, namely the brute-force algorithm that tries all exponentially many node sequences.\n\t\\vspace{1em}\\pause %Since we consider undirected graphs, the reverse of a TSP tour is also a TSP tour (I omitted these duplicates in the graphic)\n\t\n\t\\high{Good solutions for NP-complete problems in polynomial time!}\n\\end{frame}\n\n\\section{Algorithm Explanation}\n\\begin{frame}\n\\begin{center}\n\t\\huge\\high{Algorithm Explanation}\n\\end{center}\n\\end{frame}\n\n\\subsection{Algorithm Overview}\n\\begin{frame}{Algorithm Overview}\nConsider a complete undirected graph$G = (V, E)$ and its minimum spanning tree $T$.\\pause%or rather: let one of its minimum spanning trees be $T$\n\nWe are going to\n\\begin{enumerate}\n\t\\item find such a $T$\\pause\n\t\\item generate a TSP tour using $T$'s edges\n\\end{enumerate}% and we are going to argue that T has cost less than an optimal TSP tour\n\\end{frame}\n\n\\subsection{MST vs.\\ TSP tour}\n\\begin{frame}{MST vs.\\ TSP tour}\n\\begin{center}\n\t\\includegraphics[height=2cm]{complete0.png}\n\\end{center}\n\\end{frame}\n\n\\subsection{MST vs.\\ TSP tour}\n\\begin{frame}{MST vs.\\ TSP tour}\n\\begin{center}\n\t\\includegraphics[height=2cm]{complete1.png}\n\\end{center}\n\\end{frame}\n\n\\subsection{MST vs.\\ TSP tour}\n\\begin{frame}{MST vs.\\ TSP tour}\n\\begin{center}\n\t\\includegraphics[height=2cm]{complete2.png}\n\\end{center}\n\\end{frame}\n\n\\subsection{MST vs.\\ TSP tour}\n\\begin{frame}{MST vs.\\ TSP tour}\n\\begin{center}\n\t\\includegraphics[height=2cm]{complete2.png}\n\\end{center}\n\\begin{isabelle}\n\\isacommand{lemma}\\isamarkupfalse%\n\\ minimum{\\isacharunderscore}spanning{\\isacharunderscore}tree{\\isacharunderscore}le{\\isacharunderscore}OPTWEIGHT{\\isacharcolon}\\isanewline\n\\ \\ \\isakeyword{assumes}\\ {\\isacartoucheopen}minimum{\\isacharunderscore}spanning{\\isacharunderscore}tree\\ T{\\isacharprime}\\ {\\isacharparenleft}symhull\\ E{\\isacharparenright}{\\isacartoucheclose}\\isanewline\n\\ \\ \\isakeyword{assumes}\\ {\\isacartoucheopen}{\\isadigit{2}}\\ {\\isasymle}\\ card\\ V{\\isacartoucheclose}\\isanewline\n\\ \\ \\isakeyword{shows}\\ {\\isacartoucheopen}set{\\isacharunderscore}cost\\ T{\\isacharprime}\\ {\\isasymle}\\ OPTWEIGHT{\\isacartoucheclose}\n\\end{isabelle}\n\\end{frame}\n\n\\subsection{Tour Search}\n\\begin{frame}{Tour Search}\nFind an edge sequence which\\pause\n\\begin{itemize}\n\t\\item uses exactly the MST's edges, each of which twice\\pause\n\t\\item visits every node\\pause\n\t\\item starts and ends at the same node\\pause\n\\end{itemize}\nThis sequence is then called a \\textit{pretour}.\\pause\n\nIts cost is twice the cost of the MST's edges.\n%And the MST's edges are, as mentioned, less than the cost of an optimal tour.\n\\end{frame}\n\n\\subsection{Duplicate Removal}\n\\begin{frame}{Duplicate Removal}\nTransform the pretour into a TSP tour:\n%typically we require TSP tours to visits each node exactly once. Our sequence doesn't do that yet.\n\\begin{itemize}\n\t\\item remove duplicate visits, replace them by shortcuts.\\pause\n\t%That means that when the sequence goes to an already visited node, we instead jump to the next one\n\t\n\t\\high{This can be done at one go.}\n\\end{itemize}\nThese shortcuts always exist and do not increase the cost. %That is why we require a complete metric graph\n\\end{frame}\n\n\\subsection{Time Complexity}%not verified\n\\begin{frame}{Time Complexity}\\pause\n\\begin{itemize}\n\t\\item MST generation: $\\mathcal{O}(|E| \\log |E|) = \\mathcal{O}(n^2 \\log (n^2)) = \\mathcal{O}(2(n^2 \\log n)) = \\mathcal{O}(n^2 \\log n)$%We use Kruskal's, which sorts edges by comparisons\n\t\\pause\n\t\\item Tour: $\\mathcal{O}(n)$ %This is the number of edges in the MST. We discuss later how we can arrange the edges in time linear to this, to-do: Is this actually covered?\n\t\\pause\n\\end{itemize}\noverall: $\\mathcal{O}(n^2 \\log n)$\n\\end{frame}\n\n\\section{Results}\n%What I did in the setting of this algorithm\n\\begin{frame}\n\\begin{center}\n\t\\huge\\high{Results}\n\\end{center}\n\\end{frame}\n\n\\subsection{Algorithm Sketch}\n\\begin{frame}{Algorithm Sketch} %and its Specification?\n%We have a fixed graph here, OPT depends on it\n\\begin{isabelle}\n\t\\isacommand{definition}\\isamarkupfalse%\n\t\\ two{\\isacharunderscore}approx\\ \\isakeyword{where}\\isanewline\n\t\\ \\ {\\isacartoucheopen}two{\\isacharunderscore}approx\\ {\\isacharequal}\\ SPEC\\ {\\isacharparenleft}{\\isasymlambda}T{\\isachardot}\\ is{\\isacharunderscore}tour\\ T\\ {\\isasymand}\\ cost\\ T\\ {\\isasymle}\\ OPT\\ {\\isacharplus}\\ OPT{\\isacharparenright}{\\isacartoucheclose}%explain lack of multiplication\n\t\\vspace{1mm}\\pause\\\\\n\t\\isacommand{definition}\\isamarkupfalse%\n\t\\ algorithm{\\isacharunderscore}sketch\\ \\isakeyword{where}\\ {\\isacartoucheopen}algorithm{\\isacharunderscore}sketch\\ {\\isacharequal}\\isanewline\n\tdo\\ {\\isacharbraceleft}\\isanewline\n\t\\ \\ MST\\ {\\isasymleftarrow}\\ SPEC\\ {\\isacharparenleft}{\\isasymlambda}E{\\isacharprime}{\\isachardot}\\ minimum{\\isacharunderscore}spanning{\\isacharunderscore}tree\\ {\\isacharparenleft}ind\\ E{\\isacharprime}{\\isacharparenright}\\ G{\\isacharparenright}{\\isacharsemicolon}\\isanewline\n\t\\ \\ pretour\\ {\\isasymleftarrow}\\ SPEC\\ {\\isacharparenleft}{\\isasymlambda}pT{\\isachardot}\\ int{\\isacharunderscore}vertices\\ pT\\ {\\isacharequal}\\ nodes\\ G\\\\ \\ \\ \\ \\ {\\isasymand}\\ cost\\ pT\\ {\\isasymle}\\ set{\\isacharunderscore}cost\\ MST\\ {\\isacharplus}\\ set{\\isacharunderscore}cost\\ MST{\\isacharparenright}{\\isacharsemicolon}\\isanewline\n\t\\ \\ tour\\ {\\isasymleftarrow}\\ SPEC\\ {\\isacharparenleft}{\\isasymlambda}T{\\isachardot}\\ is{\\isacharunderscore}tour\\ T\\ {\\isasymand}\\ cost\\ T\\ {\\isasymle}\\ cost\\ pretour{\\isacharparenright}{\\isacharsemicolon}\\isanewline\n\t\\ \\ RETURN\\ tour\\isanewline\n\t{\\isacharbraceright}{\\isacartoucheclose}\n\\end{isabelle}\\pause\n\\vspace{-4mm}\n\\begin{theorem}\n\t\\hspace{2.6cm}$\\textit{algorithm{\\isacharunderscore}sketch} \\le \\textit{two{\\isacharunderscore}approx}$\\\\\n\\upshape if $G$ is a complete finite metric graph.\n\\end{theorem}\n\\end{frame}\n\n\\subsection{Phase I}%MST generation\n%It was already in a library, but had to be adjusted\n\\begin{frame}{Phase I}\n\n\\begin{isabelle}\n\t\\isacommand{locale}\\isamarkupfalse%\n\t\\ finite{\\isacharunderscore}weighted{\\isacharunderscore}connected{\\isacharunderscore}graph\\ {\\isacharequal}\\ finite{\\isacharunderscore}weighted{\\isacharunderscore}graph\\ {\\isacharplus}\\ connected{\\isacharunderscore}graph\\isanewline\n\t\\isakeyword{begin}\\isanewline\n\t\\isanewline\n\t\\ \\ \\isacommand{lemma}\\isamarkupfalse%\n\t\\ kruskal{\\isadigit{0}}{\\isacharunderscore}MST{\\isacharcolon}\\ {\\isacartoucheopen}s{\\isachardot}kruskal{\\isadigit{0}}\\ {\\isasymle}\\ SPEC\\ {\\isacharparenleft}{\\isasymlambda}E{\\isacharprime}{\\isachardot}\\ minimum{\\isacharunderscore}spanning{\\isacharunderscore}tree\\ {\\isacharparenleft}ind\\ E{\\isacharprime}{\\isacharparenright}\\ G{\\isacharparenright}{\\isacartoucheclose}\n\t\\isanewline\n\t\\isanewline\n\t\\isacommand{end}\\isamarkupfalse%\n\\end{isabelle}\\pause%This also includes that the forest is a tree (simple) and that it still is minimal in the symhull (quite a bit of work...)\n\n\\begin{isabelle}\n\t\\isacommand{sublocale}\\isamarkupfalse%\n\t\\ complete{\\isacharunderscore}finite{\\isacharunderscore}weighted{\\isacharunderscore}graph\\ {\\isasymsubseteq}\\\\ \\ \\ finite{\\isacharunderscore}weighted{\\isacharunderscore}connected{\\isacharunderscore}graph\n\t\\end{isabelle}\\pause\nThis required a reorganisation in the \\textit{Kruskal} AFP entry.\n\\end{frame}\n\n\\subsection{Library Connections}\n\\begin{frame}{Library Connections}\\pause\n\\vspace{3mm}\n\\begin{enumerate}\n\t\\item \\textit{Kruskal}: edges with weight field, undirected/directed\\pause\n\t%to-do: vielleicht unterringeln?\n\t% I improved this library: I had to reorganise this one to get what I want without unnecessary assumptions.\n\t\\item \\textit{DFS\\_Framework}: edges are node relations, directed\\pause\n\t\\item \\textit{Koenigsberg\\_Friendship}: edges have a label, undirected\\pause\n\t%Even worse, there are different notions of \"undirected\"\n\t%So we really have the worst case in terms of compatibility.\n\t%Koenigsberg-Friendship also forbids loops, for technical reasons\n\t%It also reinterprets..., Kruskal also has a definition \"is_path_undir\".\n\t%to-do: add Noschinski, a third notion of \"undirected\".\n\t%\"There are even more graph libraries out there: The somewhat new work by Noschinski starts anew,\n\t% with the reasonable modeling of undirected edges as two-element-sets (It sounds reasonable, and is well argued for).\n\t%Sadly, the fresh start did not really establish a new standard.\"\n\\end{enumerate}\n%Example Conversion:\\pause\n\\begin{isabelle}\n\\isacommand{lemma}\\isamarkupfalse%\n\\ ex{\\isadigit{1}}{\\isacharunderscore}the{\\isacharunderscore}path{\\isacharcolon}\\isanewline\n\\ \\ \\isakeyword{assumes}\\ {\\isacartoucheopen}is{\\isacharunderscore}path{\\isacharunderscore}undir\\ G\\ v\\ ps\\ v{\\isacharprime}{\\isacartoucheclose}\\isanewline\n\\ \\ \\isakeyword{shows}\\ {\\isacartoucheopen}{\\isasymexists}{\\isacharbang}ps{\\isacharprime}{\\isachardot}\\ map\\ fst\\ ps{\\isacharprime}\\ {\\isacharequal}\\ map\\ fst\\ ps\\ {\\isasymand}\\ is{\\isacharunderscore}path{\\isacharunderscore}undir\\ G\\ v\\ ps{\\isacharprime}\\ v{\\isacharprime}{\\isacartoucheclose}\n\\isanewline \\pause\n\\isanewline\n\t\\isacommand{definition}\\isamarkupfalse%\n\t\\ the{\\isacharunderscore}path\\ \\isakeyword{where}\\isanewline\n\t\\ \\ {\\isacartoucheopen}the{\\isacharunderscore}path\\ nodelist\\ lst\\ {\\isacharequal}\\ {\\isacharparenleft}case\\ nodelist\\ of\\isanewline\n\t\\ \\ \\ \\ {\\isacharbrackleft}{\\isacharbrackright}\\ {\\isasymRightarrow}\\ {\\isacharbrackleft}{\\isacharbrackright}\\ {\\isacharbar}\\isanewline\n\t\\ \\ \\ \\ n\\ {\\isacharhash}\\ ns\\ {\\isasymRightarrow}\\ THE\\ ps{\\isachardot}\\ map\\ fst\\ ps\\ {\\isacharequal}\\ nodelist\\\\\\ \\ \\ \\ \\ \\ {\\isasymand}\\ is{\\isacharunderscore}path{\\isacharunderscore}undir\\ G\\ n\\ ps\\ lst{\\isacharparenright}{\\isacartoucheclose}\n\\end{isabelle}\n\n\\end{frame}\n\n\\subsection{Library Connections -- Example}\n\\begin{frame}{Library Connections -- Example}\\pause\n\\begin{isabelle}\n\t\\isacommand{lemma}\\isamarkupfalse%\n\t\\ spanning{\\isacharunderscore}forest{\\isacharunderscore}symhull{\\isacharcolon}\\isanewline\n\t\\ \\ {\\isacartoucheopen}spanning{\\isacharunderscore}forest\\ F\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges\\ {\\isacharequal}\\ E{\\isasymrparr}\\ \\\\\\ \\ \\ \\ {\\isasymLongrightarrow}\\ spanning{\\isacharunderscore}forest\\ F\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges\\ {\\isacharequal}\\ symhull\\ E{\\isasymrparr}{\\isacartoucheclose}\n\\end{isabelle}\\pause\n\\begin{isabelle}\n\t\\isacommand{lemma}\\isamarkupfalse%\n\t\\ {\\isacharparenleft}\\isakeyword{in}\\ finite{\\isacharunderscore}weighted{\\isacharunderscore}graph{\\isacharparenright}\\\\\\ \\ spanning{\\isacharunderscore}forest{\\isacharunderscore}symhull{\\isacharunderscore}preimage{\\isacharcolon}\\isanewline\n\t\\ \\ \\isakeyword{assumes}\\ {\\isacartoucheopen}spanning{\\isacharunderscore}forest\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges{\\isacharequal}F{\\isasymrparr}\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\\\\\ \\ \\ \\ edges{\\isacharequal}symhull\\ E{\\isasymrparr}{\\isacartoucheclose}\\isanewline\n\t\\ \\ \\isakeyword{shows}\\ {\\isacartoucheopen}{\\isasymexists}F{\\isacharprime}{\\isachardot}\\\\\\ \\ \\ \\ spanning{\\isacharunderscore}forest\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges{\\isacharequal}F{\\isacharprime}{\\isasymrparr}\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges{\\isacharequal}E{\\isasymrparr}\\isanewline\n\t\\ \\ \\ \\ {\\isasymand}\\ edge{\\isacharunderscore}weight\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges{\\isacharequal}F{\\isacharprime}{\\isasymrparr}\\ \\\\\\ \\ \\ \\ \\ \\ {\\isacharequal}\\ edge{\\isacharunderscore}weight\\ {\\isasymlparr}nodes{\\isacharequal}V{\\isacharcomma}\\ edges{\\isacharequal}F{\\isasymrparr}{\\isacartoucheclose}\n\\end{isabelle}\n\\end{frame}\n\n\\subsection{Phase II}\n\\begin{frame}{Phase II}\n\\begin{center}\n\\includegraphics[height=7cm]{Sorted_binary_tree_preorder}\n\\end{center}\n%to-do: attribute \"By Sorted_binary_tree.svg: Milesderivative work: Pluke (talk) - Sorted_binary_tree.svg, Public Domain, https://commons.wikimedia.org/w/index.php?curid=10616003\"\n\\end{frame}\n\n\\subsection{Phase II -- Parametrisation} %to-do: replace according to e-mail\n\\begin{frame}{Phase II -- Parametrisation}\n\\vspace{8mm}\nWe use the \\textit{DFS-Framework}. This is the parametrisation:\n\\vspace{-2mm}\n\\begin{isabelle}\n\t\\isacommand{definition}\\isamarkupfalse%\n\t\\ fp{\\isadigit{0}}{\\isacharunderscore}params\\ {\\isacharcolon}{\\isacharcolon}\\ {\\isachardoublequoteopen}{\\isacharprime}v\\ fp{\\isadigit{0}}{\\isacharunderscore}param{\\isachardoublequoteclose}\\ \\isakeyword{where}\\isanewline\n\t{\\isachardoublequoteopen}fp{\\isadigit{0}}{\\isacharunderscore}params\\ {\\isasymequiv}\\ dflt{\\isacharunderscore}parametrization\\ state{\\isachardot}more\\isanewline\n\t\\ \\ {\\isacharparenleft}RETURN\\ {\\isasymlparr}\\ tour{\\isacharunderscore}list\\ {\\isacharequal}\\ {\\isacharbrackleft}{\\isacharbrackright}\\ {\\isasymrparr}{\\isacharparenright}\\isanewline\n\t\\ \\ {\\isasymlparr}on{\\isacharunderscore}discover\\ {\\isacharcolon}{\\isacharequal}\\ {\\isasymlambda}{\\isacharunderscore}\\ n\\ s{\\isachardot}\\ RETURN\\ {\\isasymlparr}tour{\\isacharunderscore}list\\ {\\isacharequal}\\ tour{\\isacharunderscore}list\\ s\\ {\\isacharat}\\ {\\isacharbrackleft}n{\\isacharbrackright}{\\isasymrparr}\\ {\\isasymrparr}{\\isachardoublequoteclose}\n\\end{isabelle}\\pause\n\\vspace{-5mm}\n\\begin{isabelle}\n\t\\isacommand{locale}\\isamarkupfalse%\n\t\\ node{\\isacharunderscore}and{\\isacharunderscore}MST{\\isacharunderscore}in{\\isacharunderscore}graph\\ {\\isacharequal}\\isanewline\n\t\\ \\ complete{\\isacharunderscore}finite{\\isacharunderscore}metric{\\isacharunderscore}graph\\ G\\ {\\isacharplus}\\isanewline\n\t\\ \\ T{\\isacharcolon}\\ tree\\ T\\isanewline\n\t\\ \\ \\isakeyword{for}\\ G{\\isacharcolon}{\\isacharcolon}{\\isacartoucheopen}{\\isacharparenleft}{\\isacharprime}v{\\isacharcolon}{\\isacharcolon}metric{\\isacharunderscore}space{\\isacharcomma}real{\\isacharparenright}\\ graph{\\isacartoucheclose}\\isanewline\n\t\\ \\ \\isakeyword{and}\\ T{\\isacharcolon}{\\isacharcolon}{\\isacartoucheopen}{\\isacharparenleft}{\\isacharprime}v{\\isacharcomma}real{\\isacharparenright}\\ graph{\\isacartoucheclose}\\ {\\isacharplus}\\isanewline\n\t\\ \\ \\isakeyword{fixes}\\ v\\isactrlsub {\\isadigit{0}}{\\isacharcolon}{\\isacharcolon}{\\isacartoucheopen}{\\isacharprime}v{\\isacartoucheclose}\\isanewline\n\t\\ \\ \\isakeyword{assumes}\\ v{\\isacharunderscore}in{\\isacharunderscore}V{\\isacharcolon}\\ {\\isacartoucheopen}v\\isactrlsub {\\isadigit{0}}\\ {\\isasymin}\\ V{\\isacartoucheclose}\\isanewline\n\t\\ \\ \\isakeyword{and}\\ mst{\\isacharcolon}\\ {\\isacartoucheopen}minimum{\\isacharunderscore}spanning{\\isacharunderscore}tree\\ T\\ G{\\isacartoucheclose}\n\\end{isabelle}\n\\end{frame}\n\n%to-do: simplification of Phase II? not really my work, but something which slowed me down...\n\n\\section{Plans}\n\\begin{frame}\n\\begin{center}\n\t\\huge\\high{Plans}%time ran out sadly...\n\\end{center}\n\\end{frame}\n\n\\subsection{Input Generation}% We need a script that generates the edge weights for us.\n\\begin{frame}{Input Generation}\nConsider a Euclidean space like $\\RR^2$, $\\RR^3$,\\dots\n\t\\begin{itemize}\n\t\t\\item A popular metric is the \\textit{Euclidean} metric.\\pause\n\t\t\n\t\t\\bad{It is often irrational, and thus difficult to store and compare with infinite precision.}\\pause\n\t\t\\item In contrast, the \\textit{maximum metric} and the \\textit{Manhattan metric} return natural numbers. %When the coordinates are natural numbers.\n\t\t%to-do: definition? small drawing?\n\t\\end{itemize}\n\t%application drilling robot\n\\end{frame}\n\n\\subsection{DFS Invariant}\n\\begin{frame}{DFS Invariant}\\pause\n\tfinish this proof\\pause\n\t\\vspace{-2mm}\n\t\\begin{isabelle}\n\t\t\\isacommand{lemma}\\isamarkupfalse%\n\t\t\\ {\\isacartoucheopen}dfs{\\isachardot}is{\\isacharunderscore}invar\\ {\\isacharparenleft}{\\isasymlambda}s{\\isachardot}\\ valid{\\isacharunderscore}graph{\\isachardot}tour\\ {\\isacharparenleft}ind{\\isacharprime}\\ {\\isacharparenleft}dom\\ {\\isacharparenleft}discovered\\ s{\\isacharparenright}{\\isacharparenright}{\\isacharparenright}\\ {\\isacharparenleft}tour{\\isacharunderscore}list\\ s{\\isacharparenright}{\\isacharparenright}{\\isacartoucheclose}\n\t\\end{isabelle}\\pause\n\tAdvancing this proof has already led to generally useful statements for the \\textit{DFS-Framework}, e.g.\\ \\pause\n\t\\begin{isabelle}\n\t\t\\isacommand{lemma}\\isamarkupfalse%\n\t\t\\ i{\\isacharunderscore}snd{\\isacharunderscore}pending{\\isacharunderscore}sane{\\isacharcolon}\\ {\\isacartoucheopen}dfs{\\isachardot}is{\\isacharunderscore}invar\\ {\\isacharparenleft}{\\isasymlambda}s{\\isachardot}\\ snd\\ {\\isacharbackquote}\\ {\\isacharparenleft}pending\\ s{\\isacharparenright}\\ {\\isasymsubseteq}\\ V{\\isacharparenright}{\\isacartoucheclose}\n\t\\end{isabelle}\\pause\n\t\\begin{isabelle}\n\t\t\\isacommand{lemma}\\isamarkupfalse%\n\t\t\\ i{\\isacharunderscore}stack{\\isacharunderscore}sane{\\isacharcolon}\\ {\\isacartoucheopen}dfs{\\isachardot}is{\\isacharunderscore}invar\\ {\\isacharparenleft}{\\isasymlambda}s{\\isachardot}\\ set\\ {\\isacharparenleft}stack\\ s{\\isacharparenright}\\ {\\isasymsubseteq}\\ V{\\isacharparenright}{\\isacartoucheclose}\n\t\\end{isabelle}\\pause\n\t\\begin{isabelle}\n\t\t\\isacommand{lemma}\\isamarkupfalse%\n\t\t\\ i{\\isacharunderscore}discovered{\\isacharunderscore}sane{\\isacharcolon}\\ {\\isacartoucheopen}dfs{\\isachardot}is{\\isacharunderscore}invar\\ {\\isacharparenleft}{\\isasymlambda}s{\\isachardot}\\ dom\\ {\\isacharparenleft}discovered\\ s{\\isacharparenright}\\ {\\isasymsubseteq}\\ V{\\isacharparenright}{\\isacartoucheclose}\n\t\\end{isabelle}\n\\end{frame}\n\n\\section{Conclusion}\n\\begin{frame}{Conclusion}\nEven with parts already verified, formal verification can require a lot of work:\n\\begin{itemize}\n\t\\item Differing graph formalisations\\pause!\\pause\n\t%in particular connecting Digraph.graph_rec to my own finite_complete_weighted...graph\n\t\\item For this particular algorithm, a lot of assumptions have to be collected and formulated.\\pause %complete_finite_weighted is actually complete_finite_weighted_simple_loopfree_metric_graph\n\t\n\t%\"Not really part of a conclusion, but I want to mention it here since I had to leave out much context earlier for brevity.\"\n\t\\item Definitions have many dependencies. %They e.g. have to fix a graph, a subtree and an arbitrary node at the same time.\n\\end{itemize}\n\\end{frame}\n\n\\section{Questions}\n\\begin{frame}\n\\begin{center}\n\t\\huge\\high{Questions}\n\\end{center}\n\\end{frame}\n\n%to-do: Problems?\n\n\\end{document}\n\n%to-do: compare to the plan from the meeting:\n%25+5min\n\n%explain in the presentation:\n\n%Bibliotheken vorstellen, Wahl begründen\n%--> Graphformalisierungen: directed vs. undirected edges in both phases. Welche gibt es alle?\n%edge labels, i.e. a third field besides start node and end node 5min\n%MST lemma zeigen, wieso nicht Graph_Theory? 5min\n%was ist der projektstatus? 5min\n%maybe a slide about the commit count?\n%add dfs-sublocale statement to plans. \"This is already proven. However, the set membership must be implemented...\"\n", "meta": {"hexsha": "d57f3f5aef6b9f06024c710f61d494145723de4a", "size": 22619, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "slides_backup/main.tex", "max_stars_repo_name": "helli/tsp-approximation", "max_stars_repo_head_hexsha": "2b0e2abc06190402f507664980c65bc8ce3fad7f", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "slides_backup/main.tex", "max_issues_repo_name": "helli/tsp-approximation", "max_issues_repo_head_hexsha": "2b0e2abc06190402f507664980c65bc8ce3fad7f", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "slides_backup/main.tex", "max_forks_repo_name": "helli/tsp-approximation", "max_forks_repo_head_hexsha": "2b0e2abc06190402f507664980c65bc8ce3fad7f", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.4945295405, "max_line_length": 434, "alphanum_fraction": 0.766523719, "num_tokens": 7460, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.34919141938779563}} {"text": "\\documentclass{fancyArticle}\n\\usepackage{lipsum}\n\n\\title{A Sample Document}\n\\author{Gary Baker}\n\\date{\\today}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\n \\lipsum[1]\n\\end{abstract}\n\\pagebreak\n\n\\section{A first section}\n\nThis\\footnote{\\lipsum[1][3]} demonstrates the footnote\nstyle. \\lipsum[4] Here's another footnote\\footnote{\\lipsum[30][1]}\nfollowed by a numbered equation.\n\n\\begin{equation}\n \\label{eq:1}\n e^{i\\pi}+1 = 0\n\\end{equation}\n\n\\lipsum[5]\n\n\\subsection{A subsection}\n\n\\lipsum[6]\n\n\\begin{equation*} \n V(l) = \\max_{\\beta\\geq 0}\\left\\{ -c(\\beta) * \\Delta + e^{-r\\Delta}\n \\left[ \\int \\frac{1}{2} (\\tanh(l/2)+1) V(l+\\hat{s}) f_\\beta(s) ds \\right]\\right\\}\n\\end{equation*}\n\n\\lipsum[7]\n\n\\section{A section with some citations}\n\n\\cite{Nash1950} and \\cite{Rubinstein1982} say some things about\nbargaining. Those citations are also clickable links to their\nrespective entries in the bibliography.\n\n\\lipsum [20-21]\n\n\\pagebreak\n\\bibliography{sampleBib.bib}\n\n\n\\end{document}", "meta": {"hexsha": "0ac0a08665b49270f2810c2f3a6f964ccf2cf33c", "size": 980, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sample.tex", "max_stars_repo_name": "ggbaker/fancy-article", "max_stars_repo_head_hexsha": "442b6966216a5d420095293dc1fe71ce428669f9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "sample.tex", "max_issues_repo_name": "ggbaker/fancy-article", "max_issues_repo_head_hexsha": "442b6966216a5d420095293dc1fe71ce428669f9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "sample.tex", "max_forks_repo_name": "ggbaker/fancy-article", "max_forks_repo_head_hexsha": "442b6966216a5d420095293dc1fe71ce428669f9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 18.8461538462, "max_line_length": 85, "alphanum_fraction": 0.7071428571, "num_tokens": 341, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.6297746213017459, "lm_q1q2_score": 0.34919141782045776}} {"text": "\\documentclass{article}\n\n\\usepackage[british]{babel}\n\\usepackage[useregional]{datetime2}\n\\DTMlangsetup[en-GB]{showdayofmonth=false}\n\n\\usepackage[utf8]{inputenc}\n\n\\usepackage{amsmath}\n\\usepackage{float}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\hypersetup{\n colorlinks=true,\n linkcolor=black,\n filecolor=magenta, \n urlcolor=blue,\n}\n\n\\title{Machine Learning}\n\\author{Josh Myers-Dean \\and Robin Cosbey}\n\\date{\\today}\n\n\\begin{document}\n\n\\maketitle\n\\newpage\n\\tableofcontents\n\\newpage\n\n\\section{Introduction} \nThe term ``machine learning'' was defined in 1959 as \\textit{a field of study that gives computers the ability to learn without being explicitly programmed.} This means that rather than giving a computer a list of instructions to follow, we take advantage of the computer's representational abilities. The computer \\textit{learns} patterns in data. Machine learning is essentially a math problem: $y = h(x)$ where $x$ is the input, $y$ is the output and $h$ represents the function (mathematical model) applied. In simple terms we are learning a mapping from $x$ to $y$. The goal of this workshop is to dive into the topics presented in \\ref{fig:flow}, show how the process of each step and provide some resources.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3in]{flowchart.jpeg}\n \\caption{General machine learning workflow. \\href{https://www.freecodecamp.org/news/every-single-machine-learning-course-on-the-internet-ranked-by-your-reviews-3c4a7b8026c0/}{Link}}\n \\label{fig:flow}\n\\end{figure}\n\n\\subsection{Unsupervised and Supervised Learning}\nWe primarily see \\textit{unsupervised} and \\textit{supervised} learning algorithms. Both are given a set of examples to learn from, known as \\textit{training data}. In the case of supervised learning, we have ground truth output associated with the input data unlike unsupervised learning in which the computer is only given the input and ``clusters'' the examples based on similarity rather than learning from the mappings provided. Most of this workshop will focus on supervised learning. \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3.1in]{supervision.png}\n \\caption{Unsupervised consists of clustering whereas supervised (in this case binary classification) consists of finding a boundary between data points. \\href{https://chatbotsmagazine.com/lets-know-supervised-and-unsupervised-in-an-easy-way-9168363e06ab}{Link}}\n\\end{figure}\n\n\\newpage\n\\subsection{Tasks} \n\nThe two general types of problems that can be solved with supervised machine learning techniques are \\textbf{classification} and \\textbf{regression}. The main difference between the two is the output: classification has a categorical (discrete) output whereas regression has a numerical (continuous) output. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3.5in]{tasks.png}\n \\caption{Basic distinction between the two types of tasks. \\href{https://medium.com/datadriveninvestor/supervised-vs-unsupervised-machine-learning-5200ffa7301a}{Link}}\n \\label{fig:tasks}\n\\end{figure}\n\n\\subsubsection{Classification}\nClassification algorithms are used when the outputs are restricted to a limited set of values. The basic form is \\textit{binary classification} as shown in \\ref{fig:tasks} in which the model assigns each input to one of two outputs. For example, if we have a model that identifies if an image contains a dog, the output would be the prediction of either `dog' or `not dog' (true or false). \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3.5in]{dogs.jpg}\n \\caption{Can you tell which photos contain a dog? Can a computer? \\href{https://www.npr.org/sections/thesalt/2016/03/11/470084215/canine-or-cuisine-this-photo-meme-is-fetching}{Link}}\n\\end{figure}\n\n\\newpage\u0002\n\\subsubsection{Regression}\nRegression involves continuous values such as temperature, length and price. You typically see regression models used for finding relationships between variables and for forecasting. How do age, gender and diet impact height? This is an example of input variables (age, gender and diet), referred to as \\textit{predictor variables}, and their relationship to a single output (height) known as the \\textit{outcome variable}. Unlike classification, we determine a trend from the training data fed into the model. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3.5in]{regression.png}\n \\caption{Regression line showing the relationship between hours studied and exam score. \\href{https://stackabuse.com/linear-regression-in-python-with-scikit-learn/}{Link}}\n\\end{figure}\n\n\\newpage\n\\section{Data}\nSo what does data look like in machine learning? \nAside from numerical data, anything that can be encoded as a numerical value or set of numerical values, such as images, words, or audio. \nWith images, we can encode each pixel into their RGB numerical value. \nAudio data can be represented numerically based on features or by way of images with what are called spectograms. \nWords can be encoded as numbers, or word frequencies throughout an entire document might be used. \nWe also use categorical data, a number that represents a group i.e. dogs, cats, or birds used primarily in classification tasks. \nTypically, this data is put into a matrix, let's examine some famous data sets where you can see real examples of data used in machine learning.\n\n\\subsection{Data Sets}\nLet's examine some data sets that are commonly used by those exploring machine learning. Probably the most famous among these data sets, is the MNIST data set; a collection of handwritten numbers. We can use this data set to classify each picture as the number it is meant to represent. The MNIST data set is quite large and machine learning models are able to get up to 99\\% accuracy classifying these images.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{mnist.png}\n \\caption{MNIST Data set \\href{http://yann.lecun.com/exdb/mnist/}{dataset linked here}.}\n\\end{figure}\n\n\\newpage\n\nAnother smaller data set is the Iris data set. In this case, we are not dealing with image data, we are given attributes. There are three kinds of iris represented in this data set and we are given four attributes of each flower: the sepal length and width, and the petal length and width. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{iris.png}\n \\caption{Iris Data set \\href{https://archive.ics.uci.edu/ml/datasets/iris}{dataset linked here}.}\n\\end{figure}\n\nThis is a great data set for beginners because there are only 150 iris samples, making it easy to test and debug. Take a look at the links included to get a feel for what data looks like in machine learning.\n\n\\subsection{Bias in Data}\nAs you begin a machine learning project, one of the first steps is to find the dataset you will be working with. As you are forming the initial questions that you hope to answer with your efforts make sure to also ask the data some questions. Before you start building a model to work with the data, you should have a firm understanding of the distribution and makeup of the data itself. Many news stories have surfaced over the years indicating the explicit, inherent biases in many publicly released face and object recognition systems and it is not hard to believe that similar stories can be found in other domains. \n\nThis analysis of your data is especially important when considering problems directly related to society. As the systems we build gain trust from the public, we, as computer scientists, can have great implications unknowingly on the world we live in. Throughout this analysis be on the lookout for an uneven distribution of data. Additionally, keep in mind that the biases present throughout history can be found implicitly in the datasets produced in those times and by using that data to train a model you are propagating those biases forward through time. Sometimes you may need to evaluate the types of problems you are looking to solve and ensure the value of this pursuit for the community.\n\n\\newpage\n\\subsection{Data Preparation}\nThe above datasets, MNIST and Iris, are \\textit{clean}. There are no missing values, or random Unicode characters that are going to trip you up when trying to parse the data. Often, data needs to be cleaned and made consistent. This process of making sure that data is clean can take up a lot of time. If an entire chunk of data is missing should it be averaged with its neighbors or be given the value of the previous data point? These are considerations a machine learning engineer will encounter. There is also the consideration of how to store input and output data as you will often train and re-train a model many times with different parameters that can be tuned called hyperparameters or maybe parse the data in a different way to see if that leads to any performance gains. Since you will want to run your model many times, you don't want to have to re-parse your cleaned data each time you want to run it as that takes a lot of time, so you will want to store your cleaned data somehow. You may also want to save various configurations of your trained model, so you will need an efficient and organized way to store trained models. You can also create data based off of your current data set, this is called data augmentation and may be something as simple as when building an image classifier reflecting some of your images horizontally or vertically to make you model more robust to image orientation. So, data preparation is a complex beast. There are many techniques that can be explored and many components to consider when preparing data.\n\nAfter our data has been cleaned we typically want to split our data into train, development (sometimes called ``validation''), and test. You'll understand more in depth as to how these data sets are used after reading the \\textit{Training} section. For now, understand that in order to train a model we will need data it can see (train), new data it can see to check its progress while training (development), and more data to see if it learned a pattern that can be applied to unseen data (test). We generally want to reserve 70\\% for our training set, 10\\% for our development set, and 20\\% for our test set. You are not limited to splitting your data in this way and can try out different configurations as well.\n\n\\subsection{Training} \nOnce we have an understanding of the data we are working with, have preprocessed the data appropriately and split the data into a train, development and test set, now it is time to design your model. More on the types of the basic machine learning and deep learning models in future sections. The training phase allows the the model to learn from the examples provided and understand the presents in the data. The development, or validation, phase is when we tune hyperparameters -- \\textit{hyperparameters} are typically related to the model architecture. Slight changes to the hyperparameters will produce different results with the model. This process repeats for a predefined amount of time. Only once we have completed training and hyperparameter tuning do we turn to the test data. The test set is used to independently assess the accuracy of the model on held out data that has not been seen previously. \n\n\\newpage\n\\section{Machine Learning \\& Deep Learning} \nThere are many differences between deep learning and machine learning algorithms. Features are pieces of information that are intended to be relevant to performing a task. For example, if we were to build an animal classifier using a machine learning algorithm a feature we may want to include is the type of coat an animal has: fur, scales, or feathers. We would want to include as many of these distinctive features as possible for each animal and then give those pieces of information to our machine learning algorithm and this would create a model that can classify animals. But with deep learning we wouldn't need to manually identify distinctive characteristics of animals, we can just give our deep learning algorithm thousands of pictures of animals. Deep learning algorithms are capable of finding which pieces of information about a picture for example, are important without human assistance. This is one benefit of deep learning algorithms.\n\nAnother difference between machine learning and deep learning algorithms is their expressiveness. Machine learning algorithms generally revolve around optimizing linear functions to fit our data. However, not all patterns found in data can be expressed by linear functions. \n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=5.0in]{linearvsnonlinear.png}\n \\caption{Non-linear models are able to model more complex patterns.}\n\\end{figure}\n\nDeep learning algorithms can consist of one or more linear or non-linear transformations. Because of this, deep learning models are able to model both linear and non-linear patterns in data. This is very powerful, and is the \\textit{secret sauce} of deep learning models. Because deep learning algorithms are able to model much more complex and nuanced patterns in data, they require a lot more data than traditional machine learning algorithms which are trying to model simpler patterns. While a machine learning model may require thousands of data points to do well, a deep learning model may require millions making deep learning much more computationally expensive.\n\n\\newpage\n\\subsection{Decision Trees} \nDecision trees are tree-like structures that can represent probability, decisions, and much more.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{decisiontree.png}\n \\caption{Decision Tree}\n\\end{figure}\n\n\\subsubsection{Random Forest}\nRandom Forest is a supervised machine learning algorithm that can be used for both classification and regression tasks that consists of a collection of decision trees, hence the name. Here is the general algorithm:\n\n\\begin{enumerate}\n \\item Split data into $n$ random sets selecting $m$ random features from each sample.\n \\item Create a decision tree using each subset of data.\n \\item Given new sample, each of these independent trees makes a prediction.\n \\item The prediction with the most \"votes\" from our trees is our overall prediction.\n\\end{enumerate}\n\nFor a regression task, we would average the predictions made by all of the decision trees. This idea of averaging decisions of multiple models is called bootstrap aggregating or bagging and can reduce variance and overfitting.\n\nEssentially, every decision tree will make a prediction and the most common prediction in among all of the decision trees will be the outcome. When using the random forest algorithm for regression, we will average the output of each decision tree. \n\n\\newpage\n\\subsection{Support Vector Machines (SVMs)} \nSupport vector machines are a simple model that do not require a large amount of data and separate classes represented in data. SVMs are simple discriminative classifiers that output an optimal decision boundary between the datapoints provided. When new datapoints are fed through the SVM, they are classified based on where they fall in relation to the decision boundary. This boundary is often referred to as a ``hyperplane.'' The margin is the distance from the boundary to the nearest datapoint. We want to find a decision boundary that allows points to be in their respective classes without crossing the boundary into other classes and a good margin is one where this separation is large for all classes represented. scikit-learn offers a SVM tool that is easy to download to test with given data. More information about SVMs and the toolkit can be found \\href{https://scikit-learn.org/stable/modules/svm.html}{here}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3.5in]{svm.jpeg}\n \\caption{Decision boundary (yellow dashed line) separating the red from blue. \\href{https://medium.com/deep-math-machine-learning-ai/chapter-3-support-vector-machine-with-math-47d6193c82be}{Link}}\n\\end{figure}\n\n\\newpage\n\\subsection{KNN} \nK-Nearest Neighbors is an supervised machine learning algorithm that can be used for classification and regression tasks. Let's focus on classification. In this case we won't do any training using our training data, we will using the position of our training data in space to classify unseen data points.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.0in]{knn.png}\n \\caption{KNN Algorithm.}\n\\end{figure}\n\n\\begin{enumerate}\n \\item Select number of neighbors to examine $k$.\n \\item Given an unseen data point, find $k$ nearest data points.\n \\item The class that appears the most of those $k$ nearest data points is the class of our unseen data point.\n\\end{enumerate}\n\nYou can experiment with different distance metrics like Euclidean distance, Manhattan distance, or Cosine similarity. Because there is no training time involved, this can be a good option when time is limited and is relatively easy to setup. K-Nearest Neighbors may not be a good idea if you have very high dimensional or feature-rich data. This is because of something called the Curse of Dimensionality; as we go into higher and higher dimensional space distance as we know it becomes more skewed and two things that are very different may seem to be \"close\" to each other in high dimensional space. For more on the Curse of Dimensionality and how higher dimensionality can affect your models read \\href{https://towardsdatascience.com/curse-of-dimensionality-2092410f3d27}{this article}.\n\n\\newpage\n\\subsection{Deep Neural Networks (DNNs)} \nDNNs are commonly used within problem spaces with a lot of available data. Once the data is preprocessed, we provide the data as input to the DNN. The data undergoes a series of non-linear transformations. This is what separates deep neural networks from one-layer, shallow, linear neural networks. These transformations are followed by a linear transformation which produces the output or result. \n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=3.5in]{nn.jpg}\n \\caption{Simple DNN with two hidden non-linear layers. \\href{https://www.digitaltrends.com/cool-tech/what-is-an-artificial-neural-network/}{Link}}\n\\end{figure}\n\n\\href{https://playground.tensorflow.org}{Tutorial here.}\n\n\\newpage\n\\subsection{Convolutional Neural Networks (CNNs)} \n\nConvolutional Neural Networks are mostly used with image data to perform image classification and other image related tasks.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{cnn.jpeg}\n \\caption{Unrolling an RNN. Each input is fed through to produce an output at the given timestep. We also pass context forward through time. \\href{https://towardsdatascience.com/a-comprehensive-guide-to-convolutional-neural-networks-the-eli5-way-3bd2b1164a53}{Link}}\n\\end{figure}\n\n\\subsection{Recurrent Neural Networks (RNNs)} \n\nRecurrent Neural Networks strength lies in finding patterns in sequential data, whether that be sentences, audio, or protein sequences. They have special cells which allow them to keep important information seen from past samples or from future samples. There are many different variations of RNNs, here is a link to some RNN tutorials written in PyTorch \\href{https://github.com/yunjey/pytorch-tutorial/tree/master/tutorials/02-intermediate}{here}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{rnn.png}\n \\caption{Unrolling an RNN. Each input is fed through to produce an output at the given timestep. We also pass context forward through time. \\href{https://medium.com/@jianqiangma/all-about-recurrent-neural-networks-9e5ae2936f6e}{Link}}\n\\end{figure}\n\n\\newpage\n\\section{Further Topics}\n\n\\subsection{Reinforcement Learning}\n\\href{https://deepmind.com/research/publications/playing-atari-deep-reinforcement-learning/}{Paper}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{rl.jpg}\n \\caption{In recent years deep learning has been introduced to reinforcement learning algorithms to improve performance. The agent chooses an action and receives updated information of the environment in order to choose a new action. Deep approaches have been used to augment the agent. \\href{https://www.kdnuggets.com/2018/03/5-things-reinforcement-learning.html}{Link}}\n\\end{figure}\n\n\\subsection{Generative Adversarial Networks}\n\\href{https://arxiv.org/abs/1406.2661}{Paper}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{gans.png}\n \\caption{We have been able to achieve incredible results using this model to generate things. This is done by training a generator and discriminator model. The generator tries to generate a realistic sample from the training set and the discriminator does its best to tell if the sample given to it is from the training set or a generated \"fake\" sample. By doing this we are able to train a generator that can create extremely good \"fakes\" or generated samples. Things it may try to \"fake\" and generate are images, audio, or video. \\href{https://skymind.ai/wiki/generative-adversarial-network-gan}{Link}}\n\\end{figure}\n\n\\newpage\n\\subsection{Few Shot Learning}\n\\href{https://openreview.net/forum?id=HkxLXnAcFQ}{Paper}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=4.5in]{fewshot.png}\n \\caption{Few shot learning works to train a model on very few examples of given classes. Many training episodes are used to understand the distinctions between the classes provided. \\href{https://medium.com/sap-machine-learning-research/deep-few-shot-learning-a1caa289f18}{Link}}\n\\end{figure}\n\\newpage\n\n\\end{document}\n", "meta": {"hexsha": "83a5d2fbbed259ff41c0d5d5d0364c9534d48e0e", "size": 21602, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "workshop_tex/main.tex", "max_stars_repo_name": "joshmyersdean/MachineLearning-Workshop", "max_stars_repo_head_hexsha": "8a94e1aa9b6bc51aeb5b3cf3dee5906b44acdb75", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-12-03T19:11:32.000Z", "max_stars_repo_stars_event_max_datetime": "2019-12-03T19:11:32.000Z", "max_issues_repo_path": "workshop_tex/main.tex", "max_issues_repo_name": "joshmyersdean/MachineLearning-Workshop", "max_issues_repo_head_hexsha": "8a94e1aa9b6bc51aeb5b3cf3dee5906b44acdb75", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "workshop_tex/main.tex", "max_forks_repo_name": "joshmyersdean/MachineLearning-Workshop", "max_forks_repo_head_hexsha": "8a94e1aa9b6bc51aeb5b3cf3dee5906b44acdb75", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-11-26T18:56:57.000Z", "max_forks_repo_forks_event_max_datetime": "2019-11-26T18:56:57.000Z", "avg_line_length": 85.3833992095, "max_line_length": 1554, "alphanum_fraction": 0.7900194426, "num_tokens": 4885, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.3491914155349654}} {"text": "\\documentclass[10pt]{article}\n\\usepackage[T1]{fontenc}\n\n% Document Details\n\\newcommand{\\CLASS}{AMATH 563}\n\\newcommand{\\assigmentnum}{Nueral Networks}\n\n\\usepackage[margin = 1in, left=0.75in,right=0.75in]{geometry}\n\\input{../../import/title.tex} % Title Styling\n\\input{../../import/styling.tex} % General Styling\n\\input{../../import/code.tex} % Code Display Setup\n\\input{../../import/math.tex} % Math shortcuts\n\n\\usepackage{dblfloatfix} % To enable figures at the bottom of page\n\n% Problem\n\\newenvironment{problem}[1]{\\vspace{2em}{\\large\\sffamily\\textbf{#1}}\\itshape\\par}{}\n\n\\usepackage{nameref}\n\\newcommand{\\vln}{\\rotatebox{90}{--}}\n\n\\begin{document}\n\n\\twocolumn[{%\n\\begin{@twocolumnfalse}\n\\maketitle\n\\vspace{2em}\n\\begin{abstract}\nWe outline a how neural networks might be applied to the study of systems governed by possibly unknown PDEs. In particular we show that future state predictions are possible for some systems, even if they are chaotic. We address how high dimensional data can be rank reduced first in order to allow for training on a smaller neural net. While some results are encouraging, we find that it is quite difficult to correctly tune the neural networks in a limited time.\n\\end{abstract}\n\n\\vspace{4em}\n%\\tableofcontents\n%\\vspace{3em}\n\\pagebreak\\end{@twocolumnfalse}\n}]\n\n\\section{Introduction and Overview}\nWith recent increases in computation power and data storage capabilities our ability effectively process data computationally has increases substantially. Machine learning uses statistical techniques to gather information about a dataset and use it to make predictions or understand trends. One class of machine learning methods, neural networks, is particularly effective given large datasets. If neural nets can be effectively used to predict solutions of PDEs then we can step away from trying to guess at the PDEs which actually describe systems and move towards generating trajectories using these neural nets.\n\n\\section{Theoretical Background}\n\\subsection{Neural Network Basics}\nSuppose we have sets \\( \\hat{X}_0 \\) and \\( \\hat{X}_N \\) with elements of size \\( m \\) and \\( n \\) respectively. Let \\( \\hat{f}:\\hat{X}_0 \\to \\hat{X}_N \\) be an injection. The goal of a neural network is to be able to accurately compute this injection to make predictions about elements of \\( \\hat{X}_0 \\) for which we do not know the actual value of \\( \\hat{f} \\).\n\nA feed forward neural network is a function of the form,\n\\begin{align}\n X_N = f_N(b_{N-1}+\\cdots f_2(b_1+ W_1f_1(b_0 + W_0x_0))) \\label{NN}\n\\end{align}\nwhere the the ``activation'' functions \\( f_j:\\RR\\to\\RR \\) are applied componentwise.\n\nEach ``layer'' is of the form,\n\\begin{align*}\n x_{j+1} = f_{j+1}j(W_j x_{j} + b_j), && j=0,1,\\ldots, N-1\n\\end{align*}\n\nNote that \\( W_j \\) can be of any shape compatible with \\( x_j \\) with the constraint that \\( W_{N-1} \\) must also give something of the dimension of \\( x_N \\).\n\nThe goal is to set the parameters \\( W_j \\) and \\( b_j \\) so that the neural network is equal to \\( \\hat{f} \\). However, it is clear that unless we know the output of \\( f \\) on all of \\( \\hat{X}_0 \\) this will not be possible even if our neural net has the same form as \\( \\hat{f} \\). Since in general we would like to be able to make predictions about unknown data we will have to settle for the neural network being an approximation to \\( \\hat{f} \\).\n\nMore specifically, let \\( X_0\\subset \\hat{X}_0 \\) and suppose the value of \\( \\hat{f}(x) \\) is known for all \\( x\\in X_0 \\). For convenience we will write \\( X_0 \\) as a matrix of size \\( n\\times t \\). Define \\( X_N\\subset \\hat{X}_N \\) as the \\( m\\times t \\) matrix found by applying \\( \\hat{f} \\) to the columns of \\( X_0 \\). We will use the data from \\( X_0 \\) to train the neural net to (hopefully) give us a good approximation for \\( \\hat{f} \\). Since the neural network can take any element of \\( \\hat{X}_0 \\) as input, we hope that it can be used for predictions about the value of \\( \\hat{f} \\) on these elements.\n\n\n\\subsection{Loss functions}\nGiven a neural network we would like to train it to be able to make predictions. In order to do this we need some way of saying what a good prediction is. In general this is done by defining a loss function which is small when the network gives good predictions, and large when it gives bad predictions.\n\nMore specifically, let \\( X_0\\in\\RR^{n} \\) and \\( X_N\\in\\RR^{m} \\) be the input data and target data respectively. Let \\( f:\\RR^{n}\\times\\RR^{Z}\\to\\RR^{m} \\) be a neural network as defined above, where \\( Z \\) is the number of free parameters. The residual error of the network for a fixed set of parameters \\( \\beta\\in\\RR^{Z} \\) is \\( Y - f(X,\\beta) \\).\n\nIdeally the residual error is zero. By picking some metric with which to measure the residual error, we have a minimization problem. In this paper we often chose to minimize the mean square error, which amounts to solving,\n\\begin{align}\n \\min_{\\beta} \\sum_{x_0\\in X_0} \\norm{\\hat{f}(x_0)-f(x_0,\\beta)}_2^2 \\label{mse}\n\\end{align}\n\n\\subsection{Optimizers}\nIn order to minimize expressions such as (\\ref{mse}) an optimization algorithm must be applied. Since the input and output lie in such high dimensional spaces, the loss function often has many local minimum or saddle points. As such, traditional methods such as gradient decent may not work very well. Instead there are many popular gradient based algorithms which introduce some stochasticity in order to avoid ``getting stuck''. Since our neural network can be expressed as the composition of fairly straightforward functions, the gradient with respect to entries of \\( \\beta \\) can be easily computed using the chain rule (done by something called back-propagation).\n\n\\subsection{SVD and Data reduction}\nWhen training a neural network the size of the input and output are determine by the data given. This means the network may have to be very large the the beginning and end nodes. On approach to reduce the size of the network is to project the input and target data into lower dimensional subspaces and then train on the projections. This is equivalent to the first and last hidden layers of the neural net being of smaller size and with fixed weights. If the weights are not fixed, the net will determine the ``best'' projection. However, since the SVD already computes the optimal low rank approximation in the 2-norm or Frobenius norm, it may be beneficial to set these weights so that the net does not have to learn them.\n\nSuppose our input data \\( X \\) is in \\( \\RR^{N} \\) and we have \\( T \\) samples. We then have a \\( N\\times T \\) data matrix. We would like to find a subspace of \\( \\RR^{N} \\) of lower dimension where our data can be approximated well. Our data has a rank \\( k \\) reduced SVD,\n\\begin{align}\n X \\approx U\\Sigma V^* \\label{SVD_eq}\n\\end{align}\nwhere \\( U \\) is \\( N\\times k \\), \\( \\Sigma \\) is \\( k\\times k \\), and \\( V^* \\) is \\( k\\times T \\). We interpret this in the following way: the columns of \\( U \\) are the dominant modes in our data, and the rows of \\( \\Sigma V^* \\) tell us how these modes vary in time. We can therefore use the columns of \\( \\Sigma V^* \\), which are of height \\( k \\), as our approximation to the input data.\n\nIn general the input and target data will not be of the same size or, for time series data, of the same dynamics. However, in the case that they are the rank reduction used on the input and output can be computed simultaneously saving a large computation. While it will no longer be the optimal reduction for each of these, it will be near optimal. In this paper we are generally building steppers, and so the input and target are of the same system at different times. We therefore compute the low dimensional spaces by taking the SVD of the full data set, and then split it into the input and target data sets.\n\n\\section{Algorithm Implementation and Development}\n\n\\subsection{Neural Network Framework}\nWe use the python libraries Keras and Tensorflow for most of our neural nets. In addition the neural network package for MATLAB is occasionally used.\n\nIn general, the setup and labeling of the data was the primary focus of this project. It is straightforward to change the net structure and tune other hyperparameters once this has been done. However, doing so is time consuming. If any of the nets work at all this should be taken as very encouraging since they were generally chosen at random without much justification.\n\n\n\\subsection{Data Generation}\nTo generate data for our systems we generally use an ODE stepper to solve a system of ODEs corresponding to the discrimination of a PDE. The target data is taken to be the training data one step forward in time (with respect to some fixed time mesh). Multiple initial conditions are generated in a variety of ways depending on\n\n\n\\subsection{Kuramoto-Sivashinsky Equation}\nWe are provided with a stepper to produce solutions to the Kuramoto-Sivashinsky equation (\\ref{KS_eqn}) with periodic boundary conditions on a given mesh (with \\( N \\) spatial points and \\( T \\) time points).\n\\begin{align}\n \\pp{u}{t} = -u \\pp{u}{x} - \\pp[2]{u}{x} - \\pp[4]{u}{x} \\label{KS_eqn}\n\\end{align}\n\nIn order to train our network to step forward in time we must generate a batch of trajectories stemming from some class of initial conditions. To do this we start with a zero initial condition and then randomly set a fixed (generally two or three) number Fourier modes to have weights with real and imaginary parts uniformly distributed on \\( [-N/2,N/2] \\). This produces periodic initial conditions which lead to fairly nice behavior. We then generate a solutions with these initial conditions and save them.\n\nThe data is then imported to Python and formatted into the training and target data sets by appropriate slicing. A neural net defined in Keras is trained to predict the value of \\( u(t+\\Delta t) \\) given the value of \\( u(t) \\).\n\n\\begin{figure}[t]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/svd_mode_1.pdf}\n \\caption{1st left singular vector reshaped}\n \\label{1mode}\n\\end{subfigure}\\hfill\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/svd_mode_100.pdf}\n \\caption{100th singular vector reshaped}\n \\label{100mode}\n\\end{subfigure}\n\\caption{SVD modes for \\( u \\) (left) and \\( v \\) (right) }\n\\label{SVD_modes}\n\\end{figure}\n\n\\begin{figure}[t]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/svd_coeff_1.pdf}\n \\caption{1st right singular vector varying in time}\n \\label{1coeff}\n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/svd_coeff_100.pdf}\n \\caption{100th right singular vector varying in time}\n \\label{100coeff}\n\\end{subfigure}\n\\caption{coefficients of SVD modes vs dataset}\n\\label{SVD_coeff}\n\\end{figure}\n\n\\subsection{ \\( \\boldsymbol\\lambda \\)-\\( \\boldsymbol\\omega \\) Reaction-Diffusion Equation}\nWe can apply similar methods to a PDE in two dimensions.\nWith,\n\\begin{align*}\n {\\bf u} = \\left[\\begin{array}{c}u \\\\ v\\end{array}\\right], &&\n {\\bf D} = \\left[\\begin{array}{cc}d_1 \\\\ & d_2 \\end{array}\\right]\n\\end{align*}\nthe Kuramoto-Sivashinsky equation is defined as,\n\\begin{align}\n \\pp{{\\bf u}}{t} =\n \\left[\\begin{array}{cc}\n \\lambda(s) & -\\omega(s) \\\\\n \\omega(s) & \\lambda(s)\n \\end{array}\\right]\n {\\bf u}\n + {\\bf D} \\nabla^2 {\\bf u} \\label{RD_eqn}\n\\end{align}\nwhere,\n\\begin{align*}\n s^2 = u^2 + v^2, &&\n \\lambda = 1-s^2, &&\n \\omega = -\\beta s^2\n\\end{align*}\n\nAgain we are given a stepper which produces trajectories on a fixed mesh. Like with the KS equation, we generate initial conditions by picking some Fourier modes to be nonzero and use these to generate trajectories which we load into Python.\n\nSince the data is much larger than before, training a neural net (we tried) is very slow. While the size of data used in industry is often massive, they also have massive computers and lots of time. We don't, so to deal with the large data set we reduce the rank of our data and train on the projections. Since the training and target data come from the same space we compute the SVD the entire dataset before splitting it into training and target data. This allows us to find a single subspace for both the training and target data, saving on computing and saving two subspaces (which would turn out to be almost identical since the vast majority of the data in each of these sets would be the same).\n\n\\begin{figure*}[h]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/sample_KS_trajectory.pdf}\n \\caption{trajectory generated by {\\tt solve\\_ivp}}\n \\label{sample_KS_trajectory}\n\\end{subfigure}\\hfill\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/predicted_KS_trajectory.pdf}\n \\caption{trajectory generated by neural net}\n \\label{predicted_KS_trajectory}\n\\end{subfigure}\n\\caption{``actual'' and neural net predicted trajectories for Kuramoto-Sivashinsky equation (\\ref{KS_eqn})}\n\\label{KS_trajectory}\n\\end{figure*}\n\nFigure~\\ref{1mode} shows the dominant left singular vector and Figure~\\ref{1coeff} shows how the first right singular vector varies in time. The ``spikes'' down clearly are related to the initial conditions of the data. However, after each spike there is an oscillation in this mode. Moreover, the mode itself has quite a bit of structure, which it seems to have inherited from the way we generated initial conditions. It is not clear whether this structure would die out if we ran the simulations for longer time. Regardless, since the structure is present in our data set we can use this to train a neural net on the rank-reduced data.\n\nWe take the rank \\( k \\) SVD (\\ref{SVD_eq}) (determined by a parameter flag) and then train the network on the weights of the basis of the subspace spanned by the first \\( k \\) columns of \\( U \\).\n\nTo test our prediction we take a trajectory not used for training, project it to the column span of \\( U \\), iteratively apply the network, and then embed back to the original space.\n\n\n\\subsection{Lorenz Equation}\nThe Lorenz equation (\\ref{lorenz_eq}) is a system of ODEs which have chaotic solutions for some parameters. It was demonstrated in \\cite{lecture_notes} that for a fixed \\( \\Delta t \\) a simple neural network can be trained to predict the position at a time \\( t+\\Delta t \\) given the position at time \\( t \\) accurately enough that the trajectory determined by iteratively applying the trained neural net matches the trajectory of a given ODE solver almost exactly. Given that the system is chaotic for these values it is somewhat surprising that the neural net manages to follow the same trajectory by only predicting one step at a time. Training data was generating using Scipy's {\\tt solve\\_ivp} with a tolerance of \\( 10^{-10} \\) and evaluating the solution along a mesh of uniformly spaced times.\n\n\\begin{align}\n \\pp{}{t} \\left[\\begin{array}{c}x\\\\y\\\\z\\end{array}\\right]\n =\n \\left[\\begin{array}{c}\n \\sigma(y-x) \\\\\n x (\\rho-z)-y \\\\\n x y-\\beta z\n \\end{array}\\right] \\label{lorenz_eq}\n\\end{align}\n\nWe explore the Lorenz equation in two further ways, namely trying to predict trajectories for varying values of the parameter \\( \\rho \\) and trying to predict when the solution will transition from one lobe to another.\n\nThe first task is straightforward. In particular, a neural net is trained on data where the input corresponds to the current position as well as the value of \\( \\rho \\), and the output is the position after a time of \\( \\Delta t \\). More specifically, we fix \\( \\sigma = 10 \\), \\( \\beta = 8/3 \\), and train on data with \\( \\rho = 10,28,40 \\). We then try to use this net to predict trajectories for \\( \\rho = 17 \\) and \\( \\rho = 35 \\).\n\nThe second tasks requires some interpretation. We decided to train a network to determine how long until a lobe switch. To train such a network requires that we know how long it will be until the solution switches lobes. To do this we first classify what it means to be at a given lobe. This is done by separating the data with plane as shown in Figure~\\ref{separating_hyperplane}. Let \\( c \\) be a normal vector for this hyperplane. Then the sign of \\( c^Tx \\) will determine which side of the plane a given point is on.\n\nWe pick \\( c \\) roughly in the direction of the vector connecting the two centers in the \\( x \\)-\\( y \\) plane. While there are probably better ways to do this, it seems like the projection to this plane provides a straightforward way to separate points circle different lobes.\n\nTherefore, given a trajectory \\( X \\), each point in the trajectory can be classified as ``left'' or ``right''. Once this is done, we know that transitions across the plane occur when the trajectory switches from ``left'' to ``right'' or from ``right'' to ``left''. We compute these transition points by taking the difference of consecutive points in \\( \\operatorname{sign}(c^TX) \\). If the difference is nonzero then the trajectory has switched lobes. Once we have these transition points it is relatively straightforward label each point in the trajectory with far each point is from the next crossover.\n\nAs a minor note, some of the data from the end of the each trajectory is discarded because it is not possible to tell when the next transition will occur without stepping out the solution further. We then train a neural network to try and predict the time until a crossover.\n\\begin{figure}[tb!]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/uv_t180.pdf}\n \\caption{original data}\n \\label{time_image}\n\\end{subfigure}\\hfill\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/svd_t180.pdf}\n \\caption{SVD (rank 100)}\n \\label{SVD_time_image}\n\\end{subfigure}\n\\caption{snapshot of \\( u \\) (left) and \\( v \\) (right) at time \\( t=180 \\)}\n\\label{RD_SVD}\n\\end{figure}\n\n\\begin{figure}[b!]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/prediction_t15.pdf}\n \\caption{actual trajectory}\n \\label{sample_RD_trajectory}\n\\end{subfigure}\\hfill\n\\begin{subfigure}{.45\\textwidth}\\centering\n \\includegraphics[width=\\textwidth]{img/svd_prediction_t15.pdf}\n \\caption{predicted trajectory}\n \\label{predicted_RD_trajectory}\n\\end{subfigure}\n\\caption{``actual'' and neural net predicted trajectories for Reaction Diffusion Equation (\\ref{RD_eqn}) at \\( t=15 \\)}\n\\label{RD_trajectory}\n\\end{figure}\n\n\\begin{figure}[t!]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n\\includegraphics[width=\\textwidth]{img/NN_lrz17_trajectory.pdf}\n\\caption{\\( \\rho = 17 \\)}\n\\end{subfigure}\\hfill\n\\begin{subfigure}{.45\\textwidth}\\centering\n\\includegraphics[width=\\textwidth]{img/NN_lrz35_trajectory.pdf} \\hfill\n\\caption{\\( \\rho = 35 \\)}\n\\end{subfigure}\n\\caption{Actual trajectory (solid) vs predicted trajectory (dotted) for variious values of \\( \\rho \\)}\n\\label{xyz_trajectory}\n\\end{figure}\n\n\n\n\n\\section{Computational Results}\n\n\\subsection{Kuramoto-Sivashinsky Equation}\n\nFigure~\\ref{sample_KS_trajectory} shows a sample trajectory generated by our ODE stepper (not part of traninig data). Figure~\\ref{predicted_KS_trajectory} shows the prediction of the neural net on the same trajectory. However, as time progresses the neural net predicted solution does not really change, while the reaction diffusion equation does continue to evolve.\n\n\n\\subsection{ \\( \\boldsymbol\\lambda \\)-\\( \\boldsymbol\\omega \\) Reaction-Diffusion Equation}\nFigure~\\ref{time_image} shows a snaptshot of a trajectory at time \\( t=180 \\). Figure~\\ref{SVD_time_image} shows the same snapshot after it has been projected into a 100 dimensional subspace determined by the SVD. While a plot of the singular values shows no clear cutoff in the rank, visually it seems that the data is fairly well represented by the first 100 modes. We proceed to train a neural network on the rank reduced data.\n\n\\begin{figure}[bt!]\\centering\n\\begin{subfigure}{.45\\textwidth}\\centering\n\\includegraphics[width=\\textwidth]{img/separating_hyerplane.pdf}\n\\caption{Projection of sample trajectory onto the \\( x \\)-\\( y \\) plane along with image of plane \\( y + 5x +0z = 0 \\).}\n\\label{separating_hyperplane}\n\\end{subfigure}\\hfill\n\\begin{subfigure}{.45\\textwidth}\\centering\n\\includegraphics[width=\\textwidth]{img/jump_predictor.pdf}\n\\caption{Actual time to jump (solid) vs. predicted time to jump (dotted) for sample trajectory .}\n\\label{jump_predictor}\n\\end{subfigure}\n\\caption{}\n\\end{figure}\n\n\\subsection{Lorenz Equation}\nSample predictions for \\( \\rho = 17 \\) and \\( \\rho = 35 \\) produced by the trained net are shown in Figure~\\ref{xyz_trajectory}. Note that we do not cross validate in the normal sense, since that would be testing how well our net can predict one step in the future. In fact, the cross validation would show a very high accuracy since we are able to predict the long term behavior relatively well despite only training n the current position.\n\nFigure~\\ref{jump_predictor} shows the results of applying the trained net to every point in a trajectory. This plot makes it clear that we are actually able to fairly effectively determine when a jump will occur given only the current position. Note here that the horizontal axis labeling is just the index of the point in the trajectory and has no impact on the neural net's prediction as it is applied independently to each point.\n\n\\pagebreak\n\\section{Summary and Conclusions}\nIt is clear that the trajectories of the Kuramoto-Sivashinsky and Reaction Difussion equations were not effectively predicted. On the other hand, solutions to the Lorenz equation, even for values of \\( \\rho \\) which the neural net had never seen, were quite successful. Finally, we were able to train a neural net to predict how long it would take a trajectory to switch lobes.\n\nDue to time constraints only simple nets were used and all hyperparameter training was done by guessing. We instead chose to focus on producing the training data in a scalable way so that in the future it would be a trivial task to generate data at large mesh sizes. With the current state of our codebase, testing new neural nets is as easy as changing a few lines in Keras. Given more time we would have liked to set up convolutional neural nets to see if the local structure of the data would be enough to determine the global behavior. In general we probably did not use enough data, and the nets we used are probably too small to be able to get very good results. However, what we did find is somewhat encouraging and given more time would be worth pursuing.\n\n\n\\bibliographystyle{plain}\n\\bibliography{hw2}\n\n\\onecolumn\n\\section{Appendix A}\n\\label{AppendixA}\nAll functions are included in the code in Appendix B as they are specific to the task performed in each of the files.\n\n%\\pagebreak\n\\section{Appendix B}\n\\lstinputlisting[]{PDECODES/KS_data_gen.m}\n\\lstinputlisting[]{python/hw2_ks.py}\n\\lstinputlisting[]{PDECODES/RD_data_gen.m}\n\\lstinputlisting[]{python/hw2_rd.py}\n\\lstinputlisting[]{python/hw2_lrz.py}\n\\lstinputlisting[]{python/lrz_classifier.py}\n\n\n\\end{document}\n", "meta": {"hexsha": "4f640311ae3686100c2f9479b48d624d5c72a19e", "size": 23246, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "amath563/hw2/hw2.tex", "max_stars_repo_name": "interesting-courses/UW_coursework", "max_stars_repo_head_hexsha": "987e336e70482622c5d03428b5532349483f87f4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-08-19T01:59:25.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-31T12:32:59.000Z", "max_issues_repo_path": "amath563/hw2/hw2.tex", "max_issues_repo_name": "interesting-courses/UW_coursework", "max_issues_repo_head_hexsha": "987e336e70482622c5d03428b5532349483f87f4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "amath563/hw2/hw2.tex", "max_forks_repo_name": "interesting-courses/UW_coursework", "max_forks_repo_head_hexsha": "987e336e70482622c5d03428b5532349483f87f4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-03-31T22:23:46.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-29T22:13:01.000Z", "avg_line_length": 74.0318471338, "max_line_length": 801, "alphanum_fraction": 0.7511829992, "num_tokens": 5922, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.554470450236115, "lm_q2_score": 0.6297745935070806, "lm_q1q2_score": 0.34919140240913726}} {"text": "\\documentclass[]{article}\n\\usepackage{proceed2e}\n\\usepackage{amssymb,amsmath,amsthm}\n\\usepackage{graphicx}\n\\usepackage{preamble}\n\\usepackage{natbib}\n\\usepackage{hyperref}\n\\usepackage{color}\n\\definecolor{mydarkblue}{rgb}{0,0.08,0.45}\n\\hypersetup{ %\n pdftitle={},\n pdfauthor={},\n pdfsubject={},\n pdfkeywords={},\n pdfborder=0 0 0,\n pdfpagemode=UseNone,\n colorlinks=true,\n linkcolor=mydarkblue,\n citecolor=mydarkblue,\n filecolor=mydarkblue,\n urlcolor=mydarkblue,\n pdfview=FitH}\n\n\n\n\\title{Bayesian Quadrature minimizes Maximum Mean Discrepancy}\n\n\\author{ {\\bf Ferenc Husz\\'{a}r} \\\\\nDepartment of Engineering\\\\\nCambridge University\\\\ \n\\texttt{fh277@cam.ac.uk}\n\\And \n{\\bf David Duvenaud } \\\\ %\\thanks{Both authors contributed equally.} \\\\ \nDepartment of Engineering\\\\ \nCambridge University \\\\\n\\texttt{dkd23@cam.ac.uk}\n} \n\n\\begin{document} \n \n\\maketitle \n \n%\\begin{abstract} \n%Maximum Mean Discrepancy is equivalent to the posterior variance of an integrated Gaussian process.\n%\\end{abstract} \n\n \n\\section{INTRODUCTION}\nA common problem in statistical machine learning is to compute expectations of functions over probability distributions of the form:\n\\begin{equation}\n\tZ_{f,p} = \\int f(x) p(x) dx \\label{eqn:integral}\n\\end{equation}\nExamples include computing marginal distributions, making predictions marginalizing over parameters, or computing the Bayes risk in a decision problem. %In this paper we assume that the distribution $p(x)$ is known in analytic form, and $f(x)$ can be evaluated at arbitrary locations.\n\n%Monte Carlo methods produce random samples from the distribution $p$ and then approximate the integral by taking the empirical mean $\\hat{Z} = \\frac{1}{N}\\sum_{n=1}^{N}f_{x_n}$ of the function evaluated at those points. This non-deterministic estimate converges at a rate $\\mathcal{O}(\\frac{1}{\\sqrt{N}})$. When exact sampling from $p$ is impossible or impractical, Markov chain Monte Carlo (MCMC) methods are often used. MCMC methods can be applied to almost any problem but convergence of the estimate depends on several factors and is hard to estimate \\citep{CowlesCarlin96}. The focus of this paper is on quasi-Monte Carlo methods that -- instead of sampling randomly -- produce a set of pseudo-samples in a deterministic fashion. These methods operate by directly minimising some sort of discrepancy between the empirical distribution of pseudo-samples and the target distribution. Whenever these methods are applicable, they achieve convergence rates superior to the $\\mathcal{O}(\\frac{1}{\\sqrt{N}})$ rate typical of random sampling.\n\n%In this paper we highlight and explore the connections between two deterministic sampling and integration methods: Bayesian quadrature (\\bq{}) \\citep{BZHermiteQuadrature,BZMonteCarlo} (also known as Bayesian Monte Carlo) and kernel herding \\citep{chen2010super}. Bayesian quadrature estimates integral \\eqref{eqn:integral} by inferring a posterior distribution over $f$ conditioned on the observed evaluations $f_{x_n}$, and then computing the posterior expectation of $Z_{f,p}$. The points where the function should be evaluated can be found via Bayesian experimental design, providing a deterministic procedure for selecting sample locations.% We call this procedure Sequential Bayesian Quadrature (\\sbq).\n\n%Herding, proposed recently by \\cite{chen2010super}, produces pseudosamples by minimising the discrepancy of moments between the sample set and the target distribution. Similarly to traditional Monte Carlo, an estimate is formed by taking the empirical mean over samples $\\hat{Z} = \\frac{1}{N}\\sum_{n=1}^{N}f_{x_n}$. Under certain assumptions, herding has provably fast, $\\mathcal{O}(\\frac{1}{N})$ convergence rates in the parametric case, and has demonstrated strong empirical performance in a variety of tasks.\n\nIn this paper, we make two main contributions. Firs, we show that the Maximum Mean Discrepancy (MMD) criterion (used to choose samples in kernel herding) is identical to the expected error in the estimate of the integral $Z_{f,p}$ under a Gaussian process prior for $f$. This expected error is the criterion being minimized when choosing samples for Bayesian quadrature (BQ). Because Bayesian quadrature assigns different weights to each of the observed function values $f(\\vx)$, we can view Bayesian quadrature as a weighted version of kernel herding. \n We show that these weights are optimal in a minimax sense over all functions in the Hilbert space defined by our kernel. This implies that Bayesian quadrature dominates kernel herding and other non-optimally weighted herding in rate of convergence.\n\nWe further show that the MMD, when using BQ weights, is submodular in the samples chosen, which implies that sequential BQ achieves the optimal rate of convergence for any sampling method.\n\n\\section{HERDING} \n\nHerding was introduced by \\cite{welling2009herding} as a method for generating pseudo-samples from a distribution in such a way that certain nonlinear moments of the sample set closely match those of the target distribution. The empirical mean $\\frac{1}{N}\\sum_{n=1}^{N}f_{x_n}$ over these pseudosamples is then used to estimate integral \\eqref{eqn:integral}.\n\n\\subsection{Maximum Mean Discrepancy}\n\nFor selecting pseudosamples, herding relies on an objective based on the maximum mean discrepancy \\citep[MMD;\\ ][]{Sriperumbudur2010}: %MMD measures the divergence between two distributions, $p$ and $q$ with respect to a class of integrand functions $\\mathcal{F}$ as follows:\n%\n\\begin{align}\n\t\\mmd_{\\mathcal{F}}\\left(p,q\\right) = \\sup_{f\\in\\mathcal{F}}\\left\\vert\\int f_x p(x) dx - \\int f_x q(x) dx \\right\\vert\n\\end{align}\n%\n%Intuitively, if two distributions are close in the MMD sense, then no matter which function $f$ we choose from $\\mathcal{F}$, the difference in its integral over $p$ or $q$ should be small. A particularly interesting case is \n%\nWhen the function class $\\mathcal{F}$ is functions of unit norm from a reproducing kernel Hilbert space (RKHS) $\\He$, the MMD between two distributions can be conveniently expressed using only expectations of the associated kernel $k(x, x')$ \\citep{Sriperumbudur2010}.\n%\n%\\begin{align}\n%MMD^2_{\\He}(p,q) =& \\sup_{\\substack{f\\in\\He\\\\\\Hnorm{f}=1}}\\left\\vert\\int f_x p(x) dx - \\int f_x q(x) dx\\right\\vert^2\\\\\n%\t=& \\Hnorm{\\mu_{p} - \\mu_{q}}^2\\\\\n%\\nonumber\t=&\\iint k(x,y) p(x) p(y) dx dy\\\\\n%\\nonumber\t-2 &\\iint k(x,y) p(x) q(y) dx dy\\\\\n%\t+ &\\iint k(x,y) q(x) q(y) dx dy,\n%\\end{align}\n%\n%where in the above formula $\\mu_{p}=\\int \\phi(\\vx)p(\\vx)d\\vx\\in\\He$ denotes the \\emph{mean element} associated with the distribution $p$.\n% For characteristic kernels, such as the Gaussian kernel, the mapping between a distribution and its mean element is bijective. As a consequence $\\mmd_{\\He}(p,q)=0$ if and only if $p=q$, making it a powerful measure of divergence.\n%\nHerding uses MMD to evaluate how well the sample set $\\{\\vx_1,\\ldots,\\vx_{N}\\}$ represents the target distribution $p$, adding points greedily according to:\n%\n%\\begin{align}\n%\t\\epsilon_{herding}&\\left(\\{\\vx_1,\\ldots,\\vx_{N}\\}\\right) = \\mmd_{\\He}\\left(p,\\frac{1}{N}\\sum_{n=1}^{N}\\delta_{x_n}\\right)\\\\\n%\\nonumber\t=&\\iint k(x,y) p(x) p(y) dx dy\\\\\n%\t\t-2 &\\frac{1}{N}\\sum_{n=1}^{N}\\int k(x,x_n) p(x) dx\n%\t\t+ \\frac{1}{N^2}\\sum_{n,m=1}^{N} k(x_n,x_m)\n%\\label{eq:mmd_assumption}\n%\\end{align}\n%\n%The herding procedure greedily minimizes its objective $\\epsilon_{herding}\\left(\\{\\vx_1,\\ldots,\\vx_{N}\\}\\right)$ , adding pseudosamples $\\vx_n$ one at a time. When selecting the $n+1$-st pseudosample:\n%\n\\begin{align}\n\\vx_{n+1} &\\leftarrow \\argmin_{\\vx \\in \\mathcal{X}} \\label{eqn:herding_criterion} \\epsilon_{herding}\\left(\\{\\vx_1,\\ldots,\\vx_{n},\\vx\\}\\right)\\\\\n\t&= \\argmax_{\\vx \\in \\mathcal{X}} 2 \\expectargs{\\vx' \\sim p}{k(\\vx, \\vx')} - \\frac{1}{n+1}\\sum_{m=1}^{n} k(\\vx,\\vx_m)\\mbox{,}\\notag\n\\end{align}\n%\nassuming $k(\\vx,\\vx) = \\mbox{const}$.\nThe formula \\eqref{eqn:herding_criterion} admits an intuitive interpretation: the first term encourages sampling in areas with high mass under the target distribution $p(\\vx)$. The second term discourages sampling at points close to existing samples. \n\n\n\n\n\\section{BAYESIAN QUADRATURE} \n\n%\\begin{figure}\n%\\centering\n%\\includegraphics[width=\\columnwidth]{figures/bq_intro4}\n%\\caption{An illustration of Bayesian Quadrature. The function $f(x)$ is sampled at a set of input locations. %This induces a Gaussian process posterior distribution on $f$, which is integrated in closed form against the target density, $p(\\vx)$. Since the amount of volume under $f$ is uncertain, this gives rise to a (Gaussian) posterior distribution over $Z_{f,p}$.}\n%\\label{fig:bq_intro}\n%\\end{figure}\n\nBayesian quadrature (\\bq) puts a prior distribution on $\\vf$, then estimates integral \\eqref{eqn:integral} by inferring a posterior distribution over the function $\\vf$, conditioned on the observations $\\vf(\\vx_n)$ at some query points $\\vx_n$. For simplicity, $\\vf$ is assigned a Gaussian process prior with kernel function $k$ and mean $0$. This assumption is very similar to the one made by kernel herding. The posterior distribution over $f$ then implies a distribution over $Z_{f,p}$. %See Figure \\ref{fig:bq_intro} for an illustration of Bayesian Quadrature.\n%\n%\\subsection{ BQ Estimator}\n%\n%Here we derive the \\bq{} estimate of \\eqref{eqn:integral}, after conditioning on function evaluations $\\vf(\\vx_1) \\dots \\vf(\\vx_N)$, denoted as $f(\\vX)$. \n The mean of this distribution, $\\expectargs{}{Z}$ is the optimal Bayesian estimator for a squared loss.\n%After conditioning on $\\vf(\\vX)$, we obtain a closed-form posterior over $\\vf$.\n%\n%\\begin{align}\n%p(\\vf(\\vx\\st)|\\vf(\\vX)) = \\N{\\vf(\\vx\\st)}{\\mf(\\vx\\st)}{\\cov(\\vx\\st,\\vx\\st')}\n%\\end{align} \n%where\n%\\begin{align}\n%\\mf(\\vx\\st) = & k(\\vx\\st, \\vX) K^{-1} \\vf(\\vX) \\\\\n%\\cov(\\vx\\st, \\vx\\st') = & k(\\vx\\st,\\vx\\st) - k(\\vx\\st, \\vX) K^{-1} k(\\vX, \\vx\\st)\n%\\end{align} \n%\n%and $K = k(\\vX, \\vX)$. \n%\nConveniently, the \\gp{} posterior allows us to compute the expectation of \\eqref{eqn:integral} in closed form: \n%\n%\\begin{align}\n%Z & = \\int f(\\vx)p(\\vx)d\\vx\n%\\end{align} \n%so we integrate over functions to get:\n\\begin{align}\n\\expectargs{\\gp}{Z} & = \\vz^T K^{-1} \\vf(\\vX)\n\\label{eq:marg_mean_symbolic}\n\\end{align} \nwhere\n\\begin{align}\nz_n & = \\int\\!\\! k(\\vx, \\vx_n) p(\\vx) d\\vx = \\expectargs{\\vx' \\sim p}{k(\\vx_n, \\vx')}.\n\\end{align}\n%\nAs in kernel herding, the desired expectation of $Z_{f,p}$ is simply a linear combination of observed function values $\\vf(\\vx)$:\n%\n\\begin{align}\n\\expectargs{\\gp}{Z} & = \\vz^T K^{-1} \\vf(\\vX) \\\\\n & = \\sum_n w_{\\bq}^{(n)} f(x_n) \\quad \\textrm{where} \\quad w_{\\bq}^{(n)} = \\sum_m \\vz_j K^{-1}_{nm}\n\\end{align} \n%\n%\\begin{align} \n%w_{\\bq}^{(n)} & = \\sum_m \\vz_j K^{-1}_{nm}\n%\\label{eq:bq_weights}\n%\\end{align}\n%\nThus, we can view the BQ estimate as a weighted version of the herding estimate.\n\n\n\\subsection{Optimal sampling for BQ}\n\nThe posterior variance $\\varianceargs{}{Z_{f,p}|f_{x_1}, \\dots, f_{x_N}}$ quantifies our uncertainty in the estimate of the integral. When selecting locations to evaluate the function $f$, minimising the posterior variance is a sensible strategy. Below, we give a closed form formula for the posterior variance of $Z_{f,p}$, conditioned on the observations $f_{x_1} \\dots f_{x_N}$, which we will denote by $\\epsilon^2_{\\bq{}}$.\n\\begin{align}\n\\epsilon^{2}_{\\bq{}} = \\varianceargs{}{Z_{f,p}|f_{x_1}, \\dots, f_{x_N}} \n% \\nonumber & = \\expectargs{f \\sim \\gp, p\\sim p(x)}{ \\left( f(\\vx) - \\mf(\\vx) \\right)\\left( f(\\vx') - \\mf(\\vx') \\right)} \\\\ \n%\\nonumber & = \\int \\Bigg( \\!\\! \\left( \\int f(\\vx) p(\\vx) d\\vx - \\int \\mf(\\vx') p(\\vx') d\\vx' \\right) \\\\ \n%\\nonumber & \\quad \\times \\left( \\int f(\\vx) p(\\vx) d\\vx - \\int \\mf(\\vx') p(\\vx') d\\vx' \\right) \\!\\! \\Bigg) p(f) df \\\\ \n%\\nonumber & = \\int\\!\\!\\! \\int\\!\\! \\int\\!\\! \\left[ f(\\vx) - \\mf(\\vx) \\right] \\left[ f((\\vx') - \\mf(\\vx') \\right] p(f) df \\\\\n%\\nonumber & \\qquad \\times p(\\vx) p(\\vx') d\\vx d\\vx' \\\\\n%\\nonumber & = \\int\\!\\! \\!\\int\\!\\! \\Cov \\left[ f((\\vx), f((\\vx') \\right] p(\\vx) p(\\vx') d\\vx d\\vx' \\\\\n%\\nonumber & = \\int\\!\\!\\! \\int\\!\\! \\left[ k(\\vx, \\vx') - k(\\vx, \\vX) K^{-1} k(\\vX, \\vx') \\right] \\\\\n%\\nonumber & \\qquad \\times p(\\vx) p(\\vx') d\\vx d\\vx' \\\\ \n%\\nonumber & = \\int\\!\\!\\! \\int\\!\\! k(\\vx, \\vx') p(\\vx) p(\\vx') d\\vx d\\vx' \\\\\n%\\nonumber & \\quad - \\left[ \\int\\!\\! k(\\vx, \\vX) p(\\vx) d\\vx \\right] K^{-1} \\left[ \\int\\!\\! k(\\vX, \\vx') p(\\vx') d\\vx' \\right] \\\\\n = \\expectargs{\\vx, \\vx' \\sim p}{k(\\vx, \\vx')} - \\vz K^{-1} \\vz\\mbox{,}\n\\label{eq:marg_var_symbolic}\n\\end{align}\n%where $\\vz_n = \\expectargs{\\vx' \\sim p}{k(\\vx_n, \\vx')}$ as before. \nPerhaps surprisingly, the posterior variance of $Z_{f,p}$ does not depend on the observed function values, only on the location $x_n$ of samples. This allows the optimal samples to be computed ahead of time, before observing any values of $f$ at all \\citep{minka2000dqr}.\n%\\begin{align}\n%\\epsilon^{2}_{\\bq{}}(\\vx_1,\\ldots,\\vx_N) = \\expectargs{\\vx, \\vx' \\sim p}{k(\\vx, \\vx')} - \\vz K^{-1} \\vz\n%\\end{align}\n%We can contrast the \\bq{} objective $\\epsilon^{2}_{\\bq{}}$ to the objective being minimized in herding, $\\epsilon^{2}_{herding}$ of equation \\eqref{eq:mmd_assumption}. \n%Just like $\\epsilon^{2}_{herding}$, $\\epsilon^{2}_{\\bq{}}$ expresses a trade-off between accuracy and diversity of samples. On the one hand, as samples get close to high density regions under $p$, the values in $\\vz$ increase, which results in decreasing variance. On the other hand, as samples get closer to each other, eigenvalues of $K$ increase, resulting in an increase in variance. \n\n\\section{RELATING $\\varianceargs{}{Z_{f,p}}$ TO $\\mmd$}\n\nThe similarity in the behaviour of $\\epsilon^{2}_{herding}$ and $\\epsilon^{2}_{\\bq{}}$ is not a coincidence, the two quantities are closely related to each other, and to \\mmd.\n\n\\begin{prop} The expected variance in the Bayesian quadrature $\\epsilon^{2}_{\\bq{}}$ is the maximum mean discrepancy between the target distribution $p$ and $q_{\\bq{}}(x) = \\sum_{n=1}^{N}w^{(n)}_{\\bq{}}\\delta_{x_n}(x)$\n\\end{prop}\n%\n%\\begin{proof} Proof omitted for space reasons.\n%The proof involves invoking the representer theorem, using bilinearity of scalar products and the fact that if $f$ is a standard Gaussian process then $\\forall g\\in\\He: \\left\\langle f,g\\right\\rangle \\sim \\mathcal{N}(0,\\Hnorm{g})$:\n%\\begin{align}\n%&\\varianceargs{}{Z_{f,p}\\vert f_{x_1}, \\dots, f_{x_N}}=\\\\\n%\t&= \\mathbb{E}_{f\\sim GP} \\left( \\int f(x) p(x) dx - \\sum_{n=1}^{N}w^{(n)}_{\\bq{}} f(x_n)\\right)^2\\\\\n%\t&= \\mathbb{E}_{f\\sim GP} \\left( \\int \\left\\langle f, \\phi (x)\\right\\rangle p(x) dx - \\sum_{n=1}^{N}w^{(n)}_{\\bq{}} \\left\\langle f, \\phi (x_n)\\right\\rangle\\right)^2\\\\\n%\t&= \\mathbb{E}_{f\\sim GP} \\left\\langle f , \\int\\phi(x) dx - \\sum_{n=1}^{N}w^{(n)}_{\\bq{}}\\phi(x_n)\\right\\rangle^2\\\\\n%\t&= \\Hnorm{\\mu_p - \\mu_{q_{\\bq{}}}}^2\\\\\n%\t&= \\mmd^2(p,q_{\\bq{}})\n%\\end{align}\n%\\end{proof}\n\nWe know that the the posterior mean $\\expectargs{\\gp}{Z_{f,p}\\vert f_1,\\ldots,f_N}$ is a Bayes estimator and has therefore the minimal expected squared error amongst all estimators. This allows us to further rewrite $\\epsilon^{2}_{\\bq{}}$ into the following minimax forms:\n%\n\\begin{align}\n\\epsilon^{2}_{\\bq{}} %&= \\sup_{\\substack{f\\in\\He\\\\\\Hnorm{f}{\\He}=1}} \\left| \\int f_x p(x) dx - \\sum_{n=1}^{N}w^{(n)}_{\\bq{}} f_{x_n}\\right|^2\\\\\n%\t&= \\inf_{\\hat{Z}:\\mathcal{X}^N\\mapsto\\mathbb{R}} \\sup_{\\substack{f\\in\\He\\\\\\Hnorm{f}{\\He}=1}} \\left| Z - \\hat{Z}\\left(f_{x_1},\\ldots,f_{x_N}\\right)\\right|^2\\\\\n\t&= \\inf_{\\bm{w}\\in\\mathbb{R}^N} \\sup_{\\substack{f\\in\\He\\\\\\Hnorm{f}{\\He}=1}} \\left| \\int f_x p(x) dx - \\sum_{n=1}^{N}w_n \tf_{x_n}\\right|^2\n\\end{align}\n%\n%Looking at $\\epsilon^{2}_{\\bq{}}$ this way, we may discover the deep similarity to the criterion $\\epsilon^2_{herding}$ that kernel herding minimises. \nOptimal sampling for Bayesian quadrature minimises the same objective as kernel herding, but with the uniform $\\frac{1}{N}$ weights replaced by the optimal weights. As a corollary\n%\n\\begin{align}\n\\epsilon^{2}_{\\bq{}}(x_1,\\ldots,x_N) \\leq \\epsilon^{2}_{KH} (x_1,\\ldots,x_N)\n\\end{align}\n%\nIt is interesting that $\\epsilon^{2}_{\\bq{}}$ has both a Bayesian interpretation as posterior variance under a Gaussian process prior, and a frequentist interpretation as a minimax bound on estimation error with respect to an RKHS. Figure \\ref{fig:bound_curve} demonstrates this closed-form, quickly shrinking bound on the error of the Bayesian estimator. This equivalence also implies that the MMD is proportional to the expected error when estimating integrals of functions drawn from a GP prior.\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{figures/bound_curve_rkhs}\n\\caption{The empirical error rate in estimating $Z_{f,p}$, for the \\sbq{} estimator, on 10 random functions drawn from the RKHS corresponding to the kernel used. Also shown is the upper bound on the error rate implied by the $\\mmd$.}\n\\label{fig:bound_curve}\n\\end{figure}\n%\n%Figure \\ref{fig:bound_curve} demonstrates the MMD bound on the error of the \\sbq{} estimator, on random functions drawn from the RKHS corresponding to the kernel used. The empirical error always falls below the bound given by the MMD.\n\n\n\\subsection{Convergence Rates}\n\nIn finite dimensional Hilbert spaces, the herding algorithm has been shown to reduce $\\mmd$ at a rate $\\mathcal{O}(\\frac{1}{N})$, which compares favourably with the $\\mathcal{O}(\\frac{1}{\\sqrt{N}})$ rate obtained by non-deterministic Monte Carlo samplers.% However, as pointed out by \\cite{bach2012equivalence}, this fast convergence is not guaranteed in infinite dimensional Hilbert spaces, such as the RKHS corresponding to the Gaussian kernel.\nwe have shown that the greedy Bayesian estimator obtains the optimal rate for any sampling method. We have empirical results demonstrating a rate of convergence much faster than $\\mathcal{O}(\\frac{1}{N})$, but this rate is unkown to the authors, and we wish to pose this question to the kernel community.\n\n\\pagebreak\n\n\\bibliographystyle{icml2012}\n\\bibliography{herding}\n\n\\end{document} \n", "meta": {"hexsha": "a038e1748fab8a563d01e4dd2f63567be329161a", "size": 18006, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/icml-workshop-submission.tex", "max_stars_repo_name": "duvenaud/herding-paper", "max_stars_repo_head_hexsha": "cecf8f1af028efd95c6262fb257f4ed9978a57e1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 17, "max_stars_repo_stars_event_min_datetime": "2015-04-16T14:19:50.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-11T22:45:05.000Z", "max_issues_repo_path": "paper/icml-workshop-submission.tex", "max_issues_repo_name": "duvenaud/herding-paper", "max_issues_repo_head_hexsha": "cecf8f1af028efd95c6262fb257f4ed9978a57e1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/icml-workshop-submission.tex", "max_forks_repo_name": "duvenaud/herding-paper", "max_forks_repo_head_hexsha": "cecf8f1af028efd95c6262fb257f4ed9978a57e1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2017-11-23T13:45:20.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-25T04:31:47.000Z", "avg_line_length": 70.6117647059, "max_line_length": 1040, "alphanum_fraction": 0.7088192825, "num_tokens": 5645, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982645, "lm_q2_score": 0.6477982315512488, "lm_q1q2_score": 0.34915238679154603}} {"text": "%==============================================================================\n\\chapter{In silico quantitative mapping of force-pCa curves to whole heart contraction and relaxation}\\label{cha:chapter8}\n%==============================================================================\n%\n%\n%\n\\begin{remark}{Outline}\n In this chapter, we focus on the assessment of the steady-state F-pCa relationship (this is traditionally used to assess the myofilament $\\Ca$ sensitivity) and its quantitative mapping to the LV contractile function. We start by investigating the F-pCa curve using the cell contraction sub-model of our full personalised SHAM rat heart contraction model (Section~\\ref{sec:cellcontr}). Next, using a combination of simulators and emulators (Section~\\ref{subsec:quantlink}), we show that changes in key features of the F-pCa curve cannot be uniquely described by changes in the sarcomere properties (Sections~\\ref{sec:changespCa50result1}--\\ref{sec:changespCa50result2}). At the same time, we show that changes in the LV function (e.g. ejection fraction, isovolumetric relaxation time, peak pressure) cannot be uniquely described by changes in the sarcomere properties (Section~\\ref{sec:changesLVfunctionresult}). By coupling these two pieces of information, we demonstrate that the mapping from the F-pCa curve to the LV function and the corresponding inverse mapping are non-unique (Section~\\ref{sec:ch8nonuniquemappings}). We then include a discussion of results and address specific limitations (Section~\\ref{sec:ch8discussion}), and we conclude with a brief summary (Section~\\ref{sec:ch8summary}).\n\\end{remark}\n\n\n%\n%\n%\n\\section{Motivation}\\label{sec:ch8motivation}\nOne potential assessment for sarcomere modulators is their impact on the F-pCa~\\cite{Walker:2010}, which is a technique that is widely used in many HF diseases including rare/genetic diseases such as dilated and hypertrophic cardiomyopathies~\\cite{Groen:2020,Bai:2013,Michael:2016,Kirschner:2005,Harris:2002}, as well as for HF with reduced and preserved ejection fraction~\\cite{Nagy:2015,Kampourakis:2018,Kieu:2019,Awinda:2021,Mamidi:2018,Sparrow:2020}. The impact on the F-pCa curve is often examined in terms of observed shifts in the curve half-maximal activation ($\\pCaf$), which is used as an index of $\\Ca$ sensitivity. Leftward shifts in the $\\pCaf$ are expected to improve contractility, whereas rightward shifts to decrease contractility. Sometimes there is an assumption that changes in $\\Ca$ sensitivity are a surrogate for changes in whole heart function and that these changes are monotonic in that, for instance, an increase in myofilament $\\Ca$ sensitivity (i.e. a leftward shift) would improve the whole heart cardiac output. In this case study, whose related manuscript is currently in submission~\\cite{Longobardi:2022}, we discuss the limitations of such an assumption and demonstrate why the dynamic whole heart behaviour cannot be predicted or assessed with a shift in the F-pCa curve alone.\n\n\\vspace{0.2cm}\nWe use a mathematical model of active tension generation at the sarcomere level in LV rat myocytes to elucidate the relationship between sarcomere properties and the steady-state F-pCa curve. We next use our previously developed multi-scale model of SHAM rat heart, which integrates the cell tension model within a $3$D biventricular rat heart contraction model (Section~\\ref{sec:ch7rat_heart_contraction_model}) and provides a quantitative link between sarcomere properties and whole heart contractile function to investigate the mapping from F-pCa curves to the LV function. In particular, using our framework, we are able to quantitatively link changes in the F-pCa curve to/from changes in the LV function for the first time to the best of our knowledge. Previously, shifts in the F-pCa curve were linked to changes in cellular tension~\\cite{Chung:2016} but not to the whole-organ dynamic contractile behaviour and without investigating the inverse mapping. We use our modelling framework to show that observations made at the sarcomere level (e.g. a leftward shift in the F-pCa curve) do not uniquely map to desired effects at the whole-organ level (e.g. an increase in EF or improved relaxation). We also show that the opposite holds: given a change in LV function, there exist many ways this could have been achieved through sarcomeric modulation, and thus the same change in LV function can correspond to different shifts in the F-pCa curve. The implications of these phenomena on using F-pCa curves to interpret whole heart dynamics are discussed in Section~\\ref{sec:ch8discussion}.\n\n\n%\n%\n%\n\\section{Methods}\\label{sec:ch8methods}\n\n\n%\n%\n%\n\\subsection{Cellular contraction model}\\label{sec:cellcontr}\nWe employed the Land et al.~\\cite{Land:2012} myocyte contraction model (Section~\\ref{sec:ch2contractionmodel}) to simulate active tension generation at the sarcomere level and isometric steady-state F-pCa relationship in the rat heart. Parameters were all set to the personalised SHAM rat model values (Table~\\ref{tab:bestfitparametersvalues}) when applicable, or to the Land et al.~\\cite{Land:2012} model baseline values when otherwise.\n\n\\vspace{0.2cm}\nWe recall that an experimentally derived F-pCa curve can be described by a Hill-type relationship between the force ($F$) and the negative logarithm of the $\\Ca$ concentration ($\\pCa$)~\\cite{Walker:2010}:\n%\n\\begin{equation}\n \\frac{F}{F_0} = \\frac{1}{1+10^{h(\\pCaf-\\pCa)}}\n\\end{equation}\n\n\\noindent\nwhere $F_{0}$ is the maximal reference force and $\\pCaf$ and $h$ are, respectively, the negative logarithm of the half-maximal effective $\\Ca$ concentration and the Hill coefficient of this relationship. It immediately follows that F-pCa curves are uniquely determined by the $\\pCaf$ and $h$ values. We have also seen that the steady-state force solution of the Land et al.~\\cite{Land:2012} model can be derived analytically, where both the $\\pCaf$ and $h$ features are expressed in terms of model parameters. In particular, we recall that the $\\pCaf$ was given as:\n%\n\\begin{align}\n & \\pCaf(\\mathbf{p}) = -\\log\\left[\\Caif\\left(\\frac{\\koff}{\\kon}\\frac{\\trpnf}{1-\\trpnf}\\right)^{1/\\ntrpn}\\right]\\,,\\quad\\text{with} \\label{eq:pCa50}\\\\\n & \\mathbf{p} := (\\Caif, \\kon, \\koff, \\trpnf, \\ntrpn) \\label{eq:pCa50params}\n\\end{align}\n\n\\vspace{0.2cm}\\noindent\nIn the next section, we shall make use of equations~\\eqref{eq:pCa50}--\\eqref{eq:pCa50params} to investigate on F-pCa curve shifts. \n\n\n%\n%\n%\n\\subsection{Non-unique mapping of changes in $\\pCaf$ to changes in sarcomere properties}\\label{sec:changespCa50}\nA given change ($\\Delta$) in the $\\pCaf$ feature value of the F-pCa curve can be written as:\n%\n\\begin{equation}\\label{eq:delta}\n \\Delta = \\pCaf^{\\text{new}} - \\pCaf\n\\end{equation}\n\n\\vspace{0.2cm}\\noindent\nwhere $\\pCaf^{\\text{new}}$ is the feature value characterising the newly observed F-pCa curve. We tested whether a shift of $\\Delta$ units in the $\\pCaf$ could be the result of unique changes in sarcomere properties. Testing for non-unique changes was carried out analytically and numerically (as visualised in Figure~\\ref{fig:2perclwrwshift}) in the case where the change was driven by perturbations in one model parameter (Section~\\ref{sec:changespCa50result1}), and numerically (as visualised in Figure~\\ref{fig:pca50isolines}) in the case where this was driven by perturbations in two model parameters (Section~\\ref{sec:changespCa50result2}).\n\n\\vspace{0.2cm}\nIn the one-parameter case, it sufficed to prove that for each parameter $p_i,\\,i=1,\\,\\dots,\\,5$ of equation~\\eqref{eq:pCa50params} there exists a scaling coefficient $\\alpha_i\\in\\mathbb{R}$ such that\n%\n\\begin{equation}\n \\mathbf{p}^{\\text{new}} := (p_1,\\,\\dots,\\,\\p_{i-1},\\,\\alpha_i\\times\\p_i,\\,p_{i+1},\\,\\dots,\\,p_5)\n\\end{equation}\n\n\\noindent\nis such that\n%\n\\begin{equation}\n \\pCaf^{\\text{new}} = \\pCaf(\\mathbf{p}^{\\text{new}})\n\\end{equation}\n\n\\vspace{0.2cm}\\noindent\nWe computed this scaling coefficient for each parameter $\\p_i$ by solving for $\\alpha_i$ equation~\\eqref{eq:delta} for a given $\\Delta$.\n\n\\vspace{0.2cm}\nIn the two-parameter case, we proceeded as follows. We allowed $2$ parameters to take equally-spaced values within a $\\pm\\SI{50}{\\percent}$ perturbation around their baseline values (Table~\\ref{tab:bestfitparametersvalues}). We then generated a $2$D uniform grid from all the combinations of $2$ parameter values and used the Land et al.~\\cite{Land:2012} cell contraction model to calculate the $\\pCaf$ value at each parameter point of the grid. By plotting the resulting $\\pCaf$ values across the grid as a heat map, we could discern regions in the $2$D parameter space that share the same $\\pCaf$ value (isolines). The same process was repeated for pairs of parameters coming from vector $\\mathbf{p}$ (equation~\\eqref{eq:pCa50params}), which was shown to regulate the $\\pCaf$ feature of the F-pCa curve (equation~\\eqref{eq:pCa50}).\n\n\n%\n%\n%\n\\subsection{The relationship between sarcomere properties and LV function}\\label{subsec:quantlink}\nTo quantitatively link sarcomere properties to the LV function, we employed the personalised model of $3$D biventricular healthy (SHAM) rat heart contraction derived in Chapter~\\ref{cha:chapter4} and extended in Chapter~\\ref{cha:chapter7}. This model integrates the Land et al.~\\cite{Land:2012} cell contraction model in the context of whole-organ simulations.\n\n\\vspace{0.2cm}\nWe have seen (Section~\\ref{sec:ch7rat_heart_contraction_model}) that this model can be represented as a multi-scale function (simulator) that maps a $16$D vector of input parameters $\\mathbf{x}$ to a scalar quantity of interest $y_j$ (e.g. $y_1=\\textrm{EDV}$, $y_2=\\textrm{ESV}$, etc.):\n%\n\\begin{align}\\label{eq:fsimul}\n f_{\\textrm{simul}}\\colon\\mathbb{R}^{16} &\\to\\mathbb{R}\\times\\mathbb{R}\\times\\dots \\\\\n \\mathbf{x} &\\mapsto (y_1,\\,y_2,\\dots) \\nonumber\n\\end{align}\n\n\\vspace{0.2cm}\\noindent\nAt the same time, we have replaced (Section~\\ref{sec:ch7training_dataset_emulators_global_sensitivity_analysis}) the simulator with a fast-evaluating surrogate model (emulator), based on Gaussian process emulation.\n\n\\vspace{0.2cm}\nWe used the simulator to investigate the dependence of LV volume features, namely EDV, ESV, SV, and EF, on model parameters. Specifically, we were interested in the dependence of the features on the $\\Caif$, $\\koff$, $\\ntrpn$ and $\\trpnf$ parameters as these directly modulated the $\\pCaf$ feature of the F-pCa curve (equation~\\eqref{eq:pCa50}). For each parameter, we performed the following operations. First, the simulator was run at $128$ input points $\\mathbf{x}_i,\\,i=1,\\dots,128$, which had all their $16$ components fixed to baseline values (Table~\\ref{tab:bestfitparametersvalues}) but the component corresponding to the examined parameter, which instead was set to $128$ equally-spaced values between $-\\SI{50}{\\percent}$ and $\\SI{50}{\\percent}$ of perturbation values from baseline. The LV volume features were then extracted from the simulated $128$ LV volume transients so that we could plot the variation of each feature as a function of the parameter variations around its baseline value.\n\n\\vspace{0.2cm}\nWe then used the emulator to perform a GPE-based GSA. We calculated the Sobol' first-order, second-order and total effects (Section~\\ref{sec:ch3globalsensitivityanalysis}) as a measure of model outputs' sensitivity to model inputs. We evaluated the impact of $\\Caif$, $\\koff$, $\\ntrpn$, $\\trpnf$ parameters into affecting the total variance of EDV, ESV, SV, EF features. Parameters whose Sobol' indices' distributions' expectation was below the threshold $0.01$ were determined to have negligible effects.\n\n\n%\n%\n%\n\\subsection{Non-unique mapping of changes in LV function to changes in sarcomere properties}\\label{sec:changesLVfunction}\nLeft ventricular EF can be continuously monitored noninvasively as an index of LV systolic function in patients enrolled for clinical trials investigating HF treating drugs. However, it is important to discern whether detected changes in the EF value directly result from a specific cellular modulation caused by the administered drug or not. From a modelling perspective and in the context of sarcomere modulators, we can test whether changes observed in the LV function at the whole-organ level, as described by the EF feature, can be uniquely explained in terms of sarcomere properties' changes.\n\n\\vspace{0.2cm}\nWe therefore proceeded as in Section~\\ref{sec:changespCa50} by generating a $2$D uniform grid of parameter values for each considered pair of parameters modulating the F-pCa curve (equation~\\eqref{eq:pCa50}). We then used the emulator to predict the EF value at each parameter point of the grid. By plotting the resulting EF values across the grid as a heat map, we could again discern regions in the $2$D parameter space that share the same EF value (isolines).\n\n\n%\n%\n%\n\\subsection{Non-unique mapping of F-pCa curve to LV function and of LV function to F-pCa curve}\nIn order to test if the mapping from the F-pCa curve to LV function is not unique, we considered all the parameter points of the $2$D grids generated in Section~\\ref{sec:changespCa50} that produced the same shift in the reference $\\pCaf$ feature value. As mentioned previously, these points all belong to isolines of the local $2$D parameter space. We then used the emulator to map the points to the corresponding organ-level EF value.\n\n\\vspace{0.2cm}\nWe also tested if the inverse mapping from the LV function to the F-pCa curve is not unique. For this purpose, we considered all the parameter points of the $2$D grids generated in Section~\\ref{sec:changesLVfunction} that produced the same shift in the reference EF feature value. We then used the cellular contraction sub-model of the full simulator to map the points to the corresponding cell-level $\\pCaf$ value.\n\n\n%\n%\n%\n\\section{Results}\n\n\n%\n%\n%\n\\subsection{Non-monotonic relationship between sarcomere parameters and LV function}\nThe variations of EDV, ESV, SV and EF features as a function of the $4$ considered F-pCa curve-modulating parameters (equation~\\eqref{eq:pCa50}) are non-linear and non-monotonic, as shown in Figure~\\ref{fig:EFvsparamsnonmonotonic}. An example of full simulator outputs comprising LV volume and pressure transients and PV loops is provided in Figure~\\ref{fig:EFvskoff} for the EF-vs-$\\koff$ case.\n\n\\begin{figure}[h!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig1.pdf}\n \\caption{The full $3$D biventricular rat heart contraction model is run at a fixed parameter set with only one parameter taking equally-spaced values in the $\\pm\\SI{50}{\\percent}$ range of perturbation from its baseline value (vertical red dashed lines). The converging mechanics simulations' output PV loops are analysed to extract the corresponding EDV, ESV, SV and EF features' values (open blue dots), given as percentages from their baseline values (horizontal red dashed lines). The process is repeated separately for each parameter regulating the $\\pCaf$ feature of the F-pCa curve. A linear regression (LR) model with second-order degree polynomials is fitted to the data (blue lines) to facilitate the visualisation of non-linear and non-monotonic relationships between the features and each of the parameters considered.}\n \\label{fig:EFvsparamsnonmonotonic}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig2.pdf}\n \\caption{The full $3$D biventricular rat heart contraction model is run at a fixed parameter set with only one parameter varying around its baseline value. The resulting LV volume and pressure transients and corresponding PV loops are plotted for some of the parameter values (full lines in blue variants) and compared to the reference parameter set mechanics solution (dashed black line). Example showing $\\koff$ parameter variation in the range obtained as a $\\pm\\SI{40}{\\percent}$ perturbation of its baseline value.}\n \\label{fig:EFvskoff}\n\\end{figure}\n\n\\vspace{0.2cm}\nThe obtained GSA Sobol' sensitivity indices are displayed as donut charts in Figure~\\ref{fig:gsarestr}. We can see that the reference thin filament $\\Ca$ sensitivity ($\\Caif$) is the most important parameter in explaining the total variance of EDV, ESV, SV and EF features. The second most important parameter is the degree of cooperativity of $\\Ca$ binding to TnC ($\\ntrpn$), followed by the fraction of bound $\\Ca$-TnC complexes for half-maximal cross-bridges activation ($\\trpnf$) and the unbinding rate of $\\Ca$ from TnC ($\\koff$).\n\n\\begin{figure}[h!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig3.pdf}\n \\caption{The impact of $\\pCaf$-modulating, sarcomere parameters on EDV, ESV, SV and EF organ-scale LV feature. The contribution of each parameter is represented by the sum of its first- and second-order effects. For each LV feature, higher-order interactions (up to the fourth order) are represented by the sum of all total effects minus the sum of all first- and second-order effects.}\n \\label{fig:gsarestr}\n\\end{figure}\n\n\n%\n%\n%\n\\subsection{$\\pCaf$ changes are non-uniquely explained by sarcomere alterations: the one-parameter case}\\label{sec:changespCa50result1}\nWe show that an observed change in $\\pCaf$ can be caused by multiple different changes in sarcomere proprieties represented by model parameters. For each parameter $p_i,\\,i=1,\\,\\dots,\\,5$ of equation~\\eqref{eq:pCa50params}, we report in Table~\\ref{tab:alphavalues} the corresponding scaling coefficient $\\alpha_i,\\,i=1,\\,\\dots,\\,5$ that would yield a shift of exactly $\\Delta$ units in the $\\pCaf$ value. This proves that an observed change in the F-pCa cannot be uniquely explained by a change in a specific sarcomere property. An example of $\\pm\\SI{2}{\\percent}$ shift from a reference $\\pCaf$ value is displayed in Figure~\\ref{fig:2perclwrwshift}.\n\n\\begin{table}[h!]\n \\myfloatalign\n \\begin{tabularx}{\\textwidth}{XX}\n \\toprule\n \\tableheadline{Parameter} & \\tableheadline{Scaling coefficient $\\alpha$} \\\\\n \\midrule\n & \\\\\n $\\Caif$ & $10^{\\Delta}$ \\\\ & \\\\\n $\\kon$ & $10^{-\\Delta\\cdot\\ntrpn}$ \\\\ & \\\\\n $\\koff$ & $10^{\\Delta\\cdot\\ntrpn}$ \\\\ & \\\\\n $\\trpnf$ & $\\dfrac{10^{\\Delta\\cdot\\ntrpn}}{\\trpnf(10^{\\Delta\\cdot\\ntrpn}-1)+1}$ \\\\ & \\\\\n $\\ntrpn$ & $\\dfrac{\\log\\left(\\frac{\\koff}{\\kon}\\frac{\\trpnf}{1-\\trpnf}\\right)}{\\Delta\\cdot\\ntrpn+\\log\\left(\\frac{\\koff}{\\kon}\\frac{\\trpnf}{1-\\trpnf}\\right)}$ \\\\ & \\\\\n \\bottomrule\n \\end{tabularx}\n \\caption{The mapping of sarcomere properties to F-pCa variations is non-unique. By scaling each parameter $p_i$ by the corresponding $\\alpha_i$ coefficient, the cellular contraction model simulates a $\\pCaf$ value which is shifted by exactly $\\Delta$ units from the reference value.}\n \\label{tab:alphavalues}\n\\end{table}\n\n\\begin{figure}[h!]\n \\myfloatalign\n \\includegraphics[width=0.8\\textwidth]{figures/chapter08/Fig4.pdf}\n \\caption{Different parameters can be individually perturbed to achieve the very same shift in the force-calcium relationship. Example showing $\\SI{2}{\\percent}$ leftwards (upper plot) and rightwards (bottom plot) shifts of the reference $\\pCaf$ value. As $\\trpnf$ and $\\ntrpn$ also regulate the F-pCa curve's Hill coefficient (equation~\\eqref{eq:h}), notice that for these two parameters the shift in the $\\pCaf$ value is also affecting the curve's slope.}\n \\label{fig:2perclwrwshift}\n\\end{figure}\n\n\n%\n%\n%\n\\subsection{$\\pCaf$ changes are non-uniquely explained by sarcomere alterations: the two-parameter case}\\label{sec:changespCa50result2}\n$\\pCaf$ evaluation on $2$-parameter grids highlighted the presence of isolines, whose parameter points induce the same $\\pCaf$ feature value (Figure~\\ref{fig:pca50isolines}A). This means that moving between any two points on two isolines will result in the same shift in $\\pCaf$, demonstrating also in the two-parameter case the non-uniqueness of changes in the half-activation of the F-pCa curve and a change in sarcomere properties.\n\n\n%\n%\n%\n\\subsubsection{Non-unique mapping of F-pCa curve to LV function}\\label{sec:fpcatolvnonuniquemapping}\nParameter points inducing the same shifts in the $\\pCaf$ feature value (isolines of Figure~\\ref{fig:pca50isolines}A) are quantitatively linked to different EF feature values (Figure~\\ref{fig:pca50isolines}B), showing that the mapping from F-pCa curve to LV function is not unique.\n\n\\begin{figure}[h!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig5.pdf}\n \\caption{For each pair of parameters regulating the $\\pCaf$ feature of the F-pCa curve, a $2$D uniform grid is constructed using a $\\pm\\SI{50}{\\percent}$ perturbation around the reference parameter set values (black dots). (A) The cell contraction model is then used to calculate the $\\pCaf$ feature value at every parameter point of the grid, and each grid is plotted as a heat map with values given as percentages from the control $\\pCaf$ value. Contour plots are finally added to highlight the presence of isolines, whose many different parameter sets induce the same shift ($-\\SI{2}{\\percent}$ dotted, $\\SI{0}{\\percent}$ full, $+\\SI{2}{\\percent}$ dashed black lines) in the control $\\pCaf$ value. (B) Parameter points from the obtained isolines are mapped by the emulator into EF feature values. These values (given as percentages of the control EF value) are represented by different colour intensities used to colour each parameter point in the isolines, showing that parameters that share the same $\\pCaf$ values are linked to different EF values.}\n \\label{fig:pca50isolines}\n\\end{figure}\n\n\n%\n%\n%\n\\subsection{LV function changes are non-uniquely explained by sarcomere alterations}\\label{sec:changesLVfunctionresult}\nEF evaluation on $2$-parameter grids highlighted the presence of isolines, where all parameter combinations along these lines result in the same predicted EF (Figure~\\ref{fig:EFisolines}A). This means that moving between any two points on each isoline will result in the same EF, demonstrating the non-uniqueness of the LV function and sarcomere properties.\n\n\n%\n%\n%\n\\subsubsection{Non-unique mapping of LV function to F-pCa curve}\\label{sec:lvtofpcanonuniquemapping}\nParameter points that give rise to the same predicted EF value (isolines of Figure~\\ref{fig:EFisolines}A) are quantitatively linked to different $\\pCaf$ values (Figure~\\ref{fig:EFisolines}B), showing that the mapping from LV function to the F-pCa curve is not unique.\n\n\\begin{figure}[h!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig6.pdf}\n \\caption{For each pair of parameters regulating the $\\pCaf$ feature of the F-pCa curve, a $2$D uniform grid is constructed using a $\\pm\\SI{50}{\\percent}$ perturbation around the reference parameter set values (black dots). (A) A trained emulator is then used to predict the $\\textrm{EF}$ feature value at every parameter point of the grid, and each grid is plotted as a heat map with values given as percentages from the control $\\textrm{EF}$ value. Contour plots are finally added to highlight the presence of isolines (full black lines), whose many different parameter sets induce the same $\\SI{0}{\\percent}$ shift in the control $\\textrm{EF}$ value. (B) Parameter points from the obtained isolines are mapped by the cell contraction model into $\\pCaf$ feature values. These values (given as percentages of the control $\\pCaf$ value) are represented by different colour intensities used to colour each parameter point in the isolines, showing that parameters that share the same $\\textrm{EF}$ values are linked to different $\\pCaf$ values.}\n \\label{fig:EFisolines}\n\\end{figure}\n\n\n%\n%\n%\n\\subsection{Non-unique mapping from/to F-pCa curve to/from LV function}\\label{sec:ch8nonuniquemappings}\nIn Sections~\\ref{sec:changespCa50result1}--\\ref{sec:changespCa50result2}, we proved that changes in the $\\pCaf$ value are not uniquely caused by changes in sarcomere properties. In Section~\\ref{sec:fpcatolvnonuniquemapping}, we also showed that changes in the EF value are not uniquely caused by changes in the $\\pCaf$ value. Because of these two findings, we can state that observed changes in the F-pCa curve cannot be uniquely mapped to changes in the LV function.\n\n\\vspace{0.2cm}\nAt the same time, in Section~\\ref{sec:changesLVfunctionresult} we proved that changes in the EF feature value are not uniquely caused by changes in sarcomere properties. In Section~\\ref{sec:lvtofpcanonuniquemapping}, we also showed that changes in the $\\pCaf$ feature value are not uniquely caused by changes in the $\\textrm{EF}$ value. Because of these two other findings, we can state that observed changes in the LV function cannot be uniquely mapped to changes in the F-pCa curve.\n\n\n%\n%\n%\n\\section{Discussion}\\label{sec:ch8discussion}\nIn this study, we have made use of mathematical models to characterise the relationship between sarcomere properties and the LV contractile function in the healthy rat heart. We have highlighted the presence of complex nonlinearities; in particular, we have demonstrated that the relationship between myofilaments' features (e.g. $\\Caif$) and PV loop characteristics (e.g. EF) is non-monotonic. As these sarcomere properties also define the steady-state F-pCa relationship in the cardiac muscle, we extended this result in terms of shifts in the F-pCa curve that are often examined when experimentally assessing the effect of sarcomere-targeting pharmacological compounds. Therefore, we have provided both analytical and simulation study evidence that alterations in the F-pCa curve can cause very different changes in whole heart function. At the same time, we have shown that observed changes in the LV function cannot be attributed to a unique modification in sarcomere properties. Although in the previous sections we have characterised the LV function using EF, the obtained results generally hold for many other clinically relevant indexes of LV systolic and diastolic function, e.g. IVRT and PeakP, as visualised in Figures~\\ref{fig:IVRTisolines}--\\ref{fig:PeakPisolines}. To design new treatment strategies, solely looking at the induced shift in the $\\pCaf$ feature value of the F-pCa curve on its own is therefore not enough to predict or to understand predictions of how this will be translated into whole-organ function changes.\n\n\\begin{figure}[ht!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig6_IVRT.pdf}\n \\caption{The same IVRT value (A) can be linked to different $\\pCaf$ values (B) (i.e. to differently shifted F-pCa curves).}\n \\label{fig:IVRTisolines}\n\\end{figure}\n\n\\begin{figure}[ht!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig6_PeakP.pdf}\n \\caption{The same PeakP value (A) can be linked to different $\\pCaf$ values (B) (i.e. to differently shifted F-pCa curves).}\n \\label{fig:PeakPisolines}\n\\end{figure}\n\n\\vspace{0.2cm}\nAlso, when studying myofilament $\\Ca$ sensitivity using F-pCa curves, simply considering the $\\pCaf$ value as a stand-alone measurement is not sufficient for translation into dynamic contraction and relaxation. This concept was previously shown for single-cell dynamics by Chung et al.~\\cite{Chung:2016} who highlighted the importance of biophysical measures of $\\koff$ and $\\kon$ to help predict tension dynamics at the cellular level. However, in this work, we expand this concept further to the whole heart level by analysing LV features and exploring the inverse mapping from LV features to F-pCa curves, highlighting the non-uniqueness of both forward and inverse mapping between F-pCa and LV features. We emphasise that, for instance, an increase in $\\Ca$ sensitivity, i.e. a leftward shift in the F-pCa curve, will have two effects. First, it will increase residual tension, decreasing EDV; second, it will increase end-systolic tension, decreasing ESV. The balance of these two effects will impact the change in SV and EF for a given change in $\\Ca$ sensitivity. This impact of changes in $\\pCaf$ depends on the starting F-pCa curve, the $\\Ca$ transient, cardiac material properties and boundary conditions. Determining these multi-scale relationships from experimental preparations remains challenging.\n\n\\vspace{0.2cm}\nThe performed GSA interpreted the variability of EDV, ESV, SV and EF in terms of the uncertainty in F-pCa curve-modulating sarcomere parameters, and it emphasised the importance of the $\\Ca$ sensitivity in affecting these PV loop characteristics. Deeper insight can be gained if we look at higher-order interactions' effects. Although these are present for all the four LV features considered, they are remarkably high only for the SV and EF features, explaining more than half of the total variance for SV and almost half of the total variance for EF. We can notice that for the EDV and ESV features, higher-order interactions' effects are instead small, and dominating effects are mostly of the first-order type. Conversely, for the SV and EF features where higher order interactions' effects are high, dominating lower-order effects are mostly of the second-order type. All these considerations can be summarised by stating that although it is possible to interpret changes in terms of the individual contribution of parameters for the EDV and ESV features, this is not the case for SV and EF features. In this sense, it is more the combined effect of the whole sarcomere to determine the LV function rather than single myofilament components.\n\n\\vspace{0.2cm}\nIndications from the GSA coupled with the information of F-pCa curve vs LV function non-monotonicity shed new light on the problem of interpreting pharmacological interventions' effects on the F-pCa curve in terms of the desired effects on whole-heart function. This concept is illustrated via the schematic in Figure~\\ref{fig:schematic}, which highlights existing feedback mechanisms that regulate contraction in the heart. Modulation of the $\\Ca$ transient can directly affect the active tension which is generated within the sarcomere, and alteration of the sarcomere generated force will eventually affect the PV loop. As the force-volume relationships at the end-diastolic and end-systolic pressures are fixed, sarcomere interventions that aim at shifting the F-pCa might result in no change in LV contractile function. For this reason, treatment strategies should aim at altering both the end-systolic and end-diastolic force-volume relationships or altering one while maintaining the other to yield an effect on EF.\n\n\\begin{figure}[ht!]\n \\myfloatalign\n \\includegraphics[width=\\textwidth]{figures/chapter08/Fig7.pdf}\n \\caption{Force-LV volume curves at the end-diastolic and end-systolic pressures (top right, orange and green lines) can be separately manipulated to modulate the LV contractile function. However, interventions on calcium transient (bottom left, blue line) and sarcomeric generated force (top left, orange and green full lines) both manifest as modifications of the former curves, which in turn causes modification of the PV loop (bottom right, blue full line). In addition, pharmacological modulations on the sarcomere might cause a shift of the force-calcium relationship (top left, orange and green dashed lines). However, this will preserve the EF (bottom right, blue dashed line) without improving LV contractile function, which was the desired outcome of the performed sarcomeric intervention.}\n \\label{fig:schematic}\n\\end{figure}\n\n\n%\n%\n%\n\\subsection{Limitations}\\label{sec:ch8limitations}\nThe limitations of this study mainly concern the adopted modelling/emulation framework, and are presented fully in Chapter~\\ref{cha:chapter9}, Section~\\ref{sec:ch9limitations}.\n\n%\n%\n%\n\\section{Summary}\\label{sec:ch8summary}\nWe have used a biophysically detailed mathematical model of a healthy rat heart contraction to quantitatively map sarcomere properties to whole heart function. Using this mapping, we demonstrated that the relationship between the F-pCa curve and LV function is non-linear and non-monotonic. This results in the non-interpretability of observed changes in the LV function in terms of unique sarcomere modulations, which highlights the need for muscle experimental findings to be put into a broader context where not only the $\\pCaf$ and Hill coefficient but also active and passive force, length and velocity dependencies, calcium transient and boundary conditions are analysed.", "meta": {"hexsha": "509de0681846afc2d511ae3a67d1420afcbc218b", "size": 32501, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "phd-thesis/chapters/chapter08.tex", "max_stars_repo_name": "stelong/phd-thesis", "max_stars_repo_head_hexsha": "25a2c45d359403bde916b9bcfb9485402b4d2a8a", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "phd-thesis/chapters/chapter08.tex", "max_issues_repo_name": "stelong/phd-thesis", "max_issues_repo_head_hexsha": "25a2c45d359403bde916b9bcfb9485402b4d2a8a", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "phd-thesis/chapters/chapter08.tex", "max_forks_repo_name": "stelong/phd-thesis", "max_forks_repo_head_hexsha": "25a2c45d359403bde916b9bcfb9485402b4d2a8a", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 108.6989966555, "max_line_length": 1591, "alphanum_fraction": 0.7747453924, "num_tokens": 8133, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3491523698675708}} {"text": "\\documentclass[dvipdfmx,uplatex,a4paper]{article}\n\\def\\vector#1{\\mbox{\\boldmath $#1$}}\n\\usepackage[hiresbb]{graphicx}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{ascmac}\n\\usepackage{braket}\n\\usepackage{circuitikz}\n\\usepackage{cite}\n\\usepackage{comment}\n\\usepackage{float}\n\\usepackage{listings}\n\\usepackage{nameref}\n\\usepackage{siunitx}\n\\usepackage{tikz}\n\\usepackage[version=3]{mhchem}\n\n\\usepackage[colorlinks=true, bookmarks=true,\nbookmarksnumbered=true, bookmarkstype=toc, linkcolor=blue,\nurlcolor=blue, citecolor=blue]{hyperref}\n\n\\makeatletter\n \\renewcommand{\\theequation}{\n \\thesubsection.\\arabic{equation}}\n \\@addtoreset{equation}{section}\n\\makeatother\n\n\\title{Solve RGB+ LEDs PWM from Chromaticity}\n\\author{CHINZEI, Kiyoyuki}\n\\date{\\today}\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nThis article\\footnote{To cite, please refer to https://github.com/kchinzei/kch-rgbw-lib/docs/rgbw\\_solver.pdf} is a private note to develop \\texttt{kch-rgbw-lib}, a TypeScript library in \\href{https://github.com/kchinzei/kch-rgbw-lib}{github} and \\href{https://www.npmjs.com/package/kch-rgbw-lib}{npm}. \\texttt{Kch-rgbw-lib} provides classes and functions for multicolor LED, including color space conversions between HSV, RGB, XYZ and xyY and calculation of accurate color composition. This article gives a general solution of multi-number (more than R-G-B) LEDs to represent composite colors. It is not intended to carry new, comprehensive, or most efficient ideas. This document is granted under MIT License.\n\\end{abstract}\n\n\\section{Define the Problem}\n\\subsection{Past works}\\label{s_intro}\nObtaining accurate colors by mixing R-G-B light sources has been utilized as color displays since 1950s. As various colors are available by LED, recent topics are solutions of color composite with additional color sources typically for OLED applications \\cite{Chi2011, Lee2014}. Usually white light sources are used as an additional light source \\cite{AN1562, Chi2011, Lee2014}. For display purposes additional colors other than R-G-B have been also used to expand the possible color ranges (gamut)\\cite{Wikipedia_multicolor}. Sharp once added yellow in Aquos flat-panel TV, but they initially researched 5-primary color display \\cite{Sharp2011}. Amber \\cite{AN2026}, turquoise, and violet can be other colors to expand the gamut.\n\n\\subsection{Given parameters and assumptions}\\label{s_assumptions}\nWe have $N \\geq 3$ color sources (LEDs) with chromaticity $(x_i, y_i)$ and maximum luminance $Y_i$, where $i=1 \\ldots N$. Our problem is to find the optimum composite output ratio (PWM output) $\\boldsymbol{\\alpha} = [\\alpha_1 \\cdots \\alpha_N]^T$ where $0 \\leq \\alpha_i \\leq 1$ to represent a given color input with chromaticity $(x, y)$ and luminance $Y$.\n\nHere, we set our goal of optimization as the following:\n\\begin{enumerate}\n \\item\\label{I_max_luminance} Under physical constraint $0 \\leq Y \\leq Y_1 + \\cdots + Y_N$,\n \\item\\label{I_solution} Obtain exact composite color to represent $(x, y, Y)$.\n \\item\\label{I_inside_gamut} If $(x, y)$ is outside the gamut of color source, the nearest color in it is used.\n It can be achieved by projecting the input to the gamut contour.\n \\item\\label{I_min_energy} When possible, minimize energy consumption $E$, \\\\\n \\begin{equation}\n \\label{E_min_energy}\n E = \\sum_1^N \\alpha_i W_i\n \\end{equation}\n where $W_i$ is the power of each LED at the maximum luminance.\n \\item\\label{I_lifetime} When possible, average turn-on time of LEDs to equalize LEDs lifetime,\n \\item\\label{I_small_alpha} When possible, set $\\alpha_i$ to null when it's very small.\n It is preferable to avoid jitter at low PWM output.\n\\end{enumerate}\n\nIt is a typical linear programming (LP) problem. When $N=3$, e.g. R-G-B color sources only, it's a deterministic and not an optimization problem. And when $N=4$, e.g. R-G-B-W LEDs, there is only one parameter to optimize, which makes the problem as simple as we don't need to use sophisticated LP solver.\n% Mathematically, it is dependent to $R = rank(\\boldsymbol{A})$, where $\\boldsymbol{A}$ is a matrix of LEDs, in Eq.~\\eqref{E_X=AY}. We first derive a general description of the problem and solve it for $R=3$, $R=4$ and $R > 4$ cases.\nWe first derive a general description of the problem and solve it for $N=3$, $N=4$ and $N > 4$ cases.\n\n\\section{General solution}\nWe use XYZ color space because a composite of color source in XYZ color space can be obtained as a simple sum of each term. In XYZ color space, our problem is to determine $\\boldsymbol{\\alpha} = [\\alpha_1 \\cdots \\alpha_N]$ which gives an equation between input color $[X, Y, Z]^T$ and color source $[X_{i}, Y_{i}, Z_{i}]^T$, $i=1 \\ldots N$;\n\n\\begin{equation}\n \\label{E_XYZcomposite}\n \\left[\n \\begin{array}{c}\n X \\\\\n Y \\\\\n Z\n \\end{array}\n \\right]\n = \\alpha_1\n \\left[\n \\begin{array}{c}\n X_1 \\\\\n Y_1 \\\\\n Z_1\n \\end{array}\n \\right]\n + \\cdots + \\alpha_N\n \\left[\n \\begin{array}{c}\n X_N \\\\\n Y_N \\\\\n Z_N\n \\end{array}\n \\right]\n\\end{equation}\n\nColor space $[X, Y, Z]^T$ is expressed by using $(x, y, Y)$:\n\n\\begin{eqnarray}\n \\label{E_xyY2XYZ}\n X & = & \\frac{x}{y} Y \\\\\n Y & = & Y \\\\\n Z & = & \\frac{1 - x - y}{y} Y\n\\end{eqnarray}\n\nUsing matrix representation, Eq.~\\eqref{E_XYZcomposite} is written as\n\n\\begin{equation}\n \\label{E_X=AY}\n \\left[ \\boldsymbol{X} \\right] =\n \\left[ \\boldsymbol{A} \\right]\n \\left[ \\boldsymbol{\\alpha} \\right]\n\\end{equation}\n\n\\noindent\nwhere\n\n\\begin{eqnarray}\n \\left[ \\boldsymbol{X} \\right] &=&\n \\left[ X \\, Y \\, Z \\right]^T \\label{E_XYZ^T} \\\\\n%\n \\left[ \\boldsymbol{A} \\right] &=&\n \\left[\n \\begin{array}{ccc}\n \\frac{x_1}{y_1} Y_1 & \\cdots & \\frac{x_N}{y_N} Y_N \\\\\n Y_1 & \\cdots & Y_N \\\\\n \\frac{1 - x_1 - y_1}{y_1}Y_1 & \\cdots & \\frac{1 - x_N - y_N}{y_N}Y_N\n \\end{array}\n \\right] \\\\\n%\n \\left[ \\boldsymbol{\\alpha} \\right] &=&\n \\left[ \\alpha_1\\, \\cdots\\, \\alpha_N \\right]^T\n\\end{eqnarray}\n\nOur goal is to solve Eq.~\\eqref{E_X=AY} with respect to $ \\boldsymbol{\\alpha} = [\\alpha_1 \\cdots \\alpha_N]^T $. To solve it, we can use the singular value decomposition (SVD) (Eq.~\\eqref{E_SVD})~\\cite{SVD_NRC}.\n\n\\begin{equation}\n \\label{E_SVD}\n \\Bigg[ \\boldsymbol{A} \\Bigg] =\n \\Bigg[ \\boldsymbol{U} \\Bigg]\n \\left[\n \\begin{array}{cccc}\n \\omega_1 & \\\\\n & \\omega_2 & & 0 \\cdots 0 \\\\\n & & \\omega_3 &\n \\end{array}\n \\right]\n \\Bigg[ \\boldsymbol{V}^T \\Bigg]\n\\end{equation}\n\n\\noindent\nwhere $ \\boldsymbol{A} $ is $ 3 \\times N $,\n$ \\boldsymbol{U} $ is $ 3 \\times 3 $,\n$ [\\omega_1 \\ddots \\omega_3, 0 \\cdots 0] $ is $ 3 \\times N $,\n$ \\boldsymbol{V}^T $ is $ N \\times N $ matrixes\n\\footnote{Many implementations of SVD return 'economy' $\\boldsymbol{V}$ in $ N \\times 3$ instead of calculating full $N \\times N$ size. Matlab, Octave without 'econ' option and \\href{https://www.npmjs.com/package/svd-js}{svd-js} javascript package in NPM with 'f' option (This is my contribution!) give full $\\boldsymbol{V}$.}\nin this specific case, since $ \\boldsymbol{A} $ is a $ 3 \\times N $ matrix and there are upto 3 $\\omega$'s. When $ N \\geq 4 $, $ [\\omega_1 \\ddots \\omega_3 ] $ is null-padded in $ 3 \\times (N-3) $.\nThe pseudo-inverse matrix $ \\boldsymbol{A}^{-1} $ is obtained by\n\n\\begin{equation}\n \\Bigg[ \\boldsymbol{A} \\Bigg]^{-1} =\n \\Bigg[ \\boldsymbol{V}_{1-3} \\Bigg]\n \\left[\n \\begin{array}{ccc}\n 1/\\omega_1 & & \\\\\n & 1/\\omega_2 & \\\\\n & & 1/\\omega_3\n \\end{array}\n \\right]\n \\Bigg[ \\boldsymbol{U}^T \\Bigg]\n\\end{equation}\n\n\\noindent\nwhere $ \\boldsymbol{V}_{1-3} $ is the first 3 columns of $ \\boldsymbol{V} $, those correspond to $ \\omega_1, \\omega_2, \\omega_3 $\n\\footnote{Many implementations of SVD do not sort $ \\boldsymbol{U}, \\boldsymbol{V}$ by $\\omega$s. Matlab and Octave do sort. Algorithm in 'Numerical Recipes in C' and \\href{https://www.npmjs.com/package/svd-js}{svd-js} in NPM do not.}.\nUsing $ \\boldsymbol{A}^{-1} $, we obtain\n\n\\begin{equation}\n \\label{E_Y=A-1X}\n \\left[ \\boldsymbol{\\alpha} \\right] =\n \\left[ \\boldsymbol{A} \\right]^{-1}\n \\left[ \\boldsymbol{X} \\right]\n\\end{equation}\n\nBy the way, what about the rest of columns in $ \\boldsymbol{V} $? They are null vectors of $ \\boldsymbol{A} $. A null vector $\\boldsymbol{n}$ of $ \\boldsymbol{A}$ is such vector that satisfies $ \\boldsymbol{A} \\boldsymbol{n} = [0]$. By denoting these columns as $ \\boldsymbol{n}_4 \\ldots \\boldsymbol{n}_N $, Eq.~\\eqref{E_Y=A-1X} can be extended as\n\n\\begin{equation}\n \\label{E_Y=A-1X+n}\n \\left[ \\boldsymbol{\\alpha} \\right] =\n \\left[ \\boldsymbol{A} \\right]^{-1}\n \\left[ \\boldsymbol{X} \\right]\n + \\beta_4 \\boldsymbol{n}_4 + \\cdots + \\beta_N \\boldsymbol{n}_N\n\\end{equation}\n\\noindent\nwhere $ \\beta_4 \\ldots \\beta_N $ are arbitrary numbers. By choosing these using other constraints, we can obtain the optimum solution.\n\n% The actual number of null vectors is determined by the rank of $\\boldsymbol{A}$.\n\n\n\n\n\\section{Solution of $N=3$ case}\n\nWhen $N=3$, Eq.~\\eqref{E_Y=A-1X} gives a deterministic solution. No optimization. However, the obtained $ \\alpha_i $ should be physically meaningful, i.e., $ 0 \\leq \\alpha_i \\leq 1 $. It can be out of range when\n\\begin{itemize}\n \\item Input color $[X]$ is out of the gamut defined by the color sources,\n \\item $Y$ of $[X]$ is greater (brighter) than the color sources.\n\\end{itemize}\n\\noindent\nWhen $ \\alpha_i < 0$, the nearest color in the gamut should be used as the input. We can also truncate such $\\alpha_i$ to 0. In this case, the output color has certain error.\n\nWhen $ \\alpha_i > 1 $, all $\\alpha$s should be normalized by the largest $\\alpha$. This way the color will be correctly obtained, but it will be darker than expected.\n\n\n\n\n\\section{Solution of $N=4$ case}\n\nWhen $N=4$, Eq.~\\eqref{E_Y=A-1X+n} is simple.\n\n\\begin{equation}\n \\label{E_Y=A-1X+1}\n \\left[ \\boldsymbol{\\alpha} \\right] =\n \\left[ \\boldsymbol{A} \\right]^{-1}\n \\left[ \\boldsymbol{X} \\right]\n + \\beta_4 \\boldsymbol{n}_4\n\\end{equation}\n\n\\noindent\nWe will determine parameter $\\beta_4$ using the assumptions in section \\ref{s_assumptions}.\nSince it is always $ 0 \\leq \\alpha_i \\leq 1$, by solving it for $\\beta$ (hereafter omitting '$_4$'), we obtain the following conditions.\n\\begin{eqnarray}\n \\beta & \\geq & \\begin{cases}\n - \\frac{b_i}{n_i} \\;\\;\\;\\; \\text{ if \\(n_i > 0\\)} \\label{E_betamin} \\\\\n \\frac{1 - b_i}{n_i} \\;\\;\\;\\; \\text{ if \\(n_i < 0\\)}\n \\end{cases} \\\\\n%\n \\beta & \\leq & \\begin{cases}\n \\frac{1 - b_i}{n_i} \\;\\;\\;\\; \\text{ if \\(n_i > 0\\)} \\\\\n - \\frac{b_i}{n_i} \\;\\;\\;\\; \\text{ if \\(n_i < 0\\)} \\label{E_betamax}\n \\end{cases} \\\\\n%\n E &=& \\sum_{i=1}^4 (\\beta n_i + b_i) W_i \\to min \\label{E_energy_n4}\n\\end{eqnarray}\n\n\\noindent\nwhere $ [b_1,\\, \\ldots{}\\, , b_4]^T = [ \\boldsymbol{A} ]^{-1} [\\boldsymbol{X}] $, $n_i$ are the elements of $\\boldsymbol{n}$. Eq.~\\eqref{E_energy_n4} is from Eq.~\\eqref{E_min_energy}. Finding the largest and smallest values of the right hand side of Eqs.~\\eqref{E_betamin} and \\eqref{E_betamax}, denoted as $\\beta_{min}$ and $\\beta_{max}$, Eqs.~\\eqref{E_betamin} and \\eqref{E_betamax} are rewritten as\n\n\\begin{equation}\n \\beta_{min} \\leq \\beta \\leq \\beta_{max}\n\\end{equation}\n\nSince Eq.~\\eqref{E_energy_n4} is rewritten as $E = s_1\\beta + s_2$, here $s_1$ and $s_2$ are constants determined by calculating the sums in Eq.~\\eqref{E_energy_n4}, optimized $\\beta$ is determined as\n\n\\begin{equation}\n \\beta = \\begin{cases}\n \\beta_{min} & \\text{if \\(s_1 = \\sum_1^4 n_i W_i > 0\\)} \\\\\n \\beta_{max} & \\text{else}\n \\end{cases}\n\\end{equation}\n\n\\subsection{When $\\beta$ is not determined}\n\nWhen $\\beta_{min} > \\beta_{max}$, there is no feasible answer. Again, there are two cases, when the input color $[\\boldsymbol{X}]$ is out of the gamut, or when $Y$ in Eq.~\\eqref{E_XYZ^T} is greater than the color sources. Since $\\alpha_i \\geq 0$ cannot compromise, we find $\\beta$ that satisfies\n\n\\begin{eqnarray}\n \\beta & \\geq & - \\frac{b_i}{n_i} \\;\\;\\;\\; \\text{ if \\(n_i > 0\\)} \\label{E_beta0min} \\\\\n \\beta & \\leq & - \\frac{b_i}{n_i} \\;\\;\\;\\; \\text{ if \\(n_i < 0\\)} \\label{E_beta0miax}\n\\end{eqnarray}\nThen we normalize the largest $\\alpha_i$ to be 1.\n\n\\subsection{When $\\alpha_i$ is small}\n\nTo implement the assumption \\ref{I_small_alpha} in section \\ref{s_assumptions}, we can introduce allowance of small $\\alpha$, $\\alpha_\\varepsilon$, and exchange Eqs.~\\eqref{E_betamin} and \\eqref{E_betamax} as\n\n\\begin{equation}\n \\beta \\geq \\frac{\\alpha_\\varepsilon - b_i}{n_i}\n\\end{equation}\n\nBut this also needs to assert if $\\beta_{min} \\leq \\beta_{max}$.\n\n\n\n\n\\section{Solution of $N > 4$ case}\nWe can optimize Eq.~\\eqref{E_min_energy} under constraints of $0 \\leq \\alpha_i \\leq 1$ for Eq.~\\eqref{E_Y=A-1X+n} using a linear programming solution\\cite{LP_NRC}.\nTo use linear programming we need to rewrite our problem into the \\textit{normal form}. Normal form of a linear programming problem is\n\n\\begin{itemize}\n \\item Objective functions are to be minimized,\n \\item All constraints are equality formulas,\n \\item All variables $ \\geq 0$.\n\\end{itemize}\n\nSince our constraints $0 \\leq \\alpha_i \\leq 1$ are unequal and $\\beta_i$ can be negative, we introduce \\textit{slack variables} $\\gamma_i, \\delta_i$ and $\\epsilon_i, \\zeta_i$ to replace $\\beta$.\n\n\\begin{itemize}\n \\item Introduce $\\gamma_i \\geq 0$ such that $b_i + \\beta_4 n_{4i} + \\cdots + \\beta_N n_{Ni} - \\gamma_i = 0$,\n\n \\item Introduce $\\delta_i \\geq 0$ such that $b_i + \\beta_4 n_{4i} + \\cdots + \\beta_n n_{Ni} + \\delta_i = 1$,\n\n \\item Replace $\\beta_i = \\epsilon_i - \\zeta_i$, such that $\\epsilon_i \\geq 0 $ and $\\zeta_i \\geq 0$.\n\\end{itemize}\n\nAfter this modification, we have $2N$ equations with $2N$ variables (for $\\gamma$ and $\\delta$) and $2(N-3)$ variables (for $\\epsilon$ and $\\zeta$). Using these replacements, the objective function (Eq.~\\eqref{E_min_energy}) and Eq.~\\eqref{E_Y=A-1X+n} are rewritten as\n\n\\begin{eqnarray}\n%\n \\label{E_min_energy5}\n \\sum_{i=1}^N (n_{4i} \\epsilon_4 - n_{4i} \\zeta_4 + \\cdots + n_{Ni} \\epsilon_N - n_{Ni} \\zeta_N + b_{i}) W_i &\\to& min \\\\\n%\n \\label{E_bmin5}\n n_{4i} \\epsilon_4 - n_{4i} \\zeta_4 + \\cdots + n_{Ni} \\epsilon_N - n_{Ni} \\zeta_N - \\gamma_i + b_i &=& 0 \\\\\n%\n \\label{E_bmax5}\n n_{4i} \\epsilon_4 - n_{4i} \\zeta_4 + \\cdots + n_{Ni} \\epsilon_N - n_{Ni} \\zeta_N + \\delta_i + b_i &=& 1\n\\end{eqnarray}\n\n\\noindent\nEq.~\\eqref{E_min_energy5} is the objective function. Solvers applicable to Eqs.~\\eqref{E_min_energy5} - \\eqref{E_bmax5} can be found in many numerical packages. \\href{https://www.npmjs.com/package/kch-rgbw-lib}{\\texttt{Kch-rgbw-lib}} uses \\href{https://www.npmjs.com/package/linear-program-parser}{linear-program-parser} and \\href{https://www.npmjs.com/package/linear-program-solver}{linear-program-solver} in npm.\n\n\\subsection{When solution infeasible}\nLinear programming solver may find it's `infeasible' - no solution to satisfy all constraints in Eqs.~\\eqref{E_bmin5} and \\eqref{E_bmax5}. Another possibility is that the solution would not converge. Yet again, there are two cases, when the input color $[X]$ is out of the gamut, or when $Y$ in Eq.~\\eqref{E_XYZ^T} is greater (brighter) than the color sources.\n\nWhen the object brightness is too large, we can find a solution by omitting Eq.~\\eqref{E_bmax5}. Physically doable solution is to normalize the maximum $Y_i$ to $1.0$.\n\nWe may also see infeasible case when one or more value in solution is very close to zero and Eq.~\\eqref{E_bmin5} fails due to numerical error. If it is the case, you can set a small negative value to the right side of Eq.~\\eqref{E_bmin5} instead of $0$.\n\n\n\n\\section{More to do}\n\nThe following sections discuss items not implemented in \\href{https://www.npmjs.com/package/kch-rgbw-lib}{\\texttt{kch-rgbw-lib}}. We leave them as `homework' fun.\n\n\\subsection{Not to compromise brightness $Y$}\nWhen $Y$ in Eq.~\\eqref{E_XYZ^T} is greater (brighter) than the color source's capacity, there is no solution. Here we suggested a (compromised) solution to normalize $Y_i$. But we can also take another strategy to maintain the goal brightness by compromising the goal color.\n\nThis strategy requires computation of gradient of $Y$ in $\\boldsymbol{\\alpha}$ space.\n\n\n\\subsection{LED's lifetime averaging}\nIn section \\ref{s_assumptions} '\\nameref{s_assumptions}' we introduced an optional condition \\ref{I_lifetime}, 'When possible, average turn-on time of LEDs to equalize LEDs lifetime'. It is equivalent to minimize the variance of $\\alpha_1, \\, \\ldots\\, , \\alpha_N$. It can be an optimization of a quadratic function.\n\nIn $N = 4$ case, we have only one variable $\\beta$ therefore we can optimize only one condition. This means we need to chose between the energy minimization and this condition.\n\n\n\n\\bibliographystyle{unsrt}\n\\bibliography{rgbw}\n\n\\section*{MIT License}\n\nCopyright (c) 2020 Kiyo Chinzei\n\nPermission is hereby granted, free of charge, to any person obtaining a copy\nof this software and associated documentation files (the \"Software\"), to deal\nin the Software without restriction, including without limitation the rights\nto use, copy, modify, merge, publish, distribute, sublicense, and/or sell\ncopies of the Software, and to permit persons to whom the Software is\nfurnished to do so, subject to the following conditions:\n\nThe above copyright notice and this permission notice shall be included in all\ncopies or substantial portions of the Software.\n\nTHE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\nIMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\nFITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\nAUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\nLIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\nOUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE\nSOFTWARE.\n\n\\end{document}\n", "meta": {"hexsha": "157ed0d3ab6ebe17ea6a58267aeab2e9169e6c36", "size": 17823, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/rgbw_solver.tex", "max_stars_repo_name": "kchinzei/kch-rgbw-lib", "max_stars_repo_head_hexsha": "4714fd3f5aa5952bde2fd2c9288d05f404194003", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2020-11-08T01:02:34.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-30T13:25:27.000Z", "max_issues_repo_path": "docs/rgbw_solver.tex", "max_issues_repo_name": "kchinzei/kch-rgbw-lib", "max_issues_repo_head_hexsha": "4714fd3f5aa5952bde2fd2c9288d05f404194003", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-19T14:43:45.000Z", "max_issues_repo_issues_event_max_datetime": "2021-06-20T15:15:55.000Z", "max_forks_repo_path": "docs/rgbw_solver.tex", "max_forks_repo_name": "kchinzei/kch-rgbw-lib", "max_forks_repo_head_hexsha": "4714fd3f5aa5952bde2fd2c9288d05f404194003", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.7828418231, "max_line_length": 730, "alphanum_fraction": 0.700555462, "num_tokens": 5801, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832058771036, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.34915236760301044}} {"text": "\\documentclass[11pt, oneside]{article}\r\n\\usepackage[margin=.9in]{geometry}\r\n\\usepackage{pgfplots}\r\n\\pgfplotsset{compat=default}\r\n\\newcommand{\\cuckoo}{{\\rm cuckoo}}\r\n\\newcommand{\\hash}{{\\rm siphash}}\r\n\\usepackage{hyperref}\r\n\\usepackage{listings}\r\n\\title{Cuckoo Cycle: \\protect\\\\ a memory bound graph-theoretic proof-of-work}\r\n\\author{John Tromp}\r\n\\begin{document}\r\n\\maketitle\r\n\r\n\\begin{abstract}\r\nWe introduce the first graph-theoretic proof-of-work system,\r\nbased on finding small cycles or other structures in large random graphs.\r\nSuch problems are trivially verifiable and arbitrarily scalable, \r\nand appear to require memory linear in graph size to solve efficiently.\r\nWe show that in random bipartite graphs with average degree 1,\r\nlength $L$ cycles occur with frequency approximately $1/L$.\r\nOn these graphs, algorithm \"lean\" finds cycles using one bit per edge, and up to one bit per node.\r\nRuntime is linear in graph size and dominated by random access latency.\r\nWe exhibit two time-memory trade-off (TMTO) algorithms\r\ndecreasing memory usage by a factor $k$ while increasing runtime by a factor $c \\cdot k$.\r\nThe constant $c$ provides a measure of memory-hardness, which is shown to be dependent\r\non cycle length, guiding the latter's choice.\r\nTrading in the other direction, algorithm \"mean\" uses a few dozen bits per edge but is 4x faster\r\nin practice, becoming memory bandwidth bound rather than latency bound.\r\nWe present performance figures for optimized CPU and GPU implementations,\r\nand discuss possible ASIC implementations.\r\n% Both algorithms are shown to parallelize well.\r\n\\end{abstract}\r\n\r\n\\section{Introduction}\r\nA {\\em proof-of-work} (PoW) system allows a verifier to check with negligible\r\neffort that a prover has expended a large amount of computational effort.\r\nOriginally introduced as a spam fighting measure, \r\nwhere the effort is the price paid by an email sender for demanding the\r\nrecipient's attention, they now form one of the cornerstones of crypto currencies.\r\n\r\nAs proof-of-work for new blocks of transactions,\r\nBitcoin~\\cite{nakamoto2009bitcoin} adopted Adam Back's hashcash~\\cite{back2002}.\r\nHashcash entails finding a nonce value such that application of a cryptographic hash function\r\nto this nonce and the rest of the block header, results in a number below a\r\ntarget threshold\\footnote{or, less accurately, results in many leading zeroes}.\r\nThe threshold is dynamically adjusted by the protocol\r\nso as to maintain an average block interval of 10 minutes.\r\n\r\nBitcoin's choice of the simple and purely compute-bound SHA256 hash function\r\nallowed for an easy migration of hash computation from \r\ndesktop processors (CPUs) to graphics-card processors (GPUs),\r\nto field-programmable gate arrays (FPGAs), and finally to custom designed\r\nchips (ASICs), with huge improvements in energy-efficiency at every step.\r\n\r\nSince Bitcoin, many other crypto-currencies have adopted hashcash, with various\r\nchoices of underlying hash function. the most well-known being {\\em scrypt} as\r\nintroduced by Tenebrix~\\cite{tenebrix2011} (since faded into obscurity)\r\nand copied by Litecoin~\\cite{litecoin2011}.\r\nScrypt, designed as a sequential memory-hard key derivation function,\r\nwas specifically chosen to resist the migration away from CPUs and be ``GPU-hostile''.\r\nHowever, to adapt to the efficient verifiability requirement of proof-of-work, its\r\nmemory footprint was severely limited, and migration slowed down only slightly.\r\n\r\nPrimecoin~\\cite{king2013} introduced the notion of a number-theoretic proof-of-work,\r\nthereby offering the first alternative to hashcash among crypto-currencies.\r\nPrimecoin identifies long chains of nearly doubled prime numbers, constrained\r\nby a certain relation to the block header.\r\nVerification of these chains, while very slow compared to bitcoin's, is much faster\r\nthan attempting to find one.\r\nThis asymmetry between proof attempt and verification is typical in non-hashcash proofs of work.\r\nRecently, two other prime-number based crypto-currencies were introduced. Riecoin is based\r\non finding clusters of prime numbers, and Gapcoin on finding large gaps between consecutive prime numbers.\r\n\r\nMomentum~\\cite{larimer2013} proposes finding birthday collisions of hash outputs,\r\nin what could well be the simplest possible asymetric proof-of-work,\r\ncombining scalable memory usage with trivial verifiability.\r\nIn section~\\ref{cycle_length_choice} we show that Momentum is in essence a special case of Cuckoo Cycle,\r\none that is particularly susceptible to time-memory trade-offs.\r\n\r\nAdam Back~\\cite{back2014} has a good overview of proof-of-work papers past and present.\r\n\r\n\\section{Motivation}\r\nCuckoo Cycle aims to be an ``egalitarian'' proof-of-work, that is,\r\nto minimize performance-per-dollar differences across hardware architectures.\r\nThis is to be achieved by making main memory latency or bandwidth the bottleneck, since\r\nDRAM latency and bandwidth vary much less across hardware architecture and process technology\r\nthan compute power.\r\n\r\nOur aim of a memory bound PoW translates to the following desirable properties:\r\n\r\n\\begin{description}\r\n\\item[MB1] a target memory footprint that exceeds a single memory chip\r\n\\item[MB2] a pattern of necessarily random memory accesses\r\n\\item[MB3] minimal computation per random memory access\\footnote{Preferably less than\r\nthe roughly $50$ nanosecond row activation delay for switching rows on a memory bank.}\r\n\\item[MB4] no feasible trade-off of memory for time\\footnote{Rather arbitrarily\r\ndefined as incurring an order of magnitude increase\r\nin $\\mbox{time}\\times\\mbox{memory}$ used per expected solution.}\r\n\\end{description}\r\n\r\nA memory bound PoW aims to take advantage of the huge economies of scale of commodity DRAM production\r\nto make DRAM chips the most cost-effective vehicle for mining.\r\nJust as SRAM is one order of magnitude faster but two orders more expensive than DRAM,\r\nso it is conceivable that development and production of custom memory chips for a memory bound PoW\r\nwill incur a sufficient cost premium to wipe out any performance gains.\r\n\r\nWe thus disagree with the premise of \\cite{poelstra2014} that PoWs should be compute bound\r\nin order to have ongoing energy costs dominate mining, which results in an expensive ASIC\r\ndesign arms race to drive up performance per Watt, rapid hardware obsolescence,\r\nand geographical centralization towards cheap electric power.\r\n\r\nWe will not strive for provable lower bounds on memory usage. Such bounds appear to be\r\nattainable only under the so-called {\\em random oracle model}, where memory tends to be used\r\nmerely as a store for chains of compute-intensive hash outputs.\r\nInstead, we present an efficient proof-finding algorithm\r\nalong with our best attempts at memory-time trade-offs,\r\nand conjecture that these cannot be significantly improved upon.\r\nLacking precise definitions and proofs of memory bound-ness, this work should be considered\r\nmore empirical than formal.\r\n\r\n\\section{Graph-theoretic proofs-of-work}\r\nWe propose to base proofs-of-work on finding certain subgraphs in large pseudo-random graphs.\r\nIn the Erd\\H{o}s-R\\'{e}nyi model, denoted $G(N,M)$, a graph is chosen uniformly at random\r\nfrom the collection of all graphs with $N$ nodes and $M$ edges. Instead, we choose edges\r\ndeterministically from the output of a keyed hash function, whose key could be chosen\r\nuniformly at random. For a well-behaved hash function, these two classes of random graphs\r\nshould have nearly identical properties.\r\n\r\nFormally, fix a keyed hash function\r\n$h: \\{0,1\\}^K \\times \\{0,1\\}^{W_i} \\rightarrow \\{0,1\\}^{W_o}$,\r\nand a small graph $H$ as a target subgraph\\footnote{hash functions generally have arbitrary length inputs,\r\nbut here we fix the input width at $W_i$ bits.}.\r\nNow pick a large number $N \\leq 2^{W_o}$ as the number of nodes,\r\nand $M \\leq 2^{W_i-1}$ as the number of edges.\r\nEach key $k \\in \\{0,1\\}^K$ generates a graph $G_k = (V,E)$ where $V=\\{v_0,\\ldots,v_{N-1}\\}$, and\r\n\\begin{equation}\r\nE=\\{(v_{h(k,2i) \\bmod N},v_{h(k,2i+1) \\bmod N}) | i \\in [0,\\ldots,M-1]\\}\r\n\\end{equation}\r\nThe inputs $i \\in [0,\\ldots,M-1]$ are also called {\\em nonces}\\footnote{These\r\n{\\em micro} nonces should be distinguished from the {\\em macro} nonce used to generate key $k$.}\r\nThe graph has a {\\em solution} if $H$ occurs as a subgraph.\r\nDenote the number of edges in $H$ as $L$.\r\nA proof of solution is an ordered list of $L$ nonces that generate the edges\r\nof $H$'s occurrence in $G_k$.\r\nSuch a proof is verifiable in time depending only on $H$ (typically linear in $L$), independent of $N$ and $M$.\r\n\r\nA simple variation generates random bipartite graphs: $G_k = (V_0 \\cup V_1,E)$ where (assuming $N$ is even)\r\n$V_0=\\{v_0,v_2,\\ldots,v_{N-2}\\}$, $V_1=\\{v_1,v_3,\\ldots,v_{N-1}\\}$, and\r\n\\begin{equation}\r\n\\label{hash_edges}\r\nE=\\{(v_{2(h(k,2i) \\bmod \\frac{N}{2})}, v_{2(h(k,2i+1) \\bmod \\frac{N}{2})+1}) | i \\in [0,\\ldots,M-1]\\}\r\n\\end{equation}\r\n\r\nThe expected number of occurrences of $H$ as a subgraph of $G$ is a function of both $N$ and $M$,\r\nand in many cases is roughly a function of $\\frac{M}{N}$ (half the average node degree).\r\nFor fixed $N$, this function is monotonically increasing in $M$.\r\nTo make the proof-of-work challenging, one chooses a value of $M$ that yields less than one\r\nexpected solution.\r\n\r\nThe simplest possible choice of subgraph is a fully connected one, or a {\\em clique}.\r\nWhile an interesting choice, akin to the number-theoretic notion of a prime-cluster\r\nas used in Riecoin, we leave its consideration to a future paper.\r\n\r\n\\section{Cuckoo Cycle}\r\nIn this paper we focus on what is perhaps the next-simplest possible choice, the {\\em cycle}.\r\nSpecifically, we propose the hash function \\hash\\ with a $K=128$ bit key, $W_i = W_o = 64$ input\r\nand output bits, $N \\leq 2^{64}$ a 2-power, $M=N/2$, and $H$ an $L$-cycle.\r\nUsing the lightweight \\hash\\-2-4 with only 6 rounds helps to attain property MB3.\r\nThe reason for calling the resulting proof-of-work Cuckoo Cycle is that\r\ninserting items in a Cuckoo hashtable naturally leads to cycle formation\r\nin random bipartite graphs.\r\n\r\n\\section{Cuckoo hashing}\r\nIntroduced by Rasmus Pagh and Flemming Friche\r\nRodler~\\cite{Pagh04cuckoohashing}, a Cuckoo hashtable consists of two\r\nsame-sized tables each with its own hash function mapping a key to a table\r\nlocation, providing two possible locations for each key.\r\nUpon insertion of a new key, if both locations are already occupied by keys,\r\nthen one is kicked out and inserted in its alternate location, possibly\r\ndisplacing yet another key, repeating the process until either a vacant\r\nlocation is found, or some maximum number of iterations is reached.\r\nThe latter is bound to happen once cycles have formed in the {\\em Cuckoo graph}.\r\nThis is a bipartite graph with a node for each location and an\r\nedge for every inserted key, connecting the two locations it can reside at.\r\nIt matches the bipartite graph defined above if the cuckoo hashtable\r\nwere based on function $h$.\r\nIn fact, the insertion procedure suggests a simple algorithm for detecting cycles.\r\n\r\n\\section{Cycle detection in Cuckoo Cycle}\r\nWe enumerate the $M$ nonces, but instead of storing the nonce itself as a key\r\nin the Cuckoo hashtable, we store the alternate key location,\r\nand forget about the nonce. We thus maintain the {\\em directed}\r\ncuckoo graph, in which the edge for a key is directed from the location where\r\nit resides to its alternate location. Moving a key to its alternate location\r\nthus corresponds to reversing its edge. The outdegree of every node in this\r\ngraph is either 0 or 1. When there are no cycles yet, the graph is a {\\em\r\nforest}, a disjoint union of trees. In each tree, all edges are directed,\r\ndirectly, or indirectly, to its {\\em root}, the only node in the tree with\r\noutdegree 0. Initially there are just $N$ singleton trees consisting of\r\nindividual nodes which are all roots.\r\nAddition of a new key causes a cycle if and only if its two endpoints are\r\nnodes in the same tree, which we can test by following the path from each\r\nendpoint to its root.\r\nIn case of different roots, we reverse all edges on the shorter of the two\r\npaths, and finally create the edge for the new key itself, thereby joining\r\nthe two trees into one. Let us illustrate this process with an actual example.\r\n\r\nThe left diagram in Figure~\\ref{cycleformation} shows the directed cuckoo graph for header ``39'' on\r\n$N=8+8$ nodes after adding edges\r\n$(2,15),(4,9),(8,5),(4,15),(12,11),(10,5)$ and $(4,13)$ (nodes\r\nwith no incident edges are omitted for clarity).\r\nIn order to add the 8th edge $(10,11)$, we follow the paths $10 \\rightarrow 5\r\n\\rightarrow 8$ and $11 \\rightarrow 12$ to find different roots $8$ and $12$.\r\nSince the latter path is shorter, we reverse it to $12 \\rightarrow 11$ so we\r\ncan add the new edge as $(11 \\rightarrow 10)$, resulting in the middle diagram.\r\nIn order to add to 9th edge\r\n$(10,13)$ we now find the path from $10$ to be the shorter one, so we reverse\r\nthat and add the new edge as $(10 \\rightarrow 13)$, resulting in the right diagram.\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}[>=stealth]\r\n\\node (2) at (1, 2) [shape=circle,draw] {2};\r\n\\node (4) at (2, 2) [shape=circle,draw] {4};\r\n\\node (8) at (3, 2) [shape=circle,draw] {8};\r\n\\node (10) at (4, 2) [shape=circle,draw] {10};\r\n\\node (12) at (5, 2) [shape=circle,draw] {12};\r\n\\node (5) at (1,-1) [shape=circle,draw] {5};\r\n\\node (9) at (2,-1) [shape=circle,draw] {9};\r\n\\node (11) at (3,-1) [shape=circle,draw] {11};\r\n\\node (13) at (4,-1) [shape=circle,draw] {13};\r\n\\node (15) at (5,-1) [shape=circle,draw] {15};\r\n\\draw [<-] (2) -- (15);\r\n\\draw [<-] (4) -- (9);\r\n\\draw [<-] (8) -- (5);\r\n\\draw [->] (4) -- (15);\r\n\\draw [<-] (12) -- (11);\r\n\\draw [->] (10) -- (5);\r\n\\draw [<-] (4) -- (13);\r\n\\end{tikzpicture}\\hspace{1cm}\\begin{tikzpicture}[>=stealth]\r\n\\node (2) at (1, 2) [shape=circle,draw] {2};\r\n\\node (4) at (2, 2) [shape=circle,draw] {4};\r\n\\node (8) at (3, 2) [shape=circle,draw] {8};\r\n\\node (10) at (4, 2) [shape=circle,draw] {10};\r\n\\node (12) at (5, 2) [shape=circle,draw] {12};\r\n\\node (5) at (1,-1) [shape=circle,draw] {5};\r\n\\node (9) at (2,-1) [shape=circle,draw] {9};\r\n\\node (11) at (3,-1) [shape=circle,draw] {11};\r\n\\node (13) at (4,-1) [shape=circle,draw] {13};\r\n\\node (15) at (5,-1) [shape=circle,draw] {15};\r\n\\draw [<-] (2) -- (15);\r\n\\draw [<-] (4) -- (9);\r\n\\draw [<-] (8) -- (5);\r\n\\draw [->] (4) -- (15);\r\n\\draw [->] (12) -- (11);\r\n\\draw [->] (10) -- (5);\r\n\\draw [<-] (4) -- (13);\r\n\\draw [<-] (10) -- (11);\r\n\\end{tikzpicture}\\hspace{1cm}\\begin{tikzpicture}[>=stealth]\r\n\\node (2) at (1, 2) [shape=circle,draw] {2};\r\n\\node (4) at (2, 2) [shape=circle,draw] {4};\r\n\\node (8) at (3, 2) [shape=circle,draw] {8};\r\n\\node (10) at (4, 2) [shape=circle,draw] {10};\r\n\\node (12) at (5, 2) [shape=circle,draw] {12};\r\n\\node (5) at (1,-1) [shape=circle,draw] {5};\r\n\\node (9) at (2,-1) [shape=circle,draw] {9};\r\n\\node (11) at (3,-1) [shape=circle,draw] {11};\r\n\\node (13) at (4,-1) [shape=circle,draw] {13};\r\n\\node (15) at (5,-1) [shape=circle,draw] {15};\r\n\\draw [<-] (2) -- (15);\r\n\\draw [<-] (4) -- (9);\r\n\\draw [->] (8) -- (5);\r\n\\draw [->] (4) -- (15);\r\n\\draw [->] (12) -- (11);\r\n\\draw [<-] (10) -- (5);\r\n\\draw [<-] (4) -- (13);\r\n\\draw [<-] (10) -- (11);\r\n\\draw [->] (10) -- (13);\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{cycle formation and detection in a Cuckoo graph}\r\n\\label{cycleformation}\r\n\\end{figure}\r\nWhen adding the 10th edge $(8,9)$, we find the paths $8 \\rightarrow 5\r\n\\rightarrow 10 \\rightarrow 13 \\rightarrow 4 \\rightarrow 15 \\rightarrow 2$ and\r\n$9 \\rightarrow 4 \\rightarrow 15 \\rightarrow 2$ with equal roots.\r\nIn this case, we can compute the length of the resulting cycle as\r\n1 plus the sum of the path-lengths to the node where the two paths join.\r\nIn the diagram, the paths join at node $4$, and the cycle length is computed as $1+4+1=6$.\r\n\r\n\\section{Union-find}\r\nThe above representation of the directed cuckoo graph is an example of\r\na {\\em disjoint-set data structure}~\\cite{wikidsds2014}, and our algorithm is\r\nclosely related to the well-known union-find algorithm, where the find operation\r\ndetermines which subset an element is in, and the union operation joins two subsets\r\ninto a single one. For each edge addition to the cuckoo graph we perform the equivalent\r\nof two find operations and one union operation.\r\nThe difference is that the union-find algorithm is free to add\r\ndirected edges between arbitrary elements. Thus it can join two subsets by adding an edge\r\nfrom one root to another, with no need to reverse any edges.\r\nOur algorithm on the other hand solves the union-find problem by maintaining\r\na direction on all union operations while keeping the maximum outdegree at 1.\r\n\r\n\\section{Cuckoo Cycle basic algorithm}\r\nThe above algorithm for inserting edges and detecting cycles forms the basis\r\nfor our basic proof-of-work algorithm.\r\nIf a cycle of length $L$ is found, then we solved the problem, and recover the proof\r\nby storing the cycle edges in a set and enumerating nonces once more to see which ones\r\ngenerate edges in the set.\r\nIf a cycle of a different length is found, then we keep the graph acyclic by ignoring the edge.\r\nThere is some risk of overlooking other $L$-cycles\r\nthrough that edge, but when the expected number of cycles is low (which is what we design for),\r\nthis ignoring of cycle forming edges hardly affects the rate of solution finding.\r\n\r\nThis algorithm is available online at \\url{https://github.com/tromp/cuckoo}\r\nas either the C-program simple\\_miner.cpp or the Java program SimpleMiner.java.\r\nA proof verifier is available as cuckoo.c or Cuckoo.java, while the repository\r\nalso has a Makefile, as well as the latest version of this paper.\r\n% `make test' tests everything.\r\n`make example' reproduces the example shown above.\r\nThe simple program uses 32 bits per node to represent the directed cuckoo graph,\r\nplus about 64KB per thread for two path-following arrays.\r\nThe left plot in Figure~\\ref{runtimes} shows both the total runtime in seconds and the runtime of just\r\nthe hash computation, as a function of (log)size. The latter is purely\r\nlinear, while the former is superlinear due to increasing memory latency\r\nas the nodes no longer fit in cache. The right plot show this more clearly\r\nas the percentage of hashing to total runtime, ending up around 5\\%.\r\n\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[xlabel={$\\mbox{log}_2(N)$}, ymode=log, ylabel={seconds}, legend pos=north west]\r\n\\addplot[color=red] coordinates {\r\n% (10,0.0000) (11,0.0000) (12,0.0000) (13,0.0001) (14,0.0001)\r\n(15,0.0002) (16,0.0004) (17,0.0008) (18,0.0017) (19,0.0034)\r\n(20,0.0068) (21,0.0139) (22,0.0271) (23,0.0542) (24,0.1084)\r\n(25,0.2166) (26,0.4336) (27,0.8658) (28,1.7322) (29,3.4719)\r\n(30,7.0389) };\r\n\\addlegendentry{hashing runtime}\r\n\\addplot[color=green] coordinates {\r\n% (10,0.0000) (11,0.0000) (12,0.0001) (13,0.0001) (14,0.0003)\r\n(15,0.0002) (16,0.0010) (17,0.0022) (18,0.0049) (19,0.0104)\r\n(20,0.0250) (21,0.0986) (22,0.2465) (23,0.5332) (24,1.1922)\r\n(25,2.5505) (26,5.3394) (27,11.0793) (28,23.1984) (29,54.6811)\r\n(30,128.1682) };\r\n\\addlegendentry{total runtime}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n% \\hspace{1cm}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[xlabel={$\\mbox{log}_2(N)$}, ylabel={\\% runtime}, legend pos=north east]\r\n\\addplot[color=blue] coordinates {\r\n% (10,38.8889) (11,33.3333) (12,45.1613) (13,39.8496) (14,44.6154)\r\n(15,96.5217) (16,45.9119) (17,38.2180) (18,35.0988) (19,32.6724)\r\n(20,27.2076) (21,14.0874) (22,11.0014) (23,10.1635) (24,9.0964)\r\n(25,8.4921) (26,8.1215) (27,7.8144) (28,7.4670) (29,6.3494)\r\n(30,5.4919) };\r\n\\addlegendentry{hashing percentage}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{Runtime and compute intensity of the basic algorithm}\r\n\\label{runtimes}\r\n\\end{figure}\r\n\r\nThe left plot in Figure~\\ref{accesses} shows the probability of finding a 42-cycle as a function\r\nof the percentage edges/nodes, while the right plot shows the average number of\r\nmemory reads and writes per edge as a function of the percentage\r\nof processed nonces (progress through main loop).\r\nBoth were determined from 10000 runs at size $2^{20}$;\r\nresults at size $2^{25}$ look almost identical.\r\nIn total the basic algorithm averages 3.3 reads and 1.1 writes per edge.\r\n\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[xlabel={$\\frac{M}{N}$ in \\%}, ylabel={probability}, legend pos=north west]\r\n\\addplot[color=blue] coordinates {\r\n(40,0) (41,0.00001) (42,0.00003) (43,0.0001) (44,0.00031) (45,0.00067)\r\n(46,0.00141) (47,0.00385) (48,0.00967) (49,0.02222) (50,0.05076)\r\n(51,0.10117) (52,0.17953) (53,0.27993) (54,0.39581) (55,0.51873)\r\n(56,0.63614) (57,0.73955) (58,0.82352) (59,0.88719) (60,0.93182)\r\n(61,0.9616) (62,0.97949) (63,0.98956) (64,0.99503) (65,0.99799)\r\n(66,0.99907) (67,0.9996) (68,0.99989) (69,0.99997) (70,0.99998) (71,1) };\r\n\\addlegendentry{has 42-cycle}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n% \\hspace{1cm}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[xlabel={\\% nonces processed}, ylabel={\\# memory accesses per nonce}, ymin=0, legend pos=north west]\r\n\\addplot[color=green] coordinates {\r\n(0,2.01) (1,2.02) (2,2.03) (3,2.04) (4,2.05) (5,2.06) (6,2.07) (7,2.08) (8,2.09) (9,2.10) (10,2.11) (11,2.12) (12,2.13) (13,2.15) (14,2.16) (15,2.17) (16,2.18) (17,2.20) (18,2.21) (19,2.22) (20,2.23) (21,2.25) (22,2.26) (23,2.28) (24,2.29) (25,2.30) (26,2.32) (27,2.33) (28,2.35) (29,2.36) (30,2.38) (31,2.40) (32,2.41) (33,2.43) (34,2.45) (35,2.46) (36,2.48) (37,2.50) (38,2.52) (39,2.54) (40,2.56) (41,2.58) (42,2.60) (43,2.62) (44,2.64) (45,2.66) (46,2.68) (47,2.71) (48,2.73) (49,2.76) (50,2.78) (51,2.81) (52,2.83) (53,2.86) (54,2.89) (55,2.92) (56,2.95) (57,2.98) (58,3.01) (59,3.05) (60,3.08) (61,3.12) (62,3.15) (63,3.19) (64,3.23) (65,3.27) (66,3.32) (67,3.36) (68,3.41) (69,3.45) (70,3.51) (71,3.56) (72,3.61) (73,3.67) (74,3.73) (75,3.80) (76,3.86) (77,3.93) (78,4.01) (79,4.09) (80,4.17) (81,4.26) (82,4.36) (83,4.46) (84,4.57) (85,4.69) (86,4.83) (87,4.97) (88,5.13) (89,5.30) (90,5.49) (91,5.71) (92,5.96) (93,6.25) (94,6.59) (95,6.99) (96,7.51) (97,8.18) (98,9.20) (99,10.93) };\r\n\\addlegendentry{reads}\r\n\\addplot[color=red] coordinates {\r\n(0,1.00) (1,1.00) (2,1.00) (3,1.00) (4,1.00) (5,1.00) (6,1.00) (7,1.00) (8,1.00) (9,1.00) (10,1.00) (11,1.00) (12,1.00) (13,1.00) (14,1.00) (15,1.00) (16,1.00) (17,1.00) (18,1.00) (19,1.00) (20,1.00) (21,1.00) (22,1.00) (23,1.01) (24,1.01) (25,1.01) (26,1.01) (27,1.01) (28,1.01) (29,1.01) (30,1.01) (31,1.01) (32,1.01) (33,1.02) (34,1.02) (35,1.02) (36,1.02) (37,1.02) (38,1.02) (39,1.02) (40,1.03) (41,1.03) (42,1.03) (43,1.03) (44,1.03) (45,1.04) (46,1.04) (47,1.04) (48,1.04) (49,1.05) (50,1.05) (51,1.05) (52,1.06) (53,1.06) (54,1.06) (55,1.07) (56,1.07) (57,1.07) (58,1.08) (59,1.08) (60,1.08) (61,1.09) (62,1.09) (63,1.10) (64,1.10) (65,1.11) (66,1.11) (67,1.12) (68,1.13) (69,1.13) (70,1.14) (71,1.14) (72,1.15) (73,1.16) (74,1.16) (75,1.17) (76,1.18) (77,1.19) (78,1.20) (79,1.21) (80,1.22) (81,1.23) (82,1.24) (83,1.25) (84,1.26) (85,1.27) (86,1.29) (87,1.30) (88,1.32) (89,1.33) (90,1.35) (91,1.37) (92,1.39) (93,1.41) (94,1.43) (95,1.46) (96,1.48) (97,1.51) (98,1.54) (99,1.58) };\r\n\\addlegendentry{writes}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{Threshold nature of solution, and increasing memory usage on threshold approach}\r\n\\label{accesses}\r\n\\end{figure}\r\n\r\n\\section{Difficulty control}\r\nThe ratio $\\frac{M}{N}$ determines a base level of difficulty,\r\nwhich may suffice for applications where difficulty is to remain fixed.\r\nRatios $\\frac{M}{N} \\geq 0.7$ are suitable when a practically guaranteed solution is desired.\r\n\r\nFor crypto currencies, where difficulty must scale in precisely\r\ncontrolled manner across a large range, adjusting the number of edges is not suitable.\r\nThe implementation default $\\frac{M}{N}=\\frac{1}{2}$ gives a solution probability of roughly $2.2\\%$,\r\nwhile the average number of cycles found increases slowly with size; from 2 at $2^{20}$\r\nto 3 at $2^{30}$.\r\n\r\nFor further control, a difficulty target $0 < T < 2^{256}$ is introduced,\r\nand we impose the additional constraint that the sha256 digest of the\r\ncycle nonces in ascending order be less than $T$, thus\r\nreducing the success probability by a factor $\\frac{2^{256}}{T}$.\r\n\r\n\\section{Edge Trimming}\r\n\\label{edge_trimming}\r\nDavid Andersen~\\cite{dga2014} suggested drastically reducing the number of edges\r\nour basic algorithm has to process, by repeatedly identifying nodes of degree one\r\nand eliminating their incident edge. Such {\\em leaf edges} can never be part of a cycle.\r\nThis works well when $\\frac{M}{N} \\leq \\frac{1}{2}$ since the expected degree of a node is then at most 1,\r\nand a significant fraction of edges are expected to be leaf edges.\r\n\r\nTrimming is implemented in our main algorithm in cuckoo\\_miner and hcuckoo\\_miner.cpp.\r\nIt maintains a set of {\\em alive} edges as a bit vector. Initially all edges are alive.\r\nIn each of a given number of trimming rounds, it shrinks this set as follows.\r\nA vector of 2-bit degree counters, one per even node, is initialized to all zeroes.\r\nNext, for all alive edges, compute its even endpoint and increase the corresponding counter,\r\ncapping the value at 2.\r\nNext, for all alive edges, compute its even endpoint and if the corresponding counter is less than 2,\r\nset the edge to be not-alive.\r\nThese steps, both of which cause the random accesses required in property MB2,\r\nare repeated for all odd endpoints.\r\n\r\nPreprocessor symbol PART\\_BITS, whose value we'll denote as $B$,\r\nallows for {\\em counter partitioning}, which trades off node counter storage for runtime,\r\nby processing nodes in multiple passes depending on the value of their $B$ least significant\r\nbits\\footnote{excluding the very least significant bit distinguishing even from odd nodes.}.\r\nThe memory usage is $M$ bits for the alive set and $N / 2^{B}$ for the counters.\r\n\r\nThe diagrams in Figure~\\ref{trimming} show two rounds of edge trimming on the earlier example. In round one\r\neven nodes 2 and 12 lose their single incident edge and in round two, odd nodes 11 and 15 lose\r\ntheir remaining single incident edge. At this point only the 6-cycle is left, so further trimming\r\nwould be pointless.\r\n\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}[>=stealth]\r\n\\node (2) at (1, 2) [shape=circle,draw] {2};\r\n\\node (4) at (2, 2) [shape=circle,draw] {4};\r\n\\node (8) at (3, 2) [shape=circle,draw] {8};\r\n\\node (10) at (4, 2) [shape=circle,draw] {10};\r\n\\node (12) at (5, 2) [shape=circle,draw] {12};\r\n\\node (5) at (1,-1) [shape=circle,draw] {5};\r\n\\node (9) at (2,-1) [shape=circle,draw] {9};\r\n\\node (11) at (3,-1) [shape=circle,draw] {11};\r\n\\node (13) at (4,-1) [shape=circle,draw] {13};\r\n\\node (15) at (5,-1) [shape=circle,draw] {15};\r\n\\draw [-] (2) -- (15);\r\n\\draw [-] (4) -- (9);\r\n\\draw [-] (8) -- (5);\r\n\\draw [-] (4) -- (15);\r\n\\draw [-] (12) -- (11);\r\n\\draw [-] (10) -- (5);\r\n\\draw [-] (4) -- (13);\r\n\\draw [-] (10) -- (11);\r\n\\draw [-] (10) -- (13);\r\n\\draw [-] (8) -- (9);\r\n\\end{tikzpicture}\\hspace{1cm}\\begin{tikzpicture}[>=stealth]\r\n\\node (2) at (1, 2) [shape=circle,draw] {2};\r\n\\node (4) at (2, 2) [shape=circle,draw] {4};\r\n\\node (8) at (3, 2) [shape=circle,draw] {8};\r\n\\node (10) at (4, 2) [shape=circle,draw] {10};\r\n\\node (12) at (5, 2) [shape=circle,draw] {12};\r\n\\node (5) at (1,-1) [shape=circle,draw] {5};\r\n\\node (9) at (2,-1) [shape=circle,draw] {9};\r\n\\node (11) at (3,-1) [shape=circle,draw] {11};\r\n\\node (13) at (4,-1) [shape=circle,draw] {13};\r\n\\node (15) at (5,-1) [shape=circle,draw] {15};\r\n\\draw [-] (4) -- (9);\r\n\\draw [-] (8) -- (5);\r\n\\draw [-] (4) -- (15);\r\n\\draw [-] (10) -- (5);\r\n\\draw [-] (4) -- (13);\r\n\\draw [-] (10) -- (11);\r\n\\draw [-] (10) -- (13);\r\n\\draw [-] (8) -- (9);\r\n\\end{tikzpicture}\\hspace{1cm}\\begin{tikzpicture}[>=stealth]\r\n\\node (2) at (1, 2) [shape=circle,draw] {2};\r\n\\node (4) at (2, 2) [shape=circle,draw] {4};\r\n\\node (8) at (3, 2) [shape=circle,draw] {8};\r\n\\node (10) at (4, 2) [shape=circle,draw] {10};\r\n\\node (12) at (5, 2) [shape=circle,draw] {12};\r\n\\node (5) at (1,-1) [shape=circle,draw] {5};\r\n\\node (9) at (2,-1) [shape=circle,draw] {9};\r\n\\node (11) at (3,-1) [shape=circle,draw] {11};\r\n\\node (13) at (4,-1) [shape=circle,draw] {13};\r\n\\node (15) at (5,-1) [shape=circle,draw] {15};\r\n\\draw [-] (4) -- (9);\r\n\\draw [-] (8) -- (5);\r\n\\draw [-] (10) -- (5);\r\n\\draw [-] (4) -- (13);\r\n\\draw [-] (10) -- (13);\r\n\\draw [-] (8) -- (9);\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{Trimming of edges which cannot be part of a cycle}\r\n\\label{trimming}\r\n\\end{figure}\r\n\r\nAfter all edge trimming rounds, the counter memory is freed, and allocated to a\r\ncustom cuckoo\\_hashtable (based on \\cite{preshing2013}) that presents the same interface as the\r\nsimple array in the basic algorithm, but gets by with much fewer locations, as long as its {\\em load},\r\nthe ratio of remaining edges to number of locations, is bounded away from 1; e.g. under 90 percent.\r\n\r\nThe number of trimming rounds, which can be set with option {\\tt -n}, defaults to\r\n$1+(B+3)*(B+4)/2$, which was determined empirically to achieve a load close to $50\\%$.\r\n\r\n\\section{Time-Memory Trade-Offs (TMTOs)}\r\nDavid Andersen also suggested an alternative method of trimming that avoids storing a bit per edge.\r\nExpanding on that idea led to the algorithm implemented in tomato\\_miner.h,\r\nwhich, unlike the main algorithm, can trade-off memory directly for runtime.\r\nOn the downside, to even achieve memory parity with the main algorithm, it already incurs a big slowdown.\r\nTo the extent that this slowdown is unavoidable, it can be called\r\nthe {\\em memory hardness} of the proof-of-work.\r\n\r\nThe TMTO algorithm selects a suitably small subset $Z$ of even vertices as a base layer,\r\nand on top of that builds a breadth-first-search\r\n(BFS) forest of depth $L/2$, i.e. half the cycle length.\r\nFor each new BFS layer, it enumerates all edges to see which ones are incident\r\nto the previous layer, adding the other endpoint.\r\nIt maintains a directed forest on all BFS nodes, like the base algorithm does on all nodes.\r\nFor increased efficiency, the base layer $Z$ is filtered for nodes with multiple incident edges.\r\nIf the graph has an $L$-cycle one of whose nodes is in $Z$, then the above procedure will find it.\r\nIf one choice of $Z$ doesn't yield a solution, then the data structures are cleared and the next\r\nsubset is tried.\r\n\r\nA variation on the above algorithm omits the filtering of $Z$, and expands\r\nthe BFS to a whole $L$ levels. This way, an $L$-cycle will be found as long as the distance from (any node in)\r\n$Z$ to the cycle is at most $L/2$. It thus has a much higher chance of finding a cycle, but requires\r\nmore space to store the significantly bigger BFS forest.\r\n\r\nFor each each value of $L \\in \\{2,4,6,8,10,12,14,16,20,24,28,32,40,48,56,64\\}$ we ran these 2 algorithms\r\non 200 graphs of size $2^{25}$ that include an $L$-cycle,\r\nchoosing subset size as a 2-power that results in a memory usage of 4MB,\r\nand analysed the distribution of number of subsets tried before\r\nfinding a solution. Since there is possible overlap between the BFS forests of different initial subsets,\r\nespecially with the second algorithm, the distributions are skewed toward lower numbers. To maximize solution\r\nfinding rate then, it pays to give up on a graph when the first few subsets tried fail to provide a solution.\r\nFor each algorithm and cycle length, we determined the minimum number of tries needed to guarantee solutions\r\nin at least 50 of the 200 graphs. In Figure~\\ref{slowdown} we plot the slowdown relative\r\nto the reference algorithm also using 4MB (2MB for edges and 2MB for nodes).\r\n\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}\r\n% \\begin{axis}[ymin=0, xtick={2,8,16,24,32,40,48,56,64}, xlabel={cycle length $L$}, ylabel={slowdown factor}, legend pos=north west]\r\n\\begin{axis}[ymin=0, xlabel={cycle length $L$}, ylabel={slowdown factor}, legend pos=north west]\r\n\\addplot [color=red] coordinates {\r\n(2,2.8) (4,6.4) (6,9.7) (8,14.5) (10,13.0) (12,25.1) (14,29.0) (16,28.5)\r\n(20,30.4) (24,46.0) (28,36.0) (32,43.3) (40,51.0) (48,82.2) (56,81.0) (64,72.7)\r\n};\r\n\\addlegendentry{BFS($L/2$)}\r\n\\addplot [color=green] coordinates {\r\n(4,14.4) (6,11.6) (8,20.7) (10,21.4) (12,13.9) (14,18.1) (16,26.8) (20,21.9)\r\n(24,18.8) (28,23.3) (32,31.8) (40,42.6) (48,35.6) (56,22.2) (64,41.0)\r\n};\r\n\\addlegendentry{BFS($L$)}\r\n%\\addplot [color=blue, mark=o] coordinates {\r\n%(2, 1.4)\r\n%};\r\n%\\addlegendentry{customized}\r\n\\addlegendentry{}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{Reduction in solution finding rate for two TMTO algorithms}\r\n\\label{slowdown}\r\n\\end{figure}\r\n\r\nThe zigzagging is caused by the current implementation being limited to 2-power sizes of\r\nboth subsets and cuckoo tables while the load of the latter is kept between 45\\% and 90.\r\n%We plan to lift this restriction in the near future and\r\n%produce smoother plots with constant load cuckoo tables.\r\nThe BFS($L$) algorithm exhibits at least one order of magnitude slowdown, that grows very slowly\r\nwith cycle length, while the BFS($L/2$) algorithm exhibits roughly linear slowdown.\r\nAssuming that these algorithms cannot be significantly improved upon,\r\nthis shows Cuckoo Cycle with larger cycle lengths satisfying property MB4,\r\n\r\n\\section{Choice of cycle length}\r\n\\label{cycle_length_choice}\r\nA cycle of length 2 means that two nonces produce identical edge endpoints---a {\\em collision} in edge space.\r\nThe Momentum proof-of-work looks for collisions on 50 bits of hash output among $2^{26}$ nonces.\r\nThis is in essence Cuckoo Cycle with $N=2^{25}+2^{25}$ nodes and cycle length $L=2$, with\r\ntwo differences.\r\n\r\nFirst, edges are generated not by equation~(\\ref{hash_edges}), but by splitting a SHA512 hash of\r\n$(k,\\mbox{nonce} / 8)$ into 8 64-bit words, taking the most significant 50 bits of the\r\n($\\mbox{nonce} \\bmod 8)th$ one, and viewing that as a pair of two 25-bit edge endpoints, appending\r\na bit to make them even and odd.\r\n\r\nSecond, the choice of $M=2^{26}$ gives a ratio $\\frac{M}{N}$ of 1 rather than $\\frac{1}{2}$ and as such\r\nprohibits the use of edge trimming.\r\n\r\nSince the extreme case of $L=2$ is so special, there is likely to be a greater variety of algorithms that are more\r\nefficient than for the general case. While we haven't found (and don't know of) a improved main algorithm,\r\nwe did find an improved BFS($L/2$) TMTO algorithm (implemented in momentomatum.cpp)\r\nthat cuts the memory usage in half, resulting in a slowdown of only $1.75$---a lack of memory-hardness.\r\n\r\nThe preceding analysis suggests that cycle length should be at least 20 to guard against the more efficient\r\nBFS($L/2$) algorithm, with an additional safety factor of 2.\r\n\r\nIn order to keep proof size manageable, the cycle length should not be too large either.\r\nWe thus consider 20-64 to be a healthy range, and suggest the use of the average of 42.\r\n\r\nThe plot below shows the distribution of cycle lengths found for sizes $2^{10},2^{15},2^{20},2^{25}$,\r\nas determined from 100000,100000,10000, and 10000 runs respectively. The tails of the distributions\r\nbeyond $L=100$ are not shown. For reference, the longest cycle found was of length 2120.\r\n\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[ymin=0, xlabel={cycle length $L$}, ylabel={probability}, legend pos=north east]\r\n\\addplot[color=orange] coordinates {\r\n(4,0.24862) (6,0.15673) (8,0.10907) (10,0.07952) (12,0.05783) (14,0.04269) (16,0.0303)\r\n(18,0.02237) (20,0.01653) (22,0.01168) (24,0.00815) (26,0.00511) (28,0.00374) (30,0.00251)\r\n(32,0.00191) (34,0.00098) (36,0.00079) (38,0.00029) (40,0.0003) (42,0.00011) (44,0.00018)\r\n(46,8e-05) (48,2e-05) (50,3e-05) };\r\n\\addlegendentry{10}\r\n\\addplot[color=green] coordinates {\r\n(4,0.24822) (6,0.16551) (8,0.12317) (10,0.09749) (12,0.08105) (14,0.07036) (16,0.05871) (18,0.05308)\r\n(20,0.04717) (22,0.04189) (24,0.03801) (26,0.03342) (28,0.03205) (30,0.02822) (32,0.02521)\r\n(34,0.02282) (36,0.0212) (38,0.01852) (40,0.01814) (42,0.01668) (44,0.01511) (46,0.01356)\r\n(48,0.01246) (50,0.01145) (52,0.0101) (54,0.0093) (56,0.00861) (58,0.00778) (60,0.00768)\r\n(62,0.00672) (64,0.00589) (66,0.00565) (68,0.00517) (70,0.00455) (72,0.00435) (74,0.00375)\r\n(76,0.00348) (78,0.00286) (80,0.00276) (82,0.0023) (84,0.00224) (86,0.00204) (88,0.00165)\r\n(90,0.00164) (92,0.00134) (94,0.00126) (96,0.00114) (98,0.00103) (100,0.00101) };\r\n\\addlegendentry{15}\r\n\\addplot[color=red] coordinates {\r\n(4,0.249) (6,0.1666) (8,0.1309) (10,0.0977) (12,0.0821) (14,0.0754) (16,0.0612) (18,0.0569)\r\n(20,0.0504) (22,0.0412) (24,0.0438) (26,0.0385) (28,0.0364) (30,0.0349) (32,0.0328) (34,0.0286)\r\n(36,0.0263) (38,0.0283) (40,0.0244) (42,0.0237) (44,0.0213) (46,0.0197) (48,0.0185) (50,0.0171)\r\n(52,0.0169) (54,0.0204) (56,0.0161) (58,0.0155) (60,0.0153) (62,0.0158) (64,0.0135) (66,0.0135)\r\n(68,0.0118) (70,0.0158) (72,0.0137) (74,0.012) (76,0.0108) (78,0.0119) (80,0.0116) (82,0.0106)\r\n(84,0.0112) (86,0.0102) (88,0.0075) (90,0.0096) (92,0.0091) (94,0.0094) (96,0.0077) (98,0.0089)\r\n(100,0.006) };\r\n\\addlegendentry{20}\r\n\\addplot[color=blue] coordinates {\r\n(4,0.2439) (6,0.1661) (8,0.1216) (10,0.1031) (12,0.0816) (14,0.0755) (16,0.0635) (18,0.055) (20,0.0511) (22,0.0451) (24,0.0427) (26,0.0369) (28,0.0375) (30,0.0336) (32,0.0302) (34,0.0297) (36,0.0264) (38,0.0268) (40,0.0254) (42,0.0215) (44,0.021) (46,0.0205) (48,0.0206) (50,0.0182) (52,0.017) (54,0.0192) (56,0.0172) (58,0.0169) (60,0.0167) (62,0.0147) (64,0.0169) (66,0.0137) (68,0.0169) (70,0.0125) (72,0.0127) (74,0.0123) (76,0.0139) (78,0.0122) (80,0.0131) (82,0.0129) (84,0.012) (86,0.0119) (88,0.0102) (90,0.0088) (92,0.0102) (94,0.0115) (96,0.0108) (98,0.0089) (100,0.0104) };\r\n\\addlegendentry{25}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{Distribution of cycle lengths in random graphs}\r\n\\label{cyclelengths}\r\n\\end{figure}\r\n\r\n\\section{Parallelization}\r\nAll our implementations allow the number of threads to be set with option {\\tt -t}.\r\nFor $0\\leq t < T$, thread $t$ processes all nonces $t \\bmod T$.\r\nParallelization in the basic algorithm presents some minor algorithmic challenges.\r\nPaths from an edge's two endpoints\r\nare not well-defined when other edge additions and path reversals are still in progress.\r\nOne example of such a path conflict is the check for duplicate edges yielding a false negative,\r\nif in between checking the two endpoints, another thread reverses a path through those nodes.\r\nAnother is the inadvertent creation of cycles when a reversal in progress hampers another thread's\r\npath following causing it to overlook root equality.\r\nThus, in a parallel implementation, path following can no longer be assumed to terminate.\r\nInstead of using a cycle detection algorithm such as~\\cite{1980-brent-cycles}, our implementation\r\nnotices when the path length exceeds MAXPATHLEN (8192 by default),\r\nand reports whether this is due to a path conflict.\r\n\r\nIn the main algorithm, cycle detection only takes a small fraction of total runtime and\r\nthe conflicts above could be avoided altogether by running the cycle detection single threaded.\r\n\r\nIn edge trimming, parallelization is achieved by partitioning the set of edges. To maintain efficient access\r\nto the bitmap of live edges, each thread handles words (of 32 edge-bits each) spaced $T$ apart.\r\n%Partitioning the bitmap over threads by whole words also avoids the need for atomic access.\r\n\r\nAtomic access is used by default for accessing the 2-bit counters. Disabling this results in a small\r\nchance of removing multiple edges incident to a node that access the counter at the same time.\r\n\r\nThe implementation further benefits from bucketing the addresses of counters to be updated or tested,\r\nbased on their most significant bits. Thus, when a bucket becomes full and is emptied by actually\r\nperforming those updates/tests, the accesses are limited to a certain address range, which turns\r\nout to reduce memory access latencies.\r\n\r\nThe plots below show the speedup over single thread performance achieved by multithreading at various\r\ngraph sizes and counter-partition levels.\r\n\r\n\\begin{figure}\r\n\\begin{center}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[ymin=0, xlabel={number of threads}, ylabel={speedup factor}, legend pos=north west]\r\n\\addplot [color=green] coordinates {\r\n(1,1.000) (2,1.921) (3,2.988) (4,3.757) (5,4.668) \r\n(6,5.347) (7,6.138) (8,6.792) (9,7.504) (10,8.108) \r\n(11,8.873) (12,9.380) (13,10.061) (14,10.543) (15,11.190) \r\n(16,11.608) (17,12.231) (18,12.535) (19,13.008) (20,13.433) \r\n};\r\n\\addlegendentry{speedup30}\r\n\\addplot [color=blue] coordinates {\r\n(1,1.336) (2,2.179) (3,3.148) (4,3.984) (5,4.781) \r\n(6,5.462) (7,6.061) (8,6.706) (9,7.354) (10,7.876) \r\n(11,8.333) (12,9.036) (13,9.457) (14,9.838) (15,10.250) \r\n(16,10.713) (17,11.048) (18,11.479) (19,11.736) (20,12.088) \r\n};\r\n\\addlegendentry{speedup30.0}\r\n\\addplot [color=purple] coordinates {\r\n(1,0.795) (2,1.416) (3,2.082) (4,2.683) (5,3.255) \r\n(6,3.762) (7,4.285) (8,4.759) (9,5.222) (10,5.639) \r\n(11,6.105) (12,6.605) (13,7.033) (14,7.400) (15,7.821) \r\n(16,8.144) (17,8.583) (18,9.014) (19,9.303) (20,9.585) \r\n};\r\n\\addlegendentry{speedup30.1}\r\n\\addplot [color=red] coordinates {\r\n(1,0.498) (2,0.919) (3,1.373) (4,1.761) (5,2.176) \r\n(6,2.538) (7,2.913) (8,3.234) (9,3.599) (10,3.883) \r\n(11,4.239) (12,4.527) (13,4.903) (14,5.234) (15,5.570) \r\n(16,5.845) (17,6.194) (18,6.511) (19,6.661) (20,7.003) \r\n};\r\n\\addlegendentry{speedup30.2}\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n% \\hspace{1cm}\r\n\\begin{tikzpicture}\r\n\\begin{axis}[ymin=0, xlabel={number of threads}, ylabel={speedup factor}, legend pos=north west]\r\n\\addplot [color=green] coordinates {\r\n(1,1.000) (2,1.752) (3,2.813) (4,3.439) (5,5.810) \r\n(6,6.762) (7,7.732) (8,8.669) (9,9.646) (10,10.460) \r\n(11,11.348) (12,12.077) (13,13.107) (14,13.663) (15,14.698) \r\n(16,15.363) (17,16.202) (18,16.623) (19,17.446) (20,18.207) \r\n};\r\n\\addlegendentry{speedup32}\r\n\\addplot [color=blue] coordinates {\r\n(1,2.337) (2,3.767) (3,5.445) (4,6.949) (5,8.319) \r\n(6,9.432) (7,10.462) (8,11.632) (9,12.648) (10,13.658) \r\n(11,14.717) (12,15.652) (13,16.461) (14,17.286) (15,18.063) \r\n(16,18.393) (17,19.149) (18,19.647) (19,20.169) (20,20.761) \r\n};\r\n\\addlegendentry{speedup32.0}\r\n\\addplot [color=purple] coordinates {\r\n(1,1.343) (2,2.404) (3,3.534) (4,4.526) (5,5.570) \r\n(6,6.410) (7,7.367) (8,8.208) (9,9.085) (10,9.932) \r\n(11,10.707) (12,11.590) (13,12.297) (14,12.954) (15,13.647) \r\n(16,14.294) (17,15.060) (18,15.718) (19,16.371) (20,16.804) \r\n};\r\n\\addlegendentry{speedup32.1}\r\n\r\n\\end{axis}\r\n\\end{tikzpicture}\r\n\\end{center}\r\n\\caption{Multi-threading speedup}\r\n\\label{speedup}\r\n\\end{figure}\r\n\r\n\\section{Choice of graph size}\r\nFor cryptocurrency purposes, the choice of Cuckoo graph size should be in accordance to its\r\nblock interval time.\r\nTo illustrate, suppose an average desktop machine needs 1 minute for a single proof attempt,\r\nand the block interval time is only 2 minutes. Then it will waste a large fraction (almost half)\r\nof its attempts, as about half the time, someone else finds a proof in under 2 minutes. To reduce such waste\r\nto a small percentage, the time for a single proof attempt should be a similarly small fraction of the block\r\ninterval time. This desirable property is known as {\\em progress-freeness}, and in our case\r\nis achieved more easily with a small graph (and hence memory) size.\r\n\r\nLarger memory sizes have two advantages. Beyond satisfying property MB1,\r\nthey also make it harder for botnets to mine without causing excessive swapping.\r\nSending a computer into swap-hell will likely alert its owner and trigger a cleanup,\r\nso botnet operators can be expected to eschew memory bound PoWs in favor of low-memory ones.\r\n\r\nWe expect these opposing goals to lead to graph sizes from $2^{28}$ to $2^{32}$, with the larger ones geared\r\nmore toward longer block interval times and faster mining hardware.\r\n\r\n\\section{Dynamic Sizing}\r\nIdeally, graph size should grow with evolving memory chip capacities, so as to preserve property MB1.\r\nAlthough these have shown remarkable adherence to Moore's Law in the past,\r\nthis cannot be relied on for the more distant future.\r\nWe therefore propose to re-evaluate the graph size every so-many difficulty adjustments.\r\nIf the difficulty target is sufficiently low, then the\r\ngraph size is deemed to have become \"too easy\" for existing hardware, and gets doubled.\r\n\r\nIn order to make this transition smoother and avoid severe loss of proof-of-work power,\r\nwe propose having a range of sizes allowed at any time,\r\nnamely $k$ consecutive 2-powers for some small number $k\\geq 2$.\r\nAs with Myriad-coin, separate difficulty controls are maintained for each size,\r\nadjusted so that each size accounts for roughly $\\frac{1}{k}$ of all blocks.\r\n\r\nDoubling graph sizes is then equivalent to disabling the smallest 2-power,\r\nand enabling a new largest one, whose initial difficulty target is twice that of the previous largest.\r\nEven if none of the hardware that was working on the smallest 2-power is repurposed for a larger size,\r\nsince this hardware only accounted for a fraction $\\frac{1}{k}$ of the rewards, the loss of\r\nproof-of-work power should be acceptable.\r\n\r\nIt remains to decide what exact form the ``difficulties too low'' condition should take.\r\n\r\n\\section{Conclusion}\r\nCuckoo Cycle is a novel graph-theoretic proof-of-work design that combines\r\nscalable memory requirements with instant verifiability,\r\nand the first where memory latency dominates the runtime.\r\n\r\nBarring any unforeseen memory-time trade-offs, it makes for a near-ideal memory bound proof-of-work\r\nwhose cost effectiveness on commodity hardware could greatly benefit decentralization of mining.\r\n% More research is also needed to determine the effectiveness of GPUs and FPGAs\r\n% at running Cuckoo Cycle.\r\n\r\n\\bibliographystyle{IEEEtran}\r\n\\bibliography{cuckoo}\r\n\r\n\\lstset{language=C,basicstyle=\\footnotesize}\r\n\\section{Appendix A: cuckoo.h}\r\n\\lstinputlisting{../src/cuckoo.h}\r\n\r\n%\\section{Appendix B: cuckoo\\_miner.h}\r\n%\\lstinputlisting{cuckoo_miner.h}\r\n\r\n%\\section{Appendix C: cuckoo\\_miner.cpp}\r\n%\\lstinputlisting{cuckoo_miner.cpp}\r\n\r\n\\end{document} \r\n", "meta": {"hexsha": "08d7b6ab274089649ab625ea4a5cc878d43e5903", "size": 46032, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "library/cuckoo/doc/cuckoo.tex", "max_stars_repo_name": "reliefs/brominer", "max_stars_repo_head_hexsha": "b37b7131de16ab199fb4bb56a578cef43f9afc30", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "library/cuckoo/doc/cuckoo.tex", "max_issues_repo_name": "reliefs/brominer", "max_issues_repo_head_hexsha": "b37b7131de16ab199fb4bb56a578cef43f9afc30", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "library/cuckoo/doc/cuckoo.tex", "max_forks_repo_name": "reliefs/brominer", "max_forks_repo_head_hexsha": "b37b7131de16ab199fb4bb56a578cef43f9afc30", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-06-04T08:50:33.000Z", "max_forks_repo_forks_event_max_datetime": "2018-06-04T08:50:33.000Z", "avg_line_length": 55.8640776699, "max_line_length": 994, "alphanum_fraction": 0.7032716371, "num_tokens": 15213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982043529716, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.3491523625378632}} {"text": "\\documentclass[aps,prl]{revtex4-2}\n\\usepackage{graphicx,amssymb,amsmath}\n\\bibliographystyle{apsrev4-2}\n\\renewcommand*{\\citenumfont}[1]{S#1}\n\\renewcommand*{\\bibnumfmt}[1]{[S#1]}\n\n\n\\begin{document}\n\n\n\\title{Supplementary Information\\\\Vicissitudes of Cities Driven by Re-distributive Growth}\n\\date{\\today}\n\\author{Gezhi Xiu, Jianying Wang, Lei Dong}\n% \\email{xiugz@pku.edu.cn}\n% \\affiliation{IRSGIS, Peking University}\n\n\\author{Yu Liu}\n\\email{liuyu@urban.pku.edu.cn}\n\\affiliation{Institute of Remote Sensing and Geographic Information Systems (IRSGIS), Peking University}\n% \\altaffiliation{CAMS (CNRS/EHESS) 190-198, avenue de France, 75244 Paris Cedex 13, France}\n\n\\pacs{} \n\n% If your reference list includes text notes as well as references,\n% include the following line; otherwise, comment it out.\n\n\n\\maketitle\n\\tableofcontents\n\\vspace{1cm}\n\n\\section{Mathematical Proofs}\n\nThe proofs presented in this section are all derived from the mean field approximations, which differ somehow from the simulated results. Their significance is the null results without random factors such as the spatial conditions.\n\n\\subsection{Zipf's law}\n\nZipf's law is the rank size distribution of cities. Denote by $q_m$ the number of cities having $m$ citizens, then $R(m) = \\int_m^\\infty q(m')dm'$ defines the rank. Empirically, $R(m)\\sim m^{-1}$. The more populated a big city is, the more it is favored by up-comers. In study of city size distributions, SYM is a realization of \\emph{the rich get richer} rule, for the city's growth rate increase proportionally with its present population. Thus, it is similar for cities size distributions to follow Zipf's law. \n\nThe wait time for the $n$th citizen of a city to be born since the $n-1$th is $\\frac{1}{\\beta_2(n-1)}$. So inversely the expected population of a city at time $t$ is $e^{t\\beta_2}$. We denote the sizes of cities at time $t$ as $n_i(t)$, for $i$ in $1,2,3,\\dots$. Meanwhile, we denote the expected population of a city initiated $t$ ago as $Z(t)$. The transition probability of $P(Z_t = j)$ is $e^{-\\beta_2 t}(1-e^{-\\beta_2 t})^{j-1}$. Using Kolmogorov’s Forward Equation\n\\begin{align}p_t'(1,j) = -\\beta_2 j p_t(1,j) + \\beta_2 (j-1) p_t(1,j-1).\\label{forward}\\end{align} When $j = 1$, we have \\[p_t'(1,1) = -\\beta_2 p_t(1,1), \\] which leads to $p_t=e^{-\\beta_2}$. When $j>1$, we differentiate $P(Z_t = j)$ to have \n\\begin{align}\n\t\\frac{dP(Z_t = j)}{dt} &= -\\beta_2e^{-\\beta_2 t} (1-e^{-\\beta_2 t})^{j-1} + e^{-\\beta_2 t}(j-1)(1-e^{-\\beta_2 t})^{j-2}\\beta_2 e^{-\\beta_2 t}\\notag\\\\\n\t&= -\\beta_2 e^{-\\beta_2 t}(1-e^{-\\beta_2 t})^{j-1} + e^{-\\beta_2 t}(j-1)(1-e^{-\\beta_2 t})^{j-2} [-(1-e^{-\\beta_2 t})\\beta_2+\\beta_2]\\notag\\\\\n\t&= -\\beta_2 e^{-\\beta_2 t}(1-e^{-\\beta_2 t})^{j-1} -\\beta_2 e^{-\\beta_2 t}(j-1)(1-e^{-\\beta_2 t})^{j-1} +\\beta_2e^{-\\beta_2 t}(j-1)(1-e^{-\\beta_2 t})^{j-2}\\notag\\\\\n\t&= -\\beta_2 j e^{-\\beta_2 t}(1-e^{-\\beta_2 t})^{j-1} +\\beta_2e^{-\\beta_2 t}(j-1)(1-e^{-\\beta_2 t})^{j-2}\n\\end{align}\nwhich copes with equation \\ref{forward}. Thus we complete the proof of the probability distribution of a city's population. The probability distribution of the region's population is easy to find according to the given result. The probability distribution of registering $j$ people in $i$ cities at time $t$, is \\[ P_t(i,j) = \\left(\\begin{array}{c}{j-1} \\\\ {i-1}\\end{array}\\right)\\left(e^{-\\beta t}\\right)^{i}\\left(1-e^{-\\beta t}\\right)^{j-i}. \\] \n\nMoreover, tuning the mechanism from pure birth to birth-death process, we receive a different scaling factor.\n\n\\subsection{Clark's law}\n\nIn this part, we use the mean-field approach to derive the spatial distribution of population within a species. This quantity can be interpreted as the spatial distribution mode of a species. The good thing about the mean-field approach is that we can use the potential concept to derive some conclusions. Here, we regard the growing mechanism as a multi-dimensional binary tree. On each dimension, the tree's $i$th layer has $i$ potential nodes. The probability on the $i$th node's generation is is the average of the two nearest nodes' potential generation probability. Basing on the homogeneity of the choice of $\\theta$, the derivation along different axis is the same. Within each city, the spatial distribution of people are captured by the introducing mechanism, $(r,\\theta)$. Clark's law\\cite{clark1951urban} and some variations for multi-centered models\\cite{griffith1981modelling} are empirical clues that correspond to such spatial distributions. Here we reformulate the Clark's law under spatial Yule principles. Since the isotropic setting, the derivation is only needed in one dimension added on a Doppler effect. When spatial constraints are neglected, the expected density distribution along an axis from the origin has an exponential form,$\\rho (R)\\sim e^{-\\alpha R}$. We start the discussion as a node being placed on a broad area. Regardless of adding nodes on else axis, the second is placed at $r$ right-side of the first with a probability of $1/2$. Along this axis, the $n$th node is placed at $k$ from the right end with $C_n^k/2^n$. Using the Stirling formula, it approximately equals to \\begin{align}\n & \\frac{n^{n+1/2}}{\\sqrt{2\\pi}k^{k+1/2}(n-k)^{n-k+1/2}}\\notag \\\\\n= & \\frac{n-k}{k+1}\\frac{(1+\\frac{1}{n-1-k})^{-k-1/2}}{(1+\\frac{1}{k})^{n-k-1/2}}\\notag \\\\\n= & \\frac{n-k}{k+1}\\frac{(1+\\frac{1}{n-1-k})^{n-k-1/2}}{(1+\\frac{1}{k})^{k+1/2}}\\notag \\\\\n\\sim & e^{-k},\n\\end{align}\nwhich turns out to be a exponential distribution. We can interpret it as the local properties of spatial Yule model is a discrete version of a maximum entropy system, since the Clark's law can also be derived by maximum entropy principle\\cite{merity2009accurate}. Recalling the simple mobility assumption as random walk in random direction, we show that individual-level diffusion process can be approximated by the sum of really simple moves.This is a non-trivial result since this is not derived by mean field approach but by random walk assumption of human mobility. To make it precise, at the early stage of the process, a new community can land near the centre of an existing one. In reality, two communities that are too adjacent are sometimes illustrated as two \\emph{districts} in the same city. In our model, a set of communities that destruct others' roundness functions the same with districts within one city. \n\nBy alternating the distributions of step length, we can reproduce other forms of people density distributions. A more skewed distribution of $r$ brings a Levy-like mobility pattern. In particular, a power law mobility distribution brings a Zipf's density distribution $\\rho '(R)\\sim R^{-\\gamma}$\\cite{PhysRevX.4.011008}, where $\\gamma$ is the scaling factor. In the following part, we focus on global characteristics to derive the area and population distributions among communities.\n\n\\subsection{Competition at the edges}\nWe consider the individual level expansion at urban edges. We denote that the distance between the first and the furthest node of a city as $O$ and $F$, respectively, and the moment that $F$ lands its first offspring as $t+\\tau$, where $t$ is the moment that $F$ is landed. We investigate the radius of the city, which is defined by the distance between $O$ and $F$. The radius of a city at time $t$, $R_t$, changes to $R_{t+\\tau}$ when the offspring is given birth. The expectation of $R_{t+\\tau}$ goes as \n\\begin{align}\n\tE R_{t+\\tau} &= \\int_0^{\\frac{\\pi}{2}} \\sqrt{(R_t+r\\cos\\theta)^2+(r\\sin\\theta)^2} d\\theta \\notag \\\\\n\t&= \\sqrt{2 R_t r} \\int_{0}^{\\frac{\\pi}{2}} \\sqrt{\\frac{R_{t}}{2 r}+\\frac{r}{2 R_{t}}+\\cos \\theta} d \\theta \\notag\\\\\n\t&= 2(R_t+r) \\mathbb{E}\\left(\\pi/4| \\frac{4R_t r}{(R_t+r)^2}\\right)\n\\end{align}where $\\mathbb{E}$ is the elliptic function. We find that it decreases as $r$ increases. So that the smaller cities have more active edging cells.\n\n\\subsection{Average Age within the Memory Kernel}\n\nWe consider the active population over the whole region as working population. The generation speed of population is $N^*\\beta_2 + k\\beta_1$. Thus the population is refreshing, leading to a constant expected age of \\[N^*/\\beta'\\] for $\\beta' := (N^*\\beta_2 + k\\beta_1)/N^*+k$. This leads to a practical implication that the working age allowed in cities shall be related with the sum of working offer from central government of a region. For instance, in the United States, the regressive value of $\\beta$ is around $0.04$. We assume that the working years of a person is 40 years, the expected $N^*$ for the United States is 1,600 units, i.e., 4 millions distributive working opportunities in all cities.\n\n\n\\section{Details on the simulations}\n\nThe simulation results presented here are obtained in the following way. As mentioned in the main text, three rules determine the dynamics of the model.\n\n1) \\textit{spatial growth rule} and \\textit{active citizen rule} both control the city generation and expansion. Instead of conducting the designed protocol, we do it in an equivalent way by stretching the timeline to events labeled in integer. At each time step, we first decide if we add a new city, with probability $p(S)$, or a new meta-population to the existing city, with probability $1 - p(S)$. The probability $p(S)$ is determined by the total ‘‘active’ population and number of cities, $\\frac{k\\beta_1}{k\\beta_1+N\\beta_2}$. If a city is to be built, the place of it is randomly chosen at an empty spot. By empty we mean the cell of the spot contains no existing node. If it is a new node to be generated, we first determine its ancestor node and land it at (r; $\\theta$) of the ancestor node, where r is constant default as 0.5 and $\\theta$ is a realization of a uniformly distributed random variable $U[0; 2\\pi]$. The nodes can survive if the target cell is not taken. The generation and spatial expansion of the city promotes spatial competition. \n\n2). Memory kernel rule captures the resources competition among cities. In the first phase of the simulation, where the total population of each city $i$ less than the population boundary $N^*$, a new generation of the node is solely accompanied with a record in the memory kernel. However, when the total population exceeds the $N^*$, a new comer would deactivate a random dweller who is previously recorded in the memory kernel. The growth of the population promotes resource competition among cities.\n\n\\section{Illustrations for the economic constrained phase}\n\n\\subsection{Turnover rate} \n\nRegional development is complex and is affected by different determinants in different time. For example, in Hebei province of China, the largest city Shijiazhuang started to develop fast since it is the crossroad of railways. We interpret this as the redistribution of social resource. There have been plenty of literature\\cite{bowles2019neolithic} that illustrate the interplay between social development and injustice. Jilin City in Jilin province used to be the capital of Jilin province until Changchun, from 1954, changed its status. Obviously, these are redistributions of social resource. In many literature, social injustice is strengthened by the technical revolutions. Stately, agricultural society is more likely to store fortune overtime than nomadic society\\cite{doi:10.1086/701789}. The Gini coefficients in ancient society is also increasing as the productivity grows\\cite{kohler2017greater}. We read this as inequality of human society is born with the desire of better life. \n\nBut what is more interesting is that, when the rich do not get richer. In this Letter we give a possible explanation that the \\emph{memory} of human society is the force of future endeavour. A city's future ability to develop is based on the active population brought by the existing active people. The limited resource is dynamically divided by increasing number of cities, so that there is, though small, but still some odd of overturn. In the following text, we first derive an approximated solution of this problem. We denote the coins share in the $k$th largest city as $n_k$, and investigate the probability that the second largest city becoming the largest, $P_{\\text{overturn}}$. In $m$ successive additions of meta-population from the moment $t_l$, the chance for $n_2(t_l+m)\\ge n_1(t_l+m)$ is equivalent to the case that the city $2$ is picked $n_1(t_l)-n_2(t_l)$ times more than city $1$. The probability of $n_1(t_l)-n_2(t_l)$ picks of city $2$ in a row is $\\frac{n_1!}{n_2!}/N^{n_1-n_2}$. Adding another step brings an probability of $n_1/N$ for $n_1$ to increase to $n_1+1$. So the process ends at $n_1(t_l)-n_2(t_l)+1$ steps is the former probability times $(n_1+1)/N$ and plus a expected latent step length. \n\n\\subsection{Urban shrinkage}\n\nThe urban development is a sequel of the spatial distribution of existing resource. Thus the preferential attachment is not only performed among people, but also on urban land-use. The concentration of urban resource result in urban shrinkage, indicating that the popular definition of resource distribution, say Gross Democratic Product (GDP), may not be the best indicator of regional fortune, since it is not a perfect indication of a place's future. Urban shrinking is widely discussed in recent years. It is always referred with demographic changes such as decreasing fertility, aging, and out-migration\\cite{haase2008urban}. In the comprehension of urban input-output framework, the governmental investment cannot follow up with the spatial growth of population. So the further investment can only go to \\emph{active} area where recent comers to the cities are mostly found. \n\nWe conduct a cell-wise analysis. The speed of a cell's losing active population equals to the speed of others' adding, $[(N^*-n)\\beta_2 +k \\beta_1]\\cdot n/N^*$; The speed of its own adding is $n\\beta_2$. Thus the equilibrium condition is \\[ [(N^*-n)\\beta_2+k\\beta_1]\\cdot \\frac{n}{N^*} = \\beta_2N^*. \\] The equilibrium $n_eq = k/\\beta$. This means that whether a spot in city will be prosperous has nothing to do with the regional resource $N^*$, but it is related to the strong force of emergence of cities.\n\n\n% \\bibliographystyle{plain}\n\\bibliography{refs.bib}\n \n\\end{document}", "meta": {"hexsha": "d302dbace97159c8ae388b439c3b1a3fbdb67277", "size": 14215, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Tex/sym_history/sym_april/SI.tex", "max_stars_repo_name": "GXIU/SpatialYule", "max_stars_repo_head_hexsha": "897e4da9f3c47bcc6e379b2473acd12e2d94442c", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Tex/sym_history/sym_april/SI.tex", "max_issues_repo_name": "GXIU/SpatialYule", "max_issues_repo_head_hexsha": "897e4da9f3c47bcc6e379b2473acd12e2d94442c", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Tex/sym_history/sym_april/SI.tex", "max_forks_repo_name": "GXIU/SpatialYule", "max_forks_repo_head_hexsha": "897e4da9f3c47bcc6e379b2473acd12e2d94442c", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 138.0097087379, "max_line_length": 1626, "alphanum_fraction": 0.7486457967, "num_tokens": 3871, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795672, "lm_q2_score": 0.66192288918838, "lm_q1q2_score": 0.3490428766721751}} {"text": "\n\\documentclass[12pt]{article}\n\\usepackage{url} \n%\\usepackage[dvips]{graphicx}\n\\usepackage[pdftex]{graphicx}\n\\usepackage[latin1]{inputenc}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{fancyhdr}\n\\usepackage{bm}\n\\usepackage{float}\n\\usepackage{color}\n\\usepackage[dvipsnames]{xcolor}\n\\usepackage{wrapfig}\n\\usepackage{multicol}\n\\usepackage[colorlinks=true,\n\t\tlinkcolor=red,\n\t\tcitecolor=blue,\n\t\turlcolor=blue]{hyperref}\n\n\\usepackage{titlesec}\n\\usepackage[titletoc]{appendix}\n%\\usepackage{appendix}\n\\setcounter{secnumdepth}{4}\n\n\\titleformat{\\paragraph}\n{\\normalfont\\normalsize\\bfseries}{\\theparagraph}{1em}{}\n\\titlespacing*{\\paragraph}\n{0pt}{3.25ex plus 1ex minus .2ex}{1.5ex plus .2ex}\n\n\\setlength {\\parindent} { 10mm} \n\\setlength{\\textheight}{230mm} \n\\setlength{\\textwidth}{160mm} \n\\setlength{\\oddsidemargin}{0mm}\n\\setlength{\\topmargin}{-10mm} \n% newcommands\n\\newcommand{\\p}{\\partial}\n\\newcommand{\\g}[1]{\\mbox{\\boldmath $#1$}}\n\\newcommand{\\vi}{\\g V_{\\! \\! i}}\n\\newcommand{\\ps}{Pfirsch-Schl\\\"{u}ter} \n\\newcommand{\\lp}{\\left(}\n\\newcommand{\\rp}{\\right)}\n\\newcommand{\\ca}[1]{\\mbox{\\cal $#1$}}\n\\newcommand{\\be}{\\begin{displaymath}}\n\\newcommand{\\ee}{\\end{displaymath}}\n\\newcommand{\\bn}{\\begin{equation}}\n\\newcommand{\\en}{\\end{equation}}\n\\newcommand{\\mygtrsim}{\\mathrel{\\mbox{\\raisebox{-1mm}{$\\stackrel{>}{\\sim}$}}}}\n\\newcommand{\\mylsim}{\\mathrel{\\mbox{\\raisebox{-1mm}{$\\stackrel{<}{\\sim}$}}}}\n\\newcommand{\\vek}{\\bf}\n\\newcommand{\\ten}{\\sf}\n\\newcommand{\\bfm}[1]{\\mbox{\\boldmath$#1$}}\n\\newcommand{\\lang}{\\left\\langle}\n\\newcommand{\\rang}{\\right\\rangle}\n\\newcommand{\\vo}[1]{\\left|\\begin {array}{l} \\mbox{} \\\\ \\mbox{} \\\\$#1$ \\end\n{array}\\right .} \n\\newcommand{\\von}[2]{\\left |\\begin {array}{l}\n\\mbox{}\\\\$#1$\\\\$#2$ \\end {array}\\right .}\n\\newcommand{\\simgt}{\\:{\\raisebox{-1.5mm}{$\\stackrel\n{\\textstyle{>}}{\\sim}$}}\\:}\n\\newcommand{\\simlt}{\\:{\\raisebox{-1.5mm}{$\\stackrel\n{\\textstyle{<}}{\\sim}$}}\\:}\n%\\renewcommand {\\baselinestretch} {1.67}\n%\\pagestyle{empty}\n\\newcommand{\\todo}[1]{\\textbf{\\textcolor{red}{TODO: #1}}}\n\\newcommand{\\remark}[1]{\\textbf{\\textcolor{red}{REMARK: #1}}}\n\n\\title{Implementation of $\\Phi_1$ in SFINCS}\n\n\\pagestyle{fancy}\n\\fancyhead{}\n\\chead{Albert Moll\\'en %850227-2019\n\\\\ Implementation of $\\Phi_1$ in SFINCS}\n\\cfoot{\\thepage}\n\\renewcommand{\\headrulewidth}{1pt}\n\\renewcommand{\\footrulewidth}{1pt}\n\\setlength{\\headheight}{28pt}\n\\setlength{\\footskip}{25pt}\n\n\\newcommand{\\red}[1]{\\textcolor{red}{#1}}\n\\newcommand{\\mE}{\\mathcal{E}}\n\\newcommand{\\energy}{\\mathcal{E}}\n\\newcommand{\\mK}{\\mathcal{K}}\n\\newcommand{\\mN}{\\mathcal{N}}\n\\newcommand{\\mD}{\\mathcal{D}}\n\\newcommand{\\ord}{\\mathcal{O}}\n\\newcommand{\\Tpe}{T_\\perp}\n\\newcommand{\\Tpa}{T_\\|}\n\\newcommand{\\vpe}{v_\\perp}\n\\newcommand{\\vpa}{v_\\|}\n\\newcommand{\\kpe}{k_\\perp}\n\\newcommand{\\kpa}{k_\\|}\n\\newcommand{\\Bv}{\\mathbf{B}}\n\\newcommand{\\Ev}{\\mathbf{E}}\n\\newcommand{\\bv}{\\mathbf{b}}\n\\newcommand{\\vv}{\\mathbf{v}}\n\\newcommand{\\cd}{\\cdot}\n\\newcommand{\\na}{\\nabla}\n\\newcommand{\\btheta}{\\bar{\\theta}}\n\\newcommand{\\phit}{\\tilde{\\phi}}\n\\newcommand{\\oert}{\\tilde{\\omega}_{Er}}\n\n\\begin{document}\n\\titlepage\n\n\\maketitle\n\n\\section*{EUTERPE old equations vs new equations}\nWe want to modify the implementation of the old EUTERPE equations \\cite{regana} in SFINCS \nto the new equations \\cite{reganaArxiv}.\n\nThe old equations for the particle trajectories and the drift-kinetic equation are\n\\begin{align}\n\\dot{\\bm{R}} & = v_\\| \\bm{b} - \\frac{\\na \\Phi_0 \\times \\bm{b}}{B} \\\\\n\\dot{v}_\\| & = - \\frac{q}{m} \\bm{b} \\cdot \\na \\Phi_1 - \\mu \\bm{b} \\cdot \\na B - \\frac{v_\\|}{B^2} \\left(\\bm{b} \\times \\na B\\right) \\cdot \\na \\Phi_0 \\\\\n\\dot{\\mu} & = 0\n\\label{eq:ParticleTrajEuterpeOld}\n\\end{align}\nand\n\\begin{multline}\n\\frac{\\p f_1}{\\p t} + \\dot{\\bm{R}} \\cdot \\na f_1 + \\dot{v}_\\| \\frac{\\p f_1}{\\p v_\\|} - C = \\\\ =\n- f_M \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) \\cdot \\na \\psi - \n\\frac{q}{m} \\frac{f_M}{v_{\\mathrm{th}}^2} \\left(v_\\| \\bm{b} + \\bm{v}_d\\right) \\cdot \\left(\\na \\Phi_0 + \\na \\Phi_1\\right).\n\\label{eq:DriftKineticEuterpeOld}\n\\end{multline}\n\nThe new equations are\n\\begin{align}\n\\dot{\\bm{R}} & = v_\\| \\bm{b} - \\frac{\\na \\Phi_0 \\times \\bm{b}}{B} \\\\\n\\dot{v}_\\| & = - \\frac{q}{m} \\bm{b} \\cdot \\na \\Phi_1 - \\mu \\bm{b} \\cdot \\na B - \\frac{v_\\|}{B^2} \\left(\\bm{b} \\times \\na B\\right) \\cdot \\na \\Phi_0 \\\\\n\\dot{\\mu} & = 0\n\\label{eq:ParticleTrajEuterpeNew}\n\\end{align}\nand\n\\begin{multline}\n\\frac{\\p f_1}{\\p t} + \\dot{\\bm{R}} \\cdot \\na f_1 + \\dot{v}_\\| \\frac{\\p f_1}{\\p v_\\|} - C = \\\\ =\n- f_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) \\cdot \\na \\psi.\n\\label{eq:DriftKineticEuterpeNew}\n\\end{multline}\nHere we have the definitions\n\\begin{equation}\n\\Phi\\left(\\psi, \\theta, \\varphi\\right) \\equiv \\Phi_0\\left(\\psi\\right) + \\Phi_1\\left(\\theta, \\varphi\\right),\n\\label{eq:Phi}\n\\end{equation}\n\\begin{equation}\n\\bm{v}_d = \\frac{m}{q} \\frac{\\mu B + v_\\|^2}{B^2} \\bm{b} \\times \\na B,\n\\label{eq:MagneticDrift}\n\\end{equation}\n\\begin{equation}\n\\bm{v}_{E1} = - \\frac{\\na \\Phi_1 \\times \\bm{b}}{B},\n\\label{eq:ElectricDrift1}\n\\end{equation}\n\\begin{equation}\nf_0 = f_M \\exp \\left(- q \\Phi_1 / T \\right) = \\frac{n_0\\left(\\psi\\right)}{\\left(2 \\pi\\right)^{3/2} v_{\\mathrm{th}}^3} \\exp \\left[- \\frac{\\left(v_\\|^2 + v_\\perp^2\\right)}{2 v_{\\mathrm{th}}^2}\\right] \\exp \\left(- q \\Phi_1 / T \\right),\n\\label{eq:f0}\n\\end{equation}\n$q = Z e$ and $v_{\\mathrm{th}}^2 = T/m$.\\\\\n\n\\noindent The only differences appear in the RHS:s of Eqs.~\\ref{eq:DriftKineticEuterpeOld} and \\ref{eq:DriftKineticEuterpeNew}:\\\\ \nFirstly, $f_M$ has been replaced by $f_0$ containing the $\\exp \\left(- q \\Phi_1 / T \\right)$ factor.\\\\ \nSecondly, some of the terms have been modified. \nWe rewrite the RHS of \\ref{eq:DriftKineticEuterpeNew}:\n\\begin{multline}\n{\\mathrm{RHS}}_{\\mathrm{NEW}} =\n- f_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) \\cdot \\na \\psi = \\\\ =\n%%\n- f_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) \\cdot \\na \\psi + \\\\ - \nf_0 \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) \\cdot \\na \\psi = \\\\ =\n%%\n- f_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_d \\cdot \\na \\psi + \\\\ - \nf_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_{E1} \\cdot \\na \\psi - f_0 \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} \\bm{v}_{E1} \\cdot \\na \\psi + \\\\ -\nf_0 \\frac{q}{T} \\Phi_1 \\frac{\\na T}{T} \\cdot\n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) = \\\\ =\n%%\n- f_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_d \\cdot \\na \\psi + \\\\ \n\\textcolor{blue}{- f_0 \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_{E1} \\cdot \\na \\psi} + \\\\ \n\\textcolor{red}{- f_0 \\frac{q}{T} \\left[\\na \\Phi_0 \\cdot \\bm{v}_{E1} + \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_d + \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_{E1} \\right]}.\n\\label{eq:DriftKineticEuterpeNewRHS}\n\\end{multline}\nSimilarly, the RHS of \\ref{eq:DriftKineticEuterpeOld} is rewritten as:\n\\begin{multline}\n{\\mathrm{RHS}}_{\\mathrm{OLD}} =\n- f_M \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\left(\\bm{v}_d + \\bm{v}_{E1}\\right) \\cdot \\na \\psi - \n\\frac{q}{m} \\frac{f_M}{v_{\\mathrm{th}}^2} \\left(v_\\| \\bm{b} + \\bm{v}_d\\right) \\cdot \\left(\\na \\Phi_0 + \\na \\Phi_1\\right) = \\\\ =\n%%\n- f_M \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_d \\cdot \\na \\psi + \\\\ \n\\textcolor{blue}{- f_M \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_{E1} \\cdot \\na \\psi} + \\\\ \n\\textcolor{red}{- f_M \\frac{q}{T} \\left[v_\\| \\bm{b} \\cdot \\na \\Phi_1 + \\bm{v}_d \\cdot \\na \\Phi_1\\right]}.\n\\label{eq:DriftKineticEuterpeOldRHS}\n\\end{multline}\nComparing ${\\mathrm{RHS}}_{\\mathrm{NEW}}$ to ${\\mathrm{RHS}}_{\\mathrm{OLD}}$ we see that, apart from $f_M \\rightarrow f_0$, only the terms in red have changed. \n\n\\subsection*{What has to be changed in SFINCS}\nThe only part of the drift-kinetic equation block we need to modify is the RHS, where we need to update the red terms and substitute $f_M \\rightarrow f_0$. SFINCS had earlier neglected the $\\bm{v}_d \\cdot \\na \\Phi_1$-term which is small in the standard $\\rho_\\ast$-expansion. \nThe RHS that was implemented is (see Matt's ISHW poster, also note that $\\bm{v}_{E} \\cdot \\na \\psi = \\bm{v}_{E1} \\cdot \\na \\psi$)\n\\begin{multline}\n{\\mathrm{RHS}}_{\\mathrm{SFINCS, OLD}} =\n- f_M \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_d \\cdot \\na \\psi + \\\\ \n\\textcolor{blue}{- f_M \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n\\bm{v}_{E} \\cdot \\na \\psi} %+ \\\\ \n\\textcolor{red}{- f_M \\frac{q}{T} v_\\| \\bm{b} \\cdot \\na \\Phi_1 }.\n\\label{eq:DriftKineticSFINCSOldRHS}\n\\end{multline}\nWe thus replace \n\\begin{equation}\n\\textcolor{red}{\nv_\\| \\bm{b} \\cdot \\na \\Phi_1\n}\n\\label{eq:ReplaceOLD}\n\\end{equation}\nwith \n\\begin{equation}\n\\textcolor{red}{\n\\na \\Phi_0 \\cdot \\bm{v}_{E} + \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_d + \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_{E}},\n\\label{eq:ReplaceNEW}\n\\end{equation}\nand make the substitution\n\\begin{equation}\n\\textcolor{red}{\nf_M \\rightarrow f_0 = f_M \\exp \\left(- q \\Phi_1 / T \\right)}.\n\\label{eq:substitution}\n\\end{equation}\n%\\remark{In EUTERPE $\\Phi_1$ is only an unknown in the quasi-neutrality equation, in the kinetic equation it is an input which means that there are no nonlinearities. It also means that the exponential in $f_0$ is not expanded in the kinetic equation. \n%Are all terms in Eq.~\\ref{eq:ReplaceNEW} feasible to implement in SFINCS? E.g. is it a problem that the $\\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_{E}$-term contains 3 factors with $\\Phi_1$?}\\\\\n\\\\\n\n\\noindent All terms which contain $\\Phi_1$ are now nonlinear. \nIt does not make sense to have both switches \\textbf{includePhi1} and \\textbf{nonlinear} still available in SFINCS, \nand consequently we will remove the \\textbf{nonlinear} switch.\n\\\\\n\n\\noindent We will also introduce to possibility to run SFINCS with an adiabatic species. \n\n\\newpage\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section*{Implementation in SFINCS}\nOf the equations implemented in SFINCS \\cite{SFINCStechnicalDoc}, the only two we need to modify are the kinetic equation\n\\begin{multline} \nR\\left(f_{1}, \\Phi_1\\right) = \nK \\left\\{\\theta\\right\\} \\frac{\\p f}{\\p \\theta} + K \\left\\{\\zeta\\right\\} \\frac{\\p f}{\\p \\zeta} + \nK \\left\\{x\\right\\} \\frac{\\p f}{\\p x} + K \\left\\{\\xi\\right\\} \\frac{\\p f}{\\p \\xi} + \n\\textcolor{Goldenrod}{K \\left\\{\\psi\\right\\} \\frac{\\p f_{M}}{\\p \\psi}} + \\\\ \n- C \\left\\{f\\right\\} - S_{1} f_{M} - S_{2} f_{M} x^2 - \\frac{Z e v}{T} x \\xi \\frac{\\left\\langle \\bm{E} \\cdot \\bm{B} \\right\\rangle B}{\\left\\langle B^2 \\right\\rangle} f_M = 0\n\\label{eq:KineticEqSFINCS}\n\\end{multline}\nand the quasineutrality equation\n\\begin{equation}\n\\sum_s Z_s \\int d^3v \\, f_s + \\lambda = 0.\n\\label{eq:QuasineutralityEqSFINCS}\n\\end{equation}\nHere $x = v/v_s = v / \\sqrt{2T/m}$ and $\\xi = v_\\| / v$.\\\\\n\n\\noindent In the implementation we need to rewrite all our equations into SFINCS units, using the following identities:\\\\\n$m = \\hat{m} \\bar{m}$, $n = \\hat{n} \\bar{n}$, $T = \\hat{T} \\bar{T}$, $\\Phi = \\hat{\\Phi} \\bar{\\Phi}$, \n$B = \\hat{B} \\bar{B}$, $B_\\zeta = \\bar{R} \\bar{B} \\hat{B}_\\zeta$, $B_\\theta = \\bar{R} \\bar{B} \\hat{B}_\\theta$, $D = \\bar{B} \\hat{D} / \\bar{R}$, \n$\\bar{v} = \\sqrt{2 \\bar{T} / \\bar{m}}$, $\\alpha = e \\bar{\\Phi} / \\bar{T}$, $\\Delta = \\bar{m} \\bar{v} / \\left(e \\bar{B} \\bar{R} \\right)$, $\\displaystyle \\frac{d X}{d \\psi} = \\frac{1}{\\hat{\\psi}_a \\bar{R}^2 \\bar{B}} \\frac{d X}{d \\psi_N}$, \n$\\hat{\\psi} = \\psi_N \\hat{\\psi}_a$, $\\displaystyle \\frac{1}{\\hat{\\psi}_a} \\frac{d X}{d \\psi_N} = \\frac{d X}{d \\hat{\\psi}}$ \nand \n$\\displaystyle \\alpha \\cdot \\Delta = \\frac{e \\bar{\\Phi}}{\\bar{T}} \\cdot \\frac{\\bar{m} \\bar{v}}{e \\bar{B} \\bar{R}} = \\frac{2^{1/2} \\bar{m}^{1/2} \\bar{\\Phi}}{\\bar{B} \\bar{R} \\bar{T}^{1/2}}$. \nFurthermore, we note that the kinetic equation is made dimensionless by multiplying with the factor \n\\begin{equation}\n \\label{eq:DimensionlessFactor}\n \\frac{\\bar{v}^3}{\\bar{n}} \\frac{\\bar{R}}{\\bar{v}} = \\frac{2 \\bar{T} \\bar{R}}{\\bar{m} \\bar{n}}.\n\\end{equation}\n\n\\subsubsection*{Newton's method}\nIn each iteration step we want to calculate the residual and Jacobian of $R\\left(\\bm{X}\\right) = 0$ with $\\bm{X} = \\left(f_{1}, \\Phi_1\\right)$. \nThe residual is $R$ itself, and the Jacobian is $\\displaystyle R' = \\frac{\\delta R\\left(\\bm{X}\\right)}{\\delta \\bm{X}}$. \nThe state-vector is updated as\n\\begin{equation}\n \\label{eq:StateVectorUpdate}\n \\bm{X}_{n+1} = \\bm{X}_{n} - \\frac{R\\left(\\bm{X}_{n}\\right)}{R'\\left(\\bm{X}_{n}\\right)}.\n\\end{equation}\n\n\\newpage\n\\subsection*{Drift-kinetic equation}\nFor the residual $R\\left(f_{1}, \\Phi_1\\right)$ the only term in the kinetic equation we need to modify is the one in yellow in Eq.~\\ref{eq:KineticEqSFINCS}. \nWe replace $f_M \\rightarrow f_0 = f_M \\exp \\left(- q \\Phi_1 / T \\right)$, \nand use that $K \\left\\{\\psi\\right\\} = \\bm{v}_E \\cdot \\na \\psi + \\bm{v}_d \\cdot \\na \\psi = \\bm{v}_{E1} \\cdot \\na \\psi + \\bm{v}_d \\cdot \\na \\psi$\nto write \n\\begin{multline}\nK \\left\\{\\psi\\right\\} \\frac{\\p f_{0}}{\\p \\psi} = \\exp \\left(- q \\Phi_1 / T \\right) \\frac{\\p f_{M}}{\\p \\psi} \\left(\\bm{v}_{E1} \\cdot \\na \\psi + \\bm{v}_d \\cdot \\na \\psi\\right) = \\\\ = \n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \n \\left(- \\frac{\\na \\Phi_1 \\times \\bm{b}}{B} \\cdot \\na \\psi + \\bm{v}_d \\cdot \\na \\psi\\right) = \\\\ =\n \\left\\| - \\frac{\\na \\Phi_1 \\times \\bm{b}}{B} \\cdot \\na \\psi = - \\frac{ \\bm{B} \\times \\na \\psi}{B^2} \\cdot \\na \\Phi_1 = \n \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] \\right\\| = \\\\ =\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\cdot \\\\\n \\left(\\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] + \\bm{v}_d \\cdot \\na \\psi\\right)\n\\label{eq:DKresidualNewTerm1}\n\\end{multline}\n(Here $D = \\na \\psi \\cdot \\na \\theta \\times \\na \\zeta$.)\nWritten like this we explicitly see the places where $\\Phi_1$ appears in $\\displaystyle K \\left\\{\\psi\\right\\} \\frac{\\p f_{0}}{\\p \\psi}$. \nFrom Eq.~\\ref{eq:DKresidualNewTerm1} we obtain the corresponding terms in the Jacobian matrix\n\\begin{multline}\n\\frac{\\delta}{\\delta \\Phi_1} \\left(K \\left\\{\\psi\\right\\} \\frac{\\p f_{0}}{\\p \\psi}\\right) = \\textcolor{orange}{\n- \\frac{q}{T}\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\cdot} \\\\ \\textcolor{orange}{\n \\left(\\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] + \\bm{v}_d \\cdot \\na \\psi\\right)} \\, + \\\\ + \\,\\textcolor{green}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\left(\\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] + \\bm{v}_d \\cdot \\na \\psi\\right)} \\, + \\\\ + \\,\n\\textcolor{brown}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\cdot} \\\\ \\textcolor{brown}{\n \\left(\\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p }{\\p \\zeta} - B_\\zeta \\frac{\\p }{\\p \\theta}\\right]\\right)}\n\\label{eq:DKJacobianNewTerm1}\n\\end{multline}\n\n\\subsubsection*{Residual}\nMany of the terms involving $\\bm{v}_d \\cdot \\na \\psi$ are almost implemented in SFINCS already except that they now contain the $\\exp \\left(- \\frac{q \\Phi_1}{T} \\right)$-factor. \nWe therefore rewrite Eq.~\\ref{eq:DKresidualNewTerm1} as \n\\begin{equation}\nK \\left\\{\\psi\\right\\} \\frac{\\p f_{0}}{\\p \\psi} = R_m + R_E\n\\label{eq:DKresidualNewTermSplit}\n\\end{equation}\nwhere\n\\begin{equation}\nR_m = \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\bm{v}_d \\cdot \\na \\psi\n\\label{eq:DKresidualNewTermRm}\n\\end{equation}\nand \n\\begin{equation}\nR_E = \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right].\n\\label{eq:DKresidualNewTermRE}\n\\end{equation}\n\n\\paragraph*{$\\bm{R_m}$}\n$R_m$ will be implemented in evaluateResidual.F90. We write the term as \n\\begin{multline}\nR_m = \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(x^2 - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\bm{v}_d \\cdot \\na \\psi + \\\\ + \\textcolor{red}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\bm{v}_d \\cdot \\na \\psi }.\n\\label{eq:DKresidualNewTermRm2}\n\\end{multline}\nNote that the first term in Eq.~\\ref{eq:DKresidualNewTermRm2} can only be implemented in evaluateResidual.F90, since it is not of the form $L\\left[\\Phi_1\\right]$ where $L\\left[\\right]$ is a linear operator.\\\\\nThe first term in Eq.~\\ref{eq:DKresidualNewTermRm2} is already implemented in evaluateResidual.F90 except for the factor \n\\[\n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) = \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}} \\right)\n\\]\nwhich has to be added.\\\\\nThe second term in Eq.~\\ref{eq:DKresidualNewTermRm2} we rewrite in SFINCS units (also considering the factor Eq.~\\ref{eq:DimensionlessFactor}) as\n\\begin{multline}\n\\textcolor{red}{\n\\left(\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\bm{v}_d \\cdot \\na \\psi\\right)_{\\mathrm{SFINCS}} = } \\\\ \\textcolor{red}{ =\n\\frac{\\alpha \\Delta}{3 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{5/2} \\hat{B}^3 } \\hat{\\Phi}_1 \\frac{\\p \\hat{T}}{\\p \\hat{\\psi}}\nx^2 \\left(P_2\\left(\\xi\\right) + 2 P_0\\right)\n\\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p \\hat{B}}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p \\hat{B}}{\\p \\theta}\\right]. \n}\n\\label{eq:Rmterm2}\n\\end{multline}\nThe first term in Eq.~\\ref{eq:DKresidualNewTermRm2} becomes\n\\begin{multline}\n\\left(\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(x^2 - \\frac{3}{2}\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\bm{v}_d \\cdot \\na \\psi \\right)_{\\mathrm{SFINCS}} = \\\\ = \\frac{1}{Z}\n\\frac{\\Delta}{3 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{1/2} \\hat{B}^3 }\nx^2 \\left(P_2\\left(\\xi\\right) + 2 P_0\\right)\n\\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p \\hat{B}}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p \\hat{B}}{\\p \\theta}\\right] \\cdot \\\\ \\cdot\n\\left[\\frac{1}{\\hat{n}} \\frac{\\p \\hat{n}}{\\p \\hat{\\psi}} + \\frac{Z \\alpha}{\\hat{T}} \\frac{\\p \\hat{\\Phi}_0}{\\p \\hat{\\psi}} + \\left(x^2 - \\frac{3}{2}\\right) \\frac{1}{\\hat{T}} \\frac{\\p \\hat{T}}{\\p \\hat{\\psi}}\\right].\n\\label{eq:Rmterm1}\n\\end{multline}\n\n\n\\paragraph*{$\\bm{R_E}$}\n$R_E$ we will instead implement in populateMatrix.F90. We write the term as \n\\begin{multline}\nR_E = \\textcolor{blue}{\n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(x^2 - \\frac{3}{2} \\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p }{\\p \\zeta} - B_\\zeta \\frac{\\p }{\\p \\theta}\\right] \\Phi_1} + \\\\ +\n\\textcolor{magenta}{\n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p }{\\p \\zeta} - B_\\zeta \\frac{\\p }{\\p \\theta}\\right]\\Phi_1 } + \\\\ +\n \\textcolor{cyan}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p }{\\p \\zeta} - B_\\zeta \\frac{\\p }{\\p \\theta}\\right] \\Phi_1}.\n\\label{eq:DKresidualNewTermRE2}\n\\end{multline}\nNote that in the code when evaluating the residual, the matrix added in populateMatrix.F90 is multiplied by the state-vector in evaluateResidual.F90 and therefore the rightmost $\\Phi_1$ should not be added inside populateMatrix.F90.\\\\\nThe first term in Eq.~\\ref{eq:DKresidualNewTermRE2} is already implemented in populateMatrix.F90 except for the factor \n\\[\n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) = \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}} \\right)\n\\]\nwhich has to be added.\\\\\nThe second term in Eq.~\\ref{eq:DKresidualNewTermRE2} we rewrite in SFINCS units (also considering the factor Eq.~\\ref{eq:DimensionlessFactor}) as\n\\begin{multline}\n\\textcolor{magenta}{\n\\left(\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] \\right)_{\\mathrm{SFINCS}} } = \\\\ = \\textcolor{magenta}{\n\\frac{Z \\alpha^2 \\Delta}{2 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{5/2} \\hat{B}^2 } \\frac{\\p \\hat{\\Phi}_0}{\\p \\hat{\\psi}} \n\\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p}{\\p \\theta}\\right] \\hat{\\Phi}_1.\n}\n\\label{eq:REterm2}\n\\end{multline}\nThe third term in Eq.~\\ref{eq:DKresidualNewTermRE2} we rewrite in SFINCS units (also considering the factor Eq.~\\ref{eq:DimensionlessFactor}) as\n\\begin{multline}\n\\textcolor{cyan}{\n\\left(\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right]\\right)_{\\mathrm{SFINCS}}} = \\\\ = \\textcolor{cyan}{\n\\frac{Z \\alpha^2 \\Delta}{2 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{7/2} \\hat{B}^2 } \\frac{\\p \\hat{T}}{\\p \\hat{\\psi}} \\hat{\\Phi}_1\n\\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p}{\\p \\theta}\\right] \\hat{\\Phi}_1.\n}\n\\label{eq:REterm3}\n\\end{multline}\nThe first term in Eq.~\\ref{eq:DKresidualNewTermRE2} in SFINCS units (also considering the factor Eq.~\\ref{eq:DimensionlessFactor}) is \n\\begin{multline}\n\\textcolor{blue}{ \\left(\n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(x^2 - \\frac{3}{2} \\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p }{\\p \\zeta} - B_\\zeta \\frac{\\p }{\\p \\theta}\\right] \\Phi_1\\right)_{\\mathrm{SFINCS}} } = \\\\ = \n\\textcolor{blue}{\n\\frac{\\alpha \\Delta}{2 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{3/2} \\hat{B}^2 } \\left[\\frac{1}{\\hat{n}} \\frac{\\p \\hat{n}}{\\p \\hat{\\psi}} + \\left(x^2 - \\frac{3}{2} \\right) \\frac{1}{\\hat{T}} \\frac{\\p \\hat{T}}{\\p \\hat{\\psi}}\\right]\n\\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p}{\\p \\theta}\\right] \\hat{\\Phi}_1.\n}\n\\label{eq:REterm1}\n\\end{multline}\n\n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\nevaluateResidual.F90\npopulateMatrix.F90\n\\end{verbatim}\n\n\n\\subsubsection*{Jacobian}\nThe Jacobian terms will be implemented in populateMatrix.F90. \nIn the code we use SFINCS units, and the Jacobian is calculated from taking the derivative of the residual in SFINCS units with respect to the state-vector in SFINCS units (also considering the factor Eq.~\\ref{eq:DimensionlessFactor}). This implies that what we are calculating here is \n\\[\n\\frac{\\delta}{\\delta \\hat{\\Phi}_1} \\left(\\hat{R}_m + \\hat{R}_E\\right),\n\\]\nwhere $\\hat{R}_m$ and $\\hat{R}_E$ are how the components of the residual are written in SFINCS. \\\\\nWe see that the \\textcolor{brown}{last term} in the Jacobian in Eq.~\\ref{eq:DKJacobianNewTerm1} corresponds to $R_E$ in Eq.~ \\ref{eq:DKresidualNewTermRE} (since the rightmost $\\Phi_1$ in the residual is not implemented in populateMatrix.F90), \nso this term is already implemented by the residual.\\\\\n%%\\remark{Is there no difference implementationwise, when $\\Phi_1$ disappears in the space derivative?}\n\n\\noindent The other two terms should only be added when 'whichMatrix==0' or 'whichMatrix==1'. \nThe \\textcolor{orange}{first term} in the Jacobian is the residual multiplied by $- q / T$. However, since the exponential is \n$ \\displaystyle\n\\exp \\left(- \\frac{q \\Phi_1}{T} \\right) = \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}} \\right), \n$\nin SFINCS the term will be implemented as \n\\begin{equation}\n \\label{eq:JacobianFirstTerm1}\n \\textcolor{orange}{- \\frac{Z \\alpha}{\\hat{T}} \\left(\\hat{R}_m + \\hat{R}_E\\right)}.\n\\end{equation}\n\n\\noindent The \\textcolor{green}{second term} in the Jacobian can be written as \n\\begin{multline}\n \\label{eq:JacobianSecondTerm1}\n \\textcolor{green}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\left(\\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] + \\bm{v}_d \\cdot \\na \\psi\\right)} = \\\\ = \\textcolor{green}{\n\\frac{1}{\\Phi_1}} \\left(\n\\textcolor{cyan}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p }{\\p \\zeta} - B_\\zeta \\frac{\\p }{\\p \\theta}\\right] \\Phi_1}\n+\n\\textcolor{red}{\n \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\bm{v}_d \\cdot \\na \\psi }\n\\right),\n\\end{multline}\nwhere the two terms inside the brackets have already been implemented in $R_E$ and $R_m$ respectively. Consequently, to obtain this term we sum these two terms written in code units, and multiply by $1 / \\hat{\\Phi}_1$.\\\\\nAlthough the \\textcolor{orange}{first term} and the \\textcolor{green}{second term} of the Jacobian consist of terms available in other terms, we need to rewrite them since we cannot access code in evaluateResidual.F90 from populateMatrix.F90, and also the residual terms in populateMatrix.F90 contain a factor $\\Phi_1$ less which instead is in the state-vector.\n\n\n% \\begin{multline}\n% K \\left\\{\\psi\\right\\} \\frac{\\p f_{M}}{\\p \\psi} =\n% %%\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\bm{v}_d \\cdot \\na \\psi \\, + \\\\ + \\,\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} + \\frac{q}{T} \\Phi_1\\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] = \\\\ = \n% %%\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} \\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\bm{v}_d \\cdot \\na \\psi \\, + \\\\ + \\,\n% \\textcolor{blue}{\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\left[\\frac{1}{n} \\frac{\\p n}{\\p \\psi} + \\left(\\frac{m v^2}{2 T} - \\frac{3}{2} \\right) \\frac{1}{T} \\frac{\\p T}{\\p \\psi}\\right] \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right]} \\, + \\\\ + \\, \n% \\textcolor{red}{\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\bm{v}_d \\cdot \\na \\psi } \\, + \\\\ + \\,\n% \\textcolor{magenta}{\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\frac{\\p \\Phi_0}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] } \\, + \\\\ + \\,\n% \\textcolor{cyan}{\n% \\exp \\left(- \\frac{q \\Phi_1}{T} \\right) f_{M} \\frac{q}{T} \\Phi_1 \\frac{1}{T} \\frac{\\p T}{\\p \\psi} \\frac{1}{B^2} D \\left[B_\\theta \\frac{\\p \\Phi_1}{\\p \\zeta} - B_\\zeta \\frac{\\p \\Phi_1}{\\p \\theta}\\right] }\n% \\label{eq:DKresidualNewTerm2}\n% \\end{multline}\n\n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\npopulateMatrix.F90\n\\end{verbatim}\n\n\n\n\\subsubsection*{Additional implementation related to the kinetic equation}\nBesides implementing the above terms, we need to remove the former term in SFINCS corresponding to $\\displaystyle \\frac{Z e}{T} f_M v_\\| \\na_\\| \\Phi_1$. \\\\\n\\\\\nWe will remove the \\textbf{nonlinear} switch from the code, since all terms related to $\\Phi_1$ are now nonlinear. \nThe \\textbf{nonlinear} switch will be incorporated into the \\textbf{includePhi1} switch.\\\\\n\\\\\nWe will replace the \\textbf{includeRadialExBDrive} switch with the \\textbf{includePhi1inKineticEquation} switch which is \\textbf{true} by default. \nThis new switched can be set to \\textbf{false} if one only wants to include $\\Phi_1$ in the quasi-neutrality equation.\n\n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\nglobalVariables.F90\npopulateMatrix.F90\npreallocateMatrix.F90\nreadInput.F90\nsfincs.F90\nsolver.F90\nvalidateInput.F90\n\\end{verbatim}\n\n\n\n\n\\newpage\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\subsection*{Quasi-neutrality equation}\nIn EUTERPE $\\Phi_1$ is calculated from quasi-neutrality by expanding the exponential, assuming adiabatic electrons and neglecting the impurities.\nHowever, this is not a generic quasi-neutrality equation and in SFINCS we can easily implement the full equation. \nTo be able to compare to results from EUTERPE we will allow for both possibilities in the code, and implement an adiabatic species. \nThe option \\textbf{quasineutralityOption = 1} corresponds to the full quasi-neutrality equation and is the default, \nwhereas \\textbf{quasineutralityOption = 2} corresponds to the EUTERPE equations.\n\n\\subsubsection*{Adiabatic species}\nWe will allow for the possibility to run SFINCS with an adiabatic species. \nThe following input parameters will be introduced (with their default values in brackets) in the \\textbf{speciesParameters} namelist:\n\\begin{verbatim}\nlogical :: withAdiabatic (.false.)\nPetscScalar :: adiabaticZ (-1)\nPetscScalar :: adiabaticMHat (5.446170214d-4)\nPetscScalar :: adiabaticNHat (1.0)\nPetscScalar :: adiabaticTHat (1.0)\n\\end{verbatim}\nNote that the \nadiabatic species will only enter into the quasi-neutrality equation, \nwe neglect its collisional impact on the kinetic species (the adiabatic species will typically be electrons, and the effect of ion-electron collisions is small compared to ion-ion collisions). \n\n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\nglobalVariables.F90\npopulateMatrix.F90\nreadInput.F90\nsfincs.F90\nvalidateInput.F90\nwriteHDF5Output.F90\n\\end{verbatim}\n\n\\subsubsection*{Full quasi-neutrality equation, quasineutralityOption = 1}\n\nFirstly, we need to make sure that the input densities fulfill quasi-neutrality (also considering the adiabatic species if it is used):\n\\begin{equation}\n \\label{eq:QuasiNeutralityInputFull}\n \\sum_s Z_s \\hat{n}_s = 0.\n\\end{equation}\n%\nThe densities can be written\n\\begin{equation}\nn_s = n_{s 0} \\left(\\psi\\right) \\, \\exp \\left(- q_s \\Phi_1 / T_s \\right) + n_{s 1},\n\\label{eq:DensityParts}\n\\end{equation}\nand the quasi-neutrality equation is \n\\begin{equation}\n\\sum_s Z_s n_s = 0.\n\\label{eq:Quasineutrality}\n\\end{equation}\nFor an adiabatic species $a$\n\\[\nn_{a 1} = 0.\n\\]\nThe velocity integration in SFINCS is done in $\\left(x, \\xi\\right) = \\left(v / v_s, v_\\| / v\\right)$, and \n\\begin{equation}\n\\int d^3 v = 2 \\pi v_{s}^3 \\int_{0}^{\\infty} dx\\, x^2 \\int_{-1}^{1} d\\xi\n\\label{eq:VelocityIntegration}\n\\end{equation}\nwhere $v_{s} = \\sqrt{2 T_s / m_s}$. \nSince \n\\begin{equation}\n \\label{eq:ns1Int}\n n_{s 1} = \\int d^3 v \\, f_{s 1},\n\\end{equation}\nwe can rewrite quasi-neutrality in form of its contribution to the residual of the full linear system as \n\\begin{equation}\nR_{QN}\\left(f_{1}, \\Phi_1\\right) = \\sum_s Z_s n_{s0} \\exp\\left(- \\frac{Z_s e \\Phi_1}{T_s}\\right) + 2\\pi \\sum_{s \\setminus a} v_s^3 Z_s \\int_0^{\\infty} dx \\, x^2 \\int_{-1}^1 d\\xi \\, f_{s 1} = 0\n\\label{eq:Quasineutrality}\n\\end{equation}\n(note that the adiabatic species is excluded in the second summation).\\\\\n\n\\noindent In SFINCS we add a Lagrange multiplier $\\lambda$, divide Eq.~\\ref{eq:Quasineutrality} by $\\bar{n}$, and use $n = \\hat{n} \\bar{n}$, $\\bar{v} = \\sqrt{2 \\bar{T} / \\bar{m}}$, $v_s / \\bar{v} = \\sqrt{\\hat{T}_s / \\hat{m}_s}$, $f_s = \\bar{n} \\hat{f}_s / \\bar{v}^3$, $ \\displaystyle\n\\exp \\left(- \\frac{q_s \\Phi_1}{T_s} \\right) = \\exp \\left(- \\frac{Z_s \\alpha \\hat{\\Phi}_1}{\\hat{T}_s} \\right) \n$, to write \n\\begin{multline}\n\\hat{R}_{QN}\\left(\\hat{f}_{1}, \\hat{\\Phi}_1, \\lambda\\right) = \\textcolor{JungleGreen}{\n\\sum_s Z_s \\hat{n}_{s0} \\exp \\left(- \\frac{Z_s \\alpha \\hat{\\Phi}_1}{\\hat{T}_s} \\right)} + \\\\ + \n\\textcolor{MidnightBlue}{\n2\\pi \\sum_{s \\setminus a} Z_s \\left(\\frac{\\hat{T}_s}{\\hat{m}_s}\\right)^{3/2} \\int_0^{\\infty} dx \\, x^2 \\int_{-1}^1 d\\xi \\, \\hat{f}_{s 1} \\, + \\, \\lambda } = 0.\n\\label{eq:QuasineutralitySFINCS}\n\\end{multline}\nThe first term in Eq.~\\ref{eq:QuasineutralitySFINCS} must be implemented in evaluateResidual.F90 because it does not include a linear operation on $\\hat{\\Phi}_1$. \nThe other terms will be implemented in populateMatrix.F90.\\\\\n\n\\noindent When implementing the Jacobian terms, we note that \n\\begin{equation}\n \\label{eq:QNJacobianf1}\n\\textcolor{MidnightBlue}{\n \\frac{\\delta \\hat{R}_{QN}}{\\delta \\hat{f}_{s 1}} = 2\\pi Z_s \\left(\\frac{\\hat{T}_s}{\\hat{m}_s}\\right)^{3/2} \\int_0^{\\infty} dx \\, x^2 \\int_{-1}^1 d\\xi},\n\\end{equation}\n\\begin{equation}\n \\label{eq:QNJacobianLambda}\n\\textcolor{MidnightBlue}{\n \\frac{\\delta \\hat{R}_{QN}}{\\delta \\lambda} = 1},\n\\end{equation}\nwhich means that these two terms are the same as the corresponding terms in the residual and are thus implemented in populateMatrix.F90.\nMoreover,\n\\begin{equation}\n \\label{eq:QNJacobianPhi1}\n\\textcolor{JungleGreen}{\n \\frac{\\delta \\hat{R}_{QN}}{\\delta \\hat{\\Phi}_1} = -\n\\sum_s \\frac{Z_s^2 \\alpha}{\\hat{T}_s} \\hat{n}_{s0} \\exp \\left(- \\frac{Z_s \\alpha \\hat{\\Phi}_1}{\\hat{T}_s} \\right)}\n\\end{equation}\nwill also be implemented in populateMatrix.F90 when 'whichMatrix==0' or 'whichMatrix==1'. \n\\\\\n\n\\noindent Note that, since the distribution function in SFINCS is stored as an expansion in Legendre polynomials in $\\xi$\n\\[\n\\hat{f}_{s 1}\\left(\\theta, \\zeta, x, \\xi\\right) = \\sum_{l=0}^{N_l} \\hat{f}_{s 1}^{\\left(l\\right)}\\left(\\theta, \\zeta, x\\right) P_l \\left(\\xi\\right),\n\\]\nand using that\n\\[\n\\int_{-1}^1 P_m\\left(\\xi\\right) P_n\\left(\\xi\\right) = \\frac{2}{2n + 1} \\delta_{m,n}\n\\]\nthe integration over $\\xi$ in Eq.~\\ref{eq:QuasineutralitySFINCS} becomes\n\\begin{equation}\n \\label{eq:xiIntegration}\n \\int_{-1}^1 d\\xi \\, f_{s 1} = \\int_{-1}^1 d\\xi \\, \\left(\\sum_{l=0}^{N_l} \\hat{f}_{s 1}^{\\left(l\\right)}\\left(\\theta, \\zeta, x\\right) P_l \\left(\\xi\\right)\\right) \\cdot P_0 \\left(\\xi\\right) = 2 \\cdot \\hat{f}_{s 1}^{\\left(0\\right)}\\left(\\theta, \\zeta, x\\right).\n\\end{equation}\n\n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\nevaluateResidual.F90\npopulateMatrix.F90\nvalidateInput.F90\n\\end{verbatim}\n\n\\subsubsection*{EUTERPE quasi-neutrality equation, quasineutralityOption = 2}\nFor the EUTERPE equations, the code must be run with \nan adiabatic species and only one kinetic species (the first) is considered in the quasi-neutrality equation. \nWe need to make sure that the input densities fulfill quasi-neutrality:\n\\begin{equation}\n \\label{eq:QuasiNeutralityInputFull}\n Z_i \\hat{n}_{i} + Z_a \\hat{n}_{a} = 0\n\\end{equation}\n(here species $i$ would be the first kinetic species and $a$ the adiabatic).\\\\\nAgain we use \n\\begin{equation}\nn_s = n_{s 0} \\left(\\psi\\right) \\, \\exp \\left(- q_s \\Phi_1 / T_s \\right) + n_{s 1},\n\\label{eq:DensityParts2}\n\\end{equation}\n\\begin{equation}\n\\sum_s Z_s n_s = 0,\n\\label{eq:Quasineutrality2}\n\\end{equation}\nbut here we Taylor expand the exponential to $1^{\\mathrm{st}}$ order and write\n\\begin{multline}\n\\label{eq:QuasineutralityExpanded}\n%\\Rightarrow \\;\\;\\; \n0 \\simeq \\sum_s Z_s \\left[n_{s 0} \\left(1 - q_s \\Phi_1 / T_s \\right) + n_{s 1}\\right] \\;\\;\\; \\Leftrightarrow \\\\\n\\sum_s Z_s \\left[n_{s 0} + n_{s 1}\\right] = \\sum_s \\frac{Z_s^2 e}{T_s} \\Phi_1 n_{s 0} .\n\\end{multline}\n%Since $n_{s 0} \\left(\\psi\\right)$ is obtained by integrating the Maxwellian $f_{M s}$ over velocity space we must have \nFrom the condition of quasi-neutral input we know that\n\\[\n\\sum_s Z_s n_{s 0} = 0,\n\\]\nwhich yields \n\\begin{equation}\n\\sum_s Z_s n_{s 1} - \\Phi_1 \\sum_s \\frac{Z_s^2 e}{T_s} n_{s 0} = 0.\n\\label{eq:Quasineutrality2}\n\\end{equation}\nWith kinetic ions, adiabatic electrons ($n_{a 1} = 0$) and neglecting other ion species we obtain \n\\begin{equation}\n\\Phi_1 = \\frac{T_a}{Z_a^2 e} \\left[\\frac{Z_i^2 T_a}{Z_a^2 T_i} n_{i 0} + n_{a 0}\\right]^{-1} Z_i n_{i 1}.\n\\label{eq:Phi1}\n\\end{equation}\n%\n%\\subsection*{Implementation in SFINCS}\n%For a first benchmark, we want to implement the same equations as EUTERPE in SFINCS. \\\\\n%\\remark{This is not a very generic quasi-neutrality equation so it is possible that we might want to change it in SFINCS later.} \\\\\n%In the code we add an adiabatic species which only enters into the quasi-neutrality equation, and neglect its collisional impact on the kinetic species (the effect of ion-electron collisions is small compared to ion-ion collisions). \n%Moreover, we will only consider the first of the kinetic species in quasi-neutrality and neglect the rest. \n%This is implemented by modifying the LHS of the row corresponding to quasi-neutrality in the block-matrix structure of Matt's ISHW poster, adding the adiabatic term to the $\\Phi_1$-column and removing all kinetic species except the first. \\\\\n% \\remark{It feels a bit weird to remove species from quasi-neutrality, even if the impurity density is small. Does this mean that we should removed the check that the input densities are quasi-neutral and instead check that $n_{i 0} \\left(\\psi\\right) = n_{e 0} \\left(\\psi\\right)$ in the input?}\\\\\nThe equation we will implement in SFINCS is thus \n\\begin{equation}\nZ_i n_{i 1} - \\Phi_1 \\left[ \\frac{Z_i^2 e}{T_i} n_{i 0} + \\frac{Z_a^2 e}{T_a} n_{a 0}\\right] = 0.\n\\label{eq:QuasineutralityIonElectrons}\n\\end{equation}\nWe note that\n\\begin{multline}\nn_s = n_{s 0} \\left(\\psi\\right) \\, \\exp \\left(- q_s \\Phi_1 / T_s \\right) + n_{s 1} = \\int d^3 v f_{Ms} \\exp \\left(- q_s \\Phi_1 / T_s \\right) + \\int d^3 v f_{1 s} = \\\\ = \nd^3 v f_{0 s} + d^3 v f_{1 s}.\n\\label{eq:distributionToDensity}\n\\end{multline}\nThe velocity integration is SFINCS is done in $\\left(x, \\xi\\right) = \\left(v / v_s, v_\\| / v\\right)$, and \n\\begin{equation}\n\\int d^3 v = 2 \\pi v_{s}^3 \\int_{0}^{\\infty} dx\\, x^2 \\int_{-1}^{1} d\\xi\n\\label{eq:VelocityIntegration2}\n\\end{equation}\n(note that $v_{s}^2 = 2 T_s/m_s$ differs from Jose's notation $v_{\\mathrm{th}}^2 = T/m$). \nUsing SFINCS normalizations $n_s = \\bar{n} \\hat{n}_s$, $T_s = \\bar{T} \\hat{T}_s$, $v_s / \\bar{v} = \\sqrt{\\hat{T}_s / \\hat{m}_s}$, \n$f_s = \\bar{n} \\hat{f}_s / \\bar{v}^3$, \nwe find \n\\begin{equation}\n\\hat{n}_s = 2 \\pi \\left(\\hat{T}_s / \\hat{m}_s\\right)^{3/2} \\int_{0}^{\\infty} dx\\, x^2 \\int_{-1}^{1} d\\xi \\hat{f}_s.\n\\label{eq:VelocityIntegrationSFINCS}\n\\end{equation}\nAlso using $\\Phi_1 = \\bar{\\Phi} \\hat{\\Phi}_1$ and $\\alpha = e \\bar{\\Phi} / \\bar{T}$ we can write Eq.~\\ref{eq:QuasineutralityIonElectrons} \n\\begin{equation}\nZ_i \\hat{n}_{i 1} - \\alpha \\hat{\\Phi}_1 \\left[ \\frac{Z_i^2}{\\hat{T}_i} \\hat{n}_{i 0} + \\frac{Z_a^2}{\\hat{T}_a} \\hat{n}_{a 0}\\right] = 0\n\\label{eq:QuasineutralityIonElectronsSFINCS}\n\\end{equation}\nand finally obtain the residual \n\\begin{equation}\n\\hat{R}_{QN}^{\\mathrm{EUTERPE}}\\left(\\hat{f}_{i 1}, \\hat{\\Phi}_1, \\lambda\\right) =\n\\left[2 \\pi Z_i \\left(\\hat{T}_i / \\hat{m}_i\\right)^{3/2} \\int_{0}^{\\infty} dx\\, x^2 \\int_{-1}^{1} d\\xi \\hat{f}_{i 1}\\right]\n- \\alpha \\hat{\\Phi}_1 \\left[ \\frac{Z_i^2}{\\hat{T}_i} \\hat{n}_{i 0} + \\frac{Z_a^2}{\\hat{T}_a} \\hat{n}_{a 0}\\right] \\, + \\, \\lambda = 0.\n\\label{eq:QuasineutralityIonElectronsSFINCS2}\n\\end{equation}\n%This is the equation we will implement in the code, but adding a $\\lambda$ to make the system square.\\\\ \n% \\remark{Is the $2\\pi$ factor correct in Eq.~\\ref{eq:QuasineutralityIonElectronsSFINCS2}? It is not in the former implementation of quasi-neutrality, but in that situation it could be divided away.}\nSince the residual has a linear operator dependence on all unknown variables, both the Jacobian and the residual can be implemented in populateMatrix.F90 with the same equations.\n\n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\npopulateMatrix.F90\nvalidateInput.F90\n\\end{verbatim}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\n\\subsection*{Additional changes}\nDecide which output fluxes should be possible to obtain, i.e. which combinations of \n\\begin{equation}\n \\label{eq:OutputFluxes}\n \\langle \\int d^3 v \\, f \\bm{v} \\cdot \\na X \\rangle,\n\\end{equation}\nshould be possible? \nHere $f$ can be $f_0$, $f_1$ or $f_0 + f_1$. \n$\\bm{v}$ can be $\\bm{v}_E$, $\\bm{v}_m$ or $\\bm{v}_E + \\bm{v}_m$. \n$X$ can be any of the radial coordinates, $r$, $r_N$, $\\psi$ or $\\psi_N$. \\\\\n\n\\noindent Allow for the output fluxes in SI units.\\\\\n\n\\noindent In all output flux quantities the substitution\n\\[\n\\textcolor{red}{\nf_M \\rightarrow f_0 = f_M \\exp \\left(- q \\Phi_1 / T \\right)}\n\\]\nmust be made. \n\\paragraph*{\\textbf{Files to change:}}\n\\begin{verbatim}\ndiagnostics.F90\n\\end{verbatim}\n\n\n\\newpage\n%\\appendix\n\\titleformat{\\section}{\\large\\bfseries}{\\appendixname~\\thesection .}{0.5em}{}\n\n\\begin{appendices}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Check of Matt's former implementation of $\\displaystyle \\frac{Z e}{T} f_M v_\\| \\na_\\| \\Phi_1$}\n\\textbf{This section is only to compare to what has already been implemented in SFINCS, to see that we understand the normalizations.}\\\\\n\\\\\nLooking at Matt's ISHW poster, since $\\Phi_1$ is an unknown this term is in the LHS of the square block matrix system. \nThe term is accessed by ``rowIndex = BLOCK\\_F'' and ``colIndex = BLOCK\\_QN''. \nWe use\n\\[\n\\na_\\| \\Phi_1 = \\bm{b} \\cdot \\na \\Phi_1 = \\frac{1}{B} \\left[ B^{\\theta} \\frac{\\p \\Phi_1}{\\p \\theta} + B^{\\zeta} \\frac{\\p \\Phi_1}{\\p \\zeta} \\right] = \n\\frac{\\bar{\\Phi}}{\\hat{B} \\bar{R}} \\left[ \\hat{B}^{\\theta} \\frac{\\p \\hat{\\Phi}_1}{\\p \\theta} + \\hat{B}^{\\zeta} \\frac{\\p \\hat{\\Phi}_1}{\\p \\zeta} \\right], \n\\]\n\\[\nf_M = n_0\\left(\\psi\\right) \\frac{m^{3/2}}{\\left(2 \\pi T\\right)^{3/2}} \\exp \\left[- \\frac{v^2}{v_{s}^2}\\right] = \n\\hat{n} \\bar{n} \\frac{\\hat{m}^{3/2}}{\\left(2 \\pi \\hat{T}\\right)^{3/2}} \\left(\\frac{\\bar{m}}{\\bar{T}}\\right)^{3/2} \\exp \\left[- x^2\\right],\n\\]\n$v_\\| = v_s x \\xi = v_s x P_1 = x P_1 \\sqrt{2 \\hat{T} / \\hat{m}} \\sqrt{\\bar{T} / \\bar{m}} $ and $x = v / v_s$. \nWith $\\alpha = e \\bar{\\Phi} / \\bar{T}$ we obtain \n\\begin{multline}\n\\frac{Z e}{T} f_M v_\\| \\na_\\| \\Phi_1 = \\frac{Z e}{\\hat{T} \\bar{T}} \\, \\hat{n} \\bar{n} \\frac{\\hat{m}^{3/2}}{\\left(2 \\pi \\hat{T}\\right)^{3/2}} \\left(\\frac{\\bar{m}}{\\bar{T}}\\right)^{3/2} \\exp \\left[- x^2\\right] \\, x P_1 \\sqrt{2 \\hat{T} / \\hat{m}} \\sqrt{\\bar{T} / \\bar{m}} \\, \\frac{\\bar{\\Phi}}{\\hat{B} \\bar{R}} \\left[ \\hat{B}^{\\theta} \\frac{\\p \\hat{\\Phi}_1}{\\p \\theta} + \\hat{B}^{\\zeta} \\frac{\\p \\hat{\\Phi}_1}{\\p \\zeta} \\right] = \\\\ =\n \\frac{Z \\alpha}{2\\pi^{3/2} } x P_1 \\exp \\left[- x^2\\right] \\, \\frac{\\hat{n} \\hat{m}}{\\hat{B} \\hat{T}^2 } \\, \\frac{\\bar{n} \\bar{m}}{\\bar{R} \\bar{T}} \\left[ \\hat{B}^{\\theta} \\frac{\\p }{\\p \\theta} + \\hat{B}^{\\zeta} \\frac{\\p }{\\p \\zeta} \\right] \\hat{\\Phi}_1.\n\\label{eq:TermNaParallelPhi1}\n\\end{multline}\n\n%\\noindent\\textcolor{red}{\n%Questions:\\\\\n%\\begin{itemize}\n%\t\\item In the code you name a quantity ``dfMdx''. I don't see the reason for doing this since the expression doesn't contain the derivative of the Maxwellian. However, because of the $x$ it will turn out to be the same.\n%\t\\item Looking in the code and comparing to what I get there seems to be a factor $1/2$ discrepancy. Can you see where it is? It seems to be because a factor $2 x$ pops out when taking the derivative of the Maxwellian.\n%\t\\item In the code I can simply neglect the bar-quantities, right?\n%\t\\item Since this is a linear term in the unknowns, I suppose it gives the same contribution to the residual matrix as to the Jacobian matrix. Is it correct understood that when I calculate the residual matrix I will substitute the term as it is, whereas when I calculate the Jacobian matrix I will have to take the derivative with respect to $\\Phi_1$ (in this case when there is no $f_1$ in the term)? E.g. would a $\\Phi_1^2$-term be $\\Phi_1^2$ in the residual but $2 \\Phi_1$ in the Jacobian?\n%\\end{itemize}\n%}\n\n\\noindent In SFINCS the kinetic equation is made dimensionless by multiplying with\n\\[\n\\frac{\\bar{v}^3 \\bar{R}}{\\bar{n} \\bar{v}} = \\frac{2 \\bar{T} \\bar{R}}{\\bar{m} \\bar{n}},\n\\]\nwhich implies that the RHS of Eq.~\\ref{eq:TermNaParallelPhi1} becomes\n\\begin{equation}\n \\frac{Z \\alpha}{\\pi^{3/2} } x P_1 \\exp \\left[- x^2\\right] \\, \\frac{\\hat{n} \\hat{m}}{\\hat{B} \\hat{T}^2 } \\, \\left[ \\hat{B}^{\\theta} \\frac{\\p }{\\p \\theta} + \\hat{B}^{\\zeta} \\frac{\\p }{\\p \\zeta} \\right] \\hat{\\Phi}_1\n\\label{eq:TermNaParallelPhi1SFINCS}\n\\end{equation}\nin the implementation.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\newpage\n% \\section{Temporary page}\n% \\subsection*{Implementation of $\\displaystyle f_0 \\frac{q}{T} \\na \\Phi_0 \\cdot \\bm{v}_{E1}$}\n% \\begin{multline}\n% \\left(f_0 \\frac{q}{T} \\na \\Phi_0 \\cdot \\bm{v}_{E1}\\right)_{\\mathrm{SFINCS}} = \\\\ =\n% \\frac{Z \\alpha^2 \\Delta}{2 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{5/2} \\hat{B}^2 \\hat{\\psi}_a} \\frac{\\p \\hat{\\Phi}_0}{\\p \\psi_N} \n% \\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p}{\\p \\theta}\\right] \\hat{\\Phi}_1\n% \\label{eq:RHSphi1term1}\n% \\end{multline}\n\n% \\subsection*{Implementation of $\\displaystyle f_0 \\frac{q}{T} \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_d$}\n% \\begin{multline}\n% \\left(f_0 \\frac{q}{T} \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_d\\right)_{\\mathrm{SFINCS}} = \\\\ =\n% \\frac{\\alpha \\Delta}{3 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{5/2} \\hat{B}^3 \\hat{\\psi}_a} \\hat{\\Phi}_1 \\frac{\\p \\hat{T}}{\\p \\psi_N}\n% x^2 \\left(P_2\\left(\\xi\\right) + 2\\right)\n% \\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p \\hat{B}}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p \\hat{B}}{\\p \\theta}\\right] \n% \\label{eq:RHSphi1term2}\n% \\end{multline}\n\n% \\subsection*{Implementation of $\\displaystyle f_0 \\frac{q}{T} \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_{E1}$}\n% \\begin{multline}\n% \\left(f_0 \\frac{q}{T} \\Phi_1 \\frac{\\na T}{T} \\cdot \\bm{v}_{E1}\\right)_{\\mathrm{SFINCS}} = \\\\ =\n% \\frac{Z \\alpha^2 \\Delta}{2 \\pi^{3/2}} \\frac{\\hat{n} \\hat{m}^{3/2} \\hat{D}}{\\hat{T}^{7/2} \\hat{B}^2 \\hat{\\psi}_a} \\frac{\\p \\hat{T}}{\\p \\psi_N} \\hat{\\Phi}_1\n% \\exp\\left(-x^2\\right) \\exp \\left(- \\frac{Z \\alpha \\hat{\\Phi}_1}{\\hat{T}}\\right) \\left[\\hat{B}_{\\theta} \\frac{\\p}{\\p \\zeta} - \\hat{B}_{\\zeta} \\frac{\\p}{\\p \\theta}\\right] \\hat{\\Phi}_1\n% \\label{eq:RHSphi1term3}\n% \\end{multline}\n \\end{appendices}\n \\newpage\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\begin{thebibliography}{99}\n\n\\bibitem{regana} J.~M.~Garc\\'{\\i}a-Rega\\~{n}a, R.~Kleiber, C.~D.~Beidler, Y.~Turkin, H.~Maa{\\ss}berg \nand P.~Helander, \n\\href{http://dx.doi.org/10.1088/0741-3335/55/7/074008}{\\em Plasma Phys.~Control.~Fusion} {\\bf 55} (2013) 074008.\n\n\\bibitem{reganaArxiv} J.~M.~Garc\\'{\\i}a-Rega\\~{n}a, C.~D.~Beidler, Y.~Turkin, R.~Kleiber, P.~Helander, H.~Maa{\\ss}berg, J. A. Alonso and J. L. Velasco, \n\\href{http://arxiv.org/abs/1501.03967}{\\em arXiv:1501.03967} (2015).\n\n\\bibitem{landremanSFINCS} Landreman~M, Smith~H~M, Moll\\'en~A and Helander~P 2014\n \\href{http://dx.doi.org/10.1063/1.4870077}{\\em Phys. Plasmas} {\\bf 21} 042503\n \n\\bibitem{SFINCStechnicalDoc} M.~Landreman, {\\em Technical Documentation for version 3 of SFINCS} (2014).\n\n%\\bibitem{simakov} A.~N.~Simakov, P.~Helander,\n% {\\em Phys. Plasmas} {\\bf 16}, 042503 (2009).\n \n%\\bibitem{nonAxis} P.~Helander, Theory of plasma confinement in non-axisymmetric magnetic fields (2013).\n\n%\\bibitem{MH} L.~Råde and B.~Westergren, Mathematics Handbook for Science and Engineering, $5^{\\mathrm{th}}$ edition, 2004. %\\vspace{-5mm}\n\n%\\bibitem{Abra} M.~Abramowitz and I.~A.~Stegun, Handbook of Mathematical Functions, $10^{\\mathrm{th}}$ printing, 1972.\n\n\\end{thebibliography}\n\n\\end{document}", "meta": {"hexsha": "57f78f2b379bad7933301160970eacdd40d215ae", "size": 50472, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/Phi1_implementation_2016-01.tex", "max_stars_repo_name": "amollen/sfincs", "max_stars_repo_head_hexsha": "a529954fd36330e1b5c816612943f39829f3542f", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 15, "max_stars_repo_stars_event_min_datetime": "2017-10-13T15:15:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T17:56:20.000Z", "max_issues_repo_path": "doc/Phi1_implementation_2016-01.tex", "max_issues_repo_name": "amollen/sfincs", "max_issues_repo_head_hexsha": "a529954fd36330e1b5c816612943f39829f3542f", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2018-01-02T09:04:48.000Z", "max_issues_repo_issues_event_max_datetime": "2021-01-28T09:53:21.000Z", "max_forks_repo_path": "doc/Phi1_implementation_2016-01.tex", "max_forks_repo_name": "amollen/sfincs", "max_forks_repo_head_hexsha": "a529954fd36330e1b5c816612943f39829f3542f", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2015-03-19T14:30:10.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-03T14:37:14.000Z", "avg_line_length": 55.4028540066, "max_line_length": 494, "alphanum_fraction": 0.6347083531, "num_tokens": 20330, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34904287667217504}} {"text": "A common operational approach that water utilities use to address water quality concerns is flushing, which \r\nis the purging of water from the distribution network via a fire hydrant or blow-off port. \r\nMany utilities flush water mains following maintenance work \r\nor in response to customer complaints. Flushing can remove the sources \r\nof poor water quality (e.g., pipe corrosion, bio-film microorganisms), as well \r\nas loose or suspended material that has accumulated in low-flow portions or \r\ndead-ends of the distribution system. It is a response option that can be undertaken \r\nrelatively quickly after a contamination incident, and it can be made more efficient through \r\nthe careful selection of where to implement flushing activities. This chapter describes \r\nthe \\code{flushing} subcommand in WST that assists in the identification of effective \r\nhydrant locations to flush in order to remove contaminated water and the valves to close \r\nin order to direct the contaminated water towards the hydrants. \r\n\r\nA flowchart representation of the \\code{flushing} subcommand is shown in Figure \\ref{fig:flushing-flowchart}. \r\nThe \\code{flushing} subcommand employs an iterative process that combines contaminant transport, impact assessment \r\nand optimization. The optimization process identifies a set of flushing activities \r\nthat are simulated in the contaminant transport process and evaluated \r\nbased upon the impact assessment process. Since the \\code{flushing} subcommand relies \r\non the \\code{tevasim} and \\code{sim2Impact} subcommands, their required input \r\nis also required for the \\code{flushing} subcommand. In addition, the sensor network \r\ndesign used to detect the contamination incident(s) and the flushing characteristics \r\nare required inputs. The utility network model is defined by a EPANET 2.00.12 INP file, \r\nwhile the rest of the input can be specified in the \\code{flushing} WST configuration file.\r\n\r\n\\begin{figure}[h]\r\n \\centering\r\n \\includegraphics[scale=0.75]{graphics/flushing_flowchart.pdf}\r\n \\caption{Flushing response simulation flowchart.}\r\n \\label{fig:flushing-flowchart}\r\n\\end{figure}\r\n\r\n\\section{Flushing Formulation}\\label{flushing_formulations}\r\nThe flushing problem formulation can be summarized \r\nas selecting a set of hydrant locations to flush and valves to close \r\nthat minimizes the average impact of all contamination incidents \r\ngiven a set of potential hydrant locations to flush and valves to close. \r\nThe mathematical formulation can be written as follows:\r\n\\begin{align}\r\n\\textrm{ minimize } \\qquad &\\dfrac{1}{A} \\sum_{a \\in {A}} d_{a,\\max} \\label{eqn:flushing} \\\\\r\n\\textrm{subject to } \\qquad &\\sum_{h\\in H}y_{h} \\leq H_{\\max} \\label{eq:flushing_2} \\\\\r\n&\\sum_{v\\in V}y_{v} \\leq V_{\\max} \\label{eq:flushing_3} \\\\ \r\n&y_{h} \\in \\{0,1\\} &&\\forall h\\in H \\label{eq:flushing_4} \\\\\r\n&y_{v} \\in \\{0,1\\} &&\\forall v\\in V \\label{eq:flushing_5}\r\n\\end{align}\r\nwhere $A$ represents a set of contamination incidents, \r\n$d_{a,\\max}$ defines the maximum impact of the contamination incident $a$, \r\n$H$ represents the set of potential hydrant locations,\r\n$y_h$ is a binary variable which is 1 if node $h$ is selected as a flushing location, \r\n$H_{\\max}$ is the maximum number of hydrant locations,\r\n$V$ represents the set of potential valve locations,\r\n$y_v$ is a binary variable which is 1 if node $v$ is selected as a valving location and \r\n$V_{\\max}$ is the maximum number of valve locations. The maximum impact of a contamination\r\nincident, $d_{a,\\max}$, is the total impact across the entire network at the end of the simulation \r\nassuming that the contaminant was not detected by a sensor, and no interventions to reduce \r\nimpacts were implemented. This value is found in the -1 entry of the impact file. \r\n\r\nFor this problem, hydrants are assumed to be located at any user-defined nodes in the network. \r\nIn addition, valves are assumed to be located on any user-defined pipes in the network. \r\n\r\n\\section{Flushing Solvers}\\label{flushing_solvers}\r\nThe flushing problem is solved through an iterative optimization process which \r\nselects different sets of hydrant and valve locations and evaluates their effectiveness\r\nin minimizing the impact of a set of contamination incidents. Two optimization \r\nmethods, an evolutionary algorithm and a network solver, are available in WST to solve this problem. \r\nEach solver is explained in more detail in the following subsections. \r\n\r\n\\subsection{Evolutionary Algorithm}\\label{coliny_ea}\r\nThe evolutionary algorithm (EA) included with DAKOTA, Coliny EA, is used in the \r\noptimization routine for the \\code{flushing} subcommand. \r\nAdditional information on DAKOTA/Coliny solvers can be found at \r\n\\url{http://dakota.sandia.gov/docs/dakota/5.2/html-ref/index.html}\r\nand in the DAKOTA user manual \\citep{DakotaUserManual}. \r\nThe random number generator used in Coliny EA is platform dependent. \r\nThis can result in slight variations in the solution.\r\n\r\nTo design an EA, the parameter space for the optimization problem \r\nis first encoded into a string of numbers. This encoded \r\nrepresentation of the problem is called a genetic string, where each \r\nelement of the genetic string represents one parameter. \r\nWhen the EA is used with the \\code{flushing} subcommand, the parameter space is defined by the number \r\nof flushing and valve closure locations. Each location is assigned a sequential \r\ninteger that represents a feasible location within the EPANET network. The final EA solution \r\nis reported based on the EPANET node/pipe IDs.\r\n\r\nThe EA has several solver options that define how the EA evolves. These options can be set in the \r\n\\code{[solver][options]} sections of the \\code{flushing} WST configuration file and are specific to the Coliny EA solver. \r\nThe EA evolves an initial genetic strings of size \\code{[population\\_size]} \r\nthat is set based on \\code{[initialization\\_type]} using the following steps:\r\n\\begin{enumerate}\r\n\\item Evaluation: Evaluate the solution for each genetic string. This involves function \r\ncalls to the \\code{tevasim} and \\code{sim2Impact} subcommands for each string to define the impact value. \r\n\\item Breeding: Select two members of the population based on fitness. The \r\nprobability of selection is based on \\code{[fitness\\_type]}. \r\n\\item Crossover: Crossover two members based on \\code{[crossover\\_type]} and \\code{[crossover\\_rate]}.\r\n\\item Mutation: Mutate the two members based on \\code{[mutation\\_type]} and \\code{[mutation\\_rate]}.\r\n\\item Steps 2-4 are repeated until the entire population has been changed.\r\n\\item Replacement: After a new population is created, the old population is \r\nreplaced by the current population while keeping the highest ranked string \r\n(elitist = 1 replacement option).\r\n\\end{enumerate}\r\nSteps 1-6 are repeated until \\code{[max\\_iterations]} or \\code{[max\\_function\\_evaluations]} criteria is met. \r\n\r\n\\subsection{Network Solver}\\label{coliny_statemachine}\r\n\r\nThe network solver used in WST is a network-constrained, derivative-free local search optimization algorithm. \r\nIt is a discrete analog-to-pattern search. The allowable moves are to adjacent nodes (or pipes), rather than moves in the \r\ncontinuous space. This approach provides local refinement of candidate solutions. The valid moves include\r\nremoving a node (or pipe) location and replacing it with one anywhere in the network. Two forms of the network solver can be \r\nused: with and without initial starting points. The initial starting points are node (or pipe) locations in the network\r\nin which the algorithm should begin its local search. If these points are not supplied to the algorithm, \r\nthen it reduces to a greedy placement algorithm. Convergence is met when no remaining moves improve the solution.\r\n\r\n\\subsection{Flushing Optimization for Large Problems}\r\n\r\nThe iterative optimization process used for flushing requires numerous \r\nsimulations of the network hydraulics and water quality. Computational run time depends on several factors, including \r\nthe size of the network model, \r\nthe number of possible contamination incidents, \r\nthe number of feasible flushing locations \r\nand the solver options. For the network solver,the computational run time also depends on the network model\r\ngeometry surrounding the local search region. For large flushing optimization problems, several techniques can be used to \r\ndecrease the computational run time. These options include running multiple instances of the underlying \r\nsimulation in parallel, setting a stop time criteria and skeletonizing the network model. \r\n\r\n\\subsubsection{Parallelization}\r\n\r\nThe Dakota coliny\\_ea and StateMachineLS solvers both include an option to set \r\nthe number of threads used to perform the simulations. \r\nThis option is specified in \\code{[solver][threads]}, as shown below. It should be \r\nset to the number of threads that are available for the simulation. The default value is 1. \r\nIf an integer greater than 1 is specified, Dakota will run that many threads.\r\nThe expected efficiency of parallelization is nearly linear with the number of \r\nthreads (up to the number of processor cores in the computer).\r\n\r\n\\begin{unknownListing}\r\nsolver:\r\n type: coliny_ea \r\n threads: 2\r\n\\end{unknownListing}\r\n\r\n\\subsubsection{Stop time criteria}\r\n\r\nIf a solution needs to be identified within a certain amount of time, a simulation stop time \r\ncriteria can be defined. This option causes the solver to terminate after a \r\nspecified time, even if the underlying algorithm has not converged to an optimal solution. \r\nThe stop time criteria is included with both the Dakota coliny\\_ea and \r\nStateMachineLS solvers, and it is specified in \\code{[solver][options][misc\\_options]}, as \r\nshown below. The max\\_time has to be given in seconds. As the optimization \r\nalgorithms only check the elapsed time once per major iteration, the actual \r\nsolver run time will be slightly longer than the specified maximum run time. \r\nWhen the optimization process is cut off prematurely, the best solution \r\nobtained so far is reported to the user.\r\n\r\n\\begin{unknownListing}\r\nsolver:\r\n type: StateMachineLS \r\n options: \r\n misc_options: \"'max_time=10'\"\r\n\\end{unknownListing}\r\n\r\n\\subsubsection{Skeletonization}\r\n\r\nTo reduce the size of the network model, possible contamination incidents and the number of \r\nfeasible flushing locations, the user can skeletonize the problem and evaluate \r\nthe results on the full network model. WST includes a utility script, spotSkeleton, \r\nthat can be used to skeletonize network models (See Executable Files Section ~\\ref{skelExecutable}). \r\nNetwork models are skeletonized based on a pipe diameter threshold. The executable, spotSkeleton, \r\ncreates a new EPANET input (inp) file and a related map file. \r\nThe map file associates the nodes in the skeletonized network model (upscaled nodes) \r\nto the nodes in the original network model (downscaled nodes). When \r\nworking with a skeletonized network model, other aspects of the flushing problem also \r\nneed to upscaled based on the skeletonization map. These include: the injection location(s) \r\nand strength of the contamination incident(s), the population at each node, the sensor \r\nplacement, the feasible flushing locations and the initial points for the optimization \r\nsolver. For example, if Nodes 1, 2, 3 and 4 in the original network model are \r\nrepresented by Node 2 in the skeletonized network model, then a sensor placed at \r\nNode 4 in the original network model should be placed at Node 2 in the skeletonized \r\nnetwork model. \r\n\r\nAfter flushing optimization is run on the skeletonized network model, the \r\nsolution can then be evaluated or refined using the original network model. \r\nTo evaluate the current solution, the locations on the skeletonized network \r\nshould be listed as the only feasible flushing locations \r\nin the original network model and the EVALUATE solver option should be used (See Example ~\\ref{flushing_ex3}). \r\nTo refine the current solution, downscale the locations on the skeletonized network\r\nand use those original network nodes as the only feasible flushing locations \r\nin a second optimization. In this refinment, the solver type and \r\nsolver options can be different for the first and second optimization.\r\n\r\n\\section{\\code{flushing} Subcommand}\r\n\r\nThe \\code{flushing} subcommand is executed using the following command line:\r\n\\begin{unknownListing}\r\nwst flushing \r\n\\end{unknownListing}\r\nwhere \\code{configfile} is a WST configuration file in the YAML format. \r\n\r\nThe \\code{---help} option prints information about this subcommand, such as usage,\r\narguments and a brief description:\r\n\\begin{unknownListing}\r\nwst flushing --help\r\n\\end{unknownListing}\r\n\r\n\\subsection{Configuration File}\r\n\r\nThe \\code{flushing} subcommand generates a template configuration file using the following command line:\r\n\r\n\\begin{unknownListing}\r\nwst flushing --template \r\n\\end{unknownListing}\r\n\r\nThe \\code{flushing} template configuration file is shown in Figure \\ref{fig:flushing_template}. \r\nBrief descriptions of the options are included in the template after the \\# sign. \r\n\r\n\\begin{figure}[p!]\r\n \\unknownInputListing{examples/flushing_config.yml}{}{1}{50}\r\n \\caption{The \\code{flushing} configuration template file.}\r\n \\label{fig:flushing_template}\r\n\\end{figure}\r\n\r\n\\subsection{Configuration Options}\r\n\r\nFull descriptions of the WST configuration options used by the \\code{flushing} subcommand are listed below.\r\n\\input{examples/flushing_config}\r\n\r\nIn addition to these standard WST configuration options, the solver block can define \r\nan evaluation option. To evaluate the flushing response without solving the \r\noptimization problem, the solver type can be set as EVALUATE. This option allows a \r\nset of flushing locations to be evaluated against a different contamination \r\nscenario than the one for which it was designed. \r\n\r\nThe solver block can also define specific options for the optimization solver. \r\nThe solver options should be modified according to the specific optimization problem. \r\nIf the options are not set in the solver block, then the default values for these options are used. \r\nThe two solvers available in the \\code{flushing} subcommand each have their own options. \r\nThe EA solver has numerous options which can be defined. Additional information on the options available\r\nfor the EA solver can found in the DAKOTA user manual \\citep{DakotaUserManual}. \r\nAn example of the EA solver options are listed below.\r\n\r\n\\begin{unknownListing}\r\nsolver:\r\n type: coliny_ea\r\n options: \r\n crossover_rate: 0.8\r\n crossover_type: uniform\r\n fitness_type: linear_rank\r\n initialization_type: unique_random\r\n max_function_evaluations: 30000\r\n max_iterations: 1000\r\n mutation_rate: 1\r\n mutation_type: offset_uniform\r\n population_size: 50\r\n seed: 11011011\r\n\\end{unknownListing}\r\n\r\nThe network solver has two options that can be set in the solver block of the configuration file. \r\n\\begin{unknownListing}\r\nsolver: \r\n type: StateMachineLS\r\n options:\r\n verbosity: 2\r\n max_fcn_evaluations: 0\r\n\\end{unknownListing}\r\n\r\n\\subsection{Subcommand Output}\r\nThe \\code{flushing} subcommand creates a YAML file called flushing\\_output.yml \r\nthat contains an optimized set of node locations (EPANET node IDs) to flush, \r\nan optimized set of pipe locations (EPANET pipe IDs) to close,\r\nthe final impact metric, the run date and CPU time. \r\nThe log file called flushing\\_output.log contains basic debugging information.\r\nA visualization YAML configuration file named flushing\\_output\\_vis.yml is also created.\r\nThe \\code{visualization} subcommand is automatically run using this YAML file.\r\n\r\n\\section{Flushing Response Examples}\\label{flushing_example}\r\nTo demonstrate the two different solvers available in the \\code{flushing} subcommand, \r\ntwo examples are presented. Both examples have the same characteristics in terms \r\nof the contamination scenario and flushing parameters. EPANET Example Network 3 (Net3.inp)\r\nis the network used and the contamination scenario is an hour long injection at \r\nnode 101 beginning at hour 3 in the simulation. A maximum of three hydrants can be flushed \r\nfor a duration of eight hours at a rate 800 gal/min. The option to close pipes/valves was \r\nnot included in these analyses. The impact metric being minimized is population exposed (PE). \r\nIn addition, the third and forth examples are provided to demonstrate the evaluate and stop time\r\ncriteria options, respectively.\r\n\r\n\\subsection{Example 1}\r\n\r\nThe first example uses the EA solver (coliny\\_ea) with the parallization option enabled \r\nand the configuration file, flushing\\_ex1.yml, is shown in Figure \\ref{fig:flushing_ex1}. This example\r\nhas the \\code{[solver][threads]} option set to 2 threads. Please note: \r\nif the computer used to execute the example only has one thread, change the \\code{[solver][threads]} \r\noption to 1 instead of 2.\r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{../../examples/flushing_ex1.yml}{}{1}{55}\r\n \\caption{The \\code{flushing} configuration file for example 1.}\r\n \\label{fig:flushing_ex1}\r\n\\end{figure}\r\n\r\nThe example can be executed using the following command line:\r\n\r\n\\begin{unknownListing}\r\nwst flushing flushing_ex1.yml\r\n\\end{unknownListing}\r\n\r\nThe YAML output file, Net3flushing\\_output.yml, for example 1 \r\nis shown in Figure \\ref{fig:flushing_ex1_yml}. The EA selected to \r\nflush nodes 113, 191 and 197 for a PE impact value of 5292. The CPU time \r\nwas approximately 6 minutes using 2 threads. \r\nSince the random number generator used in the EA solver is platform dependent, \r\nthe solution can be slightly different if the example is executed on a computer with Windows.\r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{examples/flushing/flushing_ex1_output.yml}{}{1}{11}\r\n \\caption{The \\code{flushing} YAML output file for example 1.}\r\n \\label{fig:flushing_ex1_yml}\r\n\\end{figure}\r\n\r\n% \\FloatBarrier \r\n\\subsection{Example 2}\r\n\r\nThe second example uses the network solver (StateMachineLS) \r\nwithout initial points and the configuration file, flushing\\_ex2.yml, \r\nis shown in Figure \\ref{fig:flushing_ex2}. \r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{../../examples/flushing_ex2.yml}{}{1}{45}\r\n \\caption{The \\code{flushing} configuration file for example 2.}\r\n \\label{fig:flushing_ex2}\r\n\\end{figure}\r\n\r\nThe example can be executed using the following command line:\r\n\r\n\\begin{unknownListing}\r\nwst flushing flushing_ex2.yml\r\n\\end{unknownListing}\r\n\r\nThe YAML output file, Net3flushing\\_output.yml, for example 2 is shown \r\nin Figure \\ref{fig:flushing_ex2_yml}. The network solver selected to \r\nflush nodes 101, 103 and 109 for a PE impact metric of 4919. The CPU time \r\nwas approximately 2 minutes.\r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{examples/flushing/flushing_ex2_output.yml}{}{1}{11}\r\n \\caption{The \\code{flushing} YAML output file for example 2.}\r\n \\label{fig:flushing_ex2_yml}\r\n\\end{figure}\r\n\r\nExamining the output files from the two examples shows that the optimization\r\nsolvers identified different solutions. As EAs are not guaranteed to find the \r\noptimal solution, these results are not unexpected. In addition, the CPU times \r\nto obtain the solutions are different. The EA solution took about 6 minutes \r\nto obtained using 2 threads, while the network solver solution was achieved in approximately 2 minutes.\r\n\r\n\\subsection{Example 3}\\label{flushing_ex3}\r\n\r\nThe third example uses the evaluate option and the configuration \r\nfile, flushing\\_ex3.yml, is shown in Figure \\ref{fig:flushing_ex3}. \r\nIn this example, the same contamination scenario is used but only two \r\n(2) flushing locations are evaluated in terms of reducing the PE impact \r\nmetric. The flushing locations being evaluated are nodes 101 and 127. \r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{../../examples/flushing_ex3.yml}{}{1}{44}\r\n \\caption{The \\code{flushing} configuration file for example 3.}\r\n \\label{fig:flushing_ex3}\r\n\\end{figure}\r\n\r\nThe example can be executed using the following command line:\r\n\r\n\\begin{unknownListing}\r\nwst flushing flushing_ex3.yml\r\n\\end{unknownListing}\r\n\r\nThe YAML output file, Net3flushing\\_output.yml, for example 3 is shown \r\nin Figure \\ref{fig:flushing_ex3_yml}. These two flushing locations \r\nresulted in a PE impact metric of 10,759. Compared to the results from \r\nexample 2, the PE impact metric is much larger since only two flushing \r\nlocations are used instead of three. The evaluate option can be used to compare \r\nthe impact metrics obtained from various flushing location combinations. \r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{examples/flushing/flushing_ex3_output.yml}{}{1}{11}\r\n \\caption{The \\code{flushing} YAML output file for example 3.}\r\n \\label{fig:flushing_ex3_yml}\r\n\\end{figure}\r\n\r\n\\subsection{Example 4}\r\n\r\nThe fourth example demonstrates the stop time option and uses almost the same configuration file as example 2. \r\nThe configuration file, flushing\\_ex4.yml, is shown in Figure \\ref{fig:flushing_ex4}, in which \r\nthe stop criteria option is enabled using the \\code{[solver][options][misc\\_options]} option set to \"'max\\_time=30'\".\r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{../../examples/flushing_ex4.yml}{}{1}{46}\r\n \\caption{The \\code{flushing} configuration file for example 4.}\r\n \\label{fig:flushing_ex4}\r\n\\end{figure}\r\n\r\nThe example can be executed using the following command line:\r\n\r\n\\begin{unknownListing}\r\nwst flushing flushing_ex4.yml\r\n\\end{unknownListing}\r\n\r\nThe YAML output file, Net3flushing\\_output.yml, for example 4 \r\nis shown in Figure \\ref{fig:flushing_ex4_yml}. The network solver selected to \r\nflush node 105 for a PE impact value of 9676. The CPU time was 45 seconds, \r\nwhich is greater than the stop time criteria since it is only checked periodically. \r\nThe flushing solution was different than the solution obtained from example 2, \r\nsince the stop time option was not used and it executed the optimization process to completion. \r\n\r\n\\begin{figure}[h]\r\n \\unknownInputListing{examples/flushing/flushing_ex4_output.yml}{}{1}{11}\r\n \\caption{The \\code{flushing} YAML output file for example 4.}\r\n \\label{fig:flushing_ex4_yml}\r\n\\end{figure}\r\n\r\n", "meta": {"hexsha": "3f59b8c9cd823369f2c717b3e507a4a2602622c2", "size": 22194, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/wst/flushing.tex", "max_stars_repo_name": "USEPA/Water-Security-Toolkit", "max_stars_repo_head_hexsha": "6b6b68e0e1b3dcc8023b453ab48a64f7fd740feb", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-06-10T18:04:14.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-05T18:11:40.000Z", "max_issues_repo_path": "doc/wst/flushing.tex", "max_issues_repo_name": "USEPA/Water-Security-Toolkit", "max_issues_repo_head_hexsha": "6b6b68e0e1b3dcc8023b453ab48a64f7fd740feb", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/wst/flushing.tex", "max_forks_repo_name": "USEPA/Water-Security-Toolkit", "max_forks_repo_head_hexsha": "6b6b68e0e1b3dcc8023b453ab48a64f7fd740feb", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-09-24T19:04:14.000Z", "max_forks_repo_forks_event_max_datetime": "2020-12-05T18:11:43.000Z", "avg_line_length": 53.0956937799, "max_line_length": 126, "alphanum_fraction": 0.77579526, "num_tokens": 5157, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34904287667217504}} {"text": "\n\\part{Outline of the AXFI Data Model}\n\\vspace{.5em}\n\\section{Types of Image Annotations}\n\\p{The most fundamental kinds of image annotations \nare geometric primitives such as points, lines, \nand polygons. Many other forms of annotations \nare possible, however, mostly supplemental \ndata which is associated with these primitives \nor, in some cases, with the image as a whole. \nIn general, \\AXFI{} classifies annotations into \nthe following groups:\n\n\\begin{description}\n\\item[Geometric Primitives] These annotations \ndelineate spatial/geometric regions in zero, one, \nor two dimensions (or potentially higher dimensions \nwhen working in non-\\TwoD{} contexts). At a minimum, \n\\AXFI{} data should support points, lines, \npolygons, polylines (considered a superkind of \npolygons where a polygon is a closed polyline), circles, \nand ellipses. Additionally, \\AXFI{} recognizes \ngeneric \\q{closed} and \\q{open} \\textit{curves}, which \nare nonlinear one-dimensional regions (or boundaries \nof two-dimensional regions) that are neither \nelliptical or circular arcs. Depending on \ncontext, \\AXFI{} can be extended to provide \nmore detailed subkinds of curves generated \nby different sorts of mathematical equations, \nalong with notations for the formulae \n(e.g., b-splines) that generate a particular \ncurve.\\footnote{In formal statements, \nthis and other \\sAXFI{} documentation will \nuse the term \\q{kind,} as well as \\q{superkind} \nand \\q{subkind,} to indicate groups of values/entities \nidentified via a classification. Vocabulary \nbased on \\q{kind} is preferred to \\q{type} or \n\\q{class} because of the distinct meanings these\nlatter terms have in computational contexts \nwhich are also discussed in reference to \\sAXFI{}.} \nMore information about encoding \nellipse data as well as other sorts of \ncurves is outlined below (page~\\lpageref{ph:curves}). \n\nA variation on the polygon/polygon-line alternative \nare polygon-lines demarcating spatial regions \nwhose boundaries extend to one or more sides/corners \nof the image (e.g., a \\q{quadrant} is defined \nvia one central point with line segments \nparallel to and implicitly \nextended to the edges). These poly-lines \nmay not explicitly represent the overall image \nboundary as part of their own boundary. \nConceptually, treating the image-border itself \nas a different \\textit{sort} of boundary than \nthose explicitly notated may be more accurate \nthan eliding these distinctions. This \napplies also to segment boundaries. For instance, \nthe sand/ocean border in a beach scene represents \na physical discontinuity between two different \nmaterial substances, and so it records an \nobservable detail of the depicted scene. \nOn the other hand, the edge of the sand-region \nat the bottom of the image is not a physical \nboundary, but an artifact of the camera position; \nwe assume the beach itself extends further than \nwhat the camera captures. All told, then, \nin addition to polygons (or curves or segment-boundaries) \nbeing \\textit{open} or \\textit{closed}, \\AXFI{} \nrecognizes a third form of closure dubbed \n\\q{incomplete,} meaning that a spatial region is \ngeometrically closed by the edge of the image but \nthat this closure has no observational or semantic \nsignificance. \\lAXFI{} allows a notation that \na spatial region is \\q{closed by fiat} when the closure \nresults from the intrusion of the global image boundary. \nA polygonal line may be closed by fiat when it does \nnot explicitly include lines along the global boundary, \nbut forms a closed polygon when such lines are \nincluded in practice; the result is called a \n\\textit{fiat polygon} (similarly an annotation \ncan be classified as a \\textit{fiat curve} or \n\\textit{fiat segment}).\n \n\n\\item[Segments and Regions] A \\textit{segment} is \nconsidered to be, canonically, an integral \nsubimage with a semantically precise \nseparation of \\q{foreground} from \\q{background.} \nThere may be vagueness or approximation \nin how the segment is precisely individuated \nfrom its surroundings, but these ambiguities \nare considered to be practical limitations \ndue to limited computer power, limiting pixel \nresolution, and so forth. A \\textit{region} \nor \\textit{region of interest} is similar to a \nsegment, but defined more loosely; regions \ncan have vague descriptors, and spatially \ndisconnected parts of an image could be treated \nas parts of one same region. In a photograph \nof a flock of birds, for example, there may be \nmultiple segments, each outlining one single bird. \nHowever, the flock as a whole may be outlined \nby one \\textit{region}, which could (without \nbeing deemed approximative) include some of the \nbackground sky. A \\textit{segment} can be seen \nas subkind of \\textit{region} with stricter \ngranular and topological requirements. \n\nA \\textit{partition} of an image is a segmentation \nwhich exhaustively classifies every point into \none or another segment. A \\textit{selective} \nsegmentation, unlike a complete partition, only \nisolates certain segments or regions of \ninterests. \\lAXFI{} adopts these terms \nas parameters that can characterize \nsegmentation processes, and by extension the \nresulting segments/regions. \n\n\\item[Locations] In \\AXFI{}, \\textit{locations} \nare considered to be designations of areas \nwithin an image (of varying dimensions) which \nare significant by virtue of their directional, \nmorphological, or topological relations to \nthe rest of the image, rather than by virtue \nof their intrinsic shape. Conceptually, \na \\textit{location} is in many cases similar \nto a \\textit{point}, but \\AXFI{} does not require \nlocations to be zero-dimensional. A location \nmay be designated by a small disk, or even a\nregion/segment. The distinguishing feature \nof locations is that their spatial shape or extent \nare not semantically significant; instead, \nwhat is important about locations is their \nposition in the image and how this position \nrelates to surrounding image content. For \ninstance, a location might be the point/position \nwhere two roads intersect, or it could \nbe the leftmost point on the segment-boundary of \na car's fender or a bird's wings, or the \ngeometric center of a car's body or a bird's torso.\n\n\\lAXFI{} distinguishes location-annotations from \nsecondary annotations used to identify locations \n(insofar as these may be visually distinct). For \ninstance, a position may be modeled as a single \npoint, but visually conveyed by an arrow, or by \na circle hovering above the relevant point.\n\n\\item[Focal Points] The concept of \\textit{focal \npoints} integrates locations and geometric primitives \nfor certain analytic tasks. In general, focal points \nare important for positional rather than morphological \nregions, similar to locations. However, focal points \nmay function more like segments or geometric objects \nwhen used as part of an analytic objective. For \ninstance, points embodying the center of a car's body \nor a bird's torso could be used to count the number \nof birds or cars appearing in a photograph. \nIn this case the concept of \\q{geometric center} has \nno meaningful morphological properties, so it \nis analogous to a location. On the other hand, the \ncenter may be treated as a proxy for, or a most \nsignificant component of, a segment or region; in this \nsense the point is \\textit{part of} a segment. \nThis mereological aspect makes focal points \nact conceptually more like geometric primitives than \nlike locations. In short, the classification \n\\textit{focal point} is available for spatial \nobjects which behave somewhere between locations \nand regions/points, and particularly when they \nare used in some proxying or indicative relation \nto other regions (e.g. for counting). \n \n\\item[Secondary Images] In some contexts, such \nas segmentation, image-transforms are used to \nconvey image-processing operations, in contrast \nto geometric-style annotations that can be \ndefined via vertex coordinates. A crisp \nsegment can be defined via a two-toned image \nwith the same dimensions as the annotated \nimage, but with only two logical colors \n(colors are called \\q{logical} insofar as the \nrelevant detail is how the colors compare to \none another, irregardless of the optical colors \nused to render them\\footnote{In \\sQt{}, for instance, \nthe \\textbf{QColorConstants::Color0} and \n\\textbf{QColorConstants::Color1} color values \nare considered to be special \\q{colors} \n(i.e., \\textbf{QColor} instances) which define \nthe foreground and background of a two-toned \nimage; they are not formally assigned to a \nvisual color, like black or white.}). \nA \\q{fuzzy} segment can likewise be defined \nvia a greyscale image (anything which \nis pure-background becoming either pure white, \nor pure transparent, depending on context). \nSecondary images can be used to denote annotations \nwhich are too granular to be summarized by \nany mathematical expression. In these cases, \nthe actual annotation data should identify the \nsecondary image (e.g., via a file path) and \nexplain how it relates to the primary \n(or \\q{ground}) image, whereas the secondary \nfile itself fills in the annotation details. \n\nSecondary images may be derived from ground \nimages merely to present annotations, but they \nmay also be intermediate analysands which are \nthemselves annotated. In the latter case, \nannotations on secondary images are usually \nmeaningful also as annotations on ground \nimages, so the interrelations between both \nimages should be notated in the annotation data.\n\n\\item[Proscriptive Annotations] \\lAXFI{} \nallows for the designation of certain annotations \nas \\q{proscriptive} when they do not \nformally delineate a geometric object, but \nindirectly express such an object's shape \nor how it may be derived. A canonical \nexample is the use of color --- perhaps \ncolor-enhanced secondary images --- to \ndesignate segments or regions of interest. \nFor instance, one common segmentation technique \nuses color simplification; smoothing out color \npatches can facilitate image partitioning \nby reinforcing the boundaries between different \nregions. With sufficient color manipulation, \nimage-segments can potentially be described \nchromatically: a region may for instance \nbe identified as \\q{the area all of whose \npixels display a red channel above} some \nthreshold. \\lAXFI{} data should therefore \nallow such indirect designations of \nsegments (and spatial/geometric regions in \ngeneral) to be encoded as annotations.\n\nThe concept of proscriptive annotations is not \nonly chromatic --- one can imagine other scenarios \nwhere morphological features, such as symmetries, \nmay be employed to similar effect. In describing \na chess board, for instance, a set of image-regions \n(each square on the board) can be derived \nvia translational transforms of an initial \ntwo-segment kernel (one black square and one white). \nThis representation is possible because of \ntranslational symmetries in the underlying image. \nSuch geometric transforms and symmetries can \nsometimes be used to \\q{generate} meaningful \nimage segments, so they should be notated \nin \\AXFI{} data when appropriate. \n\n\\item[Semantic Labels] Some annotations supply data \nabout other annotations (what \\AIM{} calls \n\\q{Annotation on Annotation} as opposed to \n\\q{Annotation on Image}). In general, we \ncan supply a label, description, or semantic \nclassification indicating what an image segment \nor region is \\q{about}, i.e., what it \\q{depicts} \n(a bird, a flock of birds, a car, a traffic \njam, and so forth). Such meta-annotation may be \nseen as a semantic postlude to an imaging \nprocess, but it may also be seen as providing further \ndetail about the process itself. For instance, \nsuppose segmentation isolates a bird from the \nsky: the epilogue to this operation is an ability \n(for a human user or a computer algorithm) to \nclassify the segment as \\q{bird.} However, we \ncan also say that the given fact of the segment \ncapturing the optics of a bird (with its apparent \nanatomical outline and coloration) is what \nallowed the segmentation to be possible in the \nfirst place. Therefore, the label \\q{bird} \nserves both to utilize the segmentation for \nfurther analytic/classificatory purposes and \nalso, potentially, to characterize the \ninner workings or effectiveness of \nachieving the desired processing objective. \nIt should be kept in mind, in short, that \nlabel annotations are not exclusively \nintended to be used for practical labeling \nin the contexts of tasks such as subject-marking \nimages in corpora; labeling could also \nbe used to notate how semantic properties \nof the image make segmentation (and other \nprocessing objectives) more or less \nfeasible, or computationally intensive/error-prone. \n\nThe semantics of labels themselves is outside \nthe scope of \\AXFI{}. \\lAXFI{} makes no effort \nto proscibe what sorts of terms are useful \nas labels (\\q{bird,} \\q{car,} \\q{ocean,} \n\\q{tumor,} etc.), or whether labels are \nsimple strings/words or have internal structure \nof their own. \\lAXFI{} does, however, \nmake the following minimal stipulations about \nlabels. First, \\AXFI{} distinguishes \n\\q{mass,} \\q{count,} and \\q{plural} label \ntargets --- e.g., \\textit{ocean}, \\textit{one bird}, \n\\textit{two birds}. These distinctions are \ndirectly relevant to how segments are bounded and \ncounted; for instance, labeling a body of water \nas \\textit{a lake} (e.g. on a satellite image where \nthe lake's full shore is visible) versus \\textit{ocean} \n(continuing to the horizon) implies different \ndesiderata for how the labeled region spatially extends \nin relation to the surrounding image. Secondly, \nwith respect to semantic labels, \\AXFI{} recommends \nthat applications and libraries enable \nlabels to be typed values --- instances of \napplication-specific data types --- as well \nas simple character strings (like \\q{bird.}) \nThis is consistent with \\AXFI{}'s general \napproach to application integration and \ntype systems, to be discussed further in \nPart II. \n\nIn addition to labels being second-order annotations \n--- assertions attached to an image segment, \nfor example, or any other image feature deemed \nsemantically relevant in some context --- labels \nmay be applied to the image as a whole. In this\ncontext labels would be an example of \n\\textit{metadata} annotations (discussed next), \nwhich apply to the image itself \nrather than any proper part. \n\n%\\item[Data Transformations] \n\n\\item[Image Metadata] This category \nof annotation concerns any information \n\\q{about} an image which is apart from \nthe specific image content. Such details could encompass \ncomputational/encoding details such \nas color depth, pixel dimensions, and image \nfile format; or acquisition details \nsuch as the make and settings of the camera \nwhose photograph yields the current image. \nIn theory, any information prerequisite to \ndisplaying an image may potentially \nbe relevant to how annotations are \nused and interpreted. Therefore, \n\\AXFI{} should support representations of \nimage metadata where such information is needed \nto fully specify annotation data; the metadata \nwould then, typically, be considered an \nannotation on the entire image. It is \noutside the scope of \\AXFI{} to present \nimage-related data which is applicable \nto the sharing or rendering of images but \nnot, in a particular context, important to \nannotations themselves --- unlike \\DICOM{}, \nfor instance, which is designed as a \nstandard for ensuring that images are correctly \nrendered and therefore includes detailed \nspecifications of image formats and \nprerequisites, \\AXFI{} introduces \nimage metadata only where such information \ncan be considered intrinsic to the \nspecific objectives and operations of image \nannotation. \n\n\\item[Acquisition Context] Among the class of \nannotations which apply to the image as a whole, \n\\AXFI{} recognizes a distinct kind of annotation \ncovering what \\AIM{} calls the \\q{context,} \nor the purpose and concrete goal which compels \nusers to initiate image-processing operations. \nIn general, some of this contextual data \nshould be preserved as a supplement to \nthe overall information generated by image-processing \noperations. Often image processing itself is only \none stage in a multi-faceted scientific workflow, \nand contextual information which exists prior to \nthe image-annotation step may still be important \nfor subsequent steps. For example, clinical data \nwhich motivates a radiographic study should remain \nlinked to images and annotations which result from \nthat study; this is why \\DICOMSR{} includes \nclinical as well as imaging data. \n%More details about \\AXFI{} context descriptions are provided \n%below (section ?). \n\\end{description}\n\nThis outline covers the different kinds \nof annotation data recognized by \\AXFI{}. \nThe following section will fill in some \nmissing details by addressing other \nsorts of information, apart from annotations \nthemselves, which might be included in an \n\\AXFI{} data package.}\n\n", "meta": {"hexsha": "bfe4dce314a05e3b37598f29a27a02c46746c041", "size": 16985, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "extra/papers/axfi/axfi/axfi-s1.tex", "max_stars_repo_name": "scignscape/PGVM", "max_stars_repo_head_hexsha": "e24f46cdf657a8bdb990c7883c6bd3d0a0c9cff0", "max_stars_repo_licenses": ["BSL-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "extra/papers/axfi/axfi/axfi-s1.tex", "max_issues_repo_name": "scignscape/PGVM", "max_issues_repo_head_hexsha": "e24f46cdf657a8bdb990c7883c6bd3d0a0c9cff0", "max_issues_repo_licenses": ["BSL-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "extra/papers/axfi/axfi/axfi-s1.tex", "max_forks_repo_name": "scignscape/PGVM", "max_forks_repo_head_hexsha": "e24f46cdf657a8bdb990c7883c6bd3d0a0c9cff0", "max_forks_repo_licenses": ["BSL-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.1091370558, "max_line_length": 62, "alphanum_fraction": 0.7823373565, "num_tokens": 3906, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228625116081, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34904286260507245}} {"text": "\\section{Sliding Game}\nOur metric for algorithmic performance involves running A* on the sliding game (see the Youtube video for a demonstration). The sliding game is a particularly convenient search problem for two reasons: the state representation is small and the state space is very large. It blows up exponentially with respect to the puzzle dimensions. This is particularly important since our goal was to measure the effectiveness of a parallel A* algorithm, and the expansive state space would decrease potential overhead.\n\\newline\\newline\n\\begin{tabular}{ |p{3cm}||p{3cm}|p{3cm}|p{3cm}| }\n\\hline\n\\multicolumn{4}{|c|}{Sliding Game Sequential Execution times} \\\\\n\\hline\nBoard Size& Runtime (ns)&Runtime (ms)& Runtime (s)\\\\\n\\hline\n2x3 & 35880 & 0 & 0\\\\\n3x3 & 84711 & 0 & 0\\\\\n3x4 & 258638033 & 259 & 0\\\\\n4x4 & 171623957677 & 171624 & 172\\\\\n\\hline\n\\end{tabular}\n\\newline\n\\begin{figure}[h]\n \\includegraphics[scale=0.5]{figures/4x4.png}\n \\caption{The puzzle grid used for the 4x4 measure}\n \\label{fig:gastar_ds}\n\\end{figure}\n\\newline\nThe object of the game is to order the tiles in increasing numerical order. The 0 tile represents an empty tile where adjacent tiles can \"slide\" to. Thus, every turn can result in 4 more possible states (exponential). The graph below is a visual representation of the above runtimes in log scale.\n\n\\begin{tikzpicture}[scale=0.91]\n \\begin{axis}[\n xlabel=$area$,\n ylabel=$time(ns)$,\n ymode=log\n ]\n \\addplot[mark=*,blue] plot coordinates {\n (6,35880)\n (9,84711)\n (12,258638033)\n (16,171623957677)\n };\n \\addlegendentry{Runtime}\n \n \\end{axis}\n \\node[above,font=\\large\\bfseries] at (current bounding box.north) {Sequential Execution times (log scale)};\n\\end{tikzpicture}\n\n\\section{Evaluating Performance}\nFor this problem, we evaluated our implementations by checking the wall time of the implementations. We found this to be appropriate for the serial algorithm given that we were not able to properly evaluate speedup for the GA* GPU method.\n\n\\section{Scaling the Problem Space}\nConsidering our sequential performance in the graph above, we see that the line is roughly linear. This supports our hypothesis that as the number of cells grow, the runtime grows roughly exponentially. In particular, we see that because most nodes on the frontier have 3 or 4 neighbors, the number of states being added as we go further from the initial state increases exponentially. Due to the nature of the problem, a small change in workload could result in dramatically larger runtime.\n\n\\section{Limitations of our Approach}\nThe main source of issues with our CUDA implementation came from our hash map. Our implementation of the hash map involves a table and two hash functions where the second can search for new locations in the table for a node if another element collides with it per the first hash function. Although this simplification allows us to not need sychronization in this section, it means that we need a very large hash table. If the hash table is small compared to the number of states, it is possible that an element has collisions with both the hash functions. In this case, the hash map will \\textit{lose information} that could potentially be important for the final path.\\newline\\newline\nAnother limitation of our approach was the amount of global memory that we allocate in this method. We use it for the large array \\verb|S| as well as the hash table. If we scale up the hash table size in order to reduce the likelihood of collisions, we will increase the global memory allocated. In the GPU, global memory is far less efficient to update than shared memory. \\textbf{As a result, the global memory brought upon by the hash table would likely be our main bottleneck in the CUDA implementation.}\n\n\\section{Changing the Hardware}\nAs we have discovered through this project, a GPU implementation of A* is not too pragmatic. In particular, the memory constraint of GPUs especially for search problems of this magnitude is quite restrictive. In contrast, using MPI would alleviate this. \\textbf{MPI is not restricted by a single device and can be used across a network of devices, thus relaxing the space constraints}. Furthermore, MPI is a more task oriented system whereas CUDA is more data oriented, which is more suitable for search problems.", "meta": {"hexsha": "35ac1f2d990ce2f365099813e4826721096ea414", "size": 4377, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "latex_files/Chapters/Results.tex", "max_stars_repo_name": "parallel-search/a-star", "max_stars_repo_head_hexsha": "0d8f940d9826b8abac5944c65845cb8e96f63f1e", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "latex_files/Chapters/Results.tex", "max_issues_repo_name": "parallel-search/a-star", "max_issues_repo_head_hexsha": "0d8f940d9826b8abac5944c65845cb8e96f63f1e", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "latex_files/Chapters/Results.tex", "max_forks_repo_name": "parallel-search/a-star", "max_forks_repo_head_hexsha": "0d8f940d9826b8abac5944c65845cb8e96f63f1e", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 81.0555555556, "max_line_length": 685, "alphanum_fraction": 0.7669636737, "num_tokens": 1033, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6926419958239131, "lm_q1q2_score": 0.3490265756632474}} {"text": "\\section{\u001b$@4v2?%b%G%j%s%05!G=\u001b(J}\n\n\\subsection{\u001b$@4v2?%b%G%j%s%0$N4pK\\5!G=\u001b(J}\n\n\u001b$@K\\@a$G=R$Y$k5!G=$KBP1~$9$k%=!<%9%U%!%$%k$O!\"\u001b(J\\ $geopack.l$\\ \u001b$@$G$\"$k!#\u001b(J\n\n\\subsubsection{\u001b$@4pK\\=hM}4X?t\u001b(J}\n\n\\begin{description}\n\\item[vplus {\\em vertices}]\\hfill\\\\\nvertices\u001b$@$O\u001b(J3\u001b$@@~\u001b(Jab\u001b$@$N:8\u001b(J\n\u001b$@B&$K$\"$j!\"Ii$G$\"$l$PD>@~\u001b(Jab\u001b$@$N1&B&$K$\"$k$3$H$,$o$+$k!#\u001b(J\n\\item[triangle-normal {\\em a b c}]\\hfill\\\\\n\u001b$@H?;~7W2s$j$KJB$s$@\u001b(J3\u001b$@E@\u001b(Ja,b,c \u001b$@$N:\\$kJ?LL$N@55,2=$5$l$?K!@~%Y%/%?$r5a$a$k!#\u001b(J\n\\item[vector-angle {\\em v1 v2 normal}]\\hfill\\\\\n2\u001b$@$D$NJ}8~%Y%/%?\u001b(Jv1,v2 \u001b$@$N$J$93QEY!J%i%8%\"%s!K$r5a$a$k!#\u001b(Jv1, v2, normal\u001b$@$O$$$E$l\u001b(J\n\u001b$@$bA00J$F@55,2=$5$l$F$$$kI,MW$,$\"$k!#\u001b(J\n\\item[face-normal-vector {\\em vertices \\&optional rotatep}]\\hfill\\\\\n1\u001b$@J?LL>e$N\u001b(J3\u001b$@.$ND>J}BN$N%/%i%9!#\u001b(J\n\u001b$@43>D8!::$NA0=hM}Ey$KMQ$$$k!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[object] ()\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item{minpoint}\n\\item{maxpoint}\n\\end{description}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:inner (point)] point \u001b$@$,$3$N\u001b(Jbox \u001b$@Fb$K4^$^$l$k;~\u001b(Jt,\u001b$@$=$&$G$J$$$H$-\u001b(Jnil \u001b$@!#\u001b(J\n\\item[:grow (rate)]\n\u001b$@$?$H$($P\u001b(Jrate=0.01 \u001b$@$N$H$-!\"\u001b(Jbox \u001b$@$r8=:_$NBg$-$5$+$i\u001b(J1\\%\u001b$@$U$/$i$^$;$k!#\u001b(J\n\\item[:volume ()] \u001b$@$3$N\u001b(Jbox \u001b$@$NBN@Q!#\u001b(J\n\\item[:vectors (vlist)]\nfloatvector \u001b$@$N%j%9%H\u001b(Jvlist \u001b$@$+$i!\"\u001b(Jbox \u001b$@$N\u001b(Jminpoint, maxpoint \u001b$@$r@_Dj$9$k!#\u001b(J\n\\item[:vectors2 (v1 v2)]\n2\u001b$@$D$N\u001b(Jfloatvector,v1,v2 \u001b$@$+$i\u001b(Jminpoint,maxpoint \u001b$@$r@_Dj$9$k!#\u001b(J\n\\end{description}\n\n\\end{description}\n\n\\vfill\n\\pagebreak\n{\\jLarge line}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@D>@~$N%/%i%9!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[object] ()\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[pvert] \u001b$@DL2a$9$kE@!#%5%V%/%i%9\u001b(Jedge\u001b$@$G$O!\"%(%C%8$N;OE@!#\u001b(J\n\\item[nvert] \u001b$@DL2a$9$kE@!#%5%V%/%i%9\u001b(Jedeg\u001b$@$G$O!\"%(%C%8$N=*E@!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:vertices ()] pvert \u001b$@$H\u001b(Jnvert \u001b$@$N%j%9%H!#\u001b(J\n\\item[:point (parameter)] pvert \u001b$@$r\u001b(J0.0,nvert \u001b$@$r\u001b(J1.0 \u001b$@$KBP1~$5$;$?$H$-$N\u001b(J\n\u001b$@%Q%i%a%?I=8=$+$i@~J,>e$NE@$N:BI8$r5a$a$k!#$9$J$o$A!\"\u001b(J\n$pvert=(x_{p},y_{p},z_{p})$, $nvert=(x_{n},y_{n},z_{n})$\\ \u001b$@$H$7!\"\u001b(J\n\u001b$@$3$N\u001b(J2\u001b$@E@$rDL$kD>@~$r!\"\u001b(J\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nx \\\\ y \\\\ z \\\\ 1\n\\end{array} \\right)=\\left(\n\\begin{array}{cc}\n(1-t) & t\n\\end{array} \\right) \\left(\n\\begin{array}{cc}\nx_{p} & x_{n} \\\\\ny_{p} & y_{n} \\\\\nz_{p} & z_{n} \\\\\n1 & 1\n\\end{array} \\right)\n\\end{equation}\n\u001b$@$HI=$7$?$H$-!\"\u001b(J\\ $t$\\ \u001b$@$rM?$($l$P:BI8\u001b(J\\ $(x,y,z)$\\ \u001b$@$rJV$9!#\u001b(J\n\\item[:parameter (point)] :point\u001b$@$N5U$GE@$N:BI8$+$i%Q%i%a%?$r5a$a$k!#\u001b(J\n\\item[:box ()]\n\\item[:boxtest (box)]\n:box\u001b$@$O$3$N%(%C%8$,4^$^$l$k\u001b(Jsurrounding-box \u001b$@$r:n@.$9$k!#\u001b(J:boxtest\u001b$@$O$^$:<+J,\u001b(J\n\u001b$@$N\u001b(Jsurrounding-box \u001b$@$r:n@.$7!\"B>$N\u001b(Jbox \u001b$@$H$N43>D$r8!::$9$k!#\u001b(J\n\\item[:foot(point)]\n\u001b$@E@\u001b(J\\ $point=(x_{1},y_{1},z_{1})$\\ \u001b$@$+$i$3$ND>@~$K2<$m$7$??b@~$NB-$N%Q%i%a!<%?\u001b(J\n\\ $t$\\ \u001b$@$rJV$9!#$3$l$O!\"\u001b(J\n\\begin{equation}\nt=\\frac{(x_{n}-x_{p})(x_{1}-x_{p})+(y_{n}-y_{p})(y_{1}-y_{p})\n+(z_{n}-z_{p})(z_{1}-z_{p})}\n{(x_{n}-x_{p})^{2}+(y_{n}-y_{p})^{2}+(z_{n}-z_{p})^{2}}\n\\end{equation}\n\u001b$@$K$h$jM?$($i$l$k!#!J\u001b(JRef.\u001b$@LpLn7rB@O:Cx!\"?^7A$H<0!\"\u001b(Jp.178\u001b$@!\"9VCL@~$^$G$N5wN%$rJV$9!#$3$l$O!\"\u001b(Jpoint\u001b$@$+$i$3$ND>@~$K2<$m$7$?\u001b(J\n\u001b$@?b@~$NB-$H\u001b(Jpoint\u001b$@$N5wN%$+$i5a$a$F$$$k!#\u001b(J\n\\item[:common-perpendicular(l)]\n\u001b$@D>@~\u001b(J\\ $l$\\ \u001b$@$H$N6&DL?b@~$r5a$a$k!#JV$9CM$O!\"\u001b(J2\u001b$@D>@~$H$N8rE@$N%j%9%H\u001b(J\n\u001b$@$G$\"$k!#\u001b(J2\u001b$@D>@~$,J?9T$N$H$-$O!\"\u001b(Jparallel\u001b$@$rJV$9!#$3$l$O!\"0J2<$NMM$K5a$a$F$$$k!#\u001b(J\n2\u001b$@D>@~$,7PM3$9$k\u001b(J2\u001b$@E@$r$=$l$>$l\u001b(J\\ $(x_{11},y_{11},z_{11})$\\ \n$(x_{12},y_{12},z_{12})$, $(x_{21},y_{21},z_{21})$\\ $(x_{22},y_{22},z_{22})$\n\\ \u001b$@$H$7!\"$3$N\u001b(J2\u001b$@D>@~$N6&DL?b@~\u001b(J\n\u001b$@$H$N8rE@$r\u001b(J\\ $(x_{31},y_{31},z_{31})$\\ $(x_{32},y_{32},z_{32})$\\ \u001b$@$H$9$k$H!\"\u001b(J\n\\begin{equation}\n(x_{12}-x_{11}\\ y_{12}-y_{11}\\ z_{12}-z_{11})^{T}\n(x_{32}-x_{31}\\ y_{32}-y_{32}\\ z_{32}-z_{31})=0 \\label{eq:suityoku1}\n\\end{equation}\n\\begin{equation}\n(x_{22}-x_{21}\\ y_{22}-y_{21}\\ z_{22}-z_{21})^{T}\n(x_{32}-x_{31}\\ y_{32}-y_{32}\\ z_{32}-z_{31})=0 \\label{eq:suityoku2}\n\\end{equation}\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nx_{31} \\\\ y_{31} \\\\ z_{31} \\\\ 1\n\\end{array} \\right)=\\left(\n\\begin{array}{cc}\n(1-t_{1}) & t_{1}\n\\end{array} \\right) \\left(\n\\begin{array}{cc}\nx_{11} & x_{12} \\\\\ny_{11} & y_{12} \\\\\nz_{11} & z_{12} \\\\\n1 & 1\n\\end{array} \\right)\n\\label{eq:online1}\n\\end{equation}\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nx_{32} \\\\ y_{32} \\\\ z_{32} \\\\ 1\n\\end{array} \\right)=\\left(\n\\begin{array}{cc}\n(1-t_{2}) & t_{2}\n\\end{array} \\right) \\left(\n\\begin{array}{cc}\nx_{21} & x_{22} \\\\\ny_{21} & y_{22} \\\\\nz_{21} & z_{22} \\\\\n1 & 1\n\\end{array} \\right)\n\\label{eq:online2}\n\\end{equation}\n\u001b$@$,@.N)$9$k!#\u001b(J\\ $X_{ij}=(x_{ij},y_{ij},z_{ij})^{T}$\\ \u001b$@$H$*$-!\"\u001b(J\n(\\ref{eq:suityoku1}),(\\ref{eq:suityoku2})\u001b$@<0$K\u001b(J\n(\\ref{eq:online1}),(\\ref{eq:online2})\u001b$@<0$rBeF~$9$k$H!\"\u001b(J\n\\begin{displaymath}\n(X_{12}-X_{11})^{T}(X_{12}-X_{11})t_{1}-(X_{12}-X_{11})^{T}(X_{22}-X_{21})t_{2}\n=(X_{12}-X_{11})^{T}(X_{21}-X_{11})\n\\end{displaymath}\n\\begin{displaymath}\n(X_{12}-X_{11})^{T}(X_{22}-X_{21})t_{1}-(X_{22}-X_{21})^{T}(X_{22}-X_{21})t_{2}\n=(X_{22}-X_{21})^{T}(X_{21}-X_{11})\n\\end{displaymath}\n\u001b$@$H$J$j!\"$3$l$+$i\u001b(J\\ $t_{1},t_{2}$\\ \u001b$@$r5a$a!\"$=$l$i$r\u001b(J(\\ref{eq:online1}),\n(\\ref{eq:online2})\u001b$@$KBeF~$9$k$3$H$K$h$j!\"6&DL?b@~$H$N\u001b(J2\u001b$@8rE@$,5a$^$k!#\u001b(J\n\u001b$@!J\u001b(JRef.\u001b$@LpLn7rB@O:Cx!\"?^7A$H<0!\"\u001b(Jp.178\u001b$@!\"9VCLpJs$r4^$^$J$$!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[line] (pvert nvert)\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[pface] \u001b$@%(%C%8$N:8B&$NLL\u001b(J\n\\item[nface] \u001b$@%(%C%8$N1&B&$NLL\u001b(J\n\\item[(angle float)] pface \u001b$@$H\u001b(Jnface \u001b$@$,$J$93QEY\u001b(J\n\\end{description}\n% picture= edge.tex\n% \\begin{figure}\n\\begin{minipage}[b]{16cm}\n\\begin{picture}( 391, 320)\n\\large\\tt\n\\put( 203.4, 152.8){nface}\n\\put( 102.2, 152.8){pface}\n\\put( 152.8, 51.6){pvert}\n\\put( 152.8, 253.9){nvert}\n\\thicklines\n\\put( 168.6, 101.2){\\line( 1,-1){ 84.3}}\n\\put( 168.6, 101.2){\\line(-1,-1){ 84.3}}\n\\put( 168.6, 101.2){\\vector( 0, 1){ 118.0}}\n\\put( 168.6, 219.2){\\line( 1, 1){ 84.3}}\n\\put( 84.3, 303.5){\\line( 1,-1){ 84.3}}\n\\end{picture}\n\\end{minipage}\n% \\end{figure}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:pvertex (f)]\n\\item[:nvertex (f)] f \u001b$@$O\u001b(Jpface \u001b$@$^$?$O\u001b(Jnface \u001b$@!#\u001b(Jf \u001b$@$K$H$C$F$N\u001b(Jpvert,nvert \u001b$@$r5a$a$k!#\u001b(J\n\\item[:pface ()] pface\u001b$@$rJV$9!#\u001b(J\n\\item[:nface ()] nface \u001b$@$rJV$9!#\u001b(J\n\\item[:binormal (f)]\nf \u001b$@$O\u001b(Jpface \u001b$@$^$?$O\u001b(Jnface \u001b$@!#\u001b(Jf\u001b$@$NK!@~$H$3$N%(%C%8$NN>J}$KD>8r$9$k@55,2=$5$l$?J}8~%Y\u001b(J\n\u001b$@%/%?$r7W;;$9$k!#\u001b(J\n\\item[:angle ()] angle\n\\item[:set-angle ()] \u001b$@LL4V$N3QEY!J\u001b(J2\u001b$@LL3Q!K$r7W;;$7$F\u001b(Jangle \u001b$@$KF~$l$k!#\u001b(J\n\\item[:invert ()] \u001b$@%(%C%8$NJ}8~$r5UE>$5$;$k!#$9$J$o$A!\"\u001b(Jpface \u001b$@$H\u001b(Jnface \u001b$@$r\u001b(J\n\u001b$@8r49$9$k!#\u001b(J\n\\item[:set-face (pv nv f)] pvert,nvert \u001b$@$KBP1~$9$k\u001b(Jpface \u001b$@$^$?$O\u001b(Jnface \u001b$@$r\u001b(J\n\u001b$@%;%C%H$9$k!#\u001b(J\n\\item[:distance (point)] \u001b$@E@\u001b(Jpoint\u001b$@$H!\"\u001b(Jedge\u001b$@>e$N:G6aE@$H$N5wN%$r5a$a$k!#\u001b(J\npoint\u001b$@$+$i$3$N\u001b(Jedge\u001b$@$K2<$m$7$??b@~$NB-$,$3$N\u001b(Jedge\u001b$@>e$K>h$k$H$-$O$=$NB-$H$N5wN%!\"\u001b(J\n\u001b$@$=$&$G$J$$$H$-$O!\"6a$$J}$NC@~$H$3$NJ?LL$H$N8rE@$r%Q%i%a!<%?$GJV$9!#\u001b(J\n\u001b$@$9$J$o$A!\"\u001b(J$pvert=(x_{p},y_{p},z_{p})$,\\ $nvert=(x_{n},y_{n},z_{n})$\\ \n\u001b$@$H$7!\"$3$N\u001b(J2\u001b$@E@$rDL$kD>@~$r!\"\u001b(J\n\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nx \\\\ y \\\\ z \\\\ 1\n\\end{array} \\right)=\\left(\n\\begin{array}{cc}\n(1-t) & t\n\\end{array} \\right) \\left(\n\\begin{array}{cc}\nx_{p} & x_{n} \\\\\ny_{p} & y_{n} \\\\\nz_{p} & z_{n} \\\\\n1 & 1\n\\end{array} \\right)\n\\end{equation}\n\u001b$@$H$*$$$?$H$-!\"J?LL\u001b(J\n\n\\begin{equation}\n\\left(\n\\begin{array}{cccc}\na & b & c & d\n\\end{array} \\right) \\left(\n\\begin{array}{c}\nx \\\\ y \\\\ z \\\\ 1\n\\end{array} \\right) = 0\n\\end{equation}\n\u001b$@$H$N8rE@$K$*$1$k%Q%i%a!<%?\u001b(J\\ $t$\\ \u001b$@$NCM!\"\u001b(J\n\n\\begin{equation}\nt=\\frac{ax_{p}+by_{p}+cz_{p}+d}{(ax_{p}+by_{p}+cz_{p})-(ax_{n}+by_{n}+cz_{n})}\n\\end{equation}\n\u001b$@$rJV$9!#\u001b(J\n\\item[:foot (point)]\n\u001b$@E@\u001b(J\\ $point=(x_{1},y_{1},z_{1})$\\ \u001b$@$+$iJ?LL\u001b(J\\ $ax+by+cz+d=0$\\ \u001b$@$K9_$m$7$??b@~$NB-\u001b(J\n\\ $(x_{0},y_{0},z_{0})$\\ \u001b$@$r5a$a$k!#$3$l$O!\"\u001b(J\n\\begin{equation}\n\\left(\n\\begin{array}{c}\nx_{0} \\\\ y_{0} \\\\ z_{0}\n\\end{array}\n\\right) = \\left(\n\\begin{array}{c}\nx_{1} \\\\ y_{1} \\\\ z_{1}\n\\end{array}\n\\right) + t \\left(\n\\begin{array}{c}\na \\\\ b \\\\ c\n\\end{array}\n\\right)\n\\end{equation}\n\u001b$@$H$*$-!\"$3$l$r\u001b(J\\ $ax_{0}+by_{0}+cz_{0}+d=0$\\ \u001b$@$KBeF~$7$F!\"\u001b(J\n\\begin{equation}\nt=-\\frac{ax_{1}+by_{1}+cz_{1}+d}{a^{2}+b^{2}+c^{2}}\n\\end{equation}\n\u001b$@$H%Q%i%a!<%?\u001b(J\\ $t$\\ \u001b$@$NCM$r5a$a$k$3$H$K$h$jF@$i$l$k!#\u001b(J\n\\item[:intersect-edge (edge)]\nedge\u001b$@$H$N8rE@$N%Q%i%a!<%?$H:BI8$r%j%9%H$K$7$FJV$9!#\u001b(J\n\n\\item[:init (normal apoint)]\n\u001b$@K!@~$,\u001b(Jnormal\u001b$@$G!\"E@\u001b(Japoint\u001b$@$rDL$k$h$&$JJ?LL$rDj5A$9$k!#\u001b(J\n\\end{description}\n\n\\end{description}\n\\vfill\n\\pagebreak\n{\\jLarge closed-region}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@%(%C%8$K$h$C$F0O$^$l$?J?LL$NNN0h$N%/%i%9\u001b(J\n\u001b$@LL$O4pK\\E*$KJ?LLJ}Dx<0$H%(%C%8$r0O$`%(%C%8!\"D:E@$N%k!<%W$GI=8=$5$l$k!#\u001b(J\n\u001b$@J?LLJ}Dx<0$O\u001b(Jplane \u001b$@$KJ];}$5$l!\"\u001b(Jclosed-region \u001b$@$O%k!<%W$rJ];}$9$k!#\u001b(J\nedges \u001b$@$H\u001b(Jvertices\u001b$@$OLL$rJ*BN$N30It$+$i\u001b(J\n\u001b$@D/$a$?;~!\"H?;~7W2s$j$KJB$V$h$&$J%j%9%H$G$\"$k!#JL$N8@$$J}$G$O!\"\u001b(Jedges,\nvertices\u001b$@$N=g$K%k!<%W$rIA$/$h$&$K$M$8$r2s$9$H$M$8$,?J$`J}8~$,J*BN$N30It$9\u001b(J\n\u001b$@$J$o$ALL$NK!@~J}8~$G$\"$k!#$3$N%k!<%W$K$h$C$F%(%C%8!\"D:E@$r$?$I$k$3$H$,$G$-$k\u001b(J\n\u001b$@$N$G!\"\u001b(Jedge\u001b$@$K$O%&%#%s%0>pJs$rJ];}$7$J$$!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[plane] ((normal :type floatvector) (distance :type float))\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[convexp] \u001b$@FLB?3Q7A$N$H$-\u001b(JT\n\\item[edges] \u001b$@LL$NFbIt$,%(%C%8$N:8B&$K$J$k$h$&$KJB$Y$?%(%C%8$N%j%9%H\u001b(J\n\\item[vertices] \u001b$@LL$NFbIt$,%(%C%8$N:8B&$K$J$k$h$&$KJB$Y$?D:E@$N%j%9%H!#\u001b(J\n$v_{1},v_{2},\\cdots,v_{n},v_{1})$\\ \u001b$@$H:G8e$K@hF,$NMWAG$,:F$S8=$l$k$3$H$KCm0U!#\u001b(J\n\\item[(model-normal :type floatvector)]\n\\item[(model-distance :type float))]\n\\end{description}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:box ()] \u001b$@$3$NJDNN0h$r4^$`\u001b(Jsurrounding-box \u001b$@$r:n@.$9$k!#\u001b(J\n\\item[:boxtest (box)]\n\\item[:edges ()] edges\n\\item[:vertices ()] vertices\n\\item[:insidep (point)]\npoint \u001b$@$,$3$NJDNN0hFb$K$\"$k$H$-\u001b(Jinside, \u001b$@30It$N;~\u001b(Joutside,\u001b$@6-3&>e$N;~\u001b(Jborder.\n\u001b$@FLB?3Q7A$N$H$-!\"E@$,$9$Y$F$N%(%C%8$N:8B&$K$\"$k$+$I$&$+$rD4$Y$k!#FL$G$J$$\u001b(J\n\u001b$@$H$-$OE@$,JU$r8+9~$`3QEY$NAmOB$r7W;;$7!\"\u001b(J2\u001b$@&P$N$H$-FbIt!\"\u001b(J0\u001b$@$N$H$-30It$HH=\u001b(J\n\u001b$@Dj$7$F$$$k!#\u001b(J\n\\item[:intersect-point-vector (point vnorm)]\n\u001b$@E@\u001b(Jpoint \u001b$@$+$i@55,2=%Y%/%?\u001b(Jvnorm \u001b$@$NJ}8~$K8~$+$&H>D>@~$H$N8r:9$r5a$a$k!#\u001b(J\n\u001b$@H>D>@~$HJDNN0h$,J?9T$N$H$-$O!\"7k2L$O\u001b(Jparallel\u001b$@$K$J$k!#\u001b(J\nvnorm\u001b$@$NJ}8~$HJDNN0h$NK!@~%Y%/%?$NFb@Q$,@5!JIi!K$N$H$-$O!\"\u001b(Jpoint\u001b$@$,JDNN0h\u001b(J\n\u001b$@$+$i8+$FK!@~%Y%/%?$N8~$/B&!JH?BPB&!K$K$\"$l$P!\"H>D>@~$HJDNN0h$O8r$o$i$J$$\u001b(J\n\u001b$@$N$G!\"7k2L$O\u001b(Joutside\u001b$@$K$J$k!#5U$K!\"H>D>@~$HJDNN0h$,8r$o$k$H$-$O!\"$=$N8rE@$K\u001b(J\n\u001b$@BP$7$F!\"\u001b(J:insidep\u001b$@$N7k2L\u001b(J(inside, outside, border) \u001b$@$rJV$9!#\u001b(J\n\u001b$@!J?^\u001b(J\\ref{fig:intersect-point}\\ \u001b$@;2>H!K\u001b(J\n% picture = intersect-point.tex\n\\begin{figure}\n\\begin{minipage}[b]{16cm}\n\\begin{picture}( 388, 287)(20,0)\n\\Large\\tt\n\\put( 17.0, 12.0){p= -(ax+by+cz+d)/dnm \u001b$@J,;R$OE@$NJDNN0h$+$i$N5wN%!#\u001b(J\np\u001b$@$,@5$J$i8r$o$k!#\u001b(J}\n\\put( 14.5, 45.7){dnm= au+by+cw dnm\u001b$@$,@5$J$iH>D>@~$HJDNN0h$NK!@~%Y%/%?$O\u001b(J\n\u001b$@F1$88~$-\u001b(J}\n\\put( 119.0, 113.1){ax+by+cz+d=0}\n\\put( 216.0, 198.3){(a,b,c)}\n\\put( 123.3, 232.0){(u,v,w)}\n\\put( 237.9, 253.1){(x,y,z)}\n\\thicklines\n\\put( 223.4, 255.5){\\vector(-3,-2){ 42.2}}\n\\put( 231.9, 263.9){\\line(-1,-1){ 16.9}}\n\\put( 215.0, 263.9){\\line( 1,-1){ 16.9}}\n\\put( 198.1, 145.9){\\vector( 0, 1){ 67.5}}\n\\put( 97.0, 112.1){\\line( 5,-2){ 84.3}}\n\\put( 265.6, 129.0){\\line(-2, 3){ 33.7}}\n\\put( 265.6, 129.0){\\line(-5,-3){ 84.3}}\n\\put( 147.6, 196.5){\\line( 5,-1){ 84.3}}\n\\put( 147.6, 145.9){\\line(-3,-2){ 50.6}}\n\\put( 147.6, 196.5){\\line( 0,-1){ 50.6}}\n\\end{picture}\n\\end{minipage}\n\\label{fig:intersect-point}\n\\end{figure}\n\\item[:intersect-line (p1 p2)]\n\u001b$@@~J,\u001b(Jp1-p2 \u001b$@$H$N8r:9$r5a$a$k!#8r:9$,$J$$$H$-\u001b(Jnil,\u001b$@8r:9$9$k;~!\"8rE@$N%Q%i%a%?\u001b(J\n\u001b$@$H8rE@$N:BI8$N%j%9%H$rJV$9!#\u001b(J\n\\item[:intersect-edge (e)]\n\u001b$@%(%C%8$H$N8rE@!#7k2L$O\u001b(J:intersect-line \u001b$@$HF1$8!#\u001b(J\n\\item[:intersect-face (fac)]\n\u001b$@LL\u001b(Jfac \u001b$@$H$3$NLL$H$N8r:9!#\u001b(Jt \u001b$@$^$?$O\u001b(Jnil \u001b$@!#\u001b(J\n\\item[:transform-normal (trans)]\ntrans \u001b$@$O%/%i%9\u001b(Jcoordinates \u001b$@$N%$%s%9%?%s%9$G$\"$j!\":BI8JQ49$rI=$9!#K!@~%Y%/%?\u001b(J\n\u001b$@$rJQ49$9$k!#\u001b(J\n\\item[:reset-normal ()]\nvertices\u001b$@$+$i\u001b(Jnormal\u001b$@$*$h$S\u001b(Jdistance\u001b$@$r7W;;$9$k!#\u001b(J\n\\item[:invert ()] \u001b$@$3$NLL$NFb!\"30$rH?E>$9$k!#\u001b(J\n\\item[:area ()] \u001b$@$3$NJDNN0h$NLL@Q$r7W;;$9$k!#\u001b(J\n\\item[:volume (point)] point\u001b$@$H$3$NJDNN0h$G:n$i$l$k?mBP$NBN@Q\u001b(J\n\\item[:centroid ()] \u001b$@$3$NJDNN0h$N=E?40LCV\u001b(J\n\\item[:perimeter] \u001b$@<~0OD9\u001b(J\n\\item[:init (\\&key :vertices :edges :normal :distance)]\n\\end{description}\n\n\\end{description}\n\\vfill\n\\pagebreak\n{\\jLarge face}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@B?LLBN$NLL$N%/%i%9!#\u001b(J \n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[closed-region] (convexp edges vertices)\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[holes] \u001b$@LL$K6u$$$?7j!#\u001b(J\n\\item[pbody] \u001b$@$3$NLL$,:n@.$5$l$?\u001b(Jprimitive\\ body\u001b$@$X$N%]%$%s%?!#\u001b(J\n\\item[type] \u001b$@LL$N%?%$%W\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:all-edges]\n\\item[:all-vertices] \u001b$@$3$NLL>e$NA4$F$N%(%C%8!&D:E@!\"A4$F$NFbIt%k!<%W$rJV$9!#\u001b(J\n\\item[:insidep {\\em point}]\n\\item[:area] \u001b$@LL@Q$r5a$a$k!#\u001b(J\n\\item[:centroid {\\em \\&optional reference-point}] \u001b$@LL$N=E?4$r5a$a$k!#\u001b(J\n\u001b$@7j$b9MN8$9$k!#\u001b(J\n\\item[:invert]\n\\item[:enter-hole {\\em hole}] \u001b$@7j$rEPO?$9$k!#\u001b(J\n\\item[:init {\\em \\&key :normal :distance :edges :vertices :holes}]\n\\end{description}\n\n\\end{description}\n\\vfill\n\\pagebreak\n{\\jLarge hole}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@7j$N%/%i%9!#\u001b(J \n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[closed-region] (convexp edges vertices)\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[myface] \u001b$@7j$,B0$9$k\u001b(Jface\u001b$@%*%V%8%'%/%H\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:face]\n\\item[:enter-face {\\em face}]\n\\item[:init {\\em \\&key :normal :distance :edges :vertices :face}]\n\\end{description}\n\n\\end{description}\n\\vfill\n\\pagebreak\n{\\jLarge body}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@B?LLBN$N%/%i%9!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[cascaded-coords] (parent descendants worldcoords manager)\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[faces] \u001b$@LL$N%j%9%H!#\u001b(J\n\\item[edges] \u001b$@%(%C%8$N%j%9%H!#\u001b(J\n\\item[vertices] \u001b$@D:E@$N%j%9%H!#\u001b(J\n\\item[model-vertices] \u001b$@D:E@$N%j%9%H!#\u001b(J:translate \u001b$@Ey$GF0$+$7$F$bITJQ!#\u001b(J\n\\item[box]\n\\item[convexp] \u001b$@FLB?3Q7A$N$H$-\u001b(JT\n\\item[evertedp]\n\\item[csg] body\u001b$@$N7A>u$,JQ99$5$l$?MzNr$r<($9\u001b(JCSG\u001b$@>pJs!#\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf methods}]\\hspace{1cm}\n\\begin{description}\n\\item[:newcoords {\\em rot pos}]\n\\item[:vertices]\n\\item[:faces]\n\\item[:edges]\n\\item[:box]\n\\item[:volume {\\em \\&optional (reference-point \\#f(0 0 0))}] \u001b$@BN@Q7W;;\u001b(J\n\\item[:centroid {\\em \\&optional (reference-point \\#f(0 0 0))}] \u001b$@=E?47W;;\u001b(J\n\\item[:possibly-interfering-faces {\\em box}] \u001b$@43>D$7$F$$$k2DG=@-$N$\"$kLL$r\u001b(J\nbox\u001b$@%F%9%H$GC5$9!#\u001b(J\n\\item[:common-box {\\em body}]\n\u001b$@<+J,<+?H$H\u001b(Jbody\u001b$@$H$N:G>.6&DLH\"$rJV$9!#\u001b(J\n\\item[:insidep {\\em point}]\n\\item[:intersect-face {\\em face}]\n\\item[:intersectp {\\em body}] \u001b$@<+J,<+?H$H\u001b(Jbody \u001b$@$H$N43>D%A%'%C%/$r$9$k!#\u001b(Jt\u001b$@$N$H$-\u001b(J\n\u001b$@43>D$,$\"$k!#\u001b(J\n\\item[:evert] body\u001b$@$NN\"I=$rH?E>$5$;$k!#\u001b(J\n\\item[:init {\\em \\&key :faces :edges :vertices}] :init\u001b$@$K$h$C$F:n@.$5$l$?\u001b(J\nbody\u001b$@$O!\"\u001b(J*bodies*\u001b$@%j%9%H$K\u001b(Jpush\u001b$@$5$l$k!#\u001b(J\n\\item[:translate-vertices {\\em vector}]\n\u001b$@:BI87O$H$OFHN)$K!\"D:E@!J\u001b(Jmodel-vertices\u001b$@!K$N0LCV$r0\\F0$9$k!#\u001b(J\n\\item[:rotate-vertices {\\em radian axis}]\n\u001b$@F1MM$K!\"D:E@$N0LCV$r\u001b(Jaxis \u001b$@<4!J\u001b(J:x,:y or :z\u001b$@!K$^$o$j$K2sE>$9$k!#\u001b(J\n\\item[:magnify {\\em scale}]\n\\item[:csg {\\em \\&optional newcsg}] csg\u001b$@>pJs$rJV$9!#\u001b(Jnewcsg\u001b$@$,$\"$l$P!\"\u001b(Jcsg\u001b$@$NCM\u001b(J\n\u001b$@$r99?7$7$F!\"?7$7$$CM$rJV$9!#\u001b(J\n\\end{description}\n\n\\end{description}\n\n\\clearpage\n\\subsection{\u001b$@AGN)BN$N:n@.\u001b(J}\n\n\\begin{description}\n\\item[make-plane {\\em :normal :point :distance}]\n\\item[make-prism {\\em bottom-vertices sweep \\&key name color}]\\hfill\\\\\n\u001b$@ClBN$N@8@.!#ClBN$H$O!\"\u001b(J\\ $2\\frac{1}{2}$\\ \u001b$@J*BN$N$3$H!#\u001b(J\nbottom-vertices \u001b$@$ODlLL$ND:E@\u001b(J\\ $V_{1},V_{2},\\cdots,V_{n}$\\ \n\u001b$@$N%j%9%H!#\u001b(J\\ $V_{1},V_{2},\\cdots,V_{n}$\\ \u001b$@$O!\"$3$N=g$KDlLL$r30$+$i8+$F\u001b(J\n\u001b$@H?;~7W2s$j$K2s$k!#\u001b(Jsweep \u001b$@$O!\"A]0z%Y%/%?!JNc$($P!\"\u001b(J\\#f(0 0 50)\u001b$@!K$^$?$O\u001b(J\n\u001b$@C1$KClBN$N9b$5!#:BI886E@$O!\"%o!<%k%I$N\u001b(J\\#f(0 0 0)\u001b$@!#\u001b(J\n\n% picture = make-prism.tex\n\\begin{figure}[h]\n\\begin{picture}( 286, 236)\n\\thicklines\n\\put( 21.9, 158.5){\\line( 0,-1){ 84.3}}\n\\put( 21.9, 158.5){\\line( 1, 1){ 50.6}}\n\\put( 72.5, 209.1){\\line( 5, 1){ 84.3}}\n\\put( 156.8, 226.0){\\line( 1,-1){ 50.6}}\n\\put( 21.9, 158.5){\\line( 4,-3){ 67.5}}\n\\put( 89.4, 107.9){\\line( 2, 3){ 33.7}}\n\\put( 123.1, 158.5){\\line( 5, 1){ 84.3}}\n\\put( 89.4, 107.9){\\line( 0,-1){ 84.3}}\n\\put( 123.1, 158.5){\\line( 0,-1){ 84.3}}\n\\put( 207.4, 175.4){\\line( 0,-1){ 84.3}}\n\\thinlines\n\\put( 72.5, 209.1){\\line( 0,-1){ 101.2}}\n\\put( 156.8, 226.0){\\line( 0,-1){ 101.2}}\n\\put( 156.8, 124.8){\\line(-5,-1){ 84.3}}\n\\put( 72.5, 107.9){\\line(-3,-2){ 50.6}}\n\\put( 156.8, 124.8){\\line( 3,-2){ 50.6}}\n\\thicklines\n\\put( 21.9, 74.2){\\line( 4,-3){ 67.5}}\n\\put( 123.1, 74.2){\\line(-2,-3){ 33.7}}\n\\put( 207.4, 91.1){\\line(-5,-1){ 84.3}}\n\\large\\tt\n\\put( 208.4, 75.2){V1}\n\\put( 124.1, 58.3){V2}\n\\put( 90.4, 7.7){V3}\n\\put( 6.1, 58.3){V4}\n\\put( 141.0, 108.9){Vm}\n\\put( 248.9, 127.5){sweep}\n\\put( 231.0, 92.7){\\vector( 0, 1){ 84.3}}\n\\put( 21.9, 158.5){\\line( 0,-1){ 84.3}}\n\\put( 21.9, 158.5){\\line( 1, 1){ 50.6}}\n\\put( 72.5, 209.1){\\line( 5, 1){ 84.3}}\n\\put( 156.8, 226.0){\\line( 1,-1){ 50.6}}\n\\put( 21.9, 158.5){\\line( 4,-3){ 67.5}}\n\\put( 89.4, 107.9){\\line( 2, 3){ 33.7}}\n\\put( 123.1, 158.5){\\line( 5, 1){ 84.3}}\n\\put( 89.4, 107.9){\\line( 0,-1){ 84.3}}\n\\put( 123.1, 158.5){\\line( 0,-1){ 84.3}}\n\\put( 207.4, 175.4){\\line( 0,-1){ 84.3}}\n\\thinlines\n\\put( 72.5, 209.1){\\line( 0,-1){ 101.2}}\n\\put( 156.8, 226.0){\\line( 0,-1){ 101.2}}\n\\put( 156.8, 124.8){\\line(-5,-1){ 84.3}}\n\\put( 72.5, 107.9){\\line(-3,-2){ 50.6}}\n\\put( 156.8, 124.8){\\line( 3,-2){ 50.6}}\n\\thicklines\n\\put( 21.9, 74.2){\\line( 4,-3){ 67.5}}\n\\put( 123.1, 74.2){\\line(-2,-3){ 33.7}}\n\\put( 207.4, 91.1){\\line(-5,-1){ 84.3}}\n\\put( 208.4, 75.2){V1}\n\\put( 124.1, 58.3){V2}\n\\put( 90.4, 7.7){V3}\n\\put( 6.1, 58.3){V4}\n\\put( 141.0, 108.9){Vm}\n\\put( 248.9, 127.5){sweep}\n\\put( 231.0, 92.7){\\vector( 0, 1){ 84.3}}\n\\end{picture}\n\\caption{make-prism}\n\\end{figure}\n\\item[make-cylinder {\\em radius height \\&key segments name color}]\\hfill\\\\\n\u001b$@1_Cl$N@8@.!#\u001b(Jradius \u001b$@$ODlLL$NH>7B!#\u001b(Jheight \u001b$@$O9b$5!#\u001b(Jsegment\u001b$@?t$N\u001b(Jdefault\u001b$@$O!\"\u001b(J12\u001b$@!#\u001b(J\n\u001b$@:BI886E@$O!\"DlLL$NCf?4!#\u001b(J\n\n% picture = make-cylinder.tex\n\\begin{figure}[h]\n\\begin{picture}( 388, 236)\n\\thicklines\n\\put( 185.5, 50.6){\\vector(-1,-1){ 16.9}}\n\\put( 185.5, 50.6){\\vector( 0, 1){ 16.9}}\n\\large\\tt\n\\put( 321.4, 119.0){height}\n\\put( 303.5, 50.6){\\vector( 0, 1){ 134.9}}\n\\put( 237.1, 17.9){radius}\n\\put( 185.5, 50.6){\\vector( 1, 0){ 33.7}}\n\\thinlines\n\\put( 134.9, 67.5){\\line(-1,-1){ 16.9}}\n\\thicklines\n\\put( 118.0, 168.6){\\line( 0,-1){ 134.9}}\n\\thinlines\n\\put( 219.2, 84.3){\\line(-1, 0){ 50.6}}\n\\put( 168.6, 84.3){\\line(-2,-1){ 33.7}}\n\\put( 219.2, 219.2){\\line( 0,-1){ 134.9}}\n\\thicklines\n\\put( 252.9, 185.5){\\line( 0,-1){ 134.9}}\n\\put( 236.1, 33.7){\\line( 0, 1){ 134.9}}\n\\put( 202.4, 16.9){\\line( 0, 1){ 134.9}}\n\\put( 252.9, 50.6){\\line(-1,-1){ 16.9}}\n\\put( 202.4, 16.9){\\line( 2, 1){ 33.7}}\n\\put( 151.8, 16.9){\\line( 1, 0){ 50.6}}\n\\put( 118.0, 33.7){\\line( 2,-1){ 33.7}}\n\\thinlines\n\\put( 134.9, 202.4){\\line( 0,-1){ 134.9}}\n\\thicklines\n\\put( 151.8, 16.9){\\line( 0, 1){ 134.9}}\n\\thinlines\n\\put( 168.6, 219.2){\\line( 0,-1){ 134.9}}\n\\thicklines\n\\put( 134.9, 202.4){\\line(-1,-1){ 16.9}}\n\\put( 168.6, 219.2){\\line(-2,-1){ 33.7}}\n\\put( 219.2, 219.2){\\line(-1, 0){ 50.6}}\n\\put( 252.9, 202.4){\\line(-2, 1){ 33.7}}\n\\put( 252.9, 185.5){\\line( 0, 1){ 16.9}}\n\\put( 252.9, 185.5){\\line(-1,-1){ 16.9}}\n\\put( 202.4, 151.8){\\line( 2, 1){ 33.7}}\n\\put( 202.4, 151.8){\\line(-1, 0){ 50.6}}\n\\put( 151.8, 151.8){\\line(-2, 1){ 33.7}}\n\\put( 118.0, 185.5){\\line( 0,-1){ 16.9}}\n\\thinlines\n\\put( 252.9, 67.5){\\line(-2, 1){ 33.7}}\n\\thicklines\n\\put( 185.5, 50.6){\\line( 3,-1){ 50.6}}\n\\end{picture}\n\\caption{make-cylinder}\n\\end{figure}\n\\item[make-cube {\\em xsize ysize zsize \\&key name color}]\\hfill\\\\\n\u001b$@D>J}BN$N@8@.!#\u001b(Jxsize,ysize,zsize \u001b$@$O$=$l$>$l\u001b(J3\u001b$@JU$ND9$5!#:BI886E@$O!\"\u001b(J\n\u001b$@=E?4$K0lCW!#\u001b(J\n\\begin{figure}[h]\n\\begin{picture}( 304, 152)\n\\large\\tt\n\\put( 253.9, 102.2){zsize}\n\\put( 135.9, 17.9){ysize}\n\\put( 237.1, 51.6){xsize}\n\\put( 168.6, 84.3){\\vector( 1, 0){ 33.7}}\n\\put( 168.6, 84.3){\\vector(-1,-1){ 16.9}}\n\\put( 168.6, 84.3){\\vector( 0, 1){ 33.7}}\n\\put( 134.9, 67.5){\\line( 1, 0){ 101.2}}\n\\put( 134.9, 67.5){\\line(-1,-1){ 33.7}}\n\\put( 134.9, 134.9){\\line( 0,-1){ 67.5}}\n\\thicklines\n\\put( 101.2, 33.7){\\line( 1, 0){ 101.2}}\n\\put( 236.1, 67.5){\\line(-1,-1){ 33.7}}\n\\put( 236.1, 134.9){\\line( 0,-1){ 67.5}}\n\\put( 236.1, 134.9){\\line(-1,-1){ 33.7}}\n\\put( 202.4, 101.2){\\line( 0,-1){ 67.5}}\n\\put( 101.2, 101.2){\\line( 1, 0){ 101.2}}\n\\put( 134.9, 134.9){\\line( 1, 0){ 101.2}}\n\\put( 101.2, 101.2){\\line( 1, 1){ 33.7}}\n\\put( 101.2, 101.2){\\line( 0,-1){ 67.5}}\n\\end{picture}\n\\caption{make-cube}\n\\end{figure}\n\\item[make-cone {\\em top bottom-vertices \\&key name color}]\\hfill\\\\\n\u001b$@B?3Q?m$N@8@.!#\u001b(Jtop \u001b$@$O!\"D:E@$N:BI8!#\u001b(Jbottom-vertices \u001b$@$O!\"\u001b(Jmake-prism \u001b$@$N9`;2>H!#\u001b(J\n\\begin{figure}[h]\n\\begin{picture}( 320, 169)\n\\small\\tt\n\\put( 186.5, 136.2){top}\n\\put( 237.1, 51.9){=bottom}\n\\put( 102.2, 51.9){V4}\n\\put( 135.9, 18.2){V3}\n\\put( 186.5, 18.2){V2}\n\\put( 220.2, 51.9){V1}\n\\put( 185.5, 67.5){\\line( 1,-1){ 16.9}}\n\\put( 151.8, 67.5){\\line( 1, 0){ 33.7}}\n\\put( 151.8, 67.5){\\line(-1,-1){ 16.9}}\n\\put( 168.6, 134.9){\\line( 1,-4){ 16.9}}\n\\put( 168.6, 134.9){\\line(-1,-4){ 16.9}}\n\\put( 151.8, 33.7){\\line(-1, 1){ 16.9}}\n\\put( 185.5, 33.7){\\line(-1, 0){ 33.7}}\n\\put( 202.4, 50.6){\\line(-1,-1){ 16.9}}\n\\put( 168.6, 134.9){\\line( 2,-5){ 33.7}}\n\\put( 168.6, 134.9){\\line( 1,-6){ 16.9}}\n\\put( 168.6, 134.9){\\line(-1,-6){ 16.9}}\n\\put( 134.9, 50.6){\\line( 2, 5){ 33.7}}\n\\end{picture}\n\\caption{make-cone}\n\\end{figure}\n\\clearpage\n\\item[make-solid-of-revolution {\\em points \\&key (segments 16) name key}]\n\\hfill\\\\\n\u001b$@2sE>BN$N@8@.!#\u001b(J\\ $points=(V_{1},V_{2},\\cdots,V_{n})$\\ \u001b$@$r@\\B3$7$?@^@~$r\u001b(J\nz \u001b$@<42s$j$K2sE>$7$?J*BN$r@8@.$9$k!#\u001b(Jpoints\u001b$@$N\u001b(Jy\u001b$@:BI8$O#0$K$7$F$*$/$3$H!#\u001b(J\n% picture = make-revolution\n\\begin{figure}[h]\n\\begin{picture}( 219, 169)\n\\put( 67.5, 33.7){\\line( 1, 0){ 50.6}}\n\\small\\tt\n\\put( 135.9, 35.1){V5}\n\\put( 152.8, 51.9){V4}\n\\put( 152.8, 85.7){V3}\n\\put( 135.9, 119.4){V2}\n\\put( 119.0, 136.2){V1}\n\\put( 67.5, 134.9){\\line( 1, 0){ 33.7}}\n\\thicklines\n\\put( 134.9, 50.6){\\line(-1,-1){ 16.9}}\n\\put( 134.9, 84.3){\\line( 0,-1){ 33.7}}\n\\put( 118.0, 101.2){\\line( 1,-1){ 16.9}}\n\\put( 118.0, 118.0){\\line( 0,-1){ 16.9}}\n\\put( 101.2, 134.9){\\line( 1,-1){ 16.9}}\n\\put( 16.9, 67.5){\\vector( 1, 0){ 185.5}}\n\\put( 67.5, 16.9){\\vector( 0, 1){ 134.9}}\n\\end{picture}\n\\caption{make-solid-of-revolution}\n\\end{figure}\n\\item[convex-hull-3d {\\em vertices}]\\hfill\\\\\n\u001b$@D:E@Ns$+$iFLJq$r:n@.$9$k!#\u001b(J\n\\item[make-torus {\\em points \\&key (segments 16) name color}]\n\\item[make-icosahedron] {\\em \\&optional radius}]\n\\item[make-dodecahedraon {\\em \\&optional radius}]\n\\item[make-gdome]\n\\end{description}\n\\subsection{\u001b$@J*BN4V$N=89g1i;;\u001b(J}\n\n\u001b$@=89g1i;;$K$O!\"OB\u001b(J(union)\u001b$@!\"@Q\u001b(J(intersection)\u001b$@!\":9\u001b(J(difference)\u001b$@!\"$*$h$S\u001b(J\n\u001b$@J?LL$K$h$kJ,N%\u001b(J(cut)\u001b$@$,$\"$k!#$3$l$i$r$C$F!\"9g@.$,ESCf$G<:GT$7$?$H$7$F$b\u001b(J\n\u001b$@85$NJ*BN$O2?$N1F6A$bo$KJ#;($J\u001b(J\n\u001b$@J*BN$HC1=c$JJ*BN$NOB$r$H$k$h$&$J>l9g$K$O!\"A4BN$NJ#@=$,:n$jD>$5$l$k$3$H\u001b(J\n\u001b$@$+$i=hM}$K;~4V$,$+$+$k!#\u001b(J\n\n\u001b$@8=:_$N%P!<%8%g%s$G$O!\"=89g1i;;$r;\\$9\u001b(J2\u001b$@J*BN$N$$$E$l$+$NLL$I$&$7$,F10l\u001b(J\n\u001b$@J?LL>e$K:\\$C$F$$$k!\"$\"$k$$$O!\"Hs>o$K6a$$MM$J>l9g$K$O!\"=89g1i;;$,=PMh$J$$!#\u001b(J\n\\clearpage\n\\subsection{\u001b$@J*BN4V$N@\\?($K$h$k94B+$NF3=P\u001b(J}\n\n\\subsubsection{\u001b$@%\"%k%4%j%:%`$N35MW\u001b(J}\n\u001b$@J*BN$,<+M36u4V$K$\"$k$H$9$k$H!\"$=$NJ*BN$K$OG$0U$NHy>.JQ0L$,2DG=$G$\"$k$,!\"\u001b(J\n\u001b$@B>$NJ*BN$KBP$7$FNO$rH/@8$9$k$3$H$O=PMh$J$$!#\u001b(J\n\u001b$@5U$K!\"J*BN$,==J,=E$$J*BN$K8GDj$5$l$F$$$k>l9g$O!\"$=$NJ*BN$OJQ0L$9$k$3$H$O\u001b(J\n\u001b$@=PMh$J$$$,!\"8GDj$7$F$$$kJ*BN$KBP$7$FG$0U$NNO$rH/@8$9$k$3$H$,2DG=$G$\"$k!#\u001b(J\n\u001b$@$3$NN>6KC<$N4V$N>uBV$H$7$F!\"N>B&94B+$5$l$?J*BN$N>l9g$r9M$($k$H!\"\u001b(J\n\u001b$@94B+>r7o$OHy>.JQ0L$HNO$K4X$9$kF1.JQ0L$N=89g$HH/@82DG=$JNO$N=89g$OD>8rJd6u4V$K$J$k!#\u001b(J\n\u001b$@$^$?!\"J*BN$,JRB&94B+$5$l$F$$$k>l9g$O!\"J*BN$KBP$9$k94B+>r7o$OF1.JQ0L$HH/@82DG=$JNO$N=89g$OAPBPFLB?LL?m$K$J$k!#\u001b(J\n\n\u001b$@K\\@a$G$O!\"$3$l$iJRB&94B+$dN>B&94B+$r6hJL$9$k$3$H$J$/!\"\u001b(J\n\u001b$@G$0U$N7A>u$NB?LLBN$,B>$NB?LLBN$KG$0U$N>uBV$G@\\?($7$F$$$k>l9g$K!\"\u001b(J\n\u001b$@$=$NEy2A94B+E@$N0LCV$H!\"3F!9$NE@$K$*$1$k94B+>r7o$r5a$a$k%W%m%0%i%`\u001b(J\n\u001b$@$K$D$$$F@bL@$9$k!#\u001b(J\n\u001b$@%H%C%W%l%Y%k$N$_;H$&IaDL$N%f!<%6!<$O!\"\u001b(Jc-body\u001b$@$N@bL@$N$_FI$a$P==J,$G$\"$k!#\u001b(J\n\u001b$@$3$l$i$N%W%m%0%i%`$r;H$&$?$a$K$O!\"\u001b(J\n$eus/llib/model2const.l$\\ \u001b$@$r%m!<%I$9$kI,MW$,$\"$k!#\u001b(J\n\u001b$@%W%m%0%i%`$NA4BN9=B$$r\u001b(JFig.\\ref{fig:constraint-program}\u001b$@$K<($9!#\u001b(J\n\n\\begin{figure}[h]\n\\begin{center}\n\\epsfile{file=fig-program.ps,width=8cm}\n\\end{center}\n\\caption{Structure of the Algorithm}\n\\label{fig:constraint-program}\n\\end{figure}\n\\vfill\n\\clearpage\n\\subsubsection{\u001b$@%$%s%W%j%a%s%F!<%7%g%s\u001b(J}\n\\vspace{1.0cm}\n{\\jLarge c-body}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@94B+>r7o$r3JG<$9$k%9%m%C%H$rDI2C$7$?\u001b(Jbody\u001b$@$N%5%V%/%i%9\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[body] ()\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[constraint] \u001b$@Ey2A94B+E@$H94B+>r7o\u001b(J\n\\end{description}\n\n\\item {\\jlarge \\bf methods}\n\\\\ [0.5cm]\n{\\bf :constraint (b)} self\u001b$@$H\u001b(Jb\u001b$@$NEy2A94B+E@$H94B+%Y%/%H%k$r5a$a$k\u001b(J \n\\begin{enumerate}\n\\item {\\bf if}\\ self\u001b$@$HB?LLBN\u001b(Jb\u001b$@$N3F!9$N6K>.H\"$,43>D$7$J$$\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item self\u001b$@$OB?LLBN\u001b(Jb\u001b$@$H@\\?($7$F$$$J$$\u001b(J\n\\item \u001b$@%\"%k%4%j%:%`=*N;\u001b(J\n\\end{description}\n\\item {\\bf if} b\u001b$@$+$ir7o$rLp0u$G2hLL$KI=<($9$k!#\u001b(J\nb\u001b$@$r;XDj$7$J$$$H!\"$3$l$^$G$K5a$a$?94B+$rA4$FI=<($9$k!#\u001b(J\n\u001b$@Lp0u$ND9$5$r\u001b(Jarrow-length\u001b$@$G;XDj$9$k$3$H$b=PMh$k!#\u001b(J\n\\vfill\n\\clearpage\n{\\bf :contact-vertices (b)} b\u001b$@$K@\\?($7$F$$$k\u001b(Jself\u001b$@$ND:E@!\"\u001b(J\n\u001b$@BP1~$9$k\u001b(Jb\u001b$@$N@\\?(LL!\"FL=89g$NOB=89g$GI=8=$7$?\u001b(Jself\u001b$@$N6aK5$r5a$a$k\u001b(J\n\\begin{enumerate}\n\\item cbox\\ $\\leftarrow$\\ self\u001b$@$HB?LLBN\u001b(Jb\u001b$@$N3F!9$N6K>.H\"$N6&DL=89g\u001b(J\n\\item {\\bf if}\\ cbox\u001b$@$,6u=89g\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item self\u001b$@$OB?LLBN\u001b(Jb\u001b$@$H@\\?($7$F$$$J$$\u001b(J\n\\item \u001b$@%\"%k%4%j%:%`=*N;\u001b(J\n\\end{description}\n\\item myvertices\\ $\\leftarrow$\\ cbox\u001b$@$K4^$^$l$k\u001b(Jself\u001b$@$ND:E@$N%j%9%H\u001b(J\n\\item hisfaces\\ $\\leftarrow$\\ cbox\u001b$@$K4^$^$l$k\u001b(Jb\u001b$@$NLL$N%j%9%H\u001b(J\n\\item {\\bf for\\ each}\\ v\\ $\\leftarrow$\\ myvertices\u001b$@$NA4$F$NMWAG\u001b(J\n\\begin{description}\n\\item {\\bf if}\\ v\u001b$@$,!\"\u001b(Jhisfaces\u001b$@$N$I$l$+$K@\\?($7$F$$$k\u001b(J\\ {\\bf then}\nconp\\ $\\leftarrow$\\ conp\\ $+$\\ (self\u001b$@$N@\\?(D:E@\u001b(Jv\\ \\ b\u001b$@$N@\\?(LL$N0l$D\u001b(J)\n\\end{description}\n\\item {\\bf for\\ each}\\ p\\ $\\leftarrow$\\ conp\u001b$@$NA4$F$NMWAG\u001b(J\n\\begin{description}\n\\item p\\ $\\leftarrow$\\ (self\u001b$@$N@\\?(D:E@\u001b(J\\ \\ b\u001b$@$N@\\?(LL$N0l$D\u001b(J\n\\ \\ self\u001b$@$N@\\?(D:E@$rCl9g$K!\"$=$l$rFL=89g$NOB=89g$K\u001b(J\n\u001b$@JQ49$7$F$$$k!#\u001b(J\n\\end{itemize}\n{\\bf :contact-edges (b)} b\u001b$@$K@\\?($7$F$$$k\u001b(Jself\u001b$@$NNG$NFbE@!\"\u001b(J\n\u001b$@BP1~$9$k\u001b(Jb\u001b$@$N@\\?(LL!\"FL=89g$NOB=89g$GI=8=$7$?\u001b(Jself\u001b$@$N6aK5$r5a$a$k\u001b(J\n\\begin{enumerate}\n\\item cbox\\ $\\leftarrow$\\ self\u001b$@$HB?LLBN\u001b(Jb\u001b$@$N3F!9$N6K>.H\"$N6&DL=89g\u001b(J\n\\item {\\bf if}\\ cbox\u001b$@$,6u=89g\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item self\u001b$@$OB?LLBN\u001b(Jb\u001b$@$H@\\?($7$F$$$J$$\u001b(J\n\\item \u001b$@%\"%k%4%j%:%`=*N;\u001b(J\n\\end{description}\n\\item myedges\\ $\\leftarrow$\\ cbox\u001b$@$K4^$^$l$k\u001b(Jself\u001b$@$NNG$N%j%9%H\u001b(J\n\\item hisfaces\\ $\\leftarrow$\\ cbox\u001b$@$K4^$^$l$k\u001b(Jb\u001b$@$NLL$N%j%9%H\u001b(J\n\\item {\\bf for\\ each}\\ e\\ $\\leftarrow$\\ myedges\u001b$@$NA4$F$NMWAG\u001b(J\n\\begin{description}\n\\item {\\bf for\\ each}\\ f\\ hisfaces\u001b$@$NA4$F$NMWAG\u001b(J\n\\begin{description}\n\\item {\\bf if} e\u001b$@$,J?9T$G$J$$LL\u001b(Jf\u001b$@$N$I$l$+$NNG$H@\\?($7$F$$$k!\"$+$D!\"\u001b(J\n\u001b$@@\\?(E@$,\u001b(Jconp\u001b$@$NMWAG$G$O$J$$\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item {\\bf if}\\ e\u001b$@$N\u001b(J2\u001b$@LL3Q$NJ?LL3Q$,\u001b(J\\ $\\pi$\\ \u001b$@$h$j>.$5$$\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item conp\\ $\\leftarrow$\\ conp\\ $+$\\ \n(\u001b$@@\\?($7$F$$$k\u001b(Jself\u001b$@$NNG>e$NE@\u001b(J\\ \\ b\u001b$@$N@\\?(LL$N0l$D\u001b(J\\ \\ ((\u001b$@@\\?($7$F$$$k\u001b(Jself\u001b$@$NNG\u001b(J)))\n\\end{description}\n\\item {\\bf else}\n\\begin{description}\n\\item conp\\ $\\leftarrow$\\ conp\\ $+$\\ \n(\u001b$@@\\?($7$F$$$k\u001b(Jself\u001b$@$NNG>e$NE@\u001b(J\\ \\ b\u001b$@$N@\\?(LL$N0l$D\u001b(J\n\\ \\ ((\u001b$@@\\?($7$F$$$k\u001b(Jself\u001b$@$NNG$N\u001b(Jp\u001b$@LL\u001b(J)\\ (\u001b$@@\\?($7$F$$$k\u001b(Jself\u001b$@$NNG$N\u001b(Jn\u001b$@LL\u001b(J)))\n\\end{description}\n\\end{description}\n\\end{description}\n\\end{description}\n\\item return\\ conp\n\\end{enumerate}\n\n\\end{description}\n\\vfill\n\\clearpage\n{\\Large surrounding-box}\n\\\\ [0.5cm]\n{\\jlarge \\bf methods}\n\\begin{description}\n\\item[{\\bf :contact(box)}]\nself\u001b$@$H\u001b(Jbox\u001b$@$,@\\?($7$F$$$k$+$I$&$+$r%A%'%C%/$9$k!#@\\?($7$F$$$k>l9g$O!\"\u001b(J\n\u001b$@N>l9g$O\u001b(Jnil\u001b$@$rJV$9!#\u001b(J\n\\begin{enumerate}\n\\item clearance\\ $\\leftarrow$\\ \u001b$@Hy>.%Y%/%H%k\u001b(J\n\\item v1\\ $\\leftarrow$\\ (max\\ self\u001b$@$N:G>.E@\u001b(J\\ box\u001b$@$N:G>.E@\u001b(J)\\ $-\\ clearance$\n\\item v2\\ \\ $\\leftarrow$\\ (min\\ self\u001b$@$N:GBgE@\u001b(J\\ box\u001b$@$N:G>.E@\u001b(J)\\ $+\\ clearance$\n\\item {\\bf if}\\ v1\u001b$@$,\u001b(Jv2\u001b$@$h$j>.$5$$\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item return\\ v1\u001b$@$r:G>.E@!\"\u001b(Jv2\u001b$@$r:GBgE@$H$9$kH\"\u001b(J\n\\item {\\bf else}\\ return\\ nil\n\\end{description}\n\\end{enumerate}\n\\end{description}\n\\vfill\n\\clearpage\n{\\Large edge}\n\\\\ [0.5cm]\n{\\jlarge \\bf methods}\n\\\\ [0.5cm]\n{\\bf :boxcontact(box)}self\u001b$@$,\u001b(Jbox\u001b$@$H@\\?($7$F$$$k$+$I$&$+$r%A%'%C%/$9$k!#\u001b(J\n\u001b$@@\\?($7$F$$$k$H$-$ON>.H\"\u001b(J)\n\\end{enumerate}\n{\\bf :neighborpoints(point)}self\u001b$@$N6aK5$rI=$9e$NE@$G\u001b(Jpoint\u001b$@$H$O0[$J$kE@!\"\u001b(Jself\u001b$@$N\u001b(Jpface\u001b$@>e$NE@!\"\u001b(Jself\u001b$@$N\u001b(Jnface\u001b$@>e$NE@!#\u001b(J\n\\begin{enumerate}\n\\item p1\\ $\\leftarrow$\\ (send\\ self\\ {\\bf :anothervertex}\\ point)\n\\item p2\\ \\ $\\leftarrow$\\ $point\\ +\\ pface$\u001b$@$NK!@~%Y%/%H%k\u001b(J\\ $\\times\\ \n(pvert\\ -\\ nvert)$\n\\item p3\\ \\ $\\leftarrow$\\ $point\\ +\\ nface$\u001b$@$NK!@~%Y%/%H%k\u001b(J\\ $\\times\\ \n(nvert\\ -\\ pvert)$\n\\item return\\ (p1\\ p2\\ p3)\n\\end{enumerate}\n{\\bf :contact(e)}self\u001b$@$NFbE@$HNG\u001b(Je\u001b$@$H$N@\\?(E@$r5a$a$k!#\u001b(J\n\u001b$@@\\?($7$F$$$J$$>l9g$O!\"\u001b(Jnil\u001b$@$rJV$9!#\u001b(J\n\\begin{enumerate}\n\\item {\\bf if}\\ self\u001b$@$H\u001b(Je\u001b$@$,3F!9:\\$k#2D>@~$,J?9T\u001b(J\\ {\\bf then}\\ return\\ nil\n\\begin{description}\n\\item p1\\ $\\leftarrow$\\ self\u001b$@$H\u001b(Je\u001b$@$,3F!9:\\$k#2D>@~$N6&DL?b@~$N!\"\u001b(Jself\u001b$@B&$NB-\u001b(J\n\\item p2\\ $\\leftarrow$\\ self\u001b$@$H\u001b(Je\u001b$@$,3F!9:\\$k#2D>@~$N6&DL?b@~$N!\"\u001b(Je\u001b$@B&$NB-\u001b(J\n\\item {\\bf if}\\ p1\u001b$@$H\u001b(Jp2\u001b$@$N5wN%$,Hy>.CM$h$j>.$5$$!\"\u001b(J\np1\u001b$@$,\u001b(Jself\u001b$@$NN>C.5wN%0J>eFbB&!\"$+$D!\"\u001b(J\\\\\np2\u001b$@$,\u001b(Je\u001b$@$NN>C.5wN%0J>e30B&$G$O$J$$\u001b(J\n\\ {\\bf then}\\ return\\ p1\n\\end{description}\n\\end{enumerate}\n\\begin{itemize}\n\\item self\u001b$@$H\u001b(Je\u001b$@$,=E$J$k>l9g$O!\"\u001b(Jself\u001b$@$OI,$:B>$NNG$H8r:9$9$k$N$G!\"\u001b(J\n\u001b$@@\\?(E@$r=P$9I,MW$O$J$$!#\u001b(J\n\\item p1\u001b$@$H\u001b(Jself\u001b$@$NN>C.5wN%0J2<$N>l9g$O!\"\u001b(J\nself\u001b$@$ND:E@$HNG\u001b(Je\u001b$@$N@\\?($KAjEv$9$k$N$G!\"@\\?(E@$r=P$9I,MW$O$J$$!#\u001b(J\n\\item p2\u001b$@$,NG\u001b(Je\u001b$@$NHy>.5wN%L$K~30B&$N>l9g$r4^$a$J$$$H!\"\u001b(J\n\u001b$@?tCM7W;;8m:9$G@\\?(E@$,5a$^$i$J$$$3$H$,$\"$k!#\u001b(J\n\\end{itemize}\n{\\bf :anothervertex(point)}point\u001b$@$HF10lE@$G$O$J$$\u001b(Jself\u001b$@$rC.5wN%$h$jBg$-$$\u001b(J\\ {\\bf then}\\ \n\\begin{description}\n\\item return\\ pvert\n\\item {\\bf else}\\ nvert\n\\end{description}\n\\end{enumerate}\n\\vfill\n\\clearpage\n{\\Large plane}\\\\\n\\\\ [0.5cm]\n{\\jlarge \\bf methods}\n\\\\ [0.5cm]\n{\\bf :separation(mypoints\\ hispoints)}\n\u001b$@LL\u001b(Jself\u001b$@$,\u001b(Jmypoints\u001b$@$H\u001b(Jhispoints\u001b$@$NJ,N%LL$K$J$k$+$I$&$+$r%A%'%C%/$9$k!#\u001b(J\n\u001b$@J,N%LL$K$J$k>l9g$O!\"\u001b(Jmypoints\u001b$@B&$K8~$/MM$KId9g$r9g$o$;$?!\"\u001b(Jself\u001b$@$N\u001b(J\n\u001b$@K!@~%Y%/%H%k$rJV$9!#J,N%LL$K$J$i$J$$>l9g$O!\"\u001b(Jnil\u001b$@$rJV$9\u001b(J\n\\begin{enumerate}\n\\item {\\bf if}\\ mypoints\u001b$@$NA4$F$NE@$N\u001b(Jself\u001b$@$+$i$N5wN%$,F1Id9g\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item sign\\ $\\leftarrow$\\ \u001b$@5wN%$NId9g\u001b(J\n\\item {\\bf if}\\ hispoints\u001b$@$NA4$F$NE@$N\u001b(Jself\u001b$@$+$i$N5wN%$,F1Id9g\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item return\\ sign\\ $\\times$\\ self\u001b$@$NK!@~%Y%/%H%k\u001b(J\n\\end{description}\n\\end{description}\n\\item return\\ nil\n\\end{enumerate}\n\\vspace{2.0cm}\n{\\Large closed-region}\\\\\n\\\\ [0.5cm]\n{\\jlarge \\bf methods}\n\\\\ [0.5cm]\n{\\bf :boxcontact(box)}self\u001b$@$,\u001b(Jbox\u001b$@$H@\\?($7$F$$$k$+$I$&$+$r%A%'%C%/$9$k!#\u001b(J\n\u001b$@@\\?($7$F$$$k$H$-$ON>.H\"\u001b(J)\n\\end{enumerate}\n{\\bf :contactp(p)}\u001b$@E@\u001b(Jp\u001b$@$,\u001b(Jself\u001b$@$K@\\?($7$F$$$k$+$I$&$+$r%A%'%C%/$9$k!#\u001b(J\n\u001b$@@\\?($7$F$$$k$H$-$O\u001b(Jinside\u001b$@!\"6-3&>e$N$H$-$O\u001b(Jborder\u001b$@!\"@\\?($7$F$$$J$$$H$-$O\u001b(J\noutside\u001b$@$rJV$9!#\u001b(J\n\\begin{enumerate}\n\\item d\\ $\\leftarrow$\\ p\u001b$@$H\u001b(Jself\u001b$@$H$N5wN%\u001b(J\n\\item {\\bf if}\\ d\u001b$@$,Hy>.CM$h$jBg$-$$\u001b(J\\ {\\bf then}\\ return\\ outside\n\\item {\\bf else}\n\\begin{description}\n\\item point\\ $\\leftarrow$\\ p\u001b$@$+$i\u001b(Jself\u001b$@$X9_$m$7$??b@~$NB-\u001b(J\n\\item {\\bf for each} e $\\leftarrow$ self\u001b$@$N\u001b(Jedges\n\\begin{description}\n\\item {\\bf if} (send e {\\bf :distance} point)\u001b$@$,Hy>.CM$h$j>.$5$$\u001b(J\n{\\bf then} return border\n\\end{description}\n\\item return\\ (send\\ self\\ {\\bf :insidep}\\ ip)\n\\end{description}\n\\end{enumerate}\n\\vfill\n\\clearpage\n{\\Large face}\\\\\n\\\\ [0.5cm]\n{\\jlarge \\bf methods}\n\\\\ [0.5cm]\n{\\bf :contactp(p)}\u001b$@E@\u001b(Jp\u001b$@$,\u001b(Jself\u001b$@$K@\\?($7$F$$$k$+$I$&$+$r%A%'%C%/$9$k!#\u001b(J\n\u001b$@@\\?($7$F$$$k$H$-$O\u001b(Jinside\u001b$@!\"6-3&>e$N$H$-$O\u001b(Jborder\u001b$@!\"@\\?($7$F$$$J$$$H$-$O\u001b(J\noutside\u001b$@$rJV$9!#\u001b(J\n\\begin{enumerate}\n\\item result $\\leftarrow$ (send-super {\\bf :contactp} p)\n\\item {\\bf if} result\u001b$@$,\u001b(Jinside\u001b$@$G$O$J$$\u001b(J {\\bf then} return result\n\\item {\\bf for each} h $\\leftarrow$ self\u001b$@$N\u001b(Jholes\n\\begin{description}\n\\item result $\\leftarrow$ (send h {\\bf :contactp} p)\n\\item {\\bf if} result\u001b$@$,\u001b(Jinside {\\bf then} return outside\n\\item {\\bf if} result\u001b$@$,\u001b(Jborder {\\bf then} return border\n\\item return inside\n\\end{description}\n\\end{enumerate} \n{\\bf :contact-edge(e1)}\u001b$@NG\u001b(Je1\u001b$@$H\u001b(Jself\u001b$@$,J?9T$G$J$$$H$-!\"\u001b(J\ne1\u001b$@$H\u001b(Jself\u001b$@$N$I$l$+$NNG$H$N@\\?(E@$rJV$9\u001b(J\n\\begin{enumerate}\n\\item {\\bf if}\\ e1\u001b$@$H\u001b(Jself\u001b$@$,J?9T\u001b(J\\ {\\bf then}\\ return\\ nil\n\\item {\\bf for\\ each}\\ e2\\ $\\leftarrow$\\ self\u001b$@$N\u001b(Jedges\n\\begin{description}\n\\item foot\\ $\\leftarrow$\\ (send\\ e1\\ {\\bf :contact}\\ e2)\n\\item {\\bf if}\\ foot\u001b$@$,\u001b(Jnil\u001b$@$G$J$$\u001b(J\\ {\\bf then}\\ return\\ foot \n\\end{description}\n\\end{enumerate}\n\\vfill\n\\clearpage\n{\\jLarge constrained-point}\n\\\\ [0.5cm]\n\\begin{description}\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@Ho94B+E@$N%/%i%9\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[object] ()\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[myvertex] \u001b$@Ho94B+E@$N:BI8\u001b(J\n\\item[hisface] \u001b$@Ho94B+E@$K$*$1$k94B+J*BN$NLL$N0l$D\u001b(J\n\\item[myneighborhood] \u001b$@Ho94B+E@$K$*$1$k<+J,$N6aK5\u001b(J\n\\end{description}\n\n\\end{description}\n{\\jlarge \\bf methods}\\\\\n\\begin{description}\n\\item[:to-convex\\ ()] self\u001b$@$N6aK5$rFL6aK5$NOB$KJ,2r$9$k\u001b(J \n\\begin{enumerate}\n\\item count\\ $\\leftarrow$\\ self\u001b$@$N6aK5$N\u001b(Jor-edge\u001b$@$N?t\u001b(J\n\\item {\\bf if}\\ count=0\\ {\\bf then}\n\\begin{description}\n\\item self\u001b$@$N6aK5\u001b(J\\ $\\leftarrow$\\ \n(vertex\\ hisface\\ ((myedge11\\ and\\ myedge12\\ $\\cdots$)))\n\\end{description}\n{\\bf else}\n\\begin{description}\n\\item {\\bf if}\\ count=1\\ {\\bf then}\n\\begin{description}\n\\item (send\\ self\\ {\\bf :divide1}\\ or-edge)\n\\end{description}\n\\item {\\bf else}\n\\begin{description}\n\\item (send\\ self\\ {\\bf :dividen}\\ or-edge)\n\\end{description}\n\\end{description}\n\\end{enumerate}\n\\item[:divide1\\ (or-edge)] or-edge\u001b$@$,0l$D$@$1$\"$k\u001b(Jself\u001b$@$N6aK5$r\u001b(Jor-edge\u001b$@$N\u001b(J\np-face\u001b$@$H\u001b(Jn-face\u001b$@$N3F!9$G@ZCG$7$?FL6aK5$r5a$a!\"$=$l$i$N%j%9%H$G\u001b(J\n{\\bf myneighborhood}\u001b$@%9%m%C%H$r/$7$O$\"$k\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item self\u001b$@$N\u001b(Jmyvertex\u001b$@$H\u001b(Jip\u001b$@$rN>Ce$\"$k6aK5$rFL6aK5$KJ,3d$7!\"\u001b(J\n\u001b$@$=$l$i$N%j%9%H$G\u001b(J{\\bf myneighborhood}\u001b$@%9%m%C%H$rl9g\u001b(J\n\\begin{description}\n\\item {\\bf if}\\ eset\u001b$@$KNG$,4^$^$l$k\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item {\\bf if}\\ flist\u001b$@$NMWAG?t$,\u001b(J2\u001b$@$h$jB?$$\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item n-point\\ $\\leftarrow$\\ flist\u001b$@$N@hF,$H:G8e$NMWAG$NLL$N8r@~>e\u001b(J\n\u001b$@$K$\"$j!\"6aK5$N6-3&LL>e$K$\"$kE@\u001b(J\n\\item eset\\ $\\leftarrow$\\ eset\\ $+$\\ \u001b$@@\\?(E@$H\u001b(Jn-point\u001b$@$rCr7o$r5a$a$k\u001b(J\n\\begin{enumerate}\n\\item {\\bf for\\ each}\\ $(mine,his)$\\ $\\leftarrow$\\ self\u001b$@$N9=@.MWAG$NFL6aK5$H\u001b(J\nhispoint\u001b$@$N9=@.MWAG$NFL6aK5$NA4$F$NAH9g$;\u001b(J\n\\begin{description}\n\\item {\\bf if}\\ mine\u001b$@$,LL\u001b(J\\ {\\bf then}\\ and-cond\\ $\\leftarrow$\\ \nand-cond\\ $+$\\ {\\bf face-contact}(mine\\ -1.0)\n\\end{description}\n\\begin{description}\n\\item {\\bf if}\\ his\u001b$@$,LL\u001b(J\\ {\\bf then}\\ and-cond\\ $\\leftarrow$\\ \nand-cond\\ $+$\\ {\\bf face-contact}(his\\ 1.0)\n\\end{description}\n\\begin{description}\n\\item {\\bf if}\\ \u001b$@NG$HNG$N@\\?(\u001b(J\\ {\\bf then}\\ and-cond\\ $\\leftarrow$\\ \nand-cond\\ $+$\\ {\\bf edge-edge-contact}(\u001b$@@\\?(E@\u001b(J\\ mine\\ his)\n\\item {\\bf if}\\ \u001b$@NG$HD:E@$N@\\?(\u001b(J\\ {\\bf then}\\ and-cond\\ $\\leftarrow$\\ \nand-cond\\ $+$\\ {\\bf edge-vertex-contact}(\u001b$@@\\?(E@\u001b(J\\ mine\\ his)\n\\item {\\bf if}\\ \u001b$@D:E@$HNG$N@\\?(\u001b(J\\ {\\bf then}\\ and-cond\\ $\\leftarrow$\\ \nand-cond\\ $+$\\ {\\bf edge-vertex-contact}(\u001b$@@\\?(E@\u001b(J\\ his\\ mine)\u001b$@$N$=$l$>$l$N\u001b(J\n\u001b$@MWAG$NId9g$rH?E>$7$?%j%9%H\u001b(J\n\\item {\\bf if}\\ \u001b$@D:E@$HD:E@$N@\\?(\u001b(J\\ {\\bf then}\\ and-cond\\ $\\leftarrow$\\ \nand-cond\\ $+$\\ {\\bf vertex-vertex-contact}(\u001b$@@\\?(E@\u001b(J\\ mine\\ his)\n\\end{description}\n\\item return\\ (\u001b$@@\\?(E@\u001b(J\\ and-cond)\n\\end{enumerate}\n\\begin{itemize}\n\\item \u001b$@%9%F%C%W#3$G\u001b(Jand-cond\u001b$@$KDI2C$9$k$N$O!\"DI2C$5$l$kMWAG$,\u001b(Jand-cond\u001b$@$K\u001b(J\n\u001b$@4^$^$l$F$$$J$$$H$-$N$_!#\u001b(J\n\\end{itemize}\n\\end{description}\n\\vspace{2.0cm}\n{\\bf :draw-constraint} {\\em \\&optional (length 30.0)}\\\\\n\u001b$@$3$NE@$K$*$1$k94B+$rLp0u$G2hLL$KI=<($9$k!#\u001b(J\n\u001b$@Lp0u$ND9$5$r\u001b(Jlength\u001b$@$G;XDj$9$k$3$H$b=PMh$k!#\u001b(J\n\\vfill\n\\clearpage\n{\\large \\bf \u001b$@4X?t\u001b(J}\n\\vspace{1.0cm}\\\\\n{\\bf contact-to-constraint(mycontact\\ hiscontact)}\\\\\n\u001b$@Aje$NE@$N%j%9%H\u001b(Jmycontact\u001b$@$H\u001b(J\n\u001b$@<+J,$K@\\?($7$F$$$kAje$NE@$N%j%9%H\u001b(Jhiscontact\u001b$@$+$i!\"\u001b(J\n\u001b$@<+J,$,r7o$r5a$a$k4X?t\u001b(J\\\\\n{\\bf return}\\ =\\ {\\bf ((\u001b$@@\\?(E@\u001b(J\\ \\ (($f_{11}\\ or\\ f_{12}\\ \\cdots$)\\ and\\ \n($f_{21}\\ or\\ f_{22}\\ \\cdots$)\\ $\\cdots$))\\ $\\cdots$)}\n\\begin{enumerate}\n\\item {\\bf while}\\ mycontact\u001b$@$,6u%j%9%H$G$J$$\u001b(J\\ {\\bf do}\n\\begin{description}\n\\item point\\ $\\leftarrow$\\ mycontact\u001b$@$NMWAG$N0l$D\u001b(J\n\\item {\\bf if}\\ point\u001b$@$HF1$8E@$,\u001b(Jhiscontact\u001b$@$NMWAG$K$\"$k\u001b(J\\ {\\bf then}\n\\begin{description}\n\\item anotherpoint\\ $\\leftarrow$\\ \u001b$@BP1~$9$kE@$H$=$N6aK5\u001b(J\n\\item constraints\\ $\\leftarrow$\\ constraints\\ $+$\\ (send\\ point\\ \n{\\bf :analyse-contact}\\ anotherpoint)\n\\item hiscontact\\ $\\leftarrow$\\ hiscontact\\ $-$\\ another-point\n\\end{description}\n\\item {\\bf else}\n\\begin{description}\n\\item constraints\\ $\\leftarrow$\\ constraints\\ $+$\\ \n(point\\ ((\u001b$@<+J,$ND:E@$,@\\?($7$F$$$kAjr7o$r5a$a$k!#\u001b(Jface\u001b$@$,<+?H$NLL$N$H$-$O\u001b(Jsign\u001b$@$O\u001b(J\\ $-1.0$\\ \u001b$@!\"\u001b(J\n\u001b$@Ajr7o$r5a$a$k\u001b(J\n\\begin{enumerate}\n\\item mypoints\\ $\\leftarrow$\\ (send\\ myedge\\ {\\bf :neighborpoints}\\ point)\n\\item hispoints\\ $\\leftarrow$\\ (send\\ hisedge\\ {\\bf :neighborpoints}\\ point)\n\\item {\\bf if} myedge\u001b$@$H\u001b(Jhisedge\u001b$@$,J?9T\u001b(J {\\bf then} return nil\n\\item s-plane\\ $\\leftarrow$\\ myedge\u001b$@$H\u001b(Jhisedge\u001b$@$r4^$`LL\u001b(J\n\\item return\\ (\\ (send\\ s-plane\\ {\\bf :separation}\\ mypoints\\ hispoints)\\ )\n\\end{enumerate}\n\\vfill\n\\clearpage\n{\\bf edge-vertex-contact\\ (point\\ myedge\\ hisvertex)}\n\u001b$@@\\?(E@\u001b(Jpoint\u001b$@$K$*$1$k<+J,$N6aK5$rI=$9NG\u001b(Jmyedge\u001b$@$HAjr7o$r5a$a$k\u001b(J\n\\begin{enumerate}\n\\item mypoints\\ $\\leftarrow$\\ (send\\ myedge\\ {\\bf :neighborpoints}\\ point)\n\\item hispoints\\ $\\leftarrow$\\ hisvertex\u001b$@$N:\\$C$F$$$kNG$NH?BPB&$ND:E@$N%j%9%H\u001b(J\n\\item {\\bf for\\ each}\\ f\\ $\\leftarrow$\\ myedge\u001b$@$N:\\$k#2$D$NLL\u001b(J\n\\begin{description}\n\\item c-vector\\ $\\leftarrow$\\ (send f\\ {\\bf :separation}\\ mypoints\\ hispoints)\n\\item {\\bf if}\\ c-vector\u001b$@$,\u001b(Jnil\u001b$@$G$J$$\u001b(J\\ {\\bf then}\\ \nor-cond\\ $\\leftarrow$\\ or-cond\\ $+$\\ c-vector\n\\end{description}\n\\item {\\bf for\\ each}\\ $p1,p2$\\ $\\leftarrow$\\ myedge\u001b$@>e$NE@!\"\u001b(Jhispoints\u001b$@$N\u001b(J\n\u001b$@Cf$+$i0[$J$k#2E@$NAH9g$;\u001b(J\n\\begin{description}\n\\item s-plane\\ $\\leftarrow$\\ $point,p1,p2$\\ \u001b$@$N#3E@$N:\\$kLL\u001b(J\n\\item c-vector\\ $\\leftarrow$\\ \n(send s-plane\\ {\\bf :separation}\\ mypoints\\ hispoints)\n\\item {\\bf if}\\ c-vector\u001b$@$,\u001b(Jnil\u001b$@$G$J$$\u001b(J\\ {\\bf then}\\ \nor-cond\\ $\\leftarrow$\\ or-cond\\ $+$\\ c-vector\n\\end{description}\n\\item return\\ or-cond\n\\end{enumerate}\n{\\bf vertex-vertex-contact\\ (point\\ myvertex\\ hisvertex)}\n\u001b$@@\\?(E@\u001b(Jpoint\u001b$@$K$*$1$k<+J,$N6aK5$rI=$9D:E@\u001b(Jmyvertex\u001b$@$HAjr7o$r5a$a$k\u001b(J\n\\begin{enumerate}\n\\item mypoints\\ $\\leftarrow$\\ myvertex\u001b$@$N:\\$C$F$$$kNG$NH?BPB&$ND:E@$N%j%9%H\u001b(J\n\\item hispoints\\ $\\leftarrow$\\ hisvertex\u001b$@$N:\\$C$F$$$kNG$NH?BPB&$ND:E@$N%j%9%H\u001b(J\n\\item bothpoints\\ $\\leftarrow$\\ myvertex\u001b$@$H\u001b(Jhisvertex\u001b$@$N=EJ#$r=|$$$?OB=89g\u001b(J\n\\item {\\bf for\\ each}\\ $p1,p2$\\ $\\leftarrow$\\ bothpoint\n\u001b$@$NCf$+$i0[$J$k#2E@$NAH9g$;\u001b(J\n\\begin{description}\n\\item s-plane\\ $\\leftarrow$\\ $point,p1,p2$\\ \u001b$@$N#3E@$N:\\$kLL\u001b(J\n\\item c-vector\\ $\\leftarrow$\\ \n(send s-plane\\ {\\bf :separation}\\ mypoints\\ hispoints)\n\\item {\\bf if}\\ c-vector\u001b$@$,\u001b(Jnil\u001b$@$G$J$$\u001b(J\\ {\\bf then}\\ \nor-cond\\ $\\leftarrow$\\ or-cond\\ $+$\\ c-vector\n\\end{description}\n\\item return\\ or-cond\n\\end{enumerate}\n\\vfill\n\\clearpage\n\\subsubsection{\u001b$@7W;;Nc\u001b(J}\n\u001b$@K\\%W%m%0%i%`$K$h$C$F!\"Ey2A94B+E@$N0LCV$H!\"3F!9$NE@$K$*$1$k94B+>r7o$r\u001b(J\n\u001b$@5a$a$?Nc$rB&94B+$NNc!\"\u001b(J\nFig.\\ref{fig:unilateral}\u001b$@$OJRB&94B+$NNc$G$\"$k!#\u001b(J\n\\begin{figure}[h]\n\\begin{center}\n\\epsfile{file=fig-bilateral.ps,width=15cm}\n\\end{center}\n\\caption{Bilateral constraint}\n\\label{fig:bilateral}\n\\end{figure}\n\\begin{verbatim}\n(setq x (make-cube 200 200 100))\n(setq y (make-cube 100 100 300))\n(setq ana (body- x y))\n(send y :constraint ana)\n(hidd y ana)\n(send y :draw-constraint)\n\\end{verbatim}\n\\vfill\n\\clearpage\n\\begin{figure}[h]\n\\begin{center}\n\\epsfile{file=fig-unilateral.ps,width=15cm}\n\\end{center}\n\\caption{Unilateral constraint}\n\\label{fig:unilateral}\n\\end{figure}\n\\begin{verbatim}\n(setq x (make-prism '(#f(-80 0 -50) #f(-80 -25 0) #f(-80 -100 0)\n\t\t #f(-80 -100 25) #f(-80 100 25) #f(-80 100 0)\n\t\t #f(-80 25 0))\n\t\t #f(160 0 0)))\n(setq mizo (make-prism '(#f(-100 0 -50) #f(-100 25 0) #f(-100 100 0)\n\t\t\t #f(-100 100 -100) #f(-100 -100 -100) #f(-100 -100 0)\n\t\t\t #f(-100 -25 0))\n\t\t #f(200 0 0)))\n(send x :constraint mizo)\n(hidd x mizo)\n(send x :draw-constraint mizo 20.0)\n\\end{verbatim}\n\\vfill\n\\clearpage\nFig.\\ref{fig:peg-in-hole}\u001b$@$O!\"7j$K$Z%0$rF~$l$k:n6HCf$N94B+>uBV$NNc$G$\"$k!#\u001b(J\n\\begin{figure}[h]\n\\epsfile{file=fig-peg-in-hole1.ps,width=8cm}\n\\epsfile{file=fig-peg-in-hole2.ps,width=8cm}\n\\epsfile{file=fig-peg-in-hole3.ps,width=8cm}\n\\epsfile{file=fig-peg-in-hole4.ps,width=8cm}\n\\caption{Peg in a hole}\n\\label{fig:peg-in-hole}\n\\end{figure}\n\\subsubsection{\u001b$@J88%\u001b(J}\n\u001b$@K\\@a$NFbMF$O!\"H2<$5$kMM$*4j$$$7$^$9!#\u001b(J\n\\begin{itemize}\n\\item \u001b$@HfN1@nGn5W!\">>0f=S9@!\"9b@%T\"9n!\"\u001b(J\n\u001b$@B?LLBN4V$N@\\?($K$h$k94B+>r7o$r4v2?%b%G%k$+$iF3=P$9$k0lHLE*$J%\"%k%4%j%:%`!\"\u001b(J\n\u001b$@F|K\\%m%\\%C%H3X2q;o!\"#94,!\"#49f!\"#1#9#9#1!#\u001b(J\n\\end{itemize}\n\\vfill\n\\clearpage\n\\subsection{\u001b$@J*BN4V$N@\\?($K$h$k94B+$N2r@O\u001b(J}\n\n\\subsubsection{\u001b$@%\"%k%4%j%:%`$N35MW\u001b(J}\n\u001b$@A0@a$G=R$Y$?MM$K!\"Ho94B+J*BN$H94B+J*BN$N7A>u$OB?LLBN!\"\u001b(J\n\u001b$@N>Dj$9$k$H!\"Ho94B+J*BN$KBP$9$k94B+>r7o$O0lHL$K!\"\u001b(J\n\\begin{equation}\n\\bigcap_{k=1}^{L} \\bigcap_{i=1}^{N(k)}\\ \\bigcup_{j=1}^{M(k,i)}\\ \nF_{kij}^{T}J_{k}\\left(\n\\begin{array}{c}\n\\Delta X \\\\ \\Omega\n\\end{array}\n\\right)\\geq 0\n\\label{eq:ippannokosoku}\n\\end{equation}\n\u001b$@$GM?$($i$l$k!#$3$3$K!\"\u001b(J$L$\u001b$@$OEy2A94B+E@$N?t!\"\u001b(J\n$N(k)$\u001b$@$O3FEy2A94B+E@$K$*$1$kO@M}@Q>r7o$N?t!\"\u001b(J$M(k,i)$\u001b$@$OO@M}OB>r7o$N?t$G$\"$k!#\u001b(J\n\u001b$@$^$?!\"\u001b(J$F_{kij}:3\\times 1$\u001b$@$O94B+K!@~%Y%/%H%k!\"\u001b(J\n$J_{k}:3\\times 6$\u001b$@$O!\"J*BN:BI87O$GI=$5$l$?Hy>.JQ0L\u001b(J$\\Delta X$\u001b$@!&2sE>\u001b(J$\\Omega$\u001b$@$r\u001b(J\n\u001b$@Ey2A94B+E@$r86E@$H$9$k:BI87O$GI=$5$l$?Hy>.JQ0L\u001b(J$\\Delta X_{k}$\n\u001b$@$KJQ49$9$k%d%3%S9TNs$G$\"$k!#\u001b(J\n\u001b$@J,G[B'$rMQ$$$FO@M}OB$rA0$K=P$9$H!\"\u001b(J(\\ref{eq:ippannokosoku})\u001b$@<0$O!\"\u001b(J\n\\begin{equation}\n\\bigcup_{p=1}^{Q}\\ C_{p}\\ \\left(\n\\begin{array}{c}\n\\Delta X \\\\ \\Omega\n\\end{array}\n\\right)\\geq 0\n\\label{eq:matrix-form}\n\\end{equation}\n\u001b$@$H$$$&7A$K$+$1$k!#$3$3$K!\"\u001b(J$Q=\\Pi_{k,i}M(k,i)$\u001b$@!\"\u001b(J\n$C_{p}:n\\times 6$ \u001b$@$O\u001b(J$F_{kij}^{T}J_{k}$\u001b$@$r9T%Y%/%H%k$H$9$k9TNs$G$\"$k!#\u001b(J\n$n$\u001b$@$O\u001b(J$p$\u001b$@$K0MB8$7$F7h$^$k?t$G$\"$k$,!\"$3$3$G$O4JC1$N$?$aC1$K\u001b(J$n$\u001b$@$H5-$7$?!#\u001b(J\n(\\ref{eq:ippannokosoku})\u001b$@<0$N2r$O!\"3F!9$N!\"\u001b(J\n\\begin{equation}\nC_{p}\\ \\left(\n\\begin{array}{c}\n\\Delta X \\\\ \\Omega\n\\end{array}\n\\right)\\geq 0\n\\label{eq:each-matrix-form}\n\\end{equation}\n\u001b$@$H$$$&O\"N)@~7AITEy<0$N2r$NOB=89g$K$J$k!#\u001b(J\n(\\ref{eq:each-matrix-form})\u001b$@<0$N2r$OFLB?LL?m$K$J$k$,!\"\u001b(J\n\u001b$@2r$r5a$a$k$N$KI,MW$J7W;;NL$O\u001b(J$r=rank\\ C_{p}$\u001b$@$K0MB8$9$k!#\u001b(J\n\u001b$@$=$3$G!\"$^$:9TNs\u001b(J$C_{p}$\u001b$@$r!\"\u001b(J\n\\begin{equation}\nC_{p}=\\ \\left(\n\\begin{array}{cc}\nU_{1} & U_{2}\n\\end{array}\n\\right) \\left(\n\\begin{array}{cc}\nD_{rr} & O \\\\\nO & O\n\\end{array}\n\\right) \\left(\n\\begin{array}{c}\nV_{1}^{T} \\\\\nV_{2}^{T}\n\\end{array}\n\\right)\n\\label{eq:svd}\n\\end{equation}\n\u001b$@$HFC0[CMJ,2r$9$k$3$H$K$h$j!\"\u001b(J$C_{p}$\u001b$@$NNm6u4V$rJ,N%$9$k!#\u001b(J\n\u001b$@$3$3$K!\"\u001b(J$U_{1}:n\\times r$\u001b$@!\"\u001b(J$U_{2}:n\\times (6-r)$\u001b$@!\"\u001b(J\n$V_{1}:6\\times r$\u001b$@!\"\u001b(J$V_{2}:6\\times (6-r)$\u001b$@$OD>8r9TNs!\"\u001b(J\n$D_{rr}:r\\times r$\u001b$@$OBP3Q9TNs$G$\"$k!#\u001b(J\n(\\ref{eq:each-matrix-form})\u001b$@<0$N2r$O!\"\u001b(J$C_{p}$\u001b$@$NNm6u4V$H!\"\u001b(J\n$V_{1}$\u001b$@$NNs%Y%/%H%k$GD%$i$l$k\u001b(J$r$\u001b$@OB$K$J$k!#\u001b(J\n\u001b$@8euBV$rJQ2=$5$;$kHy>.JQ0L$KBP1~$7!\"\u001b(J\n$V_{2}$\u001b$@$OA4$F$NEy2A94B+E@$H$N@\\?(>uBV$r0];}$9$kHy>.JQ0L$KBP1~$9$k!#\u001b(J\n\u001b$@$3$N<0$O!\"@\\?($K$h$j94B+$r.JQ0L$,!\"\u001b(J\n$m+2(6-r)$\u001b$@8D$NHy>.JQ0L$NHsIi#1.$N?t$N4pDl$G$\"$k!#\u001b(J\n\u001b$@$3$N$H$-!\"Ho94B+J*BN$,94B+J*BN$KBP$7$F\u001b(J\n\u001b$@H/@82DG=$JNO\u001b(J$F$\u001b$@!\"%H%k%/\u001b(J$T$\u001b$@$N=89g$O!\"\u001b(J\n(\\ref{eq:finalsolution})\u001b$@<0$NAPBPFLB?LL?m$G!\"\u001b(J\n\\begin{equation}\n\\left(\n\\begin{array}{ccc}\nV_{1}(E_{1}\\ \\cdots\\ E_{m}) & V_{2} & -V_{2}\n\\end{array}\n\\right)^{T}\n\\left(\n\\begin{array}{c}\nF \\\\ T\n\\end{array}\n\\right) \\leq 0\n\\label{eq:force}\n\\end{equation}\n\u001b$@$J$kO\"N)ITEy<0$N2r$K$J$k!#\u001b(J\n\n\u001b$@K\\@a$G$O!\"\u001b(Jc-body\u001b$@$N%a%=%C%I\u001b(J$:constraint$\u001b$@$G5a$a$?\u001b(J(\\ref{eq:ippannokosoku})\u001b$@<0\u001b(J\n\u001b$@$+$i2r$G$\"$k\u001b(J(\\ref{eq:finalsolution})\u001b$@<0$r5a$a$k%W%m%0%i%`!\"\u001b(J\n(\\ref{eq:force})\u001b$@<0$N2r$r5a$a$k%W%m%0%i%`$r@bL@$9$k!#\u001b(J\n\u001b$@$3$l$i$N%W%m%0%i%`$r;H$&$?$a$K$O!\"\u001b(J\n$eus/llib/solve-inequalities.l$\\ \u001b$@$r%m!<%I$9$kI,MW$,$\"$k!#\u001b(J\n\u001b$@K\\%\"%k%4%j%:%`$N8@8l%$%s%?!<%U%'%$%9$rMQ$$$F%j%s%/$7$F$$$k!#\u001b(J\n$C$\u001b$@$NItJ,$O\u001b(J$eus/cone$\u001b$@$H$$$&\u001b(Jdirectory\u001b$@$K$\"$j!\"\u001b(J\n\\begin{verbatim}\n % cd eus/cone\n % make cone\n % make install\n\\end{verbatim}\n\u001b$@$r\\:Y$O!\"\u001b(Jeus/llib/solve-inequalities.l\\ \u001b$@$r;2>H$5$l$?$$!#\u001b(J\n\n\u001b$@%$%s%W%j%a%s%F!<%7%g%s>e$N@)Ls$+$i!\"Ey2A94B+>r7o$N?t$O:GBg#3#2$H$J$C$F$$$k!#\u001b(J\n\\vfill\n\\clearpage\n\\subsubsection{\u001b$@%$%s%W%j%a%s%F!<%7%g%s\u001b(J}\n\\vspace{1.0cm}\n{\\jLarge c-body}\n\\\\ [0.5cm]\n\\begin{description}\n\n\\item[{\\jlarge \\bf description}]\\hspace{1cm}\n\\begin{description}\n\\item[] \u001b$@94B+>r7o$r3JG<$9$k%9%m%C%H$rDI2C$7$?\u001b(Jbody\u001b$@$N%5%V%/%i%9\u001b(J\n\\end{description}\n\n\\item[{\\jlarge \\bf super class}]\\hspace{1cm}\n\\begin{description}\n\\item[body] ()\n\\end{description}\n\n\\item[{\\jlarge \\bf slots}]\\hspace{1cm}\n\\begin{description}\n\\item[motion] \u001b$@5v$5$l$kHy>.JQ0L\u001b(J\n\\item[force] \u001b$@H/@82DG=$JNO\u001b(J\n\\end{description}\n\n\\item {\\jlarge \\bf methods}\n\\\\ [0.5cm]\n\\begin{description}\n\\item[:motion ()] self\u001b$@$KBP$9$k94B+>r7o$r2r$$$F!\"5v$5$l$kHy>.JQ0L$r5a$a$k!#\u001b(J\n\\item[:force ()] self\u001b$@$K5v$5$l$kHy>.JQ0L$+$i!\"H/@82DG=$JNO$r5a$a$k!#\u001b(J\n\\item[:draw-motion (b)] self\u001b$@$H\u001b(Jb\u001b$@$,@\\?($7$F$$$k$H$-$K!\"\u001b(Jself\u001b$@$K5v$5$l$kHy>.JQ0L\u001b(J\n\u001b$@$rI=<($9$k!#\u001b(J\n\\end{description}\n\\end{description}\n\\vfill\n\\clearpage\n\\subsubsection{\u001b$@7W;;Nc\u001b(J}\n\u001b$@K\\%W%m%0%i%`$K$h$C$F!\"\u001b(J(\\ref{eq:finalsolution})\u001b$@<0$G\u001b(J\n\u001b$@I=$5$l$k2r$r5a$a$?Nc$r0J2<$K<($9!#\u001b(J\n\nFig.\\ref{fig:trigonalprism}\u001b$@$K!\"#33QCl$,%F!<%V%k$N>e$K:\\$C$F$$$kNc$r<($9!#\u001b(J\n\u001b$@$3$N>l9g!\"2r$Or7o!\"N%C&F0:n$N0l$D!\"94B+$r0];}$9$kJB?J$H2sE>$G$\"$k!#\u001b(J\n\\begin{figure}[h]\n\\begin{center}\n\\epsfile{file=fig-onthetable-0.ps,width=7cm}\n\\epsfile{file=fig-onthetable-m1.ps,width=7cm}\\\\\n\\epsfile{file=fig-onthetable-mt.ps,width=7cm}\n\\epsfile{file=fig-onthetable-mr.ps,width=7cm}\n\\end{center}\n\\caption{Possible motions of a trigonal prism}\n\\label{fig:trigonalprism}\n\\end{figure}\n\\vfill\n\\clearpage\nFig.\\ref{fig:peg-in-a-hole}\u001b$@$K3QCl$,7j$KF~$C$F$$$kNc$r<($9!#\u001b(J\n\u001b$@$3$N>l9g!\"2r$OuBV$rJQ$($J$$F0:n$OB8:_$;$:!\"\u001b(J\n\u001b$@G$0U$NF0:n$O?^$K<($7$?MM$J#3$D$NF0:n$+$i@8@.$5$l$k!#\u001b(J\n\\begin{figure}[h]\n\\begin{center}\n\\epsfile{file=fig-peg-naname-0.ps,width=7cm}\n\\epsfile{file=fig-peg-naname-m1.ps,width=7cm}\\\\\n\\epsfile{file=fig-peg-naname-m2.ps,width=7cm}\n\\epsfile{file=fig-peg-naname-m3.ps,width=7cm}\n\\end{center}\n\\caption{Possible motions of a peg in a hole}\n\\label{fig:peg-in-a-hole}\n\\end{figure}\n\\vfill\n\\clearpage\n\\subsubsection{\u001b$@J88%\u001b(J}\n\u001b$@K\\@a$NFbMF$O!\"H2<$5$kMM$*4j$$$7$^$9!#\u001b(J\n\\begin{itemize}\n\\item \u001b$@HfN1@nGn5W!\">>0f=S9@!\"9b@%T\"9n!\"\u001b(J\n\u001b$@B?LLBN4V$N@\\?($K$h$k94B+>r7o$N9bB.2rK!$H$=$NN%C&F0:n7W2h$X$N1~MQ!\"\u001b(J\n\u001b$@F|K\\%m%\\%C%H3X2q;o!\"7G:\\M=Dj!#\u001b(J\n\\end{itemize}\n", "meta": {"hexsha": "5a2e8bcf148979fbe00743e660eaffd37e7b63db", "size": 62139, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/hirukawa/geometry.tex", "max_stars_repo_name": "snozawa/EusLisp", "max_stars_repo_head_hexsha": "021843b48ffcd029fcb6113d5022c703b6ce45fb", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 50, "max_stars_repo_stars_event_min_datetime": "2015-02-16T23:07:11.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-28T02:43:01.000Z", "max_issues_repo_path": "doc/hirukawa/geometry.tex", "max_issues_repo_name": "snozawa/EusLisp", "max_issues_repo_head_hexsha": "021843b48ffcd029fcb6113d5022c703b6ce45fb", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 286, "max_issues_repo_issues_event_min_datetime": "2015-01-04T02:47:05.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-25T16:38:43.000Z", "max_forks_repo_path": "doc/hirukawa/geometry.tex", "max_forks_repo_name": "snozawa/EusLisp", "max_forks_repo_head_hexsha": "021843b48ffcd029fcb6113d5022c703b6ce45fb", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 39, "max_forks_repo_forks_event_min_datetime": "2015-01-03T05:06:07.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-16T05:10:54.000Z", "avg_line_length": 35.3866742597, "max_line_length": 110, "alphanum_fraction": 0.5554804551, "num_tokens": 34627, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3490265692690928}} {"text": "\n\\documentclass[Thesis.tex]{subfiles}\n\\begin{document}\n\\chapter{Variational Monte Carlo}\n\\label{chp:variational-monte-carlo}\n\n\\glsresetall\n\n\\Gls{vmc} is a method for obtaining the ground state wave function of\na quantum mechanical system, and it constitutes the framework for all the calculations we\nperform in this thesis. As this method is of great significance to us, this chapter is\ndevoted to all relevant definitions, derivations and technical details required for a\nsuccessful study of the systems of interest.\n\n\\section{The Variational Principle}\n\nThe fundamental principle that enables \\gls{vmc} is the observation that the wave function that\ndescribes the ground state of a system, is the wave function which produces the lowest\nexpectation value of the energy. Read this statement a couple of times and it almost seems\nlike a circular definition, because we define the ground state as the state with the\nlowest energy. Still, this simple observation is what allows us a way to go forward.\n\nLet's formalize the above statement.\n\n\\begin{theorem}[Variational Principle]\\label{thm:variational-principle}\n Suppose a time-independent Hamiltonian, $\\hat H$, an associated set of eigenvalues,\n $E_0\\leq E_1\\leq\\dots$ and\n a set of orthonormal eigenvectors, $\\{\\ket{\\phi_n}\\}$ spanning the associated\n Hilbert space, such that the \\acrlong{tise} is satisfied,\n\n \\begin{align}\n \\hat H\\ket{\\phi_n} = E_n\\ket{\\phi_n}.\n \\end{align}\n For any arbitrary $\\ket\\psi$ in the Hilbert space, the expectation value of $\\hat H$\n for this wave function must satisfy\n \\begin{align}\n \\expval{\\hat H}&\\defeq \\expval{\\hat H}{\\psi} \\geq E_0,\n \\end{align}\n where $E_0$ is the exact ground state energy, with equality if and only if\n $\\ket\\psi=\\ket\\phi_0$.\n\\end{theorem}\n\\begin{proof}\n Because $\\{\\ket{\\phi_n}\\}$ constitutes a complete orthonormal basis of the Hilbert\n space, we can expand $\\ket\\psi$ as\n \\begin{align}\n \\ket\\psi = \\sum_n c_n\\ket{\\phi_n} \\qq{with} \\sum_n \\abs{c_n}^2 = 1.\n \\end{align}\n We then have\n \\begin{align}\n \\expval{\\hat H}{\\psi} &= \\qty(\\sum_m c^*_m\\bra{\\phi_m})\\hat H\\qty(\\sum_n\n c_n\\ket{\\phi_n})\\\\\n &= \\sum_n\\sum_m c_m^*c_n\\bra{\\phi_m}\\hat H\\ket{\\phi_n}\\\\\n &= \\sum_n\\sum_m c_m^*c_nE_n\\bra{\\phi_m}\\ket{\\phi_n}\\\\\n &= \\sum_n\\sum_m c_m^*c_nE_n\\delta_{mn}\\\\\n &= \\sum_n \\abs{c_n}^2 E_n\\\\\n &\\geq\\sum_n \\abs{c_n}^2 E_0 = E_0.\n \\end{align}\n It is clear that the equality only occurs when $c_n = \\delta_{0n}$, which implies\n $\\ket\\psi=\\ket{\\phi_0}$.\n\\end{proof}\n\nIf we require the test wave function $\\ket\\psi$ to be orthogonal to $\\ket{\\phi_0}$, then\nthe above derivation results in a similar condition for determining the first excited\nstate. By applying the same reasoning iteratively, the same can be said about any higher\nenergy state. So in principle, we can use the variational principle to first find the\nground state, then find an orthogonal state that gives the first excited energy, then find\na state orthogonal to both that gives the second excited energy etc. We will not pursue\nany excited states in this thesis, but it is useful to know that we could in principle\nextend our results if need be.\n\n\\section{The Variational Monte Carlo Algorithm}\n\nRooted in the variational principle, the general approach to finding a good estimate of\nthe ground state goes as follows:\n\n\\begin{enumerate}\n \\item Define the Hamiltonian of interest, $\\hat H$\n \\item Propose a trial wave function $\\ket{\\vb \\alpha}$ dependent on some free parameters\n $\\vb \\alpha=\\qty(\\alpha_1, \\alpha_2,\\dots, \\alpha_m)$\n \\item Return the best possible wave function $\\ket{\\vb\\alpha}$ such that $\\expval{\\hat\n H}{\\vb\\alpha}$ is minimized w.r.t the parameters $\\vb \\alpha$.\n\\end{enumerate}\nThe first step is trivial, as we shall assume that the Hamiltonians we want to examine are\ngiven in advance.\\footnote{Clearly, deriving/defining Hamiltonians that accurately describe a given\nsystem is not a trivial task in general, and might require great levels of theoretical\nwork and/or experimental data analysis. But from our perspective, this work has already\nbeen done.}\n\nThe second and third points are where we will spend our efforts, and so we devote extra\nattention to these two points in particular.\n\n\n\\section{The Trial Wave Function}\n\nCritical to the outcome of a \\gls{vmc} calculation is the quality of the proposed trail wave\nfunction. In order to have stable and sensible results we should also take care that the\nwave function obeys the requirements stated in\n\\cref{sec:requirements-of-wave-functions}. Beyond these basic assertions, it is\nimperative that the functional form of the proposed wave function is capable of expressing\nthe true underlying ground state, or at least a good approximation of it. As an example,\ntake the simple case of the idealized one-dimensional harmonic oscillator presented in\n\\cref{sec:simple-non-inter-HO}, with the ground state expressed as\n\\begin{align}\n \\phi_0(x) \\propto \\exp(- \\frac{1}{2} x^2).\n\\end{align}\nIf we proposed a trial wave function on a form\n\\begin{align}\n \\psi(x; \\alpha)\\propto \\exp(- \\alpha x^2),\n\\end{align}\nwe could expect \\gls{vmc} to produce the exact ground state after optimizing $\\alpha$. The above\nis a good trial wave function because its functional form contains the exact state.\n\nSuppose now we proposed something different, maybe because of a lack of theoretical\nunderstanding. For the sake of example, say we chose this visually similar\nfunction:\\footnote{Visually similar in the sense that plotting them both yield similar bell\nshaped curves, not in terms of their algebra.}\n\\begin{align}\n \\psi(x; \\alpha)\\propto \\frac{1}{1 + \\alpha x^2}.\n\\end{align}\nNo matter how we tune the $\\alpha$ parameter here, we will never achieve the lowest\nground state energy, simply because the ground state is not possible to express within this\nfunctional form. We can now only hope to get approximations, acting as upper bounds on the\nground state energy.\n\n\\subsection{Slater-Jastrow: Standard Approach}\n\\label{sec:slater-jastrow}\n\nIn order to come up with good trail wave functions we have a standard approach which often\nserves us well. Typically we are considering a system of many interacting particles in\nsome external potential (e.g. the quantum dots system). In such cases the typical approach\nis to build a trial wave function as a product of a single-particle part, $S$, and a correlation\npart, $J$. The two are referred to as the \\emph{Slater} factor and the\n\\emph{Jastrow} factor, respectively:\n\n\\begin{align}\n \\psi = S\\cdot J.\n\\end{align}\n\n\\subsubsection*{Slater Factor}\n\nThe idea behind the decomposition above is that we put all the physical insight we have into these\ntwo components individually. For the single particle part we include what we can deduce\nabout the system if we idealize it in some way. Typically that involves looking at the\nstates when there is only one particle, or considering some simplified interaction between\nall of them (e.g. Hartree-Fock approximation). The basis functions we obtain from such an\nanalysis are then placed in a determinant/permanent structure so as to fix the symmetry\nrequirements of fermions/bosons, respectively. This factor is then the Slater\nfactor~\\cite{Slater-1929,Dirac-1926}.\n\n\nFor clarity, we present a couple of different Slater forms. We consider the two-dimensional\nquantum dots system from \\cref{sec:quantum-dots-theory} for $N$ particles. We\ndenote the position of particle $i$ by $\\vb x_i=(x_i, y_i)$. Let $\\vb\\Phi$ be a matrix defined by $\\Phi_{i,j}\n\\defeq \\phi_{i}(\\vb x_j)$, where the $\\phi_i$ is part of the set of orbitals\n$\\{\\phi_{0,0}, \\phi_{1,0},\\phi_{0,1}, \\phi_{1,1},\\dots\\}$, as defined in\n\\cref{eq:ho-single-particle-orbitals}. Lastly, let $\\vb \\Phi^{(M)}$ denote the\nresulting matrix with $M$ rows, one for each of the $M$ lowest energy orbitals. Finally,\nif the particles are fermions we define the Slater factor as:\\footnote{Spin considerations not mentioned here, as they will not be important for the particular systems we will investigate. In general though, the structure of the Slater factor needs to account for spin in a well defined way.}\n\n\\begin{align}\n S(\\vb X)_\\text{fermions} \\defeq \\text{determinant}\\qty(\\vb\\Phi^{(M)}).\n\\end{align}\nDue to the properties of the determinant, this has the anti-symmetry required for\nfermions.\n\nIn the case of bosons, we could just change the above to\n\\begin{align}\n S(\\vb X)_\\text{bosons} \\defeq \\text{permanent}\\qty(\\vb\\Phi^{(M)}).\n\\end{align}\nHowever, there are far less computationally expensive ways to get a \\emph{symmetric}\nresult, as required for bosons. A simple product of single particle ground states is\nsufficient in this case.\n\\begin{align}\n S(\\vb X)_\\text{bosons} \\defeq \\prod_{i}^N \\phi_{0,0}(\\vb x_i).\n\\end{align}\n\nThe general structure is constant across different systems, changing only the basis\nfunctions we use to express the matrix elements.\n\nIn some cases, however, we will struggle to find basis functions that describe a meaningful\nportion of the system's behaviour in this way. In such challenging cases, the Slater factor\ncould be reduced to only being responsible for supplying the correct symmetry. In the case\nof bosons, that would mean the complete removal of the Slater factor from the trial wave\nfunction. We will see this when looking further into the liquid helium system.\n\n\n\n\\subsubsection*{Jastrow Factor}\n\nAnything not covered by this analysis is meant to be accounted for in the correlation\nterm, $J$. This is typically a function on the following generalized form, commonly called a Jastrow\nfactor~\\cite{Jastrow-1955}:\n\\begin{align}\n J(\\vb X) &= \\exp(\\sum_{i < j} U(r_{ij})),\n\\end{align}\nwhere $U(r_{ij})$ is some function dependent on the inter-particle distance only. Note that by\nconvention the Slater factor has the correct symmetry, so the correlation factor should\nalways be symmetrical in order to maintain the same total symmetry.\n\nAny variational parameters $\\vb \\alpha$ are typically introduced as part of the Jastrow\nfactor. The Slater factor represents everything we are sure should be included\nfrom a theoretical approach, while the Jastrow term should ideally account for all our ignorance.\nIt is therefore typically expressed with a few free parameters.\n\nAgain, for the sake of clarity, we state an example of a Jastrow factor. One of the most\ncommonly used is the Pade-Jastrow factor~\\cite{Drummond-Towler-Needs-2008}:\n\\begin{align}\n J(\\vb X) = \\prod_{i < j} \\exp(\\frac{\\alpha r_{ij}}{1 + \\beta r_{ij} } ).\n\\end{align}\nHere $\\beta$ is the only variational parameter, and $\\alpha$ is fixed depending on the\ndimensionality and spin of the particles. More complicated versions exist, containing\nhigher order polynomials of $r_{ij}$ in the exponential. This particular form\nwill prove to be a very good choice for the quantum dots system.\n\n\n\n\n\\section{Optimization}\n\nAt this point, we have a Hamiltonian $\\hat H$ defining our system of interest,\nand we have proposed a trial wave function $\\ket{\\vb\\alpha}$ based on the\ntheoretical intuition we have. The wave function has by design some free\nparameters $\\vb\\alpha$ which we now want to pin down to their optimal values.\n\nFrom the variational principle (\\cref{thm:variational-principle}), we know\nthat the best set of parameters are those that minimize the expected energy,\n$\\expval{\\hat H}{\\vb\\alpha}$. As such, a natural place to start would be to\ndetermine an efficient way to evaluate the energy.\n\n\\subsection{Local Energy}\n\nDenote the trial wave function as $\\psi_{\\vb\\alpha}(\\vb X)$, where $\\vb\\alpha$\nare the variational parameters and $\\vX=(\\vx_1\\ \\vx_2\\ \\dots\\ \\vx_n)^T$ is the\nmatrix of all the degrees of freedom for each particle (e.g. position\ncoordinates). The expectation value for the energy of the system is:\n\n\\begin{align}\n \\expval{\\hat H} &= \\frac{\\expval{\\hat H}{\\vb\\alpha}}{\\braket{\\vb\\alpha}}\\\\\n &= \\frac{\\int \\dd{\\vb X} \\psialpha^*\\hat H\\psialpha}{\\int\\dd{\\vX}\\abs{\\psialpha}^2}.\n % &= \\int\\dd{\\vX} P(\\vX,\\valpha)E_L(\\vX,\\valpha),\n\\end{align}\n%\nWe now restructure this integral a bit. Remembering the interpretation of\nthe wave function as a \\gls{pdf}, we can write the \\emph{probability\ndensity} for a given system configuration $\\vX$ as:\n\n\\begin{align}\n P_{\\psi_{\\vb{\\alpha}}}(\\vX) \\defeq \\frac{\\abs{\\psialpha}^2}{\\int\\dd{\\vX}\\abs{\\psialpha}^2},\\label{eq:wavefunc-probability-density-def}\n\\end{align}\nwhere we account for the possibility of $\\psialpha$ not being normalized to unity. Further, we define a quantity we will call the \\emph{local energy},\n\\begin{align}\n E_L &\\defeq \\frac{1}{\\psialpha} \\hat H\\psialpha.\\label{eq:local-energy-def}\n\\end{align}\nThe reason for these definitions is that now we can write the energy expectation as follows:\n\n\\begin{align}\n \\expval{\\hat H} &= \\int\\dd{\\vX} P_{\\psi_{\\valpha}} (\\vX) E_L(\\vX).\n\\end{align}\nWhy is this better? Because this is simply the weighted average of the local\nenergy, something which has a straightforward discrete formulation:\n\\begin{align}\\label{eq:vmc-local-energy-formulation}\n \\expval{\\hat H} &= \\lim_{N\\rightarrow \\infty} \\frac{1}{N} \\sum_{i = 1}^N E_L(\\vX_i)\n \\qq{where} \\vb X_i\\disteq P_{\\psi_{\\valpha}}.\n\\end{align}\nSo, if we can sample $\\vX$ from the distribution described by our trial wave\nfunction, we simply need to evaluate the local energy for a sufficiently large\nnumber of configurations to obtain a good estimate of the expectation value of\nthe Hamiltonian. How exactly to do the sampling is a rather involved topic, and\nwill be handled in \\cref{chp:monte-carlo}.\n\n\\subsection{Updating Parameters}\n\nNow that we can evaluate the energy, the function to minimize, we could in\nprinciple start optimizing our parameters. If the parameters are few and\ndiscrete we could simply evaluate the energy for all of them and pick the one\nwhich gives the best result. However, this approach will quickly become\nintractable for even a modest number of parameters, and more intelligent\nmethods should be pursued.\n\nPerhaps the most common optimization strategy relies on computing the gradient\nof the energy w.r.t. $\\valpha$, and then change $\\valpha$ in the opposite\ndirection of this. After all, a gradient by definition points in the direction\nof steepest (local) ascent, so going the other way will be the direction of steepest\ndescent. How exactly to make these changes is a topic of its own, and discussed in depth in \\cref{chp:machine-learning}, but the simplest implementation is as follows:\n\n\\begin{align}\n \\valpha^{(i+1)} = \\valpha^{(i)} - \\eta \\grad_{\\valpha}{\\expval{\\hat H}},\n\\end{align}\nfor a suitably chosen hyperparameter $\\eta\\in\\mathbb{R}$, typically $\\eta\\ll 1$.\nThis says that the updated parameters $\\valpha^{(i+1)}$ are equal to the\nprevious ones, $\\valpha^{(i)}$, minus a small step in the direction of the\ngradient of $\\expval{\\hat H}$ w.r.t. each parameter. After a sufficient number\nof iterations, we hope that the parameters converge to their optimal values.\n\nThis raises another question, namely how to compute\n$\\grad_{\\valpha}{\\expval{\\hat H}}$. Using the basic properties of derivatives we get (temporarily omitting the subscript on $\\psialpha$):\n\\begin{align}\n \\grad_{\\valpha}\\expval{\\hat H} &= \\grad_{\\valpha}\\qty( \\frac{\\int\\dd{\\vX}\\psi^*\\hat H\\psi}{\\int\\dd{\\vX}\\abs{\\psi}^2})\\\\\n &= \\frac{\\int\\dd{\\vX} \\abs{\\psi}^2 \\qty[ \\int\\dd{\\vX}\n \\grad_{\\valpha}(\\psi^*)\\hat H\\psi +\n \\psi^*\\hat H\\grad_{\\valpha}(\\psi)\n ]}{\\qty(\\int\\dd{\\vX}\\abs{\\psi}^2)^2}\\\\\n &\\quad{ }\\ -\n \\frac{\\qty(\\int\\dd{\\vX}\\psi^*\\hat\n H\\psi)\\qty(\\int\\dd{\\vX}\\grad_{\\valpha}(\\abs{\\psi}^2))}{\\qty(\\int\\dd{\\vX}\\abs{\\psi}^2)^2} \\\\\n &= \\expval{\\frac{\\grad_{\\valpha}(\\psi^*)\\hat H\\psi + \\psi^*\\hat H\\grad_{\\valpha}(\\psi)}{\\abs{\\psi}^2}}\n - \\expval{\\hat H}\\expval{\\frac{\\grad_{\\valpha}\\abs{\\psi}^2}{\\abs{\\psi}^2}},\n\\end{align}\nwhere we have used the fact that $\\hat H$ does not depend on $\\valpha$ in any way.\n\nTo further simplify this, we use the trick $\\flatfrac{\\grad(f)}{f}=\\grad(\\ln\nf)$, as well as the hermiticity of $\\hat H$:\n\\begin{align}\n \\grad_{\\valpha}\\expval{\\hat H}\n &= \\expval{\\frac{\\grad_{\\valpha}(\\psi^*)\\hat H\\psi + \\grad_{\\valpha}(\\psi)(\\hat H\\psi)^*}{\\abs{\\psi}^2}}\n - \\expval{\\hat H}\\expval{\\frac{\\grad_{\\valpha}\\abs{\\psi}^2}{\\abs{\\psi}^2}}\\\\\n &= \\expval{\\grad_{\\valpha}(\\ln \\psi^*) E_L + \\grad_{\\valpha}(\\ln\\psi)E_L^*}\n - 2\\expval{\\hat H}\\expval{\\grad_{\\valpha}{\\ln\\abs{\\psi}}}.\n\\end{align}\nAt this point, we are going to explicitly assume that the wave function is\nreal. We are not going to propose any complex trial wave functions in this\nthesis, so this assumption will be valid for our purposes. The final expression then becomes (using $\\expval{\\hat H}=\\expval{E_L}$):\n\n\\begin{align}\n \\grad_{\\valpha}\\expval{\\hat H}\n &= 2\\qty[\\expval{\\grad_{\\valpha}(\\ln \\psi) E_L}\n - \\expval{E_L}\\expval{\\grad_{\\valpha}{\\ln\\abs{\\psi}}}].\n \\label{eq:local-energy-gradient-alpha}\n\\end{align}\nComputationally we already know how to evaluate $\\expval{E_L}$, and we employ a completely analogous approach for the two other expectation values.\nWith this final piece of the puzzle we have everything we need in order to perform a \\gls{vmc} calculation for the ground state of a system.\n\n\\end{document}\n", "meta": {"hexsha": "a6487aef298ffc32b64245964187c750e37995db", "size": 17314, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "writing/VariationalMonteCarlo.tex", "max_stars_repo_name": "johanere/qflow", "max_stars_repo_head_hexsha": "5453cd5c3230ad7f082adf9ec1aea63ab0a4312a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2019-07-24T21:46:24.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-11T18:18:24.000Z", "max_issues_repo_path": "writing/VariationalMonteCarlo.tex", "max_issues_repo_name": "johanere/qflow", "max_issues_repo_head_hexsha": "5453cd5c3230ad7f082adf9ec1aea63ab0a4312a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 22, "max_issues_repo_issues_event_min_datetime": "2019-02-19T10:49:26.000Z", "max_issues_repo_issues_event_max_datetime": "2019-07-18T09:42:13.000Z", "max_forks_repo_path": "writing/VariationalMonteCarlo.tex", "max_forks_repo_name": "bsamseth/FYS4411", "max_forks_repo_head_hexsha": "72b879e7978364498c48fc855b5df676c205f211", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-11-04T15:17:24.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-03T16:37:38.000Z", "avg_line_length": 51.8383233533, "max_line_length": 292, "alphanum_fraction": 0.7365138039, "num_tokens": 4843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6926419767901475, "lm_q1q2_score": 0.3490265660720155}} {"text": "\\documentclass[review]{elsarticle}\n\n% Packages and macros go here\n\\usepackage{amsfonts}\n\\usepackage{amsmath}\n\\usepackage{lineno,hyperref}\n\\modulolinenumbers[5]\n\\usepackage{graphicx}\n\\usepackage{epstopdf}\n\\usepackage{algorithmic}\n\\usepackage{subcaption}\n\\captionsetup{font=small}\n\\captionsetup[sub]{font=footnotesize}\n\\usepackage{nicefrac}\n\\usepackage{units}\n\\usepackage[acronyms]{glossaries}\n\n\\journal{Elsevier}\n\n% \\title{Data recovery in computational fluid dynamics through deep image priors}\n% \\author[main]{M.~T.~Henry de Frahan\\corref{cor1}}\n% \\ead{marc.henrydefrahan@nrel.gov}\n% \\author[main]{R.~W.~Grout}\n% \\ead{ray.grout@nrel.gov}\n% \\cortext[cor1]{Corresponding author}\n% \\address[main]{High Performance Algorithms and Complex Fluids, Computational Science Center, National Renewable Energy Laboratory, 15013 Denver W Pkwy, ESIF301, Golden, CO 80401, USA}\n\n%% `Elsevier LaTeX' style\n%\\bibliographystyle{elsarticle-num}\n\\bibliographystyle{elsarticle-num-names}\n\\makeatletter\n\\providecommand{\\doi}[1]{%\n \\begingroup\n \\let\\bibinfo\\@secondoftwo\n \\urlstyle{rm}%\n \\href{http://dx.doi.org/#1}{%\n doi:\\discretionary{}{}{}%\n \\nolinkurl{#1}%\n }%\n \\endgroup\n}\n\\makeatother\n\n%=================================================================================\n% Abbreviations\n\\newacronym{dns}{DNS}{direct numerical simulations}\n\\newacronym{cfd}{CFD}{computational fluid dynamics}\n\\newacronym{pod}{POD}{proper orthogonal decomposition}\n\\newacronym{gpr}{GPR}{Gaussian process regression}\n\\newacronym{gp}{GP}{Gaussian processes}\n\\newacronym{gpu}{GPU}{graphics processing unit}\n\\makeglossaries\n\\glsdisablehyper\n\n\\begin{document}\n\\begin{frontmatter}\n\n\\title{Data recovery in computational fluid dynamics through deep image priors}\n\n\\author[main]{M.~T.~Henry de Frahan\\corref{cor1}}\n\\ead{marc.henrydefrahan@nrel.gov}\n\n\\author[main]{R.~W.~Grout}\n\\ead{ray.grout@nrel.gov}\n\n\\cortext[cor1]{Corresponding author}\n\n\\address[main]{High Performance Algorithms and Complex Fluids, Computational Science Center, National Renewable Energy Laboratory, 15013 Denver W Pkwy, ESIF301, Golden, CO 80401, USA}\n\n\\begin{abstract}\n\n One of the challenges encountered by computational simulations at\n exascale is the reliability of simulations in the face of hardware\n and software faults. These faults, expected to increase with the\n complexity of the computational systems, will lead to the loss of\n simulation data and simulation failure and are currently addressed\n through a checkpoint-restart paradigm. Focusing specifically on\n \\gls{cfd} simulations, this work proposes a method that uses a deep\n convolutional neural network to recover simulation data. This data\n recovery method (i) is agnostic to the flow configuration and\n geometry, (ii) does not require extensive training data, and (iii) is\n accurate for very different physical flows. Results indicate that\n the use of deep image priors for data recovery is more accurate than\n standard recovery techniques, such as the \\gls{gpr}, also known as\n Kriging. Data recovery is performed for two canonical fluid flows:\n laminar flow around a cylinder and homogeneous isotropic\n turbulence. For data recovery of the laminar flow around a cylinder,\n results indicate similar performance between the proposed method and\n \\gls{gpr} across a wide range of mask sizes. For homogeneous\n isotropic turbulence, data recovery through the deep convolutional\n neural network exhibits an error in relevant turbulent quantities\n approximately three times smaller than that for the\n \\gls{gpr}. Forward simulations using recovered data illustrate that\n the enstrophy decay is captured within $10\\%$ using the deep\n convolutional neural network approach. Although demonstrated\n specifically for data recovery of fluid flows, this technique can be\n used in a wide range of applications, including particle image\n velocimetry, visualization, and computational simulations of\n physical processes beyond the Navier-Stokes equations.\n\n\\end{abstract}\n\n\\begin{keyword}\ndata recovery \\sep fault tolerance \\sep Gaussian process regression \\sep deep convolutional neural network \\sep computational fluid dynamics\n\\end{keyword}\n\n\\end{frontmatter}\n\n\\glsresetall\n\n\\section{Introduction}\nAs modern computational efforts reach exascale, hardware and software\nfaults will increasingly cause difficulty in completing\nsimulations~\\cite{Brown2010}. Current research in hardware systems and\nsoftware frameworks~\\cite{Gropp2004, Hoemmen2011, Teranishi2014,\n Cappello2014, Gamell2015, Grout2017} continues to develop techniques\nto detect and anticipate system failures; however, assuming that a\nfault has been detected and signaled, the data loss from these\nfailures will require data recovery processes~\\cite{Lee2015}. This\nwork addresses this challenge. Current computational codes rely\non a checkpoint and restart paradigm to recover from faults, requiring\neither significant memory consumption for frequent checkpoints or\nlarge resimulation efforts. The ability to recover the missing data\nwithout resorting to data checkpoints has the potential to increase\nsimulation resilience.\n\nIn the context of \\gls{cfd}, data recovery has been explored using a\nvariety of machine learning approaches. Gappy \\gls{pod} has shown\nparticular success in reconstructing missing\ndata~\\cite{Everson1995,Tan2003,Venturi2004}. This approach relies on\ncombining \\gls{pod} with least-squares estimates~\\cite{Yates1933,\n Little2002} and data from \\acrlong{dns} snapshots. Venturi and\nKarniadakis~\\cite{Venturi2004} expanded on the methods proposed by\nEverson and Sirovich~\\cite{Everson1995} and used this technique to\nreconstruct missing data of unsteady flow past a cylinder. Other\napproaches rely on \\gls{gpr}, often referred to as Kriging in\ngeophysics, which is a a reconstruction technique that uses the mean and\ncovariance of the \\acrlong{gp} prior. The prior's covariance is determined\nby a kernel whose hyperparameters are optimized using training\ndata. Gunes et al.~\\cite{Gunes2006} compare \\gls{pod}-based and\n\\gls{gpr}-based solution reconstruction and show that \\gls{gpr}\ninterpolations are particularly effective for unsteady flows\n(including instability regions), whereas \\gls{pod}-based methods are\nadvantageous when the temporal resolution is high. In addition to\nthese methods, Lee et al.~\\cite{Lee2015} proposed a ``resimulation''\nmethod in which the missing data region is resimulated using\nappropriate initial and boundary conditions. This new method is\nevaluated for lid-driven cavity flows and flows past a cylinder at low\nReynolds numbers. Lee et al.~\\cite{Lee2017} combined the gap-tooth\nalgorithm, previously developed for dynamic systems~\\cite{Gear2003},\nmultiresolution information fusion, and auxiliary data to construct a\ngeneral framework for fault-tolerant \\gls{cfd}. The method is\ndemonstrated to work well for simulations of the heat equation and\nlid-driven cavity flow.\n\nRecently, the deep learning community has been proposing methods for\ndata recovery in the field of inverse image reconstruction problems,\nwhich include denoising, inpainting, and\nsuper-resolution,~\\cite{Goodfellow2014,Burger2012,Dosovitskiy2015,Lefkimmiatis2016,Ledig2017,Tai2017,Lai2017}. Deep\nconvolutional neural networks, particularly generative adversarial\nnetworks, have been very successful at solving this class of\nproblems~\\cite{Goodfellow2014}. Inpainting is of particular relevance\nto our objective of reconstructing flow solutions. The objective of\ninpainting is to fill in missing portions of a damaged image such that\nthe result is indistinguishable from the original image. Various\ngenerative adversarial neural networks have been proposed for image\ninpainting with notable\nsuccess~\\cite{Yeh2016,Denton2016,Pathak2016,Li2017a,Sasaki2017}. Generally\nspeaking, these approaches have relied on training deep neural\nnetworks with an extensive and large data set of images such that the\nnetwork learns image priors that it can use in other\nconfigurations. The deep learning methodology used in this work was\nfirst developed by Ulyanov et al.~\\cite{Ulyanov2017} for various\ninverse image reconstruction problems, including inpainting. In\ncontrast with previous image reconstruction solution with deep neural\nnetworks, Ulyanov et al.~\\cite{Ulyanov2017} showed that ``contrary to\nthe belief that learning is necessary for building good image priors,\na great deal of image statistics are captured by the structure of a\nconvolutional image generator independent of learning.'' Instead of\ntraining the neural networks with a large database of images, the\nauthors use untrained neural networks to fit single degraded images,\nusing the network weights as parameters for solving the image\nreconstruction problem.\n\nIn this work, we use deep convolutional neural networks, such as those\nproposed by Ulyanov et al.~\\cite{Ulyanov2017}, for spatial reconstruction of the flow\nsolution for simulations wherein some type of fault led to loss of\ndata, e.g.,\\,processor failure. In contrast to gappy\n\\gls{pod}~\\cite{Venturi2004}, we assume that the current gappy data are\nthe only available data for the reconstruction procedure. This\nassumption is relevant to large simulations where it is\ncomputationally expensive to reload data residing on the file system\nand the reconstruction process is restricted to data already in\nmemory. One advantage of using deep convolutional neural networks is\nthat this approach avoids eigenmode decompositions for solution\nreconstruction, which could restrict the applicability or translation of\nthe method to new configurations. As illustrated in this work, the\nmethod proposed here is not specific to the flow configuration and\ndoes not require multiple training data samples.\n\nThis paper is organized as follows. In Section\\,\\ref{sec:formulation}, we\npresent the problem formulation and define the objective function for\nthe data recovery problem. In Section\\,\\ref{sec:nn}, we detail the\narchitecture of the deep convolutional neural network used to perform\nthe data recovery process for fluid flows. In Section\\,\\ref{sec:results}, we\npresent our results by evaluating the neural network's ability to\nperform data recovery for two canonical flows: laminar flow past a\ncylinder, Section\\,\\ref{sec:cylinder}, and homogeneous isotropic turbulence,\nSection\\,\\ref{sec:hit}. These results are compared with data recovery\nperformed through \\gls{gpr}. Finally, conclusions and future work are\npresented in Section\\,\\ref{sec:ccl}.\n\n\\section{Problem formulation}\\label{sec:formulation}\n\nIn this work, we evaluate the performance of deep convolutional neural\nnetworks for data recovery in \\gls{cfd}. Deep convolutional neural\nnetworks have shown particular success for solving the image\nreconstruction problem~\\cite{Goodfellow2014}. Image reconstruction is\nanalogous to data recovery because they share a similar objective to\nprovide synthetic data that closely match the missing data. The\nimage reconstruction problem can be cast as an optimization problem:\n\\begin{align}\n \\min_x{E(x;x_0) + R(x)}\n\\end{align}\nwhere $x$ is the original image that needs to be recovered, $x_0$ is\nthe corrupted image, $E(x;x_0)$ is the task-dependent data term, and\n$R(x)$ is the image prior. In the case of inpainting, the\ntask-dependent data term is:\n\\begin{align}\n E(x;x_0) = || (x-x_0) \\circ m||^2\n\\end{align}\nwhere $\\circ$ is the Hadamart product, $m \\in \\{0,1\\}^{h\\times w}$\nrepresents the binary mask, and $h$ and $w$ are the image height and\nwidth. The image prior is usually captured through the training of\nconvolutional neural networks using a large image database. In the\napproach proposed by Ulyanov et al.~\\cite{Ulyanov2017}, $R(x)$ is replaced by a\nparameterization such that the optimization problem becomes:\n\\begin{align}\n \\min_\\theta{E(f_\\theta(z); x_0)} \\label{equ:loss}\n\\end{align}\nwhere $f$ represents the convolutional neural network with parameters\n$\\theta$ that is initialized randomly, and $z$ is a fixed input. The\nfixed input for the neural network can take many forms but is usually\nchosen to be random uniform noise or smoothly varying data.\nNote that the neural network input is \\textit{fixed}. Given a\ndeteriorated image, the neural network effectively learns, by\nbackpropagation and network parameter tuning, the encoding necessary\nto map the fixed input to an output, i.e.,\\,the \\textit{recovered}\nimage, which minimizes the loss function,~Equation\\,\\ref{equ:loss}.\n\nWe emphasize that physical constraints are not explicitly included in\nthe data recovery process. This has the advantage of enabling a\nreconstruction technique that does not depend on the physical nature\nof the problem. Higher fidelity can be achieved, however, by\nincorporating physical constraints, as suggested in~\\cite{Raissi2017,\n Sirignano2018}. The work presented here focuses on two-dimensional\nreconstruction, though there is no inherent methodological limitation\nto reconstructing three-dimensional data directly.\n\n\\section{Neural network architecture}\\label{sec:nn}\n\nThe network chosen for this work is a convolutional neural network\nthat exhibits an encoder-decoder architecture with approximately 2\nmillion tunable parameters and no skip connections,\nFigure\\,\\ref{fig:encoder_decoder}. This architecture enables the\nnetwork to encode the input in the latent space and then decode the\nlatent space representation into the reconstructed image. The\nnonlinear activation function used in the network is LeakyReLU\n\\cite{He2015}. Downsampling was performed through simple striding in\nthe convolution procedure, Figure\\,\\ref{fig:downsample}, and\nupsampling was done through nearest-neighbor upsampling,\nFigure\\,\\ref{fig:upsample}. The number of filters in the downsampling\nand upsampling layers, $n_f$, was kept fixed at 128, and the kernel\nsize, $k$, was fixed at 3. Experiments showed that using a fixed\nsmoothly varying input $z$ for the neural network imposes a smoothness\nprior, which is beneficial for data recovery for fluid flows. The\noptimization process was performed using Adam~\\cite{Kingma2014}. The\nimplementation was done in PyTorch~\\cite{Paszke2017}, and the learning\nprocess was computed on a Tesla V100 \\acrlong{gpu}. The number of\niterations for all the experiments was 2000, thereby reducing\nthe loss function by three orders of magnitude.\n\n\\begin{figure}[!tbp]%\n \\centering%\n \\begin{subfigure}[t]{1\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/encoder_decoder.pdf}%\n \\caption{Encoder-decoder architecture.}\\label{fig:encoder_decoder}%\n \\end{subfigure}\\\\[0.3cm]%\n \\begin{subfigure}[b]{0.33\\textwidth}%\n \\includegraphics[width=0.92\\textwidth]{./figs/downsample.pdf}%\n \\caption{Downsampling unit.}\\label{fig:downsample}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[b]{0.33\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/upsample.pdf}%\n \\caption{Upsampling unit.}\\label{fig:upsample}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[b]{0.15\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/legend.pdf}%\n \\vspace*{0.4cm}%\n \\caption{Legend.}\\label{fig:legend}%\n \\end{subfigure}%\n \\caption{Deep convolutional neural network used for data recovery.}\\label{fig:arch}%\n\\end{figure}%\n\n\\section{Results}\\label{sec:results}\n\nTo demonstrate the efficacy of using deep neural networks for data\nrecovery, we investigate two types of flows: laminar flow over a\ncylinder for data recovery of large flow scales and homogeneous\nisotropic turbulence for data recovery of flows spanning a wide range\nof scales. We compare the deep convolutional neural network results\nwith \\gls{gpr}. The sample points used to train the \\gls{gpr} are\nlocated in the region surrounding the mask with a depth of 10 cells,\nsimilar to~\\cite{Lee2015}. Beyond a depth of 10 cells surrounding\nthe masked regions, training the regressor becomes computationally\nintractable because the \\gls{gpr} complexity is $\\mathcal{O}(n^3)$, where\n$n$ is the number of training points. A radial basis function is used\nas the \\gls{gpr} kernel.\n\n\\subsection{Laminar flow around a cylinder}\\label{sec:cylinder}\nThe first numerical tests of the data recovery process are performed\nfor the laminar flow around a cylinder ($Re = 200$). The simulation is\nperformed using Nalu-Wind, a low Mach Navier-Stokes solver leveraging\nthe Trilinos\nlibraries\\footnote{\\url{https://github.com/Exawind/nalu-wind}}; and the\n$t=234\\,\\unit{s}$ snapshot is used for the numerical tests, at which\ntime the vortices behind the cylinder were fully developed. Masks\nsimulating data loss because of processor failure are generated in the\ncylinder wake. To capture typical domain decomposition methods for\nstructured grids, the masks are square boxes and vary in size\ndepending on the number of processors used for the simulation. The\nmask box length, $L_m$, ranged from $0.5D$ to $5D$, where $D$ is the\ncylinder diameter, and the masks are located at 40 random locations in\nthe cylinder wake, leading to 240 unique masks to be applied to the\nsimulation data. For the reconstruction process, reflection padding is\nused for the boundary conditions.\n\nAn example reconstruction is presented in\nFigure\\,\\ref{fig:illustrated_cyl} for $L_m=2D$, where the deep\nconvolutional neural network presents a slightly better reconstructed\nvelocity field than \\gls{gpr}. Specifically, the partially masked\nvortex is more accurately reconstructed using the deep convolutional\nneural network. The average $L_2$ error norm for the velocity fields\nas a function of $L_m$ is presented in\nFigure\\,\\ref{fig:cyl_error}. Both reconstruction techniques, \\gls{gpr}\nand the deep convolutional neural network, present similar error\nprofiles. At higher $L_m$ the neural network performs slightly better\nthan \\gls{gpr} for the $x$-direction velocity, whereas it performs\nsimilarly for all other lengths. Given the structured nature of the\nflow field, it is unsurprising that \\gls{gpr} performs well at\nmoderate mask sizes, given previously published\nresults~\\cite{Gunes2006}.\n\n\\begin{figure}[!tbp]%\n \\centering%\n \\begin{subfigure}[t]{0.45\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/cyl_umag0.png}%\n \\caption{Original data.}\\label{fig:cyl_umag0}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.45\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/cyl_umag0_masked.png}%\n \\caption{Deteriorated data.}\\label{fig:cyl_umag0_masked}%\n \\end{subfigure}\\\\%\n \\begin{subfigure}[t]{0.45\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/cyl_umagr.png}%\n \\caption{Deep convolutional neural network.}\\label{fig:cyl_umagr}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.45\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/cyl_umagi.png}%\n \\caption{\\gls{gpr}.}\\label{fig:cyl_umagi}%\n \\end{subfigure}%\n \\caption{Velocity magnitude for laminar flow around a cylinder where the mask box length is twice the cylinder diameter, $L_m=2D$.}\\label{fig:illustrated_cyl}%\n\\end{figure}%\n\n\\begin{figure}[!tbp]%\n \\centering%\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/cyl_error_u.png}%\n \\caption{$x$-direction velocity.}\\label{fig:cyl_error_u}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/cyl_error_v.png}%\n \\caption{$y$-direction velocity.}\\label{fig:cyl_error_v}%\n \\end{subfigure}%\n \\caption{Average $L_2$ error norm as a function of mask box length,\n $L_m$, for laminar flow past a cylinder ($Re=200$). Red squares:\n deep convolutional neural network; green diamonds:\n \\gls{gpr}.}\\label{fig:cyl_error}%\n\\end{figure}%\n\n\n\\subsection{Homogeneous isotropic turbulence}\\label{sec:hit}\n\nFor these numerical tests, we use two-dimensional slices of\nhomogeneous isotropic turbulence with a Taylor microscale Reynolds\nnumber $Re_\\lambda= \\nicefrac{\\rho_0 u' \\lambda}{\\mu}=133$, where\n$\\rho_0$ is the reference density,\n$u' = \\sqrt{\\nicefrac{\\overline{u_i u_i}}{3}}$ is the initial mean\nfluctuating velocity,\n$\\lambda = \\nicefrac{\\overline{u_1^2}}{\\overline{\\left(\\frac{\\partial\n u_1}{\\partial x_1}\\right)^2}}$ is the Taylor microscale, and\n$\\mu$ is the dynamic viscosity; a turbulent Mach number\n$M_t = \\nicefrac{u_0}{c_s} = 0.1$, where $c_s$ is the speed of sound;\nand a Prandtl number $Pr = \\nicefrac{\\mu c_p}{k} = 0.71$, where $c_p$\nis the heat capacity at constant pressure, and $k$ is the thermal\nconductivity. The reference temperature and pressure are 300~K and 1\natmosphere and the ideal gas equation of state is used to relate the\nthermodynamic quantities. The domain ranges from $[0,2\\pi]$ with\nperiodic boundary conditions. In this work, we use\nPeleC\\footnote{\\url{https://github.com/AMReX-Combustion/PeleC}}, an\nexplicit compressible Navier-Stokes flow solver based on the AMReX\nlibrary\\footnote{\\url{https://amrex-codes.github.io}}, to demonstrate\nthe data recovery process. For the reconstruction process, periodic,\ni.e.,\\,wrapped, padding was used for the boundary conditions.\n\nInitial two-dimensional data slices are generated by slicing in each\ndirection a numerical simulation of homogeneous isotropic turbulence\nat a resolution of 64 cells in each direction, leading to 192 unique\nslices. The velocities in each direction are assigned an input channel\nfor the neural network. Masks simulating data loss because of processor\nfailure are generated independently, following typical domain\ndecomposition. We explore two different parameters associated with the\nmask generation process. The first is the total percentage of missing\ndata, $f$, ranging from $6.25\\%$ to $25\\%$. The second is the length\nscale associated with each block of missing data, $L_m$, ranging from\n$3.125\\%$ to $50\\%$ of the domain length, or $0.74\\lambda$ to\n$11.87\\lambda$. For each pair of parameters $f$ and $L_m$, we randomly\ngenerate ten different masks, resulting in 130 unique masks. These are\nrandomly applied to 100 initial slices, resulting in 1300 slices\nrequiring reconstruction. For each of these deteriorated slices, the\nneural network parameters are tuned to optimize the reconstruction\nloss function, Equation\\,(\\ref{equ:loss}). The data from the resulting\nrecovered slice are then used for comparison with the original\nslice. The data recovery process is illustrated for one slice in\nFigure\\,\\ref{fig:illustrated_hit}.\n\n\\begin{figure}[!tbp]%\n \\centering%\n \\begin{subfigure}[t]{0.32\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/umag0.png}%\n \\caption{Original.}\\label{fig:original}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.32\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/umag0_masked.png}%\n \\caption{Deteriorated.}\\label{fig:masked}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.32\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/umagr.png}%\n \\caption{Recovered.}\\label{fig:result}%\n \\end{subfigure}%\n \\caption{Velocity magnitude in homogeneous isotropic turbulence\n illustrating the data recovery process, where 25\\% of the original\n data is missing and the length scale associated to each missing\n block is 6.25\\% of the domain\n ($1.5\\lambda$).}\\label{fig:illustrated_hit}%\n\\end{figure}%\n\nThe average error in $u'$ and $\\lambda$ for all the reconstructions is\napproximately three times larger for the \\gls{gpr} process compared to\nthe deep convolutional neural network. For all the slices, individual\nenergy spectra are calculated for the original data and the data\nrecovered through the deep convolutional neural network and\n\\gls{gpr}. The average energy spectrum is presented in\nFigure\\,\\ref{fig:spectra}. The average error from the \\gls{gpr}\nreconstruction increases at high wavenumbers, indicating that it not\nable to accurately capture the smallest scales of turbulence,\nFigure\\,\\ref{fig:error_spectra}. This behavior is not exhibited with\nthe deep neural network reconstruction, and the error increases\nslightly as a function of wavenumber.\n\n\\begin{figure}[!tbp]%\n \\centering%\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/spectra.png}%\n \\caption{Average energy spectrum.}\\label{fig:spectra}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/error_spectra.png}%\n \\caption{Normalized error, $e_{E_k} = \\frac{|E^h_k - E_k|}{E_k}$.}\\label{fig:error_spectra}%\n \\end{subfigure}%\n \\caption{Energy spectrum and error as a function of wavenumber $k$. Solid red: original data; dashed green: deep convolutional neural network; dot-dashed blue: \\gls{gpr}.}\\label{fig:spec}%\n\\end{figure}%\n\nVelocities in slices of the original and recovered data are used as\ninitial conditions for two-dimensional decay simulations. In these\nsimulations performed in PeleC, the initial conditions for the\nvelocity fields are taken from the (i) original data, (ii)\nrecovered data from the deep convolutional neural network, and (iii)\nrecovered data from the \\gls{gpr}. The final time for the simulations\nis $5\\tau$, where $\\tau =\n\\nicefrac{\\lambda}{u'}$. Figure\\,\\ref{fig:hit_enstrophy} illustrates\nthe decay of normalized enstrophy,\n$\\omega = \\frac{\\lambda^2}{u'^2 V}\\int_V \\left( \\nabla \\times u\n\\right)^2 \\mathrm{d}V$, for simulations where $f=25\\%$ and\n$L_m \\in [0.74\\lambda, 11.87\\lambda]$, or $3.125\\%$ to $50\\%$ of the\ndomain length. \\gls{gpr} reconstruction exhibits a significantly\ndifferent enstrophy decay from the original data,\nFigure\\,\\ref{fig:enstrophy_gp}. For the deep convolutional neural\nnetwork, the enstrophy decay is well captured at all mask sizes, with\nslightly less accuracy for large $L_m$,\nFigure\\,\\ref{fig:enstrophy_dl}. Across the range of mask sizes, the\nnormalized error at $t=1\\tau$ in kinetic energy and enstrophy is less than\n10\\% for the deep convolutional neural network and around 20\\% for the\n\\gls{gpr} reconstruction. These differences in reconstruction\nprocedures are attributed to the deep convolutional neural network's\nability to preserve the energy spectra and accurately represent all\nthe length scales.\n\n\\begin{figure}[!tbp]%\n \\centering%\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/enstrophy_result.png}%\n \\caption{Deep convolutional neural network.}\\label{fig:enstrophy_dl}%\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\includegraphics[width=\\textwidth]{./figs/enstrophy_interp.png}%\n \\caption{\\gls{gpr}.}\\label{fig:enstrophy_gp}%\n \\end{subfigure}%\n \\caption{Normalized enstrophy as a function of time where 25\\% of\n the original data is missing. Solid black: original data; solid\n red: $L_m=0.74\\lambda$; dashed green: $L_m=1.48\\lambda$;\n dot-dashed blue: $L_m=2.97\\lambda$; dotted orange:\n $L_m=5.94\\lambda$; dot-dot-dashed purple:\n $L_m=11.87\\lambda$.}\\label{fig:hit_enstrophy}%\n\\end{figure}%\n\n\n\\section{Conclusion}\\label{sec:ccl}\n\nThis work evaluated the use of deep convolutional neural networks for\ndata recovery of fluid flows in the context of data loss because of\nhardware or software faults. The method proposed here leverages an\nencoder-decoder deep convolutional neural network to transform a fixed\ninput to a recovered output using only deteriorated data and\neschewing a training database. Comparisons were performed with\n\\acrlong{gpr}, a standard data recovery algorithm often referred to as\nKriging. Data recovery was performed on two different canonical flow\nconfigurations: laminar flow around a cylinder and homogeneous\nisotropic turbulence. For data recovery of the laminar flow around a\ncylinder, results indicate similar performance between the proposed\nmethod and \\gls{gpr} across a wide range of mask sizes. For homogeneous\nisotropic turbulence, data recovery through the deep convolutional\nneural network exhibits an error in relevant turbulent quantities\napproximately three times smaller than that for the \\gls{gpr}. Forward\nsimulations using recovered data illustrate that the enstrophy decay\nis accurately captured using the deep convolutional neural network\napproach.\n\nWe emphasize that the work presented here is not necessarily beholden\nto the specific inpainting technique used. The deep learning community\nhas developed many different methods for image inpainting that can be\nused for data recovery, and it is expected that state-of-the-art\nmethodologies would perform comparably well. Rather, the use of deep\nimage priors as first proposed by Ulyanov et al.~\\cite{Ulyanov2017}\nand investigated here provides a convenient framework to perform data\nrecovery of fluid flows because it does not require pretraining the neural\nnetwork to construct image priors for different flows. It is therefore\nagnostic to the specific flow configuration, and the same framework can\nbe used for very different flows. This technique, however, does\nnecessitate the solution of an optimization problem for each data\nrecovery task. Future work will investigate alleviating this through\npartial pretraining, transfer learning, and perceptual loss\nfunctions.\n\nThis work --- including data sets, demonstration notebooks, analysis\nscripts, and figures --- can be publicly accessed at the project's GitHub\npage.\\footnote{\\url{https://github.com/NREL/deep-image-prior-cfd}} Traditional\nmachine learning algorithms were implemented through\nscikit-learn~\\cite{Pedregosa2011} and the deep learning algorithms\nthrough PyTorch~\\cite{Paszke2017}.\n\n\\section*{Acknowledgments}\nThis work was authored by the National Renewable Energy Laboratory, operated by Alliance for Sustainable Energy, LLC, for the U.S. Department of Energy (DOE) under Contract No. DE-AC36-08GO28308. Funding provided by U.S. Department of Energy Office of Science and National Nuclear Security Administration. The views expressed in the article do not necessarily represent the views of the DOE or the U.S. Government. The U.S. Government retains and the publisher, by accepting the article for publication, acknowledges that the U.S. Government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for U.S. Government purposes.\n\nThis research was supported by the Exascale Computing Project (ECP), Project Number: 17-SC-20-SC, a collaborative effort of two DOE organizations -- the Office of Science and the National Nuclear Security Administration -- responsible for the planning and preparation of a capable exascale ecosystem -- including software, applications, hardware, advanced system engineering, and early testbed platforms -- to support the nation's exascale computing imperative.\n\n\\section*{References}\n\\bibliography{library}\n\n\\end{document}", "meta": {"hexsha": "8f1f31728e92e5733a7d9fad66c0fe4c78c0a514", "size": 30605, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/report.tex", "max_stars_repo_name": "NREL/deep-image-prior-cfd", "max_stars_repo_head_hexsha": "85a86ac10bef070b1a973d2a6569849583e08d79", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2019-02-06T20:55:47.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-06T02:43:38.000Z", "max_issues_repo_path": "paper/report.tex", "max_issues_repo_name": "NREL/deep-image-prior-cfd", "max_issues_repo_head_hexsha": "85a86ac10bef070b1a973d2a6569849583e08d79", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/report.tex", "max_forks_repo_name": "NREL/deep-image-prior-cfd", "max_forks_repo_head_hexsha": "85a86ac10bef070b1a973d2a6569849583e08d79", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-01-20T10:29:08.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-24T21:55:41.000Z", "avg_line_length": 51.960950764, "max_line_length": 718, "alphanum_fraction": 0.7848064042, "num_tokens": 7914, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.34901492743438106}} {"text": "\\XtoCBlock{Sin3Gen}\r\n\\label{block:Sin3Gen}\r\n\\begin{figure}[H]\\includegraphics{Sin3Gen}\\end{figure} \r\n\r\n\\begin{XtoCtabular}{Inports}\r\nA & Amplitude\\tabularnewline\r\n\\hline\r\nf & Frequency\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n\r\n\\begin{XtoCtabular}{Outports}\r\nu & Sine wave output phase u\\tabularnewline\r\n\\hline\r\nv & Sine wave output phase v\\tabularnewline\r\n\\hline\r\nw & Sine wave output phase w\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n\\begin{XtoCtabular}{Mask Parameters}\r\nfmax & Maximum Frequency in Hz\\tabularnewline\r\n\\hline\r\nOffset & Offset\\tabularnewline\r\n\\hline\r\nts\\_fact & Multiplication factor of base sampling time (in integer format)\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n\\subsubsection*{Description:}\r\nGeneration of a 3 sine waves with amplitude (A) and frequency (f).\r\n\n% include optional documentation file\r\n\\InputIfFileExists{\\XcHomePath/Library/General/Doc/Sin3Gen_Info.tex}{\\vspace{1ex}}{}\r\n\r\n\\subsubsection*{Implementations:}\r\n\\begin{tabular}{l l}\r\n\\textbf{FiP8} & 8 Bit Fixed Point Implementation\\tabularnewline\r\n\\textbf{FiP16} & 16 Bit Fixed Point Implementation\\tabularnewline\r\n\\textbf{FiP32} & 32 Bit Fixed Point Implementation\\tabularnewline\r\n\\textbf{Float32} & 32 Bit Floating Point Implementation\\tabularnewline\r\n\\textbf{Float64} & 64 Bit Floating Point Implementation\\tabularnewline\r\n\\end{tabular}\r\n\r\n\\XtoCImplementation{FiP8}\r\n\\index{Block ID!432}\r\n\\nopagebreak[0]\r\n% Implementation details\r\n\\begin{tabular}{l l}\r\n\\textbf{Name} & FiP8 \\tabularnewline\r\n\\textbf{ID} & 432 \\tabularnewline\r\n\\textbf{Revision} & 1.0 \\tabularnewline\r\n\\textbf{C filename} & Sin3Gen\\_FiP8.c \\tabularnewline\r\n\\textbf{H filename} & Sin3Gen\\_FiP8.h \\tabularnewline\r\n\\end{tabular}\r\n\\vspace{1ex}\r\n\r\n8 Bit Fixed Point Implementation\r\n\r\n\\begin{XtoCtabular}{Controller Parameters}\r\ndelta\\_phi & Angle increment\\tabularnewline\r\n\\hline\r\noffset & Amplitude offset\\tabularnewline\r\n\\hline\r\nphi & Current angle\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n% Implementation data structure\r\n\\XtoCDataStruct{Data Structure:}\r\n\\begin{lstlisting}\r\ntypedef struct {\r\n uint16 ID;\r\n int8 *A;\r\n int8 *f;\r\n int8 u;\r\n int8 v;\r\n int8 w;\r\n int8 delta_phi;\r\n int8 offset;\r\n int8 phi;\r\n} SIN3GEN_FIP8;\r\n\\end{lstlisting}\r\n\r\n\\ifdefined \\AddTestReports\r\n\\InputIfFileExists{\\XcHomePath/Library/General/Doc/Test_Sin3Gen_FiP8.tex}{}{}\r\n\\fi\r\n\\XtoCImplementation{FiP16}\r\n\\index{Block ID!433}\r\n\\nopagebreak[0]\r\n% Implementation details\r\n\\begin{tabular}{l l}\r\n\\textbf{Name} & FiP16 \\tabularnewline\r\n\\textbf{ID} & 433 \\tabularnewline\r\n\\textbf{Revision} & 1.0 \\tabularnewline\r\n\\textbf{C filename} & Sin3Gen\\_FiP16.c \\tabularnewline\r\n\\textbf{H filename} & Sin3Gen\\_FiP16.h \\tabularnewline\r\n\\end{tabular}\r\n\\vspace{1ex}\r\n\r\n16 Bit Fixed Point Implementation\r\n\r\n\\begin{XtoCtabular}{Controller Parameters}\r\ndelta\\_phi & Angle increment\\tabularnewline\r\n\\hline\r\noffset & Amplitude offset\\tabularnewline\r\n\\hline\r\nphi & Current angle\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n% Implementation data structure\r\n\\XtoCDataStruct{Data Structure:}\r\n\\begin{lstlisting}\r\ntypedef struct {\r\n uint16 ID;\r\n int16 *A;\r\n int16 *f;\r\n int16 u;\r\n int16 v;\r\n int16 w;\r\n int16 delta_phi;\r\n int16 offset;\r\n int16 phi;\r\n} SIN3GEN_FIP16;\r\n\\end{lstlisting}\r\n\r\n\\ifdefined \\AddTestReports\r\n\\InputIfFileExists{\\XcHomePath/Library/General/Doc/Test_Sin3Gen_FiP16.tex}{}{}\r\n\\fi\r\n\\XtoCImplementation{FiP32}\r\n\\index{Block ID!434}\r\n\\nopagebreak[0]\r\n% Implementation details\r\n\\begin{tabular}{l l}\r\n\\textbf{Name} & FiP32 \\tabularnewline\r\n\\textbf{ID} & 434 \\tabularnewline\r\n\\textbf{Revision} & 1.0 \\tabularnewline\r\n\\textbf{C filename} & Sin3Gen\\_FiP32.c \\tabularnewline\r\n\\textbf{H filename} & Sin3Gen\\_FiP32.h \\tabularnewline\r\n\\end{tabular}\r\n\\vspace{1ex}\r\n\r\n32 Bit Fixed Point Implementation\r\n\r\n\\begin{XtoCtabular}{Controller Parameters}\r\ndelta\\_phi & Angle increment\\tabularnewline\r\n\\hline\r\noffset & Amplitude offset\\tabularnewline\r\n\\hline\r\nphi & Current angle\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n% Implementation data structure\r\n\\XtoCDataStruct{Data Structure:}\r\n\\begin{lstlisting}\r\ntypedef struct {\r\n uint16 ID;\r\n int32 *A;\r\n int32 *f;\r\n int32 u;\r\n int32 v;\r\n int32 w;\r\n int32 delta_phi;\r\n int32 offset;\r\n int32 phi;\r\n} SIN3GEN_FIP32;\r\n\\end{lstlisting}\r\n\r\n\\ifdefined \\AddTestReports\r\n\\InputIfFileExists{\\XcHomePath/Library/General/Doc/Test_Sin3Gen_FiP32.tex}{}{}\r\n\\fi\r\n\\XtoCImplementation{Float32}\r\n\\index{Block ID!435}\r\n\\nopagebreak[0]\r\n% Implementation details\r\n\\begin{tabular}{l l}\r\n\\textbf{Name} & Float32 \\tabularnewline\r\n\\textbf{ID} & 435 \\tabularnewline\r\n\\textbf{Revision} & 0.1 \\tabularnewline\r\n\\textbf{C filename} & Sin3Gen\\_Float32.c \\tabularnewline\r\n\\textbf{H filename} & Sin3Gen\\_Float32.h \\tabularnewline\r\n\\end{tabular}\r\n\\vspace{1ex}\r\n\r\n32 Bit Floating Point Implementation\r\n\r\n\\begin{XtoCtabular}{Controller Parameters}\r\ndelta\\_phi & Angle increment\\tabularnewline\r\n\\hline\r\noffset & Amplitude offset\\tabularnewline\r\n\\hline\r\nphi & Current angle\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n% Implementation data structure\r\n\\XtoCDataStruct{Data Structure:}\r\n\\begin{lstlisting}\r\ntypedef struct {\r\n uint16 ID;\r\n float32 *A;\r\n float32 *f;\r\n float32 u;\r\n float32 v;\r\n float32 w;\r\n float32 delta_phi;\r\n float32 offset;\r\n float32 phi;\r\n} SIN3GEN_FLOAT32;\r\n\\end{lstlisting}\r\n\r\n\\ifdefined \\AddTestReports\r\n\\InputIfFileExists{\\XcHomePath/Library/General/Doc/Test_Sin3Gen_Float32.tex}{}{}\r\n\\fi\r\n\\XtoCImplementation{Float64}\r\n\\index{Block ID!436}\r\n\\nopagebreak[0]\r\n% Implementation details\r\n\\begin{tabular}{l l}\r\n\\textbf{Name} & Float64 \\tabularnewline\r\n\\textbf{ID} & 436 \\tabularnewline\r\n\\textbf{Revision} & 0.1 \\tabularnewline\r\n\\textbf{C filename} & Sin3Gen\\_Float64.c \\tabularnewline\r\n\\textbf{H filename} & Sin3Gen\\_Float64.h \\tabularnewline\r\n\\end{tabular}\r\n\\vspace{1ex}\r\n\r\n64 Bit Floating Point Implementation\r\n\r\n\\begin{XtoCtabular}{Controller Parameters}\r\ndelta\\_phi & Angle increment\\tabularnewline\r\n\\hline\r\noffset & Amplitude offset\\tabularnewline\r\n\\hline\r\nphi & Current angle\\tabularnewline\r\n\\hline\r\n\\end{XtoCtabular}\r\n\r\n% Implementation data structure\r\n\\XtoCDataStruct{Data Structure:}\r\n\\begin{lstlisting}\r\ntypedef struct {\r\n uint16 ID;\r\n float64 *A;\r\n float64 *f;\r\n float64 u;\r\n float64 v;\r\n float64 w;\r\n float64 delta_phi;\r\n float64 offset;\r\n float64 phi;\r\n} SIN3GEN_FLOAT64;\r\n\\end{lstlisting}\r\n\r\n\\ifdefined \\AddTestReports\r\n\\InputIfFileExists{\\XcHomePath/Library/General/Doc/Test_Sin3Gen_Float64.tex}{}{}\r\n\\fi\r\n", "meta": {"hexsha": "430e016fad940d82610595d72fd13c493bc3d96c", "size": 6878, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Library/General/Doc/Sin3Gen.tex", "max_stars_repo_name": "AlexisTM/X2C", "max_stars_repo_head_hexsha": "31f39b598afe271a7fd46ef1ee9e06c410b1120c", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Library/General/Doc/Sin3Gen.tex", "max_issues_repo_name": "AlexisTM/X2C", "max_issues_repo_head_hexsha": "31f39b598afe271a7fd46ef1ee9e06c410b1120c", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Library/General/Doc/Sin3Gen.tex", "max_forks_repo_name": "AlexisTM/X2C", "max_forks_repo_head_hexsha": "31f39b598afe271a7fd46ef1ee9e06c410b1120c", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.3524904215, "max_line_length": 90, "alphanum_fraction": 0.6930793835, "num_tokens": 2099, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3490149273557987}} {"text": "\\section{Conclusion}\\label{sec:conclusion}\n\nPrice response functions provide quantitative information on the deviation from\nMarkovian behavior. They measure price changes resulting from execution of\nmarket orders. We used these functions in big data analysis for spot foreign\nexchange markets. Such a study was, to the best of our knowledge, never done\nbefore.\n\nWe analyzed price response functions in spot foreign exchange markets for\ndifferent years and different time scales. We used trade time scale and\nphysical time scale to compute the price response functions for the seven major\nforeign exchange pairs for three different years. These major pairs are highly\nrelevant in the dynamics of the market. The use of different time scales and\ncalendar years in the work had the intention to display the different behaviors\nthe price response function could take when the time parameters differ.\n\nThe price response functions were analyzed according to the time scales. On\ntrade time scale, the signals were noisier. For both time scales we observe\nthat the signal for all the pairs increases to a maximum and then starts to\nslowly decrease. However, for the year 2008 the shape of the signals is not as\nwell defined as in the other years. The increase-decrease behavior observed in\nthe spot foreign exchange market was also reported in correlated financial\nmarkets \\cite{my_paper_response_financial,Wang_2016_avg}. These results show\nthat the price response functions conserve their behavior in different years\nand in different markets. The shape of the price\nresponse functions is qualitatively explained considering an initial increase\ncaused by the autocorrelated transaction flow. To assure diffusive prices,\nprice response flattens due to market liquidity adapting to the flow in the\ninitial increase.\n\nOn both scales, the more liquid pairs have a smaller price response function\ncompared with the non-liquid pairs. As the liquid pairs have more trades during\nthe market time, the impact of each trade is reduced. Comparing years and\nscales, the price response signal is stronger in past than in recent years. As\nalgorithmic trading has gained great relevance, the quantity of trades has\ngrown in recent years, and in consequence, the impact in the response has\ndecreased.\n\nFinally, we checked the pip bid-ask spread impact in price response functions\nfor three different years. We used 46 foreign exchange pairs and grouped them\ndepending on the conditions of the corresponding year analyzed. We employ the\nyear average pip bid-ask spread of every pair for each year. For all the year\nand time scales, the price response function signals were stronger for the\ngroups of pairs with larger pip bid-ask spreads and weaker for the group of\npairs with smaller bid-ask spreads. For the average of the price response\nfunctions, it was only possible to see the increase-maximum-decrease behavior\nin the year 2015 in both scales, and in the year 2019 on trade time scale.\nHence, the noise in the cross and exotic pairs due to the lack of trading\ncompared with the majors seems stronger. A general average price response\nbehavior for each year and time scale was spotted for the groups, suggesting a\nmarket effect on the foreign exchange pairs in each year.\n\nComparing the response functions in stock and spot currency exchange markets\nfrom a more general viewpoint, we find a remarkable similarity. It triggers the\nconclusion that the order book mechanism generates in a rather robust fashion\nthe observed universal features in these two similar, yet different subsystems\nwithin the financial system.", "meta": {"hexsha": "b90bbfe7996beb0321fe8ceb3d86757bf64be699", "size": 3598, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/forex_response_spread_paper/sections/09_conclusion.tex", "max_stars_repo_name": "juanhenao21/forex", "max_stars_repo_head_hexsha": "251ccccfc9a49f546db5e325ea6b594ff035d97f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-04-01T07:22:34.000Z", "max_stars_repo_stars_event_max_datetime": "2020-04-01T07:22:34.000Z", "max_issues_repo_path": "paper/forex_response_spread_paper/sections/09_conclusion.tex", "max_issues_repo_name": "juanhenao21/forex", "max_issues_repo_head_hexsha": "251ccccfc9a49f546db5e325ea6b594ff035d97f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 18, "max_issues_repo_issues_event_min_datetime": "2020-03-17T09:30:08.000Z", "max_issues_repo_issues_event_max_datetime": "2020-03-27T08:43:29.000Z", "max_forks_repo_path": "paper/forex_response_spread_paper/sections/09_conclusion.tex", "max_forks_repo_name": "juanhenao21/forex", "max_forks_repo_head_hexsha": "251ccccfc9a49f546db5e325ea6b594ff035d97f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 63.1228070175, "max_line_length": 79, "alphanum_fraction": 0.8224013341, "num_tokens": 727, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3490149188820786}} {"text": "\\section{User Guide}\nThis section contains descriptions of how the gravity effector code works and includes descriptions of and notes on variables. It should be helpful to users who wish to use the gravity effector module.\n\n\\subsection{Code Diagram}\nThe diagram in Fig. \\ref{img:codeFlow} demonstrates the basic iterative logic of the gravity effector module. There is extensive additional code that deals with things from the messaging system to transforming the spacecraft position from one frame to another. In general, the inputs shown must be given for each gravity body to be considered. There are, however, convenience functions which add the standard values of these inputs for common celestial bodies. An example is simIncludeGravbody.createEarth() in the test\\_scenarioOrbitManeuver.py tutorial.\n\nAfter the inputs are given for each gravity body, the computeGravityInertial() method calculates the $0^{\\textrm{th}}$ degree gravity term for each body and its effects on the spacecraft. If useSphericalHarmParams is True for a given body, then computeField() is called to calculate higher degree spherical harmonics terms for that body.\n\n\\begin{figure}[H]\n\t\\centering \\includegraphics[height=1.0\\textwidth, keepaspectratio]{Figures/codeFlow.pdf}\n\t\\caption{A pseudo-code diagram demonstrating the flow of inputs and outputs in the gravity effector module.}\n\t\\label{img:codeFlow}\n\\end{figure}\n\n\\subsection{Variable Definition and Code Description}\nThe variables in Table \\ref{tabular:vars} are available for user input. Variables used by the module but not available to the user are not mentioned here. Variables with default settings do not necessarily need to be changed by the user, but may be.\n\\begin{table}[H]\n\t\\caption{Definition and Explanation of Variables Used.}\n\t\\label{tab:errortol}\n\t\\centering \\fontsize{10}{10}\\selectfont\n\t\\begin{tabular}{ | m{3cm}| m{3cm} | m{3cm} | m{6cm} |} % Column formatting, \n\t\t\\hline\n\t\t\\textbf{Variable} \t\t\t\t\t\t\t& \\textbf{LaTeX Equivalent} \t&\t\t\\textbf{Variable Type} & \\textbf{Notes}\t\t\t \\\\ \\hline\n\t\tspherHarm.maxDeg\t\t\t\t\t&$l_{\\text{max}}$\t\t \t & double & Default setting: 0\"inertial\\_state number of degree to use when calculating gravity effects using sperical harmonics.\\\\ \\hline\n\t\tradEquator\t\t\t & $R_{\\mathrm{ref}}^{l}$\t\t\t& double & [m] Default setting: 0.0. \tThis is the reference radius of the gravity body.\\\\ \\hline\n\t\tmuBody\t\t\t\t\t& $\\mu$ \t\t& double & [m3/s2] Default setting: 0.0f. This is the gravitational parameter of the body. Required Input to get any non-zero values out of the code.\\\\ \\hline\n\t\tisCentralBody & N/A & bool & Default setting: False. Determines whether the body in question is the central body and if initial spacecraft position and velocity are determined as relative or inertial.\\\\ \\hline\n\t\tisDisplayBody & N/A & bool & Default setting: False. Determines whether the body in question is the focus of the visualization\\\\ \\hline\n\t\tephemTime & N/A & double & [s] Default setting: 0. The ephemeris time for the body in question \\\\ \\hline\n\t\tephIntTime & N/A & double & [s] Default setting: 0. Required Input. The integration time associated with the ephem data. \\\\ \\hline\n\t\tbodyInMsgName & N/A & string & Required Input. The name of the message with gravity body data in the body frame. \\\\ \\hline\n\t\toutputMsgName & N/A & string & Required Input. The name of the message containing ephemeris information in display frame \\\\ \\hline\n\t\tplanetEphemName & N/A & string & Required Input. An ephemeris name for the planet (user-named). \\\\ \\hline\n\t\t\\label{tabular:vars}\n\t\\end{tabular}\n\\end{table}\n\n\\subsection{Using Central Bodies and Relative Dynamics}\n\\subsubsection{Using Central Bodies}\nIn simulations with multiple planetary bodies, using dynamics relative to a central body can improve accuracy. Generally, this is the right thing to do rather than using an absolute coordinate set. If a user has a gravBody called \\verb|earth|, the central body flag should be set to True.\n\\verb|earth.isCentralBody = True|\t\nThe dynamics will then take care of themselves, but the user needs to be careful to input initial position and velocity values as \\textit{relative to} the central body. This can be input from a set of Keplerian orbital elements using \\verb|orbitalMotion.elem2rv| as in\\\\ \\verb|Basilisk/tests/scenarios/scenarioBasicOrbit.py|. \n\nThe user should be aware that if spacecraft position and velocity are read back from a message log or plotted that the absolute position and velocity will be returned. It will take additional work to convert the outputs back to a relative form by subtracting out the central body positions and velocities. No rotation will be needed, though.\nIt is critical that the relative position and velocities are given in a frame which is linearly translated but \\textbf{not rotated} from the simulation inertial frame. There is no handling of rotated relative frames within the dynamics. The orbital element to position and velocity conversion in the section below can be used for relative dynamics inputs, as well.\n\n\\subsubsection{Not Using Central Bodies}\nIf no planets are designated as central bodies, an absolute initial position and velocity must be given. Again, \\verb|orbitalMotion.elem2rv| can be used if the orbital elements are given in a frame not rotated from the simulation inertial frame. However, now, the initial position and velocity of the central body must be accounted for. These can be retrieved from spice via the planetStates utility:\\\\\\\\\n\\verb|oe = om.ClassicElements()|\\\\\n\\verb|oe.a = orbit_a * 1000 #m, orbit semi-major axis|\\\\\n\\verb|oe.e = orbit_e #eccentricity|\\\\\n\\verb|oe.i = radians(orbit_i) #inclination, radians.|\\\\\n\\verb|oe.Omega = radians(orbit_O) # orbit RAA, radians|\\\\\n\\verb|oe.omega = radians(orbit_o) #orbit argument of periapsis, radians|\\\\\n\\verb|oe.f = radians(orbit_f) # orbit true anomaly, radians|\\\\\n\\verb|r_sc_E, v_sc_E = om.elem2rv(muEarth, oe) #get xyz coordinates from keplerian elements|\\\\\\\\\n\\verb|ephemerides = spice_interface.SpiceInterface()|\\\\\n\\verb|ephemerides.ModelTag = \"SpiceInterfaceData\"|\\\\\n\\verb|ephemerides.SPICEDataPath = splitPath[0] + '../supportData/EphemerisData/'|\\\\\n\\verb|ephemerides.outputBufferCount = 2|\\\\\n\\verb|ephemerides.planetNames = spice_interface.StringVector([\"earth\", \"sun\"])|\\\\\n\\verb|ephemerides.UTCCalInit = simStart #pick a UTC string|\\\\\n\\verb|earthPos_N, earthVel_N = planetStates.planetPositionVelocity('EARTH', simStart)|\\\\\n\\verb|r_sc_N = array(r_sc_E).flatten() + array(earthPos_N).flatten()|\\\\\n\\verb|v_sc_N = array(v_sc_E).flatten() + array(earthVel_N).flatten()|\\\\\n\\verb|scObject.hub.r_CN_NInit = array(r_sc_N)|\\\\\n\\verb|scObject.hub.v_CN_NInit = array(v_sc_N)|\\\\\\\\\nOf course, if a user has initial positions and velocities directly, those should be used. See scenarioCentralBody.py for a working example.\\\\\n\n\\subsubsection{Reference Frames}\nAn understanding of spice reference frames will help to explain the code above. The spice inertial frame is the ICRF. The ICRF is coplaner with the Earth's equator. Generally, the Earth Centered Inertial system one would give Keplerian elements in is aligned with ICRF. ICRF is referred to within spice as \"j2000\" for legacy reasons and because the J2000 system is only rotated from the ICRF by a few milliarcseconds.\n\n", "meta": {"hexsha": "2792376cca7b13e0311f20b11143de66eb23249f", "size": 7244, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/simulation/dynamics/gravityEffector/_Documentation/secUserGuide.tex", "max_stars_repo_name": "ian-cooke/basilisk_mag", "max_stars_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_stars_repo_licenses": ["0BSD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/simulation/dynamics/gravityEffector/_Documentation/secUserGuide.tex", "max_issues_repo_name": "ian-cooke/basilisk_mag", "max_issues_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_issues_repo_licenses": ["0BSD"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-03-13T20:52:22.000Z", "max_issues_repo_issues_event_max_datetime": "2019-03-13T20:52:22.000Z", "max_forks_repo_path": "src/simulation/dynamics/gravityEffector/_Documentation/secUserGuide.tex", "max_forks_repo_name": "ian-cooke/basilisk_mag", "max_forks_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_forks_repo_licenses": ["0BSD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 99.2328767123, "max_line_length": 555, "alphanum_fraction": 0.7715350635, "num_tokens": 1865, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3490149188820786}} {"text": "\\documentclass[10pt]{asme2ej}\n\n% Edited as https://www.overleaf.com/project/5f8228e5060d380001e71b21\n\n\\usepackage{epsfig} %% for loading postscript figures\n\\usepackage{placeins}\n\\usepackage[all]{xy}\n\\usepackage{dsserif}\n\\usepackage{listings}\n\\usepackage{xcolor}\n\\definecolor{codegreen}{rgb}{0,0.6,0}\n\\definecolor{codegray}{rgb}{0.5,0.5,0.5}\n\\definecolor{codepurple}{rgb}{0.58,0,0.82}\n\\definecolor{backcolour}{rgb}{0.95,0.95,0.92}\n\\lstdefinestyle{mystyle}{\n backgroundcolor=\\color{backcolour}, \n commentstyle=\\color{codegreen},\n keywordstyle=\\color{magenta},\n numberstyle=\\tiny\\color{codegray},\n stringstyle=\\color{codepurple},\n basicstyle=\\ttfamily\\footnotesize,\n breakatwhitespace=false, \n breaklines=true, \n captionpos=b, \n keepspaces=true, \n numbers=left, \n numbersep=5pt, \n showspaces=false, \n showstringspaces=false,\n showtabs=false, \n tabsize=2\n}\n\n\\lstset{style=mystyle}\n\n\\newcommand{\\catname}[1]{\\textbf{#1}}\n\\newcommand{\\Set}{\\catname{Set}}\n\\newcommand{\\Rel}{\\catname{Rel}}\n\n%% The class has several options\n% onecolumn/twocolumn - format for one or two columns per page\n% 10pt/11pt/12pt - use 10, 11, or 12 point font\n% oneside/twoside - format for oneside/twosided printing\n% final/draft - format for final/draft copy\n% cleanfoot - take out copyright info in footer leave page number\n% cleanhead - take out the conference banner on the title page\n% titlepage/notitlepage - put in titlepage or leave out titlepage\n% \n%% The default is oneside, onecolumn, 10pt, final\n\n\n\\title{Intuitionistic Logic Implemented in Scala\\\\\nSubmitted for Publication at SAI Computing Conference, London, 2021}\n\n%%% first author\n\\author{Vlad Patryshev}\t\n\n\n\n\\begin{document}\n\\UseRawInputEncoding\n\n\\maketitle \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{abstract}\n{\\it \nAn implementation of intuitionistic (Grothendieck topos-based, for a finite site) is described, with a variety of examples. \n\nThis paper demonstrates how non-Boolean logic can be implemented, to enable using all the power of Intuitionism. For that, a library was developed that works with topos logic. Sample code shows the logic for a variety of sites. \n}\n\\end{abstract}\n\n\\section{Introduction}\n\n In this paper I present a Scala implementation of topos logic (that is, an intuitionistic logic). Intuitionism includes all kinds of fuzzy and temporal logic (see TLA+). Having a code package modeling Intuitionism, one can experiment with different kinds of fuzziness and different shape s of time flow. The toposes used are Grothendieck toposes, and the sites are finite. \n \n Two categories (toposes) are chosen over which Grothendieck toposes are built. These categories are $Set$ and $Setf$, that is, a category of all (available in the code) sets, and the category of all finite sets available in the code. Since Scala does not exactly implement sets as are defined in set theories, a couple of new classes are introduced to resolve this issue.\n\nThis paper demonstrates how non-Boolean logic can be implemented, to enable using all the power of Intuitionism. For that, a library was developed that works with topos logic. Sample code shows the logic for a variety of sites.\nThe code is located at \n\\verb+https://github.com/vpatryshev/Categories+.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Non-Boolean Logic}\nLet us first introduce the idea of non-Booleanness, in a strict mathematical sense of the term. \n\n\\subsection{The Meaning of Non-Booleanness}\n\nTraditionally, as soon as we switch from the complicated logic of the real world into the realm of formal software design, we assume that just two outcomes are possible for any statement: it is either true or false. \nThis is not a feature of mathematical discourse but rather an interesting feature of the modern way of thinking. \nIn real life, we admit that there are more possibilities: something we may not know today, we may learn tomorrow. \nWe may assume something to be true with a certain probability. \nSimilarly, in science we may not know the answer, and the answer may be something we don't expect - like with the case of Continuum Hypothesis.\n\nIn intuitionistic logic, the double negation law does not have to hold. If it does, the logic is Boolean. Here we will show a way to eliminate the double negation law while keeping the remaining rules and axioms that don't depend on double negation.\nOf course, if the double negation law is included, we still have an intuitionistic logic that is also Boolean.\n\nAlso a logic may have more than two logical constants, but that does not make it non-Boolean. \nThe double negation rule may still be applicable in such a logic.\n\n\\subsubsection{Example 1. Boolean Logic, but not 2-Valued}\n\nConsider bytes and their operations: bitwise conjunction, bitwise disjunction, and negation. There are 256 different values, but we know\nthat double negation in this bitwise logic is an identity. \nSo, this 256-valued logic is Boolean.\\\\\n\n\\subsection{Dropping Booleanness}\n\nRemove the rule stating that either $P$ or $\\neg P$ is true.\nFormally, that $\\vdash P \\lor \\neg P$. Once we remove this rule, \nwe cannot apply it anymore in general settings, although it may still work in some cases.\n\nWe immediately bump into a problem: this rule was used to define\nimplication: $P\\ \\rightarrow\\ Q\\ \\equiv\\ \\neg\\ P\\ \\lor\\ Q$. \nWe must define implication differently. Here's a reasonable alternative solution:\n\n$$(P \\land Q)\\vdash R \\equiv P \\vdash ( Q \\rightarrow R)$$\n\nInformally, saying that $R$ can be deduced from a conjunction $P\\land Q$\nis the same as saying that we can deduce the implication $Q\\rightarrow R$ from $P$. \nImagine $Q$ is $\\top$. Then, from the formula\nabove, it follows that $Q\\rightarrow R$ is the same as R. On the other hand,\nif $Q$ is closer to the bottom, the situation may change. \nFor example, if $Q$ is “below” $R$, the implication $Q\\rightarrow R$ is true.\nWe can interpret the implication defined this way as the level of dependency of $R$ on $Q$.\\\\\n\nAs you see, we can define implication via conjunction, and negation is not involved.\\\\\n\nDoes this remind you currying? Given a function $f(P,Q):R$, we\ntransform it to a function $f_c(P): Q \\rightarrow R$.\\\\\n\nNow that implication is defined, we can define negation as\n$\\neg P = P\\rightarrow \\bot$. Its properties make it\nsimilar to classical negation, but the double negation law is not generally applicable anymore. Assume we use the definition above. \nIn the table below, we list some properties that are valid for this definition.\n\\FloatBarrier\n\\begin{table}[!htbp]\n\\begin{center}\n\\label{table_Sample_Statements}\n\\begin{tabular}{c l l}\n& & \\\\ \n\\hline\nStatement & Meaning \\\\\n\\hline\n$P \\land \\neg P \\vdash \\bot$ & Negation of $P$ is incompatible with $P$.\\\\\n $\\neg \\neg \\neg P \\vdash \\neg P$ & Triple negation is the same as single negation.\\\\\n $P \\vdash \\neg \\neg P$ & Double negation of $P$ is weaker than $P$.\\\\\n $\\neg P \\lor \\neg Q \\vdash \\neg (P \\land Q)$ & If not $P$ or not $Q$, then we can’t have both $P$ and $Q$\\\\\n $\\neg (P \\lor Q) \\vdash \\neg P \\land \\neg Q$ & If disjunction of $P$ and $Q$ is not true, then neither $P$ nor $Q$ is true.\\\\\n $\\neg P \\land \\neg Q \\vdash \\neg (P \\lor Q)$ & If neither $P$ nor $Q$ is true, their disjunction is not true.\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nSome properties do not apply in this kind of logic:\n\\begin{table}[!htbp]\n\\begin{center}\n\\label{table_Sample_Wrong_Statements}\n\\begin{tabular}{c l l}\n& & \\\\ \n\\hline\nWrong Statement & What’s wrong with it? \\\\\n\\hline\n$\\neg \\neg P \\vdash P$ & Negation of negation of $P$ is not strong enough to give us $P$.\\\\\n\n $\\neg (P \\land Q) \\vdash \\neg P\\lor \\neg Q$ & Even if we can’t have both at the same time,\n this does not mean that one of them is always wrong.\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsubsection{Example 2. Three Logical Values (“Ternary Logic”)}\n\nStart with three logical values, $\\top$, $\\bot$, and $?$. Define operations for them:\n\n\\begin{center}\n\\begin{tabular}{ |c|c|c| }\n\\hline\n$\\textbf{x}$ & $\\textbf{\\neg x}$ & $\\textbf{\\neg \\neg x}$ \\\\\n\\hline\n$\\top$ & $\\bot$ & $\\top$ \\\\\n$?$ & $\\bot$ & $\\top$ \\\\\n$\\bot$ & $\\top$ & $\\bot$ \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\n\\begin{center}\n\\begin{tabular}{ |c|c|c|c|c| }\n\\hline\n$\\textbf{x}$ & $\\textbf{y}$ & ${\\textbf{x} {\\land} \\textbf{y}}$ & $\\textbf{x} \\lor \\textbf{y}$ & $\\textbf{x} \\rightarrow \\textbf{y}$ \\\\\n\\hline\n$\\top$ & $\\top$ & $\\top$ & $\\top$ & $\\top$ \\\\\n$\\top$ & $?$ & $?$ & $\\top$ & $?$ \\\\\n$\\top$ & $\\bot$ & $\\bot$ & $\\top$ & $\\bot$ \\\\\n $?$ & $\\top$ & $?$ & $\\top$ & $\\top$ \\\\\n $?$ & $?$ & $?$ & $?$ & $\\top$ \\\\\n $?$ & $\\bot$ & $\\bot$ & $?$ & $\\bot$ \\\\\n$\\bot$ & $\\top$ & $\\bot$ & $\\top$ & $\\top$ \\\\\n$\\bot$ & $?$ & $\\bot$ & $?$ & $\\top$ \\\\\n$\\bot$ & $\\bot$ & $\\bot$ & $\\bot$ & $\\top$ \\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\nThis is the simplest intuitionistic logic, and it is a good tool for testing our statements.\n\nThe value ‘?,’ or ‘unknown,’ is somewhere between ‘truth’ $\\top$ and ‘false’ $\\bot$.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsubsection{Example 3. Infinite Number of Logical Values}\n\nTake $[0,1]$, the set of all real numbers between $0$ and $1$, including\n$0$ and $1$. We can turn it into logic by defining:\n\n- $\\bot = 0$\n\n- $\\top =1$\n\n- $a \\land b = glb(a,b)$\n\n- $a \\lor b = lub(a,b)$\n\nIn this example, implication is defined, as always, via the equivalence:\n$(x~\\land~y)~\\leq~z~\\equiv~x~\\leq~(y~\\rightarrow~z)$. \n\n\nWe consider the following two cases separately:\n \n\nFirst, when $y\\leq z$, \n$(x~\\land~y)~\\leq~z$ is always true, for any $x$. Hence $y~\\rightarrow~z$\ncannot be smaller than any $x$, that is, it must be the top element, $1$.\n\n\nIn the opposite case, when $y$ is bigger than $z$ (we are talking about numbers),\n$( x \\land y \\leq z) \\equiv ( x \\leq z )$, so now we will have \n$x~\\leq~z~\\equiv~x~\\leq~(y~\\rightarrow~z)$, which means that $y~\\rightarrow~z~=~z$.\n\nRemember that we defined negation, $\\neg x$, as $x\\rightarrow 0$, and we have\n$\\neg x\\ \\equiv\\ if\\ (x = 0)\\ 1\\ else\\ 0$. You can also check whether double\nnegation maps $0$ to $0$ and any non-zero value to $1$.\n\nIn this example, we started with a set of values that was popular in \"fuzzy logic,\" and we were able to build\na pretty sound logic, except that it could not be made Boolean: double negation is not an identity.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Sets and Categories}\n \\\\\n \n\\subsection{Sets}\n \\\\\nScala programming language uses a pretty loose notion of sets; they have too many methods (like \\verb+head()+), and functions are not defined as \"special sets of pairs\", as they are defined in ZFC. So we had to introduce a special class for working with sets. Class $BigSet$ is the most general class defining sets; it does not even have to be enumerable.\n\n\\begin{lstlisting}[language=Scala]\nabstract class BigSet[T] extends Set[T] {\n override def size: Int = Sets.InfiniteSize\n...\n\\end{lstlisting}\n\nThis abstract \"container\" covers all possible sets.\nRemember, sets are not typed (in a set theory).\n\n\\begin{lstlisting}[language=Scala]\nobject Sets { \n type set = Set[Any]\n\n implicit class untype[T](s: Set[T]) {\n def untyped: set = s.asInstanceOf[set]\n }\n\n def isFinite(s: Set[_]) Boolean = s.size != InfiniteSize\n val Empty: set = Set.empty[Any]\n val Unit: set = Set(Empty)\n val FiniteSets = BigSet.comprehension(isFinite)\n ...\n\\end{lstlisting}\n\nWe can now define non-enumerable sets (thus banning all those Scala/Java methods that enumerate):\n\n\\begin{lstlisting}[language=Scala]\ntrait NonEnumerableSet[T] extends Set[T] {\n private def notEnumerable = throw Stone(...)\n override def isEmpty: Boolean = notEnumerable\n def iterator: Iterator[T] = notEnumerable\n override def toArray[S >: T : ClassTag] = notEnumerable\n}\n\\end{lstlisting}\n\nThese classes will be used to properly define Grothendieck toposes over sites.\n\n\n\\section{Categories}\n\nBefore defining \\verb+Category+, let's define \\verb+Graph+:\n\n\\begin{lstlisting}[language=Scala]\ntrait Graph { graph =>\n type Node\n type Arrow\n def nodes: Set[Node]\n def arrows: Set[Arrow]\n def d0(f: Arrow): Node\n def d1(f: Arrow): Node\n...\n\\end{lstlisting}\n\n\\verb+trait Graph+ implements a pretty big number of useful methods, but what's important is that it specifies \\verb+Node+s, \\verb+Arrow+s, and their relationships.\\\\\n\nHaving a type of \\verb+Graph+s, we can define a \\verb`Category` in Scala like this:\n\n\\begin{lstlisting}[language=Scala]\nabstract class Category(override val name: String) extends Graph {\n type Obj = Node\n def id(o: Obj): Arrow\n def m(f: Arrow, g: Arrow): Option[Arrow] // composition\n \n lazy val op: Category = {\n val src = this\n new Category(s\"~$name\") {\n override def id(o: Obj): Arrow = src.id(o)\n override def m(f: Arrow, g: Arrow) = src.m(f, g)\n }\n }\n\\end{lstlisting}\n\nNote that composition of arrows is a partial function lifted to \\verb+Option+.\\\\\n\nA category does not necessarily have to be finite: here is an example of an infinite one:\n\n\\begin{lstlisting}[language=Scala]\nobject N extends BigSet[BigInt] with EnumerableSet[BigInt] {\n override def iterator: Iterator[BigInt] = new Iterator[BigInt] {...}\n override def contains(n: BigInt): Boolean = n >= 0\n}\nobject PoSet {\n lazy val ofNaturalNumbers: PoSet[BigInt] = new PoSet(N, comparator) {...}\n}\nlazy val NaturalNumbers: Category = fromPoset(\"N\", PoSet.ofNaturalNumbers)\n\\end{lstlisting}\n\nWe should never attempt to materialize such a category. \\\\\n\nThat was a countable category; we can even build an uncountable one:\n\n\\begin{lstlisting}[language=Scala]\nclass SetCategory(objects: BigSet[Set[Any]])\n extends Category(name=\"Sets\", graphOfSets(objects)) {\n type Node = set\n type Arrow = SetFunction\n\n override def d0(f: SetFunction): set = f.d0\n override def d1(f: SetFunction): set = f.d1\n override def m(f: Arrow, g: Arrow): Option[Arrow] = f compose g\n override def id(s: set): SetFunction = SetFunction.id(s)\n override def toString: String = \"Category of All Sets\"\n...\nobject Setf extends SetCategory(FiniteSets) {\n override def toString: String = \"Category of Finite Sets\"\n}\n\\end{lstlisting}\n\nBut what is \\verb+SetFunction+?\n\n\\begin{lstlisting}[language=Scala]\ncase class SetFunction(\n override val tag: String,\n override val d0: set,\n override val d1: set,\n mapping: Any => Any) extends SetMorphism { self =>\n...\ndef compose(g: SetFunction): Option[SetFunction] =\n if (d1 == g.d0) {\n Some(new SetFunction(newTag, d0, g.d1, (x: Any) => g(self(x))))\n } else None\n\\end{lstlisting}\n\n\\subsection{Samples of Categories}\nBelow are some example of categories, built using these tools:\n\n\\FloatBarrier\n\\begin{table}[!htbp]\n\\begin{center}\n\\label{table_Sample_Categories}\n\\begin{tabular}{|c|c|l|}\n\\hline\n$\\mathbb{ 0 }$ & & \n\\begin{lstlisting}[language=Scala]\nval _0_ : Cat = segment(0) \n\\end{lstlisting}\n\\\\\n\\hline\n$\\mathbb{ 1 } $ & • & \n\\begin{lstlisting}[language=Scala]\nval _1_ : Cat = segment(1) \n\\end{lstlisting}\n\\\\\n\\hline\n$\\mathbb{1}+\\mathbb{1}$ & • • & \n\\begin{lstlisting}[language=Scala]\nval _1plus1_ : Cat = discrete(Set(\"a\", \"b\")) \n\\end{lstlisting}\n\\\\\n\\hline\n$\\mathbb{2}$ & \n\\xymatrix{0 \\ar[r] & 1}\n& \n\\begin{lstlisting}[language=Scala]\nval _2_ : Cat = segment(2) \n\\end{lstlisting}\n\\\\\n\\hline\n$\\mathbb{3}$ & \n\\xymatrix{&1 \\ar[dr] \\\\\n0 \\ar[ur] \\ar[rr] && 2 }\n& \n\\begin{lstlisting}[language=Scala]\nval _3_ : Cat = segment(3) \n\\end{lstlisting}\n\\\\\n\\hline\n$\\mathbb{4}$ & \n\\xymatrix{\n&1 \\ar[drr] \\ar[r] & 2 \\ar[dr]\\\\\n0 \\ar[ur] \\ar[urr] \\ar[rrr] &&& 3\n}\n& \n\\begin{lstlisting}[language=Scala]\nval _4_ : Cat = segment(4) \n\\end{lstlisting}\n\\\\\n\\hline\n$ParallelPair$ & \n\\xymatrix{\n0 \\ar@<2pt>[r]^a \\ar@<-2pt>[r]_b & 1\n}\n& \n\\begin{lstlisting}[language=Scala]\ncategory\"ParallelPair:({0,1}, {a:0->1, b:0->1})\"\n\\end{lstlisting}\n\\\\\n\\hline\n$Pullback$ & \n\\xymatrix{\n& b \\ar[d]^{bd} \\\\\nc \\ar[r]_{cd} & d\n}\n& \n\\begin{lstlisting}[language=Scala]\ncategory\"Pullback:({b,c,d}, {bd:b->d, cd:c->d})\" \n\\end{lstlisting}\n\\\\\n\\hline\n$Square$ & \n\\xymatrix{\na \\ar[d]_{ac} \\ar[r]^{ab} \\ar[dr]|{ad} & b \\ar[d]^{bd} \\\\\nc \\ar[r]_{cd} & d\n}\n& \n\\begin{lstlisting}[language=Scala]\ncategory\"Square:({a,b,c,d},\n{ab:a->b, ac:a->c, bd:b->d, cd:c->d, ad:a->d},\n{bd o ab = ad,cd o ac = ad})\"\n\\end{lstlisting}\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Functors and Natural Transformations}\n\nDefinitions are trivial:\n\n\\begin{lstlisting}[language=Scala]\nabstract class Functor(\n val d0: Category, val d1: Category\n) extends Morphism {\n def objectsMapping(x: d0.Obj): d1.Obj\n def arrowsMapping(a: d0.Arrow): d1.Arrow\n def compose(next: Functor): Option[Functor] = {\n if (this.d1 != next.d0) None else Some { //...\n\\end{lstlisting}\n\n\n\\begin{lstlisting}[language=Scala]\nabstract class NaturalTransformation\n extends Morphism[Functor, Functor] {\n\n\n val domain: Category = d0.d0\n val codomain: Category = d1.d1 // == d0.d1\n\n def transformPerObject(x: domain.Obj): codomain.Arrow\n\\end{lstlisting}\n\nIn the picture, you see a natural transformation for category \"Pullback\":\n\n\\xymatrix{\n& F[b] \\ar[d] \\ar@{.>}[drr] \\\\\nF[c] \\ar[r] \\ar@{.>}[drr] & F[d] \\ar@{.>}[drr] && G[b] \\ar[d]\\\\\n&&G[c] \\ar[r] &G[d]\n}\n\n\\section{Copresheaves (aka Diagrams)}\n\n\\subsection{Definition}\nA $Copresheaf$, also known as $Diagram$, of category \\catname{C} in category \\catname{D} is a functor from \\catname{C} to \\catname{D}.\n\nWe will, further on, be dealing with copresheaves in \\Set, and call them $diagrams$.\n\n\\begin{lstlisting}[language=Scala]\nabstract class Diagram(val domain: Category)\n extends Functor(tag, topos.domain, Setf) { diagram =>\n\n def isElementOf(other: Diagram): Boolean =\n d0.objects.forall { o => other(o) contains (this(o)) }\n\n def isContainedIn(other: Diagram): Boolean =\n d0.objects.forall { o => this(o) subsetOf other(o) }\n\\end{lstlisting}\n\n\\subsection{Example 5, Diagrams over $\\mathbb{N}$}\nA functor $\\mathbb{N} \\rightarrow \\textbf{Sets}$ consists of sets $F[i]$ and functions $F[i]$→$F[i+1]$. (Other functions are compositions of these).\nThese diagrams model discreet “time flow” : discrete, \\cite{lamp}. \nThat’s the basic type of TLA+ book.\n\n\\xymatrix{\nF[0] \\ar[r] &F[1] \\ar[r] &F[2] \\ar[r] &{...}\n}\\\\\n\nBelow is a table of some shapes of time that we encounter in practice.\n\n\\FloatBarrier\n\n\\begin{table}[!htbp]\n\\begin{center}\n\\label{table_Sample_Diagram Categories}\n\\begin{tabular}{|c|l|}\n%\\hline\n%$\\textbf{Site}$ & $\\textbf{Concept}$ \\\\\n\\hline\n\\xymatrix{\nF[0] \\ar[r] &F[1] \\ar[r] &F[2] \\ar[r] &{...}\n} & TLA+ \\\\\n\\hline\n$\\bullet$ & Just sets\\\\\n\\hline\n\\xymatrix{0 \\ar[r] & 1}\n& Single transition (today and tomorrow)\n\\\\\n\\hline\n\\xymatrix{\n& b \\ar[d]^{bd} \\\\\nc \\ar[r]_{cd} & d\n}\n& Git Merge\n\\\\\n\\hline\n\\xymatrix{\na \\ar[d]_{ac} \\ar[r]^{ab} \\ar[dr]|{ad} & b \\ar[d]^{bd} \\\\\nc \\ar[r]_{cd} & d\n}\n& Eventual Consistency\n\\\\\n\\hline\n\\xymatrix{\n{Feb1} \\ar[r] & {Feb 2} \\ar@(ur,ul)[]_{6:00} \\ar[r] & {Feb 3}\n}\n& Groundhog day\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\FloatBarrier\n\n\\section{Logic in Grothedieck Topos}\n\nNow that we have a category of diagrams (it's a Grothendieck topos), we can dive deeper and build logic in it. We will be using points of objects to illustrate logic. Generally speaking, a Grothendieck topos is not well-pointed, though.\n\n\\subsection{Point of a Diagram}\nA point is any arrow \\xymatrix{1 \\ar[r] & D}, where $1$ is a terminal object of the category of diagrams.\n\n\\begin{lstlisting}[language=Scala]\nclass Point(val tag: Any, val mapping: Any => Any)\n\\end{lstlisting}\n\n\\subsection{Subobject Classifier}\nThis is a special object in a category of diagrams contains all the logic of a topos; we just have to discover it.\n\n\\xymatrix{&A \\ar@{>->}[d]_f \\ar[r] & 1 \\ar@{>->}[d]^{true} \\\\\n&B \\ar[r]_{\\chi_f} & \\Omega\n}\n\nHow can we build it? \n\n$\\Omega(x) \\equiv hom(hx, \\Omega)$ - by Yoneda lemma \n\n$hom(hx, \\Omega) \\equiv {p \\subset hx}$ - by definition of $\\Omega$, see above.\n\n\\begin{lstlisting}[language=Scala]\ncase class Representable(x: domain.Obj)\n extends Diagram(s\"hom($x, _)\", domain)\n//...\n\nobject Omega extends Diagram(\"Omega\", this, domain) {\n val False: Point = points.head named \"false\"\n val True: Point = points.last named \"true\"\n val conjunction: DiagramArrow = ... // OmegaxOmega(x) => Omega(x)\n val disjunction: DiagramArrow = ... // ΩxΩ(x) => Omega(x)\nal implication: DiagramArrow = ... // ΩxΩ(x) => Omega(x)\n ...\n}\nval OmegaxOmega = product2(Omega, Omega)\n\nval DiagonalOfOmega: DiagramArrow =\n buildArrow(\"Diagonal\", Omega, OmegaxOmega, _ => (s: Any) => (s, s)\n\n\\end{lstlisting}\n\n\\subsubsection{How to Build Conjunction in $\\Omega$}\n\n\\xymatrix{\n&1 \\ar@{>->}[d]_{(true,true)} \\ar[rr] && 1 \\ar@{>->}[d]^{true} \\\\\n& \\Omega\\times\\Omega \\ar[rr]_{\\land} && \\Omega\n}\n\n\\subsubsection{How to Build Implication in $\\Omega$}\n\nStart with building a subobject $\\Omega_1$ defining a partial order on $\\Omega$:\n\n\\xymatrix{\n\\Omega_1 \\ar@{>->}[r] & \\Omega\\times\\Omega \\ar@<1ex>[r]^{∧} \\ar@<-1ex>[r]_{p_1}\n& \\Omega\n}\n\nNow, implication is the arrow that classifies $\\Omega_1$:\n\n\\xymatrix{\n \\Omega_1 \\ar[d] \\ar[rr] && 1 \\ar[d]^{true} \\\\\n \\Omega \\times \\Omega \\ar[rr]_{implication} && \\Omega\n}\n\n\\begin{lstlisting}[language=Scala]\nlazy val implication: DiagramArrow = {\n val inclusion: DiagramArrow = inclusionOf(Omega_1) in OmegaxOmega iHope\n\n classifies(inclusion, \"=>\")\n}\n\\end{lstlisting}\n\n\\subsection{Predicate Logic}\n\nFirst, what is a predicate in a topos? It's just any arrow\n$D\\ \\rightarrow\\ \\Omega$.\\\\\n\nLogical operations on $\\Omega$ provide operations on predicates:\n\n\\begin{lstlisting}[language=Scala]\ntrait Predicate extends DiagramArrow { p: DiagramArrow =>\n val d0: Diagram\n val d1: Diagram = Omega\n def and(q: Predicate): Predicate = binaryOp(q, \"&\", Omega.conjunction)\n def or(q: Predicate): Predicate = binaryOp(q, \"|\", Omega.disjunction)\n def ==>(q: Predicate): Predicate = binaryOp(q, \"=>\", Omega.implication)\n ...\n}\n\ndef not(p: Predicate): Predicate = p ==> FalsePredicate\n\\end{lstlisting}\n\n\nAs an example, here's a test case from a unittest for this code:\n\n\\begin{lstlisting}[language=Scala]\ndef check(c: Category): MatchResult[Any] = {\n val topos = new CategoryOfDiagrams(c)\n import topos._\n\n for { p <- Omega.points map (_.asPredicate) } {\n (True ==> p) === p\n (False ==> p) === True\n (p ==> p) === True\n (p ==> True) === True\n not(not(not(p))) === not(p)\n\n for { q <- Omega.points map (_.asPredicate)\n r <- Omega.points map (_.asPredicate) } {\n (p and q ==> r) === (p ==> (q ==> r))\n }\n }\n}\n\\end{lstlisting}\n\n\n\n\n\\xymatrix{}\n\n\\begin{lstlisting}[language=Scala]\n\\end{lstlisting}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Conclusion}\nNeither Scala nor Java was designed to deal with problems like these: typeless sets, categories, infinite collections. \nAlso, the higher order type system in Scala 2 is not very convenient for\nthe problems like the ones I encountered implementing categories and toposes; dependent types would make more sense.\nThere's a hope that Scala 3 will help dealing with it properly.\nStrangely, implementing it in Haskell was even more challenging.\n\nIn addition to intuitionistic logic, Lawvere topologies can be implemented based on this package; and the implementation is now available in the same repository.\nUsing an older version of this software, we managed to enumerate all Lawvere topologies over finite sets.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% The bibliography is stored in an external database file\n% in the BibTeX format (file_name.bib). The bibliography is\n% created by the following command and it will appear in this\n% position in the document. You may, of course, create your\n% own bibliography by using the bibliography environment as in\n%\n% \\begin{thebibliography}{12}\n% ...\n% \\bibitem{itemreference} D. E. Knudsen.\n% {\\em 1966 World Bnus Almanac.}\n% {Permafrost Press, Novosibirsk.}\n% ...\n% \\end{thebibliography}\n\n% Here's where you specify the bibliography style file.\n% The full file name for the bibliography style file \n% used for an ASME paper is asmems4.bst.\n\\bibliographystyle{asmems4}\n\n% Here's where you specify the bibliography database file.\n% The full file name of the bibliography database for this\n% article is asme2e.bib. The name for your database is up\n% to you.\n\\bibliography{london2021}\n\n\\cite{bartosz, pierce, tt, sheaves, brief, nlab1, repo, ctgd}\n\n\\end{document}\n", "meta": {"hexsha": "6c9f8d2f36eeb66d7088cc8531333d420780fd21", "size": 24731, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "talks/London2021/london2021.tex", "max_stars_repo_name": "vpatryshev/wowiki", "max_stars_repo_head_hexsha": "1af64e2fc2aca94c763fb2fe02ef21857d7c5ed3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 15, "max_stars_repo_stars_event_min_datetime": "2020-06-20T20:23:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-10T06:46:03.000Z", "max_issues_repo_path": "talks/London2021/london2021.tex", "max_issues_repo_name": "vpatryshev/wowiki", "max_issues_repo_head_hexsha": "1af64e2fc2aca94c763fb2fe02ef21857d7c5ed3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "talks/London2021/london2021.tex", "max_forks_repo_name": "vpatryshev/wowiki", "max_forks_repo_head_hexsha": "1af64e2fc2aca94c763fb2fe02ef21857d7c5ed3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-11-11T00:55:47.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-11T00:55:47.000Z", "avg_line_length": 33.0628342246, "max_line_length": 374, "alphanum_fraction": 0.6746188994, "num_tokens": 7355, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3490149188820786}} {"text": "\\documentclass[]{AVSSimReportMemo}\n\\usepackage{AVS}\n\\usepackage{colortbl}\n\n\\newcommand{\\ModuleName}{inertial3D}\n\\newcommand{\\subject}{Guidance Module to Perform an Inertially Fixed Pointing}\n\\newcommand{\\status}{Initial Version}\n\\newcommand{\\preparer}{M. Cols}\n\\newcommand{\\summary}{Generate the reference attitude trajectory for a general 3D inertial pointing. A corrected body frame will align with the desired reference frame. }\n\n\n\\begin{document}\n\n\n\\makeCover\n\n\n%\n%\tenter the revision documentation here\n%\tto add more lines, copy the table entry and the \\hline, and paste after the current entry.\n%\n\\pagestyle{empty}\n{\\renewcommand{\\arraystretch}{2}\n\\noindent\n\\begin{longtable}{|p{0.5in}|p{4.5in}|p{1.14in}|}\n\\hline\n{\\bfseries Rev}: & {\\bfseries Change Description} & {\\bfseries By} \\\\\n\\hline\nDraft & initial copy & M. Cols \\\\\n\\hline\n\n\\end{longtable}\n}\n\n\\newpage\n\\setcounter{page}{1}\n\\pagestyle{fancy}\n\n\\tableofcontents\n~\\\\ \\hrule ~\\\\\n\n\\section{Module Input and Output}\nTable \\ref{tab:inputTable} shows the input Configuration Data of the module Inertial 3D Point.\n\\begin{table}[h!]\n\t\\centering\n\t\\caption{Input Configuration Data}\n\t\\begin{tabular}{|l|l|l|p{3in}|}\n\t\t\\hline\n\t\t\\rowcolor{BrickRed}\n\t\t\\textcolor{white}{Name} & \\textcolor{white}{Type} & \n\t\t\\textcolor{white}{Length} & \n\t\t\\textcolor{white}{Description} \\\\ \\hline\n\t\t$\\sigma_{R_0/N}$ & double [] & 3 & \n\t\tMRP attitude set of the desired reference frame with respect to the inertial frame . \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:inputTable}\n\\end{table}\n\nTable \\ref{tab:outputTable} shows the Attitude Reference output message of the module Inertial 3D Point.\n\\begin{table}[h!]\n\t\\centering\n\t\\caption{Output Attitude Reference Message}\n\t\\begin{tabular}{|l|l|l|p{3in}|}\n\t\t\\hline\n\t\t\\rowcolor{BrickRed}\n\t\t\\textcolor{white}{Name} & \\textcolor{white}{Type} & \n\t\t\\textcolor{white}{Length} & \n\t\t\\textcolor{white}{Description} \\\\ \\hline\n\t\t$\\sigma_{R/N}$ & double [] & 3 & \n\t\tMRP attitude set of the reference frame with respect to the inertial frame. \\\\ \\hline\n\t\t$\\leftexp{N} \\omega_{R/N}$ & double [] & 3 & \n\t\tAngular rate vector of the reference frame with respect to the inertial expressed in inertial frame components. \\\\ \\hline\n\t\t$\\leftexp{N} {\\dot{\\omega}_{R/N}}$ & double [] & 3 & \n\t\tAngular acceleration vector of the reference frame with respect to the inertial expressed in inertial frame components. \\\\ \\hline\n\t\\end{tabular}\n\t\\label{tab:outputTable}\n\\end{table}\n\\newpage\n\n\\section{Introduction}\nThis technical note discusses the guidance mathematics to compute a reference frame $\\mathcal{R}$ that is aligned the with an inertially fixed frame $\\mathcal{R}_0$, as shown in Figure~\\ref{fig:Fig1}.\n\\begin{figure}[htb]\n\t\\centerline{\n\t\\includegraphics{Figures/Fig3}\n\t}\n\t\\caption{Illustration of the input inertially fixed frame $\\mathcal{R}_{0}:\\{ \\hat e_{1}, \\hat e_{2}, \\hat e_{3} \\}$, the generated reference frame $\\mathcal{R}: \\{ \\hat r_{1}, \\hat r_{2}, \\hat r_{3}\\}$ and the inertial frame $\\mathcal{N}:\\{ \\hat{\\bm n}_{1}, \\hat{\\bm n}_{2}, \\hat{\\bm n}_{3} \\}$}\n\t\\label{fig:Fig1}\n\\end{figure}\n\n\\section{Reference Frame Generation}\nThe modules requires the desired reference orientation in terms of the MRP set $\\bm{\\sigma}_{R_{0}N}$. This input is only set once and does not have to be changed.\nLet us designate $\\mathcal{R}$ as the output generated reference frame. Since the fixed-pointing is inertial:\n\\begin{equation}\n\t\\bm{\\sigma}_{RN} = \\bm{\\sigma}_{R_{0}N}\n\\end{equation}\n\\begin{equation}\n\t\\bm{\\omega}_{RN} = \\dot{\\bm{\\omega}}_{RN} = 0\n\\end{equation}\n\n\\bibliographystyle{unsrt}\n\\bibliography{references}\n\n\\end{document}\n", "meta": {"hexsha": "13e893078602d216558669b9ff07b414be63c89c", "size": 3598, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/fswAlgorithms/attGuidance/inertial3D/_Documentation/AVS-Sim-Inertial3D-2016-01-15.tex", "max_stars_repo_name": "ian-cooke/basilisk_mag", "max_stars_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_stars_repo_licenses": ["0BSD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/fswAlgorithms/attGuidance/inertial3D/_Documentation/AVS-Sim-Inertial3D-2016-01-15.tex", "max_issues_repo_name": "ian-cooke/basilisk_mag", "max_issues_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_issues_repo_licenses": ["0BSD"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-03-13T20:52:22.000Z", "max_issues_repo_issues_event_max_datetime": "2019-03-13T20:52:22.000Z", "max_forks_repo_path": "src/fswAlgorithms/attGuidance/inertial3D/_Documentation/AVS-Sim-Inertial3D-2016-01-15.tex", "max_forks_repo_name": "ian-cooke/basilisk_mag", "max_forks_repo_head_hexsha": "a8b1e37c31c1287549d6fd4d71fcaa35b6fc3f14", "max_forks_repo_licenses": ["0BSD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.5961538462, "max_line_length": 297, "alphanum_fraction": 0.7156753752, "num_tokens": 1154, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.34897373067887094}} {"text": "\\documentclass{article}\n\\usepackage{graphicx}\n\\usepackage{titletoc}\n\\usepackage{titlesec}\n\\usepackage{geometry} \n\\usepackage{fontspec, xunicode, xltxtra}\n\\usepackage{float}\n\\usepackage{cite}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{listings}\n\\usepackage{titletoc}\n\\usepackage{bm}\n\\usepackage{multirow}\n\n\\geometry{left=3cm,right=3cm,top=3cm,bottom=3cm}\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\var}{var}\n\\DeclareMathOperator*{\\expec}{E}\n\n\\begin{document}\n\\title{\\textsf{Course Project for Pattern Recognition}}\n\\author{Fan JIN\\quad (2015011506)}\n\\maketitle\n\n\\tableofcontents\n\\clearpage\n\n\\section{Introduction and Preprocessing}\n{\n Here is a glance at the datasets. \n Columns with only one unique value are removed, as they have no effect as observations.\n The training set is roughly 9 times as much size as the testing set.\n\n \\begin{table}[!hbp]\n \\centering\n \\begin{tabular}{|c|c|c|c|}\n \\hline\n File & sample size (nrow) & number of features (ncol) & number features in effect \\\\\n \\hline\n 2ctrainX.txt & 2298 & 25133 & 17951 \\\\\n \\hline\n 2ctestX.txt & 256 & 25133 & 17951 \\\\\n \\hline\n 10ctrainX.txt & 16074 & 25133 & 24307 \\\\\n \\hline\n 10ctestX.txt & 1787 & 25133 & 24307 \\\\\n \\hline\n \\end{tabular}\n \\caption{Shape of datasets}\n \\end{table}\n\n \\begin{itemize}\n \\item For two-category classification, the response ``2ctrainY.txt'' takes values 1 and 2.\n \\item For ten-category one, the response ``10ctrainY.txt'' takes values ranging from 1 to 10.\n \\end{itemize}\n}\n\n\\section{Two-category classification}\n{\n \\subsection{Feature selection}\n {\n Feature selection is neccesary in this case, for we have much more features than data points, referred to as the ``large p, small n'' problem.\n\n Methods of feature selection include:\n \\begin{itemize}\n \\item Non-wrapping: Fisher-based\n \\item Non-wrapping: Correlation-based\n \\item Non-wrapping: Entropy-based\n \\item Non-wrapping: Statistical t-distribution\n \\item Non-wrapping: Principal Component Analysis (PCA)\n \\item Wrapping: Foreward-backward recursive\n \\item Stochastic: Genetic algorithm\n \\end{itemize}\n\n Here we use non-wrapping methods for feature selection. We chose Fisher-based and PCA.\n\n \\subsubsection{Fisher criterion}\n {\n The fisher criterion is defined as \n $$F = \\frac{(\\mu_1 - \\mu_2)^2}{s_1^2 + s_2^2},$$\n where $\\mu_1$, $\\mu_2$ are the means of two categories, and $s_1^2$, $s_2^2$ are the varainces, in terms of some feature.\n\n The larger the Fisher criterion, the more significant the feature is. We select 2000 features with the largest Fisher criteria. \n }\n\n \\subsubsection{the PCA}\n {\n The Principal Component Analysis performs eigendecomposition to the correlation matrix of the data points, and retains the components with principal eigenvalues.\n\n We retain 99\\% of total variance in raw data, which gives $2202$ features in total. \n }\n }\n\n \\subsection{Classifier}\n {\n \\subsubsection{Linear SVM}\n {\n The support vector machine, or SVM, is a powerful method for both classification and regression, particularly in case of ``large p, small n''.\n\n It is crucial to choose a kernel that balances between prediction accuracy and overfitting aversion. \n We tried linear, quadratic, and RBF kernels. The linear kernel turns out the best.\n }\n\n \\subsubsection{Fully connected NN}\n {\n The neural network has advantages over the linear SVM in case of nonlinearity. \n\n Since we have only limited data points, we cannot train a deep network. \n We took a simple perceptron network, which has one hidden layer with 1500 nodes. \n We tuned the hidden layer size, and $1000$ proves the best hyperparameter.\n }\n }\n\n \\subsection{Experiments}\n {\n \\begin{table}[!hbp]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n \\hline\n Dim. Reduction & Classifier & \\multicolumn{5}{|c|}{5-fold cross validation} & Avg. Accuracy \\\\\n \\hline\n Fisher & Linear SVM & 0.9848 & 0.9934 & 0.9869 & 0.9869 & 0.9956 & 0.9896 \\\\\n \\hline\n PCA & Linear SVM & 0.9544 & 0.9434 & 0.9629 & 0.9586 & 0.9498 & 0.9539 \\\\\n \\hline\n Fisher & FC NN & 0.9891 & 0.9978 & 0.9891 & 1.0000 & 0.9956 & 0.9943 \\\\\n \\hline\n PCA & FC NN & 0.8828 & 0.8652 & 0.8823 & 0.9019 & 0.8758 & 0.8816 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison}\n \\end{table}\n\n The Fisher criterion has better performance in dimensionality reduction, compared with the PCA. \n\n It is worth noting that the neural network does not always perform better than the SVM. \n It does when we use Fisher criterion for dimensionality reduction, while it does not when the PCA is used. \n This could be explained by the weak performance of the PCA, as it performs a linear transformation on the raw data.\n The data might not be normally distributed, and thus the correlation matrix fails to identify all the principal components.\n\n We tuned the hyperparamters, and found the best network size is around 1000. This attains high accuracy while avoiding overfitting.\n\n \\begin{table}[!hbp]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n Size of hidden layer & \\multicolumn{5}{|c|}{5-fold cross validation} & Avg. Accuracy \\\\\n \\hline\n 200 & 0.9934 & 0.9978 & 0.9934 & 0.9847 & 0.9934 & 0.9925 \\\\\n \\hline\n 500 & 0.9891 & 0.9978 & 0.9891 & 1.0000 & 0.9934 & 0.9939 \\\\\n \\hline\n 1000 & 0.9891 & 0.9978 & 0.9891 & 1.0000 & 0.9956 & 0.9943 \\\\\n \\hline\n 1500 & 0.9891 & 0.9978 & 0.9891 & 1.0000 & 0.9956 & 0.9943 \\\\\n \\hline\n \\end{tabular}\n \\caption{Finding the proper network size}\n \\end{table}\n }\n\n \\subsection{Prediction}\n {\n Since it has the highest accuracy in 5-fold cross validation, the fully connected NN with Fisher-based dimensionality reduction is chosen for prediction. \n The prediction is stored in ``2ctestY.txt'' in the root directory.\n }\n}\n\n\\section{Ten-category classification}\n{\n \\subsection{Feature selection}\n {\n The dataset is too large for SVD (singular value decomposition), so it would be diffucult to perform full PCA on it.\n We tried using ``arpack'' instead, but the script ran out of memory on a GPU server. \n\n Considering this, only Fisher criterion was used in this task.\n We chose 2000 features out of 24307, and trained the reduced dataset using SVM and FCNN.\n }\n\n \\subsection{Experiments}\n {\n \\begin{table}[!hbp]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|c|}\n \\hline\n Dim. Reduction & Classifier & \\multicolumn{5}{|c|}{5-fold cross validation} & Avg. Accuracy \\\\\n \\hline\n Fisher & Linear SVM & 0.9916 & 0.9878 & 0.9850 & 0.9903 & 0.9875 & 0.9884 \\\\\n \\hline\n Fisher & FC NN & 0.9891 & 0.9950 & 0.9937 & 0.9925 & 0.9934 & 0.9927 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison}\n \\end{table}\n\n The fully connected NN attains higher accuracy as the linear SVM does. \n\n \\begin{table}[!hbp]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|c|c|}\n \\hline\n Size of hidden layer & \\multicolumn{5}{|c|}{5-fold cross validation} & Avg. Accuracy \\\\\n \\hline\n 200 & 0.9934 & 0.9925 & 0.9875 & 0.9940 & 0.9931 & 0.9921 \\\\\n \\hline\n 500 & 0.9546 & 0.9928 & 0.9931 & 0.9934 & 0.9866 & 0.9841 \\\\\n \\hline\n 1000 & 0.9953 & 0.9944 & 0.9944 & 0.9937 & 0.9937 & 0.9943 \\\\\n \\hline\n 1500 & 0.9959 & 0.9940 & 0.9944 & 0.9947 & 0.9940 & 0.9946 \\\\\n \\hline\n \\end{tabular}\n \\caption{Finding the proper network size}\n \\end{table}\n\n The optimal hyperparameter for the hidden layer size is around 1000. \n\n Compared with two-category classification, the ten-category one attains slightly lower accuracy in training and cross validation.\n\n }\n\n \\subsection{Prediction}\n {\n Since it has the highest accuracy in 5-fold cross validation, the fully connected NN with Fisher-based dimensionality reduction is chosen for prediction. \n The prediction is stored in ``10ctestY.txt'' in the root directory.\n }\n}\n\n\\section{Visualization and Clustering}\n{\n \\subsection{PCA}\n {\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{./src3/pca.png}\n \\caption{Dimensionality reduction using PCA}\n \\end{figure}\n }\n\n \\subsection{t-SNE}\n {\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\linewidth]{./src3/tsne.png}\n \\caption{Dimensionality reduction using t-SNE}\n \\end{figure}\n }\n\n \\subsection{Clustering based on t-SNE}\n {\n It is astouding to find the result of PCA, although with good linearity, does not split up all the 10 categories in the 2d space.\n Only one category is distributed away from others. In comparison, the t-SNE works fine. We will use t-SNE for clustering.\n\n Since the number of categories is known, k-means seems a good choice. We fix $k=10$. \n The birch clustering algorithm is also tested. The Birch algorithm turns out better than K-means.\n\n \\begin{table}[!hbp]\n \\centering\n \\begin{tabular}{|c|c|}\n \\hline\n Method & Accuracy \\\\\n \\hline\n K-means & 0.6926 \\\\\n \\hline\n Birch & 0.7343 \\\\\n \\hline\n \\end{tabular}\n \\caption{Comparison}\n \\end{table}\n\n \\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.9\\linewidth]{./src3/kmeans.png}\n \\includegraphics[width = 0.9\\linewidth]{./src3/birch.png}\n \\caption{K-means (top); Birch (bottom)}\n \\end{figure}\n }\n}\n\n\\section{Source Code}\n{\n \\subsection{Root directory}\n {\n \\begin{itemize}\n \\item ``2ctestY.txt'': Two-category prediction. Use this file for evaluation.\n \\item ``10ctestY.txt'': Ten-category prediction. Use this file for evaluation.\n \\item ``report.pdf'': Project report.\n \\end{itemize}\n }\n\n \\subsection{``data1'' directory}\n {\n \\begin{itemize}\n \\item ``2ctrainX.txt'': Raw dataset.\n \\item ``2ctrainY.txt'': Raw dataset.\n \\item ``2ctestY.txt'': Raw dataset.\n \\item ``data1.npz'': Preprocessed file. (invalid columns removed)\n \\end{itemize}\n }\n\n \\subsection{``data2'' directory}\n {\n \\begin{itemize}\n \\item ``10ctrainX.txt'': Raw dataset.\n \\item ``10ctrainY.txt'': Raw dataset.\n \\item ``10ctestY.txt'': Raw dataset.\n \\item ``data2.npz'': Preprocessed file. (invalid columns removed)\n \\end{itemize}\n }\n\n \\subsection{``src1'' directory}\n {\n \\begin{itemize}\n \\item ``preproc.py'': Preprocessing, removing identical columns.\n \\item ``feature\\_selection'': Feature selection, generating ``fisher\\_2000.npz'' and ``pca\\_99.npz''.\n \\item ``SVM.py'': Training a SVM and use it for prediction.\n \\item ``FCNN'': Training a fully connected NN and use it for prediction.\n \\end{itemize}\n }\n\n \\subsection{``src2'' directory}\n {\n \\begin{itemize}\n \\item ``preproc.py'': Preprocessing, removing identical columns.\n \\item ``feature\\_selection'': Feature selection, generating ``fisher\\_2000.npz''.\n \\item ``SVM.py'': Training a SVM and use it for prediction.\n \\item ``FCNN'': Training a fully connected NN and use it for prediction.\n \\end{itemize}\n }\n\n \\subsection{``src3'' directory}\n {\n \\begin{itemize}\n \\item ``pca.py'': Dimensionality reduction by PCA, generating ``pca.npz''.\n \\item ``tsne.py'': Dimensionality reduction by t-SNE, generating ``tsne.npz''.\n \\item ``visualization'': Visualization after loading ``pca.pna'' and ``tsne.npz''. \n \\item ``kmeans.py'': Clustering by K-means.\n \\item ``birch.py'': Clustering by Birch.\n \\end{itemize}\n }\n}\n\n\\clearpage\n\\end{document}\n ", "meta": {"hexsha": "e8c69c5c8067481ba2b5ace2b0f27005885f991e", "size": 12942, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Project/report.tex", "max_stars_repo_name": "kingium/PatternRecognitionForUndergrads", "max_stars_repo_head_hexsha": "5cd08f3a260fae4a7edaf71599433e93484863b0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Project/report.tex", "max_issues_repo_name": "kingium/PatternRecognitionForUndergrads", "max_issues_repo_head_hexsha": "5cd08f3a260fae4a7edaf71599433e93484863b0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Project/report.tex", "max_forks_repo_name": "kingium/PatternRecognitionForUndergrads", "max_forks_repo_head_hexsha": "5cd08f3a260fae4a7edaf71599433e93484863b0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.2968299712, "max_line_length": 173, "alphanum_fraction": 0.6004481533, "num_tokens": 3580, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.5964331462646254, "lm_q1q2_score": 0.3489737222844969}} {"text": "\\documentclass{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{pgfplotstable}\n\\pgfplotsset{compat=1.15}\n\\usepackage{fancyvrb}\n\n\\title{}\n\\author{}\n\\date{}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{Introduction}\n\nUsing our DenseNet-BC we have approximately $634k$ 32FP-ops and $626k$ FP32-params. This gives us a score of approximately (rounding up) $0.078$ in total, where for the ops, we obtain $0.06$ as our score, and for the parameters, we obtain $0.018$ as our score.\n\nTo test a pre-trained network and check the score, just run profile.py. To train a network, just run train.py and then re-run profile.py. \n\nWe describe our methods and computations in the following:\n\n\\section{Data Augmentation}\n\\label{data_aug}\n\nWe first use standard Data augmentation such as flip and random crop. In addition, we propose to add other different data augmentations. We use CutMix with $\\alpha = 1$ for all training dataset, and we drop $64$ pixels randomly for each input training image. \n\n\\section{Architecture}\n\nWe use a DenseNet-BC architecture with $K = 8$, depth $D = 196$ and reduction $R = 0.5$. \n\n\\section{Computing the Number of Parameters}\n\nTo get the number of parameters of the DenseNet-BC architecture, we compute the number of parameters of each layer (or each Bottleneck layer that contains 2 Batch norm layers, 2 Convolutional layers and 2 Relu), as described in Table~\\ref{table:params}\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{c|c}\n Layer & Number of parameters \\\\\n \\hline\n Convolution & $F_{in}F_{out}S_c^2$ \\\\\n \\hline\n Batch norm & $4F_{in}$ \\\\\n \\hline\n $Bottleneck_i$ & $ 4[F_{in} + K(4 + F_{in} + i + K(1 + i)]$\\\\\n \\hline\n Linear & $F_{in}F_{out}$\\\\\n \\hline\n Transition & $F_{in}(4 + RF_{in})$ \\\\\n \\hline\n \\end{tabular}\n \\caption{The number of parameters for each layer. Note that $F_{in}$ represents the number of input feature maps, $F_{out}$ represents the number of output feature maps, $S_c^2$ represents the kernel size of a convolutional operation, $i$ represents the depth of the bottleneck inside a block that contains $\\frac{D-4}{6}$ bottleneck layers.}\n \\label{table:params}\n\\end{table}\n\n\\section{Computing the Number of Operations}\n\nto get the number of operations required to process one input through the DenseNet-BC network, we compute the number of operations of each layer (or each Bottleneck layer that contains 2 Batch norm layers, 2 Convolutional layers and 2 Relu), as described in Table~\\ref{table:Flops}\n\n\n\\begin{table}[h]\n \\centering\n \\begin{tabular}{c|c}\n Layer & Number of operations \\\\\n \\hline\n Convolution & $[\\frac{F_{in}F_{out}S_c^2}{2} + (F_{in}S_c^2 -1)F_{out}]P $ \\\\\n \\hline\n Batch norm & $3F_{in}P$ \\\\\n \\hline\n Relu & $F_{in}P$\\\\\n \\hline\n Average pooling & $S_p^2PF_{in} $ \\\\\n \\hline\n Linear & $1.5F_{in}F_{out}$ \\\\\n \\hline\n Transition & $4F_{in}P + [\\frac{F_{in}F_{out}S_c^2}{2} + (F_{in}S_c^2 -1)F_{out}]P + S_p^2PF_{out}$ \\\\\n \\hline\n $Bottleneck_i$ & Batch norm1 + Relu1 + Batch norm2 + Relu2 + Conv1 + Conv2\\\\\n & Batch norm1 = $3P(F_{in} + iK)$\\\\\n & Relu1 = $P(F_{in} + iK)$\\\\\n & Batch norm2 = $12P_{in}K$\\\\\n & Relu2 = $4P_{in}K$\\\\\n & Conv1 = $P[\\frac{(F_{in} + iK)4K}{2} + (F_{in} + ik -1)4K]$\\\\\n & Conv2 = $[\\frac{36K^2}{2} + (36K-1)K]P$\\\\\n \\hline\n \n \\hline\n \n \\hline\n \\end{tabular}\n \\caption{The number of operations for each layer. Note that $F_{in}$ represents the number of input feature maps, $F_{out}$ represents the number of output feature maps, $S_c^2$ represents the kernel size of a convolutional operation, $S_p^2$ represents the kernel size of an average pooling operation, $i$ represents the depth of the bottleneck inside a block that contains $\\frac{D-4}{6}$ bottleneck layers, $P$ represents the number of pixels of an input feature map.}\n \\label{table:Flops}\n\\end{table}\n\n\\section{Training procedure}\n\nWe train the network with the data augmentation described on Section~\\ref{data_aug} for 200 epochs, using a batch size of 32 and a softmax temperature of 6. Our learning rate starts at 0.1 and is divided by 10 at epochs 100 and 150. We then fine-tune the network by training it for an extra 5 epochs without data-augmentation. \n\n\\newpage\n\n\\input{architecture.tex}\n\n\n\\end{document} \n", "meta": {"hexsha": "7267d44ccf07c9fe1b7c217a9283d7f7fabd1215", "size": 4329, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Report/main.tex", "max_stars_repo_name": "cadurosar/Micronet", "max_stars_repo_head_hexsha": "b57df0aa765f5b503cd02f6bd367f61cbf7d1dbe", "max_stars_repo_licenses": ["ECL-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-08-08T18:42:51.000Z", "max_stars_repo_stars_event_max_datetime": "2019-09-04T02:38:49.000Z", "max_issues_repo_path": "Report/main.tex", "max_issues_repo_name": "cadurosar/Micronet", "max_issues_repo_head_hexsha": "b57df0aa765f5b503cd02f6bd367f61cbf7d1dbe", "max_issues_repo_licenses": ["ECL-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Report/main.tex", "max_forks_repo_name": "cadurosar/Micronet", "max_forks_repo_head_hexsha": "b57df0aa765f5b503cd02f6bd367f61cbf7d1dbe", "max_forks_repo_licenses": ["ECL-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-06-06T17:06:31.000Z", "max_forks_repo_forks_event_max_datetime": "2019-06-06T17:06:31.000Z", "avg_line_length": 41.2285714286, "max_line_length": 475, "alphanum_fraction": 0.6853776854, "num_tokens": 1358, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.585101139733739, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.34897371365441227}} {"text": "\\newcommand{\\createRUpd}[4]{\\fun{createRUpd}~\\var{#1}~\\var{#2}~\\var{#3}~\\var{#4}}\n\\newcommand{\\mkApparentPerformance}[3]{\\fun{mkApparentPerformance}~{#1}~\\var{#2}~\\var{#3}}\n\\newcommand{\\Q}{\\ensuremath{\\mathbb{Q}}}\n\\newcommand{\\ActiveSlotCoeff}{\\mathsf{ActiveSlotCoeff}}\n\\newcommand{\\EpochState}{\\type{EpochState}}\n\\newcommand{\\BlocksMade}{\\type{BlocksMade}}\n\\newcommand{\\RewardUpdate}{\\type{RewardUpdate}}\n\\newcommand{\\PrtclState}{\\type{PrtclState}}\n\\newcommand{\\PrtclEnv}{\\type{PrtclEnv}}\n\\newcommand{\\PoolDistr}{\\type{PoolDistr}}\n\\newcommand{\\bheader}[1]{\\fun{bheader}~\\var{#1}}\n\n\\section{Removal of the Overlay Schedule}\n\nThe overlay schedule was only used during the early days of the Shelley ledger, and can be safely removed. First, the protocol parameter $\\var{d}$ is removed, and any functions that use it are reduced to the case $\\var{d} = 0$. The function $\\fun{mkApparentPerformance}$ is reduced to one of its branches, and its first argument is dropped. It is only used in the definition of $\\fun{rewardOnePool}$, which needs to be adjusted accordingly.\n\n\\begin{figure}[htb]\n \\begin{align*}\n & \\fun{mkApparentPerformance} \\in \\unitInterval \\to \\N \\to \\N \\to \\Q \\\\\n & \\mkApparentPerformance{\\sigma}{n}{\\overline{N}} = \\frac{\\beta}{\\sigma} \\\\\n & ~~~\\where \\\\\n & ~~~~~~~\\beta = \\frac{n}{\\max(1, \\overline{N})} \\\\\n \\end{align*}\n \\caption{Function used in the Reward Calculation}\n \\label{fig:functions:rewards}\n\\end{figure}\n\nThe function $\\fun{createRUpd}$ is adjusted by simplifying $\\eta$.\n\n\\begin{figure}[htb]\n \\emph{Calculation to create a reward update}\n %\n \\begin{align*}\n & \\fun{createRUpd} \\in \\N \\to \\BlocksMade \\to \\EpochState \\to \\Coin \\to \\RewardUpdate \\\\\n & \\createRUpd{slotsPerEpoch}{b}{es}{total} = \\left(\n \\Delta t_1,-~\\Delta r_1+\\Delta r_2,~\\var{rs},~-\\var{feeSS}\\right) \\\\\n & ~~~\\where \\\\\n & ~~~~~~~\\dotsb \\\\\n & ~~~~~~~\\eta =\n \\frac{blocksMade}{\\floor{{slotsPerEpoch} \\cdot \\ActiveSlotCoeff}} \\\\\n & ~~~~~~~\\dotsb\n \\end{align*}\n\n \\caption{Reward Update Creation}\n \\label{fig:functions:reward-update-creation}\n\\end{figure}\n\n$\\fun{incrBlocks}$ gets the same treatment as $\\fun{mkApparentPerformance}$. Its invocation in $\\mathsf{BBODY}$ needs to be adjusted as well.\n\n\\begin{figure}\n \\begin{align*}\n & \\fun{incrBlocks} \\in \\KeyHash_{pool} \\to\n \\BlocksMade \\to \\BlocksMade \\\\\n & \\fun{incrBlocks}~\\var{hk}~\\var{b} =\n \\begin{cases}\n b\\cup\\{\\var{hk}\\mapsto 1\\} & \\text{if }\\var{hk}\\notin\\dom{b} \\\\\n b\\unionoverrideRight\\{\\var{hk}\\mapsto n+1\\} & \\text{if }\\var{hk}\\mapsto n\\in b \\\\\n \\end{cases}\n \\end{align*}\n\\end{figure}\n\n\\newpage\nFinally, the $\\mathsf{PRTCL}$ STS needs to be adjusted. To retire the $\\mathsf{OVERLAY}$ STS, we inline the definition of its 'decentralized' case and drop all the unnecessary variables from its environment. It is invoked in $\\mathsf{CHAIN}$, which needs to be adjusted accordingly.\n\n\\begin{figure}\n \\emph{Protocol environments}\n \\begin{equation*}\n \\PrtclEnv =\n \\left(\n \\begin{array}{r@{~\\in~}lr}\n \\var{pd} & \\PoolDistr & \\text{pool stake distribution} \\\\\n \\eta_0 & \\Seed & \\text{epoch nonce} \\\\\n \\end{array}\n \\right)\n \\end{equation*}\n \\caption{Protocol transition-system types}\n \\label{fig:ts-types:prtcl}\n\\end{figure}\n\n\\begin{figure}[ht]\n \\begin{equation}\\label{eq:prtcl}\n \\inference[PRTCL]\n {\n \\var{bhb}\\leteq\\bheader{bh} &\n \\eta\\leteq\\fun{bnonce}~(\\bhbody{bhb})\n \\\\~\\\\\n {\n \\eta\n \\vdash\n {\\left(\\begin{array}{c}\n \\eta_v \\\\\n \\eta_c \\\\\n \\end{array}\\right)}\n \\trans{\\hyperref[fig:rules:update-nonce]{updn}}{\\var{slot}}\n {\\left(\\begin{array}{c}\n \\eta_v' \\\\\n \\eta_c' \\\\\n \\end{array}\\right)}\n }\\\\~\\\\\n {\n \\vdash\\var{cs}\\trans{\\hyperref[fig:rules:ocert]{ocert}}{\\var{bh}}\\var{cs'}\n }\n \\\\~\\\\\n \\fun{praosVrfChecks}~\\eta_0~\\var{pd}~\\ActiveSlotCoeff~\\var{bhb}\n }\n {\n {\\begin{array}{c}\n \\var{pd} \\\\\n \\eta_0 \\\\\n \\end{array}}\n \\vdash\n {\\left(\\begin{array}{c}\n \\var{cs} \\\\\n \\eta_v \\\\\n \\eta_c \\\\\n \\end{array}\\right)}\n \\trans{prtcl}{\\var{bh}}\n {\\left(\\begin{array}{c}\n \\varUpdate{cs'} \\\\\n \\varUpdate{\\eta_v'} \\\\\n \\varUpdate{\\eta_c'} \\\\\n \\end{array}\\right)}\n }\n \\end{equation}\n \\caption{Protocol rules}\n \\label{fig:rules:prtcl}\n\\end{figure}", "meta": {"hexsha": "d4925d8ce5c7be4919e78901833a94aefa878518", "size": 4495, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "eras/babbage/formal-spec/remove-overlay.tex", "max_stars_repo_name": "MELD-labs/cardano-ledger", "max_stars_repo_head_hexsha": "9c3b4737b13b30f71529e76c5330f403165e28a6", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "eras/babbage/formal-spec/remove-overlay.tex", "max_issues_repo_name": "MELD-labs/cardano-ledger", "max_issues_repo_head_hexsha": "9c3b4737b13b30f71529e76c5330f403165e28a6", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "eras/babbage/formal-spec/remove-overlay.tex", "max_forks_repo_name": "MELD-labs/cardano-ledger", "max_forks_repo_head_hexsha": "9c3b4737b13b30f71529e76c5330f403165e28a6", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-03-14T16:56:46.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-14T16:56:46.000Z", "avg_line_length": 35.3937007874, "max_line_length": 440, "alphanum_fraction": 0.6100111235, "num_tokens": 1502, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7090191337850932, "lm_q2_score": 0.4921881357207956, "lm_q1q2_score": 0.3489708056480584}} {"text": "\\chapter{Overhung}\n\t\\section{Introduction}\n\t\tOverhung rotors are widely used in industrial turbo-machines. For certain gas turbines, gyroscopic effects of the disks may almost double the critical speeds of the rotor systems, compared to normal mechanical vibration systems. In addition, the asymmetry of the bearing stiffness will bring more complication to a rotor system. Even though there are many established publications about how to theoretically model an overhung rotor with anisotropic bearings (\\cite{Genta}-\\cite{Ishida},\\cite{Dimarogonas}), very few papers compared the theoretical models with experimental results. Ishida al etc.\\cite{Ishida} derived a sophisticated mathematical model to theoretically investigate the nonstationary vibration of a flexible rotor with nonlinear spring parameters during acceleration. They employed an asymptotic method to compute the first approximate solution to vibration response. Then, they calculated the amplitude variation curves of each oscillation component using complex-FFT. Furthermore, they inspected how each nonlinear component in polar format affected dynamic vibration. They proposed a unique signal processing method called complex-FFT where the rotor whirling plane is coincident with the complex plane. The whirling direction of the rotor can be judged by filtering the vibration signals at different frequencies using complex-FFT method. However, the complex-FFT doesn’t provide instrumentation phase angle information and full spectrum cascaded plots.\n\t\t\\par\n\t\tGunter etc. (\\cite{Gunter}, \\cite{Gunter 93}) performed investigations about the forward and backward modes of overhung rotors through theoretical models and Finite Element Analysis. Like the majority of the publications, they use FFT to analyze their simulation results. In 1993, Southwick, Goldman, and Muszynska (\\cite{Goldman}-\\cite{Southwick 94}) introduced the new powerful “full spectrum” plot in rotating machinery vibration and rotor dynamics. Since then, Bently Nevada Corporation has installed full spectrum cascade plots in all of their major machinery data acquisition software package entitled as ADRE system. Compared to “traditional (half) spectrum” FFT plot, full spectrum can extract more significant diagnostic information from the original signals generated by X, Y transducers, this allows engineers to determine whether the vibration response is forward or backward with respect to rotational direction of the shaft. As a powerful tool for interpreting the vibration signals of rotating machinery, the full spectrum plots display the correlation between the vibration signals from the X and Y transducers. Based on reference (\\cite{Bently}-\\cite{Southwick 94}), Bently Nevada of GE Energy builds a multi-channel signal processing and data acquisition system, the ADRE Sxp Software and the 408 DSPi (Dynamic Signal Processing Instrument) respectively. Unlike any other data acquisition systems, ADRE 408 SDPi is an extremely versatile system designed for real-time highly parallel signal processing and presentation. Cascade full spectrum plots are embedded into all ADRE software which are installed on the majority of turbo-machines in industry.\n\t\t\\par\n\t\tMuszynska (1996) \\cite{Muszynska 96} methodically investigated the dynamic behaviors of a vertically configured overhung imbalanced rotor supported by flexible anisotropic bearings by theory and experiments. She concluded that the interaction of imbalance and shaft bow causes the synchronous forced precession of the rotor to be forward or backward. Furthermore, she investigated the situation in which the mid-span rotor sections precess one way, while the outboard disk precesses the other. The phenomenon is partly attributed to the relationship between the unbalances in terms of their relative direction. Ishida et al. (2008) \\cite{Ishida 2008} theoretically investigated internal resonances near both the primary and the gravity critical speeds for a rotor system of an asymmetric shaft supported at both ends and a disk installed in the middle; nonlinearities of the system induced by bearing clearances were thoroughly explored. Nagasaka et al. (2008) \\cite{Nagasaka} did further research by studying internal resonance between the forward and backward whirl modes near both the primary and the secondary critical speeds for a simply-supported rotor system with asymmetric shaft. Based on above research, Nandakumar et al. (2010) \\cite{Nandakumar} extended their research to an overhung rotor with substantial gyroscopic effects and relatively large lateral vibration using the method of multiple scales (MMS). The authors derived the theoretical nonlinear equations of motion for an asymmetrical overhung rotor running near its gravity critical speed. The authors examined how gyroscopic effects contribute on maximum resonant amplitudes, which disclose some interesting phenomena induced solely by nonlinearities of the system.\n\t\t\\par \n\t\tYim et al. (2012) \\cite{Yim} studied the dynamic response of a flexible shaft with a disk subjected to axial forces for two overhung rotor systems using transfer matrix method. They concluded that, under the force load, the gyroscopic effect not only increases the critical axial force, but also changes the instability type from divergence to flutter. Ma et al. (2015) \\cite{Ma}experimentally investigated the vibration response of an overhung rotor under sudden unbalance excitation due to the blade loss and quantitatively assessed the impact effect. The effect of the sudden unbalance is tested in both the subcritical state and the supercritical state. The results demonstrate that the response of the system to sudden imbalance contains frequencies from the impact response as well as frequencies due to rotational speed. A flexible rotor was shown to excite more from sudden imbalance than a more rigid rotor. Ma et al. (2015) \\cite{Ma, H} presents a finite element model of the oil film instability of an overhung rotor with both parallel and angular misalignments including the gyroscopic effect. Oil film bearings are simulated using a non-linear oil-film force model, assuming short-length bearings. The validity of the model is verified by comparison to experimental results in published literature. Results show that misalignment of the coupling can delay the first mode of vibration and even reduce its amplitude.\n\t\t\\par \n\t\tUnfortunately, to the best of the authors’ knowledge, there are no publications about how to actually generate the cascade full spectrums directly from either X, Y transducers or simulation results. If we successfully solve this problem, we can predict what will happen in experiments from theoretical models. In this research, we will solve the following 4 problems. 1. We construct 3D full spectrum cascade plots using complex FFT through MATLAB programing. 2. We use tracking windows to filter the transducer data to nX components of rotor speed when the rotor starts up or runs down. 3. We experimentally compare our results with ADRE data. The results are directly comparable and the MATLAB plots, using our method, provide opportunity of further post processing full spectrum data. 4. Theoretical results of overhung rotor are converted into 3D full spectrum plots which will be compared with experimental data. The results presented in this paper match with experiments with confidence. More importantly, effects of different components of the theory are able to be determined. This allows for better diagnoses of real rotor systems. Stiffness of the bearings is determined by comparing theoretical natural frequencies to natural frequencies determined by experimentation. Using a Bode plot for each $xz$ and $yz$ plane, Figure \\ref{fig:Figure_14}, the natural frequency of the experimental apparatus is determined for each plane. The equations of motion \\ref{math:1}, with disregard for forcing and moment equations on the right hand side, are coupled with the total stiffness matrix \\ref{math:10} to provide a system of equations that is dependent only on time, speed and the unknown bearing stiffness parameters. This system of equations is used to solve for the natural frequencies in terms of the speed and the bearing stiffness using the eigenvalue problem. Different values of bearing stiffness are chosen in each plane and a Campbell diagram is used to solve for the natural frequency as a function of speed. Values of stiffness are iterated until the theoretical and experimental natural frequencies match.\\par\n\t\tSkew angle is determined by comparing theoretical and experimental angular displacement vibrations. Deducing the angular displacement from experiment is not easy to do, as transducers are observing the vibration of the shaft and not the angle of the disk. A method for determining the angle was employed in which displacements from two sets of transducers is compared. \\par\n\t\tTo the best of the authors’ knowledge, there are no papers which provide direct comparison between theoretical models and experimental results using full spectrum analysis. Experimental work sometimes uses full spectrum but most often it uses half spectrum. Analytical work that reports full spectrum does not correlate phase angel to instrument measurement.\n\t\t\\par\n\t\\section{Theoretical Model}\n\t\tThe theoretical model used to simulate the experimental apparatus is depicted in Figures \\ref{fig:Figure_1} and \\ref{fig:Figure_2}. Equations of motion \\ref{math:1} are determined using conservation of momentum and a dynamic analysis of the disk mass and inertial changes. The forcing functions on the right hand side of \\ref{math:1} include forces due to acceleration of the shaft so that the simulation can include change in speed of the shaft and gyroscopic effects, such as a start-up or run-down. Moment equations are presented as pertaining to the change in angular momentum of the disk (\\cite{Genta}, \\cite{Muszynska},\\cite{Gunter 93}).\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=0.25]{./figures/Images/Math_1}\n\t\t\t\\label{math:1}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\t\\begin{figure}[h]\n\t\t\t\n\t\t\t\\begin{subfigure}[b]{.5\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t \\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_1a}\n \t\t\t\t \\caption{Displacements, rotations, forces and moments}\n \t\t\t\t \\label{fig:Figure_1a}\n \t\t\t\t \\centering\n\t\t\t\\end{subfigure}%\n\t\t\t\\begin{subfigure}[b]{.5\\textwidth}\n \t\t\t\t\\centering\n \t\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_1b}\n \t\t\t\t\\caption{Depiction of skew angle $\\chi$}\n \t\t\t\t\\label{fig:Figure_1b}\n \t\t\t\t\\centering\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{Experimental apparatus mathematical representation}\n\t\t\t\\label{fig:Figure_1}\n\t\t\t\n\t\t\\end{figure}\n\t\t\\begin{figure}[h]\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Figure_2}\n\t\t\t\\caption{Diagram showing important parameters}\n\t\t\t\\label{fig:Figure_2}\n\t\t\t\\centering\n\t\t\\end{figure}\n\t\tThe total stiffness constants of the system are determined from geometry and properties of the rotor bearings, and divided into two main contributions from the bending of the shaft and the suspension of the bearings. These two contributions are considered independent to solve for each contribution and then are combined in series to determine the total stiffness.\\par\n\t\tCase (A) Flexible shaft with rigid bearings. Flexible Influence coefficients method in the general form of \\ref{math:2} is used to derive flexibility matrix which can be applied in both $xz$ and $yz$ plane, respectively. Since the shaft is considered to be isotropic, the matrix is identical for the xz and the $yz$ plane (\\cite{Ishida},\\cite{Dimarogonas}).\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_2}\n\t\t\t\\label{math:2}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_3}\n\t\t\t\\label{math:3}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\tCase (B) Rigid shaft with flexible bearing. The stiffness matrix is derived in the general form of \\ref{math:4} using stiffness influence coefficients method.\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_4}\n\t\t\t\\label{math:4}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_5}\n\t\t\t\\label{math:5}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\tIn order to combine the bearing stiffness with the shaft stiffness, both are added as flexibility matrices.\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_6}\n\t\t\t\\label{math:6}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\tApply to $xz$ and $yz$ plane, respectively to account for anisotropy of bearing stiffness for different planes.\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_7}\n\t\t\t\\label{math:7}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_8}\n\t\t\t\\label{math:8}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\tTotal flexibility matrices in $xz$ and $yz$ plane are:\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_9}\n\t\t\t\\label{math:9}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\tTotal stiffness matrices are shown in \\ref{math:10}, which are applied in equations of motion \\ref{math:1}.\\par\n\t\t\\begin{equation}\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Math_10}\n\t\t\t\\label{math:10}\n\t\t\t\\centering\n\t\t\\end{equation}\n\t\t\t\n\t\\section{Experimental apparatus}\n\t\tOur experimental apparatus consists of the GE (Formally Bently Nevada) RK4 rotor kit, the ADRE 408 Dspi Data asset condition monitoring equipment, and a laptop with the ADRE sxp software (Figure \\ref{fig:Figure_3}). Four eddy current displacement transducers are used to measure the vibration of the shaft. The ADRE 408, when coupled with ADRE sxp, is capable of providing real-time signal processing from the rotor system in meaningful figures.\\par\n\t\t\\begin{figure}[H]\n\t\t\t\\begin{subfigure}[b]{.5\\textwidth}\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=.95\\linewidth]{./figures/Images/Figure_3.jpg}\n\t\t\t\t\\caption{Schematic of rotor geometry}\n\t\t\t\t\\label{fig:Figure_3a}\n\t\t\t\t\\centering\n\t\t\t\\end{subfigure} %\n\t\t\t\\begin{subfigure}[b]{.5\\textwidth}\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=.95\\linewidth]{./figures/Images/Figure_3b.jpg}\n\t\t\t\t\\caption{Overview of apparatus}\n\t\t\t\t\\label{fig:Figure_3b}\n\t\t\t\t\\centering\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{Experimental apparatus}\n\t\t\t\\label{fig:Figure_3}\n\t\t\\end{figure}\n\t\t\\begin{table}[H]\n\t\t\t\\centering\n\t\t\t\\caption{Rotor parameters}\n\t\t\t\\includegraphics[scale=.25]{./figures/Images/Table_1}\n\t\t\t\\label{tab:Table_1}\n\t\t\t\\centering\n\t\t\\end{table}\n\t\\section{Identify the bearing stiffness parameters from experimental data}\n\t\tWith the experimental data we were able to identify the natural frequency in the horizontal and vertical directions from the Bode plot of each horizontal and vertical transducer. From the theoretical equations of motion, the eigenvalue problem can be used to solve for the natural frequencies of the system. Since the system includes gyroscopic moments that depend on the rotational speed of the rotor, the natural frequency is a function of the speed of the rotor. The Campbell Diagram is a plot of the natural frequency as it changes with the increasing speed of the rotor. This diagram can be used to tune the bearing stiffness. The parameters that affect the natural frequency that are not evident from the description of the system is the stiffness of the bearings in each $xz$ and $yz$ plane. To solve for the stiffness values, a Campbell Diagram is created for each value of stiffness until the point at which the natural frequency line intersects the rotor speed line is the same value as the experimental natural frequency. An example of a Campbell diagram can be seen in Figure \\ref{fig:Figure_4}. The Campbell diagram gives a good depiction of how the natural frequency changes with speed and how negative natural frequencies decrease in frequency, while the positive natural frequency increases. The stiffness of the bearings was determined to be about 91000 N/m in the xz plane and about 120000 N/m in the $yz$ plane.\\par \n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\includegraphics[scale=.75]{./figures/Images/Figure_4}\n\t\t\t\\caption{Example Campbell diagram}\n\t\t\t\\label{fig:Figure_4}\n\t\t\\end{figure}\n\t\tTo further understand the effect that stiffness anisotropy has on the system, Figure \\ref{fig:Figure_5} and \\ref{fig:Figure_6} display a cascade of amplitude plots with changing stiffness anisotropy between the horizontal and vertical planes. Viewing the theoretical data in this way allows for comparison of the two natural frequencies to experimental values just as the Campbell Diagram does. In addition, the cascaded plots depict how the shape of the amplitude plots change with varying anisotropy. Figure \\ref{fig:Figure_5} uses the amplitude of positive and negative frequencies to display the effect of anisotropy. The interaction between the first and second natural frequencies can be seen as the two get closer to one another. Negative whirling is dominant only between the two natural frequencies. When the two natural frequencies coincide, there is a highly circular positive whirl orbit. Experimental amplitude plots are used in conjunction with Figure \\ref{fig:Figure_5} \\& \\ref{fig:Figure_6} to choose the correct anisotropy at which the characteristic shape matches.\\par \n\t\tThe skew angle, $\\chi$, and the eccentricity, $\\varepsilon$, are determined next by independent methods. Eccentricity can be determined by simulating the theoretical model under varying eccentricities until the magnitude of the vibration is equivalent to that of the experimental results. The value was determined to be $9\\e{-4} m$. The skew angle can be quantified using a similar technique. But the skew angle is closely related to the amplitude of vibration in the two orthogonal directions, $x$ and $y$, as well as the disk tilting angles, $\\theta_x$ and $\\theta_y$. In order to measure the tilting angles in the experiment, there were two sets of orthogonal transducers. One before the disk on the shaft and the other after the disk. This displacement between the two sets of transducers allows for an approximation of the angles of tilt. Assuming no bending of the shaft, the angle is deduced from a pivot about Bearing B, with the angle being equal to the inverse tangent of the ratio of difference in displacement between the transducers, and the distance between the transducers. Then the simulation is run with varying values of $\\chi$ to match the amplitude of angles and displacements to the experiment. The value was determined to be 0.09 radians.\\par \n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_5}\n\t\t\t\\caption{3D cascading plot demonstrating the impact of stiffness anisotropy on positive/negative frequencies}\n\t\t\t\\label{fig:Figure_5}\n\t\t\\end{figure}\n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_6}\n\t\t\t\\caption{3D cascading plot demonstrating the impact of stiffness anisotropy on horizontal/vertical frequencies}\n\t\t\t\\label{fig:Figure_6}\n\t\t\\end{figure}\n\n\t\\section{Produce full spectrum plots from orthogonal transducers }\n\t\tThe purpose of this work was to produce figures that represent vibration amplitude data from theoretical models in a way that is comparable to figures produced by the ADRE Sxp software. This allows the analysis of theoretical models with confidence in the representation of the data with figures such as the Bode plot, the Cascade plot, and full spectrum plots. The major achievement in producing the figures was obtaining phase lag angles between two signals. Another achievement was producing spectrum plots with both positive and negative frequencies (Full Spectrum plots).\\par \n\t\tFull spectrum plots are more complete in displaying the information contained in a Fourier Transform. In the frequency domain, negative frequencies pertain to vibration of the rotor that is precessing in the opposite rotational direction than that of the rotor rotational direction. The full spectrum plot is produced by first coupling the orthogonal transducer signals into a complex signal and then performing a Fourier transform. The transform on the complex data results in frequency information both negative and positive.\\par \n\t\tNegative frequencies tell a lot about the system, especially when they are greater than their positive counterpart. When the negative frequency vibration is greater in amplitude, the precession is said to be in the “reverse” direction. This can be vital information in diagnosing a problem in a real rotor system. In the case explored here it can be a sign of both, strong gyroscopic moments, as well as anisotropy of the stiffness from the shaft or the bearings. Another example is the rub in a fluid film bearing which manifests itself as a negative precession at twice the running speed of the rotor. This problem would be more difficult to diagnose without the knowledge of negative frequencies.\\par \n\t\tIn order to verify that the MATLAB code of our method would produce accurate frequency data, the Fast Fourier Transform, or FFT, of a steady state signal was compared to the FFT produced by the ADRE 408 with ADRE Sxp software. Both the amplitude and frequency of positive and negative frequency components was verified by comparison as demonstrated by the comparison of Figure \\ref{fig:Figure_7} and \\ref{fig:Figure_8} .\\par \n\t\t\\begin{figure}[H]\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.6\\linewidth]{./figures/Images/Figure_7}\n\t\t\t\\caption{Full Spectrum plot produced with the ADRE sxp software}\n\t\t\t\\label{fig:Figure_7}\n\t\t\\end{figure}\n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.6\\linewidth]{./figures/Images/Figure_8}\n\t\t\t\\caption{Full Spectrum plot produced with using our method}\n\t\t\t\\label{fig:Figure_8}\n\t\t\\end{figure}\n\t\\section{Produce Bode plots from orthogonal transducers }\n\t\tThe process for producing phase lag angles begins with segmenting the data into windows of a fixed number of samples. This is because most rotordynamic data comes in the form of a “start-up” or a “run-down”. In both of these scenarios, the rotational speed of the rotor is continuously changing, as demonstrated by Figure \\ref{fig:Figure_9a}. This change deters capturing both time and frequency information. Segmenting the data provides small windows on which analysis can be performed as if the signal was a constant frequency within the window, as demonstrated in Figure \\ref{fig:Figure_9b}. For a segment of data in which the frequency is not changing, a Fourier Transform can be applied to obtain the frequency spectrum of the signal. In this frequency domain provided by the Fourier Transform, the phase angle of each frequency can be obtained as well as the amplitude of vibration of each frequency. In order to calculate the phase lag from one signal to another, this phase angle is compared for the same frequency of each signal in the same window in time. This process is done for each window through the entire length of data producing a phase angle for each window.\\par \n\t\tThe Bode plot is ubiquitous in rotating machinery diagnostics and can be used to characterize a system fairly thoroughly. Phase angle is a vital part of the bode plot and can shed light on many phenomenon One of the main goals in this paper is to demonstrate the ability to produce accurate phase lag information from two signals. As seen in the Bode plot of Figure \\ref{fig:Figure_10} , the MATLAB code based on complex FFT strategy produces phase lag information nearly identical to the phase lag information produced with the ADRE system using the same transducer data.\\par \n\t\t\\begin{figure}[H]\n\t\t\t\\begin{subfigure}[b]{.5\\linewidth}\t\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=1\\textwidth]{./figures/Images/Figure_9a}\n\t\t\t\t\\caption{Rotational speed change over time}\n\t\t\t\t\\label{fig:Figure_9a}\n\t\t\t\\end{subfigure}\n\t\t\t\\begin{subfigure}[b]{.5\\linewidth}\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=1\\textwidth]{./figures/Images/Figure_9b}\n\t\t\t\t\\caption{Rotational speed change over a window in time}\n\t\t\t\t\\label{fig:Figure_9b}\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{Steady-state approximation demonstration}\n\t\t\t\\label{fig:Figure_9}\n\t\t\\end{figure}\n\t\tThis ability to produce phase lag information from any transducer data allows accurate comparison of theoretical and experimental data. The MATLAB code that produces these plots also allows the ability of tuning the resolution of the phase and amplitude information, gaining accuracy in either time or frequency. The phase lag calculation accuracy is determined by the number of cycles analyzed in each window, so the larger the window, the better the phase information. But, as the window gets larger, the accuracy of speed is diminished.\\par \n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_10}\n\t\t\t\\caption{Bode plot for verification of the phase angle technique}\n\t\t\t\\label{fig:Figure_10}\n\t\t\\end{figure}\n\t\\section{Theoretical and experimental comparison}\n\t\tThe comparison of experiment and theoretical model developed in this research were performed. The geometric and other parameters of the rotor kit are shown in table 1.\\par \n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_11}\n\t\t\t\\caption{Experimental Cascade plot directly produced by the ADRE sxp software}\n\t\t\t\\label{fig:Figure_11}\n\t\t\\end{figure}\n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_12}\n\t\t\t\\caption{Re-constructed 3D experimental full-spectrum cascade plot using our strategy}\n\t\t\t\\label{fig:Figure_12}\n\t\t\\end{figure}\n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=.75\\linewidth]{./figures/Images/Figure_13}\n\t\t\t\\caption{3D theoretical full-spectrum cascade plot using our strategy}\n\t\t\t\\label{fig:Figure_13}\n\t\t\\end{figure}\n\t\tThe full-spectrum cascade plot is a vital tool for analysis in rotating machinery. The cascade plot depicts how the vibrations change with time. With the addition of the full spectrum in these cascade plots, the negative frequencies provide insight for diagnostics of certain rotor issues, such as misalignment of disk axis with the rotor axis. Just as with the FFT and Bode plots, the cascade plot produced with our MATLAB code using complex FFT strategy is compared with the cascade produced with the ADRE software to validate the method. From figure \\ref{fig:Figure_11}, \\ref{fig:Figure_12} and \\ref{fig:Figure_13}, we can see the results are similar and good enough to validate our full-spectrum strategy through experiment. Furthermore, MATLAB has the ability to display data in three dimensions as shown in the figures (\\cite{Southwick},\\cite{Southwick 94},\\cite{Dimarogonas}). With proper tuning of window size and sampling frequency, the cascade plot can display a host of rotordynamic phenomenon.\\par \n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1\\linewidth]{./figures/Images/Figure_14}\n\t\t\t\\caption{Bode plot, Experimental vs. Theoretical}\n\t\t\t\\label{fig:Figure_14}\n\t\t\\end{figure}\n\t\tThe Bode plot of Figure \\ref{fig:Figure_14} compares the startup amplitude and phase angle of the theoretical model and the experiment. Because both the experimental and the theoretical results were processed using our strategy with the same MATLAB code to produce these plots, comparisons can be confidently drawn. The phase angles are of a phase lag from the X transducer and from the Y transducer. The two transducers are 270 positive degrees apart around the shaft.\\par \n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1\\linewidth]{./figures/Images/Figure_15}\n\t\t\t\\caption{Positive/Negative frequency comparison, Experimental vs. Theoretical}\n\t\t\t\\label{fig:Figure_15}\n\t\t\\end{figure}\n\t\tObserving the comparison of positive and negative frequencies as done in Figure \\ref{fig:Figure_15} allows the detection of backward or forward whirl. When the negative frequency amplitude is greater than that of the positive frequency, the whirl is “backward”. Backward whirl describes the phenomenon of precession of the shaft vibration rotating opposite to the rotation of the shaft. Comparison to the typical horizontal vs. vertical amplitude plot shows that the crossover from positive to negative whirl occurs when either the horizontal or vertical vibration amplitudes drop.\\par \n\t\t\\begin{figure}[H]\n\t\t\t\\begin{subfigure}[b]{.5\\linewidth}\t\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=.75\\textwidth]{./figures/Images/Figure_16a}\n\t\t\t\t\\caption{Experimental 1333 RPM orbit}\n\t\t\t\t\\label{fig:Figure_16a}\n\t\t\t\\end{subfigure}\n\t\t\t\\begin{subfigure}[b]{.5\\linewidth}\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=.75\\textwidth]{./figures/Images/Figure_16b}\n\t\t\t\t\\caption{Experimental 1433 RPM orbit}\n\t\t\t\t\\label{fig:Figure_16b}\n\t\t\t\\end{subfigure}\n\t\t\t\\begin{subfigure}[b]{.5\\linewidth}\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=.75\\textwidth]{./figures/Images/Figure_16c}\n\t\t\t\t\\caption{Theoretical 1358 RPM orbit}\n\t\t\t\t\\label{fig:Figure_16c}\n\t\t\t\\end{subfigure}\n\t\t\t\\begin{subfigure}[b]{.5\\linewidth}\n\t\t\t\t\\centering\n\t\t\t\t\\includegraphics[width=.75\\textwidth]{./figures/Images/Figure_16d}\n\t\t\t\t\\caption{Theoretical 1428 RPM orbit}\n\t\t\t\t\\label{fig:Figure_16d}\n\t\t\t\\end{subfigure}\n\t\t\t\\caption{Orbit plots fo filtered experimental data compared to theoretical predictions at similar speeds}\n\t\t\t\\label{fig:Figure_16}\n\t\t\\end{figure}\n\t\tOrbit plots of Figure \\ref{fig:Figure_16} are created using experimental and theoretical data. The orbits of experimental data contain the key-phasor dot to mark the beginning of a rotation from a fixed location on the shaft. This allows the determination of the orbit rotation direction. For all of the orbit plots, the shaft was rotating in the counter-clockwise direction. Therefore, an orbit in the clockwise direction is in the “reverse” direction. Experimental data was filtered using a tracking filter for the orbit plots. Since a once per turn reference is unavailable for theoretical data, the orbits are plotted with a reference dot at the maximum $x$ value for that orbit.\\par \n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1\\linewidth]{./figures/Images/Figure_17}\n\t\t\t\\caption{Experimental 3D Orbit plot produced by our strategy}\n\t\t\t\\label{fig:Figure_17}\n\t\t\\end{figure}\n\t\t\\begin{figure}[H]\t\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=1\\linewidth]{./figures/Images/Figure_18}\n\t\t\t\\caption{Theoretical 3D Orbit plot}\n\t\t\t\\label{fig:Figure_18}\n\t\t\\end{figure}\n\t\tOrbit plots are cascaded in increasing rotational speed in the 3D orbit plots of Figures \\ref{fig:Figure_17} and \\ref{fig:Figure_18}. This cascade of orbits gives a sense of how the natural frequencies are physically manifested in these startups. This 3D orbit plot lends a more intuitive representation of the data and at the same time, important characteristics such as phase lag, amplitudes and reverse precession can be discerned from the plot. This plot was also useful in comparing the theoretical model to the experimental data, because it demonstrates magnitude as well as general shape of the response of the entire orbit in one plot. Also Figure \\ref{fig:Figure_17} and \\ref{fig:Figure_18} makes clear the effect of gyroscopic moments, as the shape of the elliptical orbits during peak amplitude vibrations are bent. The ADRE Sxp software does not produce this plot.\\par \n\t\\section{Discussion and conclusions}\n\t\t3D cascade plots (Figures \\ref{fig:Figure_12}, \\ref{fig:Figure_13}) produced using our complex FFT strategy clearly display the full vibration spectrum and complement those produced with the ADRE Sxp software (Figure \\ref{fig:Figure_11}). Both the theoretical model and the experimental results display a sudden dip in amplitude of positive frequencies between the two natural frequencies (Figures \\ref{fig:Figure_12},\\ref{fig:Figure_13}). This dip is coupled with an increase in negative frequencies, resulting in reverse orbits for a period in time. These negative orbits can be seen in Figure \\ref{fig:Figure_16}. The theoretical model suggested a strong influence from the relative phase angle of the skew to the eccentricity. Some values of phase difference offered completely positive orbits through the natural frequency, while others offered highly negative orbits. This is the greatest contribution from the skew angle, as without the consideration of it, the amount of negative to positive vibration could not be easily altered in the theoretical model.\\par \n\t\tIn this paper, the dynamic behavior of a flexible shaft with an overhung disk has been studied by employing full spectrum analysis. Anisotropic bearing stiffness and the effect of gyroscopic moments are considered in our classic theoretical model. Some of the parameters which are difficult to directly measure are estimated by combination of theoretical calculation and experimental results. Full spectrum analysis is applied to the system to reveal some interesting backward whirl vibration phenomena due to the interaction of anisotropy of the stiffness of the bearings, flexibility of the shaft and gyroscopic effect. Phenomena such as these cannot be disclosed by traditional half spectrum FFT. 3D cascade full spectrum plots from theoretical model are compared to those from experiments. Our full spectrum cascade plots using complex FFT strategy capture more detail in the vibration response of the system than the typical cascade plots. Filtered orbit plots with the key-phasor dots at different speed demonstrate forward and backward whirls. Methods for identifying the eccentricity and skew angle of the disk are presented. Also, the effect of the skew angle is determined by comparison of experiment and theory. The skew angle is determined by comparing theoretical angular amplitudes to experimental angular amplitudes. Bearing stiffness is determined through a method of matching theoretical and experimental natural frequencies. Consideration of the gyroscopic moments introduces speed as another dependent variable in the equation for natural frequency, making the process of matching an iterative one.\\par \n\t\\section{Acknowledgments}\n\t\tThe authors acknowledge the Donald E. Bently Center for Engineering Innovation at California Polytechnic State University San Luis Obispo for support of this work.\\par ", "meta": {"hexsha": "e0a8ebd1481078859ea94577d61e108f1d252266", "size": 34402, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapters/Overhung.tex", "max_stars_repo_name": "cameron1320/Cameron", "max_stars_repo_head_hexsha": "276144ab299ab00f102f3f3cc0869261a13e0059", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapters/Overhung.tex", "max_issues_repo_name": "cameron1320/Cameron", "max_issues_repo_head_hexsha": "276144ab299ab00f102f3f3cc0869261a13e0059", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapters/Overhung.tex", "max_forks_repo_name": "cameron1320/Cameron", "max_forks_repo_head_hexsha": "276144ab299ab00f102f3f3cc0869261a13e0059", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 120.2867132867, "max_line_length": 2128, "alphanum_fraction": 0.7935003779, "num_tokens": 7968, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7090191214879992, "lm_q2_score": 0.49218813572079556, "lm_q1q2_score": 0.3489707995955746}} {"text": "\\documentclass[journal = jacsat,manuscript = note]{achemso}\n\n%% The graphicx package provides the includegraphics command.\n%\\usepackage{graphicx}\n%\\usepackage[hidelinks]{hyperref}\n%\\usepackage{authblk}\n\\usepackage{natbib}\n%\\usepackage[backend=bibtex]{biblatex}\n\\usepackage[T1]{fontenc} % Use modern font encodings\n%% The amssymb package provides various useful mathematical symbols\n\\usepackage{amssymb}\n\n\\usepackage{amsmath}\n\n%% The lineno packages adds line numbers. Start line numbering with\n%% \\begin{linenumbers}, end it with \\end{linenumbers}. Or switch it on\n%% for the whole article with \\linenumbers after \\end{frontmatter}.\n%\\usepackage{lineno}\n\n%\\bibliographystyle{plain}\n%\\addbibresource{plasmid_replication.bib}\n\n\\title{How to model DNA replication in stochastic models of synthetic gene circuits (and why)}\n\n\\author{Samuel E. Clamons}\n\\author{Richard M. Murray}\n\\email{murray@cds.caltech.edu}\n\\affiliation{Caltech, Pasadena, CA, United States}\n\n%\\keywords{Keyword1, Keyword2, Keyword3}\n\n\\begin{document}\n\n\n\\begin{abstract}\nBiocircuit modeling sometimes requires explicit tracking of a self-replicating DNA species. The most obvious, straightforward way to model a replicating DNA is structurally unstable and leads to pathological model behavior. We describe a simple, stable replication mechanism with good model behavior and show how to derive it from a mechanistic model of ColE1 replication. \n\\end{abstract}\n\n\\section{Introduction}\n\nAlthough most synthetic biocircuits use DNA, models of synthetic biocircuits typically do not explicitly describe the dynamics of those DNA species. For example, the original repressilator model tracked mRNA and protein species, but not DNA \\cite{elowitz2000}, and the model for the first genetic toggle switch simply tracked ``repressor 1'' and ``repressor 2'' \\cite{gardner2000}. For many circuits under many modeling assumptions, it is sufficient to assume that all DNA species are held at a fixed concentration by the cell, by means of mysterious machinery whose details are irrelevant to understanding the circuit.\n\nIn some cases, however, it is useful or necessary to explicitly represent DNA as a dynamic species. Some circuits, for example, use DNA in different states as a dynamic component or readout (for example, integrase-based state machines \\cite{roquet2016}). Explicit representation of DNA can also be useful when DNA binding is slow relative to the other circuit processes (for example, CRISPR-based transcription factor networks under some conditions \\cite{Jones2017}). Stochastic models, in particular, are often most naturally expressed using explicit DNA species.\n\nExplicitly-modeled DNA often requires some mechanism of replication -- again particularly in stochastic models. Unfortunately, the obvious replication implementation $\\mathit{DNA} \\to \\mathit{DNA} + \\mathit{DNA}$ is a trap that leads to pathological circuit behavior. \n\n\\section{Trivially-replicating plasmids have no well-defined copy number}\n\nAn obvious way to model replication of a DNA species is the straightforward \n\n$$\\mathit{DNA} \\overset{\\alpha}{\\to} \\mathit{DNA} + \\mathit{DNA}$$\n\n\\noindent shown visually in Figure \\ref{fig:models_and_traces}A. We refer to this mechanism as ``trivial self-replication''. Although trivial replication is intuitively appealing, we strongly recommend against its use. \n\nAdding a dilution reaction $\\mathit{DNA} \\overset{\\gamma}{\\to} \\emptyset$, trivially self-replicating DNA has dynamics\n\n$$\\frac{d\\mathit{DNA}}{dt} = (\\alpha - \\gamma) \\mathit{DNA}$$\n\nNotice that at steady state, we have $\\alpha - \\gamma = 0$, \\emph{independent of DNA concentration}. In other words, the only finite, non-zero steady state of the trivial replication mechanism occurs when production is precisely balanced by dilution. If production is slightly faster than dilution, then the DNA's concentration will explode unphysically (and unbiologically) to infinity. Conversely, if dilution is slightly faster than production, $\\mathit{DNA}$ will always fall to zero concentration and die out. Such a steady state is structurally unstable\n\nIn a deterministic model of a trivially-replicating plasmid, production and dilution \\emph{can} be balanced perfectly, giving a nominally constant concentration of DNA, but this mechanism of replication rejects no disturbances--any addition or removal of DNA will remain permanently uncorrected.\n\nStochastic simulations of trivially-replicating DNA cannot even achieve this level of marginal stability. Stochastic simulation is, by its nature, noisy; a stochastically simulated, trivially replicating DNA will random walk in copy number until, practically speaking, it either dies out by wandering to zero or explodes to a concentration too large to simulate (see Figure \\ref{fig:models_and_traces}A).\n\n\\section{Zero-order replication recovers good steady-state properties}\n\nThe trivial replication mechanism is unstable because it has both production and degradation of DNA depend linearly on the concentration of DNA itself. A simple way to add stability to the replication model is to remove that dependence by conditioning replication on a ``dummy replication trigger'' produced at a constant rate (shown diagramatically in Figure \\ref{fig:models_and_traces}B):\n\n\\begin{align*}\n\t\\emptyset &\\overset{\\alpha}{\\to} R\\\\\n\t\\mathit{DNA} + R &\\overset{k}{\\to} \\mathit{DNA} + \\mathit{DNA}.\\\\\n\\end{align*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=.75]{figures/models_and_traces.png}\n\\caption{\\textbf{A)} The trivial replication mechanism, in which DNA spontaneously self-replicates. This mechanism is unstable and produces random-walking DNA concentrations. \\textbf{B)} The dummy-triggered replication mechanism, in which replication is triggered by a dummy molecule produced at a constant rate. This mechanism, coupled to dilution, is stable and rejects disturbances.}\n\\label{fig:models_and_traces}\n\\end{figure}\n\nThis mechanism has ODE dynamics\n\n\\begin{align*}\n\\frac{dR}{dt} &= \\alpha - kR*\\mathit{DNA}\\\\\n\\frac{d\\mathit{DNA}}{dt} &= kR*\\mathit{DNA} - \\gamma \\mathit{DNA}\n\\end{align*}\n\nAt steady state, $R = \\frac{\\alpha}{k*\\mathit{DNA}}$, which cancels out $\\mathit{DNA}$ in $\\mathit{DNA}$'s production term and gives $\\mathit{DNA} = \\frac{\\alpha}{\\gamma}$. This steady state is stable. As long as $k$ is fast relative to other replication dynamics ($\\alpha$ and $\\gamma$), the dummy-triggered replication mechanism emulates zero-order replication of DNA.\n\nBecause dummy-triggered replication leads to a stable steady state concentration of DNA, as shown in Figure \\ref{fig:models_and_traces}B, we recommend it over the trivial replication mechanism.\n\n\\section{Biological models of plasmid replication approximately reduce to zero-order replication}\\label{sec:reduction}\n\nDummy-triggered replication is not meant to accurately describe a real biological processes. Real cells do not control replication using consumable molecules like $R$. The critical property of the dummy-triggered replication mechanism is not its realism, but its ability to achieve the (biologically important!) property of having a defined, stable steady state.\n\nNevertheless, under a few relatively mild assumptions, at least one real-world DNA replication mechanism can be reduced to a zero-order replication mechanism equivalent to the dummy-triggered replication mechanism, . \n\nConsider the ColE1 plasmid replication system, first crystallized mathematically by Brendel \\& Perelson in 1993, shown on the left in Figure \\ref{fig:bp_diagram} \\cite{Brendel1993}. We use ColE1 as an example because it has a particularly simple and well-understood replication mechanism.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=.4]{figures/bp_diagram.pdf}\n\\caption{A mechanistic model of ColE1 (left), which can be conceptually approximated by a simplified ColE1 model (middle), which under fast RNA dynamics reduces zero-order replication equivalent to that of the dummy-triggered replication model (right).}\n\\label{fig:bp_diagram}\n\\end{figure}\n\nAlthough the details of the Brendel \\& Perelson model are a bit messy, they implement a straightforward logic:\n\n\\begin{itemize}\n\t\\item $\\mathit{DNA}$ produces an RNA species $RI$. More $\\mathit{DNA}$ leads to more $RI$. \n\t\\item Occasionally, $\\mathit{DNA}$ will spontaneously enter a primed state from which it can replicate. (Biologically, the plasmid produces an RNA primer $RNAII$ which can initiate replication.)\n\t\\item $RI$ can react with a replication-primed $\\mathit{DNA}$, un-priming it and destroying the $RI$. ($RI$ binds to $RII$ to form an inert complex that is removed by RNases.)\n\t\\item If not stopped by an $RI$, a primed $\\mathit{DNA}$ can spontaneously replicate into two non-primed $\\mathit{DNA}$. (DNA polymerase initiates replication using $RII$ as a primer.)\n\\end{itemize}\n\nWe can represent this logic more clearly with a three-species reduction of the Brendel \\& Perelson model, shown in the middle in Figure \\ref{fig:bp_diagram}, consisting of ``unprimed'' $\\mathit{DNA}$, ``primed'' $\\mathit{DNA}_p$, and feedback RNA $R$. \n\nWe can further simplify the three-species model by assuming that RNA transcription and degradation are fast compared to other replication dynamics. This should be a reasonable assumption, as both $RNAI$ and $RNAII$ have estimated half-lives of $\\approx 2$ minutes \\cite{Brendel1993}, while the plasmid's doubling time is roughly one \\emph{E. coli} cell generation time, or at least $\\approx 20$ minutes. Under this assumption, the three-species reduced ColE1 model can be approximated with one of two different single-species models, depending on the relative magnitudes of rate parameters $k_p$ and $k_{rep}$. With $k_{rep}\\gg k_p$, total DNA replication rate becomes constant with sufficiently large copy number; when $k_p\\gg k_{rep}$, replication rate is always constant. With the right parameterization, we have arrived back at a simple zero-order replication mechanism. \n\nSee supplemental notebook ``simple\\_bp\\_model\\_reduction.ipynb'' for derivations. \n\n\\section{Acknowledgments}\n\nThe authors thank John Marken and Andrew Halleran for insightful discussions and thoughtful editing. \n\n\\section{Supporting Information}\n\nFor more information and worked examples using the models described here, see our supplementary IPython notebooks at [GITHUB URL]. All notebooks use the BioSCRAPE modeling package to simulate systems with replicating DNA using both standard Gillespie stochastic simulation algorithm (SSA)\\cite{Gillespie1977} and a lineage-bjjased simulation algorithm incorporating cell growth and division.\n\nSee supplementary notebook ``model\\_comparison\\_and\\_fitting.ipynb'' for:\n\n\\begin{itemize}\n\t\\item Full descriptions of the trivial, dummy-triggered replication, full ColE1, and three-species ColE1 models.\n\t\\item Examples of replicating DNA using each of the models with the BioSCRAPE simulation package, \n\t\\item Replication of Figure \\ref{fig:models_and_traces}.\n\t\\item Comparison of model output to empirically-measured plasmid copy numbers taken from \\cite{Shao2021}, including parameter inference. \n\\end{itemize}\n\nSee supplementary notebook ``simple\\_bp\\_model\\_reduction.ipynb'' for:\n\n\\begin{itemize}\n\t\\item Derivations of the reductions made in section \\ref{sec:reduction}.\n\t\\item Random-parameter simulations evaluating the accuracy of reductions to the three-species reduced ColE1 replication model.\n\t\\item Parameter conditions for stability of the three-species reduced ColE1 model.\n\\end{itemize}\n\nSee supplementary notebook ``CRISPRlator.ipynb'' for:\n\n\\begin{itemize}\n\t\\item Worked examples of a 5-node CRISPRi-based repressilator \\cite{Clamons2017, Javier2020} implemented on a plasmid using each of the trivial, dummy-triggered replication, full ColE1, and three-species reduced ColE1 models.\n\t\\item Simulations showing synchronization of the CRISPRlator across many generations of growing and dividing cells, and dependence of that synchronization on circuit copy number, using the dummy-triggered replication model.\n\\end{itemize}\n\nSee supplementary notebook ``temporal\\_logic\\_gate.ipynb'' for:\n\n\\begin{itemize}\n\t\\item Worked examples of a population-based DNA integrase circuit capable of sensing the order of and time between arrival of two signals \\cite{hsiao2016}.\n\t\\item Demonstration of the same circuit moved to a single cell bearing the circuit on a 30 or 100 copy number plasmid (instead of in many cells bearing a single genomic copy) using dummy-triggered replication.\n\\end{itemize}\n\n\\bibliography{plasmid_replication.bib}\n\n\\end{document}\n", "meta": {"hexsha": "9613436ec90e7b7527f46095f8f7ba77479a305a", "size": 12543, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "working_copy_Clamons_plasmid_replication.tex", "max_stars_repo_name": "sclamons/plasmid_replication_modeling", "max_stars_repo_head_hexsha": "6a88ba6520e6761edb907f61623da38e7dbe3116", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "working_copy_Clamons_plasmid_replication.tex", "max_issues_repo_name": "sclamons/plasmid_replication_modeling", "max_issues_repo_head_hexsha": "6a88ba6520e6761edb907f61623da38e7dbe3116", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "working_copy_Clamons_plasmid_replication.tex", "max_forks_repo_name": "sclamons/plasmid_replication_modeling", "max_forks_repo_head_hexsha": "6a88ba6520e6761edb907f61623da38e7dbe3116", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 76.4817073171, "max_line_length": 875, "alphanum_fraction": 0.7946264849, "num_tokens": 2971, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.34889134820669665}} {"text": "\\documentclass[12pt]{article}\t\t\n%========================================================\n\\usepackage{setspace} \t\n\\usepackage{graphicx}\n\\usepackage[margin=1in]{geometry} \n\\usepackage{hyperref}\n\\usepackage{xcolor}\n\\hypersetup{\n colorlinks,\n linkcolor={red!50!black},\n citecolor={blue!50!black},\n urlcolor={blue!80!black}\n}\t\t\n\\usepackage{natbib} \t\n\\usepackage{float}\t\t\t\n\\usepackage{amsmath}\t\n\\usepackage{amssymb}\t \t\t\t\n\\usepackage{bm}\t\t\t\t\t\t\n\\usepackage{booktabs}\n\\usepackage{dcolumn}\n\\usepackage{pdflscape}\n\\usepackage{afterpage}\n\\usepackage[hang,flushmargin]{footmisc} \n\\setlength\\parindent{0pt}\t\n%========================================================\n\\newcommand{\\X}{\\mathbf{X}}\n\\newcommand{\\x}{\\mathbf{x}}\n\\newcommand{\\z}{\\mathbf{z}}\n\\newcommand{\\Y}{\\mathbf{Y}}\n\\newcommand{\\K}{\\mathbf{K}}\n\\renewcommand{\\k}{\\mathbf{k}}\n\\newcommand{\\bc}{\\mathbf{c}}\n\\renewcommand{\\r}{\\right}\n\\renewcommand{\\l}{\\left}\n\\newcommand{\\bomega}{\\bm{\\omega}}\n\\newcommand{\\bpsi}{\\bm{\\psi}}\n\\newcommand{\\bbeta}{\\bm{\\beta}}\n\\newcommand{\\balpha}{\\bm{\\alpha}}\n\\newcommand{\\bphi}{\\bm{\\phi}}\n\\newcommand{\\btheta}{\\bm{\\theta}}\n\\newcommand{\\dist}{\\buildrel\\rm d\\over\\sim}\n\\newcommand{\\ind}{\\stackrel{\\rm indep.}{\\sim}}\n\\newcommand{\\ud}{\\mathrm{d}}\n\\newcommand{\\iid}{\\stackrel{\\rm i.i.d.}{\\sim}}\n\\newcommand{\\logit}{{\\rm logit}}\n\\newcommand{\\cA}{\\mathcal{A}}\n\\newcommand{\\E}{\\mathbb{E}}\n\\newcommand{\\V}{\\mathbb{V}}\n\\newcommand{\\cJ}{\\mathcal{J}}\n\\newcommand{\\bone}{\\mathbf{1}}\n\\newcommand{\\var}{{\\rm Var}}\n\\newcommand{\\cov}{{\\rm Cov}}\n\\DeclareMathOperator*{\\argmin}{\\arg\\!\\min}\n\\DeclareMathSizes{10}{12}{10}{10}\n%========================================================\n\\begin{document}\n\n\\title{Predicting Foreign Fighter Flows to Syria Using Machine Learning: An Introduction to Kernel Regularized Hurdle Negative Binomial\\thanks{The authors are listed in alphabetical order, and contributed equally. We thank Bryce Dietrich, Chad Hazlett, Jeff Lewis, David Rapoport, Art Stein, and Barbara Walter for their feedback. We also benefited from comments at the UCLA Comparative Politics Reading Group, and the 2016 Midwestern Political Science Association Conference. The usual disclaimer applies. For replication material and our companion software---still in its developmental phase in \\texttt{R}---see \\href{https://github.com/lukesonnet/foreign_fighters}{https://github.com/lukesonnet/foreign\\_fighters}.}}\n\n\\author{George Derpanopoulos\\thanks{PhD Student, Dept. Political Science, UCLA. Email: \\href{mailto:gderpa@ucla.edu}{\\tt gderpa@ucla.edu}} \\and Luke Sonnet\\thanks{PhD Student, Dept. Political Science, UCLA. Email: \\href{mailto:luke.sonnet@gmail.com}{\\tt luke.sonnet@gmail.com}}}\n\n\t\\maketitle\n%=========================================================\n\t\\thispagestyle{empty} \n\t\\singlespacing\n\n\\begin{abstract}\t\nWhy have some countries counted hundreds of their citizens fleeing to fight in Syria, while other countries' citizens have remained bystanders? There are three methodological challenges to answering this question. First, there may be two groups of countries: one at no risk of ``supplying'' foreign fighters and another supplying some positive amount. Second, there is no clear theory to specify the functional forms linking features to foreign fighter supply. Third, existing models perform poorly out of sample or yield output that is not amenable to social-scientific interpretations. To solve these challenges, we augment a hurdle negative binomial model with two machine learning tools. Namely, we allow our features to affect the response non-parametrically by using kernel functions that represent expansions of the data. Furthermore, we add regularization terms that penalize complexity to mitigate overfitting. Our approach combines the strengths of predictive and confirmatory models: it performs similarly to state-of-the-art machine learning algorithms in prediction while providing substantively interpretable output. Applying our model to data on 163 countries, we find that populous, developed countries, with a large Sunni population and proximity to Syria supply more fighters. These results lend themselves to viewing foreign fighter supply as largely driven by structural forces.\n\\end{abstract}\n\n%=========================================================\n\t\\newpage\n\t\\setcounter{page}{1} \t\t\t\n\t\\onehalfspacing\n\n\\section{Introduction}\n\nThe ongoing Syrian Civil War has been named ``the world's largest humanitarian crisis since WWII'' \\citep[p. 1]{ECHO2015}. 230,000 dead, 850,000 injured, 4 million refugees, and 7.5 million internally displaced; these are some conservative estimates of the conflict's cost.\\footnote{All figures are the most recent estimates that could be found as of June 2015 \\citep{UNOCHA2015}.} An integral component of the conflict are the multiple rebel groups involved, like Islamic State, Jabhat al-Nusra, and the Free Syrian Army. Although rebel manpower is hard to estimate, there is consensus that a significant portion of it comes from foreign fighters \\citep{Byman2015}, ``non-state actors involved in military activity in a foreign country'' \\citep[p. 1]{Hegghammer2013}. Moreover, with an estimated 21,000 foreigners from 50 countries fighting in Syria \\citep{Neumann2015}, foreign fighter supply has reached a historical high \\citep{Hegghammer2013a}. \\\\\n\nNaturally, policy-makers are asking themselves what draws their voters to participate in a foreign conflict. Common concerns were best expressed by Britain's Prime Minister, David Cameron: ``one of the most disturbing aspects is how this conflict is sucking in our own young people, from modern, prosperous societies''. So far, public discourse has addressed these concerns by focusing on individual fighters' motivations. This has been complemented by an emerging case-study literature based on returning fighter interviews \\citep{Stenersen2011, Weggemans2014, Nilsson2015}. As a result, a popular conception of foreign fighters has emerged; that of young Muslims from poor urban environments, with ``deep-seated feelings of marginalization and exclusion'' \\citep[p. 2]{Noor2014}. \\\\\n\nWhether the inferences of this literature are accurate and generalizable is an open question. Before settling it, though, policy-makers might want to know what the role of \\textit{country-level} features is in individual fighters' calculus. In particular, we might ask to what extent can foreign fighter supply be attributed to \\textit{policy}, which is under the control of government, versus countries' \\textit{structural features}, which are sticky. To date, there is no study addressing these questions. However, the ongoing nature of the conflict, its spillovers into the region, and the continuing flow of fighters throughout the globe call for a systematic analysis of the evidence. We take a first step in that direction, thereby responding to the recommendations of policy reports, that ``strategies would benefit immensely from more evidence-based research'' \\citep[p. 17]{GCCS2014}. \\\\\n\nAn analysis of the predictors of foreign fighter supply is hindered by two classes of problems: data availability and modeling challenges. In this study, we focus on overcoming the difficulties in modeling this process. Namely, our approach addresses four sets of issues, relating to our priors about the country-level mechanism generating foreign fighter supply, the nascent state of the literature, and our dual interest in prediction as well as inference and interpretability. \\\\ \n\nFirst, our theory-motivated prior that foreign fighter supply is a two-component process---some function of countries' features should predict \\textit{whether} they supply any foreign fighters and, if so, another function should predict \\textit{how many} fighters they supply. This is motivated by qualitative evidence that supplying at least one fighter alters supply dynamics within a country; radicalization and recruitment networks form, along with policies to contain them, and a different mechanism takes over the ``scaling-up\" of supply. Such a theoretical structure suggests that a two-component mixture model is appropriate, with one component predicting assignment to supplier countries (binary response), and a second component predicting the number of fighters supplied (count response left-truncated at $1$). This directs us to the familiar hurdle model \\citep{Mullahy1986}. In addition, the count nature of the response and its large variation urge us to further refine our specification to the hurdle negative binomial model. \\\\\n\nAs with any mixture model, we can allow different features, or different functions of the same features, to enter each component. However, this brings us to a second modeling challenge: there is little theory on how a country's features combine to affect foreign fighter supply, let alone how this should vary between the model's components. The assumption behind most regression models is that the systematic component of a unit's response is a linear function of its features \\citep{King1989a}. When models are specified using vague theoretical priors, though, parametric assumptions are hard to justify \\citep{Ho2007}. Instead, model-fitting could greatly benefit from a flexible semi-parametric approach. Semi-parametric and non-parametric models are not uncommon, yet existing models do not allow us to impose the theoretically-motivated structure of the hurdle model. \\\\\n\nAnother downside of many non-/semi-parametric models is that they overfit the data, thereby yielding poor out-of-sample predictions -- a third challenge to address. This can be addressed by machine learning algorithms that penalize complexity, like LASSO and Random Forest. Regularizations based methods like the LASSO or Ridge regression use $k$-fold cross-validation to tune penalty, or regularization parameters in order to strike a better balance on the bias-variance tradeoff than classical estimators. Unfortunately, though, semi-parametric regularized models give way to a fourth challenge: they do not produce quantities of interest familiar to social scientists, like marginal effects and confidence intervals. In fact, demanding such quantities of interest requires sacrificing flexible model-fitting---for example, output from ridge regression is interpreted like that of GLMs, but the model assumes that the features affect the response linearly. \\\\\n\nIn short, we seek an algorithm that combines attractive features from multiple approaches: the intuitive structure of the hurdle model, the agnosticism of non-parametric models on how the features combine to affect the response, the good out-of-sample performance of regularized algorithms, and the interpretability of standard regression models. We bridge the generalized linear model and machine learning literatures to arrive at such an algorithm, the Kernel Regularized Hurdle Negative Binomial (KRHNB). Specifically, we derive our estimator by applying kernel expansion and regularization to a hurdle negative binomial, then develop companion software to compute parameter estimates that minimize prediction error, produce pointwise marginal effects, and get bootstrapped estimates of uncertainty. \\\\\n\nOur procedure consists of several simple steps. We begin by assuming the data is generated by a two-component mixture model: a logit predicting whether a positive count is observed and a negative binomial (left-truncated at $1$) predicting the conditional count. Then we form the sample log-likelihood function and add a regularization term (using an $L_2$ norm), thereby arriving at our target function---the penalized log-likelihood. To move away from a fully parametric form, we expand our feature matrix into a higher-dimensional space; in our application we use an infinite-dimensional space corresponding to all possible expansions of the data (e.g. polynomial, logarithmic, exponential, multiplicative). This makes our target function linear in the \\textit{mapping} of the features, instead of the features themselves. We then show that the features enter the minimum of the target function solely through inner products and thus can be substituted by positive semi-definite kernels, like the Gaussian kernel. This is known as Mercer's Theorem and enables numerical optimization of the target function. \\\\\n\nPredicted responses are derived by substituting our estimates into the hurdle negative binomial's conditional expectation function (CEF). Pointwise marginal effects for feature $j$ are computed using numerical derivation of the CEF with respect to feature $j$. Averaging these effects for each feature gives us its average pointwise marginal effect. Finally, redrawing $1,000$ samples from our data with replacement and repeating the above computations gives us non-parametric bootstrapped estimates of uncertainty---a distribution of average pointwise marginal effects for each feature. All relevant computations are performed with our companion software, written in the \\texttt{R} language.\t\\\\\n\nKRHNB has two attractive properties over the standard hurdle negative binomial. First, regularization penalizes complexity, thereby striking a better balance on the bias-variance tradeoff. Thus, our procedure is less prone to overfitting the training data and better predicts test data. Second, feature expansion implicitly allows our features to affect the systematic component of the response through any functional form. This is particularly valuable in this context; the absence of theories of country-level foreign fighter supply should deter us from making strong parametric assumptions. Nevertheless, one might ask: does our method improve over other machine learning models? After all, there are other algorithms that penalize complexity, some of which impose less structure on the data. \\\\\n\nTo answer this question, we compare our method to three popular machine learning algorithms and find that it performs better or comparably to all of them. Using a cross-section of 163 countries, and an array of 27 demographic, economic, geographic, and political features, we compute the leave-one-out cross-validation root mean square error and the mean absolute error. Higher mean absolute error rates are produced by all 3 of our benchmark models: Kernel Regularized Least Squares (KRLS) \\citep{Hainmueller2013}, KRLS truncated at 0, and a Random Forest. Only Random Forest outperforms KRHNB with respect to root mean squared error and this is largely due to its relative success in predicting a few large outliers. \\\\\n\nHaving established the merits of our approach -- theoretical motivation, flexibility, predictive power, and interpretability -- we apply it to our data, in an effort to contribute to the literature on foreign fighters. Substantively, we find that structural features dominate in predicting foreign fighter supply: populous, developed countries, that are close to Syria, and have a high concentration of Sunnis supply more fighters. Some features that respond to government policy more easily also matter: internet usage, refugee intake, government regulation of religion, and government favoritism of a particular religion positively predict foreign fighter supply. Our method also lends itself to the discovery of interesting heterogeneity in the predictive effect of our features. We find that the positive relationships of refugee intake and government regulation of religion with foreign fighter supply stem from European and Muslim-majority countries, respectively. Throughout this paper, we refer to effects in the sense of partial derivatives; we do not make causal claims.\t\\\\\n\nTo the limited extent that our research design allows us to inform policy, the implications of our findings for constraining foreign fighter flows are grim: little can be done and what can be done is costly. This is because the strongest and most accurate predictors of foreign fighter supply are structural features, while the cost of altering policies that predict supply might exceed the benefit of curbing it; in European countries, reducing refugee intake would arguably alienate the median voter, while in Muslim-majority countries, liberal government policies towards religion might be resisted. Furthermore, the relationships uncovered by this analysis say little about the general equilibrium effects of a large shift in government policy. \\\\\n\nThe remainder of this study proceeds as follows. Section \\ref{sec:background} provides background information on the Syrian Civil War and foreign fighters. Section \\ref{sec:theory} reviews the emerging literature on foreign fighter supply and synthesizes a theoretical framework to analyze the data. Section \\ref{sec:hurdles} outlines some obstacles to empirical modeling in this setting, and motivates our method. Section \\ref{sec:krhnb} presents the theory and mechanics of KRHNB. Section \\ref{sec:computation} covers issues related to modeling choices and computation. Section \\ref{sec:comparison} compares our predictions to those of other models. Section \\ref{sec:results} displays the substantive findings of our analysis and provides possible substantive interpretations. Finally, Section \\ref{sec:conclusion} summarizes, underlines the limitations of our approach, and suggests directions for future research. \n\n%=========================================================\n\\section{Foreign Fighters in Syria}\t\\label{sec:background}\n\nForeign fighters are not a novel fighting technology. They have existed since at least the Greek War of Independence in the 1820s, have participated in different types of conflicts (ethnic, political, religious), and in several regions of the world (Europe, Asia, Latin America, Africa) \\citep{Malet2010}. However, the scale of foreign fighter presence in Syria is unprecedented. Conservative estimates place it around $21,000$, far exceeding supply to conflicts in the past 40 years, like those of Afghanistan, Bosnia, or Somalia \\citep{Hegghammer2011}. Already, confirmed cases of foreign fighters have been noted in $50$ countries \\citep{Neumann2015} -- their supply is mapped in Figure \\ref{fig:ff_map}. \\\\\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width = \\linewidth]{tabs_figs/worldMap_edited.pdf}\n\t\\caption{Global Distribution of Foreign Fighter Supply, Through 2014 \\citep{Neumann2015}}\t\\label{fig:ff_map} \n\\end{figure}\n\nThe scale of the foreign fighter phenomenon has raised a number of concerns in both Syria and supplier countries. Regarding the conflict itself, there is concern that foreign fighters may swing outcomes. Indeed, there is anecdotal evidence that insurgencies with foreign fighters are more successful \\citep{Hegghammer2011, Malet2010}. Moreover, the Syrian case is complicated by the presence of multiple organizations with competing ideologies. On the one hand, groups like the Free Syrian Army (FSA) promise a secular, democratic and pro-Western government upon toppling Assad. On the contrary, groups like Islamic State (IS) strive to replace the Ba'athist regime with an Islamic caliphate. To the extent that both of these opposing camps are continuously strengthened by foreign fighters, it is unlikely that either will prevail. Hence, even if insurgents succeed in defeating the state, factional conflict will make a ceasefire unlikely.\\footnote{The dynamics are complicated further if we consider Hezbollah and other Shia or pro-government militias' support of Assad.} \t\\\\\n\nJust as many concerns have been raised in the countries where foreign fighters originate. Despite some governments taking an active stance on the conflict, they do not condone their citizens' participation, even if it is in support of an organization allied with their government (e.g. FSA).\\footnote{In order to deter their citizens from joining the conflict, some governments have passed legislation to revoke returning fighters' citizenship (e.g. Australia, Britain, Canada, USA). See \\href{http://news.nationalpost.com/news/canada/canadian-government-revoking-passports-of-citizens-trying-to-join-extremist-groups}{http://news.nationalpost.com/news/canada/canadian-government-revoking-passports-of-citizens-trying-to-join-extremist-groups}.} More problematic is the case of foreign jihadis, since a coalition of numerous countries has declared war on groups like IS, yet some jihadis are citizens of these countries. A prominent example is ``Jihadi John'', a British citizen---and IS fighter---that was bombed by air strikes funded through his taxes. Cases like this have troubled Western governments, whose citizens comprise an estimated $20\\%$ of foreign fighters in the conflict.\\footnote{This estimate is based on the data we employ \\citep{Neumann2015}. In this case, the West is defined as non-Muslim-majority countries.} \\\\\n\nIn addition to shaking the liberal-democratic foundations of these countries, the greatest concern relates to the ``veteran effect'': the risk of returning fighters carrying-out terror attacks in their home country \\citep{Hegghammer2013}. Veterans' training, experience in combat and access to networks of fighters makes them prime candidates for terrorist recruiting, with \\cite{Hegghammer2011} claiming that ``most transnational jihadi groups today are by-products of foreign fighter mobilizations'' (p. 53). Naturally, policy-makers are worried that their citizens can participate in foreign conflicts and then import violence to their home country.\t\\\\\n\nA similar pattern holds in the literature. Few studies ask what causes one to become a foreign fighter, while virtually none focuses on country-level features. Crucially, ignoring these features impedes our ability to prevent foreign fighter flows in future conflicts. Admittedly, the Syrian conflict is well underway, and thus the most efficient way to minimize its costs is to contain its spillovers. In future cases, though, given that prevention is cheaper than treatment, a more efficient strategy is to identify potential supplier countries. This will also benefit countries on the receiving end; negotiation is always preferable to conflict, but the potential inflow of fighters makes conflict more likely. \\\\\n\nThis study takes the first step towards predicting country-level foreign fighter flows. Before proceeding, though, in the next section we briefly review the emerging literature on foreign fighters and synthesize it into a broad theoretical framework. We do not put forth an argument for how foreign fighter flows take shape at the country level, let alone the individual level. We simply impose the minimum theoretical structure necessary to later enable a substantive interpretation of our findings. \n\n%=========================================================\n\\section{Predictors of Foreign Fighter Supply}\t\\label{sec:theory}\n\nThe literature on foreign fighters is still in its nascent stage, yet combines insights from a range of disciplines: political science, history, sociology, public policy, security, and counter-terrorism studies. A comprehensive review of this work is beyond our scope, and instead we focus on studies that touch on the determinants of foreign fighter supply. Unfortunately, we are aware of only one study that explores \\textit{country}-level predictors \\citep{Hewitt2009}; remaining studies and policy reports tackle the question at the individual level.\t\\\\ \n\nCrucially, although we extract the covariates that feature in this literature to synthesize a framework for country-level explanations of foreign fighter supply, we do not claim to \\textit{test} the theories posited in the literature. To make that claim would be to commit an ecological fallacy, since existing theories are about individuals, while our data covers countries. We merely borrow from individual-level theories to motivate our search for predictors at the country level. Similarly, we do not present every possible channel through which country features might operate on foreign fighter supply, as our research design does not have the power to adjudicate between competing mechanisms.\\footnote{Many of the potential mechanisms we review come from the transnational terrorism literature. For a discussion of the distinction between foreign fighting and other forms of violence, see \\cite{Hegghammer2011}.}\t\t\\\\\n\nWe divide country-level explanations of foreign fighter supply into two broad camps: structural vs. policy-related. The former involve features of a country that are ``sticky'', and respond very slowly to policy or shocks (e.g. geography, demography, development level).\\footnote{\\cite{Jackson2012} define a ``sticky'' variable as ``[a] highly autoregressive process that is slow to adapt to changes\" (p. 163). They add that ``[in] many policy areas, the influence of interest groups, the security of incumbents, the number of veto players, and bureaucratic inertia are likely to produce high values of $\\rho$ [the autocorrelation index] in any model of the policy processes'' (p. 163).} Policy-related explanations, on the other hand, point to country features that are either the direct output of government decisions (e.g. discrimination laws), or an outcome that is significantly influenced by government (e.g. respect for human rights). Although, policy and structure are interrelated---structure constrains policy, while policy can alter structure in the long-run---we adhere to this simplistic yet powerful classification for the sake of conceptual clarity.\t\\\\\n\nNumerous structural variables have figured in popular conceptions of foreign fighter-prone countries. Proximity to the conflict zone, because it decreases the travel cost incurred by fighters, along with spillovers of the conflict.\\footnote{Distance has also been used to explain the unprecedented \\textit{aggregate} flow of fighters to Syria, due to the conflict's proximity to Europe \\citep{Hegghammer2011}.} Population, since a positive (unconditional) probability of any individual becoming a foreign fighter implies that larger countries should have a higher supply. Urbanicity and population density, via the notion that radicalization networks operate more easily in urban centers and densely inhabited areas \\citep{Gibbs1989}. A right-skewed age distribution and large male population share, on account of rebel groups' preference for young male recruits.\\footnote{Nevertheless, it is worth noting that the Syrian foreign fighter movement is the first with such a high participation of females (see \\href{http://www.theguardian.com/world/2014/sep/29/schoolgirl-jihadis-female-islamists-leaving-home-join-isis-iraq-syria}{http://www.theguardian.com/world/2014/sep/29/schoolgirl-jihadis-female-islamists-leaving-home-join-isis-iraq-syria}).} The share of the country's population that identifies with a particular party in the conflict, by virtue of common nationality, ethnicity, or religion. This operates both on the supply side, with members of the diaspora having a steak in the conflict on account of their shared identity with some rebel group, and on the demand side, with recruiters manipulating the salience of identities to construct a sense of moral obligation to fight \\citep{Malet2010}.\\footnote{\\cite{Malet2015} qualifies this by arguing that shared ethnicity is not as strong of a motivating factor as shared religion.} Development, via the findings of \\cite{Hewitt2009} that more developed countries supplied more foreign fighters to Iraq.\\footnote{The authors do not offer an interpretation of their findings. However, they do review the more conventional view expressed by \\cite{Lewis2004} and \\cite{Gerges2009}; that Islamic radicalism is owed to a lack of modernization in Muslim-majority countries. Clearly, this explanation is problematic when applied to the case of Syria, as it cannot account for the large supply share held by the developed world.} Unemployment, particularly among the young and males, because they decrease the opportunity cost of participating in conflict vis-\\'a-vis recruiters' preference for that target group \\citep{RAN2014}. Homicide rates, given the notion that individuals raised in violent environments might export that violence to different theaters, especially if different forms of violence are substitutable.\t\\\\\n\nAn equally large number of policy variables has been associated with foreign fighter supply. Poor human and civil rights, repression, and censorship, due to their association with authoritarianism. Scholars have argued that aspects of liberal democracy alleviate grievances that lead to terrorism \\citep{Crenshaw1981}, but also that they decrease the cost of organizing terrorist acts and the punishment for carrying them out \\citep{Schmid1992}.\\footnote{For a review of a similar debate on the role of regime-type in terrorism, see \\cite{Chenoweth2013}.} Discriminative policies against minorities that share an identity with some party in the conflict, as they create grievances that have been argued to fuel terrorism \\citep{Piazza2011, Sageman2008}. The number of migrants and refugees, via two channels; first, their own radicalization, if they originate from countries that have a stake in the conflict (e.g. Muslim refugees and migrants in France), and second, their effect on citizens of the host country that are prone to radicalization (e.g. French citizens of Moroccan origin). Lastly, internet penetration, as most foreign fighter recruitment occurs online \\citep{Hegghammer2011}.\t\n\n%=========================================================\n\\section{Hurdles to Predicting Foreign Fighter Supply}\t\\label{sec:hurdles}\n\nAny empirical investigation of the correlates of foreign fighter supply encounters two central difficulties. The first relates to data availability, as foreign fighter supply is poorly measured. Naturally, foreign fighters do not report their participation in conflict, especially in supplier countries where returning fighters face punishment.\\footnote{Increasingly, fighters are using social media to publicize their activities. This has allowed more fine-grained measures of foreign fighter supply to develop, but not to the extent where we can use them in our analysis.} As such, any data on foreign fighter supply remains an estimate and is subject to non-random missingness and measurement error. Nevertheless, it is our conviction that the (nascent) literature on this question can gain enough from analyzing this data to warrant a full investigation. Thus, we will treat these estimates as our response variable (Section \\ref{sec:results}) and delegate improved data-collection to future work. \\\\\n\nWe instead focus on the second difficulty, which relates to prediction, inference, and interpretability. In particular, our approach is motivated by a desire to address four issues. First, we believe that foreign fighter supply at the country level can be theorized as a two-component process. \\textit{Whether} a country supplies any foreign fighters can be thought of as one component of the process; \\textit{how many} fighters it supplies can be thought of as another component. In other words, using an analogy of countries as firms and foreign fighter supply as a firm choice, one mechanism should determine which countries become suppliers of fighters and another mechanism should determine how much they scale-up their supply by. This is justified if different supply dynamics take over once a country ``decides'' to become a supplier.\\footnote{The use of ``once\" here does not imply a temporal dimension, since our model does not take account of time---in the statistical setup, the two decisions (whether to supply, how much to supply) are simultaneous.} Indeed, anecdotal evidence suggests that after radicalization and recruitment networks form, along with the policies to contain them, the mechanism underlying foreign fighter supply changes; incentives and costs for prospective fighters are fundamentally altered by the existence of previous fighters \\citep{Felter2007, Klausen2015}.\t\\\\\n\nAs such, we require a model that allows us to make a conceptual distinction between the two components of foreign fighter supply. The model should allow a country's features to differentially affect whether it becomes a supplier or not and how many fighters it supplies. More precisely, we allow for the possibility that there are different functional forms that govern the relationship between the features and the two components of the model. For example, security policies might have a small effect on whether aggrieved citizens decide to form a foreign fighter movement, but they may have a large effect on whether additional citizens join that movement---it may deter them from undertaking the potential legal costs of becoming a fighter. It is also possible that some features influence whether any foreign fighters are supplied, while having no effect on how many are supplied.\t\\\\\n\nThe above points us to the familiar hurdle model of \\cite{Mullahy1986}, a two-part mixture model combining a binary component with a truncated count component.\\footnote{The hurdle model was introduced to political science by \\cite{King1989} and \\cite{King1989a}. However, to the best of the authors' knowledge, its only application to a political science question is \\cite{Marschall2010}, which fits a Poisson process to the count component. For a comparison and application of the hurdle versus its more popular counterpart---the zero-inflated model---directed at a political science audience, see \\cite{Zorn1998}. Virtually all political science studies that employ the zero-inflated negative binomial do so to model terrorism (e.g. \\cite{Li2005}, \\cite{Burgoon2006}, and \\cite{Wilson2013}), or politically violent events more generally (e.g. \\cite{Bagozzi2015}). Note that zero-inflation is also used in other processes of discrete random variables, generating models like the zero-inflated ordered probit \\citep{Bagozzi2015a}, the middle-inflated ordered probit \\citep{Bagozzi2012}, and the baseline-inflated multinomial logit \\citep{Bagozzi2015b, Bagozzi2015c}.} Applied to the question of foreign fighter supply, the former predicts which countries are suppliers and the latter the number of fighters supplied by each country. Crucially, different features can enter each component and even if the same features are entered different marginal effects are returned. Thus, the hurdle model can handle both conceptual distinctions drawn above. Moreover, it closely follows our prior about the two-component structure of foreign fighter supply at the country level.\t\\\\\n\nTo adapt the hurdle model to the question of foreign fighter supply, appropriate statistical processes must be chosen for each component. For the binary component, the choice does not matter greatly, with the logit being the standard.\\footnote{For a theoretical overview of count models, see \\cite{Cameron2013}. For an applied overview in the context of the \\texttt{R} language, see \\cite{Zeileis2007}.} As for the count component, the choice of process should depend on the distribution of foreign fighter supply across supplier countries. A common feature of count data is over-dispersion---the variance of the distribution exceeding its mean.\\footnote{See \\cite{Cox1983} for an early discussion of problems created by over-dispersion in count models.} Indeed, as Figure \\ref{fig:ff_map} shows, this is the case with foreign fighter data---some countries supply only a handful of fighters (e.g. New Zealand), while others supply thousands (e.g. Tunisia). Since this pattern cannot arise under the canonical Poisson process, estimates from a Poisson hurdle will suffer from high variance. A popular fix to this problem is to instead fit a negative binomial distribution to the count component. This restores the good statistical properties that the Poisson process holds under no over-dispersion \\citep{Lawless1987}. \t\\\\\n\nAn altogether different concern with our empirical setting is the absence of a theory to guide functional form selection. Still at its nascent stage, the literature is very far from specifying the correct function through which country features affect foreign fighter supply. That is, even if there was consensus that the count of foreign fighters from each country is a function of, say, its poverty and unemployment rates, there would be no consensus on whether that function is linear, exponential, logarithmic, or of any other form. This is made worse by the two-component nature of our model, as the danger of misspecification is doubled. The assumption behind most regression models, not just the linear one, is that the systematic component of the response is a linear function of its features \\cite{King1989a}. However, when models are specified using vague theoretical priors, parametric assumptions may be unwarranted \\citep{Ho2007}. \t\\\\\n\nInstead, model-fitting could greatly benefit from a flexible nonparametric approach. Nonparametric models are not new to political science, but their application has been relatively limited.\\footnote{We were unable to find a single review of nonparametric methods in a political science journal or textbook. This stands in contrast to related disciplines, like economics, where nonparametric estimators are more widely employed.} We argue that our discipline has a lot to gain from drawing a closer connection between the theories posited and the functions fit to the data. It is rarely the case that our theories are developed enough to accurately test them using assumptions as strong as those of GLMs. Therefore, to conduct fairer investigations of the validity of our hypotheses, we must relax the narrow confines of parametric models. \t\t\\\\\n\nOne form-free way to include a set of features in the systematic component of the response is to map them into a high-dimensional space, such as a high order polynomial expansion of the data or even more complicated expansions. Yet, such an expansion of the features into a regression model often will make computation unfeasible. Therefore, we need a way to \\textit{implicitly} consider higher-order expansions of the features, without actually computing them. This is possible if the expanded feature matrix enters our likelihood function only as an inner product; the inner product can be substituted with an appropriate kernel matrix of the features (see next section), via Mercer's Theorem. As such, kernels allow us to consider a high-dimensional function space, thereby fitting the CEF of the hurdle negative binomial much more flexibly than its classical counterpart. In our case, we use the Gaussian kernel which corresponds to an infinite-dimensional mapping of the data, allowing for very flexible functions. \t\\\\\n\nA third issue that our approach seeks to address is overfitting---the danger of producing predictions that generalize poorly to other samples. This danger is particularly grave in settings where the data-generating process is under-theorized, as is the case with foreign fighter supply. The researcher can embark on a quest to find the best-fitting model, trying numerous different specifications in order to minimize an appropriate metric, such as mean squared error. That model, in turn, may fit the available data very well, but may not fit other samples well. Overfitting becomes even more worrisome in under-theorized settings when coupled with nonparametric methods. Algorithms such as local polynomial regression and kernel regression fit complex surfaces through the training data, but make large prediction errors when applied to test data \\citep{Mroz1999}.\\footnote{Kernel regression does not necessarily involve regularization, and thus should not be confused with kernel \\textit{regularized} regression models like KRLS and KRHNB.} In other words, their emphasis is on minimizing bias in the given sample, as opposed to generating predictions generalizable to withheld samples.\t\t\\\\\n\nTo strike a better balance on the bias-variance tradeoff, we employ regularization and cross-validation. These are standard tools in machine learning methods that penalize complexity to improve out-of-sample prediction. Regularized algorithms based on the $L_2$ penalty, such as ridge regression \\citep{Hoerl1970}, shrink the coefficients of features that do not significantly improve prediction, while others based on the $L_1$ penalty, like Lasso \\citep{Tibshirani1996}, perform well in high-dimensional settings where feature selection is desired. Cross-validation, in turn, enables the optimal tuning of the regularization parameter, especially when there are few data points in the training set. Regularization and cross-validation have spurred the development of numerous algorithms. However, until recently these tools had not been used alongside kernel expansion.\\footnote{\\cite{Zhu2005} introduce kernel logistic regression, \\cite{Shim2011} and \\cite{Shim2012} apply kernel regularization to Poisson regression, and \\cite{Hainmueller2013} present kernel regularized least squares.} This might not seem important, given that nonparametric model fitting and regularization can both be accommodated without using kernels (e.g. Random Forest). \t\\\\\n\nAvoiding kernels, though, gives rise to our fourth issue: interpretability and inference. Many machine learning methods produce output that social scientists are not accustomed to analyzing.\\footnote{For a relatively comprehensive review of interpretability issues in machine learning methods, see pp. 9-12 of \\cite{Hainmueller2013}.} For example, familiar quantities of interest, such as marginal effects and confidence intervals, are absent from algorithms like Random Forest. Similarly, other regularized algorithms, like ridge regression, do produce familiar output, but make strong parametric assumptions---namely, that the response is a linear function of the features, as in OLS.\\footnote{\\cite{Hainmueller2013} note that applying kernel expansion (with a Gaussian kernel) and regularization (with an $L_2$ norm) to a least squares problem produces an \\textit{infinite-dimensional} ridge regression model. This should be contrasted to standard ridge regression (without kernel expansion), which solves a $P$-dimensional linear problem and, hence, produces a parametric fit.} In short, kernel expansion vis-\\'a-vis regularization and cross-validation aggregates the benefits of all aforementioned models: fitting a flexible solution surface, penalizing complexity, and communicating output to social scientists. \t\t\\\\\n\nBefore presenting the mechanics of our approach, we note that machine learning methods are not new to political science. Although already in \\cite{Beck1998} and \\cite{Beck2000} efforts were made to import some of these ideas to the discipline, limitations to computing power contained their expansion. Recently, though, this containment has ceased: \\cite{Kenkel2013} introduce \\texttt{polywog}, a model that fuses basis regression with regularization, cross-validation, and bootstrapping; \\cite{Hill2014} and \\cite{Muchlinski2016} use Random Forests to model state repression and civil war onset, respectively; \\cite{Wilson2015} use KRLS as a robustness check in modeling expropriation risk in autocracies; \\cite{Green2012} and \\cite{Montgomery2015} apply Bayesian Additive Regression Trees to survey experiment data and election fraud measures, respectively; \\cite{Montgomery2015a}, \\cite{Fariss2015}, and \\cite{Jones2015} discuss possible contributions of machine learning to political science.\\footnote{These studies by no means constitute the universe of political science papers employing machine learning tools.} In the following section, we develop a method that we hope will contribute to the growing use of machine learning in political science.\\footnote{For alternative ways of motivating and deriving kernel regularized models, we refer the reader to the simple and intuitive expositions in \\cite{Hainmueller2013} and their supplementary appendix. In what follows, we only take one possible approach to the derivation, in order to expose the mechanics of our method.}\n\n%=========================================================\n\\section{The Model}\t\t\\label{sec:krhnb}\n\nIn this section we construct the likelihood for the hurdle negative binomial model, reparameterize our model in an infinite-dimensional space, demonstrate that our features only enter our penalized likelihood via inner products, and then use Mercer's theorem to rewrite the problem using Gaussian kernels rendering optimization feasible.\t\t\\\\\n\nAssume $y_i$ is a count ($y_i \\in \\{0,1,\\dots,\\infty\\}$). We begin with the general formulation of the two-component density of the hurdle model \\citep{Mullahy1986}, which combines a zero-hurdle component, right-censored at $y_i=1$, with a positive count component, left-truncated at $y_i$=1:\\footnote{For the original formulation of the hurdle model---albeit not for count variables---see \\cite{Cragg1971}. Throughout, positive counts refer to \\textit{strictly} positive counts.}\n\n\\begin{align}\n p(y_i) = \\begin{cases}\n p_0(y_i = 0) \\quad &\\text{if } y_i = 0 \\\\\n \\frac{p_1(y_i)}{1 - p_1(y_i = 0)} (1 - p_0(y_i = 0)) \\quad &\\text{if } y_i \\geq 1\n \\end{cases}\n\\end{align} \n\nAny binomial model (e.g. probit) or right-censored count model (e.g. Poisson) can be chosen for the zero hurdle component. We follow the literature in opting for the computationally simple logit. Similarly, any count model can be chosen for the positive count component. We opt for the negative binomial, because it allows us to account for over-dispersion---as we will see, a characteristic of our response variable. Before providing the likelihood for $y_i$, we note the following useful densities:\n%\n\\begin{align}\n p_0(y_i = 0) &= \\frac{1}{1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha)} \\label{eq:dens0} \\\\\n p_1(y_i) &= \\frac{\\Gamma (\\zeta + y_i) \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\l( \\frac{\\exp(\\beta_0 + \\x^\\top_i \\bbeta)}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^{y_i}}{\\Gamma(1 + y_i) \\Gamma(\\zeta)} \\label{eq:densmu} \\\\\n 1 - p_1(y_i = 0) &= 1 - \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\label{eq:denstrunc} \n\\end{align} \n\nThe first density is the likelihood of observing a zero outcome, where $\\x_i$ is a length-$P$ vector of features for observation $i$, $\\balpha$ is the parameter vector for the binary component, $\\alpha_0$ is an intercept, and $\\alpha_0 + \\x^\\top_i\\balpha$ is the linear predictor. \\\\\n\nThe second density is that of the standard negative binomial, where $\\bbeta$ is the parameter vector for the count component, $\\beta_0$ is an intercept for the count component, $\\zeta$ is the overdispersion parameter\\footnote{For completeness, we note the following properties of the negative binomial density: $\\E[y_i|\\x_i, \\bbeta] = \\exp(\\beta_0 + \\x^\\top_i\\bbeta + \\epsilon_i) = \\exp(\\beta_0 + \\x^\\top_i\\bbeta) \\exp(\\epsilon_i) = \\mu_i h_i$, where $h_i \\sim \\Gamma(\\zeta,\\zeta)$, \\; $\\E[h_i]= \\zeta/\\zeta = 1$, and $V[h_i]=1/\\zeta$. Thus, after conditioning on $\\zeta$, we obtain $\\E[y_i | \\x_i, \\bbeta, \\zeta] = \\mu_i$, as in the Poisson, and $Var[y_i | \\x_i, \\bbeta, \\zeta] = \\mu_i(1+\\mu_i/\\zeta)$, which tends to the Poisson's variance as $\\zeta \\rightarrow \\infty$.}, and $\\Gamma(\\cdot)$ is the gamma function.\\footnote{Note that $\\Gamma(n)=(n-1)!$, where $n$ is a positive integer.} The hurdle model allows us to specify each component as a function of different country features---say, $\\x_i$ for the binary component and $\\z_i$ for the count. This choice can be motivated from a substantive perspective; if theory dictates that different country features affect the likelihood of a country being a supplier (binary component), versus its likelihood of supplying a certain number of fighters conditional on being a supplier (truncated count component), including different features in each component is appropriate. However, given the nascent state of the literature on foreign fighters, we do not feel justified in making that choice, and assume that the same set of features ($\\x_i$) affects both likelihoods. Furthermore, because we allow for very flexible functional forms in both components and penalize complexity, including irrelevant variables will not cause overfitting or induce misspecification bias.\\footnote{Of course, if a feature truly has no relationship to one component of the model, including it will reduce the efficiency of our estimator. Knowing this a priori is very difficult and a benefit to regularized, flexible methods such as KRHNB is the ability to include many features and allowing the estimator to learn the appropriate features and functional form.}\t\\\\\n\nThe third density is simply the complement of the negative binomial density evaluated at zero. This term plays a crucial role in the hurdle model, since a zero count \\textit{cannot} arise from the count component, which is left-truncated at unity--hence the ``hurdle\". As such, in the formula for the density of a positive count we scale the binomial density by $1 - p_1(y_i = 0)$. This is the key difference between the hurdle and the zero-inflated model; the latter allows for zero counts to arise from \\textit{both} components.\\footnote{A case for using the hurdle is also made by \\cite{Porter2012}, but with respect to terrorism data.} We opt for the hurdle model not just because it is theoretically appropriate, as explained in Section \\ref{sec:hurdles}, but also because it is computationally more straightforward; the likelihood function is perfectly separable with respect to the two components and hence each parameter vector can be fit by independently maximizing the respective component.\\footnote{This is not the case with the zero-inflated model, whereby a mixing of zeros occurs under the two components. The computational advantage of the hurdle over the zero-inflated model becomes even larger when the \\textit{same} set of features is used in both components, as we do in our specification. This is because the mixing of zeros from the two components hinders identification of the two sets of coefficients for the features.} \t\\\\\n\nNow we can form the likelihood for observation $i$:\n\n\\begin{align}\n \\label{eq:likelihood}\n\\begin{split}\n L_i(\\cdot) &= \\l[ p_0(y_i = 0) \\r]^{1 - d_i} \\l[ \\frac{p_1(y_i)}{1 - p_1(y_ i = 0)} (1 - p_0(y_i = 0)) \\r]^{d_i}\n \\\\ d_i &= \\begin{cases}\n 0 \\text{ if } y_i = 0 \\\\\n 1 \\text{ if } y_i \\geq 1\n \\end{cases}\n\\end{split}\n\\end{align} \n\nWe use the densities in Equations~\\ref{eq:dens0}, \\ref{eq:densmu}, and \\ref{eq:denstrunc} to form the joint (sample) likelihood for $N$ observations. Taking the log, we arrive at the sample log-likelihood.\\footnote{See Appendix~\\ref{app:deriv} for the intermediate steps.} Where $\\btheta = (\\alpha_0, \\balpha^\\top, \\beta_0, \\bbeta^\\top, \\zeta)^\\top$ and $\\mathcal{D} = (\\Y, \\X)$,\n\n\\begin{align}\n \\ell_N(\\btheta | \\mathcal{D}) &= \\begin{aligned}[t]\n \\sum^N_{i=1} & - \\log \\l(1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha) \\r) + d_i \\Bigg[ \\log \\Gamma( \\zeta + y_i ) + \\zeta \\log \\zeta \\\\\n &- (\\zeta + y_i) \\log \\l( \\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta) \\r) + y_i(\\beta_0 + \\x^\\top_i \\bbeta) + (\\alpha_0 + \\x^\\top_i \\balpha) \\\\\n & - \\log \\Gamma (1 + y_i) - \\log \\Gamma (\\zeta) - \\log \\l( 1 - \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\r) \\Bigg]\n \\end{aligned} \\label{eq:sampll}\n\\end{align} \n\nNote that $\\alpha_0$ and $\\beta_0$ are intercept terms. At this stage, we reparameterize the log-likelihood, by substituting the linear predictor functions $\\alpha_0 + \\x_i^\\top\\balpha$ and $\\beta_0 + \\x_i^\\top\\bbeta$ with $\\psi_0 + \\bphi(\\x_i)^\\top\\bpsi$ and $\\omega_0 + \\bphi(\\x_i)^\\top\\bomega$, respectively, functions linear in $\\bphi(\\x_i)^\\top$, a mapping of the features. That is, the feature space $\\X \\in \\mathbb{R}^P$ is expanded onto a higher-dimensional space $\\mathbb{R}^{P'}$, where $P< 0)\\}$ and so on and so forth, where $\\x^{(j)}$ is the $j$th feature.\t\\\\\n\nThus, letting $\\bc^m = \\begin{bmatrix} c^m_1 & c^m_2 & \\dots & c^m_N\\end{bmatrix}$ where $N$ is the number of observations and $m$ represents either the first or second components, $\\psi$ or $\\omega$, letting $\\K$ be the kernel matrix of our sample, and letting $\\k_i$ be the $i$th column of $\\K$, we can rewrite our target function as:\n\n\\begin{align}\t\\label{eqn:targetFinal}\n R_N (\\btheta_{\\bc}, \\lambda_\\psi, \\lambda_\\omega | \\Y, \\K) &= \\begin{aligned}[t]\n - \\sum^N_{i=1} & \\Bigg( - \\log \\l(1 + \\exp({c^\\psi_0 + \\bc^\\psi}^\\top \\k_i) \\r) + d_i \\Bigg[ \\log \\Gamma( \\zeta + y_i ) + \\zeta \\log \\zeta \\\\\n & - (\\zeta + y_i) \\log \\l( \\zeta + \\exp({c^\\omega_0 + \\bc^\\omega}^\\top \\k_i) \\r) + y_i (c^\\omega_0 + {\\bc^\\omega}^\\top\\k_i) \\\\\n & + (c^\\psi_0 + {\\bc^\\psi}^\\top \\k_i) - \\log \\Gamma (1 + y_i) - \\log \\Gamma (\\zeta) \\\\\n & - \\log \\l( 1 - \\l( \\frac{\\zeta}{\\zeta + \\exp({\\bc^\\omega}^\\top \\k_i)} \\r)^\\zeta \\r) \\Bigg] \\Bigg) \\\\\n & + \\lambda_\\psi {\\bc^\\psi}^\\top \\K \\bc^\\psi + \\lambda_\\omega {\\bc^\\omega}^\\top \\K \\bc^\\omega\n \\end{aligned}\n\\end{align} \n\nThe resulting optimization problem does not have a closed-form solution. Therefore, we minimize the above with respect to $\\{ c^\\psi_0, \\bc^\\psi, c^\\omega_0, \\bc^\\omega, \\; \\zeta \\}$ through numerical optimization.\t\\\\ \n\nTo summarize, we receive estimates for $\\bc^\\psi$ and $\\bc^\\omega$, which act as a kind of weight for each observation $i$ in the two CEFs (describing the mean of the hurdle and the truncated count components, respectively). For example, our estimate of the probability of sending no foreign fighters, the probability in the logit component, is estimated as $$p_0(y_i = 0) = \\frac{1}{1 + \\exp(\\hat{c^\\psi_0} + {\\hat{\\bc^\\psi}}^\\top \\k_i)}$$\n\nAs a result, directly interpreting the estimated coefficients $\\hat{\\bc^\\psi}$ and $\\hat{\\bc^\\omega}$ can be very difficult for two reasons: they influence the outcome through non-linear transformations like the logistic function and are acting on $\\k_i$ instead of $\\x_i$, our features of interest. Therefore, in Section~\\ref{sec:pwmfx} we take the partial derivatives of the CEF for the outcome $y_i$ with respect to our columns of $\\X$ so that we can interpret our results using our input features.\n\n%=========================================================\n\\section{Computation \\& Quantities of Interest}\t\\label{sec:computation}\n\n%=======================\n\\subsection{Optimization}\t \n\t\nThere are three main difficulties with fitting this model. First, there is minor sensitivity to starting values in the numerical optimization. Using the BFGS algorithm limits this problem dramatically, and the sensitivity only arises when using implausible starting values.\\footnote{Implausible here means uniformly positive or negative starting values. They are implausible because the data have been scaled, thus the coefficients will generally be distributed around $0$.} However, this sensitivity could be guarded against by doing a grid search over some set of starting values. \\\\\n\nThis solution is difficult to implement because of the second problem with fitting this model numerically---speed. While we supply the analytic gradient of our target function to the BFGS algorithm, these functions are quite complicated and high-dimensional. We numerically optimize with starting values for all coefficients $\\bc = \\mathbf{0}$ and $\\zeta = 1$. From some rudimentary grid searches, this starting value has succeeded in finding the best minimum using our data and makes intuitive sense; setting $\\bc = \\mathbf{0}$ means our initial estimate of the CEF is simply the sample average over the entire feature space.\\footnote{In some instances, the algorithm gets stuck in local minima. In particular, it sometimes estimates $\\zeta < 0.0001$, which results in estimates of $y_i$ that are too large by $3$ or $4$ orders of magnitude. However, because the performance is so poor, it fails to find a way to improve it. For now, we forcibly prevent $\\zeta$ from reaching such implausibly small numbers to avoid this shortcoming of the BFGS algorithm.}\t\\\\\n\nThird, selecting the appropriate regularization parameters ($\\lambda_\\psi$, $\\lambda_\\omega$) can be difficult, as there are two parameters. (This also slows-down optimization.) The traditional approach is to use cross-validation (CV) \\citep{Stone1974} to select parameters that minimize the cross-validation RMSE \\citep{Friedman2001}. We advocate and implement in our software $k$-fold cross validation and a grid search over the two regularization parameters. For the application to foreign fighter supply, we select regularization parameters after several manual grid searches.\n\n%=======================\n\\subsection{Pointwise Marginal Effects}\n\\label{sec:pwmfx}\nThe CEF for test observation $i$ is:\n\n\\begin{align}\n \\E[y_i|\\k_i] &= p_0(y_i=0)*0 + (1-p_0(y_i=0))*\\E[\\hat{Y}_i | \\hat{Y}_i > 0] \\notag \\\\\n & = ( 1-p_0(y_i=0) ) \\frac{\\mu_i}{1-p_1(y_i=0)} \\notag \\\\\n \\widehat{\\E[y_i|\\k_i]} &= \\frac{\\exp(\\hat{c^\\psi_0} + \\hat{\\bc^\\psi}^\\top \\k_i)}{1 + \\exp(\\hat{c^\\psi_0} + \\hat{\\bc^\\psi}^\\top \\k_i)} \\frac{\\exp(\\hat{c^\\omega}_0 + \\hat{\\bc^\\omega}^\\top \\k_i)}{\\l( 1 - \\l( \\frac{\\hat{\\zeta}}{\\hat{\\zeta} + \\exp(\\hat{c^\\omega_0} + \\hat{\\bc^\\omega}^\\top \\k_i)} \\r)^{\\hat{\\zeta}} \\r)}\n\\end{align} \n\nwhere $\\mu_i$ is the mean component of the negative binomial distribution, and $\\{c^m_0, \\hat{\\bc^m}\\}, \\; m \\in \\{\\psi,\\omega\\}$ are the intercepts and coefficient vectors along with $\\hat{\\zeta}$ that minimize our target function (Equation~\\ref{eqn:targetFinal}).\t\\\\\n\nUsing this formula, we obtain a quantity of interest analogous to marginal effects in GLMs. We use numerical differentiation to take the partial derivative of $\\widehat{\\E[y_i|\\k_i]}$ with respect to each feature $\\x^{(j)}$ and evaluate it at each observation.\\footnote{Remember that $\\k_i$ is a function of the input features.} Because our CEF is non-linear, these pointwise marginal effects will vary over the feature space and provide rich detail about the shape of the CEF. Indeed, wherever an observation lives in the feature space, we will have an estimate of the slope of the CEF with respect to each feature. \\\\\n\nThis allows for us to summarize average effects and estimate heterogeneous treatment effects. For example, we can display the range of pointwise marginal effects across the training values of $\\x^{(j)}$ for feature $j$ as a histogram (see Section \\ref{sec:results}). Alternatively, one can focus on the mean of this distribution, the sample-average pointwise marginal effect of $\\x^{(j)}$, which would be analogous to the marginal effect produced by a linear model. Yet another alternative is to report the pointwise marginal effect for the ``typical\" training point---an observation with mean/median/modal values of the features. This approach is subject to the usual limitations.\\footnote{See \\cite{Gelman2006} for different approaches to summarizing predictive effects.} Our companion software allows the user to choose the effects reported. \t\\\\ \n\n%=======================\n\\subsection{Estimating Uncertainty of Sample-Average Pointwise Marginal Effects}\t\n\nAs noted in Section \\ref{sec:krhnb}, there is no closed-form solution to our optimization problem (Equation \\ref{eqn:targetFinal}). Consequently, obtaining an analytical estimate of the uncertainty of our predictions is not straightforward and we opt for a computational one.\\footnote{For one analytical approach in the context of a penalized zero-inflated negative binomial, see \\cite{Wang2014}. It is based on the sandwich estimator, whose consistency when applied to non-concave penalized likelihood problems was demonstrated in \\cite{Fan2001}.} Namely, we obtain non-parametric bootstrapped estimates of our pointwise marginal effects \\citep{Efron1979}.\\footnote{\\cite{Kenkel2013} also follow a bootstrapping approach to obtain a measure of uncertainty for their pointwise marginal effects. For an application and extension of the bootstrap to the zero-inflated negative binomial, see \\cite{Garay2011}.} This involves treating our training set as the population and repeatedly resampling from it (with replacement) to calculate sample-average pointwise marginal effects. Crucially, we hold the regularization parameters fixed across the resampled training sets and only re-fit the remaining parameters ($\\bc$ and $\\zeta$). This economizes greatly on computational time, as we do not have to execute the full optimization routine on every bootstrapped sample. However, it will result in smaller estimates of uncertainty as the variability in the regularization parameter is not incorporated \\citep{Tibshirani1996}. The resulting estimates provide a distribution of marginal effects, which can be graphed as a histogram, or summarized numerically to create a bootstrapped percentile interval.\\footnote{Our companion software provides a range of summary statistics for the bootstrapped estimates of these effects.}\n\n%=========================================================\n\\section{Performance Relative to Other Models}\t\\label{sec:comparison}\n\nBeyond getting purchase on the causes of foreign fighter supply, we are also interested in benchmarking the performance of our method against other machine learning techniques. In order to do this, we take a leave-one-out cross-validation approach: we train our method and four competing methods on all but one observation and then predict the foreign fighter supply to the withheld observation, repeating this for every observation. We take these out-of-sample predictions for each observation and their true values and calculate the root mean squared (RMSPE) and mean absolute prediction error (MAPE) of our estimated outcomes. The first method we benchmark against is KRLS. The rationale behind this choice is simple: if we cannot perform similarly or better to the method we are adding complexity to, then there is no reason for this extension beyond the ability to interpret the two components of the hurdle model. The second method is KRLS with predicted values truncated at $0$, which will uniformly improve the performance of KRLS, but will completely impair its interpretability. The third method is a random forest \\citep{Breiman2001} with $500$ trees, and the number of parameters available at each node set to $5$ -- roughly the square root of the total number of features ($27$). The last method is standard OLS, where the predictors enter as a simple linear function. \\\\\n\n\n\\begin{table}[!h] \\centering \n\t\\caption{Comparing Prediction Error} \n\t\\label{tab:compare} \n\t\\begin{tabular}{ll rr}\n\t\t\\\\ [-1.9ex]\n\t\t\\toprule\n\t\tData & Method & \\multicolumn{1}{c}{CV RMSPE} & \\multicolumn{1}{c}{CV MAPE} \\\\\n\t\t\\midrule\n\t\tForeign Fighters & KRHNB & 282.52 & \\textbf{107.91} \\\\\n\t\t& KRLS & 288.28 & 141.59 \\\\\n\t\t& KRLS (0-truncated) & 285.36 & 129.79 \\\\\n\t\t& Random Forest & \\textbf{278.08} & 131.60 \\\\\n\t\t& OLS & 299.88 & 182.86 \\\\\n\t\t\\midrule\n\t\t{\\bf \\cite{Burgoon2006}} & KRHNB & 5.81 & \\textbf{2.67} \\\\\n\t\t& KRLS & 5.71 & 2.72 \\\\\n\t\t& KRLS (0-truncated) & 5.71 & 2.71 \\\\\n\t\t& Random Forest & \\textbf{5.60} & 2.68 \\\\\n\t\t& OLS & 6.49 & 2.94 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\begin{flushleft} \\footnotesize \\textit{Notes}: Bold denotes the lowest prediction error. We use leave-one-out CV error for the foreign fighters dataset and 5-fold CV error for the Burgoon dataset, as it is much larger. Therefore, while the prediction errors for the Foreign Fighters dataset are fixed, the 5-fold CV error for the Burgoon data might vary depending on how the folds are constructed. Other results produce the same rankings. \\end{flushleft}\n\\end{table}\n\nThe results of this leave-one-out cross-validation approach are presented in the upper half of Table~\\ref{tab:compare}. KRHNB beats all other methods using MAPE and loses only to random forest on RMSPE. The source for this discrepancy can be seen easily if all of the predicted outcomes are plotted against the observed outcomes. Figure~\\ref{fig:predictions} plots the observed number of foreign fighters on the $x$-axis, and the predicted number on the $y$-axis. The key advantage that random forest has over KRHNB is that it far outperforms it when predicting the largest value, Tunisia. However, all of these methods struggle to predict countries with high levels of foreign fighter supply, presumably because the predictors that explain the difference between supplying $200$ and $2,000$ fighters are not in our data. KRHNB performs well with respect to MAPE, because for values of $y$ between $0$ and $200$, KRHNB does quite well, while KRLS and Random Forests are more likely to predict values that are too large. This can be seen more clearly when we zoom in to the predictions for outcomes less than $150$ -- the right panel of Figure~\\ref{fig:predictions}. \t\\\\\n\nTherefore, KRHNB is good at predicting whether there are any supplied foreign fighters. This means the first component is making good predictions, while the second component has little power, and most values are predicted near the mean. MAPE is much larger for random forest than KRHNB, which exhibits better fit at lower levels of foreign fighter supply. Thus, it appears that the added structure in the form of the hurdle improves fit over models that do not assume much structure, such as KRLS or random forests. However, that lack of structure does allow some improved prediction for high levels of foreign fighter supply. Nonetheless, all these methods do quite poorly in the tail of the data, indicating a lack of explanatory power for many variables, or the fundamental unpredictability that describes observations living in tails of distributions. \\\\\n\nAs an additional evaluation of our method, we replicate \\cite{Burgoon2006}, who uses a zero-inflated negative binomial model to predict cross-national time variation in terrorist attacks. As shown in the lower half of Table~\\ref{tab:compare}, we observe a similar pattern of performance on this data; KRHNB minimizes MAPE, but performs worse on RMPSE. \\\\\n\n\\begin{figure}[!htb]\n\t\\centering\n\t\\includegraphics[width=0.95\\linewidth]{tabs_figs/predictJoin.pdf}\n\t\\caption{Leave-one-out CV Predictions}\t\\label{fig:predictions} \n\t\\begin{flushleft} \\footnotesize \\textit{Notes}: Black line is where perfect predictions would lie ($Y = \\hat{Y}$). The left panel is the full dataset, while the right panel zooms in to where $Y < 150$. The values of $Y$ are jittered because of the clustering at 0. \\end{flushleft}\n\\end{figure}\n\n%=========================================================\n\\section{Results \\& Discussion}\t\\label{sec:results} \t\n\n\\subsection{Data}\n\nTo uncover predictors of foreign fighter supply, we build a design matrix of $27$ geographic, demographic, economic, and political variables for $163$ countries. All of these features have been measured prior to 2014, although issues of endogeneity are beyond the scope of this paper. Furthermore, there is some (though very little) missingness, because we aggregate over several years (where applicable) to get as complete a dataset as possible. There is some remaining missingness, which we address by taking the mean over $1,000$ datasets, imputed using Amelia II \\citep{Honaker2011}.\\footnote{This algorithm relies on the assumption that the data are missing at random, which is almost certainly invalid. However, no feature has more than $4$ missing observations, save for the measure of how many refugees are in a country (missing $10$ observations). Furthermore, only 0.7\\% of the data are missing.} In the Appendix Table~\\ref{tab:appeffectslist}, we report full results with list-wise deletion, leaving us with $147$ observations. The results are largely the same.\t\t\\\\\n\n\\begin{table}[!h]\n\t\\footnotesize\n\t\\centering\n\t\\caption{Description of Features} \t\\label{tab:data}\n\\begin{tabular}{lrrrrrl}\n\t\\\\ [-1.9ex] \\toprule\n\t& Mean & Std. Dev. & Min. & Median & Max. & Source\\\\ \n\t\\midrule\n\tForeign Fighter Supply & 114.08 & 332.34 & 0.00 & 0.00 & 2250.00 & ICSR \\citep{Neumann2015}\\\\ \n\tContiguous & 0.04 & 0.20 & 0.00 & 0.00 & 1.00 & CEPII \\\\ \n\tEurope & 0.21 & 0.41 & 0.00 & 0.00 & 1.00 & Hand Coded\\\\ \n\tDistance (km) & 5251.69 & 3743.59 & 85.94 & 4363.00 & 15630.00 & CEPII \\\\ \n\tLog Population & 16.23 & 1.51 & 13.20 & 16.15 & 21.03 & World Bank WDI \\\\ \n\tLog Pop. Density & 4.19 & 1.40 & 0.60 & 4.30 & 8.95 & World Bank WDI\\\\ \n\tYouth Bulge (15-24 pct) & 17.06 & 3.68 & 9.60 & 18.10 & 23.10 & World Bank WDI \\\\ \n\tSunni Pct & 25.45 & 35.33 & 0.07 & 5.07 & 99.40 & Pew \\\\ \n\tShia Pct & 2.61 & 10.82 & 0.00 & 0.06 & 92.22 & Pew \\\\ \n\tReligious Frac. & 0.43 & 0.24 & 0.00 & 0.44 & 0.86 & \\cite{Fearon2003a} \\\\ \n\tGovt. Reg. Religion & 2.78 & 3.24 & 0.00 & 1.39 & 10.00 & ARDA \\\\ \n\tGovt. Fav. Religion & 4.50 & 3.18 & 0.00 & 4.73 & 10.00 & ARDA \\\\ \n\tSoc. Reg. Religion & 4.22 & 3.58 & 0.00 & 3.67 & 10.00 & Freedom House \\\\ \n\tFreedom House Pol. Rights & 3.63 & 2.11 & 1.00 & 3.00 & 7.00 & Freedom House \\\\ \n\tMigrants as Pct of Pop & 7.87 & 12.06 & 0.05 & 2.94 & 74.61 & World Bank WDI\\\\ \n\tLog Refugees in Country & 8.50 & 3.04 & 0.00 & 8.85 & 14.30 & UNHCR \\\\ \n\tLog Refugees from Country & 7.35 & 2.92 & 0.00 & 7.18 & 14.75 & World Bank WDI \\\\ \n\tLog GDP pc, PPP & 9.10 & 1.24 & 6.37 & 9.26 & 11.76 & World Bank WDI \\\\ \n\tLife Expectancy & 69.89 & 9.48 & 45.55 & 72.27 & 83.83 & World Bank WDI\\\\ \n\tMale LFP & 79.42 & 7.03 & 48.40 & 79.80 & 95.90 & World Bank WDI \\\\ \n\tYouth Unemployment & 18.33 & 12.58 & 0.70 & 14.80 & 60.40 & World Bank WDI \\\\ \n\tInternet User per 100 & 40.01 & 29.46 & 0.00 & 39.20 & 95.05 & World Bank WDI \\\\ \n\tLog Homicides Per 100k & 1.46 & 1.20 & -1.61 & 1.57 & 4.50 & World Bank WDI \\\\ \n\t\\bottomrule\n\\end{tabular}\n\\begin{flushleft} \\footnotesize \\textit{Notes:} CEPII is the Centre d'\\'{E}tudes Prospectives et d'Informations Internationales \\citep{Mayer2011}. ARDA is the Association of Religion Data Archives \\citep{Finke2010}. The Pew data can be found at in a report by \\cite{Grim2012}. The Freedom House data are from \\cite{Teorell2013}. The UNHCR data can be found at \\href{http://popstats.unhcr.org/en/time\\_series}{http://popstats.unhcr.org/en/time\\_series} and are different from the World Bank summary of refugees in asylum because we exclude internally displaced persons. \\end{flushleft}\n\\end{table}\n\n\nThe full list of features can be found in Table~\\ref{tab:data}. We have data for all $163$ countries\\footnote{We exclude Macao and Puerto Rico as well. They had high rates of missingness due to many organizations not collecting data on these polities.} with populations above $500,000$ except for Syria and Iraq, where our outcome is not measured. Foreign fighter data come from the ICSR Report of 1/25/2015 \\citep{Neumann2015}, which estimates the number of all foreign fighters through the end of 2014. The estimates are either a single value, or a range. If there is a range, we take its mean, and round to the nearest integer.\\footnote{ICSR also lack data from the West Bank and Gaza. In the interest of maximizing our sample size, we split the count for Israel and assign half of those sent from Israel to the West Bank and Gaza. Table~\\ref{tab:appeffectslist} presents the results were we instead drop the West Bank and Gaza from our analysis, as well as other countries that have missingness. The results are substantively similar.} \\\\\n\n\\subsection{Main Results}\n\nWe fit KRHNB on the full data set, with $\\lambda_{\\psi}$ and $\\lambda_{\\omega}$ selected by a grid-search using leave-one-out cross validation. We use the Gaussian kernel to transform our data so that we are essentially working in an infinite-dimensional expansion of the design matrix and thus considering an infinite variety of complex functional forms. Following this optimization, we analyze the marginal effects of each feature on country-level foreign fighter supply in two ways. The first way is presented in Figure~\\ref{fig:booteffects}, which contains the sample-average marginal effect and bootstrapped confidence intervals, for both the binary and count components, as well as the two components jointly. Note that the confidence intervals are constructed using the empirical 2.5th percentile and 97.5th percentile of the bootstrapped distribution of marginal effects.\\footnote{This is sometimes known as the percentile interval, or the percentile bootstrap.} The exact results and confidence intervals for the sample-average marginal effects are presented in Table~\\ref{tab:appeffects} of the Appendix. The bootstrapped intervals are skewed away from 0 for many sample-average marginal effects in the foreign fighters application, indicating that several outliers and the complex target function may prevent the bootstrapped percentile interval from achieving nominal coverage. \\\\\n\n\\afterpage{\n\\begin{landscape}\n\\begin{figure} [!p]\n\t\\centering\n\t\\includegraphics[scale=0.9]{tabs_figs/bootEffects.pdf}\n\t\\caption{Sample-Average Pointwise Marginal Effects}\t\t\\label{fig:booteffects} \n\t\\begin{flushleft}\n\t\\footnotesize \\textit{Notes}: Pointwise marginal effects are computed numerically. All continuous features are scaled to have standard deviation of one. Dichotomous variables for Europe and Contiguous are not pointwise marginal effects, but first differences between the two values. Higher Freedom House scores indicate ``less freedom\". The outcome variable is foreign fighter supply for the first column, the probability of sending any fighters in the hurdle logit component in the second column, and the mean of the truncated negative binomial component in the third column. As a result, note that the scaling of the $x$ axes varies. \\end{flushleft}\n\\end{figure}\n\\end{landscape}\n}\n\nThe leftmost panel of Figure~\\ref{fig:booteffects} contains the estimated sample-average pointwise marginal effects for the combined model. Consistent with our usage throughout the paper, effects here mean the marginal effects our model produces, not causal effects. Furthermore, our features are scaled to have standard deviation of one so effects can be interpreted in standardized units. Most effects are near zero, showing that there is little predictive power for many of the features. Nonetheless, the bootstrapped 95 percentile intervals around seven of the features do not include 0. \\\\\n\nSubstantively, it appears that the two strongest predictors are population, and the percent of the country's population that is Sunni.\\footnote{The point estimate for contiguity is quite large, although the low number of foreign fighters from Iran, a contiguous state to Iraq, is a source of the large heterogeneity in these marginal effects.} The effect of population is not surprising, as a positive (unconditional) probability of any individual becoming a foreign fighter implies that larger countries should have a higher supply. Similarly, Sunni population share intuitively predicts participation in the conflict, as it captures the strength of identification with the opposition; virtually all groups fighting the Assad regime are Sunni. Thus, more Sunnis in a given country means that more people feel they have a steak in the conflict and recruiters can more easily manipulate the salience of Sunnis' religious identity to construct a sense of obligation to fight. Also unsurprising is the strong negative effect of distance to the conflict; increased travel costs should deter potential fighters at the margin, while dissociation from the conflict should increase with distance. Another noteworthy effect is that on life expectancy, which positively predicts foreign fighter supply. Although this seems troubling, it is in line with popular conceptions of supplier countries as developed, as well as the findings of \\cite{Hewitt2009} that more developed countries supplied more foreign fighters during Iraq's insurgency. Also consistent with conventional wisdom is the positive effect of internet usage; foreign fighters can be recruited online \\citep{Hegghammer2011}, thus countries with deeper internet penetration allow recruiters to cast their net more widely. There also is a positive relationship between government regulation of religion and foreign fighter supply, something we discuss further below. Finally, a puzzling finding is the positive effect of refugee presence. Below we show that this effect originates in European countries, hence we defer its interpretation for now. \t\t \\\\\n\nMore can be learned by disaggregating the effects into the hurdle and count components of our model. The central panel of Figure~\\ref{fig:booteffects} contains the estimated sample-average pointwise marginal effects of standardized variables in the hurdle component. These effects can be interpreted as changes in the probability that a country passes the hurdle of no supply and sends some foreign fighters. The results are quite similar to the left panel, indicating that our features are better at predicting whether or not any foreign fighters are supplied (binary component), rather than how many are supplied (count component). Six of the seven predictors that are distinguishable from zero in the combined model remain so here, and their ranking in terms of substantive significance is also similar. Sunni population share, population, and refugee presence are still the three strongest predictors, again followed by internet penetration, distance to the conflict, and life expectancy. Government regulation of religion is not distinguishable from zero in this component. In addition, there are two new results worth noting. First, GDP per capita has a positive effect. This is consistent with the positive effect of life expectancy, popular conceptions regarding foreign fighters as originating in the developed world, and prior findings on Iraq \\citep{Hewitt2009}. Second, migrant population share also has a positive effect. Yet, we hesitate to interpret this, as the effect is reversed in the count component. \t\\\\\n\nThe rightmost panel contains the estimated sample-average pointwise marginal effects of standardized variables in the count component. Only five features are clearly driving prediction of the count of foreign fighter supply. As in the combined model and hurdle component, population and Sunni population share have a positive effect distinguishable from zero. However, in contrast to the hurdle component, the effect of migrant population share is negative. Clearly, the opposite direction of the migrant effect in the two components accounts for its null effect in the combined model. Another difference from previous results is that government regulation of religion and government favoritism of religion are positive predictors.\\footnote{These indices reflect how much the government respects freedom of religion, and whether it funds and supports one religion in particular \\citep{Grim2006}.} This means that, among supplier countries, those that do not respect freedom of religion and/or fund a particular religion send more foreign fighters. Further inspection of these effects reveals that they are due to Sunni-majority countries, hence we interpret them later.\t\\\\\n\n\\subsection{Effect Heterogeneity}\n\nOur method allows for a rich exploration of marginal effect heterogeneity, by providing pointwise estimates of marginal effects for each feature. To demonstrate some of KRHNB's strength in learning from the data, Figure~\\ref{fig:histeffects} plots the distribution of pointwise marginal effects for four features. Most of the effects are near zero, indicating that much of the estimated CEF is flat across the support of the features. However, a clear difference can be seen between a feature that appears to have little to no relationship to foreign fighter supply (homicides, bottom left), and one that has a systematic relationship (Sunni population share, bottom right). Heterogeneity in the pointwise marginal effects can itself provide a lot of information about certain features of interest. Furthermore, by plotting or regressing the pointwise marginal effects on the features, we can see whether there appear to be interaction effects, or non-linear relationships \\citep{Hainmueller2013}. \\\\\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[scale=.85]{tabs_figs/histeffects.pdf}\n\t\\caption{Distribution of Pointwise Marginal Effects}\t\\label{fig:histeffects} \n\\end{figure}\n\nIn Figure~\\ref{fig:hetplot1}, we demonstrate how the pointwise marginal effects differ for countries within and outside of Europe. While internet usage has uniformly positive effects in Europe, in the rest of the world its marginal effect is clustered around zero. This could suggest that radicalization and recruiting happens mostly online in European countries, but through non-digital means in other countries (e.g. personal networks, mosques). The plot can also shed light on the puzzling result on refugee presence, as it suggests that refugee presence is associated with larger foreign fighter supply only in Western countries. A possible interpretation is that refugees' predicament -- persecution, poverty, loss of family members -- exacerbates feelings of injustice among citizens of the host country that are already at high risk of becoming foreign fighters (e.g. young Sunni unemployed males in urban centers).\\footnote{We believe it is unlikely that the positive effect of refugee presence means that refugees are more likely to fight in Syria, since there is no anecdotal evidence of refugees from Western countries becoming foreign fighters. We also note that our refugee measure predates the large influx of refugees from Syria into Europe; this rules-out the explanation that Syrians are seeking refuge in Europe, only to return to Syria as fighters.} The latter might view their participation in the Syrian conflict as a chance to address the conditions that resulted in refugees' predicament. Radicalization might be especially strong for citizens of the host country with a shared religion or ethnicity with the refugees, the more so if the refugees fled countries with similar conditions to those that sparked the Syrian conflict. This applies to refugees from a number of countries that repressed Islam(ists) -- for example, Algeria, Bosnia, Egypt, Iraq, Libya, Russia, and Tunisia -- much like the Assad regime did in Syria.\t \\\\\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[scale=.85]{tabs_figs/hetplot1.pdf}\n\t\\caption{Effects Inside and Outside Europe}\t\\label{fig:hetplot1} \n\\end{figure}\n\nThis discussion provides an opportunity to return to the effect of government regulation of religion. As previewed above, intervention in religious affairs has a positive effect on foreign fighter supply in the model's count component. Figure~\\ref{fig:hetplot2} reveals an interesting heterogeneity in this effect, by plotting the pointwise marginal effect of government regulation of religion by countries' Sunni population share with a simple LOWESS smoother.\\footnote{To see the full picture of possible interaction effects, see Appendix~\\ref{fig:interactplot} which presents an overview of heterogeneous effects.} As the Sunni share of the population increases the effect of government regulation of religion becomes stronger and more precise. That is, only in Sunni-majority supplier countries does religious intervention consistently predict a larger supply of foreign fighters. Given that virtually all foreign fighters are Sunni, this is consistent with a view of illiberal religious policies as radicalizing the population in Sunni-majority countries. This radicalization, in turn, might map onto foreign fighter supply through two possible mechanisms. One is that repressing (Sunni) Islam in countries where it is popular can create deep-seated grievances. Since the Assad regime also fought Islam, repressed Sunnis from the Muslim world might see Syria as a battleground for addressing their grievances. This could be the case in Tunisia, an overwhelmingly Sunni country that repressed Islamists until 2011, and has by far the largest per capita supply of foreign fighters. The second mechanism is that government repression of some religions or sects to benefit the official state religion might lead to exporting this attitude abroad. Thus, Sunnis from countries that enforce the dominance of Sunni Islam might consider it their duty to defend that religion wherever it is threatened, as in Syria. This could be the case of Saudi Arabia, which was founded on the radical Wahhabi (Sunni) interpretation of Islam, and has the second largest supply of foreign fighters.\t\\\\\n\nThese possible explanations are raised by our model because it is able to uncover non-linear and interactive relationships without relying on the a priori specification of complex functional forms and without overfitting. Therefore, our model provides fodder for future work that should causally identify the mechanisms that may lead from government intervention in religion to the generation of foreign fighters.\t \\\\\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[scale=.85]{tabs_figs/hetplotGovtRegSunni.pdf}\n\t\\caption{Effects of Government Regulation of Religion by Sunni Population Share}\t\\label{fig:hetplot2} \n\\end{figure}\n\nOverall, it seems that, while structural features dominate the hurdle component (distance, population, Sunni population share, life expectancy, GDP), policy variables mostly drive variation in the count component (government regulation of religion, government favoritism of a religion, migrant population share). Therefore, a tentative conclusion is that policy-makers cannot do much to prevent a foreign fighter network from emerging in their country, but, once it emerges, they can affect its size. Nevertheless, inspecting the effect of our features on the combined model, it is clear that structural features dominate in explaining foreign fighter supply, thereby limiting room for policy intervention.\n\n%=========================================================\n\\section{Conclusion}\t\\label{sec:conclusion}\n\n``The best way to reduce a foreign fighter returnee problem is to never have them go in the first place\" \\citep[p. 15]{Byman2015a}. Our analysis helps identify country features that are associated with higher foreign fighter supply. Substantively, our results suggest that countries' structural features play a larger role in shaping supply than their policies --- four of the strongest and most accurate predictors of supply are population, the share of the population that is Sunni, distance to Syria, and life expectancy. Some of these features cannot be altered by policy-makers, while others can, albeit at a slow pace and high cost. Moreover, with the exception of internet usage, all of the features that do respond to government policies -- refugee intake, regulation of religion, and favoritism of a particular religion -- are central to some countries' political culture. For example, French voters consistently prefer liberal policies towards refugees, while most Saudi citizens are likely to value state protection of Sunni Islam, and regulation of other religions. Therefore, policy-makers in such countries will hesitate to change these policies, unless the cost of doing so is less than that imposed by their foreign fighter supply. This might be unlikely, even for countries facing a high supply, like France and Saudi Arabia. In short, to the extent that our research design allows us to offer any policy recommendations, it is unclear whether policy-makers can feasibly curtail foreign fighter supply.\t\\\\\n\nOur approach is hindered by three limitations. On the empirical side, there is the inherent difficulty of measuring country-level foreign fighter supply. Fighters do not always publicize their participation in a conflict, not least because of fears of legal repercussions upon returning home. In addition, due to the nascent state of the literature, we lack a compass in our search for meaningful predictors to include in our specifications. Although we have carefully collected data on an array of economic, political, social, and demographic country features, there are many variables we have excluded from our specifications that may matter to foreign fighter supply. Future theoretical work will hopefully yield insights that can update our search for more informative predictors.\t\t\\\\\n\nOur second limitation is that our algorithm is computationally expensive. This is owed to three factors. First, our complex optimization problem (Equation \\ref{eqn:targetFinal}) involves a large number of parameters (2$n$ + 3), and requires evaluating derivatives of the Gamma function. Second, our process for selecting the regularization parameters ($\\lambda_\\psi$, $\\lambda_\\omega$), involves a grid search and leave-one-out cross validation. Third, the difficulties of deriving an analytical estimate of uncertainty for our quantities of interest force us to resort to the bootstrap, which involves repeatedly refitting the model to resampled datasets. One of the authors is also working on lowering the dimensionality of a related problem to improve speed and provide estimates of uncertainty that avoid the bootstrap.\t\\\\ \n\nOn the theoretical side, a limitation of our framework is that it ignores psychological, ideological, and organization factors that figure prominently in the terrorism literature. This is inevitable, given that such factors are only observable at the individual- or group-level, while our analysis is at the country-level. In theory, one could collect data on confirmed fighters and match them to non-fighters, in an effort to identify individual-level predictors of their decision to go to Syria. Indeed, several recent papers based on interviews of a small sample of returning fighters focus on individual features, albeit through a purely descriptive qualitative approach \\citep{Stenersen2011, Weggemans2014, Nilsson2015}. Expanding this dataset and applying classification models, such as Random Forest, is a promising avenue for future empirical research.\t\\\\\n\nIn future work, we would like to generalize the model to allow for different parameterizations of the two components. For example, if we use the procedure to model individual choice, one may want to fit a probit to the binary component, motivated by a theory that the stochastic term is normally distributed. Similarly, if there is no over-dispersion in the truncated count, we may want to make efficiency gains by fitting a Poisson.\\footnote{\\cite{Zorn1998} notes that in the hurdle model over dispersion can be largely accounted for by the binary component, and thus a Poisson model may suffice for the count component.} Yet another extension is to allow for different features to enter each component. This is appropriate for questions where the theory guiding model specification is more developed. Nonetheless, with flexible estimators it is likely better to include as many relevant features as possible into the two components. \n\nBroader extensions of our method would involve applying it to different political science questions. KRHNB can be used to predict any count processes that might have a hurdle structure, ranging from the number of terrorist attacks perpetrated in each country (e.g. \\cite{Burgoon2006}) to the number of bills passed by female legislatures in Congress (e.g. \\cite{Volden2013}). Moreover, by abandoning the hurdle structure and applying kernel regularization to different classes of GLMs (e.g. binary choice), the range of political science processes that can be modeled becomes infinite. We hope that our study will motivate political methodologists to take up this task.\t\n\n%========================================================\n\\bibliography{ff}\t\t\\bibliographystyle{chicago}\n%========================================================\n\\newpage\n\\appendix\n\\setcounter{table}{0}\n\\renewcommand{\\thetable}{A\\arabic{table}}\n\\setcounter{figure}{0}\n\\renewcommand{\\thefigure}{A\\arabic{figure}}\n\\setcounter{equation}{0}\n\\renewcommand{\\theequation}{\\Alph{section}.\\arabic{equation}}\n%========================================================\n%========================================================\n\\section{Results Table and Heterogeneous Effects}\t\\label{app:tables}\n\n\\subsection{Full Results}\n\nThis section contains two tables that have the sample-average pointwise marginal effects for two different datasets. Table~\\ref{tab:appeffects} contains the same information as Figure~\\ref{fig:booteffects} and represents the results from our main specification where we impute missing values and divide foreign fighter supply between Israel and the West Bank. \\\\\n\n\\begin{table}[!p]\n\t\\footnotesize\n\t\\centering\n\t\\caption{Average Marginal Effects on Foreign Fighter Supply} \t\\label{tab:appeffects}\n\t\\begin{tabular}{l ccc}\n\t\t\\toprule\n\t\tVariable & \\multicolumn{1}{c}{Both Components} & \\multicolumn{1}{c}{Hurdle} & \\multicolumn{1}{c}{Count} \\\\ \n\t\t\\midrule\nLife Expectancy & 8.64 & 0.02 & 6.87 \\\\ \n& [2.88, 22.87] & [0.02, 0.07] & [-2.59, 21.3] \\\\ \nContiguous & 68.3 & 0.1 & 87.89 \\\\ \n& [-117.95, 243] & [-0.16, 0.38] & [-151.81, 307.67] \\\\ \nDistance (km) & -6.99 & -0.02 & -4.36 \\\\ \n& [-23.87, -0.51] & [-0.08, -0.01] & [-21.98, 15.74] \\\\ \nInternet User per 100 & 9.48 & 0.02 & 9.37 \\\\ \n& [3.23, 36.21] & [0.01, 0.08] & [-4.11, 37.57] \\\\ \nMigrants as Pct of Pop & -3.84 & 0.02 & -20.19 \\\\ \n& [-20.76, 5.71] & [0.01, 0.08] & [-52.31, -7.87] \\\\ \nMale LFP & -6.52 & 0 & -12.23 \\\\ \n& [-26.61, 3.66] & [-0.03, 0.03] & [-46.65, 4.27] \\\\ \nYouth Unemployment & 2.73 & 0.01 & -0.12 \\\\ \n& [-9.21, 14.96] & [-0.01, 0.06] & [-29.94, 17.65] \\\\ \nFreedom House Civ. Lib. & -0.12 & 0 & -1.66 \\\\ \n& [-4.01, 12.74] & [-0.01, 0.03] & [-12.41, 15.97] \\\\ \nFreedom House Pol. Rights & 2.06 & 0.01 & -2 \\\\ \n& [-9.17, 15] & [0, 0.05] & [-23.41, 16.69] \\\\ \nSunni Pct & 15.17 & 0.05 & 12.7 \\\\ \n& [8.06, 41.84] & [0.02, 0.11] & [3.92, 38.71] \\\\ \nShia Pct & 0.19 & 0.02 & -8.48 \\\\ \n& [-11.1, 14.41] & [-0.01, 0.05] & [-25.07, 13.32] \\\\ \nGovt. Reg. Religion & 7.3 & -0.01 & 21.84 \\\\ \n& [0.34, 30.97] & [-0.02, 0.03] & [5.75, 52.54] \\\\ \nGovt. Fav. Religion & 4.45 & -0.01 & 19.11 \\\\ \n& [-3.94, 22.98] & [-0.05, 0.03] & [2.04, 46.65] \\\\ \nSoc. Reg. Religion & 3.96 & 0 & 10.8 \\\\ \n& [-5.36, 20.46] & [-0.02, 0.06] & [-12.12, 32.33] \\\\ \nYouth Bulge (15-24 pct) & -4.66 & -0.01 & -5.6 \\\\ \n& [-13.42, 9.51] & [-0.05, 0] & [-15.88, 23.18] \\\\ \nReligious Frac. & -3.8 & 0.01 & -12.71 \\\\ \n& [-22.42, 5.73] & [-0.03, 0.04] & [-38.91, 3.54] \\\\ \nLog Homicides Per 100k & -0.69 & 0 & -2.9 \\\\ \n& [-16.45, 7.97] & [-0.05, 0.01] & [-19.46, 20.01] \\\\ \nLog Pop. Density & -4.85 & -0.02 & -1.93 \\\\ \n& [-26.92, 3.47] & [-0.07, 0] & [-30.98, 15.26] \\\\ \nLog Population & 18.24 & 0.03 & 27.37 \\\\ \n& [7.31, 47.54] & [0.01, 0.08] & [13.14, 70.14] \\\\ \nLog GDP pc, PPP & 3.67 & 0.01 & 6.38 \\\\ \n& [-0.75, 20.31] & [0, 0.05] & [-7.99, 22.69] \\\\ \nLog Refugees in Country & 9.42 & 0.04 & 5.35 \\\\ \n& [1.4, 32.18] & [0.02, 0.09] & [-8.18, 36.92] \\\\ \nLog Refugees from Country & 0.55 & 0 & -1.34 \\\\ \n& [-10.9, 10.77] & [-0.03, 0.03] & [-18.47, 21.29] \\\\ \nEurope & -11.37 & -0.01 & -19.29 \\\\ \n& [-48.52, 29.19] & [-0.08, 0.09] & [-84.71, 55.31] \\\\ \n\\midrule\nN & 163 & & \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\\\ \\footnotesize \\textit{Notes}: The point estimates are the simple mean of pointwise marginal effects over the full data. The first column has the joint effect on the outcome of the full hurdle negative binomial model, while the second and third columns represent the effect on the probability of any fighters in the hurdle and the mean of the negative binomial distribution in the count component. In brackets are the 2.5th and 97.5th percentiles of the 1000 bootstrapped sample-average pointwise marginal effects. \n\\end{table}\n\nTable~\\ref{tab:appeffectslist} contains the same results, but instead on a dataset where listwise deletion is used to remove observations with missing values. Furthermore, the West Bank and Gaza observation has been dropped rather than manually imputed. The results are very similar. The only substantive difference is that the percentile interval now includes 0 for the number of refugees in the country, although the effect is still strictly positive in the hurdle component.\n\n\\begin{table}[!p]\n\t\\footnotesize\n\t\\centering\n\t\\caption{Average Marginal Effects on Foreign Fighter Supply in Listwise Deleted Dataset} \t\\label{tab:appeffectslist}\n\t\\begin{tabular}{l ccc}\n\t\t\\toprule\n\t\tVariable & \\multicolumn{1}{c}{Both Components} & \\multicolumn{1}{c}{Hurdle} & \\multicolumn{1}{c}{Count} \\\\ \n\t\t\\midrule\nLife Expectancy & 15.24 & 0.04 & 12.01 \\\\ \n& [2.71, 35.62] & [0.01, 0.06] & [-9.94, 33.48] \\\\ \nContiguous & 78.05 & 0.14 & 82.55 \\\\ \n& [-91.57, 266.88] & [-0.15, 0.34] & [-138.82, 304.3] \\\\ \nDistance (km) & -8.6 & -0.04 & 0.75 \\\\ \n& [-27.97, 4.96] & [-0.06, -0.01] & [-25.44, 35.02] \\\\ \nInternet User per 100 & 21.87 & 0.04 & 25.45 \\\\ \n& [4.38, 55.05] & [0.02, 0.07] & [-4.6, 65.7] \\\\ \nMigrants as Pct of Pop & -8.87 & 0.03 & -31.1 \\\\ \n& [-23.49, 8.55] & [0.01, 0.07] & [-67.27, -2.6] \\\\ \nMale LFP & -16.44 & 0 & -30.12 \\\\ \n& [-44.22, 0.94] & [-0.02, 0.03] & [-83.99, -1.16] \\\\ \nYouth Unemployment & 1.53 & 0.01 & -6.74 \\\\ \n& [-18.73, 18.49] & [-0.02, 0.05] & [-46.98, 25.33] \\\\ \nFreedom House Civ. Lib. & -2.27 & 0.01 & -8.71 \\\\ \n& [-7.19, 16.65] & [-0.01, 0.03] & [-20.98, 18.74] \\\\ \nFreedom House Pol. Rights & 0.65 & 0.03 & -11.18 \\\\ \n& [-17.51, 15.42] & [0, 0.04] & [-41.82, 15.94] \\\\ \nSunni Pct & 26.57 & 0.07 & 23.85 \\\\ \n& [7.78, 52.81] & [0.03, 0.09] & [0.43, 54.28] \\\\ \nShia Pct & -2.72 & 0.01 & -12.77 \\\\ \n& [-13.12, 15.71] & [-0.01, 0.04] & [-31.06, 19.03] \\\\ \nGovt. Reg. Religion & 14.75 & -0.01 & 34.68 \\\\ \n& [1.06, 43.85] & [-0.02, 0.03] & [3.94, 80.33] \\\\ \nGovt. Fav. Religion & 5.75 & -0.02 & 23.54 \\\\ \n& [-3.81, 31.51] & [-0.03, 0.03] & [-5.12, 67.13] \\\\ \nSoc. Reg. Religion & 8.3 & 0.01 & 13.15 \\\\ \n& [-4.82, 28.08] & [-0.01, 0.05] & [-13.39, 45.4] \\\\ \nYouth Bulge (15-24 pct) & -4.01 & -0.02 & -4.19 \\\\ \n& [-16.44, 15.04] & [-0.04, 0.01] & [-20.9, 34.81] \\\\ \nReligious Frac. & -6.56 & 0 & -16.27 \\\\ \n& [-32.4, 6.5] & [-0.03, 0.03] & [-58.49, 6.34] \\\\ \nLog Homicides Per 100k & 2.82 & 0 & 3.57 \\\\ \n& [-17.66, 16.3] & [-0.05, 0.01] & [-23.4, 40.09] \\\\ \nLog Pop. Density & -7.92 & -0.02 & -5.83 \\\\ \n& [-33.61, 6.05] & [-0.05, 0.01] & [-46.81, 20.93] \\\\ \nLog Population & 31.27 & 0.05 & 46.34 \\\\ \n& [7.58, 71.29] & [0, 0.06] & [14.38, 109.16] \\\\ \nLog GDP pc, PPP & 7.48 & 0.01 & 11.16 \\\\ \n& [-0.77, 30.97] & [0.01, 0.05] & [-11.26, 37.61] \\\\ \nLog Refugees in Country & 13.62 & 0.04 & 15.12 \\\\ \n& [-0.21, 47.45] & [0.01, 0.08] & [-9.63, 63.35] \\\\ \nLog Refugees from Country & -0.22 & -0.01 & 5.31 \\\\ \n& [-11.83, 19.31] & [-0.04, 0.01] & [-10.72, 47.96] \\\\ \nEurope & -11.1 & 0.01 & -25.47 \\\\ \n& [-71, 46.84] & [-0.08, 0.09] & [-135.44, 97.85] \\\\ \n\\midrule\nN & 147 & & \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\\\ \\footnotesize \\textit{Notes}: The point estimates are the simple mean of pointwise marginal effects over the full data. The first column has the joint effect on the outcome of the full hurdle negative binomial model, while the second and third columns represent the effect on the probability of any fighters in the hurdle and the mean of the negative binomial distribution in the count component. In brackets are the 2.5th and 97.5th percentiles of the 1000 bootstrapped sample-average pointwise marginal effects. \n\\end{table}\n\n\\subsection{Heterogeneous Effects}\n\nBecause we able to compute pointwise marginal effects, it is possible to explore these effects to see how they vary in different parts of the feature space. One way to do this is simply to plot the pointwise marginal effects with respect to one variable by some other variable to see an interaction effect. A more robust way to explore these effects is to regress the pointwise marginal effects of some feature and all of the features in the data. This way, we can uncover the conditional interaction or non-linearity of our features. Of course, the pointwise marginal effects can themselves be modeled using flexible models or visualized using partial residual plots. \\\\ \n\nA way to summarize all of these interaction effects is to regress the pointwise marginal effects for each feature on the full set of features. To be precise, we estimate a regression of the following form\n$$ \\frac{\\partial\\E[y_i|\\x_i]}{\\partial x^{(j)}_i} = \\bm{\\gamma}^\\top \\x_i,$$\nwhere $x^{(j)}_i$ is the $i$th observation of the $j$th feature. We do this for all features $j$. The coefficients in $\\bm{\\gamma}$ are analogous to interaction terms ($\\gamma_i \\; \\forall \\; i \\neq j$) or quadratic terms ($\\gamma_j$). Thus if we are predicting the pointwise marginal effects of life expectancy on foreign fighter supply and we estimate a large positive $\\gamma_i$ on, for example, the percent of the population that is Sunni, we then conclude that the marginal effect of life expectancy is much greater in heavily Sunni countries. Figure~\\ref{fig:interactplot} contains the results of these regressions for all features, demonstrating how the variables interact. Large blue dots indicate the two have a positive interaction effect while large red dots indicate the two have a negative interaction effect.\n\n\\begin{figure}[!h]\n\t\\centering\n\t\\includegraphics[scale=.85]{tabs_figs/interactPlot.pdf}\n\t\\caption{Interaction Effects}\t\\label{fig:interactplot}\n\t\\begin{flushleft} \\footnotesize \\textit{Notes}: Each cell is the coefficient from of a regression of one set of pointwise marginal effects on the rest of the data. A blue cell means that the two have a positive interaction while a red cell means the two have a negative interaction.\t\\end{flushleft}\n\\end{figure}\n\n%========================================================\n%========================================================\n\\section{Target Function Derivation}\t\\label{app:deriv}\n\n\\subsection{Sample Log-Likelihood}\n\nThe likelihood for observation $i$ in Equation~\\ref{eq:likelihood} can be written more explicitly with respect to the densities in Equations~\\ref{eq:dens0}, \\ref{eq:densmu}, and \\ref{eq:denstrunc}. Where $\\btheta = (\\alpha_0, \\balpha^\\top, \\beta_0, \\bbeta^\\top, \\zeta)^\\top$,\n\n\\begin{align}\n L_i(\\btheta | y_i, \\x_i) &= \\l[ p_0(y_i = 0) \\r]^{1 - d_i} \\l[ \\frac{p_1(y_i)}{1 - p_1(y_ i = 0)} (1 - p_0(y_i = 0)) \\r]^{d_i}\n \\quad \\text{;} \\quad d_i = \\begin{cases}\n 0 \\text{ if } y_i = 0 \\\\\n 1 \\text{ if } y_i \\geq 1\n \\end{cases} \\notag \\\\\n &= \\l[ \\frac{1}{1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha)} \\r]^{1 - d_i} \\\\\n &\\times \\l[ \\frac{\\Gamma (\\zeta + y_i) \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\l( \\frac{\\exp(\\beta_0 + \\x^\\top_i \\bbeta)}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^{y_i}}{\\Gamma(1 + y_i) \\Gamma(\\zeta) \\l( 1 - \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\r)} \\frac{\\exp(\\alpha_0 + \\x^\\top_i \\balpha)}{1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha)} \\r]^{d_i}\n\\end{align}\n\nNext we derive the sample log-likelihood in Equation~\\ref{eq:sampll}:\n\n\\begin{align}\n\\ell_N(\\btheta | \\Y, \\X) &= \\begin{aligned}[t]\n\\sum^N_{i=1} &(1 - d_i) \\l[ \\log 1 - \\log \\l( 1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha) \\r) \\r] + d_i \\Bigg[ \\log \\Gamma( \\zeta + y_i ) \\\\\n& + \\zeta \\log \\zeta - \\zeta \\log \\l( \\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta) \\r) + y_i (\\beta_0 + \\x^\\top_i \\bbeta) \\\\\n& - y_i \\log \\l( \\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta) \\r) - \\log \\Gamma (1 + y_i) - \\log \\Gamma (\\zeta) \\\\\n&- \\log \\l(1 - \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\r) + (\\alpha_0 + \\x^\\top_i \\balpha) \\\\\n& - \\log \\l(1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha) \\r) \\Bigg]\n\\end{aligned} \\notag \\\\\n &= \\begin{aligned}[t]\n \\sum^N_{i=1} & - \\log \\l(1 + \\exp(\\alpha_0 + \\x^\\top_i \\balpha) \\r) + d_i \\Bigg[ \\log \\Gamma( \\zeta + y_i ) + \\zeta \\log \\zeta \\\\\n &- (\\zeta + y_i) \\log \\l( \\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta) \\r) + y_i(\\beta_0 + \\x^\\top_i \\bbeta) + (\\alpha_0 + \\x^\\top_i \\balpha) \\\\\n & - \\log \\Gamma (1 + y_i) - \\log \\Gamma (\\zeta) - \\log \\l( 1 - \\l( \\frac{\\zeta}{\\zeta + \\exp(\\beta_0 + \\x^\\top_i \\bbeta)} \\r)^\\zeta \\r) \\Bigg]\n \\end{aligned}\n\\end{align} \n\n\\subsection{Using Mercer's Theorem}\n\nGiven Equation~\\ref{eqn:target}, we solve the First Order Condition (FOC) for each parameter vector in order to demonstrate our ability to use Mercer's Theorem to reduce our problem from a potentially infinite-dimensional one to a more tractable function. Where $\\btheta_{\\bphi} = (\\psi_0, \\bpsi^\\top, \\omega_0, \\bomega^\\top, \\zeta)^\\top$,\n\n\\begin{align}\n\\frac{\\partial R_N (\\btheta_{\\bphi}, \\lambda_\\psi, \\lambda_\\omega | \\Y, \\X)}{\\partial \\bpsi} &= 0 \\notag \\\\\n0 &= -\\sum^N_{i=1} \\Bigg( - \\frac{\\bphi(\\x_i)^\\top \\exp \\l( \\psi_0 + \\bphi(\\x_i)^\\top \\bpsi \\r)}{1 + \\exp \\l( \\psi_0 + \\bphi(\\x_i)^\\top \\bpsi \\r)} + d_i \\bphi(\\x_i)^\\top \\Bigg) + 2 \\lambda_\\psi \\bpsi \\notag \\\\\n\\bpsi &= \\sum^N_{i=1} \\Bigg\\{ \\frac{1}{2 \\lambda_\\psi} \\Bigg( - \\frac{\\exp \\l( \\psi_0 + \\bphi(\\x_i)^\\top \\bpsi \\r)}{1 + \\exp \\l( \\psi_0 + \\bphi(\\x_i)^\\top \\bpsi \\r)} + d_i \\Bigg) \\Bigg\\} \\bphi(\\x_i)\n\\end{align}\n\n\\begin{align}\n\\frac{\\partial R_N (\\btheta_{\\bphi}, \\lambda_\\psi, \\lambda_\\omega | \\Y, \\X)}{\\partial \\bomega} &= 0 \\notag \\\\\n0 &= \\begin{aligned}[t]\n- & \\sum^N_{i=1} d_i \\Bigg[ - \\frac{(\\zeta + y_i) \\bphi(\\x_i)^\\top \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)}{\\zeta + \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)} + y_i \\bphi(\\x_i)^\\top \\notag \\\\\n& - \\frac{-\\zeta \\l( \\frac{\\zeta}{\\zeta + \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)} \\r)^{\\zeta - 1} \\l( \\frac{- \\zeta \\bphi(\\x_i)^\\top \\exp \\l(\\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)}{\\l( \\zeta + \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r) \\r)^2} \\r)}{1 - \\l( \\frac{\\zeta}{\\zeta + \\exp \\l(\\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)} \\r)^\\zeta} \\Bigg] \\Bigg) \\\\\n&+ 2 \\lambda_\\omega \\bomega \n\\end{aligned} \\\\ \\notag \\\\\n\\bomega &= \\begin{aligned}[t]\n\\sum^N_{i=1} & \\Bigg\\{ \\frac{1}{2 \\lambda_\\omega} d_i \\Bigg[ - \\frac{(\\zeta + y_i) \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)}{\\zeta + \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)} + y_i \\\\\n& - \\frac{\\zeta \\l( \\frac{\\zeta}{\\zeta + \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)} \\r)^{\\zeta - 1} \\l( \\frac{\\zeta \\exp \\l(\\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)}{\\l( \\zeta + \\exp \\l( \\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r) \\r)^2} \\r)}{1 - \\l( \\frac{\\zeta}{\\zeta + \\exp \\l(\\omega_0 + \\bphi(\\x_i)^\\top \\bomega \\r)} \\r)^\\zeta} \\Bigg] \\Bigg\\} \\bphi(\\x_i)\n\\end{aligned}\n\\end{align} \n\nNote that in both FOCs, the terms inside $\\{\\}$ form a scalar, which we label $c^\\psi_i$ and $c^\\omega_i$. As such, we rewrite our solutions for $\\bpsi$ and $\\bomega$ like we did in the main body of the paper:\n\n\\begin{align}\n\\bpsi^* = \\sum^N_{i=1} c^\\psi_i \\bphi(\\x_i) \\\\\n\\bomega^* = \\sum^N_{i=1} c^\\omega_i \\bphi(\\x_i) \n\\end{align} \n\nWe use these solutions to rewrite Equation~\\ref{eqn:target} as Equation~\\ref{eqn:targetFinal}, our final target function.\n\n\\end{document}\n%%% Local Variables:\n%%% mode: latex\n%%% TeX-master: t\n%%% End:", "meta": {"hexsha": "9c6d4692cd2a6ad05305ad307f24347cb36474f9", "size": 113123, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/derpanopoulos_sonnet_ff.tex", "max_stars_repo_name": "lukesonnet/foreign_fighters", "max_stars_repo_head_hexsha": "88c1bfbde87b8e95c4858301d0a3e870ac59d0c3", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2016-12-27T12:50:03.000Z", "max_stars_repo_stars_event_max_datetime": "2016-12-27T12:50:03.000Z", "max_issues_repo_path": "tex/derpanopoulos_sonnet_ff.tex", "max_issues_repo_name": "lukesonnet/foreign_fighters", "max_issues_repo_head_hexsha": "88c1bfbde87b8e95c4858301d0a3e870ac59d0c3", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/derpanopoulos_sonnet_ff.tex", "max_forks_repo_name": "lukesonnet/foreign_fighters", "max_forks_repo_head_hexsha": "88c1bfbde87b8e95c4858301d0a3e870ac59d0c3", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 161.8354792561, "max_line_length": 2775, "alphanum_fraction": 0.7524287722, "num_tokens": 29204, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3488913482066966}} {"text": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\\chapter{Higher-Rank Polymorphism with the Application Mode}\n\\label{chap:BiDirectional}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n\nWe have seen in \\Cref{sec:DK} that bidirectional type checking is a useful and\nversatile tool for type checking and type inference. In traditional\nbidirectional type-checking, type information flows from functions to arguments\n(e.g., \\rref{dk-in-app} in \\Cref{sec:DK:declarative}). In this section, we\npresent a novel variant of bidirectional type checking where the type\ninformation flows from arguments to functions. This variant retains the\ninference mode, but adds a so-called \\textit{\\mode} mode. Such design can remove\nannotations that basic bidirectional type checking cannot, and is useful when\ntype information from arguments is required to type-check the functions being\napplied.\n\nWe illustrate our novel design of bidirectional type-checking using System \\ap,\na lambda calculus with implicit higher-rank polymorphism. This section first\npresents the declarative, syntax-directed type system of System \\ap\nin \\Cref{sec:AP:declarative}. The interesting aspects about the new type system\nare: 1) the typing rules, which employ a combination of the inference mode and\nthe \\textit{\\mode} mode; 2) the novel subtyping relation under an application\ncontext. Later, we prove our type system is type-safe by a type-directed\ntranslation to System F in \\Cref{sec:AP:translation}. An algorithmic type system\nis discussed in \\Cref{sec:AP:algorithm}.\n\n\n\n\\input{Gen/BiDirectional/introduction}\n\\input{Gen/BiDirectional/typesystem}\n\\input{Gen/BiDirectional/translation}\n\\input{Gen/BiDirectional/algorithm}\n\\input{Gen/BiDirectional/discussion}\n\n\n\n%%% Local Variables:\n%%% mode: latex\n%%% TeX-master: \"../Thesis\"\n%%% org-ref-default-bibliography: \"../Thesis.bib\"\n%%% End:", "meta": {"hexsha": "6258a8f70c049d1abe8288f8fe19021f387434f4", "size": 1901, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Sources/BiDirectional.tex", "max_stars_repo_name": "xnning/Dissertation", "max_stars_repo_head_hexsha": "bcdca568b3e89ba9cc0e4e9858a34e7c008a4ebb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2020-09-07T00:43:02.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-17T23:07:52.000Z", "max_issues_repo_path": "Sources/BiDirectional.tex", "max_issues_repo_name": "xnning/Dissertation", "max_issues_repo_head_hexsha": "bcdca568b3e89ba9cc0e4e9858a34e7c008a4ebb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Sources/BiDirectional.tex", "max_forks_repo_name": "xnning/Dissertation", "max_forks_repo_head_hexsha": "bcdca568b3e89ba9cc0e4e9858a34e7c008a4ebb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-09-15T08:46:35.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-15T08:46:35.000Z", "avg_line_length": 46.3658536585, "max_line_length": 80, "alphanum_fraction": 0.7317201473, "num_tokens": 433, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883592602049, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.34889134820669654}} {"text": "\\documentclass[a0]{tumposter}\n\n\\usepackage[english]{babel}\n\n\\usepackage{blindtext}\n\n\\usepackage{multicol}\n\\usepackage{amsmath}\n\\usepackage{graphicx}\n\n% for printing fontsizes\n\\usepackage{printlen}\n\\uselengthunit{mm}\n\n\\input{preamble.tex}\n\n\\usepackage[utf8]{inputenc}\n\n\\title{\n\tEarly Classification for Agricultural Monitoring \\\\ from Satellite Time Series\n\t}\n\t\n\\author{\n\tMarc Rußwurm,\\footnotemark[1] Romain Tavenard,\\footnotemark[2] Sébastien Lefèvre,\\footnotemark[2] Marco Körner\\footnotemark[1]\n\t}\n\n\\header{\n\tRemote Sensing Technology \\\\\n\tTUM Department of Civil, Geo and Environmental Engineering \\\\\n\tTechnical University of Munich\n\t}\n\t\n\\begin{document}\n\\maketitle\n\n\\renewcommand{\\subsection}[1]{\\textbf{#1}}\n\n\\begin{minipage}[t]{.65\\textwidth}\n\t\n\t\\section{Objective}\n\t\n\t\n\t\\begin{tikzpicture}[scale=16.8]\n\t\\node[label={[name=sat,text height=1.5ex,text depth=.25ex]Satellite Data}, anchor=north](x) at (-1,0){$\\M{X} = (\\V{x}_0, \\V{x}_1, \\dots , \\V{x}_T)$};\n\t\\node[below=of x, label={[yshift=1.3em, xshift=1.8em, font=\\tiny, text=white]below:ESA Sentinel 2 Satellite}](s2){\\includegraphics[width=11cm]{images/sentinel2}};\n\t\n\t\n\t\\node[below=0em of s2, text width=11cm](eqbox){\n\t\t\\small\n\t\t\\begin{itemize}\n\t\t\t\\item collected at regular temporal intervals of 2-3 days\n\t\t\t\\item measurements of 13 spectral bands\n\t\t\t\\item data available globalls\n\t\t\\end{itemize} \n\t};\n\t\n\t\\node[label={[name=cm,text height=1.5ex,text depth=.25ex]{Early} {Classification} Model}, anchor=north](f) at (0,0){ $\\yhat_t, {\\deltat} = f(\\xuptot)$};\n\t\\node[below=0em of f, text width=13cm, font=\\small](info){\n\t\t\\begin{description}\n\t\t\n\t\t\\item[$\\xuptot$] observation until $t$ \\\\\n\t\t\\item[$\\yhat_t$] class prediction scores \\\\\n\t\t\\item[$\\deltat$] probability of stopping.\n\t\t\n\t\t\\end{description}\n\t\t\n\t};\n%\t\\node[below=of f](example){\\input{images/example.tikz}};\n\t\\node[circle, below=of info, text width=11cm, fill=tumbluedark, text=white](example){Classifying a \\\\ satellite time series \\\\ {\\color{tumbluelight}\\textbf{accurately}} \\color{white} as {\\color{ecolor}\\textbf{early}} \\color{white} as possible};\n\t\n\t\n\t\\node[label={[name=ctm,text height=1.5ex,text depth=.25ex]Crop Type Labels}, anchor=north](y) at (1,0){\n\t\t$\\V{y} = \\small (y_\\text{corn}, y_\\text{barley}, \\dots) \\in \\mathbb{R}^{13}$\n\t};\n\t\\node[below=of y, label=below:crop type labels](labels){\\includegraphics[width=11cm]{images/parcels}};\n\t\\node[below=of labels, text width=11cm](lbls){\n\t\t\\small\n\t\t\\begin{itemize}\\setlength\\itemsep{.1em}\n\t\t\\item European Common Agricultural Policy (CAP)\n\t\t\\item collected yearly in entire Europe\n%\t\t\\item slowly made publicly available\n%\t\t\\item today, gathered on a national basis\n%\t\t\\item in future harmonized within Europe's INSPIRE directive\n\t\t\\end{itemize}\n\t};\n\t\n\t\\draw[very thick, -{Stealth[scale=.5]}, tumblue] (x) -- (f);\n\t\\draw[very thick, -{Stealth[scale=.5]}, tumblue] (f) -- (y);\n\t\n\t\\coordinate(bottomleftcolumn) at (example.south -| s2);\n\t\\coordinate(bottomrightcolumn) at (example.south -| lbls);\n\t\n\t\\scoped[on background layer]\n\t{\n\t\t\\node[fit=(ctm)(lbls)(bottomrightcolumn), fill=tumbluelight!20, inner sep=1em, rounded corners]{};\n\t\t\\node[fit=(f)(example)(cm), fill=tumorange!20, inner sep=1em, rounded corners]{};\n\t\t\\node[fit=(sat)(eqbox)(bottomleftcolumn), fill=tumbluelight!20, inner sep=1em, rounded corners]{};\n\t}\t\n\t\n\t\\end{tikzpicture}\n\t\n\t\n\t\\section{Method}\n\t\n\tBased on previous work (Rußwurm et al., 2019) applied to crop type mapping from remote sensing data.\n\t\n\t%A network output to indicate a probability of stopping $\\deltat$\n\t\n\t\\begin{minipage}[t]{.49\\textwidth}\n\t\t\\subsection{Mechanism}\n\t\t\n\t\t\\input{images/backprop_stopping_rule.tikz}\n\t\t\\backpropstoppingrulefull\n\t\t\n%\t\\begin{tikzpicture}\n%\t\t\\node[fill=tumbluelight!20, rounded corners](a) at (0,0){\n%\t\t\t\t\t\t\\input{images/backprop_stopping_rule.tikz}\n%\t\t\t\t\t\t\\backpropstoppingrulefull\n%\t\t\t\t\t};\n%\t\t\\node[right=of a]{\\input{images/qualitative_example.tikz}};\n%\t\\end{tikzpicture}\n%\t\\vspace{-10em}\n\n%\tprobability of not having classified before (known at training time)\n%\t$$P(t) = \\deltat \\cdot \\prod_{\\tau=0}^{t-1} 1 - p_\\tau$$\n\n\t\n\n\t\\end{minipage}\n\t\\begin{minipage}[t]{.49\\textwidth}\n\t\t\n\t\n\t\\subsection{Loss function}\n\t\n\tcomposite loss function\n\t$$\\mathcal{L}(\\V{x}, \\V{y}) = \\sum_{t=0}^T P(t;\\deltauptot) \\mathcal{L}_t(\\xuptot, \\V{y})$$\n\t\n\t\n\tA Loss function including accuracy and earliness\n\t\n\t\\begin{tikzpicture}[node distance=-.5em, draw=tumbluelight, rounded corners]\n\t\n\t%\t\\node(allloss){$\\mathcal{L}(\\V{x}, \\V{y}) = \\sum_{t=0}^T P(t;\\deltauptot) \\mathcal{L}_t(\\xuptot, \\V{y})$};\n\t%\t\n\t\\node(loss){$\\mathcal{L}_t(\\xuptot, \\V{y})$};\n\t\\node[right=of loss](equals){$=$};\n\t\\node[right=of equals, fill=accuracycolor!20, rounded corners, label={[font=\\small]Classification Loss}](classificationloss){$\\alpha \\mathcal{L}_c (\\xuptot, \\V{y})$};\n\t\\node[right=of classificationloss](minus){$-$};\n\t\\node[right=of minus, fill=earlinesscolor!20, rounded corners, label={[font=\\small]Earliness Reward}](earlinessreward){$(1 - \\alpha)\\mathcal{R}_e(t, \\ycorrect_t)$};\n\t\n\t\\node[below=1em of classificationloss, text width=10cm, xshift=-4em](clossexp){$\\mathcal{L}_c = -\\log(\\ycorrect_t)$ \\\\\n\t\t\\vspace{.5em} \n\t\t\\tiny\n\t\tcross entropy loss for accurate classifications \\par};\n\t\n\t\\node[below=1em of earlinessreward, text width=13cm, xshift=-2em](erewardexp){$\\mathcal{R}_e(t, \\ycorrect_t) = \\ycorrect_t \\left(1 - \\frac{t}{T}\\right)$ \\\\\n\t\t\\vspace{.5em}\n\t\t\\tiny\n\t\treduces loss for earlier classifications $1-\\frac{t}{T}$ if the correct class $\\ycorrect_t$ has been predicted \\par};\n\t\n\t\\draw[thick] (classificationloss) -- (clossexp);\n\t\\draw[thick] (earlinessreward) -- (erewardexp);\n\t\\end{tikzpicture}\n\t\n\t\n\t\\end{minipage}\n\n\t{\\tiny \n\tRußwurm, M., Lefèvre, S., Courty, N., Emonet, R., Körner, M., and Tavenard, R. End-to-end learning for early classification of time series. arXiv preprint arXiv:1901.10681, 2019.\n\t\\par\n\t}\n\n\t\\vspace{-1em}\n\t\\section{Application}\n\t\n\t\\begin{minipage}[t]{.5\\textwidth}\n\t\\subsection{Agriculture}\n\t\n\t\\small\n\t\n\t\\vspace{1em}\n\t\n\t\t\\textbf{Early Crop Detection}\n\t\t\\begin{itemize}\n\t\t\t\\item early assessment of cultivated crops\n\t\t\t\\item basis for early crop yield estimation\n\t\t\\end{itemize}\n\t\t\\vspace{.3em}\n\t\t\n\t\t\\textbf{Extraction of Crop Phenology}\n\t\t\\begin{itemize}\n\t\t\t\\item extraction of vegetation specific events \n\t\t\t\\item monitoring time of classification\n\t\t\t\\item regional or temporal variations\n\t\t\\end{itemize}\n\t\t\\vspace{.3em}\n\t\t\n\t\t\\textbf{Generalization}\n\t\t\\begin{itemize}\n\t\t\t\\item end-to-end trainable\n\t\t\t\\item no region-specific expert knowledge\n\t\t\t\\item applicable globally\n\t\t\\end{itemize}\n\t\n\t\\end{minipage}\n\t\\begin{minipage}[t]{.49\\textwidth}\n\t\\subsection{Dataset and Area of Interest} \\par\n\t\\begin{minipage}{.6\\textwidth}\n\t\t\\small\n\t\t\\vspace{1em}\n\t\tHollfeld region Bavaria\n\t\t\\begin{itemize}\n\t\t\t\\item 49k field parcels\n\t\t\t\\item 6 main crop types\n\t\t\t\\item covering 40km by 30 km\n\t\t\t\\item central Germany\n\t\t\\end{itemize}\n\t\t\n\t\t\\vspace{1em}\n\t\n\t\t\\tiny Challenge: Class imbalance \\par\n\t\t\\input{images/partition_histograms.tikz}\n%\t\t\\vspace{-4em}\\includegraphics[width=.3\\textwidth]{images/holl.pdf}\n\t\\end{minipage}\n\t\\begin{minipage}{.35\\textwidth}\n%\t\t\\input{images/partition_histograms.tikz}\n\t\t\\tiny {\\color{tumblue}regions with labels} and \\\\ {\\color{tumorange} location of dataset. \\par} \\par\n\t\t\\includegraphics[width=.9\\textwidth]{images/oberfrankenineurope.pdf}\n\t\t \n\t\t \\tiny partition in {\\color{traincolor} train}, {\\color{validcolor} validation}, and {\\color{evalcolor} evaluation} \\par\n\t\t\\includegraphics[width=.9\\textwidth]{images/holl.pdf}\n\t\\end{minipage}\n\t\n\t\\end{minipage}\n\n\\end{minipage}\n\\hfill\n\\begin{minipage}[t]{.32\\textwidth}\n\t\\section{Results}\n\t\n\t\\subsection{Qualitative Example} \\par\n\t{\\footnotesize Single example showing reflectance data $\\M{x}$ and predictions $\\yhat$ along with the stopping time $\\tstop \\sim \\text{Ber}(\\deltat)$. \\par}\n\t\n\t\\vspace{.5em}\n\t\\input{images/example.tikz}\n\t\\vspace{-.5em}\n\t\n\t\\subsection{Losses during Training} \\par\n\t{\\footnotesize The combined loss $L_t$, as well as earliness $L_e$ and accuracy $L_e$ losses during training. \\par}\n\n\t\\vspace{.5em}\t\n\t\\input{images/loss-accuracyplots.tikz}\n\t\\vspace{-1em}\n\n\t\\subsection{Stopping Condition Parameterization} \\par\n\t{\\footnotesize Stopping times throughout the training grouped by crop category. The parameterization of early classification is learned for different crop types at different times during training. \\par}\n\t\n\t\\input{images/trainingstoppingclasses.tikz}\n\t\n\t\\subsection{Balancing Earliness and Accuracy} \\par\n\t{\\footnotesize Evaluting the effect of the trade-off parameter $\\alpha$ on the accuracy and earliness ($\\tstop$). Runs repeated three times to evaluated the sability of the results. \\par}\n\t\n\t\\begin{table}\n\t\t\n\t\t\\scriptsize\n\t\t\\hspace{0em}\\begin{tabular}{lcccccc}\n\t\t\t\\toprule\\small\n\t\t\t\\textbf{$\\alpha$} & accuracy & $\\meantstop$ & precision & recall & $f_1$ & $\\kappa$ \\\\\n\t\t\t\\cmidrule(lr){0-0}\\cmidrule(lr){1-1}\\cmidrule(lr){2-2}\\cmidrule(lr){3-3}\\cmidrule(lr){4-4}\\cmidrule(lr){5-5}\\cmidrule(lr){6-6}\\cmidrule(lr){7-7}\n\t\t\t.0 & .25 $\\pm$ .22 & .10 $\\pm$ .17 & .19 $\\pm$ .20 & .25 $\\pm$ .17 & .16 $\\pm$ .20 & .12 $\\pm$ .19 \\\\\n\t\t\t.2 & .81 $\\pm$ .03 & .40 $\\pm$ .02 & .70 $\\pm$ .01 & .74 $\\pm$ .01 & .71 $\\pm$ .01 & .71 $\\pm$ .04 \\\\\n\t\t\t.4 & .80 $\\pm$ .09 & .47 $\\pm$ .03 & .71 $\\pm$ .02 & .74 $\\pm$ .01 & .71 $\\pm$ .02 & .71 $\\pm$ .10 \\\\\n\t\t\t.6 & .85 $\\pm$ .02 & .88 $\\pm$ .07 & .73 $\\pm$ .04 & .74 $\\pm$ .03 & .73 $\\pm$ .03 & .77 $\\pm$ .03 \\\\\n\t\t\t.8 & .84 $\\pm$ .01 & .93 $\\pm$ .05 & .72 $\\pm$ .02 & .75 $\\pm$ .01 & .73 $\\pm$ .02 & .76 $\\pm$ .02 \\\\\n\t\t\t1.0 & .83 $\\pm$ .03 & 1.00 $\\pm$ .00 & .72 $\\pm$ .03 & .75 $\\pm$ .01 & .72 $\\pm$ .03 & .75 $\\pm$ .04 \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t\n\t\n\t\t%\t\\caption{Varying the weighting factor $\\alpha$ for \\emph{early reward} loss formulation (\\cref{sec:earlynessreward}).}\n\t\t%\t\\label{tab:alpha}\n\t\t\n\t\\end{table}\n\t\n\t\n\t\\subsection{Extracting Vegetation Characteristics} \\par\n\t\n\t{\\footnotesize Stopping time per crop category reveals characteristic variations in type of vegetation confirmed by date of harvest (\\druschdatum) from local authorities. \\par} \n\t\\vspace{.5em}\n\t\\input{images/classboxplots.tikz}\n\t\n\\end{minipage}\n\n%\\frame{\n\\begin{footer}\n\t\\begin{minipage}{.15\\textwidth}\n\t\t\\textbf{ICML Workshop}\\\\\n\t\t\\includegraphics[width=5cm]{images/AI4SG}\n\t\\end{minipage}\n\t\\begin{minipage}{.275\\textwidth}\n\t\t\\textbf{Technical University of Munich}\\footnotemark[1]\\\\\n\t\tTUM Department of Civil, Geo and Env. \\\\\n\t\tRemote Sensing Technology \\\\\n\t\tArcisstr. 21, 80333 Munich, Germany\n\t\\end{minipage}\n\t\\begin{minipage}{.275\\textwidth}\n\t\t\\textbf{IRISA-Obelix}\\footnotemark[2]\\\\\n\t\tUniversité Bretagne Sud \\\\\n\t\tIRISA, UMR 6074 CNRS \\\\\n\t\tCampus de Tohannic, 56000 Vannes, France\n\t\t\n\t\\end{minipage}\n\t\\begin{minipage}{.2\\textwidth}\n\t\t\\textbf{Data \\& Code} \\\\\n\t\t%\t\t\\vspace{1em}\n\t\t{github.com/rtavenar/early\\_rnn} \\\\\n\t\t{twitter.com/MarcCoru} \\\\\n\t\twww.lmf.bgu.tum.de/vision\n\t\\end{minipage}\n\t\\begin{minipage}{.05\\textwidth}\n\t\t\\hfill\\includegraphics[width=5cm]{images/qr-code}\\\\\n\t\t\n\t\\end{minipage}\n\n\t\n\\end{footer}\n%}\n\n\\end{document}", "meta": {"hexsha": "2cf5b6b868faf8afddda497cb8744909682ccd63", "size": 10995, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "main.tex", "max_stars_repo_name": "MarcCoru/ICML19_AI4SG_Poster", "max_stars_repo_head_hexsha": "f55efb0c4286b9b28cd1716e9bd31e400c237b71", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "main.tex", "max_issues_repo_name": "MarcCoru/ICML19_AI4SG_Poster", "max_issues_repo_head_hexsha": "f55efb0c4286b9b28cd1716e9bd31e400c237b71", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "main.tex", "max_forks_repo_name": "MarcCoru/ICML19_AI4SG_Poster", "max_forks_repo_head_hexsha": "f55efb0c4286b9b28cd1716e9bd31e400c237b71", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.4194528875, "max_line_length": 245, "alphanum_fraction": 0.6866757617, "num_tokens": 3933, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.34876783486490925}} {"text": "\\chapter{Discussion}\n\\label{chapter:conclusion}\n\n %% TODO\n %% - SimGrid suppose que les consommations réseau et cpu sont linéaires\n %% et indépendantes. Ce n'est pas si simple. Tension entre\n %% mesurer/injecter et modéliser/prédire.\n %% - Ça marche mais niveau d'expertise encore assez démentiel\n %% - Résolution/automatisation de pas mal de choses en terme de\n %% modélisation de plate-forme mais il y a encore beaucoup de\n %% travail. LibSimBLAS ?\n\n \\section{Contribution}%\n\n This thesis has contributed to the improvement of experimental reproducibility in high performance computing.\n\n In Part~\\ref{part:prediction}, we described a method for predicting the performance of MPI applications through\n simulation. Using Simgrid/SMPI simulator, we managed to emulate the High Performance Linpack benchmark at scale\n by applying only a few modifications to its source code. We compared several computation and communication\n models and showed the importance of modeling both the temporal and spatial variability of the platform. In a\n thorough comparison of the simulations with real executions, we show that the prediction error remains very low,\n only a few percent, thereby demonstrating the faithfulness of this approach. Several sensibility analyses are\n then performed to quantify the effect of platform variability and showcase an important use case of simulation.\n\n The lessons learned during this work are then presented in Part~\\ref{part:experiment}. We start by describing the\n experiment engine we developed and that was used throughout this thesis. Then, we present an in-depth report of\n the many experimental biases we faced, including very unsettling phenomenons we did not anticipate. While some\n of these biases can be desirable if they are also occurring in the simulated application, most of them had to be\n suppressed through randomization. Finally, we showcase the performance non-regression test we implemented. While\n not a statistical novelty, they allowed us to detect many changes on Grid'5000 platform that affected\n significantly the performance and could harm experiments if gone unnoticed. We believe the HPC community could\n greatly benefit of such tests.\n\n \\section{Trusting our predictions}%\n \\label{sec:prediction_trust}\n\n Unlike mathematical theorems or algorithms, the correctness of a model like those of Part~\\ref{part:prediction}\n cannot be formally proven. The only sound method for validating its faithfulness is to thoroughly try to break\n it, by comparing predictions to reality while methodically changing the configurations. As presented in\n Chapter~\\ref{chapter:prediction:validation}, we did cover an extensive range of configurations in our\n validation. Before managing to systematically obtain accurate predictions, we stumbled on many problems which\n caused our predictions to be unrealistic. We had to investigate, understand and overcome these multiple\n obstacles, as reported in Part~\\ref{part:experiment}.\n\n A few weeks before the defense of this thesis, we decided to repeat the whole simulation study from scratch on\n another Grid'5000 cluster named \\gros, using 60 of its nodes. This cluster has different nodes (one Intel Xeon\n Gold 5220 processor per node with \\NSI{96}{\\giga\\byte} of memory), a different network (with\n \\NSI{25}{\\giga\\bit/\\second} Ethernet) and we used a more recent version of OpenBLAS (resulting in the use of\n AVX512 instructions by the \\dgemm function instead of AVX2). Over the course of a weekend, we calibrated the\n platform by measuring the \\dgemm and MPI durations. Then, we performed real and simulated HPL executions. At\n first, the predicted performance was too low by one order of magnitude. This was due to a mistake we made: we\n computed the linear regression of the \\dgemm model using all the terms of the polynomial, even those of degree\n one (\\ie \\(M\\), \\(N\\) and \\(K\\)) which were not statistically significant. This resulted in an overfitted model\n with spurious predictions. After fixing this issue by only considering the significant terms, we obtained\n extremely accurate predictions of HPL performance for various matrix sizes (see\n Figure~\\ref{fig:conclusion:gros_study}).\n\n \\begin{figure}[htpb]\n \\centering\n \\includegraphics[width=\\linewidth]{img/conclusion/validation_gros.pdf}\n \\caption{HPL performance: prediction vs. reality for various matrix ranks, using 60 nodes of the \\gros\n cluster.}%\n \\label{fig:conclusion:gros_study}\n \\end{figure}\n\n Similarly to what we observed in Chapter~\\ref{chapter:prediction:validation}, the predictions are more accurate\n with a complex \\dgemm model (\\ie heterogeneous, stochastic and polynomial) than with a simpler model (\\ie\n homogeneous, deterministic and linear). However, the simpler model still makes reasonably low prediction errors\n here as the \\gros cluster has less variability than the \\dahu cluster.\n\n This small additional study demonstrates clearly the level of trust we can now have in our model, but also its\n fragility, as seemingly inoffensive changes in the approach can lead to widely inaccurate predictions.\n\n \\section{Future work}%\n\n Besides the next steps already discussed in Chapter~\\ref{chapter:prediction:conclusion} and\n Section~\\ref{sec:test:conclusion}, there also remains a unification work. By going a step further in the\n automation, we could use the data produced by the non-regression tests to generate new model instances for\n Simgrid. It would then be possible to make new simulations with a platform model that reflects the latest\n changes of the real platform. Then, by implementing the same statistical test on the performance predicted by\n the simulation, we should be able to detect whether a platform change has affected the application\n performance and quantify this effect. A minor performance drop of the computation kernels could be amplified by\n the synchronization phases of the application and become very concerning at a larger scale. Conversely, it could\n also be attenuated by the global noise and go completely unnoticed.\n\n We believe that our predictions could be extremely valuable to the whole life cycle of supercomputers:\n \\begin{description}\n \\item[Design] Using simulations, manufacturers could apply co-design techniques to construct the most\n performant machines for a given set of target applications and within a given budget. This could help\n achieve more faithful results than the current techniques relying on less accurate simulations or even\n guesswork.\n \\item[Development] Simulations could also largely benefit software developers. Both debugging and tuning the\n application are more convenient and less expensive in simulation than in reality, especially if large\n scale runs are required.\n \\item[Maintenance] Whenever the platform employees need to perform some maintenance, there is a\n non-negligible risk of affecting the machine performance, as we illustrated in\n Chapter~\\ref{chapter:experiment:tests}. To verify that the performance did not change, the usual method\n is to perform large-scale runs of a benchmark such as HPL, which can largely lengthen the maintenance\n duration. A more convenient alternative would be to (1) carry small but carefully designed performance\n tests as those described in Chapter~\\ref{chapter:experiment:tests} to check if there is anything\n obviously wrong, and (2) perform large-scale simulations with the updated model and compare the new\n predictions with the previous ones.\n \\end{description}\n", "meta": {"hexsha": "6471f6903fdd7f380a40e6beacf089b4de7a6bff", "size": 8148, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapter_conclusion.tex", "max_stars_repo_name": "Ezibenroc/thesis", "max_stars_repo_head_hexsha": "62c239406d5979f1c9aabf180c8232ce2f2b1ee7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapter_conclusion.tex", "max_issues_repo_name": "Ezibenroc/thesis", "max_issues_repo_head_hexsha": "62c239406d5979f1c9aabf180c8232ce2f2b1ee7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapter_conclusion.tex", "max_forks_repo_name": "Ezibenroc/thesis", "max_forks_repo_head_hexsha": "62c239406d5979f1c9aabf180c8232ce2f2b1ee7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 78.3461538462, "max_line_length": 121, "alphanum_fraction": 0.7357633775, "num_tokens": 1650, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.34876782663137174}} {"text": "\\documentclass{report}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath,mathpazo,siunitx,xparse,tikz,enumitem,fancyhdr,pgfplots}\n\n%%Needed to properly display graphs\n\\pgfplotsset{width=10cm,compat=1.9}\n\n\\pagestyle{fancy}\n\\fancyhf{}\n\\lhead{Steven Glasford}\n\\chead{Homework 2}\n\\rhead{Page \\thepage}\n\n\\title{Homework 2}\n\\author{Steven Glasford}\n\\date{\\parbox{\\linewidth}{\\centering%\n %%Adds the last compiled date\n \\today\\endgraf\\medskip\n Math-451-M001}}\n\n%%Creates a new symbol for plus and minus together\n\\newcommand{\\rpm}{\\sbox0{$1$}\\sbox2{$\\scriptstyle\\pm$}\n \\raise\\dimexpr(\\ht0-\\ht2)/2\\relax\\box2 }\n\n%%Creates a nice format for displaying the steps taken \n\\newlist{steps}{enumerate}{1}\n\\setlist[steps, 1]{label = Step \\arabic*:}\n\n\\ExplSyntaxOn\n%%new command to round numbers\n\\newcommand*{\\prlen}[1]{%\n % round to 1 digit:\n \\pgfmathparse{round(10)/10.0}%\n %\\pgfkeys{/pgf/number format/precision=1}\n %\\pgfmathresult\n \\pgfmathprintnumber[fixed, precision=2]{\\pgfmathresult}\n}\n\\ExplSyntaxOff\n\n\n\\begin{document}\n%% adds the title to the document\n\\maketitle\n%% adds the table of contents to the document\n\\tableofcontents\n\n%%Add everything about the first problem here\n\\chapter{Hydro-Turbine Optimization}\n\\section{Introduction}\nA paper company in Maine operates a hydroelectric generating station on the Penobscot River. Water is piped through from a dam to the power station. The rate at which the water flows through the pipe varies, depending on external conditions. The power station has three different hydroelectric turbine, each with a known (and unique) power function that gives the amount of electric generated as a function of the water flow arriving at the turbine. The incoming water can be apportioned in different volumes to each turbine, so the goal is to determine how to distribute water among the turbines to give the maximum total energy production for any rate of flow. Using experimental evidence and Bernoulli's equation, the following quadratic models were determined for the power output of each turbine, along with the allowable flows of operation:\n\\begin{align*}\n KW_1 &= (-18.89+0.1277Q_1-\\num{4.08e-5}Q_1^2)(170-\\num{1.6e-6}Q_T^2)\\\\\n KW_2 &= (-24.5+0.1358Q_2-\\num{4.69e-5}Q_2^2)(170-\\num{1.6e-6}Q_T^2)\\\\\n KW_3 &= (-27.02+0.1380Q_3-\\num{3.84e-5}Q_3^2)(170-\\num{1.6e-6}Q_T^2)\n\\end{align*}\n$$250 \\leq Q_1 \\leq 1110, 250 \\leq Q_2 \\leq 1110, 250 \\leq Q_3 \\leq 1225$$\nwhere\n\n$Q_i = $ flow through turbine $i$ in cubic feet per second\n\n$KW_i = $ power generated by turbine $i$ in kilowatts\n\n$Q_T = $ total flow through the station in cubic feet per second\n\n\n\\section{Presentation of the Model}\n\\subsection{Objective}\n\\hspace{10mm} The objective of this problem is to determine the flow through each turbine needed to create the maximum amount of power.\n\\subsection{Variables and Constants}\n\n\\begin{center}\n\\begin{tabular}{c c c}\n$KW_i$ &$=$ & power generated by turbine $i$ in kilowatts.\\\\\n$Q_i$ & $=$ & flow through turbine $i$ in cubic feet per second.\\\\\n$Q_T$ & $=$ & total flow through the station in cubic feet per second.\\\\\n$P$ & $=$ & total power generated.\n\n\\end{tabular}\n\\end{center}\n\n\n\\subsection{Assumptions}\n\\begin{align*}\n KW_1 &= (-18.89+0.1277Q_1-\\num{4.08e-5}Q_1^2)(170-\\num{1.6e-6}Q_T^2)\\\\\n KW_2 &= (-24.5+0.1358Q_2-\\num{4.69e-5}Q_2^2)(170-\\num{1.6e-6}Q_T^2)\\\\\n KW_3 &= (-27.02+0.1380Q_3-\\num{3.84e-5}Q_3^2)(170-\\num{1.6e-6}Q_T^2)\n\\end{align*}\n\\begin{align*}\n 250 &\\leq Q_1 \\leq 1110\\\\\n 250 &\\leq Q_2 \\leq 1110\\\\\n 250 &\\leq Q_3 \\leq 1225\n\\end{align*}\n$$Q_T=Q_1+Q_2+Q_3$$\n$$P = KW_1+KW_2+KW_3$$\n\n\\section{Solving the Model}\nWe will be using Lagrange Multipliers to find the values for the individual flow (as functions of $Q_T$) that maximize the total energy production $KW_1+KW_2+KW_3$ subject to the constraints $Q_1+Q_2+Q_3=Q_T$ and the domain restrictions on each $Q_i$.\n\nBut first we must find the objective function, which is the function we are trying to optimize. In this case it is the function $P$.\n$$P = KW_1+KW_2+KW_3$$\nThen we must populate this function using the data we are concerned about, we are not concerned with the optimum power generated with respect to the total power outputs of each individual turbine, instead we want to optimize with respect to the flow rates to each individual turbine. This makes the power function look more like this\n\n\\begin{multiline}\n$P = (-18.89+0.1277Q_1-\\num{4.08e-5}Q_1^2)(170-\\num{1.6e-6}Q_T^2) + (-24.5+0.1358Q_2-\\num{4.69e-5}Q_2^2)(170-\\num{1.6e-6}Q_T^2) + (-27.02+0.1380Q_3-\\num{3.84e-5}Q_3^2)(170-\\num{1.6e-6}Q_T^2)$\n\\end{multiline}\n\nLagrange Multipliers have the form:\n$$\\nabla f = \\lambda \\nabla g,$$\n\nwhere $f$ is our objective function and $g$ is the constraint equation. In this problem our constraint equation is $$Q_T = Q_1 + Q_2 + Q_3$$ and $$f = P,$$ but before we continue we must replace the all of the $Q_T$ in the objective equation with the variable $Q_1, Q_2, Q_3$ which isn't difficult since the constraint equation already does this step for us.\n\n\\vspace{5mm}\n\n\\begin{tabular}{c c}\n$P =& (-18.89+0.1277Q_1-\\num{4.08e-5}Q_1^2)(170-\\num{1.6e-6}* \\\\ \n&(Q_T)^2)+ (-24.5+0.1358Q_2-\\num{4.69e-5}Q_2^2)* \\\\ \n&(170-\\num{1.6e-6}(Q_T)^2) \\\\\n&+ (-27.02+0.1380Q_3-\\num{3.84e-5}Q_3^2)*\\\\\n&(170-\\num{1.6e-6}(Q_T)^2)$\n\\end{tabular}\n\n%(170-w^2/625000)*(-(51*x^2)/1250000+(1277*x)/10000-1889/100)+(170-w^2/625000)*(-(3*z^2)/78125+(69*z)/500-1351/50)+(170-w^2/625000)*(-(469*y^2)/10000000+(679*y)/5000-49/2)\n\n\\vspace{5mm}\n\nWe shall try to find the extrema for P the old fashion way, by simply finding the gradient of P, then setting the gradient to 0 and finding min and maxes. To make our lives easier we will assume that $Q_T$ is a constant, which will end up making life much, much easier in the long run.\n$$\\nabla P = \\left\\langle\\frac{\\partial P}{\\partial Q_1},\\frac{\\partial P}{\\partial Q_2},\\frac{\\partial P}{\\partial Q_3}\\right\\rangle$$\n\nNow we must find the partials:\n\n$$\\frac{\\partial P}{\\partial Q_1} = \\dfrac{\\left(Q_T^2-106250000\\right)\\left(102Q_1-159625\\right)}{781250000000}$$\n\n$$\\frac{\\partial P}{\\partial Q_2}=\\dfrac{7\\left(Q_T^2-106250000\\right)\\left(67Q_2-97000\\right)}{3125000000000}$$\n\n$$\\frac{\\partial P}{\\partial Q_3} = \\dfrac{3\\left(Q_T^2-106250000\\right)\\left(8Q_3-14375\\right)}{195312500000}$$\n\n\\vspace{5mm}\n\nThe roots of P:\n$$\\nabla P = \\left\\langle \\dfrac{159625}{102},\\dfrac{97000}{67},\\dfrac{14375}{8}\\right\\rangle$$\nOr more approximately:\n$$\\nabla P \\approx \\left\\langle1564.95098,1447.76119,1796.875\\right\\rangle$$\n\nHowever, this extrema is outside our bounds for all of the variables so we cannot use this method of finding the gradient and the roots of the gradient. So we must now use Lagrange multipliers. \n\n\n\n\\vspace{5mm}\nLagrange multipliers are in the form $$\\nabla f = \\lambda \\nabla g,$$ where $f$ is our objective equation ($P$) and $g$ ($Q_T$) is out constraint equation and $\\lambda$ is our Lagrange multiplier. \n\nWe have already found $\\nabla P$ so we must now find the gradient of $Q_T$ The gradient of $Q_T$ \n$$\\nabla Q_T = \\left\\langle\\frac{\\partial Q_T}{\\partial Q_1},\\frac{\\partial Q_T}{\\partial Q_2},\\frac{\\partial Q_T}{\\partial Q_3}\\right\\rangle$$ which boils down to $$\\nabla Q_T = \\left\\langle1,1,1\\right\\rangle$$\n\nNow we can set the two equations equal to each other:\n\n$$\\frac{\\partial P}{\\partial Q_1} = \\dfrac{\\left(Q_T^2-106250000\\right)\\left(102Q_1-159625\\right)}{781250000000}=\\lambda$$\n$$\\frac{\\partial P}{\\partial Q_2}=\\dfrac{7\\left(Q_T^2-106250000\\right)\\left(67Q_2-97000\\right)}{3125000000000}=\\lambda$$\n$$\\frac{\\partial P}{\\partial Q_3} = \\dfrac{3\\left(Q_T^2-106250000\\right)\\left(8Q_3-14375\\right)}{195312500000} = \\lambda$$\n\nand $$Q_T=Q_1+Q_2+Q_3$$\n\nNow we want to solve for $\\lambda$\n\n$$Q_1 = \\frac{15925Q_T+781250000000\\lambda -1692031250000}{102Q_T-10837500000}$$\n$$Q_2 = \\frac{679000Q_T + 3125000000000\\lambda - 72143750000000}{469Q_T^2-49831250000}$$\n$$Q_3 = \\frac{43125Q_t^2+195312500000\\lambda - 4582031250000}{24Q_T^2-2550000000}$$\n\nPut these equations into the original $Q_T$ equation and solve for $\\lambda$\n\n$$\\lambda = \\frac{7\\left(2733Q_T-92963275\\right)\\left(Q_T^2-106250000\\right)}{4297851562500000}$$\n\nPlugging $\\lambda$ back into the original $Q_i$ equations and then simplifying we get:\n\n$$Q_1 = \\frac{7501Q_T}{22005}-\\frac{26500495}{26406}$$\n\n$$Q_2=\\frac{2176Q_T}{7335}+\\frac{1931560}{4401}$$\n\n$$Q_3=\\frac{7973Q_T}{22005}+\\frac{14911135}{26406}$$\n\nNow we graph the function between the bounds of each variable and try to find some intersection points between the $Q_i$ functions:\n\n\\begin{tikzpicture}\n \\begin{axis}[\n axis lines = left,\n ylabel = Flow,\n ]\n \n \\addplot[\n domain=225:1225,\n samples=100,\n color=green,\n ]{abs(-.3408770734*x+1003.57854276)};\n \\addlegendentry{$\\left|Q_1\\right|$}\n \\addplot [\n domain=225:1225,\n samples=100,\n color=red,\n ]{(.29665985003)*x+438.8911611};\n \\addlegendentry{$Q_2$}\n \n \\addplot[\n domain=225:1225,\n samples=100,\n color=blue,\n ]{(.36232674392)*x+564.687381656};\n \\addlegendentry{$Q_3$}\n \n \\end{axis}\n \\end{tikzpicture}\n \n % Then the optimized version of this model should be at $$790.827$$ $$Q_2=701.653$$ $$Q_3=746.24$$ for a total power generation of 16622.69864 Kilowatts.\n \n % But then I started to think about it. Logically if you have a turbine, that turbine should generate the most amount of power when it is doing the most amount of work, which would logically place the max value at the max boundary for each individual turbine, which means the max values should be at $$Q_1=1110$$ $$Q_1=1110$$ $$Q_3=1225$$ and if you put those values into the calculator you get an even higher value than you would get from the the other values. In particular you get a total power generation of 23155.92823 kilowatts, which is obviously greater than 16622.69864, so this is a bigger maximum, and a very obvious maximum.\n\n%$$Q_1 = -z-y \\pm %2500\\cdot\\sqrt{17}$$\n\n%and\n%$$Q_1 = \\dfrac{\\sqrt{2}\\sqrt{-19584Q_3^2+70380000Q_3-23919Q_2^2+69258000Q_2+15051181250} \\pm 319250}{204}$$\n\n%$$Q_2 = -Q_3-Q_1 \\pm 2500\\cdot\\sqrt{17}$$\n%and \n%$$Q_2 = \\dfrac{2\\cdot\\sqrt{14}\\sqrt{-3216Q_3^2+11557500Q_3-3417Q_1^2+10694875Q_1+2336037500} \\pm 679000}{469}$$\n\n%$$Q_3 = -Q_2-Q_1 \\pm 2500\\cdot\\sqrt{17}$$\n%and\n%$$Q_3 = \\dfrac{\\sqrt{6}\\sqrt{-469Q_2^2+1358000Q_2-408Q_1^2+1277000Q_1+535743750}\\pm 86250}{48}$$\n\n%As you may have noticed, each axis has a root something with $\\pm 2500\\cdot\\sqrt{17}$, these would be nice to use if they pass all of the constraints of the problem, however, they fail $Q_T=Q_1+Q_2+Q_3$ since you can rearrange any of those equations to be in the form $$\\pm 2500\\cdot\\sqrt{17} = Q_1 + Q_2 + Q_3$$ So these equations cannot be used if you consider that $750 \\leq Q_T \\leq 3445$, which are the combined bounds of $Q_1,Q_2,Q_3$ and $\\pm 2500\\cdot\\sqrt{17}$ is far from being in that bound.\n\n%%$$P = (-18.89+0.1277Q_1-\\num{4.08e-5}Q_1^2)(170-\\num{1.6e-6}Q_T^2) + (-24.5+0.1358Q_2-\\num{4.69e-5}Q_2^2)(170-\\num{1.6e-6}Q_T^2) + (-27.02+0.1380Q_3-\\num{3.84e-5}Q_3^2)(170-\\num{1.6e-6}Q_T^2)$$\n\n%%$$P = (-18.89+0.1277x-4.08*10^-5*x^2)(170-1.6*10^-6*(x+y+z)^2) + (-24.5+0.1358y-4.69*10^-5*y^2)(170-1.6*10^-6*(x+y+z)^2) + (-27.02+0.1380z-3.84*10^-5*z^2)(170-1.6*10^-6*(x+y+z)^2)$$\n\n%\\section{Discussion of Results}\n%\\subsection{Verification}\n%To verify that this is in fact the optimal flow rate to each turbine we will try some nearby distributions.\n\n%If we try the points (1110,1110,1224), (1109,1110,1225), (1110,1109,1225), (1109,1109,1224) which are all points a little less than the max we get the following distributions\n\n%\\begin{center}\n%\\begin{tabular}{c c}\n% $P(1110,1110,1224)=$& 23150.97440\\\\\n%% $P(1109,1110,1225)=$& 23157.57594\\\\\n% $P(1110,1109,1225)=$& 23152.82133\\\\\n% $P(1109,1109,1224)=$& 23149.51208\n%\\end{tabular}\n%\\end{center}\n\n%all of these points are less than the max, making the (1110,1110,1225) a max\n\n\\section{More Information}\n\n\\subsection{Turbine Distribution for a Predetermined Max Flow Rate}\nIf the incoming flow never goes beyond 2500 cubic feet per second, the flow must be distributed accordingly to maximize the power generated. As shown in the graph, $Q_2$ maxes at 701.653 and $Q_3$ at 74.24, we then allow the remaining water to flow through $Q_1$ this will result in the max for the situation.\n\n\n\\subsection{More Information}\n\n\\hspace{10mm} \\emph{Turbines to be used if only two are allowed to operate}\n\n\nIf we were to have 1500 cubic feet per second, I would recommend using the $Q_2$ and $Q_3$ due to these ones generating the most amount of power individually.\n\n\\subsection{Recomendation of When the Flow Rate is Near the Hypothetical Maximum Flow Rate of Each Turbine}\n\nIf there was an incoming flow rate of 3400 cubic feet per second I would recommend allowing the maximum allowed flow through both $Q_2$ and $Q_3$, and then allowing the rest of the water to go through $Q_1$, this would generate the greatest amount of power for the situation.\n\n\\section{Conclusion}\nOverall, the best point for the water distribution is when $Q_1 \\approx 790.827$, $Q_2 \\approx 701.53$ and $Q_3 = 24.24$ for $\\approx$ of 16622.984 kilowatts.\n\n\n\\vspace{5mm}\n\n\\hspace{5mm} Let's say that the company discovers that it is sometimes advantageous to only use two turbines, and the incoming flow is 1500 cubic feet per second, the company should use the turbine x and turbine y to generate the most power\n\\vspace{5mm}\n\n\\hspace{5mm} If the incoming flow is 3400 cubic feet per second, the company should take the following action.\n\n\n%%Add everything about the second problem here\n\\chapter{Advertising Budget Analysis for a Personal Computer Manufacturer}\n\\section{Introduction}\n\\emph{This is a problem can be found on page 52 number 6 of \\textit{Mathematical Modeling, Fourth Edition} by Mark M. Meerschaert.}\n\n\\vspace{5mm}\nA manufacturer of personal computers currently sells 10,000 units per month of a basic model. The cost of manufacture is \\$700/unit, and the wholesale price is \\$950. During the last quarter the manufacturer lowered the price \\$100 in a few test markets, and the result was a 50\\% increase in sales. The company has been advertising its product nationwide at a cost of \\$50,000 per month. The advertising agency claims that increasing the advertising budget by \\$10,000/month would result in a sales increase of 200 units/month. Management has agreed to consider an increase in the advertising budget to no more than \\$100,000/month.\n\n\\section{Presentation of the Model}\n\\subsection{Objective}\nWill try to find a price and the advertising budget size that will maximize the total revenue for this manufacturing company.\n\n\\subsection{Variables and Constants}\n\\begin{tabular}{c c}\n $R= $ & revenue (\\$)\\\\\n $P= $ & profit (\\$)\\\\\n $C= $ & costs (\\$)\\\\\n $U= $ & units of personal computers\\\\\n $B= $ & advertising budget (\\$/month)\\\\\n $S= $ & selling wholesale price per unit\\\\\n $D = $& price drop\\\\\n $B_i = $& budget increase\\\\\n $C_m=700=$ & cost to manufacture per unit (in dollars) \\\\\n $B_0=50000=$ & initial advertising budget size\\\\\n $B_m=100000=$ & max advertising budget\\\\\n $U_0 = 10000=$ & initial number of units sold\\\\\n $S_0 = 950=$ &initial wholesale price\n\\end{tabular}\n\n\\subsection{Assumptions}\n\\begin{center}\n\\begin{tabular}{c}\n $P=R-C$ \\\\\n $p=S_0-D$\\\\\n $R = UP$\\\\\n $C= B + C_m*U$\\\\\n $B = B_0 + B_i$\\\\\n $U=U_0+(950-S)*50 + \\frac{b-B_0}{U_0}*200$\\\\\n $B_0 \\leq B \\leq B_m$\\\\\n $700\\leq P \\leq 950$\n \n\\end{tabular}\n\\end{center}\n\n\\emph{objective:} Maximize Profits\n\n%Notice how the equation for $U$ has two main components, the next component is the increase of sets sold as the budget for advertising increases which is a linear component since for every dollar increase the number of sets grows at a known amount, in this case by 200. Then there is a quadratic function with very interesting, seemingly random coefficients. This part of the equation was found using a calculator doing linear regressions on the points:\n%\\begin{center}\n%\\begin{tabular}{c}\n % (950,10000)\\\\\n % (850,15000)\\\\\n % (750,22500)\n%\\end{tabular}\n%\\end{center}\n\n%I chose those points due to the 50\\% increase of sets purchased as a \\$100 decrease in price. I then put these points into the calculator to find a regression between all of them. I could have chosen to use a more exact regression and more points (this would also reduce the overall amount of error in the model, however I felt that this regressed version might be easier to have a calculator use later on in the problem. I chose to do a quadratic regression because the correlation between the points and the function is 1. I could have used a cubic or quadratic regression with added points, however I felt that those would have added too much complexity.\n\n\\section{Solution to Model}\nIn order to proceed we need to obtain our objective function\n\n%We will model this as a constrained optimization problem, and solve it using the method of Lagrange multipliers.\n\n%First we must make a uniform objective equation, which uses the variables $B$ and $S$:\n\n$$P=R-C$$\n\n$$\\left(10000+(950-p)50+\\frac{B-50000}{10000}\\cdot200\\right)p-\\left(B-\\left(10000+(950-p)50+\\frac{B-50000}{10000}\\cdot200\\right)700\\right)$$\n\nwhich reduces to: $$P=\\left(10000+(950-p)50+\\frac{B-50000}{10000}\\cdot 200\\right)(p-700)-B$$\n\nWithout showing it, we have determined that this problem cannot be solved by setting the gradient to zero and solving for the roots, doing that just gives us values outside our bounds. So we shall use Lagrange Multipliers\n\n$$\\nabla P = \\left\\langle-100p+91500+.01B,.02p-15\\right\\rangle$$\n\nWe solve using Lagrange by $$\\nabla f = \\lambda \\nabla g$$\n\nwhere $\\nabla f$ is our objective equation (which is $P$) and $\\nabla g$ is our constraint value: \n\n$$-100p+91500+.02B = 0$$\n$$.02p-15 = \\lambda$$\n\nSolving this for when $B$ is at a minimum (10000) \n\n$$p=935$$\n$$\\lambda = 3.7$$\nand\n$$P = 261250$$\n\nSolving for when $B$ is at a maximum we get \n\n$$B = 100000$$\n$$P(950,10000) = 2650000$$\n\n\n\n\n\\section{Sensitivity Analysis}\n%This is part 2 of the original problem\nWe will now determine the sensitivity of the decision variables (price and advertising) to price elasticity.\n\nWe will assume that $e$ is our price elasticity, we then obtain an equation $$P(p,B,e) = (10000 +e(950-p) + .02(B-50000))(p-700)-B$$\n\nNow we take the corresponding partials from the equation above:\n\n\\begin{center}\n \\begin{tabular}\n $\\dfrac{\\partial P}{\\partial p}=-e(p-700)+9000+100e(950-p)+.02a$\\\\\n $\\dfrac{\\partial P}{\\partial B}=.02p-15$\n \\end{tabular}\n\\end{center}\n\nNow we use Lagrange Multipliers method to obtain the equation:\n$$\\frac{\\partial P}{\\partial p} = 0$$\n$$\\frac{\\partial P}{\\partial B}=\\lambda$$\n\nby using $B=100000$ we get the values of $p$ and $\\lambda$:\n\n$$P = \\dfrac{275(3D+20)}{e}$$\n$$\\lambda = \\frac{1}{2}\\left(\\dfrac{3e+220}{e}\\right)$$\n\nEvaluating when $p=935$, $B=100000$ (our max point), $e=50$ \n$$S(p,e) = \\dfrac{dp}{de}\\cdot \\dfrac{e}{p}=-.12$$\n\n$$S(B,e)=\\dfrac{dB}{de}\\cdot \\dfrac{e}{p}$$\n\nWhat this means is that a 10\\% increase in the price elasticity the company would notice a 1.2\\% decrease in price but that the advertising budget would not be affected.\n\n\\vspace{5mm}\n%This is part 3 of the original book problem\n\\emph{The advertising agency estimates that the company will gain 200 new sales each time the advertising budget is increased by \\$10,000 per month. The sensitivity analysis is the following:}\n\nThis new assumption gives us the following equations:\n\n$E = $ estimate of new sales per \\$10,000 increase\n\n\\begin{center}\n \\begin{tabular}{c}\n $P(p,B,E)=(10000+50(950-p) + \\dfrac{E}{10000}(B-50000))(p-700)-B$\\\\\n $\\dfrac{\\partial P}{\\partial p}=-100p+91500+\\dfrac{E}{10000}(B-50000)$\\\\\n $\\dfrac{\\partial P}{\\partial B} = \\dfrac{E}{10000}(p-700)-1$\n \\end{tabular}\n\\end{center}\n\nBy once again using the Lagrange multiplier method, we set the first partial to zero, and the second to 1; this gives us the following:\n\n$$p = .25 +.05E$$\n$$B=100000$$\n$$\\lambda = \\dfrac{9E}{400}+\\dfrac{E^2}{20000}-1$$\n\nUsing the max point $p=935$ and $B=100000$ as well as using the point $E=200$ we obtain:\n\n$$S(p,e)=\\dfrac{dp}{de}\\cdot \\dfrac{e}{p}=-.12$$\n\n$$S(B,e)=\\dfrac{dB}{de}\\cdot \\dfrac{e}{B}=0$$\n\nWhat this means is that when a 10\\% increase in the price elasticity would result in a 1.2\\% decrease in the profit, and the advertising budget would remain.\n\n\\section{Conclusion}\n%need to describe what \\lambda is. This is part 4 of the original problem\nIn conclusion, the best price to sell the units are at \\$935 and the best size for the advertising budget is \\$100000, this would give the company a total profit of about \\$2650000.\n\nWith or without the price elasticity we discovered that a 10\\% increase in price would result in a 1.2\\% decrease in price, with no affect on the advertising budget.\n\n\n\\end{document}", "meta": {"hexsha": "9931fe5034b2eb93da81c27611b9a8b6df23c460", "size": 21025, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "HW2.tex", "max_stars_repo_name": "stevenglasford/MATH451SDSMT", "max_stars_repo_head_hexsha": "c0fa475cb08a40debda6c106e8ddf8b44dab1060", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "HW2.tex", "max_issues_repo_name": "stevenglasford/MATH451SDSMT", "max_issues_repo_head_hexsha": "c0fa475cb08a40debda6c106e8ddf8b44dab1060", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "HW2.tex", "max_forks_repo_name": "stevenglasford/MATH451SDSMT", "max_forks_repo_head_hexsha": "c0fa475cb08a40debda6c106e8ddf8b44dab1060", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.3536036036, "max_line_length": 846, "alphanum_fraction": 0.7076337693, "num_tokens": 7042, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3487603565866375}} {"text": "% !TeX spellcheck = en_US\n% !TeX root = DynELA.tex\n%\n% LaTeX source file of DynELA FEM Code\n%\n% (c) by Olivier Pantalé 2020\n%\n\\chapter{Theory}\n\n\\startcontents[chapters]\n\\printmyminitoc[1]\\LETTRINE{T}he \\DynELA~is an Explicit FEM code written in \\Cpp~using a Python's interface for creating the Finite Element Models. \n\n\\section{Introduction}\n\n\\begin{tcolorbox}[width=0.95\\textwidth,myTab,tabularx={l||C|C|C|C|C},title=Homogeneous system of units in mechanics]%,boxrule=0.5pt]\n & Dimension & SI (MKS) & SI (mm-s) & SI (mm-kg) & SI (mm-kg-s) \\\\\\hline\\hline\nLength & $L$ & $m$ & $mm$ & $mm$ & $mm$ \\\\\\hline\nWeight & $M$ & $kg$ & $T$ & $kg$ & $kg$ \\\\\\hline\nTime & $T$ & $s$ & $s$ & $ms$ & $s$ \\\\\\hline\nStress & $ML^{-1}T^{-2}$ & $Pa$ & $MPa$ & $GPa$ & $kPa$ \\\\\\hline\nForce & $MLT^{-2}$ & $N$ & $N$ & $kN$ & $mN$ \\\\\\hline\nEnergy & $ML^{2}T^{-2}$ & $J$ & $mJ$ & $J$ & $MJ$ \\\\\\hline\nDensity & $ML^{-3}$ & $kg/m^{3}$ & $T/mm^{3}$ & $kg/mm^{3}$ & $kg/mm^{3}$ \\\\\\hline\nSpeed & $LT^{-1}$ & $m/s$ & $mm/s$ & $mm/ms$ & $mm/s$ \\\\\\hline\nAngular speed & $T^{-1}$ & $s^{-1}$ & $s^{-1}$ & $ms^{-1}$ & $s^{-1}$ \\\\\\hline\nAcceleration & $LT^{-2}$ & $m/s^{2}$ & $mm/s^{2}$ & $mm/ms^{2}$ & $mm/s^{2}$ \\\\\\hline\nCouple & $ML^{2}T^{-2}$ & $N.m$ & $N.mm$ & $kN.mm$ & $mN.mm$\n\\end{tcolorbox}\n", "meta": {"hexsha": "1cda20a9edb5d7abaf98ea2c9e3d60d359f67488", "size": 1638, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Documentation/Theory-Introduction.tex", "max_stars_repo_name": "pantale/DynELA-v.-4.0", "max_stars_repo_head_hexsha": "2c366859b68df6243a1e128a7839e4fb23888820", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-12-14T20:12:21.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-21T14:47:50.000Z", "max_issues_repo_path": "Documentation/Theory-Introduction.tex", "max_issues_repo_name": "pantale/DynELA-v.-4.0", "max_issues_repo_head_hexsha": "2c366859b68df6243a1e128a7839e4fb23888820", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Documentation/Theory-Introduction.tex", "max_forks_repo_name": "pantale/DynELA-v.-4.0", "max_forks_repo_head_hexsha": "2c366859b68df6243a1e128a7839e4fb23888820", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 56.4827586207, "max_line_length": 148, "alphanum_fraction": 0.4145299145, "num_tokens": 629, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3487603565866375}} {"text": "\\section{FRP}\n\nThe most common solution for the construction of reactive systems in an imperative setting are call-back frameworks, embedded into a loop.\nThe call-backs are either used to query the state of variables, or to change them.\nThis imperative approach is well suited for rapid prototyping of small systems.\nHowever, tracing behaviors over time quickly becomes unmanageably complex for larger systems.\n\nFunctional Reactive Programming instead introduces a concrete abstraction of time that allows the programmer to safely manipulate time-varying values. \nThe key abstraction is given by a \\textit{signal}, providing the programmer with a simple type interface:\n\n\\begin{lstlisting}\n type Signal a = Time -> a\n\\end{lstlisting}\n\nFor example, the type \\texttt{Signal Image} represents a video, while \\texttt{Signal Steer} captures a steering wheel operated over time.\nTo better understand how our library works, we now introduce the basic concepts and terminology from FRP.\n\n\\subsection{Arrowized FRP}\n\nThere are many types of FRP based on different abstractions from type theory.\nExpressive abstractions, such as monads, allow for complex manipulation of signal flows~\\cite{van2014monadic}. \nHowever, for most applications they are far too expressive.\nWe instead focus on an FRP library, Yampa, which uses the arrow abstraction, or so called Arrowized FRP~\\cite{hudak2003arrows}.\nArrows generally run faster and with little need for manual optimization~\\cite{yallop2016causal}, but are fundamentally less expressive than a monadic FRP~\\cite{lindley2011idioms}.\nThis more restrictive language is in fact a benefit, as it makes it harder for the programmer to introduce errors.\nAs we will see in the sequel, Yampa is still powerful enough to write complex controllers to drive an autonomous vehicle, or even to communicate with other vehicles.\nAt the same time, the syntax is clear and accessible enough to make for an easy introduction to the FRP paradigm.\n\nAlong with signals, Yampa also introduces the abstraction of a \\textit{signal function (SF)}.\nThis is a transformer from one signal to another.\n\n\\begin{lstlisting}\n type SF a b = Signal a -> Signal b\n\\end{lstlisting}\n\n\\noindent Using the previous signals, imagine a type for a steering function, which operates based on a video stream, such as\n\n\\begin{lstlisting}\n turn :: SF Image Steer\n\\end{lstlisting}\n\n\\noindent This function processes video and uses it to decide how to steer.\nWe omit an implementation, as the details of the data transformation are not relevant to the structure of the FRP code.\n\nHaskell provides special syntax for Arrowized FRP, which mimics the structure of control flow charts.\nThe syntax provides a composition environment, in which the programmer just manages the composition of arrow functions.\nInputs are read in from the right hand side, and piped to the left hand side (\\texttt{output <- function -< input}).\nA demonstration is given in Listing~\\ref{lst:arrows}.\n\nThe example introduces \n%\n\\begin{lstlisting}\n avoid :: (Image, Steer) -> Steer\n\\end{lstlisting}\n%\na pure function that adjusts the basic steering plan based on the image to avoid any obstacles.\nIn Listing~\\ref{lst:arrows}, this \\texttt{avoid} function is lifted to the signal level using:\n%\n\\begin{lstlisting}\n arr :: (a -> b) -> SF a b\n\\end{lstlisting}\n%\nThe function \\texttt{turn} is already on the signal level (has an SF type). Hence, we do not need to lift it.\n\n\\begin{lstlisting}[float,floatplacement=h!,caption=Basic Arrowized FRP syntax,label=lst:arrows]\nmyDriver :: SF Image Steer\nmyDriver = proc image -> do\n basicSteer <- turn -< image\n adjustedSteer <- arr avoid -< (image, basicSteer)\n returnA -< adjustedSteer\n\\end{lstlisting}\n\n\n\\subsection{Stateful FRP}\n\nTo avoid obstacles on the road, we might write an \\texttt{avoid2} function as shown in Listing~\\ref{lst:loop}, which requires two images to calculate the adjusted steering command. \nFor this, we need a mechanism to maintain state between each processing step.\nTwo images would be necessary to filter noise in the image, or calculates the velocity of an approaching obstacle.\nTo implement it, we use an abstraction called \\texttt{ArrowLoop} to save the previous state of the image for the next processing step.\nThe syntax is presented in Listing~\\ref{lst:loop}.\nIntuitively, \\texttt{ArrowLoop} gives us a recursive computation, as also indicated by the \\texttt{rec} keyword\\footnote{We elide the technical details for the purposes of this presentation and refer the interested reader to~\\cite{paterson2001icfp}.}.\n\n\n\\begin{lstlisting}[float,caption=Using ArrowLoop to send feedback,label=lst:loop]\nmyDriver :: SF Image Steer\nmyDriver = proc image -> do\n rec\n oldI <- iPre null -< image\n basicSteer <- turn -< image\n adjustedSteer <- arr avoid2 -< (image, oldI, basicSteer)\n returnA -< adjustedSteer\n\\end{lstlisting}\n\nThe predefined function \\texttt{iPre} takes an initial state, in our case an empty image, and saves images for one time step, each time it is processed.\nThis way, we create a feedback loop that is then used in the updated \\texttt{avoid} function.\nAt the same time, the \\texttt{rec} keyword is used to denote a section of arrow code with mutual dependencies\\footnote{Without the keyword, there is an unresolvable dependency loop.}.\n\n\n\n \n", "meta": {"hexsha": "fc777ce0faf7b5ac237bac750dcde723933def2d", "size": 5330, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/SCAV17/secs/FRP.tex", "max_stars_repo_name": "santolucito/Haskell-TORCS", "max_stars_repo_head_hexsha": "95eef93f7089bbe95f0f28fa21f0f636d7ffc39f", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2017-03-08T14:58:48.000Z", "max_stars_repo_stars_event_max_datetime": "2020-09-14T16:49:06.000Z", "max_issues_repo_path": "paper/SCAV17/secs/FRP.tex", "max_issues_repo_name": "santolucito/Haskell-TORCS", "max_issues_repo_head_hexsha": "95eef93f7089bbe95f0f28fa21f0f636d7ffc39f", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/SCAV17/secs/FRP.tex", "max_forks_repo_name": "santolucito/Haskell-TORCS", "max_forks_repo_head_hexsha": "95eef93f7089bbe95f0f28fa21f0f636d7ffc39f", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.7722772277, "max_line_length": 251, "alphanum_fraction": 0.7782363977, "num_tokens": 1271, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102498375401, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3487603489660664}} {"text": "\\documentclass[lang=en,mode=geye,device=normal,color=blue,14pt]{elegantnote}\n\\usepackage{amsmath,amssymb}\n\\usepackage{bbm}\n\\usepackage{tcolorbox}\n\\usepackage{graphicx}\n\\usepackage{booktabs}\n \\usepackage{subfigure}\n\n\\DeclareMathOperator*{\\E}{\\mathbb{E}}\n\\DeclareMathOperator*{\\1}{\\mathbbm{1}}\n\\DeclareMathOperator*{\\R}{\\mathbbm{R}}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\n\\title{Note: Reinforcement Learning - An Introduction}\n\n%\\author{Yanqing Wu}\n%\\institute{Viwistar Robotics}\n\n% \\version{0.1.0}\n%\\date{\\today}\n\n\\begin{document}\n\\maketitle\n\n\\setlength{\\parindent}{0pt}\n\n\\newpage\n\n\\section{Chap 1: Introduction}\n\\subsection{Reinforcement Learning}\n\nRL is learning what to do, so as to maximize a numerical reward signal.\n\nRL means taking optimal action with \\textbf{long term} results or \\textbf{cumulative rewards} in mind.\n\nTwo most important distinguishing features of RL: trial-and-error search and delayed reward.\n\nRL methods generally incorporate 3 aspects: sensation, action and goal.\n\n\\begin{definition}\n\\textbf{Supervised Learning} is learning from a training set of labeled examples provided by a knowledgable external supervisor.\n\\end{definition}\n\n\\begin{itemize}\n\t\\item Example: a description of a situation\n\t\\item Label: a specification of the correct action the system should take in a situation, which is often to identify a category to which the situation belongs\n\t\\item Object of Supervised Learning: for the system to extrapolate (generalize) its responses s.t. it acts correctly in unseen situations\n\t\\item Con: inadequate for learning from \\textit{interaction}, because it is (often) impractical to obtain examples of desired behavior that are both correct and representative of all the situations in which the agent has to act\n\\end{itemize}\n\n\\begin{definition}\n\\textbf{Unsupervised Learning} is about finding structure hidden in collections of unlabeled data.\n\\end{definition}\nRL is trying to maximize a reward signal, while Unsupervised Learning is trying to find hidden structure.\n\nKey feature 1: the exploration-exploitation dilemma is one of RL's challenges.\nExploitation: to obtain reward from experienced.\nExploration: to find better actions.\n\nKey feature 2: RL explicitly considers the whole problem of a goal-directed agent interacting with an uncertain environment.\n\n\\subsection{Examples}\n\\subsection{Elements of RL}\n\nFour main subelements of a RL system: a \\textit{policy}, a \\textit{reward signal}, a \\textit{value function}, and a \\textit{model} of the environment.\n\n\\begin{definition}\n\\textbf{Policy}, defines the learning agent's way of behaving at a given time.\n\\end{definition}\n\nA policy is a mapping from perceived states of the environment, to actions to be taken, when in those states.\nA policy may be a simple function or a lookup table, or complex computation (e.g. a search process).\nPolicies may be stochastic, specifying probabilities for each action.\n\n\\begin{definition}\n\\textbf{Reward signal}, defines the goal of a RL problem.\n\\end{definition}\n\nOn each timestep, the environment sends a single number (i.e. \\textit{reward}) to the RL agent. The agent's only objective is to maximize the total reward it receives over the long run.\n\n\\begin{definition}\n\\textbf{Value function}, specifies what is good in the long run.\n\\end{definition}\n\nThe \\textit{value} of a state is the total amount of reward an agent can expect to acumulate over the future, starting from that state.\n\n!! \\textit{Rewards} determine the immediate desirability of states, \\textit{values} determines the long-term desirability of states.\n\nState Value function = Expected Return = Discounted sum of all rewards\n\n\\begin{definition}\n\\textbf{Model}, mimics the behavior of the environment.\n\\end{definition}\n\nThe model defines the reward function and transition probabilities. Models are used for \\textit{planning}.\nModel-based methods: use models and planning to solve RL problems.\nModel-free methods: explicitly trial-and-error learner (almost the opposite of planning).\n\nModern RL spans the spectrum from low-level, trial-and-error learning to high-level, deliberative planning.\n\n\\subsection{Limitations and Scope}\n\\subsection{An Extended Example: Tic-Tac-Toe}\nMinimax algorithm \\cite{wiki_minimax} (from game theory).\n\nClassical optimization methods are for sequential decision problems (e.g. dynamic programming). We need to estimate an approximate opponent model so to use dynamic programming; in such case, this is similar to RL methods.\n\nDuring playing, we need to adjust the values of the states as to make more accuracte estimates of the probabilities of winning. The current value of the earlier state is updated to be closer to the value of the later state. This can be done by moving the earlier state's value a fraction of the way toward the value of the later state.\n\n\\begin{equation}\nV(S_t) \\leftarrow V(S_t) + \\alpha[V(S_{t+1}) - V(S_t)]\n\\end{equation}\n\n$S_t$, the state before a move.\n$S_{t+1}$, the state after the move.\n$V(S_t)$, the update to the estimated value of $S_t$.\n$\\alpha$, step-size parameter, a small positive fraction, influences the rate of learning.\n\nThis update rule is an example of a temporal-difference (TD) learning method. TD means the changes are based on a difference between estimates at two consecutive times.\n\nIf $\\alpha$ is reduced properly over time, then this method converges (for any fixed opponent, it converges to the true probabilities of winning for each state); if $\\alpha$ is not reduced to zero over time, then the agent ``also plays well against opponents that slowly change their way of playing'' (?). My understanding is that, there must have some slow changes, whether the parameter or the opponent model.\n\n\\textbf{Evolutionary methods vs. Value Function methods.} Evolutionary methods ignores what happens during the games, and only the final outcome of each game is used (e.g. all of its behavior in a winning game is given credit). In contrast, value function methods evaluate individual states.\n\nIn this example, learning started with no prior knowledge except the rules of the game. In reality, prior information can be incorporated into RL in many ways for efficient learning.\nRL can be used when we have access to true state, or where some states are hidden, or when different states appear to be the same to the learner.\n\nThe player need a model of the game to see the result states in response to actions. While model-free systems cannot know how their environments will change in response to actions.\nModel-free methods are building blocks for model-based methods.\nSometimes, it is difficult to construct an accurate environment model, thus model-based methods cannot be used in such cases.\n\n\\subsection{Summary}\n\nRL is a computational approach to understanding and automating goal-directed learning and decision making.\nIt is distinguished from other computational approaches by its emphasis on learning by an agent from direct \\textbf{interaction} with its environment, without requiring exemplary supervision or complete models of the environment.\n\nRL uses the formal framework of Markov decision processes (MDP) to defined the interaction between a learning agent and its environment in terms of states, actions, and rewards.\n\nThe concepts of value and value function are key to most of the RL methods in this book.\nThe authors believe that value functions are critical for efficient search in policy space.\nThe use of value functions distinguishes RL methods from evolutionary methods.\n\n\\subsection{Early History of RL}\n\nThree threads: \n1. learning by trial and error, originated in the psychology of animal learning. Time from 1850s to early 1980s' revival of RL;\n2. optimal control problem and its solution using value functions and dynamic programming. This thread did not involve learning;\n3. temporal-difference methods.\nThese 3 threads came together in late 1980s.\nThe TD and optimal control threads were fully brought together in 1989 with Chris Watkin's development of Q-learning.\n\nThe essential idea of trial-and-error learning: `the drive to achieve some result from the environment, to control the environment toward desired ends and away from undesired ends.'\n\nTD learning methods are driven by the difference between temporally successive estimates of the same quantity.\n\nThe authors developed a method for using TD learning + trial-and-error learning, known as the \\textit{actor-critic architecture}.\n\n\\newpage\n\\textbf{Part I: Tabular Solution Methods}\n\nAlgorithms in this part are described in simplest forms: the state space and action space are small enough for the approximate value functions to be represented as arrays/tables.\nIn this case, the methods can (often) find exact solutions (i.e. optimal value function and optimal policy).\nAlgorithms in next part can only find approximate solutions but can be applied to much larger problems.\n\n\\textbf{Bandit problems}, RL problem with only a single state. Chap 2 describes general problem formulation (finite MDP) and its main ideas (including Bellman equations and value functions).\n\nThree fundamental classes of methods for solving MDP problems: dynamic programming, Monte Carlo methods and TD learning.\n\\begin{itemize}\n\\item DP: well developed, but require a complete and accurate model of the environment\n\\item MC: no model needed, but not good for incremental computation\n\\item TD: no model needed and fully incremental, but are complex\n\\end{itemize}\n\n\\newpage\n\\section{Chap. 2 Multi-armed Bandits}\n\\begin{itemize}\n\\item Evaluative: feedback depends solely on the taken action\n\\item Instructive: feedback is independent of the taken action\n\\item Nonassociative setting: no learning involved in more than one situation; no need to associate different actions with different situations/states\n\\item Associative setting: the best action depends on the situation\n\\end{itemize}\n\n\\subsection{A k-armed Bandit Problem}\n\n\\textit{k-armed bandit problem}:\nYou choose one option (action) from \\textit{k} different options (actions). The probability of choosing each option (action) is stationary. For each choice, you receive a (probably distinct) numerical reward. Your objective is to maximize the expected total reward over \\textit{j} option selections (time steps).\n\nFor now, let us define that\n\\begin{itemize}\n\\item \\textit{Value}: the expected/mean reward of a taken action\n\\item $A_t$: action selected on time step $t$\n\\item $R_t$: corresponding reward of $A_t$\n\\item $q_{*}(a)$: $q_{*}(a) \\doteq \\E[R_t|A_t=a]$. The (true) value of an arbitrary action $a$ (i.e. the expected reward given $a$ is selected)\n\\item $Q_t(a)$: the \\textit{estimated} value of action $a$ at time step $t$.\n\\end{itemize}\n\nIf we know each action values, then solving the \\textit{k}-armed bandit problem would be always select the action with highest $value$.\nIf we don't know the action values, then we can estimate them.\nOur goal is to have $Q_t(a)$ to be as close to $q_*(a)$ as possible.\n\n\\begin{definition}\n\\textbf{Greedy actions}, action(s) with highest \\textbf{estimated} value.\n\\end{definition}\n\\textit{exploiting}: select greedy action(s).\n\\textit{exploring}: select non-greedy action(s). Exploration may improve the estimate value of non-greedy actions.\n\n\\subsection{Action-value Methods}\n\n\\begin{definition}\n\\textbf{Action-value Methods}, or Q function, methods for estimating the values of actions, and using the estimates to select actions.\n\\end{definition}\n\n`Q' means the \\textbf{quality} of an action.\n\nLet see a simple action-value method: \\textit{sample-average method}, in which action-value is the mean reward for an action.\n\n$$ Q_t(a) \\doteq \\frac{\\sum_{i=1}^{t-1}R_i \\cdot \\1_{A_{i=a}}}{\\sum_{i=1}^{t-1}\\1_{A_{i=a}}}$$\n\n\\begin{itemize}\n\t\\item $\\1$: return 1 if predicate is true, otherwise 0\n\t\\item in the case when denominator is 0 (which makes the fraction mathematically undefined), $Q_t(a)$ is set as default value (e.g. 0)\n\t\\item in the case when denominator $\\rightarrow \\infty$, by the Law of Large Numbers, $Q_t(a)$ converges to $q_*(a)$\n\\end{itemize}\n\n\\begin{definition}\n\\textbf{Greedy action selection}, If there are more than one greedy action, select one of them arbitrarily.\n$$ A_t \\doteq \\argmax_{a} Q_t(a) $$\n\\end{definition}\n\n\\begin{definition}\n\\textbf{$\\varepsilon$-greedy action selection} with probability $\\varepsilon$, select an action randomly; with probability $1-\\varepsilon$, select a greedy action\n\\end{definition}\n\n\\subsection{The 10-armed Testbed}\n\nFor Exercise 2.2, note that we should first convert the sequence of actions and rewards to $Q_t(a)$, then we can decide whether the selection is greedy or exploration based on the definition of $\\varepsilon$-greedy action selection.\n\n\\subsection{Incremental Implementation}\n\nRecall that estimated action value $Q_n$, using sample-average method, is calculated as:\n$$ Q_n \\doteq \\frac{R_1 + \\cdots + R_{n-1}}{n-1} $$\n\nWe don't want to waste memory and computational resource for each new reward.\nIntuitively, to save memory, we can simplify the equation by first reconstruct the previous estimated action value $Q_{n-1}$ and then evaluate $Q_n$ with the new reward $R_n$.\nTherefore,\n$$ Q_{n+1} = \\frac{Q_n \\cdot (n-1) + R_n}{n} = \\frac{Q_n \\cdot n - Q_n + R_n}{n} = Q_n + \\frac{1}{n}[R_n - Q_n] $$\n\nThe form $Q_{n+1} = Q_n + \\frac{1}{n}[R_n - Q_n]$ is an important form occurs in this book. The general form is\n$$ NewEstimate \\leftarrow OldEstiamt + StepSize\\underbrace{[Target - OldEstimate]}_{error} $$\n\n% \\begin{tcolorbox}[width=\\textwidth,colback={grey},title={A simple bandit algorithm},colbacktitle=black,coltitle=white]}\n\\begin{tcolorbox}[width=\\textwidth,title={A simple bandit algorithm}]\n\t\\begin{align*}\n\t & \\text{Initialize, for a = 1 to }k: \\\\\n\t & \\quad Q(a) \\leftarrow 0 \\\\\n\t & \\quad N(a) \\leftarrow 0 \\\\\n\t & \\text{Loop forever: }\\\\\n\t & \\quad A \\leftarrow \\left\\{\n\t \\begin{array}{ll}\n\t \\argmax_x Q(a) & \\text{with prob. } 1 - \\varepsilon \\text{ (breaking ties randomly) }\\\\\n\t \\text{a random action} & \\text{with prob. } \\varepsilon\n\t \\end{array}\n\t\\right. \\\\\n\t & \\quad R \\leftarrow bandit(A) \\\\\n\t & \\quad N(A) \\leftarrow N(A) + 1 \\\\\n\t & \\quad Q(A) \\leftarrow Q(A) + \\frac{1}{N(A)}[R-Q(A)]\n\t\\end{align*}\n\\end{tcolorbox}\n\n\\subsection{Tacking a Nonstationary Problem}\n\nIn previous section, the reward probability distribution is stationary (i.e. doesn't change over time).\nFor nonstatoinary cases, let say we give more weight to recent rewards than to `long-past' rewards; for doing so, one popular way is to use a constant $\\alpha$ (step size).\n\nNot all choices of $\\alpha_n(a)$ guarantee convergence.\nIn \\textit{stochastic approximation theory}, two conditions required to assure convergence ``with probability 1''\\footnote{i.e. \\textit{almost sure}. No distinction between `sure' and `almost sure' in finite sample space, but becomes important in infinite sample space. see \\url{https://en.wikipedia.org/wiki/Almost_surely}}:\n\n$$ \\sum_{n=1}^\\infty \\alpha_n(a) = \\infty $$\n$$ \\sum_{n=1}^\\infty \\alpha_n^2(a) < \\infty $$\n\nThe 1st condition is required to guarantee that the steps are large enough to eventually overcome any initial conditions or random fluctuations.\nThe 2nd condition is required to guarantee that the steps eventually become small enough to assure convergence.\n\nTherefore, in sample-average method, $\\alpha_n(a) = \\frac{1}{n}$ converages, but a constant step-size parameter $\\alpha_n(a)=\\alpha$ does not converge.\n\nNote that sequences of $\\alpha$ that meet the conditions often converge very slowly and often need considerable tunning.\nThese sequences of $\\alpha$ often used in theoretical work, and are seldom used in real-world application.\n\n\\subsection{Optimistic Initial Values}\n\nHigher (more optimistic) initial values may result in better performance. More optimistic initial values ($Q_i(a)$) encourages more exploration.\nThis may only work well on stationary problems; as in nonstationary problems, the drive for exploration is inherently temporary.\n\n\\subsection{Upper-Confidence-Bound Action Selection}\n\nIn $\\varepsilon$-greedy action selection, it is not ideal to just randomly select actions; instead, we want to select among the non-greedy actions according to their potential for actually being optimal. Such taking into account both (1) how close their estimates are to being maximal, and (2) the uncertainties in those estimates.\n\n\\begin{definition}\nUpper-Confidence-Bound (UCB) Action Selection\n\n\\[\nA_t \\doteq \\argmax_a \\bigg[ Q_t(a) + c\\sqrt{\\frac{\\text{ln }t}{N_t(a)}} \\bigg]\n\\]\n\\end{definition}\n$N_t(a)$, denotes the number of times that action $a$ has been selected prior to time $t$. If $N_t(a)=0$, then $a$ is considered to be a maximizing action.\n$c$, $c > 0$ (confidence level) controls the degree of exploration\n\nThe idea of UCB action selection is that the square-root term is a measure of the uncertainty/variance in the estimate of $a$'s value.\nEach time $a$ is selected, the uncertainty is (presumably) reduced as $N_t(a)$ increments.\nEach time a non-$a$ is selected, $t$ increments but $N_t(a)$ doesn't, thus the uncertainty estimate increases.\nAll actions will eventually be selected, but actions with lower value estimates or that have been frequently selected, will be selected with decreasing frequency over time.\n\nUCB is useful in bandit problems, but is (usually) not practical in general RL settings. It is difficult to deal with (1) nonstationary problems, and (2) large state spaces.\n\n\\subsection{Gradient Bandit Algorithms}\nWe have considered methods that (1) estimate actions values; (2) select actions based on estimations.\nWe now consider a new approach for action selection: learning a numerical \\textit{preference} for each action \\textit{a}, which we denote $H_t(a) \\in \\R$.\nThe larger the preference, the more often that action is taken, but the preference has no interpretation regarding rewards.\nOnly the \\textbf{relative} preference of one action over another is important.\nThe action probabilities is determined by \\textit{soft-max distribution}.\n\n\\[\nP(A_t=a) \\doteq \\frac{e^{H_t(a)}}{\\sum_{b=1}^k e^{H_t(b)}} \\doteq \\pi_t(a)\n\\]\n\n$\\pi_t(a)$ is the probability of taking action $a$ at time $t$. The initial probability are the same for all actions.\n\nThere is a natural learning algorithm in this setting based on the idea of stochastic gradient descent.\nThis is based on updating the preference values as such, after taking action $A_t$ and obtaining reward $R_t$\n\n$$ H_{t+1}(A_t)\\doteq H_t(A_t) + \\alpha (R_t - \\overline{R_t})(1-\\pi_t(A_t)) $$\n\n$$ H_{t+1}(a) \\doteq H_t(a) - \\alpha(R_t - \\overline{R_t})\\pi_t(a)\\quad\\text{for all }a\\neq sA_t $$\n\n$\\overline{R_t} \\in \\R $ is the average of the rewards up to but not including time $t$ (with $\\overline{R_1} = R_1$)\n\nIf the reward is higher than the baseline $\\overline{R_t}$, then the probability of taking $A_t$ in the future is increased; and if the reward is below baseline, then the probability is decreased.\nThe non-selected actions move in the opposite direction.\n\nThe above update scheme is equivalent to \\textbf{stochastic gradient ascent} with bath size 1.\n\n\\subsection{Associative Search (Contextual Bandits)}\n\nFor now, we have only considered nonassociative tasks, that is, tasks in which there is no need to associate different actions with different situations.\nHowever, in the general RL task, there is more than one situation (state?), and the goal is to learn a policy, i.e., a mapping from states to optimal actions given the state.\n\nSuppose when a bandit task is selected for you, you are given some distinctive clue about its identity (but not its action values). For instance, if screen=red (actual reward is hidden from you), select arm 1; if screen=green, select arm 2... This is an example of an associative search task, it involves both trial-and-error learning to search for the best actions, and association of these actions with the situations in which they are best. This task is now called \\textit{contextual bandits} in the literature.\n\nIn this setting, we still don't have actions affecting the next situation as well as the reward. This will add another layer of complexity and lead us to full RL problems.\n\n\\begin{itemize}\n\\item (level 1) $k$-armed bandit problem: select one of $k$ actions that have fixed probabilities. For each selected action, a reward is given. The goal is to maximize the total reward after $j$ selections.\n\\item (level 2) associative search task (contextual bandits): Same as $k$-armed bandit problem, the action affects only the immediate reward. Unlike $k$-armed bandits problem, what situations/state you are facing are told and it involves learning a policy.\n\\item (level 3) full RL problem: On top of associative search task, actions affect next situation and reward.\n\\end{itemize}\n\n\\subsection{Summary}\n\nThis chapter presented several ways to balance exploration and exploitation:\n\n\\begin{itemize}\n\\item $\\varepsilon$-greedy methods choose randomly among actions a small fraction of the time to encourage exploration\n\\item UCB methods choose deterministically but favouring actions with uncertain estimates\n\\item Gradient methods don't estimate action values, but preferences, and choose actions probabilistically according to the preferences\n\\item Optimistic initialization sets optimistic initial $Q$ values in order to encourage a very active exploration in the initial phase, leading to a fast convergence with no bias\n\\end{itemize}\n\nThere is no method among the above that is best. Common ways to assess performance of these kind of algorithms are through graphs:\n\\begin{itemize}\n\\item learning curve: shows the performance of an algorithm vs. iterations for a certain parameter setting\n\\item parameter studies: summarize each algorithm by the average reward over the first 1000 steps, and show such a value for various parameter settings and algorithms.\n\\end{itemize}\n\nOne should also consider the sensitivity to parameter setting, that is an indication of \\textbf{robustness}.\n\nDespite the simplicity of methods presented in this chapter, they can be considered state of the art. More sophisticated methods usually impose additional complexity and assumptions.\n\nAnother approach to balancing exploration and exploitation in $k$-armed bandit problems is to compute a special kind of action value called \\textit{Gittins index}, which is an instance of Bayesian methods. This assume a known initial distribution over the (stationary) action values and then update the distribution exactly after each step.\nIn general, the update computations are complex, but for \\textit{conjugate priors} they are easy.\nOne way to select the best action at each time step is based on posterior probability.\nThis method, sometimes called posterior sampling or Thompson sampling.\n\n\\newpage\n\\section{Finite Markov Decision Processes}\n\n\\begin{definition}\nfinite Markov Decision Processes, involves evaluative feedback (as in bandits) and associative settings (choosing different actions in different situations)\n\\end{definition}\n\nMDPs are a classical formalization of \\textit{sequential} decision making, where actions influence both immediate rewards and subsequent situations/states (thus influence future rewards as well).\nMDPs involves delayed reward. MDPs need to consider trade-off between immediate reward and delayed reward.\n\nIn bandit problems, we estimated the value $q_{*}(a)$ of each action $a$.\n\nIn MDPs, we estimate $q_{*}(s,a)$ (value of each action $a$ in each state $s$), or $v_{*}(s)$ (value of each state given optimal action selections).\n\nMDPs are an ideal form of the RL problem where precise theoretical statements can be made.\nKey elements of RL's mathematical structure: returns, value functions, and Bellman equations.\n\n\\subsection{The Agent-Environment Interface}\n\n\\begin{itemize}\n\\item agent: learner and decision maker\n\\item environment: everything outside the agent, and which agent interacts with\n\\end{itemize}\n\n\\begin{figure}[!h]\n \\includegraphics[width=\\linewidth]{MDP_interface.PNG}\n \\caption{The agent-environment interaction in a MDP}\n \\label{fig:MDP_interface}\n\\end{figure}\n\nAt each time step $t$, the agent receives some representation of the environment's $state$, $S_t$, and on the state selects an $action$, $A_t$. One time step later, the agent receives reward $R_{t+1}$ as a result of the action, and find itself in a new state $S_{t+1}$. The trajectory/sequence is:\n\n$$ S_0, A_0, R_1, S_1, A_1, R_2, S_2, A_2, R_3, \\cdots $$\n\nIn a \\textit{finite} MDP, the state sets, action sets and reward sets are finite.\n\nFollowing defines the \\textit{dynamics} of the MDP.\n\n$$ p(s', r| s, a) \\doteq P(S_t = s', R_t = r | S_{t-1} = s, A_{t-1} = a) $$\n$$ \\sum_{s' \\in \\mathcal{S}} \\sum_{r \\in \\mathcal{R}} p(s',r|s,a) = 1, \\forall s \\in \\mathcal{S}, a \\in \\mathcal{A}(s) $$\n\n$p$ specifies a probability distribution for each choice of $s$ and $a$.\nThe probability of each possible value for $S_t$ and $R_t$ depends on the immediately preceding state and action, $S_{t-1}$ and $A_{t-1}$ (note not including earlier states and actions).\n\n\\begin{definition}\nMarkov property\\footnote{Markov property refers to the memoryless property of a stochastic process}, the future only depends on the current state, not the history.\n\\end{definition}\n\n\\begin{definition}\nstate-transition probabilities,\n$$ p(s'|s,a) \\doteq P(S_t=s'|S_{t-1}=s, A_{t-1}=a) = \\sum_{r\\in \\mathcal{R}} p(s',r|s,a) $$\n\\end{definition}\nWe can express state transition probabilities in an Matrix as shown in Figure \\ref{fig:state-trans-prob-eg}. Each row in the matrix represents the probability from moving from our original or starting state to any successor state. Sum of each row is equal to 1.\n\\begin{figure}[h!]\n \\includegraphics[width=\\linewidth]{state_transition_probability_example.jpg}\n \\caption{An example of state transition probability.}\n \\label{fig:state-trans-prob-eg}\n\\end{figure}\n\n\\begin{definition}\nexpected rewards for state-action pairs\n\n$$ r(s,a) \\doteq \\E [R_t | S_{t-1}=s, A_{t-1}=a] = \\sum_{r\\in \\mathcal{R}} \\sum_{s\\in \\mathcal{S}} p(s',r|s,a) $$\n\\end{definition}\n\n\\begin{definition}\nexpected rewards for state-action-nextState,\n\n$$ r(s,a,s') \\doteq \\E [R_t | S_{t-1}=s, A_{t-1}=a, S_t=s'] = \\sum_{r \\in \\mathcal{R}} r \\frac{p(s',r|s,a)}{p(s'|s,a)} $$\n\\end{definition}\n\n\n\\paragraph{agent-environment}\nThe agent-environment boundary represents the lit of the agent's absolute control, not of its knowledge. Anything that cannot be changed by the agent.\n\nAny problem of learning goal-directed behavior can be reduced to 3 signals passing back and forth between an agent and its environment:\n\\begin{itemize}\n\\item action: the choice made by the agent\n\\item the basis on which the choice is made\n\\item the agent's goal\n\\end{itemize}\n\n\\begin{example}\nBioreactor,\n\n\\begin{itemize}\n\\item actions (vector): target temperatures, target sitrring rates\n\\item states (vector): sensory readings, ingredients input, target chemical\n\\item rewards (number): production rate of useful chemical (target)\n\\end{itemize}\n\n\\end{example}\n\n\\begin{example}\nRobotic Arm (pick-and-place task)\n\n\\begin{itemize}\n\\item actions: voltages applied to each joint motor\n\\item states: readings of joint angles and velocities\n\\item rewards: +1 for successfully object pick-and-place; small negative reward at each time step (punish jerkiness motion)\n\\end{itemize}\n\\end{example}\n\n\n\\subsection{Goals and Rewards}\n In RL, thr purpose/goal of the agent is to maximize \\textit{cumulative reward in the long run}.\n It is important that the set-up rewards indicate what we truly want accomplished.\n In particular, rewards are not the place to tell/impart agent prior knowledge about \\textit{how} to achieve our goal.\n For example, in a chess game, the agent should only be reward for winning, instead of achieving subgoals like taking opponent's pieces.\n Better places for imparting this kind of prior knowledge in (1) initial policy; or (2) initial value function.\n \n The reward signal is a way of imparting to the agent \\textit{what} you want achieved, not \\textit{how} you want it achieved.\n\n\\subsection{Returns and Episodes}\n\nA RL agent's goal is to maximize the cumulative reward it receives in the long run.\nIn general, we seek to maximize the \\textit{expected return} $G_t$.\n\n$$ G_t \\doteq R_{t+1} + R_{t+2} + \\cdots + R_T $$\n\nwhere $T$ is a final time step.\n\nThe final time step marks the end of a subsequences.\n\n\\begin{definition}\nEpisode, a subsequence of the agent-environment interaction. Each episode ends in \\textbf{terminal state}, followed by a \\textbf{reset} to a standard starting state or to a \\textbf{sample} from a standard distribution of starting states. The next episode begins independently of how the previous one ended.\n\\end{definition}\n\nThe set of all nonterminal states, $\\mathcal{S}$. All states and the terminal state, $\\mathcal{S^+}$.\n\n\\begin{definition}\nEpisodic tasks, tasks with episodes that with terminal state.\n\\end{definition}\n\n\\begin{definition}\ncontinuing tasks, tasks without terminal state.\n\\end{definition}\n\nThis task uses discounted rewards to deal with $T=\\infty$.\n\n\\begin{definition}\ndiscounted rewards $G_t$. discount rate, $0 \\leq \\gamma \\leq 1$\n\n$$ G_t \\doteq R_{t+1} + \\gamma R_{t+2} + \\gamma^2 R_{t+3} + \\cdots = \\sum_{k=0}^{\\infty} \\gamma^k R_{t+k+1} $$\n\\end{definition}\n\nThe discount rate determines the present value of future rewards. As long as the reward sequence {$R_k$} is bounded, the infinite sum $G_t$ has a finite value.\nIf $\\gamma \\rightarrow 0$, the agent is myopic; if $\\gamma \\rightarrow 1$, the agent is farsighted.\n\nIf the reward is constant $R$, then we have the \\textit{geometric series}\n\n$$ G_t = \\sum_{k=0}^{\\infty} R\\gamma^k = R\\frac{1}{1-\\gamma} $$\n\nNote that the returns at successive time steps are related to each other (which is important for RL):\n\n$$ G_t \\doteq R_{t+1} + \\gamma R_{t+2} + \\gamma^2 R_{t+3} + \\cdots = R_{t+1} + \\gamma G_{t+1} $$\n\n\\begin{example} pole-balancing\n\\begin{itemize}\n\\item terimnal state: if the pole falls past a given angle from vertical or if the cart runs off the track\n\\item starting state: pole reset to vertical\n\\item reward (episodic): +1 for each time step before falling\n\\item reward (continuing): -1 on each falling, 0 for all other times\n\\end{itemize}\n\\end{example}\n\n\n\\begin{figure}[!h]\n \\includegraphics[width=\\linewidth]{MDP_pole.PNG}\n \\label{fig:MDP_pole}\n\\end{figure}\n\n\n\n\\subsection{Unified Notation for Episodic and Continuing Tasks}\n\nWe use $S_{t,i}$ to represents state at time $t$ of episode $i$ (same for $A_{t,i}$, $R_{t,i}$, etc).\nIn practice, we dropped the $i$ as we normally don't have to distinguish between different episodes.\nSo, now $S_t$ refers to $S_{t,i}$.\n\nWe use an $absorbing state$ to replace the terminate state. The absorbing state transits to itself and have reward = 0.\nThis way, we can have the same formula of $G_t$ for episodic and continuing tasks.\n\n\\subsection{Policies and Value Functions}\n\nValue functions (either V or Q) are \\textbf{always} conditional on some policy $\\pi$. Sometimes in literature we leave off the $\\pi$ or $*$ and just refer to $V$ and $Q$, because it’s implicit in the context, but ultimately, every value function is always with respect to some policies.\n\n\\begin{definition}\nThere are two value functions:\n\\begin{itemize}\n\\item value functions of states, or V, estimate how much expected return it is received for the agent to be in a given state;\n\\item value functions of state-action pairs, or Q, estimate how much expected return it is received to perform a given action in a given state.\n\\end{itemize}\n\\end{definition}\n\n\\begin{definition}\npolicy, $\\pi(A_t=a|S_t=s)$, a mapping from states to probabilities of selecting each action\n\\end{definition}\n\nExpectation of $R_{t+1}$ in terms of $\\pi$ and the four-argument function $p$:\n\\[\n\\E\\left[R_{t+1} \\middle|S_{t}\\right] = \\sum_{a \\in \\mathcal{A}} \\pi\\left(a \\middle| S_t\\right) \\cdot r(S_t, a) = \\sum_{a \\in \\mathcal{A}} \\pi\\left(a \\middle| S_t\\right) \\cdot \\sum_{r \\in \\mathcal{R}} r \\sum_{s' \\in \\mathcal{S}} p\\left(s', r \\middle| S_t, a\\right)\n\\]\nMy word explanation: the expectation of reward given current state $S_t$= the total sum of the probability of perform action $a$ at state $s$ $\\cdot$ the reward $r$ $\\cdot$ the probability of getting reward $r$ (when performs action $a$ at state $s$) (?)\n\n\\begin{definition}\nstate-value function for policy $\\pi$, (value function of a state $s$ under a policy $\\pi$), $v_{\\pi}(s)$, is the expected return when starting in $s$ and following $\\pi$ thereafter. For MDPs:\n\n$$ v_\\pi(s) \\doteq \\E_\\pi[G_t | S_t = s] = \\E_\\pi [ \\sum_{k=0}^{\\infty} \\gamma^k R_{t+k+1} | S_t = s], \\forall s \\in \\mathcal{S} $$\n\\end{definition}\n\n\\begin{definition}\naction-value function for policy $\\pi$, (value of taking action $a$ in state $s$ under a policy $\\pi$), $q_\\pi(s,a)$, as the expected return starting from $s$, taking the action $a$, and \\textbf{then} following policy $\\pi$:\n\n$$ q_\\pi(s,a) \\doteq \\E_\\pi [G_t | S_t = s, A_t = a] = \\E_\\pi [\\sum_{k=0}^{\\infty} \\gamma^k R_{t+k+1} | S_t = s, A_t = a] $$\n\\end{definition}\n\nThe main difference between $Q$ and $V$ then, is the $Q$-value lets you play a hypothetical of potentially taking a different action in the first time step than what the policy might prescribe and then following the policy from the state the agent winds up in.\n\\newline\n\nEquation of $v_\\pi$ in terms of $q_\\pi$ and $\\pi$:\n$$ v_\\pi(s) = \\sum_a \\pi(a|s)q_\\pi(s,a) $$\n\nMy word explanation: the value function is the total sum of probability of choosing action $a$ at state $s$ $\\times$ the action-value of taking each action.\n\\newline\nEquation of $v_\\pi$ in terms of $\\pi$ and $p$:\n\\begin{align*}\nv_\\pi(s) & \\doteq \\E_\\pi[G_t | S_t = s]\\\\\n& = \\E_\\pi[R_{t+1} + \\gamma G_{t+1} | S_t = s]\\\\\n& = \\sum_a \\pi(a|s) \\sum_{s'}\\sum_r p(s',r|s,a) [r+ \\gamma \\E_\\pi[G_{t+1} | S_{t+1}=s']]\\\\\n& = \\sum_a \\pi(a|s) \\sum_{s',r} p(s',r|s,a) [r+ \\gamma v_\\pi(s')]\n\\end{align*}\n\nEquation of $q_\\pi$ in terms of $v_\\pi$ and the four argument $p$\n$$ q_\\pi(s,a) = \\sum_{s', r} p(s',r|s,a) \\underbrace{ [r+\\gamma v_\\pi(s')] }_{G_t?} $$\n\n\\begin{definition}\nMonte Carlo methods, estimation methods that involve averaging over many random samples of actual returns.\n\\end{definition}\n\nFor example, an agent follows policy $\\pi$; and for each state $s$, the agent maintains an average of the actual returns which have followed that state, then the average converges to the state's value $v_\\pi(s)$. (Recall the Law of Large Numbers, or sample-average method in Chapter 2).\nFor another example, if separate averages are kept for each action taken in each state, then these averages will similarly converge to the action values $q_\\pi(s,a)$.\n\n\\begin{definition}\nBellman equation for $v_\\pi$:\n\n$$ v_\\pi = \\sum_a \\pi(a|s) \\sum_{s',r} p(s',r|s,a) [r+ \\gamma v_\\pi(s')] $$\n\\end{definition}\n\nIt expresses a relationship between the value of a state and the values of its successor states.\n\n\\begin{figure}[!h]\n \\centering\n \\includegraphics[]{MDP_backup.PNG}\n \\caption{Backup diagram for $v_\\pi$}\n \\label{fig:MDP_backup}\n\\end{figure}\nNote that in backup diagram, the nodes do not necessarily represent distinct states (e.g. a state might be its own successor).\n\nThe value function $v_\\pi$ is the unique solution to its Bellman equation (??). My understanding is that since $v_\\pi$ is the only solution, meaning that $v_\\pi$ must converge to a point?\n\n\\begin{definition}\nBellman equation for $q_\\pi$:\n\n$$ q_\\pi(s,a) \\doteq \\E_\\pi [G_t | S_t = s, A_t = a] = \\sum_{s',r} p(s',r|s,a)[r+\\gamma \\sum_{a'} \\pi(a'|s')q_\\pi (s',a')] $$\n\\end{definition}\n\n\\begin{figure}[!h]\n \\centering\n \\includegraphics[]{MDP_backup_Qpi.PNG}\n \\caption{Backup diagram for $q_\\pi$}\n \\label{fig:MDP_backup_qpi}\n\\end{figure}\n\n\\begin{figure}[!h]\n \\centering\n \\subfigure[]{\\includegraphics[width=\\textwidth]{MDP_1.PNG}}\n \\subfigure[]{\\includegraphics[width=\\textwidth]{MDP_2.PNG}}\n \\caption{ (a) $ v_\\pi(s)=\\sum_a \\pi(a|s)q_\\pi(s,a) $ (b) $ q_\\pi(s,a)=\\sum_{s',r}p(s',r|s,a)[r+\\gamma v_\\pi(s')] $ }\n \\label{fig:MDPMDP}\n\\end{figure}\nThe value of a state depends on the values of the actions possible in that state and on how likely each action is to be taken under the current policy.\n\n\\subsection{Optimal Policies and Optimal Value Functions}\n\nSolving a RL means finding a policy that achieves a lot of reward over the long run.\n\n\\begin{definition}\noptimal policy, $\\pi_*$\n\n$$ \\pi_* \\geq \\pi' \\iff v_\\pi(s) \\geq v_{\\pi'}(s), \\quad \\forall s \\in \\mathcal{S} $$\n\\end{definition}\n\n\\begin{definition}\noptimal state-value function, $v_*$\n\n$$ v_*(s) \\doteq \\max_\\pi v_\\pi(s), \\quad \\forall s \\in \\mathcal{S} $$\n\\end{definition}\n\n\\begin{definition}\noptimal action-value function, $q_*$\n\n$$ q_*(s) \\doteq \\max_\\pi q_\\pi(s,a), \\quad \\forall s \\in \\mathcal{s}, a \\in \\mathcal{A} $$\n\\end{definition}\n\nWe can write $q_*$ in terms of $v_*$:\n\n$$ q_*(s,a) = \\E [R_{t+1} + \\gamma v_* (S_{t+1}) | S_t = s, A_t = a] $$\n\n\\begin{definition}\nBellman optimality equation for $v_*$,\n\\begin{align*}\nv_*(s) & = \\max_{a\\in\\mathcal{A}}(s) q_{\\pi_*} (s,a) \\\\\n& = \\max_a \\E [R_{t+1} + \\gamma v_* (S_{t+1}) | S_t = s, A_t = a] \\\\\n& = \\max_a \\sum_{s',r} p(s',r|s,a)[r+\\gamma v_*(s')]\n\\end{align*}\n\\end{definition}\n\n\\begin{definition}\nBellman optimality equatoin for $q_*$,\n\\begin{align*}\nq_*(s,a) & = \\E [R_{t+1} + \\gamma \\max_{a'} q_*(S_{t+1}, a') | S_t = s, A_t = a] \\\\\n& = \\sum_{s',r} p(s',r|s,a)[r+\\gamma \\max_{a'} q_*(s',a')]\n\\end{align*}\n\\end{definition}\n\n\\begin{figure}[!h]\n \\centering\n \\includegraphics[]{MDP_optimal.PNG}\n \\caption{Backup diagram for $v_*$ and $q_*$}\n \\label{fig:MDP_optimal}\n\\end{figure}\n\nExplicitly solving the Bellman optimality equation provides one route to finding $\\pi_*$, and thus to solving the RL problem.\nHowever, this solution is rarely useful. It needs an exhaustive search, looking ahead at all possibilities, computing their probabilities of occurrence and their desirabilities in terms of expected rewards.\nThis solution relies on at least 3 assumptions:\n\\begin{itemize}\n\\item the dynamics of the environment are accurately known\n\\item computational resources are sufficient\n\\item the states have the Markov property\n\\end{itemize}\n\nIn RL, one typically has to settle for approximate solutions. Many RL methods can be taken as approximately solving the Bellman optimality equation.\n\n\\paragraph{Exercise} Consider the contining MDP shown, the only decision to be made is in the top state with two actions (left, right).\nFind optimal policy when $\\gamma$ = 0.9.\n\n\\begin{figure}[!h]\n \\centering\n \\includegraphics[]{ex3-22.PNG}\n \\label{fig:ex3-22}\n\\end{figure}\n\nRecall the definition of $v_\\pi(s)$ and $q_\\pi(s)$, we get\n\n\\begin{itemize}\n\\item Left Policy\n\\begin{itemize}\n\\item $ v(s_0) = 1 + 0.9\\cdot v(s_L) $\n\\item $ v(s_L) = 0 + 0.9\\cdot v(s_0) $\n\\item $ v(s_R) = 2 + 0.9\\cdot v(s_0) $\n\\end{itemize}\n\\item Right Policy\n\\begin{itemize}\n\\item $ v(s_0) = 0 + 0.9\\cdot v(s_R) $\n\\item $ v(s_L) = 0 + 0.9\\cdot v(s_0) $\n\\item $ v(s_R) = 2 + 0.9\\cdot v(s_0) $\n\\end{itemize}\n\\end{itemize}\nSolving above we find that all state values of Right Policy is greater than all state values of Left Policy. Thus, `right' is the optimal policy for $\\gamma$ = 0.9.\n\nIt is important to note that if we take `left' action in $s_0$, then policy $\\pi$ would never take us to state $s_R$ (same for `right' and $s_L$); however, due to the definition of $\\pi_*$, we must evaluate the value function for all states, even ones that would not be visited under a policy we are evaluating.\n\n\\subsection{Optimality and Approximation}\n\nIt is difficult to reach optimal solutions due to constraints from computational resource and memory. We can often approximate the optimal solutions.\n\n\\subsection{Summary}\n\n\\begin{itemize}\n\\item RL is about learning from interaction how to behave in order to achieve a goal.\n\\item RL \\textit{agent} and its \\textit{environment} interact over a sequence of discrete time steps.\n\\item \\textit{actions}: choices made by the agent, \\textit{states}: the basis for making the choices, \\textit{rewards}: basis for evaluating the choices.\n\\item everything inside agent is known and controllable; environment is incompletely controllable, and are partialily known\n\\item a \\textit{policy} is a stochastic rule that the agent selects actions as a function of states\n\\item agent's objective is to maximize reward it receives in the long run\n\\item when RL setup with agent, environment, states, actions, rewards, policy and formulated with well-defined transition probabilities it constitutes a Markov decision process (MDP)\n\\item a finite MDP is an MDP with finite state sets, action sets and reward sets.\n\\item \\textit{return} is the function of future rewards that the agent seeks to maximize (in expected value)\n\\item undiscounted formulation is appropriate for \\textit{episodic tasks}\n\\item discounted formulation is appropriate for \\textit{\\textbf{tabular} continuing tasks} (but not for approximate continuing tasks; see chap 10.3-4)\n\\item \\textit{value functions} $v_\\pi$ (state) and $q_\\pi$ (state-action pair) are the expected return from that state/state-action under policy $\\pi$\n\\item \\textit{optimal value functions} $v_*$ and $q_*$ are the largest expected return by any policy\n\\item A policy whose value functions are optimal is an optimal policy\n\\item !! A MDP can have many optimal policies, but can have only one unique optimal value function ($v_*$ and $q_*$)\n\\item Any policy that is \\textit{greedy} w.r.t $v_*$ and $q_*$ must be $\\pi_*$\n\\item The Bellman optimality equations are special consistency conditions that the optimal value functions must satisfy\n\\item In RL most cases, their optimal solutions cannot be found but must be approximated in some way.\n\\end{itemize}\n\n\\newpage\n\\section{Dynamic Programming}\n\nIn RL, DP refers to a collection of algorithms that can be used to compute optimal policies $\\pi_*$ given a perfect model of the environment as a Markov decision process (MDP).\nDP itself cannot be useful in RL due to its computational cost and requirement of a perfect model, but DP provides an essential foundation for the understanding of other algorithms.\nOther algorithms can be treated as attempts to achieve much the same effect as DP, only with less computation and without assuming a perfect model of the environment.\n\nStarting with this chapter, we assume the environment is a finite MDP (i.e. $S$, $A$, $R$ are finite and dynamics are given by a set of probabilities $p(s',r|s,a)$).\nA common way of obtaining approximate solutions for tasks with continuous states and actions is to quantize the state and action spaces and then apply finite-state DP methods.\nIn this chapter we show how DP can be used to compute the value functions defined in Chapter 3. As shown in Chapter 3, we can obtain $\\pi_*$ once we have found $v_*$ or $q_*$, which satisfy the Bellman optimality equations:\n\n\\begin{align*}\nv_*(s) & = \\max_a \\E [R_{t+1} + \\gamma v_*(S_{t+1}) | S_t = s, A_t = a] \\\\\n& = \\max_a \\sum_{s',r} p(s',r|s,a)[r + \\gamma v_*(s')]\n\\end{align*}\n\n\\begin{align*}\nq_*(s,a) & = \\E [R_{t+1} + \\gamma \\max_{a'} q_*(S_{t+1},a') | S_t = s, A_t = a] \\\\\n& = \\sum_{s',r} p(s',r|s,a)[r+\\gamma \\max_{a'} q_*(s',a')]\n\\end{align*}\n\n\\subsection{Policy Evaluation (Prediction)}\n\\begin{definition}\npolicy evaluation, compute the state-value function $v_\\pi$ for an arbitrary policy $\\pi$\n\\end{definition}\n\nPolicy evaluation is also called prediction problem.\n\nRecall that $\\forall s \\in \\mathcal{S}$,\n\n\\begin{align*}\nv_\\pi(s) & \\doteq \\E_\\pi [G_t | S_t = s] \\\\\n& = \\E_\\pi [R_{t+1} + \\gamma G_{t+1} | S_t = s] \\\\\n& = \\E_\\pi [R_{t+1} + \\gamma v_\\pi (S_{t+1}) | S_t = s] \\\\\n& = \\sum_a \\pi (a|s) \\sum_{s',r} p(s',r|s,a) [r+\\gamma v_\\pi (s')]\n\\end{align*}\n\nIf the environment's dynamics are completely known, then the last equation is a sysnteim of $|\\mathcal{S}|$ simultaneous linear equations.\nFor this case, iterative solution methods are most suitable. Consider a sequence of approximate value functions $v_0$, $v_1$, $v_2$, $\\cdots$, each mapping $\\mathcal{S^+}$ to $\\R$.\nThe initial approximation, $v_0$, is chosen arbitrarily, and each successive approximation is obtained by using the Bellman equation $v_\\pi$ as an update rule ($\\forall s$):\n\n\\begin{align*}\nv_{k+1} & \\doteq \\E_\\pi [R_{t+1} + \\gamma v_k (S_{t+1}) | S_t = s] \\\\\n& = \\sum_a \\pi (a|s) \\sum_{s',r} p(s',r|s,a)[r+\\gamma v_k (s')]\n\\end{align*}\n\nWe see that $v_k$ = $v_\\pi$ is a fixed point for this update rule as $k \\rightarrow \\infty$, $v_k$ converages to $v_\\pi$. This algorithm is called \\textit{iterative policy evaluation}.\nThe update is called \\textit{expected update}. It is called \\textit{expected} because the updates are based on an expectation over all possible next states rather than on a sample next state.\n\nIn this algorithm, we use one array and update values in place.\n\\begin{tcolorbox}[width=\\textwidth,title={Iterative Policy Evaluation, for estimating $V \\approx v_\\pi$}]\nInput $\\pi$ (the policy to be evaluated); $\\theta > 0$ threshold determining accuracy of estimation.\nInitialize $V(s)$ arbitrarily, for $s \\in \\mathcal{S}$, and $V$ to 0.\n\nLoop:\n\\begin{align*}\n&\\quad\\Delta \\leftarrow 0 \\\\\n&\\quad\\text{Loop for each } s \\in \\mathcal{S} \\\\\n&\\quad\\quad v \\leftarrow V(s) \\\\\n&\\quad\\quad V(s) \\leftarrow \\sum_a \\pi (a|s) \\sum_{s',r} p(s',r|s,a)[r + \\gamma V(s')] \\\\\n&\\quad\\quad \\Delta \\leftarrow \\text{max}(\\Delta, |v-V(s)|) \\\\\n&\\text{until } \\Delta < \\theta\n\\end{align*}\n\\end{tcolorbox}\n\n\nPolicy evaluation of $q_\\pi$:\n\n\\begin{align*}\nq_\\pi(s,a) & \\doteq \\E_\\pi [G_t | S_t = s, A_t = a] \\\\\n& = \\E_\\pi [R_{t+1} + \\gamma G_{t+1} | S_t = s, A_t = a] \\\\\n& = \\E_\\pi [R_{t+1} + \\gamma \\sum_{s',a'} q_\\pi (s',a') | S_t = s, A_t = a] \\\\\n& = \\sum_{s',r} p(s',r|s,a) [ r+ \\gamma \\sum_{a'} \\pi(a'|s')q_\\pi(s',a')]\n\\end{align*}\n\n\\begin{align*}\nq_{k+1}(s,a) & \\doteq \\E_\\pi [R_{t+1} + \\gamma G_{t+1} | S_t = s, A_t = a] \\\\\n& = \\sum_{s',r} p(s',r|s,a)[r+\\gamma \\sum_{a'} \\pi (a'|s')q_k(s',a')]\n\\end{align*}\n\n\\subsection{Policy Improvement}\n\n\\begin{definition}\npolicy improvement theorem. Let $\\pi$ and $\\pi'$ be any pair of deterministic policies, where $\\pi'$ is idential to $\\pi$ except $\\pi'(s) = a \\neq \\pi(s)$, s.t. $\\forall s \\in \\mathcal{S}$,\n\n$$ q_\\pi (s,\\pi'(s)) \\geq v_\\pi(s) $$\n\nThen the policy $\\pi'$ must be btter (or as good as) than $\\pi$. That is, it must obtain greater or equal expected return from all states $s \\in \\mathcal{S}$\n\n$$ v_{\\pi'}(s) \\geq v_\\pi (s) $$\n\\end{definition}\n\n\\begin{definition}\npolicy improvement, the process of making a new policy $\\pi'$ that improves on an original policy $\\pi$, by making it greedy w.r.t the value function of $\\pi$.\n\\end{definition}\n \nIf there are ties in policy improvement steps, each maximizing action can be given a portion of the probability of being selected in the new greedy policy.\n\n\\subsection{Policy Iteration}\n\nIt is trivial to see that we can ultimately reach an optimal policy through iterative improving policies and value functions \\footnote{In large-scale reinforcement learning problems, it is typically impractical to run either of these steps to convergence, and instead the value function and policy are optimized jointly. By Soft Actor-Critic: Off-Policy Maximum Entropy Deep Reinforcement Learning with a Stochastic Actor}.\n\n\\begin{definition}\npolicy iteration, the way of finding an optimal policy.\n\n$$\\pi_0 \\xrightarrow{E} v_{\\pi_0} \\xrightarrow{I} \\pi' \\xrightarrow{E} v_{\\pi'} \\xrightarrow{I} \\pi'' \\xrightarrow{E} \\cdots \\xrightarrow{I} \\pi_* \\xrightarrow{E} v_* $$\n\\end{definition}\n\n\\begin{tcolorbox}[width=\\textwidth,title={Policy Iteration (using iterative policy evaluation) for estimating $\\pi \\approx \\pi_*$}]\n1. Initialization\n\n$\\quad V(s) \\in\\R$ and $\\pi(s) \\in A(s)$ arbitrarily $\\forall s \\in \\mathcal{S}$; $V(terminal) \\doteq 0$\n\n2. Policy Evaluation\n\\begin{align*}\n&\\text{Loop:}\\\\\n&\\quad\\Delta \\leftarrow 0\\\\\n&\\quad\\text{Loop for each } s \\in \\mathcal{S}\\\\\n&\\quad\\quad v \\leftarrow V(s)\\\\\n&\\quad\\quad V(s) \\leftarrow \\sum_a \\pi (a|s) \\sum_{s',r} p(s',r|s,a)[r + \\gamma V(s')]\\\\\n&\\quad\\quad \\Delta \\leftarrow \\text{max}(\\Delta, |v-V(s)|)\\\\\n&\\text{until } \\Delta < \\theta\n\\end{align*}\n\n3. Policy Improvement\n\\begin{align*}\n&policyStable \\leftarrow true\\\\\n&\\text{For each } s \\in \\mathcal{S}\\\\\n&\\quad oldAction \\leftarrow \\pi(s)\\\\\n&\\quad \\pi(s) \\leftarrow \\argmax_a \\sum_{s',r} p(s',r|s,a)[r+\\gamma V(s')]\\\\\n&\\quad \\text{If } oldAction \\notin \\{a_i\\} \\text{, which is the all equal best solutions from $\\pi(s)$}\\\\\n& \\quad \\text{then } policyStable \\leftarrow false\\\\\n&\\text{If } policyStable \\text{, then stop and return } V \\approx v_* \\text{ and } \\pi \\approx \\pi_*\\text{; else go to 2}\t\n\\end{align*}\n\\end{tcolorbox}\n\n\\begin{tcolorbox}[width=\\textwidth,title={Policy Iteration (using iterative policy evaluation) for estimating $\\pi \\approx \\pi_*$}]\n1. Initialization\n\n$\\quad Q(s,a) \\in\\R$ and $\\pi(s) \\in A(s)$ arbitrarily $\\forall s \\in \\mathcal{S}, a \\in \\mathcal{A}$\n\n2. Policy Evaluation\n\\begin{align*}\n&\\text{Loop:}\\\\\n&\\quad\\Delta \\leftarrow 0\\\\\n&\\quad\\text{Loop for each } s \\in \\mathcal{S}, a \\in \\mathcal{A}\\\\\n&\\quad\\quad q \\leftarrow Q(s,a)\\\\\n&\\quad\\quad Q(s,a) \\leftarrow \\sum_{s',r} p(s',r|s,a)[r + \\gamma \\sum_{a'}\\pi(a'|s')Q(s',a')]\\\\\n&\\quad\\quad \\Delta \\leftarrow \\text{max}(\\Delta, |q-Q(s,a)|)\\\\\n&\\text{until } \\Delta < \\theta\n\\end{align*}\n\n3. Policy Improvement\n\\begin{align*}\n&policyStable \\leftarrow true\\\\\n&\\text{For each } s \\in \\mathcal{S}, a \\in \\mathcal{A}\\\\\n&\\quad oldAction \\leftarrow \\pi(s)\\\\\n&\\quad \\pi(s) \\leftarrow \\argmax_a Q(s,a)\\\\\n&\\quad \\text{If } oldAction \\notin \\{a_i\\} \\text{, which is the all equal best solutions from $\\pi(s)$}\\\\\n& \\quad \\text{then } policyStable \\leftarrow false\\\\\n&\\text{If } policyStable \\text{, then stop and return } Q \\approx q_* \\text{ and } \\pi \\approx \\pi_*\\text{; else go to 2}\t\n\\end{align*}\n\\end{tcolorbox}\n\n\n\\begin{definition}\n$\\varepsilon$-soft, the probability of selecting each action in each state, is at least $\\frac{\\varepsilon}{A(s)}$\n\\end{definition}\n\n\\subsection{Value Iteration}\n\nThe policy evaluation of policy iteration requires multiple sweeps through the state set and convergence occurs only in the limit.\nWe want to truncate policy evaluation without losing the convergence guarantees of policy iteration. One of the technique is value iteration.\n\n\\begin{definition}\nvalue iteration, stop policy evaluation after one sweep (one update after each state).\n\n\\begin{align*}\nv_{k+1}(s) & \\doteq max_a \\E [R_{t+1}+\\gamma v_k(S_{t+1})|S_t = s, A_t = a]\\\\\n& = max_a \\sum_{s',r} p(s',r|s,a)[r+\\gamma v_k (s')], \\forall s \\in \\mathcal{S}\n\\end{align*}\n\\end{definition}\n\n\n\n\n\n\n\n\n\n\n\\newpage\n\\section{Some Notes}\n\n\n% Please add the following required packages to your document preamble:\n\\begin{table}[]\n\\begin{tabular}{@{}ll@{}}\n\\toprule\non-policy & off-policy \\\\ \\midrule\nAgent can pick actions & Agent can't pick actions \\\\\nmost obvious setup :) & \\begin{tabular}[c]{@{}l@{}}learning with exploration\\\\ playing without exploration\\end{tabular} \\\\\nAgent always follows own policy & \\begin{tabular}[c]{@{}l@{}}Learning from expert (expert is imperfect)\\\\ Learning from sessions (recorded data)\\end{tabular} \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n% Please add the following required packages to your document preamble:\n% \\usepackage{booktabs}\n\\begin{table}[]\n\\begin{tabular}{@{}lll@{}}\n\\toprule\n & on-policy & off-policy \\\\ \\midrule\nvalue based & \\begin{tabular}[c]{@{}l@{}}Monte Carlo Learning\\\\ TD(0)\\\\ SARSA\\\\ Expected SARSA\\\\ n-Step TD/SARSA\\\\ TD(\\textbackslash{}lamda)\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}Q-Learning\\\\ DQN\\\\ Double DQN\\\\ Dueling DQN\\end{tabular} \\\\\npolicy based & \\begin{tabular}[c]{@{}l@{}}REINFORCE\\\\ REINFORCE with Advantage\\end{tabular} & \\\\\nactor-critic & \\begin{tabular}[c]{@{}l@{}}A3C\\\\ A2C\\\\ TRPO\\\\ PPO\\end{tabular} & \\begin{tabular}[c]{@{}l@{}}DDPG\\\\ TD3\\\\ SAC\\\\ IMPALA\\end{tabular} \\\\ \\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsection{Gradient Descent vs. Gradient Ascent}\n\nThe gradient of a continuous function $f$ = the vector that contains the partial derivatives $\\frac{\\partial f(p)}{\\partial x_i}$ computed at that point $p$.\nThe gradient is finite and defined if and only if all partial derivatives are also defined and finite.\nThe gradient formula:\n\n$$ \\nabla f(x) = [\\frac{\\partial f(p)}{\\partial x_1}, \\frac{\\partial f(p)}{\\partial x_2}, \\cdots, \\frac{\\partial f(p)}{\\partial x_{|x|}}]^T $$\n\nWhen using the gradient for optimization, we can either conduct gradient descent or gradient ascent.\n\n\\paragraph{Gradient Descent}\nGradient Descent is an iterative process through which we optimize the parameters of a ML model. It's particularly used in NN, but also in logistic regression and support vector machines (SVM). It is the most typical method for iterative minimization of a cost function. Its major limitation consists of its guaranteed convergence to a local, not necesarily global, minimum.\n\nA hyperparameter (i.e. pre-defined parameter) $\\alpha$ (learning rate), allows the fine-tuning of the process of decent. In particular, we may descent to a global minimum.\nThe gradient is calculated with respect to a vector of parameters for the model, typically the weight $w$. In NN, the process of applying gradient descent to the weight matrix is called backpropagation of the error.\n\nBackpropagation uses the sign of the gradient to determine whether the weights should increase or decrease. The sign of the gradient allows us to direction of the closet minimum to the cost function. For a given $\\alpha$, we iteratively optimize the vector $w$ by computing\n$$ w_{n+1} = w_n - \\alpha \\nabla_w f(w) $$\nAt step $n$, the weights of the NN are all modified by the product of the hyperparameter $\\alpha$ times the gradient of the cost function, computed with those weights.\nIf the gradient is positive, then we decrease the weights; if the gradient is negative, then we increase the weights.\n\n\\paragraph{Gradient Ascent}\nGradient ascent works in the same manner as gradient descent. The only difference is that gradient ascent maximize functions (instead of minimization).\n$$ w_{n+1} = w_n + \\alpha \\nabla_w f(w) $$\n\nGradient descent works on \\textbf{convex functions}, while gradient ascent works on \\textbf{concave functions}.\n\n\\paragraph{Summary}\n\\begin{itemize}\n\\item The gradient is the vector containing all partial derivatives of a function in a point\n\\item We can apply gradient descent on a convex function, and gradient ascent on a concave function\n\\item Gradient descent finds the nearest minimum of a function, gradient ascent finds the nearest maximum\n\\item We can use either form of optimization for the same problem if we can flip the objective function.\n\\end{itemize}\n\n\\paragraph{Gradient vs. Derivative}\n\n\\begin{itemize}\n\\item A directional derivative = a slope in an arbitrary specified direction.\n\\item A directional derivative is a rate of change of a function in any given direction.\n\\end{itemize}\n\n\\begin{itemize}\n\\item Gradient = a vector with slope of the function along each of the coordinate axes.\n\\item Gradient indicates the direction of \\textbf{greatest} change of a funciton of more than one variable.\n\\item Gradient vector can be interpreted as the `direction and rate of fastest increase'.\n\\end{itemize}\n\n\\paragraph{Differential vs. Derivative}\n\\begin{itemize}\n\\item Differential is a subfield of calculus that refers to infinitesimal difference in some varying quantity\n\\item Differential represents an equation that contains a function and one or more derivatives of that function\n\\item The function which represents the relationsihp between the dependent and the independent variables is unknown\n\\end{itemize}\n\n\\begin{itemize}\n\\item The derivative of a function is the rate of change of the output value with respect to its input value\n\\item Derivative represent the instantaneous change in the dependent variable with respect to its independent variable\n\\item The function which represents the relationship between the variables is known\n\\end{itemize}\n\n\n\\paragraph{Loss/Cost/Objective function}\n\n\\begin{itemize}\n\\item \\textbf{Loss function} is usually a function defined on a data point, prediction and label, and measures the penalty. For example: square loss in linear regression, hinge loss in SVM, 01 loss in theoretical analysis\n\\item \\textbf{Cost function} is usually more general. It might be a sum of loss functions over all training set plus some model complexity penalty. For example: MSE and SVM cost function\n\\item \\textbf{Objective function} is the most general term for any function that you optimize during training. For example, a probability of generating training set in maximum likelihood approach is a well defined objective function, but it is not a loss function nor cost function\n\\end{itemize}\n\nWe may say that, a loss function is a part of cost function which is a type of an objective function.\n\nFrom wikipedia, in mathematical optimization and decision theory, a loss function or cost function (sometimes also called an error function) is a function that maps an event or values of one or more variables onto a real number intuitively representing some \"cost\" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its opposite (in specific domains, variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized.\n\n\\paragraph{Surrogate vs. Approximation}\n\nI think surrogate and approximation can be used interchangeably.\n\nSurrogate loss function is used when the original loss function is inconvenient for calculation.\n\n\n\\bibliography{references.bib}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"hexsha": "3eb9d82302e17ee7af7fa983ef76b9e8bf9b7157", "size": 58361, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sutton_barto_notes.tex", "max_stars_repo_name": "pwyq/Notes", "max_stars_repo_head_hexsha": "242169f301c0c425633f4357260ec7c67d63c18e", "max_stars_repo_licenses": ["LPPL-1.3c"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "sutton_barto_notes.tex", "max_issues_repo_name": "pwyq/Notes", "max_issues_repo_head_hexsha": "242169f301c0c425633f4357260ec7c67d63c18e", "max_issues_repo_licenses": ["LPPL-1.3c"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "sutton_barto_notes.tex", "max_forks_repo_name": "pwyq/Notes", "max_forks_repo_head_hexsha": "242169f301c0c425633f4357260ec7c67d63c18e", "max_forks_repo_licenses": ["LPPL-1.3c"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.9111514053, "max_line_length": 575, "alphanum_fraction": 0.7325611282, "num_tokens": 15796, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352403, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.34876034896606634}} {"text": "\\documentclass{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{hyperref}\n\\usepackage{amsmath}\n\\usepackage{graphics}\n\\usepackage{graphicx}\n\\usepackage{float}\n\\graphicspath{ {images/} }\n\\title{Starlings Murmuration Simulation}\n\\author{Sankalan Pal Chowdhury, Shreshth Tuli}\n\\date{April 2018}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{Introduction}\nStarlings are a species of small birds that show tendency to migrate in flocks. These flocks are called murmurations and can contain thousands of starlings. Due to their small size and coordinated movements, they can often put up very beautiful displays while moving. In this project, we try to mathematically model this behavior of theirs and run a Computer Simulation of the same\n\\section{Modeling the Starling}\nWe seek to model each bird as an individual agent which reacts to its surroundings in order to determine its flying direction. In the most simple model, its environment consists only of other birds. The bird must therefore negotiate three types of constraints in order to determine its flying direction: External forces, Internal forces and Personal limitations.\n\\subsection{External Forces}\nExternal forces are the forces of mother nature which the bird act on the bird. We assume that the bird does not need to spend energy to generate these forces, and also cannot have any direct control over them. While there are many such forces, we seek to model only the major ones(this selection is similar to what is done for modeling airplanes):\n\\begin{itemize}\n \\item \\textbf{Gravity:} Gravity is perhaps the simplest yet most significant force to model. The concept of flight is fascinating essentially since it seems to overcome gravity.\n \\item \\textbf{Buoyancy:} Buoyancy may not be too important in air, but it is extremely easy to adjust within gravity. It can become significant for birds with very low density, so it cannot harm to model it.\n \\item \\textbf{Lift:} The lift force is the main contributor towards overcoming gravity. In fact the bird can maintain its altitude only if the sum of Lift and Buoyancy is same as Gravity.\n \\item \\textbf{Drag:} The drag force arises due to air resistance which tries to prevent the motion of any body traveling through a fluid. But for this force, the bird could keep flying without applying any effort.\n\\end{itemize}\nBirds, unlike airplanes can flap their wings in order to generate upward force. This force is more like a buoyant force, but needs the bird to spend energy to get it and can also control its magnitude to some extent. Modeling this requires a deep insight of the anatomy of the starlings which is beyond the scope of the model designers, therefore, we do not include it here.\n\\subsection{Internal Forces}\nOnce the external forces have been decided, the bird may decide to fly in a certain direction. It is somewhat misleading to call this decision a force, these are actually factors based on which the bird decides what kind of force to apply on itself(actually the bird applies the force on the environment and it gets it force by Newtons Third Law). These, however are all vector quantities and it is rather intuitive to consider them as components of the net force that the bird applies. \n\nIn the current model, we consider three forces, which are commonly used to simulate \\href{https://en.wikipedia.org/wiki/Flocking_(behavior)}{flocking behaviour}:\n\\begin{itemize}\n \\item \\textbf{Cohesion:} The cohesive force is the attractive force that a bird experiences towards other members of the flock. It ensures that the flock stays together and does not start moving in random directions.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm]{cohesion600}\n\\end{figure}\n\\item \\textbf{Separation:} The separative force acts opposite to the cohesive force and ensures that the birds keep a minimum distance between them and don't start colliding with each other\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm]{separation600}\n\\end{figure}\n \\item \\textbf{Alignment:} This force tries to coordinate the direction of motion of the entire flock. Each bird has the urge to fly in the same direction as its neighbors.\n\\end{itemize}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=10cm]{alignment600}\n\\end{figure}\n\nThe above mentioned factors present a somewhat idealistic view of how a certain bird desires to fly. However, there are other factors effecting how the bird really flies.\n\nTo start off, a bird is not likely to know about another bird which is flying right behind it. In fact, it can only see those birds which are in its \\textbf{field of view}, ie the maximum angle from its line of motion which it can see. This is actually quit large for birds since they have eyes on the sides of their heads. Further, two birds which are far off from each other are less likely to affect each other than two birds that are close by. This phenomenon is modeled by the (near)\\textbf{sightedness} of the bird. Finally, some birds may be more \\textbf{adventurous} than others, giving them urges to fly off in random directions rather than those specified by the above forces.\n\nFinally, a bird can apply only a limited amount of force, in which it would try to overcome the External forces and move in its desired direction. This limit would be determined both by an overall \\textbf{maximum acceleration} and the individual \\textbf{strength} of the bird.\n\\section{Mathematical Formulation}\nTo begin our mathematical formulation, we need to first fix a coordinate system. We fix the origin at the flock centroid(this is beneficial for bounding box creation during implementation) and the y axis upwards. The x and z axis are arbitrarily chosen since the system has cylindrical symmetry.\n\nThe mathematical values of the external forces are more or less standard and do not require much explanation\n\\begin{center}\n\n $$\\vec{F_{gravity}}=\\rho_{bird}V\\vec{g}$$\n $$\\vec{F_{buoyancy}}=-\\rho_{air}V\\vec{g}$$\n $$\\vec{F_{lift}}=C_{lift}\\rho_{air}\\vec{v}^2W\\hat{j}$$\n $$\\vec{F_{drag}}=-\\frac{1}{2} C_{drag}\\vec{v}^2A\\hat{v}$$\n $$\\vec{F_{external}}=\\vec{F_{gravity}}+\\vec{F_{buoyancy}}+\\vec{F_{lift}}+\\vec{F_{drag}}$$\n\n $$\\rho_{bird},\\rho_{air} \\text{ are the densities of the bird and air,}$$\n $$C_{lift}, C_{drag} \\text{ are the lift and drag coefficients,}$$\n $$g \\text{ is the acceleration due to gravity }=0\\hat{i}-9.8\\hat{j}+0\\hat{k}$$\n $$W, A \\text{ are wing area and cross sectional area respectively,}$$\n $$\\vec{v} \\text{ is the current velocity of the bird,}$$\n $$V \\text{ is the volume of the bird}$$\n \n\\end{center}\n\nThe internal forces are more involved. To begin with, field of view simply filters out the birds which are beyond the specified angle, so all statements made hereafter are w.r.t. the rest of the birds.\n\n\\subsection{Internal Forces}\nTwo different approaches are possible to deal with the sightedness issue. One is to say that only a fixed number of nearest neighbors affect the bird. This partially relies on the fact that the bird would not want to over complicate its life and would approximate the world by a fixed number of birds around it. However, the evidence for this is quite empirical and needs to be judge in light of the fact that the number of birds in a certain region around the bird is held more or less constant by the Cohesion and separation forces. This also does not account for the chance that the bird might look at a very distant bird and adjust itself according to it.\n\nThe other view is to limit the distance till which a bird can see. This seems more logical, except for the fact that there is unlikely to be a hard boundary at a given distance. Therefore, we choose an exponential decay function to model the sightedness of the birds:\n\\begin{center}\n$$W_i=e^{-sd_i}$$\n\\[\n d_i \\text{ is the distance to the } i^{th} \\text{ bird}\n s \\text{is the sightedness of the bird}\n\\]\n\\end{center}\nAt this point, it may seem strange that a bird would look at like a thousand other birds before deciding which direction to move, which is clearly illogical. One way of explaining this would be that the bird looks at only a certain number of birds at each moment, and the quantity mentioned here is the over time expectation of looking at each bird.\n\nThis done, we can move on to the three forces. We calculate for each bird separately(call this the target bird and then aggregate to get the net forces.\n\nIt is intuitive to keep Alignment proportional to the velocity of the target bird. As for Separation and Cohesion, they should logically be proportional to the inverse of the distance between the two birds(Actually, cohesion can be constant, allowing for the sightedness to take care of the distance factor. Ignoring this fact is more of a personal intuition). Also, the separation force needs to grow faster that the cohesion force, as we want to prevent collisions. We chose to take the simplest expressions that satisfy these constraints.\n\\begin{center}\n$$\\vec{F_{cohesion,i}}=\\frac{C_{coh}}{d_i} \\hat{d_i}$$\n$$\\vec{F_{separation,i}}=-\\frac{C_{sep}}{d_i^2} \\hat{d_i}$$\n$$\\vec{F_{align,i}}=C_{ali}\\vec{v_{i}}$$\n\n$$\\hat{d_i} \\text{ is the unit vector towards the } i^{th} \\text{ bird }$$\n$$\\vec{v_{targ}} \\text{ is the velocity of } i^{th} \\text{ bird }$$\n$$C_{coh},C_{sep},C_{ali} \\text{ are respective coefficients}$$ \n\\end{center}\n\nOnce these forces are calculated due to each bird, we need to aggregate them to get net internal force. Aggregation can be done in many ways, but the two most common ways are summation and averaging. Summation is good for aggregating quantities that are likely to add up destructively. This is clearly true for the separation force. Averaging, on the other hand, make sense for quantities that would add up constructively, like the alignment force. Cohesion force can go either way, but it is somewhat safer to consider averaging since a zero is less dangerous than an infinity. The Weights assigned to the different birds is taken care of here.\n\n\\begin{center}\n $$\\vec{F_{cohesion}}=\\frac{\\sum_iW_i\\vec{F_{cohesion,i}}}{\\sum_iW_i}$$\n $$\\vec{F_{alignment}}=\\frac{\\sum_iW_i\\vec{F_{alignment,i}}}{\\sum_iW_i}$$\n $$\\vec{F_{separation}}=\\sum_iW_i\\vec{F_{separation,i}}$$\n\\end{center}\n\nFinally, we need to add the adventure component to these forces to calculate the net internal force on the bird. Now, while an adventurous bird may not want to align itself with the other birds or get attracted to them, it will certainly not want to collide with other birds. Therefore, the random component will only replace the alignment and cohesion forces and not the separation force.\n\\begin{center}\n $$\\vec{F_{internal}}=\\vec{F_{separation}}+\\frac{1000-Adv}{1000}(\\vec{F_{cohesion}}+ \\vec{F_{alignment}})+\\frac{Adv}{1000}\\vec{R}$$\n \n $$Adv \\in \\{0,...,1000\\} \\text{ is the adventurousness parameter }$$\n $$\\vec{R} \\text{ is a suitable random vector(suitability depends on range of magnitudes)}$$\n\\end{center}\n\\subsection{Putting it Together}\nNow that we have both the external and internal force vectors, we need to put them together to calculate the acceleration, velocity and power of the bird. It is here that we need to take into account the maximum force that the bird can apply is limited.\n\nIf the bird was an inanimate object, it would simply drift in the direction of the external force. What the bird really does is try its best to change this to its desired direction of flight as much as possible. This difference is simply given by $\\vec{v_{diff}}=\\vec{v}+\\vec{F_{external}}-\\vec{F_{internal}}$. Since we can only take discrete time, the desired acceleration is $\\vec{a_{desired}}=\\vec{v_{diff}}/P$ where P is the clock period. The magnitude of $\\vec{a_{desired}}$ needs to be bounded by a $a_{bound}=a_{max} \\times strength$ where $a_{max}$ is the global maximum acceleration and strength is a property of the bird.\\newline\n\nThe bounding mechanism itself is somewhat non trivial. The naive way of doing it would be to cut off any value higher that the required bound. But that would be very unrealistic as in a real scenario, the bird would start facing difficulty gradually till it reaches a point where it can no longer fly faster. \nA good function to approximate this is the $tanh$ function, which has derivative of 1 around zero, but is bounded by 1 on the positive side. However, its derivative never falls to 0 for any finite parameter.\n\nOnce the bounding is done, we can calculate Power easily using laws of physics.\n\\begin{center}\n $$\\vec{a_{final}}=tanh(\\frac{|\\vec{a_{desired}}|}{a_{bound}})\\frac{a_{bound}}{|\\vec{a_{desired}}|}\\vec{a_{desired}}$$\n $$\\vec{v_{final}}=\\vec{v}+\\vec{v}+\\vec{F_{external}}-\\vec{a_{final}}$$\n $$Power=m\\vec{a_{final}}.\\vec{v_{final}}$$\n\\end{center}\nPosition and Energy follow trivially.\n\n\n\\pagebreak\n\\section{Implementation}\nHaving designed the model, we attempt to implement it on a computer. The aim of this part is to 1) Be able to visualize the murmurations of the starlings on a computer screen, and 2) approximate some statistics about these murmurations.\n\nThe implementation needs to proceed in two parts. First, we look at what kinds of deviations must be made from the original model in order to be able to simulate it. Second, we choose a framework to implement the design. Both of these are dependent on the characteristics of the model and/or the constraints set by the programming environment.\n\\subsection{Deviations from the Model}\nOne of the major issues we need to deal with here is the fact that in real life, Starlings are free to move anywhere in the open sky. However, our simulation only lets us look at the starlings from a fixed viewpoint, and there is little point in having the murmurations happening out of the visible region. There are multiple options to solve this:\n\\begin{itemize}\n \\item \\textbf{Moving the Viewpoint:} This approach tries to approximate the real motions of the birds as best as possible, by moving the viewpoint automatically along with the birds. The main advantage of this is the model does not need to be tampered with. However, there can be scenarios in which the motion of the view point would cancel out the motion of the birds, making them appear to be static. Also, the flock may split into two, leading to issues on deciding the viewpoint. Alternatively, the shifting of viewpoint can be left upto the user, but that is probably not very user friendly.\n \\item \\textbf{A Circular World:} To give the feeling of an infinite space, we can make the edges of the view wrap around to the opposite edge. This also would require minimal changes to the model, but its physical effect can hardly be explained. Also, it is not logical anyway to let the $-y$ direction extend to infinity.\n \\item \\textbf{Adding Imaginary Force}: To prevent the birds from escaping into infinity, we can define a potential like field that forces the birds to remain in a specified region, beyond which this field becomes too large. This can represent pressure zones or simply unfamiliar territory for the birds. However, if not tuned properly, such a field can cause the birds to actually divert to infinity.\n \\item \\textbf{Limiting Position Directly:} This would be similar to what was done for limiting the maximum acceleration, forcing the birds to stay in a fixed region. The chief downsides of this is the lack of a physical explanation and the fact that the birds might all want to escape in one direction, but get restricted by this force, making them appear as if they are all at the same point.\n\\end{itemize}\nIn our current implementation, we choose to go with a combination of the last two approaches, which tend to cancel out some of each others negative points. For the force field, we choose a force that is zero in some region around the origin and then grows as the square root of distance from origin. This choice is kind of empirical, with comparisons performed over multiple regularized polynomial fields. The modified set of equations are given below:\n\\begin{center}\n $$\\vec{v_{positional}}=-\\vec{P} * max(\\sqrt{\\frac{|\\vec{P}|}{P_{max}}}-\\sqrt{5},0)$$\n $$\\vec{v_{diff}}=\\vec{v}+\\vec{F_{external}}-\\vec{F_{internal}}-\\vec{v_{positional}}$$\n $$\\vec{P_{final}}=tanh(\\frac{|\\vec{P_{desired}}|}{20*P_{max}})\\frac{20*P_{max}}{|\\vec{P_{desired}}|}\\vec{P_{desired}}$$\n P \\text{represents position of the bird, no subscript means current position}\n\\end{center}\nThe other equations remain unchanged.\nApart from this, it turns out that the exponential sightedness model can sometimes become too much to calculate(it needs to be done O($n^2$) times in each time step, where n is the number of birds). We therefore keep an alternative of using a constant viewing range(ie a step function). In the same vein, all the other internal forces are also made toggle-able. \n\n\\subsection{Frameworks and Design Decisions}\nWhile it has not been stated previously, some decisions are implicit to our model:\n\\begin{itemize}\n \\item Since every starling has been modeled as an entity, and \\textbf{Object Oriented Paradigm} is necessary to implement these.\n \\item there is a stochastic dimension to this simulation. Every calculation therefore need to be performed on a clock edge.\n \\item Interaction between different birds is purely observational, so the all the computations are parallelisable. This makes our program fit for multi-threading.\n\\end{itemize}\n\\subsubsection{The Clock}\nWhile it is clear that a clock needs to be used, there are actually multiple options on how the clock gets used. The simplest way to go about it is to use a \\textbf{real clock} which calculates the position and updates the display in real time. In such a setting, there would be a trade off between the frequency of the clock and the number of birds. The advantage, however is that no storage needs to be done in order to create a realistic display.\n\nThe alternative to this approach is to use a \\textbf{logical clock} which can proceed at much slower pace than the actual clock, which only influences the display. This means that we can slow down the Logical clock arbitrarily, allowing us to simulate as many birds as we like(well, kind of). In such an approach, the user cannot dynamically set parameters or visualize what the current setting looks like. The main problem here though, is the fact that the positions of each bird needs to be saved into a file, and this file can get huge. \n\nSuppose we store the position and velocity of each bird at each instant as vectors of single precision floats. If we use a 50 hz clock for 2000 birds for 10 minutes we end up with $50*6*1000*30*32=1.152 Gb$ of data. Note that this data would not make any sense to a human, and if we want to use a text file, the file size blows up even more. Another consideration here is that the display of the birds itself takes up significant computational power, and that needs to be done on a real clock anyway. \n\nTherefore, we choose to stick to a real clock\n\\subsubsection{Order of Computation with Multi-threading}\nWhen talking of objects, the ideal model is that all instances of an object run their computation concurrently. This would also be closest to the real world. \n\nIn a real programming paradigm, however, to run everything concurrently is not possible(unless we have as many logical cores as birds). The closest we can get is to give each bird its own thread, but that would mean a huge number of context switches, and therefore huge slowdown. Intuitively, since the steps taken by each bird in each tick of the clock would be rather small, imposing a fixed order should not harm the model too much. \n\nIn our implementation, we use 4 threads(this number can be increased depending on how many cores are available) and within each thread, we provide two options: In one, we impose a fixed order of execution amongst the birds, while in the other, we randomize within each thread. For the non-random approach, a simple segregation scheme is used viz that from the vector of birds thread $t$ gets birds which have index $i$ such that $i\\,\\%\\,4\\,=\\,t$. Order between the threads is always random, as long as at least 4 cores are available. So there are two levels of randomization that this approach provides:\n\\begin{enumerate}\n\\item The inherent scheduler of the operating system and the seemingly random distribution of load on the physical cores, randomize the order in which the computation is done for each bird.\n\\item An array of numbers 1 to flock size is shuffled and divided into 4 sub-arrays. Each thread gets a sub-array and uses it's values as indices for execution of the computation, thus making it random.\n\\end{enumerate}\n\nAlthough the approach used still does not allow concurrent calculation for each bird (which is a limitation of the hardware) but provides the best way to model the system with finite logical cores.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=10cm]{Flock_threading}\n\\caption{Multi-threading flowchart with randomization}\n\\end{figure}\n\n\n\\subsubsection{Selected tools}\nThe calculation is implemented in \\textbf{C++} because it is perhaps one of the most widely used, rapidly evolving and fastest language for Object Oriented Programming. The display part is done in \\textbf{OpenGL} which is said to be the \\textit{industry standard for high performance graphics}. Other platforms like $Unity$ and other similar physics engines have not been used as they do not allow tuning at the root level even though they allow users to model the problem much more easily.\n\\section{Evaluation and Parameter Tuning}\nOnce we have implemented our design, we need to tune the hyper-parameters in order to get realistic displays. But before that, we need to have some metrics to evaluate the performance.\n\\subsection{Metrics for Correctness}\nIn the age of big data and machine learning, one is always talking of differentiable and convex cost functions. Luckily or unluckily, these things have not yet invaded the lives of the starlings, so we need to have heuristic measures of correctness.\n\nThe best judgment is obviously given by visualization of the simulation, to judge whether it is realistic or not. However, is is normally better to have a more quantitative measures. Following are some such possibilities:\n\\begin{itemize}\n\t\\item The most important correctness indicator is the \\textbf{visual effect} created by the starlings when in motion. A low number of birds may not give such an effect so relaxation should be for starlings when they are less than say 100.\n \\item The starlings have a diet consisting fruits and insects. Based on this, we can estimate how much Energy they can actually spend in flying. Thereafter, we can check if the \\textbf{Average Energy} and the \\textbf{Maximum Energy} spent by the starling is around this value or not\n \\item Based on the energy consumption in a finite time like 10 minutes, the amount of \\textbf{food intake} should be less than the typical food intake of a starling in a day which is around 10g. Such measurements are from online sources like \\href{https://www.tandfonline.com/doi/pdf/10.1080/00063657309476384}{this}.\n \\item The starlings can withstand only so much \\textbf{force} on themselves. If it becomes too high, the will probably get squashed. \n \\item \\textbf{Collisions} never take place in the real setting, so even if two birds get too close, we should rethink our model. \n \\item The starlings can not apply a large amount of power while in motion, so the parameters \\textbf{Average Power} and \\textbf{Maximum Power} should be in an acceptable range.\n\\end{itemize}\nBeyond these, there are things like maximum velocity and acceleration, but that have already been accounted for explicitly in our model, so it makes little sense in measuring them. Finally, we should also see what is the maximum number of birds that we are able to simulate, as that is definitely a measure of performance.\n\n\\pagebreak\n\n\\subsection{Tuning the Hyper-parameters}\n\nTuning the hyper-parameters is crucial for a more realistic simulation. Although, these parameters can be determined by complex calculations and analysis of the birds anatomical structure and the cognitive model. This becomes much more complicated when such functions are to be mapped for discrete time. A not so precise but more effective approach would be to augment the model parameters by brute-force and intelligent guesses based on the metrics of correctness as discussed earlier. This approach leads to errors in the model, but the deviation from realism is also because of modeling constraints as discussed. These approximations and relaxations allow us to take such a decision for parameter tuning. \\\\\nFor the internal forces, the parameter tuning has been based on a intuitive notion of prioritizing the various forces: separation, alignment and cohesion. As separation must dominate at lower distances and cohesion at higher, it is important that the constants used for cohesion are higher than that of separation but the dependence with distance must rise faster in the latter case. For realistic simulation, it is also important to give some randomness to the forces considering that these random forces do not dominate over the systematic ones used in the model. As these forces are based on a factor that exponentially decays or as a step function with distance, there constants must be considering the distances as well. \\\\\nFor the external forces including lift, drag, buoyancy and lift some physical aspects need to be investigated. The cross section area and the drag coefficient in the real scenario change dynamically but for the ease of implementation and their somewhat low effect on the model have been neglected. It is important to observe that the drag force depends on the airflow around the bird. A turbulent airflow is expected most of the times due to the flock around it. Thus, we use a drag coefficient somewhat in between but more the side of that for turbulent air. The decision as to how much the air is turbulent or laminar is based on quantitative data of the Reynolds number on the internet as on \\href{https://aip.scitation.org/doi/abs/10.1063/1.4807064}{\"this\"} link, and empirical observations. \n\n\\section{Conclusion}\nIn this project we have been able to approximately simulate the murmurations of starlings and also calculate some essential parameters of the motion. The fact that our outputs actually look realistic is definitely evidence in favour of the fact that the real model of flight is somewhat similar to the one that we have defined. This is therefore also a step towards understanding the behavior of starlings.\n\nFinally, our model contains a lot of hyper parameters, thus a lot of flexibility. In future, should someone be able to track the actual motion of starlings, the birds can also be made \\textit{Reinforcement Learning} agents.\n\\end{document}\n\n", "meta": {"hexsha": "db8f403c46a7796145ce3110b240a9b0c32d47e0", "size": 26906, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/main.tex", "max_stars_repo_name": "shreshthtuli/FlockSim", "max_stars_repo_head_hexsha": "4576f75c2fe61dc61311e8c582cb9c072e6a64a1", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-05-06T14:28:44.000Z", "max_stars_repo_stars_event_max_datetime": "2018-05-06T14:28:44.000Z", "max_issues_repo_path": "doc/main.tex", "max_issues_repo_name": "shreshthtuli/FlockSim", "max_issues_repo_head_hexsha": "4576f75c2fe61dc61311e8c582cb9c072e6a64a1", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/main.tex", "max_forks_repo_name": "shreshthtuli/FlockSim", "max_forks_repo_head_hexsha": "4576f75c2fe61dc61311e8c582cb9c072e6a64a1", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 118.5286343612, "max_line_length": 797, "alphanum_fraction": 0.7801233926, "num_tokens": 6237, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.550607350786733, "lm_q2_score": 0.6334102498375401, "lm_q1q2_score": 0.3487603396242106}} {"text": "\\documentclass[main.tex]{subfiles}\n\\begin{document}\n\n\\section{Physics beyond the Standard Model}\n\n\\marginpar{Sunday\\\\ 2020-7-5, \\\\ compiled \\\\ \\today}\n\nA Grand Unified Theory would unify strong, electromagnetic and weak interactions. \n\nThe electromagnetic and weak interactions were unified into a \\(SU(2)_L \\times U(1)_Y\\) theory, where the coupling constants for the two groups, \\(g_2 \\) and \\(g_1 \\), are pretty close. \nThe strong coupling, \\(g_s\\), is instead quite far from these.\n\nAll of these are running couplings: we have \n%\n\\begin{align} \\label{eq:first-order-running-coupling}\n\\dv{\\alpha _i}{\\log q^2} = b_i \\alpha_{i} + \\mathcal{O}(\\alpha_{i}^3)\n\\,,\n\\end{align}\n%\n\\todo[inline]{what's up with the \\(q^2\\)?}\nwhere \\(b_i\\) is given by: \n%\n\\begin{align}\nb_i = - \\frac{1}{4 \\pi } \\qty[ \\frac{11}{3} C(G_i) - \\sum _{f} \\frac{4}{3} T(R)_f]\n\\,,\n\\end{align}\n%\nwhere we are neglecting the contributions from scalars, and where the Casimirs in the \\(SU(N)\\) case are \\(C(SU(N)) = N\\) and \\(T(R)_{SU(N)} = 1/2\\).\n\nSo, if we label electromagnetism, weak and strong interactions with the numbers 1, 2 and 3 we find \n%\n\\begin{align}\nb_3 &= - \\frac{1}{4 \\pi } \\qty[ \\frac{11}{3} \\times 3 - \\frac{4}{3} \\qty( \\frac{1}{2} + \\frac{1}{2}) n _{\\text{fermion generations}}] \\\\\nb_2 &= - \\frac{1}{4 \\pi } \\qty[ \\frac{11}{3} \\times 2 - \\frac{4}{3} \\frac{1}{2} 4 \\frac{1}{2} n _{\\text{fermion generations}} ] \\\\\nb_1 &= - \\frac{1}{4 \\pi } \\qty[ 0 - \\frac{20}{9} n _{\\text{fermion generations}} ]\n\\,.\n\\end{align}\n\n\\todo[inline]{The calculation is not super clear: what are we counting exactly for each interaction? Are the fermions not the same?}\n\nBy integrating \\eqref{eq:first-order-running-coupling} we get the \\textbf{Renormalization Group Equations}: \n%\n\\boxalign{\n\\begin{align}\n\\frac{1}{\\alpha_{i} (q^2)} = \\frac{1}{\\alpha_{i} (M_X^2)} + b_i \\log \\qty( \\frac{M_X^2}{q^2})\n\\,.\n\\end{align}}\n\nCould we find a scale such that the three couplings become equal?\nIf it exists, such a scale would be called the \\textbf{Grand Unified Theory} Energy scale. \n\nSo, we have three equations for the values of the three couplings: \nwe can input two of the low-energy couplings and predict the GUT scale \\(M _{\\text{GUT}} \\sim \\SI{e15}{GeV}\\), the GUT coupling \\(\\alpha _{\\text{GUT}}\\), and one low energy parameter.\n\nOne possible prediction is \\(\\sin^2\\theta_{w} = e^2 / g^2\\), where \\(\\theta_{w}\\) is the weak mixing angle. \n\nThis kind of works: the three couplings come close but do not reach simultaneous equality.\nA key piece of the puzzle is the fact that the way the running coupling changes depends on the \\emph{particle content} at each energy from the electroweak scale \\(M_W \\sim \\SI{200}{GeV}\\) and the GUT scale. \n\nThe thing we try to do is to embed our gauge group into some larger symmetry group: \n%\n\\begin{align}\nSU(3)_c \\otimes SU(2)_L \\otimes U(1)_Y \\subset G _{\\text{GUT}}\n\\,.\n\\end{align}\n\nFor example, we could have \\(G _{\\text{GUT}} = SU(5)\\). This group has \\(5^2 - 1 = 24\\) generators, 12 of which are the known ones, and 12 of which would be new ones.\n\nWhat would be the charges of such a boson with respect to the known symmetries? Let us work with electric charge instead oh hypercharge directly. Then, we would have 6 bosons \\(X^{\\mu }\\) with charges \\(3, 2, 4/3\\) respectively with respect to \\(SU(3)_c\\), \\(SU(2)_L\\) and \\(U(1) _{\\text{em}}\\); and 6 bosons \\(Y^{\\mu } \\) with charges \\(3, 2, 1/3\\). \n\nThese would mediate baryon- and lepton-number violating interactions, such as \\(d_L + e^{+} \\to X^{\\mu } \\to u_L + u^{c}_{L}\\).\nThis kind of interaction would allow a process like the decay of a proton \\(p\\) into a positron and a pion: \\(p \\to e^{+} + \\pi^{0}\\). \n\nThe decay rate of this process is \\(\\Gamma \\propto M_X^{4} / m_p^{5}\\), so the lifetime of a proton would be \n%\n\\begin{align}\n\\tau_{p} \\sim \\frac{M_X^{4}}{\\alpha^2 m_p^{5}} \\sim \\SI{5e32}{yr}\n\\,,\n\\end{align}\n%\n\\todo[inline]{not what he writes, but his formulas are dimensionally inconsistent as well so I don't know}\n\nbut our experimental bounds are already higher, at \\(\\tau_{p} \\gtrsim \\SI{e33}{yr}\\). \nThe way this theory would work is by introducing a new Higgs-like field, whose VEV would be of the order of \\(M _{\\text{GUT}}\\). \n\nSo, there are two phase transitions: one at \\(E \\sim T \\sim M _{\\text{GUT}}\\), from the big symmetry group \\(G\\) to \\(G _{\\text{SM}}\\), and then one around \\(M_W\\), from \\(G _{\\text{SM}}\\) to \\(SU(3)_c \\times U(1) _{\\text{em}}\\).\n\n\\subsection{GUTs and neutrino mass}\n\nIn the Standard Model we only have \\(\\nu_{L}\\) in an isospin doublet with \\(e_L\\), and no \\(\\nu_{R}\\). \nSo, a neutrino mass term could only be written as \n%\n\\begin{align}\n\\nu_{L}^{\\alpha } \\nu_{L}^{\\beta } \\epsilon_{\\alpha \\beta }\n\\,,\n\\end{align}\n%\nwhere \\(\\alpha \\) and \\(\\beta \\) are Lorentz indices. \nLorentz transformations \\(\\Lambda \\in SO(1, 3)\\) are locally isomorphic to \\(SU(2)_L \\times SU(2)_R\\) acting on the left- and right-handed components of a spinor. Note that this \\(SU(2)\\) is \\emph{not} the isospin \\(SU(2)_L\\): it is, instead, a different way to write our Lorentz transformation. \n\nThe issue is with the way that the term \\(\\nu_{L}^{\\alpha } \\nu_{L}^{\\beta } \\epsilon_{\\alpha \\beta } \\) transforms under an \\(SU(2)_L\\) transformation: \\(\\nu_L \\) has \\(T_3 = + 1/2\\), while \\(e_L\\) has \\(T_3 = - 1/2\\). So, \\(\\nu_{L } \\nu_{L}\\) has \\(T_3 = +1\\): a component of a triplet! \n\nIf we define \\(L = (\\nu , e)_L\\) we can write a term like \\(L^{i} L^{j} \\Delta^{ij}\\), where \\(i\\) and \\(j\\) are \\(SU(2)_L\\) gauge indices, and \\(\\Delta \\) is an \\(SU(2)_L\\) triplet, which without loss of generality can be taken to be symmetric. \n\nHowever, in the SM we only have a Higgs scalar doublet: we cannot write a term like \\(LLH\\); we could write \\(LHLH\\), which would have dimension 5: we would need to introduce a new energy scale \\(M\\), so that after SSB the term looked like \n%\n\\begin{align}\n\\frac{L \\expval{H} L \\expval{H}}{M} \\implies m_\\nu \\sim \\frac{\\expval{H}^2}{M} \\sim \\frac{v^2}{M}\n\\,.\n\\end{align}\n\nIn order to comply with the bounds we have for the neutrino masses, we would need \\(M \\gg v \\sim M_W \\sim \\SI{100}{GeV}\\).\n\nAdding to the Standard Model an operator of dimension larger than 4 coming from physics above the electroweak scale makes the SM an \\emph{effective} field theory. \nWe could have \\(M\\) be the GUT scale (which would work: \\(v^2 / M _{\\text{GUT}} \\sim \\SI{10}{meV}\\)). \n\nThis would allow us to embed the SM symmetry into a parity-conserving Lagrangian, which becomes P-breaking only at low energies.\n\n\\subsubsection{Pati-Salam theory: \\(SO(10)\\)}\n\nFor example, if \\(G _{\\text{GUT}}\\) was \\(SO(10)\\) (Pati-Salam theory) we would have the subgroup \\(SU(4) \\times SU(2)_L \\times SU(2)_R\\). \n\nIn this theory, all fermions of a generation (both left and right for all: up and down quarks (three each, for color charge), electrons, neutrinos) are different components in a spinor which has dimension 16. \n\nIn this theory, the scalar coupling to the right-handed neutrino is in the form \n%\n\\begin{align}\n\\nu_{R}^{\\alpha , i} \\nu_{R}^{\\beta, j } \\epsilon_{\\alpha \\beta } \\Delta^{ij}_{R}\n\\,,\n\\end{align}\n%\nwhere \\(ij\\) are \\(SU(2)_R\\) indices, and \\(\\alpha \\beta \\) are Lorentz indices.\nThis is a Majorana mass term for the right-handed neutrino.\nNow, the VEV \\(\\expval{\\Delta_{R}}\\) will be approximately at the GUT energy scale.\n\nThen, electroweak symmetry breaking provides the Dirac term \\(\\overline{\\nu}_{L} \\nu_{R} \\expval{H}\\) at \\(\\expval{H} \\sim \\SI{100}{GeV}\\). \n\\todo[inline]{Recall how}\n\nThe neutrino mass matrix then reads \n%\n\\begin{align}\n\\left[\\begin{array}{cc}\n0 & v \\\\ \nv & V _{\\text{GUT}}\n\\end{array}\\right]\n\\,,\n\\end{align}\n%\nwhere the vector space is \\((L, R)\\). \nThe light neutrino mass eigenstate is then mostly \\(m_{\\nu_{L}} \\sim v^2 / V _{\\text{GUT}}\\) while the other one is mostly \\(m_{\\nu_{R}} sim\n V _{\\text{GUT}}\\). \n \nThis simultaneously explains the lightness of the observed neutrinos and the fact that we have only observed left-handed ones in the weak interactions we tested. \n\n\\subsubsection{Baryon and lepton number conservation}\n\nIn the SM, baryon and lepton number conservation are ``accidental'' (not imposed from on high) symmetries. \n\nIn general GUTs, they are violated in couplings of new super-heavy gauge bosons with fermions.\nSo, the new bosons interact both with leptons and quarks. \nHowever, in many GUTs the combination \\(B - L\\) is conserved. \n\nThis is not the case, for example, in \\(SO(10)\\): the term \\(\\nu_{R} \\nu_{R} \\) has \\(\\Delta L = 2\\) and \\(\\Delta B = 0\\). \n\n\\subsection{The gauge hierarchy problem}\n\nEven in low (electroweak-scale and lower) energy processes we can have GUT superheavy virtual particles appearing. These should be accounted for in the loop integrals. \n\nThe situation is different for fermions and gauge bosons on one side, versus the scalar Higgs field. \n\nFermions and gauge bosons only acquire mass after SSB: so, if we draw a diagram representing the interaction between the fermion field and the Higgs field, we must have a Higgs line as well as two fermion lines. For the gauge bosons the thing is the same, but we need two Higgs lines as well as two vector boson lines, in order to contract their Lorentz indices. \n\nSo, loop corrections will build on this diagram, but they cannot change the incoming lines:\ntherefore, their mass terms will always be proportional to \\(v\\), and never to \\(M _{\\text{GUT}}\\). \n\nThere is no such SSB protection for the mass of the Higgs boson, which already has a mass term \\(\\mu^2 \\abs{\\phi}^2\\) before SSB: GUT loop corrections can indeed affect it. \n\nLet us describe the way these perturb the mass like \\(\\mu^2 = \\mu_0^2 + \\delta \\mu^2\\). If \\(\\mu_0\\) is at the electroweak scale while \\(\\delta \\mu\\) is at the GUT scale, this breaks everything!\n\nThe VEV of our Higgs would be at the GUT scale, even though at tree level it might start off at the electroweak scale. \nThere are two ways to solve this problem: \none is to fine-tune the parameters in the Lagrangian to cancel the radiative corrections to \\(\\mathscr{L} _{\\text{Higgs}}\\). \n\nThe alternative is to introduce a cutoff scale for the corrections: this can either be achieved by supersymmetry which is broken around \\SI{100}{GeV} to \\SI{1}{TeV}, or by the introduction of new spacetime dimensions.\n\n\\subsubsection{Supersymmetry}\n\nThe idea of SUSY is to introduce a symmetry transformation mapping bosons to fermions: each SM particle would have a SUSY partner with a different spin. \n\nThese are characterized by their funny names.\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\nSM particle & SUSY partner\\\\\n\\hline\nelectron \\(e^{-}\\), spin \\(1/2\\) & selectron \\(\\widetilde{e}^{-}\\), spin \\(0\\) \\\\\n\\(W^{\\mu }\\) boson, spin \\(1\\) & \\(W\\)-ino \\(\\widetilde{W}\\), spin \\(1/2\\) \\\\\nHiggs boson \\(H\\), spin \\(1\\) & Higgsino \\(\\widetilde{H}\\), spin \\(1/2\\) \\\\\nneutrino \\(\\nu \\), spin \\(1/2\\) & sneutrino \\(\\widetilde{\\nu }\\), spin \\(0\\) \\\\\nphoton \\(\\gamma \\), spin \\(1\\) & photino \\(\\widetilde{\\gamma }\\), spin \\(1/2\\) \\\\\ngraviton, spin \\(2\\) & gravitino, spin \\(3/2\\) \n\\end{tabular}\n\\label{tab:supersymmetric-partners}\n\\caption{Supersymmetric partners.}\n\\end{figure}\n\nIf we start off with \\(\\mu_0^2 \\phi^2\\) at the correct scale, we are done: the correction \\(\\delta \\mu^2\\) vanishes for a SUSY theory, since the contributions for a fermion loop for the Higgs self-interaction diagram precisely cancel with the sfermion loops. \n\nThis works as long as SUSY is broken around the \\SI{}{TeV}, so that the electroweak breaking scale are protected (they do not vanish like the other ones). \n\nIf SUSY is an exact symmetry, then we expect the masses of the particles and of their SUSY partners to be equal. \nIf, instead, it is broken around the electroweak scale we may have \\(m_{\\widetilde{f}} - m_f \\sim M_W\\).\n\nThe important thing is then that this predicts that the SUSY partners are not very much more massive than the normal particles; and they have not been found at LHC. \n\nAs we go to higher energies in the search for new particles without finding SUSY ones we get ever stricter bounds on the SUSY breaking scale.\n\nSUSY is a discrete symmetry, an \\(R\\)-parity such that fermions have \\(R = +1\\) while bosons have \\(R = -1\\). \nThis being a symmetry means that the lightest SUSY particle cannot decay: it cannot turn into SM particles since they have a different \\(R\\), and it cannot turn into heavier SUSY particles by energy conservation. \n\nIn the Supersymmetric SM the Grand Unification works well: the couplings meet at \\(M _{\\text{GUT}}^{\\text{SUSY}} \\approx \\SI{e16}{GeV}\\). \nThis is also within our bounds for the proton lifetime.\n\nThe lightest neutralino is a good WIMP-like DM candidate. \n\n\\subsubsection{Extra dimensions}\n\nThe idea here is to suppose that our 1+3-dimensional spacetime is embedded into a higher-dimensional one, for example a 1+4-dimensional one. \nThen, suppose that only gravity ``feels'' this extra dimension: then, if would be very weak in comparison to the other interactions. \n\n\\todo[inline]{REALLY not clear what this means formally}\n\nThis way, instead of having the gravitational interaction be weak because of \\(M_P\\) being very large we could have \\(M_P\\) at the electroweak scale with \\(G_N\\) being still small; most of the gravitational flux lines could be flowing in these new spatial directions. \n\nIn these theories, every ordinary SM particle is accompanied by many more with the same quantum numbers but increasing mass --- the Kaluza-Klein modes.\nThe lightest KK mode might be a DM candidate!\n\n\\end{document}\n\n", "meta": {"hexsha": "def4d1d7eb793059a564569dbfc5f173e5e12f47", "size": 13524, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ap_second_semester/astroparticle_physics/may26.tex", "max_stars_repo_name": "jacopok/notes", "max_stars_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-10-10T13:10:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T14:52:50.000Z", "max_issues_repo_path": "ap_second_semester/astroparticle_physics/may26.tex", "max_issues_repo_name": "jacopok/notes", "max_issues_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ap_second_semester/astroparticle_physics/may26.tex", "max_forks_repo_name": "jacopok/notes", "max_forks_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-03T16:20:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-06T16:11:07.000Z", "avg_line_length": 56.35, "max_line_length": 363, "alphanum_fraction": 0.7020851819, "num_tokens": 4134, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3486032395372771}} {"text": "\\documentclass[./standalone.tex]{subfiles}\n%\\documentclass[../../../CR/pac.tex]{subfiles}\n\n\\begin{document}\n\t\n\t%% =======================================\n\t\\section{Algorithms}\n\tFor this labwork I have implemented 2 metaheuristics algorithms. The first one being the steepest hill algorithm with restarts and the second one being the tabu algorithm. \\\\\n\t\n\t\\subsection{Steepest Hill with restarts}\n\nBelow is the steepest hill algorithm without restarts. We won't detail here the auxiliary functions such as the \\textit{getDistance} method from the TSP class or the \\textit{debugSteepestRun} which only allows a few debugging prints. Furthermore some methods will be detailed in the section \\ref{neighb}. \n\t\\begin{lstlisting}[style=Java, caption={Source code of the steepest hill algorithm without restarts}]\npublic static int[] run(City[] cities, int[] citySequence, int maxMoves){\n\tint[] tmpCitySeq, res = citySequence;\n\tint nbMoves = 0; boolean stop = false;\n\t\n\t//Not essential to the algorithm but checks coherence of the algorithm:\n\tSystem.out.println(\"Random initial city sequence is: \");\n\tSystem.out.print(\"[ \");\n\tfor(int cityID : citySequence) System.out.print(cityID + \" \");\n\tSystem.out.println(\"]\");\n\t\n\t//Essential to the algorithm:\n\twhile(!stop && nbMoves < maxMoves){\n\t\ttmpCitySeq = bestNeighb(cities, res);\n\t\t\n\t\tif(TSP.getDistance(cities, tmpCitySeq) < TSP.getDistance(cities, res))\n\t\tres = tmpCitySeq;\n\t\telse\n\t\tstop = true;\n\t\t\n\t\tnbMoves++;\n\t}\n\t\n\t//Not essential to the algorithm but checks coherence of the algorithm:\n\tdebugSteepestRun(nbMoves, res);\n\treturn res;\n}\n\t\\end{lstlisting}\nThe steepest hill algorithm consists in always choosing the best neighbor and always going towards a better solution. If a better solution can't be found then the algorithm stops. The number of moves acts as a fail-safe for this algorithm (eg. if it's always possible to go up). This is actually the whole meaning of the loop quoted above from line 12 to 21. The restart version just adds another while loop over this algorithm and compares the best solutions found at each iteration between themselves to select only the best solution over the best solutions. This is a strategy to get out of local optimums.\n\n\\newpage\n\t\\subsection{Tabu}\n\\begin{lstlisting}[style=Java, caption={Source code of the tabu algorithm}]\npublic static int[] run(City[] cities, int[] citySequence, int maxMoves, int tabuSize){\n\tLinkedList tabu = new LinkedList<>();\n\tint[] tmpBestCitySeq = null, bestNonTabuNeighb, currentCitySeq = citySequence;\n\tint nbMoves = 0; boolean stop = false;\n\t\n\twhile(nbMoves < maxMoves && !stop){\n\t bestNonTabuNeighb = bestNonTabuNeighb(tabu, cities, currentCitySeq);\n\t\tif(bestNonTabuNeighb == null)\n\t\tstop = true;\n\t\telse{\n\t\t\taddTabu(currentCitySeq, tabu, tabuSize);\n\t\t\tcurrentCitySeq = bestNonTabuNeighb;\n\t\t\t\n\t\t\tif(isBetterNeighb(cities, currentCitySeq, tmpBestCitySeq))\n\t\t\ttmpBestCitySeq = currentCitySeq;\n\t\t\t\n\t\t\tnbMoves++;\n\t\t}\n\t}\n\t\n\t//Not essential to the algorithm but checks coherence of the algorithm:\n\tdebugTabu(nbMoves, currentCitySeq, tabu);\n\treturn tmpBestCitySeq;\n}\n\\end{lstlisting}\nThe steepest hill algorithm uses the random restarts strategy to get out of local optimums. The tabu algorithm on the other hand uses another philosophy: it allows itself to not always pick up a better solution. But it does always pick up the best possible neighbor around (or the least worse). In order to not backtrack and get caught in an infinite loop of backtracking back and forth to the local optimum it uses a tabu list in which already explored solutions are stored so that they are not explored again (as long as the tabu list, implemented as a FIFO, doesn't overflow in which case we drop the most ancient step). The longest the tabu list, the more freedom we give to our tabu execution to roam around the local optimum hoping to find a better local optimum. Again, as with the steepest hill, the while loop reflects this metaheuristics concept (from line 6 to line 17).\\\\\n\n\\underline{NB:} If no non tabu neighbour is found then in order to avoid a \\textit{null} value as the \\textit{currentCitySeq} (cf. line 12) we stop the algorithm and return the best solution found so far. This is the meaning of lines 8 to 9 above.\n\n\n\t%% =======================================\n\t\\newpage\n\t\\section{Neighbours and Neighbourdhoods}\n\t\\label{neighb}\n\n\t\\begin{lstlisting}[style=Java, caption={Source code of the neighbourhood generation in the Tabu class}]\npublic static LinkedList genNeighbourhood(int[] citySeq){\n\tint nbCities = citySeq.length;\n\tLinkedList res = new LinkedList<>();\n\t\n\tfor(int i = 0; i < nbCities; i++) {\n\t\tfor (int j = i + 1; j < nbCities; j++) {\n\t\t\tint[] tmp = new int[nbCities];\n\t\t\tswap(citySeq, i, j);\n\t\t\tSystem.arraycopy(citySeq, 0, tmp, 0, nbCities);\n\t\t\tres.add(tmp);\n\t\t\tswap(citySeq, i, j);\n\t\t}\n\t}\n\treturn res;\n}\t\t\n\t\\end{lstlisting}\nWhile this piece of code comes from the TSP class it is actually very similar to the \\textit{bestNeighb} static method found in the SteepestHill class. This stems from the fact that the said method kills two birds in one stone by generating the whole neighborhood of a give solution while selecting the best neighbor.\\\\\n\nWhat the above function shows is that \\underline{in the context of the TSP problem}, a neighborhood consists in the swapping of two cities from one initial sequence of cities. We can generate all the possible neighbors by generating all the possible swaps with one value then by generating all the possible swaps of the other values without including the swaps with the value already computed. Here we do this in ascending order: we generate all the possible swaps from the first city then we do this with the second city excluding the first city, etc.\\\\\n\t\n\t\n\t%% =======================================\n\t\\newpage\n\t\\section{Instances and results}\n\t\n\t\\subsection{tsp5.txt instance}\n\t\n\t\\subsubsection{Steepest Hill algorithm with restarts}\n\t\\begin{lstlisting}[style=Java, caption={Bash output of the execution of the steepest hill algorithm on the tsp5.txt instance}]\nchuxclub @ CRex2 ~/projects/wip/2021_m1csa_algo_metaheuristics (dev)\n$ java -jar tsp.jar res/tsp5.txt steepest\n\nInput maxMoves: 10\nInput maxTrials: 3\n\n######### TESTING STEEPEST HILL ALGORITHM (with restarts) ########## \n>>> INITIAL SEQUENCE OF CITIES: \n===================================================== \nVector #DataStructures.Vector@3567135c has length: 5\nVector #DataStructures.Vector@3567135c contains: \n3 4 5 2 1 \n===================================================== \nDistance: 202.72643264132455 km\n\n>>> STEEPEST HILL ALGORITHM EXECUTION (with restarts): \nTrial #1 :\nRandom initial city sequence is: \n[ 1 2 3 5 4 ]\n2 moves required to reach the following solution:\n[ 1 2 4 5 3 ]\n\nTrial #2 :\nRandom initial city sequence is: \n[ 4 2 3 5 1 ]\n3 moves required to reach the following solution:\n[ 1 2 4 5 3 ]\n\nTrial #3 :\nRandom initial city sequence is: \n[ 2 3 1 5 4 ]\n3 moves required to reach the following solution:\n[ 1 3 2 4 5 ]\n\n>>> BEST SEQUENCE OF CITIES : \n===================================================== \nVector #DataStructures.Vector@54a097cc has length: 5\nVector #DataStructures.Vector@54a097cc contains: \n1 2 4 5 3 \n===================================================== \nDistance: 194.04052963659356 km\n####################################################################\n\t\\end{lstlisting}\nWith 10 maximum moves and 3 maximum trials the best solution was reached in a fraction of a second.\n\n\\newpage\n\\subsubsection{Tabu algorithm}\n\\begin{lstlisting}[style=Java, caption={Bash output of the execution of the tabu algorithm on the tsp5.txt instance}]\nchuxclub @ CRex2 ~/projects/wip/2021_m1csa_algo_metaheuristics (dev)\n$ java -jar tsp.jar res/tsp5.txt tabu\n\nInput maxMoves: 10\nInput tabuSize: 5\n\n############################# TESTING TABU ALGORITHM ########################### \n>>> INITIAL SEQUENCE OF CITIES: \n===================================================== \nVector #DataStructures.Vector@3567135c has length: 5\nVector #DataStructures.Vector@3567135c contains: \n4 1 5 2 3 \n===================================================== \nDistance: 258.8852276594047 km\n\n>>> TABU ALGORITHM EXECUTION: \nNumber of moves used to reach last solution: 10\nLast solution reached: [ 4 2 5 3 1 ]\nCity sequences in tabu list: \n[ 3 5 4 2 1 ]\n[ 3 5 2 4 1 ]\n[ 3 4 2 5 1 ]\n[ 5 4 2 3 1 ]\n[ 2 4 5 3 1 ]\n\n>>> BEST SEQUENCE OF CITIES: \n===================================================== \nVector #DataStructures.Vector@3551a94 has length: 5\nVector #DataStructures.Vector@3551a94 contains: \n3 5 4 2 1 \n===================================================== \nDistance: 194.04052963659356 km\n################################################################################ \t\n\\end{lstlisting}\nAs with the steepest hill algorithm above, here with 10 maximum moves and a tabu list size of 5 the best solution was reached in a fraction of a second.\\\\\n\nOn the tsp5.txt we have two possible best solutions as shown above: 3 5 4 2 1 and 1 2 4 5 3 with a distance of 194.04052963659356 km. We notice that one solution is the inverted version of the other solution which trivially means that the same distance is traveled whether we do the traveling salesman circuit in one way or the other.\\\\\n\n\t\\newpage\n\t\\subsection{tsp101.txt instance}\nGiven the amount of cities (101 of them) I couldn't find no best local optimum despite the algorithm used and the amount of moves/trials/tabu size involved. But I managed once to go down to around 994km with the tabu algorithm with 1000 moves and tabu list size of 500. Recently, the steepest hill algorithm provided a solution with 1023.92 km of traveled distance with 400 moves and 100 trials. So we can conclude that both of these algorithms find more or less the same solutions. One algorithm take more memory space (the tabu algorithm) while the other seems to take more computing time (the steepest hill algorithm) meaning that the choice of one or the other algorithm depends on the application they are used for.\\\\\n\nWe also notice that the time to find local optimums given the parameters mentioned above usually spanned from 15s to 30s.\\\\\n\n\\end{document}", "meta": {"hexsha": "58088b70beee0354df0fbcbc9f38e0a84a98e849", "size": 10248, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/report/content.tex", "max_stars_repo_name": "florian-legendre/2021_m1csa_algo_metaheuristics", "max_stars_repo_head_hexsha": "95e7bc6cf63aa2c009daad1a0de22206e327d58b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/report/content.tex", "max_issues_repo_name": "florian-legendre/2021_m1csa_algo_metaheuristics", "max_issues_repo_head_hexsha": "95e7bc6cf63aa2c009daad1a0de22206e327d58b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/report/content.tex", "max_forks_repo_name": "florian-legendre/2021_m1csa_algo_metaheuristics", "max_forks_repo_head_hexsha": "95e7bc6cf63aa2c009daad1a0de22206e327d58b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.4827586207, "max_line_length": 883, "alphanum_fraction": 0.6941842311, "num_tokens": 2607, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749126, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.34860323067900834}} {"text": "\\documentclass[runningheads,a4paper]{llncs}\r\n\r\n\\usepackage{amssymb}\r\n\\setcounter{tocdepth}{3}\r\n\\usepackage{graphicx}\r\n\r\n%%%%%%%%%ADDED PACKAGES%%%%%%%%%%\r\n\\usepackage{amsmath}\r\n\\usepackage{graphicx}\r\n\\usepackage{booktabs}\r\n\\usepackage[caption=false]{subfig}\r\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\r\n\\usepackage{url}\r\n\\urldef{\\mailsa}\\path||\r\n\\urldef{\\mailsb}\\path||\r\n\\urldef{\\mailsc}\\path|simon.andermatt@unibas.ch| \r\n\\newcommand{\\keywords}[1]{\\par\\addvspace\\baselineskip\r\n\\noindent\\keywordname\\enspace\\ignorespaces#1}\r\n\r\n\r\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\r\n\\begin{document}\r\n\r\n\\mainmatter % start of an individual contribution\r\n\r\n% first the title is needed\r\n\\title{Multi-Dimensional Gated Recurrent Units\\\\\r\nfor the Segmentation of Biomedical 3D-Data}\r\n\r\n% a short form should be given in case it is too long for the running head\r\n\\titlerunning{Multi-Dimensional GRU for the Segmentation of Biomedical Data}\r\n\r\n% the name(s) of the author(s) follow(s) next\r\n%\r\n% NB: Chinese authors should write their first names(s) in front of\r\n% their surnames. This ensures that the names appear correctly in\r\n% the running heads and the author index.\r\n%\r\n\\author{Simon Andermatt \\and Simon Pezold \\and Philippe Cattin}\r\n% index{Andermatt, Simon}\r\n% index{Pezold, Simon}\r\n% index{Cattin, Philippe}\r\n%\r\n\\authorrunning{Multi-Dimensional GRU for the Segmentation of Biomedical Data}\r\n\r\n% the affiliations are given next; don't give your e-mail address\r\n% unless you accept that it will be published\r\n\\institute{Department of Biomedical Engineering, \r\nUniversity of Basel,\r\nSwitzerland\r\n\\mailsc\\\\\r\n}\r\n\r\n%\r\n% NB: a more complex sample for affiliations and the mapping to the\r\n% corresponding authors can be found in the file \"llncs.dem\"\r\n% (search for the string \"\\mainmatter\" where a contribution starts).\r\n% \"llncs.dem\" accompanies the document class \"llncs.cls\".\r\n%\r\n\r\n\\toctitle{Multi-Dimensional Gated Recurrent Units for the Segmentation of Volumetric Biomedical Data}\r\n\\tocauthor{Simon Andermatt}\r\n\\maketitle\r\n\r\n\r\n\\begin{abstract}\r\nWe present a supervised deep learning method to automatically segment 3D volumes of biomedical image data. The presented method takes advantage of a neural network with the main layers consisting of multi-dimensional gated recurrent units. We apply an on-the-fly data augmentation technique which allows for accurate estimations without the need for either a huge amount of training data or advanced data pre- or postprocessing. We show that our method performs amongst the leading techniques on a popular brain segmentation challenge dataset in terms of speed, accuracy and memory efficiency. We describe in detail advantages over a similar method which uses the well-established long~short-term~memory.\r\n\r\n\\keywords{deep learning, GRU, multi-dimensional RNN, segmentation}\r\n\\end{abstract}\r\n\r\n\r\n\\section{Introduction}\r\nWith the rapid advancements of imaging technologies, their ubiquitous availability and dropping prices, vast amounts of data are collected. This is particularly true for medical imaging. Accurate segmentation and delineation of e.g. pathologies in this medical data, however, pose real challenges as this is still mainly a manual process. In late phase drug studies with thousands of patients, multiple 3d datasets with different MR sequences are often collected per patient. If quantitative analysis of the immense amount of data is required, the time that has to be spent on the data by trained experts is enormous. A successful automated segmentation technique would decrease manual work to a minimum, cutting the costs and time spent on developing new treatments.\r\n\r\nAutomatic segmentation of biomedical volumetric data is, however, a challenging problem due to its high dimensionality, imaging noise, artifacts and other factors. Recent advances in the field of deep learning, especially the enabling effect of modern GPUs along with the advent of general purpose GPU computing, led to a revival of convolutional neural networks~\\cite{krizhevsky_imagenet_2012}. These feed-forward networks show great promise, but need a large number of layers to solve a difficult task accurately. A recurrent neural network (RNN), in contrast, can become arbitrarily deep due to its additional temporal dimension. Each timestep computed in an RNN corresponds roughly to one layer in a feed-forward network, with the weights in one RNN being the same for each timestep. This property allows defining substantially more complexity very elegantly without the need for a huge number of layers or parameters.\r\n\r\nThe multi-dimensional Long Short-Term Memory (MD-LSTM) proposed by Stollenga et al.~\\cite{stollenga_parallel_2015}, called \\textit{PyraMiD-LSTM}, applied these insights to the Long Short-Term Memory (LSTM)~\\cite{hochreiter1997long}. It defines two LSTMs for each spatial dimension, using said spatial dimension as temporal dimension. The first one processes the data along that dimension, the second one in the opposite direction. In order to make full use of the spatial information, not only the direct predecessor along the temporal direction is taken into account, but also its local neighborhood. This can be neatly expressed using convolutions. \r\n\r\nA relatively new RNN called Gated Recurrent Unit (GRU)~\\cite{cho_learning_2014} grew popular in recent years and became a strong competitor for the LSTM. It can be seen as a simplified version of the LSTM, which uses an update gate instead of a forget and input gate and combines the hidden and cell state~\\cite{olah_understanding_2015}. It has been shown that it performs comparably to the LSTM in the task of sequence modeling~\\cite{chung_empirical_2014}. Another study suggests that GRU and LSTM report similar performance on selected tasks~\\cite{greff_lstm:_2015}. An empirical search among more than 10\\,000 RNN architectures showed that on the selected tasks, although not the best performing RNN on every task, the GRU outperformed the standard LSTM architecture~\\cite{jozefowicz_empirical_2015}.\r\nA larger time dimension in an RNN can mean that larger time dependencies can be represented. The lower memory requirement of the GRU means that larger volumes can be fed into the network and larger networks can be designed for the same volume size.\r\n\r\nFor all these reasons, a modification of the GRU to be able to process volumetric data seems compelling. We propose the multi-dimensional GRU (MD-GRU), which is capable of accurate segmentation of 3d data. We hint at the theoretical memory savings compared to the MD-LSTM and show that the performance of MD-GRU is comparable if not superior. Furthermore, we show that its convergence rate, computation time and combination of fewer gates favor the MD-GRU. We apply our method on a popular brain segmentation challenge dataset, achieving a score among the top 3 best performing methods. \r\n\r\n\\section{Methods}\r\n\\subsection{Data}\r\n\\begin{figure}\r\n \\centering\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/t1.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/t1-ir.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/t2-flair.png}\r\n \\end{minipage}\r\n \r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/t1-gs.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/t1-ir-gs.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/t2-flair-gs.png}\r\n \\end{minipage}\r\n \\caption{Slice 19 of the 5th training sample. \\emph{Top row (left to right):} T1, T1\\_IR and T2\\_FLAIR. \\emph{Bottom row:} respective highpass filtered versions.}\r\n \\label{data}\r\n\\end{figure}\r\nWe used the publicly available MrBrainS~\\cite{mendrik2015mrbrains} challenge dataset, which was one of the datasets used to evaluate the PyraMiD-LSTM. The MrBrainS challenge data consists of 5 labeled samples and 15 testing samples, where each sample has a T1 weighted, T1 inversion recovery and a FLAIR scan. The additional high-resolution T1 scan was not used, as the labeling was performed on the low resolution data. The training data contained two different label maps, one for training and one for testing. The training map consists of classes for cortical gray matter (GM), basal ganglia, white matter (WM), WM lesions, cerebrospinal fluid (CSF), ventricles, cerebellum, brainstem and background. The testing map only defines classes for GM, WM and CSF, the respective classes of the training map are merged. Brainstem and cerebellum are not included in the evaluation and do therefore not appear labeled in the testing map.\r\n\r\n\r\n\\subsection{Convolutional Gated Recurrent Unit}\\label{reordering}\r\n\r\n\r\nThe standard GRU as proposed in~\\cite{cho_learning_2014} is defined as\r\n\\begin{align}\r\n r^j &= \\sigma([W_r x]^j + [U_r h_{t-1}]^j),\\\\\r\n z^j &= \\sigma([W_z x]^j + [U_z h_{t-1}]^j),\\\\\r\n \\tilde{h}^j_t &= \\phi([Wx]^j + [U(r \\odot h_{t-1})]^j),\\\\\r\n h^j_{t} &= z^j\\odot h^j_{t-1} + (1-z^j)\\odot \\tilde{h}^j_{t},\r\n\\end{align}\r\nwhere $x$ is the input data, $r^j$ is the reset gate, $z^j$ is the update gate of the hidden unit $j$ and the activation is performed in $h^j$. The operator $\\odot$ represents an elementwise multiplication. The functions $\\sigma(\\cdot)$ and $\\phi(\\cdot)$ stand for the logistic function and the hyperbolic tangent. $W$ and $U$ are the weight matrices for the current input and last step's output data respectively. Along the lines of Stollenga et al.~\\cite{stollenga_parallel_2015}, we adapt these equations to be able to process 3D volumes and introduce our convolutional GRU (C-GRU):\r\n\\begin{align}\\label{eq:cgru1}\r\n r^j &= \\sigma \\left( \\sum\\limits_i^I (x^i*w_r^{i,j}) + \\sum\\limits_k^J ( h_{t-1}^k*u_r^{k,j} ) +b^j_r\\right),\\\\\\label{eq:cgru2}\r\n z^j &= \\sigma \\left( \\sum\\limits_i^I (x^i*w_z^{i,j}) + \\sum\\limits_k^J ( h_{t-1}^k*u_z^{k,j} ) +b^j_z\\right),\\\\\\label{eq:cgru3}\r\n \\tilde{h}^j_t &= \\phi \\left( \\sum\\limits_i^I (x^i*w^{i,j}) + r^j \\odot \\sum\\limits_k^J ( h_{t-1}^k * u^{k,j} ) +b^j\\right),\\\\\r\n h^j_{t} &= z^j\\odot h^j_{t-1} + (1-z^j)\\odot\\tilde{h}^j_{t},\r\n\\end{align}\r\nwhere $*$ represents a convolution. Compared to the vanilla GRU, we introduced slight changes. We decided to use a bias $b$ on each gate. We factored $r^j$ out of the convolution operation between $u$ and $h_{t-1}$. This change was motivated by the fact that an additional convolution would require $r$ to have twice the support it needs now because of the chained convolution. Moreover, we reorder the data for each C-RNN such that the two spatial dimensions are closest to memory, and the temporal dimension is ordered according to the temporal direction, as explained in the next paragraph. We motivated that decision with faster possible processing speeds on the GPU, since all convolutions now require data that lies close in memory. The computations of one C-GRU are visualized as a computational graph in Fig.~\\ref{fig:network}a.\r\n\r\nThe MD-GRU consists of two times $D$ C-GRUs, where $D$ is the dimensionality of the image data and we need one C-GRU for each of the two directions. We set the input data of channel $i$ as $x^i \\in \\mathbb{R}^{S_1\\times \\cdots\\times S_D}$. For each spatial dimension $d$, we create the copies $x^{i,d,-1}, x^{i,d,+1} \\in \\mathbb{R}^{S_d\\times S_1\\times \\cdots\\times S_D}$ of $x$ and apply the following data transformations:\r\n\\begin{align}\r\n x^{i,d,+1}(s_d, s_1, \\dots, s_{D}) &= x^i(s_1, \\dots, s_d,\\dots, s_{D}),\\\\\r\n x^{i,d,-1}(S_d-s_d, s_1, \\dots, s_{D}) &= x^i(s_1,\\dots , s_d,\\dots, s_{D}),\r\n\\end{align}\r\nwhere $s_d$ is the index of the assigned dimension of the C-GRU and $S_d$ is the size of dimension $d$. The inverse operation is applied to $h^{j,d,+1},h^{j,d,-1} \\in \\mathbb{R}^{S_d\\times S_1\\times \\cdots\\times S_D}$ to gather the final output $h^j$:\r\n\\begin{align}\r\nh^j(s_1,\\dots, s_{D}) &= \\sum\\limits_{d=1}^D \\left ( h^{j,d,+1}(s_d, s_1, \\dots, s_{D}) + h^{j,d,-1}(S_d-s_d, s_1, \\dots, s_{D}) \\right ).\r\n\\end{align}\r\n\r\nFigure~\\ref{fig:network}b details this process for the MD-GRU. We apply the same technique for our implementation of the MD-LSTM.\r\n\\subsection{Experiments}\r\n\\subsubsection{Network}\r\nWe model our network similar to~\\cite{stollenga_parallel_2015}. We include three multi-dimensional RNN (MD-RNN) layers of 16, 32 and 64 channels which are connected with pixelwise fully connected hidden layers of 25 and 45 channels respectively, each followed by a hyperbolic tangent activation function. The last MD-RNN is attached to a pixelwise fully connected layer with $c$ channels, the same number as classes in the data. We estimate the probabilities for each class using a softmax in the last layer and consequently choose the multinomial logistic loss for the training of our network. Figure~\\ref{fig:network}c shows our network setup for the case of MD-GRU.\r\n\r\n\r\n\\begin{figure}\r\n \\centering\r\n \\subfloat[][C-GRU]{\r\n \\includegraphics[width=0.35\\textwidth]{images/cgrunotitle3.pdf}\r\n }\\qquad\r\n \\subfloat[][MD-GRU]{\r\n \\includegraphics[width=0.48\\textwidth]{images/mdgrunotitle2.pdf}\r\n }\\quad\r\n \r\n \\subfloat[][Network Architecture]{\r\n \\includegraphics[width=0.99\\textwidth]{images/network.pdf}\r\n }\r\n \\caption{(a) Directed graph denoting the computations in one C-GRU. The variables $x^{d,o}$, $h^{d,o}$ with $o \\in \\{-1,+1\\}$ represent the input and output data across all $I$ and $J$ channels respectively. The $\\circledast$ operator denotes here the sum per channel $j$ over the convolutions with each channel $i$ or $k$, as used in equations (\\ref{eq:cgru1})--(\\ref{eq:cgru3}). (b) Proposed arrangement of 6 C-GRUs in a MD-GRU for three-dimensional data. (c) Setup of our network.} \\label{fig:network}\r\n\\end{figure}\r\n\r\n\r\n\\subsubsection{Setting}\\label{setting}\r\nAll experiments were calculated on an NVIDIA GTX Titan X GPU with 12 GB global memory. Our implementation of MD-LSTM and MD-GRU relied on the fast convolution routines provided by NVIDIA's cuDNN~\\cite{chetlur_cudnn:_2014}. For other layers, the already available implementations of the CAFFE\\footnote{version 1.0.0-rc3, commit 9c46289} framework~\\cite{jia2014caffe} were used.\r\n\r\n\\subsubsection{Preprocessing}\r\nFor all volumes, unsharp masking was done using a Gaussian smoothed image ($\\sigma=5$~voxels) which was then subtracted from the original images to produce highpass filtered volumes. The original images and the highpass filtered images were normalized to $\\sigma=1$ and $\\mu=0$, assuming normally distributed values. In this way we followed a procedure similar to~\\cite{stollenga_parallel_2015}, but omitted the histogram equalization. Figure \\ref{data} shows the original and preprocessed data for training sample 5 at slice 19.\r\n\r\n\\subsubsection{Data augmentation}\r\nIn the training stage, at each iteration, a random location in the training data was selected and a deformation field was generated and applied to the subvolumes, which were then fed into the network. We used a procedure similar to~\\cite{ronneberger_u-net:_2015}, but made the grid size dependent on the data. We did not use random deformations in the feasibility study mentioned in Sec.~\\ref{feasibilitystudylabel}. For the testing phase, no deformations were applied.\r\n\r\n\\subsubsection{Training}\r\nIn three training steps we iteratively increase the subvolume size from $64\\times64\\times8$~voxels to $128\\times128\\times12$ and finally to $200\\times200\\times15$, keeping the third dimension smaller to account for the anisotropic MR volume resolution. We relied on AdaDelta~\\cite{zeiler_adadelta_2012} to omit the manual tuning of a learning rate. For the challenge, we additionally used DropConnect~\\cite{icml2013_wan13} of 0.5 on the input connections of each C-GRU to prevent overfitting. Training took around two days.\r\n\r\n\r\n\\subsubsection{Testing}\r\nIn the testing phase, we divided the volume into a grid of equally sized subvolumes of $120\\times120\\times8$, which were padded by 50, 50 and 4 voxels respectively on all sides of the volume. The padding was later used to stitch the results together using a Gaussian ($\\mu=0$, $\\sigma=(10,10,0.8)$) to produce interpolation weights, since the borders contain starting artifacts from the individual RNNs and do not contain adequate results. Since we trained for nine classes, but only four classes were needed for the final evaluation, we simply combined the binary labels for the CSF with the ventricles, the cortical GM with the basal ganglia and the WM with the WM lesions. Everything else was considered background. Testing one volume of the MRBrainS data required 32 iterations, which needed around two minutes.\r\n\r\n\r\n\\section{Results}\r\n\r\n\\subsection{Feasibility Study}\r\n\\label{feasibilitystudylabel}\r\n\\begin{table}\r\n\\caption{Feasibility study. Dice coefficients in percent for gray and white matter (GM/WM), cerebrospinal fluid (CSF) and intracranial volume (ICV).}% The case with more memory was only trained until iteration 28000}\r\n\\begin{center}\r\n\\begin{tabular}{lllll}\\toprule\r\n× & GM & WM & CSF & ICV\\\\ \\midrule\r\nMD-LSTM & \\textbf{88.09} & 90.08 & 82.62 & 97.56 \\\\ \r\nMD-GRU & 87.88 & \\textbf{90.15 } & \\textbf{83.19} & \\textbf{97.73 }\\\\ \\bottomrule\r\n\\end{tabular}\r\n\\label{feasibilityquantitative}\r\n\\end{center}\r\n\\end{table}\r\n\r\n\\begin{figure}\r\n \\centering\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.98\\linewidth]{./images/lstmfeas_new_new.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.98\\linewidth]{./images/grusfeas_new_new.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.98\\linewidth]{./images/Traindata5atslice19-Testlabel.png}\r\n \\end{minipage}\r\n \r\n \\begin{minipage}{0.80\\textwidth}\r\n \\includegraphics[width=.9\\linewidth]{./images/convergence_rate_new10000_mediumlines_withtitle6.pdf}\r\n \\end{minipage}\r\n \\caption{Feasibility study. \\emph{Top row:} slice 19 of the 5th training volume used for the evaluation. The images from left to right represent the results of the MD-LSTM, the MD-GRU and the manual labeling. \\emph{Bottom row:} convergence rates for the feasibility study of both MD-GRU and MD-LSTM.}\r\n\\label{feasibilityqualitative}\r\n\r\n\\end{figure}\r\nTo point out differences between the MD-GRU and the MD-LSTM, we ran the same setup with the multi-dimensional RNN layers either being an MD-GRU or an MD-LSTM. We used the first four volumes in the training set of the MrBrainS challenge and trained both networks for 3\\,000 iterations on the largest possible resolution which was feasible for both (limited to $192\\times192\\times14$ by our MD-LSTM implementation). On average, one training iteration for MD-GRU and MD-LSTM took 9.1 and 12.8 seconds, respectively. The Dice coefficients for CSF, GM, WM and ICV between the computed segmentation of the 5th training volume and the provided reference segmentation are shown in Table~\\ref{feasibilityquantitative} for both the MD-GRU and MD-LSTM. Slice 19 of the computed segmentations and the reference segmentation are displayed in Fig.~\\ref{feasibilityqualitative} together with a plot of a running average of 100 iterations of the loss function for each iteration of the training procedure.\r\n\r\n\r\n\\subsection{MD-GRU on MRBrainS}\r\nIn our attempt to beat the highscore of the MRBrainS challenge, we used our described data augmentation method. Each subvolume was deformed randomly throughout all three training phases. We used all provided low resolution volumes and their highpass filtered versions. Table~\\ref{mrbrainsres} lists our performance according to the Dice coefficients, 95th-percentile of the Hausdorff distance and average volume difference of the GM, WM, CSF and ICV. Nine measures were relevant for the final evaluation: Dice, modified Hausdorff distance and average volume distance in each of the categories GM, WM and CSF. The sum of the ranks in these nine categories is used as the performance score and determines the final rank. \r\nFigure~\\ref{visualresultschallenge} shows the computed segmentation at slice 19 of samples 5, 10 and 15 of the test data. \r\n\\begin{table}\r\n\\begin{center}\r\n \\caption{MrBrainS challenge. Results of the six best performing methods for GM, WM, CSF and ICV of all three used metrics (Dice, 95th-percentile of the Hausdorff distance (HD) and average volume difference (AVD)). A bold number means best out of these six. The results reflect the state on August 12, 2016.}\r\n\\label{mrbrainsres}\r\n \\scriptsize\r\n\\begin{tabular}{l c rcl rcl rcl rcl rcl}\r\n\\toprule\r\nTeam name & Rank &\\multicolumn{3}{c}{GM} & \\multicolumn{3}{c}{WM} & \\multicolumn{3}{c}{CSF} & \\multicolumn{3}{c}{ICV}\\\\%\\hline\\hline\r\n×& × & Dice & HD & AVD & Dice & HD & AVD & Dice & HD & AVD & Dice & HD & AVD \\\\\\midrule\r\nCU\\_DL2 & \\textbf{1}& \\textbf{86.15} & \\textbf{1.45} & 6.60 & \\textbf{89.46} & \\textbf{1.94} & 6.05 & \\textbf{84.25}& 2.19 & 7.69 & 98.10 & 2.75 & 1.54\\\\\r\nCU\\_DL& 2 & 86.12 & 1.47 & 6.42 & 89.39 & \\textbf{1.94} & \\textbf{5.84} & 83.96 & 2.28 & 7.44 & 97.99 & 3.16 & 1.83 \\\\\r\nMD-GRU [proposed] & 3 & 85.40 & 1.55 & \\textbf{6.09} & 88.98 & 2.02 & 7.69 & 84.13 & 2.17 & 7.44 & \\textbf{98.15} & \\textbf{2.37} & 0.86\\\\\r\nPyraMiD-LSTM2 & 4 & 84.89 & 1.67 & 6.35 & 88.53 & 2.07 & 5.93 & 83.05 & 2.30 & 7.17 & 98.04 & 2.86 & \\textbf{0.69}\\\\\r\nFBI/LMB Freiburg~\\cite{cicek_3d_2016}& 5 & 85.44 & 1.58 & 6.60 & 88.86 & 1.95 & 6.47 & 83.47 & 2.22 & 8.63 & 97.98 & 2.51 & 1.06\\\\\r\nIDSIA~\\cite{stollenga_parallel_2015} & 6 & 84.82 & 1.70 & 6.77 & 88.33 & 2.08 & 7.05& 83.72 & \\textbf{2.14} & \\textbf{7.09}& \\textbf{98.15} & 2.44 & 0.95\\\\\\bottomrule\r\n\\end{tabular}\r\n\\end{center}\r\n \\label{resultschallenge}\r\n\\end{table}\r\n\r\n\\begin{figure}\r\n \\centering\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v5-test-challres_new.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v5-test-t1.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v5-test-t1-ir.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v5-test-t2-flair.png}\r\n \\end{minipage}\r\n \r\n \\vspace{0.3em}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v10-test-challres_new.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v10-test-t1.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v10-test-t1-ir.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v10-test-t2-flair.png}\r\n \\end{minipage}\r\n \r\n \\vspace{0.3em}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v15-test-challres_new.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v15-test-t1.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v15-test-t1-ir.png}\r\n \\end{minipage}\r\n \\begin{minipage}{.24\\textwidth}\r\n \\includegraphics[width=.99\\linewidth]{./images/sl19-v15-test-t2-flair.png}\r\n \\end{minipage}\r\n \\caption{MrBrainS challenge. \\emph{Rows (top to bottom):} 5th, 10th and 15th test sample. \\emph{Columns (left to right):} slice 19 of our segmentation results, T1, T1\\_IR and T2\\_FLAIR.}\r\n \\label{visualresultschallenge}\r\n\r\n\\end{figure}\r\n\r\n\r\n\r\n\\section{Discussion}\r\n\r\nThe feasibility study has shown that MD-GRU has great potential for the segmentation of volumetric images, since it achieved comparable results to the MD-LSTM in less time with the same settings. \r\n\r\nUsing deformation as a data augmentation strategy and DropConnect for regularization in the challenge, we ranked 3rd out of 37. Unfortunately, none of the results in the top five of the challenge highscore are published so far. The 4th and 6th entries are both incarnations of the already discussed MD-LSTM, where only the latter was described in~\\cite{stollenga_parallel_2015} and the former likely contains unpublished improvements to their method. In contrast to \\cite{stollenga_parallel_2015}, we did not omit the original T1\\_IR images. Yet some obvious misclassifications could be traced back to strong bias field artifacts in the T1\\_IR images. Given the small training size, using the T1\\_IR images leads to apparent fitting to the bias field. Furthermore, we were not able to replicate the training volume size of Stollenga et al.~\\cite{stollenga_parallel_2015} due to a higher memory requirement of our implementation, since we decided to copy the input and output data for each RNN layer, as detailed in Sec.~\\ref{reordering}. This has to be kept in mind when comparing the two approaches. Relationships between areas that are located at a certain distance in the data could therefore not be modeled in our network, where~\\cite{stollenga_parallel_2015} was able to use the full spatial context in two dimensions as well as a larger third dimension. In their last training step more than half of the data was covered while we could only fit a bit more than a fifth in our memory. \r\n\r\nThe contribution on rank five was computed using the 3D U-Net~\\cite{cicek_3d_2016}. It consists of a hierarchical convolutional neural network with shortcut connections, which is trained using various on-the-fly data augmentation techniques, including the deformation strategy used in this paper. The challenge results and corresponding adaptations of the algorithm to fit the challenge data are, however, not yet published. We believe that data augmentation is key for successful applications to problems with such a small training size. \r\n\r\n\r\n \\subsubsection{Conclusion}\r\n With the MD-GRU, we combined the enormous expressive power of RNNs with a highly beneficial data augmentation strategy, resulting in a powerful supervised automatic segmentation technique. With a memory-savvy implementation that omits the initial reordering of the data, results surpassing the state of the art should be possible with MD-GRU.\r\n\r\n\r\n\r\n\\bibliographystyle{splncs03}\r\n\\bibliography{manuallib}\r\n\\end{document}\r\n", "meta": {"hexsha": "c23059ca1926fb28d6750fedd370742ddf7699b7", "size": 26276, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "dlmiapaper/paper34.tex", "max_stars_repo_name": "zubata88/dlmia16", "max_stars_repo_head_hexsha": "096fc18c40b20ed3c719851faf9773f95d1058fe", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "dlmiapaper/paper34.tex", "max_issues_repo_name": "zubata88/dlmia16", "max_issues_repo_head_hexsha": "096fc18c40b20ed3c719851faf9773f95d1058fe", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "dlmiapaper/paper34.tex", "max_forks_repo_name": "zubata88/dlmia16", "max_forks_repo_head_hexsha": "096fc18c40b20ed3c719851faf9773f95d1058fe", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 84.7612903226, "max_line_length": 1491, "alphanum_fraction": 0.7551758258, "num_tokens": 7324, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3485389718040475}} {"text": "\\section{Static Obstacle Avoidance}\n\\label{chap:StaticObstacleAvoidance}\nFor the static obstacle avoidance task, the controller must drive the vehicle in order to avoid collision with any still obstacle detected by a set of sensors mounted on the vehicle.\\\\\nBefore implementing the algorithm for obstacle avoidance, we have made a set of assumptions with the aim of simplifying this task:\n\\begin{itemize}\n \\item we have a set of sensor able to determine the position of the obstacle and to place it on the reference map, but the detection is performed only when the distance from the obstacle is less than 200 $m$;\n \\item it is always possible to avoid the obstacle by moving to the left lane;\n \\item the obstacle is never placed at less than 200 $m$ from the starting point, therefore it is always possible to avoid it without violating the zones (described below); \n \\item dimensions of the obstacle are not relevant.\n\\end{itemize}\nTo be sure that the controller tries to avoid obstacles, we have decided to implement the avoidance algorithm by giving to the controller a set of constraints on the output, in order to define a \\textit{``forbidden zone\"}, as described in subsection \\ref{sec:Safety_distance}.\\\\\nTo be precise, we have defined 5 zones nearby each obstacle:\n\\begin{itemize}\n \\item \\textbf{Zone 1}: the obstacle is detected, but the vehicle is still too far from it, so it stays in its lane;\n \\item \\textbf{Zone 2}: the vehicle is close enough to the obstacle to start the overtaking maneuver, so it can start the lane change;\n \\item \\textbf{Zone 3}: the vehicle is in the left lane, in the so called \\textit{``Safe Zone\"};\n \\item \\textbf{Zone 4}: the obstacle is passed and the vehicle can come back to the right lane;\n \\item \\textbf{Zone 5}: the vehicle is in its lane and the obstacle has been already passed, so we are in the same condition of no obstacle detected.\n\\end{itemize}\nThese zones are defined, starting from the obstacle position, in the following way:\n\\begin{itemize}\n \\item \\textbf{Zone 1} starts when the obstacle is detected, so when the distance between the vehicle and the obstacle is less than 200 $m$, as specified in the assumptions made at the start of this section;\n \\item \\textbf{Zone 2} starts 40 meters before the Zone 3. We have decided to define this zone in this way to be symmetric with respect to Zone 4, where the other lane change maneuver is performed;\n \\item \\textbf{Zone 3} is defined to be compliant with the Safety Distance described in \\ref{sec:Safety_distance}, according to equation \\ref{eq:safetyDistance}: $SafetyDistance [m] = \\left(\\frac{v[\\sfrac{km}{h}]}{10}\\right)^2$, so it is proportional to the square of the speed. Zone 3 ends 10 meters after the obstacle, to be compliant with the requirement n° 4 in section \\ref{System_Requirements};\n \\item \\textbf{Zone 4} is defined to be compliant with requirement n° 4 in Section \\ref{System_Requirements}, so it starts 10 meters after the obstacle and ends 50 meters after it, resulting in a space of 40 meters for the re-entering maneuver, thus it is symmetric with respect to the maneuver performed in Zone 2 as previously stated;\n \\item \\textbf{Zone 5} starts right after Zone 4 and it is kept until the distance from the obstacle becomes greater then 200 $m$.\n\\end{itemize}\nThe distance of 200 $m$ from the obstacle is evaluated as the Euclidean distance between the vehicle and the obstacle, while the other distances involved in the definition of the zones are evaluated as projection on the reference path, and assuming constant speed, they result in fixed points on the reference map.\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1\\textwidth]{Figures/Zones.png}\n \\caption{Zones defined for obstacle avoidance}\n \\label{fig:Zones}\n\\end{figure}\nIn Figure \\ref{fig:Zones} the aforementioned zones are depicted on a straight line scenario with a reference speed of 100 $km/h$.\\\\\\\\\nIn Zones 1 and 5, as previously said, the obstacle is too far from the vehicle to be a problem, so no procedure is applied to the controller.\\\\\\\\\nIn Zone 2, the yellow one in the figure, the vehicle should pass in the left lane. To do so, we have defined two points, represented in the figure respectively with a green circle, which is called the \\textbf{\\textit{Detection Point}}, and a red circle that is the \\textbf{\\textit{Safe Point}}.\nThe latter is defined as the projection on the left lane of the point in the reference trajectory at a distance equal to the safety distance. \\\\To let the controller move on the path defined by these two points, we have exploited the \\textit{Custom Constraint} of the MPC block in Simulink.\\\\\\\\\nAs briefly explained in Section \\ref{chap:Controller}, the linear combination of inputs and states can be set as constraints for the controller. To accomplish our task, we have defined 2 constraints on the \\textit{x} and \\textit{y} states, which result in two lines representing an upper and a lower bound for the vehicle position during the overtaking maneuver. Looking at the example in Figure \\ref{fig:Zones}, the lower bound becomes the border of the polygon defined by Zones 2, 3 and 4, while the upper bound is the left limit of the road, that is to say the left border of the left lane (straight line $y=6$ in Figure \\ref{fig:Zones}).\\\\\nFor what concerns the lower bound for constraints in Zone 2, it is completely defined by the line passing from the two aforementioned points (\\textit{Detection Point} and \\textit{Safe Point}) .\\\\\\\\\nIn Zone 3, the vehicle should stay in the left lane. To define constraints in this zone, we have set as lower bound the projection of the reference trajectory on the left lane instead of the straight line connecting points from the start (red circle) to the end (blue cross), since the former method allows to track both straight and curved paths. One limitation of the \\textit{linear} constraints is that they can only define \\textit{linear} relation between states, thus when the path is curved, we have to approximate it to a straight line, but this can cause errors because the controller tries to satisfy the constraint for all the timesteps in the prediction horizon, so resulting in a reduced or increased distance with respect to the points in the reference map. To limit this effect, we have inserted a correction factor to the constraint generated.\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1\\textwidth]{Figures/ConstraintCorrection.png}\n \\caption{Constraints generated in a curved path with 300 $m$ radius at 50 $km/h$, with and without the correction factor}\n \\label{fig:CorrectionConstraint}\n\\end{figure}\nIn Figure \\ref{fig:CorrectionConstraint} a case where the reference road is a left curve with 300 meters radius is represented. The dashed red line is the projection of the road approximated with the slope of the reference point where the controller is called, while the green dashed line is the projection evaluated with the correction factor. As it can be seen, the original projection \"collapses\" on the reference path before the corrected one does, resulting in a reduction of the distance from the reference during the predicted horizon and a consequent reduction of distance between the trajectory and starting lane, occupied by the obstacle.\\\\\nThis corrected constraint is evaluated according to the following procedure:\n\\begin{enumerate}\n \\item an approximation of the \\textit{curvature} of the road is evaluated as the mean value of the reference angle in the prediction horizon: $curvature = \\frac{\\sum_{i=0}^{p}\\theta_i-\\theta_0}{p}$;\n \\item the correction factor is: $416\\cdot\\frac{|curvature|}{V}$, where the coefficient 416 has been found empirically by simulations;\n \\item the \\textit{CorrectedSlope} is given by the road slope ($tan(\\theta_r)$) plus the correction factor;\n \\item the intercept for the constraint line is given by:\\\\\n $CorrectedIntercept = y_r + Lw/cos(\\theta_r) - CorrectedSlope\\times x_r$\\\\\n where $x_r$, $y_r$, and $\\theta_r$ are the reference trajectory values at the first timestep of the prediction horizon, and $Lw$ is the lane width;\n \\item the constraint is applied as $y > CorrectedSlope\\times x + CorrectedIntercept$.\n\\end{enumerate}\nObviously, when the reference path is straight, the correction function becomes 0 and the constraint is the simple projection of the road.\\\\\\\\\nIn Zone 4, the situation is analogous to the one in Zone 2, with the only difference that here we are \\textit{``relaxing\"} the constraints, allowing the vehicle to come back to its original position on the right lane. The bounds here are defined by the blue cross point, called \\textbf{\\textit{End Point}} and the yellow diamond point, called \\textbf{\\textit{Entry Point}} (Figure \\ref{fig:Zones}).\\\\\nTo define these bounds in a suitable form for the controller, we have transposed the information in a matrix form, describing two line equations:\n\\begin{equation}\n y > m_{lower}x + q_{lower}\n\\end{equation}\n\\begin{equation}\n y < m_{upper}x + q_{upper}\n\\end{equation}\nwhere $m$ is the slope of the constraint and $q$ is the intercept of the constraint.\nThe slope of the constraint can be evaluated by calculating the tangent of the road orientation when we are considering the upper bound of the Zone 3, while it is evaluated as the slope between the points defined above in Zones 2 and 4.\\\\\nThe intercept is given by the projection of the reference trajectory, on the left lane center line for Zone 3 and on the left road limit for the upper bound, or it is evaluated by inverting the line equation in the changing lane maneuvers.\\\\\nTo let our algorithm work on all the quadrants, we make a check of the reference trajectory orientation and we adapt the constraint generation to the case we are in. This check projects the constraints on the $x$ axis rather than $y$ if the road inclination is closer to the vertical line in the X-Y plane, moreover it defines which are the relations between the bounds and the reference trajectory, intended as \\textit{``lower than\"} or \\textit{``greater than\"}, according to the direction of the vehicle and its left.\n\\subsection{Multiple obstacles avoidance}\nThe procedure described until now shows how the controller sets constraints when an obstacle is detected, but when multiple obstacles are present, things can become harder.\\\\\nIf two obstacles are very far one from the other, at least 400 $m$, they can be treated as single obstacles in sequence, but if they are close, a new definition of the zones is needed. In detail, we have decided to control the position of the next obstacle when we reach the \\textit{End Point} of the previous. In this case, we can decide whether to come back in the right lane or to stay in the left lane and continue the overtaking maneuver. The procedure follows the rules below:\n\\begin{itemize}\n \\item if the next obstacle is more than 400 $m$ far from the previous one, it is treated as an independent single obstacle;\n \\item if the next obstacle is in the range 400 $m$ and 210 $m$ from the previous obstacle, it is not detected when the vehicle is in the \\textit{End Point} of the previous obstacle. Here the vehicle simply comes back to the right lane and when the next obstacle is detected, it is treated as a single one;\n \\item if the next obstacle is closer than 210 $m$ from the previous one, it is detected when the vehicle is in the \\textit{End Point} of the latter, so here a decision is taken:\n \\begin{enumerate}\n \\item the next \\textit{Detection Point} is after the previous \\textit{Entry Point} $\\xrightarrow{}$ the vehicle reenters in the right lane and starts the new overtaking maneuver when it reaches the next \\textit{Detection Point};\n \\item the next \\textit{Detection Point} is before the previous \\textit{Entry Point} $\\xrightarrow{}$ the vehicle stays in the left lane and sets as target the next obstacle, so then it repeats this procedure when the next \\textit{End Point} is reached.\n \\end{enumerate}\n\\end{itemize}\nTo implement this control strategy in the simulation environment, we have used a counter on the obstacles, which takes into account if an obstacle has already been passed and in that case the detection algorithm neglects it, considering the next obstacle.\\\\\nIn real-world scenarios, a counter is not a feasible solution since it is impossible to know how many obstacles will be detected during a trip, but advanced navigation and sensor systems can detect and classify obstacles, and then can understand whether they have been passed or not, so our assumption is justified.\n\n", "meta": {"hexsha": "668bc9c5feb371dcdef968d7cb41b9ce0f81151f", "size": 12646, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Documentation/Report/Chapters/9-StaticObstacleAvoidance.tex", "max_stars_repo_name": "meltinglab/dynamic-obstacle-avoidance", "max_stars_repo_head_hexsha": "2290754436864a817851c71803d5275445cbcdb1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2021-05-24T07:00:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-19T16:44:04.000Z", "max_issues_repo_path": "Documentation/Report/Chapters/9-StaticObstacleAvoidance.tex", "max_issues_repo_name": "meltinglab/dynamic-obstacle-avoidance", "max_issues_repo_head_hexsha": "2290754436864a817851c71803d5275445cbcdb1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Documentation/Report/Chapters/9-StaticObstacleAvoidance.tex", "max_forks_repo_name": "meltinglab/dynamic-obstacle-avoidance", "max_forks_repo_head_hexsha": "2290754436864a817851c71803d5275445cbcdb1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2021-04-20T19:24:23.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-12T17:58:16.000Z", "avg_line_length": 145.3563218391, "max_line_length": 858, "alphanum_fraction": 0.7753439823, "num_tokens": 2942, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5350984286266116, "lm_q1q2_score": 0.3485389718040475}} {"text": "\\section{object.size() and Strings}\n\n\\subsection{Comparison to object.size()}\n\n\\R contains a handy tool for telling you how big an already allocated \nobject is, namely the \\code{object.size()} function from the \\code{utils} \npackage. The functions in this package are essentially an extension of that \nfunction for unallocated, dense objects, provided your objects are numeric (more \non this later).\n\nSo say we have the vector \\code{x <- 1.0}. This should be using 8 bytes to \nstore that \\code{1.0} as a double, right? Well\\dots\n\\begin{lstlisting}[language=rr]\nobject.size(1.0)\n## 48 bytes\n\\end{lstlisting}\n\nSo where is all that extra space coming from? Simply put, \\R objects \nare more than just their data. They contain a great deal of very useful \nmetadata, which is where all the nice abstraction comes from. Whenever you \ncreate a vector, \\R keeps track of, for example, its length. If you \ndo not appreciate this convenience, go learn \\proglang{C} and then get back to \nme. \n\nFor vectors, this overhead is 40 bytes, regardless of the type of data. \nMatrices, unsurprisingly cost more, clocking in at 200 bytes overhead. It is \nworth noting that this overhead does not scale; it is on a per-object basis. So \nwe don't need 40 bytes for each element of a vector when just 8 would do (in the \ncase of double precision values). We need 40 plus 8 per element:\n\\begin{lstlisting}[language=rr]\n# 2 elements\n40+8*2\n## [1] 56\nobject.size(rnorm(2))\n## 56 bytes\n\n# 100.000 elements\n40+1e5*8\n## [1] 800040\nobject.size(rnorm(1e5))\n## 800040 bytes\n\\end{lstlisting}\n\nThe story is slightly more complicated for integer data (and a lot more \ncomplicated for strings; see the following section). On my machine (and \nprobably yours, but not necessarily), \\code{int}s costs 4 bytes. However, \n\\R does some aggressive allocation:\n\\begin{lstlisting}[language=rr]\nobject.size(1L:3L)\n## 56 bytes\nobject.size(1L:4L)\n## 56 bytes\n\\end{lstlisting}\nHere we see \\R allocating more bytes than it needs for integer vectors \nsometimes, choosing to allocate in 16 byte chunks rather than 8 byte chunks.\n\nThe \\pkg{memuse} package does not adjust for this overhead, because it \nhonestly just doesn't matter. This overhead is really paltry, and when you \nthink about all the abstraction it buys you, it's a hell of a bargain. If you \nhave a million R objects stored, you're wasting less than one MiB ($1024^2$ \nbytes); so you would need a billion objects to use just about a GiB ($1024^3$ \nbytes) on overhead. And if you're doing that kind of silly shit, my advice \nwould be to learn how to properly use data structures.\n\nAll that said, the main \\code{memuse()} (\\code{mu()} for short) function\nwill offer a better version of \\code{object.size()}. In previous \nversion, the package overwrote the default functionality, but as of\npackage version 3.0.0, that is no longer the case. \n\nThe \\code{object.size()} function returns an object of class \n\\code{object_size}, which is a not-so-useful S3 class provided by core \n\\R. If you instead use \\code{memuse()}, then you will get a much more useful\n\\code{memuse} object. So revisiting the code above:\n\\begin{lstlisting}[language=rr]\nmemuse(rnorm(1e5))\n## 781.289 KiB\n\\end{lstlisting}\n\n\n\n\\subsection{Strings}\n\nString objects have been avoided up until this point because they are much more \ndifficult to describe in general, unless they have a great deal of regularity \nimposed on them. In \\R, strings by default are allocated to use 56 \nbytes (not counting overhead), unless they need more. I'm not sure why this \nvalue was chosen, but 56 byte strings will allow for the storage of 7 chars \n(like \\code{a} but not \\code{aa}). Each char costs 1 byte, so there's some fat \noverhead for the strings here, and almost certainly an additional byte held out \nfor the null terminator. So for example, recall that a vector allocates 40 \nbytes of overhead, so the vector string \\code{letters} should use $56\\times 26 + \n40$ bytes. We can easily verify that this is the case:\n\\begin{lstlisting}[language=rr]\n56*26+40\n## [1] 1496\nmemuse(letters)\n## 1.461 KiB\n\\end{lstlisting}\n\nIf you have a string with more than 7 chars, \\R will allocate extra \nspace in 8-16 byte blocks. After the initial 8 byte allocation (7 chars $+$ \nnull terminator), if you need more you get an additional 8 bytes (in reality \nthis is probably a contiguous 16 byte allocation; I have not bothered to check). \n Beyond that, storage is allocate in 16 byte blocks for each string. For \nexample:\n\\begin{lstlisting}[language=rr]\nmemuse(c(paste(rep(\"a\", 7), collapse=\"\"), \"a\")) \n## 152 B\n\n memuse(c(paste(rep(\"a\", 7+1), collapse=\"\"), \"a\")) \n## 160 B\n\n memuse(c(paste(rep(\"a\", 7+8+1), collapse=\"\"), \"a\")) \n## 176 B\n\n memuse(c(paste(rep(\"a\", 7+8+16+1), collapse=\"\"), \"a\")) \n## 192 B\n\\end{lstlisting}\n\nIf you have a vector of strings with them of varying lengths, the allocation of \nindividual elements is handled on a case-by-case basis. Consider the \nfollowing:\n\\begin{lstlisting}[language=rr]\n memuse(c(paste(rep(\"a\", 7+8+16+1), collapse=\"\"), \"a\")) \n## 192 B\n\\end{lstlisting}\n\nThis object (the vector of 2 elements with first element \n``aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa'' and second element ``a'') is using 40 \nbytes for the vector, $56+8+16+16$ bytes for the first element, and 56 bytes for \nthe second.\n\nFor all of these reasons, and given the fact that I almost never (ever) deal \nwith character data, I have not bothered to make any attempt to extend, for \nexample, \\code{howmany()} or \\code{howbig()}, to incorporate strings. Deal \nwith it, nerd.\n", "meta": {"hexsha": "0b39c42b613459e26649dfe35c43bf532e174f7a", "size": 5555, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "vignettes/include/05-other.tex", "max_stars_repo_name": "cran/memuse", "max_stars_repo_head_hexsha": "22d4e53c0b1a7f9731256d0cd2c791cf8a9d1165", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 34, "max_stars_repo_stars_event_min_datetime": "2015-05-15T17:15:21.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-18T14:03:29.000Z", "max_issues_repo_path": "vignettes/include/05-other.tex", "max_issues_repo_name": "cran/memuse", "max_issues_repo_head_hexsha": "22d4e53c0b1a7f9731256d0cd2c791cf8a9d1165", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2016-09-16T08:01:25.000Z", "max_issues_repo_issues_event_max_datetime": "2019-09-04T20:08:37.000Z", "max_forks_repo_path": "vignettes/include/05-other.tex", "max_forks_repo_name": "cran/memuse", "max_forks_repo_head_hexsha": "22d4e53c0b1a7f9731256d0cd2c791cf8a9d1165", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-19T09:09:30.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-19T09:09:30.000Z", "avg_line_length": 40.5474452555, "max_line_length": 81, "alphanum_fraction": 0.7364536454, "num_tokens": 1549, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.34853896456110484}} {"text": "\\documentclass[11pt,a4paper,oneside]{report} % Single-side\n\n\\usepackage{t1enc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{enumerate}\n\\usepackage{graphics}\n\\usepackage{listings}\n\\usepackage{color}\n\\usepackage{fancyvrb}\n\\usepackage{anysize}\n\\usepackage{setspace}\n\\usepackage[center]{caption}\n\n\\usepackage{titlesec}\n\t\\titleformat{\\chapter}[hang] \n\t{\\normalfont\\huge\\bfseries}{\\chaptertitlename\\ \\thechapter:}{1em}{} \n\n\\usepackage[hidelinks]{hyperref}\n\\usepackage{textcomp}\n\\usepackage{graphicx}\n\\usepackage{svg}\n\n\\onehalfspacing\n\n\\def\\doubleunderline#1{\\underline{\\underline{#1}}}\n\\def\\dul#1{\\doubleunderline{#1}}\n\\def\\ul#1{\\underline{#1}}\n\\newcommand{\\vect}[2]{\\begin{bmatrix} #1 \\\\ #2 \\end{bmatrix}}\n\\setcounter{MaxMatrixCols}{20}\n\n\\begin{document}\n\n% Title page ==================================================================================\n\\begin{titlepage}\n\\begin{center}\n{\\huge Fast decoding of systematic Golay-24 code } \\\\\n\\vspace{0.5cm}\n{\\small Description of the algorithm used on board the SMOG-P, ATL-1 and SMOG-1 satellites } \\\\\n{\\large Timur Kristóf } \\\\\n\n\\vfill\n{\\large \\today}\n\\end{center}\n\\end{titlepage}\n\n\\chapter{Introduction}\n\nThe Golay forward error correction code was invented by Marcel Golay in 1949. Its most well-known\nuse was onboard the NASA Voyager satellites where they used the Golay(24, 12, 8) code. These days\nwe know codes with much better error correction capabilities, but the Golay code is still a good\nchoice in applications where the size of each packet is small and/or the available hardware\nresources are too modest, so there is no chance to use more complicated coding schemes.\n\nThe topic of this study is the \\emph{systematic Golay(24, 12, 8)} code, called \\emph{Golay-24} for short.\nThe aim is to present a new algorithm which can quickly and efficiently\ndecode this coding scheme even on machines with limited compute capabilities,\nsuch as embedded systems.\n\nThis study not only presents the algorithm but also proves its correctness.\n\nThe algorithm presented here is a \\emph{hard-decision} algorithm, since most of the embedded hardware\nthat I've worked with make a hard decision about the received bits, so the aim was to make decoding\nefficient on this class of hardware.\nFurther research needs to be done to determine if the ideas presented here can be utilized in soft-decision algorithms.\n\n\\section{Definitions}\n\n\\doubleunderline{A} is the Golay matrix.\n\n\\[\n \\doubleunderline{A} = \n \\begin{bmatrix}\n 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 1 \\\\\n 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 1 \\\\\n 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 \\\\\n 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 & 1 \\\\\n 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 1 & 1 & 0 \\\\\n 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 1 \\\\\n 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\\\\n 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 \\\\\n 0 & 0 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 1 & 1 & 0 \\\\\n 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\\\\n 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 0 & 1 & 1\n \\end{bmatrix}\n\\]\n\n\\doubleunderline{G} is the generator matrix of the systematic Golay(24, 12, 8) code.\n\n\\[\n \\doubleunderline{G} =\n \\begin{bmatrix}\n \\doubleunderline{I} \\\\\n \\doubleunderline{A}\n \\end{bmatrix}\n\\]\n\nIf \\underline{m} is the message (which we want to encode), and the parity vector is \\underline{p},\nthen the encoded code word is \\underline{x}.\nThe Hamming weight of the code word is $w\\{\\ul{x}\\} = w\\{\\ul{m}\\} + w\\{\\ul{p}\\}$.\n\n\\[\n \\underline{x} = \\begin{bmatrix}\n \\underline{m} \\\\\n \\underline{p}\n \\end{bmatrix} = \\doubleunderline{G} \\cdot \\underline{m} = \\begin{bmatrix}\n \\doubleunderline{I} \\\\\n \\doubleunderline{A}\n \\end{bmatrix} \\cdot \\underline{m} = \\begin{bmatrix}\n \\doubleunderline{I} \\cdot \\underline{m} \\\\\n \\doubleunderline{A} \\cdot \\underline{m}\n \\end{bmatrix} = \\begin{bmatrix}\n \\underline{m} \\\\\n \\doubleunderline{A} \\cdot \\underline{m}\n \\end{bmatrix}\n\\]\n\nIf the \\underline{x} code word is transmitted through a noisy channel, the received word might\ncontain errors. The error vector is \\underline{e} (the error vector in the received message \nis $\\underline{e}_m$ and the error vector in the received parity is $\\underline{e}_p$)\nand the received word is \\underline{r}.\n\n\\[\n \\ul{r} = \\ul{x} + \\ul{e} = \\vect{\\ul{r}_m}{\\ul{r}_p} = \\vect{\\ul{m} + \\ul{e}_m}{\\ul{p} + \\ul{e}_p}\n\\]\n\nThe Golay-24 code is capable of correcting 3 or fewer\nbit errors, and detecting 4 bit errors. So the cases we care about the most are when $w\\{\\ul{e}\\} \\leq 3$\nor $w\\{\\ul{e}\\} = 4$.\n\n\\chapter{Algorithm}\n\nThis chapter presents the decoding algorithm. We will make the assumption that $w\\{\\ul{e}\\} \\leq 4$,\nso the received vector contains at most 4 bit errors. This is reasonable because the Golay(24, 12, 8)\ncode is only capable to correct at most 3 bit errors, and detect 4 bit errors. Since the minimal\nHamming distance between each code word is 8, in case of 4 bit errors it is possible to create the\nreceived vector from multiple different code words, therefore it is not possible to tell which was\nthe original.\n\n\\section{Basics}\n\nFirst a few basic considerations are presented on which the algorithm is based.\n\n\\subsubsection{Inverse and transpose of the Golay matrix}\n\nThe \\dul{A} Golay matrix is its own inverse and transpose.\n\n\\[\n \\dul{A}^T = \\dul{A}\n\\]\n\\[\n \\dul{A}^{-1} = \\dul{A}\n\\]\n\nConsequences:\n\\begin{itemize}\n \\item \\dul{A} is an invertable linear transformation, so if $\\ul{p} = \\dul{A} \\cdot \\ul{m}$, then\n $\\dul{A} \\cdot \\ul{p} = \\dul{A} \\cdot \\dul{A} \\cdot \\ul{m} = \\dul{A} \\cdot \\dul{A}^{-1} \\cdot \\ul{m} = \\ul{m}$.\n In other words, the message can be restored from the parity vector.\n \\item We only need to store the \\dul{A} matrix in memory, and we don't need to ever invert or transpose it.\n\\end{itemize}\n\n\\subsubsection{Parity of messages with 1 Hamming weight}\n\nIf $\\ul{m} = \\ul{I}_{i}$, then $\\ul{p} = \\dul{A} \\cdot \\ul{m} = \\dul{A} \\cdot \\ul{I}_{i} = \\ul{A}_{i}$.\nSo if the message is the same as the $i$th row of the identity matrix $\\dul{I}$, then the parity vector is the\nsame as the $i$-th row of the Golay matrix $\\dul{A}$.\n\nConsequences:\n\\begin{itemize}\n \\item When changing a single bit at $\\ul{I}_{i}$ in the message, the equivalent change to the parity is $\\ul{A}_{i}$.\n \\item When changing a single bit at $\\ul{I}_{i}$ in the parity, the equivalent change to the message is $\\ul{A}_{i}$.\n\\end{itemize}\n\n\\subsubsection{Hamming weight of code words from non-zero Hamming weight messages}\n\nIf $\\ul{m} \\neq \\ul{0}$, then $w\\{ \\ul{x} \\} = w\\{ \\ul{m} \\} + w\\{ \\ul{p} \\} \\geq 8$.\n\nProof:\nWe already know that the Hamming distance between every possible code word is at least 8.\n(Not proven here, but it's a well-known fact about the Golay code.)\nThe $\\ul{m} = \\ul{0}$ message is encoded to the code word $\\ul{x} = \\ul{0}$, so\nthe minimal Hamming distance of every possible code word can only be at least 8 if\nin case of $\\ul{m} \\neq \\ul{0}$ the $w\\{ \\ul{x} \\} \\geq 8$, otherwise there\nwould be a code word $\\ul{x}$ whose Hamming distance from the $\\ul{0}$ code word is less than 8.\n\nConsequence:\n\\begin{itemize}\n \\item $\\forall \\ul{m} \\neq \\ul{0} : w\\{ \\dul{A} \\cdot \\ul{m} \\} \\geq 8$\n\\end{itemize}\n\n\\subsubsection{Lemma 1.}\n\nIf $w\\{\\ul{e}\\} \\leq 4$ and $w\\{\\dul{A} \\cdot \\ul {r_m} + \\ul{r}_p\\} \\leq 3$,\nthen $w\\{\\ul{e}_m\\} = 0$ (which is equivalent to $\\ul{e}_m = \\ul{0}$).\nSo, assuming 4 or fewer bit errors, if we calculate the parity from the received message, and\nthe Hamming distance between the calculated parity and the received parity is less than\nor equal to 3, then the received message is error-free.\n\nProof:\n\nLet's suppose that the statement is false, and $\\ul{e}_m \\neq \\ul{0}$.\nThe given expression looks like:\n\n\\[\n \\dul{A} \\cdot \\ul{r}_m + \\ul{r}_p =\n \\dul{A} \\cdot ( \\ul{m} + \\ul{e}_m ) + ( \\ul{p} + \\ul{e}_p ) =\n \\dul{A} \\cdot \\ul{m} + \\dul{A} \\cdot \\ul{e}_m + \\dul{A} \\cdot \\ul{m} + \\ul{e}_p =\n \\dul{A} \\cdot \\ul{e}_m + \\ul{e}_p\n\\]\n\nConsider the code word generated from $\\ul{e}_m$. The minimum Hamming distance between every possible code word is 8, therefore\nin case of $\\forall \\ul{e}_m \\neq \\ul{0}$ the following is true:\n$w\\{ \\ul{e}_m \\} + w\\{ \\dul{A} \\cdot \\ul{e}_m \\} \\geq 8$.\n(See \"Hamming weight of code words from non-zero Hamming weight messages\" above.)\n\nLet's define $w\\{ \\ul{e}_m \\} = N$, then because $w\\{ \\ul{e} \\} \\leq 4$ the following\nwill also be true: $w\\{ \\ul{e}_p \\} \\leq 4 - N$.\nSo, $w\\{ \\dul{A} \\cdot \\ul{e}_m \\} \\geq 8 - N$.\n\nSo according the above, we get this:\n\\[\n w\\{ \\dul{A} \\cdot \\ul{e}_m + \\ul{e}_p \\} \\geq (8 - N) - (4 - N) = 4\n\\]\n\nWhat we got is that when $\\ul{e}_m \\neq \\ul{0}$, then\n$w\\{ \\dul{A} \\cdot \\ul{e}_m + \\ul{e}_p \\} \\geq 4$ which is a contradiction, because\nwe started with the condition that it's $\\leq 3$.\n\nSo we can conclude that the statement is true, and $\\ul{e}_m = \\ul{0}$.\n\n\\subsubsection{Lemma 2.}\n\nIf $2 \\leq w\\{\\ul{e}\\} \\leq 4$ and $w\\{\\dul{A} \\cdot \\ul{r}_m + \\ul{A}_i + \\ul{r}_p\\} \\leq 2$,\nthen $\\ul{e}_m = \\ul{I}_i$. So if we assume the number of errors to be at least 2 but at most 4,\nand we correct a 1-bit error $\\ul{e}_{m, i} = \\ul{I}_i$ in the received message, then we calculate\nthe parity of the corrected message, and the Hamming distance between the calculated parity and\nthe received parity is less than or equal to 2, then the received message contains exactly 1 error,\nand that error is the $i$th row of the identity matrix, which is what was corrected.\n\nProof.\n\nConsider the $\\ul{r}_2 = \\vect{\\ul{r}_{m,2}}{\\ul{r}_{p,2}} = \\vect{\\ul{r}_m + \\ul{I}_i}{\\ul{r}_p}$ received word.\n\nBecause of Lemma 1, we know that\nif $w\\{ \\dul{A} \\cdot \\ul{r}_{m,2} + \\ul{r}_{p,2} \\} \\leq 3$, then $\\ul{e}_{m,2} = \\ul{0}$.\n\nThe above formula of $\\ul{r}_2$ yields the following result:\n\n\\[\n \\dul{A} \\cdot \\ul{r}_{m,2} + \\ul{r}_{p,2} =\n \\dul{A} \\cdot ( \\ul{r}_m + \\ul{I}_i ) + \\ul{r}_p =\n \\dul{A} \\cdot \\ul{r}_m + \\dul{A} \\cdot \\ul{I}_i + \\ul{r}_p =\n \\dul{A} \\cdot \\ul{r}_m + \\ul{A}_i + \\ul{r}_p\n\\]\n\nThe above conditions contain $w \\{ \\dul{A} \\cdot \\ul{r}_m + \\ul{A}_i + \\ul{r}_p \\} \\leq 2$,\nso for the received word $\\ul{r}_2$ the following is true: $\\ul{e}_{m,2} = \\ul{0}$.\n\nSince $\\ul{r}$ only differs from $\\ul{r}_2$ by only 1 message bit, the error of the parity is\n$\\ul{e}_p = \\ul{e}_{p,2}$. The differing message bit is at $\\ul{I}_i$.\nSo if $\\ul{e}_{m,2} = \\ul{0}$, then $\\ul{e}_m = \\ul{I}_i$.\n\n\\subsubsection{Lemma 3.}\n\nSame as Lemma 1, but the other way around:\n\nIf $w\\{\\ul{e}\\} \\leq 4$ and $w\\{\\dul{A} \\cdot \\ul {r_p} + \\ul{r}_m\\} \\leq 3$,\nthen $w\\{\\ul{e}_p\\} = 0$ (which is equivalent to $\\ul{e}_p = \\ul{0}$).\nSo, if we calculate the message from the received parity, and\nthe Hamming distance between the calculated message and the received message is less than\nor equal to 3, then the received parity is error-free. (And the received message is\n$\\dul{A} \\cdot \\ul {r_p}$.)\n\nThis can be proved in the exact same way as Lemma 1.\n\n\\subsubsection{Lemma 4.}\n\nSame as Lemma 2, but the other way around:\n\nIf $2 \\leq w\\{\\ul{e}\\} \\leq 4$ and $w\\{\\dul{A} \\cdot \\ul{r}_p + \\ul{A}_i + \\ul{r}_m\\} \\leq 2$,\nthen $\\ul{e}_p = \\ul{I}_i$. So if we assume the number of errors to be at least 2 but at most 4,\nand we correct a 1-bit error $\\ul{e}_{p, i} = \\ul{I}_i$ in the received parity, then we calculate\nthe message of the corrected parity, and the Hamming distance between the calculated message and\nthe received message is less than or equal to 2, then the received parity contains exactly 1 error,\nand that error is the $i$th row of the identity matrix, which is what we corrected.\n\nThis can be proved in the exact same way as Lemma 2.\n\n\\section{Steps of the algorithm}\n\n\\subsubsection{Step 1.}\n\nConsider the $\\ul{r}$ received vector.\nSplit $\\ul{r}$ into received message and received parity:\n\n\\[\n \\ul{r} = \\vect{\\ul{r}_m}{\\ul{r}_p}\n\\]\n\n\\subsubsection{Step 2.}\n\nCalculate the parity of the received message part: $\\ul{p}_2 = \\dul{A} \\cdot \\ul{r}_m$.\n\n\\subsubsection{Step 3.}\n\nIf $\\ul{p}_2 = \\dul{A} \\cdot \\ul{r}_m = \\ul{p}$ then the received word contains no error, so we're done.\n\n\\subsubsection{Step 4.}\n\nWhen the received message is error-free but the parity contains $w\\{\\ul{e}_p\\} \\leq 3$ errors, then according to\n\\emph{Lemma 1} we can tell this by looking at the Hamming distance between the received parity $\\ul{r}_p$ and\nthe calculated parity $\\ul{p}_2$ which is $d = w\\{\\ul{r}_p + \\ul{p}_2\\}$.\n\nIf $d \\leq 3$ then the received message is error-free, and only the received parity contained at most 3 errors.\n\n\\subsubsection{Step 5.}\n\nCalculate the message from the received parity, which will be $\\ul{m_2} = \\dul{A} \\cdot \\ul{r}_p$.\n\n\\subsubsection{Step 6.}\n\nWhen the received parity is error-free but the received message has $w\\{\\ul{e}_m\\} \\leq 3$ errors, then according to\n\\emph{Lemma 2} we can tell this by looking at the Hamming distance between the received message $\\ul{r}_m$\nand the calculated message $\\ul{m_2}$ which is $d = w\\{\\ul{r}_m + \\ul{m_2}\\}$.\n\nIf $d \\leq 3$ then the received parity is error-free, and only the received message contained at most 3 errors.\nSo the decoded message is $\\ul{m_2}$.\n\n\\subsubsection{Step 7.}\n\nSo far, the algorithm has covered every possible 1-bit error and all the 2 and 3-bit errors which were\nlimited to only the message or the parity. Let's now deal with the case where the received message contains 1 error,\nand the received parity contains 1 or 2 errors.\n\nFor $\\forall i : 1 \\leq i \\leq 12, \\ul{e}_m = \\ul{I}_i$ cases, let's look at the possible 1-bit errors of the\nreceived message, and correct them in the following manner:\n\n\\[\n \\ul{r}_{corr, i} = \\vect{\\ul{r}_{m, corr, i}}{\\ul{r}_{p, corr, i}} = \\vect{\\ul{r}_m + \\ul{I}_i}{\\ul{r}_p + \\ul{A}_i}\n\\]\n\nLet's also calculate the parity of the corrected message:\n\n\\[\n \\ul{p}_{i} = \\dul{A} \\cdot \\ul{r}_{m, corr, i}\n\\]\n\nThere are 12 possible corrections, the right one is the case when:\n\n\\[\n w\\{\\ul{p}_{i} + \\ul{r}_{p, corr, i}\\} \\leq 2\n\\]\n\nThanks to \\emph{Lemma 2}, whichever $\\ul{r}_{corr, i}$ satisfies the above formula, the correspoding $\\ul{r}_{m, corr, i}$ will be\nthe decoded message.\n\n\\subsubsection{Step 8.}\n\nLet's deal with the cases when the received parity has 1 error, and the received message has 1 or 2 errors.\n\nFor $\\forall i : 1 \\leq i \\leq 12, \\ul{e}_p = \\ul{I}_i$ cases, let's look at the possible 1-bit errors of the\nreceived parity, and correct them in the following manner:\n\n\\[\n \\ul{r}_{corr, i} = \\vect{\\ul{r}_{m, corr, i}}{\\ul{r}_{p, corr, i}} = \\vect{\\ul{r}_m + \\ul{A}_i}{\\ul{r}_p + \\ul{I}_i}\n\\]\n\nLet's calculate the message from the corrected parity:\n\n\\[\n \\ul{m}_{i} = \\dul{A} \\cdot \\ul{r}_{p, corr, i}\n\\]\n\nThere are 12 possible corrections, the right one is the case when:\n\n\\[\n w\\{\\ul{m}_{i} + \\ul{r}_{m, corr, i}\\} \\leq 2\n\\]\n\nThanks to \\emph{Lemma 4}, whichever $\\ul{r}_{corr, i}$ satisfies the above formula, the corresponding $\\ul{m}_{i}$ will be\nthe decoded message.\n\n\\subsubsection{Step 9.}\n\nIf the algorithm reached this point, it means that we've already covered all possible 1, 2 and 3 bit errors.\nSo we've detected a 4-bit error which we can't correct.\n\n\\end{document}\n\n\n", "meta": {"hexsha": "5410616b9118617258ff78c1c04c3766b57f8a6a", "size": 15437, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "proof/golay-fast-proof.tex", "max_stars_repo_name": "Venemo/golay-fast", "max_stars_repo_head_hexsha": "bb9e8145edcb8265acab6cda1d91dee599d36dd2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-07-13T12:53:38.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-13T12:53:38.000Z", "max_issues_repo_path": "proof/golay-fast-proof.tex", "max_issues_repo_name": "Venemo/golay-fast", "max_issues_repo_head_hexsha": "bb9e8145edcb8265acab6cda1d91dee599d36dd2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "proof/golay-fast-proof.tex", "max_forks_repo_name": "Venemo/golay-fast", "max_forks_repo_head_hexsha": "bb9e8145edcb8265acab6cda1d91dee599d36dd2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.1802030457, "max_line_length": 130, "alphanum_fraction": 0.6488955108, "num_tokens": 5440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.651354857898194, "lm_q1q2_score": 0.34853896093963344}} {"text": "\\subsection{Solvent Accessible Surface}\\index{SAS|ff}\r\n\\index{Solvent Accessible Surface|ff}\r\nThe solvent accessible surface is a continuous surface of the molecule which\r\ncan be reached by the center of charge of a solvent molecule. The calculation \r\nof the SAS is carried out as follows:\r\n\r\n\\begin{itemize} \r\n\\item Each atom is assigned a van der Waals' radius. VdW radii used in COSMO\r\nare given in Table~\\ref{vdw}.\r\n\\begin{table}\r\n\\caption{\\label{vdw} Van der Waals radii (\\AA ) used in COSMO}\r\n\\begin{center}\r\n\\begin{tabular}{llllllllllllll}\r\n\\hline\r\n I & R & II & R &III& R &IV & R & V & R &VI & R &VII& R \\\\\r\n\\hline\r\n H & 1.08 & \\\\\r\n Li & 1.80 & & & & & C & 1.53 & N & 1.48 & O & 1.36 & F & 1.30\\\\\r\n Na & 2.30 & & & Al& 2.05 & Si& 2.10 & P & 1.75 & S & 1.70 & Cl& 1.65\\\\\r\n K & 2.80 & Ca & 2.75 & & & & & & & & &Br & 1.80\\\\\r\n & & & & & & & & & & & &I & 2.05\\\\\r\n\\hline\r\n\\end{tabular}\r\n\\end{center}\r\n\\end{table}\r\n\r\n\\item To each radius is added a distance equal to the radius of the solvent. By\r\ndefault, this is 1.0\\AA, but may be changed by the user using \r\n\\comp{RSOLV=$n.nn$}. This gives the distance from the nucleus to the center of\r\na solvent molecule.\r\n\\item A set of points is generated on this surface. These points produce a\r\nbasic grid.\r\n\\item All points which are inside the surface of any other atom are excluded.\r\n\\item The remaining points are moved towards the center of the atom. The\r\ndistance moved is equal to the distance of the center of charge of the solvent\r\nmolecule from the center of the solvent molecule. By default, this distance is\r\nset to \\comp{RSOLV}, but may be set explicitly by keyword \\comp{RSOLV=$n.nn$}.\r\n\\item Each of the remaining points represents a small area of the solvent\r\naccessible surface. The total SAS is calculated from the number of points.\r\n\\end{itemize}\r\n\r\nFrom this definition of the SAS we see that the SAS of each atom is a surface\r\nof radius equal to the van der Waals' radius plus the radius of the solvent\r\nmolecule minus the distance of the center of charge of the solvent molecule to\r\nthe center of the solvent molecule. In other words, the radius is the VdW\r\nradius plus the distance from the surface of the solvent molecule to the center\r\nof charge of the solvent molecule. By default, this extra distance is zero.\r\nOnly that part of the atom surface which can be touched by the solvent molecule\r\nis used. This means that only those atoms on the surface of the molecule can\r\ncontribute to the SAS. Of those atoms that are on the surface of the molecule\r\nthere will be parts of the surface which cannot be reached by the solvent\r\nbecause the solvent molecule is too bulky.\r\n", "meta": {"hexsha": "846247794801869f3c6a25970f35b7484384de26", "size": 2770, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "manuals/MOPAC2000_manual/sas.tex", "max_stars_repo_name": "openmopac/MOPAC-archive", "max_stars_repo_head_hexsha": "01510e44246de34a991529297a10bcf831336038", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-16T20:53:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-16T20:54:11.000Z", "max_issues_repo_path": "manuals/MOPAC2000_manual/sas.tex", "max_issues_repo_name": "openmopac/MOPAC-archive", "max_issues_repo_head_hexsha": "01510e44246de34a991529297a10bcf831336038", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "manuals/MOPAC2000_manual/sas.tex", "max_forks_repo_name": "openmopac/MOPAC-archive", "max_forks_repo_head_hexsha": "01510e44246de34a991529297a10bcf831336038", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.2641509434, "max_line_length": 81, "alphanum_fraction": 0.6815884477, "num_tokens": 794, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7371581626286833, "lm_q2_score": 0.47268347662043286, "lm_q1q2_score": 0.34844248313045645}} {"text": "\\chapter{Generation of Band Structures}\r\n\\section{Program BZ}\r\n\\subsection{Description} \\index{BZ!description of}\r\nUnlike molecules, whose eigenvalue spectrum consists of infinitely sharp lines\r\n(Kronecker-$\\delta$ type), the eigenvalues of solids form bands. These bands\r\nare conventionally represented by Brillouin Zones (BZ). For a polymer, the BZ\r\nconsists of a line, for a layer structure, a surface, and for a solid, a\r\nthree-dimensional shape. The dimensions of the BZ are inverse distance: if two\r\npolymers have unit cells of 10 and 8 \\AA, then the ratio of their unit cell\r\ntranslation vectors is 1.25:1, and the ratio of their BZ's is 1:1.25.\r\n\r\nAlthough MOPAC can generate all the information necessary to generate\r\nband-structures, for ease of use this job is given to a special program called\r\nBZ. \\ MOPAC is designed to run in batch mode, but drawing band-structures is\r\nan interactive operation.\r\n\r\nBefore BZ can be run, a MOPAC calculation must first be run in order to\r\ngenerate the data needed by BZ. \\ Most of this data is invisible to the user,\r\nbut the user must supply to MOPAC one or two data which BZ will use. These\r\ndata are:\r\n\r\n\\begin{description}\r\n\\item[\\comp{MERS=($n_1$,$n_2$,$n_3$)}]~\\\\\r\n(See Chapter~\\ref{makpol:mers} and Section~\\ref{mopac:mers}) BZ needs to know how\r\nthe Fock matrix is organized, and \\comp{MERS} gives the number of unit cells\r\nin each direction. MOPAC uses the keyword \\comp{MERS} as the prompt to write\r\nout a file for use by BZ. Within MOPAC, \\comp{MERS} has no other use.\r\n\\index{MERS}\r\n\\item[\\comp{BCC} \\normalfont{(optional)}]~\\\\\r\nLike \\comp{MERS}, \\comp{BCC} is not used by MOPAC. \\comp{BCC} (see\r\nChapter~\\ref{makpol:bcc}) indicates that the system is Body-Centered-Cubic; in\r\nother words, that every odd unit cell is missing. If the keyword \\comp{BCC} is\r\npresent, MOPAC informs BZ that the system is Body-Centered-Cubic.\r\n\\end{description}\r\n\r\n\r\nBZ is an interactive utility program. It can perform three types of\r\ncalculation:\r\n\\begin{itemize}\r\n\\item Points in $k$-space are to be studied.\r\n\\item Lines in $k$-space are to be drawn.\r\n\\item Surfaces in $k$-space are to be drawn.\r\n\\end{itemize}\r\n\r\n\\subsection{Data input for BZ}\\index{Data! for BZ}\r\nOnce the MOPAC calculation has been completed, the band structure can be\r\ngenerated. The instruction to generate band structures is:\r\n\r\n\\comp{bz $<$filename$>$}\r\n\r\nwhere \\comp{$<$filename$>$} is the name of the file. This must be the same as\r\nthat used for the MOPAC calculation.\r\n\r\nWhen BZ starts, it invites the user to specify the type\r\nof calculation to be performed. The allowed types are:\r\n\\begin{description}\r\n\\item[0] Points in $k$-space are to be studied.\r\n\\item[1] Lines in $k$-space are to be drawn.\r\n\\item[2] Surfaces in $k$-space are to be drawn.\r\n\\end{description}\r\n\r\nInstead of supplying data at the prompt, a data set can be supplied, using the\r\n command:\r\n\r\n\\comp{bz $<$filename$>$ $<$filename2$>$}\r\n\r\nA second file, called \\comp{$<$filename2$>$.bz}, must exist. This file will\r\ncontain the data that would otherwise be supplied by the prompt.\r\n\r\n\\index{Help! with BZ}\r\nIn addition, the user is invited to enter the number `3' to request help. Help\r\ndescribes diagnostic key-words which allow an interested user to follow the\r\ncalculation.\r\n\r\nSome of the operations in BZ are very sensitive to perturbations in the Fock\r\nmatrix, specifically any perturbations which lower the symmetry. The first\r\nthing BZ does, therefore, is to restore symmetry. To do this, BZ reads in a\r\nlist of symmetry operations and symmetrizes the Fock matrix.\r\n\r\nAt the present time, the user must supply all operations of the group.\r\nEventually this will be done automatically by the program, but for now the job\r\nof defining symmetry is given to the user.\r\n\r\n\\subsubsection{Structure of $<$filename$>$.ops}\r\n\\index{Space group!operations}\r\nThis file holds the definitions of the symmetry operations for the space-group.\r\nThe format is `free format' - one or more spaces are treated as one space. The\r\nfirst line should contain one of the following two words: ``Cartesian'' or\r\n``Crystallographic''. If ``Cartesian'' is selected, then the rotation axis is\r\nrelative to the Cartesian coordinates of the atoms; if ``Crystallographic'' is\r\nselected, then the rotation axis is relative to the crystal coordinates of the\r\natoms. (In practice, ``Cartesian'' is {\\em much} easier to use.)\r\n\r\nEach operation is defined by one line and is composed of 12 data.\r\nIn order, these data are:\r\n\\begin{enumerate}\r\n\\item A `1' or `0'. `1' if the operation is composed of the inversion operation times\r\na rotation, `0' if the operation is a pure rotation.\r\n\\item The non-primitive translations. These are given as their reciprocals, thus\r\na (0.5,0.5,0.0) translation would be entered as `2~2~0'. The absence of\r\na non-primitive translation is indicated by a zero. (three numbers)\r\n\\item The rotation expressed as a fraction of a circle.\r\n\\item The axis about which the operation is performed. Units are\r\nCartesian coordinates, or, if ``Crystallographic'' is selected, the crystal\r\ncoordinates. (three numbers)\r\n\\item The center about which the operation is performed. Units are\r\nCartesian coordinates, in Angstroms. (three numbers)\r\n\\item A text string describing the operation.\r\n\\end{enumerate}\r\n\r\nThe operations list is terminated by an operation which has a `2' for the\r\ninversion option.\r\n\r\nExamples of these operations are illustrated in Table~\\ref{sgo}\r\nby the 10 classes of operation for the O$_h$ group in O$_h^7$.\r\n\r\n\r\nThe Fock matrix will be symmetrized if, and only if, every operation of\r\nthe group is represented once. For O$_h$, this would involve a total of 49\r\nlines, 48 for the operations of O$_h$, and one line to terminate the operations\r\nfile.\r\n\r\n% 24 lines, including this one\r\n\\begin{table}\r\n\\caption{\\label{sgo}\r\n Space-Group Operations of O$_h^7$}\r\n\\index{oh7@{$O_h^7$}|ff}\r\n\\begin{center}\r\n\\begin{tabular}{lllllllrrrll}\\\\ \\hline\r\ni& \\multicolumn{3}{c}{N.P.T} & Rotation & \\multicolumn{3}{c}{Axis of Rotation} &\r\n \\multicolumn{3}{c}{Center of Operation}& Name\\\\ \\hline\r\n0& 0& 0& 0& 0.0 & 0.0&0.0 &1.0 & 0.0& 0.0& 0.0 & Identity \\\\\r\n0& 0& 0& 0& 0.3333 & 1.0&1.0 &1.0 & 0.0& 0.0& 0.0 & C$_3$ \\\\\r\n0& 0& 0& 0& 0.5 & 1.0&0.0 &0.0 & 0.0& 0.0& 0.0 & C$_2$=C$_4^2$ \\\\\r\n0& 2& 2& 2& 0.5 & 1.0&0.0 &-1.0 & 0.0& 0.0& 0.0 & C$_2$ \\\\\r\n0& 2& 2& 2& 0.25 & 0.0&0.0 &1.0 & 0.0& 0.0& 0.0 & C$_4$ \\\\\r\n1& 2& 2& 2& 0.0 & 0.0&0.0 &1.0 & 0.0& 0.0& 0.0 & Inversion \\\\\r\n1& 0& 0& 0& 0.25 & 0.0&0.0 &1.0 & 0.0& 0.0& 0.0 & S$_4$ \\\\\r\n1& 0& 0& 0& 0.5 & 1.0&1.0 &0.0 & 0.0& 0.0& 0.0 & $\\sigma_d$ \\\\\r\n1& 2& 2& 2& 0.333333& 1.0&1.0 &1.0 & 0.0& 0.0& 0.0 & S$_6$ \\\\\r\n1& 2& 2& 2& 0.5 & 0.0&0.0 &1.0 & 0.0& 0.0& 0.0 & $\\sigma_h$ \\\\\r\n2& 0& 0& 0& 0.0 & 0.0&0.0 &1.0 & 0.0& 0.0& 0.0 & End \\\\ \\hline\r\n\\end{tabular}\r\n\r\nN.P.T.: Non-Primitive Translation \\end{center}\r\n\\end{table}\r\n\r\n\\subsection{Calculating Points in $k$-space}\r\n\\index{ksp@{$k$-space}!calculating points in}\r\n\r\nA point in $k$-space is specified by one number for a polymer, two\r\nnumbers for a layer system, and three numbers for a solid. All symmetry\r\npoints and lines for diamond are shown in Table~\\ref{sod} and Figure~\\ref{oh7}.\r\n% 21 lines, including this one\r\n\\begin{figure}\r\n\\begin{makeimage}\r\n\\end{makeimage}\r\n% \\setlength{\\unitlength}{0.1cm}\r\n\\begin{center}\r\n% \\begin{picture}(100,100)\r\n% \\put(62,47){$\\Gamma$}\r\n% \\put(106,42){X}\r\n% \\put(108,68){W}\r\n% \\put(107.5,58){$\\longleftarrow$Z}\r\n% \\put(96.5,54){S$\\longrightarrow$}\r\n% \\put(84,44){$\\Delta$}\r\n% \\put(100.2,59.5){U}\r\n% \\put(76,71){L}\r\n% \\put(92,71){Q}\r\n% \\put(56,80){K}\r\n% \\put(61,63){$\\Sigma$}\r\n% \\put(69,59){$\\Lambda$}\r\n% %\\put(14,38){\\psfig{figure=oh7.ps}}\r\n% \\put(14,38){\\includegraphics{oh7}}\r\n% \\end{picture}\r\n\\includegraphics{oh7}\r\n\\end{center}\r\n\\caption{\\label{oh7}Brillouin zone for Diamond}\r\n\\end{figure}\r\nA ``random'' point is indicated by the number 0.123. (Note: 0.377 = 0.500 -\r\n0.123)\r\n\r\nThese high-symmetry points have associated subgroup symmetries, thus $\\Gamma $\r\nhas the symmetry O$_h$, $\\Lambda $ has the symmetry C$_{3v}$, and X has the\r\nsymmetry D$_{4h}$.\r\n\r\n\\subsubsection{Little Groups}\\index{Little groups|(}\r\nThe symmetry operations supplied in \\comp{ $<$filename$>$.ops} are used to\r\ngenerate a little group of the space group. A ``little group'' is the\r\nsub-group of the space-group at a specific point in $k$-space. There are too\r\nmany little groups to allow all the irreducible representations to be labeled.\r\nIn addition, there is no generally-accepted convention for describing them.\r\nBecause of this, the little group is given as a group-table. Thus the little\r\ngroup for $\\Gamma$ would have the form shown in Table~\\ref{lgd}.\r\n\r\n% 22 lines, including this line\r\n\\begin{table}\r\n\\begin{center}\r\n\\caption{\\label{sod} Symmetry Points in $k$-space for Diamond}\r\n\\begin{tabular}{lllcc} \\\\ \\hline\r\n\\multicolumn{3}{c}{Point} & Little Group* & Symbol\\\\ \\hline\r\n 0.000& 0.000& 0.000 & O$_h$ & $\\Gamma$\\\\\r\n 0.500& 0.000& 0.000 & (D$_{4h}$)&X\\\\\r\n 0.500& 0.250& 0.000 & (C$_{2v}$)&W\\\\\r\n 0.500& 0.125& 0.125 & (C$_{2v}$)&U\\\\\r\n 0.375& 0.375& 0.000 & (C$_{2v}$)&K\\\\\r\n 0.250& 0.250& 0.250 & D$_{3d}$ &L\\\\\r\n 0.123& 0.000& 0.000 & C$_{4v}$ &$\\Delta $\\\\\r\n 0.500& 0.123& 0.000 & (C$_{2v}$) &Z\\\\\r\n 0.500& 0.045& 0.045 & (C$_{2v}$)&S\\\\\r\n 0.123& 0.123& 0.000 & C$_{2v}$ &$\\Sigma $\\\\\r\n 0.123& 0.250& 0.377 & C$_{2}$ &Q\\\\\r\n 0.123& 0.123& 0.123 & C$_{3v}$ &$\\Lambda $\\\\ \\hline\r\n\\end{tabular}\r\n\r\n* Point groups in parentheses only approximate little groups.\\end{center}\\end{table}\r\n\\begin{table}\r\n\\begin{center}\r\n\\caption{\\label{lgd} Little Group for $\\Gamma$ for Diamond}\r\n\\begin{tabular}{lrrrrrrrrrrrr}\\\\ \\hline\r\nLevel & Energy & $\\Gamma$ & E & C$_3$ & C$_2$=C$_4^2$ & C$_2$ & C$_4$\r\n & I & S$_4$ & $\\sigma_d$ & S$_6$\r\n&$\\sigma_h$ \\\\ \\hline\r\n 1& -58.0408& 1& 1& 1& 1& 1 &1&1&1&1&1&1 \\\\\r\n 2& -9.4687& 2& 3& 0&-1& 1&-1&3&-1&1&0&-1 \\\\\r\n 3& 0.9807& 3& 3& 0&-1&-1&1&-3&-1&1&0&1 \\\\\r\n 4& 4.1933& 4& 1& 1& 1&-1&-1&-1&1&1&-1&-1\\\\ \\hline\r\n\\end{tabular}\\end{center} \\end{table}\r\n\r\nFrom this we see that there are four different irreducible representations: 1,\r\n2, 3, and 4. These representations can be equated with those of the O$_h$\r\npoint-group, so 1 and 4, being non-degenerate, become $a$, while 2 and 3, being\r\n3-fold degenerate, become $t$. So far we have ($a$, $t$, $t$, $a$). The effect\r\nof inversion on the eigenfunctions allows us to assign {\\em gerade} and {\\em\r\nungerade}, thus ($a_g$, $t_g$, $t_u$, $a_u$), while C$_4$ resolves the\r\n``1''-``2'' nature of the representations: ($a_{1g}$, $t_{2g}$, $t_{1u}$,\r\n$a_{2u}$).\r\n\r\nInspection of the eigenvectors reveals, as expected, that the two\r\n$a$ representations can be identified with the $s$ atomic orbitals, and the two\r\n$t$ representations, with the six $p$ atomic orbitals.\r\n\r\nFor points other than $\\Gamma$, the little groups become complex. Consider a\r\npoint on $\\Delta$, the line connecting $\\Gamma$ and X, or (0,0,0) and\r\n(1/2,0,0). The little group for (0.123,0,0) has the form shown in\r\nTable~\\ref{dig}.\r\n\\begin{table}\r\n\\begin{center}\r\n\\caption{\\label{dig} Little Group for $\\Delta$ for Diamond}\r\n \\index{Diamond!little group|(}\r\n\\begin{tabular}{lrlrrrrr} \\\\ \\hline\r\nLevel&Energy&$\\Gamma$&E &C$_2$=C$_4^2$ &C$_4$&$\\sigma_d$\r\n&$\\sigma_h$ \\\\ \\hline\r\n 1& -54.2893 &1 & 1& 1& $\\theta$ & 1& $\\theta$ \\\\\r\n 2& -10.2758 &2 & 2&-2& 0 & 0& 0 \\\\\r\n 3& -10.1336 &3 & 1& 1&-$\\theta$ & 1& -$\\theta$ \\\\\r\n 4& 0.9087 &1 & 1& 1& $\\theta$ & 1& $\\theta$ \\\\\r\n 5& 1.5003 &2 & 2&-2& 0 & 0& 0 \\\\\r\n 6& 3.7878 &3 & 1& 1&-$\\theta$ & 1& -$\\theta$ \\\\ \\hline\r\n\\end{tabular}\\end{center}\\end{table}\r\nHere $\\theta$ = $e^{-ik/2\\pi}$.\r\n\r\n\r\nAs is seen, complications arise for those operations which involve\r\n\\index{Characters!complex}\\index{Complex characters} non-primitive\r\ntranslations,\\index{Translations!non-primitive} \\index{Non-primitive\r\ntranslations} e.g.\\ C$_{4}$ and $\\sigma_h$, when points other than $\\Gamma$ are\r\nstudied. The complex phase-factor can cause interpretation of the group\r\ncharacters to be difficult. One option is to correct for the phase-factor by\r\nmultiplying the character by $e^{ik/2\\pi}$. This would make the little group\r\nfor $\\Delta$ easier to read (Table~\\ref{mdig}).\r\n\r\n\\begin{table}\r\n\\begin{center}\r\n\\caption{\\label{mdig} Modified Little Group for $\\Delta$ for Diamond}\r\n\\begin{tabular}{lrlrrrrr} \\\\ \\hline\r\nLevel&Energy&$\\Gamma$&E &C$_2$=C$_4^2$ &C$_4$&$\\sigma_d$\r\n&$\\sigma_h$ \\\\ \\hline\r\n 1& -54.2893 &1 & 1& 1& 1 & 1& 1 \\\\\r\n 2& -10.2758 &2 & 2&-2& 0 & 0& 0 \\\\\r\n 3& -10.1336 &3 & 1& 1&-1 & 1& -1 \\\\\r\n 4& 0.9087 &1 & 1& 1& 1 & 1& 1 \\\\\r\n 5& 1.5003 &2 & 2&-2& 0 & 0& 0 \\\\\r\n 6& 3.7878 &3 & 1& 1&-1 & 1& -1 \\\\ \\hline\r\n\\end{tabular}\\end{center}\\end{table}\r\n\r\nHowever, for some little groups, this procedure may cause complications. Thus,\r\nat $L$, the inversion operation becomes sensitive to the point chosen, e.g.\r\n(0.25,0.25,0.25) will give a different character for the $I$ operation than\r\n(0.25,0.25,-0.25).\r\n\r\nA third option is to assume that the lowest-lying level represents the totally\r\nsymmetric representation. (Note: this is different from transforming as the\r\ntotally symmetric representation. Only one point, the $\\Gamma$-point, can ever\r\ncontain a totally symmetric representation.) By making that assumption, all\r\ncharacters can be related to those of the lowest-lying level.\r\n\r\nUsers are given key-words which select these alternatives to the basic\r\ncharacter tables when the $k$-point is specified. The key-words are:\r\n\r\n\\begin{description}\r\n\\index{One! use in BZ}\\index{BZ!use of ``one''}\r\n\\item[\\comp{one}] Assume the lowest energy level is totally symmetric. All operations\r\ninvolving non-primitive translations are then multiplied by a (complex) factor\r\nwhich will make the lowest energy level totally symmetric. Example: \\comp{0.123 0 0 one}.\r\n\r\n\\index{Byk! use in BZ}\\index{BZ!use of ``byk''}\r\n\\item[\\comp{byk}] Multiply all operations which involve non-primitive translations by\r\na phase factor (``byk''---``multiply by k'') which depends on the current value\r\nof $k$. Example: \\comp{0.123 0 0 byk}.\r\n\\end{description}\r\n\r\nUsing the points option, assignment of the symmetry of lines and points in\r\n$k$-space becomes very easy.\r\n\r\n\\subsection{Lines in $k$-space}\r\n\r\nA line in $k$-space is specified by two numbers for a polymer, four numbers\r\nfor a layer system, and six numbers for a solid. The number of points\r\ncalculated for each line is proportional to the length of the line in\r\n$k$-space. For a line of length 1.0, for example $\\Gamma$ to $\\Gamma$, 300\r\npoints would be calculated. \\index{Walks in\r\n$k$-space}\\index{ksp@{$k$-space}!walks} The data needed to specify a\r\nband-structure `walk' from $\\Gamma $ to X to W to L to $\\Gamma $ is given in\r\nTable~\\ref{walk}, and for diamond, the band-structure for this `walk' is given\r\nin Figure~\\ref{walk-pic}.\r\n\r\n\\begin{table}\r\n\\begin{center}\r\n\\caption{\\label{walk}A ``Walk'' in $k$-Space ($\\Gamma $ to X to W to L to $\\Gamma $)}\r\n\\begin{tabular}{ccccccccc} \\\\ \\hline\r\nStart && Stop&\\multicolumn{3}{c}{Start} & \\multicolumn{3}{c}{Stop} \\\\ \\hline\r\n$\\Gamma$&-&X & 0.000& 0.000& 0.000& 0.500& 0.000& 0.000 \\\\\r\nX &-&W & 0.500& 0.000& 0.000& 0.500& 0.250& 0.000 \\\\\r\nW &-&L & 0.500& 0.250& 0.000& 0.250& 0.250& 0.250 \\\\\r\nL &-&$\\Gamma$ & 0.250& 0.250& 0.250& 0.000& 0.000& 0.000 \\\\ \\hline\r\n\\end{tabular}\r\n\\end{center}\r\n\\end{table}\r\n\\begin{figure}\r\n\\begin{makeimage}\r\n\\end{makeimage}\r\n\\begin{center}\r\n\\includegraphics{diamondbs}\r\n\\end{center}\r\n\\caption{\\label{walk-pic}Band structure for Diamond}\r\n\\label{pbt}\r\n\\end{figure}\r\n\\index{Little groups|)}\r\n\\index{Diamond!little group|)}\r\n\r\n\r\n\\subsection{Surfaces in $k$-space}\r\n\\index{ksp@{$k$-space}!surfaces in}\\index{Surfaces in $k$-space} A surface is\r\ndefined by seven data. Three numbers are used to specify the center of the\r\nsurface, three are used to define a perpendicular, and one number is used to\r\ndefine the length of an edge.\r\n\r\nFor example, to generate the hexagonal face of the Brillouin Zone for diamond,\r\nthe specification would be (0.25 0.25 0.25 1 1 1 0.7071), the center of the\r\nplot being (0.25 0.25 0.25), and (1 1 1) defining the direction.\r\n\r\nTwo dimensional surfaces use 80 points in each direction, and an interpolation\r\nprocedure is used to double this number to 160 points in each direction. The\r\nlength of a side in a two-dimensional plot is defined by the user at\r\nrun-time.\r\n\r\nThe cost of generating the surface is quite large. Because of this, the data\r\nare stored on disk, once they are generated.\r\n\r\nThe user is prompted for input.\r\n\r\n\\subsection{Files used by BZ}\r\n\\index{Files used by BZ}\\index{BZ!files used by}\r\nBZ uses the following files:\r\n\\begin{description}\r\n\\item[\\comp{ $<$filename$>$.ops}] a symmetry file.\r\n\\item[\\comp{ $<$filename$>$.brz}] a Fock matrix file made by MOPAC.\r\n\\end{description}\r\n\r\nThe file \\comp{$<$filename$>$.brz} {\\em must} exist. The file\r\n\\comp{$<$filename$>$.ops}\r\n{\\em should}, but does not have to, exist. If \\comp{$<$filename$>$.ops} is\r\nwritten, the first two lines should look like this:\r\n\\begin{verbatim}\r\n0 0 0 0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 'Identity'\r\n2 0 0 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 'Terminus'\r\n\\end{verbatim}\r\nMore lines can be added, as needed, to define the other symmetry operations.\r\n", "meta": {"hexsha": "cbbbee87557473ab83db4b5cb604e7e4c22a516a", "size": 17540, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "manuals/MOPAC2000_manual/bz.tex", "max_stars_repo_name": "openmopac/MOPAC-archive", "max_stars_repo_head_hexsha": "01510e44246de34a991529297a10bcf831336038", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-12-16T20:53:27.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-16T20:54:11.000Z", "max_issues_repo_path": "manuals/MOPAC2000_manual/bz.tex", "max_issues_repo_name": "openmopac/MOPAC-archive", "max_issues_repo_head_hexsha": "01510e44246de34a991529297a10bcf831336038", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "manuals/MOPAC2000_manual/bz.tex", "max_forks_repo_name": "openmopac/MOPAC-archive", "max_forks_repo_head_hexsha": "01510e44246de34a991529297a10bcf831336038", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 44.9743589744, "max_line_length": 90, "alphanum_fraction": 0.6677879133, "num_tokens": 6109, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7371581510799253, "lm_q2_score": 0.4726834766204328, "lm_q1q2_score": 0.34844247767154934}} {"text": "% !TEX encoding = UTF-8\n% !TEX program = pdflatex\n% !TEX spellcheck = en_US\n% !TEX root = roma-demo.tex\n\n\n\\section{Introduction}\n\n\\subsection{Original problem}\n\n\\begin{frame}[t]{Code Example}\n\t\\begin{lstlisting}[language=C++, caption=Multiplication example]\n\t\tbyte mult(byte in_1, byte in_2){\n\t\t\tbyte mask, result, piv;\n\t\t\tmask = 0x01;\n\t\t\tresult = 0x00;\n\t\t\tpiv = in_1;\t\n\t\t\tfor(int i=0; i<8; i++){\n\t\t\t\tif(in_2 & mask) result^=piv;\n\t\t\t\tmask = mask << 1;\n\t\t\t\tpiv = xtime(piv);\n\t\t\t}\n\t\t\treturn result;\n\t\t}\n\t\\end{lstlisting}\n\\end{frame}\n\n\\begin{frame}[t]{Assumptions}\n\n\tLet us consider the usual linear regression model defined as: \\[y = \\beta_*^t \\mathbf{x} + \\epsilon \\text{ with }\\]\n\t\n\t\\begin{enumerate}[i.]\n\t\t\\item $\\mathbf{x} \\in \\mathbb{R}^p$ a vector of random variables normally called the input vector.\n\t\t\\item $\\epsilon \\in \\mathbb{R}$ the random noise defined as a Gaussian random variable with expectation zero and variance $\\sigma^2$, this is, $\\epsilon \\sim \\mathcal{N}(\\mu=0,\\,\\sigma^{2})$\n\t\t\\item $y \\in \\mathbb{R}$ a random variable that depends linearly on $\\mathbf{x}$.\n\t\t\\item $\\beta_* \\in \\mathbb{R}^p$ is the optimal model.\n\t\\end{enumerate} \n\n\\end{frame}\n\n\\begin{frame}[t]{Problem Formulation}\n\t\n\tThe problems we need to solve to estimate $\\beta_*$ are written as follow:\n\t\n\t\\begin{mini!}|s|[2]\n\t\t{\\beta, e}{ f(e)+ \\lambda g(\\beta)}{}{P(s, \\lambda) \\quad}\n\t\t\\addConstraint{e}{= \\mathbf{y} - X \\beta \\quad \\label{const:c1ex1}}{e \\in \\mathbb{R}^m, \\beta \\in \\mathbb{R}^p }\n\t\t\\addConstraint{h(\\beta)}{\\leq s}{ s \\in \\mathbb{R}, s \\geq 0}\n\t\t\\addConstraint{\\mathbf{L}\\leq }{A\\beta \\leq \\mathbf{U}}{\\mathbf{L,U} \\in \\mathbb{R}^m, A \\in \\mathbb{R}^{q x p}}\n\t\\end{mini!}\n\n\\end{frame}\n\n\\begin{frame}[t]{Examples}\n\t\\begin{itemize}\n\t\n\t\\item $f$ is the error function. Examples: $f(\\mathbf{e}) = {\\| \\mathbf{e} \\|}_1$ y $f(\\mathbf{e}) =\\frac{1}{2} {\\| \\mathbf{e} \\|}^2_2$, among others.\n\t\n\t\\item $h$ and $g$ are the complexity functions of the model. Examples: $g(\\mathbf{\\beta}) = {\\| \\mathbf{\\beta} \\|}_1$ y $g(\\mathbf{\\beta}) = {\\| \\mathbf{\\beta} \\|}^2_2$, $h(\\mathbf{\\beta}) = {\\| \\mathbf{\\beta} \\|}_1$ y $h(\\beta) = {\\| \\beta \\|}_0 = \\vert \\lbrace j: \\beta_j \\neq 0,\\;\\;j \\in [n] \\rbrace \\vert$, among others. \n\t\n\t\\item $A$,$\\mathbf{L}$ and $\\mathbf{U}$ allow the modelling of linear constraints over the regressors.\n\n\t\\end{itemize}\n\\end{frame}\n\n\\subsection{The Holistic Regression Problem}\n\n\\begin{frame}[t]{The Holistic Regression problem}\n\\[(R(\\lambda,s))\\;\\;\\underset{\\beta,\\mathbf{e},\\mathbf{z}}{min}\\;\\;f(\\mathbf{e}) + \\lambda g(\\beta)\\;\\;s.t.\\]\n\\[\\mathbf{y} - X \\beta = \\mathbf{e}\\]\n\\[h(\\beta) \\leq s\\]\n\\[\\mathbf{L} \\leq A\\beta \\leq \\mathbf{U}\\]\n\\[(\\beta,\\mathbf{z}) \\in H\\]\n\\[\\beta \\in \\mathbb{R}^n,\\;\\;\\mathbf{e} \\in \\mathbb{R}^m,\\;\\;\\mathbf{z} \\in {\\lbrace 0,1 \\rbrace}^n\\]\n\t\n\t\\begin{itemize}\n\t\t\\item The structure of $R(\\lambda,s)$ is too general, so the algorithms designed for $P(s, \\lambda)$ cannot be used (Specially because $H$).\n\t\t\\item For the usual optins of $f$,$g$,$h$ and for $H$ defined as affine equations and inequations in $(\\beta,\\mathbf{z})$ $R(\\lambda,s)$ is as (0-1-MICQP).\n\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[t]{Huber Function $\\rho_\\gamma(s)$}\n\t\\begin{center}\n\t\t$\\rho_\\gamma(s) =\n\t\t\\begin{cases}\n\t\t\\frac{1}{2} s^2 & 0 \\leq s \\leq \\gamma\\\\\n\t\t\\gamma s - \\frac{1}{2} \\gamma^2 & s \\geq \\gamma\n\t\t\\end{cases}\n\t\t$\n\t\t\n\t\t\\vspace{0.3cm}\n\t\t\n\t\t\\resizebox{0.58\\textwidth}{!}{\n\t\t\t\\begin{tikzpicture}\n\t\t\t\\begin{axis}[\n\t\t\txlabel=$s$,\n\t\t\tylabel={$\\rho_\\gamma(s)$},\n\t\t\tlegend pos=north west\n\t\t\t]\n\t\t\t% p_2(x)\n\t\t\t\\addplot[green, ultra thick, domain=0:3]{0.5*x^2};\n\t\t\t\\addlegendentry{$\\gamma=3$}\n\t\t\t\\addplot[blue, ultra thick, domain=0:2]{0.5*x^2};\n\t\t\t\\addlegendentry{$\\gamma=2$}\n\t\t\t\\addplot[red, ultra thick, domain=0:0.5]{0.5*x^2};\n\t\t\t\\addlegendentry{$\\gamma=0.5$}\n\t\t\t\n\t\t\t\\addplot[green, ultra thick, domain=3:6]{3*abs(x)-0.5*(3)^2};\n\t\t\t\n\t\t\t\\addplot[blue, ultra thick, domain=2:6]{2*abs(x)-0.5*(2)^2};\n\t\t\t\n\t\t\t\\addplot[red, ultra thick, domain=0.5:6]{0.5*abs(x)-0.5*(0.5)^2};\n\t\t\t\n\t\t\t\\end{axis}\n\t\t\t\\end{tikzpicture}\n\t\t}\n\t\\end{center}\n\\end{frame}\n\n\\begin{frame}[t]{The $\\epsilon$-insensitive Huber Function $g_\\gamma^\\epsilon(t)$}\n\t\\begin{center}\n\t\t\n\t\t$f^{\\epsilon}_\\gamma(\\mathbf{e}) = \\underset{i \\in [m]}{\\sum} g^{\\epsilon}_\\gamma(\\bar{\\mathbf{e}}_i)$ with: \\\\\n\t\t\n\t\t$g^{\\epsilon}_\\gamma(t) = \\left\\{\\begin{array}{lcc} 0 & si & \\vert t \\vert \\leq \\epsilon\\\\\\rho_\\gamma(\\vert t \\vert - \\epsilon) & si & \\vert t \\vert \\geq \\epsilon\\\\\n\t\t\\end{array}\n\t\t\\right.$\n\t\\end{center}\n\t\n\\end{frame}\n\n\\section{The study case}\n\n\\begin{frame}[t]{Study case}\n\t\n\t\\[(R(\\lambda,k))\\;\\;\\underset{\\beta,\\mathbf{e},\\mathbf{z}}{min}\\;\\;f^\\epsilon_\\gamma (\\mathbf{e}) + \\lambda {\\parallel \\beta \\parallel}_1\\;\\;s.t.\\;\\;(1)\\]\n\t\\[\\mathbf{y} - X \\beta = \\mathbf{e}\\;\\;(2)\\]\n\t\\[\\mbox{If}\\;\\;\\mathbf{z}_j = 1\\;\\;\\mbox{then}\\;\\;\\beta_j = 0\\;\\;\\forall j \\in [n]\\;\\;(3)\\]\n\t\\[Co(\\mathbf{z})\\leq k\\;\\;(4)\\]\n\t\\[\\underset{j \\in J1_i}{\\sum} \\mathbf{z}_j \\leq 1\\;\\;\\forall i \\in [n_1]\\;\\;(5)\\]\n\t\\[\\underset{j \\in J2_i}{\\sum} \\mathbf{z}_j = 1\\;\\;\\forall i \\in [n_2]\\;\\;(6)\\]\n\t\\[\\mathbf{z}_{j_1} = \\mathbf{z}_{j_2}\\;\\;\\forall j_1,j_2 \\in B_i\\;\\;\\forall i \\in [n_B]\\;\\;(7)\\]\n\t\\[\\beta_j \\geq 0\\;\\;\\forall j \\in J^+,\\;\\;\\beta_j \\leq 0\\;\\;\\forall j \\in J^-\\;\\;(8)\\]\n\n\\end{frame}\n\n\\begin{frame}[t]{Study case (II)}\n\t\n\t\\[(R(\\lambda,k))\\;\\;\\underset{\\beta,\\mathbf{e},\\mathbf{z}}{min}\\;\\;f^\\epsilon_\\gamma (\\mathbf{e}) + \\lambda {\\parallel \\beta \\parallel}_1\\;\\;s.t.\\;\\;(1)\\]\n\t\\[ \\cdots \\]\n\t\\[\\mathbf{z}_{j_1} + \\mathbf{z}_{j_2} \\leq 1\\;\\;\\forall (j_1,j_2) \\in Jc\\;\\;(9)\\]\n\\[\\beta \\in \\mathbb{R}^n,\\;\\;\\mathbf{e} \\in \\mathbb{R}^m,\\;\\;\\mathbf{z} \\in {\\lbrace 0,1 \\rbrace}^n\\;\\;(10)\\]\n\n\t\n\\end{frame}\n\n\\section{Roadmap}\n\n\n\\begin{frame}[t]{Road}\n\\textbf{Objective:} Find a set of solutions with values close to the optimal, this is {\\it a path of Approximate Solutions} \n\n\\begin{enumerate}\n\t\\item Formulation of $ R(\\lambda, k)$.\n\t\\item Obtaining valid values for BigM values. \n\t\tBigMs too big $\\implies$ numerical problems, inefficiency of algorithms. \n\t\tBigMs too small $ ~ implies $ good solutions removed. \n\t\t\\item {\\it Local Holistic Searches}: Distances, Neighborhoods and Complexity function are presented on $\\mathbf{z}$, which we will call Holistic, that adapt naturally to the case study, to obtain solutions locally optimal.\n\t\t\\item Algorithms for the construction of the Approximate Solutions \tPath combining (2.) and (3.)\n\\end{enumerate}\n\\end{frame}\n", "meta": {"hexsha": "3f30e8e5c199136244fa89631c7684e87f1a5b6f", "size": 6449, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "presentation/1_introduction/regression.tex", "max_stars_repo_name": "belgrades/aes", "max_stars_repo_head_hexsha": "ebd1fbf36acd8e3a787ebc0cd68f83e3784d2979", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-02-14T12:34:37.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-14T12:34:37.000Z", "max_issues_repo_path": "presentation/1_introduction/regression.tex", "max_issues_repo_name": "belgrades/aes", "max_issues_repo_head_hexsha": "ebd1fbf36acd8e3a787ebc0cd68f83e3784d2979", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "presentation/1_introduction/regression.tex", "max_forks_repo_name": "belgrades/aes", "max_forks_repo_head_hexsha": "ebd1fbf36acd8e3a787ebc0cd68f83e3784d2979", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 37.2774566474, "max_line_length": 327, "alphanum_fraction": 0.6146689409, "num_tokens": 2566, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199008363969, "lm_q2_score": 0.6757646140788308, "lm_q1q2_score": 0.34843768330007274}} {"text": "\\section{UTxO}\n\\label{sec:utxo}\n\n\\begin{figure*}[htb]\n \\emph{Functions}\n %\n \\begin{align*}\n & \\fun{isNonNativeScriptAddress} \\in \\Tx \\to \\Addr \\to \\Bool \\\\\n & \\fun{isNonNativeScriptAddress}~tx~a = \\\\\n &\\quad\\begin{cases}\n s \\in \\ScriptNonNative & a \\in \\AddrScr \\land \\fun{validatorHash}~a \\mapsto s \\in \\fun{txscripts} (\\fun{txwits}~tx) \\\\\n \\False & \\text{otherwise}\n \\end{cases}\n \\nextdef\n & \\fun{totExunits} \\in \\Tx \\to \\ExUnits \\\\\n & \\fun{totExunits}~\\var{tx} = \\sum_{\\wcard \\mapsto (\\wcard, eu) \\in \\fun{txrdmrs}~(\\fun{txwits}~tx)} eu\n \\nextdef\n & \\fun{feesOK} \\in \\PParams \\to \\Tx \\to \\UTxO \\to \\Bool \\\\\n & \\fun{feesOK}~\\var{pp}~tx~utxo~= \\\\\n &~~ \\minfee{pp}~{tx} \\leq \\txfee{txb} \\wedge (\\fun{totExunits}~tx \\neq (0, 0) \\Rightarrow \\\\\n &~~~~~~((\\forall (a, \\wcard, \\_) \\in \\fun{range}~(\\fun{txinputs_{fee}}~{txb} \\restrictdom \\var{utxo}), \\neg \\fun{isNonNativeScriptAddress}~tx~a) \\\\\n &~~~~~~\\wedge~ \\fun{adaOnly}~\\var{balance} \\\\\n &~~~~~~ \\wedge~ \\var{balance} \\geq \\txfee{txb})) \\\\\n &~~ \\where \\\\\n & ~~~~~~~ \\var{txb}~=~\\txbody{tx} \\\\\n & ~~~~~~~ \\var{balance}~=~\\fun{ubalance}~(\\fun{txinputs_{fee}}~{txb} \\restrictdom \\var{utxo})\n \\nextdef\n & \\fun{txins} \\in \\TxBody \\to \\powerset{\\TxId \\times \\Ix} \\\\\n & \\fun{txins} ~\\var{txb} = \\fun{txinputs}~txb \\cup \\fun{txinputs_{fee}}~txb\n \\nextdef\n & \\fun{txscriptfee} \\in \\Prices \\to \\ExUnits \\to \\Coin \\\\\n & \\fun{txscriptfee}~(pr_{mem}, pr_{steps})~ (\\var{mem, steps})\n = \\var{pr_{mem}}*\\var{mem} + \\var{pr_{steps}}*\\var{steps}\n \\nextdef\n &\\fun{minfee} \\in \\PParams \\to \\Tx \\to \\Coin \\\\\n &\\fun{minfee}~\\var{pp}~\\var{tx} = \\\\\n &~~(\\fun{a}~\\var{pp}) \\cdot \\fun{txSize}~\\var{tx} + (\\fun{b}~\\var{pp}) +\n \\hldiff{\\fun{txscriptfee}~(\\fun{prices}~{pp})~(\\fun{totExunits}~(\\fun{txbody}~{tx}))}\n \\end{align*}\n \\caption{Functions related to fees}\n \\label{fig:functions:utxo}\n\\end{figure*}\n\nWe have added or changed several functions that deal with fees as shown in Figure \\ref{fig:functions:utxo}.\n\n\\begin{itemize}\n \\item $\\fun{isNonNativeScriptAddress}$ is a predicate that checks\n whether an address is used as a script address with a non-native\n script.\n \\item $\\fun{totExunits}$ calculates the total $\\ExUnits$ in a transaction by summing\n the per-script units stored in the indexed redeemer structure\n \\item The predicate $\\fun{feesOK}$ checks whether the transaction is\n paying the necessary fees, and that it does it correctly. That is, it checks that:\n \\begin{enumerate}[label=({\\roman*})]\n \\item the fee amount that the transaction states it is paying suffices to cover\n the minimum fee that the transaction is obligated to pay; and if the transaction uses non-native scripts, that\n \\item the fee-marked inputs do not belong to non-native script addresses;\n \\item all the fee-marked inputs contain strictly Ada and no other kinds of token; and\n \\item the fee-marked inputs are sufficient to cover the fee amount that is stated\n in the transaction.\n \\end{enumerate}\n \\item The function $\\fun{txins}$ returns the UTxO keys of transaction inputs.\n \\item $\\fun{txscriptfee}$ calculates the fee that a transaction must pay for script\n execution based on the amount of $\\ExUnits$ it has budgeted, and the prices in the current protocol parameters\n for each component of $\\ExUnits$.\n \\item The minimum fee calculation, $\\fun{minfee}$, includes the script\n fees that the transaction is obligated to pay in order to run its scripts.\n\\end{itemize}\n\nNote that when creating a transaction, the wallet is responsible for\ndetermining the fees. Thus, it also has to execute the non-native scripts\nand include the fees for their execution.\n\n\\subsection{Combining Scripts with Their Inputs}\n\\label{sec:scripts-inputs}\n\nFigure~\\ref{fig:functions:script1} shows the functions that are needed to\nretrieve all the data that is relevant to Plutus script validation.\nThese include:\n\n\\begin{itemize}\n\\item $\\Time$ is the system time\n\\item\n $\\ScriptPurpose$ is a sum type of all parts of a transaction that may\n require a script witness to validate. Note that this contains the data\n (eg. a certificate $c \\in \\fun{txcerts}~{txb}$,\n or a transaction input $tin \\in \\fun{txcerts}~{txb}$) of the item being validated,\n not just a tag indicating the type.\n\\item\n $\\fun{indexof}$ is a helper function that finds the index of a given certificate, value, input, or\n withdrawal in a list, finite map, or set of such objects.\n It assumes there is some ordering on each of these structures.\n\\item\n $\\fun{indexedRdmrs}$ indexes the pair of a redeemer and an $\\ExUnits$ value\n for a given script by its script purpose (eg. the input, or certificate, etc.)\n\\end{itemize}\n\n\n\\begin{figure}[htb]\n \\emph{Abstract types}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}l@{}lr}\n \\var{tm}\n & \\Time\n & \\text{System time}\n \\end{array}\n \\end{equation*}\n %\n \\emph{Derived types}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}l@{\\qquad=\\qquad}lr}\n \\var{sp}\n & \\ScriptPurpose\n & \\PolicyID \\uniondistinct \\UTxOIn \\uniondistinct \\AddrRWD \\uniondistinct \\DCert\n% & \\text{item the script is validated for}\n \\end{array}\n \\end{equation*}\n %\n \\emph{Abstract functions}\n \\begin{align*}\n &\\fun{indexof} \\in \\DCert \\to \\seqof{\\DCert} \\to \\Ix\\\\\n &\\fun{indexof} \\in \\AddrRWD \\to \\Wdrl \\to \\Ix\\\\\n &\\fun{indexof} \\in \\UTxOIn \\to \\powerset{\\TxIn} \\to \\Ix\\\\\n &\\fun{indexof} \\in \\PolicyID \\to \\Value \\to \\Ix\n \\end{align*}\n %\n \\emph{Indexing functions}\n \\begin{align*}\n &\\fun{indexedRdmrs} \\in \\Tx \\to \\ScriptPurpose \\to (\\seqof{\\Data}~\\times~\\ExUnits)\\\\\n &\\fun{indexedRdmrs}~tx~sp =\n \\begin{cases}\n ([], \\var{eu}) & rdptr \\mapsto (\\Nothing, \\var{eu}) \\in \\fun{txrdmrs}~(\\fun{txwits}~{tx}) \\} \\\\\n ([d], \\var{eu}) & rdptr \\mapsto (\\var{d}, \\var{eu}) \\in \\fun{txrdmrs}~(\\fun{txwits}~{tx}) \\} \\\\\n \\Nothing & \\text{otherwise}\n \\end{cases} \\\\\n & ~~\\where \\\\\n & ~~\\quad \\var{txb} = \\txbody{tx} \\\\\n & ~~\\quad \\var{rdptr} = \\begin{cases}\n (\\mathsf{certTag},\\fun{indexof}~\\var{sp}~(\\fun{txcerts}~{txb})) & \\var{sp}~\\in~\\DCert \\\\\n (\\mathsf{wdrlTag},\\fun{indexof}~\\var{sp}~(\\fun{txwdrls}~{txb})) & \\var{sp}~\\in~\\AddrRWD \\\\\n (\\mathsf{mintTag},\\fun{indexof}~\\var{sp}~(\\fun{mint}~{txb})) & \\var{sp}~\\in~\\PolicyID \\\\\n (\\mathsf{inputTag},\\fun{indexof}~\\var{sp}~(\\fun{txinputs}~{txb})) & \\var{sp}~\\in~\\UTxOIn\n \\end{cases}\n \\end{align*}\n \\caption{Indexing script and data objects}\n \\label{fig:functions:script1}\n\\end{figure}\n\n\n\\subsection{Plutus Script Validation}\nFigure~\\ref{fig:defs:functions-valid} shows the abstract functions that are used for script validation.\n\n\\begin{itemize}\n\\item $\\fun{slotToTime}$ translates a slot number to system time if possible.\nIf it is not possible to do this translation, $\\Nothing$ is returned.\nThe reason it may not be possible to translate is that the slot number\nis too far in the future for the system to accurately\npredict the exact time to which it refers.\n\n\\item $\\fun{txInfo}$ summarizes all the necessary transaction and chain state info\n that needs to be passed to the script interpreter. The $\\Language$ argument\n is required because different languages have different expectations of the\n format and contents of the summary $\\Data$ term.\n It has a $\\UTxO$ as its argument to recover the full information of the inputs of the transaction,\n but only the inputs of the transaction are provided to scripts. For details, see~\\ref{sec:txinfo}\n\n\\item\n $\\fun{valContext}$ constructs the \\emph{validation context}. A validation context is\n a $\\Data$ term which encodes both the summary of the transaction and ledger information\n (this is supplied by the $\\fun{txInfo}$ summarization function), and the script purpose.\n\n\\item\n $\\fun{runPLCScript}$ validates Plutus scripts. It takes the following\n arguments:\n \\begin{itemize}\n \\item A cost model, that is used to calculate the $\\ExUnits$ that are needed for script execution;\n \\item A script to execute;\n \\item A list of terms of type $\\Data$ that will be passed to the script; and\n \\item the execution unit budget.\n \\end{itemize}\n It outputs the validation result.\n Note that script execution stops if the full budget has been spent before validation is complete.\n It is left abstract because it is implemented as part of the native script interpreter, not the ledger.\n The interpreter called depends on the language of the script.\n\\end{itemize}\n\n\n\\textbf{Slot to time translation.}\nOne of the inputs to scripts is the transaction validity interval (recall here that\nthey do not actually see the current slot number). The length of a\nslot may change in a future era. In this case, for a script written in a previous\nera, if we were to pass the transaction validity interval expressed as slot numbers,\nwe get that\n\n\\begin{itemize}\n \\item the script logic is expressed in terms of slot numbers\n which likely assume the slot length of the era in which it was created\n \\item the slot numbers in the validity interval of the transaction are used\n assuming slot length of the current era\n\\end{itemize}\n\nTherefore, the slot numbers inside the contract and the slot numbers in the transaction\nmap differently onto points in time. The ledger does not have access to data (or conversion functions)\nthat is needed to convert transaction validity interval slots numbers to slots numbers\nthat correspond to the contracts world view. To address this, we decided to pass\nnon-native contracts (in all languages and all eras) the system time instead of slot numbers.\nThe conversion function is implemented by consensus (see~\\cite{shelley_consensus}),\nwhich has the information to do this correctly for\n\n\\begin{itemize}\n \\item all slots prior to the current slot\n \\item a number of slots after the current slot (this number is determined by\n the consensus's forecast window)\n\\end{itemize}\n\n\\textbf{Validation context construction.}\n As additional non-native scripting languages become supported in the future, scripts of different\n languages may expect different (or differently structured) transaction and ledger data summary.\n The $\\fun{txInfo}$ function will be implemented differently\n for each new language. So, the construction is\n dependent on both the language of the script being validated and the ledger/transaction structure\n of the current era.\n\n In order to ensure that running scripts of all script languages is supported indefinitely across\n future eras, the $\\fun{txInfo}$ function must be total.\n Recall here that while \\emph{running} all scripts must be supported across\n all future ledger changes,\n it is not a requirement that every script must \\emph{validate} within the context of some transaction.\n\n The $\\fun{txInfo}$ output is computed once for the whole transaction. The output of the function\n $\\fun{valContext}$ is computed separately for each script purpose.\n The script purpose is passed to it\n to allow the script to reference itself via its hash, and to be aware of what it is validating.\n\n\\textbf{Know your contract arguments.}\n A Plutus validator script may receive either a list of three terms of type $\\Data$, in case it validates the spending of script outputs\n or two terms (redeemer and context, with no datum), for all other uses.\n Script authors must keep this in mind when writing scripts, since the ledger call to the interpreter is oblivious to what\n arguments are required.\n\n\\begin{figure*}[htb]\n \\emph{Abstract Script Validation Functions}\n %\n \\begin{align*}\n &\\fun{slotToTime} \\in \\Slot \\to \\Time^? \\\\\n &\\text{Translate slot number to system time or fail} \\\\~\\\\\n &\\fun{txInfo} \\in \\Language \\to \\UTxO \\to \\Tx \\to \\Data \\\\\n &\\text{Summarizes transaction data} \\\\~\\\\\n &\\fun{valContext} \\in \\Data \\to \\ScriptPurpose \\to \\Data \\\\\n &\\text{Pairs transaction data with a script purpose} \\\\~\\\\\n &\\fun{runPLCScript} \\in \\CostMod \\to\\ScriptPlutus \\to\n \\seqof{\\Data} \\to \\ExUnits \\to \\IsValidating \\\\\n &\\text{Validate a Plutus script, taking resource limits into account}\n \\end{align*}\n %\n \\emph{Notation}\n %\n \\begin{align*}\n \\llbracket \\var{script_v} \\rrbracket_{\\var{cm},\\var{exunits}}~\\var{d}\n &=& \\fun{runPLCScript} ~{cm}~\\var{script_v}~\\var{d}~\\var{exunits}\n \\end{align*}\n \\caption{Script Validation, cont.}\n \\label{fig:defs:functions-valid}\n\\end{figure*}\n\nFigure \\ref{fig:functions:script2} contains the functions used to\nmatch scripts with their corresponding inputs and pass them to the\nevaluator.\n\n\\begin{itemize}\n \\item $\\fun{getData}$ looks for a datum associated with a given script purpose. Note that\n only an $\\TxIn$-type script purpose can result in finding an associated datum hash.\n In no datum is found, an empty list is returned. A list containing the found datum\n is returned otherwise.\n\n \\item $\\fun{collectNNScriptInputs}$ builds a list of scripts, paired with their\n inputs. Specifically, each tuple in the list contains :\n\n \\begin{itemize}\n \\item the script;\n\n \\item the validation context, built by the $\\fun{valContext}$ function using\n the transaction summary built by $\\fun{txInfo}$, together with the current item being validated;\n\n \\item the hash of the required datum, if any (returned by the $\\fun{getData}$ function,\n wrapped in a list type instead of a $\\Data^?$ type).\n\n \\item a pair, returned by the $\\fun{indexedRdmrs}$ function, of\n \\begin{itemize}\n \\item a list of $\\Data$, containing exactly one or zero elements (this\n is not checked here, however, see~\\ref{fig:functions:script1}). This list\n contains the optional redeemer if there is one.\n \\item an $\\ExUnits$ value, which is mandatory for all scripts\n \\end{itemize}\n\n \\item $\\fun{evalScripts}$ evaluates a whole list of scripts paired with all their\n inputs by calling the native script validator function, $\\fun{runPLCScript}$\n\n \\item $\\fun{runNativeScript}$ replaces the $\\fun{validateScript}$ function,\n to diffentiate running native and non-native scripts. The definition is the\n same as in the ShelleyMA specification.\n\\end{itemize}\n\nNote that no ``checks'' are performed within these functions.\nMissing validators, missing inputs, incorrect hashes, the wrong type of script etc,\nare caught during the application of the UTXOW rule (before these functions are ever applied).\n%\n\n\\end{itemize}\n\n\\begin{figure}[htb]\n \\begin{align*}\n & \\fun{getData} \\in \\Tx \\to \\UTxO \\to \\ScriptPurpose \\to \\seqof{\\Data} \\\\\n & \\fun{getData}~{tx}~{utxo}~{sp}~=~\n \\begin{cases}\n [\\var{d}] & \\var{sp} \\mapsto (\\_, \\_, h_d) \\in \\var{utxo}, \\var{h_d}\\mapsto \\var{d} \\in \\fun{txdats}~(\\fun{txwits}~tx) \\\\\n \\epsilon & \\text{otherwise}\n \\end{cases}\n \\nextdef\n & \\fun{collectNNScriptInputs} \\in \\PParams \\to \\Tx \\to \\UTxO \\to \\seqof{(\\ScriptNonNative \\times \\seqof{\\Data} \\times \\ExUnits \\times \\CostMod)} \\\\\n & \\fun{collectNNScriptInputs} ~\\var{pp}~\\var{tx}~ \\var{utxo} ~=~ \\\\\n & ~~\\fun{toList} \\{ (\\var{script}, (\\fun{valContext}~\\var{txinfo}~\\var{sp}~++~ \\fun{getData}~tx~utxo~sp~++~\\var{d}), \\var{eu}, \\var{cm}) \\mid \\\\\n & ~~~~(\\var{sp}, \\var{scriptHash}) \\in \\fun{scriptsNeeded}~{utxo}~{tx}, \\\\\n & ~~~~\\var{scriptHash}\\mapsto \\var{script}\\in \\fun{txscripts}~(\\fun{txwits}~tx), \\\\\n & ~~~~(\\var{d}, \\var{eu}) := \\fun{indexedRdmrs}~tx~sp, \\\\\n & ~~~~\\fun{language}~{script} \\mapsto \\var{cm} \\in \\fun{costmdls}~{pp} \\} \\\\\n & \\where \\\\\n & ~~~~~~~ \\var{txinfo}~=~\\fun{txInfo}~(\\fun{language}~{script})~\\var{utxo}~\\var{tx} \\\\\n \\nextdef\n & \\fun{evalScripts} \\in \\seqof{(\\ScriptNonNative \\times \\seqof{\\Data} \\times \\ExUnits \\times \\CostMod)} \\to \\IsValidating \\\\\n & \\fun{evalScripts}~\\epsilon = \\True \\\\\n & \\fun{evalScripts}~((\\var{sc}, \\var{d}, \\var{eu}, \\var{cm});\\Gamma) =\n \\llbracket sc \\rrbracket_{cm,\\var{eu}} d \\land \\fun{evalScripts}~\\Gamma\n \\nextdef\n &\\fun{runNativeScript} \\in\\ScriptNative \\to \\Tx \\to \\IsValidating\n \\end{align*}\n \\caption{Scripts and their arguments}\n \\label{fig:functions:script2}\n\\end{figure}\n\n\\subsection{Two-Phase Transaction Validation for Non-Native Scripts}\n\\label{sec:two-phase}\n\nTransactions are validated in two phases:\nthe first phase consists of every aspect of transaction validation apart from executing the non-native scripts; and\nthe second phase involves actually executing those scripts.\nThis ensures that users pay for the computational resources that are needed to validate non-native scripts, even\nif script validation fails. %\nIn order to handle script execution, an additional transition system is used, called UTXOS.\nIt performs the appropriate UTxO state changes, based on the\nvalue of the $\\IsValidating$ tag, which it checks using the $\\fun{evalScripts}$ function.\n\nIn general, there is no way to check \\emph{a-priori} that the budget that has been supplied is sufficient for the transaction.\nThis can only be done by actually running the scripts. From the perspective of the ledger, there is no difference\nbetween a script that exhausts the $\\ExUnits$ budget during validation, and one that fails to validate.\nIf a transaction contains a failing script, the only change to the ledger that is made\nis to collect all inputs that have been marked for fees.\n\nIt is always in the interest of the slot leader to have the new block validate,\nand for it to contain only valid transactions. This motivates the\nslot leader to:\n\n\\begin{enumerate}\n \\item Correctly apply the $\\IsValidating$ tag;\n \\item Include all transactions that validate within the block,\n \\textit{even when there is a 2nd step script validation failure};\n \\item Exclude any transactions that are invalid in some way \\textit{other than 2nd step script validation failure}.\n\\end{enumerate}\n\nOne important reason for adding the validation tag\nto a transaction is that re-applying blocks will not require repeat\nexecution of scripts in the transactions inside a block, which would increase execution costs.\nIn fact, when replaying\nblocks, all the witnessing information can be thrown away.\n\n\\subsection{The UTXOS transition system}\n\\label{sec:utxo-state-trans}\n\nWe have defined a separate transition system, UTXOS, to represent the two distinct\nUTxO state changes: i) when all the scripts in a transaction validating; and\nii) when at least one fails to validate. Its transition types\nare identical to the UTXO transition (Figure\n\\ref{fig:ts-types:utxo-scripts}).\n\n\\begin{figure}[htb]\n \\emph{State transitions}\n \\begin{equation*}\n \\_ \\vdash\n \\var{\\_} \\trans{utxos}{\\_} \\var{\\_}\n \\subseteq \\powerset (\\UTxOEnv \\times \\UTxOState \\times \\Tx \\times \\UTxOState)\n \\end{equation*}\n %\n \\caption{UTxO script state update types}\n \\label{fig:ts-types:utxo-scripts}\n\\end{figure}\n\nThere are two rules, corresponding to the two possible state changes of the\nUTxO state in the UTXOS transition system (Figure~\\ref{fig:rules:utxo-state-upd}).\n%\nIn both cases, the $\\fun{evalScripts}$ function is called upon to verify that the $\\IsValidating$\ntag has been applied correctly. The function $\\fun{collectNNScriptInputs}$ is used to build\nthe inputs $\\var{sLst}$ for the $\\fun{evalScripts}$ function.\n%\nThe first rule\napplies when the validation tag is $\\True$.\nIn this case, the states of the UTxO, fee\n and deposit pots, and updates are updated exactly as in the current Shelley\n ledger spec.\n%\n The second rule\n applies when the validation tag is $\\False$.\n In this case, the UTxO state changes as follows:\n\n \\begin{enumerate}\n \\item All the\n UTxO entries corresponding to the transaction inputs selected for covering\n fees are removed;\n\n \\item The sum total of the value of the marked UTxO entries\n is added to the fee pot.\n \\end{enumerate}\n\n\n\\begin{figure}[htb]\n \\begin{equation}\n \\inference[Scripts-Yes]\n {\n \\var{txb}\\leteq\\txbody{tx} &\n \\var{sLst} := \\fun{collectNNScriptInputs}~\\var{pp}~\\var{tx}~\\var{utxo}\n \\\\\n ~\n \\\\\n \\fun{txvaltag}~\\var{tx} = \\fun{evalScripts}~\\var{sLst} = \\True\n \\\\~\\\\\n {\n \\begin{array}{r}\n \\var{slot} \\\\\n \\var{pp} \\\\\n \\var{genDelegs} \\\\\n \\end{array}\n }\n \\vdash \\var{pup} \\trans{\\hyperref[fig:rules:update]{ppup}}{\\fun{txup}~\\var{tx}} \\var{pup'}\n \\\\~\\\\\n \\var{refunded} \\leteq \\keyRefunds{pp}{txb}\n \\\\\n \\var{depositChange} \\leteq\n (\\deposits{pp}~{poolParams}~{(\\txcerts{txb})}) - \\var{refunded}\n }\n {\n \\begin{array}{l}\n \\var{slot}\\\\\n \\var{pp}\\\\\n \\var{poolParams}\\\\\n \\var{genDelegs}\\\\\n \\end{array}\n \\vdash\n \\left(\n \\begin{array}{r}\n \\var{utxo} \\\\\n \\var{deposits} \\\\\n \\var{fees} \\\\\n \\var{pup} \\\\\n \\end{array}\n \\right)\n \\trans{utxos}{tx}\n \\left(\n \\begin{array}{r}\n \\varUpdate{\\var{(\\txins{txb} \\subtractdom \\var{utxo}) \\cup \\outs{slot}~{txb}}} \\\\\n \\varUpdate{\\var{deposits} + \\var{depositChange}} \\\\\n \\varUpdate{\\var{fees} + \\txfee{txb}} \\\\\n \\varUpdate{\\var{pup'}} \\\\\n \\end{array}\n \\right) \\\\\n }\n \\end{equation}\n \\begin{equation}\n \\inference[Scripts-No]\n {\n \\var{txb}\\leteq\\txbody{tx} &\n \\var{sLst} := \\fun{collectNNScriptInputs}~\\var{pp}~\\var{tx}~\\var{utxo}\n \\\\\n ~\n \\\\\n \\fun{txvaltag}~\\var{tx} = \\fun{evalScripts}~\\var{sLst} = \\False\n }\n {\n \\begin{array}{l}\n \\var{slot}\\\\\n \\var{pp}\\\\\n \\var{poolParams}\\\\\n \\var{genDelegs}\\\\\n \\end{array}\n \\vdash\n \\left(\n \\begin{array}{r}\n \\var{utxo} \\\\\n \\var{deposits} \\\\\n \\var{fees} \\\\\n \\var{pup} \\\\\n \\end{array}\n \\right)\n \\trans{utxos}{tx}\n \\left(\n \\begin{array}{r}\n \\varUpdate{\\var{\\fun{txinputs_{fee}}~{txb} \\subtractdom \\var{utxo}}} \\\\\n \\var{deposits} \\\\\n \\varUpdate{\\var{fees} + \\fun{ubalance}~(\\fun{txinputs_{fee}}~{txb}\\restrictdom \\var{utxo})} \\\\\n \\var{pup} \\\\\n \\end{array}\n \\right)\n }\n \\end{equation}\n \\caption{State update rules}\n \\label{fig:rules:utxo-state-upd}\n\\end{figure}\n\nFigure \\ref{fig:rules:utxo-shelley} shows the $\\type{UTxO-inductive}$\ntransition rule for the UTXO transition type.\nThis rule has the following changes:\n\n\\begin{enumerate}\n \\item The transaction pays fees correctly, as defined above;\n\n \\item The end of the transaction validity interval is translatable into\n system time (ie. within the consensus's forecast window). This is checked\n by $\\fun{slotToTime}$, which returns $\\Nothing$ if the end slot is outside.\n Note that we do not need to check that the start slot can be converted to\n time, because all pasts slots can be converted into time correctly.\n\n \\item $\\fun{adaPerUTxOWord}$ is now a protocol parameter explicitly, the\n $\\fun{utxoEntrySize}$ calculation is defined differently than for ShelleyMA\n (see Section~\\ref{sec:value-size})\n\n \\item $\\fun{maxValSize}$ is now also a protocol parameter (not a constant).\n It represents a size (in bytes) of the total transaction\n size that the size of a $\\Value$ in an output can be. Otherwise, this check is\n the same as in ShelleyMA.\n\n \\item The execution unit budget for a transaction is within the maximum\n permitted number of units for a transaction;\n\n \\item The UTXOS state transition is valid (this is the transition that runs the\n non-native scripts)\n\\end{enumerate}\n\nThe resulting state transition is defined entirely by the application of the\nUTXOS rule.\n\n\\begin{figure}[htb]\n \\begin{equation}\\label{eq:utxo-inductive-shelley}\n \\inference[UTxO-inductive]\n {\n \\var{txb}\\leteq\\txbody{tx} &\n \\fun{ininterval}~\\var{slot}~(\\fun{txvldt}~{tx}) \\\\\n \\hldiff{\\var{(\\wcard, i_f)}\\leteq\\fun{txvldt}~{tx}} & \\hldiff{\\fun{slotToTime}~i_f \\neq \\Nothing} \\\\\n \\txins{txb} \\neq \\emptyset\n & \\hldiff{\\fun{feesOK}~pp~tx~utxo}\n & \\txins{txb} \\subseteq \\dom \\var{utxo}\n \\\\\n \\consumed{pp}{utxo}{txb} = ~\\produced{pp}{poolParams}~{txb}\n \\\\~\\\\\n \\mathsf{adaID}\\notin \\supp {\\fun{mint}~tx} \\\\~\\\\\n \\forall txout \\in \\txouts{txb}, \\\\\n \\fun{getValue}~txout \\geq \\fun{inject}~(\\hldiff{\\fun{utxoEntrySize}~{txout} * \\fun{adaPerUTxOWord}~pp)} \\\\~\n \\\\\n \\forall txout \\in \\txouts{txb},\\\\\n \\hldiff{\\fun{serSize}~(\\fun{getValue}~txout) ~\\leq ~ (\\fun{maxTxSize}~pp) * (\\fun{maxValSize}~pp)} \\\\~\n \\\\\n \\forall (\\wcard\\mapsto (a,~\\wcard)) \\in \\txouts{txb}, \\fun{netId}~a = \\NetworkId\n \\\\\n \\forall (a\\mapsto\\wcard) \\in \\txwdrls{txb}, \\fun{netId}~a = \\NetworkId\n \\\\\n \\fun{txsize}~{tx}\\leq\\fun{maxTxSize}~\\var{pp} \\\\\n \\hldiff{\\fun{totExunits}~{tx} \\leq \\fun{maxTxExUnits}~{pp}}\n \\\\\n ~\n \\\\\n \\hldiff{{\n \\begin{array}{c}\n \\var{slot}\\\\\n \\var{pp}\\\\\n \\var{poolParams}\\\\\n \\var{genDelegs}\\\\\n \\end{array}\n }\n \\vdash\n {\n \\left(\n \\begin{array}{r}\n \\var{utxo} \\\\\n \\var{deposits} \\\\\n \\var{fees} \\\\\n \\var{pup}\\\\\n \\end{array}\n \\right)\n }\n \\trans{utxos}{\\var{tx}}\n {\n \\left(\n \\begin{array}{r}\n \\var{utxo'} \\\\\n \\var{deposits'} \\\\\n \\var{fees'} \\\\\n \\var{pup'}\\\\\n \\end{array}\n \\right)\n }\n }}\n {\n \\begin{array}{l}\n \\var{slot}\\\\\n \\var{pp}\\\\\n \\var{poolParams}\\\\\n \\var{genDelegs}\\\\\n \\end{array}\n \\vdash\n \\left(\n \\begin{array}{r}\n \\var{utxo} \\\\\n \\var{deposits} \\\\\n \\var{fees} \\\\\n \\var{pup}\\\\\n \\end{array}\n \\right)\n \\trans{utxo}{tx}\n \\left(\n \\begin{array}{r}\n \\varUpdate{\\var{utxo'}} \\\\\n \\varUpdate{\\var{deposits'}} \\\\\n \\varUpdate{\\var{fees'}} \\\\\n \\varUpdate{\\var{pup'}}\\\\\n \\end{array}\n \\right)\n }\n \\end{equation}\n \\caption{UTxO inference rules}\n \\label{fig:rules:utxo-shelley}\n\\end{figure}\n\n\\subsection{Witnessing}\n\\label{sec:wits}\n\nBecause of two-phase transaction validation, non-native script validation is not part of phase one of transaction witnessing\n(it is done in phase 2, once the rest of the transaction is validated).\nHowever, native script validation does remain part of transaction witnessing.\nWhen witnessing a transaction in phase one, we therefore need to validate only the native scripts.\n\nWe construct the following helper functions :\n\n\\begin{itemize}\n \\item $\\fun{scriptsNeeded}$ assembles the all the $\\ScriptPurpose$ terms\n for validation of every transaction action that requires script validation,\n paired with the hashes of corresponding the witnessing scripts.\n This function collects hashes of both native and non-native scripts.\n\n \\item $\\fun{checkRedeemers}$ returns $\\True$ if, for the given script purpose,\n whenever the script it points to is non-native, the transaction contains an associated entry\n in the indexed redeemers structure.\n Recall that redeemers are optional to include, so there may be a $\\Nothing$ in place of\n a redeemer. Execution units are mandatory for all scripts, so each script purpose\n should have an associated entry specifying the $\\ExUnits$ value.\n\n \\item $\\fun{languages}$ returns the set of (non-native) script languages\n of all the scripts included in the transaction\n\\end{itemize}\n\n\\begin{figure}[htb]\n \\begin{align*}\n & \\hspace{-1cm}\\fun{scriptsNeeded} \\in \\UTxO \\to \\Tx \\to \\powerset (\\ScriptPurpose \\times \\ScriptHash) \\\\\n & \\hspace{-1cm}\\fun{scriptsNeeded}~\\var{utxo}~\\var{tx} = \\\\\n & ~~\\{ (\\var{i}, \\fun{validatorHash}~a) \\mid i \\mapsto (a, \\wcard, \\wcard) \\in \\var{utxo},\n i\\in\\fun{txinsScript}~{(\\fun{txins~\\var{txb}})}~{utxo}\\} \\\\\n \\cup & ~~\\{ (\\var{a}, \\fun{stakeCred_{r}}~\\var{a}) \\mid a \\in \\dom (\\AddrRWDScr\n \\restrictdom \\fun{txwdrls}~\\var{txb}) \\} \\\\\n \\cup & ~~\\{ (\\var{cert}, \\var{c}) \\mid \\var{cert} \\in (\\DCertDeleg \\cup \\DCertDeRegKey)\\cap\\fun{txcerts}~(\\txbody{tx}), \\\\\n & ~~~~~~\\var{c} \\in \\cwitness{cert} \\cap \\AddrScr\\} \\\\\n \\cup & ~~\\{ (\\var{pid}, \\var{pid}) \\mid \\var{pid} \\in \\supp~(\\fun{mint}~\\var{txb}) \\} \\\\\n & \\where \\\\\n & ~~~~~~~ \\var{txb}~=~\\txbody{tx}\n \\nextdef\n & \\hspace{-1cm}\\fun{checkRedeemers} \\in \\Tx \\to \\UTxO \\to (\\ScriptPurpose \\times \\ScriptHash) \\to \\Bool \\\\\n & \\hspace{-1cm}\\fun{checkRedeemers}~\\var{tx}~\\var{utxo}~(\\var{sp},\\var{h})~=~ \\forall s, (h\\mapsto s) \\in \\fun{txscripts}~(\\fun{txwits}~tx),\\\\\n & (s \\in \\ScriptNonNative~\\Rightarrow \\fun{indexedRdmrs}~tx~sp \\neq \\Nothing) \\\\\n \\nextdef\n & \\hspace{-1cm}\\fun{languages} \\in \\TxWitness \\to \\powerset{\\Language} \\\\\n & \\hspace{-1cm}\\fun{languages}~\\var{txw}~=~\n \\{\\fun{language}~s \\mid s \\in \\range (\\fun{txscripts}~{txw}) \\cap \\ScriptNonNative\\}\n \\end{align*}\n \\caption{UTXOW helper functions}\n \\label{fig:functions-witnesses}\n\\end{figure}\n\nWe have made the following changes and additions to the UTXOW preconditions:\n\n\\begin{itemize}\n\n\\item All the native scripts in the transaction validate;\n\n\\item The transaction contains exactly those scripts that are required for witnessing and no\nadditional ones;\n\n \\item The datums included in the witnesses are exactly those that are required for validating;\n\n \\item For every item that needs to be validated by a non-native script, the transaction contains\n an entry in the indexed redeemer structure (ie. the execution units for it are specified);\n\n \\item The signatures of the keys whose hashes are specified in the\n $\\fun{reqSignerHashes}$ field in a transaction\n have all indeed signed it;\n\n \\item\n The hash of the subset of protocol parameter values that have been included in the transaction body is the same as\n the hash of the same subset of protocol parameters that are currently contained in the ledger;\n\n \\item A check that ensures that the list of scripts paired with their input data (context, datum and redeemer)\n has the same number of elements as the set of scripts needed for each script purpose. Note that the same\n script may appear more than once in both the list and the set, as it can be used for different purposes, and therefore\n run separately for each, with different input data. Note also that by construction, the output of\n the $\\fun{collectNNScriptInputs}$ function should have exactly the same collection of scripts as\n the $\\fun{scriptsNeeded}$ function output - this check is in place to catch discrepancies in this construction.\n \\begin{note}\n Note that $\\fun{collectNNScriptInputs}$ is called twice, here and in UTXO. We need to avoid this.\n ALSO we need to audit the $\\fun{wppHash}$ function\n \\end{note}\n\\end{itemize}\n\nIf these conditions are all satisfied, then the resulting UTxO state change is fully determined\nby the UTXO transition (the application of which is also part of the conditions).\n\n\\begin{figure}[htb]\n \\emph{State transitions}\n \\begin{equation*}\n \\_ \\vdash\n \\var{\\_} \\trans{utxow}{\\_} \\var{\\_}\n \\subseteq \\powerset (\\UTxOEnv \\times \\UTxOState \\times \\Tx \\times \\UTxOState)\n \\end{equation*}\n %\n \\caption{UTxO with witnesses state update types}\n \\label{fig:ts-types:utxo-witness}\n\\end{figure}\n\n\\begin{figure}\n \\begin{equation}\n \\label{eq:utxo-witness-inductive-alonzo}\n \\inference[UTxO-witG]\n {\n \\var{txb}\\leteq\\txbody{tx} &\n \\var{txw}\\leteq\\fun{txwits}~{tx} \\\\\n (utxo, \\wcard, \\wcard, \\wcard) \\leteq \\var{utxoSt} \\\\\n \\var{witsKeyHashes} \\leteq \\{\\fun{hashKey}~\\var{vk} \\vert \\var{vk} \\in\n \\dom (\\txwitsVKey{txw}) \\}\\\\~\\\\\n \\hldiff{\\forall \\var{s} \\in \\range (\\fun{txscripts}~{txw}) \\cap \\ScriptNative,\n \\fun{runNativeScript}~\\var{s}~\\var{tx}}\\\\~\\\\\n \\hldiff{\\{ s \\mid (\\_, s) \\in \\fun{scriptsNeeded}~\\var{utxo}~\\var{tx}\\} = \\dom (\\fun{txscripts}~{txw})} \\\\\n \\hldiff{\\{ h \\mid (\\_ \\mapsto (a, \\_, h)) \\in \\txins{txb} \\restrictdom \\var{utxo}, \\fun{isNonNativeScriptAddress}~{tx}~{a}\\} =} \\\\\n \\hldiff{\\dom (\\fun{txdats}~{txw})} \\\\\n \\hldiff{\\forall sph \\in \\fun{scriptsNeeded}~\\var{utxo}~\\var{tx},~ \\fun{checkRedeemers}~tx~utxo~sph}\n \\\\~\\\\\n \\forall \\var{vk} \\mapsto \\sigma \\in \\txwitsVKey{txw},\n \\mathcal{V}_{\\var{vk}}{\\serialised{txbodyHash}}_{\\sigma} \\\\\n \\hldiff{\\fun{witsVKeyNeeded}~{utxo}~{tx}~{genDelegs} \\cup \\fun{reqSignerHashes}~txb \\subseteq \\var{witsKeyHashes}}\n \\\\~\\\\\n genSig \\leteq\n \\left\\{\n \\fun{hashKey}~gkey \\vert gkey \\in\\dom{genDelegs}\n \\right\\}\n \\cap\n \\var{witsKeyHashes}\n \\\\\n \\left\\{\n c\\in\\txcerts{txb}~\\cap\\DCertMir\n \\right\\} \\neq\\emptyset \\implies \\vert genSig\\vert \\geq \\Quorum \\wedge\n \\fun{d}~\\var{pp} > 0\n \\\\~\\\\\n \\var{adh}\\leteq\\fun{txADhash}~\\var{txb}\n &\n \\var{ad}\\leteq\\fun{txAD}~\\var{tx}\n \\\\\n (\\var{adh}=\\Nothing \\land \\var{ad}=\\Nothing)\n \\lor\n (\\var{adh}=\\fun{hashAD}~\\var{ad})\n \\\\~\\\\\n \\hldiff{\\fun{wppHash}~{txb}~=~\\fun{hashWitnessPPData}~\\var{pp}~(\\fun{languages}~{txw})~(\\fun{txrdmrs}~{txw})}\n \\\\~\\\\\n \\hldiff{\\|~\\fun{scriptsNeeded}~\\var{utxo}~\\var{tx}~\\|~=~\\|~\\fun{collectNNScriptInputs} ~\\var{pp}~\\var{tx}~ \\var{utxo}~\\|}\n \\\\~\\\\\n {\n \\begin{array}{r}\n \\var{slot}\\\\\n \\var{pp}\\\\\n \\var{poolParams}\\\\\n \\var{genDelegs}\\\\\n \\end{array}\n }\n \\vdash \\var{utxoSt} \\trans{\\hyperref[fig:rules:utxo-shelley]{utxo}}{tx}\n \\var{utxoSt'}\\\\\n }\n {\n \\begin{array}{r}\n \\var{slot}\\\\\n \\var{pp}\\\\\n \\var{poolParams}\\\\\n \\var{genDelegs}\\\\\n \\end{array}\n \\vdash \\var{utxoSt} \\trans{utxow}{tx} \\varUpdate{\\var{utxoSt'}}\n }\n \\end{equation}\n \\caption{UTxO with witnesses inference rules for Tx}\n \\label{fig:rules:utxow-alonzo}\n\\end{figure}\n", "meta": {"hexsha": "2f404488f58841f4b3ea6bec4f419c26f4d2b48e", "size": 33759, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "alonzo/formal-spec/utxo.tex", "max_stars_repo_name": "goolord/cardano-ledger-specs", "max_stars_repo_head_hexsha": "03a01f9ceece4ebe49fcd4f42fe94e5024dfcadd", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "alonzo/formal-spec/utxo.tex", "max_issues_repo_name": "goolord/cardano-ledger-specs", "max_issues_repo_head_hexsha": "03a01f9ceece4ebe49fcd4f42fe94e5024dfcadd", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "alonzo/formal-spec/utxo.tex", "max_forks_repo_name": "goolord/cardano-ledger-specs", "max_forks_repo_head_hexsha": "03a01f9ceece4ebe49fcd4f42fe94e5024dfcadd", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.5239852399, "max_line_length": 151, "alphanum_fraction": 0.6660742321, "num_tokens": 10541, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7310585786300049, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.3484076423105946}} {"text": "\\documentclass[11pt]{article}\n%============Macros==================%\n% you don't need to change anything. start editing from main body.\n\\usepackage{amsmath,amsfonts,amssymb,amsthm, cancel, float, enumitem}\n\\usepackage{qcircuit}\n\\usepackage[margin=1in]{geometry}\n%--------------Cosmetic----------------%\n\\usepackage{mathtools}\n\\usepackage{hyperref}\n\\usepackage{fullpage}\n\\usepackage{microtype}\n\\usepackage{xspace}\n\\usepackage[svgnames]{xcolor}\n\\usepackage[sc]{mathpazo}\n\\usepackage{enumitem}\n\\setlist[enumerate]{itemsep=1pt,topsep=2pt}\n\\setlist[itemize]{itemsep=1pt,topsep=2pt}\n%--------------Header------------------%\n\\def\\course{CS 410/510 Introduction to Quantum Computing}\n\\def\\term{Portland State U, Spring 2017}\n\\def\\prof{Lecturer: Fang Song}\n\\newcommand{\\handout}[5]{\n \\renewcommand{\\thepage}{\\arabic{page}}\n \\begin{center}\n \\framebox{\n \\vbox{\n \\hbox to 5.78in { \\hfill \\large{\\course} \\hfill }\n \\vspace{2mm}\n \\hbox to 6in { {\\Large \\hfill #5 \\hfill} }\n \\vspace{2mm}\n \\hbox to 6in { \\term \\hfill \\emph{#2}}\n \\hbox to 6in { {#3 \\hfill \\emph{#4}}}\n }\n }\n \\end{center}\n \\vspace*{4mm}\n}\n\\newcommand{\\lecture}[4]{\\handout{#1}{#2}{#3}{#4}{{Lecture #1}}}\n\n\\def\\complex{\\mathbb{C}}\n\\def\\real{\\mathbb{R}}\n\\def\\natural{\\mathbb{N}}\n\\def\\integer{\\mathbb{Z}}\n\n\\newcommand{\\norm}[1]{\\left\\lVert#1\\right\\rVert}\n\\newcommand{\\snorm}[1]{\\lVert#1\\rVert}\n\\newcommand{\\abs}[1]{\\left\\lvert #1 \\right\\rvert}\n\\newcommand{\\ceil}[1]{\\left\\lceil #1 \\right\\rceil}\n\\newcommand{\\floor}[1]{\\left\\lfloor #1 \\right\\rfloor}\n\\newcommand{\\set}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\vecb}[1]{\\boldsymbol{\\vec{#1}}}\n\\newcommand{\\conj}[1]{\\overline{#1}}\n\\newcommand{\\op}[2]{#1#2#1^\\dag}\n\n\\newcommand{\\tr}{\\operatorname{Tr}}\n\\newcommand{\\se}{\\operatorname{H}}\n\\newcommand{\\vne}{\\operatorname{S}}\n\\newcommand{\\e}{\\operatorname{E}}\n\\newcommand{\\rank}{\\operatorname{rank}}\n\\renewcommand{\\det}{\\operatorname{Det}}\n\n\\newcommand{\\bra}[1]{\\langle #1 \\rvert}\n\\newcommand{\\ket}[1]{\\lvert #1 \\rangle}\n\\newcommand{\\bret}[2]{\\langle{#1}|{#2}\\rangle}\n\\newcommand{\\kret}[2]{\\ket{#1}\\bra{#2}}\n\\newcommand{\\kera}[1]{\\kret{#1}{#1}}\n\\newcommand{\\Example}[1]{{\\bf Example #1}:}\n\n\\def\\X{\\mathcal{X}}\n\\def\\Y{\\mathcal{Y}}\n\\def\\Z{\\mathcal{Z}}\n\\def\\W{\\mathcal{W}}\n\\def\\A{\\mathcal{A}}\n\\def\\B{\\mathcal{B}}\n\\def\\V{\\mathcal{V}}\n\\def\\U{\\mathcal{U}}\n\\def\\C{\\mathcal{C}}\n\\def\\D{\\mathcal{D}}\n\\def\\E{\\mathcal{E}}\n\\def\\F{\\mathcal{F}}\n\\def\\M{\\mathcal{M}}\n\\def\\R{\\mathcal{R}}\n\\def\\P{\\mathcal{P}}\n\\def\\Q{\\mathcal{Q}}\n\\def\\S{\\mathcal{S}}\n\\def\\T{\\mathcal{T}}\n\\def\\K{\\mathcal{K}}\n\\def\\L{\\mathcal{L}}\n\n\\def\\fill{ \\hfill}\n\n%=============Main Doc=================%\n\\begin{document}\n%-----Specs: change accordingly--------%\n\\def\\lecdate{May 16, 2017} % put lecture date here\n\\def\\scribe{Scribe: Enis K. Inan} % put your name here\n\\def\\lecnum{13} % change the lecture number\n\n\\lecture{\\lecnum}{\\lecdate}{\\prof}{\\scribe}%\n\n\\begin{center}\n{\\textsc{Version: \\today}} \n\\end{center}\n\n\n\\noindent Last lecture we introduced the density operator, which made it easier for us to describe general quantum states. We will briefly review it before proceeding any further. In the old notation, we only had pure states of the form $\\ket{\\psi} = \\alpha \\ket{0} + \\beta \\ket{1}$ where $\\alpha, \\beta \\in \\complex$ and $\\abs{\\alpha}^2 + \\abs{\\beta}^2 = 1$. With our new framework, a pure state is expressed as $\\kera{\\psi}$ where for our example,\n\\begin{equation*}\n\\kera{\\psi} = \\left(\\begin{array}{cc}\n\\alpha\\conj{\\alpha} & \\alpha\\conj{\\beta} \\\\\n\\conj{\\alpha}\\beta & \\beta\\conj{\\beta}\n\\end{array}\n\\right) = \\left(\\begin{array}{cc}\n\\abs{\\alpha}^2 & \\alpha\\conj{\\beta} \\\\\n\\conj{\\alpha}\\beta & \\abs{\\beta}^2\n\\end{array}\n\\right)\n\\end{equation*}\nThe expressive power of this notation is apparent when we have general quantum states. Specifically, imagine we have a register $X$ that could be in one of the quantum states $\\ket{\\psi_1}, \\dots, \\ket{\\psi_k}$ with a probability distribution $p_1, \\dots, p_k$ where $p_i$ is the probability that $X$ is in state $\\ket{\\psi_i}$. With our new notation, we expressed our ``knowledge'' of $X$ as a density matrix $\\rho$ where\n\\begin{equation*}\n\\rho = \\sum_{i} p_i \\kera{\\psi_i}\n\\end{equation*}\nIntuitively, $\\rho$ corresponds to a weighted average of each state. There were also several important properties associated with density matrices:\n\\begin{enumerate}\n\t\\item $\\tr{(\\rho)} = 1$\n\t\n\t\\item $\\rho$ is positive semidefinite.\n\\end{enumerate}\nProperty 2 will be important later in this lecture, so keep it in the back of your head for now.\n\n\\fill\n\n\\noindent In our old notation, the physically allowable operations on $\\ket{\\psi}$ were unitary operations $U$ -- the state $\\ket{\\psi'} = U\\ket{\\psi}$ is the resulting state after applying the unitary $U$ on $\\ket{\\psi}$. In our new notation, $\\rho' = U\\rho U^\\dag$ corresponds to applying the unitary $U$ on the general quantum state described by $\\rho$ -- the result is $\\rho'$.\n\n%==============================%\n\\section{General (Physically Admissible) Quantum Operations}\n%==============================%\nThe density matrix formalism does not limit us to just unitary operations. Specifically, any \\emph{physically admissible operation} $\\Phi$ is a series of matrices $A_1, A_2, \\dots, A_k$ where $A_{i_{jk}} \\in \\complex$ and\n\\begin{equation} \\label{eq:phya:completeness}\n\\sum_{i} A^\\dag_iA_i = I \n\\end{equation}\n\\begin{equation} \\label{eq:phya:app}\n\\Phi(\\rho) = \\sum_{i} \\op{A_i}{\\rho}\n\\end{equation}\nNote that each $A_i$ in $\\Phi$ does \\emph{not} have to be a square matrix, and that unitary matrices $U$ are a special case of $\\Phi$ with $k = 1$.\n\n\\noindent Applying $\\Phi$ to $\\rho$ yields another density matrix, just like how applying unitary $U$ to $\\ket{\\psi}$ resulted in another valid quantum state. In fact, $\\Phi$ is also known as a completely positive trace preserving (CPTP) operator, which is just jargon for an operation that preserves a matrix's trace and semi-positive definitiveness. Another way to think of $\\Phi$ is as follows. Let $\\X$ and $\\Y$ denote spaces describing $m$ and $n$ qubits, respectively, and let $\\L(\\X, \\Y)$ be the set of all linear mappings from $\\X$ to $\\Y$. Let $\\D(\\X)$ be the set of all density matrices in $\\X$; define $\\D(\\Y)$ in a similar manner. Then $\\Phi : \\D(\\X) \\rightarrow \\D(\\Y)$ or, informally, $\\Phi$ is the set of all density matrix mappings from $\\X$ to $\\Y$.\n\n\\fill \n\n\\noindent \\Example{1} Here we investigate decoherence as a physically admissible operation, which is when a quantum system interacts with some environmental noise and loses its quantumness (i.e. gets measured). Note that the operation used in this example is equivalent to measurement in the standard basis $\\set{\\ket{0}, \\ket{1}}$. Here we have $A_0 = \\kera{0}$ and $A_1 = \\kera{1}$ so that\n\\begin{equation*}\n\\Phi(\\rho) = \\op{A_0}{\\rho} + \\op{A_1}{\\rho}\n\\end{equation*}\nFirs we check if $A_0$ and $A_1$ are valid, i.e. that they satisfy Eqn. \\ref{eq:phya:completeness}:\n\\begin{align*}\n\\sum_{i} A^\\dag_iA_i\n&= A^\\dag_0A_0 + A^\\dag_1A_1 \\\\\n&= \\ket{0}\\bret{0}{0}\\bra{0} + \\ket{1}\\bret{1}{1}\\bra{1} \\\\\n&= \\kera{0} + \\kera{1} \\\\\n&= I\n\\end{align*}\nwhich they do. Now assume that $\\rho=\\kera{\\psi}$ where $\\ket{\\psi} = \\alpha\\ket{0} + \\beta\\ket{1}$, i.e. that $\\rho$ corresponds to a pure state. We have,\n\\begin{align*}\n\\Phi(\\kera{\\psi}) \n&= \\op{A_0}{\\kera{\\psi}} + \\op{A_1}{\\kera{\\psi}} \\\\\n&= \\kera{0}\\kera{\\psi}\\kera{0} + \\kera{1}\\kera{\\psi}\\kera{1} \\\\\n&= \\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & 0\n\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\\abs{\\alpha}^2 & \\alpha\\conj{\\beta} \\\\\n\\conj{\\alpha}\\beta & \\abs{\\beta}^2\n\\end{array}\n\\right)\n\\left(\\begin{array}{cc}\n1 & 0 \\\\\n0 & 0\n\\end{array}\\right) + \\left(\\begin{array}{cc}\n0 & 0 \\\\\n0 & 1\n\\end{array}\\right)\n\\left(\\begin{array}{cc}\n\\abs{\\alpha}^2 & \\alpha\\conj{\\beta} \\\\\n\\conj{\\alpha}\\beta & \\abs{\\beta}^2\n\\end{array}\n\\right)\n\\left(\\begin{array}{cc}\n0 & 0 \\\\\n0 & 1\n\\end{array}\\right) \\\\\n&= \\left(\\begin{array}{cc}\n\\abs{\\alpha}^2 & 0 \\\\\n0 & \\abs{\\beta}^2\n\\end{array}\n\\right) \\\\\n&= \\abs{\\alpha}^2\\kera{0} + \\abs{\\beta}^2\\kera{1}\n\\end{align*}\nwhich is exactly what we get by doing a measurement on the standard basis! To see why, the last line shows that we can think of $\\Phi(\\kera{\\psi})$ as representing the general quantum state where we have an $\\abs{\\alpha}^2$ chance of being in $\\ket{0}$ and a $\\abs{\\beta}^2$ chance of being in $\\ket{1}$. This is the definition of measurement in the standard basis.\n\n\\fill\n\n\\noindent Another important operation is the partial trace. Imagine that we have an $m + n$ qubit-state, e.g. $\\ket{\\psi} = \\sum_{x \\in \\set{0, 1}^m}\\sum_{y \\in \\set{0, 1}^n} \\alpha_{xy}\\ket{x}\\ket{y}$. We want to examine the resulting state after we discard either $x$ or $y$. For our old notation, this is easy -- for $\\ket{\\psi}$, we can just remove $\\ket{x}$ or $\\ket{y}$. But in our new notation of density matrices, it is not as obvious. Let $\\X$ and $\\Y$ be the spaces describing $\\ket{x}$ and $\\ket{y}$, respectively, and assume without any loss of generality that we want to remove $\\ket{y}$. Then what we want is an operation $\\Phi: \\D(\\X \\otimes \\Y) \\rightarrow \\D(\\X)$ -- this is the partial trace, denoted as $\\tr_y$ for our specific case to explicitly indicate that we want to discard $\\ket{y}$.\n\n\\fill\n \n\\noindent Let's examine what $\\tr_y$ would look like when $\\X$ and $\\Y$ both describe 1-qubit each -- i.e., when we have a two-qubit state and we want to discard the second qubit. We have $\\Phi$ be the matrices:\n\\begin{equation*}\n\\begin{split}\nA_0 &= I_x \\otimes \\bra{0}_y = \\left(\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & 0 & 1 & 0\n\\end{array}\\right) \\\\\nA_1 &= I_x \\otimes \\bra{1}_y = \\left(\\begin{array}{cccc}\n0 & 1 & 0 & 0 \\\\\n0 & 0 & 0 & 1\n\\end{array}\\right)\n\\end{split}\n\\end{equation*}\nNote that $A_0$ and $A_1$ are $2 \\times 4$ matrices -- $I_x$ is the $2 \\times 2$ identity matrix. We first check that $A_0$ and $A_1$ satisfy Eqn. \\ref{eq:phya:completeness}:\n\\begin{align*}\n\\sum_{i} A^\\dag_iA_i\n&= A^\\dag_0A_0 + A^\\dag_1A_1 \\\\\n&= (I_x \\otimes \\ket{0}_y)(I_x \\otimes \\bra{0}_y)\n+ (I_x \\otimes \\ket{1}_y)(I_x \\otimes \\bra{1}_y) \\\\\n&= (I_x \\otimes \\kera{0}_y) + (I_x \\otimes \\kera{1}_y) \\\\\n&= I_x \\otimes (\\kera{0}_y + \\kera{1}_y) \\\\\n&= I_x \\otimes I_y = I_{xy}\n\\end{align*}\nNow let's do an example calculation using $\\Phi$.\n\n\\fill\n\n\\noindent \\Example{2} Let $\\rho = \\kera{0} \\otimes \\kera{1}$. Intuitively, we should get $\\kera{0}$ back after applying $\\Phi$. Let's see if this is indeed the case.\n\\begin{align*}\n\\Phi(\\rho)\n&= \\op{A_0}{\\rho} + \\op{A_1}{\\rho} \\\\\n&= (I_x \\otimes \\bra{0}_y)(\\kera{0} \\otimes \\kera{1})(I_x \\otimes \\ket{0}_y)\n+ (I_x \\otimes \\bra{1}_y)(\\kera{0} \\otimes \\kera{1})(I_x \\otimes \\ket{1}_y) \\\\\n&= \\cancelto{0}{(\\kera{0} \\otimes \\bret{0}{1}\\bra{1})}(I_x \\otimes \\ket{0}_y)\n+ (\\kera{0} \\otimes \\bret{1}{1}\\bra{1})(I_x \\otimes \\ket{1}_y) \\\\\n&= (\\kera{0} \\otimes \\bra{1})(I_x \\otimes \\ket{1}_y) \\\\\n&= \\kera{0}\n\\end{align*}\nwhich is exactly what we expected. In the general case, e.g. when we have $\\rho' = \\rho \\otimes \\ket{0}$, then $\\Phi(\\rho') = \\rho$ which is consistent with the physical meaning of $\\tr_y$ -- take away $y$ and only look at $x$. We can think of it as ``tracing out $y$'', which is where the name ``partial trace'' comes from.\n\n\\fill\n\n\\noindent Now what if the qubits $x$ and $y$ are entangled? What do we get after we apply $\\tr_y$? We will use $\\ket{\\phi^+} = \\frac{1}{\\sqrt{2}}(\\ket{00} + \\ket{11})_{xy}$ as our example.\n\n\\fill\n\n\\noindent \\Example{3} $\\tr_y(\\kera{\\phi^+}) = \\frac{1}{2}\\kera{0} + \\frac{1}{2}\\kera{1} = \\frac{1}{2}I$ which can be verified by brute-force computation using $\\Phi$, like we did in Example 2. So we see that we get a \\emph{maximally mixed state}, i.e. a density matrix where the pure states form an orthonormal basis and each state has a probability $1/n$ of occurring, where $n$ is the dimension of the matrix. Here, $n = 2$. In fact, it can be shown that $\\tr_y$ for \\emph{any} Bell state is $\\frac{1}{2}I$. Here's why this makes sense. In a Bell state, when we measure the second register, we \\emph{collapse} the resulting state to either just $\\ket{0}$ with probability $1/2$, or just $\\ket{1}$ with probability $1/2$ -- exactly our maximally mixed state. For example in $\\ket{\\phi^+}$, if we measure a $1$ in the second register, then the state collapses to $\\ket{1}_x$; otherwise, it collapses to $\\ket{0}_x$.\n\n\\fill\n\n\\noindent The next operation we will consider is an extension of Example 1. Instead of constraining the measurement to a specific basis, e.g. the standard basis, let's generalize it. Imagine that $\\Gamma$ represents the set of all possible outcomes resulting from our measurement and, for an $a \\in \\Gamma$, $M_a$ is the matrix ``capturing'' an outcome $a$. For example in Example 1, $\\Gamma = \\set{0, 1}$, and $M_0 = \\kera{0}$, $M_1 = \\kera{1}$. Then if\n\\begin{equation*}\n\\sum_{a \\in \\Gamma} M^\\dag_aM_a = I\n\\end{equation*}\nthen the set of matrices $M = \\set{M_a \\: | \\: a \\in \\Gamma}$ form a physically admissible operation $\\Phi$, i.e. $\\Phi$ is a valid measurement. Let's take an outcome $a$ and its measurement matrix $M_a$ where $M_a$ is a projecter ($M^2_a = M_a$). Then $\\tr(M_a\\rho M^\\dag_a)$ is the probability that we will observe outcome $a$ when we measure the quantum system described by $\\rho$. In Example 1, $\\tr(M_0\\rho M^\\dag_0) = \\abs{\\alpha}^2$, which is exactly the probability of measuring a $0$. Now assume that we did measure $a$. Then our state would collapse to\n\\begin{equation*}\n\\frac{M_a\\rho M^\\dag_a}{\\tr(M_a\\rho M^\\dag_a)}\n\\end{equation*}\nwhich makes sense. If $M_a$ is the measurement matrix ``capturing'' an outcome $a$ then when we see $a$, we should only have $M_a\\rho M^\\dag_a$ left in our summation for $\\Phi(\\rho)$. However $M_a\\rho M^\\dag_a$ may not be a valid density matrix, so we need to normalize its trace back down to $1$ -- this is what the $\\tr(M_a\\rho M^\\dag_a)$ in the denominator does. This corresponds to what we do in the pure state case, which is to keep only parts of the state that contain the measured outcome in its qubits ($M_a\\rho M^\\dag_a$ here), then normalize that part (divide by $\\tr(M_a\\rho M^\\dag_a)$).\n\n\\fill\n\n\\noindent The above is known as a Von-Neumann measurement. Another way to think about measurement is through positive operator valued measurements (POVMs) which are the most general class of quantum measurements. In a POVM, we do not care about the resulting state after the measurement; only the probability of a specific outcome $a$. Above, we determined this to be $\\tr(M_a\\rho M^\\dag_a)$. Using a nice property of the trace, $\\tr(AB) = \\tr(BA)$, we see that $\\tr(M_a\\rho M^\\dag_a) = \\tr(M^\\dag_aM_a \\rho ) = \\tr(E_a \\rho)$ where $E_a = M^\\dag_aM_a$. Given that $\\tr(E_a \\rho)$ is the probability that we will see outcome $a$, we can think of $E_a$ as a ``probability'' matrix for outcome $a$ that we multiply with $\\rho$ to obtain the probability that we measure $a$. Then note that the completeness condition, i.e. Eqn. \\ref{eq:phya:completeness}, can be restated as $\\sum_{a} E_a = 1$ -- intuitively, this is like $\\sum_{i} p_i = 1$ for a probability distribution. So POVM is a generalization of measurement, but in a different flavor.\n\n\\fill\n\n\\noindent It would appear on the surface that a physically admissible operator $\\Phi$ is more powerful than a unitary $U$ because $U$ is just $\\Phi$ with $k = 1$, i.e. $\\Phi$ has more matrices that we're applying on $\\rho$. Surprisingly according to Stinespring's Dilation Theorem, this is not the case - unitaries and physically admissible operators are \\emph{equally} powerful. Proving this result is beyond the scope of this class, but the basic idea is that we can simulate any circuit implementing $\\Phi$ with a corresponding unitary circuit using some ancilla bits as input to the latter. At a high-level, the diagram looks like the following: \n\\includegraphics[scale = 0.7]{phisim}\n%NOTE: Don't know how to draw the above as a circuit, spent too much time trying to figure it out.\n\n\\noindent \\Example{4} As an example, let's simulate $\\Phi$ as described in Example 1. The following circuit will do the trick:\n\n\\centerline{\n\\Qcircuit @C=1em @R=0.75em {\n\t\\lstick{\\ket{0}} & \\targ & \\qw & \\qw \\\\\n\t\\lstick{\\alpha\\ket{0}+\\beta\\ket{1}} & \\ctrl{-1} & \\qw & \\rstick{\\rho' = \\Phi(\\rho)} \n}\n}\n\n\\fill\n\n\\noindent so we see that we've introduced an extra ancillary bit and applied the CNOT gate to do the measurement, discarding the ancilla at the end.\n\n%==============================%\n\\section{Quantum Information Theory}\n%==============================%\nSo far we talked about the no-cloning theorem (1), distinguishing between two quantum states (2), entanglement (3), and the density matrix formalistm (4). An example of (2) was given in HW. 1 Problem 4. Here Alice and Bob are physically separated, each given one of the qubits of some 2-qubit state; the 2-qubit state is either State I or State II. The problem asked us to devise a strategy they could use to distinguish whether they're sharing State I or State II given that they are allowed to make only local measurements and communicate across a classical channel. An example of (3) was quantum teleportation where we showed that through entanglement, Alice can transmit a 1-qubit state to Bob.\n\n\\fill\n\n\\noindent In both classical and quantum information theory, we have three important issues that we would like to consider. These are:\n\\begin{enumerate}[start=0]\n\\item What is the information comprised of? What is its source? How do we measure the \\emph{amount} of information that's being transmitted?\n\t\\begin{itemize}\n\t\t\\item In the classical world, the source is a random variable $X$. We measure the amount of information using the Shannon entropy, $\\se(X)$.\n\t\t\n\t\t\\item In the quantum world, the source is a mixed state $\\rho$. We measure the amount of information using the Von Neumann entropy, $\\vne(\\rho)$.\n\t\\end{itemize}\n\n\\item How do we transmit information over a noiseless channel? To put another way, if we store the information on a physical device, then what is the optimal compression that's possible for storing this information source?\n\t\\begin{itemize}\n\t\t\\item Given an $m$-bit source, if we want to represent it using only $\\alpha\\bullet m$ bits where $\\alpha < 1$, then Shannon's noiseless source coding theorem says that we can only do so iff $\\alpha \\geq H(X)$\n\t\t\n\t\t\\item Given a density matrix $\\rho$ on $m$-qubits, if we want to represent it using only $\\alpha\\bullet m$ qubits where $\\alpha < 1$, then Schumacher's noiseless quantum source coding theorem says that we can only do so iff $\\alpha \\geq S(\\rho)$.\n\t\\end{itemize}\n\n\\item How do we transmit information over a noisy channel?\n\t\\begin{itemize}\n\t\t\\item We can do so using error correction codes, ECC.\n\t\t\n\t\t\\item We can do so using quantum error correction codes, QECC.\n\t\\end{itemize}\n\\end{enumerate}\n\n\\fill\n\n\\noindent Imagine that we have a classical source churning out $m$-bit binary strings that are being transmitted on a quantum channel. The information transfer can be summarized by the following steps for a given $n$-bit binary string $x$:\n\\begin{enumerate}\n\\item The classical source churns out $x$\n\n\\item Some kind of process encodes $x$ as a $\\alpha\\bullet n$ qubit quantum source $\\rho_x$, where $\\alpha \\leq 1$.\n\n\\item $\\rho_x$ is transmitted across the quantum channel.\n\n\\item $\\rho_x$ is decoded at the end to get back $x$.\n\\end{enumerate}\nThe question here is whether we can use a shorter quantum message to transmit our classical message, i.e. if we can find an $\\alpha < 1$. Unfortunately, we're stuck with $\\alpha = 1$ due to Holevo's theorem, which states that it is impossible to communicate more than $n$ bits of classical information when transmitting $n$ qubits.\n\n%==============================%\n\\section{Entropy}\n%==============================%\n\n\\subsection{Shannon Entropy}\nWe think of information as some kind of a message but, as counter-intuitive as it might be, a message may not necessarily contain any ``information''. For example, if all participating parties know the exact contents of the message that's being transmitted, then anybody who opens to read the message does not gain any new ``information'' -- they already know what to expect! So really, it is better to think of ``information'' as the \\emph{amount of uncertainty} that's present in a message. The more uncertainty we have in our message, the more ``information'' it contains. For example, an e-mailed job offer would tell you a lot of information because, in general, there are a lot of factors that determine whether you will get the job or not and hence, a lot of uncertainty on whether you will get said job.\n\n\\fill\n\n\\noindent In physics, entropy is a quantity used to capture the amount of uncertainty in a given system. Similarly, the \\emph{Shannon Entropy} quantifies the amount of uncertainty in a given piece of information -- that is why we use the term entropy. Mathematically it is defined as follows. Say we have a source that outputs a random variable $X \\in \\set{0, 1}^m$ with probability $p_x$. Then we define the Shannon Entropy $\\se(x)$ as:\n\\begin{equation} \\label{eq:def-se}\n\\se(X) = -\\sum_{x} p_x \\lg{p_x}\n\\end{equation}\nwhere we define $0\\lg{0} = 1$.\n\n\\fill\n\n\\noindent \\Example{5} Let $X \\in \\set{0, 1}$ where $p_0 = 0$ and $p_1 = 1$. Note that there is no uncertainty involved in $X$ -- we know that our source will always transmit a $1$. Thus we expect that $\\se(X) = 0$. Doing the calculation:\n\\begin{align*}\n\\se(X) \n&= -\\sum_{x} p_x \\lg{p_x} \\\\\n&= -1\\lg{1} \\\\\n&= 0 \n\\end{align*}\n\n\\fill\n\n\\noindent \\Example{6} Let $X \\in \\set{0, 1}$ where $p_0 = p_1 = 1/2$, i.e. $X$ corresponds to a fair coin. Then we see that:\n\\begin{align*}\n\\se(X) \n&= -\\sum_{x} p_x \\lg{p_x} \\\\\n&= -\\left(\n\\frac{1}{2}\\lg{\\frac{1}{2}} + \\frac{1}{2}\\lg{\\frac{1}{2}}\n\\right) \\\\\n&= -(-\\frac{1}{2} + -\\frac{1}{2}) \\\\\n&= 1\n\\end{align*}\nwhich is exactly 1-bit of information. This is the mathematical definition of a bit.\n\n\\fill\n\n\\noindent Generalizing Example 6, if we have $X \\in \\set{0, 1}^m$ and $H(X) = m$, then this corresponds to $m$ fair coins each with 1-bit of information for a total of $m$-bits of information. Note that the Shannon Entropy is non-negative, bounded by the size of %NOTE: I didn't quite hear this part in lecture, so couldn't record it.\nIt also works well in a syntactic sense in that we can take many independent and identically distributed random variables %NOTE: Didn't hear this part in lecture too, I think it was glazed over.\n\n\\subsection{Von Neumann Entropy}\nImagine that we have an $m$-qubit space $\\X$, and a source that outputs $k$ pure states with probability $p_k$ for each. Then our source is summarized by the density matrix $\\rho_x = \\sum_{i} p_i \\kera{\\psi_i}$. First, note that the quantum source subsumes the classical source as any classical source with $X \\in \\{0, 1\\}^m$ and associated probability $p_x$ is just the densitry matrix $\\rho' = \\sum_{x \\in \\set{0, 1}^m} p_x \\kera{x}$. %NOTE: In class, we had a density matrix with density matrices p1, p2, ..., pm along its diagonals as being the one that summarizes a classical source. I don't understand why this is the case. The previous sentence makes more sense to me so I included that instead. If it is incorrect, then I apologize.\nLet us try to come up with a way to describe the amount of information contained in the density matrix, i.e. its entropy. At a glance, it might seem like we could use $\\se$ defined above. Letting $\\e(\\rho_x)$ denote the entropy quantity of $\\rho_x$, we get:\n\\begin{equation*}\n\\e(\\rho_x) = -\\sum_{i} p_i \\lg{p_i}\n\\end{equation*} \nHowever this definition does not work because two different quantum sources can have the same entropy -- entropy is meant to be unique. For example, the quantum source $S_1$ that outputs $\\ket{0}$ with probability $1/2$ and $\\ket{1}$ with probability $1/2$ is certainly different than the quantum source $S_2$ that outputs $\\ket{0}$ with probability $1/2$ and $\\ket{+}$ with probability $1/2$ -- the density matrices are not the same! However the probabilities of all the possible pure states in each source are the same, so $\\e(S_1) = \\e(S_2)$. The intuition is that in the quantum world, we have different bases that we can output our information in such as the standard and Hadamard bases. Further, there is no requirement that the pure states outputted by our source be orthogonal or even form an orthonormal basis -- $S_2$ is an example of one such source. In the classical world, however, everything is in binary -- all our possible sources will output information that's the same representation, so it is enough to just examine the probabilities. But the Shannon Entropy is too limited in the quantum world because we have sources that output different kinds of pure states together.\n\n\\fill\n\n\\noindent Fortunately, there is a way we can resolve our problem and that is by using the spectral decomposition. Because $\\rho$ is positive semi-definite, it has a unique decomposition under the eigenbasis which is:\n\\begin{equation*}\n\\rho = \\sum_{i = 0}^{d-1} \\lambda_i \\kera{\\psi_i}\n\\end{equation*}\nwhere $d$ is the dimension of $\\rho$'s space, $\\lambda_i$ is an eigenvalue, and $\\ket{\\psi_i}$ is an eigenvector such that $\\rho\\ket{\\psi_i} = \\lambda_i \\ket{\\psi_i}$ and $\\bret{\\psi_i}{\\psi_j} = \\delta_{ij}$. Note that $\\lambda_i \\geq 0$. Because $\\tr(\\rho) = 1$, we see that $\\sum_{i} \\lambda_i = 1$ so the eigenvalues form a valid probability distribution. Then we define the Von-Neumann entropy $\\vne$ as:\n\\begin{equation} \\label{eq:def-vne}\n\\vne(\\rho) = -\\sum_{i = 0}^{d-1} \\lambda_i \\lg{\\lambda_i}\n\\end{equation}\n\n\\fill\n\n\\noindent \\Example{7} Consider the source summarized by $\\rho = \\kera{0}$, i.e. a source that always outputs the pure state $\\ket{0}$. We would expect that $\\vne(\\rho) = 0$ since there is no uncertainty in what state we get. This is indeed the case, as $\\vne(\\rho) = -1\\lg{1} = 0$, i.e. $\\lambda = 1$ is the only eigenvalue of $\\rho$.\n\n\\fill\n\n\\noindent \\Example{8} Consider a fair coin, i.e. the source where $\\rho = \\frac{1}{2}\\kera{0} + \\frac{1}{2}\\kera{1}$. Here, $\\vne(\\rho) = 1$ which matches the corresponding classical case. In fact, if we have a mixed state on $n$-qubits where $\\rho = \\frac{1}{2^n}I$, then $\\vne(\\rho) = n$.\n\n\\end{document}", "meta": {"hexsha": "3524de594ed70b33dff76d4329df2257ee1f9fa4", "size": 26253, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "teaching/s17_4510_qc/170516_ei_lec13.tex", "max_stars_repo_name": "fangsonghub/fangsonghub.github.io", "max_stars_repo_head_hexsha": "31a42b297a4644b307b97acd293d9111e567f5c1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "teaching/s17_4510_qc/170516_ei_lec13.tex", "max_issues_repo_name": "fangsonghub/fangsonghub.github.io", "max_issues_repo_head_hexsha": "31a42b297a4644b307b97acd293d9111e567f5c1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-05-06T23:19:12.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-06T23:19:12.000Z", "max_forks_repo_path": "teaching/s17_4510_qc/170516_ei_lec13.tex", "max_forks_repo_name": "fangsonghub/fangsonghub.github.io", "max_forks_repo_head_hexsha": "31a42b297a4644b307b97acd293d9111e567f5c1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 64.6625615764, "max_line_length": 1190, "alphanum_fraction": 0.6931017408, "num_tokens": 8255, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.34839763850849137}} {"text": "%\n% vecmat3.tex - LaTeX documentation source for vecmat3.h version 2.3.1: \n% Fast 3d vector and matrix classes using expression templates\n%\n% Copyright (c) 2007-2013 Ramses van Zon\n%\n% Permission is hereby granted, free of charge, to any person obtaining a copy\n% of this software and associated documentation files (the \"Software\"), to deal\n% in the Software without restriction, including without limitation the rights\n% to use, copy, modify, merge, publish, distribute, sublicense, and/or sell\n% copies of the Software, and to permit persons to whom the Software is\n% furnished to do so, subject to the following conditions:\n%\n% The above copyright notice and this permission notice shall be included in\n% all copies or substantial portions of the Software.\n%\n% THE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\n% IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\n% FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\n% AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\n% LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\n% OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\n% THE SOFTWARE.\n%\n\n\\documentclass[12pt,twoside]{article}\n\n\\usepackage{a4wide,fancyhdr,mathptm,charter,color}\n%\\usepackage[colorlinks]{hyperref}\n\n\\pagestyle{fancy}\n\\renewcommand{\\sectionmark}[1]{\\markright{#1}}\n\\fancyhf{}\n\\fancyhead[LE,RO]{\\bfseries\\thepage}\n\\fancyhead[LO]{\\bfseries \\hfill\\rightmark\\hfill}\n\\fancyhead[RE]{\\hfill\\bfseries\\rightmark \\hfill}\n\\fancyfoot[L]{\\textit{vecmat3} v2.3.1}\n\\fancyfoot[R]{\\hfill{\\color{white}.}\\hfill Ramses van Zon\\hfill May 15, 2013}\n\\renewcommand{\\headrulewidth}{0.5pt}\n\\renewcommand{\\footrulewidth}{0pt}\n\\addtolength{\\headheight}{2.5pt}\n\\fancypagestyle{plain}{\\fancyhead{}\\renewcommand{\\headrulewidth}{0pt}}\n\n\n\\newcommand{\\Vector}{{Vector}}\n\\newcommand{\\Matrix}{{Matrix}}\n\\newcommand{\\cxx}{C\\texttt{++}}\n\\newcommand{\\TT}{{\\tt<}T{\\tt>}}\n\n\\begin{document}\n\n\\setlength{\\parskip}{1mm}\n\n\\title{User Documentation for the \\emph{vecmat3}\\\\\\Vector\\ and\n \\Matrix\\ classes\\\\(version 2.3.1)}\n\n\\author{Ramses van Zon\\footnote{vanzonr@gmail.com}}\n\n\\date{May 15, 2013}\n\n\\maketitle\n\\begin{abstract}\nThis document describes how to use the \\cxx\\ \\Vector\\ and\n\\Matrix\\ classes defined in the header file vecmat3.h. These classes\noffer a very convenient notation for three dimensional vector and\nmatrix algebra by using overloaded operators. Furthermore, they are\nconstructed to be numerically efficient through the internal use of\nexpression templates, without interfering at the user level. As a\nresult, one can convert mathematical matrix-vector expressions\nstraightforwardly into the corresponding \\cxx\\ expression without\nhaving to worry about incurring an efficiency penalty.\n\\end{abstract}\n\n\\newpage\n\n\\renewcommand{\\contentsname}{\\vspace{-13mm}}\\markright{Contents}\n\\tableofcontents\n\n\\newpage\n\n\\section{Introduction}\nVectors and matrices are used frequently in scientific computation (as\nwell as in modeling, games and movie rendering). Unfortunately, no\nbuilt-in support for matrices and vectors exists in \\cxx. In\nprinciple, expressions involving matrices and vectors, such as\n\\[\n\\vec a=\\vec b+\\mathsf M \\cdot \\vec c\n\\]\n(with $\\vec a$, $\\vec b$ and $\\vec c$ vectors and $\\mathsf M$ a\nmatrix) can be implemented in \\cxx\\\nsuch that they strongly resemble their mathematical notations, e.g.\n\\begin{quote}\n\\tt\nVector a,b;\n\nMatrix M;\n\nVector c = a + M*b;\n\\end{quote}\nThe technique used to accomplish such notational convenience is\noperator overloading, whose straightforward implementation comes with\na high computational cost due to the creation of temporary\nobjects.\n\nA more efficient implementation is possible by using \\emph{expression\ntemplates}. Efficient matrix-vector implementations are somewhat of a\nby-product of \\cxx\\ templates, and this shows in the awkward and\ncomplicated notation needed for general matrix-vector\nmanipulations. In addition, the \\cxx\\ standard is somewhat quirky on\nwhat is and is not allowed when using templates.\n\nThese notational issues probably explain why there are far fewer\ntemplate-based implementations of matrices and vectors available.\nThis is especially problematic for small vectors and matrices of fixed\nsize, for instance three-dimensional ones. These allow additional\nefficiency gains over general-size vectors and matrices (because loops\nover indices can be replaced by explicit sums in the implementation).\nTwo known implementations are the \\texttt{TinyVector} and \\texttt{\nTinyMatrix} classes of \\texttt{Blitz++} and the ones by the same name of\n\\texttt{tvmet}. The former is not very developed, i.e., many operations\nthat one would like to have are not present, and indeed, the latter is\naimed at fixing that. Still, \\texttt{tvmet} lacks some functionality that\nbuilt-in types do have, for instance, \\texttt{TinyVector<3,double> v =\na+b;} is not possible.\n\nThis is where \\emph{vecmat3} comes in. It defines very efficient\nthree dimensional vector and matrix manipulations. The aim is to be\nable to use these vectors and matrices as if they were built-in types,\nwith which the same kind of expressions can be formed as can with\nbuilt-in types without worrying about template techniques, but also without\nsubstantial losses compared to hard-coded element-by-element\ntechniques.\n\n\\emph{vecmat3} uses expression templates and operator\noverloading. The restrictions of vecmat3 at present are that the\nelements of the vectors and matrices have to be of a single type,\nwhich is \\texttt{double} by default, and that only three-dimensional\nquantities are supported (as the name suggests).\n\n\\section*{Change history}\n\\subsection*{Changes in version 2}\n\nThe main difference between the first version of vecmat3 and the\nsecond is that the matrices and vectors no longer need to be all of\none type in a single application. In addition to the standard Vector\nand Matrix classes of type \\texttt{DOUBLE} (which defaults to double)\nas in version 1, in version 2 one also has three-dimensional\nstructures of any type \\texttt{T} at one's disposal. To be more\nprecise:\n\\begin{itemize}\n\\item \\texttt{vecmat3::Vector\\TT{}} and \\texttt{vecmat3::Matrix\\TT{}} are\n three-dimensional vector and matrix classes whose elements are of\n type \\texttt{T}. \n\n For example, one can define a 3x3 matrix of integers as\\ \\ \n \\texttt{vecmat3::Matrix~m;}\n\\item This notation reflects two changes in the code: \n\\begin{itemize}\n\\item Almost all of the \\texttt{vecmat3} code is now contained in its own\n namespace called \\texttt{vecmat3}. \n\\item The type is a template argument.\n\\end{itemize}\n\\item In version 2 the standard Vector and Matrix classes are simply\n typedef'ed as equivalent to \\texttt{vecmat3::Vector} and\n \\texttt{vecmat3::Vector}, whereas in version 1 they were the\n only vectors and matrices available.\n\\item The typedef's of Vector and Matrix will be omitted if the\n compiler flag \\texttt{NOVECMAT3DEF} is defined.\n\\end{itemize}\n\n\\subsection*{Changes between version 2 and version 2.3}\n\n(Versions 2.1 and 2.2 were internal development stages.)\n\n\\begin{itemize}\n\\item Most functionality remained the same as in version 2, except that square bracket support has been added. See Section \\ref{bracketsnowtoo}.\n\n\\item The header file now also enforces inlining the template functions for\nthe GNU (tested version 4.4.0) and the Intel compilers (tested\nversions 11 and 12), even when no optimization is used. \n\n\\item A major improvement of version 2.3 is that the library is now\ncompatible with IBM's xlC compiler, which had trouble with some\ntemplate constructions in versions 1 and 2. While I'm on the\nsubject, one should compile with \\texttt{-O4} when using the IBM\ncompilers with vecmat3 in order to get all inlining done properly (in\nthe latest version, xlC 11, the options \\texttt{-O2 -qinline=level=6} suffice).\n\\end{itemize}\n\n\\subsection*{Changes between version 2.3 and version 2.3.1}\n\nVersion 2.3.1 comes with an open-source license (MIT). Here is the\ntext of the license:\n\\begin{verbatim}\nCopyright (c) 2007-2013 Ramses van Zon\n\nPermission is hereby granted, free of charge, to any person obtaining a copy\nof this software and associated documentation files (the \"Software\"), to deal\nin the Software without restriction, including without limitation the rights\nto use, copy, modify, merge, publish, distribute, sublicense, and/or sell\ncopies of the Software, and to permit persons to whom the Software is\nfurnished to do so, subject to the following conditions:\n\nThe above copyright notice and this permission notice shall be included in\nall copies or substantial portions of the Software.\n\nTHE SOFTWARE IS PROVIDED \"AS IS\", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR\nIMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,\nFITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE\nAUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER\nLIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,\nOUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN\nTHE SOFTWARE.\n\\end{verbatim}\n\n\\section{Installation}\n\nTo use this header-only library, one merely needs to copy the header\nfile vecmat3.h to the directory of the source files that include it,\nor to a default directory where the compiler will look for header\nfiles (e.g. /usr/local/include).\n\nThe \\emph{vecmat3} library has been tested with the GNU g++ compiler version\n4.4 and up, the Intel C++ compiler version 11 and up and IBM's XL C++\ncompiler version 10 and up.\n\n\n\\section{Using the \\Vector\\ and \\Matrix\\ classes}\n\\label{use}\n\n\\subsection{Classes}\n\n\\emph{vecmat3} provides two general template classes within the\nnamespace \\texttt{vecmat3}:\n\n\\begin{quote}\\tt\n template vecmat3::Vector;\n\n template vecmat3::Matrix;\n\\end{quote}\nThe template parameter T determines the type of the vector and\nmatrix elements. Thus, for a vector of integers one uses the type\n\\texttt{vecmat3::Vector}, while for a matrix of doubles one would\nuse \\texttt{vecmat3::Matrix}.\n\nSince applications often need only one type of vector, a default\nvector type a and default matrix type are defined outside the\n\\texttt{vecmat3} namespace, as follows\n\\begin{quote}\\tt\n\ntypedef vecmat3::Vector Vector;\n\ntypedef vecmat3::Matrix Matrix;\n\\end{quote}\nHere, DOUBLE is a predefined macro that should contain the type of the\nelements of the default vectors and matrices. Thus, \\texttt{Vector v;}\ndefines a vector with elements of type DOUBLE.\n\nThe type DOUBLE can be defined in three ways: \n\\begin{enumerate}\n\\item One can write an\n\\texttt{\\#define DOUBLE } before including the\n\\texttt{vecmat3.h} header, with \\texttt{} replaced by the\ndesired type (e.g.\\ \\texttt{float} or \\texttt{double});\n\\item One can give a command line argument to the compiler to define\n DOUBLE to be \\texttt{} (e.g.\\ \\texttt{-DDOUBLE=float} for\n g++);\n\\item One can do nothing, which makes \\texttt{DOUBLE} default to\n\\texttt{double}.\n\\end{enumerate}\n\nThe definition of DOUBLE and the type definition of Vector and Matrix\nin the global namespace does pollute the global namespace, and in many\ncases is not wanted. These definitions are omitted if \\texttt{NOVECMAT3DEF} is defined.\n\n\\subsection{Header file}\n\nTo use \\emph{vecmat3}, the following general procedure should be\nfollowed: \n\n\\begin{itemize}\n \\item If the elements of the vectors and matrices are to have a\n different type than \\texttt{double}, first \\texttt{\\#define} their type as\n \\texttt{DOUBLE}, e.g.\n \\begin{quote}\n \\tt \\#define DOUBLE float\n \\end{quote}\n The type of the elements of a vector or matrix will be referred to\n as the ``value type'' in this documentation.\n \\item Include the header file vecmat3.h:\n \\begin{quote}\n \\tt \\#include \"vecmat3.h\"\n \\end{quote}\n\n \\item The class \\Vector\\ and the class \\Matrix\\ are now\n defined and instances these classes can be declared as follows:\n \\begin{quote}\\tt\n Vector a;\n\n Matrix R;\n \\end{quote}\n \\item One can explicitly use any other value type\n than DOUBLE type, e.g.\n \\begin{quote}\\tt\n vecmat3::Vector a;\n\n vecmat3::Matrix R;\n \\end{quote}\n If vectors and matrices of a specific value type are used a lot in an\n application, it may be useful to typedef them to a shorter\n notation, e.g.\n \\begin{quote}\\tt\n typedef vecmat3::Vector intVector;\n\n typedef vecmat3::Matrix intMatrix;\n\n intVector b;\n\n intMatrix S;\n \\end{quote}\n \\item Alternatively, one can have no default global \\Vector\\ and \\Matrix\\ class defined, and use only the namespace vecmat3, e.g.\n \\begin{quote}\\tt\n \\#define NOVECMAT3DEF\n\n \\#include \"vecmat3.h\" \n\n vecmat3::Vector a;\n\n vecmat3::Matrix R;\n \\end{quote}\n\n \\item In the above examples, the elements of the vectors and\n matrices are unspecified, and likely contain garbage. In the\n next section, it will be explained how to initialize these\n elements of these\n classes.\n \\item Note that currently, operations between matrices and vectors\n of different value types are not supported, even when mathematically\n this would make sense (such as for \\texttt{int} and\n \\texttt{double}). \n \\item However, it is possible to assign any kind of number to an\n element of any value type, as long as a (standard)\n conversion to that type is known to the c++ compiler. Thus,\n one may, for instance, assign an integer to an element of a\n vector, or one may multiply a vector by 2 (i.e., one may write\n \\texttt{2*v} instead of being\n forced to write \\texttt{2.0*v} or \\texttt{2.0f*v}).\n\\end{itemize}\n\n\n\\subsection{Initialization methods}\n\nThere are four ways to initialize a \\Vector\\ or \\Matrix, which we will\ndiscuss by example. In describing the initialization methods, the\ndefault Vector and Matrix types will be used; the arbitrary type\nversions \\texttt{vecmat3::Vector\\TT{}} and\n\\texttt{vecmat3::Matrix\\TT{}} have the same functionality.\n\n\\subsubsection{Initialization through constructor parameters}\n Example:\n \\begin{quote}\\tt\n Vector a(1.1, 3.0, -4.3);\n\n Matrix R(1, 2, 3,\n \n\\ \\ \\ \\ \\ \\ \\ \\ \\ 4, 5, 6,\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ 7, 8, 9);\n \\end{quote}\n defines a \\Vector\\ \\texttt a and \\Matrix\\ \\texttt R with specified\n elements. Note that the first set of three elements given to\n \\texttt R comprise the top row of \\texttt R, the second set of\n three the middle row and the last set of three the bottom row.\n\n If fewer than three or nine (for \\Vector\\ and \\Matrix,\n respectively) number are given, the unspecified elements are set\n to zero. Thus, one can define a zero \\Vector\\ and \\Matrix\\ simply by\n \\begin{quote}\\tt\n Vector a(0);\n\n Matrix R(0);\n \\end{quote}\n\n \\subsubsection{Initialization through assignment}\n Example:\n \\begin{quote}\\tt\n Vector b = a;\n\n Matrix S = R;\n \\end{quote}\n defines a \\Vector\\ \\texttt b with the same elements as \\texttt a,\n and a \\Matrix\\ \\texttt S with the elements~as~\\texttt R. \n\n The right hand sides may also be an expression involving \\Vector's\n and Matrices. The allowed expressions are explained in\n sections \\ref{operations} and \\ref{expressions}.\n%\\newpage\n \\subsubsection{Initialization through a comma separated list}\n Example:\n \\begin{quote} \\tt\n Vector a;\n\n\t a = 1.1, 3.0, -4.3;\n\n Matrix R;\n\n\t R = 1, 2, 3, \n\n \\ \\ \\ \\ 4, 5, 6, \n\n \\ \\ \\ \\ 7, 8, 9;\n \\end{quote} \n Note: this is the standard construction for \\texttt{Blitz++} and\n \\texttt{tvmet}, and is achieved though an overloaded comma\n operator. Not everybody likes overloading the comma operator,\n because it may confuse the user (more than the above methods), and\n it is somewhat less efficient than the method in 3.2.1.\n\n Furthermore, it is not possible to use this method in\n the declaration, i.e., one cannot write \\texttt{Vector\n a=1.1,3.0,-4.3;} since \\cxx\\ would consider this a\n declaration of 3.0 and -4.3 as being of type \\Vector.\n \n If not enough elements are given in the list, the remaining\n elements are set to zero. Thus, one can write\n \\begin{quote}\\tt\n b = 1;\n \\end{quote}\n to get the vector (1,0,0), and \n \\begin{quote}\\tt\n R = 2;\n \\end{quote}\n to get the matrix $\\left(\\begin{array}{ccc}2&0&0\\\\0&0&0\\\\0&0&0\\end{array}\\right)$.\n\n \\subsubsection{Initialization through member functions}\n Example:\n \\begin{quote}\\tt\n Vector a;\n\n\t a.zero();\n\n Matrix R;\n\n\t R.zero();\n \\end{quote}\n also define a \\Vector\\ and \\Matrix, respectively, with zero\n elements.\n \n\\noindent\n For a \\Matrix, there also is a member function {\\tt one()} to turn\n it into an identity matrix:\n \\begin{quote}\\tt\n Matrix S;\n\n\t S.one();\n \\end{quote}\n Furthermore, one can initialize a \\Matrix\\ also per row or column, e.g.\n \\begin{quote}\\tt\n R.setRow(0,a);\n\n R.setRow(1,Vector(0,2,0));\n\n R.setRow(2,Vector(0,2,-1));\n\n S.setColumn(0,a);\n\n S.setColumn(1,Vector(0,2,0));\n\n S.setColumn(2,Vector(0,2,-1));\n \\end{quote} \n Note that rows and columns are numbered from 0 to 2.\n\n\n \\subsubsection{Arrays}\n Example:\n \\begin{quote}\\tt\n Vector a[3];\n\n Matrix S[3];\n \\end{quote}\n Defines arrays of three Vectors and Matrices which are\n non-initialized. One can initialize these arrays as follows\n \\begin{quote}\\tt\n Vector a[3] = \\{ Vector(1,2,3), Vector(3,4,5), Vector(5,6,7) \\};\n\n Matrix S[3] = \\{ Matrix(0), Matrix(1,2,3,4,5,6,7,8,9), Matrix(2) \\} ;\n \\end{quote}\n\n\\subsection{Accessing the elements}\n\\label{bracketsnowtoo}\n\nThere are three ways to assess the elements of Vectors and Matrices:\n\n\\begin{enumerate}\n\n\\item Basic elements of Vectors and Matrices are generally accessible using the\nparenthesis notation, i.e., the elements of a \\Vector\\ \\texttt a are\n\\texttt{a(0)}, \\texttt{a(1)} and \\texttt{a(2)}, while those of a\n\\Matrix\\ \\texttt R are \\texttt{R(0,0)}, \\texttt{R(0,1)}, \\dots\n\\texttt{R(2,2)}.\n\n\\item Bracket notation can also be used, \n i.e., the elements of a \\Vector\\ \\texttt a are\n\\texttt{a[0]}, \\texttt{a[1]} and \\texttt{a[2]}, while those of a\n\\Matrix\\ \\texttt R are \\texttt{R[0][0]}, \\texttt{R[0][1]}, \\dots\n\\texttt{R[2][2]}. Note the double brackets for matrix element access.\nThis way, Vector and Matrix objects act as if they are of type \\texttt{T[3]} and \\texttt{T[3][3]}, respectively. Accessing the matrix elements in this way may be moderately slower than the parenthesis method (depending on the compiler).\n\n\\item Another way to access the elements is though the class members\nthemselves, i.e., \\texttt x, \\texttt y and \\texttt z for \\Vector, and\n\\texttt{xx}, \\texttt{xy}, \\texttt{xz}, \\texttt{yx}, \\texttt{yy},\n\\texttt{yz}, \\texttt{zx}, \\texttt{zy} and \\texttt{zz} for Matrix. This is\npotentially more efficient, but cannot be used for expressions, i.e.,\n\\texttt{(A+B).xx} is not possible, unless one writes \\texttt{Matrix(A+B).xx}.\n\\end{enumerate}\nFurthermore, the rows and columns of a Matrix can be used as if they\nwere vectors as follows\n\\begin{quote}\\tt\n Vector v = R.row(1);\n\n Vector w = R.column(2);\n\\end{quote}\n\n\n\\subsection{Operators}\n\\label{operations}\n\n\nThe available algebraic operators for the Vector and Matrix classes\nare summarized in table 1, in which '\\Vector\\TT' stands for\n'\\texttt{const Vector\\TT\\&}' or a \\Vector-valued expression, and '\\Matrix\\TT'\nstands for '\\texttt{const Matrix\\TT\\&}' or a Matrix\\TT-valued expression.\n\nIn addition, \\texttt{<<} operators are defined for output of Vectors\nand Matrices to \\texttt{ostream}s, such that\n\\begin{quote}\\tt\n Vector a(1,2,3);\n\n std::cout << a << endl;\n\\end{quote}\nwould print the numbers 1, 2 and 3 with just a space in between.\n\\begin{quote}\\tt\n Matrix M(1,2,3,4,5,6,7,8,9);\n\n std::cout << M << endl;\n\\end{quote}\nwould print a newline, the numbers 1, 2 and 3, another newline, the\nnumbers 4, 5 and 6, another newline, the numbers 7, 8 and 9 and\nfinally another newline.\n\n\\begin{table}[t]\\small\n\\begin{center}\n\\begin{tabular}{|rcl|c|c|rcl|}\n\\hline\n \\multicolumn{3}{|c|}{\\bf form} & \\bf description & \\bf example & \\multicolumn{3}{c|}{\\bf mathematically}\\\\\\hline\\hline\n%\n&\\texttt{\\textcolor{red}-}\n&\\texttt{Vector} \n& negative \n& \\tt c = -a; \n& $\\quad \\vec c$&=&$-\\vec a$\\\\\n%\n\\texttt{Vector}\n&\\textcolor{red}{\\texttt +}\n&\\texttt{Vector} \n& add \n& \\tt c = a + b; \n& $\\quad \\vec c$&=&$\\vec a +\\vec b$\\\\\n%\n\\texttt{Vector}\n&\\textcolor{red}{\\texttt -}\n&\\texttt{Vector} \n& subtract\n& \\tt c = a - b;\n& $\\quad \\vec c$&=&$\\vec a -\\vec b$\\\\\n%\n\\texttt{T}\n&\\textcolor{red}{\\texttt *}\n&\\texttt{Vector} \n& multiply with scalar \n& \\tt c = d * a; \n& $\\quad \\vec c$&=&$d\\,\\vec a$\\\\\n%\n\\texttt{Vector}\n&\\textcolor{red}{\\texttt *} \n&\\texttt{T} \n& multiply with scalar \n& \\tt c = a * d; \n& $\\quad \\vec c$&=&$\\vec a\\, d$\\\\\n%\n\\texttt{Vector}\n&\\textcolor{red}{\\texttt /}\n&\\texttt{T} \n& divide by scalar \n& \\tt c = a / d; \n& $\\quad \\vec c$&=&$\\vec a / d$\\\\\n%\n\\texttt{Vector} \n&\\textcolor{red}{\\texttt \\^{}} \n&\\texttt{Vector} \n& cross/outer product$^\\dagger$\n& \\tt c = a \\^{} b; \n& $\\quad \\vec c$&=&$\\vec a \\times \\vec b$\\\\\n%\n\\texttt{Vector}\n&\\textcolor{red}{\\texttt *}\n&\\texttt{Vector} \n& dot/inner product$^\\ddagger$\n& \\tt d = a * b; \n& $\\quad d$&=&$\\vec a \\cdot \\vec b$\\\\\n%\n\\ \\textcolor{red}( \\texttt{Vector} \n&\\textcolor{red}{\\texttt |}\n& \\texttt{Vector} \\textcolor{red})\n& dot/inner product$^\\ddagger$\n& \\tt d = (a|b); \n& $\\quad d$&=&$\\vec a \\cdot \\vec b$\\\\\n%\n\\hline\n%\n&\\texttt{\\textcolor{red}-}\n&\\texttt{Matrix} \n& negative \n& \\tt T = -S; \n& $\\quad \\mathsf T$&=&$-\\mathsf S$\\\\\n%\n\\texttt{Matrix}\n&\\textcolor{red}{\\texttt +}\n&\\texttt{Matrix} \n& add \n& \\tt T = S + R; \n& $\\quad \\mathsf T$&=&$\\mathsf S +\\mathsf R$\\\\\n%\n\\texttt{Matrix}\n&\\textcolor{red}{\\texttt -}\n&\\texttt{Matrix} \n& subtract\n& \\tt T = S - R;\n& $\\quad \\mathsf T$&=&$\\mathsf S -\\mathsf R$\\\\\n%\n\\texttt{T}\n&\\textcolor{red}{\\texttt *}\n&\\texttt{Matrix} \n& multiply with scalar \n& \\tt T = d * S; \n& $\\quad \\mathsf T$&=&$d\\,\\mathsf S$\\\\\n%\n\\texttt{Matrix} \n&\\textcolor{red}{\\texttt *} \n&\\texttt{T} \n& multiply with scalar \n& \\tt T = S * d; \n& $\\quad \\mathsf T$&=&$\\mathsf S\\, d$\\\\\n%\n\\texttt{Matrix} \n&\\textcolor{red}{\\texttt /} \n&\\texttt{T} \n& divide by scalar \n& \\tt T = S / d; \n& $\\quad \\mathsf T$&=&$\\mathsf S / d$\\\\\n%\n\\texttt{Matrix}\n&\\textcolor{red}{\\texttt *}\n&\\texttt{Matrix} \n& matrix-matrix product\n& \\tt T = S * R; \n& $\\quad \\mathsf T$&=&$\\mathsf S\\,\\mathsf R$\\\\\n%\n\\texttt{Matrix}\n&\\textcolor{red}{\\texttt *} \n&\\texttt{Vector} \n& matrix-vector product\n& \\tt c = S * a; \n& $\\quad \\vec c$&=&$\\mathsf S \\, \\vec a$\\\\\n%\n\\hline\n%\n\\texttt{Vector}\n&\\textcolor{red}{\\texttt{+=}}\n&\\texttt{Vector} \n& add \n& \\tt c += b; \n& $\\quad \\vec c$&=&$\\vec c +\\vec b$\\\\\n%\n \\texttt{Vector}\n&\\texttt{\\textcolor{red}{-=}}\n&\\texttt{Vector} \n& subtract \n& \\tt c -= b; \n& $\\quad \\vec c$&=&$\\vec c -\\vec b$\\\\\n%\n\\texttt{Vector}\n&\\texttt{\\textcolor{red}{*=}}\n&\\texttt{T} \n& multiply by scalar\n& \\tt c *= d; \n& $\\quad \\vec c$&=&$d\\,\\vec c$\\\\\n%\n\\texttt{Vector}\n&\\texttt{\\textcolor{red}{/=}}\n&\\texttt{T} \n& divide by scalar\n& \\tt c /= d; \n& $\\quad \\vec c$&=&$\\vec c/d$\\\\\n%\n\\texttt{Matrix}\n&\\texttt{\\textcolor{red}{+=}}\n&\\texttt{ Matrix} \n& add \n& \\tt T += R; \n& $\\quad \\mathsf T$&=&$\\mathsf T +\\mathsf R$\\\\\n%\n \\texttt{Matrix}\n&\\texttt{\\textcolor{red}{-=}}\n&\\texttt{ Matrix} \n& subtract \n& \\tt T -= R; \n& $\\quad \\mathsf T$&=&$\\mathsf T -\\mathsf R$\\\\\n%\n\\texttt{Matrix}&\n\\texttt{\\textcolor{red}{*=}}\n&\\texttt{T} \n& multiply by scalar\n& \\tt T *= d; \n& $\\quad \\mathsf T$&=&$d\\,\\mathsf T$\\\\\n%\n \\texttt{Matrix}\n&\\texttt{\\textcolor{red}{/=}}\n&\\texttt{T} \n& divide by scalar\n& \\tt T /= d; \n& $\\quad \\mathsf T$&=&$\\mathsf T/d$\\\\\n\\hline\n\\multicolumn{6}{l}{\n$^\\dagger$ The \\texttt \\^{} operator has rather low precedence, so\noften one has to write \\texttt{(a\\^{}b)}.\n}\\\\\\multicolumn{6}{l}{\n$^\\ddagger$ Two operators are provided for the dot product, which do\nthe exact same thing. \n}\n\\end{tabular}\n\\end{center}\n\\caption{Operators available for matrices and\n vectors with elements of type \\texttt{T}. }\n\\end{table}\n\n\n\n\\subsection{Member functions}\n\nFor any \\Vector, or \\Vector-valued expression, or for any \\Matrix, or\n\\Matrix-valued expression, the following properties are available as\nmember functions. Note that below, {\\tt T} stands for the typename of the template, while {\\tt Vector\\TT{}} and {\\tt Matrix\\TT{}} stand for {\\tt vecmat3::Vector\\TT{}} and {\\tt vecmat3::Matrix\\TT{}}, respectively. The examples all use {\\tt T=DOUBLE}.\n\n\\subsubsection{T nrm2()}\n\nThis returns the sum of the squares of the elements, which is its norm\nsquared. E.g., with {\\tt T=DOUBLE},\n\\begin{quote} \\tt\n Vector a(1,2,3.316625);\n\n Matrix R(1,2,0,\n\n\\ \\ \\ \\ \\ \\ \\ \\ \\ 2,0,2\n\t \n\\ \\ \\ \\ \\ \\ \\ \\\t\\ 1,1,1);\n\n DOUBLE d1 = a.nrm2(); // will be equal to 16.0000014\n\n DOUBLE d2 = R.nrm2(); // will be equal to 16\n\\end{quote}\n\n\\subsubsection{T nrm()}\n\nThis returns the norm of a \\Vector\\ or \\Matrix, e.g., with {\\tt T=DOUBLE},\n\\begin{quote} \\tt\n DOUBLE d3 = a.nrm(); // will be equal to 4.00000017\n\n DOUBLE d4 = R.nrm(); // will be equal to 4\n\\end{quote}\n\n\\noindent\nThe following properties are for Matrices only:\n\n\\subsubsection{T tr()}\n\nThis returns the trace of a \\Matrix, i.e., the sum of its diagonal\nelements. E.g., with {\\tt T=DOUBLE},\n\\begin{quote} \\tt\n DOUBLE d5 = R.tr(); // will be equal to 2\n\\end{quote}\n\n\\subsubsection{T det()}\n\nThis returns the determinant of a \\Matrix, e.g., with {\\tt T=DOUBLE},\n\\begin{quote} \\tt\n DOUBLE d6 = R.det(); // will be equal to -2\n\\end{quote}\n\n\\subsubsection{Vector\\TT{} row(int i)}\nThis returns the ith row of a \\Matrix.\n\n\\subsubsection{Vector\\TT{} column(int j)}\n\nThis returns the jth column of a \\Matrix.\n\n\\pagebreak[3]\n\\subsection{Non-member functions}\n\nIn the definition of the following non-member functions, the specified\nreturn type are effective ones. E.g. a return type of Matrix may\nreturn a Matrix-Expression when this is more efficient. In any case, it\ncan be treated as a Matrix in virtually all ways. Likewise, if an\nargument is of Matrix type, a Matrix expression is also allowed.\n\n\\subsubsection{Matrix\\TT{} Transpose(const Matrix\\TT{} \\& M)}\n\nReturns the transpose of the argument, which is a \\Matrix, e.g.\n\\begin{quote}\\tt\n Matrix S = Transpose(R);\n\\end{quote}\n\n\\subsubsection{Matrix\\TT{} Inverse(const Matrix\\TT{} \\& M)}\n\nReturns the inverse of the argument, which is a \\Matrix, e.g.\n\\begin{quote}\\tt\n Matrix S = Inverse(R);\n\\end{quote}\n\n\\subsubsection{Matrix\\TT{} Rodrigues(const Vector\\TT{} \\& v)}\n\nReturns the \\Matrix-valued rotation matrix for a rotation along the\naxis given by the direction of the \\Vector\\ argument, with the angle\nequal to the norm of that \\Vector, e.g.\n\\begin{quote}\\tt\n Matrix S = Rodrigues(a);\n\\end{quote}\n\n\n\\subsubsection{Matrix\\TT{} Dyadic(const Vector\\TT{} \\& a, const Vector\\TT{} \\& b)}\n\nReturns the \\Matrix-valued dyadic product of two arguments which are\n\\Vector s, e.g.\n\\begin{quote}\\tt\n Matrix S = Dyadic(a,b);\n\\end{quote}\n\n\n\\subsubsection{Vector\\TT{} MTVmult(const Matrix\\TT{} \\& M, const Vector\\TT{} \\& v)}\n\nThis simply returns Transpose(Matrix)*Vector. \n\n\\subsubsection{T dist(const Vector\\TT{} \\& a, const Vector\\TT{} \\& b)}\n\nReturns the length of the difference vector between a and b. This is a\nremnant of earlier versions of the \\Vector and \\Matrix classes, and\nbarely if at all more efficient than \\texttt{(a-b).nrm()}.\n\n\\subsubsection{T dist2(const Vector\\TT{} \\& a, const Vector\\TT{} \\& b)}\n\nReturns the square length of the difference vector between a and\nb. This is a remnant of earlier versions of the Vector and Matrix\nclasses, and barely if at all more efficient than \\texttt{(a-b).nrm2()}. \n\n\\subsubsection{T distwithshift(const Vector\\TT{}\\&a,const Vector\\TT{}\\&b,const Vector\\TT{}\\&s)}\n\nReturns the length of the difference vector between a and b shifted by\ns. This is a remnant of earlier versions of the vector and Matrix\nclasses, and barely if at all more efficient than \\texttt{(a+s-b).nrm()}.\n\n\n\\vspace{1cm}\\pagebreak[3]\nFinally, because the notation \\texttt{a*b} and \\texttt{a\\^{}b} for dot and\ncross product may be confusing, the following equivalent alternatives\nare defined:\n\\nopagebreak\n\\subsubsection{T dotProduct(const Vector\\TT{} \\& a, const Vector\\TT{} \\& b)}\n\nReturns the DOUBLE which is the dot, or inner, product of the two\n\\Vector\\ arguments. It is by definition equal to\n\\texttt{(\\texttt{Vector}|\\texttt{Vector})}.\nExample, with {\\tt T=DOUBLE}:\n\\begin{quote}\\tt\n DOUBLE d = dotProduct(a,b);\n\\end{quote}\n\n\\subsubsection{Vector\\TT{} crossProduct(const Vector\\TT{} \\& a, const Vector\\TT{} \\& b)}\n\nReturns the \\Vector\\ which is the cross, or outer, product of the two\n\\Vector\\ arguments. It is by definition equal to\n\\texttt{(\\texttt{Vector}\\^{}\\texttt{Vector})}.\nExample:\n\\begin{quote}\\tt\n Vector c = crossProduct(a,b);\n\\end{quote}\n\n\n\\section{Expressions}\n\\label{expressions}\n\nUsing the above elementary operations and functions, complex\nexpressions can be constructed just as for built-in type such as\n\\texttt{double}. To be more specific, for all operator expressions in table 1\non page 8, the arguments can be expressions themselves.\n\nFor example, one can write,\n\\begin{quote}\\tt\n Vector r[2] = \\{ Vector(1,4,5), Vector(2,3,4) \\};\n\n Vector v[2] = \\{ Vector(1,0,0), Vector(-1,0,0) \\};\n\n Vector s(7,1,0);\n\n Matrix A(1,0,0,0,1,0,0,-1,0);\n\n DOUBLE t = (r[0]+2*A*(s\\^{}r[1]) ) | (v[1]-v[0]);\n\n // alternatively:\n\n //DOUBLE t = dotProduct(r[0]+2*A*crossProduct(s,r[1]), v[1]-v[0]);\n\\end{quote}\n\nInternally, the expressions are not computed directly via temporaries,\nbut are computed only upon assignment. As a result, the definitions of\nthese operators and functions in vecmat3.h is not as simple as\ne.g. \\texttt{Vector operator+(Vector\\&,Vector\\&)}. For that reason,\nabove we used \\texttt{Vector} and \\texttt{Matrix} wherever a\n\\Vector/\\Matrix\\ or a \\Vector/\\Matrix\\ expression can occur. Never\nmind the implementation though, things work as expected.\n\n\\newpage\n\\renewcommand{\\refname}{Background references}\n\\begin{thebibliography}{9}\n\\bibitem{Lippmann} \nT. Veldhuizen, \\emph{Expression Templates} C++ Report, Vol. 7 No. 5 June\n1995, pp. 26-31. See also\nhttp://ubiety.uwaterloo.ca/\\~{}tveldhui/papers/Expression-Templates/exprtmpl.html.\\\\\nReprinted in:\nS. B. Lippmann (ed.) \\emph{C++ Gems} (Cambridge\n University Press, 1998).\n\\bibitem{VandevoordeJosuttis} D.\\ Vandevoorde and N.\\ M.\\ Josuttis,\n \\emph{C++ Templates: The Complete Guide} (Addison-Wesley, Boston,\n 2002).\n\\bibitem{blitz} http://www.oonumerics.org/blitz\n\\bibitem{tvmet} http://tvmet.sourceforge.net\n\\end{thebibliography}\n\\addcontentsline{toc}{section}{\\refname}\n\n\n\n\\end{document}\n", "meta": {"hexsha": "f23f378e014e60d82ec5ff1b02b8307c94ab263f", "size": 31011, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "vecmat3.tex", "max_stars_repo_name": "vanzonr/vecmat3", "max_stars_repo_head_hexsha": "b1c9671854f95418bd6a4c46dc6be6816ca43fff", "max_stars_repo_licenses": ["Intel", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "vecmat3.tex", "max_issues_repo_name": "vanzonr/vecmat3", "max_issues_repo_head_hexsha": "b1c9671854f95418bd6a4c46dc6be6816ca43fff", "max_issues_repo_licenses": ["Intel", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "vecmat3.tex", "max_forks_repo_name": "vanzonr/vecmat3", "max_forks_repo_head_hexsha": "b1c9671854f95418bd6a4c46dc6be6816ca43fff", "max_forks_repo_licenses": ["Intel", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.5745798319, "max_line_length": 250, "alphanum_fraction": 0.702395924, "num_tokens": 9294, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.34839763850849137}} {"text": "\\section{Introduction}\n\\label{sec:introduction}\n\nFull bayesian inference for most neural network is intractable due to the model's nonlinearity causing the true posterior distribution to be highly complex. This has led to the use of variational methods which attempt to approximate the posterior. With regard to bayesian variational inference; this paper addresses the general issue of prior selection to reduce the sensitivity BNNs have toward such initializations and the high variance that arises in the gradients when employing Monte Carlo approximation techniques.\n\nThe authors list their contributions as:\n\\begin{itemize}\n\t\\item Development of a deterministic procedure for propagating uncertain activations through neural networks with uncertain weights and ReLU or Heaviside activation functions.\n\t\\item Development of an EB method for principled tuning of weight priors during BNN training.\n\t\\item Experimental results showing the accuracy and efficiency of our method and applicability to heteroscedastic and homoscedastic regression on real datasets.\n\\end{itemize} \n \n", "meta": {"hexsha": "dc51f690608c2e82ce85279a5e47dc21d8690809", "size": 1082, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "_notes/paper-summary/DVI-2019-05-23/tex/Introduction.tex", "max_stars_repo_name": "ibrahimkakbar/ibrahimkakbar.github.io", "max_stars_repo_head_hexsha": "5e6b0ea67f5e5f8f3a7bb4394095ea7f7992673a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_notes/paper-summary/DVI-2019-05-23/tex/Introduction.tex", "max_issues_repo_name": "ibrahimkakbar/ibrahimkakbar.github.io", "max_issues_repo_head_hexsha": "5e6b0ea67f5e5f8f3a7bb4394095ea7f7992673a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_notes/paper-summary/DVI-2019-05-23/tex/Introduction.tex", "max_forks_repo_name": "ibrahimkakbar/ibrahimkakbar.github.io", "max_forks_repo_head_hexsha": "5e6b0ea67f5e5f8f3a7bb4394095ea7f7992673a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 83.2307692308, "max_line_length": 520, "alphanum_fraction": 0.8327171904, "num_tokens": 207, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.34839763850849137}} {"text": "\\documentclass[10pt]{article}\n\\usepackage[margin=1in]{geometry}\n\n\\input{global_macros}\n\\input{macros}\n\n% Enable (uncolored) cross-reference hyperlinks\n% Should always be last package loaded.\n% See: https://tex.stackexchange.com/questions/103123/links-do-not-lead-to-right-pages\n\\usepackage[colorlinks=false]{hyperref}\n\\begin{document}\n \\begin{center}\n {\\Large \\textbf{Text Classification under Negative Covariate Shift}}\n\n \\vspace{6pt}{\\large Project Proposal}\n\n \\vspace{6pt}Zayd Hammoudeh\n \\end{center}\n\n Consider the task of binary (supervised) classification. Each sample is a tuple of feature vector~${\\X \\in \\domain}$ and label~${\\y \\in \\lbls}$. Samples are generated from unknown, joint probability distribution~$\\joint$. Given hypothesis class~${\\hypoCls=\\setbuild{\\dec}{\\func{\\dec}{\\domain}{\\real}}}$ and loss function~$\\func{\\loss}{\\real \\times \\lbls}{\\real}$, the ideal classifier is:\n\n \\begin{equation}\\label{eq:IdealClassifier}\n \\dec\\opt = \\argmin_{\\dec \\in \\hypoCls} \\mathbb{E}_{(\\X, \\y) \\sim \\joint} \\sbrack{\\fLoss{\\fDec{\\X}, \\y}}\\text{.}\n \\end{equation}\n\n Since $\\joint$~is unknown, Eq.~\\eqref{eq:IdealClassifier} cannot be determined exactly. \\textit{Empirical risk minimization}, defined in Eq.~\\eqref{eq:ERM}, is used in practice for selection of the optimal learner,~${\\dec\\opt}$. Observe that the true expected loss is replaced with an empirical estimate using i.i.d.\\ training set~${\\train \\sim \\joint}$.\n\n \\begin{equation}\\label{eq:ERM}\n \\dec\\opt = \\argmin_{\\dec \\in \\hypoCls} \\sum_{(\\X_i,\\y_i) \\in \\train} \\fLoss{\\fDec{\\X_i}, \\y_i}\n \\end{equation}\n\n There are many real-world applications where $\\train$~is not an i.i.d.\\ sample. For example, given a huge (unlabeled) news corpus, a particular user may only be interested in a single topic,~$\\interest$. There are numerous irrelevant news topics,~${\\irrel_1,\\ldots,\\irrel_n}$ where~$n$ is huge. This user could label a few relevant articles she manually found as positive~($\\pCls$). She could also label any irrelevant articles she encountered negative. Such an ad-hoc labeling mechanism generally induces \\textit{covariate shift} (i.e.,~train and test marginal distributions) for the negative class. For instance, the user may fail to label documents from each negative topic~$\\irrel_i$. Similarly, the training set's negative topic prior probabilities will not match the corresponding true priors. Constructing a classifier given negative covariate shift is know as \\textit{positive, unlabeled, biased negative} (PUbN)~\\textit{learning}.\n\n There are three common approaches to PUbN learning:\n\n \\begin{enumerate}\n \\item Supervised (PN -- Positive-Negative) Learning: Ignores the bias in the negative training examples. Serves as a baseline of comparison and quantifies the effect of any correction.\n\n \\item Positive\\-/Unlabeled (PU) Learning: Premised on the idea that using the biased negative training examples can be inherently deleterious. It constructs a binary classifier strictly using the positive examples and the residual unlabeled corpus. An example where this approach was used is~\\cite{Li:2010}.\n\n \\item Positive, Unlabeled, biased Negative (PUbN) Learning: Constructs a classifier using some \\textit{a~priori} knowledge of the negative training set's bias. Similar to~\\cite{Hsieh:2019}, this project assumes knowledge of the negative training sets prior probabilities.\n \\end{enumerate}\n\n \\paragraph{Goal} This project will explore negative covariate shift's effect on news topic classification. The experiments in this section are inspired by Fei \\&~Liu~\\cite{Fei:2015} and Hsieh\\etal~\\cite{Hsieh:2019}.\n\n \\paragraph{Experimental Setup} This project will use the 20~newsgroups dataset~\\cite{20Newsgroups}. The dataset statistics are shown in Table~\\ref{tab:DatasetComparison}. We will use the latest (cleaned) version of the dataset published by Jason Rennie where duplicate and empty documents were removed.\n\n \\begin{table}[b]\n \\centering\n \\caption{Dataset statistics for 20~Newsgroups}\\label{tab:DatasetComparison}\n \\begin{tabular}{|c|c|c|}\n \\hline\n \\#~Classes & 20 \\\\\\hline\n Train Size & 13,180 \\\\\\hline\n Test Size & 5,648 \\\\\\hline\n Test Set Prior & Nearly uniform \\\\\\hline\n \\end{tabular}\n \\end{table}\n\n\n When constructing the experimental datasets, one topic will be selected as the positive class and all remaining~(19) topics will form joint form the negative class. The experiments will be divided into three categories:\n\n \\begin{itemize}\n \\item \\textit{In-Training}: All negative subclasses (${\\irrel_1,\\ldots,\\irrel_n}$) are present in both the training and test sets. The only difference is their train/test marginal distributions.\n\n \\item \\textit{Out-of-Training}: Some negative subclasses are only present in the test set. Beside that difference, the train and test marginal distributions are identical.\n\n \\item \\textit{Combined}: Synthesis of in-training and out-of-training. Bias includes missing training set subclasses and a marginal distribution shifts Most challenging case.\n \\end{itemize}\n\n \\noindent\n This procedure will be repeated with different topics as the positive class to ensure as representative of an experiment set as possible.\n\n \\paragraph{Implementation Plan} The PN, PU, and PUbN implementations will share the same standard neural network architecture implemented in PyTorch. There will be an embedding of some kind (at minimum a word embedding but potentially also character as well), a bidirectional (possibly stacked) RNN, and an output feed forward.\n\n Where the three learning classes differ is the risk estimator used in gradient descent training. PN~will use standard logistic loss. PU~will use the nnPU risk estimator by Kiryo\\etal\\ in~\\cite{Kiryo:2017}. PUbN will use~\\cite{Hsieh:2019}'s risk estimator.\n\n \\bibliographystyle{ieeetr}\n \\bibliography{bib/ref.bib}\n\\end{document}\n", "meta": {"hexsha": "c0f835d9c965996d36c65c44bb2e54f8018496b2", "size": 5997, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "project/proposal/proposal.tex", "max_stars_repo_name": "ZaydH/cis510_nlp", "max_stars_repo_head_hexsha": "e1e039ca9f228051f6a3682b3ee71665e4d693d0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "project/proposal/proposal.tex", "max_issues_repo_name": "ZaydH/cis510_nlp", "max_issues_repo_head_hexsha": "e1e039ca9f228051f6a3682b3ee71665e4d693d0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "project/proposal/proposal.tex", "max_forks_repo_name": "ZaydH/cis510_nlp", "max_forks_repo_head_hexsha": "e1e039ca9f228051f6a3682b3ee71665e4d693d0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2020-03-25T07:33:10.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-01T14:21:20.000Z", "avg_line_length": 74.037037037, "max_line_length": 943, "alphanum_fraction": 0.7470401868, "num_tokens": 1567, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.34839763043840377}} {"text": "\\mainsection{Advanced Protocols}\nThis section details the protocols needed to perform complex\nfunctionalities on LSSS style secret shared values in MAMBA.\nAs well as the documentation here further, details can be\nfound at\n\\begin{center}\n \\verb+$(HOME)/Documentation/Compiler_Documentation/index.html+\n\\end{center}\nunder the heading \\verb+Files+.\n\nThe protocols break the ``arithmetic circuit'' model of computation,\nas they make heavy use of pre-processed data and the ability\nto $\\Open$ shared values.\nIn particular we assume three lists of pre-processed data:\n\\[ \\MList, \\quad \\SList, \\quad \\BList. \\]\nAn entry on the $\\MList$ is of the form\n$(\\sshare{a}, \\sshare{b}, \\sshare{c})$ where $c=a \\cdot b \\pmod{p}$,\nan entry on the $\\SList$ is of the form\n$(\\sshare{a}, \\sshare{b})$ where $b=a^2 \\pmod{p}$,\nwhilst an entry on the $\\BList$ is of the form\n$\\sshare{b}$ where $b \\in \\{0,1\\}$.\nWe also assume a function $\\Random()$ which can generate\na random secret sharing (this can be implemented by just taking\nthe first element from a multiplication triple).\nWe add and multiply secret shared elements in what follows\nusing the notation\n\\[ \\sshare{a+b} \\asn \\sshare{a}+\\sshare{b}, \\quad\n \\sshare{a\\cdot b} \\asn \\sshare{a} \\cdot \\sshare{b}.\n\\]\n\nThe protocols in this chapter allow us to do more advanced operations,\nwithout resorting to using full blown arithmetic circuits.\nWe describe here what we have implemented as a form of\ndocumentation, both for us and for others.\nMany of the protocol ideas can be found in the five\ndocuments\n\\begin{itemize}\n \\item Unconditionally Secure Constant-Rounds\n Multi-party Computation for Equality, Comparison, Bits and Exponentiation.\n {\\em TCC 2006}, \\cite{DFKNT06}.\n \\item Improved Primitives for Secure Multiparty Integer Computation.\n {\\em SCN 2010}, \\cite{CH10}.\n \\item D9.2 of the EU project {\\em secureSCM}, \\cite{SSCM}.\n \\item Secure Computation with Fixed-Point Numbers {\\em FC 2010} \\cite{CS10}.\n \\item Secure Computation on Floating Point Numbers {\\em NDSS 2013} \\cite{ABZS13}.\n\\end{itemize}\nMany of the protocols operating on integers make use of a statistical security\nparameter $\\kappa$.\nIf the integer being operated on is $k$ bits long, then we often require $(k+\\kappa)<\\log_2 p$.\nFor ease of implementation of the protocols we recommend $k$ is\nalways a power of two, and we assume this in the write up below.\nIf this is not the case, obvious tweaks can be made to the protocols.\n\nDue to experience with the SPDZ system we prefer logarithmic\nround protocols over constant round protocols, it appears that in\npractice the logarithmic round protocols outperform the constant\nround ones.\nThe MAMBA compiler can execute non-constant round protocols, by\nflicking a compile time switch. But here we only document\nlogarithmic round protocols.\n\n\\msubsection{Basic Protocols}\n\n\\msubsubsection{$\\mathsf{Inv}(\\sshare{x})$:}\nThis produces a share $\\sshare{z}$ of $1/x \\pmod{p}$, with an\n$\\abort$ if $x=0$.\n\\begin{enumerate}\n \\item $\\sshare{a} \\asn \\Random()$.\n \\item $\\sshare{y} \\asn \\sshare{a} \\cdot \\sshare{x}$.\n \\item $y \\asn \\Open(\\sshare{y})$.\n \\item If $y=0$ then $\\abort$.\n \\item $t \\asn 1/y \\pmod{p}$.\n \\item $\\sshare{z} \\asn t \\cdot \\sshare{a}$.\n \\item Return $\\sshare{z}$.\n\\end{enumerate}\n\\paragraph{MAMBA Example:} To obtain the inverse of a \\verb|sint| or a \\verb|cint| can be done as follows:\n\\begin{lstlisting}[language={python}]\nd = sint(5)\nd_inv = AdvInteger.Inv(d)\nprint_ln(\"inverse is correct if 1: %s\", (d*d_inv).reveal())\n\\end{lstlisting}\n\n\\msubsubsection{$\\mathsf{Ran}_p^*()$:}\nThis produces a random sharing of a value $x$ and its inverse $1/x$.\nThis is faster than generating $x$, and then performing the above operaton.\n\\begin{enumerate}\n \\item Take a triple $(\\sshare{a},\\sshare{b},\\sshare{c})$ from $\\MList$.\n \\item $c \\asn \\Open(\\sshare{c})$.\n \\item If $c=0$ then return to the first step.\n \\item $\\sshare{a^{-1}} \\asn c^{-1} \\cdot \\sshare{b}$.\n \\item Output $(\\sshare{a},\\sshare{a^{-1}})$.\n\\end{enumerate}\nThis function does not exist ``as is'' in the MAMBA language, as it\nis used in place within the python compiler. Thus it is only here\nfor documentation reasons.\n\n\\msubsubsection{$\\mathsf{PreMult}(\\sshare{a_1},\\ldots,\\sshare{a_t},T)$:}\nThis computes the prefix multiplication, i.e. the values\n\\[ \\sshare{a_{i_0,i_1}} = \\bsshare{\\prod_{i=i_0}^{i_1} a_i} \\]\nwhere $(i_0,i_1) \\in T$ and $1 \\le i_0 \\le i_1 \\le t$.\n\\begin{enumerate}\n \\item For $i \\in [0,\\ldots,t]$ do.\n \\begin{enumerate}\n \\item $(\\sshare{b_i},\\sshare{b_i^{-1}}) \\asn \\mathsf{Ran}_p^*()$.\n \\end{enumerate}\n \\item For $i \\in [0,\\ldots,t]$ do.\n \\begin{enumerate}\n \\item $\\sshare{t} \\asn \\sshare{b_{i-1}} \\cdot \\sshare{a_i}$.\n \\item $\\sshare{d_i} \\asn \\sshare{t} \\cdot \\sshare{b_i^{-1}}$.\n \\item $d_i \\asn \\Open(\\sshare{d_i})$.\n \\end{enumerate}\n \\item For $(i_0,i_1) \\in T$\n \\begin{enumerate}\n \\item $d_{i_0,i_1} \\asn \\prod_{i=i_0}^{i_1} d_i$.\n \\item $\\sshare{a_{i_0,i_1}} \\asn d_{i_0,i_1} \\cdot\n \\sshare{b_{i_0-1}^{-1}} \\cdot \\sshare{b_{i_1}}$.\n \\end{enumerate}\n\\end{enumerate}\nAgain, this function does not exist ``as is'' in the MAMBA language, as it\nis used in place within the python compiler. Thus it is only here\nfor documentation reasons.\n\n\n\n\\msubsection{Bit Oriented Operations}\nThese operations refer exclusively to how to perform bit oriented operations on top of\n$\\modp$ data-types. It is not related to the \\verb|sbit| type.\n\n\\msubsubsection{$\\mathsf{OR}(\\sshare{a},\\sshare{b})$:}\nThis computes the logical OR of two input shared bits:\n\\begin{enumerate}\n \\item Return $\\sshare{a}+\\sshare{b}-\\sshare{a} \\cdot \\sshare{b}$.\n\\end{enumerate}\n\\paragraph{MAMBA Example:} To obtain the \\verb|or| of two \\verb|sint| or \\verb|cint| numbers you can:\n\\begin{lstlisting}[language={python}]\na= sint(1)\nb= sint(0)\nprint_ln(\"or is correct if 1: %s\", (a or b).reveal())\nprint_ln(\"or is correct if 1: %s\", AdvInteger.or_op(a, b).reveal())\n\\end{lstlisting}\n\n\\msubsubsection{$\\mathsf{XOR}(\\sshare{a},\\sshare{b})$:}\nThis computes the logical XOR of two input shared bits:\n\\begin{enumerate}\n \\item Return $\\sshare{a}+\\sshare{b}-2 \\cdot \\sshare{a} \\cdot \\sshare{b}$.\n\\end{enumerate}\n\\paragraph{MAMBA Example:} To obtain the \\verb|xor| of two \\verb|sint| or \\verb|cint| numbers you can:\n\\begin{lstlisting}[language={python}]\na= sint(1)\nb= sint(0)\nprint_ln(\"or is correct if 1: %s\", AdvInteger.xor_op(a, b).reveal())\n\\end{lstlisting}\n\n\\iffalse\n \\note{Nigel}{Does not seem to be used anywhere}\n \\msubsubsection{$\\mathsf{Symm}(f,\\sshare{a_1},\\ldots,\\sshare{a_t})$:}\n This takes a symmetric boolean function $f$ on $t$ binary inputs\n and evaluates it at the points $\\sshare{a_1},\\ldots,\\sshare{a_t}$,\n where we assume $a_i \\in \\{0,1\\}$.\n We first pre-process the function $f$ so that we can write\n \\[ f(x_1,\\ldots,x_t) = \\phi(1+\\sum_{i=1}^t x_i), \\]\n where the inner sum is over the integers and\n $\\phi:{1,2,\\ldots,t+1} \\longrightarrow \\{0,1\\}$.\n We then write\n \\[ \\phi(X) = \\sum_{i=0}^t \\alpha_i \\cdot X^i \\pmod{p} \\]\n using Lagrange interpolation.\n So for example if we have $f(X_1,X_2)=X_1^2+X_1 \\cdot X_2+X_2^2+1$\n then $\\phi$ is the function which maps\n $1 \\longrightarrow 1$, $2 \\longrightarrow 0$, $3 \\longrightarrow 0$.\n In which case $\\phi(X)=X^2/2-5 \\cdot X/2+3$.\n\n To compute this function we have the algorithm\n \\begin{enumerate}\n \\item $\\sshare{a} \\asn 1+ \\sum_{i=1}^t \\sshare{a_i}$.\n \\item $(\\sshare{a},\\sshare{a^2},\\ldots,\\sshare{a^t})\n \\asn \\mathsf{PreMult}(\\sshare{a},\\ldots,\\sshare{a},\n \\{(1,1),\\ldots,(1,t)\\})$.\n \\item $\\sshare{f(a_1,\\ldots,a_t)}\n \\asn \\sum_{i=0}^t \\alpha_i \\cdot \\sshare{a^i}$.\n \\end{enumerate}\n\\fi\n\n\\msubsubsection{$\\mathsf{KOp}(\\odot,\\sshare{a_1},\\ldots,\\sshare{a_k},k)$:}\nThis computes the operation $\\sshare{p}= \\bigodot_{i=1}^k \\sshare{a_i}$\ngiven a binary operator $\\odot$.\n\\begin{enumerate}\n \\item If $k>1$ then (assuming $k$ is even, if not pad)\n \\begin{enumerate}\n \\item $\\sshare{t_1} \\asn \\mathsf{KOp}(\\odot, \\sshare{a_{k/2}},\\ldots,\\sshare{a_{k/2}},k/2)$.\n \\item $\\sshare{t_2} \\asn \\mathsf{KOp}(\\odot, \\sshare{a_{k/2+1}},\\ldots,\\sshare{a_k},k/2)$.\n \\item Return $\\sshare{t_1} \\odot \\sshare{t_2}$.\n \\end{enumerate}\n \\item Else\n \\begin{enumerate}\n \\item $\\sshare{p} \\asn \\sshare{a_1}$.\n \\end{enumerate}\n \\item Return $\\sshare{p}$.\n\\end{enumerate}\n\n\\paragraph{MAMBA Example:} Note that we basically want to achieve a construction capable to call any function in an iterative fashion reducing computation time. In this sense a call to the function could be perform in the following way:\n\\begin{lstlisting}[language={python}]\n# addition\ndef addition(a, b):\n return a + b\n\nar=[1]*16\n# (k is exctracted from ar directly on the implementation)\nprint_ln(\"KOpL is correct if 32: %s\", (AdvInteger.KOpL(addition,ar)).reveal())\n\\end{lstlisting}\nThis runs in logarithmic rounds.\n\n\\msubsubsection{$\\mathsf{PreOp}(\\odot,\\sshare{a_1},\\ldots,\\sshare{a_k},k)$:}\nThis computes the prefix operator $\\sshare{p_j} = \\odot_{i=1}^j \\sshare{a_i}$,\nfor $1 \\le j \\le k$.\n\\begin{enumerate}\n \\item For $i \\in [1,\\ldots, \\log_2 k]$ do\n \\begin{enumerate}\n \\item For $j \\in [1,\\ldots,k/2^i]$ do\n \\begin{enumerate}\n \\item $y \\asn 2^{i-1}+j\\cdot 2^i$.\n \\item For $z \\in [1,\\ldots,2^{i-1}]$ do\n \\begin{enumerate}\n \\item $\\sshare{a_{y+z}} \\asn \\sshare{a_y} \\odot \\sshare{a_{y+z}}$.\n \\end{enumerate}\n \\end{enumerate}\n \\end{enumerate}\n \\item Return $(\\sshare{a_1},\\ldots,\\sshare{a_k})$.\n\\end{enumerate}\n\n\\paragraph{MAMBA Example:} Similarly, we basically want to achieve a construction capable to call any function in an iterative fashion reducing computation time. In this case, however, we return all the list of intermediate values. The function call could be performed in the following way:\n\\begin{lstlisting}[language={python}]\ndef addition(a, b):\n return a + b\n\ndef addition_triple(a, b, c):\n\t# c is a boolean parameter\n return a + b\n\ne = sint(2)\n\nar = [e]*16\n# k is stracted from ar.\nprint_ln(\"PreOpL is correct if 32: %s\", (AdvInteger.PreOpL(addition_triple,ar))[15].reveal())\nprint_ln(\"PreOpN is correct if 32: %s\", (AdvInteger.PreOpN(addition,ar))[15].reveal())\n\\end{lstlisting}\nNote that both methods are implementations of the functionality with slightly different communication and round complexity.\nWith the \\verb+PreOpL+ function corresponding to the pseudo-code above.\n\n\\msubsubsection{$\\mathsf{Sum\\mhyphen Bits}(\\sshare{x}_B)$:}\nThis outputs a shared integer $x$ in the range\n$[0,\\ldots,2^k)$ with $2^k1$ then\n \\begin{enumerate}\n \\item For $i \\in [1,\\ldots,k/2]$ do\n \\begin{enumerate}\n \\item $\\sshare{u_i}_B \\asn \\sshare{d_{2\\cdot i}}_B \\circ \\sshare{d_{2 \\cdot i-1}}_B$.\n \\end{enumerate}\n \\item $\\sshare{d}_B \\asn \\mathsf{CarryOutAux}(\\sshare{u_{k/2}}_B,\\ldots,\\sshare{u_1}_B,k/2)$.\n \\end{enumerate}\n \\item Else\n \\begin{enumerate}\n \\item $\\sshare{d}_B \\asn \\sshare{d_1}_B$.\n \\end{enumerate}\n \\item Return $\\sshare{d}_B$.\n\\end{enumerate}\n\\paragraph{MAMBA Example:} This method is thought as a subroutine for \\verb|CarryOut| and should not be used outside that context. the code is invoked in the following way:\n\\begin{lstlisting}[language={python}]\nkappa = 16\nx = [cint(i) for i in [1,0]]\nres = sint()\nAdvInteger.CarryOutAux(res, x)\n\\end{lstlisting}\n\n\n\\msubsubsection{$\\mathsf{BitAdd}((\\sshare{a_{k-1}},\\ldots,\\sshare{a_0}), (\\sshare{b_{k-1}},\\ldots,\\sshare{b_0}),k)$:}\n This function also makes use of the operator $\\circ$.\n The inputs are shared bits. The case where one set of inputs is in the clear\n is obviously more simple, and we do not detail this here.\n \\begin{enumerate}\n \\item For $i \\in [0,\\ldots,k-1]$ do\n \\begin{enumerate}\n \\item $\\sshare{d_i}_B \\asn (\\mathsf{XOR}(\\sshare{a_i}, \\sshare{b_i}),\n \\sshare{a_i}\\cdot \\sshare{b_i})$.\n \\end{enumerate}\n \\item $\\sshare{c_{k-1},t_{k-1}},\\ldots,\\sshare{c_0,t_0} \\asn \\mathsf{PreOp}(\\circ,\\sshare{d_{k-1}}_B,\\ldots,\\sshare{d_0}_B,k)$.\n \\item $\\sshare{s_0} \\asn \\mathsf{XOR}(\\sshare{a_0}, \\sshare{a_1})$.\n \\item For $i \\in [1,\\ldots,k-1]$ do\n \\begin{enumerate}\n \\item $\\sshare{s_i} \\asn \\sshare{a_i}+\\sshare{b_i}+\\sshare{c_{i-1}}-2\\cdot \\sshare{c_i}$.\n \\end{enumerate}\n \\item $\\sshare{s_k} \\asn \\sshare{c_{k-1}}$.\n \\item Return $(\\sshare{s_k},\\ldots,\\sshare{s_0})$.\n \\end{enumerate}\n There is also a variant (requiring less operations) which does an increment\n by one, i.e. $x \\asn x+1$.\n\n \\paragraph{MAMBA Example:} The addition of two numbers expressed in bits, can be performed as follows:\n \\begin{lstlisting}[language={python}]\na_bits = [sint(i) for i in [0,1,0,1,1]]\nb_bits = [sint(i) for i in [0,1,0,1,1]]\n# k is extracted from the array size\nb = AdvInteger.BitAdd(a_bits, b_bits)\nc = AdvInteger.BitIncrement(a_bits)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{BitLTFull}(\\sshare{a}_B,\\sshare{b}_B,k)$:}\n This computes the sharing of the bit $a1$, for $m=1$ use $\\mathsf{Mod2}$ below.\n \\begin{enumerate}\n \\item $\\sshare{r'}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,m,\\kappa)$.\n \\item $c \\asn \\Open(\\sshare{a}+\\signed \\cdot 2^{k-1}+\\sshare{r}+2^m \\cdot \\sshare{r'})$.\n \\item $c'\\asn c \\pmod{2^m}$.\n \\item $\\sshare{u} \\asn \\mathsf{BitLT}(c',(\\sshare{r_{m-1}},\\ldots,\\sshare{r_0}),m)$.\n \\item $\\sshare{a'} \\asn c'-\\sshare{r}+2^m \\cdot \\sshare{u}$.\n \\item Return $\\sshare{a'}$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} The \\verb|mod| to a power of $2$ of a secret shared integer register can be obtain as follows:\n \\begin{lstlisting}[language={python}]\na_prime = sint(0) # a % 2 ^ m\na = sint(100)\nk = 16 # bit length of a\nm = 2 # the power of two, i..e we want ( a mod 2^m )\nkappa = 8\nsigned = True # True/False\n\n# where a is a register of type sint\nr_dprime, r_prime, c, c_prime, u, t, c2k1 = \\\n AdvInteger.Mod2m(a_prime, a, k, m, kappa, signed)\n\\end{lstlisting}\n\n\n \\msubsubsection{$\\mathsf{Mod2}(\\sshare{a},k, \\kappa, \\signed)$:}\n \\begin{enumerate}\n \\item $\\sshare{r'}, \\sshare{r}, \\sshare{r_0} \\asn \\mathsf{PRandM}(k,1,\\kappa)$.\n \\item $c \\asn \\Open(\\sshare{a}+\\signed \\cdot 2^{k-1}+\\sshare{r}+2 \\cdot \\sshare{r'})$.\n \\item $\\sshare{a_0} \\asn c_0+\\sshare{r_0}-2 \\cdot c_0 \\cdot \\sshare{r_0}$.\n \\item Return $\\sshare{a_0}$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} You obtain the modulo two of a number as follows:\n \\begin{lstlisting}[language={python}]\na = sint(1)\na_0 = sint()\nk = 1\nkappa = 8\nsigned = False\n# y stores the result of X % 2\nAdvInteger.Mod2(a_0, A, k, kappa, signed)\n\\end{lstlisting}\n\n \\msubsubsection{Addition, Multiplication in $\\Zk$}\n Addition and multiplication $\\odot$ of two elements $\\sshare{a}, \\sshare{b}$\n to obtain $\\sshare{c}$, where $a,b,c \\in \\Zk$ is then easy to define\n by performing\n \\begin{enumerate}\n \\item $\\sshare{d} \\asn \\sshare{a} \\odot \\sshare{b}$.\n \\item $\\sshare{c} \\asn \\mathsf{Mod2m}(\\sshare{d},k',k)$,\n where $k'=k+1$ is $\\odot=+$ and $k'=2\\cdot k$ if $\\odot=\\cdot$.\n \\item Return $\\sshare{c}$.\n \\end{enumerate}\n These functions are not directly callable from MAMBA, they are\n included here purely for documentation reasons.\n\n \\msubsubsection{$\\mathsf{Pow2}(\\sshare{a},k, \\kappa)$:}\n This computes $\\sshare{2^a}$ where $a \\in [0,\\ldots,k)$\n \\begin{enumerate}\n \\item $m \\asn \\ceil{\\log_2 k}$.\n \\item $\\sshare{a_{m-1}},\\ldots,\\sshare{a_0} \\asn \\mathsf{BitDec}(\\sshare{a},m,m)$.\n \\item For $i\\in [0,\\ldots,m-1]$ do\n \\begin{enumerate}\n \\item $\\sshare{v_i} \\asn 2^{2^i} \\cdot \\sshare{a_i}+1-\\sshare{a_i}$.\n \\end{enumerate}\n \\item $\\sshare{x_0},\\ldots,\\sshare{x_{m-1}}\n \\asn \\ \\mathsf{PreMult}(\\sshare{v_0},\\ldots,\\sshare{v_{m-1}}, \\{(1,1),\\ldots,(1,m)\\})$\n \\item Return $\\sshare{x_{m-1}}$.\n \\end{enumerate}\n \\paragraph{MAMBA Example:} You can obtain the value of two raised to a secret shared number as follows:\n \\begin{lstlisting}[language={python}]\na = sint(23)\nl = 32\nkappa = 20\n# y stores the result of 2^23\ny = AdvInteger.Pow2(a, l, kappa)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{B2U}(\\sshare{a},k, \\kappa)$:}\n This converts the integer $a \\in [0,\\ldots,k)$ into unary form.\n It outputs $k$ bits, of which the last $a$ bits are zero to one,\n with the others set to zero.\n \\begin{enumerate}\n \\item $\\sshare{2^a} \\asn \\mathsf{Pow2}(\\sshare{a},k)$.\n \\item $\\sshare{r'}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,k,\\kappa)$.\n \\item $c \\asn \\Open(\\sshare{2^a}+\\sshare{r}+2^k \\cdot \\sshare{r'})$.\n \\item Let $c_{k-1},\\ldots,c_0$ be the bits of $c$.\n \\item For $i \\in [0,\\ldots,k=1]$ do\n \\begin{enumerate}\n \\item $\\sshare{x_i} \\asn c_i+\\sshare{r_i}-2 \\cdot c_i \\cdot \\sshare{r_i}$.\n \\end{enumerate}\n \\item $\\sshare{y_{k-1}},\\ldots,\\sshare{y_0} \\asn \\mathsf{PreOp}(\\mathsf{OR},\\sshare{x_{k-1}},\\ldots,\\sshare{x_0},k)$.\n \\item For $i \\in [0,\\ldots,k=1]$ do\n \\begin{enumerate}\n \\item $\\sshare{a_i} \\asn 1-\\sshare{y_i}$.\n \\end{enumerate}\n \\item Return $\\sshare{a_0},\\ldots,\\sshare{a_{k-1}}$.\n \\end{enumerate}\n Note, in the function $\\mathsf{Oblivious\\_Trunc}$ below we also require the\n value $\\sshare{2^a}$ to be returned so as to avoid recomputing it.\n\n \\paragraph{MAMBA Example:} You can transform a number into its unary form as follows:\n \\begin{lstlisting}[language={python}]\na = sint(3)\nl=5\nkappa=20\nb,c = AdvInteger.B2U(a, l, kappa)\n\\end{lstlisting}\n\n\n \\msubsubsection{$\\mathsf{TruncPr}(\\sshare{a},k,m, \\kappa)$:}\n An approximate truncation algorithm which is faster than\n a fully accurate Trunc.\n Given $a \\in \\Zk$, $m \\in [1,\\ldots,k-1]$\n this outputs $\\floor{a/2^m}+u$ where $u$ is a random\n (and unknown) bit.\n It gives the actual correct nearest integer with\n probability $1-\\alpha$, where $\\alpha$ is the distance\n between $a/2^m$ and that integer.\n \\begin{enumerate}\n \\item $\\sshare{r'}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,m,\\kappa)$.\n \\item $c \\asn \\Open(\\sshare{a}+2^{k-1}+\\sshare{r}+2^m \\cdot \\sshare{r'})$.\n \\item $c'\\asn c \\pmod{2^m}$.\n \\item $t \\asn 1/2^m \\pmod{p}$.\n \\item $\\sshare{d} \\asn t \\cdot (\\sshare{a}-c'+\\sshare{r})$.\n \\item Return $\\sshare{d}$.\n \\end{enumerate}\n \\paragraph{MAMBA Example:} A secret shared fractional register can be approximately truncated as follows:\n \\begin{lstlisting}[language={python}]\na = sint(23)\nk = 5\nm = 3\nkappa = 20\n# where b is a register of type sint\nb=AdvInteger.TruncPr(a, k, m, kappa)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{Trunc}(\\sshare{a},k,m, \\kappa, \\signed)$:}\n An exact version of $\\mathsf{Trunc}$ above, it computes the value\n$(a-a \\mod{2^m})/2^m$, where the $\\mod$ operator here produces a positive value.\n \\begin{enumerate}\n \\item $\\sshare{a'} \\asn \\mathsf{Mod2m}(\\sshare{a},k,m, \\kappa, \\signed)$.\n \\item $t \\asn 1/2^m \\pmod{p}$.\n \\item $\\sshare{d} \\asn t \\cdot (\\sshare{a}-\\sshare{a'})$.\n \\item Return $\\sshare{d}$.\n \\end{enumerate}\n Below we will give a version of $\\mathsf{Trunc}$ in which $m$ is kept secret shared.\n\n \\paragraph{MAMBA Example:} You truncate a number as follows:\n \\begin{lstlisting}[language={python}]\n# a = sint(23)\n# k = 5\n# m = 3\n# kappa = 20\n# where a is a register of type sint\na= AdvInteger.Trunc(a, k, m, kappa)\n\\end{lstlisting}\n There is also an exact version of $\\mathsf{Trunc}$ called $\\mathsf{TruncRoundNearest}$\n which rounds the value $a/2^m$ to the nearest integer.\n This is called as\n \\begin{lstlisting}[language={python}]\na= AdvInteger.TruncRoundNearest(a, k, m, kappa)\n\\end{lstlisting}\n\n\n \\msubsubsection{$\\mathsf{Oblivious\\_Trunc}(\\sshare{a},k,\\sshare{m}, \\kappa)$:}\n This does the same operation as $\\mathsf{Trunc}$ above, but $m$ is now secret shared, but guaranteed to be in the range $[0,\\ldots,k)$.\n \\begin{enumerate}\n \\item $\\sshare{x_0},\\ldots,\\sshare{x_{k-1}}, \\sshare{2^m} \\asn \\mathsf{B2U}(\\sshare{m},k)$.\n \\item $\\sshare{2^{-m}} \\asn \\mathsf{Inv}(\\sshare{2^m})$.\n \\item $\\sshare{r''}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,k,\\kappa)$.\n \\item $\\sshare{r'} \\asn \\sum_{i=0}^{k-1} 2^i \\cdot \\sshare{x_i} \\cdot \\sshare{r_i}$\n \\item $c \\asn \\Open(\\sshare{a}+\\sshare{r''} \\cdot 2^{K} +\\sshare{r})$.\n \\item For $i \\in [1,\\ldots,k=1]$ do $c_i' \\asn c \\pmod{2^i}$.\n \\item $\\sshare{c''} \\asn \\sum_{i=1}^{k-1} c_i' \\cdot (\\sshare{x_{i-1}}-\\sshare{x_i})$.\n \\item $\\sshare{d} \\asn \\mathsf{LT}(\\sshare{c''},\\sshare{r'},k)$.\n \\item $\\sshare{b} \\asn (\\sshare{a}-\\sshare{c''}+\\sshare{r'}) \\cdot \\sshare{2^{-m}} - \\sshare{d}$.\n \\item Return $\\sshare{b}$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} You truncate a number as follows:\n \\begin{lstlisting}[language={python}]\na = sint(23)\nk = 5\nm = sint(3)\nkappa = 20\n# where a is a register of type sint\nb= AdvInteger.Oblivious_Trunc(a, k, m, kappa)\n\\end{lstlisting}\n\n \\iffalse\n \\note{Nigel}{Does not seem to be used anywhere}\n \\msubsubsection{$\\mathsf{Mod}(\\sshare{a},k,x)$:}\n Given $a \\in \\Zk$ and public $x \\in [1,\\ldots,2^{k-1}-1]$\n this computes $\\sshare{a \\pmod{x}}$.\n \\begin{enumerate}\n \\item $m \\asn \\ceil{\\log_2 x }$.\n \\item $\\sshare{r'}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,m,\\kappa)$.\n \\item $c \\asn \\Open(\\sshare{a}+2^{k-1}+\\sshare{r}+x \\cdot \\sshare{r'})$.\n \\item $c' \\asn c \\pmod{x}$.\n \\item $\\sshare{v} \\asn 1 -\\mathsf{BitLT}(\\sshare{r}_B,x)$.\n \\item $\\sshare{u} \\asn \\mathsf{LTZ}(c' - \\sshare{r} + x \\cdot \\sshare{v},m)$.\n \\item $\\sshare{a'} \\asn c' - \\sshare{r} + x \\cdot ( \\sshare{v}+\\sshare{u})$.\n \\item Return $\\sshare{a'}$.\n \\end{enumerate}\n \\fi\n\n \\msubsubsection{$\\mathsf{LTZ}(\\sshare{a},k, \\kappa)$:}\n Given $a \\in \\Zk$ this tests whether $a<0$ or not,\n resulting in a shared bit.\n \\begin{enumerate}\n \\item $\\sshare{s} \\asn - \\mathsf{Trunc}(\\sshare{a},k,k-1)$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} You determine whether a number is less than zero as follows:\n \\begin{lstlisting}[language={python}]\na = sint(1)\nb = sint()\nk=80\nkappa=40\n# b stores the result of x < 0\nAdvInteger.LTZ(b, a, k, kappa)\n\\end{lstlisting}\n Like many commands in this section this can be abbreviated to\n \\begin{lstlisting}[language={python}]\nb=a<0\n\\end{lstlisting}\n In which case the default value of $\\kappa=40$ is chosen\n (when using a $128$-bit prime modulus), this default value can be\n altered by using the command\n \\begin{lstlisting}[language={python}]\nprogram.security = 100\n\\end{lstlisting}\n The default value of $k=64$ is used in this setting, and\n this can be altered by executing\n \\begin{lstlisting}[language={python}]\nprogram.bit_length = 40\n\\end{lstlisting}\n The requirement is that $k+\\kappa$ must be less than the bit length\n of the prime $p$.\n\n\n \\msubsubsection{$\\mathsf{EQZ}(\\sshare{a},k, \\kappa)$:}\n Given $a \\in \\Zk$ this tests whether $a=0$ or not,\n resulting in a shared bit.\n \\begin{enumerate}\n \\item $\\sshare{r'}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,k,\\kappa)$.\n \\item $c \\asn \\Open(\\sshare{a}+2^{k-1}+2^k \\cdot \\sshare{r'}+\\sshare{r})$.\n \\item Let $c_{k-1},\\ldots,c_0$ be the bits of $c$.\n \\item For $i \\in [0,\\ldots,k-1]$ do\n \\begin{enumerate}\n \\item $\\sshare{d_i} \\asn c_i+\\sshare{r_i}-2 \\cdot c_i \\cdot \\sshare{r_i}$.\n \\end{enumerate}\n \\item $\\sshare{z} \\asn 1- \\mathsf{KOp}(\\mathsf{OR},\\sshare{d_{k-1}},\\ldots,\\sshare{d_0},k)$.\n \\item Return $\\sshare{z}$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} You determine whether a number is equal to zero as follows:\n \\begin{lstlisting}[language={python}]\na = sint(1)\nb = sint()\nk = 80\nkappa = 40\n# b stores the result of x == 0\nAdvInteger.EQZ(b, a, k, kappa)\n\\end{lstlisting}\n\n\n \\msubsubsection{Comparison Operators:}\n We can now define the basic comparison operators on shared\n representations from $\\Zk$.\n \\begin{center}\n \\begin{tabular}{|lll|}\n \\hline\n Operator & Protocol Name & Construction \\\\\n \\hline\n $a>0$ & $\\mathsf{GTZ(\\sshare{a})}$ & $\\mathsf{LTZ}(-\\sshare{a})$ \\\\\n $a\\le0$ & $\\mathsf{LEZ(\\sshare{a})}$ & $1-\\mathsf{LTZ}(-\\sshare{a})$ \\\\\n $a\\ge0$ & $\\mathsf{GEZ(\\sshare{a})}$ & $1-\\mathsf{LTZ}(\\sshare{a})$ \\\\\n $a=b$ & $\\mathsf{EQ(\\sshare{a},\\sshare{b})}$ & $\\mathsf{EQZ}(\\sshare{a}-\\sshare{b})$ \\\\\n $ab$ & $\\mathsf{GT(\\sshare{a,\\sshare{b}})}$ & $\\mathsf{LTZ}(\\sshare{b}-\\sshare{a})$ \\\\\n $a\\le b$ & $\\mathsf{LE(\\sshare{a,\\sshare{b}})}$ & $1-\\mathsf{LTZ}(\\sshare{b}-\\sshare{a})$ \\\\\n $a\\ge b$ & $\\mathsf{GE(\\sshare{a,\\sshare{b}})}$ & $1-\\mathsf{LTZ}(\\sshare{a}-\\sshare{b})$ \\\\\n \\hline\n \\end{tabular}\n \\end{center}\n\n\n \\iffalse\n ### NPS: This does not seem to be defined, or used\n \\msubsubsection{$\\mathsf{Mod2m}(a_{prime}, \\sshare{a},k,\\sshare{m}, \\kappa)$:}\n This does the same operation as $\\mathsf{Mod2m}$ above, but $m$ is now secret shared.\n \\begin{enumerate}\n \\item $\\sshare{x_0},\\ldots,\\sshare{x_{k-1}}, \\sshare{2^m} \\asn \\mathsf{B2U}(\\sshare{m},k)$.\n \\item $\\sshare{2^{-m}} \\asn \\mathsf{Inv}(\\sshare{2^m})$.\n \\item $\\sshare{r''}, \\sshare{r}, \\sshare{r}_B \\asn \\mathsf{PRandM}(k,k,\\kappa)$.\n \\item $\\sshare{r'} \\asn \\sum_{i=0}^{k-1} 2^i \\cdot \\sshare{x_i} \\cdot \\sshare{r_i}$\n \\item $c \\asn \\Open(\\sshare{a}+\\sshare{r''}+\\sshare{r})$.\n \\item For $i \\in [1,\\ldots,k=1]$ do $c_i' \\asn c \\pmod{2^i}$.\n \\item $\\sshare{c''} \\asn \\sum_{i=1}^{k-1} c_i' \\cdot (\\sshare{x_{i-1}}-\\sshare{x_i})$.\n \\item $\\sshare{d} \\asn \\mathsf{LT}(\\sshare{c''},\\sshare{r'},k)$.\n \\item $\\sshare{b} \\asn \\sshare{c''}-\\sshare{r'}+\\sshare{2^m} \\cdot \\sshare{d}$.\n \\item Return $\\sshare{b}$.\n \\end{enumerate}\n \\fi\n\n \\paragraph{MAMBA Example:} The \\verb|mod| to a secret shared power of $2$ of a secret shared integer register can be obtain as follows:\n \\begin{lstlisting}[language={python}]\na_prime = param_a_prime # a % 2^m\na sint(2137)\nk = 16\nm = sint(2)\nkappa = 8\nsigned = True # True/False, describes a\n\n# where sb is a register of type sint\nr_dprime, r_prime, c, c_prime, u, t, c2k1 = \\\n comparison.Mod2m(a_prime, a, k, m, kappa, signed)\n\\end{lstlisting}\n\n\n \\msubsection{Arithmetic with Fixed Point Numbers}\n In this section we define basic arithmetic on fixed\n point numbers.\n We mainly follow the algorithms given in\n \\begin{itemize}\n \\item Secure Computation with Fixed-Point Numbers {\\em FC 2010} \\cite{CS10}.\n \\end{itemize}\n We define $\\Qk{f}$ as the set of rational numbers\n$\\{x \\in \\Q: x = \\overline{x} \\cdot 2^{-f}, \\overline{x} \\in \\Zk\\}$.\n We represent $x \\in \\Q$ as the integer $x \\cdot 2^f = \\overline{x} \\in \\Zk$,\n which is then represented in $\\F_p$ via the mapping used above.\n Thus $x \\in \\Q$ is in the range $[-2^e,2^e-2^{-f}]$\n where $e=k-f$.\n As we are working with fixed point numbers we assume that the\n parameters $f$ and $k$ are public.\n For our following algorithms to work (in particular fixed point\n multiplication and division) we require that\n$f\\ell$.\n \\begin{enumerate}\n \\item $\\lambda \\asn \\gamma-1$.\n \\item $\\sshare{s} \\asn \\mathsf{LTZ}(\\sshare{a},\\gamma)$.\n \\item $\\sshare{z} \\asn \\mathsf{EQZ}(\\sshare{a},\\gamma)$.\n \\item $\\sshare{a} \\asn (1- 2 \\cdot \\sshare{s}) \\cdot \\sshare{a}$.\n \\item $\\sshare{a_{\\lambda-1}},\\ldots,\\sshare{a_0}\n \\asn \\mathsf{BitDec}(\\sshare{a},\\lambda,\\lambda)$.\n \\item $\\sshare{b_0},\\ldots,\\sshare{b_{\\lambda-1}} \\asn\n \\mathsf{PreOp}(\\mathsf{OR},\\sshare{a_{\\lambda-1}},\\ldots,\\sshare{a_0},\\gamma)$.\n \\item $\\sshare{v} \\asn \\sshare{a} \\cdot\n ( 1+\\sum_{i=0}^{\\lambda-1} 2^i \\cdot (1-\\sshare{b_i}))$.\n \\item $\\sshare{p} \\asn - (\\lambda-\\sum_{i=0}^{\\lambda-1} \\sshare{b_i})$.\n \\item If $(\\gamma-1)>\\ell$ then\n \\begin{enumerate}\n \\item $\\sshare{v} \\asn \\mathsf{Trunc}(\\sshare{v},\\gamma-1,\\gamma-\\ell-1)$.\n \\end{enumerate}\n \\item Else\n \\begin{enumerate}\n \\item $\\sshare{v} \\asn 2^{\\ell-\\gamma+1} \\cdot \\sshare{v}$.\n \\end{enumerate}\n \\item $\\sshare{p} \\asn (\\sshare{p}+\\gamma-1-\\ell) \\cdot (1-\\sshare{z})$.\n \\item $\\sshare{\\err} \\asn 0$.\n \\item Return $(\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err})$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} To cast an \\verb|int| or \\verb|sint| register into a \\verb|sfloat| one, you could execute the following:\n \\begin{lstlisting}[language={python}]\nx = sfloat(5.5)\n# gamma and l are extracted from the system\ny = sfloat(x)\n\\end{lstlisting}\n\n\n \\msubsubsection{$\\mathsf{Fx2Int}(\\sshare{a},k,f)$:}\n Given a value $a \\in \\Qk{f}$ this gives the integer\n$\\floor{\\overline{a}/2^f}$.\n \\begin{enumerate}\n \\item Return $\\mathsf{Trunc}(\\sshare{a},k,f)$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} To extract the integral component of a \\verb|sfix| register, which is then encapsulated on a \\verb|sint| register,\n and taking into account what is currently implemented, you could execute the following:\n \\begin{lstlisting}[language={python}]\nx = sifx(5.5)\n# y stores 5 in a sint register\ny = AdvInteger.Trunc(x.v, x.k, x.f, x.kappa)\n\\end{lstlisting}\n \\msubsubsection{$\\mathsf{FxFloor}(\\sshare{a},k,f)$:}\n Given a value $a \\in \\Qk{f}$ this does the same, but\n gives the result as a fixed point value.\n \\begin{enumerate}\n \\item Return $2^f \\cdot \\mathsf{Trunc}(\\sshare{a},k,f)$.\n \\end{enumerate}\n \\paragraph{MAMBA Example:} To floor an \\verb|sfix| register, you could execute the following:\n \\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx = sifx(5.5)\n# k and f are extracted from x\n# y stores 5 in a sfix register\ny = mpc_math.floor_fx(x)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{Fx2FL}(\\sshare{g},\\gamma,f,\\ell,k)$:}\n Converts $g \\in \\Qk{f}$ into a floating point number\n \\begin{enumerate}\n \\item ($\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err})\n \\asn \\mathsf{Int2FL}(\\sshare{g},\\gamma,\\ell)$.\n \\item $\\sshare{p} \\asn (\\sshare{p}-f) \\cdot (1-\\sshare{z})$.\n \\item Return $(\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err})$.\n \\end{enumerate}\n \\paragraph{MAMBA Example:} To cast from \\verb|sfix| to \\verb|sfloat|, you could execute the following:\n \\begin{lstlisting}[language={python}]\n# stores 5.5 on a sfloat register\nx = sfloat(sfix(5.5))\n\\end{lstlisting}\n\n \\iffalse\n % this seems not to be used elsewhere\n \\msubsubsection{$\\mathsf{FL2Int}((\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err}),\\ell,k,\\gamma)$:}\n Note the two calls to $\\mathsf{Mod2m}$ below can be combined into\n one, as can the two calls to $\\mathsf{Pow2}$.\n The output here is in $\\Zgam$.\n \\begin{enumerate}\n \\item $\\sshare{v'},\\sshare{p'},\\sshare{z'},\\sshare{s'},\\sshare{\\err'}\n \\asn \\mathsf{FLRound}(\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err},\\mathsf{2})$.\n \\item $\\sshare{a} \\asn \\mathsf{LT}(\\sshare{p'},\\gamma-1,k)$.\n \\item $\\sshare{b} \\asn \\mathsf{LT}(\\gamma-\\ell-1,\\sshare{p'},k)$.\n \\item $\\sshare{c} \\asn \\mathsf{LTZ}(\\sshare{p'},k)$.\n \\item $\\sshare{b \\cdot c} \\asn \\sshare{b} \\cdot \\sshare{c}$.\n \\item $\\sshare{m} \\asn \\sshare{a}\\cdot (\\sshare{b} -\\sshare{b \\cdot c})\n \\cdot (\\gamma-1-\\sshare{p'})$.\n \\item $\\sshare{u} \\asn \\mathsf{Mod2m}(\\sshare{v'},\\ell,\\sshare{m})$.\n \\item $\\sshare{v'} \\asn (\\sshare{b} -\\sshare{b \\cdot c}) \\cdot ( \\sshare{u}-\\sshare{v'}) +\\sshare{v'}$\n \\item $\\sshare{2^{-p'}} \\asn \\mathsf{Pow2}(-\\sshare{c} \\cdot \\sshare{p'},\\ell)$.\n \\item $\\sshare{2^{p'}} \\asn \\mathsf{Inv}(\\sshare{2^{-p'}})$.\n \\item $\\sshare{v'} \\asn (\\sshare{c} \\cdot \\sshare{2^{p'}}+1-\\sshare{c})\n \\cdot \\sshare{v'}$.\n \\item $\\sshare{w} \\asn \\mathsf{Mod2m}(\\sshare{v'},\\ell,\n \\sshare{b \\cdot c}\\cdot(\\gamma-1))$.\n \\item $\\sshare{v'} \\asn \\sshare{b \\cdot c} \\cdot (\\sshare{w}-\\sshare{v'}) +\\sshare{v'}$.\n \\item $\\sshare{2^{p'}} \\asn \\mathsf{Pow2}(\\sshare{a} \\cdot (1-\\sshare{c})\n \\cdot \\sshare{p'},\\gamma-1)$.\n \\item $\\sshare{g} \\asn (1-\\sshare{z'}) \\cdot (1-2 \\cdot \\sshare{s'})\n \\cdot \\sshare{2^{p'}} \\cdot \\sshare{a}\n \\cdot \\sshare{v'}$.\n \\item Return $\\sshare{g}$.\n \\end{enumerate}\n \\paragraph{MAMBA Example:} To extract the integral component of a \\verb|sfloat| input,\n given the current implementation, you could execute the following:\n \\begin{lstlisting}[language={python}]\nfrom Compiler import floatingpoint\n# v, p, z, s, err are extracted from x and l, k, gamma are system parameters\nx = sfloat(5.5)\n# retunrs the floor of x\ny = floatingpoint.FLRound(x,0)\n\\end{lstlisting}\n \\fi\n\n \\msubsubsection{$\\mathsf{FL2Fx}((\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err}),\\ell,k,\\gamma,f)$:}\n\\begin{enumerate}\n \\item $\\sshare{b} \\asn \\mathsf{LT}(\\sshare{p},2^{k-1}-f,k)$.\n \\item $\\sshare{g} \\asn \\mathsf{FL2Int}((\\sshare{v},\\sshare{p}+f,\\sshare{z},\n \\sshare{s},\\sshare{\\err}),\\ell,k,\\gamma)$.\n \\item Return $\\sshare{g} \\cdot \\sshare{b}$.\n\\end{enumerate}\n\\paragraph{MAMBA Example:} To cast an \\verb|sfloat| register into a \\verb|sfix| one, you could execute the following:\n\\begin{lstlisting}[language={python}]\n# stores 5.5 on a sfix register\n# v, p, z, s, err are extracted from x and l, k, gamma are system parameters\nx = sfix(sfloat(5.5))\n\\end{lstlisting}\n\n\\msubsection{SQRT Functions}\n\\todo{These functions are only currently supported in their fixed point versions.}\nThe original description of the protocols for the fixed point square root algorithm are included in:\n\\begin{itemize}\n \\item Secure Distributed Computation of the Square Root and Applications, {\\em ISPEC 2012}\n \\cite{Liedel12}.\n\\end{itemize}\nThe floating point variant is in\n\\begin{itemize}\n \\item Secure Computation on Floating Point Numbers {\\em NDSS 2013} \\cite{ABZS13}.\\todo{This is not currently implemented}\n\\end{itemize}\nAdditionally, we provide a simplified implementation for the Square Root on fixed point variables, that is appropriate for inputs of any size. We make use Liedel's protocol, when the input's size makes it possible, as we explain later in this section.\nBoth algorithms makes use of two constants\n\\[ \\alpha = -0.8099868542, \\quad \\beta = 1.787727479. \\]\nThese are the solutions of the system of equations\n\\begin{align*}\n E(x) & = \\frac{\\alpha \\cdot x+\\beta-\\frac{1}{\\sqrt{x}}}{\\frac{1}{\\sqrt{x}}}, \\\\\n M & = \\frac{\\sqrt{3}}{3} \\cdot \\sqrt{\\frac{-\\beta}{\\alpha}}\n \\cdot \\left(\\frac{2}{3} \\cdot\\beta\n -\\frac{\\sqrt{3}}{\\sqrt{\\frac{-\\beta}{\\alpha}}}\n \\right), \\\\\n E\\left( \\frac{1}{2}\\right) & = E(1) = -M.\n\\end{align*}\n\n\\msubsubsection{$\\mathsf{ParamFxSqrt}(\\sshare{x},k,f)$:}\nThis algorithm uses the sub-algorithm $\\mathsf{LinAppSQ}$ defined below, note that \\verb|LinAppSQ| returns an scaled $1/\\sqrt{x} \\cdot 2^{f}$.\nThe algorithm only works when $3 \\cdot k -2 \\cdot f$ is less than the system precision, which is by default equal to $20$.\nIn a future release we will extend the sqrt function to cope with other input ranges.\n\\begin{enumerate}\n \\item $\\theta \\asn \\ceil{\\log_2 (k/5.4)}$.\n \\item $\\sshare{y_0} \\asn \\mathsf{LinAppSQ}(\\sshare{x},k,f)$.\n \\item $\\sshare{y_0} \\asn \\sshare{y_0} \\cdot 1/2^{f}$.\n \\item $\\sshare{g_0} \\asn \\sshare{y_0} \\cdot \\sshare{x}$.\n \\item $\\sshare{g_0} \\asn \\sshare{y_0} \\cdot 1/2^{f}$.\n %\\item $\\sshare{g_0} \\asn \\mathsf{TruncPr}(\\sshare{g_0},k,f)$.\n %\\item $\\sshare{h_0} \\asn \\mathsf{FxDiv}(\\sshare{g_0},2,k,f)$.\n \\item $\\sshare{g_0} \\asn \\sshare{y_0} \\cdot 1/2$.\n \\item $\\sshare{gh_0} \\asn \\sshare{g_0} \\cdot \\sshare{h_0}$.\n %\\item $\\sshare{gh} \\asn \\mathsf{TruncPr}(\\sshare{gh},k,f)$.\n \\item $\\sshare{g} \\asn \\sshare{g_0}$.\n \\item $\\sshare{h} \\asn \\sshare{h_0}$.\n \\item $\\sshare{gh} \\asn \\sshare{gh_0}$.\n \\item For $i \\in[1,\\ldots,\\theta-2]$ do\n \\begin{enumerate}\n \\item $\\sshare{r} \\asn 3/2-\\sshare{gh}$.\n \\item $\\sshare{g} \\asn \\sshare{g} \\cdot \\sshare{r}$.\n \\item $\\sshare{h} \\asn \\sshare{h} \\cdot \\sshare{r}$.\n %\\item $\\sshare{g} \\asn \\mathsf{TruncPr}(\\sshare{g},k,f)$.\n %\\item $\\sshare{h} \\asn sf{TruncPr}(\\sshare{h},k,f)$.\n \\item $\\sshare{gh} \\asn \\sshare{g} \\cdot \\sshare{h}$.\n %\\item $\\sshare{gh} \\asn \\mathsf{TruncPr}(\\sshare{gh},k,f)$.\n \\end{enumerate}\n \\item $\\sshare{r} \\asn 3/2-\\sshare{gh}$.\n \\item $\\sshare{h} \\asn \\sshare{h} \\cdot \\sshare{r}$.\n %\\item $\\sshare{h} \\asn \\mathsf{TruncPr}(\\sshare{h},k,f)$.\n \\item $\\sshare{H} \\asn 4 \\cdot (\\sshare{h}^2)$.\n \\item $\\sshare{H} \\asn \\sshare{H}\\cdot \\sshare{x}$.\n \\item $\\sshare{H} \\asn (3) - \\sshare{H}$.\n \\item $\\sshare{H} \\asn \\sshare{h} \\cdot \\sshare{H}$.\n \\item $\\sshare{g} \\asn \\sshare{H} \\cdot \\sshare{x}$.\n %\\item $\\sshare{g} \\asn \\mathsf{FxDiv}(\\sshare{g},2,k,f)$.\n %\\item $\\sshare{g} \\asn \\mathsf{TruncPr}(\\sshare{g},4 \\cdot k,4 \\cdot f)$.\n \\item Return $\\sshare{g}$.\n\\end{enumerate}\n\n\\msubsubsection{$\\mathsf{SimplifiedFxSqrt}(\\sshare{x}, k, f)$:}\nThis algorithm uses the sub-algorithm $\\mathsf{NormSQ}$ defined above. Among the values it returns, we base our approximation by directly using $w =2^{m/2}$. From that point it approximates the value of $\\sqrt{x}$ by calculating $\\frac{x}{2^{m/2}}$. To avoid any loss of precision, we reuse \\verb|sfix| instantiation process.\nThe algorithm work on the precision of the system and can solve values on any range. Its behaviour is still experimental. The function is designed in such a way that there is no restriction on the size $f$.\n\\begin{enumerate}\n \\item $\\theta \\asn \\max{(\\ceil{\\log_2(k)},6)}$.\n \\item $\\sshare{m_{odd}}, \\sshare{w} \\asn \\mathsf{SimplifiedNormSQ}(\\sshare{x},k)$.\n \\item $\\sshare{m_{odd}} \\asn (1 - 2 \\cdot \\sshare{m_{odd}}) \\cdot (f \\quad \\% \\quad 2)$.\n \\item $\\sshare{w} \\asn (2 \\cdot \\sshare{w} -\\sshare{w}) \\cdot (1 - \\sshare{m_{odd}}) \\cdot (f \\quad \\% \\quad 2) + \\sshare{w}$.\n \\item $\\sshare{w} \\asn sfix( \\sshare{w} \\cdot 2 ^{\\frac{f - f \\quad \\% \\quad 2}{2}})$.\n \\item $\\sshare{w} \\asn (\\sqrt{2}\\cdot \\sshare{w} - \\sshare{w})\\cdot \\sshare{m_{odd}} + \\sshare{w}$.\n \\item $\\sshare{y_0} \\asn \\frac{1}{\\sshare{w}} $.\n \\item $\\sshare{g_0} \\asn \\sshare{y_0} \\cdot \\sshare{x}$.\n \\item $\\sshare{g_0} \\asn \\sshare{y_0} \\cdot 1/2$.\n \\item $\\sshare{gh_0} \\asn \\sshare{g_0} \\cdot \\sshare{h_0}$.\n \\item $\\sshare{g} \\asn \\sshare{g_0}$.\n \\item $\\sshare{h} \\asn \\sshare{h_0}$.\n \\item $\\sshare{gh} \\asn \\sshare{gh_0}$.\n \\item For $i \\in[1,\\ldots,\\theta-2]$ do\n \\begin{enumerate}\n \\item $\\sshare{r} \\asn 3/2-\\sshare{gh}$.\n \\item $\\sshare{g} \\asn \\sshare{g} \\cdot \\sshare{r}$.\n \\item $\\sshare{h} \\asn \\sshare{h} \\cdot \\sshare{r}$.\n %\\item $\\sshare{g} \\asn \\mathsf{TruncPr}(\\sshare{g},k,f)$.\n %\\item $\\sshare{h} \\asn sf{TruncPr}(\\sshare{h},k,f)$.\n \\item $\\sshare{gh} \\asn \\sshare{g} \\cdot \\sshare{h}$.\n %\\item $\\sshare{gh} \\asn \\mathsf{TruncPr}(\\sshare{gh},k,f)$.\n \\end{enumerate}\n \\item $\\sshare{r} \\asn 3/2-\\sshare{gh}$.\n \\item $\\sshare{h} \\asn \\sshare{h} \\cdot \\sshare{r}$.\n %\\item $\\sshare{h} \\asn \\mathsf{TruncPr}(\\sshare{h},k,f)$.\n \\item $\\sshare{H} \\asn 4 \\cdot (\\sshare{h}^2)$.\n \\item $\\sshare{H} \\asn \\sshare{H}\\cdot \\sshare{x}$.\n \\item $\\sshare{H} \\asn (3) - \\sshare{H}$.\n \\item $\\sshare{H} \\asn \\sshare{h} \\cdot \\sshare{H}$.\n \\item $\\sshare{g} \\asn \\sshare{H} \\cdot \\sshare{x}$.\n %\\item $\\sshare{g} \\asn \\mathsf{FxDiv}(\\sshare{g},2,k,f)$.\n %\\item $\\sshare{g} \\asn \\mathsf{TruncPr}(\\sshare{g},4 \\cdot k,4 \\cdot f)$.\n \\item Return $\\sshare{g}$.\n\\end{enumerate}\n\n\\msubsubsection{$\\mathsf{FxSqrt}(\\sshare{x}, k \\asn \\mathtt{sfix.k}, f \\asn \\mathtt{sfix.f})$:}\nOur \\verb|FxSqrt| functionality returns the square root of any fixed point input. It receives an input value $\\sshare{x}$, from which it calculates the square root, and optional parameters regarding its bit-length and bit-wise precision. The functionality is going to make use of our \\verb|SimplifiedFxSqrt| process by default, and the somewhat more efficient Liedel's method instead when the $3 \\cdot k - 2 \\cdot f < \\mathtt{sfix.f}$ bound, provided by his paper, is met.\n\n\\begin{enumerate}\n \\item if $(3 \\cdot k - 2 \\cdot f >= \\mathtt{sfix.f})$:\n \\begin{enumerate}\n \\item Return $\\mathsf{SimplifiedFxSqrt}(\\sshare{x}, k, f)$.\n \\end{enumerate}\n \\item else:\n \\begin{enumerate}\n \\item $\\sshare{x} \\asn Trunc(\\sshare{x} \\cdot 2^{f}, sfix.k, sfix.k-sfix.f)$\n \\item Return $\\mathsf{ParamFxSqrt}(\\sshare{x}, k, f)$.\n \\end{enumerate}\n\\end{enumerate}\n\n\\paragraph{MAMBA Example:} To obtain the \\verb|sqrt| of any value, you could execute the following:\n\\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nk = 5\nf = 2\n\nx = sfix(6.25)\ny = sfix(144)\nz = sfix (257.5)\n\n# returns the sqrt of the number, i.e. 2.5\n# inputs have to be expressed such that:\n# x * 2^f \\in Z_q\n# and 3*k -2*f < sfix.f (system precision)\n# by default system precision is 20 bits.\na = mpc_math.sqrt(x, k, f)\n\n# when you don't specify k and f, the system uses\n# the default sfix values, and hence the simplified\n# version for any value range, at the cost of an\n# additional division call.\n\nb = mpc_math.sqrt(y)\nc = mpc_math.sqrt(z)\n\n\\end{lstlisting}\n\n\\msubsubsection{$\\mathsf{LinAppSQ}(\\sshare{b},k,f)$:}\nWe based this section on the contents of the original paper, \\cite{Liedel12}.\nHowever we corrected the typos from the original work, the result is as follows:\n\\begin{enumerate}\n \\item $\\alpha \\asn (-0.8099868542) \\cdot 2^k$.\n \\item $\\beta \\asn (1.787727479) \\cdot 2^{2\\cdot k}$.\n \\item $(\\sshare{c},\\sshare{v},\\sshare{m},\\sshare{W}) \\asn \\mathsf{NormSQ}(\\sshare{b},k,f)$.\n \\item $\\sshare{w} \\asn \\alpha \\cdot \\sshare{c}+\\beta$.\n \\item $\\sshare{m} \\asn \\mathsf{Mod2}(\\sshare{m},\\ceil{\\log_2 k})$.\n \\item $\\sshare{w} \\asn \\sshare{w} \\cdot \\sshare{W} \\cdot \\sshare{v}$.\n \\item $\\sshare{w} \\asn \\mathsf{FxDiv}(\\sshare{w},2^{f/2},w.k,w.f)$.\n \\item $\\sshare{w} \\asn \\mathsf{FxDiv}(\\sshare{w},2^{3 \\cdot k - 2 \\cdot f},w.k,w.f)$.\n %\\item $\\sshare{w} \\asn \\mathsf{TruncPr}(\\sshare{w},3\\cdot k, 3 \\cdot k - 2 \\cdot f)$.\n \\item $\\sshare{w} \\asn (1-\\sshare{m}) \\cdot \\sshare{w} \\cdot 2^f\n +(\\sqrt{2} \\cdot 2^f) \\cdot \\sshare{m}\\cdot \\sshare{w}$.\n \\item Return $\\sshare{w}$ % \\asn \\mathsf{TruncPr}(\\sshare{w},k,f)$.\n\\end{enumerate}\n\n\n\\msubsubsection{$\\mathsf{FLSqrt}((\\sshare{v_1},\\sshare{p_1},\\sshare{z_1},\\sshare{s_1},\\sshare{\\err_1}))$:}\n Below we let $\\ell_0$ denote the lsb of $\\ell$,\n$(v_\\alpha,p_\\alpha,z_\\alpha,s_\\alpha)$\n (resp. $(v_\\beta,p_\\beta,z_\\beta,s_\\beta)$)\n denote the floating point representation of the constant $\\alpha$ (resp. $\\beta$)\n given above,\n and $v_{\\sqrt{2}}$ and $p_{\\sqrt{2}}$ represent the $\\ell$-bit significand and exponent of\n$\\sqrt{2}$ in floating point representation.\n \\begin{enumerate}\n \\item $\\sshare{b} \\asn \\mathsf{BitDec}(\\sshare{p_1},\\ell,1)$.\n \\item $\\sshare{c} \\asn \\mathsf{XOR}(\\sshare{b},\\ell_0)$.\n \\item $\\sshare{p} \\asn 2^{-1} \\cdot (\\sshare{p_1}-\\sshare{b})\n +\\floor{\\ell/2}+\\mathsf{OR}(\\sshare{b},\\ell_0)$.\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLMult}((\\sshare{v_1},-\\ell,0,0,0),(v_\\alpha,p_\\alpha,z_\\alpha,s_\\alpha,0))$.\n \\item $(\\sshare{v_0},\\sshare{p_0},\\sshare{z_0},\\sshare{s_0},\\sshare{\\err_0})\n \\asn \\mathsf{FLAdd}((\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2}),(v_\\beta,p_\\beta,z_\\beta,s_\\beta,0))$.\n \\item $(\\sshare{v_g},\\sshare{p_g},\\sshare{z_g},\\sshare{s_g},\\sshare{\\err_g})\n \\asn \\mathsf{FLMult}((\\sshare{v_1},-\\ell,0,0,0),(\\sshare{v_0},\\sshare{p_0},\\sshare{z_0},\\sshare{s_0},\\sshare{\\err_0}))$.\n \\item $(\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h})\n \\asn (\\sshare{v_0},\\sshare{p_0}-1,\\sshare{z_0},\\sshare{s_0},\\sshare{\\err_0})$.\n \\item For $i \\in [1,\\ldots,\\ceil{\\ell/5.4}-1]$ do\n \\begin{enumerate}\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_g},\\sshare{p_g},\\sshare{z_g},\\sshare{s_g},\\sshare{\\err_g}),\n (\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h})\n )$.\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLSub}(\n (3 \\cdot 2^{\\ell-2},-(\\ell-1),0,0,0),\n (\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n )$.\n \\item $(\\sshare{v_g},\\sshare{p_g},\\sshare{z_g},\\sshare{s_g},\\sshare{\\err_g})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_g},\\sshare{p_g},\\sshare{z_g},\\sshare{s_g},\\sshare{\\err_g}),\n (\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n )$.\n \\item $(\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h}),\n (\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n )$.\n \\end{enumerate}\n \\item $(\\sshare{v_{h^2}},\\sshare{p_{h^2}},\\sshare{z_{h^2}},\\sshare{s_{h^2}},\\sshare{\\err_{h^2}})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h}),\n (\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h})\n )$.\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_1},-\\ell,0,0,0),\n (\\sshare{v_{h^2}},\\sshare{p_{h^2}},\\sshare{z_{h^2}},\\sshare{s_{h^2}},\\sshare{\\err_{h^2}})$.\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLSub}(\n (3 \\cdot 2^{\\ell-2},-(\\ell-1),0,0,0),\n (\\sshare{v_2},\\sshare{p_2}+1,\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n )$.\n \\item $(\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_h},\\sshare{p_h},\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h}),\n (\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n )$.\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_1},-\\ell,0,0,0),\n (\\sshare{v_h},\\sshare{p_h}+1,\\sshare{z_h},\\sshare{s_h},\\sshare{\\err_h})\n )$.\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n \\asn \\mathsf{FLMult}(\n (\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2})\n (2^{\\ell-1} \\cdot (1-\\sshare{c}) + v_{\\sqrt{2}} \\cdot \\sshare{c},\n -(1-\\sshare{c}) \\cdot (\\ell-1)+p_{\\sqrt{2}} \\cdot \\sshare{c},0,0,0)$.\n \\item $\\sshare{p} \\asn (\\sshare{p_2}+\\sshare{p}) \\cdot (1-\\sshare{z_1})$.\n \\item $\\sshare{v} \\asn \\sshare{v_2} \\cdot (1-\\sshare{z_1})$.\n \\item $\\sshare{\\err} \\asn \\mathsf{OR}(\\sshare{\\err_2},\\sshare{s_1})$.\n \\item Return $(\\sshare{v},\\sshare{p},\\sshare{z_1},\\sshare{s_1},\\sshare{\\err})$.\n \\end{enumerate}\n \\todo{Have we picked up $\\err$ correctly here?}\n\n \\msubsection{EXP and LOG Functions}\n \\todo{These functions are only currently supported in their fixed point versions.}\n\n A secure fixed point exponentiation and logarithm algorithm is not found anywhere, so\n this is our own one derived from the identities in the book.\n \\begin{itemize}\n \\item {\\em Computer Approximations} by Hart from 1968 \\cite{Hart:1978:CA:540084}.\n \\end{itemize}\n The floating point variants are in\n \\begin{itemize}\n \\item Secure Computation on Floating Point Numbers {\\em NDSS 2013} \\cite{ABZS13}.\n \\end{itemize}\n Once we have defined $\\mathsf{FxExp2}$ and $\\mathsf{FxLog2}$\n (resp. $\\mathsf{FLExp2}$ and $\\mathsf{FLLog2}$) we can\n define the following functions from the usual identities\n for non-secret values of the base $b$:\n \\begin{align*}\n \\log_b x & = (\\log_b 2) \\cdot \\mathsf{Log2}(x), \\\\\n x^y & = \\mathsf{Exp2}(y \\cdot \\mathsf{Log2}(x)), \\\\\n \\exp x & = \\mathsf{Exp2}(x \\cdot \\log_2 e), \\\\\n \\end{align*}\n Note, that the functions on these section require specific \\verb|sfloat| parametrization, in accordance to the algorithms in this section. They support secret shared \\verb|sfix| $x$ ad $y$, as well as public floating point or integer inputs. We can define these operations as follows:\n\n \\msubsubsection{$\\mathsf{FxExp2}(\\sshare{a},k,f)$:}\n This algorithm computes $2^a$ as a fixed point calculation.\n First takes the integer and fractional part of the input\n$|a/2^f|$, which we denote by $b$ and $c$.\n We then compute $d=2^b$, which will clearly overflow\n if $b>k-f$, but we ignore this error (if the user is stupid\n enough to put in garbage, they get garbage out).\n We then compute $e=2^c$, as $0 \\le c \\le 1$ via the\n polynomial the following polynomial\\footnote{Note polynomial\n $P_{1045}(X)$ from Hart \\cite{Hart:1978:CA:540084} is incorrect\n and does not give an accurate result}.\n %with coefficients\n %\\begin{center}\n %\\begin{tabular}{|c|c|l|}\n %\\hline\n %0 & 1 & +.10000 00077 44302 1686 \\\\\n %1 & 0 & +.69314 71804 26163 82779 5756 \\\\\n %2 & 0 & +.24022 65107 10170 64605 384 \\\\\n %3 & -1 & +.55504 06862 04663 79157 744 \\\\\n %4 & -2 & +.96183 41225 88046 23749 77 \\\\\n %5 & -2 & +.13327 30359 28143 78193 29 \\\\\n %6 & -3 & +.15510 74605 90052 57397 8 \\\\\n %7 & -4 & +.14197 84739 97656 06711 \\\\\n %8 & -5 & +.18633 47724 13796 7076 \\\\\n %\\hline\n %\\end{tabular}\n %\\end{center}\n %which gives a relative error of at most $10^{-12.11}$\n %if computed exactly.\n %The table should be read as line $(i,a,b)$ giving\n %the $i$th coefficient of the polynomial being\n %$b \\cdot 10^a$.\n Our polynomial (which we produced using a Chebyshev approximation)\n is of degree nine and has coefficients given by\n \\begin{center}\n \\begin{tabular}{|c|l|}\n \\hline\n 0 & 0.99999999999998151058451 \\\\\n 1 & 0.69314718056364205693851 \\\\\n 2 & 0.24022650683729748257646 \\\\\n 3 & 0.0555041102193305250618 \\\\\n 4 & 0.0096181190501642860210497 \\\\\n 5 & 0.0013333931011014250476911 \\\\\n 6 & 0.00015395144945146697380844 \\\\\n 7 & 0.000015368748541192116946474 \\\\\n 8 & 0.0000012256971722926501833228 \\\\\n 9 & 0.00000014433329807023165258784 \\\\\n \\hline\n \\end{tabular}\n \\end{center}\n Given $d$ and $e$ one can now compute\n$2^{|a|}=2^{b+c}=2^b \\cdot 2^c=d \\cdot e$,\n and the final dealing with the sign of $a$ can\n be done by an inversion.\n We denote by $\\mathsf{FxPol}(P_{1045},\\sshare{x},k,f)$ the evaluation\n of the polynomial $P_{1045}$ on the fixed point input $\\sshare{x}$\n where $x \\in \\Qk{f}$. This is done by Horner's rule.\n \\begin{enumerate}\n \\item $\\sshare{s}=\\mathsf{FxLTZ}(\\sshare{a})$.\n \\item $\\sshare{a} \\asn (1-2 \\cdot \\sshare{s}) \\cdot \\sshare{a}$.\n \\item $\\sshare{b} \\asn \\mathsf{Fx2Int}(\\sshare{a},k,f)$.\n \\item $\\sshare{c} \\asn \\sshare{a}-\\mathsf{Int2Fx}(\\sshare{b},k,f)$.\n \\item $\\sshare{d} \\asn \\mathsf{Int2Fx}(\\mathsf{Pow2}(\\sshare{b},k),k,f)$. [This will produce an invalid result if $b$ is too big, in which case the result cannot be held in an Fx in any case]\n \\item $\\sshare{e} \\asn \\mathsf{FxPol}(P_{1045},\\sshare{c},k,f)$.\n \\item $\\sshare{g} \\asn \\mathsf{FxMult}(\\sshare{d},\\sshare{e},k,f)$.\n \\item $\\sshare{g^{-1}} \\asn \\mathsf{FxDiv}(2^f,\\sshare{g},k,f)$.\n \\item $\\sshare{a} \\asn (1-\\sshare{s}) \\cdot g+ \\sshare{s} \\cdot \\sshare{g^{-1}}$.\n \\item Return $\\sshare{a}$.\n \\end{enumerate}\n The above works, but we have found a little numerical\n instability due to the division operation.\n \\paragraph{MAMBA Example:} To obtain $2^y$ where $y$ is secret shared you could run the following:\n \\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nsfloat.vlen = 15 # Length of mantissa in bits\nsfloat.plen = 10 # Length of exponent in bits\nsfloat.kappa = 4 # Statistical security parameter for floats\n\ny =sfix(4)\n\n# returns 2^4\n# extracts k and f from y\nexp2_y=mpc_math.exp2_fx(sfix(y))\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{FLExp2}((\\sshare{v_1},\\sshare{p_1},\\sshare{z_1},\\sshare{s_1},\\sshare{\\err_1}))$:}\nThis method assumes that $k \\le \\ell$.\nWe do not support a method if $k > \\ell$, and so if this happens\nwe will signal an error.\n\\begin{enumerate}\n \\item If $k>\\ell$ then $\\err_1 \\asn 1$.\n \\item $\\max \\asn \\ceil{\\log_2(2^{k-1}-1+\\ell)-\\ell+1}$.\n \\item $\\sshare{a} \\asn \\mathsf{LT}(\\sshare{p_1},\\max,k)$.\n \\item $\\sshare{b} \\asn \\mathsf{LT}(\\sshare{p_1},-\\ell+1,k)$.\n \\item $\\sshare{c} \\asn \\mathsf{LT}(\\sshare{p_1},-2 \\cdot \\ell+1,k)$.\n \\item $\\sshare{(1-c) \\cdot a} \\asn (1-\\sshare{c}) \\cdot \\sshare{a}$.\n \\item $\\sshare{p_2} \\asn -\\sshare{(1-c) \\cdot a} \\cdot (\\sshare{b} \\cdot \\ell +\\sshare{p_1})$.\n \\item $\\sshare{x},\\sshare{2^{p_2}} \\asn \\mathsf{Trunc}(\\sshare{v_1},\\ell,\\sshare{p_2})$.\n \\item $\\sshare{y} \\asn \\sshare{v_1}-\\sshare{x} \\cdot \\sshare{2^{p_2}}$.\n \\item $\\sshare{d} \\asn \\mathsf{EQZ}(\\sshare{y},\\ell)$.\n \\item $\\sshare{b \\cdot s_1} \\asn \\sshare{b} \\cdot \\sshare{s_1}$.\n \\item $\\sshare{(1-d) \\cdot s_1} \\asn (1-\\sshare{d}) \\cdot \\sshare{s_1}$.\n \\item $\\sshare{x} \\asn (1-\\sshare{b \\cdot s_1})\n \\cdot (\\sshare{x}-\\sshare{(1-d)\\cdot s_1})\n +\\sshare{b \\cdot s_1} \\cdot(2^\\ell-1+\\sshare{d}-\\sshare{x})$.\n \\item $\\sshare{y} \\asn \\sshare{(1-d) \\cdot s_1} \\cdot (\\sshare{2^{p_2}}-\\sshare{y})\n + (1-\\sshare{s_1})\\cdot \\sshare{y}$.\n \\item $\\sshare{w} \\asn \\sshare{(1-c) \\cdot a} \\cdot\n ((1-\\sshare{b}) \\cdot\\sshare{x}+\\sshare{b \\cdot s_1})\n \\cdot (1-2\\cdot \\sshare{s_1})-\\sshare{c} \\cdot \\sshare{s_1}$.\n \\item $\\sshare{u} \\asn \\sshare{(1-c) \\cdot a}\n \\cdot (\\sshare{b} \\cdot \\sshare{x}\n +(1-\\sshare{b}) \\cdot 2^\\ell \\cdot \\mathsf{Inv}(\\sshare{2^{p_2}})\n \\cdot \\sshare{y})\n +(2^\\ell-1)\\cdot \\sshare{c} \\sshare{s_1}$.\n \\item $\\sshare{u_\\ell},\\ldots,\\sshare{u_1} \\asn \\mathsf{BitDec}(\\sshare{u},\\ell,\\ell)$.\n \\item For $i \\in [1,\\ldots,\\ell]$ do\n \\begin{enumerate}\n \\item~ [In this loop $(cv_i,cp_i,0,0)$ represents the floating point number $2^{2^{-i}}$].\n \\item $\\sshare{a_i} \\asn 2^{\\ell-1} \\cdot (1-\\sshare{u_i})+cv_i \\cdot \\sshare{u_i}$.\n \\item $\\sshare{b_i} \\asn -(\\ell-1) \\cdot (1-\\sshare{u_i})+cp_i \\cdot \\sshare{u_i}$.\n \\end{enumerate}\n \\item $(\\sshare{v_u},\\sshare{p_i},0,0)\n \\asn \\mathsf{FLProd}((\\sshare{a_1},\\sshare{b_1},0,0), \\ldots,\n (\\sshare{a_\\ell},\\sshare{b_\\ell},0,0))$.\n [This implements a product of $\\ell$ floating point values, which is\n performed via a binary tree style method.]\n \\item $\\sshare{p} \\asn \\sshare{a} \\cdot (\\sshare{w}+\\sshare{p_u})\n +2^{k-1}\\cdot (1-\\sshare{a}) \\cdot (1-2 \\sshare{s_1})$.\n \\item $\\sshare{v} \\asn 2^{\\ell-1} \\cdot \\sshare{z_1}+(1-\\sshare{z_1}) \\cdot \\sshare{v_u}$.\n \\item $\\sshare{p} \\asn -\\sshare{z_1} \\cdot (\\ell-1)\n +(1-\\sshare{z_1}) \\cdot \\sshare{p}$.\n \\item $\\sshare{\\err} \\asn \\sshare{\\err_1} + \\mathsf{FlowDetect}(\\sshare{p})$\n \\item Return $(\\sshare{v},\\sshare{p},0,0,\\sshare{\\err})$.\n\\end{enumerate}\n\n\\msubsubsection{$\\mathsf{FxLog2}(\\sshare{a},k,f)$:}\nWe first map $a$ to a value $v$ in the interval $[1/2,1]$ by essentially\nconverting to a floating point number.\nSo we a have $a=(v/2^k) \\cdot 2^p$ where $v, p \\in \\Zk$,\nand $v/2^k \\in [1/2,1]$.\nThus we have\n$\\log_2 a = p+\\log_2 (v/2^k)$, and we then treat $v$ as a fixed\npoint number and apply the Pade approximation $P_{2524}/Q_{2524}$\nfrom Hart's book \\cite{Hart:1978:CA:540084},\nwhich produces an {\\em absolute} error of $10^{-8.32}$.\nWe denote by $\\mathsf{FxPade}(P_{2524},Q_{2524},\\sshare{x},k,f)$ the evaluation\nof the rational function $P_{2524}/Q_{2524}$ on the fixed point input $\\sshare{x}$\nwhere $x \\in \\Qk{f}$.\nThe Pade approximation is given be the rational function defined\nby the following table\n\\begin{center}\n \\begin{tabular}{|c|c|c|l|}\n \\hline\n P & 0 & 1 & -.20546 66719 51 \\\\\n P & 1 & 1 & -.88626 59939 1 \\\\\n P & 2 & 1 & +.61058 51990 15 \\\\\n P & 3 & 1 & +.48114 74609 89 \\\\\n Q & 0 & 0 & +.35355 34252 77 \\\\\n Q & 1 & 1 & +.45451 70876 29 \\\\\n Q & 2 & 1 & +.64278 42090 29 \\\\\n Q & 3 & 1 & +.1 \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\n\\begin{enumerate}\n \\item $(\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err})\n \\asn \\mathsf{Fx2FL}(\\mathsf{a},k,f,k,k)$.\n \\item $\\sshare{a} \\asn \\mathsf{FxPade}(P_{2524},Q_{2524},\\sshare{v},k,k)$.\n \\item $\\sshare{a} \\asn \\sshare{a}+\\sshare{p} + f$.\n \\item $\\sshare{a} \\asn \\sshare{a} \\cdot (1-\\sshare{z}) \\cdot (1-\\sshare{s}) \\cdot (1-\\cdot \\sshare{\\err})$.\n \\item Return $\\sshare{a}$.\n\\end{enumerate}\n\n\\paragraph{MAMBA Example:} To obtain \\verb|log2|$(x)$ where $y$ is secret shared you could run the following:\n\\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nsfloat.vlen = 15 # Length of mantissa in bits\nsfloat.plen = 10 # Length of exponent in bits\nsfloat.kappa = 4 # Statistical security parameter for floats\n\nx =sfix(4)\n# extracts k and f from y\n# returns log_2(4)\nlog2_x=mpc_math.log2_fx(sfix(x))\n\\end{lstlisting}\nNote, internally \\verb+log2_fx+ uses a polynomial to approximate the logarithm, but it uses the \\verb+sfloat+ arithmetic to do this.\nThus, you need to set up \\verb+sfloat+ and \\verb+sfix+ correctly for this to work.\nIn particular if you look deep inside the code there is the following cryptic remark in relation to \\verb+log2_fx+...\n\\begin{verbatim}\n # Note that sfloat and sfix sizes have to be parametrized correctly,\n # such that sfix k > sfix.f >= sfloat vlen. This is not the case by default.\n\\end{verbatim}\n\n\n\\msubsubsection{$\\mathsf{FLLog2}((\\sshare{v_1},\\sshare{p_1},\\sshare{z_1},\\sshare{s_1},\\sshare{\\err_1}))$:}\n In the following algorithm $(cv_i,cp_i,0,0)$ represents the floating point\n constant $(2 \\cdot \\log_2 e)/(2 \\cdot i+1)$.\n \\begin{enumerate}\n \\item $M \\asn \\ceil{ \\ell/(2 \\cdot \\log_2 3)-1/2}$.\n \\item $(\\sshare{v_2},\\sshare{p_2},0,0,0) \\asn \\mathsf{FLSub}((2^{\\ell-1},-(\\ell-1),0,0,0),(\\sshare{v_1},-\\ell,0,0,0))$.\n \\item $(\\sshare{v_3},\\sshare{p_3},0,0,0) \\asn \\mathsf{FLAdd}((2^{\\ell-1},-(\\ell-1),0,0,0),(\\sshare{v_1},-\\ell,0,0,0))$.\n \\item $(\\sshare{v_y},\\sshare{p_y},0,0,0) \\asn \\mathsf{FLDiv}((\\sshare{v_2},\\sshare{p_2},0,0,0),(\\sshare{v_3},\\sshare{p_3},0,0,0))$.\n \\item $(\\sshare{v_{y^2}},\\sshare{p_{y^2}},0,0,0) \\asn \\mathsf{FLMult}((\\sshare{v_y},\\sshare{p_y},0,0,0),(\\sshare{v_y},\\sshare{p_y},0,0,0))$.\n \\item $(\\sshare{v},\\sshare{p},0,0,0) \\asn \\mathsf{FLMult}((\\sshare{v_y},\\sshare{p_y},0,0),(cv_0,cp_0,0,0))$.\n \\item For $i \\in[1,\\ldots,M]$ do\n \\begin{enumerate}\n \\item $(\\sshare{v_y},\\sshare{p_y},0,0,0) \\asn \\mathsf{FLMult}((\\sshare{v_y},\\sshare{p_y},0,0,0),(\\sshare{v_{y^2}},\\sshare{p_{y^2}},0,0,0))$.\n \\item $(\\sshare{v_2},\\sshare{p_2},0,0,0) \\asn \\mathsf{FLMult}((\\sshare{v_y},\\sshare{p_y},0,0,0),(cv_i,cp_i,0,0,0))$.\n \\item $(\\sshare{v},\\sshare{p},0,0,0) \\asn \\mathsf{FLAdd}((\\sshare{v},\\sshare{p},0,0,0),(\\sshare{v_2},\\sshare{p_2},0,0,0))$.\n \\end{enumerate}\n \\item $(\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2}) \\asn \\mathsf{Int2FL}(\\ell,-\\sshare{p},\\ell,\\ell)$.\n \\item $(\\sshare{v},\\sshare{p},\\sshare{z},\\sshare{s},\\sshare{\\err}) \\asn \\mathsf{FLSub}((\\sshare{v_2},\\sshare{p_2},\\sshare{z_2},\\sshare{s_2},\\sshare{\\err_2}),(\\sshare{v},\\sshare{p},0,0,0))$.\n \\item $\\sshare{a} \\asn \\mathsf{EQ}(\\sshare{p_1},-(\\ell-1),k)$.\n \\item $\\sshare{b} \\asn \\mathsf{EQ}(\\sshare{v_1},2^{\\ell-1},\\ell)$.\n \\item $\\sshare{z} \\asn \\sshare{a} \\cdot \\sshare{b}$.\n \\item $\\sshare{v} \\asn \\sshare{v} \\cdot(1-\\sshare{z})$.\n \\item $\\sshare{\\err} \\asn \\mathsf{OR}(\\sshare{\\err},\\sshare{\\err_1})$.\n \\item $\\sshare{\\err} \\asn \\mathsf{OR}(\\mathsf{OR}(\\sshare{z_1},\\sshare{s_1}),\\err)$.\n \\item $\\sshare{p} \\asn \\sshare{p} \\cdot(1-\\sshare{z})$.\n \\item Return $(\\sshare{v},\\sshare{p},\\sshare{z_1},\\sshare{s_1},\\sshare{\\err})$.\n \\end{enumerate}\n\n\n \\msubsection{Trigonometic Functions}\n All three basic trigonometric functions support inputs of either\n fixed point or floating point precision.\n With the output type being equal to the input type.\n The computation of $\\sin(x)$ and $\\cos(x)$ are performed\n using polnoymial approximations, with the computation of $\\tan(x)$\n done via $\\sin(x)/\\cos(x)$.\n The basic idea for $\\sin(x)$ and $\\cos(x)$ is to first reduce\n the argument $x$ into the range $[0,\\ldots,2 \\pi)$, so\n as to obtain a new argument (which we call $y$)\n We then compute a bit $b_1$ to test as to whether\n$y \\in [0,\\pi)$ or $[\\pi,2 \\pi)$ (with $0$ being\n the former).\n We then reduce $y$ to $z$ by reducing it into the range\n$[0,\\pi)$, and compute a bit $b_2$ which says whether\n$z$ is in the range $[0,\\pi/2)$ or $[\\pi/2,\\pi)$.\n We finally reduce $z$ into the range $[0,\\pi/2)$ resulting\n in $w$.\n Then a polynomial is used to compute\n$\\sin(w)$ or $\\cos(w)$, which means\n we need to now {\\em scale} $w$ into the range $[0,1)$\n to obtain $v$.\n %We then derive the final result using the following\n %identities:\n %\\begin{align*}\n %\t\\sin(x) &= \\sin(y) = (1-2 \\cdot b_1) \\cdot \\sin(z), %\\\\\n %\t\\cos(x) &= \\cos(y) = (1-2 \\cdot b_1) \\cdot \\cos(z), %\\\\\n %\t\\sin(z) &= (1-b_2) \\cdot \\sin(w) + b_2\\cdot \\cos(w), %\\\\\n %\t\\cos(z) &= (1-b_2) \\cdot \\cos(w) - b_2\\cdot \\sin(w).\n %\\end{align*}\n For the polynomial approximations to the basic functions in\n the range $[0,\\pi/2)$, where the argument is given as $w = v \\cdot \\pi/2$\n we use the following approximations from Hart's book \\cite{Hart:1978:CA:540084}\n \\begin{align*}\n \\sin(w) & = v \\cdot P_{3307}(v^2), \\\\\n \\cos(w) & = P_{3508}(v^2).\n \\end{align*}\n Where we have\n \\begin{center}\n \\begin{tabular}{|c||c|l||c|l|}\n \\hline\n & \\multicolumn{2}{c||}{$P_{3307}$} & \\multicolumn{2}{c|}{$P_{3508}$} \\\\\n \\hline\n 0 & 1 & +.15707 96326 79489 66192 31314 989 & 0 & +.99999 99999 99999 99999 99914 771 \\\\\n 1 & 0 & -.64596 40975 06246 25365 51665 255 & 0 & -.49999 99999 99999 99999 91637 437 \\\\\n 2 & -1 & +.79692 62624 61670 45105 15876 375 & -1 & +.41666 66666 66666 66653 10411 988 \\\\\n 3 & -2 & -.46817 54135 31868 79164 48035 89 & -2 & -.13888 88888 88888 88031 01864 15 \\\\\n 4 & -3 & +.16044 11847 87358 59304 30385 5 & -4 & +.24801 58730 15870 23300 45157 \\\\\n 5 & -5 & -.35988 43235 20707 78156 5727 & -6 & -.27557 31922 39332 25642 1489 \\\\\n 6 & -7 & +.56921 72920 65732 73962 4 & -8 & +.20876 75698 16541 25915 59 \\\\\n 7 & -9 & -.66880 34884 92042 33722 & -10 & -.11470 74512 67755 43239 4 \\\\\n 8 & -11 & +.60669 10560 85201 792 & -13 & +.47794 54394 06649 917 \\\\\n 9 & -13 & -.43752 95071 18174 8 & -15 & -.15612 26342 88277 81 \\\\\n 10 & -15 & +.25002 85418 9303 & -18 & +.39912 65450 7924 \\\\\n \\hline\n \\end{tabular}\n \\end{center}\n NOTE: Polynomial tables are described by the monomial number, the degree of the approximation $p$ and its significand $s$. The coefficient of the $i$th monomial can be obtained by multiplying the significand by $10^{p_{i}}$ as follows: $s_{i} \\cdot 10^{p_{i}}$.\n\n \\msubsubsection{$\\mathsf{F\\star TrigSub}(\\sshare{x})$:}\n \\begin{enumerate}\n\n \\item $\\sshare{f} \\asn \\mathsf{F\\star Mult}(\\sshare{x},(1 /(2 \\cdot \\pi))$\n \\item $\\sshare{f} \\asn \\mathsf{F\\star Floor}(\\sshare{f})$.\n \\item $\\sshare{y} \\asn \\mathsf{F\\star Mult}(\\sshare{f},(2 \\cdot \\pi ))$.\n \\item $\\sshare{y} \\asn \\mathsf{F\\star Add}(\\sshare{x},-\\sshare{y})$.\n \\item $\\sshare{b_1} \\asn \\mathsf{F\\star GE}(\\sshare{y},(\\pi))$.\n \\item $\\sshare{f} \\asn \\mathsf{F\\star Add}(2 \\cdot \\pi,-\\sshare{y})$\n \\item $w \\asn \\mathsf{F\\star Choose}(\\sshare{f},\\sshare{y}, \\sshare{b_1})$.\n \\item $\\sshare{b_2} \\asn \\mathsf{F\\star GE}(\\sshare{2},(\\pi / 2))$.\n \\item $\\sshare{f} \\asn \\mathsf{F\\star Add}( \\pi,-\\sshare{w})$\n \\item $w \\asn \\mathsf{F\\star Choose}(\\sshare{f},\\sshare{w}, \\sshare{b_2})$.\n \\item Return $(\\sshare{w},\\sshare{b_1},\\sshare{b_2})$.\n\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} To reduce the angle you could execute the following (note that this function call is meant to be used internally):\n \\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx = sfix(4) # sfloat(4)\n# returns an angle in the [0,pi/2) interval in w and flags b1 and b2.\nw, b1, b2 = mpc_math.sTrigSub_fx(x)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{F\\star Sin}(\\sshare{x})$}\n We present these routines as generic routines given the specific helper subroutine above;\n we assume an obvious overloading/translation of arguments.\n We let $\\mathsf{F\\star Choose}(\\sshare{x},\\sshare{y},\\sshare{b})$, for a\n shared bit $b$, denote an operation which produces $\\sshare{x}$ if $\\sshare{b}=1$\n and $\\sshare{y}$ otherwise\n This is easily obtained by securely multiplying each component share\n of the data representing $\\sshare{x}$ etc by $\\sshare{b}$.\n So for fixed point representations this becomes, irrespective of the\n values $k$ and $f$,\n \\begin{enumerate}\n \\item $\\sshare{a} \\asn \\sshare{b}\\cdot \\sshare{x}+(1-\\sshare{b}) \\cdot \\sshare{y}$\n \\end{enumerate}\n For floating point representations this becomes\n \\begin{enumerate}\n \\item $\\sshare{w}, \\sshare{b_1}, \\sshare{b_2} \\asn \\mathsf{F\\star TrigSub}(\\sshare{x})$\n \\item $\\sshare{v} \\asn \\sshare{w} \\cdot (1/(\\pi/2))$.\n \\item $\\sshare{b} \\asn \\mathsf{F\\star Choose}(\\sshare{-1},\\sshare{1}, \\sshare{b_1})$.\n \\item $\\sshare{\\sin(v)} \\asn \\sshare{v} \\cdot \\mathsf{F\\star Pol}(P_{3307},\\sshare{v^2})$.\n \\item Return $(\\sshare{b} \\cdot \\sshare{\\sin(v)})$.\n \\end{enumerate}\n %\n %Given this we can now compute $\\sin(x)$ in the following lines:\n %\\begin{enumerate}\n %\\item $(\\sshare{\\sin(w)},\\sshare{\\cos(w)},\\sshare{b_1},\\sshare{b_2}) \\asn \\mathsf{F\\star TrigSub}(\\sshare{x})$.\n %\\item $\\sshare{\\sin(z)} \\asn \\mathsf{F\\star Choose}(\\sshare{\\sin(w)},\\sshare{\\cos(w)},1-\\sshare{b_2})$.\n %\\item $\\sshare{-\\sin(z)} \\asn \\mathsf{F\\star Neg}(\\sshare{\\sin(z)}$.\n %\\item $\\sshare{\\sin(x)} \\asn \\mathsf{F\\star Choose}(\\sshare{\\sin(z)},\\sshare{-\\sin(z)},1-\\sshare{b_1})$.\n %\\item Return $\\sshare{\\sin(x)}$.\n %\\end{enumerate}\n\n \\paragraph{MAMBA Example:} To obtain the \\verb|sin| of any value, you could execute the following:\n \\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx = sfix(4) # sfloat(4)\n# returns the sin of a number of any interval\ny = mpc_math.sin(x)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{F\\star Cos}(\\sshare{x})$}\n Likewise this becomes\n \\begin{enumerate}\n\n \\item $\\sshare{w}, \\sshare{b_1}, \\sshare{b_2} \\asn \\mathsf{F\\star TrigSub}(\\sshare{x})$\n \\item $\\sshare{v} \\asn \\sshare{w}$.\n \\item $\\sshare{b} \\asn \\mathsf{F\\star Choose}(\\sshare{-1},\\sshare{1}, \\sshare{b_2})$.\n \\item $\\sshare{\\cos(v)} \\asn \\mathsf{F\\star Pol}(P_{3308},\\sshare{v^2})$.\n \\item Return $(\\sshare{b} \\cdot \\sshare{\\cos(v)})$.\n\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} To obtain the \\verb|sin| of any value, you could execute the following:\n \\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx = sfix(4) # sfloat(4)\n# returns the cos of an angle on any interval\ny = mpc_math.cos(x)\n\\end{lstlisting}\n\n \\msubsubsection{$\\mathsf{F\\star Tan}(\\sshare{x})$}\n Likewise this becomes\n \\begin{enumerate}\n\n \\item $(\\sshare{w},\\sshare{b_1},\\sshare{b_2}) \\asn \\mathsf{F\\star TrigSub}(\\sshare{x})$.\n \\item $\\sshare{v} \\asn \\sshare{w} \\cdot (1/(\\pi/2))$.\n \\item $\\sshare{b} \\asn \\mathsf{F\\star Choose}(\\sshare{-1},\\sshare{1}, \\sshare{b_1})$.\n \\item $\\sshare{\\sin(v)} \\asn \\sshare{v} \\cdot \\mathsf{F\\star Pol}(P_{3307},\\sshare{v^2})$.\n \\item $\\sshare{\\sin(x)} \\asn (\\sshare{b} \\cdot \\sshare{\\sin(v)})$.\n \\item $\\sshare{v} \\asn \\sshare{w}$.\n \\item $\\sshare{b} \\asn \\mathsf{F\\star Choose}(\\sshare{-1},\\sshare{1}, \\sshare{b_2})$.\n \\item $\\sshare{\\cos(v)} \\asn \\mathsf{F\\star Pol}(P_{3308},\\sshare{v^2})$.\n \\item $\\sshare{\\sin(x)} \\asn (\\sshare{b} \\cdot \\sshare{\\cos(v)})$.\n \\item $\\sshare{\\tan(x)} \\asn \\mathsf{F\\star Div}(\\sshare{\\sin(x)},\\sshare{\\cos(x)})$.\n \\item Return $\\sshare{\\tan(x)}$.\n \\end{enumerate}\n\n \\paragraph{MAMBA Example:} To obtain the \\verb|tan| of any value, you could execute the following:\n \\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx = sfix(4) # sfloat(4)\n# returns the tan of an angle on any interval\ny = mpc_math.tan(x)\n\\end{lstlisting}\n\n \\msubsection{Inverse Trigonometric Functions}\n \\todo{Given that SCALE-MAMBA currently only supports square root operations for sfix inputs,\n inverse trigonometric functions are restricted to sfix inputs.}\n To obtain $\\arcsin$ and $\\arccos$ one makes use of the formula:\n \\begin{align*}\n \\arcsin (x) & = \\arctan \\left( \\frac{x}{\\sqrt{1-x^2}} \\right), \\\\\n \\arccos (x) & = \\frac{\\pi}{2} - \\arcsin (x).\n \\end{align*}\n Note, that $\\arcsin$ and $\\arccos$ are only defined when\n$|x|\\le 1$.\n The value of $\\arctan(x)$ is however defined for all real $x$.\n For $\\arctan(x)$ we first reduce to positive values of $x$ by using the formula\n \\[ \\arctan(-x) = - \\arctan(x). \\]\n We then reduce to the interval $[0,1)$ using the formula\n \\[ \\arctan(x) = \\frac{\\pi}{2} - \\arctan\\left(\\frac{1}{x} \\right). \\]\n The final approximation to $\\arctan(x)$ for $x \\in [0,1)$\n is obtained using the Pade approximation $P_{5102}/Q_{5102}$\nfrom Hart's book \\cite{Hart:1978:CA:540084}.\nWhere the polynomials are represented as in our earlier\ndescriptions.\n\\begin{center}\n \\begin{tabular}{|c||c|l||c|l|}\n \\hline\n & \\multicolumn{2}{c||}{$P_{5102}$} & \\multicolumn{2}{c|}{$Q_{5102}$} \\\\\n \\hline\n 0 & 5 & +.21514 05962 60244 19331 93254 468 & 5 & +.21514 05962 60244 19331 93298 234 \\\\\n 1 & 5 & +.73597 43380 28844 42408 14980 706 & 5 & +.80768 78701 15592 48851 76713 209 \\\\\n 2 & 6 & +.10027 25618 30630 27849 70511 863 & 6 & +.12289 26789 09278 47762 98743 322 \\\\\n 3 & 5 & +.69439 29750 03225 23370 59765 503 & 5 & +.97323 20349 05355 56802 60434 387 \\\\\n 4 & 5 & +.25858 09739 71909 90257 16567 793 & 5 & +.42868 57652 04640 80931 84006 664 \\\\\n 5 & 4 & +.50386 39185 50126 65579 37791 19 & 5 & +.10401 13491 56689 00570 05103 878 \\\\\n 6 & 3 & +.46015 88804 63535 14711 61727 227 & 4 & +.12897 50569 11611 09714 11459 55 \\\\\n 7 & 2 & +.15087 67735 87003 09877 17455 528 & 2 & +.68519 37831 01896 80131 14024 294 \\\\\n 8 & -1 & +.75230 52818 75762 84445 10729 539 & 1 & +.1 \\\\\n \\hline\n \\end{tabular}\n\\end{center}\n\nThe following protocol for \\verb|arcsin| implements the formulas from before.\nThe protocol and its implementation, make use of our secure implementation of Square Root.\nThis method is implemented, and it is used to derive \\verb|arccos|.\n\n\\msubsubsection{$\\mathsf{F\\star ArcSin}(\\sshare{x})$}\n\\begin{enumerate}\n \\item $\\sshare{x^2} \\asn \\mathsf{F\\star Mult}(\\sshare{x},\\sshare{x})$.\n \\item $\\sshare{-x^2} \\asn \\mathsf{F\\star Neg}(\\sshare{x^2})$.\n \\item $\\sshare{1-x^2} \\asn \\mathsf{F\\star Add}(1,\\sshare{-x^2})$.\n \\item $\\sshare{\\sqrt{1-x^2}} \\asn \\mathsf{F\\star Sqrt}(1,\\sshare{1-x^2})$.\n \\item $\\sshare{v} \\asn \\mathsf{F\\star Div}(\\sshare{x},\\sshare{\\sqrt{1-x^2}})$.\n \\item $\\sshare{y} \\asn \\mathsf{F\\star ArcTan}(\\sshare{v})$.\n \\item If $\\star=\\mathsf{L}$\n \\begin{enumerate}\n \\item $\\sshare{|x|} \\asn \\mathsf{FLAbs}(\\sshare{x})$.\n \\item $\\sshare{y_\\err} \\asn \\mathsf{FLGT}(\\sshare{|x|},1.0)$.\n \\end{enumerate}\n \\item Return $\\sshare{y}$.\n\\end{enumerate}\n\n\n\\paragraph{MAMBA Example:} To obtain the \\verb|arcSin| on the (-1,1) interval, you could execute the following:\n\\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx =sfix(0.5)\n# returns the tan of an angle on any interval\ny = mpc_math.asin(x)\n\\end{lstlisting}\n\n\n\\msubsubsection{$\\mathsf{F\\star ArcCos}(\\sshare{x})$}\n\\begin{enumerate}\n \\item $\\sshare{y} \\asn \\mathsf{F\\star ArcSin}(\\sshare{x})$.\n \\item $\\sshare{-y} \\asn \\mathsf{F\\star Neg}(\\sshare{y})$.\n \\item $\\sshare{\\pi/2-y} \\asn \\mathsf{F\\star Add}(\\pi/2,\\sshare{-y})$.\n \\item Return $\\sshare{y}$.\n\\end{enumerate}\n\n\\paragraph{MAMBA Example:} To obtain the \\verb|arcCos| of any value on the (-1,1) interval, you could execute the following:\n\\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx =sfix(0.5)\n# returns the tan of an angle on any interval\ny = mpc_math.acos(x)\n\\end{lstlisting}\n\n\\msubsubsection{$\\mathsf{F\\star ArcTan}(\\sshare{x})$}\n\\begin{enumerate}\n \\item $\\sshare{s} \\asn \\mathsf{F\\star LTZ}(\\sshare{z})$.\n \\item $\\sshare{|x|} \\asn \\mathsf{F\\star Abs}(\\sshare{x})$.\n \\item $\\sshare{b} \\asn \\mathsf{F\\star GT}(\\sshare{|x|},1.0)$.\n \\item $\\sshare{v} \\asn \\mathsf{F\\star Div}(1,\\sshare{|x|})$.\n \\item $\\sshare{v} \\asn \\mathsf{F\\star Choose}(\\sshare{|x|},\\sshare{v},1-\\sshare{b})$.\n \\item $\\sshare{v^2} \\asn \\mathsf{F\\star Mul}(\\sshare{v},\\sshare{v})$.\n \\item $\\sshare{y} \\asn \\mathsf{FxPade}(P_{5102},Q_{5102},\\sshare{v^2})$.\n \\item $\\sshare{y} \\asn \\mathsf{F\\star Mul}(\\sshare{v},\\sshare{y})$.\n \\item $\\sshare{\\pi/2-y} \\asn \\mathsf{F\\star Sub}(\\pi/2,\\sshare{y})$.\n \\item $\\sshare{y} \\asn \\mathsf{F\\star Choose}(\\sshare{y},\\sshare{\\pi/2-y},1-\\sshare{b})$.\n \\item $\\sshare{-y} \\asn \\mathsf{F\\star Neg}(\\sshare{y})$.\n \\item $\\sshare{y} \\asn \\mathsf{F\\star Choose}(\\sshare{y},\\sshare{-y},1-\\sshare{s})$.\n \\item Return $\\sshare{y}$.\n\\end{enumerate}\n\n\\paragraph{MAMBA Example:} To obtain the \\verb|arcTan| of any value, you could execute the following:\n\\begin{lstlisting}[language={python}]\nfrom Compiler import mpc_math\nx =sfix(0.5)\n# returns the tan of an angle on any interval\ny = mpc_math.atan(x)\n\\end{lstlisting}\n", "meta": {"hexsha": "cfc10db92a738f4bf5c8284e6ff92a6d97d6bfd6", "size": 113083, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Documentation/Advanced.tex", "max_stars_repo_name": "karannewatia/SCALE-MAMBA", "max_stars_repo_head_hexsha": "467b33a6c80050789204ea3ee3b5cf0113354f85", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-04-26T06:10:49.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-26T06:10:49.000Z", "max_issues_repo_path": "Documentation/Advanced.tex", "max_issues_repo_name": "karannewatia/SCALE-MAMBA", "max_issues_repo_head_hexsha": "467b33a6c80050789204ea3ee3b5cf0113354f85", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Documentation/Advanced.tex", "max_forks_repo_name": "karannewatia/SCALE-MAMBA", "max_forks_repo_head_hexsha": "467b33a6c80050789204ea3ee3b5cf0113354f85", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.305425032, "max_line_length": 472, "alphanum_fraction": 0.6211455303, "num_tokens": 43652, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7431680086124812, "lm_q2_score": 0.46879062662624377, "lm_q1q2_score": 0.3483901964460228}} {"text": "\\section{Introduction}\n\nMeasurements of anomalies in potential fields, like gravity disturbances and\ntotal-field magnetic anomalies, are widely used in geophysical exploration for\ntheir low cost of acquisition.\nThese data can be surveyed using ground, airborne, shipborne, or satellite\nsystems.\nDuring ground surveys, the data are often gathered following irregular paths or\nnetworks along the surface of the terrain, leading to highly variable\nelevations in mountainous regions.\nAirborne and satellite surveys gather data along flight lines, producing\nclosely spaced measurements along almost straight lines but with larger spacing\nbetween adjacent lines.\nMeasurement height can also change because of the vertical movement of the\naircraft.\nProcessing of the data often involves interpolation onto a regular grid at\nconstant height, both to improve visualization for interpretation purposes as\nwell as to prepare the data for further processing and modelling (e.g.,\nreduction-to-the-pole, derivative calculations, upward continuation, Euler\ndeconvolution).\n\nSeveral methods exist in the literature for interpolation in two dimensions,\nfor example continuous curvature splines in tension \\citep{smith1990},\nbi-harmonic (thin-plate) splines \\citep{sandwell1987}, and kriging\n\\citep{hansen1993}.\nThese general-purpose methods have limitations when it comes to interpolating\npotential field data, namely\n(i)~they are not able to take into account the variable height of the\nobservation points and\n(ii)~the interpolating functions are not necessarily harmonic, which\nis the underlying assumption behind many processing techniques\n(e.g., upward continuation and vertical derivatives).\n\nA widely used method for interpolating gravity and magnetic data\nis the equivalent sources technique (also known as equivalent layer, radial\nbasis functions, or Green's functions interpolation).\nFirst introduced by \\citet{dampney1969}, the method consists in fitting a model\nof finite elementary sources to the data and using this model to predict new\ndata values.\nBesides interpolation, equivalent sources have been used for\nreduction-to-the-pole of magnetic data\n\\citep{silva1986, nakatsuka2006, guspi2009}, upward\ncontinuation \\citep{emilia1973, li2010}, joint processing of gravity gradient\ndata \\citep{barnes2011}, modelling the lithospheric magnetic field\n\\citep{kother2015}, recovering the magnetic induction vector from\ntotal-field magnetic anomalies \\citep{li2020}, and more.\n\nIt is also worth mentioning the least-squares collocation method\n(LSC), which is widely used in geodesy\n\\citep[][and references therein]{tscherning2015}.\nLCS is often applied to combine and interpolate different linear functionals of\nthe disturbing gravity potential (gravity anomalies, gravity disturbances,\ndeflections of the vertical, geoid height, et cetera).\nLike equivalent sources, collocation also requires the solution of a large\nlinear system of the order of the number of observed data.\nAs such, it's practical application suffers from the same computational\nchallenges.\n\nMany variants of the equivalent sources technique have been proposed, often\nattempting to obtain faster or more accurate solutions.\nThe key factors that vary between them are: (i) the type of source, (ii)\nthe location of the sources, and (iii) the solution strategy.\n\nThe most commonly used type of source is a point mass for gravity or dipole for\nmagnetics \\citep[e.g.,~][]{vonfrese1981, silva1986, mendonca1994,\nsiqueira2017}.\nHowever, right-rectangular prisms \\citep[e.g.,][]{barnes2011, jirigalatu2019,\nli2020} and tesseroids \\citep{bouman2016} have also been used successfully.\nIn fact, even point sources with a simple inverse distance function, instead of\nactual gravity or magnetic fields, can be used as\nequivalent sources \\citep{cordell1992}.\n\nThe location of sources often follows one of two strategies.\nThe most common approach is to distribute sources on a regular grid at a\nconstant depth \\citep[e.g.,~][]{leao1989, barnes2011, oliveira2013}.\nAlternatively, sources can be placed beneath each data point\n\\citep[e.g.,~][]{cordell1992, siqueira2017}.\nSome recent work by \\citet{li2020} places the sources in two overlapping layers\nat different depths.\n\nThe coefficients of the equivalent source model are often estimated through\ndamped least-squares.\nThis imposes a heavy computational load when the number of data points is\nlarge (e.g., airborne and satellite surveys).\nTo reduce the computational load, \\citet{mendonca1994} built the solution\niteratively by incorporating one data point at a time using the ``equivalent\ndata concept''.\n\\citet{leao1989} processed the input data using a moving window, only fitting the\ndata inside the window and predicting observations at its center.\n\\citet{li2010} and \\citet{barnes2011} apply different operations to generate a\nsparse representation of the sensitive matrix (respectively, wavelet\ncompression and quadtree discretization), which significantly improves the\nspeed of the least-squares solution.\n\\citet{oliveira2013} parametrized the equivalent layer as a piecewise bivariate\npolynomial function, reducing the number of parameters in the solution.\n\\citet{siqueira2017} developed an iterative solution in which the sensitivity\nmatrix is transformed into a diagonal matrix with constant terms through the\n``excess mass criterion''.\n\\citet{jirigalatu2019} applied the Gauss-FFT method to speed up the forward\nmodelling operations and solved the least-squares problem using steepest\ndescent to avoid calculating the Hessian matrix and solving linear systems.\n\nMany of the existing methods solve under-determined problems, requiring a much\nlarger number of equivalent sources than the number of data points.\nSome achieve greater efficiency by restricting their applications\nto specific data types \\citep{siqueira2017},\ninterpolating only on regular grids \\citep{leao1989},\nor requiring already gridded data \\citep{takahashi2020},\nto name a few.\nFurthermore, many of the optimizations proposed are also complex to implement\nin a computer program, limiting their wider adoption.\n\nIn the present study,\nwe propose two strategies for reducing the computational load of\nthe equivalent sources technique:\n\n\\begin{enumerate}\n \\item Reduce the number of equivalent sources for oversampled surveys\n through a \\emph{block-averaging} strategy while maintaining the quality\n of the solution.\n \\item Fit the equivalent source model iteratively along overlapping windows\n using a \\emph{gradient boosting} algorithm \\citep{friedman2001}.\n\\end{enumerate}\n\nThe first strategy consists in dividing the survey area into horizontal blocks\nand assigning a single source to each block, located at the median horizontal\nlocation of the data points.\nFor airborne, shipborne, and satellite surveys, which are oversampled along\ntracks, this can greatly reduce the size of the inverse problem while retaining\nthe same quality of interpolation.\n\nThe gradient boosting algorithm allows us to fit the equivalent source model\niteratively by operating on individual overlapping windows.\nAs a result, our method solves several much smaller least-squares problems\ninstead of a large one.\nThis has some similarities with the strategy used by \\citet{leao1989} but\nwithout the requirement for sources and predictions to be on regular grids.\n\nThrough tests on synthetic data, we show that:\n(i)~the \\emph{block-averaged} sources are able to achieve the same accuracy as\nother traditional equivalent source layouts while using a fraction of the\nnumber of sources, and\n(ii)~the \\emph{gradient boosting} algorithm greatly reduces the computational\nmemory required to fit very large datasets without sacrificing prediction\naccuracy.\nFinally, a combination of both strategies is used to process a collection of\napproximately 1.7 million ground gravity data measurements from Australia.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Methodology}\n\n\\subsection{The equivalent sources technique}\n\nWe will follow the ``generalized equivalent sources'' of \\citet{cordell1992}\nand assume that any harmonic function $d(\\mathbf{p})$ can be approximated by a\nsum of $M$ discrete point source effects\n\n\\begin{equation}\n d(\\mathbf{p})\n =\n \\sum\\limits_{j=1}^{M} \\frac{c_j}{\\left\\lVert \\mathbf{p} - \\mathbf{q}_j\n \\right\\rVert} \\ ,\n \\label{eq:eql-forward}\n\\end{equation}\n\n\\noindent in which\n$\\mathbf{p}$ and $\\mathbf{q}_j$ are, respectively, the position vectors in a 3D\nCartesian space of data and sources,\n$c_j$ is a scalar coefficient related to the point source located at\n$\\mathbf{q}_j$,\nand $\\lVert \\cdot \\rVert$ represents the $\\text{L}_2$ norm.\nThe horizontal and vertical distribution of sources is discussed in\nsection~\\ref{sec:source_distribution}.\n\nIn case we have values of the harmonic function at $N$ discrete points\n$\\{\\mathbf{p}_1\\ \\mathbf{p}_2\\ \\ldots\\ \\mathbf{p}_N\\}$,\nwe can write a set of $N$ equations of the form\n\n\\begin{equation}\n d_i\n =\n \\sum\\limits_{j=1}^{M} \\frac{c_j}{\\left\\lVert \\mathbf{p}_i - \\mathbf{q}_j\n \\right\\rVert}\n \\quad \\forall i=1,2,\\ldots,N\n \\ ,\n \\label{eq:forward-sum}\n\\end{equation}\n\n\\noindent where $d_i$ is the calculated value at point $\\mathbf{p}_i$.\nThese equations can also be expressed in matrix form as\n\n\\begin{equation}\n \\mathbf{d} = \\mathbf{A} \\mathbf{c} \\ ,\n \\label{eq:linear-problem}\n\\end{equation}\n\n\\noindent where $\\mathbf{d}$ is a column vector containing the $N$ predicted\nvalues at the observation points,\n$\\mathbf{c}$ is a column vector containing the $M$ coefficients $c_j$,\nand $\\mathbf{A}$ is the $N \\times M$ Jacobian matrix,\nwhose elements are\n\n\\begin{equation}\n a_{ij} = \\frac{1}{\\left\\lVert\\mathbf{p}_i - \\mathbf{q}_j\\right\\rVert}\n\\end{equation}\n\nFor a given set of $N$ observed data $\\mathbf{d}^o$,\nwe can find a least-squares solution to\nEq.~\\ref{eq:linear-problem} and obtain the values of\n$\\mathbf{c}$ that best fit the observations.\nThese coefficients can, in turn, be used to predict the value of the harmonic\nfunction at any other point outside of the sources by evaluating\nEq.~\\ref{eq:eql-forward}.\nGridding and upward continuation can thus be achieved by predicting values on\npoints that fall on a regular grid or at different heights, respectively.\n\n\n\\subsection{Damped least-squares solution}\n\\label{sec:eql_inversion}\n\nWe can obtain the values of the source coefficients $\\mathbf{c}$ that best\nfit the observed field values $\\mathbf{d}^o$ by minimizing the goal function\n\n\\begin{equation}\n \\phi(\\mathbf{c}) =\n \\left[\\mathbf{d}^o - \\mathbf{A}\\mathbf{c}\\right]\\trans\n \\mathbf{W}\n \\left[\\mathbf{d}^o - \\mathbf{A}\\mathbf{c}\\right]\n + \\lambda_d\\ \\mathbf{c}\\trans\\mathbf{c}\n \\ ,\n \\label{eq:misfit-unscaled}\n\\end{equation}\n\n\\noindent where\n$\\mathbf{W}$ is a $N \\times N$ diagonal matrix of data weights and\n$\\lambda_d$ is a positive \\emph{damping} parameter with the same units as the\nJacobian matrix elements.\nThe second term on the right-hand side of Eq.~\\ref{eq:misfit-unscaled} is the\nzeroth-order Tikhonov regularization \\citep{tikhonov1977}, also known as a\ndamping regularization, that is used to stabilize the solution.\n\nThe damping parameter controls the amount of regularization that will be\napplied.\nAn overly large value would generate a smooth solution that fails to reproduce\nthe high frequency components of the data, while an overly small value would\nresult in over-fitting, thus failing to produce realistic interpolation results\n\\citep{martinez2016}.\nThe range of acceptable values for the damping parameter $\\lambda_d$ will\ndepend on the values of the Jacobian matrix $\\mathbf{A}$ and the coefficients.\nConsequently, this range will vary (often dramatically) between datasets,\nmaking it difficult to choose an appropriate value in practice.\n\nTo solve this issue, we first scale the Jacobian matrix so that its elements\nare dimensionless and each column has unit variance.\nWe define a diagonal matrix $\\mathbf{S}$\n\n\\begin{equation}\n \\mathbf{S} =\n \\begin{bmatrix}\n \\sigma_1 & 0 & \\cdots &0 \\\\\n 0 & \\sigma_2 & \\cdots &0 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & \\cdots & \\sigma_M\n \\end{bmatrix}_{M \\times M}\n ,\n\\end{equation}\n\n\\noindent in which $\\sigma_j$ is the standard deviation of the $j$-th column of\n$\\mathbf{A}$.\nWe then write the forward problem in Eq.~\\ref{eq:linear-problem} as\n\n\\begin{equation}\n \\mathbf{d}\n =\n \\mathbf{A} \\mathbf{S}\\inv \\mathbf{S} \\mathbf{c}\n =\n \\left[\n \\mathbf{A} \\mathbf{S}\\inv\n \\right]\n \\left[\n \\mathbf{S} \\mathbf{c}\n \\right]\n =\n \\mathbf{B} \\mathbf{m}\n\\end{equation}\n\n\\noindent where $\\mathbf{B} = \\mathbf{A} \\mathbf{S}\\inv$ is the scaled and\ndimensionless Jacobian matrix\nand $\\mathbf{m} = \\mathbf{S} \\mathbf{c}$ is a vector containing scaled\ncoefficients with the same units as the data.\n\nThe goal function defined in Eq.~\\ref{eq:misfit-unscaled} can be\nrewritten as\n\n\\begin{equation}\n \\phi(\\mathbf{m}) =\n \\left[\\mathbf{d}^o - \\mathbf{B}\\mathbf{m}\\right]\\trans\n \\mathbf{W}\n \\left[\\mathbf{d}^o - \\mathbf{B}\\mathbf{m}\\right]\n + \\lambda\\ \\mathbf{m}\\trans\\mathbf{m}\n \\ ,\n \\label{eq:misfit}\n\\end{equation}\n\n\\noindent where $\\lambda$ is a \\emph{dimensionless} damping parameter and\nregularization is applied on the scaled coefficients $\\mathbf{m}$ instead of\n$\\mathbf{c}$.\nUsing a dimensionless damping parameter allows us to narrow the range of values\nof $\\lambda$ that would generate the most accurate predictions, irrespective\nof the dataset and its units.\nFrom experience, we recommend searching for suitable $\\lambda$ values between\n$10^{-6}$ and $10^{4}$ varying by order-of-magnitude.\nThe choice of the damping and other hyper-parameters, like the source depth,\ncould be done through well-established statistical methods, such as\ncross-validation.\n\nThe vector of scaled coefficients $\\hat{\\mathbf{m}}$ that minimizes the goal\nfunction can be found by solving the \\emph{normal equation system}\n\\citep{menke1989}\n\n\\begin{equation}\n \\left[\n \\mathbf{B}\\trans \\mathbf{W} \\mathbf{B} + \\lambda \\mathbf{I}\n \\right]\n \\hat{\\mathbf{m}} =\n \\mathbf{B}\\trans\\mathbf{W}\n \\mathbf{d}^o.\n \\label{eq:least_squares_solution}\n\\end{equation}\n\nOnce the scaled coefficients are obtained, the estimated unscaled coefficients\n$\\hat{\\mathbf{c}}$ can be calculated by removing the scaling factor\n\n\\begin{equation}\n \\hat{\\mathbf{c}} = \\mathbf{S}\\inv \\hat{\\mathbf{m}} \\ .\n\\end{equation}\n\n\\noindent The forward modeling operations used to perform predictions\n(e.g., for interpolation and upward continuation) are left unchanged by\nusing vector $\\hat{\\mathbf{c}}$ instead of $\\hat{\\mathbf{m}}$.\n\n\n\\subsection{Gradient boosting}\n\nGradient boosting was first introduced by \\citet{friedman2001, friedman2002} as\na method for fitting additive parametric models of the form\n\n\\begin{equation}\n d = \\sum_{k=1}^K \\alpha_k f(\\mathbf{c}_k),\n\\end{equation}\n\n\\noindent where $\\alpha_k$ is a scalar coefficient called the \\emph{step-size}\nand $f$ is a function of the parameter vector $\\mathbf{c}_k$.\nFor linear problems, these additive models can be written as the matrix\nequation\n\n\\begin{equation}\n \\mathbf{d} = \\sum_{k=1}^K \\mathbf{A}_k \\mathbf{c}_k \\ .\n \\label{eq:gb-linear-model}\n\\end{equation}\n\n\\noindent Because of the linearity of the $f(\\mathbf{c}_k)$ functions, the\n$\\alpha_k$ step-size parameters can be incorporated into the parameter vector\n$\\mathbf{c}_k$.\n\nWe can transform our equivalent source problem in\nEq.~\\ref{eq:linear-problem} into an additive model by following these\nsteps:\n\n\\begin{enumerate}\n \\item Define a set of $M$ equivalent sources distributed throughout the\n survey area (see section \\ref{sec:source_distribution} for details).\n \\item Define a set of $K$ overlapping windows of equal size that cover the\n survey area.\n \\item Create $K$ separate sets of equivalent sources, one for each window.\n Each set will be formed by the portion of the original $M$ sources that\n fall inside the respective window.\n Since the windows overlap, the total number of sources from all sets will\n be greater than $M$.\n \\item Define vector $\\mathbf{c}_k$ as the $M_k$ coefficients of the\n equivalent sources of the $k$-th window.\n \\item Define matrix $\\mathbf{A}_k$ as the $N \\times M_k$ Jacobian matrix\n between the sources in the $k$-th window and all $N$ data points of the\n survey.\n \\item Model the predicted data as a superposition of the effects of the $K$\n separate sets of equivalent sources (i.e., Eq.~\\ref{eq:gb-linear-model}).\n\\end{enumerate}\n\nThe gradient boosting algorithm works by fitting each component of the\nadditive model, one at a time, to the residuals of the previous component.\n\\citet{friedman2001} demonstrates that this corresponds to a steepest-descent\noptimization in the so-called ``function space''.\nThe adaptation of the gradient boosting method to find the damped least-squares\nsolutions for the $K$ parameter vectors $\\mathbf{c}_k$ in\nEq.~\\ref{eq:gb-linear-model} is presented in\nAlgorithm~\\ref{alg:gradient_boosting}.\n\n\\begin{algorithm}[!h]\n \\DontPrintSemicolon\n \\setstretch{1.5}\n Define the residual vector $\\mathbf{r}_{0} = \\mathbf{d}^o$ \\;\n \\For{ $k = 1$ \\KwTo $K$ }{\n\n\n Calculate the $N \\times M_k$ Jacobian matrix $\\mathbf{A}_k$\n \\;\n\n $\\mathbf{B}_k = \\mathbf{A}_k \\mathbf{S}_k\\inv$\n \\nllabel{alg:scale}\n \\;\n\n $\n \\hat{\\mathbf{m}}_k = \\left[\\mathbf{B}_k\\trans \\mathbf{W}_k \\mathbf{B}_k +\n \\lambda \\mathbf{I} \\right]\\inv \\mathbf{B}_k\\trans \\mathbf{W}_k\n \\mathbf{r}_{k-1}\n $\n \\nllabel{alg:fit}\n \\;\n\n $\\hat{\\mathbf{c}}_k = \\mathbf{S}_k\\inv \\hat{\\mathbf{m}}_k$\n \\nllabel{alg:unscale}\n \\;\n\n $\\mathbf{d}_k = \\mathbf{A}_k \\hat{\\mathbf{c}}_k$\n \\nllabel{alg:predicted}\n \\;\n\n $\\mathbf{r}_k = \\mathbf{r}_{k - 1} - \\mathbf{d}_k$\n \\nllabel{alg:residual}\n \\;\n }\n \\BlankLine\n \\setstretch{1}\n \\caption{Gradient boosting solution for damped least-squares regression.}\n \\label{alg:gradient_boosting}\n\\end{algorithm}\n\nAfter all $\\mathbf{c}_k$ coefficients vectors are estimated, we can predict the\neffect of the additive equivalent source model on any point through the\nsummation\n\n\\begin{equation}\n d(\\mathbf{p}) =\n \\sum\\limits_{k=1}^K \\sum\\limits_{j=1}^{M_k}\n \\frac{{c_k}_j}{\\left\\lVert \\mathbf{p} - {\\mathbf{q}_k}_j \\right\\rVert}\n \\ ,\n \\label{eq:eql-forward-gb}\n\\end{equation}\n\n\\noindent\nin which ${c_k}_j$ is the $j$-th element of the $\\mathbf{c}_k$ vector and the\n${\\mathbf{q}_k}_j$ is the position vector of the $j$-th source of the $k$-th\nwindow.\n\nTo improve the convergence of the algorithm, \\citet{friedman2002} suggests\nintroducing randomness into the fitting process. We achieve this by randomizing\nthe order in which the $K$ windows are used in the gradient boosting algorithm.\nSection~\\ref{sec:gb_interpolation} explores the effect of randomization in the\nconvergence rate of the algorithm and the accuracy of the interpolation.\n\nThe $\\mathbf{A}_k$ matrices have only $N \\times M_k$ elements\n(where $M_k$ is the number sources on the $k-$th window), which can be\nconsiderably smaller than the $N \\times M$ elements of $\\mathbf{A}$.\nTherefore, the gradient boosting method allows us to fit\nequivalent source models that would produce Jacobian matrices that are larger\nthan the available computer memory.\nFurthermore, we can increase or decrease the size of the overlapping windows as\nneeded depending on the number of sources in the model and the available\ncomputer memory.\n\nWe can improve the efficiency of the algorithm further by:\n\n\\begin{enumerate}\n \\item Using only the $N_k$ data points that fall within the $k$-th window for\n fitting the sources (steps \\ref{alg:scale} and \\ref{alg:fit} of\n algorithm~\\ref{alg:gradient_boosting}).\n By doing so, we can replace the $N \\times M_k$ Jacobian matrix $\\mathbf{A}_k$\n with the smaller $N_k \\times M_k$ matrix $\\tilde{\\mathbf{A}}_k$.\n We still use all $N$ data points when calculating the predicted data and\n residuals (steps \\ref{alg:predicted} and \\ref{alg:residual} of\n algorithm~\\ref{alg:gradient_boosting}).\n \\item The forward modeling operation performed in step \\ref{alg:predicted}\n can be done by a summation (Eq.~\\ref{eq:forward-sum}) instead of a\n matrix-vector product, which allows us to avoid computing and storing the\n larger $N \\times M_k$ matrix $\\mathbf{A}_k$ at any point.\n\\end{enumerate}\n\nAlgorithm~\\ref{alg:gradient_boosting_window} is the final\n\\textit{gradient-boosted equivalent sources algorithm} which incorporates these\nchanges.\nFigure~\\ref{fig:gradient-boosting-schematics} shows a sketch of the algorithm\nsteps applied a set of observation points that simulate a ground survey and\nlocating one source below each data point.\n\n\\begin{algorithm}[!h]\n \\DontPrintSemicolon\n \\setstretch{1.5}\n Define the residual vector $\\mathbf{r}_{0} = \\mathbf{d}^o$ \\;\n \\For{ $k = 1$ \\KwTo $K$ }{\n\n Select weights $\\tilde{\\mathbf{W}}_k$ and residuals\n $\\tilde{\\mathbf{r}}_{k - 1}$ for data points inside the $k$-th window\n \\;\n\n Calculate Jacobian matrix $\\tilde{\\mathbf{A}}_k$ with data points and\n sources inside the $k$-th window\n \\;\n\n $\\mathbf{B}_k = \\tilde{\\mathbf{A}}_k \\mathbf{S}_k\\inv$\n \\;\n\n $\n \\hat{\\mathbf{m}}_k = \\left[\n \\mathbf{B}_k\\trans \\tilde{\\mathbf{W}}_k \\mathbf{B}_k +\n \\lambda \\mathbf{I} \\right]\\inv \\mathbf{B}_k\\trans \\tilde{\\mathbf{W}}_k\n \\tilde{\\mathbf{r}}_{k-1}\n $\n \\;\n\n $\\hat{\\mathbf{c}}_k = \\mathbf{S}_k\\inv \\hat{\\mathbf{m}}_k$\n \\;\n\n Calculate $ \\mathbf{d}_k $, where\n $\n {d_k}_i\n =\n \\sum\\limits_{j=1}^{M_k} \\dfrac{{c_k}_j}{\\left\\lVert \\mathbf{p}_i -\n {\\mathbf{q}_k}_j\n \\right\\rVert}\n \\quad \\forall\\ i=1\\ \\text{to}\\ N\n $\n \\;\n\n $\\mathbf{r}_k = \\mathbf{r}_{k - 1} - \\mathbf{d}_k$\n \\;\n }\n \\BlankLine\n \\setstretch{1}\n \\caption{Gradient-boosted equivalent sources algorithm.}\n \\label{alg:gradient_boosting_window}\n\\end{algorithm}\n\n\\begin{figure*}[tb]\n \\centering\n \\includegraphics[width=\\linewidth]{figs/gradient-boosting-schematics.pdf}\n \\caption{\n Sketch of the gradient-boosted equivalent source algorithm.\n Data points are represented by blue upwards-facing triangles,\n equivalent sources by orange dots, data residuals by red downwards-facing\n triangles, and the current window by black dashed lines.\n The algorithm starts by selecting the data and sources\n inside the first window and estimating the source coefficients\n using the selected data points.\n Then, the effect of the estimated sources is predicted on all data\n points and used to calculate the residuals.\n Another window is used to select residuals and sources and\n estimate the coefficients using the selected residuals instead of the\n original data.\n Again, the effect of the estimated sources is predicted on all data points\n and the residuals are updated.\n These steps are repeated for every window in a randomized order.\n }\n \\label{fig:gradient-boosting-schematics}\n\\end{figure*}\n\nIt is worth noting that two sets of equivalent sources obtained through two\nadjacent overlapping windows have some portion of the sources on the same\nlocations, specifically the ones that fall on the intersection between the two\nwindows.\nWe can interpret this as the gradient-boosting algorithm fitting the source\ncoefficients multiple times: one time for every window that covers each source.\nThis fact can be exploited in order to save computer memory.\nInstead of storing all of the $\\mathbf{c}_k$ vectors\n(Eq.~\\ref{eq:gb-linear-model}), we can initialize a single $\\mathbf{c}$ vector\nwith zeros, where each element represents the coefficient of each one of the\noriginal $M$ sources.\nAfter each iteration of the gradient-boosting algorithm, we add the estimated\ncoefficients $\\hat{\\mathbf{c}}_k$ to the corresponding elements of vector\n$\\mathbf{c}$.\nBecause the forward modelling function is linear, we can safely compute the\nresulting field through Eq.~\\ref{eq:eql-forward} instead of\nEq.~\\ref{eq:eql-forward-gb}.\nThis way, the memory needed to store the entire set of estimated coefficients\nis limited to a single vector of $M$ elements.\n\nOur gradient boosting algorithm for overlapping windows is similar to the\n``bootstrap inversion'' used in \\citet{vonfrese1988}, which also iteratively\nfits portions of an equivalent source model to the data residuals.\nThe key differences are that in our method:\n(i)~the sources in the overlapping portions of the windows are fitted more than\nonce, allowing the algorithm to self-correct for poor solutions to any given\nwindow;\n(ii)~we use only data points within the window when fitting, what enables the\nuse of larger datasets.\n\n\n\n\\subsection{Location of sources}\n\\label{sec:source_distribution}\n\nThe ideal number of sources and their locations, both horizontally and\nvertically, has been debated since the inception of the equivalent sources\ntechnique with \\citet{dampney1969}.\nThe choices made regarding these parameters can play an important role on the\naccuracy of the predictions and the computational resources needed to estimate\nthe source coefficients.\nAn ideal distribution of sources should simultaneously be able to reproduce the\nmeasured data on the survey points, make accurate predictions on non-surveyed\nlocations, and minimize the required computational resources.\n\nA large number of evenly distributed sources along the survey region are\ncapable of reproducing the observed data.\nNevertheless, the computational load can be prohibitive and such\nunderdetermined problems are prone to overfitting the data, leading to poor\npredictive power when interpolating and extrapolating.\nOn the other hand, using few sources will reduce the computational requirements\nbut the model may be incapable of reproducing the full spectral content of the\nmeasured data.\n\nParticular survey characteristics also play a role in the choice of equivalent\nsource distribution.\nIn a ground survey, observations are usually located along irregular paths and\nscattered points.\nThe coverage of the survey region is often uneven, leaving large areas without\nany observation.\nOn the other hand, observations from airborne surveys are located along almost\nstraight and closely spaced flight lines.\nMeasurements are usually taken at a high temporal frequency, leading to\nobservation points along the flight lines that are several times closer to each\nother than the flight line spacing.\nThis creates a bias in the sampling, which can cause aliasing artifacts in\ngridded products.\n\n\\subsubsection{Horizontal source layouts}\n\n\\begin{figure*}[tb]\n \\includegraphics[width=\\linewidth]{figs/source-layouts-schematics.pdf}\n \\caption{\n Sketch of different horizontal layouts for equivalent source models.\n Blue points represent the locations of observations and orange points\n represent the locations of equivalent sources according to different\n layout strategies.\n (a)~Set of \\SourceLayoutsSchematicsObservations{} observation points\n that simulate a ground survey.\n (b)~Location of the \\SourceLayoutsSchematicsSourceBelowData{} sources\n obtained through the \\emph{sources below data} layout.\n (c)~Location of the \\SourceLayoutsSchematicsGridSources{} sources\n obtained through the \\emph{regular grid} layout.\n (d)~Location of the \\SourceLayoutsSchematicsBlockAveragedSources{}\n sources obtained through the \\emph{block-averaged sources} layout.\n Grey dashed lines represent the spatial blocks within which the median\n observation location is calculated.\n }\n \\label{fig:source_layouts}\n\\end{figure*}\n\nThe most widely used layouts for distributing equivalent sources horizontally\nare:\n\n\\begin{enumerate}\n \\item\n \\emph{Sources below data points}: one equivalent source is placed at the\n horizontal location of each data point (Fig.~\\ref{fig:source_layouts}b).\n Therefore, the number of sources is equal to the number of observations\n ($M=N$).\n \\item\n \\emph{Regular grid}: a homogeneous distribution of point sources below the\n survey region (Fig.~\\ref{fig:source_layouts}c). A padding region is often\n added to help reduce edge effects. In practice, it often leads to\n underdetermined problems since a large number of sources is required\n ($M>N$).\n\\end{enumerate}\n\nFor ground surveys, the \\emph{regular grid} layout needs a sufficiently\nsmall grid spacing to be able to fit the observed data.\nThis creates an unnecessarily large number of sources in areas where no\nobservations exist.\nIn contrast, the \\emph{sources below data} layout is more likely to accurately\nfit the observed data with many fewer sources, reducing the computational load.\nBut when applied to airborne surveys, the \\emph{sources below data} layout may\nplace an undesirably large number of sources along the flight paths.\nThis could lead to aliasing effects on the predicted values, such as the\nstripes parallel to flight lines that are often observed when gridding airborne\nmagnetic data.\nThe \\emph{regular grid} layout can avoid this effect by evenly\ndistributing sources and using a continuous source layer (e.g.,\nright-rectangular prisms or tesseroids).\n\nWe propose a new way of distributing equivalent sources horizontally that could\nsimultaneously reduce the computational load and mitigate some of the drawbacks\nof existing layouts.\nIn the \\emph{block-averaged sources} layout,\npoint sources are placed in the average\nposition of data points that fall within specified spatial blocks\n(Fig.~\\ref{fig:source_layouts}d).\nThis is done by:\n\n\\begin{enumerate}\n \\item Dividing the survey region into rectangular blocks of equal size.\n \\item \\label{item:median-position} Computing the median horizontal position\n of the observation points that fall inside each block. Blocks without\n any observation point are omitted.\n \\item Assign one point source to each of the median horizontal positions\n calculated in step \\ref{item:median-position}.\n\\end{enumerate}\n\nThe number of sources created by this new layout will be less than the number\nof observations if the block size is chosen appropriately (i.e., making sure\nthat blocks are large enough to contain more than a single data point).\nThe overdetermined problem that arises from this layout has a lower\ncomputational load and is less prone to overfitting the data since the model\ncomplexity is lower.\nMoreover, the block averaging process can balance the spacing between sources\nalong a flight line and between adjacent lines, helping to reduce aliasing\neffects in the generated grids.\nIn Section~\\ref{sec:synthetic_distributions}, we demonstrate through tests on\nsynthetic data that the block-averaged sources layout is able to interpolate\nwith comparable accuracy to other layouts while using a fraction of the\nequivalent sources.\n\n\n\\subsubsection{Depth of sources}\n\n\\begin{figure*}[tb]\n \\includegraphics[width=\\linewidth]{figs/depth_types.pdf}\n \\caption{\n Examples of different strategies for assigning depths to equivalent\n sources.\n Here we assign one source for each\n observation point, located at the same horizontal coordinates as the\n data points.\n Source depths are\n (a)~a \\emph{constant depth} at a chosen vertical coordinate,\n (b)~a \\emph{relative depth} determined by uniformly shifting downward\n the vertical coordinate of data points,\n and\n (c)~a \\emph{variable depth} determined by shifting the vertical\n coordinates of the observation points by an amount proportional to the\n average distance to neighbouring sources.\n The distance between data points and their respective sources (a)\n depends on observation height, (b) is constant, and (c) is proportional\n to the horizontal distribution of sources.\n Notice how the closely spaced sources in the middle of the profile (c)\n are shallower than their counterparts in (b).\n }\n \\label{fig:depth_types}\n\\end{figure*}\n\nIt is widely known from potential theory that the depth of a point source\ninfluences the wavelength of the observed field at the surface.\nThis makes the source depth a key parameter affecting the outcome of\ninterpolation and other operations done with equivalent sources.\nSeveral different strategies for assigning the depths of equivalent sources\nhave been proposed in the literature.\nHere, we will highlight the following (Fig.~\\ref{fig:depth_types}):\n\n\\begin{enumerate}\n \\item\n \\emph{Constant depth}:\n The simplest option is to locate all sources at the same depth\n (Fig.~\\ref{fig:depth_types}a).\n If the measurements were taken at significantly different altitudes, some\n measurements will be more distant to the sources than others,\n which may create problems for reproducing short wavelengths in high\n altitude points.\n \\item\n \\emph{Relative depth}:\n The depths of sources are determined by shifting the vertical coordinate of\n data points downward by a fixed amount (Fig.~\\ref{fig:depth_types}b).\n The sources will not all be at the same vertical coordinate, but they will\n all be at the same vertical distance from the observation points.\n \\item\n \\emph{Variable depth}:\n The depths of sources are proportional to the horizontal distance to the\n nearest neighbouring data points or sources (Fig.~\\ref{fig:depth_types}c).\n Different variations of this strategy have been proposed before, for\n example \\citet{cordell1992}, \\citet{guspi2004}, and \\citet{guspi2009}.\n The rationale for this strategy is that if a survey has data points\n clustered in some areas, we may\n want the sources below those areas to be shallower in order to preserve the\n shorter wavelengths that can be measured.\n\\end{enumerate}\n\nOur approach to the \\emph{variable depth} strategy will be:\n\n\\begin{equation}\n z = z_{obs} + \\Delta z + \\alpha h,\n \\label{eq:variable_depth}\n\\end{equation}\n\n\\noindent\nin which $z$ is the vertical coordinate (positive downwards) of an equivalent\nsource,\n$\\Delta z$ is a relative depth shift that is the same for all sources,\n$\\alpha$ is an dimensionless depth factor,\n$h$ is the median horizontal distance to the $k$ nearest neighbouring sources,\nand\n$z_{obs}$ is a vertical observation coordinate that will depend on the\nhorizontal layout strategy.\nFor \\emph{sources below data}, it is the vertical coordinate of the data point\ncorresponding to the given source.\nFor \\emph{regular grid}, it can be interpolated from the vertical coordinates\nof all data points.\nFinally, for \\emph{block-averaged sources} it will be the median vertical\ncoordinate of the data within the corresponding block.\n\nIn Section~\\ref{sec:synthetic_distributions}, we test the effectiveness each of\nthese strategies on synthetic data.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Tests on synthetic data}\n\n\\begin{figure*}[tb]\n \\includegraphics[width=\\linewidth]{figs/synthetic-survey-layouts.pdf}\n \\caption{\n Observation heights and gravity values for the synthetic ground (a-b)\n and airborne (c-d) surveys.\n Heights are given in meters above the zero height plane.\n The synthetic gravity data are contaminated with pseudo-random Gaussian\n noise with zero mean and \\SurveyNoise{} standard deviation.\n }\n \\label{fig:synthetic-layouts}\n\\end{figure*}\n\n\\begin{figure}[tb]\n \\includegraphics[width=\\linewidth]{figs/target-grid.pdf}\n \\caption{\n Pseudo-color map of the target grid of synthetic gravity data. The grid\n is composed of \\TargetEastingSize{}$\\times$\\TargetNorthingSize{} points\n with a spacing of \\TargetSpacing{}. The grid height is \\TargetHeight{}\n above the zero height plane.\n }\n \\label{fig:synthetic-target}\n\\end{figure}\n\nWe have used synthetic gravity datasets to test the interpolation accuracy of\nthe difference horizontal and vertical source distribution strategies as well\nas the gradient-boosted equivalent sources method.\nTo generate the data, we created a model of \\NPrisms{} right-rectangular\nprisms,\ndistributed in a \\ModelEasting{}$\\times$\\ModelNorthing{} area with depths\nvarying between \\ModelDepth{} and zero.\nThe density contrast of prisms ranges from \\ModelMinDensity{} to\n\\ModelMaxDensity{}.\nThe model includes prisms of different shapes, sizes, and depths to create\ngravity disturbances with a variety of wavelengths.\n\nWe created two synthetic datasets from the model, one simulating a ground\nsurvey and another an airborne acquisition (Fig.~\\ref{fig:synthetic-layouts}).\nTo create the synthetic ground survey, we selected measurement positions from a\nportion of a public domain gravity dataset for Southern Africa, available\nthrough the NOAA National Centers for Environmental Information (NCEI).\nFor the synthetic airborne survey, we used a portion of the Great Britain\nAeromagnetic Survey acquired by Hunting Geology and Geophysics Ltd and Canadian\nAeroservices Ltd between 1955 and 1965 and made publicly available by the\nBritish Geological Survey (BGS).\nIn both cases, we rescaled the horizontal coordinates of each survey portion to\nspan an area of \\SurveyEasting{}$\\times$\\SurveyNorthing{}, matching the model\ndimensions.\nThe ground survey contains \\GroundSurveyPoints{} observations distributed at\nheights between \\GroundSurveyMinHeight{} and \\GroundSurveyMaxHeight{}\n(Fig.~\\ref{fig:synthetic-layouts}a).\nThe airborne survey has \\AirborneSurveyPoints{} observations at heights between\n\\AirborneSurveyMinHeight{} and \\AirborneSurveyMaxHeight{}\n(Fig.~\\ref{fig:synthetic-layouts}c).\n\nThe vertical component of the gravitational acceleration generated by the\nmodel was computed using the method of \\citet{nagy2000, nagy2002}\nwith recent modifications by \\citet{fukushima2020},\nas implemented in the open-source software Harmonica \\citep{harmonica2020}.\nWe generated a \\emph{target grid} of\n\\TargetEastingSize{}$\\times$\\TargetNorthingSize{} points with a spacing of\n\\TargetSpacing{} and located \\TargetHeight{} above the zero height plane\n(Fig.~\\ref{fig:synthetic-target}) to serve as a reference when calculating the\ninterpolation error.\nWe then generated synthetic ground (Fig.~\\ref{fig:synthetic-layouts}b) and\nairborne (Fig.~\\ref{fig:synthetic-layouts}d) data to which we added\npseudo-random Gaussian noise with zero mean and \\SurveyNoise{} standard\ndeviation.\n\n\n\\subsection{Source distribution strategies}\n\\label{sec:synthetic_distributions}\n\n\\begin{figure*}[p]\n \\includegraphics[width=\\linewidth]{figs/ground_survey_differences.pdf}\n \\caption{\n Pseudo-color maps of the differences between the target grid and the\n interpolated synthetic ground survey data produced by each source\n distribution strategy.\n The black dots represent the horizontal location of the synthetic data\n points. The RMS error and total number of equivalent sources is\n reported for each strategy at the top of the respective maps.\n }\n \\label{fig:ground-survey-differences}\n\\end{figure*}\n\n\\begin{figure*}[p]\n \\includegraphics[width=\\linewidth]{figs/airborne_survey_differences.pdf}\n \\caption{\n Pseudo-color maps of the differences between the target grid and the\n interpolated synthetic airborne survey data produced by each source\n distribution strategy.\n The black dots represent the horizontal location of the synthetic data\n points. The RMS error and total number of equivalent sources is\n reported for each strategy at the top of the respective maps.\n }\n \\label{fig:airborne-survey-differences}\n\\end{figure*}\n\nWe investigated the effect on interpolation accuracy of different strategies\nfor distributing the equivalent sources horizontally and vertically.\nTo do this, we used the damped least-squares solution described in\nSection~\\ref{sec:eql_inversion} (without gradient boosting) to interpolate the\nsynthetic datasets (Fig.~\\ref{fig:synthetic-layouts}) and compared the results\nagainst the target grid (Fig.~\\ref{fig:synthetic-target}).\nThis process was repeated for each combination of horizontal layout\n(\\emph{sources below data} and \\emph{block-averaged sources}) and depth type\n(\\emph{constant}, \\emph{relative}, and \\emph{variable}) and for regular grid\nsources with a constant depth, totalling 7 different combinations.\n\nEach source distribution strategy requires certain hyper-parameters to be\nchosen in order to build the set of point sources.\nFor example, using a constant depth needs the definition of the depth and using\nblock-averaged sources requires the definition of the block size.\nThe predictive capabilities of the equivalent sources depend on the choice of\nthese hyper-parameters.\nTo ensure that our comparisons are fair, we perform an exhaustive search over\ncombinations of hyper-parameter values (including the damping parameter from\nEq.~\\ref{eq:misfit}) to obtain the best prediction that can be achieved by each\nsource distribution strategy.\nHere, the best prediction is defined as the one that minimizes the root\nmean-square error (RMS) between interpolated values and the target grid\n(Fig.~\\ref{fig:synthetic-target}).\nThe parameter values used in these searches and the one producing the smallest\nRMS error are outlined in Tables~\\ref{tab:parameters-ground-survey}\nand~\\ref{tab:parameters-airborne-survey}.\n\nFig.~\\ref{fig:ground-survey-differences}\nand~\\ref{fig:airborne-survey-differences} show the differences between the\ntarget grid and the best prediction achieved by each source distribution\nstrategy for the ground and airborne synthetic surveys, respectively.\nFor the synthetic ground survey, the horizontal layouts produced similar RMS\nvalues of approximately 0.8\\mGal{} regardless of the depth type, with the\nexception of the regular grid layout which produced a larger RMS of\n\\BestGroundGridSourcesConstantDepthRms{}\\mGal{}.\nThe differences between the target grid and the interpolated values are larger\nin regions of poor data coverage.\nEdge effects are present for all strategies but are noticeably smaller for the\ncombination of block-averaged sources with a variable depth based on the\nnearest neighbour distance.\nFor the synthetic airborne survey, all strategies (including the regular grid)\nproduced similar RMS errors of approximately 0.3\\mGal{}.\nThe maps of the differences between the target grid and interpolation results\nare visually indistinguishable from each other.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\subsection{Window size and overlap in gradient boosting}\n\\label{sec:window_size_and_overlap}\n\nWe assessed the trade-offs in interpolation accuracy and computation time of\nthe gradient-boosted equivalent sources algorithm as a function of the two key\ncontrolling factors: the window size and the amount of overlap between adjacent\nwindows.\nThe comparisons were performed against a regular least-squares solution\n(Eq.~\\ref{eq:least_squares_solution}) using the synthetic airborne data\n(Fig.~\\ref{fig:synthetic-layouts}c-d).\nTo avoid biasing the results, we used the same locations of equivalent sources\nfor both the regular and gradient-boosted interpolations, namely\nblock-averaged sources with a block size of\n\\BestAirborneBlockAveragedSourcesRelativeDepthSpacing\\m{} and a\nrelative depth of\n\\BestAirborneBlockAveragedSourcesRelativeDepthDepth\\m{}.\n\n\\subsubsection{Window size}\n\\label{sec:window_size}\n\nThe size of the windows controls the size of the Jacobian matrices\n$\\tilde{\\mathbf{A}}_k$ by limiting the number of data points and equivalent\nsources used in each step of the gradient-boosting algorithm\n(Alg.~\\ref{alg:gradient_boosting_window}).\nThus, using smaller windows will reduce the total amount of computer memory\nrequired to estimate the source coefficients.\nNevertheless, smaller windows may produce less accurate interpolations by\nfailing to achieve the global minimum of the goal function in\nEq.~\\ref{eq:misfit}.\nThe window size might also impact the computation time in non-intuitive ways\nsince smaller windows generate smaller least-squares problems but also require\nmore gradient-boosting iterations.\n\nWe calculated the interpolation RMS error (between the interpolated grid and\nthe target grid in Fig.~\\ref{fig:synthetic-target}) and computation time for a\nfixed window overlap of 50\\% and several window sizes.\nTo avoid any biases introduced by the shuffling of windows, the calculations\nwere repeated using different seeds for the pseudo-random number generator used\nin the shuffling.\nFig.~\\ref{fig:gradient-boosted-comparison}a shows the RMS error and\nFig.~\\ref{fig:gradient-boosted-comparison}c shows the computation time\nrequired for estimating the source coefficients, both as functions of\nthe window size.\n\n\\begin{figure*}[tb]\n \\includegraphics[width=\\linewidth]{figs/gradient-boosted-comparisons.pdf}\n \\caption{\n Interpolation RMS error (a-b) and relative computation time (c-d) for\n regular least-squares equivalent sources (orange dashed lines) and\n gradient-boosted equivalent sources (blue dots and error bars).\n Window overlap is given as a percentage of the window size (an overlap\n of 50\\% means that two adjacent windows share an area half of the size\n of the entire window).\n For gradient-boosting, the RMS errors and computation times are the\n means (error bars are 1 standard deviation) of results using different\n seeds for the pseudo-random number generator.\n Computation time is the ratio between the time required to estimate the\n source coefficients for the gradient-boosted and the regular equivalent\n sources.\n}\n \\label{fig:gradient-boosted-comparison}\n\\end{figure*}\n\nThese results show that the interpolation error for gradient-boosting is\ngenerally larger than the error for regular equivalent sources.\nThe error decreases asymptotically to within $\\sim 40\\%$ of the regular\nequivalent sources for windows with an area greater than $\\sim 10\\%$ of the\nsurvey area.\nThe computation time similarly decreases with window size, with the\ngradient-boosting being generally faster than the regular equivalent sources\nfor windows with an area greater than $\\sim 5\\%$ of the survey area.\nAs the window size increases, both RMS error and computation time appear to\nstabilize to nearly constant levels.\n\n\\subsubsection{Window overlap}\n\nThe amount of overlap between adjacent windows plays an important role in the\nperformance of the gradient-boosted equivalent sources.\nIt controls the number of iterations and how many times a particular source is\nused in the least-squares fitting process.\nThe experiments in the previous section showed that 50\\% overlap was\nsufficient to achieve acceptable interpolation accuracy.\nHowever, we studied separately the impacts of the amount of window overlap on\nboth accuracy and computation time.\n\nWe performed a similar experiment to the one in section~\\ref{sec:window_size}\nbut this time kept the window size fixed to \\BoostOverlappingWindowSize{} and\nvaried the amount of overlap from 0\\% to 95\\% with a step size of 5\\%.\nAll other experimental procedures remained unchanged.\nFig.~\\ref{fig:gradient-boosted-comparison}b shows the RMS error and\nFig.~\\ref{fig:gradient-boosted-comparison}d shows the computation time\nrequired for estimating the source coefficients, both as functions of\nthe window overlap.\n\nOur results show that the interpolation RMS error decreases with the amount of\noverlap, reaching the same accuracy as the regular equivalent sources at\napproximately 90\\% overlap.\nOn the other hand, the computation time increases with the amount of overlap,\nbecoming larger than that of the regular equivalent sources for overlaps\ngreater than 70\\%.\nThis is expected since increasing the overlap adds iterations to the gradient\nboosting algorithm without decreasing the individual least-squares problem\nsizes to compensate.\n\n\n\\subsection{\n Interpolation with gradient boosting\n}\n\\label{sec:gb_interpolation}\n\nFinally, we applied the gradient-boosted equivalent sources to interpolate the\nsynthetic airborne survey (Fig.~\\ref{fig:synthetic-layouts}).\nAs previously, we used the block-averaged sources layout with a block size of\n\\EqlBoostAirborneSpacing{}.\nBased on the results from section \\ref{sec:window_size_and_overlap}, we adopted\na window overlap of 50\\% and a window size of \\EqlBoostAirborneWindowSize{}.\n\nWe estimated the relative depth of the sources and the damping parameter by\ncomparing the predictions against the values of the target grid.\nThe search explored \\emph{depth} values between \\EqlBoostAirborneMinDepth{} and\n\\EqlBoostAirborneMaxDepth{} and \\emph{damping} values between\n\\EqlBoostAirborneMinDamping{} and \\EqlBoostAirborneMaxDamping{} by steps of one\norder of magnitude.\nThe most accurate predictions achieved a RMS error of\n\\EqlBoostAirborneRmsScore{} with a depth of \\EqlBoostAirborneDepth{} and\na damping of \\EqlBoostAirborneDamping{}.\nIt is worth noting that the RMS error achieved by the gradient-boosted\nequivalent sources is comparable to the ones obtained by the regular equivalent\nsources in Section \\ref{sec:synthetic_distributions}.\nTo highlight the importance of randomizing the order of windows in the\ngradient-boosting iterations, we preformed the interpolation once more using\nthe same values of \\emph{damping} and \\emph{depth} but this time iterating over\nwindows in sequential order (South to North, West to East).\n\n\\begin{figure}[tb]\n \\includegraphics[width=\\linewidth]{figs/eql-boost-airborne.pdf}\n \\caption{\n Interpolation error for gradient-boosted equivalent sources using\n randomized (a) and sequential (b) window order.\n (a and b)~Pseudo-color maps of the differences between the target grid\n and the interpolated synthetic airborne survey data.\n The color scale has been cropped to the same range as\n Fig.~\\ref{fig:airborne-survey-differences}.\n (c)~Root-mean squared error after each iteration of the\n gradient-boosting algorithm.\n}\n\\label{fig:eql-boost-airborne}\n\\end{figure}\n\nFigs.~\\ref{fig:eql-boost-airborne}a-b show the differences between the target\ngrid and the interpolation results for windows in randomized and sequential\norder, respectively.\nThe differences for randomized windows resemble those for regular least-squares\nequivalent sources seen in Figs.~\\ref{fig:ground-survey-differences} and\n\\ref{fig:airborne-survey-differences}.\nOn the other hand, the differences for sequential windows show a clear\ntrend of large negative differences in the South decreasing towards the North.\nThis trend is correlated with the order in which windows are executed, with\ndifferences decreasing in absolute value towards the end of the algorithm.\nFig.~\\ref{fig:eql-boost-airborne}c shows the RMS error of the fitting process\nafter each iteration for both window orders, clearly indicating that\na randomized window order leads to faster convergence of the algorithm.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Gridding gravity data from Australia}\n\n\\begin{figure*}[p]\n \\includegraphics[width=\\linewidth]{figs/australia.png}\n \\caption{\n Pseudo-color maps of observed (a and c) and\n interpolated (b and d) gravity disturbance of Australia.\n The observed values in a and c are plotted as colored circles.\n The red rectangle marks the boundaries of the highlight maps in c\n and d.\n Observations are part of a compilation by \\citet{wynne2018} of\n over 1.7 million ground gravity measurements.\n Interpolated values were obtained through gradient-boosted equivalent\n sources and calculated on a regular grid at \\AustraliaEqlGridHeight{}\n over the WGS84 ellipsoid.\n }\n \\label{fig:australia}\n\\end{figure*}\n\nThis section will demonstrate how gradient-boosted equivalent sources can be\nused to interpolate large datasets onto regular grids at uniform height.\nFor this purpose, we selected an open-access compilation of ground gravity\nsurveys over Australia made by \\citet{wynne2018} and filtered and referenced to\nthe WGS84 ellipsoid by \\citet{australia_compilation}.\nIt contains over 1.7 million data points and covers most of the Australian\nterritory at variable point spacings.\nOur goal is to create a 1~arc-minute resolution grid of gravity disturbances at\na constant geometric height of \\AustraliaEqlGridHeight{} (the largest height of\nobservations).\n\nWe computed the gravity disturbance by removing the normal gravity of\nthe WGS84 ellipsoid from the observed gravity data (Fig.~\\ref{fig:australia}).\nHere, normal gravity was computed at each observation point through the\nclosed-form formula of \\citet{ligotze2001} using the Boule software\n\\citep{boule2020}.\nFinally, we converted the observations to planar Cartesian coordinates by\napplying a Mercator projection.\n\nWe start the interpolation process by defining a set of block-averaged sources\nusing a block size of 1.8\\km{}, resulting in a total of\n\\AustraliaEqlNSources{}~point sources.\nThe block size was chosen to match the desired resolution of the final grid\n(1~arc-minute is approximately 1.8\\km{} at the equator).\nBased on the results obtained in Section~\\ref{sec:synthetic_distributions}, we\nhave chosen to use the \\emph{relative depth} strategy.\nThe window overlap was once again fixed at 50\\%.\nTo determine the size of the windows, we calculated the amount of computer memory\nneeded to store the largest Jacobian matrix for different values of window size\n(Fig.~\\ref{fig:australia-memory-cv-error}a).\nWe have chosen a size of \\AustraliaEqlWindowSize{} in order to limit the\namount of memory needed to under 16~Gigabytes.\n\n\\begin{figure}[tbh!]\n \\makeatletter%\n \\if@twocolumn%\n \\includegraphics[width=\\linewidth]{figs/australia-memory-cv-error.pdf}\n \\else% \\@twocolumnfalse\n \\centering\n \\includegraphics[width=0.6\\linewidth]{figs/australia-memory-cv-error.pdf}\n \\fi\n \\makeatother\n \\caption{\n (a) Amount of computer memory needed to store the largest Jacobian\n matrix for different window sizes. Our implementation uses double\n precision floating point numbers (64 bits) for the Jacobian.\n (b) Root-mean square error against the observed data after each\n iteration of the gradient-boosting algorithm.\n (c) K-Fold cross-validation root-mean square errors obtained for each\n pair of damping and depth parameters. The orange star highlights the\n minimum.\n }\n \\label{fig:australia-memory-cv-error}\n\\end{figure}\n\nWe determined the depth of the sources and the damping parameter by applying\nK-Fold cross-validation through the scikit-learn library \\citep{sklearn2011}.\nThe method randomly divides the original data into $k$ sets (folds), fits the\nmodel using only data from $k - 1$ folds, and validates the model by comparing\nits predictions against the one remaining fold.\nThis process is carried out once for each one of the $k$ folds, leading to\nan estimated mean cross-validation RMS error for the model.\nTo speed up the computation, we only performed the cross-validation on a subset\nof the data corresponding to an area of\n\\AustraliaSmallAreaEastingSize{}$\\times$\\AustraliaSmallAreaNorthingSize{}\ncontaining \\AustraliaSmallAreaNPoints{} points.\nWe ran the cross-validation repeatedly for combinations of \\emph{depth},\nranging from \\AustraliaDepthMin{} to \\AustraliaDepthMax{},\nand \\emph{damping}, from \\AustraliaDampingMin{} to \\AustraliaDampingMax{} in\nsteps of one order of magnitude.\nFigure \\ref{fig:australia-memory-cv-error}c shows the resulting\ncross-validation RMS errors and highlights the minimum value of\n\\AustraliaEqlRmsScore{}, which corresponds to a relative depth of\n\\AustraliaEqlDepth{} and a damping equal to \\AustraliaEqlDamping{}.\n\nFinally, we proceeded to estimate the source coefficients using the entire\ndataset and the parameters previously determined.\nThe estimated source coefficients were then used to predict the values of the\ngravity disturbance on a regular grid of\n\\AustraliaEqlGridNLongitude{}$\\times$\\AustraliaEqlGridNLatitude{} points at\n\\AustraliaEqlGridHeight{} above the ellipsoid.\nOn a modest workstation with 16 cores and 16~Gigabytes of RAM,\nestimating the \\AustraliaEqlNSources{} coefficients with gradient-boosting took\n$\\sim 1.3$~hours and the prediction step took $\\sim 18$~minutes.\n\nFig.~\\ref{fig:australia} shows the original data distribution and the\ninterpolated grid.\nGrid points that are further than 50\\km{} from the nearest data point are\nmasked to avoid unrealistic extrapolations.\nFig.~\\ref{fig:australia-memory-cv-error}b shows the RMS error against the\nobserved data after each iteration of the algorithm.\nFig.~\\ref{fig:australia-residuals} shows the difference between the observed\nand predicted gravity disturbances.\nThe inset figure shows a histogram of these residuals, which are approximately\nnormally distributed around zero.\n\n\\begin{figure}[tb]\n \\includegraphics[width=\\linewidth]{figs/australia-residuals.png}\n \\caption{\n Residuals. Differences between the gravity disturbance data from\n Australia and the predicted values by the estimated equivalent sources\n on the same observation points. The color map has been truncated to\n improve the visualization around the largest portion of residual\n values. The inset plot shows a histogram of the residuals.\n }\n \\label{fig:australia-residuals}\n\\end{figure}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Discussion}\n\n\\subsection{Location of sources}\n\nThe results of our tests on synthetic data\n(Figs.~\\ref{fig:ground-survey-differences}\nand~\\ref{fig:airborne-survey-differences}) show that there are no significant\ndifferences in interpolation accuracy between source distribution strategies,\nboth in terms of the interpolation RMS errors and from visual inspection of the\ndifference maps.\nTherefore, we conclude that all source distribution strategies are able to\nproduce comparable interpolations.\nNevertheless, the \\emph{block-averaged sources} strategy makes use of fewer\nsources when compared with other strategies, which reduces the computational\nload of estimating the sources coefficients and forward modelling.\nTo ensure that the interpolation is able to reproduce the high frequencies in\nthe data, the block size used in the averaging should be chosen to match the\ndesired grid resolution.\n\nThe choice of source depth strategy does not appear to significantly impact the\ninterpolation RMS error.\nIn the particular case of a sparse ground survey with block-averaged sources,\nthe use of a variable depth visibly reduced edge effects and artifacts in areas\nof poor data coverage.\nAt a first glance, the choice of a depth strategy would not seem to impact\nthe computation time.\nHowever, when searching for the set of hyper-parameters that produce the most\naccurate interpolation (e.g., through cross-validation), one must solve the\ninverse problem once for every possible combination of parameters.\nA depth strategy like the \\emph{variable depth} requires a higher number of\nhyper-parameters (depth shift $\\Delta z$, depth factor $\\alpha$, and the number\nof nearest neighbours $k$ from Eq.~\\ref{eq:variable_depth}) than other\nstrategies which only require a single parameter.\nHaving more parameters means increasing the dimensions of the parameter space\nand thus increasing the number of possible combinations.\nThus, we recommend using a \\emph{constant depth} or a \\emph{relative depth}\nwhen processing large datasets in order to minimize computation time.\n\n\\subsection{Gradient boosting}\n\nFrom Fig.~\\ref{fig:gradient-boosted-comparison}a\nand~\\ref{fig:gradient-boosted-comparison}c, we can see that the\ngradient-boosted equivalent sources produce slightly less accurate\ninterpolation results but are able to achieve smaller computation times than\nregular equivalent sources.\nThe reduction of the accuracy might be due to the gradient boosting algorithm\nfailing to converge to the global minimum of the goal function.\nAs the windows increase in size, interpolation error decreases because more data\npoints are included into the least-squares fitting of the source coefficients.\nAt the same time, the fitting process becomes faster because of a reduction in\nthe number of iterations.\nOur results indicate that it is desirable to maximize the window size,\nwhich can be done up to the point that the Jacobian matrices still fit within\nthe available computer memory.\n\nThe results shown in Figs.~\\ref{fig:gradient-boosted-comparison}b\nand~\\ref{fig:gradient-boosted-comparison}d indicate that using\nwindow overlap values between 40\\% and 70\\% strike a balance between\naccuracy and computation time.\nThis corroborates our initial choice of 50\\% overlap, which is good enough for\nproducing accurate predictions in reasonable times.\n\nFinally, the results in Fig.~\\ref{fig:eql-boost-airborne} highlight the\nimportance of randomizing the order in which the overlapping windows are\niterated.\nRunning the gradient boosting algorithm sequentially produces less accurate\npredictions and decreases the convergence rate of the method.\n\n\\subsection{Australia gravity data}\n\nThe application of the gradient-boosted equivalent sources to the Australian\ngravity dataset demonstrates that the method is able to interpolate and\nupward-continue large datasets in a reasonable amount of time using only modest\ncomputational resources.\nThe resulting grid (Fig.~\\ref{fig:australia}) preserves the high resolution of\nthe original data while avoiding aliasing artifacts due to the block averaging\nof the source locations.\nSome parts of the grid are smoother and have lower amplitudes than the original\ndata (e.g., some southwestern parts), which is expected from the upward\ncontinuation that was performed to have the grid at a constant height.\nFrom the cross-validation analysis on a subset of the data, we estimate that\nthe interpolation error is approximately \\AustraliaEqlRmsScore{}.\n\nThe largest residuals in Fig.~\\ref{fig:australia-residuals} are located in\nregions with high-amplitude short-wavelength features in the observed data.\nThis is expected since the method involves some degree of smoothing because of\nthe use of damping and the source depths.\nThere are also low-amplitude long-wavelength residual signals that seem to\ncoincide with some of the windows of the gradient-boosting method.\nA possible cause of these features is inability of the equivalent-sources\nwithin a window to adequately fit the long-wavelength components of the data.\nWe note, however, that all of these long-wavelength residuals are smaller than\n1\\mGal{} in amplitude and do not represent a significant source of errors.\n\nThe elongated valley around the minimum of the cross-validation RMS errors\n(Fig.~\\ref{fig:australia-memory-cv-error}c) shows that there is ambiguity in\nthe choice of damping and source depths.\nOne could choose a large damping with a small depth or a small damping with a\nlarge depth to achieve roughly the same interpolation result.\nThis is expected since both parameters control the smoothness of the\ninterpolation.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Conclusions}\n\nThe equivalent source technique has been proven to be well suited for\ninterpolating gravity disturbances and magnetic anomalies.\nThe two main reasons that make it to stand out from other 2D interpolation\nmethods is the fact that the equivalent sources take into account the height of\nthe observations and that the interpolated values will always be harmonic\nfunctions.\nThe main challenge of using equivalent sources in practice is the high\ncomputational load of estimating the coefficients of the equivalent sources,\nspecially the computer memory needed to store the Jacobian matrix.\n\nWe present two strategies that could be simultaneously applied to interpolate\ndatasets with millions of points on modest hardware:\nblock-averaging source locations, which reduces the number of equivalent\nsources needed for the interpolation,\nand the gradient-boosted equivalent source algorithm, which breaks down the\ninverse problem into smaller sets of equivalent sources defined by overlapping\nwindows.\nBoth methods were tested against synthetic datasets in order to compare their\naccuracy and how they perform in terms of computational efficiency.\n\nOur results show that the block-averaged sources reduce the computational\nload needed to estimate source coefficients in comparison to two traditional\nstrategies (placing sources below data points or on regular grids).\nWe also show that this reduction of the number of sources does not affect\nthe accuracy of the predictions.\nThe use of block-averaged sources may also prevent aliasing of the interpolated\nvalues, specially when the observations are unevenly sampled (e.g., airborne\nand shipborne surveys).\nSpecial attention must be payed when choosing the size of the blocks for\naveraging.\nAs a thumb rule, we recommend choosing a size approximately equal to\nthe resolution of the regular grid where the values will be interpolated.\n\nTests that compared strategies for the vertical location of the\nsources showed that any one of the three strategies tested\n(\\emph{constant depth}, \\emph{relative depth} and \\emph{variable depth})\nproduces comparable accuracy of interpolation.\nNevertheless, we are more prone to recommending either the \\emph{constant\ndepth} or the \\emph{relative depth} for most applications because they involve\nless hyper-parameters that would need to be configured before the actual\ninterpolation.\n\nGradient-boosted equivalent sources were shown to heavily reduce the computer\nmemory needed to estimate source coefficients, making it possible to\ninterpolate large datasets with millions of points that would otherwise produce\nJacobian matrices larger than the available memory.\nThe interpolations obtained though this new method achieve close to the same\naccuracy than the regular equivalent sources, while reducing the computation\ntime by approximately a factor of three.\nWe also show that an overlap of 50\\% between adjacent windows achieves a good\ncompromise between accuracy and computation time.\nThe size of the overlapping windows should be chosen as the maximum value\npossible that creates Jacobian matrices that still fit into computer memory.\nMoreover, randomizing the order in which the windows are iterated increases the\nconvergence rate of the algorithm and is essential to producing accurate\npredictions.\n\nThe gradient-boosting method can be used in conjunction with any horizontal\nsource layout, depth strategy, or source type (e.g., point sources, prisms,\ntesseroids) because it does not rely on assumptions about the sources.\nFuture research should investigate the application of gradient boosting to\nother equivalent source methods.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Data and code availability}\n\nThe Python source code used to produce all results and figures presented here\nis available at\n\\url{https://doi.org/10.6084/m9.figshare.13604360} and\n\\url{https://github.com/compgeolab/eql-gradient-boosted}\nunder the BSD 3-clause open-source license.\n\nThe gradient-boosted equivalent sources implementation is based on the\nequivalent source code in the Harmonica library \\citep{harmonica2020}.\nOther software used in this study includes:\nPooch \\citep{pooch2020} for downloading and caching datasets,\nVerde \\citep{verde2018} for block reductions and coordinate manipulations,\nBoule \\citep{boule2020} for normal gravity calculations,\nxarray \\citep{xarray2017} and Numpy \\citep{numpy2020} for handling\nmultidimensional arrays and numerical computations,\nNumba \\citep{numba2015} for just-in-time compilation and parallelization,\nscikit-learn \\citep{sklearn2011} for cross-validation,\nMatplotlib \\citep{matplotlib2007} and PyGMT \\citep{pygmt2020} for generating\nthe figures and maps,\nand the Jupyter notebook programming environment \\citep{jupyter2016}.\nHarmonica, Boule, Pooch, and Verde are part of the Fatiando a Terra project\n\\citep{fatiando2013}.\n\nAll datasets used are open-access and publicly available.\nThe synthetic surveys were generated using\na public domain gravity dataset for Southern Africa distributed by the\nNOAA NCEI (\\url{https://www.ngdc.noaa.gov/mgg/gravity/gravity.html})\nand the Great Britain Aeromagnetic\nSurvey distributed by the\nBritish Geological Survey (BGS) under an Open Government License\n(\\url{https://www.bgs.ac.uk/products/geophysics/aeromagneticRegional.html}).\nThe shaded relief in Fig.~\\ref{fig:australia} is the SRTM15+ dataset by\n\\citet{tozer2019}.\nThe Australian ground gravity\ndata is based on a compilation distributed by Geoscience Australia under a\nCreative Commons Attribution 4.0 International Licence \\citep{wynne2018} which\nwas filtered and referenced to the WGS84 ellipsoid by\n\\citet{australia_compilation} and is distributed under the same license\n(\\url{https://doi.org/10.6084/m9.figshare.13643837}).\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Acknowledgements}\n\nWe are indebted to the developers and maintainers of the open-source software\nwithout which this work would not have been possible.\nWe would also like to thank Editor Frederik Simons, Assistant Editor Fern\nStorey, and two anonymous reviewers for their constructive comments.\nS.R. Soler is supported by a scholarship from CONICET, Argentina.\nThis work contains British Geological Survey materials ©~UKRI.\nS.R. Soler and L. Uieda jointly developed the initial idea, analysed the\nresults, and wrote the paper. S.R. Soler produced all results and developed the\nsoftware implementation with the assistance of L. Uieda.\n", "meta": {"hexsha": "140180c851c0c3ab98d5f86a260e935d08caa6e1", "size": 69945, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "manuscript/content.tex", "max_stars_repo_name": "compgeolab/eql-gradient-boosted", "max_stars_repo_head_hexsha": "759a0c6f34e92d83551b14c8dfaeae2a8be1e949", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 17, "max_stars_repo_stars_event_min_datetime": "2021-02-25T18:03:14.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T22:47:37.000Z", "max_issues_repo_path": "manuscript/content.tex", "max_issues_repo_name": "compgeolab/eql-gradient-boosted", "max_issues_repo_head_hexsha": "759a0c6f34e92d83551b14c8dfaeae2a8be1e949", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 24, "max_issues_repo_issues_event_min_datetime": "2021-02-25T15:21:40.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-25T10:29:33.000Z", "max_forks_repo_path": "manuscript/content.tex", "max_forks_repo_name": "compgeolab/eql-gradient-boosted", "max_forks_repo_head_hexsha": "759a0c6f34e92d83551b14c8dfaeae2a8be1e949", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2021-07-27T15:55:05.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-17T07:26:42.000Z", "avg_line_length": 47.1327493261, "max_line_length": 82, "alphanum_fraction": 0.7715490743, "num_tokens": 16558, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6859494678483918, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.34833327806942005}} {"text": "\\documentclass[a4paper]{article}\n\\usepackage{fullpage}\n\\usepackage{listings} %--- For including MATLAB code ---%\n \\usepackage{color} %red, green, blue, yellow, cyan, magenta, black, white\n \\definecolor{mygreen}{RGB}{28,172,0} % color values Red, Green, Blue\n \\definecolor{mylilas}{RGB}{170,55,241}\n \\lstset{language=Matlab,%\n basicstyle=\\footnotesize\\ttfamily,\n breaklines=true,%\n morekeywords={matlab2tikz},\n keywordstyle=\\color{blue},%\n morekeywords=[2]{1}, keywordstyle=[2]{\\color{black}},\n identifierstyle=\\color{black},%\n stringstyle=\\color{mylilas},\n commentstyle=\\color{mygreen},%\n showstringspaces=false,%without this there will be a symbol in the places where there is a space\n numbers=left,%\n numberstyle={\\tiny \\color{black}},% size of the numbers\n numbersep=9pt, % this defines how far the numbers are from the text\n emph=[1]{for,end,break},emphstyle=[1]\\color{red}, %some words to emphasise\n %emph=[2]{word1,word2}, emphstyle=[2]{style}, \n }\n%---End MATLAB code inclusion package ---%\n\n\\author{Ryan Day}\n\\title{Optimization Homework \\#4}\n\\begin{document}\n \\maketitle\n \\section{Truss Optimization}\n\n \\subsection{Scaling}\n Scaling the constraints is useful. I scaled it by dividing the constraints by $10^{2}$. This reduced the number of function calls by more than half when I implemented it. \n I tried to make the constraints the same order of magnitude as the design functions.\n The design variables are all around the same order of magnitude, so I don't see the use of scaling there.\n \n \\subsection{Matlab Code implementation}\n The matlab code is included in appendix \\ref{appendix.trussCode}.\n I used a function that took in x, then perturbed the function with a step depending on a type input (forward, central, or complex), and then calculated the gradient and constraints gradient.\n It was not too hard to implement this function. \n I simply added it to the end of obj and con in order to get the derivatvies.\n However, I had to make one change in order to get the complex function to work.\n This change was changing the inequality constraint from using the abs function to taking the square root of the value squared.\n This did the same thing as abs but could be used with complex variables.\n \n \\subsection{Expected Errors of the derivatives}\n I expected the errors of the derivative to be greatest for the forward, then central, then complex. \n The merits of the forward method is it is only takes one function call per derivative. \n Central method is slightly more accurate, but takes twice as many function calls.\n Both central and forward methods have subtractive error which makes it so you can't have the step size be too small.\n The complex step method does not have this subtractive error which makes it so you can have an extremely small step size, but then you have to make sure your function can handle complex numbers.\n In addition to this, the computations of the complex step can take longer than the forward step because of the included complex numbers.\n \n I figured out the optimal perturbation of forward and central methods by comparing it to the complex step derivative.\n I checked compared the estimated derivatives from the first iteration between the complex step with a step size of $10^{-30}$. \n I knew the complex derivative would be pretty accurate because it has no subtractive error, so you can make the step size extremely small. \n $10^{-8}$ turned out to be the ideal step size for forward and central. It had an error for forward on the order of $10^{-6}$ which was fine.\n When I tried at 1e-9 there was an error in the gradient of forward of $10^{-4}$ which was too much. \n For some reason the central wasn't converging when I scaled it, so that is why it was so much slower than the other functions.\n \n \\subsection{Table and stopping criteria}\n \\begin{center}\n \\begin{tabular}[h]{c c c c c}\n & \\# Function calls & \\# Iterations & Avg Time execution& Final Objective value \\\\\n No Derivatives supplied&287&12&0.420 &1.5932e+03\\\\\n Forward method&309&12&0.415 & 1.5932e+03\\\\\n Central method&4621&5&1.114 & 1.5932e+03\\\\ \n Complex method&287&12&0.510 & 1.5932e+03 \n \\end{tabular}\n\\end{center}\n\nThe execution time was fastest by just a little bit with the forward method. It barely beat out the fmincon with no supplied derivatives.\nThe forward method and complex method had the same number of function calls as expected since they both call the objective function just once to calculate derivatives. \nThe complex method took longer than the forward method because it had to deal with complex numbers. \nThe central method took about twice as many function calls and so took a lot longer than any other method. \nHowever, it had about half the iterations, because it goes in a more accurate direction with .\n\n\\textbf{Stopping Criterion for no derivatives supplied: }\n\nOptimization completed: The relative first-order optimality measure, 5.312214e-07,\nis less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint\nviolation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.\n\n\\textbf{Stopping Criterion for other methods:}\n\n\\textbf{Forward:}\n\nOptimization stopped because the relative changes in all elements of x are\nless than options.StepTolerance = 1.000000e-10, and the relative maximum constraint\nviolation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.\n\n\\textbf{Central:}\n\nOptimization completed: The relative first-order optimality measure, 7.486241e-07,\nis less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint\nviolation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.\n\n\\textbf{Complex:}\n\nOptimization completed: The relative first-order optimality measure, 3.928371e-08,\nis less than options.OptimalityTolerance = 1.000000e-06, and the relative maximum constraint\nviolation, 0.000000e+00, is less than options.ConstraintTolerance = 1.000000e-06.\n\n\\textbf{Commentary:}\nThe stopping criterion for forward and without supplying derivatives was that the x inputs had very small changes, signalling that there was an optimum. \nFor the complex and central step, the function output met the optimality measure. \nBecause I only looked at 5 significant digits for the optimum, these measures didn't affect the optimum at all.\n\n\\section{Automatic Differentiation}\n\\subsection{How does this method work?}\nI define the values to be of the valder class. \nAny other function that has a valder variable as part of its calculation will then take the valder variable into account and then operator overloading is used to compute derivatives for the original valder functions.\nIn this way you can get derivatives without having to change your code much.\n\\subsection{How is AD different from other numerical methods?}\nAD is different because it solves the derivatives analytically using lookup functions and operator overloading instead of numerically approximating the value.\n\\subsection{Matlab Code}\nAD code is found in appendix \\ref{appendix.adcode}\n\\appendix\n\\section{Truss Optimization}\n\\label{appendix.trussCode}\n\\lstinputlisting[language=matlab]{../OptimizeTruss/OptimizeTruss.m} \n\\section{Automatic differentiation}\n\\lstinputlisting[language=matlab]{../AutomaticDifferentiation/optimize_spring.m} \n\\label{appendix.adcode}\n\\end{document}\nxopt =\n\n Columns 1 through 5\n\n 6.3535 0.1016 6.4508 4.0730 0.1028\n\n Columns 6 through 10\n\n 0.1016 5.4464 5.2882 6.0309 0.1039\n\n\nfopt =\n\n 1.4775e+03", "meta": {"hexsha": "3754b4e329df77a6dd49f68693df6d0e3819bb49", "size": 7699, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Hw4/Hw4Writeup/main.tex", "max_stars_repo_name": "rddq/OptimizationME570", "max_stars_repo_head_hexsha": "159f0117d8d9b15e1cc4f1ff64375a7645f62681", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-09-10T03:44:49.000Z", "max_stars_repo_stars_event_max_datetime": "2019-09-10T03:44:49.000Z", "max_issues_repo_path": "Hw4/Hw4Writeup/main.tex", "max_issues_repo_name": "rddq/OptimizationME570", "max_issues_repo_head_hexsha": "159f0117d8d9b15e1cc4f1ff64375a7645f62681", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Hw4/Hw4Writeup/main.tex", "max_forks_repo_name": "rddq/OptimizationME570", "max_forks_repo_head_hexsha": "159f0117d8d9b15e1cc4f1ff64375a7645f62681", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.7898550725, "max_line_length": 217, "alphanum_fraction": 0.7573710872, "num_tokens": 1941, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.6859494550081926, "lm_q1q2_score": 0.34833327154901456}} {"text": "In the figures below we can see the surface plots of the horizontal velocity $u$, vertical velocity $v$ and mass fraction $Y$, for different values of $t$. They are all obtained using $M=256$, $N=128$ and $CFL=0.8$. Below the contour plots, and using the same parameters, we have the curve plots of the probe measurements together with the performance parameter $R$ $vs$ time. To finish, we have the $GCI$-analysis of the quantities required by the problem. At the very end, the same $GCI$-analysis will be performed to the steady state solution.\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\hspace*{\\fill}\n\\subfigure[$t=1$ s.]{\\includegraphics[scale=0.55]{u_11.eps}}\n\\hfill\n\\subfigure[$t=2$ s.]{\\includegraphics[scale=0.55]{u_12.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[$t=3$ s.]{\\includegraphics[scale=0.55]{u_13.eps}}\n\\hfill\n\\subfigure[$t=5$ s.]{\\includegraphics[scale=0.55]{u_14.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[$t=7$ s.]{\\includegraphics[scale=0.55]{u_15.eps}}\n\\hfill\n\\subfigure[$t=10$ s.]{\\includegraphics[scale=0.55]{u_16.eps}}\n\\hspace*{\\fill}\n\\caption{Profiles of horizontal velocity $u$ for $M=256$, $N=128$ and $CFL=0.8$.}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\hspace*{\\fill}\n\\subfigure[$t=1$ s.]{\\includegraphics[scale=0.55]{v_21.eps}}\n\\hfill\n\\subfigure[$t=2$ s.]{\\includegraphics[scale=0.55]{v_22.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[$t=3$ s.]{\\includegraphics[scale=0.55]{v_23.eps}}\n\\hfill\n\\subfigure[$t=5$ s.]{\\includegraphics[scale=0.55]{v_24.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[$t=7$ s.]{\\includegraphics[scale=0.55]{v_25.eps}}\n\\hfill\n\\subfigure[$t=10$ s.]{\\includegraphics[scale=0.55]{v_26.eps}}\n\\hspace*{\\fill}\n\\caption{Profiles of the vertical velocity $v$ for $M=256$, $N=128$ and $CFL=0.8$.}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\hspace*{\\fill}\n\\subfigure[$t=1$ s.]{\\includegraphics[scale=0.55]{Y_31.eps}}\n\\hfill\n\\subfigure[$t=2$ s.]{\\includegraphics[scale=0.55]{Y_32.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[$t=3$ s.]{\\includegraphics[scale=0.55]{Y_33.eps}}\n\\hfill\n\\subfigure[$t=5$ s.]{\\includegraphics[scale=0.55]{Y_34.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[$t=7$ s.]{\\includegraphics[scale=0.55]{Y_35.eps}}\n\\hfill\n\\subfigure[$t=10$ s.]{\\includegraphics[scale=0.55]{Y_36.eps}}\n\\hspace*{\\fill}\n\\caption{Profiles of the mass fraction $Y$ for $M=256$, $N=128$ and $CFL=0.8$.}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\hspace*{\\fill}\n\\subfigure[Probe 1.]{\\includegraphics[scale=0.55]{probeu.eps}}\n\\hfill\n\\subfigure[Probe 2.]{\\includegraphics[scale=0.55]{probev.eps}}\n\\hspace*{\\fill}\n\n\\hspace*{\\fill}\n\\subfigure[Probe 3.]{\\includegraphics[scale=0.55]{probeY.eps}}\n\\hfill\n\\subfigure[Performance parameter.]{\\includegraphics[scale=0.55]{Rplot.eps}}\n\\hspace*{\\fill}\n\\caption{Probe curves and performance parameter with time for $M=256$, $N=128$ and $CFL=0.8$.}\n\\end{figure}\n\n\\subsection*{GCI analysis for u at t=5}\n\nThe GCI analysis details are shown in the tables below. Note that \n\\begin{align*}\n\\beta=\\frac{GCI_{12}}{GCI_{23}}r^p,\n\\end{align*}\nand $u_{h=0}$ is obtained by Richardson extrapolation. We can see that $\\beta\\in[0.95,1.05]$ which implies that we are in the asymptotic range of convergence, and for the last mesh we have a $GCI_{12}$ value less than $0.02\\%$, the requested accuracy.\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|}\n%\\hline\n%\\multicolumn{3}{|c|}{Datos}\\\\\nM & N & $u(1,0.5)$ \\\\\n\\hline\n$16$ & $8$ & $1.543690071443089$ \\\\\n$32$ & $16$ & $ 1.335416609160283$ \\\\\n$64$ & $32$ & $ 1.322019032777453$ \\\\\n$128$ & $64$ & $ 1.319745850662477$ \\\\\n$256$ & $128$ & $ 1.319205904441201$ \\\\\n$512$ & $256$ & $ 1.319067860936907$ \\\\\n\\end{tabular}\n\\caption{GCI analysis data.}\n\\end{table}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{ut5.png}\n\\caption{GCI analysis results for the probe 1 measurement.}\n\\end{figure}\n\n\\subsection*{GCI analysis for v at t=5}\n\nWe can see that $\\beta\\in[0.95,1.05]$ which implies that we are in the asymptotic range of convergence, and for the last mesh we have a $GCI_{12}$ value less than $0.02\\%$, the requested accuracy.\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|}\n%\\hline\n%\\multicolumn{3}{|c|}{Datos}\\\\\nM & N & $v(1,1.5)$ \\\\\n\\hline\n$16$ & $8$ & $ -0.757061737618705$ \\\\\n$32$ & $16$ & $ -0.625718980721275$ \\\\\n$64$ & $32$ & $ -0.622907528189143$ \\\\\n$128$ & $64$ & $ -0.622804506934971$ \\\\\n$256$ & $128$ & $ -0.622781023934629$ \\\\\n$512$ & $256$ & $\t-0.622773391772338$ \\\\\n\\end{tabular}\n\\caption{GCI analysis data.}\n\\end{table}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{vt5.png}\n\\caption{GCI analysis results for the probe 2 measurement.}\n\\end{figure}\n\n\\subsection*{GCI analysis for Y at t=5}\n\nWe can see that $\\beta\\in[0.95,1.05]$ which implies that we are in the asymptotic range of convergence, and for the last mesh we have a $GCI_{12}$ value less than $0.4\\%$, the requested accuracy.\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|}\n%\\hline\n%\\multicolumn{3}{|c|}{Datos}\\\\\nM & N & $Y(2,0.5)$ \\\\\n\\hline\n$16$ & $8$ & $ 0.582656635075406$ \\\\\n$32$ & $16$ & $ 0.588540399303644$ \\\\\n$64$ & $32$ & $ 0.591682314985302$ \\\\\n$128$ & $64$ & $ 0.594958713346013$ \\\\\n$256$ & $128$ & $ 0.596429634107325$ \\\\\n$512$ & $256$ & $ 0.597084423776656$ \\\\\n\\end{tabular}\n\\caption{GCI analysis data.}\n\\end{table}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{Yt5.png}\n\\caption{GCI analysis results for the probe 3 measurement.}\n\\end{figure}\n\n\\subsection*{GCI analysis for R at t=5}\n\nWe can see that $\\beta\\in[0.95,1.05]$ which implies that we are in the asymptotic range of convergence, and for the last mesh we have a $GCI_{12}$ value less than $0.5\\%$, the requested accuracy.\n\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{|c|c|c|}\n%\\hline\n%\\multicolumn{3}{|c|}{Datos}\\\\\nM & N & $v(1,1.5)$ \\\\\n\\hline\n$16$ & $8$ & $ 0.168352427538196$ \\\\\n$32$ & $16$ & $ 0.162648932748133$ \\\\\n$64$ & $32$ & $ 0.161109373859052$ \\\\\n$128$ & $64$ & $ 0.160690350976423$ \\\\\n$256$ & $128$ & $ 0.160521007689289$ \\\\\n$512$ & $256$ & $ 0.160443010433551$ \\\\\n\\end{tabular}\n\\caption{GCI analysis data.}\n\\end{table}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{Rt5.png}\n\\caption{GCI analysis results for the parameter $R$.}\n\\end{figure}\n\n\\subsection*{Steady state results}\nI ran the simulation using $M=256$ and $N=128$ up to a $t=30$ to reach the steady state solution of the problem. In the following tables are the results obtained, with the GCI analysis.\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{ut30.png}\n\\caption{GCI analysis results for the probe 1 measurement.}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{vt30.png}\n\\caption{GCI analysis results for the probe 2 measurement.}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{Yt30.png}\n\\caption{GCI analysis results for the probe 3 measurement.}\n\\end{figure}\n\n\\begin{figure}[H]\n\\centering %%% not \\center\n\\includegraphics[scale=0.55]{Rt30.png}\n\\caption{GCI analysis results for the parameter $R$.}\n\\end{figure}", "meta": {"hexsha": "2651baf69ee09a3369207b5306ab9384e1c110ce", "size": 7262, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Year_1/CFD/Homework_10/Latex/Results.tex", "max_stars_repo_name": "fjcasti1/Courses", "max_stars_repo_head_hexsha": "12ab3e86a4a44270877e09715eeab713da45519d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Year_1/CFD/Homework_10/Latex/Results.tex", "max_issues_repo_name": "fjcasti1/Courses", "max_issues_repo_head_hexsha": "12ab3e86a4a44270877e09715eeab713da45519d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Year_1/CFD/Homework_10/Latex/Results.tex", "max_forks_repo_name": "fjcasti1/Courses", "max_forks_repo_head_hexsha": "12ab3e86a4a44270877e09715eeab713da45519d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.8597285068, "max_line_length": 546, "alphanum_fraction": 0.6724042963, "num_tokens": 2722, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.6859494550081925, "lm_q1q2_score": 0.3483332715490145}} {"text": "%\\documentclass[notes]{beamer} % print frame + notes\n% \\documentclass[notes=only]{beamer} % only notes\n\\documentclass{beamer} \n\n\\usepackage{wrapfig}\n\\usepackage{amsmath,amssymb}\n%\\usepackage[dvipsname]{xcolor}\n\\usepackage{tikz}\n\\usepackage{tikz-cd}\n\\usepackage{graphicx}\n\n\\usetikzlibrary{shapes.geometric,shadows,knots}\n\n\\tikzset{pics/.cd,\nopencube/.style args={#1/#2/#3}{code={\n\\coordinate (O) at (0,0,0);\n\\coordinate (A) at (0,#2,0);\n\\coordinate (B) at (0,#2,#3);\n\\coordinate (C) at (0,0,#3);\n\\coordinate (D) at (#1,0,0);\n\\coordinate (E) at (#1,#2,0);\n\\coordinate (F) at (#1,#2,#3);\n\\coordinate (G) at (#1,0,#3);\n%% Background\n\\draw[black,dotted] (O) -- (A);\n\\draw[black,dotted] (O) -- (C);\n\\draw[black,dotted] (O) -- (D);\n% Forground\n\\draw[black,dashed] (A) -- (E) -- (F) -- (B) -- cycle;\n\\draw[black,dashed] (E) -- (D) -- (G) -- (C) -- (B);\n\\draw[black,dashed] (F) -- (G);\n\n%\\draw[black,dashed, blue] (O) -- (A) -- (E) -- (D) -- cycle;\n%\\draw[black,dashed] (O) -- (A) -- (B) -- (C) -- cycle;\n%\\draw[black,dashed] (D) -- (E) -- (F) -- (G) -- cycle;\n%\\draw[black,dashed] (C) -- (B) -- (F) -- (G) -- cycle;\n%\\draw[black,dashed] (A) -- (B) -- (F) -- (E) -- cycle;\n\n}}}\n\n\\tikzset{pics/.cd,\nlinecube/.style args={#1/#2/#3/#4}{code={\n\\coordinate (OO) at (0,0,0);\n\\coordinate (AA) at (0,#2,0);\n\\coordinate (BB) at (0,#2,#3);\n\\coordinate (CC) at (0,0,#3);\n\\coordinate (DD) at (#1,0,0);\n\\coordinate (EE) at (#1,#2,0);\n\\coordinate (FF) at (#1,#2,#3);\n\n\\coordinate (GG) at (#1,0,#3);\n%% Background\n\\draw[black,dashed] (OO) -- (AA);\n\\draw[black,dashed] (OO) -- (CC);\n\\draw[black,dashed] (OO) -- (DD);\n\n\\node at (0.5*#1,0.5*#2,0.5*#3) {#4};\n\n% Foreground\n\\draw[black] (AA) -- (EE) -- (FF) -- (BB) -- cycle;\n\\draw[black] (EE) -- (DD) -- (GG) -- (CC) -- (BB);\n\\draw[black] (FF) -- (GG);\n\n%\\draw[black,dashed, blue] (O) -- (A) -- (E) -- (D) -- cycle;\n%\\draw[black,dashed] (O) -- (A) -- (B) -- (C) -- cycle;\n%\\draw[black,dashed] (D) -- (E) -- (F) -- (G) -- cycle;\n%\\draw[black,dashed] (C) -- (B) -- (F) -- (G) -- cycle;\n%\\draw[black,dashed] (A) -- (B) -- (F) -- (E) -- cycle;\n\n}}}\n\n\n\\tikzset{pics/.cd,\nshadedcube/.style args={#1/#2/#3/#4/#5}{code={\n\\coordinate (O) at (0,0,0);\n\\coordinate (A) at (0,#2,0);\n\\coordinate (B) at (0,#2,#3);\n\\coordinate (C) at (0,0,#3);\n\\coordinate (D) at (#1,0,0);\n\\coordinate (E) at (#1,#2,0);\n\\coordinate (F) at (#1,#2,#3);\n\\coordinate (G) at (#1,0,#3);\n\\draw[black,fill=#4!80] (O) -- (C) -- (G) -- (D) -- cycle;\n\\draw[black,fill=#4!30] (O) -- (A) -- (E) -- (D) -- cycle;\n\\draw[black,fill=#4!10] (O) -- (A) -- (B) -- (C) -- cycle;\n\\draw[black,fill=#4!20,opacity=0.8] (D) -- (E) -- (F) -- (G) -- cycle;\n\\draw[black,fill=#4!20,opacity=0.6] (C) -- (B) -- (F) -- (G) -- cycle;\n\\draw[black,fill=#4!20,opacity=0.8] (A) -- (B) -- (F) -- (E) -- cycle;\n\\node at (0.5*#1,0.5*#2,0.5*#3) {#5};\n}}}\n\\tikzset{pics/.cd,\ngridcube/.style args={#1/#2/#3/#4/#5/#6/#7}{code={\n\\coordinate (O) at (0,0,0);\n\\coordinate (A) at (0,#2,0);\n\\coordinate (B) at (0,#2,#3);\n\\coordinate (C) at (0,0,#3);\n\\coordinate (D) at (#1,0,0);\n\\coordinate (E) at (#1,#2,0);\n\\coordinate (F) at (#1,#2,#3);\n\\coordinate (G) at (#1,0,#3);\n\n% Foreground\n\\draw[fill=#7!20] (A) -- (E) -- (F) -- (B) -- cycle;\n\\draw[fill=#7!40] (E) -- (F) -- (G) -- (D) -- cycle;\n\\draw[fill=#7!30] (B) -- (F) -- (G) -- (C) -- cycle;\n%\\draw[black] (E) -- (D) -- (G) -- (C) -- (B);\n%\\draw[black] (F) -- (G);\n\n% lines\n\\foreach \\ll in {1,...,#4} {\n \\draw[black] (#1/#4*\\ll,#2,0) -- (#1/#4*\\ll,#2,#3) -- (#1/#4*\\ll,0,#3);\n}\n\\foreach \\ll in {1,...,#5} {\n \\draw[black] (0,#2/#5*\\ll,#3) -- (#1, #2/#5*\\ll,#3) -- (#1,#2/#5*\\ll,0);\n}\n\\foreach \\ll in {1,...,#6} {\n \\draw[black] (0,#2,#3/#6*\\ll) -- (#1,#2,#3/#6*\\ll) -- (#1,0,#3/#6*\\ll);\n}\n\n}}}\n\n%\\pgfmathsetmacro{\\xx}{0.5}\n\n\\mode\n% \\setbeameroption{hide notes}\n{\n %\\usetheme{Madrid}\n \\usetheme{metropolis}\n% CSU COLORS\n \\definecolor{csugreen}{HTML}{1C674F}\n \\definecolor{csugold}{HTML}{B78E00}\n \\definecolor{csured}{HTML}{E02F11}\n\n \\definecolor{greenmain}{HTML}{00AF64}\n \\definecolor{brick}{rgb}{.85,.1,.2}\n \\colorlet{greenstruct}{greenmain!87.5!black}\n \\usecolortheme[named=csugreen]{structure}\n % Change this in order to make the presentation look\n % different. Works just like a powerpoint template. \n % Have your RA mess around until it looks nice.\n \\setbeamercovered{dynamic}\n\n \\useinnertheme{rectangles}\n \n %Other colors.\n\n% \\definecolor{bluemain}{HTML}{0B61A4}\n% \\definecolor{orangemain}{HTML}{AA6600}\n% \\definecolor{redmain}{HTML}{E02F11}\n% \\definecolor{redlight}{HTML}{FF9B73}\n% \\definecolor{orangelight}{HTML}{FFC373}\n% \\definecolor{cmugray}{RGB}{104,104,104}\n% \\definecolor{cmulightgray}{RGB}{238,238,238}\n% \\setbeamercolor{block body}{bg=cmulightgray}\n% \\setbeamercolor{talktitle}{bg=csugreen,fg=white}\n% \\setbeamercolor{block title alerted}{bg=csugold}\n% \\setbeamercolor{block body alerted}{bg=orangelight}\n\n}\n\\renewcommand{\\footnoterule}{}\n\\renewcommand{\\hat}[1]{\\widehat{#1}}\n\\newcommand{\\sourcenum}[3]{$^{\\textcolor{bluemain} #1}$\\let\\thefootnote\\relax\n \\footnotetext{\\begin{flush#2}\\textcolor{bluemain}\n {\\tiny $^{#1}$ #3}\\end{flush#2}}} \n\\newcommand{\\source}[2]{\\let\\thefootnote\\relax\\footnotetext{\\begin{flush#1}\n \\textcolor{bluemain}{\\tiny Source: #2}\\end{flush#1}}} \n\n\n\\DeclareMathOperator{\\Span}{Span}\n\\DeclareMathOperator{\\Pres}{Pres}\n\\DeclareMathOperator{\\End}{End}\n\\newcommand{\\bmto}{\\rightarrowtail}\n\n\\usepackage{wasysym} \n\\newcommand{\\den}[1]{\\Leftcircle\\hspace*{-1mm}#1\\hspace*{-1mm} \\Rightcircle}\n\\usepackage{listings}\n\n\\begin{document}\n\n\\title{Group isomorphism is nearly-linear time for most\norders\\\\ {\\small IEEE Foundations On Computer Science FOCS 2021}}\n\\author{Heiko Dietrich\\\\ Monash University, Australia\\\\[10pt] James B. Wilson (presenting)\\\\ Colorado State University, USA}\n\\date{February 8, 2022}\n\n\\maketitle\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Motivation}\n\n\\begin{frame}[containsverbatim,fragile]{Outward Facing Motive: honest data types}\nWhere in this... \\\\\n\\begin{center}\n\\begin{tikzpicture}\n\\visible<2->{\n \\node at (0,0) {\\includegraphics[height=0.25\\textheight]{algo.jpg}};\n};\n\\visible<3->{\n\\node[rotate=30,drop shadow] at (1,1) {\\includegraphics[height=0.25\\textheight]{OIP.jpg}};\n};\n\\visible<4->{\n\\node[rotate=-30,drop shadow] at (3,0.5) {\\includegraphics[height=0.25\\textheight]{stack.png}};\n};\n\\end{tikzpicture}\n\\end{center}\n\\visible<5->{\n...do we send people to get help making this... \n\n\\includegraphics{equals.png}\n% \\begin{lstlisting}[mathescape=true,language=Java,basicstyle=\\ttfamily,commentstyle=\\color{csugreen},keywordstyle=\\color{blue}]\n% boolean equals(Object that) {\n% // can transform into ?\n% }\n% \\end{lstlisting}\n};\n\\end{frame}\n\n\n\\begin{frame}{Why groups(oids)?}\n\\begin{itemize}\n \\item \\textbf{Transitive}$\\to$ \\textbf{Partial Multiplication}\n \\begin{align*}\n trans_{xyz}:& (x\\equiv y) \\wedge (y\\equiv z) \\Rightarrow (x\\equiv z)\\\\\n *:& Eq \\times Eq \\dashrightarrow Eq\n \\end{align*}\n\\pause\n\n\\item \\textbf{Reflexive}$\\to$ \\textbf{Identity}\n \\begin{align*}\n refl_x&: x \\Rightarrow (x\\equiv x)\\\\\n trans_{xxy} &:(x\\equiv x)\\wedge (x\\equiv y) \\Rightarrow (x\\equiv y)\\\\\n \\hline\n Identity &: refl*evidence = evidence\n \\end{align*}\n\n \n \\item \\textbf{Symmetric}$\\to$ \\textbf{Inverse}\n \\begin{align*}\n sym_{xy}&:(x\\equiv y)\\Rightarrow (y\\equiv x)\\\\\n trans_{xyx} &: (x\\equiv y) \\wedge (x\\equiv y) \\Rightarrow (x\\equiv x)\\\\\n \\hline\n Inverses & : evidence * (evidence)^{-1} = refl\n \\end{align*}\n\\end{itemize}\n\\end{frame}\n \n\\begin{frame}{Anatomy of hard equality}\n \\centering\n\\begin{tikzpicture}[yscale=0.85]\n\\node at (0,0.5) {$L$};\n\\node (L) at (0,0) {\n \\begin{tikzpicture}[scale=0.5]\n \\foreach \\brk in {0,1,2} {\n \\begin{scope}[rotate=\\brk * 240]\n \\node[knot crossing, transform shape, inner sep=3pt] (k\\brk) at (0,-1) {};\n \\end{scope}\n }\n \\foreach \\brk in {0,1,2} {\n \\pgfmathparse{int(Mod(\\brk - 1,3))}\n \\edef\\brl{\\pgfmathresult}\n \\draw[thick,blue] (k\\brk) .. controls (k\\brk.4 north east) and (k\\brl.4 north west) .. (k\\brl.center);\n \\draw[thick,blue] (k\\brk.center) .. controls (k\\brk.16 south east) and (k\\brl.16 south west) .. (k\\brl);\n }\n \\end{tikzpicture}\n};\n\n\\node at (1.5,0.5) {$\\overset{?}{\\cong}$};\n\\node at (3,0.5) {$R$};\n\\node (R) at (3,0) {\n \\begin{tikzpicture}[scale=0.5]\n \\foreach \\brk in {0,1,2} {\n \\begin{scope}[rotate=\\brk * 120]\n \\node[knot crossing, transform shape, inner sep=3pt] (k\\brk) at (0,-1) {};\n \\end{scope}\n }\n \\foreach \\brk in {0,1,2} {\n \\pgfmathparse{int(Mod(\\brk - 1,3))}\n \\edef\\brl{\\pgfmathresult}\n \\draw[thick,red] (k\\brk) .. controls (k\\brk.4 north west) and (k\\brl.4 north east) .. (k\\brl.center);\n \\draw[thick,red] (k\\brk.center) .. controls (k\\brk.16 south west) and (k\\brl.16 south east) .. (k\\brl);\n }\n \\end{tikzpicture}\n};\n\n\\visible<2->{\n\\node (pL) at (-0.5,-3) {$\\pi_1(L)=\\langle x,y\\mid x^2=y^3\\rangle$};\n\\node (pR) at (4.5,-3) {$\\pi_1(R)=\\langle x,y\\mid x^2=y^{-3}\\rangle$};\n\\node[text width=4cm] at (7,-0.5) {Relax category until automorphisms computable.};\n\\draw[->,thick] (L) to (pL);\n\\draw[->,thick] (R) to (pR);\n};\n\n\\visible<3->{\n\\node (apL) at (0,-5) {$\\mathrm{Aut}_{\\chi}\\pi_1(L)$};\n\\node (apR) at (3,-5) {$\\mathrm{Aut}_{\\chi}\\pi_1(R)$};\n\\draw[->,thick] (pL) to[bend right=30] (apL);\n\\draw[->,thick] (pR) to[bend left=30] (apR);\n\\node at (2,-3) {$\\overset{?}{\\cong}$};\n\\draw[->,thick,dashed] (apL) to[bend right=30] (0.5,-7);\n\\draw[->,thick,dashed] (apR) to[bend left=30] (2.5,-7);\n\\node[color=red] at (1.5,-5) {$\\not{\\cong}_{\\chi}$};\n\\node[text width=4cm] at (7,-5.5) {Recursively refine comparing automorphisms\n with incrementally stricter properties. {\\color{red} E.g. respect crossing number $\\chi$?}};\n};\n\\end{tikzpicture}\n\n\\end{frame}\n\n\\begin{frame}{Inward facing Motive: equalivance surveys complexity}\n\\centering\n\\begin{tikzpicture}\n\n\\fill[rounded corners, black!20] (0,5.5) rectangle (11,4.5);\n\\node[text width=2cm] at (9.5,5) {Undecideable};\n\\node[text width=3cm] at (2,5) {\\textbf{FPGroupIso}\\\\ Adjan, Rabin `50's};\n\n\n\\fill[rounded corners,red!30] (0,4) rectangle (11,3);\n\\node[text width=2cm] at (10,3.5) {$\\Sigma_3^P$};\n\\node[text width=5cm] at (3,3.5) {\\textbf{PlaneGroupIso}\\\\ Dietrich et.al. STACS`21};\n\n\\fill[rounded corners,red!20] (0,3) rectangle (11,2);\n\\node[text width=2cm] at (10,2.5) {$\\Sigma_2^P$};\n\\node[text width=5cm] at (3,2.5) {\\textbf{BlackBoxGroupIso}\\\\ Babai-Szemer\\'edi FOCS`84};\n\\node[text width=5cm] at (8,2.5) {\\textbf{MatroidIso}};\n\n\\fill[rounded corners,red!10] (0,2) rectangle (11,1);\n\\node[text width=2cm] at (10,1.5) {$\\Sigma_1^P=NP$};\n\\node[text width=5cm] at (3,1.5) {\\textbf{PermGroupIso}\\\\ Luks DIMACS};\n\n\n\\fill[rounded corners,brown!20] (0,1) rectangle (11,0);\n\\node[text width=2cm] at (9,0.5) {${\\tiny DTIME}(2^{\\log^c n})$};\n\\node[text width=5cm] at (3,0.5) {\\textbf{TableGroupIso}\\\\ Tarjan};\n\\node[text width=5cm] at (8,0.5) {\\textbf{GraphIso}\\\\ Babai};\n\n\\fill[rounded corners,csugreen!20] (0,-.5) rectangle (11,-1.5);\n\\node[text width=2cm] at (9,-1) {${\\tiny DTIME}(n^2\\log^c n)$};\n\\node[text width=7cm] at (4,-1) {\\textbf{TableGroupIsoAbel}\\\\ Kavitha (nearly-linear in RAM model)};\n\n\\fill[rounded corners,csugreen!30] (0,-1.5) rectangle (11,-2.5);\n\\node[text width=2cm] at (9,-2) {${\\tiny DTIME}(n\\log^c n)$};\n\\node[text width=7cm] at (4,-2) {\\textbf{TableGroupIsoMostOrders, IsGroup}\\\\ {\\color{red}This Talk}};\n\n\n\\end{tikzpicture}\n\n\\end{frame}\n\n\\begin{frame}{Schreier-Sims}\n\n \\begin{block}{Problem: Transport}\n \\begin{description}\n \\item[Given:] A set $\\Omega$, allowed permutations $X$, $\\omega,\\omega'\\in\\Omega$\n \\item[Return:] decide if a string $g$ over $X$ maps $\\omega$ to $\\omega'$, \n written $\\omega^{g}=\\omega'$, and give all such $g$.\\footnote{Give words $W$ over $X$\n so that $\\omega^h=\\omega'$ implies $h=wg$ for a string $w$ over $W$.}\n \\end{description}\n \\end{block}\n\\end{frame}\n\n\n\n\\begin{frame}[fragile]{String Isomorphism}\n\\begin{center}\n ``Eighth'' == ``HeigHt''\n\\end{center}\n%Cycle right by 1 and replace $h$ with $H$.\n\\pause\n\\begin{block}{String Isomorphism}\n\\noindent\n\\begin{minipage}{0.6\\textwidth}\n\\noindent\n\\begin{itemize}\n\\item {\\color{csugreen}\\textbf{Given}} strings $s,t:I\\to \\Sigma$\nallowed permutations $G=\\langle g_k\\rangle\\leq \\mathrm{Sym}_{I}$, \n$H=\\langle h_k\\rangle\\leq \\mathrm{Sym}_{\\Sigma}$\n\\item {\\color{csugreen}\\textbf{Return}} strings $g=g_{a_1}\\cdots g_{a_u}$ \nand $h=h_{b_1}\\cdots h_{b_v}$ where $h(s_{i})=t_{g(i)}$; or prove impossible.\n\\end{itemize}\n\\end{minipage}\n\\begin{minipage}{0.3\\textwidth}\n\\begin{tikzcd}\n & I \\arrow[d,\"g\"]\\arrow[r, \"s\"] \n & \\Sigma\\arrow[d,\"h\"] &\\\\ \n & I\\arrow[r,\"t\"] & \\Sigma & \n\\end{tikzcd}\n\\end{minipage}\n\\end{block}\n\\pause\n\n\\textbf{Theorem.} (Babai 2016+) If $\\Sigma$ fixed, \\textsc{StringIso} \nis in Quasipolynomial $n^{O((\\log n)^c)}$-time.\n\\vfill\n(\\textsc{GraphIso}$\\leq_P$ \\textsc{StringIso})\n\n\\end{frame}\n\n\\begin{frame}{Snapshot of solving a hard isomorphism problem}\n\\centering\n\\begin{tikzpicture}\n\\visible<1->{\n\\node[fill=csugreen!20,text width=3cm] (math) at (0,5) {\\textbf{Math Facts}};\n\\node[fill=blue!20,text width=3cm] (SS) at (0,-2) {\\textbf{Schreier-Sims} `65};\n\\draw[-,dashed] (SS) to[bend right=30] (math);\n};\n\n\\visible<2->{\n\\node[fill=green!20,text width=3cm] (exp) at (0,3) {\\textbf{Exponential}\\\\ Babai-Luks STOC83};\n\\node[fill=blue!20,text width=3.5cm] (BV) at (0,0) {\\textbf{Bounded Valence}\\\\ Luks `82};\n\\node[shade, left color=blue!30, right color=red!30,text width=3cm] (color) at (4,0) {Color Refinement\\\\ Weisfeiler-Leman, Babai-Erd\\H{o}s-Selkow,$\\ldots$ 70-80's};\n\\draw[-] (exp) -- (color);\n\\draw[-] (exp) -- (BV);\n\\draw[-,dashed] (BV) to[bend right=30] (math);\n\\draw[-,dotted] (BV) -- (SS);\n\n};\n\n\\visible<3->{\n\\node[fill=red!20,text width=3cm] (CFI) at (8,-1) {\\textbf{Exceptional}\\\\ Cai-F\\\"urer-Immerman FOCS 89};\n\n\\draw[-,dashed] (CFI) to[bend right=30] (math);\n};\n\n\\visible<4>{\n\\node[fill=green!20,text width=3cm] (quasi) at (5,5) {\\textbf{Split-or-Johnson}\\\\ Babai STOC 16+};\n\n\\node[fill=red!20,text width=3cm] (Johnson) at (8,2) {\\textbf{Johnson Graphs}\\\\ Babai STOC 16+\\\\};\n\\draw[-,dotted] (Johnson) -- (CFI);\n\n\\draw[-] (quasi) -- (color);\n\\draw[-] (quasi) -- (BV);\n\\draw[-] (quasi) -- (Johnson);\n\\draw[-] (quasi) -- (CFI);\n\\draw[-,dashed] (quasi) -- (math);\n};\n\\end{tikzpicture}\n\n\\end{frame}\n\n%========================================================\n\\section{Isomorphism of Tables}\n\n\\begin{frame}[fragile]{Code Equivalence}\n\n\\[\\begin{array}{|cc|}\n \\hline \nL & i \\\\ v & e \\\\\n\\hline\n\\end{array}\n== \n\\begin{array}{|cc|}\n \\hline \nE & v \\\\ i & l \\\\\n\\hline\n\\end{array}\\]\n\\begin{block}{Code Equivalence\\footnote{Non-linear twisted, with variable alphabet.}}\n\\begin{minipage}{0.6\\textwidth}\n\\begin{itemize}\n \\item {\\color{csugreen}\\textbf{Given}} $s,t:I\\times J\\to \\Sigma$, \n (generators for) permutations \n $R\\leq \\mathrm{Sym}_{I}$, \n $C\\leq \\mathrm{Sym}_J$, \n \\& $V\\leq \\mathrm{Sym}_{\\Sigma}$ \n \\item {\\color{csugreen}\\textbf{Return}} $\\sigma\\in R$, $\\tau\\in C$, $\\mu\\in V$,\n \\[\\mu(s_{ij}) = t_{\\sigma(i)\\tau(j)}\\] \n\\end{itemize}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.3\\textwidth}\n \\begin{tikzcd}\n I \\arrow[d,\"\\sigma\"] &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax]\n \\times &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax] J \\arrow[d,\"\\tau\"]\n \\arrow[r,\"s\"] & \\Sigma\\arrow[d,\"\\mu\"]\\\\\n I &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax]\n \\times &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax] J \n \\arrow[r,\"t\"] & \\Sigma\n \\end{tikzcd}\n\\end{minipage}\n\n\\end{block}\n\nBabai-Codenotti-Grochow-Qiao $2^{O(n)}$-time bound for constant alphabet $\\Sigma$ (SODA `11) \\\\\nBuilds on Luks $2^{O(n)}$-hypergraph isomorphism, FOCS `99.\n\n\\end{frame}\n\n\n\\begin{frame}[fragile]{Algebra Isomorphism}\n\n\\[\\begin{array}{|ccccc|}\n \\hline \n1 & 2 & 3 & 4 & 5\\\\\n2 & 3 & 4 & 5 & 1\\\\\n3 & 4 & 5 & 1 & 2\\\\\n4 & 5 & 1 & 2 & 3\\\\\n5 & 1 & 2 & 3 & 4\\\\\n\\hline\n\\end{array}\n== \n\\begin{array}{|ccccc|}\n \\hline \n1 & 2 & 3 & 4 & 5\\\\\n2 & 4 & 1 & 5 & 3\\\\\n3 & 1 & 5 & 2 & 4\\\\\n4 & 5 & 2 & 3 & 1\\\\\n5 & 3 & 4 & 1 & 2\\\\\n\\hline\n\\end{array}\\]\n\\begin{block}{Algebra Isomorphism}\n\\begin{minipage}{0.6\\textwidth}\n\\begin{itemize}\n \\item {\\color{csugreen}\\textbf{Given}} $s,t:I\\times I\\to I$, \n (generators for) permutations \n $G\\leq \\mathrm{Sym}_I$, \n \\item {\\color{csugreen}\\textbf{Return}} $\\sigma\\in G$,\n \\[\\sigma(s_{ij}) = t_{\\sigma(i)\\sigma(j)}\\] \n\\end{itemize}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.3\\textwidth}\n \\begin{tikzcd}\n I\\arrow[d,\"\\sigma\"] &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax]\n \\times &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax] I \\arrow[d,\"\\sigma\"]\n \\arrow[r,\"s\"] & I\\arrow[d,\"\\sigma\"]\\\\\n I &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax]\n \\times &[-\\dimexpr\\pgfmatrixcolumnsep+0.6em\\relax] I\n \\arrow[r,\"t\"] & I\n \\end{tikzcd}\n\\end{minipage}\n\n\\end{block}\n\n\\end{frame}\n\n\\begin{frame}{Group Isomorphism Strategy}\n\n\\centering\n\\begin{tikzpicture}\n\\visible<1->{\n\\node[fill=csugreen!20,text width=3cm] (math) at (5,5) {\\textbf{Math Facts}};\n\\node[fill=blue!20,text width=3.5cm] (gt) at (0,-2) {\\textbf{Generator Test}\\\\ Tarjan, Miller STOC78};\n\\draw[-,dashed] (gt) to[bend right=30] (math);\n};\n\n\\visible<2->{\n\\node[fill=csugreen!20,text width=3cm] (data) at (0,5) {\\textbf{Data Types}};\n\\node[fill=blue!20,text width=2cm] (abel) at (0,3) {\\textbf{(Nearly) Abelian}\\\\ Kavitha `07, LeGall};\n\\node[fill=blue!20,text width=3.5cm] (ss) at (0,0) {\\textbf{Semisimplicty}\\\\ Babai-Codenotti-Qiao ICALP `11};\n\\node[fill=red!20,text width=3cm] (AT) at (8,2) {\\textbf{Adjoint Tensor}\\\\ Lewis-W. `12};\n\n\\draw[-,dashed] (abel) to[bend right=30] (math);\n\\draw[-,dashed] (ss) to[bend right=30] (math);\n\\draw[-,dashed] (abel) to[bend right=30] (data);\n\\draw[-,dashed] (ss) to[bend right=30] (data);\n\n\\draw[-,dashed] (AT) to[bend left=30] (math);\n\n};\n\n\n\\visible<3->{\n \\node[fill=red!20,text width=3cm] (TG) at (8,4) {\\textbf{Tame Genus}\\\\ Brooksbank-Maglione-W. `16}; \n\\node[fill=brown!20,text width=3cm] (BQ) at (4,3) {\\textbf{AbelTower}\\\\Babai-Qiao};\n\\node[fill=brown!20,text width=3cm] (GQ) at (4,1.5) {\\textbf{Extensions}\\\\Grochow-Qiao};\n\n \\node[shade, left color=blue!30, right color=red!30,text width=3cm] (color) at (4,-2) {\\textbf{Tensor Individualization}\\\\Li-Qiao FOCS17};\n \\node[shade, left color=blue!30, right color=red!30,text width=3cm] (cube) at (4,0) {\\textbf{Cube free}\\\\ Dietrich-W. JoA 19};\n \n\\node[fill=red!20,text width=3cm] (W) at (8,0) {\\textbf{Exceptional}\\\\ W. ToC 19};\n\\node[fill=red!20,text width=3cm] (BS) at (8,-2) {\\textbf{Exceptional}\\\\ Brachter-Schweitzer LICS20};\n%\\draw[-,dashed] (color) to[bend right=30] (data);\n\n%\\draw[-,dashed] (cube) to[bend right=30] (data);\n% \\draw[-,dashed] (W) to[bend left=30] (math);\n% \\draw[-,dashed] (TG) to[bend right=30] (math);\n\\draw[->] (AT) to (color);\n\\draw[->] (AT) to (TG);\n\\draw[->] (AT) to (W);\n\\draw[->] (ss) to (BQ);\n\\draw[->] (ss) to (GQ);\n\\draw[->] (abel) to (BQ);\n\\draw[->] (abel) to (GQ);\n};\n\n\\end{tikzpicture}\n\n \n\\end{frame}\n\n\\section{Group Isomorphism of most orders}\n\n\\begin{frame}{Divide and conquer}\n\nIsomorphism of $G=A\\times B$, i.e.\n\\[(a,b)(\\tilde{a},\\tilde{b}) = (a\\tilde{a},b\\tilde{b})\\] reduces to isomorphism\nof $A$ and $B$ in parallel.\\footnote{Technical detail: decompose maximally and\nadjust non-unique decompositions by Krull-Schmidt; see W. 2008.}\n\\pause\n\nIsomorphism of $G=A\\ltimes_{\\theta} B$, i.e.\n\\[(a,b)(\\tilde{a},\\tilde{b})=(a\\tilde{a},\\theta(\\tilde{a})(b)\\tilde{b})\\]\nreduces to isomorphism of $A$ and $B$\nsequentially, plus adjusting $\\theta$.\\footnote{Lemma~2.1}\n\n\\end{frame}\n\n% \\begin{frame}{Factorization}\n% $|A|=p_1^{e_1}\\cdots p_{\\ell}^{e_{\\ell}}$\n\n% \\begin{block}{Fact}\n% $A$ abelian implies $A=P_1\\times \\cdots \\times P_{\\ell}$ with $|P_i|=p_i^{e_i}$.\n% \\end{block}\n\n% \\begin{block}{Fact}\n% $A$ a ring implies $A=P_1\\times \\cdots \\times P_{\\ell}$ with $|P_i|=p_i^{e_i}$.\n% \\end{block}\n\n% \\begin{block}{Fact}\n% $A$ Lie algebra implies $A=P_1\\times \\cdots \\times P_{\\ell}$ with $|P_i|=p_i^{e_i}$.\n% \\end{block}\n\n\n% \\begin{block}{Almost fact}\n% $A$ a group then $A$ comes close to \n% $P_1\\times \\cdots \\times P_{\\ell}$ with $|P_i|=p_i^{e_i}$.\n% \\end{block}\n\n% \\end{frame}\n\n\n\\begin{frame}{Division Graph: Erd\\H{o}s-P\\'{a}lfy}\n\nFactor $n$ into a \\emph{graph} $\\Gamma(n)$. \\\\\nEdge $(p_i^{e_i},p_j^{e_j})$\nwhere $p_i|p_j^k-1$ for some $k\\leq e_j$, \\& symmetrically. \n\n\\begin{block}{Erd\\H{o}s-P\\'{a}lfy, 1999}\n A group $G$ of order $n$ factors as \n\\[N_1\\times \\cdots \\times N_{\\ell},\\] \n$|N_i|=n_i$, order of connected component $i$ in $\\Gamma(n)$.\n\\end{block}\n\n\\pause\n\n\\begin{block}{Example $n=1785$}\n\\begin{center}\n\\begin{tikzpicture}\n\\node[circle,draw] (3) at (0,0) {$3$};\n\\node[circle,draw] (5) at (1,0) {$5$};\n\\node[circle,draw] (7) at (0,1) {$7$};\n\\node[circle,draw] (17) at (1,1) {$17$};\n\n\\draw[-,thick] (3) to (7);\n\n\\node at (6,1) {$|G|=n\\Rightarrow N_{3\\cdot 7}\\times N_5\\times N_{17}$};\n\\end{tikzpicture}\n\\end{center}\n\\end{block}\n\n\n\\end{frame}\n \n\\begin{frame}{Extending implications}\n\nFactor $n$ into a \\emph{direct hypergraph} $\\mathcal{H}(n)$. (i) \\emph{Oriented}\nErd\\H{o}s-P\\'{a}lfy \\emph{hyper-}edges, (ii) exceptions for \nfinite nonabelian simple groups.\n\n\\begin{block}{Proposition}\n A group $G$ of order $n$ factors as \n\\[N_0\\ltimes (N_1\\ltimes \\cdots \\ltimes N_{\\ell}),\\] \n$|N_i|=n_i$, where $n_i$ is order of interconnected components of $\\mathcal{H}(n)$. \n\\end{block}\n\n\\pause\n\\begin{block}{Example $n=5,810,340$}\n\\begin{tikzpicture}\n\\node[circle,draw] (2) at (0,-1) {$2^2$};\n\\node[circle,draw] (3) at (0,1) {$3$};\n\\node[circle,draw] (5) at (-1.5,0) {$5$};\n\\node[circle,draw] (179) at (2,1) {$179$};\n\\node[circle,draw] (547) at (2,-1) {$547$};\n\n\\draw[->,thick] (2) to[bend right=30] (3);\n\\draw[->,thick] (2) to[bend left=30] (5);\n\\draw[->,thick] (2) to[bend right=20] (179);\n\\draw[->,thick] (2) to[bend right=20] (547);\n\n\\draw[->,thick] (5) to (-0.5,0) to[bend right=30] (3);\n\\draw[->,thick] (-0.5,0) to[bend left=30] (2);\n\n\\draw[->,thick] (3) to[bend left=30] (2);\n\n\n\\node at (6,1) {$|G|=n\\Rightarrow N_{60}\\ltimes (N_{179}\\times N_{547})$};\n\n\\end{tikzpicture}\n\\end{block}\n\n\n\\end{frame}\n\n\\begin{frame}{Most Orders}\n\n\\begin{center}\n\\begin{tikzpicture}\n\n\\fill[color=black!20,rounded corners] (-2.5,2) rectangle (1,-2);\n\\fill[color=blue!20,rounded corners] (1,2) rectangle (7,-2);\n\n\\node[circle,draw] (2) at (0,-1) {$p_1^{e_1}$};\n\\node[circle,draw] (3) at (0,1) {$p_2^{e_2}$};\n\\node[circle,draw] (5) at (-1.5,0) {$p_m^{e_{m}}$};\n\\node[circle,draw] (179) at (3,1) {$p_{m+1}$};\n\\node[circle,draw] (547) at (3,-1) {$p_{\\ell}$};\n\\node[circle,draw] (1949) at (4,0.5) {$p_{k}$};\n\\node[circle,draw] (7893) at (5,-0.5) {$p_{j}$};\n\n\n\n\\draw[->,thick] (2) to[bend right=30] (3);\n\\draw[->,thick] (2) to[bend left=30] (5);\n\\draw[->,thick] (2) to[bend right=20] (179);\n\\draw[->,thick] (2) to[bend right=20] (547);\n\n\\draw[->,thick] (5) to (-0.5,0) to[bend right=30] (3);\n\\draw[->,thick] (-0.5,0) to[bend left=30] (2);\n\n\\draw[->,thick] (3) to[bend left=30] (2);\n\n\\draw[->,thick] (3) to[bend right=20] (7893);\n\n\\visible<2->{\n\\node[color=red] at (-1,2.25) {Hard Group Theory};\n\\node[color=red] at (4,2.25) {Hard Number Theory};\n};\n\n\\visible<3->{\n\\node[color=csugreen] at (-1,-2.25) {Tiny Numbers};\n\\node[color=csugreen] at (4,-2.25) {Cyclic Groups};\n\\node at (2.5,-4) {$G=H\\ltimes (\\mathbb{Z}_{p_{m+1}}\\times\\cdots \\times \\mathbb{Z}_{p_{\\ell}})$};\n};\n\n\n\\end{tikzpicture} \n\\end{center}\n\n\\end{frame}\n\n% \\begin{frame}{}\n% \\begin{itemize}\n% \\item Factor $n=ab$ where prime $p|n$, $p\\geq\\log \\log n$ and not exceptional\\footnote{primes strongly connected or involved in identifying a nonabelian simple group} then $p|b$;\n% otherwise $p|a$.\n \n% \\item Factor $G=H\\ltimes_{\\theta} B$ with prime divisors $p>\\log \\log n$ dividing\n% $|B|$ but not $|H|$ (unless exceptional divisor of finite simple groups).\n% Likewise decompose $\\tilde{G}=\\tilde{H}\\ltimes_{\\tilde{\\theta}} \\tilde{B}$.\n\n% \\item Brute-force decide $H\\cong \\tilde{H}$.\n% \\item Canonical isomorphism $B\\cong \\mathbb{Z}_b\\cong \\tilde{B}$ by Group Theory.\n% \\item Adjust $\\theta$ to match $\\tilde{\\theta}$ if possible.\n\n% \\end{itemize}\n% \\end{frame}\n\n\n%==============================================================================\n\\section{Is it a Group Table?}\n\n\\begin{frame}{Promise-to-decision}\n\nA great many computational algebra are analyzed as \\textbf{promise problems}\nnot \\textbf{decision problems}.\n\nIdentity testing is needed to remove the promise; unsolvable in general (word problem), \nbut on tables at least brute-force.\n\\pause\n\n\\begin{block}{Theorem Rajagopalan-Schulman, 2000}\n Given $*:[n]\\times [n]\\to [n]$, test associativity (and other identities) in\n nearly-linear time $\\tilde{O}(n^2)$ in RAM model (constant time ops and\n memory access). Also can test if a group.\n\\end{block}\n\n\\end{frame}\n\n\\begin{frame}{RAM-to-TM}\n\nAt larger scales Turing Machine (TM) model better match to computations \nthat are communication bounded (typical in practice).\n\nRAM $\\to$ TM at most a quadratic blow-up.\n\\begin{block}{Corollary}\nNearly Quadratic-time $\\tilde{O}(n^4)$ on multi-tape Turing Machine (TM).\n\\end{block}\n\n\\pause\n\\begin{block}{Theorem Dietrich-W.}\n Given $*:[n]\\times [n]\\to [n]$,\n test if a group in time nearly-linear time\n $\\tilde{O}(n^2)$ on deterministic multi-tape TM.\n % \\footnote{One tape of input length $\\ell\\in O(n^2)$,\n % and another of length $O(\\sqrt{\\ell})$.}\n\\end{block}\n\n\\end{frame}\n\n% \\begin{frame}{IsGroup Sketch}\n% \\tikzset{\n% mybox/.style={\n% inner sep=0pt,\n% text width=5mm,\n% text height=5mm,\n% align=center,\n% }\n% }\n% \\begin{tikzpicture}\n% \\node[fill=blue!0, label=center:1,mybox] at ( 0mm, -0mm) {};\n% \\node[fill=blue!20,label=center:2,mybox] at ( 5mm, -0mm) {};\n% \\node[fill=blue!20,label=center:3,mybox] at (10mm, -0mm) {};\n% \\node[fill=blue!30,label=center:4,mybox] at (15mm, -0mm) {};\n% \\node[fill=blue!40,label=center:5,mybox] at (20mm, -0mm) {};\n% %\n% \\node[fill=blue!10,label=center:2,mybox] at ( 0mm, -5mm) {};\n% \\node[fill=blue!30,label=center:4,mybox] at ( 5mm, -5mm) {};\n% \\node[fill=blue!0, label=center:1,mybox] at (10mm, -5mm) {};\n% \\node[fill=blue!40,label=center:5,mybox] at (15mm, -5mm) {};\n% \\node[fill=blue!20,label=center:3,mybox] at (20mm, -5mm) {};\n% %\n% \\node[fill=blue!20,label=center:3,mybox] at ( 0mm,-10mm) {};\n% \\node[fill=blue!40,label=center:5,mybox] at ( 5mm,-10mm) {};\n% \\node[fill=blue!30,label=center:4,mybox] at (10mm,-10mm) {};\n% \\node[fill=blue!10,label=center:2,mybox] at (15mm,-10mm) {};\n% \\node[fill=blue!0, label=center:1,mybox] at (20mm,-10mm) {};\n% %\n% \\node[fill=blue!30,label=center:4,mybox] at ( 0mm,-15mm) {};\n% \\node[fill=blue!0, label=center:1,mybox] at ( 5mm,-15mm) {};\n% \\node[fill=blue!40,label=center:5,mybox] at (10mm,-15mm) {};\n% \\node[fill=blue!20,label=center:3,mybox] at (15mm,-15mm) {};\n% \\node[fill=blue!10,label=center:2,mybox] at (20mm,-15mm) {};\n% %\n% \\node[fill=blue!40,label=center:5,mybox] at ( 0mm,-20mm) {};\n% \\node[fill=blue!20,label=center:3,mybox] at ( 5mm,-20mm) {};\n% \\node[fill=blue!10,label=center:2,mybox] at (10mm,-20mm) {};\n% \\node[fill=blue!0, label=center:1,mybox] at (15mm,-20mm) {};\n% \\node[fill=blue!30,label=center:4,mybox] at (20mm,-20mm) {};\n% \\end{tikzpicture}\n\n% \\end{frame}\n\n\n\\begin{frame}{IsGroup}\n\n\\begin{tikzpicture}\n \\node at (0,0) {$\n \\begin{array}{|c|ccccc|}\n \\hline\n \\bullet & 1 & 2 & 3 & 4 & 5\\\\\n \\hline\n 1 & 1 & 2 & 3 & 4 & 5\\\\\n 2 & 2 & 1 & 4 & 5 & 3\\\\\n 3 & 3 & 5 & 1 & 2 & 4\\\\\n 4 & 4 & 3 & 5 & 1 & 2\\\\\n 5 & 5 & 4 & 2 & 3 & 1\\\\\n \\hline\n \\end{array} \n $};\n \\visible<2->{\n \\node at (6,0) {$\n \\rho(2) = \n \\begin{array}{|ccccc|}\n \\hline\n 1 & 2 & 3 & 4 & 5 \\\\\n \\hline \n 2 & 1 & 4 & 5 & 3 \\\\\n \\hline \\end{array} \n =\n (1,2)(3,5,4)\n $};\n };\n \\visible<3->{\n \\node at (0,-3) {\\begin{tikzpicture}\n \\node[circle,draw] (1) at (0,0) {1};\n \\node[circle,draw] (2) at (2,0) {2};\n \\draw[->,thick] (1) to[bend left=30] (2);\n \\draw[->,thick] (2) to[bend left=30] (1);\n \\end{tikzpicture}};\n \\node at (0,-4.5) {$G_{1}=\\langle (354)\\rangle$};\n };\n \\visible<4->{\n \\node at (3,-3) {\\begin{tikzpicture}\n \\node[circle,draw] (3) at (-1,0) {3};\n \\node[circle,draw] (4) at (0.5,0.7) {5};\n \\node[circle,draw] (5) at (0.5,-0.7) {4};\n\n \\draw[->,thick] (3) to[bend left=30] (4);\n \\draw[->,thick] (4) to[bend left=30] (5);\n \\draw[->,thick] (5) to[bend left=30] (3);\n \\end{tikzpicture}};\n \\node at (3,-4.5) {$G_{13}=\\{1\\}$};\n };\n \\visible<5>{\n \\node[text width=4cm] at (8,-3) {$|G|=[G:G_1][G_1:G_{13}]$\\\\ $~=2\\cdot 3=6$.\\\\ Should be 5,\\\\ not a group.};\n };\n\\end{tikzpicture}\n\n\\end{frame}\n\n \n\\begin{frame}{IsGroup}\n \\begin{tikzpicture}\n \\node at (0,0) {$\n \\begin{array}{|c|ccccc|}\n \\hline\n * & 1 & 2 & 3 & 4 & 5 \\\\\n \\hline \n 1 & 1 & 2 & 3 & 4 & 5 \\\\ \n 2 & 2 & 4 & 1 & 5 & 3 \\\\\n 3 & 3 & 5 & 4 & 2 & 1 \\\\\n 4 & 4 & 1 & 5 & 3 & 2 \\\\\n 5 & 5 & 3 & 2 & 1 & 4 \\\\\n \\hline \\end{array} \n $};\n \\visible<2->{\n \\node at (6,0) {$\n \\rho(2) = \n \\begin{array}{|ccccc|}\n \\hline\n 1 & 2 & 3 & 4 & 5 \\\\\n \\hline \n 2 & 4 & 1 & 5 & 3 \\\\\n \\hline \\end{array} \n =\n (1,2,4,5,3)\n $};\n };\n \\visible<3->{\n \\node at (0,-3) {\\begin{tikzpicture}\n \\node[circle,draw] (1) at (1,0) {1};\n \\node[circle,draw] (2) at (0.3,0.8) {2};\n \\node[circle,draw] (3) at (-0.6,0.5) {3};\n \\node[circle,draw] (4) at (-0.6,-0.5) {4};\n \\node[circle,draw] (5) at (0.3,-0.8) {5};\n\n \\draw[->,thick] (1) to[bend right=30] (2);\n \\draw[->,thick] (2) to[bend left=30] (4);\n \\draw[->,thick] (4) to[bend right=30] (5);\n \\draw[->,thick] (5) to[bend right=30] (3);\n \\draw[->,thick] (3) to[bend right=30] (1);\n \\end{tikzpicture}};\n \\node at (0,-4.5) {$|G|=[G:G_{1}|=5$};\n };\n \\visible<4>{\n \\node[text width=4cm] at (7,-3) {$\n \\begin{array}{|c|ccccc|}\n \\hline\n * & 1 & 2 & 3 & 4 & 5 \\\\\n \\hline \n \\rho(2)^0 & 1 & 2 & 3 & 4 & 5 \\\\ \n \\rho(2)^1 & 2 & 4 & 1 & 5 & 3 \\\\\n \\rho(2)^2 & 4 & 5 & 2 & 3 & 1 \\\\\n \\hline \\end{array} \n $\\\\\n\n $\\rho(2)^2=45231\\neq T_4=41532$\\\\ Not a group.};\n };\n \\end{tikzpicture}\n\\end{frame}\n\n% \\begin{frame}{IsGroup}\n% \\begin{tikzpicture}\n% \\node at (0,0) {$\n% \\begin{array}{|c|ccccc|}\n% \\hline\n% * & 1 & 2 & 3 & 4 & 5 \\\\\n% \\hline \n% 1 & 1 & 2 & 3 & 4 & 5 \\\\ \n% 2 & 2 & 4 & 1 & 5 & 3 \\\\\n% 3 & 4 & 5 & 4 & 2 & 1 \\\\\n% 4 & 4 & 1 & 5 & 3 & 2 \\\\\n% 5 & 5 & 3 & 2 & 1 & 4 \\\\\n% \\hline \\end{array} \n% $};\n% \\visible<2->{\n% \\node at (6,0) {$\n% \\rho(2) = \n% \\begin{array}{|ccccc|}\n% \\hline\n% 1 & 2 & 3 & 4 & 5 \\\\\n% \\hline \n% 2 & 4 & 1 & 5 & 3 \\\\\n% \\hline \\end{array} \n% =\n% (1,2,4,5,3)\n% $};\n% };\n% \\visible<3->{\n% \\node at (0,-3) {\\begin{tikzpicture}\n% \\node[circle,draw] (1) at (1,0) {1};\n% \\node[circle,draw] (2) at (0.3,0.8) {2};\n% \\node[circle,draw] (3) at (-0.6,0.5) {3};\n% \\node[circle,draw] (4) at (-0.6,-0.5) {4};\n% \\node[circle,draw] (5) at (0.3,-0.8) {5};\n\n% \\draw[->,thick] (1) to[bend right=30] (2);\n% \\draw[->,thick] (2) to[bend left=30] (4);\n% \\draw[->,thick] (4) to[bend right=30] (5);\n% \\draw[->,thick] (5) to[bend right=30] (3);\n% \\draw[->,thick] (3) to[bend right=30] (1);\n% \\end{tikzpicture}};\n% \\node at (0,-4.5) {$|G|=[G:G_{1}|=5$};\n% };\n% \\visible<4>{\n% \\node[text width=4cm] at (7,-3) {$\n% \\begin{array}{|c|ccccc|}\n% \\hline\n% * & 1 & 2 & 3 & 4 & 5 \\\\\n% \\hline \n% \\rho(2)^0 & 1 & 2 & 3 & 4 & 5 \\\\ \n% \\rho(2)^1 & 2 & 4 & 1 & 5 & 3 \\\\\n% \\rho(2)^2 & 4 & 5 & 2 & 3 & 1 \\\\\n% \\hline \\end{array} \n% $\\\\\n\n% $\\rho(2)^2=45231\\neq T_4=41532$\\\\ Not a group.};\n% };\n% \\end{tikzpicture}\n% \\end{frame}\n\n% \\begin{frame}{IsGroup Generalizing}\n\n% \\begin{block}{Main idea}\n% \\begin{itemize}\n% \\item Map putative generator $S$ into efficient representation of class, e.g. permutations.\n% \\item Explore representation to efficiently reproduce table\n% \\item Compare tables.\n% \\end{itemize}\n% \\end{block}\n\n% \\begin{block}{Question:}\n% Do other classes of tables permit this representation approach?\n% \\end{block}\n\n\n% \\end{frame}\n\n\\section{Summary}\n\n\\begin{frame}{Summary}\n\n\\begin{block}{IsGroup nearly linear time}\n From $\\tilde{O}(n^4)$ to $\\tilde{O}(n^2)$: \n Promise-to-decision by transferring group to permutation model.\n\\end{block}\n\n\\begin{block}{GroupIso most orders nearly linear time}\n From $n^{O(\\log n)}$ to $\\tilde{O}(n^2)$: \n Split group into\n \\begin{center} \n hard group $\\ltimes$ hard numbers \n = tiny numbers $\\ltimes$ cyclic groups.\n \\end{center}\n Then standard divide-and-conquer.\n\\end{block}\n\n\\rule{\\textwidth}{1pt}\n{\\tiny Thanks to: Newton Institute (Cambridge, UK) \nEPSRC Grant Number EP/R014604/1, Australian Research Council grant\nDP190100317, and Simons Foundation Grant 636189.}\n\\end{frame}\n\n% \\begin{frame}{}\n\n% \\begin{block}{Theorem.}\n% A positive density of positive integers $n$ has $\\Gamma(n)$ such \n% that the \n% \\end{block}\n\n% \\end{frame}\n\n% \\begin{frame}{Most integers factor like this}\n\n \n% \\begin{tikzpicture}\n% \\node[circle,draw] (3) at (0,0) {$3$};\n% \\node[circle,draw] (31) at (1,1) {$31$};\n% \\node[circle,draw] (137) at (1,-1) {$137$};\n% \\node[circle,draw] (379) at (2,-1) {$379$};\n% \\node[circle,draw] (1949) at (3,0) {$1949$};\n\n% \\draw[->,thick] (3) to[bend left=30] (0,0.5) to[bend right=30] (31);\n% \\draw[->,thick] (3) to[bend left=30] (0,0.5) to[bend right=30] (31);\n% % \\draw[->,thick] (0,0.5) to[bend left=30] (5);\n% % \\draw[->,thick] (0,0.5) to[bend right=20] (541);\n\n% % \\draw[->,thick] (5) to[bend right=30] (-0.25,-0.5) to[bend right=30] (3);\n% % \\draw[->,thick] (-0.25,-0.5) to[bend left=30] (2);\n\n% % \\draw[->,thick] (3) to[bend right=30] (2);\n\n\n% % \\node at (6,1) {$|G|=n\\Rightarrow N_{60}\\ltimes N_{541}$};\n\n% \\end{tikzpicture}\n \n% \\end{frame}\n\n\\end{document}", "meta": {"hexsha": "e3bed3426c3215f48de88141c3f718789a48379c", "size": 34982, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "FOCS21/FOCS.tex", "max_stars_repo_name": "algeboy/talks", "max_stars_repo_head_hexsha": "0787c45eb8d69ac4bbf5e23a35b9f83d910b5d73", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "FOCS21/FOCS.tex", "max_issues_repo_name": "algeboy/talks", "max_issues_repo_head_hexsha": "0787c45eb8d69ac4bbf5e23a35b9f83d910b5d73", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "FOCS21/FOCS.tex", "max_forks_repo_name": "algeboy/talks", "max_forks_repo_head_hexsha": "0787c45eb8d69ac4bbf5e23a35b9f83d910b5d73", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.5722021661, "max_line_length": 184, "alphanum_fraction": 0.5724086673, "num_tokens": 14102, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792046, "lm_q2_score": 0.6859494421679929, "lm_q1q2_score": 0.3483332650286089}} {"text": "\\documentclass[12pt]{cdblatex}\n\\usepackage{exercises}\n\\usepackage{fancyhdr}\n\\usepackage{footer}\n\n\\begin{document}\n\n% --------------------------------------------------------------------------------------------\n\\section*{Exercise 1.6 Multiple {\\tt SortOrder} lists}\n\n\\begin{cadabra}\n {D,C,B,A}::SortOrder. # first SortOrder list\n\n foo := A B C D . # cdb(ex-0106.101,foo)\n\n sort_product (foo) # cdb(ex-0106.102,foo)\n\n {V,U}::SortOrder. # second SortOrder list, all entries distinct from first list\n\n foo := U V A B C D . # cdb(ex-0106.201,foo)\n\n sort_product (foo) # cdb(ex-0106.202,foo)\n\n {A,B,C,D}::SortOrder. # all entries in this list appear in the\n # first SortOrder so they will be effectively ignored\n\n foo := U V D C B A . # cdb(ex-0106.301,foo)\n\n sort_product (foo) # cdb(ex-0106.302,foo)\n\\end{cadabra}\n\n% \\clearpage\n\n\\begin{dgroup*}[spread=2pt]\n \\Dmath*{\\cdb*{ex-0106.101}}\n \\Dmath*{\\cdb*{ex-0106.102}}\n\\end{dgroup*}\n\n\\begin{dgroup*}[spread=2pt]\n \\Dmath*{\\cdb*{ex-0106.201}}\n \\Dmath*{\\cdb*{ex-0106.202}}\n\\end{dgroup*}\n\n\\begin{dgroup*}[spread=2pt]\n \\Dmath*{\\cdb*{ex-0106.301}}\n \\Dmath*{\\cdb*{ex-0106.302}}\n\\end{dgroup*}\n\n\\end{document}\n", "meta": {"hexsha": "dda1fb06b5bb06ceaf5038e91276fa4eee01619f", "size": 1227, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "source/cadabra/exercises/ex-0106.tex", "max_stars_repo_name": "leo-brewin/cadabra-tutorial", "max_stars_repo_head_hexsha": "5b428ae158b5346315ab6c975dee9de933e5c3d7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 20, "max_stars_repo_stars_event_min_datetime": "2019-12-20T07:49:47.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-27T22:55:47.000Z", "max_issues_repo_path": "source/cadabra/exercises/ex-0106.tex", "max_issues_repo_name": "leo-brewin/cadabra-tutorial", "max_issues_repo_head_hexsha": "5b428ae158b5346315ab6c975dee9de933e5c3d7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "source/cadabra/exercises/ex-0106.tex", "max_forks_repo_name": "leo-brewin/cadabra-tutorial", "max_forks_repo_head_hexsha": "5b428ae158b5346315ab6c975dee9de933e5c3d7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-12-22T13:52:19.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-22T13:52:19.000Z", "avg_line_length": 24.54, "max_line_length": 94, "alphanum_fraction": 0.5729421353, "num_tokens": 431, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.3482894268451717}} {"text": "\\documentclass[main.tex]{subfiles}\n\\begin{document}\n\n\\marginpar{Friday\\\\ 2020-3-27, \\\\ compiled \\\\ \\today}\n\nWe can always decompose a Hermitian matrix in a basis: if \\(M = M ^\\dag\\) then \\(M = r_{\\mu } \\hat{\\Gamma}_{\\mu } \\).\nIf we choose the matrices so that \\(\\Tr [\\hat{\\Gamma}_{\\mu } \\hat{\\Gamma}_{\\nu }] = \\delta_{\\mu \\nu }\\), we can use the Euclidean scalar product. \n\n\\section{Bell nonlocality}\n\nWe will follow some of \\cite[]{brunnerBellNonlocality2014}. \n\nWhen first encountering QM, people usually think in terms of statistics. \n\nLet us make an explicit example for a hidden variable theory. \n\nLet us suppose we have a qubit which is described by the density matrix \n%\n\\begin{align}\n\\rho = \\frac{1}{2} \\qty(\\mathbb{1} + \\vec{s} \\cdot \\vec{\\sigma})\n\\,,\n\\end{align}\n%\nwhere \\(\\vec{s}\\) is a unit vector in the Bloch sphere. \nNow, we can measure spins: our observables are in the form \\(\\hat{A} = \\vec{a} \\cdot \\vec{\\sigma}\\), which means we are measuring the spin in the direction of the unit vector \\(\\vec{a}\\). \n\nThe expectation value is \n%\n\\begin{align}\n\\expval{\\hat{A}} = \\Tr (\\hat{A} \\rho ) = \\vec{a} \\cdot \\vec{s}\n= \\cos \\theta = (+1) \\mathbb{P} (+1) + (-1) \\mathbb{P}(-1)\n\\,,\n\\end{align}\n%\nwhere \\(\\theta \\) is the angle between \\(\\vec{a}\\) and \\(\\vec{s}\\). \n\nIs the output of the measurement prescribed by some hidden variable \\(\\lambda \\)?\n\nIf so, we would need to describe the state with the pair \\((\\rho , \\lambda )\\). Then, if \\(A(\\lambda )\\) is the function associating a value of \\(\\lambda \\) to its outcome we will have \n%\n\\begin{align}\n\\expval{\\hat{A}} = \\int \\dd{\\mu_{\\rho }(\\lambda )} A(\\lambda )\n\\,.\n\\end{align}\n\nIn principle \\(A(\\lambda )\\) would also depend on \\(\\rho \\), but we can rescale the measure on the space so that it doesn't. \n\nSuppose \\(\\vec{\\lambda}\\) is defined on the hemisphere \\(\\vec{\\lambda} \\cdot \\vec{s} \\geq 0 \\) and \\(\\abs{\\vec{\\lambda}} =1\\).\nThen, \\(\\dd{\\mu_{\\rho }(\\lambda )}\\) is a uniform density function defined there, and it is zero otherwise. \n\nLet us define the unit vector \\(\\vec{a}'\\) so that \n%\n\\begin{align}\nA(\\lambda ) = \\sign(\\vec{\\lambda } \\cdot \\vec{a}')\n\\,,\n\\end{align}\n%\nand so that \\(\\vec{a}'\\) is in the plane defined by \\(\\vec{s}\\) and \\(\\vec{a} \\) and the angle between \\(\\vec{s}\\) and \\(\\vec{a}'\\) is \\(\\theta'\\), defined by \n%\n\\begin{align}\n1 - \\frac{2 \\theta'}{\\pi } = \\cos \\theta \n\\,.\n\\end{align}\n\nWith this, we have a statistical model which describes quantum mechanics. \n\nThe problem comes along when we consider Bell inequalities, which deal with multiple systems. \n\nThen, in the Hidden Variable theory we will have \n%\n\\begin{align}\n\\expval{\\hat{A} \\otimes \\hat{B}} = \\int \\dd{\\mu_{\\rho } (\\lambda )} F(\\lambda )\n\\,,\n\\end{align}\n%\nbut if we assume we have locality then (as shown in Phys Rev 47, 777 (1935), the EPR paper) we must impose that there can be no causal link between events which are spacelike-separated. \nSo, under a Local Hidden Variable model we must have that the events are independent: so we must write \n%\n\\begin{align}\n\\expval{\\hat{A} \\otimes \\hat{B}} = \\int \\dd{\\mu_{\\rho } (\\lambda )} A(\\lambda ) B(\\lambda )\n\\,.\n\\end{align}\n\nReality means determinism: if we have reality and we fix \\(\\lambda \\) then the result of the measurement is also fixed. \n\nWhat Bell did was to probe that this kind of expression is incompatible with the prediction of QM: if we consider the operator \n%\n\\begin{align} \nS =\n\\expval{\\hat{A} \\otimes \\hat{B}}+\n\\expval{\\hat{A} \\otimes \\hat{B}'} +\n\\expval{\\hat{A}' \\otimes \\hat{B}} -\n\\expval{\\hat{A}' \\otimes \\hat{B}'} \n\\,,\n\\end{align}\n%\nwhere \\(\\hat{A}'\\) and \\(\\hat{B}'\\) are two different observables Alice and Bob can choose to measure randomly, and which both have eigenvalues \\(\\pm 1\\). Then, we have that its expectation value is \n%\n\\begin{align}\n\\int \\dd{\\mu (\\lambda )} \\qty[A (B + B') + A' (B - B')] \n\\,,\n\\end{align}\n%\nand this is upper-bounded by 2. \nIf we do an experiment, and measure a value which is greater than 2, then we have falsified LHV theory. \nQuantum Mechanics tells us we can indeed go beyond 2. \nIf we choose \\(\\vec{a}\\) and \\(\\vec{a}'\\) at right angles to each other, \\(\\vec{b} \\) and \\(\\vec{b}'\\) likewise, with an angle of \\(\\pi /4\\) between them, then it can be shown that if the state \\(\\rho \\) is a singlet then \n%\n\\begin{align}\n\\expval{(\\vec{a} \\cdot \\vec{\\sigma}) \\otimes (\\vec{a} \\cdot \\vec{\\sigma})} = - \\vec{a} \\cdot \\vec{b} \n\\,,\n\\end{align}\n%\nso if \\(\\vec{a} = \\vec{b} \\) we have perfect anticorrelation. So we get \n%\n\\begin{align}\n\\expval{\\abs{S}} = \\abs{- \\frac{1}{\\sqrt{2}} - \\frac{1}{\\sqrt{2}} - \\frac{1}{\\sqrt{2}} - \\frac{1}{\\sqrt{2}} } = 2 \\sqrt{2} \\approx 2.8 > 2\n\\,.\n\\end{align}\n\nSo, we either reject determinism or we reject locality. \n\nIn the orthodox interpretation of QM the theory is simply nondeterministic. \n\nThis is wonderful theoretically, but there are loopholes. \nOnly in 2015 the experiment was done in a loophole-free way. \n\n\\subsection{Loopholes}\n\n\\subsubsection{Freedom of choice}\n\nIn the experiment we discussed before Alice and Bob could choose in a random way between the two measurements. This is important since if the measurements are predetermined then that is a hidden variable. \nIf the measurements are fixed then we can describe the statistics with a hidden variable theory. \n\n\\subsubsection{Locality}\n\nThe choice of the measurement basis must be done late enough so that the events are still spacelike separated. \nSo, we should choose the measurement basis so that the photons have almost arrived when we decide. \n\n\\subsubsection{Detection loophole}\n\nPhotons are not revealed efficiently: they are lost in detection, so we may reveal only a small fraction of the total photons. \n\n\\subsubsection{Superdeterminism}\n\nIt has not been possible yet to exclude that all of nature is completely predetermined. \n\nWhen we write the correlation between two operators, we write something like \n%\n\\begin{align}\n\\expval{A \\otimes B} = \\sum _{ij} ij \\mathbb{P}_{AB} (ij)\n\\,,\n\\end{align}\n%\nbut we have that \\(\\mathbb{P} (a, b) + \\mathbb{P} (\\overline{a}, b) = \\mathbb{P}(b)\\). \nThen, we can find that \n%\n\\begin{align}\n\\expval{A \\otimes B} = 4 \\mathbb{P}(a, b) - 2 P_A(a) -2 P_B(b) +1\n\\,,\n\\end{align}\n%\nwhich is useful since it contains only positive results. \nThis allows us to write \n%\n\\begin{align}\nS_{CHSH} = 4S_{CH} + 2\n\\qquad \\text{where} \\qquad\nS_{CH} = \\mathbb{P}(a, b) \n+ \\mathbb{P} (a', b) \n+ \\mathbb{P} (a, b') \n- \\mathbb{P}(a) - \\mathbb{P}(b)\n\\,,\n\\end{align}\n%\nand \\(S_{CHSH}\\leq 2\\) is equivalent to \\(S_{CH} \\leq 0\\). The advantage is that we do not need to normalize for the total number of events: we can write this directly as \n%\n\\begin{align}\nN(a, b) \n+ N (a', b) \n+ N (a, b') \n- N(a) - N(b)\n\\leq 0\n\\,,\n\\end{align}\n%\nso we have no issue with lost photons. \n\nIf the efficiency is \\(\\eta \\) for each channel, then experimentally we will measure \n%\n\\begin{align}\nS_{CH}^{\\text{exp}} = \\eta^2 \\qty[\\mathbb{P}(a, b) \n+ \\mathbb{P} (a', b) \n+ \\mathbb{P} (a, b') ]\n- \\eta \\qty[\\mathbb{P}(a) + \\mathbb{P}(b)]\n\\,,\n\\end{align}\n%\nwhere the term multiplying \\(\\eta^2\\) is precisely \\(S_{CHSH}+ \\mathbb{P}(a) + \\mathbb{P}(b)\\). If we want to violate the inequalities, we must have \\(S_{CH}^{\\text{exp}} > 0\\): the minimum efficiency is \n%\n\\begin{align}\n\\eta_{*} = \\frac{\\mathbb{P}_{A}(a) + \\mathbb{P}_{B}(b)}{S^{Q}_{CH} + P_{A}(a) + P_{B}(b)}\n\\,,\n\\end{align}\n%\nand if we use maximally entangled singlet states we have \\(\\eta_{*} \\approx 2 \\qty(\\sqrt{2}-1) \\approx \\SI{83}{\\percent}\\). \n\nThis is combined efficiency: an emitted photon must have a \\(\\geq \\SI{83}{\\percent}\\) probability of being revealed. \n\nThe detectors used had less efficiency than this: actually, if we have a state like \\(\\cos \\theta \\ket{00} + \\sin \\theta \\ket{11 }\\) we have a lower \\(\\eta_{*}\\): the best thing is in the limit of \\(\\theta \\rightarrow 0\\), so we will need \\(\\eta_{*} \\approx 2/3 \\approx \\SI{67}{\\percent}\\). \n\nIf we do not have this efficiency, we must make a \\emph{fair sampling assumption}: the photons we do measure are a representative sample of all the photons. \n\nThere were three works published in 2015 which did this: \n\\begin{enumerate}\n \\item Nature 526, 682\n \\item PRL 115, 250401\n \\item PRL 115, 250402\n\\end{enumerate}\n\nIn the works in PRL they used photons, while the work in Nature used electron spins, entangling them with entanglement swapping. \nThe experiment using electrons did not have the detection loophole, but they are very much subject to the locality loophole. \n\nWe can further extend Bell inequalities in multipartite systems. \n\nIf we have three systems, we can measure (denoting \\(\\sigma_{x} = X\\) and so on)\n%\n\\begin{align}\nM_1 &= X_{a} X_{b} X_{c} \\\\\nM_2 &= - Y_{a} Y_{b} X_{c} \\\\\nM_3 &= - Y_{a} X_{b} Y_{c} \\\\\nM_4 &= - Y_{a} Y_{b} Y_{c}\n\\,,\n\\end{align}\n%\nand one can see that \\(M_1 M_2 M_3 = - M_4 \\), so classically if we assume we can write the expectation value as \n%\n\\begin{align}\n\\int \\dd{\\mu } A(\\lambda ) B(\\lambda ) C(\\lambda )\n\\,,\n\\end{align}\n%\nand we can violate these with the GHZ state: \n%\n\\begin{align}\n\\ket{\\text{GHZ}} = \\frac{1}{\\sqrt{2}} \\qty(\\ket{000} + \\ket{111})\n\\,,\n\\end{align}\n%\nand we can see that \n%\n\\begin{align}\n\\expval{M_i} = +1\n\\,,\n\\end{align}\n%\nfor each of these. We can write this as an inequality with \\(M_1 + M_2 + M_3 + M_4 \\leq 2\\). \n\nThe nice thing is that we can do this with perfect correlations, while in CHSH we do not have perfect correlations. \n\nOne could actually see that this can be generalized to \\(N\\) qubits: as we increase the dimension, the violation increases exponentially: \n%\n\\begin{align}\n\\frac{\\beta_{Q}}{\\beta_{c}} \\sim 2^{N}\n\\,,\n\\end{align}\n%\nwhere \\(\\beta \\) is the value we obtain for the inequality. \n\nNo one has yet done a loophole-free measurement with three subsystems, but loophole-wrought measurements have indeed been done. \n\nWe can also improve the violation s increasing the dimensionality of two systems, that is, using Q-dits.\n\nOften ``quantum nonlocality'' is discussed, but this is imprecise: we cannot really prove that quantum mechanics is nonlocal, we can only say that it either is nonlocal or nondeterministic. \n\n\\end{document}\n", "meta": {"hexsha": "9f0674d8f956186d2c8078a55571f16d8824c138", "size": 10109, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "quantum_optics/mar27.tex", "max_stars_repo_name": "jacopok/notes", "max_stars_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-10-10T13:10:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T14:52:50.000Z", "max_issues_repo_path": "quantum_optics/mar27.tex", "max_issues_repo_name": "jacopok/notes", "max_issues_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "quantum_optics/mar27.tex", "max_forks_repo_name": "jacopok/notes", "max_forks_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-03T16:20:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-06T16:11:07.000Z", "avg_line_length": 36.4945848375, "max_line_length": 291, "alphanum_fraction": 0.6762291028, "num_tokens": 3250, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5234203489363239, "lm_q2_score": 0.6654105587468141, "lm_q1q2_score": 0.3482894268451717}} {"text": "%\\documentclass[12pt]{article}\n\\documentclass[12pt,landscape]{article}\n\n\n\\include{preamble}\n\n\\newcommand{\\instr}{\\small Your answer will consist of a lowercase string (e.g. \\texttt{aebgd}) where the order of the letters does not matter. \\normalsize}\n\n\\title{Math 368 / 621 Fall \\the\\year{} \\\\ Final Examination}\n\\author{Professor Adam Kapelner}\n\n\\date{Wednesday, December 14, \\the\\year{}}\n\n\\begin{document}\n\\maketitle\n\n%\\noindent Full Name \\line(1,0){410}\n\n\\thispagestyle{empty}\n\n\\section*{Code of Academic Integrity}\n\n\\footnotesize\nSince the college is an academic community, its fundamental purpose is the pursuit of knowledge. Essential to the success of this educational mission is a commitment to the principles of academic integrity. Every member of the college community is responsible for upholding the highest standards of honesty at all times. Students, as members of the community, are also responsible for adhering to the principles and spirit of the following Code of Academic Integrity.\n\nActivities that have the effect or intention of interfering with education, pursuit of knowledge, or fair evaluation of a student's performance are prohibited. Examples of such activities include but are not limited to the following definitions:\n\n\\paragraph{Cheating} Using or attempting to use unauthorized assistance, material, or study aids in examinations or other academic work or preventing, or attempting to prevent, another from using authorized assistance, material, or study aids. Example: using an unauthorized cheat sheet in a quiz or exam, altering a graded exam and resubmitting it for a better grade, etc.\n\\\\\n\n\\noindent By taking this exam, you acknowledge and agree to uphold this Code of Academic Integrity. \\\\\n\n%\\begin{center}\n%\\line(1,0){250} ~~~ \\line(1,0){100}\\\\\n%~~~~~~~~~~~~~~~~~~~~~signature~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ date\n%\\end{center}\n\n\\normalsize\n\n\\section*{Instructions}\n\nThis exam is 105 minutes (variable time per question) and closed-book. You are allowed \\textbf{three} pages (front and back) of a \\qu{cheat sheet}, blank scrap paper and a graphing calculator. Please read the questions carefully. No food is allowed, only drinks. %If the question reads \\qu{compute,} this means the solution will be a number otherwise you can leave the answer in \\textit{any} widely accepted mathematical notation which could be resolved to an exact or approximate number with the use of a computer. I advise you to skip problems marked \\qu{[Extra Credit]} until you have finished the other questions on the exam, then loop back and plug in all the holes. I also advise you to use pencil. The exam is 100 points total plus extra credit. Partial credit will be granted for incomplete answers on most of the questions. \\fbox{Box} in your final answers. Good luck!\n\n\\pagebreak\n\n\\problem\\timedsection{14} Let $X_1, X_2, \\ldots$ be a sequence of independent rv's distributed as Deg$(\\mu / n^2)$. Let $T_n := X_1 + X_2 + \\ldots + X_n$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{18} \n\n\\begin{enumerate}[(a)]\n\\item $X_1$ has no PMF\n\\item $X_1$ has no CDF\n\\item $X_1$ is zero with probability one\n\\item The convolution of $X_1$ and $X_2$ is also degenerate\n\\item The expectation of $X_1$ is $\\mu$\n\\item The variance of $X_1$ is $\\mu^2$\n\\item $\\prob{X_1 \\geq \\mu} \\leq 0$\n\\item The chf of $X_1$ is $\\mu$\n\\item The chf of $X_1$ is $e^{it\\mu}$\n\\item The chf of $X_1$ is $\\in \\mathbb{L}^1$\n\\item The chf of $X_1$ evaluated at 0 is 1\n\\item The mgf of $X_1$ does not exist for some values of $t$ or $\\mu$\n\\item $X_n \\convd 0$\n\\item $X_n \\convp 0$\n\\item $X_n \\convLp{1} 0$\n\\item $X_n$ converges to zero in mean square\n\\item $T_n \\convd 0$\n\\item $T_n$ does not converge in probability.\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{7} Let $\\X \\sim$ Multinomial$(n, \\bv{p})$ and $\\phi_{\\X}(\\bv{t}) = \\tothepow{p_1 e^{i t_1} + p_2 e^{i t_2} + \\ldots + p_K e^{i t_K}}{n}$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{9} \n\n\\begin{enumerate}[(a)]\n\\item $\\onevec^\\top \\bv{p}$ = 1\n\\item $\\dime{\\X} = K$\n\\item $\\Xoneton$ are independent\n\\item $\\Xoneton$ are identically distributed\n\\item $X_3 \\sim \\binomial{n}{p_3}$\n\\item The chf for $X_3$ is $\\phi_{\\X}\\parens{\\bracks{t_1~t_2~0~t_4~\\ldots~t_K}^\\top}$\n\\item The chf for $X_3$ is $\\phi_{\\X}\\parens{\\bracks{~0~0~t~0~\\ldots~0}^\\top}$\n\\item The chf for $X_3$ is $\\tothepow{p_3 e^{i t} + p_4 e^{i t} \\ldots + p_K e^{i t_K}}{n}$\n\\item The chf for $X_3$ is $\\tothepow{1 + p_3 e^{i t} - p_3 }{n}$\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{9} Let $X_1, X_2, \\ldots$ be a sequence of iid rv's with mean $\\mu$ and variance $\\sigsq$ finite.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{13} \n\n\\begin{enumerate}[(a)]\n\\item $\\Xbar_n$ is exactly normally distributed\n\\item $\\Xbar_n$ is approximately normally distributed\n\\item The expectation of $\\Xbar_n$ is exactly $\\mu$\n\\item The expectation of $\\Xbar_n$ is approximately $\\mu$\n\\item The variance of $\\Xbar_n$ is exactly $\\sigsq / n$\n\\item The variance of $\\Xbar_n$ is approximately $\\sigsq / n$\n\\item $\\Xbar_n \\convd \\mu$\n\\item $\\Xbar_n \\convp \\mu$\n\\item $Z_n := \\frac{\\Xbar_n - \\mu}{\\sigma / \\sqrt{n}} \\convd \\mu$\n\\item $Z_n := \\frac{\\Xbar_n - \\mu}{\\sigma / \\sqrt{n}} \\convp \\mu$\n\\item $Z_n := \\frac{\\Xbar_n - \\mu}{\\sigma / \\sqrt{n}} \\convd \\oneoversqrt{2\\pi}e^{-z^2/2}$\n\\item Regardless of whether (h) is true or not, it is the main result of the \\qu{central limit theorem}\n\\item Regardless of whether (k) is true or not, it is the main result of the \\qu{central limit theorem}\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{19} Let $X \\sim \\chisq{k}$ and $Y~|~X = x \\sim \\uniform{0}{x}$. For this problem, you may need the following fact from Math 241: for $U \\sim \\uniform{a}{b}$, $\\var{U} = (b-a)^2 / 12$ and the following fact from this class: for $G \\sim \\gammanot{\\alpha}{\\beta}$, $\\var{G} = \\alpha / \\beta^2$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{19} \n\n\\begin{enumerate}[(a)]\n\\item $X$ and $Y$ are independent\n\\item $X \\sim \\gammanot{k/2}{1/2}$\n\\item The rv $Y$ is a compound distribution\n\\item The rv $Y$ has one parameter\n\\item $f^{old}_{X,Y}(x,y) = \\oneover{2^k \\Gammaf{k / 2}}x^{k/2 - 2} e^{-x/2}$\n\\item $f^{old}_{X,Y}(x,y) = \\oneover{2^k \\Gammaf{k / 2}}x^{k/2 - 1} y^{-1} e^{-x/2}$\n\\item $f_{X,Y}(x,y)$ is always defined\n\\item $f_{X|Y}(x,y)$ is always defined\n\\item $f_{Y|X}(y,x)$ cannot be computed given the information you have\n\\item The support of the rv $Y$ is all real numbers\n\\item The support of the rv $Y$ is all positive real numbers\n\\item The expectation of $Y$ can be computed via $\\int_\\reals y \\int_\\reals f_{X,Y}(x,y) dx dy$ \n\\item The expectation of $Y$ is $x/2$\n\\item The expectation of $Y$ is $k/2$\n\\item The expectation of $Y$ is $\\sqrt{\\pi}$\n\\item The variance of $Y$ is $k^2 / 12$ \n\\item The variance of $Y~|~X$ is $X^2 / 12$ \n\\item The variance of $Y$ is $(8k + k^2) / 12$ \n\\item The variance of $Y$ cannot be computed given the information you have\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{18} Let $X \\sim T_k$, $U = X^2$, $Y = \\mu + \\sigma X$ and $V = \\mu + \\sigma U$ where $\\mu \\in \\reals$, $\\sigma > 0$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{16} \n\n\\begin{enumerate}[(a)]\n\\item $X$ and $V$ are independent\n\\item $f^{old}_X(x) = f_X(x) = \\frac{\\Gammaf{(k+1) / 2}}{\\sqrt{k\\pi} \\Gammaf{k/2}} \\tothepow{1 + x^2 / k}{-(k+1) / 2}$\n\\item $\\support{U} = [0, \\infty)$\n\\item $\\support{V} = [0, \\infty)$\n\\item $\\expe{X} = \\mu$\n\\item $\\expe{Y} = \\mu$\n\\item $\\expe{U} = \\mu$\n\\item $\\expe{V} = \\mu$\n\\item $Y / V \\sim $ Cauchy$(\\mu,\\sigma)$\n\\item $\\var{Y} = \\sigsq$\n\\item $f^{old}_Y(y) = f_Y(y) = \\frac{\\Gammaf{(k+1) / 2}}{\\sigma\\sqrt{k\\pi} \\Gammaf{k/2}} \\tothepow{1 + (y - \\mu)^2 / (k\\sigsq)}{-(k+1) / 2}$\n\\item $f^{old}_Y(y) = f_Y(y) = \\mu + \\sigma \\frac{\\Gammaf{(k+1) / 2}}{\\sqrt{k\\pi} \\Gammaf{k/2}} \\tothepow{1 + y^2 / k}{-(k+1) / 2}$\n\\item $U \\sim F_{1,k}$\n\\item $V \\sim F_{\\sigma,k}$ if $\\mu=0$\n\\item $f^{old}_U(u) = \\frac{(1 + u / k)^{-(k+1) / 2}}{B(1/2, k/2) \\sqrt{ku} }$\n\\item $f^{old}_V(v) = \\frac{(1 + v / (k\\sigma))^{-(k+1) / 2}}{B(1/2, k/2) \\sqrt{k\\sigma v}}$ if $\\mu=0$\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{9} Let $\\Xoneton \\iid \\normnot{\\mu}{\\sigsq}$ where $\\mu$ and $\\sigsq$ are finite and let $\\Xbar = \\oneover{n} \\sum X_i$ and $S^2 = \\oneover{n - 1} \\sum (X_i - \\Xbar)^2$. Let $Z_1 = (X_1 - \\mu)/\\sigma, Z_2 = (X_2 - \\mu)/\\sigma,\\ldots, Z_n = (X_n - \\mu)/\\sigma$. Let $\\Z = \\bracks{Z_1~Z_2 ~\\ldots~ Z_n}^\\top$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{11} \n\n\\begin{enumerate}[(a)]\n\\item $\\Xbar \\sim \\normnot{\\mu}{\\sigsq / n}$\n\\item $\\Xbar \\sim \\normnot{\\mu}{\\sigsq}$\n\\item $S^2 \\sim \\chisq{n-1}$\n\\item $S^2 \\sim \\gammanot{\\frac{n-1}{2}}{\\frac{n-1}{2\\sigsq}}$\n\\item $\\Xbar$ and $S^2$ are independent\n\\item $\\bar{Z}$ and $S^2$ are independent\n\\item There exists a matrix $A$ where $S^2 = \\Z A \\Z^\\top$.\n\\item There exists a matrix $A$ where $S^2 = \\Z^\\top A \\Z$.\n\\item If $\\Z^\\top \\Z = \\Z^\\top B_1 \\Z + \\Z^\\top B_2 \\Z + \\ldots + \\Z^\\top B_k\\Z$, then $ \\Z^\\top B_1 \\Z$ is chi-squared distributed\n\\item If (e) is assumed, then Cochran's theorem can be proven\n\\item If $a>0$, then $\\prob{Z_1 > a} < \\half$.\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{19} Let $\\X \\sim \\multnormnot{n}{\\muvec}{\\Sigma}$ where $\\Sigma$ is full rank, $A \\in \\reals^{m \\times n}$, $\\bv{b} \\in \\reals^m$ and $\\Y = A\\X + \\bv{b}$. Let $\\mu_i$ denote the $i$th entry in $\\muvec$ and let $\\Sigma_{i,j}$ denote the entry on the $i$th row and $j$th column of matrix $\\Sigma$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{17} \n\n\\begin{enumerate}[(a)]\n\\item $\\expe{\\X} = \\muvec$\n\\item $\\X - \\muvec$ is a standard multivariate normal rv\n\\item The kernel of the PDF of $\\X$ is $e^{\\x^\\top \\Sigma^{-1} \\muvec - \\x^\\top \\Sigma^{-1} \\x / 2}$\n\\item $\\displaystyle\\argmax_{\\x \\in \\reals^n}\\braces{f_{\\X}(\\x)} = \\muvec$\n\\item $\\phi_{\\X}(\\bv{t}) = e^{i\\bv{t}^\\top \\muvec - \\bv{t}^\\top \\Sigma \\bv{t} / 2}$\n\\item $\\phi_{\\Y}(\\bv{t}) = e^{i\\bv{t} \\bv{b}} \\phi_{\\X}(A^\\top\\bv{t})$\n\\item $\\int_0^\\infty \\int_0^\\infty f_{\\X}(x_1,x_2) dx_1 dx_2 = 1$ if $n=2$\n\\item $X_1$ and $X_2$ would be independent if $\\muvec = \\zerovec$\n\\item $X_1$ and $X_2$ could be independent regardless of the value of $\\muvec$\n\\item $\\var{\\X + \\bv{b}} = \\Sigma\\bv{b}$ if $n=m$\n\\item $A\\X \\sim \\multnormnot{n}{\\muvec}{A\\Sigma}$\n\\item $X_3 + X_7$ is normally distributed\n\\item $X_3 / X_7$ is Cauchy distributed\n\\item $\\twovec{X_3}{X_7} \\sim \\multnormnot{2}{\\twovec{\\mu_3}{\\mu_7}}{\\twobytwomat{\\Sigma_{3,3}}{\\Sigma_{3,7}}{\\Sigma_{3,7}}{\\Sigma_{7,7}}}$\n\\item $\\var{X_1 + X_2} = \\Sigma_{1,1} + \\Sigma_{2,2} + \\Sigma_{1,2} + \\Sigma_{2,1}$\n\\item $\\X^\\top \\X \\sim \\chisq{n}$\n\\item $(\\Y - A\\muvec - \\bv{b})^\\top \\inverse{A \\Sigma A^\\top} (\\Y - A\\muvec - \\bv{b})$ is always chi-squared distributed\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\problem\\timedsection{10} Let $X, Y \\iid \\chisq{k}$ where $\\mu$ and $\\sigsq$ are finite and $a>0$.\n\n\\vspace{-0.2cm}\\benum\\truefalsesubquestionwithpoints{10} \n\n\\begin{enumerate}[(a)]\n\\item $\\prob{X > a} \\leq k / a$\n\\item $\\prob{2^X > a} \\leq \\expe{2^X} / a$\n\\item $\\prob{2^X > a} \\leq 2^k / a$\n\\item $Q[X, 1/e] \\leq ek$\n\\item $\\prob{X > b} \\leq \\var{X} / (b-k)^2$ if $b \\geq 2k$\n\\item $\\expe{\\natlog{X}} \\geq \\natlog{\\expe{X}}$\n\\item $\\expe{\\natlog{X}} \\leq \\natlog{\\expe{X}}$\n\\item If $a$ was large, the Chernoff bound for $\\prob{X > a}$ would be tighter (smaller) than the Markov bound for $\\prob{X > a}$\n\\item $\\corr{X}{Y} \\in \\bracks{-1,1}$\n\\item Computing the Cauchy-Schwartz upper bound is the most information you can provide about $\\expe{XY}$.\n\\end{enumerate}\n\\eenum\\instr\\pagebreak\n\n%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\end{document}%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n", "meta": {"hexsha": "c667e6e8756fdb0ba2fcc675b511a5dae9c3a7f1", "size": 12611, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "exams/final/final.tex", "max_stars_repo_name": "kapelner/QC_Math_621_Fall_2020", "max_stars_repo_head_hexsha": "f71dc4901d46c4aaaea3ef4795a22046e754bb11", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2020-08-25T01:46:34.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-16T01:21:21.000Z", "max_issues_repo_path": "exams/final/final.tex", "max_issues_repo_name": "kapelner/QC_Math_621_Fall_2020", "max_issues_repo_head_hexsha": "f71dc4901d46c4aaaea3ef4795a22046e754bb11", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "exams/final/final.tex", "max_forks_repo_name": "kapelner/QC_Math_621_Fall_2020", "max_forks_repo_head_hexsha": "f71dc4901d46c4aaaea3ef4795a22046e754bb11", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-09-02T18:47:18.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-17T19:42:31.000Z", "avg_line_length": 48.3180076628, "max_line_length": 877, "alphanum_fraction": 0.614225676, "num_tokens": 4548, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5234203340678567, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.3482894169515366}} {"text": "%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% Template for a conference paper, prepared for the\n%% Food and Resource Economics Department - IFAS\n%% UNIVERSITY OF FLORIDA\n%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% Version 1.0 // November 2019\n%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% Ariel Soto-Caro\n%% - asotocaro@ufl.edu\n%% - arielsotocaro@gmail.com\n%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\documentclass[10pt]{article}\n\\usepackage{format}\n\n\\usepackage{amsthm}\n\\theoremstyle{plain}\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}{Corollary}[theorem]\n\\newtheorem{lemma}[theorem]{Lemma}\n\n\\usepackage[ruled,vlined]{algorithm2e}\n\\newcommand{\\bigslant}[2]{{\\raisebox{.2em}{$#1$}\\left/\\raisebox{-.2em}{$#2$}\\right.}}\n\n\\usepackage{enumitem}\n\\setlist[itemize,1]{label=\\textbullet}\n\\setlist[itemize,2]{label=$\\circ$}\n\\setlist[itemize,3]{label=$\\ast$}\n\n%% ===============================================\n%% Setting the line spacing (3 options: only pick one)\n% \\doublespacing\n% \\singlespacing\n\\onehalfspacing\n%% ===============================================\n\n\\setlength{\\droptitle}{-5em} %% Don't touch\n\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SET THE TITLE\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n% TITLE:\n\\title{Random Trees and Effective Resistance}\n\n% AUTHORS:\n\\author{Kishlaya Jaiswal\\\\% Name author\n % \\href{mailto:kishlaya.j@gmail.com}{\\texttt{kishlaya.j@gmail.com}} %% Email author 1 \n% \\and Second Author\\\\% Name author\n% \\href{mailto:secondauthor@ufl.edu}{\\texttt{secondauthor@ufl.edu}} %% Email author 2\n% \\and Third Author\\\\% Name author\n% \\href{mailto:thirdauthor@ufl.edu}{\\texttt{thirdauthor@ufl.edu}}%% Email author 3\n%\\and Forth Author\\\\% Name author\n% \\href{mailto:forthuthor@ufl.edu}{\\texttt{forthuthor@ufl.edu}}%% Email author 4\n }\n \n% DATE:\n\\date{\\today}\n\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% ABSTRACT\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n{\\setstretch{.8}\n\\maketitle\n% %%%%%%%%%%%%%%%%%%\n\\begin{abstract}\n% CONTENT OF ABS HERE--------------------------------------\n\nWe look at a graph as an electric network to present a simple proof for the well known relation between effective resistance of an edge and the chances of that edge being in a uniformly random spanning tree.\n\n% END CONTENT ABS------------------------------------------\n\\noindent\n\\textit{\\textbf{Advisor: }%\nSamir Datta} \\\\ %% <-- Keywords HERE!\n\n\\end{abstract}\n}\n\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% BODY OF THE DOCUMENT\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n% % --------------------\n\\section{Introduction}\n% % --------------------\n\nLet $G=(V,E)$ be a given undirected graph. Sampling a uniformly random spanning tree $T$ is a well-studied problem and the question we ask is: given an edge $e \\in E$, what are the chances that a random spanning tree has this edge $e$.\n\nWe consider an electric circuit where each edge in $G$ is replaced with a resistor of $1 \\Omega$. To setup a potential difference, we supply external current $i_{ext}$ at nodes; then potential $p$ at each node is given by: $i_{ext} = Lp$ where $L$ denotes the Laplacian of the graph.\n\n\\textsl{Effective resistance} of an edge $e=(a,b)$, denoted by $R_{\\text{eff}}(e)$, is defined as potential difference across $e$ when a unit current is inducted at $a$ and taken out at $b$.\n\nSo we consider the particular vector $i_{ext} = x_e$ where $x_e(a) = 1$, $x_e(b) = -1$ and $x_e(c) = 0$ for all $c \\neq a,b$. Since $x_e \\perp \\textbf{1}$, we know that a solution for $p$ exists and can be given by $p = L^{\\dag}x_e$ where $L^{\\dag}$ denotes the psuedo-inverse of $L$. Therefore, potential difference between $a$ and $b$ is $p(a) - p(b) = x_e^T p = x_e^T L^{\\dag} x_e$ and so $R_{\\text{eff}}(e) = p(a)-p(b) = x_e^T L^{\\dag} x_e$\n\nIt turns out that:\n$$\\mathbb{P}[e \\in T] = R_{\\text{eff}}(e)$$\n\nGiven these values $\\mathbb{P}(e \\in T)$, we can use them to sample a random spanning tree as follows~\\cite{GUENOCHE1983214}:\n\n% \\begin{algorithm}[H]\n% \\SetAlgoLined\n% $T = \\{\\}$, $G_0 = G$ \\\\\n% Let $e_1, \\ldots, e_m$ be some ordering on the edges \\\\\n% \\For{$i\\gets1$ \\KwTo $m$}{\n% Let $p$ be the probability that $e_i$ is in a random spanning tree of $G_{i-1}$ \\\\\n% Flip a biased coin whose chance of heads is $p$ \\\\\n% \\eIf{heads}{\n% $T = T \\cup \\{e_i\\}$ \\\\\n% % $G_i = G_{i-1} / \\{e_i\\}$\n% $G_i = \\bigslant{G_{i-1}}{\\{e_i\\}}$\n% }{\n% $G_i = G_{i-1} \\setminus \\{e_i\\}$\n% }\n% }\n% Return $T$\n% \\caption{Sampling a uniformly random spanning tree}\n% \\end{algorithm}\n\n\\begin{algorithm}[H]\n\\SetAlgoLined\n$T = \\{\\}$ \\\\\n\\For{$e \\in E$}{\n Let $p$ be the probability that $e$ is in a random spanning tree of $G$ \\\\\n Flip a biased coin whose chance of heads is $p$ \\\\\n \\eIf{heads}{\n $T = T \\cup \\{e\\}$ \\\\\n % $G_i = G_{i-1} / \\{e_i\\}$\n $G = \\bigslant{G}{\\{e\\}}$ (contract $e$)\n }{\n $G = G \\setminus \\{e\\}$ (delete $e$)\n }\n}\nReturn $T$\n\\caption{Sampling a uniformly random spanning tree}\n\\end{algorithm}\n\n\\medskip\n\nBut directly calculating these probabilities require enumerating all spanning trees (hard), whereas computing $R_{\\text{eff}}(e)$ only involves multiplying vectors (easy).\n\nThe proof we discuss here is from ~\\cite{appralgo}. In what follows, we shall assume that the graph is connected (if the graph is not connected then no spanning tree exists).\n\n% % --------------------\n\\section{Matrix Tree Theorem}\n% % --------------------\n\nMatrix Tree Theorem~\\cite{godsil2001algebraic} counts the number of spanning trees of $G$ in terms of the Laplacian of the graph. That is, let $0 < \\lambda_1 \\leq \\cdots \\lambda_n$ be the eigen values of $L$, then \n$$\\# \\text{spanning trees of } G = \\frac1n \\lambda_1 \\ldots \\lambda_n$$\n\nWe define $$\\bar{L} = L + \\frac1n J$$\n$\\bar{L} \\textbf{1} = \\textbf{1}$ and for any other eigenvector $v$ (of $L$) of non-zero eigenvalue, since $v \\perp \\textbf{1}$, $v$ is also an eigenvector of $\\bar{L}$ with same eigenvalue. Hence $\\{1, \\lambda_1, \\ldots, \\lambda_n\\}$ are the eigenvalues of $\\bar{L}$ and so we can re-state the Matrix Tree Theorem as:\n$$\\# \\text{spanning trees of } G = \\frac1n \\det(\\bar{L})$$\n\n% % --------------------\n\\section{Main Result}\n% % --------------------\n\n\\begin{lemma}\nGiven a positive-definite symmetric matrix $M \\in \\mathbb{R}^{n \\times n}$ and $x \\in \\mathbb{R}^n$\n$$\\det(M + x x^T) = \\det(M) (1 + x^T M^{-1} x)$$\n\\end{lemma}\n\n\\begin{proof}\nSince $M$ is positive-definite symmetric, there exists a unique positive-definite symmetric matrix $M^{1/2}$ such that $(M^{1/2})^2 = M$\n\\begin{align*}\n \\det(M + x x^T) &= \\det \\left(M^{1/2} (I + M^{-1/2} x x^T M^{-1/2}) M^{1/2} \\right) \\\\\n &= \\det(M) \\det(I + M^{-1/2} x x^T M^{-1/2}) \\\\\n &= \\det(M) \\det(I + yy^T) \\\\\n\\end{align*}\nwhere $y = M^{-1/2} x$. Notice that solutions of $y^Tv = 0$ gives $n-1$ eigenvectors of $I + yy^T$ with eigenvalue $1$ and from trace computation we get the last eigenvalue is $1 + y^Ty$. Hence $\\det(I + yy^T) = 1 + y^Ty = 1 + x^T M^{-1} x$ and we are done.\n\\end{proof}\n\nSince $G$ is connected, $\\bar{L}$ is a positive definite symmetric matrix and $\\det(\\bar{L}) > 0$. We also note that Laplacian of $G - \\{e\\}$ is simply $L - x_e x_e^T$. Therefore, $\\# \\text{spanning trees of } G$ not containing $e = \\# \\text{spanning trees of } G \\setminus \\{e\\} = \\frac1n \\det(\\bar{L} - x_e x_e^T)$. Finally we have,\n\n\\begin{align*}\n \\mathbb{P}[e \\in T] &= 1 - \\mathbb{P}[e \\not \\in T] \\\\\n &= 1 - \\frac{\\det(\\bar{L} - x_e x_e^T)}{\\det(\\bar{L})} \\\\ \n &= 1 - \\frac{\\det(\\bar{L}) (1 - x_e^T \\bar{L}^{-1} x_e)}{\\det(\\bar{L})} \\\\\n &= x_e^T \\bar{L}^{-1} x_e \\\\\n &= x_e^T L^{\\dag} x_e\n\\end{align*}\nas $L = \\bar{L}$ when restricted to the subspace perpendicular to $\\textbf{1}$. Thus, $\\mathbb{P}[e \\in T] = x_e^T L^{\\dag} x_e = R_{\\text{eff}}(e)$ \\qed\n\n% % --------------------\n\\section{Further Extensions}\n% % --------------------\n\nWe can further extend this result by asking the probability of $F \\subseteq T$ where $F$ is any subset of edges. In this case~\\cite{burton1993} showed that $\\mathbb{P}[F \\subseteq T] = \\det(Y_F)$ where $Y$ is a $E \\times E$ matrix such that $Y(e,f) = x_e^T L^{\\dag}x_f$. This can be easily proved by inducting on size of $F$ and using Cauchy-Binet formula (note that the base case $|F|=1$ is what we have proved above).\n\nWhen $G$ is weighted undirected graph, in which case resistance of each edge is inverse of it's weight then we have $\\mathbb{P}[e \\in T] = w(e)R_{\\text{eff}}(e)$ where $T$ is sampled with probability proportional to $\\prod_{e \\in T} w(e)$. The proof is similar with the modification that we work with the weighted Laplacian and use Matrix Tree theorem for weighted graphs.\n\n\n\\printbibliography\n\n\\end{document}\n", "meta": {"hexsha": "e47f0ce393f73c4535c3d23c1ffdcdcf1cb55b7a", "size": 9250, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "spectral_graph_theory/Random_Trees_and_Effective_Resistance/main.tex", "max_stars_repo_name": "kishlaya/assignments", "max_stars_repo_head_hexsha": "1aa76e32d7e5059499a93359cb52118ccbf07028", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-11-17T09:28:32.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-14T17:40:34.000Z", "max_issues_repo_path": "spectral_graph_theory/Random_Trees_and_Effective_Resistance/main.tex", "max_issues_repo_name": "kishlaya/assignments", "max_issues_repo_head_hexsha": "1aa76e32d7e5059499a93359cb52118ccbf07028", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "spectral_graph_theory/Random_Trees_and_Effective_Resistance/main.tex", "max_forks_repo_name": "kishlaya/assignments", "max_forks_repo_head_hexsha": "1aa76e32d7e5059499a93359cb52118ccbf07028", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.5665024631, "max_line_length": 444, "alphanum_fraction": 0.5607567568, "num_tokens": 2858, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381667555713, "lm_q2_score": 0.6370308082623217, "lm_q1q2_score": 0.34828905627616163}} {"text": "\\chapter{Review of terms and existing solutions}\n\n\\startrelatedwork\n\nFor now the automation of different problems like community search in social networks is very relevant and progress from day to day. There are different methods for finding communities in the whole network \\cite{Newman04, Newman06, Fortunato10, Cui13} and also for finding a dense community containing all selected vertices in the network \\cite{Faloutsos06, Wiener15, Huang15, Barbieri15}. There are even algorithms for splitting selected vertices into several communities \\cite{Akoglu13} (DOT2DOT algorithm). However, almost all of these algorithms don't support noise in the query and find non-optimal subgraphs on such queries. Precisely this problem we are going to solve in this work, suggesting the new algorithm that will effectively find the needed subgraph even if there is some noise in the query.\n\n\\section{Terms and definitions}\n\n\\subsection{Graph terms}\n\nIn this chapter we will write down all non-common terms and definitions that may be helpful for the further reading.\n\n\nUnder \\textbf{$N_G(v)$} we will understand the set of the neighbors of vertex $v$ in graph $G$, i.e. the set of the vertices that are directly connected to $v$ by edge: $N_G(v) = \\{u | (v, u) \\in E(G)\\}$. If graph $G$ can be obviously recognized from the context, we can write just $N(v)$.\n\n\\textbf{$G[V]$} is called \\textbf{originated subgraph} of the graph $G$ by the set of vertices $V$ if $G[V] := (V, E[G, V])$, where $E[G, V]$~--- the subset of the set of edges of $G$, both ends of which are contained in $V$, i.e. $E[G, V] = E(G) \\cap (V \\times V)$.\n\n\\textbf{k-truss} of the graph $G$ is the subgraph $G' \\subseteq G$ containing the maximal possible number of vertices, such that for each edge $(v, u)$ the number of vertices $w$, such that edges $(w, v)$ and $(w, u)$ exists in $G'$ is at least $k$. In other words, $k-truss$ is the maximal by size subgraph $G'$, for each of which edge $(v, u)$, the $|N_{G'}(v) \\cap N_{G'}(u)| >= k$ is true.\n\n\\textbf{k-core} of the graph $G$ is called the maximal by the number of vertices subgraph $G' \\subseteq G$, so that the degree of each of its vertices is at least $k$. For the fixed $k$, by $C_k$ we will denote \\textit{$k$-core}, namely the set of the connected components of which it consists. So, $C_k = \\{H_i\\}$, where $H_i$ is the $i$-th connected component, where the degree of each vertex is at least $k$. The number $k$ we will call the \\textbf{order} of \\textit{k-core}.\n\nBy \\boldmath$\\mu(G)$\\unboldmath we will denote the minimal degree of the vertices $G$, i.e. $\\mu(G) = \\min_{v \\in V(G)} deg(v)$.\n\n\\textbf{Core decomposition} is the set of \\textit{k-core} for all possible $k$: $C = \\{C_k\\}_{k=1}^{k=k^*}$. We also need to clarify that from the definition of the \\textit{k-core} you can see that $C_1 \\supseteq C_2 \\supseteq C_3 \\ldots \\supseteq C_{k^*}$ (where $k^*$ is the maximal possible $k$ in core decomposition).\n\n\\textbf{Сore index} for the vertex $v$ is called the minimal by the size \\textit{k-core} which includes $v$, i.e. \\textit{k-core} with the maximal $k$: $c(v) = \\max(k \\in [0..k^*] | v \\in C_k)$.\n\n\\textbf{$\\gamma$-quasi-clique} of the graph $G$ is called any such subgraph $G' \\subseteq G$ that it is <>, i.e. $\\frac{2 \\cdot |E(G')|}{|V(G')| \\cdot (|V(G')| - 1)} \\ge \\gamma$.\n\n\\subsection{Social networks}\n\n\\textbf{The community} or \\textbf{The community in social network} is called the set of vertices of the social network $G$, where all vertices are united by some property or attribute. For example, <> or <>.\n\n\\textbf{The social clique} or \\textbf{clique} we will call the set of people in social network, where everyone \"knows\" (i.e. is connected by edge) each other, in other words when between any pair of distinct people there is an edge in social network.\n\n\\textbf{Social pseudoclique} or \\textbf{pseudoclique} we will call the set of people, where it is not required that each pair of distinct people is connected by edge, but this set is still densely connected. The estimation, how dense the pseudoclique is connected depends on the type of the pseudoclique and will be discussed later in the work, but in all definitions the biggest role plays the number of edges in subgraph in comparison with the number of pairs of vertices ($\\frac{2 \\cdot |E(G)|}{|V(G)| \\cdot (|V(G)| - 1)}$).\n\n\\textbf{Free-rider effect} is called the effect, appearing during the obtaining the answer for the given problem (finding the dense community in social network), when the final or intermediate answer contains unnecessary subgraphs~--- subgraphs which can be deleted without violation the optimality of the answer. Thereby, we can make the answer smaller which is one of ours main goals.\n\n\\subsection{Useful abbreviations}\n\n\\textbf{RW}~--- Random Walks. The main idea of this method is based on moving from one vertex to the neighbor one with probability proportional to the edge weight.\n\n\\textbf{RWR}~--- Random Walks with Restarts. The idea is similar to the RW, but in this case the probability to move to the initial vertex (from which we've started) from the current one exists as well.\n\n\\textbf{Smart-ST}~--- Smart Spanning Trees. The heuristic for Steiner Tree problem, which is used in Gionis et al. article \\cite{Gionis15}.\n\n\\textbf{CSP}~--- Community Search Problem. This is the problem for finding the community in the social network which contains all the selected vertices.\n\n\\textbf{NCSP}~--- Noising Community Search Problem. This is the problem for finding the community in the social network which contains most of the selected vertices, but not necessary all (not including the noise).\n\n\\section{Overview}\n\nThe problem that we're analyzing in the work is formulated as follows: given a graph $G$ and a set of selected vertices $Q \\subset V(G)$, the goal is to solve community search problem~--- to find the community which contains most of the vertices from $Q$, but not necessary all of them. Sometimes we will call vertices from $Q$ <>, <> or <>.\n\n\\subsection{Initial solutions}\n\n\\begin{enumerate}\n \\item The community search problem by the given selected vertices is researched during a long time. Even in 2004 Faloutsos et al. \\cite{Faloutsos04} suggested the algorithms for finding the dense community by $2$ selected vertices in network ($|Q| = 2$). The algorithm shows that metrics like <> and <> between two given vertices are not optimal. Instead of them, the initial graph is considered as the electric network and <> metric is used~--- setting voltage \\texttt{+1} on the first vertex-query and \\texttt{0} on the second one, we find subgraph which delivers the maximal current between vertices from query. The provided metrics works only for $|Q| = 2$, but this algorithm was the foundation in research of the community search problem. After that many authors were working on optimization for this article and were quite successful.\n\n \\item Authors of the second article that was taken for consideration \\textbf{Faloutsos06} suggest the metric function based on \\textit{random walks with restarts} (RWR) used on the weighted graph. They consider $r(i, j)$~--- the probability that starting in vertex $i$-th query vertex $q_i$ we will end in vertex $j$ using RWR, where on each step we move to the neighbor by edge vertex with probability proportional the edge weight. Also we introduce $r(Q, j)$ which is equal to the sum of $r(i, j)$ for each query vertices: $r(Q, j) = \\sum_{i = 1}^{i = |Q|}\\; r(i, j)$. The dense metric is considered as $g(H) = \\sum_{j \\in H}\\; r(Q, j)$. This method has shown quite good results compared to the previous article, because expanded the number of query vertices from $2$ to any number from $2$ to $|V(G)|$. Also, it introduced new ability of finding the subgraph containing not all vertices, but at least $k$ of them ($k$ is a parameter which is given as input). This operation was called $K\\_softAND$ and was successfully implemented in the article. Further algorithms were expanding this idea, were applying other metrics and improved the results of this algorithm, but the problem of finding the community containing not necessary all query vertices, but only a part of them (our problem), almost wasn't optimized.\n\n \\item Authors of the third article \\cite{Wiener15} suggest to use \\textit{Wiener index} as the metric for subgraph density. This metric is equal to the pairwise sum of the shortest distances between vertices from the query. Authors are trying to solve the issue of obtaining too large graph as the result of processing the query if query vertices are placed in several communities and are weakly connected between each other. To solve this issue, authors suggest to add some <> to the query which will connect communities, even if not very dense. Results has shown that this method works several times better than previous methods \\cite{Faloutsos06, Sozio10} and almost the same as the methods that based on \\textit{Steiner tree problem}. Unfortunately, the article doesn't consider more late methods based on \\textit{Steiner tree problem} which significantly improved the old results, which make this method less priority comparing to them.\n \n\\end{enumerate}\n\n\\subsection{Finding optimal pseudocliques}\n\nThe most part of all algorithms for solving the described problem are the algorithms based on the finding optimal pseudocliques with some additional heuristics. There are a lot of different pseudocliques the were considered in different articles: for example, \\textit{k-core} \\cite{Barbieri15}, \\textit{k-truss} \\cite{Huang15}, \\textit{$\\gamma$-quasi-clique} \\cite{Zhu11} or just algorithms that maximizes the edge density in the resulting subgraph \\cite{Wu15} which is almost a definition of a pseudoclique. For each of these pseudocliques the algorithms are evolving and becoming better, optimizing the previous results using new heuristics. Comparing the results of the algorithms that use different pseudocliques is quite hard and unlikely will give visible results because of the difference of the metrics that are being optimized~--- the result strongly depend on the initial graph and the queries on it. In some cases one pseudoclique will obtain results better than others, but in other cases it will work worse, so actually it's worth to compare some common performance metrics, but unfortunately it doesn't give us the whole understanding of the optimality or non-optimality of the algorithms.\n\nLet's consider several newest algorithms for the most popular pseudocliques:\n\n\\begin{enumerate}\n \\item X. Huang et al. \\cite{Huang15} choose \\textit{k-truss} pseudoclique. However, just finding the optimal \\textit{k-truss} (i.e. \\textit{k-truss} with the maximal $k$ containing all query vertices in it) is not an optimal solution, and also it is an already solved problem (even with polynomial solution). That's why the authors of the article suggest to find \\textit{k-truss} with maximal $k$ and minimal subgraph diameter, which, as they show in their article, is a NP-hard problem. However, this idea pretends to show good results, so authors made a research trying to understand how close the answer found by polynomial time may be to the theoretically optimal answer. It turned out that this problem couldn't be solved with accuracy better than in $(2 - \\varepsilon)$ times worse for each $\\varepsilon > 0$ (under the accuracy we understand the length of the diameter in the final answer). However, authors suggested the heuristic algorithm which in the worst case makes exactly $2$ times error, which shows that their algorithm is optimal for the provided problem. The algorithm is based on building the supposed maximal $\\textit{k-truss}$ with the followed iterative deleting vertices which doesn't make the answer worse and make the diameter smaller. The results obtained in this article are really good comparing to the previous articles \\cite{Sozio10, Wu15}, however, even despite of proved optimality for the provided algorithm, it is not optimal for the initial problem (finding the dense subgraph by the given query vertices), because authors found the optimal solution only for the problem provided \\textit{by themselves}.\n\n \n \\item N. Barbieri et al. \\cite{Barbieri15} is using \\textit{k-core} pseudoclique. However, even here simple finding the optimal \\textit{k-core} (i.e. \\textit{k-core} with maximal $k$ which contains all query vertices in it) is not an optimal solution, and also this problem is already solved by polynomial time \\cite{Sozio10}. That's why the authors of the article apply some heuristics targeted for minimizing the size of the resulting subgraph without loosing its optimality. These heuristics allow to reduce the problem to finding the answer in component $H^* \\subset G$, and besides it is guaranteed that all possible optimal answers for the initial problem are lying in $H^*$. After that authors bring some heuristics for minimizing the obtained subgraph $H^*$. The main statement described by the authors is not new, but it looks quite interesting for the further researches because it adds more information to the initial problem without loosing any solutions. The results of this article shows that provided method really works better and faster than previous ones \\cite{Sozio10, Cui14}. Based on all the above information, it was decided to take this article as the baseline and try to improve it, especially to expand it to our problem (finding the community containing not necessary all selected vertices).\n\\end{enumerate}\n\n\\subsection{Other methods}\n\nAs we already saw earlier, optimizing functions may be quite different. In the previous part we were considering pseudocliques, and here we're going to consider several other popular optimizing functions.\n\n\\begin{enumerate}\n \\item L. Akoglu et al. \\cite{Akoglu13} slightly change the initial problem~--- they try to find subgraph that unites not all the vertices in the query, but their groups. Actually the idea is based on splitting the query into groups and building the answer for each of the groups separately, so that in each groups the vertices are densely connects and are united by some common property. But between each other the groups may be connected not very densely. This corresponds to the splitting the query into several communities. The results of the article has shown that this method solves the problem provided by authors quite good, but however as we said before, this problem differs from ours and it's hard to compare it's results with ours. However, we still are going to compare this solution with ours, because this solution supports finding community not for all selected vertices. \n\n \\item A. Gionis et al. \\cite{Gionis15} in their article consider \\textit{linear local discrepancy} metric which is equal to the weighted difference of the number of query vertices in the resulting subgraph and the number of remaining vertices. More formally, $g(C) = \\alpha p_C - n_C$, where $p_C = |Q \\cap V(C)|$ and $n_C = |V(C) \\setminus Q|$. The algorithm that maximized this function based on the \\textit{Steiner tree problem} and \\textit{Smart-ST}s. The distinguishing feature of this algorithm is the ability of solving the problem using \\textit{local access model}, i.e. the model where we don't know the whole graph (or it is too big to save it in RAM), and API allows us only to make queries for accessing all vertex neighbors~--- $get-neighbors$ method. This model allows to solve the problem optimally even on very big social networks, such as \\textit{Twitter} or \\textit{Facebook}. Based on the definition, the resulting subgraph may not contain all vertices from query, which coincides with our research. Unfortunately, the provided metric doesn't fit our problem very good~--- it doesn't take into consideration edge density of the resulting subgraph, it looks only the ratio of the vertices. Also it's possible that the answer will contain too few vertices from the initial query~--- it also doesn't work for us.\n\\end{enumerate}\n\n\\section{Final requirements for our work}\n\nLet's sum up everything described above. Most of the current solutions solves CSP quite optimal~--- each of the solutions uses it's own metric and obtains quite good results. However, as we can see, solutions for NCSP (which includes noise into consideration) are quite rare, despite it is more useful problem in real life. We've noted consideration of NCSP problem in articles C. Faloutsos \\& H. Tong \\textbf{[3]} and A. Gionis et al. \\cite{Gionis15}, however the last problem is not based on solving NCSP (but solves it at the same time). So, the goal of our article will be to build the algorithm that focuses on NCSP solving and obtains better results than the current ones. Here are some requirements for our algorithm:\n\n\\begin{itemize}\n \\item The algorithm should obtain better results than the current ones \\cite{Faloutsos06, Gionis15, Barbieri15};\n \\item The algorithm should be quite optimal, ideally not loosing the competition with other algorithms in terms of working time;\n \\item It would be an advantage to support backwards compatibility~--- if the user wants to find subgraph that contains \\textit{all} query vertices, it should be possible to be done.\n\\end{itemize}\n", "meta": {"hexsha": "57e66feb81723481eaae1124e83dd5cb00755d74", "size": 17452, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "pdf/articles/article-1/chapters-eng/chapter1.tex", "max_stars_repo_name": "DimaPhil/master-thesis", "max_stars_repo_head_hexsha": "a7a398fc91ee2d274f37dfeaa8e6264904f84e6d", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "pdf/articles/article-1/chapters-eng/chapter1.tex", "max_issues_repo_name": "DimaPhil/master-thesis", "max_issues_repo_head_hexsha": "a7a398fc91ee2d274f37dfeaa8e6264904f84e6d", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "pdf/articles/article-1/chapters-eng/chapter1.tex", "max_forks_repo_name": "DimaPhil/master-thesis", "max_forks_repo_head_hexsha": "a7a398fc91ee2d274f37dfeaa8e6264904f84e6d", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 176.2828282828, "max_line_length": 1641, "alphanum_fraction": 0.7707999083, "num_tokens": 4164, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.3482890393314951}} {"text": "\\chapter{The object classifier}\\label{chap:univalent}\n\nIn this chapter we establish notation, and we highlight the basic results concerning fiberwise transformations and fiberwise equivalences, which we will use for the descent theorems \\cref{thm:descent,thm:rcoeq_cartesian}. Of particular importance are the following theorems:\n\\begin{enumerate}\n\\item The Fundamental Theorem of Identity Types (\\cref{thm:id_fundamental}), which establishes that a type family $B$ over $A$ with $b:B(a)$ for a given $a:A$ is fiberwise equivalent to the identity type $a=x$ if and only if its total space is contractible. This result appears in \\cite{hottbook} as Theorem 5.8.2, which contains other equivalent conditions as well.\n\\item \\cref{thm:pb_fibequiv}, in which we establish that for any fiberwise map\n\\begin{equation*}\ng:\\prd{x:A}P(x)\\to Q(f(x)),\n\\end{equation*}\nthe commuting square\n\\begin{equation*}\n\\begin{tikzcd}[column sep=large]\n\\sm{x:A}P(x) \\arrow[r,\"{\\total[f]{g}}\"] \\arrow[d,swap,\"\\proj 1\"] & \\sm{y:B}Q(y) \\arrow[d,\"\\proj 1\"] \\\\\nA \\arrow[r,swap,\"f\"] & B,\n\\end{tikzcd}\n\\end{equation*}\nwhere $\\total[f]{g}$ is defined as $\\lam{(x,p)}(f(x),g(x,y))$, is a pullback square if and only if $g$ is a fiberwise equivalence\\index{fiberwise equivalence}. As a consequence, we obtain that a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nA \\arrow[d,swap,\"f\"] \\arrow[r] & B \\arrow[d,\"g\"] \\\\\nX \\arrow[r,swap,\"h\"] & Y\n\\end{tikzcd}\n\\end{equation*}\nis a pullback square if and only if the induced fiberwise transformation\n\\begin{equation*}\n\\prd{x:X} \\fib{f}{x}\\to\\fib{g}{h(x)}\n\\end{equation*}\nis a fiberwise equivalence. Our main reference \\cite{hottbook} does not present many results of homotopy pullbacks, although the material we present about homotopy pullbacks is surely well-known. The connection between pullbacks and fiberwise equivalences has an important role in the descent theorem\\index{descent} in \\cref{chap:descent}, which is why we devote a section to this result.\n\\item \\cref{thm:classifier}, in which we establish that the universe is an object classifier. This result appears in \\cite{RijkeSpitters} as Theorem 2.31, and in \\cite{hottbook} as Theorem 4.8.4\n\\end{enumerate}\n\n\\section{Notation and preliminary results}\n\nWe work in Martin-L\\\"of dependent type theory with $\\Pi$-types, $\\Sigma$-types and cartesian products, coproducts $A+B$ equipped with $\\inl:A\\to A+B$ and $\\inr:B\\to A+B$ for any two types $A$ and $B$, an empty type $\\emptyt$, a unit type $\\unit$ equipped with $\\ttt:\\unit$, a type $\\bool$ of booleans equipped with $\\btrue,\\bfalse:\\bool$, a type $\\N$ of natural numbers equipped with $0:\\N$ and $\\suc:\\N\\to\\N$, and identity types. \n\n\\begin{rmk}\nAs usual, we write $\\idfunc[A]:A\\to A$ for the \\define{identity function} $\\lam{x}x$ on $A$, and we write $g\\circ f:A\\to C$ for the \\define{composite function} $\\lam{x}g(f(x))$ of $f:A\\to B$ and $g:B\\to C$. For any two types $A$ and $B$, and any $b:B$, we write\n\\begin{equation*}\n\\const_b : A\\to B\n\\end{equation*}\nfor the \\define{constant function} $\\lam{x}b$. Sometimes we also write $\\lam{\\nameless}b$ for the constant function.\n\nIn the case of $\\Sigma$-types, the empty type $\\emptyt$, and the unit type, we use the following notation to define functions by pattern-matching:\n\\begin{align*}\n\\lam{(x,y)}f(x,y) & : \\prd{t:\\sm{x:A}B(x)} P(t) \\\\\n\\lam{\\ttt}y & : \\prd{t:\\unit}P(t).\n\\end{align*}\nFor instance, the first and second projection maps \n\\begin{align*}\n\\proj 1 & : (\\sm{x:A}B(x))\\to A \\\\\n\\proj 2 & : \\prd{p:\\sm{x:A}B(x)}B(\\proj 1)\n\\end{align*}\nare defined as $\\proj 1\\defeq \\lam{(x,y)}x$ and $\\proj 2\\defeq\\lam{(x,y)}y$.\nWe use similar notation for definitions by iterated pattern-matching. For instance, given a dependent function $f:\\prd{x:A}{y:B(x)}{z:C(x,y)} P((x,y),z)$ we obtain the function\n\\begin{equation*}\n\\lam{((x,y),z)} f(x,y,z): \\prd{t:\\sm{s:\\sm{x:A}B(x)}C(s)}P(t).\n\\end{equation*}\n\\end{rmk}\n\nGiven a type $A$ in context $\\Gamma$, the \\define{identity type} of $A$ at $a:A$ is the inductive type family \n\\begin{equation*}\n\\Gamma,x:A\\vdash a =_A x~\\mathrm{type}\n\\end{equation*}\nwith constructor\n\\begin{equation*}\n\\Gamma \\vdash \\refl{a} : a=_A a.\n\\end{equation*}\nThe induction principle for the identity type of $A$ at $a$ asserts that for any type family\n\\begin{equation*}\n\\Gamma,x:A,\\alpha: a=_A x\\vdash P(x,\\alpha)~\\mathrm{type}\n\\end{equation*}\nthere is a term\n\\begin{equation*}\n\\ind{a=} : P(a,\\refl{a})\\to \\prd{x:A}{\\alpha:a=_A x}P(x,\\alpha)\n\\end{equation*}\nin context $\\Gamma$, satisfying the computation rule\n\\begin{equation*}\n\\ind{a=}(p,a,\\refl{a})\\jdeq p.\n\\end{equation*}\n\nA term of type $a=_A x$ is also called an \\define{identification} of $a$ with $x$, or a \\define{path} from $a$ to $x$.\nThe induction principle for identity types is sometimes called \\define{identification elimination} or \\define{path induction}. Occasionally, we also write $\\idtypevar{A}$ for the identity type on $A$. \n\nMoreover, we assume that there is a universe $\\UU$ with a universal family $\\mathrm{El}$ over $\\UU$, that is closed under the type forming operations. For example, there is a map\n\\begin{equation*}\n\\check{\\idtypevar{}}:\\prd{A:\\UU}\\mathrm{El}(A)\\to\\mathrm{El}(A)\\to\\UU\n\\end{equation*}\nsatisfying\n\\begin{equation*}\n\\mathrm{El}(\\check{\\idtypevar{}}(A,x,y))\\jdeq (x=_{\\mathrm{El}(A)} y),\n\\end{equation*}\nestablishing that the universe is closed under identity types.\n\nGiven a type $A$ the \\define{concatenation} operation\n\\begin{equation*}\n\\concat : \\prd{x,y,z:A} (\\id{x}{y})\\to(\\id{y}{z})\\to (\\id{x}{z})\n\\end{equation*}\nis defined by $\\concat(\\refl{x},q)\\defeq q$. We will usually write $\\ct{p}{q}$ for $\\concat(p,q)$. \nThe concatenation operation satisfies the unit laws\n\\begin{align*}\n\\leftunit(p) & : \\ct{\\refl{x}}{p}=p \\\\\n\\rightunit(p) & : \\ct{p}{\\refl{y}}=p.\n\\end{align*}\n\nThe \\define{inverse operation} \n\\begin{equation*}\n\\invfunc:\\prd{x,y:A} (x=y)\\to (y=x)\n\\end{equation*}\nis defined by $\\invfunc(\\refl{x})\\defeq\\refl{x}$. We will usually write $p^{-1}$ for $\\invfunc(p)$.\nThe inverse operation satisfies the inverse laws\n\\begin{align*}\n\\leftinv(p) & : \\ct{p^{-1}}{p} = \\refl{y} \\\\\n\\rightinv(p) & : \\ct{p}{p^{-1}} = \\refl{x}.\n\\end{align*}\n\nThe \\define{associativity operation}, which assigns to each $p:x=y$, $q:y=z$, and $r:z=w$ the \\define{associator}\n\\begin{equation*}\n\\assoc(p,q,r) : \\ct{(\\ct{p}{q})}{r}=\\ct{p}{(\\ct{q}{r})}\n\\end{equation*}\nis defined by $\\assoc(\\refl{x},q,r)\\defeq \\refl{\\ct{q}{r}}$.\n\nGiven a map $f:A\\to B$, the \\define{action on paths} of $f$ is an operation\n\\begin{equation*}\n\\apfunc{f} : \\prd{x,y:A} (\\id{x}{y})\\to(\\id{f(x)}{f(y)})\n\\end{equation*}\ndefined by $\\ap{f}{\\refl{x}}\\defeq\\refl{f(x)}$. \nMoreover, there are operations\n\\begin{align*}\n\\apid_A & : \\prd{x,y:A}{p:\\id{x}{y}} \\id{p}{\\ap{\\idfunc[A]}{p}} \\\\\n\\apcomp(f,g) & : \\prd{x,y:A}{p:\\id{x}{y}} \\id{\\ap{g}{\\ap{f}{p}}}{\\ap{g\\circ f}{p}}\n\\end{align*}\ndefined by $\\apid_A(\\refl{x})\\defeq \\refl{\\refl{x}}$ and $\\apcomp(f,g,\\refl{x})\\jdeq \\refl{\\refl{g(f(x))}}$, respectively.\nIt can be shown easily that the action on paths of a map preserves the groupoid operations, and that the groupoid laws are also preserved.\n\n\\begin{defn}\nLet $A$ be a type, and let $B$ be a type family over $A$. The \\define{transport} operation\n\\begin{equation*}\n\\tr_B:\\prd{x,y:A} (\\id{x}{y})\\to (B(x)\\to B(y))\n\\end{equation*}\nis defined by $\\tr_B(\\refl{x}) \\defeq \\idfunc[B(x)]$. \n\\end{defn}\n\n\\begin{defn}\\label{defn:apd}\nGiven a dependent function $f:\\prd{a:A}B(a)$ and a path $p:\\id{x}{y}$ in $A$, the \\define{dependent action on paths}\n\\begin{equation*}\n\\apdfunc{f} : \\prd{x,y:A}{p:x=y}\\id{\\tr_B(p,f(x))}{f(y)}\n\\end{equation*}\nis defined by $\\apd{f}{\\refl{x}}\\defeq \\refl{f(x)}$.\n\\end{defn}\n\n\\begin{defn}\nLet $f,g:\\prd{x:A}P(x)$ be two dependent functions. The type $f\\htpy g$ of \\define{homotopies}\\index{homotopy|textbf} from $f$ to $g$ is defined as\n\\begin{equation*}\nf\\htpy g \\defeq \\prd{x:A} f(x)=g(x).\n\\end{equation*}\n\\end{defn}\n\nCommutativity of diagrams is stated using homotopies. For instance, a triangle\n\\begin{equation*}\n\\begin{tikzcd}[column sep=tiny]\nA \\arrow[dr,swap,\"f\"] \\arrow[rr,\"h\"] & & B \\arrow[dl,\"g\"] \\\\\n& X\n\\end{tikzcd}\n\\end{equation*}\nis said to commute if it comes equipped with a homotopy $H:f\\htpy g\\circ h$, and a square\n\\begin{equation*}\n\\begin{tikzcd}\nA \\arrow[d,\"i\"'] \\arrow[r,\"g\"] & X \\arrow[d,\"f\"] \\\\\nB \\arrow[r,swap,\"h\"] & Y\n\\end{tikzcd}\n\\end{equation*}\nis said to commute if it comes equipped with a homotopy $H:h\\circ i\\htpy f\\circ g$. \n\nThe reflexivity, inverse, and concatenation operations on homotopies are defined pointwise.\nWe will write $H^{-1}$ for $\\lam{x}H(x)^{-1}$, and $\\ct{H}{K}$ for $\\lam{x}\\ct{H(x)}{K(x)}$.\nThese operations satisfy the groupoid laws (phrased appropriately as homotopies). Apart from the groupoid operations and their laws, we will occasionally need \\emph{whiskering} operations and the naturality of homotopies.\n\n\\begin{defn}\\label{defn:htpy_whisering}\nWe define the following \\define{whiskering}\\index{homotopy!whiskering operations|textbf}\\index{whiskering operations!of homotopies|textbf} operations on homotopies:\n\\begin{enumerate}\n\\item Suppose $H:f\\htpy g$ for two functions $f,g:A\\to B$, and let $h:B\\to C$. We define\n\\begin{equation*}\nh\\cdot H\\defeq \\lam{x}\\ap{h}{H(x)}:h\\circ f\\htpy h\\circ g.\n\\end{equation*}\n\\item Suppose $f:A\\to B$ and $H:g\\htpy h$ for two functions $g,h:B\\to C$. We define\n\\begin{equation*}\nH\\cdot f\\defeq\\lam{x}H(f(x)):h\\circ f\\htpy g\\circ f.\n\\end{equation*}\n\\end{enumerate}\n\\end{defn}\n\nWe will frequently make use of commuting cubes. The commutativity of a cube is stated using the whiskering operations on homotopies.\n\n\\begin{defn}\\label{defn:cube}\nA \\define{commuting cube}\\index{commuting cube|textbf}\n\\begin{equation*}\n\\begin{tikzcd}\n& A_{111} \\arrow[dl] \\arrow[dr] \\arrow[d] \\\\\nA_{110} \\arrow[d] & A_{101} \\arrow[dl] \\arrow[dr] & A_{011} \\arrow[dl,crossing over] \\arrow[d] \\\\\nA_{100} \\arrow[dr] & A_{010} \\arrow[d] \\arrow[from=ul,crossing over] & A_{001} \\arrow[dl] \\\\\n& A_{000},\n\\end{tikzcd}\n\\end{equation*}\nconsists of \n\\begin{enumerate}\n\\item types\n\\begin{equation*}\nA_{111},A_{110},A_{101},A_{011},A_{100},A_{010},A_{001},A_{000},\n\\end{equation*}\n\\item \\begin{samepage}%\nmaps\n\\begin{align*}\nf_{11\\check{1}} & : A_{111}\\to A_{110} & f_{\\check{1}01} & : A_{101}\\to A_{001} \\\\\nf_{1\\check{1}1} & : A_{111}\\to A_{101} & f_{01\\check{1}} & : A_{011}\\to A_{010} \\\\\nf_{\\check{1}11} & : A_{111}\\to A_{011} & f_{0\\check{1}1} & : A_{011}\\to A_{001} \\\\\nf_{1\\check{1}0} & : A_{110}\\to A_{100} & f_{\\check{1}00} & : A_{100}\\to A_{000} \\\\\nf_{\\check{1}10} & : A_{110}\\to A_{010} & f_{0\\check{1}0} & : A_{010}\\to A_{000} \\\\\nf_{10\\check{1}} & : A_{101}\\to A_{100} & f_{00\\check{1}} & : A_{001}\\to A_{000},\n\\end{align*}\n\\end{samepage}%\n\\item homotopies\n\\begin{align*}\nH_{1\\check{1}\\check{1}} & : f_{1\\check{1}0}\\circ f_{11\\check{1}} \\htpy f_{10\\check{1}}\\circ f_{1\\check{1}1} & H_{0\\check{1}\\check{1}} & : f_{0\\check{1}0}\\circ f_{01\\check{1}} \\htpy f_{00\\check{1}}\\circ f_{0\\check{1}1} \\\\\nH_{\\check{1}1\\check{1}} & : f_{\\check{1}10}\\circ f_{11\\check{1}} \\htpy f_{01\\check{1}}\\circ f_{\\check{1}11} & H_{\\check{1}0\\check{1}} & : f_{\\check{1}00}\\circ f_{10\\check{1}} \\htpy f_{00\\check{1}}\\circ f_{\\check{1}01} \\\\\nH_{\\check{1}\\check{1}1} & : f_{\\check{1}01}\\circ f_{1\\check{1}1} \\htpy f_{0\\check{1}1}\\circ f_{\\check{1}11} & H_{\\check{1}\\check{1}0} & : f_{\\check{1}00}\\circ f_{1\\check{1}0} \\htpy f_{0\\check{1}0}\\circ f_{\\check{1}10},\n\\end{align*}\n\\item and a homotopy \n\\begin{align*}\nC & : \\ct{(f_{\\check{1}00}\\cdot H_{1\\check{1}\\check{1}})}{(\\ct{(H_{\\check{1}0\\check{1}}\\cdot f_{1\\check{1}1})}{(f_{00\\check{1}}\\cdot H_{\\check{1}\\check{1}1})})} \\\\\n& \\qquad \\htpy \\ct{(H_{\\check{1}\\check{1}0}\\cdot f_{11\\check{1}})}{(\\ct{(f_{0\\check{1}0}\\cdot H_{\\check{1}1\\check{1}})}{(H_{0\\check{1}\\check{1}}\\cdot f_{\\check{1}11})})}\n\\end{align*}\nfilling the cube.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{defn}\nLet $f:A\\to B$ be a function. We say that $f$ has a \\define{section}\\index{section!of a map|textbf} if there is a term of type\\index{sec(f)@{$\\sections(f)$}|textbf}\n\\begin{equation*}\n\\sections(f) \\defeq \\sm{g:B\\to A} f\\circ g\\htpy \\idfunc[B].\n\\end{equation*}\nDually, we say that $f$ has a \\define{retraction}\\index{retraction} if there is a term of type\\index{retr(f)@{$\\retractions(f)$}|textbf}\n\\begin{equation*}\n\\retractions(f) \\defeq \\sm{h:B\\to A} h\\circ f\\htpy \\idfunc[A].\n\\end{equation*}\nIf $f$ has a retraction, we also say that $A$ is a \\define{retract}\\index{retract!of a type} of $B$.\n\\end{defn}\n\n\\begin{defn}\nWe say that a function $f:A\\to B$ is an \\define{equivalence}\\index{equivalence|textbf}\\index{bi-invertible map|see {equivalence}} if it has both a section and a retraction, i.e.~if it comes equipped with a term of type\\index{is_equiv@{$\\isequiv$}|textbf}\n\\begin{equation*}\n\\isequiv(f)\\defeq\\sections(f)\\times\\retractions(f).\n\\end{equation*}\nWe will write $\\eqv{A}{B}$\\index{equiv@{$\\eqv{A}{B}$}|textbf} for the type $\\sm{f:A\\to B}\\isequiv(f)$.\n\\end{defn}\n\nClearly, if $f$ is \\define{invertible}\\index{invertible map} in the sense that it comes equipped with a function $g:B\\to A$ such that $f\\circ g\\htpy\\idfunc[B]$ and $g\\circ f\\htpy\\idfunc[A]$, then $f$ is an equivalence. We write\\index{has_inverse@{$\\hasinverse$}|textbf}\n\\begin{equation*}\n\\hasinverse(f)\\defeq\\sm{g:B\\to A} (f\\circ g\\htpy \\idfunc[B])\\times (g\\circ f\\htpy\\idfunc[A]).\n\\end{equation*}\nThe section of an equivalence is also a retraction (and vice versa), so we define the \\define{inverse} of an equivalence to be its section. It follows immediately that the inverse of any equivalence is again an equivalence.\\index{equivalence!invertibility of} The identity function $\\idfunc[A]$ on a type $A$ is an equivalence since it is its own section and its own retraction.\n\nIt is straightforward to show that for any two functions $f,g:A\\to B$, we have\n\\begin{equation*}\n(f\\htpy g)\\to (\\isequiv(f)\\leftrightarrow\\isequiv(g)).\n\\end{equation*}\nGiven a commuting triangle\n\\begin{equation*}\n\\begin{tikzcd}[column sep=tiny]\nA \\arrow[rr,\"h\"] \\arrow[dr,swap,\"f\"] & & B \\arrow[dl,\"g\"] \\\\\n& X.\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\htpy g\\circ h$, we have:\n\\begin{enumerate}\n\\item If the map $h$ has a section, then $f$ has a section if and only if $g$ has a section.\n\\item If the map $g$ has a retraction, then $f$ has a retraction if and only if $h$ has a retraction.\n\\item (The \\define{3-for-2 property} for equivalences.) If any two of the functions\n\\begin{equation*}\nf,\\qquad g,\\qquad h\n\\end{equation*}\nare equivalences, then so is the third.\n\\end{enumerate}\n\nIn the following theorem we characterize the identity type of a $\\Sigma$-type as a $\\Sigma$-type of identity types.\n\n\n\\begin{prp}[Theorem 2.7.2 of \\cite{hottbook}]\\label{thm:eq_sigma}\nLet $B$ be a type family over $A$, let $s:\\sm{x:A}B(x)$, and consider the dependent function\\index{pair_eq@{$\\paireq$}|textbf}\n\\begin{equation*}\n\\paireq_s:\\prd{t:\\sm{x:A}B(x)} (s=t)\\to \\sm{\\alpha:\\proj 1(s)=\\proj 1(t)} \\tr_B(\\alpha,\\proj 2(s))=\\proj 2(t)\n\\end{equation*}\ndefined by $\\paireq_s(\\refl{s}) \\defeq (\\refl{\\proj 1(s)},\\refl{\\proj 2(s)})$. Then $\\paireq_{s,t}$ is an equivalence for every $t:\\sm{x:A}B(x)$.\\index{Sigma type@{$\\Sigma$-type}!identity types of|textit}\\index{identity type!of a Sigma-type@{of a $\\Sigma$-type}|textit}\n\\end{prp}\n\nWe include the proof mainly to introduce some more notation.\n\n\\begin{proof}\nThe maps in the converse direction\\index{eq_pair@{$\\eqpair$}}\n\\begin{equation*}\n\\eqpair_{s,t} : \\Big(\\sm{p:\\proj 1(s)=\\proj 1(t)}\\id{\\tr_B(p,\\proj 2(s))}{\\proj 2(t)}\\Big)\\to(\\id{s}{t})\n\\end{equation*}\nis defined by\n\\begin{equation*}\n\\eqpair_{(x,y),(x',y')}(\\refl{x},\\refl{y})\\defeq \\refl{(x,y)}.\n\\end{equation*}\nThe proofs that the function $\\eqpair_{s,t}$ is indeed an inverse of $\\paireq_{s,t}$ are also by induction.\n\\end{proof}\n\n\\begin{defn}\nWe say that a type $A$ is \\define{contractible}\\index{contractible!type|textbf} if there is a term of type\n\\begin{equation*}\n\\iscontr(A) \\defeq \\sm{c:A}\\prd{x:A}c=x.\n\\end{equation*}\nGiven a term $(c,C):\\iscontr(A)$, we call $c:A$ the \\define{center of contraction}\\index{center of contraction|textbf} of $A$, and we call $C:\\prd{x:A}a=x$ the \\define{contraction}\\index{contraction} of $A$.\n\\end{defn}\n\nSuppose $A$ is a contractible type with center of contraction $c$ and contraction $C$. Then the type of $C$ is (judgmentally) equal to the type\n\\begin{equation*}\n\\const_c\\htpy\\idfunc[A].\n\\end{equation*}\nIn other words, the contraction $C$ is a \\emph{homotopy} from the constant function to the identity function.\n\n\\begin{defn}\nConsider a type $A$ with a base point $a:A$. We say that $A$ satisfies \\define{singleton induction}\\index{singleton induction|textbf} if for every type family $B$ over $A$, the map\n\\begin{equation*}\n\\evpt:\\Big(\\prd{x:A}B(x)\\Big)\\to B(a)\n\\end{equation*}\ngiven by $f\\mapsto f(a)$ has a section.\n\\end{defn}\n\n\\begin{prp}\\label{thm:contractible}\nA type $A$ is contractible if and only if it satisfies singleton induction.\n\\end{prp}\n\n\\begin{eg}\nBy definition the unit type\\index{unit type!contractibility} $\\unit$ satisfies singleton induction, so it is contractible.\n\\end{eg}\n\n\\begin{rmk}\nFor any family $P:\\Big(\\sm{x:A}B(x)\\Big)\\to\\UU$ there is a map\n\\begin{equation*}\n\\evpair : \\Big(\\prd{t:\\sm{x:A}B(x)}P(t)\\Big)\\to \\prd{x:A}{y:B(x)}P(x,y)\n\\end{equation*}\nthat evaluates $f:\\prd{t:\\sm{x:A}B(x)}P(t)$ at pairs $(x,y)$. In other words, $\\evpair$ is defined by $\\lam{f}{x}{y}f(x,y)$. By the induction principle for $\\Sigma$-types, this map has a section. It is easy to show that $\\evpair$ is in fact an equivalence. \n\nSimilarly, there is a map\n\\begin{equation*}\n\\evrefl : \\Big(\\prd{x:A}{p:a=x}B(x,p)\\Big)\\to B(a,\\refl{a})\n\\end{equation*}\ngiven by $\\lam{f}f(a,\\refl{a})$, for any type family $B:\\prd{x:A} (a=x)\\to\\UU$. By path induction, this map has a section, and again it is easy to show that this map is in fact an equivalence. \n\\end{rmk}\n\n\\begin{prp}[Lemma 3.11.8 in \\cite{hottbook}]\\label{thm:total_path}\nFor any $x:A$, the type\n\\begin{equation*}\n\\sm{y:A}x=y\n\\end{equation*}\nis contractible.\\index{identity type!contractibility of total space|textit}\n\\end{prp}\n\n\\begin{proof}\nWe have the term $(x,\\refl{x}):\\sm{y:A}x=y$, and both maps in the composite\n\\begin{equation*}\n\\begin{tikzcd}[column sep=large]\n\\prd{t:\\sm{y:A}x=y}B(t) \\arrow[r,\"\\evpair\"] & \\prd{y:A}{p:x=y}B((y,p)) \\arrow[r,\"\\evrefl\"] & B((x,\\refl{x}))\n\\end{tikzcd}\n\\end{equation*}\nhave sections, so the composite has a section. The composite is $\\evpt$, so we see that the asserted type satisfies singleton induction.\n\\end{proof}\n\n\\begin{defn}\nLet $f:A\\to B$ be a function, and let $b:B$. The \\define{fiber}\\index{fiber|textbf}\\index{homotopy fiber|see {fiber}} of $f$ at $b$ is defined to be the type\n\\begin{equation*}\n\\fib{f}{b}\\defeq\\sm{a:A}f(a)=b.\n\\end{equation*}\n\\end{defn}\n\n\\begin{eg}[Lemma 4.8.1 of \\cite{hottbook}]\\label{eg:fib_proj}\nConsider a type family $B$ over $A$. Then the map\n\\begin{equation*}\nB(a)\\to \\fib{\\proj 1}{a}\n\\end{equation*}\ngiven by $b\\mapsto ((a,b),\\refl{a})$ is an equivalence. In other words, the fibers of the projection function $\\proj 1 : \\big(\\sm{x:A}B(x)\\big)\\to A$ are just the fibers of the family $B$.\n\\end{eg}\n\n\\begin{defn}\nWe say that a function $f:A\\to B$ is \\define{contractible}\\index{contractible!map|textbf} if there is a term of type\n\\begin{equation*}\n\\iscontr(f)\\defeq\\prd{b:B}\\iscontr(\\fib{f}{b}).\n\\end{equation*}\n\\end{defn}\n\nWe cite Chapter 4 of \\cite{hottbook} for the following result, although it is well-known that it can be proven directly and without the use of function extensionality.\n\n\\begin{prp}[Chapter 4 in \\cite{hottbook}]\\label{thm:contr_equiv}\nA function is an equivalence if and only if it is contractible.\\index{contractible!map!is an equivalence|textit}\n\\end{prp}\n\n\\section{The Fundamental Theorem of Identity Types}\nConsider a family\n\\begin{equation*}\nf : \\prd{x:A}B(x)\\to C(x)\n\\end{equation*}\nof maps. Such $f$ is also called a \\define{fiberwise map} or \\define{fiberwise transformation}.\n\n\\begin{defn}[Definition 4.7.5 of \\cite{hottbook}]\nWe define the map\n\\begin{equation*}\n\\total{f}:\\sm{x:A}B(x)\\to\\sm{x:A}C(x).\n\\end{equation*}\nby $\\lam{(x,y)}(x,f(x,y))$.\n\\end{defn}\n\n\\begin{lem}[Theorem 4.7.6 of \\cite{hottbook}]\\label{lem:fib_total}\nFor any fiberwise transformation $f:\\prd{x:A}B(x)\\to C(x)$, and any $a:A$ and $c:C(a)$, there is an equivalence\n\\begin{equation*}\n\\eqv{\\fib{f(a)}{c}}{\\fib{\\total{f}}{\\pairr{a,c}}}.\n\\end{equation*}\n\\end{lem}\n\n\\begin{eg}\nThere are equivalences\n\\begin{equation*}\n\\eqv{\\fib{(\\apfunc{f})_{x,y}}{q}}{\\fib{\\delta_f}{(x,y,q)}}\n\\end{equation*}\nfor any $q:f(x)=f(y)$, because the triangle\n\\begin{equation*}\n\\begin{tikzcd}[column sep=-1em]\nA \\arrow[rr,\"{\\lam{x}(x,x,\\refl{x})}\"] \\arrow[dr,swap,\"\\delta_f\"] & & \\sm{x,y:A}x=y \\arrow[dl,\"\\total{\\total{\\apfunc{f}}}\"] \\\\\n\\phantom{\\sm{x,y:A}x=y} & \\sm{x,y:A}f(x)=f(y)\n\\end{tikzcd}\n\\end{equation*}\ncommutes, and the top map is an equivalence.\n\\end{eg}\n\n\\begin{prp}[Theorem 4.7.7 of \\cite{hottbook}]\\label{thm:fib_equiv}\nLet $f:\\prd{x:A}B(x)\\to C(x)$ be a fiberwise transformation. The following are logically equivalent:\n\\begin{enumerate}\n\\item For each $x:A$, the map $f_x:B(x)\\to C(x)$ is an equivalence. In this case we say that $f$ is a \\define{fiberwise equivalence}.\n\\item The map $\\total{f}:\\sm{x:A}B(x)\\to\\sm{x:A}C(x)$ is an equivalence.\n\\end{enumerate}\n\\end{prp}\n\nThe following theorem is the key to many results about identity types, which we will use instead of the \\emph{encode-decode method}\\index{encode-decode method} of \\cite{LicataShulman}. We refer to it as the \\define{Fundamental Theorem of Identity Types}.\n\n\\begin{thm}[Theorem 5.8.2 of \\cite{hottbook}]\\label{thm:id_fundamental}\nLet $A$ be a type with $a:A$, and let $B$ be a type family over $A$ with $b:B(a)$.\nThen the following are logically equivalent:\n\\begin{enumerate}\n\\item The canonical family of maps\n\\begin{equation*}\n\\ind{a{=}}(b):\\prd{x:A} (a=x)\\to B(x)\n\\end{equation*}\nis a fiberwise equivalence.\n\\item The total space\n\\begin{equation*}\n\\sm{x:A}B(x)\n\\end{equation*}\nis contractible.\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nBy \\autoref{thm:fib_equiv} it follows that the fiberwise transformation $\\ind{a{=}}(b)$ is a fiberwise equivalence if and only if it induces an equivalence\n\\begin{equation*}\n\\eqv{\\Big(\\sm{x:A}a=x\\Big)}{\\Big(\\sm{x:A}B(x)\\Big)}\n\\end{equation*}\non total spaces. We have that $\\sm{x:A}a=x$ is contractible. Now it follows by the 3-for-2 property of equivalences, applied in the case\n\\begin{equation*}\n\\begin{tikzcd}\n\\sm{x:A}a=x \\arrow[rr,\"\\total{\\ind{a{=}}(b)}\"] \\arrow[dr,swap,\"\\eqvsym\"] & & \\sm{x:A}B(x) \\arrow[dl] \\\\\n& \\unit & \\phantom{\\sm{x:A}a=x}\n\\end{tikzcd}\n\\end{equation*}\nthat $\\total{\\ind{a{=}}(b)}$ is an equivalence if and only if $\\sm{x:A}B(x)$ is contractible.\n\\end{proof}\n\nObserve that in the proof of \\cref{thm:id_fundamental} we haven't used the actual definition of the fiberwise transformation. Indeed, for any fiberwise transformation\n\\begin{equation*}\nf:\\prd{x:A}(a=x)\\to B(x)\n\\end{equation*}\nwe have that $f$ is a fiberwise equivalence if and only if the total space of $B$ is contractible.\n\nSince retracts of contractible types are again contractible, it follows that the only retract of the identity type is the identity type itself:\n\n\\begin{cor}\\label{cor:id_fundamental_retr}\nLet $a:A$, and let $B$ be a type family over $A$. If each $B(x)$ is a retract of $\\id{a}{x}$, then $B(x)$ is equivalent to $\\id{a}{x}$ for every $x:A$.\n\\end{cor}\n\nAs a first application of the fundamental theorem we give a quick new proof that equivalences are embeddings. The proof of the corresponding theorem in \\cite{hottbook} is more involved.\n\n\\begin{defn}\nAn \\define{embedding}\\index{embedding|textbf} is a map $f:A\\to B$ satisfying the property that\n\\begin{equation*}\n\\apfunc{f}:(\\id{x}{y})\\to(\\id{f(x)}{f(y)})\n\\end{equation*}\nis an equivalence for every $x,y:A$. We write $\\isemb(f)$ for the type of witnesses that $f$ is an embedding.\n\\end{defn}\n\n\\begin{prp}[Theorem 2.11.1 in \\cite{hottbook}]\n\\label{cor:emb_equiv} \nAny equivalence is an embedding.\\index{embedding!equivalences are embeddings|textit}\\index{equivalence!is an embedding|textit}\n\\end{prp}\n\n\\begin{proof}\nLet $e:\\eqv{A}{B}$ be an equivalence, and let $x:A$. By \\autoref{thm:id_fundamental} it follows that\n\\begin{equation*}\n\\apfunc{e} : (\\id{x}{y})\\to (\\id{e(x)}{e(y)})\n\\end{equation*}\nis an equivalence for every $y:A$ if and only if the total space\n\\begin{equation*}\n\\sm{y:A}e(x)=e(y)\n\\end{equation*}\nis contractible for every $y:A$. Now observe that $\\sm{y:A}e(x)=e(y)$ is equivalent to the fiber $\\fib{e}{e(x)}$, which is contractible by \\cref{thm:contr_equiv}.\n\\end{proof}\n\n\\begin{defn}\nA type $A$ is said to be a \\define{proposition} if there is a term of type\n\\begin{equation*}\n\\isprop(A)\\defeq\\prd{x,y:A}\\iscontr(x=y).\n\\end{equation*}\nFurthermore, we write $\\prop\\defeq\\sm{X:\\UU}\\isprop(X)$ for the type of all small propositions.\n\\end{defn}\n\nWe will often use either of the following characterizations of propositions.\n\n\\begin{lem}[Lemma 3.11.10 and Exercise 3.5 of \\cite{hottbook}]\\label{lem:prop_char}\nFor any type $A$ the following are equivalent:\n\\begin{enumerate}\n\\item $A$ is a proposition.\n\\item $A$ is \\define{proof irrelevant} in the sense that $\\prd{x,y:A}x=y$.\n\\item $A\\to\\iscontr(A)$. \n\\end{enumerate}\n\\end{lem}\n\n\\begin{eg}\\label{eg:prop_contr}\nAny contractible type is a proposition. The empty type is a proposition by a direct application of the induction principle of the empty type. Furthermore, any retract of a proposition is again a proposition. In particular, propositions are closed under equivalences.\n\\end{eg}\n\n\\begin{lem}\\label{lem:id_fib}\nConsider a function $f:A\\to B$, and let $(a,p),(a',p'):\\fib{f}{b}$ for some $b:B$. Then the canonical map\n\\begin{equation*}\n((a,p)=(a',p'))\\to \\fib{\\apfunc{f}}{\\ct{p}{p'^{-1}}}\n\\end{equation*}\nis an equivalence.\n\\end{lem}\n\n\\begin{proof}\nBy \\cref{thm:fib_equiv} it suffices to show that the type\n\\begin{equation*}\n\\sm{y:A}{q:f(y)=b}{p:\\proj 1(s)=y} \\ap{f}{p}=\\ct{\\proj 2(s)}{q^{-1}}\n\\end{equation*}\nis contractible, which is immediate by two applications of \\cref{thm:total_path}.\n\\end{proof}\n\n\\begin{prp}[Lemma 7.6.2 of \\cite{hottbook}]\\label{thm:prop_emb}\nA map is an embedding if and only if its fibers are propositions.\n\\end{prp}\n\n\\begin{proof}\nIf $f$ is an embedding, then the fibers of $\\apfunc{f}$ are contractible by \\cref{thm:contr_equiv}. Thus it follows by \\cref{lem:id_fib} that the fibers of $f$ are propositions.\n\nConversely, if the fibers of $f$ are propositions, then we have by \\cref{lem:id_fib} an equivalence\n\\begin{equation*}\n\\eqv{((x,p)=(y,\\refl{f(y)}))}{\\fib{\\apfunc{f}}{p}}\n\\end{equation*}\nfor any $p:f(x)=f(y)$, which shows that the fibers of $\\apfunc{f}$ are contractible. Thus $f$ is an embedding by \\cref{thm:contr_equiv}.\n\\end{proof}\n\n\\begin{defn}\nA type family $B$ over $A$ is said to be a \\define{subtype} of $A$ if for each $x:A$ the type $B(x)$ is a proposition.\n\\end{defn}\n\n\\begin{cor}\\label{thm:subtype}\nA type family $B$ over $A$ is a subtype if and only if the projection map\n\\begin{equation*}\n\\proj 1 : \\Big(\\sm{x:A}B(x)\\Big)\\to A\n\\end{equation*}\nis an embedding.\n\\end{cor}\n\n\\begin{proof}\nImmediate by \\cref{eg:fib_proj,thm:prop_emb}.\n\\end{proof}\n\n\\section{Function extensionality}\n\\begin{prp}[Theorem 4.9.5 of \\cite{hottbook}]\\label{thm:funext_wkfunext}\nThe following are equivalent:\n\\begin{enumerate}\n\\item The \\define{function extensionality principle}\\index{function extensionality}: For every type family $B$ over $A$, and any two dependent functions $f,g:\\prd{x:A}B(x)$, the canonical map\\index{htpy_eq@{$\\htpyeq$}|textbf}\n\\begin{equation*}\n\\htpyeq(f,g) : (\\id{f}{g})\\to (f\\htpy g)\n\\end{equation*}\nby path induction (sending $\\refl{f}$ to $\\lam{x}\\refl{f(x)}$) is an equivalence. We will write $\\eqhtpy$\\index{eq_htpy@{$\\eqhtpy$}} for its inverse.\n\\item The \\define{weak function extensionality principle}\\index{weak function extensionality} holds: For every type family $B$ over $A$ one has\\index{contractible!weak function extensionality}\n\\begin{equation*}\n\\Big(\\prd{x:A}\\iscontr(B(x))\\Big)\\to\\iscontr\\Big(\\prd{x:A}B(x)\\Big).\n\\end{equation*}\n\\end{enumerate}\n\\end{prp}\n\nFrom now on we will assume that function extensionality holds.\n\n\\begin{cor}[Theorem 7.1.9 of \\cite{hottbook}]\\label{thm:prop_pi}\nFor any type family $B$ over $A$ one has\n\\begin{equation*}\n\\Big(\\prd{x:A}\\isprop(B(x))\\Big)\\to \\isprop\\Big(\\prd{x:A}B(x)\\Big).\n\\end{equation*}\nIn particular, if $B$ is a proposition, then $A\\to B$ is a proposition for any type $A$.\n\\end{cor}\n\nWe show in this section that a map $f:A\\to B$ is an equivalence if and only if for any type family $P$ over $B$, the precomposition map\n\\begin{equation*}\n\\blank\\circ f: \\Big(\\prd{y:B}P(y)\\Big)\\to\\Big(\\prd{x:A}P(f(x))\\Big)\n\\end{equation*}\nis an equivalence. \nIn the proof we use the notion of \\emph{path-split} maps, which was introduced in \\cite{RijkeShulmanSpitters}.\n\n\\begin{defn}\nWe say that a map $f:A\\to B$ is \\define{path-split}\\index{path-split|textbf} if $f$ has a section, and for each $x,y:A$ the map\n\\begin{equation*}\n\\apfunc{f}(x,y):(x=y)\\to (f(x)=f(y))\n\\end{equation*}\nalso has a section. We write $\\pathsplit(f)$\\index{path_split(f)@{$\\pathsplit(f)$}|textbf} for the type\n\\begin{equation*}\n\\sections(f)\\times\\prd{x,y:A}\\sections(\\apfunc{f}(x,y)).\n\\end{equation*}\n\\end{defn}\n\nWe will also use the notion of \\emph{half-adjoint equivalences}, which were introduced in \\cite{hottbook}.\n\n\\begin{defn}[Definition 4.2.1 of \\cite{hottbook}]\nWe say that a map $f:A\\to B$ is a \\define{half-adjoint equivalence}\\index{half-adjoint equivalence|textbf}, in the sense that there are\n\\begin{align*}\ng & : B \\to A\\\\\nG & : f\\circ g \\htpy \\idfunc[B] \\\\\nH & : g\\circ f \\htpy \\idfunc[A] \\\\\nK & : G\\cdot f \\htpy f\\cdot H.\n\\end{align*}\nWe write $\\halfadj(f)$\\index{half_adj(f)@{$\\halfadj(f)$}|textbf} for the type of such quadruples $(g,G,H,K)$.\n\\end{defn}\n\nFurthermore, we will need `type theoretic choice'. \n\n\\begin{prp}[Theorem 2.15.7 of \\cite{hottbook}]\\label{thm:choice}\nLet $C(x,y)$ be a type in context $\\Gamma,x:A,y:B(x)$. Then the map\n\\begin{equation*}\n\\varphi:\\Big(\\prd{x:A}\\sm{y:B(x)}C(x,y)\\Big)\\to \\Big(\\sm{f:\\prd{x:A}B(x)}\\prd{x:A}C(x,f(x))\\Big)\n\\end{equation*}\ngiven by $\\lam{h}(\\lam{x}\\proj 1(h(x)),\\lam{x}\\proj 2(h(x)))$ is an equivalence.\n\\end{prp}\n\n\\begin{cor}\nFor type $A$ and any type family $C$ over $B$, the map\n\\begin{equation*}\n\\Big(\\sm{f:A\\to B} \\prd{x:A}C(f(x))\\Big)\\to\\Big(A\\to\\sm{y:B}C(x)\\Big)\n\\end{equation*}\ngiven by $\\lam{(f,g)}{x}(f(x),g(x))$ is an equivalence.\n\\end{cor}\n\n\\begin{prp}\\label{prp:equiv_precomp}\nFor any map $f:A\\to B$, the following are equivalent:\n\\begin{enumerate}\n\\item $f$ is an equivalence.\n\\item $f$ is path-split.\n\\item $f$ is a half-adjoint equivalence.\n\\item For any type family $P$ over $B$ the map\n\\begin{equation*}\n\\Big(\\prd{y:B}P(y)\\Big)\\to\\Big(\\prd{x:A}P(f(x))\\Big)\n\\end{equation*}\ngiven by $s\\mapsto s\\circ f$ is an equivalence.\n\\item For any type $X$ the map\n\\begin{equation*}\n(B\\to X)\\to (A\\to X)\n\\end{equation*}\ngiven by $g\\mapsto g\\circ f$ is an equivalence. \n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nTo see that (i) implies (ii) we note that any equivalence has a section, and its action on paths is an equivalence by \\cref{cor:emb_equiv} so again it has a section.\n\nTo show that (ii) implies (iii), assume that $f$ is path-split. Thus we have $(g,G):\\sections(f)$, and the assumption that $\\apfunc{f}:(x=y)\\to (f(x)=f(y))$ has a section for every $x,y:A$ gives us a term of type\n\\begin{equation*}\n\\prd{x:A}\\fib{\\apfunc{f}}{G(f(x))}.\n\\end{equation*}\nBy \\cref{thm:choice} this type is equivalent to\n\\begin{equation*}\n\\sm{H:\\prd{x:A}g(f(x))=x}\\prd{x:A}G(f(x))=\\ap{f}{H(x)},\n\\end{equation*}\nso we obtain $H:g\\circ f\\htpy \\idfunc[A]$ and $K:G\\cdot f\\htpy f\\cdot H$, showing that $f$ is a half-adjoint equivalence.\n\nTo show that (iii) implies (iv), suppose that $f$ comes equipped with $(g,G,H,K)$ witnessing that $f$ is a half-adjoint equivalence. Then we define the inverse of $\\blank\\circ f$ to be the map\n\\begin{equation*}\n\\varphi:\\Big(\\prd{x:A}P(f(x))\\Big)\\to\\Big(\\prd{y:B}P(y)\\Big)\n\\end{equation*}\ngiven by $s\\mapsto \\lam{y}\\tr_P(G(y),sg(y))$. \n\nTo see that $\\varphi$ is a section of $\\blank\\circ f$, let $s:\\prd{x:A}P(f(x))$. By function extensionality it suffices to construct a homotopy $\\varphi(s)\\circ f\\htpy s$. In other words, we have to show that\n\\begin{equation*}\n\\tr_P(G(f(x)),s(g(f(x)))=s(x)\n\\end{equation*}\nfor any $x:A$. Now we use the additional homotopy $K$ from our assumption that $f$ is a half-adjoint equivalence. Since we have $K(x):G(f(x))=\\ap{f}{H(x)}$ it suffices to show that\n\\begin{equation*}\n\\tr_P(\\ap{f}{H(x)},sgf(x))=s(x).\n\\end{equation*}\nA simple path-induction argument yields that\n\\begin{equation*}\n\\tr_P(\\ap{f}{p})\\htpy \\tr_{P\\circ f}(p)\n\\end{equation*}\nfor any path $p:x=y$ in $A$, so it suffices to construct an identification\n\\begin{equation*}\n\\tr_{P\\circ f}(H(x),sgf(x))=s(x).\n\\end{equation*}\nWe have such an identification by $\\apd{H(x)}{s}$.\n\nTo see that $\\varphi$ is a retraction of $\\blank\\circ f$, let $s:\\prd{y:B}P(y)$. By function extensionality it suffices to construct a homotopy $\\varphi(s\\circ f)\\htpy s$. In other words, we have to show that\n\\begin{equation*}\n\\tr_P(G(y),sfg(y))=s(y)\n\\end{equation*}\nfor any $y:B$. We have such an identification by $\\apd{G(y)}{s}$. This completes the proof that (iii) implies (iv).\n\nNote that (v) is an immediate consequence of (iv), since we can just choose $P$ to be the constant family $X$.\n\nIt remains to show that (v) implies (i). Suppose that\n\\begin{equation*}\n\\blank\\circ f:(B\\to X)\\to (A\\to X)\n\\end{equation*}\nis an equivalence for every type $X$. Then its fibers are contractible by \\cref{thm:contr_equiv}. In particular, choosing $X\\jdeq A$ we see that the fiber\n\\begin{equation*}\n\\fib{\\blank\\circ f}{\\idfunc[A]}\\jdeq \\sm{h:B\\to A}h\\circ f=\\idfunc[A]\n\\end{equation*}\nis contractible. Thus we obtain a function $h:B\\to A$ and a homotopy $H:h\\circ f\\htpy\\idfunc[A]$ showing that $h$ is a retraction of $f$. We will show that $h$ is also a section of $f$. To see this, we use that the fiber\n\\begin{equation*}\n\\fib{\\blank\\circ f}{f}\\jdeq \\sm{i:B\\to B} i\\circ f=f\n\\end{equation*}\nis contractible (choosing $X\\jdeq B$). \nOf course we have $(\\idfunc[B],\\refl{f})$ in this fiber. However we claim that there also is an identification $p:(f\\circ h)\\circ f=f$, showing that $(f\\circ h,p)$ is in this fiber, because\n\\begin{align*}\n(f\\circ h)\\circ f & \\jdeq f\\circ (h\\circ f) \\\\\n& = f\\circ \\idfunc[A] \\\\\n& \\jdeq f\n\\end{align*}\nNow we conclude by the contractibility of the fiber that there is an identification $(\\idfunc[B],\\refl{f})=(f\\circ h,p)$. In particular we obtain that $\\idfunc[B]=f\\circ h$, showing that $h$ is a section of $f$.\n\\end{proof}\n\n\\section{Homotopy pullbacks}\n\\subsection{Cartesian squares}\n\nRecall that a square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"g\"] & X\n\\end{tikzcd}\n\\end{equation*}\nis said to \\define{commute}\\index{commuting square|textbf} if there is a homotopy $H:f\\circ p\\htpy g\\circ q$. \n\n\\begin{defn}\\label{defn:cospan}\nA \\define{cospan}\\index{cospan|textbf} consists of three types $A$, $X$, and $B$, and maps $f:A\\to X$ and $g:B\\to X$. Given a type $C$, a \\define{cone}\\index{cone!on a cospan|textbf} on the cospan $A \\stackrel{f}{\\rightarrow} X \\stackrel{g}{\\leftarrow} B$ with \\define{vertex} $C$\\index{vertex!of a cone|textbf} consists of maps $p:C\\to A$, $q:C\\to B$ and a homotopy $H:f\\circ p\\htpy g\\circ q$ witnessing that the square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\ncommutes. We write\\index{cone(C)@{$\\cone(\\blank)$}|textbf}\n\\begin{equation*}\n\\cone(C)\\defeq \\sm{p:C\\to A}{q:C\\to B}f\\circ p\\htpy g\\circ q\n\\end{equation*}\nfor the type of cones with vertex $C$.\n\\end{defn}\n\nGiven a cone with vertex $C$ on a span $A\\stackrel{f}{\\rightarrow} X \\stackrel{g}{\\leftarrow} B$ and a map $h:C'\\to C$, we construct a new cone with vertex $C'$ in the following definition.\n\n\\begin{defn}\nFor any cone $(p,q,H)$ with vertex $C$ and any type $C'$, we define a map\\index{cone map@{$\\conemap$}|textbf}\n\\begin{equation*}\n\\conemap(p,q,H):(C'\\to C)\\to\\cone(C')\n\\end{equation*}\nby $h\\mapsto (p\\circ h,q\\circ h,H\\cdot h)$. \n\\end{defn}\n\n\\begin{defn}\nWe say that a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\circ p\\htpy g\\circ q$ is a \\define{pullback square}\\index{pullback square|textbf}, or that it is \\define{cartesian}\\index{cartesian square|textbf}, if it satisfies the \\define{universal property} of pullbacks\\index{universal property!of pullbacks}, which asserts that the map\n\\begin{equation*}\n\\conemap(p,q,H):(C'\\to C)\\to\\cone(C')\n\\end{equation*}\nis an equivalence for every type $C'$. \n\\end{defn}\n\nWe often indicate the universal property with a diagram as follows:\n\\begin{equation*}\n\\begin{tikzcd}\nC' \\arrow[drr,bend left=15,\"{q'}\"] \\arrow[dr,densely dotted,\"h\"] \\arrow[ddr,bend right=15,swap,\"{p'}\"] \\\\\n& C \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\n& A \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nsince the universal property states that for every cone $(p',q',H')$ with vertex $C'$, the type of pairs $(h,\\alpha)$ consisting of $h:C'\\to C$ equipped with $\\alpha:\\conemap((p,q,H),h)=(p',q',H')$ is contractible by \\cref{thm:contr_equiv}.\n\n\n\\begin{prp}\\label{thm:pullback_up}\nConsider a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\circ p\\htpy g\\circ q$\nThen the following are equivalent:\\index{universal property!of pullbacks (characterization)|textit}\n\\begin{enumerate}\n\\item The square is a pullback square.\n\\item For every type $C'$ and every cone $(p',q',H')$ with vertex $C'$, the type of quadruples $(h,K,L,M)$ consisting of\n\\begin{align*}\nh & : C'\\to C \\\\\nK & : p\\circ h \\htpy p' \\\\\nL & : q\\circ h \\htpy q' \\\\\nM & : \\ct{(H\\cdot h)}{(g\\cdot L)} \\htpy \\ct{(f\\cdot K)}{H'}\n\\end{align*}\nis contractible.\n\\end{enumerate}\n\\end{prp}\n\n\\begin{rmk}\nThe homotopy $M$ in \\cref{thm:pullback_up} witnesses that the square\n\\begin{equation*}\n\\begin{tikzcd}\nf\\circ p\\circ h \\arrow[r,\"f\\cdot K\"] \\arrow[d,swap,\"H\\cdot h\"] & f\\circ p' \\arrow[d,\"{H'}\"] \\\\\ng\\circ q\\circ h \\arrow[r,swap,\"g\\cdot L\"] & g\\circ q'\n\\end{tikzcd}\n\\end{equation*}\nof homotopies commutes.\n\\end{rmk}\n\n\\subsection{The unique existence of pullbacks}\n\n\\begin{defn}\nLet $f:A\\to X$ and $B\\to X$ be maps. Then we define\n\\begin{align*}\nA\\times_X B & \\defeq \\sm{x:A}{y:B}f(x)=g(y) \\\\\n\\pi_1 & \\defeq \\proj 1 & & : A\\times_X B\\to A \\\\\n\\pi_2 & \\defeq \\proj 1\\circ\\proj 2 & & : A\\times_X B\\to B\\\\\n\\pi_3 & \\defeq \\proj 2\\circ\\proj 2 & & : f\\circ \\pi_1 \\htpy g\\circ\\pi_2.\n\\end{align*}\nThe type $A\\times_X B$ is called the \\define{canonical pullback}\\index{canonical pullback|textbf} of $f$ and $g$.\n\\end{defn}\n\nNote that $A\\times_X B$ depends on $f$ and $g$, although this dependency is not visible in the notation.\n\n\\begin{prp}[Exercise 2.11 of \\cite{hottbook}]\nGiven maps $f:A\\to X$ and $g:B\\to X$, the commuting square\\index{canonical pullback|textit}\n\\begin{equation*}\n\\begin{tikzcd}\nA\\times_X B \\arrow[r,\"\\pi_2\"] \\arrow[d,swap,\"\\pi_1\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X,\n\\end{tikzcd}\n\\end{equation*}\nis a pullback square.\n\\end{prp}\n\nIn the following lemma we establish the uniqueness of pullbacks up to equivalence via a \\emph{3-for-2 property} for pullbacks.\n\n\\begin{lem}\\label{lem:pb_3for2}\\index{pullback!3-for-2 property|textit}\\index{3-for-2 property!of pullbacks|textit}%\nConsider the squares\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] & {C'} \\arrow[r,\"{q'}\"] \\arrow[d,swap,\"{p'}\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X & A \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith homotopies $H:f\\circ p \\htpy g\\circ q$ and $H':f\\circ p'\\htpy g\\circ q'$.\nFurthermore, suppose we have a map $h:C'\\to C$ equipped with\n\\begin{align*}\nK & : p\\circ h \\htpy p' \\\\\nL & : q\\circ h \\htpy q' \\\\\nM & : \\ct{(H\\cdot h)}{(g\\cdot L)} \\htpy \\ct{(f\\cdot K)}{H'}.\n\\end{align*}\nIf any two of the following three properties hold, so does the third:\n\\begin{samepage}%\n\\begin{enumerate}\n\\item $C$ is a pullback.\n\\item $C'$ is a pullback.\n\\item $h$ is an equivalence.\n\\end{enumerate}%\n\\end{samepage}%\n\\end{lem}\n\n\\begin{proof}\nThe type of triples $(K,L,M)$ is equivalent to the type of identifications\n\\begin{equation*}\n\\conemap((p,q,H),h)=(p',q',H').\n\\end{equation*}\nLet $D$ be a type, and let $k:D\\to C'$ be a map. We observe that\n\\begin{align*}\n\\conemap((p,q,H),(h\\circ k)) & \\jdeq (p\\circ (h\\circ k),q\\circ (h\\circ k),H\\circ (h\\circ k)) \\\\\n& \\jdeq ((p\\circ h)\\circ k,(q\\circ h)\\circ k, (H\\circ h)\\circ k) \\\\\n& \\jdeq \\conemap(\\conemap((p,q,H),h),k) \\\\\n& = \\conemap((p',q',H'),k).\n\\end{align*}\nThus we see that the triangle \n\\begin{equation*}\n\\begin{tikzcd}[column sep=-1em]\n(D\\to C') \\arrow[rr,\"{h\\circ \\blank}\"] \\arrow[dr,swap,\"{\\conemap(p',q',H')}\"] & & (D\\to C) \\arrow[dl,\"{\\conemap(p,q,H)}\"] \\\\\n& \\cone(D) & \\phantom{(D\\to C')}\n\\end{tikzcd}\n\\end{equation*}\ncommutes. Therefore it follows from the 3-for-2 property of equivalences that if any two of the following properties hold, then so does the third:\n\\begin{enumerate}\n\\item The map $\\conemap(p,q,H):(D\\to C)\\to \\cone(D)$ is an equivalence,\n\\item The map $\\conemap(p',q',H'):(D\\to C')\\to \\cone(D)$ is an equivalence,\n\\item The map $h\\circ\\blank : (D\\to C')\\to (D\\to C)$ is an equivalence.\n\\end{enumerate}\nThus the 3-for-2 property for pullbacks follows from the fact that $h$ is an equivalence if and only if $h\\circ\\blank : (D\\to C')\\to (D\\to C)$ is an equivalence for any type $D$.\n\\end{proof}\n\n\n\\begin{defn}\nGiven a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\circ p \\htpy g \\circ q$, we define the \\define{gap map}\\index{gap map|textbf}\\index{pullback!gap map|textbf}\n\\begin{equation*}\n\\gap(p,q,H):C \\to A\\times_X B\n\\end{equation*}\nby $\\lam{z}(p(z),q(z),H(z))$. Furthermore, we will write\\index{is_pullback@{$\\ispullback$}|textbf}\n\\begin{equation*}\n\\ispullback(f,g,H)\\defeq \\isequiv(\\gap(p,q,H)).\n\\end{equation*}\n\\end{defn}\n\n\\begin{prp}\\label{thm:is_pullback}\nConsider a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\circ p \\htpy g \\circ q$. The following are equivalent:\n\\begin{enumerate}\n\\item The square is a pullback square\n\\item There is a term of type\n\\begin{equation*}\n\\ispullback(p,q,H)\\defeq \\isequiv(\\gap(p,q,H)).\n\\end{equation*}\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nNote that there are homotopies\n\\begin{align*}\nK & : \\pi_1\\circ \\gap(p,q,H) \\htpy p \\\\\nL & : \\pi_2\\circ \\gap(p,q,H) \\htpy q \\\\\nM & : \\ct{(\\pi_3\\cdot \\gap(p,q,H))}{(g\\cdot L)} \\htpy \\ct{(f\\cdot K)}{H}.\n\\end{align*}\ngiven by \n\\begin{align*}\nK & \\defeq \\lam{z}\\refl{p(z)} \\\\\nL & \\defeq \\lam{z}\\refl{q(z)} \\\\\nM & \\defeq \\lam{z}\\ct{\\rightunit(H(z))}{\\leftunit(H(z))^{-1}}.\n\\end{align*}\nTherefore the claim follows by \\cref{lem:pb_3for2}.\n\\end{proof}\n\n\\subsection{Fiberwise equivalences}\n\n\\begin{prp}\\label{thm:pb_fibequiv}\nLet $f:A\\to B$, and let $g:\\prd{a:A}P(a)\\to Q(f(a))$ be a fiberwise transformation\\index{fiberwise transformation|textit}. The following are equivalent:\n\\begin{enumerate}\n\\item The commuting square\n\\begin{equation*}\n\\begin{tikzcd}[column sep=large]\n\\sm{a:A}P(a) \\arrow[r,\"{\\total[f]{g}}\"] \\arrow[d,swap,\"\\proj 1\"] & \\sm{b:B}Q(b) \\arrow[d,\"\\proj 1\"] \\\\\nA \\arrow[r,swap,\"f\"] & B\n\\end{tikzcd}\n\\end{equation*}\nis a pullback square.\n\\item $g$ is a fiberwise equivalence.\\index{fiberwise equivalence|textit}\n\\end{enumerate}\n\\end{prp}\n\n\\begin{proof}\nThe gap map factors as follows\n\\begin{equation*}\n\\begin{tikzcd}[column sep=-2em]\n\\sm{x:A}P(x) \\arrow[dr,swap,\"\\total{g}\"] \\arrow[rr,\"\\gap\"] & & A \\times_B \\Big(\\sm{y:B}Q(y)\\Big) \\\\\n\\phantom{A \\times_B \\Big(\\sm{y:B}Q(y)\\Big)} & \\sm{x:A}Q(f(x)) \\arrow[ur,swap,\"{\\gap'\\,\\defeq\\,\\lam{(x,q)}(x,(f(x),q),\\refl{f(x)})}\"]\n\\end{tikzcd}\n\\end{equation*}\nSince $\\gap'$ is an equivalence, it follows by \\cref{thm:fib_equiv} that the gap map is an equivalence if and only if $g$ is a fiberwise equivalence.\n\\end{proof}\n\n\\begin{lem}\nConsider a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\circ p\\htpy g\\circ q$, and consider the fiberwise transformation\n\\begin{equation*}\n\\fibf{(f,q,H)} : \\prd{a:A} \\fib{p}{a}\\to \\fib{g}{f(a)}\n\\end{equation*}\ngiven by $\\lam{a}{(c,u)}(q(c),\\ct{H(c)^{-1}}{\\ap{f}{u}})$. Then there is an equivalence\n\\begin{equation*}\n\\eqv{\\fib{\\gap(p,q,H)}{(a,b,\\alpha)}}{\\fib{\\fibf{(f,q,H)}(a)}{(b,\\alpha^{-1})}}\n\\end{equation*}\n\\end{lem}\n\n\\begin{proof}\nTo obtain an equivalence of the desired type we simply concatenate known equivalences:\n\\begin{align*}\n\\fib{h}{(a,b,\\alpha)} & \\jdeq \\sm{z:C} (p(z),q(z),H(z))=(a,b,\\alpha) \\\\\n& \\eqvsym \\sm{z:C}{u:p(z)=a}{v:q(z)=b}\\ct{H(z)}{\\ap{g}{v}}=\\ct{\\ap{f}{u}}{\\alpha} \\\\\n& \\eqvsym \\sm{(z,u):\\fib{p}{a}}{v:q(z)=b} \\ct{H(z)^{-1}}{\\ap{f}{u}}=\\ct{\\ap{g}{v}}{\\alpha^{-1}} \\\\\n& \\eqvsym \\fib{\\varphi(a)}{(b,\\alpha^{-1})}\\qedhere\n\\end{align*}\n\\end{proof}\n\n\\begin{cor}\\label{cor:pb_fibequiv}\nConsider a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X\n\\end{tikzcd}\n\\end{equation*}\nwith $H:f\\circ p\\htpy g\\circ q$. The following are equivalent:\n\\begin{enumerate}\n\\item The square is a pullback square.\\index{pullback square!characterized by fiberwise equivalence|textit}\n\\item The induced map on fibers\n\\begin{equation*}\n\\fibf{(p,q,H)}:\\prd{x:A}\\fib{p}{x}\\to \\fib{g}{f(x)}\n\\end{equation*}\nis a fiberwise equivalence.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{cor}\\label{cor:pb_equiv}\nConsider a commuting square\n\\begin{equation*}\n\\begin{tikzcd}\nC \\arrow[r,\"q\"] \\arrow[d,swap,\"p\"] & B \\arrow[d,\"g\"] \\\\\nA \\arrow[r,swap,\"f\"] & X.\n\\end{tikzcd}\n\\end{equation*}\nand suppose that $g$ is an equivalence. Then the following are equivalent:\n\\begin{enumerate}\n\\item The square is a pullback square.\n\\item The map $p:C\\to A$ is an equivalence.\\index{equivalence!pullback of|textit}\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\nIf the square is a pullback square, then by \\cref{thm:pb_fibequiv} the fibers of $p$ are equivalent to the fibers of $g$, which are contractible by \\cref{thm:contr_equiv}. Thus it follows that $p$ is a contractible map, and hence that $p$ is an equivalence.\n\nIf $p$ is an equivalence, then by \\cref{thm:contr_equiv} both $\\fib{p}{x}$ and $\\fib{g}{f(x)}$ are contractible for any $x:X$. It follows that the induced map $\\fib{p}{x}\\to\\fib{g}{f(x)}$ is an equivalence. Thus we apply \\cref{cor:pb_fibequiv} to conclude that the square is a pullback.\n\\end{proof}\n\n\\section{The univalence axiom}\n\nThe univalence axiom characterizes the identity type of the universe. It is considered to be an \\emph{extensionality principle}\\index{extensionality principle!types} for types. In the following theorem we introduce the univalence axiom and give two more equivalent ways of stating this.\n\n\\begin{prp}\\label{thm:univalence}\nThe following are equivalent:\n\\begin{enumerate}\n\\item The \\define{univalence axiom}\\index{univalence axiom|textbf}: for any $A:\\UU$ the map\\index{equiv_eq@{$\\equiveq$}|textbf}\n\\begin{equation*}\n\\equiveq\\defeq \\ind{A=}(\\idfunc[A]) : \\prd{B:\\UU} (\\id{A}{B})\\to(\\eqv{A}{B}).\n\\end{equation*}\nis a fiberwise equivalence.\\index{identity type!universe} If this is the case, we write\n$\\eqequiv$\\index{eq equiv@{$\\eqequiv$}|textbf}\nfor the inverse of $\\equiveq$.\n\\item The type\n\\begin{equation*}\n\\sm{B:\\UU}\\eqv{A}{B}\n\\end{equation*}\nis contractible for each $A:\\UU$.\n\\item The principle of \\define{equivalence induction}\\index{equivalence induction}\\index{induction principle!for equivalences}: for every $A:\\UU$ and for every type family\n\\begin{equation*}\nP:\\prd{B:\\UU} (\\eqv{A}{B})\\to \\type,\n\\end{equation*}\nthe map\n\\begin{equation*}\n\\Big(\\prd{B:\\UU}{e:\\eqv{A}{B}}P(B,e)\\Big)\\to P(A,\\idfunc[A])\n\\end{equation*}\ngiven by $f\\mapsto f(A,\\idfunc[A])$ has a section.\\qedhere\n\\end{enumerate}\n\\end{prp}\n\nIt is a trivial observation, but nevertheless of fundamental importance, that by the univalence axiom the identity types of $\\UU$ are equivalent to types in $\\UU$, because it provides an equivalence $\\eqv{(A=B)}{(\\eqv{A}{B})}$, and the type $\\eqv{A}{B}$ is in $\\UU$ for any $A,B:\\UU$. Since the identity types of $\\UU$ are equivalent to types in $\\UU$, we also say that the universe is \\emph{locally small}.\n\n\\begin{defn}\\label{defn:ess_small}\n\\begin{enumerate}\n\\item A type $A$ is said to be \\define{essentially small}\\index{essentially small!type|textbf} if there is a type $X:\\UU$ and an equivalence $\\eqv{A}{X}$. We write\\index{ess_small(A)@{$\\esssmall(A)$}|textbf}\n\\begin{equation*}\n\\esssmall(A)\\defeq\\sm{X:\\UU}\\eqv{A}{X}.\n\\end{equation*}\n\\item A map $f:A\\to B$ is said to be \\define{essentially small}\\index{essentially small!map|textbf} if for each $b:B$ the fiber $\\fib{f}{b}$ is essentially small.\nWe write\\index{ess_small(f)@{$\\esssmall(f)$}|textbf}\n\\begin{equation*}\n\\esssmall(f)\\defeq\\prd{b:B}\\esssmall(\\fib{f}{b}).\n\\end{equation*}\n\\item A type $A$ is said to be \\define{locally small}\\index{locally small!type} if for every $x,y:A$ the identity type $x=y$ is essentially small.\nWe write\\index{loc_small(A)@{$\\locsmall(A)$}|textbf}\n\\begin{equation*}\n\\locsmall(A)\\defeq \\prd{x,y:A}\\esssmall(x=y).\n\\end{equation*}\n\\item Similarly, a map $f:A\\to X$ is said to be \\define{locally small} if $\\delta_f:A\\to A\\times_X A$ is essentially small.\n\\end{enumerate}\n\\end{defn}\n\n\\begin{lem}\\label{lem:isprop_ess_small}\nThe type $\\esssmall(X)$ is a proposition for any type $X$.\\index{essentially small!is a proposition|textit}\n\\end{lem}\n\n\\begin{proof}\nLet $X$ be a type. Our goal is to show that the type\n\\begin{equation*}\n\\sm{Y:\\UU}\\eqv{X}{Y}\n\\end{equation*}\nis a proposition. Suppose there is a type $X':\\UU$ and an equivalence $e:\\eqv{X}{X'}$, then the map\n\\begin{equation*}\n(\\eqv{X}{Y})\\to (\\eqv{X'}{Y})\n\\end{equation*}\ngiven by precomposing with $e^{-1}$ is an equivalence. This induces an equivalence on total spaces\n\\begin{equation*}\n\\eqv{\\Big(\\sm{Y:\\UU}\\eqv{X}{Y}\\Big)}{\\Big(\\sm{Y:\\UU}\\eqv{X'}{Y}\\Big)}\n\\end{equation*}\nHowever, the codomain of this equivalence is contractible by \\cref{thm:univalence}. Thus it follows that the asserted type is a proposition.\n\\end{proof}\n\n\\begin{cor}\nFor each function $f:A\\to B$, the type $\\esssmall(f)$ is a proposition, and for each type $X$ the type $\\locsmall(X)$ is a proposition.\n\\end{cor}\n\n\\begin{proof}\nThis follows from the fact that propositions are closed under dependent products, established in \\cref{thm:prop_pi}.\n\\end{proof}\n\n\\begin{rmk}\nThe property of essentially smallness is preserved by $\\Pi$, $\\Sigma$, and $\\mathsf{Id}$. Of course, any contractible type is essentially small, and so is any small type. The property of essentially smallness is preserved by $\\Sigma$ and $\\mathsf{Id}$, and the exponent $X^A$ of a locally small type $X$ by an essentially small type $A$ is again locally small. Furthermore, any proposition is locally small, and any universe is locally small with respect to itself. \n\\end{rmk}\n\n\\begin{defn}\nConsider two functions $f:A\\to X$ and $g:B\\to X$. We define the type \n\\begin{equation*}\n\\hom_X(f,g)\\defeq \\sm{h:A\\to B} f\\htpy g\\circ h.\n\\end{equation*}\n\\end{defn}\n\nIn other words, the type $\\hom_X(f,g)$ is the type of functions $h:A\\to B$ equipped with a homotopy witnessing that the triangle\n\\begin{equation*}\n\\begin{tikzcd}[column sep=tiny]\nA \\arrow[dr,swap,\"f\"] \\arrow[rr,\"h\"] & & B \\arrow[dl,\"g\"] \\\\\n& X\n\\end{tikzcd}\n\\end{equation*}\n\n\\begin{lem}\nLet $P$ and $Q$ be two type families over $X$, and write $\\proj 1^P$ and $\\proj 1^Q$ for their first projections, respectively. Then the map\n\\begin{equation*}\n\\tottriangle:\\Big(\\prd{x:X} P(x)\\to Q(x)\\Big)\\to \\hom_X(\\proj 1^P,\\proj 1^Q)\n\\end{equation*}\ngiven by $\\tottriangle(f)\\defeq (\\total{f},\\lam{(x,y)}\\refl{x})$, is an equivalence.\n\\end{lem}\n\n\\begin{cor}\\label{cor:fib_triangle}\nFor any two maps $f:A\\to X$ and $g:B\\to X$, the map\n\\begin{equation*}\n\\fibtriangle : \\hom_X(f,g) \\to \\prd{x:X}\\fib{f}{x}\\to \\fib{g}{x}\n\\end{equation*}\ngiven by $\\lam{(h,H)}{x}{(a,p)}(h(a),\\ct{H(a)^{-1}}{p})$ is an equivalence.\n\\end{cor}\n\n\\begin{thm}\\label{thm:fam_proj}\nFor any small type $A:\\UU$ there is an equivalence\n\\begin{equation*}\n\\eqv{(A\\to \\UU)}{\\Big(\\sm{X:\\UU} X\\to A\\Big)}.\n\\end{equation*}\n\\end{thm}\n\n\\begin{proof}\nNote that we have the function\n\\begin{equation*}\n\\varphi :\\lam{B} \\Big(\\sm{x:A}B(x),\\proj 1\\Big) : (A\\to \\UU)\\to \\Big(\\sm{X:\\UU}X\\to A\\Big).\n\\end{equation*}\nThe fiber of this map at $(X,f)$ is by univalence and function extensionality equivalent to the type\n\\begin{equation*}\n\\sm{B:A\\to \\UU}{e:\\eqv{(\\sm{x:A}B(x))}{X}} \\proj 1\\htpy f\\circ e.\n\\end{equation*}\nBy \\cref{cor:fib_triangle} this type is equivalent to the type\n\\begin{equation*}\n\\sm{B:A\\to \\UU}\\prd{a:A} \\eqv{B(a)}{\\fib{f}{a}},\n\\end{equation*}\nand by `type theoretic choice', which was established in \\cref{thm:choice}, this type is equivalent to\n\\begin{equation*}\n\\prd{a:A}\\sm{X:\\UU}\\eqv{X}{\\fib{f}{a}}.\n\\end{equation*}\nWe conclude that the fiber of $\\varphi$ at $(X,f)$ is equivalent to the type $\\esssmall(f)$. However, since $f:X\\to A$ is a map between small types it is essentially small. Moreover, since being essentially small is a proposition by \\cref{lem:isprop_ess_small}, it follows that $\\fib{\\varphi}{(X,f)}$ is contractible for every $f:X\\to A$. In other words, $\\varphi$ is a contractible map, and therefore it is an equivalence.\n\\end{proof}\n\n\\begin{rmk}\nThe inverse of the map\n\\begin{equation*}\n\\varphi : (A\\to \\UU)\\to \\Big(\\sm{X:\\UU}X\\to A\\Big).\n\\end{equation*}\nconstructed in \\cref{thm:fam_proj} is the map $(X,f)\\mapsto \\fibf{f}$.\n\\end{rmk}\n\n\\section{The object classifier}\n\n\\begin{defn}\nLet $p:E\\to B$ and $p':E'\\to B'$ be maps. A morphism from $p'$ to $p$ is a triple $(f,g,H)$ consisting of maps $f:B'\\to B$ and $g:E'\\to E$ and a homotophy $H:f\\circ p'\\htpy p\\circ g$ witnessing that the square\n\\begin{equation}\\label{eq:morphism_arrow}\n\\begin{tikzcd}\nE' \\arrow[r,\"g\"] \\arrow[d,swap,\"p'\"] & E \\arrow[d,\"p\"] \\\\\nB' \\arrow[r,\"f\"] & B\n\\end{tikzcd}\n\\end{equation}\ncommutes. We write $\\mathsf{hom}(p',p)$ for the type of such triples $(f,g,H)$, and sometimes we write $\\mathsf{hom}_f(p',p)$ for the type of pairs $(g,H)$. A morphism $(f,g,H)$ is said to be \\define{cartesian} if the square in \\cref{eq:morphism_arrow} is cartesian. We write $\\mathsf{cart}(p',p)$ for the type of cartesian morphisms from $p'$ to $p$, and we write $\\mathsf{cart}_f(p',p)$ for the type of triples $(g,H,t)$ for the type of triples, where $t:\\ispullback(p',g,H)$. \n\\end{defn}\n\n\\begin{defn}\\label{defn:object_classifier}\nA morphism $p:E\\to B$ is said to be an \\define{object classifier} if the type $\\mathsf{cart}(p',p)$ is a proposition for each $p':E'\\to B'$. If $p:E\\to B$ is an object classifier, we also write \n\\begin{equation*}\n\\mathsf{is\\usc{}classified}(p')\\defeq \\mathsf{hom}(p',p).\n\\end{equation*}\n\\end{defn}\n\nOur goal in this section is to show that a univalent universe is an object classifier. \n\n\\begin{prp}\\label{thm:pb_fibequiv_complete}\nLet $\\alpha:I\\to J$ be a map, and let $A:I\\to\\UU$ and $B:J\\to\\UU$ be type families.\nThen the map\n\\begin{equation*}\n\\Big(\\prd{i:I}A_i\\to B_{\\alpha(i)}\\Big) \\to \\mathsf{hom}_\\alpha(\\proj 1^A,\\proj 1^B)\n\\end{equation*}\ngiven by $\\lam{f}(\\total[\\alpha]{f},\\lam{(x,y)}\\refl{x})$ is an equivalence. Furthermore, the map\n\\begin{equation*}\n\\Big(\\prd{i:I}A_i\\eqvsym B_{\\alpha(i)}\\Big) \\to \\mathsf{cart}_\\alpha(\\proj 1^A,\\proj 1^B)\n\\end{equation*}\ngiven by $\\lam{e}(\\total[\\alpha]{e},\\lam{(x,y)}\\refl{x},t)$ where $t$ is the term constructed in \\cref{thm:pb_fibequiv}, is an equivalence. \n\\end{prp}\n\n\\begin{proof}\nWe have the equivalences\n\\begin{align*}\n\\prd{i:I}A_i\\to B_{\\alpha(i)} & \\eqvsym \\prd{i:I}{a:A_i}\\sm{j:J}{\\gamma : \\alpha(i)=j} B_j \\\\\n& \\eqvsym \\prd{i:I}{a:A_i}\\sm{j:J}{b:B_j}\\alpha(i)=j \\\\\n& \\eqvsym \\prd{(i,a):\\sm{i:I}A_i}\\sm{(j,b):\\sm{j:J}B_j}\\alpha(i)=j \\\\\n& \\eqvsym \\sm{f:\\big(\\sm{i:I}A_i\\big)\\to\\big(\\sm{j:J}B_j\\big)}\\alpha\\circ \\proj 1^A\\htpy \\proj 1^B\\circ f \\\\\n& \\jdeq \\mathsf{hom}_\\alpha(\\proj 1^A,\\proj 1^B).\n\\end{align*}\nIt is easy to check that this composite is the asserted map. The second claim follows from \\cref{thm:pb_fibequiv}.\n\\end{proof}\n\n\\begin{cor}\\label{cor:sq_fib}\nConsider a diagram of the form\n\\begin{equation*}\n\\begin{tikzcd}\nA \\arrow[d,swap,\"f\"] & B \\arrow[d,\"g\"] \\\\\nI \\arrow[r,swap,\"\\alpha\"] & J.\n\\end{tikzcd}\n\\end{equation*}\nThen the map\n\\begin{equation*}\n\\mathsf{hom}_\\alpha(f,g)\\to \\Big(\\prd{i:I}\\fib{f}{i}\\to\\fib{g}{\\alpha(i)}\\Big)\n\\end{equation*}\ngiven by $\\lam{(h,H)}{i}{(a,p)}(h(a),\\ct{H(a)^{-1}}{\\ap{\\alpha}{p}})$ is an equivalence.\n\\end{cor}\n\n\\begin{thm}\\label{thm:classifier}\nLet $f:A\\to B$ be a map, and let $\\UU$ be a univalent universe with universal family $\\mathrm{El}$ over $\\UU$. Then there is an equivalence\n\\begin{equation*}\n\\eqv{\\esssmall(f)}{\\mathsf{cart}(f,\\proj 1^{\\mathrm{El}})}.\n\\end{equation*}\nIn particular, the type $\\mathsf{cart}(f,\\proj 1^{\\mathrm{El}})$ is a proposition for each map $f$, so the universe is an object classifier in the sense of \\cref{defn:object_classifier}.\n\\end{thm}\n\n\\begin{proof}\nFrom \\cref{cor:sq_fib} we obtain that the type of pairs $(\\tilde{F},H)$ is equivalent to the type of fiberwise transformations\n\\begin{equation*}\n\\prd{b:B}\\fib{f}{b}\\to F(b).\n\\end{equation*}\nBy \\cref{cor:pb_fibequiv} the square is a pullback square if and only if the induced map\n\\begin{equation*}\n\\prd{b:B}\\fib{f}{b}\\to F(b)\n\\end{equation*}\nis a fiberwise equivalence. Thus the data $(F,\\tilde{F},H,pb)$ is equivalent to the type of pairs $(F,e)$ where $e$ is a fiberwise equivalence from $\\fibf{f}$ to $F$. By \\cref{thm:choice} the type of pairs $(F,e)$ is equivalent to the type $\\esssmall(f)$. \n\\end{proof}\n\n\\begin{rmk}\nFor any type $A$ (not necessarily small), and any $B:A\\to \\UU$, the square\\index{Sigma-type@{$\\Sigma$-type}!as pullback of universal family|textit}\n\\begin{equation*}\n\\begin{tikzcd}[column sep=6em]\n\\sm{x:A}B(x) \\arrow[d,swap,\"\\proj 1\"] \\arrow[r,\"{\\lam{(x,y)}(B(x),y)}\"] & \\sm{X:\\UU}X \\arrow[d,\"\\proj 1\"] \\\\\nA \\arrow[r,swap,\"B\"] & \\UU\n\\end{tikzcd}\n\\end{equation*}\nis a pullback square. Therefore it follows that for any family $B:A\\to\\UU$ of small types, the projection map $\\proj 1:\\sm{x:A}B(x)\\to A$ is an essentially small map.\nTo see that the claim is a direct consequence of \\cref{thm:pb_fibequiv} we write the asserted square in its rudimentary form:\n\\begin{equation*}\n\\begin{tikzcd}[column sep=6em]\n\\sm{x:A}\\mathrm{El}(B(x)) \\arrow[d,swap,\"\\proj 1\"] \\arrow[r,\"{\\lam{(x,y)}(B(x),y)}\"] & \\sm{X:\\UU}\\mathrm{El}(X) \\arrow[d,\"\\proj 1\"] \\\\\nA \\arrow[r,swap,\"B\"] & \\UU.\n\\end{tikzcd}\n\\end{equation*}\n\\end{rmk}\n\nIn the following theorem we show that a type is locally small if and only if its diagonal is classified by $\\UU$.\n\n\\begin{thm}\nLet $A$ be a type. The following are equivalent:\n\\begin{enumerate}\n\\item $A$ is locally small.\\index{locally small|textit}\n\\item The diagonal $\\delta_A : A\\to A\\times A$ is classified by $\\UU$.\\index{diagonal!of a type|textit}\n\\end{enumerate}\n\\end{thm}\n\n\\begin{proof}\nThe identity type $x=y$ is the fiber of $\\delta_A$ at $(x,y):A\\times A$. Therefore it follows that $A$ is locally small if and only if the diagonal $\\delta_A$ is essentially small.\nNow the result follows from \\cref{thm:classifier}.\n\\end{proof}\n", "meta": {"hexsha": "4125c93ce81d2ef607348948d926dcf6e4f1ee5d", "size": 59494, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "univalent.tex", "max_stars_repo_name": "EgbertRijke/dissertation", "max_stars_repo_head_hexsha": "f2c087ba8983205d3dd336bbc194be5b7218c2c5", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-07-06T10:37:12.000Z", "max_stars_repo_stars_event_max_datetime": "2018-07-06T10:37:12.000Z", "max_issues_repo_path": "univalent.tex", "max_issues_repo_name": "EgbertRijke/dissertation", "max_issues_repo_head_hexsha": "f2c087ba8983205d3dd336bbc194be5b7218c2c5", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "univalent.tex", "max_forks_repo_name": "EgbertRijke/dissertation", "max_forks_repo_head_hexsha": "f2c087ba8983205d3dd336bbc194be5b7218c2c5", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 44.431665422, "max_line_length": 480, "alphanum_fraction": 0.682589841, "num_tokens": 21682, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.546738151984614, "lm_q2_score": 0.6370307875894138, "lm_q1q2_score": 0.34828903556393925}} {"text": "\\section{Protocol Parameters}\n\\label{sec:protocol-parameters}\n\n\\subsection{Updatable Protocol Parameters}\n\\label{sec:updatable-protocol-parameters}\n\nThe Shelley protocol parameters are listed in Figure~\\ref{fig:defs:protocol-parameters}.\nSome of the Shelley protocol parameters are common to the Byron era,\nspecifically, the common ones are $\\var{a}$, $\\var{b}$, $\\var{maxTxSize}$, and\n$\\var{maxHeaderSize}$ (see the document~\\cite{byron_ledger_spec}).\n\nThe type $\\Ppm$ represents the names of the protocol parameters,\nand $\\mathsf{T_{ppm}}$ is the type of the protocol parameter $\\var{ppm}$.\nThe type $\\PParams$ is a finite map containing all the Shelley parameters,\nindexed by their names.\nWe will explain the significance of each parameter as it comes up in\nthe calculations used in transition rules.\nThe type $\\PParamsUpdate$ is similar to $\\PParams$, but is\na partial mapping of the protocol parameters. It is used in the update\nsystem explained in Section~\\ref{sec:update}.\n\nThe type $\\Coin$ is defined as an alias for the integers.\nNegative values will not be allowed in UTxO outputs or reward accounts,\nand $\\Z$ is only chosen over $\\N$ for its additive inverses.\n\nSome helper functions are defined in Figure~\\ref{fig:defs:protocol-parameters-helpers}.\nThe $\\fun{minfee}$ function calculates the minimum fee that must be paid by a transaction.\nThis value depends on the protocol parameters and the size of the transaction.\n\nTwo time related types are introduced, $\\Epoch$ and $\\type{Duration}$.\nA $\\type{Duration}$ is the difference between two slots, as given by $\\slotminus{}{}$.\n\nLastly, there are two functions, $\\fun{epoch}$ and $\\fun{firstSlot}$ for converting\nbetween epochs and slots and one function $\\fun{kesPeriod}$ for getting the cycle of a slot.\nNote that $\\Slot$ is an abstract type, while the constants are integers.\nWe use multiplication and division symbols on these distinct types\nwithout being explicit about the types and conversion.\n\n\\begin{figure*}[htb]\n \\emph{Abstract types}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}lr}\n \\var{p} & \\Ppm & \\text{protocol parameter}\\\\\n \\var{dur} & \\Duration & \\text{difference between slots}\\\\\n \\var{epoch} & \\Epoch & \\text{epoch} \\\\\n \\var{kesPeriod} & \\KESPeriod & \\text{KES period} \\\\\n \\end{array}\n \\end{equation*}\n %\n \\emph{Derived types}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}l@{\\qquad=\\qquad}lr}\n \\var{pp}\n & \\PParams\n & \\Ppm \\to \\mathsf{T_{ppm}}\n & \\text{protocol parameters}\n \\\\\n \\var{ppup}\n & \\PParamsUpdate\n & \\Ppm \\mapsto \\mathsf{T_{ppm}}\n & \\text{protocol parameter update}\n \\\\\n \\var{coin}\n & \\Coin\n & \\Z\n & \\text{unit of value}\n \\\\\n \\var{pv}\n & \\ProtVer\n & \\N\\times\\N\n & \\text{protocol version}\n \\end{array}\n \\end{equation*}\n %\n \\emph{Protocol Parameters}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}lr}\n \\var{a} \\mapsto \\Z & \\PParams & \\text{min fee factor}\\\\\n \\var{b} \\mapsto \\Z & \\PParams & \\text{min fee constant}\\\\\n \\var{maxBlockSize} \\mapsto \\N & \\PParams & \\text{max block body size}\\\\\n \\var{maxTxSize} \\mapsto \\N & \\PParams & \\text{max transaction size}\\\\\n \\var{maxHeaderSize} \\mapsto \\N & \\PParams & \\text{max block header size}\\\\\n \\var{keyDecayRate} \\mapsto \\nonnegReals & \\PParams & \\text{stake credential decay rate}\\\\\n \\var{poolDeposit} \\mapsto \\Coin & \\PParams & \\text{stake pool deposit}\\\\\n \\var{E_{max}} \\mapsto \\Epoch & \\PParams & \\text{epoch bound on pool retirement}\\\\\n \\var{n_{opt}} \\mapsto \\Npos & \\PParams & \\text{desired number of pools}\\\\\n \\var{a_0} \\mapsto \\posReals & \\PParams & \\text{pool influence}\\\\\n \\tau \\mapsto \\unitInterval & \\PParams & \\text{treasury expansion}\\\\\n \\rho \\mapsto \\unitInterval & \\PParams & \\text{monetary expansion}\\\\\n \\var{d} \\mapsto \\{0,~0.1,~0.2,~\\ldots,~1\\} & \\PParams & \\text{decentralization parameter}\\\\\n \\var{extraEntropy} \\mapsto \\Seed & \\PParams & \\text{extra entropy}\\\\\n \\var{pv} \\mapsto \\ProtVer & \\PParams & \\text{protocol version}\\\\\n \\var{minUTxOValue} \\mapsto \\Coin & \\PParams & \\text{minimum allowed value of a new \\TxOut}\\\\\n \\var{minPoolCost} \\mapsto \\Coin & \\PParams & \\text{minimum allowed stake pool cost}\\\\\n \\end{array}\n \\end{equation*}\n %\n \\emph{Accessor Functions}\n %\n \\begin{center}\n \\fun{a},\n \\fun{b},\n \\fun{maxBlockSize},\n \\fun{maxTxSize},\n \\fun{maxHeaderSize},\n \\fun{keyDeposit},\n \\fun{poolDeposit},\n \\fun{emax},\n \\fun{nopt},\n \\fun{influence},\n \\fun{tau},\n \\fun{rho},\n \\fun{d},\n \\fun{extraEntropy},\n \\fun{pv},\n \\fun{minUTxOValue},\n \\fun{minPoolCost}\n \\end{center}\n %\n \\emph{Abstract Functions}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}lr}\n (\\slotminus{}{}) & \\Slot \\to \\Slot \\to \\Duration\n & \\text{duration between slots}\n \\end{array}\n \\end{equation*}\n %\n \\caption{Definitions Used in Protocol Parameters}\n \\label{fig:defs:protocol-parameters}\n\\end{figure*}\n\n\\subsection{Global Constants}\n\\label{sec:global-constants}\n\nIn additon to the updatable protocol parameters defined in\nSection~\\ref{sec:updatable-protocol-parameters},\nthere are ten parameters which cannot be changed by the update\nsystem in Section~\\ref{sec:update}.\nWe call these the global constants, as changing these values can only\nbe done by updating the software, i.e. a soft or a hard fork.\nFor the software update mechanism, see Section~\\ref{sec:software-updates}.\n\nThe constants $\\SlotsPerEpoch$ and $\\SlotsPerKESPeriod$\nrepresent the number of slots in an epoch/KES period (for a brief explanation\nof a KES period, see Section \\ref{sec:crypto-primitives-shelley}).\nThe constants $\\StabilityWindow$ and $\\RandomnessStabilisationWindow$ concern the chain stability.\nThe maximum number of time a KES key can be evolved before a pool operator\nmust create a new operational certificate is given by $\\MaxKESEvo$.\n\\textbf{Note that if } $\\MaxKESEvo$\n\\textbf{is changed, the KES signature format may have to change as well.}\n\nThe constant $\\Quorum$ determines the quorum amount needed for votes on the\nprotocol parameter updates and the application version updates.\n\nThe constant $\\MaxMajorPV$ provides a mechanism for halting outdated nodes.\nOnce the major component of the protocol version in the protocol parameters\nexceeds this value, every subsequent block is invalid.\nSee Figures~\\ref{fig:funcs:chain-helper} and~\\ref{fig:rules:chain}.\n\nThe constant $\\MaxLovelaceSupply$ gives the total number of lovelace in the system,\nwhich is used in the reward calculation.\nIt is always equal to the sum of the values in the UTxO, plus the sum of the\nvalues in the reward accounts, plus the deposit pot, plus the fee pot,\nplus the treasury and the reserves.\n\nThe constant $\\ActiveSlotCoeff$ is the value $f$ from the\nPraos paper \\cite{ouroboros_praos}.\n\nLastly, $\\NetworkId$ determines what network, either mainnet or testnet, is expected.\nThis value will also appear inside every address, and transactions\ncontaining addresses with an unexpected network ID are rejected.\n\n\\begin{figure*}[htb]\n \\emph{Global Constants}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}lr}\n \\SlotsPerEpoch & \\N & \\text{- slots per epoch} \\\\\n \\SlotsPerKESPeriod & \\N & \\text{- slots per KES period} \\\\\n \\StabilityWindow & \\Duration &\n \\begin{array}{r}\n \\text{- window size for chain growth} \\\\\n \\text{guarantees, see}\\text{ in \\cite{ouroboros_praos}}\n \\end{array} \\\\\n \\RandomnessStabilisationWindow & \\Duration &\n \\begin{array}{r}\n \\text{- duration needed for epoch}\\\\\n \\text{nonce stabilization}\\\\\n \\end{array} \\\\\n \\MaxKESEvo & \\N & \\text{- maximum KES key evolutions}\\\\\n \\Quorum & \\N & \\text{- quorum for update system votes}\\\\\n \\MaxMajorPV & \\N & \\text{- all blocks are invalid after this value}\\\\\n \\MaxLovelaceSupply & \\Coin & \\text{- total lovelace in the system}\\\\\n \\ActiveSlotCoeff & (0, 1] & \\text{ - }f\\text{ in \\cite{ouroboros_praos}}\\\\\n \\NetworkId & \\Network & \\text{- the network, mainnet or testnet}\\\\\n \\end{array}\n \\end{equation*}\n %\n \\caption{Global Constants}\n \\label{fig:defs:global-constants}\n\\end{figure*}\n\n\\begin{figure*}[htb]\n \\emph{Helper Functions}\n %\n \\begin{align*}\n \\fun{minfee} & \\in \\PParams \\to \\Tx \\to \\Coin & \\text{minimum fee}\\\\\n \\fun{minfee} & ~\\var{pp}~\\var{tx} =\n (\\fun{a}~\\var{pp}) \\cdot \\fun{txSize}~\\var{tx} + (\\fun{b}~\\var{pp})\n \\\\\n \\\\\n \\fun{epoch} & \\in ~ \\Slot \\to \\Epoch & \\text{epoch of a slot}\n \\\\\n \\fun{epoch} & ~\\var{slot} = \\var{slot}~/~\\SlotsPerEpoch\n \\\\\n \\\\\n \\fun{firstSlot} & \\in ~ \\Epoch \\to \\Slot\n & \\text{first slot of an epoch}\n \\\\\n \\fun{firstSlot} & ~\\var{e} = \\var{e}~\\cdot~\\SlotsPerEpoch\n \\\\\n \\\\\n \\fun{kesPeriod} & \\in ~ \\Slot \\to \\KESPeriod & \\text{KES period of a slot}\n \\\\\n \\fun{kesPeriod} & ~\\var{slot} = \\var{slot}~/~\\SlotsPerKESPeriod\n \\end{align*}\n %\n \\caption{Helper functions for the Protocol Parameters}\n \\label{fig:defs:protocol-parameters-helpers}\n\\end{figure*}\n\n\\clearpage\n", "meta": {"hexsha": "776c72b80ce4c7560b1b7c9a89453cfa862a2261", "size": 9202, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "shelley/chain-and-ledger/formal-spec/protocol-parameters.tex", "max_stars_repo_name": "ConsultD/cardano-ledger-specs", "max_stars_repo_head_hexsha": "76365a530d064c9c74fbc93e6c4f751ee18004c3", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "shelley/chain-and-ledger/formal-spec/protocol-parameters.tex", "max_issues_repo_name": "ConsultD/cardano-ledger-specs", "max_issues_repo_head_hexsha": "76365a530d064c9c74fbc93e6c4f751ee18004c3", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "shelley/chain-and-ledger/formal-spec/protocol-parameters.tex", "max_forks_repo_name": "ConsultD/cardano-ledger-specs", "max_forks_repo_head_hexsha": "76365a530d064c9c74fbc93e6c4f751ee18004c3", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.8270042194, "max_line_length": 100, "alphanum_fraction": 0.6689850033, "num_tokens": 2756, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7490872131147276, "lm_q2_score": 0.46490157137338844, "lm_q1q2_score": 0.34825182247274916}} {"text": "\\section{Power Reactor Terminology}\n\\begin{labeling}\n \\item [\\underline{Coolant}:] Material used to remove heat from core, to\n heat water, to push a turbine, etc.\n \\item [\\underline{Steam or Coolant Loops}:] Number of heat transfer mechanisms.\n Must be at least 1.\n \\item [\\underline{Moderator}:] Material that slows down neutrons. Not all\n reactors are moderated. Sometimes this is the same as the coolant. Typically\n low-Z materials.\n \\item [\\underline{Neutron Energies}:]\n \\begin{labeling}\n \\item [\\underline{Fast}:] At or near fission neutron birth energies $> 1$ MeV\n \\item [\\underline{Epithermal}:] Neutrons in the process of slowing down\n $[10^{-5}, 1)$ MeV.\n \\item [\\underline{Thermal}:] Neutrons in thermal equilibrium with the moderator,\n $< 10^{-5}$ MeV. Maxwellian distribution\n % FIXME Add image of maxwell distribution for linaer and log.\n \\end{labeling}\n \\item [\\underline{Fuel Production}:]\n \\begin{labeling}\n \\item [\\underline{Burner}:] If destroies more atomes of fuel at discharge.\n \\item [\\underline{Breeder}:] If creates more atoms of fuel at discharge\n \\end{labeling}\n\\end{labeling}\n\n% FIXME Add single loop core image\n% FIXME Add two loop core image\n\n\\section{Neutron Multiplication}\nReactors are controlled by how many neutron are in teh core at any given time.\nThis is because neutrons cause fission, which releases energy and more neutrons.\n\n\\section{Prompt Neutron Multiplication}\n``Prompt'' means that neutrons born in a given generation are destroyed or lost before the\nnext generation, i.e. instantly.\n\nChange in \\# neutrons = Sources - Losses\n\nSymbolically,\n\\[ \\frac{dN}{dt} = \\nu \\Sigma_f\\phi - \\Sigma_a\\phi -DB^2\\phi \\]\n\nLet's go through this term-by-term.\n\nNote: The book uses $\\Phi$ for $\\phi$\n\n\\section{Difussion Theory Terms}\n\\begin{labeling}\n \\item [$N$:] Neutron density [n/cm$^3$]\n \\item [$t$:] Time [sec]\n \\item [$\\nu$:] Number of neutrons born per fission, 2 - 5 typically [unitless]\n \\item [$\\Sigma_f$:] macroscopic fission cross section, [1/cm]\n \\item [$\\phi$:] neutron flux, [n/cm$^2$ s]\n \\item [$\\Sigma_a$:] Macroscopic absorption cross-section\n \\item [$D$:] Diffission coefficient, [cm], related to neutron mean free path\n $\\lambda_{\\mathrm{tr}}$ by\n \\[ D = \\frac{\\lambda_{\\mathrm{tr}}}{3} = \\frac{1}{3\\Sigma_{\\mathrm{tr}}} = \\frac{1}{3\\Sigma_s(1 -\\frac{2}{3A})} \\]\n \\item [$B$:] Buckling [1/cm]\n \\begin{itemize}\n \\item \\mathbf{Geometric Buckling} $B_g$ depends on the shape of the core, see table 4-2.\n \\item \\mathbf{Material Buckling} $B_m$'s is derived from the criticality constraint\n such that $B_m^2 = \\frac{\\nu\\Sigma_f - \\Sigma_a}{D}$ for critical systems.\n \\end{itemize}\n\\end{labeling}\n\nSo, $\\frac{dN}{dt}$ is the growth rate.\n\n$\\nu\\Sigma_f\\phi$ is the rate of neutron birth.\n\n$\\Sigma_a\\phi$ is the rate at which neutrons are destroyed.\n\n$DB^2\\phi$ is the \\textit{Leakage}, or the rate at which neutrons leave the system entirely.\n\n\\underline{On the board:} Units for each term. Ensure that they are the same.\n", "meta": {"hexsha": "285a536b09bcb88057093c823a8e69c650034dbd", "size": 3226, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "notes/n03.tex", "max_stars_repo_name": "scopatz/rxps", "max_stars_repo_head_hexsha": "bef0045f12215ebe0ac15f9e944047eaefb0ec36", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notes/n03.tex", "max_issues_repo_name": "scopatz/rxps", "max_issues_repo_head_hexsha": "bef0045f12215ebe0ac15f9e944047eaefb0ec36", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notes/n03.tex", "max_forks_repo_name": "scopatz/rxps", "max_forks_repo_head_hexsha": "bef0045f12215ebe0ac15f9e944047eaefb0ec36", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.5945945946, "max_line_length": 122, "alphanum_fraction": 0.6602603844, "num_tokens": 912, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961016, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.34815112699611533}} {"text": "\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%\n\\documentclass[12pt]{amsart}\n\\usepackage{amssymb, amstext, amscd, amsmath}\n% \\usepackage[notcite,notref]{showkeys}\n% \\input{amsmacros2e}\n% %%%%%%%%%%%%%%%%%%%%%%% begin macros\n%%%%%%%%%%%%%%%%%%%%%%\n%\n% Cites in bold rather than roman.\n\\makeatletter\n\\def\\@cite#1#2{{\\m@th\\upshape\\bfseries%\n[{#1\\if@tempswa{\\m@th\\upshape\\mdseries, #2}\\fi}]}}\n\\makeatother\n%\n% Change first-level `enumerate' numbering\n%style from arabic to roman numeral.\n%\n\\renewcommand{\\labelenumi}{(\\roman{enumi}) }\n%\n% normal style theorems,\n%numbered within the section\n\\theoremstyle{plain}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{cor}[thm]{Corollary}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{lem}[thm]{Lemma}\n%\n% Theorem style with roman text, numbered within section\n\\theoremstyle{definition}\n\\newtheorem{rem}[thm]{Remark}\n\\newtheorem{defn}[thm]{Definition}\n\\newtheorem{note}[thm]{Note}\n\\newtheorem{eg}[thm]{Example}\n%\n\\newcommand{\\Prf}{\\noindent\\textbf{Proof.\\ }}\n\\newcommand{\\bx}{\\hfill$\\blacksquare$\\medbreak}\n\\newcommand{\\upbx}{\\vspace{-2.5\\baselineskip}\\newline\\hbox{}%\n\\hfill$\\blacksquare$\\newline\\medbreak}\n\\newcommand{\\eqbx}[1]{\\medbreak\\hfill\\(\\displaystyle #1\\)\\bx}\n% for proofs ending in a one line equation\n% Useful shortforms\n\\newcommand{\\BH}{{\\B(\\H)}}\n\\newcommand{\\bsl}{\\setminus}\n\\newcommand{\\ca}{\\mathrm{C}^*}\n\\newcommand{\\cstar}{\\mathrm{C}^*}\n\\newcommand{\\cenv}{\\mathrm{C}^*_{\\text{env}}}\n\\newcommand{\\rip}{\\rangle}\n\\newcommand{\\ol}{\\overline}\n\\newcommand{\\td}{\\widetilde}\n\\newcommand{\\wh}{\\widehat}\n\\newcommand{\\sot}{\\textsc{sot}}\n\\newcommand{\\wot}{\\textsc{wot}}\n\\newcommand{\\wotclos}[1]{\\ol{#1}^{\\textsc{wot}}}\n\n% Blackboard bold letters\n\\newcommand{\\bbA}{{\\mathbb{A}}}\n\\newcommand{\\bbB}{{\\mathbb{B}}}\n\\newcommand{\\bbC}{{\\mathbb{C}}}\n\\newcommand{\\bbF}{{\\mathbb{F}}}\n\\newcommand{\\bbI}{{\\mathbb{I}}}\n\\newcommand{\\bbD}{{\\mathbb{D}}}\n\\newcommand{\\bbN}{{\\mathbb{N}}}\n\\newcommand{\\bbQ}{{\\mathbb{Q}}}\n\\newcommand{\\bbR}{{\\mathbb{R}}}\n\\newcommand{\\bbT}{{\\mathbb{T}}}\n\\newcommand{\\bbZ}{{\\mathbb{Z}}}\n\n% Capital script letters\n \\newcommand{\\A}{{\\mathcal{A}}}\n \\newcommand{\\B}{{\\mathcal{B}}}\n \\newcommand{\\C}{{\\mathcal{C}}}\n \\newcommand{\\D}{{\\mathcal{D}}}\n \\newcommand{\\E}{{\\mathcal{E}}}\n \\newcommand{\\F}{{\\mathcal{F}}}\n \\newcommand{\\G}{{\\mathcal{G}}}\n\\renewcommand{\\H}{{\\mathcal{H}}}\n \\newcommand{\\I}{{\\mathcal{I}}}\n \\newcommand{\\J}{{\\mathcal{J}}}\n \\newcommand{\\K}{{\\mathcal{K}}}\n\\renewcommand{\\L}{{\\mathcal{L}}}\n \\newcommand{\\M}{{\\mathcal{M}}}\n \\newcommand{\\N}{{\\mathcal{N}}}\n\\renewcommand{\\O}{{\\mathcal{O}}}\n\\renewcommand{\\P}{{\\mathcal{P}}}\n \\newcommand{\\Q}{{\\mathcal{Q}}}\n \\newcommand{\\R}{{\\mathcal{R}}}\n%\\renewcommand{\\S}{{\\mathcal{S}}}\n \\newcommand{\\T}{{\\mathcal{T}}}\n \\newcommand{\\U}{{\\mathcal{U}}}\n \\newcommand{\\V}{{\\mathcal{V}}}\n \\newcommand{\\W}{{\\mathcal{W}}}\n \\newcommand{\\X}{{\\mathcal{X}}}\n \\newcommand{\\Y}{{\\mathcal{Y}}}\n \\newcommand{\\Z}{{\\mathcal{Z}}}\n\n%Greek Letters\n\\newcommand{\\eps}{\\varepsilon}\n%\\renewcommand{\\phi}{\\varphi}\n\n%Gothic letters\n\\newcommand{\\fA}{{\\mathfrak{A}}}\n\\newcommand{\\germA}{{\\mathfrak{A}}}\n\\newcommand{\\germB}{{\\mathfrak{B}}}\n\\newcommand{\\fB}{{\\mathfrak{B}}}\n\\newcommand{\\fC}{{\\mathfrak{C}}}\n\\newcommand{\\fD}{{\\mathfrak{D}}}\n\\newcommand{\\fE}{{\\mathfrak{E}}}\n\\newcommand{\\fF}{{\\mathfrak{F}}}\n\\newcommand{\\fG}{{\\mathfrak{G}}}\n\\newcommand{\\fH}{{\\mathfrak{H}}}\n\\newcommand{\\fI}{{\\mathfrak{I}}}\n\\newcommand{\\fJ}{{\\mathfrak{J}}}\n\\newcommand{\\fK}{{\\mathfrak{K}}}\n\\newcommand{\\fL}{{\\mathfrak{L}}}\n\\newcommand{\\fM}{{\\mathfrak{M}}}\n\\newcommand{\\fN}{{\\mathfrak{N}}}\n\\newcommand{\\fO}{{\\mathfrak{O}}}\n\\newcommand{\\fP}{{\\mathfrak{P}}}\n\\newcommand{\\fQ}{{\\mathfrak{Q}}}\n\\newcommand{\\fR}{{\\mathfrak{R}}}\n\\newcommand{\\fS}{{\\mathfrak{S}}}\n\\newcommand{\\fT}{{\\mathfrak{T}}}\n\\newcommand{\\fU}{{\\mathfrak{U}}}\n\\newcommand{\\fV}{{\\mathfrak{V}}}\n\\newcommand{\\fW}{{\\mathfrak{W}}}\n\\newcommand{\\fX}{{\\mathfrak{X}}}\n\\newcommand{\\fY}{{\\mathfrak{Y}}}\n\\newcommand{\\fZ}{{\\mathfrak{Z}}}\n\n%Roman letters for math\n\\newcommand{\\rN}{\\mathrm{N}}\n\n% Text used in equations\n\\newcommand{\\foral}{\\text{ for all }}\n\\newcommand{\\qand}{\\quad\\text{and}\\quad}\n\\newcommand{\\qif}{\\quad\\text{if}\\quad}\n\\newcommand{\\qfor}{\\quad\\text{for}\\quad}\n\\newcommand{\\qforal}{\\quad\\text{for all}\\quad}\n\\newcommand{\\qforsome}{\\quad\\text{for some}\\quad}\n\\newcommand{\\qwhere}{\\quad\\text{where}\\quad}\n\n%Operators\n\\newcommand{\\ad}{\\operatorname{ad}}\n\\newcommand{\\Alg}{\\operatorname{Alg}}\n\\newcommand{\\alg}{\\operatorname{alg}}\n\\newcommand{\\ball}{\\operatorname{ball}}\n\\newcommand{\\dirlim}{\\varinjlim}\n\\newcommand{\\dist}{\\operatorname{dist}}\n\\newcommand{\\dom}{\\operatorname{Dom}}\n\\newcommand{\\frRep}{\\operatorname{Rep}_{\\operatorname{fr}}}\n\\newcommand{\\Id}{\\operatorname{Id}}\n\\newcommand{\\LL}{\\operatorname{L}}\n\\newcommand{\\Lat}{\\operatorname{Lat}}\n\\newcommand{\\nil}{\\operatorname{nil}}\n\\newcommand{\\PI}{\\operatorname{PI}}\n\\newcommand{\\Prim}{\\operatorname{Prim}}\n\\newcommand{\\rad}{\\operatorname{rad}}\n\\newcommand{\\ran}{\\operatorname{Ran}}\n\\renewcommand{\\dom}{\\operatorname{Dom}}\n\\newcommand{\\rank}{\\operatorname{rank}}\n\\newcommand{\\spn}{\\operatorname{span}}\n\\newcommand{\\sumoplus}{\\operatornamewithlimits{\\sum\\oplus}}\n\\newcommand{\\supr}{\\operatorname{sup}}\n\\newcommand{\\Spec}{\\operatorname{Spec}}\n\\newcommand{\\rep}{\\operatorname{rep}}\n\\newcommand{\\ba}{\\backslash}\n\\newcommand{\\om}{\\omega}\n\\newcommand{\\trace}{\\operatorname{trace}}\n\n\n%Commands specific to this paper\n\\def\\bra#1{\\langle #1|}\n\\def\\ket#1{|#1 \\rangle}\n\\def\\one{{\\mathchoice{\\rm 1\\mskip-4mu l}{\\rm 1\\mskip-4mu l}{\\rm 1\\mskip-4.5mu l}{\\rm\n1\\mskip-5mu l}}}\n\\newcommand{\\fgeeplus}{\\bbF^+\\!(G)}\n\\newcommand{\\fgeeplusnot}{\\bbF_+^+\\!(G)}\n\\newcommand{\\inftree}{\\tau_G}\n\\newcommand{\\Lg}{{\\mathfrak{L}}_G}\n\\newcommand{\\Rg}{{\\mathfrak{R}}_G}\n\\newcommand{\\fixed}{\\B(\\K)^\\Phi}\n\\newcommand{\\sumin}{\\sum_{i=1}^n}\n\\newcommand{\\ai}{A_i}\n\\newcommand{\\si}{S_i}\n\\newcommand{\\aistar}{A_i^*}\n\\newcommand{\\sistar}{S_i^*}\n\\newcommand{\\cp}{\\Phi}\n\\newcommand{\\bofk}{\\B(\\K)}\n\\newcommand{\\bofh}{\\B(\\H)}\n\\newcommand{\\rowt}{(T_e)_{e\\in E(G)}}\n\\newcommand{\\rows}{(S_e)_{e\\in E(G)}}\n\\newcommand{\\rowl}{(L_e)_{e\\in E(G)}}\n\\newcommand{\\fnplus}{{\\bbF}_n^+}\n\\newcommand{\\mulam}{\\nu_{\\lambda,i}}\n\\newcommand{\\nulam}{\\nu_{\\lambda,i}}\n\\newcommand{\\ank}{\\fA (n_k)}\n\\newcommand{\\flgee}{\\fL_G}\n\\newcommand{\\frgee}{\\fR_G}\n\\newcommand{\\alphuw}{\\alpha_w^u}\n\\newcommand{\\alphvw}{\\alpha_w^v}\n\\newcommand{\\fomg}{\\bbF_\\omega(G)}\n\\newcommand{\\fomgk}{\\bbF_{\\omega(G)}^{k}}\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%% end of macros\n%%%%%%%%%%%%%%%%%%%\n\n\\begin{document}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%\n\\title[]{A unified framework for graph algebras and quantum causal histories}\n%\\thanks{}\n%\n%\n\\author[D.W. Kribs]{David~W.~Kribs}\n%\n\\address{Department of Mathematics and Statistics, University of\nGuelph, Guelph, Ontario, Canada N1G 2W1} \\address{Institute for\nQuantum Computing, University of Waterloo, Waterloo, ON, CANADA\nN2L 3G1} \\address{Perimeter Institute for Theoretical Physics, 31\nCaroline St. N., Waterloo, ON, CANADA N2L 2Y5}\n%\n%\n\\begin{abstract}\nWe present a mathematical framework that unifies the quantum\ncausal history formalism from theoretical high energy physics and\nthe directed graph operator framework from the theory of operator\nalgebras. The approach involves completely positive maps and\ndirected graphs and leads naturally to a new class of operator\nalgebras.\n\\end{abstract}\n\n\\thanks{2000 {\\it Mathematics Subject Classification.} 46L05, 47L75, 81P68, 83C45.}\n\\thanks{{\\it Key words and phrases.} directed graph, completely positive\nmap, partial isometry, quantum operation, quantum causal\nhistory.}\n%\\date{November, 2004.}\n\\maketitle\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction} \\label{S:intro}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nIn this paper we present a new operator theoretic framework that\nprovides a unified approach for recent studies in theoretical high\nenergy physics and contemporary operator algebra theory. More\nspecifically, this approach involves completely positive maps and\ndirected graphs, and includes the quantum causal history formalism\nfrom recent work towards a quantum theory of gravity on the one\nhand and the graph-operator framework from the theory of directed\ngraph operator algebras on the other. We also define a new class\nof operator algebras that is naturally defined through this\napproach.\n\nThe basic physical properties that a quantum theory of gravity\nmust satisfy motivated F. Markopoulou to invent a formalism called\n``Quantum Causal Histories'' \\cite{HMS,M02,M00C,M00B,M00A}. A\nsecondary goal of this paper is to introduce this formalism to the\nbroader mathematical community. Fundamental examples include\ncausal evolutions of spin networks \\cite{Gupta,MS97} and quantum\ncomputers \\cite{M00C}. The basic definitions have been refined\nthrough a series of papers and now a clean mathematical definition\nis emerging \\cite{HMS}. Mathematically, and somewhat roughly\nspeaking at this point, a {\\it quantum causal history} (QCH) is\ngiven by a directed graph with a finite-dimensional Hilbert space\nfor each vertex and a quantum operation associated with each\ndirected edge. The vertex spaces correspond to events, or\nobservables, within a local history and the quantum operations\nindicate causal links between pairs of related events. As\ndescribed below, the QCH framework incorporates further structure\nmotivated by the characterization of evolution within quantum\nsystems.\n\nOn the other hand, Cuntz and Krieger \\cite{CK} were motivated by a\nproblem in dynamical systems to initiate the study of operator\nalgebras associated with directed graphs. Over the past two\ndecades the study of directed graph operator algebras and related\ntopics has taken on a life of its own and now, it is fair to say,\nplays a central role in the modern theory of operator algebras. We\nmention the articles \\cite{BP} -- \\cite{Spi} as entrance points\ninto the extensive literature on the subject.\n\nIn $\\S~2$ we recall some basic properties of completely positive\nmaps and quantum operations. We define the ``CP -- directed graph\nframework'' and associated $\\ca$-algebras in $\\S~3$, and show how\ngraph algebras fit into the framework. In $\\S~4$ we discuss in\nsome detail the QCH formalism, draw a connection with quantum\ncomputing, and prove a theorem for the QCH $\\ca$-algebras.\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Completely Positive Maps} \\label{S:cpmaps}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nGiven a Hilbert space $\\H$ we let $\\B(\\H)$ be the set of bounded\noperators that act on $\\H$. A {\\it completely positive} (CP) map\nis a linear map $\\Phi : \\B(\\H_1)\\rightarrow\\B(\\H_2)$ such that the\n``ampliated'' maps\n\\[\n \\one_k \\otimes \\Phi : \\M_k \\otimes \\B(\\H_1) \\longrightarrow\n\\M_k \\otimes \\B(\\H_2)\n\\]\nare positive for $k\\geq 1$. (Here $\\M_k$ denotes the set of\n$k\\times k$ complex matrices and $\\one_k$ denotes the identity\noperator, the ``maximally mixed state'', inside $\\M_k$.) For basic\nmathematical properties of CP maps see \\cite{Paulsentext2} and\nphysical motivations see \\cite{Kraustext}.\n\nA fundamental technical device in the study of CP maps is the {\\it\noperator-sum representation} given by the theorem of Choi\n\\cite{Choi} and Kraus \\cite{Kraus}. For every CP map $\\Phi$ on\nfinite-dimensional space, there is a set of {\\it noise operators}\n(or {\\it errors}) $\\{A_i\\}\\subseteq \\B(\\H_1,\\H_2)$ such that\n\\begin{eqnarray}\\label{opsum}\n\\Phi(\\rho) &=& \\sum_i A_i\\, \\rho\\, A_i^* \\quad\\forall\n\\rho\\in\\B(\\H_1).\n\\end{eqnarray}\nThe map is {\\it unital} if also $\\sum_i A_i A_i^* = \\one_{\\H_2}$.\nIt is always possible to choose a family of noise operators with\ncardinality at most $\\dim(\\H_1)\\dim(\\H_2)$. On infinite\ndimensional space not all CP maps have such a form, and when they\ndo the sum in (\\ref{opsum}) converges in the strong operator\ntopology. For brevity, we assume all the CP maps considered here\nhave a representation as in (\\ref{opsum}).\n%The family $\\{A_i\\}$ can be chosen with cardinality\n%$|\\{A_i\\}|\\leq \\dim(\\H_1) \\dim(\\H_2)$ and is of course non-unique.\n%(However, the algebra generated by the $A_i$ is strictly an\n%artifact of $\\Phi$, it is independent of the choice of noise\n%operators \\cite{Kchannel}.)\n\nA {\\it quantum operation} (or {\\it quantum evolution}, or {\\it\nquantum channel}) is a CP map $\\Phi : \\B(\\H_1)\\rightarrow\\B(\\H_2)$\nthat is also trace preserving. When $\\Phi$ is represented as in\n(\\ref{opsum}), trace preservation is equivalent to the identity\n\\begin{eqnarray}\\label{tracepreserve}\n \\sum_i A_i^* A_i = \\one_{\\H_1}.\n\\end{eqnarray}\nThus, a quantum operation $\\Phi$ is a map that satisfies\n(\\ref{opsum}) and (\\ref{tracepreserve}) for some set of operators\n$\\{A_i\\}$. Equivalently, a quantum operation is a CP map such that\nits associated {\\it dual map}, denoted by $\\Phi^\\dagger:\n\\B(\\H_2)\\rightarrow\\B(\\H_1)$, is unital. (Recall that the dual map\nfor a map $\\Phi$ is defined via the equation\n$\\trace(\\rho\\,\\Phi^\\dagger(\\sigma)) =\n\\trace(\\Phi(\\rho)\\,\\sigma)$.)\n\nThe ideal physical examples of quantum operations are {\\it unitary\nmaps} as they characterize evolution of states within a closed\nquantum system. Such a map is of the form $\\Phi(\\rho) = U \\rho\\,\nU^*$ for some unitary operator $U$. When evolution occurs in an\nopen system (i.e., when the system of interest is exposed to an\nexternal environment) quantum operations have the more general\nform given by (\\ref{opsum}) and (\\ref{tracepreserve}). See\n\\cite{Ksurvey} for further discussions and references.\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{The CP -- Directed Graph Framework} \\label{S:cpgraph}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nLet $E=(E^0,E^1,r,s)$ be a (countable) directed graph with\nvertices $x\\in E^0$, directed edges $e\\in E^1$ and range and\nsource maps $r,s: E^1\\rightarrow E^0$ giving the initial ($s(e)$)\nand final ($r(e)$) vertices of an edge $e$. When $e\\in E^1$\nsatisfies $s(e)=x$ and $r(e)=y$, we shall write $e=(x,y)$.\n\nSuppose we have a Hilbert space $\\{\\H(x):x\\in E^0\\}$ for each\nvertex and a family of CP maps $\\Psi=\\{\\Phi_e:e\\in E^1\\}$ with\ndomains and ranges that satisfy\n\\[\n(\\dagger)\\left\\{\n\\begin{array}{lll}\n(i) & \\dom(\\Phi_e) = \\B(\\H(s(e))) & \\mbox{$\\forall\\, e \\in E^1$} \\\\\n(ii) & \\ran(\\Phi_e) \\subseteq \\B(\\H(r(e))) & \\mbox{$\\forall\\, e\\in\nE^1$}\n\\end{array}\\right.\n\\]\n\nGiven such a family of spaces and maps, define the Hilbert space\n$\\H = \\oplus_{x\\in E^0} \\H(x)$ and let $P_x$ be the projection of\n$\\H$ onto $\\H(x)$.\n\n\\begin{defn}\nGiven a directed graph $E$, let $\\{\\H(x):x\\in E^0\\}$ be Hilbert\nspaces and let $\\Psi=\\{\\Phi_e: e\\in E^1\\}$ be a family of CP maps\nthat satisfy $(\\dagger)$. Suppose $\\Phi_e = \\{A_{e,i}:i\\in\\I_e\\}$\nis an operator-sum representation of $\\Phi_e$ for each $e\\in E^1$.\nWe can naturally regard the $A_{e,i}$ as operators acting on $\\H$.\nDefine $\\fA_\\Psi$ to be the $\\ca$-algebra generated by all\noperators $A_{e,i}$ and vertex projections $P_x$; so that,\n\\[\n\\fA_\\Psi = \\ca \\big( \\{ P_x, A_{e,i} : x\\in E^0,\\, e\\in E^1, \\,\ni\\in\\I_e\\}\\big).\n\\]\n\\end{defn}\n\nThe choice of noise operators that represent a given CP map in\n(\\ref{opsum}) is of course not unique. However, as our notation\nsuggests the algebras $\\fA_\\Psi$ are independent of these choices.\n\n\\begin{prop}\nLet $E$ be a directed graph and let $\\Psi= \\{\\Phi_e:e\\in E^1\\}$ be\na family of CP maps that satisfy $(\\dagger)$. Then the algebra\n$\\fA_\\Psi$ is independent of the choice of operators $\\{A_{e,i}\\}$\nthat represent the maps $\\Phi_e$ as in (\\ref{opsum}).\n\\end{prop}\n\n\\Prf Suppose that $\\{A_{e,i}\\}_i$ and $\\{A_{e,j}^\\prime\\}_j$\nrepresent $\\Phi_e$ via equation (\\ref{opsum}). By possibly\nincluding zero operators we may assume the cardinality of these\ntwo sets is the same. Then from the structure theory for CP maps,\nthere is a scalar unitary matrix $U=(u_{ij})$ such that\n\\[\nA_{e,i} = \\sum_j u_{ij} A_{e,j}^\\prime \\quad \\forall i.\n\\]\nIt follows that the algebras generated by\n$\\{P_x,A_{e,i}\\}_{x,e,i}$ and $\\{P_x,A_{e,j}^\\prime\\}_{x,e,j}$\ncoincide, and the result follows.\n \\bx\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Graph Algebras}\\label{sS:graphalg}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe now discuss one of the motivating special cases for this\nframework. Let $E=(E^0,E^1,r,s)$ be a directed graph. Consider\nfamilies of operators $\\{P_x,S_e: x\\in E^0, e\\in E^1\\}$, where the\n$P_x$ are projections and the $S_e$ are partial isometries (or\nequivalently, unitary operators restricted to a subspace), that\nact on the same Hilbert space and satisfy:\n\\[\n(\\ddagger)\\left\\{\n\\begin{array}{lll}\n(i) & S_{e}^{*}S_e = P_{s(e)} & \\mbox{$\\forall\\, e \\in E^1$} \\\\\n(ii) & S_{e}S_e^* \\leq P_{r(e)} & \\mbox{$\\forall\\, e\\in E^1$}\n\\end{array}\\right.\n\\]\nThen the structure of $E$ determines the relations satisfied by\n$\\{P_x,S_e\\}$ in the sense that the initial projection for each\n$S_e$ is equal to the projection for the source vertex of $e$ and\nthe range projection for each $S_e$ is supported on the projection\nfor the range vertex of $e$.\n\nThe relations $(\\ddagger)$ provide the fundamental base case for\ninvestigations into operator algebras associated with directed\ngraphs. In the most general context, a {\\it graph algebra} is an\noperator algebra generated by a family $\\{S_e,P_x\\}$. There are a\nnumber of refinements and generalizations of the formulation\n$(\\ddagger)$. In many instances the $S_e$ are assumed to have\nmutually orthogonal ranges. The projections $P_x$ are typically\nassumed to have mutually orthogonal ranges as well, or sometimes\njust mutually commuting ranges. There are also topological graph\ngeneralizations wherein the vertices and edges are locally compact\nspaces and the range and source maps are continuous maps. However,\nin every setting the motivating case is the same: A Hilbert space\n$\\H(x)$ associated with every vertex $x$ in $E$ and for every\ndirected edge $e=(x,y)$ a partial isometry $S_e$ that maps from\n$\\H(x)$ to $\\H(y)$.\n\nIf $\\{S_e,P_x\\}$ satisfy $(\\ddagger)$, observe that for each\n$e=(x,y)$ the operator $S_e$ defines a unitary from\n$\\H(x)=P_{x}\\H$ into $\\H(y)=P_{y}\\H$ and a unitary CP map\n$\\Phi_e:\\B(\\H(x))\\rightarrow\\B(\\H(y))$ via\n\\[\n\\Phi_e(\\rho) = S_e^\\prime\\, \\rho\\, (S_e^\\prime)^* \\qwhere\nS_e^\\prime = S_e|_{\\H(x)}.\n\\]\nThus the corresponding algebra $\\fA_\\Psi$ defines a graph algebra,\nand so graph algebras form a subclass of the algebras $\\fA_\\Psi$.\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{The Quantum Causal History Formalism} \\label{S:qch}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\nThe mathematical formalism for QCH's has undergone a series of\nrefinements since being introduced in \\cite{M00B}. The\npresentation below is most closely related to the recent\nformulation of Hawkins, Markopoulou and Sahlmann \\cite{HMS}. The\nnomenclature we use is slightly different than \\cite{HMS}, we do\nthis to mesh with the graph algebra terminology. We shall focus on\nthe mathematical aspects and touch on the physical motivations for\nvarious constraints.\n\nTo define a QCH then, we begin with a graph $E=(E^0,E^1,r,s)$,\nwhich may also be interpreted as a partial order when there are no\nloops. This graph represents a causal set wherein the vertices\ncorrespond to a set of local events in the universe and vertices\nlinked by directed edges indicate causal relations between events.\nFrom the postulates of quantum mechanics, events are represented\nby density operators on Hilbert space. Recent work in string\ntheory and loop quantum gravity (see \\cite{M02}) suggests that any\nfinite region of space should contain a finite amount of\ninformation. Thus, each of the event spaces is assumed to be\nfinite-dimensional. Since causality can be interpreted as\ntransferring information from one event to another and because, by\ndefinition, a QCH describes local causality at the quantum level,\na causal relation given by a directed edge $e=(x,y)\\in E^1$\ncorresponds to a quantum operation $\\Phi(x,y) : \\B(\\H(x))\n\\rightarrow \\B(\\H(y))$ between event spaces.\n\nThus, at its mathematical core, a QCH consists of a directed\ngraph, with a finite-dimensional Hilbert space for each vertex,\nand a quantum operation for each directed edge. There are further\nconstraints within a QCH and we discuss them now briefly.\n\nFirst some terminology. Given $x,y\\in E^0$ write $x\\leq y$ when\n$x$ precedes $y$ as an event. In this case there is a\nfuture-directed curve from $x$ to $y$ and this is represented by a\ndirected edge $e=(x,y)\\in E^1$. If $x\\leq y$ or $y\\leq x$ then $x$\nand $y$ are {\\it related} and otherwise they are {\\it spacelike\nseparated} and we use $x\\sim y$ to denote this. A path in $E$\ncorresponds to a {\\it future-directed path} through the events in\nthe history. Such a path is {\\it future (past) inextendible} if\nthere is no event in $E$ which is in the future (past) of the\nentire path. Loops in $E$ correspond to {\\it closed timelike\ncurves}. From the finiteness assumption discussed above, $E$ is\n{\\it locally finite} in the sense that for any $x,y\\in E^0$ there\nare at most finitely many $z\\in E^0$ such that $x\\leq z \\leq y$.\nGiven $x,y\\in E^0$, there is also no generality lost in assuming\nthere is at most one edge $e=(x,y)$ in $E$ from $x$ to $y$. (If\n$s(e)=x=s(f)$ and $r(e)=y=r(f)$ then the operations associated\nwith these edges could be combined to form a single operation that\nencodes the relevant causal structure from event $x$ to event\n$y$.)\n\nAn {\\it acausal set} $\\xi\\subseteq E^0$ is defined by the property\nthat $x\\sim y$ whenever $x,y\\in\\xi$. Such a set is a {\\it complete\nfuture} for an event $x$ if $\\xi$ intersects any future\ninextendible future-directed path that starts at $x$. A {\\it\ncomplete past} is defined analogously. The composite state space\nfor $x\\sim y$ (the physical existence of which is guaranteed by\nquantum mechanics) is $\\H(\\{x,y\\}) = \\H(x) \\otimes \\H(y)$ and more\ngenerally $\\H(\\xi) = \\otimes_{x\\in\\xi} \\H(x)$. For $x\\in E^0$\nwrite $\\A(x)$ for the matrix algebra $\\B(\\H(x))$ and similarly\ndefine $\\A(\\xi)=\\otimes_{x\\in\\xi}\\A(x)$ for a set $\\xi\\subseteq\nE^0$. Given an acausal set $\\xi$ and an event $x\\in \\xi$, there is\na natural unital embedding $\\iota_x: \\A(x)\\hookrightarrow\\A(\\xi)$\nand we shall write $\\A(x)\\subseteq\\A(\\xi)$.\n\nIf $\\xi$ and $\\zeta$ are acausal sets such that $\\xi$ is a\ncomplete past for $\\zeta$ and $\\zeta$ is a complete future for\n$\\xi$, then we write $\\xi \\preceq \\zeta$ and say that\n$(\\xi,\\zeta)$ form a {\\it complete pair}. Such a pair represents\nan evolution in a closed quantum system, hence woven into the\nfabric of the QCH there should be a unitary operator\n$U(\\xi,\\zeta): \\H(\\xi) \\rightarrow \\H(\\zeta)$. Such an operator\ndetermines a unitary map (an isomorphism) $\\Phi(\\xi,\\zeta):\n\\A(\\xi)\\rightarrow\\A(\\zeta)$ via\n\\[\n\\Phi(\\xi,\\zeta)(\\rho) = U(\\xi,\\zeta)\\, \\rho\\, U(\\xi,\\zeta)^*\n\\quad\\forall \\,\\rho\\in\\A(\\xi).\n\\]\nNote the restriction of $\\Phi(\\xi,\\zeta)$ (respectively\n$\\Phi(\\xi,\\zeta)^\\dagger$) to $\\A(x)\\subseteq\\A(\\xi)$ for\n$x\\in\\xi$ (respectively $\\A(z)\\subseteq\\A(\\zeta)$ for $z\\in\\zeta$)\nis a $\\ast$-homomorphism. This gives the structure of a QCH at the\nglobal level, but does not indicate how the isomorphisms\n$\\Phi(\\xi,\\zeta)$ should depend on the individual causal relations\nbetween events in $\\xi$ and $\\zeta$. This is the role played by\nthe operations $\\Phi(x,y)$ on individual edges.\n\nWe now give a precise mathematical definition of a QCH. We note\nthat the maps in the definition below have directions reversed\nfrom the presentation in \\cite{HMS}. Here we take the dual\napproach so the ``directions'' of the maps are in line with the\ngraph structure. Recall that if $\\iota_A: \\A_1\\rightarrow\\A_2$ and\n$\\iota_B: \\B_1\\rightarrow\\B_2$ are $\\ast$-monomorphisms and $\\Psi:\n\\A_2\\rightarrow\\B_2$ is a map, then the {\\it reduction} of $\\Psi$\nto $\\A_1\\mapsto\\B_1$ is the map $\\Phi = \\iota_B^\\dagger \\circ \\Psi\n\\circ i_A$.\n\n\\begin{defn}\nA {\\it quantum causal history} consists of a directed graph $E$\nwith a Hilbert space $\\{\\H(x): x\\in E^0\\}$ for each event and a\nquantum operation $\\Phi (x,y) : \\A(x)\\rightarrow\\A(y)$ for each\npair of related events $x\\leq y$ such that the following axioms\nare satisfied:\n\n$(i)$ (Extension) For all $y\\in E^0$ and $\\zeta\\subseteq E^0$ a\ncomplete future of $y$, there is a homomorphism $\\Phi_F(y,\\zeta) :\n\\A(y)\\rightarrow\\A(\\zeta)$ such that $\\Phi_F(y,\\zeta)^\\dagger$ is\na quantum operation and for all $z\\in\\zeta$, the reduction of\n$\\Phi_F(y,\\zeta)^\\dagger$ to $\\A(z)$ is $\\Phi(y,z)^\\dagger$.\nLikewise, for all $y\\in E^0$ and $\\xi\\subseteq E^0$ a complete\npast of $y$, there is a quantum operation\n$\\Phi_P(\\xi,y):\\A(\\xi)\\rightarrow\\A(y)$ such that\n$\\Phi_P(\\xi,y)^\\dagger$ is a homomorphism and for all $x\\in\\xi$\nthe reduction of $\\Phi_P(\\xi,y)^\\dagger$ to $\\A(x)$ is\n$\\Phi(x,y)^\\dagger$.\n\n$(ii)$ (Spacelike Commutativity) If $x\\sim y\\in E^0$ and\n$\\zeta\\subseteq E^0$ is a complete future of $x$ and $y$, then the\nimages of $\\Phi_F(x,\\zeta)$ and $\\Phi_F(y,\\zeta)$ commute inside\n$\\A(\\zeta)$. Likewise, if $y\\sim z \\in E^0$ and $\\xi\\subseteq\nE^0$ is a complete past of $y$ and $z$, then the images of\n$\\Phi_P(\\xi,y)^\\dagger$ and $\\Phi_P(\\xi,z)^\\dagger$ commute inside\n$\\A(\\xi)$.\n\n$(iii)$ (Composition) If $\\zeta\\subseteq E^0$ is a complete future\nof $x$ and a complete past of $y$, then $\\Phi(x,y) =\n\\Phi_P(\\zeta,y)\\circ\\Phi_F(x,\\zeta)$.\n\\end{defn}\n\n%\\begin{rem}\n%With the QCH formalism and the DGO framework in hand, the overlap\n%between the two is now evident. Each partial isometry $S_e$ in a\n%DGO family acts on $\\H = \\oplus_{x\\in E^0} \\H(x)$, but it only\n%acts non-trivially on the vertex space $\\H(s(e))$ where it acts as\n%a unitary from $\\H(s(e))$ into $\\H(r(e))$. As such, $S_e$\n%determines a unitary map with initial observable space $\\H(s(e))$\n%and final observable space supported on $\\H(r(e))$. Thus, when a\n%DGO family satisfies the further constraints discussed above, the\n%resulting construct may be interpreted as a QCH. Hence certain DGO\n%families define QCH's and vice versa. It is conceivable that the\n%overlap could expand, and further connections come to light, as\n%both the QCH and DGO frameworks undergo refinements.\n%\\end{rem}\n\nIf $\\xi\\preceq\\zeta$ form a complete pair within a given QCH, it\nis proved in \\cite{HMS} that there is a unique unitary map\n$\\Phi(\\xi,\\zeta):\\A(\\xi)\\rightarrow\\A(\\zeta)$ such that the\nreduction of $\\Phi(\\xi,\\zeta)$ to $\\A(x)\\mapsto\\A(y)$ is\n$\\Phi(x,y)$ for all $x\\in\\xi$ and $y\\in\\zeta$. Thus the\nisomorphisms $\\Phi(\\xi,\\zeta)$ discussed above may be built up\nfrom the individual edge maps $\\Phi(x,y)$ and hence the edge maps\nare the fundamental building blocks for a QCH.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{eg}\\label{S:qcomp}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n({\\bf Quantum Computers}) As discussed in \\cite{M00C}, the basic\nmodel for a quantum computer fits into the QCH formalism.\nSpecifically, each quantum algorithm may be interpreted as a QCH\nvia its ``circuit-gate'' presentation. (See \\cite{Ksurvey} for a\nbrief mathematical introduction to quantum algorithms and\nreferences.) The QCH for a given algorithm has vertex spaces all\nequal to $\\bbC^2$. The directed edges correspond to the choice of\nunitary gates within the algorithm and the vertex spaces encode\nthe intermediate states of the quantum bits of information (the\n`qubits'). The structure of the associated directed graph is the\nsame as the circuit-gate diagram, with the circuits labelled as\nvertices and the gates labelled as directed edges.\n\\end{eg}\n\nWe finish by proving that the algebras $\\fA_\\Psi$ associated with\nQCH's are familiar objects from operator theory. For basic\nproperties of AF-algebras we point the reader to the text\n\\cite{byeg}.\n\n\\begin{thm}\\label{qchopalg}\nLet $\\fA_\\Psi$ be the $\\ca$-algebra associated with a given\nquantum causal history. Then $\\fA_\\Psi$ is an AF-algebra.\n\\end{thm}\n\n\\Prf It is enough to prove that every finite set of elements of\n$\\fA_\\Psi$ can be approximated by elements lying in a\nfinite-dimensional subalgebra. But elements of the form $A =\nA_1\\cdots A_n$, where each $A_k=A_{e,i}$ or $A_k = A_{e,i}^*$ for\nsome $e$ and $i$, span a dense subspace of $\\fA_\\Psi$. (Note that\nthe vertex projections are obtained via equation\n(\\ref{tracepreserve}).) Hence it is enough to show that each\nfinite set of such elements lies in a finite-dimensional\nsubalgebra. Suppose $\\F$ is such a set. Note that each $A_{e,i}$\nbelongs to $\\B(\\H(x),\\H(y))$ for some $x,y\\in E^0$, and so the\nsame is true for every $A_{e,i}^*$ and all elements $A\\in\\F$.\nThus, as $\\dim\\H(x)<\\infty$ for all $x\\in E^0$, it follows that\nthe algebra generated by $\\F$ is a finite-dimensional algebra\nwhich is also a subalgebra of $\\fA_\\Psi$, and the result follows.\n\\bx\n\n\\begin{rem}\nWhile graph algebras and quantum causal histories provided the\ninitial impetus for the CP -- directed graph framework presented\nhere, it is evident that this structure admits other\npossibilities. Indeed, there are many CP maps that are neither\nunitary maps nor quantum operations, and presumably the\n$\\ca$-algebras $\\fA_\\Psi$ would go beyond the graph algebra and\nQCH subclasses in general. We have also not considered here the\nvarious possibilities for non-selfadjoint algebras defined by this\nframework. We plan to undertake investigations of these algebras\nelsewhere and hope this paper motivates others to do the same.\n\\end{rem}\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\section{Conclusion} \\label{S:conclusion}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n%We have illustrated a connection between the quantum causal\n%history formalism from quantum gravity and the directed graph\n%operator framework from operator algebras. This motivates a number\n%of new problems and possible avenues of research. For instance,\n%this connection begs the question as to whether there is a\n%physical interpretation of graph algebras within a quantum theory\n%of gravity. On the other hand, a natural mathematical\n%generalization of the graph-operator framework would be to allow\n%quantum operations (or even completely positive maps) instead of\n%only partial isometries on the directed edges and to consider the\n%corresponding operator algebras. Further, Example~\\ref{S:qcomp}\n%suggests an enticing link between graph algebras and quantum\n%computing. We plan to undertake these investigations elsewhere and\n%we hope this paper motivates others to do the same.\n\n\n\n\n\n\n\n\\vspace{0.05in}\n\n\n{\\noindent}{\\it Acknowledgements.}\n%I am grateful to the referee\n%for helpful suggestions.\nI would like to thank Fotini Markopoulou\nfor enlightening conversations and Eli Hawkins for helpful\ncomments on an early draft. I am also grateful to colleagues at\nthe Institute for Quantum Computing and the Perimeter Institute\nfor interesting discussions. This work was partially supported by\nan NSERC grant.\n\n\n\\begin{thebibliography}{99}\n\n\n\\bibitem{HMS} E. Hawkins, F. Markopoulou, H. Sahlmann,\n\\textit{Evolution in quantum causal histories}, Class. Quant.\nGrav. {\\bf 20} (2003), 3839.\n\n\\bibitem{M02} F. Markopoulou,\n\\textit{Planck-scale models of the universe}, invited contribution\nto ``Science \\& Ultimate Reality: From Quantum to Cosmos\n(Explorations Celebrating the Vision of John Archibald Wheeler,''\narxiv.org/gr-qc/0210086.\n\n\\bibitem{M00C} F. Markopoulou,\n\\textit{An insider's guide to quantum causal histories}, Nucl.\nPhys. Proc. Suppl. {\\bf 88} (2000), 308-313.\n\n\\bibitem{M00B} F. Markopoulou,\n\\textit{Quantum causal histories}, Class. Quant. Grav. {\\bf 17}\n(2000), 2059-2072.\n\n\\bibitem{M00A} F. Markopoulou,\n\\textit{The internal description of a causal set: What the\nuniverse looks like from the inside}, Commun. Math. Phys. {\\bf\n211} (2000), 559-583.\n\n\\bibitem{Gupta} S. Gupta,\n\\textit{Causality in spin foam models}, Phys. Rev. D {\\bf 61}\n(2000), 064014.\n\n\\bibitem{MS98} F. Markopoulou, L. Smolin,\n\\textit{Quantum geometry with intrinsic local causality}, Phys.\nRev. D {\\bf 58} (1998), 084032.\n\n\\bibitem{MS97} F. Markopoulou, L. Smolin,\n\\textit{Causal evolution of spin networks}, Nucl. Phys. B {\\bf\n508} (1997), 409-430.\n\n\\bibitem{CK} J. Cuntz, W. Krieger,\n\\textit{A class of $\\ca$-algebras and topological Markov chains,}\nInvent. Math. \\textbf{56} (1980), 251-268.\n\n\\bibitem{BP} T. Bates, D. Pask,\n\\textit{Flow equivalence of graph algebras}, Ergod. Theory \\& Dyn.\nSys. {\\bf 24} (2004), 367-382.\n\n\\bibitem{BPRS} T. Bates, D. Pask, I. Raeburn, W. Szymanski,\n\\textit{The $\\ca$-algebras of row-finite graphs}, New York J.\nMath. {\\bf 6} (2000), 307-324.\n\n\\bibitem{BJKR} O. Bratteli, P.E.T. Jorgensen, K.H. Kim, F. Roush,\n\\textit{Decidability of the isomorphism problem for stationary\nAF-algebras}, Ergod. Theory \\& Dyn. Sys. {\\bf 21} (2001),\n1625-1655.\n\n\\bibitem{Bre} B. Brenken,\n\\textit{$\\ca$-algebras of infinite graphs and Cuntz-Krieger\nalgebras}, Can. Math. Bull. {\\bf 45} (2002), 321-336.\n\n\\bibitem{DK} K.R. Davidson, E. Katsoulis,\n\\textit{Nest representations of directed graph algebras},\narxiv.org/math.OA/0410449.\n\n\\bibitem{DKP} K.R. Davidson, E. Katsoulis, D.R. Pitts,\n\\textit{The structure of free semigroup algebras}, J. Reine Angew.\nMath. {\\bf 533} (2001), 99-125.\n\n\\bibitem{Dun} B. Duncan,\n\\textit{Universal operator algebras of directed graphs}, Houston\nJ. Math., to appear.\n\n\\bibitem{ELQ} R. Exel, M. Laca, J. Quigg,\n\\textit{Partial dynamical systems and $\\ca$-algebras generated by\npartial isometries}, J. Operator Theory {\\bf 47} (2002), 169-186.\n\n\\bibitem{EL} R. Exel, M. Laca,\n\\textit{Cuntz-Krieger algebras for infinite matrices}, J. Reine\nAngew. Math. \\textbf{512} (1999), 119-172.\n\n\\bibitem{FR} N. Fowler, I. Raeburn,\n\\textit{The Toeplitz algebra of a Hilbert bimodule}, Indiana Univ.\nMath. J. \\textbf{48} (1999), 155--181.\n\n%\\bibitem{FMR} N. Fowler, P. Muhly, I. Raeburn,\n%\\textit{Representations of Cuntz-Pimsner algebras}, Indiana Univ.\n%Math. J. \\textbf{52}, 2003, 569-605.\n\n\\bibitem{HPP} A. Hopenwasser, J.R. Peters, S.C. Power,\n\\textit{Subalgebras of graph $\\ca$-algebras},\narxiv.org/math.OA/0409075.\n\n\\bibitem{aHR} A. an Huef, I. Raeburn,\n\\textit{The ideal structure of Cuntz-Krieger algebras}, Ergod.\nTheory \\& Dyn. Sys. {\\bf 17} (1997), 611-624.\n\n\\bibitem{JP} F. Jaeck, S.C. Power,\n\\textit{Hyper-reflexivity of free semigroupoid algebras},\narxiv.org/math.OA/0311201.\n\n\\bibitem{JK2} M.T. Jury, D.W. Kribs,\n\\textit{Partially isometric dilations of noncommuting $n$-tuples\nof operators}, Proc. Amer. Math. Soc. {\\bf 133} (2005), 213-222.\n\n%\\bibitem{JK1} M.T. Jury, D.W. Kribs,\n%\\textit{Ideal structure in free semigroupoid algebras from\n%directed graphs}, J. Operator Theory, to appear.\n\n\\bibitem{KQR} S. Kaliszewski, J. Quigg, I. Raeburn,\n\\textit{Skew products and crossed products by coactions}, J.\nOperator Theory \\textbf{46} (2001), 411-433.\n\n\\bibitem{KK} E. Katsoulis, D.W. Kribs,\n\\textit{Isomorphisms of algebras associated with directed graphs},\nMath. Ann. {\\bf 330} (2004), 709-728.\n\n\\bibitem{Kat} T. Katsura,\n\\textit{A class of $\\ca$-algebras generalizing both graph algebras\nand homeomorphism $\\ca$-algebras. I. Fundamental results}, Trans.\nAmer. Math. Soc. {\\bf 356} (2004), 4287-4322.\n\n\\bibitem{KS} D.W. Kribs, B. Solel,\n\\textit{A class of limit algebras associated with directed\ngraphs}, arxiv.org/math.OA/0411379.\n\n\\bibitem{KP04} D.W. Kribs, S.C. Power,\n\\textit{The $H^\\infty$ algebras of higher rank graphs},\narxiv.org/math.OA/0409432.\n\n\\bibitem{KP} D.W. Kribs, S.C. Power,\n\\textit{Free semigroupoid algebras}, J. Ramanujan Math. Soc. {\\bf\n19} (2004), 117-159.\n\n%\\bibitem{KP2} D.W. Kribs, S.C. Power,\n%\\textit{Partly free algebras}, Operator Theory: Advances and\n%Applications, Birkhauser-Verlag Basel/Switzerland, {\\bf 149}\n%(2004), 381-393.\n\n%\\bibitem{Kchannel} D.W. Kribs,\n%\\textit{Quantum channels, wavelets, dilations, and representations\n%of $\\O_n$}, Proc. Edin. Math. Soc., {\\bf 46} (2003), 421-433.\n\n%\\bibitem{KPR} A. Kumjian, D. Pask, I. Raeburn,\n%\\textit{Cuntz-Krieger algebras of directed graphs},\n%Pacific J. Math. \\textbf{184}, 1998, 161--174.\n\n\\bibitem{KumP1} A. Kumjian, D. Pask,\n\\textit{Higher rank graph $\\ca$-algebras}, New York J. Math.\n\\textbf{6} (2000), 1-20.\n\n\\bibitem{KumP2} A. Kumjian, D. Pask, I. Raeburn, J. Renault,\n\\textit{Graphs, groupoids, and Cuntz-Krieger algebras}, J. Funct.\nAnal. \\textbf{144} (1997), 505--541.\n\n\n\n\n\n%\\bibitem{Mu} P.S. Muhly,\n%\\textit{A finite dimensional introduction to operator algebra},\n%A. Katavolos (ed.), Operator Algebras and\n%Application, Kluwer Academic Publishers, 1997, 313--354.\n\n\\bibitem{MT} P.S. Muhly, M. Tomforde, \\textit{Topological\nquivers}, arxiv.org/math.OA/0312109.\n\n\\bibitem{MS1} P.S. Muhly, B. Solel,\n\\textit{Tensor algebras, induced representations, and the Wold\ndecomposition}, Can. J. Math. {\\bf 51} (1999), 850--880.\n\n\\bibitem{MS2} P.S. Muhly, B. Solel,\n\\textit{Tensor algebras over $\\ca$-correspondences:\nrepresentations, dilations and $\\ca$-envelopes} J. Funct. Anal.\n\\textbf{158} (1998), 389--457.\n\n%\\bibitem{Pim}M. Pimsner,\n%\\textit{A class of $C\\sp *$-algebras generalizing both\n%Cuntz-Krieger algebras and crossed products by $\\bbZ$}, Free\n%probability theory (Waterloo, ON, 1995), 189--212, Fields Inst.\n%Commun., \\textbf{12}, Amer. Math. Soc., Providence, RI, 1997.\n\n\\bibitem{Rcbms} I. Raeburn,\nCBMS conference on `graph $\\ca$-algebras' notes, in preparation,\n2004.\n\n\\bibitem{RST} I. Raeburn, A. Sims, T. Yeend,\n\\textit{Higher-rank graphs and their $\\ca$-algebras}, Proc.\nEdinburgh Math. Soc. {\\bf 46} (2003), 99-115.\n\n%\\bibitem{Sol} B. Solel,\n%\\textit{You can see the arrows in a quiver operator algebra},\n%e-print arxiv math.OA/0309420, preprint, 2003.\n\n\\bibitem{Spi} J. Spielberg,\n\\textit{A functorial approach to the $\\ca$-algebras of a graph},\nInternat. J. Math. {\\bf 13} (2002), 245-277.\n\n\n\\bibitem{Paulsentext2} V. Paulsen,\n\\textit{Completely bounded maps and operator algebras,} Cambridge\nUniversity Press, Cambridge, United Kingdom, 2002.\n\n\\bibitem{Kraustext} K. Kraus,\n\\textit{States, Effects and Operations: Fundamental Notions of\nQuantum Theory,} Lecture Notes in Physics, vol. 190, Berlin:\nSpringer-Verlag, 1983.\n\n\\bibitem{Choi} M.D. Choi,\n\\textit{Completely positive linear maps on complex matrices,} Lin.\n\\ Alg.\\ Appl. \\ \\textbf{10} (1975), 285-290.\n\n\\bibitem{Kraus} K. Kraus,\n\\textit{General state changes in quantum theory,} Ann. \\ Physics \\\n\\textbf{64} (1971), 311-335.\n\n\\bibitem{Ksurvey} D.W. Kribs,\n\\textit{A quantum computing primer for operator theorists}, Lin.\nAlg. Appl., to appear.\n\n\\bibitem{byeg} K.R. Davidson,\n\\textit{$\\ca$-algebras by example}, Fields Institute Monographs,\n{\\bf 6}, Amer. Math. Soc., Providence, 1996.\n\n\\end{thebibliography}\n\n\\end{document}\n", "meta": {"hexsha": "d3e05abafa7efda1e58a468fc9b9d2df43bfe1be", "size": 38714, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "benchmark/src/test-data/0501/math0501087/math0501087.tex", "max_stars_repo_name": "e-sim/pdf-text-extraction-benchmark", "max_stars_repo_head_hexsha": "42eede9867e5795a6fc040b0a7ce92da3ddd3120", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-08-23T19:07:01.000Z", "max_stars_repo_stars_event_max_datetime": "2018-08-23T19:07:01.000Z", "max_issues_repo_path": "benchmark/src/test-data/0501/math0501087/math0501087.tex", "max_issues_repo_name": "e-sim/pdf-text-extraction-benchmark", "max_issues_repo_head_hexsha": "42eede9867e5795a6fc040b0a7ce92da3ddd3120", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "benchmark/src/test-data/0501/math0501087/math0501087.tex", "max_forks_repo_name": "e-sim/pdf-text-extraction-benchmark", "max_forks_repo_head_hexsha": "42eede9867e5795a6fc040b0a7ce92da3ddd3120", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.2239108409, "max_line_length": 84, "alphanum_fraction": 0.6996952007, "num_tokens": 12491, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3481511269961152}} {"text": "\\title{Lattice methods and the nuclear few- and many-body problem}\n\\author{Dean Lee}\n\\institute{Dean Lee \\at Department of Physics, \nNorth Carolina State University, Raleigh, NC 27695, USA, \\email{dean\\_lee@ncsu.edu}}\n\\maketitle\n\n\\abstract{This chapter builds upon the review of lattice methods and effective\nfield theory of the previous chapter. We begin with a brief overview of lattice calculations using chiral effective field theory and some recent applications. We then describe several methods for computing scattering on the lattice. After that we focus on the main goal, explaining the theory and algorithms\nrelevant to lattice simulations of nuclear\nfew- and many-body systems. We discuss the exact equivalence\nof four different lattice formalisms, the Grassmann path integral, transfer matrix operator, Grassmann path\nintegral with auxiliary fields, and transfer matrix operator with auxiliary\nfields. Along with our analysis we include several coding examples and a number of exercises for the calculations of few- and many-body systems at leading order in chiral effective field theory.}\n\n\n\\section{Introduction}\nThis chapter builds upon the general overview of lattice methods for effective field theory of the previous chapter. We discuss the theory and algorithms used in lattice simulations of nuclear\nfew and many body systems. We show the exact equivalence of the Grassmann path integral, transfer matrix operator, Grassmann path\nintegral with auxiliary fields, and transfer matrix operator with auxiliary\nfields. Along with our analysis we include several coding examples and a number of exercises for the calculations of few- and many-body systems at leading order in chiral effective field theory.\n\n\nEffective field theory (EFT) provides a theoretical framework for organizing low-energy\ninteractions in powers of particle\nmomenta. \\ Chiral\neffective field theory applies this framework to the low-energy interactions of protons and neutrons while explicitly including the interactions of pions \\cite{Weinberg:1990rz,Weinberg:1991um,Ordonez:1992xp,Ordonez:1993tn,vanKolck:1994yi,Epelbaum:1998hg,Epelbaum:1998ka,Bedaque:2002mn,Epelbaum:2008ga}.\nPions are qualitatively different from other mesons since they become massless in the limit of massless quarks, thereby producing long-range exchange interactions. The low-energy expansion of chiral EFT is organized\nin powers of $Q$, where $Q$ denotes the typical momentum of the nucleons as well as explicit factors of the pion mass. \\ The most important interactions are called\nleading order (LO) or $O(Q^{0})$. \\ The next most important contributions\nare \nnext-to-leading order (NLO) or $O(Q^{2})$. \\ The terms after this are\nnext-to-next-to-leading order (NNLO) or $O(Q^{3})$, and so on. \n\nLattice EFT refers generally to lattice simulations based upon the framework of effective\nfield theory. There are a few reviews in the literature which discuss current methods used\nin lattice effective field theory \\cite{Lee:2008fa,Drut:2012a} as well as the discussion in the previous chapter of this volume. Many different phenomena can be studied\nin\nlattice EFT using the same lattice action. \\ In principle all systematic\nerrors are introduced up front when defining the low-energy effective theory,\nas opposed to the particular computational scheme used to calculate\nobservables. \\ \n\nLattice EFT has been aided by efficient lattice methods developed for lattice QCD and condensed matter applications. \\ The methods include\nMarkov Chain Monte Carlo techniques, auxiliary fields\n\\cite{Hubbard:1959ub,Stratonovich:1958}, pseudofermion methods\n\\cite{Weingarten:1980hx}, and non-local updating schemes such as the hybrid Monte\nCarlo algorithm\\cite{Scalettar:1986uy,Gottlieb:1987mq,Duane:1987de}. \\ Lattice EFT\nwas\nfirst used in studies of infinite nuclear matter \\cite{Muller:1999cp} and\ninfinite neutron matter with and without explicit pions\n\\cite{Lee:2004si,Lee:2004qd,Lee:2005is,Lee:2005it}. \\ The method has also been used\nto\nstudy light nuclei in pionless EFT \\cite{Borasoy:2005yc} and chiral EFT at\nleading order \\cite{Borasoy:2006qn}. There have been further studies of neutron matter \n\\cite{Borasoy:2007vi,Borasoy:2007vk,Wlazlowski:2014jna} and light nuclei \\cite{Epelbaum:2009zs,Epelbaum:2009pd}, and there have been several applications to nuclear structure and nuclear clustering \\cite{Epelbaum:2011md,Epelbaum:2012qn,Epelbaum:2012iu,Lahde:2013uqa,Epelbaum:2013paa,Elhatisari:2016owd} as well as recent work on nuclear scattering and reactions~\\cite{Rupak:2013aue,Rupak:2014xza,Elhatisari:2015iga}.\n\n\\section{Recent Applications}\n\n\nWe review here several recent applications of lattice effective field theory to nuclear systems. In Ref.~\\cite{Epelbaum:2013paa},\nthe first {\\it ab initio} evidence is presented for a tetrahedral alpha-cluster structure\nof the ground state of $^{16}$O. The first excited $0^+$ state of $^{16}$O is found to be a planar or square arrangement of alpha clusters. The evidence for these geometric arrangements come from the strong overlap between nuclear states and initial state configurations with these alpha-cluster geometries. \n\n\nIn Table~\\ref{oxygen1} we presented the energies of the low-lying even parity states of oxygen-16. The columns labeled ``LO(2N)'' and ``NNLO(2N)'' show\nthe energies at each order using the two-nucleon force only. The column labeled\n``+3N'' also includes the 3NF, which first appears \nat NNLO. The column ``+4N$_\\mathrm{eff}$'' includes an ``effective''\n4N force, and the column ``Exp'' gives\nthe empirical energies. This ``effective'' 4N force was introduced in Ref.~\\cite{Lahde:2013uqa} as a proxy measure of unknown systematic errors responsible for overbinding in lattice chiral effective field theory calculations with increasing numbers of nucleons. This tendency towards overbinding has also been noted in other nuclear structure calculations \\cite{Ekstrom:2015rta,Hagen:2015yea}. \n\\begin{table}[h]\n\\centering\n\\caption{Lattice results and experimental energies for the lowest even-parity\nstates of $^{16}$O in MeV. \nThe errors include statistical\nMonte Carlo errors and \nuncertainties due to the extrapolation to infinite Euclidean time. \n\\label{oxygen1}}\n\\vspace{.5cm}\n\\begin{tabular}{c | r | r r r | r}\n$J_n^p$ & \\multicolumn{1}{c |}{LO (2N)} & \\multicolumn{1}{c}{NNLO (2N)} \n& \\multicolumn{1}{c}{+3N} & \\multicolumn{1}{c |}{+4N$_\\mathrm{eff}$} & \\multicolumn{1}{c}{Exp}\n \\\\ \\hline\\hline\n$0^+_1$ & $-147.3(5)$ & $-121.4(5)$ & $-138.8(5)$ & $-131.3(5)$ & $-127.62$\n\\\\\n$0^+_2$ & $-145(2)$ & $-116(2)$ & $-136(2)$ & $-123(2)$ & $-121.57$ \\\\\n$2^+_1$ & $-145(2)$ & $-116(2)$ & $-136(2)$ & $-123(2)$ & $-120.70$\n\\end{tabular}\n\\end{table}\n\nIn order to understand the source of this overbinding, the problem was revisited again in Ref.~\\cite{Elhatisari:2016owd}. In that work numerical evidence from {\\it ab initio} lattice simulations showed that the problem appears related to the fact that the nuclear forces reside near a quantum phase transition. Using lattice effective field theory, Monte Carlo simulations were performed for systems with up to twenty nucleons. For even and equal numbers of protons and neutrons, a first-order transition was found at zero temperature from a Bose-condensed gas of alpha particles to a nuclear liquid. Whether one has an alpha-particle gas or nuclear liquid is determined by the strength of the alpha-alpha interactions, and the alpha-alpha interactions depend on the strength and locality of the nucleon-nucleon interactions. This insight is useful in improving calculations of nuclear structure and important astrophysical reactions involving alpha capture on nuclei. These findings also provide a tool to probe the structure of alpha cluster states such as the Hoyle state responsible for the production of carbon in red giant stars and point to a connection between nuclear states and the universal physics of bosons at large scattering length.\n\nProcesses such as the scattering of alpha particles,\nthe triple-alpha reaction, and\nalpha capture play an important role in stellar nucleosynthesis. In\nparticular, alpha capture on carbon determines the ratio of carbon to oxygen\nduring helium burning and impacts the following carbon, neon, oxygen,\nand silicon burning stages. In these\nreactions the elastic scattering of alpha particles\nand alpha-like nuclei (nuclei with even and equal numbers of protons\nand neutrons) are important for understanding background and resonant\nscattering contributions. In Ref.~\\cite{Elhatisari:2015iga}\nthe first {\\it ab initio}\ncalculations of the scattering of two alpha particles were performed using a technique called the adiabatic projection method. These\ncalculations represent a significant algorithmic improvement since the calculations presented in\n Ref.~\\cite{Elhatisari:2015iga}\nscale roughly quadratically with the number of nucleons and opens a gateway to scattering and reactions involving heavier nuclei. \\\\ \n\n\n\n\n\n\\section{Scattering on the lattice}\n\nAt any given order in the chiral EFT expansion, there will be short-range interaction coefficients which depend on the chosen regularization of the large-momentum divergences. On the lattice this regularization is provided by the lattice spacing, unless some additional regularization\nis applied to the lattice interactions.\nIn order to set the values of the short-range two-nucleon interaction\ncoefficients, we make a comparison of nucleon-nucleon scattering on\nthe lattice with experimental scattering data. The extension to three-nucleon\ninteraction coefficients is also required at NNLO, and that procedure\non the lattice has been discussed in Ref.~\\cite{Epelbaum:2009zs}\n\n\nAs discussed in the previous chapter, L\\\"{u}scher \\cite{Luscher:1985dn,Luscher:1986pf,Luscher:1991ux} has shown that the finite-volume energy levels for a two-body system in a periodic cubic box are related to the infinite-volume scattering matrix. \\ While the method is very useful\nat low momenta, it can become less accurate at higher momenta and higher orbital angular momenta. \\ Also spin-orbit\ncoupling and partial-wave mixing are difficult to measure accurately using\nL\\\"{u}scher's method due to scattering artifacts produced by the\ncubic periodic boundary. \\ An alternative approach has been developed to measure\nphase shifts for particles on the lattice using a spherical wall boundary \\cite{Borasoy:2007vy,Carlson:1984}.\n\\ \n\nIn this approach, a hard spherical wall boundary is imposed on the relative separation between\nthe two particles. This wall is placed at some chosen radius $R_{\\text{wall}}$, and it removes copies of the interactions produced by the periodic lattice.\n\\ Working in the center-of-mass frame, we solve the time-independent Schr{\\\"o}dinger equation as a function of the relative separation between the particles and compute spherical standing waves which vanish at $r=R_{\\text{wall}}$.\nAt values of $r$ beyond the range of the interaction, the spherical standing\nwaves can be written as a superposition of products of spherical harmonics\nand spherical Bessel functions,\\begin{equation}\n\\left[ \\cos\\delta_{\\ell}\\cdot j_{\\ell}(kr)-\\sin\\delta_{\\ell}\\cdot y_{\\ell}(kr)\\right]\nY_{\\ell,\\ell_{z}}(\\theta,\\phi). \\label{wavefunction}%\n\\end{equation}\nHere $k$ is the relative momentum between the scattering particles,\nand $\\delta_{\\ell}$ is the phase shift for partial wave $\\ell$. \\ We can extract\n$k$ from the energy of the standing wave, and the phase shift $\\delta_{\\ell}$\nis determined by setting the wave function in Eq.~(\\ref{wavefunction}) to\nzero at the wall boundary.\n\nWhen the total intrinsic spin of the two nucleons is nonzero,\nspin-orbit coupling generates mixing between partial waves. $\\ $In this case\nthe standing wave at the wall boundary is decomposed into spherical harmonics\nand coupled-channel equations are solved to extract the phase shifts and\nmixing angles. The spherical wall method\nwas used to calculate phase shifts and mixing angle for low-energy nucleon-nucleon\nscattering \\cite{Borasoy:2007vi}.\nRecently the spherical wall approach has been improved\nin accuracy and computational efficiency \\cite{Lu:2015riz}.\nIn the improved approach one projects onto spherical harmonics\n$Y_{\\ell,\\ell_z}$ with angular momentum quantum numbers $\\ell,\\ell_z$. In this manner one constructs radial position states for a given partial wave,\n\\begin{equation}\n|r\\rangle^{\\ell,\\ell_z} = \\sum_{{\\bf r'}}Y_{\\ell,\\ell_z}({\\bf\\hat{r}'})\\delta_{r,|{\\bf r'}|}|{\\bf r'} \\rangle.\n\\end{equation} We require that $r$ is less than half the box length $L/2$.\n Using this technique we are essentially constructing a radial position basis for each partial wave. \n\nIt is also useful to introduce auxiliary potentials in the region lying just in front\nof the spherical wall boundary \\cite{Lu:2015riz}. The auxiliary potential\nis a spherical attractive well that is positioned in front\nof the spherical wall boundary. We can tune to any scattering energy by\nadjusting the depth of the well. For systems with partial wave mixing due\nto spin-orbit coupling, we also include a Hermitian but imaginary off-diagonal\nauxiliary potential\nfor the two coupled channels. This breaks time reversal\nsymmetry, and the resulting standing wave solutions now have both real and imaginary\nparts that are linearly independent. From the real and imaginary solutions\none can determine the scattering phase shifts and mixing angle at any\ngiven value of the scattering energy.\n\nThis spherical wall approach has been used together with a technique called the adiabatic projection method to study nuclear scattering and reactions on the lattice. The adiabiatic projection method \\cite{Pine:2013zja,Elhatisari:2014lka,Elhatisari:2015iga,Rokash:2015hra,Elhatisari:2016owd} is a general framework that produces a low-energy effective theory for\nclusters of particles which becomes exact in the limit of large projection\ntime.\nFor the case of two-cluster scattering, we consider a set of two cluster states\n$|{\\bf R}\\rangle$ labeled by the spatial separation vector {\\bf R}. The initial\nwave\nfunctions are wave packets which, for large $|{\\bf R}|$, factorize into a\nproduct of two\nindividual clusters,\n%\n\\begin{equation}\n|{\\bf R}\\rangle=\\sum_{{\\bf r}} |{\\bf r}+{\\bf R}\\rangle_1\\otimes|{\\bf r}\\rangle_2.\n\\label{eqn:single_clusters}\n\\end{equation}\n%\nThe summation over $\\bf {r}$ is required to produce states with \ntotal momentum equal to zero. We bin the initial cluster states together\naccording to radial distance and angular momentum. In this manner, we form\nradial \nposition states with projected angular momentum quantum numbers, which we\nlabel $|R\\rangle^{\\ell,\\ell_z}$. \n\nThe next step is to multiply by powers of the transfer matrix in order to\nform ``dressed'' cluster\nstates. This produces states that approximately span the set of low-energy cluster-cluster scattering\nstates in our periodic box. We discuss the transfer matrix formalism in detail later in this chapter. After $n_t$ time steps, we have the dressed cluster\nstates \n%\n\\begin{equation}\n\\vert R\\rangle^{\\ell,\\ell_z}_{n_t} = M^{n_t}|R\\rangle^{\\ell,\\ell_z}.\n\\end{equation}\n%\nThese dressed cluster states are then used to compute matrix\nelements of the transfer matrix $M$,\n%\n\\begin{equation}\n\\left[M_{n_t}\\right]^{\\ell,\\ell_z}_{R',R} =\\ ^{\\ell,\\ell_z}_{\\!\\!\\!\\!\\!\\quad{n_t}}\\langle\nR'\\vert M \\vert R\\rangle^{\\ell,\\ell_z}_{n_t}.\n\\label{Hmatrix}\n\\end{equation}\n%\nSince such states are not orthogonal, we also compute a norm\nmatrix\n% \n\\begin{equation}\n\\left[N_{n_t}\\right]^{\\ell,\\ell_z}_{R',R} =\\ ^{\\ell,\\ell_z}_{\\!\\!\\!\\!\\!\\quad{n_t}}\\langle\nR'\\vert R\\rangle^{\\ell,\\ell_z}_{n_t}. \n\\label{eqn:norm}\n\\end{equation}\n%\nThe ``radial adiabatic transfer matrix'' is defined as the matrix product\n%\n\\begin{equation}\n\\left[ {M^a_{n_t}} \\right]^{\\ell,\\ell_z}_{R',R} = \n\\left[N_{n_t}^{-\\frac{1}{2}}M_{n_t}\nN_{n_t}^{-\\frac{1}{2}} \\right]^{\\ell,\\ell_z}_{R',R},\n\\label{eqn:Adiabatic-Hamiltonian}\n\\end{equation}\n%\nand the scattering phase shifts can then be determined from the standing waves\nof the radial adiabatic transfer matrix. \n\n\\section{Lattice formalisms}\n\nThroughout our discussion of the lattice formalism we use dimensionless\nparameters and operators corresponding with physical values times\nthe\nappropriate power of the spatial lattice spacing $a$. In our notation the\nthree-component integer vector ${\\bf n}$ labels the lattice sites of a\nthree-dimensional periodic lattice with dimensions $L^{3}$. The spatial\nlattice unit vectors are denoted \n$\\mathbf{\\hat{l}}$ = $\\mathbf{\\hat{1}}$,\n$\\mathbf{\\hat{2}}$, $\\mathbf{\\hat{3}}$.\nWe use $n_t$ to label lattice steps in the temporal direction, and $L_{t}$\ndenotes the total number of lattice time steps. \\ The temporal lattice spacing\nis given by $a_{t}$, and $\\alpha_{t}=a_{t}/a$ is the ratio of the temporal\nto\nspatial lattice spacing. \\ We also define $h=\\alpha_{t}/(2m)$, where $m$\nis\nthe nucleon mass in lattice units. In Fig.~\\ref{formalism} we show a diagram\nof the four different but exactly equivalent lattice formulations that we\ndiscuss, the Grassmann path integral, transfer matrix operator, Grassmann\npath integral with auxiliary fields, and transfer matrix operator with auxiliary\nfields.\n\n\\begin{figure}\n[pb]\n\\begin{center}\n\\includegraphics[width=3in\n]%\n{Chapter6-figures/formalism.png}%\n\\caption{A schematic diagram of the different lattice formulations, namely, the Grassmann path integral, transfer matrix operator, Grassmann\npath integral with auxiliary fields, and transfer matrix operator with auxiliary\nfields.}%\n\\label{formalism}%\n\\end{center}\n\\end{figure}\n%EndExpansion\n\n\\subsection{Grassmann path integral}\n\nWe define the lattice action starting from the lattice Grassmann path integral\naction without auxiliary fields.\n\\ This\nis the simplest formulation in which to derive the lattice Feynman rules.\n We let $c$ and $c^*$ be anticommuting Grassmann fields for the nucleons.\n\\ In our notation $c$ is a column vector composed of the spin-isospin nucleon\ndegrees of freedom $c_i$, while $c^*$ is a row vector of the components\n$c^*_i$. The Grassmann fields are periodic with respect to\nthe spatial extent of the $L^{3}$ lattice,%\n\\begin{equation}\nc_i({\\bf n}+L\\hat{1},n_t)=c_i({\\bf n}+L\\hat{2},n_t)=c_i({\\bf n}+L\\hat{3},n_t)=c_i({\\bf n},n_t),\n\\end{equation}%\n\\begin{equation}\nc_i^{\\ast}({\\bf n}+L\\hat{1},n_t)=c_i^{\\ast}({\\bf n}+L\\hat{2}%\n,n_t)=c_i^{\\ast}({\\bf n}+L\\hat{3},n_t)=c_i^{\\ast}({\\bf n},n_t),\n\\end{equation}\nand antiperiodic along the temporal direction,%\n\\begin{equation}\nc_i({\\bf n},n_t+L_{t})=-c_i({\\bf n},n_t),\n\\end{equation}%\n\\begin{equation}\nc_i^{\\ast}({\\bf n},n_t+L_{t})=-c_i^{\\ast}({\\bf n},n_t).\n\\end{equation}\nWe write $DcDc^{\\ast}$ as shorthand for the integral measure,%\n\\begin{equation}\nDcDc^{\\ast}=\\prod_{{\\bf n},n_t,i}dc_{i}({\\bf n}%\n,n_t)dc_{i}^{\\ast}({\\bf n},n_t).\n\\end{equation}\nWe use the usual convention for Grassmann integration,%\n\\begin{equation}\n\\int dc_{i}({\\bf n},n_t)=\\int dc_{i}^{\\ast}({\\bf n},n_t)=0\\text{,}%\n\\end{equation}%\n\\begin{equation}\n\\int dc_{i}({\\bf n},n_t)c_{i}({\\bf n},n_t)=\\int dc_{i}^{\\ast}(\\vec\n{n},n_t)c_{i}^{\\ast}({\\bf n},n_t)=1\\text{ \\ (no sum on }i\\text{)}.\n\\end{equation}\nWe consider the Grassmann path integral%\n\\begin{equation}\n\\mathcal{Z}=\\int DcDc^{\\ast}\\exp\\left[ -S\\left(c^{\\ast},c\\right) \\right]\n, \\label{defining_Z}%\n\\end{equation}\nwhere the lattice action can be broken into a free part and interacting part,\n\\begin{equation}\nS(c^*,c)=S_{\\text{free}}(c^{\\ast},c)+S_\\text{int}(c^{\\ast},c).\n\\label{path_nonaux}%\n\\end{equation}\nThe free part is the free non-relativistic nucleon action, which is \n\\begin{align}\nS_{\\text{free}}(c^{\\ast},c) & =\\sum_{{\\bf n},n_t\n} c_{}^{\\ast}({\\bf n},n_t) \\left[ c({\\bf n},n_t+1)-c({\\bf n},n_t)\\right]\n+\\alpha_t \\sum_{n_t} K^{(n_t)}(c^*,c),\n\\end{align}\nwhere\n\\begin{align}\nK^{(n_t)}(c^*,c) =\\sum_{k=0,1,2,\\cdots} (-1)^k \\frac{w_k}{2m} \\sum_{{\\bf\nn},{\\bf \\hat{l}}} c^{\\ast}({\\bf n},n_t) \\left[ c({\\bf n}+k{\\bf\\hat{l}},n_t)\n+ c({\\bf n}-k{\\bf \\hat{l}},n_t)\\right], \n\\end{align}\nand the hopping coefficients $w_k$ correspond to a hopping parameter expansion\nof the squared momentum,\n\\begin{align}\nP^2({\\bf p})=2\\sum_{k=0,1,2,\\cdots}\\sum_{l=1,2,3}%\n(-1)^{k}w_{k}\\cos\\left(kp_{l}\\right).\n\\end{align}\nThe hopping coefficients are chosen to match the continuum relation \n\\begin{align}\nP^2({\\bf p})={\\bf p}^2,\n\\end{align}\nup to some chosen level of lattice discretization error. The hopping coefficients\n$w_k$ for a few different lattice actions are shown in Table~\\ref{hopping_coeff}.\n\n\\begin{table}[tbh]\n\\caption{Hopping coefficients $w_k$ for several lattice actions}%\n\\label{hopping_coeff}%\n\\begin{center}\n\\begin{tabular}{p{2cm}p{3cm}p{3cm}p{3cm}}\n\\hline\\noalign{\\smallskip}\ncoefficient& standard & $O(a^{2})$-improved & $O(a^{4})$-improved \\\\\n\\noalign{\\smallskip}\\svhline\\noalign{\\smallskip}\n$w_{0}$ & $1$ & $5/4$ & $49/36$\\\\\n$w_{1}$ & $1$ & $4/3$ & $3/2$\\\\\n$w_{2}$ & $0$ & $1/12$ & $3/20$\\\\\n$w_{3}$ & $0$ & $0$ & $1/90$\\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\subsection{Transfer matrix operator}\n\n\nLet $a_{i}({\\bf n})$ and\n$a_{i}^{\\dagger}({\\bf n})$ denote fermion annihilation and creation operators\nfor the nucleon component $i$ at lattice site ${\\bf n}$. The shorthand $a_{}({\\bf\nn})$ represents a column vector of nucleon components $a_{i}({\\bf n})$, and\n$a^{\\dagger}({\\bf n})$ represents a row vector of components $a^{\\dagger}_{i}({\\bf\nn})$. We can write any Grassmann path\nintegral with instantaneous interactions as the trace of a product of\noperators using the identity \\cite{Creutz:1988wv,Creutz:1999zy}%\n\\begin{align}\n& {\\rm Tr}\\left\\{ \\colon F_{L_{t}-1}\\left[ a_{}^{\\dagger}({\\bf n}^{\\prime}),a({\\bf\nn})\\right] \\colon\\times\\cdots\\times\\colon\nF_{0}\\left[ a_{}^{\\dagger}({\\bf n}^{\\prime}),a({\\bf n})\\right]\n\\colon\\right\\} \\nonumber\\\\\n& =\\int DcDc^{\\ast}\\exp\\left\\{ \\sum_{n_t=0}^{L_{t}-1}\\sum_{{\\bf n},i}%\nc_{i}^{\\ast}({\\bf n},n_t)\\left[ c_{i}({\\bf n},n_t)-c_{i}({\\bf n}%\n,n_t+1)\\right] \\right\\} \\nonumber\\\\\n& \\qquad\\qquad\\qquad\\times\\prod_{n_t=0}^{L_{t}-1}F_{n_t}\\left[\nc_{}^{\\ast}({\\bf n}^{\\prime},n_t),c({\\bf n},n_t)\\right]\n ,\n\\label{correspondence}%\n\\end{align}\nwhere $c_{i}({\\bf n},L_{t})=-c_{i}({\\bf n},0)$.\n\nLet us define the free non-relativistic lattice Hamiltonian\n\\begin{equation}\nH_{\\rm free}(a^{\\dagger},a) =\\sum_{k=0,1,2,\\cdots} (-1)^k \\frac{w_k}{2m}\n\\sum_{{\\bf n},{\\bf \\hat{l}}} a^{\\dagger}({\\bf n}) \\left[ a({\\bf n}+k{\\bf\\hat{l}})\n+ a({\\bf n}-k{\\bf \\hat{l}})\\right].\n\\end{equation}\nWe write the interaction term as $H_{\\rm int}(a^{\\dagger},a)$, so that our\ntotal Hamiltonian is\n\\begin{equation}\nH(a^{\\dagger},a) = H_{\\rm free}(a^{\\dagger},a) + H_{\\rm int}(a^{\\dagger},a).\n\\end{equation}Using the correspondence Eq.~(\\ref{correspondence}), we can\nrewrite the path\nintegral $\\mathcal{Z}$ defined in Eq.~(\\ref{defining_Z}) as a transfer-matrix\npartition function,%\n\\begin{equation}\n\\mathcal{Z}={\\rm Tr}\\left( M^{L_{t}}\\right) ,\n\\end{equation}\nwhere $M$ is the normal-ordered transfer matrix operator%\n\\begin{equation}\nM=:\\exp\\left[ -H(a^{\\dagger},a)\\alpha_{t}\\right] :. \\label{transfer_noaux}%\n\\end{equation}\nRoughly speaking, the transfer matrix operator is the exponential of the\nHamiltonian operator over one Euclidean lattice time step.\n\\ In order to satisfy the identity Eq.~(\\ref{correspondence}), the exact\ndefinition of the transfer matrix is the normal-ordered exponential as defined\nin Eq.~(\\ref{transfer_noaux}).\n\nIn this transfer matrix formalism, one can do simulations of nucleons using Monte Carlo, and this would essentially be a lattice version of diffusion or Green's function Monte Carlo \\cite{Carlson:2014vla}. Visually one can view the nucleons as interacting with each other while diffusing in space with each time step, as indicated in Fig.~\\ref{worldlines}. At leading order in chiral effective field theory, the\ninteractions include two independent $S$-wave contact interactions and the exchange of pions. We discuss these interactions in detail in the following.\n\\begin{figure}[ptb]%\n\\centering\n\\sidecaption\n\\includegraphics[\nheight=7.00cm\n]%\n{Chapter6-figures/worldlines.png}%\n\\caption{A sketch showing nucleons which evolve with each time step. At leading order in chiral effective field theory, the interactions include\ntwo contact\ninteractions and the exchange of pions.}%\n\\label{worldlines}%\n\\end{figure}\n\n\n\\subsection{Grassmann path integral with auxiliary field}\n\nWe assume that there exists an integral relation that allows us to write\n$\\exp\\left[-S_{\\rm int}(c^*,c)\\right]$ as an integral over auxiliary fields.\n The purpose of the auxiliary field transformation is to decouple the interactions\namong the nucleons. Instead the interactions will be between the nucleons\nand the auxiliary fields. \n\nWe illustrate using the interactions that appear at leading order in\nchiral effective field theory. For pedagogical purposes we discuss the simplest possible implementation of the leading order action on the lattice. We first consider a zero-range contact interaction\nwhich is independent of\nnucleon spin and isospin.\nThe action has the form \n\\begin{equation}\nS^{C}_\\text{int}(c^{\\ast},c) = \\alpha_{t}\\frac{C}{2} \\sum_{{\\bf n},n_t} \\left[c^{\\ast}({\\bf\nn},n_t)c({\\bf n},n_t)\\right]^2.\n\\end{equation}\nWe can write this as\n\\begin{equation}\n\\exp\\left[-S^C_{\\rm int}(c^*,c)\\right] = \\int Ds \\; \\exp\\left[-S_{ss}(s)\n- S_{s}(c^*,c,s)\\right]\n\\label{aux}\n\\end{equation}\nfor auxiliary field $s({\\bf n},n_t)$, \nwhere\n\\begin{align}\nS_{ss}(s) = \\frac{1}{2} & \\sum_{{\\bf n},n_t} s^{2}({\\bf n},n_t),\n \\\\\nS_{s}(c^*,c,s) = \\sqrt{-C\\alpha_{t}} & \\sum_{{\\bf n},n_t} s({\\bf n},n_t)c^{\\ast}({\\bf\nn},n_t)c({\\bf n},n_t). \n\\end{align}\nIn our definition of the integration measure $Ds$, we include a factor of\n$1/\\sqrt{2\\pi}$ for each degree of freedom.\n\nNext we consider an isospin-dependent contact interaction \n\\begin{equation}\nS^{C'}_\\text{int}(c^{\\ast},c) = \\alpha_{t}\\frac{C'}{2} \\sum_{{\\bf n},n_t,I}\n\\left[c^{\\ast}({\\bf\nn},n_t)\\tau_I c({\\bf n},n_t)\\right]^2,\n\\end{equation}\nwhere $\\tau_I$ for $I=1,2,3$ are the Pauli matrices in isospin space. Then\nwe can use\n\\begin{equation}\n\\exp\\left[-S^{C'}_{\\rm int}(c^*,c)\\right] = \\int \\prod_I Ds_I \\exp\\left[-S_{s_Is_I}(s_I)\n- S_{s_I}(c^*,c,s_{I})\\right]\n\\label{aux2}\n\\end{equation}\nfor auxiliary fields $s_I({\\bf n},n_t)$ where\n\\begin{align}\nS_{s_Is_I}(s_I)=\\frac{1}{2} & \\sum_{{\\bf n},n_t,I} s_I^{2}({\\bf n},n_t),\n\\\\\nS_{s_I}(c^*,c,s_{I}) = \\sqrt{-C'\\alpha_{t}} & \\sum_{{\\bf n},n_t,I} s_{I}({\\bf\nn},n_t)c^{\\ast}({\\bf\nn},n_t)\\tau_Ic({\\bf n},n_t). \n\\end{align}\n\nFinally we work with the one-pion exchange potential (OPEP). In this case\nthe pion acts much like the auxiliary fields. However there are also spatial\ncorrelations in the quadratic part of the pion action and a gradient coupling\nbetween the pions and nucleons.\n The one-pion exchange interaction on the lattice can written as\n\\begin{equation}\n\\exp\\left[-S^{\\rm OPEP}_{\\rm int}(c^*,c)\\right] = \\int \\prod_I D\\pi_I\n\\exp\\left[-S_{\\pi_I\\pi_I}(\\pi_I) - S_{\\pi_I}(c^*,c,\\pi_{I})\\right].\n\\end{equation}\nThe free pion action is\n\\begin{align}\nS_{\\pi_I\\pi_I}(\\pi_I)= & \\frac{1}{2}\\alpha_{t}m^2_{\\pi} \\sum_{{\\bf n},n_t,I}\\pi^2_{I}({\\bf\nn},n_t) \\\\\n& +\\frac{1}{2}\\alpha_{t}%\n\\sum_{k=0,1,2,\\cdots}(-1)^k w_k \\sum_{{\\bf n},n_t,I,{\\bf \\hat{l}}} \\pi_{I}({\\bf\nn},n_t)\n\\left[ \\pi_{I}({\\bf n}+k{\\bf \\hat{l}},n_t) + \\pi_{I}({\\bf n}-k{\\bf \\hat{l}},n_t)\n\\right],\n\\end{align}\nwith the coefficient $w_k$ as defined in Table~\\ref{hopping_coeff} and $m_{\\pi}$ is the pion mass. At leading order we do not\nconsider any isospin-breaking effects. The pion coupling to the nucleon is\\begin{align}\nS_{\\pi_I}(c^*,c,\\pi_{I}) = \\frac{g_A \\alpha_t}{2 f_{\\pi}} \\sum_{{\\bf n},n_t,l,I}\n\\Delta_k \\pi_{I}({\\bf\nn},n_t)c^{\\ast}({\\bf\nn},n_t)\\sigma_k\\tau_Ic({\\bf n},n_t), \n\\end{align}\nwhere $\\sigma_l$ for $l=1,2,3$ are the Pauli matrices in spin space and \n\\begin{equation}\n\\Delta_l \\pi_{I}({\\bf\nn},n_t)=\\frac{1}{2}\\sum_{k=1,2,\\cdots} (-1)^{k-1} o_k \\left[ \\pi_{I}({\\bf\nn}+k{\\bf \\hat{l}},n_t) - \\pi_{I}({\\bf\nn}-k{\\bf \\hat{l}},n_t)\\right],\n\\end{equation}\nwith coefficients $o_k$ corresponding to a hopping parameter expansion of the\nmomentum,\n\\begin{align}\nP(p_l)=\\sum_{k=1,2,\\cdots}%\n(-1)^{k-1} o_{k}\\sin\\left(kp_{l}\\right).\n\\end{align}\nHere\n$g_A$ is the axial-vector coupling constant, and $f_{\\pi}$ is the pion\ndecay constant. The hopping coefficients can be chosen to match the continuum result\n\\begin{equation}\nP(p_l)= p_l.\n\\end{equation}\nThe hopping coefficients $o_k$ for a few different lattice actions are shown\nin Table~\\ref{hopping_coeff_2}.\n\\begin{table}[tbh]\n\\caption{Hopping coefficients $o_k$ for several lattice actions.}%\n\\begin{center}\n\\label{hopping_coeff_2}%\n\n\\begin{tabular}{p{2cm}p{3cm}p{3cm}p{3cm}}\n\\hline\\noalign{\\smallskip}\ncoefficient& standard & $O(a^{2})$-improved & $O(a^{4})$-improved \\\\\n\\noalign{\\smallskip}\\svhline\\noalign{\\smallskip}\n$o_{1}$ & $1$ & $4/3$ & $3/2$\\\\\n$o_{2}$ & $0$ & $1/6$ & $3/10$\\\\\n$o_{3}$ & $0$ & $0$ & $1/30$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\subsection{Transfer matrix operator with auxiliary field}\nUsing the equivalence in Eq.~(\\ref{correspondence}), we can write $\\mathcal{Z}$\nas the trace of a product of transfer matrix operators which depend on the\nauxiliary field,\n\\begin{equation}\n\\mathcal{Z} = \\int Ds \\prod_I \\left(Ds_I D\\pi_I\\right) \\; \n\\exp{\\left[-S_{ss}(s)\n-S_{s_Is_I}(s_I)\n-S_{\\pi_I\\pi_I}(\\pi_I)\\right]}\n {\\rm Tr}\\left\\{ M^{(L_t-1)}\\cdots M^{(0)}\\right\\}.\n\\end{equation}\nThe transfer matrix at time step $n_t$ is given by\n\\begin{equation}\nM^{(n_t)}=\\colon\\exp\\left[ -H^{(n_t)}(a^{\\dagger},a,s,s_I,\\pi_I)\\alpha_{t}\n\\right]\n\\colon,\n\\end{equation}\nwhere\n\\begin{equation}\nH^{(n_t)}(a^{\\dagger},a,s,s_I,\\pi_I)\\alpha_{t}=H_{\\text{free}}(a^{\\dagger},a)\\alpha_{t}+S^{(n_t)}_s(a^{\\dagger},a,s)+S^{(n_t)}_{s_I}(a^{\\dagger},a,s_{I})+S^{(n_t)}_{\\pi_I}(a^{\\dagger},a,\\pi_{I}),\n\\end{equation}\nand\n\\begin{equation}\nS^{(n_t)}_{s}(a^{\\dagger},a,s) = \\sqrt{-C\\alpha_{t}} \\sum_{\\bf n} s({\\bf\nn},n_t)a^{\\dagger}({\\bf\nn})a({\\bf n}), \n\\end{equation}\n\\begin{equation}\nS^{(n_t)}_{s_I}(a^{\\dagger},a,s_{I}) = \\sqrt{-C'\\alpha_{t}} \\sum_{{\\bf n},I}\ns_{I}({\\bf\nn},n_t)a^{\\dagger}({\\bf\nn})\\tau_Ia({\\bf n}),\n\\end{equation}\n\\begin{equation}\nS^{(n_t)}_{\\pi_I}(a^{\\dagger},a,\\pi_{I}) = \\frac{g_A \\alpha_t}{2 f_{\\pi}}\n\\sum_{{\\bf n},k,I}\n\\Delta_k \\pi_{I}({\\bf\nn},n_t)a^{\\dagger}({\\bf\nn})\\sigma_k\\tau_I a({\\bf n}).\n\\end{equation}\n\n\\section{Projection Monte Carlo}\nLet us consider a system with $A$ nucleons. We can create a general single-nucleon\nstate using \ncreation operators acting on the vacuum with coefficient function $f({\\bf\nn})$. We write $f({\\bf n})$ as a column vector in the space of nucleon spin and isospin components, and the single-nucleon\nstate can be written as\\begin{equation}\n\\left| f \\right> = \\sum_{\\bf n}a^{\\dagger}({\\bf n})f({\\bf n}) \\left| 0 \\right>.\n\\end{equation}\nFor our projection Monte Carlo calculation we take our A-body initial\nstate\nto be a Slater determinant of single nucleon states,\n\\begin{equation}\n\\left| f_1,\\cdots ,f_A \\right> = \\left[\\sum_{\\bf n}a^{\\dagger}({\\bf n})f_1({\\bf\nn}) \\right] \\cdots \\left[\\sum_{\\bf n}a^{\\dagger}({\\bf n})f_A({\\bf n}) \\right]\n\\left| 0 \\right>.\n\\end{equation} We use the same construction for the $A$-body final state.\n\nFor the purposes of coding the projection Monte Carlo calculation, it is\nconvenient to view the identical nucleons as having a hidden index $j=1,\\cdots\n,A$ that makes all of the nucleons distinguishable. If we antisymmetrize\nall physical states over this extra index then all physical observables are\nexactly recovered. So our initial state $\\left| f_1,\\cdots ,f_A \\right>$\nbecomes\n\\begin{align}\n \\frac{1}{\\sqrt{A!}} & \\sum_{P} \\left[\\sum_{\\bf n}a^{\\dagger}_{[P(1)]}({\\bf n})f_1({\\bf\nn}) \\right] \\cdots \\left[\\sum_{\\bf n}a^{\\dagger}_{[P(A)]}({\\bf n})f_A({\\bf\nn}) \\right]\n\\left| 0 \\right> \\nonumber \\\\\n & =\\frac{1}{\\sqrt{A!}} \\sum_{P'} {\\rm sgn}(P') \\left[\\sum_{\\bf n}a^{\\dagger}_{[1]}({\\bf\nn})f_{P'(1)}({\\bf\nn}) \\right] \\cdots \\left[\\sum_{\\bf n}a^{\\dagger}_{[A]}({\\bf n})f_{P'(A)}({\\bf\nn}) \\right]\n\\left| 0 \\right>,\n\\end{align}\nwhere the summations are over all permutations $P$, and ${\\rm sgn}$ is\nthe sign of the permutation.\nWith these hidden indices our normal-ordered auxiliary-field transfer matrix\n$M^{(n_t)}$ becomes \n\\begin{equation}\n\\left[1-H^{(n_t)}(a_{[1]}^{\\dagger},a_{[1]},s,s_I,\\pi_I)\\alpha_{t}\\right]\n\\cdots \\left[1-H^{(n_t)}(a^{\\dagger}_{[A]},a_{[A]},s,s_I,\\pi_I)\\alpha_{t}\\right]\n\\end{equation}\nWe see that the higher-order powers of the exponential vanish due to normal ordering.\n\nIn the projection Monte Carlo calculation we compute the amplitude\n\\begin{equation}\nZ(n_t) = \\left< f_1,\\cdots ,f_A \\right| M^{(n_t-1)} \\cdots M^{(0)} \\left|\nf_1,\\cdots ,f_A \\right>\n\\end{equation}\nfor $n_t=L_t$ and $n_t=L_t-1$. In the limit of large $L_t$ the amplitudes will be dominated\nby the state with the lowest energy $E_0$ and nonzero overlap with $\\left|\nf_1,\\cdots ,f_A \\right>$. In this limit the ratio $Z(n_t)/Z(n_t-1)$\nwill converge to $\\exp(-E_0 \\alpha_t)$ from above.\n\nEach nucleon evolves as a\nparticle in a fluctuating background of auxiliary fields and pion\nfields. \\ The original interactions are reproduced after integrating over\nthe fluctuating auxiliary\nand pion fields. For a simulation with $A$ nucleons, the amplitude for\na given configuration of pion and auxiliary fields is proportional to the\ndeterminant of an $A\\times A$ matrix ${\\bf M}$. \\ The\nentries of ${\\bf M}_{ij}$ are single nucleon worldline amplitudes for a nucleon\nstarting at state $\\left| f_j \\right> $ at $t=0$ and ending at state $\\left|\nf_i \\right> $ at $t=t_{f}=L_t\\alpha_t$. This is shown in Fig. \\ref{hsworldlines}.\n\n\\begin{figure}[ptb]%\n\\centering\n\\sidecaption\n\\includegraphics[\nheight=7.00cm\n]\n{Chapter6-figures/HSworldlines.png}%\n\\caption{A sketch showing the worldline for a single nucleon with a background\nof pion fields and\nauxiliary fields.}%\n\\label{hsworldlines}%\n\\end{figure}\n\nIn the following we show sample code \\ref{label_1} in the Fortran programming language which calculates the \nauxiliary-field transfer matrix multiplications on the left starting from the \nsingle-nucleon initial states. We show only the terms which arise from \nthe free-nucleon transfer matrix and the auxiliary field $s$. \n\n\\begin{lstlisting}[language=Fortran,caption=Sample code calculating \nthe auxiliary-field transfer matrix multiplications on the left starting from the single-nucleon initial states,label={label_1}]\nDO nt = nt1+1, nt2 \n DO np = 0,num-1 \n DO nz = 0,L-1; DO ny = 0,L-1; DO nx = 0,L-1; DO ni = 0,1; DO ns = 0,1\n \n zvecs(nx,ny,nz,nt,ns,ni,np) = zvecs(nx,ny,nz,nt-1,ns,ni,np) &\n * (1.D0-6.D0*w0_N*h+CDSQRT(-c0*atovera*(1.D0,0.D0))*s(nx,ny,nz,nt-1))\n \n zvecs(nx,ny,nz,nt,ns,ni,np) = zvecs(nx,ny,nz,nt,ns,ni,np) &\n + w1_N*h*zvecs(MOD(nx+1,L),ny,nz,nt-1,ns,ni,np) &\n + w1_N*h*zvecs(MOD(nx-1+L,L),ny,nz,nt-1,ns,ni,np) &\n + w1_N*h*zvecs(nx,MOD(ny+1,L),nz,nt-1,ns,ni,np) &\n + w1_N*h*zvecs(nx,MOD(ny-1+L,L),nz,nt-1,ns,ni,np) &\n + w1_N*h*zvecs(nx,ny,MOD(nz+1,L),nt-1,ns,ni,np) &\n + w1_N*h*zvecs(nx,ny,MOD(nz-1+L,L),nt-1,ns,ni,np)\n \n IF (improveN >= 1) THEN \n zvecs(nx,ny,nz,nt,ns,ni,np) = zvecs(nx,ny,nz,nt,ns,ni,np) & \n - w2_N*h*zvecs(MOD(nx+2,L),ny,nz,nt-1,ns,ni,np) &\n - w2_N*h*zvecs(MOD(nx-2+L,L),ny,nz,nt-1,ns,ni,np) &\n - w2_N*h*zvecs(nx,MOD(ny+2,L),nz,nt-1,ns,ni,np) &\n - w2_N*h*zvecs(nx,MOD(ny-2+L,L),nz,nt-1,ns,ni,np) &\n - w2_N*h*zvecs(nx,ny,MOD(nz+2,L),nt-1,ns,ni,np) &\n - w2_N*h*zvecs(nx,ny,MOD(nz-2+L,L),nt-1,ns,ni,np)\n END IF\n \n IF (improveN == 2) THEN\n zvecs(nx,ny,nz,nt,ns,ni,np) = zvecs(nx,ny,nz,nt,ns,ni,np) &\n + w3_N*h*zvecs(MOD(nx+3,L),ny,nz,nt-1,ns,ni,np) &\n + w3_N*h*zvecs(MOD(nx-3+L,L),ny,nz,nt-1,ns,ni,np) &\n + w3_N*h*zvecs(nx,MOD(ny+3,L),nz,nt-1,ns,ni,np) &\n + w3_N*h*zvecs(nx,MOD(ny-3+L,L),nz,nt-1,ns,ni,np) & \n + w3_N*h*zvecs(nx,ny,MOD(nz+3,L),nt-1,ns,ni,np) &\n + w3_N*h*zvecs(nx,ny,MOD(nz-3+L,L),nt-1,ns,ni,np)\n END IF\n \n END DO; END DO; END DO; END DO; END DO\n END DO\nEND DO\n\n\\end{lstlisting}\n\n\nSimilarly, we now show sample code \\ref{label_2} which calculates \nthe auxiliary-field transfer matrix multiplications on the right starting from \nthe single-nucleon final states. Again we present only the terms arising from the free-nucleon transfer matrix and the auxiliary field $s$. \n\n\\begin{lstlisting}[language=Fortran,caption=Sample code calculating \nthe auxiliary-field transfer matrix multiplications on the right starting\nfrom the single-nucleon final states,label={label_2}]\nDO nt = nt2,nt1+1,-1\n DO np = 0,num-1 \n DO nz = 0,L-1; DO ny = 0,L-1; DO nx = 0,L-1; DO ni = 0,1; DO ns = 0,1\n \n zdualvecs(nx,ny,nz,nt-1,ns,ni,np) &\n = zdualvecs(nx,ny,nz,nt,ns,ni,np) &\n * (1.D0-6.D0*w0_N*h+CDSQRT(-c0*atovera*(1.D0,0.D0))*s(nx,ny,nz,nt-1))\n \n zdualvecs(nx,ny,nz,nt-1,ns,ni,np) &\n = zdualvecs(nx,ny,nz,nt-1,ns,ni,np) & \n + w1_N*h*zdualvecs(MOD(nx+1,L),ny,nz,nt,ns,ni,np) &\n + w1_N*h*zdualvecs(MOD(nx-1+L,L),ny,nz,nt,ns,ni,np) &\n + w1_N*h*zdualvecs(nx,MOD(ny+1,L),nz,nt,ns,ni,np) &\n + w1_N*h*zdualvecs(nx,MOD(ny-1+L,L),nz,nt,ns,ni,np) &\n + w1_N*h*zdualvecs(nx,ny,MOD(nz+1,L),nt,ns,ni,np) &\n + w1_N*h*zdualvecs(nx,ny,MOD(nz-1+L,L),nt,ns,ni,np) \n \n IF (improveN >= 1) THEN\n zdualvecs(nx,ny,nz,nt-1,ns,ni,np) &\n = zdualvecs(nx,ny,nz,nt-1,ns,ni,np) &\n - w2_N*h*zdualvecs(MOD(nx+2,L),ny,nz,nt,ns,ni,np) &\n - w2_N*h*zdualvecs(MOD(nx-2+L,L),ny,nz,nt,ns,ni,np) &\n - w2_N*h*zdualvecs(nx,MOD(ny+2,L),nz,nt,ns,ni,np) &\n - w2_N*h*zdualvecs(nx,MOD(ny-2+L,L),nz,nt,ns,ni,np) &\n - w2_N*h*zdualvecs(nx,ny,MOD(nz+2,L),nt,ns,ni,np) &\n - w2_N*h*zdualvecs(nx,ny,MOD(nz-2+L,L),nt,ns,ni,np) \n END IF\n \n IF (improveN == 2) THEN\n zdualvecs(nx,ny,nz,nt-1,ns,ni,np) &\n = zdualvecs(nx,ny,nz,nt-1,ns,ni,np) &\n + w3_N*h*zdualvecs(MOD(nx+3,L),ny,nz,nt,ns,ni,np) &\n + w3_N*h*zdualvecs(MOD(nx-3+L,L),ny,nz,nt,ns,ni,np) &\n + w3_N*h*zdualvecs(nx,MOD(ny+3,L),nz,nt,ns,ni,np) &\n + w3_N*h*zdualvecs(nx,MOD(ny-3+L,L),nz,nt,ns,ni,np) &\n + w3_N*h*zdualvecs(nx,ny,MOD(nz+3,L),nt,ns,ni,np) &\n + w3_N*h*zdualvecs(nx,ny,MOD(nz-3+L,L),nt,ns,ni,np) \n END IF\n \n END DO; END DO\n END DO; END DO; END DO\nEND DO\n\n\\end{lstlisting}\n\nIn the following we show sample code \\ref{label_3} where these transfer matrix product multiplications are called as subroutines and used to compute the determinant and inverse of the matrix of single-nucleon amplitudes {\\bf M}. \n\n\\begin{lstlisting}[language=Fortran,caption=Sample code where transfer matrix product multiplications are called and used to compute the determinant and inverse of the matrix of single-nucleon\namplitudes.,label={label_3}]\nCALL getzvecs(s,sI,zvecs,zwave,Lt,0,pion,ztau2x2,n_f) \nCALL getzdualvecs(s,sI,zdualvecs,zdualwave,Lt,0,pion,ztau2x2,n_f)\nCALL getinvcorr(zvecs,zdualvecs,zldeter,zcorrmatrix,zcorrinv,Lt)\n\naldeterabs = DBLE(zldeter)\nzdeterphase = CDEXP((0.D0,1.D0)*DIMAG(zldeter))\nact = bose - aldeterabs\n\\end{lstlisting}\n\n\\section{Importance sampling}\n\nWe do importance sampling according to the positive measure \n\\begin{equation}\n|Z(L_t)|\\exp{\\left[-S_{ss}(s)\n-S_{s_Is_I}(s_I)\n-S_{\\pi_I\\pi_I}(\\pi_I)\\right]},\n\\end{equation} \nand use hybrid Monte Carlo to do global updates of the auxiliary and pion\nfields. The hybrid Monte Carlo~(HMC) algorithm\n\\cite{Scalettar:1986uy,Gottlieb:1987mq,Duane:1987de} is efficient in quickly generating decorrelated configurations for\neach\nauxiliary and pion field. Here we describe the\nupdating algorithm for the $s$ field. The updating of the $s_I$ and $\\pi_I$\nfields proceed in a very similar fashion. In\ngeneral terms, the HMC algorithm can be described by means of a probability\nweight $P(s)$\n\\begin{equation}\nP(s)\\propto\\exp[-V(s)],\n\\end{equation}\nwhere $V(s)$ is in general a non-local function of the field $s({{\\bf n}%\n,n_t}),$ and a molecular dynamics~(MD) Hamiltonian,\n\\begin{equation}\n\\quad\\quad H(s,p)\\equiv\\frac{1}{2}\\sum_{{\\bf n},n_t%\n}\\left[p_s({{\\bf n},n_t})\\right] ^{2}+V(s).\n\\end{equation}\nClassical Hamiltonian dynamics is introduced by defining\nthe\nmomentum $p_s({{\\bf n},n_t})$ conjugate to $s({{\\bf n},n_t})$.\n\nGiven an arbitrary initial configuration $s^{0}({{\\bf n},n_t})$, the\nconjugate momentum is chosen from a random Gaussian distribution according\nto\n\\begin{equation}\nP[p_s^{0}({\\bf n},n_t)]\\propto\\exp\\left\\{ -\\frac{1}{2}\\left[ p_s^{0}({\\bf\nn},n_t)\\right] ^{2}\\right\\} , \\label{Gdistr}%\n\\end{equation}\nafter which the Hamiltonian equations of motion are integrated numerically\nwith a small but nonzero step size $\\varepsilon_{\\mathrm{step}}$. This method begins\nwith a\n\\textquotedblleft half-step\\textquotedblright\\ forward in the conjugate\nmomentum,\n\\begin{equation}\n\\tilde{p}_s^{0}({\\bf n},n_t)=p_s^{0}({\\bf n},n_t)-\\frac{\\varepsilon\n_{\\mathrm{step}}}{2}\\left[ \\frac{\\partial V(s)}{\\partial s({\\bf n},n_t%\n)}\\right] _{s=s^{0}},\n\\end{equation}\nfollowed by repeated updates of $s$ and $\\tilde{p}_s$ according to\n\\begin{equation}\ns^{i+1}({\\bf n},n_t)=s^{i}({\\bf n},n_t)+\\varepsilon_{\\mathrm{step}}%\n\\tilde{p}^{i}_s({\\bf n},n_t),\\quad\\quad\\tilde{p}^{i+1}_s({\\bf n},n_t%\n)=\\tilde{p}^{i}_s({\\bf n},n_t)-\\varepsilon_{\\mathrm{step}}\\left[\n\\frac{\\partial V(s)}{\\partial s({\\bf n},n_t)}\\right] _{s=s^{i+1}},\n\\end{equation}\nfor a specified number of steps $N_{\\mathrm{step}}$. This is followed by an additional half-step\nbackward in $\\tilde{p}_s$ given by\n\\begin{equation}\np_s^{N_{\\mathrm{step}}}({\\bf n},n_t)=\\tilde{p}_s^{N_{\\mathrm{step}}}(\\vec\n{n},n_t)+\\frac{\\varepsilon_{\\mathrm{step}}}{2}\\left[ \\frac{\\partial\nV(s)}{\\partial s({\\bf n},n_t)}\\right] _{s=s^{0}}.\n\\end{equation}\n\nFor algorithmic efficiency the length of such an MD \\textquotedblleft trajectory\\textquotedblright%\n\\ should be taken large enough to ensure decorrelation between\nsuccessive configurations of the auxiliary field. The evolved configuration\nis\nthen subjected to a \\textquotedblleft Metropolis test\\textquotedblright%\n\\ against a random number $r\\in\\lbrack0,1)$. The new configuration is accepted if\n\\begin{equation}\nr<\\exp\\left[ -H(s^{N_{\\mathrm{step}}},p^{N_{\\rm step }}_s)+H(s^{0}%\n,p_s^{0})\\right] .\n\\end{equation}\nIt\nshould be noted that although $H$ is in principle conserved in the MD\nevolution, the truncation error of the leapfrog method introduces a systematic\nerror. The Metropolis test eliminates the need for extrapolation in\n$\\varepsilon_{\\mathrm{step}}$. \n\nIn our case $\\exp[-V(s)]$ has the form \n\\begin{equation}\n|Z(L_t)|\\exp{\\left[-S_{ss}(s)\n-S_{s_Is_I}(s_I)\n-S_{\\pi_I\\pi_I}(\\pi_I)\\right]},\n\\end{equation} \nwhere $Z(L_t)$ is the determinant\nof an $A\\times A$ matrix of single-nucleon amplitudes {\\bf M}. The derivative of\n$V_{}$ is then computed using%\n\\begin{align}\n\\frac{\\partial V_{}(s)}{\\partial s({\\bf n},n_t)} & =\\frac{\\partial\nS_{ss}(s)}{\\partial s({\\bf n},n_t)}- \\frac{\\partial {\\rm Re} \\left[\n\\ln \\left( \\det\\mathbf{M} \\right) \\right]}{\\partial s(\\vec\n{n},n_t)} \\nonumber\\\\\n & =\\frac{\\partial\nS_{ss}(s)}{\\partial s({\\bf n},n_t)}- {\\rm Re} \\left[ \\frac{1}{\\det\\mathbf{M}}\\sum\n_{k,l}\\frac{\\partial\\det\\mathbf{M}}{\\partial \\mathbf{M}\n_{kl}}\\frac{\\partial \\mathbf{M}_{kl}}{\\partial s(\\vec\n{n},n_t)} \\right] \\nonumber\\\\\n& =\\frac{\\partial S_{ss}(s)}{\\partial s({\\bf n},n_t)}- {\\rm Re} \\left[\n\\sum_{k,l}\n\\mathbf{M}^{-1} _{lk}\\frac{\\partial \\mathbf{M}\n_{kl}}{\\partial s({\\bf n},n_t)} \\right].\n\\end{align}\n\nIn the following we show sample code \\ref{label_4} calculating the\nquadratic part of the action due to the auxiliary fields and pion fields,\n\\begin{equation}\n\\frac{1}{2}\\sum_{{\\bf n},n_t}\\left[p_s({{\\bf n},n_t})\\right]^{2}+\\frac{1}{2}\\sum_{{\\bf n},n_t,I}\\left[ p_{s_I}({{\\bf n},n_t})\\right]^{2}+\\frac{1}{2}\\sum_{{\\bf n},n_t,I}\\left[p_{\\pi_I}({{\\bf n},n_t})\\right]^{2}+S_{ss}(s) + S_{s_Is_I}(s_I) + S_{\\pi_I\\pi_I}(\\pi_I).\n\\end{equation}\nIn the code we have found it convenient to rescale the pion field by a factor of $\\sqrt{q_{\\pi}}$ where \n\\begin{equation}\nq_{\\pi}=\\alpha_{t} \\left( m_{\\pi}^2 + 6w_0 \\right).\n\\end{equation}\n\n\\begin{lstlisting}[language=Fortran,caption=Sample code calculating the\nquadratic part of the action due to the auxiliary fields and pion fields.,label={label_4}]\nbose = 0.D0\nDO nt = 0,Lt-1\n DO nz = 0,L-1; DO ny = 0,L-1; DO nx = 0,L-1\n bose = bose &\n + s(nx,ny,nz,nt)**2.D0/2.D0 &\n + p_s(nx,ny,nz,nt)**2.D0/2.D0\n DO iso = 1,3\n bose = bose &\n + sI(nx,ny,nz,nt,iso)**2.D0/2.D0 &\n + p_sI(nx,ny,nz,nt,iso)**2.D0/2.D0 &\n + pion(nx,ny,nz,nt,iso)**2.D0/2.D0 &\n + atovera/qpi3*pion(nx,ny,nz,nt,iso)*( &\n - w1_P*pion(MOD(nx+1,L),ny,nz,nt,iso) &\n - w1_P*pion(nx,MOD(ny+1,L),nz,nt,iso) &\n - w1_P*pion(nx,ny,MOD(nz+1,L),nt,iso) &\n + w2_P*pion(MOD(nx+2,L),ny,nz,nt,iso) &\n + w2_P*pion(nx,MOD(ny+2,L),nz,nt,iso) &\n + w2_P*pion(nx,ny,MOD(nz+2,L),nt,iso) &\n - w3_P*pion(MOD(nx+3,L),ny,nz,nt,iso) &\n - w3_P*pion(nx,MOD(ny+3,L),nz,nt,iso) &\n - w3_P*pion(nx,ny,MOD(nz+3,L),nt,iso)) &\n + p_pion(nx,ny,nz,nt,iso)**2.D0/2.D0\n END DO\n END DO; END DO; END DO\nEND DO\n\n\\end{lstlisting} \n\nIn following we show sample code \\ref{label_5} which calculates \n\\begin{equation}\n\\left[ \\frac{\\partial V(s)}{\\partial s({\\bf n},n_t%\n)}\\right] _{s=s^{0}}\n\\end{equation}\nand uses it to compute the\n half-step\\ forward in the conjugate\nmomentum,\n\\begin{equation}\n\\tilde{p}_s^{0}({\\bf n},n_t)=p_s^{0}({\\bf n},n_t)-\\frac{\\varepsilon\n_{\\mathrm{step}}}{2}\\left[ \\frac{\\partial V(s)}{\\partial s({\\bf n},n_t%\n)}\\right] _{s=s^{0}}.\n\\end{equation}\n\n\\begin{lstlisting}[language=Fortran,caption=Sample code computing derivative with respect to the auxiliary field and half-step forward in the conjugate momentum.,label={label_5}]\n\nDO npart1 = 0,n_f-1; DO npart2 = 0,n_f-1\n zdcorrmatrix(npart2,npart1) = 0.D0\n DO ni = 0,1; DO ns = 0,1\n zdcorrmatrix(npart2,npart1) = &\n zdcorrmatrix(npart2,npart1) + &\n zdualvecs(nx,ny,nz,nt+1,ns,ni,npart2) &\n *zvecs(nx,ny,nz,nt,ns,ni,npart1) &\n *CDSQRT(-c0*atovera*(1.D0,0.D0))/L**3\n END DO; END DO\nEND DO; END DO\n \ndVds(nx,ny,nz,nt) = s(nx,ny,nz,nt) \n\nDO npart1 = 0,n_f-1; DO npart2 = 0,n_f-1\n dVds(nx,ny,nz,nt) = dVds(nx,ny,nz,nt) &\n - DBLE(zdcorrmatrix(npart2,npart1) &\n *zcorrinv(npart1,npart2))\nEND DO; END DO\n \np_sHMC(nx,ny,nz,nt,0) = &\n p_s(nx,ny,nz,nt) - 0.5D0*eHMC*dVds(nx,ny,nz,nt)\n \n\\end{lstlisting}\n\nIn following code \\ref{label_6} we show an example code which performs the Metropolis test%\n\\ against a random number $r\\in\\lbrack0,1)$, with the new configuration being accepted if\n\\begin{equation}\nr<\\exp\\left[ -H(s^{N_{\\mathrm{step}}},p^{N_{\\rm step }}_s)+H(s^{0}%\n,p_s^{0})\\right].\n\\end{equation}\n\n\\begin{lstlisting}[language=Fortran,caption=Sample code which performs the Metropolis acceptance test,label={label_6}]\nIF (ntrial .eq. 1 .or. grnd() .lt. DEXP(-actnew+act)) THEN\n\n accept = accept + 1.\n\n DO nt = 0,Lt-1\n DO nz = 0,L-1; DO ny = 0,L-1; DO nx = 0,L-1 \n s(nx,ny,nz,nt) = snew(nx,ny,nz,nt)\n END DO; END DO; END DO\n END DO\n \n DO nt = 0,Lt-1\n DO nz = 0,L-1; DO ny = 0,L-1; DO nx = 0,L-1 \n DO iso = 1,3\n sI(nx,ny,nz,nt,iso) = sInew(nx,ny,nz,nt,iso)\n pion(nx,ny,nz,nt,iso) = pionnew(nx,ny,nz,nt,iso)\n END DO\n END DO; END DO; END DO\n END DO\n \n aldeterabs = aldeternewabs\n zdeterphase = zdeternewphase\n \nEND IF\n\\end{lstlisting}\n\nAlthough the Monte Carlo importance sampling uses only the absolute value of the amplitude, the complex phase\nof the amplitude is treated as an observable and is collected with each\nconfiguration of the auxiliary and pion fields.\n\n\\section{Exercises}\n\n\\begin{prob}\n\\label{prob6.1}\nWrite a lattice hybrid Monte Carlo code which performs updates of the lattice action according to only the quadratic part of the action due to the auxiliary fields and pions,\n\\begin{equation}\n\\frac{1}{2}\\sum_{{\\bf n},n_t}\\left[p_s({{\\bf n},n_t})\\right]^{2}+\\frac{1}{2}\\sum_{{\\bf\nn},n_t,I}\\left[ p_{s_I}({{\\bf n},n_t})\\right]^{2}+\\frac{1}{2}\\sum_{{\\bf\nn},n_t,I}\\left[p_{\\pi_I}({{\\bf n},n_t})\\right]^{2}+S_{ss}(s) + S_{s_Is_I}(s_I)\n+ S_{\\pi_I\\pi_I}(\\pi_I).\n\\end{equation} \nVerify that the change in the action produced by the hybrid Monte Carlo update is scaling quadratically in the step size, $\\varepsilon_{\\mathrm{step}}$, in the limit $\\varepsilon_{\\mathrm{step}} \\rightarrow 0$ with $N_{\\rm step}\\varepsilon_{\\mathrm{step}}$ held fixed. \n\\end{prob} \n\n\\begin{prob}\n\\label{prob6.2}\nWrite a function or subroutine that generates initial/final single-nucleon states on the lattice corresponding to a Slater-determinant state with one neutron spin-up and one neutron spin-down, both with zero momentum. \\end{prob} \n\n\\begin{prob}\n\\label{prob6.3}\nWrite a function or subroutine that generates initial/final single-nucleon\nstates on the lattice corresponding to a Slater-determinant state with one proton spin-up and one neutron spin-up, both with zero momentum.\\end{prob}\n\n\\begin{prob}\n\\label{prob6.4}\nWrite a function or subroutine that generates initial/final single-nucleon\nstates on the lattice corresponding to a Slater-determinant state of four\nnucleons --- proton spin-up, proton spin-down, neutron spin-up, and neutron\nspin-down --- each with zero momentum. \\end{prob}\n\n\\begin{prob}\n\\label{prob6.5}\nWrite a function or subroutine that extends the sample code \\ref{label_1} to repeatedly multiply the auxiliary-field transfer matrix on the left starting from the initial single-nucleon wave functions. Include the contributions from the auxiliary fields $s$ and $s_I$ as well as the pion\n field $\\pi_I$. \n\\end{prob}\n\n\\begin{prob}\n\\label{prob6.6}\nWrite a function or subroutine that extends the sample code \\ref{label_2} to\nrepeatedly multiply the auxiliary-field transfer matrix on the right starting from the final\nsingle-nucleon wave functions. Include the contributions from the auxiliary\nfields $s$ and $s_I$ as well as the pion\n field $\\pi_I$. \n\\end{prob}\n\n\\begin{prob}\n\\label{prob6.7}\nUse the Slater-determinant states constructed in Probs.~\\ref{prob6.2}, \\ref{prob6.3}, \\ref{prob6.4} as initial and final states. In each case apply the functions or subroutines written in Prob.~\\ref{prob6.5} and Prob.~\\ref{prob6.6} with all coupling constants set to zero. Verify that in each case the initial/final state is the ground state of the non-interacting system with energy equal to zero.\n\\end{prob}\n\n\\begin{prob}\n\\label{prob6.8}\nUse the Slater-determinant states constructed in Probs.~\\ref{prob6.2}, \\ref{prob6.3},\n\\ref{prob6.4} as initial and final states. Using the functions or subroutines written in Prob.~\\ref{prob6.5} and Prob.~\\ref{prob6.6}, extend the sample code \\ref{label_5} to compute the derivatives of $V(s)$ with respect to $s({\\bf n},n_t)$, $s_I({\\bf n},n_t)$, and $\\pi_I({\\bf n},n_t)$.\n\\end{prob}\n\n\\begin{prob}\n\\label{prob6.9}\nTake the code you have written for Prob.~\\ref{prob6.8} and complete the remaining steps needed to do hybrid Monte Carlo updates for $s$, $s_I$, and $\\pi_I$. Verify that the change in the action produced by the hybrid Monte Carlo update\nis scaling quadratically in $\\varepsilon_{\\mathrm{step}}$ in the limit\n$\\varepsilon_{\\mathrm{step}} \\rightarrow 0$ with $N_{\\rm step}\\varepsilon_{\\mathrm{step}}$\nheld fixed.\n\\end{prob} \n\n\\begin{prob}\n\\label{prob6.10}\nTake the code you have written for Prob.~\\ref{prob6.9} and complete the remaining\nsteps needed to calculate the energy of the ground state by computing the ratio of the amplitudes $Z(L_t)/Z(L_t-1)$.\n\\end{prob}\n\n\\section{Codes and Benchmarks}\n\nComplete verisons of the codes discussed in this chapter and developed in the exercises can be found online via \\href{https://github.com/ManyBodyPhysics/LectureNotesPhysics/tree/master/Programs/Chapter6-programs}{this link}. In order to run the codes, one must first copy the corresponding initial/final wavefunctions (waveinit\\_1S0.f90, waveinit\\_3S1.f90, or waveinit\\_He4.f90) into the file waveinit.f90 used by the main program nuclei.f90. The number of nucleons is controlled by the parameter n\\_f in input.f90 and must correspond to the number of nucleons in waveinit.f90. \n\nAs an example we show the beginning of the input file input.f90 for a two nucleon state with spatial lattice spacing $a = 1/(100\\;{\\rm MeV})$, temporal lattice spacing $a_t = 1/(150\\;{\\rm MeV})$, box size $L = 4a$, and Euclidean time extent $L_t = 6a_t$. We use an $O(a^{4})$-improved lattice action for the nucleon hopping coefficients, $O(a^{0})$-improved lattice action for the pion hopping coefficients, and $O(a^{0})$-improved lattice action for the pion-nucleon coupling. The coefficient of the $^1S_0$ contact interaction is tuned to the physical $^1S_0$ $n-p$ scattering length and is $-5.615\\times 10^{-5}\\;{\\rm MeV}^{-2}$. The coefficient of the $^3S_1$ contact interaction is tuned to the deuteron binding energy at infinite volume and is $-6.543\\times 10^{-5}\\;{\\rm MeV}^{-2}$.\n\n\\begin{lstlisting}[language=Fortran,caption=Parameter declarations at the beginning of the file input.f90.]\n parameter(n_f = 2)\n parameter(L = 4)\n parameter(Lt = 6)\n parameter(cutoff = 100.D0, temporalcutoff = 150.D0)\n parameter(improveN = 2)\n parameter(improveP = 0)\n parameter(improveD = 0)\n parameter(c1S0_phys = -5.615D-5)\n parameter(c3S1_phys = -6.543D-5)\n \n\\end{lstlisting}\n\nUsing these values for the parameters of the lattice action, we now present some benchmark values which can be used to test the nuclear lattice simulations in the two-nucleon system. The values presented in these benchmarks are computed using exact calculations of the two-nucleon transfer matrix. They provide a useful independent check that there are no errors in the Monte Carlo simulations. In Table~\\ref{1S0} we show the energies for the $^1S_0$ spin combination\nof two nucleons. The initial state is one\nneutron spin-up and one neutron spin-down, both at zero momentum, for $L=4a$\nand various values of\n$L_t$. The energies are extracted by computing the\nratio of amplitudes $Z(L_t)/Z(L_t-1)$ and setting equal to $\\exp(-E\\alpha_t)$. \n\n\\begin{table}[tbh]\n\\caption{Benchmark energies for the $^1S_0$ spin combination\nof two nucleons. The initial state is one\nneutron spin-up and one neutron spin-down, both at zero momentum, for $L=4a$\nand various values of $L_t$.}%\n\\begin{center}\n\\begin{tabular}{p{1.5cm}p{1.5cm}}\n\\hline\\noalign{\\smallskip}\n$L_t$ & ${\\rm energy (MeV)}$ \\\\\n\\noalign{\\smallskip}\\svhline\\noalign{\\smallskip}\n$2$ & $-1.0915$ \\\\\n$4$ & $-1.3987$ \\\\\n$6$ & $-1.6209$ \\\\\n$8$ & $-1.7929$ \\\\ \n$10$ & $-1.9296$ \\\\\n$12$ & $-2.0398$ \\\\\n$14$ & $-2.1291$ \\\\\n$16$ & $-2.2018$ \\\\\n$18$ & $-2.2610$ \\\\\n$20$ & $-2.3094$ \\\\\n\\noalign{\\smallskip}\n\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{center}\n\\label{1S0}%\n\\end{table}\n\nWe show the energies for the $^3S_1$ spin combination\nof two nucleons in Table~\\ref{3S1}. The initial state is one\nproton spin-up and one neutron spin-up, both at zero momentum, for $L=4a$\nand various values of\n$L_t$. The energies are extracted by computing the\nratio of amplitudes $Z(L_t)/Z(L_t-1)$ and setting equal to $\\exp(-E\\alpha_t)$. \n\n\n\\begin{table}[tbh]\n\\caption{Benchmark energies for the $^3S_1$ spin combination\nof two nucleons. The initial state is one\nproton spin-up and one neutron spin-up, both at zero momentum, for $L=4a$\nand various values of\n$L_t$.}%\n\\begin{center}\n\\begin{tabular}{p{1.5cm}p{1.5cm}}\n\\hline\\noalign{\\smallskip}\n$L_t$ & ${\\rm energy (MeV)}$ \\\\\n\\noalign{\\smallskip}\\svhline\\noalign{\\smallskip}\n$2$ & $-1.4446 $ \\\\\n$4$ & $-2.0400$ \\\\\n$6$ & $-2.4774$ \\\\\n$8$ & $-2.8331$ \\\\ \n$10$ & $-3.1341$ \\\\\n$12$ & $-3.3925$ \\\\\n$14$ & $-3.6151$ \\\\\n$16$ & $-3.8069 $ \\\\\n$18$ & $-3.9718$ \\\\\n$20$ & $-4.1132$ \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n\\end{tabular}\n\\end{center}\n\\label{3S1}%\n\\end{table}\n\n\n\\begin{acknowledgement}\nThe author is grateful for discussions with Amy Nicholson and Morten\nHjorth-Jensen. He is also greatly indebted to his collaborators Jose\nAlarc{\\'o}n, Dechuan Du, Serdar Elhatisari, Evgeny Epelbaum, Nico\nKlein, Hermann Krebs, Timo L{\\\"a}hde, Ning Li, Bing-nan Lu, Thomas\nLuu, Ulf-G. Mei{\\ss}ner, Alexander Rokash, and Gautam Rupak. Partial\nfinancial support provided by the U.S. Department of Energy\n(DE-FG02-03ER41260). Computational resources were provided by the\nJ\\\"{u}lich Supercomputing Centre.\n\\end{acknowledgement}\n\n\\bibliographystyle{spphys} \n\\bibliography{lnp}\n", "meta": {"hexsha": "2efc64d23a295ff27f16e8987f9324038b543203", "size": 56834, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/src/chapter6.tex", "max_stars_repo_name": "cpmoca/LectureNotesPhysics", "max_stars_repo_head_hexsha": "8e9f8c5d7f163ea10b14002850f7c79acc4513df", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 24, "max_stars_repo_stars_event_min_datetime": "2016-11-22T09:42:49.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-11T01:33:46.000Z", "max_issues_repo_path": "doc/src/chapter6.tex", "max_issues_repo_name": "cpmoca/LectureNotesPhysics", "max_issues_repo_head_hexsha": "8e9f8c5d7f163ea10b14002850f7c79acc4513df", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/src/chapter6.tex", "max_forks_repo_name": "cpmoca/LectureNotesPhysics", "max_forks_repo_head_hexsha": "8e9f8c5d7f163ea10b14002850f7c79acc4513df", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 25, "max_forks_repo_forks_event_min_datetime": "2016-05-24T22:54:07.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-20T00:08:19.000Z", "avg_line_length": 48.1644067797, "max_line_length": 1249, "alphanum_fraction": 0.7071647253, "num_tokens": 18923, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.3481511269961152}} {"text": "\\chapter{Bundle Adjustment}\n\\label{ch:bundle_adjustment}\n\n\\definecolor{lgray}{gray}{0.95}\n\nSatellite position and orientation errors have a direct effect on the\naccuracy of digital elevation models produced by the Stereo Pipeline.\nIf they're not corrected, these uncertainties will result in\nsystematic errors in the overall position and slope of the \\ac{DEM}. Severe\ndistortions can occur as well, resulting in twisted or ``taco shaped''\n\\acp{DEM}, though in most cases these effects are quite subtle and hard to\ndetect.\n\nThe Stereo Pipeline includes a powerful suite of tools for correcting\ncamera position and orientation errors using a process called\n\\emph{bundle adjustment}. Bundle adjustment is the process of\nsimultaneously adjusting the properties of many cameras and the 3D\nlocations of the objects they see in order to minimize the error\nbetween the estimated, back-projected pixel location of the 3D\nobjects and their actual measured location in the captured images.\n\nThat complex process can be boiled down to this simple idea: bundle\nadjustment ensures that observations in multiple different images of a\nsingle ground feature are self-consistent. If they are not consistent,\nthen the position and orientation of the cameras as well as the 3D\nposition of the feature must be adjusted until they are. This\noptimization is carried out along with thousands (or more) of similar\nconstraints involving many different features observed in other\nimages. Bundle adjustment is very powerful and versatile: it can\noperate on just two overlapping images, or on thousands.\n\n\\begin{figure}[bt]\n \\centering\n \\includegraphics[width=8cm]{images/ba_orig}\n \\includegraphics[width=8cm]{images/ba_adjusted}\n \\caption{Bundle adjustment is illustrated here using a color-mapped,\n hill-shaded DEM mosaic from Apollo 15 Orbit 33 imagery. (a)\n Prior to bundle adjustment, large discontinuities can exist between\n overlapping DEMs made from different images. (b) After bundle\n adjustment, DEM alignment errors are minimized, and no longer visible.}\n \\label{fig:bundle_adjustment}\n\\end{figure}\n\nBundle adjustment can also take advantage of \\acp{GCP}, which are\n3D locations of features that are known a priori (often by measuring\nthem by hand in another existing \\ac{DEM}). \\acp{GCP} can improve the internal\nconsistency of your \\ac{DEM} or align your \\ac{DEM} to an existing data\nproduct. Finally, even though bundle adjustment calculates the\nlocations of the 3D objects it views, only the final properties of\nthe cameras are recorded for use by the Ames Stereo Pipeline. Those\nproperties can be loaded into the \\texttt{stereo} program which\nuses its own method for triangulating 3D feature locations.\n\nWhen using the Stereo Pipeline, bundle adjustment is an optional step\nbetween the capture of images and the creation of \\acp{DEM}. The bundle\nadjustment process described below should be completed prior to\nrunning the \\texttt{stereo} command.\n\nAlthough bundle adjustment is not a required step for generating\n\\acp{DEM}, it is {\\em highly recommended} for users who plan to\ncreate \\acp{DEM} for scientific analysis and publication. Incorporating\nbundle adjustment into the stereo work flow not only results in\n\\acp{DEM} that are more internally consistent, it is also the correct\nway to co-register your \\acp{DEM} with other existing data sets and\ngeodetic control networks.\n\nAt the moment however, Bundle Adjustment does not automatically work\nagainst outside DEMs from sources such as laser altimeters. Hand\npicked \\acp{GCP} are the only way for \\acp{ASP} to register to those\ntypes of sources.\n\n\\subsection{A deeper understanding}\n\nIn bundle adjustment the position and orientation of each camera\nstation are determined jointly with the 3D position of a set of image\ntie-points points chosen in the overlapping regions between\nimages. Tie points, like they sound, tie individual camera images\ntogether. Their physical manifestation would be a rock or small crater\nthan can be observed across multiple images.\n\nTie-points can be automatically extracted using Vision Workbench's\nInterest Point module or through a number of outside methods such as\nthe famous SURF\\citep{surf08}. Creating a tie point is a three step\nprocess. First, all images are processed for their natural {\\it\n 'interesting'} points. In most algorithms, an interest point is\ndefined as a place where image gradients accumulate together into a\npeak. These interesting points are then described by the texture that\nsurrounds them. Finally the described interesting points are matched\nacross the multiple images. A single matched pair of interest points\nis now a tie point to use in bundle adjustment. In application, there\nis also a little filtering of the tie points with RANSAC\n\\citep{fischler81}.\n\n\\begin{figure}[b!]\n \\begin{center}\n \\includegraphics[trim=20mm 20mm 20mm 15mm,clip,width=6in]{images/ba_feature_observation.pdf}\n \\end{center}\n \\caption{ A feature observation in bundle adjustment, from \\citet{moore09} }\n \\label{fig:ba_feature}\n\\end{figure}\n\nOur bundle adjustment approach follows the method described\nin~\\cite{triggs00} and determines the best camera parameters that\nminimize the projection error given by ${\\bf \\epsilon} =\n\\sum_k\\sum_j(I_k-I(C_j, X_k))^2$ where $I_k$ are the tie points on the\nimage plane, $C_j$ are the camera parameters, and $X_k$ are the 3D\npositions associated with features $I_k$. $I(C_j, X_k)$ is an image\nformation model (i.e. forward projection) for a given camera and 3D\npoint. To recap, it projects the 3D point, $X_k$, into the camera with\nparameters $C_j$. This produces a predicted image location for the 3D\npoint that is compared against the observed location, $I_k$. We reduce\nthis error with the Levenberg-Marquardt algorithm (LMA). Speed is\nimproved by using sparse methods as described in \\citet{hartley04}.\n\nEven though the arithmetic for bundle adjustment sounds clever. There\nare faults with the base implementation. Imagine a case where all\ncameras and 3D points were collapsed into a single point. If you\nevaluate the above cost function, you'll find that the error is indeed\nzero. Sadly, this is not the correct solution if the images were taken\nfrom orbit. Another example is if a translation was applied equally to\nall 3D points and camera locations. This again would not effect the\ncost function. This fault comes from bundle adjustment's inability to\ncontrol scale and translation of the solution. It will correct\ngeometric shape of the problem. Yet it can not guarantee that solution\nwill have correct scale and translation.\n\nWe attempt to fix this problem by adding two additional cost functions\nto bundle adjustment. First of which is ${\\bf \\epsilon} =\n\\sum_j(C_j^{initial}-C_j)^2$. This constrains camera parameters to\nstay relatively close to their initial values. Second, a small handful\nof 3D ground control points can be chosen by hand and added to the\nerror metric as ${\\bf \\epsilon} = \\sum_k(X_k^{gcp}-X_k)^2$ to\nconstrain these points to known locations in the planetary coordinate\nframe. A physical example of a ground control point could be the\nlocation of a lander that has a well known location. In the cost\nfunctions discussed above, errors are weighted by the inverse\ncovariance of the measurement that gave rise to the constraint.\n\nLike other iterative optimization methods, there are several\nconditions that will cause bundle adjustment to terminate. When\nupdates to parameters become insignificantly small or when the error,\n${\\bf \\epsilon}$, becomes insignificantly small, then the algorithm\nhas converged and the result is most likely as good as it will get.\nHowever, the algorithm will also terminate when the number of\niterations becomes too large, in which case bundle adjustment may or\nmay not have finished refining the parameters of the cameras.\n\n\\section{Performing bundle adjustment with isis\\_adjust}\n\nFirst off, let it be known that USGS's \\ac{ISIS} has its own bundle\nadjustment software called \\texttt{jigsaw}. It has a long history and\nis supported by a team of skilled developers. Despite this, Ames\nStereo Pipeline provides an alternative called \\texttt{isis\\_adjust}\nthat has its own benefits.\n\nLike \\texttt{jigsaw}, the \\texttt{isis\\_adjust} program is designed to\nperform bundle adjustment on images supported by \\ac{ISIS} 3\nsoftware package. The \\texttt{isis\\_adjust} program does not\ndiscriminate based on camera type. It can perform bundle adjustment\non images from line-scan imagers like \\ac{MOC}, \\ac{LROC} NAC,\n\\ac{HiRISE}, and the \\ac{CTX}. The \\texttt{isis\\_adjust} program can\nalso perform bundle adjustment on traditional frame cameras\n(e.g. Apollo Metric Camera). Theoretically it should also work with\npush-frame imagers like the \\ac{THEMIS} VIS, and \\ac{LROC} WAC, though\nthis is untested.\n\nThe \\texttt{isis\\_adjust} program works by first converting all\npixel measurements in an image to measurements defined on the ideal\nfocal plane using millimeters and the \\ac{ET}. The \\ac{ET} is the\nabsolute second at which that pixel measurement was recorded on the\ncamera. For a frame camera, all of the pixels are captured at the\nsame time so the \\ac{ET} will be identical for all measurements on\nthe image. For pushbroom, pushframe, or other cameras which build\nup their `image' over time, different parts of the image will have\ndifferent ET values. For example, on a MOC image between the first\nand last line about 5 seconds of \\ac{ET} will have elapsed.\n\nWhen \\texttt{isis\\_adjust} calculates the partial derivatives of the\nforward projection of a point, it uses an ideal pinhole camera model.\nThe properties of this model are defined as properties of the subject\ncamera at the specified \\ac{ET} for the current measure plus the\ncorrection function, $f(t)$, that \\texttt{isis\\_adjust} is solving\nfor. Many forms of $f(t)$ could be used; the only limit is the\nnumber of parameters in the equations. Some initial work hints that\nanything greater than a second order polynomial becomes an ill-posed\nproblem, but we hope to investigate this further in the future.\n\nOur \\texttt{isis\\_adjust} implements multiple bundle adjustment\nalgorithms which can be selected from the command line. It implements\nthe standard sparse algorithm that can be found in literature. It also\nimplements our research work into robust cost functions. This allows\nthe algorithm to ignore measurements that it thinks to be outliers.\n\nFinally, \\texttt{isis\\_adjust} stores its solution for a camera as a\ndelta that can be added to the camera's original SPICE in a separate\nfile. Those files have the extension \\texttt{*.isis\\_adjust}. This is\nless than ideal and one feature that we value in \\ac{ISIS}'s\n\\texttt{jigsaw}. Their software overwrites the SPICE information in\nthe cube file. This means that \\texttt{jigsaw} results can be used in\nother \\ac{ISIS} programs like \\texttt{cam2map}. This is not the case\nwith \\texttt{isis\\_adjust}'s results.\n\n\\subsection{Options}\n\nThe following is a listing and explanation of the options that can be\ngiven to \\texttt{isis\\_adjust} on the command line. These options are\nnot required.\n\n\\begin{description}\n\n\\item[\\texttt{-\\/-cnet|-c \\textit{control-network-file}}] \\hfill \\\\\n \\emph{Optional.} This option will force {\\tt isis\\_adjust} to\n use a pre-built built control network. This control network can\n either be in the \\ac{USGS} \\ac{ISIS} ``cnet'' format or in the binary Vision\n Workbench format.\n\n If no control network is supplied using this option, {\\tt\n isis\\_adjust} will look for match files in the current working\n directory with base filenames that match the input images. The\n \\texttt{isis\\_adjust} program will then create its own control\n network file and save it as \\texttt{isis\\_adjust.cnet}.\n\n\\item[\\texttt{-\\/-cost-function L1|L2|Cauchy|Huber|PseudoHuber(=L2)} ] \\hfill \\\\\n Sets the cost function used for bundle adjustment. Default is \\texttt{L2}\n which is the normal squared error. The full list of available options\n are:\n\n \\begin{description}\n \\item[L1] Proportional Error\n \\item[L2] Squared Error and Default Option.\n \\item[Cauchy]\n \\item[Huber]\n \\item[PseudoHuber]\n \\end{description}\n\n The options towards the end of the list are robust cost functions\n that deal better with non-ideal data that has outliers. These robust\n cost functions are performed by post-weighting the original errors yet\n still using equations derived for squared error.\n\n\\item[\\texttt{-\\/-bundle-adjuster Ref|Sparse|RobustRef|RobustSparse|RobustSparseKGCP(=Sparse)}] \\hfill \\\\\n Sets the bundle adjustment code to be used. The standard to use is\n \\texttt{Sparse}, which is a traditional squared error derived method that\n utilizes sparse matrices to obtain speed. Here are the complete list\n of options:\n\n \\begin{description}\n \\item[Ref] Reference implementation that doesn't use sparse methods.\n \\item[Sparse] Default implementation.\n \\item[RobustRef]\n \\item[RobustSparse]\n \\item[RobustSparseKGCP]\n \\end{description}\n\n The ending methods are experimental student-t derived bundle\n adjustment algorithms. Using the experimental robust algorithm\n overrides the {\\tt cost-function} option. The last two bundle\n adjustment algorithms are still experimental and are a work in\n progress.\n\n\\item[\\texttt{-\\/-disable-camera-const}] \\hfill \\\\\n \\emph{Optional.} This disables the camera constraint error. Useful\n for debugging and just exploring what are the effects of this cost\n function.\n\n\\item[\\texttt{-\\/-disable-gcp-const}] \\hfill \\\\\n \\emph{Optional.} This disables the \\ac{GCP} constraint error even\n if \\acp{GCP} are provided. Useful for debugging and just exploring\n what the effects are of \\acp{GCP}.\n\n\\item[\\texttt{-\\/-gcp-scalar \\textit{multiplier(=1)}}] \\hfill \\\\\n \\emph{Optional.} Sets the multiplier that is used to adjust the\n sigma (or uncertainty) of the \\aclp{GCP}. The sigmas of\n \\aclp{GCP} are defined in the \\ac{GCP} data file, so this\n option is useful when debugging for universally scaling \\ac{GCP} sigmas\n up or down.\n\n\\item[\\texttt{-\\/-lambda|-l \\textit{float}}] \\hfill \\\\\n \\emph{Optional.} This sets the starting value for $\\lambda$: the\n parameter in the Levenberg Marquardt (LMA) optimization algorithm\n that selects between Gauss-Newton optimization and gradient\n descent. This parameter evolves over time on its own, but this\n argument can be used to override its initial value. \\emph{This is an\n advanced setting, not recommended for normal use.}\n\n\\item[\\texttt{-\\/-min-matches \\textit{integer(=5)}}] \\hfill \\\\\n Set the minimum number of tie-points that are required between a\n pair of images for them to be included in the control network. This\n option is useful for eliminating tie-points from image pairs that\n have only a handful of poor or erroneous matches.\n\n\\item[\\texttt{-\\/-max-iterations \\textit{integer(=25)}}] \\hfill \\\\\n Sets the maximum number of iterations for bundle adjustment. The\n number of required iterations will vary by problem size, so this\n parameter allows the user to decide how much time they're willing\n to dedicate to the correction of the data. We have found that 20\n iterations suffices for small problems with 10 or fewer images, and\n tens or hundreds of iterations may be required for problems with\n hundreds or thousands of images.\n\n\\item[\\texttt{-\\/-poly-order \\textit{integer(=0)}}] \\hfill \\\\\n \\emph{Optional.} Sets the order of the polynomial that is used for\n adjustment. Using zero means only apply offset to the camera\n parameters that are not time dependent. That setting is recommend\n for frame cameras. Linescan imagers should using either a first\n order polynomial or a second order. Increasing this number too high can\n lead to a problem that is ill-defined and would prevent the algorithm\n from converging on a solution. \\emph{Initial work suggest that anything\n beyond a 2nd order polynomial would be ill-defined. 3rd order may\n work with a good dose of ground control points.}\n\n\\item[\\texttt{-\\/-position-sigma \\textit{float(=100)}}] \\hfill \\\\\n Sets the sigma (or uncertainty) of the spacecraft position in\n units of meters.\n\n\\item[\\texttt{-\\/-pose-sigma \\textit{float(=0.1)}}] \\hfill \\\\\n Sets the sigma (or uncertainty) of the spacecraft pose in units\n of radians.\n\n\\item[\\texttt{-\\/-report-level|-r \\textit{integer(=10)}}] \\hfill \\\\\n \\emph{Optional.} Sets the report level for the final bundle\n adjustment report. This report is saved as\n \\texttt{isis\\_adjust.report}. Report levels available are:\n\n \\begin{description}\n \\item[0 - CommandLine Error and Final report]\n \\item[10 - CommandLine Iteration Error (default)]\n \\item[20 - Write Report file]\n \\item[25 - \\textnormal{\\emph{In development}}]\n \\item[30 - Write Stereo Triangulation Error]\n \\item[35 - \\textnormal{\\emph{In development}}]\n \\item[100 - Debug, Write Error Vectors (big human readable)]\n \\item[110 - Debug, Write Jacobian Matrix (massive human readable)]\n \\end{description}\n\n\\item[\\texttt{-\\/-robust-threshold \\textit{float(=10)}}] \\hfill \\\\\n Sets the robust threshold; an additional parameter specifically for\n the \\texttt{PseudoHuber}, \\texttt{Huber}, and \\texttt{Cauchy} arguments to the \\texttt{-\\/-cost-function} option.\n\n\\item[\\texttt{-\\/-save-iteration-data|-s}] \\hfill \\\\\n \\emph{Optional.} Use to write {\\tt bundlevis} visualization files.\n\n\\item[\\texttt{-\\/-seed-with-previous}] \\hfill \\\\\n \\emph{Optional.} Loads up the previous {\\tt isis\\_adjust} session's\n adjustment file and uses them as a starting point for this session.\n\n\\item[\\texttt{-\\/-write-isis-cnet-also}] \\hfill \\\\\n \\emph{Optional.} Write an \\ac{ISIS} \\ac{PVL} style control network file to\n \\texttt{isis\\_adjust.net}. The output file is very large compared to\n the binary output, \\texttt{isis\\_adjust.cnet}, but is human readable\n and compatible with the \\ac{ISIS} 3 \\texttt{qnet} tool.\n\n\\item[\\texttt{-\\/-write-kml [0|1(=0)]}] \\hfill \\\\\n \\emph{Optional.} Providing this option with a zero will have the\n program write a \\ac{KML} file showing the location of all the \\acp{GCP}\n and be colored according to their final error. Providing this\n option with a one will have this perform as before, but also have it\n write all of the 3D point estimates. This is useful for debugging and\n for having a quick visualization of where stress points might exist in\n a bundle adjustment problem when there are many cameras.\n\n\\item[\\texttt{-\\/-help|-h}] \\hfill \\\\\n Provides a shortened list of the above.\n\n\\end{description}\n\n\\section{Visualizing bundle adjustment with bundlevis}\n\nThe \\texttt{bundlevis} program is used to visualize the process of\nbundle adjustment. It will show an animated, fully interactive 3D\nscene containing all the 3D points and cameras across all iterations\nof bundle adjustment. This tool is used to quickly determine if\nbundle adjustment was successful. If something does go wrong,\n\\texttt{bundlevis} can be a powerful debugging tool for identifying\nthe problem.\n\n\\begin{figure}[b!]\n \\begin{center}\n \\includegraphics[width=5in]{images/bundlevis_apollo.png}\n \\end{center}\n \\caption{ A screenshot of \\texttt{bundlevis} visualizing the bundle\n adjustment of imagery from the Apollo 15 Metric Camera (orbit\n 33). }\n \\label{fig:bundlevis}\n\\end{figure}\n\nOnce \\texttt{bundlevis} has loaded the data from a bundle adjustment run,\nthe user can click on and inspect the position of 3D points that are\nin purple and play back the iterations using the keyboard. Double\nclicking on a camera will cause lines to be drawn to each point viewed\nby the camera. Clicking on a 3D point causes lines to be drawn to all\ncameras that view the point.\n\nBundle adjustment can fail in a variety of different ways, but users\nshould be aware of two common failure modes. The first is segmentation;\nwhere tie points will split into two or more distinct groups,\nproducing cliffs between or clumps among points. This is usually\ncaused by insufficient matches between a pair of images in your\ncontrol network. You may need to choose some tie-points between\nthese images by hand or add additional images that overlap with the\nproblem area.\n\nThe second common sign of a failure is a point cloud explosion (the\nresulting terrain looks unrecognizably noisy). This most often results\nfrom a high number of outlying, bad tie-point measurements that the\nbundle adjustment algorithm can't recover from. These bad constraints\ncan be removed by hand or mitigated using one of the robust cost modes\n(e.g. by using the \\texttt{-\\/-cost-function} argument for\n\\texttt{isis\\_adjust}).\n\n\\subsection{Options}\n\nThe following is a listing and explanation of the options\nthat can be given to \\texttt{bundlevis} from the command line.\n\n\\begin{description}\n\n\\item[\\texttt{-\\/-camera-iteration-file|-c \\textit{bundlevis-camera-iteration-file}}] \\hfill \\\\\n \\emph{Optional.} Supply a camera iteration file that was produced by\n \\texttt{isis\\_adjust}. \\texttt{bundlevis} will only draw cameras\n if you supply a camera iteration file.\n\n\\item[\\texttt{-\\/-points-iteration-file|-p \\textit{bundlevis-point-iteration-file}}] \\hfill \\\\\n \\emph{Optional.} Supply a point iteration file that was produced by\n \\texttt{isis\\_adjust}. \\texttt{bundlevis} will only draw 3D points\n if you supply a point iteration file.\n\n\\item[\\texttt{-\\/-control-network-file|-n \\textit{Vision-Workbench-binary-control-network-file}}] \\hfill \\\\\n \\emph{Optional.} Supply a control network file that was produced by\n {\\tt isis\\_adjust}. This allows {\\tt bundlevis} to show the\n relationship between points and cameras when used in\n conjunction with \\texttt{-\\/-camera-iteration-file} and\n \\texttt{-\\/-points-iteration-file}.\n\n\\item[\\texttt{-\\/-additional-pnt-files \\textit{bundlevis-point-iteration-files}}] \\hfill \\\\\n \\emph{Optional.} Supply additional points to be animated alongside\n the camera and 3D points. The files given must be in the same\n format as a \\texttt{bundlevis} point iteration file and have the same number\n of iterations.\n\n\\item[\\texttt{-\\/-fullscreen}] \\hfill \\\\\n \\emph{Optional.} Displays \\texttt{bundlevis} using the entire screen;\n otherwise the program loads in a window. \\emph{The fullscreen option\n does not work correctly with dual screen systems.}\n\n\\item[\\texttt{-\\/-stereo}] \\hfill \\\\\n \\emph{Optional.} Render the 3D scene in red/blue anaglyph mode.\n\n\\item[\\texttt{-\\/-show-moon}] \\hfill \\\\\n \\emph{Optional.} Draws a wireframe sphere with a radius of 1737.3~km\n that represents the Moon.\n\n\\item[\\texttt{-\\/-show-mars}] \\hfill \\\\\n \\emph{Optional.} Draws a wireframe sphere with a radius of 3397~km\n that represents Mars.\n\n\\item[\\texttt{-\\/-show-earth}] \\hfill \\\\\n \\emph{Optional.} Draws a wireframe sphere that represents the Earth.\n\n\\end{description}\n\n\\subsection{Controls}\n\nOnce \\texttt{bundlevis} is running, there are several controls that\ncan be used to interface with the program. There are the playback\ncontrols, jump-to-frame controls, and the mouse.\n\n\\paragraph{Playback Controls}\n\nPlayback controls are similar to those in the popular Winamp program;\narranged on the keyboard like the controls on a tape deck.\n\n\\newenvironment{myindentpar}[1]\n {\\begin{list}{}\n {\\setlength{\\leftmargin}{#1}}\n \\item[]\n }\n {\\end{list}}\n\n\\begin{myindentpar}{3cm}\n\\begin{description}\n \\item[Z] Step back one iteration\n \\item[X] Play\n \\item[C] Pause\n \\item[V] Stop \\emph{(which is the same as Pause except that\n bundlevis goes back to iteration 0)}\n \\item[B] Step forward one iteration\n\\end{description}\n\\end{myindentpar}\n\n\\paragraph{Jump-to-Frame Controls}\nJump-to-Frame controls are the numbers \\textbf{1-9} along the top of\nthe keyboard. Pressing \\textbf{1} will display the very first\niteration. Pressing \\textbf{9} will display the very last\niteration. Pressing \\textbf{2} through \\textbf{8} will display\niterations that are somewhere in between based on the value of the\nnumber. Finally, pressing \\textbf{0} will cause \\texttt{bundlevis} to\ndisplay the points at their very last iteration with an additional\ntail pointing back to the starting position of the points in the first\niteration.\n\n\\paragraph{Mouse}\nThe mouse is used to move around the model, and has the same controls\nfound in many 3D environments (e.g. \\texttt{osgviewer}). Moving the\nmouse with the \\textbf{left mouse button} held down will cause the\nmodel to rotate. Moving with the \\textbf{right mouse button} pressed\nwill zoom, and moving with the \\textbf{middle mouse button} will\ntranslate. Alternatively for systems where the mouse is\nbutton-challenged, \\textbf{option + mouse} is translation and\n\\textbf{command + mouse} is zoom.\n\nDouble clicking with the mouse on a point or camera will allow the\nuser to query entities in the model. A double click will cause the\npoint number and camera number to be printed to the terminal. The\nnumber identifier for a given camera or point will also appear when\nthe viewer is zoomed in on that entity.\n\n\\section{Examples of Use}\n\\label{sec:ba_example}\n\n\\subsection{Processing Mars Orbital Camera}\n\nWhat follows is an example of bundle adjustment using two \\ac{MOC}\nimages of the south Cydonia region. We use images M10/00254 and\nR09/01059. These images are available from NASA's \\ac{PDS} (the\n\\ac{ISIS} \\texttt{mocproc} program will operate on either the IMQ\nor IMG format files, we use the \\texttt{.imq} below in the example).\nFor reference, the following \\ac{ISIS} commands are how to convert\nthe \\ac{MOC} images to \\ac{ISIS} cubes.\n\n\\begin{verbatim}\n ISIS 3> mocproc from= m1000254.imq to= m1000254.cub mapping=no\n ISIS 3> mocproc from= r0901059.imq to= r0901059.cub mapping=no\n\\end{verbatim}\n\nYou will note that the resulting images are not map projected. Bundle\nadjustment requires the ability to project arbitrary 3D points into\nthe camera frame. The process of map projecting an image dissociates\nthe camera model from the image. Map projecting can be perceived as\nthe generation of a new infinitely large camera sensor that is\nperfectly parallel to the surface of its subject ( and thus spherical\n). That makes it extremely hard to project a random point into\nthe camera's original model. The math would follow the transformation from\nprojection into the camera frame, then projected back down to surface\nthat ISIS uses, then finally up into the infinitely large sensor. The\n\\texttt{isis\\_adjust} program does not support this.\n\nAt this point, we need to automatically generate tie-points between\nthese two images. This can be done using the \\texttt{ipfind} and\n\\texttt{ipmatch} utilities. These tools do not reliably (currently)\nwork with photometrically calibrated images, so we must first convert\nthese images to a standard format using the \\ac{ISIS} program\n\\texttt{isis2std} or \\texttt{gdal\\_translate}. The utility\n\\texttt{gdal\\_translate} is the preferred option as it is able to\nproduce lossless compressed images.\n\n\\begin{verbatim}\n ISIS 3> isis2std from= m1000254.cub to= m1000254.tif format=TIFF\n ISIS 3> isis2std from= r0901059.cub to= r0901059.tif format=TIFF\n\\end{verbatim}\n\nor ...\n\n\\begin{verbatim}\n ISIS 3> gdal_translate -of GTiff -scale -ot Byte\n -co COMPRESS=LZW -co TILED=yes m1000254.cub m1000254.tif\n ISIS 3> gdal_translate -of GTiff -scale -ot Byte\n -co COMPRESS=LZW -co TILED=yes r0901059.cub r0901059.tif\n\\end{verbatim}\n\nHere is how to process those newly created TIFF files for tie-points\nusing the Interest Point Module tools from Vision Workbench.\n\n\\begin{verbatim}\n > ipfind m1000254.tif r0901059.tif -g 1.2\n > ipmatch m1000254.tif r0901059.tif -d -r homography -i 30\n\\end{verbatim}\n\n\\begin{center}\n\\fcolorbox{black}{lgray}{ \\begin{minipage}{5.5in}\n\n Be aware that the tie-point tools available in Vision Workbench do\n not always produce enough matches for bundle adjustment. An\n alternative would be to use outside code such as SURF. We have\n included a patch at the end of the book that converts SURF output\n to a Vision Workbench style match file. In extreme cases, users\n can be forced to make measurements themselves. This can be\n performed with \\ac{ISIS}'s \\texttt{qnet} progam. \\\\ \\\\ For this\n example we found that \\texttt{ipfind} and \\texttt{ipmatch} work\n great. Expect to find approximately 100 matched points. Your\n results will be slightly different due to the random nature of\n RANSAC.\n\n\\end{minipage}}\n\\end{center}\n\nFinally it is time to start bundle adjustment. There are many options\nthat can be used at this stage. We have chosen those required to\ncreate visualization data for \\texttt{bundlevis}. We have also set\nthe maximum iterations to 100 and chose the option to create a\ndetailed report file of \\texttt{isis\\_adjust}'s results.\n\n\\begin{verbatim}\n ISIS 3> isis_adjust *.cub -s --max 30 -r 50\n\\end{verbatim}\n\nBefore you hit enter, notice that we are not feeding the interest\npoint match files to \\texttt{isis\\_adjust}. Instead, we only provide\nthe camera files. During run time, the program will see that it\ndoesn't have any image measurements in the form of a control\nnetwork. In that case it will attempt to build its own by searching\nthe current working directory for \\texttt{*.match} files.\n\nNow you have permission to run \\texttt{isis\\_adjust}. You'll see the\ncommand will produce considerable debugging output and will place many\noutput files in the current working directory. If you look through the\noutput in the terminal or alternatively in the output report file,\n\\texttt{isis\\_adjust.report}, you'll see that the problem did not\nconverge but reduced most of the error in the first 40 iterations\n(again, your results may vary slightly). The lack of convergence is\nworrisome and a few (like 3) ground control points would probably help\nconsiderably. Notice that the error improved only slightly after\nthose first 40 iterations, but the shape of the pointcloud and the\ncamera paths changed considerably. This untwisting of the cameras can\nbe seen next in \\texttt{bundlevis}.\n\nVisualizing all of the data that was exported for \\texttt{bundlevis}\ncan be carried out as follows:\n\n\\begin{verbatim}\n > bundlevis -p iterPointsParam.txt -c iterCameraParam.txt \\\n -n isis_adjust.cnet\n\\end{verbatim}\n\nPress escape to exit out of \\texttt{bundlevis} when finished. You may\nalso want to try viewing the data with a wire-frame of Mars to give\nsome perspective. Note, you will have to zoom in very far since the\nsize of a MOC frame is quite small relative to the size of Mars!\n\n\\begin{verbatim}\n > bundlevis -p iterPointsParam.txt -c iterCameraParam.txt \\\n -n isis_adjust.cnet --show-mars\n\\end{verbatim}\n\nProducing a DEM using the newly created corrections is the same as\ncovered in the Tutorial on page \\pageref{ch:tutorial}, with one small\ndifference: \\texttt{stereo} needs to know of the existence of the\ncorrection files, \\texttt{m1000254.isis\\_adjust} and\n\\texttt{r0901059.isis\\_adjust}.\n\n\\begin{verbatim}\n ISIS 3> stereo m1000254.cub r0901059.cub m1000254.isis_adjust \\\n r0901059.isis_adjust MOC_RESULTS/M1000254_R0901059\n\\end{verbatim}\n\nThe two new arguments (\\texttt{*.isis\\_adjust}) provide\n\\texttt{stereo} with the necessary corrections. When providing outside\ncamera models for images, they are always the 3rd and 4th argument for\n\\texttt{stereo}. This is also how \\texttt{stereo} can be made to\noperate on images from standard consumer cameras.\n\n\\subsection{Processing with Ground Control Points}\n\nGround control point files describe a single point in the world\nthat is seen by 1 or more cameras. How they are measured in the\nfirst place is up to the user. We use a manual process of comparing\neach image to a respected map projected image and then recording\nthe latitude, longitude, and altitude of the point(s). The maps to\nregister against can be anything, but it is recommended to register\nagainst a product with a high amount of cartographic stability and\naccuracy. For terrestrial work, we would use a \\ac{USGS} product\nthat can provide imagery that is registered to LIDAR height\nmeasurements.\n\nUnlike match files, ground control points must specifically be given\nto \\texttt{isis\\_adjust} from the command line, but in no particular\norder. Ground control point files are written with the extension\n\\texttt{.gcp}. Below is an example of a ground control point file that\nwas created to control a series of Apollo Metric Camera images from\nseveral Apollo 15 orbits.\n\n\\begin{verbatim}\n -52.8452 27.2561 1735999 300 300 500\n sub4-AS15-M-2086.cub 210.9 3565.0\n sub4-AS15-M-2087.cub 1476.9 3579.0\n sub4-AS15-M-2088.cub 2798.9 3586.8\n sub4-AS15-M-2089.cub 4133.5 3588.6\n sub4-AS15-M-2344.cub 906.9 3874.8\n sub4-AS15-M-2345.cub 2204.2 3913.9\n sub4-AS15-M-2482.cub 939.8 4348.0\n sub4-AS15-M-2483.cub 2282.0 4340.7\n sub4-AS15-M-2484.cub 3642.1 4330.9\n\\end{verbatim}\n\nThe first line of a \\texttt{.gcp} file is like a header line and\nis different from the remaining lines. The first line defines the\nworld location of the ground control point, and the rest of the\nlines define the image locations of the ground control points. Here\nare what the columns mean for the first line.\n\n\\begin{myindentpar}{2cm}\n\\begin{description}\n \\item[Column 1:] Longitude in degrees\n \\item[Column 2:] Latitude in degrees\n \\item[Column 3:] Radius in meters\n \\item[Column 4:] Sigma (or uncertainty) in meters for Local X axis\n \\item[Column 5:] Sigma (or uncertainty) in meters for Local Y axis\n \\item[Column 6:] Sigma (or uncertainty) in meters for Local Z axis\n\\end{description}\n\\end{myindentpar}\n\nThe other lines describe where this \\ac{GCP} is found in each image:\n\n\\begin{myindentpar}{2cm}\n\\begin{description}\n \\item[Column 1:] Image name\n \\item[Column 2:] Sample (X) image measurement\n \\item[Column 3:] Line (Y) image measurement\n\\end{description}\n\\end{myindentpar}\n\nMake a {\\tt .gcp} file for every ground control point, then be sure\nto feed them as an input to {\\tt isis\\_adjust}. Remember that you\ncan scale the sigma of all ground control points by using the {\\tt\n-\\/-gcp-scalar} flag. This can save time by allowing you to make\nadjustments without needing to edit all of the files individually.\n\n\\subsection{Sharing Data with ISIS 3's qnet program}\n\n\\ac{ISIS} contains a program called \\texttt{qnet} whose purpose is\nto create and edit \\ac{ISIS} style control network files. To share\na control network with \\texttt{qnet}, you will need to save our\ncontrol network in the \\ac{ISIS} format. If bundle adjustment has\nalready been performed once and if we want to simply convert the\ncontrol network for use in \\texttt{qnet}, you can use this command\nto save an \\ac{ISIS} style control network:\n\n\\begin{verbatim}\n ISIS 3> isis_adjust -c isis_adjust.cnet --write-isis-cnet-also *.cub\n\\end{verbatim}\n\nOtherwise if this is the first time performing bundle adjustment and a\ncontrol network does not already exist, use:\n\n\\begin{verbatim}\n ISIS 3> isis_adjust --write-isis-cnet-also *.cub\n\\end{verbatim}\n\nThere should now be an \\texttt{isis\\_adjust.net} file in the project's\ndirectory. It will be quite a bit larger than the other control\nnetwork file since it is stored as ASCII text, but it can be read\nand edited with a text editor. Before starting \\texttt{qnet}, there\nis one additional preparation that must be performed. \\ac{ISIS}'s\n\\texttt{qnet} requires a text file listing of all the cubes used\nby the control network. Here's how to create one:\n\n\\begin{verbatim}\n > ls *.cub > list_of_cubes.lis\n\\end{verbatim}\n\nNow, start up \\texttt{qnet} without any command line arguments. Click\nFile$\\rightarrow$Open. It will first ask for the list of cubes. Refer\nit to the newly created \\texttt{list\\_of\\_cubes.lis}. Next it will ask for\nthe control network. Give it \\texttt{isis\\_adjust.net}.\n\n\\begin{center}\n\\fcolorbox{black}{lgray}{ \\begin{minipage}{5.5in}\n\n At this time \\texttt{qnet} does not work with the Apollo Metric\n Camera's cube files. When loading the text file listing of cubes\n it will issue an error about invalid serial numbers for the listed\n cube files. % (03/11/09)\n\n\\end{minipage}}\n\\end{center}\n\nWhen finished, save the new control network file. Here's how to use\nthe new control network in \\texttt{isis\\_adjust}:\n\n\\begin{verbatim}\n ISIS 3> isis_adjust -c the_new_control_network.net *.cub\n\\end{verbatim}\n\nTake note that to distinguish \\ac{ISIS} style control network files\nfrom Stereo Pipeline style control network files is by the file\nextension. \\ac{ISIS} control networks have the extension of\n\\texttt{.net} and can be read with a text editor. Stereo Pipeline's\ncontrol networks have the extension \\texttt{.cnet} and are binary\nformat files.\n", "meta": {"hexsha": "1d208c138057f7b77aa461f55f952930eff3cc02", "size": 36743, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/book/bundle_adjustment.tex", "max_stars_repo_name": "nasa/StereoPipeline", "max_stars_repo_head_hexsha": "8b9c0bcab258c41d10cb2973d97722765072a7bf", "max_stars_repo_licenses": ["NASA-1.3"], "max_stars_count": 29, "max_stars_repo_stars_event_min_datetime": "2015-05-06T01:28:21.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-19T22:55:29.000Z", "max_issues_repo_path": "docs/book/bundle_adjustment.tex", "max_issues_repo_name": "imagineagents/StereoPipeline", "max_issues_repo_head_hexsha": "8b9c0bcab258c41d10cb2973d97722765072a7bf", "max_issues_repo_licenses": ["NASA-1.3"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/book/bundle_adjustment.tex", "max_forks_repo_name": "imagineagents/StereoPipeline", "max_forks_repo_head_hexsha": "8b9c0bcab258c41d10cb2973d97722765072a7bf", "max_forks_repo_licenses": ["NASA-1.3"], "max_forks_count": 27, "max_forks_repo_forks_event_min_datetime": "2015-01-15T04:20:50.000Z", "max_forks_repo_forks_event_max_datetime": "2020-01-10T01:31:17.000Z", "avg_line_length": 46.5101265823, "max_line_length": 117, "alphanum_fraction": 0.765941812, "num_tokens": 9509, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3481511269961152}} {"text": "\\documentclass{article}\n\\usepackage[a4paper, left=25.4mm, top=25.4mm, right=25.4mm, bottom=25.4mm]{geometry}\n\\usepackage[shortlabels]{enumitem}\n\\usepackage{adjustbox}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{authoraftertitle}\n\\usepackage{blindtext}\n\\usepackage{booktabs}\n\\usepackage{bussproofs}\n\\usepackage{cancel}\n\\usepackage{color}\n\\usepackage{fancyhdr}\n\\usepackage{graphicx}\n\\usepackage{latexsym}\n\\usepackage{listings}\n\\usepackage{multicol}\n\\usepackage{newpxtext}\n%\\usepackage{newpxmath}\n\\usepackage{subcaption}\n\\usepackage{textcomp}\n\\usepackage{textgreek}\n\\usepackage{vwcol}\n\\newcommand\\tab[1]{\\hspace*{#1}}\n\\newcommand\\lowermidtilde{\\raisebox{-0.8ex}{\\textasciitilde}}\n\\newcommand\\midtilde{\\raisebox{-0.6ex}{\\textasciitilde}}\n\\newcommand{\\partfarrow}{\\rightharpoonup}\n\\newcommand{\\partf}{\\rightharpoonup\\!\\!\\!\\!\\!\\!\\!\\!\\!\\raisebox{2.5pt}{$\\rightharpoonup$}}\n\\renewcommand{\\thesubsection}{\\arabic{subsection}}\n\\renewcommand{\\headrulewidth}{0pt}\n\\renewcommand{\\ttdefault}{pcr}\n\\renewcommand{\\sfdefault}{lmss}\n\\renewcommand\\qedsymbol{Q.E.D.}\n\\definecolor{gray}{rgb}{0.5,0.5,0.5}\n\\lstset{\n basicstyle=\\ttfamily,\n commentstyle=\\color{gray}\\ttfamily,\n keywordstyle=\\ttfamily\\bfseries,\n columns=flexible,\n numbers=left,\n numberstyle=\\tiny,\n firstnumber=0,\n showstringspaces=false,\n}\n\\setlist{ \n listparindent=\\parindent,\n align=parleft,\n}\n\n\\author{Victor Zhao\\\\xz398@cam.ac.uk}\n\n\\begin{document}\n\\centering\n\\section*{Types\\\\CST Part II Paper 8 \\& 9}\n\\MyAuthor\n\n\\justifying\n\n\\subsection{Simply-Typed $\\lambda$-Calculus}\n\n\\subsection*{Syntax}\n\n\\hspace{-1em}\n\\begin{tabular}{lccl}\nTypes & $T$ & $::=$ & $1\\ |\\ 0\\ |\\ T_1\\times T_2\\ |\\ T_1+T_2\\ |\\ T_1\\to T_2$\\\\[0.5em]\nTerms & $e$ & $::=$ & $x\\ |\\ \\langle\\rangle\\ |\\ \\langle e_1,e_2\\rangle\\ |\\ \\textsf{fst }e\\ |\\ \\textsf{snd }e\\ |\\ \\textsf{L }e\\ |\\ \\textsf{R }e\\ |\\ \\textsf{case}(e,\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2)$\\\\ \n & {} & $|$ & $\\lambda x:T.\\ e\\ |\\ e_1\\ e_2\\ |\\ \\textsf{abort}$\\\\[0.5em]\nValues & $v$ & $::=$ & $\\langle\\rangle\\ |\\ \\langle v_1,v_2\\rangle\\ |\\ \\lambda x:T.\\ e\\ |\\ \\textsf{L }v\\ |\\ \\textsf{R }v$\\\\[0.5em] \nContexts & $\\Gamma$ & $::=$ & $\\cdot\\ |\\ \\Gamma,x:T$\n\\end{tabular} \n\n\\subsection*{Typing rules}\n\n(\\textsc{I}: introduction rule, \\textsc{E}: elimination rule, \\textsc{Hyp}: hypothesis)\n\\medbreak\n\\noindent\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{1I}}\n\\UnaryInfC{$\\Gamma\\vdash\\langle\\rangle:1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e_1:T_1$}\n\\AxiomC{$\\Gamma\\vdash e_2:T_2$}\n\\RightLabel{\\textsc{$\\times$I}}\n\\BinaryInfC{$\\Gamma\\vdash\\langle e_1, e_2\\rangle:1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e:T_1\\times T_2$} \n\\RightLabel{\\textsc{$\\times$E$_1$}}\n\\UnaryInfC{$\\Gamma\\vdash\\textsf{fst }e:T_1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e:T_1\\times T_2$} \n\\RightLabel{\\textsc{$\\times$E$_2$}}\n\\UnaryInfC{$\\Gamma\\vdash\\textsf{snd }e:T_2$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$x:T\\in\\Gamma$} \n\\RightLabel{\\textsc{Hyp}}\n\\UnaryInfC{$\\Gamma\\vdash x:T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma,x:T\\vdash e:T'$} \n\\RightLabel{\\textsc{$\\to$I}}\n\\UnaryInfC{$\\Gamma\\vdash\\lambda x:T.\\ e:T\\to T'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e_1:T\\to T'$}\n\\AxiomC{$\\Gamma\\vdash e_2:T$}\n\\RightLabel{\\textsc{$\\to$E}}\n\\BinaryInfC{$\\Gamma\\vdash e_1\\ e_2:T'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Gamma\\vdash e:T_1$} \n\\RightLabel{\\textsc{$+$I$_1$}}\n\\UnaryInfC{$\\Gamma\\vdash\\textsf{L }e:T_1+T_2$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e:T_2$} \n\\RightLabel{\\textsc{$+$I$_2$}}\n\\UnaryInfC{$\\Gamma\\vdash\\textsf{R }e:T_1+T_2$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Gamma\\vdash e:T_1+T_2$}\n\\AxiomC{$\\Gamma,x:X\\vdash e_1:T$}\n\\AxiomC{$\\Gamma,x:X\\vdash e_2:T$}\n\\RightLabel{\\textsc{$+$E}}\n\\TrinaryInfC{$\\Gamma\\vdash\\textsf{case}(e,\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2):T$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n(No introduction for 0)$\\qquad$\n\\AxiomC{$\\Gamma\\vdash e:0$} \n\\RightLabel{\\textsc{0E}}\n\\UnaryInfC{$\\Gamma\\vdash\\textsf{abort }e:T$}\n\\DisplayProof\n\n\\subsection*{Operational semantics}\n\n(No rule for unit)$\\qquad$\n\\AxiomC{$e_1\\leadsto e_1'$}\n\\RightLabel{\\textsc{Pair1}}\n\\UnaryInfC{$\\langle e_1,e_2\\rangle\\leadsto\\langle e_1',e_2\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e_2\\leadsto e_2'$}\n\\RightLabel{\\textsc{Pair2}}\n\\UnaryInfC{$\\langle v,e_2\\rangle\\leadsto\\langle v,e_2'\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{Proj1}}\n\\UnaryInfC{$\\textsf{fst }\\langle v_1,v_2\\rangle\\leadsto v_1$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{Proj2}}\n\\UnaryInfC{$\\textsf{snd }\\langle v_1,v_2\\rangle\\leadsto v_2$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{$e\\leadsto e'$} \n\\RightLabel{\\textsc{Proj3}}\n\\UnaryInfC{$\\textsf{fst }e\\leadsto\\textsf{fst }e'$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{$e\\leadsto e'$} \n\\RightLabel{\\textsc{Proj4}}\n\\UnaryInfC{$\\textsf{snd }e\\leadsto\\textsf{snd }e'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$} \n\\RightLabel{\\textsc{Sum1}}\n\\UnaryInfC{$\\textsf{L }e\\leadsto\\textsf{L }e'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e\\leadsto e'$} \n\\RightLabel{\\textsc{Sum2}}\n\\UnaryInfC{$\\textsf{R }e\\leadsto\\textsf{R }e'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$} \n\\RightLabel{\\textsc{Case1}}\n\\UnaryInfC{$\\textsf{case}(e,\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2)\\leadsto\\textsf{case}(e',\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2)$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\RightLabel{\\textsc{Case2}}\n\\UnaryInfC{$\\textsf{case}(\\textsf{L }v,\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2)\\leadsto[v/x]e_1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\RightLabel{\\textsc{Case3}}\n\\UnaryInfC{$\\textsf{case}(\\textsf{R }v,\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2)\\leadsto[v/y]e_2$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e_1\\leadsto e_1'$}\n\\RightLabel{\\textsc{App1}}\n\\UnaryInfC{$e_1\\ e_2\\leadsto e_1'\\ e_2$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e_2\\leadsto e_2'$}\n\\RightLabel{\\textsc{App2}}\n\\UnaryInfC{$v\\ e_2\\leadsto v\\ e_2'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\RightLabel{\\textsc{Fn}}\n\\UnaryInfC{$(\\lambda x:T.\\ e)\\ v\\leadsto[v/x]e$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$} \n\\RightLabel{\\textsc{Abort}}\n\\UnaryInfC{$\\textsf{abort }e\\leadsto\\textsf{abort }e'$}\n\\DisplayProof\\newpage %=========================================================\n\n\\subsection{Polymorphic $\\lambda$-Calculus (System F)}\n\n\\subsection*{Syntax}\n\n\\hspace{-1em}\n\\begin{tabular}{lccl}\nTypes & $T$ & $::=$ & $\\alpha\\ |\\ T_1\\to T_2\\ |\\ \\forall\\alpha.\\ T\\ |\\ \\exists\\alpha.\\ T$\\\\[0.5em]\nTerms & $e$ & $::=$ & $x\\ |\\ \\lambda x:T.\\ e\\ |\\ e_1\\ e_2\\ |\\ \\Lambda\\alpha.\\ e\\ |\\ e\\ T\\ |\\ \\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},e_\\text{impl})$\\\\\n{} & {} & $|$ & $\\textsf{let pack}(\\alpha,x)=e_\\text{impl}\\textsf{ in }e_\\text{use}$\\\\[0.5em]\nValues & $v$ & $::=$ & $\\lambda x:T.\\ e\\ |\\ \\Lambda\\alpha.\\ e\\ |\\ \\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},v_\\text{impl})$\\\\[0.5em] \nType Contexts & $\\Theta$ & $::=$ & $\\cdot\\ |\\ \\Theta,\\alpha$\\\\[0.5em] \nTerm Contexts & $\\Gamma$ & $::=$ & $\\cdot\\ |\\ \\Gamma,x:T$\n\\end{tabular} \n\n\\subsection*{Well-formedness of types}\n\n\\vspace{0.5em}\n\\AxiomC{$\\alpha\\in\\Theta$} \n\\UnaryInfC{$\\Theta\\vdash\\alpha\\textsf{ type}$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Theta\\vdash T_1\\textsf{ type}$} \n\\AxiomC{$\\Theta\\vdash T_2\\textsf{ type}$} \n\\BinaryInfC{$\\Theta\\vdash T_1\\to T_2\\textsf{ type}$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Theta,\\alpha\\vdash T\\textsf{ type}$} \n\\UnaryInfC{$\\Theta\\vdash\\forall\\alpha.\\ T\\textsf{ type}$}\n\\DisplayProof\n\n\\subsection*{Well-formedness of term contexts}\n\n\\vspace{0.5em}\n\\AxiomC{{\\color{white} $\\Theta\\vdash\\Gamma$}} %placeholder \n\\UnaryInfC{$\\Theta\\vdash\\cdot\\textsf{ ctx}$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Theta\\vdash\\Gamma\\textsf{ ctx}$} \n\\AxiomC{$\\Theta\\vdash T\\textsf{ type}$} \n\\BinaryInfC{$\\Theta\\vdash\\Gamma,x:T\\textsf{ ctx}$}\n\\DisplayProof\n\n\\subsection*{Typing rules}\n\n\\vspace{0.5em}\n\\AxiomC{$x:T\\in\\Gamma$} \n\\RightLabel{\\textsc{Hyp}}\n\\UnaryInfC{$\\Theta;\\Gamma\\vdash x:T$}\n\\DisplayProof$\\qquad$ %==========================================================\n\\AxiomC{$\\Theta\\vdash T\\textsf{ type}$} \n\\AxiomC{$\\Theta;\\Gamma,x:T\\vdash e:T'$} \n\\RightLabel{\\textsc{$\\to$I}}\n\\BinaryInfC{$\\Theta;\\Gamma\\vdash\\lambda x:T.\\ e:T\\to T'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Theta;\\Gamma\\vdash e_1:T\\to T'$}\n\\AxiomC{$\\Theta;\\Gamma\\vdash e_2:T$}\n\\RightLabel{\\textsc{$\\to$E}}\n\\BinaryInfC{$\\Theta;\\Gamma\\vdash e_1\\ e_2:T'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Theta,\\alpha;\\Gamma\\vdash e:T$} \n\\RightLabel{\\textsc{$\\forall$I}}\n\\UnaryInfC{$\\Theta;\\Gamma\\vdash\\Lambda\\alpha.\\ e:\\forall\\alpha.\\ T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Theta;\\Gamma\\vdash e:\\forall\\alpha.\\ T'$}\n\\AxiomC{$\\Theta\\vdash T\\textsf{ type}$}\n\\RightLabel{\\textsc{$\\forall$E}}\n\\BinaryInfC{$\\Theta;\\Gamma\\vdash e\\ T:[T/\\alpha]T'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Theta,\\alpha_\\text{abs}\\vdash T_\\text{sig}\\textsf{ type}$} \n\\AxiomC{$\\Theta\\vdash T_\\text{conc}\\textsf{ type}$}\n\\AxiomC{$\\Theta;\\Gamma\\vdash e_\\text{impl}:[T_\\text{conc}/\\alpha_\\text{abs}]T_\\text{sig}$} \n\\RightLabel{\\textsc{$\\exists$I}}\n\\TrinaryInfC{$\\Theta;\\Gamma\\vdash\\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},e_\\text{impl}):\\exists\\alpha_\\text{abs}.\\ T_\\text{sig}$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Theta;\\Gamma\\vdash e_\\text{impl}:\\exists\\alpha_\\text{abs}.\\ T_\\text{sig}$} \n\\AxiomC{$\\Theta,\\alpha;\\Gamma,x:[\\alpha_\\text{abs}/\\alpha]T_\\text{sig}\\vdash e_\\text{use}:T_\\text{use}$}\n\\AxiomC{$\\Theta\\vdash T_\\text{use}\\textsf{ type}$} \n\\RightLabel{\\textsc{$\\exists$E}}\n\\TrinaryInfC{$\\Theta;\\Gamma\\vdash\\textsf{let pack}(\\alpha,x)=e_\\text{impl}\\textsf{ in }e_\\text{use}:T_\\text{use}$}\n\\DisplayProof\n\n\\subsection*{Operational semantics}\n\n(\\textsc{Cong}: congruence rule, \\textsc{Eval}: evaluation rule)\n\\medbreak\n\\noindent\n\\AxiomC{$e_1\\leadsto e_1'$}\n\\RightLabel{\\textsc{CongFun}}\n\\UnaryInfC{$e_1\\ e_2\\leadsto e_1'\\ e_2$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e_2\\leadsto e_2'$}\n\\RightLabel{\\textsc{CongFunArg}}\n\\UnaryInfC{$v\\ e_2\\leadsto v\\ e_2'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\RightLabel{\\textsc{FunEval}}\n\\UnaryInfC{$(\\lambda x:T.\\ e)\\ v\\leadsto[v/x]e$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$}\n\\RightLabel{\\textsc{CongForall}}\n\\UnaryInfC{$e\\ T\\leadsto e'\\ T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\RightLabel{\\textsc{ForallEval}}\n\\UnaryInfC{$(\\Lambda\\alpha.\\ e)\\ T\\leadsto[T/\\alpha]e$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e_\\text{impl}\\leadsto e'_\\text{impl}$}\n\\RightLabel{\\textsc{CongExists}}\n\\UnaryInfC{$\\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},e_\\text{impl})\\leadsto\\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},e'_\\text{impl})$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e_\\text{impl}\\leadsto e'_\\text{impl}$}\n\\RightLabel{\\textsc{CongExistsUnpack}}\n\\UnaryInfC{$\\textsf{let pack}(\\alpha,x)=e_\\text{impl}\\textsf{ in }e_\\text{use}\\leadsto\\textsf{let pack}(\\alpha,x)=e'_\\text{impl}\\textsf{ in }e_\\text{use}$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\RightLabel{\\textsc{ExistsEval}}\n\\UnaryInfC{$\\textsf{let pack}(\\alpha,x)=\\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},v_\\text{impl})\\textsf{ in }e_\\text{use}\\leadsto [T_\\text{conc}/\\alpha,v_\\text{impl}/x]e_\\text{use}$}\n\\DisplayProof\n\n\\subsection*{Church encodings}\n\n\\subsubsection*{Pairs}\n\n\\hspace{-1em}\n\\begin{tabular}{lcl}\n$T_1\\times T_2$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\forall\\alpha.\\ (T_1\\to T_2\\to\\alpha)\\to\\alpha$\\\\[0.5em]\n$\\langle e_1,e_2\\rangle$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda k:T_1\\to T_2\\to\\alpha.\\ k\\ e\\ e'$\\\\[0.5em]\n\\textsf{fst }$e$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$e\\ T_1\\ (\\lambda x:T_1.\\ \\lambda y:T_2.\\ x)$\\\\[0.5em]\n\\textsf{snd }$e$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$e\\ T_2\\ (\\lambda x:T_1.\\ \\lambda y:T_2.\\ y)$\n\\end{tabular}\n\n\\subsubsection*{Sums}\n\n\\hspace{-1em}\n\\begin{tabular}{lcl}\n$T_1+T_2$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\forall\\alpha.\\ (T_1\\to\\alpha)\\to(T_2\\to\\alpha)\\to\\alpha$\\\\[0.5em]\n\\textsf{L }$e$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda f:T_1\\to\\alpha.\\ \\lambda g:T_2\\to\\alpha.\\ f\\ e$\\\\[0.5em]\n\\textsf{R }$e$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda f:T_1\\to\\alpha.\\ \\lambda g:T_2\\to\\alpha.\\ g\\ e$\\\\[0.5em]\n\\end{tabular}\\\\\n$\\textsf{case}(e,\\textsf{L }x\\to e_1,\\textsf{R }y\\to e_2):T\\ \\triangleq\\ e\\ T\\ (\\lambda x:T_1\\to T.\\ e_1)\\ (\\lambda y:T_2\\to T.\\ e_2)$\n\n\\subsubsection*{Existential types}\n\n$\\exists\\alpha.\\ T_\\text{sig}\\ \\triangleq\\ \\forall\\beta.\\ (\\forall\\alpha.\\ T_\\text{sig}\\to\\beta)\\to\\beta$\\medbreak\n\\noindent\n$\\textsf{pack}_{\\alpha_\\text{abs}.T_\\text{sig}}(T_\\text{conc},e_\\text{impl})\\ \\triangleq\\ \\Lambda\\beta.\\ \\lambda k:\\forall\\alpha_\\text{abs}.\\ T_\\text{sig}\\to\\beta.\\ k\\ T_\\text{conc}\\ e_\\text{impl}$\\medbreak\n\\noindent\n$\\textsf{let pack}(\\alpha,x)=e_\\text{impl}\\textsf{ in }e_\\text{use}:T_\\text{use}\\ \\triangleq\\ e_\\text{impl}\\ T_\\text{use}\\ (\\Lambda\\alpha.\\ \\lambda x:T_\\text{sig}.\\ e_\\text{use})$\n\n\\subsubsection*{Booleans}\n\n\\hspace{-1em}\n\\begin{tabular}{lcl}\n\\textsf{bool} & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\forall\\alpha.\\ \\alpha\\to\\alpha\\to\\alpha$\\\\[0.5em]\n\\textsf{True} & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda x:\\alpha.\\ \\lambda y:\\alpha.\\ x$\\\\[0.5em]\n\\textsf{False} & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda x:\\alpha.\\ \\lambda y:\\alpha.\\ y$\\\\[0.5em]\n\\end{tabular}\\\\\n\\textsf{if }$e$\\textsf{ then }$e_1$\\textsf{ else }$e_2:T\\ \\triangleq\\ e\\ T\\ e_1\\ e_2$\n\n\\subsubsection*{Natural numbers}\n\n\\hspace{-1em}\n\\begin{tabular}{lcl}\n$\\mathbb{N}$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\forall\\alpha.\\ \\alpha\\to(\\alpha\\to\\alpha)\\to\\alpha$\\\\[0.5em]\n\\textsf{zero} & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda z:\\alpha.\\ \\lambda s:\\alpha\\to\\alpha.\\ z$\\\\[0.5em]\n\\textsf{succ}$(e)$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda z:\\alpha.\\ \\lambda s:\\alpha\\to\\alpha.\\ s\\ (e\\ \\alpha\\ z\\ s)$\\\\[0.5em]\n\\end{tabular}\\\\\n$\\textsf{iter}(e,\\textsf{zero}\\to e_\\textsf{z},\\textsf{succ}(x)\\to e_\\textsf{s}):T\\ \\triangleq\\ e\\ T\\ e_\\textsf{z}\\ (\\lambda x:T.\\ e_\\textsf{s})$\n\n\\subsubsection*{Lists}\n\n\\hspace{-1em}\n\\begin{tabular}{lcl}\n\\textsf{list }$T$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\forall\\alpha.\\ \\alpha\\to(T\\to\\alpha\\to\\alpha)\\to\\alpha$\\\\[0.5em]\n$[]$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda n:\\alpha.\\ \\lambda c:T\\to\\alpha\\to\\alpha.\\ n$\\\\[0.5em]\n$e::e'$ & \\hspace{-0.5em}$\\triangleq$ & \\hspace{-0.5em}$\\Lambda\\alpha.\\ \\lambda n:\\alpha.\\ \\lambda c:T\\to\\alpha\\to\\alpha.\\ c\\ e\\ (e'\\ \\alpha\\ n\\ c)$\\\\[0.5em]\n\\end{tabular}\\\\\n$\\textsf{fold}(e,[]\\to e_\\textsf{n},x::r\\to e_\\textsf{c}):T'\\ \\triangleq\\ e\\ T'\\ e_\\textsf{n}\\ (\\lambda x:T.\\ \\lambda r:T'.\\ e_\\textsf{c})$\n\n\\newpage\n\n\\subsection{Monadic $\\lambda$-Calculus for State}\n\n\\subsection*{Syntax}\n\n\\hspace{-1em}\n\\begin{tabular}{lccl}\nTypes & $T$ & $::=$ & $1\\ |\\ \\mathbb{N}\\ |\\ T_1\\to T_2\\ |\\ \\textsf{ref }T\\ |\\ \\textsf{M }T$\\\\[0.5em]\nPure Terms & $e$ & $::=$ & $\\langle\\rangle\\ |\\ n\\ |\\ \\lambda x:T.\\ e\\ |\\ e_1\\ e_2\\ |\\ l\\ |\\ \\{t\\}$\\\\[0.5em]\nImpure Terms & $t$ & $::=$ & $\\textsf{new }e\\ |\\ !e\\ |\\ e:=e'\\ |\\ \\textsf{let }x=e;t\\ |\\ \\textsf{return }e$\\\\[0.5em]\nValues & $v$ & $::=$ & $\\langle\\rangle\\ |\\ n\\ |\\ \\lambda x:T.\\ e\\ |\\ l\\ |\\ \\{t\\}$\\\\[0.5em]\nStores & $\\sigma$ & $::=$ & $\\cdot\\ |\\ \\sigma,l:v$\\\\[0.5em]\nContexts & $\\Gamma$ & $::=$ & $\\cdot\\ |\\ \\Gamma,x:T$\\\\[0.5em]\nStore Typings & $\\Sigma$ & $::=$ & $\\cdot\\ |\\ \\Sigma,l:T$ \n\\end{tabular} \n\n\\subsection*{Typing rules}\n\n\\subsubsection*{Pure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$x:T\\in\\Gamma$} \n\\RightLabel{\\textsc{Hyp}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash x:T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{1I}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash\\langle\\rangle:1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{$\\mathbb{N}$I}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash n:\\mathbb{N}$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Sigma;\\Gamma,x:T\\vdash e:T'$} \n\\RightLabel{\\textsc{$\\to$I}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash\\lambda x:T.\\ e:T\\to T'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e_1:T\\to T'$}\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e_2:T$}\n\\RightLabel{\\textsc{$\\to$E}}\n\\BinaryInfC{$\\Sigma;\\Gamma\\vdash e_1\\ e_2:T'$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$l:T\\in\\Sigma$} \n\\RightLabel{\\textsc{RefBar}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash l:\\textsf{ref }T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Sigma;\\Gamma\\vdash t\\div T$} \n\\RightLabel{\\textsc{MI}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash \\{t\\}:\\textsf{M }T$}\n\\DisplayProof\n\n\\subsubsection*{Impure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e:T$} \n\\RightLabel{\\textsc{RefI}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash \\textsf{new }e\\div\\textsf{ref }T$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e:\\textsf{ref }T$} \n\\RightLabel{\\textsc{RefGet}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash !e\\div T$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e:\\textsf{ref }T$}\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e':T$}\n\\RightLabel{\\textsc{RefSet}}\n\\BinaryInfC{$\\Sigma;\\Gamma\\vdash e:=e'\\div 1$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e:T$} \n\\RightLabel{\\textsc{MRet}}\n\\UnaryInfC{$\\Sigma;\\Gamma\\vdash \\textsf{return }e\\div T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Sigma;\\Gamma\\vdash e:\\textsf{M }T$}\n\\AxiomC{$\\Sigma;\\Gamma,x:T\\vdash t\\div T'$}\n\\RightLabel{\\textsc{MLet}}\n\\BinaryInfC{$\\Sigma;\\Gamma\\vdash \\textsf{let }x=e;t\\div T'$}\n\\DisplayProof\n\n\\subsubsection*{Store and configuration}\n\n\\vspace{0.5em}\n\\AxiomC{{\\color{white} $\\Sigma\\vdash\\cdot$}} %placeholder \n\\RightLabel{\\textsc{StoreNil}}\n\\UnaryInfC{$\\Sigma\\vdash\\cdot:\\cdot$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{$\\Sigma\\vdash\\sigma':\\Sigma'$} \n\\AxiomC{$\\Sigma;\\cdot\\vdash v:T$} \n\\RightLabel{\\textsc{StoreCons}}\n\\BinaryInfC{$\\Sigma\\vdash(\\sigma',l:v):(\\Sigma',l:T)$}\n\\DisplayProof$\\quad$ %==========================================================\n\\AxiomC{$\\Sigma\\vdash\\sigma:\\Sigma$} \n\\AxiomC{$\\Sigma;\\cdot\\vdash t\\div T$} \n\\RightLabel{\\textsc{ConfigOK}}\n\\BinaryInfC{$\\langle\\sigma;t\\rangle:\\langle\\Sigma;T\\rangle$}\n\\DisplayProof\n\n\\subsection*{Operational semantics}\n\n\\subsubsection*{Pure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$e_1\\leadsto e_1'$}\n\\UnaryInfC{$e_1\\ e_2\\leadsto e_1'\\ e_2$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e_2\\leadsto e_2'$}\n\\UnaryInfC{$v\\ e_2\\leadsto v\\ e_2'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\UnaryInfC{$(\\lambda x:T.\\ e)\\ v\\leadsto[v/x]e$}\n\\DisplayProof\n\n\\subsubsection*{Impure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\sigma;\\textsf{new }e\\rangle\\leadsto\\langle\\sigma;\\textsf{new }e'\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$l\\notin\\text{dom}(\\sigma)$}\n\\UnaryInfC{$\\langle\\sigma;\\textsf{new }v\\rangle\\leadsto\\langle(\\sigma,l:v);\\textsf{return }l\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\sigma;!e\\rangle\\leadsto\\langle\\sigma;!e'\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$l:v\\in\\text{dom}(\\sigma)$}\n\\UnaryInfC{$\\langle\\sigma;!l\\rangle\\leadsto\\langle\\sigma;\\textsf{return }v\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e_1\\leadsto e_1'$}\n\\UnaryInfC{$\\langle\\sigma;e_1:=e_2\\rangle\\leadsto\\langle\\sigma;e_1':=e_2\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e_2\\leadsto e_2'$}\n\\UnaryInfC{$\\langle\\sigma;v:=e_2\\rangle\\leadsto\\langle\\sigma;v:=e_2'\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} %placeholder\n\\UnaryInfC{$\\langle(\\sigma,l:v,\\sigma');l:=v\\rangle\\leadsto\\langle(\\sigma,l:v',\\sigma');\\textsf{return }\\langle\\rangle\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\sigma;\\textsf{return }e\\rangle\\leadsto\\langle\\sigma;\\textsf{return }e'\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\sigma;\\textsf{let }x=e;t\\rangle\\leadsto\\langle\\sigma;\\textsf{let }x=e';t\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} %placeholder\n\\UnaryInfC{$\\langle\\sigma;\\textsf{let }x=\\{\\textsf{return v}\\};t\\rangle\\leadsto\\langle\\sigma;[v/x]t\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\langle\\sigma;t_1\\rangle\\leadsto\\langle\\sigma';t_1'\\rangle$}\n\\UnaryInfC{$\\langle\\sigma;\\textsf{let }x=\\{t_1\\};t_2\\rangle\\leadsto\\langle\\sigma';\\textsf{let }x=\\{t_1'\\};t_2\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\n\\subsection{Monadic $\\lambda$-Calculus for I/O}\n\n\\subsection*{Syntax}\n\n\\hspace{-1em}\n\\begin{tabular}{lccl}\nTypes & $T$ & $::=$ & $1\\ |\\ \\mathbb{N}\\ |\\ T_1\\to T_2\\ |\\ \\textsf{M}_\\textsf{IO}\\ T$\\\\[0.5em]\nPure Terms & $e$ & $::=$ & $\\langle\\rangle\\ |\\ n\\ |\\ \\lambda x:T.\\ e\\ |\\ e_1\\ e_2\\ |\\ l\\ |\\ \\{t\\}$\\\\[0.5em]\nImpure Terms & $t$ & $::=$ & $\\textsf{print }e\\ |\\ \\textsf{let }x=e;t\\ |\\ \\textsf{return }e$\\\\[0.5em]\nValues & $v$ & $::=$ & $\\langle\\rangle\\ |\\ n\\ |\\ \\lambda x:T.\\ e\\ |\\ \\{t\\}$\\\\[0.5em]\nOutput Tokens & $\\omega$ & $::=$ & $\\cdot\\ |\\ n::\\omega$\\\\[0.5em]\nContexts & $\\Gamma$ & $::=$ & $\\cdot\\ |\\ \\Gamma,x:T$\n\\end{tabular} \n\n\\subsection*{Typing rules}\n\n\\subsubsection*{Pure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$x:T\\in\\Gamma$} \n\\RightLabel{\\textsc{Hyp}}\n\\UnaryInfC{$\\Gamma\\vdash x:T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{1I}}\n\\UnaryInfC{$\\Gamma\\vdash\\langle\\rangle:1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} % placeholder\n\\RightLabel{\\textsc{$\\mathbb{N}$I}}\n\\UnaryInfC{$\\Gamma\\vdash n:\\mathbb{N}$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$\\Gamma,x:T\\vdash e:T'$} \n\\RightLabel{\\textsc{$\\to$I}}\n\\UnaryInfC{$\\Gamma\\vdash\\lambda x:T.\\ e:T\\to T'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e_1:T\\to T'$}\n\\AxiomC{$\\Gamma\\vdash e_2:T$}\n\\RightLabel{\\textsc{$\\to$E}}\n\\BinaryInfC{$\\Gamma\\vdash e_1\\ e_2:T'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash t\\div T$}\n\\RightLabel{\\textsc{MI}}\n\\UnaryInfC{$\\Gamma\\vdash \\{t\\}:\\textsf{M}_\\textsf{IO}\\ T$}\n\\DisplayProof\n\n\\subsubsection*{Impure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$\\Gamma\\vdash e:\\mathbb{N}$} \n\\RightLabel{\\textsc{MPrint}}\n\\UnaryInfC{$\\Gamma\\vdash \\textsf{print }e\\div 1$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e:T$} \n\\RightLabel{\\textsc{MRet}}\n\\UnaryInfC{$\\Gamma\\vdash \\textsf{return }e\\div T$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\Gamma\\vdash e:\\textsf{M}_\\textsf{IO}\\ T$}\n\\AxiomC{$\\Gamma,x:T\\vdash t\\div T'$}\n\\RightLabel{\\textsc{MLet}}\n\\BinaryInfC{$\\Gamma\\vdash \\textsf{let }x=e;t\\div T'$}\n\\DisplayProof\n\n\\subsection*{Operational semantics}\n\n\\subsubsection*{Pure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$e_1\\leadsto e_1'$}\n\\UnaryInfC{$e_1\\ e_2\\leadsto e_1'\\ e_2$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e_2\\leadsto e_2'$}\n\\UnaryInfC{$v\\ e_2\\leadsto v\\ e_2'$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $[]$}} %placeholder\n\\UnaryInfC{$(\\lambda x:T.\\ e)\\ v\\leadsto[v/x]e$}\n\\DisplayProof\n\n\\subsubsection*{Impure terms}\n\n\\vspace{0.5em}\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\omega;\\textsf{print }e\\rangle\\leadsto\\langle\\omega;\\textsf{print }e'\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} %placeholder\n\\UnaryInfC{$\\langle\\omega;\\textsf{print }n\\rangle\\leadsto\\langle(n::\\omega);\\textsf{return }\\langle\\rangle\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\omega;\\textsf{return }e\\rangle\\leadsto\\langle\\omega;\\textsf{return }e'\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$e\\leadsto e'$}\n\\UnaryInfC{$\\langle\\omega;\\textsf{let }x=e;t\\rangle\\leadsto\\langle\\omega;\\textsf{let }x=e';t\\rangle$}\n\\DisplayProof\\bigbreak %========================================================\n\\noindent\n\\AxiomC{{\\color{white} $\\langle\\rangle$}} %placeholder\n\\UnaryInfC{$\\langle\\omega;\\textsf{let }x=\\{\\textsf{return v}\\};t\\rangle\\leadsto\\langle\\omega;[v/x]t\\rangle$}\n\\DisplayProof$\\qquad$ %=========================================================\n\\AxiomC{$\\langle\\omega;t_1\\rangle\\leadsto\\langle\\omega';t_1'\\rangle$}\n\\UnaryInfC{$\\langle\\omega;\\textsf{let }x=\\{t_1\\};t_2\\rangle\\leadsto\\langle\\omega';\\textsf{let }x=\\{t_1'\\};t_2\\rangle$}\n\\DisplayProof\\newpage %=========================================================\n\n\\end{document}\n", "meta": {"hexsha": "3b993cace876689c4c117ec469e3c668ec27caf1", "size": 28783, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Types Cheat Sheet.tex", "max_stars_repo_name": "VictorZXY/cst-part-ii-types-cheat-sheet", "max_stars_repo_head_hexsha": "62d5b1a5c8477f0c1a0b61d3e9adf662c70015de", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Types Cheat Sheet.tex", "max_issues_repo_name": "VictorZXY/cst-part-ii-types-cheat-sheet", "max_issues_repo_head_hexsha": "62d5b1a5c8477f0c1a0b61d3e9adf662c70015de", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Types Cheat Sheet.tex", "max_forks_repo_name": "VictorZXY/cst-part-ii-types-cheat-sheet", "max_forks_repo_head_hexsha": "62d5b1a5c8477f0c1a0b61d3e9adf662c70015de", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 44.6248062016, "max_line_length": 209, "alphanum_fraction": 0.5638397665, "num_tokens": 10337, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3481511269961152}} {"text": "\\documentclass[../PHYS306Notes.tex]{subfiles}\n\n\\begin{document}\n\\section{Lecture 6}\n\\subsection{Lecture Notes - Charged Particles in EM Fields}\n\\subsubsection{Newton's Law in the presence of EM fields}\nNewton says:\n\\[m\\ddot{\\v{r}} = q(\\v{E} + \\dot{\\v{r}}\\times \\v{B})\\] for an electric field $\\v{E}$ and magnetic field $\\v{B}$. This is just the sum of the Coloumb force and Lorentz force.\n\n\\subsubsection{Scalar and Vector Potentials}\nWe will now apply the ideas of vector potential (for the magnetic field) from PHYS 301 to this situation. Recall that we defined the vector potential $\\v{A}$ as \\[\\v{B} = \\curl{\\v{A}}\\].\nWe also recall Faraday's Law, which states (combined with the definition of the potential above) that:\n\\[\\curl{\\v{E}} = -\\dpd{\\v{B}}{t} = -\\dpd{\\curl{\\v{A}}}{t}\\]\nInterchanging the order of taking the curl and taking the time derivative, we see that:\n\\[\\curl{\\left(\\v{E} + \\dpd{\\v{A}}{t}\\right)} = \\v{0}\\]\nSo there exists a scalar potential $V$ such that:\n\\[-\\grad{V} = \\v{E} + \\dpd{\\v{A}}{t}\\]\nWhich comes from the fact that a curl of a gradient is zero. We can rewrite this to say that:\n\\[\\v{E} = -\\grad{V} - \\dpd{\\v{A}}{t}\\]\nNow, we construct $\\LL$ such that we get the Lorentz Force. Generalize $U = qV$ to get:\n\\[U' = qV - q\\dot{\\v{r}}\\cdot\\v{A}\\]\nAnd we use $\\LL' = T - U'$ as our Lagrangian.\n\nNext week Monday, we will look at symmetry properties of the Lagrangian and how we can use this to derive conservation laws. On Wednesday we will look at the method of using Lagrange multipliers for constrained systems. On Friday we will review damped oscillators before moving into a discussion of coupled oscillators.\n\n\n\\end{document}", "meta": {"hexsha": "625cfe37dab5219ba2de9ad412e77be6619e4755", "size": 1658, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Lecture-6/Lecture-Notes-6.tex", "max_stars_repo_name": "RioWeil/PHYS306-notes", "max_stars_repo_head_hexsha": "9394a8cd986722b6fdcb57c8846c6b0d52c23188", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Lecture-6/Lecture-Notes-6.tex", "max_issues_repo_name": "RioWeil/PHYS306-notes", "max_issues_repo_head_hexsha": "9394a8cd986722b6fdcb57c8846c6b0d52c23188", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Lecture-6/Lecture-Notes-6.tex", "max_forks_repo_name": "RioWeil/PHYS306-notes", "max_forks_repo_head_hexsha": "9394a8cd986722b6fdcb57c8846c6b0d52c23188", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.4074074074, "max_line_length": 319, "alphanum_fraction": 0.6996381182, "num_tokens": 519, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961013, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.34815112699611517}} {"text": "\\documentclass{llncs}\n\\usepackage[dvips,final]{graphics}\n\\usepackage{amsmath}\n\\usepackage{times}\n%\n%\n%\n\\begin{document}\n\\mainmatter\n\n%\n\\title{An Image Inpainting Technique based on the Fast Marching Method}\n\\titlerunning{An Image Inpainting Technique}\n\\author{Alexandru Telea}\n\n\\authorrunning{Alexandru Telea}\n\\tocauthor{Alexandru Telea (Eindhoven University of Technology)}\n\n\\institute{Department of Mathematics and Computer Science,\\\\\nEindhoven University of Technology, \\\\\nDen Dolech 2,Eindhoven 5600 MB, The Netherlands, \\\\\n\\email{alext@win.tue.nl},\n\\texttt{http://www.win.tue.nl/$\\sim$alext}}\n%\n%\n\\maketitle\n%\n%\n\\begin{abstract} \n% \n Digital inpainting provides a means for reconstruction of small\ndamaged portions of an image. Although the inpainting basics are\nstraightforward, most inpainting techniques published in\nthe literature are complex to understand and implement.\nWe present here a new algorithm for digital inpainting based on the fast\nmarching method for level set applications. Our algorithm is very\nsimple to implement, is fast, and produces nearly identical results as\nmore complex, and usually slower, known methods. Source code is available online.\n\\end{abstract}\n%\n%\n\\section{Introduction}\n%\n%\n Digital inpainting, the technique of reconstructing small damaged portions\nof an image, has received considerable attention in the last years.\nDigital inpainting serves a wide range of\napplications, such as removing text and logos from still images or videos, reconstructing\nscans of deteriorated images by removing scratches or stains, or creating\nartistic effects.\n Most inpainting methods work as follows. First, the\nimage regions to be inpainted are selected, usually in a\nmanual way. Next, color information is propagated inwards from the region\nboundaries, i.e. the known image information is used to fill-in the missing areas. \nIn order to produce a perceptually plausible reconstruction, an inpainting\ntechnique should attempt to continue the \n\\textit{isophotes} (lines of equal gray value) as smoothly as possible inside the\nreconstructed region. In other words, the missing region should be inpainted\nsuch that the inpainted gray value and gradient extrapolate\nthe gray value and gradient outside this region. \n \nSeveral inpainting methods are based on the above ideas.\nIn \\cite{bertalmio1,bertalmio2}, the image smoothness\ninformation, estimated by the image Laplacian, is propagated along the\nisophotes directions, estimated by the image gradient rotated with 90\ndegrees. The \\textit{Total Variational} (TV) model~\\cite{chan} uses an\nEuler-Lagrange equation coupled with anisotropic diffusion to maintain the\nisophotes' directions. The \\textit{Curvature-Driven Diffusion} (CCD)\nmodel~\\cite{chan2} enhances the TV method to drive diffusion along the isophotes' directions\nand thus allows inpainting thicker regions. All above methods essentially \nsolve a partial differential equation (PDE) that describes\nthe color propagation inside the missing region, subject to various\nheuristics that attempt to preserve the\nisophotes' directions. Preserving isophotes is, however desirable, never\nperfectly attained in practice. The main problem is that both isophote\nestimation and information propagation are subject to numerical diffusion.\nDiffusion is desirable as it stabilizes the PDEs to be solved, but leads\ninevitably to a cetain amount of blurring of the inpainted area.\n\nA second type of methods~\\cite{oliveira}\nrepeatedly convolves a simple 3x3 filter over the missing regions to\ndiffuse known image information to the missing pixels.\n \nHowever impressive, the above methods have several drawbacks that preclude\ntheir direct use in practice. The PDE-based methods require implementing non-trivial iterative numerical\nmethods and techniques, such as anisotropic diffusion and multiresolution schemes \\cite{bertalmio1}.\nLittle or no information is given on practical implementation details such as various thresholds or\ndiscretization methods, although some steps are mentioned as numerically unstable.\nMoreover, such methods are quite slow, e.g. a few minutes for the relatively\nsmall inpainting region shown in Fig.~\\ref{fig:example3}.\nIn contrast, the convolution-based method described in~\\cite{oliveira} \nis simple to implement and fast. However, this method has no provisions for\npreserving the isophotes' directions.\nHigh-gradient image areas must be manually selected before inpainting and\nseparately treated in order not to be blurred. \n\n We propose a new inpainting algorithm based on propagating an image smoothness estimator along the\nimage gradient, similarly to~\\cite{bertalmio1}. We estimate the image\nsmoothness as a weighted average over a known image neighborhood of the pixel to\ninpaint. We treat the missing regions as level sets and use\nthe fast marching method (FMM) described in~\\cite{sethian} to propagate the\nimage information. Our approach has several advantages:\n\\begin{itemize}\n \\item it is very simple to implement (the complete pseudocode is given here).\n \\item it is considerably faster than other inpainting methods\n --- processing an 800x600 image (Fig.~\\ref{fig:example3}) takes under 3 seconds on a 800 MHz PC.\n \\item it produces very similar results as compared to the other methods.\n \\item it can easily be customized to use different local inpainting strategies.\n\\end{itemize}\n%\n%\n In Section~\\ref{sec:method}, we describe our method. Section~\\ref{sec:discussion}\npresents several results, details our method's advantages and limitations\nin comparison to other methods, and discusses possible enhancements. Source\ncode of a sample method implementation is available online at the address\nlisted at the end of the paper.\n%\n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{1.0\\textwidth}{!}{\\includegraphics{EPS/example3.eps}}\n\t\\caption{800x600 image inpainted in under 3 seconds}\n\t\\label{fig:example3} \n \\end{figure} \n%\n% \n%\n%\n\\section{Method}\n\\label{sec:method}\n%\nThis section describes our inpainting method. First,\nwe introduce the mathematical model we base our inpainting\non (Sec.~\\ref{sec:model}). Next, we describe how the missing regions are\ninpainted using the FMM (Sec.~\\ref{sec:fmm}). Finally, we detail the\nimplementation of inpainting one point on the missing region's boundary\n(Sec.~\\ref{sec:inp_function}).\n%\n\\subsection{Mathematical model}\n\\label{sec:model}\n%\n To explain our method, consider Fig.~\\ref{fig:method}, in which one must\ninpaint the point $p$ situated on the boundary $\\partial\\Omega$ of the\nregion to inpaint $\\Omega$. Take a small neighborhood $B_{\\varepsilon}(p)$\nof size $\\varepsilon$ of the known image around $p$\n(Fig.~\\ref{fig:method}~a). As described in~\\cite{bertalmio1,oliveira,chan},\nthe inpainting of $p$ should be determined by the values\nof the known image points close to $p$, i.e. in $B_{\\varepsilon}(p)$. \nWe first consider gray value images, color images being a natural extension (see Sec.~\\ref{sec:implem}).\n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{0.65\\textwidth}{!}{\\includegraphics{EPS/method.eps}}\n\t\\caption{The inpainting principle}\n\t\\label{fig:method} \n \\end{figure} \n%\nFor $\\varepsilon$ small enough, we consider a first order approximation $I_q(p)$\nof the image in point $p$, given the image $I(q)$ and gradient $\\nabla I(q)$\nvalues of point $q$ (Fig~\\ref{fig:method}~b):\n%\n%\n\\begin{equation}\n I_q(p) = I(q) + \\nabla I(q) (p-q)\n\\label{eqn:linear}\n\\end{equation}\n%\n%\nNext, we inpaint point $p$ as function of all points $q$ in\n$B_{\\varepsilon}(p)$ by summing the estimates of all points $q$, \nweighted by a normalized weighting function $w(p,q)$:\n%\n%\n\\begin{equation}\n I(p) = \\frac{\\sum_{q \\in B_{\\varepsilon}(p)} w(p,q) [I(q) + \\nabla I(q) (p-q)]}\n {\\sum_{q \\in B_{\\varepsilon}(p)} w(p,q)}\n\\label{eqn:sum}\n\\end{equation}\n%\n%\nThe weighting function $w(p,q)$, detailed in Sec.~\\ref{sec:inp_function}, \nis designed such that the inpainting of $p$ propagates the gray value as well \nas the sharp details of the image over $B_{\\varepsilon}(p)$. \n%\n\\subsection{Adding inpainting to the FMM}\n\\label{sec:fmm}\n%\n%\n Section~\\ref{sec:model} explained how to inpaint\na point on the unknown region's boundary $\\partial\\Omega$ as a function of\nknown image pixels only. To inpaint the whole $\\Omega$, we iteratively apply\niteratively apply Equation~\\ref{eqn:sum} to all the discrete pixels of\n$\\partial\\Omega$, in increasing distance from $\\partial\\Omega$'s initial\nposition $\\partial\\Omega_i$, and advance the boundary inside $\\Omega$ until the whole\nregions has been inpainted (see pseudocode in Fig.~\\ref{fig:pseudo}). \nInpainting points in increasing\ndistance order from $\\partial\\Omega_i$ ensures that areas closest to known\nimage points are filled in first, thus mimicking manual inpainting\ntechniques~\\cite{bertalmio1,bertalmio2}. \n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{0.45\\textwidth}{!}{\\includegraphics{EPS/pseudo.eps}}\n\t\\caption{Inpainting algorithm}\n\t\\label{fig:pseudo} \n \\end{figure} \n%\n\nImplementing the above requires thus a method that propagates\n$\\partial\\Omega$ into $\\Omega$ by advancing the pixels of $\\partial\\Omega$\nin order of their distance to the initial boundary $\\partial\\Omega_i$. \nFor this, we use the fast marching method (FMM) developed by Sethian et al.\n\\cite{sethian}. In brief, the FMM is an algorithm that solves the Eikonal equation:\n%\n%\n\\begin{equation}\n |\\nabla T| = 1 \\mbox{~~~~~on $\\Omega$,~~~~~with $T=0$ on $\\partial\\Omega$}\n\\label{eqn:eikonal}\n\\end{equation}\n%\nThe solution $T$ of Eqn.~\\ref{eqn:eikonal} is the distance map of the $\\Omega$ pixels to the boundary\n$\\partial\\Omega$. The level sets, or isolines, of $T$ \nare exactly the successive boundaries $\\partial\\Omega$ of the shrinking\n$\\Omega$ that we need for inpainting. The normal $N$ to\n$\\partial\\Omega$, also needed for inpainting, is exactly $\\nabla T$. The FMM\nguarantees that pixels of $\\partial\\Omega$ are always processed in\nincreasing order of their distance-to-boundary $T$~\\cite{fmm_book}, i.e.\nthat we always inpaint the closest pixels to the known image area first. \n\nWe prefer the FMM over other 'distance transform' (DT) methods that compute the distance map $T$ to a boundary $\\partial\\Omega$\n(e.g.~\\cite{chamfer1,chamfer2,roerdinck}). The FMM's main advantage is that it\n\\textit{explicit} maintains the narrowband that\nseparates the known from the unknown image area \\textit{and} specifies which is the next pixel to\ninpaint. Other DT methods compute the distance map $T$ but do not maintain an\nexplicit narrowband. Adding a narrowband structure to these method would\ncomplicate their implementation, whereas the FMM provides this structure by\ndefault.\n\nTo explain in detail our use of the FMM --- and since the FMM is not\nstraightforward to implement from the reference\nliterature~\\cite{sethian,fmm_book} --- we provide next its complete pseudocode. \n The FMM maintains a so-called \\textit{narrow band} of pixels, which is\nexactly our inpainting boundary $\\partial\\Omega$. For every image pixel, we store\nits value $T$, its image gray value $I$ (both represented as floating-point\nvalues), and a flag $f$ that may have three values:\n\\begin{itemize}\n \\item\\textit{BAND}: the pixel belongs to the narrow band. Its $T$ value\n undergoes update.\n \\item\\textit{KNOWN}: the pixel is outside $\\partial\\Omega$, in the known\n image area. Its $T$ and $I$ values are known.\n \\item\\textit{INSIDE}: the pixel is inside $\\partial\\Omega$, in the region to inpaint. Its $T$\n and $I$ values are not yet known.\n\\end{itemize}\n%\n%\n The FMM has an initialization and propagation phase, as follows.\nFirst, we set $T$ to zero on and outside the boundary\n$\\partial\\Omega$ of the region to inpaint and to some large value (in practice $10^6$)\ninside, and initialize $f$ over the whole image as explained\nabove. All $BAND$ points are inserted in a heap \\texttt{NarrowBand}\nsorted in ascending order of their $T$ values. \nNext, we propagate the $T$, $f$, and $I$ values using the code\nshown in Fig.~\\ref{fig:fmm_code}.\n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{0.75\\textwidth}{!}{\\includegraphics{EPS/code.eps}}\n\t\\caption{Fast marching method used for inpainting}\n\t\\label{fig:fmm_code} \n \\end{figure} \n%\n%\nStep 1 extracts the $BAND$ point with the smallest $T$. Step 2 marches\nthe boundary inwards by adding new points to it. Step 3 performs the\ninpainting (see Sec.~\\ref{sec:inp_function}). Step 4 propagates the value\n$T$ of point $(i,j)$ to its neighbors $(k,l)$ by solving the finite difference discretization of\nEqn.~\\ref{eqn:eikonal} given by:\n%\n\\begin{equation}\n \\mbox{max}(D^{-x}T,-D^{+x}T,0)^2+\\mbox{max}(D^{-y}T,-D^{+y}T,0)^2 = 1\n\\label{eqn:quad}\n\\end{equation}\n%\nwhere $D^{-x}T(i,j) = T(i,j)-T(i-1,j)$ and $D^{+x}T(i,j) = T(i+1,j)-T(i,j)$\nand similarly for $y$. Following the upwind idea of Sethian~\\cite{sethian},\nwe solve Eqn.~\\ref{eqn:quad} for $(k,l)$'s four quadrants and retain\nthe smallest solution. Finally, step 5 (re)inserts $(k,l)$\nwith its new $T$ in the heap.\n\n%\n%\n\\subsection{Inpainting one point} \n\\label{sec:inp_function}\n%\n%\nWe consider now how to inpaint a newly discovered point $(k,l)$, as function of the\n$KNOWN$ points around it, following the idea described in Sec.~\\ref{sec:model}\n(step 3 in Fig.~\\ref{fig:fmm_code}, detailed in Fig.~\\ref{fig:inpaint_code})).\n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{0.7\\textwidth}{!}{\\includegraphics{EPS/inpaint_code.eps}}\n\t\\caption{Inpainting one point}\n\t\\label{fig:inpaint_code} \n \\end{figure} \n%\nWe iterate over the $KNOWN$ points in the neighborhood $B_{\\varepsilon}$ of\nthe current point $(i,j)$ and compute $I(i,j)$ following\nEqn.~\\ref{eqn:sum}. The image gradient $\\nabla I$ (\\texttt{gradI} in the\ncode) is estimated by central differences. As stated in Sec.~\\ref{sec:model}, the design of the\nweighting function $w(p,q)$ is crucial to propagate the sharp image details and the smooth\nzones as such into the inpainted zone. We design $w(p,q) = dir(p,q) \\cdot\ndst(p,q) \\cdot lev(p,q)$ as a product of three factors:\n%\n%\n%\\begin{equation}\n% w(p,q)=dir(p,q) \\cdot dst(p,q) \\cdot lev(p,q)\n%\\end{equation}\n%\n%\n\\begin{eqnarray*}\n dir(p,q) &=& \\frac{p-q}{||p-q||} \\cdot N(p)\\\\\n dst(p,q) &=& \\frac{d_0^2}{||p-q||^2} \\\\\n lev(p,q) &=& \\frac{T_0}{1+|T(p)-T(q)|} \n\\end{eqnarray*}\n%\n%\nThe \\textit{directional} component $dir(p,q)$ ensures that the\ncontribution of the pixels close to the normal direction\n$N=\\nabla T$ (\\texttt{gradT} in the code), i.e. close to the\nFMM's information propagation direction, is higher than for those further from\n$N$. The \\textit{geometric distance} component $dst(p,q)$ decreases the\ncontribution of the pixels geometrically farther from $p$. \nThe \\textit{level set} distance\ncomponent $lev(p,q)$ ensures that pixels close to the contour\nthrough $p$ contribute more than further pixels. Both $dst$ and $lev$ are\nrelative with respect to the reference distances $d_0$ and $T_0$. In practice, we set\n$d_0$ and $T_0$ to the inter-pixel distance, i.e. to 1.\nOverall, the above factors model the manual inpainting heuristics~\\cite{bertalmio1} that\ndescribe how to paint a point by strokes bringing color from a small region\naround it.\n\nFor $\\varepsilon$ up to about\n6 pixels, i.e. when inpainting thin regions, $dst$ and $lev$ have\na weak effect. For thicker regions to inpaint, such as\nFig.~\\ref{fig:example2}~d where we used an $\\varepsilon$ of 12 pixels, \nusing $dst$ and $lev$ provides better\nresults than using $dir$ alone. The above is clearly visible in\nFig.~\\ref{fig:ellipse}, on a test image taken from~\\cite{bertalmio1}, where\nthe missing ring shaped region is over 30 pixels thick.\nFigure~\\ref{fig:ellipse}~c shows, on an image detail, the effect of $dir$ alone. The results are\nsomewhat less blurry when $dir$ and $dst$ (Fig.~\\ref{fig:ellipse}~d) or\n$dir$ and $lev$ (Fig.~\\ref{fig:ellipse}~e) are used together. The inpainting\nis visually the best when all three components are used (Fig.~\\ref{fig:ellipse}~f).\n%\n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{0.81\\textwidth}{!}{\\includegraphics{EPS/ellipse.eps}}\n\t\\caption{Thick region to inpaint (a) and result (b). Effect of weighting functions: \n direction (c), direction and geometric distance (d), direction and level set distance (e),\n direction, geometric, and level set distance (f)}\n\t\\label{fig:ellipse} \n \\end{figure} \n%\n%\n%\n%\n\\subsection{Implementation details}\n\\label{sec:implem}\n%\n%\n Several implementation details are important. First, we compute the\nboundary normal $N=\\nabla T$ by numerical derivation of the field $T$\ncomputed by the FMM. Derivating $T$ on the fly as it is computed by the\nFMM is unstable, since we are not guaranteed that a large enough\nneighborhood around the current point contains only $KNOWN$ points.\nWe first run the FMM outside the initial inpainting boundary\n$\\partial\\Omega$ and obtain the distance field $T_{out}$. Since we use only\nthose points closer to $\\partial\\Omega$ than $\\varepsilon$, we run the FMM\noutside $\\partial\\Omega$ only until we reach $T > \\varepsilon$. This\nrestricts the FMM computations to a band of thickness $\\varepsilon$ around\n$\\partial\\Omega$, thus speeding up the process. Next, we run the FMM\ninside $\\partial\\Omega$ and obtain $T_{in}$. The field $T$ over the whole\nimage is given by:\n%\n\\begin{equation}\nT(p) = \\left\\{\n \\begin{array}{rcl}\n &T_{in}&(p)~~~\\mbox{if}~p \\in \\Omega\\\\\n -&T_{out}&(p)~~~\\mbox{if}~p \\notin \\Omega.\n\\end{array}\n\\right.\n\\end{equation}\n%\nNext, we smooth $T$ by a 3x3 tent filter, and then compute $\\nabla T$ by\ncentral differences. \n\n The value of $\\varepsilon$ giving the size of $B$ usually ranges from 3 to 10\npixels. This corresponds with the 'thickness' of the regions to inpaint,\nwhich is usually less than 15 pixels. Higher values blur the sharp details\nto be reconstructed by inpainting, although they are useful when inpainting\nthicker regions. \n\n The test \\texttt{f(k,l)!=OUTSIDE} in Fig.~\\ref{fig:inpaint_code} that\nrestricts $B_{\\varepsilon}$ to the known image points can be changed to\n\\texttt{f(k,l)==KNOWN}. The results are visually identical, as\n$B_{\\varepsilon}$ contains very few $BAND$ pixels. However, one would use the second test\nif the initial $\\partial\\Omega$ corresponds to unknown image pixels.\n\nThe \\texttt{NarrowBand} sorted heap (Sec.~\\ref{sec:fmm}) is straightforwardly implemented using\nthe C++ STL \\texttt{multimap} container~\\cite{stl}. Finally, for color (RGB) images we apply the presented \nmethod separately for each color channel.\n%\n\\section{Discussion}\n\\label{sec:discussion}\n%\n%\n We have compared our inpainting method with the methods presented\nby Bertalmio et al. in~\\cite{bertalmio1} and Oliveira et al. in~\\cite{oliveira}, further denoted by\nBSCB and OBMC, by running it on the same input images (see Figures~\\ref{fig:example1}\nand~\\ref{fig:example2}~a-c). For BSCB we used the implementation publicly available\nat~\\cite{implem}, whereas we reimplemented OBMC ourselves. \nOur method produced visually nearly identical results with BSCB. Compare, for example, the\ninpaintings in Fig.~\\ref{fig:example1}~c,d (Fig.~\\ref{fig:example1}~e,f in\ndetail) and Fig.~\\ref{fig:example2}~b,c (Fig.~\\ref{fig:example2}~g,h in detail).\nIn contrast, the results of OBMC (shown in~\\cite{oliveira}) \nwere visibly more blurry for regions thicker than 6 pixels. \n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{1.05\\textwidth}{!}{\\includegraphics{EPS/example1.eps}}\n\t\\caption{Lincoln 'cracked photo' inpainting}\n\t\\label{fig:example1} \n \\end{figure} \n%\n% \n The running time for our method was in all cases much lower than for\nBSCB. Our C++ implementation took under 3 seconds on an 800 MHz PC for a 800x600\ncolor image with about 15\\% pixels to inpaint (Fig.~\\ref{fig:example3}).\nOn the same input, the original BSCB method, published in~\\cite{bertalmio1}, is reported to take under 5 minutes on a\n300 MHz PC. The BSCB implementation we used~\\cite{implem}, which is\nhowever mentioned to be unoptimized by its authors, took between 2.5 and 3 minutes on the 800\nMHz PC, depending on its various parameter settings.\nIn contrast, OBMC takes in all cases about the same time as our method.\nThe above matches the fact that both OBMC and our method are linear\nin the inpainted region's size. \n\nThe main limitation of our method (applicable to the methods BSCB and\nOBMC mentioned here too) is the blurring produced when inpainting\nregions thicker than 10-15 pixels, especially visible when sharp isophotes\nintersect the region's boundary almost tangentially. See for example the\ninpainting in Fig.~\\ref{fig:example2}~d,e (in\ndetail in Fig.~\\ref{fig:example2}~i,j). The above is caused by the linear\nand local character of our method. Techniques using an explicit nonlinear\nand/or global image model, such as the TV~\\cite{chan} and the CDD~\\cite{chan2} methods,\nachieve better results, at the cost of considerably more complex implementations.\n%\n\t\\begin{figure}[h] \\centering\n\t\\resizebox{1.05\\textwidth}{!}{\\includegraphics{EPS/example2.eps}}\n\t\\caption{Inpainting examples. Damaged photo (a), inpainting by method BSCB (b), \n our method (c), and close-ups (f,g,h).\n Damaged photo (d), distance-weighted inpainting (e), and close-ups (i,j)}.\n\t\\label{fig:example2} \n \\end{figure} \n\nOverall, the presented inpainting method is simple to implement (our complete C++ code is \nabout 500 lines), is fast, and easy to customize for different inpainting strategies.\nWe plan to extend the method by developing new inpainting functions that\nare better able to preserve the isophotes' directions. One such way is to\nintegrate anisotropic diffusion, e.g. following~\\cite{bertalmio1}, in the FMM boundary\nevolution in order to reduce the blurring for inpainting thick regions. A\nsecond extension would be to modulate the evolution speed of the FMM, now\nequal to 1, by the image anisotropy, i.e. let inpainting 'work more' on the\nhigh detail areas than on the smooth regions.\n%\n%\n\\section*{Acknowledgements}\n%\n We are indebted to prof. J. J. van Wijk from the Department of\nMathematics and Computer Science of the Eindhoven University of Tehcnology\nfor his numerous suggestions on how to improve this paper.\n%\n\\section*{Web Information}\n%\n Source code of a sample C++ implementation of the inpainting method\ndescribed here is available at\\\\\n\\texttt{http://www.acm.org/jgt/papers/Telea03.html}\n%\n%\n%\n%-------------------------------------------------------------------------------\n%\n\\begin{thebibliography}{13}\n%\n\\bibitem {bertalmio1}\n {\\sc M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester},\n {\\em Image Inpainting}, Proc. SIGGRAPH 2000, ACM Press, pp. 417-424.\n\\bibitem {bertalmio2}\n {\\sc M. Bertalmio, A. L. Bertozzi, and G. Sapiro},\n {\\em Navier-Stokes, Fluid Dynamics, and Image and Video Inpainting},\n Proc. ICCV 2001, IEEE CS Press, vol. 1, pp. 1335-1362.\n\\bibitem{chamfer1}\n {\\sc G. Borgefors},\n {\\em Distance transformations in arbitrary images},\n Comp. Vision, Graphics, and Image Proc., 27(3), pp. 321-345, 1984.\n\\bibitem{chamfer2}\n {\\sc G. Borgefors},\n {\\em Distance transformations in digital images},\n Comp. Vision, Graphics, and Image Proc., 34(3), pp. 344-371, 1986.\n\\bibitem {oliveira}\n {\\sc M. Oliveira, B. Bowen, R. McKenna, and Y.-S. Chang},\n {\\em Fast Digital Image Inpainting},\n Proc. VIIP 2001 (Marbella, Spain), pp. 261-266.\n\\bibitem {chan}\n {\\sc T. Chan, J. Shen}, \n {\\em Mathematical Models for Local Deterministic Inpaintings},\n Tech. report CAM 00-11, Image Processing Research Group, UCLA, 2000.\n\\bibitem {chan2}\n {\\sc T. Chan, J. Shen},\n {\\em Non-texture Inpainting by Curvature-Driven Diffusions (CDD)},\n Tech. report CAM 00-35, Image Processing Research Group, UCLA, 2000.\n\\bibitem{roerdinck}\n {\\sc A. Meijster, J. Roerdink, W. Hesselink},\n {\\em A general algorithm for computing distance transforms in linear time},\n Math. Morph. and its Appls. to Image and Signal Proc., pp. 331--340, Kluwer, 2000.\n\\bibitem {fmm_book}\n {\\sc J. A. Sethian},\n {\\em Level Set Methods and Fast Marching Methods},\n Cambridge Univ. Press, 2nd edition, 1999.\n\\bibitem {sethian}\n {\\sc J. A. Sethian},\n {\\em A Fast Marching Level Set Method for Monotonically Advancing Fronts},\n Proc. Nat. Acad. Sci. vol. 93, nr. 4, pp. 1591-1595, 1996. \n\\bibitem{implem}\n {\\sc W. Yung, A. J. Shankar},\n {\\em Image Inpainting Implementation Software}, \\texttt{www.bantha.org/$\\sim$aj/inpainting}\n\\bibitem{stl}\n {\\sc D. R. Musser, A. Saini},\n {\\em STL Tutorial and Reference Guide: C++\n Programming with the Standard Template Library},\n Addison-Wesley Professional Computing Series, 1996.\n \n\n\\end{thebibliography}\n\n\n%-------------------------------------------------------------------------------\n%\n%\n%\n%\n\\end{document}\n%\n\n", "meta": {"hexsha": "859fc89760d32ac84cf7cb2c753410c3b7732060", "size": 24704, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Volume_09/Number_1/Telea2004/sources/pres.tex", "max_stars_repo_name": "kyeonghopark/jgt-code", "max_stars_repo_head_hexsha": "08bbcc298e12582e32cb56a52e70344c57689d73", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 415, "max_stars_repo_stars_event_min_datetime": "2015-10-24T17:37:12.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-18T04:09:07.000Z", "max_issues_repo_path": "Volume_09/Number_1/Telea2004/sources/pres.tex", "max_issues_repo_name": "kyeonghopark/jgt-code", "max_issues_repo_head_hexsha": "08bbcc298e12582e32cb56a52e70344c57689d73", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 8, "max_issues_repo_issues_event_min_datetime": "2016-01-15T13:23:16.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-27T01:49:50.000Z", "max_forks_repo_path": "Volume_09/Number_1/Telea2004/sources/pres.tex", "max_forks_repo_name": "kyeonghopark/jgt-code", "max_forks_repo_head_hexsha": "08bbcc298e12582e32cb56a52e70344c57689d73", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 77, "max_forks_repo_forks_event_min_datetime": "2015-10-24T22:36:29.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-24T01:03:54.000Z", "avg_line_length": 44.9981785064, "max_line_length": 127, "alphanum_fraction": 0.7420660622, "num_tokens": 7000, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043816, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.34815111900864}} {"text": "\\documentclass[12]{scrartcl}\n\\usepackage{amssymb,amsmath,gensymb,dsfont,calc,multicol,fullpage}\n\\makeatletter\n\\newcommand\\Aboxed[1]{\n \\@Aboxed#1\\ENDDNE}\n\\def\\@Aboxed#1\\ENDDNE{%\n &\n \\settowidth\\@tempdima{$\\displaystyle#1{}$}\n \\setlength\\@tempdima{\\@tempdima+\\fboxsep+\\fboxrule}\n \\kern-\\@tempdima\n \\boxed{#1#2}\n}\n\\makeatother\n\n\\begin{document}\n\n\\title{Homework 26, Section 4.9: 1, 2, 6, 7, 12, 18}\n\\author{Alex Gordon}\n\\date{\\today}\n\\maketitle\n\\section*{Homework}\n\\subsection*{1. A)}\n\\ \\ N \\ M \\\\\n$\\begin{bmatrix} .7 & .6 \\\\ .3 & .4 \\end{bmatrix} $ News and Music\n\\subsection*{1. B)}\n$\\begin{bmatrix} 1\\\\ 0 \\end{bmatrix} $\n\\subsection*{1. C)}\nApproximately 33%\n\\subsection*{2. A)}\n1, 2, and 3 respectively at the tops and it goes to 1, 2, and 3 on the side respectively. \n$\\begin{bmatrix} .6 & .2 & .2 \\\\ .2 & .6 & .2 \\\\ .2 & .2 & .6 \\end{bmatrix} $\n\\subsection*{2. B)}\nApproximately .28\n\\subsection*{6.}\n$\\begin{bmatrix} 4/7 \\\\ 3/7 \\end{bmatrix} $ is the steady state vector. \n\\subsection*{7.}\n$\\begin{bmatrix} 1/4 \\\\ 1/2 \\\\ 1/4 \\end{bmatrix} $ is the steady state vector. \n\\subsection*{12.}\nEach food will be preferred equally, because $\\begin{bmatrix} 1/3 \\\\ 1/3 \\\\ 1/3 \\end{bmatrix} $ of the equal numbers in the steady state vector. \n\\subsection*{18.}\nIf alpha = beta = 0, then $\\begin{bmatrix} 1\\\\ 0 \\end{bmatrix} $ and $\\begin{bmatrix} 0\\\\ 1 \\end{bmatrix} $ are the steady state vectors. If it is not 0, then it should equal $\\frac{1}{\\alpha + \\beta} \\begin{bmatrix} \\beta \\\\ \\alpha \\end{bmatrix}$\n\n\n\\end{document}", "meta": {"hexsha": "341dd6116e68340e4d555c32dbce20cc497cef0b", "size": 1554, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "LinearAlgebra/Homework26.tex", "max_stars_repo_name": "alexggordon/latex", "max_stars_repo_head_hexsha": "7dd945f33490e6585e26cff39d9cf6ad8f582a0e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "LinearAlgebra/Homework26.tex", "max_issues_repo_name": "alexggordon/latex", "max_issues_repo_head_hexsha": "7dd945f33490e6585e26cff39d9cf6ad8f582a0e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LinearAlgebra/Homework26.tex", "max_forks_repo_name": "alexggordon/latex", "max_forks_repo_head_hexsha": "7dd945f33490e6585e26cff39d9cf6ad8f582a0e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.3181818182, "max_line_length": 253, "alphanum_fraction": 0.6473616474, "num_tokens": 633, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.34815111799038173}} {"text": "\\section{Results}\n\\label{section:results}\n\nIn order to demonstrate the performance of our method, we conducted three sets\nof tests.\nIn the first we simulated observables from a set of stellar parameters for a\nfew hundred stars using the MIST stellar evolution models and the\ngyrochronology model of equation \\ref{eqn:gyro}.\nThe ages predicted with our model were compared to the true parameters used to\ngenerate the data.\nIn the second we tested our model by measuring the ages of individual stars in\nthe NGC 6819 open cluster, and in the third we tested our model on \\kepler\\\nasteroseismic stars.\n\n\\subsection{Test 1: simulated stars}\nFor the first test we drew masses, ages, bulk metallicities, distances and\nextinctions at random for 1000 stars from the following uniform distributions:\n\\begin{eqnarray}\n& \\mathrm{EEP} \\sim U(198, 480) \\\\\n% & M \\sim U(0.5, 1.5)~[M_\\odot] \\\\\n& t \\sim U(0.5, 14)\\mathrm{~[Gyr]} \\\\\n& [Fe/H] \\sim U(-0.2, 0.2) \\\\\n& D \\sim U(10, 1000)~\\mathrm{[pc]} \\\\\n& A_V \\sim U(0, 0.1).\n\\end{eqnarray}\n\\teff, \\logg, \\fhat, parallax, and apparent magnitudes $B$, $V$, $J$, $H$, $K$,\n\\gaia\\ $G$, $G_{BP}$ and $G_{RP}$ were generated from these\nstellar parameters using the MIST stellar evolution models.\nWe added a small amount of noise to the `observed' stellar properties in order\nto reflect optimistic observational uncertainties for isochrone-dating.\nWe added Gaussian noise with a standard deviation of 25 K to \\teff, 0.01 dex\nto \\feh\\ and \\logg, and 10 mmags to $B$, $V$, $J$, $H$, and $K$ magnitudes.\nThese are just one choice of uncertainties that we could have adopted and are\nextremely optimistic.\nThe uncertainties on predicted ages {\\it will} depend strongly on all\nobservational uncertainties, however, since this analysis is designed to show\nthe relative improvement in stellar age precision when {\\it rotation periods}\nare included, we chose to use best-case spectroscopic parameters.\nThe noise added to Gaia $G$-band photometry ranged from\n0.3 mmag for stars brighter than 13th magnitude, to 10 mmag for stars\naround 20th magnitude \\citep{evans2017, brown2018}.\nNoise added to \\gaia\\ $G_{BP}$ and $G_{RP}$ bands ranged from 2 mmag for stars\nbrighter than 13th magnitude to 200 mmag for stars fainter than 17th.\nUnphysical combinations of stellar parameters were discarded, resulting in a\nfinal sample size of 841 simulated stars.\nFigure \\ref{fig:CMD_age} shows the position of these stars on an HRD\n(with \\logg\\ on the y-axis instead of luminosity to improve the visibility of\nthe MS), colored by their age.\nRotation periods for these stars were generated using the gyrochronology\nrelation described in equation \\ref{eqn:gyro}.\nWe added 5\\% Gaussian noise to all stellar rotation periods to represent\nrealistic measurement uncertainties of 5\\%.\nThe median uncertainty on rotation periods calculated from \\kepler\\ light\ncurves, provided in the \\citet{mcquillan2014} catalog is 1\\%.\nHowever, the \\citet{aigrain2015} injection and recovery study showed that true\nrotation period uncertainties are often slightly larger than this, and noise\ndistribution of rotation periods can be highly non-Gaussian\n\\citep[\\eg][]{aigrain2015, angus2018}.\n\\begin{figure}\n \\caption{\n The simulated star sample plotted on an HRD, colored by age\n (top panel) and rotation period (bottom panel).\n HRD positions were calculated using MIST isochrones via the {\\tt\n isochrones.py} {\\it Python} package and rotation periods were generated\n using equation \\ref{eqn:gyro}.\n This figure was generated in a Jupyter notebook available at\n \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/Simulate_data.ipynb}\n}\n \\centering\n \\includegraphics[width=1\\textwidth]{simulated_CMD}\n\\label{fig:CMD_age}\n\\end{figure}\nFigure \\ref{fig:rotation_model} shows the rotation periods of 841\nstars generated from the gyrochronology model.\n\\begin{figure}\n \\caption{\nData simulated from the rotation period model.\n Late F, GK and early M dwarfs (stars with 0.56 $<$ \\gcolor\\ $<$ 2.7 follow\n the Praesepe-calibrated gyrochronology relation (dashed gray lines), with\n the exception of old, slowly\n rotating stars with large Rossby numbers whose rotation periods are fixed\n at 2$\\times$ their convective overturn time.\n The rotation periods of early F (\\gcolor $<$ 0.56), late M dwarfs (\\gcolor\n $<$ 2.7) and subgiants (EEP $\\gtrsim$ 420) were generated\n from a log-normal distribution with standard deviation given by equation\n \\ref{eqn:gyro}.\n% The top panel shows the rotation periods vs. B-V colors of simulated stars,\nThe top panel shows the rotation periods vs. \\gcolor\\ colors of simulated stars,\n colored by their age and the bottom panel shows the same stars colored\n by their equivalent evolutionary phase (EEP).\n % 9, 11, and 13 (rotation periods rise with age).\n The gray lines describe the mean gyrochronology model at ages 1,\n 3, 5, 7, 9, 11, and 13 (rotation periods rise with age).\n This figure was generated in a Jupyter Notebook, available at\n \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/Simulate_data.ipynb}\n}\n \\centering\n \\includegraphics[width=1.\\textwidth]{rotation_model_praesepe}\n\\label{fig:rotation_model}\n\\end{figure}\n\nWe took two approaches to inferring the ages of these simulated stars:\nfirstly using isochrone fitting {\\it only}, and secondly using isochrone\nfitting {\\it combined with} a gyrochronology model (\\sd).\nSince the posterior PDFs of stars are often multimodal, we found that the\nchoice of initial positions of the {\\tt emcee} walkers influenced the final\noutcome because walkers occasionally got stuck in local minima.\nWe found that the following set of initial parameters worked well, though not\nperfectly: EEP = 330, $t = 9.56$ Gyr, $[M/H] = -0.05$, $D = 269$ pc and $A_V =\n0.0$\\footnote{These are the default initial parameters provided in \\sd.}.\nFigure \\ref{fig:simulation_results} shows the results of combining\ngyrochronology with isochrone fitting for the simulated sample.\nThe stars' true ages are plotted against their predicted ages, with ages\ninferred with gyrochronology and isochrone fitting\nin color, and ages inferred using isochrone fitting only plotted in light\ngrey.\nThe different panels show the results for different types of stars: FGK\ndwarfs that are still undergoing magnetic braking, FGK dwarfs that have ceased\nmagnetic braking (their Rossby number is around 2), M dwarfs, and evolved\nstars.\nThe selection criteria for these groups are in the panel headings.\nThe FGK dwarfs with low Rossby numbers showed the largest improvement: the\nmedian age precision for this group (defined as the standard deviation of the\nposterior as a percentage of the median age) was 8\\% when using a combination\nof isochrones and gyrochronology, and 22\\% using isochrone fitting alone.\nThis equates to an almost 3$\\times$ improvement in age precision.\nThe age RMS of this group was 0.8 Gyr using isochrones and gyrochronology, and\n2 Gyr using isochrones only.\nDespite the fact that stars with Rossby numbers of 2 have stopped spinning\ndown and their rotation periods no longer evolve with age, the ages of these\nstars can still be relatively precisely constrained with isochrone fitting.\nThe median age precision for stars with large Rossby numbers was 11\\% with\ngyrochronology and isochrone fitting, and 13\\% with isochrone fitting only.\n% This is because their rotation periods are still directly related to their\n% age, but also {\\it indirectly} related to their age via EEP and metallicity,\n% as shown in equation \\ref{eqn:gyro}.\n% When including rotation periods to infer the ages of stars with large Rossby\n% numbers, age precision improves from 21\\% to 11\\% and the age RMS improves\n% from 2.5 to 0.9 Gyr.\n% Age precision also increases for F dwarfs when incorporating their rotation\n% periods, even though they are not directly related to their ages.\n% This is because rotation period relates to color via the variance model in\n% equation \\ref{eqn:gyro}.\nThe precision of M dwarf ages did improve overall when their rotation periods\nwere included, but this improvement was entirely driven by the early M dwarfs.\nThe precision of this group improved overall from 33\\% to 22\\% and RMS from\n5.4 Gyr to 4.6 Gyr.\nThe precision of ages inferred for evolved stars changed very little when\nrotation periods were included in the inference process.\nThis is because the variance of the rotation-age relation was inflated by a\nlarge amount in the gyrochronology model (equation \\ref{eqn:gyro}), making\nrotation periods almost entirely uninformative for this group.\nThe median age precision of subgiants from both gyrochronology and isochrone\nfitting and isochrone fitting alone was 7\\%.\n\n\\begin{figure}\n \\caption{\nThe true vs. predicted ages of simulated stars.\n Ages calculated by combining gyrochronology\n and isochrone fitting with \\sd\\ are shown in color and ages calculated with\n isochrone fitting only are shown in gray.\nThe different panels show the results for stars\n with \\gcolor\\ $<$ 2.2 (FGK dwarfs) that are still braking magnetically\n ($Ro < 2$), stars with \\gcolor\\ $<$ 2.2 that have stopped braking\n magnetically ($Ro \\geq 2$),\n stars with 2.2 $<$ \\gcolor\\ (M dwarfs), and evolved stars (EEP $>$ 420).\nGyrochronology is highly effective for FGK stars and ages inferred with both\n gyrochronology and isochrone fitting are more accurate and precise than\n ages inferred via isochrone fitting only for this group.\nNeither gyrochronology nor isochrone fitting can provide precise ages for\n M dwarfs, so the ages of these stars are imprecise regardless of\n age-dating method.\n This figure was generated in a Jupyter Notebook available at\n \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/Results_plots.ipynb}.\n}\n \\centering\n \\includegraphics[width=1\\textwidth]{simulation_results}\n\\label{fig:simulation_results}\n\\end{figure}\n\nAn important caveat associated with these results is that they strongly depend\non the uncertainties adopted for all observables used in the analysis: \\teff,\n\\feh, \\logg, $G$, $G_BP$, $G_RP$, $J$, $H$, $K$, $B$, $V$, and rotation\nperiod.\nChanging the uncertainties on these observables will affect the uncertainties\non inferred ages in different ways.\nThis demonstration is not intended to reflect the typical age uncertainties\nthat will result for all stars in reality, it merely exemplifies the increase\nin stellar age precision that results from one specific choice of\nuncertainties.\nNeedless to say, estimating the uncertainties on observables accurately can be\nas important as accurately measuring the observables themselves.\n\n% Figure \\ref{fig:all_iso_gyr} shows the inferred vs. true ages of all simulated\n% stars, colored by their \\gaia\\ \\gcolor\\ age.\n% The ages of FGK stars (\\gcolor\\ $<$ 2.2) were more precisely recovered with\n% gyrochronology and isochrone fitting than with isochrone fitting alone.\n% In both cases M dwarfs ages were not precisely recovered.\n% \\begin{figure}\n% \\caption{\n% Inferred vs. true ages for all simulated stars using isochrones plus\n% gyrochronology via the \\sd\\ code (left) and isochrones only (right).\n% Stellar ages are colored by \\gaia\\ \\gcolor\\ color.\n% The ages of FGK stars (\\gcolor\\ $<$ 2.2)\n% were more precisely recovered with \\sd\\ than with isochrones only.\n% In both cases M dwarfs ages were not precisely recovered.\n% Points in the lower right of the left panel are either M dwarfs or\n% evolved (EEP $>$ 400).\n% In both panels, the ages of FGK stars were slightly underestimated.\n% }\n% \\centering\n% \\includegraphics[width=1\\textwidth]{all_iso_gyro}\n% \\label{fig:all_iso_gyro}\n% \\end{figure}\n\n% Figure \\ref{fig:eep} shows the inferred equivalent evolutionary phase (EEP)\n% for the simulated star sample.\n% EEP is a dimensionless number that describes evolutionary stage and, combined\n% with age and metallicity, determines stellar mass.\n% An interesting result of inferring stellar properties with both isochrones\n% {\\it and} rotation periods is an increase in EEP precision, .\n% The reason for this is partly that EEP and age are correlated, so shrinking\n% the age posterior also shrinks the EEP posterior, and partly because rotation\n% period is a function of HRD position.\n% As shown in equation \\ref{eqn:gyro}, rotation period does not just depend on\n% color and age, but also on EEP and [Fe/H].\n% \\begin{figure}\n% \\caption{\n% Inferred EEP for the simulated star sample with isochrones plus\n% gyrochronology (left panel) and isochrone fitting only (right panel).\n% Rotation periods provide information about position on the HRD via\n% equation \\ref{eqn:gyro} and therefore improve all stellar parameters, not\n% just age.\n% This figure was generated in a Jupyter Notebook available at\n% \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/Results_plots.ipynb}.\n% }\n% \\centering\n% \\includegraphics[width=1\\textwidth]{eep}\n% \\label{fig:eep}\n% \\end{figure}\n% In general, the posterior PDFs of stellar properties, inferred using\n% isochrones or stellar evolution models, are multimodal.\n% The stars with under-predicted EEP in both panels of figure \\ref{fig:eep} are\n% slightly evolved, with EEPs $>$ 400.\n% These slightly evolved stars have two high probability modes: one at their\n% true EEP and age values, and another at a lower EEP and corresponding lower\n% age.\n% In other words, these are subgiants being mistaken for dwarfs.\n% Rotation periods do not carry information for subgiants, so including the\n% rotation periods in the analysis of these stars doesn't break the degeneracy,\n% and this is why stars with under-predicted EEPs are present in both panels.\n% Stars with over-predicted EEPs in the right-hand panel of figure \\ref{fig:eep}\n% are usually dwarfs being confused for subgiants.\n% The rotation periods of dwarfs are informative, so when the rotation periods\n% are included in the analysis for these stars, the degeneracy is broken, the\n% amplitude of the secondary mode is reduced, and their EEP is correctly\n% inferred.\n\n% Figure \\ref{fig:precision} shows the simulated stars on an HRD, with\n% points colored by the precision of their predicted ages\n% (defined as the standard deviation of the age\n% posterior PDF, as a percentage of the median age).\n% The top panel shows the precision of ages calculated using both gyrochronology\n% and isochrone fitting via \\sd\\ and the bottom panel shows the precision of\n% ages calculated with isochrone fitting only.\n% Although these are empirical uncertainties, calculated via MCMC which only\n% approximates the age posterior PDFs, they nevertheless show that combining\n% gyrochronology and isochrone fitting improves age precision on the MS,\n% particularly at low masses.\n% \\begin{figure}\n% \\caption{\n% Simulated stars on an HRD, colored by their relative age precision\n% using gyrochronology and isochrone fitting via \\sd\\ (top panel) and\n% isochrone fitting only (bottom panel).\n% Combining gyrochronology with isochrone fitting significantly improves stellar\n% age precision on the MS.\n% Isochrone fitting provides precise ages for hot stars and subgiants and the\n% rotation periods of these stars are relatively uninformative, so\n% gyrochronology does not significantly improve their age precision.\n% The ages of late M dwarfs are highly imprecise because their ages are not well\n% determined by either their rotation periods or their position on the HRD\n% or CMD.\n% This figure was generated in a Jupyter Notebook available at\n% \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/Results_plots.ipynb}.\n% }\n% \\centering\n% \\includegraphics[width=1\\textwidth]{precision_plot}\n% \\label{fig:precision}\n% \\end{figure}\n\nThis simulation experiment was designed to show the theoretical improvement in\nage precision when gyrochronology is incorporated into isochrone fitting.\nHowever, it does not demonstrate the accuracy of this method because the test\ndata were simulated from the same model used to infer ages.\nThe results of this experiment are therefore extremely accurate by design.\nWhen applying this method to real data, the results will only be accurate if\nthe model is accurate.\nIn other words, \\sd, like any age-dating method, provides model-dependent\nages.\nStellar ages calculated with \\sd\\ depend on both the accuracy of the MIST\nmodels {\\it and} the accuracy of the gyrochronology model (equation\n\\ref{eqn:gyro}).\nIn order to test the accuracy of \\sd, we applied it to real data, as described\nin the following section.\n\n\\subsection{Test 2: Open clusters}\nIn order to test our model on real stars with known ages, we selected a sample\nof stars in the 2.5 Gyr NGC 6819 cluster.\nWe compiled \\kepler-based rotation periods \\citep{meibom2015}, \\Gaia\\\nphotometry and \\gaia\\ parallaxes for members of the NGC 6819 cluster.\nFigure \\ref{fig:NGC6819} shows the period-color relation of this\ncluster and figure \\ref{fig:NGC6819_results} shows the results of inferring\nthe ages of individual cluster members using a combination of gyrochronology\nand isochrone fitting (via \\sd) and isochrone fitting alone.\nThe ages of F stars in this cluster (\\gcolor\\ $\\sim$ 5.5-6.5) were relatively\nprecisely constrained by isochrone fitting alone because, at 2.5 Gyr, they are\napproaching the MS turnoff.\nFor these hot stars, ages inferred with gyrochronology and isochrones were\nsimilar to ages inferred with isochrones and similarly precise, showing that\nisochrones provide a lot of age information for these stars and rotation\nperiods do not add significantly more information.\nThe G and early K stars in this cluster (\\gcolor\\ $\\lesssim$ .65) were not\nprecisely recovered from isochrone fitting alone -- the isochrone-only age\nposteriors tend towards the prior which is a uniform distribution between 0\nand 13.8 Gyrs.\nThe median age of stars in the cluster was 4.27 $\\pm$ 0.48 Gyr when only\nisochrone fitting was used.\nIn contrast, including gyrochronology when inferring the ages of G and K stars\nin this cluster significantly improved age precision.\nThe median age of stars in this cluster was 2.63 $\\pm$ 0.16 Gyr when ages were\ninferred with a combination of isochrone fitting and gyrochronology, using the\nnewly calibrated Praesepe-based gyrochronology model.\nThe previously-calibrated \\citet{angus2015} model resulted in a median stellar\nage of 2.66 $\\pm$ 0.21 Gyr, which is still consistent with the established\ncluster age of 2.5 Gyr.\nThe median age of stars in the cluster using uncorrected photometry was\nslightly underestimated at 1.86 $\\pm$ 0.22 Gyr.\nThis suggests that, despite the fact that V-band extinction is marginalized\nover during the inference process, correcting for extinction {\\it before} ages\nare estimated will reduce bias introduced by dust.\n\n\\begin{figure}\n \\caption{\n The \\kepler-based rotation periods of members of the 2.5 Gyr NGC 6819 open\n cluster.\n The raw \\gcolor\\ colors are shown in red and the dust-corrected colors are\n shown in black.\n The dashed line shows a gyrochronology model that was fit to the Praesepe\n cluster and the Sun in this work, interpolated to 2.5 Gyrs.\n The solid blue line shows a previously calibrated gyrochronology model\n \\citep{angus2015}.\n This figure was generated in a Jupyter Notebook available at:\n \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/NGC6819.ipynb}\n }\n \\centering\n \\includegraphics[width=1\\textwidth]{NGC6819}\n\\label{fig:NGC6819}\n\\end{figure}\n\n\\begin{figure}\n \\caption{\n The inferred ages of members of the NGC 6819 open cluster as a function of\n their \\gcolor\\ color.\n Ages of stars inferred using a combination of isochrone fitting and\n gyrochronology (Praesepe and Sun calibration) with dereddened \\gaia\\ $G$,\n $G_{BP}$, and $G_{RP}$ photometry (black circles) and uncorrected, raw,\n photometry (red squares).\n % Black circles and red squares show the ages of stars inferred using a\n % combination of isochrone fitting and gyrochronology, with a gyrochronology\n % relation that was calibrated to Praesepe and the Sun. Black circles show\n % ages inferred with dereddened \\gaia\\ $G$, $G_{BP}$ and\n % $G_{RP}$ photometry and red squares show ages inferred with uncorrected,\n % raw, photometry.\n Even though V-band extinction is marginalized over in the inference\n process, reddening can still bias ages.\n Blue triangles, pointing up, show ages inferred using isochrone fitting\n and gyrochronology, with the \\citet{angus2015} gyrochronology model.\n Orange triangles, pointing down, show ages inferred using isochrone\n fitting only.\n The ages of F stars (stars bluer than 0.7) were precisely constrained by\n isochrones and including gyrochronology makes little difference to their\n inferred ages.\n The age precision of G and K dwarfs (stars redder than 0.7) was improved\n by including gyrochronology.\n The median age of stars inferred using the gyrochronology model\n calibrated to Praesepe and the Sun (black circles) was 2.65 $\\pm$ 0.13\n which is consistent with the established cluster age (2.5 Gyr).\n This figure was generated in a Jupyter Notebook available at:\n \\url{https://github.com/RuthAngus/stardate/blob/master/paper/code/NGC6819.ipynb}\n}\n \\centering\n \\includegraphics[width=1\\textwidth]{NGC6819_results}\n\\label{fig:NGC6819_results}\n\\end{figure}\n\nWe found that 5\\% rotation period uncertainties resulted in the most accurate\nages for NGC 6819.\nThe uncertainties on the measured rotation periods, provided in\n\\citet{meibom2015} and shown in figure \\ref{fig:NGC6819}, were likely\nunderestimated for some stars.\nUnderestimated rotation period uncertainties can result in inaccurate age\nestimates.\nThis raises the question: how should uncertainties on rotation periods be\nestimated?\nThe likelihood is weighted by the inverse variance, so uncertainties on the\nrotation period control the relative information provided by gyrochronology,\nisochrones, and the prior.\nIf rotation period uncertainties are either too large or too small, the\nresulting age estimate will be imprecise and/or inaccurate.\nIt is difficult to measure uncertainties on rotation periods directly:\nstandard techniques such as Lomb-Scargle periodograms and autocorrelation\nfunctions do not provide them.\n% In general, formal rotation period uncertainties should probably be calculated\n% empirically via simulations as in the \\citet{aigrain2015} study.\n\\racomment{Ideally, rotation period uncertainties should capture {\\it both} the\nmeasurement precision, {\\it and} the physical uncertainty introduced by the\nlatitudinal movement of star spots on the surface of a differentially rotating\nstar.\nFor example, \\citet{donahue1996} demonstrated that the seasonal variation in\nmeasurements of G and K star rotation periods is a function of period, $\\Delta\n\\mathrm{P_{rot}} \\propto \\mathrm{P_{rot}}^{1.3\\pm0.1}$.\nThis variation is presumably caused by a latitudinal drift in the dominant\nactive regions, which traces the stellar cycle over several years, in\ncombination with latitudinal differential rotation.\nThis suggests that latitudinal spot drifting does not significantly\naffect stellar rotation periods when stars are young, for example the scatter\nof rotation periods about the mean gyrochronology model in Praesepe is only\naround 5\\%.\nHowever it is likely that this effect will become more important at older\nages.\n% The impact of this phenomenon on gyrochronology remain to be seen, and an\n% in-depth study is warranted.\nA thorough exploration of how rotation period uncertainties, from both\nmeasurement uncertainty and physical variation, affect stellar ages via\ngyrochronology is key to understanding the power of gyrochronology as an\nage-dating method.\nFor now, we leave this exploration for a future study.}\n\n\\subsection{Test 3: Kepler asteroseismic stars}\n\\racomment{In order to test our method in the regime where both isochrone\nfitting and gyrochronology become important, we recovered the ages of the 21\nasteroseismic stars analysed in \\citet{vansaders2016}.\nThese 21 stars were observed in \\kepler's short cadence mode and are a mixture\nof dwarfs and subgiants.\nTheir asteroseismic ages were calculated from the analysis of the frequencies\nof individual oscillation modes \\citep{mathur2012, metcalfe2014,\nsilvaaguirre2015, ceillier2016}, and their rotation periods from their\n\\kepler\\ light curves \\citep{garcia2014}.\nWe crossmatched these stars with the \\Gaia\\ catalog to obtain parallaxes and\napparant magnitudes in the \\Gaia\\ G, $\\mathrm{G_{BP}}$ and $\\mathrm{G_{RP}}$\nband passes.\nWe also added J, H and K 2MASS magnitudes from the Kepler input catalog\n\\citep{brown2011},\n% asteroseismic parameters $\\Delta_\\nu$ and $\\nu_{\\mathrm{max}}$ from\n% \\citet{ceillier2017} and \\citet{serenelli2017},\nand used spectroscopic effective temperatures, spectroscopic metallicities and\nrotation periods reported in table 1 of \\citet{vansaders2016}.\nThe \\gaia\\ photometry is extremely precise, and we found that artificially\ninflating the uncertainties on \\gaia\\ apparent magnitudes by a factor of 10\nsubstantially improved the quality of fit, both in terms of MCMC convergence\nand agreement with asteroseismic age measurements.\nIn figure \\ref{fig:astero} we show the ages of these 21 stars inferred using\nisochrone fitting and gyrochronology, against their asteroseismic ages,\ncalculated using the Asteroseismic Modeling Portal (AMP) \\citep{metcalfe2009,\nmetcalfe2012, metcalfe2014}.\nIn this figure, the colored symbols show ages inferred using a combination of\ngyrochronology and isochrone fitting, implemented with the \\sd\\ {\\it Python}\npackage.\nThe black and grey triangles show ages inferred from gyrochronology only,\nwhere the mass and color of the stars were not inferred, but fixed to be the\nasteroseismic mass and the \\gaia\\ \\gcolor\\ color.\nThese gyrochronal ages were calculated with age as the only free parameter,\nwithout marginalizing over stellar mass, \\gcolor\\ color or extinction.\nAs a result, these gyrochronal ages are {\\it not the same} as the ages given\nby the maximum of the gyrochronal likelihood function used in the combined\nage model, they simply represent an approximation to the gyrochronal age.\nGrey triangles are shown for stars where gyrochronology is not applicable\nbecause the stars are either too metal poor, too metal rich, or too evolved.\nBlack triangles are shown for stars where gyrochronology {\\it is} applicable.\nThe white circles show ages inferred from isochrones only.\nDashed lines connect the three different age measurements for the same\nstars.\n}\n\n\\racomment{\nAlthough the ages of all 21 stars shown in figure \\ref{fig:astero} were\ninferred with a joint isochronal and gyrochronal model, most (all but 8) were\neither too evolved, too metal poor, or too metal rich for gyrochronology to\ncontribute any information to the ages.\nThese metal poor/rich or evolved stars lie in a regime where the variance on\ntheir rotation period was artificially inflated because the gyrochronology\nrelations are not thoroughly understood or well calibrated.\nThe rotation periods of the remaining eight stars {\\it did} contribute to\ntheir inferred ages to some degree, however isochrones still dominated the age\ninformation for some of them because these stars are relatively old and/or\nrelatively hot.\n% Isochrones are information-rich for these hot and old stars, and this is\n% exactly where gyrochronology is at its most information-poor.\n}\n\n\\racomment{\nIn general, there is relatively poor agreement between the asteroseismic ages\nand the ages inferred using \\sd.\nMuch of this discrepancy is driven by differences in the isochronal ages,\nwhich is likely attributable to differences between the MIST stellar evolution\nmodels and those used in the AMP analysis: a combination of the Aarhus stellar\nevolution code \\citep[ASTEC][]{christensen-dalsgaard2008a} and the adiabatic\npulsation code \\citep[ADIPLS][]{christensen-dalsgaard2008b}.\nWe compared non-rotating, Solar-metallicity MIST isochrones for middle-aged\nstars with Solar-metallicity BaSTI isochrones \\citep{pietrinferni2004,\nhidalgo2018} and found that, for stars between 4 and 8 Gyrs, in the same\neffective temperature range as the asteroseismic stars, the age discrepancy\nbetween the two sets of models can be as large as 1-2 billion years.\nThe MIST isochrones lie above the BaSTI isochrones on the HR diagram, leading\nto a systematic underprediction of ages.\n}\n\n\\racomment{\nThe gyrochronal ages, where gyrochronology is applicable, do not show\nexcellent agreement with the asteroseismic ages either.\nThe four hot stars to the left in figure \\ref{fig:astero} are rotating more\nslowly than predicted by the Praesepe-based gyrochronology models, and as a\nresult their gyro-ages are older than their asteroseismic ones.\nFor these hot stars, two out of four have ages that are still consistent, or\nclose to consistent, with their asteroseismic ages.\nThe third star from the left is an anomalously slow rotator for its age and\nmass and \\citet{vansaders2016} also found this star to be surprisingly slowly\nrotating.\nIn contrast, the star with a black triangle symbol (indicating that\ngyrochronology is applicable) furthest to the right is {\\it rapidly} rotating\nfor its mass and age, even when weakened braking is taken into account.\n% Both stars are around Solar mass (1.06 \\pm 0.02 and 1.0 \\pm 0.03 M$_\\odot$\n% from left to right) and have asteroseismic ages older than the Sun (6.82 \\pm\n% 0.28 and 7.28 \\pm 0.51 Gyr from left to right), and yet both rotate more\n% rapidly than the Sun (23.2 \\pm 7.4).\nThis star is around Solar mass (1.0 $\\pm$ 0.03 M$_\\odot$) and has an\nasteroseismic age older than the Sun (7.28 $\\pm$ 0.51 Gyr), yet rotates with a\nperiod of only 19.8 $\\pm$ 1.3 days.\nThis is KIC 9098294, a single-lined spectroscopic binary with an orbital\nperiod of around 20 days (Latham, private communication).\nIt is the only clear SB1 in the \\citet{vansaders2016} sample, although some\nothers do have binary companions with long orbital periods, for which tidal\ninteractions are not expected to be strong.\n}\n\n\\begin{figure}\n \\caption{ A comparison of stellar ages inferred using\n asteroseismic modeling with ages inferred using a combination of isochrone\n fitting and gyrochronology.\nColored circles show ages inferred using isochrone fitting and gyrochronology\ncombined via the \\sd\\ software package.\nBlack triangles show the ages of all stars inferred via gyrochronology only\n and white circles show ages inferred via isochrone-fitting only.\n% This plot shows that, for the majority of stars in this sample, isochrone\n% fitting dominates the age information and gyrochronology only contributes\n% significantly for two or three stars.\n% This is a consequence of many of these stars being hot old, extremely metal\n% rich, extremely metal poor, or evolved.\n }\n \\centering\n \\includegraphics[width=1.2\\textwidth]{asteroseismic_results_nosdss_gaia}\n\\label{fig:astero}\n\\end{figure}\n", "meta": {"hexsha": "ed449b1c0392845dda109208e47e5adc058cfc03", "size": 31084, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/results.tex", "max_stars_repo_name": "john-livingston/stardate", "max_stars_repo_head_hexsha": "5c0d45c1e2eb9ec5b6c57aeacbcb301304065bbc", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-02-19T13:46:46.000Z", "max_stars_repo_stars_event_max_datetime": "2020-03-31T23:46:36.000Z", "max_issues_repo_path": "paper/results.tex", "max_issues_repo_name": "john-livingston/stardate", "max_issues_repo_head_hexsha": "5c0d45c1e2eb9ec5b6c57aeacbcb301304065bbc", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2019-02-21T21:37:05.000Z", "max_issues_repo_issues_event_max_datetime": "2019-09-09T10:38:15.000Z", "max_forks_repo_path": "paper/results.tex", "max_forks_repo_name": "john-livingston/stardate", "max_forks_repo_head_hexsha": "5c0d45c1e2eb9ec5b6c57aeacbcb301304065bbc", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2019-02-11T02:46:32.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T22:16:53.000Z", "avg_line_length": 55.1134751773, "max_line_length": 93, "alphanum_fraction": 0.7804658345, "num_tokens": 7811, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6959583124210896, "lm_q2_score": 0.5, "lm_q1q2_score": 0.3479791562105448}} {"text": "%!TEX root = ../PhD_thesis__Lilian_Besson\n\n% ----------------------------------------------------------------------\n\\chapter{Improving Spectrum Usage of IoT Networks with Selfish MAB Learning}\n\\label{chapter:4}\n\n\\graphicspath{{2-Chapters/4-Chapter/Images/}}\n\n\n\\abstractStartChapter{}%\n%\nAfter detailing in Part~\\ref{part:Introduction} the MAB model, we now come back to our main question of interest.\nIn this chapter, we focus on wireless networks following the hypotheses common to present and future Internet of Things (IoT) networks exposed in Chapter~\\ref{chapter:1}.\nOur goal is to show that the IoT devices can automatically learn to increase their battery life and their successful transmission rates, without changing anything on the IoT standard side.\n%\nWe propose two models of IoT networks, composed of many independent IoT end-devices, that can use low-cost Reinforcement Learning (RL) algorithms in order to learn how to improve their spectrum access.\n%\nDecentralized RL for IoT lets the devices use acknowledgements sent back by their Base Station as a reward, instead of sensing feedback like for OSA.\nWe consider many independent ``dynamic'' devices, each communicating with a small probability at every instant.\nSimulations show that dynamic devices can greatly improve their spectrum efficiency,\nby using MAB algorithms like \\UCB.\n% in the case of a non-uniform distribution of the static devices in the channels\nWe also developed a proof-of-concept using USRP platforms, for a real-world validation of this approach.\n% Numerical simulations justify the interest of the approach,\n% of the advantage of using MAB algorithms for the first model.\n%\n% On a simple example, we present in Section~\\ref{sec:4:gnuradio} a demonstration that implements and validates the proposed approach, using real wireless radio hardware.\n% The first model\n% % without retransmission of packets,\n% is interesting for its simplicity,\n% and because considering no retransmission can improve the battery life of IoT devices, at the cost of a lower Quality of Service (QoS).\n%\nIn a second step, we consider a second model where a dynamic device has to (try to) retransmit a message, in case of a failed first transmission, up-to a fixed number of retransmissions.\nWe compare heuristics based on \\UCB, and simulations confirm that the non-naive heuristics also significantly improve the network efficiency.\n% Interestingly, the simplest heuristic is found to be the most efficient one, as its simplicity implies a faster learning and convergence time.\n%\n% The second model is more interesting if the main target is an improvement of the QoS, rather than a longer battery life.\n\n\\minitocStartChapter{}\n\n% ----------------------------------------------------------------------------\n% \\section{Motivations for Selfish MAB Learning for IoT Networks}\n\\section{Introduction and motivations for MAB learning for IoT Networks}\n\\label{sec:4:motivations}\n% ----------------------------------------------------------------------\n\nAfter the first chapters that presented the model of MAB, we go back to the initial problems studied in this thesis, and thus we focus on Internet of Things (IoT) networks.\n%\nAs explained in the introduction in Chapter~\\ref{chapter:1},\nunlicensed bands are more and more used and considered for mobile and LAN (Local Area Network) and for Internet of Things communication standards.\n% (Wi-Fi, LTE-U), (IoT) standards for short-range (ZigBee, Z-Wave, Bluetooth) and long-range (LoRaWAN, SIGFOX, Ingenu, Weightless) communications \\cite{Centenaro16}.\nThis heavy use of unlicensed bands, in particular with the expected exponential growth of the number of IoT devices, will cause performance drop, due to radio collisions that could even compromise IoT promises.\n\nEfficient Medium Access Control (MAC) policies allow devices to avoid interfering traffic and can significantly reduce the spectrum contention problem in unlicensed bands.\nAs end-devices battery life is a key constraint of IoT networks,\nand as IoT networks are decentralized, because the devices initiate transmissions,\nthis leads to IoT protocols using as low signaling overhead as possible and simple ALOHA-based mechanisms.\n%\nIn this chapter, we analyze the performance of Multi-Armed Bandits (MAB) algorithms, that could be used in combination with a time-frequency slotted ALOHA-based protocol.\nWe highlight that even without changing anything on the level of IoT standards, our proposal is just an add-on capability that can be used on a unit-per-unit basis.\nWe consider the \\UCB \\cite{Auer02}, and the Thompson-Sampling (TS) algorithms \\cite{Thompson33,AgrawalGoyal11,\nKaufmann12Thompson}, for the first model. For the demonstration as well as for the second model, without loss of generality, we preferred to focus on heuristics based on the simplest algorithm (\\ie, \\UCB), to give a clear presentation of the different ideas explored to solve the problem of learning in order to retransmit efficiently.\n\n% As detailed in Chapter~\\ref{chapter:1},\n% MAB learning has already been proposed in Cognitive Radio (CR) \\cite{Mitola99,Haykin05}, and in particular, for sensing-based Dynamic Spectrum Access (DSA) in licensed bands \\cite{Jouini10}.\n% For example,\n% proof-of-concepts like \\cite{kumar2016two} have proven the capability of such approaches on real radio signals for OSA,\n% and \\cite{Maghsudi16} shows how MAB learning can be applied for small cell management in licensed 5G networks.\n% Some analysis on real radio measurements made for HF ionospheric channels have also proven that solutions based on MAB learning is appropriate and solves efficiently this kind of decision-making problems on real-world wireless signals \\cite{Melian15}.\n% Recent works show that stationary MAB algorithms work well to solve reinforcement learning models that represent accurately real-world radio problems.\n% Recently, TS and \\UCB{} algorithms have been used for improving the spectrum access in (unlicensed) Wi-Fi networks, for instance by \\cite{Toldov16} or \\cite{Wilhelmi19collaborative,Wilhelmi19potential}.\n% % However, even with only one dynamic user using the learning algorithm, the background traffic or the traffic of the other devices is never really stationary or \\iid{}.\n\nWe present in Section~\\ref{sec:4:firstModel} how the MAB algorithms can be used in a unlicensed but frequency- and time-slotted IoT network.\nSeveral devices are using bandit algorithms, and the assumptions made by the stochastic bandit algorithms are not satisfied: as several agents learn simultaneously and their activation processes are random, their behavior is not stationary.\nAs far as we know, we provide the first practical study to confirm robustness of the use of stochastic bandit algorithms for decision making in IoT networks with a large number of intelligent devices in the network, which makes the environment ``strongly not stationary''.\nThis specific context makes it very hard to give mathematical proofs of convergence and of efficiency of bandit algorithms (that is why we relax the hypothesis and only consider up-to $M \\leq K$ players in Chapter~\\ref{chapter:5}).\nWe then validate the model with a hardware implementation on real radio signals, detailed in Section~\\ref{sec:4:gnuradio}.\n%\nWe conclude this chapter by presenting in Section~\\ref{sec:4:retransmissions} an extension of this model to take into account another aspect of the ALOHA protocol, that is the possibility for dynamic devices to retransmit their packets if the \\emph{Ack} was not received.\n\n\n\\paragraph{Publications.}\n%\nThis chapter is mainly based on our articles \\cite{Bonnefoi17,Besson2018ICT,Besson2019WCNC,Bonnefoi2019WCNC,MoyBesson2019,MoyBesson2019Annales}.\n\n\n% ----------------------------------------------------------------------------\n\\section[Selfish learning for many dynamic devices in an IoT network]{Selfish learning for many dynamic devices with low activation probabilities in an IoT network}\n\\label{sec:4:firstModel}\n% ----------------------------------------------------------------------\n\n% - ``Multi-Armed Bandit Learning in IoT Networks and non-stationary settings'', see https://hal.inria.fr/hal-01575419\n\n\\input{2-Chapters/4-Chapter/CrownCom_17/IoT_slotted.tex}\n\n\n\\newpage % WARNING ?\n% ----------------------------------------------------------------------------\n% \\section{Test-bed implementation of the first model for real-world validation}\n\\section{Proof-of-concept of our model for real-world validation}\n\\label{sec:4:gnuradio}\n% ----------------------------------------------------------------------\n\n% - ``MALIN: Multi-Arm bandit Learning for Iot Networks with GRC: A TestBed Implementation and Demonstration that Learning Helps'', demo at ICT, and the companion paper ``GNU Radio Implementation of MALIN: \"Multi-Armed bandits Learning for Internet-of-things Network\"'', see https://hal.inria.fr/hal-02006825\n\n\\input{2-Chapters/4-Chapter/IEEE_WCNC_2019__DemoICT/IEEE_WCNC_2019__DemoICT.tex}\n\n\n\n\\newpage % WARNING ?\n% ----------------------------------------------------------------------------\n\\section{Extending the model to account for retransmissions}\n\\label{sec:4:retransmissions}\n% ----------------------------------------------------------------------\n\n% - ``Upper-Confidence Bound for Channel Selection in LPWA Networks with Retransmissions'', see \\texttt{https://perso.crans.org/besson/articles/BMBM\\_\\_IEEE\\_WCNC\\_2019.pdf}\n\n\\input{2-Chapters/4-Chapter/IEEE_WCNC__2019__Paper__BMBM/IEEE_WCNC__2019__Paper__BMBM.tex}\n\n\n\\newpage % WARNING ?\n% ----------------------------------------------------------------------------\n\\section{Conclusion -- Towards theoretical guarantees}\n% \\section{Conclusion}\n\\label{sec:4:conclusion}\n% ----------------------------------------------------------------------\n\n% % We give in this section conclusions about the two models presented above, one for long-life IoT devices (without retransmission) and for improved QoS devices (\\ie, with retransmissions), and about our demonstration, as well as directions of future works, some of which are currently being studied by researchers of our team.\n\n% % \\subsection*{Conclusions}\n\n% % ----------------------------------------------------------------------\n% % \\paragraph{Summary of the first model.}\n% \\label{sub:41:conclusion}\n\n% In Section~\\ref{sec:4:firstModel}, we proposed an evaluation of the performance of MAB learning algorithms in IoT networks,\n% with a focus on the convergence of algorithms, in terms of successful transmission rates, when the proportion of intelligent dynamic devices changes.\n% Concretely, increasing this probability allows to insert more devices in the same network, while maintaining a good Quality of Service.\n% Similarly, if the number of devices remain constant, increasing the successful transmission rate directly extends the IoT devices battery life, as they suffer less from failed transmissions.\n% We show that \\UCB{} and TS have near-optimal performance, even when their underlying \\iid{} assumption (see Chapter~\\ref{chapter:2}) is violated by the presence of many ``intelligent'' end-devices which follow a random activation process.\n% %\n% This is both surprising and encouraging, it shows that applying bandit algorithms tailored for a stochastic model is still useful in broader settings.\n% The fully \\emph{decentralized} application of classic stochastic MAB algorithms are almost as efficient as the best possible centralized policy in this setting, after a short learning period, even though the dynamic devices \\emph{cannot} communicate with each other, and \\emph{do not} know the system parameters.\n\n\n% % ----------------------------------------------------------------------\n% % \\paragraph{Conclusions taken from the demonstration.}\n% \\label{sub:42:conclusionFromDemonstration}\n\n% We presented in Section~\\ref{sec:4:gnuradio} a demonstration, showed in $2018$ at the ICT conference \\cite{Besson2018ICT}, and further detailed in the companion paper \\cite{Besson2019WCNC}.\n% We gave all the necessary details on both the PHY and the MAC layer, as well as details on the User Interface developed for the demo.\n% Results obtained in practice were discussed, to highlight the interest of using learning algorithms for radio online optimization problem, and especially multi-armed bandit learning algorithms.\n% %\n% By using such low-cost algorithms, we demonstrated empirically that a dynamically re-configurable device can learn on its own to favor a certain channel, if the environment traffic is not uniform between the $K$ different channels, by using the acknowledgement (\\Ack) feedback sent from the base station.\n\n\n% % ----------------------------------------------------------------------\n% % \\paragraph{Conclusion about this second model.}\n% \\label{sub:43:conclusion}\n\n% In Section~\\ref{sec:4:retransmissions}, we presented an extension of our model of LPWA networks based on an ALOHA protocol, slotted both in time and frequency.\n% % , in which dynamic IoT devices can again use machine learning algorithms, to improve their Packet Loss Ratio (PLR) when accessing the network.\n% If the priority is the Quality of Service, for instance with renewable energy capabilities, this second model is more appropriate.\n% The main novelty of this model is to address the packet retransmissions upon radio collision, by using a Multi-Armed Bandit framework.\n% We presented and evaluated several heuristics that try to learn how to transmit and retransmit in a smarter way, by using the \\UCB{} algorithm for channel selection for first transmission, and different proposals based on \\UCB{} for the retransmissions upon collisions.\n% %\n% We showed that incorporating learning for the transmission is needed to achieve optimal performance, with significant gain in terms of successful transmission rate in networks with a large number of devices (up-to $30\\%$ in the example network).\n% Our simulations show that each of our proposed heuristic outperforms a naive random access scheme.\n% Surprisingly, the main take-away message is that a simple \\UCB{} learning approach, that retransmit in the same channel, turns out to perform as well as more complicated heuristics.\n\n\n% % ----------------------------------------------------------------------\n% \\subsection*{Future works}\n% \\label{sub:4:futureWorks}\n\n% % Future works related to this chapter include the following directions.\n\n\n% % \\paragraph{Possible extensions of the first model.}\n\n% The first model presented in Section~\\ref{sec:4:firstModel} could easily be generalized with two probabilities $p_S$ and $p_D$, if static and dynamic devices have different transmission pattern, and less easily with one probability per device. Also, other emission pattern could be considered, instead of a Bernoulli process for each user.\n% In this whole Chapter~\\ref{chapter:4}, we prefer to consider that all devices have the same activation probability, to keep the notation as simple as possible.\n% %\n% Moreover, for sake of simplicity we supposed that all devices use the same standard.\n% Future works could consider more realistic interference scenarios and IoT networks, with, \\eg, non-slotted time, more than one base station etc.\n\n% Another extension could be to not have a Bernoulli process (or any random process), but a fixed rate of transmission, \\eg, one transmission a day.\n% So additionally to deciding the channel for communication (\\ie, \\emph{where} to communicate), each device has to also decide \\emph{when} to communicate.\n% % This is another direction of research, that we will investigate in the future.\n% However, this clearly leads to a much larger action space, as there are many time slots in one day (for example), and thus we believe that as soon as the action space becomes too large in this extension, the simple MAB-based learning approach could be no longer appropriate.\n% It is well-known in the MAB literature that the larger the action space, the slower is the convergence speed of any stationary MAB algorithms.\n% It could be interesting to study the possible application of \\emph{contextual} MAB \\cite{Li10,Luo18} or structured MAB \\cite{Combes17} models and algorithms for this extension.\n\n% % We will investigate this behavior in order to understand it better theoretically.\n% % We will also experiment more with adversarial algorithms, to confirm that they work less efficiently than stochastic bandit algorithms in our non-stochastic setting.\n\n% % \\paragraph{Extensions of the demonstration.}\n\n% Possible future extensions of our demonstration include the following points.\n% We could consider more dynamic devices (\\eg, $100$) but it would either cost more in terms of equipment, or in terms of software engineering to simulation more devices with the same card.\n% We could also implement a real-world IoT communication protocol (like the LoRaWAN standard), which we prefer not to do as it would cost a significant effort of development.\n% Finally, we could also study the interference in case of other gateways located nearby, and this could be done without needing a lot of new hardware (using one extra USRP card to simulate another gateway).\n% %\n% % Pub pour notre autre article\n% % We are also interested in studying the possible gain of using a learning step when the transmission model follows ALOHA-like retransmissions, and this is presented in the next Section~\\ref{sec:4:retransmissions}.\n\n\n% % \\paragraph{Possible extensions for the second model.}\n\n% Finally, the utility and impact of the proposed approaches for LPWA networks motivates us to address several subjects as future works. Among them, the non-stationarity of the channel occupancy caused by the learning policy employed by the IoT devices.\n% %\n% For that end, modifications of MAB algorithms have been proposed, such as Sliding-Window-\\UCB{} or Discounted-\\UCB{} \\cite{Garivier11UCBDiscount}\n% or more recently M-\\UCB{} \\cite{CaoZhenKvetonXie18},\n% or more recently GLR-\\UCB{} \\cite{Besson2019GLRT} which is presented in Chapter~\\ref{chapter:6},\n% that nevertheless have not been explored for the targeted problem.\n% Chapter~\\ref{chapter:6} is focusing on this direction, but we did not have enough time to explore the possible applications of MAB policies designed for non-stationary problems to the model with retransmissions presented in this Chapter.\n\n% In order to validate our results in a realistic experimental setting and not only with simulations, future works include a hardware implementation of the analyzed models to complete our demonstration \\cite{Besson2019WCNC}.\n% Note that Julio César Mango-Vasquez is currently working on this direction, for the EPHYL project, in collaboration with Carlos Faouzi Bader at IETR and CentraleSupélec campus of Rennes.\n% % and Christophe Moy at IETR and University Rennes 1.\n% %\n% A hardware demonstrator could be also benefit to study other settings by removing some hypotheses, for instance by studying a similar model in non-slotted time.\n\n\n% \\subsection*{Summary of this chapter}\n\nWe focused in this chapter on models of IoT networks, and we proposed to use classical stationary multi-armed bandit learning algorithms implemented in a selfish and decentralized manner by each of the dynamic devices in the IoT network.\nWe presented two models of wireless IoT networks, without relying on the feedback provided by spectrum sensing, and inspired by the ALOHA protocol. We proposed two versions, with or without retransmissions of up-link packages in case of collisions.\n% In both cases, we try to model the existing standards, like the LoRa standard, and we demonstrated the efficiency of the proposed MAB-based approach in both numerical simulations and empirical measurements on real wireless radio signals.\n%\nWe conclude that this learning-based approach is efficient, as it allows the IoT devices to automatically and independently increase their successful transmission rates.\n\nIt is also quite surprising that stochastic MAB algorithms can be of any use in such non-stationary applications.\nUnfortunately, it turned out to be of extreme difficulty to analyze analytically the considered model with thousands of independent devices, all communicating and learning in their own (random) time scales.\nThat is why we focus on two different simplifications of this model in the next chapters, for which we are able to provide a rigorous theoretical analysis.\n% %\n% We can also highlight that the two models we studied are complementary:\n% the first model (without retransmission of packets), is interesting for its simplicity, and focused on improving the battery life of IoT devices,\n% % , at the cost of a lower Quality of Service (QoS),\n% %\n% while the second model is more interesting if the main target is an improvement of the Quality of Service (QoS).\n% % , rather than a longer battery life.\n\n\n\\paragraph{Multi-Players MAB.}\n%\nOn the one hand, we are actually able to analyze a simpler model, if we assume to have at most $M \\leq K$ devices with a transmission probability of $p=1$.\nInstead of the experiment-driven direction pursued in this chapter, another possibility is to consider a \\emph{multi-players MAB} model to describe our problem.\n%\nThe main difference between the two models is the fact that in Chapter~\\ref{chapter:4}, $M \\gg K$ devices transmit their messages at every time step, by following a random activation process (with a fixed transmission probability $p < 1$).\nIf static and dynamic devices that have to transmit at a fixed time are denoted \\emph{active devices},\nthen their random activation patter makes the number of active devices an (unpredictable) random variable.\nAnalyzing multi-players MAB models under this hypothesis is much harder, and is left as a future work.\n%\nWe study the case of $M \\leq K$ devices learning independently to play a $K$-armed bandit in the next Chapter~\\ref{chapter:5}.\n\n% In that case, the static and dynamic devices effect is decoupled, and arms only model the availability of the channels in the absence of dynamic devices: they are \\iid{} with mean $\\mu_i = 1 - p S_i$.\n% Moreover, dynamic devices are usually assumed to be able to \\emph{sense} a channel before sending \\cite{Zhao10}, and so communicate only if no static device is detected on the channel.\n% The smart devices try to learn the arms with highest means, while coordinating to choose different arms, \\ie, avoid collisions in their choice, in a decentralized manner.\n% However, in this model it is assumed that the multiple agents can know that they experienced a collision with another agent, which is non-realistic for our problem at stake, as our model of smart device cannot do sensing nor differentiate collisions between smart and non-smart devices.\n\n\n\\paragraph{Non-stationary MAB.}\n%\nOn the other hand, while it is hard to analyze the models of this chapter because of the unpredictable behaviors of the IoT devices' activation patterns and the evolving number of active devices, we are also able to analyze a simpler model, if we focus on a single player accessing a network which is assumed to be \\emph{piece-wise stationary},\nthat are bandit problems which are stationary on ``long enough'' intervals, of unknown locations and lengths.\n%\nWe then study this second direction in Chapter~\\ref{chapter:6}.\n\n\n\\newpage\n% ----------------------------------------------------------------------------\n\\section{Appendix}\n\\label{sec:4:appendix}\n\n\\subsection{Proof of Proposition~\\ref{prop:41:Lagrangian}}\n\\label{sec:4:proofLagrangian}\n% ----------------------------------------------------------------------------\n\n% We include here a missing proof from Section~\\ref{sec:4:firstModel}.\n\n\\input{2-Chapters/4-Chapter/CrownCom_17/IoT_slotted_appendix.tex}\n\n\n\\newpage\n\\subsection{Illustration of the GNU Radio Companion Flowcharts}\n\\label{sec:4:IllustrationFlowcharts}\n% ----------------------------------------------------------------------------\n\n\\input{2-Chapters/4-Chapter/IEEE_WCNC_2019__DemoICT/IEEE_WCNC_2019__DemoICT_appendix.tex}\n", "meta": {"hexsha": "9ca8ff915f511f66072090065ae7222e9b8f6535", "size": 23996, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "2-Chapters/4-Chapter/chapter4.tex", "max_stars_repo_name": "Naereen/phd-thesis", "max_stars_repo_head_hexsha": "0fa93ca0d738771f4215bc4aeb66157f2026ba00", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-11-18T12:22:18.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-09T19:29:48.000Z", "max_issues_repo_path": "2-Chapters/4-Chapter/chapter4.tex", "max_issues_repo_name": "Naereen/phd-thesis", "max_issues_repo_head_hexsha": "0fa93ca0d738771f4215bc4aeb66157f2026ba00", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2019-11-18T09:19:15.000Z", "max_issues_repo_issues_event_max_datetime": "2020-07-08T14:13:08.000Z", "max_forks_repo_path": "2-Chapters/4-Chapter/chapter4.tex", "max_forks_repo_name": "Naereen/phd-thesis", "max_forks_repo_head_hexsha": "0fa93ca0d738771f4215bc4aeb66157f2026ba00", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2020-05-28T20:56:13.000Z", "max_forks_repo_forks_event_max_datetime": "2020-09-13T11:11:57.000Z", "avg_line_length": 82.1780821918, "max_line_length": 344, "alphanum_fraction": 0.7519586598, "num_tokens": 5095, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.3478862186402445}} {"text": "\\chapter{Mass Relations and Stoichiometry}\n\\section{Atomic Mass}\nTo determine the weight of an atom, scientists developed a small unit at which\nsuch quantities may be measured. This unit is know as the AMU, or\n\\textit{atomic mass unit}.\n\n12 AMUs are defined as the weight of one atom of the Carbon-12 isotope. That is\nto say that $1amu$ is exactly the weight of 1/12th of one atom of Carbon-12.\n\n\\subsection{Masses of Elements}\nHowever, the numbers on the periodic table are not so round? Why might this be?\nWell, since we are accounting for isotopes, we must apply a weighted average to\nall naturally occuring isotopes. For example:\n\n\\begin{table}[h]\n\\begin{tabular}{lll}\n\\hline\n & Abundance & \\textit{Mass (amu)} \\\\ \\hline\noxygen-16 & 99.762\\% & 15.994 \\\\\noxygen-17 & 0.038\\% & 16.999 \\\\\noxygen-18 & 0.200\\% & 17.999\n\\end{tabular}\n\\end{table}\n\nThe calculation for the relative \\textit{amu} of Oxygen is as follows:\n$(0.99762*15.994) + (0.00038*16.999) + (0.00200*17.999) = 15.9994amu$\n\n\\section{The Mole}\nAvagadro's Number (or $N_A$) is thought of a quantity, usually of atoms. The\nnumber of that quantity is $6.022e23$. Therefore, since this is a quantity, the\nunit is $anything/mole$.\n\n\\subsection{Molar Mass}\n\\subsubsection{Of Elements}\nTo calculate the molar mass of an element: simply look to the periodic table!\nIn the upper-left of the element box, you will find the relative molar mass of\nthe element. That is to say, the \\textit{amu} of an element is the weight in\ngrams of one mole of that element. For example, $1.008mol$ of \\ce{H} weighs\n\\ce{1g}.\n\n\\subsubsection{Of Compounds}\nSimply put, the molar mass of a compound is the sum of all its parts, multiplied\nindividiually by each of the relative quantities.\n\n\\section{Percent Composition}\n\\textbf{Percent composition} of an element is the percent by mass of itself\nwithin a compound. For example, the molar mass of \\ce{Cu2S} is 159.2. The mass\namount of \\ce{Cu}\n\n\\subsection{Empirical Formulae}\nAn \\textit{emperical formula} consists of the sumbols for the elements combined\nin a compound, with subscripts showing the smallest whole-number mole ratio of\nthe different atoms in the compound.\n\nFor example, the emperical formula \\ce{P2O5} might lead to a chemical formula of\n\\ce{P4O10}, or similarily, the emp. formula \\ce{C3H6O} may lead to the formula\n\\ce{C6H18O2}.\n\n\\subsubsection{Calculating}\nTo calculate an emperical formula, you must first start with the percentages of\nan atom or the mass of the atom present. Once these are determined, you may\nthen convert to moles, and simplify the ratio by dividing by the smallest amount\nand rounding. If you cannot round cleanly, multiple the whole ratio by a\nfactor.\n\nThe steps for calculating this emperical formula follow below:\n\n\\begin{enumerate}\n \\item Assume $100g$s was sampled, unless given other measurements.\n \\item Convert the grams to moles, unless given moles.\n \\item Simplify the ratio of moles by dividing by the smallest term.\n \\item Multiply by factors to arrive at cleaner ratios, if not already clean.\n\\end{enumerate}\n\n\\section{Stoichiometry}\nStoichiometry involves the mass relationships between reactants and products in\na chemical reaction. IT allows you to determine the amount of additional\nreactants or products given the chemical equation and just one amount.\n\nStoichiometry relies on the ratios of moles between different compounds in a\nchemical reaction. Consider the following reaction: \\ce{Al2(SO4)3 + 3Ca(OH)2 ->\n2Al(OH)3 + 3CaSO4}. It has the following mass relations:\n\n\\begin{frame}\n \\centering\n {\\huge $\\frac{1mol \\ce{Al2(SO4)3}}{3mol \\ce{Ca(OH)2}}$, $\\frac{1mol\n \\ce{Al2(SO4)3}}{2mol \\ce{Al(OH)3}}$, $\\frac{1mol \\ce{Al2(SO4)3}}{3mol\n \\ce{CaSO4}}$}\n\\end{frame}\n\nTo work with these mass relations, you can convert from mass $\\rightarrow$ molar\nmass $\\rightarrow$ moles $\\rightarrow$ mole-to-mole ratio $\\rightarrow$ moles\n$\\rightarrow$ molar mass $\\rightarrow$ mass.\n\n\\subsection{Limiting Reagent}\nSometimes when doing stoichometry, one reagent appears in a limiting amount such\nthat you cannot fully complete the reaction. These substances are known as\n\\textit{limiting reagents}. The substances that is not completely used up in a\nreaction is referred to as an \\textit{excess reagent (reactant)}.\n\nTo determine the limiting reagent, just do the reaction with both against one of\nthe products and figure out which makes \\textit{less}. The excess reagent will\nbe what is left over. To determine the quantity of the excess reagent, use the\nquantity of the limiting reagent that you originally plugged in and wire that\nthrough.\n\n\\subsubsection{Percent Yield Calculations}\nSometimes, you are given an amount of which is supposed to occur (via\nstoichiometry) and an amount that you actually produced. To calculate this, you\nuse the following:\n\n{\\huge $100 * \\frac{actual}{theoretical}$}\n", "meta": {"hexsha": "5b4595b11ab3246114462eb307600bf65a742fb6", "size": 4893, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "2015/chemistry/sections/stoichiometry.tex", "max_stars_repo_name": "ttaylorr/midterms", "max_stars_repo_head_hexsha": "fdde0fd1a66eb5242d0dfa04a5201c3ab6d6b7eb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2015-01-06T06:55:26.000Z", "max_stars_repo_stars_event_max_datetime": "2015-01-06T06:55:26.000Z", "max_issues_repo_path": "2015/chemistry/sections/stoichiometry.tex", "max_issues_repo_name": "ttaylorr/midterms", "max_issues_repo_head_hexsha": "fdde0fd1a66eb5242d0dfa04a5201c3ab6d6b7eb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2015/chemistry/sections/stoichiometry.tex", "max_forks_repo_name": "ttaylorr/midterms", "max_forks_repo_head_hexsha": "fdde0fd1a66eb5242d0dfa04a5201c3ab6d6b7eb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.3008849558, "max_line_length": 80, "alphanum_fraction": 0.7572041692, "num_tokens": 1369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.34786381905871405}} {"text": "\\documentclass[a4paper]{article}\n%\\usepackage[ngerman]{babel}\n\\usepackage[T1]{fontenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{textcomp}\n\\usepackage{geometry}\n\\geometry{ left=2cm, right=2cm, top=2cm, bottom=3cm, bindingoffset=5mm}\n\\usepackage{graphicx}\n\\usepackage{xcolor}\n\\usepackage{hyperref}\n\\usepackage{longtable}\n\\usepackage{amstext}\n\\usepackage{array}\n\\usepackage{amsmath}\n\\newcolumntype{L}{>{$}l<{$}}\n\\usepackage{tabularx, ragged2e}\n\\usepackage{helvet}\n\\renewcommand{\\familydefault}{\\sfdefault}\n\\usepackage{lastpage}\n\\usepackage{todonotes}\n\\usepackage{titlesec}\n\\titleformat*{\\section}{\\large\\bfseries}\n\\usepackage{listings}\n\\usepackage{color}\n\n\\usepackage{tikz}\n%\\newcommand{\\tikzmark}[2]{\\tikz[overlay, remember picture] \\node[inner sep=0pt, outer sep=0pt, anchor=base] (#1) {#2};}\n\\usetikzlibrary{tikzmark}\n\n\n\\definecolor{mygreen}{rgb}{0.18, 0.545, 0.341}\n\\definecolor{mygray}{rgb}{0.5,0.5,0.5}\n\\definecolor{myblue}{rgb}{0.53,0.61,0.85}\n\n\\lstset{\n keywordstyle=\\color{mygreen},\n commentstyle=\\color{mygray},\n numbers=left,\n numbersep=5pt, \n numberstyle=\\scriptsize\\color{mygray}\n }\n\n\\usepackage{amsmath,amssymb}\n\n\\DeclareRobustCommand{\\bbone}{\\text{\\usefont{U}{bbold}{m}{n}1}}\n\n\\DeclareMathOperator{\\EX}{\\mathbb{E}}% expected value\n\n\\date{}\n\\author{}\n\\usepackage{fancyhdr}\n\\pagestyle{fancy}\n\\fancyhf{}\n\\fancyhead[R]{Felix Burk\\\\ Pascal Huszár}\n\\fancyhead[L]{Reinforcment Learning \\\\ Summer Term 2021 }\n\\fancyfoot[R]{page \\thepage \\text{ }/ \\pageref*{LastPage}}\n%\\fancyfoot[LE]{Seite \\thepage \\text{ }von \\pageref{LastPage}}\n\\renewcommand{\\headrulewidth}{0.5pt}\n\n\\usepackage{amsmath}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\n\n\\title{\\textbf{Exercise 09}}\n\n\\begin{document}\n\t\\maketitle \n\t\\thispagestyle{fancy}\n\t\n \\section*{Task 1 - REINFORCE on the Cart-Pole}\n \\subsection*{a)}\n $\\frac{\\delta}{\\delta \\theta_a}\\frac{se^{s*\\theta_a^T+s*\\theta_b^T)}}{(e^{\\theta_a^T*s}+e^{\\theta_b^T*s})^2}$\n \t\\subsection*{b)}\n \t$\\pi(a|s,\\theta) = \\frac{e^{\\theta_a^T*s}}{e^{\\theta_b^T*s}+e^{\\theta_c^T*s}}$ where $b,c$ are all possible actions and $a \\in \\{b,c\\}$ \\\\\n \t$\\nabla_\\theta log(\\pi_\\theta(s,a)) = \\nabla_\\theta log(\\frac{e^{\\theta_a^T*s}}{e^{\\theta_b^T*s}+e^{\\theta_c^T*s}})$ \\\\\n \t$= \\nabla_\\theta log(e^{\\theta_a^T*s})-\\nabla_\\theta log(e^{\\theta_b^T*s}+e^{\\theta_c^T*s})$ \\\\\n \t$= \\nabla_\\theta \\theta_a^T*s*log(e)-\\nabla_\\theta log(e^{\\theta_b^T*s}+e^{\\theta_c^T*s})$ \\\\\n \t$= s -\\nabla_\\theta log(e^{\\theta_b^T*s}+e^{\\theta_c^T*s})$\n \t$= s -\\frac{\\nabla_\\theta log(e^{\\theta_b^T*s}+e^{\\theta_c^T*s})}{e^{\\theta_b^T*s}+e^{\\theta_c^T*s}}$ \\\\\n \t$= s - \\frac{e^{\\theta_b^T*s}}{e^{\\theta_b^T*s}+e^{\\theta_c^T*s}}$ \n \t$= \\frac{e^{\\theta_c^T*s}*s}{e^{\\theta_b^T*s}+e^{\\theta_c^T*s}}$\n \t$= \\frac{1}{1+e^{\\theta_c^T*s}}*s$ \\\\\n \tFor the case $a=c$, $a=b$ is equivalent.\n\n\n\\end{document}", "meta": {"hexsha": "92bef9ea13d64a8332ae7a60ead0b770b78470f3", "size": 2829, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ex09-pg/ex09.tex", "max_stars_repo_name": "FlxB2/rl-course", "max_stars_repo_head_hexsha": "0db6632fbb2e9f90977b5b3422494279477cb8a8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ex09-pg/ex09.tex", "max_issues_repo_name": "FlxB2/rl-course", "max_issues_repo_head_hexsha": "0db6632fbb2e9f90977b5b3422494279477cb8a8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ex09-pg/ex09.tex", "max_forks_repo_name": "FlxB2/rl-course", "max_forks_repo_head_hexsha": "0db6632fbb2e9f90977b5b3422494279477cb8a8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.5172413793, "max_line_length": 140, "alphanum_fraction": 0.6783315659, "num_tokens": 1153, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.34786381905871405}} {"text": "\\chapter{Course Introduction}\n\\label{cha:intro}\n\n\\sloppy\n%\\lecture{1 --- Wednesday, February 19th}\n%{Spring 2020}{Rasmus Kyng}{Course Introduction}\n\n\\section{Overview}\n This course will take us quite deep into modern approaches to\n graph algorithms using convex optimization techniques.\n%\n By studying convex optimization through the lens of graph algorithms,\n we'll try to develop an understanding of fundamental\n phenomena in optimization.\n %\n Much of our time will be devoted to flow problems on graphs.\n We will not only be studying these problems for their own sake,\n but also because they often provide a useful setting for thinking more broadly about optimization.\n\n The course will cover some traditional discrete approaches to various graph\n problems, especially flow problems, and then contrast these approaches\n with modern, asymptotically faster methods based on combining convex\n optimization with spectral and combinatorial graph theory.\n\n\\section{Electrical Flows and Voltages - a Graph Problem from Middle School?}\n\nWe will dive right into graph problems by considering how electrical\ncurrent moves through a network of resistors.\n\nFirst, let us recall some middle school physics.\nIf some of these things don't make sense two you, don't worry, in less\nthan paragraph from here, we'll be make to safely doing math.\n\nRecall that a typical battery that buy from Migros has two\nendpoints, and produces what is called a \\emph{voltage\n difference} between these endpoints.\n\nOne end of the battery will have a positive charge (I think that means an excess of\npositrons\\footnote{I'm joking, of course! Try Wikipedia if you want to know\nmore. However, you will not need it for this class.}), and the other a\nnegative charge.\nIf we connect the two endpoints with a wire, then a current will flow\nfrom one end of the battery to the other in an attempt to even out\nthis imbalance of charge.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.5\\linewidth]{fig/lecture1_battery9volts.png}\n \\captionof{figure}{A 9 volts battery with a wire attached.}\n \\label{fig:battery-volt}\n\\end{figure}\n\nWe can also imagine a kind of battery that tries to send a certain\namount of current the wires between its endpoints, e.g. 1 unit of charge per\nunit of time.\nThis will be a little more convenient to work with, so let us focus on\nthat case.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.5\\linewidth]{fig/lecture1_battery1ampere.png}\n \\captionof{figure}{A 1 ampere battery with a wire attached.}\n \\label{fig:battery-current}\n\\end{figure}\n\nA \\emph{resistor} is a piece of wire that connects two\npoints $u$ and $v$, and is completely described by a single number $r$\ncalled its \\emph{resistance}.\n\nIf the voltage difference between the endpoints of the resistor is\n$x$, and the resistance is $r$ then this will create a flow of charge per unit of time of $f = x / r$.\nThis is called Ohm's Law.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.5\\linewidth]{fig/lecture1_ohmslawx-vs-f.png}\n \\captionof{figure}{Ohm's Law for a resistor with resistance $r = 1$.}\n \\label{fig:ohmslaw}\n\\end{figure}\n\nSuppose we set up a bunch of wires that route electricity from our\ncurrent source $s$ to our current sink $t$ in some pattern:\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.5\\linewidth]{fig/lecture1_graphpath.png}\n \\captionof{figure}{A path of two resistors.}\n \\label{fig:graphpath}\n\\end{figure}\n\nWe have one unit of charge flowing out of $s$ per unit of time, and\none unit coming into $t$.\nBecause charge is conserved, the current flowing into any other point\n$u$ must equal the amount flowing out of it.\nThis is called Kirchoff's Current Law.\n\nTo send one unit of current from $s$ to $t$, we must be sending it\nfirst form $s$ to $u$ and then from $u$ to $t$.\nSo the current on edge $(s,u)$ is 1 and the current on $(u,t)$ is 1.\nBy Ohm's Law, the voltage difference must also be 1 across each of the two\nwires.\nThus if the voltage is $x$ at $s$, it must be $x+1$ at $u$ and $x+2$\nat $t$. What is $x$? It turns out it doesn't matter: We only care\nabout the differences. So let us set $x = 0$.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.5\\linewidth]{fig/lecture1_graphpath-labelled.png}\n \\captionof{figure}{A path of two resistors.}\n \\label{fig:graphpathlabelled}\n\\end{figure}\n\n\nLet us try one more example:\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.5\\linewidth]{fig/lecture1_graphtriangle.png}\n \\captionof{figure}{A network with three resistors.}\n \\label{fig:graphtriangle}\n\\end{figure}\n\nHow much flow will go directly from $s$ to $t$ and how much via $u$?\n\nWell, we know what the net current flowing into and out of each vertex\nmust be, and we can use to set up some equations.\nLet us say the voltage at $s$ is $x_s$, at $u$ is $x_u$ and at $t$ is $x_t$.\n\\begin{tight_itemize}\n\\item Net current at $s$: $-1 = (x_s-x_t) + (x_s-x_u)$\n\\item Net current at $u$: $\\phantom{-}0 = (x_u-x_s) + (x_u-x_t)$\n\\item Net current at $t$:\\, $\\phantom{-}1 = (x_t-x_s)+(x_t-x_u)$\n\\end{tight_itemize}\nThe following is a solution: $x_s = 0$, $x_u = \\frac{1}{3}$, $x_t\n= \\frac{2}{3}$.\nAnd as before, we can shift all the voltages by some constant $x$ and\nget another solution $a = x+0$, $x_u = x+\\frac{1}{3}$, $x_t\n= x+\\frac{2}{3}$. You might want to convince yourself that these are the only solutions.\n\n\\paragraph{Electrical flows in general graphs.}\nDo we know enough to calculate the electrical flow in some other\nnetwork of resistors?\nTo answer this, let us think about the network\nas a graph.\nConsider a undirected graph $G = (V,E)$ with $\\abs{V} = n$ vertices and\n$\\abs{E} = m$ edges, and let us assume $G$ is connected.\nLet's associate a resistance\n$\\rr(e) > 0$ with every edge $e \\in E$.\n\nTo keep track of the direction of the flow on each edge, it will be\nuseful to assign an arbitrary direction to every edge. So let's do\nthat, but remember that this is just a bookkeeping tool that helps\nus track where flow is going.\n\nA \\emph{flow} in the graph is a vector $\\ff: \\R^E$.\n%\nThe \\emph{net flow} of $\\ff$ at a vertex $u \\in V$ is defined as\n$\\sum_{v \\to u} \\ff(v,u) - \\sum_{u \\to v} \\ff(u,v).$\n% As a second accounting trick, we can define $\\ff(v,u) \\defeq\n% -\\ff(u,v)$,\n% and then write the net flow as $\\sum_{(u,v)\\in E} \\ff(u,v)$.\n\nWe say a flow routes the demands $\\dd \\in \\R^V$ if the net flow at\nevery vertex $v$ is $\\dd(v)$.\n\nWe can assign a voltage to every vertex $\\xx \\in R^V$.\nOhm's Law says that the electrical flow induced by these voltages will\nbe $\\ff(u,v) = \\frac{1}{\\rr(u,v)} (\\xx(u) - \\xx(v))$.\n\nSay we want to route unit of current from vertex $s \\in V$ to vertex\n$t \\in V$.\nAs before, we can write an equation for every vertex saying that the\nvoltage differences must produce the desired net current:\n\\begin{tight_itemize}\n\\item Net current at $s$:\n $\\phantom{----,,,}-1 = \\sum_{(s,v)} \\frac{1}{\\rr(s,v)} (\\xx(s) - \\xx(v))$\n\\item Net current at $u \\in V\\setminus\\setof{s,t}$:\n $\\phantom{..} 0 = \\sum_{(u,v)} \\frac{1}{\\rr(u,v)} (\\xx(u) - \\xx(v))$\n\\item Net current at $t$: $\\phantom{------,} 1=\\sum_{(t,v)} \\frac{1}{\\rr(t,v)} (\\xx(t) - \\xx(v))$\n\\end{tight_itemize}\nThis gives us $n$ constraints, exactly as many as we have voltage\nvariables.\nHowever we have to be a little careful when trying to conclude that a\nsolution exists, yielding voltages $\\xx$ that gives induce an\nelectrical flow routing the desired demand.\n\n\nYou will prove in the exercises (Week 1, Exercise 3) that a solution $\\xx$ exists. The proof requires two\nimportant observations: Firstly that the graph is connected, and\nsecondly that summed over all vertices, the net demand is zero,\ni.e. as much flow is coming into the network as is leaving it.\n\n\\paragraph{The incidence matrix and the Laplacian matrix.}\nTo have a more compact notation for net flow constraints, we also\nintroduce the \\emph{edge-vertex incidence matrix} of the graph,\n$\\BB \\in \\R^{V \\times E}$.\n\\[\n \\BB(v,e) =\n \\begin{cases}\n 1 & \\text{ if } e = (u,v) \\\\\n -1 &\\text{ if } e = (v,u) \\\\\n 0 &\\text{ o.w.}\n \\end{cases}\n\\]\n%\nNow we can express the net flow constraint that $\\ff$ routes $\\dd$ by\n\\[\n \\BB \\ff = \\dd\n .\n\\]\nThis is also called a conservation constraint. In our examples so far,\nwe have $\\dd(s) = -1$, $\\dd(t) = 1$ and $\\dd(u) = 0$ for all $u\\in V\\setminus\\setof{s,t}$.\n\nIf we let $\\RR = \\diag_{e \\in E} \\rr(e)$\nthen Ohm's law tells us that $\\ff = \\RR^{-1}\\BB^{\\trp} \\xx$.\nPutting these observations together, we have $\\BB \\RR^{-1}\\BB^{\\trp}\n\\xx= \\dd$.\nThe voltages $\\xx$ that induce $\\ff$ must solve this system of linear\nequations, and we can use that to compute both $\\xx$ and $\\ff$.\nIt is exactly the same linear\nequation as the one we considered earlier.\nWe can show a that for a connected graph, a solution $\\xx$\nexists if and only if the\nflow into the graph equals the net flow out, which we can express as\n$\\sum_{v} \\dd(v) = 0$ or $\\vecone^\\trp \\dd = 0$.\nYou will show this as part of Exercise 3.\nThis also implies that an electrical flow routing\n$\\dd$ exists if and only if the net\nflow into the graph equals the net flow out, which we can express as\n$\\vecone^\\trp \\dd = 0$.\n\n\nThe matrix $\\BB \\RR^{-1}\\BB^{\\trp}$ is called the \\emph{Laplacian} of\nthe graph and is usually denoted by $\\LL$.\n\n\\paragraph{An optimization problem in disguise.}\nSo far, we have looked at electrical voltages and flows as arising\nfrom a set of linear equations -- and it might not be apparent that\nthis has anything to do with optimization.\nBut transporting current through a resistor requires energy, which will be\ndissipated as heat by the resistor (i.e. it will get hot!).\nIf we send a current of $f$ across a resistor with a potential drop of\n$x$, then the amount of energy spent per unit of time by the resistor will be\n$f \\cdot x$. This is called Joule's Law.\nApplying Ohm's law to a resistor with resistance $r$,\nwe can also express this energy per unit of time\nas $f \\cdot x = x^2 / r = r \\cdot f ^2$.\nSince we aren't bothering with units, we will even forget about time,\nand refer to these quantities as ``energy'', even though a physicist\nwould call them ``power''.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{fig/lecture1_flow-vs-energy.png}\n \\captionof{figure}{Energy has a function of flow in a resistor with\n resistance $r = 1$.}\n \\label{fig:flow-vs-energy}\n\\end{figure}\n\nNow, another interesting question would seem to be: If we want to find\na flow routing a certain demand $\\dd$, how should have flow behave in\norder to minimize the the electrical energy spent routing the flow?\nWe can phrase this as an optimization problem:\n\\begin{align*}\n \\min_{\\ff \\in \\R^E} & \\sum_e \\rr(e) \\ff(e)^2\n \\\\\n \\textrm{s.t. } & \\BB \\ff = \\dd .\n\\end{align*}\nWe call this problem \\emph{electrical energy-minimizing flow}.\nIt turns out, that the flow $\\ff^*$ that minimizes the electrical\nenergy among all flows that satisfy $\\BB \\ff = \\dd$ is precisely the\nelectrical flow.\n\n\\paragraph{A pair of problems.}\nWhat about our voltages, can we also get them from some optimization\nproblem?\n%\nWell, we can work backwards from the fact that our voltages solve the\nequation $\\LL \\xx = \\dd$.\n%\nConsider the function $c(\\xx) = \\frac{1}{2} \\xx^{\\trp} \\LL \\xx -\n\\xx^{\\trp}\\dd$.\nWe should ask ourselves some questions about this function $c : \\R^V\n\\to \\R$. Is it continuous and continuously differentiable? The answer to this is\nyes, and that is not hard to see.\nDoes the function have a minimum?\nThis is maybe not immediately clear, but the minimum does indeed exist.\n\nWhen this is minimized, the derivative of $c(\\xx)$ with respect to each coordinate of $\\xx$\nmust be zero. This condition yields exactly the system of linear\nequations $\\LL \\xx = \\dd$.\nYou will confirm this in Exercise 4.\n\nBased on our derivative condition for the optimum, we can also express the electrical voltages as the solution to\nan optimization problem, namely\n\\begin{align*}\n\\min_{\\xx \\in \\R^V} & \\frac{1}{2} \\xx^{\\trp} \\LL \\xx -\n\\xx^{\\trp}\\dd\n\\end{align*}\nAs you are probably aware, having the derivative of each coordinate equal zero is not a\nsufficient condition for being at the optimum of a\nfunction\\footnote{Consider the function in one variable $c(x) =\n x^3$.}.\nIt is also interesting to know whether \\emph{all} solutions to $\\LL\n\\xx = \\dd$ are in fact minimizers of $c$. The answer is\nyes, and will see some very general tools for proving statements like\nthis in Chapter~\\ref{sec:cvopt}.\n% Although, we have not yet proven the electrical flow minimizes\n% electrical energy $\\sum_e \\rr(e) \\ff(e)^2$, I have promised you that\n% this is true.\n% Let us denote the electrical flow and voltages by $\\fftil$ and $\\xxtil$,\n% and recall that $\\fftil = \\RR^{-1} \\BB ^\\trp \\xxtil$.\n% Based on this, we can make an interesting observation:\n% \\[\n% \\sum_e \\rr(e) \\ff(e)^2\n% =\n% \\ff^{\\trp} \\RR \\ff\n% =\n % \\]\n % \\frac{1}{2} \\xxtil^{\\trp} \\LL \\xxtil - \\xxtil^{\\trp}\\dd\n % =\n\nAltogether, we can see that routing electrical current through a\nnetwork of resistors leads to a \\emph{pair} of optimization problems,\nlet's call them $\\ff^*$ and $\\xx^*$,\nand that the solutions to the two problems are related, in our case\nthrough the equation $\\ff^* = \\RR^{-1}\\BB^{\\trp} \\xx^*$.\nIn Exercise 5, you will explore this relationship more.\n\\\\\n\nThis turns out to be an instance of a much broader phenomenon, known\nas Lagrangian duality, which allows us to learn a lot about many\noptimization problems by studying two related pairs of problems.\n\n\\paragraph{Solving $\\LL \\xx = \\dd$.}\nGiven a graph $G$ with resistances for the edges, and some net flow\nvector $\\dd$, how quickly can we compute $\\xx$?\n%\nBroadly speaking, there are two very different families of algorithms\nwe could use to try to solve this problem.\n\nEither, we could solve the linear equation using something like\n\\emph{Gaussian Elimination} to compute an exact solution.\n\nAlternatively,\nwe could start with a guess at a solution, e.g. $\\xx_0\n= \\veczero$, and then we could try to make a change to $\\xx_0$ to reach\na new point $\\xx_1$ with a lower value of $c(\\xx) = \\frac{1}{2} \\xx^{\\trp} \\LL \\xx -\n\\xx^{\\trp}\\dd$, i.e. $c(\\xx_1) < c(\\xx_0)$.\nIf we repeat a process like that for enough steps, say $t$, hopefully we\neventually reach $\\xx_t$ with $c(\\xx_t)$ close to $c(\\xx^*)$, where\n$\\xx^*$ is a minimizer of $c(\\xx)$ and hence $\\LL \\xx^* = \\dd$.\nNow, we also need to make sure that $c(\\xx_t) \\approx c(\\xx^*)$\nimplies that $\\LL \\xx_t \\approx \\dd$ in some useful sense.\n\nOne of the most basic algorithms in this framework of ``guess and adjust''\nis called \\emph{Gradient Descent}, which we will study in two weeks.\nThe rough idea is the following: if we make a very small step from $\\xx$ to\n$\\xx + \\ddelta$, then a multivariate Taylor expansion suggests that\n$c(\\xx + \\ddelta) - c(\\xx) \\approx \\sum_{v \\in V} \\ddelta(v)\n\\frac{\\partial c(\\xx)}{\\partial\\xx(v)} $.\n\nIf we are dealing with smooth convex function,\nthis quantity is negative if we let $\\ddelta(v) = -\\epsilon \\cdot \\frac{\\partial\n c(\\xx)}{\\partial\\xx(v)}$ for some small enough $\\epsilon$ so the\napproximation holds well.\nSo we should be able to make progress by taking a small step in this\ndirection.\nThat's Gradient Descent!\nThe name comes from the vector of partial\nderivatives, which is called the gradient.\n% But first we need to become familiar with some basic terminology about\n% optimization and some convex geometry.\n% As we will see later in this course, understanding electrical problems\n% from an optimization perspective is crucial to developing fast\n% algorithms.\n\nAs we will see later in this course, understanding electrical problems\nfrom an optimization perspective is crucial to developing fast\nalgorithms for computing electrical flows and voltages, but to do very\nwell, we also need to borrow some ideas from Gaussian Elimination.\n\nWhat running times do different approaches get?\n\\begin{enumerate}\n\\item Using Gaussian Elimination, we can find $\\xx$ s.t. $\\LL \\xx =\n \\dd$ in $O(n^3)$ time and with asymptotically faster algorithms based on matrix\n multiplication, we can bring this down to roughly $O(n^{2.372})$.\n\\item Meanwhile Gradient Descent will get a running time of $O(n^3m)$\n or so -- at least this is a what a simple analysis suggests.\n\\item However, we can do much better: By combining ideas from both\n algorithms, and a bit more,\n we can get $\\xx$ up to very high\n accuracy in time $O(m \\log^c n)$ where $c$ is some small constant.\n % This type of result was orignally shown by Spielman and Teng in a\n % breakthrough result in 2004.\n\\end{enumerate}\n\n\n\\section{Convex Optimization}\n\nRecall our plot in Figure~\\ref{fig:flow-vs-energy} of the energy required to\nroute a flow $f$ across a resistor with resistance $r$, which was\n$\\energy(f) = r\\cdot f^2$.\n%\nWe see that the function has a special structure: the graph of the function sits\nbelow the line joining any two points $(f, \\energy(f))$ and $(g,\\energy(g))$.\nA function $\\energy : \\R \\to \\R$ that has this property is said to be convex.\n\nFigure~\\ref{fig:flow-vs-energy-convex} shows the energy as a function of flow, along with two\npoints $(f, \\energy(f))$ and $(g,\\energy(g))$. We see the function sits below the\nline segment between these points.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{fig/lecture1_flow-vs-energy-convex.png}\n \\captionof{figure}{Energy has a function of flow in a resistor with\n resistance $r = 1$. The function is convex.}\n \\label{fig:flow-vs-energy-convex}\n\\end{figure}\nWe can also interpret this condition as saying that for all $\\theta \\in [0,1]$\n\\[\n \\energy(\\theta f + (1-\\theta)g) \\leq \\theta \\energy(f) +\n (1-\\theta) \\energy(g)\n .\n \\]\nThis immediately generalizes to functions $\\energy:\\R^m \\to \\R$.\n\nA \\emph{convex set} is a subset of $S \\subseteq \\R^m$ s.t.\nif $\\ff, \\gg \\in S$ then for all $\\theta \\in [0,1]$ we have\n$\\theta \\ff + (1-\\theta) \\gg \\in S$.\n\nFigure~\\ref{fig:convex_sets} shows some examples of sets that are and\naren't convex.\n\n\\begin{figure}[t]\n\\begin{centering}\n \\includegraphics[trim = 0mm 0mm 0mm 0mm, height=60mm]{fig/lec1_convex_sets.pdf}\n \\caption{A depiction of convex and non-convex sets. The sets $A$ and $B$ are convex since the straight line between any two points inside them is also in the set. The set $C$ is not convex.}\\label{fig:convex_sets}\n \\end{centering}\n\\end{figure}\n\n\nConvex functions and convex sets are central to optimization,\nbecause for most problems of minimization a convex function over a\nconvex set, we can develop fast algorithms \\footnote{\n There are some convex optimization problems that are NP-hard.\n That said, polynomial time algorithms exist for almost any convex\n problem you can come up with.\n The most general polynomial time algorithm for convex optimization\n is probably the\n \\href{https://en.wikipedia.org/wiki/Ellipsoid_method}{Ellipsoid Method}.\n}.\n\n\nSo why convex functions and convex sets?\nOne important reason is that\nfor a convex function defined over a convex feasible set,\nany local minimum is also a global minimum,\nand this fact makes searching for an optimal solution\ncomputationally easier.\nIn fact, this is closely related to why Gradient Descent works well\non many convex functions.\n\n\nNotice that the set $\\setof{ \\ff: \\BB \\ff = \\dd}$ is convex, i.e. the\nset of all flows that route a fixed demand $\\dd$ is convex.\nIt is also easy to verify that $\\energy(\\ff) = \\sum_e \\rr(e) \\ff(e)^2$\nis a convex function, and hence finding an electrical flow is an\ninstance of convex minmization:\n\n\n\n\\section{More Graph Optimization Problems}\n\n\\paragraph{Maximum flow.}\nAgain, let $G=(V,E)$ be an undirected, connected graph with $n$\nvertices and $m$ edges.\nSuppose we want to find a flow\n$\\ff \\in \\R^E$ that routes $\\dd$, but instead of trying to minimize\nelectrical energy, we try to pick an $\\ff$ that minimizes the largest\namount of flow on any edge, i.e. $\\max_e \\abs{\\ff_e}$ -- which we\nalso denote by $\\norm{\\ff}_{\\infty}$.\nWe can write this problem as\n\\begin{align*}\n\\min_{\\ff \\in \\R^E } & \\norm{\\ff}_{\\infty} \\\\\n\\textrm{s.t. } & \\BB \\ff = \\dd\n\\end{align*}\nThis problem is known as the Minimum Congested Flow Problem\\footnote{This\n version is called undirected, because the graph is undirected, and\n \\emph{uncapacitated} because we are aiming for the same\n bound on the flow on all edges.}.\nIt is equivalent to the more famous Maximum Flow Problem.\n\nThe behavior of this kind of flow is very different than electrical\nflow. Consider the question of whether a certain demand can be routed\n$\\norm{\\ff}_{\\infty} \\leq 1$.\nImagine sending goods from a source $s$ to a destination $t$ using a\nnetwork of train lines that all have the\nsame capacity and asking whether the network is able to route the\ngoods at the rate you want: This boils down to whether routing exists\nwith $\\norm{\\ff}_{\\infty} \\leq 1$, if we set it up right.\n\nWe have a very fast, convex optimization-based algorithm for Minimum Congested Flow:\nIn $m \\epsilon^{-1} \\log^{O(1)} n$ time, we can find a flow $\\fftil$\ns.t. $\\BB \\fftil = \\dd$ and $\\norm{\\fftil}_{\\infty} \\leq\n(1+\\epsilon)\\norm{\\ff^*}_{\\infty}$, where $\\ff^*$ is an optimal solution, i.e. an\nactual minimum congestion flow routing $\\dd$.\n\nBut what if we want $\\epsilon$ to be very small, e.g. $1/m$? Then this running\ntime isn't so good anymore.\nBut, in this case, we can use another algorithm, that finds an optimal\nflow $\\ff^*$ \\emph{exactly}, in time\\footnote{And there's even a paper on arXiv.org that brings this further\n down to $m^{11/8} \\log^{O(1)} n$.} $m^{10/7} \\log^{O(1)} n $.\n\nJust as the electrical flow problem had a dual voltage problem, so\nmaximum flow has a dual voltage problem, which is know as the\n$s$-$t$ minimum cut problem.\n\n\\paragraph{Maximum flow, with directions and capacities.}\nWe can make the maximum flow problem harder by introducing directed\nedges: To do so, we allow edges in both directions to exist between a\nvertex to exist, and we require that that flow on a directed edge is\nalways non-negative. So now $G=(V,E)$ is a directed graph.\nWe can also make the problem harder by introducing capacities.\nWe define a capacity vector $\\cc \\in \\R^E \\geq \\veczero$ and try to minimize $\\norm{\\CC^{-1} \\ff}_{\\infty}$, where $\\CC =\n\\diag_{e \\in E}\\cc(e)$.\nThen our problem becomes\n\\begin{align*}\n\\min_{\\ff \\in \\R^E } & \\norm{\\CC^{-1} \\ff}_{\\infty} \\\\\n \\textrm{s.t. } & \\BB \\ff = \\dd\\\\\n & \\ff \\geq \\veczero.\n\\end{align*}\nFor this capacitated, directed maximum flow problem, our best\nalgorithms run in about $O( m \\sqrt{n} )$ time\\footnote{Provided the\n capacities are integers satisfying a condition like $\\cc \\leq n^{100} \\vecone$.}, even if we are willing to\naccept fairly low accuracy solution.\nIf the capacities are allowed to be exponentially large, the best\nrunning time we can get is $O(m n)$.\nFor this problem, we do not yet know how to improve over classical\ncombinatorial algorithms using convex optimization.\n\n\\paragraph{Multi-commodity flow.}\nWe can make the even harder still, by simultaneously trying to route\nto types of flow (imagine pipes with Coke and Pepsi).\nOur problem now looks like\n\\begin{align*}\n\\min_{\\ff_1, \\ff_2 \\in \\R^E } & \\norm{\\CC^{-1} (\\ff_1 + \\ff_2)}_{\\infty} \\\\\n \\textrm{s.t. } & \\BB \\ff_1 = \\dd_1 \\\\\n & \\BB \\ff_2 = \\dd_2\\\\\n & \\ff_1, \\ff_2 \\geq \\veczero.\n\\end{align*}\nSolving this problem to high accuracy is essentially as hard as\nsolving a general linear program! We should see later in the course\nhow to make this statement precise.\n\nIf we in the above problem additionally require that our flows must be\ninteger valued, i.e. $\\ff_1, \\ff_2 \\in \\N_{0}$, then the problem becomes NP-complete.\n\n\\paragraph{Random walks in a graph.}\nGoogle famously uses\\footnote{At least they did at some point.} the\nPageRank problem to help decide how to rank their search results.\nThis problem essentially boils down to computing the \\emph{stable\ndistribution} of a random walk on a graph.\nSuppose $G=(V,E)$ is a directed graph where each edge outgoing edge\n$(v,u)$, which we will define as going from $u$ to $v$, has a\ntransition probability $p_{(v,u)} > 0$ s.t. $\\sum_{z \\leftarrow u}\np_{(z,u)} = 1$.\nWe can take a step of a random walk on the vertex set by starting at some vertex\n$u_0 = u$, and then randomly picking one of it the outgoing edges\n$(v,u)$ with probability $p_{(v,u)}$ and move to\nthe chosen vertex $u_1 = v$. Repeating this procedure, to take a step from\nthe next vertex $u_1$, gives us a\n\\emph{random walk} in the graph, a sequence of vertices $u_0, u_1, u_2\n\\ldots, u_k$.\n\nWe let $\\PP \\in \\R^{V \\times V}$ be the matrix of transition\nprobabilities given by\n\\[\n \\PP_{vu} =\n \\begin{cases}\n p_{(v,u)} & \\text{ for } (u,v) \\in E \\\\\n 0 & \\text{ o.w.}\n \\end{cases}\n\\]\n\nAny probability distribution over the vertices can be specified by a vector\n$\\pp \\in \\R^V$ where $\\pp \\geq\n\\veczero$ and $\\sum_v \\pp(v) = 1$.\nWe say that probability distribution $\\ppi$ on the vertices is a \\emph{stable distribution} of the random walk\nif $\\ppi = \\PP \\ppi$.\nA strongly connected graph always has exactly one stable distribution.\n\nHow quickly can we compute the stable distribution of a general random\nwalk? Under some mild conditions on the stable\ndistribution\\footnote{Roughly something like $\\max_{v} 1/\\ppi(v) \\leq n^{100}$.}, we can find\na high accuracy approximation of $\\ppi$ in time $O(m \\log^cn )$ for\nsome constant $c$.\n\nThis problem does not easily fit in a framework of convex\noptimization, but nonetheless, our fastest algorithms for it use ideas\nfrom convex optimization.\n\n\\section*{Topics in this Course}\nIn this course, we will try to address the following questions.\n\\begin{enumerate}\n\\item What are the fundamental tools of fast convex optimization?\n\\item What are some problems we can solve quickly on graphs using optimization?\n\\item What can graphs teach us about convex optimization?\n\\item What algorithm design techniques are good for getting algorithms\n that quickly find a crude approximate solution? And what techniques\n are best when we need to get a highly accurate answer?\n\\item What is special about flow problems?\n\\end{enumerate}\n\n% \\pagebreak\n% \\section{Exercises}\n\n% NEW\n% \\begin{enumerate}\n% \\item sub-level set of a convex function is a convex set (show).\n% \\item is it true that a function whose sub-level sets are all convex\n% is necessarily convex?\n% \\item \\todo{exercise A}\n% \\begin{itemize}\n% \\item kernel of the Laplacian\n% \\item show that solutions (voltage and flow) exist when\n% graph is connected and $\\dd^\\trp \\vecone = 0$.\n% \\item prove matrix symmetric, consider quadratic form,\n% show positive semi-definite.\n% \\end{itemize}\n% \\item \\todo{exercise B}\n% define gradient. do the voltage gradient computation\n% \\item \\todo{exercise C: more duality}\n% {\\color{olive}\n% \\\\\n% Recall $\\ff^* = \\RR^{-1}\\BB^{\\trp} \\xx^*$ and $\\BB \\ff^* = \\dd$,\n% we can conclude\n% $(\\ff^* )^\\trp R \\ff^* = (\\ff^* )^\\trp \\BB^\\trp \\xx^* = \\dd^\\trp \\xx^* = (\\xx^*)^\\trp \\LL\n% \\xx^*$\n% (Maybe don't directly point this out to students!)\n\n% now we can see that\n% \\begin{align*}\n% \\max_{\\xx \\in \\R^V} \\xx^{\\trp}\\dd - \\frac{1}{2} \\xx^{\\trp} \\LL \\xx\n% &=\n% \\min_{\\ff \\in \\R^E} \\frac{1}{2}\\sum_e \\rr(e) \\ff(e)^2 \\\\\n% & \\qquad \\textrm{ s.t. }\n% \\BB \\ff = \\dd\n% .\n% \\end{align*}\n\n% Maybe also point out to students that the stuff they have proved can\n% easily be generalized a larger class of quadratic problems.\n% }\n% \\item GRADED? gaussian elimination for Laplacians is closed ?\n% \\end{enumerate}\n\n\n%%% Local Variables:\n%%% mode: latex\n%%% TeX-master: \"agao21_script\"\n%%% TeX-engine: luatex\n%%% End:\n", "meta": {"hexsha": "80d3bfecf1ff9708575d975201ab9b4d46856ddd", "size": 27649, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "agao21_script/lecture1_mod.tex", "max_stars_repo_name": "csssaz/agao21_script", "max_stars_repo_head_hexsha": "51044f4775e5e20d2c5fc5c0d035363e5beb66be", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-03-15T09:04:46.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-26T05:39:09.000Z", "max_issues_repo_path": "agao21_script/lecture1_mod.tex", "max_issues_repo_name": "csssaz/agao21_script", "max_issues_repo_head_hexsha": "51044f4775e5e20d2c5fc5c0d035363e5beb66be", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "agao21_script/lecture1_mod.tex", "max_forks_repo_name": "csssaz/agao21_script", "max_forks_repo_head_hexsha": "51044f4775e5e20d2c5fc5c0d035363e5beb66be", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2021-03-11T12:35:23.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-13T06:04:51.000Z", "avg_line_length": 41.7028657617, "max_line_length": 232, "alphanum_fraction": 0.714456219, "num_tokens": 7985, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.640635868562172, "lm_q1q2_score": 0.34777769092573024}} {"text": "\\documentclass[12pt]{amsart}\n\\usepackage[margin=0.5in]{geometry} \n % see geometry.pdf on how to lay out the page. There's lots.\n\\usepackage{bsymb}\n\\usepackage{../unitb}\n\\usepackage{calculational}\n\\usepackage{ulem}\n\\usepackage{hyperref}\n\n\\newcommand{\\REQ}{\\text{REQ}}\n\n\\begin{document}\n % \\input{main9/machine_m0.tex}\n\\begin{machine}{m0}\n\n \\newset{\\REQ}\n\nWe need a variable to keep track of all the requests to mutate the\ndata structure.\n\n \\[ \\variable{ req,req_0 : \\set [\\REQ] } \\]\n\n... and new events:\n% % \\hide{\n \\newevent{req}{req}\n \\newevent{handle}{handle}\n % \\newevent{req}{req\\_b} \n % \\newevent{handle}{handle\\_b} \n\n\\[\\param{req}{r : \\REQ }\\]\n\\with{sets}\n\\begin{align*}\n & \\evguard{req}{grd0}{ \\neg r \\in req } \\\\\n & \\evbcmeq{req}{act0}{ req }{ req \\bunion \\{ r \\} } \\\\\n & \\evbcmeq{req}{act1}{ req_0 }{ req } \n\\end{align*}\n\\[\\param{handle}{r : \\REQ }\\]\n\\begin{align*}\n \\evguard{handle}{grd0}{ r \\in req } \\\\\n \\evbcmeq{handle}{act0}{ req }{ req \\setminus \\{ r \\} } \\\\\n \\evbcmeq{handle}{act1}{ req_0 }{ req }\n\\end{align*}\n\n% \\section{Requirements}\n% \\dummy{ R : \\set[\\REQ] }\n\\begin{align*}\n \\constraint{co0}{ req_0' = req \\1\\lor (req_0' = req_0 \\1\\land req' = req) }\n\\end{align*}\n\\begin{align*}\n & \\progress{prog0}\n { \\neg req = \\emptyset }\n { \\neg req_0 \\subseteq req }\n \\refine{prog0}{ensure}{handle}{ using \\ref{handle} }\n & \\progress{prog1}\n { V = req}\n { req \\subset V \n \\1\\lor req = \\emptyset \\1\\lor \\neg req \\subseteq req_0}\n \\refine{prog1}{ensure}{handle}{ using \\ref{handle} }\n\\end{align*}\n\\begin{align*}\n \\dummy{ R,V : \\set [\\REQ] } \\\\\n \\cschedule{handle}{m0:sch0}{ \\neg req = \\emptyset } \\\\\n \\initialization{m0:in0}{ req = \\emptyset }\n\\end{align*}\n\\end{machine}\n\n% \\input{main9/machine_m1}\n\\begin{machine}{m1}\n \\refines{m0}\n\\[ \\indices{handle}{ b : \\Bool } \\]\n\\[ \\dummy{ b : \\Bool } \\]\n\\[ \\variable{ ch : \\Bool } \\]\n\\begin{align*}\n & \\cschedule{handle}{m1:sch0}{ b = ch } \\\\\n & \\witness{handle}{b}{b = ch} \\\\\n & \\progress{m1:prog1}{ b = ch }{ b = ch }\n\\end{align*}\n\\replace{handle}{m1:sch0}{m1:prog1}\n \\begin{liveness}{m1:prog1}\n \\progstep{\\true}{req = \\emptyset \\1\\lor \\neg req \\subseteq req_0}\n {induction}{}{ \\var{req}{down}{\\emptyset} }\n \\begin{flatstep}\n \\progstep\n {V = req}\n {req \\subset V\n \\1\\lor req = \\emptyset \\1\\lor \\neg req \\subseteq req_0}\n {discharge}{}{}\n \\begin{step}\n \\trstep{handle}{ \\index{b}{ \\true } }\n { req = V \\1\\land \\neg req = \\emptyset }\n \\safstep\n { V = req }\n { req \\subset V \\1\\lor \\neg req \\subseteq req_0 }\n {}\n \\end{step}\n \\end{flatstep}\n \\end{liveness}\n\\end{machine}\n\n% % \\input{main9/machine_m2}\n% \\begin{machine}{m2}\n% \\refines{m0}\n\n% We now partitial $req$ into requests for operation A ($reqA$) and\n% requests for operation B ($reqB$).\n\n% \\[ \\variable{ reqA, reqB : \\set [\\REQ] } \\]\n% \\begin{align*}\n% \\invariant{m1:inv0}{ reqA \\bunion reqB = req } \\\\\n% \\invariant{m1:inv1}{ reqA \\binter reqB = req } \n% \\end{align*}\n% And consequ\n% \\begin{align*}\n% \\initialization{m1:in0}{ reqA = \\emptyset \\land reqB = \\emptyset }\n% \\end{align*}\n% \\end{machine}\n\n\\end{document}\n", "meta": {"hexsha": "0d72475262e41cba8ffd441eee69516f7eac5552", "size": 3171, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Tests/lock-free deque/main9.tex", "max_stars_repo_name": "literate-unitb/literate-unitb", "max_stars_repo_head_hexsha": "0d843456dc103bb09babc5b12855435d2e10f534", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2017-07-27T11:05:56.000Z", "max_stars_repo_stars_event_max_datetime": "2018-01-20T14:53:33.000Z", "max_issues_repo_path": "Tests/lock-free deque/main9.tex", "max_issues_repo_name": "unitb/literate-unitb", "max_issues_repo_head_hexsha": "0d843456dc103bb09babc5b12855435d2e10f534", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 32, "max_issues_repo_issues_event_min_datetime": "2017-06-25T03:53:02.000Z", "max_issues_repo_issues_event_max_datetime": "2017-06-25T04:28:38.000Z", "max_forks_repo_path": "Tests/lock-free deque/main9.tex", "max_forks_repo_name": "literate-unitb/literate-unitb", "max_forks_repo_head_hexsha": "0d843456dc103bb09babc5b12855435d2e10f534", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 26.6470588235, "max_line_length": 77, "alphanum_fraction": 0.5935036266, "num_tokens": 1206, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3477776834764115}} {"text": "\\title {\\TeX\\ macros for proof boxes}\n\\author {Paul Taylor}\n\n\\documentclass{article}\n\\usepackage{boxproof}\n\\usepackage{a4wide}\n\\usepackage{daymonthyear}\n\n\\def\\meta#1{\\mbox{$\\langle\\hbox{#1}\\rangle$}}\n\\def\\macrowitharg#1#2{{\\tt\\string#1\\bra\\meta{#2}\\ket}}\n\n{\\escapechar-1 \\xdef\\bra{\\string\\{}\\xdef\\ket{\\string\\}}}\n\n\\def\\intro#1{{#1}{\\cal I}}\n\\def\\elim#1{{#1}{\\cal E}}\n\n\\showboxbreadth 999\n\\showboxdepth 999\n\\tracingoutput 1\n\n\n\\let\\imp\\to\n\\def\\elim#1{{{#1}{\\cal E}}}\n\\def\\intro#1{{{#1}{\\cal I}}}\n\\def\\lt{<}\n\\def\\eqdef{=}\n\\def\\eps{\\mathrel{\\epsilon}}\n\\def\\biimplies{\\leftrightarrow}\n\\def\\flt#1{\\mathrel{{#1}^\\flat}}\n\\def\\setof#1{{\\left\\{{#1}\\right\\}}}\n\\let\\implies\\to\n\\def\\KK{{\\mathsf K}}\n\\let\\squashmuskip\\relax\n\n%=======================================================================\n\\begin{document}\n\\maketitle\n\n\\section{Introduction}\nThe proof\n\\begin{proofbox}\n \\(\\\"1\"\\:\\alpha\\leftrightarrow\\psi(x,\\top)\\=\\\\\n \\:\\Some\\beta.\\psi(x,\\beta)\\=\\mathsf{total}\\\\\n \\[\\exists\\beta\\kern-1em\\:\\psi(x,\\beta)\\\\\n \\(\\:\\beta\\=\\\\\n \\:\\beta=\\top\\=(*)\\\\\n \\:\\psi(x,\\!\\top)\\=\\mathsf{subs}\\\\\n \\:\\alpha\\=\\elim\\leftrightarrow(\\ref{1})\\\\\n \\*\\:\\alpha\\=\\\\\n \\:\\psi(x,\\!\\top)\\=\\elim\\leftrightarrow(\\ref{1})\\\\\n \\:\\beta=\\top\\=\\mathsf{func}\\\\\n \\:\\beta\\=(*)\\\\\n \\)\\:\\alpha=\\beta\\=\\intro\\leftrightarrow\\\\\n \\:\\psi(x,\\alpha)\\=\\mathsf{subs}\\\\\n \\] \\:\\psi(x,\\alpha)\\=\\elim\\exists\\\\\n \\* \\:\\psi(x,\\alpha)\\=\\\\\n \\(\\:\\alpha\\=\\\\\n \\:\\alpha=\\top\\=(*)\\\\\n \\:\\phi(x,\\!\\top)\\=\\mathsf{subs}\\\\\n \\*\\:\\psi(x,\\!\\top)\\=\\\\\n \\:\\alpha=\\top\\=\\mathsf{func}\\\\\n \\:\\alpha\\=(*)\\\\\n \\)\\:\\alpha\\leftrightarrow\\psi(x,\\!\\top)\\=\\intro\\leftrightarrow\\\\\n \\) \\:\\psi(x,\\alpha)\\leftrightarrow(\\alpha\\leftrightarrow\\psi(x,\\!\\top))\n \\=\\intro\\leftrightarrow\\\\\n\\end{proofbox}\n is produced by\n\\begin{verbatim}\n\\begin{proofbox}\n \\(\\\"1\"\\:\\alpha\\leftrightarrow\\psi(x,\\top)\\\\\n \\:\\Some\\beta.\\psi(x,\\beta) \\=\\mathsf{total}\\\\\n \\[\\exists\\beta\\kern-1em\\:\\psi(x,\\beta)\\\\\n \\(\\:\\beta\\\\\n \\:\\beta=\\top \\=(*)\\\\\n \\:\\psi(x,\\!\\top) \\=\\mathsf{subs}\\\\\n \\:\\alpha \\=\\elim\\leftrightarrow(\\ref{1})\\\\\n \\*\\:\\alpha\\\\\n \\:\\psi(x,\\!\\top) \\=\\elim\\leftrightarrow(\\ref{1})\\\\\n \\:\\beta=\\top \\=\\mathsf{func}\\\\\n \\:\\beta\\=(*)\\\\\n \\)\\:\\alpha=\\beta \\=\\intro\\leftrightarrow\\\\\n \\:\\psi(x,\\alpha) \\=\\mathsf{subs}\\\\\n \\] \\:\\psi(x,\\alpha) \\=\\elim\\exists\\\\\n \\* \\:\\psi(x,\\alpha)\\\\\n \\(\\:\\alpha\\\\\n \\:\\alpha=\\top \\=(*)\\\\\n \\:\\phi(x,\\!\\top) \\=\\mathsf{subs}\\\\\n \\*\\:\\psi(x,\\!\\top)\\\\\n \\:\\alpha=\\top \\=\\mathsf{func}\\\\\n \\:\\alpha \\=(*)\\\\\n \\)\\:\\alpha\\leftrightarrow\\psi(x,\\!\\top) \\=\\intro\\leftrightarrow\\\\\n \\) \\:\\psi(x,\\alpha)\\leftrightarrow\n (\\alpha\\leftrightarrow\\psi(x,\\!\\top)) \\=\\intro\\leftrightarrow\\\\\n\\end{proofbox}\n\\end{verbatim}\n\n Syntax as follows:\n each line is of the form\n \\begin{center}\n \\meta{variables}\n \\meta{name}\n \\verb/\\:/ \\meta{formula}\n \\verb/\\=/ \\meta{reason}\n \\verb/\\-/ \\meta{use}\n \\verb/\\\\/\n \\end{center}\n where\n \\begin{itemize}\n \\item \\meta{variables} is something like ``$x,y$'' ---\n it's for variables declared at the beginning of $\\intro\\forall$- and\n $\\elim\\exists$-boxes.\n \\item \\meta{name} is a command \\verb/\\label{fred}/ which\n defines \\verb/fred/ to be the label text, which may be used anywhere\n as \\verb/\\ref{fred}/ --- see {\\em The \\LaTeX book.}\n Local labels are also available, using \\macrowitharg\\lbl{name}\n or \\verb/\\\"/\\meta{name}\\verb/\"/; these obey the scoping rules of\n the boxes.\n You may also refer to the previous line as \\verb/\\ref{-}/.\n \\item \\meta{formula} is the proposition being asserted.\n \\item \\meta{reason} is \\verb/\\intro\\land(\\ref{john},\\ref{mary})/\n or \\verb/\\elim\\forall(\\ref{jim})/.\n \\item \\meta{use} is provided for linear logic, to record the step\n which uses this one. How this accords with theory I don't yet know.\n \\end{itemize}\n Note that the parts are separated by \\verb/\\:/, \\verb/\\=/ and \\verb/\\\\/;\n these correspond to\n \\begin{center}\n {\\bf let } \\meta{name} = \\meta{expression} : \\meta{type}\n \\end{center}\n in a declarative language.\n The \\verb/\\:/, \\verb/\\=/ and \\verb/\\-/ fields are optional and may occur\n in any order. If any of them is repeated the last is taken.\n If none of them is present the \\meta{variables} field is also ignored.\n\n Proof {\\em boxes\\/} are ``wrapped up'' as follows:\n \\begin{itemize}\n \\item the whole proof in \\verb/\\begin{proofbox}/...\\verb/\\end{proofbox}/;\n \\item single-column boxes ($\\intro\\forall$, $\\intro\\imp$, $\\elim\\exists$),\n in \\verb/\\[/...\\verb/\\]/.\n \\item multiple-column boxes are of two kinds:\n \\begin{itemize}\n \\item separate ($\\intro\\land$) boxes: \\verb/\\(/...\\verb/\\*/...\\verb/\\)/.\n \\item stuck together ($\\elim\\lor$) boxes:\n \\verb/\\(/...\\verb/\\+/...\\verb/\\)/.\n \\end{itemize}\n \\end{itemize}\n You may put more than two columns in \\verb/\\(/...\\verb/\\)/ and even mix\n the \\verb/\\+/ and \\verb/\\*/ separators.\n\n The whole proof is enclosed in \\verb/\\proofbox/...\\verb/\\endproofbox/\n or \\verb/\\begin{proofbox}/...\\verb/\\end{proofbox}/, but the \\LaTeX\\\n environment form {\\em must not\\/} be used for nested boxes.\n\n If the proof occurs in paragraph mode (ie in vertical or\n unrestricted horizontal mode) then it is set as a display, using the\n full width of the page. Otherwise it uses only the required width.\n\n A lot of the internals are potentially configurable, but there is not\n yet a user interface suitable for doing this. This will be provided\n in the next version.\n\n\n%=======================================================================\n\\section{Redefinable macros}\n WARNING: most of these commands will be hidden and replaced with\n optional arguments to \\verb/\\proofbox/ in a future version.\n Do not rely on them.\n\n We provide three different ways of numbering the lines of the proof:\n \\begin{itemize}\n \\item \\verb/\\runningproofline/: a global running sequence (default),\n \n \\item \\verb/\\nestedproofline/: a hierarchical system with dots,\n\n \\item \\verb/\\nestedproofline/: a fully hierarchical system which\n also includes the column number\n (\\verb/\\proof@columns/) as a letter (ASCII quote plus number).\n\n \\end{itemize}\n \\verb/\\theproofline/ is the default.\n\n The macro \\verb/\\proofboxmakelabel#1/ is used to print the line label.\n We only put it in the leftmost box.\n It is printed in small non-ranging Arabic numerals\n ({\\the\\scriptfont1 0123456789}).\n Right-justify it in \\verb/\\prooflinenowidth/ if it will fit,\n otherwise let it stick out on the right, {\\em i.e.}~left-justify it.\n\n Kill the numbers altogether with \\verb/\\proofboxnonumbers/.\n\n How to make the left column of the proof box:\n use the variables field, a space if necessary and the line label.\n\n How to make the middle column of the proof box:\n left justify the formula field.\n\n How to make the right column of the proof box:\n use the reason and use fields.\n\n\n Make the four edges of a rectangular box and the separator\n between \\verb/\\+/ columns.\n\n\n Use dotted lines: \\verb/\\dottedproofbox/.\n\n Leave the boxes open at the bottom: \\verb/\\openproofbox/\n\n%=======================================================================\n\\section{Miscellaneous logical notations}\n\nThese macros are now in my \\texttt{logicsym.sty}\n\n Print the names of the introduction and elimination rules, for example:\n \\begin{quote}\n \\verb/\\elim\\forall/ $\\elim\\forall$ \\qquad\n \\verb/\\intro\\land/ $\\intro\\land$ \\qquad\n \\end{quote}\n Recall that in \\TeX\\ the logical connectives and quantifiers are called\n \\begin{center}\n \\verb/\\lor/ $\\lor$\\quad\n \\verb/\\land/ $\\land$\\quad\n \\verb/\\lnot/ $\\lnot$\\quad\n \\verb/\\forall/ $\\forall$\\quad\n \\verb/\\exists/ $\\exists$\n \\end{center}\n The following provide macros for the \\verb/\\implies/ {\\em relation\\/}\n and for the binary {\\em operation\\/} which yields the abstract\n \\verb/\\implic/ation between formulae.\n The point is that \\TeX\\ spaces them and breaks the lines differently:\n \\begin{center}\n \\verb/A\\implies B/ $A\\implies B$\\quad{\\em versus\\/}\\quad\n \\verb/A\\implic B/ $A\\implic B$\n \\end{center}\n There are forward and reverse, single and Double versions.\n\n Handle the spacing after a variable (and optionally its type)\n bound by a quantifier symbol. For example\n \\begin{quote}\n \\verb/\\All x:X. \\phi(x)/\\quad prints as\\quad $\\All x:X.\\phi(x)$\\quad\n instead of\\quad $\\forall x:X.\\phi(x)$\n \\end{quote}\n We provide some commonly used forms; \\verb/\\iota/ ($\\iota$) is Russell's\n description operator and should really be inverted.\n There are several notations for substitution.\n After writing $a[x:=b]$ throughout my book I~thought I~might change to\n $[b/x]^*a$.\n This macro reads the source in the first form and prints in the second.\n If you use it you can, like me, defer the decision about\n which notation to use until the final stages, doing\n \\begin{quote}\n \\verb/\\renewcommand{\\Subst}{\\plainsubstitution}/\n \\end{quote}\n if you finally decide on making substitution act on the right.\n This is already an improvement on the literal text, because it\n automatically enlarges the brackets according to the text inside.\n \\verb/\\Subst/ itself is (following my book)\n defined in terms of the action of a context morphism (\\verb/\\CtxtMor/)\n on a term. Again you can do\n \\begin{quote}\n \\verb/\\renewcommand{\\CtxtMor}{\\plaincontextmorphism}/\n \\end{quote}\n for something simpler.\n This macro interprets its argument as a comma-separated list\n of items in the form $x:=b$, which it switches to $b/x$.\n The simple versions.\n\n%=======================================================================\n\\section{Some very easy logic exercises}\n\nThe following examples are taken from Krysia Broda's\n{\\em Solutions to Problems~5\\/} (KB-Logic-B1-90) and took me a little\nunder an hour to type in.\n\npage 1: (a)\n\n\\begin{proofbox}\n \\: P\\land Q \\\\\n \\: P \\= \\elim\\land\\\\\n\\end{proofbox}\n\n\\begin{proofbox}\n \\lbl{1}\\: P \\\\\n \\[\n \\: Q \\\\\n \\: P \\= (\\ref{1}) \\\\\n \\]\n \\: Q\\to P \\= \\intro\\to \\\\\n\\end{proofbox}\n\n\npage 2: (c)\n\n\\begin{proofbox}\n \\lbl{2}\\: P\\\\\n \\[\n \\lbl{3}\\: Q\\\\\n \\: P\\land Q\\= \\intro\\land(\\ref{2},\\ref{3}) \\\\\n \\]\n \\: Q\\to(P\\land Q) \\= \\intro\\to \\\\\n\\end{proofbox}\n\n(g)\n\n\\begin{proofbox}\n \\lbl{6}\\: P\\to(Q\\to R)\\\\\n \\[\n \\lbl{4}\\: P\\to Q\\\\\n \\[\n \\lbl{5}\\: P\\\\\n \\lbl{8}\\: Q \\= \\elim\\to(\\ref{4},\\ref{5})\\\\\n \\lbl{7}\\: Q\\to R \\= \\elim\\to(\\ref{6},\\ref{5})\\\\\n \\: R \\= \\elim\\to(\\ref{7},\\ref{8})\\\\\n \\]\n \\: P\\to R \\= \\intro\\to \\\\\n \\]\n \\: (P\\to Q)\\to(P\\to R) \\= \\intro\\to \\\\\n\\end{proofbox}\n\npage 3: (h)\n\n\\begin{proofbox}\n \\lbl{10}\\: P\\to(Q\\to R)\\\\\n \\[\n \\lbl{9}\\: P\\land Q\\\\\n \\lbl{11}\\: P \\= \\elim\\land1(\\ref{9})\\\\\n \\lbl{12}\\: Q\\to R \\= \\elim\\to(\\ref{10},\\ref{11})\\\\\n \\lbl{13}\\: Q \\=\\elim\\land2(\\ref{9})\\\\\n \\: R \\=\\elim\\to(\\ref{12},\\ref{13})\\\\\n \\]\n \\: P\\land Q\\to R \\= \\intro\\to \\\\\n\\end{proofbox}\n\n(i)\n\n\\begin{proofbox}\n \\lbl{16}\\: P\\land Q\\to R \\\\\n \\[\n \\lbl{14}\\: P \\\\\n \\[\n \\lbl{15}\\: Q \\\\\n \\lbl{17}\\: P\\land Q \\= \\intro\\land(\\ref{14},\\ref{15})\\\\\n \\: R \\= \\elim\\to(\\ref{16},\\ref{17})\n \\]\n \\: Q\\to R \\= \\intro\\to \\\\\n \\]\n \\: P\\to(Q\\to R) \\= \\intro\\to \\\\\n\\end{proofbox}\n\n(j)\n\n\\begin{proofbox}\n \\lbl{18}\\: P\\to Q\\\\\n \\lbl{20}\\: \\lnot Q\\\\\n \\[\n \\lbl{19}\\: P \\\\\n \\lbl{21}\\: Q \\= \\elim\\to(\\ref{18},\\ref{19})\\\\\n \\: \\bot \\= \\elim\\lnot(\\ref{20},\\ref{21})\\\\\n \\]\n \\: \\lnot P \\= \\intro\\lnot \\\\\n\\end{proofbox}\n\npage 5: (k)\n\n\\begin{proofbox}\n \\lbl{22}\\: \\lnot P\\\\\n \\[\n \\lbl{23}\\: P\\\\\n \\[\n \\: \\lnot Q\\\\\n \\: \\lnot P \\land P \\= \\intro\\land(\\ref{22},\\ref{23})\\\\\n \\: \\bot \\= \\elim\\lnot(\\ref{22},\\ref{23})\\\\\n \\]\n \\: \\lnot\\lnot Q \\= \\intro\\lnot\\\\\n \\: Q \\= \\lnot\\lnot\\\\\n \\]\n \\: P\\to Q \\= \\intro\\to \\\\\n\\end{proofbox}\n\npage 8: (o)\n\n\\begin{proofbox}\n \\lbl{24}\\: P\\to Q \\\\\n \\[\n \\lbl{26}\\: \\lnot Q \\\\\n \\[\n \\lbl{25}\\: P \\\\\n \\lbl{27}\\: Q \\= \\elim\\to(\\ref{24},\\ref{25})\\\\\n \\: \\bot \\= \\elim\\lnot(\\ref{26},\\ref{27})\\\\\n \\]\n \\: \\lnot P \\= \\intro\\lnot\\\\\n \\]\n \\: \\lnot Q\\to \\lnot P \\= \\intro\\to \\\\\n\\end{proofbox}\n\n(p) The $\\lnot\\lnot$ rule is unnecessary!\n\n\\begin{proofbox}\n \\lbl{28}\\: P\\to Q \\\\\n \\[\n \\: \\lnot\\lnot P\\\\\n \\lbl{29}\\: P \\= \\lnot\\lnot\\\\\n \\[\n \\lbl{31}\\: \\lnot Q \\\\\n \\lbl{30}\\: Q \\= \\elim\\to(\\ref{28},\\ref{29})\\\\\n \\: Q\\land\\lnot Q \\= \\intro\\land(\\ref{30},\\ref{31})\\\\\n \\: \\bot \\= \\elim\\lnot\\\\\n \\]\n \\: \\lnot\\lnot Q \\= \\intro\\lnot\\\\\n \\]\n \\: \\lnot\\lnot P\\to\\lnot\\lnot Q \\= \\intro\\to \\\\\n\\end{proofbox}\n\n%=======================================================================\n\\section{Some more exercises}\n\n\\begin{proofbox}\n \\: P\\lor\\lnot P \\\\\n \\(\n \\: P \\\\\n \\: Q\\to P \\=\\hbox{by (e)}\\\\\n \\: (P\\to Q)\\lor(Q\\to P) \\= \\intro\\lor \\\\\n \\+\n \\: \\lnot P\\\\\n \\: P\\to Q \\=\\hbox{by (k)}\\\\\n \\: (P\\to Q)\\lor(P\\to Q) \\= \\intro\\lor \\\\\n \\)\n \\: (P\\to Q)\\lor(P\\to Q) \\= \\elim\\lor \\\\\n\\end{proofbox}\n\n\n\n\\begin{proofbox}\n \\lbl{35}\\: P\\lor Q \\\\\n \\[\n \\lbl{30}\\: (\\lnot P)\\land(\\lnot Q) \\\\\n \\(\n \\lbl{31} \\: P \\\\\n \\lbl{32} \\: \\lnot P \\=\\elim\\land1(\\ref{30})\\\\\n \\: \\bot \\= \\elim\\lnot(\\ref{32},\\ref{31})\\\\\n \\+\n \\lbl{33} \\: Q \\\\\n \\lbl{34} \\: \\lnot Q \\= \\elim\\land2(\\ref{30})\\\\\n \\: \\bot \\= \\elim\\lnot(\\ref{33},\\ref{34})\\\\\n \\)\n \\: \\bot \\= \\elim\\lor(\\ref{35})\\\\\n \\]\n \\: \\lnot((\\lnot P)\\land(\\lnot Q)) \\= \\intro\\lnot \\\\\n\\end{proofbox}\n\n\n\\begin{proofbox}\n\\lbl{51} x \\: \\forall x'.x' l_t \\\\\n FN &= \\hat{l}_t < l_t\n \\end{aligned}\n\\end{equation}\n\nThese metrics are then utilized to generate slightly more sophisticated evaluation metrics across trials, including:\n\n\\begin{equation}\n \\begin{aligned}\n \\text{Correctly Detected Trials (CDT)} &= \\sum^{T}_{t = 1} \\mathbbm{1}( (FP_{t} = 1) \\ \\& \\ (TP_t >= 1)) \\\\\n \\text{Trial Failures} &= (t_a - \\argmin_t(l_t = 1)) \\\\\n \\text{False Positive Trial \\%} &= \\frac{FP}{T} \\\\\n \\text{False Negative Trial \\%} &= \\frac{FN}{T}\n \\end{aligned}\n\\end{equation} \n\nIn summary, the primary goal of this family of evaluation metrics is to minimize the number of false positives. This is far more important to PAR than the other metrics. Trial Failures exclusive looks at how many trials too early you declared novelty.\n\n\\section{Sample-based Evaluation Metrics: Reaction Performance}%\n\\label{sec:sample_based_evaluation_metrics_reaction_performance}\n\nThe next family of evaluation metrics concerns sample-based evaluation metrics. These metrics conducted over batches (unless specified otherwise) and then averaged. The evaluation metrics are defined below.\n\n\n\\begin{equation}\n \\begin{aligned}\n FP &= (\\hat{y}_{\\theta ti} = k+1) \\ \\& \\ (y_{ti} \\neq k+1)) \\\\\n FN &= (\\hat{y}_{\\theta ti} \\neq k+1) \\ \\& \\ (y_{ti} = k+1))\n \\end{aligned}\n\\end{equation}\n\nThese can be computed in any way, but in particular there are several metrics that compare these FP, FN values in the pre- and post-novelty regime. The formulas are the same for the baseline model, but $\\theta$ is replaced by $\\phi$.\n\n\\begin{equation}\n \\begin{aligned}\n \\text{Accuracy} &= \\frac{1}{T} \\sum^{T}_{t=1} \\frac{1}{n} \\sum^{n}_{i=1} \\mathbbm{1}(\\hat{y}_{\\theta ti} = y_{ti}) \\\\\n \\text{Baseline Accuracy} &= \\frac{1}{T} \\sum^{T}_{t=1} \\frac{1}{n} \\sum^{n}_{i=1} \\mathbbm{1}(\\hat{y}_{\\phi ti} = y_{ti}) \\\\\n \\end{aligned}\n\\end{equation}\n\nPer trial asymptotic performance is defined as the ``measure in the change in accuracy over post-novelty rounds''. No notation or formulae of any kind accompany this statement. However, some additional metrics include:\n\n\\begin{equation}\n \\begin{aligned}\n \\text{Novelty Response Performance (NRP)} &= \\frac{( \\text{Accuracy} | t \\geq t_a)}{( \\text{Accuracy} | t < t_a)} \\\\\n \\text{Overall Performance Task Improvement (OPTI)} &= \\frac{( \\text{Accuracy} | t \\geq t_a)}{( \\text{Baseline Accuracy} | t \\geq t_a)} \\\\\n \\text{Asymptotic Performance Task Improvement (APTI)} &= \\frac{( \\text{Accuracy per Trial} | t \\geq t_a)}{( \\text{Baseline Accuracy per trial} | t \\geq t_a)} \\\\\n \\text{Area Under Activity Monitor Operating Characteristic (AUAMOC)} &= \\text{AUROC of TP and FP}\n \\end{aligned}\n\\end{equation}\n\nIn words, \\textbf{NRP} is the ratio of the average batch accuracy in batches after novelty over the average batch accuracy in batches before novelty - we want this as high as possible. \\textbf{OPTI} is the ratio of post-novelty accuracy in our model over post-novelty accuracy of the baseline. We also want this to be as high as possible. \\textbf{APTI} is a more granular \\textbf{OPTI} that makes use of per-trial differences. \\textbf{AUAMOC} is a simple AUROC curve plotted for global TP and FP, \\textbf{threshold based on anomaly score} \n\n\n\\section{Oustanding items}%\n\\label{sec:oustanding_items}\n\n\\begin{enumerate}\n \\item Top-k classification with anomaly score not a probability - how do they intend to do this?\n\\end{enumerate}\n\n\n\n\\end{document}\n", "meta": {"hexsha": "2bac9add388b21bfe770639a35ae87466540fd4b", "size": 6841, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "summaries/summ_1/eval_mets.tex", "max_stars_repo_name": "SamShowalter/CS272-NLP", "max_stars_repo_head_hexsha": "6914064018fbcd416c2f9b24ee105319d02b3cb2", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "summaries/summ_1/eval_mets.tex", "max_issues_repo_name": "SamShowalter/CS272-NLP", "max_issues_repo_head_hexsha": "6914064018fbcd416c2f9b24ee105319d02b3cb2", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "summaries/summ_1/eval_mets.tex", "max_forks_repo_name": "SamShowalter/CS272-NLP", "max_forks_repo_head_hexsha": "6914064018fbcd416c2f9b24ee105319d02b3cb2", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 60.0087719298, "max_line_length": 614, "alphanum_fraction": 0.7159771963, "num_tokens": 1900, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6406358411176238, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3477776760270927}} {"text": "\\chapter{Background}\n\\label{ch:02-background}\n\nWe can make human language manageable to computers by using a sort of different methods and techniques. This is the bread and butter of any NLP researcher. Here, following the NLP community, we will focus only in the techniques derived from the machine learning field. In this chapter, we will present some abstract machine learning models follow by some applications in NLP. \n\n\\section{Machine Learning}\n\n\\textit{Machine learning} is the branch of computer science that deals with programs that can improve with experience (i. e., learn). Machine learning is divided into three main subareas: \\textit{supervised learning}, \\textit{reinforcement learning} and \\textit{unsupervised learning}.\n\nWe will work with supervised learning only. In this setting, we assume that the process of interest is defined by an unknown function $g:X\\rightarrow Y$. We try to approximate $g$ by changing the parameters of a function $f$ through an optimization process. The optimization is done by defining an \\textit{error function} that evaluates how well $f$ approximates $g$ in the part of $g$ that we have access: the training data $D = \\{(\\vect{x}^{(1)}, \\vect{y}^{(1)}), \\dots ,(\\vect{x}^{(N)}, \\vect{y}^{(N)})\\}$ (where $g(\\vect{x}^{(i)})=\\vect{y}^{(i)}$).\n\n\n$f$ can be a function from any family of models. One family that is having a lot of success for language tasks is the \\textit{neural network} family.\n\n\\section{Neural Networks}\n\nA neural network is a non-linear function $f(\\vect{x}; \\vect{\\theta})$. It is defined by a collection of parameters $\\vect{\\theta}$ and a collection of non-linear transformations. It is usual to represent $f$ as a compositions of functions:\n\n\\begin{align}\nf(\\vect{x}; \\vect{\\theta}) &= f^{(2)}(f^{(1)}(\\vect{x}; \\vect{W}_1, \\vect{b}_1); \\vect{W}_2, \\vect{b}_2)\\\\\n&= softmax(\\vect{W}_2 (\\sigma(\\vect{W}_1\\vect{x} + \\vect{b}_1)) + \\vect{b}_2)\n\\end{align}\n\n\nThe output of these intermediary functions are referred as \\textit{layers}. So in the example above, $\\vect{x}$ (the output of the identity function) is the \\textit{input layer}, $f^{(1)}(\\vect{x}; \\vect{W}_1, \\vect{b}_1)$ is the \\textit{hidden layer} and $f^{(2)}(f^{(1)}(\\vect{x}; \\vect{W}_1, \\vect{b}_1); \\vect{W}_2, \\vect{b}_2)$ is the \\textit{output layer}. Since each layer is a vector, we normally speak about the \\textit{dimension} of a layer. For historical reasons we also say that each entry on a layer is a \\textit{node} or a \\textit{neuron}. Models with a large number of hidden layers are called \\textit{deep models}, for this reason the name \\textit{deep learning} is used. \n\n\\par A neural network is a function approximator: it can approximate any Borel measurable function from one finite dimensional space to another with any desired nonzero amount of error. This theoretical result is know as the \\textit{universal approximation theorem} \\cite{Cybenko}. Without entering in the theoretical concepts, it suffice to note that the family of Borel mensurable functions include all continuous functions on a closed and bounded subset of $\\mathbb{R}^n$.\n\n\n\nDifferent deep learning architectures are used in NLP: \\textit{convolutional architectures} have a good performance in tasks were it is required to find a linguistic indicator regardless of its position (e.g., document classification, short-text categorization, sentiment classification, etc); high quality word embeddings can be achieved with models that are a kind of \\textit{feedforward neural network} \\cite{Mikolov23}. But for a variety of works in natural language we want to capture regularities and similarities in a text structure. That is way \\textit{recurrent} and \\textit{recursive} models have been widely used in the field. Here we are focused on generative models and since recurrent models have been producing very strong results for language modeling \\cite{goldberg15}, we will concentrate on them.\n\n\n\\section{Recurrent Neural Network}\n\\label{sec:RNN}\n\n\n\\textit{Recurrent neural network} (RNN) is a family of neural network specialized in sequential data $\\vect{x}^{(1)}, \\dots, \\vect{x}^{(\\tau)}$. As a neural network, a RNN is a parametrized function that we use to approximate one hidden function from the data. What makes RNN unique is a recurrent definition of one of its hidden layer:\n\n\\begin{equation}\n\\vect{h}^{(t)} = g(\\vect{h}^{(t-1)}, \\vect{x}^{(t)}; \\vect{\\theta})\n\\end{equation}\n\n$\\vect{h}^{(t)}$ is called \\textit{state}, \\textit{hidden state}, or \\textit{cell}.\n\n\n\\par This recurrent equation can be unfolded for a finite number of steps $\\tau$. For example, when $\\tau =3$:\n\n\\begin{align}\n\\vect{h}^{(3)}& = g(\\vect{h}^{(2)}, \\vect{x}^{(3)}; \\vect{\\theta})\\\\\n & = g(g(\\vect{h}^{(1)}, \\vect{x}^{(2)}; \\vect{\\theta}), \\vect{x}^{(3)}; \\vect{\\theta})\\\\\n & = g(g(g(\\vect{h}^{(0)}, \\vect{x}^{(1)}; \\vect{\\theta}), \\vect{x}^{(2)}; \\vect{\\theta}), \\vect{x}^{(3)}; \\vect{\\theta})\\\\\n\\end{align}\n\nUsing a concrete example consider the following classification model define by the equations:\n\n\\begin{equation}\nf(\\vect{x}^{(t)}, \\vect{h}^{(t-1)}; \\vect{V}, \\vect{W}, \\vect{U}, \\vect{c}, \\vect{b}) = \\vect{\\hat{y}}^{(t)}\n\\end{equation}\n \\vspace{0.2cm}\n\\begin{equation}\n\\vect{\\hat{y}}^{(t)} = softmax(\\vect{V} \\vect{h}^{(t)} + \\vect{c})\n\\end{equation}\n\\vspace{0.2cm}\n \\begin{equation}\n\\vect{h}^{(t)} = g(\\vect{h}^{(t-1)}, \\vect{x}^{(t)}; \\vect{W},\\vect{U}, \\vect{b})\n\\end{equation}\n\\vspace{0.2cm}\n\\begin{equation}\n\\vect{h}^{(t)} = \\sigma(\\vect{W} \\vect{h}^{(t-1)} + \\vect{U} \\vect{x}^{(t)} + \\vect{b})\n\\end{equation}\n\n\nThis kind of model can create an output $\\vect{\\hat{y}}^{(t)}$ at each time $t$, or the model can produce a single output $\\vect{\\hat{y}}$ after processing an entire input sequence. This choice depends on the learning problem that is being modeled.\n\n\nWith the model's prediction at hand, we can use a loss function (like cross entropy for the classification problem) and apply the back-propagation algorithm to optimize the model. These models look complex, but it quite straightforward to compute the gradients \\cite[p.~374]{DeepLearningbook}.\n\nAlthough this kind of deep learning model is very useful, it presents a severe flaw. When computing the gradients there is a lot of repeated matrix multiplication using the recurrent weight matrix (in the example above, the matrix $\\vect{W}$). Depending on some configurations of this matrix \\textit{the gradients may vanish or explode exponentially with respect to the number of time steps}.\n\nThus, handing long-term dependencies became a problem when using RNNs. Different solutions were proposed, the most effective results came from some modifications of this model. We will present the two most famous modifications: \\textit{the gated recurrent unit} and \\textit{the long short-term memory}. \n\n\n\\section{Gated Recurrent Unit}\n\\label{sec:GRU}\n\nTo capture long-term dependencies on a RNN the authors of the paper \\cite{ChungGCB14} proposed a new architecture called \\textit{gated recurrent unit} (GRU). This model was constructed to make each hidden state $\\vect{h}^{(t)}$ to adaptively capture dependencies of different time steps. It work as follows, at each step $t$ one candidate for hidden state is formed:\n\n\\begin{equation}\n\\vect{\\widetilde{h}}^{(t)} = tahn(\\vect{W} (\\vect{h}^{(t-1)} \\odot \\vect{r}^{(t)}) + \\vect{U} \\vect{x}^{(t)} + \\vect{b})\n\\end{equation}\n\nwhere $\\vect{r}^{(t)}$ is a vector with values in $[0, 1]$ called a \\textit{reset gate}, i.e., a vector that at each entry outputs the probability of reseting the corresponding entry in the previous hidden state $\\vect{h}^{(t-1)}$. Together with $\\vect{r}^{(t)}$ we define an \\textit{update gate}, $\\vect{u}^{(t)}$. It is also a vector with values in $[0, 1]$. Intuitively we can say that this vector decides how much on each dimension we will use the candidate update. Both $\\vect{r}^{(t)}$ and $\\vect{u}^{(t)}$ are defined by $\\vect{h}^{(t-1)}$ and $\\vect{x}^{(t)}$; they also have specific parameters:\n\n\\begin{equation}\n\\vect{r}^{(t)} = \\sigma(\\vect{W}_{r} \\vect{h}^{(t-1)} + \\vect{U}_{r} \\vect{x}^{(t)} + \\vect{b}_{r})\n\\end{equation}\n\n\n\\begin{equation}\n\\vect{u}^{(t)} = \\sigma(\\vect{W}_{u} \\vect{h}^{(t-1)} + \\vect{U}_{u} \\vect{x}^{(t)} + \\vect{b}_{u})\n\\end{equation}\n\nAt the end the new hidden state $\\vect{h}^{(t)}$ is defined by the recurrence:\n\n\\begin{equation}\n\\vect{h}^{(t)} = \\vect{u}^{(t)} \\odot \\vect{\\widetilde{h}}^{(t)} + (1 - \\vect{u}^{(t)}) \\odot \\vect{h}^{(t-1)} \n\\end{equation}\n\nNote that the new hidden state combines the candidate hidden state $\\vect{\\widetilde{h}}^{(t)}$ with the past hidden state $\\vect{h}^{(t-1)}$ using both $\\vect{r}^{(t)}$ and $\\vect{u}^{(t)}$ to adaptively copy and forget information.\n\n\\section{Long Short-Term Memory}\n\\label{sec:LSTM}\n\n\\textit{Long short-term memory} (LSTM) is one of the most applied versions of the RNN family of models. Historically it was developed before the GRU model, but conceptually we can think in the LSTM as an expansion of the model presented in the last session. Because of notation differences they can look different. LSTM is also based on parametrized gates; in this case three: the \\textit{forget gate}, $\\vect{f}^{(t)}$, the \\textit{input gate}, $\\vect{i}^{(t)}$, and the \\textit{output gate}, $\\vect{o}^{(t)}$. The gates are defined only by $\\vect{h}^{(t-1)}$ and $\\vect{x}^{(t)}$ with specific parameters:\n\n\n\\begin{equation}\n\\vect{f}^{(t)} = \\sigma(\\vect{W}_{f} \\vect{h}^{(t-1)} + \\vect{U}_{f} \\vect{x}^{(t)} + \\vect{b}_{f})\n\\end{equation}\n\n\\begin{equation}\n\\vect{i}^{(t)} = \\sigma(\\vect{W}_{i} \\vect{h}^{(t-1)} + \\vect{U}_{i} \\vect{x}^{(t)} + \\vect{b}_{i})\n\\end{equation}\n\n\\begin{equation}\n\\vect{o}^{(t)} = \\sigma(\\vect{W}_{o} \\vect{h}^{(t-1)} + \\vect{U}_{o} \\vect{x}^{(t)} + \\vect{b}_{o})\n\\end{equation}\n\nIntuitively $\\vect{f}^{(t)}$ should control how much informative will be discarded, $\\vect{i}^{(t)}$ controls how much information will be updated, and $\\vect{o}^{(t)}$ controls how munch each component should be outputted. A candidate cell, $\\tilde{\\vect{c}}^{(t)}$ is formed:\n\n\\begin{equation}\n\\tilde{\\vect{c}}^{(t)} = tahn(\\vect{W} \\vect{h}^{(t-1)} + \\vect{U} \\vect{x}^{(t)} + \\vect{b})\n\\end{equation}\n\nAnd a new cell $\\vect{c}^{(t)}$ is formed by forgetting some information of the previous cell $\\tilde{\\vect{c}}^{(t-1)}$ and by adding new values from $\\tilde{\\vect{c}}^{(t)}$ (scaled by the input gate)\n\n\\begin{equation}\n\\vect{c}^{(t)} = \\vect{f}^{(t)} \\odot \\vect{c}^{(t-1)} + \\vect{i}^{(t)} \\odot \\tilde{\\vect{c}}^{(t)}\n\\end{equation}\n\nThe new hidden state, $\\vect{h}^{(t)}$, is formed by filtering $\\vect{c}^{(t)}$:\n\n\\begin{equation}\n\\vect{h}^{(t)} = \\vect{o}^{(t)} \\odot tanh(\\vect{c}^{(t)})\n\\end{equation}\n\nUntil now we have presented general deep learning theory, now we will focus on the specificities of these models applied to natural language problems.\n\n\n\\section{Language model}\n\nWe call \\textit{language model} a probability distribution over sequences of tokens in a natural language.\n\n\\[\nP(x_1,x_2,x_3,x_4) = p\n\\]\n\nThis model is used for different NLP tasks such as speech recognition, machine translation, text auto-completion, spell correction, question answering, summarization and many others.\n\nThe classical approach to a language model was to use the chain rule of probability and a Markovian assumption, i.e., for a specific $n$ we assume that:\n\n\\begin{equation}\nP(x_1, \\dots, x_T) = \\prod_{t=1}^{T} P(x_t \\vert x_1, \\dots, x_{t-1}) = \\prod_{t=1}^{T} P(x_{t} \\vert x_{t - (n+1)}, \\dots, x_{t-1})\n\\end{equation} \n\n\nThis gave raise to models based on $n$-gram statistics. The choice of $n$ yields different models; for example, the \n\\textit{unigram} language model ($n=1$) is defined as: \n\\begin{equation}\nP_{uni}(x_1, x_2, x_3, x_4) = P(x_1)P(x_2)P(x_3)P(x_4)\n\\end{equation}\n\nwhere $P(x_i) = count(x_i)$ and $count$ is a function that counts tokens occurrence in a corpus.\\\\\n\nSimilarly the \\textit{bigram} language model ($n=2$) is defined as: \n\\begin{equation}\nP_{bi}(x_1,x_2,x_3,x_4) = P(x_1)P(x_2\\vert x_1)P(x_3\\vert x_2)P(x_4\\vert x_3)\n\\end{equation} \nwhere\n\\begin{equation}\nP(x_i\\vert x_j) = \\frac{count(x_i, x_j)}{count(x_j)}\n\\end{equation} \n\nWith these basic statistics we can already define useful language models. It is observed that higher $n$-grams yields better performance. This comes with a price though, higher $n$-grams requires great amounts of memory \\cite{Heafield}. For this motive $n$-grams based language models that are trained on large corpora uses at most $5$-grams. \n\nSince \\cite{Mikolov11} the landscape has change, instead of using one approach that is specific for the language domain, we can use a general model for sequential data prediction, a RNN. The RNN's ability to deal with unrestricted size sequence input permits to abandon the $n$-gram model's restricted context assumption.\n\nTo understand the language model task as a machine learning task, some details should be clear. \n\nFirst, the learning task is to estimate the probability distribution \n\n\\begin{equation}\n\\label{languagedistri}\nP(x_{n} = \\text{word}_{j^{*}} | x_{1}, \\dots ,x_{n-1})\n\\end{equation}\n\nfor any $(n-1)$-sequence of words $x_{1}, \\dots ,x_{n-1}$.\n\n\nSecond, the function $f$ being used to approximate \\ref{languagedistri} is trained on a corpus in the following way: each word $x_t$ of the corpus will be used as input to $f$ (in the form of an one-hot vector), and the immediate subsequent word, say $x_{t+1}$, will be used as a target. The training is done by minimizing the cross entropy loss between the model's output and the probability distribution given by the target.\n\nOne example is in order. Suppose our corpus $\\corpus$ is compose only by the lines below:\n\n\\begin{quote}\nYes, here we go again, give you more, nothing lesser\\\\\nBack on the mic is the anti-depressor\\\\\nAd-Rock, the pressure, yes, we need this\\\\\nThe best is yet to come, and yes, believe this\\\\\n\\end{quote}\n\n\nFrom this corpus we can construct a vocabulary list $\\Vocab$ as follows: after preprocessing the text we create a list of the most frequent words (in this case we can take all words) with the size $V$ (here $V=27$). Hence, we can treat each word in the text either by an index referring to the word position on $\\Vocab$ or as a one-hot vector that codifies this index (e.g., \"the\" would be identified with $0$, \"yes\" with $1$, \"we\" with $2$, and so on).\n\nThen, the dataset is the collection of words \n\\[\nD = \\{(, \\text{Yes}), (\\text{Yes}, \\text{here}), (\\text{here}, \\text{we}),\\dots,(\\text{believe}, \\text{this}), (\\text{this}, )\\}\n\\]\nwhere $$ is the \"end of sentence\" token (also a member of $\\Vocab$).\n\nA simple recurrent language model $f(\\vect{x}^{(t)}, \\vect{\\theta})$ is defined by the following equations:\n\n\\begin{equation}\n\\vect{e}^{(t)} = \\vect{E}\\vect{x}^{(t)}\n\\end{equation}\n\\vspace{0.2cm}\n\\begin{equation}\n\\vect{h}^{(t)} = \\sigma(\\vect{W}\\vect{h}^{(t-1)}+ \\vect{U}\\vect{e}^{(t)}+ \\vect{b})\n\\end{equation}\n\\vspace{0.2cm}\n\\begin{equation}\nf(\\vect{x}^{(t)}, \\vect{\\theta}) = \\vect{\\hat{y}}^{(t)} = softmax(\\vect{V}\\vect{h}^{(t)} + \\vect{c})\n\\end{equation}\n\nwhere $\\vect{E} \\in \\mathbb{R}^{d,V}$ is the matrix of word embeddings, $\\vect{x}^{(t)} \\in \\mathbb{R}^{V}$ is one-hot word vector at time step $t$, $\\vect{y}^{(t)} \\in \\mathbb{R}^{V}$ is the ground truth at time step $t$ (also a one-hot word vector) and $d$ is the size of the word embeddings.\n\nFor each word $x_t$ let $j_t$ be the index of the subsequent word, so at each time $t$ the point-wise loss is:\n\n\\begin{align}\n\\label{lossCE}\nL^{(t)}(\\vect{\\theta}) &= CrossEntropy(\\vect{y}^{(t)},\\vect{\\hat{y}}^{(t)})\\\\\n &= - \\log({\\vect{\\hat{y}}^{(t)}}_{j_t})\\\\\n &= - \\log P(x_{t+1} = \\text{word}_{j_t}|x_{1}, \\dots, x_{t})\\\\\n &= - \\log P(x_{t+1}|x_{1}, \\dots, x_{t})\n\\end{align}\n\nThe loss $L$ is the mean of all point-wise losses\n\n\\begin{equation}\nL(\\vect{\\theta})=\\frac{1}{T}\\sum_{t=1}^{T}L^{(t)}(\\vect{\\theta})\n\\end{equation}\n\nWith the loss function defined, we apply some optimization algorithm like \\textit{stochastic gradient descent} to choose the optimal parameters for the language model:\n\n\\begin{equation}\n\\vect{\\theta}^{*} = \\argmin_{\\vect{\\theta}} L(\\vect{\\theta})\n\\end{equation}\n\nBecause of the historical connections with information theory, the \\textit{perplexity} ($PP$) metric is often used to evaluate a language model. This metric can be thought as the weighted average branching factor of a language.\n\nGiven $\\corpus = x_1, x_2, \\dots, x_T$, we define the perplexity of $\\corpus$ ($PP(\\corpus)$) as:\n\n\\begin{align}\nPP(\\corpus) &= P(x_1, x_2, \\dots, x_T)^{-\\frac{1}{T}}\\\\\n &= \\sqrt[T]{\\frac{1}{P(x_1, x_2, \\dots, x_T)}}\\\\\n &= \\sqrt[T]{\\prod_{i=1}^{T}\\frac{1}{P(x_i \\vert x_1,\\dots, x_{i-1})}}\n\\end{align}\n\nUsing \\ref{lossCE} we can relate cross entropy loss and perplexity:\n\n\\begin{align}\n L(\\vect{\\theta}) &=\\frac{1}{T} \\sum_{t=1}^{T} L^{(t)}(\\vect{\\theta})\\\\\n &=\\frac{1}{T} \\sum_{t=1}^{T} - \\log P(x_{t+1}|x_{1}, \\dots, x_{t})\\\\\n &=\\frac{1}{T} \\sum_{t=1}^{T} \\log ((\\frac{1}{P(x_{t+1}|x_{1}, \\dots, x_{t})})\\\\\n &= \\log\\left( \\sqrt[T]{\\prod_{i=1}^{T}\\frac{1}{P(x_i \\vert x_1,\\dots, x_{i-1})}} \\right)\\\\\n &= \\log(PP(\\corpus))\n\\end{align}\n\nHence,\n\n\\begin{equation}\n2^{L(\\vect{\\theta})} = PP(\\corpus)\n\\end{equation}\n\nThus, by finding the parameters that minimize the cross entropy error we are also minimizing the perplexity of the language model. \n\n\\section{Sequence-to-Sequence}\n\\label{sec:Seq2seq}\n\nThere is one powerful application of RNN based language model. The authors from \\cite{Sustskever} used two RNNs to create an end-to-end translation model that is now know as \\textit{the encoder-decoder} or \\textit{the sequence-to-sequence} (seq2seq) architecture. This architecture is define as follows: let $\\vect{x}^{(1)}, \\dots, \\vect{x}^{(n)}$ be a source sentence in the one-hot representation, let $\\vect{y}^{(1)}, \\dots, \\vect{y}^{(m)}$ be the target sentence also in the one-hot format. $f_{enc}$ (the \\textit{encoder}) is a RNN with the sole purpose of creating a vector representation of input language's sequences. $f_{dec}$ (the \\textit{decoder}) is a language model for the target language. These models are trained together mapping source sentences to target sentences.\n\nFor example, suppose the training pair $(\\vect{x}^{(1)}, \\dots, \\vect{x}^{(n)}, \\vect{y}^{(1)}, \\dots, \\vect{y}^{(m)})$ is (\"Nas tardes de fazenda há muito azul demais\", \"In the farm’s afternoons there is too much blue\"). Here we want the model to translate one Portuguese sentence to an English one. We first encode the Portuguese sentence in the vector $\\vect{s}$, i.e.,\n\n\\begin{equation}\n\\vect{s} = f_{enc}(\\vect{x}^{(n)}, \\vect{h}^{(n-1)})\n\\end{equation}\n\nThen using the control English sentence as a target, at each time $t$ we compute the cross entropy error between the decoder prediction $f_{dec}(\\vect{y}^{(t)}, \\vect{\\tilde{h}}^{(t-1)})$ and the target $\\vect{y}^{(t+1)}$ ($\\vect{y}^{(0)}$ is the $$ token). The decoder uses the vector representation of the source sentence $\\vect{s}$ as an initial hidden state (i.e., $\\vect{\\tilde{h}}^{(0)} = \\vect{s}$). The goal of this model is to to approximate the probability distribution over the tokens from the target language given the sentence of the source language, i.e.,\n\n\\begin{equation}\n\\vect{\\tilde{h}}^{(t)} = f_{dec}(\\vect{y}^{(t)}, \\vect{\\tilde{h}}^{(t-1)})\n\\end{equation}\n\n\\begin{equation}\n\\label{decpred}\np(y_t | y_1, \\dots, y_{t-1}, x_1, \\dots, x_{n}) = softmax(\\vect{W}_{s} \\vect{\\tilde{h}}^{(t)} + \\vect{b}_s)\n\\end{equation}\n\nOne limitation of this architecture is that the source sentence, in some cases, has more features than the decoder embedding $\\vect{s}$ can properly summarize. To address that some attention mechanisms were introduced.\n\n\\section{Attention}\n\\label{sec:Attention}\n\nThe attention-based models are models built on top of the seq2seq architecture. The encoding part continues the same as before, but now at each time $t$ a context vector $\\vect{c}^{(t)}$ is defined to capture relevant source-side information to help the prediction of the current target word $\\vect{y}^{(t)}$. Once $\\vect{c}^{(t)}$ is constructed the attention hidden state is defined as: \n\n\\begin{equation}\n\\vect{\\tilde{h}}^{(t)} = tahn(\\vect{W}_c[\\vect{c}^{(t)};\\vect{h}^{(t)}])\n\\end{equation}\n\nWith the attention hidden state defined, the model's prediction is the same as the one defined in \\ref{decpred}.\n\nThe core of this technique is the definition of $\\vect{c}^{(t)}$. There are different strategies available, here we will focus only on one: \\textit{global attention}.\n\nLet $\\vect{a} \\in \\mathbb{R}^{m,n}$. We will use this matrix as an alignment matrix, i.e., at the end of the training $\\vect{a}_{ts}$ should reflect the probability of the source representation $\\vect{h}^{(s)}$ be relevant for the output $\\hat{y}^{(t)}$. We define $\\vect{a}_{ts}$ as\n\n\n\\begin{equation}\n\\vect{a}_{ts} = \\frac{exp(score(\\vect{\\tilde{h}}_t,\\vect{h}_s))}{\\sum_j exp(score(\\vect{\\tilde{h}}_t,\\vect{h}_j))}\n\\end{equation}\n\nWhere $score$ is a content-based function that can have different implementations: \n\n\\begin{equation}\nscore(\\vect{\\tilde{h}}_t,\\vect{h}_s) = \\begin{cases}\n\\vect{\\tilde{h}}_t ^{\\top}\\vect{h}_s\\\\\n\\vect{\\tilde{h}}_t ^{\\top}\\vect{W}_a \\vect{h}_s\\\\\n\\vect{v}_a ^{\\top}tahn(\\vect{W}_a[\\vect{\\tilde{h}}_t;\\vect{h}_s])\\\\\n\\end{cases}\n\\end{equation}\n\nAt the end, a global context vector $\\vect{c}^{(t)}$ is computed as the weighted average, according to $\\vect{a}_t$ over all source states:\n\n\\begin{equation}\n\\vect{c}^{(t)} = \\sum_{s} \\vect{a}_{ts}\\vect{h}^{(s)}\n\\end{equation}\n\n\n\n", "meta": {"hexsha": "c9d1d8a38f1e9a7bee255501fa1df70b9d4dd285", "size": 21572, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/chapters/ch02-background.tex", "max_stars_repo_name": "felipessalvatore/quali", "max_stars_repo_head_hexsha": "500dc69a10ccd0320cc90c8c51dc13bb8097584e", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/chapters/ch02-background.tex", "max_issues_repo_name": "felipessalvatore/quali", "max_issues_repo_head_hexsha": "500dc69a10ccd0320cc90c8c51dc13bb8097584e", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/chapters/ch02-background.tex", "max_forks_repo_name": "felipessalvatore/quali", "max_forks_repo_head_hexsha": "500dc69a10ccd0320cc90c8c51dc13bb8097584e", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.8108882521, "max_line_length": 815, "alphanum_fraction": 0.6913128129, "num_tokens": 6812, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791787121629465, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34754702457537057}} {"text": "\\section{Learning Motion Primitive using Kinesthetic Demonstrations}\n\\label{sec:Learning motion primitive}\nThe main idea of the work is to infer the relevant set of features,\nwhich describe the demonstrated motion primitive.\nBased on the learnt relevant features, during the execution phase \nthe robot tries to predict the goal of the motion primitive, starting \nfrom a new initial condition.\n\nThe demonstrations consist of recording two states of the world\ncalled as the start state and the end state.\nSo each demonstrations have 2 snapshots of the world in the initial state and the final state.\nBased on these snapshots we try to infer the intent of the demonstrations. \n\nThe motion primitive in our representation consist of pre-conditions and the post-conditions.\nThe execution of each motion primitive depends on the features used to explain it.\nEach motion primitive is defined by a set of features which are a subset of the complete feature space, \nwhich are also relevant for the robot to execute the respective motion primitive.\n\n\\subsection{Features}\nFeatures are the basic building block, which are used to describe a motion primitive.\nA Feature $f$ can be defined as any quantitative parameter of the world $W$.\nFor example the pose of the robot, color of the box, distance between box and robot,\ndisplacement of the tooltip in time.\n\nFeatures are broadly classified into :\n\\begin{enumerate}\n \\item features describing object properties $f(O)$\n \\begin{enumerate}\n \\item features describing robot properties $f(O_r)$ (eg : pose of the robot)\n \\item features describing environment properties $f(O_e)$(eg: color of the box)\n \\end{enumerate}\n \\item features describing relation between object properties $g(O_1, O_2)$\n \\begin{enumerate}\n \\item relation between different objects $g(O_1, O_2)$ (eg : distance between box and robot)\n \\item relation between same object in time $g(O_{1i}, O_{1f}) $(eg : displacement of tooltip)\n \\end{enumerate}\n\\end{enumerate}\n\n\nThus the collection of featues can be defined as \n\\begin{equation}\n \\begin{aligned}\n F &= f(O) + g(O_1 , O_2) \\\\\n &= (f(O_r) + f(O_e)) + (g(O_1, O_2) + g(O_{1i}, O_{1f}))\n \\end{aligned}\n\\end{equation}\n\nAs explained in section \\ref{sec:Proposed motion primitive}\nthe proposed motion primitive is defined using pre and post conditions of the features.\n\\begin{equation}\n \\begin{aligned}\n & \\text{Motion primitive} := (f_s, f_e ) \\\\\n & where \\nonumber \\\\ \n & f_s = \\text{features in start of demonstrations} \\nonumber\\\\\n & f_e = \\text{features in end of demonstrations} \\nonumber\n \\end{aligned}\n\\end{equation}\n\n\n\n\\subsection{Modelling Motion Primitive}\n\nThe learning problem can be considered as a supervised learning problem, where \nthe model has to learn from the set of demonstrations to predict the output\n$f_e$ when a new input $f_s$ is provided as explained in figure \\ref{model}.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.40]{images/model.png}\n\\caption{Model of motion primitive}\n\\label{model}\n\\end{figure}\n\nWhen the number of training set is limited,learning on\n all the features of the world is not a feasible solutions.\n The approach taken here is to find the relevant features on which the \nlearning can be made most effectively.\nThe prespective adopted in finding the relevant features is that in multiple\ndemonstrations of the same action, some features will share similar values \nin all the demonstrations. Since we are demonstrating a single action\nthere has to be consistency in some features of the action.\nWe need to identify these consistent features and these become the relevant\nfeatures. The consistency increases the relevance of the feature to\nsucessfully predict from new unseen start states.\n\nWe will consider a bivariate probability density function $\\phi_i$ by\nconsidering 2 variables, the value of a feature in start of demonstration and\nits value at the end of the demonstration.\n\nLet $s_i$ be the $i^{th}$ feature of $f_s$, \nand $e_i$ be the $i^{th}$ feature of $f_e$, \\\\\nThe bivariate distribution is given by :\n\\begin{equation}\n \\phi_i(s_i , r_i) = \\eta_i I_i ( \\lfloor s_i \\rfloor, \\lfloor e_i \\rfloor)\n\\end{equation}\nwhere operator $\\lfloor . \\rfloor$ is a quantization operator, that returns the bin unit in \nthe histogram $I_i$, that corresponds to the feature.\n\nThe bi-variate distribution is an appropriate criteria to comment on the\nrelevance of the feature. Using the distribution we can conclude on the\nconvergence of the feature. A feature which has converged, the final values\nwill lie on a straight line in the distribution. If all the final values lie on\na straight line it ensures that whatever maybe the initial value the end values\nalways remain constant. The constant value in all the demonstration concludes\nthat the corresponding feature is relevant with respect to the intent of the\nmotion primitive.\nThis is explained in the figure \\ref{fig:feature distribution}. Higher the\ndegree of convergence, higher the relevance of the feature for the motion\nprimitive.\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\includegraphics[scale=0.5]{images/arm_joint_1JoinPDF.png} \n \\caption{Distribution of the feature describing arm joint 1.}\n \\label{sub fig 1}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.4\\textwidth}\n \\includegraphics[scale=0.5]{images/d_linear_m0_gripperJoinPDF.png} \n \\caption{Distribution of feature describing distance of object 0 and tooltip. }\n \\label{sub fig 2}\n \\end{subfigure}\n \\caption[Bivariate distribution of features]{An example of bivariate\n distributions of two features for the \\textit{move} arm relative to object motion\n primitive.The x-axis is the end values of demonstration while the \n y-axis is the start value of the demonstration. The dots are the \n feature values across 12 demonstrations.\n Based on the distribution in \\ref{sub fig 1}, we can say that\n there is no certaininty in the end values. This \n results in lower relvance to the motion primitive . Based on the\n distribution in \\ref{sub fig 2},we can say that the end states are\n more concentrated.This ensures that for any value in the start state\n we are sure that the end values always remain same.\n This results in a higher relevance with respect to the motion\n primitive.}\\label{fig:feature distribution}\n\\end{figure}\n\nTo calculate this relevance in the distribution, we use two different \nmeasuring methods. 1) Entropy 2) Conditional entropy.\nWe compute the \nentropy $H_i$ and conditional entropy $CH_i$ of the bi-variate distribution $\\phi_i(s_i, r_i)$.\n\nEntropy of a discrete random variable is given by,\n\\begin{equation}\n H(X) = - \\sum P(X) \\log P(X)\n\\end{equation}\n\n\nFor a pair of discrete random variables X and Y, which are co related \nconditional entropy of X given Y $h(X|Y)$ is given by :\n\\begin{equation}\n H(X|Y) = - \\sum _{k = -K}^{K} p_X(x|y) p_Y(y|x) \\log \\frac{p_Y(y|x)}{p_X(x|y) p_Y(y|x)}\n\\end{equation}\n\n\nBased on the entropy or conditional entropy we define the model relevance of a motion primitive.\n\\begin{equation}\n p(f | \\theta ) = \\prod_i e^{-E_i}\n\\end{equation}\nwhere $\\theta$ is the set of all $\\phi$, and $E_i$ can be $H_i$ or $CH_i$\n\n\n\n\\subsection{Expert knowledge base}\nIn our work we create the knowledge base based on expert knowledge of the tasks being performed.\nAn individual subset of feature space $F$ is called as a template $t$. $t_i \\subset F $ .\nFor example the template $t_1$ contains all the features describing the joint angles of the robot, template $t_2$ contains features which describes \ndistance of tooltip to manipulated object.\n\nA collection of the templates is called as a knowledge base $K$.\n\n\n$K = t_1 \\and t_2 \\ldots t_n $\n\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.50]{images/feature_space.png}\n\\caption[Feature space venn diagram]{Relation between feature space F, knowledge base K and the templates t}\n\\label{}\n\\end{figure}\nThus for each template $t_i$ we have created a low dimension subset of the feature space $F$.\nOur aim is not to find minimum set of features but to find relevant set of features which can describe an action.\n\n\nThe novelty in our approach is the representation of the expert knowledge base.\nThe knowledge base has been structured using the effect metrics. \nThe templates in the knowledge base are organised on the basis of the effect metrics.\nThe structured nature of the knowledge base helps in determining the relevant features with\nless number of demonstrations.\n\\subsubsection{Effect Metrics}\nEffects are defined as changes to the robot-world relationship and/or to positions, orientations\nand states of external objects and robot \\cite{alissandrakis_action_2006}\n\nIn this work we only consider changes to the robot-world relationship and changes to positions, orientations and states of the robot. Based on the orientation and position of the \nrobot in environment 2 types of effect metrics can be used, \\textit{position} and \\textit{orientation}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.70]{images/position_effect_metrics.png}\n\\caption[Position effect metrics]{Position effect metrics.\n To measure the change in the relation of change in positions between \nrobot and environment objects \n\\textit{relative displacement, absolute position, relative position and mirror position}\n effect metrics can be used. First row shows the demonstration and\n their resulting effects. The second rows represents the\n corresponding object (in different workspace ) how it needs to be\n moved (from dashed to solid outline) by an imitator to match the \ncorresponding effects according to the metric. The grey triangle\n are superimposed to show the relative position of the objects are \nthe same in final state. \\cite{alissandrakis_action_2006} }\n\\label{position effect metrics}\n\\end{figure}\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.70]{images/angular_effect_metrics.png}\n\\caption[Orientation effect metrics]{Orientation effect metrics. To measure the change in the relation of change in orientation between robot and environment objects \\textit{relative rotation, absolute orientation, relative orientation and mirror rotation} effect metrics can be used.First row shows the demonstration and their resulting effects. The second rows represents the corresponding object (in different workspace) how it needs to be moved. (from dashed to solid outline) by an imitator to match the corresponding effects according to the metric. The guide lines are superimposed to show the relative orientation of the objects are the same in final state. \\cite{ alissandrakis_action_2006}}\n\\label{orientation effect metrics}\n\\end{figure}\n\nThe different position metrics are \n\\begin{itemize}\n\t\\item relative displacement\n\t\\item absolute position\n\t\\item relative position\n\t\\item mirror position\n\\end{itemize}\n\nthe different orientation metrics are \n\\begin{itemize}\n\t\\item relative rotation\n\t\\item absolute orientation\n\t\\item relative orientation\n\t\\item mirror rotation\n\\end{itemize}\n\nDepending on the effect metric the same demonstration can be interpreted as different. \nThe example in figure \\ref{effect metrics} explains this.\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.8]{images/effect_position.png}\n\\caption[Effect metrics in imitation]{The figure illustrates three examples\n of the imitation based on a single demonstration. The left box is the\n demonstrated action. The right box is the possible final state based on\n the position metrics used. The grey triangles are superimposed to show\n the relative distance is maintained \\cite{alissandrakis_action_2006}}\n\\label{effect metrics}\n\\end{figure}\n\nBased on the effect metrics the knowledge base is organized in different templates which describe an action.\nThe templates developed in this work are categorized based on the effect metrics.\n\n\\subsection{Selecting template from the knowledge base}\nFor computing the relevance, we use a 3 stage procedure:\n\\begin{enumerate}\n \\item Comparing mean of start value and end value of the feature. $\\rightarrow$ To determine that the feature has changed.\n (This was used because features which were not changed in demonstration also have low entropy. So to remove these features this condition \n check was introduced.)\n \\item Comparing the standard deviation of start value and end value of the featue $\\rightarrow$ To determing the feature shows convergence in information.\n (This was used because there were features which changed and had low entropy but they were divergent in nature. So to remove the effect of \n these features this condition check was introduced as explained in figure \\ref{fig:box plot})\n \\item Calculating the entropy/conditional entropy of the final value of features whose values have changed.\n\\end{enumerate}\n\n\\begin{algorithm}[H]\n \\KwData{\\\\ $f_s$ : start value of the features \\\\\n $f_e$ : end value of the features\\\\\n K : knowledge base }\n \\KwResult{\\\\ relevance $L$ of each Template }\n R=0\\;\n \\For{each template $t$ in K}{\n \\For{each feature $f$ in $t$ }{\n read start values $s$ of feature $f$ from $f_s$\\;\n read end values $e$ of featue $f$ from $f_e$\\;\n \\If{mean(s) equals mean(e)}{\n $ R += 1$\\;\n }\n \\If{standard deviation(s) is less or equal to standard deviation(e)}{\n $ R += 1$\\;\n }\n $R += \\text{entropy(e)} or\\text{ conditional entropy}(e|s)$\\;\n }\n }\n \\caption{Algorithm for computing relevance of the templates in knowledge base}\n\\end{algorithm}\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[scale=0.25]{images/boxplot_same_mean.png} \n \\caption{}\n \\label{sub box 1}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[scale=0.25]{images/boxplot_noconvergence.png} \n \\caption{}\n \\label{sub box 2}\n \\end{subfigure}\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[scale=0.25]{images/boxplot_converged.png} \n \\caption{}\n \\label{sub box 3}\n \\end{subfigure}\n \\caption[Box plot of features]{Box plots of features. Each figure has 2 box plots.\nPlot I is of the start value of the feature while plot F is of the end value\nof the feature. Figure \\ref{sub box 1} is box plot of a featue whose mean values \nhave not changed for both start and end values of the feature .\nFigure \\ref{sub box 2} is the box plot of a feature whose mean value \nhas changed but the final values has not converged. \nFigure \\ref{sub box 3} is the box plot of a feature whose mean value \nhas changed and the final values have converged.} \\label{fig:box plot}\n\\end{figure}\n\n\nFor each $t_i \\in K $, we compute a score $\\beta_i$ that combines model fitting and the number of \nfeatures in the template.\n\\begin{equation}\n \\beta_i = -2 \\log (p (f | \\theta_i)) - \\alpha_i L_i\n\\end{equation}\nwhere $\\theta_i$ are the distributions related to the features of $t_i$\nand $L_i$ is the number of features in the template.\nThe first term of equation computes relevance and the second term, weighted by $\\alpha$, encourages the usage of templates\nconsisting of a large number of features $L_i$ .\n\n\nThe most relevant template $T^*$ is selected as :\n\\begin{equation}\n T^* = \\operatornamewithlimits{argmin}_i (\\beta_i)\n\\end{equation}\n", "meta": {"hexsha": "f91b19288cff1b49b648e3bb41210fa3e3dcdb20", "size": 15465, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/rnd_report_tex/Content.tex", "max_stars_repo_name": "deebuls/RecommenderSystemInRobotics", "max_stars_repo_head_hexsha": "2badd51dc0f196374a9195e346d85354b1e2e108", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tex/rnd_report_tex/Content.tex", "max_issues_repo_name": "deebuls/RecommenderSystemInRobotics", "max_issues_repo_head_hexsha": "2badd51dc0f196374a9195e346d85354b1e2e108", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2015-04-10T16:33:27.000Z", "max_issues_repo_issues_event_max_datetime": "2015-04-19T09:30:04.000Z", "max_forks_repo_path": "tex/rnd_report_tex/Content.tex", "max_forks_repo_name": "deebuls/RecommenderSystemInRobotics", "max_forks_repo_head_hexsha": "2badd51dc0f196374a9195e346d85354b1e2e108", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.1493902439, "max_line_length": 701, "alphanum_fraction": 0.7523440026, "num_tokens": 3742, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786991753931, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3475470179294239}} {"text": "\n% \\documentclass[draft,11pt]{article}\n\\chapter{Solving Laplacian\n Linear Equations}\n\n%\\allowdisplaybreaks\n\n%%% For this lecture\n%\\newcommand\\symset{S}\n%\\newcommand\\psdset{S_+}\n%\\newcommand\\pdset{S_{++}}\n\n%\\newcommand\\symsetn{\\symset^n}\n%\\newcommand\\psdsetn{\\psdset^n}\n%\\newcommand\\pdsetn{\\pdset^n}\n\n\n%\\newcommand{\\vcliq}[2]{\\textsc{Clique}\\!\\left({#1,#2}\\right)}\n%\\newcommand{\\vstar}[2]{\\textsc{Star}\\!\\left({#1,#2}\\right)}\n\n%%% added by Hongjie\n%\\newcommand{\\Hongjie}[1]{{\\color{red} Hongjie: #1}}\n%\\newcommand{\\vcliqsp}[2]{\\textsc{CliqueSample}\\!\\left({#1,#2}\\right)}\n\n%%% layout and code\n%\\usepackage[vlined, ruled]{algorithm2e}\n%\\usepackage{float}\n\n%\\begin{document}\n\\sloppy\n%\\lecture{9 --- Wednesday, April 22nd}\n%{Spring 2020}{Rasmus Kyng, Scribe: Hongjie Chen}{Solving Laplacian\n% Linear Equations}\n\n% \\todo{introduce stopped martingale}\n\n% \\todo{remarks, Tropp and Matrix Freedman and things to know}\n\n% \\paragraph{outline}\n% \\begin{itemize}\n% \\item objective\n% \\item precond\n% \\item algo\n% \\item martingale\n% \\item mat conc\n% \\end{itemize}\n\n\n\n\\section{Solving Linear Equations Approximately}\n\nGiven a Laplacian $\\LL$ of a connected graph and a demand vector\n$\\dd \\perp \\vecone$, we want to find $\\xx^*$ solving the linear equation\n$\\LL \\xx^* = \\dd$.\nWe are going to focus on fast algorithms for finding approximate (but\nhighly accurate) solutions.\n\nThis means we need a notion of an approximate solution.\nSince our definition is not special to Laplacians, we state it more\ngenerally for positive semi-definite matrices.\n\\begin{definition}\n Given PSD matrix $\\MM$ and $\\dd \\in \\ker(\\MM)^{\\perp}$, let\n $\\MM \\xx^* = \\dd$.\n We say that $\\xxtil$ is an $\\epsilon$-approximate solution to the\n linear equation $\\MM \\xx = \\dd$ if\n \\[\n \\norm{\\xxtil - \\xx^*}_{\\MM}^2 \\leq \\epsilon \\norm{\\xx^*}_{\\MM}^2.\n \\]\n\\end{definition}\n\n\\begin{remark}\n The requirement $\\dd \\in \\ker(\\MM)^{\\perp}$ can be removed,\n but this is not important for us.\n \\end{remark}\n\n\\begin{theorem}[Spielman and Teng (2004) \\cite{st04}]\nGiven a Laplacian $\\LL$ of a weighted undirected graph $G = (V,E,\\ww)$\nwith $\\abs{E}=m$ and $\\abs{V} = n$ and a demand vector $\\dd \\in \\R^V$,\nwe can find $\\xxtil$ that is an $\\epsilon$-approximate solution to\n$\\LL \\xx = \\dd$, using an algorithm that takes time\n$O(m \\log^c n \\log(1/\\epsilon))$ for some fixed constant $c$ and succeeds with probability $1 - 1/n^{10}$.\n\\end{theorem}\nIn the original algorithm of Spielman and Teng, the exponent on the\nlog in the running time was $c \\approx 70$.\n\nToday, we are going to see a simpler algorithm. But first, we'll look\nat one of the key tools behind all algorithms for solving Laplacian\nlinear equations quickly.\n\n\n\\section{Preconditioning and\nApproximate Gaussian Elimination}\nRecall our definition of two positive semi-definite matrices being\napproximately equal.\n\\begin{definition}[Spectral approximation]\n Given $\\AA, \\BB \\in \\psdsetn$, we say that\n \\[\n \\AA \\approx_{K} \\BB\n \\text{ if and only if }\n \\frac{1}{1+ K} \\AA \\preceq \\BB \\preceq (1+ K) \\AA.\n \\]\n\\end{definition}\nSuppose we have a positive definite matrix $\\MM \\in \\pdsetn$ and want to\nsolve a linear equation $\\MM \\xx =\\dd$.\nWe can do this using gradient descent or accelerated gradient\ndescent, as we covered in Graded Homework 1.\nBut if we have access to an easy-to-invert matrix that happens to also\nbe a good spectral approximation of $\\MM$, then we can use this to\nspeed up the (accelerated) gradient descent algorithm.\nAn example of this would be that we have a factorization\n$\\matlow\\matlow^{\\trp} \\approx_{K} \\MM$, where $\\matlow$ is\nlower triangular and sparse, which means we can invert it quickly.\n\nThe following lemma, which you will prove in Problem Set 6, makes this\npreconditioning precise.\n\\begin{lemma}\n \\label{lem:cholprecond}\n Given a matrix $\\MM \\in \\pdsetn$, a vector $\\dd$ and a decomposition $\\MM\n \\approx_{K} \\matlow\\matlow^{\\trp}$,\n we can find $\\xxtil$ that $\\epsilon$-approximately solves $\\MM \\xx =\n \\dd$, using\n$O( (1+ K)\\log(K/\\epsilon)(T_{\\text{matvec}} + T_{\\text{sol}} + n) ) $ time.\n\\begin{itemize}\n\\item $T_{\\text{matvec}}$ denotes the time required to compute $\\MM\n \\zz$ given a vector $\\zz$, i.e. a ``matrix-vector multiplication''.\n\\item $T_{\\text{sol}}$ denotes the time required to compute\n $\\matlow^{-1}\\zz$ or $(\\matlow^{\\trp})^{-1}\\zz$ given a vector\n $\\zz$.\n\\end{itemize}\n\\end{lemma}\n\\paragraph{Dealing with pseudo-inverses.}\nWhen our matrices have a null space, preconditioning becomes slightly\nmore complicated, but as long as it is easy to project to the\ncomplement of the null space, there's no real issue. The following\ndescribes precisely what we need (but you can ignore the null-space\nissue when first reading these notes without losing anything\nsignificant).\n\\begin{lemma}\n \\label{lem:cholprecondpinv}\n Given a matrix $\\MM \\in \\psdsetn$, a vector $\\dd \\in \\ker(\\MM)^{\\perp}$ and a decomposition $\\MM\n \\approx_{K} \\matlow\\calDD\\matlow^{\\trp}$, where $\\matlow$ is invertible,\n we can find $\\xxtil$ that $\\epsilon$-approximately solves $\\MM \\xx =\n \\dd$, using\n${O( (1+ K)\\log(K/\\epsilon)(T_{\\text{matvec}} + T_{\\text{sol}} +\nT_{\\text{proj}} + n) ) }$ time.\n\\begin{itemize}\n\\item $T_{\\text{matvec}}$ denotes the time required to compute $\\MM\n \\zz$ given a vector $\\zz$, i.e. a ``matrix-vector multiplication''.\n\\item $T_{\\text{sol}}$ denotes the time required to compute\n $\\matlow^{-1}\\zz$ and $(\\matlow^{\\trp})^{-1}\\zz$ and $\\calDD^{+}\\zz$ given a vector\n $\\zz$.\n\\item $T_{\\text{proj}}$ denotes the time required to compute\n $\\proj_{\\MM} \\zz$ given a vector $\\zz$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{theorem}[Kyng and Sachdeva (2015) \\cite{ks16}]\n \\label{thm:apxgauss}\nGiven a Laplacian $\\LL$ of a weighted undirected graph $G = (V,E,\\ww)$\nwith $\\abs{E}=M$ and $\\abs{V} = n$,\nwe can find a decomposition\n$\\matlow\\matlow^{\\trp} \\approx_{0.5} \\LL $, such that\n$\\matlow$ has number of non-zeroes $\\nnz(\\matlow) = O(m \\log^3 n)$,\nwith probability at least $1 - 3/n^{5}$.\nin time $O(m \\log^3 n)$.\n\\end{theorem}\nWe can combine Theorem~\\ref{thm:apxgauss} with\nLemma~\\ref{lem:cholprecondpinv} to get a fast algorithm for solving\nLaplacian linear equations.\n% ...\n% \\begin{corollary}\n% Given a decomposition $\\matlow\\matlow^{\\trp} \\approx_{0.5} \\LL $,\n% \\end{corollary}\n\\begin{corollary}\n \\label{cor:precondsolver}\n Given a Laplacian $\\LL$ of a weighted undirected graph $G = (V,E,\\ww)$\nwith $\\abs{E}=m$ and $\\abs{V} = n$ and a demand vector $\\dd \\in \\R^V$,\nwe can find $\\xxtil$ that is an $\\epsilon$-approximate solution to\n$\\LL \\xx = \\dd$, using an algorithm that takes time $O(m \\log^3 n\n\\log(1/\\epsilon))$ and succeeds with probability $1 - 1/n^{10}$.\n\\end{corollary}\n\\begin{proof}[Proof sketch]\n\n First we need to get a factorization that confirms to\n Lemma~\\ref{lem:cholprecondpinv}.\n The decomposition $\\matlow\\matlow^{\\trp}$ provided by\n Theorem~\\ref{thm:apxgauss} can be rewritten as\n $\\matlow\\matlow^{\\trp} = \\matlowtil \\calDD (\\matlowtil)^{\\trp}$\n where\n$\\matlowtil$ is equal to $\\matlow$ except $\\matlow(n,n) = 1$\nand we let $\\calDD$ be the identity matrix, except $\\calDD (n,n)\n= 0$.\nThis ensures $\\calDD^{\\pinv} = \\calDD$ and that $\\matlowtil$ is\ninvertible and lower triagular with $O(m \\log^3 n)$ non-zeros.\nWe note that the inverse of an invertible lower or upper triangular\n matrix with $N$ non-zeros can be applied in time $O(N)$ given an\n adjacency list representation of the matrix.\n Finally, as $\\ker(\\matlow\\matlow^{\\trp}) = \\Span\\setof{\\vecone}$,\nwe have $\\proj_{ \\matlowtil \\calDD (\\matlowtil)^{\\trp}}\n= \\II -\\frac{1}{n} \\vecone \\vecone^{\\trp}$, and this projection matrix can be\napplied in $O(n)$ time.\nAltogether, this means that $T_{\\text{matvec}} + T_{\\text{sol}} +\nT_{\\text{proj}} = O(n)$, which suffices to complete the proof.\n\\end{proof}\n\n\\section{Approximate Gaussian Elimination Algorithm}\nRecall \\emph{Gaussian Elimination / Cholesky decomposition} of a graph Laplacian $\\LL$.\nWe will use $\\AA(:,i)$ to denote the the $i$th\ncolumn of a matrix $\\AA$.\nWe can write the algorithm as\n% \\begin{algorithm}[h]\n% \\begin{algorithmic}\n% \\For{$i = 1$ to $i = n-1$}\n% \\State $\\ll_i = \\frac{1}{\\sqrt{\\SS_{i-1}(i,i)}} \\SS_{i-1}(:,i)$\n% \\State $\\SS_{i} = \\SS_{i-1} - \\ll_i \\ll_i^{\\trp}.$\n% \\EndFor\n% \\State $\\ll_n = \\veczero_{n \\times 1}$\n% \\State $\\matlow=\\begin{bmatrix} \\ll_1 \\cdots \\ll_n \\end{bmatrix}$\n% \\end{algorithmic}\n% \\end{algorithm}\n\n\\begin{algorithm}[H]\n\\label{alg:ge}\n\\caption{Gaussian Elimination / Cholesky Decomposition}\n\\KwIn{Graph Laplacian $\\LL$}\n\\KwOut{Lower triangular $\\matlow$ s.t. $\\matlow\\matlow^{\\trp} = \\LL$}\nLet $\\SS_0 = \\LL$ \\;\n \\For{$i = 1$ to $i = n-1$}{\n$\\ll_i = \\frac{1}{\\sqrt{\\SS_{i-1}(i,i)}} \\SS_{i-1}(:,i)$\\;\n$\\SS_{i} = \\SS_{i-1} - \\ll_i \\ll_i^{\\trp}.$\\\n}\n$\\ll_n = \\veczero_{n \\times 1}$\\;\n\\Return{$\\matlow=\\begin{bmatrix} \\ll_1 \\cdots \\ll_n \\end{bmatrix}$}\\;\n\\end{algorithm}\n% Now, for $i = 1$ to $i = n-1$ we define\n% \\begin{align*}\n% \\ll_i &= \\frac{1}{\\sqrt{\\SS_{i-1}(i,i)}} \\SS_{i-1}(:,i), \\\\\n% \\SS_{i} &= \\SS_{i-1} - \\ll_i \\ll_i^{\\trp}.\n% \\end{align*}\n% Finally, we let $\\ll_n = \\veczero_{n \\times 1}$. It follows that\n% $\\matlow=\\begin{bmatrix} \\ll_1 \\cdots \\ll_n \\end{bmatrix}$ is lower\n% triangular and $\\LL=\\matlow\\matlow^\\trp$.\nWe want to introduce some notation that will help us describe and\nanalyze a faster version of Gaussian elimination -- one that uses sampling\nto create a sparse approximation of the decomposition.\n\nConsider a Laplacian $\\SS$ of a graph $H$ and a vertex $v$ of $H$.\nWe define $\\vstar{v}{\\SS}$ to be the Laplacian of the subgraph of $H$\nconsisting of edges incident on $v$.\nWe define\n\\[\n\\vcliq{v}{\\SS} = \\vstar{v}{\\SS} - \\frac{1}{\\SS(v,v)} \\SS(:,v) \\SS(:,v)^{\\trp}\n\\]\nFor example, suppose\n\\[\n \\LL =\n\\left(\n\\begin{array}{ccc}\nW & -\\aa^\\trp \\\\\n-\\aa& \\diag(\\aa) + \\LL_{-1}\n\\end{array} \\right)\n\\]\nThen\n\\[\n\\vstar{1}{\\LL}\n=\n\\left(\n\\begin{array}{ccc}\nW & -\\aa^\\trp \\\\\n-\\aa& \\diag(\\aa)\n\\end{array} \\right)\n\\text{ and }\n\\vcliq{1}{\\LL}\n=\n\\left(\n\\begin{array}{ccc}\n0 & \\veczero \\\\\n\\veczero & \\diag(\\aa) - \\frac{1}{W} \\aa \\aa^\\trp\n\\end{array} \\right)\n\\]\nwhich is illustrated in Figure \\ref{fig:schurclique}.\n%\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1\n \\textwidth]{fig/lecture7_schur-clique.jpeg}\n \\caption{Gaussian Elimination:\n $\\vcliq{1}{\\LL} = \\vstar{1}{\\LL} - \\frac{1}{\\LL(1,1)} \\LL(:,1)\n \\LL(:,1)^{\\trp}$.}\n \\label{fig:schurclique}\n \\end{figure}\nIn Chapter~\\ref{cha:ge}, we proved that $\\vcliq{v}{\\SS}$ is a graph Laplacian --\nit follows from the proof of Claim~\\ref{clm:optimgaussclosed} in that chapter.\nThus we have that following.\n \\begin{claim}\n\\label{clm:optimgaussclosedagain}\nIf $\\SS$ is the Laplacian of a connected graph, then\n$\\vcliq{v}{\\SS}$ is a graph Laplacian.\n% \\begin{enumerate}\n% \\item $\\vcliq{v}{\\SS}$ is a graph Laplacian.\n% \\item\n% \\end{enumerate}\n\\end{claim}\n\nNote that in Algorithm~\\ref{alg:ge}, we have $\\ll_i\\ll_i^\\trp = \\vstar{v_i}{\\SS_{i-1}}-\\vcliq{v_i}{\\SS_{i-1}}$. The update rule can be rewritten as\n\\[ \\SS_{i} = \\SS_{i-1} - \\vstar{v_i}{\\SS_{i-1}} +\n \\vcliq{v_i}{\\SS_{i-1}}, \\]\n\nThis also provides way to understand why\nGaussian Elimination is slow in some cases.\nAt each step, one vertex is eliminated, but a clique is added to the\nsubgraph on the remaining vertices, making the graph denser.\nAnd at the $i$th step, computing $\\vstar{v_i}{\\SS_{i-1}}$\ntakes around $\\deg(v_i)$ time, but computing $\\vcliq{v_i}{\\SS_{i-1}}$\nrequires around $\\deg(v_i)^2$ time.\nIn order to speed up Gaussian\nElimination, the algorithmic idea of \\cite{ks16}\n%\\cite{?}\n is to plug in a sparser appproximate of the intended clique instead\n of the entire one.\n\nThe following procedure $\\vcliqsp{v}{\\SS}$ produces a sparse\napproximation of $\\textsc{clique}(v,\\SS)$.\nLet $V$ be the vertex set of the graph associated with $\\SS$ and $E$\nthe edge set.\nWe define $\\bb_{i,j} \\in \\R^V$ to be the vector with\n\\[\n \\bb_{i,j}(i) = 1\n \\text{ and }\n \\bb_{i,j}(j) = -1\n \\text{ and }\n \\bb_{i,j}(k) = 0\n \\text{ for }\n k \\neq i,j.\n\\]\nGiven weights $\\ww \\in \\R^E$ and a vertex $v \\in V$,\nwe let\n\\[\n \\ww_v = \\sum_{(u,v) \\in E} \\ww(u,v)\n .\n \\]\n\\begin{algorithm}[H]\n\\label{alg:cliquesamp}\n\\caption{$\\vcliqsp{v}{\\SS}$}\n\\KwIn{Graph Laplacian $\\SS \\in \\R^{V \\times V}$, of a graph with edge\n weights $\\ww$, and vertex $v \\in V$}\n\\KwOut{$\\YY_v \\in \\R^{V \\times V}$ sparse approximation of\n $\\textsc{clique}(v,\\SS)$ }\n $\\YY_v \\gets \\matzero_{n \\times n} $\\;\n \\ForEach{Multiedge $e\n = (v,i)$ from $v$ to a neighbor $i$}{\n Randomly pick a neighbor $j$ of $v$ with probability $\\frac{\\ww(j,v)}{\\ww_v}$\\;\n If $i \\neq j$, let $\\YY_v \\gets \\YY_v + \\frac{\\ww(i,v)\\ww(j,v)}{\\ww(i,v)+\\ww(j,v)} \\bb_{i,j}\\bb_{i,j}^\\trp$\\;\n}\n\\Return{$\\YY_v$}\\;\n\\end{algorithm}\n% Let $\\YY_v = \\matzero_{n \\times n}$. For each neighbor $i$ of $v$, randomly pick a neighbor $j$ of $v$ with probability $\\frac{\\ww(j,v)}{\\ww_v}$ where $\\ww_v = \\sum_{u \\sim v} \\ww(u,v)$ is the sum of weights of edges incident to $v$. If $i\\neq j$, then draw an edge with weight $\\ww(i,v)\\ww(j,v)/(\\ww(i,v)+\\ww(j,v))$ between $i$ and $j$, i.e. update $\\YY_v$ by\n% \\[ \\YY_v \\gets \\YY_v + \\frac{\\ww(i,v)\\ww(j,v)}{\\ww(i,v)+\\ww(j,v)} \\bb_{i,j}\\bb_{i,j}^\\trp. \\]\n%\n\\begin{remark}\n We can implement each sampling of a neighbor $j$ in $O(1)$ time\n using a classical algorithm known as Walker's method (also known as the Alias\n method or Vose's method).\n This algorithm requires an additional $O(\\deg_{\\SS}(v))$ time to\n initialize a data structure used for sampling.\n Overall, this means the total time for $O(\\deg_{\\SS}(v))$ samples is still $O(\\deg_{\\SS}(v))$.\n\\end{remark}\n\n% %TODO this paragraph sometimes causes horizontal \"badness\" failure\n% In what sense does $\\vcliqsp{v}{\\SS}$ produce a sparse approximation\n% of the clique $\\vcliq{v}{\\SS}$ created by Gaussian elimination?\n% This is a crucial question, and the next lemma gives part of the\n% answer: Namely that the output in expectation equals the clique.\n\\begin{lemma}\\label{lem:cliquesample_expectation}\n $\\E{\\YY_v} = \\vcliq{v}{\\SS}$.\n\\end{lemma}\n\\begin{proof}\n Let $\\CC=\\vcliq{v}{\\SS}$. Observe that both $\\E{\\YY_v}$ and $\\CC$ are Laplacians. Thus it suffices to verify $\\Ex{\\YY_v(i,j)}=\\CC(i,j)$ for $i\\neq j$.\n \\[ \\CC(i,j) = -\\frac{\\ww(i,v)\\ww(j,v)}{\\ww_v}, \\]\n \\[ \\Ex{\\YY_v(i,j)} = -\\frac{\\ww(i,v)\\ww(j,v)}{\\ww(i,v)+\\ww(j,v)} \\left( \\frac{\\ww(j,v)}{\\ww_v} + \\frac{\\ww(i,v)}{\\ww_v} \\right) = -\\frac{\\ww(i,v)\\ww(j,v)}{\\ww_v} = \\CC(i,j). \\]\n\\end{proof}\n\\begin{remark}\n Lemma \\ref{lem:cliquesample_expectation} shows that $\\vcliqsp{v}{\\LL}$ produces the original $\\vcliq{v}{\\LL}$ in expectation.\n\\end{remark}\n\nNow, we define \\emph{Approximate Gaussian Elimination}.\n% Let $\\SS_0 = \\LL$. Generate a radom permutation $\\pi$ on $[n]$. For $i = 1$ to $i = n-1$ we define\n% \\begin{align*}\n% \\ll_i &= \\frac{1}{\\sqrt{\\SS_{i-1}(\\pi(i),\\pi(i))}} \\SS_{i-1}(:,\\pi(i)), \\\\\n% \\SS_{i} &= \\SS_{i-1} - \\vstar{\\pi(i)}{\\SS_{i-1}} + \\vcliqsp{\\pi(i)}{\\SS_{i-1}}.\n % \\end{align*}\n\n\\begin{algorithm}[H]\n\\label{alg:apxge}\n\\caption{Approximate Gaussian Elimination / Cholesky Decomposition}\n\\KwIn{Graph Laplacian $\\LL$}\n\\KwOut{Lower triangular\\footnote{$\\matlow$ is not actually lower\n triangular. However, if we let $\\PP_\\pi$ be the permutation matrix\n corresponding to $\\pi$, then $\\PP_\\pi \\matlow$ is lower\n triangular. Knowing the ordering that achieves this is enough to\n let us implement forward and backward substitution for solving\n linear equations in $\\matlow$ and $\\matlow^\\trp$.}\n $\\matlow$ as\n given in Theorem~\\ref{thm:apxgauss}}\nLet $\\SS_0 = \\LL$\\;\nGenerate a random permutation $\\pi$ on $[n]$\\;\n\\For{$i = 1$ to $i = n-1$}{\n$\\ll_i = \\frac{1}{\\sqrt{\\SS_{i-1}(\\pi(i),\\pi(i))}} \\SS_{i-1}(:,\\pi(i)) $\\;\n$\\SS_{i} = \\SS_{i-1} - \\vstar{\\pi(i)}{\\SS_{i-1}} + \\vcliqsp{\\pi(i)}{\\SS_{i-1}}$\\\n}\n$\\ll_n = \\veczero_{n \\times 1}$\\;\n\\Return{$\\matlow=\\begin{bmatrix} \\ll_1 \\cdots \\ll_n \\end{bmatrix}$ and\n$\\pi$}\\;\n\\end{algorithm}\n% $\\PP_\\pi$ be the permutation matrix corresponding to $\\pi$. It follows that $\\matlow =\n% \\PP_\\pi \\begin{bmatrix} \\ll_1 \\cdots \\ll_n \\end{bmatrix}$ is lower\n% triangular and $\\LL=\\matlow\\matlow^\\trp$.\nNote that if we replace\n$\\vcliqsp{\\pi(i)}{\\SS_{i-1}}$ by $\\vcliq{\\pi(i)}{\\SS_{i-1}}$ at each\nstep, then we can recover Gaussian Elimination, but with a random\nelimination order.\n\n\\section{Analyzing Approximate Gaussian Elimination}\n\nIn this Section, we're going to analyze Approximate Gaussian\nElimination, and see why it works.\n\nUltimately, the main challenge in proving Theorem~\\ref{thm:apxgauss}\nwill be to prove for the output $\\matlow$\nof Algorithm~\\ref{alg:apxge} that with high probability\n\\begin{equation}\n \\label{eq:relerrgoal}\n0.5 \\LL\n\\preceq\n\\matlow \\matlow^{\\trp}\n\\preceq 1.5 \\LL\n.\n\\end{equation}\nWe can reduce this to proving that with high probability\n\\begin{equation}\n\\label{eq:spectralnormgoal}\n \\norm{\\LL^{+/2} (\\matlow \\matlow^{\\trp} - \\LL) \\LL^{+/2}} \\leq 0.5\n\\end{equation}\nUltimately, the proof is going to have a lot in common with our proof\nof Matrix Bernstein in Chapter~\\ref{cha:randmat}.\nOverall, the lesson there was that when we have a sum of independent,\nzero-mean random matrices, we can show that the sum is likely to have\nsmall spectral norm if the spectral norm of each random matrix is\nsmall, and the matrix-valued variance is also small.\n\nThus, to replicate the proof, we need control over\n\\begin{enumerate}\n\\item The \\emph{sample norms}.\n\\item The \\emph{sample variance}.\n\n\\end{enumerate}\n\nBut, there is seemlingly another major obstacle: We are trying to\nanalyze a process where the samples are far from independent.\nEach time we sample edges, we add new edges to the remaining graph,\nwhich we will the later sample again. This creates a lot of\ndependencies between the samples, which we have to handle.\n\nHowever, it turns out that independence is more than what is needed to\nprove concentration. Instead, it suffices to have a sequence of random\nvariables such that each is mean-zero in expectation, conditional on\nthe previous ones. This is called a martingale difference sequence.\nWe'll now learn about those.\n\n\\subsection{Normalization, a.k.a. Isotropic Position}\n\nSince our analysis requires frequently measuring matrices after right\nand left-multiplication by $\\LL^{\\pinv/2}$, we reintroduce the\n``normalizing map'' $\\Phi : \\R^{n\\times n} \\to\n\\R^{n\\times n} $\ndefined by\n\\[\n \\Phi(\\AA) = \\LL^{\\pinv/2}\\AA\\LL^{\\pinv/2}.\n \\]\nWe previously saw this in Chapter~\\ref{cha:randmat}.\n\n\\subsection{Martingales}\n\nA scalar martingale is a sequence of random variables $Z_0, \\ldots,\nZ_k$, such that\n\\begin{align}\n\\label{eq:martingale}\n \\E{Z_i \\mid Z_0, \\ldots, Z_{i-1}} = Z_{i-1}\n.\n\\end{align}\nThat is, conditional on the outcome of all the previous random\nvariables, the expectation of $Z_{i}$ equals $Z_{i-1}$.\nIf we unravel the sequence of conditional expectations, we\nget that \\emph{without conditioning}\n$\\E{Z_k}=\\E{Z_0}$.\n\nTypically, we use martingales to show a statement along like ``$Z_k$\nis concentrated around $\\E{Z_k}$''.\n\nWe can also think of a martingale in terms of the sequence of changes\nin the $Z_i$ variables.\nLet $X_i = Z_i - Z_{i-1}$.\nThe sequence of $X_i$s is called a martingale difference sequence.\nWe can now state the martingale condition as\n\\[\n \\E{X_i \\mid Z_0, \\ldots, Z_{i-1}} = 0\n.\n\\]\nAnd because $Z_0$ and $X_{1}, \\ldots, X_{i-1}$ completely determine\n$Z_1, \\ldots, Z_{i-1}$, we could also write\nthe martingale condition equivalently as\n\\[\n \\E{X_i \\mid Z_0, X_1, \\ldots, X_{i-1}} = 0\n .\n\\]\nCrucially, we can write\n\\[\nZ_k = Z_0 + \\sum_{i=1}^k Z_i - Z_{i-1} = Z_0 + \\sum_{i=1}^k X_i\n\\]\nand when we are trying to prove concentration, the martingale difference property\nof the $X_i$'s is often ``as good as'' independence, meaning that\n$\\sum_{i=1}^k X_i$ concentrates similarly to a sum of independent\nrandom variables.\n\n\n\n\n\\paragraph{Matrix-valued martingales.}\nWe can also define matrix-valued martingales.\nIn this case, we replace the martingalue condition of\nEquation~\\eqref{eq:martingale}, with the condition that the whole matrix\nstays the same in expectation.\nFor example, we could have a sequence of random matrices $\\ZZ_0, \\ldots,\n\\ZZ_k \\in \\R^{n \\times n}$, such that\n\\begin{align}\n\\label{eq:matmartingale}\n \\E{\\ZZ_i \\mid \\ZZ_0, \\ldots, \\ZZ_{i-1}} = \\ZZ_{i-1}\n.\n\\end{align}\n\\begin{lemma}\\label{lem:ApxGE_martingale}\n Let $\\LL_i = \\SS_i + \\sum_{j=1}^i \\ll_j\\ll_j^\\trp$ for $i=1,...,n$ and $\\LL_0=\\SS_0=\\LL$. Then\n \\[ \\E{\\LL_i | \\text{all random variables before } \\vcliqsp{\\pi(i)}{\\SS_{i-1}} } = \\LL_{i-1}. \\]\n\\end{lemma}\n\\begin{proof}\n Let's only consider $i=1$ here as other cases are similar.\n \\[ \\LL_0 = \\LL = \\ll_1\\ll_1^\\trp + \\vcliq{v}{\\LL} + \\LL_{-1} \\]\n \\begin{align*}\n \\LL_1 &= \\ll_1\\ll_1^\\trp + \\vcliqsp{v}{\\LL} + \\LL_{-1} \\\\\n \\E{\\LL_1 | \\pi(1)}\n &= \\ll_1\\ll_1^\\trp + \\E{\\vcliqsp{v}{\\LL}|\\pi(1)} + \\LL_{-1} \\\\\n &= \\ll_1\\ll_1^\\trp + \\vcliq{v}{\\LL} + \\LL_{-1} \\\\\n &= \\LL_0\n \\end{align*}\n where we used Lemma~\\ref{lem:cliquesample_expectation} to get\n $\\E{\\vcliqsp{v}{\\LL}|\\pi(1)} = \\vcliq{v}{\\LL} $.\n\\end{proof}\n\\begin{remark}\n $\\sum_{j=1}^i \\ll_j\\ll_j^\\trp$ can be treated as what has already been eliminated by (Approximate) Gaussian Elimination, while $\\SS_i$ is what still left or going to be eliminated.\n In Approximate Gaussian Elimination, $\\LL_n=\\sum_{i=1}^n\\ll_i\\ll_i^\\trp$ and our goal is to show that $\\LL_n\\approx_K\\LL$.\n Note that $\\LL_i$ is always equal to the original Laplacian $\\LL$ for all $i$ in Gaussian Elimination.\n Lemma \\ref{lem:ApxGE_martingale} demonstrates that $\\LL_0,\\LL_1,...,\\LL_n$ forms a matrix martingale.\n\\end{remark}\n\nUltimately, our plan is to use this matrix martingale structure to show\nthat ``$\\LL_n$ is concentrated around $\\LL$'' in some appropriate\nsense.\nMore precisely, the spectral approximation we would like to show can\nbe established by showing that ``$\\Phi(\\LL_n)$ is concentrated around $\\Phi(\\LL)$''\n\n\\subsection{Martingale Difference Sequence as Edge-Samples}\nWe start by taking a slightly different view of the observations we\nused to prove Lemma~\\ref{lem:ApxGE_martingale}.\nRecall that $\\LL_i = \\SS_i + \\sum_{j=1}^i \\ll_j\\ll_j^\\trp$,\nand $\\LL_{i-1} = \\SS_{i-1} + \\sum_{j=1}^{i-1} \\ll_j\\ll_j^\\trp$\nand\n\\[\n \\SS_{i} = \\SS_{i-1} - \\vstar{\\pi(i)}{\\SS_{i-1}} +\n \\vcliqsp{\\pi(i)}{\\SS_{i-1}}\n . \\]\nPutting these together, we get\n\\begin{align}\n \\LL_i - \\LL_{i-1}\n &=\n \\ll_i\\ll_i^\\trp\n +\n \\vcliqsp{\\pi(i)}{\\SS_{i-1}}\n -\n \\vstar{\\pi(i)}{\\SS_{i-1}}\n \\nonumber\n \\\\\n \\label{eq:apxgaussdiffseq}\n &=\n \\vcliqsp{\\pi(i)}{\\SS_{i-1}}\n -\n \\vcliq{\\pi(i)}{\\SS_{i-1}}\n\\\\\n &=\n \\vcliqsp{\\pi(i)}{\\SS_{i-1}}\n -\n \\E{\\vcliqsp{\\pi(i)}{\\SS_{i-1}} \\mid \\text{preceding samples}}\n \\tag*{by Lemma~\\ref{lem:cliquesample_expectation}.}\n\\end{align}\nIn particular, recall that by Lemma~\\ref{lem:cliquesample_expectation},\nconditional on the randomness before\nthe call to $\\vcliqsp{\\pi(i)}{\\SS_{i-1}}$, we\nhave\n\\[\n\\E{\\vcliqsp{\\pi(i)}{\\SS_{i-1}} \\mid \\text{preceding samples}}= \\vcliq{\\pi(i)}{\\SS_{i-1}}\n\\]\nAdopting the notation of Lemma~\\ref{lem:cliquesample_expectation}\nwe write\n\\[\n \\YY_{\\pi(i)} = \\vcliqsp{\\pi(i)}{\\SS_{i-1}}\n\\]\nand we further introduce notation each multi-edge sample for\n$e \\in \\vstar{\\pi(i)}{\\SS_{i-1}}$, as $\\YY_{\\pi(i),e}$, denoting the random\nedge Laplacian sampled when the algorithm is processing multi-edge\n$e$.\nThus, conditional on preceding samples, we have\n\n\\begin{equation}\n \\label{eq:cliquesamplesum}\n \\YY_{\\pi(i)} = \\sum_{e \\in\n \\vstar{\\pi(i)}{\\SS_{i-1}}}\\YY_{\\pi(i),e}\n\\end{equation}\nNote that even the number of multi-edges in\n$\\vstar{\\pi(i)}{\\SS_{i-1}}$ depends on the preceding samples.\nWe also want to associate zero-mean variables with each edge.\nConditional on preceding samples, we also define\n\\[\n \\XX_{i,e} = \\Phi\\left( \\YY_{\\pi(i),e} - \\E{\\YY_{\\pi(i),e}} \\right)\n \\text{ and }\n \\XX_{i} = \\sum_{e \\in\n \\vstar{\\pi(i)}{\\SS_{i-1}}}\\XX_{i,e}\n\\]\nand combining this with Equations~\\eqref{eq:apxgaussdiffseq} and \\eqref{eq:cliquesamplesum}\n\\[\n \\XX_{i}\n = \\Phi(\\YY_{\\pi(i)} - \\E{\\YY_{\\pi(i)}})\n = \\Phi(\\LL_i - \\LL_{i-1})\n\\]\nAltogether, we can write\n\\[\n \\Phi\\left( \\LL_n -\\LL \\right)\n =\n \\sum_{i=1}^n \\Phi(\\LL_i - \\LL_{i-1})\n =\n \\sum_{i=1}^n \\XX_i\n =\n \\sum_{i=1}^n \\sum_{e \\in\n \\vstar{\\pi(i)}{\\SS_{i-1}}}\\XX_{i,e}\n\\]\nNote that the $\\XX_{i,e}$ variables form a martingale difference\nsequence, because the linearity of $\\Phi$ ensures they are zero-mean\nconditional on preceding randomness.\n\n\\subsection{Stopped Martingales}\nUnfortunately, directly analyzing the concentration properties of\nthe $\\LL_i$ martingale that we just introduced turns out to be\ndifficult.\nThe reason is that we're trying to prove some very delicate\nmultiplicative error guarantees.\nAnd, if we analyze $\\LL_i$, we find that the multiplicative error is\nnot easy to control, \\emph{after it's already gotten big}.\nBut that's not really what we care about anyway: We want to say it\nnever gets big in the first place, with high probability.\nSo we need to introduce another martingale, that lets us ignore the\nbad case when the error has already gotten too big.\nAt the same time, we also need to make sure that statements about our\nnew martingale can help us prove guarantees about $\\LL_i$.\nFortunately, we can achieve both at once.\nThe technique we use is related to the much broader topic of\nmartingale \\emph{stopping times}, which we only scratch the surface of\nhere. We're also going to be quite informal about it, in the interest\nof brevity.\nLecture notes by Tropp \\cite{tropp19} give a more formal introduction for\nthose who are interested.\n\n\n\nWe define the stopped martingale sequence $\\LLtil_i$ by\n\\begin{equation}\n \\label{eq:stoppedmartingale}\n \\LLtil_i\n =\n \\begin{cases}\n \\LL_i & \\text{ if for all } j < i \\text{ we have }\n \\LL_i \\preceq 1.5 \\LL\n \\\\\n \\LL_{j^*} & \\text{ for } j^* \\text{ being the least $j$ such that } \\LL_{j} \\not\\preceq 1.5 \\LL\n \\end{cases}\n\\end{equation}\n\n\n\nFigure~\\ref{fig:stopmart} shows the $\\LLtil_i$ martingale getting\nstuck at the first time $\\LL_{j^*} \\not\\preceq 1.5 \\LL$.\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width=1\n \\textwidth]{fig/lec9_stoppedmartingale.png}\n \\caption{Gaussian Elimination\n :\n $\\vcliq{1}{\\LL} = \\vstar{1}{\\LL} - \\frac{1}{\\LL(1,1)} \\LL(:,1)\n \\LL(:,1)^{\\trp}$.}\n \\label{fig:stopmart}\n \\end{figure}\n\nWe state the following without proof:\n\\begin{claim}\n\\noindent\n \\begin{enumerate}\n \\item The sequence $\\setof{\\LLtil_i}$ for $i = 0, \\ldots, n$ is a\n martingale.\n \\item $\\norm{\\LL^{+/2} (\\LLtil_i - \\LL) \\LL^{+/2}} \\leq 0.5$\n implies $\\norm{\\LL^{+/2} (\\LL_i - \\LL) \\LL^{+/2}} \\leq 0.5$\n \\end{enumerate}\n\\end{claim}\nThe martingale property also implies that the unconditional\nexpectation satisfies $\\E{\\LLtil_n} = \\LL$.\nThe proof of the claim is easy to sketch: For Part 1, each difference is\nzero-mean if the condition has not been violated, and is identically\nzero (and hence zero-mean) if it has been violated.\nFor Part 2, if the martingale $\\setof{\\LLtil_i}$ has stopped,\nthen $\\norm{\\LL^{+/2}\n (\\LLtil_i - \\LL) \\LL^{+/2}} \\leq 0.5$ is false, and the implication\nis vacuosly true.\nIf, on the other hand, the martingale has not stopped, the\nquantities are equal, because $\\LLtil_i = \\LL_i$, and again it's easy\nto see the implication holds.\n\nThus, ultimately, our strategy is goin to be to show that\n$\\norm{\\LL^{+/2} (\\LLtil_i - \\LL) \\LL^{+/2}} \\leq 0.5$\nwith high probability.\nExpressed using the normalizing map $\\Phi(\\cdot)$,\nour goal is to show that with high probability\n\\[\n \\norm{\\Phi(\\LLtil_n - \\LL)} \\leq 0.5.\n\\]\n\n\\paragraph{Stopped martingale difference sequence.}\nIn order to prove the spectral norm bound, we want to express the\n$\\setof{\\LLtil_i}$ martingale in terms of a sequence of martingale\ndifferences.\nTo this end, we define\n$\\XXtil_i = \\Phi(\\LLtil_i - \\LLtil_{i-1})$.\nThis ensures that\n\\begin{equation}\n \\label{eq:stoppedmartingale}\n \\XXtil_i\n =\n \\begin{cases}\n \\XX_i& \\text{ if for all } j < i \\text{ we have }\n \\LL_i \\preceq 1.5 \\LL\n \\\\\n \\matzero & \\text{ otherwise }\n \\end{cases}\n\\end{equation}\nWhenever the modified martingale $\\XXtil_i$ has not yet stopped, we\nalso introduce individual modified edge samples $\\XXtil_{i,e} =\n\\XX_{i,e}$.\nIf the martingale \\emph{has} stopped, i.e. $\\XXtil_i = \\matzero$,\nthen we can take these edge samples $\\XXtil_{i,e}$\nto be zero.\nWe can now write\n\\[\n \\Phi\\left( \\LLtil_n -\\LL \\right)\n =\n \\sum_{i=1}^n \\Phi(\\LLtil_i - \\LLtil_{i-1})\n =\n \\sum_{i=1}^n \\XXtil_i\n =\n \\sum_{i=1}^n \\sum_{e \\in\n \\vstar{\\pi(i)}{\\SS_{i-1}}}\\XXtil_{i,e}\n .\n\\]\nThus, we can see that Equation~\\eqref{eq:spectralnormgoal} is implied by\n\\begin{equation}\n \\label{eq:zeromeanspectralnormgoal}\n \\norm{\\sum_{i=1}^n \\XXtil_i} \\leq 0.5.\n\\end{equation}\n\n\\subsection{Sample Norm Control}\nIn this Subsection, we're going to see that the norms of each\nmulti-edge sample is controlled throughout the algorithm.\n\\begin{lemma}\\label{lem:cliquesample_bounded_norm}\n Given two Laplacians $\\LL$ and $\\SS$ on the same vertex\n set.\\footnote{$\\LL$ can be regarded as the original Laplacian we\n care about, while $\\SS$ can be regarded as some intermediate\n Laplacian appearing during Approximate Gaussian Elimination.} If\n each multiedge $e$ of $\\vstar{v}{\\SS}$ has bounded norm in the following sense,\n \\[ \\norm{\\LL^{\\pinv/2}\\ww_{\\SS}(e)\\bb_e\\bb_e^{\\trp}\\LL^{\\pinv/2}} \\leq R, \\]\n then each possible sampled multiedge $e'$ of $\\vcliqsp{v}{\\SS}$ also satisfies\n \\[ \\norm{\\LL^{\\pinv/2}\\ww_{\\mathrm{new}}(e')\\bb_{e'}\\bb_{e'}^{\\trp}\\LL^{\\pinv/2}} \\leq R. \\]\n\\end{lemma}\n\\begin{proof}\n Let $\\ww=\\ww_{\\SS}$ for simplicity. Consider a sampled edge between $i$ and $j$ with weight $\\ww_{\\mathrm{new}}(i,j) = \\ww(i,v)\\ww(j,v)/(\\ww(i,v)+\\ww(j,v))$.\n \\begin{align*}\n \\norm{\\LL^{\\pinv/2} \\ww_{\\mathrm{new}}(i,j) \\bb_{ij}\\bb_{ij}^{\\trp}\\LL^{\\pinv/2}}\n &= \\ww_{\\mathrm{new}}(i,j) \\norm{\\LL^{\\pinv/2} \\bb_{ij}\\bb_{ij}^{\\trp}\\LL^{\\pinv/2}} \\\\\n &= \\ww_{\\mathrm{new}}(i,j) \\norm{ \\LL^{\\pinv/2}\\bb_{ij} }^2 \\\\\n &\\leq \\ww_{\\mathrm{new}}(i,j) \\left( \\norm{\\LL^{\\pinv/2}\\bb_{iv}}^2 + \\norm{\\LL^{\\pinv/2}\\bb_{jv}}^2 \\right) \\\\\n &= \\frac{\\ww(j,v)}{\\ww(i,v)+\\ww(j,v)} \\norm{\\LL^{\\pinv/2} \\ww(i,v)\\bb_{iv}\\bb_{iv}^{\\trp}\\LL^{\\pinv/2}} + \\\\\n & ~\\quad \\frac{\\ww(i,v)}{\\ww(i,v)+\\ww(j,v)} \\norm{\\LL^{\\pinv/2} \\ww(j,v)\\bb_{jv}\\bb_{jv}^{\\trp}\\LL^{\\pinv/2}} \\\\\n &\\leq \\frac{\\ww(j,v)}{\\ww(i,v)+\\ww(j,v)} R + \\frac{\\ww(i,v)}{\\ww(i,v)+\\ww(j,v)} R \\\\\n &= R\n \\end{align*}\n The first inequality uses the triangle inequality of effective\n resistance in $\\LL$, in that effective resistance is a distance as\n we proved in Chapter~\\ref{cha:pinver}. The second inequality just uses the conditions of this lemma.\n\\end{proof}\n\\begin{remark}\n Lemma \\ref{lem:cliquesample_bounded_norm} only requires that each single multiedge has small norm instead of that the sum of all edges between a pair of vertices have small norm. And this lemma tells us, after sampling, each multiedge in the new graph still satisfies the bounded norm condition.\n\\end{remark}\nFrom the Lemma, we can conclude that each edge sample $\\YY_{\\pi(i),e}$\nsatisfies\n$\\norm{\\Phi(\\YY_{\\pi(i),e})} \\leq R$\nprovided the assumptions of the\nLemma hold. Let's record this observation as a Lemma.\n\\begin{lemma}\n If for all $e \\in \\vstar{v}{\\SS_i}$,\n \\[\n \\norm{\\Phi(\\ww_{\\SS_i }(e)\\bb_{e} \\bb_{e}^{\\trp})}\n \\leq\n R\n .\n \\]\n then all $e \\in \\vstar{\\pi(i)}{\\SS_i}$,\n \\[\n \\norm{\\Phi(\\YY_{\\pi(i),e})} \\leq R\n .\n \\]\n\\end{lemma}\n\n\\paragraph{Preprocessing by multi-edge splitting.}\nIn the original graph of Laplacian $\\LL$ of graph $G = (V,E,\\ww)$, we\nhave for each edge $\\hat{e}$ that\n\\[\n \\ww(\\hat{e}) \\bb_{\\hat{e}}\\bb_{\\hat{e}}^{\\trp} \\preceq \\sum_e \\ww(e) \\bb_{e}\\bb_{e}^{\\trp}= \\LL\n\\]\nThis also implies that\n\\[\n \\norm{\\LL^{\\pinv/2} \\ww(\\hat{e}) \\bb_{\\hat{e}}\\bb_{\\hat{e}}^{\\trp} \\LL^{\\pinv/2} }\\leq 1.\n\\]\nNow, that means that if we split every original edge $e$ of the graph\ninto $K$ multi-edges $e_1, \\ldots e_K$, with a fraction $1/K$ of the\nweight, we get a new graph $G' = (V, E', \\ww')$ such that\n\\begin{claim}\n \\label{clm:edgesplit}\n \\noindent\n \\begin{enumerate}\n \\item $G'$ and $G$ have the same graph Laplacian.\n \\item $\\abs{E'} = K\\abs{E}$\n \\item For every multi-edge in $G'$\n \\[\n \\norm{\\LL^{\\pinv/2} \\ww'(e) \\bb_{e}\\bb_{e}^{\\trp} \\LL^{\\pinv/2}\n }\\leq 1/K.\n \\]\n \\end{enumerate}\n\\end{claim}\nBefore we run Approximate Gaussian Elimination, we are going to do\nthis multi-edge splitting to ensure we have control over multi-edge\nsample norms.\n%\nCombined with Lemma~\\ref{lem:cliquesample_bounded_norm} immediately\nestablishes the next lemma, because we start off with all multi-edges\nhaving bounded norm and only produce multi-edges with bounded norm.\n\\begin{lemma}\n \\label{lem:edgesampnorm}\n When Algorithm~\\ref{alg:apxge} is run on the (multi-edge) Laplacian of $G'$,\n arising from splitting edges of $G$ into $K$ multi-edges, the every\n edge sample $\\YY_{\\pi(i),e}$ satisfies\n \\[\n \\norm{\\Phi(\\YY_{\\pi(i),e})} \\leq 1/K\n .\n \\]\n\\end{lemma}\nAs we will see later $K = 200 \\log^2 n$ suffices.\n\n\\subsection{Random Matrix Concentration from Trace Exponentials}\nLet us recall how matrix-valued variances come into the picture when\nproving concentration following the strategy from Matrix Bernstein in\nChapter~\\ref{cha:randmat}.\n\nFor some matrix-valued random variable $\\XX \\in \\symsetn$,\nwe'd like to show $\\Pr[ \\norm{\\XX} \\leq 0.5 ]$.\nUsing Markov's inequality, and some observations about matrix\nexponentials and traces, we saw that for all $\\theta > 0$,\n\\begin{equation}\n \\label{eq:masterprob}\n \\Pr[ \\norm{\\XX} \\geq 0.5 ]\n \\leq\n \\exp(-0.5\\theta ) \\left(\\E{\\trace{ \\exp\\left(\\theta \\XX\\right)}}\n +\n \\E{\\trace{ \\exp\\left(-\\theta \\XX\\right)}}\n \\right)\n .\n\\end{equation}\nWe then want to bound $\\E{\\trace{ \\exp\\left(\\theta \\XX\\right)}}$\nusing Lieb's theorem.\nWe can handle $\\E{\\trace{ \\exp\\left(-\\theta \\XX\\right)}}$ similarly.\n\\begin{theorem}[Lieb]\n \\label{thm:Lieb}\n Let $f:S^n_{++}\\to\\R$ be a matrix function given by\n \\[ f(\\AA) = \\trace{\\exp\\left(\\HH+\\log(\\AA)\\right)} \\]\n for some $\\HH\\in S^n$. Then $-f$ is convex (i.e. $f$ is concave).\n\\end{theorem}\nAs observed by Tropp, this is useful for proving matrix concentration statements.\nCombined with Jensen's inequality, it gives that for a random matrix\n$\\XX \\in \\symsetn $ and a fixed $\\HH \\in \\symsetn$\n\\[\n \\E{\\trace{ \\exp\\left(\\HH+\\XX\\right)}}\n \\leq\n \\trace{ \\exp\\left(\\HH+\\log(\\E{\\exp(\\XX)})\\right)}\n .\n\\]\nThe next crucial step was to show that it suffices to obtain an upper\nbound on the matrix $\\E{\\exp(\\XX)}$ w.r.t the Loewner order.\nUsing the following three lemmas, this conclusion is an immediate\ncorollary.\n\\begin{lemma}\\label{lem:trexpmono}\n If $\\AA\\preceq\\BB$, then $\\trace{\\exp(\\AA)}\\leq\\trace{\\exp(\\BB)}$.\n % i.e. $\\ZZ \\mapsto \\trace{\\exp(\\ZZ)}$ is monotone increasing.\n\\end{lemma}\n\\begin{lemma}\\label{lem:logmono}\n If $0\\prec\\AA\\preceq\\BB$, then $\\log(\\AA)\\preceq\\log(\\BB)$.\n\\end{lemma}\n\\begin{lemma}\\label{lem:ineq_log}\n $\\log(\\II+\\AA) \\preceq \\AA$ for $\\AA\\succ-\\II$.\n\\end{lemma}\n%\n\\begin{corollary}\n \\label{cor:trexpub}\n For a random matrix\n $\\XX \\in \\symsetn $ and a fixed $\\HH \\in \\symsetn$,\n if $\\E{\\exp(\\XX)} \\preceq \\II + \\UU$ where $\\UU \\succ -\\II$, then\n\\[\n \\E{\\trace{ \\exp\\left(\\HH+\\XX\\right)}}\n \\leq\n \\trace{ \\exp\\left(\\HH+\\UU\\right)}\n .\n\\]\n\\end{corollary}\n\n% \\begin{corollary}\\label{cor:trexpmono}\n% If $\\AA\\preceq\\BB$, then $\\trace{\\exp(\\AA)}\\leq\\trace{\\exp(\\BB)}$,\n% i.e. $\\XX \\mapsto \\trace{\\exp(\\XX)}$ is monotone increasing.\n% \\end{corollary}\n% \\begin{lemma}\\label{lem:logmono}\n% If $0\\prec\\AA\\preceq\\BB$, then $\\log(\\AA)\\preceq\\log(\\BB)$.\n% \\end{lemma}\n\n\\subsection{Mean-Exponential Bounds from Variance Bounds}\nTo use Corollary~\\ref{cor:trexpub}, we need to construct useful upper\nbounds on $\\E{\\exp(\\XX)}$.\nThis can be done, starting from the following lemma.\n\\begin{lemma}\\label{lem:ineq_exp}\n $\\exp(\\AA) \\preceq \\II + \\AA + \\AA^2$ for $\\|\\AA\\|\\leq1$.\n\\end{lemma}\nIf $\\XX$ is zero-mean and $\\norm{\\XX} \\leq 1$,\nthis means that $\\E{\\exp(\\XX)} \\preceq \\II + \\E{\\XX^2}$,\nwhich is how we end up wanting to bound the matrix-valued variance $\\E{\\XX^2}$.\nIn the rest of this Subsection, we're going to see the matrix-valued variance of the stopped\nmartingale is bounded throughout the algorithm.\n\nFirstly, we note that for a single edge sample $\\XXtil_{i,e}$,\nby Lemma~\\ref{lem:edgesampnorm}, we have that\n\\[\n \\norm{\\XXtil_{i,e}} \\leq\n \\norm{\\Phi\\left( \\YY_{\\pi(i),e} - \\E{\\YY_{\\pi(i),e}} \\right)}\n \\leq 1/K,\n\\]\nusing that $\\norm{\\AA-\\BB} \\leq \\max( \\norm{\\AA}, \\norm{\\BB} )$, for\n$\\AA, \\BB \\succeq \\matzero$, and $\\norm{\\E{\\AA}} \\leq \\E{\\norm{\\AA}}$\nby Jensen's inequality.\n\nThus, if $0 < \\theta \\leq K$, we have that\n\\begin{align}\n \\label{eq:edgelogexp}\n \\E{ \\exp(\\theta \\XXtil_{i,e}) \\mid \\text{preceding samples} }\n &\\preceq\n \\II +\n \\E{(\\theta \\XXtil_{i,e})^2 \\mid \\text{preceding samples} }\n \\\\ \\nonumber &\\preceq\n \\II +\n \\frac{1}{K}\\theta^2\\cdot\n \\E{\\Phi(\\YY_{\\pi(i),e}) \\mid \\text{preceding samples} }\n\\end{align}\n% \\begin{lemma}\n% \\label{lem:edgelogexp}\n% \\[\n% \\log \\Ex{\\XX_{v_i,d_i-1}}\\exp(\\theta\\XX_{v_i,d_i}) \\preceq\n% \\theta^2 R \\cdot\n% \\Ex{\\XX_{v_i,d_i-1}}\\Phi(\\YY_{v_i,d_i})\n% \\]\n% \\end{lemma}\n\n\n\n\\subsection{The Overall Mean-Trace-Exponential Bound}\n\nWe will use $\\Ex{( 0$, is there a subset $N' \\subseteq \\{1, \\dots, n\\}$ such that $\\Sigma_{i \\in N'} v_i < V$ (the objects in $N'$ can be placed in the knapsack) and $\\Sigma_{i \\in N'} s_i \\geq S$ (their price exceeds the given limit $S$)?\n \\end{itemize}\n\n \\solution\\\\\n The complete proof consists of two steps. Given any candidate solution (purported certificate) of this problem which divides the $n$ independent tasks into two bins, we can easily compute the total execution time of tasks in each bin, and then check if both of them $\\leq D$. Since the solution can be verified in linear time, by definition\\autocite[1049]{Cormen09} it is an NP problem.\\\\\n\n After Stephen Cook proved 3-SAT to be the first known NP-complete problem in 1971\\autocite{Cook71}, the next year Richard Karp further proved that another 21 common computational problems are all NP-complete\\autocite{Karp72}, one of which is the knapsack problem. To prove NP-hardness, we will show that the knapsack problem is reducible to the two-machine scheduling problem.\\\\\n\n For every instance of the knapsack problem, construct $n + 2$ independent tasks with execution time $\\{a_1, \\dots, a_{n+2}\\}$ and a deadline $D = \\Sigma_{i=1}^{n} a_i + 2$. Let\n \\begin{align*}\n a_i &= v_i + s_i \\,,\\, 1 \\le i \\le n\\\\\n a_{n+1} &= \\Sigma_{i \\in P} v_i + \\Sigma_{i \\in P} s_i + 2\\\\\n a_{n+2} &= \\Sigma_{i=1}^{n} a_i - \\Sigma_{i \\in P} v_i - \\Sigma_{i \\in P} s_i + 2\n \\end{align*}\n\n where $P$ denotes the set of items placed in the knapsack. It is evident that this mapping takes polynomial time in the size of input. Now let's prove that this translation is indeed a reduction.\\\\\n\n $\\Rightarrow$ First, suppose that the knapsack problem has a true instance, which means there is indeed a subset of items $P \\subseteq \\{1, \\dots, n\\}$ such that\n \\begin{align*}\n \\Sigma_{i \\in P} v_i &< V\\\\\n \\Sigma_{i \\in P} s_i &\\geq S\n \\end{align*}\n\n Let\n \\begin{align*}\n M_1 &= \\{a_i \\mid i \\in P\\} \\cup \\{a_{n+2}\\}\\\\\n M_2 &= \\{a_i \\mid i \\notin P\\} \\cup \\{a_{n+1}\\}\n \\end{align*}\n\n Then we have\n \\begin{align}\n \\text{the sum of } M_1 &= \\Sigma_{i \\in P} v_i + \\Sigma_{i \\in P} s_i + \\Sigma_{i=1}^{n} a_i - \\Sigma_{i \\in P} v_i - \\Sigma_{i \\in P} s_i + 2 = \\Sigma_{i=1}^{n} a_i + 2\\\\\n \\text{the sum of } M_2 &= \\Sigma_{i=1}^{n} a_i - \\Sigma_{i \\in P} v_i - \\Sigma_{i \\in P} s_i + \\Sigma_{i \\in P} v_i + \\Sigma_{i \\in P} s_i + 2 = \\Sigma_{i=1}^{n} a_i + 2\n \\end{align}\n\n From (1) and (2), it's clear that we have $M_1=M_2=D$, also notice that $M_1 \\cup M_2 $ is the set of all tasks, therefore we have found a feasible schedule for two identical machines.\\\\\n\n $\\Leftarrow$ Conversely, suppose that the translated two-machine scheduling problem has a true instance, so the set of $n + 2$ independent tasks can be divided into two subsets, which can be scheduled on two machines $M_1$ and $M_2$, respectively, by the deadline $D = \\Sigma_{i=1}^{n} a_i + 2$.\\\\\n\n Since execution time must be nonnegative, and notice that $a_{n+1} + a_{n+2} = \\Sigma_{i=1}^{n} a_i + 4>D$, this means $a_{n+1}$ and $a_{n+2}$ must not be executed on the same machine. Without loss of generality, assume that $a_{n+2}$ is scheduled on $M_2$, then the execution time of the rest of the tasks scheduled on $M_2$ must sum up to\n \\begin{align*}\n D - a_{n+2} = \\Sigma_{i=1}^{n} a_i + 2 - a_{n+2} = \\Sigma_{i \\in P} v_i + \\Sigma_{i \\in P} s_i\n \\end{align*}\n\n As per our reduction assumption, it imples that this subset $P \\subseteq \\{1, \\dots, n\\}$ has the property\n \\begin{align*}\n \\Sigma_{i \\in P} v_i &< V\\\\\n \\Sigma_{i \\in P} s_i &\\geq S\n \\end{align*}\n\n Thus, it makes a true instance of the knapsack problem as well.\\\\\n\n Now we have \\textsc{Knapsack} $\\leq_p$ \\textsc{Two-machine scheduling}, which implies that the two-machine scheduling problem is at least as hard as the knapsack. This completes our proof that two-machine scheduling is also NP-complete.\\\\\n\n The optimization variant of this is the so-called \\textit{makespan} problem, in our case with two machines denoted by $P2||C_{max}$\\autocite{Schuurman00}, which can be stated as: Given the execution time $a_1, a_2 \\dots, a_n$ of $n$ independent tasks, find a way to schedule the $n$ tasks on two identical parallel machines in a non-preemptive fashion, so as to minimize the maximum task completion time. In other words, we would like to do our best to equally assign the $n$ tasks to both machines, such that the total execution time on the machine that finishes last is minimized. This variant is NP-hard.\\autocite{Karp72}\n\n \\section{Approximate Maximum Clique}\n Let $G = (V,E)$ be an undirected graph. For integer $k \\geq 1$ define $G^{(k)} = (V^{(k)}, E^{(k)})$ such that $V^{(k)} = \\{(v_1, v_2, \\dots, v_k) | v_i \\in V, 1 \\leq i \\leq k\\}$ and $((v_1, v_2, \\dots, v_k), (w_1 ,w_2 ,\\dots ,w_k)) \\in E^{(k)}$ iff either $(v_i, w_i) \\in E$ or $v_i = w_i$ for all $1 \\leq i \\leq k$.\n \\begin{enumerate}[leftmargin=*]\n \\item Prove $|C^{(k)}| = |C|^k$, where $C^{(k)}$ and $C$ are the maximum cliques of $G^{(k)}$ and $G$, respectively.\n \\item Argue that the existence of an approximation algorithm for finding the maximum clique with a constant approximation ratio implies the existence of a polynomial-time approximation scheme for finding the maximum clique.\n \\end{enumerate}\n\n \\solution\n \\begin{enumerate}[leftmargin=*, topsep=0pt]\n \\item Let $V' = \\{(v_1, v_2, \\dots, v_k) | v_i \\in C, 1 \\leq i \\leq k\\} \\subseteq V^{(k)}$, where $C$ is the maximum clique of $G$, so every pair of vertices $(v_i, v_j)$ is adjacent, clearly $V'$ is a clique of $G^{(k)}$. Since each element $v_i$ in the $k$-tuple $(v_1, v_2, \\dots, v_k)$ comes from $C$, the size of $V'$ is $|C|^k$. Therefore, the size of the maximum clique of $G^{(k)}$ is at least $|C|^k$. We will prove by induction that this is indeed the size of the maximum clique.\n\n In the base case, when $k = 1$, $G^{(1)} = G$, we have $|C^{(1)}| = |C|^1$. Now suppose that $|C^{(k)}| = |C|^k$ is true for $1 \\leq i \\leq k$. Let $C^{(k+1)} = \\{(p_i, v_1, v_2, \\dots, v_k)\\}$ where $p_i$ is some vertex in $G$.\n\n Since the adjacency condition of $(v_1, v_2, \\dots, v_k)$ and $(w_1, w_2, \\dots, w_k)$ only requires the adjacency of $v_i$ and $w_i$, the choice of vertices at index $i$ is independent of vertices at any other index. Hence, if we remove the prefix vertex $p_i$ from every tuple in $C^{(k+1)}$, the remaining $k$-tuples must form a maximum clique of $G^{(k)}$, otherwise we can always do better. By induction hypothesis, it has size $|C|^k$.\n\n In order to make $C^{(k+1)}$ a clique of $G^{(k+1)}$, we must choose a set of prefixes $\\{p_i\\} \\subseteq V$ such that every pair of vertices $(p_i, p_j)$ is connected by an edge. Moreover, to make it a maximum clique, we also want to maximize the number of prefixes we choose. In other words, we are trying to find a maximum set of mutually adjacent vertices from $G$ and prepend them to every $k$-tuple in $C^{(k)}$, so we should choose the set of prefix vertices $\\{p_i\\} = C$. Therefore, the size of $C^{(k+1)}$ is $|C|^k \\times |C| = |C|^{k+1}$.\n\n \\item Given a $c$-approximation algorithm for finding the maximum clique, we can apply it on $G^{(k)}$ and obtain a clique of size $n$, while the true maximum clique of $G^{(k)}$ has size $|C|^k$.\n\n By definition of an approximation ratio, $|C|^k/n \\leq c$, so we have $|C|/n^{1/k} \\leq c^{1/k}$. For any $\\varepsilon > 0$, we can choose a positive integer $k > \\log_{1+\\varepsilon}{c}$, then $|C|/n \\leq |C|/n^{1/k} \\leq c^{1/k} < 1 + \\varepsilon$. Since it is polynomial in the size of input, so we have a polynomial-time approximation scheme for finding the maximum clique.\n \\end{enumerate}\n\n \\section{Linear Inequality Feasibility}\n Given a set of $m$ linear inequalities over $n$ variables, the linear inequality feasibility problem asks whether there exists a set of the instances that satisfies all the linear inequalities simultaneously.\n \\begin{enumerate}[leftmargin=*]\n \\item Prove that we can use an algorithm for linear programming to solve linear inequality feasibility problems. The number of variables and constraints used in the linear programming problem must be polynomial in $n$ and $m$.\n \\item Prove that we can use an algorithm for the linear inequality feasibility problem to solve linear programming problems. The number of variables and linear inequalities must be polynomial in the number of variables and constraints of the linear program.\n \\end{enumerate}\n\n \\solution\n \\begin{enumerate}[leftmargin=*, topsep=0pt]\n \\item Given a set of linear inequalities, we can simply formulate the equality feasibility problem as a linear program. First, we can set the objective function to be a constant such as 0 because there's nothing to maximize or minimize in this case. Next, we need to transform those linear inequalities into their slack form, and then apply the simplex algorithm. If the linear program returns a feasible solution, it implies that all the linear inequalities can be satisfied at the same time. Otherwise, those linear inequalities are not feasible. The variables and constraints in the linear program are exactly the same as in the equality feasibility problem so they are polynomial in $n$ and $m$.\n\n \\item Given a linear program in standard form, we want to maximize the objective function $\\Sigma_{j=1}^{n} c_jx_j$ with $n$ variables, subject to $m$ constraints $\\Sigma_{j=1}^{n} a_{ij}x_j \\leq b_i, 1 \\leq i \\leq m$. Suppose we have an algorithm for the linear inequality feasibility problem, then we can apply it on the $m$ constraints of the linear program. If the algorithm fails to find a solution for the linear inequality feasibility problem, it means that those inequalities can not be satisfied, so the linear program should return infeasible. If the algorithm does find an assignment of the $n$ variables that satisfies all the linear inequalities, then the linear program is also feasible.\n\n In order to solve the linear program and maximize the objective function, in each step we first calculate the value of our objective function using the assignment values returned by that algorithm, then we update the linear program by introducing a new constraint. For example, if the algorithm finds an assignment $\\overline{x}$ which gives an objective function value of $\\Sigma_{j=1}^{n} c_jx_j = k$, we will add another constraint $\\Sigma_{j=1}^{n} c_jx_j > k$ to the original linear program. As long as this updated linear program is feasible, it must be equivalent to the original linear program because both the objective function and all other constraints are not affected. If we iteratively run this step, we are getting closer to the optimal solution. We will repeat this step until the algorithm detects that the updated constraints are not feasible anymore, in which case we stop and return the last calculated objective function value as the optimal solution.\n\n Whenever the algorithm returns a feasible assignment so that all constraints are satisfied, it represents a point that's somewhere in the simplex. The basic idea is to update that point until it reaches a local maximum, which is also a global maximum because the simplex is convex. Since we are only adding 1 more constraint in each step, we can optimistically assume that the number of linear inequalities is polynomial in the number of constraints. If that's not true, we can modify the constraint to be $\\Sigma_{j=1}^{n} c_jx_j > k + \\alpha$, where $\\alpha$ is a reasonable step size to choose in different trials. Unless the original linear program has an unbounded solution, we will eventually converge to an optimal solution.\n \\end{enumerate}\n\n \\section{Ghostbusters and Ghosts}\n A group of $n$ ghostbusters is battling $n$ ghosts. Each buster carries a proton pack, which shoots a stream at a ghost, eradicating it. A stream goes in a straight line and terminates when it hits the ghost. The ghostbusters decide upon the following strategy: They will pair off with the ghosts, forming $n$ ghostbuster-ghost pairs, and then simultaneously each ghostbuster will shoot a stream at his chosen ghost. As we all know, it is \\textit{very} dangerous to let streams cross, and so the busters must choose pairings for which no streams will cross. Assume that the position of each buster and each ghost is a fixed point in the plane and that no three positions are colinear.\n \\begin{enumerate}[leftmargin=*]\n \\item Argue that there exists a line passing through one Ghostbuster and one ghost such that the number of Ghostbusters on one side of the line equals the number of ghosts on the same side. Describe how to find such a line in $O(n \\log n)$ time.\n \\item Give an $O(n^2 \\log n)$-time algorithm to pair Ghostbusters with ghosts such that no streams cross.\n \\end{enumerate}\n\n \\solution\n \\begin{figure}[H]\n \\centering\n \\begin{tikzpicture}\n \\draw[black, dotted, thick]\n (-0.5,0) -- (5.5,0)\n (0,-0.5) -- (0,3.5);\n \\draw[->](0,0) -- (5,1);\n \\draw[->](0,0) -- (4,2);\n \\draw[->](0,0) -- (5,2);\n \\draw[->](0,0) -- (3,3);\n \\draw[->](0,0) -- (3,2);\n \\draw[->](0,0) -- (1,2);\n \\draw[color=black] (-0.25,-0.25) node {$P_0$};\n \\draw (5,1) node;\n \\draw (4,2) node;\n \\draw (5,2) node;\n \\draw (3,3) node;\n \\draw (3,2) node;\n \\draw (1,2) node;\n \\draw [fill=black] (5,1) circle (1.0pt);\n \\draw [fill=black] (4,2) circle (1.0pt);\n \\draw [fill=black] (5,2) circle (1.0pt);\n \\draw [fill=black] (3,3) circle (1.0pt);\n \\draw [fill=black] (3,2) circle (1.0pt);\n \\draw [fill=black] (1,2) circle (1.0pt);\n \\draw [fill=ffqqqq] (0,0) circle (1.5pt);\n \\end{tikzpicture}\n \\caption{Graham Scan}\n \\end{figure}\n\n \\begin{enumerate}[leftmargin=*, topsep=0pt]\n \\item First we pick the leftmost bottom point $P_0$ as the origin. For the remaining points, we can easily sort them by polar angle in counterclockwise order. Concretely, the cross product $(P_1 - P_0) \\times (P_2 - P_0)$ gives us a comparison relationship between any two points $P_1$ and $P_2$, which can be applied to any comparison-based sorting algorithm such as merge sort.\n\n Given the sorted points by polar angle with respect to $P_0$, we then visit them one by one in this order, and count the number of ghostbusters and ghosts along the way. We would stop at a point $P_k$ when the following two conditions are met:\n\n \\begin{itemize}[leftmargin=*, topsep=0pt]\n \\item Either $P_0$ is a ghostbuster and $P_k$ is a ghost, or $P_0$ is a ghost and $P_k$ is a ghostbuster.\n \\item The number of ghostbusters and ghosts visited so far are equal (including $P_0$ and $P_k$).\n \\end{itemize}\n\n It follows that the line passing through $P_0$ and $P_k$ is the solution. Since the number of ghostbusters and ghosts are equal, we are guaranteed to find such a line. It takes $O(n\\log n)$ to sort and $O(n)$ to scan, so the running time is $O(n\\log n)$ in total.\n\n \\item Now we have paired one ghostbuster with one ghost in $O(n\\log n)$ time, applying this algorithm recursively on both sides of this line will pair all the ghostbusters with ghosts.\n\n That is, for the subset of points on each side, we choose the origin with the minimum coordinate, sort the remaining points on this side and then scan through them until we find a new line. During each recursion step, work done on one side of the line is independent from the other, so that no lines will ever cross. We need to find $n$ lines, so this divide-and-conquer approach runs in $O(n^2\\log n)$ time.\n \\end{enumerate}\n\n \\section{Divide and Conquer Prefix Computations}\n We use again divide and conquer method to perform prefix computation (with $\\circ$ as binary operator) on a sequence $S$ of size $n$. This time the running time will be $O(\\log \\log n)$ time using $O(n/\\log \\log n)$ processors. This algorithm would be optimal, faster than the one mentioned in the book, but also using more processors. To obtain such an algorithm we proceed as follows:\n\n \\begin{enumerate}[leftmargin=*]\n \\item[a] Divide the sequence $S$ into $n^{1/2}$ subsequences of size $n^{1/2}$ each. Perform prefix computation on each sequence recursively using $n^{1/2}$ processors. Let the result of the computation on the $i$-th sequence be $s(i,1), s(i,2), \\dots, s(i,n^{1/2})$.\n \\item[b] Perform a prefix computation on the sequence $s(1,n^{1/2}), s(2,n^{1/2}), \\dots, s(n^{1/2}-1,n^{1/2})$ using $n$ processors. Let the result of this computation be $s^\\prime(1,n^{1/2}), s^\\prime(2,n^{1/2}), \\dots, s^\\prime(n^{1/2}-1,n^{1/2})$.\n \\item[c] For all $1 \\leq i < n^{1/2}$ use the binary operator $\\circ$ to combine $s^\\prime(i,n^{1/2})$ with all the elements of $s(i+1,1),s(i+1,2),\\dots,s(i+1,n^{1/2})$ using $n^{1/2}$ processors per subsequence.\n \\end{enumerate}\n\n \\begin{enumerate}[leftmargin=*]\n \\item Show that the algorithm described above uses $n$ processors to run in $O(\\log \\log n)$ time.\n \\item \\label{l} Show how the number of processors can be reduced to $O(n/\\log \\log n)$ while maintaining the $O(\\log \\log n)$ running time, thus achieving the optimal cost $O(n)$.\n \\item Discuss the implications of the result obtained in Question \\ref{l} on related computations such as array packing.\n \\end{enumerate}\n\n \\solution\n \\begin{enumerate}[leftmargin=*, topsep=0pt]\n \\item In step 1 we recursively perform prefix computation on each subsequence of size $n^{1/2}$ using $n^{1/2}$ processors. Suppose our recursive algorithm on the entire sequence $S$ has running time $t(n)$, then the recursion call on each subsequence takes $t(n^{1/2})$. Since this is done in parallel, this step takes $t(n^{1/2})$.\n\n In step 2, the sequence $s(1,n^{1/2}), s(2,n^{1/2}), \\dots, s(n^{1/2}-1,n^{1/2})$ contains every last element of a subsequence except the first one, so there are only $n^{1/2}-1$ elements, but we have $n$ processors available. We can take advantage of all of them by letting $n^{1/2}$ processors work on each element simultaneously, that is, for some element $s(i,n^{1/2})$, we use $n^{1/2}$ processors in parallel to perform binary operations on $s(i,n^{1/2})$ with all other elements $s(j,n^{1/2})$, where $j \\neq i, 1 \\leq j \\leq n^{1/2}-1$. Since we have $n$ processors so that all elements can update in parallel, this step takes $O(1)$. However, this does not work for all binary operations, in fact, the binary operation must be both associative and commutative, such as addition, multiplication and union, but not concatenation because concurrent writes would mess up the order. Besides, this speed up is possible only with the Combining CRCW PRAM model such that concurrent reads and writes can be effectively reconciled.\n\n In step 3, for each subsequence we use $n^{1/2}$ processors to combine its elements with the last element in the preceding subsequence. Each element can be mapped to a processor, so this single operation is done in parallel, this step takes $O(1)$ time.\n\n Now combining these results we have $t(n) = t(n^{1/2}) + O(1)$. To derive the running time from this equation, we substitute $n$ with $e^m$, so $t(e^m) = t(e^{m/2}) + O(1)$. Let $s(m) = t(e^m)$, then $s(m) = s(m/2) + O(1)$. According to the Master's Theorem, this implies that\n \\begin{equation*}\n s(m) = O(\\log m) \\hspace{3mm} \\Rightarrow \\hspace{3mm} t(n) = t(e^m) = s(m) = O(\\log m) = O(\\log \\log n)\n \\end{equation*}\n\n \\item If we want to reduce the number of processors to $O(n/\\log \\log n)$, the previous partition will not work, we need a new partition. Besides, we want to follow the previous three steps, but need to make some modifications.\n\n In concrete, we will divide $S$ into $n/\\log \\log n$ subsequences or groups, each group has size $\\log \\log n$. Since the number of groups is equal to the number of processors, each group can only have one processor, so in step 1, prefix computation must be executed sequentially within each group, while all groups do this in parallel. Hence, this step takes $O(\\log \\log n)$.\n\n In step 2, the sequence will be $s(1,\\log \\log n), s(2,\\log \\log n), \\dots, s(n/\\log \\log n-1,\\log \\log n)$ because we only have $n/\\log \\log n$ groups, processors and last elements. To perform prefix computation on this sequence, we will adopt the algorithm in Question 1. As we have shown in Question 1, that algorithm runs in $O(\\log \\log n)$ time on $n$ elements using $n$ processors, so here with $n/\\log \\log n$ elements and processors, it will take $O(\\log \\log(n/\\log \\log n))$ time to run. By removing the trivial terms, we have\n \\begin{equation*}\n O(\\log \\log(n/\\log \\log n)) = O(\\log(\\log n-\\log(\\log \\log n))) = O(\\log \\log n)\n \\end{equation*}\n\n In step 3, for the same reason as in step 1, the combining operation has to be executed sequentially for each group due to the lack of processors. As a result, each group takes $O(\\log \\log n)$ to run, while different groups run in parallel with each other. This step takes $O(\\log \\log n)$.\n\n In summation, the total running time is $3 \\times O(\\log \\log n) = O(\\log \\log n)$ which is identical to the previous one, but the cost is optimal as we are now using less than $n$ processors.\n\n \\item The result obtained in Question 2 has several implications.\n\n We have seen that it is possible to devise an optimal parallel algorithm that takes a suboptimal algorithm as an intermediate step. The divide and conquer approach allows us to divide a problem into many subproblems which can be solved in parallel, and the way we partition the data can sometimes make a difference. In addition, different PRAM models may lead to different optimal algorithms to solve the same problem, we are free to choose among them depending on the scenario. This gives us a lot more flexibility from a practical perspective since the number of processors and hardware is often fixed in real world. For example, compared to the other optimal algorithm discussed in the class notes, this one is also optimal but much faster, so it can be very useful if we are strict on the running time and have sufficient processors that support CRCW PRAM models. It has also proved the possibility of applications being highly scalable based on commutative binary operations such as prefix sum. For example, the array packing problem uses addition as the underlying binary operation, so that as long as the hardware requirement is satisfied, it can easily scale up to boost performance, especially when the input array size is huge.\n \\end{enumerate}\n\n \\section{Descending in a Hypercube}\n Let $N = 2^g$ data be stored in the processors of a hypercube, one value per processor (so that the value $x_i$ is stored in $P_i$, $0 \\leq i < N$). A technique with wide applicability to hypercube algorithms is called \\textsc{Descend} and consists of $g$ iterations. During the $j$-th iteration a basic binary operation \\textsc{Operation}$(P_i, P_l)$ is performed on data in cores whose indices differ by $2^j$. Here is the pseudocode for this process, with $i_{g-1}i_{g-2}\\dots i_{1}i_{0}$ the binary representation of index $i$:\n\n \\newpage\n\n \\begin{algorithm}\n % \\renewcommand\\thealgorithm{} % suppress the caption number\n \\caption{\\textsc{Descend}}\n \\label{ad}\n \\begin{algorithmic}[1]\n \\For{$j \\leftarrow g - 1$ \\textbf{to} $0$} \\Comment{iterate over each bit}\n \\For{$i \\leftarrow 0$ \\textbf{to} $2^{g}-1$} \\Comment{iterate over each core}\n \\If{$i_j = 0$} \\textsc{Operation}$(P_i, P_{i+2^j})$\n \\hspace{2mm}\\raisebox{.5\\baselineskip}[0pt][0pt]{$\\left.\\rule{0pt}{1.0\\baselineskip}\\right\\}\\ \\mbox{in parallel}$} \\Comment{operate on adjacent cores}\n \\end{algorithmic}\n \\end{algorithm}\n\n \\noindent If \\textsc{Operation} requires constant time then \\textsc{Descend} runs in $O(\\log N)$ time. The dual to \\textsc{Descend} is \\textsc{Ascend}, in which $j$ in the outer loop varies from $0$ to $g-1$.\n\n \\begin{enumerate}[leftmargin=*]\n \\item Show how \\textsc{Descend} can be used to obtain an algorithm that computes the sum of all the values $x_i$ and places the result in $P_0$.\n \\item Use either \\textsc{Ascend} or \\textsc{Descend} to broadcast the datum held by some processor $P_k$ to all the other processors.\n \\item Suggest other problems that can be solved using these paradigms.\n \\end{enumerate}\n\n \\solution\\\\\n If two integers differ by $2^k$, then in their binary representation, all bits must be the same except the $k$-th bit where they differ by 1. That is, in the $j$-th iteration of the \\textsc{Descend} algorithm, the binary operation is applied on any pair of processors whose indices differ only at the $j$-th bit. A bit can be either 0 or 1, only 2 possible values. Therefore, in each iteration, all processors are divided into halves by some bit, one half will exchange data with the other half via the binary operation. After $g$ iterations, we would have divided the set of processors bit by bit in every possible way, so that every processor has talked to all the other processors exactly once, either directly or indirectly.\n\n \\begin{enumerate}[leftmargin=*]\n \\item[1.] Since the addition operation is commutative, it doesn't matter if we loop from the highest bit or the lowest bit, so we don't mind whether the \\textsc{Descend} or \\textsc{Ascend} algorithm is being used. In order to place the final result in $P_0$ whose index is a bunch of zeros in binary format, during the $j$-th step we must store the sum of $P_i$ and $P_{i+2^j}$ into $P_i$ because $i_j=0$. Then after $g$ iterations, the value in each processor will be added exactly once to $P_0$. So we define\n \\begin{enumerate}\n \\item[] \\textsc{Operation}($P_i$, $P_{i+2^j}$) to be \\fbox{$P_i$ = $P_i$ + $P_{i+2^j}$}\n \\end{enumerate}\n As an example, we can graphically illustrate how it works in a 3 dimensional hypercube. In the figure below, processors $P_0,P_1,\\dots,P_7$ correspond to the 8 corner vertices of the cube, which are represented in their binary format as $(0,0,0),(0,0,1),\\dots,(1,1,1)$. For simplicity, suppose that initially each processor has value $x_i=1$, so we expect $P_0$ to be 8 in the end.\n\n In the first iteration the algorithm looks at the highest bit, which is 0 for vertices on the left side and 1 for vertices on the right side. So as to perform the binary operation, values on the right side are added to those on the left. Vertices whose values have been updated are colored blue. Likewise, the second round looks at the middle bit, so values on the upper side are added to those on the lower side, and so forth. When the algorithm finishes, $P_0(0,0,0)$ has been involved in all iterations and colored blue for three times, hence it has the sum of all values and we color it red. In the end, we managed to compute the sum of 8 numbers in merely 3 steps.\n \\end{enumerate}\n\n % tex code generated by GeoGebra\n \\begin{figure}[H]\n \\centering\n \\begin{tikzpicture}[line cap=round,line join=round,>=triangle 45,x=1cm,y=1cm]\n \\clip(-1,-8.5) rectangle (16,6.6);\n \\draw [line width=0.4pt] (3.5,-7.5)-- (5.5,-5.5);\n \\draw [line width=0.4pt] (7.5,-7.5)-- (9.5,-5.5);\n \\draw [line width=0.4pt] (3.5,-3.5)-- (5.5,-1.5);\n \\draw [line width=0.4pt] (7.5,-3.5)-- (9.5,-1.5);\n \\draw [line width=0.4pt] (0,0)-- (2,2);\n \\draw [line width=0.4pt] (4,0)-- (6,2);\n \\draw [line width=0.4pt] (0,4)-- (2,6);\n \\draw [line width=0.4pt] (4,4)-- (6,6);\n \\draw [line width=0.4pt] (8,0)-- (10,2);\n \\draw [line width=0.4pt] (12,0)-- (14,2);\n \\draw [line width=0.4pt] (8,4)-- (10,6);\n \\draw [line width=0.4pt] (12,4)-- (14,6);\n \\draw (0.7150220913107512,-0.25439503619441556) node[anchor=north west] {(a) right to left};\n \\draw (8.676730486008838,-0.216132368148914) node[anchor=north west] {(b) top to bottom};\n \\draw (4.332842415316643,-7.715615305067217) node[anchor=north west] {(c) back to front};\n \\draw [line width=0.4pt] (0,4)-- (4,4);\n \\draw [line width=0.4pt] (2,6)-- (6,6);\n \\draw [line width=0.4pt] (0,0)-- (4,0);\n \\draw [line width=0.4pt] (2,2)-- (6,2);\n \\draw [line width=0.4pt] (0,4)-- (0,0);\n \\draw [line width=0.4pt] (2,2)-- (2,6);\n \\draw [line width=0.4pt] (6,2)-- (6,6);\n \\draw [line width=0.4pt] (4,4)-- (4,0);\n \\draw [line width=0.4pt] (14,6)-- (14,2);\n \\draw [line width=0.4pt] (8,4)-- (12,4);\n \\draw [line width=0.4pt] (10,6)-- (14,6);\n \\draw [line width=0.4pt] (8,0)-- (12,0);\n \\draw [line width=0.4pt] (10,2)-- (14,2);\n \\draw [line width=0.4pt] (8,4)-- (8,0);\n \\draw [line width=0.4pt] (12,0)-- (12,4);\n \\draw [line width=0.4pt] (10,6)-- (10,2);\n \\draw [line width=0.4pt] (5.5,-1.5)-- (9.5,-1.5);\n \\draw [line width=0.4pt] (3.5,-3.5)-- (7.5,-3.5);\n \\draw [line width=0.4pt] (9.5,-5.5)-- (9.5,-1.5);\n \\draw [line width=0.4pt] (7.5,-7.5)-- (3.5,-7.5);\n \\draw [line width=0.4pt] (5.5,-5.5)-- (5.5,-1.5);\n \\draw [line width=0.4pt] (3.5,-3.5)-- (3.5,-7.5);\n \\draw [line width=0.4pt] (9.5,-5.5)-- (5.5,-5.5);\n \\draw [line width=0.4pt] (7.5,-3.5)-- (7.5,-7.5);\n \\draw [->,line width=0.4pt,dash pattern=on 1pt off 1pt] (3.5,2.9) -- (0.5,2.9);\n \\draw [->,line width=0.4pt,dash pattern=on 1pt off 1pt] (9.622754360465116,3.5725472383720938) -- (9.622754360465116,0.5725472383720931);\n \\draw [->,line width=0.4pt,dash pattern=on 1pt off 1pt] (7,-3) -- (5,-5);\n \\begin{scriptsize}\n \\draw [fill=ffqqqq] (3.5,-7.5) circle (2.5pt);\n \\draw[color=ffqqqq] (4.59499263622975,-7.193846949327817) node {(0,0,0) = 8};\n \\draw [fill=qqqqff] (7.5,-7.5) circle (2.5pt);\n \\draw[color=qqqqff] (8.100883652430046+0.5,-7.093846949327817-0.1) node {(1,0,0) = 4};\n \\draw [fill=qqqqff] (7.5,-3.5) circle (2.5pt);\n \\draw[color=qqqqff] (8.100883652430046+0.5,-3.0953981385729055-0.1) node {(1,1,0) = 2};\n \\draw [fill=qqqqff] (3.5,-3.5) circle (2.5pt);\n \\draw[color=qqqqff] (4.09499263622975+0.5,-3.0953981385729055-0.1) node {(0,1,0) = 4};\n \\draw [fill=black] (5.5,-5.5) circle (2.5pt);\n \\draw[color=black] (6.0979381443298974+0.2,-5.085056876938985-0.1) node {(0,0,1) = 4};\n \\draw [fill=black] (9.5,-5.5) circle (2.5pt);\n \\draw[color=black] (10.103829160530191+0.2,-5.085056876938985-0.1) node {(1,0,1) = 2};\n \\draw [fill=black] (9.5,-1.5) circle (2.5pt);\n \\draw[color=black] (10.103829160530191+0.2,-1.0866080661840742-0.1) node {(1,1,1) = 1};\n \\draw [fill=black] (5.5,-1.5) circle (2.5pt);\n \\draw[color=black] (6.0979381443298974+0.2,-1.0866080661840742-0.1) node {(0,1,1) = 2};\n \\draw [fill=qqqqff] (0,0) circle (2.5pt);\n \\draw[color=qqqqff] (1.1023564064801179,0.30563598759048614) node {(0,0,0) = 2};\n \\draw [fill=black] (4,0) circle (2.5pt);\n \\draw[color=black] (4.595729013254787+0.5,0.40563598759048614-0.1) node {(1,0,0) = 1};\n \\draw [fill=black] (4,4) circle (2.5pt);\n \\draw[color=black] (4.595729013254787+0.5,4.404084798345398-0.1) node {(1,1,0) = 1};\n \\draw [fill=qqqqff] (0,4) circle (2.5pt);\n \\draw[color=qqqqff] (0.6023564064801179+0.5,4.404084798345398-0.1) node {(0,1,0) = 2};\n \\draw [fill=qqqqff] (2,2) circle (2.5pt);\n \\draw[color=qqqqff] (2.6053019145802656+0.2,2.4144260599793173-0.1) node {(0,0,1) = 2};\n \\draw [fill=black] (6,2) circle (2.5pt);\n \\draw[color=black] (6.598674521354934+0.2,2.4144260599793173-0.1) node {(1,0,1) = 1};\n \\draw [fill=black] (6,6) circle (2.5pt);\n \\draw[color=black] (6.598674521354934+0.2,6.412874870734228-0.1) node {(1,1,1) = 1};\n \\draw [fill=qqqqff] (2,6) circle (2.5pt);\n \\draw[color=qqqqff] (2.6053019145802656+0.2,6.412874870734228-0.1) node {(0,1,1) = 2};\n \\draw [fill=qqqqff] (8,0) circle (2.5pt);\n \\draw[color=qqqqff] (9.10162002945508,0.30563598759048614) node {(0,0,0) = 4};\n \\draw [fill=qqqqff] (12,0) circle (2.5pt);\n \\draw[color=qqqqff] (12.59499263622975+0.5,0.40563598759048614-0.1) node {(1,0,0) = 2};\n \\draw [fill=black] (12,4) circle (2.5pt);\n \\draw[color=black] (12.59499263622975+0.5,4.404084798345398-0.1) node {(1,1,0) = 1};\n \\draw [fill=black] (8,4) circle (2.5pt);\n \\draw[color=black] (8.60162002945508+0.5,4.404084798345398-0.1) node {(0,1,0) = 2};\n \\draw [fill=qqqqff] (10,2) circle (2.5pt);\n \\draw[color=qqqqff] (10.604565537555228+0.2,2.4144260599793173-0.1) node {(0,0,1) = 4};\n \\draw [fill=qqqqff] (14,2) circle (2.5pt);\n \\draw[color=qqqqff] (14.597938144329897+0.2,2.4144260599793173-0.1) node {(1,0,1) = 2};\n \\draw [fill=black] (14,6) circle (2.5pt);\n \\draw[color=black] (14.597938144329897+0.2,6.412874870734228-0.1) node {(1,1,1) = 1};\n \\draw [fill=black] (10,6) circle (2.5pt);\n \\draw[color=black] (10.604565537555228+0.2,6.412874870734228-0.1) node {(0,1,1) = 2};\n \\end{scriptsize}\n \\end{tikzpicture}\n \\vspace{-9mm}\n \\caption{Parallel sum in a 3-hypercube}\n \\end{figure}\n\n \\begin{enumerate}[leftmargin=*]\n \\item[2.] No matter which bit we start from, the \\textsc{Descend} or \\textsc{Ascend} algorithm always divides the processors by a different dimension in each round, then the binary operation is performed on pairs of processors, one from each group. Suppose that we use the \\textsc{Ascend} algorithm that loops from the $0$-th bit, further assume that $P_k$ has the datum set $\\{x\\}$ while other processors have no data ($\\emptyset$) initially. In order to broadcast, we can simply define\n \\begin{enumerate}\n \\item[] \\textsc{Operation}($P_i$, $P_{i+2^j}$) to be \\fbox{$P_i \\bigcup P_{i+2^j}$}\n \\end{enumerate}\n\n In the first round, $P_k$ sends $\\{x\\}$ to its counterpart $\\bar{P_{k}}$ in the other group via the binary operation, this implies that $P_k$ and $\\bar{P_{k}}$ can only differ by 1 at the $0$-th bit. On the other hand, the rest of the processors are empty, so the binary operation has no effect on them. When this round finishes, now 2 processors know about $\\{x\\}$. In the second round, we have a new partition of processors. Since $P_k$ and $\\bar{P_{k}}$ differs only at the $0$-th bit but now we have divided the processors by another bit, they cannot be counterparts this time or in any future rounds. Hence, they will both have a new counterpart who can receive the datum $\\{x\\}$. When this round finishes, now 4 processors know about $\\{x\\}$, and so forth...\n\n Given that any two processors can be counterparts in only one round, we will always have new pairs of counterparts in each round. As a result, after the $j$-th iteration, $2^j$ processors will know about $\\{x\\}$, and finally the datum will be propagated across the entire interconnection network.\n\n \\item[3.] The \\textsc{Descend} and \\textsc{Ascend} paradigm can be a powerful tool in many scenarios.\n\n If the binary operation is set intersection and each processor has a set, we can find the common values and remove duplicates. If the binary operation is set union, we can also find all distinct values or aggregate the data. For other binary operations such as the max()/min() function or multiplication, we can compute the maximum, minimum and product of numbers in an array or other data structures. It is also possible to use logical gates in parallel, where each processor handles a condition that is either true or false. For instance, we can check if the constraints of a linear program are violated, or if an interpretation of the 3-SAT formula is satisfiable.\n \\end{enumerate}\n\n \\vspace{8mm}\n \\printbibliography\n\n \\vfill\n \\doclicenseThis\n\n\\end{document}\n", "meta": {"hexsha": "e6fc1cd99badf5686c18f5f15f15ef37b1b605c8", "size": 38570, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "misc/advanced_algorithms.tex", "max_stars_repo_name": "neo-mashiro/BU", "max_stars_repo_head_hexsha": "2d2e789342ada0da6c1676e93d7d4e4839bdd6ef", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "misc/advanced_algorithms.tex", "max_issues_repo_name": "neo-mashiro/BU", "max_issues_repo_head_hexsha": "2d2e789342ada0da6c1676e93d7d4e4839bdd6ef", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-09-25T14:29:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-25T14:29:00.000Z", "max_forks_repo_path": "misc/advanced_algorithms.tex", "max_forks_repo_name": "neo-mashiro/DOC", "max_forks_repo_head_hexsha": "2d2e789342ada0da6c1676e93d7d4e4839bdd6ef", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 91.8333333333, "max_line_length": 1243, "alphanum_fraction": 0.6852735286, "num_tokens": 12112, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041652, "lm_q2_score": 0.6791786926816161, "lm_q1q2_score": 0.34754701460645027}} {"text": "\n\n% ==================\n%\n% =-------------------\n\\section{Simulation of the Multidomain Model}\\label{sec:solver_multidomain_model}\n\nThe multidomain model is an alternative to the fiber based electrophysiology model discussed in the last section.\nAs introduced in \\cref{sec:multidomain_model}, it does not explicitly resolve muscle fibers, but considers activity in the muscle domain in a homogenized view.\nOn every point in the 3D muscle mesh, separate values $V_m^k$ of the transmembrane potential exist for every MU $k \\in \\{1,\\dots,N_\\text{MU}\\}$, in addition to the value $\\phi_e$ for the extracellular electric potential. \nThe computational domain considers the muscle volume $\\Omega_M$ and the body domain $\\Omega_B$, which represents adipose tissue on top of the muscle.\nTo solve the multidomain model, a large linear system has to be solved in every timestep, as described in \\cref{sec:discretization_body_domain}.\n\nIn the following, \\cref{sec:multidomain_components} discusses the model setup for a scenario with four MUs. \\Cref{sec:multidomain_simulation_emg} demonstrates a larger simulation scenario with 25 MUs, which can be used to simulate surface EMG signals. We discuss differences between the multidomain approach and the fiber based electrophysiology model in \\cref{sec:multidomain_differences}.\n\n\\subsection{Components of the Computational Model}\\label{sec:multidomain_components}\n\nIn the following, we consider a multidomain simulation with muscle and body fat domains and four MUs. We use a muscle mesh with $16 \\times 16 \\times 74 = \\num{18944}$ elements and linear finite element ansatz functions and a fat mesh with $32 \\times 4 \\times 74 = \\num{9472}$ elements, which are partitioned to 128 subdomains for 128 processes. \n\nThe scenario uses the following electric conduction tensors $\\bfsigma_i$ and $\\bfsigma_e$ for the intra-cellular domain and the extracellular domain, respectively:\n\\begin{align}\\label{eq:multidomain_isotropic_e}\n \\bfsigma_i &= \\mat{\n 8.93 & 0 & 0\\\\\n 0& 0& 0\\\\\n 0& 0& 0}\\SI{}{\\milli\\siemens\\per\\centi\\meter}, & \\bfsigma_e &= \\mat{6.7 & 0 & 0\\\\\n 0& 6.7& 0\\\\\n 0& 0& 6.7}\\SI{}{\\milli\\siemens\\per\\centi\\meter}.\n\\end{align}\n%\nIn this scenario, we use the subcellular model of Hodgkin and Huxley \\cite{Hodgkin1952} and solve it using Heun's method. We discretize the multidomain equations using the Crank-Nicolson scheme with $\\theta=\\frac12$.\nWe solve the resulting linear system of equations by a GMRES solver with the parallel incomplete LU factorization preconditioner \\emph{Euclid} \\cite{euclid} from the HYPRE package \\cite{falgout2002hypre}. A tight residual norm tolerance of \\num{1e-15} is used in the abortion criterion of the GMRES solver. Such a low tolerance is required, as, for higher tolerances, spurious artificial stimulations can be observed. Timestep widths of $\\dt_\\text{0D}=\\dt_\\text{multidomain}=\\dt_\\text{splitting}=\\SI{1e-3}{\\ms}$ are used. \nThe computation for a simulation end time of $t_\\text{end}=\\SI{20}{\\ms}$ in this scenario takes approximately \\SI{27}{\\min}.\n\n%MU 0 around fiber 864, fr_max: 0.050, stddev: 0.10, r: 40.0 μm, Cm: 0.6 uF/cm^2, tstart: 0.000 s, f: 24.000 Hz\n%MU 1 around fiber 343, fr_max: 0.061, stddev: 0.12, r: 40.7 μm, Cm: 0.6 uF/cm^2, tstart: 0.010 s, f: 22.937 Hz\n%MU 2 around fiber 265, fr_max: 0.072, stddev: 0.14, r: 41.3 μm, Cm: 0.6 uF/cm^2, tstart: 0.020 s, f: 21.895 Hz\n\n%scenario_name: 4mus, n_subdomains: 4 1 32, n_ranks: 128, end_time: 20\n%dt_0D: 1e-03 multidomain solver: 1000 it. of gmres (10000 it. of gmres), lumped mass matrix: False, initial guess: previous solution\n%dt_multidomain: 1e-03 multidomain preconditioner: euclid (euclid), symmetric precond.: True\n%dt_splitting: 1e-03 theta: 1.0, solver tolerances, abs: 1e-15, rel: 1e-15\n%fiber_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/left_biceps_brachii_33x33fibers.bin\n%fat_mesh_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/left_biceps_brachii_33x33fibers.bin_fat.bin\n%cellml_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/hodgkin_huxley_1952.c\n%firing_times_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/MU_firing_times_always.txt\n%********************************************************************************\n%128 ranks, partitioning: x4 x y1 x z32\n %sampling 3D mesh with stride 2 x 2 x 20\n %linear 3D mesh nodes global: 17 x 17 x 75 = 21675, local: 4 x 17 x 3 = 204\n %linear 3D mesh elements global: 16 x 16 x 74 = 18944, local: 4 x 16 x 3 = 192\n %fat mesh, n points total: 12375 (33 x 5 x 75), (per process: 4 x 5 x 3 = 60)\n \nIn the multidomain model, we have to specify which point in the 3D mesh belongs to which MU to which extent. This is achieved by the relative occupancy factors $f_r^{k}$ for MU $k$.\nIn our implementation, the occupancy factors are computed by Python code in the settings file of the simulation.\nThe location of a MU in the 3D domain is specified by choosing a 1D muscle fiber, which is considered to be the center of the MU territory. This is possible, as the nodes in the structured 3D mesh can also be interpreted as a set of adjacent 1D fibers.\n\nFor every MU $k$, the occupancy factors $f_r^k(x,y,z)$ in every muscle cross-section with fixed $z$ coordinate are defined by a radial function $f(|(x,y)^\\top|/d(z))$, which reaches a configurable maximum value at the location of the specified fiber. The argument of the radial function is scaled by the diameter $d(z)$ of the muscle at the considered cross-section. The factor $f_r^k(x,y,z)$ is constant in longitudinal direction of the muscle ($z$ axis). Before the simulation, all factors $f_r^k$ are scaled, such that the maximum of their sum is equal to one:%\n\\begin{align*}\n \\max\\limits_{(x,y,z)\\in\\Omega_M} \\sum_{k=1}^{N_\\text{MU}} f_r^k(x,y,z) = 1.\n\\end{align*}\n\n% multidomain fr factors\n\\begin{figure}\n \\centering%\n \\begin{subfigure}[t]{0.23\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_fr0_cropped.png}%\n \\caption{$f_r^1$}%\n \\label{fig:fr0}%\n \\end{subfigure}\n \\,\n \\begin{subfigure}[t]{0.23\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_fr1_cropped.png}%\n \\caption{$f_r^2$}%\n \\label{fig:fr1}%\n \\end{subfigure}\n \\,\n \\begin{subfigure}[t]{0.23\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_fr2_cropped.png}%\n \\caption{$f_r^3$}%\n \\label{fig:fr2}%\n \\end{subfigure}\n \\,\n \\begin{subfigure}[t]{0.23\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_fr3_cropped.png}%\n \\caption{$f_r^4$}%\n \\label{fig:fr3}%\n \\end{subfigure}\n \\caption{Simulation of electrophysiology with the multidomain model: Value of the occupancy factors $f_r^k$ for MUs 1 to 4. The color coding encodes the maximum value by red color and the decreasing value along the radius by increased transparency and the color transition to blue color. The locations where the factor vanishes, $f_r^k=0$, correspond to full transparency.}%\n \\label{fig:multidomain_fr}%\n\\end{figure}%\n\n\\Cref{fig:multidomain_fr} shows the MU occupancy factors for the four MUs in the considered example scenario. It can be seen that the individual MU territories only occupy a small fraction of the muscle domain and are centered around fibers in longitudinal direction of the muscle.\n\nResults of the simulation at $t=\\SI{14}{\\ms}$ are given in \\cref{fig:multidomain_4mus}. In the considered scenario, the first and second MU are stimulated at $t=\\SI{0}{\\ms}$ and $t=\\SI{10}{\\ms}$, respectively. At the time of the displayed images, MUs 3 and 4 have not yet been stimulated. \n\nUpon stimulation, we prescribe the membrane voltage $V_m$ for one timestep as $\\SI{20}{\\milli\\volt}$ at the stimulated nodes in the mesh of the respective MU. In this scenario, the stimulated nodes are located in the middle of the muscle in longitudinal direction in three adjacent cross-sectional layers of mesh elements.\n\nThe upper two images in \\cref{fig:multidomain_4mus} show the locations of the propagated action potential fronts at $t=\\SI{14}{\\ms}$ for MU 1 and MU 2, given by the values of $V_m^k$. While the action potentials span the entire cross-section of the muscle domain, they contribute to the EMG value scaled by their locally varying occupancy factor $f_r^k$.\n\n% multidomain Vm MUs\n\\begin{figure}\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_mu1.png}\\\\[4mm]\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_mu2.png}\\\\[4mm]\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_mu3.png}\\\\[4mm]\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_mu4.png}%\n \\caption{Simulation of electrophysiology with the multidomain model: Transmembrane potential $V_m^k$ of MUs 1 to 4 at $t=\\SI{14}{\\ms}$.}%\n \\label{fig:multidomain_4mus}%\n\\end{figure}%\n\nAs the $V_m^k$ values of all MUs $k\\in \\{1,\\dots,N_\\text{MU}\\}$ are strongly coupled, the active MUs, MU 1 and MU 2, influence the $V_m^k$ scalar fields of the inactive MUs, MU 3 and MU 4. The lower two images in \\cref{fig:multidomain_4mus} show the computational domain of the muscle with several layers of 3D elements removed. \nIt can be seen that, at some regions in the interior of the domain, the values of $V_m^3$ and $V_m^4$ correspond to the negated value of $V_m^2$ with a smaller absolute value. Note the different color scales for $V_m^1$, $V_m^3$ and $V_m^4$ in these images.\n\n\\Cref{fig:multidomain_4mus_body} shows the values of the extracellular potential $\\phi_e$ on the muscle domain. The contributions from the two active MUs can be seen. \n\\Cref{fig:multidomain_4mus_phi_e_points} gives an impression of the used mesh and shows the value of $\\phi_e$ for all nodes. It can be seen that the $\\phi_e$ values span a larger value range in the interior of the domain than on the boundary, as previously shown in \\cref{fig:multidomain_4mus_body}. Correspondingly, the color coding in \\cref{fig:multidomain_4mus_phi_e_points} uses a larger range than in \\cref{fig:multidomain_4mus_body}.\n\n\\Cref{fig:multidomain_4mus_emg} shows the EMG values $\\phi_b$ on the surface of the body domain mesh. The effect of the body domain is revealed by comparing the electric potential on the boundary of the muscle mesh in \\cref{fig:multidomain_4mus_body} with the values in \\cref{fig:multidomain_4mus_emg}. The signals get locally smoothed by the fat layer.\n\n\n% multidomain phi_e\n\\begin{figure}\n \\centering%\n \\begin{subfigure}[t]{\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_body.png}%\n \\caption{Extracellular potential $\\phi_e$ on the surface of the muscle domain.}%\n \\label{fig:multidomain_4mus_body}%\n \\end{subfigure} \\\\\n \\begin{subfigure}[t]{\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_phi_e_points.png}%\n \\caption{Extracellular potential $\\phi_e$ at points of the 3D muscle mesh.}%\n \\label{fig:multidomain_4mus_phi_e_points}%\n \\end{subfigure} \\\\\n \\begin{subfigure}[t]{\\textwidth}%\n \\centering%\n \\includegraphics[width=\\textwidth]{images/results/application/multidomain_4mus_emg.png}%\n \\caption{EMG signal $\\phi_b$ on the surface of the body fat domain. The comparison with (a) shows the effect of the fat layer.}%\n \\label{fig:multidomain_4mus_emg}%\n \\end{subfigure}\n \\caption{Simulation of electrophysiology with the multidomain model: Simulation results at $t=\\SI{14}{\\ms}$ for a scenario with 4 MUs.}%\n \\label{fig:multidomain_4mus_2}%\n\\end{figure}%\n\n\n\n\\subsection{Simulation of EMG Signals}\\label{sec:multidomain_simulation_emg}\n\nIn the second scenario, we simulate a higher number of 25 MUs. The MUs are activated at random times within the first \\SI{20}{\\ms}. We use a mesh with $12 \\times 12 \\times 74 = \\num{10656}$ elements in the muscle domain and $24 \\times 4 \\times 74 = 7104$ elements in the body domain. Compared to the mesh in the previous scenario, the spatial resolution in radial direction is chosen slightly coarser, to speed up the computation.\nThe other model parameters, discretization schemes and solvers are chosen as before.\n\nIn this scenario, the stimulated nodes are no longer located in the middle of the muscle, but randomly varied by up to \\SI{10}{\\percent} of the muscle length using a uniform distribution. This approach to model the neuromuscular junctions is analog to the approach in \\cref{sec:simfiber_mu} for the fiber based electrophysiology. \n\\Cref{fig:multidomain_25mus_snapshot} shows the membrane voltage $V_m^1$ of MU 1 shortly after the MU has been activated. Because of the different locations of the stimulated points, no uniform action potential \\say{front} as in \\cref{fig:multidomain_4mus} is seen. Instead, a characteristic 2D MU action potential forms. \n\nThe spike of the depolarized membrane voltage at every stimulated point propagates along the fiber direction and additionally diffuses in transverse direction. This yields the cone-like structures of lower $V_m$ values as seen in \\cref{fig:multidomain_25mus_snapshot}. The origin of the transverse propagation is the electric conduction in the extracellular space, which is governed by the isotropic conduction tensor $\\bfsigma_e$ in \\cref{eq:multidomain_isotropic_e}. This isotropic conduction is strongly coupled to the directed action potential propagation within every MU compartment.\n\nThe resulting EMG values $\\phi_e$ on the muscle boundary and $\\phi_b$ on the skin surface are given in \\cref{fig:multidomain_25mus2_emg} and \\cref{fig:multidomain_25mus2_body}, respectively. The fat layer again smooths out the signal, as observed in the last section and in \\cref{sec:simfiber_fat} for the fiber based electrophysiology model.\n\nThe two blue vertical stripes in \\cref{fig:multidomain_25mus2_emg} with lower $\\phi_e$ values correspond to the action potentials of multiple MUs at the respective location. It can be seen that the resulting EMG signal varies more in longitudinal direction than in transverse direction. This can be explained by the wide MU territories in this scenario.\n\n% multidomain Vm MU\n% multidomain phi_e\n\\begin{figure}\n \\centering%\n \\begin{subfigure}[t]{\\textwidth}%\n \\centering%\n \\includegraphics[width=11cm]{images/results/application/multidomain_25mus_snapshot.png}%\n \\caption{Transmembrane voltage $V_m^1$ of the first MU.}%\n \\label{fig:multidomain_25mus_snapshot}%\n \\end{subfigure} \\\\[4mm]\n \\begin{subfigure}[t]{\\textwidth}%\n \\centering%\n \\includegraphics[width=12cm]{images/results/application/multidomain_25mus2_emg.png}%\n \\caption{Extracellular potential $\\phi_e$ at the surface of the 3D muscle mesh.}%\n \\label{fig:multidomain_25mus2_emg}%\n \\end{subfigure} \\\\[4mm]\n \\begin{subfigure}[t]{\\textwidth}%\n \\centering%\n \\includegraphics[width=12cm]{images/results/application/multidomain_25mus2_body.png}%\n \\caption{EMG signal $\\phi_b$ at the surface of the body fat domain.}%\n \\label{fig:multidomain_25mus2_body}%\n \\end{subfigure}\n \\caption{Simulation of electrophysiology with the multidomain model: Simulation results at $t=\\SI{20}{\\ms}$ for a scenario with 25 MUs.}%\n \\label{fig:multidomain_25mus2}%\n\\end{figure}%\n\n\n\\subsection{Comparison of the Fiber Based Electrophysiology Model and the Multidomain Model}\\label{sec:multidomain_differences}\n\nEMG signals on the upper arm can be simulated by both the fiber based electrophysiology model, as demonstrated in \\cref{sec:results_fiber_based_electrophysiology} and by the multidomain model, as shown in the previous sections. The two approaches have several similarities and differences.\n\nBoth model approaches have in common that they are based on biophysical principles. They both involve a detailed subcellular model, which describes the biochemical processes on the muscle fiber membranes. The subcellular model is solved at discrete points in the 3D domain and the model instances are coupled to a description of electric volume conduction in the muscle domain and the body fat layer. Both the fiber based approach and the multidomain model are, thus, multi-scale descriptions. \n\nBoth models also resolve the physiological structure of muscle activity given by multiple MUs. Action potential propagation is computed separately for different MUs. In a comprehensive simulation of the neuromuscular system, motor neuron models can be coupled to drive the activation of the MUs.\n\nThe two domains of the electrically active muscle tissue and the passive layer of skin and adipose tissue are also considered in both modeling approaches. In the fiber based approach, electric volume conduction is described by the 3D bidomain equation \\cref{eq:bidomain1} for the muscle domain and a strongly coupled 3D Laplace equation \\cref{eq:body} for the body domain.\nThe multidomain approach for electric volume conduction generalizes the bidomain model and yields the bidomain equation as a special case, if only one MU is considered. The Laplace equation for the body domain is coupled in the same way as in the fiber based model. In summary, both approaches are very similar regarding the 3D electric conduction part.\n\nThe major difference between the models is, that the multidomain approach considers only 3D domains, whereas the fiber based approach resolves individual 1D muscle fibers. Another difference lies in the coupling between the model components. For the fiber based approach, action potential propagation on the 1D fibers is unidirectionally coupled to the 3D volume conduction part of the model. In the multidomain model, all components are bidirectionally coupled. This allows, e.g., to simulate externally applied stimulations by active electrodes on the skin surface. The effects of the external currents on the electric potentials in the 3D muscle volume and down to the 0D subcellular behavior can only be described accurately by the multidomain approach.\n\nThe difference in coupling between the action potential propagation model part and the electric volume conduction in the extracellular space can also be seen in the computed EMG signals. A comparison of EMG simulations using the fiber based model, e.g., \\cref{fig:emg273529b}, and the multidomain model, e.g., \\cref{fig:multidomain_25mus2_emg}, shows that the multidomain approach yields less sharp artifacts in the 2D EMG signal on the muscle surface than the fiber based method. In the fiber based model, the action potentials of individual fibers are visible in the signal. In the multidomain simulations, the regions of similar activity are more clustered in the resulting EMG signals.\n\nOther differences between the two model approaches exist in terms of the computational performance properties of their solvers. In the fiber based approach, action potential propagation can be computed independently for all muscle fibers, which enables large speedups by parallelization and makes large problem sizes with realistic numbers of muscle fibers feasible. For example, \\cref{sec:effects_of_the_mesh_width_emg} presents the simulation of \\num{270000} muscle fibers with \\num{27000} compute cores.\nIn the multidomain approach, on the other hand, a large linear system of equations has to be solved in every timesteps. This can also be parallelized, but requires communication between the involved processes, which limits the parallel scalability for large problem sizes.\n\nIn the multidomain model, the computational effort increases, in good approximation, linearly with both the number of MUs and the number of nodes in the mesh. In the fiber based approach, the amount of computational work mainly corresponds to the number of fibers, not to the number of MUs. \nThe 3D problem and, thus, the mesh width of the 3D mesh, typically plays a minor role in the total runtime for the fiber based approach, as the 3D problem is only solved according to the desired EMG sampling frequency. In the multidomain approach, no separate timestep widths can be chosen for the computations of the extracellular and body domain electric potentials, $\\phi_e$ and $\\phi_b$, as they are computed as a solution of the same linear system of equations.\n\nFor example, the computation of \\SI{24}{\\ms} of the multidomain scenario in \\cref{sec:multidomain_simulation_emg} with 25 MUs and 126 processes has a runtime of approximately $\\SI{106}{\\min}$. The fiber based approach with the same 3D mesh and the same parallel partitioning with 126 processes has a total runtime of \\SI{6}{\\s} for 169 fibers or \\SI{20}{\\s} for 1369 fibers. A scenario with 169 fibers leads to a fiber spacing that corresponds to the 3D mesh width in the compared multidomain scenario. The speedup between the models in this case is approximately \\num{1000}.\nNote that only the computation of the fiber based approach is highly optimized in this work, and a better performance of the multidomain solver could be achieved in future work. However, the structural properties of the models facilitate highly parallel simulations only for the fiber based approach.\n\nAs a result, if the simplifications of a unidirectional coupling of the extracellular potential $\\phi_e$ from the muscle fibers to the 3D volume can be tolerated, the fiber based approach should be used, as it exhibits significantly lower runtimes. The fiber based approach is (considering the current implementation) the only possible choice for scenarios with at least two of the three requirements (i) long simulation time spans in the range of seconds, (ii) large number of MUs in the range of multiple dozens, and (iii) finely resolved 3D meshes in the range of several $\\num{1e6}$ degrees of freedom.\n \nThe multidomain approach, on the other hand, can describe phenomena that are not accurately captured by the fiber based model, as described earlier.\nMoreover, the multidomain model is potentially easier to handle for more irregular geometries, where only a structured 3D mesh and no physiologically oriented fibers are given. Another advantage of the multidomain approach is its ability to fine tune the MU territories. \nThe multidomain model can also possibly simulate a given MU distribution with the same accuracy with less 3D points than the fiber based approach. However, investigations in this direction are subject of future research.\nIf large runtimes are not an issue, the multidomain approach can be used to yield more physically accurate results than the fiber based approach, ultimately advancing the means to describe the neuromuscular system as detailed and accurately as possible.\n\n\n% common\n% biophysically based, MUs, subcellular models\n% 3D bidomain in both\n\n% differences\n% coupling uni->bi\n% computation\n\n\n% fibers and multidomain\n%\\begin{figure}[H]\n% \\centering%\n% \\begin{subfigure}[t]{0.48\\textwidth}%\n% \\centering%\n% \\includegraphics[width=\\textwidth]{images/results/application/2_multidomain.png}%\n% \\caption{Multidomain}%\n% \\label{fig:2_multidomain}%\n% \\end{subfigure}\n% \\quad\n% \\begin{subfigure}[t]{0.48\\textwidth}%\n% \\centering%\n% \\includegraphics[width=\\textwidth]{images/results/application/2_fibers.png}%\n% \\caption{Fibers}%\n% \\label{fig:2_fibers}%\n% \\end{subfigure} \n% \\caption{Fibers and Multidomain}%\n% \\label{fig:multidomain_fibers}%\n%\\end{figure}%\n\n\n\\begin{reproduce_no_break}\n The two simulations in this section with 4 and 25 MUs, respectively, which are visualized in \\cref{fig:multidomain_4mus,fig:multidomain_4mus_2,fig:multidomain_25mus2}, can be executed by the following commands:\n \\begin{lstlisting}[columns=fullflexible,breaklines=true,postbreak=\\mbox{\\textcolor{gray}{$\\hookrightarrow$}\\space}]\n cd $\\$$OPENDIHU_HOME/examples/electrophysiology/multidomain/multidomain_with_fat/build_release\n mpirun -n 128 multidomain_with_fat ../settings_multidomain_with_fat.py 4mus.py --n_subdomains 8 1 16\n mpirun -n 126 ./multidomain_with_fat_emg ../settings_multidomain_with_fat.py all_active.py --n_subdomains 6 1 21\n \\end{lstlisting}\n For other available numbers of processes, the subdomains at the end of the commands have to be adjusted.\n\\end{reproduce_no_break}\n\n% ----------------\n%\n% =================\n\n%-----\n% ==================\n%\n% =-------------------\n%-----\n\n\\section{Simulation of Coupled Electrophysiology and Solid Mechanics}\\label{sec:coupled_electrophysiology_and_solid_mechanics}\n\nSimulating muscle contraction with a detailed model, which accurately describes motor recruitment,\nyields the basis for new insights into the neuromuscular orchestration of processes that lead to muscle force generation.\n\nWe couple the two model approaches for electrophysiology, the fiber based model presented in \\cref{sec:results_fiber_based_electrophysiology} and the multidomain model presented in \\cref{sec:solver_multidomain_model}, with a solid mechanics model. \nIn \\cref{sec:solver_solid_mechanics}, we demonstrated the solver for nonlinear hyperelasticity models in simulations of the passive behavior of muscle tissue. The current section aims at simulating active muscle contraction.\n\n\\Cref{sec:fiber_based_contraction} couples the fiber based electrophysiology model with a model of muscle contraction. \\Cref{sec:prestress_contraction} discusses an algorithm to add prestress to the description. \\Cref{sec:multidomain_contraction} demonstrates the coupling of the multidomain model with the model of muscle contraction. \\Cref{sec:volume_coupling_contraction} and \\cref{sec:surface_coupling_contraction} describe simulations using the numerical coupling library preCICE.\n\n\\subsection{Fiber Based Electrophysiology and Muscle Contraction}\\label{sec:fiber_based_contraction}\n%-----\n\nWe begin with coupling the fiber based electrophysiology solver with the solid mechanics model to simulate muscle contraction as a result of the activation of muscle fibers.\nWe use the subcellular model of Shorten et al. \\cite{Shorten2007}. It computes the microscopic activation parameter $\\gamma \\in [0,1]$, which is related to the concentration of attached cross-bridges in the sarcomeres. The parameter $\\gamma$ is mapped and homogenized from the 0D subcellular points to $\\bar{\\gamma}$ on the 3D mesh. In the macroscopic 3D mechanics description, the factor is multiplied with a maximum active stress parameter $S_\\text{max,active}$ and a force-velocity characteristic $f_l(\\lambda_f)$, as described in \\cref{sec:material_nonlinear_model}.\nThe 3D mechanics model updates the geometry of the 3D domain and transfers the fiber stretch value $\\lambda_f$ and the contraction velocity $\\dot{\\lambda}_f$ back to the subcellular model.\n\nIn this scenario, we aim to simulate a rapid and strong contraction of the biceps muscle.\nThe scenario contains 169 fibers, which are associated with 15 MUs. This association is generated by method 1 in \\cref{sec:method1_assignment}. All MUs are subsequently activated in a ramp in the first $\\SI{1.4}{\\s}$. \n\nThe muscle geometry is fixed at its lower end and no external forces are considered in this scenario. The dynamic formulation with the transversely isotropic Mooney-Rivlin material is used, as described in \\cref{sec:material_nonlinear_model}.\nThe 3D muscle mesh contains $2 \\times 3 \\times 18 = 108$ elements with quadratic finite element ansatz functions and 1295 nodes in total and is partitioned into subdomains for four processes. Time step widths of $\\dt_\\text{0D} = \\dt_\\text{1D} = \\dt_\\text{splitting} = \\SI{1e-4}{\\ms}$ and $\\dt_\\text{3D}=\\SI{1}{\\ms}$ are used. The used numerical solvers and other settings of the electrophysiology and contraction models are equal to the described scenarios in \\cref{sec:simfiber_mu,sec:effects_of_the_mesh_width_emg} and \\cref{sec:comparison_linear_nonlinear}.\n\n% normal muscle contraction\n%0/4 : This is opendihu 1.2, built Apr 16 2021, C++ 201402, GCC 10.2.0, current time: 2021/4/17 19:56:16, hostname: ipvs-epyc1, n ranks: 4\n%0/4 : Open MPI v3.1.6, package: Open MPI maierbn@ipvs-epyc1 Distribution, ident: 3.1.6, repo rev: v3.1.6, Mar 18, 2020\n%0/4 : File \"../settings_biceps_contraction.py\" loaded.\n%0/4 : ---------------------------------------- begin python output ----------------------------------------\n%Loading variables from \"15mus.py\".\n%scenario_name: 15mus, n_subdomains: 2 1 2, n_ranks: 4, end_time: 4000.0\n%dt_0D: 1e-04, diffusion_solver_type: cg\n%dt_1D: 1e-04, potential_flow_solver_type: gmres\n%dt_splitting: 1e-04, emg_solver_type: cg, emg_initial_guess_nonzero: False\n%dt_3D: 1e+00, paraview_output: True\n%output_timestep: 1e+00 stimulation_frequency: 0.1 1/ms = 100.0 Hz\n%fast_monodomain_solver_optimizations: True, use_analytic_jacobian: True, use_vc: True\n%fiber_file: ../../../../input/left_biceps_brachii_13x13fibers.bin\n%fat_mesh_file: ../../../../input/left_biceps_brachii_13x13fibers.bin_fat.bin\n%cellml_file: ../../../../input/new_slow_TK_2014_12_08.c\n%fiber_distribution_file: ../../../../input/MU_fibre_distribution_15MUs_13x13fibers.txt\n%firing_times_file: ../../../../input/MU_firing_times_always.txt\n%********************************************************************************\n%prefactor: sigma_eff/(Am*Cm) = 0.0132 = 3.828 / (500.0*0.58)\n%diffusion solver type: cg\n%n fibers: 169 (13 x 13), sampled by stride 2 x 2\n%n points per fiber: 1481, sampled by stride 40\n%4 ranks, partitioning: x2 x y1 x z2\n%13 x 13 = 169 fibers, per partition: 6 x 12 = 72\n%per fiber: 1D mesh nodes global: 1481, local: 760\n %sampling 3D mesh with stride 2 x 2 x 40 \n %quadratic 3D mesh nodes global: 5 x 7 x 37 = 1295, local: 2 x 7 x 18 = 252\n %quadratic 3D mesh elements global: 2 x 3 x 18 = 108, local: 1 x 3 x 9 = 27\n%number of degrees of freedom:\n %1D fiber: 1481 (per process: 760)\n %0D-1D monodomain: 82936 (per process: 42560)\n %all fibers 0D-1D monodomain: 14016184 (per process: 3064320)\n %3D bidomain: 1295 (per process: 252)\n %total: 14017479 (per process: 3064572)\n\n% contraction fibers\n\\begin{figure}\n \\centering%\n \\begin{subfigure}[t]{0.3\\textwidth}%\n \\centering%\n \\includegraphics[height=9cm]{images/results/application/contraction_fibers_044.png}%\n \\caption{}%\n \\label{fig:contraction_fibers_044}%\n \\end{subfigure} \\,\n \\begin{subfigure}[t]{0.18\\textwidth}%\n \\centering%\n \\includegraphics[height=9cm]{images/results/application/contraction_fibers_844b.png}%\n \\caption{}%\n \\label{fig:contraction_fibers_844b}%\n \\end{subfigure}\\,\n \\begin{subfigure}[t]{0.25\\textwidth}%\n \\centering%\n \\includegraphics[height=9cm]{images/results/application/contraction_fibers_1684b.png}%\n \\caption{}%\n \\label{fig:contraction_fibers_1684b}%\n \\end{subfigure}\\,\n \\begin{subfigure}[t]{0.2\\textwidth}%\n \\centering%\n \\includegraphics[height=9cm]{images/results/application/contraction_fibers_2084b.png}%\n \\caption{}%\n \\label{fig:contraction_fibers_2084b}%\n \\end{subfigure}\n \\caption{Simulation of fiber based electrophysiology and muscle contraction: Activation of the muscle fibers and overall deformation at various timesteps.}%\n \\label{fig:contraction_fibers_1}%\n\\end{figure}%\n\n\\Cref{fig:contraction_fibers_1} shows the fibers of the contracting muscle at four different timesteps between $t=\\SI{44}{\\ms}$ and $t=\\SI{2.084}{\\s}$. The fibers are colored according to the resulting activation parameter $\\gamma$, which is a measure for the generated force on the sarcomere level. Between $t=\\SI{44}{\\ms}$ and $t=\\SI{844}{\\ms}$, shown in \\cref{fig:contraction_fibers_044,fig:contraction_fibers_844b}, the smallest MUs are activated, which, in this example, are mainly located on the left-hand side. As a consequence, the muscle domain initially bends slightly to the left. As more MUs become active at $t=\\SI{1684}{\\ms}$, depicted in \\cref{fig:contraction_fibers_1684b}, the deformation increases and the bending direction is reversed. However, the fibers on the left-hand side still exhibit the highest $\\gamma$ value, as they have been stimulated most often at that time.\nAt $t=\\SI{1684}{\\ms}$, visualized in \\cref{fig:contraction_fibers_2084b}, almost all fibers have a $\\gamma$ value close to one, corresponding to full activation.\n\n% contracted state\n\\begin{figure}\n \\centering%\n \\begin{subfigure}[t]{0.31\\textwidth}%\n \\centering%\n \\includegraphics[height=77mm]{images/results/application/contraction_fibers.png}%\n \\caption{Action potentials given by the membrane voltage $V_m$ (in millivolts) on the muscle fibers.}%\n \\label{fig:contraction_fibers}%\n \\end{subfigure}\\,\n \\begin{subfigure}[t]{0.31\\textwidth}%\n \\centering%\n \\includegraphics[height=8cm]{images/results/application/contraction_lambda.png}%\n \\caption{Stretch parameter $\\lambda$ of the deformed 3D muscle domain (red mesh) in comparison to the reference configuration (yellow mesh).}%\n \\label{fig:contraction_lambda}%\n \\end{subfigure}\n \\begin{subfigure}[t]{0.31\\textwidth}%\n \\centering%\n \\includegraphics[height=8cm]{images/results/application/contraction_active_stress.png}%\n \\caption{Value of the active second Piola-Kirchhoff stress.}%\n \\label{fig:contraction_active_stress}%\n \\end{subfigure}\n \\caption{Simulation of fiber based electrophysiology and muscle contraction: Simulation results at $t=\\SI{2084}{\\ms}$.}%\n \\label{fig:contraction_at_end}%\n\\end{figure}%\n\n\\Cref{fig:contraction_at_end} shows several variables at the simulation end time of $t=\\SI{2084}{\\ms}$. \\Cref{fig:contraction_fibers} visualizes the transmembrane voltage $V_m$ on the muscle fibers. Action potentials can be seen on almost all fibers, as the whole muscle is activated at this time. \\Cref{fig:contraction_lambda} shows a comparison between the reference configuration given by the yellow mesh and the current configuration given by the red mesh. The muscle domain is colored according to the stretch $\\lambda$, which has a nearly constant value of $\\lambda \\approx \\SI{85}{\\percent}$ at the end time of this scenario. A similar visualization is given in \\cref{fig:contraction_active_stress} for the active stress in the muscle.\n\nBecause of the high level of activation and the corresponding active stress distribution in the muscle, our mechanics solver only converges up to the shown simulation time of \\SI{2084}{\\ms} in this scenario. The aim of the scenario is to simulate the contraction of a fully activated muscle. Other scenarios, where the activation is applied more slowly, allow for a convergence of the mechanics solver during longer simulation time spans.\n\nThe presented scenario showed that a fully activated biceps muscle contracted to about \\SI{85}{\\percent} of its original length. However, in reality, larger contractions are possible. In the shown scenario, the muscle was initially in a stress-free configuration. More realistic scenarios can incorporate pretension forces, where the undeformed reference configuration is subject to a constant stress level in the muscle's direction of the line of action. This is considered in the next scenario.\n\n% muscle contraction\n%\\begin{figure}\n% \\centering%\n% \\includegraphics[width=0.5\\textwidth]{images/results/application/neuromuscular_muscle_contraction_traction.png}%\n % \\caption{contraction traction}%\n% \\label{fig:neuromuscular_muscle_contraction_traction}%\n%\\end{figure}%\n\n\\begin{reproduce}\n The simulation in this section can be run as follows:\n \\begin{lstlisting}[columns=fullflexible,breaklines=true,postbreak=\\mbox{\\textcolor{gray}{$\\hookrightarrow$}\\space}]\n cd $\\$$OPENDIHU_HOME/examples/electrophysiology/fibers/fibers_contraction/no_precice/build_release\n mpirun -n 4 ./biceps_contraction ../settings_biceps_contraction.py ramp.py\n \\end{lstlisting}\n\\end{reproduce}\n\n\\subsection{Simulation of Prestress}\\label{sec:prestress_contraction}\n% compressing muscle by external force does not work -> need activation \n\nTo obtain more realistic ranges of muscle contraction, a nonzero, constant prestress can be considered in the undeformed configuration of the muscle. \nIn our solid mechanics formulation, the reference configuration always has zero stress. Thus, we need to construct a separate, first reference configuration of a shorter muscle geometry. We stretch it to the original muscle length by applying external forces. The resulting, second configuration resembles the original muscle geometry and has the desired prestress characteristics.\n\nThe detailed steps of this algorithm are visualized in \\cref{fig:neuromuscular_prestretch} and are described in the following.\nWe begin with the given geometry of the muscle with body fat layer, which is shown as black wireframe mesh in \\cref{fig:neuromuscular_prestretch_1}. In a first static simulation step, a constant active stress $\\alpha_\\text{pre}\\,S_\\text{max,active}$ is prescribed in the entire muscle volume. The resulting muscle deformation is computed, using the usual nonlinear hyperelastic muscle material. $S_\\text{max,active}$ refers to the maximum active stress value as used in the mechanics model description in \\cref{eq:active_stress_term}. The result of this first step is a shortened muscle with the same volume as the original geometry. \\Cref{fig:neuromuscular_prestretch} shows the result by the yellow volume for $\\alpha_\\text{pre}=0.3$. It can be seen that the length of the muscle has shortened by approximately \\SI{13}{\\percent}.\n\nIn the second step, we reuse the computed deformed geometry of the first step as new stress-free reference configuration and re-extend it by applying a constant surface load $F_\\text{pre}$ pointing to the bottom on the lower face in the setting of \\cref{fig:neuromuscular_prestretch}. The value of $F_\\text{pre}$ corresponding to $\\alpha_\\text{pre}$ has to be estimated by numerical experiments.\n\nThis step is again solved as a static problem. The result is a similar muscle geometry as the original one, and contains prestress according to the applied force. The muscle volume is exactly preserved due to the incompressible material formulation. \\Cref{fig:neuromuscular_prestretch_2} shows the starting point for the second step by the black wireframe mesh and the resulting geometry for a total applied force of $F_\\text{pre}=\\SI{30}{\\newton}$ by the red volume. The comparison of the original, black mesh in \\cref{fig:neuromuscular_prestretch_1} with the red volume in \\cref{fig:neuromuscular_prestretch_2} shows a good match of the geometry.\nFor the subsequent dynamic simulations of, e.g., muscle contraction, the surface load has to be constantly applied. It corresponds to the tendon forces and the loads of the musculoskeletal system acting on the muscle.\n\nThe active stress parameter $\\alpha_\\text{pre}$ and the corresponding preload force $F_\\text{pre}$ can be chosen according to the desired amount of prestress. However, the higher these values are chosen, the more difficult is it for the nonlinear solid mechanics solvers to converge to a solution. Especially for irregular or large mechanics meshes, a lower stress factor of, e.g., $\\alpha_\\text{pre}=0.1$ has to be chosen. To improve convergence, we apply the load in the second step of the algorithm incrementally by several load steps. In addition, reducing the number of unknowns and increasing the mesh width in the mechanics problem can help, as this improves the conditioning of the problem. \n \n% contracted state\n\\begin{figure}\n \\centering%\n \\hfill\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\centering%\n \\includegraphics[height=12cm]{images/results/application/neuromuscular_prestretch_1.png}%\n \\caption{In the first step, the original mesh (black wireframe) is contracted by an artificial active stress $\\alpha_\\text{pre}\\,S_\\text{max,active}$ to yield a shortened muscle (yellow mesh).}%\n \\label{fig:neuromuscular_prestretch_1}%\n \\end{subfigure}\\hfill\n \\begin{subfigure}[t]{0.48\\textwidth}%\n \\centering%\n \\includegraphics[height=12cm]{images/results/application/neuromuscular_prestretch_2.png}%\n \\caption{In the second step, the mesh is extended again by an external surface load. The black wireframe corresponds to the yellow volume in (a), the red volume is the resulting geometry.}%\n \\label{fig:neuromuscular_prestretch_2}%\n \\end{subfigure}\n \\hfill\n \\caption{Simulation of biceps muscle geometry with prestress: The two steps of the algorithm to generate a reference geometry with prestress, shown with the geometry of the tendons for reference.}%\n \\label{fig:neuromuscular_prestretch}%\n\\end{figure}%\n\n\n\\subsection{Coupling of the Multidomain Model and Solid Mechanics Model with Prestress}\\label{sec:multidomain_contraction}\n\nIn the following, we present a scenario that uses the prestress algorithm of the last section and couples the multidomain and mechanics models to simulate surface EMG signals on the skin surface over a contracting muscle.\n\nWe choose $\\alpha_\\text{pre}=0.1$ and apply the prestretch force $F_\\text{pre}=\\SI{10}{\\newton}$ in three load steps. The multidomain model considers 5 MUs with stimulation frequencies between \\SI{7}{\\hertz} and \\SI{24}{\\hertz} and a 3D mesh of $8 \\times 8 \\times 28 = 1792$ elements. We execute the simulation with four processes. All other parameters and settings of the multidomain model and the solid mechanics model that are not explicitly mentioned in the following are chosen the same as in \\cref{sec:multidomain_components} and \\cref{sec:comparison_linear_nonlinear}.\n\nFor the discretization of the mechanics model, we use a coarser mesh than for the multidomain model. Furthermore, we use quadratic elements instead of linear elements. The Python implementation of the settings script of this example contains functionality to create the mechanics mesh by subsampling the multidomain mesh with specified factors. In the current scenario, we set these factors for the $x$, $y$ and $z$ directions to 0.7, 0.7, and 0.3, respectively. As a result, we get meshes with $5 \\times 7 \\times 9 = 315$ elements for the muscle and $5 \\times 1 \\times 4 = 20$ elements for the body fat domain. \\Cref{fig:multidomain_prestretch5} visualizes all meshes used in this scenario: The orange muscle mesh and the red body mesh are used for the multidomain model, and the yellow mesh is used for the solid mechanics model.\n\n\n% multidomain prestretch\n\\begin{figure}\n \\centering%\n \\hfill\n \\begin{subfigure}[t]{0.31\\textwidth}%\n \\centering%\n \\includegraphics[height=87mm]{images/results/application/multidomain_prestretch5.png}%\n \\caption{Multidomain meshes of the muscle domain (orange), body fat domain (red) and the coarser mesh used for the solid mechanics model (yellow).}%\n \\label{fig:multidomain_prestretch5}%\n \\end{subfigure}\n \\begin{subfigure}[t]{0.31\\textwidth}%\n \\centering%\n \\includegraphics[height=95mm]{images/results/application/multidomain_prestretch6.png}%\n \\caption{Reference configuration (yellow mesh) and current configuration of the muscle colored according to the distribution of the second Piola-Kirchhoff stress.}%\n \\label{fig:multidomain_prestretch6}%\n \\end{subfigure}\\qquad\n \\begin{subfigure}[t]{0.31\\textwidth}%\n \\centering%\n \\includegraphics[height=87mm]{images/results/application/multidomain_prestretch2.png}%\n \\caption{Electric potential $\\phi_b$ in the body domain (green color scale, in millivolts) and active stress in the interior of the muscle (blue-red color scale, in \\SI{}{\\newton\\per\\centi\\meter\\squared}).}%\n \\label{fig:multidomain_prestretch2}%\n \\end{subfigure}\n \\hfill\n \\caption{Simulation of muscle contraction based on the multidomain model with prestressed muscle geometry: Used meshes and simulation results at $t=\\SI{920}{\\ms}$ of a scenario of the multidomain electrophysiology model coupled to the solid mechanics model.}%\n \\label{fig:multidomain_prestretch}%\n\\end{figure}%\n\n\\Cref{fig:multidomain_prestretch6,fig:multidomain_prestretch2} depict results of the simulation at time $t=\\SI{920}{\\milli\\second}$. \n\\Cref{fig:multidomain_prestretch6} shows the reference geometry by the yellow wireframe after applying the prestress. The muscle is colored according to the value of the second Piola-Kirchhoff stress. During this dynamic simulation, the muscle bends elastically slightly to the left and right, as it is only fixed at its bottom in \\cref{fig:multidomain_prestretch}. This explains the stress distribution at the snapshot for $t=\\SI{920}{\\milli\\second}$ in \\cref{fig:multidomain_prestretch6}, where higher stresses occur on the right-hand side.\n\n\\Cref{fig:multidomain_prestretch2} shows the electric potential $\\phi_b$ of the body domain by the green color scale on the left of the image. The visible part of the fat layer shows two action potentials, visualized by the two dark green stripes.\n\nMoreover, \\cref{fig:multidomain_prestretch2} displays the total active stress $\\bfS_\\text{active}$ in the interior of the muscle by the color scale that ranges from blue to red color. In the multidomain model, $\\bfS_\\text{active}$ is calculated as a weighted sum over the contributions $\\bfS_\\text{active}^k$ of the MU compartments, scaled by the occupancy factors $f_r^k$ (cf. \\cref{sec:multidomain_model}):\n\\begin{align*}\n \\bfS_\\text{active} = \\sum\\limits_{k=1}^{N_\\text{MU}} f_r^k\\,\\bfS_\\text{active}^k.\n\\end{align*}\nThe muscle domain in \\cref{fig:multidomain_prestretch2} is cut open, such that interior distribution at the cut plane can be seen. The image shows two regions of higher active stress, which run vertically through the muscle, given by red color. They are a result on the location of the MUs in this scenario. The legend shows that the active stress inside the muscle is below $\\SI{0.4}{\\newton\\per\\centi\\meter\\squared}$, while the maximum active stress parameter is chosen as $S_\\text{max,active}=\\SI{7.3}{\\newton\\per\\centi\\meter\\squared}$. This low activation level is a result of the chosen MU recruitment. As a result, the muscle only slightly contracts, as can be seen in \\cref{fig:multidomain_prestretch6}.\n\nIn summary, both the fiber based electrophysiology model and the multidomain model can be coupled with the nonlinear solid mechanics model to simulate muscle contraction, as presented in \\cref{sec:fiber_based_contraction} and in this section. \nThe computational efficiency considerations discussed in the comparison of the fiber based and multidomain approaches in \\cref{sec:multidomain_differences} also apply to coupled simulations with muscle contraction. For longer simulation times, the fiber based approach in \\cref{sec:fiber_based_contraction} is, therefore, favored.\n\n%0/4 : This is opendihu 1.2, built Apr 17 2021, C++ 201402, GCC 10.2.0, current time: 2021/4/18 17:13:03, hostname: ipvs-epyc1, n ranks: 4 \n%0/4 : Open MPI v3.1.6, package: Open MPI maierbn@ipvs-epyc1 Distribution, ident: 3.1.6, repo rev: v3.1.6, Mar 18, 2020\n%0/4 : File \"../settings_multidomain_prestretch.py\" loaded. \n%0/4 : ---------------------------------------- begin python output ---------------------------------------- \n%Loading variables from \"multidomain.py\". \n%scenario_name: multidomain, n_subdomains: 2 1 2, n_ranks: 4, end_time: 4000.0 \n%dt_0D: 1e-03 multidomain solver: gmres, lumped mass matrix: False \n%dt_multidomain: 1e-03 multidomain preconditioner: euclid, symmetric precond.: True\n%dt_splitting: 1e-03 theta: 1.0, solver tolerances, abs: 1e-15, rel: 1e-15\n%dt_elasticity: 1e+00 elasticity solver: lu, preconditioner: none\n%fiber_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/left_biceps_brachii_9x9fibers.bin\n%fat_mesh_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/left_biceps_brachii_9x9fibers.bin_fat.bin\n%cellml_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/hodgkin_huxley-razumova.cellml\n%firing_times_file: /data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/MU_firing_times_always.txt\n%********************************************************************************\n%4 ranks, partitioning: x2 x y1 x z2\n %sampling 3D mesh with stride 1 x 1 x 50 \n %linear 3D mesh nodes global: 9 x 9 x 29 = 2349, local: 4 x 9 x 14 = 504\n %linear 3D mesh elements global: 8 x 8 x 28 = 1792, local: 4 x 8 x 14 = 448\n %quadratic 3D mesh nodes global: 9 x 9 x 29 = 2349, local: 4 x 9 x 14 = 504\n %quadratic 3D mesh elements global: 4 x 4 x 14 = 224, local: 2 x 4 x 7 = 56\n %fat mesh, n points total: 2465 (17 x 5 x 29), (per process: 4 x 5 x 14 = 280)\n %sub-sampling 3D elasticity mesh with factors 0.7, 0.7, 0.3 \n %elasticity quadratic 3D meshes:\n %muscle: nodes global: 5 x 7 x 9 = 315, local: 2 x 7 x 4 = 56\n %quadratic elements global: 2 x 3 x 4 = 24, local: 1 x 1 x 2 = 2\n %fat and skin layer: nodes global: 11 x 3 x 9 = 297, local: 2 x 3 x 4 = 24\n %quadratic elements global: 5 x 1 x 4 = 20, local: 1 x 1 x 2 = 2\n\n%Debugging output about compartment firing: Taking input from file \"/data/scratch/sgs/maierbn/opendihu/examples/electrophysiology/input/MU_firing_times_always.txt\"\n%First stimulation times\n %Time MU compartments\n %0.00 0 [0]\n %0.00 1 [1]\n %0.00 2 [2]\n %0.00 3 [3]\n%stimulated MUs: 4, not stimulated MUs: 0\n%duration of assembling this list: 0.000 s\n\n%Load relative factors, f_r, from file \"compartments_relative_factors.left_biceps_brachii_9x9fibers.bin.5_mus_stride_1x1x50_partitioning_2x1x2\"\n%MU 0, maximum fr: 0.1999999999999993\n%MU 1, maximum fr: 0.4426237526712864\n%MU 2, maximum fr: 0.6900999803960008\n%MU 3, maximum fr: 0.9425257326752092\n%MU 4, maximum fr: 1.1999999999999993\n\n\n\n% prestretch\n%\\begin{figure}\n% \\centering%\n% \\includegraphics[width=0.5\\textwidth]{images/results/application/fibers_muscle_contraction_quantities.png}%\n% \\caption{prestretch}%\n% \\label{fig:prestrech1b}%\n%\\end{figure}%\n\n\\begin{reproduce_no_break}\n The simulation can be run with the following commands. Instead of four processes also other numbers are possible. A lot of parameters can be fine-tuned in the \\code{../variables/multidomain.py} settings file.\n \\begin{lstlisting}[columns=fullflexible,breaklines=true,postbreak=\\mbox{\\textcolor{gray}{$\\hookrightarrow$}\\space}]\n cd $\\$$OPENDIHU_HOME/examples/electrophysiology/multidomain/multidomain_prestretch/build_release\n mpirun -n 4 ./multidomain_prestretch ../settings_multidomain_prestretch.py multidomain.py\n \\end{lstlisting}\n\\end{reproduce_no_break}\n\n% --------\n%\n% f==============\n\n% ==================\n%\n% =-------------------\n%-----\n\\subsection{Coupling of Solid Mechanics Models using the Software preCICE}\\label{sec:volume_coupling_contraction}\n\nOne problem of multi-scale simulations with solid mechanics models is the limited amount of parallelism, \nif a coarse mechanics mesh with a low number of elements is chosen. The domain can only be partitioned into as many subdomains as there are elements in the 3D mechanics mesh. While this is not an issue for small scale simulations like the ones shown in the previous sections, it prohibits exploitation of High Performance Computing resources, e.g., if numerous muscle fibers are considered as in \\cref{sec:effects_of_the_mesh_width_emg}.\n\nThe reason for the limited parallelism lies in the partitioning scheme, where every node in the 3D domain corresponds to the subdomain of exactly one process, regardless of the mesh. OpenDiHu does not allow to partition, e.g., the finely resolved 1D muscle fiber meshes differently than the coarse 3D mechanics mesh. However, this restriction can be circumvented by using multiple OpenDiHu programs with different partitioning schemes and by performing the data transfer between the meshes using an external coupling software.\n\nWe provide support for the black-box coupling library preCICE \\cite{precice}. This open source library allows mapping data between different meshes, can communicate values between subdomains that reside on different processors, and implements implicit numerical coupling schemes with quasi-Newton methods. The implementation is known to scale well on small-scale clusters and supercomputers.\nThe preCICE library targets a minimally-invasive approach, where the user application implements a preCICE adapter. Multiple, potentially different solver codes can be coupled numerically and compute individual model parts of a joint multi-physics simulation. Moreover, preCICE has an active and growing community where experiences and codes are shared, and open source adapters are available for several popular solvers.\n\nThis makes the library suited for our use case. We provide two different types of preCICE adapters in OpenDiHu, one for surface coupling of 2D meshes and one for volume coupling of 3D meshes. These adapters integrate with the structure of nested solvers and can be positioned anywhere in the solver tree (cf. \\cref{fig:solver_tree_multidomain_spindles}). The meshes and variables that are exposed to preCICE can be configured in the settings file.\n\nIn the current section, we show how to use the volume coupling adapter to resolve the initially stated issue of limited scalability for coupled simulations with electrophysiology and mechanics models. Subsequently, the next section presents a simulation that uses surface coupling. Details can also be found in \\cite{hlrs2021}.\n\nWe simulate muscle contraction and surface EMG of the biceps muscle using the fiber based electrophysiology model. To fully exploit the capabilities of an 18-core Intel Core i9-10980XE processor, we compute the electrophysiology model using 16 processes and the mechanics model using 2 processes. The data mapping between the differently partitioned 3D meshes is performed by preCICE.\n\n\\Cref{fig:precice_muscle_force} shows the structure of the simulation components with the used meshes and the exchanged variables in this simulation. Two different OpenDiHu programs are executed at the same time, given by the gray boxes. The program corresponding to the left box solves the electric conduction problem, given by the bidomain equation \\cref{eq:bidomain1} on the 3D domain and the action potential propagation model, given by the monodomain equation \\cref{eq:monodomain} on a large number of 1D muscle fiber meshes. \nThe 3D and the 1D mesh in this program are partitioned into 16 subdomains for the 16 processes.\n\n\\Cref{fig:precice_muscle_force} visualizes the meshes and their partitioning to the different processes by the colored inset images. It can be seen that the fibers meshes and the 3D mesh used in the OpenDiHu program in the left box have corresponding subdomains.\n\nThe second OpenDiHu program visualized by the right box in \\cref{fig:precice_muscle_force} only solves the solid mechanics problem using a coarse 3D mesh. The mesh of this problem is partitioned to two processes, as shown by the image. \n\nThe three model parts are numerically coupled and need to exchange several variables. The action potential propagation model, shown at the lower left of \\cref{fig:precice_muscle_force}, computes the transmembrane voltage $V_m$ and the activation parameter $\\gamma$ and maps them from the 0D points on the fibers to the 3D mesh using the mapping scheme described in \\cref{sec:data_mapping_between_meshes}. The activation parameter $\\gamma$ is needed in the solid mechanics model. It is transferred between the two OpenDiHu programs using the functionality of preCICE. After the solid mechanics solver has computed a new deformation of the coarse solid mechanics mesh, preCICE maps the node positions to the finer 3D mesh in the left program. The geometries of the 3D and 1D meshes in the left program are updated accordingly.\nThe preCICE couplings in this example use serial explicit coupling and radial basis functions for the data mapping.\n\n% precice coupling scheme\n\\begin{figure}\n \\centering%\n \\def\\svgwidth{0.9\\textwidth}\n \\input{images/results/application/precice_scheme1.pdf_tex}%\n \\caption{Simulation of muscle contraction: Structure of a coupled simulation with the coupling library preCICE on 18 processes, consisting of the two independent OpenDiHu programs indicated by the gray boxes. The program in the left box solves the electric conduction model using the shown 3D mesh and the action potential propagation model using the shown 1D fiber meshes. Both meshes are partitioned to 16 subdomains as shown by the colors. The program in the right box solves the mechanics problem on a coarse 3D mesh, which is partitioned into 2 subdomains. The arrows between the models indicate the exchanged variables. The coupling within the left gray box is implemented in OpenDiHu, the coupling between the gray boxes is realized using preCICE.}%\n \\label{fig:precice_scheme1}%\n\\end{figure}\n\nThe presented scheme in \\cref{fig:precice_scheme1} allows us to simulate muscle contraction and surface EMG signals. The volume coupling with preCICE is configured between the two 3D meshes. Even for scenarios where EMG signals are not of interest and only the muscle contraction resulting from the activated muscle fibers should be simulated, the presented approach can be used.\n\nAn alternative approach, where preCICE instead couples directly between the fiber meshes and the solid mechanics 3D mesh is also implemented. This approach neither includes the electric conduction model nor the fine 3D mesh for the left program.\nHowever, the mapping between the solid mechanics mesh and the set of 1D fiber meshes is more costly than the mapping between the two 3D meshes, as the fibers contain more data points in total than the 3D mesh of the electric conduction problem. A quantitative analysis of this effect is subject to work in progress.\n\nApart from ensuring better parallel scalability, the OpenDiHu model setup using preCICE also allows to exchange the solid mechanics solver by a different solver code, e.g., a commercial solver. The black-box approach of preCICE allows to exchange the mechanics solver without any changes to the electrophysiology simulation, contributing to the extensibility goal of combining modular model components.\n\n\\begin{reproduce_no_break}\n The two programs with preCICE coupling can be used as follows. Note that the compilation of preCICE has to be enabled in the \\code{user-variables.scons.py} configuration file for the \\code{scons} build system in the \\code{$\\$$OPENDIHU_HOME} directory.\n \\begin{lstlisting}[columns=fullflexible,breaklines=true,postbreak=\\mbox{\\textcolor{gray}{$\\hookrightarrow$}\\space}]\n cd $\\$$OPENDIHU_HOME/examples/electrophysiology/fibers/fibers_contraction/with_precice_volume_coupling/build_release\n mpirun -n 2 ./muscle_contraction ../settings_muscle_contraction.py ramp.py\n mpirun -n 16 ./fibers_with_3d ../settings_fibers_with_3d.py ramp.py\n \\end{lstlisting}\n\\end{reproduce_no_break}\n\n%-----\n\\subsection{Simulation of a Muscle-Tendon Complex using Surface Coupling with preCICE}\\label{sec:surface_coupling_contraction}\n\nIn all previously presented simulations of muscle contraction, the biceps muscle was considered in isolation. In the following, we present a physiologically more correct scenario that includes a 3D description of the tendon mechanics.\nThe simulation consists of four individual solvers in OpenDiHu for the distal tendon, the two proximal tendons, and the muscle belly.\nThe coupling library preCICE is used to numerically couple the parts.\n\nAn advantage of simulations of an entire muscle-tendon complex is their more realistic line of action of the muscle force, compared with a model of the muscle belly without tendons. The goal of the simulation described in this section is to predict the progression of the total muscle force as result of MU recruitment.\n\nFor the simulation of the muscle contraction part, we couple the fiber based electrophysiology model with the nonlinear solid mechanics model as described in \\cref{sec:fiber_based_contraction}. The electrophysiology part of the muscle uses the subcellular model of Shorten et al. \\cite{Shorten2007}. \nThe solid mechanics description of the three tendons uses the hyperelastic Saint-Venant Kirchhoff material, which is the extension of the linear elastic formulation given in \\cref{sec:material_linear_model} to the geometrically nonlinear regime.\nThe proximal tendons are fixed at their insertion points to the skeletal system. We apply corresponding Dirichlet boundary conditions. At the lower end of the distal tendon, a downwards pulling force is applied. We gradually increase the value of this force in the corresponding Neumann boundary conditions from zero up to the maximum value \\SI{100}{\\newton} during the first \\SI{100}{\\ms} of the simulation.\n\nThe muscle fibers are associated with 10 MUs and activated in a ramp during the first $\\SI{1.8}{\\s}$. After each MU has been activated for the first time, it fires with a MU specific frequency between \\SI{7.66}{\\hertz} and \\SI{23.92}{\\hertz} plus a random jitter value of \\SI{10}{\\percent}. This setup replicates the progressive recruitment scenario in \\cite{Klotz2020}.\n\nThe four simulation programs are connected using an implicit Neumann-Dirichlet multi-coupling scheme in preCICE with a constant relaxation factor of 0.5. At the interfaces between the muscle and the tendons, the implicit numerical coupling ensures continuity for the displacements, velocities and stresses.\nThe tendon solvers send their computed displacement and velocity values to the muscle model, where the corresponding Dirichlet boundary conditions are applied. The muscle model computes traction forces by integrating the stress values over the surface and sends the values to the tendon models, where corresponding Neumann boundary conditions are applied. \n\nWe configure preCICE to use Gaussian radial basis functions for the consistent mapping of the variables between the surface meshes of the muscle and the tendons. \nAn error threshold of $\\eps=0.1$ for the coupled displacement values is used to terminate the implicit coupling scheme. As a consequence, the scheme requires approximately two iterations per timestep on average to reach the error threshold. The coupling step is repeated with a timestep width of $\\dt_\\text{coupling} = \\SI{1}{\\ms}$.\n\nWe simulate two scenarios of this model. The first scenario considers a high spatial resolution of 1089 muscle fibers and a simulation time of approximately \\SI{1}{\\s}, while the second scenario considers only 81 fibers but a longer simulation time span of \\SI{10}{\\s}. \n\nIn the first scenario, we use a 3D mesh with $9\\times 9 \\times 21=1701$ nodes, which are partitioned into 160 subdomains.\nThe meshes of the three tendons each consists of 125 nodes and are each partitioned to four subdomains. We run the computation using 172 processes on four compute nodes of the supercomputer Hawk at the High Performance Computing Center Stuttgart. The hardware is described in more detail in \\cref{sec:effects_of_the_mesh_width_emg}. The simulation time span of $\\SI{1}{\\s}$ has a runtime of approximately $\\SI{7}{\\hour}$ $\\SI{20}{\\minute}$.\n\n\\Cref{fig:precice_muscle_force} presents the simulation results of this scenario at the simulation time $t=\\SI{1.05}{\\s}$. \\Cref{fig:precice_activated_muscles_1} shows the muscle fibers, which are attached to the tendons at both ends. Several action potentials can be seen on the fibers. \\Cref{fig:precice_activated_muscles_2} displays the distribution of active stresses in the 3D mesh at the same simulation time. One vertical red line of higher active stress values can be seen at the foreside of the muscle belly, which corresponds to a region of higher muscle activity. This muscle activity results from a MU that is activated early on in the simulation scenario. A corresponding active fiber at that location can also be identified in \\cref{fig:precice_activated_muscles_1}.\n\\Cref{fig:precice_activated_muscles_3} visualizes the parallel partitioning of the 3D domains of muscle and tendons into 160 subdomains for the muscle mesh and 4 subdomains for each of the three tendon meshes.\n\n% contracted state\n\\begin{figure}\n \\centering%\n \\begin{subfigure}[t]{0.34\\textwidth}%\n \\centering%\n \\includegraphics[height=97mm]{images/results/application/precice_activated_muscles1.png}%\n \\caption{Muscle fibers colored by the value of the transmembrane potential $V_m$.}%\n \\label{fig:precice_activated_muscles_1}%\n \\end{subfigure}\\,\n \\begin{subfigure}[t]{0.28\\textwidth}%\n \\centering%\n \\includegraphics[height=97mm]{images/results/application/precice_activated_muscles2.png}%\n \\caption{Active stress in the 3D muscle mesh.}%\n \\label{fig:precice_activated_muscles_2}%\n \\end{subfigure}\n \\begin{subfigure}[t]{0.28\\textwidth}%\n \\centering%\n \\includegraphics[height=97mm]{images/results/application/precice_activated_muscles3.png}%\n \\caption{Parallel partitioning scheme.}%\n \\label{fig:precice_activated_muscles_3}%\n \\end{subfigure}\n \\caption{Simulation of a muscle-tendon complex. Muscle and tendon geometries of the scenario with 1089 muscle fibers, embedded in the skeletal system comprising the ulna bone (lower end) and humerus bone (upper end), result of the simulation at $t=\\SI{1.05}{\\s}$.}%\n \\label{fig:precice_muscle_force}%\n\\end{figure}%\n\nThe second scenario uses a coarser 3D mesh with 525 nodes and a parallel partitioning into eight subdomains. We simulate the resulting muscle force, measured at the top insertion point of the proximal tendons, over a longer time period of \\SI{10}{\\s}. \\Cref{fig:precice_muscle_force0} shows the resulting relative force progression over time. The plot shows the total force as a moving average function over $\\SI{0.1}{\\s}$. It can be seen that the force initially increases, as more and more MUs get activated. A short delay between the onset of the last MU at $\\SI{1.8}{\\s}$ and the maximum force at \\SI{2.39}{\\s} can be seen. The muscle force exhibits large oscillations during the period of high muscle activation. They result from the lower firing frequencies of the later activated, large MUs and their higher contribution to the overall activity, compared to the smaller MUs. \n\n% muscle force\n\\begin{figure}\n \\centering%\n \\includegraphics[width=0.7\\textwidth]{images/results/application/precice_muscle_force.pdf}%\n \\caption{Simulation of muscle force in a muscle-tendon complex. Resulting relative muscle force of the biceps muscle with attached tendons using the second scenario with 81 muscle fibers.}%\n \\label{fig:precice_muscle_force0}%\n\\end{figure}\n\n\\Cref{fig:precice_muscle_force0} also shows that the generated muscle force rapidly decreases after the maximum is reached. This is a result from muscle fatigue, which is described by the Shorten subcellular model. The observed decrease to below \\SI{60}{\\percent} after \\SI{10}{\\s} can also be found in experimental studies of healthy subjects, e.g., in \\cite{Enoka2008}. \n\nIn conclusion, several biophysical simulation scenarios of MU activity induced muscle contraction have been presented in the previous sections. OpenDiHu allowed us to couple the computationally efficient fiber based electrophysiology description with the solid mechanics model of muscle deformation, as well as the biophysically more accurate multidomain model. An algorithm to include prestresses was presented and the coupling software preCICE was used to numerically couple individual parts of the multi-scale model. \n\nThe last presented scenario simulated the generated force of a muscle-tendon complex for a simulation time span of \\SI{10}{\\s}. \nIt can be used in the future to test hypotheses on the influence of various processes along the pathway from MU recruitment over muscle activation to force generation and macroscopic deformation. The simulated force progression related to the maximum voluntary contraction force is a macroscopic quantity, which can be easily measured in in vivo studies. Thus, a connection between the simulation domain and the experimental domain is given, and the microscopic subcellular processes in the muscle fibers are linked to a quantifiable outcome that can be compared with experiments.\n\n\\begin{reproduce_no_break}\n The simulations in this section were carried out on the supercomputer Hawk in Stuttgart. The job scripts can be found in the repository at \\\\ \\href{https://github.com/dihu-stuttgart/performance}{github.com/dihu-stuttgart/performance} in the directory \\code{opendihu/15_precice_biceps/with_electrophysiology}. To run similar simulations on other computers, run commands that are similar to the following (adjust the numbers of processes):\n \\begin{lstlisting}[columns=fullflexible,breaklines=true,postbreak=\\mbox{\\textcolor{gray}{$\\hookrightarrow$}\\space}]\n cd $\\$$OPENDIHU_HOME/examples/electrophysiology/fibers/fibers_contraction/with_tendons_precice/multiple_tendons_with_electrophysiology\n mpirun -n 1 muscle_electrophysiology_precice settings_muscle.py ramp.py\n mpirun -n 1 tendon_linear_precice_dynamic settings_tendon_bottom.py\n mpirun -n 1 tendon_linear_precice_dynamic settings_tendon_top_a.py\n mpirun -n 1 tendon_linear_precice_dynamic settings_tendon_top_b.py\n \\end{lstlisting}\n\\end{reproduce_no_break}\n\n", "meta": {"hexsha": "21a890a2e3a45ee70b47a3cc25805dbcccca6a25", "size": 69918, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "document/08_results_1b.tex", "max_stars_repo_name": "maierbn/phd_thesis_source", "max_stars_repo_head_hexsha": "babee64f01f15d93cb75140eb8c8424883b33c6c", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-05T19:00:04.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-05T19:00:04.000Z", "max_issues_repo_path": "document/08_results_1b.tex", "max_issues_repo_name": "maierbn/phd_thesis_source", "max_issues_repo_head_hexsha": "babee64f01f15d93cb75140eb8c8424883b33c6c", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "document/08_results_1b.tex", "max_forks_repo_name": "maierbn/phd_thesis_source", "max_forks_repo_head_hexsha": "babee64f01f15d93cb75140eb8c8424883b33c6c", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 98.8939179632, "max_line_length": 892, "alphanum_fraction": 0.7705168912, "num_tokens": 18138, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34754701128347687}} {"text": "\\section*{Exercise 20.3-2}\n\\subsection*{Modify vEB trees to support keys that have associated satellite data}\n\nFor both of the keys in the vEB(2) trees (meaning leaves), which are not summaries, we define an additional field, holding the associated satellite data.\n\\\\\nFurthermore, we also define this new field for every $min$ in the (entire) tree, that is not a summary, since the elements stored in $min$ does not appear in the clusters.\n\\\\\nThis way, the new field holds the associated satellite data for every key in the entire tree, but it will not affect running time, since the satellite data are found together with the keys.", "meta": {"hexsha": "0efcbd6b04f5762b4d8476fc320412b02cf290c2", "size": 632, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Uge3/Ex.20.3-2.tex", "max_stars_repo_name": "pdebesc/AADS", "max_stars_repo_head_hexsha": "a26e24d18adee973d3ce88bdfd96d857ec472fdf", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Uge3/Ex.20.3-2.tex", "max_issues_repo_name": "pdebesc/AADS", "max_issues_repo_head_hexsha": "a26e24d18adee973d3ce88bdfd96d857ec472fdf", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Uge3/Ex.20.3-2.tex", "max_forks_repo_name": "pdebesc/AADS", "max_forks_repo_head_hexsha": "a26e24d18adee973d3ce88bdfd96d857ec472fdf", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 79.0, "max_line_length": 189, "alphanum_fraction": 0.7768987342, "num_tokens": 147, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.347535856635085}} {"text": "%# -*- coding: utf-8-unix -*-\n% !TEX program = xelatex\n% !TEX root = ../thesis.tex\n% !TEX encoding = UTF-8 Unicode\n\n\\chapter{Procedures for evaluating the state of networks}\n\\label{appendixA}\n\\begin{algorithm}[H]\n\t\\caption{Algorithm for evaluating the state of networks}\n\t\\label{algo:evaluating}\n\t\\begin{algorithmic} \n\t\t\\State \\bf{INPUT : Two-layer Network G(V, E), interval (m, n), repeat\\_number} \\normalfont \\\\\n\t\t\\Comment{In this study, The number of nodes(V/2) on each layer is $2048$, The number of edges is different according to network structures, and the interval of both parameters is $m = n = 40$}\n\t\t\\State \\bf{OUTPUT : array Average\\_States} \\normalfont \n\t\t\\Comment{Array for Average States according to all parameters}\n\t\t\\State N = len(V), P = linspace(0, 1, m), V = linspace(0, 1, n), M = len(P), N = len(V)\n\t\t\\State set array Average\\_States[M][N]\n\t\t\\State set temp\\_Array[i]\n\t\t\\ForEach {$p \\in \\mathcal P $}\n\t\t\t\\ForEach {$v \\in \\mathcal V $}\n\t\t\t\t\\For{$i$ to repeat\\_number} \n\t\t\t\t\\Comment{In this work, simulations are repeated 100times for average values}\n\t\t\t\t\t\\State \\Call{Doing\\_Interacting\\_Dynamics}{G(V,E), p, v}\n\t\t\t\t\t\\State temp\\_Array = GET\\_STATE(G(V,E))\n\t\t\t\t\t\\Comment{The summation of all nodes' states }\n\t\t\t\t\\EndFor\n\t\t\t\t\\State Average\\_States[p][v] = SUM(temp\\_Array)/repeat\\_number\n\t\t\t\\EndFor\n\t\t\\EndFor\n\t\t\\State \\Return {array Average\\_States[M][N]}\\\\\n\t\t\n\t\t\\Function {Doing\\_Interacting\\_Dynamics}{G(V,E), p, v}\n\t\t \\For{$i$ to $100$steps}\n\t\t \\Comment{100 steps for opinion evolutions}\n\t\t\t\t\\State \\Call{Doing\\_layer\\_A\\_dynamics}{G(V, E), p, sequential}\n\t\t\t\t\\State \\Call{Doing\\_layer\\_B\\_dynamics}{G(V,E), v}\n\t\t\t\\EndFor\n\t\t\t\\State \\Return {G(V, E)}\n\t\t\\EndFunction\n\t\t\\Comment{In this function, the updating rule are assumed as O(o, o) $\\to$ D(o), other updating rules are presented in Appendix \\ref{appendixC}}\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\n\\chapter{Procedures for two-layer network dynamics}\n\\label{appendixB}\n\\begin{breakablealgorithm}\n\t\\caption{Algorithm for layer A dynamics}\n\t\\label{algo:layer A}\n\t\\begin{algorithmic} \n\t\\Function {Doing\\_layer\\_A\\_dynamics}{G(V, E), p, sequential}\n\t\t\\State in\\_edge\\_list = GET\\_INTERNAL\\_EDGE\\_LIST(G(V,E))\n\t\t\\State ex\\_edge\\_list = GET\\_EXTERNAL\\_EDGE\\_LIST(G(V,E))\\\\\n\t\t\\Comment{List of links between two nodes such as (i, j)}\n\t\t\\ForEach {edge $\\in$ in\\_edge\\_list}\n\t\t \\State (a, b) = \\Call{Two\\_Node\\_Change}{edge[0], edge[1]}\n\t\t \\State G(V, E)[edge[0]][state] = a\n\t\t \\State G(V, E)[edge[1]][state] = b\n\t\t\\EndFor \n\t\t\\ForEach {edge $\\in$ ex\\_edge\\_list}\n\t\t\t\\State a = \\Call{One\\_Node\\_Change}{edge[0], edges[1]}\n\t\t\t\\State G(V, E)[edge[0]][state] = a \n\t\t\\EndFor\n\t\t\\State \\Return {G(V, E)} \n\t\\EndFunction\n\t\t\n\t\\Function{One\\_Node\\_Change}{edge[0], edges[1], G(V, E)}\n\t \\Comment{In the case that the state of only one node is changed when two nodes interact: external edges, simultaneous updating rules}\t\n\t\t\\If {edge[0][state] * edge[1][state] > 0}\n\t\t\t\\State node\\_state(a, b) = {Persuasion\\_Process}{edge[0], edge[1]}\n\t\t\t\\State a = node\\_state(a, b)[0]\n\t\t\\EndIf\n\t\t\\If {edge[0][state] * edge[1][state] < 0}\n\t\t\t\\If {edge[0] * edge[1] = -1}\n\t\t\t\t\\If {random z > (1-p)/2} \n\t\t\t\t\t\\Comment{0 0}\n\t\t\t\\State node\\_state(a, b) = \\Call{Persuasion\\_Process}{edge[0], edge[1]}\n\t\t\\EndIf\n\t\t\\If {edge[0][state] * edge[1][state] < 0}\n\t\t\t\\If {edge[0][state] * edge[1][state] = -1}\n\t\t\t\t\\If {random z > (1-p)/2} \n\t\t\t\t\\Comment{00 and edge[1]>0}\n\t\t\t\\State node\\_state(a, b) = (\\Call{move\\_right}{edge[0]}, \\Call{move\\_right}{edge[1]})\n\t\t\\EndIf\n\t\t\\If {edge[0]<0 and edge[1]<0}\n\t\t\t\\State node\\_state(a, b) = (\\Call{move\\_left}{edge[0]}, \\Call{move\\_left}{edge[1]})\n\t\t\\EndIf\n\t\t\\State \\Return {node\\_state(a, b)}\n\t\\EndFunction\n\t\n\t\\Function {Compromise\\_Process}{edge[0], edge[1]}\n\t\\Comment{The states of nodes become moderate}\n\t\t\\If {edge[0]>0 and edge[1]<0}\n\t\t\t\\State node\\_state(a, b) = (\\Call{move\\_left}{edge[0]}, \\Call{move\\_right}{edge[1]})\n\t\t\\EndIf\n\t\t\\If {edge[0]<0 and edge[1]>0}\n\t\t\t\\State node\\_state(a, b) = (\\Call{move\\_right}{edge[0]}, \\Call{move\\_left}{edge[1]})\n\t\t\\EndIf\n\t\t\\State \\Return {node\\_state(a, b)}\n\t\\EndFunction\n\t\t\n\t\\Function {move\\_left}{a}\n\t\\Comment{the state of a node is changed into -1 or unchanged}\n\t\t\\State set Min = -2\n\t\t\\If {a = Min}\n\t\t\t\\State a = a\n\t\t\\Else\n\t\t\t\\State a = a - 1\n\t\t\\EndIf\n\t\t\\State \\Return {a}\n\t\\EndFunction\n\t\n\t\\Function {move\\_right}{a}\n\t\\Comment{the state of a node is changed into +1 or unchanged}\n\t\t\\State set Max = +2\n\t\t\\If {a = Max}\n\t\t\t\\State a = a\n\t\t\\Else\n\t\t\t\\State a = a + 1\n\t\t\\EndIf\n\t\t\\State \\Return {a}\n\t\\EndFunction\n\t\\end{algorithmic}\n\\end{breakablealgorithm}\n\n\\begin{algorithm}[H]\n\t\\caption{Algorithm for layer B dynamics}\n\t\\label{algo:layer B}\n\t\\begin{algorithmic} \n\t\t\\State set A = [for node i in layer A]\n\t\t\\State set B = [for node j in layer B]\n\t\t\\Function {Doing\\_layer\\_B\\_dynamics}{G(V,E), v, sequential}\n\t\t\t\\State same = 0\n\t\t\t\\State opposite = 0\n\t\t\t\\ForEach {node\\_i $\\in \\mathcal B $}\n\t\t\t\t\\State neighbor\\_list = GET\\_NEIGHBOR\\_NODES(node\\_i)\n\t\t\t\t\\ForEach{node\\_j $\\in$ neighbor\\_list}\n\t\t\t\t\t\\If {node\\_i * node\\_j > 0}\n\t\t\t\t\t\t\\State same += 1\n\t\t\t\t\t\t\\Comment {the number of the same orientation node}\n\t\t\t\t\t\\Else\n\t\t\t\t\t\t\\State opposite += 1\n\t\t\t\t\t\t\\Comment {the number of the opposite orientation node}\n\t\t\t\t\t\\EndIf\n\t\t\t\t\\EndFor\n\t\t\t\t\\If {opposite = 0}\n\t\t\t\t\t\\State prob = 0\n\t\t\t\t\\Else\n\t\t\t\t\t\\If {v = 0}\n\t\t\t\t\t\t\\State prob = 1\n\t\t\t\t\t\\Else\n\t\t\t\t\t\t\\State prob = ${{opposite / len(neighbor\\_list)}^{1 / v}} {\\cdot} {len(neighbor\\_list) / opposite}$\n\t\t\t\t\t\\EndIf\n\t\t\t\t\\EndIf\n\t\t\t\t\\If {prob < random z}\n\t\t\t\t\t\\State G(V, E)[node\\_i][state] = -G(V, E)[node\\_i][state]\n\t\t\t\t\\EndIf\n\t\t\t\\EndFor\n\t\t\t\\State \\Return G(V, E)\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{algorithm}\n\n\\chapter{Procedures for applying updating rules on a two-layer network}\n\\label{appendixC}\n\\begin{breakablealgorithm}\n\t\\caption{Updating rules of the layers' order}\n\t\\label{layers_order}\n\t\\begin{algorithmic} \n\t\t\\Function{Layer\\_Order}{layer\\_order}\n\t\t\t\\State set A = [for node i in layer A]\n\t\t\t\\State set B = [for node j in layer B]\n\t\t\t\\If {layer\\_order = sequential order}\n\t\t\t\t\\If{Layer A $\\to$ Layer B}\n\t\t\t\t\t\\State \\Call{Doing\\_layer\\_A\\_dynamics}{G(V, E), p}\n \t\\State \\Call{Doing\\_layer\\_B\\_dynamics}{G(V, E), v}\n \t\\EndIf\n\t\t\t\t\\If{Layer A $\\leftarrow$ Layer B}\n\t\t\t \t\\State \\Call{Doing\\_layer\\_B\\_dynamics}{G(V, E), v}\n\t\t\t \t\t\\State \\Call{Doing\\_layer\\_A\\_dynamics}{G(V, E), p}\n\t\t\t\t\\EndIf\n\t\t\t\\EndIf\n\t\t\t\\If{layer\\_order = simultaneous order}\n\t\t\t\t\\State copy\\_G(V, E) = DEEPCOPY(G(V, E))\n\t\t\t\t\\State A\\_G(V, E) = \\Call{Doing\\_layer\\_A\\_dynamics}{copy\\_G(V, E), p}\n\t\t\t\t\\State B\\_G(V, E) = \\Call{Doing\\_layer\\_B\\_dynamics}{copy\\_G(V, E), v}\n\t\t\t\t\\ForEach {node\\_i $\\in \\mathcal A $}\n\t\t\t\t\t\\State G(V, E)[node\\_i][state] = A\\_G(V, E)[node\\_i][state]\n\t\t\t\t\\EndFor\n\t\t\t\t\\ForEach {node\\_j $\\in \\mathcal B $}\n\t\t\t\t\t\\State G(V, E)[node\\_j][state] = B\\_G(V, E)[node\\_j][state]\n\t\t\t\t\\EndFor\n\t\t\t\\EndIf\n\t\t\t\\If{layer\\_order = random order} \n\t\t\t\t\\State node\\_i = CHOOSE\\_ONE\\_NODE(V)\n\t\t\t\t\\If{node\\_i $\\in \\mathcal A $}\n\t\t\t\t\t\\State \\Call{Doing\\_layer\\_A\\_dynamics}{copy\\_G(V, E), p}\n\t\t\t\t\\EndIf\n\t\t\t\t\\If{node\\_i $\\in \\mathcal B $}\n\t\t\t\t\t\\State \\Call{Doing\\_layer\\_B\\_dynamics}{copy\\_G(V, E), v}\n\t\t\t\t\\EndIf\n\t\t\t\\EndIf\n\t\t\\EndFunction\n\t\\end{algorithmic}\n\\end{breakablealgorithm}\n\n\n\\begin{breakablealgorithm}\n\t\\caption{Updating rules of the nodes' order}\n\t\\label{nodes_order}\n\t\\begin{algorithmic} \n\t\t\\State in\\_edge\\_list = GET\\_INTERNAL\\_EDGE\\_LIST(G(V,E))\n\t\t\\State ex\\_edge\\_list = GET\\_EXTERNAL\\_EDGE\\_LIST(G(V,E))\\\\\n\t\t\\Comment{List of links between two nodes such as (i, j)}\n\t\t\\If {sequential order}\n\t\t\t\\ForEach {edge $\\in$ in\\_edge\\_list}\n\t\t\t\\Comment{internal edges}\n\t\t\t\t\\If {edge$\\in$ Layer A}\n\t\t\t\t\\Comment{When the layer A includes the edge}\n\t\t\t\t\t\\State (a, b) = \\Call{Two\\_Node\\_Change}{edge[0], edge[1], G(V, E)}\n\t\t\t\t\t\\State G(V, E)[edge[0]][state] = a\n\t\t\t\t\t\\State G(V, E)[edge[1]][state] = b\n\t\t\t\t\\EndIf\n\t\t\t\t\\If {edge$\\in$ Layer B}\n\t\t\t\t\\Comment{When the layer B includes the edge}\n\t\t\t\t\t\\State \\Call{Doing\\_layer\\_B\\_dynamics}{G(V, E), v, sequential}\n\t\t\t\t\\EndIf\n\t\t\t\\EndFor \n\t\t\t\\ForEach {edge $\\in$ ex\\_edge\\_list}\n\t\t\t\\Comment{external edge}\n\t\t\t\t\\State \\Call{One\\_Node\\_Change}{edge[0], edges[1], G(V, E)}\n\t\t\t\t\\State G(V, E)[edge[0]][state] = a\n\t\t\t\\EndFor\n\t\t\t\\State \\Return {G(V, E)}\n\t\t\\EndIf\n\t\t\\If {simultaneous order}\n\t\t\tcopy\\_G(V, E) = DEEPCOPY(G(V, E))\n\t\t\t\\ForEach {edge $\\in$ in\\_edge\\_list}\n\t\t\t\t\\If {edge$\\in$ Layer A}\n\t\t\t\t\t\\State previous\\_state = copy\\_G(V, E)\n\t\t\t\t\t\\State a = \\Call{One\\_Node\\_Change}{edge[0], edge[1], copy\\_G(V, E)}\n\t\t\t\t\t\\State G(V, E)[edge[0]][state] = a\n\t\t\t\t\t\\State copy\\_G(V, E) = previous\\_state\n\t\t\t\t\\EndIf\n\t\t\t\t\\If {edge$\\in$ Layer B}\n\t\t\t\t\t\\State previous\\_state = copy\\_G(V, E)\n\t\t\t\t\t\\State G(V, E)[edge[0][state]\n\t\t\t\t\t\\State = \\Call{Doing\\_layer\\_B\\_dynamics}{copy\\_G(V, E), v, sequential}[edge[0]][state]\n\t\t\t\t\t\\State G(V, E)[edge[1][state]\n\t\t\t\t\t\\State = \\Call{Doing\\_layer\\_B\\_dynamics}{copy\\_G(V, E), v, sequential}[edge[1]][state]\n\t\t\t\t\t\\State copy\\_G(V, E) = previous\\_state\n\t\t\t\t\\EndIf\n\t\t\t\\EndFor \n\t\t\t\t\\ForEach {edge $\\in$ ex\\_edge\\_list}\n\t\t\t\t\t\\State a = \\Call{One\\_Node\\_Change}{edge[0], edges[1], copy\\_G(V, E)}\n\t\t\t\t\t\\State G(V, E)[edge[0]][state] = a\n\t\t\t\t\\EndFor\n\t\t\t\t\\State \\Return {G(V, E)}\n\t\t\\EndIf\n\t\t\\If {random order}\n\t\t\t\\While{len(edge\\_list) = 0}\n\t\t\t\t\\State SELECT edge IN edge\\_list\n\t\t\t\t\\Comment{select an edge among the edge lists until all edges are selected}\n\t\t\t\t\\If {edge $\\in$ in\\_edge\\_list}\n\t\t\t\t\t\\If {edge$\\in$ Layer A}\t\n\t\t\t\t\t\t\\State (a, b) = \\Call{Two\\_Node\\_Change}{edge[0], edge[1], G(V, E)}\n\t\t\t\t\t\t\\State G(V, E)[edge[0]][state] = a\n\t\t\t\t\t\t\\State G(V, E)[edge[1]][state] = b\n\t\t\t\t\t\\EndIf\n\t\t\t\t\t\\If {edge$\\in$ Layer B}\t\n\t\t\t\t\t\t\\State \\Call{Doing\\_layer\\_B\\_dynamics}{G(V, E), v, sequential}\n\t\t\t\t\t\\EndIf\n\t\t\t\t\\EndIf\n\t\t\t\t\\If {edge $\\in$ ex\\_edge\\_list}\n\t\t\t\t\t\\State a = \\Call{One\\_Node\\_Change}{edge[0], edges[1], G(V, E)}\n\t\t\t\t\t\\State G(V, E)[edge[0]][state] = a\n\t\t\t\t\\EndIf\n\t\t\t\\EndWhile\n\t\t\t\\State \\Return {G(V, E)}\n\t\t\\EndIf\n\t\\end{algorithmic}\n\\end{breakablealgorithm}\n\n\n\\begin{breakablealgorithm}\n\t\\caption{Updating rules of the edges' order(Simultaneous)}\n\t\\label{edges_order}\n\t\\begin{algorithmic} \n\t\t\\ForEach {node i $ \\in \\mathcal V $}\n\t\t\t\\State (a, b, c) = \\Call{Calculate\\_Prob}{G(V, E), p, node i}\n\t\t\t\\State set random z\n\t\t\t\\Comment{0 < z < 1}\n\t\t\t\\If {z > a + b}\n\t\t\t\t\\State s = COMPROMISE\\_FUNC(node i)\n\t\t\t\t\\Comment {the state of node i becomes moderate, the state of node i is returned}\n\t\t\t\\EndIf\n\t\t\t\\If {z < a}\n\t\t\t\t\\State s = UNCHANGE(node i)\n\t\t\t\t\\Comment {the state of node i is unchanged, the state of node i is returned}\n\t\t\t\\EndIf\n\t\t\t\\If {z > a and z < a + b}\n\t\t\t \t\\State s = PERSUASION\\_FUNC(node i)\n\t\t\t \t\\Comment {the state of node i becomes extreme, the state of node i is returned}\n\t\t\t\\EndIf\n\t\t\t\\State G(V, E)[node i][state] = s\n\t\t\\EndFor\n\t\t\\State \\Return {G(V, E)}\n\t\t\n\t\t\\Function{Calculate\\_Prob}{G(V, E), p, node i}\n\t\t\t\\State same = GET\\_SAME\\_ORIENTATION\\_NUM(node i)\n\t\t\t\\State opposite = GET\\_OPPOSITE\\_ORIENTATION\\_NUM(node i)\n\t\t\t\\For{$n$ to same}\n\t\t\t\t\\For{$m$ to opposite}\n\t\t\t\t\t\\State n\\_combi = $_{same} C_n$\n\t\t\t\t\t\\State m\\_combi = $_{opposite} C_m$\n\t\t\t\t\t\\If {$n = m$}\n\t\t\t\t\t\t\\State $unchange += {p^{n+opposite-m}} * {(1 - p)^{same-n+m}} * n\\_combi * m\\_combi$\n\t\t\t\t\t\\EndIf\n\t\t\t\t\t\\If{$n > m$}\n\t\t\t\t\t\t\\State $persuasion += {p^{n+opposite-m}} * {(1 - p)^{same-n+m}} * n\\_combi * m\\_combi$\n\t\t\t\t\t\\EndIf\n\t\t\t\t\t\\If{$n < m$}\n\t\t\t\t\t\t\\State $compromise += {p^{n+opposite-m}} * {(1 - p)^{same-n+m}} * n\\_combi * m\\_combi$\n\t\t\t\t\t\\EndIf\n\t\t\t\t\\EndFor\n\t\t\t\\EndFor\n\t\t\t\\State \\Return {unchange, persuasion, compromise}\n\t\t\\EndFunction\t\t\t\n\t\\end{algorithmic}\n\\end{breakablealgorithm}\n\n\n\n\\chapter{Selecting keynodes by single and multiple indicators}\n\\label{appendixD}\n\\begin{algorithm}[!htb]\n\t\\caption{Algorithm for ranking by single and multiple indicators}\n\t\\label{algo:keynodes}\n\t\\begin{algorithmic} %每行显示行号\n\t\t\\State \\bf{INPUT : Network G(V, E), selected\\_layer, parameters $(p, v)$, indicators} \\normalfont \n\t\t\\State \\bf{OUTPUT : array critical\\_nodes\\_order} \\normalfont \n \\State set N = len(V), M = len(indicators)\n\t\t\\State set array node\\_centrality[N][M]\n\t\t\\State set array node\\_rank[N][M]\n\t\t\\State set array temp[N]\n\t\t\\State set array critical\\_nodes\\_order[N]\n\t\t\\State node\\_centrality = Calculate\\_Centrality(G(V, E), selected\\_layer)\n\t\t\\State node\\_rank = Get\\_Rank(node\\_centrality)\n \\ForEach {$i \\in \\mathcal V $}\n \\For{$m \\gets 0$ to $M-1$}\n\t\t\t\t\\State temp[i] += node\\_rank[i][m]\n\t\t \t\\EndFor\n\t\t\\EndFor\n \\ForEach {$i \\in \\mathcal V $}\n\t\t\t\\State critical nodes order.append(temp.index(min(temp)))\n\t\t\t\\State temp[temp.index(min(temp))] = $2*N$ or $\\inf$\n\t\t\\EndFor\n\t\t\\State \\Return array critical\\_nodes\\_order[N]\n\t\\end{algorithmic}\n\\end{algorithm}\n", "meta": {"hexsha": "f57d52e57a893993a0bfdfc044d5e9ba092354f9", "size": 13587, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/app_code.tex", "max_stars_repo_name": "leighsix/thesisgit", "max_stars_repo_head_hexsha": "77d3c41a7c045c656d2e28dc4e5cfea418632d62", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tex/app_code.tex", "max_issues_repo_name": "leighsix/thesisgit", "max_issues_repo_head_hexsha": "77d3c41a7c045c656d2e28dc4e5cfea418632d62", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-09-21T09:57:36.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-19T19:06:17.000Z", "max_forks_repo_path": "tex/app_code.tex", "max_forks_repo_name": "leighsix/thesisgit", "max_forks_repo_head_hexsha": "77d3c41a7c045c656d2e28dc4e5cfea418632d62", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.8384615385, "max_line_length": 194, "alphanum_fraction": 0.6214764113, "num_tokens": 5064, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.34734535367853997}} {"text": "\\chapter{parReset}\n\n\\section{Introduction}\n\nA linearized process may have parameters that are initialized, changed and used, and subsequently ignored until they are initialized again.\nParameters that often follow this pattern are \\emph{control flow parameters}.\n\nDuring the period in between the last change or use of a parameter and its subsequent initialization, a parameter could have different values.\nThese values contribute to the size of the state space of the process \\emph{without} adding any new behavior!\nIt may therefore be advantageous to detect from which moment the value of a parameter is no longer used and set it to a default value instead.\n\n\\section{Algorithm}\n\nThe algorithm is a generalization of an existing algorithm \\cite{van2009state}.\nIt consists of two phases, a preparation phase and an iteration phase.\n\n\\subsection{Preparation phase}\n\nConsider all possible pairs of summands of the LPE (including symmetric pairs).\nOf a given summand pair $(s, t)$, let $t$ be a \\emph{successor} of $s$ if $s$ contains a recursive process instantiation and if the following equation is satisfiable:\n\\begin{align*}\nc_s \\land {c_t}[p \\rightarrow v_s(p) \\;|\\; p \\in P]\n\\end{align*}\n\nwhere\n\n\\begin{itemize}\n\\item $c_s$ and $c_t$ are the guards of summands $s$ and $t$, respectively;\n\\item $P$ is the set of all process parameters;\n\\item $v_s$ is a function that yields the expression that summand $s$ assigns to a given process parameter in its recursive process instantiation.\n\\end{itemize}\n\nDuring the preparation phase, we determine all successors of each summand of the LPE.\nThis gives an overapproximation of the control flow graph of the LPE.\n\n\\subsection{Iteration phase}\n\nThis phase follows these steps:\n\n\\begin{enumerate}\n\n\\item For each summand $s$, create a set $R_s$ that contains all process parameters.\nThis means that, initially, we assume that all process parameters are used by one or more of the successors of $s$.\n\n\\item For each summand $s$, set the value of $R_s$ to $\\bigcup\\limits_{t \\in S_s}^{} r(t)$ where $S_s$ is the set of all successors of $s$ (as determined during the previous phase) and where $r$ is the function\n\\begin{align*}\nr(t) = \\left( \\text{vars}(c_t) \\cup \\bigcup\\limits_{x \\in R_t}^{} \\text{vars}(v_t(x)) \\right) \\setminus C_t\n\\end{align*}\n\nwhere\n\n\\begin{itemize}\n\\item $\\text{vars}(c_t)$ gives the free variables in $c_t$, the guard of summand $t$;\n\\item $v_t$ is a function that yields the expression that summand $t$ assigns to a given process parameter in its recursive process instantiation;\n\\item $\\text{vars}(v_t)$ gives the free variables in $v_t$;\n\\item $C_t$ is the set of the communication variables used by summand $t$.\n\\end{itemize}\n\n\\item Repeat the previous step until the new value of $R_s$ is the same as before for all summands $s$.\n\n\\item For each summand $s$ with $R_s \\subset P$, choose some $\\rho = [x \\rightarrow h_x \\;|\\; x \\in P \\setminus R_s]$ so that $({c_s} \\land {c_t}[p \\rightarrow v_s(p) \\;|\\; p \\in P])[\\rho]$ is unsatisfiable for all $t \\notin S_s$, where\n\n\\begin{itemize}\n\\item $P$ is the set of all process parameters of the LPE;\n\\item $h_x$ is a closed expression of the same sort as $x$;\n\\item $c_s$ and $c_t$ are the guards of summands $s$ and $t$, respectively.\n\\end{itemize}\n\nFor each process parameter $x \\in P \\setminus R_s$, change the recursive process instantiation of $s$ so that $x$ is assigned $v_s(x)[\\rho]$.\nNote that the choice of $\\rho$ ensures that the set of summands that are successors of $s$ can only decrease.\n\nIn practice, it is expensive to generate a suitable $\\rho$, and only the initial values of the parameters in $P \\setminus R_s$ are tried (that is, $\\rho = [x \\rightarrow v_0(x) \\;|\\; x \\in P \\setminus R_s]$).\nIf this does not satisfy the further requirements, the summand is left unchanged.\n\n\\end{enumerate}\n\n\\section{Example}\n\nConsider the following LPE:\n\n\\begin{lstlisting}\n//Process definition:\nPROCDEF example[A :: Int, B](x, y :: Int)\n = A ? i [[x==0]] >-> example[A, B](1, i)\n + A ? i [[x==1 && i==y]] >-> example[A, B](2, y)\n + B [[x==2]] >-> example[A, B](3, y)\n + B [[x==3]] >-> example[A, B](0, y)\n ;\n\n//Initialization:\nexample[A, B](0, 0);\n\\end{lstlisting}\n\nFinding the successors of each summand is easy: each summand has exactly one successor, namely the next one, except in case of the fourth summand, where the first summand is the successor.\n\nIt is also obvious that $x$ will always be in $R_s$ for each summand $s$, because each summand uses $x$ in its guard.\n\nProcess parameter $y$ will always be in $R_{s_1}$, where $s_1$ is the first summand, because $y$ is used in the guard of $s_1$'s successor (the second summand).\nAfter a few iterations, however, $y$ is removed from $R_{s_2}$ to $R_{s_4}$.\nThis means that $y$ is assigned a default value in the corresponding summands.\nDepending on the mood of the SMT solver, this could give\n\n\\begin{lstlisting}\n//Process definition:\nPROCDEF example[A :: Int, B](x, y :: Int)\n = A ? i [[x==0]] >-> example[A, B](1, i)\n + A ? i [[x==1 && i==y]] >-> example[A, B](2, 0)\n + B [[x==2]] >-> example[A, B](3, 0)\n + B [[x==3]] >-> example[A, B](0, 0)\n ;\n\n//Initialization:\nexample[A, B](0, 0);\n\\end{lstlisting}\n\n\n\n\n", "meta": {"hexsha": "be1461c62a55b456b4d46a732d9531f2c99bc77b", "size": 5197, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "_tex/lpeopsDoc/parReset.tex", "max_stars_repo_name": "Sercammus/TxsLpeOps", "max_stars_repo_head_hexsha": "3354f2762cf195e571f4c05040ec500165969359", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_tex/lpeopsDoc/parReset.tex", "max_issues_repo_name": "Sercammus/TxsLpeOps", "max_issues_repo_head_hexsha": "3354f2762cf195e571f4c05040ec500165969359", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_tex/lpeopsDoc/parReset.tex", "max_forks_repo_name": "Sercammus/TxsLpeOps", "max_forks_repo_head_hexsha": "3354f2762cf195e571f4c05040ec500165969359", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.6722689076, "max_line_length": 236, "alphanum_fraction": 0.7119492015, "num_tokens": 1506, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.689305616785446, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.34734535367853997}} {"text": "\\documentclass[12pt]{article}\n\\usepackage{lipsum}\n\\usepackage{authblk}\n\\usepackage{fancyhdr}\n\n\\usepackage{amssymb}\n\\usepackage{amsmath}\n\n\\usepackage[english]{babel}\n\\usepackage{tikz}\n\\usetikzlibrary{positioning}\n\\usepackage[colorlinks=true,\n urlcolor=blue,\n linkcolor=black,\n citecolor=black]{hyperref}\n\\usepackage[doublespacing]{setspace}\n\\usepackage{geometry}\n \\geometry{\n a4paper,\n right=25mm,\n left=25mm,\n bottom=25mm,\n top=25mm\n }\n\n\\usepackage{amsmath}\n\\usepackage{listings}\n\n\\pagestyle{fancy}\n\\setlength{\\headheight}{15pt}\n\n\\usepackage{algpseudocode}\n\\usepackage{algorithm}\n\n%\\usepackage{cite}\n\\usepackage[nottoc,numbib]{tocbibind}\n\\usepackage{natbib}\n\n\\usepackage{../presentation/colordef}\n\\usepackage{../presentation/lvblisting}\n\n\\usepackage{wrapfig}\n\\usepackage{booktabs}\n%\\raggedbottom\n\\usepackage{tikz}\n\\usetikzlibrary{shapes.geometric, arrows}\n\n\\tikzstyle{io} = [trapezium, trapezium left angle=70,\n trapezium right angle=110, minimum width=3cm,\n minimum height=0.5cm, text centered,\n draw=black, fill=blue!30]\n\\tikzstyle{process} = [rectangle, minimum width=3cm,\n minimum height=0.5cm, text centered,\n draw=black, fill=orange!30]\n\\tikzstyle{decision} = [rectangle, minimum width=3cm,\n minimum height=0.5cm, text centered,\n draw=black, fill=green!30]\n\\tikzstyle{algo} = [circle, minimum width=0.5cm, draw=black, fill=orange!30]\n\\tikzstyle{arrow} = [thick,->,>=stealth]\n\n\\renewenvironment{abstract}{%\n\n\\begin{center}\n\\begin{minipage}{0.9\\textwidth}\n\\rule{\\textwidth}{1pt}}\n{\\par\\noindent\\rule{\\textwidth}{1pt}\n\\end{minipage}\n\\end{center}}\n\n\\begin{document}\n\n\\title{Numerical Methods for solving Eigenvalue-Problems}\n\\author{Thomas Siskos}\n\\date{}\n\n\\begin{titlepage}\n \\begin{center}\n\n \\includegraphics[scale=1.25]{../presentation/hulogo.pdf} \\par\n {\\scshape\\LARGE Humboldt Universit{\\\"a}t zu Berlin \\par}\n\n {\\scshape\\Large Seminar Paper\\par}\n\n {\\huge\\bfseries Numerical Methods for solving Eigenvalue-Problems\\par}\n\n\\vspace{1cm}\n\n {\\Large\\itshape Thomas Siskos (580726)\\par}\n\n {\\Large\\scshape Numerical Introductory Course\\par}\n\n \\vfill\n Supervised by: \\par\n {\\Large Prof. Dr. Brenda L{\\'o}pez Cabrera \\par}\n \\vfill\n {\\large \\today\\par}\n \\end{center}\n\n\\end{titlepage}\n\n\\tableofcontents\n\\newpage\n\\listofalgorithms\n\\listoffigures\n\\listoftables\n\\newpage\n\n\\section{Motivation}\n\n\\begin{singlespacing}\n\\begin{abstract}\n\\centerline{\\textbf{Abstract}} \\\\\n\\small\nEigenvalues and eigenvectors are often the solution to multidimensional optimization problems, however computing them by hand for anything but trivial matrices is most of the time infeasible or inpractical. To this extend we would like to deploy an automated procedure which yields the correct eigenvectors and eigenvalues. We demonstrate the relevance of eigenvalues and eigenvectors by revising two applications from statistics, Principal Component Analysis and Fisher's Linear Discriminant Analysis, which we follow up by investigating four algorithms suited for eigenvalue problems. Finally we provide a compound solution that takes advantage of each algorithm's strengths.\n\\end{abstract}\n\\vspace{3mm}\n\\end{singlespacing}\n\nFor many statistical applications eigenvectors provide a formidable solution. Be it dimensionality reduction in terms of a Principal Component Analysis or classification by Fisher's Linear Discriminant Analysis, both come in the guise of optimization problems. But what are eigenvalues and eigenvectors?\n\nIf $A$ is an $n \\times n$ matrix, $v$ is a non-zero vector and $\\lambda$ is a scalar, such that\n\n\\begin{equation}\n\\label{eigenvalue-def}\nAv = \\lambda v\n\\end{equation}\n\nthen $v$ is called an \\textit{eigenvector} and $\\lambda$ is called an \\textit{eigenvalue} of the matrix $A$.\nAn eigenvalue of A is a root of the characteristic equation,\n\n\\begin{equation}\n\\label{eigenvalue-solve}\ndet\\left(A - \\lambda I \\right) = 0.\n\\end{equation}\n\nEach \\textit{eigenvalue} is associated with an \\textit{eigenvector}. Geometrically speaking, we require a vector which, when multiplied by matrix $A$, will not get rotated but only elongated by the \\textit{eigenvalue} $\\lambda$ \\citep[p. 13]{NME}.\n\nWhen confronted with a high-dimensional data matrix $X \\in \\mathbb{R}^{n \\times m}$ an analyst often wishes to find a lower-dimensional representation, while conserving as much of the data's structure as possible. One way of achieving this goal is to choose a standardized linear combination of features that aim to maximize the variance of the projection $\\delta^{\\prime} X$. We can formalize this as\n\n\\begin{equation}\n\t\\label{pca_obj}\n \tmax\\ \\delta^{\\prime} Var \\left(X\\right) \\delta \\;\\; s.t. \\;\\; \\sum \\delta_i^2 = 1.\n\\end{equation}\n\nThe solution to this optimization problem is given by\n\\begin{equation}\n\\label{pca_sol}\n\tY = \\Gamma^{\\prime} \\left(X - \\mu\\right)\n\\end{equation}\n\nwhere $Y \\in \\mathbb{R}^{n \\times m}$ is the matrix of rotations,\n\t $\\Gamma \\in \\mathbb{R}^{m \\times m}$ is the matrix of eigenvectors,\n\t $\\mu \\in \\mathbb{R}^m$ is the vector of sample means \\citep[p. 234]{MVA}.\n\nA second use for Eigenvalues lies in Fisher's Linear Discriminant Analysis for classification problems. In this setting we would like to find the rotation that is best suited to separate the classes. To this effect we try to maximize the between-class-variance while simoultaneously trying to minimize the within-class variance. This is equivalent to maximizing the so called Rayleigh-coefficient\n\n\\begin{equation}\n\\label{LDA}\nmax J = \\frac{J_b(w)}{J_w(w)} = \\frac{w^{\\prime} S_{b} w}{w^{\\prime} S_{w} w}.\n\\end{equation}\n\nThe between and within class variance matrices are defined as $S_b = \\sum\\limits_{c=1}^{C}(\\mu_c - \\mu)(\\mu_c - \\mu)^{\\prime}$ and $S_w = \\sum\\limits_{c=1}^{C}\\sum\\limits_{i \\in c}(x_i - \\mu_c)(x_i - \\mu_c)^{\\prime}$. The solution to this maximization problem is given by\n\\begin{equation}\nS_B^{\\frac{1}{2}} S_W^{-1} S_B^{\\frac{1}{2}} v = \\lambda v,\n\\end{equation}\n\nwhich again is an eigenvalue problem of a square, symmetric matrix and can be readily computed \\citep[p. 219]{duda}.\n\nIn the following we will try to extract the eigenvalues of matrices by reformulating any complicated eigenvalue problem into a straightforward one by diagonalizing the matrix in question, without altering the eigenvalues we would like to compute. In section two we provide the mathematical tools, which allow us to do so. We follow these justifications by proposing two main algorithms for computing eigenvalues, first the Jacobi-Method for symmetric matrices, then the QR-Method for arbitrary square matrices in section 3. Additionally, for the QR-Method we define two extensions which try to increase the initial QR-algorithm's speed.\nFor all algorithms we provide implementations in the \\texttt{Python}-programming-language, all plots have been produced with the \\texttt{matplotlib}-module and the custom algorithms rely solely on modules from \\texttt{Python}'s \\texttt{scipy} environment for scientific computing \\citep{python, matplotlib, scipy}. All scripts are available \\href{https://github.com/thsis/NIS18}{online}. In section 4 we will analyse the implemented routines by critically reflecting upon the accuracy of the obtained results as well as their efficiency. In the final section we provide a final algorithm which combines the strengths of the defined procedures by chosing the algorithm that is most fit for the underlying problem.\n% ==============================================================================\n\\section{Similarity Transformations}\n\nIn general we want to reformulate the eigenvalue problem of a complicated matrix into an eigenvalue problem of a simple matrix, which yields the same eigenvalues. Simple matrices in our case will be diagonal matrices, since with them it is possible to identify their eigenvalues as entries on the main diagonal. A transformation that conserves the eigenvalues of a matrix is called a \\textit{similarity transformation}.\n\nTwo $n \\times n$ matrices $A$ and $B$ are called \\textit{similar} if there exists an invertible matrix $P$ such that\n\n\\begin{equation}\n\\label{similarity}\nA = P^{-1} B P.\n\\end{equation}\n\nIt is obvious that the similarity relationship is commutative as well as transitive. If $A$ and $B$ are similar, it holds that\n\n\\begin{align*}\nB - \\lambda I &= P^{-1} B P - \\lambda P^{-1} I P \\notag \\\\\n &= A - \\lambda I.\n\\end{align*}\n\n Hence $A$ and $B$ have the same eigenvalues. This fact also follows immediately from the transitivity of the similarity relationship and the fact that a matrix is similar to the diagonal matrix formed from its eigenvalues, as stated in the spectral-decomposition. Important types of similarity transformations are based around orthogonal matrices. If $Q$ is orthogonal and\n\n$$ A = Q^{\\prime} B Q, $$\n\n$A$ and $B$ are called \\textit{orthogonally similar} \\citep{NLA}. We will use \\textit{orthogonal similarity transformations} to diagonalize matrices we wish to know the eigenvalues of. For our custom implementations we will make use of Givens-Rotations and Householder Reflectors (for a detailed discussion consider the appendix).\n\n% ==============================================================================\n\\section{Algorithms}\n% ------------------------------------------------------------------------------\n\\subsection{Jacobi Method}\n\nThe \\textit{Jacobi-Method} for computing the eigenvalues of a symmetric matrix $A \\in \\mathbb{R}^{n \\times n}$ deploys a sequence of orthogonal similarity transformations that eventually results in\n$$ A = P \\Lambda P^{-1} \\Leftrightarrow \\Lambda = P^{-1} A P,$$\n\nwhere $\\Lambda$ is diagonal and $P$ consists of a sequence of matrix multiplications $P = \\prod\\limits_{k=1}^{K} V_{p_k, q_k}(\\theta_k)$ and $V_{p_k, q_k}(\\theta_k)$ is a matrix that performs a Givens-Rotation. Such a rotation is performed by an identity matrix with four of its elements replaced, such that\n\\small\n\\begin{equation}\n\\label{givens_rotator}\nV_{pq}(\\theta) = \\begin{bmatrix}\n 1 \\\\\n & \\ddots \\\\\n & & \\cos\\theta & & \\sin\\theta \\\\\n & & & \\ddots \\\\\n & & -\\sin\\theta & & \\cos\\theta \\\\\n & & & & & \\ddots \\\\\n & & & & & & 1 \\\\\n \\end{bmatrix}\n\\end{equation}\n\\normalsize\n\\newpage\nThe \\textit{Jacobi iteration} is defined as\n\n\\begin{equation}\nA^{(k)} = V^{\\prime}_{p_k, q_k}(\\theta_k) A^{(k-1)} V_{p_k, q_k}(\\theta_k),\n\\end{equation}\n\nwhere $p_k, q_k$ and $\\theta_k$ are chosen such that $A^{(k)}$ resembles more a diagonal matrix than $A^{(k-1)}$. Specifically they will be chosen as to reduce the sum of squares of the off-diagonal elements. Also, note that $A^{(k)}$ and $A^{(k-1)}$ are similar, since $V^{\\prime}_{p_k, q_k}(\\theta_k)$ is an orthogonal matrix for every step $k$. The benefit of matrices as in (\\ref{givens_rotator}) is, that it is easy to choose an angle $\\theta_k$ in order to introduce a zero in a single multiplication step. Here we use the rotations in the context of a similarity transformation, so it is a little more complicated.\n\nWe require that $a_{pq}^{(k)} = 0$, this implies\n\\begin{equation}\n\\label{theta-troubles}\na^{(k-1)}_{pq} (\\cos^2\\theta - \\sin^2\\theta) + \\left( a^{(k-1)}_{pp} - a^{(k-1)}_{qq} \\right) \\cos\\theta \\sin\\theta = 0.\n\\end{equation}\nWe can use the trigonometric identities\n\\begin{align*}\n\\cos(2\\theta) &= \\cos^2 \\theta \\sin^2 \\theta \\\\\n\\sin(2\\theta) &= 2 \\cos\\theta \\sin\\theta,\n\\end{align*}\n\nin (\\ref{theta-troubles}) we have\n$$\\tan(2\\theta) = \\frac{2a^{(k-1)}_{pq}}{a^{(k-1)}_{pp} - a^{(k-1)}_{qq}}.$$\n\nFrom this we can retrieve the angle and obtain the rotation matrix in each iteration \\citep[p. 128]{NLA}.\n\nThe algorithm converges if the off-diagonal elements are sufficiently small. The best index pair at a given iteration is the pair $(p, q)$ that satisfies\n\n\\[\n|a^{(k-1)}_{pq}| = \\mathop{\\max_{i precision$}\n \\State Find indices $i$, $j$ of largest value in lower triangle of $abs(L)$\n \\State $L_{max} \\gets L_{i,j}$\n \\State $\\theta \\gets \\frac{1}{2}\\cdot \\arctan(\\frac{2A_{i, j}}{A_{i, i}-A_{j, j}})$\n \\State $V \\gets I$\n \\State $V_{i, i}, V_{j, j} \\gets \\cos \\theta$; $V_{i, j}, V_{j, i} \\gets -\\sin \\theta,\\; \\sin \\theta$\n \\State $A \\gets V^{\\prime} A V$; $U \\gets UV$\n\n \\EndWhile\\\\\n \\Return $diag(A),\\; U$\n\\end{algorithmic}\n\\end{algorithm}\n\n% ------------------------------------------------------------------------------\n\\subsection{QR-Method}\n\nThe most widely used algorithm to extract eigenvalues is the so called \\textit{QR}-method. The most important advantage of the \\textit{QR}-method over the \\textit{Jacobi}-method is that it can be applied to non-symmetric matrices. Note however, that it is simpler for symmetric matrices, since the eigenvalues are real-valued.\n\nThe \\textit{QR}-method to extract the eigenvalues of a square matrix $A \\in \\mathbb{R}^{n \\times n}$ is performed by first computing the titular \\textit{QR} decomposition of $A$.\n\n\\begin{equation}\n\\label{qr_a}\nA = QR,\n\\end{equation}\n\nwhere $Q$ is an orthogonal and $R$ is an upper triangular matrix. Then define the \\textit{QR} iteration as\n\n\\begin{equation}\n\\label{qr-method}\n A^k = Q_{k-1}^{\\prime} A_{k-1} Q_{k-1} = R_{k-1}Q_{k-1}\n\\end{equation}\n\nNote hereby that all matrices in the sequence $\\{A_k\\}$ share the same eigenvalues, since this procedure is a similarity transformation due to $Q$'s orthogonality \\citep[p. 121]{NME}. Additionally, for practical applications it is preferable to compute the \\textit{QR}-iteration in the way shown at the rightmost part of equation (\\ref{qr-method}). Although, mathematically, each statement is exactly identical there is a non-negligible difference due to computational imperfections and limited machine precision. The reason is that the computation of $Q_{k-1}^{\\prime} A_{k-1} Q_{k-1}$, obviously, requires two matrix multiplications whereas the result of $R_{k-1}Q_{k-1}$ can be readily obtained by one. When combining multiple steps over a long sequence of \\textit{QR} iterations the additional computations lead to additional rounding errors, which can have an influence on the accuracy of the obtained results. Besides, less computations lead of course to a faster procedure in general.\n\n\\begin{figure}\n\\begin{center}\n\\label{qrm1-plot}\n\\caption{\\href {https://github.com/thsis/NIS18/tree/master/media/plots}{Progress basic QR-Method} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n \\includegraphics[scale=0.6]{../media/plots/qrm1.png}\n\\end{center}\n\\end{figure}\n\nFigure \\ref{qrm1-plot} visualizes the progress of the basic \\textit{QR}-method on the same $5 \\times 5$ matrix as in Figure \\ref{j-plot}. Compared to the \\textit{Jacobi}-method it does not explicitly pick a single element that will be eliminated per iteration. Instead, the \\textit{QR}-method extracts the eigenvalues by a process that is called \\textit{chasing}. By that we mean that alternating steps are being performed, which create non-zero eintries in positions $(i+2, i)$, $(i+3, i)$ and $(i+3, i+1)$ and restore them to zero, as the nonzero entries are moved farther down the matrix \\citep[p. 130]{NLA}. We can also see that compared to the \\textit{Jacobi}-Method, so far, the \\textit{QR}-algorithm lacks in speed. Where the \\textit{Jacobi}-method was almost done diagonalizing the matrix in iteration 10, the basic \\textit{QR}-algorithm still had multiple non-zero entries left. Thus we would like to make minor improvements on the algorithm's efficiency.\n\\begin{algorithm}\n\\caption{\\href {https://github.com/thsis/NIS18/blob/master/algorithms/eigen.py}{\\texttt{QRM1}} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n\\label{qr1-meth}\n \\begin{algorithmic}[1]\n \\Require square matrix $A$\n \\Statex \\textbf{initialize: } $conv \\gets False$\n \\While{not $conv$}\n \\State $Q, R \\gets$ QR-Factorization of $A$\n \\State $A \\gets RQ$\n \\If{$A$ is diagonal}\n \\State $conv \\gets \\texttt{True}$\n \\EndIf\n \\EndWhile\\\\\n \\Return $diag\\left(A\\right),\\; Q$\n \\end{algorithmic}\n\\end{algorithm}\n\n\n\\subsubsection{Hessenberg Variant}\n\nIn order to speed up the \\textit{QR}-method it is advisable to transform the matrix to its upper \\textit{Hessenberg} form. A matrix $A$ is of upper \\textit{Hessenberg} form if it is upper triangular except for the first subdiagonal, which may be non-zero. In particular $a_{ij} = 0\\; \\forall i > j + 1$:\n\\small\n$$\n\\begin{bmatrix}\nX & X & X & \\dots & X & X\\\\\nX & X & X & \\dots & X & X\\\\\n0 & X & X & \\dots & X & X\\\\\n0 & 0 & X & \\dots & X & X \\\\\n\\vdots & \\vdots & & \\ddots & \\vdots & \\vdots\\\\\n0 & 0 & 0 & \\dots & X & X \\\\\n\\end{bmatrix}$$\n\\normalsize\nA matrix can be reduced to \\textit{Hessenberg} form in a finite number of similarity transformations using Householder transformations or Givens rotations. For symmetric matrices the transformation into a \\textit{Hessenberg}-form results in a tridiagonal matrix. But even for non-symmetric matrices, the \\textit{Hessenberg}-form allows a large saving in subsequent computations. After the transformation we can deploy the previously defined \\textit{QR}-method \\citep[p. 130]{NLA}.\n\n\\begin{figure}\n\\centering\n\\caption{\\href {https://github.com/thsis/NIS18/tree/master/media/plots}{Progress Hessenberg-QR-Method} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n \\label{qr2-plot}\n \\includegraphics[scale=0.6]{../media/plots/qrm2.png}\n\\end{figure}\n\nFigure \\ref{qr2-plot} visualizes the progress of the \\textit{Hessenberg} variant of the \\textit{QR}-method. In order to make it comparable to the previous algorithms the same $5 \\times 5$-dimensional matrix is being evaluated. We can readily see, that the transformation to \\textit{Hessenberg}-form results in a tridiagonal matrix. This facilitates computations and explains the vastly improved resulting matrix after 10 iterations compared to the basic \\textit{QR}-method. However, it still does not match the progress of the \\textit{Jacobi}-method after the same number of iterations.\n\n\n\\begin{algorithm}\n\\caption{\\href {https://github.com/thsis/NIS18/blob/master/algorithms/eigen.py}{\\texttt{QRM2}} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n\\label{qr2-meth}\n\\begin{algorithmic}[1]\n \\Require square matrix $A$\n \\State $A \\gets \\texttt{hessenberg(}A\\texttt{)}$\n \\State continue with: \\Call {QRM1} A\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Accelerated Variant}\n\nWe could already improve the \\textit{QR}-method and cut down on computational cost. However, we still cannot match the results of the \\textit{Jacobi} method. To this effect we present one additional adjustment on the \\textit{QR}-method that aims to improve convergence speed. The general idea is, that we deliberately create an additional zero entry on the main diagonal by subtracting a scalar on each element, perform the \\textit{QR}-iteration and finally undo the subtraction. In particular, we define\n\n\\begin{equation}\n\\label{qrm3-prop}\nT^{m} = \\begin{bmatrix}\n\n\\alpha^{m}_1 & \\beta^{m}_1 & 0 & 0 & \\dots & 0 \\\\\n\\beta^{m}_1 & \\alpha^{m}_2 & \\beta^{m}_2 \\\\\n0 & \\beta^{m}_2 & \\alpha^{m}_3 & \\beta^{m}_3 & & \\vdots \\\\\n & & \\ddots & \\ddots & \\ddots \\\\\n & & & & \\beta^{m}_{n-2} & \\alpha^{m}_{n-1} & \\beta^{m}_{n-1} \\\\\n0 & & & & & \\beta^{m}_{n-1} & \\alpha^{m}_n \\\\\n\n\\end{bmatrix}\n\\end{equation}\n\n\n\\begin{align*}\nT^{m} &= T - t _{n, n} I \\\\\nT^{m} &= QR \\\\\nT^{m+1} &= T^{m} + t _{n, n} I\n\\end{align*}\n\nAfter this we can define the accelerated iteration step as\n\\begin{align}\n\\label{qrm3}\nR_m &= Q^{\\prime}_m\\left(T^m - \\alpha_n^{m}I\\right) \\\\\nT^{m+1} &= Q^{\\prime}_m\\left(T^m - \\alpha_n^{m}I\\right)Q_m + \\alpha_n^{m}I \\\\\n &= Q^{\\prime}_m T^m Q_m\n\\end{align}\n\nAgain $T^{m+1}$ is similar to $T^m$.\n\n\\begin{figure}[H]\n\\centering\n\\caption{\\href {https://github.com/thsis/NIS18/tree/master/media/plots}{Progress Accelerated QR-Method} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n \\label{qr3-plot}\n \\includegraphics[scale=0.6]{../media/plots/qrm3.png}\n\\end{figure}\n\nFigure \\ref{qr3-plot} visualizes the progress of the accelerated \\textit{QR}-method. For comparability, the matrix used, is the same $5 \\times 5$ matrix as before. Most notably is, that the accelerated method still performs worse than the \\textit{Jacobi} method. So far results seem similar compared to the \\textit{Hessenberg} variant of the \\textit{QR}-method. The case can be made that the accelerated method performs considerably better than the basic \\textit{QR}-method and slightly better than the \\textit{Hessenberg} variant after 10 iterations. Howbeit, this claim warrants further analysis.\n\n\\begin{algorithm}\n\\begin{algorithmic}[1]\n\\caption{\\href {https://github.com/thsis/NIS18/blob/master/algorithms/eigen.py}{\\texttt{QRM3}} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n\n\\Require square matrix $A \\in \\mathbb{R}^{p \\times p}$\n\\State $T \\gets \\texttt{hessenberg}(A),\\ conv \\gets False$\n\\While{not $conv$}\n \\State $Q, R \\gets$ QR-Factorization of $T - t_{p-1, p-1} I$\n \\State $T \\gets RQ + t_{p-1, p-1}I$\n \\If{$T$ is diagonal}\n \\State $conv \\gets True$\n \\EndIf\n\\EndWhile\\\\\n\\Return $diag\\left(T\\right),\\; Q$\n\\end{algorithmic}\n\\end{algorithm}\n\n% ==============================================================================\n\\section{Analysis}\n\nIn order to test the quality of our implemented algorithms we will follow a two pronged approach. First, we will scrutinize the quality of the obtained results. Once we can be sure that the implemented routines are reliable or, to be more precise, under which conditions our routines are reliable we can analyze their convergence behavior and compare the efficiency accross algorithms in terms of computational cost.\n% ------------------------------------------------------------------------------\n\\subsection{Accuracy}\n\nIf we wish to investigate the accuracy of our algorithms, we first need to define what we mean by that. We set up an environment of unit tests that compare the absolute difference between the vector of true eigenvalues of a matrix and the vector of computed eigenvalues by our custom algorithms. If any element of this $L_1$-norm is larger than a threshold of $10^{-5}$ we consider the test as failed. In order to obtain matrices with known eigenvalues we will reverse the spectral decomposition, which states that every square, symmetric matrix $A \\in \\mathbb{R}^{m \\times m}$ can be written as\n\\begin{equation}\n\\label{spectral-decomp}\nA = \\Gamma \\Lambda \\Gamma^{\\prime} = \\sum\\limits_{j=1}^m \\lambda_j \\gamma_j \\gamma^{\\prime}_j,\n\\end{equation}\nwhere $\\Lambda = diag(\\lambda_1, \\dots, \\lambda_m)$ is a diagonal matrix with the eigenvalues of $A$ on its main diagonal and where $\\Gamma=(\\gamma_1, \\dots, \\gamma_m)$ is the matrix containing the associated eigenvectors \\citep{MVA}. This means we can draw a random vector of eigenvalues $\\lambda_{true}$ and a random orthogonal matrix to construct a test matrix $A_{test}$ \\citep{lapack}. Once the matrix $A_{test}$ is initialized, we can plug it into the eigenvalue-routine we wish to test.\n\n\n\\begin{table}[b]\n\\centering\n\\caption{\\href {https://github.com/thsis/NIS18/tree/master/tests/tests_eigen.py}{Failed unit tests accross matrix-sizes} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n\\label{acc-res}\n\\scriptsize\n\\begin{tabular}{ccrrrrr}\n\\toprule\nAlgorithm & Maximum number & \\multicolumn{5}{c}{Dimension}\\\\\n\\cline{3-7}\n & of iterations & 3 & 4 & 5 & 6 & 7 \\\\\n\\toprule\n\\bottomrule\n\\textit{Jacobi} & - & 0 & 0 & 0 & 0 & 0 \\\\\n\\textit{QRM1} & 10 & 785 & 952 & 994 & 999 & 1000 \\\\\n & 100 & 110 & 224 & 349 & 474 & 565 \\\\\n & 1000 & 7 & 16 & 33 & 38 & 45 \\\\\n & 10000 & 1 & 1 & 1 & 1 & 3 \\\\\n & 100000 & 0 & 0 & 0 & 0 & 0 \\\\\n\\textit{QRM2} & 10 & 826 & 975 & 997 & 1000 & 1000 \\\\\n & 100 & 111 & 191 & 360 & 467 & 602 \\\\\n & 1000 & 9 & 16 & 22 & 35 & 58 \\\\\n & 10000 & 0 & 0 & 3 & 2 & 0 \\\\\n & 100000 & 0 & 0 & 0 & 0 & 0 \\\\\n\\textit{QRM3} & 10 & 251 & 616 & 896 & 983 & 997 \\\\\n & 100 & 29 & 52 & 116 & 222 & 311 \\\\\n & 1000 & 6 & 2 & 8 & 19 & 21 \\\\\n & 10000 & 0 & 0 & 1 & 1 & 3 \\\\\n & 100000 & 0 & 0 & 0 & 0 & 0 \\\\\n\\bottomrule\n\\end{tabular}\n\\normalsize\n\\end{table}\n\n\nThe number of failures out of 1000 unit tests on matrices of different dimensions are catalogued in Table \\ref{acc-res}. Most notably, we see that the \\textit{Jacobi}-method outperforms any of the \\textit{QR}-methods in terms of accuracy. This is mostly due to its convergence criterion being defined differently than the convergence criterion of the \\textit{QR}-methods. Simply put, the \\textit{Jacobi}-method will always converge in this setup. Also keep in mind, that since the \\textit{Jacobi}-method is only allowed for symmetric matrices, it essentially eliminates two non-zero off-diagonal elements at once. Comparing only the \\textit{QR}-methods amongst each other one can clearly see, that the quality of the obtained results increases, the more willing an analyst is to wait. Convergence is slower for the \\textit{QR}-methods and a premature stop leads to poor eigenvalues. One the other hand, if the \\textit{QR}-methods converge, they unanimously pass all tests. It would therefore be possible to always achieve an arbitrary grade of precision by allowing for a higher number of iterations. For the highest possible accuracy one can set the maximum number of iterations to infinity.\n\n% ------------------------------------------------------------------------------\n\\subsection{Efficiency}\n\n\n\\begin{figure}[b]\n\\centering\n\\caption{\\href {https://github.com/thsis/NIS18/blob/master/tests/tests_eigen.py}{Unit-tests: Iterations} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n \\label{algo_efficiency}\n \\includegraphics[width=\\textwidth, height=6.5cm]{../media/plots/iterations_boxplot.png}\n\\end{figure}\n\nNow that we can be confident that each of the routines is able to offer accurate results we are interested in which method is most appropriate for which kind of matrix. Especially, we are interested in convergence speed. Figure \\ref{algo_efficiency} shows the distribution of steps performed before the algorithms reached convergence or their maximum allowed number of iterations. Again, the \\textit{Jacobi}-method outperforms every \\textit{QR}-method. For matrices of small dimensions, i.e. eleven or less, the accelerated \\textit{QR}-method outperforms the other \\textit{QR}-methods. However, for larger matrices the \\textit{Hessenberg} variant seems to be slightly more appropriate, since the accelerated method tends to have a wider range of needed steps. Additionally, the accelerated method's distribution has a wider tail whenever the dimension of the test matrices is larger than eleven. This means that high waiting times, operating on the same matrix, are more likely to occur using the \\textit{accelerated} variant compared to the \\textit{Hessenberg} variant or even the basic variant. This can only mean that the supplementary additions and subtractions of the accelerated method provide a questionable aid. In some cases they do seem to help for small matrices, however when confronted with larger matrices the cost of the additions and subtractions outweighs the benefit of the facilitated \\textit{QR}-step. Surprisingly, there seems to be no striking difference between the distributions of the basic and the \\textit{Hessenberg} variants. Looking closely, the only effect of performing a \\textit{Hessenberg} transformation prior to running a \\textit{QR}-algorithm on a test matrix is to mitigate outliers in terms of computation steps needed before convergence. And even that is true, only for large matrices. This can only mean that the cost of computing the \\textit{Hessenberg}-form of the matrix is roughly as high as the reduction of steps needed before convergence is reached.\n\n% ==============================================================================\n\\section{Conclusion}\n\\begin{figure}\n\\centering\n\\caption{\\href {https://github.com/thsis/NIS18/tree/master/algorithms/eigen.py}{Decision process of the final eigenvalue routine} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n\\begin{tikzpicture}\n\\node (in1) [io] {Input: matrix A};\n\\node (dec_sym) [decision, below of=in1, yshift=-0.25cm] {Is A symmetric?};\n\\node (algo1) [algo, below of=dec_sym, xshift=-2cm, yshift=-0.5cm] {\\texttt{Jacobi}};\n\\node (dec_small) [decision, below of=dec_sym, xshift=2cm, yshift=-0.5cm] {rows of A $\\leq$ 11?};\n\\node (algo2) [algo, below of=dec_small, xshift=-1cm, yshift=-0.5cm] {\\texttt{QRM3}};\n\\node (algo3) [algo, below of=dec_small, xshift=1cm, yshift=-0.5cm] {\\texttt{QRM2}};\n\n\\draw [arrow] (in1) -- (dec_sym);\n\\draw [arrow] (dec_sym) -- node[anchor=east] {yes} (algo1);\n\\draw [arrow] (dec_sym) -- node[anchor=west] {no} (dec_small);\n\\draw [arrow] (dec_small) -- node[anchor=east] {yes} (algo2);\n\\draw [arrow] (dec_small) -- node[anchor=west] {no} (algo3);\n\\end{tikzpicture}\n\\end{figure}\n\nFinally, we would like to present a compound solution which combines the strengths of all presented algorithms and automatically picks the procedure that fits best to the problem. For obvious reasons, it is best to chose the \\textit{Jacobi}-method, whenever applicable. That is, when the input-matrix is symmetric. Whenever the matrix is not symmetric, a second decision based on the matrix' size will determine which \\textit{QR}-method will be used. If the matrix' dimension is smaller than eleven it will chose the accelerated \\textit{QR} method. Otherwise, the \\textit{Hessenberg} variant of the \\textit{QR}-method will be deployed.\n\nWe have thus demonstrated the practical importance of eigenvalues in the context of Principal Component Analysis and Fisher's Linear Discriminant Analysis. For computing the eigenvalues we have shown various numerical procedures. We examinated the \\textit{Jacobi}- and the \\textit{QR}-method, as well as a selection of simple improvements towards the basic \\textit{QR}-method. We have shown that the implemented routines can achieve an arbitrary grade of precision and provided a guided procedure for when to use which of the provided algorithms.\n\n\n% ==============================================================================\n\\newpage\n\\section{Appendix}\n\\subsection{Householder-Reflections}\n\nOur goal is to diagonalize a matrix in order to programmatically extract its eigenvalues. In section 2 we have seen that there exist such transformations that conserve the eigenvalues of a given matrix. However, we require transformations that, additionally, eliminate non-zero entries on the off-diagonal elements of a particular matrix. A greedy technique, that eliminates all but the first elements of a vector is proposed in the form of Householder-Reflections.\n\nLet $u$ and $v$ be orthonormal vectors and let $x$ be a vector in the space spanned by $u$ and $v$, such that\n$$x = c_1 u + c_2 + v$$\nfor some scalars $c_1$ and $c_2$. The vector\n$$\\tilde{x}=-c_1 u + c_2 v$$\nis a \\textit{reflection} of x through the line difined by the vector u. Now consider the matrix\n\n\\begin{equation}\nP = I - 2 uu^{\\prime}.\n\\end{equation}\nNote that\n\\begin{align*}\nPx &= c_1 u + c_2 v - 2c_1 uuu^{\\prime} - 2 c_2 v uu^{\\prime} \\\\\n &= c_1 u + c_2 v - 2c_1 u^{\\prime}uu - 2 c_2 u^{\\prime} v u \\\\\n &= -c_1 u + c_2 v\\\\\n &= \\hat{x}.\n\\end{align*}\n\nThe matrix $P$ is called a reflector. The usefulness of Householder-Reflections stems from the fact that it is easy to transform a vector of the form\n\n$$x = (x_1, x_2, \\dots, x_n)$$\n\ninto a vector\n$$\\hat{x} = (\\hat{x}_1, 0, \\dots, 0).$$\n\nIf $Qx = \\hat{x}$, then $||x||_2 = ||\\hat{x}||_2$ and thus $\\hat{x_1} = \\pm ||x||_2$, since it is the only non-zero entry. To construct the reflector let\n\\begin{equation}\n\\label{house-con}\nv = (x_1 + sign(x_1)||x||_2, x2, \\dots, x_n)\n\\end{equation}\nand $u = \\frac{v}{||v||_2}$ \\citep{NLA}. We use the $sign$-function, which simply returns the sign of its argument in order to avoid the numerical problem known as \\textit{catastrophic cancellation}. It can occur when adding two very close, but different, floating point numbers of differing signs. In some unfortunate cases both of these numbers get represented by the same computer number and, because of their opposing signs cancel each other out. In our case this would mean, that we reflect the vector onto the origin. Fortunately, by making use of the \\textit{sign} function we can make sure that both summands will share the same sign, thus mitigating any concerns about catastrophic cancellation.\n\nWe use reflectors to compute the so called $QR$ factorization of an aribitrary square matrix $A \\in \\mathbb{R}^{n \\times n}$.\n\n\\begin{equation}\n\\label{QR-prop}\nA = QR\n\\end{equation}\n\nwhere $Q$ is orthogonal and $R$ is upper triangular. We use Householder transformations to reflect the $i^{th}$ column and produce zeros below the $(i, i)$ element. The QR-factorization of a matrix $A \\in \\mathbb{R}^5$ would therefore consist of five Householder-reflections with $Q=P_5 P_4 P_3 P_2 P_1$. The number of computations for the \\textit{QR} factorization in this fashion is $2n^3 / 3$ multiplications and $2n^3 / 3$ additions \\citep[p. 110]{NLA}.\n\\newpage\n% ------------------------------------------------------------------------------\n\\subsection{Givens-Rotations}\n\n\\begin{wrapfigure}[7]{r}{0.35\\textwidth}\n\\centering\n\\caption{\\href {https://github.com/thsis/NIS18/blob/master/media/plots}{Rotation of \\textit{x}} \\protect\\includegraphics[scale=0.05]{qletlogo.pdf}}\n\\includegraphics[scale=0.5]{../media/plots/givens.png}\n\\end{wrapfigure}\n\nAnother way of forming the \\textit{QR}-factorization is by using orthogonal transformations which rotate a vector in a way such that a specific element becomes 0 and only one other element in the vector being changed. These transformations are called \\textit{Givens transformations, Givens Rotations} or \\textit{Jacobi transformations}\n\nUsing orthogonal transformations we can also rotate a vector in such a way that a specified element becomes 0 and only one other element in the vector is changed. The basic idea can be seen in a two-dimensional space. We wish to rotate the vector $x = (x_1, x_2)$ to $\\tilde{x} = (\\tilde{x_1}, 0)$ as with a reflector.\n\nIt is easy to see that the orthogonal matrix\n$$Q=\\begin{bmatrix}\n\\cos\\theta & \\sin\\theta \\\\\n-\\sin\\theta & \\cos\\theta\n\\end{bmatrix}$$\n\nperforms the desired rotation, if $\\cos\\theta = \\frac{x_1}{||x||_2}$ and $\\sin\\theta = \\frac{x_2}{||x||_2}$\n\nIn general, we can construct an orthogonal $matrix V_{pq}$, that will transform the vector $$x = (x_1,\\dots, x_p, \\dots x_q, \\dots, x_n)$$ to $$\\tilde{x} = (x_1,\\dots, \\tilde{x}_p, \\dots 0, \\dots, x_n)$$. The matrix that does this is\n\\scriptsize\n\\begin{equation}\n\\label{givens-rotation}\nV_{pq}(\\theta) = \\begin{bmatrix}\n 1 \\\\\n & \\ddots \\\\\n & & \\cos\\theta & & \\sin\\theta \\\\\n & & & \\ddots \\\\\n & & -\\sin\\theta & & \\cos\\theta \\\\\n & & & & & \\ddots \\\\\n & & & & & & 1 \\\\\n \\end{bmatrix}\n\\end{equation}\n\\normalsize\n\twhere $\\cos\\theta = \\frac{x_p}{||x||}$ and $\\sin\\theta = \\frac{x_q}{||x||}.$\n\nA rotation matrix is therefore the same as an identitiy matrix, in which we change four elements \\citep[p. 105]{NME}. We will use Givens rotations primarily in the Jacobi-Method.\n\\newpage\n\\subsection{Code}\n\\subsubsection{Eigenvalue Routines}\n \\lstinputlisting[language=Python]{../algorithms/helpers.py}\n \\lstinputlisting[language=Python]{../algorithms/eigen.py}\n \\newpage\n\\subsubsection{Analysis: Figures}\n \\lstinputlisting[language=Python]{../analysis/analysis.py}\n \\newpage\n\\subsubsection{Analysis: Unit tests}\n \\lstinputlisting[language=Python]{../tests/tests_eigen.py}\n \\newpage\n% ==============================================================================\n\\bibliographystyle{apalike}\n\\bibliography{references}\n\n\\end{document}\n", "meta": {"hexsha": "3e798586c6f158a435d040758de0c57a7f5944fe", "size": 38217, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "term-paper/term-paper.tex", "max_stars_repo_name": "thsis/NIS18", "max_stars_repo_head_hexsha": "1f2a7be1ab209fa7c0a25cb8eace744336b07c1f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "term-paper/term-paper.tex", "max_issues_repo_name": "thsis/NIS18", "max_issues_repo_head_hexsha": "1f2a7be1ab209fa7c0a25cb8eace744336b07c1f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "term-paper/term-paper.tex", "max_forks_repo_name": "thsis/NIS18", "max_forks_repo_head_hexsha": "1f2a7be1ab209fa7c0a25cb8eace744336b07c1f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.4421221865, "max_line_length": 1996, "alphanum_fraction": 0.6869456001, "num_tokens": 10665, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.689305616785446, "lm_q1q2_score": 0.34734535367853997}} {"text": "% Note on compiling: requires external programs dot (from the graphvix package\n% on Debian/Ubuntu) and dot2tex.\n% You need to pass the --shell-escape option to pdflatex\n% (or use the Makefile, which required latexmk).\n%\\documentclass[aspectratio=169,handout]{beamer}\n%\\documentclass[aspectratio=169,handout,hyphens]{beamer} % hyphens option for url package which beamer loads.\n\\documentclass[aspectratio=169,hyphens]{beamer} % hyphens option for url package which beamer loads.\n\n\\def\\UrlFont{\\small\\tt}\n\n\\usepackage{minted}\n\\usepackage{color}\n%\\usepackage[pgf,dot]{dot2texi}\n%\\usepackage{tikz}\n%\\usetikzlibrary{shapes,arrows}\n\n\\usetheme{Pittsburgh}\n\\usecolortheme{beaver}\n% \\useoutertheme{infolines}\n\n\\title{Constraint Programming in Haskell}\n\\subtitle{Melbourne Haskell Users Group}\n\\author{David Overton}\n\\date{29 October 2015}\n\n\\AtBeginSection[]\n{\n\t\\begin{frame}\n\t\t\\frametitle{Table of Contents}\n\t\t\\tableofcontents[currentsection]\n\t\\end{frame}\n}\n\n%\\lstnewenvironment{code}{\\lstset{language=Haskell,basicstyle=\\small}}{}\n\n\\newminted[code]{haskell}{fontsize=\\small}\n\\newmint[hask]{haskell}{fontsize=\\small}\n\\newminted{prolog}{fontsize=\\small}\n\n\\definecolor{mygreen}{rgb}{0,0.6,0}\n\\definecolor{mygrey}{rgb}{0.5,0.5,0.5}\n\\definecolor{mymauve}{rgb}{0.58,0,0.82}\n\n\\newtheorem{observation}[theorem]{Observation}\n\n\\newcommand\\myheading[1]{%\n \\par\\bigskip\n {\\large\\color{blue}#1}\\par\\smallskip}\n\\begin{document}\n\n\\frame{\\titlepage}\n\n\\section{Constraint programming}\n\n\\begin{frame}\n \\frametitle{Constraint programming}\n Constraint programming is a declarative programming paradigm for solving constraint satisfaction problems.\n \\pause\n \\begin{itemize}\n \\item A set of \\emph{constraint variables} over a \\emph{domain}, e.g. Booleans, integers, reals, finite domain.\n \\pause\n \\item A set of \\emph{constraints} between those variables.\n \\pause\n \\item A \\emph{solver} to find solutions to the constraints, i.e. assignments of variables to values in the domain such that all constraints are satisfied.\n \\end{itemize}\n \\pause\n Applications: planning, scheduling, resource allocation, computer graphics, digital circuit design, programming language analysis, \\ldots\n\\end{frame}\n\n\\section{Constraint logic programming}\n\n\\begin{frame}\n \\frametitle{Constraint logic programming}\n \\begin{itemize}\n \\item Constraint programming and logic programming work well together.\n \\pause\n \\item Many Prolog implementations have built in constraint solvers.\n \\pause\n \\item Basic idea:\n \\begin{itemize}\n \\item add constraints to the constraint store\n \\pause\n \\item constraint solver works behind the scenes to propagate constraints\n \\pause\n \\item use Prolog's backtracking search mechanism to generate solutions\n \\pause\n \\end{itemize}\n \\item Advantages over pure logic programming:\n \\begin{itemize}\n \\item ``constrain-and-generate'' rather than ``generate-and-test''\n \\pause\n \\item constraint solver can greatly reduce the search space required compared to Prolog's built-in depth-first-search\n \\pause\n \\item much more powerful than relying on just unification and backtracking\n \\end{itemize}\n \\end{itemize}\n\\end{frame}\n\n\\section{Finite domain constraints}\n\n\\begin{frame}[fragile]\n \\frametitle{Finite domain constraints}\n\\begin{itemize}\n \\item One of the most widely used varieties of constraint solver.\n \\pause\n \\item Variables range over a finite domain of integers.\n \\pause\n \\item Simple equality and inequality constraints: $=$, $\\neq$, $<$, $>$, $\\leq$, $\\geq$\n \\pause\n \\item Also simple arithmetic expressions: $+$, $-$, $\\times$, \\texttt{abs}\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\frametitle{Arc consistency}\nSolver uses an \\emph{arc consistency algorithm}, e.g.\\ AC-3\n\\begin{itemize}\n \\item Constraint store holds the set of constraints to be checked.\n \\pause\n \\item For each constraint, the domains of the variables involved are checked to\n ensure they are consistent with the contraint.\n \\pause\n \\item Any values in the domains that break consistency are removed.\n \\pause\n \\item If the domain of a variable changes then all other constraints involving that variable are\n rechecked.\n\\end{itemize}\n\\pause\n\\begin{example}\n\\begin{displaymath}\n \\begin{array}{c}\n x \\in \\{1, 2, 3\\} ~~\\land~~ y \\in \\{1, 2, 3\\} \\\\\n \\pause\n \\textrm{add constraint~} x < y \\\\\n \\pause\n \\Rightarrow ~~~ x \\in \\{1, 2\\} ~~\\land~~ y \\in \\{2, 3\\} \\\\\n \\pause\n \\textrm{add constraint~} y = 2 \\\\\n \\pause\n \\Rightarrow ~~~ x \\in \\{1\\} ~~\\land~~ y \\in \\{2\\}\n \\end{array}\n\\end{displaymath}\n\\end{example}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\frametitle{Example: $n$ queens in SWI-Prolog}\n\n\\begin{columns}[t]\n\\column[c]{0.35\\paperwidth}\n\\begin{minted}[fontsize=\\small]{prolog}\nn_queens(N, Qs) :-\n length(Qs, N),\n Qs ins 1..N,\n safe_queens(Qs).\n\nsafe_queens([]).\nsafe_queens([Q|Qs]) :-\n safe_queen(Qs, Q, 1),\n safe_queens(Qs).\n\nsafe_queen([], _, _).\nsafe_queen([Q|Qs], Q0, D0) :-\n Q0 #\\= Q,\n abs(Q0 - Q) #\\= D0,\n D1 #= D0 + 1,\n safe_queen(Qs, Q0, D1).\n\\end{minted}\n\\column[c]{0.5\\paperwidth}\n\\includegraphics[width=0.45\\paperwidth]{8queens.png}\n\\end{columns}\n\\end{frame}\n\n\\section{Constraint programming in Haskell}\n\n\\subsection{Basic equality and inequality}\n\n\\begin{frame}[fragile]\n \\frametitle{Constraint programming in Haskell}\nHow can we do something similar in Haskell?\\pause{}\n\\textbf{Use a monad!}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\frametitle{Example: $n$ queens in SWI-Prolog and Haskell}\n\n\\begin{columns}[t]\n\\column[t]{0.33\\paperwidth}\n\\begin{minted}[fontsize=\\footnotesize]{prolog}\nn_queens(N, Qs) :-\n length(Qs, N),\n Qs ins 1..N,\n safe_queens(Qs).\n\nsafe_queens([]).\nsafe_queens([Q|Qs]) :-\n safe_queen(Qs, Q, 1),\n safe_queens(Qs).\n\nsafe_queen([], _, _).\nsafe_queen([Q|Qs], Q0, D0) :-\n Q0 #\\= Q,\n abs(Q0 - Q) #\\= D0,\n D1 #= D0 + 1,\n safe_queen(Qs, Q0, D1).\n\\end{minted}\n\\pause\n\\column[t]{0.59\\paperwidth}\n\\begin{minted}[fontsize=\\footnotesize]{haskell}\nnQueens :: Int -> FD [FDExpr]\nnQueens n = do\n qs <- news n (1, n)\n safeQueens qs\n return qs\n\nsafeQueens :: [FDExpr] -> FDConstraint\nsafeQueens [] = return ()\nsafeQueens (q : qs) = do\n safeQueen qs q 1\n safeQueens qs\n\nsafeQueen :: [FDExpr] -> FDExpr -> FDExpr -> FDConstraint\nsafeQueen [] _ _ = return ()\nsafeQueen (q : qs) q0 d0 = do\n q0 #\\= q \n abs (q0 - q) #\\= d0\n safeQueen qs q0 (d0 + 1)\n\\end{minted}\n\\end{columns}\n\\end{frame}\n\\begin{frame}[fragile]\n\\begin{itemize}\n \\item List monad provides backtracking / search / multiple solutions.\n \\pause\n \\item Wrap it in a state monad transformer to keep track of the constraint store.\n \\pause\n\\end{itemize}\n\\begin{code}\ntype FD a = StateT FDState [] a\n\\end{code}\n\\pause\n\\begin{code}\ntype FDConstraint = FD ()\n\\end{code}\n\\pause\n\\begin{code}\n-- Run the monad to obtain a list of solutions.\nrunFD :: FD a -> [a]\nrunFD fd = evalStateT fd initState\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n\\begin{code}\nnewtype FDVar = FDVar { _unwrapFDVar :: Int } deriving (Ord, Eq)\n\ntype VarSupply = FDVar\n\ndata Domain\n = Set IntSet\n | Range Int Int\n\ndata VarInfo = VarInfo { _delayedConstraints :: !FDConstraint\n , _domain :: !Domain }\n\ntype VarMap = Map FDVar VarInfo\n\ndata FDState = FDState { _varSupply :: !VarSupply, _varMap :: !VarMap }\n\ninitState :: FDState\ninitState = FDState { _varSupply = FDVar 0, _varMap = Map.empty }\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n\\begin{code}\nnewVar :: ToDomain a => a -> FD FDVar\nnewVar d = do\n v <- use varSupply\n varSupply . unwrapFDVar += 1\n let vi = initVarInfo & domain .~ toDomain d\n varMap . at v ?= vi\n return v\n\nnewVars :: ToDomain a => Int -> a -> FD [FDVar]\nnewVars n d = replicateM n (newVar d)\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n\\begin{code}\n-- Look up the current domain of a variable.\nlookup :: FDVar -> FD Domain\nlookup x =\n use $ varMap . ix x . domain\n\n-- Update the domain of a variable and fire all delayed constraints\n-- associated with that variable.\nupdate :: FDVar -> Domain -> FDConstraint\nupdate x i = do\n vi <- use $ varMap . ix x\n varMap . ix x . domain .= i\n vi ^. delayedConstraints\n\n-- Add a new constraint for a variable to the constraint store.\naddConstraint :: FDVar -> FDConstraint -> FDConstraint\naddConstraint x constraint =\n varMap . ix x . delayedConstraints %= (>> constraint)\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n\\begin{code}\n-- Useful helper function for adding binary constraints between FDVars.\ntype BinaryConstraint = FDVar -> FDVar -> FDConstraint\naddBinaryConstraint :: BinaryConstraint -> BinaryConstraint\naddBinaryConstraint f x y = do\n let constraint = f x y\n constraint\n addConstraint x constraint\n addConstraint y constraint\n\n-- Constrain two variables to have the same value.\nsame :: FDVar -> FDVar -> FDConstraint\nsame = addBinaryConstraint $ \\x y -> do\n xv <- lookup x\n yv <- lookup y\n let i = xv `intersection` yv\n guard $ not $ Domain.null i\n when (i /= xv) $ update x i\n when (i /= yv) $ update y i\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n\\begin{code}\n-- Constrain two variables to have different values.\ndifferent :: FDVar -> FDVar -> FDConstraint\ndifferent = addBinaryConstraint $ \\x y -> do\n xv <- lookup x\n yv <- lookup y\n guard $ not (isSingleton xv) || not (isSingleton yv) || xv /= yv\n when (isSingleton xv && xv `isSubsetOf` yv) $\n update y (yv `difference` xv)\n when (isSingleton yv && yv `isSubsetOf` xv) $\n update x (xv `difference` yv)\n\n-- Constrain a list of variables to all have different values.\nvarsAllDifferent :: [FDVar] -> FDConstraint\nvarsAllDifferent (x:xs) = do\n mapM_ (different x) xs\n varsAllDifferent xs\nvarsAllDifferent _ = return ()\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\frametitle{Labelling}\n\\emph{Labelling} is used to obtain valid solutions for a set of variables.\nThe embedded list monad allows us to search for and return all possible solutions.\n\\begin{code}\n-- Label variables using a depth-first left-to-right search.\nvarsLabelling :: [FDVar] -> FD [Int]\nvarsLabelling = mapM label where\n label var = do\n vals <- lookup var\n val <- lift $ elems vals\n var `hasValue` val\n return val\n\\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\nWe now have enough to solve Sudoku!\n\\begin{code}\nsudoku :: [Int] -> [[Int]]\nsudoku puzzle = runFD $ do\n vars <- newVars 81 (1, 9)\n zipWithM_ (\\x n -> when (n > 0) (x `hasValue` n)) vars puzzle\n mapM_ varsAllDifferent (rows vars)\n mapM_ varsAllDifferent (columns vars)\n mapM_ varsAllDifferent (boxes vars)\n varsLabelling vars\n\nrows, columns, boxes :: [a] -> [[a]]\nrows = chunk 9\ncolumns = transpose . rows\nboxes = concat . map (map concat . transpose) . chunk 3 . chunk 3 . chunk 3\n\nchunk :: Int -> [a] -> [[a]]\nchunk _ [] = []\nchunk n xs = ys : chunk n zs where\n (ys, zs) = splitAt n xs\n\\end{code}\n\\end{frame}\n\n\\subsection{Arithmetic expressions}\n\n\\begin{frame}[fragile]\n \\frametitle{Arithemtic expressions}\n \\begin{itemize}\n \\item So far we have seen how to declare contraint variables and define simple equality constraints\n between them.\n \\pause\n \\item We also want to be able to write constraints involving simple arithmetic expressions.\n \\end{itemize}\n\\end{frame}\n\\begin{frame}[fragile]\n \\begin{code}\ndata FDExpr\n = Int !Int\n | Var !FDVar\n | Plus !FDExpr !FDExpr\n | Minus !FDExpr !FDExpr\n | Times !FDExpr !FDExpr\n | Negate !FDExpr\n | Abs !FDExpr\n | Signum !FDExpr\n \\end{code}\n\\pause\n \\begin{code}\n\n-- Num instance allows us to use the usual arithmetic operators\n-- and integer literals\ninstance Num FDExpr where\n (+) = Plus\n (-) = Minus\n (*) = Times\n negate = Negate\n abs = Abs\n signum = Signum\n fromInteger = Int . fromInteger\n \\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\begin{code}\n-- Define new variables and return as expressions\nnew :: ToDomain a => a -> FD FDExpr\nnew d = newVar d <&> Var\n\nnews :: ToDomain a => Int -> a -> FD [FDExpr]\nnews n d = replicateM n $ new d\n \\end{code}\n \\pause\n \\begin{code}\n\n-- Interpret an FDExpr and return an FDVar representing it\ninterpret :: FDExpr -> FD FDVar\ninterpret (Var v) = return v\ninterpret (Int i) = newVar [i]\ninterpret (Plus e0 e1) = interpretBinary (+) e0 e1\ninterpret (Minus e0 e1) = interpretBinary (-) e0 e1\ninterpret (Times e0 e1) = interpretBinary (*) e0 e1\ninterpret (Negate e) = interpretUnary negate e\ninterpret (Abs e) = interpretUnary abs e\ninterpret (Signum e) = interpretUnary signum e\n \\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\begin{minted}[fontsize=\\footnotesize]{haskell}\ninterpretBinary :: (Int -> Int -> Int) -> FDExpr -> FDExpr -> FD FDVar\ninterpretBinary op e0 e1 = do\n v0 <- interpret e0\n v1 <- interpret e1\n d0 <- lookup v0\n d1 <- lookup v1\n v <- newVar [n0 `op` n1 | n0 <- elems d0, n1 <- elems d1]\n let pc = constrainBinary (\\n n0 n1 -> n == n0 `op` n1) v v0 v1\n nc0 = constrainBinary (\\n0 n n1 -> n == n0 `op` n1) v0 v v1\n nc1 = constrainBinary (\\n1 n n0 -> n == n0 `op` n1) v1 v v0\n addConstraint v0 $ pc >> nc1\n addConstraint v1 $ pc >> nc0\n addConstraint v $ nc0 >> nc1\n return v\n\nconstrainBinary :: (Int -> Int -> Int -> Bool) -> FDVar -> FDVar -> FDVar -> FDConstraint\nconstrainBinary pred v v0 v1 = do\n d <- lookup v\n d0 <- lookup v0\n d1 <- lookup v1\n let d' = toDomain [n | n <- elems d, n0 <- elems d0, n1 <- elems d1, pred n n0 n1]\n guard $ not $ Domain.null d'\n when (d' /= d) $ update v d'\n \\end{minted}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\begin{code}\ninfix 4 #\\=\n(#\\=) :: FDExpr -> FDExpr -> FDConstraint\na #\\= b = do\n v0 <- interpret a\n v1 <- interpret b\n v0 `different` v1\n\nallDifferent :: [FDExpr] -> FDConstraint\nallDifferent = varsAllDifferent <=< mapM interpret\n\nlabelling :: [FDExpr] -> FD [Int]\nlabelling = varsLabelling <=< mapM interpret\n \\end{code}\n\\end{frame}\n\n\\begin{frame}[fragile]\n \\frametitle{Example: $n$ queens in SWI-Prolog and Haskell}\n\n\\begin{columns}[t]\n\\column[t]{0.33\\paperwidth}\n\\begin{minted}[fontsize=\\footnotesize]{prolog}\nn_queens(N, Qs) :-\n length(Qs, N),\n Qs ins 1..N,\n safe_queens(Qs).\n\nsafe_queens([]).\nsafe_queens([Q|Qs]) :-\n safe_queen(Qs, Q, 1),\n safe_queens(Qs).\n\nsafe_queen([], _, _).\nsafe_queen([Q|Qs], Q0, D0) :-\n Q0 #\\= Q,\n abs(Q0 - Q) #\\= D0,\n D1 #= D0 + 1,\n safe_queen(Qs, Q0, D1).\n\\end{minted}\n\\pause\n\\column[t]{0.59\\paperwidth}\n\\begin{minted}[fontsize=\\footnotesize]{haskell}\nnQueens :: Int -> FD [FDExpr]\nnQueens n = do\n qs <- news n (1, n)\n safeQueens qs\n return qs\n\nsafeQueens :: [FDExpr] -> FDConstraint\nsafeQueens [] = return ()\nsafeQueens (q : qs) = do\n safeQueen qs q 1\n safeQueens qs\n\nsafeQueen :: [FDExpr] -> FDExpr -> FDExpr -> FDConstraint\nsafeQueen [] _ _ = return ()\nsafeQueen (q : qs) q0 d0 = do\n q0 #\\= q \n abs (q0 - q) #\\= d0\n safeQueen qs q0 (d0 + 1)\n\\end{minted}\n\\end{columns}\n\\end{frame}\n\\begin{frame}[fragile]\n \\begin{verbatim}\n SEND\n + MORE\n -------\n MONEY\n \\end{verbatim}\n\n \\begin{code}\nsendMoreMoney = runFD $ do\n vars@[s, e, n, d, m, o, r, y] <- news 8 (0, 9)\n s #\\= 0\n m #\\= 0\n allDifferent vars\n\n 1000 * s + 100 * e + 10 * n + d\n + 1000 * m + 100 * o + 10 * r + e\n #== 10000 * m + 1000 * o + 100 * n + 10 * e + y\n\n labelling vars\n \\end{code}\n\\end{frame}\n\n\\section{Conclusion}\n\\begin{frame}\n \\frametitle{Consclusion}\n \\begin{itemize}\n \\item Haskell can do constraint logic programming -- all you need is monads.\n \\pause\n \\item Advantages of Haskell\n \\begin{itemize}\n \\pause\n \\item Awesomeness of Haskell.\n \\pause\n \\item Type safety.\n \\pause\n \\item Leverage libraries, such as monad combinators, in a very natural way.\n \\pause\n \\end{itemize}\n \\item Disadvantages\n \\begin{itemize}\n \\pause\n \\item Not full Prolog, e.g. missing unification between terms, multi-moded predicates.\n \\pause\n \\item Some Prolog implementations have very powerful and efficient built-in solvers, which Haskell can't use.\n \\end{itemize}\n \\end{itemize}\n \\pause\n Github repository: \\url{https://github.com/dmoverton/finite-domain}\n\\end{frame}\n\\end{document}\n", "meta": {"hexsha": "a11e94e2634738803dbfb958a95fd148cbfd5a00", "size": 16900, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "mhug-talk/constraints.tex", "max_stars_repo_name": "dmoverton/finite-domain", "max_stars_repo_head_hexsha": "ae4afe755c1c1880d1283b9fdae90c1dc4a03f10", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 14, "max_stars_repo_stars_event_min_datetime": "2016-08-07T21:57:08.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-22T21:37:31.000Z", "max_issues_repo_path": "mhug-talk/constraints.tex", "max_issues_repo_name": "dmoverton/finite-domain", "max_issues_repo_head_hexsha": "ae4afe755c1c1880d1283b9fdae90c1dc4a03f10", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2016-12-07T01:46:45.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-02T07:21:00.000Z", "max_forks_repo_path": "mhug-talk/constraints.tex", "max_forks_repo_name": "dmoverton/finite-domain", "max_forks_repo_head_hexsha": "ae4afe755c1c1880d1283b9fdae90c1dc4a03f10", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2015-10-29T11:16:22.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-03T21:59:24.000Z", "avg_line_length": 27.9801324503, "max_line_length": 162, "alphanum_fraction": 0.6447337278, "num_tokens": 5160, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290806, "lm_q2_score": 0.6893056104028799, "lm_q1q2_score": 0.34734535046232556}} {"text": "\\documentclass[12pt]{article}\n\\input{physics1}\n\\begin{document}\n\n\\section*{NYU Physics I---Problem Set 9}\n\nDue Thursday 2018 November 08 at the beginning of lecture.\n\n\\paragraph{\\problemname~\\theproblem}\\refstepcounter{problem}%\nWhat is the pressure gradient (units of $\\Pa\\,\\m^{-1}$) in air at sea level at STP?\nWhat is the pressure gradient in water?\nWhat is the pressure gradient in liquid mercury?\nWhat would be the pressure gradient in rock, if we treated the rock like a fluid?\n\nCompute the ratio of the pressure gradient in mecury to the pressure gradient\nin water. How does this ratio compare to the ratio of $33\\,\\ft$ to $760\\,\\mm$?\nWhy did I ask that?\n\n\\paragraph{\\problemname~\\theproblem}\\refstepcounter{problem}%\nIn a bungy jump, the bungy cord has a rest (unstretched) length of\n$\\ell_0 = 5\\,\\m$, and has a spring constant $k$ such that it stretches\nby 1~m for every 400~N of force.\n\n\\textsl{(a)} If an adult of mass $M=80\\,\\kg$ jumps off of a very tall\nbridge at time $t=0$ with this bungy cord attached between her or\nhimself and also the bridge, to what maximum distance $h_\\mathrm{max}$\nbelow the bridge will he or she fall? (You might use energy\nconservation). Give your answer in terms of the symbols $M$, $\\ell_0$,\n$k$, and $g$ as well as numerically. That is, we want the numerical\nand symbolic answers both.\n\n\\textsl{(b)} What is the maximum tension $T$ in the bungy? Again, give\nboth answers.\n\n\\textsl{(c)} If you stiffen the cord (increase $k$), but keep\nthe natural length and adult's mass fixed, do you increase or decrease the maximum\ntension?\n\n\\textsl{(d)} What acceleration $a$ does the adult feel at the maximum\nextension of the bungy (that is, at the bottom)? Again, give your\nanswer both symbolically (in terms of the same symbols) and numerically.\n\n\\paragraph{\\problemname~\\theproblem:}\\refstepcounter{problem}%\nHere we consider the construction and tuning of a standard grand piano.\nEach string of the piano has a mass $M$, a length $L$, and a tension $T$.\n\n\\textsl{(a)} Use dimensional analysis to estimate the natural angular\nfrequency $\\omega$ of a piano string with these properties. That is,\nwhat combination has units of frequency?\n\n\\textsl{(b)} Look up the natural frequency of a guitar or piano string\nin its lowest harmonic. You might have to look up ``standing wave'' or\nsomething like that, and you might also have to look up ``transverse\nwave speed'' in a string. A string fixed at both ends (like a piano\nstring) is different from an open organ pipe! How does what you find\nin your research compare to what you found in part \\textsl{(a)}?\n\n\\textsl{(c)} Look inside a piano at or near middle C. Roughly what are\nthe diameters of the strings? And what are the lengths of the strings?\nUse these quantities and the density of steel to estimate the masses\nof the strings.\n\n\\textsl{(d)} Given what you know about the piano---the number of keys,\nthe number of strings per key (which isn't one for most keys), and the\nrange of frequencies and string lengths, estimate \\emph{very roughly}\nthe total force compressing a piano frame, in Newtons. That is,\nestimate the total of all the tension forces. Do you understand why\npiano frames and harps are so heavy?\n\n\\paragraph{Problem~\\theproblem:}\\refstepcounter{problem}%\nEverything submerged in the Earth's atmosphere is subject to a buoyant\nforce from the air. In the following, use a sensible (reasonably\naccurate) measure of the density of air at STP.\n\n\\textsl{(a)} When you measure your weight on a standard bathroom\nscale, you are measuring the \\emph{normal force} between yourself and\nthe floor. This normal force opposes the \\emph{combination} of\ngravity and buoyancy. What is the correction to your weight coming\nfrom buoyancy, roughly? Express it as a \\emph{fraction} of the\ngravitational force. Is this correction positive or negative---that\nis, does it increase or decrease the weight measured by the scale?\n\n\\textsl{(b)} Look up the ``volume'' of the Goodyear blimp model GZ-22.\nImagine that it is floating in an air atmosphere at STP, and that the\ngas inside the blimp is \\emph{also} at STP. What is the approximate\nbuoyant force on the blimp if it is filled with helium? What about if\nit were filled with hydrogen (molecular hydrogen)? Compare these\nnumbers with the gross weight and capacity of the blimp.\n\n\\textsl{(c)} Will the buoyant force increase, decrease, or stay the\nsame as you decrease the temperature?\nAssume that the blimp contents are always at the same pressure as the\nexterior air (and therefore the volume of the blimp must change).\n\n\\paragraph{Extra Problem (will not be graded for credit):}%\nA child of mass $m$ sits exactly on top of a hemispherical mound of\nfrictionless ice of radius $R$. If the child is displaced a tiny (ie,\nsmall relative to $R$) horizontal distance $x$ from the top of the\nmound of ice, what is the $x$-component $F_x$ of the net force on the\nchild? Write down the differential equation relating the $x(t)$ to\nits second derivative (with respect to time). Use the small-angle\napproximation to get rid of trigonometric functions! What functions\n$x(t)$ solve your equation? Try to be as general as possible.\n\\emph{Hint: Try exponentials!}.\n\n\\paragraph{Extra Problem (will not be graded for credit):}%\nLook up the Youngs modulus for steel, and estimate the spring\npotential energy stored in the middle C string, using what you figured\nout in the piano problem. Now estimate the total mechanical energy\nstored in the tuned piano!\n\n\\end{document}\n", "meta": {"hexsha": "e49981cca0e6dd6ae09d802d090730d26fad6986", "size": 5496, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/physics1_ps09.tex", "max_stars_repo_name": "davidwhogg/Physics1", "max_stars_repo_head_hexsha": "6723ce2a5088f17b13d3cd6b64c24f67b70e3bda", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-11-13T03:48:56.000Z", "max_stars_repo_stars_event_max_datetime": "2017-11-13T03:48:56.000Z", "max_issues_repo_path": "tex/physics1_ps09.tex", "max_issues_repo_name": "davidwhogg/Physics1", "max_issues_repo_head_hexsha": "6723ce2a5088f17b13d3cd6b64c24f67b70e3bda", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 29, "max_issues_repo_issues_event_min_datetime": "2016-10-07T19:48:57.000Z", "max_issues_repo_issues_event_max_datetime": "2019-01-29T22:47:25.000Z", "max_forks_repo_path": "tex/physics1_ps09.tex", "max_forks_repo_name": "davidwhogg/Physics1", "max_forks_repo_head_hexsha": "6723ce2a5088f17b13d3cd6b64c24f67b70e3bda", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.6371681416, "max_line_length": 83, "alphanum_fraction": 0.7641921397, "num_tokens": 1422, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6893056104028797, "lm_q1q2_score": 0.34734535046232545}} {"text": "\\documentclass[10pt,twocolumn]{article}\n\\usepackage{cite}\n\\usepackage{amsmath,amssymb,amsfonts,amsthm}\n\\usepackage{multicol}\n\\setlength{\\columnsep}{0.75cm}\n\\usepackage{caption}\n\\usepackage{graphicx}\n\\usepackage{csquotes}\n\\usepackage{todonotes}\n\n\\def\\BibTeX{{\\rm B\\kern-.05em{\\sc i\\kern-.025em b}\\kern-.08em\n T\\kern-.1667em\\lower.7ex\\hbox{E}\\kern-.125emX}}\n\\usepackage{hyperref}\n\\usepackage[margin=1.75cm]{geometry}\n\\usepackage[hpos=0.72\\paperwidth,vpos=0.97\\paperheight,angle=0,scale=0.8]{draftwatermark}\n\\SetWatermarkText{Preprint}\n\\SetWatermarkLightness{0.5}\n\\SetWatermarkText{Preprint of preliminary work}\n\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{corollary}{Corollary}[theorem]\n\\newtheorem{lemma}[theorem]{Lemma}\n\n\n\\title{Optimally Reliable \\& Cheap Payment Flows on the Lightning Network}\n\n\n\\author{Rene Pickhardt \\& Stefan Richter\\thanks{The authors have contributed to this work in equal measure. The order merely reflects the fact that this line of research was initiated by Rene Pickhardt.}}\n\\begin{document} \n\\maketitle\n\n\n\n\n\n\n \n\\begin{abstract}\n Today, payment paths in Bitcoin's Lightning Network are found by searching for shortest paths on the fee graph.\n We enhance this approach in two\n dimensions. Firstly, we take into account the probability of a payment\n actually being possible due to the unknown balance distributions in the\n channels. Secondly, we use minimum cost flows as a proper generalization of\n shortest paths to multi-part payments (MPP).\n\n In particular we show that under plausible assumptions about the balance\n distributions we can find the most likely MPP for any given set of senders, recipients and amounts by solving for a (generalized) integer\n minimum cost flow with a separable and convex cost function. Polynomial time\n exact algorithms as well as approximations are known for this optimization\n problem.\n \n\n We present a round-based algorithm of min-cost flow computations for\n delivering large payment amounts over the Lightning Network. This algorithm\n works by updating the probability distributions with the information gained\n from both successful and unsuccessful paths on prior rounds. In all our\n experiments a single digit number of rounds sufficed to deliver payments \n of sizes that were close to the total local balance of the sender.\n Early experiments indicate that our approach increases the size of payments that can be reliably delivered by several orders of magnitude compared to the current state of the art.\n\n We observe that finding the cheapest multi-part payments is an NP-hard problem\n considering the current fee structure and propose dropping the base fee to make\n it a linear min-cost flow problem. Finally, we discuss possibilities for\n maximizing the probability while at the same time minimizing the fees of a\n flow. While this turns out to be a hard problem in general as well --- even in\n the single path case --- it appears to be surprisingly tractable in practice.\n \n\\end{abstract}\n\n%==========================================================================\n\\section{Introduction}\nThe Lightning Network is a payment channel network using source-based onion routing to deliver payments from senders to recipients.\nA necessary condition for a single onion package to be delivered successfully is that the onion follows a path with sufficient liquidity.\nIn this context sufficient liquidity does not just mean that the publicly known channel capacities of the channels on the path between sender and recipient are larger than the payment amount. Rather, every node along the path has to own enough of the channel capacity as their local balance to be able to forward the amount to the next hop.\nAs broadcasting the balance values would hinder the scalability of the Lightning Network they are generally kept private and thus unknown to other nodes.\nCurrently the sender node mitigates this uncertainty by entering a trial-and-error loop for delivering payments.\nHowever, past experiments have demonstrated that payments are often failing, in particular when the amounts to be delivered are increasing\\cite{DBLP:journals/corr/abs-1911-09432,DBLP:journals/corr/abs-2006-14358,lange2021impact, pickhardt2021security}.\n\nCurrent implementations largely find candidate paths for the trial-and-error-loop by solving\nshortest path problems or generalizations like $k$-shortest paths on the\nweighted channel graph, where the weights correspond to the routing fees charged by nodes forwarding a payment along a channel.\\footnote{We also observe a combination of the fee function with features like a penalty for longer CLTV values, prior experiences of using the channel and a bias against smaller channels.}\nThat approach tries to find the cheapest payment path for the sender but does not systematically factor in success probabilities. As a consequence, the payment loop might try a large number of cheapest but unreliable paths before timing out instead of using slightly more expensive but vastly more reliable paths. It also does not produce an optimal split of the payment into multiple paths.\n\nSoftware implementations of the Lightning Network protocol have mainly focused on three strategies for handling the uncertainty of sending a payment.\n\\begin{enumerate}\n\\item Incentivizing the path finding algorithm to favor larger channels.\\footnote{\\url{https://lists.ozlabs.org/pipermail/c-lightning/2021-May/000203.html}}%clightning dev\n\\item Ad-hoc splitting of large payment amounts into smaller ones after failed attempts using a technique called multi-part payments (MPP).\n\\item Using provenance scores of nodes and channels and other data collected during operation of a node to estimate which nodes and channels might be reliable.\n\\end{enumerate}\n\nIn this work we are developing a general technique that is achieving the effects of these rather ad-hoc techniques in a systematic fashion.\nThis text largely builds upon and extends prior research which pointed out that\nthe uncertainty of channel balance values can be mathematically modeled to\narrive at a probabilistic path finding scheme\\cite{pickhardt2021security}. While\nthis earlier work demonstrated a significant reduction in failed attempts while\ndelivering payments it still mostly kept the perspective of a single path finding\nproblem.\n\nIt has long been a folklore view that delivering payments on the Lightning Network can be modeled as a network flow problem.\nIn what follows we show that the discovery of the most likely multi-path payment is equivalent to solving a (generalized) min-cost flow problem in which the negative logarithms of channel success probabilities are considered as the cost of using a channel.\nThe channel success probabilities are priors that have to be estimated, through sampling for example.\nUnder the assumption of an independent uniform balance distribution on each channel as the prior, finding the most probable multi-part payment for a given amount from a sender to a recipient can be usefully modeled as solving an integer min-cost flow problem with a separable convex cost function.\nWhile in general min-cost flow problems are complex\noptimization problems, the above mentioned subclass is known to have polynomial\ntime solutions with a runtime complexity of $O(m \\cdot \\log(U) \\cdot S(n,m))$\nwhere $n$ is the number of nodes, $m$ is the number of edges on the network, $U$\nis the amount to be delivered, and $S(n,m)$ is the time it takes to obtain a\nsolution to the single source shortest path problem\\cite{Minoux1986,ahuja1993network}. This\nis typically done using Dijkstra's Algorithm in time $O(m+n) \\cdot \\log(n)$, so\nthat we arrive at a total runtime of $O(\\log(U)\\cdot(m^2+mn)\\cdot\\log(n))$.\n\nIf the balance values were known, the decision if a payment between two nodes can be conducted could be arrived at by finding a max-flow / min-cut and comparing it to the amount that is to be delivered.\nGiven the uncertainty of channel balances, the decision problem is much harder to solve and still involves a trial-and-error loop.\nWe introduce a practical round-based algorithm that can either quickly deliver the payment or decide with a high probability that the min-cut is lower than the payment amount that is supposed to be delivered between sender and receiver.\nIt starts by creating the most likely MPP split as the solution of the min-cost flow problem and sending out the partial payments.\nThe algorithm reduces its uncertainty of the balance values by learning from the failures and successes.\nThis is done by updating the prior probabilities after the failing onions have returned.\nFinally it creates another candidate MPP for the residual amount by solving a min-cost flow on the updated graph.\n\n\\section{Payments as Integer Flows}\n\\label{flows}\nLet $G = (V,E)$ be a directed graph and $u: E \\longrightarrow \\mathbb{N}$ a function assigning capacity values to all edges in the graph.\nFor every node $v\\in V$ let $b_v \\in \\mathbb{Z}$ denote its \\emph{excess, supply or demand}. Typically $b_v$ will be $0$ except for the source node $s$ (with supply $b_v>0$) and the destination $d$ (with demand $b_v<0$). We call a function $f: E \\longrightarrow \\mathbb{N}_{0}$ a \\emph{flow} if the following conditions hold:\n\\begin{enumerate}\n\\item \\textbf{capacity constraint}: For every edge $e \\in E$ we have: $$0 \\leq f(e) =: f_{e} \\leq u_{e} := u(e).$$\n\\item \\textbf{flow conservation}: For every node $i\\in V$ we have: $$\\sum_{(i,j)\\in E} f_{ij} - \\sum_{(j,i)\\in E} f_{ji} = b_i.$$\n\\end{enumerate}\n\n\\subsection{Flows on a Known Balance Graph}\nAssuming the balance values of every payment channel of the Lightning Network were publicly known, one could create a flow network called \\emph{balance graph} as follows:\nFor any given payment channel between two nodes $i$ and $j$ with capacity $u$ we know that the balance $u_j$ of node $j$ plus the balance $u_i$ of node $i$ must be equal to the capacity $u$.\nOn the balance graph we add two directed edges for the payment channel between the nodes $i,j$.\nFirst we add the edge $(i,j)$ with a capacity of $u(i,j) = u_i$ and then we add another edge in the opposing direction $j,i$ with a capacity of $u(j,i)=u_j$.\\footnote{In reality, channel reserves would need to be accounted for by lowering the capacities to the spendable balances.}\n\nObserve that in this balance graph, a set of payment paths from $s$ to $t$ determines a flow simply by summing up the amount sent through any edge.\\footnote{We abstract from the fact that the amount sent through a payment path diminishes along the path by the fees collected; we assume the total fees are included until the end. See Section~\\ref{generalized} for a discussion of the impact of this relaxation.}\nIn fact, the converse is also true: It is easy to see that any flow can be decomposed into paths from $s$ to $t$ and cycles in linear time~\\cite[p.~79ff]{ahuja1993network}.\nSince cycles do not change supply or demand at any node, we can safely ignore them.\nIn this way, we can represent any MPP split as a flow and vice versa.\nWe emphasize that the paths of the decomposed flow that lead to the MPP split do not need to be disjoint.\n\nGiven the balance values, we could decide the maximum possible amount that can be sent for any given node pair $s,t \\in V$ by computing the minimal $s$--$t$-cut.\nUsing, say, the Ford-Fulkerson algorithm\\cite{ford_fulkerson_1956} one could compute a max-flow and disect it into a series of paths.\nThese paths could then be used to construct several onions for a multi-part payment.\n\nIn a preliminary test we use two different prior distributions to generate two static balance graphs and check on each how often the min-cut between arbitrarily chosen pairs of sender and receiver is actually determined by the minimum of the local outbound capacity of the sender and the local inbound capacity of the recipient.\nIn Figure~\\ref{fig:mincut} we can see that for both of these distributions --- which have been observed on the entire Lightning Network and an active core subset, respectively --- in only about $5\\%$ of the payment pairs is the max-flow smaller than the maximum amount locally sendable by the payer and the maximum amount locally receivable by the target node.\n\\begin{figure}[htpb]\n \\center\n \\includegraphics[width=0.45\\textwidth]{img/maxFlowFraction.png}\n \\caption{Showing the percentage of payment pairs where the maximal payable amount is actually lower than the upper bound given by the local balance known to both sender and recipient.}\n \\label{fig:mincut}\n\\end{figure}\n\nThe fact that with publicly known balance values in $19$ out of $20$ payment pairs the amount that can be delivered is as high as the local limits of the payment pair is in stark contradiction to the currently observed and reported \\cite{DBLP:journals/corr/abs-1911-09432,DBLP:journals/corr/abs-2006-14358,lange2021impact, pickhardt2021security} success rates. In fact, these are declining heavily with amounts larger than $100,000$ satoshi (1 BTC = 100,000,000 sat) for which the delivery should be almost always possible.\nWe conjecture that this is due to the fact that in reality, the balance values are not publicly known.\nThis forces us to take the total channel capacities $u$ as capacities on our flow network in both directions.\nNote that finding a max-flow on this network is not sufficient for deciding if a payment can be made.\nHowever, we can generalize the flow model to the case of uncertain balance values and will return to this question in Section~\\ref{rounds}.\n\n\\subsection{Uncertainty Networks}\nEarlier research~\\cite{pickhardt2021security} has introduced a mathematical framework for handling uncertain balance values in payment channels with the goal of making path finding decisions that maximize the success probability of payment paths.\nWe recall that --- given a prior belief about the balance uncertainty via a probability distribution $P_e$ and a random variable $X_e$ --- the channel success probability for a payment of size $f_e$ for a channel $e$ is expressed as $P_e(X_e\\geq f_e)$.\nWhereas in~\\cite{pickhardt2021security} the goal was maximizing the path success probabilities, here we aim to maximize the success probability for the entire flow. Assuming the channel balances to be independently distributed, this combined success probability is simply the product of all channel success probabilities:\n\n\\[\nP(f)=\\prod_{e\\in E}P_e\\left(X_{e} \\geq f_e\\right)\n\\]\n\nAny flow $f$ that maximizes the success probability $P(f)$ is also minimizing $-\\log\\left(P(f)\\right)$ and vice versa.\n%The following sentence is a mathematical beauty and it would be really sad to have it removed (: \nUsing the fact that the logarithm is a group homomorphism from the multiplicative group of positive real numbers to the additive group of real numbers we can write:\n\\[\n-\\log\\left(\\prod_{e\\in E}P_e(X_{e} \\geq f_e)\\right) = \\sum_{e\\in E}-\\log\\left(P_e(X_e \\geq f_e)\\right)\n\\]\nThe right hand side of the equation has the form of a separable cost function $C$ for the flow $f$ from the theory of min-cost flows:\n\\[\nC(f) := \\sum_{e\\in E}-\\log\\left(P_e(X_e \\geq f_e)\\right)\n\\]\nGiven fixed probability distributions for the channel balances, finding the most likely flow is therefore equivalent to finding a flow of minimum cost on this \\emph{uncertainty network}. Any such flow can then be disected into the most likely multi-path payment as in the previous section.\n\nIn general, finding optimal solutions to the min-cost flow problem with non-linear costs is NP-hard~\\cite{guisewite1990minimum}. Fortunately, in the special case of integer flows and capacities together with a separable convex cost function a polynomial algorithm has been introduced by~\\cite{Minoux1986}.\nSince our flows are integer-valued and the cost function is separable we need to understand when a cost function arising from channel success probabilities is convex in order to be able to apply such an algorithm.\nBecause the cost function $C$ is separable we can test convexity independently for any given channel $e$ and the resulting cost function $c_e(f_e):=-\\log\\left(P_e(X_e\\geq f_e)\\right)$\nAfter simplifying by assuming a flow value $x:=f(e)$ and defining $p(x):=P_e(X_x\\geq x)$ we get:\n\\[\nc_e(x)=-\\log(p(x))\n\\]\nAssuming this function is twice differentiable, it is convex iff its second derivative is be nonnegative on the entire domain.\nThe first derivative is:\n\\[\nc_e'(x) = -\\frac{p(x)'}{p(x)}\n\\]\nand the second derivative is:\n\\[\nc_e''(x) = \\frac{(p'(x))^2-p(x)p''(x)}{{p^2(x)}} \\geq 0\n\\]\nIn particular we see that the negative log probabilities result in a convex cost function iff the following inequality holds:\n\\[\n(p'(x))^2 \\geq p(x)p''(x)\n\\]\nIn the uniform case $p(x)=\\frac{u+1-x}{u+1}$ (cf.~\\cite{pickhardt2021security}) we have $p'(x)=\\frac{-1}{u+1}$ and $p''(x)=0$ demonstrating that the resulting cost function $c_e(x)$ is convex on its entire domain.\nThis indicates that the polynomial algorithm can be used to find the flow that minimizes:\n\\[\nC(f)=\\sum_{e\\in E}-\\log\\left(\\frac{u_{e}+1-f_e}{u_{e}+1}\\right)\n\\]\nThus the most probable multi-part split for delivering the amount $U$ can be found by solving a min-cost flow problem and disecting the flow into paths.\n\nWe do not explicitly handle fees in this model, but observe that if we can find a flow that includes an upper bound to the total fees, the real payment success probability will be at least as high as the one predicted by this model, since the transported amount is only falling along the paths and our probability function is monotonic.\n\n\\subsection{Maximizing Success Probabilities vs Minimizing Fees}\n\\label{fees}\nThe current routing fee function on the Lightning network is a separable cost function depending only on the flow across each channel.\nHowever, it is easy to see that the function is not convex at the transition between flow $0$ (cost $0$) and flow $1$ (cost base fee plus unit flow cost), whenever the base fee is larger than the proportional unit flow cost. In fact, a cost function of this form is often called a \\emph{concave fixed-charge cost} in the literature. Unfortunately, finding the flow that minimizes a cost function of this form is a strongly NP-hard problem as shown in~\\cite{guisewite1990minimum} by reduction from 3-SAT to a min-cost flow problem with only fixed-charge costs.\n\nOn the other hand, if the Lightning Network community were to drop the base fee, the separable cost function would become linear in the flow value of each arc.\nFinding an MPP split that minimizes routing fees could easily be done by solving the linear min-cost flow problem using any of a number of algorithms~\\cite{ahuja1993network}.\nHowever, we note that minimizing the routing fees alone tends to\nsaturate the full capacity of cheap channels. Such paths are highly improbable to succeed since they can only do so when the entire balance is on the right side of the channel (even ignoring channel reserves). In our opinion that makes optimizing purely for fees a poor choice in general. On the other hand, only maximizing the success probability might allow routing node operators and liquidity providers to game the algorithm and extract unlimited fees.\n\nSo in practice, it should be our goal to both minimize fees and maximize success probabilities.\nNaturally, these goals can be contradictory, since node operators\ncan and will choose fees freely. Two ways of expressing this goal might be\n\\begin{enumerate}\n\\item to minimize fees with a side constraint of achieving a certain minimal probability bound, or\n\\item to maximize success probability with a side constraint of staying below a certain maximal fee bound.\n\\end{enumerate}\n\nUnfortunately, both of these problems are weakly NP-hard via the following argument:\nFirst, observe that adding unit capacities to the problems and looking for a flow of size 1 makes them instances of the so-called constrained shortest-path problem.\nThen,~\\cite[p.~798]{ahuja1993network} shows that this subclass of constrained min-cost flow problems is already NP-hard by reduction from the Knapsack problem.\n\nFortunately, the picture is not quite as bleak as it looks on first sight.\nFirst, the reduction only implies weak NP-hardness, meaning that we could find a polynomial algorithm whenever $U=O(n^k)$ for some $k$.\nStrictly speaking, the flow size $U$ is always bounded by a constant in our applications, since the total number of bitcoin is limited. Looking into the theory of Lagrangian relaxation~\\cite[p.~598ff]{ahuja1993network} methods, however, gives us immediate practical results instead of just hope.\n\nIn fact, the two cases enumerated above collapse into one when we try to find bounds for them by applying a simple one-dimensional Lagrange multiplier, that is we try to minimize:\n\\[\n\\sum_{e\\in E}-\\log(\\frac{c_{e}+1-f_e}{c_{e}+1})+\\mu\\cdot f_e\\cdot fee(e)\n\\]\nthe linear combination of both costs, with a suitable multiplicative constant\n$\\mu$.\nBy calculating this combined min-cost flow (note that the linear combination of\nthe two cost functions remains convex), not only do we get a feasible flow of size $U$, but because of the \\emph{Lagrangian bounding principle}~\\cite[p. 605f]{ahuja1993network}, whatever total fee $x$ and success probability $p$ we achieve, we are guaranteed that this combination is optimal on both sides.\nThat is, there is no solution with cheaper total fees than $x$ and at least probability $p$, and there is no solution with higher success probability than $p$ that still only costs at most $x$.\nSo in case we are not satisfied with the parameters of the solution we got, finding an adequate solution is reduced to either increasing $\\mu$ and getting a potentially cheaper, but less probable solution, or decreasing $\\mu$ and receiving a more reliable but also more expensive solution.\n\n\\subsection{Generalized Flows with Losses}\n\\label{generalized}\nSo far we have ignored the fact that every node on every path takes some part of the payment as a fee, which means that the total amount of flow gets smaller towards the target.\nThis observation is best described by a slightly more general model called \\emph{generalized (minimum cost) flows with gains and losses}. In this formulation, the flow conservation condition for any node $i\\in V$ is changed to $$\\sum_{(i,j)\\in E} f_{ij} - \\sum_{(j,i)\\in E} \\gamma_{ji}f_{ji} = b_i.$$\nThus, when we send 1 unit of flow along an edge $(i,j)$, $\\gamma_{ij}$ units of flow arrive at node $j$. The edge multipliers $\\gamma_e$ are positive rational numbers\\footnote{In the Lightning Network, these correspond to the proportional part of the fee. Again, including the base fee makes solving the problem infeasible, which is why we propose abolishing it.} and the edges are called \\emph{gainy} ($\\gamma_e > 1$) or \\emph{lossy} ($\\gamma_e < 1$) accordingly.\nNotice that until now, this formulation still depends on us knowing the exact supply/demand amounts at the source and destination nodes. This is especially troubling here, because we cannot just use an upper bound to the supply as before: there might not be a solution that uses the corresponding exact amount of fees. Therefore we introduce a cost-free high-capacity gainy (say, $\\gamma_s=2$) self-loop at the source node and set the source excess to zero. This allows for introduction of an arbitrary amount of flow at the source. Then, we aim to minimize the convex cost function as before under the remaining flow and capacity constraints.\n\nThe generalized flow problem is clearly a proper generalization of the min-cost flow problem outlined above. Unfortunately, it also appears to be harder to solve. The algorithm that we implemented for the min-cost flow problem does not seem to carry over to this more general problem. We did find pointers to some approaches~\\cite{tsengbertsekas} that might be worth exploring. So far, we have been reluctant to invest too much effort in this direction, because in our application, the fees are generally expected to be a small fraction of the total flow. Thus it is doubtful if the greater computational effort will be worth the slightly more favorable probability/fee result.\n\n\n\\section{Payment Algorithm}\n\\label{rounds}\nOnce we are able to efficiently compute minimum cost flows optimizing either for success probabilities, fees or both using the Lagrange relaxation we naturally arrive at a round-based payment algorithm that can be used by a node $s$ that wishes to send an amount $U$ to a destination $d$.\nFor now, we assume a Lagrange-style combination between channel success probabilities and the linear fee rate function, as we believe this achieves the most useful results.\n\nThe round-based algorithm is initialized by the sending node $s$ in the following way:\nIt starts by creating a new uncertainty network model $N$ of the Lightning Network which encodes the initial uncertainty and information $s$ is gaining about the balance distribution on the network during the rounds.\n%TODO: put back in? Thus the network will be updated during the process of delivering the payment during the following rounds.\nIn order to deliver the full amount, the node $s$ will have to solve a minimum cost flow problem, send out onions and update the uncertainty network based on the successes and failures of the onions in each of the rounds.\nThe uncertainty network $N$ consists of all the nodes that are on the same connected component as $s$ on the channel graph.\nThe edges of $N$ correspond to payment channels on the Lightning Network.\nIf $s$ has no further knowledge about the channels, a directed arc for both directions of each payment channel is added to the uncertainty network.\nThe capacities of the edges are set to the capacities of the payment channel (possibly deducting channel reserves).\n\nNotice for example that in the local channels of $s$ the balance values are known and there is no uncertainty.\nThus the capacity for those channels is set to the local balance value $u$ as this is the most that can currently be sent on those channels.\nThe probability is set to $1 = P(X\\geq a | X=u)$ for any amount $a$ between $0$ and $u$.\nThis results in a negative log probability of $0$ and thus makes it very cheap for the minimum cost flow computation to utilize the liquidity in this channel. In particular since the node $s$ also does not have to pay any fees to itself.\n\nSimilarly, the receiving destination node $d$ could tell the sending node about the state of its local channels and this knowledge could also be incorporated into the graph by creating edges with 0 log probabilities and decreased capacities.\\footnote{Communicating this information in invoices is currently not part of the protocol but routing hints in the invoices might be extended to encode such information.} In Figure~\\ref{fig:mincut} we have demonstrated that for about $95\\%$ of all payment pairs the amount that can be delivered through the network is actually limited by the local outbound capacity of the sender and the local inbound capacity of the receiving node which yields another motivation for this information to be shared.\n\nAfter the setup phase the round-based phase starts. Here the algorithm iterates over the following steps until either the entire amount is delivered or the minimum cost flows become too unlikely or cannot be computed for the residual amount, which means the minimum $s$--$d$-cut has been discovered.\\footnote{In such cases it seems reasonable that $s$ open a new payment channel with $d$ for at least the remainder amount.}\nThe round starts with $s$ computing a minimum cost flow for the amount $U$ to $d$ following the optimization goal.\nThe flow is then decomposed into paths and cycles. Note that cycles cannot appear in our application as long as we do not allow negative cost edges. If they appeared, they would indicate profitable rebalancing options that may or may not be reachable for $s$. Since negative fees are not allowed in the BOLT standards, we can safely ignore cycles for now.\n\nThe node then conducts a multi-part payment by concurrently sending out onions along the resulting paths.\nIn practice one has to chose a decomposition of the flow into paths that does not create more HTLCs on a remote channel than the HTLC limit encoded into the protocol permits.\nThis engineering challenge as well as others like the question of channel reserves are ignored here for simplicity of the presentation.\nDespite the fact that the most likely flow was used, some of the onions will not reach the target in most cases. So the sending node gathers information from the error codes of failed attempts as previously described in~\\cite{tikhomirov2020probing} as well as information from the paths that have not returned an error to update the probabilities as described in~\\cite{pickhardt2021security}.\nThis step decreases the uncertainty of the channel balances and is crucial for the improvement and different results in the next round which is why we explicitly explain how the knowledge is updated in several cases.\n\\begin{enumerate}\n\\item If an onion with the amount $h$ has not been returned, we assume it has arrived at the destination. Thus all channels across the path have now locked some liquidity into an HTLC.\\@ In our uncertainty network we thus reduce the capacity $u$ of each involved channel by the amount $h$ that was sent along that channel on that path. This changes our future success probabilities for the amount $a$ to be $P(X\\geq a+h|X\\geq h)$ which corresponds to a change from $\\frac{u+1-a}{u+1}$ to $\\frac{(u-h)+1-a}{(u-h)+1}$ in the uniform case. For any value of $a$ and positive $h$ the second fraction is smaller that the first one. This leads to lower probabilities which in turn yields higher costs to use those channels again in follow up rounds.\n\\item If an onion of size $h$ fails we learn the following new information:\n \\begin{enumerate}\n \\item On every channel up to the failed channel there has been sufficient liquidity to forward the amount $h$. In future rounds we can use the conditional probability $P(X\\geq a|X\\geq h)$. For $a<=h$ this term will be $1$, resulting in log costs of $0$. For $a>h$ the conditional probability is uniform again. It is important to notice that the conditional probability will again lead to a convex cost function.\n \\item Assuming the failure code is related to liquidity issues for the failed channel we know that the amount $h$ was not available. Thus we are able to reduce the capacity to $h-1$ and change the probability for the future rounds from $\\frac{u+1-a}{u+1}$ to $\\frac{h-a}{h}$ in the uniform case, or $P(X\\geq a|X < h)$ in general. This probability decrease will result in higher costs for utilizing this channel.\n \\item If on the other hand the failure code is related to an issue with the next hop, for example a downtime, the node can update its local view by temporarily removing the failed node with all its channels.\n \\end{enumerate}\n\\end{enumerate}\n\nNote that both successful and failing onions would also allow us to update the knowledge about the balances in the reverse direction.\nOnce all the knowledge from the partial attempts of the round has been learned --- which is reflected by the update of the probabilities and cost functions --- the algorithm enters the next round.\nFirst, the sum of all failed onions is computed as the residual amount that still needs to be delivered.\nWe enter the next round with that amount on the updated uncertainty network by computing a new minimum cost flow to generate the next optimal MPP split.\n\n\n\\section{Example}\nLet us go through an example that illustrates why finding a min-cost flow is necessary instead of just adding optimal paths.\nIt also serves to better understand the round-based algorithm.\nIn order to simplify the example, we ignore fees and channel reserves and optimize purely for probability here.\nWe also hide the nonintuitive negative log probabilities by writing down the corresponding max-probable flow with probabilities as edge weights.\nFor the channel graph depicted in Figure~\\ref{fig:counterexample},\n\\begin{figure}[htpb]\n \\center\n \\includegraphics[width=0.45\\textwidth]{img/counterexample.png}\n \\caption{Example Channel Graph on which the $2$-flow with maximal probability is not an extension of the $1$-flow from $s$ to $d$ with maximal probability. The success probabilities for sending $i$ sat are depicted as $p_i$. The channels capacities are the bold numbers along the edges.}\n \\label{fig:counterexample}\n\\end{figure}\nassuming uniform probability distributions we compute the following probabilities for the $1$-flows (paths delivering 1 sat):\n\\[\n\\begin{aligned}\np([s,A,B,d]) = & \\frac{2}{3}\\cdot\\frac{2}{3}\\cdot\\frac{4}{5}=\\frac{16}{45} &= 0.35\\overline{5} \\\\\np([s,X,Y,d]) = & \\frac{1}{2}\\cdot\\frac{7}{8}\\cdot\\frac{4}{5}=\\frac{28}{80} &= 0.35 \\\\\np([s,X,B,d]) = & \\frac{1}{2}\\cdot\\frac{9}{10}\\cdot\\frac{4}{5}=\\frac{36}{100} &= 0.36 \\\\\n\\end{aligned}\n\\]\n\nThis indicates that $s,X,B,d$ is the minimum cost $1$-flow.\nThe $(s,X)$ arc is obviously saturated so that a $2$-flow extending the $1$-flow would have to go via the $(s,A)$ arc.\nOne can easily compute the probability of the resulting $2$-flow $f^2$ if the next sat is also using the $(B,d)$ channel and being merged with the min-cost $1$-flow as\n\\[\np(f^2) = \\frac{1}{2}\\cdot \\frac{9}{10}\\cdot \\frac{2}{3}\\cdot \\frac{2}{3}\\cdot \\frac{3}{5} = \\frac{3}{25}=0.12\n\\]\n\nHowever if we look at the $2$-flow $g^2$ that sends $1$ sat along $s,A,B,d$ and $1$ along $s,X,Y,d$ we get $p(g^2)=\\frac{16}{45}\\cdot\\frac{28}{80} = 0.124\\overline{4}$\nwhich is also the min-cost $2$-flow in this graph.\nThis example shows that finding a min-cost flow cannot in general be done by computing the most likely path for a single sat and extending it with the next most likely $1$-sat-paths.\\footnote{This simple $+1$-algorithm could actually be rescued so that it would be able to compute the min-cost flow. However both versions would also be quite slow as they would be linear in the amount that was to be sent --- which is exponential in the input size}\n\nExtending our example and assuming we want to send a total of 3 sat we start again by computing the min-cost flow $f^3$ which can be disected into two paths $l_1= s,X,Y,d$ with an amount of $1$ and another path $l_2=s,A,B,d$ with an amount of $2$.\nAfter sending out the onions we might have learned that the onion along $l_1$ has been successful, but the one with $2$ sat along $l_2$ has failed because $B$ did not have enough liquidity to forward the onion to $d$ on the $(B,d)$ channel.\nFor the second phase of the algorithm we now compute the min-cost flow on a graph where we know that we can deliver 2 sat with perfect certainty to $B$.\nThis updated uncertainty network is depicted in Figure~\\ref{fig:round2}\n\\begin{figure}[htpb]\n \\center\n \\includegraphics[width=0.45\\textwidth]{img/round2.png}\n\\caption{The uncertainty network of $s$ after HTLCs of value $1$ are locked in along the path $s,X,Y,d$ and a 2-sat onion along $s,A,B,d$ failed because of missing liquidity on the $B,d$ channel. Saturated edges are removed. The black labels on the edges depict the uncertainty range of the balance or a single number if the balance is known to $s$. The $p_i$ express the success probability for sending a further $i$ sat along an edge given the updated knowledge. }\n\n \\label{fig:round2}\n\\end{figure}\n\nThe flow that sends the full residual amount of $2$ sat along $s,A,B,X,Y,d$ has a probability of $\\frac{8}{10}\\cdot\\frac{5}{7}\\cdot\\frac{1}{2}=0.286...$ while the flow that sends $1$ sat along the $B,d$ channel has a probability of $\\frac{1}{2}\\cdot\\frac{9}{10}\\cdot\\frac{6}{7}\\cdot\\frac{3}{4}=0.289...$ telling us that for the residual amount we should make another split payment sending one sat along each of the paths $q_1=s,A,B,d$ and $q_2=s,A,B,X,Y,d$.\n\nFinally, if the path $q_1$ locks in and $q_2$ returns an error at any of the channels $(B,X),(X,Y)$ or $(Y,d)$ we would know that we cannot deliver the full payment as the min-cut in the network on the balance graph had the value $2$.\nIf however $q_2$ and $q_1$ both lock in we have successfully delivered the payment.\nIf both $q_1$ and $q_2$ return an error we know the min-cut between $s$ and $d$ on the balance graph was $1$ (as that had been locked-in in the first round and no further HTLCs have locked in).\nFinally if $q_1$ returns an error and $q_2$ locks in we will have to enter the third round.\nIn the third round there is only 1 sat to be delivered on a single possible path $q_2$ which, given our knowledge, has a success probability of $\\frac{8}{9}\\cdot\\frac{5}{6}\\cdot\\frac{2}{3}=0.494\\ldots$\\footnote{the numerical similarity to the twitter account mentioned in the acknowledgements is completely coincidental as we used that graph even before we had the discussion with the individual.}.\n\n\n\\section{Anecdotal Lab Report}\n\\textbf{Disclaimer:} We stress that this document is a preprint. In particular this section cannot be considered a proper evaluation. For that we would also have to test the algorithms on the Lightning Network mainnet.\nIn addition to the fundamental complications arising through the base fee\\footnote{which should be easily avoidable at the moment by incorporating some buffer in the fee size, as long as it remains lightly used}, such a real-world test requires overcoming several engineering challenges that we did not have the time to address yet:\n\nA practical implementation needs to automatically answer questions that might arise in case of hanging HTLCs, if channels become inactive, or amounts reach HTLC limits of channels, for example.\nAlso the implementation of the min-cost flow solver would have to be engineered to have a much faster runtime than our experimental version so that it would actually be feasible to use it on the real Lightning Network.\nHowever we felt the need to share some preliminary research progress after some test results from our simulated setting indicated an improvement of reliability of several orders of magnitude over the currently reported statistics for payment delivery.\nIn particular, an anecdotal lab report --- while highly unusual --- seemed appropriate for this particular situation in order to inform the developer community as early as possible about the potential need for a protocol upgrade removing the base fee.\n\n\\subsection{Simulation}\nSince the computation with base fees is not feasible\\footnote{and the fees are also currently low enough to not impact the results too much}, we have ignored the base fees in all our computations.\nIn fact, we started experimenting by not even optimizing for low fees at all but just for high probabilities.\n\nWe took a recent snapshot from the channel graph of the Lightning Network that was observed via the gossip protocol on one of our long running mainnet nodes.\nWe then created a static balance graph instantiation of the simulated network by splitting the channel capacity into two balance values uniformly at random independently for each channel.\n\nWe created a Python-based min-cost solver following the algorithm described in~\\cite[p.~556ff]{ahuja1993network} and a Scala version later on that turned out to be faster by a factor of about $3$--$4$.\nWhile implementing this algorithm we made some mistakes early on that accidentally led to the discovery of a heuristic that, on our snapshot Lightning Network graph, reliably produced results with less than 1\\% deviation from the optimal cost in less than 1\\% of the runtime.\nBecause the exact algorithm takes more than 30 minutes even in the faster Scala implementation, the following results have been mostly obtained with this heuristic, which typically takes about 6 seconds to run in Scala. So it is notable that the optimal results would be even better, although negligibly so.\n\nWe picked a medium-sized Lightning node that got randomly assigned a local balance of $0.42$ BTC and tried to send $0.4$ BTC to another node $3$ hops distant using the round-based algorithm described in Section~\\ref{rounds}.\nThe remote node had a total capacity of roughly $1.5$ BTC and more than $0.4$ BTC inbound liquidity. \nAssuming no routing hints from the recipient we started the first round computing a flow that was disected into several hundred small paths.\nSending them out we where able to deliver almost $75\\%$ of the amount that we wanted to deliver at once.\nWe updated the graph with the insights from the successes and failures and started the second round for the residual amount of roughly $0.1$ BTC.\nIn this round the min-cost solver on the graph with less uncertainty suggested a split of about $100$ paths.\nAfter sending out the payments we observed that the residual amount was only about $0.009$ Bitcoin.\nIn the third round, again on the updated graph, the min-cost solver suggested to send about 15 concurrent payments, of which all but one where successful.\nWe entered the fourth and final round with an amount of $30,000$ sat ($0.0003$ BTC) remaining.\nOwing to the learned data, the heuristic of our min-cost solver sent the full amount on a single path with $8$ hops, because it had already gained enough certainty for all but one channel (of size 1 BTC) along that path that it could forward $30,000$ sat.\nThus with a $99.97\\%$ probability the $30,000$ sat path settled on the selected $8$ hops path and the payment was delivered in full.\nWhile sending out all the onions we tracked the total routing fees to be $814$ sat.\n\nWe also repeated the experiment with reduced initial uncertainty by assuming the recipient node had initially communicated to the sender on which channels it could receive what amounts in the invoice.\nIn this case and on the same graph the algorithm delivered the final payment in the third round already.\n\nWe repeated the experiment a couple of times with different amounts and different instantiations of the simulated balance graph, resulting in similar results with every run. We therefore believe it reasonable to expect that we would see very similar results on the actual Lightning Network even though the unknown balance graph of the mainnet is constantly changing (potentially making some of our learned knowledge invalid).\n\nWe also used the above combination of balance graph, source and destination node for some experiments with the Lagrangian relaxation. This time, we allowed multiple parallel channels between nodes, as they are actually observed in the Lightning Network. Also, instead of going through the rounds of the payment algorithm, we just looked at the results of a single min-cost flow calculation, with a payment amount of 9.2 million sat (0.092 BTC). In this setting\\footnote{without using any knowledge about the source's or the destinations' channels}, optimizing for reliability only ($\\mu=0$) yields a probability of $P=0.16$, with total fees of 697 sat (excluding base fees). On the other hand, choosing $\\mu=100$ means optimizing almost exclusively for fees. This brings the total fees down to $16$ sat, but, as expected from the arguments in Section~\\ref{fees}, the success probability drops to $P=1.1\\cdot10^{-11}$. Table~1 shows that when we lower $\\mu$ through multiple orders of magnitude, the success probability increases drastically, while the fees are only rising moderately. Unfortunately, we increasingly observed numerical instabilities while decreasing $\\mu$. This lead to our algorithm not terminating beyond $\\mu=0.01$.\n\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}[h]{l|l|r}\n $\\mu$ & P & fee(sat) \\\\\n \\hline\n 100 & $1.1\\cdot10^{-11}$ & 16 \\\\\n 10 & $2.3\\cdot 10^{-5}$ & 16 \\\\\n 1 & $0.0097$ & 18 \\\\\n 0.1 & 0.044 & 24 \\\\\n 0.01 & 0.056 & 28 \\\\\n 0 & 0.16 & 697\n\\end{tabular}\n\\caption{Results of varying $\\mu$ on a fixed payment pair}\n\\end{center}\n\\label{table:mu}\n\\end{table}\n\nAgain, all numbers are from the heuristic, but the exact algorithm performed nearly identically in all our samples, including the numerical instabilities.\n\n\\subsection{Source code}\nWe could not find any preexisting open source software implementing the solution of the integer minimum cost flow problem for arbitrary separable convex cost functions.\nWe therefore share the source code of the described algorithms and methods as well as the latex sources of this document with an open MIT license at: \\url{https://github.com/renepickhardt/mpp-splitter}.\nThis repository consists of Scala- and Python-based example implementations of the exact min-cost flow algorithm described in~\\cite[p.~556ff]{ahuja1993network} for a separable convex cost function.\nIt also includes a minimalistic simulation framework in Python to test the practicality of the round-based payment loop.\nThe Scala version includes example code to demonstrate the usage of the Lagrangian relaxation.\n\n\n\\section{Advanced Applications}\n\n\n\\subsection{Multiple Senders and/or Receivers}\n\nNotice that our definition of a flow in Section~\\ref{flows} allows for an arbitrary number of both sources and sinks, that is, nodes with non-zero excess. This means that while a min-cost flow calculation might be computationally expensive, it can result in an optimized flow for multiple payments and/or channel balancing efforts at the same time. With respect to the runtime of the algorithm we have implemented\\footnote{which relies on single source shortest path calculations}, a more complex flow will take longer to optimize in practice, even though it will still respect the same worst-case runtime bounds.\nWe expect that entities like Lightning Service Providers (LSP) or trampoline routers, who need to find paths for many payments, will find this aspect helpful.\nOne could imagine a permanently running min-cost flow calculation loop that keeps learning about the network and sending out remainder amounts as in Section~\\ref{rounds}, but can always add additional payments in the next round.\nIt bears mentioning that in such multi-purpose rounds, the minimum cost is always optimized globally, which could lead to some payments being cheaper at the expense of others.\nThis needs to be accounted for when, e.g., routing payments for multiple clients.\n\n\n\\subsection{Optimal Channel Rebalancing}\n\\label{rebalancing}\nIt is well-known in the community that routing nodes can contribute to the overall payment reliability in the Lightning Network by using various channel rebalancing techniques. A recent survey paper~\\cite{papadis2020blockchain} in particular describes channel rebalancing via off-chain circular onions. This can happen proactively or lazily at routing time via a technique called just in time routing (JIT-routing)\\cite{Pickhardt2019}. To our knowledge, rebalancing has so far only been considered one channel pair at a time.\nWe observe that a node $i$ might want to rebalance several channels at once by shifting excess balance from source channels to target channels where more liquidity would be demanded.\nIn the uncertainty network, we can then assign the supply for rebalancing to the channel partner nodes of the corresponding outbound supply channels and remove these edges from the graph.\nBecause we have to account for the fees on inbound channels, the construction is a little more involved for the channels that demand extra balance: For every incoming edge $(j,i)$ that demands a balance increase, we create a copy $(j,k)$\n(with the same capacity and cost function) leading to a new node $k$ that is assigned the demand.\nFinally, we can compute a multi-source-multi-sink min-cost flow in order to shift the liquidity and conduct a multi-channel rebalancing.\nAs rebalancing is rarely time critical, one might prefer a high value of $\\mu$ in the min-cost flow computation that favors low fees over a high success rate.\nA node might even decide to stop the rebalancing operation before all of the onions have been successfully delivered: it has just engaged into a cheap opportunity for rebalancing; if delivering the remaining amounts turns out to be too expensive in the next min-cost flow calculation, it might prefer to stay with this improved but not perfect balance according to its own rebalancing strategy.\n\nAt first sight, rebalancing seems most interesting for nodes that engage more in routing than sending or receiving payments.\nHowever, we want to stress that for LSPs who conduct several payments per second, it might be very reasonable to combine the rebalancing and payment aspect, and suggest two ideas.\nFirst, we recall the global uncertainty of a node is always decreasing while delivering payments, so learning this knowledge could help find opportunities for engaging in rebalancing operations.\nSecond, an LSP might aim to keep its channels balanced according to a certain strategy.\nInstead of allowing itself to use all channels for making a payment, an LSP could restrict itself only to the channels where it has too much liquidity and exclude other channels from the computation.\nThis min-cost flow might be more expensive and less likely, but it might increase the chances for the node to forward payments; potentially earning a fee and increasing its overall reliability for other nodes might make such a trade-off worthwhile.\n\n\n\\section{Limitations}\n\n\\begin{enumerate}\n\\item As discussed in Section~\\ref{fees}, the currently adopted base fees in the Lightning Network make computing a min-cost flow NP-hard whenever the cost function includes these fees. Thus we have inquired about the motivation for the inclusion of a base fee online and received a response from the developer who appears to be responsible for this decision.\\footnote{Rusty Russell's answer to the question: Why was the base\\_fee for the routing fee calculation of the Lightning Network included? \\url{https://bitcoin.stackexchange.com/a/107341/81940}}\nWe are under the impression that the base fee was a rather ad-hoc and arbitrary choice and is not of significant importance to the Lightning Network protocol.\nEven if it were too difficult to change the protocol we see clear incentives for node operators setting the base fee to zero:\nNodes who want to conduct path finding might ignore base fees in their path computation in the future and thus ignore channels with a non-zero base fee.\nFurthermore, we note that our approach favors channels with large capacity, whose operators might therefore be able to demand higher fees.\nWe conjecture that this will give rise to a more balanced fee market, which should be in the interest particularly of node operators who have invested significant liquidity.\n\\item We made the crucial assumption that channel balances are uniformly distributed. While this was confirmed by prior research~\\cite{pickhardt2021security}, the situation could evolve over time, making our priors less suitable.\nIf, say, we assumed a normal distribution, the negative log probabilities would not be convex on the entire domain and the min-cost flow problem might not be solvable in polynomial time.\nHowever, we note for future research that there are practical solutions\nfor this, like limiting the domain by limiting the allowed channel capacity. Moreover, in~\\cite{pickhardt2021security} it has also been shown in the single path case that adopting a rebalancing protocol which changed the prior to a normal distribution but computing paths with a uniform prior still performed well.\nIn fact, we conjecture that we cannot do better than assume uniformity unless we have knowledge of the parameters of the actual distribution.\n\\item While the min-cost flow problem admits a polynomial time solution in the case of a convex cost function (without base fees), it is still computationally intensive, with a runtime that is quadratic in the number of channels. Remember also that we might have to solve several of these problems per payment round in order to find a suitable Lagrange multiplier in the trade-off between reliability and fees.\nOur prototype implementation is currently not optimized for speed; on the full Lightning Network graph, running times can easily reach 30 minutes and more, depending on the degree of precision. However, we have achieved preliminary experimental results on a promising heuristic that seems to very favorably trade reductions in runtime for only a slight deterioration from the optimal solution of the minimum cost flows.\nUsing heuristics like this, algorithmic methods including approximation and parallelization, as well as optimized code, we estimate that on the currently public Lightning Network it should be feasible to achieve sub-second runtimes on commodity hardware.\nHowever, we feel that this research is still too early for publication.\n\\end{enumerate}\n\n\\section{Future work}\nBeyond the optimization steps required for practical usability and extensive tests on the actual Lightning Network that we have\nhinted at above, we see some additional directions for further research:\n\n\\begin{enumerate}\n\\item Recall from Section~\\ref{fees} that when optimizing for both fees and reliability one has to find a suitable value for $\\mu$. We hope that improving on the current Lightning Network user experience will be possible with some experimentation, user interface design, and drawing from the extensive literature on Lagrangian relaxation.\n\n\\item In the round-based payment algorithm described in \\ref{rounds}, the optimal delay spent waiting for further responses after the first failed onion has returned and before entering the next round with updated knowledge remains an open question. In our experiments and simulation we have assumed waiting for the status of all onions to be resolved. We defer investigation of this question to practical experimentation but point out that it does not appear too critical since we can always incorporate information that arrives too late for one round in any later ones. In the extreme case, for every returned onion a new round could be entered.\n\n\\item It is conceptionally straight forward to extend the probabilities with provenance scores from operating nodes on the Lightning Network.\nInstead of just looking at the channel balance distribution one could create a joint distribution from, e.g., estimating the nodes' uptime.\n\n\\item Large channels could become the equivalent to the autobahn and attract a lot of traffic. Given the limitations of concurrent payments on the Lightning Network, the need for congestion control mechanisms might arise.\n\n\\item The payment planning and execution algorithm accumulates knowledge about the actual channel success probability distributions. However, we know that in practice those distributions do not stay constant. In fact, they change with every payment that is conducted through a particular channel or via the rebalancing behavior of node operators.\nFurther research into the dynamics of the money flow through the Lightning Network will help estimate how long to rely on the knowledge gained from previous payment attempts. This knowledge can then be adequately discounted or forgotten in future rounds.\n\n\\item The promising idea of adding redundant liquidity during the path finding phase of MPP splits was introduced by~\\cite{DBLP:journals/corr/abs-1910-01834}. It would be very interesting to study if we could find optimal redundant overpayments so that we can expect to finish within one round of sending out multi-part payments with high probability.\n\n\\item In spite of the arguments given in Section~\\ref{generalized} that led to prioritizing other aspects first, we still believe generalized flows to be an interesting future research direction, especially in light of multiple-source-multiple-sink applications (cf. Section~\\ref{rebalancing}). In this setting, constructions like the self-loop we described might allow for more flexibility: One can specify some sources and destinations with exact excess requirements as well as leave others open in order to give the optimization more leeway.\n\n\\end{enumerate}\n\n\\section{Acknowledgements}\nThis research was partially sponsored by the Norwegian University of Science and Technology (NTNU).\nWe want to thank David Karger and MIT open courseware for sharing a lecture series about min-cost flows\\footnote{\\url{https://www.youtube.com/playlist?list=PLaRKlIqjjguDXlnJWG2T7U52iHZl8Edrc}} that has been very useful for us in creating our reference implementation and understanding the theory of min-cost flows.\nWe are grateful to Ravindra Ahuja, Thomas Magnanti and James Orlin for their exceptional textbook~\\cite{ahuja1993network} which contains so much of the knowledge we have been adapting to our use case here.\nWe also thank Twitter user \\texttt{@four9four} for discussing our preliminary results after the Lightning hacksprint in April 2021.\nThanks to GB for polishing the graphics.\nWe thank Rene's co-authors Andreas M. Antonopoulos and Olaoluwa Osuntokun for accepting Rene's decision to temporarily prioritize this research over their Lightning Network book project.\nFinally our gratitude goes to Christian Decker with whom we had several discussions and who provided valuable feedback along the course of this research.\nIf you like the idea of decentralized and independent research and development of the Lightning Network feel free to support future work by contributing via \\url{https://donate.ln.rene-pickhardt.de}.\n\n\n\n\\bibliography{mppSplitting}\n\\bibliographystyle{plain}\n\n\\begin{appendix}\n \\section{Twitter-based TL;DR}\n While putting all together we actually shared the method and results in 6 Tweets:\n \\\\\\\\\n\n \\textbf{March 17th 2021}\\footnote{\\url{https://twitter.com/renepickhardt/status/1372169686251626499}} \\\\\n Over the last year I have been making quite some discoveries about \\#bitcoin Payment Pathfinding on the \\#LightningNetwork .\\\\A paper which introduces a probabilistic approach of modeling the uncertainty of remote channel balances is out and discussed on \\url{https://lists.linuxfoundation.org/pipermail/lightning-dev/2021-March/002984.html}\\\\\\\\\n \n \\textbf{April 22nd 2021}\\footnote{\\url{https://twitter.com/renepickhardt/status/1385144337907044352}} \\\\\n Multipathfinding \n\\begin{enumerate}\n\\item capacities are integers (Satoshis)\n\\item channel success probabilities -log(1-x/c) are convex functions\n\\item Solving integer minimum cost flows with separable convex cost objective polynomially: \\url{https://link.springer.com/chapter/10.1007\\%2FBFb0121104}\n\\end{enumerate}\n Kudos @stefanwouldgo 4 digging this out\n\\\\\\\\\n \n \\textbf{May 26th 2021}\\footnote{\\url{https://twitter.com/renepickhardt/status/1397559345139888137}}\\\\\n most likely MPP-split to deliver 92 mBTC from my lightning node (03efc...) to (022c6...).\n We split into 11 onions!\n\\begin{figure}[htpb]\n \\center\n \\includegraphics[width=0.45\\textwidth]{img/optflow.jpeg}\n \\label{fig:optflow}\n\\end{figure}\\\\\nWith knowledge of my own balance \\& routing hints in the invoice the total likelihood to deliver all 11 onions is $64.84\\%$\nNote the 6 hop path with 4 WUMBO channels!\n\\\\\\\\\n\n\\textbf{June 6th 2021}\\footnote{\\url{https://twitter.com/renepickhardt/status/1401514950984712198}}\\\\\nCouldn't go to Miami so I coded up my algorithm!\\\\\\\\\nIn a simulated network with realistic balance values a node having 0.42 BTC could send 0.4 BTC to a remote node (that could receive up to 1.59 BTC) with no direct channel\\\\\\\\\nIt took just 4 attempts to deliver and 814 sat in fees\\\\\\\\\n\n\\textbf{June 8th 2021}\\footnote{\\url{https://twitter.com/renepickhardt/status/1402264479677693958}}\\\\\non other good news:\n\\begin{enumerate}\n\\item this yields another test vector for unit tests\n\\item the previous / non optimal algorithm might for other reasons (that go beyond a tweet) actually be better suitable for the lightning network after all\n\\item either way the issue seems fixable (:\n\\end{enumerate}\n\n \\textbf{July 5th 2021}\\footnote{\\url{https://twitter.com/stefanwouldgo/status/1412158904008646660}}\\\\\n Lightning Routing IS NP hard, though.\n\n It's funny 'cause it's true.\n\n\n\n\\end{appendix}\n\\end {document}\n", "meta": {"hexsha": "f864a53267a49b5bcd0cdd6c74653ecd7e8bb673", "size": 59855, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/mppSplitting.tex", "max_stars_repo_name": "drmartinberger/mpp-splitter", "max_stars_repo_head_hexsha": "74b7634bb4c21e62c24865906a83fd391e4e06bf", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper/mppSplitting.tex", "max_issues_repo_name": "drmartinberger/mpp-splitter", "max_issues_repo_head_hexsha": "74b7634bb4c21e62c24865906a83fd391e4e06bf", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/mppSplitting.tex", "max_forks_repo_name": "drmartinberger/mpp-splitter", "max_forks_repo_head_hexsha": "74b7634bb4c21e62c24865906a83fd391e4e06bf", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 105.9380530973, "max_line_length": 1234, "alphanum_fraction": 0.7888229889, "num_tokens": 13550, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34722653010511845}} {"text": "% Created 2019-08-27 mar 16:52\n\\documentclass[presentation,aspectratio=1610]{beamer}\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage{fixltx2e}\n\\usepackage{graphicx}\n\\usepackage{longtable}\n\\usepackage{float}\n\\usepackage{wrapfig}\n\\usepackage{rotating}\n\\usepackage[normalem]{ulem}\n\\usepackage{amsmath}\n\\usepackage{textcomp}\n\\usepackage{marvosym}\n\\usepackage{wasysym}\n\\usepackage{amssymb}\n\\usepackage{hyperref}\n\\tolerance=1000\n\\usepackage{khpreamble}\n\\usepackage{amssymb}\n\\DeclareMathOperator{\\shift}{q}\n\\DeclareMathOperator{\\diff}{p}\n\\usetheme{default}\n\\author{Kjartan Halvorsen}\n\\date{2018-01-19}\n\\title{Computerized Control - LTIs, impulse response, difference equations}\n\\hypersetup{\n pdfkeywords={},\n pdfsubject={},\n pdfcreator={Emacs 25.3.50.2 (Org mode 8.2.10)}}\n\\begin{document}\n\n\\maketitle\n\n\n\\section{Intro: Discrete-time signals}\n\\label{sec-1}\n\n\\begin{frame}[label=sec-1-1]{The discrete causal linear time-invariant system}\n\\begin{center}\n\\begin{tikzpicture}[node distance=20mm, anchor=north]\n\\node[coordinate] (input) {};\n\\node[rectangle, draw, right of=input, inner sep=3mm] (lti) {g};\n\\node[coordinate, right of=lti] (output) {};\n\\draw[->] (input) -- node[near start, above] {$u(k)$} (lti);\n\\draw[->] (lti) -- node[near end, above] {$y(k)$} (output);\n\\end{tikzpicture}\n\\end{center}\n\n\\[ y(k) = g \\ast u = \\sum_{n=0}^\\infty g(n) u(k-n) \\]\n\nIf input signal is a pulse (delta-function)\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{axis}[\n width=14cm,\n height=2.5cm,\n xlabel={$k$},\n ylabel={$u(k)$},\n xmin=-2.5,\n xmax=10.5,\n]\n\n\\addplot+[black, ycomb, domain=-2:10, samples=13,variable=k] { (k==0)}; \n\n\\end{axis}\n\\end{tikzpicture}\n\\end{center}\n\n\\vspace*{-5mm}\n\\[ y(k) = \\sum_{n=0}^\\infty g(n) \\delta(k-n) = g(k) \\]\n\\end{frame}\n\n\\begin{frame}[label=sec-1-2]{Linearity, time invariance and the pulse response}\nThe input signal\n\n\\begin{center}\n\\begin{tikzpicture}\n\\small\n\\begin{axis}[\n width=14cm,\n height=3.5cm,\n xlabel={$k$},\n ylabel={$u(k)$},\n xmin=-0.5,\n xmax=10.5,\n ytick = {-1, 0, 0.6, 2},\n]\n\n\\addplot+[black, ycomb, domain=-2:10, samples=13,variable=k] { 0.6*(k==0) + 2*(k==1) - 1*(k==2)}; \n\n\\end{axis}\n\\end{tikzpicture}\n\\end{center}\n\n\\vspace*{-5mm}\n\n\nCan be written \n\\[u(k) = 0.6\\delta(k) + 2\\delta(k-1) - \\delta(k-2) \\]\nSince the system's response to a pulse is given by $g(k)$, the output signal is\n\\[ y(k) = ?\\]\n\\end{frame}\n\n\\begin{frame}[label=sec-1-3]{Linearity, time invariance and the pulse response}\nThe input signal\n\n\\begin{center}\n\\begin{tikzpicture}\n\\small\n\\begin{axis}[\n width=14cm,\n height=3.5cm,\n xlabel={$k$},\n ylabel={$u(k)$},\n xmin=-0.5,\n xmax=10.5,\n ytick = {-1, 0, 0.6, 2},\n]\n\n\\addplot+[black, ycomb, domain=-2:10, samples=13,variable=k] { 0.6*(k==0) + 2*(k==1) - 1*(k==2)}; \n\n\\end{axis}\n\\end{tikzpicture}\n\\end{center}\n\n\\vspace*{-5mm}\n\n\nCan be written \n\\[u(k) = 0.6\\delta(k) + 2\\delta(k-1) - \\delta(k-2) \\]\nSince the system's response to a pulse is given by $g(k)$, the output signal is\n\\[ y(k) = 0.6g(k) + 2g(k-1) - g(k-2) \\]\n\\end{frame}\n\n\n\\begin{frame}[label=sec-1-4]{The output of a causal, linear discrete-time system is a weighted sum of previous input}\n\\[ y(k) = g \\ast u = \\sum_{n=0}^\\infty g(n) u(k-n) \\]\nThe \\alert{weighting function} $g(k)$ is the \\alert{pulse response} of the system.\n\nWhat if the weighting function looks like this\n\n\\begin{center}\n\\begin{tikzpicture}\n\\small\n\\begin{axis}[\n width=14cm,\n height=3.5cm,\n xlabel={$k$},\n ylabel={$g(k)$},\n xmin=-0.5,\n xmax=10.5,\n ytick = {0, 1},\n]\n\n\\addplot+[black, ycomb, domain=-2:10, samples=13,variable=k] { (k==4)}; \n\n\\end{axis}\n\\end{tikzpicture}\n\\end{center}\n\n\\[y(k) = \\]\n\\end{frame}\n\n\\begin{frame}[label=sec-1-5]{Exercise: Impulse response}\n\\end{frame}\n\n\n\n\\section{The shift operator}\n\\label{sec-2}\n\\begin{frame}[label=sec-2-1]{The shift operator}\n\\begin{itemize}\n\\item For difference equations the shift operator \\(\\shift\\) is very useful.\n\\item The shift operator is defined for double-infinite sequences $x_k$, i.e. the sequence $x_k$ must be infinitely long both for negative and positive $k$.\n\\item The operator shifts the sequence ahead one step:\n\\[ \\shift x_k = x_{k+1} \\]\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[label=sec-2-2]{The difference equation is a representation of a discrete-time dynamical systems}\n\\begin{center}\n\\begin{tikzpicture}[node distance=25mm]\n\\node[rectangle, draw, minimum height=10mm, minimum width=14mm] (sys) {$G(\\shift)$};\n\\node[coordinate, left of=sys] (input) {};\n\\node[coordinate, right of=sys] (output) {};\n\n\\draw[->] (input) -- node [near start, above] {$u(k)$} (sys);\n\\draw[->] (sys) -- node [near end, above] {$y(k)$} (output);\n\n\\end{tikzpicture}\n\\end{center}\n\n\\[ y_{k+n} + a_1 y_{k+n-1} + \\cdots + a_n y_k = b_0 u_{k+m} + b_1 u_{k+m-1} + \\cdots + b_m u_k \\]\n\n\\[ \\left( \\shift^n + a_1 \\shift^{n-1} + \\cdots + a_n \\right) y(k) = \\left( b_0 \\shift^m + b_1\\shift^{m-1} + \\cdots + b_m \\right) u(k) \\]\n\n\\[ y(k) = \\frac{b_0 \\shift^m + b_1\\shift^{m-1} + \\cdots + b_m}{ \\shift^n + a_1 \\shift^{n-1} + \\cdots + a_n} u(k) = \\frac{B(\\shift)}{A(\\shift)} u(k) = G(\\shift) u(k) \\]\n\\end{frame}\n\n\\section{First order system and pulse response}\n\\label{sec-3}\n\n\\begin{frame}[label=sec-3-1]{First order systems}\n\\begin{center}\n\\begin{tikzpicture}[node distance=20mm, anchor=north]\n\\node[coordinate] (input) {};\n\\node[rectangle, draw, right of=input, inner sep=3mm] (lti) {$G(q)=\\frac{q-1}{q}$};\n\\node[coordinate, right of=lti] (output) {};\n\\draw[->] (input) -- node[near start, above] {$u(k)$} (lti);\n\\draw[->] (lti) -- node[near end, above] {$y(k)$} (output);\n\\end{tikzpicture}\n\\end{center}\n\nThe system with pulse-transfer operator $G(q)=\\frac{q-1}{q}$ corresponds to the difference equation\n\\[ y(k) = G(q)u(k) \\Leftrightarrow y(k) = \\frac{q-1}{q} u(k) \\]\n\\[ y(k+1) = ?\\]\n\\end{frame}\n\n\\begin{frame}[label=sec-3-2]{First order systems}\n\\begin{center}\n\\begin{tikzpicture}[node distance=20mm, anchor=north]\n\\node[coordinate] (input) {};\n\\node[rectangle, draw, right of=input, inner sep=3mm] (lti) {$G(q)=\\frac{q-1}{q}$};\n\\node[coordinate, right of=lti] (output) {};\n\\draw[->] (input) -- node[near start, above] {$u(k)$} (lti);\n\\draw[->] (lti) -- node[near end, above] {$y(k)$} (output);\n\\end{tikzpicture}\n\\end{center}\n\nThe system with pulse-transfer operator $G(q)=\\frac{q-1}{q}$ corresponds to the difference equation\n\\[ y(k) = G(q)u(k) \\Leftrightarrow y(k) = \\frac{q-1}{q} u(k) \\]\n\\[ y(k+1) = u(k+1)-u(k), \\quad \\text{i.e.~a discrete-time differentiator}\\]\n\\end{frame}\n\n\\begin{frame}[label=sec-3-3]{First order systems}\n\\begin{center}\n\\begin{tikzpicture}[node distance=20mm, anchor=north]\n\\node[coordinate] (input) {};\n\\node[rectangle, draw, right of=input, inner sep=3mm] (lti) {$G(q)=\\frac{q}{q-a}$};\n\\node[coordinate, right of=lti] (output) {};\n\\draw[->] (input) -- node[near start, above] {$u(k)$} (lti);\n\\draw[->] (lti) -- node[near end, above] {$y(k)$} (output);\n\\end{tikzpicture}\n\\end{center}\n\nThe system with pulse-transfer operator $G(q)=\\frac{q}{q-a}$ corresponds to the difference equation\n\\[ y(k) = G(q)u(k) \\Leftrightarrow y(k) = \\frac{q}{q-a} u(k) \\]\n\\[ y(k+1) = ?\\]\n\\end{frame}\n\n\\begin{frame}[label=sec-3-4]{First order systems}\n\\begin{center}\n\\begin{tikzpicture}[node distance=20mm, anchor=north]\n\\node[coordinate] (input) {};\n\\node[rectangle, draw, right of=input, inner sep=3mm] (lti) {$G(q)=\\frac{q}{q-a}$};\n\\node[coordinate, right of=lti] (output) {};\n\\draw[->] (input) -- node[near start, above] {$u(k)$} (lti);\n\\draw[->] (lti) -- node[near end, above] {$y(k)$} (output);\n\\end{tikzpicture}\n\\end{center}\n\nThe system with pulse-transfer operator $G(q)=\\frac{q}{q-a}$ corresponds to the difference equation\n\\[ y(k) = G(q)u(k) \\Leftrightarrow y(k) = \\frac{q}{q-a} u(k) \\]\n\\[ y(k+1) = ay(k) + u(k+1). \\quad \\text{If $a=1$, the system is a discrete-time integrator}\\]\n\\end{frame}\n\n\\begin{frame}[label=sec-3-5]{Pulse-response of a first order system}\n\\[ y(k+1) = ay(k) + u(k+1) \\]\n\\end{frame}\n\n\\begin{frame}[label=sec-3-6]{Pulse response of a first order system}\n\\[ y(k+1) = ay(k) + u(k+1) \\]\n\nPair the pulse response to each of the values of $a$\n\\[ \\text{I)}\\; a=1 \\qquad \\text{II)}\\; a=2 \\qquad \\text{III)}\\; a = 0.5 \\qquad \\text{IV)}\\; a=-0.9 \\]\n\n\\begin{tikzpicture}\n\\small\n\\begin{axis}[\nwidth=7cm,\nheight=2.5cm,\nxlabel={$k$},\nylabel={$g(k)$},\nxmin=-3.5,\nxmax=10.5,\nytick = {-1,0,1},\nymin = -1.2, ymax=1.2,\n]\n\\addplot+[black, ycomb, domain=-3:10, samples=14,variable=k] { (k>=0)*pow(1,k)};\n\\end{axis}\n\n\\begin{axis}[\nxshift=7cm,\nwidth=7cm,\nheight=2.5cm,\nxlabel={$k$},\nylabel={$g(k)$},\nxmin=-3.5,\nxmax=10.5,\nytick = {0},\nytick = {-1,0,1},\nymin = -1.2, ymax=1.2,\n]\n\\addplot+[black, ycomb, domain=-3:10, samples=14,variable=k] { (k>=0)*pow(-0.9,k)};\n\\end{axis}\n\n\\begin{axis}[\nxshift=0cm,\nyshift=-2.5cm,\nwidth=7cm,\nheight=2.5cm,\nxlabel={$k$},\nylabel={$g(k)$},\nxmin=-3.5,\nxmax=10.5,\nytick = {0},\nytick = {-1,0,8},\nymin = -0.2, ymax=8.2,\n]\n\\addplot+[black, ycomb, domain=-5:8, samples=14,variable=k] { (k>=0)*pow(2,k) };\n\\end{axis}\n\n\\begin{axis}[\nxshift=7cm,\nyshift=-2.5cm,\nwidth=7cm,\nheight=2.5cm,\nxlabel={$k$},\nylabel={$g(k)$},\nxmin=-3.5,\nxmax=10.5,\nytick = {0},\nytick = {-1,0,1},\nymin = -1.2, ymax=1.2,\n]\n\\addplot+[black, ycomb, domain=-5:8, samples=14,variable=k] { (k>=0)*pow(0.5,k)};\n\\end{axis}\n\n\n\\end{tikzpicture}\n\\end{frame}\n% Emacs 25.3.50.2 (Org mode 8.2.10)\n\\end{document}", "meta": {"hexsha": "224b9952be1d398d8f33971b57941e7ddbc0df22", "size": 9259, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "discrete-time-systems/slides/problem-solving-sampling-poles-etc.tex", "max_stars_repo_name": "kjartan-at-tec/mr2007-computerized-control", "max_stars_repo_head_hexsha": "16e35f5007f53870eaf344eea1165507505ab4aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-11-07T05:20:37.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-22T09:46:13.000Z", "max_issues_repo_path": "discrete-time-systems/slides/problem-solving-sampling-poles-etc.tex", "max_issues_repo_name": "kjartan-at-tec/mr2007-computerized-control", "max_issues_repo_head_hexsha": "16e35f5007f53870eaf344eea1165507505ab4aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2020-06-12T20:44:41.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-12T20:49:00.000Z", "max_forks_repo_path": "discrete-time-systems/slides/problem-solving-sampling-poles-etc.tex", "max_forks_repo_name": "kjartan-at-tec/mr2007-computerized-control", "max_forks_repo_head_hexsha": "16e35f5007f53870eaf344eea1165507505ab4aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-14T03:55:27.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-14T03:55:27.000Z", "avg_line_length": 26.9156976744, "max_line_length": 167, "alphanum_fraction": 0.641213954, "num_tokens": 3616, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6442251201477016, "lm_q1q2_score": 0.34722653010511845}} {"text": "\\section{delimited continuations}\n\\label{sec:cont-delim-cont}\nContinuatioins\nA conditional banch selects a continuation from the two possible\nfutures; rasing an exception discards. Traditional way to handle\ncontinuations explicitly in a program is to transform a program into\ncps style. Continuation captured by call/cc is the {\\bf whole} continuation\nthat includes all the future computation.. In practice, most of the\ncontinuations that we want to manipulate are only a part of\ncomputation. Such continuations are called {\\bf delimited continuations} or\n{\\bf partial continuations}.\n\n\n\\begin{enumerate}\n\\item cps transform \\\\\n there are multiple ways to do cps transform, here are two.\n\n \n \\begin{bluetext}\n------------------------------------------\n [x] --> x\n [\\x. M] --> \\k . k (\\x . [M])\n [M N] --> \\k. [M] (\\m . m [N] k)\n------------------------------------------\n\n\n------------------------------------------\n [x] --> \\k . k x\n [\\x. M] --> \\k. k (\\x.[M])\n [M N] --> \\k. [M] (\\m . [N] (\\n. m n k))\n------------------------------------------\n\n\n[callcc (\\k. body)] = \\outk. (\\k. [body] outk) (\\v localk. outk v)\n \n \\end{bluetext}\n\n \n\\item experiment\n\n\\begin{alternate}\n#load \"delimcc.cma\";;\n\\end{alternate}\n\\begin{alternate}\nDelimcc.shift;;\n- : 'a Delimcc.prompt -> (('b -> 'a) -> 'a) -> 'b = \n\\end{alternate}\n\n\\begin{bluetext}\nreset (fun () -> M ) --> push_prompt p (fun () -> M )\nshift (fun k -> M) --> shift p (fun k -> M )\n\\end{bluetext}\nin racket you should have \\textit{(require racket/control)}\nand then \\textit{(reset expr ...+)}\n\\textit{(shift id expr ...+)}\n\n\n\\begin{ocamlcode}\nmodule D = Delimcc\n(** set the prompt *) \nlet p = D.new_prompt ()\nlet (reset,shift),abort = D.(push_prompt &&& shift &&& abort ) p;;\nlet foo x = reset (fun () -> shift (fun cont -> if x = 1 then cont 10 else 20 ) + 100 )\n\\end{ocamlcode}\n\n\\begin{alternate}\nfoo 1 ;;\n- : int = 110\nfoo 2 ;;\n- : int = 20\n5 * reset (fun () -> shift (fun k -> 2 * 3 ) + 3 * 4 );;\n- : int = 30\nreset (fun () -> 3 + shift (fun k -> 5 * 2) ) - 1 ;;\n- : int = 9\n\\end{alternate}\n\\begin{bluetext}\nval p : '_a D.prompt = \nval reset : (unit -> '_a) -> '_a = \nval shift : (('_a -> '_b) -> '_b) -> '_a = \nval abort : '_a -> 'b = \n\\end{bluetext}\n\n\\begin{ocamlcode}\nlet p = D.new_prompt ()\nlet (reset,shift),abort = D.(push_prompt &&& shift &&& abort ) p;;\n\\end{ocamlcode}\n\n\\begin{alternate}\nreset (fun () -> if (shift (fun k -> k(2 = 3))) then \"hello\" else \"hi \") ^ \"world\";;\n- : string = \"hi world\"\nreset (fun () -> if (shift (fun k -> \"laji\")) then \"hello\" else \"hi \") ^ \"world\";;\n- : string = \"lajiworld\"\nreset (fun _ -> \"hah\");;\n- : string = \"hah\"\n\\end{alternate}\n\n\n\\begin{ocamlcode}\nlet make_operator () = \n let p = D.new_prompt () in \n let (reset,shift),abort = D.(push_prompt &&& shift &&& abort) p in \n p,reset,shift,abort\n\\end{ocamlcode}\n\nDelimited continuations seems not able to handle answer type polymorphism.\n\n\\begin{bluetext}\nexception Str of [`Found of int | `NotFound] \n\\end{bluetext}\n\n\\begin{ocamlcode}\nlet times lst = \n let rec times_aux lst = match lst with \n | [] -> 1 \n | 0 :: xs -> shift (fun _ -> 0 )\n | x :: xs -> begin \n (* printf \"entering %d\\n\" x ; *)\n let v = x * times_aux xs in \n (* printf \"exiting %d\\n\" x ; *)\n v\n end in \n reset (fun () -> times_aux lst )\n\\end{ocamlcode}\n\nStore the continuation, the type system is not friendly to the\ncontinutations, but fortunately we have \\textit{side effects} at hand, we can\nstore it. (This is pretty hard in Haskell )\n\n\\begin{ocamlcode}\nlet p,reset,shift,abort = make_operator() in \n let c = ref None in \n begin \n reset (fun () -> 3 + shift (fun k -> c:= Some k ; 0) - 1) ; \n Option.get (!c) 20 \n end ;;\n Characters 81-139:\n reset (fun () -> 3 + shift (fun k -> c:= Some k ; 0) - 1) ; \n ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^\n Warning 10: this expression should have type unit.\n \\end{ocamlcode}\n\\begin{ocamlcode} \n- : int = 22\n\\end{ocamlcode}\n\\begin{ocamlcode}\nlet cont = \n let p,reset,shift,abort = make_operator() in \n let c = ref None in \n let rec id lst = match lst with \n | [] -> shift (fun k -> c:=Some k ; [] )\n |x :: xs -> x :: id xs in \n let xs = reset (fun () -> id [1;2;3;4]) in \n xs, Option.get (!c);;\n\\end{ocamlcode}\n\\begin{ocamlcode}\nval cont : int list * (int list -> int list) = ([], )\n\\end{ocamlcode}\n\\begin{alternate}\n# let a,b = cont ;;\nval a : int list = []\n\nval b : int list -> int list = \n# b [];;\n- : int list = [1; 2; 3; 4]\n\\end{alternate}\n\n\n\n\\begin{ocamlcode}\ntype tree = Empty | Node of tree * int * tree \nlet walk_tree = \n let cont = ref None in \n let p,reset,shift,abort = make_operator() in \n let yield n = shift (fun k -> cont := Some k; print_int n ) in \n let rec walk2 tree = match tree with \n |Empty -> ()\n |Node (l,v,r) -> \n walk2 l ;\n yield v ; \n walk2 r in \n fun tree -> (reset (fun _ -> walk2 tree ), cont);;\n\\end{ocamlcode}\n\\begin{ocamlcode}\nval walk_tree : tree_t -> unit * ('_a -> unit) option Batteries.ref =\n\\end{ocamlcode}\n\n\\begin{alternate}\n# let _, cont = walk_tree tree1 ;;\n1val cont : ('_a -> unit) option Batteries.ref = {contents = Some }\n# Option.get !cont ();;\n2- : unit = ()\n# Option.get !cont ();;\n3- : unit = ()\n# Option.get !cont ();;\n- : unit = ()\n# Option.get !cont ();;\n- : unit = ()\n\\end{alternate}\n\nIt's quite straightforward to implement yield using delimited\ncontinuation, since each time shifting will escape the control, and you store the continuation, later it can be resumed.\n\n\n\\begin{bluetext}\n(** defer the continuation *) \nshift (fun k -> fun () -> k \"hello\")\n\\end{bluetext}\n\nBy wrapping continuations, we can \\textbf{access the information outside} of the enclosing\nreset while staying within reset lexically.\n\nsuppose this type check\n\n\\begin{alternate}\n let f x = reset (fun () -> shift (fun k -> fun () -> k \"hello\") ^ \"world\" ) x\n f : unit -> string \n\\end{alternate}\n\n\\item Answer type modification (serious)\n in the following context,\n \\verb|reset (fun () -> [...] ^ \"word\" )|, the value returned by\n reset appears to be a string. An answer type is a type of the enclosing\n \\emph{reset}.\n\n\\item reorder delimited continuations \\\\\n if we apply a continuation at the tail position, the captured computation is simply\n resumed. If we apply a continuation at the non-tail position, we can perform\n additional computation after resumed computation finishes.\n\n Put differently, we can switch the execution order of the surrounding context.\n\n\\begin{ocamlcode}\nlet p,reset,shift,abort = make_operator () in\n reset (fun () -> 1 + (shift (fun k -> 2 * k 3 )));;\n\\end{ocamlcode}\n\\begin{ocamlcode}\n- : int = 8 \n\\end{ocamlcode}\n\n\\begin{ocamlcode}\nlet p,reset,shift,abort = make_operator () in \n let either a b = shift (fun k -> k a ; k b ) in \n reset (fun () -> \n let x = either 0 1 in \n print_int x ; print_newline ());;\n \\end{ocamlcode}\n\\begin{ocamlcode} \n 0\n 1 \n\\end{ocamlcode}\n\\item useful links \\\\\n \\href{http://blog.fitzell.ca/2009/01/seaside-partial-continuations.html}{sea\n side} \\\\\n \\href{http://pllab.is.ocha.ac.jp/~asai/cw2011tutorial/}{shift and\n reset tutorial} \\\\\n \\href{http://pllab.is.ocha.ac.jp/~asai/cw2011tutorial/main-e.pdf}{shift\n reset tutorial} \\\\\n \\href{http://docs.racket-lang.org/reference/cont.html#(part._.Classical_.Control_.Operators)}{racket\n control operators} \\\\\n \\href{http://okmij.org/ftp/continuations/caml-shift.pdf}{caml-shift-paper.pdf} \\\\\n \\href{http://okmij.org/ftp/continuations/caml-shift-talk.pdf}{caml-shift-talk} \\\\\n\n\\end{enumerate}\n", "meta": {"hexsha": "f07d3d2d74b112f429bdd93f7de8e2376307b705", "size": 7622, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "library/delimcc.tex", "max_stars_repo_name": "mgttlinger/ocaml-book", "max_stars_repo_head_hexsha": "09a575b0d1fedfce565ecb9a0ae9cf0df37fdc75", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 142, "max_stars_repo_stars_event_min_datetime": "2015-01-12T16:45:40.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-15T00:47:37.000Z", "max_issues_repo_path": "library/delimcc.tex", "max_issues_repo_name": "mgttlinger/ocaml-book", "max_issues_repo_head_hexsha": "09a575b0d1fedfce565ecb9a0ae9cf0df37fdc75", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-10-09T13:53:43.000Z", "max_issues_repo_issues_event_max_datetime": "2018-12-03T04:15:48.000Z", "max_forks_repo_path": "library/delimcc.tex", "max_forks_repo_name": "mgttlinger/ocaml-book", "max_forks_repo_head_hexsha": "09a575b0d1fedfce565ecb9a0ae9cf0df37fdc75", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 17, "max_forks_repo_forks_event_min_datetime": "2015-02-10T18:12:15.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-21T06:57:32.000Z", "avg_line_length": 28.9809885932, "max_line_length": 120, "alphanum_fraction": 0.6078457098, "num_tokens": 2400, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.538983220687684, "lm_q2_score": 0.6442251201477016, "lm_q1q2_score": 0.3472265301051184}} {"text": "\\documentclass[10pt,conference,compsocconf]{IEEEtran}\n\n\\usepackage{hyperref}\n\\usepackage{graphicx}\t% For figure environment\n\\usepackage{amssymb}\n\\usepackage{authblk}\n\\usepackage{color}\n\\usepackage{graphicx}\n\\graphicspath{ {images/} }\n\\usepackage[skip=2pt]{caption} % example skip set to 2pt\n\n\\title{Project 1 on Machine Learning team Yoor}\n\n\\author[1]{Sergei Volodin}\n\\author[1]{Baran Nama}\n\\author[1]{Omar Mehio}\n\\affil[1]{EPFL}\n\\affil[ ]{\\textit {\\{sergei.volodin,baran.nama,omar.mehio\\}@epfl.ch}}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nA classification dataset from the Large Hadron Collider simulations is being studied. First, the data is thoroughly explored using visual aids.\nAfter that, several basic Machine Learning methods are applied on preprocessed data.\nResults are evaluated using cross-validation.\nModel overview is given for each considered algorithm and the best model is chosen.\n\\end{abstract}\n\n\\section{Introduction}\nThe Higgs boson is a famous elementary particle which was first predicted in 1960s and then discovered in 2012 \\cite{higgs}. Its famousness is due to two facts, first being the collosal amount of effort put into construction of the Large Hadron Collider and conducting the ATLAS experiment and the second being the fact that the Higgs boson is considered to be connected to the fact that particles have a mass.\n\nThe ATLAS experiment consists of protons colliding at near-relative speed. After collision, the resulting particles sometimes contain the Higgs boson. Itself, it is not detectable by LHC. However, it is possible to detect the particles that it is decaying to.\n\nThe data being studied comprises of $N=250 000$ objects (train) each having 30 features: $\\{(x_i,y_i)\\}_{i=1}^N$, $x_i\\in\\mathbb{R}^D$, $D=30$. Each object represents a collision of a stream of protons. The data was not obtained during the ATLAS experiment, but rather from the simulation \\cite{data}. Features represent properties of detected particles. It is required to determine if the particles represent the Higgs boson. The dataset has two classes: signal (+1) and background (-1).\n\nThe following paper claims that it is possible to use simple methods, such as Linear and Logistic Regressions to classify the data. In the following sections, the data is thoroughly studied and then the model is chosen based on reasoning and cross-validation.\n\\section{Models and Methods}\n\\begin{table}[!htb]\n\t\\caption{Feature processing tricks and linear regression}\t\\label{tab:feature_proc}\n\t\\centering\\begin{tabular}{|lc|}\\hline\n\t\t{\\bf Feature trick added} & {\\bf Test accuracy}\\\\\\hline\n\t\tConstant feature & 0.745\\\\\n\t\tStandardization & 0.745\\\\\n\t\tImputation & 0.747\\\\\n\t\tBinarization & 0.747\\\\\n\t\tDegree 3 poly & 0.785\\\\\\hline\n\t\\end{tabular}\n\\end{table}\n\n\\begin{figure}[!htb]\n\t\\centering \\includegraphics[width=100px]{../src/analysis/xy_tr_imp}\n\t\\centering \\includegraphics[width=100px]{../src/analysis/hist_te_imp}\n\t\\centering \\includegraphics[width=100px]{../src/analysis/hist-scatter-tr-imp-0}\n\t\\centering \\includegraphics[width=110px]{../src/analysis/correlation}\n\t\\caption{Exploratory data analysis using charts}\n\t\\label{fig:data}\n\\end{figure}\n\nFirst, the exploratory data analysis shows that most of the features (except one) are real-valued (Figure \\ref{fig:data}, 1-2). Moreover, it can also be seen that the test distribution does not differ from the training one (Figure \\ref{fig:data}, 1-2). Scatter plots (Figure \\ref{fig:data}, 3) show dependencies between features.\nTo further explore linear interactions among our data, we computed the correlation matrix (Figure \\ref{fig:data}, 4).\nSome of the significant dependencies we observed include the relationship among following tuples ({\\tt DER\\_prodeta\\_jet\\_jet}, {\\tt DER\\_mass\\_jet\\_jet}) and ({\\tt PRI\\_jet\\_subleading\\_phi}, {\\tt DER\\_sum\\_pt}).\n\nEach of the respective tuples have a $-1$ and $+1$ correlation coefficient. This implies that for each column in the tuple we can eliminate the other element since we are able to deduce the same amount of knowledge from either one of them. In practice this didn't turn out to be efficient as our prediction with deducted features resulted in lower accuracy. \n\nTo begin with, a simple linear regression model is used on raw data (with a constant feature added), with 5-fold cross-validation to control overfitting. The results of each subsample are averaged to provide train and test accuracy. However, the model gives unsatisfying results on the hold-out dataset. As a means to deal with this issue, feature imputation and engineering is used: missing features are imputed with mean values (and ``missing value'' column is added), categorical features are binarized. Moreover, it has been discovered that some of the features represent particle's mass, momentum or energy. Using the fact that squaring these quantities would make linear combinations of them meaningful in terms of physics ($E^2=m^2+p^2$), we introduce the polynomial basis starting from the degree of 2. Table \\ref{tab:feature_proc} shows the respective effects of these tricks.\n\n\\begin{figure}[!htb]\n\t\\centering \\includegraphics[width=110px]{linear_5}\n\t\\centering \\includegraphics[width=110px]{ridge_accuracy_deg6}\n\t\\caption{Degree 6 basis, MSE and Accuracy metrics}\n\t\\label{fig:deg6}\n\\end{figure}\n\nApplying ridge regression, it might be seen that it does not overfit in terms of accuracy (but does in terms of MSE \\ref{fig:deg6}) on the dataset given even for the 6th degree, meaning that there is no significant difference between the test and the train accuracies. Applying it using different degrees, one obtains results shown in the Table \\ref{tab:degrees}.\n\n\\begin{table}[!htb]\n\t\\caption{Polynomial basis and linear regression, $\\lambda=0$}\t\\label{tab:degrees}\n\t\\centering\\begin{tabular}{|lcccc|}\\hline\n\t\t{\\bf Degree} & {\\bf Test accuracy} & {\\bf MSE Test} & {\\bf MSE Train} & {$\\lambda$ best}\\\\\n\t\t1 & 0.7470 & 0.3378 & 0.3377 & $10^{-6}$\\\\\n\t\t2 & 0.7737 & 0.3483 & 0.3158 & $10^{-2}$\\\\\t\t\n\t\t3 & 0.7850 & 0.3060 & 0.3040 & $10^{-4}$\\\\\n\t\t4 & 0.7937 & 0.3030 & 0.3010 & 0.0004\\\\\n\t\t5 & 0.7971 & 0.3050 & 0.3010 & 0.0006\\\\\\hline\n\t\\end{tabular}\n\\end{table}\n\nMoreover, as can be seen in the figures (Table \\ref{tab:degrees}), while train MSE is continuously decreasing because of over-fitting, there is no significant change in test error after third degree polynomial basis. Therefore, feature extraction using polynomial basis does not seem to work for higher degrees. Furthermore, the accuracy of the model in Kaggle is not as good as what we expected (below 0.8), so that we decided to switch to logistic regression since it uses a loss function tailored for the purpose of classification.\n\nTraining logistic regression with gradient descent on the dataset, we have found out that it is crucial to implement the Newton's method since gradient update rule is taking too much time. This is due to the fact that the final iterations are going with a small step size since setting the step size of gradient descent to a larger value resulted in a numerical overflow.\n\nMoreover, we discovered that the magnitude $\\gamma$ of the gradient step depends on the size of the dataset. Therefore, we tried using batched version of stochastic gradient descent for logistic regression as the optimizer to stabilize the size of the dataset the used for training. However, applying this batched version to the dataset was costly since working with a large matrix in numpy is faster than working with $k$ smaller matrices. Therefore, we used the standard Netwon algorithm and tuned the $\\gamma$ manually for the final model.\n\nThe Figure \\ref{fig:logreg} (1) shows logistic regression accuracy for different regularization parameters $\\lambda$. For this experiment, a constant number of iterations $20$ was chosen. The $\\gamma$ parameter equals $\\gamma=10^{-1}$. The chart indicates that the logistic regression does not overfit. Therefore, we have chosen the first in magnitude value of $\\lambda=2\\times 10^{-4}$ which allowed us to use $\\gamma=0.1$ without numerical overflows in loss. The figure \\ref{fig:logreg} (2) indicates the training schedule. Total number of $N=70$ iterations allowed us to bring the training error to 0. The solution has scored 0.82058 on Kaggle \\cite{kaggle} (first part of test).\n\nDespite the fact that the gradient at the final point was close to $0$ compared to the previous values, the value of the loss did not equal to $0$. Moreover, it can be seen that the train and test errors were similar. These two facts indicate that linear model has a high bias and low variance when applied to this dataset.\n\n\\begin{figure}[!htb]\n\t\\centering \\includegraphics[width=110px]{logistic_lambda_cv_netwon_nobatch}\n\t\\centering \\includegraphics[width=110px]{logistic_01}\n%\t\\centering \\includegraphics[width=200px]{logistic_02}\n\t\n\t\\caption{Degree 5 basis, Logistic regression accuracy}\n\t\\label{fig:logreg}\n\\end{figure}\n\n\\section{Results}\nWe have presented an application of linear, ridge and logistic regression to the dataset. The best model was the logistic one with best accuracy of 0.82 trained with Newton's method. All of the linear models do not overfit on the dataset even when using a polynomial basis.\n\\section{Discussion}\nDespite thorough analysis, our project lacks further consideration in the following directions. First, it can be seen that some of the features do not look like the Gaussian distribution. It might prove beneficial to split such features into two using a certain threshold. Moreover, it might be useful to apply bagging techniques, as well as to use more sophisticated nonlinear classifiers, such as neural networks.\n\\section{Summary}\nWe have shown that it is possible to detect the Higgs boson using linear methods and feature augmentation. Such a simple solution gives fairly good performance and places the team above average on the Kaggle submission. However, this approach is not optimal in terms of model selection, since a linear model seem to not capture all of the specifics of data since it produces high bias.\n\\begin{thebibliography}{99}\n\\bibitem{higgs} \\href{https://en.wikipedia.org/wiki/Higgs\\_boson}{en.wikipedia.org/wiki/Higgs\\_boson}\n\\bibitem{data} \\href{http://higgsml.lal.in2p3.fr/files/2014/04/documentation\\_v1.8.pdf}{Experiment and feature description}\n\\bibitem{kaggle} \\href{http://kaggle.com/c/epfml-higgs}{EPFL ML Project 1}\n\\end{thebibliography}\n\n\\end{document}\n", "meta": {"hexsha": "b925a4980ada09ad98efc6252d9ce12126d02b4e", "size": 10411, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "report/report.tex", "max_stars_repo_name": "EPFLMachineLearningTeamYoor/Project1", "max_stars_repo_head_hexsha": "a0558be0d1ff39424bcd33184fc4e3f64292f602", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "report/report.tex", "max_issues_repo_name": "EPFLMachineLearningTeamYoor/Project1", "max_issues_repo_head_hexsha": "a0558be0d1ff39424bcd33184fc4e3f64292f602", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "report/report.tex", "max_forks_repo_name": "EPFLMachineLearningTeamYoor/Project1", "max_forks_repo_head_hexsha": "a0558be0d1ff39424bcd33184fc4e3f64292f602", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 85.3360655738, "max_line_length": 885, "alphanum_fraction": 0.7778311401, "num_tokens": 2642, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.34722652274188975}} {"text": "\\lab{GeoPandas}{GeoPandas}\n\n\n\\objective{GeoPandas is a package designed to organize and manipulate geographic data, It combines the data manipulation tools of pandas with the geometric capabilities of the Shapely package. In this lab, we explore the basic data structures of GeoSeries and GeoDataFrames and their functionalities.}\n\n\n\\section*{Installation} %============================\nGeoPandas is a new package designed to combine the functionality of pandas with Shapely, a package used for geometric manipulation.\nUsing GeoPandas with geographic data is very useful as it allows the user to not only compare numerical data, but also geometric attributes.\nGeoPandas can be installed via \\li{pip}:\n\\begin{lstlisting}\n>>> pip install geopandas\n\\end{lstlisting}\nHowever, Geopandas can be notoriously difficult to install.\n%It is very finicky, which is likely a by-product of it still being in development\nThis is especially the case if the python environment is not carefully maintained.\nSome of its dependencies can also be very difficult to install on certain systems.\nBecause of this, using Colab for this lab is recommended; its environment is set up in a way that makes GeoPandas very easy to install.\nOtherwise, the GeoPandas documentation contains some additional options that can be used if installation difficulties occur: \\url{https://geopandas.org/install.html}.\n\n\\section*{GeoSeries} %============================\n\nA GeoSeries is a pandas Series where each entry is a set of geometric objects.\nThere are three classes of geometric objects inherited from the Shapely package:\n\\begin{enumerate}\n\\item Points / Multi-Points\n\\item Lines / Multi-Lines\n\\item Polygons / Multi-Polygons\n\\end{enumerate}\nA point is used to identify objects like coordinates, where there is one small instance of the object.\nA line could be used to describe objects such as roads.\nA polygon could be used to identify regions, such as a country.\nMultipoints, multilines, and multipolygons contain lists of points, lines, and polygons, respectively.\n\nSince each object in the GeoSeries is also a Shapely object, the GeoSeries inherits many methods and attributes of Shapely objects.\nSome of the key attributes and methods are listed in Table \\ref{shapely-methods}.\nThese attributes and methods can be used to calculate distances, find the sizes of countries, and determine whether coordinates are within country's boundaries.\nThe example below uses the attribute \\li{bounds} to find the maximum and minimum coordinates of Egypt in a built-in GeoDataFrame.\n\n\\begin{table}[ht]\n\\begin{center}\n\\begin{tabular}{c|c}\nMethod/Attribute & Description\\\\\n\\hline\n\\li{distance(other)}&returns minimum distance from GeoSeries to \\li{other}\\\\\n\\li{contains(other)}&returns \\li{True} if shape contains \\li{other}\\\\\n\\li{intersects(other)}&returns \\li{True} if shape intersects \\li{other}\\\\\n\\li{area}&returns shape area\\\\\n\\li{convex_hull}&returns convex shape around all points in the object\\\\\n\\li{bounds} & returns the bounding x- and y-coordinates of the object\n\\end{tabular}\n\\end{center}\n\\caption{Attributes and Methods for GeoSeries}\n\\label{shapely-methods}\n\\end{table}\n\n\\begin{lstlisting}\n>>> import geopandas as gpd\n>>> world = gpd.read_file(gpd.datasets.get_path('naturalearth_lowres'))\n# Get GeoSeries for Egypt\n>>> egypt = world[world['name']=='Egypt']\n\n# Find bounds of Egypt\n>>> egypt.bounds\n minx miny maxx maxy\n47 24.70007 22.0 36.86623 31.58568\n\\end{lstlisting}\n\n\\section*{Creating GeoDataFrames} %============================\n\nThe main structure used in GeoPandas is a GeoDataFrame, which is similar to a pandas DataFrame.\nA GeoDataFrame has one special column called \\li{geometry}, which must be a GeoSeries.\nThis GeoSeries column is used when a spatial method, like \\li{distance()}, is used on the GeoDataFrame.\n\nA GeoDataFrame can be made from a pandas DataFrame.\nAt least one of the columns in the DataFrame should contain geometric information.\nThis column containing geometric information can be converted to a GeoSeries using the \\li{apply()} method.\nAt this point, the Pandas DataFrame can be cast as a GeoDataFrame.\nAssign which column will be the \\li{geometry} using either the \\li{geometry} keyword in the constructor or the \\li{set_geometry()} method afterwards.\n\n\\begin{lstlisting}\n>>> import pandas as pd\n>>> import geopandas as gpd\n>>> from shapely.geometry import Point, Polygon\n\n# Create a Pandas DataFrame\n>>> df = pd.DataFrame({'City': ['Seoul', 'Lima', 'Johannesburg'],\n... 'Country': ['South Korea', 'Peru', 'South Africa'],\n... 'Latitude': [37.57, -12.05, -26.20],\n... 'Longitude': [126.98, -77.04, 28.04]})\n\n# Create geometry column\n>>> df['Coordinates'] = list(zip(df.Longitude, df.Latitude))\n\n# Make geometry column Shapely objects\n>>> df['Coordinates'] = df['Coordinates'].apply(Point)\n\n# Cast as GeoDataFrame\n>>> gdf = gpd.GeoDataFrame(df, geometry='Coordinates')\n\n# Equivalently, specify the geometry after construction\n# Note that set_geometry() returns a new GeoDataFrame\n>>> gdf = gpd.GeoDataFrame(df)\n>>> gdf = gdf.set_geometry('Coordinates')\n\n# Display the GeoDataFrame\n>>> gdf\n City Country Latitude Longitude Coordinates\n0 Seoul South Korea 37.57 126.98 POINT (126.98000 37.57000)\n1 Lima Peru -12.05 -77.04 POINT (-77.04000 -12.05000)\n2 Johannesburg South Africa -26.20 28.04 POINT (28.04000 -26.20000)\n\n# Create a polygon with all three cities as points\n>>> city_polygon = Polygon(list(zip(df.Longitude, df.Latitude)))\n\\end{lstlisting}\n\nA \\li{GeoDataFrame} can also be made directly from a dictionary.\nIf the dictionary already contains geometric objects, the corresponding column can be directly set as the \\li{geometry} in the constructor.\nOtherwise, a column containing geometry data can be created as in the above example and then set as the \\li{geometry} with the \\li{set_geometry()} method.\n\n\\begin{lstlisting}\n# Both of these methods create the same GeoDataFrame as above\n# Directly create the GeoDataFrame from the dictionary\n>>> gdf = gpd.GeoDataFrame({'City': ['Seoul', 'Lima', 'Johannesburg'],\n... 'Country': ['South Korea', 'Peru', 'South Africa'],\n... 'Latitude': [37.57, -12.05, -26.20],\n... 'Longitude': [126.98, -77.04, 28.04]})\n# Create geometry column and set as the geometry\n>>> gdf['Coordinates'] = list(zip(df.Longitude, df.Latitude))\n>>> gdf['Coordinates'] = df['Coordinates'].apply(Point)\n# inplace=True modifies gdf itself rather than returning a copy\n>>> gdf.set_geometry('Coordinates', inplace=True)\n\n# Equivalently, using a dictionary that already contains geometry objects\n>>> gdf = gpd.GeoDataFrame({'City': ['Seoul', 'Lima', 'Johannesburg'],\n... 'Country': ['South Korea', 'Peru', 'South Africa'],\n... 'Coordinates': [Point(126.98,37.57),\n... Point(-77.04,-12.05), Point(28.04,-12.05)]},\n... geometry='Coordinates')\n\\end{lstlisting}\n\n\\begin{info}\nLongitude is the angular measurement starting at the Prime Meridian, $0\\degree$, and going to $180\\degree$ to the east and $-180\\degree$ to the west.\nLatitude is the angle between the equatorial plane and the normal line at a given point; a point along the Equator has latitude $0$, the North Pole has latitude $+90\\degree$ or $90\\degree N$, and the South Pole has latitude $-90\\degree$ or $90\\degree S$.\n\\end{info}\n\n\\section*{Plotting GeoDataFrames} %============================\n\nInformation from a GeoDataFrame is plotted based on the geometry column.\nData points are displayed as geometry objects.\nThe following example plots the shapes in the \\li{world} GeoDataFrame.\n\n\\begin{lstlisting}\n# Plot world GeoDataFrame\n>>> world.plot()\n\\end{lstlisting}\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=.55]{figures/worldmap.pdf}\n\\end{center}\n\\label{world-map}\n\\caption{World map}\n\\end{figure}\n\nMultiple GeoDataFrames can be plotted at once.\nThis can be done by by setting one GeoDataFrame as the base of the plot and ensuring that each layer uses the same axes.\nIn the following example, the file \\li{airports.csv}, containing the coordinates of world airports, is loaded into a GeoDataFrame and plotted on top of the boundary of the \\li{world} GeoDataFrame.\n\n\\begin{lstlisting}\n# Set outline of world countries as base\n>>> fig,ax = plt.subplots(figsize=(10,7), ncols=1, nrows=1)\n>>> base = world.boundary.plot(edgecolor='black', ax=ax, linewidth=1)\n\n# Load airport data and convert to a GeoDataFrame\n>>> airports = pd.read_csv('airports.csv')\n>>> airports['Coordinates'] = list(zip(airports.Longitude, airports.Latitude))\n>>> airports['Coordinates'] = airports.Coordinates.apply(Point)\n>>> airports = gpd.GeoDataFrame(airports, geometry='Coordinates')\n\n# Plot airports on top of world map\n>>> airports.plot(ax=base, marker='o', color='green', markersize=1)\n>>> ax.set_xlabel('Longitude')\n>>> ax.set_ylabel('Latitude')\n>>> ax.set_title('World Airports')\n\\end{lstlisting}\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=.6]{figures/airports.pdf}\n\\end{center}\n\\label{airport-map}\n\\caption{Airport map}\n\\end{figure}\n\n\\begin{problem}\nRead in the file \\li{airports.csv} as a pandas DataFrame.\nCreate three convex hulls around the three sets of airports listed below.\nThis can be done by passing in lists of the airports' coordinates to a \\li{shapely.geometry.Polygon} object.\n\nCreate a new GeoDataFrame with these three Polygons as entries.\nPlot this GeoDataFrame on top of an outlined world map.\n\\begin{itemize} %These could also be listed by 4-character ICAO codes, but listing the names makes the problem more exciting/realistic. Probably.\n\t\\item Maio Airport, Scatsta Airport, Stokmarknes Skagen Airport, Bekily Airport, K. D. Matanzima Airport, RAF Ascension Island\n\t\\item Oiapoque Airport, Maio Airport, Zhezkazgan Airport, Walton Airport, RAF Ascension Island, Usiminas Airport, Piloto Osvaldo Marques Dias Airport\n\t\\item Zhezkazgan Airport, Khanty Mansiysk Airport, Novy Urengoy Airport, Kalay Airport, Biju Patnaik Airport, Walton Airport\n\\end{itemize}\n\\end{problem}\n\n\\section*{Working with GeoDataFrames} %============================\n\nAs previously mentioned, GeoDataFrames contain many of the functionalities of pandas DataFrames.\nFor example, to create a new column, define a new column name in the GeoDataFrame with the needed information for each GeoSeries.\n\n\\begin{lstlisting}\n# Create column in the world GeoDataFrame for gdp_per_capita\n>>> world['gdp_per_cap'] = world.gdp_md_est / world.pop_est\n\\end{lstlisting}\n\nGeoDataFrames can utilize many pandas functionalities, and they can also be parsed by geometric manipulations.\nFor example, a useful way to index GeoDataFrames is with the \\li{cx} indexer.\nThis splits the GeoDataFrame by the coordinates of each geometric object.\nIt is used by calling the method \\li{cx} on a GeoDataFrame, followed by a slicing argument, where the first element refers to the longitude and the second refers to latitude.\n\n\\begin{lstlisting}\n# Create a GeoDataFrame containing the northern hemisphere\n>>> north = world.cx[:, 0:]\n\n# Create a GeoDataFrame containing the southeastern hemisphere\n>>> south_east = world.cx[0:, :0]\n\\end{lstlisting}\n\nGeoSeries objects in a GeoDataFrame can also be dissolved, or merged, together into one GeoSeries based on their geometry data.\nFor example, all countries on one continent could be merged to create a GeoSeries containing the information of that continent.\nThe method designed for this is called \\li{dissolve}.\nIt receives two parameters, \\li{by} and \\li{aggfunc}.\n\\li{by} indicates which column to dissolve along, and \\li{aggfunc} tells how to combine the information in all other columns.\nThe default \\li{aggfunc} is \\li{first}, which returns the first application entry.\nIn the following example, we use \\li{sum} as the \\li{aggfunc} so that each continent is the combination of its countries.\n\n\\begin{lstlisting}\n>>> world = world[['continent', 'geometry', 'gdp_per_cap']]\n\n# Dissolve world GeoDataFrame by continent\n>>> continent = world.dissolve(by = 'continent', aggfunc='sum')\n\\end{lstlisting}\n\n\n\\section*{Projections and Coloring} %============================\n\nWhen plotting, GeoPandas uses the CRS (coordinate reference system) of a GeoDataFrame.\nThis reference system indicates how coordinates should be spaced on a plot.\nTwo of the most commonly used CRSs are EPSG:4326 and EPSG:3395.\nEPSG:4326 is the standard latitude-longitude projection used by GPS.\nEPSG:3395, also known as the Mercator projection, is the standard navigational projection.\n\nWhen creating a new GeoDataFrame, it is important to set the \\li{crs} attribute of the GeoDataFrame.\nThis allows any plots to be shown correctly.\nFurthermore, GeoDataFrames being layered need to have the same CRS.\nTo change the CRS, use the method \\li{to_crs()}.\n\n\\begin{lstlisting}\n# Check CRS of world GeoDataFrame\n>>> print(world.crs)\nepsg:4326\n\n# Change CRS of world to Mercator\n# inplace=True ensures that we modify world instead of returning a copy\n>>> world.to_crs(3395, inplace=True)\n>>> print(world.crs)\nepsg:3395\n\\end{lstlisting}\n\nGeoPandas accepts many different CRSs; a reference can be found at \\url{www.spatialreference.org}.\n% I think this is helpful information, but I'm not sure\nAdditionally, inspecting a given CRS object in the terminal without using \\li{print()} or \\li{str()} can be used to get additional information about a specific CRS:\\footnote{This can also be accomplished using \\li{print(repr(crs))}.}\n\\begin{lstlisting}[mathescape]\n>>> world.crs\n\nName: WGS 84 / World Mercator\nAxis Info [cartesian]:\n- E[east]: Easting (metre)\n- N[north]: Northing (metre)\nArea of Use:\n- name: World between 80$^\\circ$S and 84$^\\circ$N.\n- bounds: (-180.0, -80.0, 180.0, 84.0)\nCoordinate Operation:\n- name: World Mercator\n- method: Mercator (variant A)\nDatum: World Geodetic System 1984\n- Ellipsoid: WGS 84\n- Prime Meridian: Greenwich\n\\end{lstlisting}\n\nGeoDataFrames can also be plotted using the values in the the other attributes of the GeoSeries.\nThe map plots the color of each geometry object according to the value of the column selected.\nThis is done by passing in the parameter \\li{column} into the \\li{plot()} method.\n\n\\begin{lstlisting}\n>>> fig, ax = plt.subplots(1, figsize=(10,4))\n# Plot world based on gdp\n>>> world.plot(column='gdp_md_est', cmap='OrRd', legend=True, ax=ax)\n>>> ax.set_title('World Map based on GDP')\n>>> ax.set_xlabel('Longitude')\n>>> ax.set_ylabel('Latitude')\n>>> plt.show()\n\\end{lstlisting}\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=.7]{figures/gdp.pdf}\n\\end{center}\n\\caption{World Map Based on GDP}\n\\label{figure:gdp_map}\n\\end{figure}\n\n\n\\begin{problem}\nThe file \\li{county_data.gpkg.zip} contains information about US counties.\\footnote{Source: \\url{http://www2.census.gov/geo/tiger/GENZ2016/shp/cb_2016_us_county_5m.zip}}\nAfter unzipping, use the command \\li{gpd.read_file('county_data.gpkg')} to create a GeoDataFrame of this information.\nEach county's shape is stored in the \\li{geometry} column.\nUse this to plot all US counties two times, first using the default CRS and then using EPSG:5071.\n\nNext, create a new GeoDataFrame that merges all counties within a single state.\nDrop regions with the following STATEFP codes: 02, 15, 60, 66, 69, 72, 78.\nPlot this GeoDataFrame to see an outline of all 48 contiguous states.\nEnsure a CRS of EPSG:5071.\n\\label{problem:us_counties}\n\\end{problem}\n\\begin{info}\n\\li{.gpkg} files are actually structured as a directory that contains several files that each contain parts of the data.\nFor instance, \\li{county_data.gpkg} consists of the files \\li{county_data.cpg}, \\li{county_data.dbf}, \\li{county_data.prj}, \\li{county_data.shp}, and \\li{county_data.shx}.\nBe sure that these files are placed directly in the first level of folders, and not in further subdirectories.\\\\\nTo use this file in Google Colab, upload the zipped file and extract it with the following code:\n\\begin{lstlisting}\ncounty = files.upload()\n!unzip county_data.gpkg.zip\n\\end{lstlisting}\nIt then can be loaded:\n\\begin{lstlisting}\ncounty_df = gpd.read_file('county_data.gpkg')\n\\end{lstlisting}\n\\end{info}\n\n\\section*{Merging GeoDataFrames} %============================\n\nJust as multiple pandas DataFrames can be merged, multiple GeoDataFrames can be merged with attribute joins or spatial joins.\nAn attribute join is similar to a merge in pandas.\nIt combines two GeoDataFrames on a column (not the geometry column) and then combines the rest of the data into one GeoDataFrame.\n\n\\begin{lstlisting}\n>>> world = gpd.read_file(geopandas.datasets.get_path('naturalearth_lowres'))\n>>> cities = gpd.read_file(geopandas.datasets.get_path('naturalearth_cities'))\n\n# Create subsets of the world and cities GeoDataFrames\n>>> world = world[['continent', 'name', 'iso_a3']]\n>>> cities = cities[['name', 'iso_a3']]\n\n# Merge the GeoDataFrames on their iso_a3 code\n>>> countries = world.merge(cities, on='iso_a3')\n\\end{lstlisting}\n\nA spatial join merges two GeoDataFrames based on their geometry data.\nThe function used for this is \\li{sjoin}.\n\\li{sjoin} accepts two GeoDataFrames and then direction on how to merge.\nIt is imperative that two GeoDataFrames have the same CRS.\nIn the example below, we merge using an \\li{inner} join with the option \\li{intersects}.\nThe \\li{inner} join means that we will only use keys in the intersection of both geometry columns, and we will retain only the left geometry column.\n\\li{intersects} tells the GeoDataFrames to merge on GeoSeries that intersect each other.\nOther options include \\li{contains} and \\li{within}.\n\n\\begin{lstlisting}\n# Combine countries and cities on their geographic location\n>>> countries = gpd.sjoin(world, cities, how='inner', op='intersects')\n\\end{lstlisting}\n\n\\begin{problem}\nLoad in the file \\li{nytimes.csv}\\footnote{Source: \\url{https://raw.githubusercontent.com/nytimes/covid-19-data/master/us-counties.csv}} as a DataFrame.\nThis file includes county-level data for the cumulative cases and deaths of Covid-19 in the US, starting with the first case in Snohomish County, Washington, on January 21, 2020.\nBegin by converting the \\li{date} column into a \\li{DatetimeIndex}.\n\nNext, use county FIPS codes to merge your GeoDataFrame from Problem \\ref{problem:us_counties} with the DataFrame you just created.\nA FIPS code is a 5-digit unique identifier for geographic locations.\nIgnore rows in the Covid-19 DataFrame with unknown FIPS codes as well as all data from Hawaii and Alaska.\n\nNote that the \\li{fips} column of the Covid-19 DataFrame stores entries as floats, but the county GeoDataFrame stores FIPS codes as strings, with the first two digits in the \\li{STATEFP} column and the last three in the \\li{COUNTYFP} column.\n\nOnce you have completed the merge, plot the cases from March 21, 2020 on top of your state outline map from Problem \\ref{problem:us_counties}, using the CRS of EPSG:5071.\nFinally, print out the name of the county with the most cases on March 21, 2020 along with its case count.\n\\label{problem:merge_covid}\n\\end{problem}\n\n\\section*{Logarithmic Plotting Techniques} %============================\n\nThe color scheme of a graph can also help to communicate information clearly.\nA good list of available colormaps can be found at \\url{https://matplotlib.org/3.2.1/gallery/color/colormap_reference.html}.\nNote also that you can reverse any colormap by adding \\li{_r} to the end.\nThe following example demonstrates some plotting features, using country GDP as in Figure \\ref{figure:gdp_map}.\n\n\\begin{lstlisting}\n>>> fig, ax = plt.subplots(figsize=(15,7), ncols=1, nrows=1)\n>>> world.plot(column='gdp_md_est', cmap='plasma_r',\n...\t\t\tax=ax, legend=True, edgecolor='gray')\n\n# Add title and remove axis tick marks\n>>> ax.set_title('GDP on Linear Scale')\n>>> ax.set_yticks([])\n>>> ax.set_xticks([])\n>>> plt.show()\n\\end{lstlisting}\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=.75]{figures/world_linear.pdf}\n\\end{center}\n\\caption{World map showing country GDP}\n\\label{figure:linear_world}\n\\end{figure}\n\nSometimes data can be much more informative when plotted on a logarithmic scale.\nSee how the world map changes when we add a \\li{norm} argument in the code below.\nDepending on the purpose of the graph, Figure \\ref{figure:log_world} may be more informative than Figure \\ref{figure:linear_world}.\n\n\\begin{lstlisting}\n>>> from matplotlib.colors import LogNorm\n>>> from matplotlib.cm import ScalarMappable\n>>> fig, ax = plt.subplots(figsize=(15,6), ncols=1, nrows=1)\n\n# Set the norm using data bounds\n>>> data = world.gdp_md_est\n>>> norm = LogNorm(vmin=min(data), vmax=max(data))\n\n# Plot the graph using the norm\n>>> world.plot(column='gdp_md_est', cmap='plasma_r', ax=ax,\n...\t\t\tedgecolor='gray', norm=norm)\n\n# Create a custom colorbar\n>>> cbar = fig.colorbar(ScalarMappable(norm=norm, cmap='plasma_r'),\n...\t\t\tax=ax, orientation='horizontal', pad=0, label='GDP')\n\n>>> ax.set_title('Country Area on a Log Scale')\n>>> ax.set_yticks([])\n>>> ax.set_xticks([])\n>>> plt.show()\n\\end{lstlisting}\n\n\\begin{figure}[H]\n\\begin{center}\n\\includegraphics[scale=.65]{figures/world_log.pdf}\n\\end{center}\n\\caption{World map showing country GDP using a log scale}\n\\label{figure:log_world}\n\\end{figure}\n\n\\begin{problem}\nAs in Problem \\ref{problem:merge_covid}, plot your state outline map from Problem \\ref{problem:us_counties} on top of a map of the Covid-19 cases from March 21, 2020.\nThis time, however, use a log scale.\nUse EPSG:5071 for the CRS.\nPick a good colormap (the counties with the most cases should generally be darkest) and be sure to display a colorbar.\n\\end{problem}\n\n\\begin{problem}\nIn this problem, you will create an animation of the spread of Covid-19 through US counties from January 21, 2020 to June 21, 2020.\nUse a log scale and a good colormap, and be sure that you're using the same norm and colorbar for the whole animation.\nUse EPSG:5071 for the projection.\n\nAs a reminder, below is a summary of what you will need in order to animate this map.\nYou may also find it helpful to refer to the animation section included with the Volume 4 lab manual.\n\n%------------------------------------------- Is that ok to say?\n\\begin{enumerate}\n\t\\item Set up your figure and norm. Be sure to use the highest case count for your \\li{vmax} so that the scale remains uniform.\n\t\\item Write your \\li{update} function. This should plot the cases from a given day.\n\t\\item Set up your colorbar. Do this outside the \\li{update} function to avoid adding a new colorbar each day.\n\t\\item Create the animation and embed it.\n\\end{enumerate}\n\\end{problem}\n", "meta": {"hexsha": "3394f2b3103f71caf8365c937e7e1d756387ba24", "size": 22669, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "DataScienceEssentials/Pandas4/Pandas4.tex", "max_stars_repo_name": "chrismmuir/Labs-1", "max_stars_repo_head_hexsha": "13c23611b90d73b0c2c7d275bce9808f829009f2", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": 190, "max_stars_repo_stars_event_min_datetime": "2015-07-17T01:57:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-15T19:16:19.000Z", "max_issues_repo_path": "DataScienceEssentials/Pandas4/Pandas4.tex", "max_issues_repo_name": "chrismmuir/Labs-1", "max_issues_repo_head_hexsha": "13c23611b90d73b0c2c7d275bce9808f829009f2", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": 184, "max_issues_repo_issues_event_min_datetime": "2015-07-16T17:56:06.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-06T23:47:14.000Z", "max_forks_repo_path": "DataScienceEssentials/Pandas4/Pandas4.tex", "max_forks_repo_name": "chrismmuir/Labs-1", "max_forks_repo_head_hexsha": "13c23611b90d73b0c2c7d275bce9808f829009f2", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": 76, "max_forks_repo_forks_event_min_datetime": "2015-08-06T02:53:11.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-27T11:08:57.000Z", "avg_line_length": 47.4246861925, "max_line_length": 301, "alphanum_fraction": 0.7432617231, "num_tokens": 5739, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5273165382362518, "lm_q2_score": 0.6584175072643413, "lm_q1q2_score": 0.34719444064477467}} {"text": "%!TEX root = thesis-kdyoung.tex\n\n\\chapter{Computational Experiments}\n\\label{chap:exp}\n\nAll experiments were run on a personal computer with\nan Intel i5 5575R CPU (four cores each with 2.8GHz clock speed) and 16GB of main memory.\nThe scripts for the MIPs and the primary script used \nto iterate through the Benders\ndecomposition algorithm was written in Python 3.4.\nWe employed Gurobi 7.0.2 as the solver\nused for all MIP formulations.\nAs mentioned previously, \\chuffed was the solver\nwe chose to solve all constraint programs with.\nMiniZinc version 2.1.2 was used to compile the models\ninto the \\chuffed FlatZinc format.\nUnless stated otherwise, all tests were run\nwith a time limit of 30 minutes (1800 seconds).\n\nWe chose to only test \\fsbf{2} and \\scbf{2} for the\ncomputational experiments of the previous work.\nThis is a result of our encoding of the \\ssbf{2} model\nreturning infeasible solutions in our preliminary tests.\nDue to time constraints we were unable to discern whether\nthe problem was our encoding of the model\nor if the \\ssbf{2} is ill-formed.\n\nWe now provide a brief outline of the computational\nresults reported in this chapter.\nFirst a description of the data used for testing\nis given in Section \\ref{sec:exp:data}.\nAs the MIPs presented in \\authciteb{Esmaeilbeigi2016}\nhave been untested, we investigate the best combination\nof valid inequalities to include for these models\nin Section \\ref{sec:exp:mip}.\nIn Section \\ref{sec:exp:cuts} we report on the best combination\nof Benders cuts to use for our decomposition.\nWe provide a detailed investigation into the best configuration\nof the CP sub-problem in Section \\ref{sec:exp:cp}.\nOnce the superior sub-problem formulation is determined, \nwe close in Section \\ref{sec:exp:benchmarkExp} of\nthis chapter by reporting on the results of our best\nsolution methodology on a set of benchmark instances.\n\n\\section{Data}\n\\label{sec:exp:data}\nThe primary dataset available for the \\sua{} is\nthe SBF generated by \\citeauthor{Scholl2013}\nHowever, this dataset was created for the type-1 case in\nparticular and so is not immediately applicable to\nthe \\sua{2}.\nThe SBF is divided into two halves, the SBF1 and SBF2, \nwhere each half consists of a total of 1076 instances.\nWe first provide a brief explanation of how SBF1\nwas generated.\n\nThe basis of the SBF1 are a set of 269 instances\nfor the \\sab{1} containing between 4 and 297 tasks.\nWhen creating instances for the \\sua{1}, the cycle time,\ntask times and precedence relations are all transferred from\nthe original instances.\nWe also require values of the forward and backward setup times \\ap \nso to create these the authors consider the average processing\ntime among all tasks, denoted $t_{av}$.\nIn practice the setup times are smaller than the execution time\nof a task, so a parameter $\\alpha$ is used when randomly generating\nthe setups.\nAn upper bound on the possible setup is defined as $\\alpha\\cdot t_{av}$\nwhere $\\alpha\\in\\{\\:0.25,\\:0.50,\\:0.75,\\:1.00\\:\\}$.\nFor example, when $\\alpha=1.00$, the largest possible setup time\nis 50\\% of the average task time.\nThe 269 input instances are duplicated for each $\\alpha$ value\nand so 1076 instances of \\sua{1} are generated for SBF1.\n\nThe SBF2 was generated in almost exactly the same way as SBF1.\nHowever in this case,\ncare was taken when randomly generating the setup times to ensure\nthat at least one optimal solution to the original \\sab{1} instance\nis still feasible in the new \\sua{1} instance.\nThis solution will necessarily be the optimal solution\nof the \\sua{2} as noted by \\citeauthor{Scholl2013}\nWhen storing the instance data, the authors also recorded the\noptimal number of stations for the original \\sab{1} instance.\n\nThe data we chose to run all our tests on was adapted from the SBF2\ndataset.\nAs the optimal solutions were recorded,\nwe could take the optimal number of stations as input\nand then solve for the cycle time.\nThis simple change allowed 1076 instances for testing our formulations.\n\\authciteb{Esmaeilbeigi2016} considered a subset of the SBF2 instances\nwhich they divided into 3 classes.\nWe considered the same subset of instances for the type-2 problem\nwhich are detailed in Table \\ref{tab:exp:dataSBF2}.\nIn this table we give the number of tasks, stations, precedence\nrelations and order strength of the precedence graph.\nIn total 396 instances were considered in our tests.\n\nThe order strength of an instance is a measure of its complexity\nwhere higher values correspond to more difficult instances.\nHowever, we note that in the extreme, when $OS=1$, there is only one \nprecedence-feasible task sequence and thus the problem becomes trivial.\n\n\\begin{table}[tpb]\n\t\\centering\n\t\\caption{Dataset summary}\n\t\\vspace{2mm}\n\t\\begin{tabular}{llrrrrr}\n\t\t\\toprule\n\t\tClass & Creator & \\#Inst. & $n$ & $m$ & $|E|$ & Order Strength $(OS)$ \\\\\n\t\t\\midrule\\midrule\n\t\t1 & & 108 & 7-21 & 2-8 & 6-27 & \\\\[1mm]\n\t\t & {\\tt mertens} & 6 & 7 & 2-6 & 6 & medium (52.40)\\\\\n\t\t & {\\tt bowman8} & 1 & 8 & 5 & 8 & high (75.00)\\\\\n\t\t & {\\tt jaeschke} & 5 & 9 & 3-8 & 11 & high (83.33)\\\\\n\t\t & {\\tt jackson} & 6 & 11 & 3-8 & 13 & medium (58.18)\\\\\n\t\t & {\\tt mansoor} & 3 & 11 & 2-4 & 11 & medium (60.00)\\\\\n\t\t & {\\tt mitchell} & 6 & 21 & 3-8 & 27 & high (70.95)\\\\\\midrule\n\t\t2 & & 112 & 25-30 & 3-14 & 32-40 & \\\\[1mm]\n\t\t & {\\tt roszieg} & 6 & 25 & 4-10 & 32 & high (71.67)\\\\\n\t\t & {\\tt heskia} & 6 & 28 & 3-8 & 40 & low (22.49)\\\\\n\t\t & {\\tt buxey} & 7 & 29 & 7-13 & 36 & medium (50.74)\\\\\n\t\t & {\\tt sawyer30} & 9 & 30 & 5-14 & 32 & low (44.83)\\\\\\midrule\n\t\t3 & & 176 & 32-58 & 3-31 & 38-82 & \\\\[1mm]\n\t\t & {\\tt lutz1} & 6 & 32 & 6-11 & 38 & high (83.47)\\\\\n\t\t & {\\tt gunther} & 7 & 35 & 7-14 & 43 & medium (59.50)\\\\\n\t\t & {\\tt kilbrid} & 10 & 45 & 3-10 & 62 & low (44.60)\\\\\n\t\t & {\\tt hahn} & 5 & 53 & 4-8 & 82 & high (83.82)\\\\\n\t\t & {\\tt warnecke} & 16 & 58 & 14-31 & 70 & medium (59.10)\\\\\\midrule\n\t\tOverall & & 396 & 7-58 & 2-31 & 6-82 & \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:dataSBF2}\n\\end{table}\n\n% \\section{Preliminary Tests}\n% \\label{sec:exp:prelim}\n\n\\section{MIP Formulations}\n\\label{sec:exp:mip}\nWe first report on how the MIP formulations presented\nin Chapter \\ref{chap:mip} performed.\nIn Table \\ref{tab:exp:resultsFSBF} and \\ref{tab:exp:resultsSCBF} we give the results\nof \\fsbf{2} and \\scbf{2} respectively.\nEach is tested on each combination of the valid inequalities from\nSection \\ref{sec:mip:validIneqs} against\nall 3 classes of the subset\nof instances from SBF2.\nThe following datapoints are summarized for each test:\nthe number of nodes explored by \\gurobi's \\bab tree,\nthe average gap for all instances,\nthe number of instances where no feasible solution was found,\nthe fraction of instances found to optimality out of the whole class,\nthe percentage of optimal instances and finally\nthe average runtime taken by \\gurobi.\n\n\\begin{table}[tpb]\n\t\\caption{FSBF-2 formulations tested on classes 1,2 and 3}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{clrrrrrr}\n\t\t\\toprule\n\t\tClass & Ineqs. & \\#Nodes & \\%Gap & \\#No.s. & \\#Opt. & \\%Opt. & Rt.(s) \\\\\\midrule\\midrule\n\t\t1 & -- & 40,181 & 0.56 & 0 & 98/108 & 90.74 & 172.90 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}) & 43,437 & 0.16 & 0 & \\bf{101}/108 & 93.52 & 166.57 \\\\\n\t\t & (\\ref{eq:mip:valIneq3}) & 50,987 & 1.14 & 0 & 84/108 & 77.78 & 628.47 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}),(\\ref{eq:mip:valIneq3}) & 51,734 & 1.12 & 0 & 85/108 & 78.73 & 645.02 \\\\\\midrule\n\t\t2 & -- & 171,847 & 14.81 & 0 & 0/112 & 0.00 & 1801.08 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}) & 168,899 & 13.76 & 0 & 0/112 & 0.00 & 1800.66 \\\\\n\t\t & (\\ref{eq:mip:valIneq3}) & 205,752 & 18.04 & 0 & 0/112 & 0.00 & 1801.52 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}),(\\ref{eq:mip:valIneq3}) & 202,980 & 17.90 & 0 & 0/112 & 0.00 & 1801.27 \\\\\\midrule\n\t\t3 & -- & 48,927 & 19.75 & 9 & 7/176 & 3.98 & 1763.45 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}) & 46,060 & 19.56 & 9 & 7/176 & 3.98 & 1759.67 \\\\\n\t\t & (\\ref{eq:mip:valIneq3}) & 62,845 & 25.64 & 16 & 0/176 & 0.00 & 1800.23 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}),(\\ref{eq:mip:valIneq3}) & 63,002 & 24.72 & 15 & 0/176 & 0.00 & 1801.12 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsFSBF}\n\\end{table}\n\n\\begin{table}[tpb]\n\t\\caption{SCBF-2 formulations tested on classes 1,2 and 3}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{clrrrrrr}\n\t\t\\toprule\n\t\tClass & Ineqs. & \\#Nodes & \\%Gap & \\#No.s. & \\#Opt. & \\%Opt. & Rt.(s) \\\\\\midrule\\midrule\n\t\t1 & -- & 221,842 & 1.62 & 0 & \\bf{92}/108 & 86.11 & 374.88 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}) & 244,811 & 1.97 & 0 & 84/108 & 77.78 & 403.17 \\\\\n\t\t & (\\ref{eq:mip:valIneq3}) & 238,515 & 1.75 & 0 & 86/108 & 76.93 & 377.05 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}),(\\ref{eq:mip:valIneq3}) & 215,866 & 1.75 & 0 & 86/108 & 76.93 & 380.82 \\\\\\midrule\n\t\t2 & -- & 651,880 & 18.01 & 28 & 0/112 & 0.00 & 1800.33 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}) & 701,617 & 19.75 & 31 & 0/112 & 0.00 & 1800.47 \\\\\n\t\t & (\\ref{eq:mip:valIneq3}) & 686,566 & 18.98 & 30 & 0/112 & 0.00 & 1801.03 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}),(\\ref{eq:mip:valIneq3}) & 612,442 & 18.57 & 31 & 0/112 & 0.00 & 1800.99 \\\\\\midrule\n\t\t3 & -- & 301,055 & 10.09 & 109 & 0/176 & 0.00 & 1800.88 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}) & 326,284 & 12.20 & 117 & 0/176 & 0.00 & 1801.00 \\\\\n\t\t & (\\ref{eq:mip:valIneq3}) & 324,878 & 12.15 & 112 & 0/176 & 0.00 & 1801.21 \\\\\n\t\t & (\\ref{eq:mip:valIneq1}),(\\ref{eq:mip:valIneq3}) & 319,908 & 12.06 & 117 & 0/176 & 0.00 & 1801.01 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsSCBF}\n\\end{table}\n\nBy inspecting the data we find that using the \nvalid inequality (\\ref{eq:mip:valIneq1})\nwithout (\\ref{eq:mip:valIneq3}) is the best \nformulation of \\fsbf{2}.\nAs for \\scbf{2}, we found that the best formulation\nwas when neither of the two valid inequalities were included in\nthe program.\n\nFor both the MIPs, optimality was proven for almost none of the instances of\nclass 2 and 3.\nAs a result, all the average runtime values are close\nto the time limit of 30 minutes.\nIn the case of \\scbf{2}, no feasible solution was found for 66.5\\% of the instances\nin class 3.\nEven for the smallest class of data, where there is no more\nthan 21 tasks, neither MIP was able to prove\noptimality for all instances.\nWe note that when calculating the gap value, we only average\nthe instances where a feasible solution was found.\nWith a 30 minute time limit, both MIPs could not achieve\nan average gap value less than 10\\% for class 2 and 3.\n\nHere we tested the full mixed-integer programming approach\non the smallest subset of instances of the SBF2.\nThe poor results we have found make it clear that these programs\nare unsatisfactory exact solution methods for tackling\nthe \\sua{2}.\n\n\\section{Cuts}\n\\label{sec:exp:cuts}\nThe cutting strategy used in the iteration of the Benders\nalgorithm can have a large impact on its overall performance.\nEven when either the master or sub-problem is exceptionally\ndifficult to solve, if the Benders cuts added to the master\nallow us to find optimality in very few iterations then the problem\ncan still be tractable.\n\n\\subsection{Nogood Cuts}\n\\label{sec:exp:NGcuts}\nIn the preliminary testing phase, we considered feasibility sub-problems\nusing the nogood cuts presented in Section \\ref{sec:bend:NGcuts} of the \nprevious chapter.\nThese results are briefly summarized in Table \\ref{tab:exp:resultsNGcut}.\nEven on the smallest class 1, the Benders\nalgorithm failed to prove optimality for 66 of the instances tested.\nDue to this, we felt that further investigation into the use\nof nogood cuts was not the most fruitful research direction. \nNote that we did not make use of the possible parallel computing capabilities\ndetailed earlier in Section \\ref{sec:bend:SPfeas},\nwhere the feasibility sub-problems could be run concurrently.\nThis potential avenue was not explored as the percentage of computation\ntime spent solving the sub-problems was dwarfed by the time spent solving\nthe master problem; as we will see in the coming results.\n\nFor all remaining experiments optimality sub-problems were used\nunless stated otherwise.\n\n\\begin{table}[tpb]\n\t\\caption{Benders algorithm tested on class 1 with nogood cuts}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{ccrrrr}\n\t\t\\toprule\n\t\tCuts & Class & \\#No sol. & \\#Opt. & \\%Opt. & Runtime(s) \\\\\\midrule\\midrule\n\t\t$\\mathcal{C}_{ng}$ & 1 & 66 & 42/108 & 38.89 & 1456.57 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsNGcut}\n\\end{table}\n\n\\subsection{Cutting Procedure}\n\\label{sec:exp:cuttngProcedure}\nIn Tables \\ref{tab:exp:resultsCutsClass1} and \\ref{tab:exp:resultsCutsClass2}\nwe provide the primary comparison of the possible cutting strategies\nthat we considered.\nThe first table compares the 9 possible cut combinations on the smallest\nclass of data (class 1).\nThe variation in the results of Table \\ref{tab:exp:resultsCutsClass1},\nmakes it difficult to discern a\nclear best cutting procedure.\nSo we further tested 6 cut combinations\non class 2 in the Table \\ref{tab:exp:resultsCutsClass2}.\nThe choice of solving technology for the sub-problems \nwill only effect the time taken to solve all scheduling problems\nbut not the cutting procedure.\nThus we arbitrarily chose to test all combinations of cuts using MIP\nsub-problems.\n\nThe two tables comparing the cut combinations contain\nthe following data:\naverage number of nodes explored by \\gurobi's\n\\bab tree (total combined between the \\rmp{} and sub-problems),\naverage number iterations of the Benders algorithm,\naverage number of cuts generated,\naverage Benders gap,\ntotal number of instances where no feasible solution was found,\npercentage of the instances proved optimal and\nthe average runtime of all instances.\n\nBy inspecting the results of Table \\ref{tab:exp:resultsCutsClass1}\nit is difficult to reason about which cutting strategy is superior\nas all instances were proved optimal for every combination.\nThese results immediately tell us that the Benders decomposition\nwe present here has outperformed the pure MIP formulations\npreviously presented by the literature.\nWe note that when the global bound in not enforced after\na globally feasible solution is found, the Benders gap\nremains substantially large and so it is difficult\nto measure the convergence of the Benders algorithm.\nThus for the remaining experiments we run, the global bound\nwill always be used unless stated otherwise.\nThe $3^{\\text{rd}}$ version of the inference cut ($\\mathcal{C}_{iii}$) appears to\nbe a key ingredient in the Benders algorithm terminating efficiently,\nbut to confirm this suspicion we test our solution\nmethodology on class 2.\n\n\\begin{table}[tpb]\n\t\\caption{Benders cutting strategies compared on class 1}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{cccccrrrrrrr}\n\t\t\\toprule\n\t\t\\multicolumn{5}{c}{Cuts and Bound} & & & & & & & \\\\\\cmidrule(rl){1-5}\n\t\t$\\mathcal{C}_{i}$ & $\\mathcal{C}_{ii}$ & $\\mathcal{C}_{iii}$ & $\\mathcal{C}_{l}$ & $\\mathcal{C}_{gb}$& \\#Nodes & \\#Its. & $|\\mathcal{C}|$ & \\%Gap & \\#No.s. & \\%Opt. & Rt.(s) \\\\\\midrule\\midrule\n\t\t\\checkmark & & & & & 4,718 & 18 & 35 & 0.00 & 0 & 100.00 & 5.8 \\\\\n\t\t & \\checkmark & & & & 2,845 & 15 & 28 & 0.00 & 0 & 100.00 & 4.1 \\\\\n\t\t & & \\checkmark & & & 170 & 4 & 8 & 0.00 & 0 & 100.00 & 1.1 \\\\\\midrule\n\t\t\\checkmark & & & & \\checkmark & 3,790 & 16 & 32 & 0.00 & 0 & 100.00 & 2.9 \\\\\n\t\t & \\checkmark & & & \\checkmark & 2,064 & 12 & 26 & 0.00 & 0 & 100.00 & 1.9 \\\\\n\t\t & & \\checkmark & & \\checkmark & 128 & 3 & 9 & 0.00 & 0 & 100.00 & 0.4 \\\\\\midrule\n\t\t\\checkmark & & & \\checkmark & \\checkmark & 766 & 17 & 34 & 0.00 & 0 & 100.00 & 2.6 \\\\\n\t\t & \\checkmark & & \\checkmark & \\checkmark & 896 & 17 & 34 & 0.00 & 0 & 100.00 & 2.8 \\\\\n\t\t & & \\checkmark & \\checkmark & \\checkmark & 56 & 4 & 12 & 0.00 & 0 & 100.00 & 0.5 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsCutsClass1}\n\\end{table}\n\nIn Table \\ref{tab:exp:resultsCutsClass2} we can see that\nclass 2 proved more challenging for the Benders decomposition\nto solve the problem as in many cases optimality was\nproven for only $\\sim30\\%$ of the instances.\nThese results make it clear that $\\mathcal{C}_{iii}$\nis the strongest infer cut as it is able to\nsolve almost twice as many instances to optimality\nthat the other 2 infer cuts.\nFurther to this, it needed half as many iterations\nand similarly half as many cuts in total.\nFor the instances where the optimal solution was not found,\n$\\mathcal{C}_{iii}$ provided the lowest Benders gap values\nas well.\n\nWe must also examine whether including the logic cut, $\\mathcal{C}_{l}$,\nprovided a benefit.\nWhen considering the logic cut, a global bound on the cycle time is instituted\nas an upper bound on the load of each station's sub-problem,\npossibly leading to feasibility sub-problems, rather than optimality.\nThe logic cut was formulated to provide the \\rmp{} with detailed information\nabout the assignment variables whilst also reducing the time\nneeded to solve a sub-problem; however it also has a notable drawback.\nWhen a logic cut is generated by a sub-problem, that sub-problem\nhas not been solved to optimality.\nThis results in us not being able to generate an optimality cut for\nthis sub-problem, such as $\\mathcal{C}_{iii}$.\nThe results of Table \\ref{tab:exp:resultsCutsClass2} indicate\nthat on average, not being able to perform an infer cut has not outweighed the beneficial\ninformation provided by the logic cut.\n\n\\begin{table}[tpb]\n\t\\caption{Benders cutting strategies compared on class 2}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{cccccrrrrrrr}\n\t\t\\toprule\n\t\t\\multicolumn{5}{c}{Cuts and Bound} & & & & & & & \\\\\\cmidrule(rl){1-5}\n\t\t$\\mathcal{C}_{i}$ & $\\mathcal{C}_{ii}$ & $\\mathcal{C}_{iii}$ & $\\mathcal{C}_{l}$ & $\\mathcal{C}_{gb}$ & \\#Nodes & \\#Its. & $|\\mathcal{C}|$ & \\%Gap & \\#No.s. & \\%Opt. & Rt.(s) \\\\\\midrule\\midrule\n\t\t\\checkmark & & & & \\checkmark & 1,522k & 99 & 396 & 11.63 & 1 & 31.25 & 1339.7 \\\\\n\t\t & \\checkmark & & & \\checkmark & 1,430k & 99 & 399 & 10.79 & 1 & 32.14 & 1348.5 \\\\\n\t\t & & \\checkmark & & \\checkmark & 1,024k & 47 & 196 & 8.42 & 1 & \\bf{60.71} & 959.9 \\\\\\midrule\n\t\t\\checkmark & & & \\checkmark & \\checkmark & 918k & 112 & 439 & 11.38 & 1 & 33.04 & 1318.4 \\\\\n\t\t & \\checkmark & & \\checkmark & \\checkmark & 964k & 114 & 443 & 11.33 & 1 & 31.25 & 1342.3 \\\\\n\t\t & & \\checkmark & \\checkmark & \\checkmark & 898k & 68 & 225 & 9.03 & 1 & 58.04 & 956.8 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsCutsClass2}\n\\end{table}\n\nHence, for all remaining experiments we used the infer cut $\\mathcal{C}_{iii}$\nin combination with the global bound $\\mathcal{C}_{gb}$ as the cutting\nprocedure, unless stated otherwise.\n\nWe mention here that combining different \ntypes of infer cuts together in a single cutting procedure\nwas not explored.\nThis is because every inference made by $\\mathcal{C}_{i}$ is \ncontained within $\\mathcal{C}_{ii}$ and every inference made by $\\mathcal{C}_{ii}$\nis similarly contained within $\\mathcal{C}_{iii}$.\nThus combining these infer cuts will not lead to a\nsubstantial reduction in runtime.\n\n\\section{CP Sub-Problem}\n\\label{sec:exp:cp}\nThe CP solver \\chuffed provides the\nuser with numerous input parameters to\ntailor the propagation and search procedure \nto the problem at hand.\nIn this section, we report on the comparisons\nbetween the model formulation\nand search strategies in order to find the best suited\nconfiguration for the \\sua{2} instances.\n\nAs we are only concerned with optimizing the sub-problems'\nsolving time, the runtime values we present here will\nonly average the time used by the CP model, \n\\ie excluding the runtime of the master problem.\nSimilarly, this also applies to the average number of nodes\nexplored by the CP sub-problems..\n\n% The upper bound cut reduces the space complexity of the sub-problems\n% very helpful for cp solver whose difficultly is very closely\n% related to the inference that can be done on the finite domains. \n\n\\subsection{Model Formulation}\n\\label{sec:exp:cpForm}\nTwo main types of CP models were considered, so we\nfirst check which of these is superior before moving\non to test other input parameters.\n$CP_1$ represents the model detailed in Section \\ref{sec:bend:cpModel}\nfrom the previous chapter.\n$CP_2$ represents the same model but with the additional decision\nvariable $\\sigma_{ij}$ and the \\cumu global constraint.\n\nIn Table \\ref{tab:exp:resultsCPmodel}, we compare these two models \nagainst each other on the class 1 and\nclass 2 instances where $\\alpha$ is fixed to $1.00$.\nThe search strategy used for these tests was simply the default\nsearch of \\chuffed.\nThese tests indicate that there is a reduction\nin sub-problem runtime when the \\cumu constraint\nis used in the CP formulation.\nFor all remaining experiments where CP sub-problems\nwere used, $CP_2$ was the formulation utilized\nunless stated otherwise.\n\n\\begin{table}[tpb]\n\t\\caption{CP formulations compared on classes 1 and 2 when $\\alpha=1.00$}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{ccrrrrrr}\n\t\t\\toprule\n\t\tClass & Model & \\#SP Nodes & \\%Gap & \\%Opt. & SP Runtime(s) \\\\\\midrule\\midrule\n\t\t1 & $CP_1$ & 32,755 & 0.00 & 100.00 & 3.83 \\\\\n\t\t & $CP_2$ & 29,225 & 0.00 & 100.00 & \\bf{2.84} \\\\\\midrule\n\t\t2 & $CP_1$ & 2,768,871 & 15.45 & 25.00 & 386.71 \\\\\n\t\t & $CP_2$ & 2,503,183 & 14.03 & 25.00 & \\bf{329.60} \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsCPmodel}\n\\end{table}\n\n\\subsection{Search Strategy}\n\\label{sec:exp:cpSearch}\nEarlier in Section \\ref{sec:bend:cpForm}\nof the previous Chapter, we detailed a range of possible search procedures\nwhich could be employed when solving our constraint program.\nIn Table \\ref{tab:exp:resultsCPsearch} we present the results\nof testing the CP sub-problems on a selection of these procedures.\nThe names of the searches refer to the following procedures:\n{\\tt def} is the default search, {\\tt start} is the basic search on start time variables,\n{\\tt start\\_ff\\_Then\\_sigma} is a sequential search of start then $\\sigma$ variables. \nFour priority searches were considered which only differ by the variable selection\nstrategy used to choose the next sequential procedure.\n% ; these are \n% {\\tt priority\\_io} using {\\tt input\\_order} variable selection, \n% {\\tt priority\\_sm} using {\\tt smallest},\n% {\\tt priority\\_sml} using {\\tt smallest\\_largest} and\n% {\\tt priority\\_ff} using {\\tt first\\_fail}.\n\nFrom the results of Table \\ref{tab:exp:resultsCPsearch}, \nwe can immediately conclude that the default search\nis inferior to the search strategies we have tailored for\nthe \\sua{2}.\nAlthough, among the search procedures we devised, there\nis very little variations in the results.\nEach procedure was able to find the optimal solution\nfor the same number of instances,\nwhile the runtime was very similar.\nThe basic search {\\tt start} needed to explore the fewest\nnumber of nodes in the search tree, however the priority search\n{\\tt priority\\_ff} provided the best average runtime.\nFor all remaining tests where CP sub-problems\nwere utilized, the priority based search strategy using \nthe {\\tt first\\_fail} variable selection was used.\n\n\n\\begin{table}[tpb]\n\t\\caption{CP search strategies compared on class 2 when $\\alpha=1.00$}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{clrrrrrr}\n\t\t\\toprule\n\t\tClass & Search & \\#SP Nodes & \\%Gap & \\%Opt. & SP Runtime(s) \\\\\\midrule\\midrule\n\t\t2 & {\\tt default}\t\t\t& 2,503k & 14.03 & 25.00 & 329.60 \\\\\n\t\t & {\\tt start}\t\t\t\t& 1,884k & 8.97 & 42.86 & 246.57 \\\\\n\t\t & {\\tt start\\_ff\\_Then\\_sigma}\t& 2,391k & 9.69 & 42.86 & 254.23 \\\\\n\t\t & {\\tt priority\\_io}\t\t& 2,448k & 9.12 & 42.86 & 253.70 \\\\\n\t\t & {\\tt priority\\_sm}\t\t& 2,408k & 9.06 & 42.86 & 252.91 \\\\\n\t\t & {\\tt priority\\_sml}\t\t& 2,470k & 9.04 & 42.86 & 252.63 \\\\\n\t\t & {\\tt priority\\_ff}\t\t& 2,426k & 8.90 & 42.86 & \\bf{245.74} \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsCPsearch}\n\\end{table}\n\n\\section{Benchmark Experiments}\n\\label{sec:exp:benchmarkExp}\nWe close this chapter by comparing the best\nconfiguration of our logic-based Benders decomposition\nagainst the results of the MIP formulations for all benchmark\ninstances from the subset of SBF2 we have used.\nBefore presenting this final result, we must first\ndecide on the best sub-problem formulation for our\nBenders decomposition.\n\nEarlier in Section \\ref{sec:lit:cpChuffed}, we gave a\nbrief description of the CP solver, \\chuffed, we chose to utilize when\ntackling the constraint programs we formulated.\nAs mentioned in that section, \\chuffed does not currently have\nthe ability to be run in parallel.\nThis handicap has hampered its ability to effectively compete\nwith \\gurobi, which can easily run in parallel on all four cores\nof the computer we used for testing.\n\nIn Table \\ref{tab:exp:resultsSPform} we present a comparison\nof our Benders decomposition algorithm tested on the class 2\ninstances when using MIP and CP sub-problems.\nThere are many more nodes explored when using CP sub-problems\ndue to the different search procedure of a CP solver\nwhen compared to a MIP solver.\nThe results of the two formulations are similar, but we can\nconclude that our best configuration of the CP model\nhas provided a reduction in runtime when compared to the MIP\nformulation.\n\n\\begin{table}[tpb]\n\t\\caption{MIP and CP sub-problem formulations compared on class 2}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{ccrrrrrr}\n\t\t\\toprule\n\t\tClass & Sub-Problem & \\#SP Nodes & \\%Gap & \\%Opt. & SP Runtime(s) \\\\\\midrule\\midrule\n\t\t2 & \\spmip{} & 630k & 8.42 & 60.71 & 173.05 \\\\\n\t\t & \\spcp{} & 2,489k & 8.21 & 61.61 & \\bf{165.26} \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsSPform}\n\\end{table}\n\nNow using the best formulation of our Benders decomposition\nwe present the results of our final benchmark experiments \non all 3 classes from the subset of the SBF2 instances.\nThese results can be found in Table \\ref{tab:exp:resultsBestBenders}\nwhere we provide the average number of iterations,\ncuts, average gap, number optimal and runtime breakdown.\nThe instances from classes 2 and 3 proved to be more\nchallenging than the smaller instances from class 1.\nMost notably, the total runtime taken by\nthe Benders algorithm is almost entirely due to the complexity of the \\rmp{}.\n\nMore cuts are to be expected when the difficultly of the problem\nis increased, but we also mention that for the instances of class 3\nthere is also a larger number of stations.\nThis will lead to more cuts being made for every iteration of the algorithm.\n\n\\begin{table}[tpb]\n\t\\caption{Benchmark results of our final Benders decomposition}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{crrrrrr}\n\t\t\\toprule\n\t\tClass & \\#Its. & \\#Cuts. & \\%Gap & \\#Opt. & RMP.Runtime(s) & SP.Runtime(s) \\\\\\midrule\\midrule\n\t\t1 & 3 & 9 & 0.00 & 108/108 & 0.36 (87.95\\%) & 0.04 (10.98\\%) \\\\\n\t\t2 & 47 & 196 & 8.21 & 69/112 & 781.29 (82.33\\%) & 165.26 (17.42\\%) \\\\\n\t\t3 & 92 & 468 & 15.76 & 27/176 & 1462.01 (88.00\\%) & 173.63 (10.45\\%) \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsBestBenders}\n\\end{table}\n\nTable \\ref{tab:exp:resultsBenchmark} compares how our final results perform\nwhen compared to the pure MIP formulations of the \\sua{2}.\nFor each class of instances our solution methodology beats the best formulation\nof the MIPs.\nIn the case of class 2 where the Benders decomposition could not prove optimality for\nall instances, the average Benders gap at termination is superior to\nthe best average gap found by \\gurobi when solving either \\fsbf{2} and \\scbf{2}.\nFor class 3, our algorithm could only find the optimal solution for 15.3\\% of the 176 instances.\nAlso the \\scbf{2} model returned superior average gap values for class 3, even though\nit was unable to find any optimal solutions.\nAlthough the gap value of \\scbf{2} for the class 3 instances\nmay be misleading, as it only calculated from\nthe 59 of 176 instances where a feasible solution was found.\n\n\\begin{table}[tpb]\n\t\\caption{Benders decomposition compared to pure MIP formulations}\n\t\\centering\n\t\\vspace{2mm}\n\t\\begin{tabular}{crrrrrrrr}\n\t\t\\toprule\n\t\t & \\multicolumn{4}{c}{Benders decomposition} & \\multicolumn{2}{c}{\\fsbf{2}} & \\multicolumn{2}{c}{\\scbf{2}} \\\\\n\t\t \\cmidrule(lr){2-5} \\cmidrule(lr){6-7} \\cmidrule(lr){8-9}\n\t\tClass & \\%Gap & \\#Opt. & \\%Opt. & Rt.(s) & \\%Gap & \\%Opt. & \\%Gap & \\%Opt. \\\\\\midrule\\midrule\n\t\t1 & 0.00 & 108/108 & 100.00 & 0.41 & 0.16 & 93.52 & 1.62 & 77.78 \\\\\n\t\t2 & 8.21 & 69/112 & 61.61 & 948.93 & 13.76 & 0.00 & 18.01 & 0.00 \\\\\n\t\t3 & 15.76 & 27/176 & 15.34 & 1661.93 & 19.56 & 3.98 & 10.09 & 0.00 \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\label{tab:exp:resultsBenchmark}\n\\end{table}\n\nOur results indicate that the \\sua{2} is a difficult problem to solve\nwhen the instance size becomes large.\nHowever, we can see from these results that decomposing the problem using a \nlogic-based Benders decomposition framework provides\nthe most optimal solutions when compared to the other available exact\nsolution procedures.\n\n% \\section{Summary}\n% \\label{sec:exp:summary}\n\n% blah\n", "meta": {"hexsha": "b69f29cb4ed5d42e780035be988058d5f941660c", "size": 28670, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Assembly Task Scheduling/sualbsp-2-master/thesis/chap_experiments.tex", "max_stars_repo_name": "BillChan226/ODA-Multi-Manipulator", "max_stars_repo_head_hexsha": "6863558a9fdb946fb16c67a7660172154274cfe0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Assembly Task Scheduling/sualbsp-2-master/thesis/chap_experiments.tex", "max_issues_repo_name": "BillChan226/ODA-Multi-Manipulator", "max_issues_repo_head_hexsha": "6863558a9fdb946fb16c67a7660172154274cfe0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Assembly Task Scheduling/sualbsp-2-master/thesis/chap_experiments.tex", "max_forks_repo_name": "BillChan226/ODA-Multi-Manipulator", "max_forks_repo_head_hexsha": "6863558a9fdb946fb16c67a7660172154274cfe0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.0786163522, "max_line_length": 195, "alphanum_fraction": 0.7138472271, "num_tokens": 9208, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6584174938590245, "lm_q1q2_score": 0.34719442379402826}} {"text": "\\chapter{Shape-from-Shading}\n\\label{ch:sfs}\n\nASP provides a tool, named \\texttt{sfs}, that can improve the level of\ndetail of DEMs created by ASP or any other source using\n\\textit{shape-from-shading} (SfS). The tool takes as input one or more\ncamera images, a DEM at roughly the same resolution as the images, and\nreturns a refined DEM.\n\n\\texttt{sfs} works only with ISIS cub images. It has been tested\nthoroughly with Lunar LRO NAC datasets, and some experiments were done\nwith Mars HiRISE images and with pictures from Charon, Pluto's moon. As\nseen later in the text, it returns reasonable results on the Moon as far\nas $85^\\circ$ South.\n\nCurrently, \\texttt{sfs} is computationally expensive, and is practical\nonly for DEMs whose width and height are several thousand pixels. It can be \nsensitive to errors in the position and orientation of the cameras, the\naccuracy of the initial DEM, and to the value of the two weights it uses.\nYet, with some effort, it can work quite well. \n\nA tool named \\texttt{parallel\\_sfs} is provided (section \\ref{psfs}) \nthat parallelizes \\texttt{sfs} using multiple processes (optionally on\nmultiple machines) by splitting the input DEM into tiles with padding,\nrunning \\texttt{sfs} on each tile, and then blending the results. \n\nThe \\texttt{sfs} program can model position-dependent albedo, different\nexposure values for each camera, shadows in the input images, and regions\nin the DEM occluded from the Sun. It can refine the positions and orientations\nof the cameras.\n\nThe tool works by minimizing the cost function\n\\begin{equation}\\label{cost}\n%\\begin{multline}\\label{cost}\n\\int\\!\\! \\int \\! \\sum_k \\left[ I_k(\\phi)(x, y) - T_k A(x, y)\n R_k(\\phi)(x, y) \\right]^2\\, \n% R_k(\\phi)(x, y) \\right]^2\\, \\\\\n+ \\mu \\left\\|\\nabla^2 \\phi(x, y) \\right\\|^2 \n+ \\lambda \\left[ \\phi(x, y) - \\phi_0(x, y) \\right]^2\n\\, dx\\, dy.\n\\end{equation}\n%\\end{multline}\n\nHere, $I_k(\\phi)(x, y)$ is the $k$-th camera image interpolated at\npixels obtained by projecting into the camera 3D points from the terrain\n$\\phi(x, y)$, $T_k$ is the $k$-th image exposure, $A(x, y)$ is the\nper-pixel albedo, $R_k(\\phi)(x, y)$ is the reflectance computed from the\nterrain for $k$-th image, $\\left\\|\\nabla^2 \\phi(x, y) \\right\\|^2 $ is the sum\nof squares of all second-order partial derivatives of $\\phi$, $\\mu > 0$\nis a smoothing term, and $\\lambda > 0$ determines how close we should\nstay to the input terrain $\\phi_0$ (smaller $\\mu$ will show more detail\nbut may introduce some artifacts, and smaller $\\lambda$ may allow for\nmore flexibility in optimization but the terrain may move too far from the input). \n\nWe use either the regular Lambertian reflectance model,\nor the Lunar-Lambertian model \\cite{mcewen1991photometric}, more specifically as given in\n\\cite{lohse2006derivation} (equations (3) and (4)).\nAlso supported is the Hapke model, \n\\cite{johnson2006spectrophotometric}, \\cite{fernando2013surface},\n\\cite{hapke2008bidirectional}, \\cite{hapke1993opposition}.\nCustom values for the coefficients of these models can be passed to the program.\n\n\\section{How to get good test imagery}\n\nWe obtain the images from\n\\url{http://wms.lroc.asu.edu/lroc/search} (we search for EDR images of\ntype NACL and NACR). \n\nA faster (but not as complete) interface is provided by\n\\url{http://ode.rsl.wustl.edu/moon/indexproductsearch.aspx}. The related\nsite\n\\url{http://ode.rsl.wustl.edu/moon/indextools.aspx?displaypage=lolardr}\ncan provide LOLA datasets which can be used as (sparse) ground truth.\n\nWe advise the following strategy for picking images. First choose a small\nlongitude-latitude window in which to perform a search for imagery. Pick\ntwo images that are very close in time and with a big amount of overlap\n(ideally they would have consecutive orbit numbers).\nThose can be passed to ASP's \\texttt{stereo} tool to create an initial DEM. Then, search for \nother images close to the center of the maximum overlap of the first\ntwo images. Pick one or more of those, ideally with different illumination\nconditions than the first two. Those (together with one of the first two images) can be used for SfS.\n\nTo locate the area of spatial overlap, the images can be map-projected (either\nwith \\texttt{cam2map} with a coarse resolution) or with\n\\texttt{mapproject}, using for example the LOLA DEM as the terrain to project onto, \nor the DEM obtained from running \\texttt{stereo} on those images. Then the images can be overlayed\nin \\texttt{stereo\\_gui}. A good sanity check is to examine the shadows in\nvarious images. If they point in different directions in the images and perhaps\nalso have different lengths, that means that illumination conditions are\ndifferent enough, which will help constrain the \\texttt{sfs} problem better.\n\n\\section{Running sfs at 1 meter/pixel using a single image}\n\nIn both this and the next sections we will work with LRO NAC images taken\nclose to the Lunar South Pole, at a latitude of $85^\\circ$ South (the tool was\ntested on equatorial regions as well). We will use four images,\nM139939938LE, M139946735RE, M173004270LE, and M122270273LE.\n\nWe first retrieve the data sets.\n\\begin{verbatim}\n wget http://lroc.sese.asu.edu/data/LRO-L-LROC-2-EDR-V1.0/\\\nLROLRC_0005/DATA/SCI/2010267/NAC/M139939938LE.IMG\n wget http://lroc.sese.asu.edu/data/LRO-L-LROC-2-EDR-V1.0/\\\nLROLRC_0005/DATA/SCI/2010267/NAC/M139946735RE.IMG\n wget http://lroc.sese.asu.edu/data/LRO-L-LROC-2-EDR-V1.0/\\\nLROLRC_0009/DATA/SCI/2011284/NAC/M173004270LE.IMG\n wget http://lroc.sese.asu.edu/data/LRO-L-LROC-2-EDR-V1.0/\\\nLROLRC_0002/DATA/MAP/2010062/NAC/M122270273LE.IMG\n\\end{verbatim}\n\nThen we convert them to ISIS cubes, initialize the SPICE kernels, and\nperform radiometric calibration and echo correction. Here are the steps, \nillustrated on the first image:\n\\begin{verbatim} \n lronac2isis from = M139939938LE.IMG to = M139939938LE.cub\n spiceinit from = M139939938LE.cub\n lronaccal from = M139939938LE.cub to = M139939938LE.cal.cub\n lronacecho from = M139939938LE.cal.cub to = M139939938LE.cal.echo.cub\n\\end{verbatim}\nWe rename, for simplicity, the obtained four processed datasets to\nA.cub, B.cub, C.cub, and D.cub.\n\nThe first step is to run stereo to create an initial guess DEM. We\npicked for this the first two of these images. These form a stereo pair,\nthat is, they have a reasonable baseline and sufficiently close times of\nacquisition (hence very similar illuminations). These conditions are\nnecessary to obtain a good stereo result.\n\\begin{verbatim}\nparallel_stereo --job-size-w 1024 --job-size-h 1024 A.cub B.cub \\\n --left-image-crop-win 0 7998 2728 2696 \\\n --right-image-crop-win 0 9377 2733 2505 \\\n --threads 16 --corr-seed-mode 1 --subpixel-mode 3 \\\n run_full1/run\n\\end{verbatim}\nNext we create a DEM at 1 meter/pixel, which is about the resolution\nof the input images. We use the stereographic projection since this\ndataset is very close to the South Pole. Then we crop it to the region\nwe'd like to do SfS on.\n\\begin{verbatim}\n point2dem -r moon --stereographic --proj-lon 0 \\\n --proj-lat -90 run_full1/run-PC.tif\n gdal_translate -projwin -15471.9 150986 -14986.7 150549 \\\n run_full1/run-DEM.tif run_full1/run-crop-DEM.tif\n\\end{verbatim}\nThis creates a DEM of size $456 \\times 410$ pixels.\n\nThen we run \\texttt{sfs}:\n\\begin{verbatim}\n sfs -i run_full1/run-crop-DEM.tif A.cub -o sfs_ref1/run \\\n --reflectance-type 1 \\\n --smoothness-weight 0.08 --initial-dem-constraint-weight 0.0001 \\\n --max-iterations 10 --use-approx-camera-models \\\n --use-rpc-approximation --crop-input-images \n\\end{verbatim}\n\nThe smoothness weight is a parameter that needs tuning. If it is too\nsmall, SfS will return noisy results, if it is too large, too much\ndetail will be blurred. Here we used the Lunar Lambertian model. The meaning of the other\n\\texttt{sfs} options can be looked up in section \\ref{sfs}.\n\nWe show the results of running this program in figure\n\\ref{fig:sfs1}. The left-most figure is the hill-shaded original DEM,\nwhich was obtained by running:\n\\begin{verbatim}\n hillshade --azimuth 300 --elevation 20 run_full1/run-crop-DEM.tif \\\n -o run_full1/run-crop-hill.tif \n\\end{verbatim}\nThe second image is the hill-shaded DEM obtained after running\n\\texttt{sfs} for 10 iterations.\n\nThe third image is, for comparison, the map-projection of A.cub onto the\noriginal DEM, obtained via the command:\n\\begin{verbatim}\n mapproject --tr 1 run_full1/run-crop-DEM.tif A.cub A_map.tif --tile-size 128\n\\end{verbatim}\nThe forth image is the colored absolute difference between the original\nDEM and the SfS output, obtained by running:\n\\begin{verbatim}\n geodiff --absolute sfs_ref1/run-DEM-final.tif run_full1/run-crop-DEM.tif\n colormap --min 0 --max 2 --colormap-style binary-red-blue \\\n run-DEM-final__run-crop-DEM-diff.tif\n\\end{verbatim}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=7in]{images/sfs1.jpg}\n\\caption[sfs]{An illustration of \\texttt{sfs}. The images are, from\n left to right, the original hill-shaded DEM, the hill-shaded DEM obtained\nfrom \\texttt{sfs}, the image A.cub map-projected onto the original DEM,\nand the absolute difference of the original and final DEM, where the brightest\nshade of red corresponds to a 2 meter height difference.}\n\\label{fig:sfs1}\n\\end{center}\n\\end{figure}\n\nIt can be seen that the optimized DEM provides a wealth of detail and\nlooks quite similar to the input image. It also did not diverge\nsignificantly from the input DEM. We will see in the next section that\nSfS is in fact able to make the refined DEM more accurate than the\ninitial guess (as compared to some known ground truth), though that is\nnot guaranteed, and most likely did not happen here where just one image\nwas used.\n\n\\section{SfS with multiple images in the presence of shadows}\n\nIn this section we will run \\texttt{sfs} with multiple images. We would\nlike to be able to see if SfS improves the accuracy of the DEM rather\nthan just adding detail to it. We evaluate this using the following\n(admittedly imperfect) approach. We resample the original images by a\nfactor of 10, run stereo with them, followed by SfS using the stereo\nresult as an initial guess and with the resampled images. As ground\ntruth, we create a DEM from the original images at 1 meter/pixel, which\nwe bring closer to the initial guess for SfS using\n\\texttt{pc\\_align}. We would like to know if running SfS brings us even\ncloser to this ``ground truth'' DEM.\n\nThe most significant challenge in running SfS with multiple images is\nthat shape-from-shading is highly sensitive to errors in camera position\nand orientation. The \\texttt{sfs} tool can improve these by floating\nthem during optimization and by using a coarse-to-fine scheme, where\nthe problem is first solved using subsampled images and terrain then\nit is successively refined.\n\nIf possible, it may still be desirable to bundle-adjust the cameras first\n(section \\ref{bundleadjust}). It is important to note that bundle adjustment may\nfail if the images have sufficiently different illumination, as it will\nnot be able to find matches among images. A solution to this is discussed\nin section \\ref{sfs-lola}.\n\nTo make bundle adjustment and stereo faster, we first crop the images,\nsuch as shown below (the crop parameters can be determined via\n\\texttt{stereo\\_gui}).\n\\begin{verbatim}\n crop from = A.cub to = A_crop.cub sample = 1 line = 6644 nsamples = 2192 nlines = 4982\n crop from = B.cub to = B_crop.cub sample = 1 line = 7013 nsamples = 2531 nlines = 7337\n crop from = C.cub to = C_crop.cub sample = 1 line = 1 nsamples = 2531 nlines = 8305\n crop from = D.cub to = D_crop.cub sample = 1 line = 1 nsamples = 2531 nlines = 2740\n\\end{verbatim}\nThen we bundle-adjust and run stereo\n\\begin{verbatim}\n bundle_adjust A_crop.cub B_crop.cub C_crop.cub D_crop.cub \\\n --min-matches 1 -o run_ba/run\n stereo A_crop.cub B_crop.cub run_full2/run --subpixel-mode 3 \\\n --bundle-adjust-prefix run_ba/run\n\\end{verbatim}\n\nThis will result in a point cloud, \\verb#run_full2/run-PC.tif#, which will\nlead us to the ``ground truth'' DEM. As mentioned\nbefore, we'll in fact run SfS with images subsampled by a factor of\n10. Subsampling is done by running the ISIS \\texttt{reduce} command\n\\begin{verbatim}\n for f in A B C D; do \n reduce from = ${f}_crop.cub to = ${f}_crop_sub10.cub sscale = 10 lscale = 10\n done\n\\end{verbatim}\n\nWe run bundle adjustment and stereo with the subsampled images using\ncommands analogous to the above:\n\\begin{verbatim}\n bundle_adjust A_crop_sub10.cub B_crop_sub10.cub C_crop_sub10.cub D_crop_sub10.cub \\\n --min-matches 1 -o run_ba_sub10/run --ip-per-tile 100000\n stereo A_crop_sub10.cub B_crop_sub10.cub run_sub10/run --subpixel-mode 3 \\\n --bundle-adjust-prefix run_ba_sub10/run\n\\end{verbatim}\nWe'll obtain a point cloud named \\verb#run_sub10/run-PC.tif#.\n\nWe'll bring the ``ground truth'' point cloud closer to the initial guess\nfor SfS using \\texttt{pc\\_align}:\n\\begin{verbatim}\n pc_align --max-displacement 200 run_full2/run-PC.tif run_sub10/run-PC.tif \\\n -o run_full2/run --save-inv-transformed-reference-points\n\\end{verbatim}\nThis step is extremely important. Since we ran two bundle adjustment\nsteps, and both were without ground control points, the resulting clouds\nmay differ by a large translation, which we correct here. Hence we would like to \nmake the ``ground truth'' terrain aligned with the datasets on which we \nwill perform SfS. \n\nNext we create the ``ground truth'' DEM from the aligned high-resolution\npoint cloud, and crop it to a desired region:\n\\begin{verbatim}\n point2dem -r moon --tr 10 --stereographic --proj-lon 0 --proj-lat -90 \\\n run_full2/run-trans_reference.tif\n gdal_translate -projwin -15540.7 151403 -14554.5 150473 \\\n run_full2/run-trans_reference-DEM.tif run_full2/run-crop-DEM.tif\n\\end{verbatim}\nWe repeat the same steps for the initial guess for SfS:\n\\begin{verbatim}\n point2dem -r moon --tr 10 --stereographic --proj-lon 0 --proj-lat -90 \\\n run_sub10/run-PC.tif\n gdal_translate -projwin -15540.7 151403 -14554.5 150473 \\\n run_sub10/run-DEM.tif run_sub10/run-crop-DEM.tif\n\\end{verbatim}\nAfter this, we run \\texttt{sfs} itself. Since our dataset has many shadows, we found\nthat specifying the shadow thresholds for the tool improves the\nresults. The thresholds can be determined using \\texttt{stereo\\_gui}.\nThis can be done by turning on shadow-threshold mode from the GUI menu, \nand then clicking on a few points in the shadows. Then the thresholded images\ncan be visualized/updated from the menu as well, and this process can be iterated.\n\\begin{verbatim}\n sfs -i run_sub10/run-crop-DEM.tif A_crop_sub10.cub C_crop_sub10.cub \\\n D_crop_sub10.cub -o sfs_sub10_ref1/run --threads 4 \\\n --smoothness-weight 0.12 --initial-dem-constraint-weight 0.0001 \\\n --reflectance-type 1 --float-exposure \\\n --float-cameras --use-approx-camera-models \\\n --max-iterations 10 --use-approx-camera-models \\\n --use-rpc-approximation --crop-input-images \\\n --bundle-adjust-prefix run_ba_sub10/run \\\n --shadow-thresholds \"0.00162484 0.0012166 0.000781663\"\n\\end{verbatim}\nWe compare the initial guess to \\texttt{sfs} to the ``ground truth'' DEM\nobtained earlier and the same for the final refined DEM using\n\\texttt{geodiff} as in the previous section. Before SfS:\n\n\\begin{verbatim}\n geodiff --absolute run_full2/run-crop-DEM.tif run_sub10/run-crop-DEM.tif\n gdalinfo -stats run-crop-DEM__run-crop-DEM-diff.tif | grep Mean= \n\\end{verbatim}\nand after SfS:\n\\begin{verbatim}\n geodiff --absolute run_full2/run-crop-DEM.tif sfs_sub10_ref1/run-DEM-final.tif\n gdalinfo -stats run-crop-DEM__run-DEM-final-diff.tif | grep Mean=\n\\end{verbatim}\n\nThe mean error goes from 2.64~m to 1.29~m, while the standard deviation\ndecreases from 2.50~m to 1.29~m. Visually the refined DEM looks more detailed\nas well as seen in figure \\ref{fig:sfs2}. The same experiment can be\nrepeated with the Lambertian reflectance model (reflectance-type 0), and\nthen it is seen that it performs a little worse. \n\nWe also show in this figure the first of the images used for SfS,\n\\verb#A_crop_sub10.cub#, map-projected upon the optimized DEM. Note that\nwe use the previously computed bundle-adjusted cameras when\nmap-projecting, otherwise the image will show as shifted from its true\nlocation:\n\\begin{verbatim}\n mapproject sfs_sub10_ref1/run-DEM-final.tif A_crop_sub10.cub A_crop_sub10_map.tif \\\n --bundle-adjust-prefix run_ba_sub10/run\n\\end{verbatim}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=7in]{images/sfs2.jpg}\n\\caption[sfs]{An illustration of \\texttt{sfs}. The images are, from\n left to right, the hill-shaded initial guess DEM for SfS, the hill-shaded DEM obtained\nfrom \\texttt{sfs}, the ``ground truth'' DEM, and the first of the\nimages used in SfS map-projected onto the optimized DEM.}\n\\label{fig:sfs2}\n\\end{center}\n\\end{figure}\n\n\\section{Dealing with large camera errors and LOLA comparison}\n\\label{sfs-lola}\n\nSfS is very sensitive to errors in camera positions and\norientations. These can be optimized as part of the problem, but if they\nare too far off, the solution will not be correct. In the previous\nsection we used bundle adjustment to correct these errors, and then we\npassed the adjusted cameras to \\texttt{sfs}. However, bundle adjustment may\noften fail, simply because the illumination conditions can be very\ndifferent among the images, and interest point matching may not succeed.\n\nThe option \\texttt{-\\/--coarse-levels \\it{int}} can be passed to\n\\texttt{sfs}, to solve for the terrain using a multi-resolution\napproach, first starting at a coarse level, where camera errors have\nless of an impact, and then jointly optimizing the cameras and\nthe terrain at ever increasing levels of resolution. Yet, this may still\nfail if the terrain does not have large and pronounced features on the\nscale bigger than the errors in the cameras.\n\nThe approach that we found to work all the time is to manually select\ninterest points in the images, as the human eye is much more skilled at\nidentifying a given landmark in multiple images, even when the lightning\nchanges drastically. Picking about 4 landmarks in each image is\nsufficient. Ideally they should be positioned far from each other, to\nimprove the accuracy.\n\nBelow is one example of how we manually select interest points, run SfS,\nand then how we compare to LOLA, which is an independently acquired\nsparse dataset of 3D points on the Moon. According to\n\\cite{smith2011results}, the LOLA accuracy is on the order of 1~m. To\nensure a meaningful comparison of stereo and SfS with LOLA, we resample\nthe LRO NAC images by a factor of 4, making them nominally 4\nm/pixel. This is not strictly necessary, the same exercise can be\nrepeated with the original images, but it is easier to see the\nimprovement due to SfS when comparing to LOLA when the images are\ncoarser than the LOLA error itself.\n\nWe work with the same images as before. To resample them, we do:\n\\begin{verbatim}\n for f in A B C D; do \n reduce from = ${f}_crop.cub to = ${f}_crop_sub4.cub sscale=4 lscale=4\n done\n\\end{verbatim}\n\nWe run \\texttt{stereo} and \\texttt{point2dem} to get a first cut DEM. We don't do bundle\nadjustment at this stage yet. \n\\begin{verbatim}\n stereo A_crop_sub4.cub B_crop_sub4.cub run_stereo_noba_sub4/run --subpixel-mode 3\n point2dem --stereographic --proj-lon -5.7113451 --proj-lat -85.000351 \\\n run_stereo_noba_sub4/run-PC.tif --tr 4 \n\\end{verbatim}\n\nWe would like now to manually pick interest points for the purpose of\ndoing bundle adjustment. We found it it much easier to locate the landmarks if we \nfirst map-project the images, which brings them all into the same\nperspective. We then pick interest points in \\texttt{stereo\\_gui}, and then project\nthem back into the original cameras and do bundle adjustment. Here are\nthe steps:\n\\begin{verbatim}\n for f in A B C D; do \n mapproject --tr 4 run_stereo_noba_sub4/run-DEM.tif ${f}_crop_sub4.cub \\\n ${f}_crop_sub4_v1.tif --tile-size 128\n done\n stereo_gui A_crop_sub4_v1.tif B_crop_sub4_v1.tif C_crop_sub4_v1.tif \\\n D_crop_sub4_v1.tif run_ba_sub4/run\n\\end{verbatim}\nInterest points are selected by zooming and right-clicking with the\nmouse, one point at a time, from left to right, and then saving them. An\nillustration is shown in Figure \\ref{fig:sfs3}.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=7in]{images/sfs3.jpg}\n\\caption[sfs]{An illustration of how interest points are picked manually for the purpose of bundle adjustment and then SfS.}\n\\label{fig:sfs3}\n\\end{center}\n\\end{figure}\n\nThen bundle adjustment happens:\n\\begin{verbatim}\n P='A_crop_sub4_v1.tif B_crop_sub4_v1.tif' # to avoid long lines below\n Q='C_crop_sub4_v1.tif D_crop_sub4_v1.tif run_stereo_noba_sub4/run-DEM.tif'\n bundle_adjust A_crop_sub4.cub B_crop_sub4.cub C_crop_sub4.cub D_crop_sub4.cub \\\n -o run_ba_sub4/run --mapprojected-data \"$P $Q\" \\\n --min-matches 1\n\\end{verbatim}\n\nA good sanity check to ensure that at this stage cameras are aligned properly is to map-project\nusing the newly obtained camera adjustments and then overlay the obtained images in the GUI.\nThe features in all images should be perfectly on top of each other.\n\\begin{verbatim}\n for f in A B C D; do \n mapproject --tr 4 run_stereo_noba_sub4/run-DEM.tif ${f}_crop_sub4.cub \\\n ${f}_crop_sub4_v2.tif --tile-size 128 --bundle-adjust-prefix run_ba_sub4/run\n done\n\\end{verbatim}\nThis will also show where the images overlap, and hence on what portion of the DEM we\ncan run SfS.\n\nThen we run stereo, followed by SfS. \n\\begin{verbatim}\n stereo A_crop_sub4.cub B_crop_sub4.cub run_stereo_yesba_sub4/run \\\n --subpixel-mode 3 --bundle-adjust-prefix run_ba_sub4/run\n point2dem --stereographic --proj-lon -5.7113451 --proj-lat -85.000351 \\\n run_stereo_yesba_sub4/run-PC.tif --tr 4\n gdal_translate -srcwin 138 347 273 506 run_stereo_yesba_sub4/run-DEM.tif \\\n run_stereo_yesba_sub4/run-crop1-DEM.tif \n sfs -i run_stereo_yesba_sub4/run-crop1-DEM.tif A_crop_sub4.cub \\\n C_crop_sub4.cub D_crop_sub4.cub -o sfs_sub4_ref1_th_reg0.12_wt0.001/run \\\n --shadow-thresholds '0.00149055 0.00138248 0.000747531' \\\n --threads 4 --smoothness-weight 0.12 --initial-dem-constraint-weight 0.001 \\\n --reflectance-type 1 --float-exposure --float-cameras --max-iterations 20 \\\n --use-approx-camera-models --use-rpc-approximation --crop-input-images \\\n --bundle-adjust-prefix run_ba_sub4/run\n\\end{verbatim}\n\nWe fetch the portion of the LOLA dataset around the current DEM from the\nsite described earlier, and save it as\n\\verb#RDR_354E355E_85p5S84SPointPerRow_csv_table.csv#. It is necessary\nto align our stereo DEM with this dataset to be able to compare\nthem. We choose to bring the LOLA dataset into the coordinate system\nof the DEM, using:\n\\begin{verbatim}\n pc_align --max-displacement 280 run_stereo_yesba_sub4/run-DEM.tif \\\n RDR_354E355E_85p5S84SPointPerRow_csv_table.csv -o run_stereo_yesba_sub4/run \\\n --save-transformed-source-points\n\\end{verbatim}\n\nThen we compare to the aligned LOLA dataset the input to SfS and its output:\n\\begin{verbatim}\n geodiff --absolute -o beg --csv-format '1:lon 2:lat 3:radius_km' \\\n run_stereo_yesba_sub4/run-crop1-DEM.tif run_stereo_yesba_sub4/run-trans_source.csv\n geodiff --absolute -o end --csv-format '1:lon 2:lat 3:radius_km' \\\n sfs_sub4_ref1_th_reg0.12_wt0.001/run-DEM-final.tif \\\n run_stereo_yesba_sub4/run-trans_source.csv\n\\end{verbatim}\n\nWe see that the mean error between the DEM and LOLA goes down, after SfS,\nfrom 1.14~m to 0.90~m, while the standard deviation decreases from\n 1.18~m to 1.06~m.\n\n\\section{Running SfS with an external initial guess DEM}\n\nSometimes it is convenient to run SfS with a DEM not created using ASP's\n\\texttt{stereo}. For example, for the Moon, the LOLA gridded DEM is\navailable. It is somewhat noisy and the nominal resolution is on the\norder of 10 m/pixel, but it is available even for permanently shadowed\nregions.\n\nThe main challenge in such a situation is that the images, such as\ncoming from LRO NAC, may not be aligned well to this external DEM and\namong themselves, and then SfS will fail. To get an initial alignment,\nwhat worked for us to coarsen both this DEM and the images to about 40\nmeters/pixel (using the ISIS \\texttt{reduce} command and\n\\texttt{dem\\_mosaic} with the options to blur and change the grid size),\nand then run SfS over a reasonably large area (say about $500 \\times\n500$ pixels) with pronounced terrain and where images have a lot of\noverlap with the option \\texttt{-\\/-float-all-cameras} (and of course\n\\texttt{-\\/-float-exposure}, and an appropriate \\texttt{-\\/-shadow-threshold}). \nThen, the SfS program will find\nadjustments to the cameras, writing them in the output directory. The\n\\texttt{mapproject} tool can be used to map-project the coarse images\nonto the input DEM using \\texttt{-\\/-bundle-adjust-prefix} pointing to\nthe \\texttt{sfs} output prefix. This should be used to verify that images\nare now aligned correctly, for example by overlaying them in\n\\texttt{stereo\\_gui}. If so, these adjustments can be used as input for\nSfS with images at finer levels of resolution (after appropriately renaming the\nadjustment files and using the \\texttt{-\\/-bundle-adjust-prefix} option of \\texttt{sfs}).\n\nHere, a higher value can be used for \\texttt{-\\/-initial-dem-constraint-weight}\nto ensure we cameras have more motivation to align to the terrain.\n\nWhen it comes to running SfS with many large high-resolution\nimages, one runs very fast into memory constraints. It is then necessary\nto parallelize the problem using \\texttt{parallel\\_sfs}, which will run\n\\texttt{sfs} on multiple tiles. Yet, the camera adjustments need to be\ndetermined before running this tool, and then kept fixed when this tool\nis run, as otherwise each single tile will optimize its cameras\nindependently, and as result there will be discontinuities at tile\nboundaries. \n\nIf the camera adjustments are determined by first running \\texttt{sfs}\non a clip representative of the entire terrain, far from that clip the\ncameras will start to disagree. For that reason, \\texttt{sfs} makes it\npossible to optimize the cameras on an entire collection of clips,\nchosen so they are reasonably spread out over the entire terrain, \nand that each camera image is covered by at least a handful of such clips.\nThe clips can be passed to \\texttt{sfs} as a quoted string via the \\texttt{-i} option. \nAs before, at the end the \\texttt{mapproject}\nprogram can be used to verify that the camera images mapprojected using the\nobtained camera adjustments are perfectly on top of each other, and if not,\nmore clips can be added to the joint optimization problem. When a good\nenough set of camera adjustments is obtained, \\texttt{parallel\\_sfs} can\nbe run as before. \n\n\\section{Insights for getting the most of SfS}\n\\label{sfs:insights}\n\nHere are a few suggestions we have found helpful when running \\texttt{sfs}:\n\n\\begin{itemize}{}\n\n\\item First determine the appropriate smoothing weight $\\mu$ by running a\nsmall clip, and using just one image. A value between 0.06 and 0.12 seems to work \nall the time with LRO NAC, even when the images are subsampled. The other weight, $\\lambda,$ can\nbe set to something small, like $0.0001.$ This can be increased to $0.001$ if noticing\nthat the output DEM strays too far. \n\n\\item As stated before, more images with more diverse illumination conditions\nresult in more accurate terrain. Ideally there should be at least 3 images, with the shadows\nbeing, respectively, on the left, right, and then perhaps missing or small. \n\n\\item Bundle-adjustment for multiple images is crucial, to eliminate\n camera errors which will result in \\texttt{sfs} converging to a local\n minimum. This is described in section \\ref{sfs-lola}.\n\n\\item Floating the albedo (option \\texttt{-\\/-float-albedo}) can\n introduce instability and divergence, it should be avoided unless\n obvious albedo variation is seen in the images.\n\n\\item Floating the DEM at the boundary (option\n \\texttt{-\\/-float-dem-at-boundary}) is also suggested to be avoided.\n\n\\item Overall, the best strategy is to first use SfS for a single image\n and not float any variables except the DEM being optimized, and then\n gradually add images and float more variables and select whichever\n approach seems to give better results.\n\n\\item If an input DEM is large, it may not be completely covered by a\nsingle set of imagery with various illumination conditions. It should\nthen be broken up into smaller regions (with overlap), the SfS problem\ncan be solved on each region, and then every output terrain can be\ntransformed using \\texttt{pc\\_align} into LOLA's global coordinate\nsystem, where they can be mosaicked together using\n\\texttt{dem\\_mosaic}. Or, \\texttt{sfs} can be run not on one clip, but\non an entire collection of clips covering this area to get the\nadjustments, and then \\texttt{parallel\\_sfs} can be run as described in\nthe previous section. \n\n The easier case is when at least the two images in the stereo pair\n cover the entire terrain. Then, portions of this terrain can be used\n as an initial guess for each SfS sub-problem (even as the other images\n used for SfS change), the results can be mosaicked, and the alignment\n to LOLA can happen just once, after mosaicking. This approach is\n preferable, if feasible, as alignment to LOLA is more accurate if the\n terrain to align is larger in extent.\n\n\\item The \\texttt{mapproject} program can be used to map-project each image\nonto the resulting SfS DEM (with the camera adjustments solved using\nSfS). These orthoimages can be mosaicked using \\texttt{dem\\_mosaic}. If the\n\\texttt{-\\/-max} option is used with this tool, it create a mosaic with\nthe most illuminated pixels from this image. If during SfS the camera\nadjustments were solved accurately, this mosaic should have little or no\nblur. An alterantive is to use the \\texttt{-\\/-block-max} option\nwhich will pick the most lit of the images per each block, with the latter\nbeing specified via \\texttt{-\\/-block-size}. \n\\end{itemize}\n", "meta": {"hexsha": "1673d673103dafb0995dae21db815621bf10a3f5", "size": 30437, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/book/sfs.tex", "max_stars_repo_name": "fenglang12345/asp20", "max_stars_repo_head_hexsha": "565894567b58ecb72385e7bb698e318a734bed5a", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/book/sfs.tex", "max_issues_repo_name": "fenglang12345/asp20", "max_issues_repo_head_hexsha": "565894567b58ecb72385e7bb698e318a734bed5a", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/book/sfs.tex", "max_forks_repo_name": "fenglang12345/asp20", "max_forks_repo_head_hexsha": "565894567b58ecb72385e7bb698e318a734bed5a", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-02-04T14:34:23.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-04T14:34:23.000Z", "avg_line_length": 50.0608552632, "max_line_length": 124, "alphanum_fraction": 0.7573348227, "num_tokens": 8373, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241911813151, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34716737140727527}} {"text": "\\section{Interpretation and Discussion of the Results}\n%In this section, first the challenges of the given problem are summarized and an ideal model to solve the problem is defined. Second, all properties of the ideal model are discussed with the relation to selected models and their performance. Last but not the least, conclusion is made on which of the three models is the best for classification of glass fragments.\n\n\\subsection{Classification Challenges and the Ideal Classifier}\nBuilding a good model to accurately predict glass fragments from a data set of $214$ data points turned out to be demanding. The core challenges were:\n\n%\\vspace{10pt}\n\\begin{enumerate}\n \\item Little amount of data for 6-class classification problem\n \\item Skewed class distribution \n \\item Class overlap\n\\end{enumerate}\n%\\vspace{10pt}\n\nIn the light of the model being used in criminal investigation processes, an ideal model would be transparent in a way that the decision-making is comprehensible for humans. Furthermore, the model should be well-suited to tackle the challenges mentioned.\n\n\n%Before a discussion of particular models, it is important to highlight the main challenges of the given classification problem and based on it define ideal model. First, certain classes of glass, such as $1$, $2$ and $3$, had a large overlap given its similar features. This implies a need for a complex model capable of handling such overlap and still being able to generalize well. Second, distribution of classes within the training and test split of data was skewed. Thus, minority classes might be very hard to predict. This goes hand in hand with the size of the training data-set which was only 149 records. Therefore, optimal model should be able to learn even from a small training data-set. Last but not the least, since the output of the model might be used as part of crime analysis, ideal model should be also interpretable, i.e. it should be clear how it decides.\n\n\\subsection{Comparison of Model Performances}\nThe three models evaluated within this project can be ranked by their performance (measured by the expected out-of-sample accuracy) as follows:\n\n\\begin{enumerate}\n \\item \\class{RandomForestClassifier()}\n \\item \\class{NeuralNetworkClassifier()}\n \\item \\class{DecisionTreeClassifier()}\n\\end{enumerate}\n\nLooking at the individual classification reports, all models were able to almost perfectly predict class $7$ which goes hand in hand with the findings from the EDA and Figure \\ref{pca}. In contrast, the prediction of the overlapping classes 1, 2 and 3 turned out to be generally challenging. In order to classify these classes correctly, the model needed to establish a complex decision boundary, that at the same time generalises well. The \\class{RandomForestClassifier()} did the best job at separating the classes 1 and 2 from each other (Table \\ref{random_forest_evaluation}), giving one indication of its overall good performance. The \\class{DecisionTreeClassifier()} struggled the most at separating the overlapping classes (Table \\ref{dt_evaluation}). \nAnother noticeable, common pattern from the classification reports is that the minority classes were generally more difficult to predict. That is reasonable, since a lack of training examples makes it difficult to learn class-specific properties.\n\n\\subsection{Interpretation of Model Performances}\nThe \\class{RandomForestClassifier()}, being an ensemble of a large number of, uncorrelated decision trees, as well as the \\class{NeuralNetworkClassifier()} are complex models, with longer training times, but more accurate predictions. This explains, why both models outperform the simpler \\class{DecisionTreeClassifier()}. \nThe fact that the \\class{RandomForestClassifier()} gives better results than the \\class{NeuralNetworkClassifier()} is probably due to the fact that neural networks usually need larger amounts of training data to give good results. In fact, neural networks perform best on complex classification problems with large amount of data to learn from. Since this classification problem is quite the opposite of that, the neural network is not performing as well as the \\class{RandomForestClassifier()}.\n\n%- 1. difference between high performant, but slow/ less performant, but fast\n%- 2. neural net not as good, because usually needs more data points\n\n%- 3. however, when choosing a model, dt might stil be intersting, since transparent model (main advantage not only speed, but also interpretability)\n\n\\begin{comment}\nFirst, all models were able to almost perfectly predict class $7$ which goes hand in hand with the findings from EDA and figure $2$ where even using only first two principal components, it was possible to see the clear separation of class $7$ from others. Second, more challenging proved to be a prediction of classes $1$ and $2$. In order to classify them correctly, the model needed to establish complex decision boundary. From this perspective, it was expected that \\class{DecisionTreeClassifier()} will perform the worst as it can only divide feature space with straight boundaries. This expectation was wrong as \\class{NeuralNetwork()} (both implementations) had similar results. One possible explanation is that \\class{NeuralNetwork()} got stuck in a local minimum during training and even advanced optimizer (Adam) did not help to overcome this. More interestingly, \\class{RandomForestClassifier()} performed way better than the other two models. Its core advantage is concealed in its ability to rely on a majority vote of a large and diverse set of \\class{DecisionTreeClassifier()} trained only on a subsample of original data. Thanks to this diversity, it is capable of fitting complex patterns but even more importantly generalize well. In other words, it is less likely to make an error due to small variation in data.\n\n\\subsection{Learning from a small data set}\nIn the given data-set, there were three minor classes, namely $3$, $5$ and $6$. From EDA (Figure 3), it was clear that it will be very hard to classify them correctly as they all overlapped with the other majority classes. This proved to be a problem especially for \\class{DecisionTreeClassifier()}. A good explanation can be found from its visualization and following the splits of class 3 for example. In majority of cases (3 out of 5), it ends being in a leaf node either with class 1 or 2 with which it has a large overlap. Further, the nodes are almost pure, thus the model has no further incentive to split it. In addition, since these classes have more records, they are also more likely to be predicted. Finally, compare to other two methods, \\class{NeuralNetwork()} is parametric and given the chosen architecture, it has a large number of parameters to be tuned. Therefore, it also needs a larger training data-set in order for it to work well. This might be a possible explanation why it outperformed by \\class{RandomForestClassifier()} for prediction of classes $1$ and $2$.\n\n%\\subsection{Interpretability}\n%\\class{DecisionTreeClassifier()}'s core advantage compare to the other two models is that its decision can be simply translated into just if-else statements. For example, can see that within its first two levels (root is level 1), it only uses first or second principal component to decide. This in fact makes sense as the first two components explain most of the variability of the given features and thus it is a good way to separate the classes. On the opposite site, trying to understand how a \\class{RandomForestClassifier()} decided is way more difficult and infeasible as it would require to go through all the 100 trees.\n\\end{comment}\n\n\\subsection{Conclusion}\nThe analysis has shown that a model chosen solely on the basis of best performance (\\class{RandomForestClassifier()}) has an expected out-of-sample performance of over 80\\%, which is a solid result given the challenges of the problem at hand. However, in the light of the model being used in assisting criminal investigations, transparency in the way decision are being made might be relevant, i.e. in court trials. Thus, if interpretability of the model is a core requirement for the final model, a \\class{DecisionTreeClassifier()} should be considered.", "meta": {"hexsha": "422af1f19b83b7ad2c35b1b6a3b03f9f128997c2", "size": 8237, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "report/06_results_discussion.tex", "max_stars_repo_name": "jonas-mika/ml-project", "max_stars_repo_head_hexsha": "c052c33010033cd9fd596eb5ac4d270d1bf98ee3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "report/06_results_discussion.tex", "max_issues_repo_name": "jonas-mika/ml-project", "max_issues_repo_head_hexsha": "c052c33010033cd9fd596eb5ac4d270d1bf98ee3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "report/06_results_discussion.tex", "max_forks_repo_name": "jonas-mika/ml-project", "max_forks_repo_head_hexsha": "c052c33010033cd9fd596eb5ac4d270d1bf98ee3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-29T17:23:15.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-29T17:23:15.000Z", "avg_line_length": 158.4038461538, "max_line_length": 1330, "alphanum_fraction": 0.7983489134, "num_tokens": 1714, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6261241842048093, "lm_q1q2_score": 0.34716736753900884}} {"text": "\\documentclass{article}\n\\usepackage[T1]{fontenc}%\n\\usepackage[utf8]{inputenc}%\n\\usepackage{lmodern}%\n\\usepackage{textcomp}%\n\\usepackage{lastpage}%\n\\input{common_symbols_and_format.tex}\n\n\n\\begin{document}\n\\logo\n\\rulename{Simple Moving Averages}\n\\tblofcontents\n\n\\ruledescription{\nThese representations have the position dependent on comparing two moving average windows one is short and the other is Long when the short window average length $\\averagelengthshort$ is greater than the long one average length $\\averagelengthlong$ so the model will go for the long position and when the short $\\averagelengthshort$ is less than the long one $\\averagelengthlong$, the position will be short}\n\\\\\n\n\\ruleparameters\n{Long Window size}{10}{This is the number of time steps over which exponential contributions are sourced}{$\\averagelengthlong$}\n{Short Window Size}{5}{This is the decay factor that reduces older contributions from the price series}{$\\averagelengthshort$}\n\\stoptable\n\\section{Equation}\n\\begin{equation}\n\\sum_{\\dummyiterator}^{\\averagelengthlong} \\price_{\\currenttime - \\dummyiterator} \\\\\n\\label{eq1}\n\\end{equation}\n\n\\begin{equation}\n\\sum_{\\dummyiterator}^{\\averagelengthshort} \\price_{\\currenttime - \\dummyiterator} \\\\\n\\label{eq2}\n\\end{equation}\n\n\\[ \\position_\\currenttime = \\begin{cases} \\mbox{1,} & \\mbox{if } equ.\\ref{eq2}>equ.\\ref{eq1} \\\\ \\mbox{0,} & \\mbox{if } equ.\\ref{eq1}>equ.\\ref{eq2} \\end{cases} \\]\n\n\n\\hspace{200mm}\n\nwhere $\\price_\\currenttime$ is the price at time $\\currenttime$,\n\n\\hspace{200mm}\n\\hspace{200mm}\n\n\\keyterms\n\\furtherlinks %The footer\n\\end{document}\n", "meta": {"hexsha": "9d27b534beb8292f8fe98a502959edc509627d94", "size": 1581, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/strategies/tex/SimpleMovingAverage.tex", "max_stars_repo_name": "SaurabhBansalInferstat/infertrade", "max_stars_repo_head_hexsha": "ee72d72cec54b2d29243f23dd0b6d9caf5dc49e8", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-08-07T14:40:51.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-07T14:40:51.000Z", "max_issues_repo_path": "docs/strategies/tex/SimpleMovingAverage.tex", "max_issues_repo_name": "GirijaDas9/infertrade", "max_issues_repo_head_hexsha": "9245eaae202aea997e62d4eeb299f70516da35e0", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/strategies/tex/SimpleMovingAverage.tex", "max_forks_repo_name": "GirijaDas9/infertrade", "max_forks_repo_head_hexsha": "9245eaae202aea997e62d4eeb299f70516da35e0", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.6382978723, "max_line_length": 408, "alphanum_fraction": 0.7621758381, "num_tokens": 467, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6261241842048093, "lm_q1q2_score": 0.34716736753900884}} {"text": "\\documentclass[a4paper, 11 pt, article, accentcolor=tud7b]{tudreport}\n\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath}\n\\usepackage{placeins}\n\\usepackage{tabularx}\n\\usepackage{subcaption}\n\n\\title{CNuVS Exercise 9}\n\\author{Nils Rollshausen, Daniel Drodt}\n\\subtitle{Nils Rollshausen, Daniel Drodt}\n\n\\begin{document}\n\t\\maketitle\n\t\\section{TCP Congestion control}\n\t\\subsection*{a) Congestion control vs flow control}\n\tFlow control intends to protect the receiver from overload while congestion control protects the network. TCP specifically uses a window-based approach to congestion control.\n\t \n\t\\subsection*{b) Congestion window}\n\tThe amount added to the congestion window with each ack is given as $inc = MSS \\cdot \\frac{MSS}{window}$. So, on the first acknowledgement, we increment the window by $1320 \\cdot \\frac{1320}{13200} = 132$ bytes to $13332$ bytes. On the second ack, we increment by $1320 \\cdot \\frac{1320}{13332} = 130.69$ byte to a new congestion window of $13462.69$ byte.\n\t\n\t\\subsection*{c) Slow start}\n\tAfter establishing a connection or pausing the connection when congestion is detected, it would take a long time for TCP to incrementally adjust the congestion window to its proper value. To speed this process up, the congestion window is increased exponentially (as opposed to linearly in the regular congestion avoidance phase) until the desired window size is reached (half the previous window if a packet loss has been detected) or a packet is lost.\n\t\n\t\\section{Three-Way-Handshake}\n\t\n\t\\subsection*{a) Handshake}\n\tThe three-way-handshake consists of the initiating party sending a SYN packet to the receiving party. The receiving party responds to the SYN request with a SYN-ACK packet acknowledging the receipt of the SYN request. Finally, the initiating party acknowledges the receipt of the SYN-ACK with an ACK packet.\n\t\n\t\\subsection*{b) TCP Segment}\n\t\n\tIn addition to the fields already given, the TCP header contains the following values:\n\t\n\t\\begin{table}[h]\n\t \\centering\n\t \\begin{tabular}{|l|l|}\n\t \\hline\n\t Source port & 0x076c \\\\ \\hline\n\t Destination port & 0x07cd \\\\ \\hline\n\t Data offset & 0x5 \\\\ \\hline\n\t Checksum & 0x3c8f \\\\ \\hline\n\t \\end{tabular}\n\t \\caption{Remaining TCP header fields}\n\t\\end{table}\n\t\n\tThe application data is \\verb|0x4e6f 7274 6865 726e 204c 6967 6874 7300|. The checksum is calculated as the inverse two's-complement sum as follows: \n\t\n\t\\begin{align*}\n\t 0xFFFF - (&0x076c + 0x07cd + 0x0000 + 0x1337 + 0x0000 + 0x0000 \\\\\n\t &+ 0x5020 + 0x5000 + 0x0000 + 0x0000 + 0x4e6f + 0x7274 \\\\\n\t &+ 0x6865 + 0x726e + 0x204c + 0x6967 + 0x6874 + 0x7300) \\\\\n\t = &0xFFFF - 0xC370 = 0x3C8F\n\t\\end{align*}\n\t\n\t\\newpage\n\t\n\t\\subsection*{c) TCP header fields}\n\t\n\t\\begin{table}[h]\n\t \\centering\n\t \\begin{tabularx}{\\linewidth}{|l|X|}\n\t \\hline\n\t Source port & The port from which the packet is sent \\\\ \\hline\n\t Destination port & The port to which the packet is sent \\\\ \\hline\n\t Sequence number & Sequence number of the packet, corresponds to bytes already sent \\\\ \\hline\n\t Acknowledgement number & If the ACK flag is set, this acknowledges the receipt of all bytes up to the acknowledgement number \\\\ \\hline\n\t Data offset & The length of the header in 32 bit words, used to determine where the payload starts \\\\ \\hline\n\t Reserved & Reserved bits intended for future extensions of the protocol \\\\ \\hline\n\t URG & Indicates that the Urgent Data Pointer field is valid and should be considered \\\\ \\hline\n\t ACK & Indicates that the acknowledgement number is valid \\\\ \\hline\n\t PSH & Indicated that received data should be pushed to the receiving process immediately \\\\ \\hline\n\t RST & Resets the connection \\\\ \\hline\n\t SYN & Indicates a connection request \\\\ \\hline\n\t FIN & Indicates a connection termination \\\\ \\hline\n\t Window Size & The amount of bytes the sender of the packet is currently willing to receive, starting at the value in the acknowledgement field \\\\ \\hline\n\t Checksum & The 16bit twos-complement checksum over the entire packet \\\\ \\hline\n\t Urgent Pointer & If URG is set, this indicates that the first n bytes from the message start are urgent \\\\ \\hline\n\t \\end{tabularx}\n\t \\caption{TCP header fields}\n\t\\end{table}\n\t\n\t\\subsection*{d) Retransmission}\n\t\\begin{figure}[h]\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[width=\\textwidth]{retrans1.png}\n \\caption{Lost ACK}\n \\label{fig:1}\n \\end{subfigure}\n %\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[width=\\textwidth]{retrans2.png}\n \\caption{Premature Timeout}\n \\label{fig:2}\n \\end{subfigure}\n %\n \\begin{subfigure}[b]{0.3\\textwidth}\n \\includegraphics[width=\\textwidth]{retrans3.png}\n \\caption{Cumulative ACK}\n \\label{fig:2}\n \\end{subfigure}\n \\end{figure}\n\tWhen an ACK packet is lost, the sender would, after a timeout, consider the packet whose acknowledgement got dropped to be lost and re-send it. The receiver recognizes the re-sent packet as a duplicate, discards it, and responds with an up-to-date acknowledgement.\n\t\n\t\\FloatBarrier\n\t\n\t\\section{RED, State Chart \\& Nagle}\n\t\n\t\\subsection*{a) Random Early Detection}\n\tRandom Early Detection is necessary for routers and other network equipment to signal to end nodes that they are running close to their maximum capacity. As the routers should be agnostic to the choice of transport protocol, there is no protocol-inherent mechanism for them to communicate their capacity. As a solution, routers communicate implicitly by randomly dropping packets when running close to maximum load to reduce the network load before actually running completely out of capacity, which would lead to heavy congestion.\n\t\n\t\\subsection*{TIME\\_WAIT and ESTABLISHED}\n\tThe TIME\\_WAIT state is used by the party terminating a connection to ensure that the other party was able to terminate the connection properly (i.e. it received the acknowledgement to its FIN). A duplicate received FIN in this state indicates that the previous ACK was lost and should be retransmitted. After a sufficient amount of time passes, it can be assumed that no such duplicate FIN will arrive and the remote party was able to terminate the connection. \\\\ \\medskip\n\tThe ESTABLISHED state is the normal state of a TCP connection in which both parties have established the connection and are able to send and receive data.\n\t\n\t\\subsection*{c) Nagle's Algorithm}\n\tNagle's Algorithm deals with the problem network congestion caused by acknowledging small amounts of data and thus creating very high network overhead for comparatively little data. It states that data should only be sent if either the amount of data to send is larger or equal to the maximum segment size or there is no unacknowledged data packet still in-flight to the receiver. Using this algorithm, non-full packets are only sent when it is assumed that the receiver is currently idle.\n\t\n\t\\end{document}\n", "meta": {"hexsha": "a4641b89e697ec966edbae00fc8e85ff5fdc9a95", "size": 6948, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ex9/ex9.tex", "max_stars_repo_name": "rec0de/CNuVS19", "max_stars_repo_head_hexsha": "52d07fe5c4380af707c63f718aa9533044224a3e", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "ex9/ex9.tex", "max_issues_repo_name": "rec0de/CNuVS19", "max_issues_repo_head_hexsha": "52d07fe5c4380af707c63f718aa9533044224a3e", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ex9/ex9.tex", "max_forks_repo_name": "rec0de/CNuVS19", "max_forks_repo_head_hexsha": "52d07fe5c4380af707c63f718aa9533044224a3e", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 58.3865546218, "max_line_length": 532, "alphanum_fraction": 0.7443868739, "num_tokens": 1809, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6261241842048092, "lm_q1q2_score": 0.3471673675390088}} {"text": "\\documentclass[11pt,a4paper]{article}\n\\usepackage[english]{babel}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath}\n\\usepackage{fourier}\n\\usepackage{yfonts}\n\\usepackage[hmargin=3cm,vmargin=2.4cm]{geometry}\n\\usepackage[colorlinks=true]{hyperref}\n\\usepackage{fancyhdr}\n\n\\pagestyle{fancy}\n\\fancyhead{}\n\\fancyhead[L]{\\href{https://www-ljk.imag.fr/membres/Pierre.Karpman/cry\\_eng2017\\_td\\_hf.pdf}{https://www-ljk.imag.fr/membres/Pierre.Karpman/cry\\_eng2017\\_td\\_hf.pdf}}\n\n\n\\DeclareMathOperator\\hash{\\mathcal{H}}\n\\DeclareMathOperator\\E{\\mathcal{E}}\n\\DeclareMathOperator\\comp{\\textswab{f}}\n\\DeclareMathOperator\\cat{\\mathrm{CAT}}\n\\DeclareMathOperator\\pmm{\\mathrm{PM}}\n\\DeclareMathOperator\\smm{\\mathrm{SM}}\n\\DeclareMathOperator\\sand{\\mathrm{SANDWICH}}\n\n\\title{Crypto Engineering (GBX9SY03)\\\\\nTD Hash functions}\n\\date{2017-10-18}\n\n\\begin{document}\n\n\\maketitle{}\n\n\\subsection*{Exercise 1: Multicollisions for Merkle-Damg\\aa rd hash functions}\nIn 2004, Joux showed a simple attack illustrating the fact that Merkle-Damg\\aa rd hash functions were not ``ideal''. This attack consists in computing a collision on many\n(more than two) messages, i.e. finding $m_0$, $m_1$, $\\ldots$, $m_q$ that all have the same hash, more efficiently than what is possible for a random oracle.\\\\\n\\emph{In this exercice, we can safely ignore padding issues.}\n\n\\paragraph{Q. 1:} We assume that the expected number of collisions in the elements of two lists $L_0$ and $L_1$ of random $n$-bit elements is $\\approx \\#L_0\\times\\#L_1/2^n$.\nLet $\\hash : \\{0,1\\}^* \\rightarrow \\{0,1\\}^n$ be a random oracle; what is the expected complexity of finding an $r$-collision for $\\hash$?\\\\\n{\\footnotesize Hint: try to find the optimal balance in the list sizes for the case $r = 3$, and generalize the formula.}\n\n\\paragraph{Q. 2:} Recall the structure of a Merkle-Damg\\aa rd hash function\n$\\hash : \\{0,1\\}^* \\rightarrow \\{0,1\\}^n$ based on a compression function $\\comp : \\{0,1\\}^n \\times \\{0,1\\}^b \\rightarrow \\{0,1\\}^n$.\nLet $m$, $m'$ be two messages such that $|m| = |m'| = b$.\nExplicit the relation between $\\hash(m)$, $\\hash(m||m')$ and $\\comp$.\n\n\\paragraph{Q. 3} Take $\\hash$ as in Q.~2. Let $m_0$ and $m'_0$ be two one-block messages colliding through $\\hash$ (i.e. $h_0 :=\n\\hash(m_0) = \\comp(\\text{IV}, m_0) = \\hash(m'_0) = \\comp(\\text{IV}, m'_0)$). Assuming $\\comp$ is ideal, how efficiently can you compute a collision\n($m_1$, $m'_1$) for $\\comp(h_0, \\cdot)$? Once you know such a collision, how many messages colliding with $\\hash(m_0||m_1)$ can you easily (i.e. in\nconstant time) create? Conclude about the cost of\ncomputing a $2^r$-collision for $\\hash$ and why Merkle-Damg\\aa rd hash functions are not ideal.\n\n\\paragraph{Q. 4} The \\emph{concatenation combiner} is a simple construction taking two hash functions $\\hash_1$ and $\\hash_2$ and defined as\n$\\cat_{\\hash_1,\\hash_2}(m) := \\hash_1(m)||\\hash_2(m)$. Assuming $\\hash_1$ and $\\hash_2$ have an output size of $n$ bits and follow the Merkle-Damg\\aa rd construction,\nhow efficiently can you compute a collision for $\\cat_{\\hash_1, \\hash_2}$? Is it possible to significantly improve the collision-resistance of SHA-1 by using\n$\\cat_{\\text{SHA-1}, \\text{MD5}}$?\n\n\n\\subsection*{Exercise 2: Davies-Meyer fixed-points}\nIn this exercise, we will see one reason why \\emph{Merkle-Damg\\aa rd strengthening} (adding the length of a message in its padding) is necessary\nin some practical constructions.\n\n\\paragraph{Q. 1} Recall the ``Davies-Meyer'' construction of a compression function $\\comp$ from a block cipher~$\\E$.\n\n\\paragraph{Q. 2} Considering the feed-forward structure of Davies-Meyer, under what conditions would you obtain a fixed-point for such a compression function?\n\n\\paragraph{Q. 3} Show how to compute the (unique) fixed-point of $\\comp(\\cdot,m)$ for a fixed $m$. Given $h$, is it easy to find $m$ such that it is a fixed-point,\nif $\\E$ is an ideal block cipher?\n\n\\paragraph{Q. 4} A \\emph{semi-freestart collision attack} for a Merkle-Damg\\aa rd hash function $\\hash$ is a triple $(h, m, m')$ s.t. $\\hash_h(m) = \\hash_h(m')$,\nwhere $\\hash_h$ denotes the function $\\hash$ with its original IV replaced by $h$.\nShow how to use a fixed-point to efficiently mount such an attack for Davies-Meyer + Merkle-Damg\\aa rd, when strengthening is not used.\n\n\\paragraph{Note:} Fixed-points of the compression function can be useful to create the \\emph{expandable messages} used in second preimage attacks on Merkle-Damg\\aa rd.\n\n\n\\subsection*{Exercise 3: Meet-in-the-middle preimage attack on BRSS/PGV-13 + MD}\nBRSS/PGV-13 is an alternative to Davies-Meyer, defined as $\\comp(h,m) = \\E(m,h) \\oplus c$ for a cipher $\\E$ and with $c$ a constant.\nIt can be shown in the ideal cipher model that\na Merkle-Damg\\aa rd function with such a compression function is secure up to the birthday bound for both collision \\emph{and} preimage attacks (Black \\& al., 2010).\n\n\\paragraph{Q. 1} If $\\E$ is ideal, what is the complexity, given $h$ and $t$, of finding $m$ such that $\\comp(h, m) = t$? Conclude about the preimage security\nof $\\comp$ itself.\n\n\\paragraph{Q. 2} Show how to compute a two-block preimage for $\\hash$ with the above compression function, using a\nmeet-in-the-middle attack, and roughly evaluate its complexity (both time and memory).\n\n\\paragraph{Q. 3} Give a rough explanation of how the attack of Q.~2 is prevented when using a Davies-Meyer compression function.\n\n\\subsection*{Exercise 4: Hash-based message-authentication codes}\n\n\\paragraph{Q. 1} Recall the definition of a message-authentication code (MAC), existential forgery, and universal forgery.\n\n\n\\paragraph{Q. 2} Let us first assume that $\\hash$ is a random oracle. Explain (roughly) why the ``prefix-MAC'' construction\n$\\pmm_{\\hash}(k, m) := \\hash(k||m)$ is secure? Is there a difference with the ``suffix-MAC'' variant $\\smm_{\\hash}(k, m) := \\hash(m||k)$?\n\n\\paragraph{Q. 3} Now assume that $\\hash$ is a Merkle-Damg\\aa rd hash function. Suppose I know $m$ and its tag $t := \\pmm_{\\hash}(k, m)$,\nand that the size of $k$ is known. How easily can I compute another message and its corresponding tag under $\\pmm_{\\hash}(k, \\cdot)$? Is this\nMAC secure against existential forgery?\n\n\\paragraph{Q. 4} Still assuming that $\\hash$ is a Merkle-Damg\\aa rd function, show how collisions on $\\hash$ lead to an existential forgery attack\nof $\\smm_{\\hash}$. What is the expected complexity of this attack for an otherwise secure $\\hash$? Is this better than what you would expect for a ``good'' MAC?\n\n\\paragraph{Q. 5} Is it reasonable to instantiate prefix/suffix-MAC with SHA-3? With SHA-256? With SHA-512/256?\n\n\\paragraph{Note:} It can be proven (Yasuda, 2007) that, using appropriate padding rules, the ``Sandwich-MAC'' construction $\\sand_{\\hash}(k,m) := \\hash(k||p||m||p'||k)$\n(where $p$ and $p'$ denote padding) is secure, without requiring $\\hash$ to be a random oracle (in particular, it can be built with a Merkle-Damg\\aa rd construction).\n\n\\end{document}\n", "meta": {"hexsha": "9c29f5960f9573abb20c94224fd394fc132e6485", "size": 6927, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "CRY-M2-ENG/2017/TD/td_hf.tex", "max_stars_repo_name": "P1K/LectureNotes", "max_stars_repo_head_hexsha": "7a2a2e912afe9b1eff01430194761f4ae1bc65c5", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-10-18T06:00:24.000Z", "max_stars_repo_stars_event_max_datetime": "2017-10-18T06:00:24.000Z", "max_issues_repo_path": "CRY-M2-ENG/2017/TD/td_hf.tex", "max_issues_repo_name": "P1K/LectureNotes", "max_issues_repo_head_hexsha": "7a2a2e912afe9b1eff01430194761f4ae1bc65c5", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "CRY-M2-ENG/2017/TD/td_hf.tex", "max_forks_repo_name": "P1K/LectureNotes", "max_forks_repo_head_hexsha": "7a2a2e912afe9b1eff01430194761f4ae1bc65c5", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 62.9727272727, "max_line_length": 173, "alphanum_fraction": 0.7278764256, "num_tokens": 2117, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34716736367074236}} {"text": "\\documentclass[a4paper]{letter}\r\n\\usepackage{hyperref}\r\n\\usepackage{color}\r\n\r\n\\title{Tutorial in Computational Science using Visual Python}\r\n\r\n\\begin{document}\r\n\\definecolor{code}{cmyk}{1,0,1 ,.5}\r\n\r\n\r\n\r\n\\section{Introduction to Visual python: Simple motion}\r\nThe first section of the tutorial introduces you to programming in visual python by stepping \r\nthrough a program to display a ball bouncing around inside a box. \r\nThis program will later be a modified to become a simulation of an ideal gas.\r\n\\subsection{Your First Program}\r\nStart IDLE by clicking on the snake icon on the panel at the bottom of your desktop. \r\nA window labeled 'Untitled' should appear. \r\nThis is the window in which you will type your program.\r\nType the following statements \r\n(You can select the text below and then paste it into IDLE using the middle mouse button.):\r\n{\\color{code}\r\n\\begin{verbatim}\r\nfrom visual import * # This instructs Python to use the Visual graphics module\r\n \r\nwallR = box(pos=vector(6,0,0),size=vector(0.2,4,4),color=color.green) \r\n#opens a window for displaying graphics and creates a thin green slab \r\n#in it which we can refer to later as wallR\r\nball = sphere(pos=vector(-5,0,0), radius=0.5, color=color.red)\r\n#creates a red sphere and names it ball\r\n\\end{verbatim}}\r\nIDLE will colour different sections of the program to make it easier to read.\r\nFor example comment lines, which begin with the \\# symbol are red.\r\nThe functions {\\color{code}sphere} and {\\color{code}box} are 'constructors' for 3D objects. \r\nThe position of an object's centre is specified in three dimensions \r\nusing a vector.\r\n\r\nSave your program as 'firstprogram.py' by choosing 'save as' from the file menu.\r\n\r\n\\subsection{Running the Program}\r\nNow run your program by choosing 'Run program' from the 'Run' menu. \r\nWhen you run the program, two new windows appear. \r\nThere is a window titled 'VPython,' in which you should see a red sphere, \r\nand a window titled 'Output'. \r\n\r\nIn the VPython window, hold down the middle mouse button and move the mouse. \r\nYou should see that you are able to zoom into and out of the scene. \r\nNow try holding down the right mouse button . \r\nYou should find that you are able to rotate your view of the scene. \r\n\r\nYou can stop the program by choosing 'Stop program' from the run menu.\r\n\r\n\\subsection{Objects and attributes}\r\nSphere and box are types of objects that visual python recognises and displays. \r\nThey have some attributes associated with them (such as \r\nposition({\\color{code}pos}), colour({\\color{code}color}) and \r\n{\\color{code}radius}) which can be set when you first define and name the object, \r\notherwise the default values will be used. \r\n\r\nYou can change the radius attribute of {\\color{code}ball} after it has been constructed \r\nwith the statement:\r\n{\\color{code}\\begin{verbatim}\r\nball.radius=0.5 \r\n\\end{verbatim}}\r\nYou can also add any new attributes you want to associate with the object, eg: \r\n{\\color{code}\\begin{verbatim}\r\nball.velocity=vector(2,.1,0) \r\n\\end{verbatim}}\r\n\r\n\\subsection{Moving Objects}\r\nTo move objects you will need a program loop that repeatedly updates the position. \r\nThis can be done with a while loop that repeats the indented lines of code following \r\nthe while statement, as long as the while condition remains true.\r\n\r\nAdd the following while loop to the bottom of your program to make the ball move according to the velocity you gave it earlier:\r\n{\\color{code}\\begin{verbatim}\r\ntimestep=0.05 \r\nball.velocity=vector(2,.1,0) \r\nwhile (1==1):\r\n rate(20) \r\n ball.pos=ball.pos + ball.velocity*timestep \r\n\\end{verbatim}}\r\nIn Python {\\small\\color{code}==} stands for 'is equal to' \r\nso the statement {\\small\\color{code}1==1} always has a value of true,\r\n and code inside the while loop will repeat forever. \r\nThe symbol {\\small\\color{code}=} is used for assignment, to set a new value for a variable. \r\n\r\nIn the above example the value of {\\color{code}ball.pos + ball.velocity*timestep} \r\nis calculated and then assigned as the new value for ball.pos. \r\nPython knows how to add vectors and how to multiply them by a scalar so you don't have \r\nto specify this element by element. \r\n\r\nThe rate statement specifies how many times each second the loop will be executed. \r\nIt allows you to control the animation speed so that the ball moves at the same speed \r\non faster computers.\r\n\r\nRun your program by choosing 'Run program' from the 'Run' menu. \r\nYou should observe that the ball moves to the right. \r\nYou can change how fast either by changing the rate or the timestep. \r\n\r\n\\subsection{Making the ball bounce: Logical tests}\r\nTo make the ball bounce off the wall, we need to detect a collision between the ball and the wall. \r\nA simple approach is to compare the x coordinate of the ball to the x coordinate of the wall, \r\nand reverse the x component of the ball's velocity if the ball has moved too far to the right. \r\nThe components of vectors are attributes which can be referred to individually as .x, .y and .z.\r\nThe logical test we would use to detect a collision and reverse the velocity might look like: \r\n{\\color{code}\\begin{verbatim}\r\nif ball.x > wallR.x: \r\n ball.velocity.x = -ball.velocity.x \r\n\\end{verbatim}}\r\nThe indented line after the if statement will be executed only if the \r\nlogical test in the previous line gives 'true' for the comparison. \r\nIf the result of the logical test is false \r\n(that is, if the x coordinate of the ball is not greater than the x coordinate of the wall), \r\nthe indented line will be skipped. \r\nSince we want this logical test to be performed every time the ball is moved, \r\nwe need to indent both of these lines so they are inside the while loop. \r\nYour program should now look like this: \r\n{\\color{code}\\begin{verbatim}\r\nfrom visual import * \r\n\r\n# create wall\r\nwallR = box(pos=vector(6,0,0), size=vector(0.2,4,4), color=color.green) \r\n#create a ball and define its velocity\r\nball = sphere(pos=vector(-5,0,0), radius=0.5, color=color.red) \r\nball.velocity = vector(2,.1,0) \r\n# set the timestep between position updates\r\ntimestep = 0.05 \r\nwhile (1==1): \r\n#the code within this while loop will be repeated 100 times per second\r\n rate(100) \r\n ball.pos = ball.pos + ball.velocity*timestep #move ball\r\n if ball.pos.x > wallR.pos.x: #check for collision with wall\r\n ball.velocity.x = -ball.velocity.x #reflect velocity at collision\r\n\\end{verbatim}}\r\nRun your program. \r\nYou should observe that the ball moves to the right, bounces off the wall,\r\nand then moves to the left, continuing off into space. \r\nNote that our test is not very sophisticated. \r\n{\\color{code}ball.x} is at the center of the ball \r\nand wallR.x is at the center of the wall so the ball penetrates the wall before it bounces. \r\n\r\nAdjust the position test so that the ball bounces when the edge of the balls reaches \r\nthe edge of the wall (ie when {\\color{code}ball.x + ball.radius == wallR.x-wallR.size.x/2}). \r\nFor a more realistic bounce you should reflect the position of the ball as well as its velocity. \r\nYou can add another wall at the left side of the display, and make the ball bounce off that wall also. \r\n\r\nYour program should now look something like the following: \r\n{\\color{code}\\begin{verbatim}\r\nfrom visual import * \r\n# create walls\r\nwallR = box(pos=vector(6,0,0), size=vector(0.2,4,4), color=color.green) \r\nwallL = box(pos=vector(-6,0,0), size=vector(0.2,4,4), color=color.green)\r\n# create a ball and define its velocity\r\nball = sphere(pos=vector(-5,0,0), radius=0.5, color=color.red) \r\nball.velocity = vector(2,.1,0) \r\n# set the maximum and minimum position for the centre of the ball\r\nmax_xpos=wallR.pos.x-ball.radius-wallR.size.x/2 \r\nmin_xpos=wallL.pos.x+ball.radius+wallL.size.x/2 \r\n# set the timestep between position updates\r\ntimestep = 0.05 \r\n\r\nwhile (1==1): \r\n#the code within this while loop will be repeated 100 times per second\r\n rate(100) \r\n ball.pos = ball.pos + ball.velocity*timestep #move ball\r\n if ball.x > max_xpos: #check for collision with right wall\r\n ball.velocity.x = -ball.velocity.x #reflect velocity\r\n ball.pos.x=2*max_xpos-ball.pos.x #reflect position\r\n if ball.x < min_xpos: #check for collision with left wall\r\n ball.velocity.x = -ball.velocity.x #reflect velocity\r\n ball.pos.x=2*min_xpos-ball.pos.x #reflect position\r\n\\end{verbatim}}\r\nThis program makes a ball bounce backward and forward between two parallel walls. \r\nThe ball continues to bounce even when it has passed the top of the walls. \r\n\r\nYou should be able to add and extend walls now so that the ball bounces inside a box. \r\nYou will want to have an invisible front wall so that you can see inside. \r\nPlay around with different starting positions and velocities for the ball, \r\nand sizes for the ball and box. \r\n\r\nIf you have trouble, or are short of time you can copy the example solution from the link below.\r\nThe next section is based on the solution to this exercise.\r\n%Example solution \\href{bounce2.py}{bounce2.py}\r\n%\\href{bounce2.html}{( html format )}\r\n\\href{bounce2.html}{ Example Solution }\r\n\r\n\\section{Simulating particle motion in a gas}\r\nIn this section you will extend the bouncing ball program so that it simulates the motion \r\nof atoms in a gas. \r\nThere will need to be many atoms in the box and they should bounce off each other as well\r\n as off the sides of the box.\r\nWe will need to set the initial distribution of atom positions and velocities, \r\nbut these distributions may evolve as the simulation progresses.\r\n\r\n\\subsection{Random numbers}\r\nYou can make the velocity of the ball different each time the program is run \r\nby using a random number generator. \r\nThe random number generator is not part of standard python and needs to be \r\nimported from the random module. \r\nThe random module contains random number generator for several different distributions. \r\nYou can import just a uniform distribution random number generator by inserting a line at \r\nthe beginning of the program (following the line {\\color{code}from visual import *}): \r\n{\\color{code}\\begin{verbatim}\r\nfrom random import uniform \r\n\\end{verbatim}}\r\nThe function {\\color{code}uniform(-1,1)} will give a random number between -1 and 1. \r\nYou can set a random ball velocity by replacing the line setting the ball's velocity with: \r\n{\\color{code}\\begin{verbatim}\r\nmaxv=2. \r\nball.velocity=maxv*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\n\\end{verbatim}} \r\n\\subsection{Multiple balls: Using lists}\r\nTo simulate a gas we will need many particles inside the box. \r\nOne easy way of dealing with many particles is to put them into a list.\r\n \r\nReplace the section of code in which you created the ball \r\nand gave it velocity with the following code \r\n(it should be after the section defining the walls):\r\n{\\color{code}\\begin{verbatim}\r\nside=wallR.pos.x\r\nballradius=0.4\r\nthk=wallR.size.x\r\nmaxpos=side-thk/2-ballradius \r\nmaxv=2.0 \r\n\r\nball1=sphere(color=color.green,radius=ballradius) \r\nball1.pos=maxpos*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\nball1.velocity=maxv*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\n\r\nball2=sphere(color=color.green,radius=ballradius) \r\nball2.pos=maxpos*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\nball2.velocity=maxv*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\n\r\nball3=sphere(color=color.green,radius=ballradius) \r\nball3.pos=maxpos*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\nball3.velocity=maxv*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1))\r\n\r\nballs=[ball1,ball2,ball3] \r\n\\end{verbatim}}\r\n\r\nThis gives a list containing three balls. Any type of object can go into a list. \r\nIn fact a list can be made up of several different types of object. \r\nIf you run the code now you will see three balls within the box but the section \r\nof code to move the ball no longer works.\r\n\r\nTo move all the balls in a list you use a {\\color{code}for} loop. \r\nIn python a {\\color{code}for} loop cycles through the elements of a list.\r\nThe loop to move all the balls in the list would look like:\r\n{\\color{code}\\begin{verbatim}\r\nwhile (1==1): \r\n for ball in balls: \r\n ball.pos = ball.pos + ball.velocity*timestep \r\n if ball.pos.x > maxpos: \r\n ball.velocity.x = -ball.velocity.x \r\n ball.pos.x=2*maxpos-ball.pos.x \r\n . . . \r\n\\end{verbatim}}\r\nThe code within the for loop is indented by two levels \r\nsince it is also inside the while loop (with code inside the if indented an additional level). \r\nYou can indent a whole section by selecting it and then choosing 'indent section' \r\nfrom the format menu.\r\n(Try running the program now to see three balls bouncing within your box)\r\n\r\nA for loop can also be used to set up the list of balls more efficiently. \r\nTo add each ball to the list {\\color{code}balls} use {\\color{code}balls.append(ball)}.\r\nThe construction of a for loop in python is slightly different from many \r\nother programming languages which cycle through a set of numbers. \r\nTo get the more usual counted for loop you use {\\color{code}range} to create a \r\nlist of integers between 0 and {\\color{code}no\\_particles-1}.\r\n\r\nRelplace the part of the code which created a list of three balls with this for loop \r\nwhich creates a list of ten balls. \r\nThe for loop which moves the balls in the list should still work perfectly. \r\n{\\color{code}\\begin{verbatim}\r\nno_particles=10 \r\nside=wallR.pos.x\r\nballradius=0.4\r\nthk=wallR.size.x\r\nmaxpos=side-thk/2-ballradius \r\nmaxv=2.0 \r\nballs=[]\r\nfor i in range(no_particles): \r\n ball=sphere(color=color.green,radius=ballradius) \r\n ball.pos=maxpos*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\n ball.velocity=maxv*vector(uniform(-1,1),uniform(-1,1),uniform(-1,1)) \r\n balls.append(ball) \r\n\\end{verbatim}}\r\n\r\n\\subsection{Particle interactions}\r\nYou should now have a program that shows many particles bouncing around inside a box. \r\nAt the moment the balls are simply passing through each other \r\n(make the ball radius large to see this more clearly). \r\nFor this to be a realistic model of a gas the particles need to interact in some fashion. \r\n\r\nDifferent 'interaction potentials' could be used depending on the type of molecule making up \r\nthe gas. \r\nFor an ideal gas an appropriate interaction is 'hard sphere collisions' \r\n--- the balls bounce off each other the same way they bounce off the walls. \r\n\r\n\\subsubsection{Detecting a collision}\r\nAt each time step we need to check whether any particles have collided. \r\nFor each pair of particles we need to check whether the distance between \r\nthem is less than the sum of their radii. \r\nFor example add the following lines to the end of your program, \r\nafter you have updated the position of all the particles. \r\nThe indentation should place it within the while loop but outside the other for loop.\r\n{\\color{code}\\begin{verbatim}\r\n for ball in balls: \r\n for otherball in balls:\r\n if not ball == otherball:\r\n distance=mag(otherball.pos-ball.pos) \r\n if distance<(ball.radius+otherball.radius): \r\n print 'collision' , distance\r\n\\end{verbatim}}\r\n{\\color{code}mag} is a function that returns the size (magnitude) of a vector. \r\n\r\nRun your program.\r\nNotice that every time a pair of balls collide in the display window the word 'collision' \r\nis printed twice to the output window. \r\nThis is because the loop goes through every pair of balls twice. \r\nTo check each pair only once the second loop should only check against balls with \r\nhigher index than the first. To do this replace the collision detection code with: \r\n{\\color{code}\\begin{verbatim}\r\n for i in range(no_particles): \r\n for j in range(i+1,no_particles): \r\n distance=mag(balls[i].pos-balls[j].pos) \r\n if distance<(balls[i].radius+balls[j].radius): \r\n print 'collision', distance \r\n\\end{verbatim}}\r\nNow when you run the program, 'collision' should be written once for each collision. \r\n \r\n\\subsubsection{Exchanging momentum: more vector calculations}\r\nWhen two objects collide elastically (without losing energy) in one dimension they swap momentum. \r\nIf their masses are equal this means they should swap velocity. \r\nSince we are modelling the interaction between particles as hard spheres, \r\nwith no friction, the force at impact is only along the line joining the centres of the balls. \r\nThe balls exchange momentum in this direction, \r\nleaving the perpendicular velocity components unchanged. \r\nThe result is similar to the way billiard balls collide.\r\nThe section of code that detects a collision and swaps the velocity component of the two balls \r\nin the collision direction looks like:\r\n{\\color{code}\\begin{verbatim}\r\n for i in arange(no_particles): \r\n for j in arange(i+1,no_particles): \r\n distance=mag(balls[i].pos-balls[j].pos) \r\n if distance<(balls[i].radius+balls[j].radius): \r\n direction=norm(balls[j].pos-balls[i].pos) \r\n vi=dot(balls[i].velocity,direction) \r\n vj=dot(balls[j].velocity,direction) \r\n exchange=vj-vi \r\n balls[i].velocity=balls[i].velocity+exchange*direction \r\n balls[j].velocity=balls[j].velocity-exchange*direction \r\n\\end{verbatim}}\r\nThe function {\\color{code}norm} returns a vector with the same direction as the input \r\nbut with its magnitude normalised to one. \r\nThe function {\\color{code}dot} returns the dot product of the two input vectors, \r\nin this case the component of the ball's velocity that is in the collision direction. \r\n\r\nYou should now have a program that shows ten spheres bouncing around inside a box \r\nand bouncing off each other. \r\nYou might find that balls occasionally get stuck together, \r\nparticularly if three or more balls collide at the same time. \r\nThis problem is reduced by making the time step smaller. \r\nWe can also make the collisions more accurate, and stop balls sticking together, \r\nby adjusting the position of the balls after a collision, in much the same way we did \r\nfor collisions with the sides of the box. \r\n{\\color{code}\\begin{verbatim}\r\n overlap=2*ballradius-distance \r\n balls[i].pos=balls[i].pos - overlap*direction \r\n balls[j].pos=balls[j].pos + overlap*direction \r\n\\end{verbatim}}\r\n%An example solution is at \\href{manybounce1.py}{manybounce1.py}\r\n% \\href{manybounce1.html}{(html version)}\r\n\\href{manybounce1.html}{Example solution} \r\n\r\n\\subsection{Program speed}\r\nThe program you have written shows a simple model of the behaviour of particles within a gas. \r\nPlay around with changing the number and size of particles and their average velocities. \r\nAs you add more particles the speed of the program slows down significantly. \r\nDoubling the number of particles may increase the time to run the program by four times. \r\nThis is because the number of steps to check for collisions increases as the square of the \r\nnumber of particles. \r\nMany computational science models which are based on interacting particles have this $N^2$ speed dependence. \r\nModels with large numbers of particles need high speed computers and efficient programs. \r\n\r\nIncreasing the speed of a simulation program can make up a large part of the work of a \r\ncomputational scientist. \r\nTactics for improving the efficiency of a program include:\r\n\\begin{itemize}\r\n\\item{removing unneccessary calculations (eg the calculation of the distance between particles involves calculating a square root whch is quite slow. The comparison of particle separation for detecting collisions could be made just as easily with the square of the distance which is much quicker to calculate.)}\r\n\\item{Saving results of calculations instead of repeating them. }\r\n\\item{Using more efficient data constructions (eg the elements of lists in python are not necessarily all the same data type. This makes them flexible for many different programming tasks but the program takes longer to access this data. Python arrays have each element the same data type and are faster to work with.) }\r\n\\item{Improved mathematical algorithms which allow larger step sizes for the same accuracy.}\r\n\\item{Minimise the number of particles needed by the model}\r\n\\item{Include only the most important interactions.} \r\n\\end{itemize}\r\n\r\n%\\href{quickbounce1.py}{quickbounce1.py} is a faster version of the above example program.\r\n\\href{quickbounce1.html}{Here} is a faster version of the ideal gas simulation program.\r\n\r\n\\subsection{Extracting useful information}\r\nThe visualisation of a model can be useful in itself to get insight into its behaviour. \r\nBut the program might be more useful as a computational science 'experiment' if we \r\ncould extract from it predictions of gas behaviour that could be measured in a \r\nphysical experiment or were relevant in some real world application. \r\nAlternatively we could use the results to confirm some theory about ideal gasses. \r\n\r\nTo do either of these we need to extract the sort of information that can normally \r\nbe measured since you don't get to watch the particles of a real gas bouncing around.\r\n\r\nExamples of some measurements we could extract from our simulation:\r\n\\begin{itemize}\r\n\\item{measure how the pressure on the walls varies with gas temperature and density}\r\n\\item{distribution of particle velocities}\r\n\\item{collision frequency}\r\n\\end{itemize}\r\n\\section{Velocity distribution: using graphs}\r\nWe gave our particles a uniform inititial velocity distribution for each direction. \r\nA statistical analysis suggests that the 'most likely' velocity distribution for each \r\ncomponent will have more low energy particles and a long tail of high energy particles. \r\nDoes our gas evolve to this 'most likely' velocity distribution.\r\n\r\n\\subsection{Graphing in visual python}\r\nWe will need the graphing functions contained in the visual.graph module, so insert a line \r\nat the beginning of your program\r\n{\\color{code}\\begin{verbatim}\r\nfrom visual.graph import *\r\n\\end{verbatim}}\r\nSeveral types of graph are available. \r\nWe will use a histogram that displays how many particles fit into each velocity range. \r\nThis graph is set up before the main while loop with:\r\n{\\color{code}\\begin{verbatim}\r\ngraphwindow=gdisplay(xtitle='v_x',ytitle='N',ymax=no_particles/2)\r\nvelocity_dist=ghistogram(bins=arange(0,2*maxv,maxv/5))\r\n\\end{verbatim}}\r\nvelocity\\_dist defines a histogram with a set of ten velocity 'bins' into which the list of \r\nparticle velocities will be sorted. \r\nYou might want to adjust the number of bins depending on the number of particles in your gas. \r\nThe graph will be displayed in a new graphing window. \r\n\r\nTo plot the distribution of particle speeds in the x direction you would form a list and use \r\nthis list as data for the histogram\r\n{\\color{code}\\begin{verbatim}\r\nvlist=[]\r\nfor ball in balls:\r\n vlist.append(abs(ball.velocity.x))\r\nvelocity_dist.plot(data=vlist)\r\n\\end{verbatim}}\r\nYou can put these lines within the while loop and see the velocity distribution evolve. \r\nUnless you have many particles in your gas the velocity distribution will jump around alot. \r\nYou can reduce this problem by averaging the distribution over time using:\r\n{\\color{code}\\begin{verbatim}\r\nvelocity_dist=ghistogram(bins=arange(0,2*maxv,maxv/5),accumulate=1,average=1)\r\n\\end{verbatim}}\r\n(You may find that displaying the graph slows down the simulation considerably. \r\nIf this is a problem, turn your while loop into a for loop so that it runs for a limited time \r\nand put all the graphing parts of the program after the loop finishes.)\r\n\r\nIf you are using the quick version of the code the velocity is stored as in varray, \r\nwhich can be fed directly to the histogram as:\r\n{\\color{code}\\begin{verbatim}\r\n velocity_dist.plot(data=varray[:,0])\r\n\\end{verbatim}}\r\n\\subsection{Compare with expected result}\r\nThe expected 'most likely' velocity distribution can be plotted for comparison as a curve set \r\nup with:\r\n{\\color{code}\\begin{verbatim}\r\nexpected_distribution=gcurve(color=color.green)\r\n\\end{verbatim}}\r\nand then plotted point by point with:\r\n{\\color{code}\\begin{verbatim}\r\nfor vx in arange(0,2*maxv,maxv/20):\r\n expected_distribution.plot(pos=(vx,.3*no_particles*exp(-vx**2/maxv**2*3/2)))\r\n\\end{verbatim}}\r\n(plot only once - so just before while loop. You may need to rescale this graph if you \r\nused a different number of velocity bins)\r\n\r\n% An example of graphing with the quick program is \r\n%\\href{quickbouncevdist.py}{quickbouncevdist.py}\r\n\\href{quickbouncevdist.html}{Here} is an example of graphing with the quick program.\r\n%An example of graphing with the slower program is\r\n%\\href{manybouncevdist.py}{manybouncevdist.py}\r\n\\href{manybouncevdist.html}{Here} is an example of graphing with the slower program.\r\n\r\nSome questions that you could use this program to investigate:\r\n\\begin{itemize}\r\n\\item {Is the initial velocity distribution different from the final velocity distribution?}\r\n\\item{Does the final velocity distribution match the expected result?}\r\n\\item{Does changing the initial velocity distribution effect the final distribution?}\r\n\\item{Consider the ways this simulation is different from a real gas. \r\nWhat changes might you need to make to model a real gas more closely? \r\nHow might this effect the behaviour of the simulation?}\r\n\\end{itemize}\r\n\\section{More complicated simulations}\r\nYou could extend this simulation to investigate more complex situations.\r\n\\begin{itemize}\r\n\\item{Mixing of two ideal gasses (diffusion)(eg look at \\href{quicktwogas.html}{quicktwogas})}\r\n\\item{Heat conduction}\r\n\\item{Other interaction potentials including some attraction may give phase change to \r\nsolid or liquid phases}\r\n\\item{Include effects of earth's gravity to get height variations in pressure and \r\ntemperature - or better simulation of liquid/gas interface}\r\n\\item{Velocity distribution when a gas includes molecules of different masses} \r\n\\item{Movable or elastic walls (eg look at \\href{quickbounce2.html}{quickbounce2})}\r\n\\end{itemize}\r\n%\\section{Other situations that can be modelled using similar techniques}\r\n%The technique of following individual particles, calculating their accelerations \r\n%due to their position and the position of other particles around them can be used \r\n%to simulate a wide variety of situations.\r\n%\\begin{itemize}\r\n%\\item{Molecular dynamics and computational chemistry can determine the \r\n%shape of molecules, their resonant frequencies and their potential reactions. \r\n%The most difficult part is to determine the interaction potentials which can depend on \r\n%relative positions of more than two particles, and also on angles between them.}\r\n%\\item{Gravitational interactions determine the orbits of planets in the solar system. \r\n%Precise calculations of these orbits determine whether the solar system is stable or chaotic, \r\n%whether orbits may change drastically at some future time, give clues as to how the solar \r\n%system form, and whether any asteroid is likely to hit the earth. \r\n%With large objects like planets their shape and rotation may need to be taken into account.}\r\n%\\item{Animal behaviour. Animals use the positions of other animals and objects around them to decide how they \r\n%should move in order to acheive their goals. The goals might include getting food, fleeing predators, staying \r\n%in a group and avoiding collisions. This simulations apply to people and traffic flow and can be used to design\r\n%adequate escape routes from burning buildings and sinking ships, road systems that minimise congestions, or to \r\n%suggest driving techniques that avoid collisions.}\r\n\r\n\\end{document} ", "meta": {"hexsha": "5643ea569d554a689594584adbb00e6e5a7cbceb", "size": 27544, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/simple.tex", "max_stars_repo_name": "stoiver/python_gas_tutorial", "max_stars_repo_head_hexsha": "0e931681507bb1f37ac662fca91e4e89c0893e73", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-04-14T22:29:15.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-23T17:08:31.000Z", "max_issues_repo_path": "tex/simple.tex", "max_issues_repo_name": "stoiver/python_gas_tutorial", "max_issues_repo_head_hexsha": "0e931681507bb1f37ac662fca91e4e89c0893e73", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/simple.tex", "max_forks_repo_name": "stoiver/python_gas_tutorial", "max_forks_repo_head_hexsha": "0e931681507bb1f37ac662fca91e4e89c0893e73", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2017-02-19T06:25:13.000Z", "max_forks_repo_forks_event_max_datetime": "2019-05-05T19:39:24.000Z", "avg_line_length": 52.265654649, "max_line_length": 321, "alphanum_fraction": 0.7482210282, "num_tokens": 6415, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467904168342855}} {"text": "\\chapter{Applications I: Pure Binary Arithmetic}\\label{arithchapter}\n\n[PLACEHOLDER]\n\n% [TODO I need to clean up this chapter. This paper this chapter is\n% based on presented the definitions in Prolog. I've translated the\n% code to miniKanren, but much of the English still refers to Prolog's\n% SLD resolution strategy and incomplete, depth-first search. I still\n% need to clean up this wording, since miniKanren uses a complete,\n% interleaving search strategy. I also need to integrate the text from\n% the intro and conclusion sections into the rest of the chapter, and\n% the thesis intro and conclusion chapters, as appropriate]\n\n% %%% This could bite me when it comes to Slatex. Better disable at end of this chapter.\n% %\\DefineShortVerb{\\|}\n\n% %\\begin{abstract}\n% We present \\emph{decidable} logic programs for addition, multiplication,\n% division with remainder, exponentiation, and logarithm with remainder over the\n% \\emph{unbounded} domain of natural numbers. Our predicates represent\n% \\emph{relations} without mode restrictions or annotations. They are\n% fully decidable under the common, Depth-First-Search-like, SLD resolution strategy of\n% Prolog, or under an interleaving refinement of DFS\\@, such as that used by miniKanren. We prove that\n% the evaluation of our arithmetic goals always terminates, given arguments \n% that share no logic variables. Further, the (possibly infinite) set of\n% solutions for a goal denotes exactly the\n% corresponding mathematical relation. (For SLD without interleaving, and for\n% some infinite solution sets, only half of the relation's domain may\n% be covered.) We define predicates to handle unary (for illustration)\n% and binary representations of natural numbers, and prove termination\n% and completeness of these predicates. Our predicates are written in pure miniKanren, without\n% \\scheme|conda|/\\scheme|condu|, \\scheme|varo|, \\scheme|project|, or other non-logical operators. The purity and\n% minimalism of our approach allows us to declare arithmetic in other\n% logic systems, such as Haskell type classes.\n% %\\end{abstract}\n\n% %% \\begin{keywords}\n% %% declarative arithmetic, binary, logic programming, termination\n% %% \\end{keywords}\n\n\n% \\section{Introduction}\n\n% Logic programming is said to be programming with relations, but\n% arithmetic is often dealt with in a non-relational, restricted way.\n% For example, Prolog's built-in \\verb|is/2| predicate for evaluating\n% arithmetic expressions does not allow free (unbound) logic variables\n% in the expressions. Whereas the goal \\verb|Z is 6*7|\n% succeeds, binding \\verb|42| to~\\verb|Z|, the related goal \\verb|42 is X*Y| is\n% considered erroneous because its multiplicands `are not sufficiently\n% instantiated.' Multiplication is not treated as a ternary relation\n% between multiplicands and the product because of mode restrictions on\n% the first two arguments. Constraint logic programming (CLP) overcomes\n% this drawback to some extent~\\cite{apt03principles}; for example,\n% disjunctive Datalog~\\cite{Eiter:1997:DD} treats arithmetic\n% relationally. Indeed, relational handling of arithmetic was one of\n% the motivations for CLP\\@. Unfortunately, this flexibility has a\n% price: CLP restricts the\n% arithmetic domain to be finite and changes the evaluation mode\n% of the logic programming system away from Kowalski's `predicate as\n% function' model~\\cite{Kowalski74a}.\n\n% We present fully relational arithmetic on the \\emph{unbounded} domain\n% of binary natural numbers for conventional\n% (SLD~\\cite{lloyd:lp}, or SLD with\n% interleaving~\\cite{backtracking}) logic\n% programming systems. We define predicates for addition,\n% multiplication, division with remainder, and logarithm with\n% remainder. These predicates express the remaining arithmetic\n% operations, including subtraction and exponentiation. These so-called base\n% predicates have \\emph{no} mode restrictions or annotations \n% on their arguments, and are\n% implemented in a \\emph{pure} logic system without\n% \\scheme|conda|/\\scheme|condu|, \\scheme|varo|, \\scheme|project|, or negation. Furthermore,\n% each base predicate terminates under SLD evaluation (and re-evaluation, upon\n% backtracking), provided that the predicate's arguments share no\n% logic variables. The stream of answers produced by (re-)evaluating\n% each arithmetic predicate under SLD with interleaving~\\cite{backtracking}\n% covers exactly the corresponding\n% mathematical relation; under SLD resolution and some infinite domains,\n% only half of the relation is covered.\n\n\n% In particular, we define the following decidable predicates:\n% %\n% \\begin{description}\n% \\item[\\scheme|addo|] such that \\scheme{(addo x y z)} can be used to add two numbers\n% \\scheme{x} and \\scheme{y} to get \\scheme{z}, to subtract \\scheme{x} or \\scheme{y} from \\scheme{z}, to decompose\n% \\scheme{z} into summands, or to compare two naturals. For example, \n% we can determine that the triple $(1,2,3)$ is in the ternary addition\n% relation%\n% \\footnote{We mean a relation that relates triples of numbers $(x,y,z)$\n% so that $x+y=z$.}\n% by\n% evaluating either the goal\n% \\scheme{(addo bn1 bn2 bn3)}%\n% \\footnote{\\bn{3} means the representation of the binary numeral 3, which in\n% miniKanren is encoded as \\schemeresult{(l l)} (\\S\\ref{s:binary}).}\n% or the goal:\n\n% \\schemedisplayspace\n\n% \\begin{schemebox}\n% (exist (x y z)\n% (addo x y z)\n% (== bn1 x)\n% (== bn2 y)\n% (== bn3 z)).\n% \\end{schemebox}\n\n% By evaluating \\scheme{(addo x bn3 bn2)} we determine, also in finite\n% time, that the addition relation does \\emph{not} include\n% \\emph{any} triple $(n,3,2)$.\n\n% \\item[\\scheme|mulo|] such that \\scheme{(mulo x y z)} can be used, inter alia, to\n% multiply \\scheme{x} and~\\scheme{y}, to factor~\\scheme{z},\n% or to generate a stream of triples related\n% by multiplication.\n\n% \\item[\\scheme|divo|] such that \\scheme{(divo n m q r)}\n% succeeds if and only if $n = m \\cdot q + r < m\\cdot(q+1)$.%\n% \\footnote{This equation implies that \\scheme{divo} fails when $m$ is\n% \\bn0. Hence \\scheme{mulo} is not reducible to \\scheme{divo}. Besides,\n% the former is simpler and is part of the implementation of the latter.}\n% For instance, the goal \\scheme{(divo bn1 bn0 q r)} to relate the \n% divisor zero to a non-zero dividend\n% fails instantly, without trying to enumerate all natural numbers.\n% The goal \\scheme{(divo bn5 m bn1 r)} finds all \n% numbers $m$ that divide 5, perhaps unevenly, \n% with a quotient of 1. These values are $5$, $4$, and $3$. Finally,\n% \\scheme{(divo bn5 m bn7 r)} fails in finite time rather than diverging.\n\n% \\item[\\scheme|logo|] such that \\scheme{(logo n b q r)} succeeds if\n% and only if $n = b^q + r < b^{q+1}$.\n% We can use \\scheme{logo} to perform exponentiation, find logarithms, \n% and find $n$-th roots.\n% \\end{description}\n% %\n% We prove that these base predicates are decidable and\n% that the arithmetic relations over natural numbers form their\n% universal model\n% \\begin{comment}\n% \\oleg{\\url{http://www.inf.ed.ac.uk/teaching/courses/fpls/note15.pdf}}\n% \\end{comment}\n% (see the\n% \\emph{faithfulness} property introduced in~\\S\\ref{s:unary}).\n\n\n% \\subsection{Challenges}\n\n% We require the predicates to be both effective and efficient.\n% First, the\n% evaluation of base arithmetic goals must terminate. Put\n% differently, it must be effectively computable whether a\n% tuple of naturals is included in or \\emph{excluded} from a\n% base arithmetic relation (addition, multiplication, division with\n% remainder, and logarithm with remainder).\n% Further, these computations must finish\n% without taking an exponential amount of time or space with respect to\n% the `search depth' (corresponding, in the case of binary numbers, \n% to the logarithm of the largest\n% number appearing in the computation). In other words, we wish to\n% maintain the efficiency of depth-first search (DFS) (or DFS with\n% interleaving \\cite{backtracking,kanrensite}) of the and-or tree expressing the\n% solution space of our base goals.\n\n% The main challenge in defining our predicates is that the domains of\n% natural numbers, and of arithmetic relations in general, are\n% infinite---enumerating them is not an option. \n% The incompleteness of DFS (used in SLD resolution) immediately\n% presents a problem. For example, assuming that \\scheme{geno} is a predicate\n% that generates all (ground) natural numbers in sequence, one may be\n% tempted to implement \\scheme{mulo} (separating the generation and testing for\n% clarity) as\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mulo\n% (lambda (x y z)\n% (exist (a b c)\n% (geno a)\n% (== x a)\n% (geno b)\n% (== y b)\n% (geno c)\n% (== z c)\n% (project (a b)\n% (== (* a b) c)))))\n% \\end{schemedisplay}\n\n% In addition to its obvious inefficiency, \n% this implementation%\n% \\footnote{Bra{\\ss}el, Fischer, and Huch \\citep{numbers-Curry} describe the drawbacks of this\n% residuation-based approach in functional logic programming.}\n% often diverges under\n% DFS\\@. For example, evaluating \\scheme{(mulo x y 1)}\n% instantiates \\scheme{a} and \\scheme{b} to \\verb|0| and \\scheme{c} to \\verb|1|, causing the goal\n% \\scheme{(== (* a b) c)} to fail; after backtracking into \\scheme{(geno b)}, \\scheme{mulo}\n% keeps forever instantiating \\scheme{b} to\n% larger and larger positive numbers, each time resulting in failure of\n% the goal \\scheme{(== (* 0 b) 1)}.\n\n% One may attempt to fix this problem by using a complete search\n% strategy, such as breadth-first search (BFS) or iteratively-deepening\n% DFS\\@. Even ignoring efficiency concerns, this\n% implementation of \\scheme{mulo} is still unacceptable: a complete search strategy\n% will find a solution \\emph{if it exists}, but if no solution exists\n% the search will continue forever. For example, although BFS finds the\n% instantiation of \\scheme{y} that satisfies \\mbox{\\scheme|(mulo 1 y 2)|}, the goal\n% \\mbox{\\scheme|(mulo 2 y 1)|} still diverges. Thus we must devise a termination\n% criterion for \\scheme{mulo}.\n\n% Devising a termination criterion may seem easy. For example, when\n% searching for \\scheme{x} and \\scheme{y} that satisfy the goal\n% \\scheme{(mulo x y 5)}, we\n% only need to examine \\scheme{x} and \\scheme{y} values up to~$5$---since\n% the search space is finite, the evaluation of the goal\n% certainly terminates. The problem arises when determining whether the\n% third argument in a specific use of \\scheme{mulo} is instantiated to a\n% ground numeral. This task is trivial if we use the `impure'\n% non-logical features provided by Prolog's reflection facilities, such\n% as the infamous \\verb|var/1| predicate. Even if \\verb|var/1| were absent from\n% Prolog, it could be emulated using cuts and negation. We\n% disavow such tools---we aim to implement our predicates in a pure\n% subset of Prolog, without cuts, reflection, or any way to distinguish\n% a logic variable. This aim for purity is a\n% challenge that in return makes our approach most\n% elucidating and extensible.\n\n% The final challenge is that the binary representation of a number\n% may not be structurally part of that of its successor.\n% For example, the binary numeral 111 (in decimal, 7)\n% is not structurally part of its successor 1000. This lack of\n% structural inclusion prevents straightforward structural recursion.\n\n\n% \\subsection{Termination and Solvability}\n\n% These challenges make the arithmetic predicates tricky to code.\n% It is not obvious that the resulting predicates have the\n% claimed properties, in particular, that they terminate in all\n% modes. We therefore devote much of the paper to proofs. A typical\n% termination theorem we prove assures that evaluating or re-evaluating\n% \\mbox{\\scheme|(addo x y z)|} terminates, provided that the terms initially associated\n% with \\scheme{x}, \\scheme{y}, and \\scheme{z} share no logic variables. Successive\n% re-evaluations of this goal recursively enumerate the stream of unique\n% triples $(x,y,z)$ of potentially non-ground terms whose denotation\n% (\\S\\ref{s:solution-sets-of-addition}) is the domain $\\set{(u,v,w) \\in x\\times\n% y\\times z : u+v=w}$. Thus, membership and non-membership are\n% computable for base arithmetic relations.\n\n% We guarantee termination only for stand-alone base arithmetic goals but\n% not their conjunctions (see \\S\\ref{s:solution-set}). This\n% non-compositionality is expected, since conjunctions of arithmetic\n% goals can express Diophantine equations; were such conjunctions\n% guaranteed to terminate, we would be able to solve Hilbert's 10th\n% problem, which is undecidable~\\cite{hilbertstenth}. We also do not\n% guarantee termination if the goal's arguments share variables. Such a\n% goal can be expressed by conjoining a sharing-free base goal and\n% equalities.\n\n% We proceed as follows.\n% In~\\S\\ref{s:unary} we define addition and multiplication predicates\n% for a \\emph{unary} representation of natural numbers. \n% We introduce solution sets, \\S\\ref{s:solution-set}, to carefully\n% establish termination for these predicates, laying a\n% foundation for our analysis of binary predicates.\n% In~\\S\\ref{s:binary} we introduce our representation of binary\n% numerals, and in \\S\\ref{s:addsub} and~\\S\\ref{s:mult}\n% we define predicates for binary addition and\n% subtraction, and multiplication, respectively.\n% In~\\S\\ref{s:div-exp-log} we briefly describe our predicates \n% for binary division, exponentiation, and logarithm. For lack of\n% space, we relegate our pure Prolog implementation of exponentiation,\n% logarithm, and non\\hyp interleaving binary multiplication to\n% the accompanying source\n% code,\\footnote{\\url{http://okmij.org/ftp/Prolog/Arithm/}}\n% along with additional proofs, tests, examples, and discussion.\n% We review related work in~\\S\\ref{s:related}, and\n% conclude in~\\S\\ref{s:conclusion}.\n% The full version of this\n% paper\\footnote{\\url{http://okmij.org/ftp/Prolog/Arithm/arithm.pdf}}\n% includes appendices outlining proofs of the properties of solution sets.\n% We have also implemented\n% declarative arithmetic as Haskell type-level relations (type classes)\n% \\cite{lightweight-resources}.\n\n\n% \\section{Predicates for Unary Arithmetic}\n% \\label{s:unary}\n\n% We begin with unary numerals.\n% Unary is simpler than binary, since every unary numeral is\n% structurally part of its successor, so we may use structural\n% recursion. However, membership and non-membership still need to\n% be decidable for the base arithmetic predicates, whose domains are\n% infinite however they are represented.\n% Hence, the unary case already presents our main challenges.\n% The unary case also lets us introduce and illustrate\n% most of our terminology in this section.\n\n% We represent unary numerals as lists of atoms \\schemeresult{u}: \\scheme{`()} denotes zero,\n% \\scheme{`(u)} denotes one, \\scheme{`(u u)} denotes two, etc.%\n% \\footnote{We could just as easily use a more common representation \n% $z$, $s(z)$, $s(s(z))$, etc. We chose lists for consistency with our\n% binary representation.} Throughout this section\n% we use the shorthand \\bn{n} to indicate our representation of\n% the number $n$---that is, the list of \\schemeresult{u}'s of length~$n$.\n\n% Although miniKanren is not statically typed, we assume an \n% implicit type of `unary numerals' and an implicit\n% typing of logic variables. For example, the implicit type of the term\n% \\scheme{`(u u . ,x)} is `unary numeral'; the logic variable \\scheme{x} has\n% the same type. The type predicate for unary numerals can be expressed\n% as a generator:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define genuo\n% (lambda-e (t)\n% (`(()))\n% (`((u . ,x)) (genuo x))))\n% \\end{schemedisplay}\n\n% \\noindent A term \\scheme{t} has the type of `unary numeral' if the goal \\scheme{(genuo t)}\n% succeeds. When providing denotations for terms and goals we\n% use the implicit type of a logic variable to characterize the \n% variable's domain.\n\n% We can add unary numerals with a special case of the\n% \\scheme{appendo} predicate.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define addo\n% (lambda-e (n m s)\n% (`(() ,x ,x))\n% (`((u . ,x) ,y (u . ,z)) (addo x y z))))\n% \\end{schemedisplay}\n\n% \\noindent We can use \\scheme{addo} to add numerals: the goal\n% \\scheme{(addo bn1 bn2 x)} unifies \\scheme{x} with \\bn3. We can also use \\scheme{addo} for\n% subtraction: both \\scheme{(addo x bn1 bn3)} and\n% \\scheme{(addo bn1 x bn3)} unify \\scheme{x} with \\bn2, and the evaluation of\n% \\scheme{(addo x bn3 bn1)} fails finitely. We can further use \\scheme{addo} to\n% decompose a number into its summands: for example, \\scheme{(addo x y bn3)}\n% has four solutions, in which \\scheme{x} and \\scheme{y} are unified in turn with\n% $\\bn0$ and $\\bn3$, $\\bn1$ and $\\bn2$, $\\bn2$ and $\\bn1$, and $\\bn3$\n% and $\\bn0$. Finally, passing three distinct uninstantiated logic\n% variables to \\scheme{addo} lets us enumerate the domain of addition over the\n% natural numbers. The evaluation of the query \\scheme{(addo x y z)} in miniKanren yields an infinite set of solutions; here are the first three, assuming a {\\em depth-first} search strategy:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (run3 (q)\n% (exist (x y z)\n% (addo x y z)\n% (== `(,x ,y ,z) q))) $\\Rightarrow$ \n% \\end{schemedisplay}\n% \\nspace\n% \\begin{schemeresponse}\n% ((() _$_{_{0}}$ _$_{_{0}}$)\n% ((u) _$_{_{0}}$ (u . _$_{_{0}}$))\n% ((u u) _$_{_{0}}$ (u u . _$_{_{0}}$)))\n% \\end{schemeresponse}\n\n% \\noindent This stream of solutions represents the infinite addition\n% relation \\emph{in two different ways}. First, the goal produces an\n% infinite number of solutions; we can always re-evaluate the\n% goal to get another solution. Second, each solution represents\n% infinitely many triples, all members of\n% the addition relation. For example, the second solution\n% compactly represents the infinitely many triples of naturals $(x,y,z)$\n% for which $x$ is one and $z$ is the successor of $y$. Each instantiation\n% of the free logic variable \\schemeresult|_$_{_{0}}$| to a member of its domain\n% (corresponding to its implicit type) yields a new triple of naturals\n% that is a member of the addition relation. We will make extensive use of\n% this compact representation of infinite domains when proving\n% termination properties of our predicates.\n\n% \\subsection{Solution Sets}\\label{s:solution-set}\n\n% To formulate propositions about \\scheme{addo} and other predicates, we\n% introduce the notion of \\emph{solution sets}, which accounts for the\n% search strategy used to run a logic program (unlike other procedural\n% notions of solutions). We use miniKanren syntax.\n% We identify a bound logic variable with the term it is bound to,\n% so all variables are free for us, as for Lloyd \\cite{lloyd:lp}.\n% Our goals are all pure (contain no \\scheme|varo|, \\scheme|project|,\n% negation, or \\scheme|conda|/\\scheme|condu|).\n\n% We assume an \\emph{idempotent} notion of substitution\n% \\cite{FBaade01}: a substitution~$\\theta$\n% is a finite map $\\set{x_i=t_i}$ from logic variables~$x_i$ to\n% terms~$t_i$ such that no $t_j$ contains any~$x_i$.\n% \\begin{comment}\n% The finite set $\\set{x_i}$ is\n% the domain $\\dom\\theta$.\n% and the set of variables occurring in \n% $\\set{t_i}$ is the set $\\fv\\theta$ of free variables of $\\theta$.\n% \\end{comment}\n% We write the application of the substitution $\\theta$ to a term $t$ as\n% $t\\theta$; this application easily extends to tuples of\n% terms and sets of tuples.\n\n% \\begin{definition}[Conjunction of substitutions]\\label{defn:conjunction}\n% We define the \\emph{conjunction} $\\theta\\xi$ of two \n% substitutions $\\theta$ and $\\xi$ by treating them\n% as sets of equations $\\set{x_i = t_i}$: we combine both\n% sets (the result may contain two equations for the same $x_i$) and use\n% unification to solve the resulting equations\n% \\cite[\\S2.2.3]{FBaade01}. Our conjunction of substitutions is thus\n% commutative and associative. Because of the unification our\n% conjunction is partial: if unification fails we call\n% the original substitutions contradictory.\n% \\end{definition}\n\n% \\begin{comment}\n% Notion of instantiatedness and instantiatedness tree from Mercury\n% documentation.\n% \\url{http://www.cs.mu.oz.au/research/mercury/information/doc-release/mercury_ref/Insts-modes-and-mode-definitions.html#Insts-modes-and-mode-definitions}\n% \\end{comment}\n\n% \\newcommand{\\bind}{\\star}\n% \\newcommand{\\plus}{\\oplus}\n% \\newcommand{\\equal}{=}\n% We can interpret a goal~$g$ as a function from a substitution\n% to a (finite or infinite) stream of substitutions. Both SLD and\n% SLD-interleaving \\cite{backtracking} interpret (right-associative and non-commutative)\n% disjunction and conjunction by\n\n% \\hspace{2.9cm}\\scheme|(conde (g1) (g2))|$(\\theta) \\equal g_{_{1}}(\\theta) \\plus g_{_{2}}(\\theta)$,\n\n% \\hspace{3.2cm}\\scheme|(exist () g1 g2)|$(\\theta) \\equal g_{_{1}}(\\theta) \\bind g_{_{2}}$\n\n% % we use ; and , rather than \\land and \\lor for consistency with Prop 2\n% where the \\emph{bind} operation $\\bind$ is defined recursively by\n% \\begin{align*}\n% \\boldleftparen \\boldrightparen &\\bind f \\equal \\boldleftparen \\boldrightparen, \\\\\n% \\boldleftparen \\theta\\ \\centerdot\\ \\vec\\theta \\boldrightparen &\\bind f \\equal (f \\theta) \\plus (\\vec\\theta \\bind f).\n% \\end{align*}\n% The difference between SLD and SLD-interleaving is that $\\plus$ above is\n% defined as stream concatenation in SLD but stream \n% interleaving in SLD-interleaving.\n\n% \\begin{definition}[Solution sequence and set]\n% Given an $n$-ary predicate \\scheme{g} and $n$ terms $t_1,\\ldots,t_n$ that may\n% contain logic variables, a \\emph{solution}\n% of a goal {\\rm \\scheme|(g t1 ... tn)|} is an $n$-tuple\n% $(t^{\\prime}_1,\\ldots,t^{\\prime}_n)$ where each $t^{\\prime}_i$ instantiates $t_i$\n% after the evaluation of the goal succeeds\n% using SLD or SLD-interleaving strategy.\n% A solution can be represented as $(t_1,\\ldots,t_n)\\theta$ for some\n% substitution $\\theta$.\n% A \\emph{solution sequence} is\n% a sequence of solutions obtained by evaluating and\n% successively re-evaluating a goal.\n% The sequence is a \\emph{solution set} if no two of its members\n% unify with each other.\n% \\end{definition}\n% A goal that fails has the empty solution set. A goal whose evaluation\n% or re-evaluation does not terminate does not have a solution\n% sequence.\n% In contrast, the goal \\mbox{\\scheme|(genuo x)|} does have a solution sequence (which\n% can be infinite)\n% because its evaluation and re-evaluation always terminates.\n% We assume SLD as the default solution strategy.\n% If we add the clause \\scheme{(addo x '() x)} as a well-meaning optimization ($x+0=x$)\n% after the first clause of \\scheme{addo} above, the solution sequence may not\n% necessarily be a solution set: for example, the goal \\scheme{(addo x y bn0)}\n% will have $(\\bn0,\\bn0,\\bn0)$ in duplicate.\n\n\n% The notion of solution sequence is constructive, `proof-theoretic,' to\n% be distinguished from a model of a logic program: each\n% solution has been actually \\emph{derived}, in finite time, from\n% the facts and rules at hand using the given solution strategy. A term\n% in a solution may contain free logic variables. We can prove\n% the following properties of solution sets\n% (see Appendix~A of our full paper).\n\n% \\begin{comment}\n% \\oleg{define the monotonicity property: what happens with the solution\n% sequence if we add one more clause to the definition of the goal,\n% after the existing clauses. Note starvation: under SLD, solution set\n% remains the same if it is already infinite.}\n% \\end{comment}\n\n% \\begin{proposition}\\label{p:ss-monotonicity}\n% If a goal {\\rm \\scheme|(g t1 ... tn)|} has a (finite or infinite) solution set \n% $\\set{ (t_1,\\ldots,t_n)\\theta_i }$\n% and $\\xi$ is a substitution that contradicts only finitely many \n% $\\theta_i$, then the goal \n% \\mbox{{\\rm \\scheme|(g t1$\\xi$ ... tn$\\xi$)|}} has a solution set\n% $\\set{ (t_1,\\ldots,t_n)({\\theta_i}\\xi) }$,\n% omitting the elements where $\\theta_i$ and $\\xi$ are contradictory.\n% \\end{proposition}\n\n% The solution set of a conjunction of goals \n% \\scheme{(g1 t11 ... t1n)} and \\scheme{(g2 t21 ... t2n)} is \n% (as in a natural database join)\n% the set of tuples $\\set{ (t_{11},\\ldots,t_{1n},t_{21},\\ldots,t_{2n}) }$\n% after the evaluation of the conjunction has succeeded.\n\n% \\begin{proposition}\\label{p:ss-conj-finite}\n% If the goals $g_1$ and $g_2$ have a finite solution set, \n% then the conjunction of $g_1$ and $g_2$ has a finite solution set.\n% \\end{proposition}\n% This proposition is a corollary of the previous proposition. It\n% does not generally hold if the solution set of one of the conjuncts is infinite. For example, \n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mfo\n% (lambda-e (n m)\n% (`(() (u . ,y)) (genuo y))\n% (`((u . ,x) ,y) (mfo `(u . ,x) y))))\n% \\end{schemedisplay}\n\n% \\noindent The goal \\scheme{(mfo x y)}, with the variables \\scheme{x} and \\scheme{y} free, has the infinite solution set \n% $\\set{(\\bn0,\\bn{i}): i \\in \\mathbb{N}^+}$. However, the (left or right)\n% conjunction of this goal with \\mbox{\\scheme|(== x bn1)|} diverges and has no solution set.\n% Similarly, the conjunction of \\mbox{\\scheme|(mfo x y)|} with \\mbox{\\scheme|(== x y)|} has no solution set\n% either.\n% Prop.\\,\\ref{p:ss-monotonicity} applies to neither conjunction because \n% the substitutions $\\{x=\\bn1\\}$ and $\\{x=y\\}$ both\n% contradict infinitely many solution-set substitutions (in fact, all\n% of them).\n% The latter conjunction is equivalent to the goal\n% \\scheme{(mfo x x)}; such sharing of variables among the arguments of\n% a goal is lethal to the termination guarantees below (such as\n% Prop.\\,\\ref{p:add-unground}).\n\n% \\begin{proposition}\\label{p:ss-conj-infinite}\n% If a goal {\\rm \\scheme|(g1 t1 ... tn)|} has a finite solution set \n% $\\set{(t_1,\\ldots,t_n)\\theta_i}$ and the goal\n% \\mbox{{\\rm \\scheme|(g2 $t^{\\prime}_1$$\\theta_1$ ... $t^{\\prime}_m$$\\theta_1$)|}} has an infinite solution set\n% $\\set{(t_1^{\\prime}\\theta_1,\\ldots,t_m^{\\prime}\\theta_1)\\xi_j}$, then\n% the conjunction of $g_1$ and $g_2$ has the infinite solution set\n% $\\set{(t_1,\\ldots,t_n,t_1^{\\prime},\\ldots,t_m^{\\prime})\\penalty\\binoppenalty(\\theta_1\\xi_j)}$.\n% \\end{proposition}\n% This proposition describes the incompleteness of SLD\\@: its underlying\n% depth-first search becomes trapped exploring the leftmost \n% infinite branch of the search tree. However, using \n% the SLD-interleaving strategy, we can strengthen the proposition:\n\n% \\begin{proposition}\\label{p:ss-conj-interleaving}\n% If a goal\n% \\mbox{{\\rm \\scheme|(g1 t1 ... tn)|}} has a (finite or infinite) solution set \n% $\\set{(t_1,\\ldots,t_n)\\theta_i}$ and the goal\n% \\mbox{{\\rm \\scheme|(g2 $t^{\\prime}_1$$\\theta_i$ ... $t^{\\prime}_m$$\\theta_i$)|}} has a non-empty (finite or \n% infinite) solution set $\\set{(t_1^{\\prime}\\theta_i,\\ldots,t_m^{\\prime}\\theta_i)\\xi_{ij}}$\n% for each~$i$, then\n% the conjunction of $g_1$ and $g_2$ has the solution set\n% $\\set{ (t_1,\\ldots,t_n,t_1^{\\prime},\\ldots,t_m^{\\prime})(\\theta_i\\xi_{ij}) }$.\n% \\end{proposition}\n% The proof is based on the laws of fair conjunction and disjunction in\n% \\cite{backtracking}; details are given in\n% Appendix~B of our full paper.\n% We use analogous properties for disjunctions of goals.\n\n% \\subsection{Properties of Addition: Solution Sets of Addition}\\label{s:solution-sets-of-addition}\n\n% If $t$ is a term and $n$ is a natural number, then we write\n% \\mbox{\\scheme|`(u$^{n}$ . ,t)|} to mean \\mbox{\\scheme|`(u ... u . ,t)|} where \\mbox{\\scheme|`u ... `u|}\n% consists of $n$ occurrences of~\\scheme{`u}.\n\n% One can easily prove the following propositions:\n% \\begin{proposition}\\label{p:add-ground-XY}\n% The goal {\\rm \\scheme|(addo x y z)|}, where \\scheme{x} and \\scheme{y} are instantiated to ground\n% numerals and \\scheme{z} is free, has a singleton solution set\n% unifying \\scheme{z} with the\n% numeral that is the sum of those corresponding to \\scheme{x} and \\scheme{y}.\n% \\end{proposition}\n% That is, if the first two arguments of \\scheme{addo} are instantiated to\n% numerals, the goal is decidable and has one solution.\n% \\begin{proposition}\\label{p:add-ground-Z}\n% The goal {\\rm \\scheme|(addo x y z)|} where \\scheme{z} is instantiated to a ground numeral\n% has a finite solution set.\n% \\end{proposition}\n% The proof is an easy induction on the third argument.\n\n% \\begin{proposition}\\label{p:add-ground-X}\n% The goal {\\rm \\scheme|(addo x y z)|}, where \\scheme{x} is a ground numeral \\bn{n} and \\scheme{y} and \\scheme{z} are\n% free, has the singleton solution set\n% $\\{ (\\bn{n},g,${\\rm \\scheme|`(u$^n$ . ,g)|}$) \\}$ where \\scheme{g}\n% is a free logic variable.\n% \\end{proposition}\n% The proof is by induction on \\scheme{x}. The proposition easily extends to\n% the case where \\scheme{y} and \\scheme{z} are arbitrary terms, using\n% Prop.\\,\\ref{p:ss-monotonicity}. \n\n% \\begin{proposition}\\label{p:add-free}\n% The goal {\\rm \\scheme|(addo x y z)|}, where the arguments are distinct\n% free logic variables, has an infinite solution set\n% $\\{ (\\bn{n},g,${\\rm \\scheme|`(u$^n$ . ,g)|}$) : n\\in\\mathbb{N} \\}$ \n% where $g$ is a free logic variable. In each solution,\n% $x$ is unified to a ground numeral. \n% \\end{proposition}\n% The proof is by induction on $x$ and soundness of SLD resolution.\n\n% \\begin{definition}[Denotation of arithmetic solutions]\n% The \\emph{denotation} $\\denot\\cdot$ of an arithmetic term or solution set is defined as follows.\n% \\begin{align*}\n% \\denot {\\bn n} &= \\{n\\}\\\\\n% \\denot t &= \\set{\\, n \\in \\mathbb{N} : n \\ge m \\,} \\notag\\\\[-\\jot]\n% &\\qquad\\text{where $t$ is the non-ground term \\emph{$\\boldleftparen \\sfusymbol^m\\ \\centerdot\\ x\\boldrightparen$}} \\notag\\\\[-\\jot]\n% &\\qquad\\text{where $x$ is a free logic variable of the type of unary numerals}\\\\\n% \\denot{(t_1,\\ldots,t_n)} &= \\denot{t_1} \\times \\dotsb \\times \\denot{t_n} \\\\\n% \\denot S &= \\cup_{s \\in S}\\denot {s}\n% \\quad\\text{for a solution set $S$ (whose elements are all disjoint)}\n% \\end{align*}\n% \\end{definition}\n\n% \\noindent\n% The previous propositions along with Prop.\\,\\ref{p:ss-monotonicity} let\n% us prove:\n% \\begin{proposition}\\label{p:add-unground}\n% The goal {\\rm \\scheme|(addo x y z)|} where the arguments have no shared logic variables has\n% a solution set with \\scheme{x} always unified to a ground numeral. \n% The denotation of the solution set is\n% $\\set{(u,v,w) \\in \\denot{x} \\times \\denot{y} \\times \\denot{z} : u+v=w}$.\n% \\end{proposition}\n% The proof simply invokes Prop.\\,\\ref{p:add-ground-Z} or\n% Prop.\\,\\ref{p:add-ground-X} if $z$ or $x$ is ground.\n% If neither $z$ nor $x$ is ground, then the arguments of the goal,\n% since they share no logic variables,\n% contradict only finitely many solutions in the set of\n% Prop.\\,\\ref{p:add-free}, so we invoke Prop.\\,\\ref{p:ss-monotonicity}.\n\n\n% The proposition states that the predicate \\scheme{addo} is fully decidable:\n% for any arguments sharing no free variables, the predicate\n% decides if the denotation of these arguments is in the domain\n% of addition \\emph{or not}. Furthermore, for any subset of the domain\n% of addition of the form\n% $\\set{(u,v,w) \\in \\denot{x} \\times \\denot{y} \\times \\denot{z} : u+v=w}$,\n% there is a goal with exactly this denotation.\n% We call such a predicate \\emph{faithful}.\n\n\n% \\subsection{Multiplication}\\label{s:unary-mul}\n\n% Multiplication of unary numbers may seem as trivial as\n% addition. (We will be developing several versions of multiplication,\n% so we label the \\scheme{mulo} predicates with a numeric version \n% to distinguish them.)\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mul1o\n% (lambda-e (n m p)\n% (`(() __ ()))\n% (`((u . ,x) ,y ,z)\n% (exist (z1)\n% (mul1o x y z1)\n% (addo z1 y z)))))\n% \\end{schemedisplay}\n\n% \\noindent This predicate directly encodes the inductive definition of\n% multiplication: \\mbox{$0 \\cdot x = 0$,} $(x+1)\\cdot y = x\\cdot y + y$. Indeed, \n% the goal \\scheme{(mul1o bn3 bn2 x)} has the singleton solution set with $x$\n% unified with $\\bn6$. However, the goal \\scheme{(mul1o x bn2 bn6)}\n% diverges after producing the first solution, the goal\n% \\scheme{(mul1o x bn2 bn5)} diverges without producing anything, and\n% \\scheme{(mul1o x y bn6)} overflows the stack after producing three solutions.\n% % -- $x=\\bn1,y=\\bn6$, $x=\\bn2,y=\\bn3$, $x=\\bn3,y=\\bn2$ -- \n\n% The problem is left-recursion in the second clause: the goal\n% \\scheme{(mul1o x bn2 z)} with free $x$ and~$z$ requires evaluating\n% \\scheme{(mul1o x^ bn2 z^)} again with free \\scheme{x^} and~\\scheme{z^}. Reordering\n% goals in the body of the second clause eliminates left-recursion:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mul2o\n% (lambda-e (n m p)\n% (`(() __ ()))\n% (`((u . ,x) ,y ,z)\n% (exist (z1)\n% (addo z1 y z)\n% (mul2o x y z1)))))\n% \\end{schemedisplay}\n\n% \\noindent Now \\scheme{(mul2o x bn2 bn6)} has a singleton solution set and\n% \\scheme{(mul2o x bn2 bn5)} fails finitely.\n% However, whereas before\n% \\scheme{(mul1o bn3 bn2 x)} had a singleton solution set (with $x=\\bn6$), \n% this version \\scheme{mul2o} diverges after the\n% first solution, so the goal has no solution set.\n\n% Interestingly, simply swapping the arguments to \\scheme{addo} fixes the\n% problems.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mul3o\n% (lambda-e (n m p)\n% (`(() __ ()))\n% (`((u . ,x) ,y ,z)\n% (exist (z1)\n% (addo y z1 z)\n% (mul3o x y z1)))))\n% \\end{schemedisplay}\n\n% \\noindent Now \\mbox{\\scheme|(mul3o x bn2 bn6)|} and \\mbox{\\scheme|(mul3o bn3 bn2 x)|} both have singleton\n% solution sets, and \\mbox{\\scheme|(mul3o x bn2 bn5)|} fails finitely. \n% The reason is important. Evaluating\n% \\mbox{\\scheme|(mul2o bn3 bn2 z)|} requires evaluating\n% \\mbox{\\scheme|(addo z1 bn2 z)|} whereas evaluating\n% \\mbox{\\scheme|(mul3o bn3 bn2 z)|} requires evaluating\n% \\mbox{\\scheme|(addo bn2 z1 z)|}.\n% Although both addition goals denote the same relation, the former\n% has the infinite solution set\n% $\\set{ (\\bn0,\\bn2,\\bn2), \\penalty\\binoppenalty (\\bn1,\\bn2,\\bn3),\n% \\penalty\\binoppenalty (\\bn2,\\bn2,\\bn4),\\ldots}$\n% whereas the latter has the singleton solution set\n% $\\{ (\\bn2,g,$\\scheme|`(u u . ,g)|$) \\}$ (Prop.\\,\\ref{p:add-ground-X}).\n% That makes all the difference, as we prove below.\n\n% However, \\scheme{mul3o} is not perfect. Evaluating\n% \\scheme{(mul3o x y bn6)} overflows the stack,\n% because it requires evaluating\n% \\mbox{\\scheme|(addo y z1 bn6)|}, which gives $z_1=\\bn6, y=\\bn0$ as one\n% solution. This solution causes a recursive call\n% \\mbox{\\scheme|(mul3o x^ y bn6)|}, same as the original call. Thus we must treat\n% zero multiplicands separately,\n% taking care to avoid overlapping solutions so that the solution\n% sequence remains a solution set.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mulo\n% (lambda-e (n m p)\n% (`(() __ ()))\n% (`((u . __) () ()))\n% (`((u . ,x) (u . ,y) ,z)\n% (exist (z1)\n% (addo `(u . ,y) z1 z)\n% (mulo x `(u . ,y) z1)))))\n% \\end{schemedisplay}\n\n% \\noindent This pattern of fixing one problem only to see another problem emerge\n% is quite common, which is why we need proofs.\n\n% \\begin{comment}\n% In conducting the proofs below, we will appeal to the size-change\n% termination principle (Jones et al, GPCE 2002) ``a program terminates\n% on all inputs if every valid infinite call sequence would, if\n% executed, cause an infinite decrease in some parameter values''\n% trivially extended to logic programs (exploration of search trees).\n% \\oleg{use that paper and a paper at FLOPS2006 for relevant literature\n% on termination analysis of logic programs. Mention Mercury's\n% termination checker.}\n% \\end{comment}\n\n% \\begin{proposition}\\label{p:mul-ground-XY}\n% The goal {\\rm \\scheme|(mulo x y z)|}, where $x$ and $y$ are instantiated to ground\n% numerals and $z$ is free, has the singleton solution set that\n% unifies $z$ with the\n% numeral that is the product of the numerals for $x$ and $y$.\n% \\end{proposition}\n% The proof is an induction on $x$ using Prop.\\,\\ref{p:ss-conj-finite}\n% and Prop.\\,\\ref{p:add-ground-X}.\n% \\begin{comment}\n% \\oleg{or should we appeal to the size-changing principle?}\n% That is, if the first two arguments of \\verb|mul/3| are instantiated to\n% numerals, the goal is decidable and has at most one solution.\n% \\end{comment}\n\n% \\begin{proposition}\\label{p:mul-ground-Z}\n% The goal {\\rm \\scheme|(mulo x y z)|}, where $z$ is instantiated to a ground numeral,\n% has a finite solution set.\n% \\end{proposition}\n% The proof depends on Prop.\\,\\ref{p:ss-conj-finite} and\n% Prop.\\,\\ref{p:add-ground-Z}:\n% each solution of\n% \\scheme{(addo `(u . ,y) z1 z)} instantiates\n% $z_1$ to a ground numeral smaller than the numeral for~$z$. \n% \\begin{comment}\n% The latter\n% fact follows from the soundness of addition.\n% \\end{comment}\n\n% \\begin{proposition}\\label{p:mul-ground-Y}\n% The goal {\\rm \\scheme|(mulo x y z)|}, where $y$ is instantiated to a positive \n% ground numeral \\bn{n} and $x$ and $z$ are free, has the solution set\n% $\\set{ (\\bn{i},\\bn{n},\\bn{i\\cdot n}) : i \\in \\mathbb{N} }$. If $y$ is \\bn0,\n% the solution set is finite: $\\{ (\\bn0,\\bn0,\\bn0), (${\\rm \\scheme|`(u . ,x)|}$,\\bn0,\\bn0) \\}$.\n% \\end{proposition}\n% The proof is by induction on $y$ and Prop.\\,\\ref{p:add-ground-X}.\n\n\n% \\begin{proposition}\\label{p:mul-unground-interleave}\n% Under the SLD-interleaving strategy,\n% the goal {\\rm \\scheme|(mulo x y z)|}, where the arguments share no logic variables,\n% has a solution set that denotes\n% $\\set{ (u,v,w) \\in \\denot{x} \\times \\denot{y} \\times \\denot{z} :\n% u\\cdot v=w}$.\n% \\end{proposition}\n% The proof depends on Props.\\,\\ref{p:add-unground}, \\ref{p:mul-ground-Y}\n% and~\\ref{p:ss-conj-interleaving}.\n% Thus under SLD with interleaving, our \\scheme{mulo} predicate is faithful\n% to the multiplication\n% relation on naturals.\n\n% Without interleaving, the goal \\scheme{(mulo x y z)} has the solution set:\n\n% $\\{ (\\bn0, g,\\bn0), (${\\rm \\scheme|`(u . ,g$^{\\prime}$)|}$,\\bn0,\\bn0), (\\bn1,\\bn1,\\bn1), (\\bn2,\\bn1,\\bn2),(\\bn3,\\bn1,\\bn3),\\ldots \\}$.\n\n% \\noindent The denotation of this solution set\n% obviously does not cover the entire\n% multiplication relation: the second argument gets `stuck' on~$1$.\n% Under SLD\\@, then, our predicate \\scheme{mulo} covers only an infinitesimal part of\n% the domain of multiplication. We can do better: we define\n% a predicate \\scheme{semimulo} that\n% has the \\emph{same} termination properties as \\scheme{mulo} but covers \\emph{half} of the\n% domain of multiplication, namely $y \\le x$, whichever arguments are instantiated.\n\n% We first define the predicate \\scheme{lesso}, corresponding to the less-than\n% relation.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define lesso\n% (lambda-e (n m)\n% (`(() (__ . __)))\n% (`((u . ,x) (u . ,y)) (lesso x y))))\n% \\end{schemedisplay}\n\n% \\noindent The \\scheme{semimulo} predicate is then as follows.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define semimulo\n% (lambda-e (n m p)\n% (`(() __ ()))\n% (`((u . __) () ()))\n% (`((u . ,x) (u . ,y) ,z) (lesso x z) (lesso y `(u . ,x))\n% (mulo `(u . ,x) `(u . ,y) z))))\n% \\end{schemedisplay}\n\n% \\begin{proposition}\n% The goal {\\rm \\scheme|(semimulo x y z)|}, where the arguments are free logic variables,\n% has a solution set that denotes\n% $\\set{ (u,v,w) \\in \\mathbb{N}^3: v \\le u,\\, u\\cdot v=w }$.\n% \\end{proposition}\n% The proof depends on the goal \\scheme{(lesso x z)}, which asserts the trivial inequality\n% $x < (x+1)\\cdot (y+1)$ for all $x,y \\in \\mathbb{N}$.\n% With free $x$ and $z$, the goal\n% \\scheme{(lesso x z)} has an infinite solution set whose solutions each\n% instantiate $x$ to a ground numeral.\n% The goal \\scheme{(lesso y `(u . ,x))}\n% then has a \\emph{finite} solution set that grounds $y$ as well.\n% The last goal has thus a singleton\n% solution set, by Prop.\\,\\ref{p:mul-ground-XY}.\n\n% These attempts to define decidable multiplication even for the\n% seemingly trivial unary case show the difficulties that become more\n% pronounced as we move to binary arithmetic. We rely on a\n% finite representation of infinite domains, precise instantiatedness\n% analysis, and reasoning about SLD using search trees.\n\n\n% \\section{Binary Numerals}\\label{s:binary}\n\n% We represent numerals as lists of binary digits in little-endian order\n% (least significant bit first), with zero represented as the empty\n% list. A zero bit is denoted by \\schemeresult{o} (lower-case `oh') and a one bit is denoted\n% by \\schemeresult{l} (lower-case `el')---to distinguish a\n% number from its representation. For example, the terms \\scheme{`()}, \\scheme{`(l)},\n% \\scheme{`(o l)}, \\scheme{`(l l)}, \\scheme{`(o o l)} represent $0$ through~$4$.\n% The last bit of\n% a positive numeral \\emph{must} be~\\schemeresult{l}. Our code below takes special\n% care to maintain this well-formedness condition.\n% It is trivial to convert\n% between this and Prolog's native representations of integers.\n\n% We often use the auxiliary one-clause\n% predicates \\scheme{zeroo}, \\scheme{poso}, and \\scheme{gtlo}:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define zeroo\n% (lambda-e (n)\n% (`(()))))\n\n% (define poso\n% (lambda-e (n)\n% (`((__ . __)))))\n\n% (define gtlo\n% (lambda-e (n)\n% (`((__ __ . __)))))\n% \\end{schemedisplay}\n\n% \\noindent The goal \\scheme{(zeroo n)} succeeds if $n$ is zero. The goal \\scheme{(poso n)}\n% succeeds if $n$ is positive. The goal \\scheme{(gtlo n)} succeeds\n% if $n$ is at least two.\n\n% The predicate \\scheme{genbo} below expresses the\n% implicit type of binary numerals.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define genbo\n% (lambda-e (n)\n% (`(()))\n% (`((l . ,x)) (genbo x))\n% (`((o . ,x)) (poso x) (genbo x))))\n% \\end{schemedisplay}\n\n% \\noindent The presence of \\scheme{(poso x)} in the last clause ensures that the last\n% bit of a positive numeral is~\\schemeresult{l}. The code below contains similar\n% guarding occurrences of \\scheme{poso}.\n\n% Recall that one challenge of binary arithmetic is that a numeral\n% is not structurally part of its\n% successor. However, such a notion of inclusion exists (with\n% the attendant induction principle) if we consider the `length' of a binary\n% number, i.e., the number of bits in its binary representation. \n% More precisely, the length~$\\norm{n}$ of a numeral~$n$\n% is $\\floor{\\log_2 n} + 1$ if $n>0$, and $0$ if $n=0$. We define \\scheme{lesslo} by\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define lesslo\n% (lambda-e (n m)\n% (`(() (__ . __)))\n% (`((__ . ,x) (__ . ,y)) (lesslo x y))))\n% \\end{schemedisplay}\n\n% \\noindent It has the meaning and form of the unary \\scheme{lesso}\n% in~\\S\\ref{s:unary-mul}---and the same termination properties---but\n% compares the length of binary numbers rather their magnitude.\n\n% \\section{Addition and Subtraction}\\label{s:addsub}\n\n% % http://en.wikipedia.org/wiki/Adder_(electronics)\n\n% % full adder: one-bit adder\n% % what below is full_adder is nbit adder (multiple-bit) adder\n% % of the ripple-carry adder type\n\n% Our treatment of addition is inspired by hardware \n% full-adders and multi-bit adders, \n% as found in a digital computer's arithmetic logic\n% unit \\citep{hennessy-computer}. A one-bit full-adder\n% \\mbox{\\scheme|(full1-addero cin a b s cout)|} relates two input bits $a,b$ and the incoming\n% carry bit $c_{\\mathrm{in}}$ with the sum bit~$s$ and the outgoing carry \n% bit $c_{\\mathrm{out}}$, according\n% to the equation $c_{\\mathrm{in}} + a + b = s + 2 c_{\\mathrm{out}}$. In miniKanren,\n% we define \\scheme{full1-addero} by enumerating eight facts:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define full1-addero\n% (lambda-e (cin a b s cout)\n% (`(o o o o o))\n% (`(o o l l o))\n% (`(o l o l o))\n% (`(o l l o l))\n% (`(l o o l o))\n% (`(l o l o l))\n% (`(l l o o l))\n% (`(l l l l l))))\n% \\end{schemedisplay}\n\n% The multi-bit adder \\mbox{\\scheme|(fulln-addero cin a b s)|} relates\n% the incoming carry bit \\scheme{cin} (either \\schemeresult|o| or \\schemeresult|l|), \n% two binary numbers $a,b$, and their sum~$s$, according to the equation $c_{\\mathrm{in}} + a + b = s$.\n% It is defined by recursively combining one-bit adders\n% as in a ripple-carry adder of digital logic,\n% only in our case, the summands' bitwidths need not be the same or limited.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define fulln-addero\n% (lambda-e (cin a b r)\n% (`(o ,a () ,a))\n% (`(o () ,b ,b) (poso b))\n% (`(l ,a () ,r) (fulln-addero 'o a '(l) r))\n% (`(l () ,b ,r) (poso b) (fulln-addero 'o '(l) b r))\n% (`(,cin (l) (l) (,r1 ,r2)) (full1-addero cin 'l 'l r1 r2))\n% (`(,cin (l) (,ba . ,bd) (,ra . ,rd)) (poso bd) (poso rd)\n% (exist (cout)\n% (full1-addero cin 'l ba ra cout)\n% (fulln-addero cout '() bd rd)))\n% (`(,cin ,a (l) ,r) (gtlo a) (gtlo r)\n% (fulln-addero cin '(l) a r))\n% (`(,cin (,aa . ,ad) (,ba . ,bd) (,ra . ,rd))\n% (poso ad) (poso bd) (poso rd)\n% (exist (cout)\n% (full1-addero cin aa ba ra cout)\n% (fulln-addero cout ad bd rd)))))\n% \\end{schemedisplay}\n\n% \\noindent The first four clauses above deal with the cases\n% of a summand being zero. The next three clauses handle\n% the cases of a summand being one. The last clause adds\n% numbers at least two bits wide. We take\n% care to keep clauses from overlapping, so the solution\n% sequence of a \\scheme{fulln-addero} goal is a solution set. The\n% splitting of the cases and the many occurrences of \\scheme{poso} are\n% necessary to keep all numerals in a solution `well-typed':\n% If logic variables in a solution are instantiated according to their \n% implicit types, we never see a list whose last element unifies with \\schemeresult{o}.\n% \\begin{comment}\n% i.e., satisfy the typing predicate \\verb|genb/1|. If free logic variables in a\n% solution are instantiated according to their implicit types, \n% all the numerals are well-formed---we never have a finite list \n% whose last element unifies with \\verb|o|.\n% \\end{comment}\n\n% This multi-bit adder expresses binary addition, subtraction, and\n% ordering:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define addo\n% (lambda (a b c)\n% (fulln-addero 'o a b c)))\n\n% (define subo\n% (lambda (a b c)\n% (addo b c a)))\n\n% (define lesso\n% (lambda (a b)\n% (exist (x)\n% (poso x)\n% (addo a x b))))\n% \\end{schemedisplay}\n\n% \\begin{comment}\n% The latter says that the number $a$ is less than the number $b$ if\n% there exists a positive number $x$ that can be added to $a$ to produce\n% $b$.\n% \\end{comment}\n\n% Our predicate \\scheme{addo} for binary numerals satisfies\n% Prop.\\,\\ref{p:add-ground-XY} (by induction on \\mbox{$4 c_{\\mathrm{in}} + 3a + b$}) and\n% Prop.\\,\\ref{p:add-ground-Z}.\n% However, due to carry propagation (clauses 4 and 6 of \\scheme{fulln-addero}),\n% Prop.\\,\\ref{p:add-ground-X} and Prop.\\,\\ref{p:add-free} no longer\n% accurately describe the solution sets of binary \\scheme{addo}:\n% the goal \\mbox{\\scheme|(addo x y z)|}, with either $x$ or $y$ ground and the other\n% arguments being distinct free variables, has an infinite solution\n% set.\n\n% Prop.\\,\\ref{p:add-unground} holds (without $x$ being always\n% ground) only in the case of SLD with interleaving;\n% for SLD\\@, we can develop a binary addition\n% predicate that covers half of the domain of addition,\n% as in~\\S\\ref{s:unary-mul}.\n% The proofs begin by unrolling the recursion in clauses 3, 4,\n% and~7 of \\scheme{fulln-addero} (splitting cases for clauses 3 and~4),\n% so that \\scheme{fulln-addero} invokes itself recursively only for shorter\n% arguments.\n% \\begin{comment}\n% \\oleg{For the full proof, we need\n% one more proposition about the interleaving, about adding\n% to the disjunction.\n% Actually, the proposition is as follows: limited completeness of\n% SLD with interleaving: if a goal $\\mathtt{g}(\\vec t)$ has a \n% (possibly infinite) solution set, and $\\mathtt{g}(\\vec t)\\theta$\n% has a non-empty solution set, the solution set of the latter goal is\n% included in the solution set of the former goal. That is not the\n% full completeness since it does not address the divergence. That's\n% why there is a premise that a goal must have a solution set to\n% start with. To prove that the solution set exists, we can prove that\n% given any particular `depth' the strategy will find a fact there.\n% }\n% \\end{comment}\n\n% \\section{Multiplication}\\label{s:mult}\n\n% Binary multiplication may seem an obvious generalization of unary\n% multiplication. The most complex case is to multiply an odd number by\n% a positive one, $(2n + 1) m = 2 (n m) + m$, because it\n% involves addition. One may think that the\n% approach in \\scheme{mulo} in~\\S\\ref{s:unary-mul} will work here. Alas, binary\n% addition does not satisfy Prop.\\,\\ref{p:add-ground-X}, so we must\n% turn to a much less obvious solution.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define mulo\n% (lambda-e (n m p)\n% (`(() ,m ()))\n% (`(,n () ()) (poso n))\n% (`((l) ,m ,m) (poso m))\n% (`((o . ,nd) ,m (o . ,pd)) (poso m) (poso nd) (poso pd) (mulo nd m pd))\n% (`((l . ,nd) ,m ,p) (poso m) (poso nd) (gtlo p)\n% (exist (p1)\n% (lessl3o p1 p `(l . ,nd) m)\n% (mulo nd m p1)\n% (addo `(o . ,p1) m p)))))\n% \\end{schemedisplay}\n\n% \\noindent This solution relies on a seemingly contrived predicate \\scheme{lessl3o}:\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define lessl3o\n% (lambda-e (p1 p n m)\n% (`(() (__ . __) __ __))\n% (`((__ . ,p1d) (__ . ,pd) () (__ . ,md)) (lessl3o p1d pd '() md))\n% (`((__ . ,p1d) (__ . ,pd) (__ . ,nd) m) (lessl3o p1d pd nd m))))\n% \\end{schemedisplay}\n\n% \\noindent The goal \\mbox{\\scheme|(lessl3o p1 p n m)|} relates four numerals such that\n% \\mbox{$\\norm{p_1} < \\min(\\norm{p},\\penalty\\binoppenalty \\norm{n} + \\norm{m} + 1)$}.\n% As long as the arguments of the goal, \\emph{however instantiated}, \n% share no logic variables, the goal has a solution set. In any solution,\n% $p_1$ is a numeral whose bits may be free but whose length is fixed.\n% We call such numerals \\emph{L-instantiated}.\n% Moreover, whenever $p_1$, $p$, or both $n$ and $m$ are\n% L-instantiated, the solution set is finite.\n\n% In the code for binary \\scheme{mulo} above, \\scheme{lessl3o} occurs in a clause that is\n% selected when multiplying an odd number $2 n + 1$ (where $n>0$) by a positive\n% number~$m$ to yield~$p$. Under these conditions, clearly\n% $\\floor{\\log_2 n m}$ is less than both $\\floor{\\log_2 p}$\n% and $\\floor{\\log_2(2n+1)} + \\floor{\\log_2 m} + 2$, so the\n% constraint imposed by \\scheme{lessl3o} does not affect the declarative meaning\n% of \\scheme{mulo}.\n% The guarantee\n% of \\scheme{lessl3o} mentioned above lets us prove\n% Prop.\\,\\ref{p:mul-ground-XY} and Prop.\\,\\ref{p:mul-ground-Z} for binary\n% multiplication. For Prop.\\,\\ref{p:mul-ground-Z},\n% \\scheme{lessl3o} ensures that $p_1$ is L-instantiated,\n% so we can use induction to prove that \\mbox{\\scheme|(mulo n m p)|} with an\n% L-instantiated $p$ has a finite solution set. \n\n% The important role of \\scheme{lessl3o} can be informally explained as follows.\n% Given the goal \\mbox{\\scheme|(mulo x y z)|}, when $x$ and $y$ are instantiated but $z$\n% is free, or when $z$ is instantiated but $x$ and $y$ are free, the\n% search space is finite even though the free variables can be\n% instantiated an infinite number of ways: the length of the\n% product limits the lengths of the multiplicands, and vice versa.\n% The predicate \\scheme{lessl3o} enforces these limits, even\n% though our code cannot distinguish these two cases by\n% determining which arguments are instantiated.\n\n% The \\scheme{mulo} predicate also satisfies\n% Prop.\\,\\ref{p:mul-unground-interleave} (the proof again relies on the\n% properties of \\scheme{lessl3o}). As in the unary case, using SLD without\n% interleaving, we cannot cover the whole domain of natural\n% multiplication---but we can cover half of it. The technique is\n% essentially the same as explained for unary\n% multiplication.\n\n\n% \\section{Division, Exponentiation, and Logarithm}\\label{s:div-exp-log}\n\n% Our predicate for division with remainder\n% \\mbox{\\scheme|(divo n m q r)|} relates four natural\n% numbers such that $n = m \\cdot q + r < m\\cdot(q+1)$. The implementation is\n% quite complex so as to finitely fail as often as possible: for example, not only\n% when the divisor $m$ is zero, but also in the case\n% \\mbox{\\scheme|(divo `(o . ,x) '(o l) q '(l))|} with \\emph{free} $x$ and $q$, as\n% no even number divided by~$2$ gives the remainder~$1$.\n% Our algorithm is akin to long division as taught in elementary school, only\n% done right-to-left. At each step, we determine at\n% least one bit of the quotient in finite time.\n\n% First, we handle the easy case when the divisor~$m$ is\n% bigger than the dividend~$n$, so $q=0$ and $n=r$.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define divo\n% (lambda-e (n m q r)\n% (`(,n ,m () ,r) (== n r) (lesso n m))\n% ...))\n% \\end{schemedisplay}\n\n% \\noindent Otherwise, $q$ must be positive.\n% The second easy case is when $n$ is at least~$m$\n% and has the same length as~$m$.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define divo\n% (lambda-e (n m q r)\n% ...\n% (`(,n ,m (l) ,r) (samelo n m) (addo r m n) (lesso r m))\n% ...))\n% \\end{schemedisplay}\n\n% \\noindent This code relies on the auxiliary predicate \\scheme{samelo}, which holds if its\n% two arguments are binary numerals with the same number of digits.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define samelo\n% (lambda-e (n m)\n% (`(() ()))\n% (`((__ . ,x) (__ . ,y)) (samelo x y))))\n% \\end{schemedisplay}\n\n% The main case of division is when $n$ has more digits than~$m$.\n% The key is to represent $n = m \\cdot q + r < m\\cdot(q+1)$ as the conjunction\n% of the two relations\n% \\begin{align*}\n% 2^{l+1}r_1 &= m q_2 + r - n_2\n% &&\\text{and}&\n% n_1 &= m q_1 + r_1\n% \\end{align*}\n% where\n% \\begin{align*}\n% n &= 2^{l+1} n_1 + n_2, & q &= 2^{l+1} q_1 + q_2, & l &= \\norm{r},\n% \\end{align*}\n% and $n_2$ and~$q_2$ are at most $l+1$ long.\n% Given $l$, then, we can decide the first\n% relation. Because $0 \\le r_1 < m$, we can invoke\n% \\mbox{\\scheme|(divo n1 m q1 r1)|} recursively to decide the second\n% relation. That gives us a convenient induction principle:\n% either $n_1$ is zero and so $q_1$ and $m q_2 + r - n_2$ are both zero,\n% or $n_1$ is positive and shorter than~$n$. These two cases correspond\n% to the disjunction below.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define divo\n% (lambda-e (n m q r)\n% ...\n% (`(,n ,m ,q ,r) (lesslo m n) (lesso r m) (poso q)\n% (exist (n1 n2 q1 q2 q2m q2md r1 rd) \n% (splito n r n1 n2)\n% (splito q r q1 q2)\n% (conde\n% ((== '() n1)\n% (== '() q1)\n% (subo n2 r q2m)\n% (mulo q2 m q2m))\n% ((poso n1)\n% (mulo q2 m q2m)\n% (addo q2m r q2md)\n% (subo q2md n2 rd)\n% (splito rd r r1 '())\n% (divo n1 m q1 r1)))))))\n% \\end{schemedisplay}\n\n% \\noindent The calls to \\scheme{lesslo} and \\scheme{lesso} at the beginning of this code\n% ensure that \\scheme{m} and then \\scheme{r} are both L-instantiated.\n% The code also relies on the predicate \\scheme{splito} to `split' a binary\n% numeral at a given length: The goal\n% \\mbox{\\scheme|(splito n r n1 n2)|} holds if \n% $n = 2^{l+1} n_1 + n_2$ where $l = \\norm{r}$ and $n_2 < 2^{l+1}$. The goal should be\n% invoked only when $r$ is L-instantiated, which is the case in the\n% \\scheme{divo} code above. The goal has a finite solution set, in which\n% $n_2$ is L-instantiated in every solution.\n\n% \\schemedisplayspace\n% \\begin{schemedisplay}\n% (define splito (n r n1 n2)\n% (lambda-e\n% (`(() __ () ()))\n% (`((o ,b . ,n) () (,b . ,n) ()))\n% (`((l . ,n) () ,n (l)))\n% (`((o ,b . ,n) (__ . ,r) ,n1 ()) (splito `(,b . ,n) r n1 '()))\n% (`((l . ,n) (__ . ,r) ,n1 (l)) (splito n r n1 '()))\n% (`((,b . ,n) (__ . ,r) ,n1 (,b . ,n2)) (poso n2) (splito n r n1 n2))))\n% \\end{schemedisplay}\n\n% Our predicate\n% \\mbox{\\scheme|(logo n b q r)|}\n% relates four numbers such that\n% $n = b^q + r < b^{q+1}$, so it implements\n% exponentiation, logarithm, and\n% $n$-th root.\n% Our implementation uses an\n% upper bound on $r$, namely $r b < n (b-1)$, and upper and lower bounds on $q$,\n% namely $(\\norm{b}-1) q < \\norm{n}$ and $\\norm{n}-1 < \\norm{b} (q+1)$. These\n% bounds constrain the search just as \\scheme{lessl3o} does for\n% binary multiplication in~\\S\\ref{s:mult}. Because the base-$2$\n% case is so simple, we treat it separately.\n\n\n% \\section{Conclusions}\\label{s:conclusion}\n\n% [TODO Integrate this text into rest of chapter, and thesis conclusion,\n% as appropriate.]\n\n% In a pure logic programming system,\n% we have declared decidable arithmetic of unrestricted unary and\n% binary natural numbers: addition, multiplication, division with\n% remainder, exponentiation, and logarithm with remainder.\n% The declared relations have unlimited domain and\n% are free from any mode restrictions or annotations. We have proven\n% that our arithmetic predicates are fully decidable and faithfully\n% represent the corresponding arithmetic relations. Our technique can\n% be easily extended to full integers (i.e., a tuple of the sign and a\n% natural number).\n\n% The gist of our approach is to limit the search space by a\n% balancing act of computing \\emph{bidirectional} bounds from arguments of unknown\n% instantiatedness. For example, the key to decidable\n% multiplication is to limit the search using bounds that\n% the inputs place on each other, without\n% testing whether any input is instantiated. We\n% also rely on the ability to finitely represent infinite domains\n% using logic variables, so that goals with infinite denotations\n% may have only a finite solution set.\n% Our notion of solution sets and the associated proof techniques\n% are not specific to arithmetic and SLD\\@; rather, they appear\n% applicable to logic programming in other domains using a variety\n% of search strategies.\n\n% %\\UndefineShortVerb{\\|}\n\n% \\newpage\n\n% % [fix typography: remove sfu, bind definition should use bold parens and dot]\n\n% % [should include expo and logo code. Make sure logo code is the corrected code.]\n\n% % [need to test all of this]\n\n% % [I dumped the two proofs in the appendix--refer the reader to the published paper, or the full version on Oleg's site]\n\n% % [should I build up the definition of divo one function at a time? Probably not as important if we use the lambda-e macro.]\n\n% % [need to discuss varo, project, conda, condu, somewhere, preferably before this chapter]\n\n% % [Careful with the use of 'we'. Should refer to authors, not 'we', when referring to work I wasn't part of (for example, Ken and Oleg's arithmetic system using Haskell typeclasses).]\n\n% % [is it necessary or desirable to include all of the proofs? Or can I point to the full paper?]\n\n% % [the *full* paper, with proof outlines, wasn't presented at FLOPS. the full paper is on Oleg's website (see footnote)]\n\n% % [does this code include the fix for logo?]\n\n% % [check against canonical miniKanren arithmetic code]\n\n% % [can we write a relational quicksort program, like in Apt's paper?\n% % See page 3 of Apt's paper 'Declarative Programming in Prolog']\n\n% % \\noindent [also, can we write the trans relation w/out negation? see\n% % page 15 of Apt's paper]\n\n% % [no constraints (other than equality constraints via unification).]\n\n% % [individual operations are decidable]\n\n% % Interesting examples (need to translate to chapter's conventions and names):\n\n% % factoring\n\n% % \\schemedisplayspace\n% % \\begin{schemedisplay}\n% % (run* (q)\n% % (exist (x y)\n% % (mulo x y '(o o l l))\n% % (== `(,x ,y) q)))\n% % \\end{schemedisplay}\n% % \\begin{schemeresponse}\n% % (((l) (o o l l))\n% % ((o o l l) (l))\n% % ((o l) (o l l))\n% % ((o o l) (l l))\n% % ((l l) (o o l))\n% % ((o l l) (o l)))\n% % \\end{schemeresponse}\n\n% % division\n\n% % \\schemedisplayspace\n% % \\begin{schemedisplay}\n% % (run* (t)\n% % (exist (x q r)\n% % (divo '(o l l) x q r)\n% % (== `(,x ,q ,r) t)))\n% % \\end{schemedisplay}\n% % \\begin{schemeresponse}\n% % (((_.0 _.1 _.2 _.3 . _.4) () (o l l))\n% % ((l l l) () (o l l))\n% % ((o l l) (l) ())\n% % ((l o l) (l) (l))\n% % ((o o l) (l) (o l))\n% % ((l) (o l l) ())\n% % ((l l) (o l) ())\n% % ((o l) (l l) ()))\n% % \\end{schemeresponse}\n", "meta": {"hexsha": "2ab54a09496537b677b366d9ce36e3e00c43b481", "size": 60254, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "oldarith.tex", "max_stars_repo_name": "holtzermann17/dissertation-single-spaced", "max_stars_repo_head_hexsha": "aca0e56a33916596c98709308342d9ccabd4718b", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 50, "max_stars_repo_stars_event_min_datetime": "2015-01-11T21:22:55.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-10T12:49:11.000Z", "max_issues_repo_path": "oldarith.tex", "max_issues_repo_name": "holtzermann17/dissertation-single-spaced", "max_issues_repo_head_hexsha": "aca0e56a33916596c98709308342d9ccabd4718b", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-08-08T18:10:18.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-09T02:33:25.000Z", "max_forks_repo_path": "oldarith.tex", "max_forks_repo_name": "holtzermann17/dissertation-single-spaced", "max_forks_repo_head_hexsha": "aca0e56a33916596c98709308342d9ccabd4718b", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2017-07-29T13:58:01.000Z", "max_forks_repo_forks_event_max_datetime": "2018-09-14T05:01:31.000Z", "avg_line_length": 43.4419610671, "max_line_length": 192, "alphanum_fraction": 0.6908752946, "num_tokens": 18277, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467904083364927}} {"text": "\\chapter{How Cameras Work}\n\nLet's say it is a sunny day and you are standing in field a few meters\nfrom a cow. You use the camera on your phone to take a picture of the\ncow. How does that whole process work?\n\n\\section{The Light That Shines On the Cow}\n\nThe sun is a sphere of hot gas. About 70\\% of the gas is\nhydrogen. About 28\\% is helium. There's also a little carbon, nitroge,\nand oxygen.\n\nGradually, the sun is converting hydrogen into helium through a\nprocess known as ``nuclear fusion''. (We will talk more about nuclear\nfusion in a later chapter.) A lot of heat is created in this\nprocess. The heat makes the gases glow.\n\nHow does heat make things glow? The heat pushes the electrons into\nhigher orbitals. When the fall back down to a lower orbital, they\nrelease a photon of energy, which travels away from the atam as an\nelectromagnetic wave.\n\nHeat isn't the only way to push the electrons into a higher\norbital. For example, a flourescent lightbulb is filled with gas.\nWhen we pass electricity through the gas, its electrons are moved to a\nhigher orbital. When they fall, light is created.\n\nWhat is the frequency of the wave that the photon travels on?\nDepending on what orbital it falls from and how far it falls, the\nphoton created has different amounts of energy. The amount of energy\ndetermines the frequency of the electromagnetic wave.\n\n\\begin{mdframed}[style=important, frametitle={Formula for enegy of a photon}]\n\nIf you want to know the amount of energy $E$ in a photon, here is the formula:\n\n$$E = \\frac{h c}{\\lambda}$$\n\nwhere $c$ is the speed of light, $\\lambda$ is the wavelength of the\nelectromagnetic wave, and $h$ Planck's constant: $6.63 \\times 10^{-34} m^2 kg/s$\n\nFor example, a red laser light has a wavelength of about 630 nm. So the energy in each photon is:\n\n$$\\frac{(300 \\times 10^6) (6.63 \\times 10^{-34})}{630 \\times 10^{-9}} = 3.1 \\times 10^{-19} \\text{ joules}$$\n\n\\end{mdframed}\n\nIn the sun, there are several kinds of molecules and each has a few\ndifferent orbitals that the electrons can live in. Thus, the light\ncoming from the sun is made up of electromagnetic waves of many\ndifferent frequencies.\n\nWe can see some of these frequencies as different colors, but some are\ninvisible to humans, for example ultraviolet and infrared.\n\n\\section{Light Hits the Cow}\n\nWhen these photons from the sun hit the cow, the hide and hairs of the\ncow will absorb some of the photons. These photons will become heat\nand make the cow feel warm. Some of the photons will not be absorbed\n-- they will leave the cow. When you say ``I see the cow,'' what you are\nreally saying is ``I see some photons that were not absorbed by the cow.''\n\nDifferent materials absorb different amounts of each wavelength. A\nplant, for example, absorbs large percentage of all blue and red\nphotons that hit it, but it absorbs only a small percentage of the\ngreen photons that hit it. Thus we say ``That plant is green.''\n\nWhite things absorb very small percentages of photons of any visible\nwavelength. Black things absorb very \\emph{large} percentages of\nphotons of any visible wavelength.\n\nBefore we go on, let's review: The sun creates photons that travel as\nelectromagnetic waves of assorted wavelengths to the cow. Many of\nthose photons are absorbed, but some are not. Some of those photons\nthat are not absorbed go into the lens of our camera.\n\n\\section{Pinhole camera}\n\nThe simplest cameras have no lenses. They are just a box. The box has\na tiny hole that allows photons to enter. The side of the box\nopposite the hole is flat and covered with film or some other\nphoto-sensitive material.\n\nThe photons entering the box continue in the same direction they were\ngoing when they passed through the hole. Thus, the photons that\nentered from high, hit the back wall low. The photons that came from\nthe left, hit the back wall on the right. Thus the image is projected\nonto the back wall rotated 180 degrees: What was up is down, what was\non the left is on the right.\n\nFIXME: picture here\n\n\\begin{Exercise}[title={Height of the image}, label=image_height]\n\nLet's say that that the pinhole is exactly the same height as the\nshoulder of the cow and that the shoulder is directly above one hoof.\nThan the pinhole, the shoulder, and the hoof form a right triangle.\n\nNow, let's say that the camera is being held perpendicular to the\nground. Now, the pinhole, the image of the shoulder, and the image of\nthe hoof on the back wall of the camera also form a right triangle.\n\nThese two triangles are similar.\n\nThe shoulder is 2 meters from the hoof. The cow is standing 3 meters\nfrom the camera. The distance from the pinhole to the back wall of\nthe camera is 3 cm. How tall is the image of the cow on the back wall\nof the camera?\n\n\\end{Exercise}\n\\begin{Answer}[ref=image_height]\n\nThe two triangles are similar, one is 2 m and 3m. The other is $x$ cm and 3 cm.\n\nThe image of the cow is 2 cm tall.\n\n\\end{Answer}\n\n\\section{Lenses}\n\nQuick review: A photon leaves the sun in some random direction. It\ntravels 150 million km from the sun and hits a cow. It is not\nabsorbed by the cow, and heads off in a new direction. It passes\nthrough the pinhole and hits the back wall of the camera. That seems\nincredibly improbable, right?\n\nIt actually is kind of improbable, especially if there isn't a lot of\nlight -- like you are taking the picture at dusk. To increase the\nodds, we added a \\newterm{lens} to the camera.\n\nIf you focus a lens on a wall, and then you draw a dot on the\nwall. The lens is designed such that all the photons from the dot that\nhit the lens get redirected to the same spot on the back wall of the\ncamera -- regardless of which path it took to get to the lens.\n\nFIXME: illustration here\n\nNote that the image still gets flipped. There is a \\newterm{ focal\npoint } that all the photons pass through.\n\nFIXME: illustration here\n\nThe distance from the lens to its focal point is called the lens's\n\\newterm{focal length}. Telephoto lenses, that let you take big\npictures of things that are far away, have long focal lengths.\nWide-angle lenses have short focal lengths.\n\n\\section{Sensors}\n\nThe camera on your phone has a sensor on the back wall of the\ncamera. The sensor is broken up into tiny rectangular regions called\npixels. When you say a sensor is 6000 by 4000 pixels, we are saying\nthe sensor is a grid of 24,000,000 pixels: 6000 pixels wide and\n4000 pixels tall.\n\nEach pixel has three types of cavities that take in photos. One of the\ncavities measures the amount of short wavelength light, like blues and\nviolets. One of the cavities measures the long wavelength light, like\nreds and oranges. One of the cavities measures the intensity of\nwavelengths in the middle, like greens.\n\nThus, if your camera has a resolution of $6000 \\times 4000$, the image\nis 72,000,000 numbers: Every one of the 24,000,000 pixels yeilds three\nnumbers: intensity of long wavelength, mid wavelength, and long\nwavelength light. We call these numbers ``RGB'' for Red, Green, and\nBlue.\n\n\n\n", "meta": {"hexsha": "a4bc9fe893fb6c2028f1f76d63b56f34a274a916", "size": 6993, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Modules/Oscillations/camera-en_US.tex", "max_stars_repo_name": "hillegass/sequence", "max_stars_repo_head_hexsha": "b7b4896d804c49cbc93fe86a0d2fce531afbcc1f", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 10, "max_stars_repo_stars_event_min_datetime": "2020-06-13T17:19:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-05T00:43:44.000Z", "max_issues_repo_path": "Modules/Oscillations/camera-en_US.tex", "max_issues_repo_name": "hillegass/sequence", "max_issues_repo_head_hexsha": "b7b4896d804c49cbc93fe86a0d2fce531afbcc1f", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Modules/Oscillations/camera-en_US.tex", "max_forks_repo_name": "hillegass/sequence", "max_forks_repo_head_hexsha": "b7b4896d804c49cbc93fe86a0d2fce531afbcc1f", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-05T00:43:58.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-05T00:43:58.000Z", "avg_line_length": 40.8947368421, "max_line_length": 108, "alphanum_fraction": 0.7670527671, "num_tokens": 1749, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467903998386998}} {"text": "\\documentclass[main.tex]{subfiles}\n\\begin{document}\n\n\\marginpar{Tuesday\\\\ 2020-4-21, \\\\ compiled \\\\ \\today}\n\nNow we have a dilemma: beauty versus pragmatism.\n\nHow do we provide a mass to the \\(W\\) vector boson?\n\n% We could introduce an \\(SU(2)\\) symmetry. \n\nAs we were discussing, we could add a mass term and write \n%\n\\begin{align}\n\\mathscr{L} = \\mathscr{L} _{\\text{Yang-Mills}} + M^2 W^{+}_{\\mu } W^{\\mu, -}\n\\,,\n\\end{align}\n%\nbut this term \\emph{brutally} breaks the symmetry.\nThis is explicit breaking of the symmetry. \n\nHowever, if we do not break the symmetry we cannot reproduce the data\\dots or can we?\n\n\\subsection{Spontaneous Symmetry Breaking}\n\nWe introduce \\textbf{Spontaneous Symmetry Breaking}. \nThis is a phenomenon which is not exclusive to HEP: it happens in ferromagnets, for example.\nThey start off (above the Curie temperature) with the spins pointing in uniformly distributed directions: the situation is symmetric.\nAs we cool them, at a certain stage all the spins align.\nWe cannot predict the direction along which they will align (since the initial state is symmetric), but once they do the symmetry is \\emph{spontaneously broken}.\n\nLet us take the standard approach (also done in the Theoretical Physics course \\cite[sec.\\ 5.2]{tissinoTheoreticalPhysicsNotes2020}) introduce the Lagrangian of a complex scalar field \\(\\phi \\): \n%\n\\begin{align}\n\\mathscr{L} = \\qty(\\partial^{\\mu } \\phi^{*}(x)) \\qty(\\partial_{\\mu } \\phi (x)) \n\\underbrace{- \\mu^2 \\abs{\\phi (x)}^2\n- \\lambda^{4} \\abs{\\phi (x)}^{4}}_{- V(\\phi )}\n\\,,\n\\end{align}\n%\nand to find the ground state we can move to the Hamiltonian: \n%\n\\begin{align}\n\\mathscr{H} = \\pi(x) \\phi (x) - \\mathscr{L}\n\\,.\n\\end{align}\n\nThe parameter \\(\\lambda \\) must be \\(>0\\), while \\(\\mu^2\\) has no constraints.\n\nIt is a critical parameter: if \\(\\mu^2\\) is positive we are in the symmetric phase of the system, and the vacuum is only \\(\\phi = 0\\): the VEV is \\(\\bra{0} \\phi \\ket{0} = 0\\). \n\nIf, on the other hand, \\(\\mu^2\\) is negative we transition to a new phase which is not symmetric: the vacuum becomes a whole circle, whose VEV is \\(v = \\sqrt{- \\mu^2 / 2 \\lambda }\\). The whole region parametrized as \\(v e^{i \\theta }\\), with \\(\\theta \\in \\mathbb{R}\\), provides equivalent vacua:\na specific ground state is not symmetric under the whole symmetry of the Lagrangian.\n\nWhy do we only have terms in \\(\\phi^2\\) and \\(\\phi^{4}\\)? we basically constructed the simplest potential which has the properties we want.\n\nIf \\(\\mu^2>0\\) we can perturb around the state without the quartic term, which is described by the KG equation.\n\nIf, instead, we want to perturb around a vacuum in the \\(\\mu^2<0\\) case we need to distinguish two directions, since the Hessian of the potential has two different eigenvalues, one positive and one equal to zero. We parametrize \n%\n\\begin{align}\n\\phi = \\frac{1}{\\sqrt{2}} \\qty(v + \\sigma (x) + i\\eta (x))\n\\,,\n\\end{align}\n%\nwhere \\(\\sigma \\) and \\(\\eta \\) are both real. \n\nThe Lagrangian can then be rewritten as \n%\n\\begin{align}\n\\mathscr{L} = \\frac{1}{2} \\partial^{\\mu } \\sigma \\partial_{\\mu } \\sigma \n+ \\frac{1}{2} \\partial^{\\mu } \\eta \\partial_{\\mu } \\eta \n+ 2 \\lambda v^2 \\frac{1}{2} \\sigma^2 \n- \\frac{\\lambda}{4} \\qty(\\sigma^2 + \\eta^2)^2\n- \\lambda v \\sigma \\qty(\\sigma^2 + \\eta^2)\n\\,,\n\\end{align}\n%\nso we have two real fields, one massive (\\(\\sigma \\)) and one massless (\\(\\eta \\)). The massless field is called a \\textbf{Goldstone boson}.\n\nIf we compute the VEV of \\(\\phi \\) in this configuration we get \\(v / \\sqrt{2 }\\). \n\nIn general, the Golstone theorem states that for each broken symmetry generator we get a massless boson. \n\nIt seems that instead of solving the problem of our massless gauge bosons we have created another problem, predicting more massless particles; but in fact our two problems solve each other.\n\nWe must break a different symmetry, though: a gauge local symmetry, instead of a global one.\nIf we explicitly break a \\(U(1)\\) global symmetry, we have a Goldstone boson.\nIf we break a \\(U(1)\\) local symmetry, instead, we get a massless vector boson and a massless scalar.\nThis spontaneous breaking of local gauge symmetry is the \\textbf{Higgs mechanism}.\n\n\\subsubsection{SQED Higgs mechanism}\n\nWe will see this explicitly, with the breaking of the scalar-QED \\(U(1)\\) symmetry: let us consider a Lagrangian \n%\n\\begin{align}\n\\mathscr{L} = \\qty(\\DD^{\\mu } \\phi )^{*} \\DD_{\\mu } \\phi \n- \\mu^2 \\abs{\\phi}^2 - \\lambda \\abs{\\phi }^4\n- \\frac{1}{4} F^{\\mu \\nu } F_{\\mu \\nu }\n\\,,\n\\end{align}\n%\nwhere \\(\\DD_{\\mu } = \\partial_{\\mu } + iq A_{\\mu} \\).\nThis is similar to a Lagrangian describing a fermion, but we substitute it with a complex scalar field with charge \\(q\\). \n\nThis Lagrangian has a \\(U(1)\\) gauge symmetry, whose action is \n%\n\\begin{align}\n\\phi &\\to e^{-iq \\alpha (x)} \\phi \\\\\n\\phi^{*} &\\to e^{iq \\alpha (x)} \\phi^{*} \\\\\nA_{\\mu } &\\to A_{\\mu } + \\partial_{\\mu } \\alpha \n\\,.\n\\end{align}\n\nIf \\(\\mu^2>0\\) we have no issues: the only minimum is the global one at \\(\\phi = A_{\\mu } = 0\\). This is the only configuration which is Lorentz invariant: any other vector would define a direction, breaking Lorentz symmetry. \n\nThe last point also holds if \\(\\mu^2 < 0\\); so the possible vacua are those defined as \\(\\phi = v e^{i \\theta } / \\sqrt{2}\\) and \\(A^{\\mu } = 0 \\).\nLet us perturb as we did before: if we define \n%\n\\begin{align}\n\\phi (x) = \\frac{1}{\\sqrt{2}} \\qty(v + \\sigma (x) + i \\eta (x))\n\\,,\n\\end{align}\n%\nthe Lagrangian can be rewritten as \n%\n\\begin{align}\n\\begin{split}\n\\mathscr{L} &= \\frac{1}{2} \\partial^{\\mu } \\sigma \\partial_{\\mu }\\sigma \n- \\frac{1}{2} (2 \\lambda v^2) \\sigma^2(x) \n- \\frac{1}{4} F^{\\mu \\nu } F_{\\mu \\nu } \\\\\n&\\phantom{=}\\ \n+ \\frac{1}{2} (q v)^2 A_{\\mu } A^{\\mu }\n+ \\frac{1}{2} \\partial^{\\mu }\\eta \\partial_{\\mu } \\eta \n+ qv A^{\\mu } \\partial_{\\mu } \\eta \n+ \\text{higher-order interactions} \n\\,.\n\\end{split}\n\\end{align}\n\nThis Lagrangian describes a real scalar \\(\\sigma \\) with mass \\(\\sqrt{2 \\lambda v^2}\\); a massless real scalar \\(\\eta \\); a massive vector boson and the interactions between them. \n\nLet us count degrees of freedom: we started with a complex scalar \\(\\phi \\) (2 dof) and a massless vector (2 dof); now we have two real scalars (1 dof each) and a massive vector (3 dof): we have gone from a total of four to a total of five! \nSo, there must be an unphysical field between these. \n\nIn fact, we can do a gauge transformation (going to the \\emph{unitary gauge}) so that \\(\\phi \\) is real everywhere: this removes the need for \\(\\eta \\). \nAfter this manipulation, the Lagrangian will read \n%\n\\begin{align}\n\\begin{split}\n\\mathscr{L} &= \\frac{1}{2} \\partial^{\\mu } \\sigma \\partial_{\\mu } \\sigma \n- \\frac{1}{2} (2 \\lambda v^2) \\sigma^2 \n- \\frac{1}{4} F_{\\mu \\nu } F^{\\mu \\nu } \\\\\n&\\phantom{=}\\ \n+ \\frac{1}{2} (qv)^2 A_{\\mu} A^{\\mu } \n+ \\text{higher-order interactions}\n\\,,\n\\end{split}\n\\end{align}\n%\nso we are left with a massive scalar \\(\\sigma \\) (1 dof) and a massive vector \\(A_{\\mu } \\) (3 dof): the total is 4 again! \n\nIt is a ``transmutation'' of degrees of freedom: one dof has gone from the scalar to the vector making it massive. \n\nThis is the \\textbf{Higgs mechanism}.\n\n% The wonderful thing is that the two dof of the massless vector ad the single degree of freedom of the massless scalar couple to give a massive vector with three degrees of freedom.\n\n\n\\end{document}\n", "meta": {"hexsha": "a63600b400ae699dcc0902e52d605bf63a3ac347", "size": 7345, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ap_second_semester/astroparticle_physics/apr21.tex", "max_stars_repo_name": "jacopok/notes", "max_stars_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-10-10T13:10:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T14:52:50.000Z", "max_issues_repo_path": "ap_second_semester/astroparticle_physics/apr21.tex", "max_issues_repo_name": "jacopok/notes", "max_issues_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ap_second_semester/astroparticle_physics/apr21.tex", "max_forks_repo_name": "jacopok/notes", "max_forks_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-03T16:20:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-06T16:11:07.000Z", "avg_line_length": 42.9532163743, "max_line_length": 295, "alphanum_fraction": 0.6778761062, "num_tokens": 2321, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3467903998386998}} {"text": "\\documentclass[12pt]{cdblatex}\n\\usepackage{bssn-eqtns}\n\n\\begin{document}\n\n\\section*{PhysRevD.62.044034 equation (20)}\n\n\\begin{cadabra}\n from shared import *\n import cdblib\n\n jsonfile = 'bssn-eqtns-20.json'\n cdblib.create (jsonfile)\n\n # --------------------------------------------------------------------------\n\n DGiBarDt := \\partial_{t}{GammaBar^{i}}. # cdb(eq20.00,DGiBarDt)\n DGiBarDt := - 2 ABar^{i j} \\partial_{j}{N}\n + 2 N ( GammaBar^{i}_{j k} ABar^{k j}\n - (2/3) gBar^{i j} \\partial_{j}{trK}\n + 6 ABar^{i j} \\partial_{j}{\\phi}). # cdb(eq20.01,DGiBarDt)\n\n substitute (DGiBarDt,defGammaBarU) # cdb(eq20.02,DGiBarDt)\n\n distribute (DGiBarDt)\n DGiBarDt = product_sort (DGiBarDt) # cdb(eq20.03,DGiBarDt)\n\n canonicalise (DGiBarDt) # cdb(eq20.04,DGiBarDt)\n # cdb(eq20.99,DGiBarDt)\n\n cdblib.put ('DGiBarDt',DGiBarDt,jsonfile)\n\\end{cadabra}\n\n\\begin{dgroup*}\n \\begin{dmath*}\n \\cdb{eq20.00} = \\Cdb*{eq20.01}\n = \\Cdb*{eq20.02}\n = \\Cdb*{eq20.03}\n \\end{dmath*}\n\\end{dgroup*}\n\n\\end{document}\n", "meta": {"hexsha": "2bf4c71a0844053fbc22d3a5434ab9bfccec8813", "size": 1261, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "bssn/cadabra/bssn-eqtns-20.tex", "max_stars_repo_name": "leo-brewin/adm-bssn-numerical", "max_stars_repo_head_hexsha": "9e32c201272e9a41e7535475fe381e450b99b058", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-25T11:36:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-25T11:36:06.000Z", "max_issues_repo_path": "bssn/cadabra/bssn-eqtns-20.tex", "max_issues_repo_name": "leo-brewin/adm-bssn-numerical", "max_issues_repo_head_hexsha": "9e32c201272e9a41e7535475fe381e450b99b058", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "bssn/cadabra/bssn-eqtns-20.tex", "max_forks_repo_name": "leo-brewin/adm-bssn-numerical", "max_forks_repo_head_hexsha": "9e32c201272e9a41e7535475fe381e450b99b058", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.3255813953, "max_line_length": 82, "alphanum_fraction": 0.4829500397, "num_tokens": 415, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032313, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.346769911690831}} {"text": "\\documentclass[twoside]{MATH77}\n\\usepackage{multicol}\n\\usepackage[fleqn,reqno,centertags]{amsmath}\n\\begin{document}\n\\begmath 19.4 Converting Codes to Different Versions\n\n\\silentfootnote{$^\\copyright$1997 Calif. Inst. of Technology, \\thisyear \\ Math \\`a la Carte, Inc.}\n\n\\subsection{Purpose}\n\nThis program, {\\tt m77con}, converts codes from one version to another.\nIt is used to convert between different precisions, and to support the\nspecial comments used in some of the Fortran codes for input to a program\nthat converts programs from Fortran to C. Support is provided for\nprecision greater than double for systems that support such a feature.\nSupport is provided for conversion between real programs and integer or\ncomplex versions of a program, and also partial double precision and\npartial quadruple precision. Most MATH77 codes contain the comments only\nfor conversion between the single and double precision cases. Although\ndesigned for use with MATH77 codes, {\\tt m77con} could be used in other\napplications that want a portable means to support multiple versions. Any\nversion of a code can be used as input to {\\tt m77con} for the purpose of\ngetting another version.\n\n\\subsection{Usage}\n\n One uses this program by creating a control file named {\\tt\nm77job}, and executing the program {\\tt m77con} from the command\nline. The first part of this section describes the control file which\ndefines what version is desired. The second part describes the\ncomments in the code which indicate how different versions are to be\nobtained.\n\n\\subsubsection{Transforming an Instrumented Code}\n\nIn the descriptions below, items in brackets are optional; $L_i$ is\nused for letters (frequently the first letter in the name of a\nprogram, which indicates the precision); and $T_i$ is used for\nletters associated with types. Types supported include: S, D, Q, I,\nC, Z, and W, which are used for Single precision, Double precision,\nQuadruple precision (or at least some precision greater than\ndouble), Integer, Complex, double precision complex, and quadruple\nprecision complex respectively.\n\n``C...'' is a comment and can appear anywhere in the control file. A\ncomment of the form ``C$>>$yyyy$-$mm$-$dd...'' is treated as a special\ncase. It inserts a date stamp in the file being processed, which is\nused to track changes in MATH77 codes. Details on this can be found in\nthe code listing for {\\tt m77con}.\n\n``MAKE $L_{out}$ [with type $T_{out}$] [from $L_{in}$ [with type\n$T_{in}$]]'' is used to define a letter in the names of the output\nand input files and types associated with these files. $L_{out}$ is\na letter used in output file names. $T_{out}$ is the type for the\noutput file. If $T_{out}$ is not given it is assumed to be $L_{out}$\nif $L_{out}$ is an allowed type, is assumed to be S or D, for\n$L_{out}$ = P or X respectively, and otherwise an error results. (P\nis assumed to be used for partial double precision, and X for partial\nquadruple precision.) $L_{in}$ and $T_{in}$ are defined as for\n$L_{out}$ and $T_{out}$, except they apply to input files. If the\n``from'' clause is missing, $L_{in}$ is D, except when $L_{out}$ is\nD, in which case $L_{in}$ is S. The case of letters is not\nsignificant, except for $L_{out}$. The case used for $L_{out}$\ndetermines the case used when substituting $L_{out}$ or $L_{in}$ into\na file name, and thus the case is significant on systems where case\nis significant in a file name.\n\n``$T$ Defined by: $L_{exp}$, {\\em keyword}, {\\em type\\_conv}, {\\em\nmach\\_const}\\/''\\newline\n``$T$ Defined by: {\\em keyword}, {\\em type\\_conv},\n{\\em conj}, {\\em imag\\_part}\\/''\\newline\nOne of these lines is needed only if the input or output code makes\nuse of a nonstandard type, namely, Q, Z, or W. Note that W makes use\nof Q, and thus if it is used, Q must also be defined. The first type\nof line is used to define real (as opposed to complex) types. $L_{exp}$\nis the letter used to define the exponent part of floating point\nconstants, {\\em keyword} is the text used to declare a variable of\ntype $T$, {\\em type\\_conv} is the name of the function used for type\nconversion to type $T$, {\\em mach\\_const} is the name of the function\nused for getting machine constants, {\\em conj} and {\\em imag\\_part}\nare the functions used to conjugate or get the imaginary part of a\ncomplex expression, respectively. The standard S, D, and C types are\ninitialized as if one had input:\\newline ``S Defined by: E, real,\nreal, r1mach'',\\newline ``D Defined by: D, double precision, dble,\nd1mach'', and \\newline ``C Defined by: complex, cmplx, conjg, aimag''\n\nNote that at most one embedded blank is allowed inside {\\em keyword},\nand if such a blank is desired it must be included on this line.\nInternally, `I' defines {\\em keyword} as ``integer'', and {\\em\ntype\\_conv} as ``int''.\n\n``VERIFY path'' if present will compare the lines of the result files\nwith lines in the files of the same name as the result files preceded\nby ``path'', and when lines are different, both lines will be printed.\nThe text in ``path'' is taken exactly as given, and is considered\nterminated by either a blank or an end of line. This comparison is\nnot sensitive to changes in the special comment lines described in\nthe following subsection, or to differences in the number of\nblanks.\n\n``SET $A_1$, $A_2$, ... $A_n$'' is used to set values for the\nmetavariables defined by the assignments $A_i$. These assignments, which\nsometimes are necessary to get the version of a code desired, and the\nmetavariables set by these assignments, are described in the following\nsubsection. One can have any number of ``SET...'' lines.\n\n``STOP ...'' acts like an end of file, i.e. no more lines are read in\nthe control file.\n\n``FILE {\\em name}[[,] {\\em ext}]'' is used to specify the name of a file\nto be processed. A ``?'' in name may be used to parameterize the name if\ndesired. $L_{in}$ is substituted for the ``?'' (if any) and a file of\nthis name is processed as the input file. Then $L_{out}$ is substituted\nfor ``?'' to obtain the name of the output file. If the ``,~{\\em ext}\\/''\nis present and one of the first two characters in {\\em ext} is ``.'' this\nname is further modified by searching backward from the end of ``{\\em\nname}\\/'' looking for the first character in ``{\\em ext}\\/''. When this\ncharacter is found, this character and the following characters in {\\em\nname} are replaced by ``{\\em ext}\\/''. If the ``,~{\\em ext} is present\nand neither of the first two characters is a ``.'', {\\em ext} is used as\nthe name of the output file. The input and output files can have the same\nname. If the ``?'' is missing, there is no need to have had a ``MAKE...''\nline. Any characters except ``,'' can appear in ``{\\em name}\\/'' or\n``{\\em ext}\\/'', as long as the results will serve as valid file names.\nExamples and Remarks, below, gives samples of FILE statements.\n\nProcessing a ``FILE...'' line involves carrying out conversions on\nthe specified input file and placing the result in the output file.\nIf a ``VERIFY...'' line is present, the output file will be compared\nwith the file specified there. After processing a ``FILE ...'' line,\nthe next line from the control file, {\\tt m77job}, is read, and\nactions are taken as requested there. All metavariables defined in\nprevious SET instructions become undefined if there is a new SET\ninstruction; otherwise the old values are kept.\n\n\\subsubsection{Instrumenting a Code}\n\nWe use the notation:\n\n\\begin{tabular}{l@{ }l@{ }ll}\n$L$,&$L_1$,&... $L_i$ & Letters\\newline\\\\\n$T$,&$T_1$,&... $T_i$ & Letters used to denote types\\newline\\\\\n$I$,&$I_1$,&... $I_i$ & Integer constants\\newline\\\\\n$V$,&$V_1$,&... $V_i$ & Metavariables\\newline\\\\\n$E$,&$E_1$,&... $E_i$ & Expressions\\newline\\\\\n$A$,&$A_1$,&... $A_i$ & Assignments\\newline\\\\\n\\end{tabular}\n\nThe types are: S, D, Q, I, C, Z, and W, which are used for Single\nprecision, Double precision, Quadruple precision (or at least some\nprecision greater than double), Integer, Complex, double precision\ncomplex, and quadruple precision complex respectively. An ``$L$''\nequal to one of these letters is used for a code of the corresponding\ntype. Other letters are used for mixtures of types. In particular,\nP (Partial double) is used for a code that uses some D in a code that\nis primarily S (perhaps to save on storage), and X (eXtended) is used\nfor a code that is primarily in D, but uses some Q. No other\nprecisions are supported.\n\nThere are integer, logical, and string metavariables and constants.\nAn integer constant is defined as in Fortran. The logical constants\nare .T.\\, .F.\\, .U.\\, .D.\\, .S.\\, .I.\\, and .C.\\ Of course .T.\\ is\nalways true, and .F.\\ is false; .U.\\ is undefined (see below); .D.\\,\n.S.\\, and .I.\\ have a value of .T.\\ if the upper case of the letter\n$L_{out}$ in a MAKE command matches the letter between the dots, has a\nvalue of .F. if $L_{out}$ is defined but does not match, and otherwise\nhas the value .U.; and .C.\\ has the value .T.\\ if a version of the\ncode used as input to the Fortran to C converter is being generated,\nand is .F. otherwise. Initially .C.\\ =.F.; its value can be set to\n.T.\\ with the statement ``SET .C.'' in {\\tt m77job}, and can be reset\nto .F.\\ with the statement ``SET $\\sim $.C.''. (With T=.T., F=.F.,\nand U=.U., we have, U$|$F=.U., U$|$T=.T., U\\&F=.F., U\\&T=.U., $\\sim\n$U=.U. If U is on either side of a relational operator, the result is\n.U.)\n\nA string is defined as an arbitrary sequence of characters enclosed by\neither apostrophes ({\\tt '}), or quotes ({\\tt \"}), has the same form as a\nmetavariable (see below), or is .X., .Y., or .N.\\ \\ \\ .X. gives the\nstring consisting of the letter used for floating point exponents in the\noutput, .Y.\\ gives the string consisting of the letter $T_{out}$, and\n.N.\\ gives the string consisting of the letter $L_{out}$. If a\nlogical value of a string is required by the context, a string has the\nvalue .U.\n\nA metavariable is any number of nonoperators, not all of which are\ndigits, and not one of the letters ``TFUXN'' preceded and followed by\na ``.''. Metavariables once defined remain defined until the end of\nthe file. Those metavariables defined in {\\tt m77job} remain defined\npast the end of file, until the next ``SET...'' line read in {\\tt\nm77job}. Note that metavariables appear only in ``C++...'' lines\nand in lines immediately following ``C++ Substitute for ...'' lines.\n\nExpressions may use the following operators, all of which have the same\nmeaning and precedence as their Fortran counterparts.\n\n\\begin{tabular}{ll}\nArithmetic:& + \\hspace{8pt} $-$ \\hspace{8pt} * \\hspace{8pt} /\n \\hspace{8pt} **\\\\\nRelational:& $<$ \\hspace{8pt} $>$ \\hspace{8pt} == \\hspace{8pt} /=\n \\hspace{8pt} $<=$ \\hspace{8pt} $>=$\\\\\nAlt. Relational:& = \\hspace{8pt} $\\sim $=\\\\\nLogical:& $|$ (or) \\hspace{8pt} \\& (and) \\hspace{8pt} $\\sim $ (not)\\\\\nConcatenation:& // \\\\\nGrouping letters:& $\\{$ \\hspace{8pt} $\\}$\\\\\nOther:& ( \\hspace{8pt} ) \\hspace{8pt} ,\\\\\n\\end{tabular}\n\nThe ``$\\{$'' and ``$\\}$'' are used to define a special operand. If\nthe letter $L_{out}$ from the ``MAKE...'' line, see ``Transforming an\nInstrumented Code'' above, appears inside the \\{...\\}, then this\noperand has the value true, and otherwise has the value false. The\nletters inside the \\{...\\} may optionally be separated by commas.\nThus \\{SP\\} is true if $L_{out}$ is either S or P, and is false\notherwise.\n\nWe recommend using ``=='' rather than ``='' in relational\nexpressions. ``='' is allowed for compatibility with comments\nwritten for an earlier processor. The first ``='' in an assignment\nstatement is part of the assignment statement, not part of an\nexpression.\n\nAn expression can be formed as one would expect using the operators\nthat are available. A diagnostic is given if one tries to mix types\nin a way that make no sense. Integer and logical are converted to\nstrings when they appear as an operand of ``//''.\n\nAs assignment has the form: $V = E$, or simply $V$. The latter form is\nequivalent to writing $V$=.T.\n\nThree kinds of comments in the source code initiate special processing by\nthe program. (Not counting the ``C$<<$...'' comments which are used for\ndate stamps.) Those headed by ``C++'' are intended primarily to support\ndifferent versions of a program. Those headed by ``C$--$'' are intended\nprimarily to support different precisions, which involves changing types in\nFortran statements. Those headed by ``C\\%\\%'' are used to convert the\ncode to the form needed for the Fortran to C conversion program. The\nactions for ``C++...'' lines precede other actions, and thus a ``C++...''\nline could result in turning off a ``C$--$...'' line. Also, if there was\nno ``MAKE...'' line, type conversion is turned off, and ``C$--$...'' lines\nare simply treated like any other comments. These comments allow either\ncase for the ``C'' and also allow an ``*'' in column one.\n\n``C++...'' lines have the form:\n\n\\begin{tabular}{@{}l@{ }l@{ }l}\n(1) & ``C++ & Code for $E$ is active''\\\\\n(2) & ``C++ & Code for $E$ is inactive''\\\\\n(3) & ``C++ & END''\\\\\n(4) & ``C++ & Of next $I$ lines, only the first $E$ are active''\\\\\n(5) & ``C++ & Default $A_1$, $A_2$, ... $A_i$'', or\\\\\n & ``C++L & Default $A_1$, $A_2$, ... $A_i$'', or\\\\\n & ``C++(E) & Default $A_1$, $A_2$, ... $A_i$''\\\\\n(6) & ``C++ & Current has A''\\\\\n(7) & ``C++ & Substitute for $V_1$, $V_2$, ... $V_i$ below''\\\\\n(8) & ``C++ & Replace $A_1$, $A_2$, ... $A_i$''\\\\\n(9) & ``C++ & [With first index $E_1$,]\\\\\n & & Save data by elements if $E_2$''\n\\end{tabular}\n\nA line of form (1) or (2) initiates a scope of applicability that is\nterminated by the next line of form (1), (2), or (3). In either case\n(1) or (2), if $E$=.U., no change is made in this line or in the\nlines in its scope.\n\nIn the case of (1), if $E$=.T. there is no change. If $E$=.F.,\nthe word ``active'' is changed to ``inactive'' and in the following\nscope, columns 1--71 are shifted to columns 2--72 and a C placed in\ncolumn 1. An error message is printed if the original line was not\nblank in column 72. (Only the first 72 characters of input lines are\nread.)\n\nIn the case of (2), if $E$=.F. there is no change. If $E$=.T. the\nword ``inactive'' is replaced by ``active'' and in the following\nscope, columns 2--72 are copied to columns 1--71, and a blank placed\nin column 72.\n\nNo other ``C++...'' line may appear between a line of type (1) or (2)\nand the next line of type (1), (2), or (3). No code changing action\ncan be active when the end of a Fortran program unit is reached.\n\nExamples from AMACH of lines of type (1), (2) and (3) are,\n\n\\begin{verbatim}\nc++ Code for SYS = IEEE is ACTIVE\n...\nc++ Code for SYS = AMDAHL is INACTIVE\nC ...\nc++ END\n\\end{verbatim}\n\nIn case (4), one must have $I$ (an integer constant) $>$ 0, and $E$\nmust have an integer value $ <= I$. The following $I$ lines in the\ncode must be valid Fortran statements if column~1 is set to a blank\nand there must be $I$ lines preceding the last line of the Fortran\nprogram unit. This causes the action of storing a ` ' in column~1 of\nthe following $E$ lines, and then storing a `C' in column~1 for the\nremaining $I - E$ lines (neither shifts the line).\n\nAn example from DIVA of using this feature is (the first two lines\ngiven here are one line in the code):\n\n\\begin{verbatim}\nc++ Of next 23 lines, only the first\n KDIM+MAXORD are active\n data B(1) / 5.00000...0D-1 /\n ...\n data B(22) / 1.97628...5D-3 /\nC data B(23) / 1.81159...2D-3 /\n\\end{verbatim}\n\nIn case (5) when no L or (E) is present each assignment, $A_i$, is\nexecuted if and only if the $V_i$ on the left of the ``='' in the $A_i$ is\nnot already defined. (Recall that if $A_i$ does not contain an ``=''\nsign, it is as if it had the form ``$V_i$ = .T.''.) $V_i$ may have been\ndefined either from a ``SET...'' line in {\\tt m77job} or from an\nearlier ``C++'' line. When L is present, L must be a sequence of\nletters and the assignments will be processed as described for the\ncase when no L is present if and only if the current $L_{out}$ matches\none of the letters in L. When (E) is present, the line is processed as\nabove if E is a logical expression with the value .T., else the line rest\nof the line is ignored.\n\nThe following example from DRDIVA illustrates defining a default\nvalue for NDIG of 4 for type of ``S'' or ``P'', and a default value for NDIG\nof 10 in other cases. If NDIG has been set in {\\tt m77job}, then\nthe value set there will be used.\n\n\\begin{verbatim}\nc++SP Default NDIG = 4\nc++ Default NDIG = 10\n\\end{verbatim}\n\nIn case (6), the A must have an ``='' sign. If the $V$ on the left\nside of the ``='' in A is not defined this works just like a\n``C++ Default A'' line. If $V$ is defined, then the text on the\nright of the ``='' is replaced by the current value of $V$.\n\nThe following line from AMACH shows that AMACH is currently\nconfigured for SYS = IEEE. If SYS is not set in {\\tt m77job}, this\nvalue for SYS will continue to be used. Otherwise the new value will\nreplace the {\\tt IEEE} on this line.\n\n{\\tt c++ CURRENT HAS SYS = IEEE}\n\nIn case (7), the following line is examined for each of the $V_i$'s in\nturn. If such a metavariable name is found followed by an ``='' sign, and\nan integer value, the integer is replaced by the integer value of $V_i$.\nIf the ``='' sign is followed by a logical value, this value is replaced\nby the corresponding Fortran logical constant for $V_i$. A type mismatch\nin either of these cases results in a diagnostic, with no change. A\nvalue of .U. in this case is treated as a mismatch.\n\nThe following from DRDIVA illustrates using this feature.\n\n\\begin{verbatim}\nc++ Substitute for NDIG below\n parameter (NDIG = 10)\n\\end{verbatim}\n\nIn case (8), the left side of each $A_i$ must be a constant character\nstring, i.e. either {\\tt '}xx...x{\\tt '} or {\\tt \"}xx...x{\\tt \"}, and\nthe right side, $E_i$, must be an expression that evaluates to a\nstring value. Occurrences (case of letters must match) of the\n``xx...x'' on the left of $A_i$ are replaced by the value for $E_i$.\nThe replacement is made on ``C++ Replace...'' lines, and other lines\nnot starting with ``C++''. If this line precedes a ``C$--$...'' line\nof type (1), see below, then the replacement applies starting with\nthe first line of the file else it applies starting with the current\nline.\n\nThe following from DRDRAN illustrates this feature.\n\n\\begin{verbatim}\nc++S Default NDIG = 6\nc++ Default NDIG = 12\nc++ Replace \"f15.12\" = \"f\"//NDIG+3//\".\"//NDIG\n...\n print'(7x,i7,'',''i5,10x,f15.12)',K...\n\\end{verbatim}\n\nCase (9) is intended primarily to avoid unnecessary mess in Fortran code\nthat would have too many continuation lines for a data statement in\nFortran when declared with a single array. The C converter we are using\ngenerates much better code when such data statements are declared with a\nsingle array instead of a separate statement for each element. $E_1$ must\nbe a integer expression, and $E_2$ must be logical. If the part\ncontaining $E_1$ is missing the action is as if $E_1$ were present with a\nvalue of 1. The data statement following this must contain exactly one\ndata item per line. When $E_2$ = .T., the first line and every following\nline that is either a continuation line, or is a data statement for the\nsame name, is changed (if necessary) to be a data statement for a single\nelement of the array. The index used for the array element is $E_1$ for\nthe first data statement and is incremented by one for the following\nstatements. If $E_1$ is .F. the first data statement line uses only the\narray name, and following lines that are either continuation lines, or\ndata statements for the same name are changed to be continue statements.\nThere is no limit on the number of continued lines.\n\nThe ``C$--$...'' lines have the form:\n\n\\begin{tabular}{@{}lll}\n(1) & ``C$-$$-$$L$ & Replaces ``?'': $V_1$, $V_2$, ...''\\\\\n(2) & ``C$-$$-$$T$ & (Type)Replaces ``?'': $V_1$, $V_2$, ...''\\\\\n(3) & ``C$-$$-$\\& & $v_1$, $v_2$, ...''\\\\\n(4) & ``C$-$$-T$ & Next line special: $S_1$, $S_2$, ...''\\\\\n(5) & ``C$-$$-$ & Begin mask code changes''\\\\\n(6) & ``C$-$$-$ & End mask code changes''\\\\\n\\end {tabular}\n\nAt most one of (1) or (2) can occur in a file, and all lines of type\n(3) must follow immediately after a (1) (2) or (3). (No embedded\ncomments.) In case (1) $L$ denotes the value of $L_{out}$ for the\ncurrent version of the code. (See the description of the ``MAKE...''\nline in Usage above.) If $L$ = $L_{out}$, then no names are changed.\nWhen $L=L_{in}$ the $L$ in this statement is replaced by $L_{out}$.\nEach of the $V_i$, or $v_i$, must be Fortran names with one of the\nletters replaced by a ``?''. These names with the ``?'' replaced by\n$L_{in}$ are searched for and replaced as described below, starting\nat the beginning of the file. A line of type (2) must follow\nimmediately after a line of type (1) or (3). The action for this\nkind of line is similar to that for type (1), but with $L$,\n$L_{out}$, and $L_{in}$, replaced by $T$, $T_{out}$, and $T_{in}$\nrespectively.\n\nThe following excerpt from DWCOMP illustrates lines of type (1) and\n(3).\n\n\\begin{verbatim}\nc--D replaces \"?\": ?WCOMP,?WATAN,...\nc--& ?WSQRT,?WEXP,?WSIN,?WCOS,...\n\\end{verbatim}\n\nCase (4) is used to indicate either that something different from the\nstandard conversion is to be used on the next line, or that\nconversion to or from integer or one of the complex types is allowed.\nConversions to or from integer types occur only on lines so marked.\nFor a conversion to or from a code corresponding to the first letter\nof one of the pairs, (C, S), (Z, D), (W, Q), the conversion is done\nas if the second letter applied, except for lines marked in this\nmanner. $T$ indicates the type to assume for the input on the\nfollowing line. The $T$ is replaced by the value used for the type of\nthe output on the next line. The $S_i$ is either a single $L_i$, or\nan expression of the form $L_i$ $=>$ $T_i$. If $L_{out}$ is the same\nas any of these $L_i$, then the following line is treated as special\nbased on this $S_i$. If there is no ``$=>$ $T_i$'' part then\npresumably $L_i$ is I, C, Z, or W, and the following line is\nmodifiable to suit these types. If the ``$=>$ $T_i$'' part is\npresent, then the following line is to be converted as if the output\ntype were $T_i$. If none of the $L_i$ match the $L_{out}$ from the\n``MAKE...'' line, the following line is treated just like any other\nline.\n\nThe following from DRDIVA illustrates specifying metavariables with a\nhigher than usual precision when $L_{out}$ is ``P'' or ``X''. In all\nother cases the declaration here would be what one would expect\nfrom the letter specifying the precision.\n\n\\begin{verbatim}\nc--D Next line special: P=>D, X=>Q\n double precision TSPECS(4), Y(IYDIM)...\n\\end{verbatim}\n\n?VECPR is one of the few codes in MATH77 which allows all the real\nprecisions, and also allows integer. The first pair of lines below\nget converted to the second pair when DVECPR is converted to IVECPR.\n\n\\begin{verbatim}\nc--D Next line special: I\n double precision V(N)\n\nc--I Next line special: I\n integer V(N)\n\\end{verbatim}\n\nLines of type (5) and (6) mark the beginning and end, respectively,\nof a set of lines in which all code modifications due to ``C$--$''\ncommands will be inhibited.\n\nThe ``C\\%\\%'' lines are documented here so that those looking at the\nlibrary source will understand their purpose. These lines only have a\nspecial meaning when they are not being treated in a special way for some\nother reason. In particular, if a ``C++ END'' statement would be\nacceptable, then such lines are not treated in a special way. When .C.\\ =\n.T., the ``C'' is deleted from column one in any sequence of consecutive\nlines containing ``C\\%\\%'', and a C is inserted at the start of the\nfirst line (one line only) that does not start in this way. If .C.\\ =\n.F., and the lines are not being treated in a special way for some other\nreason, then consecutive lines starting with ``\\%\\%'' have a C inserted\nin column 1, and the next line has the character in column 1 deleted.\nThese lines provide a way of indicating changes required to get the code\nconverted to C, without using the ``C++'' mechanism, which makes the code\ndifficult to read when used frequently.\n\n\\subsection{Examples and Remarks}\n\nIf one wanted to get the partial double precision version of the\nintegration program DIVA, Chapter~14.1, one could set {\\tt m77job} to\nthe following for a unix machine. (For a PC replace the {\\tt .f}\nwith {\\tt.for}.)\n\n{\\tt \\begin{tabbing}\n MAKE p\\\\\n FILE ?iva.f\\\\\n FILE ?ivag.f \\\\\n FILE ?ivadb.f\n\\end{tabbing}}\n\nTo convert {\\tt amach} for a VAX running UNIX, {\\tt m77job} would\ncontain the following to obtain the output file {\\tt amach.f} from\nthe input file {\\tt amach}.\n\n{\\tt \\begin{tabbing}\n SET SYS = VAX\\\\\n FILE amach.f\n\\end{tabbing}}\n\n\\subsection{Functional Description}\n\nThe program {\\tt m77con} converts between different versions of Fortran~77\nsource code that satisfy conventions used for MATH77. It combines\nfunctionality from earlier programs: ``The Specializer''\n\\cite{Krogh:1974:ALS}, ``MARVEL'' \\cite{Lawson:1980:MAR}, and ``CHGTYP''\n\\cite{Lawson:1988:CHG}.\n\nThe code is processed a line at a time, first doing actions on\n``C++...'' lines, then for other lines doing the actions called for\nby ``C++...'' lines (except for replacing strings), then doing\nactions on ``C$-$$-$...'' lines, and then doing other replacements.\nProcessing restarts with the first line after initial processing of\n``C$-$$-$'' lines of types 1--3.\n\nExcept for strings defined by ``C++ Replace...'' lines, all\ncomparisons done by this code are case insensitive, all replacements\nare done in the same case as for the text found, and blanks are\nremoved before comparisons are made in the source code.\n\nLet ``in'' denote a string, keyword, variable name, or function name\nwe are searching for, and ``out'' denote the corresponding\nreplacement. When conversion results in substitutions, if ``out'' is\nlonger than ``in'', characters to the right of ``in'' are shifted\nright to leave enough space to insert ``out''. A diagnostic\nresults if in the process nonblank characters are shifted beyond\ncolumn 72 (and thus lost). If ``out'' is shorter than ``in'', then\n``out'' is padded with blanks so as to occupy the same space as\n``in''.\n\nWhen type conversion is on (as it is if there is a ``MAKE...'' line\nin the control file), then all lines, except those with ``++'' or\n``$--$'' in columns 2 and 3, are examined in turn for the following:\n\nIf the ``in'' {\\em keyword} is found, it is replaced by the ``out''\n{\\em keyword}.\n\nThen ``in'' default function names for type conversion and getting\nmachine constants (see ``$T$ Defined by:...'' lines in Usage above)\nare searched for, and when found are replaced by the corresponding\nnames of the ``out'' functions. When searching for names (here and\nbelow) they are only found if they are preceded and followed by a\nnonletter or nondigit. Function names which do not have ``mach'' as\ntheir last four letters are found only if followed by a `('.\n\nThen the $V_i$'s and $v_i$'s generated from the ``C$--$L...'',\n``C$--$T...'', and ``C$--$\\&'' lines are searched for. When found,\n$L_{out}$ (or $T_{out}$ for names defined on ``C$--$T (Type)...''\nlines or ``C$--$\\&'' lines following such lines) replaces the letter\nin the position of the ``?'', preserving the case of the letter that\nwas there.\n\nThen, those strings defined by the ``C++ Replace...'' lines are\nsearched for and replaced if found.\n\nThen, floating point constants are converted. These are assumed to\nconsist of (not a letter).(0 or more digits)(the letter used for\nfloating point constants in the current version)(an optional + or\n$-$)(a digit). If such a string is found, the letter is replaced as\nspecified.\n\nWhether type conversion is on or not, lines consisting of only ``END'' are\nsearched for. When found, {\\tt m77con} checks if there are any ``C++...''\nactions still active or if the processing is not in ``usual precision''\nmode. In either of these cases a diagnostic is given and the processing\nreturns to normal mode.\n\n\\bibliography{math77}\n\\bibliographystyle{math77}\n\n\\subsection{Error Procedures and Restrictions}\n The program prints a variety of error messages on the standard\noutput. An error in the control file causes the program to stop. In\nmost cases of errors in the special comments of the files being\nprocessed, processing continues. After 10 comparison errors in\na single file, no more comparison errors are printed for that file.\n\n\\subsection{Supporting Information}\n\nThe source language is ANSI Fortran~77.\n\nDesign and code due to F. T. Krogh, JPL, October~1994. Multiple minor\nmodifications through April~1996.\n\n\\end{multicols}\n\\end{document}\n", "meta": {"hexsha": "c6682834bcfcb0e681d8fd1b8534b3770f7c24c3", "size": 28804, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/doctex/ch19-04.tex", "max_stars_repo_name": "jacobwilliams/math77", "max_stars_repo_head_hexsha": "b562d09e191e99eba8a5bedfec45acf7461203b1", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2016-01-04T03:17:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-25T19:17:42.000Z", "max_issues_repo_path": "doc/doctex/ch19-04.tex", "max_issues_repo_name": "jacobwilliams/math77", "max_issues_repo_head_hexsha": "b562d09e191e99eba8a5bedfec45acf7461203b1", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2019-01-17T02:48:32.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-21T16:04:58.000Z", "max_forks_repo_path": "doc/doctex/ch19-04.tex", "max_forks_repo_name": "jacobwilliams/math77", "max_forks_repo_head_hexsha": "b562d09e191e99eba8a5bedfec45acf7461203b1", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2016-01-07T09:26:45.000Z", "max_forks_repo_forks_event_max_datetime": "2018-11-25T05:32:54.000Z", "avg_line_length": 47.767827529, "max_line_length": 98, "alphanum_fraction": 0.7046937925, "num_tokens": 8151, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073802837478, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.34675854675216977}} {"text": "% rubber: module xelatex\n\\documentclass[english,final,compress]{beamer}\n%\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage{fixltx2e}\n\\usepackage{graphicx}\n\\usepackage{longtable}\n\\usepackage{float}\n\\usepackage{wrapfig}\n\\usepackage{soul}\n\\usepackage{textcomp}\n\\usepackage{marvosym}\n\\usepackage{wasysym}\n\\usepackage{latexsym}\n\\usepackage{amssymb}\n\\usepackage{hyperref}\n\\tolerance=1000\n\\mode{\\usetheme{Bonn}}\n\\usepackage{fontspec}\n\\defaultfontfeatures{Mapping=tex-text}\n\\usepackage{xunicode}\n\\usepackage{xltxtra}\n\\setmainfont[Scale=0.86,Mapping=tex-text]{News Gothic MT}\n\\setsansfont[Scale=0.86,Mapping=tex-text]{News Gothic MT}\n\\setmonofont[Scale=0.86,Mapping=tex-text]{Andale Mono}\n\\usepackage{polyglossia}\n\\setdefaultlanguage{english}\n\\usepackage[EU1]{fontenc}\n\\usepackage{listings}\n\\usepackage{color}\n\\usepackage{csquotes}\n\\usepackage{mdwtab}\n\\usepackage{booktabs}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{amstext}\n\\usepackage{amsthm}\n\\usepackage{listings}\n\\renewcommand\\maketitle{\\frame[plain]{\\titlepage}\\addtocounter{framenumber}{-1}}\n\\providecommand{\\alert}[1]{\\textbf{#1}}\n\n\\setbeamercolor{insult}{fg=black,bg=red!30}\n\\setbeamercolor{noinsult}{fg=black,bg=green!30}\n\\setbeamerfont*{myserif}{family=\\rmfamily,size=\\scriptsize}\n\n\\title{Tutorial Machine Learning in Python}\n\\author{Andreas M\\\"{u}ller, Hannes Schulz, Nenad Bire\\v sev and Sven \n Behnke\\\\[5mm]\\includegraphics[width=.2\\linewidth]{style/Logo_UBo_h24_4c-crop}}\n\\date{GK Bionik Tutorial 2012}\n\\begin{document}\n\n\\maketitle\n\n\n\\lstset{%\n basicstyle=\\small\\tt,\n keywordstyle=\\color{beamer@bonnblue}\\bfseries, % style for keywords\n numbers=none, % where to put the line-numbers\n numberstyle=\\tiny, % the size of the fonts that are used for the line-numbers\n showspaces=false, % show spaces adding particular underscores\n showstringspaces=false, % underline spaces within strings\n showtabs=false, % show tabs within strings adding particular underscores\n tabsize=2, % sets default tabsize to 2 spaces\n captionpos=b, % sets the caption-position to bottom\n breaklines=true, % sets automatic line breaking\n breakatwhitespace=false, \n}\n\\newcommand\\fenc{f_{\\mathrm{enc}}}\n\\newcommand\\fdec{f_{\\mathrm{dec}}}\n\\newcommand\\Wand{\\ensuremath{W_{\\mathbf{and}}}}\n\\newcommand\\Wor{\\ensuremath{W_{\\mathbf{or}}}}\n\\newcommand{\\w}[1]{\\ensuremath{\\mathbf{#1}}}\n\\newcommand\\loss{\\ell}\n\n\\frame[plain]{\\frametitle{Outline}\\tableofcontents\\addtocounter{framenumber}{-1}}\n\\section{Introduction to Python}\n\n%\\begin{frame}\n% \\frametitle{Machine Learning Overview}\n% \\begin{itemize}\n% \\item Intro: Sven\n% \\item Python: Hannes\n% \\item Before Lunch: Unsupervised Learning (Hannes)\n% \\item After Lunch: Supervised Learning (Nenad LinReg, Andy LogReg/kNN)\n% \\end{itemize}\n%\\end{frame}\n\n\n\\begin{frame}\n \\frametitle{A Short Introduction to Python}\n \\begin{itemize}\n \\item Please log in, using:\n \\begin{description}\n \\item[Username] \\alert{gkbionik}\n \\item[Password] \\alert{tut0rial} (with a zero instead of the \n ``o''!)\n \\end{description}\n \\end{itemize}\n\\end{frame}\n\n\n\\section{Unsupervised Learning}\n\n\\subsection{PCA}\n\n\\input{pca.tex}\n\n\\subsection{k-Means}\n\n\\input{kmeans.tex}\n\n\\section{Supervised Learning}\n\n\n\\begin{frame}\n \\frametitle{Supervised Learning -- General}\n \\begin{itemize}\n\t\\item Task: Learn the function $ y = f(x) $ which predicts the output $y$ for the given input $x$, knowing the desired output\n \\item Each example in data is a tuple of the input and desired output (target)\n \\end{itemize}\n\\end{frame}\n\n\n\\begin{frame}\n \\frametitle{Example: Supervised Learning}\n \\begin{itemize}\n \\item Input Data: 40 examples of persons (age, height, smoker). \n\t\\item Targets: Weight of the person (desired output)\n\t\\item Goal: Learn a function which predicts the weight for the new person\n\t\tknowing the age, height, nationality of person.\n\n \\end{itemize}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Training / Test data}\n \\begin{itemize}\n\t\\item Learning is done on the training data, for which we know the input and targets\n\t\\item To test if the model learned to predict the output, we use test data. \t\t\n \\end{itemize}\n\\end{frame}\n\n\\subsection{Linear Regression}\n\n\\input{linreg.tex}\n\n\\subsection{Classification}\n\\begin{frame}\n \\frametitle{Classification}\n \\begin{itemize}\n \\item Predict to which class a data point belongs.\n \\item Training data are pairs $\\left( (x_0, y_0), \\cdots, (x_N, y_N)\n \\right), x_i \\in \\mathbb{R}^n, y_i \\in \\{0, \\cdots, k\\}$\n \\item Classical example: Spam / Ham.\n \\item All classes known beforehand.\n \\item Other examples: Digit recognition, cancer benign/malignant, \\ldots\n \\end{itemize}\n\\end{frame}\n\n\\subsection{Logistic Regression}\n\n\\input{logreg.tex}\n\n\\subsection{$k$ Nearest Neighbors}\n\n\\input{knn.tex}\n\n\\end{document}\n", "meta": {"hexsha": "9e802b19e3cd2c615a4a1f447da497b6e7dd8239", "size": 4969, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "presentation/presentation.tex", "max_stars_repo_name": "wagglefoot/Target-Oxbridge-ML-workshop", "max_stars_repo_head_hexsha": "d2b547169c45f985f8f6e501c837921950565dee", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 16, "max_stars_repo_stars_event_min_datetime": "2015-01-28T15:12:06.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-27T17:29:54.000Z", "max_issues_repo_path": "presentation/presentation.tex", "max_issues_repo_name": "kbastani/tutorial_ml_gkbionics", "max_issues_repo_head_hexsha": "2317d3137cc463716b73932a70083aef70fd8eca", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "presentation/presentation.tex", "max_forks_repo_name": "kbastani/tutorial_ml_gkbionics", "max_forks_repo_head_hexsha": "2317d3137cc463716b73932a70083aef70fd8eca", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2015-08-15T08:13:19.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-11T19:58:53.000Z", "avg_line_length": 29.4023668639, "max_line_length": 126, "alphanum_fraction": 0.7166431878, "num_tokens": 1507, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352403, "lm_q2_score": 0.6297746074044135, "lm_q1q2_score": 0.3467585374639343}} {"text": "\\documentclass[11pt, oneside]{article} \t% use \"amsart\" instead of \"article\" for AMSLaTeX format\n\n\n% \\usepackage{draftwatermark}\n% \\SetWatermarkText{Draft}\n % \\SetWatermarkScale{5}\n% \\SetWatermarkLightness {0.85} \n% \\SetWatermarkColor[rgb]{0.7,0,0}\n\n\n\\usepackage{geometry} \t\t% See geometry.pdf to learn the layout options. There are lots.\n\\geometry{letterpaper} \t\t% ... or a4paper or a5paper or ... \n%\\geometry{landscape} \t\t% Activate for for rotated page geometry\n%\\usepackage[parfill]{parskip} \t\t% Activate to begin paragraphs with an empty line rather than an indent\n\\usepackage{graphicx}\t\t\t\t% Use pdf, png, jpg, or eps� with pdflatex; use eps in DVI mode\n\t\t\t\t\t\t\t\t% TeX will automatically convert eps --> pdf in pdflatex\t\t\n\\usepackage{amssymb}\n\\usepackage{mathrsfs}\n\\usepackage{hyperref}\n\\usepackage{url}\n\\usepackage{authblk}\n\\usepackage{amsmath}\n\\usepackage{graphicx}\n\\usepackage{fixltx2e}\n\\usepackage{hyperref}\n\\usepackage{alltt}\n\\usepackage{color}\n\\usepackage{bigints}\n\n\\newcommand{\\argmax}{\\operatornamewithlimits{argmax}}\n\\newcommand{\\argmin}{\\operatornamewithlimits{argmin}}\n\n\n\\title{What is Rejection Sampling?}\n\\author{David Meyer \\\\ dmm@\\{1-4-5.net,uoregon.edu,brocade.com,...\\}}\n\n% \\date{September 30, 2015}\t\t\t\t\t\t\t% Activate to display a given date or no date\n\n\n\\begin{document}\n\\maketitle\n\n\\section{Introduction} \n\\label{sec:intro}\nSuppose that we want to sample from a distribution $f(x)$ that is difficult or impossible to sample from directly. Instead of trying to sample from $f(x)$, can we use a simpler distribution $q(x)$ from which sampling is easier? Perhaps, and the idea behind Rejection Sampling (aka Acceptance-rejection sampling) is to sample from $q(x)$ and apply some rejection/acceptance criterion such that the samples that are accepted are distributed according to $f(x)$.\n\n\\subsection{Envelope distribution and rejection criterion}\n\nIn order to be able to reject samples from $q(x)$ that aren't (approximately) sampled from $ f(x)$, $q(x)$ must “cover” or envelop the distribution $f(x)$. This is generally done by choosing a constant $c > 1$ such that $cq(x) > f(x)$, for all $x$. For this reason $cq(x)$ is often called the \\emph{envelope distribution}. A common criterion for accepting samples from $x \\sim q(x)$ is based on the ratio of the target distribution to that of the envelope distribution. The samples are accepted if\n\n\\begin{flalign}\n\\frac{f(x)}{cq(x)} > u\n\\end{flalign}\n\n\\noindent\nwhere $u \\sim Unif(0,1)$, and rejected otherwise. If the ratio is close to one, then $f(x) $ must have a large amount of probability mass around $x$ and that sample should be more likely accepted. If the ratio is small, then it means that $f(x)$ has low probability mass around $x$ and we should be less likely to accept the sample. See Figure~\\ref{fig:rgr}. I think...\n\n\n\n\\begin{figure}\n\\center{\\includegraphics[scale=0.75]{images/rejectionsamplingcriterion.png}}\n\\caption{Rejection Sampling with a Normal proposal distribution}\n\\label{fig:rgr}\n\\end{figure}\n\n\n\n\\end{document} \n\n", "meta": {"hexsha": "74a9932988436c4d7e558e0a56421df4fca6069a", "size": 3066, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "_my_stuff/papers/ml/rs/rs.tex", "max_stars_repo_name": "davidmeyer/davidmeyer.github.io", "max_stars_repo_head_hexsha": "14f01e0a50b9c643b5176a10c840f270b9da7bc1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "_my_stuff/papers/ml/rs/rs.tex", "max_issues_repo_name": "davidmeyer/davidmeyer.github.io", "max_issues_repo_head_hexsha": "14f01e0a50b9c643b5176a10c840f270b9da7bc1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "_my_stuff/papers/ml/rs/rs.tex", "max_forks_repo_name": "davidmeyer/davidmeyer.github.io", "max_forks_repo_head_hexsha": "14f01e0a50b9c643b5176a10c840f270b9da7bc1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.8, "max_line_length": 500, "alphanum_fraction": 0.7322243966, "num_tokens": 850, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.3467585374639343}} {"text": "\\chapter{The GNK value: A new solution}\n\\label{cha:new_solution}\n\nIn the previous chapter we introduced several different solution concepts, each stemming from different ethical principles and perspectives.\nIn this chapter we detail a new approach that combines some of the features of the solution concepts outlaid in the previous chapter.\nThe primary motivation for this new solution, is to attempt to distil the intuition behind the Nash bargaining solution concepts (per section \\ref{sec:solutions_bargaining}) into a description of total allocation that is suitable for arbitrary number of players - rather than just two.\nOur solution extends the work of others to the space of generalised non-cooperative games, which are then suitable for application in various electrical network contexts.\n\nOur solution relates to many of the concepts of the previous chapter: where we have a fundamentally coalitional scheme (per Coalitional Game Theory), that rewards based upon disagreement points (Bargaining Theory), which is partially informed by how much individual's participation influences the group's wellbeing (reminiscent of VCG), and attempts to describe normative trading between large numbers of participants (Marginalism).\n\nWe call our new solution concept the \\textit{Generalized Neyman and Kohlberg Value} or \\textit{GNK value} for short, and is computed and directly compared against other solutions concepts of the previous chapter in the context of electricity allocation.\nBy this comparison it is seen that the different solution concepts give different outcomes, and we discuss these differences in light of our ethical considerations (from Chapter \\ref{cha:background}).\n\nThe material from this chapter extends from work which was originally submitted to AAMAS, and was accepted as an extended abstract: \\\\\n\\-\\hspace{5mm}``The Generalized N\\&K Value: An Axiomatic Mechanism for Electricity Trading''\\\\ International Conference on Autonomous Agents and Multiagent Systems\\\\ (AAMAS) 2018 (accessible: \\href{ifaamas.org/Proceedings/aamas2018/pdfs/p1883.pdf}{ifaamas.org/Proceedings/aamas2018/pdfs/p1883.pdf})\n\nThis chapter consists of the following parts:\n\n\\begin{itemize}\n\\item\tIn sections \\ref{sec:prelude_gnk} and \\ref{the_value_def2}, we introduce and define the GNK value, relating it to historical roots and similar solution concepts.\n\\item\tIn section \\ref{more_involved}, we consider how the GNK value can be computed to derive financial and electrical outcomes on a DC electricity network, against LMP and VCG.\n\\item\tIn section \\ref{sec:features}, we point-by-point discuss the features GNK, LMP and VCG as they are expressed in the context of an example electricity network.\n%\\item\tin section \\ref{sec:scaling} we address the possible ways of remedying the GNK value for computation in larger network contexts, particularly via the use of sampling to approximate its value, and by the adoption of a proxy for the GNK inner optimisation (which we denote as the M-GNK).\n%\\item\tin section \\ref{sec:results_and_evaluation_of_GNK}, we detail computation and the efficiency of computation of the sampled M-GNK value for larger networks, and discuss its shortcommings for an example computed larger network.\n%\\item\tin section \\ref{sec:GNK_extensions_discussion} and the concluding section \\ref{sec:GNK_value_discussion}, we discuss possible extensions to the GNK concept and compare it point-by-point with ethical qualities.\n\\end{itemize}\n\nWe identify that the GNK value is difficult to compute for large numbers of players, and so in the next chapter \\ref{sec:scaling} we address and consider the GNK value at scale against ethical criteria.\n\n%\\begin{itemize}\n%\\item Derive the GNK value (section \\ref{the_value_def2})\n%\\item Apply it to DC networks (section \\ref{more_involved})\n%\\item We discuss some of its qualities (section \\ref{sec:features})\n%\\item Before discussing techniques to scale it (section \\ref{sec:scaling})\n%\\item Then evaluating it at scale (section \\ref{sec:results_and_evaluation_of_GNK})\n%\\item And then conclude with an ethical evaluation (section \\ref{sec:GNK_extensions_discussion})\n%\\end{itemize}\n\n\n%derive the GNK value (section \\ref{the_value_def2}) before applying it to DC networks (section \\ref{more_involved}), we discuss its qualities (section \\ref{sec:features}), before discussing techniques to scale it (section \\ref{sec:scaling}), before evaluating it at scale (section \\ref{sec:results_and_evaluation_of_GNK}) and concluding with ethical evaluation (section \\ref{sec:GNK_extensions_discussion})\n\n\n%Particularly:\n%\\begin{itemize}\n%\\item\tin sections \\ref{sec:prelude_gnk},\\ref{sec:introduction_gnk_sec},\\ref{the_value_def2}, We introduce and define the GNK value at a conceptual and mathematical level, relating it to historical roots and similar solution concepts.\n%\\item\tin sections \\ref{more_involved} we consider how the GNK value can be applied to derive financial and electrical transactions on a small example electricity network, and outlay the procedures for its computation alongside LMP and VCG in this context.\n%\\item\tin section \\ref{sec:features} we point-by-point discuss the features GNK, LMP and VCG as they are expressed in the context of the small example DC electricity network. particularly we pay attention to the fact that the GNK value is computationally difficult to compute for larger networks.\n%\\item\tin section \\ref{sec:scaling} we address the possible ways of remedying the GNK value for computation in larger network contexts, particularly via the use of sampling to approximate its value, and by the adoption of a proxy for the GNK inner optimisation (which we denote as the M-GNK).\n%\\item\tin section \\ref{sec:results_and_evaluation_of_GNK}, we detail computation and the efficiency of computation of the sampled M-GNK value for larger networks, and discuss its shortcommings for an example computed larger network.\n%\\item\tin section \\ref{sec:GNK_extensions_discussion} and the concluding section \\ref{sec:GNK_value_discussion}, we discuss possible extensions to the GNK concept and compare it point-by-point with ethical qualities.\n%\\end{itemize}\n\n%After this chapter, the next Chapter \\ref{chap:stratified_sampling_chapter} details our investigation into novel sampling methods which are which are used to scale the GNK value in section \\ref{sec:scaling}.\n\n\\section{Introduction to the GNK value}\\label{sec:prelude_gnk}\\label{sec:introduction_gnk_sec}\n\n\nHow should we model ideal competition? In the two player case, existing bargaining solutions (such as Nash's) seem rather difficult to surpass as they describe a singular and axiomatic outcome that is both cooperatively Pareto optimal and also accounts for anti-cooperative strategising.\n\nIn the context of Nash bargaining with endogenous disagreement point (per section \\ref{subsec:nash_bargaining_endogenous}), the disagreement point was interpreted as a unique point defined by a minimax equilibrium in the payoff-advantage of strategies.\nHowever there exists a problem extending this scheme directly to three or more players, as there may not exist a unique minimax equilibrium in payoff-advantages for strategies in a game of more than two players.\nThe question then is how to logically and consistently extend Nash's bargaining solution with endogenous disagreement point to an arbitrary number of players.\n\nWhile it may be possible to arbitrarily choose one of those possible minimax equilibria in a 3+ person game as the disagreement point, \ninstead we consider all the possible divisions of players between two groups and then consider all the two-player minimax payoff advantages between them. We then integrate this information via Shapley Value axioms to form a unique outcome, that does not depend on any arbitrary choice.\nIn this way our new solution allocates outcomes in proportion to the aggregate leverage that all the individuals - and groups of individuals - could hypothetically posses in bargaining for outcomes they desire.\nIn the following sections of this chapter (sections \\ref{the_value_def2} to \\ref{more_involved}) we give all the details of this process.% Particularly we note that this process of how to reasonably extend Nash's bargaining solution to multiple players has a history.\n\nThe fundamental idea behind our approach was first detailed by \\cite{values3}.\nSo far as we know, Harsanyi's solution concepts have never been applied to electricity networks, as there exists a particular problem in doing so, particularly that Harsanyi's solution concepts apply to non-cooperative games but cannot apply to generalised non-cooperative games.\nThe details of this problem and our novel remedy are given in following section \\ref{the_value_def3}.\n\nBut briefly, in electricity network contexts the mutual interactions of participants can be limited by the physical constraint of the network, for instance: network participants cannot simply draw or push power to/from the network without restraint, as doing so would lead to damage to the network.\nAnd these limitations on the space of possible mutual actions is best modelled by a generalised non-cooperative game, and unfortunately in the context of such a generalised game there is no unique minimax point between two players.\nOur principle and novel development in this chapter is the provision of a remedy, such that Harsanyi's solution can then be applied.\nThe remedy is to take the expected outcome on a coin-flip on who chooses actions first in the minimax strategies - and as we shall see - this turns out to be a unique value that satisfies all required properties, and thus leads to a coherent outcome.\n\nThe resulting outcome we call the \\textit{Generalized Neyman and Kohlberg Value} or the \\textit{GNK value} for short, \nas Harsanyi's solution was also axiomatically derived by \\cite{value2,KOHLBERG2018139}.\nThis solution concept is shown to apply for all transferable utility (TU) generalised non-cooperative games, and directly equivalent to the Nash bargaining with endogenous disagreement point under transferable utility (TU) between two players (such as per section \\ref{sec:nash_bargaining_exogenous}, as in the example in that section).\n\nThe GNK value is thus flexible enough to extend to many contexts, but we focus particularly on the specific case of allocating monetary payments over Optimal Power Flow (OPF) instances under the DC approximation - as we will explain.\n\nLet us derive the GNK value (in section \\ref{the_value_def2}) before giving details of its application to DC networks (in section \\ref{more_involved}), discussing it (section \\ref{sec:features}) and scaling it (section \\ref{sec:scaling}).\n\n\\section{Deriving the GNK value}\\label{the_value_def2}\n\nWe begin by presenting the axiomatic foundations of the GNK value, in a similar manner as \\cite{value2}'s exposition.\nWe begin by defining the GNK value to be the integration of \\emph{threat} values between possible coalitions; defined via Shapley Value axioms.\nWe then describe the \\emph{threat} or \\emph{advantage} of a coalition $v(S)$ in the context of a \\textit{generalized non-cooperative game} (which is our key point of novelty in the solution concept).\nAnd then we clarify how the GNK value relates to other prominent solution concepts in non-cooperative games.\n\n\\subsection{Axiomatic foundations and the \\textit{Value}}\\label{the_value_def}\n\n%There have been many attempts to answer the question of what cooperative outcome \\textit{should} occur in the context of a non-cooperative TU game.\n%One well known answer is the \\textit{Nash bargaining solution} between two players \\cite{nash2}, which Harsanyi extended to arbitrary numbers of players.\n%Building on this, Harsanyi's solution \\cite{values3} was derived from a simple set of axioms by Neyman and Kohlberg ~\\cite{value2}; \n%our axiomatic derivation of a value for games with generalized action spaces mirrors the steps in theirs.\n\nWe begin by considering \\cite{KOHLBERG2018139}'s \\textit{coalitional game of threats}, \nwhich is a coalitional game defined by a pair $\\langle N,v \\rangle$ in which:\n\\begin{itemize}\n\\item\t$N=\\{1,\\dots,n\\}$ is a finite set of \\textit{players} or \\textit{agents}, and\n\\item\t$v:2^N\\rightarrow \\mathbb{R}$ is a \\textit{characteristic function} with \n\\begin{equation}\nv(S)=-v(N\\setminus S) \\label{myeq2} \\quad \\forall S\\subseteq N.\n\\end{equation}\n\\end{itemize}\nThe intuition for \\eqref{myeq2} is that the characteristic function of this game is a measure of the strength of the bargaining position (the `threat' or `advantage') that a coalition, $S$, has over its complement, $N\\setminus S$.\nThis contrasts with classical cooperative game theory games, where the characteristic function $v(\\emptyset)=0$ and equation \\ref{myeq2} does not generally hold (see section \\ref{sec:cooperative_game_theory_part}).\n\nNeyman and Kohlberg's key result was to prove that if $\\mathbb{D}$ is the set of all such games, then there exists a unique mapping $\\varphi:\\mathbb{D}\\rightarrow\\mathbb{R}^n$ that satisfies the following four axioms:\n\n\\begin{itemize}\n\\item\t\\textbf{Efficiency}: $\\sum_i\\varphi(\\langle N,v\\rangle)_i = v(N)\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad~~\\refstepcounter{equation}(\\theequation)\\label{myeq}$\n\\item\t\\textbf{Symmetry}: If two players $i$ and $j$ are substitutes, such that if\\\\ $v(S\\cup i)=v(S\\cup j)~~\\forall S\\subseteq N\\setminus\\{i,j\\}$, then $\\varphi(\\langle N,v\\rangle)_i = \\varphi(\\langle N,v\\rangle)_j$\n\\item\t\\textbf{Null Player}: If a player $i$ is a null player (i.e.\\ $v(S\\cup i)=v(S)~~\\forall S\\subseteq N$) then $\\varphi(\\langle N,v\\rangle)_i=0$\n\\item\t\\textbf{Additivity}: for any $v_1$ and $v_2$, $\\varphi(\\langle N,v_1+v_2\\rangle)=\\varphi(\\langle N,v_1 \\rangle) + \\varphi(\\langle N,v_2\\rangle)$\n\\end{itemize}\n\nLetting agent $i$'s element of $\\varphi$ be denoted by $\\varphi_i$, this mapping is:\n\\begin{equation}\\label{da_value_eq} \n\\varphi_i(\\langle N,v\\rangle)\n= \\frac{1}{n}\\sum_{k=1}^n v_{i,k} \n= \\frac{1}{n}\\sum_{k=1}^n \\frac{1}{\\binom{n-1}{k-1}} \\sum_{\\substack{S:i\\in S \\\\ |S|=k}}v(S) \n\\end{equation}\nWhere $v_{i,k}$ is the average value of $v(S)$ for all coalitions of size $k$ that include $i$.\nThis mapping gives a distribution of the total surplus $v(N)$ among the players, and \\cite{KOHLBERG2018139} appropriately call this unique mapping the `Shapley Value' of the game of threats as it mirrors the classic \\textit{Shapley Value} of cooperative game theory.\n\nIndeed \\cite{KOHLBERG2018139} have shown that for any game of threats $\\langle N,v\\rangle$ there is a classic cooperative game $\\langle N,v'\\rangle$ where the two Shapley Values are the same.\nIt is possible to map a game of threats $v$ to a cooperative game $v'$ via relation:\n\\begin{equation}\\label{convert1}\nv'(S)=\\frac{1}{2}v(S)+\\frac{1}{2}v(N)\n\\end{equation}\n%Where the Shapley value is hence given by the classic expression:\n%\\begin{equation}\\label{eq:shapley_value}\n% \\varphi_i(\\langle N,v\\rangle)= \\frac{1}{n}\\sum_{S\\subseteq N\\setminus\\{i\\}} \\binom{n-1}{|S|}^{-1} \\left(v'(S\\cup\\{i\\})-v'(S)\\right) \n%\\end{equation}\n%Which identifies the Shapley value of a player in a cooperative game as its average over marginal contributions; which we can similarly be expressed by averages over marginal contributions by player and size:\n%\\begin{equation}\\label{eq:shapley_value2}\n%\\hat{v}_{i,k} = \\frac{1}{\\binom{n-1}{k}}\\sum_{S\\subset N\\setminus \\{ i\\} , |S|=k} %\\frac{(n-|S|-1)!\\,|S|!}{(n-1)!}\n%(v'(S\\cup\\{i\\})-v'(S))\n%\\end{equation}\n%\\begin{equation}\\label{shap2} \\varphi_i(\\langle N,v\\rangle) = \\frac{1}{n}\\sum_{k=0}^{n-1}\\hat{v}_{i,k} \\end{equation}\n%These multiple formulations of the Shapley value will be useful to us in Section \\ref{} where we use these expressions in different methods of sampling the GNK value.\n\nA central question in \\cite{KOHLBERG2018139}'s coalitional game of threats, is what the `threat' of a coalition of players $v(S)$ should be (such as on an electricity network), and one way of considering this is in relation to the actions that the coalition could exert and their consequences (positive and negative) over other players; such as may be described in non-cooperative game theory.\n\n\\subsection{Defining threats in games with general action spaces}\\label{the_value_def3}\n\nIn this subsection we define the characteristic function $v(S)$, in the context of a \\textit{generalized non-cooperative game}.\nA generalised non-cooperative game is a game where the strategies available to one player may be restricted by the strategy choice of others.\nSuch games were introduced by \\cite{Debreu01101952} and the problem of finding equilibria in such games has been a topic of further research \\citep{Facchinei2007,fischer2014}.\n\nIn more detail, a generalised non-cooperative game consists of a triplet $G = \\langle N,A,u \\rangle$ in which:\n\\begin{itemize}\n\\item\t$N=\\{1,\\dots,n\\}$ is a finite set of players,\n\\item\t$A\\subseteq \\prod_{i\\in N}A^i$ is a set of all possible joint strategies, where $A^i$ denotes the set of strategies available to player $i\\in N$, and $A$ is a subset of their product space\n\\item\t$\\{u_i(a) : A\\rightarrow \\mathbb{R}\\}_{i\\in N}$ is a set of functions of each player's payoff/utility when joint strategy $a\\in A$ is executed.\n\\end{itemize}\n\nIn this context, we wish to describe the payoff `threat' or `advantage' $v(S)$ of a coalition $S\\subseteq N$ (letting $A^S=\\prod_{i\\in S}A^i$), taking into account the constraints that apply to the joint action space. \nA key contribution in our research is the following construction of the coalitional game of threat's characteristic function. \nDenoting $(x,y)\\in A$ as a partition of a joint action between two coalitions $S$ and $N\\setminus S$, \nthe characteristic function for the game of threats with generalised action spaces is given by:\n\\begin{align}\n\\label{knvalue1}\nv(S) = &\n\\frac{1}{2}\\min_{\\substack{y\\in A^{N\\setminus S} \\\\ \\text{s.t.}\\exists x,(x,y)\\in A}} \n\\max_{\\substack{x\\in A^S \\\\ \\text{s.t.}(x,y)\\in A}}\n\t\\left(\\sum_{i\\in S} u_i(x,y) - \\sum_{i\\in N\\setminus S}u_i(x,y)\\right)\\nonumber\\\\\n+&\n\\frac{1}{2}\\max_{\\substack{x\\in A^S \\\\ \\text{s.t.}\\exists y,(x,y)\\in A}}\n\\min_{\\substack{y\\in A^{N\\setminus S} \\\\ \\text{s.t.}(x,y)\\in A}}\n\t\\left(\\sum_{i\\in S} u_i(x,y) - \\sum_{i\\in N\\setminus S} u_i(x,y) \\right)\n\\end{align}\n\nThe requisite condition $v(S)=-v(N\\setminus S)$, as given in~\\eqref{myeq2}, is immediately satisfied irrespective of the structure of strategy space $A$, insofar as the $\\max$ and $\\min$ terms are defined.\nThus, \\eqref{knvalue1} is a feasible representation of the competitive advantage (or threat) that a coalition has over its complement in a generalised strategy space.\nWith the characteristic function \\eqref{knvalue1}, the formulation of $\\varphi$ (per \\eqref{da_value_eq}) defines the GNK value.\nThis is a novel extension of existing work to the space of generalised games (see Section~\\ref{relating_to_the_old}).\n\n\\subsection{Understanding the GNK value}\\label{the_value_def4}\n\nIn the characteristic function~\\eqref{knvalue1}, the inner term:\n\\[\n\\sum_{i\\in S} u_i(x,y) - \\sum_{i\\in N\\setminus S} u_i(x,y)\n\\] \nis the sum of payoffs that the coalition $S$ receives, \nminus the sum of payoffs that the complement $N\\setminus S$ receives, \nunder the joint strategy $(x,y)\\in A$, we call this the \\textit{payoff advantage} to $S$.\n\nThe first line of $v(S)$ in~\\eqref{knvalue1} is half the payoff advantage achieved if, the players in $S$ collectively choose their strategies to maximise the payoff advantage knowing that the players in $N\\setminus S$ will subsequently choose their strategies to minimise it - and thus this dynamic constitutes a bilevel optimisation problem.\nThen the second line of~\\eqref{knvalue1} is an additional half of the payoff advantage achieved if the ordering of choice were reversed, with $N\\setminus S$ choosing first.\nIn this way, \\eqref{knvalue1} can be interpreted as the expectation of Nash equilibrium payoff advantage of $S$ over its complement under a fair coin-toss of who chooses their strategies first.\n\nIn this formulation $v(N) = \\max_{a\\in A} (\\sum_{i\\in N} u_i(a))$ is the maximum achievable sum of payoffs that the players can achieve, and the GNK value $\\varphi$, splits all of this amount between the players (by the efficiency axiom).\nThe allocation of utility that the GNK value allocates can be realised by having the players execute the strategies that achieve this maximal sum, and then enacting appropriate utility transfers between the players.\nIn this way the GNK value can be seen as a method of allocating a Pareto optimal outcome and budget-balanced payments between players, to attain utility proportional to their competitive advantages.\n\n\\subsection{Relation to other solution concepts}\\label{relating_to_the_old}\n\nThe GNK value is closely related to several other solution concepts, and even equivalent to them under certain conditions.\n\nMost immediately, the GNK value is identical to \\cite{value2}'s Value when the strategy space $A$ represents a mixed strategy game that is not generalised; that is when the strategy space, $A$, is an unconstrained combination of strategies for all agents (ie. $A = \\prod_{i\\in N}A^i$).\nTo see this, we observe that the two halves of \\eqref{knvalue1} are equal in the absence of joint action constraints (%proof in Appendix \\ref{appendix1}, or \nvia direct application of von Neumann's minimax theorem\\footnote{see Lemma 1 of \\cite{value2}}), \nand hence the characteristic value reduces to that used in Neyman and Kohlberg's original definition:\n\\begin{equation}\\label{knvalue2}v_o(S) = \\max_{x\\in A^S}\\min_{y\\in A^{N\\setminus S}} \\left(\\sum_{i\\in S} u_i(x,y) - \\sum_{i\\in N\\setminus S} u_i(x,y) \\right).\\end{equation}\n%\nThat is, Neyman and Kohlberg's Value is the formulation of $\\varphi$ (per \\eqref{da_value_eq}) \nwith $v_o(S)$ (per \\eqref{knvalue2}).\nUnfortunately Neyman and Kohlberg's Value cannot be directly applied to generalised games because the required condition $v_o(S)=-v_o(N\\setminus S)$ can fail to hold in that case. \n\nNeyman and Kohlberg's Value (and the GNK value) are also directly conceptually related to \\cite{values3}'s solution in this context,\nwhile in the 2-player context, it is identical to \\cite{kalai1,Kalai2010}'s \\textit{coco-value} in the context of complete information \nand also identical to \\cite{nash2}'s bargaining solution in the context of transferable utility (see \\cite{value2}, or section \\ref{subsec:nash_bargaining_endogenous}).\nIt also shares a conceptual similarity with \\cite{aumann1961core}'s $\\alpha$ and $\\beta$ core solution concepts, and \\cite{1944}'s historic formulation :\n\\begin{equation}\\label{knvalue3}v_m(S) = \\max_{x\\in A^S}\\min_{y\\in A^{N\\setminus S}} \\sum_{i\\in S} u_i(x,y).\\end{equation}\nIn this way the GNK value can be seen as a conceptual continuation of historic solution concepts, and can be judged according to how well it derives outcomes in application contexts.\n\nAs a simple example demonstration, it is possible to see how the GNK value is identical to Nash bargaining in the context of transferable utility games (such as in the example matrix game \\ref{eq:example_game1} of section \\ref{subsec:nash_bargaining_endogenous}), we calculate all the terms $v(S)$ for coalitions of players $1$ and $2$, particularly:\n\n\\begin{align*}\nv(\\{1,2\\})=&\\max\\max\\begin{bmatrix}3 & -3\\\\ -3 & 3\\end{bmatrix}=3\\\\\nv(\\{1\\})=&\\frac{1}{2}\\min\\max\\begin{bmatrix}1 & 1\\\\ -1 & -1\\end{bmatrix}+\\frac{1}{2}\\max\\min\\begin{bmatrix}1 & -1\\\\ 1 & -1\\end{bmatrix}=1\\\\\nv(\\{2\\})=&\\frac{1}{2}\\min\\max\\begin{bmatrix}1 & -1\\\\ 1 & -1\\end{bmatrix}+\\frac{1}{2}\\max\\min\\begin{bmatrix}1 & 1\\\\ -1 & -1\\end{bmatrix}=-1\\\\\nv(\\emptyset)=&-\\max\\max\\begin{bmatrix}3 & -3\\\\ -3 & 3\\end{bmatrix}=-3\\\\\n\\end{align*}\nFrom this:\n$$ \\varphi_1(\\langle N,v\\rangle)=\\frac{1}{2}(v(\\{1,2\\})+v(\\{1\\}))=\\frac{1}{2}(3+1)=2 $$\n$$ \\varphi_2(\\langle N,v\\rangle)=\\frac{1}{2}(v(\\{1,2\\})+v(\\{2\\}))=\\frac{1}{2}(3-1)=1 $$\nWhich matches exactly the result in section \\ref{subsec:nash_bargaining_endogenous}. In section \\ref{subsec:nash_bargaining_endogenous} we considered that the minimax in the payoff advantage matrix (ie. $d$ in that section) was the nash equilibrium point of threat in bargaining between the players in choosing a subsequent point on the Pareto frontier (see Figure \\ref{fig:graph1_utilities}) that divides the maximum possible sum revenue (ie $s$ in that section).\nThe GNK value also directly encodes this same logic as $v(\\{1\\})=-v(\\{2\\})$ is the minimax in the payoff advantage between the player (in this ungeneralised game), and $v(\\{1,2\\})$ is the maximum sum revenue which is split. This result is the same as we would get by the \\textit{coco-value}, and working with Neyman and Kohlberg's value (equation \\ref{knvalue2}).\n\nIn the following section we describe the setup for an application and evaluation of the GNK value in the context of DC-approximated electrical networks.\n\n\\section{GNK value computation on DC powerflows}\\label{more_involved}\n\nBecause of its flexibility the GNK value has the potential to be used in a large range of different contexts,\nhowever in this section we focus solely on the development of a simple case --- the pricing of the immediate consumption and generation of power on a meshed network under DC approximation, where all participants have linear utilities over their own power.\nAlthough this construction simplifies away some key technical problems in power networks, it allows us to clarify the analysis of the GNK value and its features.\n\nIn this section, we consider the DC network model, discuss how to calculate the GNK value as well as LMP and VCG for it, before in the next section \\ref{sec:features} we discuss the features of these mechanisms with an example.\n\n\\subsection{Network model}\\label{sec:the_setup}\n\n\nWe begin by setting out the elements of an electricity network under DC approximation:\n\\begin{itemize}\n \\item A set of buses $B$ with, for all $i\\in B$:\n \\begin{itemize} \n \\item Power consumption at each bus $p_i$, and \n \\item A bus voltage phase-angle $\\theta_i$,\n \\end{itemize}\n \\item Lines $C\\subseteq B\\times B$, with, for all $(i,j)\\in C$: \n \\begin{itemize} \n \\item Line susceptance $b_{i,j}$, and \n \\item Power flow $p_{i,j}$ (power from bus $i$ to $j$), with $p_{i,j}=-p_{j,i}$. \n \\end{itemize}\n\\end{itemize}\nIn this context, the DC approximated powerflow constraints (per \\cite{Wang1}) are expressed as follows:\n\\begin{equation}\n\\label{dcopf1}\n\\begin{aligned}\n\\text{DC-powerflow} \\quad& \\\\\n\\text{Variables:} \\quad& p_{i\\in B},\\ \\theta_{i\\in B},\\ p_{(i,j)\\in C} \\\\\n\\text{constraints:} \\quad& p_i^{l}\\le p_i \\le p_i^{u} \\\\\n&p_{i,j}^l \\le p_{i,j} \\le p_{i,j}^u \\\\\n&p_j = \\sum_{(i,j)\\in C}p_{i,j}\\\\\n&p_{i,j} = -b_{i,j}(\\theta_i - \\theta_j)\n\\end{aligned}\n\\end{equation}\nwhere $p_i^{l}$, $p_i^{u}$, $p_{i,j}^l$, $p_{i,j}^u$ are the upper and lower bounds on power consumption/generation and line limits, respectively.\n\nWe can eliminate redundant variables, such as $\\theta_i$ and $p_{i,j}$, and to ease presentation, and use the abstract functions $h_j$ and $g_k$ (for indices $j,k$) to represent the remaining linear functions:\n\n\\begin{equation}\n\\label{dcopf2}\n\\begin{aligned}\n\\text{DC-powerflow}\\\\\n\\text{Variables:}\\quad & p_{i\\in B} \\\\\n\\text{constraints:}\\quad & h_j(p_1,p_2,\\dots)=0\\quad \\forall j\\\\\n& g_k(p_1,p_2,\\dots)\\le 0 \\quad \\forall k\n\\end{aligned}\n\\end{equation}\n\nIn this DC powerflow network the participants on each bus are treated as players in a game.\nFor simplicity, we have one player per bus (i.e.~$N=B$), and the power consumption of that bus is the respective player's strategy space (i.e.\\ $A_i=[p_i^l,p_i^u]$).\nThen the DC constraints define the space of jointly executable strategies --- forming the generalised strategy space $A$.\n\nWe further assume that there is a linear utility (or payoff) associated with the power consumption of each player, denoted $u_i(p_i)$ for player $i$, which makes all the components of a generalised game.\\footnote{the Linear utility assumption simplifies the mathematics, as it makes the objective function linear, intuitively consumers should value consuming more power more, and generators cost producing more power more with some gradient.}\nWe now consider how to calculate the GNK value against LMP and VCG for such a generalised game.\n\n\n\n\n\n\n\\subsection{Computing the GNK value}\\label{subsec:gnk_compute1}\n\nThe GNK value is difficult to solve because of the bilevel structure of \\eqref{knvalue1} which must be computed for each of the possible coalitions of network participants.\nEven though we have modelled our example network with a set of linear utility functions and linear constraints (as given in section \\ref{sec:the_setup}), equation \\eqref{knvalue1} is still quite difficult to solve as it constitutes a linear bilevel program (LBP) which are a class of problems known by to be NP-hard. \\citep{DBLP:journals/tec/SinhaMD18,Ben-Ayed:1990:CDB} \n\nThere exist a range of techniques which can be used to solve LBPs, such as summarised by \\cite{DBLP:journals/tec/SinhaMD18,S.Dempe.Optimisations}.\nSome of the many methods include: vertex enumeration processes \\citep{Bialas:1984:TLP:2784019.2784026,Shi:2005:EKA:2641854.2642183,LIU1995644}; penalty method schemes \\citep{KleinertSchmidt2019,ONAL1993126,dempe_optimisation111};\ncutting plane approaches \\citep{cuttingplane1};\nbranch-and-bound/cut methods \\citep{SHI200551,Hansen:1992:NBR:141164.141181,Audet2007};\nand approximating algorithms \\citep{Pineda2018,rnnlbp1,genetic_algirthm_blp}.\n\nOne well known way of addressing LBPs involves converting the inner optimisation constraints into KKT conditions (introduced by \\cite{kuhn1951nonlinear}), and then converting the complementarity conditions into disjunctive constraints with binary variables - see \\cite{Fortuny-Amat1981,Pineda2018}.\nIn this way, a bilevel program is converted into a mixed integer linear program, which is then directly amenable to standard optimisation software.\nThis method was chosen, and the SCIP Optimisation Suite was employed to compute the GNK value for an example network as described in the next section \\ref{sec:example_network}.\n\nKKT conditions are a well known set of algebraic tests which imply that the function under consideration is locally optimal (maximal or alternatively minimal) with respect to its variables under a set of constraint functions (with some regularity assumptions on those functions).\nKKT are well documented, and extend the method of Lagrange multipliers.\n\nSpecifically for maximising an objective function $f(\\mathbf{x})$ subject to multiple constraints:\n\\begin{equation}\\label{eq:KKT_condition1} g_k(\\mathbf{x})\\le 0 \\end{equation}\n\\begin{equation}\\label{eq:KKT_condition2} h_j(\\mathbf{x})=0 \\end{equation}\nThen $\\mathbf{x}*$ is a local maximum if the following KKT conditions are true:\n\\begin{equation}\\label{eq:KKT_conditions3} \\nabla f(\\mathbf{x}*) - \\sum_k\\lambda_k\\nabla g_k(\\mathbf{x}*) - \\sum_j\\mu_k\\nabla h_k(\\mathbf{x}*) =0\\end{equation}\nand equations \\ref{eq:KKT_condition1} and \\ref{eq:KKT_condition2} hold for $\\mathbf{x}*$, and for all $i$ that $\\lambda_i\\ge 0$ and $\\lambda_ig_i(\\mathbf{x}*)=0$.\n\nBy KKT conditions, we can convert the DC powerflow constraints given by equations \\ref{dcopf2} into a set of equations for maximising an objective function $f(p_{i\\in B})$:\n\n\\begin{equation}\\forall i~~\\frac{\\partial f}{\\partial p_i}(p_{i\\in B})=\\sum_j\\mu_j\\frac{\\partial h_j}{\\partial p_i}(p_{i\\in B}) + \\sum_k\\lambda_k\\frac{\\partial g_k}{\\partial p_i}(p_{i\\in B})\\end{equation}\n\\begin{equation}\\forall j~~ h_j(p_{i\\in B})=0\\end{equation}\n\\begin{equation}\\forall k~~ g_k(p_{i\\in B})\\le 0\\end{equation}\n\\begin{equation}\\forall k~~ \\lambda_k \\ge 0\\end{equation}\n\\begin{equation}\\label{eq:complementarity_constraint_KKT}\\forall k~~ \\lambda_kg_k(p_{i\\in B}) = 0\\end{equation}\n\nHence the reformulation of our LBPs in equation \\ref{knvalue1} involves transforming the inner maximisation/minimisation constraints into KKT conditions.\nThe sets of variable values which satisfy the KKT conditions are called KKT points, if we denote the set of values of an maximised objective function $f(p_{i\\in B})$ at the KKT points as $\\mathcal{KKT}(f(p_{i\\in B}))$.\nThen reformulation of the inner part of \\eqref{knvalue1} to involve KKT conditions is as follows:\n\n\\begin{equation}\n\\label{kkt_optimization1}\n\\begin{aligned}\nv(S) =& \n \\frac{1}{2}\\max_{\\substack{p_i \\\\ i\\in S}} \\min\\left[-\\mathcal{KKT}\\left(-\\sum_{i\\in S} u_i(x,y) + \\sum_{i\\in N\\setminus S}u_i(x,y)\\right)\\right] +\\\\\n&\\frac{1}{2}\\min_{\\substack{p_i \\\\ i\\notin S}}\\max\\left[\\mathcal{KKT}\\left(\\sum_{i\\in S} u_i(x,y) - \\sum_{i\\in N\\setminus S}u_i(x,y)\\right)\\right]\n\\end{aligned}\n\\end{equation}\n\n\nBy reformulating the inner maximisations/minimisations of \\eqref{knvalue1} in this way we replaced the inner minimisations/maximisations in the space $(x,y)\\in A$ with minimisations/maximisations over KKT points in the same space.%\\footnote{By doing this we tacitly we assume that $u_i(p_i)$ is continuously differentiable, and satisfy some regularity conditions.}\n\nAs the constraints under DC-approximation are linear and hence define a convex polygon,\n%if we assume that the functions $u_i(p_i)$ are concave then there is either: only ever a single KKT point for all sets of outer variables, or (if weakly-concave) then any of the KKT points will be equal to the global minima (maxima).\n%Hence the inner maximization (minimization) over the KKT points can be ignored.\nand as the functions $u_i(p_i)$ are linear (per the assumption of linear utility in section \\ref{sec:the_setup}), then there will only be a single maximum/minimum value of these inner minimisations/maximisations - which will be the global maximum/minimum value.\\footnote{as linear optimisation problem has a unique solution, although there may be multiple points which attain this maximum/minimum value}\nIn this way the inner maximisation (minimisation) over the KKT points can be ignored.\n\nIt was also realised that a binary reformulation of the complementary slackness conditions (equation \\ref{eq:complementarity_constraint_KKT}) would increase computational efficiency, and so we transformed these complementarity constraints into disjunctive binary constraints.\n\nSpecifically, for each complementary slackness condition $\\lambda_kg_k(p_{i\\in B}) = 0$ we introduced a binary variable $Z_k$ to indicate whether $\\lambda_k$ or $g_k(p_{i\\in B})$ was zero and then introduced large numbers $\\bar{\\lambda}_k$ and $ \\underline{g_k} $ such as to make the complementary slackness condition equivalent to: $(1-Z_k)\\bar{\\lambda}_k \\ge \\lambda_k \\ge 0$ and $\\underline{g_k}Z_k\\le g_k(p_{i\\in B})\\le 0$.\nWhere $\\bar{\\lambda}_k$ and $\\underline{g_k}$ are the estimated upper and lower bounds on the KKT multipliers and constraint functions respectively.\n\nthe resulting KKT conditions for maximising and objective function $f(p_{i\\in B})$ under this complementary slackness conditions are as follows:\n\n\\begin{equation}\\forall i~~\\frac{\\partial f}{\\partial p_i}(p_{i\\in B})=\\sum_j\\mu_j\\frac{\\partial h_j}{\\partial p_i}(p_{i\\in B}) + \\sum_k\\lambda_k\\frac{\\partial g_k}{\\partial p_i}(p_{i\\in B})\\end{equation}\n\\begin{equation}\\forall j~~ h_j(p_{i\\in B})=0\\end{equation}\n\\begin{equation}\\forall k~~ g_k(p_{i\\in B})\\le 0\\end{equation}\n\\begin{equation}\\forall k~~ (1-Z_k)\\bar{\\lambda}_k \\ge \\lambda_k \\ge 0\\end{equation}\n\\begin{equation}\\forall k~~ \\underline{g_k}Z_k\\le g_k(p_{i\\in B}) \\le 0\\end{equation}\n\nif we denote the set of values of an maximised objective function $f(p_{i\\in B})$ that satisfy these new KKT conditions $\\mathbb{KKT}(f(p_{i\\in B}))$.\nThen reformulation of the inner part of \\eqref{knvalue1} to involve KKT conditions is as follows:\n\nThe resulting reformulation is as follows:\n\n\\begin{equation}\n\\label{optimization_eq1}\n\\begin{aligned}\nv(S) =\n&\\frac{1}{2}\\max_{\\substack{p_i \\\\ i\\in S}} \\left[-\\mathbb{KKT}\\left(-\\sum_{i\\in S} u_i(x,y) + \\sum_{i\\in N\\setminus S}u_i(x,y)\\right)\\right] +\\\\\n&\\frac{1}{2}\\min_{\\substack{p_i \\\\ i\\notin S}}\\left[\\mathbb{KKT}\\left(\\sum_{i\\in S} u_i(x,y) - \\sum_{i\\in N\\setminus S}u_i(x,y)\\right)\\right]\n\\end{aligned}\n\\end{equation}\n\nThis reformulation renders the LBP into a mixed integer program which is directly amenable for calculation by optimisation solvers, and the SCIP optimisation suite was used in our case.\n%Using the SCIP optimisation suite the GNK transfers were calculated for an example DC network in the next section \\ref{sec:example_network} particularly as shown in Figures \\ref{fig:1d} and \\ref{fig:1c} show the utility transfers and the post-transfer utilities under GNK.\n\n\n\\subsection{Computing the LMP transfers}\\label{subsec:LMP_compute1}\n\nComputing the Locational Marginal Price (LMP) transfers for DC electricity networks is a process described in various literature - such as by \\cite{lmp1,lmp2}.\n\nParticularly we consider optimising the sum of utility: $\\sum_{i\\in B} u_i(p_i)$ subject to the DC powerflow constraints in equations \\ref{dcopf1}.\nIn this context the power conservation constraints on each bus are the constraints $p_j = \\sum_{(i,j)\\in C}p_{i,j}$, and the lagrange multipliers associated with these constraints are the marginal prices for power on each of the respective busses.\nThis process is an application of the more general marginal price calculation procedure discussed previously in section \\ref{subsec:marginal_price_sketch}.\nWe utilised the SCIP optimisation suite\\footnote{development reported by \\cite{MaherFischerGallyetal.2017}}.\n\n\\subsection{Computing the VCG imputations}\\label{subsec:VCG_compute1}\n\nFrom section \\ref{sec:solutions_VCG} the VCG payment that a participant makes is the difference between the sum of other's utility at the social optimum point $x^*$, and the utility that others would have if the participant were not present and the optimisation were only over the remaining participants, as seen in equation \\ref{eq:VCG_payment_rule}.\n\nIn this way the socially optimum value $x^*$ needs to be computed, and then additionally an additional optimisation problem for each player - where we assume that the excluded participant has power zero $p_i=0$. Thus in our context, equation \\ref{eq:VCG_payment_rule} becomes:\n\\begin{equation}\\label{eq:VCG_calculation_DC} d_i=\\argmax_{\\sum_{j} u_j(p_j)}\\sum_{j\\ne i}u_j(p_j) - \\argmax_{\\sum_{j\\ne i} u_j(p_j), p_i=0}\\sum_{j\\ne i}u_j(p_j)\\end{equation}\n\nThe first term in this equation is the sum of utilities at the point which maximises the sum of utilities minus the player $i$.\nThe second term in this equation is the sum of other's utilities (ie. excluding player $i$) at the point which maximises the sum of other's utilities in the context that the players power is zero.\nThe first part of the equation is common to all player's contributions $d_i$, but the second negative part is unique for each.\nIn this way, if there are $n$ participants, there are $n+1$ comparable OPF optimisation problems which need to be solved to calculate VCG payments.\nThe SCIP optimisation suite was used solving these optimisation problems.\n\n\\subsection{Computing the Shapley Value imputations}\\label{subsec:Shap_compute1}\n\nFrom section \\ref{subsec:the_shapley_value} the Shapley Value utility imputations that a participant receives is the average marginal contribution that it adds to a characteristic function under ambiguity of the join ordering.\nIn this way we need to consider a characteristic function in the context of DC networks:\n\\begin{equation*}v_{shap}(S) = \\max\\sum_{j}u_j(p_j)~~~~\\text{s.t.}~~\\forall i\\notin S~~p_i=0\\end{equation*}\nThis characteristic function describes the utility that a coalition could achieve by themselves, absent any consumption/generation from those not in the coalition - it is one possible way of creating a characteristic function for DC networks.\n\nThe utilities under the Shapley Value (by efficiency axiom) sum to give the value of the grand coalition $v_{shap}(N)$ which occurs at the optimal operating point $x^*$.\nThus the Shapley Value can be implementing by enacting electrical outcomes described by $x^*$ and conducting budget balanced utility transfers between participants.\nTo calculate the Shapley Value imputations, the maximal sum of utilities under OPF for all the possible $2^n-1$ coalitions need computed, and the SCIP optimisation suite was used for these calculations.\n\n\n\n\n\\section{Some features of GNK, in context of an example}\\label{sec:features}\\label{sec:example_network}\n\nIn the previous section \\ref{more_involved} we detailed a procedure to calculate the financial payments and dispatched powers under the GNK value as well as LMP, VCG and Shapley Value, so that now we can compute and compare them with an example.\nIn this section we are thus able to witness and discuss the characteristics of the GNK value, in the context of an example 5-bus network shown in Figure \\ref{fig:example1}, with parameters given in Table~\\ref{tab:example1}.\nIn this example we calculate and subsequently consider the features of the GNK value against LMP, VCG and Shapley Value against parameter $p_1^l$ (the generator capacity in the network).\nThe results of these calculations of the financial payments are plotted against $p_1^l$ in Figures \\ref{fig:1d}, \\ref{fig:1f}, \\ref{fig:1h} and \\ref{fig:1j} for GNK, LMP, VCG and Shapley Value respectively. Which in addition to the utilities derived from power consumption/generation (Figure \\ref{fig:1b}) form the post payment utility imputations as Figures \\ref{fig:1c}, \\ref{fig:1e}, \\ref{fig:1g} and \\ref{fig:1i}.\n\n%For GNK, LMP and VCG, the power allocations and the utilities associated with these power allocations are plotted against $p_1^l$ (the generator capacity in the network) and are shown in Figures \\ref{fig:1a} and \\ref{fig:1b}.\n%And through subsequent pairs of figures we can see how the different mechanisms allocate payments and utilities in the context of constrained resources - Figures \\ref{fig:1c}, \\ref{fig:1d} for GNK, Figures \\ref{fig:1e}, \\ref{fig:1f} for LMP, Figures \\ref{fig:1g}, \\ref{fig:1h} for VCG.\n\n\n\\input{figs/Electro_table.tex}\n\n\n\n\n\\iffigures\n% \\input{graph0.tex}\n\\input{figs/GNK_graph1.tex}\n\\fi\n\nIn Figure~\\ref{fig:1a}, increasing the generator capacity from $0$ shows that power is initially consumed entirely by the consumer at bus 2, \nwho uses all $p_1^l$kW of power and values it at a rate of 1.9 units of utility. This continues until the power constraint on line (1,2) binds, at 70kW.\nThen the consumer at bus 3 begins to be supplied with power, who values it at a rate of 1.8 units of utility until its consumption is maximised at 100kW. \nThis dynamic is then repeated for the agent with the next-highest marginal utility for power (given by the utility function coefficients in Table~\\ref{tab:example1}), until the respective line constraints are also met.\n\nIn the interval of the first 70 units of generator capacity (ie. $p_1^l \\in [0,-70]$), the LMP price for this power, for both the generator at bus 1 and the consumer at bus 2, is given by its marginal value 1.9 (as the line constraint is not active). This corresponds to the slope of the black line in Figure~\\ref{fig:1f} (or \\ref{fig:1b}) over this interval. \nMore generally, the full set of the LMP transfers plotted in Figure~\\ref{fig:1f} are given by the Lagrange multiplier for power conservation in the OPF optimisation multiplied by the power consumed at that bus, and that as the generation capacity increases the marginal price of power changes, and the gradients of the lines change in discrete steps as the utility functions in the example network are linear.\n\nAdditionally, the VCG payments are plotted against $p_1^l$ in Figures \\ref{fig:1h} and resulting utilities in Figure \\ref{fig:1g}.\nThe VCG payments are similar to the LMP payments, except that instead of payment proportional to the marginal unit of electricity, each player is compensated at his/her marginal cost of their participation.\nVCG takes into account the marginal effect that each of the player's participation has upon each other at the social optimum\n\nIn contrast VCG and LMP, the GNK value takes into account the full bargaining position of each agent and also every possible coalition of agents when determining transfers, which are based on the utilities (or costs, in the case of the generator) of all agents in the system, and not just the marginal value of participation or electrical supply.\nThe GNK value is plotted against $p_1^l$ for our example in Figure~\\ref{fig:1c}, and the resulting transfers (GNK value less utility) are plotted in Figure~\\ref{fig:1d}.\n\nIn contrast to VCG, LMP and GNK, the Shapley Value exhibits character traits that are similar to LMP and VCG, particularly as the Shapley value is composed of many more marginal contributions than those ones inherent to VCG, which is in turn proportional to the marginalism inherent in LMP.\nFurthermore it has a superficial similarity to the GNK value as they both extend from the same axioms - either exhibited in section \\ref{the_value_def}, or by conversion to Shapley value by equation \\ref{convert1} thus having exactly the same axioms as in section \\ref{subsec:the_shapley_value}.\nThe Shapley Value results are plotted against $p_1^l$ for our example in Figure~\\ref{fig:1i}, and the resulting transfers (GNK value less utility) are plotted in Figure~\\ref{fig:1j}.\n\n\nFrom these graphs we can witness some of the qualities of the GNK value against LMP and VCG and the Shapley Value:\n\n% \\begin{itemize}\n% \\item GNK is continuous in the parameters of the network\n% \\end{itemize}\n\n\\subsubsection*{The GNK value, VCG and Shapley Value imputations are continuous in the parameters of the network}\nThe GNK value has some evident continuity properties, as can be seen from equation \\ref{knvalue1}, in which the minimax characteristic function, $v(S)$, always changes continuously with the utility functions $u$.\nThis continuity property is proven in the Appendix \\ref{appendix:continuity_of_GNK}, together with some associated monotonicity properties.\n%, and are generally continuous with continuous deformations of the strategy space $A$.\nThis continuity is similarly witnessed in VCG imputations, but notably not in the context of LMP payments.\n\nIt is seen that LMP features discontinuous changes in financial transfers and this can clearly be seen from the jagged edges in Figure \\ref{fig:1f}, where the payments received by generator 1 drop sharply with increasing generator capacity.\nThis happens because the change in generator capacity is changing the parameters of the network itself and its feasibility region, which yields discontinuous changes in network operating point, and thus marginal prices.\nThis dynamic under LMP might be seen to lead to a somewhat perverse incentive to produce less power than what is socially optimal, and in contrast, the utilities under the GNK value (Figure \\ref{fig:1c}), VCG (Figure \\ref{fig:1g}) and Shapley Value (Figure \\ref{fig:1i}) which feature no such drops or discontinuities.\nIn a power systems context under LMP, these discontinuities are known to occur precisely in the event of network \\emph{congestion}\nwhich is one known cause of the volatility experienced in electricity markets - see \\cite{RePEc:aen:journl:2006v27-02-a09}. \nIn contrast, the post-payment utilities under the GNK value and VCG are always continuous with network parameters.\n\n\n%In the electrical context, \n%$v(S)$ will change continuously when the utilities of the participants and/or the network constraint functions change continuously. \n%In these cases, because $v(S)$ will change continuously, the GNK value $\\varphi$ will also change continuously. \n\n% \\begin{itemize}\n% \\item the GNK value is always budget balanced\n% \\end{itemize}\n\\subsubsection*{The GNK and Shapley Value payments are always budget balanced}\nThe transactions under VCG are not necessarily budget-balanced and can yield a surplus or deficit, and transactions under LMP can yield a budget surplus (they are weakly budget balanced \\cite{Wu1996}[Fact 4]), whereas payments under the GNK value and Shapley Value are strongly budget-balanced and result in no surplus or deficit.\nThis is an outcome of the GNK value's axiomatic derivation, as given by equation \\eqref{myeq}, which is reflective of the Shapley Value axioms, per section \\ref{subsec:the_shapley_value}.\nUnder LMP, each participant is credited or debited at the effective rate of supply for their location but there is no guarantees that the total payments should add to zero.\nThis can be seen by inspecting the region $x>300$ in Figure~\\ref{fig:1f}, where generator 1 is credited $\\$56$ while the consumers are debited at $\\$133.0$, $\\$160.0$, $\\$119.0$, and $\\$64.0$ respectively, leading to a budget surplus of $\\$420$.\nThe surplus of $\\$420$ comes particularly from the existence of congestion in the example network which is well known to introduce so-called `congestion-rents'.\nAdditionally VCG payments are known not to be budget balanced generally and may yield a surplus or deficit depending on various conditions, as considered in section \\ref{sec:solutions_VCG}.\n\n\n%In a power systems context, budget-balancedness of payments is desirable in that any revenue/deficit collected is independent of the network operating conditions.\n\n% \\begin{itemize}\n% \\item the GNK value can offset those that do not receive/generate power\n% \\end{itemize}\n\\subsubsection*{The GNK value (but not VCG or LMP) can offset those that do not receive or generate power}\nThe GNK value can allocate payments between parties such that the consumers that receive power compensate those that are excluded from receiving it.\nThis can be seen from Figures \\ref{fig:1a} and \\ref{fig:1c} particularly in the region where $x<50$.\nIn this region there is only sufficient power to supply consumer 2 (who has the highest utility for that power) whereas consumers 3, 4 and 5 who would otherwise be in a position to receive that power are\ncompensated such as to be barely worse off (as can be seen from \\ref{fig:1c}).\n\nFor instance at $x=50$, generator 1 produces $50$kW which is consumed entirely by consumer 2; the utilities of the participants before transfers are: $-10, 95, 0, 0, 0$ respectively (which can be seen from Figure \\ref{fig:1b}).\nHowever under the GNK value, consumer 2 must pay both the generator and also the other consumers for its right-of-way to consumption.\n\nThe utilities after the transfers of the participants are: $0.5, 23.83, 21.33, 20.08, 19.25$ respectively (which can be seen from Figure \\ref{fig:1c}).\nIn a power systems context, this is likely to be seen as a desirable quality as it may correspond to people's intuitions about the fair allocation of resources.\nFor example, distribution network feeders that have a high penetration of PV systems have been identified by \\cite{feeder1} to experience voltage rise problems at \ntimes of high-supply/low-demand, particularly at the feeder extremities. \nIn these settings, the inverters of PV owners at the bottom are unable to inject their power into the network and also typically get no compensation for essentially a forced curtailment of their electricity generation.\n\nIn this context under LMP, curtailed generators that dont inject their power get no reward, and under VCG curtailed generators will only be allocated utility if their presence or absence would make a difference to the network operating point, which is not assured.\nUnder Shapley Value, it is witnessed that there is some rather small offset for those that do not receive power, though this is mostly determined by marginal coalitional considerations that exclude most other consumers.\n\n% \\begin{itemize}\n% \\item the GNK value is not incentive compatible\n% \\end{itemize}\n\\subsubsection*{The GNK value is not incentive compatible}\nUnlike VCG, the GNK value, LMP and Shapley Value are not \\emph{incentive compatible} in the sense that is often referred-to in Mechanism Design (see Section \\ref{sec:solutions_VCG}).\nSpecifically, the payments between parties are potentially subject to strategic manipulation if the agents are freely able to report their utility.\nIn the GNK value this can be seen in \\eqref{knvalue1}, or more easily in its reduced form, \\eqref{knvalue2}, where the payoff advantage of a coalition $v(S)$ is based on its reported utilities in minimax strategies which may not be actualised; the same consideration holds for the Shapley Value.\nBecause of this consideration, misreporting the utilities of these unactualised events may change the $v(S)$ and hence the GNK value itself.\nAdditionally, LMP is known not to be incentive compatible, and there is further work to understand exactly how consequential this would likely be - such as by \\cite{8054716}.\nIn section \\ref{sec:GNK_extensions_discussion} we continue discussion about this point.\n\n% \\begin{itemize}\n% \\item the GNK value is computationally difficult\n% \\end{itemize}\n\\subsubsection*{The GNK value and Shapley Value are computationally difficult}\nThe GNK value is more difficult than the Shapley Value, which is more difficult than VCG, which is more difficult than LMP, to compute.\nThis can be seen via \\eqref{da_value_eq} where calculating the GNK value exactly requires calculating $v(S)$ for all the $2^n-1$ possible coalitions $S$, and each calculation of $v(s)$ is an NP-hard bilevel optimisation problem.\nThe Shapley Value also requires calculating $v_{shape}(S)$ for all the $2^n-1$ possible coalitions $S$, but the calculation of $v_{shape}(S)$ is not necessarily NP-hard problem, as DC OPF problem can be a linear optimisation.\nConversely VCG calculation for $n$ agents requires $n+1$ OPF optimisations (which are potentially linear in the DC case) per equation \\ref{eq:VCG_calculation_DC}, where as LMP calculation requires exactly one OPF optimisation (as identified in section \\ref{subsec:LMP_compute1}).\n\nThe computational difficulty of the GNK value is reflected in a later Figure \\ref{fig:performance_graph2} where it is seen that even using sampling to approximate the GNK value to sufficient accuracy is witnessed to be a double-exponentially complex process. \n\n\n\\section{Summary}\n\nIn this Chapter we have introduced the GNK value from its axioms and we have considered its conceptual inheritance from other solution concepts.\nIn order to apply the GNK value to the pricing of immediate power generation and consumption in small DC networks we formally introduced the elements of the DC network and have detailed a computation methodology for the GNK value in that context.\nWe then compared the GNK against LMP, VCG and Shapley Value on a small example network to examine their features, particularly we identified that the GNK value has some nice budget-balance and continuity properties, however we also realised that the GNK is not incentive compatable and that it is difficult to calculate for larger electricity networks.\n\nIn the next chapter we consider the ways in which the computational difficultly of calculating the GNK value on larger networks can be ameliorated.\nThe two primary techniques to ameliorate this difficulty are principally by sampling, and also by the adoption of a proxy for the inner optimisations.\nWe discuss these to directions in the next chapter \\ref{sec:scaling} before actually applying these techniques for an application of GNK to larger network in section \\ref{sec:results_and_evaluation_of_GNK} where we discuss in section \\ref{sec:GNK_value_discussion}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"hexsha": "37cbfabfe704c7429edadac43fe87de8fe8999d2", "size": 55408, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Thesis/chapters/new_solution.tex", "max_stars_repo_name": "Markopolo141/Thesis_code", "max_stars_repo_head_hexsha": "df7cffff8127641b0fed0309adf38cfc9372e618", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Thesis/chapters/new_solution.tex", "max_issues_repo_name": "Markopolo141/Thesis_code", "max_issues_repo_head_hexsha": "df7cffff8127641b0fed0309adf38cfc9372e618", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Thesis/chapters/new_solution.tex", "max_forks_repo_name": "Markopolo141/Thesis_code", "max_forks_repo_head_hexsha": "df7cffff8127641b0fed0309adf38cfc9372e618", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 97.3778558875, "max_line_length": 464, "alphanum_fraction": 0.7773787179, "num_tokens": 14440, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3467585374639343}} {"text": "% Features to discuss in depth:\nThis section gives a basic introduction of SymPy, and lists its features.\nA few features---assumptions, simplification, calculus, polynomials, printers,\nsolvers, and matrices---are core components of SymPy and are discussed in\ndepth. Many other features are discussed in depth in the supplementary\nmaterial.\n\n\\subsection{Basic Usage}\n\\label{sec:basic-usage}\n\\input{basic_usage.tex}\n\n\\subsection{List of Features}\n\nAlthough SymPy's extensive feature set cannot be covered in depth in this\npaper, bedrock areas, that is, those areas that are used throughout the\nlibrary, are discussed in their own\nsubsections below. Additionally, Table~\\ref{features-table} gives a compact listing\nof all major capabilities present in the SymPy codebase. This grants a\nsampling from the breadth of topics and application domains that SymPy\nservices. Unless stated otherwise, all features noted in\nTable~\\ref{features-table} are symbolic in nature. Numeric features are\ndiscussed in Section~\\ref{sec:numerics}.\n\n\\begin{longtable}[htbc]{>{\\raggedright}p{0.30\\linewidth}p{0.63\\linewidth}}\n\\caption{SymPy Features and Descriptions.\\label{features-table}}\\\\\n\\toprule\n\\textbf{Feature (submodules)} & \\textbf{Description} \\\\\n\\midrule\nCalculus (\\texttt{sympy.core}, \\texttt{sympy.calculus}, \\texttt{sympy.integrals}, \\texttt{sympy.series}) & Algorithms for computing derivatives, integrals, and limits.\\\\\n\nCategory Theory (\\texttt{sympy.categories}) & Representation of objects, morphisms, and diagrams. Tools\nfor drawing diagrams with Xy-pic~\\cite{rose1999xy}.\\\\\n\nCode Generation (\\texttt{sympy.printing}, \\texttt{sympy.codegen}) & Generation of compilable and executable code in a\nvariety of different programming languages from expressions directly. Target\nlanguages include C, Fortran, Julia, JavaScript,\nMathematica, MATLAB and Octave, Python, and Theano.\\\\\n\nCombinatorics \\& Group Theory (\\texttt{sympy.combinatorics}) & Permutations, combinations,\npartitions, subsets, various permutation groups (such as polyhedral, Rubik,\nsymmetric, and others), Gray codes~\\cite{Nijenhuis1978combinatorial},\nand Prufer sequences~\\cite{biggs1976graph}.\\\\\n\nConcrete Math (\\texttt{sympy.concrete}) & Summation, products, tools for determining whether summation\nand product expressions are convergent, absolutely convergent, hypergeometric,\nand for determining other properties; computation of Gosper's normal form~\\cite{Petkovsek1997AeqB} for two univariate polynomials.\\\\\n\nCryptography (\\texttt{sympy.crypto}) & Block and stream ciphers, including shift, Affine,\nsubstitution, Vigen\\`{e}re's, Hill's, bifid, RSA, Kid RSA,\nlinear-feedback shift registers, and Elgamal encryption.\\\\\n\nDifferential Geometry (\\texttt{sympy.diffgeom}) & Representations of manifolds, metrics, tensor\nproducts, and coordinate systems in Riemannian and pseudo-Riemannian\ngeometries~\\cite{sussman2013functional}.\\\\\n% TODO: Someone verify that this is a good summary of the diffgeom module\n\nGeometry (\\texttt{sympy.geometry}) & Representations of 2D geometrical entities, such as lines and\ncircles. Enables queries on these entities, such as asking the area of an\nellipse, checking for collinearity of a set of\npoints, or finding the intersection between objects.\\\\\n\nLie Algebras (\\texttt{sympy.liealgebras}) & Representations of Lie algebras and root systems.\\\\\n\nLogic (\\texttt{sympy.logic}) & Boolean expressions, equivalence testing, satisfiability, and normal\nforms.\\\\\n\nMatrices (\\texttt{sympy.matrices}) & Tools for creating matrices of symbols and expressions.\nBoth sparse and dense representations, as well as symbolic linear\nalgebraic operations (e.g., inversion and factorization), are\nsupported.\\\\\n\nMatrix Expressions (\\texttt{sympy.matrices.expressions}) & Matrices with symbolic dimensions (unspecified entries).\nBlock matrices.\\\\\n\nNumber Theory (\\texttt{sympy.ntheory}) & Prime number generation, primality testing, integer\nfactorization, continued fractions, Egyptian fractions, modular arithmetic,\nquadratic residues, partitions, binomial and multinomial coefficients,\nprime number tools, hexidecimal digits of $\\pi$, and integer factorization. \\\\\n\nPlotting (\\texttt{sympy.plotting}) & Hooks for visualizing expressions via matplotlib~\\cite{Hunter:2007}\nor as text drawings when lacking a graphical back-end. 2D function plotting,\n3D function\nplotting, and 2D implicit function plotting are supported.\\\\\n\nPolynomials (\\texttt{sympy.polys}) & Polynomial algebras over various coefficient domains.\nFunctionality ranges from simple operations (e.g., polynomial division) to\nadvanced computations (e.g., Gr\\\"obner bases~\\cite{Adams1994intro} and multivariate\nfactorization over algebraic number domains).\\\\\n\nPrinting (\\texttt{sympy.printing}) & Functions for printing SymPy expressions in the terminal with ASCII\nor Unicode characters and converting SymPy expressions to \\LaTeX{} and\nMathML.\\\\\n\nQuantum Mechanics (\\texttt{sympy.physics.quantum}) & Quantum states, bra--ket notation, operators, basis sets,\nrepresentations, tensor products, inner products, outer products, commutators,\nanticommutators, and specific quantum system implementations.\\\\\n\nSeries (\\texttt{sympy.series}) & Series expansion, sequences, and limits of sequences.\nThis includes Taylor, Laurent, and Puiseux series as well as special series, such\nas Fourier and formal power series.\\\\\n\nSets (\\texttt{sympy.sets}) & Representations of empty, finite, and infinite sets (including\nspecial sets such as the natural, integer, and complex numbers). Operations\non sets such as union, intersection, Cartesian product, and building sets from\nother sets are supported.\\\\\n\nSimplification (\\texttt{sympy.simplify}) & Functions for manipulating and simplifying expressions.\nIncludes algorithms for simplifying hypergeometric functions, trigonometric\nexpressions, rational functions, combinatorial functions, square root\ndenesting, and common subexpression elimination.\\\\\n\nSolvers (\\texttt{sympy.solvers}) & Functions for symbolically solving equations, systems\nof equations, both linear and non-linear, inequalities, ordinary differential\nequations, partial differential equations, Diophantine equations, and\nrecurrence relations.\\\\\n\nSpecial Functions (\\texttt{sympy.functions}) & Implementations of a number of well known special functions,\nincluding Dirac delta, Gamma, Beta, Gauss error functions, Fresnel integrals,\nExponential integrals, Logarithmic integrals, Trigonometric integrals, Bessel,\nHankel, Airy, B-spline, Riemann Zeta, Dirichlet eta, polylogarithm, Lerch\ntranscendent, hypergeometric, elliptic integrals, Mathieu, Jacobi polynomials,\nGegenbauer polynomial, Chebyshev polynomial, Legendre polynomial, Hermite\npolynomial, Laguerre polynomial, and\nspherical harmonic functions.\\\\\n\nStatistics (\\texttt{sympy.stats}) & Support for a random variable type as well as the ability to\ndeclare this variable from prebuilt distribution functions such as\nNormal, Exponential, Coin, Die, and other custom distributions~\\cite{rocklin2012symbolic}.\\\\\n\nTensors (\\texttt{sympy.tensor}) & Symbolic manipulation of indexed objects.\\\\\n\nVectors (\\texttt{sympy.vector}) & Basic operations on vectors and differential calculus with respect\nto 3D Cartesian coordinate systems.\\\\\n\\bottomrule\n\n\\end{longtable}\n\n%% Assumptions\n\\subsection{Assumptions}\n\\label{sec:assumptions}\n\\input{assumptions.tex}\n\n\\subsection{Simplification}\n\n\\input{simplification}\n\n\\subsection{Calculus}\n\\label{sec:calculus}\n\\input{calculus}\n\n\\subsection{Polynomials}\n\\input{polys}\n\n\\subsection{Printers}\n\n\\input{printers.tex}\n\n% Solvers (regular equations, maybe also mention other types of solvers like ODEs/recurrence/Diophantine)\n\\subsection{Solvers}\n\\input{solvers.tex}\n\n% Matrices (worth emphasizing that they are symbolic)\n\\subsection{Matrices}\n\n\\input{matrices}\n", "meta": {"hexsha": "21d8b6aeca7868d174758ba8902c602ee743aafa", "size": 7773, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "features.tex", "max_stars_repo_name": "ProgZone/sympy-paper", "max_stars_repo_head_hexsha": "b3b85809cc92d1fd588971f944abda9fa995a426", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 40, "max_stars_repo_stars_event_min_datetime": "2016-03-27T06:55:25.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-22T18:42:39.000Z", "max_issues_repo_path": "features.tex", "max_issues_repo_name": "ProgZone/sympy-paper", "max_issues_repo_head_hexsha": "b3b85809cc92d1fd588971f944abda9fa995a426", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 205, "max_issues_repo_issues_event_min_datetime": "2016-03-17T03:08:15.000Z", "max_issues_repo_issues_event_max_datetime": "2020-10-01T17:09:29.000Z", "max_forks_repo_path": "features.tex", "max_forks_repo_name": "ProgZone/sympy-paper", "max_forks_repo_head_hexsha": "b3b85809cc92d1fd588971f944abda9fa995a426", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 37, "max_forks_repo_forks_event_min_datetime": "2016-03-17T16:02:22.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-15T15:06:54.000Z", "avg_line_length": 46.8253012048, "max_line_length": 169, "alphanum_fraction": 0.8007204426, "num_tokens": 1929, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.3467585374639343}} {"text": "% Copyright 2021 Google LLC\n%\n% Use of this source code is governed by an MIT-style\n% license that can be found in the LICENSE file or at\n% https://opensource.org/licenses/MIT.\n\n%!BIB program = biber\n%!TeX program = lualatex\n%!TeX spellcheck = en-US\n\n\\documentclass[hctr2.tex]{subfiles}\n\\begin{document}\n\\section{Injectivity of H onto polynomials}\\label{injective}\nTo demonstrate injectivity, \nthe following algorithm recovers \\(T\\) and \\(M\\) given\na binary string \\(X\\) of length \\(\\abs{X} = n (1 + \\deg(H(T, M)))\\)\nrepresenting the coefficients of \nthe polynomial \\(H(T, M)\\) in binary form,\nstarting with the greatest nonzero power; thus\n\\(13\\hpoly^3\\) would encode as \\(1011 \\Concat 0^{4n -4}\\).\n\\begin{algorithmic}[1]\n \\Procedure{GetTM}{$X$}\n \\State \\textbf{assert} \\(\\abs{X} \\bmod n = 0\\)\n \\State \\textbf{assert} \\(\\abs{X} \\geq 2n\\)\n \\State \\textbf{assert} \\(X[\\abs{X} - n; n] = 0^n\\)\n \\State \\(t \\gets \\fromint^{-1}_{n-1}(X[1;n-1])\\)\n \\State \\textbf{assert} \\(t > 0\\)\n \\State \\(t \\gets t-1\\)\n \\State \\(w \\gets n(1 + \\ceil{t/n})\\)\n \\If{\\(X[0; 1] = 0\\)}\n \\State \\textbf{assert} \\(w + n \\leq \\abs{X}\\)\n \\State \\(M \\gets X[w;\\abs{X}-w-n]\\)\n \\Else\n \\State \\textbf{assert} \\(w + 2n \\leq \\abs{X}\\)\n \\State \\textbf{assert} \\(X[\\abs{X}-2n+1; n-1] \\neq 0^{n-1}\\)\n \\State \\(i \\gets \\abs{X} - n - 1\\)\n \\While{\\(X[i; 1] = 0\\)}\n \\State \\(i \\gets i - 1\\)\n \\EndWhile\n \\State \\(M \\gets X[w;i - w]\\)\n \\EndIf\n \\State \\(T \\gets X[n;t]\\)\n \\State \\textbf{return} \\(T, M\\)\n \\EndProcedure\n\\end{algorithmic}\n\n\\end{document}\n", "meta": {"hexsha": "1e5b6e93b63a5e628011772f9951db1b53525d81", "size": 1615, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/injective.tex", "max_stars_repo_name": "google/hctr2", "max_stars_repo_head_hexsha": "1913bddafea9797108446d4284a1e3f8e43a1b5a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2021-11-05T14:23:38.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-30T17:24:43.000Z", "max_issues_repo_path": "paper/injective.tex", "max_issues_repo_name": "google/hctr2", "max_issues_repo_head_hexsha": "1913bddafea9797108446d4284a1e3f8e43a1b5a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 17, "max_issues_repo_issues_event_min_datetime": "2021-11-01T21:35:34.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-14T23:41:06.000Z", "max_forks_repo_path": "paper/injective.tex", "max_forks_repo_name": "google/hctr2", "max_forks_repo_head_hexsha": "1913bddafea9797108446d4284a1e3f8e43a1b5a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2021-11-04T17:58:46.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-28T13:29:03.000Z", "avg_line_length": 33.6458333333, "max_line_length": 68, "alphanum_fraction": 0.5789473684, "num_tokens": 601, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3467585298119606}} {"text": "\\section{A simple multi-skill contact center}\n\\label{sec:SimpleMSK}\n\nThis section presents an example of a contact center with three\ncontact types, two\nagent groups, and three two-hours periods. Contacts arrive\nfollowing a non-homogeneous Poisson process\nwith randomized arrival rate\n$B\\lambda_{k, p}$ for contact type~$k$ during period~$p$.\nAs in the previous section,\n$B$ is a gamma busyness factor with parameters\n$(\\alpha_0, \\alpha_0)$.\n\nWhen a contact arrives, an agent is selected from a group depending on\nthe contact type. Contacts of type~0 can only be served by agents in\ngroup~0 while contacts of type~2 can only be served by agents in\ngroup~1. Contacts of type~1 are served by agents in group~0, or\nin group~1 if no agent is free in group~0.\nFor contacts arriving during period~$p$,\nservice times are i.i.d.\\ exponential random variables with\nmean~$1/\\mu_p$.\nAgents in groups are not differentiated, and the total number of\nagents can change from periods to periods.\n\nA contact that cannot be served immediately is added to a waiting\nqueue corresponding to its type. After a patience time, if service\nhas not started, the contact abandons. For contacts arriving during\nperiod~$p$,\npatience times are i.i.d.\\ exponential variables with mean~$1/\\nu_p$.\n\nWe are interested in the overall service level defined\nby~(\\ref{eq:slab}), and the occupancy ratio\nof the first agent group defined by\n\\begin{equation}\no_i=\\frac{E\\left[\\int_0^{t_P} \\Nb[i](t)\\\n dt\\right]}{E\\left[\\int_0^{t_P} (N_i(t)+\\Ng[i](t))\n dt\\right]}\\label{eq:occi}\n\\end{equation}\nwhere $\\Nb[i](t)$ corresponds to the number of busy agents in\ngroup~$i$, $N_i(t)$ corresponds to the total number of agents in group~$i$,\nand $\\Ng[i](t)$ corresponds to the number of ghost agents in\ngroup~$i$, at time~$t$.\nWe also estimate $\\E[\\Sg[k, p](s)]$ for $k=0,\\ldots,K-1$ and\n$p=0,\\ldots,P-1$, the number of contacts meeting the service level\ntarget.\n\n\\subsection{Implementing the model}\n\nListing~\\ref{lst:SimpleMSK} presents the code implementing this\nmodel.\n\n\\lstinputlisting[\ncaption={A simple multi-skill contact center},%\nemph={main,simulateOneDay,MyContactFactory,MyContactMeasures},\nlabel=lst:SimpleMSK\n]\n{SimpleMSK.java}\n\nAs with previous examples, a class representing the simulator is\ndefined. The \\texttt{main} method constructs the simulator, runs a\nsimulation, and prints some statistics.\n\nFields are declared for contact center's components, statistical\ncounters, etc.\nFor the number of served and\nabandoned contacts, simple integers are sufficient,\nbut a matrix is needed to get the number of contacts meeting service\nlevel target, for each contact type and main period.\n\nAs in previous examples, a constructor is responsible for creating all\narrival processes, agent groups, waiting queues, etc.\nThe period-change event has a preliminary period of length~0,\ntwo-hours main periods, and a wrap-up period with\nrandom length.\nFor each contact type, a factory and an arrival process are\nconstructed. In contrast with the previous examples, the\nfactory requires the contact type identifier to be given as an\nargument. In this example,\nall the arrival processes use the same period-change event to be\nnotified when a new period starts. Some arrival processes could of\ncourse use different period-change events.\nConstructing the agent groups only\nrequires the period-change event for $N_i(t)$ to be automatically updated,\nand an array containing the number of agents for each period.\n\nService and patience times are generated using\nrandom variate generators adapted for multiple periods. Such\ngenerators use a\nperiod-change event to determine the current period and select a\nperiod-specific generator to get random values. The generic way to\nconstruct them is to create a random variate generator for each\nperiod and give the resulting array of generators, with a\nperiod-change event, to the\nconstructor of \\texttt{Multi\\-Period\\-Gen}. For some distributions,\nincluding exponential,\nhelper methods such as \\texttt{create\\-Exponential}\nare available to construct the generators more\nconveniently.\n\nBecause each waiting queue and agent group is\nfunctionally identical, it is the task of the router to decide which contacts to\nsend to which agents or\nqueues, and from which queues contacts must be removed.\nFor the router to be constructed, a type-to-group map and a\ngroup-to-type map are needed. The selected\n\\texttt{Single\\-FIFO\\-Queue\\-Router} class affects how these\nstructures are used. Here, because we model a multi-skill contact\ncenter, the data structures for the routing policy become important\nand can greatly affect the performance of the contact center.\nThe choice of the subclass of \\texttt{Router} affects how\ncontacts interact with the center.\n\nThe \\texttt{simulate\\-One\\-Day} method must initialize several arrays\nof elements, including arrival processes, agent groups, and counters.\nThe same value of $B$ is used for every arrival process, because the\nbusyness is not specific to a contact type.\nAs in the previous example, the period-change event and the\narrival processes are started before the simulation starts.\n\nOne factory object has been constructed for each arrival\nprocess, the only difference between these objects being the value of\ntheir \\texttt{type} field. This\navoids the necessity of one factory class for each contact type,\nwhich greatly improves scalability.\nThe factory constructs a contact of the\nappropriate type and generates a service and a patience times.\n\nWhen a contact of type~0 arrives, the router takes the element~0 of\nthe type-to-group map, which corresponds to an ordered list containing\nthe\nagent group~0 only. Let $\\Nf[i](t)$ be the number of idle agents in\ngroup~$i$ available to serve contacts.\nIf $\\Nf[0](t)>0$, the contact is served\nimmediately. Otherwise, it is added to waiting queue~0. Contacts of\ntype~2 are treated similarly. For contacts of type~1, the router\nobtains an ordered list containing~0 and~1. If $\\Nf[0](t)>0$, the contact\nis served immediately by an agent in group~0. Otherwise, it overflows\nto the next agent\ngroup in the list: if $\\Nf[1](t)>0$, the contact is served by an\nagent in group~1.\nOtherwise, it is added at the end of the queue~1.\n\nWhen an agent within group~0 becomes free or is added, the router uses\nthe group-to-type map to obtain the ordered list $\\{1, 0\\}$. The\nchosen\nrouter selects the queued contact with the longest waiting time rather\nthan using the order induced by the list. If the waiting queues\naccessible for agents in group~0 contain no contact, the agent\nremains free until new arrivals occur. Similar routing\nhappens for agents in group~1. This is equivalent to managing a\nsingle FIFO queue by merging all\nper-type waiting queues, sorting contacts by increasing arrival\ntimes, and removing the first contact the free agent can serve.\n\nIf \\texttt{Queue\\-Priority\\-Router} was used as in previous examples,\nthe algorithm for agent selection would be the same, but waiting\nqueues would be scanned sequentially rather than considered as a\nsingle FIFO queue. For example, if an agent in\ngroup~0 became free, the queue priority router would check for\ncontacts in queue~0 only when queue~1 is empty.\n\nEach contact exiting the system is notified to the registered\nexited-contact listener. The \\texttt{blocked} method does nothing\nbecause\nthe capacity of the contact center is infinite; no contact is\nblocked. When a contact leaves the queue without service, it is\ncounted as having abandoned.\nIf its waiting time is greater than or equal\nto $s$, a contact having abandoned after the acceptable waiting time\nis also counted. When a\ncontact is served, a new service is counted. If its waiting time is\nsmall enough, it is also counted as a contact meeting service level\ntarget.\n\nFor a contact to be counted in \\texttt{numGoodSLKP}, the main\nperiod of its arrival must be\ndetermined. Arrivals occur in periods~$1,\\ldots,P$,\ncorresponding to main periods~$0,\\ldots,P-1$.\nIf the main period index, i.e., the period index minus one, is\nnegative or greater than or equal to~$P$,\nthe arrival occurred during the preliminary or\nwrap-up periods, and the event is ignored. When the arrival occurs\nduring a main period, the appropriate element of the matrix is\nincremented.\n\n\\lstinputlisting[caption={Results of the program \\texttt{SimpleMSK}},%\nlabel=res:SimpleMSK,language={},float=htb,breaklines,prebreak={\\char92}\n ]{SimpleMSK.res}\n\nAfter the simulation stops, the \\texttt{stop} method of\n\\texttt{Period\\-Change\\-Event} is called,\ncomputed observations are added to collectors.\nTo keep the program simple, we estimate the occupancy ratio from\ntimes~0 to~$T$ rather than~$t_P$. Getting the correct ratio would\nrequire the creation of a custom period-change listener to get\n$\\int_0^{t_P}\\Nb[0](t)\\ dt$ and $\\int_0^{t_P} (N_0(t)+\\Ng[0](t)) dt$ at\ntime~$t_P$, i.e., at the beginning of wrap-up period.\nThis will be done in a further example, in sections~\\ref{sec:occcor}\nand~\\ref{sec:Blend}.\nListing~\\ref{res:SimpleMSK} displays the results of the program when\nperforming 1000 independent runs.\n\n\n\\subsection{Adding a contact-by-contact trace}\n\nFor debugging or advanced statistical processing using tools such as\nSAS or R, it may be\nneeded to get a contact-by-contact trace of the simulation. This can be done\neasily by using observers, as shown in\nListing~\\ref{lst:SimpleMSKWithTrace}.\nThis example is a simplified version of the trace facility available\nin the generic simulator of call centers supporting blend and\nmulti-skill systems.\nIt shows how it is possible to extract information from a contact\nobject, and other parts of the simulator.\n\n\\lstinputlisting[\ncaption={Contact-by-contact trace added to the simple multi-skill contact\n center example},%\nemph={main,simulateOneDay,ContactTrace},\nlabel=lst:SimpleMSKWithTrace\n]\n{SimpleMSKWithTrace.java}\n\nThis program behaves the same as the program in the preceding subsection,\nbut it creates a text file named \\texttt{contactTrace.log} with one line\nfor each processed contact, whether it has abandoned or was served.\nOne can then open the resulting (large) trace file with any text\neditor, or make a program to parse it.\nListing~\\ref{res:SimpleMSKWithTrace} displays ten lines of the trace\nproduced by the program. We cannot display the full trace, because\nits size is 2MB, even though we have simulated only five days.\n\nThe trace contains the following fields:\n\\begin{description}\n\\item[Step] The index of the replication the contact arrived in.\n\\item[Type] The type of the contact.\n\\item[Period] The period the contact arrived in.\n\\item[ArvTime] The simulation time of the contact's arrival.\n\\item[QueueTime] The waiting time in queue of the contact.\n\\item[Outcome] The outcome of the contact, can be \\texttt{Served} or \\texttt{Abandoned}.\n\\item[Group] The group index of the agent who has served the contact, or\n $-1$ for abandoned contacts.\n\\item[SrvTime] The service time of the contact, or \\texttt{NaN} for\n unserved contacts.\n\\end{description}\n\n\\lstinputlisting[caption={Sample trace produced by the program \\texttt{SimpleMSKWithTrace}},%\nlabel=res:SimpleMSKWithTrace,language={},float=htb\n ]{SimpleMSKTrace.res}\n\nThe program defines a class named \\texttt{SimpleMSKWithTrace} which\nextends \\texttt{SimpleMSK} to inherit the simulation logic of the\nexample in the previous subsection, but it adds a new exited-contact\nlistener to the router for logging contacts.\nThis trace manager, which is initialized at the beginning of the\nsimulation, writes a trace entry to a file for any exiting contact.\nIt does not have any effect on how the contacts are managed by the\nsimulator.\n\nThe new subclass defines three fields: the trace manager, the output\ntrace file, and the current\nreplication number.\nThe first two fields are initialized by the constructor, which also\nattach the trace manager to the router, defined in the superclass\n\\texttt{Simple\\-MSK}.\nThis way, every contact processed by the router is broadcast to the contact\ntrace manager.\nMoreover, one could disable the contact-by-contact trace simply by unregistering\nthe trace manager from the router.\n\nThe last field is updated after each replication by an overridden\n\\texttt{simulate\\-One\\-Day} method, and is used when formatting trace\nentries.\n\nThe overridden \\texttt{simulate} method opens the trace file at\nthe beginning of the simulation, calls the \\texttt{simulate} method\nfrom the superclass to\nperform the simulation, and closes the trace at the end.\nWe put the call to \\texttt{close} into a\n\\texttt{finally} block that will be called even if an unexpected\nexception is thrown by the simulator.\nThis prevents any loss of information caused by an unflushed buffer.\n\nThe main part of the program is the\n\\texttt{Trace\\-Manager} inner class which is an exited-contact listener\nproducing the data written into the trace.\nA trace manager has an associated print writer used to format trace\nentries into the output file.\n\nLog lines are constructed by the \\texttt{dequeued} and \\texttt{served}\nmethods of \\texttt{Trace\\-Manager}, using the \\texttt{format} method\nof \\texttt{Print\\-Writer}.\nNote how the arrival time, the service time, the waiting time, the\ncontact type, etc.\\ can all be extracted from the \\texttt{Contact}\nobject.\nThe \\texttt{format} method fills a user-provided pattern\nby using the values given by the remaining arguments. The print stream\ntakes the system's current locale and the line separator into account.\n\nArrival times and time durations are formatted with a fixed\nnumber \\texttt{timePrecision} of decimal digits of precision, which is\nset to 3 in this program.\nThis allows for a better visual formatting of the data in the plain\ntext file.\nOne could easily use another format, e.g., XML, or use JDBC to\nwrite data into a database instead of a file.\n\n\\subsection{Rerouting queued contacts}\n\nAgents may sometimes be allowed to serve some types of contacts only\nafter the contacts have waited for some time in queue.\nFor example, we can modify the model of this example to allow agents\nin group~1 to serve contacts of type~1 only after these contacts have\nwaited more than 12s.\nThis can be done by customizing the router as in\nListing~\\ref{lst:SimpleMSKWithRerouting} for \\emph{contact rerouting},\ni.e., the router can, after a delay elapses, reprocess a queued\ncontact for a new agent selection with different criteria.\nNote that the router in ContactCenters also supports \\emph{agent\n rerouting} which is not covered in this example but\nworks in a way similar to contact rerouting:\nif an agent stays free after the end of a service for a certain delay,\nthe router can reprocess the agent to do a new contact selection, with\ndifferent criteria.\nOf course, agent rerouting requires the agent groups to keep track of\nindividual agents.\n\n\\lstinputlisting[\ncaption={Contact rerouting added to the simple multi-skill contact\n center example},%\nemph={main,MyRouter},\nlabel=lst:SimpleMSKWithRerouting\n]\n{SimpleMSKWithRerouting.java}\n\nThe customized router, named \\texttt{My\\-Router}, extends the\n\\texttt{Single\\-FIFO\\-Queue\\-Router} used by the original example.\nAs a result, the agent and contact selections are performed the same\nway as in the original example, but we use a different type-to-group\nmap which allows incoming contacts of type~1 to be served by agents in\ngroup~0 only.\n\nContact rerouting works as follows.\nWhen a new contact is notified to the router, an agent is selected by\nthe \\texttt{select\\-Agent} methods which is given the new contact as\nits argument. This method must construct and return an end-service\nevent representing the served contact, or \\texttt{null} if the contact\ncannot be served.\nWhen the contact cannot be served, it is added into a waiting queue by\nthe \\texttt{select\\-Waiting\\-Queue} method of the router.\nIf the contact can be queued, the router obtains its initial\n\\emph{rerouting delay}, i.e., the time after which the queued contact\nis reprocessed if still in queue.\nIf this delay is negative, no rerouting happens, which is the default.\nWhen a positive rerouting delay elapses for a queued contact, the\nrouter passes this queued contact to a second \\texttt{select\\-Agent}\nmethod which accepts the dequeue event along with the number of\nreroutings done. For the first rerouting, this corresponds to 0.\nThis new method returns an end-service event or \\texttt{null}, exactly\nas the ordinary \\texttt{select\\-Agent} method.\nOf course, the scheme for agent selection implemented by this method\nshould differ from the scheme implemented by the ordinary\n\\texttt{select\\-Agent} method.\nIf the contact still cannot be served, it remains in queue, and a new\nrerouting delay is required.\nThe process continues until a negative delay is obtained, the contact\nis served, or abandons.\n\nThe initial rerouting delay is obtained\nusing \\texttt{get\\-Rerouting\\-Delay} with\n$-1$ as the number of preceding reroutings. By\ndefault, this method always returns $-1$, which results in no rerouting.\nIn our case, we return \\texttt{DELAY} if the number of reroutings done\nis $-1$, or $-1$ otherwise.\nThe \\texttt{select\\-Agent} method for rerouting applies the overflow\nrouting of the\noriginal example, which allows contacts of type~1 to be routed to\nagents in group~1 when no agent is available in group~0.\n\nWe also need to override contact selection in order to prevent an\nagent in group~1 to dequeue a contact of type~1 if its waiting time is\nsmaller than \\texttt{DELAY}. We therefore override the\n\\texttt{select\\-Contact}\nmethod of \\texttt{Router}, which accepts an agent group containing the\nfree agents, and returns the dequeue event representing the\ncontact to serve.\nOur method scans the waiting queues accessible to the free agent, and\nrecords the waiting queue whose first contact has the smallest enqueue\ntime. However, if the free agent is in group~1, the method filters out\nqueue~1 if the first contact has a waiting time smaller than\n\\texttt{DELAY}.\nAfter the queue is selected, the first contact is removed, and\nreturned. The contact is removed with dequeue type\n\\texttt{DEQUEUETYPE\\_BEGINSERVICE} which is used to represent the\nbeginning of the service for a queued contact.\n\nListing~ \\ref{res:SimpleMSKWithRerouting} shows that the modified\nprogram produces slightly different results: the number of contacts\nhaving abandoned is higher and the service level is smaller than with\nthe original model. The simulator behaves as we would expect:\nsome contacts of type~0 have to wait 12s longer in queue before they\ncan be served, and some of them abandon while they were served in the\noriginal setting.\n\n\\lstinputlisting[caption={Results of \\texttt{SimpleMSKWithRerouting}},%\nlabel=res:SimpleMSKWithRerouting,language={},float=htb,breaklines,prebreak={\\char92}\n ]{SimpleMSKWithRerouting.res}\n", "meta": {"hexsha": "37aabafc605a3c2d8b67cb72edbe088505e9ef47", "size": 18757, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/examples/simplemsk.tex", "max_stars_repo_name": "umontreal-simul/contactcenters", "max_stars_repo_head_hexsha": "f5ddb4a0a4b30dbf436ac36e6d97facce2f3576d", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/examples/simplemsk.tex", "max_issues_repo_name": "umontreal-simul/contactcenters", "max_issues_repo_head_hexsha": "f5ddb4a0a4b30dbf436ac36e6d97facce2f3576d", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/examples/simplemsk.tex", "max_forks_repo_name": "umontreal-simul/contactcenters", "max_forks_repo_head_hexsha": "f5ddb4a0a4b30dbf436ac36e6d97facce2f3576d", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.8606356968, "max_line_length": 93, "alphanum_fraction": 0.7889854454, "num_tokens": 4578, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.3467585298119606}} {"text": "\\subsection{Bernstein inequalities}\\label{subsec:bernstein_inequalities}\n\n\\begin{definition}\\label{def:real_function_space_operators}\n Consider the \\hyperref[def:function]{operator} \\( T: C([a, b]) \\to C([a, b]) \\).\n\n \\begin{thmenum}\n \\thmitem{def:real_function_space_operators/positive} If \\( f([a, b]) \\subseteq [0, \\infty) \\) implies \\( T(f)([a, b]) \\subseteq [0, \\infty) \\), we say that \\( T \\) is \\term{positive}.\n\n \\thmitem{def:real_function_space_operators/monotone} If \\( f(x) \\leq g(x) \\) for all \\( x \\in [a, b] \\) implies \\( T(f)(x) \\leq T(g)(x) \\) for all \\( x \\in [a, b] \\), we say that \\( T \\) is \\term{monotone}.\n \\end{thmenum}\n\\end{definition}\n\n\\begin{definition}\\label{def:periodic_function_space}\\mcite[44]{Николов2020Лекции}\n We denote by \\( \\tilde{C}([a, b]) \\) the subspace of \\( C([a, b]) \\) consisting of all continuous functions in \\( [a, b] \\) which are periodic with minimal period \\( b - a \\).\n\\end{definition}\n\n\\begin{definition}\\label{def:approximation_error}\\mcite[44]{Николов2020Лекции}\n We introduce two operators.\n\n \\begin{thmenum}\n \\thmitem{def:approximation_error/algebraic} The \\term{algebraic approximation error}\n \\begin{balign*}\n E_n: C([a, b]) \\to [0, \\infty] \\\\\n E_n(f) \\coloneqq \\inf_{p \\in \\pi_n} \\norm{f - p}.\n \\end{balign*}\n\n \\thmitem{def:approximation_error/trigonometric} The \\term{trigonometric approximation error}\n \\begin{balign*}\n \\tilde{E}_n: C([a, b]) \\to [0, \\infty] \\\\\n \\tilde{E}_n(f) \\coloneqq \\inf_{p \\in \\tau_n} \\norm{f - p}.\n \\end{balign*}\n \\end{thmenum}\n\\end{definition}\n\n\\begin{theorem}[Jackson's trigonometric theorem]\\label{thm:jacksons_trigonometric_theorem}\\mcite[47]{Николов2020Лекции}\n For \\( f \\in \\tilde{C}[-\\pi, \\pi] \\) we have\n \\begin{equation*}\n \\tilde{E}_n(f) \\leq \\frac {6^{k+1}} {n^k} \\omega\\left(f^{(k)}, \\frac 1 n \\right).\n \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}[Szego’s inequality]\\label{thm:szegos_trigonometric_inequality}\\mcite[55]{Николов2020Лекции}\n For any nonnegative integer \\( n \\) and any \\( s \\in S_{\\tau_n} \\), we have\n \\begin{equation}\\label{eq:thm:szegos_trigonometric_inequality}\n [s'(\\theta)]^2 + n^2 s^2 (\\theta) \\leq n^2 \\quad\\forall \\theta \\in [-\\pi, \\pi].\n \\end{equation}\n\\end{theorem}\n\\begin{proof}\n Fix \\( n = 1, 2, \\ldots \\) and \\( \\alpha \\in [-1, 1] \\).\n\n For brevity, denote \\( c(\\theta) \\coloneqq \\cos(n \\theta) \\). Let \\( \\theta_s \\) and \\( \\theta_c \\) be numbers in \\( [-\\pi, \\pi] \\) such that\n \\begin{equation*}\n s(\\theta_s) + c(\\theta_c) = \\alpha.\n \\end{equation*}\n\n We will show that\n \\begin{equation}\\label{eq:thm:szegos_trigonometric_inequality/abs}\n \\abs{s'(\\theta_s)} \\leq \\abs{c'(\\theta_c)}\n \\end{equation}\n\n This will, in turn, imply that\n \\begin{equation*}\n [s'(\\theta_s)]^2\n \\leq\n [c'(\\theta_c)]^2\n =\n n^2 [\\sin(n \\theta_c)]^2\n \\reloset {\\ref{thm:trigonometric_identities/pythagorean_identity}}\n =\n n^2 [1 - \\cos(n \\theta_c)^2]\n =\n n^2 [1 - s(\\theta_s)]\n \\end{equation*}\n which is equivalent to \\eqref{eq:thm:szegos_trigonometric_inequality}.\n\n Now we will prove \\eqref{eq:thm:szegos_trigonometric_inequality/abs}. If \\( \\theta_c \\) is a critical point of \\( c \\), i.e. if \\( r'(\\theta_c) = 0 \\), then \\( c'(\\theta_c) = s'(\\theta_s) \\) and \\eqref{eq:thm:szegos_trigonometric_inequality/abs} holds. Suppose that \\( \\theta_c \\) is not a critical point. Denote by\n \\begin{equation*}\n \\theta_n \\coloneqq \\tfrac k n \\pi, k = -n, -n+1, \\ldots, n-2, n-1.\n \\end{equation*}\n the extrema of \\( c(\\theta) \\) in \\( [-\\pi, \\pi) \\).\n\n Define the auxiliary function\n \\begin{equation*}\n r(\\theta) \\coloneqq c(\\theta) - s(\\theta - \\theta_c + \\theta_s).\n \\end{equation*}\n\n We now have \\( r(\\theta_c) = c(\\theta_c) - s(\\theta_s) = 0 \\).\n\n Furthermore, since \\( \\norm{s} = 1 \\), then \\( \\abs{s(\\theta)} \\leq 1 \\) for all \\( \\theta \\in [-\\pi, \\pi) \\). Therefore, \\( r(\\theta_k) \\leq 0 \\) for all odd \\( k \\) and \\( r(\\theta_k) \\geq 0 \\) for all even \\( k \\).\n\n If \\( \\theta_c \\) coincides with any of the extrema \\( \\theta_k \\), then \\( r(\\theta_c) \\) holds trivially. Suppose that \\( \\theta_c \\) is between \\( \\theta_{k-1} \\) and \\( \\theta_k \\). Without loss of generality, assume that \\( k \\) is even.\n\n By the intermediate value theorem, there exists a zero of \\( r \\) between \\( r(\\theta_c) \\) and \\( r(\\theta_k) \\). If \\( r(\\theta_c) < 0 \\), then \\( \\theta_c \\) is a local minimum and hence there exists a point \\( \\theta_{c'} \\) between \\( r(\\theta_{k-1}) \\) and \\( \\theta_c \\) such that \\( r(\\theta_{c'}) = r(\\theta_c) = 0 \\). But this would imply that \\( r \\) has more than \\( 2n \\) different roots in the interval \\( [-\\pi, \\pi) \\), which is a contradiction.\n\\end{proof}\n\n\\begin{corollary}[Bernstein's trigonometric inequality]\\label{thm:bernsteins_trigonometric_inequality}\\mcite[53]{Николов2020Лекции}\n For any nonnegative integer \\( n \\) and any \\( s \\in \\tau_n \\) we have\n \\begin{equation}\\label{eq:thm:bernsteins_trigonometric_inequality}\n \\abs{s'(\\theta)} \\leq n \\norm{s} \\quad\\forall \\theta \\in [-\\pi, \\pi].\n \\end{equation}\n\\end{corollary}\n\\begin{proof}\n The case \\( n = 0 \\) is trivial. If \\( n > 0 \\), for any \\( s \\in \\tau_n \\), we can apply \\eqref{eq:thm:szegos_trigonometric_inequality} to \\( \\frac s {\\norm s} \\) to obtain\n \\begin{equation}\n [s'(\\theta)]^2 + n^2 s^2 (\\theta) \\leq \\norm{s}^2 n^2 \\quad\\forall \\theta \\in [-\\pi, \\pi].\n \\end{equation}\n\n \\eqref{eq:thm:bernsteins_trigonometric_inequality} follows directly.\n\\end{proof}\n\n\\begin{theorem}[Bernstein's trigonometric theorem]\\label{thm:bernsteins_trigonometric_theorem}\\mcite[55]{Николов2020Лекции}\n Let \\( f \\in \\tilde(C)[-\\pi, \\pi] \\) and\n \\begin{equation*}\n \\tilde{E}_n(f) \\leq \\frac A {n^{k+\\alpha}} \\quad n = 0, 1, 2, \\ldots,\n \\end{equation*}\n where \\( A \\in \\BbbR \\) and \\( \\alpha \\in (0, 1) \\).\n\n Then \\( f \\in C^{(k)}[-\\pi, \\pi] \\) and \\( f^{(k)} \\) is \\( \\alpha \\)-H\\\"older.\n\\end{theorem}\n\\begin{proof}\n Since \\( \\tilde{E}_n(f) \\) is bounded by \\( \\frac A {n^{k+\\alpha}} \\) on a compact interval, there exists a sequence \\( \\{ s_k \\}_{k=0}^\\infty \\) such that\n \\begin{equation*}\n \\norm{f - s_n} \\leq A {n^{k+\\alpha}}.\n \\end{equation*}\n\n Define the sequence of polynomials\n \\begin{equation*}\n v_j \\coloneqq \\begin{cases}\n s_1, &j = 0, \\\\\n s_{2^j} - s_{2^{j-1}} &j > 0.\n \\end{cases}\n \\end{equation*}\n\n It is now clear that\n \\begin{equation*}\n \\norm{f - \\sum_{j=0}^n v_j} \\xrightarrow[]{j \\to \\infty} 0\n \\end{equation*}\n because\n \\begin{equation*}\n \\abs{f(\\theta) - \\sum_{j=0}^n v_j (\\theta)}\n =\n \\abs{f(\\theta) - s_{2^n}(\\theta)}\n \\leq\n A {n^{k+\\alpha}}.\n \\end{equation*}\n\n For each term of the series, we have\n \\begin{equation*}\n \\abs{v_j(\\theta)}\n \\leq\n \\abs{f(\\theta) - s_{2^j}(\\theta)} + \\abs{f(\\theta) - s_{2^{j-1}}(\\theta)}\n \\leq\n \\frac A {2^{j(k + \\alpha)}} + \\frac A {2^{(j - 1) (k + \\alpha)}}.\n \\end{equation*}\n\n By setting \\( B \\coloneqq A (2^{k + \\alpha}) \\), we obtain\n \\begin{equation*}\n \\abs{v_j(\\theta)} \\leq \\frac B {2^{j(k + \\alpha)}}.\n \\end{equation*}\n\n By \\fullref{thm:bernsteins_trigonometric_inequality},\n \\begin{equation*}\n \\norm{v_j^{(r)}} \\leq 2^{jr} \\norm{v_j} \\leq \\frac B {2^{j(k - r + \\alpha)}}.\n \\end{equation*}\n\n Then\n \\begin{equation*}\n \\sum_{j=0}^\\infty v_j^{(r)} (\\theta)\n \\end{equation*}\n converges uniformly, therefore\n \\begin{equation*}\n f^{(r)}(\\theta) = \\sum_{j=0}^\\infty v_j^{(r)} (\\theta).\n \\end{equation*}\n\\end{proof}\n\n\\begin{theorem}[Bernstein's algebraic inequality]\\label{thm:bernsteins_algebraic_inequality}\\mcite[59]{Николов2020Лекции}\n For any nonnegative integer \\( n \\) and any \\( p \\in \\pi_n \\) and \\( x \\in (a, b) \\) we have\n \\begin{equation*}\n \\abs{p'(x)} \\leq n \\frac 1 {(b - a)(b - x)} \\norm{p}.\n \\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}[Bernstein's algebraic theorem]\\label{thm:bernsteins_algebraic_theorem}\\mcite[60]{Николов2020Лекции}\n Let \\( f \\in C[a, b] \\) and\n \\begin{equation*}\n E_n(f) \\leq \\frac A {n^{k+\\alpha}} \\quad n = 0, 1, 2, \\ldots,\n \\end{equation*}\n where \\( A \\in \\BbbR \\) and \\( \\alpha \\in (0, 1) \\).\n\n Then \\( f \\in C^{(k)}(a, b) \\) and \\( f^{(k)} \\) is \\( \\alpha \\)-H\\\"older in every \\( [a_1, b_1] \\) such that \\( a_1 > a \\) and \\( b_1 < b \\).\n\\end{theorem}\n", "meta": {"hexsha": "38ce11d78fc0252bf0563d4083b3960520d58a80", "size": 8300, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/bernstein_inequalities.tex", "max_stars_repo_name": "v--/anthology", "max_stars_repo_head_hexsha": "89a91b5182f187bc1aa37a2054762dd0078a7b56", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/bernstein_inequalities.tex", "max_issues_repo_name": "v--/anthology", "max_issues_repo_head_hexsha": "89a91b5182f187bc1aa37a2054762dd0078a7b56", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/bernstein_inequalities.tex", "max_forks_repo_name": "v--/anthology", "max_forks_repo_head_hexsha": "89a91b5182f187bc1aa37a2054762dd0078a7b56", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.2291666667, "max_line_length": 463, "alphanum_fraction": 0.613373494, "num_tokens": 3239, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.550607350786733, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.34675852817569885}} {"text": "\n\n \\filetitle{datdiff}{Number of periods between two dates with check for date frequency}{dates/datdiff}\n\n\t\\paragraph{Syntax}\\label{syntax}\n\n\\begin{verbatim}\nD = datdiff(D1,D2)\n\\end{verbatim}\n\n\\paragraph{Input arguments}\\label{input-arguments}\n\n\\begin{itemize}\n\\itemsep1pt\\parskip0pt\\parsep0pt\n\\item\n \\texttt{D1}, \\texttt{D2} {[} numeric {]} - IRIS dates of vectors of\n IRIS dates.\n\\end{itemize}\n\n\\paragraph{Output arguments}\\label{output-arguments}\n\n\\begin{itemize}\n\\itemsep1pt\\parskip0pt\\parsep0pt\n\\item\n \\texttt{D} {[} numeric {]} - Number of periods between \\texttt{D1} and\n \\texttt{D2}, positive for \\texttt{D1} greater than \\texttt{D2},\n negative for \\texttt{D1} smaller than \\texttt{D2}, or NaN for dates of\n different frequencies.\n\\end{itemize}\n\n\\paragraph{Description}\\label{description}\n\n\\paragraph{Example}\\label{example}\n\n\\begin{verbatim}\nd1 = mm(2010,12);\nd2 = mm(2011,12);\n\ndatdiff(d1,d2)\nans =\n -12\n\ndatdiff(d2,d1)\nans =\n 12\n\nd3 = yy(2011);\ndatdiff(d1,d3)\nans =\n NaN\n\\end{verbatim}\n\n\n", "meta": {"hexsha": "79f56736a7d6ef1c588141e53bfe7d7d44450afb", "size": 1015, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "-help/dates/datdiff.tex", "max_stars_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_stars_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-12-06T13:38:38.000Z", "max_stars_repo_stars_event_max_datetime": "2017-12-06T13:38:38.000Z", "max_issues_repo_path": "-help/dates/datdiff.tex", "max_issues_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_issues_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2017-03-28T08:13:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-02T10:40:25.000Z", "max_forks_repo_path": "-help/dates/datdiff.tex", "max_forks_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_forks_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-17T07:06:39.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-17T07:06:39.000Z", "avg_line_length": 18.7962962963, "max_line_length": 105, "alphanum_fraction": 0.7133004926, "num_tokens": 348, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073507867328, "lm_q2_score": 0.6297746004557471, "lm_q1q2_score": 0.34675852434971205}} {"text": "\\documentclass[11pt]{article}\n\\usepackage{euler}\n\\usepackage{xltxtra}\n\\usepackage{amsmath}\n%\\usepackage{amssymb}\n\\usepackage[pdfborder={0 0 0}]{hyperref}\n\\usepackage{listings}\n%\\setmainfont[Mapping=tex-text]{Ubuntu}\n%{FreeSerif}\n%{DejaVu Serif}\n\n\\date{26th of June 2012}\n\\author{Lukas Prokop}\n\\title{Software paradigms exam 27.6.2011}\n\n% Definitions by Daniel Gruß\n% https://github.com/lava2/swp/blob/master/commands.tex\n% various operators\n\\DeclareMathOperator {\\id}{id}\n\\DeclareMathOperator*{\\ggt}{ggT}\n\\DeclareMathOperator*{\\kgv}{kgV}\n\\DeclareMathOperator {\\ord}{ord}\n\\DeclareMathOperator {\\fmod}{mod}\n\\DeclareMathOperator*{\\trace}{tr}\n\\DeclareMathOperator {\\nil}{nil}\n\\DeclareMathOperator*{\\supp}{Supp}\n\\DeclareMathOperator {\\lm}{lm}\n\\DeclareMathOperator {\\lt}{lt}\n\\DeclareMathOperator {\\lc}{lc}\n\\DeclareMathOperator*{\\rest}{rest}\n\\DeclareMathOperator*{\\tmult}{\\:{mult}\\:}\n\\DeclareMathOperator*{\\tplus}{\\:{plus}\\:}\n\\DeclareMathOperator*{\\tminus}{\\:{minus}\\:}\n\\DeclareMathOperator*{\\tnull}{\\:{null}\\:}\n\\DeclareMathOperator*{\\teins}{\\:{eins}\\:}\n\\DeclareMathOperator*{\\tif}{\\:{if}\\:}\n\\DeclareMathOperator*{\\telse}{\\:{else}\\:}\n\\DeclareMathOperator*{\\tthen}{\\:{then}\\:}\n\\DeclareMathOperator*{\\tlt}{\\:{lt?}\\:}\n\\DeclareMathOperator*{\\teq}{\\:{eq?}\\:}\n\\DeclareMathOperator*{\\tumult}{\\:\\u{mult}\\:}\n\\DeclareMathOperator*{\\tuplus}{\\:\\u{plus}\\:}\n\\DeclareMathOperator*{\\tuminus}{\\:\\u{minus}\\:}\n\\DeclareMathOperator*{\\tunull}{\\:\\u{null}\\:}\n\\DeclareMathOperator*{\\tueins}{\\:\\u{eins}\\:}\n\\DeclareMathOperator*{\\tuif}{\\:\\u{if}\\:}\n\\DeclareMathOperator*{\\tuelse}{\\:\\u{else}\\:}\n\\DeclareMathOperator*{\\tuthen}{\\:\\u{then}\\:}\n\\DeclareMathOperator*{\\tult}{\\:\\u{lt?}\\:}\n\\DeclareMathOperator*{\\tueq}{\\:\\u{eq?}\\:}\n\\DeclareMathOperator {\\End}{end}\n\\DeclareMathOperator*{\\rang}{rang}\n\\DeclareMathOperator*{\\dist}{d}\n\\DeclareMathOperator*{\\real}{Re}\n\\DeclareMathOperator*{\\imag}{Im}\n\\DeclareMathOperator*{\\sign}{sign}\n\\DeclareMathOperator*{\\weight}{weight}\n\n% graphics and sectioning\n\\newcommand{\\skizze}[1]{$ $ \\begin{center}\\includegraphics[scale = 0.33]{images/#1.png}\\end{center}}\n\\newcommand{\\mychapter}[1]{\\chapter{#1}}\n\\newcommand{\\mysection}[1]{\\section{#1}}\n\\newcommand{\\mysubsection}[1]{\\subsection{#1}}\n\\newcommand{\\separate}{\\begin{center}*\\:*\\:*\\end{center}}\n\n% definition and proof tools\n\\newcommand{\\proofdone}{\\hfill \\ensuremath{\\Box}}\n\\newcommand{\\mathdone}{\\tag*{\\ensuremath{\\Box}}}\n\n\\newcommand{\\defas}{\\mathrel{\\mathop:}=}\n\\newcommand{\\asdef}{=\\mathrel{\\mathop:}}\n\n\\newcommand{\\there}{``$\\Rightarrow$'' \\;}\n\\newcommand{\\back}{``$\\Leftarrow$'' \\;}\n\n\\newcommand{\\then}{\\: \\Rightarrow \\:}\n\\newcommand{\\ifonlyif}{\\: \\Leftrightarrow \\:}\n\\renewcommand{\\u}[1]{\\underline{#1}}\n\\newcommand{\\tu}[1]{\\text{\\underline{#1}}}\n\n\\newcommand{\\tobe}{\\overset{!}{=}}\n\\newcommand{\\isit}{\\overset{?}{=}}\n\n% referencing\n\\newcommand{\\tagstar}{\\tag*{$\\star$}}\n\\newcommand{\\refstar}{\\ensuremath{(\\star)}}\n\\newcommand{\\alrefstar}{\\tag*{(\\star)}}\n\\newcommand{\\tagstarn}[1]{\\tag*{$\\star_{#1}$}}\n\\newcommand{\\refstarn}[1]{\\ensuremath{(\\star_{#1})}}\n\\newcommand{\\alrefstarn}[1]{\\tag*{(\\star_{#1})}}\n\n% mathtext\n\\newcommand{\\sonst}{\\text{sonst}}\n\\newcommand{\\const}{\\mathrm{const}}\n\n% braces\n\\newcommand{\\abs}[1]{\\left\\lvert{#1}\\right\\rvert}\n\\newcommand{\\norm}[1]{\\left\\lVert{#1}\\right\\rVert}\n\\newcommand{\\angled}[1]{\\left\\langle{#1}\\right\\rangle}\n\\newcommand{\\tangled}[1]{\\langle{#1}\\rangle}\n\\newcommand{\\braced}[1]{\\left( {#1} \\right)}\n\\newcommand{\\sbraced}[1]{\\left\\{#1\\right\\}}\n\\newcommand{\\bbraced}[1]{\\left[#1\\right]}\n\n\\newcommand{\\bbinom}[2]{\\genfrac[]{0pt}{0}{#1}{#2}}\n\\newcommand{\\cbinom}[2]{\\genfrac{\\{}{\\}}{0pt}{0}{#1}{#2}}\n\\newcommand{\\ebinom}[2]{\\genfrac{}{}{0pt}{0}{#1}{#2}}\n\n\\newcommand{\\gausslo}[1]{\\left\\lfloor {#1} \\right\\rfloor}\n\\newcommand{\\gausshi}[1]{\\left\\lceil {#1} \\right\\rceil}\n\\newcommand{\\gaussround}[1]{\\left\\lfloor {#1} \\right\\rceil}\n\n% under/overlining\n\\newcommand{\\ov}[1]{\\overline{#1}}\n\\newcommand{\\un}[1]{\\underline{#1}}\n\\newcommand{\\vek}[1]{\\vec{#1}}\n\n% relations\n\\newcommand{\\sm}{\\setminus}\n\\newcommand{\\se}{\\subseteq}\n\\newcommand{\\nse}{\\nsubseteq}\n\\newcommand{\\sne}{\\nsubseteq}\n\\newcommand{\\ise}{\\supseteq}\n\\newcommand{\\nise}{\\nsupseteq}\n\\newcommand{\\sneq}{\\subsetneqq}\n\\newcommand{\\isneq}{\\supsetneqq}\n\n\\newcommand{\\subideal}{\\trianglelefteq}\n\\newcommand{\\isubideal}{\\trianglerighteq}\n\n\\newcommand{\\divs}{\\mathrel\\vert}\n\\newcommand{\\ndivs}{\\nmid}\n\\newcommand{\\iso}{\\simeq}\n\n\\newcommand{\\nequiv}{\\not\\equiv}\n\n% sets\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\F}{\\mathbb{F}}\n\\newcommand{\\K}{\\mathbb{K}}\n\\newcommand{\\primes}{\\mathbb{P}}\n\\newcommand{\\potenz}{\\mathcal{P}}\n\n\\newcommand{\\dotcup}{\\ensuremath{\\mathaccent\\cdot\\cup}}\n\n\\newcommand{\\prob}{\\mathbb{P}}\n\\newcommand{\\erw}{\\mathbb{E}}\n\\newcommand{\\variance}{\\mathbb{V}}\n\n% greek symbols\n\\newcommand{\\fee}{\\varphi}\n\\newcommand{\\eps}{\\varepsilon}\n\n% various tools\n\\newcommand{\\where}{\\mathrel\\big\\vert}\n\\newcommand{\\limited}{\\big\\vert}\n\n% complexity theory\n\\newcommand{\\ordn}{\\mathcal{O}}\n\\newcommand{\\oh}{\\mathcal{O}}\n\n% analysis\n\\newcommand{\\de}{\\, \\dd}\n\\newcommand{\\dd}{\\mathrm{d}}\n\n\\renewcommand{\\thesubsection}{\\arabic{subsection}}\n\\newcommand{\\rbr}[1]{\\left(#1\\right)}\n\\newcommand{\\ebr}[1]{\\left[#1\\right]}\n\\newcommand{\\sbr}[1]{\\left<#1\\right>}\n\\newcommand{\\gbr}[1]{\\left\\{#1\\right\\}}\n\\renewcommand{\\det}[1]{\\text{det}\\left(#1\\right)}\n\\renewcommand{\\u}[1]{\\underline{#1}}\n\\renewcommand{\\o}[1]{\\overline{#1}}\n\\renewcommand{\\b}[1]{\\textbf{#1}}\n\n\\newcommand{\\E}{\\mathbb{E}}\n\\renewcommand{\\P}{\\mathbb{P}}\n%\\newcommand{\\I}{\\mathbb{I}}\n\\newcommand{\\V}{\\mathbb{V}}\n\\newcommand{\\dx}{\\text{d}x}\n\\newcommand{\\D}{\\mathbb{D}}\n\\newcommand{\\kA}{\\mathcal{A}}\n\\newcommand{\\kC}{\\mathcal{C}}\n\\newcommand{\\kP}{\\mathcal{P}}\n\n\\newcommand{\\Lra}{\\:\\Leftrightarrow\\:}\n\\newcommand{\\lra}{\\:\\leftrightarrow\\:}\n\\newcommand{\\Ra}{\\:\\Rightarrow\\:}\n\\newcommand{\\ra}{\\:\\rightarrow\\:}\n\\newcommand{\\La}{\\:\\Leftarrow\\:}\n\\newcommand{\\la}{\\:\\leftarrow\\:}\n\n\\newcommand{\\limi}[1]{\\lim_{#1 \\rightarrow \\infty}}\n\\newcommand{\\mceil}[1]{\\left\\lceil #1 \\right\\rceil}\n\\newcommand{\\mfloor}[1]{\\left\\lfloor #1 \\right\\rfloor}\n\n\\DeclareMathOperator{\\ggT}{ggT}\n\\DeclareMathOperator{\\kgV}{kgV}\n\n\\DeclareMathOperator{\\RR}{R}\n\\DeclareMathOperator{\\notdivides}{\\nmid}\n\n\\newcommand{\\lng}[1]{\\mathcal{#1}}\n\\newcommand{\\kwu}[1]{\\text{\\:\\u{#1}\\:}}\n\\newcommand{\\kw}[1]{\\text{\\:{#1}\\:}}\n\\newcommand{\\I}[3]{I_{\\lng{#1}}\\left(#2,\\tu{#3}\\right)}\n\\newcommand{\\Iu}[3]{I_{\\lng{#1}}\\left(#2,#3\\right)}\n\n\\newcommand{\\twedge}{\\text{$\\wedge$}}\n\\newcommand{\\tforall}{\\text{$\\forall$}}\n\\newcommand{\\texists}{\\text{$\\exists$}}\n\\newcommand{\\tvee}{\\text{$\\vee$}}\n\\newcommand{\\tto}{\\text{$\\to$}}\n\\newcommand{\\tneg}{\\text{$\\neg$}}\n% End Definitions\n\n\\DeclareMathOperator*{\\s}{s}\n\\DeclareMathOperator*{\\binStack}{binStack}\n\\DeclareMathOperator*{\\add}{add}\n\\DeclareMathOperator*{\\len}{len}\n\\newcommand{\\Null}{\\text{null}}\n\\newcommand{\\set}[1]{\\left\\{#1\\right\\}}\n\n\\lstdefinelanguage{EXP}%\n {keywords={if,then,else},%\n% emph={plus,minus,mult,add0,add1,sub,build,rest,first,second,reverse,depth,max,nth,lt?,eq?,is0?,is1?,isEmpty?,atom?},\n moreemph={null,eins,0,1,[]},% epsilon is missing\n emphstyle=\\emph,\n mathescape={true}%\n }%\n\\newcommand{\\NE}{\\text{NE:}\\hspace{5pt}}\n\\newcommand{\\NR}{\\text{NR:}\\hspace{5pt}}\n\n\\begin{document}\n\\maketitle\n\\tableofcontents\n\\newpage\n%\n\\section{Exercise 1}\n%\n%\\begin{table}\n \\begin{tabular}{lcccc}\n & S & A & B & C \\\\ \\hline\n \\textsc{FIRST} & $\\set{\\u{a}}$ & $\\set{\\u{a}}$ & $\\set{\\u{b}, \\u{c}, \\u{a}}$ & $\\set{\\u{c}, \\varepsilon}$ \\\\\n \\textsc{FOLLOW} & $\\set{\\$}$ & $\\set{\\$, \\u{b}, \\u{c}, \\u{a}}$ & $\\set{\\$, \\u{b}, \\u{c}, \\u{a}}$ & $\\set{\\u{a}}$ \\\\\n \\end{tabular}\n%\\end{table}\n%\n\\section{Exercise 2a}\n%\n%\\begin{table}\n \\begin{tabular}{lcccc}\n & $\\u{a}$ & $\\u{b}$ & $\\u{c}$ & \\$ \\\\\n \\hline\n S & $S \\to A$ & $S \\to A$ & $S \\to A$ & \\\\\n A & $A \\to AB, A \\to \\u{a}$ & & & \\\\\n B & $B \\to CA$ & $B \\to \\u{b}$ & $B \\to CA$ & \\\\\n C & $C \\to \\varepsilon$ & & $C \\to \\u{c}D, C \\to \\u{c}$ & \\\\\n D & & $D \\to \\u{b}$ & $D \\to \\u{c}$ & \\\\\n \\end{tabular}\n%\\end{table}\n\n$(A, \\u{a})$ contains a left recursion.\n$(C, \\u{c})$ contains an ambiguity.\n%\n\\section{Exercise 2b}\n%\n\\begin{align*}\n S &\\to IR \\\\\n R &\\to CIR \\\\\n R &\\to \\varepsilon \\\\[7pt]\n I &\\to \\u{a}J \\\\\n J &\\to \\u{b}J \\\\\n J &\\to \\varepsilon \\\\[7pt]\n C &\\to \\u{c}D \\\\\n C &\\to \\varepsilon \\\\\n D &\\to \\u{c} \\\\\n D &\\to \\u{b} \\\\\n D &\\to \\varepsilon\n\\end{align*}\n%\n%\\begin{table}\n \\begin{center}\n \\begin{tabular}{lcccc}\n & $\\u{a}$ & $\\u{b}$ & $\\u{c}$ & \\$ \\\\\n \\hline\n S & $S \\to IR$ & & & \\\\\n R & $R \\to CIR$ & & $R \\to CIR$ & $R \\to \\epsilon$ \\\\\n I & $I \\to \\u{a}J$ & & & \\\\\n J & $J \\to \\varepsilon$ & $J \\to \\u{b}J$ & $J \\to \\varepsilon$ & $J \\to \\varepsilon$ \\\\\n C & $C \\to \\varepsilon$ & & $C \\to \\u{c}D$ & \\\\\n D & $D \\to \\varepsilon$ & $D \\to \\u{b}$ & $D \\to \\u{c}$ & \\\\\n \\end{tabular}\n \\end{center}\n%\\end{table}\n%\n\\section{Exercise 2c}\n%\n\\begin{center}\n \\begin{tabular}{lll}\n \\textsc{Input} & \\textsc{Stack} & \\textsc{Comment} \\\\\n $\\u{\\$abcbba}$ & $S$ & $S \\to IR$ \\\\\n $\\u{\\$abcbba}$ & $RI$ & $I \\to \\u{a}J$ \\\\\n $\\u{\\$abcbba}$ & $RJ\\u{a}$ & $J \\to \\u{b}J$ \\\\\n $\\u{\\$abcbb}$ & $RJ\\u{b}$ & $J \\to \\u{b}J$ \\\\\n $\\u{\\$abcb}$ & $RJ\\u{b}$ & $J \\to \\varepsilon$ \\\\\n $\\u{\\$abc}$ & $R$ & $R \\to CIR$ \\\\\n $\\u{\\$abc}$ & $RIC$ & $C \\to \\u{c}D$ \\\\\n $\\u{\\$abc}$ & $RID\\u{c}$ & $D \\to \\u{b}$ \\\\\n $\\u{\\$ab}$ & $RI\\u{b}$ & $I \\to \\u{a}J$ \\\\\n $\\u{\\$a}$ & $RJ\\u{a}$ & $J \\to \\varepsilon$ \\\\\n $\\u{\\$}$ & $R$ & $R \\to \\varepsilon$ \\\\\n $\\u{\\$}$ & & accepted \\\\\n \\end{tabular}\n\\end{center}\n%\n\\section{Exercise 3}\n%\n\\lstset{language=EXP}\n\\begin{lstlisting}\nfactorial(x) =\n if eq?(x, 0) then eins\n else if eq?(x, eins) then eins\n else mult(factorial(minus(x, eins)), x)\n\\end{lstlisting}\n\nProof by complete induction.\n\n\\subsection{Induction hypothesis}\n\\[\n \\forall \\omega, \\omega(\\u{x}) \\leq n, \\omega(\\u{x}) \\in ℕ:\n I(\\delta, \\omega, \\u{factorial(x)}) = \\omega(\\u{x})!\n\\]\n\n\\subsection{Induction base 0}\n\\[\n \\omega(\\u{x}) = 0\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{if eq?(x, 0) then eins else if eq?}}\\ldots)\n\\] \\[\n \\NR I(\\delta, \\omega, \\u{\\text{eq?(x, 0)}})\n = \\operatorname{eq?}(I(\\delta, \\omega, \\u{x}), I(\\delta, \\omega, \\u{0}))\n = \\operatorname{eq?}(\\omega(\\u{x}), 0)\n = T\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{eins}}) = 1 = 0!\n\\]\n\n\\subsection{Induction base 1}\n%\n\\[\n \\omega(\\u{x}) = 1\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{if eq?(x, 0) then eins else if eq?}}\\ldots)\n\\] \\[\n \\NR I(\\delta, \\omega, \\u{\\text{eq?(x, 0)}})\n = \\operatorname{eq?}(I(\\delta, \\omega, \\u{x}), I(\\delta, \\omega, \\u{0}))\n = \\operatorname{eq?}(\\omega(\\u{x}), 0)\n = F\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{if eq?(x, 1) then eins else mult}}\\ldots)\n\\] \\[\n \\NR I(\\delta, \\omega, \\u{\\text{eq?(x, eins)}})\n = \\operatorname{eq?}(I(\\delta, \\omega, \\u{x}), I(\\delta, \\omega, \\u{\\text{eins}}))\n = \\operatorname{eq?}(\\omega(\\u{x}), 1)\n = T\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{eins}}\\ldots) = 1 = 1!\n\\]\n\n\\subsection{Induction step}\n%\n\\[\n \\omega(\\u{x}) = n + 1 \\qquad n \\geq 2\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{if eq?(x, 0) then eins else if eq?}}\\ldots)\n\\] \\[\n \\NR I(\\delta, \\omega, \\u{\\text{eq?(x, 0)}})\n = \\operatorname{eq?}(I(\\delta, \\omega, \\u{x}), I(\\delta, \\omega, \\u{0}))\n = \\operatorname{eq?}(\\omega(\\u{x}), 0)\n = F\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{if eq?(x, 1) then eins else mult}}\\ldots)\n\\] \\[\n \\NR I(\\delta, \\omega, \\u{\\text{eq?(x, eins)}})\n = \\operatorname{eq?}(I(\\delta, \\omega, \\u{x}), I(\\delta, \\omega, \\u{\\text{eins}}))\n = \\operatorname{eq?}(\\omega(\\u{x}), 1)\n = F\n\\] \\[\n I(\\delta, \\omega, \\u{\\text{mult(factorial(minus(x, eins)), x)}})\n = \\operatorname{mult}(I(\\delta, \\omega, \\u{\\text{factorial(minus(x, eins))}}), I(\\delta, \\omega, \\u{x}))\n\\] \\[\n \\NE \\omega'(\\u{x})\n = I(\\delta, \\omega, \\u{\\text{minus(x, eins)}})\n = \\operatorname{minus}(I(\\delta, \\omega, \\u{x}), I(\\delta, \\omega, \\u{\\text{eins}}))\n\\] \\[\n = \\operatorname{minus}(\\omega(\\u{x}), 1)\n = \\operatorname{minus}(n + 1, 1) = n\n\\] \\[\n \\operatorname{mult}(n!, n+1) = (n+1)! \\qquad \\text{ corresponds to hypothesis}\n\\]\n\n%\n\\section{Exercise 4a}\n%\n\\lstset{language=prolog}\n\\begin{lstlisting}\nodd(binStack(V, R)) :- odd(R).\nodd(binStack(s(0), null)).\n\\end{lstlisting}\n%\n\\section{Exercise 4b}\n%\nTables~\\ref{tab:4b-def} and~\\ref{tab:4b-deriv}.\n\\begin{table}[h]\n \\begin{tabular}{cl}\n Q & $\\neg \\len(\\binStack(0, \\binStack(0, \\binStack(0, \\Null))), \\s(\\s(\\s(0))))$ \\\\\n C1 & $\\add(X, 0, X)$ \\\\\n C2 & $\\add(X, \\s(Y), \\s(Z)) \\lor \\neg \\add(X, Y, Z)$ \\\\\n C3 & $\\len(\\Null, 0)$ \\\\\n C4 & $\\len(\\binStack(V, R), E) \\lor \\neg \\len(R, S) \\lor \\neg \\add(\\s(0), S, E)$\n \\end{tabular}\n \\caption{Rules of exercise 4b in logical notation}\n \\label{tab:4b-def}\n\\end{table}\n%\n\\begin{table}[h]\n \\begin{tabular}{cl}\n Q & $\\neg \\len(\\binStack(0, \\binStack(0, \\binStack(0, \\Null))), \\s(\\s(\\s(0))))$ \\\\\n C4 & $\\len(\\binStack(V, R), E) \\lor \\neg \\len(R, S) \\lor \\neg \\add(\\s(0), S, E)$ \\\\\n $\\Theta$ & $\\{ V = 0, R = \\binStack(0, \\binStack(0, null)), E = \\s(\\s(\\s(0))) \\}$ \\\\\n R & $\\neg \\len(\\binStack(0, \\binStack(0, \\Null)), S_1) \\lor \\neg \\add(\\s(0), S_1, \\s(\\s(\\s(0))))$ \\\\[10pt]%\n\n C4 & $\\len(\\binStack(V, R), E) \\lor \\neg \\len(R, S) \\lor \\neg \\add(\\s(0), S, E)$ \\\\\n $\\Theta$ & $\\{ V_2 = 0, R_2 = \\binStack(0, \\Null), E_2 = S_1 \\}$ \\\\\n R & $\\neg \\len(\\binStack(0, \\Null), S_2) \\lor \\neg \\add(\\s(0), S_2, S_1)\n \\lor \\neg \\add(\\s(0), S_1, \\s(\\s(\\s(0))))$ \\\\[10pt]%\n\n C3 & $\\len(\\Null, 0)$ \\\\\n $\\Theta$ & $\\{ S_3 = 0 \\}$ \\\\\n R & $\\neg \\add(\\s(0), 0, S_2) \\lor \\neg \\add(\\s(0), S_2, S_1) \\lor \\neg \\add(\\s(0), S_1, \\s(\\s(\\s(0))))$ \\\\[10pt]%\n\n C1 & $\\add(X, 0, X)$ \\\\\n $\\Theta$ & $\\{ S_2 = \\s(0), X = S_2 \\}$ \\\\\n R & $\\neg \\add(\\s(0), \\s(0), S_1) \\lor \\neg \\add(\\s(0), S_1, \\s(\\s(\\s(0))))$ \\\\[10pt]%\n\n C2 & $\\add(X, \\s(Y), \\s(Z)) \\lor \\neg \\add(X, Y, Z)$ \\\\\n $\\Theta$ & $\\{ X_2 = \\s(0), Y_2 = 0, Z_2 = A, S_1 = \\s(A) \\}$ \\\\\n R & $\\neg \\add(\\s(0), 0, A) \\lor \\neg \\add(\\s(0), \\s(A), \\s(\\s(\\s(0))))$ \\\\[10pt]%\n\n C1 & $\\add(X, 0, X)$ \\\\\n $\\Theta$ & $\\{ X_3 = \\s(0), A = X_3 \\}$ \\\\\n R & $\\neg \\add(\\s(0), \\s(\\s(0)), \\s(\\s(\\s(0)))$ \\\\[10pt]%\n\n C2 & $\\add(X, \\s(Y), \\s(Z)) \\lor \\neg \\add(X, Y, Z)$ \\\\\n $\\Theta$ & $\\{ X_4 = \\s(0), Y_3 = \\s(0), Z_3 = \\s(\\s(0)) \\}$ \\\\\n R & $\\neg \\add(\\s(0), \\s(0), \\s(\\s(0)))$ \\\\[10pt]%\n\n C2 & $\\add(X, \\s(Y), \\s(Z)) \\lor \\neg \\add(X, Y, Z)$ \\\\\n $\\Theta$ & $\\{ X_5 = \\s(0), Y_4 = 0, Z_4 = \\s(0) \\}$ \\\\\n R & $\\neg \\add(\\s(0), 0, \\s(0))$ \\\\[10pt]%\n\n C1 & $\\add(X, 0, X)$ \\\\\n $\\Theta$ & $\\{ X_6 = \\s(0) \\}$ \\\\\n R & empty, query got successfully derivated. \\\\\n \\end{tabular}\n \\caption{Derivation of the query in exercise 4b}\n \\label{tab:4b-deriv}\n\\end{table}\n%\n\\section{Exercise 5}\n%\n\\[\n \\pi[\\text{add}](\\pi(a), \\pi(b)) = \\operatorname{add}(a, b)\n\\]\n%\n\\lstset{language=EXP}\n\\begin{lstlisting}\nadd(a, b) =\n if eq?(mod(a, 2), 1) then\n if eq?(mod(b, 2), 1) then\n sub(add(a, b), 1)\n else\n if gt?(a, b) then\n sub(a, b)\n else\n add(sub(b, a), 1)\n else\n if eq?(mod(b, 2), 1) then\n if gt?(a, b) then\n add(sub(a, b), 1)\n else\n sub(b, a)\n else\n add(a, b)\n\\end{lstlisting}\n\n\\begin{enumerate}\n \\item Sign of first parameter is positive, second parameter the same\n \\item Sign of first parameter is positive, second parameter is negative\n \\item Sign of first parameter is negative, second parameter is positive\n \\item Sign of first parameter is negative, second parameter the same\n\\end{enumerate}\n\n\\textbf{Case both parameters are positive:} \\\\\nIf both parameters are positive, the modulo value of both values by 2 is 1.\nTherefore the very last line of the function is evaluated. For decoded\nvalues the result is $a+b$, which equals to $\\pi(a) + \\pi(b)$ for encoded\nvalues as far as the sum of two even numbers is even (which represents\na positive number).\n\n\\end{document}\n", "meta": {"hexsha": "0fdafae0b8bc04d4e56cb11720a567c1e4007f14", "size": 16495, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "pub/swp_prev_exam.tex", "max_stars_repo_name": "meisterluk/tug_lp", "max_stars_repo_head_hexsha": "eaf7e0a9bfaa91400248f7231c6891531ee71275", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "pub/swp_prev_exam.tex", "max_issues_repo_name": "meisterluk/tug_lp", "max_issues_repo_head_hexsha": "eaf7e0a9bfaa91400248f7231c6891531ee71275", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "pub/swp_prev_exam.tex", "max_forks_repo_name": "meisterluk/tug_lp", "max_forks_repo_head_hexsha": "eaf7e0a9bfaa91400248f7231c6891531ee71275", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.9052224371, "max_line_length": 120, "alphanum_fraction": 0.5624734768, "num_tokens": 6675, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.596433160611502, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3467084396262222}} {"text": "% from latexml\n\\documentclass{article}\n\\usepackage{xcolor}\n\\usepackage{cancel}\n\\usepackage{siunitx}\n\\usepackage{float}\n\\usepackage{multirow}\n\\usepackage{booktabs}\n\n\\DeclareSIQualifier\\polymer{pol}\n\\DeclareSIQualifier\\catalyst{cat}\n\\DeclareSIQualifier\\isotropic{i} %?\n\n\\DeclareSIUnit\\abbra{ a }\n\\DeclareSIUnit\\abbrb{ \\abbra } % abbreviation\n\\DeclareSIUnit\\abbrc{ \\abbrb } % abbreviation twice\n\\DeclareSIUnit\\abbrd{ \\abbre } % before defined\n\\DeclareSIUnit\\abbre{ e }\n\n\\def\\mabbra{a}\n\\DeclareSIUnit\\mabbrb{\\mabbra}\n\n\\DeclareSIPrefix\\killer{\\kilo}{3}\n\n\\usepackage{amsmath}\n\\begin{document}\n{\n\\color{red}\nSome text \\\\\n\\SI{4}{\\metre\\per\\sievert} \\\\\nMore text \\\\\n\\SI[color = blue]{4}{\\metre\\per\\sievert} \\\\\nStill red here!\n\\numlist[color = blue]{1;2;3;4}\\\\\nStill red here!\\\\\n}\n\n\\si{m^2.s}\\\\\n\\si{\\mu m^2}\\\\\n\n\\SI{0.094}{\\pi . \\milli\\meter . \\milli\\radian}\\\\\n\n\\SI{0.094}{\\frac{1}{3} . \\milli\\meter . \\milli\\radian}\\\\\n\n\\SI{0.094}{\\pi \\per \\milli\\meter . \\milli\\radian\\tothe{3}}\\\\\n\\section{Numbers}\n\\subsection{General}\n%\\num{\\sqrt{2}}\\\\\n%\\num{{a word}}\\\\\n\\num{12345,67890}\\\\\n\\num{1+-2i}\\\\\n\\num{.3e45}\\\\\n\\num{1.654 x 2.34 x 3.430}\\\\\n\\num{\\pi}\\\\\n\\num{2\\pi}\\\\\n\\num{\\pi/3}\\\\\n\n\\num{123}\\\\\n\\num{1234}\\\\\n\\num{12345}\\\\\n\\num{0.123}\\\\\n\\num{0,1234}\\\\\n\\num{.12345}\\\\\n\\num{3.45d-4}\\\\\n\\num{-e10}\\\\\n\n\\num{123e4}\\\\\n%\\num{123e4(3)}\\\\\n\\num{123(3)e4}\\\\\n\n\\num{123\\pm2}\\\\\n\\num{123\\pm2i}\\\\\n\\num{123+234i}\\\\\n%\\num{123+234}\\\\\n%\\num{123e2+234e3i}\\\\\n\\num{123+234ie3}\\\\\n\\num{123(1)+234(1)ie3}\\\\\n%\\num{123e2(1)+234e3i(1)}\\\\\n%\\num{+234(1)ie3}\\\\ %comes out weird?\n\\num{+3i}\\\\\n\\num{+3ie4}\\\\\n\nPretty nonsensical stuff?\n\\num{1.\\pi e+3}\\\\\n\\def\\dig{1234}\\\\\n\\num{\\dig.\\dig}\\\\\n\\def\\odd{\\xi}\\\\\n\\def\\odder{\\odd}\\\\\n\\num[input-symbols=\\xi]{3\\xi}\\\\\n\\num[input-symbols=\\xi]{3\\odd}\\\\\n%\\num[input-symbols=\\odd]{3\\odd}\\\\\n\\num[input-symbols=\\odd, input-protect-tokens=\\odd]{3\\odd}\\\\\n\\num[input-symbols=\\odd, input-protect-tokens=\\odd]{3\\odder}\\\\\n\\num[input-symbols=\\odder, input-protect-tokens=\\odder]{3\\odder}\\\\\n%\\num[input-symbols=\\odder, input-protect-tokens=\\odd]{3\\odder}\\\\\n%\\num[input-symbols=\\odd, input-protect-tokens=\\odder]{3\\odder}\\\\\n\n\\num{1.23(1)}\\\\\n%\\num{1.23(0.01)}\n\\num{1.23\\pm0.01}\\\\\n%\\num{1.23(1.0)}\\\\\n%\\num{1.23(\\pm1)}\\\\\n\\num{1.23(\\pi)}\\\\\n\n\\subsection{Parsing numbers}\n\n\\subsubsection{input-digits, input-decimal-markers, input-signs, input-exponent-markers}\n\n\\subsubsection{input-symbols, input-ignore}\n\\subsubsection{input-comparators}\n\n\\num{< 10} \\\\\n%\\SI{>> 5}{\\metre} \\\\\n\\num{\\le 0.12}\\\\\n\n\\subsubsection{input-open-uncertainty, input-close-uncertainty, input-uncertainty-signs}\n\\num{9.99(9)}\\\\\n\\num{9.99 +- 0.09}\\\\\n\\num{9.99 \\pm 0.09}\\\\\n\\num{123 +- 4.5}\\\\\n\\num{12.3 +- 6}\\\\\n\n\\subsubsection{input-complex-roots}\n\\num{9.99 + 88.8i} \\\\\n\\num{9.99 + i88.8}\\\\\n\n\\subsubsection{input-protect-tokens}\n\n\\subsubsection{parse-numbers}\n\\num[parse-numbers = false]{\\sqrt{2}}\\\\\n\n\\subsection{Post-processing numbers}\n\\subsubsection{round-mode, round-precision}\n{\n\\num{1.23456} \\\\\n\\num{14.23} \\\\\n\\num{0.12345(9)} \\\\\n\\sisetup{\n round-mode = places,\n round-precision = 3\n}%\n\\num{1.23456} \\\\\n\\num{14.23} \\\\\n\\num{0.12345(9)} \\\\\n\\sisetup{\n round-mode = figures,\n round-precision = 3\n}%\n\\num{1.23456} \\\\\n\\num{14.23} \\\\\n\\num{0.12345(9)}\\\\\n}\n\\subsubsection{round-integer-to-decimal}\n{\n\\num[round-mode = figures]{1} \\\\\n\\num[round-mode = places]{1} \\\\\n\\sisetup{round-integer-to-decimal}\n\\num[round-mode = figures]{1} \\\\\n\\num[round-mode = places]{1}\\\\\n}\n\n\\subsubsection{round-minimum}\n{\n\\sisetup{round-mode = places}%\n\\num{0.0055} \\\\\n\\num{0.0045} \\\\\n\\sisetup{round-minimum = 0.01}%\n\\num{0.0055} \\\\\n\\num{0.0045}\\\\\n}\n\n\\subsubsection{round-half}\n{\n\\sisetup{round-mode = places, round-half = up}%\n\\num{0.055} \\\\\n\\num{0.045} \\\\\n\\sisetup{round-half = even}%\n\\num{0.055} \\\\\n\\num{0.045}\n}\n\n\\subsubsection{add-decimal-zero, add-integer-zero}\n{\n\\num{123.} \\\\\n\\num{456} \\\\\n\\num{.789} \\\\\n\\sisetup{\n add-decimal-zero = false,\n add-integer-zero = false,\n}%\n\\num{123.} \\\\\n\\num{456} \\\\\n\\num{.789}\\\\\n}\n\n\\subsubsection{minimum-integer-digits}\n\\num{123} \\\\\n\\num[minimum-integer-digits = 1]{123} \\\\\n\\num[minimum-integer-digits = 2]{123} \\\\\n\\num[minimum-integer-digits = 3]{123} \\\\\n\\num[minimum-integer-digits = 4]{123}\\\\\n\n\\subsubsection{explicit-sign, retain-explicit-sign}\n\\num{+345} \\\\\n\\num[retain-explicit-plus]{+345} \\\\\n\\num[explicit-sign = -]{345}\\\\\n\\num[explicit-sign = -]{+345}\\\\\n\n\\subsubsection{retain-unity-mantissa, retain-zero-exponent}\n\\num{1e4} \\\\\n\\num[retain-unity-mantissa = false]{1e4} \\\\\n\\num{444e0} \\\\\n\\num[retain-zero-exponent = true]{444e0}\\\\\n\n\\subsubsection{scientific-notation, fixed-exponent}\n{\n\\num{0.001}\\\\\n\\num{0.0100} \\\\\n\\num{1200}\\\\\n\\sisetup{scientific-notation = true}%\n\\num{0.001}\\\\\n\\num{0.0100} \\\\\n\\num{1200}\\\\\n\\sisetup{scientific-notation = engineering}%\n\\num{0.001}\\\\\n\\num{0.0100} \\\\\n\\num{1200}\\\\\n\\sisetup{\nfixed-exponent = 2,\nscientific-notation = fixed,\n}%\n\\num{0.001}\\\\\n\\num{0.0100} \\\\\n\\num{1200}\\\\\n}\n\n\\subsubsection{omit-uncertainty}\n\\num{0.01(2)} \\\\\n\\num[omit-uncertainty]{0.01(2)}\\\\\n\n\\subsection{Printing numbers}\n\\subsubsection{group-digits, group-four-digits,group-seperator}\n\\num{12345.67890}\\\\\n\\num[group-digits= false]{12345.67890}\\\\\n\\num[group-digits= decimal]{12345.67890} \\\\\n\\num[group-digits= integer]{12345.67890}\\\\\n\n\\num[group-digits= false]{12345.67890}\\\\\n\\num[group-digits= decimal]{12345.67890} \\\\\n\\num[group-digits= integer]{12345.67890}\\\\\n\n\\num{1234567890.1234567890}\\\\\n\\num[group-four-digits]{1234567890.1234567890}\\\\\n\n\\num{12345} \\\\\n\\num[group-separator = {,}]{12345} \\\\\n\\num[group-separator = \\text{~}]{12345}\\\\\n\n\\subsubsection{group-minimum-digits}\n\\num{1234} \\\\\n\\num[group-minimum-digits = 4]{1234} \\\\\n\\num{1234.5678} \\\\\n\\num[group-minimum-digits = 4]{1234.5678}\\\\\n\n\\subsubsection{output-complex-root,output-decimal-marker,copy-complex-root,copy-decimal-marker}\n\\num{1.23} \\\\\n\\num[output-decimal-marker = {,}]{1.23} \\\\\n\\num{1+2i} \\\\\n\\num[output-complex-root = \\text{\\ensuremath{i}}]{1+2i} \\\\\n\\num[output-complex-root = j]{1+2i} \\\\\n\\num[copy-complex-root]{1+2j} \\\\\n\\num[copy-decimal-marker]{555,555}\\\\\n\n\\subsubsection{complex-root-position}\n\\num{67-0.9i} \\\\\n\\num[complex-root-position = before-number]{67-0.9i} \\\\\n\\num[complex-root-position = after-number]{67-0.9i}\\\\\n\n\\subsubsection{exponent-base, exponent-product}\n\\num[exponent-product = \\times]{1e2} \\\\\n\\num[exponent-product = \\cdot]{1e2} \\\\\n\\num[exponent-base = 2]{1e2}\\\\\n\n\\subsubsection{output-exponent-marker}\n\\num[output-exponent-marker = \\text{e}]{1e2} \\\\\n\\num[output-exponent-marker = \\ensuremath{\\mathrm{E}}]{1e2}\\\\\n\n\\subsubsection{separate-uncertainty,uncertainty-separator,output-open-uncertainty,output-close-uncertain\nty}\n{\n\\num{1.234(5)} \\\\\n\\num{1.234\\pm 0.005} \\\\\n\\num[separate-uncertainty = true]{1.234(5)} \\\\\n\\num[separate-uncertainty = true]{1.234\\pm 0.005} \\\\\n\\sisetup{\n output-open-uncertainty = [,\n output-close-uncertainty = ],\n uncertainty-separator= {\\,}\n}\n\\num{1.234(5)}\\\\\n}\n\\num{8.2(13)} \\\\\n\\num{8.2\\pm1.3} \\\\\n\\num[separate-uncertainty]{8.2(13)}\\\\\n\\num[separate-uncertainty]{8.2\\pm1.3} \\\\\n\n\\num{1.234(5) x \\pi} \\\\\n\\num[separate-uncertainty = true]{1.234(5) x \\pi} \\\\\n\n\\num{1.2 +- 0.001}\\\\\n\\num[separate-uncertainty]{1.2 +- 0.001}\\\\\n\n\\subsubsection{bracket-numbers, open-bracket, close-bracket}\n{\n\\num{1 e10} \\\\\n\\num{2i e10} \\\\\n\\num{1+2i e10} \\\\\n\\num[bracket-numbers = false]{1+2i e10} \\\\\n\\sisetup{\n open-bracket = \\{,\n close-bracket = \\},\n}\n\\num{1+2i e10}\\\\\n}\n\n\\subsubsection{negative-color}\n\\num{-15673} \\\\\n\\num[negative-color = red]{-15673}\n\n\\subsubsection{bracket-negative-numbers}\n\\num{-15673} \\\\\n\\num[bracket-negative-numbers]{-15673} \\\\\n%\\SI{-10}{\\metre} \\\\\n%\\SI[bracket-negative-numbers]{-10}{\\metre}\n\n\\subsection{Multi-part Numbers}\n\n\\subsubsection{input-product,input-quotient}\n\\num{1 x 2 x 3} \\\\\n\\num{1e4 x 2(3) x 3/4} \\\\\n\\num[input-product=*]{4 * 5 * 6} \\\\\n\\num{ 1 / 2e4 } \\\\\n\\num{ 1e2 / 3e4 }\\\\\n\n\\subsubsection{output-product, output-quotient}\n\\num[output-product = \\cdot]{4.87 x 5.321 x 6.90545} \\\\\n\\num[output-quotient = \\text{ div }]{1 / 2}\\\\\n\n\\subsubsection{quotient-mode}\n\\num{1 / 2e4} \\\\\n\\num[quotient-mode = fraction]{1 / 2e4}\\\\\n\n\\subsubsection{fraction-function}\n{\n\\sisetup{quotient-mode= fraction}\n\\num{1 / 1}\\\\\n\\num[fraction-function= \\dfrac]{1 / 2}\\\\\n%\\num[fraction-function= \\sfrac]{1 / 3}\\\\\n\\num[fraction-function= \\tfrac]{1 / 4}\\\\\n}\n\n\\subsection{Lists and ranges of numbers}\n\\subsubsection{list-final-separator,list-pair-separator,list-separator}\n\\numlist{0.1;0.2;0.3}\\\\\n\\numlist[list-separator = {; }]{0.1;0.2;0.3}\\\\\n\\numlist[list-final-separator = {, }]{0.1;0.2;0.3} \\\\\n\\numlist[\nlist-separator\n= { and },\nlist-final-separator = { and finally }\n]{0.1;0.2;0.3} \\\\\n\\numlist{0.1;0.2} \\\\\n\\numlist[list-pair-separator = {, and }]{0.1;0.2}\\\\\n\n\\subsection{range-phrase}\n\n\\numrange{5}{100} \\\\\n\\numrange[range-phrase = --]{5}{100}\\\\\n\n\\subsection{Angles}\n\\subsubsection{number-angle-product}\n\\ang{2.67} \\\\\n\\ang[number-angle-product = \\,]{2.67}\n\\subsubsection{arc-separator}\n\\ang{6;7;6.5} \\\\\n\\ang[arc-separator = \\,]{6;7;6.5}\n\\subsubsection{add-arc-degree-zero,add-arc-minute-zero,add-arc-second-zero}\n\\ang{-1;;} \\\\\n\\ang{;-2;} \\\\\n\\ang{;;-3} \\\\\n{\n\\sisetup{add-arc-degree-zero}\n\\ang{-1;;} \\\\\n\\ang{;-2;} \\\\\n\\ang{;;-3} \\\\\n}\n{\n\\sisetup{add-arc-minute-zero}\n\\ang{-1;;} \\\\\n\\ang{;-2;} \\\\\n\\ang{;;-3} \\\\\n}\n{\n\\sisetup{add-arc-second-zero}\n\\ang{-1;;} \\\\\n\\ang{;-2;} \\\\\n\\ang{;;-3}\\\\\n}\n\\ang[add-arc-minute-zero,add-arc-second-zero]{45.697}\\\\\n\n\\subsubsection{angle-symbol-over-decimal}\n\\ang{45.697} \\\\\n\\ang{6;7;6.5} \\\\\n\\ang[angle-symbol-over-decimal]{45.697} \\\\\n\\ang[angle-symbol-over-decimal]{6;7;6.5}\n\n% \\ang{10}\\\\\n% \\ang{12.3}\\\\\n% \\ang{4,5}\\\\\n% \\ang{1;2;3}\\\\\n% \\ang{;;1}\\\\\n% \\ang{+10;;}\\\\\n% \\ang{-0;1;}\\\\\n\n\n\\section{Units}\n\\def\\showunit#1{#1 & \\texttt{$\\backslash$#1} & \\si{\\csname#1\\endcsname}}\n\\begin{table}\n\\caption{SI base units}\n\\centering\n\\begin{tabular}{lll}\\hline\\hline\nUnit & Macro & Symbol \\\\\\hline\n\\showunit{ampere}\\\\\n\\showunit{candela}\\\\\n\\showunit{kelvin}\\\\\n\\showunit{kilogram}\\\\\n\\showunit{metre}\\\\\n\\showunit{mole}\\\\\n\\showunit{second}\\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Coherent derived units}\n\\centering\n\\begin{tabular}{llllll}\\hline\\hline\nUnit & Macro & Symbol &\nUnit & Macro & Symbol \\\\\\hline\n\\showunit{becquerel} & \\showunit{newton} \\\\\n\\showunit{degreeCelsius} & \\showunit{ohm} \\\\\n\\showunit{coulomb} & \\showunit{pascal} \\\\\n\\showunit{farad} & \\showunit{radian} \\\\\n\\showunit{gray} & \\showunit{siemens} \\\\\n\\showunit{hertz} & \\showunit{sievert} \\\\\n\\showunit{henry} & \\showunit{steradian} \\\\\n\\showunit{joule} & \\showunit{tesla} \\\\\n\\showunit{katal} & \\showunit{volt} \\\\\n\\showunit{lumen} & \\showunit{watt} \\\\\n\\showunit{lux} & \\showunit{weber} \\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Non-SI units}\n\\centering\n\\begin{tabular}{lll}\\hline\\hline\nUnit & Macro & Symbol \\\\\\hline\n\\showunit{day}\\\\\n\\showunit{degree}\\\\\n\\showunit{hectare}\\\\\n\\showunit{hour} \\\\\n\\showunit{litre}\\\\\n\\showunit{liter}\\\\\n\\showunit{arcminute}\\\\\n\\showunit{minute}\\\\\n\\showunit{arcsecond}\\\\\n\\showunit{tonne}\\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Expermental Non-SI units}\n\\centering\n\\begin{tabular}{lll}\\hline\\hline\nUnit & Macro & Symbol \\\\\\hline\n\\showunit{astronomicalunit}\\\\\n\\showunit{atomicmassunit}\\\\\n\\showunit{bohr}\\\\\n\\showunit{clight}\\\\\n\\showunit{dalton}\\\\\n\\showunit{electronmass}\\\\\n\\showunit{electronvolt}\\\\\n\\showunit{elementarycharge}\\\\\n\\showunit{hartree}\\\\\n\\showunit{planckbar}\\\\\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Other non-SI units}\n\\centering\n\\begin{tabular}{lll}\\hline\\hline\nUnit & Macro & Symbol \\\\\\hline\n\\showunit{angstrom}\\\\\n\\showunit{bar}\\\\\n\\showunit{barn}\\\\\n\\showunit{bel}\\\\\n\\showunit{decibel}\\\\\n\\showunit{knot}\\\\\n\\showunit{mmHg}\\\\\n\\showunit{nauticalmile}\\\\\n\\showunit{neper}\\\\\n\\end{tabular}\n\\end{table}\n\n\\def\\showprefix#1{#1 & \\texttt{$\\backslash$#1} & \\si{\\csname#1\\endcsname} & \\si[prefixes-as-symbols=false]{\\csname#1\\endcsname}}\n\\begin{table}\n\\caption{Other non-SI units}\n\\centering\n\\begin{tabular}{llllllll}\\hline\\hline\nUnit & Macro & Symbol & Power\nUnit & Macro & Symbol & Power \\\\\\hline\n\\showprefix{yocto} & \\showprefix{deca}\\\\\n\\showprefix{zepto} & \\showprefix{hecto}\\\\\n\\showprefix{atto} & \\showprefix{kilo}\\\\\n\\showprefix{femto} & \\showprefix{mega}\\\\\n\\showprefix{pico} & \\showprefix{giga}\\\\\n\\showprefix{nano} & \\showprefix{tera}\\\\\n\\showprefix{micro} & \\showprefix{peta}\\\\\n\\showprefix{milli} & \\showprefix{exa}\\\\\n\\showprefix{centi} & \\showprefix{zetta}\\\\\n\\showprefix{deci} & \\showprefix{yotta}\\\\\n\\end{tabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Abbreviated units}\n\\centering\n\\begin{tabular}{lll}\\hline\\hline\nUnit & Macro & Symbol \\\\\\hline\n\\showunit{fg}\\\\\n\\showunit{pg}\\\\\n\\showunit{ng}\\\\\n\\showunit{ug}\\\\\n\\showunit{mg}\\\\\n\\showunit{g}\\\\\n\\showunit{kg}\\\\\n\\showunit{amu}\\\\\\hline\n\\showunit{pm}\\\\\n\\showunit{nm}\\\\\n\\showunit{um}\\\\\n\\showunit{mm}\\\\\n\\showunit{cm}\\\\\n\\showunit{dm}\\\\\n\\showunit{m}\\\\\n\\showunit{km}\\\\\\hline\n\\showunit{as}\\\\\n\\showunit{fs}\\\\\n\\showunit{ps}\\\\\n\\showunit{ns}\\\\\n\\showunit{us}\\\\\n\\showunit{ms}\\\\\n\\showunit{s}\\\\\\hline\n\\showunit{fmol}\\\\\n\\showunit{pmol}\\\\\n\\showunit{nmol}\\\\\n\\showunit{umol}\\\\\n\\showunit{mmol}\\\\\n\\showunit{mol}\\\\\n\\showunit{kmol}\\\\\\hline\n\\showunit{pA}\\\\\n\\showunit{nA}\\\\\n\\showunit{uA}\\\\\n\\showunit{mA}\\\\\n\\showunit{A}\\\\\n\\showunit{kA}\\\\\\hline\n\\showunit{ul}\\\\\n\\showunit{ml}\\\\\n\\showunit{l}\\\\\n\\showunit{hl}\\\\\n\\showunit{uL}\\\\\n\\showunit{mL}\\\\\n\\showunit{L}\\\\\n\\showunit{hL}\\\\\\hline\n\\showunit{mHz}\\\\\n\\showunit{Hz}\\\\\n\\showunit{kHz}\\\\\n\\showunit{MHz}\\\\\n\\showunit{GHz}\\\\\n\\showunit{THz}\\\\\\hline\n\\showunit{mN}\\\\\n\\showunit{N}\\\\\n\\showunit{kN}\\\\\n\\showunit{MN}\\\\\\hline\n\\showunit{Pa}\\\\\n\\showunit{kPa}\\\\\n\\showunit{MPa}\\\\\n\\showunit{GPa}\\\\\\hline\n\\showunit{mohm}\\\\\n\\showunit{kohm}\\\\\n\\showunit{Mohm}\\\\\\hline\n\\showunit{pV}\\\\\n\\showunit{nV}\\\\\n\\showunit{uV}\\\\\n\\showunit{mV}\\\\\n\\showunit{V}\\\\\n\\showunit{kV}\\\\\\hline\n\\showunit{W}\\\\\n\\showunit{uW}\\\\\n\\showunit{mW}\\\\\n\\showunit{kW}\\\\\n\\showunit{MW}\\\\\n\\showunit{GW}\\\\\\hline\n\\showunit{J}\\\\\n\\showunit{kJ}\\\\\\hline\n\\showunit{eV}\\\\\n\\showunit{meV}\\\\\n\\showunit{keV}\\\\\n\\showunit{MeV}\\\\\n\\showunit{GeV}\\\\\n\\showunit{TeV}\\\\\\hline\n\\showunit{kWh}\\\\\\hline\n\\showunit{F}\\\\\n\\showunit{fF}\\\\\n\\showunit{pF}\\\\\\hline\n\\showunit{K}\\\\\\hline\n\\showunit{dB}\\\\\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Binary prefixes}\n\\centering\n\\begin{tabular}{llllllll}\\hline\\hline\nUnit & Macro & Symbol & Power \\\\\\hline\n\\showprefix{kibi} \\\\\n\\showprefix{mebi} \\\\\n\\showprefix{gibi} \\\\\n\\showprefix{tebi} \\\\\n\\showprefix{pebi} \\\\\n\\showprefix{exbi} \\\\\n\\showprefix{zebi} \\\\\n\\showprefix{yobi} \\\\\n\\end{tabular}\n\\end{table}\n\n%\\subsection{Creating Units}\n%\\subsection{Loading additional units}\n\n\\subsection{Using units}\n% Note that \\kilogram is defined as \\kilo\\gram, and acts sorta like a macro\n\\si{\\kilogram}\n\\si{\\kilo\\gram}\n% Not allowed.\n%\\si{\\kilo\\kilogram}\n%\\si{\\kilo\\kilo\\gram}\n{\n% and note that regular macros ARE expanded!\n\\def\\killermeater{\\kilo\\meter}\n\\si{\\killermeater}\n}\n{\n% BUT, it prefers its OWN definitions!\n\\def\\gram{Grr}\n\\si{\\kilo\\gram}\n}\n\n% However, it's happy with various indirect definitions.\n% Ie. units, prefixes (at least) act like regular macros within \\si.\n\\si{\\abbra}\\\\\n\\si{\\abbrb}\\\\\n\\si{\\abbrc}\\\\\n\\si{\\abbrd}\\\\\n\\si{\\abbre}\\\\\n\n\\si{\\mabbra}\\\\\n\\si{\\mabbrb}\\\\\n\n\\si{\\killer\\meter}\\\\\n\n% Where can highlight go?\n\\si{\\highlight{red}\\kilo\\gram\\metre\\per\\second} \\\\\n\\si{\\kilo\\highlight{red}\\gram\\metre\\per\\second} \\\\\n\\si{\\kilo\\gram\\highlight{red}\\metre\\per\\second} \\\\\n\\si{\\kilo\\gram\\metre\\highlight{red}\\per\\second} \\\\\n\\si{\\kilo\\gram\\metre\\per\\highlight{red}\\second} \\\\\n\n\\si{\\cancel\\kilo\\gram\\metre\\per\\second} \\\\\n\\si{\\kilo\\cancel\\gram\\metre\\per\\second} \\\\\n\\si{\\kilo\\gram\\cancel\\metre\\per\\second} \\\\\n\\si{\\kilo\\gram\\metre\\cancel\\per\\second} \\\\\n\\si{\\kilo\\gram\\metre\\per\\cancel\\second} \\\\\n\n\\subsubsection{forbid-literal-units, inter-unit-product}\n\\si{\\farad\\squared\\lumen\\candela} \\\\\n\\si[inter-unit-product = \\ensuremath{{}\\cdot{}}]\n{\\farad\\squared\\lumen\\candela}\\\\\n\n\\subsubsection{per-mode, per-symbol, bracket-unit-denominator}\n\\si{\\joule\\per\\mole\\per\\kelvin} \\\\\n\\si{\\metre\\per\\second\\squared} \\\\\n\\si[per-mode=fraction]{\\joule\\per\\mole\\per\\kelvin} \\\\\n\\si[per-mode=fraction]{\\joule\\raiseto{-1}\\mole\\per\\kelvin} \\\\\n\\si[per-mode=fraction]{\\metre\\per\\second\\squared}\\\\\n\\si{\\ampere\\per\\mole\\second} \\\\\n\\si[per-mode = reciprocal-positive-first]{\\ampere\\per\\mole\\second}\n\n{\n\\sisetup{per-mode = symbol}%\n\\si{\\joule\\per\\mole\\per\\kelvin} \\\\\n\\si{\\metre\\per\\second\\squared} \\\\\n\\si[per-symbol = \\text{~div~}]{\\joule\\per\\mole\\per\\kelvin} \\\\\n\\si[bracket-unit-denominator = false]{\\joule\\per\\mole\\per\\kelvin}\\\\\n}\n\\si[per-mode=repeated-symbol]{\\joule\\per\\mole\\per\\kelvin}\\\\\n{\n\\sisetup{per-mode = symbol-or-fraction}%\n\\( \\si{\\joule\\per\\mole\\per\\kelvin} \\)\n\\[ \\si{\\joule\\per\\mole\\per\\kelvin} \\]\n\\si{\\joule\\per\\mole\\per\\kelvin} \\\\\n\\(\n\\displaystyle\n\\si{\\joule\\per\\mole\\per\\kelvin}\n\\)\n\\[\n\\textstyle\n\\si{\\joule\\per\\mole\\per\\kelvin}\n\\]\n}\n\n\\subsubsection{sticky-per}\n\\si{\\pascal\\per\\gray\\henry} \\\\\n\\si[sticky-per]{\\pascal\\per\\gray\\henry}\\\\\n\n\\subsubsection{power-font}\n\\si{\\metre\\per\\second\\squared} \\\\\n\\si[power-font = unit]{\\metre\\per\\second\\squared}\\\\\n\n\\subsubsection{literal-superscript-as-power}\n\\si{m.s^{2}} \\\\\n\\si[literal-superscript-as-power = false]{m.s^{2}}\\\\\n\n\\subsubsection{qualifier-mode, qualifier-phrase}\n\\si{\\kilogram\\polymer\\squared\\per\\mole\\catalyst\\per\\hour} \\\\\n\\si[qualifier-mode = brackets]\n{\\kilogram\\polymer\\squared\\per\\mole\\catalyst\\per\\hour} \\\\\n\\si[qualifier-mode = subscript]\n{\\kilogram\\polymer\\squared\\per\\mole\\catalyst\\per\\hour} \\\\\n\\si[qualifier-mode = space]\n{\\kilogram\\polymer\\squared\\per\\mole\\catalyst\\per\\hour} \\\\\n\\si[qualifier-mode = text]\n{\\deci\\bel\\isotropic}\\\\\n\n\\si[qualifier-mode = phrase]\n{\\kilogram\\polymer\\squared\\per\\mole\\catalyst\\per\\hour} \\\\\n\\si[qualifier-mode = phrase, qualifier-phrase = { by }]\n{\\kilogram\\polymer\\squared\\per\\mole\\catalyst\\per\\hour} \\\\\n\n\\subsubsection{prefixes-as-symbols}\n\\si{\\milli\\litre\\per\\mole\\deci\\ampere} \\\\\n%%\\SI{10}{\\kilo\\gram\\squared\\deci\\second} \\\\\n\\si[prefixes-as-symbols=false]{\\milli\\litre\\per\\mole\\deci\\ampere}\\\\\n\\si[prefixes-as-symbols=false]{\\kilo\\gram\\squared\\deci\\second}\\\\\n\\si{\\mega\\gram\\squared\\deci\\second}\\\\\n\\si[prefixes-as-symbols=false]{\\mega\\gram\\squared\\deci\\second}\\\\\n\\si{\\micro\\gram\\squared\\deci\\second}\\\\\n\\si[prefixes-as-symbols=false]{\\micro\\gram\\squared\\deci\\second}\\\\\n\\si{\\per\\mega\\gram\\squared\\deci\\second}\\\\\n\\si[prefixes-as-symbols=false]{\\per\\mega\\gram\\squared\\deci\\second}\\\\\n\\si{\\per\\micro\\gram\\squared\\deci\\second}\\\\\n\\si[prefixes-as-symbols=false]{\\per\\micro\\gram\\squared\\deci\\second}\\\\\n%%\\SI[prefixes-as-symbols=false]{10}{\\kilo\\gram\\squared\\deci\\second}\\\\\n\n\\subsubsection{parse-units}\n%%\\SI{300}{\\MHz} \\\\\n%%\\SI[parse-units = false]{300}{\\MHz}\\\\\n\n\\subsection{Numbers with units}\n\\subsubsection{allow-number-unit-breaks}\n\\subsubsection{number-unit-product}\n\\SI{2.67}{\\farad} \\\\\n\\SI[number-unit-product = \\ ]{2.67}{\\farad} \\\\\n\\SI[number-unit-product = ]{2.67}{\\farad}\\\\\n\n\\subsubsection{multi-part-units}\n{\n\\sisetup{separate-uncertainty}%\n\\SI{12.3(4)}{\\kilo\\gram} \\\\\n\\SI[multi-part-units = brackets]{12.3(4)}{\\kilo\\gram} \\\\\n\\SI[multi-part-units = repeat]{12.3(4)}{\\kilo\\gram}\\\\\n\\SI[multi-part-units = single]{12.3(4)}{\\kilo\\gram}\\\\\n}\n\n{\n\\sisetup{separate-uncertainty,bracket-numbers = false}%\n\\num{1.234(5)e-4} \\\\\n\\SI{1.234(5)e-4}{\\metre}\\\\\n}\n\\subsubsection{product-units}\n\\SI{2 x 3 x 4}{\\metre} \\\\\n\\SI[product-units = brackets]{2 x 3 x 4}{\\metre}\\\\\n\\SI[product-units = brackets-power]{2 x 3 x 4}{\\metre}\\\\\n\\SI[product-units = power]{2 x 3 x 4}{\\metre}\\\\\n\\SI[product-units = repeat]{2 x 3 x 4}{\\metre}\\\\\n\\SI[product-units = single]{2 x 3 x 4}{\\metre}\\\\\n\n\\subsubsection{list-units,range-units}\n\\SIlist{2;4;6;8}{\\tesla} \\\\\n\\SIlist[list-units = brackets]{2;4;6;8}{\\tesla} \\\\\n\\SIlist[list-units = repeat]{2;4;6;8}{\\tesla} \\\\\n\\SIlist[list-units = single]{2;4;6;8}{\\tesla} \\\\\n\\SIrange{2}{4}{\\degreeCelsius} \\\\\n\\SIrange[range-units = brackets]{2}{4}{\\degreeCelsius} \\\\\n\\SIrange[range-units = repeat]{2}{4}{\\degreeCelsius} \\\\\n\\SIrange[range-units = single]{2}{4}{\\degreeCelsius}\\\\\n\n\\subsubsection{exponent-to-prefix}\n\\SI{1700}{\\g} \\\\\n\\SI{1.7e3}{\\g} \\\\\n{\n\\sisetup{exponent-to-prefix}%\n\\SI{1700}{\\g} \\\\\n\\SI{1.7e3}{\\g} \\\\\n}\n{\n\\sisetup{fixed-exponent = 3, scientific-notation = fixed}%\n\\SI{1700}{\\g} \\\\\n\\SI{1.7e3}{\\g}\\\\\n}\n\n\\subsubsection{just degree}\n\n\\si{\\degree}\n\n\\section{Tabular material}\n% \\begin{tabular}{S}\n% 2.3456 \\\\\n% \\end{tabular}\n% \\end{document}\n\n\n\\begin{table}[H]\n\\caption{Standard behaviour of the \\texttt{S} column type.}\n\\label{tab:S:standard}\n\\centering\n\\begin{tabular}{S}\n\\toprule\n{Some Values} \\\\\n\\midrule\n2.3456 \\\\\n34.2345 \\\\\n-6.7835 \\\\\n90.473 \\\\\n5642.5 \\\\\n1.2e3 \\\\\ne4 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n%\\end{document}\n\n\n\n\\begin{table}[H]\n\\caption{Detection of surrounding material in an \\texttt{S} column.}\n\\label{tab:S:extras}\n\\centering\n\\begin{tabular}{S[color=orange]}\n\\toprule\n{Some Values} \\\\\n\\midrule\n12.34 \\\\\n\\color{purple} 975,31 \\\\\n44.268 \\textsuperscript{\\emph{a}} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[H]\n\\caption{Controlling complex alignment with the tablenum macro.}\n\\label{tab:tablenum}\n\\centering\n\\begin{tabular}{lr}\n\\toprule\nHeading & Heading \\\\\n\\midrule\nInfo & More info \\\\\nInfo & More info \\\\\n\\multicolumn{2}{c}{\\tablenum[table-format = 4.4]{12,34}} \\\\\n\\multicolumn{2}{c}{\\tablenum[table-format = 4.4]{333.5567}} \\\\\n\\multicolumn{2}{c}{\\tablenum[table-format = 4.4]{4563.21}} \\\\\n\\bottomrule\n\\end{tabular}\n\\hfil\n\\begin{tabular}{lr}\n\\toprule\nHeading & Heading \\\\\n\\midrule\n\\multirow{2}*{\\tablenum{88,999}} & aaa \\\\\n& bbb \\\\\n\\multirow{2}*{\\tablenum{33,435}} & ccc \\\\\n& ddd \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n\\caption{Units in tables.}\n\\label{tab:s:demo}\n\\begin{tabular}{s}\n\\toprule\n\\multicolumn{1}{c}{Unit} \\\\\n\\midrule\n\\metre\\squared\\per\\second \\\\\n\\pascal \\\\\nm.s^{-1} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n\\caption{The \\texttt{s} column processes everything.}\n\\label{tab:s:processing}\n\\sisetup{color = orange}\n\\begin{tabular}{ss}\n\\toprule\n{Unit}\n& \\multicolumn{1}{c}{Unit}\\\\\n\\midrule\n{\\si{m^3}} & \\multicolumn{1}{c}{\\si{m^3}} \\\\\n\\kilogram\n& \\kilogram \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-parse-only}\n\\begin{table}[H]\n\\centering\n\\caption{Parsing without aligning in an \\texttt{S} column.}\n\\label{tab:S:parse}\n\\begin{tabular}{SS[table-parse-only]}\n\\toprule\n{Decimal-centred} &\n{Simple centring} \\\\\n\\midrule\n12.345 & 12.345 \\\\\n6,78 & 6,78 \\\\\n-88.8(9) & -88.8(9) \\\\\n4.5e3 & 4.5e3 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-number-alignment}\n\\begin{table}[H]\n\\caption{Aligning the \\texttt{S} column.}\n\\label{tab:S:align}\n\\centering\n\\sisetup{\ntable-figures-integer = 2,\ntable-figures-decimal = 4\n}\n\\begin{tabular}{\nS\nS[table-number-alignment = center]\nS[table-number-alignment = left]\nS[table-number-alignment = right]\n}\n\\toprule\n{Some Values} & {Some Values} & {Some Values} & {Some Values} \\\\\n\\midrule\n2.3456 & 2.3456 & 2.3456 & 2.3456\\\\\n34.2345 & 34.2345 & 34.2345 & 34.2345\\\\\n56.7835 & 56.7835& 56.7835 & 56.7835\\\\\n90.473 & 90.473 & 90.473 & 90.473\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-figures-decimal, table-figures-exponent,table-figures-integer,table-figures-\nuncertainty}\n\n\\begin{table}[H]\n\\caption{Reserving space in \\texttt{S} columns.}\n\\label{tab:S:space}\n\\sisetup{\ntable-number-alignment = center,\ntable-figures-integer = 2\n}\n\\centering\n\\begin{tabular}{\nS\nS[table-number-alignment = right]\nS[table-figures-uncertainty = 1]\nS[separate-uncertainty, table-figures-uncertainty = 1]\nS[table-sign-mantissa]\nS[table-figures-exponent = 1]\n}\n\\toprule\n{Values}\n& {Values}\n& {Values}\n& {Values}\n& {Values}\n& {Values} \\\\\n\\midrule\n2.3 & 2.3 & 2.3(5) & 2.3(5) & 2.3 & 2.3e8\\\\\n34.23 & 34.23 & 34.23(4) & 34.23(4) & 34.23 & 34.23\\\\\n56.78 & 56.78 & 56.78(3) & 56.78(3) & -56.78 & 56.78e3\\\\\n3,76 & 3,76 & 3,76(2) & 3.76(2) & +-3.76 & e6\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-comparator}\n\\begin{table}[H]\n\\caption{Reserving space for comparators in \\texttt{S} columns.}\n\\label{tab:S:comparators}\n\\sisetup{\ntable-number-alignment = center,\ntable-figures-integer = 2,\ntable-figures-decimal = 2,\ntable-figures-exponent = 2,\n}\n\\centering\n\\begin{tabular}{\nS\nS[table-comparator = true]}\n\\toprule\n{Values}\n& {Values} \\\\\n\\midrule\n2 .3 & < 2.3e8\\\\\n34.23 & = 34.23 \\\\\n56.78 & >= 56.78e3\\\\\n3,76 & \\gg e6 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-format}\n\\begin{table}[H]\n\\caption{Using the \\texttt{table-format} option.}\n\\label{tab:S:format}\n\\centering\n\\begin{tabular}{\nS\nS[table-format = 2.2]\nS[table-format = 2.2(1)]\nS[table-format = +2.2]\nS[table-format = 2.2e1]\n}\n\\toprule\n{Values}\n& {Values}\n& {Values}\n& {Values}\n& {Values} \\\\\n\\midrule\n2.3 & 2.3 & 2.3(5) & 2.3 & 2.3e8 \\\\\n34.23 & 34.23 & 34.23(4) & 34.23 & 34.23 \\\\\n56.78 & 56.78 & 56.78(3) & -56.78 & 56.78e3 \\\\\n3,76 & 3,76 & 3.76(2) & +-3.76 & e6 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\subsubsection{table-space-text-pre, table-space-text-post}\n\\begin{table}[H]\n\\caption{Text before and after numbers.}\n\\label{tab:S:ends}\n\\centering\n\\sisetup{\ntable-number-alignment = center,\ntable-figures-integer = 2,\ntable-figures-decimal = 4,\ntable-space-text-pre\n= now~,\ntable-space-text-post =\n\\textsuperscript{\\emph{a}}\n}\n\\begin{tabular}{S}\n\\toprule\n{Values} \\\\\n\\midrule\n2.3456 \\\\\n34.2345 \\textsuperscript{\\emph{a}}\\\\\n56.7835 \\\\\nnow~ 90.473 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-align-comparator, table-align-exponent, table-align-uncertainty}\n\\begin{table}[H]\n\\centering\n\\caption{The \\texttt{table-align-exponent} option}\n\\label{tab:align:exp}\n\\sisetup{table-format = 1.3e2, table-number-alignment = center}\n\\begin{tabular}{SS[table-align-exponent = false]}\n\\toprule\n{Header} & {Header} \\\\\n\\midrule\n1.2e3 & 1.2e3 \\\\\n1.234e56 & 1.234e56 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n\\caption{The \\texttt{table-align-uncertainty} option}\n\\label{tab:align:uncert}\n\\sisetup{\nseparate-uncertainty,\ntable-format = 1.3(1),\n}\n\\begin{tabular}{SS[table-align-uncertainty = false]}\n\\toprule\n{Header} & {Header} \\\\\n\\midrule\n1.2(1) & 1.2(3) \\\\\n1.234(5) & 1.234(5) \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}[H]\n\\centering\n\\caption{The \\texttt{table-align-comparator} option}\n\\label{tab:align:comp}\n\\sisetup{table-format = >2.2}\n\\begin{tabular}{SS[table-align-comparator = false]}\n\\toprule\n{Header} & {Header} \\\\\n\\midrule\n> 1.2 & > 1.2 \\\\\n< 12.34 & < 12.34 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-omit-exponent}\n\\begin{table}[H]\n\\centering\n\\caption{The \\texttt{table-omit-exponent} option}\n\\label{tab:exp:omit}\n\\begin{tabular}{\nS[table-format = 1.1e1]\nS[fixed-exponent = 3, table-format = 2.1, table-omit-exponent]\n}\n\\toprule\n{Header} & {Header / \\num{e3}} \\\\\n\\midrule\n1.2e3 & 1.2e3 \\\\\n3e2 & 3e2 \\\\\n1.0e4 & 1.0e4 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-align-text-pre,table-align-text-post}\n% \\begin{table}[H]\n% \\caption{Closing notes up to text.}\n% \\label{tab:S:notes}\n% \\newrobustcmd\\NoteMark[1]{%\n% \\textsuperscript{\\emph{#1}}%\n% }\n% \\centering\n% \\sisetup{\n% table-number-alignment = center,\n% table-figures-integer = 2,\n% table-figures-decimal = 4,\n% table-space-text-pre\n% = \\NoteMark{a}\n% }\n% \\begin{tabular}{\n% S\n% S[table-align-text-pre = false]\n% }\n% \\toprule\n% {Values}\n% & {Values} \\\\\n% \\midrule\n% 2.3456 &\n% 2.3456 \\\\\n% \\NoteMark{a} 4.234 & \\NoteMark{a} 4.234 \\\\\n% \\NoteMark{b}\n% .78\n% & \\NoteMark{b}\n% .78 \\\\\n% \\NoteMark{d} 88\n% & \\NoteMark{d} 88\n% \\\\\n% \\bottomrule\n% \\end{tabular}\n% \\hfil\n% \\sisetup{table-space-text-post = \\NoteMark{a}}\n% \\begin{tabular}{\n% S\n% S[table-align-text-post = false]\n% }\n% \\toprule\n% {Values}\n% & {Values} \\\\\n% \\midrule\n% 2.3456\n% & 2.3456 \\\\\n% 34.234 \\NoteMark{a} & 34.234 \\NoteMark{a} \\\\\n% 56.78\n% \\NoteMark{b} & 56.78 \\NoteMark{b} \\\\\n% 90.4\n% \\NoteMark{c} & 90.4\n% \\NoteMark{c} \\\\\n% 88\n% \\NoteMark{d} & 88\n% \\NoteMark{d} \\\\\n% \\bottomrule\n% \\end{tabular}\n% \\end{table}\n\n\\subsubsection{table-auto-round}\n\\begin{table}[H]\n\\centering\n\\caption{The \\texttt{table-auto-round} option.}\n\\label{tab:S:auto}\n\\sisetup{\ntable-number-alignment = center,\ntable-figures-integer = 1,\ntable-figures-decimal = 3\n}\n% Notice the overfull hbox which results with\n% the first column\n\\begin{tabular}{\nS\nS[table-auto-round]\n}\n\\toprule\n{Header} & {Header} \\\\\n\\midrule\n1.2 & 1.2 \\\\\n1.2345 & 1.2345 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{parse-numbers}\n\\begin{table}[H]\n\\caption{Aligning without parsing.}\n\\label{tab:S:nonparsed}\n\\sisetup{\nparse-numbers = false,\ntable-figures-integer = 2,\ntable-figures-decimal = 3\n}\n\\centering\n\\begin{tabular}{\nS\nS[table-number-alignment = center]\nS[table-number-alignment = right]\nS[table-number-alignment = left]\n}\n\\toprule\n{Some values}\n& {Some values}\n& {Some values}\n& {Some values} \\\\\n\\midrule\n2.35 &\n2.35 &\n2.35 &\n2.35 \\\\\n34.234 &\n34.234 &\n34.234 & 34.234 \\\\\n56.783 &\n56.783 &\n56.783 & 56.783 \\\\\n3,762 &\n3,762 &\n3,762 &\n3.762 \\\\\n\\sqrt{2} & \\sqrt{2} & \\sqrt{2} & \\sqrt{2} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-text-alignment}\n\\begin{table}[H]\n\\caption{Aligning text in \\texttt{S} columns.}\n\\label{tab:S:text}\n\\sisetup{\ntable-number-alignment = center,\ntable-figures-integer = 4,\ntable-figures-decimal = 4\n}\n\\centering\n\\begin{tabular}{\nS\nS[table-text-alignment = left]\nS[table-text-alignment = right]\n}\n\\toprule\n{Values}\n& {Values}\n& {Values} \\\\\n\\midrule\n992.435 & 992.435 & 992.435 \\\\\n7734.2344 & 7734.2344 & 7734.2344 \\\\\n56.7834 &\n56.7834 &\n56.7834 \\\\\n3,7462 &\n3,7462 &\n3,7462 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-unit-alignment}\n\\begin{table}[H]\n\\centering\n\\caption{Alignment options in \\texttt{s} columns.}\n\\label{tab:s:align}\n\\begin{tabular}\n{\ns[table-unit-alignment = right]\ns\ns[table-unit-alignment = left]\n}\n\\toprule\n{Right-aligned} &\n{Centred text} &\n{Left-aligned} \\\\\n\\midrule\n\\metre\\per\\second & \\metre\\per\\second & \\metre\\per\\second \\\\\n\\kilogram\n& \\kilogram\n& \\kilogram\n\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\subsubsection{table-alignment}\n\n\\subsubsection{table-column-width}\n\\begin{table}[H]\n\\centering\n\\caption{Fixed-width columns.}\n\\label{tab:width:fixed}\n\\begin{tabular}\n{\ns\ns[table-column-width = 2 cm]\nS\nS[table-column-width = 2 cm]\n}\n\\toprule\n{Flexible} &\n{Fixed}\n&\n{Flexible} &\n{Fixed}\n\\\\\n\\midrule\n\\metre\\per\\second & \\metre\\per\\second & 1.23 & 1.23 \\\\\n\\kilogram\\candela & \\kilogram\\candela & 45.6 & 45.6 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n% \\begin{table}[H]\n% \\centering\n% \\caption{Right-aligning under a heading.}\n% \\label{tab:width:special}\n% \\settowidth\\mylength{Long header}\n% \\sisetup{\n% table-format\n% = 4\n% ,\n% table-number-alignment = center\n% ,\n% table-column-width\n% = \\mylength ,\n% input-decimal-markers =\n% ,\n% input-symbols\n% = .\n% ,\n% }\n% \\begin{tabular}{S}\n% \\toprule\n% {Long header} \\\\\n% \\midrule\n% 12.33 \\\\\n% 2\n% \\\\\n% 1234\n% \\\\\n% \\bottomrule\n% \\end{tabular}\n% \\end{table}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\si{kg.m.s^{-1}}\\\\\n\\si{\\kilogram\\metre\\per\\second} \\\\\n\\si[per-mode=symbol]{\\kilogram\\metre\\per\\second} \\\\\n\\si[per-mode=symbol]{\\kilogram\\metre\\per\\ampere\\per\\second}\n\n\\numlist{10;20;30}\\\\\n\\SIlist{0.13;0.67;0.80}{\\milli\\metre} \\\\\n\\numrange{10}{20}\\\\\n\\SIrange{0.13}{0.67}{\\milli\\metre}\\\\\n\n\n\\numlist{10;30;50;70}\n\n\\numrange{10}{30}\n\n\n\\si{kg.m/s^2} \\\\\n\\si{g_{polymer}~mol_{cat}.s^{-1}}\n\n\\si{\\kilo\\gram\\metre\\per\\square\\second}\n\\si{\\gram\\per\\cubic\\centi\\metre}\n\\si{\\square\\volt\\cubic\\lumen\\per\\farad}\n\\si{\\metre\\squared\\per\\gray\\cubic\\lux}\n\\si{\\henry\\second}\n\n\\SI[mode=text]{1.23}{J.mol^{-1}.K^{-1}}\\\\\n\\SI{.23e7}{\\candela}\\\\\n\\SI[per-mode=symbol]{1.99}[\\$]{\\per\\kilogram}\\\\\n\\SI[per-mode=fraction]{1,345}{\\coulomb\\per\\mole}\\\\\n\n\\SIlist{10;30;45}{\\metre}\n\n\\si{\\square\\becquerel} \\\\\n\\si{\\joule\\squared\\per\\lumen} \\\\\n\\si{\\cubic\\lux\\volt\\tesla\\cubed}\n\n\\si{\\henry\\tothe{5}} \\\\\n\\si{\\raiseto{4.5}\\radian}\n\n\\si{\\joule\\per\\mole\\per\\kelvin} \\\\\n\\si{\\joule\\per\\mole\\kelvin} \\\\\n\\si{\\per\\henry\\tothe{5}} \\\\\n\\si{\\per\\square\\becquerel}\n\n\\si{\\kilogram\\of{metal}} \\\\\n\\SI[qualifier-mode = brackets]\n{0.1}{\\milli\\mole\\of{cat}\\per\\kilogram\\of{prod}}\n\n\\si[per-mode = fraction]\n{\\cancel\\kilogram\\metre\\per\\cancel\\kilogram\\per\\second} \\\\\n\\si{\\highlight{red}\\kilogram\\metre\\per\\second} \\\\\n\\si[unit-color = purple]\n{\\highlight{red}\\kilogram\\metre\\per\\second}\n\n\n\\si{\\kilo} \\\\\n\\si{\\micro} \\\\\n\\si[prefixes-as-symbols = false]{\\kilo}\n\n\\si{\\kilo\\gram\\micro} \\\\\n\\SI{10}{\\micro}\n\n\n\\begin{table}\n\\caption{Standard behaviour of the \\texttt{S} column type.}\n\\label{tab:S:standard}\n\\centering\n\\begin{tabular}{S}\n\\toprule\n{Some Values} \\\\\n\\midrule\n2.3456 \\\\\n34.2345 \\\\\n-6.7835 \\\\\n90.473 \\\\\n5642.5\n\\\\\n1.2e3 \\\\\ne4 \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Detection of surrounding material in an \\texttt{S}\ncolumn.}\n\\label{tab:S:extras}\n\\centering\n\\begin{tabular}{S[color=orange]}\n\\toprule\n{Some Values} \\\\\n\\midrule\n12.34 \\\\\n\\color{purple} 975,31 \\\\\n44.268 \\textsuperscript{\\emph{a}} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{Controlling complex alignment with the tablenum macro.}\n\\label{tab:tablenum}\n\\centering\n\\begin{tabular}{lr}\n\\toprule\nHeading & Heading \\\\\n\\midrule\nInfo & More info \\\\\nInfo & More info \\\\\n\\multicolumn{2}{c}{\\tablenum[table-format = 4.4]{12,34}}\n\\\\\n\\multicolumn{2}{c}{\\tablenum[table-format = 4.4]{333.5567}} \\\\\n\\multicolumn{2}{c}{\\tablenum[table-format = 4.4]{4563.21}}\n\\bottomrule\n\\end{tabular}\n\\hfil\n\\begin{tabular}{lr}\n\\toprule\nHeading & Heading \\\\\n\\midrule\n\\multirow{2}*{\\tablenum{88,999}} & aaa \\\\\n& bbb \\\\\n\\multirow{2}*{\\tablenum{33,435}} & ccc \\\\\n& ddd \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Units in tables.}\n\\label{tab:s:demo}\n\\begin{tabular}{s}\n\\toprule\n\\multicolumn{1}{c}{Unit} \\\\\n\\midrule\n\\metre\\squared\\per\\second \\\\\n\\pascal \\\\\nm.s^{-1} \\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\n\\end{document}\n", "meta": {"hexsha": "6045338bed37403bfe44ec419ad4b13557fe307a", "size": 33610, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tests/integration/packages/siunitx.tex", "max_stars_repo_name": "kenjikun/engrafo", "max_stars_repo_head_hexsha": "2ac87e215daea32699aa7f888f0405936d2ef452", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 836, "max_stars_repo_stars_event_min_datetime": "2017-10-23T10:16:18.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T00:31:05.000Z", "max_issues_repo_path": "tests/integration/packages/siunitx.tex", "max_issues_repo_name": "kenjikun/engrafo", "max_issues_repo_head_hexsha": "2ac87e215daea32699aa7f888f0405936d2ef452", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 894, "max_issues_repo_issues_event_min_datetime": "2017-10-23T09:27:08.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-29T19:06:07.000Z", "max_forks_repo_path": "tests/integration/packages/siunitx.tex", "max_forks_repo_name": "kenjikun/engrafo", "max_forks_repo_head_hexsha": "2ac87e215daea32699aa7f888f0405936d2ef452", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 62, "max_forks_repo_forks_event_min_datetime": "2017-10-23T19:29:09.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-23T12:19:08.000Z", "avg_line_length": 21.39401655, "max_line_length": 128, "alphanum_fraction": 0.6673013984, "num_tokens": 13225, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.34670843128633855}} {"text": "\\section{Modules}\n\n\\begin{itemize}\n\t\n\t\\item Modules provide a way to store data structures and procedures\n\t\n\t\\item Example:\\\\\n\t\\gcl{0}{\\Module ~ UniqueNumberAllocator}\n\t\\gcl{1}{\\Export ~ Acquire, Reset}\n\t\\gcl{1}{\\Import ~ Choose}\n\t\\gcl{1}{}\n\t\\gcl{1}{\\Var ~ u : \\textbf{set} \\left[0, N\\right)}\n\t\\gcl{1}{}\n\t\\gcl{1}{\\Procedure ~ Acquire(\\Result ~ t) \\defeq}\n\t\\gcl{2}{Choose(\\left[0, N\\right) - $ u $, t); u := u \\cup \\left\\lbrace t \\right\\rbrace]}\n\t\\gcl{1}{}\n\t\\gcl{1}{\\Procedure ~ Reset() \\defeq u := \\left\\lbrace \\right\\rbrace}\n\t\\gcl{1}{}\n\t\\gcl{1}{\\Procedure ~ Choose(\\Value ~ s; \\Result ~ e) \\defeq e : [s \\ne \\left\\lbrace \\right\\rbrace, e \\in s]}\n\t\\gcl{1}{}\n\t\\gcl{1}{\\Initially ~ u = \\left\\lbrace \\right\\rbrace}\n\t$ \\End $\n\t\\item Syntax\n\t\n\t\\begin{itemize}\n\t\t\n\t\t\\item Modules are declared with $ \\Module $ and have a unique name\n\t\t\n\t\t\\item Module-level variables are declared in the $ \\Var $ clause and given a type\n\t\t\n\t\t\\item The initial condition of module variables is given the predicate in the $ \\Initially $ clause\n\t\t\n\t\t\\item Modules may define procedures which make use of its variables\n\t\t\n\t\t\\item Modules list which procedures are exported publicly with an $ \\Export $ clause (if there is no $ \\Export $ clause, all procedures are exported)\n\t\t\n\t\t\\item The variables and procedures of another module may be used if they are included in the $ \\Import $ clause\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item Imported variables must be redeclared exactly as in their source module\n\t\t\t\n\t\t\t\\item Imported procedures must be redeclared: the original declaration must refine the redeclaration\n\t\t\t\n\t\t\t\\item Imported procedures cannot refer to the local variables of the module they are imported into\n\t\t\t\n\t\t\t\\item Circular import/export is not well defined\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\\end{itemize}\n\t\n\\end{itemize}\n\n\\subsection{Module Refinement}\n\n\\begin{itemize}\n\t\n\t\\item A module $ M' $ refines some module $ M $ with exported procedures $ E $, imported procedures $ I $ and initialisation condition $ init $ when\n\t\n\t\\begin{itemize}\n\t\t\n\t\t\\item $ M' $ has the same local and imported variables as $ M $\n\t\t\n\t\t\\item The exported procedures $ E' $ refine those in $ E $ (there may be more procedures in $ E' $, but not fewer)\n\t\t\n\t\t\\item The imported procedures $ I' $ refine those in $ I $ (there may be fewer procedures in $ I' $ but not more)\n\t\t\n\t\t\\item The initialisation $ init' $ is stronger than $ init $ -- i.e. $ init' \\entails init $\n\t\t\n\t\\end{itemize}\n\t\n\t\\item To refine modules with different variables, data refinement is required\n\t\n\\end{itemize}\n\n\\newpage\n\n\\subsection{Data Refinement}\n\n\\begin{itemize}\n\t\n\t\\item The local state of a module cannot be accessed from the outside, so it may be changed provided the difference cannot be detected by use of the exported procedures\n\t\n\t\\item Rule 1: \\textbf{Introducing new variables}\n\t\n\t\\begin{itemize}\n\t\t\n\t\t\\item Relationships between new and existing variables are maintained via a \\textbf{coupling invariant} $ CI $\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item E.g. $ CI \\defeq p = q + r $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\t\\item The initialisation $ init $ becomes $ init \\land CI $\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item E.g. if initialisation was $ p = 1 $, it would become $ p = 1 \\land p = q + r $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\t\\item Any specification $ w : [P, Q] $ becomes $ w, c : [P \\land CI, Q \\land CI] $ where $ c $ is the list of new variables\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item E.g. $ p : [p > 0, p < p_0] $ becomes $ p, q, r : [p > 0 \\land p = q + r, p < p_0 \\land p = q + r] $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\t\\item Every assignment in the module $ w := E $ becomes $ w, c := E, F $ provided that $ CI \\entails CI[w, c \\backslash E, F] $\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item E.g. $ p := p + 1 $ becomes $ p, q := p + 1, q + 1 $, or alternately $ p, r := p + 1, r + 1 $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\t\\item Every guard in the module $ G $ becomes $ G' $ provided that $ CI \\entails (G \\iff G') $\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item $ G' \\defeq CI \\land G $ is always suitable\n\t\t\t\n\t\t\t\\item E.g. the guard $ p > 0 $ could become $ p > 0 \\land p = q + r $, or alternately $ p = q +r \\implies p > 0 $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\\end{itemize}\n\t\n\t\\item Rule 2: \\textbf{Removing an existing variable}\n\t\n\t\\begin{itemize}\n\t\t\n\t\t\\item Only \\textbf{auxiliary variables} may be removed, i.e. they must only appear in:\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item Assignments\n\t\t\t\n\t\t\t\\item Specifications which modify only auxiliary variables\n\t\t\t\n\t\t\\end{itemize}\n\t\n\t\t\\item The initialisation $ init $ becomes $ \\exists a \\cdot init $ where $ a $ is the auxiliary variable\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item The `one-point rule' may be used to remove the existential quantifier: $ \\exists x \\cdot P \\land x = n \\equiv P[x \\backslash n] $\n\t\t\t\n\t\t\t\\item E.g. given initialisation $ p = 1 \\land p = q + r $, it would become $ \\exists p \\cdot p = 1 \\land p = q + r \\equiv q + r = 1 $\n\t\t\t\n\t\t\\end{itemize}\n\t\n\t\t\\item All specifications $ w, a : [P, Q] $ become $ w : [\\exists a \\cdot P, \\forall a_0 \\cdot P[w, a \\backslash w_0, a_0] \\implies (\\exists a \\cdot Q)] $\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item A similar one-point rule may be used to remove the universal quantifier:\\\\\n\t\t\t$~~~~ \\forall x \\cdot P \\land x = n \\implies Q \\equiv P[x \\backslash n] \\implies Q[x \\backslash n] $\n\t\t\t\n\t\t\t\\item E.g. $ p, q, r : [p > 0 \\land p = q + r, p < p_0 \\land p = q + r] $ can be refined to:\\\\\n\t\t\t\\form{q, r : [\\exists p \\cdot p > 0 \\land p = q + r, \\forall p_0 \\cdot p_0 > 0 \\land p_0 = q_0 + r_0 \\implies (\\exists p \\cdot p < p_0 \\land p = q + r)]}\n\t\t\t\\hint{\\refsto}{Apply $ \\exists $ one-point rule to pre and postconditions}\n\t\t\t\\form{q, r : [q + r > 0, \\forall p_0 \\cdot p_0 > 0 \\land p_0 = q_0 + r_0 \\implies q + r < p_0]}\n\t\t\t\\hint{\\refsto}{Apply $ \\forall $ one-point rule to postcondition}\n\t\t\t\\form{q, r : [q + r > 0, q_0 + r_0 > 0 \\implies q + r < q_0 + r_0]}\n\t\t\t\n\t\t\\end{itemize}\n\t\n\t\t\\item Any assignment $ w, a := E, F $ where $ E $ contains no variables from $ a $ can be replaced by $ w := E $\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item E.g. $ p, q := p + 1, q + 1 $ can be replaced by $ q := q + 1 $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\t\\item Normally the coupling invariant $ CI $ relates each concrete state to a unique abstract state (e.g. $ p = q + r $), in which case the following rule applies:\n\t\t\n\t\t\\begin{itemize}\n\t\t\t\n\t\t\t\\item Given abstract variables $ a $ and concrete variables $ c $, if $ CI \\defeq a = f(c) \\land P(c) $ then a guard $ G $ may be replaced by $ G[a \\backslash f(c)] \\land P(c)$, or simply by $ G[a \\backslash f(c)] $\n\t\t\t\n\t\t\t\\item E.g. $ p > 0 \\land p = q + r $ can be replaced by $ q + r > 0 \\land q + r = q + r \\equiv q + r > 0 $\n\t\t\t\n\t\t\\end{itemize}\n\t\t\n\t\\end{itemize}\n\t\n\\end{itemize}", "meta": {"hexsha": "661ebb325211adcd3083fce537032e0085c57f01", "size": 6651, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "CSSE3100/modules.tex", "max_stars_repo_name": "mcoot/CourseNotes", "max_stars_repo_head_hexsha": "c643f46e32cdf4c567bf73d4a23784c834278803", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "CSSE3100/modules.tex", "max_issues_repo_name": "mcoot/CourseNotes", "max_issues_repo_head_hexsha": "c643f46e32cdf4c567bf73d4a23784c834278803", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "CSSE3100/modules.tex", "max_forks_repo_name": "mcoot/CourseNotes", "max_forks_repo_head_hexsha": "c643f46e32cdf4c567bf73d4a23784c834278803", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.4611398964, "max_line_length": 218, "alphanum_fraction": 0.6248684408, "num_tokens": 2242, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177488, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3467084229464548}} {"text": "%auto-ignore\n\\providecommand{\\MainFolder}{..}\n\\documentclass[\\MainFolder/Text.tex]{subfiles}\n\\begin{document}\n\n\\section{Homological perturbation lemma}\n\n\n\\begin{Remark}[Origin of $\\BV$-formalism in physics]\nin $\\BRST$ we have clasical fields $x$ and ghosts $c$ (and Lagrange multiplier $\\lambda$ and $\\bar{c}$ for gauge fixing)\nin $\\BV$-formalism add anti-fields and anti-ghosts\n$$ \\int_{V_0} \\Vol(G) \\int_{V_0/G} f(v) e^{S_{\\BRST}}$$\nHaving $S$, they use the \nresolution of the integration\n$$(V_{\\BV}\\coloneqq T^*[-1]V_{\\BRST},) \\rightarrow (V_{\\BRST},\\int,Q,\\Action_{\\BRST} =)\\rightarrow (V_0,\\int,\\Action_0)$$\n-and $\\Action_0$ gets written as $\\Action_{\\BRST} = \\Action_0 + Q(\\Psi)$, where $\\Psi$ is a $Q$-exact term depending on $\\lambda$, $\\bar{c}$ and $c$ (and Faddeev Poppov determinant and gauge fixing) called the gauge fixing fermion. Now, eventhough $\\int_{V_0} e^{\\Action_0}$ does not exist, $\\int_V e^{\\Action_{\\BRST}}$ exists and does not depend $\\Psi$. To remove the choice completely, one considers the $\\BV$-formalism on odd cotangent bundle $V_{\\BV} = T^*[1] V_{0}$ with the canonical odd symplectic form, where there is a well-defined notion of integration over a Lagrangian subspace $L \\subset V_{\\BV}$ which does not depend on $L$ if the integrand is $\\BV$-closed. The choice of $\\Psi$ then corresponds to the choice of $L$. Having $V_0 = \\Harm \\oplus C$ of ``harmonic'' and the rest and. This amounts \n$$ Z: $$\n$\\BRST$ extends the space of fields \n$$ $$\n$$ \\int_V f(v) e^{\\Action(v)} $$\n\\end{Remark}\n\n\nIn \\cite{Doubek2018}, they consider multilinear operations $l_{kg}: V^{\\otimes k} \\rightarrow \\R$ for $k\\ge 1$, $g\\ge 0$ on an odd symplectic vector space~$V$ and write down an action $\\Action\\in \\Fun(V)$ similar to \\eqref{Eq:MyAction} with~$\\MC_{kg}$ replaced with~$l_{kg}^+$. Notice that whereas they have $l^+_{kg}(v,\\dotsc,v)$ for a ``field'' $v\\in V$, we have $\\MC_{kg}(v_1,\\dotsc,v_k)$ for a ``string of fields'' $v_1\\dotsb v_k\\in \\CycB(V)$. They consider the Schwarz's canonical $\\BV$-operator on $\\Fun(V)$ and show that $\\Action$ satisfies the quantum master equation if and only if $(l_{kg})$ satisfy the relation of a quantum $\\LInfty$-algebra. This is equivalent to the notion of a loop homotopy algebra from \\cite{Markl1997} and to string brackets in closed string field theory from \\cite{Zwiebach1992}.\n\nIn order to transform \\eqref{Eq:QME} and \\eqref{Eq:Twist} to the convention of \\cite{Doubek2018}, we make the substitutions $\\BVOp \\mapsto \\BVOp$ and $S \\mapsto S = \\hbar S$.\n\n\nA deformation retract \\eqref{Eq:DefRetr} induces a deformation retract\n\\begin{equation*}\n\\begin{tikzcd}\n\\bigl(\\CycC(V),\\OPQ_{110}\\bigr) \\arrow[loop left]{l}{K_\\CycC}\\arrow[shift left]{r}{P_\\CycC} & \\arrow[shift left]{l}{I_\\CycC} \\bigl(\\CycC(V'),\\OPQ_{110}'\\bigr),\n\\end{tikzcd}\n\\end{equation*}\nwhich further induces a deformation retract\n\\begin{equation*}\n\\begin{tikzcd}\n\\bigl(\\Fun(B(V)),\\hat{\\OPQ}_{110}\\bigr) \\arrow[loop left]{l}{K}\\arrow[shift left]{r}{P} & \\arrow[shift left]{l}{I} \\bigl(\\Fun(B(V')),\\hat{\\OPQ}_{110}'\\bigr).\n\\end{tikzcd}\n\\end{equation*}\nWe can take $P$ and $P$, resp.~$I$ and $I$ to be the natural extensions of $\\iota^*$, resp.~$\\pi^*$. In~\\cite{Doubek2018}, they write down formulas for $K_F$ given $K$ using a tensor trick due to Eilenberg Mac-Lane (the same method may apply to get $K$ from $\\Htp$ too). Note that any surjective quasi-isomorphism over $\\R$ is a deformation retract.\\ToDo[caption={injectiv qi},noline]{Are injective quasi-isomorphisms sections of deformatino retractions?} They idea of \\cite{Doubek2018} would be to view $\\BVOp$ and $\\BVOp^\\MC$ as perturbations of $\\{\\FreeAction,\\cdot\\} = \\hat{\\OPQ}_{110}$ and apply the Homological Perturbation Lemma from \\cite{Crainic2004} to obtain deformation retracts\n\\begin{equation*}\\begin{tikzcd}[execute at end picture={\n\\draw[->,dashed] (3.5,1.5) to[out=0,in=90] node[midway,right,xshift=.5cm]{$\\delta^{(1)} = \\BVOp_0$} (5,.75) to[out=-90,in=0] (3.5,0);\n\\draw[->,dashed] (3.5,1.5) to[out=0,in=0] node[pos=0.8,right,xshift=.3cm]{$\\delta^{(2)} = \\BVOp_0 + \\{\\IntAction,\\cdot\\}$} (3.5,-1.5);\n}]\n\\bigl(\\Fun(B(V)),\\hat{\\OPQ}_{110}\\bigr) \\arrow[loop left]{l}{K}\\arrow[shift left]{r}{P} & \\arrow[shift left]{l}{I} \\bigl(\\Fun(B(V')),\\hat{\\OPQ}_{110}'\\bigr) \\\\\n\\bigl(\\Fun(B(V)),\\BVOp \\bigr) \\arrow[loop left]{l}{K^{(1)}}\\arrow[shift left]{r}{P^{(1)}} & \\arrow[shift left]{l}{I^{(1)}} \\bigl(\\Fun(B(V')),\\BVOp^{(1)}\\bigr)\\\\\n\\bigl(\\Fun(B(V)),\\BVOp^\\MC \\bigr) \\arrow[loop left]{l}{K^{(2)}}\\arrow[shift left]{r}{P^{(2)}} & \\arrow[shift left]{l}{I^{(2)}} \\bigl(\\Fun(B(V')),\\BVOp^{(2)}\\bigr)\n\\end{tikzcd}\\end{equation*}\nwhere\n\\begin{align*} \n \\BVOp^{(i)} &= \\hat{\\OPQ}_{110}' + P(\\Id - \\delta^{(i)}K)^{-1}\\delta^{(i)} I = \\hat{\\OPQ}_{110}' + P \\delta^{(i)} I + P \\delta^{(i)} K \\delta^{(i)} I + \\dotsb \\\\\n I^{(i)} &= I + K(\\Id - \\delta^{(i)}K)^{-1}\\delta^{(i)} I = I + K \\delta^{(i)} I + K \\delta^{(i)} K \\delta^{(i)} I + \\dotsb \\\\\n P^{(i)} &= P + P(\\Id-\\delta^{(i)}K)^{-1}\\delta^{(i)} K = P + P \\delta^{(i)} K + P \\delta^{(i)} K \\delta^{(i)} K + \\dotsb\\\\\n K^{(i)} &= K + K(\\Id-\\delta^{(i)}K)^{-1}\\delta^{(i)}K = K + K\\delta^{(i)} K + K \\delta^{(i)} K \\delta^{(i)} K + \\dotsb\n\\end{align*}\nThe prerequisite for this is ``smallness'' of $\\delta^{(i)}$, i.e., that $(\\Id - \\delta^{(i)} K)$ is invertible, plus another condition which guarantees $P(A K^2 A + A K + K A)I = 0$ where $A= (\\Id - \\delta K)\\delta$. In the case of a special deformation retract, it holds automatically.\n\nIn the special case of being the deformation retract onto the Hodge decomposition \n\n$$ \\BVOp^{(1)} = P\\BVOp_0 I,\\quad \\BVOp^{(2)}=\\BVOp^{(1)} + \\{W,\\cdot\\}^{(1)}\\quad\\text{and}\\quad P^{(2)}_{\\Fun} = Z $$\n\nand define the \\emph{effective action} \n$$W \\coloneqq \\log\\bigl(P'(e^{\\IntAction})\\bigr) \\in \\Fun(\\CycB(V'))$$\nand the \\emph{path integral}\n$$P^{(2)}_{\\Fun} \\coloneqq L_{\\bigl(P'(e^{\\IntAction})\\bigr)^{-1}} \\circ P'_{\\Fun} \\circ L_{P'(e^{\\IntAction})}: \\Fun(\\CycB(V)) \\rightarrow \\Fun(\\CycB(V')).$$\n\n\n\\begin{Question}\nCan one choose the data and apply the Lemma to get\n$$ \\BVOp^{(1)} = \\BVOp_0', \\quad W = \\Action_{\\HTP_* \\MC},\\quad P^{(1)} = e^\\HTP,\\quad P^{(2)}= e^{\\HTP^\\MC} $$\n\\end{Question}\n\n\nThe article of \\cite{Doubek2018} contains references of other authors on how to get the path integral.\n\nThe Chern-Simons is \n\\end{document}\n", "meta": {"hexsha": "105ee311ab82c853d86811985eb8fb220c2a87f2", "size": 6312, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Subfiles/BV_deRham.tex", "max_stars_repo_name": "p135246/phd-thesis", "max_stars_repo_head_hexsha": "0e124466a3d0ff988c012225400fadb0b170aa9e", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Subfiles/BV_deRham.tex", "max_issues_repo_name": "p135246/phd-thesis", "max_issues_repo_head_hexsha": "0e124466a3d0ff988c012225400fadb0b170aa9e", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Subfiles/BV_deRham.tex", "max_forks_repo_name": "p135246/phd-thesis", "max_forks_repo_head_hexsha": "0e124466a3d0ff988c012225400fadb0b170aa9e", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 79.8987341772, "max_line_length": 815, "alphanum_fraction": 0.6498732573, "num_tokens": 2366, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102956, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3466676208393454}} {"text": "\n\\subsection{Symmetric}\n\nWe have a document we want to be able to transfer on an insecure medium.\n\nWe use a key to encrypt the file, and a key to decrypt the file.\n\nWith symmetric encryption these are the same key.\n\n", "meta": {"hexsha": "6e1b6d804f781818022f7ecb246f06e1de627b60", "size": 216, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/pug/theory/statistics/encryptionModern/02-01-symmetric.tex", "max_stars_repo_name": "adamdboult/nodeHomePage", "max_stars_repo_head_hexsha": "266bfc6865bb8f6b1530499dde3aa6206bb09b93", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/pug/theory/statistics/encryptionModern/02-01-symmetric.tex", "max_issues_repo_name": "adamdboult/nodeHomePage", "max_issues_repo_head_hexsha": "266bfc6865bb8f6b1530499dde3aa6206bb09b93", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-03-03T12:36:56.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-01T22:16:09.000Z", "max_forks_repo_path": "src/pug/theory/statistics/encryptionModern/02-01-symmetric.tex", "max_forks_repo_name": "adamdboult/nodeHomePage", "max_forks_repo_head_hexsha": "266bfc6865bb8f6b1530499dde3aa6206bb09b93", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.6, "max_line_length": 72, "alphanum_fraction": 0.7638888889, "num_tokens": 49, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.3466676072981253}} {"text": "\\chapter{Physical Background }\n\\label{chapter:Introduction}\n\\textit{In this chapter raise and explains the basis of fundamental physics in semiconductor and quantum structures, implemented to understand the results in this work.}\n\\vfill\n\\minitoc\n\\newpage\n\n\\lettrine[lines=3, lraise=.1, nindent=0mm, slope=0mm]{\\textbf{Q}}{uantum mechanics} is basically electron behavior that exhibits many phenomena non explained by classical regime. Quantum structures (QS) are artificially systems conformed by semiconductors where electrons exhibit their quantum nature, this is a great platform to study and create quantum devices. Nowadays, the progress in creation of QS consist in precisely deposition of thin films, in which electrons show fundamentally new electrical and optical properties\\cite{sundram1991structures}. Most of these properties consist in quantum behavior as the energy confinement, which is the principal interest to study the electron and their consequent interactions which generates analogous hydrogen atom in semiconductors. Therefore, the interest to studying QS was increasing for many years ago and nowadays, those continue considering an excellent research area. \n\nIn this chapter, it presents the fundamental concepts to describe the physical phenomena resultant in this work, without intention to replicate concepts and models already explained in publications with major impact. Therefore, the purpose is to present an own interpretation to highlight the great obtained results.\n\n\\section{Semiconductor Bandstructure}\n\\label{sec:chapter-1-semiconductor}\n\\vspace{-10mm}\nTo starting with physical background to understand the QS, it has to start with understand the band structure of semiconductors. The band structure describes the electron behavior in a solid, therefore, we will need to invoke the \\sch equation to describe it behavior. But, due inside a solid around $10^{23}$ valence electrons contribute to the bonding in each cubic centimeter, this results in a many-body complex problem\\cite{piprek2017handbook}, then the general hamiltonian for a solid has the form\\cite{alloul2010introduction,cardona2005fundamentals}: \n\n\\begin{equation}\n\\begin{split}\n\tH = &\\dfrac{1}{2M}\\sum\\limits_{i=1}^{N_{n}} \\bff{P}_{j}^{2} + \\dfrac{1}{2m_{0}} \\sum\\limits_{j=1}^{N_{e}} \\bff{p}_{j}^{2} + \\dfrac{Z^{2}}{2} \\sum\\limits_{i,j=1,i\\neq j}^{N_{n}} V_{c}\\left(\\bff{R}_{i}-\\bff{R}_{j}\\right)-Z\\sum\\limits_{i=1}^{N_{n}}\\sum\\limits_{j=1}^{N_{e}}V_{c}\\left(\\bff{r}_{j}-\\bff{R}_{i}\\right) \\\\\n\t & + \\dfrac{1}{2} \\sum\\limits_{i,j=1,i\\neq j}^{N_{e}} V_{c} \\left(\\bff{r}_{i}-\\bff{r}_{j}\\right).\n\\end{split}\n\\label{eq:chapter-1-solid-hamiltonian}\n\\end{equation}\n\nWhere $N_{n}$ is the number of atomic nuclei, $N_{e}$ is the number of electrons with mass $m_{0}$, asumming that the nuclei are the same mass it's consider $M$ and charge $Z_{e}$. As is obviously, this Hamiltonian is so complicated, the sum of five terms which consists in : kinetic enrgies to electrons and nuclei, the nucleus-nucleus, nucleus-electron and electron-electron Coulomb interactions, also $\\bff{R}_{i}$ are the positions of the nuclei and $\\bff{r}_{j}$ are the position of the electrons, the operators $\\bff{P}$ and $\\bff{p}$ are momentum operators to nuclei and electrons respectively. Finally, the consider of Coulomb potential $V_{c}$\\cite{alloul2010introduction}. \n\n\nFortunately, the QS are formed by crystalline materials, the Bloch theorem provide the most important tool to develop required equations. The Bloch theorem establish a periodic potential $U(\\rv)$ for electrons, this due the material is periodic (definition of crystal structure) and the \\sch equation it can describe in terms of single electron as: \n\n\n\\begin{equation}\n\t\\left[-\\dfrac{\\hbar^2}{2m_{0}}\\nabla^2 + U(\\rv)\\right]\\psi (r)=\\senergy\\psi(\\rv)\n\t\\label{eq:chapter-1-first-sch}\n\\end{equation}\n\nThe principal reason that the periodic potential in a crystal structure is highly important is their translational invariance concept and the consequent symmetry operations that are possible in a crystalline solid. The symmetry concept, as a tool to understand solids, is discussed with major detail in the next chapter. In according to Bloch's theorem we can associate a wave vector $\\boldsymbol{k}$ with each energy state, $E_{n}(\\boldsymbol{k})$. Thus, it is useful to display the energies $E_{n}(\\bf{k})$ as a function of the wave vector $\\boldsymbol{k}$. This result also knowing as dispersion relation, but in general terms is the electron band structure of the given solid\\cite{piprek2017handbook}. \n\nEven if, the calculation of electron band structure in solids are very complex by the distance between atoms that composes it's, and the Bloch's theorem provide the most important tool to reduce the problem to crystalline structures, exists several methods to calculate the realistic bandstructure for semiconductors that are categorized in two groups: Atomistic methods\\footnote{In this category can include the ab initio methods, these are the most complex methods due to propose solutions of the many-body problem} (Tight-binding, orthogonalized plane wave methods) and Perturbative methods ($\\boldsymbol{k}\\bigcdot \\boldsymbol{p}$)\\footnote{In fact, both TB and $\\boldsymbol{k}\\bigcdot \\boldsymbol{p}$ also consider in same kind, because both are \\emph{semi-empirical} methods due to they consider experimental parameters.}. \nThese two main categories with theirs respective methods have special characteristics which becomes in the reasons to choose them. The reasons have to do with to the described bandstructure, this mean, in case of Atomistic methods the entire bands (valence and conduction) can describe, but in case of perturbative methods are reserve to near bandedge bandstructures. So, each of these methods can be chosen and enhanced as the system to study requires. \nWe won't enter in discussion about of which of these methods are the best, the reasons are simple, each method is powerful, and we must be remembered that the complexity of solutions requires that these are solved by numerical techniques therefore this convert it to good approximations. \nThen, basically the electron behavior inside a semiconductor consist in solutions of the appropriate Schrödinger equation\\cite{boer2018semiconductor}. \n\\subsection{Valence and Conduction Bands}\n\\label{subsec:chapter-1-valence-and-conduction-bands}\n\\vspace{-10mm}\nThe most important characteristic of semiconductors even we can call as the fingerprint of these, is their bands structure, this characteristic sort out the solids as insulators, metals, and semiconductors. \nThese are the reason of many mechanisms and phenomena by which study this structures. In general, the bands of semiconductors composed by valence and conduction bands separated by a region known as bandgap. \nThe bandgap, which is proportional to separation energy of valence and conduction bands also is called as forbidden region, this is because doesn't exist electron states, therefore this gap energy determine the electron conduction in a semiconductor and the difference they have with the insulators and metals as shows \\Cref{fig:subsubsection-1.1.1-solid-types}. \n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{../figures/chapter-1/solid-sort/build/solid-sort}\n\t\\caption{Band energy diagram for insulators (left), semiconductors(center) and metals (right). The principal difference is the gap energy, for insulators this is longer than semiconductors, although in semiconductors gap energy depends on materials, finally in metals doesn't exist gap energy instead exist an overlap bands characterize these. Dashed line determine Fermi's level. }\n\t\\label{fig:subsubsection-1.1.1-solid-types}\n\\end{figure}\n\nSo, the bandgap determines many characterizes and functionalities in semiconductors. The bandgap energy classifies semiconductors in direct and indirect semiconductors, but doesn't only depend on this energy, in reality the band structure is the liable for this. As before mentioned, it is so difficult to describe electrons behaviors over the solids due to the many body interactions that exists in its, and therefore the Scr\\\"odinger equation that's describe electrons behaviors is complicatedly to solve. Fortunately, the semiconductor structures have a one of the most important characterize and his has to do with their atomic structure, that's periodically arrangement of atoms.\nThis periodicity is the key to propose solutions and describe the semiconductor band structure. \nStarting with describe bulk semiconductors, for example GaAs which consists in with the family III-V cube semiconductor so that their lattice structure consist in a two sublattices correspond to each atom which it conform as shows in \\Cref{fig:subsubsection-1.1.1-bulk-1}. For this case, when the atoms in two sublattice are diferent, the crystal structure is then called \\emph{zinc-blende}\\cite{vurgaftman2020bands}.\n\nTo calculate bandstructures of bulk semiconductors it's important to define specific symmetry direction, this mean that it's not possible to plot dispersion relation. For each three-dimensional wave vector $\\boldsymbol{k}$, then the plot energy as a function of $\\boldsymbol{k}$ is along of different high-symmetry directions\\cite{piprek2017handbook}. \nIn this work the structures to study are composed of semiconductors III-V, being GaAs the bulk in each structure. GaAs is a direct semiconductor so that the $\\left[001\\right]$ direction is the high-symetry direction then is denoted by $\\Gamma$ point (at $\\boldsymbol{k}=0$)\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{../figures/chapter-1/bulk-1/build/bulk-1}\n\t\\caption{\n\t\t GaAs crystal lattice, where the each sublattice correspond of each atom species Ga and As. }\n\t\\label{fig:subsubsection-1.1.1-bulk-1}\n\\end{figure}\n\n\nAs before mentioned several times, it's very complex to compute the Schrödinger equation in the solids. The most ``exact'' compute is employed by DFT theory, these calculations commonly are called atomistic even some semiempirical models can consider as atomistic, but the semiemprical models are good approximations in comparison with the DFT theory. So, which is the reason to call ``exact'' solutions to the DFT results? The answer leads us to great discussion and it's not intended to get into controversy, but in general the DFT calculations have the capacity to calculate in terms of electrons interaction and the empirical methods are based in potential choice.\\\\\n \nWe will, don't into details about band calculations theories and models, but we will make a general reference to the importance in this work. The models more performed in semiconductor heterostructures as GaAs/AlGaAs are semiempirical, this is because DFT theory and their derived models are very limited to carried out in large structures, their electron interaction nature need high computational perform. So that in comparison with empirical models where the main role is the potential of semiconductor structures, this reduces computational reduce. So, the most models used in semiconductor band calculations are empirical models, these models are distinguished by low computational requires for this reason are considered like approximations. The importance to discuss these concepts will take relevance when we discuss the physics model proposed in this work. \n\n\\begin{figure}[h!]\\label{fig:subsubsection-1.1.1-GaAsbands-1}\n\t\\centering\n\t\\begin{subfigure}{\\textwidth}\n\t\\includegraphics[width=\\linewidth]{../figures/chapter-1/bands/build/bands01}\n\t\\phantomsubcaption\\label{subfig:subsubsection-1.1.1-GaAsbands-1-a)}\n\t\\phantomsubcaption\\label{subfig:subsubsection-1.1.1-GaAsbands-1-b)}\n\\end{subfigure}\n\t\\caption{Band structure of GaAs, \\subref{subfig:subsubsection-1.1.1-GaAsbands-1-a)} hows the zoom around of $\\Gamma$ to denote the direct band gap and the electrons energy needed to jump from valence to conduction band. \\subref{subfig:subsubsection-1.1.1-GaAsbands-1-b)} denotes the two directions to dispersion of the bands corresponds to Brillouin zone: $\\Gamma\\to\\mathrm{X}$ and $\\Gamma\\to\\mathrm{L}$.\\cite{fox2002optical}}\n\\end{figure}\n\nThe \\Cref{fig:subsubsection-1.1.1-GaAsbands-1} shows the results of calculations of TB model as discus it in \\cite{vogl1983asemiempirical} and the code was impmemented by R. Muller \\cite{rpmuller2017}. The model purposes by Vogl et al. take into account small number of localized pseudo-orbitals and based the empirical parameters to substituted on TB Hamiltonian. \nThe importance to get bandstructure it's based importance to study optical properties of solid structures, if it doesn't exist band electrons it's like look a place without map, so, the band structure far from being a complex tool it's the key to get the information to investigate the optical properties. \n\nAs shown in \\Cref{subfig:subsubsection-1.1.1-GaAsbands-1-a)} the GaAs bandstructure shows that it's a direct semiconductor as a previously mentioned, this gives way to get electron transitions from VB to CB and the energy to success this it. In \\Cref{subfig:subsubsection-1.1.1-GaAsbands-1-b)} it's plotted, the band dispertion around $\\Gamma$ point, it's the most symmetry point. It is well-known that the band dispersion increasing $\\boldsymbol{k}$ along two different directions of the Brillouin zone, from $\\boldsymbol{k}=(0,0,0)$ to X point $\\boldsymbol{k}=(2\\pi a_{L})(1,0,0)$ and L point $\\boldsymbol{k}=(2\\pi a_{L})(1,1,1)$. So, these figures, are the typical representation of direct gap to III-V semiconductors around of $\\boldsymbol{k}=0$, then, is obviously that the shape of dispersion is parabolic. For the GaAs bandstructure calculations shown in \\Cref{fig:subsubsection-1.1.1-GaAsbands-1} doesn't take into account the contribution of spin\\footnote{This spin contribution as called as split-off (so) hole band.}, therefore it's focus on three bands dispersion correspond to a single $s$-like conduction band and two $p$-like valence bands. Is important to say that the characteristic curvature $E\\!\\!-\\!\\!\\boldsymbol{k}$ of dispersion bands corresponds to an electron ($e$) in case of positive curvature while the negative curvature correspond to holes states; heavy ($hh$) and light hole ($lh$) bands, so, this denotes that the transitions, are of dipole nature\\cite{fox2002optical,cardona2005fundamentals}. \n\\begin{figure}[h!]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\linewidth]{../figures/chapter-1/bands/build/bands02}\n\t\\caption{Two typical transitions for GaAs near $\\boldsymbol{k}=0$. The first one correspond to the heavy-hole and second one to the light-hole. }\n\t\\label{fig:subsubsection-1.1.1-GaAsbands-2}\n\\end{figure}\nAll the above disputed is significant to refers of one of the most indispensable quantum mechanisms in solids, this is absorption. The electron absorption, specifically interband absorption, give way to a fundamental physical process that involves the principle of many basic studies of semiconductors, applications, and the importance to understand the electron behavior in a semiconductor structures disputed in this work. Then the \\Cref{fig:subsubsection-1.1.1-GaAsbands-2} schematizes the two typical transitions in GaAs bulk, this transitions are near to k=0, so that is called interband absorption. Is important to remark that the interband transitions are observed in all solids, but the mechanisms are different dependently of their bandstructure, for what, being repetitive when mentioning that the bandstructure is the key to study solids. \n\nIn case of GaAs the interband tranisitions are called as direct transitions, this is because their bandstructure proofs GaAs is a direct Gap semiconductor. This process is determined by quantum mechanical rate $W_{i\\to f}$ for exciting and electron in a initial quantum state $\\psi_{i}$ to final state $\\psi_{f}$ by absorption of a photon of angular frequency $\\omega$\\cite{fox2002optical}. As is very know, this is given by Fermi's golden rule. Later this is disputed according to highlight the model and results obtained. It has been mentioned that the direct transitions are of dipole nature, as before mentioned the CB is type $s$-like while the VB is $p$-like, then it've electric-dipole allowed transitions $p\\to s$. The excitation of electron in CB carries to leave an initial state unoccupied, this is called as a hole creation, then the electron in the final state and the hole is considered as \\textbf{electron-hole pair}. The next part subject this theme with major focus.\n\n\n\\subsection{Excitons}\n\\label{subsec:chapter-1-excitons}\n\\vspace{-10mm}\nThe importance to study bandstrcuture of semiconductors is very clear so far, so that could be said that absorption process is the source of optical properties of solids. It's due to this that the importance to study of semiconductors in the optoelectronics applications. But, this process give rise to formation of one of the most important excitations in the crystal structures. The photon absorption process carries to an electron is excited from CB to VB, this generates an empty location in VB which has positive charge. This positive empty location called as hole, therefore the electron and hole have opposite charge then it's to be expected that they are attracted, so, this creates a bound state called an exciton\\cite{leonard2017exciton}. \n\nFrom \\Cref{fig:subsubsection-1.1.1-GaAsbands-2} and \\Cref{fig:subsubsection-1.1.1-x-1} is clearly that excitons are commonly presented in direct band gap semiconductors as GaAs, this was denoted by absorption experiments, after mentioned in \\Cref{subsec:chapter-3-pl} this is the cause in photoluminiscence mechanism.\n\n\\begin{figure}[b]\n\t\\centering\n\t\\includegraphics[width=0.5\\linewidth]{../figures/chapter-1/exciton-1/build/x-1}\n\t\\caption{Qualitative scheme of exciton creation in GaAs as direct gap.}\n\t\\label{fig:subsubsection-1.1.1-x-1}\n\\end{figure}\n\n\\section{Semiconductor Low-Dimensional Structures}\n\\label{sec:chapter-1-low-dimensional-structures}\n\\vspace{-10mm} \nThe previous section engaged to explain the principles of semiconductors, this is the bandstructure, the importance of these is practically the fingerprint of all semiconductor, without bandstructure the understanding of these would be improbable. The first approximations were based in GaAs bulk, their cubic symmetry practically defines their nature and consequently the physical effects as excitons existence. But, what happens if joined several semiconductors with same symmetry and structural parameters? The bulk properties and physical properties are the same?. \nThe answers they are well-known, when two materials with relatively same structural parameters, as lattice constant can create a heterojunction, the union of several heterojunction make up a heterostructure.\\footnote{The samples studied in this work are heterostructures, for this reason, and by nomenclature it's we refer to that way.}\\\\*\n\\begin{figure}[h]\n\t\\centering\n\t\\includegraphics[width=0.65\\textwidth]{../figures/chapter-1/heterostructures/out/hs-01}\n\t\\caption{General scheme of GaAs/\\algaas heterostructure, at top show the scheme of atomic arranged of this heterjunction, the dashed lines are the matched between two dissimilar materials. In bottom shows the band-edge profile.}\n\t\\label{fig:subsection-1.2-heterostructure}\n\\end{figure}\nThe \\Cref{fig:subsection-1.2-heterostructure} is a general scheme of a heterostructure, in this case is presents three species of atoms Al, As, Ga. These atoms can locate in columns III-V of the periodic table, hence its name of III-V semiconductors. The principal characteristic of these atoms is that it can create matched structures as GaAs, AlAs and ternary alloys as \\algaas with specific Al concentration. The matched semiconductors produce a material with new properties based principally in the difference of bandgap which involves the alloys. \\\\* \nAs can see in \\Cref{fig:subsection-1.2-heterostructure} it's consisting a GaAs/\\algaas heterostructure, these interface is well-matched due to the lattice parameters is relatively equals, therefore and thanks to powerful growth technics as MBE it's possible to get high-quality quantum structures. \n\nAlso, the heterostructure composed by two semiconductors with different band gaps generate a discontinuity in either the conduction or the valence band can be represented by a constant potential term\\cite{harrison2016quantum}. The theory to treatment the electron behavior in these structures, is relatively simple if we consider the above. \nAlthough, in this chapter doesn't have intention to explore the theory of electron behavior in that, worth noting that it get one-dimensional potential $V(z)$ to both bands, so the Schr\\\"odinger equation can solve simple. \n\n\n\n\\subsection{Quantum wells}\n\\label{subsection:chapter-1-quantum-wells}\n\\vspace{-10mm} \nDoubtless the creation or growth of heterostructures increased the interest in the study of quantum structures, the interactions, and physical behavior of light-matter they would not have been possible without these. The major relevance is due the quantum confinement, the junction of semiconductors results in an interest quantum structures with specific dimensions. From 3D bulk the dimensions reduce to 2D,1D and 0D dimensional structures. Therefore each of that has interest properties and their correspond applications. The \\Cref{fig:subsection-1.2-heterostructures} schematics the low dimensional heterostructures from 3D bulk, the first low dimensional from 3D to 2D is the Quantum Wells, then from 2D to 1D it have the Quantum Wires finally with 0D have the Quantum Dots.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{../figures/chapter-1/heterostructures/out/lds-00}\n\t\\caption{Heterostructures from bulk (3D), to Quantum Wells (2D), Quantum Wires (1D) and Quantum Dots (0D). }\n\t\\label{fig:subsection-1.2-heterostructures}\n\\end{figure}\nThe quantum confinement so is the principal reason to study that structures, the electron behavior which exhibits in it should can to help understand a great variety of quantum mechanical phenomena as electron interaction on a crystal. \nSuppose a heterostructure composed with a two semiconductor alloys as sandwich, this 2D quantum structure is called a Single Quantum Well (SQW). \\\\*\nThis dissimilar semiconductors in terms of their potential energy ($V(z)$) can be schematized as \\Cref{fig:subsection-1.2-single-quantum-well-scheme}. The Gap difference of \\algaas and GaAs is due to $x$ Al concentration in \\algaas therefore it obtains a one dimensional potential profile, with that can confinement electrons in a 2D plane along $z$ direction. \nAll of these carries to quantum mechanics formalism, the electron behavior should be obeyed these rules. If we have an electron closed in two potential barriers an $L$ distance, the wave which describe it will be spatially confined.\nSo if we confined many electrons in these potential, we have two important physical aspects: the first one is knowing as Pauli's exclusion principle, which as of its Fermion nature prevents carriers with the same spin occupying the same region in of space\\cite{harrison2016quantum,pauli1925zusammenhang}, the second one and one of the most relevant in the birth of the quantum mechanics; the Heisenberg's uncertainty principle. \nThat last, it can say that is the consequence of quantum confinement due to the space reduction of the electrons is expected that momentum increases by an amount of the order $\\hbar/L$. Therefore, the energy of that confined particles increases, and it's referred to as confinement energy\\cite{cardona2005fundamentals}. \\\\*\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.65\\textwidth]{../figures/chapter-1/heterostructures/out/qw1}\n\t\\caption{GaAs/\\algaas Single Quantum Well }\n\t\\label{fig:subsection-1.2-single-quantum-well-scheme}\n\\end{figure}\nThen the quantum confinement is our started point to understand the optical properties in QWs. As is referred in the figure, the uni-dimensional potential profile can well describe by top conduction- and bottom valence-bands, the band offset in these two of correspond gap energy between that, while Al concentration increases their bandgap and the band offset ($\\mathrm{Q_{c}}$ to CB and $\\mathrm{Q_{v}}$ to VB) also to. \n\nIt's so clearly that the QWs have the potential to presents amazing quantum properties, even if all of these are very important we focus on the optical properties, basically our interest is the light-matter interaction through its result mechanisms. \n \n\\subsection{Preliminary approach of Quantum Confinement effect in QWs}\n\\label{subsection:chapter-1-preliminary-approach-of-quantum-confinment-effect-in-qws}\n\\vspace{-10mm} \nAs the title describes, here it will try to explain as the first approach the quantum confinement effect in QWs. If it starts with the scoop, which it can be reduced the electrons space, then this mean that in reciprocal space it has two components $k_{x}$ and $k_{y}$. Then say in crystal symmetry properties to the case of GaAs/\\algaas QWs it's $\\Gamma$ the central point, as long as $x < 0.4$\\footnote{It will be explained in the next section, although it's due to the Gap go from direct to indirect, shortly the symmetry $\\Gamma\\to X$.} the bandstructure depends on confinement energ, so say which the bandstructure depends on confinement energy. \nIn this case, the electronic properties in comparison with a bulk semiconductor properties can solve trough particle-in-a-box as textbook problem as first approach. Nevertheless, even if usually can solver without much mathematical formalism is very essential that it dedicates a chapter with their solution, this is because will employ a physical formalism exclusively to QWs structures. In general way, as it before mentioned the Schrödinger equation solution is the fundamental pillar to understand, where it's taken into account which in a crystal the periodic potential is the key. Here are important remarks before to continue, when it has a heterostructure starting with the bulk model it's clearly that the system doesn't same, the Quantum Mechanics which is behind take into account the symmetry properties, then it can be developed a Hamiltonian to understand that system. In the next chapter will be explained details and the formalism both physical and mathematical to solve and discus it's. The model which give the tools to get the solutions is called as Effective Mass Approximation, thus their correspond Schrödinger equation is\\cite{harrison2016quantum,chuang1995physics,singh2003electronic,bastard1990wave,fox2002optical,davies1998physics}: \n\\begin{equation}\\label{eq:chapter-1-ema-schroedinger}\n\t-\\dfrac{\\hbar^{2}}{2m^{*}}\\dfrac{\\partial^{2}}{\\partial {z}^{2}}\\psi(z)+V(z)\\psi(z)=E\\psi(z),\n\\end{equation}\n\nwhere the $m^{*}$ is the effective mass in each material, and $V(z)$ is the potential profile got by heterostructure materials properties. Therefore, that differential equation can solve as in textbooks explained\\cite{de2014introduccion,griffiths2018introduction,sakurai1995modern,cohen2019quantum,chuang1995physics,harrison2016quantum,fox2002optical,bastard1990wave}. The idea is thinking as a one particle in a finite potential well, where is well important established the boundary conditions and solve the Schrödinger equation in each part of single QW, this means that need to create a potential function. Then it's can obtain the Eigenfunctions and their correspond Eigenenergies.\n\n\\begin{figure}\n\t\\begin{subfigure}{\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=0.65\\textwidth]{../figures/chapter-1/heterostructures/out/qw2}\n\t\t\\phantomsubcaption\\label{subfig:subsection-1.2-single-quantum-well-scheme2-a)}\n\t\t\\phantomsubcaption\\label{subfig:subsection-1.2-single-quantum-well-scheme2-b)}\n\t\\end{subfigure}\n\t\\caption{General scheme of typical Schrödinger's equation solutions to one-dimensional potential as \\subref{subfig:subsection-1.2-single-quantum-well-scheme2-a)} where the Eigenenergies of both electron and holes are denoted with same color depending on $n$ value.\\subref{subfig:subsection-1.2-single-quantum-well-scheme2-b)} It's plot, of the subbands in the same case of \\subref{subfig:subsection-1.2-single-quantum-well-scheme2-a)} to both particles. }\n\t\\label{fig:subsection-1.2-single-quantum-well-scheme2}\n\\end{figure}\n\nThe principal idea doesn't is reproducing something which is very well known, the objective of this part is established the scoop of the next chapter. Therefore, before to continue, we will finish with to explain the dispersion in-plane of single QW. As in the QW the one-dimensional potential set up the 1D confinement, is important doesn't confuse which the QW is a 2D structure, but their confinement is along of $z$ direction this mean 1D. Then, the particle can motion in the $x-y$ plane. By this reason, even if consider 3D Schrödinger equation and the above is considered it obtain \\Cref{eq:chapter-1-ema-schroedinger}\ntherefore, the solutions in the one-dimensional potential produce discrete states of energy $E_{z}=E_{n}$\\cite{harrison2016quantum}, where $n$ is the energy level it which produce subbands as shows in \\Cref{fig:subsection-1.2-single-quantum-well-scheme2}. In contrast, before it called as ``energy bands'' in the bulk case, now due to the quantum confinement gets subbands to both conduction- and valence bands.\n\nThese subbands are the result of the sum of $E_{z}$ and $E_{x,y}$, which are the 1D confinement energy and the in-plane momentum $k_{x,y}$ then\\cite{harrison2016quantum}:\n\\begin{equation}\\label{eqn:chapter-1-total-enery-ema-aprox}\t\t\n\tE = E_{n} + \\dfrac{\\hbar^{2}|\\boldsymbol{k}_{x,y}|^{2}}{2m^{*}}.\n\\end{equation} \nFrom equation the effective mass $m^*$ depends on particle, i.e the effective mass to electrons in CB and the holes in VB. So, the most relevant in the solutions is the energy $E_{n}$ (\\Cref{subfig:subsection-1.2-single-quantum-well-scheme2-a)}) is discrete, this is the quantum confinement in the low-dimensional heterostructures. \n\n\n\\section{Summary}\n\\vspace{-10mm} \nIn this chapter, was exposed the generalities of semiconductor band structure and low-dimensional heterostructures, highlighting or taking in major GaAs/\\algaas that's the semiconductors of major importance in this work. \nThe band structure interpretation usually be so hard, and their calculations even more, but the impact and relevance in optical properties of semiconductors starts from that interpretation, from here arises the mathematical arsenal to right physical interpretation. Another significant concept which was treatment as first approach is the effective mass concept, even if, when solved, the bulk Hamiltonian it considers the mass as constant parameter or depending on semiconductor material, contrary in low-dimensional structures have an important role. \\\\*\n\nIn generally, the band structure of semiconductors is the key to understand quantum properties of solids, in this work the relevant is the light-matter. Remember that light-mater interaction in solids can be studied by process resulting in it, as absorption, reflection, transmission, diffraction, scattering, and others\\cite{rivera2020light}. Although, the light-matter interactions are fundamentally quantum electrodynamical, also, can be studied in quantum way through before mentioned process. Firstly, is the photon absorption process can help to understand or calculate the fundamental parameter in semiconductors as is bandgap energy. \nThis parameter is the started point in the study of semiconductors, this is the start point in the map called bandstructure, if it ignores the gap value in the semiconductor to study it couldn't possibly get the principal optical properties of that. \\\\*\nThen, the bandstructure of semiconductors is the map to understand them, without these routes or fundamental parameters couldn't have quantum devices. \n \n", "meta": {"hexsha": "f3ed8950f34fe78d10b9938bcee85464f409f0f9", "size": 31980, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "phd-thesis/files/chapter-1-introduction.tex", "max_stars_repo_name": "lflmgroup/ruco-phd-project", "max_stars_repo_head_hexsha": "b26156508d1aff06825360470608abf8559b3388", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "phd-thesis/files/chapter-1-introduction.tex", "max_issues_repo_name": "lflmgroup/ruco-phd-project", "max_issues_repo_head_hexsha": "b26156508d1aff06825360470608abf8559b3388", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "phd-thesis/files/chapter-1-introduction.tex", "max_forks_repo_name": "lflmgroup/ruco-phd-project", "max_forks_repo_head_hexsha": "b26156508d1aff06825360470608abf8559b3388", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 159.9, "max_line_length": 1529, "alphanum_fraction": 0.7967166979, "num_tokens": 7749, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3466676072981252}} {"text": "\\documentclass{article}\n\n\\title{FreeCat design document}\n\\author{Morgan Thomas}\n\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\n\\begin{document}\n\n\\maketitle\n\nProgramming languages can usually be classified as typed or untyped. Untyped languages are usually relatively simple and flexible, but less capable of automatically finding errors in programs. Typed languages are usually more capable of automatically finding errors in programs, but this usually comes at the expense of flexibility and expressive capability. Expressive capability of typed languages can be increased, but this usually comes at the expense of increasing their complexity.\n\nFreeCat is a programming language, a system of logic and computer algebra, and an approach to type theory and the foundations of math. It aims to chart a new programming language design path, where the guiding hope is that we can have automatic error finding, language simplicity, flexibility, and expressive capability, all at the same time, all to a degree which is robust and state of the art. FreeCat is based on the hypothesis that we can do this by following a language design approach based on type theory, when we do not try to rule out logical inconsistency in the core type theory.\n\nBy definition, a system is logically inconsistent when it can prove that some statement is both true and false. In the case of type theory based programming languages such as FreeCat, this definition depends on the Curry-Howard correspondence, under which types are considered as statements, and values are considered as proofs of the statements which are their types. The statement that FreeCat is logically inconsistent means in other words that in FreeCat, for any type it is possible to construct a value inhabiting that type.\n\nProgramming languages to which FreeCat is similar include Haskell, PureScript, ML, Agda, Coq, Cayenne, and Idris. FreeCat follows a novel variant of a well-established type theory based approach to programming language design.\n\nWhat's novel (as far as I'm aware) about FreeCat is that it accepts the possibility of logical inconsistency, and it incorporates the assumption that logical inconsistency will be possible, starting from the design of the core type theory. In contrast, similar languages I'm aware of have mechanisms for ruling out inconsistency incorporated into the core type theory, even in cases where the final language is inconsistent. The motivation for designing FreeCat in this way is the hope that the resulting language will be significantly simpler and easier to understand, compared to alternatives, while also being more flexible and expressive.\n\nThis document describes FreeCat from a theoretical perspective. There are two main target audiences. First, programming language theorists who want to understand FreeCat's theoretical contributions. Second, people who want to understand the implementation of FreeCat, for example in order to contribute to it.\n\nFreeCat is an open source, non-profit project whose primary aim is to give people a new way to develop software. The hope is that FreeCat will help people reduce their costs for developing software, while achieving higher quality.\n\nA further, related aim of this project is to describe a new approach to understanding the foundations of math. The hope is that people will find this approach simpler, conceptually clearer, and capable of expressing more concepts, compared to alternatives. The capability to express more concepts comes from not imposing restrictions designed to prevent logical inconsistency. This approach is limited in scope to constructive math, and more specifically it is limited to countably infinite objects and (partial) computable functions. These limitations can probably be removed if one gives up the goal of being able to implement the system as a computer programming language.\n\nFreeCat doesn't aim to be the best approach to solving everybody's problem. For now the focus is on making it a good approach to solving some problems.\n\nThe current implementation of FreeCat can be considered the reference implementation in progress. This implementation provides the most complete definition and explanation of FreeCat. This implementation doesn't focus on performance optimization. The plan is to keep the reference implementation as simple as possible, optimizing for ease of human comprehension. Once the reference implementation is mature, it would provide a good basis for writing a performance-optimizing implementation.\n\nThe conception and justification of FreeCat is rooted in a holistic, interdisciplinary approach to technical analysis. This holistic methodology assumes that there are deep connections between fundamental problems in different technical disciplines. It assumes in essence that we can clarify our thinking about technical problems in general by developing theories which can systematize solutions to technical problems in general.\n\nThis design document aims to give a self-contained conceptual explanation of FreeCat. The Context section is not self-contained. The other sections should explain FreeCat in a self-contained way which does not require understanding of the Context section. It's hard to state exact prerequisites for reading this document, but practically necessary prerequisites include some familiarity with functional programming and programming language theory concepts.\n\nPrior experience in any of the following areas is helpful, but not necessarily required, for understanding this document: type theory, foundations of math, mathematical logic, philosophical logic, category theory, abstract/universal algebra, holistic analysis.\n\n\\section{Context}\n\nThis section gestures at some of the areas of thought which are closely related to FreeCat. The main aim is to give an idea of where FreeCat's intellectual DNA comes from. For people familiar with these fields, hopefully it will help to contextualize FreeCat. None of this section is necessary for understanding FreeCat. It aims to provide perspectives which are useful for understanding FreeCat and its motivations.\n\nTODO: sprinkle some citations\n\n\\subsection{Symbolic computation}\n\nSymbolic computation is one conceptual paradigm from which FreeCat is derived. A prototypical example of a symbolic computation system is Lisp. In Lisp, the most fundamental data structures are symbols and conses. These two simple data structures suffice for basic untyped symbolic computation. Typed symbolic computation languages include ML, Haskell, PureScript, and so forth. In these languages the notion of algebraic data types replaces the notions of symbols and conses.\n\nFreeCat is another typed symbolic computation language. The basic and only data structure is the (symbolic) expression. The simplest types of expressions are symbols and function applications. Constructors are functions with no reduction rules. Instead of algebraic data types, users are given the ability to declare symbols to have arbitrary types. They can use this feature to declare types, values belonging to types, and constructors.\n\n\\subsection{Type theory and set theory}\n\nType theory is a field of math and a way of describing the foundations of math. Type theory is closely related to set theory. Set theory is the study of sets. Sets are collections of objects: e.g., the set of all even integers, the set of all unicorns, the set of all sets.\n\nType theory is basically a theoretical approach to studying sets, which can be distinguished from classical set theory. Classical set theory, as exemplified by ZFC, was originally motivated in essence by the desire to provide theoretical foundations for math. Type theory today is most interesting for its applications to technology. Many computer programming languages and software verification systems are based closely on type theory. Examples include Haskell, ML, Coq, Agda, and Idris. The demonstrated applicability of type theory to technology distinguishes it from classical set theory. The reason for the distinction is that type theory has been formulated in such a way that computers can feasibly process it.\n\nThere are other interesting differences between type theory and classical set theory. In classical set theory, every object is an element of infinitely many sets, whereas in many versions of type theory (FreeCat included), every object is an element of at most one set (i.e. type).\n\nAlso, classical set theory includes the principle of extensionality, which states that if two sets have exactly the same elements, then they are the same set. Type theory frequently does not include the principle of extensionality. In most computer implementations of type theory, the principle of extensionality is not included, or it's trivially true because no two distinct sets have any elements in common.\n\nHere is a further important difference between type theory and classical set theory. Classical set theory is based on classical logic, most often first-order logic. The classical approach to logic and set theory defines logical rules of inference, such as those of first-order logic, and axioms about sets, such as those of ZFC. When combined they let you do math, once you've defined every mathematical concept in terms of sets. Type theory, on the other hand, simply defines rules for talking about types, and the rules of logic can be derived from these rules via the Curry-Howard isomorphism. Whereas classical set theory is founded on logic, type theory stands alone and can be used as a foundation for logic.\n\nIn short, type theory is a way of thinking about sets, which can form the basis of computer programming languages, which often relaxes or trivializes the principle of extensionality, and which unifies logic with set theory via the Curry-Howard isomorphism.\n\n\\subsection{Constructive math}\n\nConstructive math is a style of math in which one restricts one's attention to mathematical objects which can be constructed in some suitable sense. For example, if you are doing math which only pertains to a finite number of things which could fit in the universe, then in a robust sense the math you are doing is constructive.\n\nIntuitionism is a subfield or paradigm of constructive math. Speaking generally and simplistically, intuitionists think that math properly concerns objects which can be constructed mentally. Intuitionists think that if an object exists for the purposes of math, then we can mentally construct (conceptualize, visualize) the object to any desired degree of detail. Whether you agree or disagree with intuitionists, their idea of mental construction of mathematical objects provides an accessible way of thinking about constructive math.\n\nConstructive math does not necessarily restrict its attention to finite objects only. There are a wide variety of ways of understanding the notion of constructive math, and the widest variety is in the ways of understanding constructive math which assume that constructive math can study infinite things.\n\nThe varieties of type theory which are most relevant to technology are constructive type theories. What it means for a type theory to be constructive is a subtle question to which many answers are possible. For purposes of intuition, I lean on the intuitionist notion of mental construction. Constructive type theory, I'll say, is only concerned with objects which can be mentally constructed to any desired degree of detail, as well as realized in computer implementations. Roughly, constructive type theory (conceived this way) can talk about finite objects, countably infinite types, and infinite objects which can be described by means of (partial) computable functions. For example, constructive type theory can talk about the type of integers, and the type of (partial computable) functions from integers to integers.\n\nThe real numbers are an example of something constructive type theory can't talk about. There are uncountably many real numbers. If you had a computer with infinite memory and infinite computation time, such a computer could not represent and do arithmetic on all real numbers. That statement assumes the hypothetical infinite computer is built in something resembling a conventional manner: something like an infinitely large circuit board whose circuits are of uniform density. Actual computers have only finite memory and computation time. Uncountably infinite sets like the real numbers are far beyond what actual computers can fathom. Therefore they are beyond the realm of constructive type theory as I'm conceiving it.\n\n\\subsection{Inconsistent math}\n\nFreeCat is a system of inconsistent math. ``Inconsistent math'' refers to any style of math where contradictions can be true, or in other words where the same statement can be both true and false. Inconsistent math is a fairly obscure and unpopular subfield of math, which I believe is mostly studied and practiced by philosophers.\n\nInconsistent math is unpopular probably in large part because most people assume that it is absurd and impossible for any statement (or at least any statement of math) to be provably both true and false. FreeCat starts by rejecting that assumption, following other scholars including C.E. Mortensen, Mark Colyvan, Zach Weber, Graham Priest, Jc Beall, and David Ripley, to name a few.\n\n\\subsection{Functional programming}\n\nFreeCat is a functional programming language. Functional programming is a way of thinking about software engineering. It is a very broad topic which today has increasing importance to practical software engineers.\n\nFunctional programming languages can be classified as pure or impure. In pure functional languages, there is no notion of mutable state in the semantics of the core language. Pure functional languages ask the programmer to think about a world in which there is no change. Impure functional languages lack this property.\n\nFunctional programming languages can be classified as typed or untyped, depending on whether or not they have a type system as part of the semantics of the core language.\n\nFreeCat is a pure typed functional programming language. This puts it in the same class as programming languages including Agda, Cayenne, Coq, Haskell, Idris, and PureScript.\n\nAll of these languages are based on type theory. Type theory provides the underlying mathematical theories which are used to construct the languages mentioned in the previous paragraph. As far as I understand, in every case mentioned, the underlying type theories are constructive. This makes sense because constructive type theories can talk about what computers can talk about, whereas non-constructive type theories more or less by definition (depending on your definition) can talk about some things which computers can't talk about.\n\nMost of the type theory literature is concerned with logically consistent type theories. A logically consistent type theory is one in which it is not possible to prove that the same statement is both true and false. To prove a statement is in other words to construct an inhabitant of the type which is the statement under the Curry-Howard correspondence.\n\nPractical typed functional programming languages tend to be logically inconsistent. This means that in these languages it's possible to prove any statement, or in other words (under the Curry-Howard correspondence), for any type it's possible to construct a value inhabiting it. Haskell, Idris, PureScript, Agda, and Coq all have this property, either by the nature of the type system (as with Haskell), or by importing a module (as with PureScript), or in the case where you make assumptions which are logically inconsistent (as with Agda and Coq).\n\nThere are some good reasons to make a practical programming language inconsistent or potentially inconsistent. For example, a language is potentially inconsistent if you are able to tell the implementation to assume that something is true when you haven't proven it. One important reason to want this capability is to be able to compile a program which is missing a proof you don't yet understand how to construct or don't have time to construct.\n\nAnother reason to allow a programming language to be inconsistent is that the restrictions which type theorists impose to avoid logical inconsistency tend to prevent some ideas from being expressed.\n\nThere is an interesting disconnect here. Type theorists usually study consistent type theories. Designers of practical typed functional programming languages tend to construct inconsistent languages, and there seem to be some good reasons for that. Yet language designers who rely on type theory have tended to base their work on consistent type theories, even when the languages they design are inconsistent in the end. Arguably this leads to a worst of both worlds situation, where you have restrictions on what you can express which are intended to prevent inconsistency, but you also have the potential of inconsistency.\n\nFreeCat is based on the approach of rethinking type theory from the beginning with the assumption that it's not necessary to take steps to prevent the possibility of inconsistency. Of course it will be useful to have tools to try to discover inconsistency, or to rule out inconsistency in cases where this is practical and desirable. But FreeCat's core type theory doesn't contain restrictions designed to prevent paradoxes. This simplifies the core type theory and language design, and the hypothesis is that this will lead to substantial additional simplifications further down the road.\n\nThe previous statements locate FreeCat in the space of conceptual paradigms discussed so far. FreeCat is a pure typed functional programming language and an approach to the foundations of math. It is founded on a form of inconsistent constructive type theory.\n\n\\subsection{Category theory}\n\nCategory theory is a field of math with close connections to logic, type theory, and the foundations of math, among many other fields. Category theory is a valuable tool for interdisciplinary/holistic technical analysis. Category theory provides a useful perspective on the core concepts of FreeCat. FreeCat, in turn, provides a useful perspective on the core concepts of category theory.\n\nFreeCat has the core concepts of types and functions, which correspond to category theory's notions of objects and morphisms. All FreeCat constructions can be considered to occur in a category of all FreeCat types, within which it is possible to construct an infinite variety of categories.\n\nFrom this perspective, FreeCat defines a form of inconsistent constructive category theory. This form of category theory abandons the usual paradox-avoiding restrictions of category theory, with the consequence that categories such as the category of all categories and the category of all groups can be discussed.\n\n\\subsection{Abstract/universal algebra}\n\nAbstract algebra is a field of math which (basically) studies mathematical objects considered as sets of abstract objects with operations defined on them, with the sets and operations usually required to satisfy axioms.\n\nUniversal algebra is a field of math which tries to generalize the patterns of abstract algebra in the form of general theories of algebras and concepts which help in studying algebras.\n\nFreeCat is conceptually connected to abstract algebra and universal algebra. FreeCat provides a system for defining sets (types) and operations (functions) on them. FreeCat can be viewed as a general system for constructing algebras. As such, FreeCat provides a way of thinking about abstract/universal algebra, while abstract/universal algebra provides a way of thinking about FreeCat.\n\n\\section{Basic Concepts}\n\nThe most basic concept in FreeCat is the concept of an object. Everything FreeCat can talk about is an object. ``Object'' is the root node of FreeCat's conceptual taxonomy.\n\nTypes are a fundamental class of object. Every object is an instance of some type. There is a type called Type. Every type is an instance of Type. In particular, Type is an instance of itself.\n\nObjects in FreeCat are represented by expressions. For all practical purposes, in FreeCat, objects can be equated with expressions.\n\nFunctions are a fundamental class of object. A function maps objects of some type to objects of other types. The type of a function's output can be dependent on the object it receives as input. In other words, FreeCat is a dependently typed language.\n\nFunctions are defined by pattern matching equations. These equations express rules stating that expressions of given shapes can be replaced by other, dependent expressions. A function can have different equations defined for it in different contexts. A lambda expression can be thought of as having exactly one pattern matching equation.\n\nA constructor is a function which has no pattern matching equations defined. Whether a function is a constructor or not can be dependent on the context.\n\nFreeCat expressions are evaluated following the rule of eager evaluation, where subexpressions of an expression are completely evaluated, left to right, before performing any evaluation step on the outermost expression. Pattern matching definitions are tried in the order they were declared, with the first match being used.\n\nEvery expression has a type, which is intrinsic to it. The type is itself an expression, whose type is Type. The type of an expression of a specific form will depend on the context in which the expression originally occurs. A symbol name, such as ``foo'' or ``Type,'' may denote different symbols in different contexts. As such, syntactically identical expressions may be semantically different expressions.\n\nThere is a function digestExpr of type $\\text{Context} \\to \\text{RawExpr} \\to \\text{FreeCat Expr}$. (Here FreeCat denotes the FreeCat monad, which accounts for error handling and other issues.) A RawExpr or raw expression is the syntax of an expression without any contextual semantic interpretation applied to it. Given a context and a raw expression, digestExpr gives you an expression which is the interpretation of that raw expression in that context (or else it throws an error, etc.) This function, digestExpr, describes the one-to-many, context-dependent correspondence between syntactic or raw expressions, and semantic expressions, i.e. expressions.\n\nThere are the following kinds of expressions: symbols, applications, lambdas, and (dependent and non-dependent) function types.\n\nA context can be represented as a sequence of declarations. There are the following kinds of declarations: type declarations and equation declarations.\n\nType declarations assert that a given symbol has a given type. Whenever you have an expression denoting a type, you can declare a symbol to have that type, as long as you haven't previously declared a type for that symbol in the current context.\n\nEquation declarations, also called pattern matching definitions or reduction rules, assert that during evaluation, expressions following the left hand side pattern should be replaced by the right hand side expression. As usual, the right hand expression may be dependent on variables in the left hand pattern.\n\nPatterns are like expressions, but they're restricted to consist of symbols and applications. The first symbol in the pattern is called the lead symbol. An equation declaration's left hand pattern must have a lead symbol whose type has been declared previously.\n\n\\section{Syntax}\n\nTODO\n\n\\section{Semantics}\n\nSemantics is the study of meaning. Whereas the syntax of FreeCat tells you what FreeCat expressions and declarations look like, the semantics of FreeCat tells you how to interpret FreeCat expressions and declarations. Specifically, it gives you a computer-implementable theory of how to interpret FreeCat expressions and declarations.\n\nThis section gives a high level overview of the semantics of FreeCat. For a complete definition of the semantics of FreeCat, see the code, and in particular the file src/FreeCat/Core.hs.\n\nA context in FreeCat is represented by a sequence of declarations: type declarations and equation declarations.\n\nThere are basically two aspects to the semantics of FreeCat: the semantics of contexts, and the semantics of expressions. The semantics of contexts tells you how to interpret a representation of a context (i.e., a series of declarations). The semantics of expressions tells you how to interpret an expression relative to a context.\n\nThe semantics of contexts is basically characterized by a function digestContext of type $\\text{RawContext} \\to \\text{FreeCat Context}$. Here RawContext is the type of uninterpreted, syntactic sequences of declarations. Context is the type of contexts, and FreeCat is the FreeCat monad, which deals with error conditions and other things.\n\nThe semantics of expressions is basically characterized by two functions: digestExpr and evaluate.\n\ndigestExpr has type $\\text{Context} \\to \\text{RawExpr} \\to \\text{FreeCat Expr}$. It takes as input a context and an uninterpreted syntactic expression, i.e. a RawExpr. It produces as output an interpreted semantic expression, i.e. an Expr (or it throws an error, etc.)\n\nevaluate has type $\\text{Context} \\to \\text{Expr} \\to \\text{FreeCat Expr}$. It takes as input a context and an expression, and produces as output an expression. The output expression is produced from the input expression by repeatedly reducing all reducible subexpressions, in depth first left to right order, so that in the end there are no reducible subexpressions. A reducible subexpression is either a function application whose head (i.e., left-hand subexpression) is a lambda expression, or a function application matching the left hand pattern of some equation declaration in the current context.\n\nLet's now unpack the types Context and Expr and related types. Both of these types involve the type Symbol, which we'll look at first.\n\nA Symbol basically consists of the following data: its name (a string), its type (an Expr), its native context (the Context in which it first appeared), and its equations (a list of Equation objects representing equation declarations). Symbols are considered to be the same symbol when they have the same name and native context. However, different occurrences in different contexts of the same symbol may have different equations.\n\nAn Equation is of the form $a_1 : t_1, ..., a_n : t_n. p \\Rightarrow d$, where $a_1,...,a_n$ are symbols, $t_1,...,t_n$ are Expr of type Type, $p$ is a Pattern, and $d$ is an Expr. Each Equation is associated with a Context, called its evaluation context.\n\nTODO\n\n\\end{document}\n", "meta": {"hexsha": "f5bb6eb705ad631c29ed99fc22e1175aab08225b", "size": 26617, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/design-doc/design-doc.tex", "max_stars_repo_name": "morganthomas/freecat", "max_stars_repo_head_hexsha": "a608a22e44f01cb61db73adfd5aa96d26b8c51e6", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-12-02T20:17:29.000Z", "max_stars_repo_stars_event_max_datetime": "2017-12-02T20:17:29.000Z", "max_issues_repo_path": "doc/design-doc/design-doc.tex", "max_issues_repo_name": "morganthomas/freecat", "max_issues_repo_head_hexsha": "a608a22e44f01cb61db73adfd5aa96d26b8c51e6", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 37, "max_issues_repo_issues_event_min_datetime": "2017-12-02T18:10:06.000Z", "max_issues_repo_issues_event_max_datetime": "2018-02-24T17:34:46.000Z", "max_forks_repo_path": "doc/design-doc/design-doc.tex", "max_forks_repo_name": "morganthomas/freecat", "max_forks_repo_head_hexsha": "a608a22e44f01cb61db73adfd5aa96d26b8c51e6", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 142.3368983957, "max_line_length": 823, "alphanum_fraction": 0.8167336665, "num_tokens": 5067, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266116, "lm_q2_score": 0.6477982315512488, "lm_q1q2_score": 0.34663581577017116}} {"text": "\\newcommand{\\TOMNetTrimap}{TOM-Net$^{\\text{+Trimap}}$\\xspace}\n\\newcommand{\\TOMNetBg}{TOM-Net$^{\\text{+Bg}}$\\xspace}\n\\chapter{Learning Transparent Object Matting}\n\\label{ch:tomnet}\n\n\\section{Introduction}\nImage matting refers to the process of extracting the foreground matte of an image by locating the region of the foreground object and estimating the opacity of each pixel inside the foreground region. The foreground object can then be composited onto a new background image using the \\emph{matting equation}~\\cite{smith1996blue} \n\\begin{equation}\n \\label{eq:alphamatte}\n C = F + (1-\\alpha)B, \\quad \\alpha \\in [0, 1],\n\\end{equation}\nwhere $C$ denotes the composited color, $F$ the foreground color, $B$ the background color, and $\\alpha$ the opacity.\n\nImage matting has been widely used in image editing and film production. However, most of the existing methods are tailored for opaque objects, and cannot handle transparent objects whose appearance depends on how light is refracted from the background.\n\nTo model the effect of refraction, Zongker~\\etal~\\cite{zongker1999environment} introduced \\emph{environment matting} as \n\\begin{equation}\n \\label{eq:em_general}\n C = F + (1-\\alpha)B + \\Phi, \\quad \\alpha \\in [0, 1],\n\\end{equation}\nwhere $\\Phi$ is the contribution of environment light caused by refraction or reflection at the foreground object. Besides estimating the foreground shape, environment matting also describes how objects interact with the background. \n\n\\begin{figure}[t] \\centering\n \\includegraphics[width=\\textwidth]{ch-tomnet/images/Method/Intro_network_intro_v3}\n \\caption[Learning transparent object matting]{Learning transparent object matting. Given an image of a transparent object as input, our model can estimate the environment matte (consisting of an object mask, an attenuation mask, and a refractive flow field) in a feed-forward pass. The transparent object can then be composited onto new background images with the extracted matte.}\n \\label{fig:intro}\n\\end{figure}\n\nMany efforts~\\cite{chuang2000environment,wexler2002image,peers2003wavelet,zhu2004frequency,duan2011flexible,duan2015compressive} have been devoted to improving the seminal work of~\\cite{zongker1999environment}. The resulting methods often require either a huge number of input images to achieve a higher accuracy, or the use of specially designed patterns to reduce the number of required images. They are in general all very computational expensive.\n\nIn this work, we focus on environment matting for transparent objects. It is highly ill-posed, if not impossible, to estimate an accurate environment matte for transparent objects from a single image with an arbitrary background. Given the huge solution space, there exist multiple objects and backgrounds which can produce the same refractive effect. In order to make the problem more tractable, we simplify our problem to estimating an environment matte that can produce visually realistic refractive effect from a single image, instead of estimating a highly accurate refractive flow. We define the environment matte in our model as a triplet consisting of an object mask, an attenuation mask, and a refractive flow field. Realistic refractive effect can then be obtained by compositing the transparent object onto new background images (see \\fref{fig:intro}). We then show that the performance of the proposed method can be improved when a trimap or a background image is available.\n\nInspired by the great successes of convolutional neural networks (CNNs) in high-level computer vision tasks, we propose a convolutional neural network, called TOM-Net, for simultaneous learning of an object mask, an attenuation mask, and a refractive flow field from a single image with an arbitrary background. The key contributions of this work can be summarized as follows:\n\n\\begin{itemize}\n \\item We introduce a simple and efficient model for transparent object matting as simultaneous estimation of an object mask, an attenuation mask, and a refractive flow field.\n \\item We propose a convolutional neural network, TOM-Net, to learn an environment matte of a transparent object from a single image. To the best of our knowledge, TOM-Net is the first CNN that is capable of learning transparent object matting.\n \\item We create a large-scale synthetic dataset and a real dataset as a benchmark for learning transparent object matting. Our TOM-Net has produced promising results on both the synthetic and real datasets.\n \\item We propose two additional convolutional neural networks, denoted as \\TOMNetTrimap and \\TOMNetBg, for handling the cases where a trimap or a background image is available, respectively.\n\\end{itemize} \n\nPreliminary results of this chapter were published in~\\cite{chen2018tomnet,chen2019learning}.\nOur code, trained models, and datasets can be found at \\url{https://guanyingc.github.io/TOM-Net}.\n\n\\section{Related Work}\n\\label{sec:related_work}\nIn this section, we briefly review representative works on environment matting and recent works on CNN based image matting. \n\n\\paragraph{Environment matting}\n\\label{par:Environment Matting}\nZongker~\\etal~\\cite{zongker1999environment} introduced the concept of environment matting, and assumed each foreground pixel being originated from a single rectangular region of the background. They obtained the environment matte by identifying the corresponding background region for each foreground pixel using three monitors and multiple images. Chuang~\\etal~\\cite{chuang2000environment} extended~\\cite{zongker1999environment} in two different ways. First, they replaced the single rectangular supporting area for a foreground pixel with multiple 2D oriented Gaussian strips. This makes it possible for their method to model the effects of color dispersion, multiple mapping, and glossy reflection. Second, they simplified the environment matting equation by assuming the object being colorless and perfectly transparent. This allows them to achieve real time capture environment matting (RTCEM). The environment matte was then extracted with one image taken in front of a pre-designed pattern. However, RTCEM requires background images to segment the transparent objects, and depends on a time-consuming off-line processing. \n\nWexler~\\etal~\\cite{wexler2002image} introduced a probabilistic model based method which assumes each background point has a probability to make contribution towards the color of a certain foreground point. Their approach does not require pre-designed patterns during data acquisition, but it still needs multiple images and can only model thin transparent objects. Peers and Dutr{\\'e}~\\cite{peers2003wavelet} used a large number of wavelet basis backgrounds to obtain the environment matte, and their method can also model the effect of diffuse reflection. Based on the fact that a signal can be decomposed uniquely in the frequency domain, Zhu and Yang~\\cite{zhu2004frequency} proposed a frequency-based approach to extract an accurate environment matte. They used Fourier analysis to solve the decomposition problem. Both~\\cite{peers2003wavelet} and~\\cite{zhu2004frequency} require a large number of images to extract the matte (\\eg,~\\cite{peers2003wavelet} needs $2,400$ images and~\\cite{zhu2004frequency} needs $4,096$ images for an image of size $1024\\times 1024$), making them not very practical. Recently, compressive sensing theory has been applied to environment matting to reduce the number of images required. Duan~\\etal~\\cite{duan2011fast} applied this theory in the spatial domain and Qian~\\etal~\\cite{qian2015frequency} applied it in the frequency domain. However, the number of images needed is still in the order of hundreds. In contrast, our work can estimate an environment matte from a single image in a fast feed-forward computation without the need for pre-designed patterns or additional background images.\n\n\\begin{sidewaystable}[htbp]\n \\caption[Comparison of different environment matting methods]{Comparison of different environment matting methods. $k$ indicates the image size and mapping type stands for how a foreground point is composited by the point(s) in the background image.}\n \\label{tab:related_work}\n \\input{ch-tomnet/tables/comparisons}\n\\end{sidewaystable}\n\nYeung~\\etal~\\cite{yeung2011matting} proposed an interactive way to estimate an environment matte given an image containing a transparent object. Their method requires users to manually mark the foreground and background in the image, and models the refractive effect using a thin-plate-spline transformation. Their method does not produce an accurate environment matte, but instead a visually pleasing refractive effect. Our method shares the same spirit, but does not involve any human interaction. \n\n\\Tref{tab:related_work} shows a comparison of different environment matting methods. Compared with other methods, our method requires only a single image and can extract a matte in $0.5$ second without the need for any predefined backgrounds.\n\n\\paragraph{CNN based image matting}\nAlthough the potential of CNN on transparent object matting has not yet been explored, some existing work have adopted CNNs for solving the traditional image matting problem. Shen~\\etal~\\cite{shen2016deep} introduced a CNN for image matting of color portrait images. Cho~\\etal~\\cite{cho2016natural} proposed a network to predict a better alpha matte by taking the matting results of the traditional method and normalized color images as input. \nSome deep learning methods~\\cite{xu2017deep,zhang2019late,lu2019indices} have been introduced to estimate an alpha matte given an image and its trimap. \nHowever, none of these methods can be applied directly to the task of transparent object matting as object opacity alone is not sufficient to model the refractive effect. \n\n\\section{Matting Formulation}\n\\label{sec:formulation}\nAs a transparent object may have multiple optical properties (\\eg, color attenuation, translucency, and reflection), estimating an accurate environment matte for a generic transparent object from a single image is very challenging. \nFollowing the work of~\\cite{chuang2000environment}, we cast environment matting to a refractive flow estimation problem by assuming that each foreground pixel only originates from one point in the background due to refraction. Compared to the seminal work of~\\cite{zongker1999environment}, which models each foreground pixel as a linear combination of a patch in the background, our formulation is more tractable and can be easily encoded using a CNN.\n\nIn~\\cite{zongker1999environment}, the per-pixel environment matting is obtained through leveraging color information from multiple background images. Given a set of pre-designed background patterns, matting is formulated as\n\\begin{equation}\n \\label{eq:em_origin}\n C = F + (1-\\alpha)B + \\sum_{i=1}^{k} R_i \\mathcal{M}(\\mathbf{T}_i, \\mathbf{A}_i),\n\\end{equation}\nwhere $F$, $B$ and $\\alpha$ denote the ambient illumination, background color and opacity, respectively. The last term in \\eref{eq:em_origin} accounts for the environment light accumulated from $k$ pre-designed background images ($k=3$ in~\\cite{zongker1999environment}). $R_i$ is a factor describing the contribution of light emanating from the $i$-$th$ background image $\\mathbf{T}_i$. $\\mathcal{M}(\\mathbf{T}_i, \\mathbf{A}_i)$ denotes the average color of a rectangular region $\\mathbf{A}_i$ on the background image $\\mathbf{T}_i$. \n\nTo obtain an environment matte, the transparent object is placed in front of the monitor(s), and multiple pictures of the object are captured with the monitor(s) displaying different background patterns\\footnote{For an image of size $512\\times 512$, $18$ pictures and around $20$ minutes processing time are needed.}. \nGenerally, a surface point receives light from multiple directions, especially for a diffuse surface. When it comes to a perfectly transparent object, however, a surface point will only receive light from one direction as determined by the law of refraction. Consider a single background image as the only light source (\\ie, no ambient illumination), the problem can be modeled as\n\\begin{equation}\n \\label{eq:em_simplify1}\n C = (1-\\alpha)B + R \\mathcal{M}(\\mathbf{T}, P),\n\\end{equation}\nwhere $\\mathcal{M}(\\mathbf{T}, P)$ is a bilinear sampling operation at location $P$ on the background image $\\mathbf{T}$. Further, by assuming a colorless transparent object, $R$ becomes a light attenuation index $\\rho$ (a scalar value). The formulation in \\eref{eq:em_simplify1} can be simplified to\n\\begin{equation}\n \\label{eq:em_simplify2}\n C = (1-\\alpha)B + \\rho \\mathcal{M}(\\mathbf{T}, P),\n\\end{equation}\nwhere $\\rho \\in [0, 1]$ denotes the attenuation index.\n\nHere, we use refractive flow to model the refractive effect of a transparent object. The refractive flow of a foreground pixel is defined as the offset between the foreground pixel and its refraction correspondence on the background image. \n\nWe further introduce a binary foreground mask to define the object region in the image. The matting equation can now be rewritten as\n\\begin{equation}\n \\label{eq:em_simplify3}\n C = (1 - m) B + m\\rho \\mathcal{M}(\\mathbf{T}, P),\n\\end{equation}\nwhere $m \\in\\{0, 1\\}$ denotes background ($m = 0$) or foreground ($m = 1$). The matte can then be estimated by solving $m$, $\\rho$ and $P$ for each pixel in the input image containing the transparent \nobject\\footnote{For an image with $n$ pixel, we have $7$ unknowns ($3$ for $B$, $2$ for $P$, $1$ for $m$, and $1$ for $\\rho$) for each pixel, resulting in a total of $7n$ unknowns.}.\n\n\n\\section{Learning Transparent Object Matting}\n\\label{sec:method}\nIn this section, we present a two-stage deep learning framework, called TOM-Net, for learning transparent object matting (see \\fref{fig:networkStructure}). The first stage, denoted as CoarseNet, is a multi-scale encoder-decoder network that takes a single image as input, and predicts an object mask, an attenuation mask, and a refractive flow field simultaneously. CoarseNet is capable of predicting a robust object mask. However, the estimated attenuation mask and refractive flow field lack local structural details. \nTo overcome this problem, we introduce the second stage of TOM-Net, denoted as RefineNet, to achieve a sharper attenuation mask and a more detailed refractive flow field. RefineNet is a residual network~\\cite{he2016deep} that takes both the input image and the output of CoarseNet as input. After training, our TOM-Net can predict an environment matte from a single image in a fast feed-forward pass.\n\n\\begin{figure}[tbp] \\centering\n \\includegraphics[width=\\textwidth]{ch-tomnet/images/Method/TOMNet_framework.pdf}\n \\caption[Network architecture of TOM-Net]{Network architecture of TOM-Net. The upper subnetwork is the CoarseNet and the bottom subnetwork is the RefineNet. (Cross-link and multi-scale outputs are not shown for simplicity.)} \\label{fig:networkStructure}\n\\end{figure}\n\n\\subsection{Encoder-Decoder for Coarse Prediction}\n\\label{sub:Encoder-decoder Net for Coarse Prediction}\nThe first stage of our TOM-Net (\\ie, CoarseNet) is based on mirror-link CNN introduced in~\\cite{shi2016learning}. Mirror-link CNN was proposed to learn non-Lambertian object intrinsic decomposition. Its output consists of an albedo map, a shading map, and a specular map. It shares a similar output structure with our transparent object matting task (\\ie, three output branches sharing the same spatial dimensionality). Therefore, it is reasonable for us to adapt mirror-link CNN for our CoarseNet. \n\nThe mirror-link CNN adapted for our CoarseNet consists of one shared encoder and three distinct decoders. The encoder contains six down-sampling convolutional blocks, leading to a down-sampling factor of $64$ in the bottleneck layer. Features in the encoder layers are connected to the decoder layers having the same spatial dimensions through skip connections~\\cite{ronneberger2015u}. Cross-links~\\cite{shi2016learning} are introduced to make different decoders share the same input in each layer, so that decoders can better utilize the correlation between different predictions.\n\nLearning with multi-scale loss has been proven to be helpful in dense prediction tasks (\\eg,~\\cite{eigen2014depth,fischer2015flownet}). Since we formulate the problem of transparent object matting as refractive flow estimation, which is a dense prediction task, we augment our mirror-link CNN with multi-scale loss similar to~\\cite{fischer2015flownet}. \nWe use four different scales in our model, where the first scale starts from the decoder features with a down-sampling factor of $8$ and the largest scale has the same spatial dimensions as the input.\n\nIn contrast to the recent two-stage framework for image matting~\\cite{xu2017deep}, our TOM-Net has a shared encoder and three parallel decoders to accommodate different outputs. Besides, we augment our CoarseNet with multi-scale loss and cross-link. Moreover, TOM-Net is trained from scratch while the encoder in~\\cite{xu2017deep} is initialized with the pre-trained VGG16.\n\n\\newcommand{\\mathL}{\\mathcal{L}}\n\\subsection{Loss Function for Coarse Stage}\n\\label{sub:Loss Function for CoarseNet}\nCoarseNet takes a single image as input and predicts the environment matte as a triplet consisting of an object mask, an attenuation mask, and a refractive flow field. The learning of CoarseNet is supervised by the ground-truth matte using an object \\textbf{m}ask \\textbf{s}egmentation loss $\\mathL_{ms}$, an \\textbf{a}ttenuation \\textbf{r}egression loss $\\mathL_{ar}$, and a refractive \\textbf{f}low \\textbf{r}egression loss $\\mathL_{fr}$. Besides, the predicted matte is expected to render an image as close to the input image as possible when applied to the ground-truth background based on \\eref{eq:em_simplify3}. Hence, in addition to the supervision of the matte, we also take \\textbf{i}mage \\textbf{r}econstruction loss $\\mathL_{ir}$ into account (bilinear sampling is implemented following~\\cite{jaderberg2015spatial}). Note that the ground-truth background is only used to calculate the reconstruction error during training but not needed during testing. CoarseNet can therefore be trained by minimizing \n\\begin{align}\n \\mathL^c = \\alpha^c_{ms} \\mathL_{ms} + \\alpha^c_{ar} \\mathL_{ar} + \\alpha^c_{fr} \\mathL_{fr} + \\alpha^c_{ir} \\mathL_{ir},\n\\end{align}\nwhere \n$\\alpha^c_{ms}, \\alpha^c_{ar}, \\alpha^c_{fr}, \\alpha^c_{ir}$ are weights for the corresponding loss terms. \n\n\\paragraph{Object mask segmentation loss}\n\\label{par:Object Mask Classification Loss}\nObject mask segmentation is simply a spatial binary classification problem. The output of the object mask decoder has a dimension of $2\\times H\\times W$, where $H$ and $W$ denote the height and width of the input. We normalize the output with {\\em softmax} and compute the loss using the binary cross-entropy function\n\\begin{equation}\n \\mathL_{ms} = -\\frac{1}{HW} \\sum_{ij} (\\tilde{M}_{ij}\\log(P_{ij}) + (1-\\tilde{M}_{ij}) \\log(1-P_{ij})),\n\\end{equation}\nwhere $\\tilde{M}_{ij} \\in \\{0,1\\}$ and $P_{ij}\\in [0,1]$ represent ground truth and normalized foreground probability of the pixel at $(i, j)$, respectively.\n\n\\paragraph{Attenuation regression loss} \nThe predicted attenuation mask has a dimension of $1\\times H\\times W$. The value of this mask is in the range of $[0, 1]$, where $0$ indicates no light can pass and $1$ indicates the light will not be attenuated. \nWe adopt a mean square error (MSE) loss\n\\begin{equation}\n \\mathL_{ar} = \\frac{1}{HW} \\sum_{ij} (A_{ij}-\\tilde{A}_{ij})^2,\n\\end{equation}\nwhere $A_{ij}$ is the predicted attenuation index and $\\tilde{A}_{ij}$ the ground truth at $(i, j)$.\n\\paragraph{Refractive flow regression loss}\n\\label{par:Disparity Smoothness Loss}\nThe predicted refractive flow field has a dimension of $2\\times H\\times W$, where we have one channel for the horizontal displacement and another for the vertical displacement. We normalize the refractive flow with $tanh$ activation and multiply it by the width of the input, such that the output is constrained in the range of $[-W, W]$.\nWe adopt an average end-point error (EPE) loss\n\\begin{equation}\n \\mathL_{fr} = \\frac{1}{HW} \\sum_{ij} \\sqrt{(F^x_{ij}-\\tilde{F}^x_{ij})^2 + (F^y_{ij}-\\tilde{F}^y_{ij})^2},\n\\end{equation}\nwhere $(F^x, F^y)$ and $(\\tilde{F}^x, \\tilde{F}^y)$ denote the predicted flow and the ground truth, respectively.\n\n\\paragraph{Image reconstruction loss}\n\\label{par:Image Reconstruction Loss}\nWe use MSE loss to measure the dissimilarity between the reconstructed image and the input image. \nDenoting the reconstructed image by $I$ and the ground-truth image (\\ie, the input image) by $\\tilde{I}$, the reconstruction loss is given by\n\\begin{equation}\n \\mathL_{ir} = \\frac{1}{HW} \\sum_{ij} \\Vert I_{ij}-\\tilde{I}_{ij}\\Vert_2^2.\n\\end{equation}\n\n\\paragraph{Implementation details}\n\\label{par:Implementation Details}\nIn all experiments, we empirically set $\\alpha^c_{ms}=0.1, \\alpha^c_{ar}=1, \\alpha^c_{fr}=0.01,$ and $\\alpha^c_{ir}=1$. The loss weights for different scales are $\\frac{1}{2^{(4 -\\text{s})}}$, where $s \\in\\{1,2,3,4\\}$ denotes the scale. %\nCoarseNet contains $8$ million parameters and it takes about $2.5$ days to train with Adam optimizer~\\cite{kingma2014adam} on a single NVIDIA Titan X Pascal GPU. We first train the CoarseNet from scratch until convergence and then train the RefineNet. \n\n\\subsection{Residual Learning for Matte Refinement}\n\\label{par:Residual Learning for Matte Refinement}\nAs the attenuation mask and the refractive flow field predicted by the CoarseNet lack structural details, a refinement stage is needed to produce a detailed matte. Observing that residual learning is particularly suitable for tasks whose input and output are largely similar~\\cite{kim2016accurate,Nah_2017_CVPR}, we propose a residual network, denoted as RefineNet, to refine the matte predicted by the CoarseNet. \nSimilar strategy has also been successfully applied to progressively refine the estimated optical flow in~\\cite{ilg2017flownet}.\n\nWe concatenate the input image and the output of the CoarseNet to form the input of the RefineNet. As the object mask predicted by the CoarseNet is already plausible, the RefineNet only outputs an attenuation mask and a refractive flow field. The parameters of the CoarseNet are fixed when training the refinement stage. \n\n\\paragraph{Loss for the refinement stage}\n\\label{par:Loss for Refinement}\nThe overall loss for the refinement stage is\n\\begin{align}\n \\mathL^r = \\alpha^r_{ar} \\mathL_{ar} + \\alpha^r_{fr} \\mathL_{fr} ,\n\\end{align}\nwhere $\\mathL_{ar}$ is the refinement attenuation regression loss, $\\mathL_{fr}$ the refinement flow regression loss, and $\\alpha^r_{ar}$, $\\alpha^r_{fr}$ their weights. The definitions of these two losses are identical to those defined in the first stage. \nWe found that adding the image reconstruction loss in the refinement stage did reduce the image reconstruction error during training, but was not helpful in preserving sharp edges of the refractive flow field (\\eg, mouth of a glass). This could be explained by the fact that a lower image reconstruction loss does not guarantee a better refractive flow field. As the matte estimated by the CoarseNet has already achieved a small reconstruction error, simultaneously optimizing the flow regression loss and image reconstruction loss in the refinement stage may compromise the flow estimation.\nSince our goal in the refinement stage is to estimate a more detailed matte, we remove the image reconstruction loss to make our network focus on reducing the flow regression loss.\n\n\\paragraph{Implementation details}\n\\label{par:Implementation Details}\nWe set $\\alpha^r_{ar}=1$, $\\alpha^r_{fr}=1$ for the refinement. RefineNet contains $1$ million parameters and it takes about $2$ days to train with Adam optimizer on a single NVIDIA Titan X Pascal GPU. RefineNet is randomly initialized during training.\n\\subsection{Improvement with Trimap and Background Image}\n\\label{ssec:trimap}\nAs the problem of transparent object matting from a single image is highly ill-posed, we investigate how to reinforce our framework by utilizing additional information. \nIn particular, we consider the cases where a trimap or a background image is available.\nOur framework can be easily extended to make use of these additional information by taking the concatenation of the input image and the background image (or trimap) as input, while keeping the overall network architecture unchanged.\n\n\\paragraph{\\TOMNetTrimap}\nTrimap can provide a rough location of the transparent object to help the model better locate the transparent object. The trimap used in this work is a single channel image with $3$ different values, where values $0$, $1$, and $2$ indicate background, unknown, and foreground regions, respectively. \nDuring training, we randomly generate trimaps based on the ground-truth object mask. We first perform random erosion and cropping on the object mask to form the known (rough) foreground region. The unknown region is then generated by subtracting the foreground region from a tight bounding box of the object mask, leaving the rest of the regions as the background region.\nThe variant model, denoted as \\TOMNetTrimap, takes both the input image and trimap as input, giving rise to an input channel number of $4$ in the first convolutional layer. \n\n\\paragraph{\\TOMNetBg}\nGiven the background image, the model can easily identify the accurate location of the transparent object based on the difference of the input and background images. Moreover, having access to the background image allows the model to better estimate the refractive flow field.\nThe variant model, denoted as \\TOMNetBg, takes both the input and background images as input, giving rise to an input channel number of $6$ in the first convolutional layer.\n\n\\TOMNetTrimap and \\TOMNetBg are trained with the same procedure as \\hbox{TOM-Net}.\nOur experimental results show that with the additional information, our framework can achieve better results on both synthetic and real dataset.\n\n\\section{Dataset for Learning and Evaluation}\n\\label{sec:dataset}\nCurrently there is no off-the-shelf dataset for transparent object matting. \nOne potential direction is to create a real dataset with ground-truth mattes (\\ie, object masks, attenuation masks, and refractive flow fields) for training. \nHowever, it is almost impossible for human to manually label the refractive flow field of the transparent object.\nOne may consider estimating the mattes using existing transparent object matting methods and using them as the ground truth for training.\nHowever, it is very difficult and tedious as traditional methods require a large number of images and/or a long processing time for each object. \nBesides, there is no publicly available code for transparent object matting.\nTo bypass this problem, we created a large-scale synthetic dataset using \\emph{POV-Ray}~\\cite{povray} to render images of synthetic transparent objects.\nBesides, we captured a real dataset for evaluation. We will show that our TOM-Net trained on the synthetic dataset can generalize well to real world objects, demonstrating its good transferability.\n\n\\subsection{Synthetic Dataset} \n\\label{sub:Synthetic Dataset}\n\n\\begin{figure}[tbp] \\centering\n \\input{ch-tomnet/figures/syn_data_samples}\n \\caption[Examples of synthetic data]{Examples of synthetic data. Top to bottom: examples of \\emph{Glass}, \\emph{Glass with Water}, \\emph{Lens} and \\emph{Complex}, respectively. First three columns: background image, rendered image, refractive flow visualization (sparse). Last three columns: ground-truth refractive flow field, object mask, attenuation mask.} \n \\label{fig:syn_data_samples}\n\\end{figure}\n\nWe used a large number of background images and 3D models to render our training samples. We randomly changed the pose of the models, as well as the viewpoint and focal length of the camera in the rendering process to avoid overfitting to a fixed setting.\n\n\\paragraph{Background images}\n\\label{par:Backgrounds Images}\nWe employed two types of background images, namely scene images and synthetic patterns. For scene images, we randomly sampled images from the Microsoft COCO~\\cite{lin2014microsoft} dataset\\footnote{Other large-scale datasets like ImageNet~\\cite{deng2009imagenet} can also be used.}. The background images for the synthetic training set are sampled from COCO Train2014 and Test2015, while that for the synthetic test dataset are from COCO Val2014, giving rise to $100K$ scene images in total. \nFor synthetic patterns, we rendered $40K$ patterns of size $512\\times 512$ using \\emph{POV-Ray} built-in textures. \n\n\\begin{table} \\centering\n \\caption[Statistics of our synthetic datasets]{Statistics of our synthetic datasets.}\n \\label{tab:synth}\n \\resizebox{0.7\\textwidth}{!}{\n \\Large\n \\begin{tabular}{l|*{4}{c}|c}\n \\toprule\n Type & \\emph{Glass} & \\emph{Glass with Water} & \\emph{Lens} & \\emph{Complex} & Total \\\\\n \\midrule\n Synthetic Train & $52K$ & $26K$ & $20K$ & $80K$ & $178K$\\\\\n Synthetic Test & $250$ & $250$ & $200$ & $200$ & $900$\\\\\n \\bottomrule\n \\end{tabular}\n }\n\\end{table}\n\n\\paragraph{Transparent objects}\n\\label{par:Transparent Object}\nWe divided common transparent objects into four categories, namely \\emph{Glass}, \\emph{Glass with water}, \\emph{Lens}, and \\emph{Complex} shape (see \\fref{fig:syn_data_samples} for examples). We constructed parametric 3D models for the first three categories, and generated a large number of models using random parameters. For complex shapes, we constructed parametric 3D models for basic shapes like sweeping-spheres and squashed surface of revolution (SOR) parts, and composed a larger number of models using these basic shapes. We generated $178K$ 3D models in total, with each model assigned a random refractive index $\\lambda \\in [1.3, 1.5]$. The distribution of these models in four categories is shown in \\Tref{tab:synth}. \n\n\\paragraph{Ground-truth matte generation}\n\\label{par:Ground Truth Generation}\nWe obtained the ground-truth object mask of a model by rendering it in front of a black background image and setting its color to white. Similarly, we obtained the ground-truth attenuation mask of a model by simply rendering it in front of a white background image. Finally, we obtained the ground-truth refractive flow field (see \\fref{fig:syn_data_samples}) of a model by rendering it in front of a sequence of Gray-coded patterns. Technical details for the data rendering can be found at \\url{https://github.com/guanyingc/TOM-Net_Rendering}.\n\n\n\\paragraph{Data augmentation}\nTo improve the diversity of the training data and narrow the gap between real and synthetic data, extensive data augmentation was carried out on-the-fly. \nFor an image of size $512\\times 512$ with color intensity normalized to $[0, 1]$, we randomly performed color (brightness, contrast and saturation) augmentation (in a range of $[-0.2, 0.2]$), image scaling (in a range of $[0.875, 1.05]$), noise perturbation (in a range of $[-0.05, 0.05]$), and horizontal/vertical flipping. Besides, we also blurred the object boundary to make the synthetic data visually more natural. A patch with a size of $448\\times 448$ was then randomly cropped from an augmented image and used as input to train CoarseNet. To speed up the training and save memory, a smaller patch with a size of $384\\times 384$ was used to train RefineNet after the training of CoarseNet.\n\n\\begin{figure}[t] \\centering\n \\makebox[0.16\\textwidth]{\\footnotesize \\emph{Glass}} \n \\makebox[0.16\\textwidth]{\\footnotesize \\emph{Glass with Water}} \n \\makebox[0.16\\textwidth]{\\footnotesize \\emph{Lens}} \n \\makebox[0.16\\textwidth]{\\footnotesize \\emph{Complex}}\n \\makebox[0.16\\textwidth]{\\footnotesize \\emph{Complex}}\n \\\\\n \\input{ch-tomnet/figures/real_data_samples}\n \\caption[Sample images in real dataset]{Sample images in real dataset. The first row shows the background images and the second row shows the images of transparent objects.} \\label{fig:real_sample}\n\\end{figure}\n\n\\begin{table}[t] \\centering\n \\caption[Statistics of our real dataset]{Statistics of our real dataset. The first and second rows show the number of objects and the number of backgrounds used during data acquisition, respectively. The last row shows the number of captured samples. Note that the category of \\emph{Glass with Water} are created by filling five of the glasses with different amount of water, and some backgrounds are shared between different shape categories.}\n \\label{tab:real_data}\n \\resizebox{0.6\\textwidth}{!}{\n \\Large\n \\begin{tabular}{l|*{4}{c}}\n \\toprule\n & \\emph{Glass} & \\emph{Glass with Water} & \\emph{Lens} & \\emph{Complex} \\\\% & Total \n \\midrule %\n \\# Objects & $7$ & ($5$ glasses used) & $1$ & $6$ \\\\% & 14 \n \\# Backgrounds & $60$ & $38$ & $4$ & $18$ \\\\% & 60 \n \\midrule %\n \\# Samples & $470$ & $103$ & $61$ & $242$ \\\\% & 876 \n \\bottomrule\n \\end{tabular}\n }\n\\end{table}\n\\subsection{Real Dataset}\n\\label{sub:Real Dataset}\nTo validate the transferability of TOM-Net, we introduce a real dataset, which was captured using $14$ objects\\footnote{The objects consist of $7$ glasses, $1$ lens and $6$ complex objects. Glasses with water are implicitly included.} and $60$ background images, resulting in a dataset of $876$ images. Note that the background images for real data have not been used in the synthetic training or test dataset. The data distribution is summarized in \\Tref{tab:real_data}. During the data capturing process, the objects were placed under different poses, with the distances between the camera, object, and background uncontrolled. \\fref{fig:real_sample} shows some sample images from the real dataset. Note that we do not have the ground-truth matte for the real dataset. We instead captured images of the backgrounds without the transparent objects to facilitate evaluation.\n\nFollowing previous works, the transparent objects were captured in front of a monitor displaying different background images.\nDue to the sampling problem, there may exist Moir\\'{e}-effect in the captured image. We carefully adjusted the focal length and shutter speed to remove the Moir\\'{e}-effect during the data capturing.\n\n\\section{Experimental Results}\n\\label{sec:experiments}\nIn this section, we present experimental results and analysis.\nWe performed ablation study for TOM-Net, and evaluated our approach on both synthetic and real data. For synthetic data, we evaluated end-point error (EPE) for refractive flow fields, intersection over union (IoU) for object masks, mean square error (MSE) for attenuation masks and image reconstruction results, respectively. For real data, due to the absence of ground-truth matte, evaluation on the absolute error with respect to the ground truth is not possible. \nInstead, we reconstructed the input images using the estimated mattes and background images, and then evaluated the PSNR and SSIM metrics~\\cite{wang2004image} between each pair of input image (\\ie, photograph) and reconstructed image (\\ie, composite). In addition, a user study was conducted to validate the realism of TOM-Net composites.\n\nWe showcased an application of image editing of transparent object by manipulating the extracted matte, and analyzed typical failure cases. We also investigated how the performance of our method can be improved when a trimap or a background image is available.\n\n\\subsection{Ablation Study for Network Architecture}\n\\label{sub:Network Analysis}\n\n\\begin{table} \\centering\n \\caption[Ablation study for TOM-Net]{Ablation study for TOM-Net. F, A, I, and M are short for flow, attenuation, image reconstruction, and object mask, respectively. (The first value for EPE is measured on the whole image and the second measured within the object region. A-MSE and I-MSE are computed on the whole image.)}\n \\label{tab:self_compare}\n \\input{ch-tomnet/tables/ablation_study}\n\\end{table}\n\nWe quantitatively analyzed different components of TOM-Net using synthetic dataset\\footnote{Complex shape is excluded in experiments here to speed up training.}. We first verified the effectiveness of \\emph{refractive flow regression loss} ($\\mathL^{c}_{fr}$), \\emph{cross-link}, \\emph{multi-scale loss} and \\emph{image reconstruction loss} ($\\mathL^{c}_{ir}$) in the coarse stage by removing each of them from \\emph{CoarseNet} during training. We then validated the effectiveness of \\emph{RefineNet} in recovering details of the refractive flow field. RefineNet was evaluated by adding it to a trained CoarseNet and was trained while fixing the parameters of CoarseNet. \nFor comparison, we also included a naive baseline, denoted as \\emph{Background}, by considering a zero matte case (\\ie, whole image as object mask, no attenuation, and no refractive flow) where the reconstructed image is the same as the background image. The quantitative results are summarized in \\Tref{tab:self_compare} and the qualitative comparisons are shown in \\fref{fig:syn_ablation_study}.\nOverall, the baseline \\emph{Background} was outperformed by all TOM-Net variants with a large margin for all the evaluation metrics, which clearly shows that TOM-Net can successfully learn the matte. \n\n\\paragraph{Effectiveness of refractive flow regression loss} Comparing experiments with IDs 1 \\& 5 in \\Tref{tab:self_compare}, it can be clearly seen that the CoarseNet trained with the refractive flow regression loss significantly outperformed that without it in refractive flow estimation. This result indicates that image reconstruction loss alone is not enough to supervise the learning of refractive flow. \\fref{fig:syn_ablation_study} (a \\& e) qualitatively show that the refractive flow regression loss improved the performance of refractive flow estimation.\n\n\\begin{figure}[tbp] \\centering\n \\input{ch-tomnet/figures/syn_ablation_study_larger}\n \\caption[Qualitative comparison of different model variants]{Qualitative comparison of different model variants. The first row shows a sample of \\emph{Glass with Water} from the synthetic test dataset. The second and third rows show the estimated refractive flow fields and attenuation masks by different variants, respectively. (\\textbf{CNet} and \\textbf{RNet} are short for CoarseNet and RefineNet.)} \\label{fig:syn_ablation_study}\n\\end{figure}\n\n\\paragraph{Effectiveness of cross-link} Comparing experiments with IDs 2 \\& 5 in \\Tref{tab:self_compare}, we can see that augmenting the decoders of CoarseNet with cross-link helped improve the performance in all metrics, suggested that utilizing correlation is helpful for the matte estimation.\n\\fref{fig:syn_ablation_study} (b \\& e) qualitatively show the results without and with cross-link during training.\n\n\\paragraph{Effectiveness of multi-scale loss} Comparing experiments with IDs 3 \\& 5 in \\Tref{tab:self_compare}, we can see that multi-scale loss boosted performance of CoarsNet in all of the evaluation metrics, particularly the attenuation mask MSE (see \\fref{fig:syn_ablation_study} (c \\& e) for qualitative comparison).\n\n\\paragraph{Effectiveness of image reconstruction loss} Comparing experiments with IDs 4 \\& 5 in \\Tref{tab:self_compare}, we can see that adding image reconstruction loss in the coarse stage slightly improved the performance of refractive flow estimation and was very effective for reducing the image reconstruction error (see \\fref{fig:syn_ablation_study} (d \\& e) for qualitative comparison).\n\n\\begin{figure}[tbp] \\centering\n \\input{ch-tomnet/figures/refine_vis}\n \\caption[Visualization of the effectiveness of the refinement stage on real data]{Visualization of the effectiveness of the refinement stage on real data. After refinement, the refractive flow and attenuation mask have more clear structural details (\\eg, glass mouth).} \\label{fig:refine}\n\\end{figure}\n\n\\paragraph{Effectiveness of RefineNet}\nComparing experiments with IDs 5 \\& 6 in \\Tref{tab:self_compare}, we can clearly see that RefineNet can significantly improve the refractive flow estimation. \\fref{fig:syn_ablation_study} (e \\& f) and \\fref{fig:refine} show that RefineNet can infer sharp details on both the synthetic and real data based on the outputs of CoarseNet, demonstrating the effectiveness of the RefineNet. We also found that image reconstruction loss is not helpful for refractive flow estimation in the refinement stage (experiments with IDs 6 \\& 7 in \\Tref{tab:self_compare}). This is reasonable since the matte produced by CoarseNet already gives a small image reconstruction error, and further reducing the image reconstruction error does not guarantee a better refractive flow field. \n\n\\subsection{Evaluation on Synthetic Data}\n\\label{sub:Results on Synthetic data}\n\\begin{table*}[htbp] \\centering\n \\caption[Quantitative results on the synthetic test dataset]{Quantitative results on the synthetic test dataset. (The first value for EPE is measured on the whole image and the second measured within the object region. A-MSE and I-MSE are computed on the whole image.)}\n \\input{ch-tomnet/tables/syn_quant_larger}\n \\label{tab:quant_synth}\n\\end{table*}\n\nQuantitative results for synthetic test dataset are presented in \\Tref{tab:quant_synth}. We compared TOM-Net against \\emph{Background} and CoarseNet. Here, to accelerate training convergence, we first trained CoarseNet from scratch using our synthetic dataset excluding the complex shape subset. The trained CoarseNet was then fine-tuned using the entire training set including complex shapes, followed by training of RefineNet on the entire training set with random initialization. Similar to previous experiments, TOM-Net outperformed \\emph{Background} by a large margin, and slightly outperformed CoarseNet in both EPE and MSE, which implies more local details can be learned by RefinedNet. \n\nThe average IoU for object mask estimation is $0.96$, indicates that TOM-Net can robustly segment the transparent object given only a single image as input.\nAlthough TOM-Net is not expected to learn highly accurate refractive flow, the average EPE errors ($2.7/18.6$)\\footnote{The first value is measured on the whole image and the second measured within the object region.} are very small compared with the size of the input image ($448\\times 448$). In this sense, our predicted flow is capable of producing visually plausible refractive effect. The errors of complex shape category are larger than that of others, because complex shapes contain more sharp regions that will induce more errors. \n\\Fref{fig:qual_synth} and \\Fref{fig:qual_synth2} show the qualitative results on five synthetic objects. The objects in the first four examples come from the test set where each example shows a specific object category. Although the background images and objects in the test set never appear in the training set, TOM-Net can still predict robust matte.\n The last row (\\ie, \\fref{fig:qual_synth2}~(e)) shows a sample of \\emph{complex dog}, which was rendered using a 3D dog model. The pleasing result on the \\emph{complex dog} demonstrates that our model can generalize well from simple shapes to complex shapes.\n\n\\Fref{fig:qual_synth} shows that the reconstructed images using the estimated mattes (column 2) are very close to the input images (column 2), which empirically verifies that our simplified matting equation \\eref{eq:em_simplify3} is sufficiently accurate for this problem.\n\n\\begin{figure}[htbp] \\centering\n \\input{ch-tomnet/figures/syn_qual_larger}\n \\caption[Qualitative results on synthetic data (part $1$)]{Qualitative results on synthetic data (part $1$). For each example, the first column shows the input image and background. The second column shows the reconstructed image and reconstruction error map. The last three columns show the ground truth matte and estimation. Quantitative results are shown below each example. Dark region in GT flow indicates no valid flow.} \\label{fig:qual_synth}\n\\end{figure}\n\n\\begin{figure}[htbp] \\centering\n \\input{ch-tomnet/figures/syn_qual_larger2}\n \\caption[Qualitative results on synthetic data (part $2$)]{Qualitative results on synthetic data (part $2$). For each example, the first column shows the input image and background. The second column shows the reconstructed image and reconstruction error map. The last three columns show the ground truth matte and estimation. Quantitative results are shown below each example. Dark region in GT flow indicates no valid flow.} \\label{fig:qual_synth2}\n\\end{figure}\n\n\\clearpage\n\\subsection{Evaluation on Real Data}\n\\label{sub:Results}\n\\begin{table}[htbp] \\centering\n \\caption[Quantitative results on real data]{Quantitative results on real data. (Value the higher the better.)}\n \\resizebox{0.9\\textwidth}{!}{ \n \\huge\n \\begin{tabular}{c|*{2}{c}|*{2}{c}|*{2}{c}|*{2}{c}|*{2}{c}}\n \\toprule\n \\multirow{2}{*}{} & \\multicolumn{2}{c}{\\emph{Glass}} \n & \\multicolumn{2}{c}{\\emph{Glass with Water}} \n & \\multicolumn{2}{c}{\\emph{Lens}} \n & \\multicolumn{2}{c}{\\emph{Complex}} \n & \\multicolumn{2}{c}{Avg} \\\\\n & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM \\\\\n \\midrule\n Background & 22.05 & 0.894 & 20.75 & 0.886 & 18.60 & 0.860 & 16.85 & 0.816 & 19.56 & 0.864 \\\\ \n CoarseNet & 25.09 & 0.921 & 23.53 & 0.911 & 21.13 & 0.895 & 17.89 & 0.835 & 21.91 & 0.891 \\\\ \n TOM-Net & 25.06 & 0.920 & 23.53 & 0.911 & 20.89 & 0.893 & 17.88 & 0.835 & 21.84 & 0.890 \\\\ \n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:real_quant}\n\\end{table}\n\nWe evaluated TOM-Net on our captured real dataset, which consists of $876$ images of real objects. The results are shown in \\Tref{tab:real_quant}. The average PSNR and SSIM are above $21.0$ and $0.89$ respectively. The values are a bit lower for complex shapes, due to the opaque base of complex objects as well as the sharp regions of the objects that might induce large errors. After training, TOM-Net generalized well to common real transparent objects (see \\fref{fig:real_qualitative}). It is worth to note that during training, each sample contains only one object, while TOM-Net can predict reliable matte for images containing multiple objects (see \\fref{fig:real_qualitative} (c)), which indicates the transferability and robustness of TOM-Net.\n\n\\begin{figure}[htbp] \\centering\n \\input{ch-tomnet/figures/real_qual}\n \\caption[Qualitative results on real data]{Qualitative results on real data. The PSNR and SSIM between input photographs and reconstructed images are shown below each example. The last column shows the composites on novel backgrounds given the estimated matte.} \n \\label{fig:real_qualitative}\n\\end{figure}\n\n\\paragraph{User study}\n\\label{par:User Study}\nRemember that our goal is to estimate an environment matte that can produce visually realistic refractive effect from the input image, instead of estimating the highly accurate refractive flow.\nA user study was carried out to validate the realism of TOM-Net composites. $69$ subjects participated in our user study. At the beginning, we showed each participant photographs of the transparent objects that will be seen during the user study. The objects consisted of $3$ different glasses, $1$ glass with water, $1$ lens, and $1$ complex shape. $40$ samples, including $20$ photographs\\footnote{glass $\\times$12, glass \\& water $\\times$4, lens $\\times$2, and complex shape $\\times$2.} and the corresponding $20$ TOM-Net composites, were then randomly presented to each subject. When showing each sample, we also showed the corresponding background image to the subject for reference. We provided $3$ options for each sample: (P) {\\em photograph}, (C) {\\em composite}, (N) {\\em not distinguishable}.\n\\Tref{tab:user_study} shows the statistics of the user study. The $69$ participants produced $1,380$ votes for the $20$ real photographs, and $1,380$ votes for the $20$ composites, respectively. The P:C:N ratios are $850:455:75$ and $827:482:71$ for photographs and composites respectively. The per-category ratios also follow a similar trend, indicating close chance of photographs and composites to be considered real, which further demonstrates TOM-Net can produce realistic matte. \n\n\\begin{table}[tbp] \\centering\n \\caption[User study results]{User study results. P, C, and N are short for votes for photograph, composite, and not distinguishable.}\n \\large\n \\resizebox{0.8\\textwidth}{!}{\n \\begin{tabular}{c|*{3}{c}|*{3}{c}|*{3}{c}|*{3}{c}|*{3}{c}}\n \\toprule\n \\multirow{2}{*}{} & \\multicolumn{3}{c}{\\emph{Glass}} \n & \\multicolumn{3}{c}{\\emph{Glass with Water}} \n & \\multicolumn{3}{c}{\\emph{Lens}} \n & \\multicolumn{3}{c}{\\emph{Complex}} \n & \\multicolumn{3}{c}{All} \\\\\n & P & C & N & P & C & N & P & C & N & P & C & N & P & C & N \\\\ \\midrule\n Photographs & 522 & 275 & 31 & 163 & 97 & 16 & 74 & 48 & 16 & 91 & 35 & 12 & 850 & 455 & 75 \\\\\n Composites & 531 & 266 & 31 & 145 & 113 & 18 & 73 & 52 & 13 & 78 & 51 & 9 & 827 & 482 & 71 \\\\\n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:user_study}\n\\end{table}\n\n\\begin{figure}[htbp] \\centering\n \\includegraphics[width=0.8\\textwidth]{ch-tomnet/images/user_study/user_study_larger} \\\\\n \\makebox[0.4\\textwidth]{\\footnotesize Photograph} \n \\makebox[0.4\\textwidth]{\\footnotesize Composite} \n \\caption[Comparison of the photograph and composite]{Comparison of the photograph and composite. The first row shows the predicted matte, which is estimated by taking the photograph as input to our method. The second row compares the photograph and composite. When looking at the photograph and composite simultaneously, users can easily spot some imperfections of the composites (mostly in the boundary region).}\n \\label{fig:sup_user_study}\n\\end{figure}\n\nAlthough we stress that TOM-Net can produce visually realistic composites, the results are still less than perfect. When looking at the real image and our composite side-by-side, users can spot some imperfections of the composite (mostly in the boundary region, see \\fref{fig:sup_user_study}). Therefore, we did not include such a user study by showing the real image and our composite side-by-side. Otherwise, the result will be biased. In the future, we will strengthen our approach to produce more realistic composites, so that the real image and our composite are indistinguishable even when showing them side-by-side.\n\n\\subsection{Transparent Object Editing by Manipulating Environment Matte}\n\\label{sec:edit_flow}\n\n\\begin{figure}[htbp] \\centering\n \\makebox[0.132\\textwidth]{\\scriptsize Background} \n \\makebox[0.132\\textwidth]{\\scriptsize Input} \n \\makebox[0.132\\textwidth]{\\scriptsize Rec. Image} \n \\makebox[0.132\\textwidth]{\\scriptsize Rec. Error} \n \\makebox[0.132\\textwidth]{\\scriptsize Refractive Flow} \n \\makebox[0.132\\textwidth]{\\scriptsize Object Mask} \n \\makebox[0.132\\textwidth]{\\scriptsize Attenuation Mask} \n \\\\\n \\input{ch-tomnet/figures/sup_editFlow_example}\n \\makebox[\\textwidth]{\\footnotesize (a) An example result on \\emph{Glass with Water}. Reconstruction error: PSNR=25.69, SSIM=0.95}\n \\\\\n \\input{ch-tomnet/figures/sup_scaleFlow}\n \\vspace{-0.2em}\\makebox[\\textwidth]{\\footnotesize (b) Rescaling the magnitude of the estimated refractive flow field.}\n \\\\\n \\vspace{0.4em}\n \\input{ch-tomnet/figures/sup_editFlow} \n \\makebox[\\textwidth]{\\footnotesize (c) Translate, rotate or rescale the environment matte.}\n \\caption[Image editing by manipulating the predicted environment matte]{Various novel composites of a \\emph{Glass with Water} shape obtained by manipulating the predicted environment matte.} \\label{fig:sup_edit}\n\\end{figure}\n\nGiven a single image as input, our TOM-Net can estimate the environment matte as a triplet (consisting of an object mask, an attenuation mask, and a refractive flow field) in a fast feed-forward pass (see \\fref{fig:sup_edit} (a) for an example).\nNote that the goal of the proposed TOM-Net is to extract an environment matte that can produce realistic refractive effect from a single image, instead of estimating a highly accurate environment matte. \nThe reconstructed image in \\fref{fig:sup_edit} (a) looks realistic but does not have the same refractive effect as the original input, as the refractive effect of the estimated matte seems stronger. \nBy decreasing the magnitude of the estimated refractive flow field\\footnote{We simply multiply the refractive flow field by a scaling factor ($<1$).}, we can produce a similar refractive effect as the input image (see \\fref{fig:sup_edit} (b)). When the scaling factor becomes $0.6$, the reconstructed image achieved the lowest reconstruction error, with an improvement of $1.49$ and $0.01$ in PSNR and SSIM, respectively.\nApart from rescaling the magnitude of the refractive flow field to adjust the refractive effect of the object, more interesting composites can be obtained by translating, rotating and rescaling the environment matte (see \\fref{fig:sup_edit} (c)). \n\n\\subsection{Failure Cases}\n\\label{sec:Failure Cases}\n\n\\begin{figure}[htbp] \\centering\n \\makebox[0.05\\textwidth]{\\footnotesize } \n \\makebox[0.18\\textwidth]{\\footnotesize Input} \n \\makebox[0.18\\textwidth]{\\footnotesize Refractive Flow} \n \\makebox[0.18\\textwidth]{\\footnotesize Object Mask} \n \\makebox[0.18\\textwidth]{\\footnotesize Attenuation Mask} \\\\\n \\input{ch-tomnet/figures/sup_failure}\n \\caption[Failure cases on real data]{Two failure cases on real data. In (a), our model fails to estimate the upper-part of the matte as there is no visual clue to find the object. In (b), the bottom part of the estimated matte is incomplete as the background image is heavily cluttered and the bottom part of the object is very dark.}\n \\label{fig:sup_fail}\n\\end{figure}\n\nOur model can robustly estimate environment matte for different transparent objects in front of different backgrounds, however, when there is no visual clue for the objects or the image is too cluttered to separate the object from the background, our model may fail. \\Fref{fig:sup_fail} shows two failure cases of our model on real data. In \\fref{fig:sup_fail} (a), our model fails to extract the upper-part of the environment matte for the transparent glass due to the lack of visual clue. \nIn \\fref{fig:sup_fail} (b), although our model is still able to estimate a reasonable matte, the bottom part of the estimated matte is incomplete due to the very cluttered background. \n\n\n\\subsection{Improvement with Trimap and Background Image}\n\\begin{table*}[htbp] \\centering\n \\caption[Quantitative comparison on the synthetic test dataset]{Quantitative comparison between TOM-Net, \\TOMNetTrimap and \\TOMNetBg on the synthetic test dataset.}\n \\input{ch-tomnet/tables/syn_quant_trimap_bg_larger}\n \\label{tab:stereo_quant_synth}\n\\end{table*}\n\n\\begin{table}[htbp] \\centering\n \\caption[Quantitative comparison on real data]{Quantitative comparison between TOM-Net, \\TOMNetTrimap and \\TOMNetBg on real data.}\n \\resizebox{0.8\\textwidth}{!}{ \n \\huge\n \\begin{tabular}{l|*{2}{c}|*{2}{c}|*{2}{c}|*{2}{c}|*{2}{c}}\n \\toprule\n \\multirow{2}{*}{} & \\multicolumn{2}{c}{\\emph{Glass}} \n & \\multicolumn{2}{c}{\\emph{Glass with Water}} \n & \\multicolumn{2}{c}{\\emph{Lens}} \n & \\multicolumn{2}{c}{\\emph{Complex}} \n & \\multicolumn{2}{c}{Average} \\\\\n & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM & PSNR & SSIM \\\\\n \\midrule\n Background & 22.05 & 0.894 & 20.75 & 0.886 & 18.60 & 0.860 & 16.85 & 0.816 & 19.56 & 0.864 \\\\ \n TOM-Net & 25.06 & 0.920 & 23.53 & 0.911 & 20.89 & 0.893 & 17.88 & 0.835 & 21.84 & 0.890 \\\\ \n \\TOMNetTrimap & 25.48 & 0.924 & 23.77 & 0.914 & 23.98 & 0.913 & 20.88 & 0.868 & 23.53 & 0.905\\\\ \n\\TOMNetBg & 26.10 & 0.931 & 24.58 & 0.922 & 25.52 & 0.924 & 22.23 & 0.884 &\\textbf{24.61} & \\textbf{0.915} \\\\ \n \\bottomrule\n \\end{tabular}\n }\n \\label{tab:real_quant_stereo}\n\\end{table}\n\n\\begin{figure*} \\centering\n \\input{ch-tomnet/figures/qual_mono_trimap_stereo_larger}\n \\caption[Qualitative comparison on real data]{Qualitative comparison between TOM-Net, \\TOMNetTrimap and \\TOMNetBg on real data.\n For each testing object, the input to the model is shown on the first column, and the results of TOM-Net (up), \\TOMNetTrimap (middle) and \\TOMNetBg (bottom) are shown on the rest of the columns. Note that for \\TOMNetTrimap and \\TOMNetBg, we do not show the input image for simplicity. The PSNR and SSIM between input photographs and reconstructed images are shown right after the error maps.}\n \\label{fig:qual_stereo_mono}\n\\end{figure*}\n\nAt test time, the input trimaps for \\TOMNetTrimap were generated in the same way adopted in the training (as described in \\Sref{ssec:trimap}), except that the foreground regions were obtained by performing erosion operation on the ground-truth object mask with a fixed (rather than a random) kernel size of $10$ pixels for evaluation.\n\\Tref{tab:stereo_quant_synth} shows the quantitative comparisons between TOM-Net, \\TOMNetTrimap and \\TOMNetBg on the synthetic test dataset. \nAs expected, with the access to the additional information, both \\TOMNetTrimap and \\TOMNetBg performed better than TOM-Net. Due to the fact that a background image contains more useful information than a trimap, \\TOMNetBg achieved the best results.\n\n\\Tref{tab:real_quant_stereo} presents the quantitative comparison on real data. Compared with TOM-Net, \\TOMNetTrimap and \\TOMNetBg achieved an improvement of $1.69$ and $2.77$ in average PSNR and an improvement of $0.015$ and $0.024$ in average SSIM, respectively. \n\\fref{fig:qual_stereo_mono} shows the qualitative comparison on real data, where the foreground region of the trimap was marked by the user. It can be seen that with the additional information, \\TOMNetTrimap and \\TOMNetBg can identify the transparent object from the cluttered background more accurately than TOM-Net and model the opaque base of the transparent object (\\fref{fig:qual_stereo_mono} (a)). As a result, the environment matte predicted by \\TOMNetTrimap and \\TOMNetBg can produce more realistic composites and achieve lower reconstruction errors, clearly demonstrating the effectiveness of our framework in handling cases where a trimap or a background image is available.\n\n\\section{Discussion}\n\\label{sec:discussion}\n\n\\begin{figure*}[t] \\centering\n \\input{ch-tomnet/figures/limitation}\n \\caption[Results on colored object and objects under natural illumination]{Qualitative results of TOM-Net on colored transparent object (first row) and objects under natural illuminations (last four rows).}\n \\label{fig:limitation}\n\\end{figure*}\n\n\\subsection{Limitations}\nAlthough our method can produce plausible results for transparent object matting, there do exist limitations that require further study.\nFirst, our model assumes objects to be colorless so that the attenuation property of an object can be depicted as a scalar value $\\rho$ in our formulation. However, this is not applicable to colored transparent objects, as shown in see \\fref{fig:limitation}~(a). \nAlthough our method can estimate a reasonably good object mask and refractive flow field for the \\emph{Glass with Water}, the estimated attenuation mask cannot model the colored effect of the object.\n\nSecond, our model assumes a single planar background (following most of the previous works) as the only light source and simplifies the interaction between object and background image to a point-to-point (single) mapping. However, more complicated effects exist in the real world, such as specular highlights, translucency, multi-mapping (\\ie, refraction and reflection happen simultaneously at a surface point), and color dispersion (\\ie, different color components may have different supporting background regions). \n\\fref{fig:limitation}~(b)-(e) show four example results of TOM-Net on transparent objects under different types of natural illuminations. Regardless of the fact that TOM-Net can estimate a plausible object mask and refractive flow field, the composites do not look very realistic. \nThis is because our current formulation does not consider the more sophisticated refractive properties of a transparent object under natural illumination like complex interaction with environment light, specular highlight, Fresnel effect, and acoustic shadow.\n\n\\subsection{Colored Objects and Specular Highlights}\nHere we sketch the potential solutions to colored transparent objects as well as the cases when specular highlights appear on transparent objects. In \\Sref{sec:formulation}, we simplified matting equation as \\eref{eq:em_simplify3}.\nTo handle colored objects, the scalar attenuation index $\\rho$ should be expanded to a color attenuation 3-vector $R$, in which each value corresponds to an attenuation index for a specific color channel. The matting equation then becomes \n\\begin{equation}\n \\label{eq:color_2}\n C = (1 - m) B + mR \\circ \\mathcal{M}(\\mathbf{T}, P),\n\\end{equation}\nwhere $\\circ$ represents element-wise multiplication.\n\nConsider a white near point light source, we can simplify the specular highlight effect with a specular highlight component $S$, then the generalized matting equation can be written as\n\\begin{equation}\n \\label{eq:color_2}\n C = (1 - m) B + mR \\circ \\mathcal{M}(\\mathbf{T}, P) + S,\n\\end{equation}\nwhere $S$ is a 3-vector containing three identical values.\nThe problem of transparent object matting now becomes simultaneously estimating an object mask, a color attenuation mask, a refractive flow field, and a specular highlight mask from a single image, while more efforts are needed to implement them for practical use and we leave this as our future work.\n\n\\subsection{Difficulty in Comparison with Previous Works}\nCurrently, it is not trivial to have a fair comparison with existing methods. On one hand, applying our method on the data used in the previous methods is difficult. Most of the previous methods require multiple images of the transparent object captured in front of pre-designed patterns, which are not publicly available and lack enough textures for our method to estimate the refractive effect of the transparent object. \nThe single image based methods RTCEM~\\cite{chuang2000environment} and~\\cite{yeung2011matting} have additional requirements. In particular, RTCEM~\\cite{chuang2000environment} requires the object to be captured in front of a coded-pattern (also not publicly available), and the background image is needed to segment the foreground object. \n\\cite{yeung2011matting} requires human interaction to segment the foreground object and model the object's refractive effect with thin-plate-spline transformation. The data used in~\\cite{yeung2011matting} does not follow our assumption that the light comes from a single background image, thus it cannot be directly processed by our method.\nOn the other hand, there are no public implementations for the previous methods, and even if there were, those methods cannot be applied to our dataset which is created for single image transparent object matting.\n\nDifferent from the previous methods, our method aims to estimate the foreground mask, attenuation mask, and refractive flow field from a single natural image. Since our code and datasets have been made publicly available, it will ease the comparison for the future work. We believe our work can serve as a baseline and provide meaningful insight for future researches in this area.\n\n\\subsection{Generalization to Real Data}\n As it is very difficult and time consuming to create a large scale real dataset for training, we use synthetic data for training and evaluate its performance on real data. \nIt is well-known that there is a domain gap between the synthetic and real data, and a model trained on synthetic data may not generalize well to real data.\nWe hypothesize that the reasons why our method works well on real data are as follows.\nFollowing previous works, the real transparent objects are captured in front of a monitor.\nUnder our assumption, the captured images and the rendered images are very similar. Moreover, extensive data augmentation is performed to close the gap between the synthetic and real data.\n\nTo further improve the generalization ability of our method on real data, we will explore the idea of exploiting real data for self-supervised training or fine-tuning in the future.\n\n\\subsection{Design of the Network Architecture}\nTo better recover the details of the refractive flow field and attenuation mask, we propose a two-stage network architecture for this problem. \nOur results show that RefineNet can effectively improve the results of CoarseNet.\nHowever, our two-stage network requires stepwise training (\\ie, we first trained CoarseNet until convergence and then trained RefineNet) which requires longer training time.\nAn interesting future direction is to develop a more efficient single-stage network that achieves comparable performance as the two-stage network. \n\n\\section{Conclusion}\n\\label{sec:conclusion}\nWe have introduced a simple and efficient model for transparent object matting, and proposed a CNN architecture, called TOM-Net, that takes a single image as input and predicts environment matte as an object mask, an attenuation mask, and a refractive flow field in a fast feed-forward pass. We created a large-scale synthetic dataset and a real dataset as a benchmark for learning transparent object matting. We have also shown that TOM-Net can perform better by incorporating a trimap or a background image in the input. Promising results have been achieved on both synthetic and real data, which clearly demonstrate the feasibility and effectiveness of the proposed approach. \n", "meta": {"hexsha": "528bdb814867b64e687c75d9af8a546145bbcef2", "size": 66390, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ch-tomnet/tomnet.tex", "max_stars_repo_name": "guanyingc/HKU-PhD-Thesis-LaTex", "max_stars_repo_head_hexsha": "b243a0517ed2e802f480d87925d110f15f030ff6", "max_stars_repo_licenses": ["Unlicense", "MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-05-06T01:05:00.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-06T01:05:00.000Z", "max_issues_repo_path": "ch-tomnet/tomnet.tex", "max_issues_repo_name": "guanyingc/HKU-PhD-Thesis-LaTex", "max_issues_repo_head_hexsha": "b243a0517ed2e802f480d87925d110f15f030ff6", "max_issues_repo_licenses": ["Unlicense", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ch-tomnet/tomnet.tex", "max_forks_repo_name": "guanyingc/HKU-PhD-Thesis-LaTex", "max_forks_repo_head_hexsha": "b243a0517ed2e802f480d87925d110f15f030ff6", "max_forks_repo_licenses": ["Unlicense", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 111.5798319328, "max_line_length": 1629, "alphanum_fraction": 0.7713812321, "num_tokens": 16218, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737344123242, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3466190405799409}} {"text": "\\chapter{Solving the \\acrshort{ECP} and the \\acrshort{D-ECP}}\n\\label{ch:solving}\n\nWe now present some techniques for calculating exactly and approximating both\nthe \\acrshort{ECP} and the \\acrshort{D-ECP} (\\autoref{sub:the_problem_definition}).\n\n\\section{Exact Solutions}%\n\\label{sec:exact-solutions}\n\nWe start with our exact algorithms that are based on MIPs.\n\n\\subsection{A MIP Model for the \\acrshort{ECP}}%\n\\label{sub:a_mip_model_for_the_ecp}\n\nLet $G$ be an \\emph{interaction graph} for contents $\\mathcal{C} $ and\nthreads $T \\in \\mathcal{T}_{C}, \\; C \\in \\mathcal{C} $ for which we want to\nsolve the \\acrshort{ECP}. Fix\n$\\alpha \\in [0, 1]$. Let $\\mathcal{\\hat{C}} \\subseteq \\mathcal{C} $ be the\nset of controversial contents and $E_k$ the set\nof all edges of thread $T_k$ associated to a controversial content, i.e. $T_{k}\n\t\\in \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}}$; let also $E^{+}_k $\nand $E^{-}_k $ be the set of positive and negative edges in $T_k$, respectively.\n\nThe following \\acrshort{MIP} model is able to solve the \\acrshort{ECP} on $G$\nfor values of $\\alpha \\leq 0.5$.\n\n\\begin{alignat}{3}\n\t\\label{eq:ecp-exact1}\n\t\\text{maximize} & & \\sum_{ T_{k} \\in \\mathcal{T}_{C}, \\; C \\in\n\t\t\\mathcal{\\hat{C}} } \\big( \\sum^{}_{ij \\in E^{+}_k } x_{ij}\n\t^{k} - \\sum_{ij \\in E^{-}_k } x_{ij} ^{k} \\big) \\\\\n\t\\label{eq:ecp-v1}\n\t\\text{subject to} & \\quad & x _{ij}^{k} \\leq y_i & \\quad \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-v2}\n\t & & x _{ij}^{k} \\leq y_j & \\quad \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-t1}\n\t & & x _{ij}^{k} \\leq z_k & \\quad \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-e1}\n\t & & x _{ij} ^{k} \\geq - 2 + y_i + y_j + z_k & \\quad \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-alpha-constraint1}\n\t & & \\sum^{}_{ij \\in E_k^{-} } x_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\tx_{ij} ^{k} \\leq 0 & \\quad \\forall T_{k} \\in \\mathcal{T} _{C}, C \\in\n\t\\hat{\\mathcal{C}} \\\\\n\t\\label{eq:ecp-vertex-def1}\n\t & & y _{i} \\in \\{0, 1\\} & \\quad \\forall i \\in V \\\\\n\t\\label{eq:ecp-edge-def1}\n\t & & 0 \\leq x _{ij} ^{k} \\leq 1 & \\quad \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-thread-def1}\n\t & & 0 \\leq z _{k} \\leq 1 & \\quad \\forall T_{k} \\in \\mathcal{T} _{C}, C \\in\n\t\\hat{\\mathcal{C}}\n\\end{alignat}\n\nThe \\acrshort{MIP} model introduces variables $x$, $y$ and $z$.\n\\begin{itemize}\n\t\\item $y$ variables are associated to vertices\n\t (\\autoref{eq:ecp-vertex-def1}). Intuitively $y_i = 1$ means that the\n\t vertex $v_{i} $ is part of the set $U \\subseteq V$ considered for the\n\t score.\n\t\\item $x$ variables are associated to edges (\\autoref{eq:ecp-edge-def1}).\n\t A value of $x_{ij}^{k} = 1$ should be interpreted as the fact that\n\t the edge $e_{ij} \\in E_k$ is contributing to the score,\n\t i.e. $T_k \\in \\mathcal{S}_{C} (U)$.\n\t\\item $z$ variables are associated to threads\n\t (\\autoref{eq:ecp-thread-def1}). A value greater than 0\n\t is generally associated to \\emph{non-controversial} threads and\n\t \\emph{controversial} threads have value $z_k = 0$.\n\\end{itemize}\n\nWe will now show that the \\acrshort{ECP} can be solved through\nMIP~\\eqref{eq:ecp-exact1}-\\eqref{eq:ecp-thread-def1}.\n\n\\begin{theorem}\n\t\\label{th:ecp-mip}\n\tLet $G = \\{G_k = (V,E_k) \\}_k$ be an \\emph{Interaction Graph} and $\\alpha \\in\n\t\t[0, 0.5]$. Then\n\n\t\\begin{equation}\n\t\t\\max_{U \\subseteq V} \\xi(U) = \\operatorname{OPT}(\\operatorname{MIP}),\n\t\\end{equation}\n\twhere $\\operatorname{OPT}(\\operatorname{MIP})$ denotes the optimal solution to\n\tMIP~\\eqref{eq:ecp-exact1}-\\eqref{eq:ecp-thread-def1}.\n\\end{theorem}\n\n\\begin{proof}\n\tWe will show the equality by first proving that $RHS \\geq LHS$ and then\n\tthat $LHS \\geq RHS$.\n\n\t\\begin{claim}\n\t\t\\label{th:claim-v-b-xi}\n\t\tFor any $U \\subseteq V$, the\n\t\tMIP~\\eqref{eq:ecp-exact1}-\\eqref{eq:ecp-thread-def1} achieves value at\n\t\tleast $\\xi(U)$.\n\t\\end{claim}\n\n\t\\begin{proof}\n\t\tLet $E_k[U]$ the set of edges induced by $U$ in thread $T_k$. We\n\t\tconstruct a MIP solution as follows:\n\n\t\t\\begin{alignat}{2}\n\t\t\t\\label{eq:ecpp-y}\n\t\t\ty_i & = \\begin{cases}\n\t\t\t\t1, & \\text{if } v_{i} \\in U, \\\\\n\t\t\t\t0, & \\text{otherwise,}\n\t\t\t\\end{cases} & \\forall v_i \\in V \\\\\n\t\t\t\\label{eq:ecpp-z}\n\t\t\tz_k & = \\begin{cases}\n\t\t\t\t1, & \\text{if } T_{k} \\in \\mathcal{S}_C(U), \\; C \\in\n\t\t\t\t\\mathcal{\\hat{C}} , \\\\\n\t\t\t\t0, & \\text{otherwise,}\n\t\t\t\\end{cases} & \\forall T_k \\in \\mathcal{T}_C, C \\in \\mathcal{C} \\\\\n\t\t\t\\label{eq:ecpp-x}\n\t\t\tx_{ij}^{k} & = \\begin{cases}\n\t\t\t\t1, & \\text{if } e_{ij} \\in E_{k}[U], \\; T_{k} \\in \\mathcal{S}_C(U), \\; C \\in\n\t\t\t\t\\mathcal{\\hat{C}}, \\\\\n\t\t\t\t0, & \\text{otherwise.}\n\t\t\t\\end{cases} & \\forall e_{ij}^k \\in E_k\n\t\t\\end{alignat}\n\n\t\tTo satisfy \\eqref{eq:ecp-v1}-\\eqref{eq:ecp-e1} we need that\n\t\t\\begin{equation}\n\t\t\t\\label{eq:ecpp-x-iff}\n\t\t\tx_{ij}^{k} = 1 \\iff y_i = 1 \\land y_j = 1 \\land z_k = 1.\n\t\t\\end{equation}\n\t\tThis is always true since we defined $x_{ij}^{k}$ to be $1$ only and if it\n\t\tis associated to an edge induced in a $T_k \\in \\mathcal{S}_C(U), \\; C \\in\n\t\t\t\\mathcal{\\hat{C}} $.\n\n\t\tLet us now consider a thread $T_k \\in \\mathcal{S}_C(U)$. Then\n\n\t\t\\begin{equation}\n\t\t\t\\eta(T_k[U]) \\leq \\alpha \\implies \\frac{|E^{-}_{k}[U]|}{|E_{k}[U]|} \\leq\n\t\t\t\\alpha \\implies |E^{-}_{k}[U]| - \\alpha |E_{k}[U]| \\leq 0.\n\t\t\\end{equation}\n\t\tso \\eqref{eq:ecp-alpha-constraint1} is satisfied. It is easy to see that\n\t\tif $T_k \\not\\in \\mathcal{S}_C(U)$ then $x_{ij}^{k} = 0$ for all $ij \\in E_{k}$ and\n\t\tthe constraint is also satisfied.\n\n\t\tFinally, any edge contributing to $\\xi(U)$ will also\n\t\tequally contribute to the objective function.\n\t\\end{proof}\n\n\t\\begin{claim}\n\t\t\\label{th:claim-xi-b-v}\n\t\tGiven a feasible solution of\n\t\tMIP~\\eqref{eq:ecp-exact1}-\\eqref{eq:ecp-thread-def1} with value $v$ we can\n\t\tconstruct $U \\; s.t. \\; \\xi(U) \\geq v$.\n\t\\end{claim}\n\n\t\\begin{proof}\n\t\tWe define $U \\coloneqq \\{ v_{i} \\; s.t. \\; y_i = 1\\}$. Again, by\n\t\t\\eqref{eq:ecp-v1}-\\eqref{eq:ecp-e1} we have\n\t\t\\eqref{eq:ecpp-x-iff}, so\n\t\t\\begin{gather}\n\t\t\tx_{ij}^{k} = 1 \\implies z_k = 1, \\\\\n\t\t\t\\label{eq:ecpp-z-iff-xij}\n\t\t\tz_k = 1 \\implies x_{i'j'}^{k} = 1 \\; \\forall i'j'\n\t\t\t\\in E_k[U],\n\t\t\\end{gather}\n\t\tmeaning that if $z_k = 1$ then for all the edges $e_{ij} \\in E_k$ induced\n\t\tby $U$ we will have $x_{ij}^{k} = 1$ (i.e.\\ they will contribute to the\n\t\tobjective).\n\t\tLet us now consider $T_k \\; s.t. \\; z_k = 1$. Because of\n\t\t\\eqref{eq:ecp-alpha-constraint1} and \\eqref{eq:ecpp-z-iff-xij} we have\n\n\t\t\\begin{equation}\n\t\t\t\\label{eq:ecpp-nc-t}\n\t\t\t|E^{-}_{k}[U]| - \\alpha |E_{k}[U]| \\leq 0 \\implies\n\t\t\t\\frac{|E^{-}_{k}[U]|}{|E_{k}[U]|} \\leq \\alpha.\n\t\t\\end{equation}\n\t\ti.e. $\\eta(T_k[U]) \\leq \\alpha$. So $T_k \\in\n\t\t\t\\mathcal{S}_C(U), \\; C \\in \\mathcal{\\hat{C}}$, i.e. $z_k = 1 \\implies T_k\n\t\t\t\\in \\mathcal{S}_C(U), \\; C \\in \\mathcal{\\hat{C}}$, thus $T_k[U]$ contributes to\n\t\t$\\xi(U)$; more specifically any edge contributing to the objective function equally\n\t\tcontributes to $\\xi(U)$.\n\n\t\tNow suppose there exists $T_k \\in \\mathcal{S}_C(U) \\; s.t.\n\t\t\t\\; z_k = 0$. Then $ x_{ij}^{k} = 0$ by \\eqref{eq:ecp-t1}. Since $\\alpha \\leq 0.5$,\n\n\t\t\\begin{align}\n\t\t\t\\eta(T_k[U]) \\leq \\alpha & \\implies \\frac{|E^{-}_{k}[U]|}{|E_{k}[U]|}\n\t\t\t\\leq 0.5 \\\\\n\t\t\t & \\implies |E^{-}_{k}[U]| \\leq 0.5 \\cdot (|E^{+}_{k}[U]| + |E^{-}_{k}[U]|) \\\\\n\t\t\t & \\implies 0.5 \\cdot |E^{+}_{k}[U]| - 0.5 \\cdot |E^{-}_{k}[U]| \\geq 0 \\\\\n\t\t\t\\label{eq:ecpp-nc-contr}\n\t\t\t & \\implies |E^{+}_{k}[U]| - |E^{-}_{k}[U]|\n\t\t\t\\geq 0.\n\t\t\\end{align}\n\t\tConsequently $T_k$ will contribute positively to $\\xi(U)$, i.e.\n\t\tin the subgraph induced on $T_k$ by $U$\n\t\tthe number of positive edges is greater or equal than the\n\t\tnumber of negative edges.\n\n\t\tMore generally, due to \\eqref{eq:ecp-v1}-\\eqref{eq:ecp-e1} we have\n\t\t\\begin{gather*}\n\t\t\tz_k = c > 0 \\implies x_{i'j'}^{k} = c \\; \\forall i'j'\n\t\t\t\\in E_k[U],\n\t\t\t% \\\\\n\t\t\t% x_{ij}^{k} = c \\iff z_k = c \\land y_i = 1 \\land y_j = 1,\n\t\t\\end{gather*}\n\t\tmeaning that if $z_k = c$ all and only the variables $x_{ij}^{k}$\n\t\tassociated to edges $e_{ij} \\in E_k$ induced by $U$\n\t\twill get value $c$ (any other $x_{ij}^{k}$ will get value $0$ for\n\t\t\\eqref{eq:ecp-v1}-\\eqref{eq:ecp-v2}).\n\n\t\tCombining this result with \\eqref{eq:ecp-alpha-constraint1} we get\n\t\tagain \\eqref{eq:ecpp-nc-t} and, consequently, \\eqref{eq:ecpp-nc-contr}.\n\t\tTherefore, the contribution of $T_k$ associated to $z_k > 0$ will be\n\n\t\t\\begin{equation}\n\t\t\t\\sum^{}_{ij \\in E^{+}_k } x_{ij}\n\t\t\t\t^{k} - \\sum_{ij \\in E^{-}_k } x_{ij} ^{k} = z_k (|E^{+}_{k}[U]| -\n\t\t\t|E^{-}_{k}[U]|) \\geq 0.\n\t\t\\end{equation}\n\t\tSince $z_k \\in [0, 1]$ the contribution of the same thread in $\\xi(U)$\n\t\twill be greater (if $z_k \\in [0, 1)$) or equal (if $z_k = 1$) to the\n\t\tcontribution of thread $T_k$ in the objective function.\n\n\t\t% can be similarly shown to be\n\t\t% \\emph{non-controversial}; furthermore its positive contribution in the\n\t\t% objective function will be $\\leq$ to $T_k$ contribution in $\\xi(U)$, so\n\t\t% $\\xi(U) \\geq v$.\n\t\\end{proof}\n\n\tThis concludes the proof for \\autoref{th:ecp-mip}.\n\\end{proof}\n\n\\subsubsection{A MIP Model for $\\alpha > 0.5$}\n\nPreviously, we solved the \\acrshort{ECP} for $\\alpha \\in [0, 0.5]$. Solving the problem for $\\alpha \\in [0, 1]$ requires the definition of\nadditional variables and constraints.\n\n\\begin{equation}\n\t\\label{eq:ecp-exact2}\n\t\\text{maximize}\\quad \\sum_{ T_{k} \\in \\mathcal{T}_{C}, \\; C \\in\n\t\t\\mathcal{\\hat{C}} } \\big( \\sum^{}_{ij \\in E^{+}_k } x_{ij}\n\t^{k} - \\sum_{ij \\in E^{-} _k} x_{ij} ^{k} \\big)\n\\end{equation} \\begin{center} subject to \\end{center}\n\\begin{alignat}{3}\n\t\\label{eq:ecp-v12}\n\t & & x _{ij}^{k} \\leq y_i \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-v22}\n\t & & x _{ij}^{k} \\leq y_j \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-t2}\n\t & & x _{ij}^{k} \\leq z_k \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-e2}\n\t & & x _{ij} ^{k} \\geq - 2 + y_i + y_j + z_k \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-a-alpha-constraint}\n\t & & -N_k z_k < \\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij\n\t\t\\in E_k}\n\ta_{ij} ^{k} \\leq M_{k} (1 - z_{k}) \\quad\\quad & \\forall T_{k} \\in\n\t\\mathcal{T} _{C}, C \\in \\mathcal{\\hat{C}} \\\\\n\t\\label{eq:ecp-a-ij-g-i-j2}\n\t & & a_{ij}^{k} \\geq -1 + y_i + y_j \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-a-ij-l-i2}\n\t & & a_{ij}^{k} \\leq y_i\\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-a-ij-l-j2}\n\t & & a_{ij}^{k} \\leq y_j \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-a-domain-2}\n\t & & 0 \\leq\n\ta_{ij}^{k} \\leq 1 \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-vertex-def2}\n\t & & y _{i} \\in \\{0, 1\\} \\quad\\quad & \\forall i \\in V \\\\\n\t\\label{eq:ecp-edge-def2}\n\t & & 0 \\leq x _{ij} ^{k} \\leq 1 \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:ecp-z-domain-2}\n\t & & z _{k} \\in \\{0, 1\\} \\quad\\quad & \\forall T_{k} \\in \\mathcal{T} _{C}, C \\in\n\t\\hat{\\mathcal{C}}\n\\end{alignat}\n\nWhere are $N_k$ and $M_k$ are constants of value $\\alpha (|E_k^{+}| + 1)$ and $(1 -\n\t\\alpha ) (|E^{-}_k| + 1)$, respectively.\n\nNote also that \\eqref{eq:ecp-a-alpha-constraint} involves a strict inequality,\nwhich is not allowed by the definition of\n\\acrshort{MIP} or \\acrshort{LP}. However, this constraint can easily be\ntransformed into a valid\nand equivalent formulation by the means of a small $\\epsilon > 0$ (generally\nwe can choose $\\epsilon < \\alpha \\cdot 10^{-10}$), so that\nit becomes\n\\begin{equation*}\n\t-N_k z_k \\leq \\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij\n\t\t\\in E_k}\n\ta_{ij} ^{k} - \\epsilon \\leq M_{k} (1 - z_{k}) \\quad\\quad \\forall T_{k} \\in\n\t\\mathcal{T} _{C}, C \\in \\mathcal{\\hat{C}}.\n\\end{equation*}\nWe will however use \\eqref{eq:ecp-a-alpha-constraint} for simplifying notation.\n\nThis problem requires the introduction of variables $a_{ij}^{k}$,\nassociated to edges \\eqref{eq:ecp-a-domain-2}, which\ngenerally take value one if they are associated to an edge induced by the\nset of vertices considered as solution (i.e. $y_i$ with value one). Note that,\nin order to contribute to the score, $x_{ij}^{k}$ also require that the\ncorresponding thread $T_k \\in S_C(U)$. The\nother variables have the same meaning as in MIP~\\eqref{eq:ecp-exact1}-\\eqref{eq:ecp-thread-def1}.\n\n\\begin{theorem}\n\t\\label{th:ecp-mip}\n\tLet $G = \\{G_k = (V,E_k) \\}_k$ be an \\emph{Interaction Graph} and $\\alpha \\in\n\t\t[0, 1]$. Then\n\n\t\\begin{equation}\n\t\t\\max_{U \\subseteq V} \\xi(U) = \\text{OPT(MIP)}\n\t\\end{equation}\n\n\twhere $OPT(MIP)$ denotes the optimal solution to\n\tMIP~\\eqref{eq:ecp-exact2}-\\eqref{eq:ecp-z-domain-2}.\n\\end{theorem}\n\\begin{proof}\n\tWe will prove the theorem by showing that $LHS \\geq RHS$ and that $LHS \\leq\n\t\tRHS$.\n\t\\begin{claim}\n\t\t\\label{th:claim-v-b-xi2}\n\t\tFor any $U \\subseteq V$, the\n\t\tMIP~\\eqref{eq:ecp-exact2}-\\eqref{eq:ecp-z-domain-2} gets value $\\geq \\xi(U)$.\n\t\\end{claim}\n\n\tIt is easy to see that by choosing $x_{ij}^{k}, \\; y_i, \\; z_k$ as in\n\tClaim~\\ref{th:claim-v-b-xi} and\n\t\\begin{equation*}\n\t\ta_{ij}^{k} = \\begin{cases}\n\t\t\t1 & \\text{if } e_{ij} \\in E_k[U] \\\\\n\t\t\t0 & \\text{otherwise}\n\t\t\\end{cases}\n\t\\end{equation*}\n\tall the constraints of the new formulation are satisfied and\n\tClaim~\\ref{th:claim-v-b-xi2} is consequently proved for\n\tMIP~\\eqref{eq:ecp-exact2}-\\eqref{eq:ecp-z-domain-2}.\n\n\tWe will instead focus on proving the analogous of Claim~\\ref{th:claim-xi-b-v}.\n\t\\begin{claim}\n\t\t\\label{th:claim-xi-b-v2}\n\t\tGiven a feasible solution of\n\t\tMIP~\\eqref{eq:ecp-exact2}-\\eqref{eq:ecp-z-domain-2} with value $v$ we can\n\t\tconstruct $U \\; s.t. \\; \\xi(U) \\geq v$ for any $\\alpha$.\n\t\\end{claim}\n\n\t\\begin{proof}\n\t\tLet $U \\coloneqq \\{ v_i \\; s.t. \\; y_i = 1\\} $. We will not prove\n\t\tsome results in Claim~\\ref{th:claim-xi-b-v} which still hold.\n\n\t\tDue to \\eqref{eq:ecp-a-ij-g-i-j2}-\\eqref{eq:ecp-a-ij-l-j2} we\n\t\thave that\n\t\t\\begin{equation}\n\t\t\ta_{ij}^{k} = 1 \\iff y_i = 1 \\land y_j = 1\n\t\t\\end{equation}\n\t\ti.e. all and only the edges induced by $U$ will have the corresponding\n\t\t$a_{ij}^{k} = 1$. Therefore,\n\t\t\\begin{align}\n\t\t\t\\label{eq:ecpp-a-E-sets}\n\t\t\t|E^{-}_{k}[U]| & = \\sum^{}_{ij \\in E^-_k} a_{ij}^{k}, &\n\t\t\t|E_{k}[U]| & = \\sum^{}_{ij \\in E_k } a_{ij}^{k}.\n\t\t\\end{align}\n\t\tConsider now a thread $T_k \\in \\mathcal{S}_C(U)$. By definition, we\n\t\thave that\n\t\t\\begin{align}\n\t\t\t\\eta(T_k) \\leq \\alpha & \\implies \\frac{|E^{-}_{k}[U]|}{|E_{k}[U]|} \\leq \\alpha \\\\\n\t\t\t\\label{eq:ecpp-nc-thread2}\n\t\t\t & \\implies |E^{-}_{k}[U]| - \\alpha \\cdot\n\t\t\t|E_{k}[U]| \\leq 0.\n\t\t\\end{align}\n\t\tNow suppose $z_k = 0$. This means that\n\t\t\\eqref{eq:ecp-a-alpha-constraint} resolves to\n\t\t\\begin{equation*}\n\t\t\t0 < \\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij\n\t\t\t\t\\in E_k}\n\t\t\ta_{ij} ^{k} \\leq M_{k}\n\t\t\\end{equation*}\n\t\tWhich is not satisfied due to \\eqref{eq:ecpp-nc-thread2} and\n\t\t\\eqref{eq:ecpp-a-E-sets}, since they imply that $\\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij\n\t\t\t\t\\in E_k}\n\t\t\ta_{ij} ^{k} \\leq 0$. This justifies\n\t\t\\eqref{eq:ecp-a-alpha-constraint}.\n\n\t\tThus, $z_k = 1$. This\n\t\tmeans that the constraint resolves to\n\t\t\\begin{equation*}\n\t\t\t-N_k = -\\alpha (|E^{+}_{k}[U]| + 1)< \\sum^{}_{ij \\in E^-_k} a_{ij}^{k} -\n\t\t\t\\alpha \\sum^{}_{ij \\in E_k} a_{ij} ^{k}\n\t\t\\end{equation*}\n\t\twhich is satisfied since\n\t\t\\begin{align}\n\t\t\t\\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\t\t\ta_{ij} ^{k} = & |E^{-}_{k}[U]| - \\alpha \\cdot |E_{k}[U]| \\\\\n\t\t\t= & (1- \\alpha)|E^{-}_{k}[U]| - \\alpha \\cdot |E^{+}_{k}[U]| \\\\\n\t\t\t\\geq & - \\alpha \\cdot |E^{+}_{k}[U]| \\\\\n\t\t\t> & - N_k.\n\t\t\\end{align}\n\n\t\tConsequently, because of \\eqref{eq:ecpp-z-iff-xij}, we have that\n\t\t\\begin{equation}\n\t\t\tT_k \\in \\mathcal{S}_C(U) \\iff x_{ij}^{k} = 1 \\; \\forall \\;i, j \\in U\n\t\t\\end{equation}\n\t\tmeaning that each \\emph{non-controversial} $T_k$ will contribute to\n\t\t$\\xi$ and $v$ with the same score, i.e.\n\t\t\\begin{equation*}\n\t\t\t\\sum^{}_{ij \\in E^{+}_k } x_{ij}\n\t\t\t\t^{k} - \\sum_{ij \\in E^{-} _k} x_{ij} ^{k} = |E^{-}_{k}[U]| -\n\t\t\t|E_{k}[U]|.\n\t\t\\end{equation*}\n\n\t\tNow consider $T_k$ \\emph{controversial} and suppose $z_k = 1$. This means that\n\t\t\\eqref{eq:ecp-a-alpha-constraint} resolves to\n\t\t\\begin{equation*}\n\t\t\t-N_k < \\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij\n\t\t\t\t\\in E_k}\n\t\t\ta_{ij} ^{k} \\leq 0,\n\t\t\\end{equation*}\n\t\twhere the second inequality is not satisfied due to \\eqref{eq:ecpp-a-E-sets} and by\n\t\tdefinition of \\emph{controversial}. So $z_k$ must be $0$.\n\t\tThen, we have\n\t\t\\begin{align*}\n\t\t\t\\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij\n\t\t\t\t\\in E_k}\n\t\t\ta_{ij} ^{k} = & |E^{-}_{k}[U]| - \\alpha \\cdot |E_{k}[U]| \\\\\n\t\t\t= & (1- \\alpha)|E^{-}_{k}[U]| - \\alpha \\cdot |E^{+}_{k}[U]| \\\\\n\t\t\t\\leq & (1- \\alpha )\\cdot |E^{-}_{k}[U]| \\\\\n\t\t\t\\leq & M_k,\n\t\t\\end{align*}\n\t\tthus satisfying \\eqref{eq:ecp-a-alpha-constraint}. Therefore, $x_{ij}^{k} = 0 \\; \\forall \\;i, j, \\in V$ (because of\n\t\t\\eqref{eq:ecp-t2}).\n\t\tThis means that the contribution of a controversial $T_k$ is the same in $\\xi$ and $v$ and,\n\t\tin general, $\\xi(U) = v$, proving the claim.\n\t\\end{proof}\n\n\tDue to Claims~\\ref{th:claim-v-b-xi2} and \\ref{th:claim-xi-b-v2} the theorem\n\tis proved.\n\\end{proof}\n\n\\subsection{A MIP Model for the \\acrshort{D-ECP}}%\n\\label{sub:a_mip_model_for_the_d_ecp}\n\nSimilarly to the \\acrshort{ECP}, here we propose a \\acrshort{MIP} model for\nfinding a solution for the \\acrshort{D-ECP}, $\\alpha \\in [0, 1]$.\n\nFor simplifying notation we define $E_{k} \\coloneqq E(T_{k}), T_{k} \\in\n\t\\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}}$.\n\n\\begin{equation}\n\t\\label{eq:d-ecp-objective}\n\t\\text{maximize} \\; \\sum_{ T_{k} \\in \\mathcal{T}_{C}, \\; C \\in\n\t\t\\mathcal{\\hat{C}} } \\big( \\sum^{}_{ij \\in E^+_k} x_{ij}\n\t^{k} - \\sum_{ij \\in E^-_k} x_{ij} ^{k} \\big)\n\\end{equation}\n\\begin{center} subject to \\end{center}\n\\begin{alignat}{3}\n\t\\label{eq:d-ecp-a-ij-l-bi}\n\t & & a_{ij}^{k} \\leq b_{i} \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-a-ij-l-bj}\n\t & & a_{ij}^{k} \\leq b_{j} \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-a-ij-g-ijk}\n\t & & a _{ij} ^{k} \\geq - 1 + b_i + b_j \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-alpha-constraint}\n\t & & -N_{k} z_k < \\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\ta_{ij} ^{k} \\leq M_k (1 - z_k) \\quad\\quad & \\forall T_{k} \\in \\mathcal{T} _{C}, C \\in\n\t\\hat{\\mathcal{C}} \\\\\n\t\\label{eq:d-ecp-edge-charikar1}\n\t & & x _{ij}^{k} \\leq y_i \\quad\\quad & \\forall ij \\in E_{k} \\\\\n\t\\label{eq:d-ecp-edge-charikar2}\n\t & & x _{ij} ^{k} \\leq y_j \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-vertex-charikar1}\n\t &\n\t & \\sum^{}_{i \\in V} y_i = 1\n\t\\quad\\quad\n\t & \\\\\n\t\\label{eq:d-ecp-vertex-l-b}\n\t & & y_i \\leq b_i \\quad\\quad & \\forall i \\in V \\\\\n\t\\label{eq:d-ecp-vertex-g-bi-yj}\n\t & & y_i \\geq -1 + b_i + y_j \\quad\\quad & \\forall i,j \\in V \\\\\n\t\\label{eq:d-ecp-x-l-sum1}\n\t &\n\t & x_{ij}^{k} \\geq -2 +\n\ta_{ij} ^{k} + z_k + y_i \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-x-l-sum2}\n\t &\n\t & x_{ij}^{k} \\geq -2 +\n\ta_{ij} ^{k} + z_k + y_j \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-x-l-a}\n\t & & x_{ij} ^{k} \\leq a_{ij} ^{k} \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-x-l-z}\n\t & & x_{ij} ^{k} \\leq z_k \\quad\\quad & \\forall T_{k} \\in \\mathcal{T} _{C}, C \\in\n\t\\hat{\\mathcal{C}} \\\\\n\t\\label{eq:d-ecp-a-ij}\n\t & & a _{ij} ^{k} \\in \\{0, 1\\} \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-b-i}\n\t & & b _{i} \\in \\{0, 1\\} \\quad\\quad & \\forall i \\in V \\\\\n\t\\label{eq:d-ecp-y-i}\n\t & & y _{i} \\geq 0 \\quad\\quad & \\forall i \\in V \\\\\n\t\\label{eq:d-ecp-x-ij}\n\t & & x _{ij} ^{k} \\geq 0 \\quad\\quad & \\forall ij \\in E_k \\\\\n\t\\label{eq:d-ecp-z-k}\n\t & & z _{k} \\in \\{0, 1\\} \\quad\\quad & \\forall T_{k} \\in \\mathcal{T} _{C}, C \\in\n\t\\hat{\\mathcal{C}}\n\\end{alignat}\n\nWhere are $N_k$ and $M_k$ are constants of value $\\alpha (|E_k^{+}| + 1)$ and $(1 -\n\t\\alpha ) ( |E^{-}_k| + 1)$, respectively.\nAgain the strict inequality of \\eqref{eq:d-ecp-alpha-constraint} can be\ntransformed into a valid constraint as explained in\n\\autoref{sub:a_mip_model_for_the_ecp}.\n% In this formulation we introduce variables $b_i$, associated to vertices, that\n% take the meaning of variables $y_i$ in\n\n\\begin{theorem}\n\t\\label{th:d-ecp-mip}\n\tLet $G = \\{G_k = (V,E_k) \\}_k$ be an \\emph{Interaction Graph} and $\\alpha \\in\n\t\t[0, 1]$.\n\n\t\\begin{equation}\n\t\t\\max_{U \\subseteq V} \\psi(U) = \\text{OPT(MIP)}\n\t\\end{equation}\n\n\twhere $OPT(MIP)$ denotes the optimal solution to\n\tMIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k}.\n\\end{theorem}\n\n\\begin{proof}\n\tSimilarly to \\autoref{th:ecp-mip} we will prove this equality by $2$\n\tinequalities: $RHS \\geq LHS$ and $LHS \\geq RHS$\n\n\t\\begin{claim}\n\t\tFor any $U \\subseteq V$, the\n\t\tMIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k} gets value at least\n\t\t$\\psi(U)$.\n\t\\end{claim}\n\n\t\\begin{proof}\n\t\tLet $c = 1 / |U|$. We construct a feasible solution for\n\t\tMIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k} as follows:\n\n\t\t\\begin{align}\n\t\t\t\\label{eq:ecpp-y}\n\t\t\ty_i & = \\begin{cases}\n\t\t\t\tc, & \\text{if } v_{i} \\in U, \\\\\n\t\t\t\t0, & \\text{otherwise,}\n\t\t\t\\end{cases} \\quad & \\forall v_i \\in U \\\\\n\t\t\tb_i & = \\begin{cases}\n\t\t\t\t1, & \\text{if } v_{i} \\in U, \\\\\n\t\t\t\t0, & \\text{otherwise,}\n\t\t\t\\end{cases}\\quad & \\forall v_i \\in U \\\\\n\t\t\t\\label{eq:ecpp-z}\n\t\t\tz_k & = \\begin{cases}\n\t\t\t\t1, & \\text{if } T_{k} \\in \\mathcal{S}_C(U), \\; C \\in\n\t\t\t\t\\mathcal{\\hat{C}}, \\\\\n\t\t\t\t0, & \\text{otherwise,}\n\t\t\t\\end{cases} \\quad & \\forall T_k \\in \\mathcal{T}_C, C \\in\n\t\t\t\\mathcal{\\hat{C}} \\\\\n\t\t\t\\label{eq:ecpp-x}\n\t\t\ta_{ij}^{k} & = \\begin{cases}\n\t\t\t\t1, & \\text{if } e_{ij} \\in E_{k}[U], \\; T_{k} \\in \\mathcal{T} _C, \\; C \\in\n\t\t\t\t\\mathcal{\\hat{C}}, \\\\\n\t\t\t\t0, & \\text{otherwise,}\n\t\t\t\\end{cases} \\quad & \\forall e_{ij} \\in E_k, T_k \\in \\mathcal{T}_C, C\n\t\t\t\\in \\mathcal{\\hat{C}} \\\\\n\t\t\tx_{ij}^{k} & = \\begin{cases}\n\t\t\t\tc, & \\text{if } e_{ij} \\in E_{k}[U], \\; T_{k} \\in \\mathcal{S}_C(U), \\; C \\in\n\t\t\t\t\\mathcal{\\hat{C}}, \\\\\n\t\t\t\t0, & \\text{otherwise.}\n\t\t\t\\end{cases} & \\forall e_{ij} \\in E_k, T_k \\in \\mathcal{T}_C, C\n\t\t\t\\in \\mathcal{\\hat{C}}\n\t\t\\end{align}\n\n\t\t\\eqref{eq:d-ecp-a-ij-l-bi}-\\eqref{eq:d-ecp-a-ij-g-ijk} are easily\n\t\tsatisfied since $a_{ij}^{k} = 1 \\iff b_i = 1 \\land b_j = 1$, meaning\n\t\tthat an edge is induced (thus $a_{ij}^{k} = 1$) if and only if $ v_i,\n\t\t\tv_j \\in U$. The same idea applies to\n\t\t\\eqref{eq:d-ecp-edge-charikar1}-\\eqref{eq:d-ecp-edge-charikar2}.\n\t\t\\eqref{eq:d-ecp-vertex-l-b} is trivial and\n\t\t\\eqref{eq:d-ecp-x-l-a}-\\eqref{eq:d-ecp-x-l-z} hold since\n\t\t$\\mathcal{S}_C(U) \\subseteq \\mathcal{T}_C $. It is also easy to see that\n\t\t\\eqref{eq:d-ecp-vertex-g-bi-yj} is satisfied since we defined $y_i$\n\t\tand $b_i$ s.t. $b_i = 1$ if and only if $y_i = c$. Furthermore, since\n\t\t$y_i = c$ if and only if $\n\t\t\ty_i \\in U$ then\n\t\t\\begin{equation*}\n\t\t\t\\sum^{}_{i \\in V} y_i = \\sum^{}_{i \\in U} c = 1\n\t\t\\end{equation*}\n\t\tand \\eqref{eq:d-ecp-vertex-charikar1} is also satisfied.\n\n\t\tLet us know consider $T_k \\in \\mathcal{S}_C(U)$ which by definition\n\t\timplies $z_k = 1$. Then,\n\t\t\\begin{align}\n\t\t\t\\eta(T_k[U]) \\leq \\alpha & \\implies\n\t\t\t\\frac{|E^{-}_{k}[U]|}{|E_{k}[U]|} \\leq \\alpha \\\\\n\t\t\t & \\implies |E^{-}_{k}[U]| - \\alpha\n\t\t\t(|E_{k}[U]|) \\leq 0.\n\t\t\\end{align}\n\n\t\tThus, due to the definition of $a_{ij}^k$, the second inequality of \\eqref{eq:d-ecp-alpha-constraint} is true;\n\t\tthe first one is also satisfied since\n\t\t\\begin{align}\n\t\t\t\\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\t\t\ta_{ij} ^{k} & = |E^{-}_{k}[U]| - \\alpha \\cdot |E_{k}[U]| \\\\\n\t\t\t & = (1- \\alpha)|E^{-}_{k}[U]| - \\alpha \\cdot\n\t\t\t|E^{+}_{k}[U]| \\\\\n\t\t\t & \\geq - \\alpha \\cdot |E^{+}_{k}[U]| \\\\\n\t\t\t & > N_k.\n\t\t\\end{align}\n\n\t\t\\eqref{eq:d-ecp-x-l-sum1}-\\eqref{eq:d-ecp-x-l-sum2} are true\n\t\tsince in this case (for $z_k = 1$) by definition $a_{ij}^{k} = 1\n\t\t$ implies $x_{ij}^{k} = c$. If instead $T_k \\not\\in S_c$ and $z_k = 0$ we\n\t\thave that\n\n\t\t\\begin{align}\n\t\t\t\\eta(T_k[U]) > \\alpha & \\implies\n\t\t\t\\frac{|E^{-}_{k}[U]|}{|E_{k}[U]|} > \\alpha \\\\\n\t\t\t & \\implies |E^{-}_{k}[U]| - \\alpha (|E_{k}[U]|) > 0\n\t\t\\end{align}\n\n\t\tand consequently the first inequality of\n\t\t\\eqref{eq:d-ecp-alpha-constraint} is satisfied. Also\n\t\t\\begin{align}\n\t\t\t\\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\t\t\ta_{ij} ^{k} & = |E^{-}_{k}[U]| - \\alpha \\cdot |E_{k}[U]|\n\t\t\t\\\\ & =\n\t\t\t(1- \\alpha)|E^{-}_{k}[U]| - \\alpha \\cdot |E^{+}_{k}[U]| \\\\\n\t\t\t & \\leq (1 -\\alpha) \\cdot |E^{-}_{k}[U]| \\\\\n\t\t\t & < M_k.\n\t\t\\end{align}\n\n\t\tThus, also the second inequality is true. Also,\n\t\t\\eqref{eq:d-ecp-x-l-sum1}-\\eqref{eq:d-ecp-x-l-sum2} are trivially\n\t\tsatisfied.\n\n\t\t\\bigskip\n\t\tConsequently an edge contributing to $\\psi(U)$ will also count in the\n\t\tobjective function by $c$. So, for a given thread $T_k \\in \\mathcal{S}_C, C \\in \\mathcal{\\hat{C}}$\n\n\t\t\\begin{align*}\n\t\t\t\\sum^{}_{ij \\in E^+_k} x_{ij} ^{k} - \\sum_{ij \\in E^{-}\n\t\t\t(T_{k})} x_{ij} ^{k} & = \\sum^{}_{ij \\in E^{+}_k[U] } c - \\sum_{ij \\in E^{-}\n\t\t\t_k[U]} c \\\\\n\t\t\t & = c (|E^{+}_{k}[U]| - |E^{-}_{k}[U]|) \\\\\n\t\t\t & = \\frac{|E^{+}_{k}[U]| -\n\t\t\t|E^{-}_{k}[U]|}{|U|}.\n\t\t\\end{align*}\n\n\t\tThus, for each thread we have the same contribution to both $\\psi(U)$\n\t\tand the objective function of\n\t\tMIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k}. Therefore, the sum through\n\t\tall the threads will correspond as well.\n\t\\end{proof}\n\n\t\\begin{claim}\n\t\tGiven a feasible solution of\n\t\tMIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k} with value $v$ we can\n\t\tconstruct $U \\; s.t. \\; \\psi(U) \\geq v$.\n\t\\end{claim}\n\n\t\\begin{proof}\n\n\t\tLet us define $U \\coloneqq \\{ v_i \\; s.t. \\; y_i \\neq 0\\}$; consider\n\t\t$v_i$ s.t. $y_i \\neq 0$ (if no such vertex exists then the proof is\n\t\ttrivial) and let $c \\coloneqq y_i$. By\n\t\t\\eqref{eq:d-ecp-vertex-g-bi-yj} and \\eqref{eq:d-ecp-vertex-l-b}\n\t\twe have that\n\t\t\\begin{gather}\n\t\t\t\\label{eq:d-ecpp-y-in}\n\t\t\t\\forall v_j \\in V, \\; y_j \\in \\{ 0, c\\}, \\\\\n\t\t\t\\forall i, j \\in V, \\; b_i = 1 \\land b_j = 1 \\implies y_i = y_j =\n\t\t\tc.\n\t\t\\end{gather}\n\n\t\tFurthermore, due to \\eqref{eq:d-ecp-a-ij-l-bi}-\\eqref{eq:d-ecp-a-ij-g-ijk} we\n\t\thave that\n\t\t\\begin{equation}\n\t\t\t\\label{eq:d-ecpp-a-iff}\n\t\t\ta_{ij}^{k} = 1 \\iff b_i = 1 \\land b_j = 1.\n\t\t\\end{equation}\n\n\t\tNow consider some $x_{ij}^{k} > 0 $. By \\eqref{eq:d-ecp-x-l-a}\n\t\t$x_{ij}^{k} > 0$ implies $a _{ij}^{k} = 1$ and, thanks to\n\t\t\\eqref{eq:d-ecp-x-l-z}, $x_{ij}^{k} > 0$ implies $z_k = 1$. Thus, combining\n\t\tthis to the results in \\eqref{eq:d-ecpp-y-in} and\n\t\t\\eqref{eq:d-ecpp-a-iff}\n\t\t\\begin{gather}\n\t\t\t\\label{eq:d-ecpp-x-impl}\n\t\t\tx_{ij}^{k} > 0 \\implies a_{ij}^{k} = 1 \\land z_k =1, \\\\\n\t\t\t\\label{eq:d-ecpp-z-impl}\n\t\t\tz_k = 1 \\implies x_{i'j'}^{k} = c, \\; \\forall i', j' \\in U, \\\\\n\t\t\t\\label{eq:d-ecpp-za-impl}\n\t\t\tz_k = 1 \\land a_{ij}^{k} = 1 \\implies x_{ij}^{k} = c.\n\t\t\\end{gather}\n\t\tThis means that if exists $ x_{ij}^{k} > 0$ then all the variables\n\t\t$x_{i'j'}^{k}$ associated to edges induced by $U$ have value $c$.\n\t\tAlso, since we have that $z_k = 1$, \\eqref{eq:d-ecp-alpha-constraint}\n\t\twill correspond to\n\n\t\t\\begin{equation}\n\t\t\t\\sum^{}_{ij \\in E^-_k} a_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\t\t\ta_{ij} ^{k} \\leq 0.\n\t\t\\end{equation}\n\t\tWe showed in \\eqref{eq:d-ecpp-x-impl} and \\eqref{eq:d-ecpp-za-impl} that $a_{ij}^{k} = 1 \\land z_k\n\t\t\t= 1$ if and only if $x_{ij}^{k} > 0 $. In other other words, the edges contributing to the objective\n\t\tfunction are part of a \\emph{non-controversial} subgraph, i.e. the\n\t\tcorresponding thread $T_k \\in\n\t\t\t\\mathcal{S}_C(U)$ and it will contribute to $\\psi(U)$.\n\n\t\tNow suppose exists $ T_k \\in \\mathcal{S}_C(U), \\; C \\in \\mathcal{\\hat{C}},\n\t\t\t\\; s.t. \\; z_k = 0$. By definition of $\\mathcal{S}_C(U)$ we have that\n\n\t\t\\begin{align}\n\t\t\t\\eta(T_k[U]) \\leq \\alpha & \\implies |E^{-}_{k}[U]| - \\alpha |E_{k}[U]|\n\t\t\t\\\\\n\t\t\t & \\implies \\sum^{}_{ij \\in E^-_k}\n\t\t\ta_{ij}^{k} - \\alpha \\sum^{}_{ij \\in E_k}\n\t\t\ta_{ij} ^{k} \\leq 0,\n\t\t\\end{align}\n\t\tbecause $a_{ij}^{k} = 1$ for all edges induced by $U$. But, if $z_k =\n\t\t\t0$ then constraint \\eqref{eq:d-ecp-alpha-constraint} is violated,\n\t\tand this would be a \\emph{contradiction}. So no such $T_k$ exists and $T_k\n\t\t\t\\in \\mathcal{S}_C(U) \\iff z_k = 1$.\n\n\t\tThis means that a thread contributing to the objective function of\n\t\tMIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k} also counts towards\n\t\t$\\psi(U)$. Due to \\eqref{eq:d-ecpp-z-impl} we can then write, for\n\t\t$T_k \\in \\mathcal{S}_C(U)$\n\n\t\t\\begin{align*}\n\t\t\t\\sum^{}_{ij \\in E^+_k} x_{ij} ^{k} - \\sum_{ij \\in E^{-}\n\t\t\t(T_{k})} x_{ij} ^{k} & = \\sum^{}_{ij \\in E^{+}_k[U] } c - \\sum_{ij \\in E^{-}\n\t\t\t_k[U]} c \\\\\n\t\t\t & = c (|E^{+}_{k}[U]| - |E^{-}_{k}[U]|) \\\\\n\t\t\t & = \\frac{|E^{+}_{k}[U]| -\n\t\t\t|E^{-}_{k}[U]|}{|U|}.\n\t\t\\end{align*}\n\n\t\tThus, each threads equally contributes to $\\psi(U)$ and the objective\n\t\tfunction of MIP~\\eqref{eq:d-ecp-objective}-\\eqref{eq:d-ecp-z-k}.\n\t\tTherefore, $\\psi(U) \\geq v$.\n\t\\end{proof}\n\n\tThis concludes the proof of the theorem.\n\n\\end{proof}\n\n\\section{Heuristics}%\n\\label{sub:approximation_algorithms}\n\nWe now present some heuristic algorithms for solving the \\acrshort{ECP} and\n\\acrshort{D-ECP}. We start by describing how $\\xi(U)$ and $\\psi(U)$ can be\ncomputed in practice.\n\nLet $\\textsc{Score}_{\\xi} (U)$ and $\\textsc{Score}_{\\psi} (U)$ be the functions\ncomputing the \\emph{Echo Chamber Score} and \\emph{Densest-Echo Chamber Score}\nof $U$, respectively. These subroutines iterate over the edges of the vertices\nin $U$, ignoring those that are not induced by $U$, and counting for each\nthread $T \\in \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}} $ the number of edges\nand negative edges to see which are \\emph{controversial}, then calculating\ntheir contributions (\\autoref{alg:score_xi} shows in detail\n$\\textsc{Score}_\\xi$; $\\textsc{Score}_\\psi$ can simply be computed as\n$\\textsc{Score}_\\xi(U)/|U|$). Note that this algorithm operates also on the\nweights $w_{ij}^{k}$ of the edges. This is something that is \"implicitely\" done\nin the \\acrshort{MIP}s, as the sums iterate over positive and negative edges.\n\n\\begin{algorithm}\n\t\\SetAlgoLined\n\t\\KwIn{Interaction graph $G = \\{G_k = (V,E_k) \\}_k$, a set of users $U\n\t\t\t\\subseteq V$, $\\alpha \\in [0, 1]$}\n\t\\KwResult{$\\xi(U)$}\n\t$N^{+} (T) \\leftarrow 0, \\; N^{-} (T) \\leftarrow 0\\; $ for all threads $T\n\t\t\\in \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}} $ \\;\n\n\t\\ForEach{$v_{i} \\in U$}{\n\t\t$S_i \\leftarrow$ edges starting from $v_{i} $ \\;\n\t\t\\ForEach{$e_{ij} \\in S_i$ \\textbf{if} $v_{j} \\in U$}{\n\t\t\t$T_{ij} \\leftarrow$ thread of $e_{ij} $ \\;\n\t\t\t$w_{ij} \\leftarrow$ weight of $e_{ij} $ \\;\n\n\t\t\t\\uIf{$w_{ij} \\geq 0$}{\n\t\t\t\t$N^{+}(T_{ij} ) \\leftarrow N^{+}(T_{ij} ) + 1$ \\;\n\t\t\t}\\Else{\n\t\t\t\t$N^{-}(T_{ij} ) \\leftarrow N^{-}(T_{ij} ) + 1$ \\;\n\t\t\t}\n\n\t\t}\n\t}\n\n\t$\\xi(U) \\leftarrow 0$ \\;\n\t$\\eta(T) \\leftarrow\\frac{N^{-}(T)}{(N^{-}(T) + N^{+} (T))} \\; $ for all threads $T\n\t\t\\in \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}} $ \\;\n\t\\ForEach{$T\\in \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}}$ \\textbf{if}\n\t\t$ \\eta(T) \\leq \\alpha $}{\n\t\t$\\xi(U) \\leftarrow \\xi(U) + N^{+}(T) - N^{-}(T)$\n\t}\n\t\\Return $\\xi(U)$ \\;\n\n\t\\caption{The $\\textsc{Score}_{\\xi} $ subroutine}\n\t\\label{alg:score_xi}\n\\end{algorithm}\n\nWe present our algorithms focusing on \\acrshort{ECP} in detail. They can generally be\nadapted for solving the \\acrshort{D-ECP} by replacing calls to\n$\\textsc{Score}_{\\xi}$ with $\\textsc{Score}_{\\psi}$.\n\n\\subsection{The $\\beta$-Algorithm}%\n\\label{ssub:the_beta_approach}\n\nThe $\\beta$-algorithm is an heuristic for the \\acrshort{ECP}. The\n$\\beta$-algorithm (\\autoref{alg:algorithm_beta}) constructs a set of users $U$ by\niteratively adding the node which increases the most the score or removing from\n$U$ the one which contributes the least, stopping when the score cannot be\nincreased by adding a node. The frequencies of addition and removal are regulated\nthrough a parameter $\\beta \\in [0, 1]$ (for smaller values a higher density is to be expected,\ngenerally).\n\n$U$ is initialized by sampling one node from the graph. There are two possible\napproaches in doing that: one is uniformly;\nthe other is using probabilities proportional to the number\nof positive edges each node has.\n\n\\begin{algorithm}\n\t% \\KwIn{Interaction graph G, $\\alpha \\in [0,1]$} \\;\n\t\\SetAlgoLined\n\t% \\KwResult{Write here the result }\n\t$U = \\{$ a single random node $\\}$\\;\n\t$\\xi(U) = 0$ \\;\n\t\\While{ $\\exists \\; v_{j} \\; s.t. \\; \\textsc{Score}_{\\xi} (U \\cup \\; \\{\n\t\t\tv_{j} \\} )> \\xi(U)$}{\n\t\t$N(U) \\leftarrow $ neighbours of vertices in $U$ in the graph $G$ \\;\n\t\tFlip a coin which gives head with probability $\\beta$ \\;\n\t\tIf head\\: {\n\t\t\t$U \\leftarrow U \\cup \\; \\{ \\arg\\max_{v_{j} \\in N(U)}\n\t\t\t\t\\textsc{Score}_{\\xi} (U \\cup \\;\n\t\t\t\t\\{ v_{j} \\}) \\}$ \\;\n\t\t}\n\n\t\telse\\: {\n\t\t\t$U \\leftarrow U \\setminus \\{ \\arg\\max_{v_{j} \\in U }\n\t\t\t\t\\textsc{Score}_{\\xi} (U \\setminus \\{ v_{j} \\}) \\}$ \\;\n\t\t}\n\n\t}\n\t\\Return $\\textsc{Score}_{\\xi} (U)$ \\;\n\t\\caption{$\\beta$-algorithm}\n\t\\label{alg:algorithm_beta}\n\\end{algorithm}\n\nIn addition, one may also want to ignore a node when it is removed for the\nnext iterations, in order to prevent the algorithm from repeatedly adding\nand taking out from $U$ the same vertex.\n\nThe result is clearly dependent on the choice of the initial node. For this\nreason the process should be repeated for different initial nodes.\n\nOne of the limitations of this approach is that the algorithm will only find\nsets of users that are connected in the original graph. This is due to the fact\nthat it will never add a node which is not connected to any of the vertices in\n$U$, as it produces an increase of the score equal to $0$.\n\n\\subsection{Peeling Algorithm}%\n\\label{ssub:peeling_algorithm}\n\nInspired to the greedy algorithm proposed in \\cite{charikar2000greedy}, the\npeeling\nalgorithm starts by considering a set $U = V$,\niteratively removing the worst nodes (\\autoref{alg:algorithm_peeling}).\n\n\\begin{algorithm}\n\t\\SetAlgoLined\n\t% \\KwResult{Write here the result }\n\t$U = V$\\;\n\t$S = \\textsc{Score}_{\\xi}(U)$ \\;\n\t\\While{$U \\neq \\emptyset$ }{\n\t\t$v = \\arg\\max_{v_{j} \\in U }\n\t\t\t\\textsc{Score}_{\\xi} (U \\setminus \\{ v_{j} \\})$ \\; $U \\leftarrow U \\setminus \\{ v \\}$ \\;\n\n\t\t$S_i = \\textsc{Score}_{\\xi} (U) $ \\;\n\t}\n\t\\Return $\\arg\\max_{i}S_i$ \\;\n\n\t\\caption{Peeling algorithm}\n\t\\label{alg:algorithm_peeling}\n\\end{algorithm}\n\nIf many nodes produce the same score, then one of them is\nrandomly selected (or, alternatively, the one which has the highest fraction of\nnegative edges).\n\n\\subsection{Rounding Algorithm}%\n\\label{ssub:rounding_algorithm}\n\nThis algorithm reconstructs a solution starting from the results of the\nrelaxation of the exact models and is again inspired by the algorithm for\nreconstructing the exact solution from the \\acrshort{LP} model in\n\\cite{charikar2000greedy}.\n\nMore specifically, our relaxation of\nMIP~\\eqref{eq:ecp-exact2}-\\eqref{eq:ecp-z-domain-2} replaces constraints\n\\eqref{eq:ecp-vertex-def2} and \\eqref{eq:ecp-z-domain-2} with\n\\begin{gather}\n\t0 \\leq y_i \\leq 1, \\\\\n\t0 \\leq z_i \\leq 1.\n\\end{gather}\nWe now have to solve an \\acrshort{LP} problem.\n\nLet $\\tilde{E}$ be the sequence of edges ordered in descending order by $x_{ij}\n\t\t^{k} $. The algorithm (\\autoref{alg:algorithm_rounding}) iterates over\nthe edges in $\\tilde{E}$, adding them to a \\emph{dummy} graph $\\hat{G}$, also eventually\nadding incident nodes if not already present. At each iteration it computes\nthe score of the vertices in the graph $\\hat{G}$ and the score\nof the vertices of each component in the graph, keeping track of the best\nresult.\n\n\\begin{algorithm}\n\t\\SetAlgoLined\n\tSolve the relaxation of MIP~\\eqref{eq:ecp-exact2}-\\eqref{eq:ecp-z-domain-2}\n\t\\;\n\t$\\hat{G} \\leftarrow $ empty graph \\;\n\t$\\hat{V} \\leftarrow $ vertices of $\\hat{G}$ \\;\n\t$S = 0$\n\n\t\\ForEach{ $e_{ij}^{k} \\in \\tilde{E}$ in descending order of $x_{ij}^{k}$ }{\n\t\t$\\hat{V} \\leftarrow \\hat{V} \\bigcup \\{ v_{i} \\}$ \\textbf{if} $v_i\n\t\t\t\\not\\in \\hat{V}$ \\;\n\t\t$\\hat{V} \\leftarrow \\hat{V} \\bigcup \\{ v_{j} \\}$ \\textbf{if} $v_j\n\t\t\t\\not\\in \\hat{V}$ \\;\n\n\t\t$S \\leftarrow \\max(S, \\; \\textsc{Score}_{\\xi}(\\hat{V}) )$\n\n\t\t\\ForEach{component $C$ in $\\hat{G}$}{\n\t\t\t$S \\leftarrow \\max(S, \\; \\textsc{Score}_{\\xi}(C) )$\n\t\t}\n\n\t}\n\n\t\\Return S \\;\n\t\\caption{Rounding algorithm}\n\t\\label{alg:algorithm_rounding}\n\\end{algorithm}\n\nThe motivation for the algorithm can be seen in\nFigures~\\ref{fig:rounding-original}-\\ref{fig:rounding-relaxed}: the problem\nrelaxation involves a solution whose value assigned to the edges can be used to\nfind subgraphs with many positive edges by using each separate component as set\nof users $U$.\n\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\t\\tikzfig{tex/tikz/rounding_original_t1}\n\t\t\t\\caption{$T_1$}\n\t\t\t\\label{fig:rounding-original-t1}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\t\\tikzfig{tex/tikz/rounding_original_t2}\n\t\t\t\\caption{$T_2$}\n\t\t\t\\label{fig:rounding-original-t2}\n\t\t\\end{subfigure}\n\t\\end{center}\n\t\\caption{Example original \\emph{Interaction Graph} $G$}\n\t\\label{fig:rounding-original}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\t\\tikzfig{tex/tikz/rounding_integer_t1}\n\t\t\t\\caption{$T_1$}\n\t\t\t\\label{fig:rounding-integer-t1}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\t\\tikzfig{tex/tikz/rounding_integer_t2}\n\t\t\t\\caption{$T_2$}\n\t\t\t\\label{fig:rounding-original-t2}\n\t\t\\end{subfigure}\n\t\\end{center}\n\t\\caption{Exact solution of the example in \\autoref{fig:rounding-original},\n\t\t$\\alpha = 0.4$}\n\t\\label{fig:rounding-integer}\n\\end{figure}\n\\begin{figure}\n\t\\begin{center}\n\t\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\t\\scalebox{0.8}{\n\t\t\t\t\\tikzfig{tex/tikz/rounding_relaxed_t1}\n\t\t\t}\n\t\t\t\\caption{$T_1$, where $z_1 = 0.66$}\n\t\t\t\\label{fig:rounding-relaxed-t1}\n\t\t\\end{subfigure}\n\t\t\\begin{subfigure}[b]{0.4\\textwidth}\n\t\t\t\\centering\n\t\t\t\\scalebox{0.8}{\n\t\t\t\t\\tikzfig{tex/tikz/rounding_relaxed_t2}\n\t\t\t}\n\t\t\t\\caption{$T_2$, where $z_2 = 1.0$}\n\t\t\t\\label{fig:rounding-relaxed-t2}\n\t\t\\end{subfigure}\n\t\\end{center}\n\t\\caption{Solution of the relaxation of $G$ of\n\t\t\\autoref{fig:rounding-original}, $\\alpha = 0.4$}\n\t\\label{fig:rounding-relaxed}\n\\end{figure}\n\nWhile one may think from these examples that the relaxation trivially assigns\nnon-zero values only to positive edges, \\autoref{fig:rounding-original2} shows\na case in which a negative edge, $e_{31}$, gets the value of $1$. Furthermore,\nin this example the algorithm is able to reconstruct the exact solution of the problem.\n\n\\begin{figure}\n\t\\centering\n\t\\tikzfig{tex/tikz/rounding_original2}\n\t\\caption[Example of rounding algorithm finding the exact solution]{Another\n\t\t\\emph{Interaction graph} example, with only one thread. In\n\t\tthis case the rounding algorithm is able to find the exact solution by\n\t\tselecting all the nodes except for $v_4$. In the result of the\n\t\trelaxation all the edges except for $e_{42}$ get the value of $1$.}%\n\t\\label{fig:rounding-original2}\n\\end{figure}\n\n\\section{Alternative Formulations}%\n\\label{sec:alternative-formulations}\n\nDue to the intrinsic complexity of the problems\n(\\autoref{ch:complexity}) we define variants of\nthe \\acrshort{ECP} and \\acrshort{D-ECP} problems, for some of which we are also able to find an exact solution.\n\nFor these new problems we need to define new graphs, obtained by preprocessing\nthe \\emph{interaction graph}.\n\n\\subsection{The \\acrlong{PA} Graph}%\n\\label{sub:pa-graph}\n\nLet $G = \\{G_k = (V,E_k) \\}_k$ be the \\emph{interaction graph}, let $E_k$ and\n$E^-_k$ denote the edges and negative edges in thread $T_k$,\nrespectively. We define $\\delta(v_{i}, v_{j})$ and\n$\\delta^{-} (v_{i}, v_{j})$ to be the sum of the edges and negative edges, respectively,\nassociated to controversial contents between vertices $v_{i} $ and $v_{j}\n$, i.e.\\\n\n\\begin{align}\n\t\\label{eq:}\n\t\\delta(v_{i}, v_{j}) & = \\sum^{}_{T_k \\in \\mathcal{T}_C, C \\in\n\t\\mathcal{\\hat{C}} } ( \\sum^{}_{e_{ij} \\in E_k} w_{ij} + \\sum^{}_{e_{ji} \\in\n\tE_k} w_{ji} ),\n\t\\\\\n\t\\delta^-(v_{i}, v_{j}) & = \\sum^{}_{T_k \\in \\mathcal{T}_C, C \\in\n\t\\mathcal{\\hat{C}} } ( \\sum^{}_{e_{ij} \\in E^-_k} w_{ij} + \\sum^{}_{e_{ji} \\in\n\tE^-_k} w_{ji} ).\n\\end{align}\n\n\\bigskip\n\nThe \\acrfull{PA} graph $G_P = (V_{P}, E_{P}) $ is constructed as follows from\n$G$:\n\n\\begin{itemize}\n\t\\item For any vertex $v_{i} \\in V$ add a corresponding vertex in $V_{P} $.\n\t\\item For any pair of vertices $v_i, v_j$ in $G$ let $\\eta(v_i,v_j)\n\t\t \\coloneqq \\frac{\\delta^{-} (v_i,v_j)}{\\delta (v_i,v_j)} $. If\n\t $\\eta(v_i,v_j) \\leq \\alpha $, add a positive edge between $v_{i} $ and\n\t $v_{j} $ in $G_{P} $. If, instead, $\\eta(v_i,v_j) > \\alpha $ or $\\delta(v_{i}, v_{j}) =\n\t\t 0$ then don't add any edge between the two vertices.\n\t % \\item don't add any edge otherwise between $v_{i} $ and $v_{j} $\n\\end{itemize}\n\nThe problem then is finding the Densest Subgraph of $G_P$, i.e., if $E_{P} [U]$\nis the set of edges induced on $G_P$ by $U \\subseteq V$, finding $U$ maximizing\n\n\\begin{equation}\n\t\\xi(U) = \\frac{|E_{P} [U]|}{|U|}.\n\\end{equation}\n\n\\subsection{The \\acrlong{TPA} Graph}%\n\\label{ssub:the_tpa_graph}\n\nDifferently from the previous method, in this case edges are aggregated\nseparately for each thread.\n\n\\bigskip\n\nMore specifically, given an \\emph{interaction graph} $G = \\{G_k = (V,E_k) \\}_k$,\nlet $E(T)$ and\n$E^-(T)$ denote the edges and negative edges in thread $T$,\nrespectively. We define $\\delta_{T}(v_{i}, v_{j})$ and\n$\\delta^{-} _{T}(v_{i}, v_{j})$ to be the sum of the edges and negative edges, respectively,\nassociated to thread $T$ between vertices $v_{i} $ and $v_{j} $, being $T \\in\n\t\\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}}$, i.e.\n\n\\begin{align}\n\t\\label{eq:}\n\t\\delta_T(v_{i}, v_{j}) & = \\sum^{}_{e_{ij} \\in E(T)} w_{ij} + \\sum^{}_{e_{ji} \\in\n\tE(T)} w_{ji},\n\t\\\\\n\t\\delta^-_T(v_{i}, v_{j}) & = \\sum^{}_{e_{ij} \\in E^-(T)} w_{ij} + \\sum^{}_{e_{ji} \\in\n\tE^-(T)} w_{ji}.\n\\end{align}\n\nWe will produce a graph, the \\acrfull{TPA} Graph\n\\begin{equation*}\n\tG_{TP} = (V_{TP}, E_{TP}),\n\\end{equation*}\nthat, differently from the \\acrshort{PA} Graph, is a multiplex\ngraph (each layer representing a thread). The construction of the\n\\acrshort{TPA} Graph is as follows:\n\n\\begin{itemize}\n\t\\item For any vertex $v_{i} \\in V$ add a corresponding vertex in $V_{P} $.\n\t\\item For any thread $T \\in\n\t\t \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}}$ we add a layer $T$ to\n\t $G_{TP}$.\n\t\\item For any thread $T \\in\n\t\t \\mathcal{T}_{C}, C \\in \\mathcal{\\hat{C}}$ and pair of vertices\n\t $v_i, v_j$ in $G$, let $\\eta_{T}(v_i,v_j)\n\t\t \\coloneqq \\frac{\\delta^{-} _{T}(v_i,v_j)}{\\delta _{T}(v_i,v_j)} $. If\n\t $\\eta_{T}(v_i,v_j) \\leq \\alpha $ add a positive edge between $v_{i} $ and\n\t $v_{j} $ in $G_{TP} $, in the layer associated to thread $T$.\n\t % \\item don't add any edge otherwise between $v_{i} $ and $v_{j} $\n\\end{itemize}\n\nWe can then solve on $G_{TP}$:\n\n\\begin{enumerate}\n\t\\item The Densest Subgraph Problem (or, equivalently, the\n\t \\acrshort{DCS}-MM), which we will refer to as the Densest\n\t \\acrlong{TPA} (D-\\acrshort{TPA}) Problem\n\t (\\autoref{sub:densest_subgraphs}).\n\t\\item The \\acrshort{O2BFF} Problem, more specifically \\acrshort{O2BFF}-AM,\n\t which we will denote as \\acrshort{O2BFF} \\acrlong{TPA}\n\t (\\acrshort{O2BFF}-\\acrshort{TPA}) Problem\n\t (\\autoref{sub:the_o_2_bff_problem}).\n\\end{enumerate}\n", "meta": {"hexsha": "e2415ed73bb5b6a2b8b10d32d0e3cefaa3b88765", "size": 50755, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/methods.tex", "max_stars_repo_name": "morpheusthewhite/master-thesis", "max_stars_repo_head_hexsha": "2ab4c0509a119d7b5f332b842a4101470a884351", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-06-15T14:01:29.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-15T14:01:29.000Z", "max_issues_repo_path": "tex/methods.tex", "max_issues_repo_name": "morpheusthewhite/master-thesis", "max_issues_repo_head_hexsha": "2ab4c0509a119d7b5f332b842a4101470a884351", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/methods.tex", "max_forks_repo_name": "morpheusthewhite/master-thesis", "max_forks_repo_head_hexsha": "2ab4c0509a119d7b5f332b842a4101470a884351", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.0354924579, "max_line_length": 240, "alphanum_fraction": 0.5182937642, "num_tokens": 17464, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979746, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.34661903402188093}} {"text": "% language=uk\n\n\\environment luatex-style\n\n\\startcomponent luatex-math\n\n\\startchapter[reference=math,title={Math}]\n\n\\topicindex {math}\n\nThe handling of mathematics in \\LUATEX\\ differs quite a bit from how \\TEX82 (and\ntherefore \\PDFTEX) handles math. First, \\LUATEX\\ adds primitives and extends some\nothers so that \\UNICODE\\ input can be used easily. Second, all of \\TEX82's\ninternal special values (for example for operator spacing) have been made\naccessible and changeable via control sequences. Third, there are extensions that\nmake it easier to use \\OPENTYPE\\ math fonts. And finally, there are some\nextensions that have been proposed or considered in the past that are now added\nto the engine.\n\n\\section{Math styles}\n\n\\subsection{\\lpr {mathstyle}}\n\n\\topicindex {math+styles}\n\nIt is possible to discover the math style that will be used for a formula in an\nexpandable fashion (while the math list is still being read). To make this\npossible, \\LUATEX\\ adds the new primitive: \\lpr {mathstyle}. This is a \\quote\n{convert command} like e.g. \\prm {romannumeral}: its value can only be read,\nnot set.\n\nThe returned value is between 0 and 7 (in math mode), or $-1$ (all other modes).\nFor easy testing, the eight math style commands have been altered so that the can\nbe used as numeric values, so you can write code like this:\n\n\\starttyping\n\\ifnum\\mathstyle=\\textstyle\n \\message{normal text style}\n\\else \\ifnum\\mathstyle=\\crampedtextstyle\n \\message{cramped text style}\n\\fi \\fi\n\\stoptyping\n\nSometimes you won't get what you expect so a bit of explanation might help to\nunderstand what happens. When math is parsed and expanded it gets turned into a\nlinked list. In a second pass the formula will be build. This has to do with the\nfact that in order to determine the automatically chosen sizes (in for instance\nfractions) following content can influence preceding sizes. A side effect of this\nis for instance that one cannot change the definition of a font family (and\nthereby reusing numbers) because the number that got used is stored and used in\nthe second pass (so changing \\type {\\fam 12} mid|-|formula spoils over to\npreceding use of that family).\n\nThe style switching primitives like \\prm {textstyle} are turned into nodes so the\nstyles set there are frozen. The \\prm {mathchoice} primitive results in four\nlists being constructed of which one is used in the second pass. The fact that\nsome automatic styles are not yet known also means that the \\lpr {mathstyle}\nprimitive expands to the current style which can of course be different from the\none really used. It's a snapshot of the first pass state. As a consequence in the\nfollowing example you get a style number (first pass) typeset that can actually\ndiffer from the used style (second pass). In the case of a math choice used\nungrouped, the chosen style is used after the choice too, unless you group.\n\n\\startbuffer[1]\n [a:\\mathstyle]\\quad\n \\bgroup\n \\mathchoice\n {\\bf \\scriptstyle (x:d :\\mathstyle)}\n {\\bf \\scriptscriptstyle (x:t :\\mathstyle)}\n {\\bf \\scriptscriptstyle (x:s :\\mathstyle)}\n {\\bf \\scriptscriptstyle (x:ss:\\mathstyle)}\n \\egroup\n \\quad[b:\\mathstyle]\\quad\n \\mathchoice\n {\\bf \\scriptstyle (y:d :\\mathstyle)}\n {\\bf \\scriptscriptstyle (y:t :\\mathstyle)}\n {\\bf \\scriptscriptstyle (y:s :\\mathstyle)}\n {\\bf \\scriptscriptstyle (y:ss:\\mathstyle)}\n \\quad[c:\\mathstyle]\\quad\n \\bgroup\n \\mathchoice\n {\\bf \\scriptstyle (z:d :\\mathstyle)}\n {\\bf \\scriptscriptstyle (z:t :\\mathstyle)}\n {\\bf \\scriptscriptstyle (z:s :\\mathstyle)}\n {\\bf \\scriptscriptstyle (z:ss:\\mathstyle)}\n \\egroup\n \\quad[d:\\mathstyle]\n\\stopbuffer\n\n\\startbuffer[2]\n [a:\\mathstyle]\\quad\n \\begingroup\n \\mathchoice\n {\\bf \\scriptstyle (x:d :\\mathstyle)}\n {\\bf \\scriptscriptstyle (x:t :\\mathstyle)}\n {\\bf \\scriptscriptstyle (x:s :\\mathstyle)}\n {\\bf \\scriptscriptstyle (x:ss:\\mathstyle)}\n \\endgroup\n \\quad[b:\\mathstyle]\\quad\n \\mathchoice\n {\\bf \\scriptstyle (y:d :\\mathstyle)}\n {\\bf \\scriptscriptstyle (y:t :\\mathstyle)}\n {\\bf \\scriptscriptstyle (y:s :\\mathstyle)}\n {\\bf \\scriptscriptstyle (y:ss:\\mathstyle)}\n \\quad[c:\\mathstyle]\\quad\n \\begingroup\n \\mathchoice\n {\\bf \\scriptstyle (z:d :\\mathstyle)}\n {\\bf \\scriptscriptstyle (z:t :\\mathstyle)}\n {\\bf \\scriptscriptstyle (z:s :\\mathstyle)}\n {\\bf \\scriptscriptstyle (z:ss:\\mathstyle)}\n \\endgroup\n \\quad[d:\\mathstyle]\n\\stopbuffer\n\n\\typebuffer[1]\n\n% \\typebuffer[2]\n\nThis gives:\n\n\\blank $\\displaystyle \\getbuffer[1]$ \\blank\n\\blank $\\textstyle \\getbuffer[1]$ \\blank\n\nUsing \\prm {begingroup} \\unknown\\ \\prm {endgroup} instead gives:\n\n\\blank $\\displaystyle \\getbuffer[2]$ \\blank\n\\blank $\\textstyle \\getbuffer[2]$ \\blank\n\nThis might look wrong but it's just a side effect of \\lpr {mathstyle} expanding\nto the current (first pass) style and the number being injected in the list that\ngets converted in the second pass. It all makes sense and it illustrates the\nimportance of grouping. In fact, the math choice style being effective afterwards\nhas advantages. It would be hard to get it otherwise.\n\n\\subsection{\\lpr {Ustack}}\n\n\\topicindex {math+stacks}\n\nThere are a few math commands in \\TEX\\ where the style that will be used is not\nknown straight from the start. These commands (\\prm {over}, \\prm {atop},\n\\prm {overwithdelims}, \\prm {atopwithdelims}) would therefore normally return\nwrong values for \\lpr {mathstyle}. To fix this, \\LUATEX\\ introduces a special\nprefix command: \\lpr {Ustack}:\n\n\\starttyping\n$\\Ustack {a \\over b}$\n\\stoptyping\n\nThe \\lpr {Ustack} command will scan the next brace and start a new math group\nwith the correct (numerator) math style.\n\n\\section{Unicode math characters}\n\n\\topicindex {math+\\UNICODE}\n\\topicindex {\\UNICODE+math}\n\nCharacter handling is now extended up to the full \\UNICODE\\ range (the \\type {\\U}\nprefix), which is compatible with \\XETEX.\n\nThe math primitives from \\TEX\\ are kept as they are, except for the ones that\nconvert from input to math commands: \\type {mathcode}, and \\type {delcode}. These\ntwo now allow for a 21-bit character argument on the left hand side of the equals\nsign.\n\nSome of the new \\LUATEX\\ primitives read more than one separate value. This is\nshown in the tables below by a plus sign in the second column.\n\nThe input for such primitives would look like this:\n\n\\starttyping\n\\def\\overbrace{\\Umathaccent 0 1 \"23DE }\n\\stoptyping\n\nThe altered \\TEX82 primitives are:\n\n\\starttabulate[|l|l|r|c|l|r|]\n\\DB primitive \\BC min \\BC max \\BC \\kern 2em \\BC min \\BC max \\NC \\NR\n\\TB\n\\NC \\prm {mathcode} \\NC 0 \\NC 10FFFF \\NC = \\NC 0 \\NC 8000 \\NC \\NR\n\\NC \\prm {delcode} \\NC 0 \\NC 10FFFF \\NC = \\NC 0 \\NC FFFFFF \\NC \\NR\n\\LL\n\\stoptabulate\n\nThe unaltered ones are:\n\n\\starttabulate[|l|l|r|]\n\\DB primitive \\BC min \\BC max \\NC \\NR\n\\TB\n\\NC \\prm {mathchardef} \\NC 0 \\NC 8000 \\NC \\NR\n\\NC \\prm {mathchar} \\NC 0 \\NC 7FFF \\NC \\NR\n\\NC \\prm {mathaccent} \\NC 0 \\NC 7FFF \\NC \\NR\n\\NC \\prm {delimiter} \\NC 0 \\NC 7FFFFFF \\NC \\NR\n\\NC \\prm {radical} \\NC 0 \\NC 7FFFFFF \\NC \\NR\n\\LL\n\\stoptabulate\n\nFor practical reasons \\prm {mathchardef} will silently accept values larger\nthat \\type {0x8000} and interpret it as \\lpr {Umathcharnumdef}. This is needed\nto satisfy older macro packages.\n\nThe following new primitives are compatible with \\XETEX:\n\n% somewhat fuzzy:\n\n\\starttabulate[|l|l|r|c|l|r|]\n\\DB primitive \\BC min \\BC max \\BC \\kern 2em \\BC min \\BC max \\NC \\NR\n\\TB\n\\NC \\lpr {Umathchardef} \\NC 0+0+0 \\NC 7+FF+10FFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Umathcharnumdef}\\rlap{\\high{5}} \\NC -80000000 \\NC 7FFFFFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Umathcode} \\NC 0 \\NC 10FFFF \\NC = \\NC 0+0+0 \\NC 7+FF+10FFFF \\NC \\NR\n\\NC \\lpr {Udelcode} \\NC 0 \\NC 10FFFF \\NC = \\NC 0+0 \\NC FF+10FFFF \\NC \\NR\n\\NC \\lpr {Umathchar} \\NC 0+0+0 \\NC 7+FF+10FFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Umathaccent} \\NC 0+0+0 \\NC 7+FF+10FFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Udelimiter} \\NC 0+0+0 \\NC 7+FF+10FFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Uradical} \\NC 0+0 \\NC FF+10FFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Umathcharnum} \\NC -80000000 \\NC 7FFFFFFF \\NC \\NC \\NC \\NC \\NR\n\\NC \\lpr {Umathcodenum} \\NC 0 \\NC 10FFFF \\NC = \\NC -80000000 \\NC 7FFFFFFF \\NC \\NR\n\\NC \\lpr {Udelcodenum} \\NC 0 \\NC 10FFFF \\NC = \\NC -80000000 \\NC 7FFFFFFF \\NC \\NR\n\\LL\n\\stoptabulate\n\nSpecifications typically look like:\n\n\\starttyping\n\\Umathchardef\\xx=\"1\"0\"456\n\\Umathcode 123=\"1\"0\"789\n\\stoptyping\n\nThe new primitives that deal with delimiter|-|style objects do not set up a\n\\quote {large family}. Selecting a suitable size for display purposes is expected\nto be dealt with by the font via the \\lpr {Umathoperatorsize} parameter (more\ninformation can be found in a following section).\n\nFor some of these primitives, all information is packed into a single signed\ninteger. For the first two (\\lpr {Umathcharnum} and \\lpr {Umathcodenum}), the\nlowest 21 bits are the character code, the 3 bits above that represent the math\nclass, and the family data is kept in the topmost bits This means that the values\nfor math families 128--255 are actually negative. For \\lpr {Udelcodenum} there\nis no math class. The math family information is stored in the bits directly on\ntop of the character code. Using these three commands is not as natural as using\nthe two- and three|-|value commands, so unless you know exactly what you are\ndoing and absolutely require the speedup resulting from the faster input\nscanning, it is better to use the verbose commands instead.\n\nThe \\lpr {Umathaccent} command accepts optional keywords to control various\ndetails regarding math accents. See \\in {section} [mathacc] below for details.\n\nThere are more new primitives and all of these will be explained in following\nsections:\n\n\\starttabulate[|l|l|]\n\\DB primitive \\BC value range (in hex) \\NC \\NR\n\\TB\n\\NC \\lpr {Uroot} \\NC 0 + 0--FF + 10FFFF \\NC \\NR\n\\NC \\lpr {Uoverdelimiter} \\NC 0 + 0--FF + 10FFFF \\NC \\NR\n\\NC \\lpr {Uunderdelimiter} \\NC 0 + 0--FF + 10FFFF \\NC \\NR\n\\NC \\lpr {Udelimiterover} \\NC 0 + 0--FF + 10FFFF \\NC \\NR\n\\NC \\lpr {Udelimiterunder} \\NC 0 + 0--FF + 10FFFF \\NC \\NR\n\\LL\n\\stoptabulate\n\n\\section{Cramped math styles}\n\n\\topicindex {math+styles}\n\\topicindex {math+spacing}\n\\topicindex {math+cramped}\n\n\\LUATEX\\ has four new primitives to set the cramped math styles directly:\n\n\\starttyping\n\\crampeddisplaystyle\n\\crampedtextstyle\n\\crampedscriptstyle\n\\crampedscriptscriptstyle\n\\stoptyping\n\nThese additional commands are not all that valuable on their own, but they come\nin handy as arguments to the math parameter settings that will be added shortly.\n\nIn Eijkhouts \\quotation {\\TEX\\ by Topic} the rules for handling styles in scripts\nare described as follows:\n\n\\startitemize\n\\startitem\n In any style superscripts and subscripts are taken from the next smaller style.\n Exception: in display style they are in script style.\n\\stopitem\n\\startitem\n Subscripts are always in the cramped variant of the style; superscripts are only\n cramped if the original style was cramped.\n\\stopitem\n\\startitem\n In an \\type {..\\over..} formula in any style the numerator and denominator are\n taken from the next smaller style.\n\\stopitem\n\\startitem\n The denominator is always in cramped style; the numerator is only in cramped\n style if the original style was cramped.\n\\stopitem\n\\startitem\n Formulas under a \\type {\\sqrt} or \\prm {overline} are in cramped style.\n\\stopitem\n\\stopitemize\n\nIn \\LUATEX\\ one can set the styles in more detail which means that you sometimes\nhave to set both normal and cramped styles to get the effect you want. If we\nforce styles in the script using \\prm {scriptstyle} and \\lpr {crampedscriptstyle}\nwe get this:\n\n\\startbuffer[demo]\n\\starttabulate\n\\DB style \\BC example \\NC \\NR\n\\TB\n\\NC default \\NC $b_{x=xx}^{x=xx}$ \\NC \\NR\n\\NC script \\NC $b_{\\scriptstyle x=xx}^{\\scriptstyle x=xx}$ \\NC \\NR\n\\NC crampedscript \\NC $b_{\\crampedscriptstyle x=xx}^{\\crampedscriptstyle x=xx}$ \\NC \\NR\n\\LL\n\\stoptabulate\n\\stopbuffer\n\n\\getbuffer[demo]\n\nNow we set the following parameters\n\n\\startbuffer[setup]\n\\Umathordrelspacing\\scriptstyle=30mu\n\\Umathordordspacing\\scriptstyle=30mu\n\\stopbuffer\n\n\\typebuffer[setup]\n\nThis gives:\n\n\\start\\getbuffer[setup,demo]\\stop\n\nBut, as this is not what is expected (visually) we should say:\n\n\\startbuffer[setup]\n\\Umathordrelspacing\\scriptstyle=30mu\n\\Umathordordspacing\\scriptstyle=30mu\n\\Umathordrelspacing\\crampedscriptstyle=30mu\n\\Umathordordspacing\\crampedscriptstyle=30mu\n\\stopbuffer\n\n\\typebuffer[setup]\n\nNow we get:\n\n\\start\\getbuffer[setup,demo]\\stop\n\n\\section{Math parameter settings}\n\n\\topicindex {math+parameters}\n\nIn \\LUATEX, the font dimension parameters that \\TEX\\ used in math typesetting are\nnow accessible via primitive commands. In fact, refactoring of the math engine\nhas resulted in many more parameters than were accessible before.\n\n\\starttabulate\n\\DB primitive name \\BC description \\NC \\NR\n\\TB\n\\NC \\lpr {Umathquad} \\NC the width of 18 mu's \\NC \\NR\n\\NC \\lpr {Umathaxis} \\NC height of the vertical center axis of\n the math formula above the baseline \\NC \\NR\n\\NC \\lpr {Umathoperatorsize} \\NC minimum size of large operators in display mode \\NC \\NR\n\\NC \\lpr {Umathoverbarkern} \\NC vertical clearance above the rule \\NC \\NR\n\\NC \\lpr {Umathoverbarrule} \\NC the width of the rule \\NC \\NR\n\\NC \\lpr {Umathoverbarvgap} \\NC vertical clearance below the rule \\NC \\NR\n\\NC \\lpr {Umathunderbarkern} \\NC vertical clearance below the rule \\NC \\NR\n\\NC \\lpr {Umathunderbarrule} \\NC the width of the rule \\NC \\NR\n\\NC \\lpr {Umathunderbarvgap} \\NC vertical clearance above the rule \\NC \\NR\n\\NC \\lpr {Umathradicalkern} \\NC vertical clearance above the rule \\NC \\NR\n\\NC \\lpr {Umathradicalrule} \\NC the width of the rule \\NC \\NR\n\\NC \\lpr {Umathradicalvgap} \\NC vertical clearance below the rule \\NC \\NR\n\\NC \\lpr {Umathradicaldegreebefore}\\NC the forward kern that takes place before placement of\n the radical degree \\NC \\NR\n\\NC \\lpr {Umathradicaldegreeafter} \\NC the backward kern that takes place after placement of\n the radical degree \\NC \\NR\n\\NC \\lpr {Umathradicaldegreeraise} \\NC this is the percentage of the total height and depth of\n the radical sign that the degree is raised by; it is\n expressed in \\type {percents}, so 60\\% is expressed as the\n integer $60$ \\NC \\NR\n\\NC \\lpr {Umathstackvgap} \\NC vertical clearance between the two\n elements in a \\prm {atop} stack \\NC \\NR\n\\NC \\lpr {Umathstacknumup} \\NC numerator shift upward in \\prm {atop} stack \\NC \\NR\n\\NC \\lpr {Umathstackdenomdown} \\NC denominator shift downward in \\prm {atop} stack \\NC \\NR\n\\NC \\lpr {Umathfractionrule} \\NC the width of the rule in a \\prm {over} \\NC \\NR\n\\NC \\lpr {Umathfractionnumvgap} \\NC vertical clearance between the numerator and the rule \\NC \\NR\n\\NC \\lpr {Umathfractionnumup} \\NC numerator shift upward in \\prm {over} \\NC \\NR\n\\NC \\lpr {Umathfractiondenomvgap} \\NC vertical clearance between the denominator and the rule \\NC \\NR\n\\NC \\lpr {Umathfractiondenomdown} \\NC denominator shift downward in \\prm {over} \\NC \\NR\n\\NC \\lpr {Umathfractiondelsize} \\NC minimum delimiter size for \\type {\\...withdelims} \\NC \\NR\n\\NC \\lpr {Umathlimitabovevgap} \\NC vertical clearance for limits above operators \\NC \\NR\n\\NC \\lpr {Umathlimitabovebgap} \\NC vertical baseline clearance for limits above operators \\NC \\NR\n\\NC \\lpr {Umathlimitabovekern} \\NC space reserved at the top of the limit \\NC \\NR\n\\NC \\lpr {Umathlimitbelowvgap} \\NC vertical clearance for limits below operators \\NC \\NR\n\\NC \\lpr {Umathlimitbelowbgap} \\NC vertical baseline clearance for limits below operators \\NC \\NR\n\\NC \\lpr {Umathlimitbelowkern} \\NC space reserved at the bottom of the limit \\NC \\NR\n\\NC \\lpr {Umathoverdelimitervgap} \\NC vertical clearance for limits above delimiters \\NC \\NR\n\\NC \\lpr {Umathoverdelimiterbgap} \\NC vertical baseline clearance for limits above delimiters \\NC \\NR\n\\NC \\lpr {Umathunderdelimitervgap} \\NC vertical clearance for limits below delimiters \\NC \\NR\n\\NC \\lpr {Umathunderdelimiterbgap} \\NC vertical baseline clearance for limits below delimiters \\NC \\NR\n\\NC \\lpr {Umathsubshiftdrop} \\NC subscript drop for boxes and subformulas \\NC \\NR\n\\NC \\lpr {Umathsubshiftdown} \\NC subscript drop for characters \\NC \\NR\n\\NC \\lpr {Umathsupshiftdrop} \\NC superscript drop (raise, actually) for boxes and subformulas \\NC \\NR\n\\NC \\lpr {Umathsupshiftup} \\NC superscript raise for characters \\NC \\NR\n\\NC \\lpr {Umathsubsupshiftdown} \\NC subscript drop in the presence of a superscript \\NC \\NR\n\\NC \\lpr {Umathsubtopmax} \\NC the top of standalone subscripts cannot be higher than this\n above the baseline \\NC \\NR\n\\NC \\lpr {Umathsupbottommin} \\NC the bottom of standalone superscripts cannot be less than\n this above the baseline \\NC \\NR\n\\NC \\lpr {Umathsupsubbottommax} \\NC the bottom of the superscript of a combined super- and subscript\n be at least as high as this above the baseline \\NC \\NR\n\\NC \\lpr {Umathsubsupvgap} \\NC vertical clearance between super- and subscript \\NC \\NR\n\\NC \\lpr {Umathspaceafterscript} \\NC additional space added after a super- or subscript \\NC \\NR\n\\NC \\lpr {Umathconnectoroverlapmin}\\NC minimum overlap between parts in an extensible recipe \\NC \\NR\n\\LL\n\\stoptabulate\n\nEach of the parameters in this section can be set by a command like this:\n\n\\starttyping\n\\Umathquad\\displaystyle=1em\n\\stoptyping\n\nthey obey grouping, and you can use \\type {\\the\\Umathquad\\displaystyle} if\nneeded.\n\n\\section{Skips around display math}\n\n\\topicindex {math+spacing}\n\nThe injection of \\prm {abovedisplayskip} and \\prm {belowdisplayskip} is not\nsymmetrical. An above one is always inserted, also when zero, but the below is\nonly inserted when larger than zero. Especially the later makes it sometimes hard\nto fully control spacing. Therefore \\LUATEX\\ comes with a new directive: \\lpr\n{mathdisplayskipmode}. The following values apply:\n\n\\starttabulate[|c|l|]\n\\DB value \\BC meaning \\NC \\NR\n\\TB\n\\NC 0 \\NC normal \\TEX\\ behaviour \\NC \\NR\n\\NC 1 \\NC always (same as 0) \\NC \\NR\n\\NC 2 \\NC only when not zero \\NC \\NR\n\\NC 3 \\NC never, not even when not zero \\NC \\NR\n\\LL\n\\stoptabulate\n\n\\section{Font|-|based Math Parameters}\n\n\\topicindex {math+parameters}\n\nWhile it is nice to have these math parameters available for tweaking, it would\nbe tedious to have to set each of them by hand. For this reason, \\LUATEX\\\ninitializes a bunch of these parameters whenever you assign a font identifier to\na math family based on either the traditional math font dimensions in the font\n(for assignments to math family~2 and~3 using \\TFM|-|based fonts like \\type\n{cmsy} and \\type {cmex}), or based on the named values in a potential \\type\n{MathConstants} table when the font is loaded via Lua. If there is a \\type\n{MathConstants} table, this takes precedence over font dimensions, and in that\ncase no attention is paid to which family is being assigned to: the \\type\n{MathConstants} tables in the last assigned family sets all parameters.\n\nIn the table below, the one|-|letter style abbreviations and symbolic tfm font\ndimension names match those using in the \\TeX book. Assignments to \\prm\n{textfont} set the values for the cramped and uncramped display and text styles,\n\\prm {scriptfont} sets the script styles, and \\prm {scriptscriptfont} sets the\nscriptscript styles, so we have eight parameters for three font sizes. In the\n\\TFM\\ case, assignments only happen in family~2 and family~3 (and of course only\nfor the parameters for which there are font dimensions).\n\nBesides the parameters below, \\LUATEX\\ also looks at the \\quote {space} font\ndimension parameter. For math fonts, this should be set to zero.\n\n\\def\\MathLine#1#2#3#4#5%\n {\\TB\n \\NC \\llap{\\high{\\tx #2\\enspace}}\\ttbf \\string #1 \\NC \\tt #5 \\NC \\NR\n \\NC \\tx #3 \\NC \\tt #4 \\NC \\NR}\n\n\\starttabulate[|l|l|]\n\\DB variable / style \\BC tfm / opentype \\NC \\NR\n\\MathLine{\\Umathaxis} {} {} {AxisHeight} {axis_height}\n\\MathLine{\\Umathoperatorsize} {6} {D, D'} {DisplayOperatorMinHeight} {\\emdash}\n\\MathLine{\\Umathfractiondelsize} {9} {D, D'} {FractionDelimiterDisplayStyleSize} {delim1}\n\\MathLine{\\Umathfractiondelsize} {9} {T, T', S, S', SS, SS'}{FractionDelimiterSize} {delim2}\n\\MathLine{\\Umathfractiondenomdown} {} {D, D'} {FractionDenominatorDisplayStyleShiftDown}{denom1}\n\\MathLine{\\Umathfractiondenomdown} {} {T, T', S, S', SS, SS'}{FractionDenominatorShiftDown} {denom2}\n\\MathLine{\\Umathfractiondenomvgap} {} {D, D'} {FractionDenominatorDisplayStyleGapMin} {3*default_rule_thickness}\n\\MathLine{\\Umathfractiondenomvgap} {} {T, T', S, S', SS, SS'}{FractionDenominatorGapMin} {default_rule_thickness}\n\\MathLine{\\Umathfractionnumup} {} {D, D'} {FractionNumeratorDisplayStyleShiftUp} {num1}\n\\MathLine{\\Umathfractionnumup} {} {T, T', S, S', SS, SS'}{FractionNumeratorShiftUp} {num2}\n\\MathLine{\\Umathfractionnumvgap} {} {D, D'} {FractionNumeratorDisplayStyleGapMin} {3*default_rule_thickness}\n\\MathLine{\\Umathfractionnumvgap} {} {T, T', S, S', SS, SS'}{FractionNumeratorGapMin} {default_rule_thickness}\n\\MathLine{\\Umathfractionrule} {} {} {FractionRuleThickness} {default_rule_thickness}\n\\MathLine{\\Umathskewedfractionhgap} {} {} {SkewedFractionHorizontalGap} {math_quad/2}\n\\MathLine{\\Umathskewedfractionvgap} {} {} {SkewedFractionVerticalGap} {math_x_height}\n\\MathLine{\\Umathlimitabovebgap} {} {} {UpperLimitBaselineRiseMin} {big_op_spacing3}\n\\MathLine{\\Umathlimitabovekern} {1} {} {0} {big_op_spacing5}\n\\MathLine{\\Umathlimitabovevgap} {} {} {UpperLimitGapMin} {big_op_spacing1}\n\\MathLine{\\Umathlimitbelowbgap} {} {} {LowerLimitBaselineDropMin} {big_op_spacing4}\n\\MathLine{\\Umathlimitbelowkern} {1} {} {0} {big_op_spacing5}\n\\MathLine{\\Umathlimitbelowvgap} {} {} {LowerLimitGapMin} {big_op_spacing2}\n\\MathLine{\\Umathoverdelimitervgap} {} {} {StretchStackGapBelowMin} {big_op_spacing1}\n\\MathLine{\\Umathoverdelimiterbgap} {} {} {StretchStackTopShiftUp} {big_op_spacing3}\n\\MathLine{\\Umathunderdelimitervgap} {} {} {StretchStackGapAboveMin} {big_op_spacing2}\n\\MathLine{\\Umathunderdelimiterbgap} {} {} {StretchStackBottomShiftDown} {big_op_spacing4}\n\\MathLine{\\Umathoverbarkern} {} {} {OverbarExtraAscender} {default_rule_thickness}\n\\MathLine{\\Umathoverbarrule} {} {} {OverbarRuleThickness} {default_rule_thickness}\n\\MathLine{\\Umathoverbarvgap} {} {} {OverbarVerticalGap} {3*default_rule_thickness}\n\\MathLine{\\Umathquad} {1} {} {} {math_quad}\n\\MathLine{\\Umathradicalkern} {} {} {RadicalExtraAscender} {default_rule_thickness}\n\\MathLine{\\Umathradicalrule} {2} {} {RadicalRuleThickness} {}\n\\MathLine{\\Umathradicalvgap} {3} {D, D'} {RadicalDisplayStyleVerticalGap} {default_rule_thickness+abs(math_x_height)/4}\n\\MathLine{\\Umathradicalvgap} {3} {T, T', S, S', SS, SS'}{RadicalVerticalGap} {default_rule_thickness+abs(default_rule_thickness)/4}\n\\MathLine{\\Umathradicaldegreebefore}{2} {} {RadicalKernBeforeDegree} {}\n\\MathLine{\\Umathradicaldegreeafter} {2} {} {RadicalKernAfterDegree} {}\n\\MathLine{\\Umathradicaldegreeraise} {2,7}{} {RadicalDegreeBottomRaisePercent} {}\n\\MathLine{\\Umathspaceafterscript} {4} {} {SpaceAfterScript} {script_space}\n\\MathLine{\\Umathstackdenomdown} {} {D, D'} {StackBottomDisplayStyleShiftDown} {denom1}\n\\MathLine{\\Umathstackdenomdown} {} {T, T', S, S', SS, SS'}{StackBottomShiftDown} {denom2}\n\\MathLine{\\Umathstacknumup} {} {D, D'} {StackTopDisplayStyleShiftUp} {num1}\n\\MathLine{\\Umathstacknumup} {} {T, T', S, S', SS, SS'}{StackTopShiftUp} {num3}\n\\MathLine{\\Umathstackvgap} {} {D, D'} {StackDisplayStyleGapMin} {7*default_rule_thickness}\n\\MathLine{\\Umathstackvgap} {} {T, T', S, S', SS, SS'}{StackGapMin} {3*default_rule_thickness}\n\\MathLine{\\Umathsubshiftdown} {} {} {SubscriptShiftDown} {sub1}\n\\MathLine{\\Umathsubshiftdrop} {} {} {SubscriptBaselineDropMin} {sub_drop}\n\\MathLine{\\Umathsubsupshiftdown} {8} {} {SubscriptShiftDownWithSuperscript} {\\emdash}\n\\MathLine{\\Umathsubtopmax} {} {} {SubscriptTopMax} {abs(math_x_height*4)/5}\n\\MathLine{\\Umathsubsupvgap} {} {} {SubSuperscriptGapMin} {4*default_rule_thickness}\n\\MathLine{\\Umathsupbottommin} {} {} {SuperscriptBottomMin} {abs(math_x_height/4)}\n\\MathLine{\\Umathsupshiftdrop} {} {} {SuperscriptBaselineDropMax} {sup_drop}\n\\MathLine{\\Umathsupshiftup} {} {D} {SuperscriptShiftUp} {sup1}\n\\MathLine{\\Umathsupshiftup} {} {T, S, SS,} {SuperscriptShiftUp} {sup2}\n\\MathLine{\\Umathsupshiftup} {} {D', T', S', SS'} {SuperscriptShiftUpCramped} {sup3}\n\\MathLine{\\Umathsupsubbottommax} {} {} {SuperscriptBottomMaxWithSubscript} {abs(math_x_height*4)/5}\n\\MathLine{\\Umathunderbarkern} {} {} {UnderbarExtraDescender} {default_rule_thickness}\n\\MathLine{\\Umathunderbarrule} {} {} {UnderbarRuleThickness} {default_rule_thickness}\n\\MathLine{\\Umathunderbarvgap} {} {} {UnderbarVerticalGap} {3*default_rule_thickness}\n\\MathLine{\\Umathconnectoroverlapmin}{5} {} {MinConnectorOverlap} {0}\n\\LL\n\\stoptabulate\n\nNote 1: \\OPENTYPE\\ fonts set \\lpr {Umathlimitabovekern} and \\lpr\n{Umathlimitbelowkern} to zero and set \\lpr {Umathquad} to the font size of the\nused font, because these are not supported in the \\type {MATH} table,\n\nNote 2: Traditional \\TFM\\ fonts do not set \\lpr {Umathradicalrule} because\n\\TEX82\\ uses the height of the radical instead. When this parameter is indeed not\nset when \\LUATEX\\ has to typeset a radical, a backward compatibility mode will\nkick in that assumes that an oldstyle \\TEX\\ font is used. Also, they do not set\n\\lpr {Umathradicaldegreebefore}, \\lpr {Umathradicaldegreeafter}, and \\lpr\n{Umathradicaldegreeraise}. These are then automatically initialized to\n$5/18$quad, $-10/18$quad, and 60.\n\nNote 3: If \\TFM\\ fonts are used, then the \\lpr {Umathradicalvgap} is not set\nuntil the first time \\LUATEX\\ has to typeset a formula because this needs\nparameters from both family~2 and family~3. This provides a partial backward\ncompatibility with \\TEX82, but that compatibility is only partial: once the \\lpr\n{Umathradicalvgap} is set, it will not be recalculated any more.\n\nNote 4: When \\TFM\\ fonts are used a similar situation arises with respect to \\lpr\n{Umathspaceafterscript}: it is not set until the first time \\LUATEX\\ has to\ntypeset a formula. This provides some backward compatibility with \\TEX82. But\nonce the \\lpr {Umathspaceafterscript} is set, \\prm {scriptspace} will never be\nlooked at again.\n\nNote 5: Traditional \\TFM\\ fonts set \\lpr {Umathconnectoroverlapmin} to zero\nbecause \\TEX82\\ always stacks extensibles without any overlap.\n\nNote 6: The \\lpr {Umathoperatorsize} is only used in \\prm {displaystyle}, and is\nonly set in \\OPENTYPE\\ fonts. In \\TFM\\ font mode, it is artificially set to one\nscaled point more than the initial attempt's size, so that always the \\quote\n{first next} will be tried, just like in \\TEX82.\n\nNote 7: The \\lpr {Umathradicaldegreeraise} is a special case because it is the\nonly parameter that is expressed in a percentage instead of as a number of scaled\npoints.\n\nNote 8: \\type {SubscriptShiftDownWithSuperscript} does not actually exist in the\n\\quote {standard} \\OPENTYPE\\ math font Cambria, but it is useful enough to be\nadded.\n\nNote 9: \\type {FractionDelimiterDisplayStyleSize} and \\type\n{FractionDelimiterSize} do not actually exist in the \\quote {standard} \\OPENTYPE\\\nmath font Cambria, but were useful enough to be added.\n\n\\section {Nolimit correction}\n\n\\topicindex {math+limits}\n\nThere are two extra math parameters \\lpr {Umathnolimitsupfactor} and \\lpr\n{Umathnolimitsubfactor} that were added to provide some control over how limits\nare spaced (for example the position of super and subscripts after integral\noperators). They relate to an extra parameter \\lpr {mathnolimitsmode}. The half\ncorrections are what happens when scripts are placed on above and below. The\nproblem with italic corrections is that officially that correction italic is used\nfor above|/|below placement while advanced kerns are used for placement at the\nright end. The question is: how often is this implemented, and if so, does the\nkerns assume correction too. Anyway, with this parameter one can control it.\n\n\\starttabulate[|l|ck1|ck1|ck1|ck1|ck1|ck1|]\n \\NC\n \\NC \\mathnolimitsmode0 $\\displaystyle\\int\\nolimits^0_1$\n \\NC \\mathnolimitsmode1 $\\displaystyle\\int\\nolimits^0_1$\n \\NC \\mathnolimitsmode2 $\\displaystyle\\int\\nolimits^0_1$\n \\NC \\mathnolimitsmode3 $\\displaystyle\\int\\nolimits^0_1$\n \\NC \\mathnolimitsmode4 $\\displaystyle\\int\\nolimits^0_1$\n \\NC \\mathnolimitsmode8000 $\\displaystyle\\int\\nolimits^0_1$\n \\NC \\NR\n \\TB\n \\BC mode\n \\NC \\tttf 0\n \\NC \\tttf 1\n \\NC \\tttf 2\n \\NC \\tttf 3\n \\NC \\tttf 4\n \\NC \\tttf 8000\n \\NC \\NR\n \\BC superscript\n \\NC 0\n \\NC font\n \\NC 0\n \\NC 0\n \\NC +ic/2\n \\NC 0\n \\NC \\NR\n \\BC subscript\n \\NC -ic\n \\NC font\n \\NC 0\n \\NC -ic/2\n \\NC -ic/2\n \\NC 8000ic/1000\n \\NC \\NR\n\\stoptabulate\n\nWhen the mode is set to one, the math parameters are used. This way a macro\npackage writer can decide what looks best. Given the current state of fonts in\n\\CONTEXT\\ we currently use mode 1 with factor 0 for the superscript and 750 for\nthe subscripts. Positive values are used for both parameters but the subscript\nshifts to the left. A \\lpr {mathnolimitsmode} larger that 15 is considered to\nbe a factor for the subscript correction. This feature can be handy when\nexperimenting.\n\n\\section {Math italic mess}\n\n\\topicindex {math+italics}\n\nThe \\lpr {mathitalicsmode} parameter can be set to~1 to force italic correction\nbefore noads that represent some more complex structure (read: everything\nthat is not an ord, bin, rel, open, close, punct or inner). We show a Cambria\nexample.\n\n\\starttexdefinition Whatever #1\n \\NC \\type{\\mathitalicsmode = #1}\n \\NC \\mathitalicsmode#1\\ruledhbox{$\\left|T^1\\right|$}\n \\NC \\mathitalicsmode#1\\ruledhbox{$\\left|T\\right|$}\n \\NC \\mathitalicsmode#1\\ruledhbox{$T+1$}\n \\NC \\mathitalicsmode#1\\ruledhbox{$T{1\\over2}$}\n \\NC \\mathitalicsmode#1\\ruledhbox{$T\\sqrt{1}$}\n \\NC \\NR\n\\stoptexdefinition\n\n\\start\n \\switchtobodyfont[cambria]\n \\starttabulate[|c|c|c|c|c|c|]\n \\Whatever{0}%\n \\Whatever{1}%\n \\stoptabulate\n\\stop\n\nThis kind of parameters relate to the fact that italic correction in \\OPENTYPE\\\nmath is bound to fuzzy rules. So, control is the solution.\n\n\\section {Script and kerning}\n\n\\topicindex {math+kerning}\n\\topicindex {math+scripts}\n\nIf you want typeset text in math macro packages often provide something \\type\n{\\text} which obeys the script sizes. As the definition can be anything there is\na good change that the kerning doesn't come out well when used in a script. Given\nthat the first glyph ends up in an \\prm {hbox} we have some control over this.\nAnd, as a bonus we also added control over the normal sublist kerning. The \\lpr\n{mathscriptboxmode} parameter defaults to~1.\n\n\\starttabulate[|c|l|]\n\\DB value \\BC meaning \\NC \\NR\n\\TB\n\\NC \\type {0} \\NC forget about kerning \\NC \\NR\n\\NC \\type {1} \\NC kern math sub lists with a valid glyph \\NC \\NR\n\\NC \\type {2} \\NC also kern math sub boxes that have a valid glyph \\NC \\NR\n\\NC \\type {2} \\NC only kern math sub boxes with a boundary node present\\NC \\NR\n\\LL\n\\stoptabulate\n\nHere we show some examples. Of course this doesn't solve all our problems, if\nonly because some fonts have characters with bounding boxes that compensate for\nitalics, while other fonts can lack kerns.\n\n\\startbuffer[1]\n $T_{\\tf fluff}$\n\\stopbuffer\n\n\\startbuffer[2]\n $T_{\\text{fluff}}$\n\\stopbuffer\n\n\\startbuffer[3]\n $T_{\\text{\\boundary1 fluff}}$\n\\stopbuffer\n\n\\unexpanded\\def\\Show#1#2#3%\n {\\doifelsenothing{#3}\n {\\small\\tx\\typeinlinebuffer[#1]}\n {\\doifelse{#3}{-}\n {\\small\\bf\\tt mode #2}\n {\\switchtobodyfont[#3]\\showfontkerns\\showglyphs\\mathscriptboxmode#2\\relax\\inlinebuffer[#1]}}}\n\n\\starttabulate[|lBT|c|c|c|c|c|]\n \\NC \\NC \\Show{1}{0}{} \\NC\\Show{1}{1}{} \\NC \\Show{2}{1}{} \\NC \\Show{2}{2}{} \\NC \\Show{3}{3}{} \\NC \\NR\n \\NC \\NC \\Show{1}{0}{-} \\NC\\Show{1}{1}{-} \\NC \\Show{2}{1}{-} \\NC \\Show{2}{2}{-} \\NC \\Show{3}{3}{-} \\NC \\NR\n \\NC modern \\NC \\Show{1}{0}{modern} \\NC\\Show{1}{1}{modern} \\NC \\Show{2}{1}{modern} \\NC \\Show{2}{2}{modern} \\NC \\Show{3}{3}{modern} \\NC \\NR\n \\NC lucidaot \\NC \\Show{1}{0}{lucidaot} \\NC\\Show{1}{1}{lucidaot} \\NC \\Show{2}{1}{lucidaot} \\NC \\Show{2}{2}{lucidaot} \\NC \\Show{3}{3}{lucidaot} \\NC \\NR\n \\NC pagella \\NC \\Show{1}{0}{pagella} \\NC\\Show{1}{1}{pagella} \\NC \\Show{2}{1}{pagella} \\NC \\Show{2}{2}{pagella} \\NC \\Show{3}{3}{pagella} \\NC \\NR\n \\NC cambria \\NC \\Show{1}{0}{cambria} \\NC\\Show{1}{1}{cambria} \\NC \\Show{2}{1}{cambria} \\NC \\Show{2}{2}{cambria} \\NC \\Show{3}{3}{cambria} \\NC \\NR\n \\NC dejavu \\NC \\Show{1}{0}{dejavu} \\NC\\Show{1}{1}{dejavu} \\NC \\Show{2}{1}{dejavu} \\NC \\Show{2}{2}{dejavu} \\NC \\Show{3}{3}{dejavu} \\NC \\NR\n\\stoptabulate\n\nKerning between a character subscript is controlled by \\lpr {mathscriptcharmode}\nwhich also defaults to~1.\n\nHere is another example. Internally we tag kerns as italic kerns or font kerns\nwhere font kerns result from the staircase kern tables. In 2018 fonts like Latin\nModern and Pagella rely on cheats with the boundingbox, Cambria uses staircase\nkerns and Lucida a mixture. Depending on how fonts evolve we might add some more\ncontrol over what one can turn on and off.\n\n\\def\\MathSample#1#2#3%\n {\\NC\n #1 \\NC\n #2 \\NC\n \\showglyphdata \\switchtobodyfont[#2,17.3pt]$#3T_{f}$ \\NC\n \\showglyphdata \\switchtobodyfont[#2,17.3pt]$#3\\gamma_{e}$ \\NC\n \\showglyphdata \\switchtobodyfont[#2,17.3pt]$#3\\gamma_{ee}$ \\NC\n \\showglyphdata \\switchtobodyfont[#2,17.3pt]$#3T_{\\tf fluff}$ \\NC\n \\NR}\n\n\\starttabulate[|Tl|Tl|l|l|l|l|]\n \\FL\n \\MathSample{normal}{modern} {\\mr}\n \\MathSample{} {pagella} {\\mr}\n \\MathSample{} {cambria} {\\mr}\n \\MathSample{} {lucidaot}{\\mr}\n \\ML\n \\MathSample{bold} {modern} {\\mb}\n \\MathSample{} {pagella} {\\mb}\n \\MathSample{} {cambria} {\\mb}\n \\MathSample{} {lucidaot}{\\mb}\n \\LL\n\\stoptabulate\n\n\\section {Unscaled fences}\n\n\\topicindex {math+fences}\n\nThe \\lpr {mathdelimitersmode} primitive is experimental and deals with the\nfollowing (potential) problems. Three bits can be set. The first bit prevents an\nunwanted shift when the fence symbol is not scaled (a cambria side effect). The\nsecond bit forces italic correction between a preceding character ordinal and the\nfenced subformula, while the third bit turns that subformula into a ordinary so\nthat the same spacing applies as with unfenced variants. Here we show Cambria\n(with \\lpr {mathitalicsmode} enabled).\n\n\\starttexdefinition Whatever #1\n \\NC \\type{\\mathdelimitersmode = #1}\n \\NC \\mathitalicsmode1\\mathdelimitersmode#1\\ruledhbox{\\showglyphs\\showfontkerns\\showfontitalics$f(x)$}\n \\NC \\mathitalicsmode1\\mathdelimitersmode#1\\ruledhbox{\\showglyphs\\showfontkerns\\showfontitalics$f\\left(x\\right)$}\n \\NC \\NR\n\\stoptexdefinition\n\n\\start\n \\switchtobodyfont[cambria]\n \\starttabulate[|l|l|l|]\n \\Whatever{0}\\Whatever{1}\\Whatever{2}\\Whatever{3}%\n \\Whatever{4}\\Whatever{5}\\Whatever{6}\\Whatever{7}%\n \\stoptabulate\n\\stop\n\nSo, when set to 7 fenced subformulas with unscaled delimiters come out the same\nas unfenced ones. This can be handy for cases where one is forced to use \\prm\n{left} and \\prm {right} always because of unpredictable content. As said, it's an\nexperimental features (which somehow fits in the exceptional way fences are dealt\nwith in the engine). The full list of flags is given in the next table:\n\n\\starttabulate[|c|l|]\n\\DB value \\BC meaning \\NC \\NR\n\\TB\n\\NC \\type{\"01} \\NC don't apply the usual shift \\NC \\NR\n\\NC \\type{\"02} \\NC apply italic correction when possible \\NC \\NR\n\\NC \\type{\"04} \\NC force a ordinary subformula \\NC \\NR\n\\NC \\type{\"08} \\NC no shift when a base character \\NC \\NR\n\\NC \\type{\"10} \\NC only shift when an extensible \\NC \\NR\n\\LL\n\\stoptabulate\n\nThe effect can depend on the font (and for Cambria one can use for instance \\type {\"16}).\n\n\\section {Math spacing setting}\n\n\\topicindex {math+spacing}\n\nBesides the parameters mentioned in the previous sections, there are also 64 new\nprimitives to control the math spacing table (as explained in Chapter~18 of the\n\\TEX book). The primitive names are a simple matter of combining two math atom\ntypes, but for completeness' sake, here is the whole list:\n\n\\starttwocolumns\n\\startlines\n\\lpr {Umathordordspacing}\n\\lpr {Umathordopspacing}\n\\lpr {Umathordbinspacing}\n\\lpr {Umathordrelspacing}\n\\lpr {Umathordopenspacing}\n\\lpr {Umathordclosespacing}\n\\lpr {Umathordpunctspacing}\n\\lpr {Umathordinnerspacing}\n\\lpr {Umathopordspacing}\n\\lpr {Umathopopspacing}\n\\lpr {Umathopbinspacing}\n\\lpr {Umathoprelspacing}\n\\lpr {Umathopopenspacing}\n\\lpr {Umathopclosespacing}\n\\lpr {Umathoppunctspacing}\n\\lpr {Umathopinnerspacing}\n\\lpr {Umathbinordspacing}\n\\lpr {Umathbinopspacing}\n\\lpr {Umathbinbinspacing}\n\\lpr {Umathbinrelspacing}\n\\lpr {Umathbinopenspacing}\n\\lpr {Umathbinclosespacing}\n\\lpr {Umathbinpunctspacing}\n\\lpr {Umathbininnerspacing}\n\\lpr {Umathrelordspacing}\n\\lpr {Umathrelopspacing}\n\\lpr {Umathrelbinspacing}\n\\lpr {Umathrelrelspacing}\n\\lpr {Umathrelopenspacing}\n\\lpr {Umathrelclosespacing}\n\\lpr {Umathrelpunctspacing}\n\\lpr {Umathrelinnerspacing}\n\\lpr {Umathopenordspacing}\n\\lpr {Umathopenopspacing}\n\\lpr {Umathopenbinspacing}\n\\lpr {Umathopenrelspacing}\n\\lpr {Umathopenopenspacing}\n\\lpr {Umathopenclosespacing}\n\\lpr {Umathopenpunctspacing}\n\\lpr {Umathopeninnerspacing}\n\\lpr {Umathcloseordspacing}\n\\lpr {Umathcloseopspacing}\n\\lpr {Umathclosebinspacing}\n\\lpr {Umathcloserelspacing}\n\\lpr {Umathcloseopenspacing}\n\\lpr {Umathcloseclosespacing}\n\\lpr {Umathclosepunctspacing}\n\\lpr {Umathcloseinnerspacing}\n\\lpr {Umathpunctordspacing}\n\\lpr {Umathpunctopspacing}\n\\lpr {Umathpunctbinspacing}\n\\lpr {Umathpunctrelspacing}\n\\lpr {Umathpunctopenspacing}\n\\lpr {Umathpunctclosespacing}\n\\lpr {Umathpunctpunctspacing}\n\\lpr {Umathpunctinnerspacing}\n\\lpr {Umathinnerordspacing}\n\\lpr {Umathinneropspacing}\n\\lpr {Umathinnerbinspacing}\n\\lpr {Umathinnerrelspacing}\n\\lpr {Umathinneropenspacing}\n\\lpr {Umathinnerclosespacing}\n\\lpr {Umathinnerpunctspacing}\n\\lpr {Umathinnerinnerspacing}\n\\stoplines\n\\stoptwocolumns\n\nThese parameters are of type \\prm {muskip}, so setting a parameter can be done\nlike this:\n\n\\starttyping\n\\Umathopordspacing\\displaystyle=4mu plus 2mu\n\\stoptyping\n\nThey are all initialized by \\type {initex} to the values mentioned in the table\nin Chapter~18 of the \\TEX book.\n\nNote 1: for ease of use as well as for backward compatibility, \\prm {thinmuskip},\n\\prm {medmuskip} and \\prm {thickmuskip} are treated especially. In their case a\npointer to the corresponding internal parameter is saved, not the actual \\prm\n{muskip} value. This means that any later changes to one of these three\nparameters will be taken into account.\n\nNote 2: Careful readers will realise that there are also primitives for the items\nmarked \\type {*} in the \\TEX book. These will not actually be used as those\ncombinations of atoms cannot actually happen, but it seemed better not to break\northogonality. They are initialized to zero.\n\n\\section[mathacc]{Math accent handling}\n\n\\topicindex {math+accents}\n\n\\LUATEX\\ supports both top accents and bottom accents in math mode, and math\naccents stretch automatically (if this is supported by the font the accent comes\nfrom, of course). Bottom and combined accents as well as fixed-width math accents\nare controlled by optional keywords following \\lpr {Umathaccent}.\n\nThe keyword \\type {bottom} after \\lpr {Umathaccent} signals that a bottom accent\nis needed, and the keyword \\type {both} signals that both a top and a bottom\naccent are needed (in this case two accents need to be specified, of course).\n\nThen the set of three integers defining the accent is read. This set of integers\ncan be prefixed by the \\type {fixed} keyword to indicate that a non-stretching\nvariant is requested (in case of both accents, this step is repeated).\n\nA simple example:\n\n\\starttyping\n\\Umathaccent both fixed 0 0 \"20D7 fixed 0 0 \"20D7 {example}\n\\stoptyping\n\nIf a math top accent has to be placed and the accentee is a character and has a\nnon-zero \\type {top_accent} value, then this value will be used to place the\naccent instead of the \\prm {skewchar} kern used by \\TEX82.\n\nThe \\type {top_accent} value represents a vertical line somewhere in the\naccentee. The accent will be shifted horizontally such that its own \\type\n{top_accent} line coincides with the one from the accentee. If the \\type\n{top_accent} value of the accent is zero, then half the width of the accent\nfollowed by its italic correction is used instead.\n\nThe vertical placement of a top accent depends on the \\type {x_height} of the\nfont of the accentee (as explained in the \\TEX book), but if value that turns out\nto be zero and the font had a \\type {MathConstants} table, then \\type\n{AccentBaseHeight} is used instead.\n\nThe vertical placement of a bottom accent is straight below the accentee, no\ncorrection takes place.\n\nPossible locations are \\type {top}, \\type {bottom}, \\type {both} and \\type\n{center}. When no location is given \\type {top} is assumed. An additional\nparameter \\nod {fraction} can be specified followed by a number; a value of for\ninstance 1200 means that the criterium is 1.2 times the width of the nucleus. The\nfraction only applies to the stepwise selected shapes and is mostly meant for the\n\\type {overlay} location. It also works for the other locations but then it\nconcerns the width.\n\n\\section{Math root extension}\n\n\\topicindex {math+radicals}\n\nThe new primitive \\lpr {Uroot} allows the construction of a radical noad\nincluding a degree field. Its syntax is an extension of \\lpr {Uradical}:\n\n\\starttyping\n\\Uradical \n\\Uroot \n\\stoptyping\n\nThe placement of the degree is controlled by the math parameters \\lpr\n{Umathradicaldegreebefore}, \\lpr {Umathradicaldegreeafter}, and \\lpr\n{Umathradicaldegreeraise}. The degree will be typeset in \\prm\n{scriptscriptstyle}.\n\n\\section{Math kerning in super- and subscripts}\n\nThe character fields in a \\LUA|-|loaded \\OPENTYPE\\ math font can have a \\quote\n{mathkern} table. The format of this table is the same as the \\quote {mathkern}\ntable that is returned by the \\type {fontloader} library, except that all height\nand kern values have to be specified in actual scaled points.\n\nWhen a super- or subscript has to be placed next to a math item, \\LUATEX\\ checks\nwhether the super- or subscript and the nucleus are both simple character items.\nIf they are, and if the fonts of both character items are \\OPENTYPE\\ fonts (as\nopposed to legacy \\TEX\\ fonts), then \\LUATEX\\ will use the \\OPENTYPE\\ math\nalgorithm for deciding on the horizontal placement of the super- or subscript.\n\nThis works as follows:\n\n\\startitemize\n \\startitem\n The vertical position of the script is calculated.\n \\stopitem\n \\startitem\n The default horizontal position is flat next to the base character.\n \\stopitem\n \\startitem\n For superscripts, the italic correction of the base character is added.\n \\stopitem\n \\startitem\n For a superscript, two vertical values are calculated: the bottom of the\n script (after shifting up), and the top of the base. For a subscript, the two\n values are the top of the (shifted down) script, and the bottom of the base.\n \\stopitem\n \\startitem\n For each of these two locations:\n \\startitemize\n \\startitem\n find the math kern value at this height for the base (for a subscript\n placement, this is the bottom_right corner, for a superscript\n placement the top_right corner)\n \\stopitem\n \\startitem\n find the math kern value at this height for the script (for a\n subscript placement, this is the top_left corner, for a superscript\n placement the bottom_left corner)\n \\stopitem\n \\startitem\n add the found values together to get a preliminary result.\n \\stopitem\n \\stopitemize\n \\stopitem\n \\startitem\n The horizontal kern to be applied is the smallest of the two results from\n previous step.\n \\stopitem\n\\stopitemize\n\nThe math kern value at a specific height is the kern value that is specified by the\nnext higher height and kern pair, or the highest one in the character (if there is no\nvalue high enough in the character), or simply zero (if the character has no math kern\npairs at all).\n\n\\section{Scripts on horizontally extensible items like arrows}\n\n\\topicindex {math+scripts}\n\\topicindex {math+delimiters}\n\\topicindex {math+extensibles}\n\nThe primitives \\lpr {Uunderdelimiter} and \\lpr {Uoverdelimiter} allow the\nplacement of a subscript or superscript on an automatically extensible item and\n\\lpr {Udelimiterunder} and \\lpr {Udelimiterover} allow the placement of an\nautomatically extensible item as a subscript or superscript on a nucleus. The\ninput:\n\n% these produce radical noads .. in fact the code base has the numbers wrong for\n% quite a while, so no one seems to use this\n\n\\startbuffer\n$\\Uoverdelimiter 0 \"2194 {\\hbox{\\strut overdelimiter}}$\n$\\Uunderdelimiter 0 \"2194 {\\hbox{\\strut underdelimiter}}$\n$\\Udelimiterover 0 \"2194 {\\hbox{\\strut delimiterover}}$\n$\\Udelimiterunder 0 \"2194 {\\hbox{\\strut delimiterunder}}$\n\\stopbuffer\n\n\\typebuffer will render this:\n\n\\blank \\startnarrower \\getbuffer \\stopnarrower \\blank\n\nThe vertical placements are controlled by \\lpr {Umathunderdelimiterbgap}, \\lpr\n{Umathunderdelimitervgap}, \\lpr {Umathoverdelimiterbgap}, and \\lpr\n{Umathoverdelimitervgap} in a similar way as limit placements on large operators.\nThe superscript in \\lpr {Uoverdelimiter} is typeset in a suitable scripted style,\nthe subscript in \\lpr {Uunderdelimiter} is cramped as well.\n\nThese primitives accepts an option \\type {width} specification. When used the\nalso optional keywords \\type {left}, \\type {middle} and \\type {right} will\ndetermine what happens when a requested size can't be met (which can happen when\nwe step to successive larger variants).\n\nAn extra primitive \\lpr {Uhextensible} is available that can be used like this:\n\n\\startbuffer\n$\\Uhextensible width 10cm 0 \"2194$\n\\stopbuffer\n\n\\typebuffer This will render this:\n\n\\blank \\startnarrower \\getbuffer \\stopnarrower \\blank\n\nHere you can also pass options, like:\n\n\\startbuffer\n$\\Uhextensible width 1pt middle 0 \"2194$\n\\stopbuffer\n\n\\typebuffer This gives:\n\n\\blank \\startnarrower \\getbuffer \\stopnarrower \\blank\n\n\\LUATEX\\ internally uses a structure that supports \\OPENTYPE\\ \\quote\n{MathVariants} as well as \\TFM\\ \\quote {extensible recipes}. In most cases where\nfont metrics are involved we have a different code path for traditional fonts end\n\\OPENTYPE\\ fonts.\n\n\\section {Extracting values}\n\n\\topicindex {math+codes}\n\nYou can extract the components of a math character. Say that we have defined:\n\n\\starttyping\n\\Umathcode 1 2 3 4\n\\stoptyping\n\nthen\n\n\\starttyping\n[\\Umathcharclass1] [\\Umathcharfam1] [\\Umathcharslot1]\n\\stoptyping\n\nwill return:\n\n\\starttyping\n[2] [3] [4]\n\\stoptyping\n\nThese commands are provides as convenience. Before they came available you could\ndo the following:\n\n\\starttyping\n\\def\\Umathcharclass{\\directlua{tex.print(tex.getmathcode(token.scan_int())[1])}}\n\\def\\Umathcharfam {\\directlua{tex.print(tex.getmathcode(token.scan_int())[2])}}\n\\def\\Umathcharslot {\\directlua{tex.print(tex.getmathcode(token.scan_int())[3])}}\n\\stoptyping\n\n\\section{fractions}\n\n\\topicindex {math+fractions}\n\nThe \\prm {abovewithdelims} command accepts a keyword \\type {exact}. When issued\nthe extra space relative to the rule thickness is not added. One can of course\nuse the \\type {\\Umathfraction..gap} commands to influence the spacing. Also the\nrule is still positioned around the math axis.\n\n\\starttyping\n$$ { {a} \\abovewithdelims() exact 4pt {b} }$$\n\\stoptyping\n\nThe math parameter table contains some parameters that specify a horizontal and\nvertical gap for skewed fractions. Of course some guessing is needed in order to\nimplement something that uses them. And so we now provide a primitive similar to the\nother fraction related ones but with a few options so that one can influence the\nrendering. Of course a user can also mess around a bit with the parameters\n\\lpr {Umathskewedfractionhgap} and \\lpr {Umathskewedfractionvgap}.\n\nThe syntax used here is:\n\n\\starttyping\n{ {1} \\Uskewed / {2} }\n{ {1} \\Uskewedwithdelims / () {2} }\n\\stoptyping\n\nwhere the options can be \\type {noaxis} and \\type {exact}. By default we add half\nthe axis to the shifts and by default we zero the width of the middle character.\nFor Latin Modern The result looks as follows:\n\n\\def\\ShowA#1#2#3{$x + { {#1} \\Uskewed / #3 {#2} } + x$}\n\\def\\ShowB#1#2#3{$x + { {#1} \\Uskewedwithdelims / () #3 {#2} } + x$}\n\n\\start\n \\switchtobodyfont[modern]\n \\starttabulate[||||||]\n \\NC \\NC\n \\ShowA{a}{b}{} \\NC\n \\ShowA{1}{2}{} \\NC\n \\ShowB{a}{b}{} \\NC\n \\ShowB{1}{2}{} \\NC\n \\NR\n \\NC \\type{exact} \\NC\n \\ShowA{a}{b}{exact} \\NC\n \\ShowA{1}{2}{exact} \\NC\n \\ShowB{a}{b}{exact} \\NC\n \\ShowB{1}{2}{exact} \\NC\n \\NR\n \\NC \\type{noaxis} \\NC\n \\ShowA{a}{b}{noaxis} \\NC\n \\ShowA{1}{2}{noaxis} \\NC\n \\ShowB{a}{b}{noaxis} \\NC\n \\ShowB{1}{2}{noaxis} \\NC\n \\NR\n \\NC \\type{exact noaxis} \\NC\n \\ShowA{a}{b}{exact noaxis} \\NC\n \\ShowA{1}{2}{exact noaxis} \\NC\n \\ShowB{a}{b}{exact noaxis} \\NC\n \\ShowB{1}{2}{exact noaxis} \\NC\n \\NR\n \\stoptabulate\n\\stop\n\n\\section {Last lines}\n\n\\topicindex {math+last line}\n\nThere is a new primitive to control the overshoot in the calculation of the\nprevious line in mid|-|paragraph display math. The default value is 2 times\nthe em width of the current font:\n\n\\starttyping\n\\predisplaygapfactor=2000\n\\stoptyping\n\nIf you want to have the length of the last line independent of math i.e.\\ you don't\nwant to revert to a hack where you insert a fake display math formula in order to\nget the length of the last line, the following will often work too:\n\n\\starttyping\n\\def\\lastlinelength{\\dimexpr\n \\directlua {tex.sprint (\n (nodes.dimensions(node.tail(tex.lists.page_head).list))\n )}sp\n\\relax}\n\\stoptyping\n\n\\section {Other Math changes}\n\n\\subsection {Verbose versions of single|-|character math commands}\n\n\\topicindex {math+styles}\n\n\\LUATEX\\ defines six new primitives that have the same function as\n\\type {^}, \\type {_}, \\type {$}, and \\type {$$}:\n\n\\starttabulate[|l|l|]\n\\DB primitive \\BC explanation \\NC \\NR\n\\TB\n\\NC \\lpr {Usuperscript} \\NC duplicates the functionality of \\type {^} \\NC \\NR\n\\NC \\lpr {Usubscript} \\NC duplicates the functionality of \\type {_} \\NC \\NR\n\\NC \\lpr {Ustartmath} \\NC duplicates the functionality of \\type {$}, % $\n when used in non-math mode. \\NC \\NR\n\\NC \\lpr {Ustopmath} \\NC duplicates the functionality of \\type {$}, % $\n when used in inline math mode. \\NC \\NR\n\\NC \\lpr {Ustartdisplaymath} \\NC duplicates the functionality of \\type {$$}, % $$\n when used in non-math mode. \\NC \\NR\n\\NC \\lpr {Ustopdisplaymath} \\NC duplicates the functionality of \\type {$$}, % $$\n when used in display math mode. \\NC \\NR\n\\LL\n\\stoptabulate\n\nThe \\lpr {Ustopmath} and \\lpr {Ustopdisplaymath} primitives check if the current\nmath mode is the correct one (inline vs.\\ displayed), but you can freely intermix\nthe four mathon|/|mathoff commands with explicit dollar sign(s).\n\n\\subsection{Script commands \\lpr {Unosuperscript} and \\lpr {Unosubscript}}\n\n\\topicindex {math+styles}\n\\topicindex {math+scripts}\n\nThese two commands result in super- and subscripts but with the current style (at the\ntime of rendering). So,\n\n\\startbuffer[script]\n$\n x\\Usuperscript {1}\\Usubscript {2} =\n x\\Unosuperscript{1}\\Unosubscript{2} =\n x\\Usuperscript {1}\\Unosubscript{2} =\n x\\Unosuperscript{1}\\Usubscript {2}\n$\n\\stopbuffer\n\n\\typebuffer\n\nresults in \\inlinebuffer[script].\n\n\\subsection{Allowed math commands in non|-|math modes}\n\n\\topicindex {math+text}\n\\topicindex {text+math}\n\nThe commands \\prm {mathchar}, and \\lpr {Umathchar} and control sequences that are\nthe result of \\prm {mathchardef} or \\lpr {Umathchardef} are also acceptable in\nthe horizontal and vertical modes. In those cases, the \\prm {textfont} from the\nrequested math family is used.\n\n\\section{Math surrounding skips}\n\n\\topicindex {math+spacing}\n\nInline math is surrounded by (optional) \\prm {mathsurround} spacing but that is fixed\ndimension. There is now an additional parameter \\lpr {mathsurroundskip}. When set to a\nnon|-|zero value (or zero with some stretch or shrink) this parameter will replace\n\\prm {mathsurround}. By using an additional parameter instead of changing the nature\nof \\prm {mathsurround}, we can remain compatible. In the meantime a bit more\ncontrol has been added via \\lpr {mathsurroundmode}. This directive can take 6 values\nwith zero being the default behaviour.\n\n\\start\n\n\\def\\OneLiner#1#2%\n {\\NC \\type{#1}\n \\NC \\dontleavehmode\\inframed[align=normal,offset=0pt,frame=off]{\\mathsurroundmode#1\\relax\\hsize 100pt x$x$x}\n \\NC \\dontleavehmode\\inframed[align=normal,offset=0pt,frame=off]{\\mathsurroundmode#1\\relax\\hsize 100pt x $x$ x}\n \\NC #2\n \\NC \\NR}\n\n\\startbuffer\n\\mathsurround 10pt\n\\mathsurroundskip20pt\n\\stopbuffer\n\n\\typebuffer \\getbuffer\n\n\\starttabulate[|c|c|c|pl|]\n\\DB mode \\BC x\\$x\\$x \\BC x \\$x\\$ x \\BC effect \\NC \\NR\n\\TB\n\\OneLiner{0}{obey \\prm {mathsurround} when \\lpr {mathsurroundskip} is 0pt}\n\\OneLiner{1}{only add skip to the left}\n\\OneLiner{2}{only add skip to the right}\n\\OneLiner{3}{add skip to the left and right}\n\\OneLiner{4}{ignore the skip setting, obey \\prm {mathsurround}}\n\\OneLiner{5}{disable all spacing around math}\n\\OneLiner{6}{only apply \\lpr {mathsurroundskip} when also spacing}\n\\OneLiner{7}{only apply \\lpr {mathsurroundskip} when no spacing}\n\\LL\n\\stoptabulate\n\n\\stop\n\nMethod six omits the surround glue when there is (x)spacing glue present while\nmethod seven does the opposite, the glue is only applied when there is (x)space\nglue present too. Anything more fancy, like checking the begining or end of a\nparagraph (or edges of a box) would not be robust anyway. If you want that you\ncan write a callback that runs over a list and analyzes a paragraph. Actually, in\nthat case you could also inject glue (or set the properties of a math node)\nexplicitly. So, these modes are in practice mostly useful for special purposes\nand experiments (they originate in a tracker item). Keep in mind that this glue\nis part of the math node and not always treated as normal glue: it travels with\nthe begin and end math nodes. Also, method 6 and 7 will zero the skip related\nfields in a node when applicable in the first occasion that checks them\n(linebreaking or packaging).\n\n% \\section{Math todo}\n%\n% The following items are still todo.\n%\n% \\startitemize\n% \\startitem\n% Pre-scripts.\n% \\stopitem\n% \\startitem\n% Multi-story stacks.\n% \\stopitem\n% \\startitem\n% Flattened accents for high characters (maybe).\n% \\stopitem\n% \\startitem\n% Better control over the spacing around displays and handling of equation numbers.\n% \\stopitem\n% \\startitem\n% Support for multi|-|line displays using \\MATHML\\ style alignment points.\n% \\stopitem\n% \\stopitemize\n\n\\subsection {Delimiters: \\type{\\Uleft}, \\prm {Umiddle} and \\prm {Uright}}\n\n\\topicindex {math+delimiters}\n\nNormally you will force delimiters to certain sizes by putting an empty box or\nrule next to it. The resulting delimiter will either be a character from the\nstepwise size range or an extensible. The latter can be quite differently\npositioned that the characters as it depends on the fit as well as the fact if\nthe used characters in the font have depth or height. Commands like (plain \\TEX\ns) \\type {\\big} need use this feature. In \\LUATEX\\ we provide a bit more control\nby three variants that supporting optional parameters \\type {height}, \\type\n{depth} and \\type {axis}. The following example uses this:\n\n\\startbuffer\n\\Uleft height 30pt depth 10pt \\Udelimiter \"0 \"0 \"000028\n\\quad x\\quad\n\\Umiddle height 40pt depth 15pt \\Udelimiter \"0 \"0 \"002016\n\\quad x\\quad\n\\Uright height 30pt depth 10pt \\Udelimiter \"0 \"0 \"000029\n\\quad \\quad \\quad\n\\Uleft height 30pt depth 10pt axis \\Udelimiter \"0 \"0 \"000028\n\\quad x\\quad\n\\Umiddle height 40pt depth 15pt axis \\Udelimiter \"0 \"0 \"002016\n\\quad x\\quad\n\\Uright height 30pt depth 10pt axis \\Udelimiter \"0 \"0 \"000029\n\\stopbuffer\n\n\\typebuffer\n\n\\startlinecorrection\n\\ruledhbox{\\mathematics{\\getbuffer}}\n\\stoplinecorrection\n\nThe keyword \\type {exact} can be used as directive that the real dimensions\nshould be applied when the criteria can't be met which can happen when we're\nstill stepping through the successively larger variants. When no dimensions are\ngiven the \\type {noaxis} command can be used to prevent shifting over the axis.\n\nYou can influence the final class with the keyword \\type {class} which will\ninfluence the spacing. The numbers are the same as for character classes.\n\n\\subsection{Fixed scripts}\n\nWe have three parameters that are used for this fixed anchoring:\n\n\\starttabulate[|c|l|]\n\\DB parameter \\BC register \\NC \\NR\n\\NC $d$ \\NC \\lpr {Umathsubshiftdown} \\NC \\NR\n\\NC $u$ \\NC \\lpr {Umathsupshiftup} \\NC \\NR\n\\NC $s$ \\NC \\lpr {Umathsubsupshiftdown} \\NC \\NR\n\\LL\n\\stoptabulate\n\nWhen we set \\lpr {mathscriptsmode} to a value other than zero these are used\nfor calculating fixed positions. This is something that is needed for instance\nfor chemistry. You can manipulate the mentioned variables to achive different\neffects.\n\n\\def\\SampleMath#1%\n {$\\mathscriptsmode#1\\mathupright CH_2 + CH^+_2 + CH^2_2$}\n\n\\starttabulate[|c|c|c|p|]\n\\DB mode \\BC down \\BC up \\BC example \\NC \\NR\n\\TB\n\\NC 0 \\NC dynamic \\NC dynamic \\NC \\SampleMath{0} \\NC \\NR\n\\NC 1 \\NC $d$ \\NC $u$ \\NC \\SampleMath{1} \\NC \\NR\n\\NC 2 \\NC $s$ \\NC $u$ \\NC \\SampleMath{2} \\NC \\NR\n\\NC 3 \\NC $s$ \\NC $u + s - d$ \\NC \\SampleMath{3} \\NC \\NR\n\\NC 4 \\NC $d + (s-d)/2$ \\NC $u + (s-d)/2$ \\NC \\SampleMath{4} \\NC \\NR\n\\NC 5 \\NC $d$ \\NC $u + s - d$ \\NC \\SampleMath{5} \\NC \\NR\n\\LL\n\\stoptabulate\n\nThe value of this parameter obeys grouping but applies to the whole current\nformula.\n\n% if needed we can put the value in stylenodes but maybe more should go there\n\n\\subsection{Penalties: \\lpr {mathpenaltiesmode}}\n\n\\topicindex {math+penalties}\n\nOnly in inline math penalties will be added in a math list. You can force\npenalties (also in display math) by setting:\n\n\\starttyping\n\\mathpenaltiesmode = 1\n\\stoptyping\n\nThis primnitive is not really needed in \\LUATEX\\ because you can use the callback\n\\cbk {mlist_to_hlist} to force penalties by just calling the regular routine\nwith forced penalties. However, as part of opening up and control this primitive\nmakes sense. As a bonus we also provide two extra penalties:\n\n\\starttyping\n\\prebinoppenalty = -100 % example value\n\\prerelpenalty = 900 % example value\n\\stoptyping\n\nThey default to inifinite which signals that they don't need to be inserted. When\nset they are injected before a binop or rel noad. This is an experimental feature.\n\n\\subsection{Equation spacing: \\lpr {matheqnogapstep}}\n\nBy default \\TEX\\ will add one quad between the equation and the number. This is\nhard coded. A new primitive can control this:\n\n\\startsyntax\n\\matheqnogapstep = 1000\n\\stopsyntax\n\nBecause a math quad from the math text font is used instead of a dimension, we\nuse a step to control the size. A value of zero will suppress the gap. The step\nis divided by 1000 which is the usual way to mimmick floating point factors in\n\\TEX.\n\n\\subsection {Tracing}\n\n\\topicindex {math+tracing}\n\nBecause there are quite some math related parameters and values, it is possible\nto limit tracing. Only when \\type {tracingassigns} and|/|or \\type\n{tracingrestores} are set to~2 or more they will be traced.\n\n\\subsection {Math options}\n\nThe logic in the math engine is rather complex and there are often no universal\nsolutions (read: what works out well for one font, fails for another). Therefore\nsome variations in the implementation will be driven by options for which a new\nprimitive \\lpr {mathoption} has been introduced (so that we don't end up with\nmany new commands). The approach of options also permits us to see what effect a\nspecific solution has.\n\n\\subsubsection {\\type {\\mathoption old}}\n\nThis option was introduced for testing purposes when the math engine got split\ncode paths and it forces the engine to treat new fonts as old ones with respect\nto italic correction etc. There are no guarantees given with respect to the final\nresult and unexpected side effects are not seens as bugs as they relate to font\nproperties.\n\n\\startbuffer\n\\mathoption old 1\n\\stopbuffer\n\nThe \\type {oldmath} boolean flag in the \\LUA\\ font table is the official way to\nforce old treatment as it's bound to fonts.\n\n\\subsubsection {\\type {\\mathoption noitaliccompensation}}\n\nThis option compensates placement for characters with a built|-|in italic\ncorrection.\n\n\\startbuffer\n{\\showboxes\\int}\\quad\n{\\showboxes\\int_{|}^{|}}\\quad\n{\\showboxes\\int\\limits_{|}^{|}}\n\\stopbuffer\n\n\\typebuffer\n\nGives (with computer modern that has such italics):\n\n\\startlinecorrection[blank]\n \\switchtobodyfont[modern]\n \\startcombination[nx=2,ny=2,distance=5em]\n {\\mathoption noitaliccompensation 0\\relax \\mathematics{\\getbuffer}}\n {\\nohyphens\\type{0:inline}}\n {\\mathoption noitaliccompensation 0\\relax \\mathematics{\\displaymath\\getbuffer}}\n {\\nohyphens\\type{0:display}}\n {\\mathoption noitaliccompensation 1\\relax \\mathematics{\\getbuffer}}\n {\\nohyphens\\type{1:inline}}\n {\\mathoption noitaliccompensation 1\\relax \\mathematics{\\displaymath\\getbuffer}}\n {\\nohyphens\\type{1:display}}\n \\stopcombination\n\\stoplinecorrection\n\n\\subsubsection {\\type {\\mathoption nocharitalic}}\n\nWhen two characters follow each other italic correction can interfere. The\nfollowing example shows what this option does:\n\n\\startbuffer\n\\catcode\"1D443=11\n\\catcode\"1D444=11\n\\catcode\"1D445=11\nP( PP PQR\n\\stopbuffer\n\n\\typebuffer\n\nGives (with computer modern that has such italics):\n\n\\startlinecorrection[blank]\n \\switchtobodyfont[modern]\n \\startcombination[nx=2,ny=2,distance=5em]\n {\\mathoption nocharitalic 0\\relax \\mathematics{\\getbuffer}}\n {\\nohyphens\\type{0:inline}}\n {\\mathoption nocharitalic 0\\relax \\mathematics{\\displaymath\\getbuffer}}\n {\\nohyphens\\type{0:display}}\n {\\mathoption nocharitalic 1\\relax \\mathematics{\\getbuffer}}\n {\\nohyphens\\type{1:inline}}\n {\\mathoption nocharitalic 1\\relax \\mathematics{\\displaymath\\getbuffer}}\n {\\nohyphens\\type{1:display}}\n \\stopcombination\n\\stoplinecorrection\n\n\\subsubsection {\\type {\\mathoption useoldfractionscaling}}\n\nThis option has been introduced as solution for tracker item 604 for fuzzy cases\naround either or not present fraction related settings for new fonts.\n\n\\stopchapter\n\n\\stopcomponent\n", "meta": {"hexsha": "5f82eb142d40e9c3bc0c4c4bb9f85f442f32865c", "size": 66829, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/texmf-context/doc/context/sources/general/manuals/luatex/luatex-math.tex", "max_stars_repo_name": "kensh/pandoc_resume", "max_stars_repo_head_hexsha": "74ea55abde343f4e6e07fa6cd94694816e6e3cc4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tex/texmf-context/doc/context/sources/general/manuals/luatex/luatex-math.tex", "max_issues_repo_name": "kensh/pandoc_resume", "max_issues_repo_head_hexsha": "74ea55abde343f4e6e07fa6cd94694816e6e3cc4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/texmf-context/doc/context/sources/general/manuals/luatex/luatex-math.tex", "max_forks_repo_name": "kensh/pandoc_resume", "max_forks_repo_head_hexsha": "74ea55abde343f4e6e07fa6cd94694816e6e3cc4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.5933715743, "max_line_length": 160, "alphanum_fraction": 0.6933367251, "num_tokens": 19369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290152, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3466060044399067}} {"text": "% !TeX root = ../main.tex\n% Add the above to each chapter to make compiling the PDF easier in some editors.\n\n\\chapter{Imperative B-Trees in Isabelle}\\label{chapter:imp-set}\n\nIn the previous chapter, we have seen\nan abstract definition of B-trees and the reasoning\nbehind its correct implementation of the set interface.\nHowever, the specification would not yield \nefficiently executable code.\n% main difference: specify allocation of data\nThe reason is that the abstract specification utilizes\nthe persistent data types node and list.\nThis way, trivially, no data is unknowingly\nmodified or corrupted,\nhowever this is at the cost of computational efficiency.\nA small change in a list or node\nrequires allocating complete copies of all objects that have to be updated,\nrecursively propagating through the whole data structure.\n\nIn order to obtain an efficient implementation\non ephemeral data structures common to imperative languages,\nwe specify imperative code and show that\nit refines the abstract specification.\nThe imperative code can then be translated to\nthe languages Scala \\parencite{OderskyScala} and SML \\parencite{DBLP:books/daglib/0069232}\nusing the code generator\nfor Isabelle/HOL \\parencite{Haftmann07acode}.\nHowever, code extraction is not limited\nto these specific languages\nsince the code generator is extensible.\n\n\\section{Refinement to Imperative/HOL}\n\nThe proofs of correct behavior for our imperative\ncode are stated in separation logic\nan extension of Hoare Logic invented by Reynolds \\parencite{DBLP:conf/lics/Reynolds02}.\nIt was formalized in Isabelle/HOL by Lammich \\parencite{DBLP:journals/jar/Lammich19}\nand comes with a framework that simplifies reasoning about\nits expressions.\n\n\\subsection{Separation Logic}\n\nSeparation logic provides a way to reason about mutable resources\nthat lie in separated parts of an external heap,\nan abstracted memory device.\nThe assumptions on the state of the heap are called \\textit{assertions}.\nThey are stated as formulae that hold for specific heap states.\nThe basic assertions used in this work are the following:\n\n\\begin{itemize}\n \\item \\textit{emp} holds for the empty heap\n \\item \\textit{true} and \\textit{false} hold for every and no heap respectively\n \\item $\\uparrow(P)$ holds if the heap is empty and predicate $P$ holds\n \\item $a \\mapsto_a as$ holds if the heap at position $a$ is reserved and contains\n an array representing $as$.\n \\item $a \\mapsto_r x$ holds if the heap at position $a$ is reserved and contains\n value $x$ where $x$ is of some type $'a::heap$\n \\item $\\exists_A x.\\ P\\ x$ holds if there exists some $x$ such that predicate P\n holds on the heap for given $x$.\n \\item $P_1 * P_2$ holds if each assertion holds on its part of the heap\n and the areas of the heap described by each assertion are non-overlapping\n\\end{itemize}\n\nThe last assertion is key to enabling reasoning about local\nparts of the heap.\nWith the tools provided by separation logic,\nwe will prove our imperative implementations functionally correct\nby showing that they refine the abstract implementation in \\Cref{chapter:abs-set}.\n\nIn general, refinement relates a concrete data structure\nto an abstract structure via some refinement assertion.\nExamples for such a refinement assertion are the id assertion and the\nlist assertion.\nThe latter relates lists, given a refinement\nassertion between the elements of the first and second list.\n\n\\begin{lstlisting}[mathescape=true, language=Isabelle]\ndefinition id_assn a b = $\\uparrow$(a = b)\n\nfun list_assn :: ('a $\\Rightarrow$ 'c $\\Rightarrow$ assn) $\\Rightarrow$ 'a list $\\Rightarrow$ 'c list $\\Rightarrow$ assn where\n list_assn P [] [] = emp\n| list_assn P (x#xs) (y#ys) = P x y * list_assn P xs ys\n| list_assn _ _ _ = false\n\\end{lstlisting}\n\nWe can then specify Hoare Triples on assertions,\nto encapsulate statements about imperative programs.\n\n\\begin{definition}{Hoare Triple}\n $\\langle$ P $\\rangle$ c $\\langle \\lambda r. $ Q r $\\rangle$ holds iff \\\\\n if assertion $P$ holds on some heap before operation $c$\n is executed on the heap,\n and operation $c$ returns some $r$, then assertion $Q$ holds\n for $r$ and on the heap modified by $c$\n\\end{definition}\n\nA simple example Hoare Triple is \n$\\langle$ id\\_assn $a$ $b$ $\\rangle$ return $b$ $\\langle \\lambda r.$ id\\_assn $a$ $r$ $\\rangle$.\nIn the refinement process, we will use more complex relationship assertions\nbetween some abstract $b$ and some refined $bi$.\nWe then show that the value $r$ returned\nby the imperative program \\textit{imp\\_op} satisfies the same relationship\nto \\textit{op} $b$ where \\textit{op} is the refined abstract specification.\nFurther we use the notation $\\langle\\ P\\ \\rangle\\ c\\ \\langle \\lambda r.\\ Q\\ r\\ \\rangle_t$\nas a shorthand for $\\langle\\ P\\ \\rangle\\ c\\ \\langle \\lambda r.\\ Q\\ r\\ * true\\ \\rangle$.\nSince $true$ holds for any heap, this part of the assertion\nmay be used to subsume all temporary and discarded variables.\n\nRefining all set operations from \\Cref{chapter:abs-set}\nwill yield imperative code that operates on the heap\nwith the option to use efficient destructive updates.\nWe do not need to show that the operations themselves\nsatisfy the requirements of the set interface.\nInstead we merely need to show that the imperative code\nacts on a concretized version of the B-trees\nthe same way as the abstract operations on the abstract version.\nThe abstract idea of this approach is discussed in more detail\nin the work of Lammich \\parencite{DBLP:journals/jar/Lammich19}\n\n% TODO note what to introduce here\n% abstract idea: implementation that has same effect on concrete ds\n% as abstract function on abstract ds\n% TODO short intro hoare triples and notation here\n% * and \\uparrow\n\n\\subsection{Refinement of abstract lists}\n\nIn the imperative collection framework by Lammich\n\\parencite{DBLP:journals/jar/Lammich19},\nthe abstract data type list is usually\nrefined to a dynamic array like data structure.\nIt comprises an array and a natural number,\nwhere the latter denotes the current number of elements\nstored in the array.\nB-tree nodes do not always contain a constant number of elements,\nwhich is why this property should be transferred to\nthe data structure refining key-value lists. \n\n% TODO intended physical storing of B-tree on pages of memory\n% --> partially filled array, non-dynamic!\nHowever, dynamic arrays are designed to grow and shrink\nas elements are inserted and deleted.\nB-trees were invented to be stored on slow hard disks.\nAs one disk access loads as much data as fits into a page of main memory,\nnodes of roughly this size make the most efficient use of an access\n\\parencite{DBLP:journals/acta/BayerM72}.\n%TODO give clear reason\nTherefore the data structure to store the node content\nshould reserve arrays of fixed size, independent\nof the actual number of keys and children contained.\n\nBased on the definition of dynamic arrays in the \nimperative collection framework,\nwe introduce a simpler data structure,\nthe \\textit{partially filled array},\nthat keeps count of currently inserted elements,\nbut does neither grow nor shrink.\nA list $ls$ is represented by a partially filled array $(a,n)$\nif the array $a$ represents a list $ls'$, of which the first $n$\nelements form list $ls$.\nFormally,\n\n\\begin{lstlisting}[mathescape=true, language=Isabelle]\n\ndatatype 'a pfarray = 'a array $\\times$ nat\n\ndefinition is_pfa :: nat $\\Rightarrow$ 'a list $\\Rightarrow$ 'a pfarray where\n is_pfa c ls (a,n) =\n $\\exists_A$ ls'. a $\\mapsto_a$ ls' * $\\uparrow$(c = length ls' $\\wedge$ n $\\le$ c $\\wedge$ ls = (take n ls'))\n\\end{lstlisting}\n\nwhere $c$ is the $capacity$ of the array,\nthe actually allocated size of the array on the heap.\nAll references to lists in the abstract data type $btree$\nwill in the imperative version be refined to partially filled arrays.\n\nHowever one should note that\nfrom the set operation specification in \\Cref{chapter:abs-set} alone\nit is not determined which operations are supposed to be conducted\nin-place and which will require copying data.\nTo obtain the most efficient code we therefore manually\ndefine the operations in \\Cref{lst:pfarray-def} to replace complex composite\noperations based on list construction and catenation.\n\n% abstract function has length, insertion, take/drop\n% --> concrete implementations\n\\begin{figure}\n\\begin{lstlisting}[mathescape=true, language=Isabelle,label={lst:pfarray-def},\n caption={Important Partially Filled Array functions for Insertion.}]\ndefinition pfa_length $\\equiv$ $\\lambda$(a,n). return n\n\ndefinition pfa_get $\\equiv$ $\\lambda$(a,n) i. Array.nth a i (* returns the nth element in array a *)\n\ndefinition pfa_set $\\equiv$ $\\lambda$(a,n) i x. do {\n Array.upd i x a; (* sets the element at position i in a to value x *)\n return (a,n)\n}\n\ndefinition pfa_shrink k $\\equiv$ $\\lambda$(a,n). return (a,k)\n\ndefinition pfa_shrink_cap k $\\equiv$ $\\lambda$(a,n). do {\n a' $\\leftarrow$ array_shrink a k; (* returns an array with the given actual size, potentially reallocated*)\n return (a',min k n)\n}\n\ndefinition pfa_drop $\\equiv$ $\\lambda$(src,sn) si (dst,dn). do {\n blit src si dst 0 (sn-si);\n return (dst,(sn-si))\n}\n\ndefinition pfa_insert $\\equiv$ $\\lambda$(a,n) i x. do {\n array_shr a i 1; (* shifts elements from index i on to the right by 1 *)\n Array.upd i x a;\n return (a,n+1)\n}\n\ndefinition pfa_ensure $\\equiv$ $\\lambda$(a,n) k. do {\n a' $\\leftarrow$ array_ensure a k default; (* returns an array with given minimal size, potentially reallocated *)\n return (a',n)\n}\n\ndefinition pfa_insert_grow $\\equiv$ $\\lambda$(a,n) i x. do {\n a' $\\leftarrow$ pfa_ensure (a,n) (n+1);\n a'' $\\leftarrow$ pfa_insert a' i x;\n return a''\n}\n\n$\\dots$\n\\end{lstlisting}\n\\end{figure}\n\n% TODO also include what the rule gives us?\n\nDuring the splitting of nodes, a part of the overflowing\nlist is copied into another array.\nThe operation to copy data is based on the function $blit$,\nthat copies a slice of an array to a slice of another array.\nThe function already existed in the standard collection and could be reused\nto define the \\textit{pfa\\_drop} function in \\Cref{lst:pfarray-def}, which in this implementation copies data to a new array.\n\nThe $blit$ function is however not capable of correctly copying data\nwithin the array, as required for efficient in-place insertion and deletion.\nA suitable function, called $sblit$, was hence added in the course of this project.\nThe relevant addition is to differentiate between copying elements\nfrom higher to lower indices or from lower to higher indices.\nDepending on the direction of the copy, it is relevant\nin which order to copy singleton elements so that in case\nof overlapping ranges no elements are overwritten before being copied.\n\nBy adding a function that copies elements in reverse order\nand conditionally calling either copy function,\nloss-free in-place copying is achieved.\nThe resulting function is conveniently agnostic to the direction\nof the copying, just as the respective implementations\nin target languages for code extraction.\\footnote{\n See for example \\href{https://caml.inria.fr/pub/docs/manual-ocaml/libref/Array.html}{the specification of \\texttt{Array.blit} in Ocaml} or\n the \\href{https://smlfamily.github.io/Basis/array-slice.html\\#SIG:ARRAY_SLICE.copy:VAL}{\\texttt{ArraySlice.copy} specification in SML}.\n}\nThe \\textit{pfa\\_insert} function in \\Cref{lst:pfarray-def} makes\nimplicit use of in-place copying by calling a function to shift elements\nto the right by one.\nThis is achieved by copying the slice of elements\nfrom position $i$ to the end of the list\nto the slice $i+1$ to the end of the list.\nThe element to be inserted is then placed in the remaining free spot.\n\nFor all functions in \\Cref{lst:pfarray-def},\nappropriate Hoare Triples were derived.\nAll triples are straightforward to derive but the reader is encouraged\nto look them up in the proof documents.\nTo roughly sketch the obtained lemmas, the two examples\n\\textit{pfa\\_drop} and \\textit{pfa\\_shrink} are investigated below.\nThe most important difference is whether\nthe operations are destructive (in-place) or not.\nFor most operations, the approach that required the least\nmemory allocations was chosen.\\footnote{\n The only exception is node$_i$, which in the case of overflow\n currently requires temporary allocation of an array that holds more than $2k$ elements.\n}\n\n\\begin{lemma}\n $k \\le \\length s \\wedge (\\length s - k) \\le dn \\Longrightarrow$ \\\\\n \\begin{center}\n $\\langle \\pfarraycap$ sn s si * $\\pfarraycap$ dn d di $\\rangle$ \\\\\n $\\pfadrop$ si k di \\\\\n $\\langle\\lambda di'. \\pfarraycap$ sn s si * $\\pfarraycap$ dn ($\\drop$ k s) di' $\\rangle$\n \\end{center}\n\\end{lemma}\n\nThe array $si$ stays untouched.\nThis fact is expressed by the fact that the term about its relationship to $s$\nis still available in the postcondition of the triple.\nThe only thing that changed is the content of $di$.\\footnote{\n In the actual proof, where possible, we even show that the array itself has not changed\n (i.e. has not been reallocated).\n The correct notation does however not yield high readability\n and is hence abstracted here.\n}\nThe $take$ function was refined differently.\nThe cheapest and in our context consistent way to\nimplement $take$ is to\nreduce the number of elements knowingly stored in the array\nfrom $n$ to $k$.\nThe result is that the original array gets modified and we loose any information\nabout the content that is written beyond given $k$.\n\n\\begin{lemma}\n $k \\le \\length s\\Longrightarrow$ \\\\\n \\begin{center}\n $\\langle \\pfarraycap$ sn s si$\\rangle$\n $\\pfashrink$ si k\n $\\langle\\lambda si'. \\pfarraycap$ sn ($\\take$ k s) si' $\\rangle$\n \\end{center}\n\\end{lemma}\n\nWith these operations and suitable Hoare-Triples on them,\nwe have all the tools required to refine the abstract set implementation.\n% TODO more about sblit?\n\n\\section{Set-Operations}\n\nHaving covered the refinement for the list data type,\nthe B-tree data type itself can be refined.\nRather than actually storing nodes within lists,\nthe B-tree nodes are supposed to be stored in the heap,\nand the node list will only contain pointers to the corresponding nodes.\nPointers are represented by the type \\textit{ref} in Imperative/HOL.\nLeaf nodes contain no information and consequently don't have to be stored\non the heap, therefore we refine them using null pointers.\nSince Imperative/HOL does not support null pointers \\parencite{DBLP:conf/tphol/BulwahnKHEM08},\nand we do not want to explicitly store \\textit{Leaf} objects on the heap,\nB-tree nodes are refined by heap reference \\textit{options} instead.\nThus \\textit{None} represents leafs or empty trees,\nwhile \\textit{Some} heap reference points to refinements of internal nodes.\n\n\\begin{lstlisting}[mathescape=true, language=Isabelle]\ndatatype 'a btnode =\n Btnode (('a btnode ref option*'a) pfarray) ('a btnode ref option)\n\\end{lstlisting}\n\nNote the similarity to the definition in \\Cref{lst:btree-def},\nonly that \\textit{btree} was replaced by \\textit{btnode ref option},\n\\textit{list} was replaced by \\textit{pfarray},\nand there is no leaf case anymore.\n\nAn abstract B-tree is represented by such a physical node\nif all references on subtrees represent the corresponding\nabstract subtrees and all keys are the same.\nThe precise refinement assertion can be seen in \\Cref{lst:imp-btree-refine}.\n\n\\begin{figure}\n\\begin{lstlisting}[mathescape=true, language=Isabelle, caption={The refinement assertion for B-trees},\n label={lst:imp-btree-refine}]\nfun btree_assn :: nat $\\Rightarrow$ 'a::heap btree $\\Rightarrow$ 'a btnode ref option $\\Rightarrow$ assn where\n btree_assn k Leaf None = emp |\n btree_assn k (Node ts t) (Some a) = ($\\exists_A$ tsi ti tsi'.\n a $\\mapsto_r$ Btnode tsi ti\n * btree_assn k t ti\n * is_pfa (2*k) tsi' tsi\n * list_assn ((btree_assn k) $\\times_a$ id_assn) ts tsi'\n ) |\n btree_assn _ _ _ = false\n\nabbreviation blist_assn k $\\equiv$ list_assn ((btree_assn k) $\\times_a$ id_assn)\n\n\\end{lstlisting}\n\\end{figure}\n\nIt seems a little bit awkward that the definition\nincludes two relationships regarding only the list of elements in the subtree.\nThe reason is that there are two steps of refinements going on.\nFirst, all elements in the list are refined to their imperative counterparts.\nThis fact is expressed by the \\textit{list\\_assn} term in the assumption.\nSecond, the list itself is refined to an array.\nThis is expressed in \\textit{is\\_pfa} term.\nSince the list assumption makes recursive use of the\nB-tree assertion, we again cover up usage of the term\nby an abbreviation.\n\nUsing the parameters of \\textit{is\\_pfa},\nnote that we can specify that every node of a B-tree should have\na capacity of $2k$.\n\n\\subsection{The split function}\n\\label{sec:imp-split}\n\nTo implement any set operations we again need a split operation.\nHowever, what split function exactly will be used can be abstracted again.\nThe only important thing is that it refines an abstract split function.\nHowever we have some freedom on how to specify this refinement relationship.\n\nImperative split functions should be efficient and as such should not actually\nsplit the array in half.\nThe operation should rather return the index of the correct\nsubtree-separator pair.\nThe refinement condition derived from this relationship is as follows.\n\n\\begin{lstlisting}[mathescape=true, language=Isabelle]\ndefinition split_relation xs (ls,rs) i = i $\\le$ length xs $\\wedge$ ls = take i xs $\\wedge$ rs = drop i xs\n\\end{lstlisting}\n\nA very useful alternative statement of the relationship follows automatically.\nThis allows us to simplify terms that only contain $xs$\nto the two sublists when we obtain this relationship in proofs.\n\n\\begin{lemma}\n $\\splitrelation xs\\ (ls,rs)\\ i = (xs = ls@rs \\wedge i = \\length ls)$\n\\end{lemma}\n\nFrom this, we characterize the desired imperative\nsplit function \\textit{imp\\_split} by its Hoare-Triple.\n\n\\begin{align*}\n \\langle & \\pfarraycap c\\ xs\\ (a,n) \\rangle \\\\\n & \\impsplit\\ (a,n)\\ p \\\\\n\\langle\\lambda i. & \\pfarraycap c\\ xs\\ (a,n)\\\\\n& * \\uparrow(\\splitrelation xs\\ (\\splitfun xs\\ p)\\ i)\\rangle_t\n\\end{align*}\n\nJust as with the abstract implementation,\nthe imperative refinements can be built with this characterization alone.\nHowever this time we are actually interested in finding an\nefficient split and will spend some time finding\na good split function.\n%TODO move to after the set operations?\n\nIn the imperative context, we prefer not to directly\nrefine the recursive function from \\Cref{fig:linear_split}\nbut we embark on the exercise of using a while loop.\nThe implementation in \\Cref{lst:imp-linear-split}\nnot more efficient than the functional linear search,\nbut suitable for getting to know\nthe usage of while loops in imperative HOL.\nEvery iteration, an index on the array is moved forward by one.\nThe main part is done in the loop head - if we\nare within the array bounds,\nwe obtain the current separator and check if it is\nsmaller than the partitioning element.\nWe return the first index at which the separator is greater or equal.\n\n\\begin{figure}\n\\begin{lstlisting}[mathescape=true, language=Isabelle, caption={The imperative linear split},\n label={lst:imp-linear-split}]\ndefinition lin_split :: ('a::heap $\\times$ 'b::{heap,linorder}) pfarray $\\Rightarrow$ 'b $\\Rightarrow$ nat Heap where\n lin_split (a,n) p = do { \n i $\\leftarrow$ while \n ($\\lambda$i. if i s then \n return (mid+1, high) \n else\n return (mid,mid) \n }) \n (0::nat,n); \n return low' \n }\n\\end{lstlisting}\n\\end{figure}\n\nSince pairs as list elements (i.e. pairs of trees and separators)\nwere expected to add another layer of complexity,\na binary split algorithm on normal lists has been derived first.\nThe split function for lists of pairs was then defined\nand could be tackled using the same loop invariant.\nNote that the binary split works on two indices\nas states, the upper bound $h$ and the lower bound $l$.\n\n\\begin{itemize}\n \\item $\\forall j 0 \\wedge \\sorted (\\inorder t) \\Longrightarrow$ \\\\\n\\begin{align*}\n \\langle &\\btreeassn k\\ t\\ ti\\rangle \\\\\n &\\insertfun k\\ x\\ ti \\\\\n \\langle \\lambda ri. &\\btreeassn k\\ (\\insertfun k\\ x\\ t)\\ ri\\rangle_t\n\\end{align*}\n\\label{thm:imp-ins-rule}\n\\end{theorem}\n\n% TODO say something about k?\nNote how we have the additional assumption that $k$ is positive.\nThis was only required to show correct order in the abstract setting.\nHere, the capacity of the root node needs to be specified at initialization.\nAnd since the capacity has to be $2k$ and the node has to hold at least one element\na positive $k$ is required such that no contradiction appears.\n\nUntil now we have only shown that the concrete\nimplementations refine the abstract interpretations.\nWhat does this mean concretely?\nDo we have to resort to an informal argument to show that the result\nsatisfies the invariants and the set interface?\nThis is not the case.\n\nThe final theorem that proves functional correctness of the imperative\nimplementations with respect to the set interface follows.\n\n\\begin{corollary} $k > 0 \\Longrightarrow$\n\\begin{align*}\n \\langle &\\btreeassn k\\ t\\ ti \\wedge \\btree_k t \\rangle \\\\\n &\\insertfun k\\ x\\ ti \\\\\n \\langle \\lambda ri. \\exists_A r. &\\btreeassn k\\ r\\ ri \\wedge \\btree_k r \\wedge \\inorder r = \\inslist x\\ (\\inorder t)\\rangle_t\n\\end{align*}\n\\end{corollary}\n\nAnd indeed, this can be derived automatically with a single command invocation,\nunfolding \\Cref{def:btree-def} and using \\Cref{thm:imp-ins-rule}, \\Cref{thm:ins-set} and \\Cref{thm:ins-invar}.\\footnote{\n Technically we also need the fact that ins$_{list}$ preserves sortedness.\n This was already proven by Nipkow \\parencite{DBLP:conf/itp/Nipkow16}\n and is not considered here.\n}\n\n\n\\subsection{Deletion}\n\nAn imperative version of the deletion operation has been specified.\nIt required some additional operations on partially filled arrays but\nposed no fundamental challenges.\nDue to its length it is not shown here but may be looked up in the\nactual proof documents.\n\nHowever, the proof of its refining property was not conducted.\nWe assume that this proof would be structured similar to\nthe previous proofs.\nThe only benefit would be exploring usage of the new\nheap rules introduced for the additional array operations.\nThe main difference to the verification of the\ninsertion function is that the refined functions\nare undefined for many cases.\nHence it is highly important\nto strongly tie the refined and imperative version\nin order to discharge void cases.\n", "meta": {"hexsha": "032a1eba840374d9366318b0ff15e0a3b877aa33", "size": 37286, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapters/03_imperative.tex", "max_stars_repo_name": "nielstron/btrees-thesis", "max_stars_repo_head_hexsha": "14b6d8a4819378140e5a977e5278ae0a48057f6f", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapters/03_imperative.tex", "max_issues_repo_name": "nielstron/btrees-thesis", "max_issues_repo_head_hexsha": "14b6d8a4819378140e5a977e5278ae0a48057f6f", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapters/03_imperative.tex", "max_forks_repo_name": "nielstron/btrees-thesis", "max_forks_repo_head_hexsha": "14b6d8a4819378140e5a977e5278ae0a48057f6f", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.4186575654, "max_line_length": 143, "alphanum_fraction": 0.7253124497, "num_tokens": 9512, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.34660600397108476}} {"text": "\n\\section*{Problem 2}\n\n\\begin{problem}{A.}\n\\vspace{2mm}\n\nWe have a hypothetical crystal, whose refractive index depends on both the polarisation and direction of propagation of an incoming EM wave.\n\\\\\nLet us choose a Cartesian reference frame, where the boundary between the medium we are investigating and vacuum is $x=0$ plane. The incident wave, lies in the xy plane, having a wave vector $\\vec{k}_i$, making an angle $\\theta_i$ with the $x-\\mathrm{axis}$ (As shown in the figure). Let us define two polarisations, S and P. Where S polarisation is when the wave is polarised perpendicular to the incident $xy$ plane and P is the polarisation when the electric field of the wave lies in the xy plane. Let us assume two real and positive refractive indices for S and P waves $\\eta_S$ and $\\eta_P$ repectively. Without loss of generality assume $\\eta_P>\\eta_S$.\n\\\\\nSay the incoming wave is linearly polarised confined in the $xy$ plane, with its electric field making an angle $\\frac{\\pi}{4}$ with the $z$ axis. This wave is a mixture of both $S$ and $P$ polarisation. It splits into two at different angles $\\theta_{t\\pm}=\\theta_t \\pm \\alpha$, where $\\vec{k}_{t+}$ corresponds to S, and $\\vec{k}_{t-}$ corresponds to P. \n\\end{problem}\n\\begin{center}\n \n\n\\tikzset{every picture/.style={line width=0.75pt}} %set default line width to 0.75pt \n\n\\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]\n%uncomment if require: \\path (0,606); %set diagram left start at 0, and has height of 606\n\n%Shape: Rectangle [id:dp41394729732408253] \n\\draw [draw opacity=0][fill={rgb, 255:red, 155; green, 155; blue, 155 } ,fill opacity=0.35 ] (251.13,82.37) -- (485.49,82.37) -- (485.49,289) -- (251.13,289) -- cycle ;\n%Shape: Axis 2D [id:dp26172173283960065] \n\\draw [line width=1.5] (153.17,162) -- (515.81,162)(250,15) -- (250,289) (508.81,157) -- (515.81,162) -- (508.81,167) (245,22) -- (250,15) -- (255,22) ;\n%Straight Lines [id:da8337450025284996] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (119.17,285.2) -- (247.17,163.2) ;\n%Curve Lines [id:da1405822884062613] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (217.13,164) .. controls (212.3,164.95) and (226.77,183.09) .. (227.5,180.02) ;\n%Straight Lines [id:da479198997901781] \n\\draw [line width=2.25] (162.74,243.46) -- (197.17,211) ;\n\\draw [shift={(200.81,207.57)}, rotate = 136.68] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (11.43,-5.49) -- (0,0) -- (11.43,5.49) -- cycle ;\n%Straight Lines [id:da6299546920152124] \n\\draw [line width=0.75] [dash pattern={on 4.5pt off 4.5pt}] (248.32,161.72) -- (444.81,58.57) ;\n%Straight Lines [id:da3883728648151408] \n\\draw [line width=0.75] [dash pattern={on 4.5pt off 4.5pt}] (247.17,163.2) -- (444.81,87.93) ;\n%Straight Lines [id:da4998060156867363] \n\\draw [line width=0.75] [dash pattern={on 4.5pt off 4.5pt}] (248.32,161.72) -- (445.81,118.93) ;\n%Curve Lines [id:da6470768122125261] \n\\draw [line width=1.5] [dash pattern={on 5.63pt off 4.5pt}] (379.8,92.91) .. controls (395.81,95.93) and (391.81,139.93) .. (390.81,127.93) ;\n%Curve Lines [id:da8038047661899634] \n\\draw [line width=1.5] [dash pattern={on 5.63pt off 4.5pt}] (329.8,131.91) .. controls (338.32,133.52) and (339.19,144.75) .. (337.99,153.1) .. controls (336.93,160.44) and (334.27,165.54) .. (333.81,159.93) ;\n%Straight Lines [id:da4217023866157301] \n\\draw [line width=2.25] (345.56,110.14) -- (390.35,87.19) ;\n\\draw [shift={(394.8,84.91)}, rotate = 152.87] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10,-4.8) -- (0,0) -- (10,4.8) -- cycle ;\n%Straight Lines [id:da49843178394801124] \n\\draw [line width=2.25] (352.81,139.93) -- (403.95,127.14) ;\n\\draw [shift={(408.81,125.93)}, rotate = 165.96] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10,-4.8) -- (0,0) -- (10,4.8) -- cycle ;\n%Shape: Circle [id:dp03983391745542253] \n\\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=1 ] (244.51,161.93) .. controls (244.51,158.46) and (247.33,155.64) .. (250.81,155.64) .. controls (254.28,155.64) and (257.1,158.46) .. (257.1,161.93) .. controls (257.1,165.41) and (254.28,168.22) .. (250.81,168.22) .. controls (247.33,168.22) and (244.51,165.41) .. (244.51,161.93) -- cycle ;\n%Shape: Cube [id:dp6684229072869232] \n\\draw [fill={rgb, 255:red, 0; green, 0; blue, 0 } ,fill opacity=0.22 ] (244.28,398.8) -- (295.28,347.8) -- (511.28,347.8) -- (511.28,473.8) -- (460.28,524.8) -- (244.28,524.8) -- cycle ; \\draw (511.28,347.8) -- (460.28,398.8) -- (244.28,398.8) ; \\draw (460.28,398.8) -- (460.28,524.8) ;\n%Straight Lines [id:da7498729323670823] \n\\draw (123.68,487.8) -- (123.68,371.8) ;\n\\draw [shift={(123.68,368.8)}, rotate = 90] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n%Straight Lines [id:da8208999367418137] \n\\draw (123.68,487.8) -- (290.88,487.8) ;\n\\draw [shift={(293.88,487.8)}, rotate = 180] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n%Straight Lines [id:da7174552504343465] \n\\draw (123.68,487.8) -- (177.03,420.16) ;\n\\draw [shift={(178.88,417.8)}, rotate = 128.26] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n%Straight Lines [id:da21779076116998253] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (123.68,487.8) -- (279.98,447.55) ;\n\\draw [shift={(282.88,446.8)}, rotate = 165.56] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n%Straight Lines [id:da4538137858169531] \n\\draw (175.88,474.8) -- (230,459.61) ;\n\\draw [shift={(232.88,458.8)}, rotate = 164.32] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n%Curve Lines [id:da5782358459753356] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (151.28,452.8) .. controls (164.88,452.8) and (169.88,484.8) .. (162.88,487.8) ;\n\n%Straight Lines [id:da995607271595156] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (460.28,524.8) -- (459.96,555.61) -- (459.88,562.6) ;\n%Straight Lines [id:da22758558142862872] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (511.28,473.8) -- (511.12,489.63) -- (510.97,503.6) -- (510.88,511.6) ;\n%Straight Lines [id:da17388485750558225] \n\\draw [dash pattern={on 4.5pt off 4.5pt}] (462.41,541.68) -- (509.16,494.92) ;\n\\draw [shift={(511.28,492.8)}, rotate = 135] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n\\draw [shift={(460.28,543.8)}, rotate = 315] [fill={rgb, 255:red, 0; green, 0; blue, 0 } ][line width=0.08] [draw opacity=0] (10.72,-5.15) -- (0,0) -- (10.72,5.15) -- (7.12,0) -- cycle ;\n% Text Node\n\\draw (102.68,377.2) node [anchor=north west][inner sep=0.75pt] {${y}$};\n% Text Node\n\\draw (295.88,491.2) node [anchor=north west][inner sep=0.75pt] {${z}$};\n% Text Node\n\\draw (291.68,439.2) node [anchor=north west][inner sep=0.75pt] {${x}$};\n% Text Node\n\\draw (212.68,434.2) node [anchor=north west][inner sep=0.75pt] {${k_{i}}$};\n% Text Node\n\\draw (168.68,398.2) node [anchor=north west][inner sep=0.75pt] {${E_{i}}$};\n% Text Node\n\\draw (170.68,441.2) node [anchor=north west][inner sep=0.75pt] {$\\phi$};\n% Text Node\n\\draw (487.78,521.7) node [anchor=north west][inner sep=0.75pt] {${d}$};\n% Text Node\n\\draw (516.8,169.31) node [anchor=north west][inner sep=0.75pt] {${X}$};\n% Text Node\n\\draw (226.8,19.31) node [anchor=north west][inner sep=0.75pt] {${Y}$};\n% Text Node\n\\draw (379.8,136.31) node [anchor=north west][inner sep=0.75pt] {${k_{t-}}$};\n% Text Node\n\\draw (403.8,106.31) node [anchor=north west][inner sep=0.75pt] {$-\\alpha $};\n% Text Node\n\\draw (393.62,86.13) node [anchor=north west][inner sep=0.75pt] [rotate=-338.35] {$+\\alpha $};\n% Text Node\n\\draw (364.8,68.31) node [anchor=north west][inner sep=0.75pt] {${k_{t+}}$};\n% Text Node\n\\draw (349.06,143.73) node [anchor=north west][inner sep=0.75pt] {${\\theta }_{t}$};\n% Text Node\n\\draw (191.8,167.4) node [anchor=north west][inner sep=0.75pt] {${\\theta _{i}}$};\n% Text Node\n\\draw (181.7,234.84) node [anchor=north west][inner sep=0.75pt] [rotate=-314.46] {${k_{i}}$};\n% Text Node\n\\draw (228.8,142.4) node [anchor=north west][inner sep=0.75pt] {$Z$};\n\n\n\\end{tikzpicture}\n\n\\end{center}\n\\newpage\n\\begin{subpr}{A1. \\hfill 1.5 pts.} Find $n_S$ and $n_P$ in terms of $\\theta_t$, $\\theta_i$, $\\alpha$. Assume that $n_S=n_0+\\Delta n$ and $n_P=n_0-\\Delta n$, where $\\Delta n/n_0 <<1$ and we only look for first-order terms in $\\Delta n/n_0$.\n\\end{subpr}\n\\begin{problem}{}\n Now assume the incidence to be normal, and the electric field still makes an angle $\\pi/4$ with the $z$-axis (As can be seen in the figure). The crystal has a thickness $t>>\\lambda$. \n\\end{problem}\n\\begin{subpr}{A2. \\hfill 1 pts.} Find the values of $t$ for which the light exiting crystal is either circularly polarised or it is linearly polarised, but rotated by and angle $\\pi/2$ with respect to the incident polarisation. You may neglect the difference between Reflection coefficients for $S$ and $P$.\n\\end{subpr}\n\n\\begin{problem}{B.}\n Let us investigate a plane EM wave, of frequency $\\omega$ travelling in a static uniform magnetic field $B_0 \\hat{z}$, where $\\hat{z}$ is the unit vector $z$ of the chosen Cartesian frame of reference. $B_0$ is much stronger than the field of the wave and the wave propagation is also along $\\hat{z}$. \n\\\\\n This medium contains $N_e$ bound electrons per unit volume, each obeying the classical equation of motion.\n \\begin{equation}\ne(\\vec{E}+\\frac{\\vec{v}}{c}\\times \\vec{B}) +m_e \\omega^2 r= -m_e \\ddot{\\vec{r}}, \\vec{v}=\\frac{d\\vec{r}}\n{dt}\\end{equation}\nWhere $m_e$ and $-e$ are the mass and charge of electron respectively.\n\\end{problem}\n\\begin{subpr}{B1. \\hfill 4.5 pts.} Prove that the wave propagation depends on polarisation, by calculating the refractive index for circular polarisation (either left handed or right).\n\\end{subpr}\n\\begin{problem}{}\nNow consider the propagation of a linearly polarised in the same wave. Asuume the field at $z=0$ to be given by $\\vec{E}_i (z=0,t)=E_i e^{-i\\omega t} \\hat{x}$, and a relatively weak magnetic field such that $\\omega >> \\omega_c$ and higher order terms of $\\omega_c/\\omega$ can be neglected.\n\\end{problem}\n\\begin{subpr}{B2. \\hfill 3 pts.} Calculate the electric field at $z=l$, and find the angle by which the polarization has rotated.\n\\end{subpr}\n\\clearpage", "meta": {"hexsha": "04a96f60ea7cc7d52e55b59f5eea3c639a991eb2", "size": 10763, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/Round-3/Problems/2.tex", "max_stars_repo_name": "GRAMOLY-ORG/FIZIKA", "max_stars_repo_head_hexsha": "b6f920b15ea598de3777f386aa80edbb49657af8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-04T07:21:50.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-04T07:21:50.000Z", "max_issues_repo_path": "tex/Round-3/Problems/2.tex", "max_issues_repo_name": "GRAMOLY-ORG/FIZIKA", "max_issues_repo_head_hexsha": "b6f920b15ea598de3777f386aa80edbb49657af8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/Round-3/Problems/2.tex", "max_forks_repo_name": "GRAMOLY-ORG/FIZIKA", "max_forks_repo_head_hexsha": "b6f920b15ea598de3777f386aa80edbb49657af8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 75.7957746479, "max_line_length": 661, "alphanum_fraction": 0.6448945461, "num_tokens": 4324, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3466059957112453}} {"text": "\\chapter{Bidirectional \\kl(tit){PCUIC}}\n\\label{chap:bidir-pcuic}\n\n\\margintoc\n\nAs we have seen in \\cref{sec:tech-pcuic}, there is much more to the real \\kl{Coq} than\n\\kl{CCω}.\nThe ideas exposed in the previous chapter nevertheless\nscale very well to these extensions.\nThere are two areas, though, where some care needs to be taken.\nThe first is cumulativity, which in particular forces us to reconsider the statement\nof the completeness and uniqueness properties, see \\cref{sec:bidir-pcuic-cumulativity}.\nBut the main one is the introduction of inductive types. In particular, there is a subtle\ninterplay with cumulativity in the treatment of pattern-matching.\nWorking on the formalized proof of completeness in \\kl{MetaCoq} led to the discovery of\nan incompleteness bug in the kernel of \\kl{Coq} linked to this.\nIn \\cref{sec:bidir-pcuic-inductives} we show how the bidirectional setting adapts to\ninductive types, and try and give an intuition of the origin of the completeness issue.\n\nWe do not give precise proofs in this chapter,\ninstead relying on the formalization in \\kl{MetaCoq} described in \\arefpart{metacoq}.\n\n % The first area of difference are the universes. While on paper those are simply integer, to handle typical ambiguity and polymorphic (co)-inductive types, PCUIC uses algebraic universes, containing level variables, algebraic $\\vee$ and $+1$ operators, and a special level for the sort Prop. Moreover, those universes are cumulative, that is they behave as if smaller universes were included in larger ones. The precise handling of the algebraic universes is abstracted away in MetaCoq, and quite similar in the directed and undirected systems, so it did not prove too difficult to handle. Cumulativity, however, introduces some not-so-small differences with the previous presentation, so we spend some time on it in \\cref{sec:pcuic-cumul}.\n\n % The second is the addition of new base type and term constructors. We describe the treatment of inductive types in \\cref{sec:pcuic-indu}. Co-inductive types and records behave very similarly to inductive types at the level of typing, so we do not dwell on them. The difference lies mainly at the level of reduction/conversion, but as our type system treats those as black boxes the differences have a negligible impact.\n \n \n\\section{Cumulativity}\n\\label{sec:bidir-pcuic-cumulativity}\n\n\\begin{marginfigure}\n \\begin{mathpar}\n \\inferdef{CheckCum}\n {\\inferty{\\Gamma}{t}{T} \\\\ T \\cum T'}{\\checkty{\\Gamma}{t}{T'}}\n \\label{rule:bd-check-cum}\n \\end{mathpar} \n\\end{marginfigure}\nThe introduction of the more liberal cumulativity rules in the undirected system\nof course calls for an update to the computation rules.\nThe change to \\ruleref{rule:bd-check} is direct: simply replace conversion with cumulativity,\nas done in \\ruleref{rule:bd-check-cum} opposite.\nAs for the constrained inference rules, they do not even need any modification.\nIntuitively, this is because there is no reason to degrade a type to a larger one,\nunless it is forced by a given target type in the checking judgment.\n\nThe statement of completeness also needs to account for cumulativity,\nand becomes the following one.\n\n\\begin{theorem}[Completeness, with cumulativity]\n \\label{thm:comp-cumul}\n If $\\Gamma \\vdash t \\ty T$, then $\\inferty{\\Gamma}{t}{T'}$ is derivable\n for some $T'$ such that $T' \\cum T$.\n\\end{theorem}\n\nThis also means that in the setting of \\kl{PCUIC},\n\\kl{uniqueness of types} up to \\kl{conversion} is not true any more.\nFor instance, we both have $\\Gamma \\vdash \\uni[0] \\ty \\uni[1]$ and $\\Gamma \\vdash \\uni[0] \\ty \\uni[2]$, but $\\uni[1]$ and $\\uni[2]$ are not convertible. In that context, however,\nthe type $\\uni[1]$ still has a special property: it is minimal among all types, what\nwe call a \\kl{principal type}.\n\n\\begin{definition}[Principal type]\n The type $T$ is a \\intro{principal type} for term $t$ – in a context $\\Gamma$ –\n if $\\Gamma \\vdash t \\ty T$ and for any $T'$ such that $\\Gamma \\vdash t \\ty T'$,\n we have $T \\cum T'$.\n\\end{definition}\n\nThe existence of such a principal type is the same as \\kl{uniqueness of types} \nup to cumulativity. Moreover, even in the cumulative setting, \\cref{thm:unique-inf}%\n\\sidenote{Uniqueness of inferred types up to joinability.}\nstays true. Intuitively, this is because it only relies on properties of reduction, but not of\nconversion. Thus, following the same proof as that of \\cref{thm:unique-undir},%\n\\sidenote{Uniqueness of types for undirected typing.}\nwe obtain that inferred types are principal.\n\n\\begin{theorem}[Inferred types are principal]\n \\label{thm:princ-types}\n If $\\Gamma$ is well-formed and $\\inferty{\\Gamma}{t}{T}$,\n then $T$ is a \\kl{principal type} for $t$ in $\\Gamma$.\n\\end{theorem}\n \n\\begin{proof}\n If $\\Gamma \\vdash t \\ty T'$, then by completeness there exists some $T''$ such that\n $\\inferty{\\Gamma}{t}{T''}$, and moreover $T'' \\cum T'$.\n But by \\cref{thm:unique-inf}, $T \\conv T'' \\cum T'$ and thus $T \\cum T'$, and $T$ is thus indeed a principal type for $t$ in $\\Gamma$.\n\\end{proof}\n\nThe existence of \\kl{principal types} is not so easy to prove directly, as it more or less\namounts to showing correctness and completeness of the bidirectional system at once.\nNevertheless, it is useful, because it in particular means that any well-typed term $t$\nhas an unambiguous smallest universe, which can be obtained as the \\kl{principal\ntype} of its \\kl{principal type}. This means that there is a good separation between irrelevant \npropositions – those terms whose smallest universe is $\\Prop$ – and relevant terms\n– those whose smallest universe is some $\\uni[i]$ –, and that this stays true even in\npresence of cumulativity, and even if $\\Prop \\cum \\uni[i]$. If this were not the case,\nthe erasure of propositional content – which is one of the important use cases of $\\Prop$ –\nwould not make sense.\n\n\\section{Inductive Types}\n\\label{sec:bidir-pcuic-inductives}\n\n\\subsection{An example: the pair type}[The pair type]\n\n\\begin{figure}\n \n \\begin{mathpar}\n \\inferdef{PairTy}{\n \\pinferty{\\uni}{\\Gamma}{A}{\\uni[i]} \\\\ \n \\pinferty{\\uni}{\\Gamma, x : A}{B}{\\uni[j]}}\n {\\inferty{\\Gamma}{\\Sb x : A .\\ B}{\\uni[\\umax{i}{j}]}}\n \\label{rule:pair-type-bd} \\and\n \\inferdef{Pair}{\n \\pinferty{\\uni}{\\Gamma}{A}{\\uni[i]} \\\\\n \\pinferty{\\uni}{\\Gamma, x : A}{B}{\\uni[j]} \\\\\n \\checkty{\\Gamma}{t}{A} \\\\\n \\checkty{\\Gamma}{u}{\\subs{B}{x}{t}}}\n {\\inferty{\\Gamma}{\\pair[A][x.B]{t}{u}}{\\Sb x : A .\\ B}}\n \\label{rule:pair-bd} \\and\n \\inferdef{PairInd}{\n \\pinferty{\\Sb}{\\Gamma}{s}{\\Sb x : A .\\ B} \\\\\n \\pinferty{\\uni}{\\Gamma, z : \\Sb x : A .\\ B}{P}{\\uni} \\\\\n \\checkty{\\Gamma, y_1 : A, y_2 : \\subs{B}{x}{y_1}}{b}\n {\\subs{P}{z}{\\pair[A][x.B]{y_1}{y_2}}}}\n {\\inferty{\\Gamma}{\\ind{\\Sb}{s}{z.P}{y_1.y_2.b}}{\\subs{P}{z}{s}}}\n \\label{rule:pair-ind-bd} \\and\n \n \\inferdef{PairInf}{\n \\inferty{\\Gamma}{t}{T} \\\\ T \\red \\Sb x : A .\\ B}\n {\\pinferty{\\Sb}{\\Gamma}{t}{\\Sb x : A .\\ B}}\n \\label{rule:sig-inf} \n \\end{mathpar}\n\n \\caption{Bidirectional pair type}\n \\label{fig:bidir-pair}\n\\end{figure}\n\nTo set ideas straight, let us look at how we can adapt the dependent pair type of\n\\cref{fig:sig} to the bidirectional setting: see \\cref{fig:bidir-pair}.\nTo obtain these rules, first notice that all undirected typing rules for the\npair type (\\cref{fig:sig})\nmust become inference rules if we want the resulting system to be complete.\nThe question therefore is once again to choose modes for the premises.\nRules \\nameref{rule:pair-type-bd} and \\nameref{rule:sig-inf} are\nvery similar to the rule for Π-types, there is not much surprise there.\n\n\\ruleref{rule:pair-bd} shows why we insisted in the undirected system\non recording the types $A$ and $B$ in the pair. Indeed, they are needed to\nknow which type to infer for the pair. Without the annotation, one could infer a\ntype $A$ for $t$ and a type $B'$ for $u$, but there are potentially many incomparable types $B$ that would be correct for the whole pair, depending on which instances of $t$ in $B'$ are abstracted to $x$. We only know that $B'$ is $\\subs{B}{x}{t}$,\nbut this is not enough to inambiguously determine $B$.\nThis impossibility to invert a substitution is a general source of need\nfor annotations, which is not specific to pair types!\n\nFinally, \\ruleref{rule:pair-ind-bd} is the most complex.\nIn presentations of \\kl{recursors}, often the predicate appears first, then the branches,\nand finally the scrutinee. But this is not possible here, as the parameters of the inductive\ntype are needed to construct the context in which the predicate is typed.\nInstead, those parameters can be inferred from the scrutinee.\nThus, a type for the scrutinee is first obtained using a new constrained inference judgment,\nforcing the inferred type to be a Σ-type, but leaving its parameters free.\nNext, these parameters can be used to construct the context to type the predicate.\nAnd finally, once the predicate is known to be well-formed,\nit can be used to type-check the branch.\n\nThis same approach can be readily extended to the other inductive types of\n\\cref{sec:tech-cic}, with recursion or indices posing no specific problems.\n%, see \\cref{fig:bidir-indu-other}.\n\n\\subsection{Polymorphic inductive types}\n\nThe account of general inductive types in \\kl{PCUIC} is slightly different from\nthe one we just gave. The reason for this is that giving a general account of rules\nwhich infer type levels like our \\ruleref{rule:pair-type-bd} is not easy.\nIndeed, the parameters of an inductive type can\nbe of a type much more complex than simply $\\uni$, and in that general setting deciding which\ntype variable can be inferred is a non-trivial problem.\nInstead, the polymorphic inductive types as implemented in \\kl{Coq} store explicit universe\nlevels on inductive types and constructors. The pair type of \\cref{fig:bidir-pair},\nfor instance, would contain universe levels $i,j$, so that both $A$ and $B$\nwould be checked rather than having their level inferred.\nThe rule for the type constructor in that context is given opposite.\n\\begin{marginfigure}\n \\begin{mathpar}\n \\inferrule{\n \\checkty{\\Gamma}{A}{\\uni[i]} \\\\ \n \\checkty{\\Gamma, x : A}{B}{\\uni[j]}}\n {\\inferty{\\Gamma}{\\Sb\\ulev{i,j} x : A .\\ B}{\\uni[\\umax{i}{j}]}}\n \\end{mathpar}\n\\end{marginfigure}\nThis makes the treatment of complex inductive types possible by using checking uniformly –\nrather than relying on constrained inference to infer universe levels –\nat the cost of possibly needless annotations, as here with Σ-types.\nThis is mostly invisible for the end user though, as she does very seldom write universe\nlevels thanks to \\kl{typical ambiguity} anyway.\n\nIn the same spirit, pattern-matching in \\kl{Coq} – and its counterpart in \\kl{PCUIC} –\nalso stores enough information to easily reconstruct the context\nin which the predicate and branches are typed. This information consists in universe levels\n– for polymorphic inductive types – and parameters of the inductive type.\nThus, the actual typing rule for pattern-matching in the case of Σ-types\nis closer to the following one:\n\n\\begin{mathpar}\n \\inferrule{\n \\pinferty{\\Sb}{\\Gamma}{s}{\\Sb\\ulev{i,j} x : A .\\ B} \\\\\n i \\le i' \\\\ j \\le j' \\\\ A \\cum A' \\\\ B \\cum B' \\\\\n \\pinferty{\\uni}{\\Gamma, z : \\Sb x : A' .\\ B'}{P}{\\uni} \\\\\n \\checkty{\\Gamma, y_1 : A', y_2 : \\subs{B'}{x}{y_1}}{b}\n {\\subs{P}{z}{\\pair[A'][x.B']{y_1}{y_2}}}}\n {\\inferty{\\Gamma}{\\match{\\Sb}{i',j'}{A',B'}{s}{z.P}{y_1.y_2.b}}{\\subs{P}{z}{s}}}\n\\end{mathpar} \n\nNote that the domain and codomain are compared using \\kl{cumulativity}. This is crucial\nto retain \\kl{subject reduction}. Indeed, reduction of the scrutinee might make its inferred\ntype decrease. For instance, suppose we have a polymorphic inductive $I\\ulev{i}$ with a single\nconstructor $c$ such that $A : \\uni[i] \\vdash c\\ulev{i}(A)$. Now consider\n \\[ \\left( \\l y : I\\ulev{1}.\\ y \\right)\\ c\\ulev{0}(\\Nat)\n \\ored c\\ulev{0}(\\Nat) \\]\nthe redex infers type $I\\ulev{1}$, while the reduct infers\n$I\\ulev{0}$. Thus, if such a term is plugged as scrutinee in a pattern-matching,\nthe whole term is still typeable after the reduction of\nthe scrutinee because we allow inequalities rather than equalities between levels.\n\nBut here lies a subtle issue: in pen-and-paper accounts of recursors,\nthe predicate and branches are often\nrepresented respectively as Π-types and λ-abstractions. This is also how previous versions of\n\\kl{Coq} represented pattern-matching.%\n\\sidenote{Until version 8.14 to be precise.}\nBut recall that in \\kl{PCUIC}, cumulativity is\nequivariant on the domain of Π-types. This led to an implementation that wrongly compared\nthe universe levels using equality rather than inequality, leading to a completeness\nbug that manifested as a failure of subject reduction in situations such as the one above.%\n\\sidenote{A precise description of the problem in the kernel and an example similar to the\n one above are given in issue \\coqIssue{13495}.}\nThis prompted subsequent work, both on the theory of \\kl{PCUIC} and on the implementation, to\nremove the use of Π- and λ-abstractions completely from pattern-matching%\n\\sidenote{This was carried out by Pierre-Marie Pédrot starting with pull-request \\coqPR{13563},\nfollowing ideas that had been laid down earlier by Hugo Herbelin in the\n\\href{https://github.com/coq/ceps/blob/master/text/inductive-branch-predicate-representation-and-reduction.md}{\\kl{Coq} enhancement\nproposal \\#34}.},\nmaking both the implementation less ad-hoc, and the theory cleaner.\nA detailed summary has been given in \\sidetextcite{Sozeau2022}.\n\nFurther investigations in that area might still be valuable though, in particular in order\nto determine what kind of annotations are actually needed for pattern-matching, both\nin theory and in practice. Can we give a presentation of polymorphic inductive types\nthat is as lightweight as pair types in \\cref{fig:bidir-pair}?\nThe bidirectional presentation is valuable there, because now\nit is clear what the specification of an alternative syntax is:\nit should remain complete, in the sense of \\cref{thm:comp-cumul}.", "meta": {"hexsha": "19050b1e53d7017454535350bf4fd8d959617587", "size": 14128, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Manuscript/bidir-pcuic.tex", "max_stars_repo_name": "MevenBertrand/PhD-Thesis", "max_stars_repo_head_hexsha": "5bb9852b747bf0700d7c60b74dc64e11372478f8", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Manuscript/bidir-pcuic.tex", "max_issues_repo_name": "MevenBertrand/PhD-Thesis", "max_issues_repo_head_hexsha": "5bb9852b747bf0700d7c60b74dc64e11372478f8", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2022-03-22T14:04:27.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-22T18:26:29.000Z", "max_forks_repo_path": "Manuscript/bidir-pcuic.tex", "max_forks_repo_name": "MevenBertrand/PhD-Thesis", "max_forks_repo_head_hexsha": "5bb9852b747bf0700d7c60b74dc64e11372478f8", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 56.7389558233, "max_line_length": 743, "alphanum_fraction": 0.7352066818, "num_tokens": 3963, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3466059957112453}} {"text": "\\documentclass[main.tex]{subfiles}\n\\begin{document}\n\n\\marginpar{Thursday\\\\ 2020-8-27, \\\\ compiled \\\\ \\today}\n\nWe then expand the electron and photon energy distributions \\emph{after} the scattering up to second order in \\(\\Delta \\): using the fact that \\(\\epsilon' - \\epsilon = \\Delta \\Theta \\) (\\(\\Delta \\) \\emph{times} \\(\\Theta \\), not the variation of \\(\\Theta \\)) we find\n%\n\\begin{align}\nf(E') &= f(E - \\Delta \\Theta ) \\approx \\qty(1 + \\Delta + \\frac{\\Delta^2}{2}) f(E) \\\\\nn(\\epsilon ') &= n( \\epsilon + \\Delta \\Theta ) \\approx n(\\epsilon ) + \\pdv{n}{\\epsilon } \\Delta \\Theta \n+ \\pdv[2]{n}{\\epsilon } \\frac{(\\Delta \\Theta )^2}{2 }\n\\,,\n\\end{align}\n%\nwhere we have used the explicit expression we have for the thermal distribution \\(f(E) \\sim e^{- E/ \\Theta }\\), which gives us a factor \\(-1/\\Theta\\) each we differentiate with respect to \\(E\\). \n\nIf we define the usual rescaled energy \\(x = h \\nu / k_B T = \\epsilon / \\Theta \\) we can express the latter expansion as \n%\n\\begin{align}\nn(x') \\approx n(x) + \\Delta \\pdv{n}{x} + \\frac{\\Delta^2}{2} \\pdv[2]{n}{x}\n\\,.\n\\end{align}\n\nWe will assume that the medium is \\textbf{homogeneous}, \\textbf{isotropic} and \\textbf{infinite}, and that the photon distribution is \\textbf{isotropic}. The latter point means that \\(n\\) is not a function of \\(\\Omega \\). \nThis will be close to true as long as the optical depth is large, since then we will see many interactions which will spread the photons. \n\nWe can then take out of the Boltzmann integral all the terms depending only on \\(n\\), since they have no more angular dependence. This yields the following expression (up to second order in \\(\\Delta \\)): \n%\n\\begin{align}\n\\pdv{n}{t} = \\qty[n' + n(n+1)]\n\\underbrace{\\int \\dd[3]{p} \\dd{\\Omega } \\dv{\\sigma }{\\Omega } f(E) \\Delta }_{I_1 }\n+ \n\\qty[\\frac{n''}{2}\n+ \\qty(n' + \\frac{n}{2})\n\\qty(n+1)]\n\\underbrace{\\int \\dd[3]{p} \\dd{\\Omega } \\dv{\\sigma }{\\Omega }\nf(E) \\Delta^2 }_{I_2 }\n\\,.\n\\end{align}\n\nHere we are denoting \\(n' = \\pdv*{n}{x}\\) for simplicity. \nLet us then try to evaluate the integrals \\(I_1 \\) and \\(I_2 \\), starting from the latter. It depends on \\(\\Delta^2\\), which is given by \n%\n\\begin{align}\n\\Delta^2 = x^2 \\beta^2 (\\vec{\\Omega}' - \\vec{\\Omega})^2 \\cos^2 \\theta_v\n\\,,\n\\end{align}\n%\nso it evaluates to \n%\n\\begin{align}\nI_2 = \n\\int \\dd[3]{p} \\dd{\\Omega } \\dv{\\sigma }{\\Omega } f(E) \\Delta^2\n= x^2 \\int f(E) p^2 \\beta^2 \\cos^2 \\theta _v \\dd{p} \\sin \\theta _v \\dd{\\theta _v} \\dd{\\phi _v} \\int \\dd{\\Omega } \\dv{\\sigma }{\\Omega } (\\vec{\\Omega}' - \\vec{\\Omega})^2\n\\,,\n\\end{align}\n%\nwhere \\(\\theta _v\\) and \\(\\phi _v\\) are the two angles defining the unit vector \\(\\hat{p}\\), so that \\(\\dd[3]{p} = p^2 \\sin \\theta _v \\dd{p } \\dd{\\theta _v} \\dd{\\phi _v}\\). We choose them by placing the vector \\(\\vec{\\Omega}' - \\vec{\\Omega}\\) on the \\(z\\) axis, so that the angle \\(\\theta _v\\) is also the one appearing in \\((\\vec{\\Omega}' - \\vec{\\Omega}) \\cdot \\vec{p} = p \\abs{\\vec{\\Omega}' - \\vec{\\Omega}} \\cos \\theta _v\\). \n\nThe angular integrals in \\(\\dd{\\phi _v}\\) and \\(\\dd{\\theta _v}\\) are standard, while for the integral in the modulus of \\(p\\) we need to use the actual expression of the Maxwellian: substituting \\(\\beta = p / mc\\) and then \\(z= \\beta / \\sqrt{\\Theta }\\) we get \n%\n\\begin{align}\nI_2 \n&= 2 \\pi x^2m^3 c^3 \\frac{2}{3} (2 \\pi m k_B T)^{-3/2} n_e \n\\int_0^{\\infty } \\dd{\\beta } \\beta^{4} \\exp(- \\frac{\\beta^2}{2 \\Theta })\n\\int \\dd{\\Omega } \\dv{\\sigma }{\\Omega } \\qty(\\vec{\\Omega}' - \\vec{\\Omega})^2 \\\\\n&= 2 \\pi x^2m^3 c^3 \\frac{2}{3} \\Theta^{5/2} (2 \\pi m k_B T)^{-3/2} n_e \n\\underbrace{\\int_0^{\\infty } \\dd{z } z^{4} \\exp(- \\frac{z^2}{2})}_{= 3 \\sqrt{2 \\pi } / 2}\n\\int \\dd{\\Omega } \\dv{\\sigma }{\\Omega } \\qty(\\vec{\\Omega}' - \\vec{\\Omega})^2 \\\\\n&= x^2 \\Theta n_e \n\\int \\dd{\\Omega } \\dv{\\sigma }{\\Omega } \\qty(\\vec{\\Omega}' - \\vec{\\Omega})^2\n\\,,\n\\end{align}\n%\nso we are left with only the integral in the photon space: in order to evaluate it we need to substitute the expression of the modulus of the difference of the unit vectors and the differential cross section: \n%\n\\begin{align}\n\\qty(\\vec{\\Omega}' - \\vec{\\Omega})^2 = 2 (1 - \\cos \\theta )\n\\qquad \\text{and} \\qquad\n\\dv{\\sigma }{\\Omega } = \\frac{r_0^2}{2} \\qty(1 + \\cos^2 \\theta )\n\\,,\n\\end{align}\n%\nso, also substituting the explicit expression for the radius \\(r_0 \\) we get \n%\n\\begin{align}\nI_2 &= n_e x^2 \\Theta \\int \\sin \\theta \\dd{\\theta } \\underbrace{\\dd{\\phi }}_{= 2 \\pi } \\frac{3 \\sigma _T}{16 \\pi } 2 (1 + \\cos^2 \\theta ) (1 - \\cos \\theta ) \\\\\n&= 2 n_e x^2 \\Theta \\sigma _T \\frac{3}{16 \\pi } 2 \\pi \\underbrace{\\int_{-1}^{1} (1+\\mu^2) (1 - \\mu ) \\dd{\\mu }}_{= 8/3} \\\\\n&= 2 \\sigma _T x^2 n_e \\Theta \n\\,.\n\\end{align}\n\nThe integral \\(I_1 \\) looks similar, and one might try to evaluate it similarly. This turns out to be tricky, and there is a better indirect way to do it. \n\nThe photon number density in phase space is given by \n%\n\\begin{align}\n\\frac{ \\dd{N}}{ \\dd{V} \\dd[3]{p}} = \\frac{2}{h^3} n\n\\,,\n\\end{align}\n%\nso, since the momentum of a photon is proportional to its energy and also to \\(x = h \\nu / k_B T\\) we can write \n%\n\\begin{align}\n\\dv{N}{V} \\propto \\int_0^{\\infty } \\dd{x} n x^2\n\\,.\n\\end{align}\n\n\\todo[inline]{If I'm interpreting this correctly, the slides use \\(x\\) with two different meanings in the same equation: one is position, and the other is the normalized energy. Really confusing if it is the case.}\n\nSince the medium is assumed to be homogeneous and isotropic, the photon number \\emph{density} is conserved: this means that \n%\n\\begin{align}\n\\dv{}{t} \\qty(\\dv{N}{V}) = 0 \n\\implies\n\\dv{}{t} \\int_0^{\\infty } \\dd{x} n x^2 = \\int_0^{\\infty } \\dd{x} \\pdv{n}{t} x^2 = 0\n\\,.\n\\end{align}\n\nNow, for reasons which will become clear in a moment, we make an assumption: suppose that there is a function \\(j(x, t )\\) such that the derivative of the photon density can be written as \n%\n\\begin{align}\n\\pdv{n}{t} = - \\frac{1}{x^2} \\pdv{[x^2 j(x, t)]}{x}\n\\,.\n\\end{align}\n\nIf this is the case, then the integral which we have just seen shown to equal zero can be written as \n%\n\\begin{align}\n\\int_0^{\\infty } \\dd{x} \\pdv{n}{t} x^2 = -\\int_0^{\\infty } \\pdv{[x^2 j(x, t)]}{x} = \\eval{- x^2 j(x, t) }_{0}^{\\infty } = 0\n\\,.\n\\end{align}\n\nThis gives us the boundary conditions for \\(j\\), which will have to be well-behaved at zero (non diverging) and at infinity (going to zero at least as \\(1/x^2\\)). \nNow, the Boltzmann equation's dependence on \\(n\\) and its derivatives is in the form \n%\n\\begin{align}\n\\pdv{n}{t} = C_1 (x ) n'' + C_2 (n, x) n' + C_3 (n, x)\n\\,,\n\\end{align}\n%\nand if we compare this expression with the one which defines \\(j\\)\\footnote{Seeing that \n%\n\\begin{align}\n\\pdv{n}{t} = - \\frac{1}{x^2} \n\\pdv{[x^2 j(x, t)]}{x}\n= - \\frac{2j}{x} - \\pdv{j}{x}\n\\,.\n\\end{align}\n%\n} we can prove that its dependence on \\(n\\) and \\(x\\) looks like \n%\n\\begin{align}\nj = g(x) \\qty[n' + h (n, x)]\n\\,,\n\\end{align}\n%\nfor some functions \\(g\\) and \\(h\\). In general \\(h\\) is a function of both \\(n\\) and \\(x\\), but we will assume it only depends on \\(n\\). \n\n\\todo[inline]{Why though? Is there a physical reason for it?}\n\nNow, we substitute this general expression for \\(j\\) back into the Boltzmann equation: this yields \n%\n\\begin{align}\n\\begin{split}\n&\\frac{1}{2} I_2 n'' + \\qty[(1+ n) I_2 + I_1 ]n' + \\qty(I_1 + \\frac{I_2}{2}) n (1+n) =\n\\\\\n&\\phantom{=}\\ = - g(x) n'' - \\qty( \\frac{2 g(x)}{x} + g(x) \\pdv{h}{n} + \\pdv{g}{x}) n' - \\qty(h \\frac{2 g(x)}{x} + g(x) \\pdv{h}{x} + h \\pdv{g}{x})\n\\end{split}\n\\,,\n\\end{align}\n%\nfrom which, identifying the terms, we can recover the expressions for \\(g\\), \\(I_1 \\) and \\(h\\): \n%\n\\begin{align}\ng &= - \\frac{I_2}{2} = - x^2 n_e \\sigma_T \\Theta \\\\\nI_1 &= - 2 \\frac{g}{x} - \\pdv{g}{x} - \\frac{I_2}{2} = (4-x) x n_e \\sigma_T \\Theta \\\\\nh &= n (1 + n)\n\\,.\n\\end{align}\n\nThis means that the form of \\(j\\) is \n%\n\\begin{align}\nj= g(x) \\qty[n' + h (n, x)] = - x^2 n_e \\sigma_T \\Theta (n' + n + n^2 )\n\\,,\n\\end{align}\n%\nwhich we can finally substitute into the Kompaneets equation: \n%\n\\begin{align}\n\\pdv{n}{t} = - c \\frac{1}{x^2} \\pdv{[x^2 j]}{x}\n&= n_e \\sigma_T c \\frac{\\Theta }{x^2} \\pdv{[x^{4} (n' +n + n^2)]}{x} \\\\\n\\pdv{n}{t_s} &= \\frac{\\Theta}{x^2} \\pdv{[x^{4} (n' + n + n^2)]}{x}\n\\,,\n\\end{align}\n%\nwhere we calculate time in units of the mean time between subsequent scatterings, \\(T = \\frac{1}{n_e \\sigma _T c} \\), so that \\(t_s = t / T\\).\n\\todo[inline]{There is now a \\(c\\) which was not there before\\dots}\n\nThis is the \\textbf{Kompaneets equation}. \nIn general this nonlinear equation must be solved numerically, it is an equation of the Fokker-Planck type, whose solution exhibit two main effects: \n\\begin{enumerate}\n \\item drift of the photon distribution to higher energies, this is called the \\textbf{secular change}, and is due to the fact that \\(I_1 \\sim 4 k_B T - h \\nu \\) (as long as the photons start out with energies lower than \\(4 k_B T\\));\n \\item broadening of the photon distribution, due to random walk effects: this is related to the integral \\(I_2 \\). \n\\end{enumerate}\n\n\\subsection{Stationary solutions}\n\nWe expect that, after the photons have interacted with the medium for a long time, their variation in time will become ever smaller, approaching a stationary solution which will satisfy \n%\n\\begin{align}\n0= \\pdv{[x^{4} (n' + n + n^2)]}{x}\n\\,,\n\\end{align}\n%\nmeaning that \n%\n\\begin{align}\nn' + n + n^2 = \\frac{A}{x^{4}}\n\\,.\n\\end{align}\n\nThe constant \\(A\\) must equal zero, since otherwise the solution will not be regular at zero: recall that \\(x^2 j = x^{4 }( n' + n +n^2)\\) must vanish at zero and infinity. \nThen, we can separate variables in order to integrate \\(n' + n + n^2 = 0\\): this yields \n%\n\\begin{align}\n\\log \\frac{n}{1 + n} = - x - \\mu \n\\,\n\\end{align}\n%\nfor a constant \\(\\mu \\),\nmeaning that \n%\n\\begin{align}\nn = \\frac{1}{ \\exp( x + \\mu ) - 1}\n\\,,\n\\end{align}\n%\nwhich is nice!\nWe found a Bose-Einstein distribution with an arbitrary chemical potential \\(\\mu \\): photons are at equilibrium with another chemical species. \n\nThe reason the photons do not achieve a Planck distribution (which is a Bose-Einstein distribution with \\(\\mu = 0\\)) is that their number must be conserved under our assumptions. \n\n\\todo[inline]{To understand better: the chemical potential quantifies how much the (energy?) changes if we change the number of particles, right?}\n\nThe Kompaneets equation is a \\textbf{parabolic} PDE, like the Fourier heat transport equation. \nThis means that we need to provide boundary conditions not only in the time domain, but also in the frequency domain for \\(x \\to 0\\) and \\(x \\to \\infty \\). We must set the distribution to zero in both regions, in order to avoid photons escaping through the boundary. \n\nIf the number of photons we initially insert in the model is smaller than the number of photons which would be contained in a blackbody at that temperature, then the evolution works out smoothly towards the blackbody distribution.\n\nOn the other hand, if we insert \\emph{more} photons than those which would be contained in the Planckian the numerical solution goes nuts. \n\n\\end{document}\n", "meta": {"hexsha": "633fa81e6327a3959660a59ef4b851846f6b4303", "size": 11003, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ap_second_semester/radiative_processes/apr16.tex", "max_stars_repo_name": "jacopok/notes", "max_stars_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-10-10T13:10:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T14:52:50.000Z", "max_issues_repo_path": "ap_second_semester/radiative_processes/apr16.tex", "max_issues_repo_name": "jacopok/notes", "max_issues_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ap_second_semester/radiative_processes/apr16.tex", "max_forks_repo_name": "jacopok/notes", "max_forks_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-03T16:20:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-06T16:11:07.000Z", "avg_line_length": 43.1490196078, "max_line_length": 427, "alphanum_fraction": 0.642188494, "num_tokens": 3839, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.34660599571124523}} {"text": "\\documentclass[Physics.tex]{subfiles}\r\n\\begin{document}\r\n\\chapter{Superposition}\r\nSuperposition is a property of linear systems where the net response at a given point and time caused by two or more stimuli is the sum of the responses which would have been caused by each stimulus individually.\r\n\r\nWhen contextualised to waves, the \\sldef{superposition principle} is as such: When two waves of the same kind meet at a point in space, the resultant displacement at that point is the vector sum of the displacements that the two waves would separately produce at that point.\r\n\\section{Interference}\r\n\\sldef{Interference} is the superposing of 2 or more waves of the same type, interacting according to the principle of superposition. \\sldef{Observable interference} is the superposing of 2 or more coherent waves to produce regions of maxima and minima in space, according to the principle of superposition.\r\n\r\n\\sldef{Coherent sources} are sources with the same frequency and a constant phase difference.\r\n\r\nFor interference to be observable, the sources must be coherent; the waves must have approximately equal amplitudes, and they must be polarised in the same plane, or unpolarised.\r\n\r\n\\sldef{Constructive interference} occurs when two or more waves arrive at the screen in phase with each other, such that the amplitude of the resultant wave is the sum of the amplitudes of the resultant waves.\r\n\r\n\\sldef{Destructive interference} occurs when two or more waves arrive at the screen \\(\\pi\\) out of phase with each other, such that the amplitude of the resultant wave is the minimum possible value.\r\n\r\nInterfering waves produce interference patterns with alternating maxima and minima. The zero order maximum is the brightest maximum; if there are multiple maxima with the highest amplitude, the zero order can be chosen as any. Successive maxima away from the 0 order are numbered the 1st, 2nd, 3rd and so on. Minima have the same order as the adjacent maximum away from the 0 order. This means that there is no 0 order minimum.\r\n\\subsection{One-source interference i.e. diffraction}\r\n\\sldef{Diffraction} is the bending or spreading of waves when they travel through a small opening. It is a special case of interference, and it occurs due to the Huygens-Fresnel principle, where each point along a slit is considered a source of spherical waves.\r\n\r\nIn diffraction, waves from the Huygens sources interfere with each other and produce a pattern with alternating maxima and minima, with minima at angles \\(\\theta\\) satisfying the equation \\begin{equation}a\\sin\\theta = m\\lambda\\end{equation} where \\(a\\) is the slit width and \\(m\\in\\mathbb{Z}^+\\) is the order of the minimum.\r\n\r\nGenerally, diffraction can only be observed when the size of the opening is approximately the same order as the wavelength or smaller.\r\n%\r\n%The intensity \\(I\\) of a diffraction pattern at distance \\(d \\propto\\sin\\theta\\) is \\[I = I_0\\left(\\frac{\\sin( \\pi a\\lambda^{-1}\\sin\\theta)}{\\pi a\\lambda^{-1}\\sin\\theta}\\right)^2\\] where \\(a\\) is the slit width.\r\n\\subsection{Two-source interference}\r\nTwo point sources of spherical waves can produce an interference pattern on a screen, with alternating maxima and minima. The zero order maximum occurs at the point on the screen that is equidistant from both sources (assuming the sources are in phase).\r\n\r\nThe separation between successive maxima is estimated by \\begin{equation}\\Delta x = \\frac{\\lambda D}{a}\\end{equation} for \\(D \\gg a\\), where \\(D\\) is the distance from the point sources to the screen, and \\(a\\) is the separation of the two point sources. If \\(D \\approx a\\) or \\(D < a\\), then Pythagoras' theorem must be used.\r\n\r\nThis phenomenon is investigated in Young's double slit experiment, where the two point sources are replaced by a monochromatic source of light passing through a single slit that acts as a point source, followed by a double slit that acts as two coherent point sources and causes light to diffract so that they can overlap and interfere.\r\n\r\nFor such a setup to produce a visible pattern, the double slits must be small enough for diffraction to occur; the single slit must be small enough so the light reaching the double slits is coherent; and the slit-screen separation must be at least one order larger than the slit separation so that the diffracted waves can overlap and interfere.\r\n\r\nThe path difference \\(\\Delta x\\) between the two waves reaching a maximum and minimum in two-source interference can be expressed in terms of the order of the maximum \\(n\\) or minimum \\(n\\). If the two sources are in phase, then for maxima \\begin{align}\\Delta x &= n\\lambda\\\\\\intertext{and for minima}\\Delta x &= (n - \\frac{1}{2})\\lambda\\end{align} and vice versa for sources in antiphase.\r\n%\r\n%The intensity \\(I\\) of a two-source interference pattern at distance \\(d \\propto \\sin\\theta\\) is \\[I = I_{0}\\left(\\frac{\\pi a\\sin\\theta}{\\lambda}\\right)\\] \\(I_0\\) can be expressed in terms of an unknown constant where required.\r\n\r\nTwo-source interference can be demonstrated with a ripple tank. Two dippers connected to a bar connected to a vibrator are set into vertical vibrations with the same amplitude and frequency and in phase. Each dipper produces circular waves that have the same frequency, which spread out and overlap, interfering constructively and destructively, and an interference pattern where there are points with water waves of maximum amplitude and points with minimum amplitude is formed.\r\n\r\nTwo-source interference can also be demonstrated with a microwave. Microwaves from a transmitter pass through two slits on an aluminium plate that act as coherent sources of microwaves. Microwaves emerging from the slits diffract and overlap, resulting in constructive and destructive interference. A microwave detector is moved along a line and the current registered will vary alternately from a maximum to a minimum, showing constructive and destructive interference. Since the wavelength of microwaves is about \\SI{3}{\\centi\\metre}, the slit size should be about \\SI{3}{\\centi\\metre}, and the slit separation should be about \\SI{30}{\\centi\\metre}, about \\(10\\lambda\\). The interference can be observed about \\SI{1}{\\metre} away.\r\n\\subsection{Many-source interference}\r\nA \\sldef{diffraction grating} is a plate with a large number of parallel, identically spaced slits of the same width. These multiple slits act as multiple coherent sources.\r\n\r\nMultiple sources of waves can produce an interference pattern similar to the previous two, with alternating maxima and minima. The zero order beam is the one parallel to the original beam of light.\r\n\r\nThe angle \\(\\theta\\) from the normal to the grating at which each maximum (or each beam) occurs satisfies the equation \\begin{equation}d\\sin\\theta = m\\lambda\\end{equation} where \\(d\\) is the slit separation, and \\(m\\) is the order of the maximum.\r\n\r\nThe maximum observable order for a planar screen parallel to the grating occurs at \\[0 < \\theta < \\SI{90}{\\degree} \\iff \\sin\\theta < 1\\] From the equation above, it can be derived that \\begin{equation}m < \\frac{d}{\\lambda}\\end{equation}\r\n\\section{Standing waves}\r\nWhen two waves of the same type, amplitude and frequency travel in opposite directions to each other and overlap, a standing wave will be formed due to superposition.\r\n\r\nStanding waves have points where constructive and destructive interference always occurs; they are known as \\sldef{antinodes} and \\sldef{nodes} respectively. Nodes are halfway between antinodes, and vice versa. Nodes have the minimum amplitude and intensity, while antinodes have the maximum amplitude and intensity.\r\n\r\nIn stationary waves, each point between consecutive nodes are in phase with each other i.e. they will reach the amplitude and equilibrium at the same time. Points to the between a node and the next node to the left are in antiphase with points between that node and the node to the right.\r\n\r\nThe distance between consecutive nodes or consecutive antinodes is half the wavelength of the original waves.\r\n\r\nStanding waves can be produced when a reflector reflects waves back in the direction they came from. In reality, however, reflected waves will not have the same amplitude, so perfect standing waves are not formed.\r\n%\r\n%The displacement-distance and time graph of a standing wave is given by \\(y = 2y_0\\sin kx\\sin\\omega t\\), where \\(k\\) is the wavenumber and \\(y_0\\) is the amplitude of the original waves. From this equation it can be seen that the amplitude at any point is \\(y\\prime_0 = 2y_0\\sin kx\\), where \\(x\\) is the distance from any node.\r\n\\subsection{Harmonics}\r\nA \\sldef{harmonic} of a wave is a frequency that is an integer multiple of the fundamental frequency.\r\n\r\nStanding waves can be produced with nodes at both ends of the wave. This occurs e.g. in a string fixed at both ends that is then plucked. The longest possible wavelength occurs when the two end nodes are consecutive nodes i.e. the distance \\(L\\) between the two nodes is half the wavelength. This mode is known as the 1st harmonic or the fundamental. The frequency, which is the fundamental frequency, is then \\[f_1 = \\frac{v}{2L}\\] The next possible standing wave occurs when there is one node between the two end nodes i.e. the distance \\(L\\) is the wavelength. This mode is the 2nd harmonic or the 1st overtone and has frequency \\[f_2 = \\frac{v}{L} = 2f_1\\] Therefore, the frequency and wavelength of the \\(n\\)th harmonic or \\((n-1)\\)th overtone are \\begin{align*}f_n &= nf_1\\\\\\lambda_n &= \\frac{2L}{n}\\end{align*}\r\n\r\nStanding waves can be produced with a node at one end and an antinode at the other. This occurs in a sound wave that enters an air column with one end closed, for example. The longest possible wavelength occurs also when the end node and antinode are adjacent i.e. \\[L = \\frac{1}{4}\\lambda\\] Thus the 1st harmonic or fundamental \\[f_1 = \\frac{v}{4L}\\] The next possible standing wave occurs when there is one node between the node and antinode i.e. \\[L = \\frac{3}{4}\\lambda\\] This is the \\emph{3rd} harmonic (since \\(f\\) is thrice that of \\(f_1\\)) or 1st overtone; \\[f_3 = 3\\frac{v}{4L} = 3f_1\\] Therefore, the frequency and wavelength of the \\(n\\)th harmonic or \\(\\frac{1}{2}(n-1)\\)th overtone are \\begin{align*}f_n &= nf_1\\\\\\lambda_n &= \\frac{4L}{n}\\end{align*}\r\n\r\nStanding waves can be produced with antinodes at both ends. This occurs in a sound wave that enters an air column open at both ends. The modes of operation for this are identical to standing waves with nodes as ends.\r\n\r\nIn general, closed ends are usually nodes as particles cannot vibrate there; conversely, open ends are antinodes as particles must vibrate there.\r\n\r\nFor sound waves, pressure nodes are displacement antinodes, and vice versa. At a pressure node, a microphone or ear will not detect sound as they sense pressure variations, not displacement.\r\n\r\nThe air at the end(s) of a pipe are generally free to move and so naturally vibrations will extend into the air outside the pipe i.e. the antinode is slightly beyond the open end. The extra length it extends is called the \\sldef{end correction}, and it is added to the length of the pipe i.e. \\[f_1 = \\frac{v}{4(L+e)}\\] for a pipe with one open end, and \\[f_1 = \\frac{v}{2(L+2e)}\\] for a pipe open at both ends.\r\n\\end{document}", "meta": {"hexsha": "d7598fdb7edf6f170033c45517a8ecab562689c3", "size": 11270, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "TeX/Physics/ch10_superposition.tex", "max_stars_repo_name": "oliverli/A-Level-Notes", "max_stars_repo_head_hexsha": "5afdc9a71c37736aacf3ae1db9d0384cdb6a0348", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-08-05T11:44:33.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-05T11:44:33.000Z", "max_issues_repo_path": "TeX/Physics/ch10_superposition.tex", "max_issues_repo_name": "oliverli/A-Level-Notes", "max_issues_repo_head_hexsha": "5afdc9a71c37736aacf3ae1db9d0384cdb6a0348", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "TeX/Physics/ch10_superposition.tex", "max_forks_repo_name": "oliverli/A-Level-Notes", "max_forks_repo_head_hexsha": "5afdc9a71c37736aacf3ae1db9d0384cdb6a0348", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 146.3636363636, "max_line_length": 818, "alphanum_fraction": 0.7714285714, "num_tokens": 2696, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.600188359260205, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.34660599571124523}} {"text": "\\documentclass[simplex.tex]{subfiles}\n% NO NEED TO INPUT PREAMBLES HERE\n% packages are inherited; you can compile this on its own\n\n\\onlyinsubfile{\n\\title{NeuroData SIMPLEX Report: Reduced Dimension Clustering}\n}\n\n\\begin{document}\n\\onlyinsubfile{\n\\maketitle\n\\thispagestyle{empty}\n\nThe following report documents the progress made by the labs of Randal~Burns and Joshua~T.~Vogelstein at Johns Hopkins University towards goals set by the DARPA SIMPLEX grant.\n\n%%%% Table of Contents\n\\tableofcontents\n\n%%%% Publications\n\\bibliographystyle{IEEEtran}\n\\begin{spacing}{0.5}\n\\section*{Publications, Presentations, and Talks}\n%\\vspace{-20pt}\n\\nocite{*}\n{\\footnotesize\t\\bibliography{simplex}}\n\\end{spacing}\n%%%% End Publications\n}\n\n\\subsection{Reduced Dimension Clustering}\n\nWe develop a new statistical method, named ``automatic repulsive\nclustering'' (ARC, to handle the clustering problem when the\ndimensionality outgrows the number of the data points. The standard\ndimension reduction tool such as principle component analysis (PCA) is\nnot optimal for finding the subspace best suitable for clustering.\nTherefore, we seek the method to do dimension reduction and clustering\nat the same time: we develop a regularization to encourage good\nseparation of the clusters, forcing the subspace to an angle so that\neach point can belong to a cluster completely. The plots below shows the\ndifferent performances in clustering a equally proportioned 3\ncomponents: the subspace found by PCA leads to incorrect clustering of\nthe data, due to the orientation of the space; whereas ARC successfully\nidentifies the ideal clustering direction in one dimensional space,\nhence generating satisfactory clustering result.\n\n\n\\begin{figure}[h!]\n\\begin{cframed}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{./figs/pcaBased.png}\n\\includegraphics[width=0.45\\textwidth]{./figs/ARC.png}\n\\caption{PCA (left) versus Reduced Dimension Clustering (right).}\n\\label{fig:arc}\n\\end{cframed}\n\\end{figure}\n\n\n\\end{document}\n", "meta": {"hexsha": "34865f25b34dd757a0b46afbfc5490f5559d05fc", "size": 1984, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Reporting/reports/2016-12Q4/reducedDimensionClustering.tex", "max_stars_repo_name": "openconnectome/SIMPLEX_Q2", "max_stars_repo_head_hexsha": "f10a6c4b9548670f9bf8e177914aa8d25fa1230b", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Reporting/reports/2016-12Q4/reducedDimensionClustering.tex", "max_issues_repo_name": "openconnectome/SIMPLEX_Q2", "max_issues_repo_head_hexsha": "f10a6c4b9548670f9bf8e177914aa8d25fa1230b", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Reporting/reports/2016-12Q4/reducedDimensionClustering.tex", "max_forks_repo_name": "openconnectome/SIMPLEX_Q2", "max_forks_repo_head_hexsha": "f10a6c4b9548670f9bf8e177914aa8d25fa1230b", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 33.0666666667, "max_line_length": 175, "alphanum_fraction": 0.7963709677, "num_tokens": 490, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3466059874514057}} {"text": "\\section{Coin and Multi-Asset Token algebras}\n\\label{sec:coin-ma}\n\nIn this chapter we introduce the concept of a \\emph{Token algebra},\nwhich is an abstraction used to generalize this specification to two\ndifferent eras depending on the Token algebra chosen.\n\n\\begin{definition}[Token algebra]\n A \\emph{Token algebra} is a partially ordered commutative monoid\n $T$, written additively, (i.e. a commutative monoid together with a\n partial order, such that addition is monotonic in both variables)\n together with the functions and properties as described in Figure\n \\ref{fig:TokenAlgebra}.\n\\end{definition}\n\n%%\n%% Figure ValMonoid and its Functions\n%%\n\\begin{figure}[htb]\n %\n \\emph{Functions}\n %\n \\begin{align*}\n & \\fun{coin} ~\\in~ T \\to \\Coin\\\\\n & \\text{Return the Ada contained inside the $\\mathsf{ValType}$ element}\n \\nextdef\n %\n & \\fun{inject} ~\\in~ \\Coin \\to T\\\\\n & \\text{Create a $\\mathsf{ValType}$ element containing only this amount of Ada}\n \\nextdef\n %\n & \\fun{policies} ~\\in~ T \\to \\powerset{\\PolicyID} \\\\\n & \\text{The set of policy IDs required for minting}\n \\nextdef\n %\n & \\fun{size} ~\\in~ T \\to \\MemoryEstimate\\\\\n & \\text{Return the size, in words, of a $\\mathsf{ValType}$ element}\n \\end{align*}\n %\n \\emph{Properties}\n %\n \\begin{align*}\n & \\fun{coin} \\circ \\fun{inject} = \\fun{id}_{\\Coin}\n \\end{align*}\n \\caption{Additional functions and properties required for a Token algebra}\n \\label{fig:TokenAlgebra}\n\\end{figure}\n\nA Token algebra is precisely the structure required to generalize the\n$\\Coin$ type of the Shelley specification in transaction outputs for\nthis ledger. We can then describe the ShelleyMA\ntransaction processing rules without fixing a concrete Token algebra.\n\nDepending on the Token algebra chosen, we obtain distinct ledgers. In\nparticular, we get\n\n\\begin{itemize}\n \\item the Allegra ledger rules with $\\Coin$, and\n \\item the Mary ledger rules with $\\Value$ (defined below).\n\\end{itemize}\n\nWhen multi-asset support on the ledger is introduced, Ada ($\\Coin$) will still be\nthe most common type of asset on the ledger, as the ledger rules enforce that\nsome quantity of it (specified via\nthe $\\fun{coinsPerUTxOWord}$ protocol parameter) must\nbe contained in every UTxO on the ledger.\nIt is the only\ntype of asset used for all non-UTxO ledger accounting, including deposits,\nfees, rewards, treasury, and the proof of stake protocol. For this reason, not\nall occurrences of $\\Coin$ inside a transaction or in the ledger state can or\nshould be replaced by the chosen Token algebra.\n\nBelow we give the definitions of all the functions that must be defined on\n$\\Coin$ and $\\Value$ in order for them to have the structure of a Token algebra.\nIn Section \\ref{sec:other-valmonoids} we give several other types which we can meaningfully\nsupport the definition of the required functions (addition, size, etc.), including\nan optimized representation that more accurately represents the implementation\nused in the Haskell implementation of the multi-asset type.\nThese types are convertible to and from the $\\Value$ type, and appear in other\nparts of the system as a multi-asset representation which is more suitable in particular\nuse cases.\n\n\\subsection{$\\Coin$ as a Token algebra}\n\nThis section defines the Token algebra structure for the $\\Coin$ type,\nsee Figure \\ref{fig:coin}. The structure of a partially ordered monoid\nis inherited from the (additive) integers.\n\nFor $\\Coin$, no policies are associated with $\\Coin$, since minting of Ada is not allowed.\n\n%%\n%% Figure Coin Functions\n%%\n\\begin{figure}[htb]\n \\begin{align*}\n & \\fun{coin} = \\fun{id}_{\\Coin}\n \\nextdef\n %\n & \\fun{inject} = \\fun{id}_{\\Coin}\n \\nextdef\n %\n & \\fun{policies}~v = \\emptyset\n \\nextdef\n %\n & \\fun{size}~v = 0\n \\end{align*}\n \\caption{The Token algebra structure of $\\Coin$}\n \\label{fig:coin}\n\\end{figure}\n\n\n\\subsection{Multi-assets and the Token algebra $\\Value$}\n\nElements of $\\Value$ represent heterogeneous collections of assets,\nboth user-defined and Ada. The Mary era ledger uses $\\Value$ as its Token algebra\nin order to support multi-assets.\n$\\Value$ and its Token algebra structure are given in Figure \\ref{fig:defs:value}.\n\n\\begin{figure*}[t!]\n \\emph{Derived types}\n %\n \\begin{equation*}\n \\begin{array}{r@{~\\in~}l@{\\qquad=\\qquad}lr}\n \\var{aname} & \\AssetName & \\mathsf{ByteString} \\\\\n \\var{pid} & \\PolicyID & \\ScriptHash \\\\\n \\var{adaID} & \\AdaIDType & \\{~\\mathsf{AdaID}~\\} \\\\\n \\var{aid} & \\AssetID & \\AdaIDType \\uniondistinct (\\PolicyID \\times \\AssetName) \\\\\n \\var{quan} & \\Quantity & \\Z \\\\\n \\var{v}, \\var{w} & \\Value & \\AssetID \\mapsto_0 \\Quantity\n \\end{array}\n \\end{equation*}\n %\n \\emph{Token algebra structure of $\\Value$}\n %\n \\begin{align*}\n & \\fun{coin}~\\var{v} = \\var{v}~\\mathsf{AdaID}\n \\nextdef\n %\n & \\fun{inject}~c = \\mathsf{AdaID}~\\mapsto_0~\\var{c}\n \\nextdef\n %\n & \\fun{policies}~v = \\{~\\var{pid}~\\vert~(\\var{pid,~\\wcard}) \\in \\supp v~\\}\n \\nextdef\n %\n & \\fun{size} ~~~ \\text{see Section \\ref{sec:value-size}}\n \\end{align*}\n \\caption{$\\Value$ and its Token algebra structure}\n \\label{fig:defs:value}\n\\end{figure*}\n\n\\begin{itemize}\n \\item $\\PolicyID$ identifies monetary policies. A policy ID $\\var{pid}$ is associated with a script\n $s$ such that $\\fun{hashScript}~s=pid$. When a transaction attempts to create or destroy assets\n that fall under the policy ID $\\var{pid}$,\n $s$ verifies that the transaction\n respects the restrictions that are imposed by the monetary policy.\n See sections \\ref{sec:transactions} and \\ref{sec:utxo} for details.\n\n \\item $\\AssetName$ is a byte string used to distinguish different assets with the same $\\PolicyID$.\n Each $aname$ identifies a particular kind of asset out of all the assets under the\n $\\var{pid}$ policy (but not necessarily among assets under other policies).\n The maximum length of this\n byte string is 32 bytes (this is not explicitly enforced in this specification).\n\n \\item $\\AssetID$ is either $\\mathsf{AdaID}$ or a pair of a policy ID and an asset name.\n It is a unique and permanent\n identifier of an asset. That is, there are is no mechanism to change it or\n any part of it for any assets.\n\n Mary MA assets are fungible with each other if and only if they have to the same $\\AssetID$.\n The reason the unique identifier is a pair of two elements (except for the non-mintable Ada case) is to allow\n minting arbitrary collections of unique assets under a single policy.\n\n \\item $\\mathsf{AdaID}$ is a special asset ID for Ada, different than all other asset IDs.\n It is a term of the single-term type $\\AdaIDType$.\n It does not include a policy, so instead, the validation outcome in the presence\n of Ada in the $\\fun{mint}$ field of the transaction is specified in the UTXO\n ledger rule. The rule disallows the $\\fun{mint}$ field to contain Ada.\n\n \\item $\\Quantity$ is an integer type that represents an amount of a specific $\\AssetName$. We associate\n a $q\\in\\Quantity$ with a specific asset to track how much of that asset is contained in a given asset value.\n\n \\item $\\Value$ is the multi-asset type that is used to represent\n a collection of assets, including Ada. This type is a finitely supported map.\n\n If $\\var{aid}$ is an $\\AssetID$ and $v \\in \\Value$,\n the quantity of assets with that assed ID contained in $v$ is $v~\\var{aid}$.\n Elements of $\\Value$ are sometimes also referred to as\n \\emph{asset bundles}.\n\\end{itemize}\n\nTo give $\\Value$ the structure of a partially ordered monoid, we define the\nrequired operations\npointwise, in accordance with the usual definitions of these operations on\nfinitely supported maps. See Figure \\ref{fig:pointwise}.\n\n\\begin{figure*}[t!]\n \\begin{align*}\n v + w &= \\{~ aid \\mapsto v~\\var{aid} + w~\\var{aid} ~\\vert~ \\var{aid} \\in \\fun{dom}~v \\cup \\fun{dom}~w ~\\} \\\\\n v \\leq w &\\Leftrightarrow \\forall~\\var{aid} \\in \\AssetID, v~\\var{aid} \\leq w~\\var{aid}\n \\end{align*}\n \\caption{Pointwise operations on Value}\n \\label{fig:pointwise}\n\\end{figure*}\n\n\\subsection{Special Ada representation}\nAlthough all assets are native on the Cardano ledger (ie. the accounting and\ntransfer logic for them is done directly by the ledger), Ada is still treated in a\nspecial way by the ledger rules, and is the most common type of asset on the ledger.\nIt can\nbe used for special purposes (such as fees) for which other assets cannot be used.\nThe underlying consensus algorithm relies on Ada in a way that\ncannot be extended to user-defined assets.\nAda can also neither be minted nor burned.\n\nNote that in the $\\Value$ definition above, we pick a special asset ID for Ada, that\nis not part of the type which represents the asset IDs for all other assets.\nCombining the asset name and policy for Ada gives a type-level guarantee that there is exactly\none kind of asset that is associated with it in any way, rather than\nderiving this guarantee as an emergent property of minting rules.\n\nAdditionally, not giving Ada an actual policy ID\n(that could have a hash-associated policy) eliminates the possibility\ncertain cryptographic attacks.\nWe sometimes refer to Ada as the primary or principal currency. Ada does not,\nfor the purposes of the Mary ledger specification, have a $\\PolicyID$ or an $\\AssetName$.\n\n\\subsection{Fixing a Token algebra}\n\nFor the remainder of this specification, let $\\ValMonoid$ be an\narbitrary but fixed Token algebra. As described above, choosing\n$\\Coin$ results in the Allegra ledger, and choosing $\\Value$ results\nin the Mary ledger.\n", "meta": {"hexsha": "34896122c06d8d23ceb5934fed9275cac9c0820b", "size": 9641, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "eras/shelley-ma/formal-spec/value.tex", "max_stars_repo_name": "MELD-labs/cardano-ledger", "max_stars_repo_head_hexsha": "9c3b4737b13b30f71529e76c5330f403165e28a6", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 108, "max_stars_repo_stars_event_min_datetime": "2019-03-24T02:26:41.000Z", "max_stars_repo_stars_event_max_datetime": "2021-10-30T05:27:16.000Z", "max_issues_repo_path": "eras/shelley-ma/formal-spec/value.tex", "max_issues_repo_name": "MELD-labs/cardano-ledger", "max_issues_repo_head_hexsha": "9c3b4737b13b30f71529e76c5330f403165e28a6", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1266, "max_issues_repo_issues_event_min_datetime": "2019-03-18T20:23:28.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-04T12:50:51.000Z", "max_forks_repo_path": "eras/shelley-ma/formal-spec/value.tex", "max_forks_repo_name": "MELD-labs/cardano-ledger", "max_forks_repo_head_hexsha": "9c3b4737b13b30f71529e76c5330f403165e28a6", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 86, "max_forks_repo_forks_event_min_datetime": "2019-03-29T06:53:05.000Z", "max_forks_repo_forks_event_max_datetime": "2021-10-04T17:17:15.000Z", "avg_line_length": 41.2008547009, "max_line_length": 115, "alphanum_fraction": 0.7165231822, "num_tokens": 2621, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7154239957834733, "lm_q2_score": 0.48438008427698437, "lm_q1q2_score": 0.3465371353713757}} {"text": "% !TeX root = ../main.tex\n% !TEX spellcheck = en_GB\n\n\\chapter{Design}\n\\label{ch:Design}\n\n\n\\section{Matlab models}\n\\subsection{Speaker}\n\n\\subsection{Drive unit}\n\n\\subsection{Closed Box}\nThis modelling is primarily based on \\citetitle[chap.~2.6,~p.~49-52]{Elektroakustik}\n\n\\subsubsection{Physical parameters}\n\\begin{itemize}\n\t\\item Volume of box $V_B$\n\t\\item Density of air $\\rho_0$ \\footnote{\\href{https://en.wikipedia.org/wiki/Density_of_air}{Wikipedia, Density of air}}\n\t\\item Speed of sound $c$\n\\end{itemize}\n\n\\begin{equation}\n\tC_{AB} = \\frac{V_B}{\\rho c^2}\n\\end{equation}\n\n\\subsection{Crossover filter}\n\n\\FloatBarrier", "meta": {"hexsha": "b9a210b66a53bfd3a4d844f20263725d39371302", "size": 620, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Report/Report/design.tex", "max_stars_repo_name": "lsangild/ETLYAK", "max_stars_repo_head_hexsha": "cf26993e2642b0f5529e7ec1f309446ee5c3cbfa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Report/Report/design.tex", "max_issues_repo_name": "lsangild/ETLYAK", "max_issues_repo_head_hexsha": "cf26993e2642b0f5529e7ec1f309446ee5c3cbfa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Report/Report/design.tex", "max_forks_repo_name": "lsangild/ETLYAK", "max_forks_repo_head_hexsha": "cf26993e2642b0f5529e7ec1f309446ee5c3cbfa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.3793103448, "max_line_length": 120, "alphanum_fraction": 0.7306451613, "num_tokens": 199, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.66192288918838, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3464639096279216}} {"text": "%% LyX 2.3.3 created this file. For more info, see http://www.lyx.org/.\n%% Do not edit unless you really know what you are doing.\n\\documentclass[12pt,letterpaper,reqno,english]{amsart}\n\\usepackage[T1]{fontenc}\n\\synctex=-1\n\\usepackage{babel}\n\\usepackage{amstext}\n\\usepackage{amsthm}\n\\usepackage[unicode=true,pdfusetitle,\n bookmarks=true,bookmarksnumbered=false,bookmarksopen=false,\n breaklinks=false,pdfborder={0 0 1},backref=false,colorlinks=false]\n {hyperref}\n\n\\makeatletter\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% LyX specific LaTeX commands.\n\\pdfpageheight\\paperheight\n\\pdfpagewidth\\paperwidth\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Textclass specific LaTeX commands.\n\\theoremstyle{definition}\n \\newtheorem{example}{\\protect\\examplename}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% User specified LaTeX commands.\n\\usepackage{tikz}\n\\usepackage{textgreek, enumitem}\n\\usepackage[T1]{fontenc}\n\\usetikzlibrary{decorations.pathreplacing, calc,fit,shapes, positioning,arrows, patterns}\n\n\n%\n\\@ifundefined{definecolor}\n {\\usepackage{color}}{}\n\\usepackage{amsfonts}\\setcounter{MaxMatrixCols}{30}\n\\usepackage{ifpdf} % part of the hyperref bundle\n\\ifpdf % if pdflatex is used\n % set fonts for nicer pdf view\n \\IfFileExists{lmodern.sty}\n {\\usepackage{lmodern}}{}\n\\fi % end if pdflatex is used\n\n\n\\providecommand{\\U}[1]{\\protect \\rule{.1in}{.1in}}\n\n\n\\providecommand{\\possessivecite}[1]{\\citeauthor{#1}'s\\nolinebreak[2]\n(\\citeyear{#1})}\n\n\n\\renewcommand{\\baselinestretch}{1.35}\n\\setlength{\\topmargin}{0.0in}\n\\setlength{\\textheight}{8.0in}\n\\setlength{\\evensidemargin}{0.20in}\n\\setlength{\\oddsidemargin}{0.20in}\n\\setlength{\\textwidth}{6.1in}\n\n\n\\usepackage{bbold}\n\\DeclareMathOperator{\\EX}{\\mathbb{E}}\n\\DeclareMathOperator{\\E}{\\mathbb{E}}\n\\DeclareMathOperator{\\Prob}{\\mathbb{P}}\n\\DeclareMathOperator{\\R}{\\mathbb{R}}\n\\DeclareMathOperator{\\N}{\\mathbb{N}}\n\\DeclareMathOperator{\\1}{\\mathbb{1}}\n\\DeclareMathOperator{\\0}{\\mathbb{0}}\n\n\\DeclareMathOperator{\\CU}{\\mathcal{U}}\n\\DeclareMathOperator{\\CG}{\\mathcal{G}}\n\\DeclareMathOperator{\\CQ}{\\mathcal{Q}}\n\n\\DeclareMathOperator{\\val}{val}\n\\DeclareMathOperator{\\marg}{marg}\n\\DeclareMathOperator{\\supp}{supp}\n\n\\newcommand{\\euler}{\\mathrm{e}}\n\n\\usepackage{setspace}\n%\\onehalfspacing\n\n\\makeatother\n\n\\providecommand{\\examplename}{Example}\n\n\\begin{document}\n\\title{TA Assignment Problem}\n\\author{Marcin Pęski}\n\\begin{abstract}\nThe goal is to formulate a TA assignment problem and discuss some\nproblems with tryig to solve it using stability-like notions. \n\\end{abstract}\n\n\\date{\\today}\n\\maketitle\n\n\\section{Model}\n\nLet $S$ be the set of students (i.e., candidate TAs). Let $C$ be\nthe set of courses. A (TA) assignment is a mapping $\\phi:C\\rightarrow\\R_{+}^{S}$\nWe refer to $\\phi\\left(c\\right)$ as the assignment to course $c$.\nIf $\\phi$ is an assignment, then for each $s$, define $\\phi\\left(s\\right)\\in\\R_{+}^{C}$\nby $\\phi\\left(s,c\\right)=\\phi\\left(c,s\\right)$. We refer to $\\phi\\left(s\\right)$\nas the assignment of student $s$. Let $\\Phi$ be the space of the\nassignments.\n\nLet $A_{S}=\\R_{+}^{S}$ be the space of possible assignments to courses.\nAnalogously, let $A_{C}=\\R_{+}^{C}$ be the space of possible assignments\nfor students. \n\nEach course $c$ has a subset of assignments $F_{c}\\subseteq A_{S}$\nthat are feasible for it. Examples of feasibility constraints include:\n\\begin{itemize}\n\\item budget: Each course has a budget of $h_{c}$ TA hours that need to\nbe allocated and assignment $\\phi\\left(c\\right)\\in A_{S}$ is feasible\nfor $c$ if and only if \n\\[\n\\sum_{s}\\phi\\left(c,s\\right)=h_{c},\n\\]\n\\item minimum assignment hours: there is a constraint $\\tau_{c}$ such that\nan assignment is feasible only if for each $s$, either $\\phi\\left(c,s\\right)=0$\nor $\\phi\\left(c,s\\right)\\geq\\tau_{c}$. \n\\end{itemize}\nSimilarly, each student $s$ has a subset $A_{s}\\subseteq A_{C}$\nof assignments that are feasible for them. Examples of student feasibility\nconstraints:\n\\begin{itemize}\n\\item maximum assignment: each student $s$ cannot work for more than $\\kappa_{s}$\nhours: \n\\[\n\\sum_{c}\\phi\\left(c,s\\right)\\leq\\kappa_{s},\n\\]\n\\item minimum assignment: student $s$ must work for at least $l_{s}$ hours,\n\\item balance across semesters: There is a set $M$ of ``semesters'' and\neach course $c$ is associated with a subset $M_{c}\\subseteq M$ of\n``semesters'' in which this course is taught. The student $s$ assigned\nhours in semester $m$ are equal to \n\\[\nh_{s}\\left(m;\\phi\\right)=\\sum_{c:m\\in M_{c}}\\frac{1}{\\left|M_{c}\\right|}\\phi\\left(c,s\\right).\n\\]\nIt is required that the student assignment across semesters is balanced:\nfor some constant $\\rho>1$, for each $m,m^{\\prime}$, \n\\[\n\\frac{h_{s}\\left(m,\\phi\\right)}{h_{s}\\left(m^{\\prime},\\phi\\right)}\\leq\\rho.\n\\]\n\\end{itemize}\nAn assignment $\\phi$ is \\emph{feasible} if $\\phi\\left(i\\right)\\in A_{i}$\nfor each $i\\in S\\cup C$. \n\nEach student $s$ has preferences over possible assignments represented\nby utility function $u_{s}:A_{C}\\rightarrow\\R$. Similarly, each course\n$c$ has preferences over assignments represented by $u_{c}:A_{S}\\rightarrow\\R$.\n\nAn assignment is individually rational (for students) if none of the\nstudents is better off by dropping some courses from its assignment\n(the logic is that students cannot be made to work for a course, and\na typical TA contract is written separately for each course. If TAs\nsign up for a bundle, i.e., for each of their assignment, the individual\nrationality would require that each student prefers their assignment\nto \\textbf{0} assignment). Formally, for each $s$, each $D\\subseteq C$,\n$u\\left(\\phi\\left(s\\right)\\right)\\geq u\\left(\\phi_{D}\\left(s\\right)\\right)$,\nwhere $\\phi_{D}$ is an assignment obtained from $\\phi$ by zeroing\nall courses not in $D$ $\\phi_{D}\\left(c,s\\right)=\\begin{cases}\n\\phi\\left(c,s\\right) & c\\in D\\\\\n0 & \\text{otherwise}\n\\end{cases}$. \n\nLet $\\Phi^{*}$ be the set of feasible and individually rational assignments. \n\nA social planner has preferences over $\\Phi^{*}$. \n\n\\section{Core and stability}\n\nSuppose that the social planner would like its assignment to be stable,\nin some sense. It is not clear what exactly stable assignment means\n- as there are many options available. Perhaps the most straightforward\nwould be t consider individually rational assignments that are not\nblocked by 2-agent (student and course) coalitions. But, with many-to-many\nassignments, 2-agent coalitions may not be sufficient and one may\nwant to think about some sort of core-type concept. \n\nInstead of focusing on a stability notion, I want to illustrate the\nproblem with few examples why the any stability-like notion can be\nproblematic. \n\n\\subsection{No externalities\\label{subsec:No-externalities}}\n\nSuppose that student preferences are additive: for each $s$, there\nexist constants $\\nu_{s}\\left(c\\right)$ such that for each $a\\in A_{C}$,\n\\[\nu_{s}\\left(a\\right)=\\sum_{c}a_{c}\\nu_{s}\\left(c\\right).\n\\]\nAdditionally, suppose that all assignments of student $s$ are subject\nto maximum constraint $\\kappa_{s}$. \n\nSimilarly, suppose that course preferences are additive: for each\n$c$, there exist constants $\\nu_{c}\\left(s\\right)$ such that for\neach $a\\in A_{S}$,\n\\[\nu_{c}\\left(a\\right)=\\sum_{s}a_{s}\\nu_{c}\\left(s\\right).\n\\]\nAdditionally, suppose that all assignments of course $c$ are subject\nto maximum constraint $h_{s}$. \n\nIn this environment, one can show that the core is non-empty. Moreover,\ncore allocations can be found by a version of Gale-Shapley algorithm.\n(Should I add this?)\n\n\\subsection{Courses that span across semesters}\n\nIt is well-known that any type of externality (or, more precisely,\ncomplementarities) in many-to-many matching problems leads to problems\nwith the existence of a stable outcome (or core).\n\nOne source of externality that reduces the use of stability as a satisfactory\nsolution concept is given by courses that span across multiple semesters. \n\\begin{example}\n\\label{exa:There-are-three}There are three students $x,y,z$, and\nfour courses $aF,aS,bY,cY$. Courses $bY$ and $cY$ are offered across\ntwo semesters, the other two courses are offered only in one semester.\nEach course needs exactly 1 TA, and each student wants at most 1 course\nper semester. All allocations must be either 1 or 0. (In this example,\npartial allocations would not help anyway.) Student preferences over\nbundles of courses are \n\\begin{align*}\nx: & \\left\\{ aF,aS\\right\\} >bY>aF>\\emptyset,\\\\\ny: & aF>cY>\\emptyset,\\\\\nz: & cY>aS>\\emptyset,\n\\end{align*}\nEssentially, student $x$ really likes course $aF$, but also needs\nmoney and would rather TA for $aF$ and $aS$ than for the whole-year\ncourse $Y$, but is willing to TA for $aF$ only if that's the only\noption. Course preferences are \n\\begin{align*}\naF: & x>y>\\emptyset\\\\\naS: & z>x>\\emptyset,\\\\\nbY: & x>\\emptyset,\\\\\ncY: & y>z>\\emptyset.\n\\end{align*}\nSo, $x$ and $aF$ are top matches for e\\textbackslash each other,\nbut otherwise, there are no obvious matches. \n\nWe show that there is no core. If $y$ is assigned to $aF$, then\n$z$ must be assigned to $cY$ (as that is the top assignment for\n$z$ and the second best assignment for $cY$ after $y$ is taken).\nBut then, given that the best remaining for $aS$ is $x$, $x$ and\npair $\\left\\{ aF,aS\\right\\} $ are a blocking coalition (as the student\nand all courses prefer to rematch with each other). \n\nIf $y$ is assigned to $cY$, then $z$ must be assigned with $aS$\n(as $z$ is the top choice for $aS$ and $aS$ is the top choice for\n$z$ aparat from $cY$ which prefers $y$. But then, since $aS$ prefers\n$z$, $bY$ is the best option for $x$. As a consequence $aF$ is\navailable for $y$ who prefers it to $cY$.\n\nIf $y$ is not assigned to anybody, then $z$ gets $cY$, but then\n$y$ and and $cY$ is a blocking pair. \n\\end{example}\n\n\\subsection{Preferences over the number of assignments}\n\nAnother source of problematic externality is when either students\n(or, resp., courses) have preferences over the total number of assigned\ncourses (resp. students). In fact, a reinterpretation of Example \\ref{exa:There-are-three}\nillustrates a difficulty:\n\\begin{example}\nThere are three students $x,y,z$, and four courses $a,a^{\\prime},b,c$.\nEach student has up to 2h of work available. Courses $b$ and $c$\nrequire 2h of work and they only want to hire the same TA. Courses\n$a,a^{\\prime}$ require 1h of work each. Otherwise, the preferences\nare similar to Example \\ref{exa:There-are-three}: for students \n\\begin{align*}\nx: & \\left\\{ a,a^{\\prime}\\right\\} >b>a>\\emptyset,\\\\\ny: & a>c>\\emptyset,\\\\\nz: & c>a^{\\prime}>\\emptyset,\n\\end{align*}\nand for courses:\n\\begin{align*}\na: & x>y>\\emptyset\\\\\na^{\\prime}: & z>x>\\emptyset,\\\\\nb: & x>\\emptyset,\\\\\nc: & y>z>\\emptyset.\n\\end{align*}\nThere is no stable matching.\n\\end{example}\n\n\\subsection{Minimum assignment}\n\nThere are $n\\geq1$ students and $n+1$ courses. Each student has\n$n+1$ hours and each course needs $n$ hours. The student $s$ utility\nfrom assignment is linear in assignments (like in Section \\ref{subsec:No-externalities}),\nwith coefficients equal to \n\\[\nv_{s}\\left(c\\right)=\\begin{cases}\n1+\\varepsilon & c=s\\\\\n1 & c\\neq s.\n\\end{cases}.\n\\]\nCourse $c$ utility from the assignment $a\\in A_{S}$ depends on the\nnumber of non-zero assignments and it equal to \n\\[\nu_{c}\\left(a\\right)=\\varepsilon1_{s=c}+\\begin{cases}\n1 & \\left|\\left\\{ s:a_{s}>0\\right\\} \\right|=1\\\\\n1-\\varepsilon & \\left|\\left\\{ s:a_{s}>0\\right\\} \\right|=2\\\\\n1-10\\left|\\left\\{ s:a_{s}>0\\right\\} \\right| & \\left|\\left\\{ s:a_{s}>0\\right\\} \\right|>2.\n\\end{cases}\n\\]\nThe first term is a small boost of the utility when the student has\nthe same index as he course. \n\nThen, the unique core allocation (or stable matching under any reasonable\ndeifnition) is when each student is matched for $n$ hours with the\ncourse with the same index and for remaining $1$ hours is assigned\nto course $n+1$. \n\nOf course, the above assignment is Pareto-optimal, but wildly inefficient. \n\\end{document}\n", "meta": {"hexsha": "0c5ff4662f3953ec4ef3c5eb1026f6e27e61b3cd", "size": 11751, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "theory/taallocation.tex", "max_stars_repo_name": "michaelpetersubc/TA-algorithm", "max_stars_repo_head_hexsha": "8bb52be8bb53ce29768f9acf80c8a8b77a07b05f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "theory/taallocation.tex", "max_issues_repo_name": "michaelpetersubc/TA-algorithm", "max_issues_repo_head_hexsha": "8bb52be8bb53ce29768f9acf80c8a8b77a07b05f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "theory/taallocation.tex", "max_forks_repo_name": "michaelpetersubc/TA-algorithm", "max_forks_repo_head_hexsha": "8bb52be8bb53ce29768f9acf80c8a8b77a07b05f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-27T17:23:27.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-27T17:23:27.000Z", "avg_line_length": 37.0694006309, "max_line_length": 93, "alphanum_fraction": 0.7248744788, "num_tokens": 3550, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.3464638956647564}} {"text": "\\section{Introduction} \\label{sec:introduction}\nWorst-case optimal join (WCOJ) algorithms, e.g. Leapfrog Triejoin, in the last few years turned conventional thinking about join processing\non its head because these multi-join algorithms have provable lower complexity than classical binary joins,\ni.e. join algorithms that join just two tables at-a-time.\nIn the areas of data warehousing and OLAP, this finding does not have much impact, though,\nsince the join patterns most commonly encountered there are primary-foreign-key joins, short (PK-FK joins),\nwhere the join shapes a tree or snowflake and contains no cycles.\nThe computational complexity of PK-FK joins is by definition linear in size of the inputs.\nIn these \\textit{conventional} cases, binary joins are worst-case optimal already, e.g. hash joins.\n\nHowever, analytical graph queries often use FK-FK joins which can grow over linearly in the size of their inputs,\nand often contain cycles.\nFor these use-cases, binary joins often exhibit highly suboptimal run-times because they generate a rapidly increasing set of\nintermediary results, e.g. when navigating a social graph with an out-degree in the hundreds.\nMany of these intermediary results are eliminated in later joins, e.g. a join that closes a cycle.\nHence, an algorithm which avoids generating these results in the first place is to perform much better and\ncloser to the optimal possible performance given by the output size.\n\\textsc{WCOJ} avoid many of the intermediary results and are guaranteed to reach the best possible run time in terms of\nthe output size~\\cite{agm}.\n\nBecause of the frequent presence of cyclic join-patterns in graph-pattern matching, we believe that worst-case optimal join algorithms\ncould be a useful addition to (analytical) graph database systems.\nTherefore, we aim to integrate a scalable, \\textsc{WCOJ} algorithm in Spark which is used by some modern graph engines~\\cite{caps,gcore,\ngraphframe}.\n\nThe rest of our introduction is structures as follows.\n\\Cref{subsec:graph-pattern-matching} defines the term graph pattern matching, its translation into datalog and\nrelational queries and two examples of cyclic graph pattern used in practice.\nWe aim to give the reader an intuitive understanding of why \\textsc{WCOJ}s are superior to binary joins for graph pattern matching\nin~\\cref{subsec:intuitive-example}.\n\\Cref{subsec:graphs-on-spark} motivates our choice to use Spark as the base for our thesis.\nNext, we state our research questions and contributions in~\\cref{subsec:research-questions-and-contributions}.\nFinally in~\\cref{subsec:thesis-idea}, we outline the main ideas behind the thesis and their connections.\n\n\\subsection{Graph pattern matching}\\label{subsec:graph-pattern-matching}\nGraph pattern matching is the problem of finding all instances of a specific subgraph in a graph.\nThe subgraph to find is described as a pattern or query.\nIn this thesis, we use datalog queries to define subgraph queries.\n\nFor example, \\cref{eqn:triangle} shows the datalog query describing a triangle.\n\n\\begin{equation}\n \\textit{triangle(a, b, c) $\\leftarrow$ R(a, b), S(b, c), T(c, a)} \\label{eqn:triangle}\n\\end{equation}\n\nHere we join three atoms $S, R$ and $T$ with two attributes each $(a, b)$, $(b, c)$ and $(a, c)$ respectively.\nThe task of enumerating all triangles within the three atoms can be also be described as finding all possible\nbindings for the join variables $a, b$ and $c$ within them.\n\nThe translation from datalog queries to graph patterns is straightforward.\nAn attribute or a variable refers to a vertice in a graph and an atom to an edge.\nA depiction of the subgraph pattern described by~\\cref{eqn:triangle} is shown in~\\cref{fig:pattern-triangle}.\n\n\\begin{figure}\n \\centering\n \\includesvg[width=0.2\\textwidth]{svg/triangle}\n \\caption{Depiction of the triangle subgraph query.}\n \\label{fig:pattern-triangle}\n\\end{figure}\n\nIn relational terms, a graph pattern matching query is an n-ary, conjunctive, self-equijoin on the edge relationship of the graph.\nIn this thesis, all join queries discussed belong to this subcategory of possible join queries.\nOther join queries can be useful to describe more complex graph patterns, e.g. disjunction for two edges of which only one needs to\nexist or negation to exclude instances that have too many connections.\nSome techniques used in this work can be extended to cover these cases, we mention related literature but do not focus\nour efforts on these extensions.\n\nGraph pattern matching is fundamental to analytical graph analysis workloads~\\cite{fraud-detection,flake2002,\nbodaghi2018automobile,newman2004detecting}.\nWe show two graph patterns which are used in practice below and explain the use-cases.\n\n\\Cref{fig:pattern-diamond} shows the diamond query which is used by Twitter to recommend their users new people to follow.\nThe idea is that if user $a$ is following multiple accounts $c_1, \\dots, c_k$ who all follow a person $b$ then it is likely that\n$b$ would be interesting to follow for $a$ as well.\nIn the figure, we see the diamond query for $k = 2$.\nThis is the diamond query as discussed in most papers in academia~\\cite{olddog,myria-detailed,mhedhbi2019}, although,\nTwitter uses $k = 3$ in production~\\cite{twitter-diamond}.\n\n\\begin{figure}\n \\centering\n \\includesvg[width=0.2\\textwidth]{svg/diamond}\n \\caption{The dimaond query is used by Twitter. The vertices are users and the edges follower relationships.\n In the example, they could recommend $A$ to follow $D$ because $A$ follows $B_1$ and $B_2$ which both follow $C$.\n }\n \\label{fig:pattern-diamond}\n\\end{figure}\n\nOur second concrete use-case example is the n-cycle.\nAs explained in~\\cite{fraud-detection}, cycles can be used to detect bank fraud.\nA typical bank-fraud often involves so-called \\textit{fraud-rings}.\nThese are two or more people who combine their legitimate contact information in new ways to craft multiple false identities.\nFor example, two people share real phone numbers and addresses to craft four fake identities; all combinations possible with two pieces\nof information.\nThey open accounts under wrong names with real contact information, use these accounts normally to build trust with the bank and\nbuild up bigger credit lines.\nAt a certain date, they max out all credit lines and disappear.\nThe phone numbers are dropped and the actual people living at the addresses deny ever knowing the identities that opened the accounts.\n\nThis scheme can be detected using graph pattern matching.\nLet us assume, we have a graph database in which customer of the bank, their addresses and phone numbers are all vertices and the\nrelationship of an address or phone number belonging to a customer are edges.\nThen, the case described above forms an 8-cycle of 4 persons (fake identities) connected by the shared use of phone numbers and\naddresses.\nThe imagined cycle is shown in~\\cref{fig:graph-pattern-example-bank-fraud}.\n\n\\begin{figure}\n \\centering\n \\includesvg[width=0.2\\textwidth]{svg/bank-fraud}\n \\caption{Schematics of a bank fraud ring.\n 2 fraudsters share their phone numbers and addresses (labelled $P$ and $A$) to create four\n fake customers ($C$ vertices).\n }\n \\label{fig:graph-pattern-example-bank-fraud}\n\\end{figure}\n\n\n\\subsection{Binary joins vs \\textsc{WCOJ}s: an intuitive example} \\label{subsec:intuitive-example}\nWe introduce the triangle query and possible binary join plans.\nThen we point out the general problem of binary join plans on this query and the idea of how \\textsc{WCOJ}s can improve the situation.\nNext, we give a concrete example of a database instance to illustrate the aforementioned problem.\nWe conclude our motivation to use worst-case optimal joins by reporting multiple papers that show that these joins are highly beneficial\nto graph pattern matching queries in practice.\n\nThe simplest example of a cyclical join query enumerates all triangles in a graph.\nIt is shown in~\\cref{eqn:triangle} and \\cref{fig:pattern-triangle}.\n\nTraditionally, this would be processed by using multiple binary joins:\n\\begin{equation}\n R \\bowtie S \\bowtie T\n\\end{equation}\nThe join above can be solved in 3 different orders: $ (R \\bowtie S) \\bowtie T$, $ (R \\bowtie T) \\bowtie S$ and\n$ R \\bowtie (T \\bowtie S)$.\nIndependent of the chosen order, database instances exist where the intermediary result size is in $\\mathcal{O}(N^2)$ with\n\\textit{N}= |\\textit{R}| = |\\textit{S}| = |\\textit{T}|.\nHowever, it is provable that the output of this query is guaranteed to be in $\\mathcal{O}(n^{3/2})$~\\cite{agm,skew-strikes-back}\nfor any database instance.\nHence, binary joins materialize huge intermediary results after processing parts of the query,\nwhich are much bigger than the final result.\n\nThe described problem is a fundamental issue with traditional binary join plans~\\cite{agm,skew-strikes-back}.\nWe call these plans also \\textit{join-at-a-time} approach because they process whole joins at the time.\n\nFortunately, worst-case optimal join algorithms can materialize cyclic joins with memory usage linear to their output size\nby solving the join \\textit{variable-at-a-time} which avoids materializing big intermediary results~\\cite{lftj,nprr}.\n\nIn variable-at-a-time the algorithm finds a binding for the first variable $a$, then one for $b$ and\nfinally one for $c$.\nAfter this, it emits the tuple as part of the output.\nThen it finds further bindings via backtracking until they enumerated the whole join when all bindings for $a$ have been explored.\n\n% Example concrete\nA simple example graph database instance gives an idea of why a variable-at-a-time approach is beneficial for cyclic queries.\nIn \\cref{fig:edge-rel-example}, we see an edge relationship.\nIt is repeated three times labelled with different attributes to ease the understanding of the following explanation;\nhowever, in a system's implementation, only one table exists and is used by all joins as input.\n\n\\begin{figure}\n \\centering\n \\subfloat{\n $R$\n \\begin{tabular}{rr}\n \\toprule\n a & b \\\\\\midrule\n 1 & 2 \\\\\n 2 & 7 \\\\\n 2 & 8 \\\\\n 2 & 9 \\\\\n 2 & 10 \\\\\n 3 & 2 \\\\\n 4 & 2 \\\\\n 5 & 2 \\\\\n 6 & 11 \\\\\n 11 & 12 \\\\\n 12 & 6 \\\\\\bottomrule\n \\end{tabular}\n }\n \\hspace{0.2\\textwidth}\n \\subfloat{\n $S$\n \\begin{tabular}{rr}\n \\toprule\n b & c \\\\\\midrule\n 1 & 2 \\\\\n 2 & 7 \\\\\n 2 & 8 \\\\\n 2 & 9 \\\\\n 2 & 10 \\\\\n 3 & 2 \\\\\n 4 & 2 \\\\\n 5 & 2 \\\\\n 6 & 11 \\\\\n 11 & 12 \\\\\n 12 & 6 \\\\\\bottomrule\n \\end{tabular}\n }\n \\hspace{0.2\\textwidth}\n \\subfloat{\n $T$\n \\begin{tabular}{rr}\n \\toprule\n c & a \\\\\\midrule\n 1 & 2 \\\\\n 2 & 7 \\\\\n 2 & 8 \\\\\n 2 & 9 \\\\\n 2 & 10 \\\\\n 3 & 2 \\\\\n 4 & 2 \\\\\n 5 & 2 \\\\\n 6 & 11 \\\\\n 11 & 12 \\\\\n 12 & 6 \\\\\\bottomrule\n \\end{tabular}\n }\n \\caption{\n Three aliases to an edge relationship which contains three triangles, the permutations of \\{6, 11, 12\\},\n and one skewed value.\n }\n \\label{fig:edge-rel-example}\n\\end{figure}\n\nA binary join plan which joins $R$ and $S$ via $b$ first produces $16 + 3$ intermediary results;\n4 times 4 results for $b = 2$ and one for 6, 11, 12 each.\nThe next join reduces these 16 results to the three triangle instances; all permutations of the set \\{6, 11, 12\\}.\n\nA variable-at-a-time approach finds 4 bindings for $a$, namely $2, 6, 11, 12$;\nthe intersections of both columns labelled $a$.\n\nIntersecting both columns of $b$ values we notice $2, 6, 11, 12$ could be possible bindings for $b$.\nWhen we fix an $a$ value these four possibilities are reduced to the $b$ values which exist for this\n$a$ value in the leftmost table.\nSo once we fixed a binding for $a$, we find one possible binding for $b$ each;\nexcept the binding $a = 2$ for which we cannot find a matching $b$ value.\n\nFinally, we find all three instances of the triangle by completing the three $a, b$ bindings with\nthe matching $c$ binding;\nonly one exists for each $a, b$ binding.\n\nWe can drastically reduce the workload by formulating the join as a problem of\nfinding variable bindings using information from all parts of the join, instead of, using only one constraint at the time\nand building it join-by-join.\n\nWe do not claim that the example above illustrates the generality of why binary join plans are provable worse than\n\\textsc{WCOJ}s.\nClearly, the example does not show an intermediary result of $N^2$ as $N = 11$ and the intermediary result has the size of 16.\nHowever, we note that even in such a simple example all possible binary join orders produce an intermediary result of size 16.\nWhile all possible variable orderings for a variable-at-a-time approach eliminate the skewed value (2) after finding no binding\nfor the second variable.\nA more general but less concrete example is explained in~\\cite{skew-strikes-back}.\n\n% WCOJ's in practice\nIn practice, these worst-case optimal join algorithms are highly beneficial for cyclic queries in analytical graph\nworkloads in an optimized, single machine system~\\cite{lftj,olddog}.\n\\cite{olddog} compares a system using \\textsc{WCOJ}s against multiple general-purpose database\nsystems using binary joins and some graph pattern matching engines on 15 datasets and 7 queries and\nfinds that worst-case optimal joins can beat all other systems in the vast majority of queries\nand datasets, often by the order of magnitudes or even being the only system to finish within 30 minutes.\n\nLater worst-case optimal joins have been applied successfully to a distributed shared-nothing settings~\\cite{myria-detailed,\nammar2018distributed};\nwe describe these systems in more detail in~\\cref{subsec:myria} and~\\ref{subsec:wcoj-timely-data-flow}.\n\n\\subsection{Graphs on Spark}\\label{subsec:graphs-on-spark}\nSpark is an attractive target for big graph processing, due to its generality, widespread acceptance in the industry, the ability to use\ncloud hardware and its fault tolerance by design.\nFor example, GraphFrames~\\cite{graphframe}, GraphX~\\cite{graphx} (a Pregel~\\cite{pregel} implementation) or graph query languages\nas \\mbox{G-CORE}~\\cite{gcore} and \\mbox{openCypher} with `Cyper for Apache Spark' or \\textsc{CAPS}~\\cite{caps} all aim to ease graph\nprocessing on Spark.\nThe last two technologies translate their graph specific operations to the relational interface of Spark (SparkSQL)\nto profit from Spark's relational query optimizer Catalyst~\\cite{spark-sql}.\nMoreover, they allow the user to formulate graph pattern matching queries naturally.\n\nHence, we believe that the \\textsc{WCOJ}s, with their efficiency for analytical graph queries, are a valuable addition to Spark's\nbuilt-in join algorithms in general and these graph-on-spark systems in particular.\nIdeally, they are integrated such that they can be naturally used in the ecosystem of Catalyst.\nThis would allow easier use in SQL like graph languages as \\textsc{G-CORE} or Cypher for graph pattern matching.\n\n\\subsection{Research questions and contributions}\\label{subsec:research-questions-and-contributions}\nWe identify two challenging, novel directions for our research.\nFirst, all papers about \\textsc{WCOJ} focus on queries widely used in graph pattern matching, e.g. clique finding or path queries.\nAs explained above, graph pattern matching uses only self-joins on a single relationship with two attributes\nnamely the edge relationship of the graph.\nHowever, all systems use worst-case optimal joins developed for general n-ary joins.\nThis raises the question if and how \\textsc{WCOJ}s can be specialized for graph pattern matching.\n\nSecond, while the communication costs for worst-case optimal joins in MapReduce like systems\\footnote{\nAn excellent definition of the term MapReduce like systems is given in~\\cite{shares}}\nis well-understood~\\cite{shares,shares-skew,shares-proof,shares-skew-proof},\ntheir scalability has not been studied in depth.\nGiven that the only integration in a MapReduce like system exhibits a speedup of 8 on 64 nodes over two workers (an efficiency of 0.125)\n~\\cite{myria-detailed},\nwe find that designing a scalable, distributed \\textsc{WCOJ} for a MapReduce like system is an unsolved challenge.\n\nIt is time to investigate how these algorithms scale in the provable most widely used, general-purpose big data processing engine: Spark.\nTo the best of our knowledge, this is also the first time a worst-case optimal join is integrated with an industrial-strength cluster\ncomputing model.\nWe detail our research questions below.\n\n\\begin{enumerate}\n \\item Can we gain performance in \\textsc{WCOJ}s by specializing them to graph pattern matching?\n \\begin{enumerate}\n \\item How much performance can we gain by using compressed sparse row representations as backing data structure to \\textsc{WCOJ}s?\n \\item Can we find a more suitable algorithm to build intersections for graph-pattern matching than the complex n-ary approach\n proposed originally?\n \\end{enumerate}\n \\item How well do \\textsc{WCOJ}s scale in Spark when used for graph pattern matching?\n \\begin{enumerate}\n \\item How well does a previously proposed, optimal partitioning scheme, named Shares, scale?\n We explain Shares in detail in~\\cref{subsubsec:shares}.\n \\item How to integrate scalable work-stealing into Spark to counter tuple replication and skew?\n \\end{enumerate}\n\\end{enumerate}\n\nTowards answering our research questions, we make the following contributions.\n\\begin{enumerate}\n \\item We integrate a sequential, general worst-case optimal join into Spark.\n This implementation serves as a baseline for our \\textsc{WCOJ} optimized to graph pattern matching.\n \\item We design and implement GraphWCOJ which is a worst-case optimal join specialized to graph pattern matching.\n It is backed by a compressed sparse row representation of the graph which reduces its memory footprint and speeds up execution by up\n to 11 times over a normal \\textsc{LFTJ} because it acts as an index. % NUMBER\n Furthermore, we exploit the typical low out-degree of most graphs to by specializing the \\textsc{LFTJ} for small intersections.\n \\item We analyse how many tuples Shares replicates for typical graph pattern matching queries.\n From this analysis and the fact that Shares is an optimal partitioning scheme, we conclude that replication is inevitable for complex\n graph-pattern matching queries.\n Therefore, we cache the graph in the memory of all workers.\n \\item Based on a replicated edge relationship, we design \\textit{logical} Shares.\n This is an approach where the graph is fully replicated but we use Shares partitioning to divide work between executors.\n We measure a speedup of 13 on 64 workers for some queries and beat an existing implementation of (physical) Shares which reaches\n a speedup of 8 for the same number of machines.\n The results show that Shares is good in dealing with skew but requires too much replicated work to scale well.\n \\item Therefore, we abandon static partitioning and design a \\textsc{WCOJ} that applies work-stealing.\n We show that work-stealing can scale linearly on some input queries and beats \\textit{logical} Shares for all levels of parallelism\n \\item for 3-cliques and 5-cliques on three different datasets.\n \\item We run experiments on 5 datasets, 6 queries, for up to 382 workers using Spark's build-in hash join,\n a general Leapfrog Triejoin and our specialized GraphWCOJ.\n\\end{enumerate}\n\n\\subsection{Thesis overview} \\label{subsec:thesis-idea}\nIn this section, we outline the main ideas, motivation and decisions taken in this thesis.\nWe summarize the whole thesis as a graph in~\\cref{fig:thesis-overview}.\nWe see the background that motivates our decision in the corners and our system in the centre.\nThe edges show connections between different ideas and components.\nWe give an overview of the thesis in the next paragraphs.\n\n\\begin{figure}\n \\centering\n \\includesvg[width=\\textwidth]{svg/thesis-overview}\n \\caption{\n Main ideas and components of the thesis.\n Background and related work shown in the corners.\n The center shows the main component of our parallized worst-case optimal join\n }\n \\label{fig:thesis-overview}\n\\end{figure}\n\nAs mentioned in~\\cref{subsec:graphs-on-spark}, Spark is a good platform for our work because many graph pattern matching systems use\nit, like neo4j's Cypher on Apache Spark (CAPS) and LDBC's G-CORE~\\cite{caps,gcore}.\nIn particular, they build on top of Spark's structured query execution offered by Catalyst.\nCatalyst is designed to be easily extendable and allows to introduce new operators, such as a worst-case optimal join,\nwithout modifying the core of Spark.\nThis is even possible in a way such that these new operators can be used with a native, unchanged installation of Spark.\nWe describe Catalyst query compilation process in~\\cref{subsubsec:catalyst}.\nOur integration is detailed in~\\cref{sec:spark-integration}.\n\nNormally, Spark achieves parallelism and distributed algorithms by partitioning data over all workers.\nWhen data from different tables needs to be joined, Spark repartitions the data such that each worker can process parts of the join locally.\nHowever, shuffling is an expensive operation in Spark, involving disk writes and reads, which should be avoided if possible.\nMoreover, we can show that a communication-optimal partitioning scheme degenerates into a full broadcast for bigger graph pattern queries;\nwe explain this in detail in~\\cref{subsubsec:shares}.\n\nGiven this finding, we design our system to build on a replicated and cached edge relationship on each worker.\nAs a short study of us reveals, most large graph problems described in literature, have edge structures that would fit in\nmain-memory (see~\\cref{subsec:graph-analysis});\nand our compressed storage format further helps to keep memory usage under control.\nThis has a few distinct advantages.\n\nFirst, the broadcast can be done once at system startup.\nThen, we can reuse it for any graph pattern matching query;\nall of them need to join over the edge relationship many times.\nTherefore, we can answer many graph pattern matching queries without shuffling data.\nWe explain the integration of replicated edge relationships into Spark in~\\cref{subsec:spark-integration-graphWCOJ} and introduce\nthe necessary background in~\\cref{subsubsec:broadcast-variables}.\n\nFurthermore, such reuse helps to amortize the non-trivial setup costs for worst-case optimal joins;\nthey require their input data to be sorted.\n\nFinally, a replicated data structure allows us to use dynamic work-sharing schemes as work-stealing cheaply without relocating\ndata.\nNormally, partitioning is done completely statically in Spark.\nHowever, this is problematic given that many real-world graphs are highly skewed, e.g. power-law graphs such as many follower graphs\n(Facebook, Twitter) or web graphs.\nWe find that this skew can easily lead to bad load-balancing with the static partitioning of Spark.\nWe explain how to integrate work-stealing with Spark in~\\cref{subsec:work-stealing}.\n\nUsing a fully replicated edge relationship and potentially work-stealing leads to the necessity to build data-partitioning into\nthe worst-case optimal join operators.\nThis is because Spark would normally have an operator work on all local data and archive parallelism via physically partitioning the\ndata over multiple workers.\nWe call partitioning built into our operators \\textit{logical} partitioning.\nThe concept and its implementation are described in~\\cref{sec:worst-case-optimal-join-parallelization}.\n\nWe choose to use the Leapfrog Triejoin~\\cite{lftj} as a basis for our system;\nthis choice is motivated in~\\cref{subsec:worst-case-optimal-join-algorithm}.\nThis join requires its input relationships to be presented in a sorted data structure which is searchable for upper bounds in\n$\\mathcal{O} (\\log N)$.\nFurthermore, it mainly uses intersections to compute the join.\nThe algorithm is explained in detail in~\\cref{subsubsec:leapfrog-triejoin}.\n\nWe specialize the Leapfrog Triejoin to graph pattern matching by introducing a compressed sparse row representation~(\\textsc{CSR}, see\n\\cref{subsec:csr-background}) as backing data structure for the input relationships.\n\\textsc{CSR} can compress the graph edge relationship by a compression factor of nearly 2.\nAdditionally, we show that it speeds up the \\textsc{WCOJ} execution to be backed by a \\textsc{CSR} because this representation\nacts like an index.\n\nAnother graph specific optimization we apply to \\textsc{LFTJ} is that we change the intersection building algorithm for one\nthat is specialized in small intersections.\nThis is motivated by the fact that real-world graphs have normally small average out degrees.\nHence, the intersection of multiple adjacency lists is predictably small.\nWe discuss both specializations to the Leapfrog Triejoin algorithm in~\\cref{sec:graphwcoj}.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n% Also introduce cached edge table, ref related work for why communication does not work\n% and goals for explanation of why we think caching is helpful\n% use mcsherry, other guy here? --> read them\n\n% These academic systems are not very usable nor used, nor is LogicBlox on the market as a database system.\n% For all practical senses and purposes, there are no %systems available that implement WCOJs. Apache Spark is currently the most popular\n% analytical data processing system. It does not implement WCOJs yet and has %multiple popular graph processing APIs or subsystems, among\n% which GraphFrames, CAPS (neo4j's Cypher on Apache Spark) and the recent LDBC effort to implement the G-%CORE query language on Apache\n% Spark. All of these APIs could potentially benefit greatly from a WCOJ algorithm.\n\n%Spark offers a well optimized Relational interface [SparkSQL] [Catalsyst]\n%Relational interfaces rely on JOINS which have different characteristics different for graphs than for traditional star or snowflake schemes.\n% - they are cyclic for important graph algorithms (cluster, ?subgraphing?)\n% - large intermediary results which make the queries really expensive for the CPU as well as the in memory\n% - work is done for nothing because most of the intermediary results are filtered out later\n% - they are highly selective (paths starting from a specific node)\n% - allowing to safe work when all \"filters\" are applied simultaneously\n% - allowing for big jumps on sorted keys with a seek operation, naturally, applied by LFTJ\n% - therefore, they are a prime area of application for a new class of join algorithms with worst-case guarantees, which guarantee that no big intermediary results build up\n%because the evaluate multiple joins as once only materializing results that fulfil all join filters.\n% - Furthermore, they are naturally suited for highly selective queries because of using an O(log(N)) seek a method to jump over \"uninteresting\" parts of a sorted result. \n", "meta": {"hexsha": "de634280c8b56fc8611eda69bc4707f6d29df8d4", "size": 27101, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "introduction.tex", "max_stars_repo_name": "PerFuchs/master-thesis", "max_stars_repo_head_hexsha": "85386c266fecf72348114bcbafeeb896a9e74601", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-11-02T20:23:03.000Z", "max_stars_repo_stars_event_max_datetime": "2019-11-02T20:23:03.000Z", "max_issues_repo_path": "introduction.tex", "max_issues_repo_name": "PerFuchs/master-thesis", "max_issues_repo_head_hexsha": "85386c266fecf72348114bcbafeeb896a9e74601", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "introduction.tex", "max_forks_repo_name": "PerFuchs/master-thesis", "max_forks_repo_head_hexsha": "85386c266fecf72348114bcbafeeb896a9e74601", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.0435729847, "max_line_length": 172, "alphanum_fraction": 0.7690860116, "num_tokens": 6409, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318337259584, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.34646265258178166}} {"text": "%$Id: Oral.tex 82 2007-06-08 14:32:56Z jlconlin $\n%$Author: jlconlin $\n%$Date: 2007-06-08 08:32:56 -0600 (Fri, 08 Jun 2007) $\n%$Revision: 82 $\n\n\\documentclass[11pt]{article}\n\\usepackage{geometry}\n\\usepackage[algo2e, ruled, linesnumbered]{algorithm2e}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\n\\SetKwComment{Comment}{$\\triangleright$ }{}\n\\dontprintsemicolon\n\n%\\renewcommand{\\span}{\\mathrm{span}}\n\n\\author{Jeremy Conlin}\n\\title{Oral Thesis Prospectus}\n\\date{June 6, 2007}\n\n\\begin{document}\n\\maketitle\n\\abstract{\n The power method, or source iteration, has been the staple of Monte Carlo particle transport eigenvalue calculations for more than fifty years. This method is easy to implement and provides the dominant eigenvalue and eigenfunction, however, this method is slow. The rate of convergence is proportional to the dominance ratio, the ratio of the second largest eigenvalue to the largest eigenvalue.\n\n For this thesis, I propose using an alternative to the power method, Arnoldi's Method of Minimized Iterations. Arnoldi's method is an improvement over the power method; it uses information from all previous iterations---instead of only the last iteration---to calculate the eigenfunction. \n \n Preliminary results indicate that Arnoldi's method is less noisy than the Power method and the eigenvalue and the eigenfunction are converged more quickly. However, the RMS error of the eigenfunction---compared to a deterministic SN calculation---calculated with Arnoldi's method is larger than the RMS error from a power method calculation for the same number of Monte Carlo histories. Other results show that Arnoldi's method can calculate higher eigenmodes which the power method is unable to do without additional calculations.\n\n Several questions must be explored before the use Arnoldi's method in Monte Carlo particle transport can be accepted. Most importantly, how can the uncertainty be calculated? Without a measure of the statistical uncertainty, the answer is meaningless. A simple solution is to restart the method after a few iterations and find the mean and variance of the eigenvalues after each restart. I believe there is a better and more efficient way; this is one aspect of my thesis research.\n\n Arnoldi's method requires the calculation of an inner product between two functions. In Monte Carlo particle transport, these functions are fission sources. In initial studies, these fission sources were spatially binned; the height of these bins are the elements of a vector. Discretizing a fission source in this manner introduces bias to the calculation. The inner product of two vectors is a simple dot product. Is there a way to calculate the inner product of two fission sources without resorting to binning the sources? Can we use basis functions other than histogram step functions to expand the fission source? These are additional questions I plan on addressing in my thesis.\n\n All calculations so far have been performed in 1-D homogeneous slab geometries. What will happen in heterogeneous and multi-dimensional geometries? Is Arnoldi's method still valid and does it still offer the same benefits? Although unknown for certain at this time, I propose the answer is yes. The reduced noise when using Arnoldi's method may lead to dramatic improvements in calculating eigenfunctions of geometries where highly absorbing materials are separated by highly scattering materials (e.g. the eigenvalue of the world type problems). No mathematical obstacles---other than lack of experience of the author---immediately present themselves as problems to using Arnoldi's method in higher dimensions. \n}\n\n%\\end{document}\n\\section{Power Method---Source Iteration}\nThe particle transport equation\n\\begin{equation}\n \\mathbf{\\Omega}\\cdot\\mathbf{\\nabla}\\psi(\\mathbf{r},\\mathbf{\\Omega})+\\Sigma_t\\psi(\\mathbf{r},\\mathbf{\\Omega}) = \\Sigma_s\\psi(\\mathbf{r},\\mathbf{\\Omega}) + \\frac{1}{k}\\nu\\Sigma_f\\psi(\\mathbf{r},\\mathbf{\\Omega}),\n\\end{equation}\ncan be written in operator form\n\\begin{subequations}\\begin{align}\n (\\mathbf{L} + \\mathbf{C} - \\mathbf{S})\\psi &= \\frac{1}{k}\\mathbf{F}\\psi \\\\\n \\mathbf{T}\\psi &= \\frac{1}{k}\\mathbf{F}\\psi. \\label{eq:Opr}\n\\end{align}\\end{subequations}\nwhere $\\mathbf{L}, \\mathbf{C},$ and $\\mathbf{S}$ are the leakage, collision, and scattering operators respectively; and $\\mathbf{F}$ is the fission operator. The operator $\\mathbf{T}$ is the transport-collision operator. The left-hand side represents the neutron loss mechanisms and the right-hand side represents the neutron gain mechanism.\n\nLet us define\n\\begin{align}\n q &\\equiv \\mathbf{F}\\psi, \\\\\n \\mathbf{A} &\\equiv \\mathbf{F}\\,\\mathbf{T}^{-1}\n\\end{align}\nand manipulate Eq. \\ref{eq:Opr} to find\n\\begin{equation}\n q = \\frac{1}{k}\\mathbf{A}q. \\label{eq:evalue}\n\\end{equation}\nThis is a standard eigenvalue problem with eigenvalue $k$ and eigenvector $q$. The linear operator $\\mathbf{A}$ first transports source particles and then computes a fission source from the resulting particle population.\n\nWe can use Eq. \\ref{eq:evalue} in an iterative manner\n\\begin{equation}\n q^{(n+1)} = \\frac{1}{k^{(n)}}\\mathbf{A}q^{(n)}. \n\\end{equation}\ngiven an initial eigenvector $q$ and eigenvalue $k$ (usually $k=1$.) Each iteration consists of calculating a new estimate of the eigenvector and eigenvalue,\n\\begin{equation}\n k^{(n+1)} = \\frac{Aq^{(n)}}{q^{(n)}} = \\frac{q^{(n+1)}}{q^{(n)}}.\n\\end{equation}\n\n\\section{Krylov Subspaces}\nA careful reader will notice Power Method generates a series of vectors\n\\begin{equation}\n \\{v, Av, A^2v, \\ldots, A^{m-1}v\\} = \\{q_0, q_1, q_2, \\ldots, q_{m-1}\\}.\n\\end{equation}\nThe vectors $\\{q\\}$ form a basis of the Krylov subspace denoted\n\\begin{equation}\n \\mathcal{K}_m(A,v) = \\mathrm{span} \\{q_0, q_1, \\ldots, q_{m-1}\\}.\n\\end{equation}\nAny Monte Carlo based particle transport code will create a Krylov subspace. One starts with the initial source distribution, $q_0$, and transports it defined by the operator, $A$. This generates a new source distribution, $q_1$ which is then transported. \n\nPower Method has been used for greater than 50 years, but isn't there a better way? Deterministic numerical methods researchers have found eigenvalue methods that are faster than Power Method. Can these be applied in a Monte Carlo code? The answer is a resounding ``We think so.''\n\n\\section{Minimized Iteration---Arnoldi's Method}\nIn contrast to the Power Method, Arnoldi's Method uses all the vectors previously calculated; after $k$ steps, we have $k+1$ vectors $\\{q, Aq, A^2,\\ldots, A^kq\\}$. In practice, these vectors usually are an ill-conditioned basis for the space they span. The vector $A^jq$ points more and more in the direction of a dominant eigenveactor for $A$ as $j$ is increased. \\marginpar{Why is this a problem? Is it because we get little information about the other vectors?}\n\nArnoldi's method creates an orthonormal set of basis vectors $\\{q_1, \\ldots, q_{k+1}\\}$ that spans the same Krylov subspace, $\\mathcal{K}(A, q)$. This orthonormal basis can be built up one vector at a time. At the $k+1$st step we could simply multiply $A^{k-1}q$ by $A$ to get $A$. It is inefficient to raise the matrix to a high poer, but we dont' have to. To get $q_{k+1}$ we operate on $q_k$ by $A$ and orthogonalize against the previous vectors. \n\nThe details of the Arnoldi process follow. We begin with the normalization of the original vector\n\\begin{equation}\n q_1 = q/\\|q\\|_2. \\label{eq:begin}\n\\end{equation}\nAt all subsequent steps we take\n\\begin{equation}\n \\tilde{q}_{k+1} = Aq_k - \\sum_{j=1}^k q_j h_{jk}. \\label{eq:mvOrtho}\n\\end{equation}\nwhere\n\\begin{equation}\n h_{jk} = \\langle Aq_k, q_j \\rangle.\n\\end{equation}\nLastly, the vector is normalized\n\\begin{equation}\n q_{k+1} = \\frac{\\tilde{q}_{k+1}}{h_{k+1,k}},\n\\end{equation}\nwhere\n\\begin{equation}\n h_{k+1,k} = \\|\\tilde{q}_{k+1}\\|_2. \\label{eq:end}\n\\end{equation}\n\nEqs. \\ref{eq:begin}-\\ref{eq:end} are summarized in Algorithm \\ref{alg:Arnoldi}. The matrix/operator $A$ itself is never used, only the matrix-vector product (see line \\ref{algln:MonteCarlo}). This is what allows us to use this method in a Monte Carlo code. The Monte Carlo particle transport solves $\\tilde{q}_{k+1} = Aq_k$.\n\n\\begin{algorithm2e}\n \\label{alg:Arnoldi}\n \\SetVline\n \\caption{Arnoldi Process (freely borrowed from Watkins(2002))}\n $q_1 = q/\\left\\|q\\right\\|_2$ \\;\n \\For{$k=1,\\ldots,m-1$}{\n $\\tilde{q}_{k+1} \\gets Aq_k$\\; \\nllabel{algln:MonteCarlo}\n \\For(\\Comment*[f]{Orthogonalize}){$j=1, \\ldots, k$}{ \n $h_{jk} \\gets \\left$\\;\n $q_{k+1} \\gets q_{k+1} - q_jh_{jk}$}\n $h_{k+1,k} \\gets \\left\\|q_{k+1}\\right\\|_2$\\;\n \\If(\\Comment*[f]{Span \\{$q_1, \\ldots, q_k$\\} is invariant under $A$}){$h_{k+1,k} = 0$}{\n quit.}\n $q_{k+1} \\gets q_{k+1}/h_{k+1,k}$\\;\n }\n\\end{algorithm2e}\n\n\\subsection{Finding Eigenvalues with Arnoldi's Method}\nThe orthonormal vectors can be combined into a single matrix\n\\begin{equation}\n Q_m = \\left[q_1 \\cdots q_m\\right] \\in \\mathbb{C}^{n\\times m}.\n\\end{equation}\nThe inner products \n\\begin{equation}\n h_{jk} = \\langle Aq_k, q_j\\rangle\n\\end{equation}\nform an upper Hessenberg matrix, \n\\begin{equation}\n H = \n \\begin{bmatrix}\n h_{11} & h_{12} & \\cdots & h_{1,m-1} & h_{1m} \\\\\n h_{21} & h_{22} & \\cdots & h_{2,m-1} & h_{2m} \\\\\n 0 & h_{32} & \\cdots & h_{3,m-1} & h_{3m} \\\\\n \\vdots & \\ddots & \\ddots & & \\vdots \\\\\n 0 & \\cdots & 0 & h_{m,m-1} & h_{mm} \\\\\n 0 & 0 & \\cdots & 0 & h_{m+1,m}\n \\end{bmatrix}\n \\in \\mathbb{C}^{m+1 \\times m}.\n\\end{equation}\n\nWe can rewrite Eq. \\ref{eq:mvOrtho} in terms of $A$, $Q$, and $H$,\n\\begin{equation}\n AQ_m = Q_{m+1}H_{m+1,m}, \\label{eq:MatrixRep}\n\\end{equation}\nor by separating the last column of $Q_{m+1}$ and the bottom row of $H_{m+1,m}$\n\\begin{equation}\n AQ_m = Q_{m}H_{m} + q_{m+1}h_{m+1,m}e_m^T \\label{eq:MRepmthIter}\n\\end{equation}\nwhere $e_m$ is the $m$th standard basis vector in $\\mathbb{R}^m$. Writing the equation in this way shows how the result of Arnoldi's process proceeds at each step.\n\nIf the vectors $q_j, \\;j = 1, \\ldots, k$ are linearly independent then $h_{m+1,m} \\neq 0$. If they are dependent, $h_{m+1,m} = 0$ and Eq. \\ref{eq:MatrixRep} becomes\n\\begin{subequations}\\begin{gather}\n AQ_m = Q_mH_m \\\\\n H_m = Q_m^{-1}AQ_m\n\\end{gather}\\end{subequations}\nand the Krylov space spanned by the columns of $Q$, $\\mathcal{K}_m(q,A)$, is invariant under $A$; that is for any vector $x \\in \\mathcal{K}_m$, $Ax$ is also in $\\mathcal{K}_m$. It can be shown that the eigenvalues of $H_m$ are the eigenvalues of $A$ and the eigenvectors of $A$ are related to the eigenvectors of $H$ as\n\\begin{equation}\n x = Q_mv\n\\end{equation}\nwhere $x$ is an eigenvector of $A$ and $v$ is the associated eigenvector of $H_m$. \n\nIf carried out to completion, Arnoldi's method computes a similarity transform of the matrix $A$ to an upper Hessenberg matrix, $H$. However, unless a very clever choice of the initial vector $q_1$ is used, one rarely finds an invariant subspace and must estimate the eigenvalues and eigenvectors of A from just a few steps of the Arnoldi process.\n\nWe can show that the eigenvalues of $H$ are estimates of the eigenvalues of $A$ with some residual error at any step of the Arnoldi process. We begin by finding an eigenpair of $H$, $(\\mu,v)$. If we multiply Eq. \\ref{eq:MRepmthIter} by $v$ we obtain\n\\begin{subequations}\\begin{gather}\n AQ_mv = Q_{m}H_{m}v + q_{m+1}h_{m+1,m}e_m^Tv \\\\\n AQ_mv = \\mu Q_mv + q_{m+1}h_{m+1,m}e_m^Tv \\\\[4mm] \\label{eq:FullResidual}\n Ax = \\mu x + r.\n\\end{gather}\\end{subequations}\nHere $x = Q_mv$ and $r = q_{m+1}h_{m+1,m}e_m^Tv$, the residual vector. \n\nWe can quantify how close the eigenpair $(\\mu,x)$ is to a true eigenpair of $A$ by taking the 2-norm of the difference $Ax-\\mu x$\n\\begin{equation}\n \\|Ax - \\mu x\\|_2 = |h_{m+1,m}||e_m^Tv|\n\\end{equation}\nwhere we have implicitly remembered $\\|q_{m+1}\\|_2 = 1$. This difference is called the residual. A small residual guarantees that $(\\mu, x)$ is an exact eigenpair of a matrix that is close to $A$ thus one can monitor the residual to determine convergence of the calculation.\n\n\\subsection{Restarted Arnoldi}\nPreviously I mentioned that unless a careful choice of the initial vector $q_1$ is made, one rarely finds an invariant subspace. One can make a careful choice by taking a few Arnoldi iterations and use the estimated eigenvector as the initial vector for a new calculation. This is how Arnoldi's method is restarted. Arnoldi's method can be restarted many times, each time with a better estimate of the eigenvector.\n\nRestarting Arnoldi's method has several benefits including memory management and processing time of which I will not focus on at this time. Performing restarts is a naive and simple way of duplicating the cycle in the power method. The calculation of the eigenvalue and eigenvector after each restart can be compared to the eigenvalues and eigenvectors of the power method. These eigenpairs can be evaluated to find their mean and variance, just like with the power method. \n\n\\section{Questions to Answer}\n\\begin{enumerate}\n \\item \\emph{What is the uncertainty? Is the answer biased?} This is the question of greatest import. A Monte Carlo calculation is useless without some measure of its statistical uncertainty. The easiest way to get this is to do restarted Arnoldi and find the mean and variance of of the eigenvalues and eigenvectors after each restart. I think there is a better way. There must also be some analysis of the bias (if one exists) in the answer.\n \\item \\emph{Is there a better way to evaluate the inner product $\\langle q_k, q_j \\rangle$?} We know the definition of the inner product\n \\begin{equation}\n \\langle q_k, q_j \\rangle \\equiv \\int q_j(x)q_k(x) dx. \\label{eq:InnerProduct}\n \\end{equation}\n For vectors, this is a simple dot-product. While this is simple, it requires biasing our results by spatial binning the fission source. It should be possible to sample from a (semi-)continuous source distribution $q_j$ and finding the corresponding value from $q_k$\n \\begin{equation}\n \\langle q_k, q_j \\rangle = \\frac{1}{N}\\sum_{i=1}^N q_k(x_i).\n \\end{equation}\n This is more of a Monte Carlo way of doing things and should reduce the bias in the calculation.\n \\item \\emph{Beyond Histograms---Other Functional Expansions} Our fission source vectors are histogram sources\n \\begin{equation}\n q(x) = \\sum_{i=1}^B a_if_i(x)\n \\end{equation}\n where $B$ is the number of spatial bins and\n \\begin{equation}\n f_i(x) = \\left \\{\n \\begin{array}{ccc}\n 1 &,& x_{i} \\leq x < x_{i+1} \\\\\n 0 &,& \\mathrm{otherwise}\n \\end{array}\\right. .\n \\end{equation}\n We can, however choose a different set of basis vectors, $f_i(x)$, about which we can expand. One choice to pursue is a linear inspace across each spatial bin, but I'm not going to show how this is done.\n\n\\end{enumerate}\n\n\\section{Tasks to Complete}\n\\begin{itemize}\n \\item Heterogeneity\n \\item Multi-dimensional\n\\end{itemize}\n\n\n\\end{document}\n", "meta": {"hexsha": "e8cdfaf6e6def9dd2b3de50e6d802e4a96c1bf1a", "size": 15299, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Prospectus1/Oral.tex", "max_stars_repo_name": "jlconlin/PhDThesis", "max_stars_repo_head_hexsha": "8e704613721a800ce1c59576e94f40fa6f7cd986", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Prospectus1/Oral.tex", "max_issues_repo_name": "jlconlin/PhDThesis", "max_issues_repo_head_hexsha": "8e704613721a800ce1c59576e94f40fa6f7cd986", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Prospectus1/Oral.tex", "max_forks_repo_name": "jlconlin/PhDThesis", "max_forks_repo_head_hexsha": "8e704613721a800ce1c59576e94f40fa6f7cd986", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 66.807860262, "max_line_length": 722, "alphanum_fraction": 0.7157984182, "num_tokens": 4578, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525098, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.34646265258178166}} {"text": "\\chapter{Synthesis in Example of a Magnetic Bearing on an Overhung Rotor}\\label{MagExample}\nAnalysis and modeling techniques of the previous chapters will now be put to use in a practical example. The goal of which will be to reduce vibration on an overhung disk rotor system with the use of an Active Magnetic Bearing(AMB). An experimental test rig, not unlike the system used in the experimental example of \\S\\ref{ExperimentalPlots}, will be used to calibrate a finite element theoretical model. Then, the theoretical model will be extended to include an AMB near the overhung disk. The model will be evaluated for stability, and parameters of the control algorithm for the AMB will be varied to attempt to eliminate modes and stabilize the system.\n\\section{Physical System Description}\nThe rotor system of interest is depicted in Figure \\ref{fig:OverhungDiagram}. Geometric parameters are listed in Table \\ref{tab:GeometricParametersofOverhung}. The springs at nodes 1 \\& 4 are intended to represent bushings, portion at node 6 is the rotating disk, and nodal numbers are indications of how the rotor will be discretized for the finite element model.\n\\begin{figure}\n\t\\centering\n\t\\def\\svgwidth{250pt}\n\t\\import{figures/}{OverhungDiagram.pdf_tex}\n\t\\caption{Overhung rotor system diagram.}\n\t\\label{fig:OverhungDiagram}\n\\end{figure}\n\\begin{table}\n\t\\centering\n\t\\caption{Geometric parameters of the overhung rotor system.}\n\t\\label{tab:GeometricParametersofOverhung}\n\t\\begin{tabular}{cccccc}\n\t\t$ L[m] $&$ a[m] $&$ b[m] $&$ l_d[m] $&$ d_d[m] $&$ d_s[m] $\\\\\\hline\n\t\t$ 0.5 $&$ 0.23 $&$ 0.13 $&$ 0.025 $&$ 0.075 $&$ 0.01 $\n\t\t\\end{tabular}\n\\end{table}\n\\section{Experimental Results}\nThe rotor system was tested in a start-up from slow roll to $ 3000[RPM] $. Data shown here is taken from $ 1000[RPM] $ to $ 2000[RPM] $. A set of orthogonal eddy current position sensors placed near the disk on the outboard side were used to measure the position of the shaft throughout the start-up. Position data was recorded at a sampling rate of $ 128000[Hz] $ with no processing applied, this sampling rate is much higher than the required rate to avoid aliasing. The resulting 3D orbit of the start-up is shown in Figure \\ref{fig:MagExampleOrbit3D}. Of note is the necking in the 3D orbit during the natural frequency that is indicative of high anisotropy inducing a negative whirl during the first natural frequency. Also resulting from the experiment is the full spectrum cascade plot of Figure \\ref{fig:MagExampleCascade}, in which it is evident that synchronous vibration dominates the spectra. Though, for the production of the bode diagram (Figure \\ref{fig:MagExampleBode}) there was a benefit in clarity from filtering the data to synchronous speed.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.5\\linewidth]{./figures/MagExpExampleOrbit3D.png}\n\t\\caption{3D Orbit of the experimental overhung rotor system.}\n\t\\label{fig:MagExampleOrbit3D}\n\\end{figure}\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=.8\\linewidth]{./figures/MagExampleCascade.png}\n\t\\caption{Cascade of the experimental overhung rotor system.}\n\t\\label{fig:MagExampleCascade}\n\\end{figure}\n\\begin{figure}[!htb]\n\t\\def\\width{.7\\linewidth}\n\t\\def\\height{.4\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={ylabel style={xshift=0},xlabel style={yshift=0}}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{}{./figures/MagExampleBode.tex}\n\t\\caption{Bode diagram of the experimental overhung rotor filtered to 1X.}\n\t\\label{fig:MagExampleBode}\n\\end{figure}\n\\section{Theoretical Model}\nTo create a finite element model for this rotor system the shaft will be discretized into 7 elements a disk at node 6 and bearings at nodes 1 and 4, as depicted in Figure \\ref{fig:OverhungDiagram}. In the experiment, a small length of shaft continued after the overhung disk and has been included in this model. The AMB will be included as a nodal point element with with stiffness and damping to be derived in \\S\\ref{Active Magnetic Bearing}. Parameter values for the finite element model are provided in Table \\ref{tab:OverhungParameters}. Discovered values such as $ \\eta_v $ and the stiffnesses of bearing are listed here, but the process for their determination is discussed.\n\\begin{table}\n\t\\caption{Properties of disks, shaft elements, and bearings of the theoretical model.} \\label{tab:MagTheoryRotorTable}\n\t\\centering\n\t\\label{tab:OverhungParameters}\n\t\\begin{tabular}{rcccccc}\n\t\t\t\t\t\t\t&$\\rho\\left[\\frac{kg}{m^3}\\right]$\t&$r[m]$\t\t\t\t\t&$\\nu$\t\t\t\t&$E[Pa]$\t\t\t&$ \\eta_v[s] $\t&$ \\eta_h $\t\\\\\\hline\n\t\t\\textbf{Shaft}\t\t&$7850$\t\t\t\t\t&$0.005$\t\t\t\t&$0.3$\t\t\t\t&$\\num{210E9}$\t\t&$ 0.0002 $\t\t&$ 0 $\t\t\\\\[-.2em]\n\t\t\t\t\t\t\t&$\\rho\\left[\\frac{kg}{m^3}\\right]$\t&$r[m]$\t\t\t\t\t&$l[m]$\t\t\t\t&\t\t\t\t\t&\t\t\t\t&\t\t\t\\\\\\hline\n\t\t\\textbf{Disks}\t\t&$7850$\t\t\t\t\t&$0.0375$\t\t\t\t&$0.025$\t\t\t&\t\t\t\t\t&\t\t\t\t&\t\t\t\\\\[-.2em]\n\t\t\t\t\t\t\t&$k_y\\left[\\frac{N}{m}\\right]$\t\t&$k_z\\left[\\frac{N}{m}\\right]$\t\t&$c_y\\left[\\frac{Ns}{m}\\right]$&$c_z\\left[\\frac{Ns}{m}\\right]$&\t\t\t\t&\t\t\t\\\\\\hline\n\t\t\\textbf{Bearing A}\t&$1.7\\e{5}$\t\t\t\t&$2.2\\e{5}$\t\t\t\t&$68$\t\t\t\t&$88$\t\t\t\t&\t\t\t\t&\t\t\t\\\\[-.5em]\n\t\t\\textbf{Bearing B}\t&$2.04\\e{5}$\t\t\t&$2.64\\e{5}$\t\t\t&$81.6$\t\t\t\t&$105.6$\t\t\t&\t\t\t\t&\t\t\t\\\\\n\t\\end{tabular}\n\\end{table}\nFirst the model is formed to match the experimental results. Known parameters, such as beam lengths, beam diameters, density of the material, and geometry of the disk and rotor are used to begin construction of the model. Then, guesses are made for the stiffnesses in the rotor bearings. The first natural frequency is calculated with the resulting model and its value compared to the experimentally found natural frequency from the bode diagram (fig.\\ref{fig:MagExampleBode}). Stiffness are then adjusted to better match the natural frequency, and this is repeated until the natural frequency of the model matches the experiment. After this process, the stiffness was determined to be around $ 2\\e{5} \\left[\\frac{N}{m}\\right]$.\\par\nIt is evident by inspection of the Bode diagram for the experimental system (fig.\\ref{fig:MagExampleBode}), and the 3D orbit, that there is anisotropy in the system, leading to the dip in amplitude of one plane of vibration. It is also known from inspection of the frequency spectrum in the cascade of figure \\ref{fig:MagExampleCascade} that in this speed range the orbit is in the opposite direction of the rotation--a phenomena only possible with anisotropy of the stiffness. Figures \\ref{fig:HorVertStiffAniCompare} \\& \\ref{fig:PosNegStiffAniCompare} demonstrate this affect anisotropy has on both the real coordinates, as well as positive and negative whirl amplitudes. In the real coordinates, cross coupling of from gyroscopic moments causes an interaction of the two peaks in amplitude. As the stiffness anisotropy approaches a value of 1, the vibration of the more flexible plane will split into two peaks. One peak coinciding with that planes natural frequency and one plane coinciding with the orthogonal natural frequency. During this change in anisotropy of stiffness, the peaks of both positive and negative amplitude spectrums split to coincide with either plane's natural frequency. When the two orthogonal planes have widely separated natural frequencies, the negative amplitude spectrum is greater than the positive amplitude spectrum for the range of speed between the two natural frequencies. \\par \n\\begin{figure}\n\\begin{subfigure}{\\textwidth/2}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{./figures/MagExampleHorVertStiffAniCompare.png}\n\t\\caption{Comparison of v \\& w as stiffness anisotropy is changed.}\n\t\\label{fig:HorVertStiffAniCompare}\n\\end{subfigure}\n\\begin{subfigure}{\\textwidth/2}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{./figures/MagExamplePosNegStiffAniCompare.png}\n\t\\caption{Comparison of positive and negative orbit amplitudes as stiffness anisotropy is changed.}\n\t\\label{fig:PosNegStiffAniCompare}\n\\end{subfigure}\n\\end{figure}\nUsing the bode diagram, the stiffness anisotropy is adjusted until the shapes of the amplitudes and phases match the experimental results of figure \\ref{fig:ExpExampleBode}. Then damping is added to the system to appropriately match the experimental results. The resulting bode diagram is Figure \\ref{fig:MagTheoryBode}. Stiffnesses were determined to be $ k_y=1.7\\e{5}[N/m]\\ \\&\\ k_z=2.2\\e{5}[N/m] $.\n\\begin{figure}[!htb]\n\t\\def\\width{.6\\linewidth}\n\t\\def\\height{.4\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={ylabel style={yshift=.5em},xlabel style={yshift=0}},\n\tevery axis plot/.style={line width=1.4pt}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{figures/}{MagTheoryBode.tex}\n\t\\caption{Bode Diagram of overhung system without AMB.}\n\t\\label{fig:MagTheoryBode}\n\\end{figure}\nAnd now the resulting system is further described with an expanded speed range of $ 20000[RPM] $. This resulting model is described with the new Campbell diagram of Figure \\ref{fig:MagTheoryCampbell}, the roots locus of Figure \\ref{fig:MagTheoryRootLocus}, and the first three mode shapes (Figures \\ref{fig:MagTheoryShape1}, \\ref{fig:MagTheoryShape2}, \\& \\ref{fig:MagTheoryShape3}). Note that the first mode is conical in shape, but the disk is far from an antinode at node 6 so it does not experience significant gyroscopic moments. This idea is supported by the campbell diagram, fig. \\ref{fig:MagTheoryCampbell}, as the first mode natural frequency does not change significantly over the speed range. On the other hand, the second mode has its antinode atode 6 so the disk experience maximum gyroscopic moments and the second mode critical speed is much more dependent on speed.\n\\begin{figure}[!htb]\n\t\\def\\width{.6\\linewidth}\n\t\\def\\height{.4\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={ylabel style={yshift=.5em},xlabel style={yshift=0}},\n\tevery axis plot/.style={line width=1.4pt}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{figures/}{MagTheoryCampbell.tex}\n\t\\caption{Campbell Diagram of the overhung system without AMB.}\n\t\\label{fig:MagTheoryCampbell}\n\\end{figure}\n\\begin{figure}[!htb]\n\t\\def\\width{.6\\linewidth}\n\t\\def\\height{.4\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={ylabel style={yshift=.0},xlabel style={yshift=0}},\n\tevery axis plot/.style={line width=1.4pt}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{}{./figures/MagExampleRootsLocus.tex}\n\t\\caption{Roots Locus of overhung system without AMB.}\n\t\\label{fig:MagTheoryRootLocus}\n\\end{figure}\n\\begin{figure}\n\t\\def\\cs{.29}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={yticklabel style={xshift=0,yshift=0},minor tick num=2}, grid style={line width=.1pt, draw=gray!20},major grid style={line width=.2pt,draw=gray!50},ticks=none,minor tick style={draw=none},\tevery axis plot/.style={line width=.9pt},}%, every axis/.style={hide axis}}% \n\t\\begin{subfigure}{\\cs\\textwidth}\n\t\t\\centering\n\t\t\\def\\width{\\linewidth}\n\t\t\\def\\height{\\linewidth}\n\t\t\\import{}{./figures/MagTheoryShape1.tex}\n\t\t\\caption{Mode Shape 1.}\n\t\t\\label{fig:MagTheoryShape1}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\cs\\textwidth}\n\t\t\\centering\n\t\t\\def\\width{\\linewidth}\n\t\t\\def\\height{\\linewidth}\n\t\t\\import{figures/}{MagTheoryShape2.tex}\n\t\t\\caption{Mode Shape 2.}\n\t\t\\label{fig:MagTheoryShape2}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\cs\\textwidth}\n\t\t\\centering\n\t\t\\def\\width{\\linewidth}\n\t\t\\def\\height{\\linewidth}\n\t\t\\import{}{./figures/MagTheoryShape3.tex}\n\t\t\\caption{Mode Shape 3.}\n\t\t\\label{fig:MagTheoryShape3}\n\t\\end{subfigure}\n\\end{figure}\n\\subsection{Active Magnetic Bearing}\\label{Active Magnetic Bearing}\nMagnetic pole-rotor relationship will be derived based on the detailed derivation in \\cite{das2008vibration}, with influence from \\cite{childs1993turbomachinery}. Assumptions made in this model are as follows:\n\\begin{itemize}\n\t\\item Air gap between the magnet pole face and the rotor is vanishingly small compared to the diameter of the shaft.\\\n\t\\item Flux leakage from the magnetic pole is negligible.\n\t\\item Curvature of rotor surface under the pole face is ignored.\n\t\\item A linear relationship of flux density and magnetic field is assumed.\n\t\\item Hysteresis of the magnetic field is negligible.\n\\end{itemize}\nThese assumptions lead to a magnetic force due to coil current in the relationship of\n\\begin{equation}\\label{key}\nF_m=\\frac{-k_mi^2}{l_g^2}\n\\end{equation}\nwhere, $ k_m=\\frac{\\mu_0A_pN^2}{4} $, and $ \\mu_0=4\\pi\\e{-7} $ is the absolute permeability in free air, $ N $ is the number of coil turns, $ A_p $ is the pole face area, $ i $ is the supplied electrical current, and $ l_g $ is the air gap.\\par \nConsider a set of magnetic pole pairs in each the $ y\\ \\&\\ z $ directions (i.e. One on the left and one on the right; one on top and one on bottom), where each pole pair has its own electrical circuit. There are four total magnets in this model, two in each direction, and eight total poles where two form a magnet. All poles in the neutral position of the system will have a nominal air gap of $ g_0 $ and will be supplied by a bias current of $ i_0 $. Deviations of the current from this bias will be considered as $ i_y $ and $ i_z $. Deviations of the position from this neutral position are the displacements of the rotor at this beam axis location, $ v,\\ \\&\\ w $. Therefore, total current will be $ (i_0\\pm i_y)\\ \\&\\ (i_0\\pm i_z) $ for opposing poles in the $ y\\ \\&\\ z $ directions respectively. Similarly, total gaps are given by $ (g_0\\pm v)\\ \\&\\ (g_0\\pm w) $. Using these new definitions for the gap and current, and summing forces in the $ y\\ \\&\\ z $ directions leads to the total forces\n\\begin{equation}\\label{eq:MagneticForceNL}\nF_y=K_m\\left\\{\\left(\\frac{i_0+i_y}{g_0+v}\\right)^2-\\left(\\frac{i_0-i_y}{g_0-v}\\right)^2\\right\\}\\ \\&\\ F_z=K_m\\left\\{\\left(\\frac{i_0+i_z}{g_0+w}\\right)^2-\\left(\\frac{i_0-i_z}{g_0-w}\\right)^2\\right\\}\n\\end{equation}\nwhere, $ K_m=k_m\\cos(\\alpha) $, and $ \\alpha $ is half of the angle between the poles of a magnet. Linearizing the magnetic force equations \\eqref{eq:MagneticForceNL}, while assuming the operating point for the system is where all positions and control currents are zero, results in\n\\begin{equation}\\label{eq:MagneticForce}\nF_y=k_ii_y+k_yv,\\ \\&\\ F_z=k_ii_z+k_zw\n\\end{equation}\nwhere, $ k_i=4K_m\\frac{i_0}{g_0^2} $ is the current stiffness developed by the bias current, and $ k_y=k_z=k_s=-4K_m\\frac{i_0^2}{g_0^3} $.\n\\subsubsection{Proportional Derivative Control}\nA control algorithm is used to control the current sent to each pair of poles based on the position (proportional) and velocity (derivative) of the rotor. It is assumed that each set of pole pairs will receive opposite currents to act as a unit. Current control is given by \n\\begin{equation}\\label{eq:ControlCurrent}\ni_y=-k_g(k_pv+k_v\\dot{v}),\\ \\&\\ i_z=-k_g(k_pw+k_v\\dot{w})\n\\end{equation}\nwhere, $ k_g $ is the power amplifier gain, $ k_p $ is the proportional gain, and $ k_v $ is the derivative gain. The total linearized force becomes\n\\begin{equation}\\label{key}\nF_y=-(k_gk_ik_p-k_s)v-k_gk_ik_v\\dot{v},\\ \\&\\ F_z=-(k_gk_ik_p-k_s)w-k_gk_ik_v\\dot{w}\n\\end{equation}\nso then the stiffness of the AMB is $ k_{mag}=(k_gk_ik_p-k_s) $ and the damping is $ d_{mag}=k_gk_ik_v $. As an equation of nodal stiffness and damping in the finite element system it can be represented by\n\\begin{equation}\\label{eq:DiskNodeEquationofMotion}\n\\bunderline{\\mathbf{D}}^m\\dot{\\vec{\\mathbf{q}}}_k+\\bunderline{\\mathbf{K}}^m\\vec{\\mathbf{q}}_k=0\n\\end{equation}\nwhere,\n\\begin{equation*}\n\\def\\cs{2em}\n\\begin{array}{cc}\n\\bunderline{\\mathbf{K}}^m=\\left[\\def\\arraystretch{.8}\\arraycolsep=0pt\\begin{array}{cccccc}\n\\makebox[\\cs]{$k_{mag}$}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}\\\\\n0&k_{mag}&0&0&0&0\\\\\n0&0&0&0&0&0\\\\\n0&0&0&0&0&0\\\\\n0&0&0&0&0&0\\\\\n0&0&0&0&0&0\n\\end{array}\\right] & \\bunderline{\\mathbf{D}}^m=\\left[\\def\\arraystretch{.8}\\arraycolsep=0pt\\begin{array}{cccccc}\n\\makebox[\\cs]{$d_{mag}$}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}&\\makebox[\\cs]{0}\\\\\n0&d_{mag}&0&0&0&0\\\\\n0&0&0&0&0&0\\\\\n0&0&0&0&0&0\\\\\n0&0&0&0&0&0\\\\\n0&0&0&0&0&0\n\\end{array}\\right]\n\\end{array}\n\\end{equation*}\n\n\\section{Addition of Magnetic Bearing to the Rotor Model}\nIn order to measure the effectiveness of the AMB, the Stability of the theoretical model before the addition is shown in Figure \\ref{fig:MagTheoryStability}. The magnetic bearing added is modeled after a magnetic bearing that is currently in the lab at California Polytechnic State University. This theoretical exercise is intended to be followed by experimental verification not included in this work. The parameters used are listed in \\ref{tab:MagBearingParameters} as well as the control values whose determinations will be evaluated.\\par \nFirst the axial position of the AMB must be determined. By inspection of the mode shapes, it is evident that in the first mode (the mode we are most concerned about suppressing) a shape exists with increasing amplitude toward the end of the beam after the second bearing. It is also known that the source of vibration, the rotating disk, is located at nodal index six. With both these pieces of information, node seven is chosen as a starting point for the AMB for two reasons: having the AMB closer to the source of vibration reduces the phase lag between the source and the bearing, increasing the effectiveness of control; amplitude of vibration, according to the mode shape, is higher on the outboard side of the disk and placing the AMB on this side will minimize more vibration. Additionally, this location should significantly suppress the second mode of vibration as node seven is near the maximum amplitude of that mode as well.\\par \nIn order to choose a value for bias current, $ i_b $, and provide upper range estimates of the controller outputs, $ k_v,\\ \\&\\ k_p $, the maximum expected current output of the control loop must be determined. $ i_b $ will be set so that under peak power output, in the expected speed range, the control current will push the total current output beyond the system limits.\\par \nKnowing that the amplitude of vibration, $ A $, in the experimental system peaks at about 10 mils, or $ 2.54\\e{-4}[m] $, the greatest possible velocity for synchronous vibration is determined using the simple equation $ vel=(A/2)\\omega $, where $ \\omega $ is the whirl speed in rad/s. Under the assumption of synchronous vibration, $ \\omega $ during the natural frequency is then $ 167.6[Rad/s] $. Leading to a velocity, $ vel $, of $ 0.02[m/s] $. Also, another possible peak velocity occurs in the upper speed range. Knowing that the amplitude of vibration after the first natural frequency is equal to the eccentricity of the unbalance, with an eccentricity of $ 1\\e{-5} $, and an upper speed of $ 15000[RPM] $, the velocity is calulated to be $ 0.008[m/s] $. Since the estimated velocity during the natural frequency is higher, it will be used to limit the output of the AMB controller. Total voltage supply of most digital to analog converters is limited to $ 10[V] $. So the control voltage is calculated to be within this range for the given velocity and positions that will be seen under synchronous vibration. This results in a limitation of the term $ k_v $, which is proportional to the voltage control signal that is sent to the amplifier for conversion to control current. A value of $ 480[Vs/m] $ would max the converter, so the value to be determined must be less than this. With a similar, but separate, evaluation of the displacement leads to a maximum allowable value of $ 500000[V/m] $ for $ k_p $--a value not anticipated to be necessary. The maximum force due to unbalance is $ \\epsilon m_d\\Omega^2 $, where $ \\epsilon $ is the eccentricity, and $ m_d $ is the mass of the disk. At the maximum speed expected of $ 15000[RPM] $, the force will be $ 24[N] $. To counteract this force with a single coil would require $ 1.25[A] $, or $ 0.625[A] $ per coil in a opposing pair. The amplifier gain is assumed to be programmable to $ k_g=1[A/V] $, this leads to a reasonable choice of bias current at $ 0.5[A] $ to maintain a good resolution on the voltage output of the controller. Under the operating conditions described, the control voltage should not exceed $ 1.25[V] $, and will rest at an output of $ 0.5[V] $. Now in the next section, control parameters $ k_v\\ \\&\\ k_p $ will be determined to maximize stability of the system while also minimizing vibration.\\par\n To determine best derivative control constant ($ k_v $), the proportional control constant ($ k_p $) was set to zero. $ k_v $ is increased until the first mode on the Roots locus(fig. \\ref{fig:MagTheoryRootLocus}) moves away from the imaginary axis, becoming more damped. The resulting movement of the Roots on the Roots Locus is given in Figure \\ref{fig:MagTheoryRootLocusControllerTune}. Note that because of bias current flowing through the bearing, a baseline stiffness of $ k_S $ is present in the system even withou $ k_p $. None of the modes that appear on the roots locus of the system with $ k_p=0,\\ \\&\\ k_v=10[\\frac{Vm}{s}] $ control algorithm cross the real plane--this demonstrates the stability of the new system. Stability of the system with the AMB addition is confirmed in the stability plot of Figure \\ref{fig:MagTheoryStabilityWith}. In fact from the roots locus with $ k_v=1 $ to the roots locus with $ k_v=10 $ the first mode of vibration is completely relegated to the imaginary axis and not reaching break-away for this entire speed range--this indicates that the first mode has been over-damped. With additional damping through the increase of $ k_v $ it would be possible to over-damp the second mode as well, but in the speed range given, it would not improve the performance.\\par \n Proportional control, $ k_p $, was added after the ideal derivative control was determined, but it did not improve the performance of the controller. It is the case that the stiffness of the bearing inherent to the bias current is sufficient. In any case, proportional control in this scenario is only adding stiffness to the system, and with the objective being to minimize vibration, $ k_p $ is not effective. So the optimal control is determined to be with $ k_p=0,\\ \\&\\ k_v=10[Vs/m] $. The remaining parameters for this resulting controller are listed in table \\ref{tab:MagBearingParameters}. It is worthwhile to note that this optimal control is standing on the basis that the feedback is of synchronous vibration only. In a scenario where there is significant sub or super-synchronous vibrations, this controller may exceed its voltage limit. It is recommended that the feedback signal be filtered to match rotor speed to ensure this scenario does not take place. Furthermore, without at least low-pass filtering of the feedback signal the control would certainly provide out of range signals due to the volatile nature of derivatives of discrete signals.\\par \n The frequency spectrum is provided showing the result of the AMB application in the bode diagram of Figure \\ref{fig:MagTheoryBodeCompare}. Certainly it can be concluded that the AMB is successfully performing the desired task of reducing the vibration while also inproving the stability of the system.\\par \nThe result from this synthesis exercise can be implemented on the actual experimental test rig with the AMB set to the control parameters suggested. The power of the finite element method in this application is the ability to move components around with ease. For instance, with the changing of just two parmeters in the input file for this model, a new simulation is created for complete levitation of the overhung rotor. The AMB is put in place of bearing b and the resulting frequency spectrum is plotted in Figure \\ref{fig:MagLevTheoryBodeCompare}\n\\begin{table}\n\t\\centering\n\t\\caption{Active Magnetic Bearing Parameters.}\n\t\\begin{tabular}{cccccccc}\n\t\t$\\alpha[rad]$&$g_0[m]$&$i_0[A]$&$k_p[\\frac{V}{m}]$&$k_v[\\frac{Vs}{m}]$&$k_g[\\frac{A}{V}]$&$ N[\\#]$&$A_p[m^2]$\\\\\\hline\n\t\t$\\frac{\\pi}{8}$&$2.5\\e{-3}$&$0.5$&$0$&$10$&$1$&$800$&$\\frac{5}{100*100}$\n\t\\end{tabular}\n\t\\label{tab:MagBearingParameters}\n\\end{table}\n\\begin{figure}\n\t\\begin{subfigure}{\\textwidth/2}\n\t\t\\def\\width{.8\\linewidth}\n\t\t\\def\\height{.4\\linewidth}\n\t\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={ylabel style={yshift=-20},xlabel style={yshift=35}},every x tick scale label/.style={at={(xticklabel* cs:1,.3cm)},anchor=near xticklabel},every axis plot/.style={line width=1.4pt}}%, every axis/.style={hide axis}}%\n\t\t\\centering\n\t\t\\import{figures/}{MagTheoryStability.tex}\n\t\t\\caption{Stability plot of rotor without AMB, threshold of stability:$ \\ 4646[RPM] $.}\n\t\t\\label{fig:MagTheoryStability}\n\t\\end{subfigure}\n\t\\begin{subfigure}{\\textwidth/2}\n\t\t\\def\\width{.8\\linewidth}\n\t\t\\def\\height{.35\\linewidth}\n\t\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right}, every axis/.style={xlabel style={yshift=-2em},xlabel style={yshift=25,at={(axis description cs:0.5,1.05)},anchor=north}},every x tick scale label/.style={at={(xticklabel* cs:1,.3cm)},anchor=near xticklabel},every axis plot/.style={line width=1.4pt}}%, every axis/.style={hide axis}}%\n\t\t\\centering\n\t\t\\import{figures/}{MagTheoryStabilityWith.tex}\n\t\t\\caption{Stability plot of rotor with AMB, indicates complete stability through speed range.}\n\t\t\\label{fig:MagTheoryStabilityWith}\n\t\\end{subfigure}\n\\end{figure}\n\\begin{figure}[!htb]\n\t\\def\\width{.7\\linewidth}\n\t\\def\\height{.5\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right},every axis plot/.style={line width=1.4pt}}%, every axis/.style={ylabel style={yshift=0em},xlabel style={yshift=0}}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{figures/}{MagTheoryRootLocusControllerTune.tex}\n\t\\caption{Roots locus of Overhung rotor system with varying $ k_v $.}\n\t\\label{fig:MagTheoryRootLocusControllerTune}\n\\end{figure}\n\\begin{figure}[!htb]\n\t\\def\\width{.6\\linewidth}\n\t\\def\\height{.4\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right},every axis plot/.style={line width=1.3pt}}%, every axis/.style={ylabel style={yshift=0em},xlabel style={yshift=0}}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{figures/}{MagTheoryBodeCompare.tex}\n\t\\caption{Bode diagram at node 6 comparing the rotor without AMB(solid) and with AMB(dashed).}\n\t\\label{fig:MagTheoryBodeCompare}\n\\end{figure}\n\\begin{figure}[!htb]\n\t\\def\\width{.6\\linewidth}\n\t\\def\\height{.4\\linewidth}\n\t\\def\\sep{3em}\n\t\\pgfplotsset{every picture/.style={trim axis left, trim axis right},every axis plot/.style={line width=1.3pt}}%, every axis/.style={ylabel style={yshift=0em},xlabel style={yshift=0}}}%, every axis/.style={hide axis}}%\n\t\\centering\n\t\\import{figures/}{MagLevTheoryBodeCompare.tex}\n\t\\caption{Bode diagram at node 6 comparing the rotor without AMB(solid) and with AMB(dashed) for complete levitation at node 4.}\n\t\\label{fig:MagLevTheoryBodeCompare}\n\\end{figure}", "meta": {"hexsha": "ba8ab1d238062b89c8ce2c220f84f35fa9580f33", "size": 27081, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapters/MagExample.tex", "max_stars_repo_name": "cameron1320/Cameron", "max_stars_repo_head_hexsha": "276144ab299ab00f102f3f3cc0869261a13e0059", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapters/MagExample.tex", "max_issues_repo_name": "cameron1320/Cameron", "max_issues_repo_head_hexsha": "276144ab299ab00f102f3f3cc0869261a13e0059", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapters/MagExample.tex", "max_forks_repo_name": "cameron1320/Cameron", "max_forks_repo_head_hexsha": "276144ab299ab00f102f3f3cc0869261a13e0059", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 102.1924528302, "max_line_length": 2383, "alphanum_fraction": 0.7535172261, "num_tokens": 7797, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736783928749127, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.34646264377790675}} {"text": "% !TEX root = ../zeth-protocol-specification.tex\n\n\\section{Processing $\\zethTx$}\\label{zeth-protocol:process-tx}\n\nWhen a $\\zethTx$ is mined (hence assuming that $\\ethVerifyTx(\\zethTx)$ returns $\\true$), the state transition specified by the $\\mix$ function of $\\mixer$ is executed.\n\nTo preserve the soundness of \\zeth, and make sure that no $\\zparty{U}$ is able to create value by double spending $\\zethnotes$, various checks need to be satisfied. The function $\\zethVerifyTx$ is defined as the function that returns $\\true$ if all the checks are satisfied, and $\\false$ otherwise.\n\nIf $\\zethVerifyTx(\\zethTx)$ returns $\\true$, then $\\mix$ modifies the ``World state'' $\\wstate$ to account for the spent $\\zethnotes$ and the newly generated ones. However, if $\\zethVerifyTx(\\zethTx)$ returns $\\false$, then the state transition ends.\n\n\\begin{notebox}\n Even if $\\zethVerifyTx(\\zethTx)$ returns $\\false$, $\\wstate$ is modified since the \\ethereum~balances of the transaction originator is decremented by the sum of $\\txDefaultGas$ and the gas consumed by the $\\zethVerifyTx$ function, and the balance of the \\ethereum~account of the miner gets incremented by the same amount.\n\\end{notebox}\n\nThus, $\\mix$ proceeds as follows:\n\\begin{enumerate}\n \\item Check that all the values of the primary inputs' ($\\inp{\\mix}.\\primInp$) entries are elements of the scalar field over which the zk-proof is generated:\n \\[\n \\inp{\\mix}.\\primInp \\in \\FFx{\\rCURVE}^*\n \\]\n \\item Unpack the nullifiers, signature hash and public values (see~\\cref{instantiation:statement:pack} for the definitions of the $\\unpack{}{}$ functions):\n \\begin{align*}\n \\nf{i} &= \\unpack{\\inp{\\mix}.\\primInp.\\nfs{i}, \\inp{\\mix}.\\primInp.\\resbits}{\\nf{}}\\ \\forall i \\in [\\jsin] \\\\\n \\vin &= \\decode{\\unpack{(), \\inp{\\mix}.\\primInp.\\resbits}{\\vin}}{\\NN} \\\\\n \\vout &= \\decode{\\unpack{(), \\inp{\\mix}.\\primInp.\\resbits}{\\vout}}{\\NN} \\\\\n \\hsig &= \\unpack{\\inp{\\mix}.\\primInp.\\hsig, \\inp{\\mix}.\\primInp.\\resbits}{\\hsig}\n \\end{align*}\n \\item Check the validity of the $\\zethTx$ object ($\\zethVerifyTx$):\n \\begin{enumerate}\n \\item Check that $\\inp{\\mix}.\\primInp.\\hsig$ is correctly computed, i.e.~check that the following equation holds (to prevent transaction malleability, see~\\cref{appendix:trnm}):\n \\[\n \\hsig = \\crhhsig{\\inp{\\mix}.\\primInp.\\nfs{}, \\inp{\\mix}.\\otsvk}\n \\]\n \\item\\label{zeth-protocol:process-tx:verify-zkp} Check that $\\zkp$ is a valid zk-SNARK proof for $\\inp{\\mix}.\\primInp$, i.e.~check that:\n \\[\n \\zksnark.\\verifier(\\pparams_{\\zksnark}, \\zkp, \\inp{\\mix}.\\primInp) = \\true\n \\]\n \\item Check that none of the nullifiers in $\\inp{\\mix}.\\primInp.\\nfs{}$ have already been used, i.e.~check that:\n \\[\n \\nf{i} \\not\\in \\nullifierset, \\forall i \\in [\\jsin]\n \\]\n where $\\nullifierset$ is the set of all nullifiers that are ``declared'' on $\\mixer$.\n \\item\\label{zeth-protocol:process-tx:check-otsig} Check that $\\inp{\\mix}.\\otssig$ is a valid signature of the $\\ethereum{}$ sender's address $\\addr$ (see~\\cref{zeth-protocol:create-tx}) and the attributes of $\\inp{\\mix}$, to prevent transaction malleability (see~\\cref{appendix:trnm}), i.e.~check that:\n \\begin{align*}\n & \\otsigscheme.\\verify(\\inp{\\mix}.\\otsvk, \\msg, \\inp{\\mix}.\\otssig) = \\true \\\\\n & \\text{where} \\ \\datatobesigned = \\addr \\concat \\inp{\\mix}.\\primInp \\concat \\inp{\\mix}.\\zkp \\concat \\inp{\\mix}.\\ciphers, \\\\\n & \\text{and} \\ \\msg = \\crhots{\\datatobesigned}\n \\end{align*}\n \\item Check that $\\inp{\\mix}.\\primInp.\\mkroot$ corresponds to a valid state of the Merkle tree held on $\\mixer$, i.e.~check that:\n \\[\n \\inp{\\mix}.\\primInp.\\mkroot \\in \\rootset'\n \\]\n where $\\rootset'$ is the set of all Merkle roots corresponding to one of the states of the Merkle tree.\n \\item Check that $\\vin$ corresponds to the value $\\val$ of the transaction object, i.e.~check that:\n \\[\n \\vin = \\zethTx.\\val\n \\]\n \\end{enumerate}\n \\item If all checks above pass, i.e.~if $\\zethVerifyTx(\\zethTx)$ returns $\\true$, then the following additional modifications are made in $\\wstate$:\n \\begin{enumerate}\n \\item Add the commitments $\\inp{\\mix}.\\primInp.\\cms{}$ to the Merkle tree held on $\\mixer$.\n \\item $\\rootset' \\gets \\rootset' \\cup \\smallset{\\mkroot'}$, where $\\mkroot'$ is the Merkle root of the Merkle tree after insertion of the commitments $\\inp{\\mix}.\\primInp.\\cms{}$ in the Merkle tree.\n \\item $\\nullifierset \\gets \\nullifierset \\cup \\indexedset{\\nf{i}}{i \\in [\\jsin]}$, i.e.~the nullifiers $\\nfs{}$ become ``declared''.\n \\item Modify the \\ethereum~balances according to the public values:\n \\begin{itemize}\n \\item $\\wstate[\\eparty{S}.\\addr].\\balance = \\wstate[\\eparty{S}.\\addr].\\balance - \\vin$\n \\item $\\wstate[\\eparty{S}.\\addr].\\balance = \\wstate[\\eparty{S}.\\addr].\\balance + \\vout$\n \\item $\\mixer.\\balance = \\mixer.\\balance + \\vin$\n \\item $\\mixer.\\balance = \\mixer.\\balance - \\vout$\n \\end{itemize}\n \\item Emit an event (\\cref{sssec:ethereum-events}) $\\evMixOut$ of type \\mixEventDType, containing the new root $\\mkroot'$ of the Merkle tree of commitments, the nullifiers $\\indexedset{\\nf{i}}{i \\in [\\jsin]}$, commitments to the newly created \\zethnotes $\\inp{\\mix}.\\primInp.\\cms{}$, and the corresponding ciphertexts $\\inp{\\mix}.\\primInp.\\ciphers$.\n \\end{enumerate}\n\\end{enumerate}\n\n\\begin{remark}\\label{zeth-protocol:process-tx:dispatch-call}\n In some deployments, verification of the zk-SNARK proof $\\zkp$ may be delegated to an external mechanism (in such a way that integrity of the system can still be guaranteed), and $\\zkp$ may not appear as public data on-chain. For example, where multiple \\zeth~transactions are aggregated by a system such as that described in \\cite{rondelet2020zecale}, the original zk-SNARK proofs become auxiliary inputs to a ``wrapping'' SNARK, which checks their validity via a single proof verification. A modified version of the \\mix~function receives \\mix~parameters from a specific contract (known to behave correctly with respect to the delegation protocol) without $\\inp{\\mix}.\\zkp$.\n\n In this case, the value of $\\datatobesigned$ in \\cref{zeth-protocol:process-tx:check-otsig} may be replaced by:\n \\[\n \\datatobesigned = \\addr \\concat \\inp{\\mix}.\\primInp \\concat \\inp{\\mix}.\\ciphers, \\\\\n \\]\n and the equivalent change must be made when generating the \\mix~parameters, as described in \\cref{zeth-protocol:mix-inp:otsig-for-dispatch-call}. The transaction, as presented to an aggregator, is malleable since the $\\groth$ proofs can be modified in a way that preserves the validity. However, once aggregated, the transaction proof data for the transaction does not appear on chain (it is an auxiliary input to some externally generated ``wrapping'' poof). Hence, the transaction can only be identified by the remaining public data, which is protected by the one-time signature $\\otsSigma$, and hence non-malleable.\n\n The external scheme used to verify the zk-SNARK proof must specify the exact requirements of the contact and how it should be modified, including any further checks that must be carried out. Thus, this modification is not part of the core \\zeth~protocol described in this document, but an augmentation forming part of an external protocol. However, for completeness, we briefly describe a \\emph{dispatch} entry point in the proof-of-concept \\mixer~implementation which supports delegation of proof verification, as described above. The \\emph{dispatch} entry point performs the following checks:\n \\begin{itemize}\n \\item Check that the \\mixer~has been deployed with the (immutable) address of a trusted contract, permitted to call this entry point. Otherwise, abort.\n \\item Check that the caller $msg.sender$ matches the permitted caller set at deployment time, otherwise abort.\n \\item Perform all checks related to the \\mix~parameters, except \\cref{zeth-protocol:process-tx:verify-zkp}, with the modification to \\cref{zeth-protocol:process-tx:check-otsig} described in this remark.\n \\end{itemize}\n After these checks, the state-transition continues as normal.\n\n The \\zeth~client implementation is also augmented to include a flag to enable the corresponding change described in \\cref{zeth-protocol:mix-inp:otsig-for-dispatch-call} (to generate a signature on the modified $\\datatobesigned$). Naturally, the \\emph{dispatch} entry point can only be used with parameters generated using this flag (otherwise the signature check will fail).\n\n Such modifications \\MUSTNOT{} be implemented except as described by the secure external scheme for delegating proof verification.\n\\end{remark}\n", "meta": {"hexsha": "0741d1e1e0e1e68483303a0f5872e7aa375ddeaa", "size": 9164, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapters/chap02-sec05.tex", "max_stars_repo_name": "clearmatics/zeth-specifications", "max_stars_repo_head_hexsha": "ba29c67587395f5c7b26b52ee7ab9cba12f1cc6b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-04-29T18:22:00.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-29T18:22:00.000Z", "max_issues_repo_path": "chapters/chap02-sec05.tex", "max_issues_repo_name": "clearmatics/zeth-specifications", "max_issues_repo_head_hexsha": "ba29c67587395f5c7b26b52ee7ab9cba12f1cc6b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 13, "max_issues_repo_issues_event_min_datetime": "2020-10-27T10:41:50.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-16T10:57:05.000Z", "max_forks_repo_path": "chapters/chap02-sec05.tex", "max_forks_repo_name": "clearmatics/zeth-specifications", "max_forks_repo_head_hexsha": "ba29c67587395f5c7b26b52ee7ab9cba12f1cc6b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-07-26T04:51:29.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-26T04:51:29.000Z", "avg_line_length": 95.4583333333, "max_line_length": 678, "alphanum_fraction": 0.6745962462, "num_tokens": 2476, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7981867777396212, "lm_q2_score": 0.43398146480389854, "lm_q1q2_score": 0.3463982669905446}} {"text": "\\documentclass{article}\n\n\\usepackage{amsmath,amssymb}\n\\usepackage[dvipdfmx]{hyperref,graphicx}\n\\usepackage{listings}\n\n\\lstset{%\n language={lisp},\n basicstyle={\\small\\ttfamily},%\n identifierstyle={\\small},%\n commentstyle={\\small\\itshape},%\n keywordstyle={\\small\\bfseries},%\n ndkeywordstyle={\\small},%\n stringstyle={\\small\\ttfamily},\n frame={tb},\n keepspaces=true,\n breaklines=true,\n columns=[l]{fullflexible},%\n numbers=left,%\n xrightmargin=0zw,%\n xleftmargin=3zw,%\n numberstyle={\\scriptsize},%\n stepnumber=1,\n numbersep=1zw,%\n lineskip=-0.5ex%\n}\n\n\\title{Typing Rule of Baremetalisp}\n\\author{Yuuki Takano\\\\ ytakano@wide.ad.jp}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{Introduction}\n\nIn this paper, I will formally describe the typing rule of Baremetalisp,\nwhich is a well typed Lisp for trusted execution environment.\n\n\\begin{table}[tb]\n \\centering\n \\caption{Notation}\n \\label{tab:notation}\n \\begin{tabular}{rl}\n $A \\Rightarrow B$ & logical implication (if A then B)\\\\\n $e$ & expression \\\\\n $z$ & integer literal such as 10, -34, 112 \\\\\n $x$ & variable \\\\\n $t$ & type variable \\\\\n $D$ & type name of user defined data \\\\\n $L$ & label of user defined data \\\\\n $E$ & effect \\\\\n $E_\\mathcal{T}: T \\rightarrow E$ & effect of type \\\\\n $\\mathcal{T}$ & type \\\\\n $C$ & type constraint \\\\\n $io(C): C \\rightarrow \\mathtt{Bool}$ & does $C$ contain $\\mathtt{IO}$ functions? \\\\\n $\\Gamma$ & context \\\\\n $\\mathcal{P}$ & pattern \\\\\n $\\mathcal{P}_{let}$ & pattern of let expression \\\\\n $\\mathcal{T}_1 \\equiv_\\alpha \\mathcal{T}_2$ & $\\mathcal{T}_1$ and $\\mathcal{T}_2$ are $\\alpha$-equivalent \\\\\n $\\mathcal{S} : t \\rightarrow \\mathcal{T}$ & substitution from type variable to type\\\\\n $\\mathcal{T} \\cdot \\mathcal{S}$ & apply $\\mathcal{S}$ to $\\mathcal{T}$ \\\\\n $\\mathcal{X}$ & set of $t$ \\\\\n $FV_\\mathcal{T} : \\mathcal{T} \\rightarrow \\mathcal{X}$ & function from $\\mathcal{T}$ to its free variables\\\\\n $FV_\\Gamma : \\Gamma \\rightarrow \\mathcal{X}$ & function from $\\Gamma$ to its free variables\\\\\n $Size : L \\rightarrow \\mathtt{Int}$ & the number of labels $L$'s type has \\\\\n $\\Gamma \\vdash e : \\mathcal{T}\\ |_\\mathcal{X}\\ C$ & $e$'s type is deduced as $\\mathcal{T}$ from $\\Gamma$ \\\\\n & under constraint $C$ and type variables $\\mathcal{X}$\n \\end{tabular}\n\\end{table}\n\n\\begin{figure}[tb]\n \\centering\n \\begin{tabular}{rrll}\n $\\mathcal{C}$ & := & $\\mathcal{T} = \\mathcal{T}, \\mathcal{C}$ & \\bf{type constraint} \\\\\n & $|$ & $\\varnothing$ \\\\ \\\\\n\n $\\Gamma$ & := & & \\bf{context} \\\\\n & & $x: \\mathcal{T}, \\Gamma$ & type of variable \\\\\n & $|$ & $L: \\mathcal{T}, \\Gamma$ & type of label \\\\\n & $|$ & $L_{nth}: \\mathcal{T}, \\Gamma$ & n-th type of label's element \\\\\n & $|$ & $\\varnothing$ \\\\ \\\\\n\n $E$ & := & $\\mathtt{Pure}\\ |\\ \\mathtt{IO}$ & \\bf{effect} \\\\ \\\\\n\n $\\mathcal{T}$ & := & & \\bf{type} \\\\\n & & $\\mathtt{Int}$ \\\\\n & $|$ & $\\mathtt{Bool}$ \\\\\n & $|$ & $'(\\mathcal{T})$ & list type \\\\\n & $|$ & $[\\mathcal{T}+]$ & tuple type \\\\\n & $|$ & $D$ & user defined type \\\\\n & $|$ & $(D\\ \\mathcal{T}+)$ & user defined type with type arguments \\\\\n & $|$ & $(E\\ (\\rightarrow\\ (\\mathcal{T}*)\\ \\mathcal{T}))$ & function type \\\\\n & $|$ & $t$ & type variable \\\\ \\\\\n\n $\\mathcal{P}$ & := & & \\bf{pattern} \\\\\n & & $x$ & variable \\\\\n & $|$ & $L$ & label \\\\\n & $|$ & $(L\\ \\mathcal{P}+)$ & label with patterns \\\\\n & $|$ & $'()$ & empty list \\\\\n & $|$ & $[\\mathcal{P}+]$ & tuple \\\\ \\\\\n\n $\\mathcal{P}_{let}$ & := & & \\bf{patten for let} \\\\\n & & $x$ & variable \\\\\n & $|$ & $(L\\ \\mathcal{P}_{let}+)$ & label with patterns \\\\\n & $|$ & $[\\mathcal{P}_{let}+]$ & tuple \\\\\n \\end{tabular}\n \\caption{Syntax}\n \\label{fig:syntax}\n\\end{figure}\n\n\\section{Notation and Syntax}\n\nTable~\\ref{tab:notation} and Fig.~\\ref{fig:syntax} shows notation used in this paper\nand syntax for the typing rule, respectively.\n\n\\begin{lstlisting}[caption=Example of variable and type,label=src:vars]\n(defun add (a b) (Pure (-> (Int Int) Int))\n (+ a b))\n\\end{lstlisting}\n\n$x$ is a variable.\nFor example, $x \\in \\{a, b\\}$ in Listing~\\ref{src:vars}.\n$\\mathcal{T}$ is a type.\nFor example, $\\mathcal{T} \\in \\{\\mathtt{Int}, (\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int})\\}$\nin Listing~\\ref{src:vars}.\n$(\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int})$ is a function type which takes 2 integer values\nand return 1 integer value.\n$\\mathtt{Pure}$ in Listing~\\ref{src:vars} denotes the effect of the function but I just ignore it now.\nFunction effects will be described in Sec.~\\ref{sec:effect}.\n$\\mathcal{T}$ can be other forms as described in Fig.~\\ref{fig:syntax} such as\n$\\mathtt{Bool}$, $'(\\mathtt{Int})$, $[\\mathtt{Bool}\\ \\mathtt{Int}]$, $(\\mathrm{List}\\ a)$, $(\\mathrm{List}\\ \\mathtt{Int})$.\n$C$ is a type constraint, which is a set of pairs of types.\nFor example, $C = \\{(\\rightarrow\\ (t_1\\ t_2)\\ t) = (\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int})\\}$ deduced from\nListing~\\ref{src:vars} means $(\\rightarrow\\ (t_1\\ t_2)\\ t)$ and $(\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int})$ are\nsemantically equal and every type variable in $C$, $t_1, t_2, t$, is thus $\\mathtt{Int}$.\n$\\Gamma$ is a map from variable and label to type.\nFor example, $\\Gamma = \\{a : t_1, b : t_2, + : (\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int})\\}$\nin Listing~\\ref{src:vars}.\n$\\Gamma$ is called context generally, thus I call $\\Gamma$ context in this paper.\n\n\\begin{lstlisting}[caption=Example of user defined data type,label=src:cons]\n(data (List a)\n (Cons a (List a))\n Nil)\n\\end{lstlisting}\n\n$t$ is a type variable.\nFor example, $t \\in \\{a\\}$ in Listing~\\ref{src:cons}.\n$L$ is a label for user defined type.\nFor example, $L \\in \\{\\mathrm{Cons}, \\mathrm{Nil}\\}$ in Listing~\\ref{src:cons}.\n$D$ is user defined data.\nFor example, $D \\in \\{\\mathrm{List}\\}$ in Listing~\\ref{src:cons}.\n$\\Gamma$ will hold mapping from labels in addition to variables.\nFor example,\n$\\Gamma = \\{\\mathrm{Cons} : (\\mathrm{List}\\ a), \\mathrm{Nil} : (\\mathrm{List}\\ a), \\mathrm{Cons}_{1st} : a, \\mathrm{Cons}_{2nd} : (\\mathrm{List}\\ a)\\}$\nin Listing~\\ref{src:cons}.\n\n$FV_\\mathcal{T}$ and $FV_\\Gamma$ are functions, which take $\\mathcal{T}$ and $\\Gamma$ and return free variables.\nFor example, $FV_\\mathcal{T}((\\rightarrow\\ (t_1\\ t_2)\\ t)) = \\{t_1, t_2, t\\}$ and\n\\begin{equation*}\n \\begin{aligned}\n &FV_\\Gamma(\\{a : t_1, b : t_1, + : (\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int})\\}) \\\\\n &=\\{FV_\\mathcal{T}(t_1), FV_\\mathcal{T}(t_1), FV_\\mathcal{T}((\\rightarrow\\ (\\mathtt{Int}\\ \\mathtt{Int})\\ \\mathtt{Int}))\\} \\\\\n &=\\{t_1, t_2\\}.\n \\end{aligned}\n\\end{equation*}\n$\\mathcal{T}_1 \\equiv_\\alpha \\mathcal{T}_2$ denotes that $\\mathcal{T}_1$ and $\\mathcal{T}_2$ are $\\alpha$-equivalent,\nwhich means $\\mathcal{T}_1$ and $\\mathcal{T}_2$ are semantically equal.\nFor example, $(\\rightarrow\\ (t_1\\ t_2)\\ t) \\equiv_\\alpha (\\rightarrow\\ (t_{10}\\ t_{11})\\ t_{12})$.\n$\\mathcal{S}$ is a substitution, which is a map from type variable to type,\nand it can be applied to $\\mathcal{T}$ as $\\mathcal{T} \\cdot \\mathcal{S}$.\nFor example, if $\\mathcal{S}(t_1) = [\\mathtt{Bool}\\ \\mathtt{Int}], \\mathcal{S}(t_2) = (\\mathrm{List}\\ t_3)$ then\n$(\\rightarrow\\ (t_1\\ t_2)\\ t) \\cdot \\mathcal{S} = (\\rightarrow\\ ([\\mathtt{Bool}\\ \\mathtt{Int}]\\ (\\mathrm{List}\\ t_3))\\ t)$.\n\n\\begin{lstlisting}[caption=Example of pattern matching,label=src:match]\n(data Dim2 (Dim2 Int Int))\n\n(data (Maybe t)\n (Just t)\n Nothing)\n\n(defun match-let (a) (Pure (-> ((Maybe Dim2)) Int))\n (match a\n ((Just val)\n (let (((Dim2 x y) val))\n (+ x y)))\n (Nothing\n 0)))\n\\end{lstlisting}\n$\\mathcal{P}$ and $\\mathcal{P}_{let}$ are pattern in match and let expressions.\nFor example, in listings~\\ref{src:match}, $(\\mathrm{Just}\\ val)$ and Nothing at line 9 and 12 are from $\\mathcal{P}$\nand $(\\mathrm{Dim2}\\ x\\ y)$ at line 10 is from $\\mathcal{P}_{let}$.\n$Size$ is a function which takes a label and return the number of labels the label's type has.\nFor example, $Size(\\mathrm{Just}) = Size(\\mathrm{Nothing}) = 2$ because Maybe type has 2 labels\nand $Size(\\mathrm{Dim2}) = 1$ because Dim2 type has 1 label in listings~\\ref{src:match}.\n\n\\begin{figure}[tb]\n \\centering\n \\begin{tabular}{rlrl}\n $\\Gamma \\vdash \\mathtt{true} : \\mathtt{Bool}\\ |_\\varnothing\\ \\varnothing$ & (T-True) &\n $\\Gamma \\vdash \\mathtt{false} : \\mathtt{Bool}\\ |_\\varnothing\\ \\varnothing$ & (T-False) \\vspace{5mm} \\\\\n\n $\\dfrac{x : T \\in \\Gamma}{\\Gamma \\vdash x : T\\ |_\\varnothing\\ \\varnothing}$ & (T-Var) &\n $\\Gamma \\vdash z : \\mathtt{Int}\\ |_\\varnothing\\ \\varnothing$ & (T-Num) \\vspace{5mm} \\\\\n\n $\\dfrac{x : T' \\in \\Gamma \\hspace{5mm} T' \\cdot S \\equiv_\\alpha T}{\\Gamma \\vdash x : T\\ |_{FV_\\mathcal{T}(T)}\\ \\varnothing}$ & (T-VarPoly) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma_0 \\vdash \\mathcal{P}_{let} : \\mathcal{T}_0\\ |_{\\mathcal{X}_0}\\ C_0 \\hspace{5mm}\n \\Gamma \\vdash e_1 : \\mathcal{T}_1\\ |_{\\mathcal{X}_1}\\ C_1 \\hspace{5mm}\n \\Gamma, \\Gamma_0 \\vdash e_2 : \\mathcal{T}_2\\ |_{\\mathcal{X}_2}\\ C_2\\\\\n &\\mathcal{X}_0 \\cap \\mathcal{X}_1 \\cap \\mathcal{X}_2 = \\varnothing \\hspace{5mm}\n C = C_0 \\cup C_1 \\cup C_2 \\cup \\{ \\mathcal{T}_0 = \\mathcal{T}_1 \\}\n \\end{aligned}\n }{\n \\Gamma \\vdash (\\mathtt{let1}\\ \\mathcal{P}_{let}\\ e_1\\ e_2) : \\mathcal{T}_2\\ |_{\\mathcal{X}_0 \\cup \\mathcal{X}_1 \\cup \\mathcal{X}_2}\\ C\n }$} & (T-Let1) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash e_1 : \\mathcal{T}_1\\ |_{\\mathcal{X}_1}\\ C_1 \\hspace{5mm} \\Gamma \\vdash e_2 : \\mathcal{T}_2\\ |_{\\mathcal{X}_2}\\ C_2 \\hspace{5mm} \\Gamma \\vdash e_3 : \\mathcal{T}_3\\ |_{\\mathcal{X}_3}\\ C_3 \\\\\n &\\mathcal{X}_1 \\cap \\mathcal{X}_2 \\cap \\mathcal{X}_3 = \\varnothing \\hspace{5mm} C = C_1 \\cup C_2 \\cup C_3 \\cup \\{ \\mathcal{T}_1 = \\mathtt{Bool}, \\mathcal{T}_2 = T_3 \\}\n \\end{aligned}\n }{\n \\Gamma \\vdash (\\mathtt{if}\\ e_1\\ e_2\\ e_3) : \\mathcal{T}_2\\ |_{\\mathcal{X}_1 \\cup \\mathcal{X}_2 \\cup \\mathcal{X}_3}\\ C\n }$} & (T-If) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash e_1 : \\mathcal{T}_1\\ |_{\\mathcal{X}_1}\\ C_1 \\hspace{5mm}\n \\Gamma \\vdash e_2 : \\mathcal{T}_2\\ |_{\\mathcal{X}_2}\\ C_2 \\land \\cdots \\land \\Gamma \\vdash e_n : \\mathcal{T}_n\\ |_{\\mathcal{X}_n}\\ C_n \\\\\n &\\{t\\} \\cap FV_\\Gamma(\\Gamma) = \\varnothing \\hspace{5mm} \\{t\\} \\cap \\mathcal{X}_1 \\cap \\cdots \\cap \\mathcal{X}_n = \\varnothing\\\\\n &\\mathcal{X} = \\{t\\} \\cup \\mathcal{X}_1 \\cup \\cdots \\cup \\mathcal{X}_n \\hspace{5mm} E = E_\\mathcal{T}(\\mathcal{T}_1) \\\\\n &C = C_1 \\cup \\cdots \\cup C_n \\cup \\{ \\mathcal{T}_1 = (E\\ (\\rightarrow\\ (\\mathcal{T}_2\\ \\cdots\\ \\mathcal{T}_n)\\ t)) \\}\n \\end{aligned}\n }{\n \\Gamma \\vdash (e_1\\ e_2\\ \\cdots\\ e_n) : t\\ |_\\mathcal{X}\\ C\n }$} & (T-App) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash e_0 : \\mathcal{T}_0\\ |_{\\mathcal{X}_0}\\ C_0 \\\\\n &\\Gamma, \\Gamma_1 \\vdash e_1 : \\mathcal{T}_{e1}\\ |_{\\mathcal{X}_{e1}}\\ C_{e1} \\land \\cdots \\land \\Gamma, \\Gamma_n \\vdash e_n : \\mathcal{T}_{en}\\ |_{\\mathcal{X}_{en}}\\ C_{en} \\\\\n &\\Gamma_1 \\vdash \\mathcal{P}_1 : \\mathcal{T}_{p1}\\ |_{\\mathcal{X}_{p1}}\\ C_{p1} \\land \\cdots \\land \\Gamma_n \\vdash \\mathcal{P}_{pn} : \\mathcal{T}_{pn}\\ |_{\\mathcal{X}_{pn}}\\ C_{pn} \\\\\n &\\mathcal{X}_0 \\cap \\mathcal{X}_{e1} \\cap \\cdots \\cap \\mathcal{X}_{en} \\cap \\mathcal{X}_{p1} \\cap \\cdots \\cap \\mathcal{X}_{pn} = \\varnothing \\\\\n &\\mathcal{X} = \\mathcal{X}_0 \\cup \\mathcal{X}_{e1} \\cup \\cdots \\cup \\mathcal{X}_{en} \\cup \\mathcal{X}_{p1} \\cup \\cdots \\cup \\mathcal{X}_{pn} \\\\\n &\\begin{aligned}\n C =\\ &C_0 \\cup C_{e1} \\cup \\cdots \\cup C_{en} \\cup C_{p1} \\cup \\cdots \\cup C_{pn} \\cup \\\\\n &\\{\\mathcal{T}_0 = \\mathcal{T}_{p1}, \\cdots, \\mathcal{T}_0 = \\mathcal{T}_{pn}\\} \\cup\n \\{\\mathcal{T}_{e1} = \\mathcal{T}_{e2}, \\cdots, \\mathcal{T}_{e1} = \\mathcal{T}_{en}\\}\n \\end{aligned}\n \\end{aligned}\n }{\n \\Gamma \\vdash (\\mathtt{match}\\ e_0\\ (\\mathcal{P}_1\\ e_1)\\ \\cdots\\ (\\mathcal{P}_n\\ e_n)) : T_{e1}\\ |_\\mathcal{X}\\ C\n }$} & (T-Match)\n \\end{tabular}\n \\caption{Typing rule (1/2)}\n \\label{fig:typing1}\n\\end{figure}\n\n\\begin{figure}[tb]\n \\centering\n \\begin{tabular}{rlrl}\n $\\Gamma \\vdash\\ '() :\\ '(T)\\ |_{\\{T\\}}\\ \\varnothing$ & (T-Nil) &\n $\\dfrac{L : \\mathcal{T}' \\in \\Gamma \\hspace{5mm} \\mathcal{T}' \\cdot \\mathcal{S} \\equiv_\\alpha \\mathcal{T}}{\\Gamma \\vdash L : \\mathcal{T}\\ |_{FV_\\mathcal{T}(\\mathcal{T})}\\ \\varnothing}$ & (T-Label0) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash e_1 : T_1\\ |_{\\mathcal{X}_1}\\ C_1 \\land \\cdots \\land \\Gamma \\vdash e_n : T_n\\ |_{\\mathcal{X}_n}\\ C_n \\\\\n &\\mathcal{X}_1 \\cap \\cdots \\cap \\mathcal{X}_n = \\varnothing \\hspace{5mm}\n \\mathcal{X} = \\mathcal{X}_1 \\cup \\cdots \\cup \\mathcal{X}_n \\hspace{5mm}\n C = C_1 \\cup \\cdots \\cup C_n\n \\end{aligned}\n }{\\Gamma \\vdash [e_1\\ \\cdots\\ e_n] : [T_1\\ \\cdots\\ T_n]\\ |_\\mathcal{X}\\ C}$\n } & (T-Tuple) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash e_1 : T_1\\ |_{\\mathcal{X}_1}\\ C_1 \\land \\cdots \\land \\Gamma \\vdash e_n : T_n\\ |_{\\mathcal{X}_n}\\ C_n \\\\\n &\\mathcal{X}_1 \\cap \\cdots \\cap \\mathcal{X}_n = \\varnothing \\hspace{5mm}\n \\mathcal{X} = \\mathcal{X}_1 \\cup \\cdots \\cup \\mathcal{X}_n \\\\\n &C = C_1 \\cup \\cdots \\cup C_n \\cup \\{T_1 = T_2, \\cdots, T_1 = T_n \\}\n \\end{aligned}\n }{\\Gamma \\vdash\\ '(e_1\\ \\cdots\\ e_n) :\\ '(T_1)\\ |_\\mathcal{X}\\ C}$\n } & (T-List) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash e_1 : \\mathcal{T}_1\\ |_{\\mathcal{X}_1}\\ C_1 \\land \\cdots \\land\n \\Gamma \\vdash e_n : \\mathcal{T}_n\\ |_{\\mathcal{X}_n}\\ C_n \\\\\n &L : \\mathcal{T}_0' \\in \\Gamma \\hspace{5mm} \\mathcal{T}_0' \\cdot \\mathcal{S}\\equiv_\\alpha \\mathcal{T}_0 \\hspace{5mm} FV(\\mathcal{T}_0) \\cap \\mathcal{X}_1 \\cap \\cdots \\cap \\mathcal{X}_n = \\varnothing \\\\\n &FV_\\mathcal{T}(\\mathcal{T}_0) \\cap FV_\\Gamma(\\Gamma) = \\varnothing \\hspace{5mm}\n \\mathcal{X} = FV(\\mathcal{T}_0) \\cup \\mathcal{X}_1 \\cup \\cdots \\cup \\mathcal{X}_n \\\\\n &L_{1st} : T_1' \\in \\Gamma \\land \\cdots \\land L_{nth} : T_n' \\in \\Gamma \\\\\n &C = C_1 \\cup \\cdots \\cup C_n \\cup \\{T_1' \\cdot \\mathcal{S} = \\mathcal{T}_1, \\cdots, T_n' \\cdot \\mathcal{S} = \\mathcal{T}_n\\} \\\\\n \\end{aligned}\n }{\n \\Gamma \\vdash (L\\ e_1\\ \\cdots\\ e_n) : \\mathcal{T}_0\\ |_{\\mathcal{X}}\\ C\n }$} & (T-Label) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma, x_1 : t_1, \\cdots, x_n : t_n \\vdash e : \\mathcal{T}_0\\ |_{\\mathcal{X}}\\ C_0 \\hspace{5mm} \\neg io(C)\\\\\n &C = \\{\\mathcal{T} = (\\mathtt{Pure}\\ (\\rightarrow\\ (t_1\\ \\cdots\\ t_n)\\ \\mathcal{T}_0))\\} \\cup C_0\n \\end{aligned}\n }\n {\n \\Gamma \\vdash (\\mathtt{lambda}\\ (x_1\\ \\cdots\\ x_n)\\ e) : \\mathcal{T}\\ |_\\mathcal{X}\\ C\n }$\n } & (T-Lambda) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma, x_1 : t_1, \\cdots, x_n : t_n \\vdash e : \\mathcal{T}_0\\ |_{\\mathcal{X}}\\ C_0 \\hspace{5mm}\\\\\n &E = E_\\mathcal{T}(\\mathcal{T}) \\hspace{5mm} (E = \\mathtt{Pure}) \\Rightarrow \\neg io(C)\\\\\n &C = C_0 \\cup \\{\\mathcal{T} = (E\\ (\\rightarrow (\\mathcal{T}_1\\ \\cdots\\ \\mathcal{T}_n)\\ \\mathcal{T}_0)) \\}\n \\end{aligned}\n }{\n \\Gamma \\vdash (\\mathtt{defun}\\ \\mathrm{name}\\ (x_1\\ \\cdots\\ x_n)\\ \\mathcal{T}\\ e) : \\mathcal{T}\\ |_\\mathcal{X}\\ C\n }$\n } & (T-Defun)\n \\end{tabular}\n \\caption{Typing rule (2/2)}\n \\label{fig:typing2}\n\\end{figure}\n\n\n\\begin{figure}[tb]\n \\centering\n \\begin{tabular}{rlrl}\n $\\Gamma \\vdash \\mathtt{true} : \\mathtt{Bool}\\ |_\\varnothing\\ \\varnothing$ & (P-True) &\n $\\Gamma \\vdash \\mathtt{false} : \\mathtt{Bool}\\ |_\\varnothing\\ \\varnothing$ & (P-False) \\vspace{5mm} \\\\\n\n $\\dfrac{x : T \\in \\Gamma}{\\Gamma \\vdash x : T\\ |_\\varnothing\\ \\varnothing}$ & (P-Var) &\n $\\Gamma \\vdash z : \\mathtt{Int}\\ |_\\varnothing\\ \\varnothing$ & (P-Num) \\vspace{5mm} \\\\\n\n $\\Gamma \\vdash\\ '() :\\ '(T)\\ |_{\\{T\\}}\\ \\varnothing$ & (P-Nil) &\n $\\dfrac{L : \\mathcal{T}' \\in \\Gamma \\hspace{5mm} \\mathcal{T}' \\cdot \\mathcal{S} \\equiv_\\alpha \\mathcal{T}}{\\Gamma \\vdash L : \\mathcal{T}\\ |_{FV_\\mathcal{T}(\\mathcal{T})}\\ \\varnothing}$ & (P-Label0) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash \\mathcal{P}_1 : \\mathcal{T}_1\\ |_{\\mathcal{X}_1}\\ C_1 \\land \\cdots \\land\n \\Gamma \\vdash \\mathcal{P}_n : \\mathcal{T}_n\\ |_{\\mathcal{X}_n}\\ C_n \\\\\n &L : \\mathcal{T}_0' \\in \\Gamma \\hspace{5mm} \\mathcal{T}_0' \\cdot \\mathcal{S}\\equiv_\\alpha \\mathcal{T}_0 \\hspace{5mm} FV(\\mathcal{T}_0) \\cap \\mathcal{X}_1 \\cap \\cdots \\cap \\mathcal{X}_n = \\varnothing \\\\\n &FV_\\mathcal{T}(\\mathcal{T}_0) \\cap FV_\\Gamma(\\Gamma) = \\varnothing \\hspace{5mm}\n \\mathcal{X} = FV(\\mathcal{T}_0) \\cup \\mathcal{X}_1 \\cup \\cdots \\cup \\mathcal{X}_n \\\\\n &L_{1st} : T_1' \\in \\Gamma \\land \\cdots \\land L_{nth} : T_n' \\in \\Gamma \\\\\n &C = C_1 \\cup \\cdots \\cup C_n \\cup \\{T_1' \\cdot \\mathcal{S} = \\mathcal{T}_1, \\cdots, T_n' \\cdot \\mathcal{S} = \\mathcal{T}_n\\} \\\\\n &Size(L) = 1\\ \\mbox{for only}\\ P_{let}\n \\end{aligned}\n }{\n \\Gamma \\vdash (L\\ \\mathcal{P}_1\\ \\cdots\\ \\mathcal{P}_n) : \\mathcal{T}_0\\ |_{\\mathcal{X}}\\ C\n }$} & (P-Label) \\vspace{5mm} \\\\\n\n \\multicolumn{3}{r}{\n $\\dfrac{\n \\begin{aligned}\n &\\Gamma \\vdash \\mathcal{P}_1 : \\mathcal{T}_1\\ |_{\\mathcal{X}_1}\\ C_1 \\land \\cdots \\land\n \\Gamma \\vdash \\mathcal{P}_n : \\mathcal{T}_n\\ |_{\\mathcal{X}_n}\\ C_n \\\\\n &\\mathcal{X}_1 \\cap \\cdots \\cap \\mathcal{X}_n = \\varnothing \\hspace{5mm}\n \\mathcal{X} = \\mathcal{X}_1 \\cup \\cdots \\cup \\mathcal{X}_n \\hspace{5mm}\n C = C_1 \\cup \\cdots \\cup C_n\n \\end{aligned}\n }{\n \\Gamma \\vdash [\\mathcal{P}_1\\ \\cdots\\ \\mathcal{P}_n] : [\\mathcal{T}_1 \\cdots \\mathcal{T}_n]\\ |_{\\mathcal{X}}\\ C\n }$} & (P-Tuple) \\\\\n \\end{tabular}\n \\caption{Typing rule of pattern}\n\\end{figure}\n\n\\section{Typing Rule}\n\nIn this section, I will introduce the typing rule of Baremetalisp.\nBefore describing the rule, I introduce an assumption\nthat there is no variable shadowing to make it simple.\nThis means that every variable should be properly $\\alpha$-converted\nby using the De Bruijn index technique or variable shadowing should be handled\nwhen implementing the type inference algorithm.\n\nFig.~\\ref{fig:typing1} and \\ref{fig:typing2} are the typing rule of\nexpressions and function definitions.\n\n\\section{Effect}\n\\label{sec:effect}\n\n\\end{document}", "meta": {"hexsha": "97be5c4084a1926d42c98c5c7b668faf92d9fa89", "size": 20142, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "specification/typing.tex", "max_stars_repo_name": "syohex/blisp", "max_stars_repo_head_hexsha": "3a884c44a74586d23aba17c60f24c5be80e476aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 115, "max_stars_repo_stars_event_min_datetime": "2020-11-20T13:15:44.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-29T23:01:16.000Z", "max_issues_repo_path": "specification/typing.tex", "max_issues_repo_name": "syohex/blisp", "max_issues_repo_head_hexsha": "3a884c44a74586d23aba17c60f24c5be80e476aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2021-11-06T23:47:16.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-11T04:47:58.000Z", "max_forks_repo_path": "specification/typing.tex", "max_forks_repo_name": "syohex/blisp", "max_forks_repo_head_hexsha": "3a884c44a74586d23aba17c60f24c5be80e476aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2020-07-31T11:12:47.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-29T01:57:03.000Z", "avg_line_length": 50.608040201, "max_line_length": 221, "alphanum_fraction": 0.5518319929, "num_tokens": 7354, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6926419831347361, "lm_q2_score": 0.5, "lm_q1q2_score": 0.34632099156736806}} {"text": "% !TEX root = ../00_thesis.tex\n%-------------------------------------------------------------------------------\n\\section{Designing \\triscale}\n\\label{sec:triscale}\n%-------------------------------------------------------------------------------\n\nIn this section, we first describe the data analysis performed by \\triscale and how the analysis procedure is linked to the design of experiments~(\\cref{subsec:metrics} to \\cref{subsec:repeatability}).\nWe then illustrate how the formalism introduced by \\triscale allows to unambiguously describe an entire performance evaluation with only a handful of parameters~(\\cref{subsec:parameters}).\nThereafter, we detail the robust and non-parametric statistical methods used by \\triscale~(\\cref{subsec:triscale_stats}), and discuss how the framework assists a user in deciding the required time span for a series of runs~(\\cref{subsec:network_profiling}).\nWe finally show how \\triscale helps assessing the reproducibility of experiments by computing a variability score~(\\cref{subsec:reproducibility}).\n\n%-------------------------------------------------------------------------------\n\\subsection{Runs and Metrics}\n\\label{subsec:metrics}\n\n\\afterpage{\n\\begin{figure}\n \\centering\n \\begin{subfigure}{\\linewidth}\n \\centering\n \t\\href{\\triscalefig{Figure-3a}}{\n \\includegraphics[scale=1]{Figures/plot_example_metric.pdf}}\n \\caption{\\raggedright Raw data (one-way delay) and metric data (95th percentile).}\n \\label{fig:analysis_metric}\n \\end{subfigure}\n\n \\vspace{5pt}\n\n \\begin{subfigure}{0.45\\linewidth}\n \\centering\n \t\\href{\\triscalefig{Figure-3b}}{\n \\includegraphics[scale=1]{Figures/plot_example_KPI.pdf}}\n \\caption{Runs' metric data and corresponding KPI value.}\n \\label{fig:analysis_kpi}\n \\end{subfigure}\n %\n \\hfill\n \\begin{subfigure}{0.45\\linewidth}\n \\centering\n \t \\href{\\triscalefig{Figure-3c}}{\n \\includegraphics[scale=1]{Figures/plot_example_var_score.pdf}}\n \\caption{Series' KPI data and corresponding variability score.}\n \\label{fig:analysis_score}\n \\end{subfigure}\n \\caption{Example plots produced by \\triscale during the data analysis.\n \\Cref{fig:analysis_metric}: computation of the metric (95th percentile on one-way delay) with convergence test (confidence 95\\%, tolerance 1\\%).\n \\Cref{fig:analysis_kpi}: computation of the KPI (75th percentile with 75\\% confidence).\n \\Cref{fig:analysis_score}: computation of the variability score (25-75th percentiles with 75\\% confidence).\n \\capt{Sample data from the case study (\\cref{sec:triscale_eval}) for the \\textit{FillP} congestion-control scheme.}}\n \\label{fig:analysis_examples}\n\\end{figure}\n}\n\nMetrics in \\triscale evaluate a performance dimension across a run, for example, the mean throughput achieved by a congestion-control scheme over 30\\s runtime of a full-throttle flow.\nComputing a metric takes the following inputs.\n\n\\fakepar{Inputs}\n\\custommini{%\n \\begin{itemize}\n \\item The metric \\emph{measure}; \\eg mean, maximum, \\etc\n \\item The \\emph{convergence} requirements\\\\\n \\begin{tabular}{@{\\quad}l@{\\quad}r@{\\;\\;:\\;\\;}ll@{\\quad}l}\n \\{ & expected & \\multicolumn{2}{@{}l}{true/false,} &\\\\\n & confidence & $C$ &(\\textit{default: 95\\%}), &\\\\\n & tolerance & $t$ &(\\textit{default: 5\\%}) &\\}\n \\end{tabular}\n \\item The raw data of the run.\n \\end{itemize}\n}%\n\n\\noindent\nIn general, any measure can be used. The current implementation (\\cref{sec:triscale_implementation}) supports the arithmetic mean, the minimum, the maximum, and any percentile.\n\n\\fakepar{Procedure}\nIf the run is expected to converge, \\triscale starts by performing a convergence test, whose purpose is to assess whether the metric has reached a stable value by the end of the run (and hence if it is a good estimate of the long-running performance).\n%\nTo test this, \\triscale computes metric values over a sliding window of the raw data points. For each window, whose size is fixed to half of the data points, one metric value is computed, starting with the first half of the data.\nThe window repeatedly slides by 100th of the number of samples until all data points are used, leading to a set of 100 metric values.%\n\\triscale performs its convergence test~(detailed in \\cref{subsec:triscale_stats}) on the metric values.\nThis procedure tests the convergence of the \\emph{metric} (not the raw data) and damps the impact of transient behavior on the convergence test.\nIf the test is passed, \\triscale returns the median of metric values as run metric.\nIf convergence is not expected, \\triscale simply computes the run metric over the whole raw data.\n\n% To test this, \\triscale divides the raw data in 200 chunks. First, it considers the first 100 chunks (\\ie the first half of the data) and computes a first value of the measure. Then, \\triscale adds one chunk of data and recomputes a new measure value (\\ie using the first 101 chunks). The process repeats until all chunks are used, leading to a set of 100 measure values. \\triscale performs its convergence test~(detailed in \\cref{subsec:triscale_stats}) on the measure data.\n% This procedure (i)~tests what we are interested in (\\ie the convergence of the \\emph{measure}, not of the protocol in general) and (ii)~smooths the effects of ramp-up time by computing the first measure with already half of the run data.\n% If the test is passed, \\triscale returns the median of the measure data as the metric measure for the run.\n\n\\begin{remark}\n If there are less than 100 raw data points, \\triscale reduces the number of windows to the number of data points.\n \\triscale (arbitrarily) sets a minimum of 20 data points for a convergence test.\n\\end{remark}\n\n\\fakepar{Outputs}\n\\custommini{%\n\\begin{itemize}\n \\item\n The result of the convergence test (if performed),\n \\item\n The metric value for the run,\n \\item\n Textual logs; plot of the input data and metric (\\Cref{fig:analysis_metric}).\n\n\\end{itemize}\n}\n\n\\fakepar{Link to the experiment design}\nThe computation of \\triscale metrics is linked to the definition of the \\emph{runtime}; \\ie how long a run should be.\nIf the evaluation scenario is terminating (\\eg transmit 1\\MB through a link), the runtime must be long enough to complete the task.\nIf the evaluation is ``long-running'' (\\eg one-way delay in a full-throttle flow), the runtime must be long enough for the metric (the one-way delay) to converge (convergence test details in \\cref{subsec:triscale_stats}).\n\\triscale can analyze preliminary experiments to estimate the required runtime: by performing increasing long runs and test for convergence~(illustrated in \\cref{sec:triscale_eval}).\n\n%-------------------------------------------------------------------------------\n\\subsection{Series and KPIs}\n\\label{subsec:KPIs}\n%-------------------------------------------------------------------------------\n\n\\triscale's key performance indicators (KPIs) evaluate performance dimensions across a series of runs.\nPerforming multiple runs allows to mitigate the inherent variability of the experimental conditions.\nKPIs capture this variability by estimating percentiles of the (unknown) metric distributions.\nConcretely, a \\triscale KPI is a one-sided CI of one percentile; \\eg a lower-bound for the 75th percentile of the delay metric estimated with a 75\\% confidence level.\nComputing a KPI takes the following inputs.\n\n\\fakepar{Inputs}\n\\custommini{%\n \\begin{itemize}\n \\item\n The {KPI} definition\\\\\n \\begin{tabular}{@{\\quad}l@{\\quad}r@{\\;\\;:\\;\\;}l@{\\quad}l}\n \\{ & percentile & $p$ &\\\\\n & confidence & $C$&\\}\n \\end{tabular}\n \\item\n The metric data (computed from a series of runs).\n\\end{itemize}}\n\n\n\\fakepar{Procedure}\nTo compute the KPI (\\ie to compute a CI for a given percentile), \\triscale uses the Thompson's method (\\cref{subsec:triscale_stats}), which requires the input data to be \\iid.\nThus, \\triscale starts by performing an independence test~(\\cref{subsec:triscale_stats})\non the metric data before computing the KPI.\n\n\\fakepar{Outputs}\n\\custommini{\n\\begin{itemize}\n \\item The result of the independence test,\n \\item The KPI value for the series of runs,\n \\item Textual logs; plot of the metric data and corresponding KPI (\\Cref{fig:analysis_kpi}).\n\\end{itemize}\n}\n\n\\fakepar{Link to the experiment design}\nThe computation of \\triscale KPIs is linked to the definition of the number of runs in a series (\\emph{\\#\\,runs}) and the series time span (\\emph{span}).\nThe minimal number of runs in a series directly follows from the definition of the KPI; \\ie the percentile to estimate $p$ and the desired confidence level $C$.\nThe series time span refers to the time interval used for scheduling the runs in a series (\\ie when to run the experiment).\nThis is important because networks often feature time-dependent conditions; for example, there may be systematically more cross-traffic during daytime than nighttime. Failing to account for such dependencies may bias the results and yield wrong conclusions.\n\\triscale helps the experimenter handling this problem with a dedicated analysis module called ``network profiling'' (described in~\\cref{subsec:network_profiling}).\n\n\n\n%-------------------------------------------------------------------------------\n\\subsection{Sequels and Variability Score}\n\\label{subsec:repeatability}\n%-------------------------------------------------------------------------------\n\n\\emph{Sequels} are repetitions of series of runs.\n\\triscale's variability score evaluates the variations of KPI values across sequels, which enable to detect long-term variations of KPIs and ultimately quantify the reproducibility of an experiment.\n\nConcretely, a variability score is a two-sided CI, \\ie a symmetric pair of percentiles. For example, a 75\\% confidence interval is defined by the 25-75th percentiles of the delay KPIs from all sequels. Again, we attach a confidence value to the confidence interval, or equivalently, to the percentiles.\n\n\\fakepar{Inputs}\n\\custommini{%\n\\begin{itemize}\n \\item\n The {variability score} definition\\\\\n \\begin{tabular}{@{\\quad}l@{\\quad}r@{\\;\\;:\\;\\;}l@{\\quad}l}\n \\{ & percentile & $p$ (or 1-$p$) &\\\\\n & confidence & $C$&\\}\n \\end{tabular}\n \\item\n The KPI values of each sequel.\n\\end{itemize}}\n\n\\fakepar{Procedure}\nThe procedure is the same as for the KPI: The Thompson's method requires the input data to be \\iid~(\\cref{subsec:triscale_stats}), thus \\triscale performs an independence test on the KPI data before computing the variability score.\n\n\\fakepar{Outputs}\n\\custommini{%\n\\begin{itemize}\n \\item\n The result of the independence test,\n \\item\n The variability score value for the entire sequels,\n \\item\n Textual logs; plot of the KPI data and corresponding variability score (\\Cref{fig:analysis_score}).\n\\end{itemize}}\n\n\\fakepar{Link to the experiment design}\nThe computation of the variability score is linked to the definition of the number of series (\\emph{\\#series}).\nThe minimal number of series directly follows from the definition of the variability score; \\ie the percentile to estimate $p$ and the desired confidence level $C$.\n\n%-------------------------------------------------------------------------------\n\\subsection{Formalism Brings Conciseness}\n\\label{subsec:parameters}\n%-------------------------------------------------------------------------------\n\n\\afterpage{\n\\begin{landscape}\n \\begin{table}\n \\centering\n \\caption{Exemplary evaluation parameters of typical networking use cases.\n \\capt{$^{*}$\\triscale returns the minimal number of runs (\\#runs) and series (\\#series) based on the definition of KPI and variability score, respectively.}}\n \\input{\\PathTab/triscale_parameters.csv}\n \\label{table:triscale_param}\n \\end{table}\n\\end{landscape}\n}\n\n\\triscale formalizes the definition of the evaluation objectives. For each performance dimension, the experimenter defines a metric and convergence requirements~(\\cref{subsec:metrics}), a KPI~(\\cref{subsec:KPIs}), and a variability score~(\\cref{subsec:repeatability}).\n\\triscale links these objectives with the experiment design, resulting in four additional parameters: the number of runs per series (\\emph{\\#\\,runs}), the number of series (\\emph{\\#\\,series}), the length of a run (\\emph{runtime}), and the time span of a series (\\emph{span}).\n\nThanks to this formalism, \\triscale meets the \\feature{Conciseness} requirement:\nAltogether, these 12 parameters are sufficient to \\emph{formally describe the entire performance evaluation} such that it can (eventually) be reproduced.\nIn particular, since the data analysis in \\triscale is automated and deterministic, documenting these parameters guarantees computational reproducibility (the ability to recreate the results when all raw data are available~\\cite{liu19computational}).\n\n\\Cref{table:triscale_param} shows a few examples of concrete parameter settings for typical networking evaluation objectives.\nFor example, evaluating the latency of a real-time protocol requires high confidence levels for extreme percentiles.\nThis very quickly increases the number of runs that one must perform:\\\\\n\\inlineitem\n at least 90 runs for estimating the 95th percentile with 99\\% confidence;\\linebreak\n\\inlineitem\n at least 299 runs for estimating the 99th percentile with 95\\% confidence.\\linebreak\nThis illustrates that it is ``easier'' to increase the confidence level of an estimate than to estimate a more extreme percentile with the same confidence level.\nNote that both \\emph{\\#runs} and \\emph{\\#series} are only derived based on the definition of the KPI and variability score; these parameters are not influenced by the runtime or the time span of an experiment.\n\nThe second use case in \\Cref{table:triscale_param} (bottom rows) illustrates two different perspectives on ``averages'', using delay as an example:\n\\\\\n% \\begin{itemize}\n % \\item\n \\inlineitem\n If the metric is the median and the KPI the 90th percentile, one can conclude that 90\\% of the runs have a median delay equal or better than the KPI value.\n\\\\\n % \\item\n \\inlineitem\n If the metric is the 90th percentile and the KPI the median, one can conclude that, in half of the runs, the 90th percentile of the delay in the run is equal or better than the KPI.\n% \\end{itemize}\n\\\\\nBoth are ``averages'' but with different meanings and different requirements in terms of number of runs.\n\n\n%-------------------------------------------------------------------------------\n\\subsection{Statistics in \\triscale}\n\\label{subsec:triscale_stats}\n%-------------------------------------------------------------------------------\n\n\\triscale uses carefully chosen statistical methods.\nAs discussed in~\\cref{sec:stats}, networking performance evaluations should focus on statistics that are both \\emph{robust} (\\ie that can tolerate outliers) and \\emph{non-parametric} (\\ie that do not make any assumption on the nature of the data distribution).\nThis section describes the three statistical methods used in \\triscale. We first present the convergence test used in the computation of metrics (\\cref{subsec:metrics});\nThis test is based on the \\mbox{Theil-Sen} linear regression~\\cite{theil1992RankInvariant, sen1968Estimates}.\nWe then introduce the computation of confidence intervals using Thompson's method~\\cite{thompson1936Confidence}, which requires the data to be \\iid. Thus, to verify this assumption, \\triscale integrates an independence test that we present last.\n\n%-------------------------------------------------------------------------------\n\\fakepar{Convergence test}\n\\label{subsec:test_convergence}\nWhen an evaluation aims to estimate the ``long-running'' performance (\\ie the expected performance if the run would run “forever”), one must verify whether the runs are long enough to produce reliable estimates.\n\n\\squarepar{%\n To verify this, \\triscale implements a convergence test based on the Theil-Sen linear regression~\\cite{theil1992RankInvariant, sen1968Estimates}.\n This approach computes the slope of the regression line as the median of all slopes between paired values.\n A $C$\\% confidence interval (CI) for the slope is defined as the interval containing the middle $C$\\% of slopes between single pairs.\n \\triscale convergence test is passed if the $C$\\% CI for the regression is included in the tolerance value ($\\pm$\\,$t$\\%).\n To test the convergence of a run, \\triscale uses the confidence $C$ and tolerance $t$ parameters specified in the evaluation objectives (\\Cref{sec:triscale_overview});\n otherwise defaults to 95\\% and 5\\%.}\n\nSuch a test is sensitive to the scale of the input data.\nTo remove this dependency, \\triscale first maps the data to $[-1, 1]$ using a linear transformation then performs the convergence test on the scaled data.\nHence, the convergence test becomes dimensionless and the same tolerance value can be used for different evaluations without introducing bias.\nAn example of the Theil-Sen slope (brown, solid), its CI (light orange, solid), and tolerance (black, dotted) is shown in \\Cref{fig:analysis_metric}.\n\n%-------------------------------------------------------------------------------\n\\fakepar{Confidence intervals}\n\\triscale defines KPIs and variability scores based on CIs for distribution percentiles, which can be computed using a robust and non-parametric approach based on Thompson's method~\\cite{thompson1936Confidence}, later shown to be valid for any independent samples of a continuous distribution~\\cite{david2005Order}.\n\nLet us denote by $P_p$, the $p$-th percentile of a distribution and $\\mathbb{P}(X)$ the probability of an event $X$.\nBy definition, every data sample $x$ is smaller than $P_p$ with probability $p$ (and larger with probability $1-p$).\nFor a sorted list of \\iid samples $x_i$ (where $i = 1 .. N$), the probability that $P_p$ lies between two consecutive samples follows the binomial distribution~\\cite{thompson1936Confidence}:\n\\begin{equation}\n \\mathbb{P}(x_k \\leq P_p \\leq x_{k+1}) = \\binom{N}{k} p^k(1-p)^{N-k}, \\quad k = 0 .. N\n\\end{equation}\nwhere we assume $x_0 \\rightarrow - \\infty$ and $x_{N+1} \\rightarrow +\\infty$. From this result, it follows that the probability of $P_p$ to be larger than any sample $x_m$ (where $1\\leq m < N$) can be computed as:\n\\begin{align}\n \\nonumber\n \\mathbb{P}(x_m \\leq P_p)\n &= \\mathbb{P}(x_{N-m+1} \\geq P_{1-p})\\\\\n \\label{eq:lb}\n &= 1 - \\sum_{k=0}^{m-1} \\binom{N}{k} p^k(1-p)^{N-k}\n\\end{align}\n\\Cref{eq:lb} provides the upper- and lower-bound required for computing of CIs.\nFurther, one can derive the minimum number of samples $N$ needed to compute a CI for any percentile $p$ with any confidence level $C$~\\cite{schmid2014measuring}:\n\\begin{equation}\n\\label{eq:minN}\n \\cref{eq:lb} \\quad \\Rightarrow \\quad N \\;\\geq\\; \\frac{\\log(1-C)}{\\log(1-p)}\n\\end{equation}\nwhich defines the minimum number of runs and series required based on the definitions of the KPIs and the variability scores.\nIf the probability distribution is discrete, \\cref{eq:lb} becomes an inequality ($\\mathbb{P}(x_m \\leq P_p) \\geq \\ldots$~) that provides a safe (\\ie conservative) estimate of which sample $x_m$ is the bound of the CI of interest~\\cite{david2005Order}.\n\n\nThis approach provides robust estimates for distribution percentiles and \\emph{does not make any assumption on the nature of the underlying distribution}.\nIt does, however, require that the data samples are \\iid. \\triscale checks whether this hypothesis holds using an independence test, described below.\n\n%-------------------------------------------------------------------------------\n\\fakepar{Independence test}\nEstimating the percentile of a distribution requires often (if not always) that the samples are \\iid (\\cref{sec:stats}); this is also the case for Thompson's method~\\cite{thompson1936Confidence}.\n\\triscale implements an empirical independence test to verify whether the \\iid assumption holds.\n\n\\begin{remark}\n Generally, independence results from the experiment design. For networking experiments, however, it is generally not possible to guarantee independence, \\eg the experimental conditions cannot be fully controlled and may be correlated.\n In such cases, it is common to empirically check whether the data are correlated. If the effective dependence between data samples is sufficiently low, it is considered safe to treat the samples as \\iid.\n\\end{remark}\n\nThis independence test is applied to the metric data (resp. KPI data) before the computation of a KPI (resp. a variability score).\nThis poses the particular challenge that the number of data samples may be very small (\\eg 3 or 5 KPI values). \\triscale's independence test must therefore not be too strict.\n\nThe test is divided in two steps. First, \\triscale tests whether the data appear \\emph{weakly stationary} (\\ie no trend and constant autocorrelation structure~\\cite{brockwell1991Time}). \\triscale verifies this empirically using its convergence test with a confidence of 50\\% and tolerance of 10\\%; these ``loose'' parameters are used to compensate for (very) small sample sizes.\nSecond, \\triscale computes the \\emph{sample autocorrelation coefficients}, denoted by $\\widehat{\\rho_k}$, which measure the linear dependency between values of a weakly stationary data series.\nA series of size $N$ is \\iid with 95\\% probability if $|\\widehat{\\rho_k}| \\leq 1.95/\\sqrt{N}$ for $k \\geq 1$~\\cite{brockwell1991Time}.\n\n\\fakeparQM{What if the tests fail}\nThe experimenter is responsible for designing the evaluation in such a way that the collected data will (likely) pass the tests.\n\\triscale facilitates this by guiding the choice of runtime to pass the convergence test and informing about any network time dependencies~(\\cref{subsec:network_profiling}) to pass the independence test.\nYet, the data may still be correlated or unstable, leading to failing tests (see examples in ~\\cref{sec:triscale_eval}).\nEven in such cases, the data still contain useful information.\n\\triscale metrics, KPIs, or variability scores can be computed, however since the corresponding hypotheses do not hold, the statistics are \\emph{only descriptive}~(\\cref{sec:stats}); they do not predict the expected performance, and in particular they cannot (and should not be used to) assess the reproducibility of the experiment.\n\n%-------------------------------------------------------------------------------\n\\subsection{Network Profiling}\n\\label{subsec:network_profiling}\n\n\\begin{figure}\n \\centering\n \t\\href{\\triscalefig{Figure-4}}{\n \\includegraphics[scale=1]{Figures/plot_flocklab_autocorr.pdf}}\n\n \\caption{Autocorrelation plot for the wireless link quality on FlockLab, based on the raw data collected by the testbed maintainers~\\cite{jacob2019datasetLQE} (data from August 2019).\n \\capt{The dataset contains one test every two hours. The first peak at lag 12 (\\ie 24h) reveals the daily seasonal component. The data also show another at lag 84; which corresponds to one week. Indeed, there is less interference in the weekends than on weekdays: this creates a weekly seasonal component.}}\n \\label{fig:flocklab_autocorr}\n\\end{figure}\n\n\\triscale assists the user in deciding on the time span for a series of runs, \\ie when should the run be performed in a series. This is important to avoid biasing the evaluation results with time dependencies in the experimental conditions.\nIndeed, it is common for real-world networks to exhibit periodic patterns.\nFor example, there may be a lot more cross-traffic (\\ie interference) at specific times. In the statistics literature, these patterns are called \\emph{seasonal components}.\nNeglecting these may result in biased experiments leading to wrong conclusions, as illustrated in the case study below.\n\n\\squarepar{%\n \\fakepar{Case study: low-power wireless}\n We run a simple evaluation of Glossy~\\cite{ferrari2011Glossy}, a low-power wireless protocol based on synchronous transmissions~(\\cref{sec:ST}). Glossy includes as parameter the number of retransmissions of a packet, called $N$. We investigate the impact of two values of $N$ on the reliability of Glossy, measured as the packet reception ratio (PRR). We define our KPI as the median with 95\\% confidence level.\n Refer to \\cref{append:triscale_artifacts} for the complete case study.%\n}\n\nWe collect data using the FlockLab testbed~\\cite{lim2013FlockLab}.\nThis testbed is located in an office building, where we expect more interference during daytime than nighttime.\nThus, we schedule a series of 24 runs randomly within one day.\\linebreak\n \\inlineitem $N=1$ leads to a PRR of 88\\%,\\\\\n \\inlineitem $N=2$ leads to a PRR of 84\\%.\\\\\nIn other words, it appears that doing two retransmissions (instead of one) reduces reliability.\n\nThe experiment leads to this (incorrect) conclusion because we have neglected a second seasonal component of the FlockLab testbed: there is a weekly time dependence, revealed by \\Cref{fig:flocklab_autocorr}.\nTo account for this dependency, one must schedule runs with a span of at least one week. When comparing again the performance of Glossy but with tests spanning over a week\\\\\n \\inlineitem $N=1$ leads to a PRR of 80\\%,\\\\\n \\inlineitem $N=2$ leads to a PRR of 88\\%,\\\\\nwhich better matches our knowledge about the performance of Glossy.\n\n\\fakepar{Conclusion}\nThis simple example illustrates that using a high confidence level is not enough to avoid drawing wrong conclusions due to the variability in the experimental conditions.\nOn a real network, short-term variations are unpredictable and (often) unavoidable. This is why it is important to perform multiple runs in a series: it increases the chances to do the experiment in the whole range of favorable to unfavorable conditions.\n\nHowever, we illustrated that systematic patterns are also present. In other words, there are times where there is consistently more or less interference. Knowing about these dependencies is important to\n ensure fairness in the comparison between protocols, and\n enable reproducibility of the evaluations.\nThe series span must be long enough such that it does not matter when the series actually starts (\\eg a weekend or a weekday)\n\n\\squarepar{%\n \\triscale integrates a network profiling function that analyzes link quality data (\\eg available at~\\cite{jacob2019datasetLQE}) and searches for seasonal components in the link quality data. This helps the experimenter detecting (sometimes unexpected) time dependencies, thus choosing a suitable time span for series of runs.}\n\n\n%-------------------------------------------------------------------------------\n\\subsection{Assessing Reproducibility}\n\\label{subsec:reproducibility}\n\nReproducibility refers to the ability of obtaining ``the same'' results when performing ``the same'' experiment.\nIn statistics, such property can be investigated using \\textit{equivalence testing}~\\cite{lakens2017Equivalence}, which checks whether the values of some parameter of interest (\\eg the median) obtained for different samples are sufficiently close to be considered ``the same''.\nUnfortunately, there is no general way to define ``sufficiently close''; one must define in advance a threshold for the equivalence test based on expertise.\nThen, how to assess reproducibility of networking experiments? How to design a ``reproducibility test'' that fairly adapts to different networking contexts and different metrics?\nAfter some failed attempts, we conclude that defining a generic threshold for equivalence testing in networking might not be possible. But it may not be necessary.\n\nWe argue that the most important is to confidently estimate the variability of the results, which \\triscale computes with its variability score~(\\cref{subsec:repeatability}).\nThis score \\emph{quantifies reproducibility}: the larger the score, the less reproducible the results are.\nShall a binary cut between ``reproducible'' and ``not reproducible'' be desired, a threshold can be set based on the variability score; \\eg ``Results are said reproducible when the variability score is less than 20\\mbps.''.\nSuch a threshold can only be context-specific; thus, deciding on threshold values relates more to benchmarking and therefore goes beyond the scope of \\triscale~(see discussion in \\cref{sec:going_further}).\n", "meta": {"hexsha": "2ab72eabd9540c4a3d212294f2b68f6a05a4ced0", "size": 28278, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "20_TriScale/4_triscale.tex", "max_stars_repo_name": "romain-jacob/doctoral-theis", "max_stars_repo_head_hexsha": "fd21e9f0cddeda91821eb061c9ab12df9f610da9", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "20_TriScale/4_triscale.tex", "max_issues_repo_name": "romain-jacob/doctoral-theis", "max_issues_repo_head_hexsha": "fd21e9f0cddeda91821eb061c9ab12df9f610da9", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "20_TriScale/4_triscale.tex", "max_forks_repo_name": "romain-jacob/doctoral-theis", "max_forks_repo_head_hexsha": "fd21e9f0cddeda91821eb061c9ab12df9f610da9", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 70.8721804511, "max_line_length": 477, "alphanum_fraction": 0.7332555343, "num_tokens": 6810, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.63341027751814, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.3463095645783293}} {"text": "\n\n\\section{Contractual Money}\n \\label{sec:contractual}\n\n In our opinion, the overwhelming majority of use-cases for public blockchains (even those that claim to provide a general-purpose decentralized world computer) are for financial applications, which do not require Turing-completeness. For instance, if an oracle writes down non-financial data into the blockchain~(such as temperature), this data is usually used further in a financial\n contract. Another trivial observation we make is that many applications use digital tokens with mechanics different from the native token.\n\nFor an application developer, the Ergo Platform offers custom tokens~(which are first-class citizens) and a domain-specific language for writing box protecting\n conditions in order to implement flexible and secure financial applications.\n Ergo applications are defined in terms of protecting scripts built into boxes, which may also contain data involved in the execution.\n We use the term {\\em contractual money} to define Ergs (and secondary tokens) whose usage is bounded by a contract. This applies to all tokens on the platform in existence because any box with its contents~(Ergs, tokens, data) is bounded by a contract.\n \n However, we can distinguish between two types of contractual Ergs. The first, called {\\em free Ergs}, are the ones that could change their contracts easily and have no restrictions on the outputs or the other inputs of a spending transaction. The second type is {\\em bounded Ergs}, whose contracts require the spending transaction to have input and output boxes with specific properties.\n \n For example, if a box $A$ is protected by just a public key~(so providing a signature against a spending transaction is enough in order to destroy the box), the public key owner can spend $A$ and transfer the Ergs to any arbitrary output box. Thus, the Ergs within $A$ are free. \n% to change the contract. \nIn contrast, imagine a box $B$ protected by a combination of a public key and a condition that demands the spending transaction to create an output box with the same amount of Ergs as in $B$ and whose guarding script has the hash \\texttt{rBMUEMuPQUx3GzgFZSsHmLMBouLabNZ4HcERm4N} (in Base58 encoding). In this case, the Ergs in $B$ are bounded Ergs.\n \n Similarly, we can define free and bounded tokens. An Ergo contract can have several hybrids such as bounded Ergs and free tokens or both bounded under one public key and free under another.\n\n\\subsection{Preliminaries For Ergo Contracts}\n\n While in Bitcoin, a transaction output is protected by a program in a stack-based language named {\\em Script}, in Ergo a box is protected by a logic formula which combines predicates over a context with cryptographic statements provable via zero-knowledge protocols using AND, OR, and $k$-out-of-$n$ connectives. The formula is represented as a typed direct\n acyclic graph, whose serialized form is written in a box. To destroy a box, a spending transaction needs to provide arguments (which include zero-knowledge proofs) satisfying the formula.\n\n However, in most cases, a developer is unlikely to develop contracts in terms of graphs. Instead, he would like to use a high-level language such as ErgoScript, which we provide with the reference client. \n \n Writing scripts in ErgoScript is easy. As an\n example, for a one-out-of-two signature, the protecting script would be ${pk_1 \\|pk_2}$, which means ``prove knowledge of\n a secret key corresponding to the public key $pk_1$ or knowledge of a secret key corresponding to public key $pk_2$''. We have\n two separate documents for help in developing contracts with ErgoScript: the ``ErgoScript Tutorial''~\\cite{ergoTutorial}\n and the ``Advanced ErgoScript Tutorial''~\\cite{ergoAdvTutorial}. Thus, we do not get into the details of developing contracts with ErgoScript. Rather, we provide a couple of motivating examples in the following sections.\n\nTwo more features of Ergo shaping contracting possibilities are:\n\n \\begin{itemize}\n \\item {\\em Data Inputs: }\n To be used in a transaction, a box need not be destroyed but can instead be read-only. In the latter case, we refer to the box as being part of the {\\em data input} of the transaction. Thus, a transaction gets two box sets as its arguments, the inputs and\n data inputs, and produces one box set named {\\em outputs}. Data inputs are useful for oracle applications and interacting contracts.\n\n \\item {\\em Custom Tokens: }\n A transaction can carry many tokens as long as the estimated complexity for processing them does not exceed a limit, a parameter that is set by miner voting. A transaction can also issue a single token with a unique identifier which is equal to the identifier of a first~(spendable) input box of the transaction. The identifier is unique assuming the collision resistance of an underlying hash function.\n The amount of the tokens issued could be any number within the range $[1, 9223372036854775807]$. The weak preservation rule is followed for tokens, which requires that the total amount of any token in a transaction's outputs should be no more\n than the total amount of that token in the transaction's inputs~(i.e., some amount of token could be burnt). In contrast, the strong reservation rule is followed for Ergs, which requires that the total amount of Ergs in the inputs and outputs must be the same.\n \\end{itemize}\n\n\\subsection{Contract Examples}\n\\label{sec:examples}\n\n In this section, we provide some examples which demonstrate the superiority of Ergo contracts compared to Bitcoin's. The examples include betting on oracle-provided data, non-interactive mixing, atomic swaps, complementary currency, and an initial coin offering implemented on top of the Ergo blockchain.\n\n \\subsubsection{An Oracle Example}\n \\label{sec:platform}\n\n Equipped with custom tokens and data inputs, we can develop a simple oracle example which also shows some design patterns that we discovered while playing with Ergo contracts. Assume that Alice and Bob want to bet on tomorrow's weather by putting money into a box that becomes spendable by Alice if tomorrow's temperature is more than 15 degrees, and spendable by Bob otherwise. To deliver the temperature into the blockchain, a trusted oracle is needed.\n\n In contrast to Ethereum with its long-lived accounts, where a trusted oracle's identifier is usually known in advance, delivering data with one-time boxes is more tricky. For starters, a box protected by the oracle's key cannot be trusted, as anyone can create such a box. It is possible to include signed data into a box and check the oracle's signature in the contract (we have such an example), but this is quite involved. Instead, a solution with custom\n tokens is very simple.\n\n Firstly, a token identifying the oracle should be issued. In the simplest case, the amount of this token could be one. We call such a token {\\em a singleton token}. The oracle creates a box containing this token along with its data (i.e., the temperature) in register $R_4$ and the UNIX epoch time in register $R_5$.\n In order to update the temperature, the oracle destroys this box and creates a new one with the updated temperature.\n\n Assume that Alice and Bob know the oracle's token identifier in advance. With this knowledge, they can jointly create a box with a contract that requires first (read-only) data input to contain the oracle's token. The contract extracts the temperature and time from the data input\n and decides who gets the payout. The code is as simple as following:\n\n \\begin{algorithm}[H]\n \\caption{Oracle Contract Example}\n \\label{alg:oracle}\n \\begin{algorithmic}[1]\n \\State val dataInput = CONTEXT.dataInputs(0)\n \\State val inReg = dataInput.R4[Long].get\n \\State val inTime = dataInput.R5[Long].get\n \\State val inToken = dataInput.tokens(0).\\_1 == tokenId\n \\State val okContractLogic = (inTime $>$ 1556089223) \\&\\&\n \\State\\hspace{\\algorithmicindent}\\hspace{\\algorithmicindent} ((inReg $>$ 15L \\&\\& pkA) $||$ (inReg $\\le$ 15L \\&\\& pkB))\n \\State inToken \\&\\& okContractLogic\n \\end{algorithmic}\n \\end{algorithm}\n\n This contract shows how a singleton token could be used for authentication. As a possible alternative, the oracle\n can put the time and temperature into a box along with a signature on this data. However, this requires signature verification, which is more complex and expensive compared to\n the singleton token approach. Also, the contract shows how read-only data inputs could be useful for contracts which need to access data stored in some other box in the state. Without data inputs, an oracle must issue one spendable box for each\n pair of Alice and Bob. With data inputs, the oracle issues only a single box.\n\n\\subsubsection{A Mixing Example}\n \\label{sec:platform}\n\n Privacy is important for a digital currency but implementing it can be costly or require a trusted setup. Thus, it is desirable to find a cheaper way for coin mixing. As a first step towards that, we offer a non-interactive mixing protocol between two users Alice and Bob that works as follows:\n \\begin{enumerate}\n \\item{} Alice creates a box which demands the spending transaction to satisfy certain conditions. After that, Alice only listens to the blockchain; no interaction with Bob is needed.\n \\item{} Bob creates a transaction spending Alice's box along with one of his own to generate two outputs having identical scripts but different data. Each of Alice and Bob may spend only one of the two outputs but an observer decide which output belongs to whom because they look indistinguishable.\n \\end{enumerate}\n\n For simplicity, we do not consider fee in the example. The idea of mixing is similar to non-interactive Diffie-Hellman key exchange. First, Alice generates a secret value $x$~(a huge number) and publishes the corresponding public value $gX = g^x$. She requires Bob to generate a secret number $y$, and to include into each output two\n values $c_1$, $c_2$, where one value is equal to $g^y$ and the other is equal to $g^{xy}$. Bob uses a random coin to choose meanings for $\\{c_1, c_2\\}$. Without access to the secrets, an external observer cannot guess with probability better than $\\frac{1}{2}$ whether $c_1$ is equal to $g^y$ or to $g^{xy}$. This is assuming that the cryptographic primitive we use has a certain property, that the Decision Diffie-Hellman (DDH) problem is hard. To destroy an output box, a proof should be given that either $y$ is known such that $c_2 = g^y$, or $x$ is known such that $c_2 = c_1^x$.\n The contract of Alice's box checks that $c_1$ and $c_2$ are well-formed. The code snippets for Alice's coin and the mixing transaction's output are provided in Algorithms \\ref{alg:alice} and \\ref{alg:mixing-out} respectively. Since ErgoScript currently doesn't have support for proving knowledge of some $x$ such that $c_2 = {c_1}^x$ for arbitrary $c_1$, we will prove a slightly longer statement that is supported, namely, proving knowledge of $x$ such that $gX = g^x$ and $c_2 = {c_1}^x$. This is called proveDHTuple.\n\n \\begin{algorithm}[H]\n \\caption{Alice's Input Script}\n \\label{alg:alice}\n \\begin{algorithmic}[1]\n \\State val c1 = OUTPUTS(0).R4[GroupElement].get\n \\State val c2 = OUTPUTS(0).R5[GroupElement].get\n \\State\n \\State OUTPUTS.size == 2 \\&\\&\n \\State OUTPUTS(0).value == SELF.value \\&\\&\n \\State OUTPUTS(1).value == SELF.value \\&\\&\n \\State blake2b256(OUTPUTS(0).propositionBytes) == fullMixScriptHash \\&\\&\n \\State blake2b256(OUTPUTS(1).propositionBytes) == fullMixScriptHash \\&\\&\n \\State OUTPUTS(1).R4[GroupElement].get == c2 \\&\\&\n \\State OUTPUTS(1).R5[GroupElement].get == c1 \\&\\& \\{\n \\State\\hspace{\\algorithmicindent} proveDHTuple(g, gX, c1, c2) $||$\n \\State\\hspace{\\algorithmicindent} proveDHTuple(g, gX, c2, c1)\n \\State \\}\n \\end{algorithmic}\n \\end{algorithm}\n\n \\begin{algorithm}[H]\n \\caption{Mixing Transaction Output Script}\n \\label{alg:mixing-out}\n \\begin{algorithmic}[1]\n \\State val c1 = SELF.R4[GroupElement].get\n \\State val c2 = SELF.R5[GroupElement].get\n \\State proveDlog(c2) $||$ // either c2 is $g^y$\n \\State proveDHTuple(g, c1, gX, c2) // or c2 is $u^y = g^{xy}$\n \\end{algorithmic}\n \\end{algorithm}\n\n We refer the reader to \\cite{ergoAdvTutorial} for a proof of indistinguishability of the outputs and details on why Alice and Bob can spend only their respective coins.\n\n\n\\subsubsection{More Examples}\n\n In this section, we briefly shed light on a few more examples along with links to the documents providing the details and code.\n\n\\paragraph{Atomic Swap}\nCross-chain atomic swap between Ergo and any blockchain that supports payment to either SHA-256 or Blake2b-256 hash preimages and time-locks can be done in a similar way to that proposed for Bitcoin~\\cite{Nol13}. An Ergo alternative implementation is provided in~\\cite{ergoTutorial}. As Ergo also has custom tokens, atomic exchange on the single Ergo blockchain (Erg-to-token or token-to-token) is also possible. An implementation for this can also be found in~\\cite{ergoTutorial}.\n\n\\paragraph{Crowdfunding}\n\n We consider the simplest crowdfunding scenario. In this example, a crowdfunding project with a known public key is considered successful if it can collect unspent outputs with a total value not less than a certain amount before a certain height. A project backer creates an output box protected by the following statement: the box can be spent\n if the spending transaction has the first output box protected by the project's key and amount no less than the target amount.\n Then the project can collect (in a single transaction) the biggest backer output boxes with a total value not less than the target amount~(it is possible to collect up to ~22,000 outputs, which is\n enough even for a big crowdfunding campaign). For the remaining outputs, it is possible to construct follow-up transactions. The code can be found in~\\cite{ergoTutorial}.\n\n\\paragraph{The Local Exchange Trading System}\n\n Here we briefly demonstrate a Local Exchange Trading System (LETS) in Ergo. In such a system, a member of a community may issue community currency via personal debt. For example, if Alice with zero balance is buying something for $5$\n community tokens from Bob, whose balance is zero as well, her balance after the trade would be $-5$ tokens, and\n Bob's balance would be $5$ tokens. Then Bob can buy something using his $5$ tokens, for example, from Carol.\n Usually, in such systems, there is a limit on negative balances (to avoid free-riding).\n\n Since a digital community is vulnerable to Sybil attacks~\\cite{sybilDef}, some mechanism is needed to prevent such attacks where Sybil nodes create debts.\n The simplest solution is to use a committee of trusted managers that approve new members of the community. A trust-less but more complex solution is to use security deposits made in Ergs. For simplicity, we consider the approach with the committee here.\n\n This example contains two interacting contracts. A {\\em management contract} maintains a list of community members, and a new member can be added if some management condition is satisfied (for example, a threshold\n signature is provided). A new member is associated with a box containing a token that identifies the member. This box, which contains the {\\em member contract}, is protected by a special exchange script that requires the spending transaction to do a fair exchange.\n We skip the corresponding code, which can be found in a separate article~\\cite{letsTutorial}.\n \n What this contract shows, in contrast to the previous example, is that instead of storing the members list, only a short digest of an authenticated AVL+ tree can be included in the box. This allows a reduction in storage requirements for the state. A transaction doing lookup or modification of the member list should provide a proof for AVL+ tree lookup or modification operations. Thus, saving space in the state storage leads to bigger transactions, but this scalability problem is easier to solve.\n\n\\paragraph{Initial Coin Offering}\n\n We discuss an Initial Coin Offering (ICO) example that shows how multi-stage contracts can be created in Ergo. Like most ICOs, our example has three stages. In the first stage, the project raises money in Ergs. In the second stage, the project issues a new token, whose amount equals the number of nanoErgs raised in the first stage. In the third stage, the investors can withdraw issued tokens.\n\n Note that the first and third stages have many transactions on\n the blockchain, while a single transaction is enough for the second stage. Similar to the previous example, the ICO contract uses an AVL+ tree to store the list of (investor, amount) pairs. The complete code is available at~\\cite{icoTutorial}.\n\n\n\\paragraph{More Examples}\n\n We have even more examples of Ergo applications in \\cite{ergoTutorial, ergoAdvTutorial}. These examples include time-controlled emission, cold wallets contracts, rock-paper-scissors game, and many others.\n", "meta": {"hexsha": "0e30bddf0473061b8dca681af6ef04d77d105f07", "size": 17101, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "papers/whitepaper/money.tex", "max_stars_repo_name": "scasplte2/ergo", "max_stars_repo_head_hexsha": "9964f415526f491a4837774d80b59792e1e2b8bb", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 424, "max_stars_repo_stars_event_min_datetime": "2017-07-17T12:33:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T13:33:57.000Z", "max_issues_repo_path": "papers/whitepaper/money.tex", "max_issues_repo_name": "scasplte2/ergo", "max_issues_repo_head_hexsha": "9964f415526f491a4837774d80b59792e1e2b8bb", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 886, "max_issues_repo_issues_event_min_datetime": "2017-07-20T21:59:30.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T10:21:25.000Z", "max_forks_repo_path": "papers/whitepaper/money.tex", "max_forks_repo_name": "scasplte2/ergo", "max_forks_repo_head_hexsha": "9964f415526f491a4837774d80b59792e1e2b8bb", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 131, "max_forks_repo_forks_event_min_datetime": "2017-07-19T12:46:49.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-22T01:08:16.000Z", "avg_line_length": 98.2816091954, "max_line_length": 586, "alphanum_fraction": 0.7703058301, "num_tokens": 3937, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102775181399, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.34630956457832923}} {"text": "\\subsection{Pulsar Accelerations}\n\n\\textbf{key}: \\texttt{/pulsar}\n\n\\subsubsection{Datasets}\n\n\\begin{center}\n\\begin{table}[H]\n\\begin{tabular}{ | c | c | c | c | }\n \\hline\n Variable & Dataset Name & Supplementary Datasets & attributes \\\\\n \\hline\\hline\n Radial distance & \\texttt{r\\(^*\\)} & & \\texttt{unit} \\\\\n \\hline\n Spin period & \\texttt{P\\(^*\\)} & \\texttt{\\(\\Delta\\)P} & \\texttt{unit}\\\\\n \\hline\n Spin period derivative & \\texttt{Pdot\\_meas\\(^*\\)} &\n \\texttt{\\(\\Delta\\)Pdot\\_meas\\(^*\\)} & \\texttt{unit} \\\\\n \\hline\n Orbital period & \\texttt{Pb\\(^*\\)} & \\texttt{\\(\\Delta\\)Pb} & \\texttt{unit}\\\\\n \\hline\n Orbital period derivative & \\texttt{Pbdot\\_meas\\(^*\\)} &\n \\texttt{\\(\\Delta\\)Pbdot\\_meas\\(^*\\)} & \\texttt{unit} \\\\\n \\hline\n Dispersion Measure & \\texttt{DM} & \\texttt{\\(\\Delta\\)DM} & \\texttt{unit} \\\\\n \\hline\n Pulsar identifier & \\texttt{id} & & \\\\\n \\hline\n\\end{tabular}\n\\caption*{\n Pulsars can be fit on the timing solutions of both the isolated\n pulsar spin (P, Pdot\\_meas) and the binary systems orbit (Pb, Pbdot).\n The period, derivative and corresponding errors are required for either.\n Optionally, the dispersion measure (DM) can be used to provide a better\n acceleration constraint, where available.\n}\n\\end{table}\n\\end{center}\n\n\\subsubsection{Attributes}\n\n\n\\begin{center}\n\\begin{table}[H]\n\\begin{tabular}{ | c | c | }\n \\hline\n Attribute & Description \\\\\n \\hline\\hline\n \\texttt{source} & Literature source(s) of data \\\\\n \\hline\n \\texttt{m} & Mean stellar mass of tracer stars [\\(M_\\odot\\)] \\\\\n \\hline\n\\end{tabular}\n\\end{table}\n\\end{center}\n", "meta": {"hexsha": "eeea83b5152c7b60c0ee1cde1214b3f570bf9a27", "size": 1640, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/source/raws/DPC/DP_pulsar.tex", "max_stars_repo_name": "pjs902/GCfit", "max_stars_repo_head_hexsha": "f26f004bb9caf7429cbae23a6ca559fad42f3498", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-22T17:54:07.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-22T17:54:07.000Z", "max_issues_repo_path": "docs/source/raws/DPC/DP_pulsar.tex", "max_issues_repo_name": "pjs902/GCfit", "max_issues_repo_head_hexsha": "f26f004bb9caf7429cbae23a6ca559fad42f3498", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2022-02-09T16:04:32.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-29T17:44:09.000Z", "max_forks_repo_path": "docs/source/raws/DPC/DP_pulsar.tex", "max_forks_repo_name": "pjs902/GCfit", "max_forks_repo_head_hexsha": "f26f004bb9caf7429cbae23a6ca559fad42f3498", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-07T21:21:24.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-07T21:21:24.000Z", "avg_line_length": 29.2857142857, "max_line_length": 80, "alphanum_fraction": 0.6268292683, "num_tokens": 553, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.63341026367784, "lm_q2_score": 0.546738151984614, "lm_q1q2_score": 0.3463095570113093}} {"text": "\\hypertarget{premium-allocation-and-default-priority-rules}{%\n\\section{Premium Allocation and Default Priority\nRules}\\label{premium-allocation-and-default-priority-rules}}\n\nNov 25, 2018\n\nDerived from MM\\_2RM\n\n\\hypertarget{three-important-differentials-homogeneous-case}{%\n\\subsection{Three important differentials (homogeneous\ncase)}\\label{three-important-differentials-homogeneous-case}}\n\nFrom Tasche and other references. Let \\(X_u=X=u_1X_1 + u_2X_2\\) be a\nhomogeneous portfolio.\n\nThe deriviative of the survival function \\(S_u(t)=\\text{Pr}(X_u>t)\\) is\n\\[\n\\nabla_u S(t) = \\text{E}[X_i 1_{\\{X=t\\}}] = \\text{E}[X_i \\mid X=t] f_u(t)\n\\] where \\(f_u\\) is the sensity of \\(X_u\\).\n\nThe deriviative of value at risk \\(Q(u)\\) for \\(X_u\\) at a fixed\nthreshold \\(p\\) is \\[\n\\nabla_u Q(u) = E[X_i\\mid \\sum u_ix_i = Q(u,p)]\n\\]\n\nThe deriviative of tail value at risk \\(T\\) is \\[\n\\nabla_u T(u) = E[X_i\\mid \\sum u_ix_i>Q(u,p)]\n\\]\n\n\\hypertarget{method-a}{%\n\\subsection{Method A}\\label{method-a}}\n\nFor M\\textbf{A}jor :-)\n\nHomogeneous portfolio \\(X_u=X=u_1X_1 + u_2X_2\\).\n\nRegulatory capital measure \\(\\alpha\\), typically VaR or TVaR.\n\\textbf{Assume \\(\\alpha=\\)TVaR} at a threshold \\(p\\) close to 1. Thus\n\\(\\alpha(X_u) = \\text{E}[X\\mid X > F_u^{-1}(p)]\\). Total assets\n\\(\\alpha\\) have a natural co-measure allocation given by\n\\(\\alpha_i(X_u)= \\text{E}[u_iX_i \\mid X_u > F_u^{-1}(p)] = u_i \\partial T/\\partial u_i\\).\n\nDistortion \\(g\\) determines market pricing for risk transfer, via\n\\(E_g(X) = \\int g(S_X(t))dt\\).\n\nPrice is determined by using \\(g\\) to price\n\\(X\\wedge \\alpha(X):=\\text{min}(X, \\alpha(X))\\).\n\nThe resulting pricing functional \\(\\pi\\) can be written in several ways.\nLet \\(F_u^{-1}\\) be the quantile function (i.e.~value at risk) of\n\\(X_u\\). Then \\[\n\\begin{align}\n\\pi(X_u) &= \\int_0^{\\alpha(X)} g(S_u(t))dt \\\\\n&= \\text{E}[(X_u\\wedge \\alpha(X_u)) g'(S_u(X)] \\\\\n&= \\int_0^1 (F_u^{-1}(p)\\wedge \\alpha(X_u)) g'(1-p)dp. \\\\\n\\end{align}\n\\]\n\nWe both agree that, subject to certain continuity and differentiability\nassumptions, \\[\n\\begin{align}\n\\frac{\\partial\\pi}{\\partial u_i}(u) &= \\text{E}\\left[\\left(X_i 1_{\\{X\\le \\alpha\\}} + E[X_i\\mid X > F_u^{-1}(p)]1_{\\{X > \\alpha\\}}\\right) g'(S_u(X_u) \\right] \\\\\n&= \\text{E}[ X_i g'(S(X)) \\mid X \\le \\alpha](1-g(S(\\alpha)))+ \\text{E}[X_i \\mid X > F_u^{-1}(p)]g(S(\\alpha)).\\\\\n\\end{align}\n\\] The premium for line \\(i\\) is \\(u_i\\partial\\pi / \\partial u_i\\) in\nthe usual way. Euler implies the premium ``adds-up'' because the\nportfolio is homogeneous.\n\nThe result can be seen in a couple of ways. From the first defintion of\n\\(\\pi\\) as an \\(\\int_0^\\alpha\\) and the three differentials noted above\nwe get \\[\n\\begin{align}\n\\nabla_u\\pi &= \\int_0^{\\alpha(X)} \\nabla_u(g(S_u(t)))dt + g(S_u(\\alpha))\\nabla_u\\alpha(u) \\\\\n&=\\int_0^{\\alpha(X)} g'(S_u(t))\\text{E}[X_i 1_{\\{X=t\\}}] dt + g(S_u(\\alpha))E[X_i\\mid X> F_u^{-1}(p) ] \\\\\n&=\\int_0^{\\alpha(X)} g'(S_u(t))\\text{E}[X_i \\mid X=t] f_u(t) dt + g(S_u(\\alpha))E[X_i\\mid X> F_u^{-1}(p) ] \\\\\n&=\\text{E}\\left[\\left(X_i 1_{\\{X\\le \\alpha\\}} + E[X_i\\mid X> F_u^{-1}(p)]1_{\\{X > \\alpha\\}}\\right) g'(S_u(X)) \\right]\n\\end{align}\n\\] where the last line follows from the conditional expectation (tower\nproperty) formula. Notice that the default expectation is take wrt\n\\(X>F_u^{-1}(p)\\) whereas the outside integral is split at tvar.\n\nAlternatively using the third definition of \\(\\pi\\) \\[\n\\begin{align}\n\\nabla_u\\pi\n&= \\nabla_u \\int_0^1 (F_u^{-1}(p)\\wedge \\alpha(u)) g'(1-p)dp \\\\\n&= \\int_0^1 \\left(\\nabla_u F_u^{-1}(p) 1_{\\{X_u\\le \\alpha\\}} +\n \\nabla_u \\alpha(u) 1_{\\{X_u > \\alpha\\}}\\right) g'(1-p)dp \\\\\n&= \\int_0^{F(\\alpha)} \\text{E}[X_i \\mid X=F_u^{-1}(p) ] g'(1-p)dp + \\text{E}[X_i \\mid X> F_u^{-1}(p)] g(S(\\alpha)) \\\\\n&= \\int_0^{\\alpha} \\text{E}[X_i \\mid X=t ] g'(S(t))f(t)dt + \\text{E}[X_i \\mid X> F_u^{-1}(p)] g(S(\\alpha)) \\\\\n&= \\text{E}\\left[ \\text{E}[X_i \\mid X ]1_{\\{X\\le\\alpha\\}} g'(S(X_u)) \\right]\n+ \\text{E}[X_i \\mid X> F_u^{-1}(p)] g(S(\\alpha)) \\\\\n&= \\text{E}\\left[ X_i 1_{\\{X\\le\\alpha\\}} g'(S(X_u)) \\right]\n+ \\text{E}[X_i \\mid X> F_u^{-1}(p)] g(S(\\alpha)) \\\\\n\\end{align}\n\\] It is important that \\(g'(1-p)\\) \\emph{does not depend on \\(u\\)}. The\ndependence on \\(u\\) has been subsumed into the re-ordering defined by\n\\(F_u^{-1}\\). The fourth line follows from the third via substituting\n\\(F(t)=p\\).\n\n\\hypertarget{properties-of-method-a}{%\n\\subsubsection{Properties of Method A}\\label{properties-of-method-a}}\n\n\\begin{itemize}\n\\tightlist\n\\item\n \\textbf{Assumes and relies critically on homogeneity}\n\\item\n Result is independent of the default rule\n\\item\n Produces core of fuzzy game and only measure consistent with\n performance management. (In fuzzy game approach there is an assumption\n that the underlying risks can be fractionally quota shared, not that\n they are homogeneous themselves.)\n\\end{itemize}\n\n\\hypertarget{method-b}{%\n\\subsection{Method B}\\label{method-b}}\n\nPortfolio \\(X(u)\\) that may or may not be homogeneous. An inhomogenous\nexample is \\(X(u)=X_1(u_1) + X_2(u_2)\\) where \\(X_i(u_i)\\) is a compound\nPoisson distribution with expected claim count \\(u_i\\).\n\nRegulatory capital \\(\\alpha\\) determined by TVaR. Same distortion \\(g\\)\ndetermines cost of risk transfer.\n\nThe insurer sells the residual value of the risk when supported by\nassets of \\(a\\) to the capital markets for a premium \\(P(a)\\). The sale\nis structured by tranche, so the premium is an integral of the premium\ndensity. For a layer attaching at \\(x\\) (i.e.~a layer \\(x\\) to \\(x+dx\\))\nthe expected loss cost is \\(S(x)dx\\). As is well known, the expected\nlosses upto total capital \\(a\\) are given by \\[\n\\int_0^a S(x)dx.\n\\] The distortion function \\(g\\) prices the layer at \\(x\\) as\n\\(g(S(x))\\). Therefore the total premium is \\[\nP(a) = \\int_0^a g(S(x))dx.\n\\] The premium density is therefore \\[\nP'(x) = \\frac{dP}{da} = g(S(x)).\n\\]\n\nThe insurer now faces the problem of allocating the premium\n\\(P(\\alpha(X))\\) back to lines 1 and 2. Use a co-measure approach---this\nis exavtly what our spreadsheet did in the CAS sessions! We have a\nstable world, with \\(u_1,u_2\\) are fixed. The pricing distortion\nfunction \\(g\\) combined with fixed \\(X\\) gives a probability distortion\n\\(g'(S(X))\\), i.e.~and \\(E_{\\Bbb{Q}}[X] := E[Xg'(S(X))]\\). Standard\nfinance theory says that in this situation, with a linear pricing rule,\nwe price any payoff \\(Y\\) by \\$\nE\\_\\{\\Bbb{Q}\\}{[}Y{]}=E{[}Yg'(S(X)){]}\\$.\n\nWe have to specify the payoff to line \\(i\\) when the insurer only has\nassets \\(a\\). Consider two possibilities.\n\n\\begin{enumerate}\n\\def\\labelenumi{\\arabic{enumi}.}\n\\tightlist\n\\item\n The two lines have equal priority in event of default. The payment to\n line \\(i\\) is \\[\n X_i(a):=\n \\begin{cases}\n X_i & X \\le a \\\\\n X_i\\frac{a}{X} & X > a \\\\\n \\end{cases}\n = X_i \\frac{X\\wedge a}{X}.\n \\]\n\\item\n Line 2 has lower priority than line 1. In this case the payout to line\n 1 is \\(X_1\\wedge a\\) and the payout to line 2 is \\[\n X_2(a):=\n \\begin{cases}\n X_2 & X \\le a \\\\\n a-X_1 & X_1 \\le a < X \\\\\n 0 & X_1 > a.\n \\end{cases}\n \\]\n\\end{enumerate}\n\nIn the case of equal priority the expected recovery to line \\(i\\) when\nthe insurer has assets \\(a\\) is \\[\n\\begin{align}\n\\text{E}[X_i(a)] &= \\text{E}[X_i \\mid X \\le a ]F(a) + a \\text{E}\\left[\\frac{X_i}{X} \\mid X > a \\right] S(a) \\\\\n&= \\text{E}\\left[ X_i\\frac{X\\wedge a}{X} \\right].\n\\end{align}\n\\] From the second expression it is obvious this is an additive\nallocation---in all cases, not just when \\(X\\) is a homogeneous family.\n\nFinance theory now tells us the price of the contingent payout\n\\(X_i(\\alpha)\\) is \\[\n\\begin{align}\nP_i(u)=P_i(\\alpha(X_u) &= \\text{E}_{\\Bbb{Q}}\\left[ X_i\\frac{X\\wedge a}{X} \\right] \\\\\n&= \\text{E}\\left[ X_i\\frac{X\\wedge a}{X} g'(S(X)) \\right] \\\\\n&= \\text{E}\\left[ \\text{E}\\left[ X_i\\frac{X\\wedge a}{X} g'(S(X)) \\mid X \\right] \\right] \\\\\n&= \\text{E}\\left[ \\text{E}\\left[\\frac{X_i}{X}\\mid X \\right] (X\\wedge a) g'(S(X)) \\right] \\\\\n&= \\text{E}\\left[ \\text{E}\\left[ X_i \\mid X \\le \\alpha \\right] g'(S(X)) \\right] (1-g(S(\\alpha)))\n + a\\text{E}\\left[ \\text{E}\\left[\\frac{X_i}{X}\\mid X>\\alpha \\right] g'(S(X)) \\right] g(S(\\alpha)) \\\\\n&= \\text{E}\\left[ X_i 1_{\\{X \\le \\alpha \\}} g'(S(X)) \\right]\n + a\\text{E}\\left[ \\frac{X_i}{X} g'(S(X)) 1_{\\{ X>\\alpha\\}} \\right] \\\\\n&= \\text{E}_{\\Bbb{Q}}\\left[ X_i 1_{\\{X \\le \\alpha \\}} + a \\frac{X_i}{X} 1_{\\{ X>\\alpha\\}} \\right] \\\\\n\\end{align}\n\\]\n\nThe price component when the insurer remains solvent is the same as for\nMethod A. However the value of the payment made in the event of\ninsolvency is different---and depends on the explicit default\nassumption: \\[\n\\begin{align}\n\\text{Method A: } & \\text{E}[X_i | X > \\alpha]g(S(\\alpha) \\\\\n\\text{Method B: } & a\\text{E}\\left[ \\frac{X_i}{X} g'(S(X)) \\mid X>\\alpha \\right] g(S(\\alpha))\n= a\\text{E}\\left[ \\text{E}[\\frac{X_i}{X} \\mid X] g'(S(X)) 1_{\\{ X>\\alpha\\}} \\right] \\\\\n\\end{align}\n\\]\n\nIn method A the expectation is the allocation of total assets \\(\\alpha\\)\nto line \\(i\\).\n\nSimilar calculations can be performed for the case Line 2 has lower\npriority.\n\n\\hypertarget{properties-of-method-b}{%\n\\subsubsection{Properties of Method B}\\label{properties-of-method-b}}\n\n\\begin{itemize}\n\\tightlist\n\\item\n Does \\textbf{not} assume or rely on homogeneity\n\\item\n Result dependents particular the default rule\n\\end{itemize}\n", "meta": {"hexsha": "a05d737e83f185a73986159b98657260d53235ab", "size": 9236, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "examples/dm.tex", "max_stars_repo_name": "mynl/aggregate", "max_stars_repo_head_hexsha": "48ab306fb9d19f08d6d42112490fc305c376ca8d", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-01-07T13:42:57.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-23T19:46:55.000Z", "max_issues_repo_path": "examples/dm.tex", "max_issues_repo_name": "mynl/aggregate", "max_issues_repo_head_hexsha": "48ab306fb9d19f08d6d42112490fc305c376ca8d", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "examples/dm.tex", "max_forks_repo_name": "mynl/aggregate", "max_forks_repo_head_hexsha": "48ab306fb9d19f08d6d42112490fc305c376ca8d", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.3318777293, "max_line_length": 160, "alphanum_fraction": 0.6474664357, "num_tokens": 3351, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3463095570113093}} {"text": "Having outlined a variety of models and metrics tracked for the\npurposes of their objective comparison, we proceed to present and discuss our\nresults in the next sections.\n\n\n\\subsection{Results of Decoupled Sampling}\n\\label{sec:modelres}\n\nWe begin by comparing the performance of a diverse set of surrogate families on previously generated samples of the expensive MC TBR model.\nThrough the four experimental cases described\nin~\\cref{sec:experiment-methodology}, we aim to study properties of the\nconsidered models in terms of regression performance, training and prediction\ntime.\n\n\n\\subsubsection{Hyperparameter Tuning}\n\nThe first two experiments perform Bayesian optimisation to maximise~$R^2$ in\na cross-validation setting as a function of model hyperparameters. While in the\nfirst experiment we limit training and test sets to the scope of four selected\nslices of the feature space, in the second experiment we lift this restriction\nto examine surrogate capability to model a more complex domain.\n\nThe results displayed in~\\cref{fig:exp1-time-vs-reg} (and listed\nin~\\cref{tbl:exp1-detailed-results} in the Appendix) indicate that in the first\nexperiment, GBTs clearly appear to be the most accurate as\nwell as the fastest surrogate family in terms of mean prediction time. Following\nthat, we note that ERTs, SVMs and ANNs also achieved satisfactory results with respect to both examined metrics.\nWhile the remainder of tested surrogate families does not exhibit problems in\ncomplexity, its regression performance falls below average.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[b]{0.333\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp1_slice0}\n\t\t\\caption{Run 2, batches 0-2}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.333\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp1_slice1}\n\t\t\\caption{Run 2, batches 100-102}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.333\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp1_slice2}\n\t\t\\caption{Run 2, batches 200-202}\n\t\\end{subfigure}\n\t\\caption{20~best-performing surrogates per each considered family, plotted in\n\t\tterms of complexity (as~$\\overline{t}_{\\text{pred.}}$) and regression\n\t\tperformance (as~$R^2$) on selected slices of run~2, evaluated in\n\texperiment~1. Here, batches refer to subsets of training and test datasets that\n\tmay be matched to slices using~\\cref{tbl:slices}.}\n\t\\label{fig:exp1-time-vs-reg}\n\\end{figure}\n\nThe results of the second experiment, shown in~\\cref{fig:exp2-time-vs-reg} (and listed\nin~\\cref{tbl:exp2-detailed-results} in the Appendix),\nseem to confirm our expectations. Compared to the previous case, we observe\nthat many surrogate families consistently achieved worse regression\nperformance and prediction times. The least\naffected models appear to be GBTs, ANNs and ERTs, which are known to be capable of capturing relationships\ninvolving mixed feature types that were deliberately withheld in the first\nexperiment. With only negligible differences, the first two of these families\nappear to be tied for the best performance as well as the shortest prediction\ntime. We observe that ERTs and RBFs also\ndemonstrated satisfactory results, relatively outperforming the remaining surrogates in\nterms of regression performance, and in some cases also in prediction time.\n\n\\begin{wrapfigure}[10]{r}{0.333\\textwidth}\n\t\\centering\n\t\\vspace{-2ex}\n\t\\includegraphics[width=\\linewidth]{exp2_time_vs_reg}\n\t\\caption{Results of experiment~2, plotted analogously\n\tto~\\cref{fig:exp1-time-vs-reg}.}\n\t\\label{fig:exp2-time-vs-reg}\n\\end{wrapfigure}\n\nFollowing both hyperparameter tuning experiments, we conclude that while domain\nrestrictions employed in the first case have proven effective in improving the\nregression performance of some methods, this result has fluctuated considerably\ndepending on the selected slices. Furthermore, in all instances the best\nresults were achieved by families of surrogates that were nearly unaffected by\nthis modification.\n\n\n\\subsubsection{Scaling Benchmark}\n\nIn the third experiment we examine surrogate scaling properties by correlating\nmetrics of interest with training set size. Firstly, the results shown \nin~\\cref{fig:scaling-r2} (and listed in~\\cref{tbl:exp3-detailed-results-r2} in\nthe Appendix) suggest that the most accurate families from the previous experiments\nconsistently maintain their relative advantage over others, even as we introduce\nmore training points. While such families achieve nearly comparable\nperformance on the largest dataset, in the opposite case tree-based approaches\nclearly outperform ANNs. This can be observed\nparticularly on sets of sizes up to~\\num{6000}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[b]{0.333\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{scaling_metric_r2}\n\t\t\\caption{Regression performance (as $R^2$)}\n\t\t\\label{fig:scaling-r2}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.333\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{scaling_time_train}\n\t\t\\caption{Complexity (as~$\\overline{t}_{\\text{trn.}}$)}\n\t\t\\label{fig:scaling-trn}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.333\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{scaling_time_pred}\n\t\t\\caption{Complexity (as~$\\overline{t}_{\\text{pred.}}$)}\n\t\t\\label{fig:scaling-pred}\n\t\\end{subfigure}\n\t\\caption{Various metrics collected during experiment 3 (scaling\n\tbenchmark) displayed as a function of training set size.}\n\t\\label{fig:scaling}\n\\end{figure}\n\nNext, we examine scaling behaviour in terms of the mean training time (displayed\nin~\\cref{fig:scaling-trn} and listed in~\\cref{tbl:exp3-detailed-results-t-train} in\nthe Appendix). Consistent with our expectation, the shortest times\nwere achieved by instance-based learning methods (e.g. KNN, IDW) that\nare trained trivially at the expense of increased lookup complexity later during prediction.\nFurthermore, we observe that the majority of tree-based algorithms also perform\nand scale well, unlike RBFs and GPR which appear to behave superlinearly. We note that ANNs,\nwhich are the only family to utilise parallelisation during training, show an\ninverse scaling characteristic. Our conjecture is that this effect may be caused\nby a constant multi-threading overhead that possibly dominates the training process\non relatively small training sets.\n\nFinally, we study scaling with respect to the mean prediction time (shown\nin~\\cref{fig:scaling-pred} and listed in~\\cref{tbl:exp3-detailed-results-t-pred} in\nthe Appendix). Our initial observation is that all tested\nfamilies with the exception of previously mentioned instance-based\nmodels offer desirable characteristics overall. Analogous to previous\nexperiments, GBTs, ABTs and ANNs appear to be tied, as they not only exhibit\ncomparable times but also similar scaling slopes. Following that, we notice a\nclear hierarchy of ERTs, SVMs, GPR and RBFs, trailed by IDW and KNNs.\n\n\n\\subsubsection{Model Comparison}\n\nIn the fourth experiment proposed\nin~\\cref{sec:experiment-methodology}, we exploit previously collected information\nto produce surrogates with desirable properties for practical use. We\naim to create models that yield: (a)~the best regression performance regardless\nof other features, (b)~acceptable performance with the shortest mean\nprediction time, or (c)~acceptable performance with the smallest training set.\nTo this end, we trained 8~surrogates that are presented in~\\cref{fig:reg-performance}\nand~\\cref{tbl:exp4-detailed-results}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model6}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model7}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model1}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model3}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model4}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model5}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model2}\n\t\\end{subfigure}\\hfill%\n\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{exp4_model8}\n\t\\end{subfigure}\n\t\\caption{Regression performance of models 1-4 (row 1, from the left) and 5-8\n\t\t(row 2) trained in experiment~4 (model comparison), viewed\n\t\tas true vs.~predicted TBR on a test set of a selected cross-validation\n\t\tfold. Points are coloured by density.}\n\t\\label{fig:reg-performance}\n\\end{figure}\n\nHaving selected ANNs, GBTs, ERTs, RBFs and SVMs based on the results of\nexperiments~2-3, we utilised the best-performing hyperparameters.\nIn pursuit of goal~(a), the best approximator (no.~1,\nANN) achieved~$R^2=\\num{0.998}$ and mean prediction\ntime~$\\overline{t}_{\\text{pred.}}=\\SI{1.124}{\\micro\\second}$. These correspond\nto a standard error~$S=\\num{0.013}$ and a relative speedup~$\\omega=\\num{6916416} \\times$\nwith respect to the MC TBR evaluation baseline measured during run~1\n(see~\\cref{tbl:sampling-runs} for details). Satisfying\ngoal~(b), the fastest model (no.~2, ANN) achieved~$R^2=\\num{0.985}$,\n$\\overline{t}_{\\text{pred.}}=\\SI{0.898}{\\micro\\second}$, $S=\\num{0.033}$\nand~$\\omega=\\num{8659251} \\times$.\nWhile these surrogates\nwere trained on the entire available set of~\\num{500000} datapoints, to satisfy\ngoal~(c) we also trained a more simplified model (no.~4, GBT)\nthat achieved~$R^2=\\num{0.913}$,\n$\\overline{t}_{\\text{pred.}}=\\SI{6.125}{\\micro\\second}$, $S=\\num{0.072}$ and $\\omega=\\num{1269777} \\times$\nwith a set of size only~\\num{10000}.\n\n\\begin{table}[h]\n\t\\centering\n\t\\sisetup{round-mode=places,round-precision=3,detect-weight=true,detect-family=true}\n\t\\setlength\\tabcolsep{2pt}\n\t{\\scriptsize\n\t\t\\begin{tabular}{lrrrrrrrr}\n\t\t\\toprule\n\t\t{} & {} & \\multicolumn{4}{c}{Regression performance} &\n\t\t\\multicolumn{3}{c}{Complexity}\\\\\n\t\t\\cmidrule(lr){3-6}\n\t\t\\cmidrule(lr){7-9}\n\t\tModel & $|\\mathcal{T}|$ & MAE [TBR] & $S$ [TBR] & $R^2$ [rel.] & $R^2_{\\text{adj.}}$ [rel.]\n\t\t\t\t\t\t& $\\overline{t}_{\\text{trn.}}$ [\\si{\\milli\\second}] &\n\t\t$\\overline{t}_{\\text{pred.}}$ [\\si{\\milli\\second}] & $\\omega$ [rel.]\\\\\n\t\t\\midrule\n\t\t\\input{tbl/table11}\n\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\t\\caption{Results of experiment~4. Here, figures are reported over 5~cross-validation folds,\n\t\t$|\\mathcal{T}|$~denotes cross-validation set size ($\\times 10^3$)\n\t\tand $\\omega$ is a relative speedup with respect to\n\t\t$\\overline{t}_{\\text{eval.}}=\\SI{7.777049573054314}{\\second}$\n\t\tmeasured in the MC TBR model during run~1 (see~\\cref{tbl:sampling-runs}).\n\t\tThe best-performing metrics are highlighted in bold.}\n\t\\label{tbl:exp4-detailed-results}\n\\end{table}\n\nOverall we found that due to their superior performance, boosted tree-based\napproaches seem to be advantageous for fast surrogate modelling on relatively small training\nsets (up to the order of~$10^4$). Conversely, while neural networks perform\npoorly in such a setting, they dominate on larger training sets (at least of the\norder of~$10^5$) both in terms of regression performance and mean prediction time.\n\n\\begin{wrapfigure}[10]{r}{0.4\\textwidth}\n\t\\centering\n\t\\vspace{-16ex}\n\t\\includegraphics[width=\\linewidth]{fig5_sintoy}\n\t\\caption{Sinusoidal toy TBR theory over two continuous parameters,\n\t$n=1$.}\n\t\\label{fig:sintoy}\n\\end{wrapfigure}\n\n\\subsection{Results of Adaptive Sampling}\n\\label{sec:adaptiveres}\nIn order to test our QASS prototype, several functional toy theories for TBR were developed as alternatives to the expensive MC model. By far the most robust of these was the following sinusoidal theory with adjustable wavenumber parameter $n$:\n\n\\begin{equation}\n\t\\text{TBR} = \\frac{1}{|C|}\\sum_{i \\in C} \\left[1 + \\sin(2\\pi n (x_i - 1/2)) \\right]\n\\end{equation}\n\nplotted in~\\cref{fig:sintoy} for $n=1$ and two continuous parameters $C$.\nANNs trained on this model demonstrated similar performance to those on the expensive\nMC model. QASS performance was verified by training a $\\text{1h3f}(256)$ ANN on\nthe sinusoidal theory for varied quantities of initial, incremental, and MCMC\ncandidate samples. Although the scope of this project did not include thorough\nsearches of this hyperparameter domain, sufficient runs were made to identify\nsome likely trends.\n\nAn increase in initial samples with increment held constant had a strong impact\non final surrogate precision, an early confirmation of basic functionality. An\nincrease in MCMC candidate samples was seen to have a positive but very weak\neffect on final surrogate precision, suggesting that the runtime of MCMC on each\niteration can be limited for increased efficiency. The most complex dynamics\narose with the adjustment of sample increment, shown in~\\cref{fig:qassincr}. For\neach tested initial sample quantity $N$, the optimal number of step samples was seen to be well-approximated by $\\sqrt{N}$. The plotted error trends suggest that\nincremental samples larger than this optimum give slower model improvement on both the training and evaluation sets, and a larger minimum error on the evaluation set. This performance distinction is predicted to be even more significant when trained on the expensive MC model, where the number of sample evaluations will serve as the primary bottleneck for computation time.\n\\begin{figure}[h!]\n \\centering\n \\vspace{-1ex}\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1.1\\linewidth]{fig6a_qassincrsamp.pdf}\n \\end{subfigure}%\n \\hfill%\n \\begin{subfigure}[t]{0.5\\textwidth}\n \\centering\n \\includegraphics[width=1.1\\linewidth]{fig6b_qassincrtime.pdf}\n \\end{subfigure}\n \\caption{QASS absolute training error over total sample quantity (left) and number of iterations (right). MAE represents surrogate error on the adaptively-sampled training/test set, and E\\_MAE on the independent evaluation sets.}\n \\label{fig:qassincr}\n \\vspace{-1ex}\n\\end{figure}\n\nThe plateau effect in surrogate error on the evaluation set, seen\nin~\\cref{fig:qassincr}, was universal to all configurations and thought to\nwarrant further investigation. At first this was suspected to be a residual\neffect of retraining the same ANN instance without adjustment to data\nnormalisation. A ``Goldilocks scheme'' for checking normalisation drift was\nimplemented and tested, but did not affect QASS performance. Schemes in which\nthe ANN is periodically retrained were also discarded, as the retention of\nnetwork weights from one iteration to the next was demonstrated to greatly\nbenefit QASS efficiency. Further insight came from direct comparison between\nQASS and a baseline scheme with uniformly random incremental samples, shown\nin~\\cref{fig:qasssampling}.\n\n\\begin{figure}[h]\n\\centering\n\\begin{minipage}{.48\\textwidth}\n \\centering\n \\includegraphics[width=1.1\\linewidth]{fig7_qasssampling.pdf}\n \\captionof{figure}{Absolute training error for QASS, baseline scheme, and mixed scheme.}\n \\label{fig:qasssampling}\n\\end{minipage}%\n\\hfill%\n\\begin{minipage}{.48\\textwidth}\n \\centering\n \\includegraphics[width=1.1\\linewidth]{fig8_qasssampling100k.pdf}\n \\captionof{figure}{Absolute training error for QASS and baseline scheme, with 100k initial samples.}\n \\label{fig:qasssampling100k}\n\\end{minipage}\n\\end{figure}\n\nSuch tests revealed that while QASS has unmatched performance on its own\nadaptively-sampled training set, it is outperformed by the baseline scheme on\nuniformly-random evaluation sets. We suspected that while QASS excels in\nlearning the most strongly peaked regions of the TBR theory, this comes at the\nexpense of precision in broader, smoother regions where uniformly random\nsampling suffices. Therefore a mixed scheme was implemented, with half MCMC\nsamples and half uniformly random samples incremented on each iteration, which\nis also shown in~\\cref{fig:qasssampling}. An increase in initial sample size was observed to also resolve precision in these smooth regions of the toy theory, as the initial samples were obtained from a uniform random distribution. As shown in ~\\cref{fig:qasssampling100k}, with~\\num{100000} initial samples it was possible to obtain a ${\\sim}40\\%$ decrease in error as compared to the baseline scheme, from 0.0025 to 0.0015 mean averaged error. Comparing at the point of termination for QASS, this corresponds to a ${\\sim}6\\%$ decrease in the number of total samples needed to train a surrogate with the same error. \n\n\n", "meta": {"hexsha": "eb8e52632bf2bf553a34c848427b273c8d951952", "size": 16634, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "final_report/results.tex", "max_stars_repo_name": "ukaea-group-project/Documentation", "max_stars_repo_head_hexsha": "fcc642a2969e86c532d03254bb4b162e9f23ee01", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-06-22T11:25:42.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-22T11:25:42.000Z", "max_issues_repo_path": "final_report/results.tex", "max_issues_repo_name": "ukaea-group-project/Documentation", "max_issues_repo_head_hexsha": "fcc642a2969e86c532d03254bb4b162e9f23ee01", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "final_report/results.tex", "max_forks_repo_name": "ukaea-group-project/Documentation", "max_forks_repo_head_hexsha": "fcc642a2969e86c532d03254bb4b162e9f23ee01", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-06-04T14:53:55.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-04T14:53:55.000Z", "avg_line_length": 48.9235294118, "max_line_length": 617, "alphanum_fraction": 0.7759408441, "num_tokens": 4584, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381372136564, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.3463095438717233}} {"text": "\\documentclass[reqno]{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{lmodern}\n\\usepackage{blindtext}\n\\usepackage{amsmath}\n\\usepackage{relsize} %bigger math symbols\n\\usepackage{bm} %bolder text\n\\usepackage[a4paper, inner=1.7cm, outer=2.7cm, top=3cm, bottom=3cm, bindingoffset=1.2cm]{geometry}\n\\usepackage{tcolorbox}\n\\usepackage{graphicx}\n\\usepackage{wrapfig}\n\\usepackage{enumitem}\n\\newlist{terms}{description}{1}\n\\begin{document}\n\\title{\\textbf{Scalable Reward Distribution with Compounding Stakes}}\n\\author{Richard Pardoe \\thanks{rick@liquity.org}}\n\\date{December 2020\\\\v1.0}\n\\maketitle\n\\begin{abstract}\n Scalability of Ethereum dApps is constrained by the gas costs of writing to storage. \n In previous work, Batog $\\cite{Batog}$ and Solmaz $\\cite{Solmaz}$ solved the problem of scalable reward distribution in a staking pool, distributing rewards in proportion to a staker’s share of total stakes with a ``pull'' based approach. Only a single running sum need be updated at each reward, sidestepping the need for an expensive $O(n)$ storage update for n stakers. However, their work stopped short of solving the problem for a compounding stake.\n Here we derive formulas for a scalable implementation of a compounding, decreasing Stability Pool deposit and its corresponding ETH gain. Thus, Liquity tracks the deposits and rewards of every depositor with only $O(1)$ complexity for each liquidation.\n\\end{abstract}\n\n\\section{Introduction}\nThe Liquity protocol uses a Stability Pool as its primary mechanism to liquidate collateralized debt positions (called ``troves'') that fall below the minimum collateralization ratio of 110$\\%$ $\\cite{Whitepaper}$. Stability deposits (provided in LUSD) absorb and cancel the debt (in LUSD) from defaulted troves: an amount of LUSD in the Stability Pool corresponding to the debt of the liquidated trove is burned from the pool’s balance, offsetting the debt. In return, Stability Pool depositors are rewarded with the acquisition of collateral (in ETH) from liquidated positions at a significant discount. The proportion of a stability depositor’s current deposit to the total LUSD in the pool determines the collateral share it receives from the liquidation. \n\nThe fact that liquidated troves decrease all deposits in the Stability Pool leads to an implementation challenge: Due to Ethereum's block gas limit, it becomes impossible to iterate over all deposits in the Stability Pool for a large number of deposits. Such an operation has computational complexity of $O(n)$ and does not scale. \n\nThis paper presents a method to perform liquidations against the Stability Pool in a scalable manner with only $O(1)$ complexity. The method is ``pull-based\": Instead of adjusting all stability deposits upon every liquidation, the update is deferred to the moment at which an owner withdraws or tops up their deposit.\n\nThe methods presented in this paper are generalizable and can be used for other dApps running on Ethereum.\n\n\\section{Derivation}\nFor a pull-based implementation in Liquity’s Stability Pool, we must factor out the initial deposit, and track reward terms that are independent from the individual deposit values. We derive formulas for a compounding LUSD deposit and corresponding ETH gain as functions of the initial deposit, and involving a product $P$ and a sum $S$: the former is a function of the LUSD losses, and the latter is a function of $P$, the ETH rewards, and the total deposits.\n\n\\bigskip\n\\raggedright %align to the left side\n\n\\textbf{Terms}\n\\begin{terms}\n \\item $d_i:$ A given user’s LUSD deposit at liquidation event \\textit{i}\\\\\n \\item $Q_i:$ The LUSD debt absorbed by the Stability Pool from liquidation \\textit{i}\\\\\n \\item $E_i:$ The ETH sent to the Stability Pool from liquidation \\textit{i}\\\\\n \\item $D_i:$ Total LUSD deposits at liquidation \\textit{i}\n\\end{terms}\n\n\\bigskip\nLet $d_0$ represent the user’s initial Stability Pool deposit. After the first liquidation, their new\ndeposit $d_0$ is given by:\\\\\n\\begin{equation} \n d_1=d_0-LUSDLoss\n\\end{equation}\n\n\\bigskip\nLiquidation assigns an LUSD loss to the deposit proportional to the prior share of total deposits:\n\\begin{equation} \\label{eq:2}\n d_1=d_0-Q_1\\frac{d_0}{D_0}\n\\end{equation}\n\\begin{equation} \n d_1=d_0\\left(1-\\frac{Q_1}{D_0}\\right)\n\\end{equation}\n\n\\bigskip\nSince the deposit compounds, the deposit value after liquidation 2 is a function of the previous deposit value:\n\\begin{equation} \n d_2=d_1\\left(1-\\frac{Q_2}{D_1}\\right)\n\\end{equation}\n\n\\bigskip\nand substituting equation (\\ref{eq:2}):\n\\begin{equation} \n d_2=d_0 \\left(1-\\frac{Q_2}{D_1}\\right) \\left(1-\\frac{Q_1}{D_o} \\right)\n\\end{equation}\n\n\\bigskip\nSimilarly:\n\\begin{equation} \n d_3=d_2\\left(1-\\frac{Q_3}{D_2}\\right)\n\\end{equation}\n\\begin{equation} \n d_3=d_0\\left(1-\\frac{Q_3}{D_2}\\right)\\left(1-\\frac{Q_2}{D_1}\\right)\\left(1-\\frac{Q_1}{D_0}\\right)\n\\end{equation}\n\n\\bigskip\nAnd the general case for a compounded deposit:\n\\begin{equation} \\label{eq:8}\n d_n=d_0\\prod^n_{i=1}{\\left(1-\\frac{Q_i}{D_{i-1}}\\right)}\n\\end{equation}\n\\subsection{Compounded deposit for initial deposit made at $t>0$}\nFor a deposit made between liquidations $[t, t+1]$, which is withdrawn between liquidations $[s, s+1]$, rewards are earned from liquidations $t+1, t+2, \\ldots, s$.\n\nWe take a snapshot of the product at liquidation $t$, and calculate the compounded stake at liquidation $n$, with $t < n \\leq s$, as:\n\n\\begin{equation} \n d_n=d_t\\prod^n_{i=t+1}{\\left(1-\\frac{Q_i}{D_{i-1}}\\right)}=d_t\\frac{\\displaystyle\\prod^n_{i=1}{\\left(1-\\frac{Q_i}{D_{i-1}}\\right)}}{\\displaystyle\\prod^t_{i=1}\\left(1-\\frac{Q_i}{D_{i-1}}\\right)}\n\\end{equation}\nThis assumes that every multiplicand is non-zero, i.e. that $Q_i < D_{i-1}$ for all $i$ (see section 3.1 for the case where $Q_i = D_{i-1}$ and the pool is emptied).\n\n\\bigskip\nLabelling the product term $P_k$, i.e: $P_k = \\displaystyle\\prod^k_{i=1}{\\left(1-\\frac{Q_i}{D_{i-1}}\\right)}$, yields:\n\n\\begin{equation} \\label{eq:10}\nd_n=d_t\\frac{P_n}{P_t}\n\\end{equation}\n\n\\bigskip\nWhere $P_n$ is the product at liquidation $n$, and $P_t$ is the snapshot of the product at the time when the user made the deposit.\n\n\\bigskip\n\\subsection{Corresponding ETH Gain}\n\n\\bigskip\nThe LUSD deposit earns an ETH gain at each liquidation.\nAt each liquidation $i$, the user’s LUSD deposit effectively decreases. Thus we can write the\nuser’s cumulative ETH gain $e_{n}$, from a series of \\textit{n} liquidations, as:\n\\begin{equation} \ne_n=d_0\\left(\\frac{E_1}{D_0}\\right)+d_1\\left(\\frac{E_2}{D_1}\\right)+d_2\\left(\\frac{E_3}{D_2}\\right)+...+d_{n-1}\\left(\\frac{E_n}{D_{n-1}}\\right)\n\\end{equation}\n\n\\bigskip\nThen, using our expression for the deposit from equation (\\ref{eq:8}):\n\\begin{equation} \ne_n=d_0\\left(\\frac{E_1}{D_0}\\right)+d_0\\left(\\frac{E_2}{D_1}\\right)\\prod^1_{i=1}\\left(1-\\frac{Q_i}{D_{i-1}}\\right)+d_0\\left(\\frac{E_3}{D_2}\\right)\\prod^2_{i=1}\\left(1-\\frac{Q_i}{D_{i-1}}\\right)+...+d_0\\left(\\frac{E_n}{D_{n-1}}\\right)\\prod^{n-1}_{i=1}\\left(1-\\frac{Q_i}{D_{i-1}}\\right)\n\\end{equation}\n\n\\bigskip\nAnd factoring out the initial deposit, $d_0$:\n\\begin{equation} \\label{eq:13}\n e_n=d_0\\sum^n_{k=1}\\left[\\frac{E_k}{D_{k-1}}\\prod^{k-1}_{i=1}\\left(1-\\frac{Q_i}{D_{i-1}}\\right)\\right] = d_0\\sum^n_{k=1}\\left[\\frac{E_k}{D_{k-1}}P_{k-1}\\right]\n\\end{equation}\n\n\\bigskip\nWhere the initial product term $\\displaystyle\\prod^0_{i=1}(...)=1$.\n\n\\bigskip\nThus, we have the depositor's cumulative ETH gain $e_n$ as a function of the initial deposit $d_0$, and the rewards and total deposits at each liquidation.\\\\\nLet the summation $\\displaystyle\\sum^n_{k=1}[...]$ be denoted $S_n$.\n\n\\bigskip\n\\subsection{ETH gain for initial deposit made at $t>0$}\n\n\\bigskip\nFor a deposit made between liquidations $[t, t+1]$, which is withdrawn between liquidations $[s, s+1]$, rewards are earned from liquidations $t+1, t+2, \\ldots, s$.\n\nWe take a snapshot of the product at liquidation $t$, and calculate the compounded stake at liquidation $n$, with $t < n \\leq s$, as:\n\n\\bigskip\nTo account for the deposit entering after liquidations have already begun, we correct the product terms in (\\ref{eq:13}) to $\\frac{P_n}{P_t}$, as per (\\ref{eq:10}):\n\\begin{equation} \n e_n = d_t\\sum^n_{k=t+1}\\left[\\frac{E_k}{D_{k-1}}\\frac{P_{k-1}}{P_t}\\right]\n\\end{equation}\n\n\\bigskip\nWe take a snapshot of the sum at liquidation $t$, $S_t$. Then, the ETH gain earned by the deposit at liquidation $n$ is:\n\\begin{equation} \ne_n=d_t\\frac{S_n-S_t}{P_t}\n\\end{equation}\n\n\\bigskip\n\\section{Basic Implementation}\n\\underline{Making a deposit:}\n\nRecord deposit: deposit[user] = $d_t$\\\\\nUpdate total deposits: $D = D + d_t$\\\\\nRecord product snapshot: $P_t = P$\\\\\nRecord sum snapshot: $S_t = S$\n\n\\bigskip\n\\underline{Upon each liquidation yielding LUSD debt ‘Q’ offset with the Stability Pool and ETH gain ‘E’:}\n\nUpdate $S$: $S = S + \\frac{E}{D}P$ (\\textit{intuition: a deposit’s marginal ETH gain is equal to the deposit $\\times$ ETH per unit staked $\\times$ current correction factor)}\\\\\nUpdate $P$: $P = P(1 - \\frac{Q}{D})$\\\\\nUpdate total deposits: $D = D - Q$\n\n\\bigskip\n\\underline{Withdrawing the deposit and ETH gain:}\n\nCompute final compounded LUSD deposit $d$: $d = d_t\\frac{P}{P_t}$\\\\\nCompute final corresponding cumulative ETH gain $e$: $e = d_t\\frac{S - S_t}{P_t}$\\\\\nSend $d$ and $e$ to user\\\\\nUpdate deposit: deposit[user] = 0\\\\\nUpdate total deposits: $D = D - d$\\\\\n\n\\bigskip\n\\section{Practical Implementation in Liquity}\nTwo further considerations are needed for our implementation:\n\\begin{enumerate}\n \\item Liquidations that completely empty the Stability Pool\n \\item How to handle the eternally decreasing product $P$, without truncating to 0\n\\end{enumerate}\n\n\\bigskip\n\\subsection{Liquidations that completely empty the Pool}\n\n\\bigskip\n\\textbf{Problem}: Pool-emptying should reduce all deposits to 0, but we should not set $P$ to 0. Doing so would break deposit computations for all future deposits: as $P$ is a running product, all snapshots and future values would be 0.\n\n\\bigskip\n\\textbf{Solution}: Complete pool-emptying is handled by tracking a “current epoch” variable. The ETH gain reward sum $S$ for each epoch is stored in a mapping.\n\n\\bigskip\nUpon a pool-emptying liquidation, $S$ is first updated as usual, for the current epoch. Then, the current epoch is incremented by 1, and $P$ and $S$ terms are reset for the new epoch. By definition, deposits that were made in past epochs will have been completely cancelled with liquidated debt, and so reduced to 0.\n\n\\bigskip\n\\underline{Making a deposit:}\n\nRecord a snapshot of the current epoch for the deposit\n\n\\bigskip\n\\underline{Liquidation that empties the Pool:}\n\nCompute the latest value of $S$, and store it for the current epoch.\nIncrease the epoch by 1, and reset the product and sum ($P$ = 1, $S$ = 0).\n\n\\bigskip\n\\underline{Withdrawing Deposit:}\n\nWhen users withdraw, check their epoch snapshot against the current epoch: If equal, deposit and ETH gain are computed as normal.\n\n\\bigskip\nIf the current epoch is greater than the deposit’s epoch snapshot, then the deposit was made before the pool-emptying liquidation. Therefore:\n\\begin{itemize}\n \\item The \\textbf{compounded deposit} is 0, as the deposit has been fully used to absorb debt.\n \\item The user’s \\textbf{ETH gain} is computed using the $S$ sum that corresponds to the epoch in which their deposit was made.\n\\end{itemize}\n\n\\bigskip\n\\subsection{Eternally decreasing the product $P$, without truncating to 0}\n\n\\bigskip\nAt liquidation $i$, $P$ is multiplied by some new term $0 < p_i <1$. Thus in theory, $P$ is always decreasing, but should never reach 0. \n\n\\bigskip\n\\textbf{Problem:} We cannot represent an arbitrarily small value in Solidity. Eventually, division will truncate a small value to 0. If so, all future values of $P$ would be 0, and deposit computation would break.\n\n\\bigskip\n\\textbf{Solution:} use a “current scale” that allows $P$ to decrease indefinitely, but never reach 0. Each scale represents a division by 1e18. \n\n\\bigskip\nUpon a liquidation that would otherwise truncate $P$ to 0, $S$ is first updated as usual, for the current scale. Then, the current scale is incremented by 1, and $P$ is updated and scaled by 1e18.\n\n\\bigskip\n\\underline{Making a deposit:}\n\nRecord a snapshot of the current scale for the deposit\n\n\\bigskip\n\\underline{Liquidation:}\n\nCompute the latest value of $S$, and store it for the current scale.\nLiquidation $L_i$ causes the product $P$ to be multiplied by some new product factor $p_i$. If $P$ $\\times$ $p_i$ would be truncated to 0, instead do: $P = (P \\times p_i \\times 1e18)$, and increment the current scale by 1.\n\n\\bigskip\n\\underline{Withdrawing a deposit:}\n\nFirst, compute the number of scale changes made during the deposit’s lifetime, i.e. (currentScale - scaleSnapshot).\n\n\\bigskip\nIf the number of scale changes is 0, compute the deposit and ETH gain as normal.\n\n\\bigskip\nOtherwise:\n\n\\bigskip\n\\textbf{Compounded deposit:} If a scale change in $P$ was made during the deposit's lifetime, account for it in the deposit computation. If the deposit has decreased by a factor of $<$ 1e-18 (i.e. if more than one scale change was made) just return 0.\n\n\\bigskip\n\\textbf{ETH Gain:} Since the deposit may span up to one scale change, so too does the reward. In this case, obtain the ETH gain using the $S$ sums from the two consecutive scales that the deposit spans. The reward from the second sum is scaled by 1e-18. The rewards from both scales are added to make the final ETH gain.\n\n\\bigskip\n\\section{Putting it all together}\n\n\\bigskip\nThe implementation relies on a nested mapping: epochToScaleToSum. \n\n\\bigskip\nThe inner mapping stores the sum $S$ at different scales, for a given epoch. The outer mapping stores the ($scale \\Rightarrow sum$) mappings for each epoch.\n\n\\bigskip\nThis allows us to track the ETH reward terms at each scale and epoch.Thus we can correctly compute the ETH rewards for a compounding decreasing stake, taking account of periodic Pool-emptying, and getting around the limitations of Solidity arithmetic.\n\n\\bigskip\nAll implementation logic above is combined in PoolManager.sol, in the functions:\n\n\\bigskip\n$\\_getCompoundedLUSDDeposit()$\n\n$\\_getDepositorETHGain()$\n\n$\\_updateRewardSumAndProduct()$\n\n\\section{Acknowledgements}\nWe would like to thank Daniel Simon, Robert Lauko, Bingen Eguzkitza and Onur Solmaz for the helpful discussion on reward distributions and Bojan Peček for his help with preparing this paper.\n\n\\begin{thebibliography}{9}\n\\bibitem{Batog} \nB. Batog, L. Boca, N. Johnson.\n\\textit{Scalable Reward Distribution on the Ethereum Blockchain, 2018}. \nhttp://batog.info/papers/scalable-reward-distribution.pdf\n\n\\bibitem{Solmaz} \nO. Solmaz. \n\\textit{Scalable Reward Distribution with Changing Stake Sizes”, 2019}. \nhttps://solmaz.io/2019/02/24/scalable-reward-changing/\n\n\\bibitem{Whitepaper} \nR. Lauko, R. Pardoe. \n\\textit{Liquity: Decentralized Borrowing Protocol (Whitepaper), 2020}. \nhttps://docsend.com/view/bwiczmy\n\n\n\\end{thebibliography}\n\n\\end{document}", "meta": {"hexsha": "c42012752a2bb21559e84e2a09c76c09b382e694", "size": 14993, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "papers/Scalable_Reward_Distribution_with_Compounding_Stakes.tex", "max_stars_repo_name": "EthereumGeeks/dev", "max_stars_repo_head_hexsha": "daf2d0fb3418cac564461d03c1a9fed4fdec3589", "max_stars_repo_licenses": ["CC-BY-4.0", "MIT"], "max_stars_count": 133, "max_stars_repo_stars_event_min_datetime": "2021-04-06T09:59:40.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T17:53:12.000Z", "max_issues_repo_path": "papers/Scalable_Reward_Distribution_with_Compounding_Stakes.tex", "max_issues_repo_name": "EthereumGeeks/dev", "max_issues_repo_head_hexsha": "daf2d0fb3418cac564461d03c1a9fed4fdec3589", "max_issues_repo_licenses": ["CC-BY-4.0", "MIT"], "max_issues_count": 228, "max_issues_repo_issues_event_min_datetime": "2021-04-05T14:19:34.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-31T08:01:38.000Z", "max_forks_repo_path": "papers/Scalable_Reward_Distribution_with_Compounding_Stakes.tex", "max_forks_repo_name": "sparklingGibbon/dev", "max_forks_repo_head_hexsha": "daf2d0fb3418cac564461d03c1a9fed4fdec3589", "max_forks_repo_licenses": ["CC-BY-4.0", "MIT"], "max_forks_count": 169, "max_forks_repo_forks_event_min_datetime": "2021-04-06T09:55:59.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T08:20:06.000Z", "avg_line_length": 47.5968253968, "max_line_length": 760, "alphanum_fraction": 0.748882812, "num_tokens": 4433, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328917, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3462784684960046}} {"text": "\\documentclass{article}\n\\usepackage{amsmath,amssymb}\n\\newcommand{\\stepthecounter}[2]{\\stepcounter{#1} #2 \\texttt{.}}\n\\begin{document}\n\\section{Seminaive Evaluation}\n\\subsection{Non-Distributed Algorithm}\n\\newcounter{mycounter}\n\\[\n\\begin{array}{ll}\n\\stepthecounter{mycounter}{\\themycounter} & ARGUMENTS : (P, I) \\\\\n\\stepthecounter{mycounter}{\\themycounter} & RETURN : P(I) \\\\\n\\stepthecounter{mycounter}{\\themycounter} & P^{'} := \\{ rule : rule \\in rules(P) \\land idb(P) \\cap body(rule) = \\emptyset \\} \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\forall S \\in idb(P) : S := \\emptyset \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\forall S \\in idb(P) : \\Delta^{1}_{S} := P'(I)(S) \\\\\n\\stepthecounter{mycounter}{\\themycounter} & i := 1 \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\texttt{DO} \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\texttt{FOR } S \\in idb(P) \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\texttt{BEGIN} \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\quad P_{S} := \\{ rule : rule \\in rules(P) \\land S = head(rule) \\} \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\quad T := \\bigcup_{rule \\in P_{S}} body(rule) \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\quad S^{i} := S^{i-1} \\cup \\Delta^{i}_{S} \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\quad \\Delta^{i+1}_{S} := P^{i}_{S}(I, T^{i-1}, T^{i}, \\Delta^{i}_{T}) - S \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad \\texttt{END} \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\quad i := i + 1 \\\\\n\\stepthecounter{mycounter}{\\themycounter} & \\texttt{UNTIL } \\forall S \\in idb(P) : \\Delta^{i}_{S} = \\emptyset \\\\\n\\end{array}\n\\]\n\\newline\n% @@TODO: also do \\subsection{Distributed Blocking Algorithm}\n\\subsection{Distributed Streaming Algorithm}\n% @@TODO\n\\[\n\\begin{array}{l}\n\\texttt{REMOVE I AS ARGUMENT AND USE IO DIRECTIVES HERE} \\\\\nARGUMENTS : (P, I) \\\\\nRETURN : P(I) \\\\\nP^{'} := \\{ rule : rule \\in rules(P) \\land idb(P) \\cap body(rule) = \\emptyset \\} \\\\\n\\forall S \\in idb(P) : S := \\emptyset \\\\\n\\forall S \\in idb(P) : \\Delta^{1}_{S} := P'(I)(S) \\\\\n\\texttt{CONSUME AND MERGE ALL INPUT RELATIONS HERE} \\\\\ni := 1 \\\\\n\\texttt{DO} \\\\\n\\quad \\texttt{FOR } S \\in idb(P) \\\\\n\\quad \\texttt{BEGIN} \\\\\n\\quad \\quad P_{S} := \\{ rule : rule \\in rules(P) \\land S = head(rule) \\} \\\\\n\\quad \\quad T := \\bigcup_{rule \\in P_{S}} body(rule) \\\\\n\\quad \\quad S^{i} := S^{i-1} \\cup \\Delta^{i}_{S} \\\\\n\\quad \\quad \\Delta^{i+1}_{S} := P^{i}_{S}(I, T^{i-1}, T^{i}, \\Delta^{i}_{T}) - S \\\\\n\\quad \\quad \\texttt{PRODUCE DELTA i + 1 HERE} \\\\\n\\quad \\quad \\texttt{CONSUME AND MERGE WITH DELTA i + 1 HERE} \\\\\n\\quad \\texttt{END} \\\\\n\\quad i := i + 1 \\\\\n\\texttt{UNTIL } \\forall S \\in idb(P) : \\Delta^{i}_{S} = \\emptyset \\\\\n\\end{array}\n\\]\n\\end{document}\n", "meta": {"hexsha": "7002e658801fc18423b3689e32cde4cb7b313b9c", "size": 2732, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "kafka/lh/notes/notes.tex", "max_stars_repo_name": "kjezek/souffle", "max_stars_repo_head_hexsha": "3ec9442c288281c4486ab205094b129b0adf0be7", "max_stars_repo_licenses": ["UPL-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "kafka/lh/notes/notes.tex", "max_issues_repo_name": "kjezek/souffle", "max_issues_repo_head_hexsha": "3ec9442c288281c4486ab205094b129b0adf0be7", "max_issues_repo_licenses": ["UPL-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-11-20T22:48:09.000Z", "max_issues_repo_issues_event_max_datetime": "2019-11-20T22:48:09.000Z", "max_forks_repo_path": "kafka/lh/notes/notes.tex", "max_forks_repo_name": "kjezek/souffle", "max_forks_repo_head_hexsha": "3ec9442c288281c4486ab205094b129b0adf0be7", "max_forks_repo_licenses": ["UPL-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.9298245614, "max_line_length": 127, "alphanum_fraction": 0.6383601757, "num_tokens": 1036, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6076631698328916, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3462784684960045}} {"text": "\\achapter{Polymorphic Universes}\n\\aauthor{Matthieu Sozeau}\n\n\\label{Universes-full}\n\\index{Universes!presentation}\n\n\\asection{General Presentation}\n\n\\begin{flushleft}\n \\em The status of Universe Polymorphism is experimental.\n\\end{flushleft}\n\nThis section describes the universe polymorphic extension of Coq.\nUniverse polymorphism allows writing generic definitions making use of\nuniverses and reuse them at different and sometimes incompatible levels.\n\nA standard example of the difference between universe \\emph{polymorphic} and\n\\emph{monomorphic} definitions is given by the identity function:\n\n\\begin{coq_example*}\nDefinition identity {A : Type} (a : A) := a.\n\\end{coq_example*}\n\nBy default, constant declarations are monomorphic, hence the identity\nfunction declares a global universe (say \\texttt{Top.1}) for its\ndomain. Subsequently, if we try to self-apply the identity, we will get\nan error:\n\n\\begin{coq_eval}\nSet Printing Universes.\n\\end{coq_eval}\n\\begin{coq_example}\nFail Definition selfid := identity (@identity).\n\\end{coq_example}\n\nIndeed, the global level \\texttt{Top.1} would have to be strictly smaller than itself\nfor this self-application to typecheck, as the type of \\texttt{(@identity)} is\n\\texttt{forall (A : Type@{Top.1}), A -> A} whose type is itself \\texttt{Type@{Top.1+1}}.\n\nA universe polymorphic identity function binds its domain universe level\nat the definition level instead of making it global.\n\n\\begin{coq_example}\nPolymorphic Definition pidentity {A : Type} (a : A) := a.\nAbout pidentity.\n\\end{coq_example}\n\nIt is then possible to reuse the constant at different levels, like so:\n\n\\begin{coq_example}\nDefinition selfpid := pidentity (@pidentity).\n\\end{coq_example}\n\nOf course, the two instances of \\texttt{pidentity} in this definition\nare different. This can be seen when \\texttt{Set Printing Universes} is\non:\n\n\\begin{coq_example}\nPrint selfpid.\n\\end{coq_example}\n\nNow \\texttt{pidentity} is used at two different levels: at the head of\nthe application it is instantiated at \\texttt{Top.3} while in the\nargument position it is instantiated at \\texttt{Top.4}. This definition\nis only valid as long as \\texttt{Top.4} is strictly smaller than\n\\texttt{Top.3}, as show by the constraints. Note that this definition is\nmonomorphic (not universe polymorphic), so in turn the two universes are\nactually global levels.\n\nInductive types can also be declared universes polymorphic, on universes\nappearing in their parameters or fields. A typical example is given by\nmonoids:\n\n\\begin{coq_example}\nPolymorphic Record Monoid := { mon_car :> Type; mon_unit : mon_car; \n mon_op : mon_car -> mon_car -> mon_car }.\nPrint Monoid.\n\\end{coq_example}\n\nThe \\texttt{Monoid}'s carrier universe is polymorphic, hence it is\npossible to instantiate it for example with \\texttt{Monoid} itself.\nFirst we build the trivial unit monoid, in \\texttt{Set}:\n\\begin{coq_example}\nDefinition unit_monoid : Monoid := \n {| mon_car := unit; mon_unit := tt; mon_op x y := tt |}.\n\\end{coq_example} \n\nFrom this we can build a definition for the monoid of\n\\texttt{Set}-monoids (where multiplication would be given by the product\nof monoids).\n\n\\begin{coq_example*}\nPolymorphic Definition monoid_monoid : Monoid.\n refine (@Build_Monoid Monoid unit_monoid (fun x y => x)).\nDefined.\n\\end{coq_example*}\n\\begin{coq_example}\nPrint monoid_monoid.\n\\end{coq_example} \n\nAs one can see from the constraints, this monoid is ``large'', it lives\nin a universe strictly higher than \\texttt{Set}.\n\n\\asection{\\tt Polymorphic, Monomorphic}\n\\comindex{Polymorphic}\n\\comindex{Monomorphic}\n\\optindex{Universe Polymorphism}\n\nAs shown in the examples, polymorphic definitions and inductives can be\ndeclared using the \\texttt{Polymorphic} prefix. There also exists an\noption \\texttt{Set Universe Polymorphism} which will implicitly prepend\nit to any definition of the user. In that case, to make a definition\nproducing global universe constraints, one can use the\n\\texttt{Monomorphic} prefix. Many other commands support the\n\\texttt{Polymorphic} flag, including:\n\n\\begin{itemize}\n\\item \\texttt{Lemma}, \\texttt{Axiom}, and all the other ``definition''\n keywords support polymorphism.\n\\item \\texttt{Variables}, \\texttt{Context}, \\texttt{Universe} and\n \\texttt{Constraint} in a section support polymorphism. This means\n that the universe variables (and associated constraints) are\n discharged polymorphically over definitions that use them. In other\n words, two definitions in the section sharing a common variable will\n both get parameterized by the universes produced by the variable\n declaration. This is in contrast to a ``mononorphic'' variable which\n introduces global universes and constraints, making the two\n definitions depend on the \\emph{same} global universes associated to\n the variable.\n\\item \\texttt{Hint \\{Resolve, Rewrite\\}} will use the auto/rewrite hint\n polymorphically, not at a single instance.\n\\end{itemize}\n\n\\asection{Global and local universes}\n\nEach universe is declared in a global or local environment before it can\nbe used. To ensure compatibility, every \\emph{global} universe is set to\nbe strictly greater than \\Set~when it is introduced, while every\n\\emph{local} (i.e. polymorphically quantified) universe is introduced as\ngreater or equal to \\Set.\n\n\\asection{Conversion and unification}\n\nThe semantics of conversion and unification have to be modified a little\nto account for the new universe instance arguments to polymorphic\nreferences. The semantics respect the fact that definitions are\ntransparent, so indistinguishable from their bodies during conversion.\n\nThis is accomplished by changing one rule of unification, the\nfirst-order approximation rule, which applies when two applicative terms\nwith the same head are compared. It tries to short-cut unfolding by\ncomparing the arguments directly. In case the constant is universe\npolymorphic, we allow this rule to fire only when unifying the universes\nresults in instantiating a so-called flexible universe variables (not\ngiven by the user). Similarly for conversion, if such an equation of\napplicative terms fail due to a universe comparison not being satisfied,\nthe terms are unfolded. This change implies that conversion and\nunification can have different unfolding behaviors on the same\ndevelopment with universe polymorphism switched on or off.\n\n\\asection{Minimization}\n\nUniverse polymorphism with cumulativity tends to generate many useless\ninclusion constraints in general. Typically at each application of a\npolymorphic constant $f$, if an argument has expected type\n\\verb|Type@{i}| and is given a term of type \\verb|Type@{j}|, a $j \\le i$\nconstraint will be generated. It is however often the case that an\nequation $j = i$ would be more appropriate, when $f$'s\nuniverses are fresh for example. Consider the following example:\n\n\\begin{coq_eval}\nSet Printing Universes.\n\\end{coq_eval}\n\\begin{coq_example}\nDefinition id0 := @pidentity nat 0.\nPrint id0.\n\\end{coq_example}\n\nThis definition is elaborated by minimizing the universe of id to level\n\\Set~while the more general definition would keep the fresh level i\ngenerated at the application of id and a constraint that $\\Set \\le i$.\nThis minimization process is applied only to fresh universe\nvariables. It simply adds an equation between the variable and its lower\nbound if it is an atomic universe (i.e. not an algebraic \\texttt{max()}\nuniverse).\n\nThe option \\texttt{Unset Universe Minimization ToSet} disallows\nminimization to the sort $\\Set$ and only collapses floating universes\nbetween themselves.\n\n\\asection{Explicit Universes}\n\nThe syntax has been extended to allow users to explicitly bind names to\nuniverses and explicitly instantiate polymorphic definitions.\n\n\\subsection{\\tt Universe {\\ident}.\n \\comindex{Universe}\n \\label{UniverseCmd}}\n\nIn the monorphic case, this command declare a new global universe named\n{\\ident}. It supports the polymorphic flag only in sections, meaning the\nuniverse quantification will be discharged on each section definition\nindependently.\n\n\\subsection{\\tt Constraint {\\ident} {\\textit{ord}} {\\ident}.\n \\comindex{Constraint}\n \\label{ConstraintCmd}}\n\nThis command declare a new constraint between named universes. \nThe order relation can be one of $<$, $\\le$ or $=$. If consistent, \nthe constraint is then enforced in the global environment. Like\n\\texttt{Universe}, it can be used with the \\texttt{Polymorphic} prefix\nin sections only to declare constraints discharged at section closing time.\n\n\\begin{ErrMsgs}\n\\item \\errindex{Undeclared universe {\\ident}}.\n\\item \\errindex{Universe inconsistency}\n\\end{ErrMsgs}\n\n\\subsection{Polymorphic definitions}\nFor polymorphic definitions, the declaration of (all) universe levels\nintroduced by a definition uses the following syntax:\n\n\\begin{coq_example*}\nPolymorphic Definition le@{i j} (A : Type@{i}) : Type@{j} := A.\n\\end{coq_example*}\n\\begin{coq_example}\nPrint le.\n\\end{coq_example}\n\nDuring refinement we find that $j$ must be larger or equal than $i$, as\nwe are using $A : Type@{i} <= Type@{j}$, hence the generated\nconstraint. At the end of a definition or proof, we check that the only\nremaining universes are the ones declared. In the term and in general in\nproof mode, introduced universe names can be referred to in\nterms. Note that local universe names shadow global universe names.\nDuring a proof, one can use \\texttt{Show Universes} to display\nthe current context of universes.\n\nDefinitions can also be instantiated explicitly, giving their full instance:\n\\begin{coq_example}\nCheck (pidentity@{Set}).\nUniverses k l.\nCheck (le@{k l}).\n\\end{coq_example}\n\nUser-named universes are considered rigid for unification and are never\nminimized.\n\n\\subsection{\\tt Unset Strict Universe Declaration.\n \\optindex{StrictUniverseDeclaration}\n \\label{StrictUniverseDeclaration}}\n\nThe command \\texttt{Unset Strict Universe Declaration} allows one to\nfreely use identifiers for universes without declaring them first, with\nthe semantics that the first use declares it. In this mode, the universe\nnames are not associated with the definition or proof once it has been\ndefined. This is meant mainly for debugging purposes.\n\n%%% Local Variables: \n%%% mode: latex\n%%% TeX-master: \"Reference-Manual\"\n%%% End: \n", "meta": {"hexsha": "f47973601b3619ace5627ed83c259d6d17b37f82", "size": 10201, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "presentations/coq-workshop-2014-coq/doc/refman/Universes.tex", "max_stars_repo_name": "JasonGross/test-broken-tar", "max_stars_repo_head_hexsha": "6b52b8532879df53386b0f5413485888a1aa886a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "presentations/coq-workshop-2014-coq/doc/refman/Universes.tex", "max_issues_repo_name": "JasonGross/test-broken-tar", "max_issues_repo_head_hexsha": "6b52b8532879df53386b0f5413485888a1aa886a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "presentations/coq-workshop-2014-coq/doc/refman/Universes.tex", "max_forks_repo_name": "JasonGross/test-broken-tar", "max_forks_repo_head_hexsha": "6b52b8532879df53386b0f5413485888a1aa886a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.6401515152, "max_line_length": 88, "alphanum_fraction": 0.7849230468, "num_tokens": 2542, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.34627845961784715}} {"text": "\\documentclass[11pt]{article}\n\\bibliographystyle{alpha}\n\n\\usepackage{todonotes}\n\n\\usepackage{fullpage}\n% Uncomment this when debuging labels. This will display labels\n% citations etc when they are refered. Which makes some debuggin\n% easier. Comment it for the final version.\n\n%\\usepackage{showkeys}\n\\input{preamble}\n\\renewcommand{\\R}{\\mathcal{R}} %these two are very specific to thsi\n\\renewcommand{\\F}{\\mathcal{F}} %file. was just lazy\n\n\\newcommand{\\mr}{\\mathcal{M}_\\mathcal{R}} %M_r\n\n\\title{Fast Integer Multiplication Using Modular Arithmetic \\footnote{A preliminary version appeared in the proceedings of the 40th ACM Symposium on Theory of Computing, 2008.}}\n\\author{Anindya De\\thanks{Research done while the author was at the Dept\n of Computer Science and Engineering, IIT Kanpur}\\\\\n Computer Science Division\\\\\n University of California at Berkeley\\\\\n Berkeley, CA 94720, USA\\\\\n {\\tt anindya@eecs.berkeley.edu}\\\\\n %\n \\and\n Piyush P Kurur%\n \\thanks{Research supported through Research I Foundation project\n NRNM/CS/20030163},\\\\ %\n Dept. of Computer Science and Engineering\\\\%\n Indian Institute of Technology Kanpur\\\\%\n Kanpur, UP, India, 208016\\\\\n {\\tt ppk@cse.iitk.ac.in}\\\\%\n \\and%\n Chandan Saha\\\\%\n Dept. of Computer Science and Engineering\\\\%\n Indian Institute of Technology Kanpur\\\\%\n Kanpur, UP, India, 208016\\\\%\n {\\tt csaha@cse.iitk.ac.in}%\n \\and Ramprasad Saptharishi%\n \\thanks{Research done while visiting IIT Kanpur %\n under Project FLW/DST/CS/20060225}\\\\%\n Chennai Mathematical Institute\\\\%\n Plot H1, SIPCOT IT Park\\\\%\n Padur PO, Siruseri, India, 603103\\\\%\n {\\tt ramprasad@cmi.ac.in}\n}\n\n\\date{}\n\n\\begin{document}\n\\maketitle\n\\begin{abstract}\nWe give an $N\\cdot \\log N\\cdot 2^{O(\\log^*N)}$ time algorithm to\nmultiply two $N$-bit integers that uses modular arithmetic for\nintermediate computations instead of arithmetic over complex numbers\nas in F\\\"{u}rer's algorithm, which also has the same and so far the\nbest known complexity. The previous best algorithm using modular\narithmetic (by Sch{\\\"{o}}nhage and Strassen) has complexity $O(N \\cdot\n\\log N \\cdot \\log\\log N)$. The advantage of using modular arithmetic\nas opposed to complex number arithmetic is that we can completely\nevade the task of bounding the truncation error due to finite\napproximations of complex numbers, which makes the analysis relatively\nsimple. Our algorithm is based upon F\\\"{u}rer's algorithm, but uses\nFFT over multivariate polynomials along with an estimate of the least\nprime in an arithmetic progression to achieve this improvement in the\nmodular setting. It can also be viewed as a $p$-adic version of\nF\\\"{u}rer's algorithm.\n\\end{abstract}\n\n\\section{Introduction}\n\nComputing the product of two $N$-bit integers is nearly a ubiquitous\noperation in algorithm design. Being a basic arithmetic operation, it\nis no surprise that multiplications of integers occur as intermediate\nsteps of computation in algorithms from every possible domain of\ncomputer science. But seldom do the complexity of such multiplications\ninfluence the overall efficiency of the algorithm as the integers\ninvolved are relatively small in size and the multiplications can\noften be implemented as fast hardware operations. However, with the\nadvent of modern cryptosystems, the study of the bit complexity of\ninteger multiplication received a significant impetus. Indeed, large\ninteger multiplication forms the foundation of many modern day\npublic-key crystosytems, like RSA, El-Gamal and Elliptic Curve\ncrytosystems. One of the most notable applications is the RSA\ncryptosystem, where it is required to multiply two primes that are\nhundreds or thousands of bits long. The larger these primes the harder\nit is to factor their product, which in turn makes the RSA extremely\nsecure in practice.\n\nIn this paper, our focus is more on the theoretical aspects of integer\nmultiplication, it being a fundamental problem in its own right. This\nis to say, we will be concerned with the asymptotic bit complexity of\nmutiplying two $N$-bit integers with little emphasis on optimality in\npractice. We begin with a brief account of earlier work on integer\nmultiplication algorithms.\n\n\\subsection{Previous Work}\n\nThe naive approach to multiply two $N$-bit integers leads to an\nalgorithm that uses $O(N^2)$ bit operations. Karatsuba \\cite{K63}\nshowed that some multiplication operations of such an algorithm can be\nreplaced by less costly addition operations which reduces the overall\nrunning time of the algorithm to $O(N^{\\log_23})$ bit\noperations. Shortly afterwards, this result was improved by Toom\n\\cite{T63} who showed that for any $\\varepsilon>0$, integer\nmultiplication can be done in $O(N^{1+\\varepsilon})$ time. This led to\nthe question as to whether the time complexity can be improved further\nby replacing the term $O(N^{\\epsilon})$ by a poly-logarithmic\nfactor. In a major breakthrough, Sch\\\"{o}nhage and\nStrassen~\\cite{SS71} gave two efficient algorithms for multiplying\nintegers using fast polynomial multiplication. One of the algorithms\nachieved a running time of $O(N\\cdot \\log N\\cdot \\log\\log N\\ldots\n2^{O(\\log^*N)})$ using arithmetic over complex numbers (approximated\nto suitable precisions), while the other used arithmetic modulo\ncarefully chosen integers to improve the complexity further to\n$O(N\\cdot \\log N\\cdot \\log\\log N)$ bit operations. The modular\nalgorithm remained the best for a long period of time until a recent\nremarkable result by F\\\"{u}rer \\cite{F07} (see also\n\\cite{F09}). F\\\"{u}rer gave an algorithm that uses arithmetic over\ncomplex numbers and runs in $N\\cdot\\log N\\cdot 2^{O(\\log^\\ast N)}$\ntime. Till date this is the best time complexity known for integer\nmultiplication and indeed our result is inspired by F\\\"{u}rer's\nalgorithm.\n\nFurther details on other approaches and enhancements to previous\ninteger multiplication algorithms can be found in \\cite{F09}.\n\n\\subsection{The Motivation}\n\nSch\\\"{o}nhage and Strassen introduced two seemingly different\napproaches to integer multiplication -- using complex and modular\narithmetic. F\\\"{u}rer's algorithm improves the time complexity in the\ncomplex arithmetic setting by cleverly reducing some costly\nmultiplications to simple shift operations. However, the algorithm\nneeds to approximate the complex numbers to certain precisions during\ncomputation. This introduces the added task of bounding the total\ntruncation errors in the analysis of the algorithm. On the contrary,\nin the modular setting the error analysis is virtually absent or\nrather more implicit, which in turn simplifies the overall\nanalysis. In addition, modular arithmetic gives a discrete approach to\na discrete problem like integer multiplication. Therefore, it seems\nnatural to ask whether we can achieve a similar improvement in time\ncomplexity of this problem in the modular arithmetic setting. In this\nwork, we answer this question affirmatively. We give an $N\\cdot\n\\log{N}\\cdot 2^{O(\\log^{*}{N})}$ time algorithm for integer\nmultiplication using only modular arithmetic, thus matching the\nimprovement made by F\\\"{u}rer.\n\n\\subsection*{Overview of our result}\n\nAs is the case in both Sch\\\"{o}nhage-Strassen's and F\\\"{u}rer's\nalgorithms, we start by reducing the problem to polynomial\nmultiplication over a ring $\\mathcal{R}$ by properly encoding the\ngiven integers. Polynomials can be multiplied efficiently using\nDiscrete Fourier Transforms (DFT). However, in order that we are able\nto use Fast Fourier Transform (FFT), the ring $\\mathcal{R}$ should\nhave some special roots of unity. For instance, to multiply two\npolynomials of degree less than $M$ using FFT, we require a\n\\emph{principal} $2M$-th root of unity (see\nDefinition~\\ref{def-principal-root} for principal roots). One way to\nconstruct such a ring in the modular setting is to consider rings of\nthe form $\\mathcal{R} = \\Z/(2^M + 1) \\Z$ as in Sch\\\"{o}nhage and\nStrassen's work ~\\cite{SS71}. In this case, the element $2$ is a\n$2M$-th principal root of unity in $\\mathcal{R}$. This approach can be\nequivalently viewed as attaching an `artificial' root to the ring of\nintegers. However, this makes the size of $\\mathcal{R}$ equal to $2^M$\nand thus a representation of an arbitrary element in $\\mathcal{R}$\ntakes $M$ bits. This means an $N$-bit integer is encoded as a\npolynomial of degree $M$ with every coefficient about $M$ bits long,\nthereby making $M \\approx \\sqrt{N}$ as the optimal choice. Indeed, the\nchoice of such an $\\mathcal{R}$ is the basis of Sch\\\"{o}nhage and\nStrassen's modular algorithm in which they reduce multiplication of\n$N$-bit integers to multiplication of $\\sqrt{N}$-bit integers and\nachieve a complexity of $O(N \\cdot \\log N \\cdot \\log \\log N)$ bit\noperations.\n\nNaturally, such rings are a little too expensive in our setting. We\nwould rather like to find a ring whose size is bounded by some\npolynomial in $M$ and which still contains a principal $2M$-th root of\nunity. In fact, it is this task of choosing a suitable ring that poses\nthe primary challenge in adapting F\\\"{u}rer's algorithm and making it\nwork in the discrete setting.\n\nWe choose the ring to be $\\mathcal{R} = \\Z/p^c\\Z$, for a prime $p$ and\na constant $c$ such that $p^c = \\mathsf{poly}(M)$. The ring\n$\\Z/p^c\\Z$, has a principal $2M$-th root of unity if and only if $2M$\ndivides $p-1$, which means that we need to find a prime $p$ from the\narithmetic progression $\\inbrace{1 + i\\cdot 2M}_{i>0}$. To make this\nsearch computationally efficient, we also need the degree of the\npolynomials, $M$ to be sufficiently small. This we can achieve by\nencoding the integers as multivariate polynomials instead of\nunivariate ones. It turns out that the choice of the ring as\n$\\mathcal{R} = \\Z/p^c\\Z$ is still not quite sufficient and needs a\nlittle more refinement. This is explained in Section \\ref{sec:ring}.\n\nThe use of multivariate polynomial multiplications along with a small\nbase ring are the main steps where our algorithm differs from earlier\nalgorithms by Sch\\\"{o}nhage-Strassen and F\\\"{u}rer. Towards\nunderstanding the notion of \\emph{inner} and \\emph{outer} DFT in the\ncontext of multivariate polynomials, we also present a group theoretic\ninterpretation of DFT. The use of inner\nand outer DFT plays a central role in both F\\\"{u}rer's as well as our\nalgorithm. Arguing along the line of F\\\"{u}rer \\cite{F07}, we show that\nrepeated use of efficient computation of inner DFT's using some\nspecial roots of unity in $\\mathcal{R}$ reduces the number of\n`bad multiplications' (in comparison to Sch\\\"{o}nhage-Strassen's algorithm)\nand makes the overall process efficient, thereby leading to an $N\\cdot \\log{N}\\cdot 2^{O(\\log^{*}{N})}$\ntime algorithm.\n\n\n\\section{The Basic Setup}\n\n\\subsection{The Underlying Ring} \\label{sec:ring}\n\nRings of the form $\\mathcal{R} = \\Z/(2^M + 1) \\Z$ have the nice\nproperty that multiplications by powers of $2$, the $2M$-th principal\nroot of unity, are mere shift operations and are therefore very\nefficient. Although by choosing the ring $\\mathcal{R} = \\Z/p^c\\Z$ we\nensure that the ring size is small, it comes with a price:\nmultiplications by principal roots of unity are no longer just shift\noperations. Fortunately, this can be redeemed by working with rings of\nthe form $\\mathcal{R} = \\Z[\\alpha]/(p^c, \\alpha^m + 1)$ for some $m$\nwhose value will be made precise later. Elements of $\\mathcal{R}$ are\nthus $m-1$ degree polynomials over $\\alpha$ with coefficients from\n$\\Z/p^c\\Z$. By construction, $\\alpha$ is a $2m$-th root of unity and\nmultiplication of any element in $\\mathcal{R}$ by any power of\n$\\alpha$ can be achieved by shift operations --- this property is\ncrucial in making some multiplications in the FFT less costly (see\nSection~\\ref{fourier_analysis}).\n\nGiven an $N$-bit number $a$, we encode it as a $k$-variate polynomial\nover $\\mathcal{R}$ with degree in each variable less than $M$. The\nparameters $M$ and $m$ are powers of two such that $M^k$ is roughly\n$\\frac{N}{\\log^2N}$ and $m$ is roughly $\\log{N}$. The parameter $k$\nwill ultimately be chosen a constant (see Section\n\\ref{complexity_section}). We now explain the details of this encoding\nprocess.\n\n\\subsection{Encoding Integers into $k$-variate\n Polynomials}\\label{encoding_section}\n\nGiven an $N$-bit integer $a$, we first break these $N$ bits into $M^k$\nblocks of roughly $\\frac{N}{M^k}$ bits each. This corresponds to\nrepresenting $a$ in base $q = 2^{\\frac{N}{M^k}}$. Let $a = a_0 +\n\\ldots + a_{M^k-1}q^{M^k - 1}$, where every $a_i < q$. The number $a$\nis converted into a polynomial as follows:\n\\begin{enumerate}\n\\item Express $i$ in base $M$ as $i = i_1 + i_2M + \\cdots +\n i_kM^{k-1}$. \\label{base_M_item}\n\\item Encode each term $a_iq^i$ as the monomial $a_i\\cdot\n X_1^{i_1}X_2^{i_1}\\cdots X_k^{i_k}$. As a result, the number $a$\n gets converted to the polynomial $\\sum_{i=0}^{M^k - 1}\n a_i\\cdot X_1^{i_1}\\cdots X_k^{i_k}$.\n\\end{enumerate}\n\nFurther, we break each $a_i$ into $\\frac{m}{2}$ equal sized blocks\nwhere the number of bits in each block is $u = \\frac{2N}{M^k\\cdot m}$.\nEach coefficient $a_i$ is then encoded as a polynomial in $\\alpha$ of\ndegree less than $\\frac{m}{2}$. The polynomials are then padded with\nzeroes to stretch their degrees to $m$. Thus, the $N$-bit number $a$\nis converted to a $k$-variate polynomial $a(X)$ over\n$\\Z[\\alpha]/(\\alpha^m + 1)$.\\\\\n\nGiven integers $a$ and $b$, each of $N$ bits, we encode them as\npolynomials $a(X)$ and $b(X)$ and compute the product polynomial. The\nproduct $a\\cdot b$ can be recovered by substituting $X_s =\nq^{M^{s-1}}$, for $1\\leq s\\leq k$, and $\\alpha = 2^u$ in the\npolynomial $a(X)\\cdot b(X)$. The coefficients in the product\npolynomial could be as large as $M^k\\cdot m\\cdot 2^{2u}$ and hence it\nis sufficient to do arithmetic modulo $p^c$ where $p^c > 2M^k\\cdot\nm\\cdot 2^{2u}$. Our choice of\nthe prime $p$ ensures that $c$ is in fact a constant (see\nSection~\\ref{complexity_section}). We summarize this discussion as a lemma.\n\n\\begin{lemma}\\label{lem:encoding-time}\n Multiplication of two $N$-bit integers reduces to\n multiplication of two $k$-variate polynomials, with degree in each\n variable bounded by $M$, over the ring $\\Z[\\alpha]/(p^c,\\alpha^m +\n 1)$ for a prime $p$ satisfying $p^c > 2M^k\\cdot m \\cdot 2^{2u}$,\n where $u=\\frac{2N}{M^km}$. Furthermore, the reduction can be\n performed in $O(N)$ time.\n\\end{lemma}\n\n\n\\subsection{Choosing the Prime}\\label{prime_section}\n\nThe prime $p$ should be chosen such that the ring $\\Z/p^c\\Z$ has a\n\\emph{principal} $2M$-th root of unity, which is required for\npolynomial multiplication using FFT. A principal root of unity is\ndefined as follows.\n\n\\begin{definition}\\label{def-principal-root}\n\\emph{\\textsf{(Principal root of unity)}} An $n$-th root of unity\n$\\zeta\\in \\mathcal{R}$ is said to be primitive if it generates a\ncyclic group of order $n$ under multiplication. Furthermore, it is\nsaid to be principal if $n$ is coprime to the characteristic of\n$\\mathcal{R}$ and $\\zeta$ satisfies $\\sum_{i=0}^{n-1}\\zeta^{ij}=0$ for\nall $0< j < n$.\n\\end{definition}\n\n\\noindent\nIn $\\Z/p^c\\Z$, a $2M$-th root of unity is principal if and only if\n$2M\\mid p-1$ (see also Section~\\ref{Qp_section}). As a result, we need\nto choose the prime $p$ from the arithmetic progression $\\inbrace{1 +\n i\\cdot 2M}_{i> 0}$, which is potentially the main bottleneck of our\napproach. We now explain how to circumvent this problem. \\\\\n\n\nAn upper bound for the least prime in an arithmetic progression is\ngiven by the following theorem by Linnik \\cite{L44}:\n\n\\begin{theorem}\\label{linnik_theorem}\n\\emph{\\textsf{(Linnik)}} There exist absolute constants $\\ell$ and $L$\nsuch that for any pair of coprime integers $d$ and $n$, the least\nprime $p$ such that $p\\equiv d\\bmod{n}$ is less than $\\ell n^L$.\n\\end{theorem}\n\nHeath-Brown \\cite{B92} showed that the \\emph{Linnik constant} $L\\leq\n5.5$ (a recent work by Xylouris \\cite{X09} showed that $L \\leq\n5.2$). Recall that $M$ is chosen such that $M^k$ is\n$O\\inparen{\\frac{N}{\\log^2N}}$. If we choose $k=1$, that is if we use\nunivariate polynomials to encode integers, then the parameter $M =\nO\\inparen{\\frac{N}{\\log^2N}}$. Hence the least prime $p\\equiv\n1\\pmod{2M}$ could be as large as $N^L$. Since all known deterministic\nsieving procedures take at least $N^L$ time this is clearly infeasible\n(for a randomized approach see Section~\\ref{ERH_section}). However, by\nchoosing a larger $k$ we can ensure that the least prime $p\\equiv\n1\\pmod{2M}$ is $O(N^\\varepsilon)$ for some constant $\\varepsilon <\n1$. Since primality testing is in deterministic polynomial\\footnote{a\n subexponential algorithm would suffice} time\\cite{AKS04}, we can find the least prime $p\\equiv 1 \\pmod{2M}$ in $o(N)$\ntime.\n\n\n\\begin{lemma}\\label{prime_time}\nIf $k$ is any integer greater than $L+1$, then $M^L =\nO\\inparen{N^{\\frac{L}{L+1}}}$ and hence the least prime $p\\equiv\n1\\pmod{2M}$ can be found in $o(N)$ time.\n\\end{lemma}\n\n\\subsubsection*{Choosing the Prime Randomly}\\label{ERH_section}\n\nTo ensure that the search for a prime $p\\equiv 1\\pmod{2M}$ does not\naffect the overall time complexity of the algorithm, we considered\nmultivariate polynomials to restrict the value of $M$; an alternative\nis to use randomization.\n\n\\begin{proposition} \\label{prop:randomprime}\nAssuming ERH, a prime $p\\equiv 1\\pmod{2M}$ can be computed by a\nrandomized algorithm with expected running time $\\tilde{O}(\\log^3 M)$.\n\\end{proposition}\n\\begin{proof}\nTitchmarsh \\cite{Titchmarsh} (see also Tianxin \\cite{Tianxin})\nshowed, assuming ERH, that the number of primes less than $x$ in the\narithmetic progression $\\{ 1 + i \\cdot 2M\\}_{i > 0}$ is given by,\n\\begin{equation*}\n\\pi(x,2M) = \\frac{Li(x)}{\\varphi(2M)} + O(\\sqrt{x} \\log x)\n\\end{equation*}\nfor $2M \\leq \\sqrt{x} \\cdot (\\log x)^{-2}$, where $Li(x) =\n\\Theta(\\frac{x}{\\log x})$ and $\\varphi$ is the Euler totient\nfunction. In our case, since $M$ is a power of two, $\\varphi(2M) = M$,\nand hence for $x \\geq 4M^2 \\cdot \\log^6 M$, we have $\\pi(x, 2M) =\n\\Omega\\inparen{\\frac{x}{M\\log x}}$. Therefore, for an $i$ chosen\nuniformly randomly in the range $1 \\leq i \\leq 2M \\cdot \\log^6 M$, the\nprobability that $i\\cdot 2M + 1$ is a prime is at least $\\frac{d}{\\log\n x}$ for a constant $d$. Furthermore, primality test of an $O(\\log\nM)$ bit number can be done in $\\tilde{O}(\\log^2 M)$ time using\nRabin-Miller primality test \\cite{M76, R80}. Hence, with $x = 4M^2\n\\cdot \\log^6 M$, a suitable prime for our algorithm can be found in\nexpected $\\tilde{O}(\\log^3 M)$ time.\n\\end{proof}\n\n\\noindent \\textbf{Remark - } We prefer to use Linnik's theorem (Theorem\n\\ref{linnik_theorem}) instead of Proposition \\ref{prop:randomprime}\nwhile choosing the prime in the progression $\\{ 1 + i \\cdot 2M\\}_{i > 0}$\nso as to make the results in this paper independent of the ERH and the usage\nof random bits.\n\n\\subsection{Finding the Root of Unity}\\label{root_section}\n\nWe require a principal $2M$-th root of unity $\\rho(\\alpha)$ in\n$\\mathcal{R}$ to compute the Fourier transforms. This root\n$\\rho(\\alpha)$ should also have the property that its\n$\\inparen{\\frac{M}{m}}$-th power is $\\alpha$, so as to make some\nmultiplications in the FFT efficient (see Section\n\\ref{fourier_analysis}). The root $\\rho(\\alpha)$ can be computed by\ninterpolation in a way similar to that in F\\\"{u}rer's algorithm\n\\cite[Section 3]{F07}, except that we need a principal $2M$-th root of\nunity $\\omega$ in $\\Z/p^c\\Z$ to start with.\n\nTo obtain such a root, we first obtain a $2M$-th root of unity\n$\\omega_1$ in $\\Z/p\\Z$. A generator $\\zeta$ of $\\mathbb{F}_p^{\\times}$\ncan be computed by brute force, as $p$ is sufficiently small, and\n$\\omega_1 = \\zeta^{(p-1)/2M}$ is a principal $2M$-th root of unity in $\\Z/p\\Z$.\nA principal $2M$-root of unity\n$\\omega_1$ must be a root of the polynomial $f(x) = x^M+1$ in $\\Z/p\\Z$. Having obtained $\\omega_1$,\nwe use Hensel Lifting \\cite[Theorem 2.23]{Zuckerman}.\n\n\\begin{lemma}[\\textsf{Hensel Lifting}]\n Let $\\omega_s$ be a root of $f(x) = x^M + 1$ in $\\Z/p^s\\Z$. Then\n there exists a unique root $\\omega_{s+1}$ in $\\Z/p^{s+1}\\Z$ such\n that $\\omega_{s+1}\\equiv \\omega_s\\pmod{p^s}$ and $f(\\omega_{s+1}) =\n 0\\pmod{p^{s+1}}$. This unique root is given by $\\omega_{s+1} =\n \\omega_s - \\frac{f(\\omega_s)}{f'(\\omega_s)}$.\\end{lemma}\n\n%%% HANDLING TODO PRES 5\n\n\\noindent\nIt is clear from the above lemma that we can compute a $2M$-th root of\nunity $\\omega = \\omega_c$ in $\\Z/p^c\\Z$. We will need the following\nwell-known and useful fact about principal roots of unity in any ring.\n\n\\begin{lemma}\\cite[Lemma 2.1]{F09}\\label{lem:Furer-principal-root}\n If $M$ is a power of $2$ and $\\omega^{M} = -1$ in an arbitrary ring\n $\\mathcal{R}$. If $M$ is relatively prime to the characteristic of\n $\\mathcal{R}$, then $\\omega$ is a principal $2M$-th root of unity in\n $\\mathcal{R}$.\n\\end{lemma}\n\n\\noindent\nHence it follows that the root $\\omega$ of $f(x) = x^M + 1$ in\n$\\Z/p^c\\Z$ is a principal $2M$-th root of unity in $\\Z/p^c\\Z$.\nFurthermore, $\\zeta^{(p-1)/2M} = \\omega_1 \\equiv \\omega\\bmod p$. Since\n$\\zeta$ is a generator of $\\mathbb{F}_p^\\times$, different powers of\n$\\zeta$ must generate the group $\\mathbb{F}_p^\\times$ and hence must\nbe distinct modulo $p$. Hence it follows that different powers of\n$\\omega$ must be distinct modulo $p$ as well. Therefore, the\ndifference between any two of them is a unit in $\\Z/p^c\\Z$ and this\nmakes the following\ninterpolation feasible in our setting. \\\\\n\n\\paragraph{Finding $\\rho(\\alpha)$ from $\\omega$:} Since $\\omega$ is a\nprincipal $2M$-th root of unity, $\\gamma = \\omega^{\\frac{2M}{2m}}$ is\na principal $2m$-th root of unity in $\\Z/p^c\\Z$. Notice that,\n$\\alpha^m + 1$ uniquely factorizes as, $\\alpha^m + 1 = (\\alpha -\n\\gamma)(\\alpha - \\gamma^3) \\ldots (\\alpha - \\gamma^{2m-1})$. The\nideals generated by $(\\alpha - \\gamma^i)$ and $(\\alpha - \\gamma^j)$\nare mutually coprime as $\\gamma^i -\\gamma^j$ is a unit for $i\\neq j$\nand is contained in the ideal generated by $(\\alpha - \\gamma^i)$ and\n$(\\alpha - \\gamma^j)$. Therefore, using Chinese Remaindering,\n$\\alpha$ has the direct sum representation $(\\gamma, \\gamma^3, \\ldots,\n\\gamma^{2m-1})$ in $\\mathcal{R}$. Since we require\n$\\rho(\\alpha)^{\\frac{2M}{2m}} = \\alpha$, it is sufficient to choose a\n$\\rho(\\alpha)$ whose direct sum representation is $(\\omega, \\omega^3,\n\\ldots, \\omega^{2m-1})$. Now use Lagrange's formula to interpolate\n$\\rho(\\alpha)$ as,\n\\begin{equation*}\n \\rho(\\alpha) = \\sum_{i=1, \\text{ } i \\text{ odd}}^{2m-1}{\\omega^i \\cdot \\prod_{j=1, \\text{ } j \\neq i, \\text{ } j \\text{ odd}}^{2m-1}{\\frac{\\alpha - \\gamma^j}{\\gamma^i - \\gamma^j}}}\n\\end{equation*}\nThe inverses of the elements $\\gamma^i - \\gamma^j$ in $\\Z/p^c\\Z$ can\nbe easily computed in $\\text{poly}(\\log p)$ time. It is also clear\nthat $\\rho(\\alpha)^M$, in the direct-sum representation, is\n$(\\omega^M, \\omega^{3M}, \\cdots, \\omega^{(2m -1)M})$ which is the element $-1$\nin $\\R$. Hence $\\rho(\\alpha)$ is\na principal $2M$-th root of unity (by Lemma~\\ref{lem:Furer-principal-root}).\\\\\n\nBesides finding a generator $\\zeta$ of $\\Z/p\\Z$ by brute-force (which\ncan be performed in $O(p)$ time), all other computations can be done\nin $\\text{poly}(\\log p)$ time. We summarize this as a lemma.\n\n\\begin{lemma}\\label{lem:root-time}\n A principal $2M$-th root of unity $\\rho(\\alpha)\\in \\R$ such that\n $\\rho(\\alpha)^{2M/2m} = \\alpha$ can be computed in deterministic\n time $O(p \\cdot \\mathrm{poly}(\\log(p)))$ (which is $o(N)$ if $p =\n o(N)$).\n\\end{lemma}\n\n% \\subsection*{old}\n\n% We require a principal $2M$-th root of unity $\\rho(\\alpha)$ in\n% $\\mathcal{R}$ to compute the Fourier transforms. This root\n% $\\rho(\\alpha)$ should also have the property that its\n% $\\inparen{\\frac{M}{m}}$-th power is $\\alpha$, so as to make some\n% multiplications in the FFT efficient (see Section\n% \\ref{fourier_analysis}). The root $\\rho(\\alpha)$ can be computed by\n% interpolation in a way similar to that in F\\\"{u}rer's algorithm\n% \\cite[Section 3]{F07}, except that we need a principal $2M$-th root of\n% unity $\\omega$ in $\\Z/p^c\\Z$ to start with. To obtain such a root, we\n% first obtain a $(p-1)$-th root of unity $\\zeta$ in $\\Z/p^c\\Z$ by\n% lifting a generator of $\\mathbb{F}_p^{\\times}$. The\n% $\\inparen{\\frac{p-1}{2M}}$-th power of $\\zeta$ gives us the required\n% $2M$-th root of unity $\\omega$. A generator of $\\mathbb{F}_p^{\\times}$\n% can be computed by brute force, as $p$ is sufficiently small. Having\n% obtained a generator, we use Hensel Lifting \\cite[Theorem\n% 2.23]{Zuckerman}.\n\n% \\begin{lemma}[\\textsf{Hensel Lifting}]\n% Let $\\zeta_s$ be a primitive $(p-1)$-th root of unity in\n% $\\Z/p^s\\Z$. Then there exists a unique primitive $(p-1)$-th root of\n% unity $\\zeta_{s+1}$ in $\\Z/p^{s+1}\\Z$ such that $\\zeta_{s+1}\\equiv\n% \\zeta_s\\pmod{p^s}$. This unique root is given by $\\zeta_{s+1} =\n% \\zeta_s - \\frac{f(\\zeta_s)}{f'(\\zeta_s)}$ where $f(x) = x^{p-1} - 1$.\n% \\end{lemma}\n\n% \\noindent It can be shown that the root $\\zeta$ in $\\Z/p^c\\Z$ thus\n% obtained by lifting is principal. Furthermore, different powers of\n% $\\zeta$ are distinct modulo $p$. Therefore, the difference between any\n% two of them is a unit in $\\Z/p^c\\Z$ and this makes the following\n% interpolation feasible in our setting.\n\n% \\paragraph{Finding $\\rho(\\alpha)$ from $\\omega$:} Since\n% $\\omega$ is a principal $2M$-th root of unity, $\\gamma =\n% \\omega^{\\frac{2M}{2m}}$ is a principal $2m$-th root of unity in\n% $\\Z/p^c\\Z$. Notice that, $\\alpha^m + 1$ uniquely factorizes as,\n% $\\alpha^m + 1 = (\\alpha - \\gamma)(\\alpha - \\gamma^3) \\ldots (\\alpha -\n% \\gamma^{2m-1})$, and the ideals generated by $(\\alpha - \\gamma^i)$ in\n% $\\mathcal{R}$ are mutually coprime as $\\gamma^i - \\gamma^j$ is a unit\n% for $i \\neq j$. Therefore, using Chinese Remaindering, $\\alpha$ has\n% the direct sum representation $(\\gamma, \\gamma^3, \\ldots,\n% \\gamma^{2m-1})$ in $\\mathcal{R}$. Since we require\n% $\\rho(\\alpha)^{\\frac{2M}{2m}} = \\alpha$, it is sufficient to choose a\n% $\\rho(\\alpha)$ whose direct sum representation is $(\\omega, \\omega^3,\n% \\ldots, \\omega^{2m-1})$. Now use Lagrange's formula to interpolate\n% $\\rho(\\alpha)$ as,\n% \\begin{equation*}\n% \\rho(\\alpha) = \\sum_{i=1, \\text{ } i \\text{ odd}}^{2m-1}{\\omega^i \\cdot \\prod_{j=1, \\text{ } j \\neq i, \\text{ } j \\text{ odd}}^{2m-1}{\\frac{\\alpha - \\gamma^j}{\\gamma^i - \\gamma^j}}}\n% \\end{equation*}\n% The inverses of the elements $\\gamma^i - \\gamma^j$ can be easily computed in $\\Z/p^c\\Z$.\n\n\\section{Fourier Transform}\n\\subsection{Inner and Outer DFT} \\label{sec:inoutDFT}\n\nSuppose that $a(x) \\in \\mathcal{S}[x]$ is a polynomial of degree less\nthan $2M$, where $\\mathcal{S}$ is a ring containing a $2M$-th principal\nroot of unity $\\rho$. Let us say that we want to compute the\n$2M$-point DFT of $a(x)$ using $\\rho$ as the root of unity. In other\nwords, we want to compute the elements $a(1), a(\\rho), \\ldots,\na(\\rho^{2M-1})$ is $\\mathcal{S}$. This can be done in two steps.\n\n\\paragraph{Step $1$:}\nCompute the following polynomials using $\\alpha = \\rho^{2M/2m}$.\n\\begin{eqnarray*}\na_0(x) &=& a(x) \\mod (x^{2M/2m} - 1) \\\\\na_1(x) &=& a(x) \\mod (x^{2M/2m} - \\alpha) \\\\\n\t\t\t &\\vdots& \\\\\na_{2m-1}(x) &=& a(x) \\mod (x^{2M/2m} - \\alpha^{2m-1}),\n\\end{eqnarray*}\nwhere $\\deg(a_j(x)) < \\frac{2M}{2m}$ for all $0 \\leq j < 2m$.\n\n\\paragraph{Step $2$:}\nNote that, $a_j(\\rho^{k \\cdot 2m + j}) = a(\\rho^{k \\cdot 2m + j})$ for\nevery $0 \\leq j < 2m$ and $0 \\leq k < \\frac{2M}{2m}$. Therefore, all\nwe need to do to compute the DFT of $a(x)$ is to evaluate the\npolynomials $a_j(x)$ at appropriate powers of $\\rho$.\\\\\n\n\\noindent The idea is to show that both Step $1$ and Step $2$ can be\nperformed by computation of some `smaller' DFTs. Let us see how.\n\n\\paragraph{Performing Step $1$:}\nThe crucial observation here is the following. Fix an integer $\\ell$\nin the range $[0, \\frac{2M}{2m} - 1]$. Then the $\\ell^{th}$\ncoefficients of $a_0(x), a_1(x), \\ldots, a_{2m-1}(x)$ are exactly\n$e_\\ell(1), e_\\ell(\\alpha), \\ldots, e_\\ell(\\alpha^{2m-1})$,\nrespectively, where $e_\\ell(y)$ is the polynomial,\n\n\\begin{equation*}\ne_\\ell(y) = \\sum_{j=0}^{2m-1}{a_{j \\cdot \\frac{2M}{2m} + \\ell} \\cdot y^j}.\n\\end{equation*}\nBut then, finding $e_\\ell(1), e_\\ell(\\alpha), \\ldots,\ne_\\ell(\\alpha^{2m-1})$ is essentially computing the $2m$-point DFT of\n$e_\\ell(y)$ using $\\alpha$ as the $2m^{th}$ root of unity. Therefore,\nall we need to do to find $a_0(x), \\ldots, a_{2m-1}(x)$ is to compute\nthe DFTs of $e_\\ell(y)$ for all $0 \\leq \\ell < \\frac{2M}{2m}$. These\n$\\frac{2M}{2m}$ many $2m$-point DFTs are called the \\emph{inner} DFTs.\n\n\\paragraph{Performing Step $2$:} In order to find $a_j(\\rho^{k \\cdot\n 2m + j})$, for $0 \\leq k < \\frac{2M}{2m}$ and a fixed $j$, we first\ncompute the polynomial $\\tilde{a}_j(x) = a_j(x \\cdot \\rho^j)$ followed\nby a $\\frac{2M}{2m}$-point DFT of $\\tilde{a}_j(x)$ using $\\rho^{2m}$\nas the root of unity. These $2m$ many $\\frac{2M}{2m}$-point DFTs ($j$\nrunning from $0$ to $2m - 1$) are called the \\emph{outer} DFTs. The\npolynomials $\\tilde{a}_j(x)$ can be computed by multiplying the\ncoefficients of $a_j(x)$ by suitable powers of $\\rho$. Such\nmultiplications are termed as \\emph{bad} multiplications (as they would result in recursive calls to integer\nmultiplication). \\\\\n\n\\noindent The above discussion is summarized in the following lemma.\n\n\\begin{lemma} \\label{lem:inoutDFT} \\emph{\\textsf{(DFT time = Inner\n DFTs + Bad multiplications + Outer DFTs)}} \\\\ Time taken to\n compute a $2M$-point DFT over $\\mathcal{S}$ is sum of:\n\\begin{enumerate}\n\\item Time taken to compute $\\frac{2M}{2m}$ many $2m$-point inner DFTs over $\\mathcal{S}$ using $\\alpha$ as the $2m$-th root of unity.\n\\item Time to do $2M$ multiplications in $\\mathcal{S}$ by powers of\n $\\rho$ (bad multiplications).\n\\item Time taken to compute $2m$ many $\\frac{2M}{2m}$-point outer DFTs over $\\mathcal{S}$ using $\\rho^{2m}$ as the $\\frac{2M}{2m}$-th root of unity.\n\\end{enumerate}\n\\end{lemma}\n\n\\subsection{Analysis of the FFT}\\label{fourier_analysis}\n\nWe are now ready to analyse the complexity of multiplying the two\n$k$-variate polynomials $a(X)$ and $b(X)$ (see Section\n\\ref{encoding_section}) using Fast Fourier Transform. Treat $a(X)$ and\n$b(X)$ as univariate polynomials in variable $X_k$ over the ring\n$\\mathcal{S} = \\mathcal{R}[X_1, \\ldots, X_{k-1}]$. We write $a(X)$ and\n$b(X)$ as $a(X_k)$ and $b(X_k)$, respectively, where $\\deg(a(X_k))$\nand $\\deg(b(X_k))$ are less than $M$. Multiplication of $a(X)$ and\n$b(X)$ can be thought of as multiplication of the univariates $a(X_k)$\nand $b(X_k)$ over $\\mathcal{S}$. Also note that, the root\n$\\rho(\\alpha)$ (constructed in Section \\ref{root_section}) is a\nprimitive $2M$-th root of unity in $\\mathcal{S} \\supset\n\\mathcal{R}$. Denote the multiplication complexity of $a(X_k)$ and\n$b(X_k)$ by $\\mathcal{F}(2M, k)$.\n\nMultiplication of $a(X_k)$ and $b(X_k)$ using FFT involves computation\nof three $2M$-point DFTs over $\\mathcal{S}$ and $2M$ pointwise (or\ncomponentwise) multiplications in $\\mathcal{S}$. Let $\\mathcal{D}(2M,\nk)$ be the time taken to compute a $2M$-point DFT over\n$\\mathcal{S}$. By Lemma \\ref{lem:inoutDFT}, the time to compute a DFT\nis the sum of the time for the inner DFTs, the bad multiplications and\nthe outer DFTs. Let us analyse these three terms separately. We will\ngo by the notation in Section \\ref{sec:inoutDFT}, using $\\mathcal{S} =\n\\mathcal{R}[X_1, \\ldots, X_{k-1}]$ and $\\rho = \\rho(\\alpha)$.\n\n\\paragraph{Inner DFT time:} Computing a $2m$-point DFT requires $2m\n\\log (2m)$ additions in $\\mathcal{S}$ and $m \\log (2m)$\nmultiplications by powers of $\\alpha$. The important observation here\nis: since $\\mathcal{R} = \\Z[\\alpha]/(p^c, \\alpha^m + 1)$,\nmultiplication by a power of $\\alpha$ with an element in $\\mathcal{R}$\ncan be readily computed by simple cyclic shifts (with possible\nnegations), which takes only $O(m \\cdot \\log p)$ bit operations. An\nelement in $\\mathcal{S}$ is just a polynomial over $\\mathcal{R}$ in\nvariables $X_1, \\ldots, X_{k-1}$, with degree in each variable bounded\nby $M$. Hence, multiplication by a power of $\\alpha$ with an element\nof $\\mathcal{S}$ can be done using $\\mathcal{N}_{\\mathcal{S}} =\nO(M^{k-1} \\cdot m \\cdot \\log p)$ bit operations. A total of $m \\log\n(2m)$ multiplications takes $O(m \\log m \\cdot\n\\mathcal{N}_{\\mathcal{S}})$ bit operations. It is easy to see that $2m\n\\log (2m)$ additions in $\\mathcal{S}$ also require the same order of\ntime.\n\nSince there are $\\frac{2M}{2m}$ many $2m$-point DFTs, the total time\nspent in the inner DFTs is $O(2M \\cdot \\log m \\cdot\n\\mathcal{N}_{\\mathcal{S}})$ bit operations.\n\n\\paragraph{Bad multiplication time:}\nSuppose that two arbitrary elements in $\\mathcal{R}$ can be multiplied\nusing $\\mathcal{M}_{\\mathcal{R}}$ bit operations. Mulitplication in\n$\\mathcal{S}$ by a power of $\\rho$ amounts to $c_{\\mathcal{S}} =\nM^{k-1}$ multiplications in $\\mathcal{R}$. Since there are $2M$ such\nbad multiplications, the total time is bounded by $O(2M \\cdot\nc_{\\mathcal{S}} \\cdot \\mathcal{M}_{\\mathcal{R}})$.\n\n\\paragraph{Outer DFT time:}\nBy Lemma \\ref{lem:inoutDFT}, the total outer DFT time is $2m \\cdot\n\\mathcal{D}\\left(\\frac{2M}{2m}, k \\right)$.\n\n\\paragraph{Total DFT time:}Therefore, the net DFT time is bounded as,\n\\begin{eqnarray*}\n\\mathcal{D}(2M, k) &=& O\\left(2M \\cdot \\log m \\cdot \\mathcal{N}_{\\mathcal{S}} +\n 2M \\cdot c_{\\mathcal{S}} \\cdot \\mathcal{M}_{\\mathcal{R}}\\right) +\n2m \\cdot \\mathcal{D}\\left(\\frac{2M}{2m}, k\\right) \\\\\n&=& O\\left(2M \\cdot \\log m \\cdot \\mathcal{N}_{\\mathcal{S}} +\n 2M \\cdot c_{\\mathcal{S}} \\cdot \\mathcal{M}_{\\mathcal{R}}\\right) \\cdot\n\\frac{\\log 2M}{ \\log 2m} \\\\\n&=& O\\left(M^k \\log M \\cdot m \\log p + \\frac{M^k \\log M}{\\log m} \\cdot\n \\mathcal{M}_{\\mathcal{R}}\\right),\n\\end{eqnarray*}\nputting the values of $\\mathcal{N}_{\\mathcal{S}}$ and $c_{\\mathcal{S}}$.\n\n\\paragraph{Pointwise multiplications:}\nFinally, FFT does $2M$ pointwise multiplications in\n$\\mathcal{S}$. Since elements of $\\mathcal{S}$ are $(k-1)$-variate\npolynomials over $\\mathcal{R}$, with degree in every variable bounded\nby $M$, the total time taken for pointwise multiplications is $2M\n\\cdot \\mathcal{F}(2M, k-1)$ bit operations.\n\n\\paragraph{Total polynomial multiplication time:}\nThis can be expressed as,\n\n\n\\begin{eqnarray} \\label{eqn_FFT_complexity}\n\\mathcal{F}(2M, k) &=& O\\left(M^k \\log M \\cdot m \\log p + \\frac{M^k \\log M}{\\log m} \\cdot \\mathcal{M}_{\\mathcal{R}}\\right) + 2M \\cdot \\mathcal{F}(2M, k-1) \\nonumber \\\\\n&=& O\\left(M^k \\log M \\cdot m \\log p + \\frac{M^k \\log M}{\\log m} \\cdot \\mathcal{M}_{\\mathcal{R}}\\right),\n\\end{eqnarray}\nas $k$ is a constant. \\\\\n\n% \\subsection*{Inverse FFT}\n\n\n% \\todo{Pres 9}Computing the inverse of a fourier transform with $\\rho$ as the root\n% of unity just amounts to yet another fourier transform with\n% $\\rho^{-1}$ as the root of unity. This can be seen by observing that a\n% fourier transform can be thought of as a matrix multiplication. If $a(x) = a_0 + a_1 x + \\cdots a_{2M-1}x^{2M-1}$ and $b_i = a(\\rho^i)$ for $i=0\\cdots (2M-1)$, then\n% \\begin{eqnarray*}\n% \\insquar{\\begin{array}{c}b_0\\\\b_1\\\\b_2\\\\ \\vdots \\\\b_{2M-1}\\end{array}} & = & \\insquar{\\begin{array}{cccc}\n% 1 & 1 & \\cdots & 1\\\\\n% 1 & \\rho & \\cdots & \\rho^{2M-1}\\\\\n% 1 & \\rho^2 & \\cdots & \\rho^{2(2M-1)}\\\\\n% \\vdots & \\vdots & \\ddots & \\vdots \\\\\n% 1 & \\rho^{2M-1}& \\cdots & \\rho^{(2M-1)(2M-1)}\n% \\end{array}}\\insquar{\\begin{array}{c}a_0\\\\a_1\\\\a_2\\\\ \\vdots \\\\a_{2M-1} \\end{array}}.\n% \\end{eqnarray*}\n% Since $\\rho$ is a principal root of unity, it is easy to see that\n% \\begin{eqnarray*}\n% \\insquar{\\begin{array}{c}a_0\\\\a_1\\\\a_2\\\\ \\vdots \\\\a_{2M-1}\\end{array}} & = & \\frac{1}{N}\\cdot\\insquar{\\begin{array}{cccc}\n% 1 & 1 & \\cdots & 1\\\\\n% 1 & \\rho^{-1} & \\cdots & \\rho^{-(2M-1)}\\\\\n% 1 & \\rho^{-2} & \\cdots & \\rho^{-2(2M-1)}\\\\\n% \\vdots & \\vdots & \\ddots & \\vdots \\\\\n% 1 & \\rho^{-(2M-1)}& \\cdots & \\rho^{-(2M-1)(2M-1)}\n% \\end{array}}\\insquar{\\begin{array}{c}b_0\\\\b_1\\\\b_2\\\\ \\vdots \\\\b_{2M-1} \\end{array}}.\n% \\end{eqnarray*}\n% which is just another fourier transform with $\\rho^{-1}$ as a root of unity instead. \\\\\n\n\n\nWe now present an equivalent group theoretic interpretation of the\nabove process of polynomial multiplication, which is a subject of\ninterest in itself.\n\n\\subsection{A Group Theoretic Interpretation}\n\nA convenient way to study polynomial multiplication is to interpret it\nas multiplication in a \\emph{group algebra}.\n\n\\begin{definition}\n\\emph{\\textsf{(Group Algebra)}}\n Let $G$ be any group. The \\emph{group algebra} of $G$ over a ring $R$ is\n the set of formal sums $\\sum_{g \\in G} \\alpha_g g$ where $\\alpha_g\n \\in R$ with addition defined point-wise and multiplication defined\n via convolution as follows\n $$ \\left(\\sum_g \\alpha_g g\\right) \\left(\\sum_h\n \\beta_h h\\right) = \\sum_{u}\\inparen{\\sum_{gh=u} \\alpha_g \\beta_h}u $$\n\\end{definition}\n\nIn this section, we study the Fourier transform over the group algebra\n$R[E]$ where $E$ is an \\emph{additive abelian group}. Most of this,\nalbeit in a different form, is well known but is provided here for\ncompleteness \\cite[Chapter 17]{Igor}.\n\nIn order to simplify our presentation, we will fix the base ring to be\n$\\C$, the field of complex numbers. Let $n$ be the \\emph{exponent} of\n$E$, that is the maximum order of any element in $E$. A similar\napproach can be followed for any other base ring as long as it has a\nprincipal $n$-th root of unity.\n\nWe consider $\\C[E]$ as a vector space with basis $\\{ x \\}_{x \\in E}$\nand use the Dirac notation to represent elements of $\\C[E]$ --- the\nvector $\\ket{x}$, $x$ in $E$, denotes the element $1 . x$ of $\\C[E]$.\n\nMultiplying univariate polynomials over $R$ of degree less than $n$\ncan be seen as multiplication in the group algebra $R[G]$ where $G$ is\nthe cyclic group of order $2n$. Say $a(x) = a_0 + a_1 x +\n\\cdots + a_dx^d$ and $b(x)= b_0 + b_1x \\cdots + b_dx^d$ (with $d 2M^k\\cdot m\\cdot 2^{2u}$ where $u = \\frac{2N}{M^km}$. This\n is to prevent overflows during modular arithmetic (see\n Section~\\ref{encoding_section}).\n\\end{enumerate}\n\\noindent\nIt is straightforward to check that $k > L+1$ and $c > 5(k+1)$ satisfy\nthe above constraints. Since $L\\leq 5.2$, it is sufficient to choose\n$k = 7$ and $c = 42$.\\\\\n\n\nLet $T(N)$ denote the time complexity of multiplying two $N$ bit\nintegers. This consists of:\n\n\\begin{itemize}\n\\item Time required to pick a suitable prime $p$,\n\\item Computing the root $\\rho(\\alpha)$,\n\\item Encoding the input integers as polynomials,\n\\item Multiplying the encoded polynomials,\n\\item Evaluating the product polynomial.\n\\end{itemize}\n\nAs argued before, the prime $p$ can be chosen in $o(N)$ time. To\ncompute $\\rho(\\alpha)$, we need to lift a generator of\n$\\mathbb{F}_p^{\\times}$ to $\\Z/p^c\\Z$ followed by an interpolation. Since $c$\nis a constant and $p$ is a prime of $O(\\log N)$ bits, the time\nrequired for Hensel Lifting and interpolation is $o(N)$.\n\nThe encoding involves dividing bits into smaller blocks, and\nexpressing the exponents of $q$ in base $M$\n(Section~\\ref{encoding_section}) and all these take $O(N)$ time since\n$M$ is a power of $2$. Similarly, evaluation of the product polynomial\ntakes linear time as well. Therefore, the time complexity is dominated\nby the time taken for polynomial multiplication.\n\n\\subsubsection*{Time complexity of Polynomial Multiplication}\n\nFrom Equation~\\ref{eqn_FFT_complexity}, the complexity of polynomial multiplication is given by,\n\\[\n\\mathcal{F}(2M,k) = O\\inparen{M^k\\log M\\cdot m\\cdot \\log p + \\frac{M^k\\log\n M}{\\log m}\\cdot \\mathcal{M}_{\\mathcal{R}}}.\n\\]\n\n\\begin{proposition}\\cite{scho_complex}\n If $\\mathcal{M}_{\\mathcal{R}}$ denotes the complexity of\n multiplication in $\\mathcal{R}$, then $\\mathcal{M}_{\\mathcal{R}} =\n T\\left(O(\\log^2{N})\\right)$ where $T(x)$ denotes the complexity of\n multiplying two $x$-bit integers.\n\\end{proposition}\n\\begin{proof}\n Elements of $\\mathcal{R}$ can be viewed as polynomials in $\\alpha$\n over $\\Z/p^c\\Z$ with degree at most $m$. Given two such polynomials\n $f(\\alpha)$ and $g(\\alpha)$, encode them as follows: Replace\n $\\alpha$ by $2^d$, transforming the polynomials $f(\\alpha)$ and\n $g(\\alpha)$ to the integers $f(2^d)$ and $g(2^d)$ respectively. The\n parameter $d$ is chosen such that the coefficients of the product\n $h(\\alpha) = f(\\alpha) g(\\alpha)$ can be recovered from the product\n $f(2^d)\\cdot g(2^d)$. For this, it is sufficient to ensure that the\n maximum coefficient of $h(\\alpha)$ is less than $2^d$. Since $f$\n and $g$ are polynomials of degree $m$, we would want $2^d$ to be\n greater than $m\\cdot p^{2c}$, which can be ensured by choosing $d =\n O\\left(\\log{N}\\right)$. The integers $f(2^d)$ and $g(2^d)$ are\n bounded by $2^{md}$, which is of $O(\\log^2 N)$ bits. The product\n $f(\\alpha)\\cdot g(\\alpha)$ can be decoded from the integer product\n $f(2^d)\\cdot g(2^d)$ by splitting the bits into $(2m-1)$ blocks of\n $d$ bits each (one for each coefficient of $\\alpha^i$) to obtain a\n polynomial in $\\Z[\\alpha]$ and reducing it modulo $p^c$ and\n $\\alpha^m + 1$. Reducing modulo $(\\alpha^m + 1)$ can be performed in\n $O(md) = O(\\log^2N)$ time. Dividing by $p^c$, which has $O(\\log N)$\n bits, can be performed in the same time as multiplying $O(\\log N)$\n bit integers using standard techniques (see for example\n \\cite[Chapter 4]{Knuth}). Since $T(N) = \\Omega(N)$, we have that\n $\\mathcal{M}_{\\mathcal{R}} = T(O(\\log^2 N)) + O(\\log N \\cdot\n T(O(\\log N))) = T(O(\\log^2 N))$ bit operations.\n\\end{proof}\n\nTherefore, the complexity of our integer multiplication algorithm\n$T(N)$ is given by,\n\\begin{eqnarray*}\nT(N) & = & O(\\mathcal{F}(2M, k)) = O\\inparen{M^k\\log M\\cdot m\\cdot \\log p + \\frac{M^k\\log\n M}{\\log m} \\cdot \\mathcal{M}_{\\mathcal{R}}}\\\\\n& = & O\\inparen{N\\log N + \\frac{N}{\\log N\\cdot \\log\\log N} \\cdot T(O(\\log^2N))}\n\\end{eqnarray*}\n\n\\noindent Solving the above recurrence leads to the following theorem.\n\n\\begin{theorem} \\label{thm:mainthmx}\nGiven two $N$ bit integers, their product can be computed using\n$N\\cdot \\log N\\cdot 2^{O(\\log^*N)}$ bit operations.\n\\end{theorem}\n\n\\subsubsection*{Performing on multi-tape turing machines}\n\nThe upper-bound presented in Theorem \\ref{thm:mainthmx} holds for\nmulti-tape turing machines. The only part of the algorithm that\nwarrants an explanation is regrouping of the terms in preparation for\nthe inner and outer DFTs. For the inner DFT, we are given $\\ket{f} =\n\\sum_{a,b} f_{b,a}\\ket{x_b + a}$ and we wish to write down $\\ket{f_b}\n= \\sum_{a} f_{b,a}\\ket{a}$ for each $b\\in B$. The following discussion\nessentially outlines how this can be performed on a multi-tape turing\nmachine using $O(N\\log m)$ bit operations. (Recall that the group $E =\n(\\Z/2M\\Z)^k$ and $A = (\\Z/2m\\Z)^k$ and both $M$ and $m$ are powers of\n$2$). \\\\\n\nWe may assume that the coefficients are listed according to the\nnatural lexicographic order of $(\\Z/2M\\Z)^k$. To ease the\npresentation, we first present the approach for $k=1$. It would be\nstraightforward to generalize this to larger $k$ by repeated\napplications of this approach.\n\n\nGiven as input is a sequence of coefficients $f_g$ for each $g\\in\n\\tilde{E} = \\Z/2M\\Z$ in the natural lexicographic order on one of the\ntapes of the turing machine. We can then ``shuffle'' the input tape\nusing two passes to order the coefficients according to\n$\\inbrace{0,M,1,M+1,\\dots, M-1,2M-1}$ (by copying the first half with\nappropriate blanks, and copying the second half on the second\npass). This results in regrouping the inputs as various cosets of\n$\\Z/2\\Z$. By considering two successive elements as a block and\nrepeating the shuffling, we obtain various cosets of\n$Z/4\\Z$. Repeating this process $\\log(2m)$ times regroups the elements\naccording to various cosets of $\\Z/2m\\Z$.\\\\\n\nSuppose that $E = (\\Z/2M\\Z)^k$ and that the coefficients are ordered\naccording to the natural lexicographic order. By repeating the same\nshuffling process for $\\log(2m)$ steps, the coefficients are regrouped\naccording to the cosets of $\\inbrace{0}^{k-1} \\times (\\Z/2m\\Z)$ and\nhence there are $M/m$ many groups. By considering every $2m$\nsuccessive coefficients as one block, each of the $M/m$ groups can\nbe thought of as coefficients ordered according to the natural\nlexicographic order of $(\\Z/2M\\Z)^{k-1}$. By repeating this for\n$(k-1)$ more steps, we can reorder the coefficients according to the\ncosets of $(\\Z/2m\\Z^k)$.\n\nThe procedure totally requires $k\\log(2m)$ passes over the tapes and\nhence can be performed in $O(N\\log m)$ time on a $3$-tape\\footnote{an\n input tape, an output tape, and a third tape for a counter} turing\nmachine. With the above discussion, it can be easily seen that the\nupper-bound holds for multi-tape turing machines.\n\n\\begin{theorem}\nGiven two $N$ bit integers, their product can be computed using\n$N\\cdot \\log N\\cdot 2^{O(\\log^*N)}$ bit operations on a multi-tape turing machine.\n\\end{theorem}\n\n\n\\section{A Comparison with F\\\"{u}rer's Algorithm} \\label{Qp_section}\n\nOur algorithm can be seen as a $p$-adic version of F\\\"{u}rer's integer\nmultiplication algorithm, where the field $\\C$ is replaced by $\\Q_p$,\nthe field of $p$-adic numbers (for a quick introduction, see Baker's\nonline notes \\cite{Baker}). Much like $\\C$, where representing a\ngeneral element (say in base $2$) takes infinitely many bits,\nrepresenting an element in $\\Q_p$ takes infinitely many $p$-adic\ndigits. Since we cannot work with infinitely many digits, all\narithmetic has to be done with finite precision. Modular arithmetic in\nthe base ring $\\Z[\\alpha]/(p^c, \\alpha^m + 1)$, can be viewed as\narithmetic in the ring $\\Q_p[\\alpha]/(\\alpha^m + 1)$ keeping a\nprecision of $\\varepsilon = p^{-c}$.\n\nArithmetic with finite precision naturally introduces some errors in\ncomputation. However, the nature of $\\Q_p$ makes the error analysis\nsimpler. The field $\\Q_p$ comes with a norm $\\abs{\\ \\cdot\\ }_p$ called\nthe $p$-adic norm, which satisfies the stronger triangle inequality\n$\\abs{x+y}_p \\leq \\max\\inparen{\\abs{x}_p, \\abs{y}_p}$ \\cite[Proposition\n 2.6]{Baker}. As a result, unlike in $\\C$, the errors in computation\ndo not compound.\\\\\n\nRecall that the efficiency of FFT crucially depends on a special\nprincipal $2M$-th root of unity in $\\Q_p[\\alpha]/(\\alpha^m + 1)$. Such\na root is constructed with the help of a primitive $2M$-th root of\nunity in $\\Q_p$. The field $\\Q_p$ has a primitive $2M$-th root of\nunity if and only if $2M$ divides $p-1$ \\cite[Theorem\n 5.12]{Baker}. Also, if $2M$ divides $p-1$, a $2M$-th root can be\nobtained from a $(p-1)$-th root of unity by taking a suitable power. A\nprimitive $(p-1)$-th root of unity in $\\Q_p$ can be constructed, to\nsufficient precision, using Hensel Lifting starting from a generator\nof $\\mathbb{F}_p^{\\times}$.\n\n\\section{Conclusion}\\label{conclusions_section}\n\nAs mentioned earlier, there has been two approaches to multiplying\nintegers - one using arithmetic over complex numbers and the other\nusing modular arithmetic. Using complex numbers, Sch\\\"{o}nhage and\nStrassen \\cite{SS71} gave an $O(N \\cdot \\log N \\cdot \\log\\log N\\ldots\n2^{O(\\log^* N)})$ algorithm. F\\\"{u}rer \\cite{F07} improved this\ncomplexity to $N\\cdot\\log N \\cdot2^{O(\\log^*N)}$ using some special\nroots of unity. The other approach, that is modular arithmetic, can be\nseen as arithmetic in $\\Q_p$ with certain precision. A direct\nadaptation of the Sch\\\"{o}nhage-Strassen's algorithm in the modular\nsetting leads to an $O(N \\cdot \\log N \\cdot \\log\\log N\\ldots\n2^{O(\\log^* N)})$ time algorithm. In this work, we show that by\nchoosing an appropriate prime and a special root of unity, a running\ntime of $N\\cdot \\log N \\cdot 2^{O(\\log^*N)}$ can be achieved through\nmodular arithmetic as well. Therefore, in a way, we have unified the\ntwo paradigms. The important question that remains open is:\n\\begin{itemize}\n\\item Can $N$-bit integers be multiplied using $O(N \\cdot \\log N)$ bit operations?\n\\end{itemize}\n\n\\noindent Even an improvement of the complexity to $O(N \\cdot \\log N\n\\cdot \\log^{*}N)$ operations will be a significant step forward\ntowards answering this question.\n\n\\section*{Acknowledgements}\n\nWe are greatly thankful to the anonymous reviewers for their detailed\ncomments that have improved the presentation of the paper\nsignificantly.\n\n\n\\bibliography{references}\n\\end{document}\n", "meta": {"hexsha": "2cd8077a5e68b8e50dad76e1500097feff82b3d8", "size": 63497, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "contents/research/publication/Journal/2013-04-18-Fast-integer-multiplication/intMult.tex", "max_stars_repo_name": "piyush-kurur-pages/website", "max_stars_repo_head_hexsha": "246dfa730328b45b65840ebed3293e96c497aa86", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-04-16T09:55:17.000Z", "max_stars_repo_stars_event_max_datetime": "2017-04-16T09:55:17.000Z", "max_issues_repo_path": "contents/research/publication/Journal/2013-04-18-Fast-integer-multiplication/intMult.tex", "max_issues_repo_name": "piyush-kurur-pages/website", "max_issues_repo_head_hexsha": "246dfa730328b45b65840ebed3293e96c497aa86", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "contents/research/publication/Journal/2013-04-18-Fast-integer-multiplication/intMult.tex", "max_forks_repo_name": "piyush-kurur-pages/website", "max_forks_repo_head_hexsha": "246dfa730328b45b65840ebed3293e96c497aa86", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-11-10T22:18:56.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-10T22:18:56.000Z", "avg_line_length": 48.8814472671, "max_line_length": 183, "alphanum_fraction": 0.7036237933, "num_tokens": 20548, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819591324416, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.34605352095346736}} {"text": "%\\documentclass[12pt]{article}\n\\documentclass[aip,jcp,preprint,superscriptaddress,floatfix]{revtex4-1}\n\\usepackage{url,graphicx,tabularx,array,geometry,amsmath,listings}\n\\setlength{\\parskip}{2ex} %--skip lines between paragraphs\n\\setlength{\\parindent}{20pt} %--don't indent paragraphs\n\n\\usepackage[colorinlistoftodos,prependcaption,textsize=tiny]{todonotes} %--TODO notes\n\n\\setlength{\\headheight}{-50pt}\n\\setlength{\\textheight}{700pt}\n\\setlength{\\textwidth}{500pt}\n\\setlength{\\oddsidemargin}{-10pt}\n\\setlength{\\footskip}{50pt}\n\\usepackage{graphicx}% Include figure files\n\\usepackage{bm}\n\\usepackage{url}\n\\usepackage[colorlinks = true,\n linkcolor = blue,\n urlcolor = blue,\n citecolor = blue,\n anchorcolor = blue]{hyperref}\n\\graphicspath{{./Figures/}}\n\n%-- Commands for header\n\\renewcommand{\\title}[1]{\\textbf{\\large{#1}}\\\\}\n\\renewcommand{\\line}{\\begin{tabularx}{\\textwidth}{X>{\\raggedleft}X}\\hline\\\\\\end{tabularx}\\\\[-0.5cm]}\n\\newcommand{\\leftright}[2]{\\begin{tabularx}{\\textwidth}{X>{\\raggedleft}X}#1%\n& #2\\\\\\end{tabularx}\\\\[-1cm]}\n\n%\\linespread{2} %-- Uncomment for Double Space\n\\begin{document}\n\n\\title{\\center{Monte Carlo Simulation of the Lennard Jones Fluid in the Canonical Ensemble}}\n\\rule{\\textwidth}{1pt}\n\\leftright{The Molecular Sciences Software Institute}{Eliseo Marin-Rimoldi and John D.~Chodera} %-- left and right positions in the header\n\n\\bigskip\n\n\\section{Introduction}\n\\subsection{Monte Carlo Integration}\n\nIn statistical mechanics, we are interested in computing averages of\nthermodynamic properties as a function of atom positions and momenta~\\cite{Tuckerman.Book,Hill.Book,McQuarrie.Book}.\nA thermodynamic average depending only on configurational properties can be computed using the following\nexpectation value integral\n\\begin{equation}\n\t\\left = \\int_V Q\\left(\\textbf{r}^N\\right)\n\t\\rho\\left(\\textbf{r}^N\\right) d\\textbf{r}^N\n\t\\label{eq.statMechAverage}\n\\end{equation}\n$\\textbf{r}^N$ is a $3N$ dimensional vector containing the positions of the $N$ atoms,\nwhere $Q(\\textbf{r}^N)$ is the thermodynamic quantity of interest that depends only on the configuration $\\textbf{r}^N$, \n$\\rho\\left(\\textbf{r}^N\\right)$ is the probability\ndensity whose functional form depends on the statistical mechanical ensemble\nof interest, and $V$ defines the volume of configuration space over which $\\rho$ has support. \nNote that the integrals over momenta have been factored out, as\nthey can be evaluated\nanalytically. The integral (Eq.~\\ref{eq.statMechAverage}) is very hard to compute\neven for small atomic systems.\nFor instance, a monoatomic system of 10 atoms leads to a 30-dimensional\nintegral. Consequently, we need to resort to a numerical integration scheme if we want to\nstudy atomic systems.\n\nMonte Carlo methods are numerical techniques\nfrequently used to estimate complex multidimensional\nintegrals which otherwise\n could not be performed~\\cite{Liu.Book,Newman.Book}.\n For instance, the integral of the function $f(\\textbf{x})$,\n where $\\textbf{x} \\in \\Re^{M}$, is approximated as\n \\begin{equation}\n\t I = \\int_V f(\\textbf{x}) \\, d\\textbf{x} = \\int_V\n\t \\frac{f(\\textbf{x})}{h(\\textbf{x})} h(\\textbf{x}) \\, d\\textbf{x} \n\t = \\left< \\frac{f(\\textbf{x})}{h(\\textbf{x})} \\right>_{h(\\textbf{x})}\n\t \\label{eq.averageGeneral}\n \\end{equation}\n The idea of Monte Carlo integration is to estimate the expectation value\n $\\left<\\frac{f(\\textbf{x})}{h(\\textbf{x})}\\right>_{{h(\\textbf{x})}}$ by\n generating random samples of \\textbf{x} from the probability density $h(\\textbf{x})$. \n\n\\subsection{Importance Sampling}\n\nIn Equation \\ref{eq.averageGeneral}, we are free to chose the probability\ndistribution\n$h(\\textbf{x})$. The simplest case is to uniformly generate $\\textbf{x}$\nin the volume $V$. In this way, $h(\\textbf{x})$ becomes constant as\n\\begin{equation}\n\th(\\textbf{x}) = \\frac{1}{V}\n\t\\label{eq.uniformDist}\n\\end{equation}\nUsing this sampling density $h(\\textbf{x})$, the integral (Eq.~\\ref{eq.averageGeneral}) becomes\n\\begin{equation}\n\tI = \\int_V f(\\textbf{x}) \\, d\\textbf{x} \\approx \\frac{V}{N} \\sum_{i=1}^N\n\tf(\\textbf{x}_i)\n\t\\label{eq.averageUniform}\n\\end{equation}\nwhere N is the total number of random samples and $f(\\textbf{x}_i)$ is the\nintegrand evaluated using the $i^{th}$ sample. While using a\nuniform sampling density often works sufficiently well for simple unidimensional cases, it generally fails to produce useful estimates\nfor complex problems.\n\nThe problem at hand involves the evaluation of $3N$-dimensional integral Eq.~\\ref{eq.statMechAverage}, which is dominated by a small region of configuration space~\\cite{Tuckerman.Book,Hill.Book,McQuarrie.Book}. \nUsing a uniform probability distribution $h(\\textbf{r}^N)$ over the configuration space hypervolume $V^{3N}$ to generate\nrepresentative samples of this subset is not efficient,\nas most states generated this way would have a low weight.\n\nA solution to this problem is to sample positions $\\textbf{r}^N$ from the desired equilibrium probability density $\\rho\\left(\\textbf{r}^N\\right)$:\n\\begin{eqnarray}\n\t\\textbf{r}^N &\\sim& \\rho(\\textbf{r}^N)\n\\end{eqnarray}\nThis is a way to generate\nrelevant configurations more frequently than configurations that have low\nprobability.\nMathematically, we set $h(\\textbf{r}^N)=\\rho\\left(\\textbf{r}^N\\right)$.\nThis idea is known as \\textit{importance sampling}.\n\nCombining Eqs.~\\ref{eq.statMechAverage} and \\ref{eq.averageGeneral}\nand the condition $h(\\textbf{r}^N)=\\rho\\left(\\textbf{r}^N\\right)$,\nwe find that\n\\begin{equation}\n\t%\\left = \\frac{\\int Q e^{-\\beta U\\left(\\textbf{r}^N\\right)}\n\t%d\\textbf{r}^N}{\\int e^{-\\beta U\\left(\\textbf{r}^N\\right)}\n\t%d\\textbf{r}^N}\n\t\\left \\approx \\frac{1}{N} \\sum_{i=1}^N Q\\left(\\textbf{r}_i^N\\right) .\n\t\\label{eq.importanceSamplingAverage}\n\\end{equation}\nThus, we can get thermodynamic properties by simply computing an unweighted sample average, given\nthat we perform importance sampling from $h(\\textbf{r}^N)=\\rho\\left(\\textbf{r}^N\\right)$.\n\n\\subsection{Detailed Balance Condition}\n\nThe question now becomes how to generate such atomic positions $\\textbf{r}^N$\n(or \\textit{states}) distributed\naccording to $\\rho\\left(\\textbf{r}^N\\right)$. In 1953, Metropolis, Rosenbluth, Rosenbluth, and Teller introduced a\nsolution based on Markov chains~\\cite{Metropolis.JCP.21.1087.1953}. \nThey proposed to use the detailed balance\ncondition in order to ensure proper configurational sampling from\nthe statistical mechanical distribution of interest. In order to generate\na new configuration $n$ from an old configuration $m$,\nthe detailed balance condition is\n\n\\begin{equation}\n\t\\rho_m\\left(\\textbf{r}^N\\right) \\alpha \\left(m \\rightarrow n \\right)\n\tP_{acc} \\left(m \\rightarrow n \\right) =\n\t\\rho_n\\left(\\textbf{r}^N\\right) \\alpha \\left(n \\rightarrow m \\right)\n\tP_{acc} \\left(n \\rightarrow m \\right) \n\t \\label{eq.detailedBalance}\n\\end{equation}\n\nWhere $\\rho_m\\left(\\textbf{r}^N\\right)$ is the probability of observing\nstate $m$,\n$\\alpha \\left(m \\rightarrow n \\right)$ is the probability of\nattempting to generate a new state $n$ starting from a state $m$ and \n$P_{acc} \\left(n \\rightarrow m \\right)$ is the probability of accepting \nsuch transition. Basically, the condition of detailed balance tells us that the\n``flux'' of transitions\nfrom state $m$ to state $n$ equals the flux from state $n$ to state $m$ at\nequilibrium.\n\nThere are many ways to satisfy Eq.~\\ref{eq.detailedBalance} by construction of different acceptance probabilities. \nWhile Metropolis et.~al proposed a choice that both satisfies Eq.~\\ref{eq.detailedBalance} and maximizes the average acceptance probability $\\left< P_{acc} \\right>$~\\cite{Metropolis.JCP.21.1087.1953}, Hastings generalized this to the case where proposal probabilities are not symmetric, such that $\\alpha \\left(m \\rightarrow n \\right) \\ne \\alpha \\left(n \\rightarrow m \\right)$, producing the \\emph{Metropolis-Hastings}~\\cite{Hastings.Biometrika.57.97.1970} acceptance criteria:\n\\begin{equation}\n\tP_{acc}(m \\rightarrow n) = \\text{min} \\left[1,\\frac{\\alpha \\left(n\n\t\t\\rightarrow m \\right)}{\\alpha \\left(m \\rightarrow n \\right)}\n\t\t\\frac{\\rho_n\\left(\\textbf{r}^N\\right)}{\\rho_m\\left(\\textbf{r}^N\\right)}\n\t\\right]\n\t\\label{eq.MetropolisHastings} .\n\\end{equation}\nThis algorithm is one of a general class of \\emph{Markov chain Monte Carlo (MCMC)} algorithms that generate Markov chains to sample a desired target density, and a great deal of the MCMC literature is valuable for molecular simulations~\\cite{Liu.Book}.\n\n\\section{Canonical Ensemble Monte Carlo of a Lennard Jones Fluid}\n\nAssume we have $N$ monoatomic particles that interact using the Lennard-Jones (LJ) pairwise potential:\n\\begin{equation}\nU(r) = 4 \\epsilon \\left[\\left(\\frac{\\sigma}{r}\\right)^{12} -\\left(\\frac{\\sigma}{r}\\right)^{6} \\right] \n\\end{equation}\nwhere $r$ is the interparticle distance, $\\sigma$ is the distance where the interaction energy is zero, and $\\epsilon$ is the well depth. \nFor simulating argon, for example, a common choice is $\\sigma = 3.4$~\\AA and $\\epsilon/k_B = 120$~K.\n\nOur goal is to generate a set of states of $N$ LJ particles distributed according to the canonical (NVT) ensemble\n\\begin{eqnarray}\n\t\\rho_n \\left(\\textbf{r}^N ; \\beta \\right) &=& Z(\\beta)^{-1} e^{-\\beta U \\left( \\textbf{r}^N \\right)} \\label{eq:nvtDist} \\\\\n\tZ(\\beta) &\\equiv& \\int_V e^{-\\beta U \\left( \\textbf{r}^N \\right)} \\, d\\textbf{r}^N \n\\end{eqnarray}\nwhere $U \\left( \\textbf{r}^N \\right)$ is the potential energy of the system, $\\beta = (k_B T)^{-1}$ is the inverse temperature, $k_B$ is the Boltzmann constant, and $T$ is the absolute temperature. \n\nNote that $U \\left( \\textbf{r}^N \\right)$ is given by\n\\begin{equation}\n\tU \\left( \\textbf{r}^N \\right) = \\sum_{i < j} U \\left( r_{ij} \\right) \n\t\\label{eq.LennardJonesTotalPotential}\n\\end{equation}\nwhere $r_{ij} \\equiv || \\textbf{r}^N_j - \\textbf{r}^N_i ||_2$ is the interparticle separation distance.\n\nSubstituting Eq.~\\ref{eq:nvtDist} into Eq.~\\ref{eq.MetropolisHastings} and assuming $\\alpha \\left( n \\rightarrow m \\right) = \\alpha \\left( m \\rightarrow n \\right)$, we obtain\n\\begin{equation}\n\tP_{acc}(m \\rightarrow n) = \\text{min} \\left[\n\t\t1,e^{-\\beta \\Delta U}\n\t\\right]\n\t\\label{eq.detailedBalanceNVT}\n\\end{equation}\nwhere $\\Delta U \\equiv U(\\textbf{r}^N_m) - U(\\textbf{r}^N_n)$ is the difference in potential energy of the system\nbetween the new state $n$ and the old state $m$. Note that the argument\nof the energy $\\textbf{r}^N$ has been dropped for clarity.\n\n\\subsection{Flow of Calculations in a Metropolis Monte Carlo simulation}\n\\label{sec.flow}\n\nThe following workflow can be used to implement the Metropolis algorithm to sample the canonical ensemble of configurations of LJ particles \\cite{Shell.Notes, Maginn.Notes}:\n\n\\begin{enumerate}\n\\setlength{\\itemsep}{0em} %--don't put a huge amount of space between items in a list\n\\item Generate an inital system state $m$.\n\\item Choose an atom with uniform probability from $\\{1, \\ldots, N\\}$ from old state $m$.\n\\item Propose a new state $n$ by translating a LJ particle by a uniform random displacement $\\Delta r \\sim U(-\\Delta x, +\\Delta x)$ in each dimension.\n\tThe displacement scale $\\Delta x$ should not be too large as this would\n\tlikely result in particle overlaps, but should not be too small\n\tas this would result in a slow sampling of configurational space.\n\tMore on this below.\n\\item The difference in energy between the new and old states is computed.\n\tNote that you do not need to compute the \\textit{total} system energy\n\tdifference, as all particles but one remain at the same position. \n\tIt is enough to get difference in energy of the selected molecule \n\tin the new and old states.\n\\item The new state is accepted or rejected using the Metropolis criterion.\n\tPractically, this can be implemented as follows. \n\tIf a move from $m$ to $n$ is ``downhill'', $\\beta \\Delta U \\leq 0$,\n\tthe move is always accepted. For ``uphill'' moves, a random\n\tnumber $\\zeta$ is generated uniformly on (0,1). \n\tIf $\\zeta < \\exp[-\\beta {\\Delta U}]$, the move is\n\taccepted. Otherwise, the move is rejected. \n\tIf a non-symmetric proposal is used, this acceptance scheme will have to be modified to implement Eq.~\\ref{eq.MetropolisHastings}.\n\\end{enumerate}\n\n\\subsection{Technical considerations}\n\n\\textbf{Initial configuration.} \nWhile computed equilibrium expectations should be independent of the starting configuration, from a practical standpoint, we have to select an initial configuration that is not so atypical of equilibrium configurations that relaxation to equilibrium will require an enormous amount of simulation time. \nFor this particular example, a good way to start is to place atoms in a 3D lattice. \nDuring the Metropolis Monte Carlo simulation at temperatures and box volumes (or pressures) typical of the liquid state, the lattice will ``melt'' and liquid configurations will be obtained. \nOnce a liquid configuration has been generated from a simulation, you can always save a ``snapshot'' from this simulation as a starting point for a new simulation of the liquid states, and relaxation to equilibrium will generally be much more rapid as long as the conditions are similar.\n\n\\textbf{Random number generation.} \nComputers cannot generate truly random numbers.\nInstead, they rely on \\emph{pseudorandom number generators} (PRNGs) that aim to produce random numbers with the desired statistical properties and long recurrence times between repeats of the same random number sequence.\nUsing a low-quality random number generator can lead to simulation artifacts that can lead to incorrect physical behavior~\\cite{Ferrenberg.PRL.69.3382.1992,Cerutti.JCTC.4.1669.2008}.\nWe recommend you avoid writing your own PRNGs, as this can lead to inadvertent implementation of an ill-conceived algorithm~\\cite{RANDU}.\nInstead, rely on high-quality, well-understood PRNGs and implementations that have are well-supported, such as the Mersenne Twister~\\cite{Mersenne.Twister,Matsumoto.ACMTransModelComputSimul.8.3.1998} implementation provided by \\href{https://docs.scipy.org/doc/numpy/reference/routines.random.html}{\\tt numpy.random}.\n\n\\textbf{Equilibration.} \nWhen the initial configuration is highly atypical compared to true samples from the equilibrium density---such as the initial 3D lattice conditions compared to a true disordered liquid state---it may require very long simulation times for the bias in equilibrium averages computed over the entire trajectory to become small compared to the statistical error~\\cite{Chodera.JCTC.12.1799.2016}.\nIt is therefore common practice to discard some initial part of the simulation to \\emph{equilibration} and to average over the subsequent \\emph{production} region to minimize this bias at the cost of potentially increasing statistical error by including less data in the average.\nWhile common practice traditionally had selected an arbitrary initial portion of the simulation to equilibrium, modern best practice recommends the use of an automated approach for selecting the optimal equilibration/production split point in a manner that maximizes the number of statistically uncorrelated samples in the production part of the trajectory~\\cite{Chodera.JCTC.12.1799.2016}.\nTo do this, you can use the \\href{http://pymbar.readthedocs.io/en/latest/timeseries.html#automatically-identifying-the-equilibrated-production-region}{\\tt pymbar.timeseries.detectEquilibration} function from the \\href{http://pymbar.org/}{\\tt pymbar} module to analyze an array containing timeseries data for your observable (such as energies, box volumes, or densities)~\\cite{Chodera.JCTC.12.1799.2016}.\n\n\\textbf{System size and periodic boundary conditions.} \nA typical simulation of a Lennard-Jones fluid is carried out anywhere from 216 to 10,000 particles. \nThis amount of particles is far away of being representative of a bulk liquid. \nTo get around this problem, we employ a trick called periodic boundary conditions. \nThe primary simulation box is surrounded\nby copy images of itself. For a cubic simulation, there would be 26 \nimages around the central box. This trick has two implications\n\\begin{enumerate}\n\\setlength{\\itemsep}{0em} %--don't put a huge amount of space between items in a list\n\t\\item If the position of a particle (i.e.\\ Cartesian coordinates) is\n\t\toutside the simulation box after a particle translation, \n\t\tan identical particle should enter the box through the\n\t\topposite face of the box, as shown in Figure ~\\ref{fig:pbc} \n\t\\item When computing distances $r_{ij}$ used in the evaluation of the LJ potential (Eq.~\\ref{eq.LennardJonesTotalPotential}), we use the \\textit{minimum image distance}. See Figure ~\\ref{fig:minimumImage}.\n\\end{enumerate}\n\n\\begin{figure}[t]\n \\includegraphics[scale = 0.5]{pbc.eps}\n \\caption{Molecule one is displaced outside the bounds of the central box and placed back\n in through the oposite side of the box. Image from Allen and Tildesley. ~\\cite{Allen.Book}}\n \\centering\n \\label{fig:pbc}\n\\end{figure}\n\n\\begin{figure}[t]\n \\includegraphics[scale = 0.5]{minimumImage.eps}\n \\caption{Representation of the minimum image distance. Molecule 1 does not interact with\n molecule 4 located in the simulation box because the distance between particles is\n greater than the cutoff. However, molecule 1 does interact with\n the image of molecule 4 located in box E. Image from Allen and\n Tildesley ~\\cite{Allen.Book}}\n \\centering\n \\label{fig:minimumImage}\n\\end{figure}\n\n\\textbf{Maximum displacement $\\Delta x$.} \nAs noted above, the Metropolis Monte Carlo algorithm requires translating a selected LJ particle by a random perturbation. \nThis displacement should not be too large as this would result in particle overlaps and low acceptance rates; on the other hand, it should not be so small as to result in inefficient sampling of configuration space.\nA common practice is to adjust the maximum particle displacement $\\Delta x$ during an explicit equilibration phase in order to achieve $\\sim$50\\% acceptance translation rates over a recent window of $\\sim N$ Monte Carlo trial moves. \n\n\\textbf{Energy truncation and tail corrections.} \nIf two particles are separatedby more than a certain distance, we typically truncate their interaction energy if $r > r_c$, where $r_c$ denotes the \\emph{cutoff distance}. \nTruncating interactions removes contribution to the potential energy that might be non negligible and can lead to significant artifacts, such as significantly perturbed densities when a barostat is used to sample the NPT ensemble due to neglected long-range dispersion interactions. \nWe can estimate the truncated interactions by incorporating an energy \ncorrection, known as the tail or long range correction.\nFor the Lennard-Jones fluid, we assume that we have an homogeneous liquid at $r>r_c$ to obtain the correction for neglecting this contribution for all interacting pairs of particles~\\cite{Shirts.JPCB.111.13052.2007}:\n\\begin{equation}\n\tU_\\mathrm{correction} = \\frac{8 \\pi N^2}{3 V} \\epsilon \\sigma^3\n\t\\left[\\frac{1}{3} \\left(\\frac{\\sigma}{r_c} \\right)^9 \n\t- \\left(\\frac{\\sigma}{r_c} \\right)^3 \\right]\n\\end{equation}\nFor a Lennard-Jones fluid, it is common to set $r_c \\sim 3 \\sigma$, since the pair interaction at this separation is small, $U(3 \\sigma) \\approx 4 [(3)^{-12} - (3)^{-6} \\approx - 0.0055 \\epsilon$, or about 0.5\\% of the well depth $\\epsilon$.\nYou will want to verify that your computed properties are relatively insensitive to the choice of cutoff $r_c$ so that a too-short cutoff does not induce artifacts in computed physical properties.\n\n\\textbf{Units.}\nDealing with unit-bearing quantities---such as energies, distances, masses, and physical constants---in your code can be tricky.\nErrors can easily creep in when one part of the code assumes one set of implicit units while another part of the code assumes a different set of implicit units, and error like this have led to billion-dollar accidents (such as the loss of the Mars Polar Lander due to one part of the code using metric units while the other used English units~\\cite{MarsPolarLander}).\nTo avoid costly mistakes that complicate debugging, we recommend one of two approaches:\n(1) use a unit library such as \\href{http://pint.readthedocs.io/}{\\tt pint} or \\href{http://docs.openmm.org/7.1.0/userguide/library.html#units-and-dimensional-analysis}{\\tt simtk.unit} that automatically handles unit coversions,\nor (2) select a single compatible unit system (e.g.~mks or {\\AA}ngstroms-amu-picoseconds) and explicitly document the units for all unit-bearing constants and quantities;\nThe first option is highly preferred, though care must still be paid to ensuring unit-bearing quantities possess the appropriate dimensionality and some considerations must be made for performance impact in parts of the code that must be performant since unit conversion incurs non-negligible overhead.\n\n\\textbf{Compatible constants.}\nWhile some constants are exact, many physical constants are known only to a number of significant digits.\nAll values must be rounded to some number of significant figures when used in computer simulations, and these numbers have finite-precision floating-point representation that further reduces their accuracy.\nPerhaps surprisingly, choices of how many significant figures that each constant should be rounded to can have an impact on the resulting simulations if care is not take to ensure that constants frequently used together are not rounded in a self-consistent manner.\nFor example, a major difference between potential energies computed in CHARMM and AMBER arises from an early arbitrary choice of the number of significant digits to round a set of critical Coulomb energy coefficients~\\cite{Shirts.JCAMD.31.147.2017}.\nFortunately, the National Institute of Standards and Technology (NIST) provides up-to-date guidelines on consistent sets of physical parameters in the \\href{http://ws680.nist.gov/publication/get_pdf.cfm?pub_id=920686}{2014 CODATA Self-Consistent Physical Constants} (updated every four years).\nIt is highly recommended that the CODATA parameters be adopted unless there is strong need to reproduce flawed results from a specific molecular simulation package.\n\n\n\\textbf{Pressure, the virial and pressure tail corrections.} To compute the system pressure,\nwe use the virial theorem\n\n\\begin{equation}\n P = \\frac{1}{3V} \\left< 3 N k_B T + \\sum_{i < j} \\textbf{f}_i \\cdot \\textbf{r}_i \\right>\n\\end{equation}\n\nWhere\n\n\\begin{equation}\n \\sum_{i < j} \\textbf{f}_i \\cdot \\textbf{r}_i = - \\sum_{i < j} \\frac{dU\\left(r_{ij} \\right)}{dr} r_{ij} = W\n\\end{equation}\n\nThe term $W$ is known as the virial and it is a pairwise sum and it can be computed\nalongside the energies. Note that the negative derivative of the energy is the force.\nFor the LJ potential, this is\n\n\\begin{equation}\n \\mathbf{f}\\left(r_{ij} \\right) = -\\frac{dU \\left(r_{ij} \\right)}{dr} = - \\frac{48 \\epsilon}{r^2_{ij}} \\left[\\left(\\frac{\\sigma}{r_{ij}}\\right)^{12} -\\frac{1}{2}\\left(\\frac{\\sigma}{r_{ij}}\\right)^{6} \\right] \\mathbf{r}_{ij}\n\\end{equation}\n\nThe effect of the truncation of the interactions must be also taken into account for the pressure\ncomputation using a correction term\n\n\\begin{equation}\n P_{tail} = \\frac{16 \\pi N^2}{3 V^2} \\epsilon \\sigma^3\n \\left[\\frac{2}{3} \\left(\\frac{\\sigma}{r_c} \\right)^9 \n - \\left(\\frac{\\sigma}{r_c} \\right)^3 \\right]\n\\end{equation}\n\n\n\n\n\\textbf{Reduced units.}\nLennard-Jones fluids have the surprisingly pleasant behavior of possessing universal behavior when expressed in terms of \\emph{reduced units} as\n\n\\begin{equation}\n\tU^*\\left(r_{ij} \\right) = 4 \\left[\\left(\\frac{1}{r^*_{ij}}\\right)^{12} -\\left(\\frac{1}{r^*_{ij}}\\right)^{6} \\right]\n\t\\label{eq.lennardjonesReduced}\n\\end{equation}\n\nwhere \n\n\\begin{equation}\n\tU^* = \\frac{U}{\\epsilon}\n\\end{equation}\n\nand \n\n\\begin{equation}\n\tr^* = \\frac{r}{\\sigma}\n\\end{equation}\n\nThat is, when plotted in reduced units, all Lennard-Jones fluids exhibit the same universal behavior despite the exact choices of $\\epsilon$ and $\\sigma$ used in the simulation. See Table \\ref{table:reducedUnits} for a list\nof variables in reduced units.\nUsing reduced units for input and output will allow you to compare your results with others.\n\n\\begin{table}[t]\n\\centering\n \\begin{tabular}{|c c|} \n \\hline\n Quantity & Expression \\\\ [0.5ex] \n \\hline\\hline\n Length & $L^*=L / \\sigma$ \\\\ \n Density & $\\rho^* = N \\sigma^3 / V$ \\\\\n Energy & $U^* = U / \\epsilon$ \\\\\n Pressure & $P^* = P \\sigma^3 / \\epsilon$ \\\\\n Temperature & $T^* = k_B T / \\epsilon$ \\\\\n Volume & $V^* = V / \\sigma^3$ \\\\\n Time & $t^* = t \\sqrt{\\frac{\\epsilon}{m \\sigma^2}}$ \\\\\n \\hline\n \\end{tabular}\n \\caption{Conversion factors between real and reduced units.}\n \\label{table:reducedUnits}\n\\end{table}\n\n\\subsection{Reference calculations}\n\nSee the following resources for NIST Lennard-Jones fluid reference data:\n\\begin{itemize}\n\\item \\href{https://mmlapps.nist.gov/srs/LJ_PURE/mc.htm}{Reference thermodynamic data ($N=500$, $T^*=0.90$ and $0.90$, $r_c = 3\\sigma$)}\n\\item \\href{https://www.nist.gov/mml/csd/chemical-informatics-research-group/lennard-jones-fluid-reference-calculations}{Reference snapshot potential energies and virials}\n\\end{itemize}\n\n\\section{Concrete tasks}\n\n\\textbf{Implement one (or more) methods to generate initial configurations. }\nYou will want implement more than one method to generate an initial\nconfiguration. This will allow you to test if your code produces the same\nfinal answers with different initial conditions. Among the possible options\nare random placement of particles in the box, placement of particles in a\nlattice or start a simulation using an old configuration.\n\n\\textbf{Implement the pairwise and long tail correction equations. } \nThese are the LJ potential, force (or virial) expression, energy and \npressure long tail corrections. Use reduced units, \nas in Equation ~\\ref{eq.lennardjonesReduced} to facilitate comparison with\nothers.\n\n\\textbf{Implement a function that computes the total energy of the system. } \nCompare to the NIST reference snapshot potential energies and virials.\n\n\\textbf{Implement the Metropolis algorithm and compare to the NIST\nbenchmark. } See Section ~\\ref{sec.flow} as\na reference. This is the part of the code where you will implement\nfeatures such as periodic boundary conditions, minimum image distance, \nparticle translation and the acceptance criteria. You can use the state point\n$T^* = 0.9$, $\\rho^* = 0.9$, and $r_c = 3\\sigma$ and compare to the\nNIST benchmark for thermodynamic data.\n\n\\textbf{Check if your energies are consistent with the NIST benchmark. }\n\n\n\\newpage\n%\n\\bibliographystyle{aip.bst}\n\\bibliography{references.bib}\n\n\\end{document}\n", "meta": {"hexsha": "d10f400baa5c3f4b95d200955247381d05ba2e7b", "size": 26729, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "handouts/01_MC_NVT.tex", "max_stars_repo_name": "MolSSI-SSS/MM_SSS_Lesson_Plan", "max_stars_repo_head_hexsha": "98af78796fbeca5924cbf800e573594cc24d96fc", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-07-19T02:17:44.000Z", "max_stars_repo_stars_event_max_datetime": "2020-07-19T02:17:44.000Z", "max_issues_repo_path": "handouts/01_MC_NVT.tex", "max_issues_repo_name": "MolSSI-SSS/MM_SSS_Lesson_Plan", "max_issues_repo_head_hexsha": "98af78796fbeca5924cbf800e573594cc24d96fc", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-07-20T20:23:32.000Z", "max_issues_repo_issues_event_max_datetime": "2017-07-21T20:44:29.000Z", "max_forks_repo_path": "handouts/01_MC_NVT.tex", "max_forks_repo_name": "dgasmith/MM_SSS_Lesson_Plan", "max_forks_repo_head_hexsha": "98af78796fbeca5924cbf800e573594cc24d96fc", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 60.2004504505, "max_line_length": 477, "alphanum_fraction": 0.7547607468, "num_tokens": 7271, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185205547239, "lm_q2_score": 0.611381973294151, "lm_q1q2_score": 0.34605352001778306}} {"text": "\\chapter{Non-deterministic Classical Processes}\\label{sec:main}\nIn this section, we will discuss our main contribution: an extension of \\cp\nwhich allows for races while still excluding deadlocks. \nWe have seen in \\cref{sec:cp-example} how \\cp excludes deadlocks, but how exactly\ndoes \\cp exclude races?\nLet us return to our first example from \\cref{sec:introduction}, to the\ninteraction between John, Mary and the store.\n\\[\n \\begin{array}{c}\n \\tm{(\\piPar{%\n \\piSend{x}{\\sliceofcake}{\\piSend{x}{\\nope}{\\store}}\n }{%\n \\piPar{\\piRecv{x}{y}{\\john}}{\\piRecv{x}{z}{\\mary}}\n })}\n \\\\[1ex]\n \\rotatebox[origin=c]{270}{$\\Longrightarrow^{\\star}$}\n \\\\[1ex]\n \\tm{(\\piPar{\\store}{\\piPar{\\piSub{\\sliceofcake}{y}{\\john}}{\\piSub{\\nope}{z}{\\mary}}})}\n \\quad\n \\text{or}\n \\quad\n \\tm{(\\piPar{\\store}{\\piPar{\\piSub{\\nope}{y}{\\john}}{\\piSub{\\sliceofcake}{z}{\\mary}}})}\n \\end{array}\n\\]\nRaces occur when more than two processes attempt to communicate simultaneously\nover the \\emph{same} channel. However, the \\text{Cut} rule of \\cp requires that\n\\emph{exactly two} processes communicate over each channel:\n\\begin{center}\n \\cpInfCut\n\\end{center}\nWe could attempt write down a protocol for our example, stating that the store\nhas a pair of channels $\\tm{x}, \\tm{y} : \\ty{\\cake}$ with which it communicates\nwith John and Mary, taking \\cake to be the type of interactions in which cake\n\\emph{may} be obtained, i.e.\\ of both \\sliceofcake and \\nope, and state that the\nstore communicates with John \\emph{and} Mary over a channel of type \\ty{\\cake\n \\parr \\cake}.\nHowever, this \\emph{only} models interactions such as the following:\n\\begin{prooftree}\n \\AXC{$\\seq[{ \\john }]{ \\Gamma, \\tmty{x}{\\cake^\\bot} }$}\n \\AXC{$\\seq[{ \\mary }]{ \\Delta, \\tmty{y}{\\cake^\\bot} }$}\n \\SYM{\\tens}\n \\BIC{$\\seq[{ \\cpSend{y}{x}{\\john}{\\mary} }]{\n \\Gamma, \\Delta, \\tmty{y}{\\cake^\\bot \\tens \\cake^\\bot} }$}\n \\AXC{$\\seq[{ \\store }]{ \\Theta, \\tmty{x}{\\cake}, \\tmty{y}{\\cake} }$}\n \\SYM{\\parr}\n \\UIC{$\\seq[{ \\cpRecv{y}{x}{\\store} }]{\n \\Theta, \\tmty{y}{\\cake \\parr \\cake} }$}\n \\NOM{Cut}\n \\BIC{$\\seq[{ \\cpCut{y}{\\cpSend{y}{x}{\\john}{\\mary}}{\\cpRecv{y}{x}{\\store}} }]{\n \\Gamma, \\Delta, \\Theta }$}\n\\end{prooftree}\nNote that in this interaction, John will get whatever the store decides to send\non \\tm{x}, and Mary will get whatever the store decides to send on \\tm{y}.\nThis means that this interactions gives the choice of who receives what \\emph{to\nthe store}. This is not an accurate model of our original example, where\nthe choice of who receives the cake is non-deterministic and depends on factors\noutside of any of the participants' control!\nAnd to make matters worse, the term which models our example is entirely\ndifferent from the one we initially wrote down in the \\textpi-calculus!\n\nThe ability to model racy behaviour, such as that in our example, is essential\nto describing the interactions that take place in realistic concurrent systems.\nTherefore, we would like to extend \\cp to allow such races.\nSpecifically, we would like to do it in a way which mirrors the way in which the\n\\textpi-calculus handles non-determinism.\nWe will base our extension on \\rcp, a subset of \\cp which we introduced in\n\\cref{sec:background}.\nWe have chosen to do this to keep our discussion as simple as possible.\nFurthermore, as compatibility with the \\textpi-calculus is of interest, we will\nuse the reduction system without commuting conversions, which we introduced in\n\\cref{sec:cppi}.\n\nThis chapter proceeds as follows.\nIn \\cref{sec:nc-terms-and-types}, we introduce the extensions to the terms,\nstructural congruence and types of \\cp.\nIn \\cref{sec:nc-typing-clients-and-servers}, we introduce the typing rules for\n\\nodcap. \nIn \\cref{sec:nc-running-clients-and-servers}, we introduce the reduction rules\nfor \\nodcap.\nIn \\cref{sec:nc-properties}, we prove that our extension preserves the\nmeta-theoretical of \\cp.\nFinally, in \\cref{sec:nc-local-choice}, we discuss the relation between\nnon-determinism in \\nodcap and the non-determinism introduced by the addition of\nnon-deterministic local choice.\n\n\\section{Terms and types}\\label{sec:nc-terms-and-types}\nLet us return, briefly, to our example.\n\\[\n \\tm{(\\piPar{%\n \\piSend{x}{\\sliceofcake}{\\piSend{x}{\\nope}{\\store}}\n }{%\n \\piPar{\\piRecv{x}{y}{\\john}}{\\piRecv{x}{z}{\\mary}}\n })}\n\\]\nIn this interaction, we see that the channel \\tm{x} is used only as a way to\nconnect the various clients, John and Mary, to the store.\nThe \\emph{real} communication, sending the slice of cake and disappointment,\ntakes places on the channels \\tm{\\sliceofcake}, \\tm{\\nope}, \\tm{y} and \\tm{z}.\n\nInspired by this, we add two new constructs to the term language of \\cp: sending\nand receiving on a \\emph{shared} channel.\nThese actions are marked with a \\tm{\\star} in order to distinguish them\nsyntactically from ordinary sending and receiving.\nTo group clients, we add another form of parallel composition, which we refer to\nas \\emph{pooling}. \n\\input{def-nc-terms}\nAs before, round brackets denote input, square brackets denote output.\nNote that \\tm{\\ncCnt{x}{y}{P}}, much like \\tm{\\cpSend{x}{y}{P}{Q}}, is a bound\noutput---this means that both client creation and server interaction bind a new\nname.\n\nIn \\rcp, we terms are identified up to the commutativity and associativity of\nparallel composition. In \\nodcap, we add another form of parallel composition,\nand therefore must extend our structural congruence:\n\\input{def-nc-equiv}\nWe add axioms for the commutativity and associativity of pooling.\nWe do not add an axiom for \\ncEquivPoolAss2, as it follows from\n\\cref{def:nc-equiv}, see \\cref{thm:nc-pool-assoc2}.\nIt should be noted that \\tm{\\cpCut{x}{P}{Q}} is considered a single,\n\\emph{atomic} construct.\nTherefore you \\emph{cannot} use \\ncEquivPoolAss1 to rewrite\n\\tm{\\cpCut{x}{P}{\\ncPool{Q}{R}}} to \\tm{\\cpCut{x}{\\ncPool{P}{Q}}{R}}.\nWe do, however, add two axioms which relate cuts and pool.\nWe call these \\emph{extrusion}, because they closely resemble the\n\\textpi-calculus axiom for scope extrusion.\nWe add both \\ncRedKappaPool1 and \\ncRedKappaPool2, as these relate two different\nconstructs, and therefore we cannot use the one to derive the other. \n\\input{thm-nc-pool-assoc2}\nFurthermore, the extensions to structural congruence preserve symmetry.\n\\input{thm-nc-symmetry}\nWe can make another observation from our examples.\nIn every example in which a server interacts with a pool of clients, and which\ndoes not deadlock, there are \\emph{exactly} as many clients as there are\nserver interactions.\nTherefore, we add two new \\emph{dual} types for client pools and servers, which\ntrack how many clients or server interactions they represent.\n\\input{def-nc-types}\n\\input{def-nc-negation}\nWith these new types, duality remains an involutive function.\n\\input{thm-nc-negation-involutive}\n\n\\section{Typing clients and servers}\\label{sec:nc-typing-clients-and-servers}\nWe have to add typing rules to associate our new client and server\ninteractions with their types.\nThe definition for environments will remain unchanged, but we will extend the\ndefinition for the typing judgement.\nTo determine the new typing rules, we essentially have to answer the question\n``What typing constructs do we need to complete the following proof?''\n\\begin{prooftree}\n \\AXC{$\\seq[{ \\john }]{ \\Gamma, \\tmty{y}{\\cake^\\bot} }$}\n \\noLine\\UIC{$\\smash{\\vdots}\\vphantom{\\vdash}$}\n \\AXC{$\\seq[{ \\mary }]{ \\Delta, \\tmty{y'}{\\cake^\\bot} }$}\n \\noLine\\UIC{$\\smash{\\vdots}\\vphantom{\\vdash}$}\n \\AXC{$\\seq[{ \\store }]{ \\Theta, \\tmty{z}{\\cake}, \\tmty{z'}{\\cake} }$}\n \\noLine\\UIC{$\\smash{\\vdots}\\vphantom{\\vdash}$}\n \\noLine\\TIC{$\\seq[{\n \\cpCut{x}{\\ncPool{\\ncCnt{x}{y}{\\john}}{\\ncCnt{x}{y'}{\\mary}}}{\n \\ncSrv{x}{z}{\\ncSrv{x}{z'}{\\store}}} }]{\n \\Gamma, \\Delta, \\Theta }$}\n\\end{prooftree}\nIdeally, we would still like the composition of the client pool and the server\nto be a cut. This seems reasonable, as the left-hand side of the term above has\n2 clients, and the right-hand side has two server interactions, so \\tm{x} is\nused at type \\ty{\\take[2]{\\cake^\\bot}} on the left, and as \\ty{\\give[2]{\\cake}}\non the right.\n\\begin{prooftree}\n \\AXC{$\\seq[{ \\john }]{ \\Gamma, \\tmty{y}{\\cake^\\bot} }$}\n \\noLine\\UIC{$\\smash{\\vdots}\\vphantom{\\vdash}$}\n \\AXC{$\\seq[{ \\mary }]{ \\Delta, \\tmty{y'}{\\cake^\\bot} }$}\n \\noLine\\UIC{$\\smash{\\vdots}\\vphantom{\\vdash}$}\n \\noLine\\BIC{$\\seq[{ \\ncPool{\\ncCnt{x}{y}{\\john}}{\\ncCnt{x}{y'}{\\mary}} }]{\n \\Gamma, \\Delta, \\tmty{x}{\\take[2]{\\cake^\\bot}} }$}\n\n \\AXC{$\\seq[{ \\store }]{ \\Theta, \\tmty{z}{\\cake}, \\tmty{z'}{\\cake} }$}\n \\noLine\\UIC{$\\smash{\\vdots}\\vphantom{\\vdash}$}\n \\noLine\\UIC{$\\seq[{ \\ncSrv{x}{z}{\\ncSrv{x}{z'}{\\store}} }]{\n \\Theta, \\tmty{x}{\\give[2]{\\cake}} }$}\n\n \\NOM{Cut}\n \\BIC{$\\seq[{\n \\cpCut{x}{\\ncPool{\\ncCnt{x}{y}{\\john}}{\\ncCnt{x}{y'}{\\mary}}}{\n \\ncSrv{x}{z}{\\ncSrv{x}{z'}{\\store}}} }]{\n \\Gamma, \\Delta, \\Theta }$}\n\\end{prooftree}\nWe will define the typing judgement, and then discuss servers and clients, the\ntwo sides of the above cut, describe the rules we add, and show how they allow\nus to complete our proof.\n\\input{def-nc-typing-judgement}\n\\input{fig-nc-typing-judgement}\n\n\\subsection{Clients and pooling}\\label{sec:clients-and-pooling}\nA client pool represents a number of independent processes, each wanting to\ninteract with the server. Examples of such a pool include John and Mary from our\nexample, customers for online stores in general, and any number of processes\nwhich interact with a single, centralised server.\n\nWe introduce two new rules: one to construct clients, and one to pool them\ntogether. The first rule, $(\\take[1]{})$, marks interaction over some channel as\na client interaction. It does this by receiving a channel \\tm{y} over a\n\\emph{shared} channel \\tm{x}. The channel \\tm{y} is the channel across which the\nactual interaction will eventually take place.\nThe second rule, \\textsc{Pool}, allows us to pool together clients. This is\nimplemented, as in the \\textpi-calculus, using parallel composition.\n\\begin{center}\n \\ncInfTake1\n \\ncInfPool\n\\end{center}\nUsing these rules, we can derive the left-hand side of our proof by marking John\nand Mary as clients, and pooling them together.\n\\begin{prooftree}\n \\AXC{$\\seq[{ \\john }]{ \\Gamma, \\tmty{y}{\\cake^\\bot} }$}\n \\SYM{(\\take[1]{})}\n \\UIC{$\\seq[{ \\ncCnt{x}{y}{\\john} }]{ \\Gamma, \\tmty{z}{\\take[1]{\\cake^\\bot}} }$}\n\n \\AXC{$\\seq[{ \\mary }]{ \\Delta, \\tmty{y'}{\\cake^\\bot} }$}\n \\SYM{(\\take[1]{})}\n \\UIC{$\\seq[{ \\ncCnt{x}{y'}{\\mary} }]{ \\Delta, \\tmty{y'}{\\take[1]{\\cake^\\bot}} }$}\n\n \\NOM{Pool}\n \\BIC{$\\seq[{ \\ncPool{\\ncCnt{x}{y}{\\john}}{\\ncCnt{x}{y'}{\\mary}} }]{\n \\Gamma, \\Delta, \\tmty{x}{\\take[2]{\\cake^\\bot}} }$}\n\\end{prooftree}\n\n\\subsection{Servers and contraction}\\label{sec:servers-and-contraction}\nDual to a pool of clients is a server. Our interpretation of a server is a\nprocess which offers up some number of interdependent interactions of the same\ntype. Examples include the store from our example, which gives out slices of\ncake and disappointment, online stores in general, and any central server which\ninteracts with some number of client processes.\n\nWe introduce two new rules to construct servers. The first rule, $(\\give[1]{})$,\nmarks a interaction over some channel as a server interaction. It does this by\nsending a channel \\tm{y} over a \\emph{shared} channel \\tm{x}. The channel \\tm{y}\nis the channel across which the actual interaction will eventually take place.\nThe second rule, \\textsc{Cont}, short for contraction, allows us to contract\nseveral server interactions into a single server. This allows us to construct a\nserver which has multiple interactions of the same type, across the same shared\nchannel.\\footnote{%\n While it ultimately does not matter whether $(\\give[1]{})$ and $(\\take[1]{})$\n are implemented with a send or a receive action, it feels more natural to have\n the server do the sending.\n Clients indicate their interest in interacting with the server by connecting\n to the shared channel, but it is up to the server to decide \\emph{when} to\n interact with each channel.}\n\\begin{center}\n \\ncInfGive1\n \\ncInfCont\n\\end{center}\nUsing these rules, we can derive the right-hand side of our proof, by marking\neach of the store's interactions as server interactions, and then contracting\nthem.\n\\begin{prooftree}\n \\AXC{$\\seq[{ \\store }]{ \\Theta, \\tmty{z}{\\cake}, \\tmty{z'}{\\cake} }$}\n \\SYM{(\\give[1]{})}\n \\UIC{$\\seq[{ \\ncSrv{x'}{z'}{\\store} }]{\n \\Theta, \\tmty{z}{\\cake}, \\tmty{x'}{\\give[1]{\\cake}} }$}\n \\SYM{(\\give[1]{})}\n \\UIC{$\\seq[{ \\ncSrv{x}{z}{\\ncSrv{x'}{z'}{\\store}} }]{\n \\Theta, \\tmty{x}{\\give[1]{\\cake}}, \\tmty{x'}{\\give[1]{\\cake}} }$}\n \\NOM{Cont}\n \\UIC{$\\seq[{ \\ncSrv{x}{z}{\\ncSrv{x}{z'}{\\store}} }]{\n \\Theta, \\tmty{x}{\\give[2]{\\cake}} }$}\n\\end{prooftree}\nThus, we complete the typing derivation of our example.\n\n\\section{Running clients and servers}\\label{sec:nc-running-clients-and-servers}\nOnce we have a client/server interaction, how do we run it? Ideally, we would\nsimply use the reduction rule closest to the one used in the \\textpi-calculus. \n\\[\n \\reducesto\n {\\tm{\\cpCut{x}{\\ncCnt{x}{y}{P}}{\\ncSrv{x}{z}{R}}}}\n {\\tm{\\cpCut{y}{P}{\\cpSub{y}{z}{R}}}}\n\\]\nHowever, our case is complicated by the fact that in \\tm{\\cpCut{x}{P}{Q}} the\nname restriction is an inseparable part of the composition, and therefore has to\nbe part of our reduction rule. \nBecause of this, the above reduction can only apply in the singleton case.\nIf the client pool contains more than one client, such as in the term below,\nthen there is no way to isolate a single client together with the server,\nbecause \\tm{x} occurs in both \\tm{\\ncCnt{x}{y}{P}} and \\tm{\\ncCnt{x}{z}{Q}}.\n\\[\n \\tm{\\cpCut{x}{\\ncPool{\\ncCnt{x}{y}{P}}{\\ncCnt{x}{z}{Q}}}{\\ncSrv{x}{w}{R}}}\n \\centernot\\Longrightarrow\n\\]\nTherefore, we add a second reduction rule, which handles communication between a\none client in a pool of multiple clients and a server.\n\\[\n \\reducesto\n {\\tm{\\cpCut{x}{\\ncPool{\\ncCnt{x}{y}{P}}{Q}}{\\ncSrv{x}{z}{R}}}}\n {\\tm{\\cpCut{x}{Q}{\\cpCut{y}{P}{\\cpSub{y}{z}{R}}}}}\n\\]\nLastly, because we have added another form of parallel composition, we add\nanother congruence rule, to allow for reduction inside client pools.\n\\input{def-nc-term-reduction}\n\nThe rules \\ncRedBetaStar1 and \\ncRedBetaStar{n+1} seem like the elimination\nrules for a list-like construct. This may come as a surprise, as our client\npools are built up like binary trees, and the typing rules for both sides are\ntree-like, with $(\\take[1]{})$ and $(\\give[1]{})$ playing the role of leaves,\nand \\textsc{Pool} and \\textsc{Cont} merging two trees with $m$ and $n$ leaves\ninto one with $m+n$ leaves.\nHowever, the server process imposes a sequential ordering on its interactions,\nand it is because of this that we have to use list-like elimination rules.\n\nSo where does the non-determinism in \\nodcap come from? Let us say we have a\nterm of the following form:\n\\[\n \\tm{\n \\cpCut{x}\n {\\ncPool{\\ncCnt{x}{y_1}{P_1}}{\\dots \\mid \\ncCnt{x}{y_n}{P_n}}}\n {\\ncSrv{x}{y}{Q}}\n }\n\\]\nBecause pooling is commutative and associative, we can rewrite this term to\nbring any client in the pool to the front, before applying \\ncRedBetaStar{n+1}.\nThus, like in the \\textpi-calculus, the non-determinism is introduced by the\nstructural congruence.\n\nDoes this mean that, for an arbitrary client pool \\tm{P} in\n\\tm{\\cpCut{x}{P}{\\ncSrv{x}{z}{Q}}}, every client in that pool is competing for\nthe server interaction on \\tm{x}?\nNot necessarily, as some portion of the clients can be blocked on an external\ncommunication. For instance, in the term below, clients\n$\\tm{\\ncCnt{x}{y_{n+1}}{P_{n+1}}} \\dots \\tm{\\ncCnt{x}{y_m}{P_m}}$ are blocked\non a communication on the external channel \\tm{a}.\n\\[\n \\arraycolsep=0pt\n \\tm{\n \\begin{array}{lrl}\n \\nu x.& ((&\\; \\ncPool{\\ncCnt{x}{y_1}{P_1}}{\\dots\\mid\\ncCnt{x}{y_n}{P_n}}\\\\\n &\\mid&\\; \\cpWait{a}{\\ncPool{\\ncCnt{x}{y_{n+1}}{P_{n+1}}}{\\dots \\mid \\ncCnt{x}{y_m}{P_m}}}\\;)\\\\\n &\\mid&\\; \\ncSrv{x}{y_1}{\\dots\\ncSrv{x}{y_m}{Q}}\\;)\n \\end{array}}\n\\]\nIf we reduce this term, then only the clients\n$\\tm{\\ncCnt{x}{y_1}{P_1}} \\dots \\tm{\\ncCnt{x}{y_n}{P_n}}$\nwill be assigned server interactions, and we end up with the following canonical\nform term. \n\\[\n \\arraycolsep=0pt\n \\tm{\n \\begin{array}{lrl}\n \\nu x.& (&\\; \\cpWait{a}{\\ncPool{\\ncCnt{x}{y_{n+1}}{P_{n+1}}}{\\dots\\mid\\ncCnt{x}{y_m}{P_m}}}\\\\\n &\\mid&\\; \\ncSrv{x}{y_{n+1}}{\\dots\\ncSrv{x}{y_m}{Q}}\\;)\n \\end{array}}\n\\]\nThis matches the reduction behaviour of the \\textpi-calculus, and it fits with\nout notion of computation with processes.\n\n\\section{Properties of \\nodcap}\\label{sec:nc-properties}\nIn this section, we will revisit the proofs for three important properties of\n\\rcp, namely preservation, progress, and termination, and show that our\nextensions preserve these properties.\n\n\\subsection{Preservation}\nPreservation is the fact that term reduction preserves typing. There are two\nproofs involved in this. First, we show that structural congruence preserves\ntyping.\n\\input{thm-nc-preservation-equiv}\n\\input{fig-nc-preservation-equiv}\nSecondly, we prove that term reduction preserves typing.\n\\input{thm-nc-preservation}\n\\input{fig-nc-preservation-1}\n\n\\subsection{Canonical forms and progress}\nIn this section, we will extend the definition of canonical forms and the proof\nof progress progress given in \\cref{sec:cppi}.\n\n\\subsubsection{Canonical forms}\nFirst, we extend the definitions of actions with our actions for client and\nserver creation. \n\\input{def-nc-action}\nSecondly, as we can reduce inside client pools, we will add pooling to our\ndefinition of evaluation prefixes.\n\\input{def-nc-evaluation-prefixes}\nThe definition for the maximum evaluation prefix is unchanged.\n\nThere are some subtleties to our definition of canonical forms. The type system\nfor \\cp guarantees that all links directly under an evaluation context act on a\nbound channel. Not so for \\nodcap.\n\\begin{scprooftree}\n \\AXC{}\n \\NOM{Ax}\n \\UIC{$\\seq[{ \\cpLink{x}{y} }]{ \\tmty{x}{\\take[m]{A}}, \\tmty{y}{\\give[m]{A^\\bot}} }$}\n \\AXC{$\\seq[{ P }]{ \\Gamma, \\tmty{x}{\\take[n]{A}} }$}\n \\NOM{Pool}\n \\BIC{$\\seq[{ \\ncPool{\\cpLink{x}{y}}{P} }]{ \\Gamma, \\tmty{x}{\\take[m+n]{A}}, \\tmty{y}{\\give[m]{A^\\bot}} }$}\n\\end{scprooftree}\nThere is no way to sensibly reduce this link.\nFurthermore, in \\cp, if two processes act on the same channel, then they must be\non different sides of the cut introducing that channel. The addition of shared\nchannels and client pools invalidates this property.\nTherefore, we will have to be more careful about the way we define canonical\nforms.\nWe restate the definition of canonical forms below. The additions have been\nitalicised.\n\\input{def-nc-canonical-forms}\n\n\\subsection{Evaluation contexts}\nEvaluation contexts are one-holed term contexts under which reduction can take\nplace. Since we have added another congruence rule, stating that reduction can\ntake place inside client pools, we extend our definition of evaluation contexts\nto match this.\n\\input{def-nc-evaluation-contexts}\nWe also restate \\cref{thm:cp-display-cut-1}, and prove that our extension\npreserves the property.\n\\input{thm-nc-display-cut-1}\nFurthermore, it will be useful to prove a similar lemma, which shows that we can\npush any pooling downwards under an evaluation context.\n\\input{thm-nc-display-pool-1}\n\n\\subsection{Progress}\nProgress is the fact that every term is either in some canonical form, or can be\nreduced further.\nFirst, we will restate \\cref{thm:cp-progress-link} and \\cref{thm:cp-progress-beta},\nwhich relate evaluation prefixes and evaluation contexts, and show that our\nextension preserves these properties. \n\\input{thm-nc-progress-link}\n\\input{thm-nc-progress-beta}\nIn essence, \\cref{thm:nc-progress-link} and \\cref{thm:nc-progress-beta} cover\nthe cases in which either \\cpRedAxCut1 or a \\textbeta-reduction rule will be\napplied.\nHowever, after applying \\cref{thm:nc-progress-beta}, we cannot immediately apply\n\\ncRedBetaStar{n+1}. For that, we must uncover at least one layer of pooling.\nWe prove a lemma which states that if we have an interaction on a shared channel\n\\tm{x}, we can push all pooling rules which pool clients communicating on \\tm{x}\ninwards. \n\\input{thm-nc-progress-shared}\nFinally, we are ready to extend our proof of progress. The overall structure of\nthe proof remains the same, though the addition of pooling makes the wording\nslightly more subtle.\n\\input{thm-nc-progress}\n\n\\subsection{Termination}\nTermination is the fact that if we iteratively apply progress to obtain a\nreduction, and apply that reduction, we will eventually end up with a term in\ncanonical form.\nWe restate its proof here for the sake of completeness, but its wording is\nunchanged, modulo references to figures.\n\\input{thm-nc-termination}\n\n\\section{\\nodcap and non-deterministic local choice}\\label{sec:nc-local-choice}\nIn \\cref{sec:local-choice}, we discussed the non-deterministic local choice\noperator, which is used in several extensions of \\piDILL and\n\\cp~\\parencite{atkey2016,caires2014,caires2017}.\nThis operator is admissible in \\nodcap.\nWe can derive the non-deterministic choice \\tm{P+Q} by constructing the\nfollowing term:\n\\[\n \\arraycolsep=0pt\n \\tm{\n \\begin{array}{lrlrl}\n \\nu x.&(( & \\; \\ncCnt{x}{y}{\\cpInr{y}{\\cpHalt{y}}} \\\\\n &\\mid& \\; \\ncCnt{x}{z}{\\cpInr{z}{\\cpHalt{z}}} \\; )\\\\\n &\\mid& \\; \\ncSrv{x}{y}{\\ncSrv{x}{z}{}}\\text{case}\\;y\\;\n &\\;\\{& \\; \\cpWait{y}{\\cpCase{y}{\\cpWait{y}{P}}{\\cpWait{y}{P}}}\\\\\n &&& ;& \\; \\cpWait{y}{\\cpCase{y}{\\cpWait{y}{Q}}{\\cpWait{y}{Q}}}\\;\\}\\;)\n \\end{array}\n }\n\\]\nThe term is a cut between two processes.\nBoth sides are well-typed, see~\\cref{fig:nc-local-choice}.\n\\input{fig-nc-local-choice}\nLet us unpack what each side is doing. \nOn the left-hand side, we have a pool of two processes,\n\\tm{\\ncCnt{x}{y}{\\cpInr{y}{\\cpHalt{y}}}} and \\tm{\\ncCnt{x}{z}{\\cpInr{z}{\\cpHalt{z}}}}.\nEach makes a choice---the first sends \\tm{\\text{inl}}, the second sends \\tm{\\text{inr}}.\nOn the right-hand side, we have a server with both \\tm{P} and \\tm{Q}.\nThis server has two channels on which a choice is offered, \\tm{y} and\n\\tm{z}. However, the choice on \\tm{z} does not affect the outcome of the\nprocess.\nWhen these clients and the server are put together, the choices offered by the\nserver will be non-deterministically lined up with the clients which make\nchoices, and either \\tm{P} or \\tm{Q} will run.\n\nWhile there is a certain amount of overhead involved in this encoding, it scales\nlinearly in terms of the number of processes.\nThe reverse---encoding the non-determinism present in \\nodcap using\nnon-deterministic local choice---scales exponentially, as with the \\textpi-calculus.\n\nNonetheless, it is worrying that we duplicate each program \\tm{P} and \\tm{Q} in\norder to encode non-deterministic local choice.\nHowever, we can replace the each term of the form \\tm{\\cpCase{y}{\\cpWait{y}{P}}{\\cpWait{y}{P}}}\nwith \\tm{\\cpCut{w}{\\cpCase{z}{\\cpWait{z}{\\cpHalt{w}}}{\\cpWait{z}{\\cpHalt{w}}}}{\\cpWait{w}{P}}}.\nThis process is also well-typed, see below.\n\\begin{scprooftree}\n \\AXC{}\n \\SYM{(\\one)}\n \\UIC{$\\seq[{ \\cpHalt{w} }]{ \\tmty{w}{\\one} }$}\n \\SYM{(\\bot)}\n \\UIC{$\\seq[{ \\cpWait{z}{\\cpHalt{w}} }]{ \\tmty{w}{\\one}, \\tmty{z}{\\bot} }$} \n \\AXC{}\n \\SYM{(\\one)}\n \\UIC{$\\seq[{ \\cpHalt{w} }]{ \\tmty{w}{\\one} }$}\n \\SYM{(\\bot)}\n \\UIC{$\\seq[{ \\cpWait{z}{\\cpHalt{w}} }]{ \\tmty{w}{\\one}, \\tmty{z}{\\bot} }$} \n \\SYM{(\\with)}\n \\BIC{$\\seq[{\n \\cpCase{z}{\\cpWait{z}{\\cpHalt{w}}}{\\cpWait{z}{\\cpHalt{w}}}\n }]{ \\tmty{w}{\\one}, \\tmty{z}{\\bot\\with\\bot} }$} \n \\AXC{$\\seq[{ P }]{ \\Gamma }$}\n \\SYM{(\\bot)} \n \\UIC{$\\seq[{ \\cpWait{w}{P} }]{ \\Gamma, \\tmty{w}{\\bot} }$}\n \\NOM{Cut}\n \\BIC{$\\seq[{\n \\cpCut{w}{\\cpCase{z}{\\cpWait{z}{\\cpHalt{w}}}{\\cpWait{z}{\\cpHalt{w}}}}{\\cpWait{w}{P}}\n }]{ \\Gamma, \\tmty{z}{\\bot\\with\\bot} }$} \n\\end{scprooftree}\n%%% Local Variables:\n%%% TeX-master: \"main\"\n%%% End:\n", "meta": {"hexsha": "baa39fbb5f4bccb315e43ab527105d409e956ce2", "size": 23932, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/thesis/cpnd.tex", "max_stars_repo_name": "wenkokke/nodcap", "max_stars_repo_head_hexsha": "fb5e78d6182276e4d93c4c0e0d563b6b027bc5c2", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2018-09-05T08:58:11.000Z", "max_stars_repo_stars_event_max_datetime": "2019-09-24T20:16:35.000Z", "max_issues_repo_path": "doc/thesis/cpnd.tex", "max_issues_repo_name": "pepijnkokke/nodcap", "max_issues_repo_head_hexsha": "fb5e78d6182276e4d93c4c0e0d563b6b027bc5c2", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/thesis/cpnd.tex", "max_forks_repo_name": "pepijnkokke/nodcap", "max_forks_repo_head_hexsha": "fb5e78d6182276e4d93c4c0e0d563b6b027bc5c2", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-09-05T08:58:13.000Z", "max_forks_repo_forks_event_max_datetime": "2018-09-05T08:58:13.000Z", "avg_line_length": 46.7421875, "max_line_length": 108, "alphanum_fraction": 0.7040364366, "num_tokens": 7528, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548782017745, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.346005816693504}} {"text": "% ==============================================================================\n%\n% T H E O R E T I C A L B A C K G R O U N D\n%\n% ==============================================================================\n\\chapter{Theoretical Background} % ------------------------------------------- %\n\\label{ch:theory}\n% ---------------------------------------------------------------------------- %\n% ==============================================================================\n%\n% O V E R V I E W\n%\n% ==============================================================================\n\nBefore delving into the design process, some aspects of the underlying\ntheory should be understood. Whilte this chapter is not intended to be a\ncomprehensive treatise on the subject, it shall serve as a short refresher. At\nits end, the reader should have sufficient insight to understand the basic\nmotivation of our project from a theoretical point of view.\n\n% ==============================================================================\n%\n% D S P C H A I N\n%\n% ==============================================================================\n\\section{The Digital Signal Processing Chain}% <<< --------------------------- %\n\\label{sec:dsp_chain}\n% ---------------------------------------------------------------------------- %\n\nDigitally acquiring a signal generally requires at least the following steps:\n\\begin{itemize}\\tightlist\n \\item\n Passing the signal through an analog low-pass filter.\n \\item\n Sampling and quantizing the filtered signal.\n\\end{itemize}\n\nThe resulting sequence of values can then be further digitally\nprocessed. The necessary building blocks for this process are portrayed in\nFigure~\\ref{fig:dspChain:blocks}.\n\n\\begin{figure}\n \\centering\n \\input{images/dspChain/dspChain.tikz}\n \\caption[The DSP Chain]{%\n The basic building blocks of the DSP chain from its analog\n input to its digitally processed output. From left to right: The\n analog low-pass filter (\\emph{LP}), the analog-to-digital converter\n (\\emph{ADC}), and an arbitrary digital signal processing system for\n further processing of the ADC's output (\\emph{DSP}).%\n }\n \\label{fig:dspChain:blocks}\n\\end{figure}\n\nOf particular interest for our application is what happens in the ADC. The\nquantization process converts a value-continuous signal into a value-discreet\none, with its resolution being a specification of the ADC. As an example,\nthe ADC in our system has a resolution of \\num{14}\\,bits, meaning it\ncan divide its valid input range into \\num{16384} values. Given an input\nrange of \\SI{2}{\\volt_\\mathrm{PP}}, this equates to a resolution of roughly\n\\SI{122}{\\micro\\volt} (in theory). This quantization process is the source of\nwhat is generally known as \\emph{quantization noise}.\n\nBesides the quantization, the other step happening in the ADC is sampling; a\ntime-continuous signal is converted into a series of time-discreet values. The\ntime between those values is known as \\emph{sampling time}, its inverse is the\n\\emph{sampling frequency}. Note that usually these are constant, at least\nduring the time where the signal is measured. This need not strictly be the\ncase in theory though. In our system, this sampling frequency is a fixed\nproperty of the ADC, and is \\SI{125}{\\mega\\hertz}.\n\nThe sampling step lies at the core of the problem our project intends to\naddress: \\emph{aliasing}. Therefore, we will take a closer look at a few\nconsequences of the sampling process, and how they are relevant to this\nproject.\n\nDescriptively, the sampling process can be thought of as looking at a\nsignal at specific points in time and capturing its value at those\npoints. Mathematically, this amounts to multiplying the signal with a series\nof Dirac pulses in the time domain, and convolving with a series of Dirac\npulses in the frequency domain\\footnote{%\n \\emph{Pro memoria}: A series of Dirac pulses in the time domain has as its\n spectrum a series of Dirac pulses as well.%\n}.\nThis convolution in the frequency domain lies at the heart of the problem of\naliasing, because it results in the incoming signal's spectrum being repeated\nat intervals of $f_s$ (see stage 3 in Figure~\\ref{fig:dspChain:signals}). This\nis no problem as long as the spectrum of the incoming signal fits within the\nboundaries set by this repetition. But if the spectrum of the incoming signal\nis too broad, two or more recurrences of the spectrum will overlap. This\neffect is highlighted in Figure~\\ref{fig:aliasing:band}.\n\n\\begin{figure}[p]\n \\centering\n \\input{images/dspChain/timeDomain.tikz}\n \\input{images/dspChain/freqDomain.tikz}\n \\caption[Signals Passing Through the DSP Chain (Simplified)]{%\n Simplified time-domain (top) and frequency-domain (bottom) view\n of the signal at different stages on its way through the DSP\n chain. The circled numbers correspond to the stages as outlined\n in Figure~\\ref{fig:dspChain:blocks}. Stage 1 is the signal before\n passing through the input low-pass filter, with a significant amount\n of high-frequency noise. The low-pass filter removes any frequency\n components above ${f_s}/{2}$ in an ideal scenario (in reality, it\n merely attenuates them, as we will see later), resulting in the signal\n at stage 2.\\protect\\newline\n After having been filtered, the ADC samples and quantizes the\n signal, yielding a sequence of values, schematically portrayed in the\n rightmost picture for stage 3. Note that due to the sampling process,\n the spectrum of the filtered signal is repeated at intervals of\n $f_s$. This is the source of the issue of \\emph{aliasing}.%\n }\n \\label{fig:dspChain:signals}\n\\end{figure}\n\\begin{figure}[p]\n \\centering\n \\input{images/aliasing/band.tikz}\n \\caption[Aliasing Illustrated via Signal Frequency Band]{%\n Simplified view of a signal which does not produce aliasing between\n its recurrences in the frequency spectrum (top), contrasted with a\n signal whose frequency band has components above half the sampling\n frequency, resulting in aliasing; its spectral copies overlap\n (highlighted areas in the bottom plot).%\n }\n \\label{fig:aliasing:band}\n\\end{figure}\n\nThis overlap results in two primary problems:\n\\begin{itemize}\\tightlist\n \\item\n The digital signal may not be unambiguously reconstructable into an\n analog signal, if that is intended.\n \\item\n Frequencies may occur in the digital signal stream which are not\n actually present in the original signal. This problem is often\n referred to as the \\emph{folding back} of frequency components. See\n Figure~\\ref{fig:aliasing:dirac} for an illustration of how this might\n look. This problem is of particular interest to our application, as\n we will see later.\n\\end{itemize}\n\n\\begin{figure}\n \\centering\n \\input{images/aliasing/dirac.tikz}\n \\caption[Aliasing With Harmonic Signals]{%\n Example of two harmonic signals being sampled. In the top plot, the\n signal's frequency is below half the sampling frequency and there is\n no aliasing. The signal can be reconstructed without error. In the\n bottom plot, the signal's frequency is above half the sampling\n frequency. Consequently, the copies of the signal's frequency spectrum\n centered around the sampling frequency and its negative alias back\n into the band between $-f_s/2$ and $f_s/2$. If this signal is\n reconstructed, the resulting signal would have a frequency of $f_s -\n f_{\\mathrm{sig}}$ instead of $f_{\\mathrm{sig}}$.%\n }\n \\label{fig:aliasing:dirac}\n\\end{figure}\n\n\nOnce a signal has left the ADC and is handed down the DSP chain for further\nprocessing, the primary problem becomes one of resources, particularly in\nreal-time applications. In most systems, the available hardware is a fixed\nconstraint and depending on what sort of processing is to be conducted on the\ndigital data stream, the available resources may or may not suffice. If\navailable resources are found to be insufficient for real-time processing of\nthe data stream, one may choose to\n\\begin{itemize}\\tightlist\n \\item\n not process the data in real time,\n \\item\n reduce the complexity of the computations, or\n \\item\n reduce the amount of data to be processed through \\emph{downsampling}\n of the signal.\n\\end{itemize}\nThe last case is the route which is chosen in our application. The main\nconstraint on the STEMlab is that the data being generated cannot be moved off\nthe device in real time. Therefore the amount of data must be reduced before\nit can be moved off the device to a computer for viewing or further processing\n(see Section~\\ref{sec:requirements}).\n\nBecause downsampling a signal is in essence nothing more than the sampling of\na signal which has already been sampled, a lot of the considerations\nwhich are valid for the step from an analog to a digital signal as\noutlined above are either very similar or even identical. Specifically, the\nsame considerations for aliasing still apply: If the signal which is to\nbe downsampled has frequency components above $f_\\mathrm{s,downsampled}/2$,\naliasing will occur. And since the signal coming out of the ADC has the analog\nsignal's spectrum (filtered by the analog lowpass before the ADC) recurring at\nintervals of the ADC's sampling frequency, this is always the case.\n\nTherefore, the sampled signal must be filtered through a low-pass filter\nbefore being downsampled, just as the original analog signal was low-pass\nfiltered before being passed into the ADC. In light of the signal to be\ndownsampled being a \\emph{digital} signal instead of an analog one, that\nlow-pass filter must naturally be a digital filter as well. Designing such\ndigital low-pass filters is the core mission of this project.\n\nThe key properties of such filters which are relevant to our application are\nits transition band width (filter steepness) and its aliasing attenuation.\nThe aliasing attenuation refers to the fact that when a filter is being used\nfor downsampling, copies of its frequency response will be created at\nintervals of the lower sampling rate (analogous to the sampling process\nproducing spectral copies of a signal when sampling an analog signal).\n\nThe stopband components of these copies overlap with the intended passband,\nleading to aliasing (it should be noted that this phenomenon is also present\nin the case of the analog input filter for the DSP chain at the ADC's\noutput). This effect is portrayed in Figure~\\ref{fig:aliasing:iirCopies}. The\ntop plot shows the filter's frequency response along with four copies to\nillustrate the overlap effect. The bottom plot shows the aliasing effect more\nclearly by removing the spectral copies and retaining the aliased components.\n\nThe overlapping parts of the spectrum are composed of spectral copies both to\nthe right and left side of the original. Therefore, the aliased regions are\nalternately flipped around the vertical axis. This creates in essence the same\neffect as if the paper were folded along multiples of the lower sampling\nrate over the frequency range of the central copy (in the case of our\nexample: $0.2f_s$, $0.4f_s$, $0.6f_s$ and $0.8f_s$) like an accordion. This is\nwhere the term \\emph{folding back} originates.\n\n\\begin{figure}\n \\centering\n \\input{images/aliasing/iirCopies.tikz}\n \\caption[Folding Back of Stopband Components Into Passband]{%\n The phenomenon of folding back when downsampling, illustrated for a\n lowpass IIR filter with a cutoff frequency of $0.2\\cdot f_s$ for a\n downsampling ratio of $R=5$. The downsampling process produces copies\n of the filter's frequency response at intervals of the lower sampling\n frequency, visible in the top plot. The stopbands of these copies\n then overlap with the intended passband. The bottom plot shows a\n close-up view with the spectral copies for clarity.%\n }\n \\label{fig:aliasing:iirCopies}\n\\end{figure}\n\n%>>>\n% ==============================================================================\n%\n% D I G I T A L F I L T E R S\n%\n% ==============================================================================\n\\section{Digital Filters}% <<< ----------------------------------------------- %\n\\label{sec:digital_filters}\n% ---------------------------------------------------------------------------- %\n\nAhead of studying any potential details of digital filters, it is helpful to\nhave an overview of the major types of filters and some of their key\nproperties. Digital filters can be distinguished by several characteristics;\ncommon ways to categorize them are by topology, impulse response and their\nfrequency response. There are two to three commonly used types of digital\nfilters, depending on how they are grouped: Infinite impulse response (IIR)\nfilters and finite impulse response (FIR) filters. A third group, cascaded\nintegrator-comb (CIC) filters, is variously thought of as a special type of\nFIR filter (similarity of its transfer functions), a subgroup of IIR filtersk\n(it contains feedback and feedforward paths) or a new type of filter\naltogether, depending on which source one chooses to consult. While our\nsystem uses FIR and CIC filters, a brief overview of IIR filters is still\npresented here, for the sake of completeness and some context.\n\n\n% ==============================================================================\n%\n% I I R F I L T E R S\n%\n% ==============================================================================\n\\subsection{IIR Filters} %<<< ------------------------------------------------ %\n\\label{subsec:iir_filters}\n% ---------------------------------------------------------------------------- %\n\nInfinite impulse response filters are so named because their impulse response\ncontinues into perpetuity, never reaching zero. In practice, the response\nusually comes sufficiently close to zero at a certain point that it can be\nconsidered zero for most intents and purposes.\n\nIIR filters have feedback paths, resulting in a filter response equation\nwith non-zero denominator components. Their basic building blocks are\ndelay elements, multipliers and adders, as seen in the example of\nFigure~\\ref{fig:filtertopologies:iir}. Their transfer functions have the form\n\\begin{equation}\n \\label{eq:iir_filter}\n H(z) = \\frac{%\n \\sum_{k=0}^N b_k \\cdot z^{-k}}{%\n 1 + \\sum_{i=0}^M a_i \\cdot z^{-i}}\n\\end{equation}\n\nIIR filters generally require a lower order (and therefore fewer resources) to\napproximate a certain frequency response specification than FIR filters do\n(particularly the constraint of a narrow transition band), but this comes at a\ncost: IIR filters have a non-linear phase response; linear-phase responses\ncan only be approximated. Furthermore, IIR filters are not guaranteed to be\nBIBO stable due to their feedback paths.\n\n\\begin{figure}\n \\centering\n \\input{images/filtertopologies/iir.tikz}\n \\caption[IIR Filter: Biquad]{Example of an IIR filter topology for a biquad}\n \\label{fig:filtertopologies:iir}\n\\end{figure}\n\nSome of the generally used types of IIR filters are:\n\\begin{itemize}\\tightlist\n \\item\n Butterworth filter: Named after the British engineer and physicist\n Stephen Butterworth (1885 -- 1958), who first described it in\n 1930. Characterized by a very flat passband (no passband ripple).\n \\item\n Chebyshev filter (type I and II): Named after Russian mathematician\n Pafnuty Chebyshev (1821 -- 1894). They are steeper than Butterworth\n filters, at the cost of suffering from ripple in the passband (type I)\n or stopband (type II).\n \\item\n Bessel filter: Named for the German mathematician Friedrich Bessel\n (1784 -- 1846). Optimized to have a maximally linear phase response in\n order to minimize the distortion of signals passing through the\n filter.\n \\item\n Elliptical filters: Also known as Cauer filters, after the German\n mathematician Wilhelm Cauer (1900 -- 1945), or Zolotarev filter, after\n Russian mathematician Yegor Zolotarev (1847 -- 1878). Characterized by\n equiripple in the bassband and stopband and a very narrow transition\n band compared to other filters of the same order.\n\\end{itemize}\n\n%>>>\n% ==============================================================================\n%\n% F I R F I L T E R S\n%\n% ==============================================================================\n\\subsection{FIR Filters} %<<< ------------------------------------------------ %\n\\label{subsec:FIR_filters}\n% ---------------------------------------------------------------------------- %\n\nFIR filters are characterized by an impulse response which decays to zero in\nfinite time (see Figure~\\ref{fig:filter_specs:coefs}, unlike IIR filters. The\nfilter response is characterized by Equation~\\ref{eq:fir_filter}:\n\\begin{equation}\n \\label{eq:fir_filter}\n H(z) = \\sum_{k=0}^{N} b_k \\cdot z^{-k}\n\\end{equation}\n\nFigure~\\ref{fig:filtertopologies:fir} shows one possible topology for\nimplementing a FIR filter, the so-called direct form. As can be seen, the\nbasic building blocks of a FIR filter are delay elements, multipliers and\nadders, same as for IIR filters.\n\n\\begin{figure}[b]\n \\centering\n \\input{images/filtertopologies/fir.tikz}\n \\caption[FIR Filter Topology Example]\n {One possible topology for a FIR filter (direct form)}\n \\label{fig:filtertopologies:fir}\n\\end{figure}\n\nFIR filters have several advantages:\n\\begin{itemize}\\tightlist\n \\item\n They are inherently BIBO stable because they lack feedback paths.\n \\item\n They can be easily designed to have a linear phase response,\n preventing signal distortion due to different group delays for signal\n components of different frequencies.\n \\item\n The shape of their frequency response can be very finely tuned. This\n makes them ideally suited for certain purposes, such as compensation\n filters (see Section~\\ref{subsec:CIC_filters}).\n \\item\n Implementation is usually rather straightforward.\n\\end{itemize}\n\nTheir main disadvantage is that due to the lack of feedback, they\ngenerally require comparatively high filter orders for narrow transition band\nwidths. Illustratively, this can be understood by the following considerations:\n\\begin{itemize}\\tightlist\n \\item\n The frequency response of an ideal low-pass filter is a rectangle.\n \\item\n The inverse Fourier transform of a rectangle is an infinitely long\n $sinc$ function.\n \\item\n Therefore, the impulse response of the ideal brick wall filter\n (rectangular) would have an infinite number of taps.\n \\item\n Truncation of the number of taps leads to a deviation of the filter's\n frequency response from the brick wall filter. As the number of taps\n (and therefore the FIR filter's impulse response) is reduced, its\n frequency response deviates more and more from the brick wall filter,\n resulting in a flatter transition between the passband and the\n stopband as well as the introduction of ripple.\n\\end{itemize}\nThis process is illustrated in simplified form in\nFigure~\\ref{fig:brick_wall_vs_FIR}. The FIR filter's transition band width is\nparticularly important for our application in order to reduce aliasing\neffects, as will be shown later.\n\n\\begin{figure}\n \\centering\n \\input{images/brickwallVsFIR/bwVsFIR.tikz}\n \\caption[Brick Wall Filter vs. FIR Filter (simplified)]{%\n The effect of truncating a $sinc$ function in the time domain on its\n spectrum (simplified)%\n }\n \\label{fig:brick_wall_vs_FIR}\n\\end{figure}\n\nDesigning FIR filters is usually performed by specifying\ncertain desired characteristics of the filter's frequency\nresponse. Figure~\\ref{fig:filter_specs:freqResponse} shows one possible way of\ndoing this for FIR filters by specifying four parameters:\n\\begin{itemize}\\tightlist\n \\item\n pass band ripple: $A_\\mathrm{P}$\n \\item\n stop band attenuation: $A_\\mathrm{St}$\n \\item\n pass band edge frequency: $F_\\mathrm{P}$\n \\item\n stop band edge frequency: $F_\\mathrm{St}$\n\\end{itemize}\nThe resulting transition band width $F_\\mathrm{Tb}$ is the difference between\nthe passband edge frequency and the stopband edge frequency, and serves as a\nuseful indicator of how many coefficients (i.e. resources) the filter will end\nup using. Narrower transition bands tend to require a higher filter order, and\ntherefore more resources. Coefficient counts of several hundred are not\nuncommon for steep FIR filters. Other sets of parameters can be used to\ndesign filters, but these are the ones used in this project, therefore the\nemphasis on them.\n\nFigure~\\ref{fig:filter_specs:coefs} shows the resulting impulse response\n(coefficient set) for a FIR filter designed by using the four above\nmentioned parameters, with values given by Equations~\\ref{eq:filter_specs:ap}\nthrough \\ref{eq:filter_specs:fst} handed to one of Matlab's FIR filter design\nalgorithms.\n\\begin{align}\n A_\\mathrm{P} &= \\SI{2}{\\dB} \\label{eq:filter_specs:ap}\\\\\n A_\\mathrm{St} &= \\SI{60}{\\dB} \\label{eq:filter_specs:ast}\\\\\n F_\\mathrm{P} &= 0.3 \\cdot f_s \\label{eq:filter_specs:fp}\\\\\n F_\\mathrm{St} &= 0.4 \\cdot f_s \\label{eq:filter_specs:fst}\n\\end{align}\n\n\\begin{figure}\n \\centering\n \\input{images/filterSpecs/freqResponse.tikz}\n \\caption[Specifying FIR Filter Constraints]{\n Specifications in the frequency domain and the resulting filter's\n frequency response as designed by Matlab.%\n }\n \\label{fig:filter_specs:freqResponse}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\input{images/filterSpecs/coefs.tikz}\n \\caption[Impulse Response of a FIR Filter]{\n Impulse response (coefficients) for the filter from\n Figure~\\ref{fig:filter_specs:freqResponse} with the parameters\n as given by Equations~\\ref{eq:filter_specs:ap} through\n \\ref{eq:filter_specs:fst} passed to one of Matlab's FIR filter design\n algorithms, resulting in a set of \\num{39} coefficients. Note that\n the coefficients to the left and right of these values are zero, hence\n \\emph{finite} impulse response filters.%\n }\n \\label{fig:filter_specs:coefs}\n\\end{figure}\n\nOne particular form of a FIR filter is the so-called half-band\nfilter. Half-band filters are used for downsampling by a ratio of\n$R=2$. They are characterized by a point-symmetric frequency response\nacross the $(f_s/4,0.5)$ point. Their advantage lies in the efficiency of\ntheir coefficient structure: Each second coefficient is zero, and all the\nnon-zero coefficient are symmetrical around the center of the impulse\nresponse. For higher downsampling rates, multiple half-band filters can\nbe cascaded. Figure~\\ref{fig:fir:halfband_linear_example} shows the amplitude\nfrequency response and the coefficient set of an example filter.\n\n\\begin{figure}\n \\centering\n \\input{images/halfband/halfbandLinearExample.tikz}\n \\caption[Half-band Filter Frequency Response]{%\n The frequency response and coefficient set of a half-band filter.\n The frequency response is the amplitude plotted linearly, not the\n magnitude plotted logarithmically, in order to emphasize the symmetry.\n The coefficient set is symmetrical around its midpoint. Also, every\n second coefficient outside the central peak is zero.%\n }\n \\label{fig:fir:halfband_linear_example}\n\\end{figure}\n%>>>\n% ==============================================================================\n%\n% C I C F I L T E R S\n%\n% ==============================================================================\n\\clearpage\n\\subsection{CIC Filters} %<<< ------------------------------------------------ %\n\\label{subsec:CIC_filters}\n% ---------------------------------------------------------------------------- %\n\nCIC filters offer a very computationally efficient way to decimate or\ninterpolate, though their design parameters are limited. They were first\nintroduced in 1981 in \\cite{1163535} by Eugene B. Hogenauer. They can be\nimplemented both as decimation filters (reduction in sampling rate) and\ninterpolation filters (increase in sampling rate).\n\n\\subsubsection{General Description}\n\\label{subsubsec:cic:general_description}\n\nA CIC filter is a cascade of integrator and comb stages, with\neither a sampling rate compressor (in case of a decimator)\nor a sampling rate expander (in case of an interpolator)\nbetween the integrator and comb sections. A single integrator\nstage is shown in Figure~\\ref{fig:filtertopologies:integrator}, and\nFigure~\\ref{fig:filtertopologies:comb} shows a single comb stage in\nfeedforward form. Figure~\\ref{fig:filtertopologies:cic} depicts a complete CIC\ndecimator with three stages.\n\n\\begin{figure}\n \\centering\n \\begin{minipage}[t][][b]{0.45\\textwidth}\n \\centering\n \\input{images/filtertopologies/integrator.tikz}\n \\caption[Integrator Stage]{A single integrator stage}\n \\label{fig:filtertopologies:integrator}\n \\end{minipage}\n \\begin{minipage}[t][][b]{0.45\\textwidth}\n \\centering\n \\input{images/filtertopologies/comb.tikz}\n \\caption[Comb Stage]{A single comb stage in feedforward form}\n \\label{fig:filtertopologies:comb}\n \\end{minipage}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\input{images/filtertopologies/cic.tikz}\n \\caption[CIC Filter Topology]\n {CIC decimation filter topology with three integrator and comb stages}\n \\label{fig:filtertopologies:cic}\n\\end{figure}\n\nThe integrator stages run at $f_\\mathrm{s}$ and have a transfer function of\n\\begin{equation}\n \\label{eq:cic:integrator_stage}\n H_\\mathrm{I}(z) = \\frac{1}{1-z^{-1}}\\text{,}\n\\end{equation}\nwhile the comb stages run at the reduced frequency of $f_\\mathrm{s}/R$ and\nhave the transfer function\n\\begin{equation}\n \\label{eq:cic:comb_stage}\n H_\\mathrm{C}(z) = 1 - z^{-RM}\n\\end{equation}\nwhere $M$ is the \\emph{differential delay}, one of the filter's design\nparameters.\n\nThe transfer function of a complete CIC filter (referenced to the high\nsampling rate $f_\\mathrm{s}$) consisting of $N$ stages is deduced by\nmultiplying the transfer functions of the $N$ cascaded integrator and comb\nstages, yielding\n\\begin{equation}\n \\label{eq:cic:complete}\n H_\\mathrm{CIC}(z) = H_\\mathrm{I}^N(z) \\cdot H_\\mathrm{C}^N(z) =\n \\frac{\\left(1 - z^{-RM}\\right)^N}{\\left( 1 - z^{-1} \\right)^N} =\n \\left[\\sum_{k = 0}^{RM-1} z^{-k}\\right]^N\n\\end{equation}\n\nLooking at the last form of the CIC filter's transfer function, it becomes\nevident that it can also be thought of as a FIR filter with unitary\ncoefficients. Of particular note is the fact that this is so despite each\nstage having feedback or feedforward paths and the integrator stages having\npoles at $f = 0$ (i.e. the integrators by themselves are not in fact BIBO\nstable, even though the complete system is). The fact that the resulting\nfilter has no poles can be intuitively understood by looking at the frequency\nresponses of the integrator and comb stages, and finally their cascade (see\nSection~\\ref{subsubsec:cic:frequency_characteristics}).\n\nCIC filters are well-suited to large reductions in sampling rates because they\nare very economical in their resource usage. This economy is based on six\nprimary factors~\\cite{1163535}:\n\\begin{itemize}\\tightlist\n \\item\n The filter requires no multipliers.\n \\item\n There are no filter coefficients to store.\n \\item\n The amount of storage needed for intermediate results is reduced by\n running the comb stages at a lower sampling rate. A conventional FIR\n filter topology implementing the same transfer function would require\n more resources for storing its intermediate results because the entire\n filter would run at the incoming sampling rate.\n \\item\n The topology of the filter has a high degree of regularity;\n consisting of two primary building blocks. This lends itself well to\n optimization.\n \\item\n The control logic can be kept simple.\n \\item\n The same filter design can be used for a large range of rate change\n factors $R$, requiring minimal adaption in circuitry. This effect can\n be seen in the frequency responses plotted in the top plot of\n Figure~\\ref{fig:cic:freq_responses:var}.\n\\end{itemize}\n\nHowever, CIC filters do suffer from some drawbacks. The two primary ones are:\n\\begin{itemize}\\tightlist\n \\item\n For large rate change factors $R$, the register growth of the filter\n can become very high. See Section~\\ref{subsubsec:cic:register_growth}.\n \\item\n A CIC filter has only three design parameters determining\n its frequency response: Rate change factor $R$, differential\n delay $M$, and the number of stages $N$. The amount\n of fine-tuning which can be conducted on the filter's\n frequency response is therefore extremely limited (more in\n Section~\\ref{subsubsec:cic:frequency_characteristics}).\n\\end{itemize}\n\nAs can be seen in Equation~\\ref{eq:cic:integrator_stage}, the integrator\nstages have unity feedback coefficients. In the case of CIC decimators, the\nregisters of the integrators will therefore suffer from register overflow.\nThis causes no harm as long as two conditions are fulfilled:\n\\begin{itemize}\\tightlist\n \\item\n The filter's implementation is based on two's complement or another\n number system allowing wrap-around between its most positive and most\n negative numbers.\n \\item\n The maximum magnitude which is expected at the output is within the\n range of that number system.\n\\end{itemize}\n\nA numerical example to demonstrate this effect and better explain the inner\nworkings of a CIC filter can be found in Appendix~\\ref{sec:app:cic_simu},\nstarting on page~\\pageref{sec:app:cic_simu}.\n\n% ==============================================================================\n%\n% F R E Q U E N C Y C H A R A C T E R I S T I C S\n%\n% ==============================================================================\n\\subsubsection{Frequency Characteristics} % ---------------------------------- %\n\\label{subsubsec:cic:frequency_characteristics}\n% ---------------------------------------------------------------------------- %\n\nThis section presents some of the more important frequency characteristics of\nthe CIC filter. We will start with some considerations about how the\nintegrators and comb sections interact in the frequency domain to create the\nCIC filter's frequency response.\n\nAs shown in the top plot in Figure~\\ref{fig:cic:freq_responses}, an integrator\nis in essence a lowpass filter, with a pole at $f = 0$. A comb filter is a\nfilter which attenuates one specific frequency component along with its\nmultiples (in a notch comb filter; there is also the inverse concept of a peak\nfilter which only lets a certain frequency and multiples of it pass). It is\nalso evident that comb filters have no poles (a fact which can be deduced from\nEquation~\\ref{eq:cic:comb_stage} as well, of course).\n\nThe bottom plot from Figure~\\ref{fig:cic:freq_responses} is the result of\ncascading integrators and combs. The integrator's pole at $f = 0$ compensates\nfor the comb section's zero at the same location, leading to a significant,\nbut finite, DC gain of the CIC filter.\n\n\\begin{figure}\n \\centering\n \\input{images/cic/cicFreqResponses.tikz}\n \\caption[Frequency Responses for Integrators, Combs and CIC Filters]{%\n Frequency responses for integrators, combs and their combination\n into a three-stage CIC filter with a rate change factor of \\num{9}\n and a differential delay of \\num{1}. Note that \\num{4.5} lobes\n fit into the plot for the comb filter, due to $R\\cdot M = 9$ (the\n order of the comb filter).%\n }\n \\label{fig:cic:freq_responses}\n\\end{figure}\n\n\nOne drawback of CIC filters is that they have no clearly defined\npassband as such. Rather, their frequency response starts dropping off\nright as the frequency axis goes beyond zero. This effect ( also\nreferred to as \\emph{passband droop} or \\emph{passband attenuation}) is\nvisible in the bottom plot of Figure~\\ref{fig:cic:freq_responses} and in\nFigure~\\ref{fig:cic:freq_responses:passband:attenuation} in more detail. Since\nCIC filters lack a clearly defined transition band edge, defining the\nfrequency band which is to be used, i.e. the actual passband, is a design\ndecision and can vary even when using the same filter, depending on the\napplication.\n\nThe amount of passband droop is constant for a given product\nof the differential delay $M$ and the cutoff frequency\n$f_\\mathrm{c}$, where $f_\\mathrm{c}$ is a fraction of the\nlower sampling rate (i.e. a fraction of the first lobe's\nwidth). Figure~\\ref{fig:cic:freq_responses:passband:attenuation} highlights\nthis effect for two different filters. Table~\\ref{tab:cic:pb_attenuation}\nin Appendix~\\ref{sec:app:cic_filter_tables} on\npage~\\pageref{tab:cic:pb_attenuation} contains a list with more values for\nsome common configurations.\n\nBecause of the passband droop, a CIC filter by itself is rarely a viable\nsolution. Rather, it is generally deployed as the first element in a chain of\nfilters, where the later stages are FIR filters. Due to the CIC filter's\nfrugality in terms of resource usage, it is ideally suited as an initial stage\nwhere the most samples per time need to be processed. The fact that FIR\nfilters need to perform many more computations (and more complex ones) per\nsample is then no longer as much of a problem, since the FIR filters run\nat lower sampling frequencies and therefore have many more clock cycles\navailable to compute each output. Also, because the frequency response\nof a FIR filter can be very finely tuned to a desired profile, they\ncan be used to compensate for the CIC filter's passband droop; this is\ngenerally known as a \\emph{CIC compensation filter}. More on the topic in\nSection~\\ref{subsubsec:cic:compensators}.\n\n\\begin{figure}\n \\centering\n \\input{images/cic/cicPassbandAttenuation.tikz}\n \\caption[CIC Filter: Passband and Aliasing Attenuation]{%\n Passband attenuation for two CIC filters with $R=9$,\n $N=4$ and $M=1$ and $M=2$, respectively. The attenuation\n is identical for the bandwidth-differential delay product,\n which is $1/8$ for the two configurations. The\n attenuation is \\SI{-3.65}{\\dB} in both cases; the value\n can be found in Table~\\ref{tab:cic:pb_attenuation} on\n page~\\pageref{tab:cic:pb_attenuation}.%\n }\n \\label{fig:cic:freq_responses:passband:attenuation}\n\\end{figure}\n\nAnother effect which must be taken into consideration when designing CIC\nfilters is the amount of aliasing which occurs from the stopband into the\npassband. A region of width $f_\\mathrm{c}$ above and below each $M$th null is\nfolded back into the filter's passband. This effect is highlighted in\nFigure~\\ref{fig:cic:freq_responses:passband:aliasing}. The gravity of this\neffect depends on the width of the cutoff frequency $f_\\mathrm{c}$ as\nwell as the differential delay $M$. Table~\\ref{tab:cic:pb_attenuation} in\nAppendix~\\ref{sec:app:cic_filter_tables} contains some values for common\nranges for M and $f_\\mathrm{c}$.\n\n\\begin{figure}\n \\centering\n \\input{images/cic/cicPassbandAliasing.tikz}\n \\caption[CIC Filter: Passband and Aliasing Attenuation]{%\n Passband aliasing for a CIC filter with $R = 9$, $N=4$\n and $M=1$ and a cutoff frequency of $f_\\mathrm{c} = 0.25$,\n referenced to the lower sampling frequency $f_\\mathrm{s,low}$.\n The region of width $f_\\mathrm{c}$ around every $M$th null\n is folded back into the passband. The regions beyond\n that are of course folded back as well, but are cut\n off by the compensator, see Figure~\\ref{fig:cic:cfir}. The\n resulting passband aliasing attenuation is \\SI{41.8}{\\dB},\n as indicated in Table~\\ref{tab:cic:pb_aliasing} on\n page~\\pageref{tab:cic:pb_aliasing}.%\n }\n \\label{fig:cic:freq_responses:passband:aliasing}\n\\end{figure}\n\nAs mentioned, the CIC filter has only three design parameters: Its rate\nchange factor $R$, the differential delay $M$ and the number of stages $N$.\nThe influence of these parameters on the CIC filter's frequency response is\nportrayed in Figure~\\ref{fig:cic:freq_responses:var}. Some things of note are:\n\\begin{itemize}\\tightlist\n \\item\n Increasing $R$ increases the amount of nulls and the filter's overall\n gain.\n \\item\n Increasing $M$ also increases the number of nulls as well as the\n filter's gain. Note that for CIC decimators, the region around every\n $M$th null is folded back into the passband. For practical purposes,\n $M$ is usually set to \\num{1} or \\num{2}, see \\cite{1163535}.\n \\item\n Adding more stages leads to a high increase in filter gain, since $N$\n occurs in the exponent of the filter's transfer function. It does not,\n however, change the number or placement of the nulls.\n\\end{itemize}\n\n\\begin{figure}\n \\centering\n \\input{images/cic/cicFreqResponsesVar.tikz}\n \\caption[Influence of Design Parameters on Frequency Response]{%\n The influence of the design parameters $R$, $M$ and $N$ on a CIC\n filter' frequency response. Increasing $R$ and $M$ leads to an\n increased number of nulls, as visible in the top two plots, as\n well as an increase in the DC gain. Adding more stages does not\n change the location of the nulls, but does add significant DC\n gain.%\n }\n \\label{fig:cic:freq_responses:var}\n\\end{figure}\n\n% ==============================================================================\n%\n% C O M P E N S A T O R S\n%\n% ==============================================================================\n\\subsubsection{Compensators} % ----------------------------------------------- %\n\\label{subsubsec:cic:compensators}\n% ---------------------------------------------------------------------------- %\n\nIn order to achieve a flat passband, the CIC\nfilter's attenuation in the frequency range observed in\nFigure~\\ref{fig:cic:freq_responses:passband:attenuation} can be compensated\nwith a filter whose frequency response has the opposite shape. Operated in a\ncascade, the two filters create a frequency response with a flat passband and\na sharp drop-off into the stopband. Figure~\\ref{fig:cic:cfir} shows an example\nof such a system, with frequency responses of a CIC filter, its compensator,\nand the resulting cascade.\n\nThe compensation filter not only serves to compensate for the passband, but is\nalso responsible for the transition band width of the cascade. Due to their\nflexibility, FIR filters are generally employed for this purpose. As mentioned\nin the previous section, the sharpness of their transition band can be\ncontrolled by adjusting their kernel size. If more filters follow after\nthe compensator in the filter chain, the compensator's transition band\nneed not be very narrow. A filter with a few dozen coefficients in size\nis often sufficient in such cases (the filter used for the example in\nFigure~\\ref{fig:cic:cfir} has \\num{50} coefficients).\n\nThe design of CIC compensators is usually left up to software algorithms, for\nexample with Matlab. For a more detailed introduction to the topic, including\nexample code and more elaborate explanations, Altera's Application Note\nfrom~\\cite{altera:an455} is warmly recommended.\n\n\\begin{figure}\n \\centering\n \\input{images/cic/cfirDemo.tikz}\n \\caption[CIC Compensator]{%\n Frequency behavior of a CIC filter, its compensator, and the cascade\n of the two. Note the spectral copies of the compensator around the\n nulls of the CIC filter, i.e. the multiples of its outgoing sampling\n rate.%\n }\n \\label{fig:cic:cfir}\n\\end{figure}\n% ==============================================================================\n%\n% R E G I S T E R G R O W T H\n%\n% ==============================================================================\n\\subsubsection{Register Growth}\n\\label{subsubsec:cic:register_growth}\n\nAs shown by Hogenauer in~\\cite{1163535}, the maximum register growth is\n\\begin{equation}\n \\label{eq:cic:maximum_register_growth}\n G_\\mathrm{max} = (R \\cdot M)^N\n\\end{equation}\nThe most significant bit $B_\\mathrm{max}$ of the output register as well as\nfor all stages (both the integrators and the comb stages) of the filter is\ndetermined to be\n\\begin{equation}\n \\label{eq:cic:maximum_register_growth:bit_width}\n B_\\mathrm{max} = \\lceil N \\log_2 RM + B_\\mathrm{in} - 1 \\rceil\n\\end{equation}\nwhere $B_\\mathrm{in}$ is the bit width of the input register. For high rate\nchange factors, these values can become very large. A filter with three\nstages, a differential delay of \\num{1}, a rate change of \\num{128} and an\ninput width of \\num{16}\\,bits yields \\num{36}\\,bits output width at full\nprecision.\n% ==============================================================================\n%\n% T R U N C A T I O N A N D R O U N D I N G E R R O R S\n%\n% ==============================================================================\n\\subsubsection{Errors Due to Truncation and Rounding}\n\\label{subsubsec:cic:truncation_and_rounding}\n\nIn practical cases, it is often not feasible to retain full precision; in such\nsituations, either truncation or rounding may be used at each filter stage to\nreduce register widths and keep resource usage within certain limits. For this\npurpose, it is necessary to know the system function from the $j$th stage up\nto and including the last:\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:system_function}\n H_j(z) = \\left\\lbrace\n \\begin{aligned}\n H_I^{N-j+1}H_C^N &\n = \\sum_{k=0}^{(RM-1)N+j-1} h_j[k]z^{-k} &\n \\quad j &\n = 1, 2, \\cdots, N \\\\\n H_C^{j-N} &\n = \\hspace{1.1em}\\sum_{k=0}^{2N+1-j} h_j[k]z^{-kRM} &\n \\quad j &\n = N+1, \\cdots, 2N\n \\end{aligned}\n \\right.\n\\end{equation}\nwhere\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:system_function}\n h_j[k] = \\left\\lbrace\n \\begin{aligned}\n \\sum_{l=0}^{\\lfloor k/(RM) \\rfloor} (-1)^l\n {{N}\\choose{l}}{{N-j+k-RMl}\\choose{k = RMl}} &\n j &\n = 1, 2, \\cdots N \\\\\n (-1)^k{{2N+1-j}\\choose{k}} &\n j &\n = N+1, \\cdots 2N\n \\end{aligned}\n \\right.\n\\end{equation}\nare the impulse response coefficients. These functions are derived by\nHogenauer in~\\cite{1163535}.\n\nIn a filter with $N$ stages, there are $2N+1$ error sources in the case of\nlimited precision: Each stage, and the output register. Each error source is\npresumed to have white noise characteristics, i.e. its noise is uncorrelated\nto its input as well as other error sources. The error at the $j$th source is\nassumed to have a uniform probability distribution with a width of\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:probability_distribution}\n E_j = \\left\\lbrace\n \\begin{aligned}\n 0 & \\quad\\text{without truncation or rounding}\\\\\n 2^{B_j} & \\quad\\text{otherwise}\n \\end{aligned}\n \\right.\n\\end{equation}\nwhere the number of bits discarded at the $j$th error source is $B_j$. The mean\nof this error is\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:mean}\n \\mu_j = \\left\\lbrace\n \\begin{aligned}\n \\frac{1}{2}E_j & \\quad\\text{for truncation}\\\\\n 0 & \\quad\\text{otherwise}\n \\end{aligned}\n \\right.\n\\end{equation}\nand the variance comes out to\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:variance}\n \\sigma_j^2 = \\frac{1}{12}E_j^2.\n\\end{equation}\n\nThe total mean error at the filter's output due to the $j$th stage is\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:total_mean_error_jth_stage}\n \\mu_{T_j} = \\mu_jD_j\n\\end{equation}\nwhere\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:mean_error_gain}\n D_j = \\left\\lbrace\n \\begin{aligned}\n (RM)^N & \\quad j = 1\\\\\n 0 & \\quad j = 2, 3, \\cdots, 2N\\\\\n 1 & \\quad j = 2N+1\n \\end{aligned}\n \\right.\n\\end{equation}\nis the \\emph{mean error gain} for the $j$th error source. Note that only the\nfirst and the last error source contribute to the filter's mean error at the\noutput. This is because the sum of the impulse response coefficients is zero\nfor all other stages. Consequently, whether one chooses to truncate or round\nis without consequence except in the case of the first and last error\nsources. In an analogous manner, the total variance computes to\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:total_variance_jth_stage}\n \\sigma_{T_j}^2 = \\sigma_j^2F_j^2\n\\end{equation}\nwhere\n\\begin{equation}\n \\label{eq:cic:truncation_rounding:variance_error_gain}\n F_j = \\left\\lbrace\n \\begin{aligned}\n \\sum_k h_j^2[k] & \\quad j = 1, 2, \\cdots, 2N\\\\\n 1 & \\quad j = 2N+1\n \\end{aligned}\n \\right.\n\\end{equation}\nis called the \\emph{variance error gain} for the $jth$ error source. We can\nnow compute the global mean error and variance of the filter:\n\\begin{align}\n \\label{eq:cic:truncation_rounding:global:mean_error}\n \\mu_T &= \\sum_{j = 1}^{2N+1} \\mu_{T_j} = \\mu_{T_1} + \\mu_{T_{2N+1}}\\\\\n \\label{eq:cic:truncation_rounding:global:variance}\n \\sigma_{T}^2 &= \\sum_{j=1}^{2N+1} \\sigma_{T_j}\n\\end{align}\nThese equations are used to calculate the properties of the CIC filter as\ndeployed in our design in Section~\\ref{subsec:fpga:errors_in_cic_filter}.\n\n% ==============================================================================\n%\n% C I C F I L T E R S : S U M M A R Y\n%\n% ==============================================================================\n\\subsubsection{Summary}\n\\label{subsubsec:cic:summary}\n\nIn conclusion, the key properties of CIC filters are:\n\\begin{itemize}\\tightlist\n \\item\n They can be implemented both as decimators and interpolators.\n \\item\n Neither multipliers nor storage for coefficients are needed.\n \\item\n CIC decimation filters have a high gain, leading to significant\n register growth. Truncation or rounding can be used to limit the\n resource usage, both at the filter's output and internally.\n \\item\n The three design parameters are the rate change $R$, the differential\n delay $M$ and the number of stages $N$.\n \\item\n The presence of passband droop requires a compensation filter to\n achieve a flat passband response.\n\\end{itemize}\n%>>>\n\n%>>>\n\n% ==============================================================================\n%\n% M U L T I - S T A G E F I L T E R D E S I G N S\n%\n% ==============================================================================\n\\section{Multi-Stage Filter Designs} % <<< ----------------------------------- %\n\\label{sec:multi_stage_filter_designs}\n% ---------------------------------------------------------------------------- %\n\nAt first sight, the most obvious way to implement a downsampling\nsystem might appear to be to design one filter for each desired rate\nchange factor. However, this would be highly impractical. Instead,\nmulti-stage designs are usually used in practice. An in-depth discussion\nof their advantages and drawbacks was offered by Crochiere and Rabiner\nin~\\cite{crochiere-rabiner:multirate-dsp}. A few aspects of multi-stage filter\ndesign which are relevant to our application shall be presented here.\n\nTo reduce aliasing effects in the passband, it is generally desirable\nto keep the width of the transition band roughly constant in relation\nto the width of the passband (visible in the filter's flank in\nFigure~\\ref{fig:aliasing:iirCopies}). As the downsampling ratio increases\nand the passband width decreases, the transition band therefore becomes\nprogressively narrower, necessitating higher filter orders in a single-stage\ndesign.\n\nThis effect is illustrated in Figure~\\ref{fig:fdesign:tbw_width_Rvar},\ncomparing two filters with a transition band $1/5$ as wide as the passband for\ndownsampling ratios of \\num{2} and \\num{4}. The filter for $R=4$ requires\n\\num{60} coefficients, compared to \\num{30} coefficients for the filter\ndesigned for $R=2$. The other specifications (passband ripple, stop band\nattenuation) are identical. As an extreme case, a filter designed by Matlab\nwith the same parameters for a downsampling ratio of $R=625$ is \\num{8860}\ncoefficients in size.\n\nUsing multi-stage designs helps to avoid the need to implement filters with\nsuch large kernels. When cascading filters, it is the last stage of the chain\nwhich defines the overall passband and transition band width. The same overall\ntransition band in absolute terms can be achieved with smaller filters in\nmulti-stage designs, because a filter's transition band width is relative to\nthe sampling rate at which it is running. This is shown in the bottom plot in\nFigure~\\ref{fig:fdesign:tbw_width_Rvar}.\n\n\\begin{figure}\n \\centering\n \\input{images/fdesign/tbwDemoRVar.tikz}\n \\caption\n [Frequency Response of Multi-Stage Vs. Single-Stage Design]{%\n Two filters are used here: Both have a transition band $1/5$ as wide\n as their passbands. The top filter is designed for a downsampling\n ratio $R=2$ and has a coefficient count of $N=30$. The second filter\n is designed for $R=4$ and has \\num{60} coefficients. Cascading two of\n the top filters into a two-stage design, depicted in the bottom plot,\n results in the same overall passband and transition band width as the\n single-stage design, but with fewer computations per time.%\n }\n \\label{fig:fdesign:tbw_width_Rvar}\n\\end{figure}\n\nSince cascading multiple filters does increase coefficient count (and\nstorage), one might be inclined to think that not much has been won. Indeed,\nthe overall number of coefficients is identical for both filters in\nFigure~\\ref{fig:fdesign:tbw_width_Rvar} (though, this obviously need not be\nso in other examples). However: Merely \\num{30} multipliers and \\num{29}\nadders (those of the first filter in the cascade) run at the incoming\nsampling frequency in the case of the cascade, while the components of\nthe second filter run at half that. In the case of the single-stage\nfilter, all its \\num{60} multipliers and \\num{59} adders run at the\nfull sampling frequency. Distributing the calculations over two stages has\ntherefore yielded an overall reduction in needed computation power. As rate\nchange factors increase, the benefits of multi-stage designs become even\nmore pronounced~\\cite{crochiere-rabiner:multirate-dsp}. \n\nIn general, the earlier stages in a cascade tend to have higher downsampling\nratios but wider transition bands, and the later stages have lower\ndownsampling ratios and narrower transition bands (normalized to their\nrespective sampling frequencies). This minimizes computations across the\noverall design by using smaller filters for high-frequency calculations, and\ngiving the larger filters more time to compute their outputs. Multiple stages\nhaving the same downsampling ratios is also a common approach, but earlier\nstages having lower downsampling ratios than later stages is uncommon.\n\nA further advantage of cascading filters is that successive stages can be used\nto shape the overall frequency response, as seen in the case of the CIC\ncompensator in Section~\\ref{subsubsec:cic:compensators} A drawback of cascades\nis that the ripple in their passband shows additive behavior, so the stages in\na cascade of filters have more stringent ripple requirements in the passband\nthen a single-stage. However, the cost for this is usually offset by the\nadvantages of multi-stage designs.\n\nWhen designing multi-stage filters, it can happen that the transition band of\nan earlier stage overlaps with the spectral copy of a later stage running at a\nreduced sampling rate. In that case, the stopband response of the cascade can\nhave peaks exceeding the desired overall stopband attenuation. To prevent\nthis, the following condition must be satisfied:\n\\begin{equation}\n \\label{eq:cascade:transition_band_overlap}\n f_\\mathrm{st,1} < \\frac{f_\\mathrm{s,1} - f_\\mathrm{st,2}}{R_1}\n\\end{equation}\n\\begin{conditions}\n f_\\mathrm{s,1} & high sampling rate \\\\\n f_\\mathrm{st,1} & stopband frequency of first filter \\\\\n f_\\mathrm{st,2} & stopband frequency of second filter \\\\\n R_1 & rate reduction in first filter \\\\\n\\end{conditions}\nFigure~\\ref{fig:fdesign:cascade:good_vs_bad} shows some examples for this\ncondition being broken or fulfilled.\n\n\\begin{figure}\n \\centering\n \\input{images/fdesign/cascadeDemo.tikz}\n \\caption[Cascade: Transition Band Overlap]{%\n Comparison of two cascades: The first cascade (blue) has\n sufficient distance between the start of its stopband ($0.65\n \\cdot f_\\mathrm{s1}$) and the start of the transition band of\n the second stage's first copy around $f_\\mathrm{s2}$ ($0.675 \\cdot\n f_\\mathrm{s1}$). The second cascade (orange) has a peak in its\n stopband because the transition band of its first stage overlaps with\n the copy of the second stage ($0.8 \\cdot f_\\mathrm{s1}$ vs. $0.6\n \\cdot f_\\mathrm{s1}$). \\emph{Note:} All frequencies are normalized to\n the high sampling rate $f_\\mathrm{s1}$.%\n }\n \\label{fig:fdesign:cascade:good_vs_bad}\n\\end{figure}\n\nOne last effect of note when cascading filters concerns stopband attenuation:\nWhen cascading two filters with different stopband attenuations, two things\ncan happen:\n\\begin{itemize}\\tightlist\n \\item\n The second filter attenuates more strongly than the first one. This\n results in peaks above the second filter's stopband attenuation\n in the regions where the spectral copies of the second filter's\n passband are located. This can be seen in the top plot in\n Figure~\\ref{fig:fdesign:cascade:ast_demo}.\n \\item\n The first filter attenuates more strongly than the first one. In that\n case, the stopband region of the cascade right next to its transition\n band is less strongly attenuated than the stopband regions farther\n away from the edge. This case is shown in the middle plot in\n Figure~\\ref{fig:fdesign:cascade:ast_demo}.\n\\end{itemize}\nNeither of these two effects is usually desired. In both cases, the resources\ninvested into the steeper filter's stronger attenuation are wasted by the\nother filter's weaker stopband attenuation. It therefore makes more sense for\nthe various stages in a cascade to have the same stopband attenuation.\n\n\\begin{figure}\n \\centering\n \\input{images/fdesign/astDemo.tikz}\n \\caption[Cascade: Stopband Attenuation]{%\n Cascading two filters with different stopband attenuations: If a\n filter with stronger stopband attenuation is cascaded after a filter\n with weaker attenuation, the resulting cascade has peaks above the\n second filter's stopband attenuation (top plot). If the stronger\n filter is first in the cascade, the drop-off in the stopband right\n next to the transition band is weaker. The two filters by themselves\n are shown in the bottom plots.%\n }\n \\label{fig:fdesign:cascade:ast_demo}\n\\end{figure}\n\n%>>>\n\n%^^A vim: foldenable foldcolumn=4 foldmethod=marker foldmarker=<<<,>>>\n", "meta": {"hexsha": "dbe49f70b65498bdf030eb4197b20ccbd1385df3", "size": 59021, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/report/chunks/theory.tex", "max_stars_repo_name": "alpenwasser/pitaya", "max_stars_repo_head_hexsha": "a6ced99408171ffcd96c9444adfe30d2ba699f48", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2017-03-22T15:26:34.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-15T20:19:03.000Z", "max_issues_repo_path": "doc/report/chunks/theory.tex", "max_issues_repo_name": "alpenwasser/pitaya", "max_issues_repo_head_hexsha": "a6ced99408171ffcd96c9444adfe30d2ba699f48", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/report/chunks/theory.tex", "max_forks_repo_name": "alpenwasser/pitaya", "max_forks_repo_head_hexsha": "a6ced99408171ffcd96c9444adfe30d2ba699f48", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.2734241908, "max_line_length": 80, "alphanum_fraction": 0.6563765439, "num_tokens": 15594, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.3460058095032024}} {"text": "\\input{common/prelude.tex}\n\n% Spacing for logo\n\\addtolength{\\wpXoffset}{5.5cm}\n\\addtolength{\\wpYoffset}{13.1cm}\n\n\\begin{document}\n\n\\CenterWallPaper{0.3}{common/anchor-logo.png}\n\n\n{\\huge \\bfseries Control.Lens (viewing) \\\\[0.2cm]}\n\n\\HRule%\n\n\\begin{multicols}{2}\n\n\\begin{box1}\n\\subsection *{Getting with Getters}\n\nAny function $(s \\to a)$ can be flipped into continuation passing style, $(a \\to r)\n\\to s \\to r$ and decorated with \\textbf{Const} to obtain:\n\n\\begin{minted}{haskell}\ntype Getting r s a =\n (a -> Const r a) -> s -> Const r s\n\\end{minted}\nA \\textbf{Getter} describes how to retrieve a single value in a way that can be composed\nwith other \\textbf{LensLike} constructions.\n\nWhen you see this in a type signature it indicates that you can pass the\nfunction a \\textbf{Lens}, \\textbf{Getter}, \\textbf{Traversal}, \\textbf{Fold},\n\\textbf{Prism}, \\textbf{Iso}, or one of the indexed\nvariants, and it will just ``do the right thing''.\n\\end{box1}\n\n\\begin{box2}\n\\subsection *{Safe head}\nPerform a safe head of a \\textbf{Fold} or \\textbf{Traversal} or retrieve\n\\textbf{Just} the result from a \\textbf{Getter} or \\textbf{Lens}.\n\n$ (\\ \\hat{}\\,?) \\equiv flip\\;preview $\n\n\\begin{minted}{haskell}\n(^?) :: s -> Getting (First a) s a -> Maybe a\n\n> Right 4 ^?_Left\nNothing\n> \"world\" ^? ix 3\nJust `l'\n\\end{minted}\n\\end{box2}\n\n\\begin{box1}\n\n\\subsection *{Viewing lenses}\n\nView the value pointed to by a \\textbf{Getter} or \\textbf{Lens} or the result\nof folding over all the results of a \\textbf{Fold} or \\textbf{Traversal} that\npoints at a monoidal values.\n\nThis is the same operation as \\textbf{view} with the arguments flipped.\n\n\\begin{minted}{haskell}\n(^.) :: s -> Getting a s a -> a \n\n> (0, -5)^._2.to abs\n5\n> [\"a\", \"b\", \"c\"] ^. traversed\n\"abc\"\n\\end{minted}\n\\end{box1}\n\n\\begin{box2}\n\\subsection*{Using MonadState}\n\nUse the target of a \\textbf{Lens}, \\textbf{Iso}, or \\textbf{Getter} in the\ncurrent state, or use a summary of a \\textbf{Fold} or \\textbf{Traversal} that\npoints to a monoidal value.\n\n\\begin{minted}{haskell}\nuse :: MonadState s m => Getting a s a -> m a\n\n> evalState (use _1) (1,2)\n1\n> evalState (uses _1 length) (\"hello\",\"\")\n5\n\\end{minted}\n\\end{box2}\n\n\\columnbreak\n\n\\begin{box1}\n\\subsection *{Folding Foldables}\n\\begin{minted}{haskell}\ntype Fold s a =\n forall m. Monoid m => Getting m s a\n\\end{minted}\nA \\textbf{Fold s a} is a generalization of something \\textbf{Foldable}. It\nallows you to extract multiple results from a container. Every \\textbf{Getter}\nis a valid \\textbf{Fold} that simply doesn't use the Monoid it is passed.\n\nIf there exists a \\textbf{foo} method that expects a\n\\textbf{Foldable (f a)}, then there should be a \\textbf{fooOf} method that\ntakes a \\textbf{Fold s a} and a value of type \\textbf{s}.\n\n\\end{box1}\n\n\\begin{box2}\n\\subsection *{Extracting lists from Folds}\n\nExtract a list of the targets of a \\textbf{Fold}, an infix version of\n\\textbf{toListOf}.\n\n$ toList\\;xs \\equiv xs\\;\\hat{}.. folded $\n\n\\begin{minted}{haskell}\n(^..) :: s -> Getting (Endo [a]) s a -> [a] \n\n> [[1,2],[3]] ^.. traverse . traverse\n[1,2,3]\n> (1,2) ^.. both\n[1,2]\n\\end{minted}\n\\end{box2}\n\n\\begin{box1}\n\\subsection *{Checking for matches}\n\nCheck to see if this \\textbf{Fold} or \\textbf{Traversal} matches 1 or more entries. For the\nopposite, use \\textbf{hasn't}.\n\n\\begin{minted}{haskell}\nhas :: Getting Any s a -> s -> Bool\n\n> has (element 0) []\nFalse\n> has _Right (Left 12)\nFalse\n> hasn't _Right (Left 12)\nTrue\n\\end{minted}\n\\end{box1}\n\n\n\\begin{box2}\n\\subsection*{Indexed Getters}\nFor most operations, there is an indexed variant which will work as expected if\nthe underlying target supports a notion of \\textbf{Indexing}.\n\n\\begin{minted}{haskell}\n> [\"ab\", \"c\"] ^@.. itraversed <.> itraversed \n[((0,0),'a'),((0,1),'b'),((1,0),'c')]\n> \"hello\" ^@.. itraversed . indices even\n[(0,'h'),(2,'l'),(4,'o')]\n\n> ifind (\\i k -> i > k) [1,2,2,2]\nJust (3,2)\n\\end{minted}\n\\end{box2}\n\\end{multicols}\n\\newpage\n\n{\\huge \\bfseries Control.Lens (setting) \\\\[0.2cm]}\n\n\\HRule%\n\n\\begin{multicols}{2}\n\n\\begin{box1}\n\\subsection *{Modifying records with Setters}\nA \\textbf{Setter s t a b} is a generalization of fmap from \\textbf{Functor}. It\nallows you to map into a structure and change out the contents, but it isn't\nstrong enough to allow you to enumerate those contents. Starting with\n$fmap :: Functor f \\Rightarrow (a \\to b) \\to f a \\to f b$ we monomorphize the type\nto obtain $(a \\to b) \\to s \\to t$ and then decorate it with Identity to obtain:\n\n\\begin{minted}{haskell}\ntype Setter s t a b =\n (a -> Identity b) -> s -> Identity t\n\\end{minted}\n\nEvery \\textbf{Traversal} is a valid \\textbf{Setter}, since \\textbf{Identity} is\n\\textbf{Applicative}.\n\n\\end{box1}\n\n\\begin{box2}\n\\subsection *{Modifying with a function}\n\nModifies the target of a \\textbf{Lens} or all of the targets of a\n\\textbf{Setter} or \\textbf{Traversal} with a user supplied function.\n\nThis is an infix version of \\textbf{over}.\n\n\\begin{minted}{haskell}\n(%~) :: Profunctor p\n => Setting p s t a b -> p a b -> s -> t\n\n> traverse %~ even $ [1,2,3]\n[False,True,False]\n\\end{minted}\n\\end{box2}\n\n\\begin{box1}\n\\subsection *{Modifying with a constant value}\n\nReplace the target of a \\textbf{Lens} or all of the targets of a\n\\textbf{Setter} or \\textbf{Traversal} with a constant value.\n\n\\begin{minted}{haskell}\n(.~) :: ASetter s t a b -> b -> s -> t\n\n> [1,2,3] & element 0 .~ 3\n[3,2,3]\n> [1,2,3] & traversed . filtered odd .~ 0\n[0,2,0]\n\\end{minted}\n\\end{box1}\n\n\\begin{box2}\n\\subsection *{Prisms and Isos}\n\nAn \\textbf{Iso} is a pair of inverse functions. You can invert an \\textbf{Iso} with \\textbf{from}.\n\n\\textbf{Prism}s can be thought of as \\textbf{Iso}s that can fail in one\ndirection. You can invert a \\textbf{Prism} with \\textbf{re}.\n\n\\begin{minted}{haskell}\ntype Prism s t a b \n forall p f. (Choice p, Applicative f) =>\n p a (f b) -> p s (f t)\ntype Prism' s a = Prism s s a a\n\nprism :: (b -> t)\n -> (s -> Either t a)\n -> Prism s t a b\nprism' :: (a -> s)\n -> (s -> Maybe a)\n -> Prism' s a\n\n> 5^.re _Left ^?! _Left\n5\n> _Left # 1\nLeft 1\n\ntype Iso s t a b =\n forall p f. (Profunctor p, Functor f) =>\n p a (f b) -> p s (f t)\ntype Iso' s a = Iso s s a a\n\niso :: (s -> a) -> (b -> t) -> Iso s t a b\nfrom :: AnIso s t a b -> Iso b a t s\n\n> 'a' ^. from enum\n97\n> 97 ^. enum :: Char\n'a'\n\n> Map.empty & at \"hi\"\n> . non Map.empty\n> . at \"world\" ?~ \"!\"\nfromList [(\"hi\",fromList [(\"world\",\"!\")])]\n\\end{minted}\n\n\\end{box2}\n\n\\end{multicols}\n\n\\begin{box2}\n\\subsection *{Some setting operators}\n\\begin{tabular}{ l l l l l }\n\\textbf{Operator} & \\textbf{W/result} & \\textbf{W/state} & \\textbf{W/both} & \\textbf{Action}\\\\\n\\hline\n\\verb!+~ ! & \\verb!<+~ ! & \\verb!+= ! & \\verb!<+= ! & Add to target(s)\\\\\n\\verb!-~ ! & \\verb!<-~ ! & \\verb!-= ! & \\verb!<-= ! & Subtract from target(s)\\\\\n\\verb!*~ ! & \\verb!<*~ ! & \\verb!*= ! & \\verb!<*= ! & Multiply target(s)\\\\\n\\verb!//~! & \\verb!~! & \\verb!<<>~! & \\verb!<>=! & \\verb!<<>=! & mappend to the target monoidal value(s)\\\\\n\\end{tabular}\n\\end{box2}\n\n\n\\end{document}\n", "meta": {"hexsha": "497206eb454476bd2055fdde8ca97a290a317392", "size": 7841, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "lens.tex", "max_stars_repo_name": "caiorss/haskell-cheat-sheets", "max_stars_repo_head_hexsha": "48b1980ef99bf723ac6e001b33f0dccc90386ba7", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-03-07T09:52:45.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-07T09:52:45.000Z", "max_issues_repo_path": "lens.tex", "max_issues_repo_name": "caiorss/haskell-cheat-sheets", "max_issues_repo_head_hexsha": "48b1980ef99bf723ac6e001b33f0dccc90386ba7", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "lens.tex", "max_forks_repo_name": "caiorss/haskell-cheat-sheets", "max_forks_repo_head_hexsha": "48b1980ef99bf723ac6e001b33f0dccc90386ba7", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.0379310345, "max_line_length": 135, "alphanum_fraction": 0.6106363984, "num_tokens": 2751, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.6513548511303336, "lm_q1q2_score": 0.34600580231290085}} {"text": "\\setlength{\\footskip}{8mm}\r\n\\chapter{ Methodology } \\label{methodology}\r\nThe chapter describes the methodology for the design and implementation of the system. \r\n\r\n\\section{Camera Calibration}\r\nSince I did not calibrate the camera I followed the \r\nfollowing steps to get the intrinsic and extrinsic parameters of the camera. \r\n\r\n\\begin{figure}\r\n\\centering\r\n\\includegraphics[width=110mm]{figures/camera-car-coordinate-system3.pdf}\r\n\\caption{The relationship between world and car coordinate system.}\r\n\\label{fig:camera_car_coordinate}\r\n\\end{figure}\r\n\r\n\\subsection{Intrinsic Parameters}\r\n\\begin{itemize}\r\n \\item Focal Length: Every picture taken by a digital camera contains some meta \r\ndata such as camera model, dimensions, color space, focal length, etc. The focal \r\nlength obtained from the meta data is in millimeters where as the focal length \r\nthat is needed in the camera matrix must be in pixels. Using the following \r\nequation\\fullcite{FocalLength} I was able to convert the focal length from \r\nmillimeters to pixels. \r\n\\[ \\text{focal length in pixels} =\r\n \\text{image width in pixels} \\times \r\n \\frac{\\text{focal length in mm}} {\\text{CCD width in in mm}} \r\n\\]\r\n \r\nThe value of CCD can be obtained from the camera specifications. \r\n\\item Principal Point: I assumed the principal point $(p_{x}, p_{y})$ to be at \r\nthe center of the image. \r\n\\item Skew: Skewing arise when the $x$ and $y$ axes of the pixel element are not \r\nperpendicular to each other. This is a very rare condition and it is safe to \r\nassume $s = 0$. \r\n\\end{itemize}\r\n\r\n\\subsection{Extrinsic Parameters}\r\nAs described in Section \\ref{sec:camera_calibration} the extrinsic parameters \r\nconsists of the rotational matrix $\\mat{R}$ and translation vector $\\vec{t}$. \r\n\r\nEstimating the rotational matrix $\\mat{R}$ without camera calibration is hard, since it depends on the yaw, pitch and roll of the camera. Thus, I conducted a simple experiment to estimate the rotational matrix. \r\n\r\nAssume three points $\\vec{X_{1}}, \\vec{X_{2}}, \\vec{X_{3}}$ in the homogenous \r\nworld coordinate system such that $\\vec{X_{1}} = \r\n\\begin{bmatrix}0&\\beta&0&1\\end{bmatrix}^{T}$, $\\vec{X_{2}} = \r\n\\begin{bmatrix}1&\\beta&0&1\\end{bmatrix}^{T}$, $\\vec{X_{3}} = \r\n\\begin{bmatrix}0&\\beta&1&1\\end{bmatrix}^{T}$ where $\\beta$ is a large number so \r\nthat the point $\\vec{X_{1}}$ will be projected on the horizontal row of the \r\nimage. Using Equation \\ref{eq:camera_equation} and \\ref{eq:camera_equation_2} \r\nthe points $X_{1}, X_{2}, X_{3}$ are projected onto the image. Thus, \r\n\r\n\\begin{center}\r\n\\begin{tabular}{c c c}\r\n $\\vec{x_{1}} = \\mat{K}[\\mat{R}\\mid\\vec{t}]\\vec{X_{1}}$, & \r\n $\\vec{x_{2}} = \\mat{K}[\\mat{R}\\mid\\vec{t}]\\vec{X_{2}}$ &\r\n $\\vec{x_{3}} = \\mat{K}[\\mat{R}\\mid\\vec{t}]\\vec{X_{3}}$\r\n\\end{tabular}\r\n\\end{center}\r\n\r\nFrom the above equation it can be seen that $\\vec{x_{1}}, \\vec{x_{2}}, \\vec{x_{3}}$ depends on \r\n$\\mat{R}$, which needs to be estimated. Since we know that $\\vec{x_{1}}$ should be on \r\nthe horizontal row and at the center of the lane and $\\vec{x_{2}}, \\vec{x_{3}}$ being \r\nslightly to the right and top of $\\vec{x_{1}}$. $\\mat{R}$ is adjusted until the condition is \r\nmet. \r\n\r\nFigure \\ref{fig:rotation_matrix}(a) shows the three points $\\vec{x_{1}}$(red), \r\n$\\vec{x_{2}}$(green), $\\vec{x_{3}}$(blue) being off to the right. Figure \r\n\\ref{fig:rotation_matrix}(b) shows the results of tuning the rotational matrix \r\n$\\mat{R}$ so that the thee points are in the center of the lane.\r\n\r\n\\begin{figure}\r\n \\centering\r\n \\begin{tabular}{c c}\r\n \\includegraphics[width=80mm]{figures/rotation_after.png} &\r\n \\includegraphics[width=80mm]{figures/rotation_before.png} \\\\\r\n (a) & (b) \\\\\r\n \\end{tabular}\r\n \\caption{Estimating the Rotation Matrix by using 3 points}\r\n \\label{fig:rotation_matrix}\r\n\\end{figure}\r\n\r\nThe translation vector $\\vec{t}$ is the distance from the world to camera \r\ncoordinate system. This is the distance between the two wheels in the $y$ axis \r\nand the distance from the ground to the camera in the $z$ axis.\r\n\r\n\\section{Road Lane Extraction}\r\nTo detect the road lane marking I used the ``Road Paint Detection using Matched\r\nFilters\" algorithm \\fullcite{huang2008rss}\r\n\r\n\\subsection{Calculating lane width in pixels}\r\nThe first step is to construct the matched filter. The matched filter varies \r\ndepending on the lane width at each row of the image. Therefore in order to \r\nconstruct the matched filter the lane width at each row of the image must be \r\ncalculated. Given that the camera matrix has already been determined as \r\ndescribed in Section \\ref{sec:camera_calibration}. A point in the image is \r\nrelated to a point in the real world by:\r\n\\begin{equation}\\vec{x} \\sim \\mat{P}\\vec{X} \\end{equation} \\label{eq:camera_eq}\r\nAs shown in Figure \\ref{fig:camera_car_coordinate} the $z$ co-ordinate points \r\nupward to the sky therefore the $x,y$ plane is the ground plane. Assuming that \r\nthe ground is flat $Z$ can be estimated to be $0$. \r\n\\[\r\n\\begin{bmatrix}x\\\\y\\\\1\\end{bmatrix} \\sim \r\n\\mat{P} \\begin{bmatrix}X\\\\Y\\\\0\\\\1\\end{bmatrix}\r\n\\]\r\n\\begin{eqnarray*}\r\n\\begin{bmatrix}x\\\\y\\\\1\\end{bmatrix} & = & \r\n\\begin{bmatrix}\r\np_{11} & p_{12} & p_{13} & p_{14} \\\\\r\np_{21} & p_{22} & p_{23} & p_{24} \\\\\r\np_{31} & p_{32} & p_{33} & p_{34}\r\n\\end{bmatrix}\r\n\\begin{bmatrix}X\\\\Y\\\\0\\\\1\\end{bmatrix}\r\n\\end{eqnarray*}\r\n\\begin{eqnarray*}\r\n\\begin{bmatrix}x\\\\y\\\\1\\end{bmatrix} & = & \r\n\\begin{bmatrix}\r\n Xp_{11} & Yp_{12} & \\cancelto{0}{p_{13}} & p_{14} \\\\\r\n Xp_{21} & Yp_{22} & \\cancelto{0}{p_{23}} & p_{24} \\\\\r\n Xp_{31} & Yp_{32} & \\cancelto{0}{p_{33}} & p_{34}\r\n\\end{bmatrix} \\\\\r\n\\end{eqnarray*}\r\nSince we are using homogenous coordinates therefore, \r\n\\begin{eqnarray*}\r\ny = \\frac {\\mat{P_{21}}X + \\mat{P_{22}}Y + \\mat{P_{24}}} \r\n {\\mat{P_{31}}X + \\mat{P_{32}}Y + \\mat{P_{34}}}\r\n\\end{eqnarray*}\r\nIn practice the width of the lane marking would not be uniform though out the \r\nentire row of the image. The lane marking that appeared directly in front of the \r\ncamera would be slightly wider ($\\sim$ 0.1 cm) than the lane marking that were \r\nat the sides. For simplification I've assumed that the width of the lane marking \r\nis uniform throughout the row. Therefore $X = 0$\r\n\r\n$Y$ which is the distance from the \r\nrear of the car can be calculated for every row of the image. \r\n\\begin{eqnarray*}\r\nY = \\frac{\\mat{P_{24}} - y\\mat{P_{34}}} \r\n {y\\mat{P_{32}} - \\mat{P_{22}}}\r\n\\end{eqnarray*}\r\nThe width of lane marking in pixels can be calculated by: \r\n\\begin{eqnarray*}\r\n\\mathrm{width(pixels)} = \r\n\\frac {\\mat{P_{1}}\\begin{bmatrix}W&Y&0&1\\end{bmatrix}^{\\mathrm{T}}}\r\n {\\mat{P_{3}}\\begin{bmatrix}W&Y&0&1\\end{bmatrix}^{\\mathrm{T}}}\r\n- \r\n\\frac {\\mat{P_{1}}\\begin{bmatrix}0&Y&0&1\\end{bmatrix}^{\\mathrm{T}}}\r\n {\\mat{P_{3}}\\begin{bmatrix}0&Y&0&1\\end{bmatrix}^{\\mathrm{T}}} \r\n\\end{eqnarray*}\r\nwhere $W$ in the lane width on the ground in centimeters. \r\n\r\n\\subsection{Matched Filter Convolution}\r\nA matched filter is created after obtaining the lane width in pixel of a row.\r\nRows with width less than one pixels are not are not considered. Since these \r\nrows are either above the horizon or are rows with no lane markings on them.\r\n\r\nThe size of the matched filter is calculated by the following:\r\n\\[ \\textrm{Kernel Size} = (\\textrm{Lane Width} \\times 2) + 1\\] \r\nThis ensures that the size of the kernel is always odd. The kernel needs to be \r\nodd so that they are equal negative signal on both side of the positive \r\nsignal. \r\n\r\nFigure \\ref{fig:kernel} shows a kernel for a lane width marking of size 10 \r\npixels. The sum of the kernel is always zero so that it has a DC response of \r\nzero. \r\n\\begin{figure}\r\n\\centering\r\n\\includegraphics[width=100mm]{figures/kernel}\r\n\\caption{Matched filter kernel for a lane width of 10 pixels.}\r\n\\label{fig:kernel}\r\n\\end{figure}\r\n\r\n\\subsection{Non-Maximum Suppression}\r\nThe results from ``Matched Filter Convolution\" consists of the detected lane \r\nmarking and noise. To simplify the processing I used the ``Non-Maximum \r\nSuppression'' technique to transform a lane marking which has some width to a \r\nlane marking which which is one pixel wide. The ``Non-Maximum Suppression\" used \r\nhere is similar to the one used in edge detectors except that in this case it is \r\napplied to each row of the image and only take the magnitude of the pixels into \r\nconsideration. \r\n\r\nFigure \\ref{fig:non_maxima} shows the effect of applying non-maximum suppression \r\nto an input signal. \r\n\r\n\\begin{figure}\r\n\\centering\r\n\\includegraphics[width=100mm]{figures/non_maxima_suppression.png}\r\n\\caption{The effect of applying Non-Maximum Suppression to an input signal.}\r\n\\label{fig:non_maxima}\r\n\\end{figure}\r\n\r\n\\section{Connected Component}\r\nAfter obtaining the results from non-maxima suppression. The pixels that are\r\nclose to each other and are likely to belong to the same lane marking needs to\r\nbe connected. For this I've used the connected component labeling algorithm \r\n\\fullcite{ConnectedComponent} with slight modification. In the connected \r\ncomponent algorithm the connectivity is either 4-connected or 8-connected. \r\n\r\nIn the 4-connected, connected component labeling algorithm a point $p$ at \r\ncoordinate $(x,y)$ is considered connected if there is another point in it's 4 \r\nneighbors which are:\r\n$$(x+1, y), (x-1, y), (x, y+1), (x, y-1)$$ \r\nIn the case of the 8-connected the four diagonals neighbors are taken into account as well.\r\n$$(x+1, y+1), (x+1, y-1), (x-1, y+1), (x-1, y-1)$$ \r\n \r\n \r\n\\begin{figure}\r\n \\centering\r\n \\begin{tabular}{c c}\r\n \\includegraphics[width=100mm]{figures/connected_component1.png} &\r\n \\includegraphics[width=60mm]{figures/pixel_for_connected_component.png} \\\\\r\n (a) & (b) \\\\\r\n \\end{tabular}\r\n \\caption{Connectivity for Connected Component}\r\n \\label{fig:connected_component}\r\n\\end{figure}\r\n \r\nFigure \\ref{fig:connected_component}(b) shows a sample of road lane marking that \r\nneeds to be connected. It can be seen that a road lane marking will only have \r\none pixel at each row and the closest pixel belonging to the same marking will \r\nbe either on the rows above or below or both. Figure \r\n\\ref{fig:connected_component}(a) shows the connectivity used to connect the \r\nindividual pixels of the road lane marking. The wideness of the connectivity is \r\nto ensure that all pixel in the lane marking gets connected since the distance \r\nbetween any two neighboring pixel in the lane marking varies. \r\n\r\n\\section{Image Plane To Ground Plane Transformation}\r\nIn order to do further post processing the detected road lanes needs to be \r\ntransformed from the image plane to the ground plane (bird-eye view). \r\n\r\nEquation ~\\ref{eq:camera_eq} already describes the relationship between a point in \r\nthe image plane and a point in the real world. Assuming that the ground is flat \r\nwould result in $Z = 0$. Hence equation ~\\ref{eq:camera_eq} can be written as: \r\n\\begin{eqnarray*}\r\n\\begin{bmatrix}x\\\\y\\\\1\\end{bmatrix} & = & \r\n\\begin{bmatrix}\r\np_{11} & p_{12} & p_{13} & p_{14} \\\\\r\np_{21} & p_{22} & p_{23} & p_{24} \\\\\r\np_{31} & p_{32} & p_{33} & p_{34}\r\n\\end{bmatrix}\r\n\\begin{bmatrix}X\\\\Y\\\\0\\\\1\\end{bmatrix}\r\n\\end{eqnarray*}\r\n\\begin{eqnarray*}\r\n& = & \\begin{bmatrix}\r\n p_{11} & p_{12} & p_{14} \\\\\r\n p_{21} & p_{22} & p_{24} \\\\\r\n p_{31} & p_{32} & p_{34} \r\n\\end{bmatrix}\\begin{bmatrix}X\\\\Y\\\\1\\end{bmatrix}\\\\\r\n & = & \r\n\\mathrm{H}\\begin{bmatrix}X\\\\Y\\\\1\\end{bmatrix}\r\n\\end{eqnarray*}\r\nTherefore in order to calculate the ground point we can take the inverse of and \r\nmultiply it with the image point. \r\n\\begin{eqnarray*}\r\n\\mathrm{H}^{-1} \\begin{bmatrix}x\\\\y\\\\1\\end{bmatrix} & = & \r\n\\begin{bmatrix}X\\\\Y\\\\1\\end{bmatrix}.\r\n\\end{eqnarray*}\r\n\r\n\\section{Estimating Consistent Lane Marking}\r\n\r\nEach lane marking is just a set of points, these points can be represented in \r\neither the image plane or the ground plane. The transformation was described in \r\nthe previous section. After transforming the lane marking to the ground plane. A \r\nsimple RANSAC line-fitting algorithm is applied to each lane marking this helps \r\nin removing any outliers that may be present in any lane marking. \r\n\r\n\\subsection{Lateral Distance Estimation}\r\nOn the roads they are both straight and curved lane marking. \r\nSometimes even straight lane marking appear as curved due to the orientation of \r\nthe camera. As of this writing I've assumed all lane marking on to be a line. \r\n\r\nIn order to find the lateral distance from the vehicle to each side \r\nof the lane all the dash lane marking that are consistent must be grouped \r\ntogether to form just one line. The following steps describes how consistent \r\nlane marking are grouped together. \r\n\r\n\\begin{enumerate}\r\n\\item Randomly select two lane marking.\r\n\\item If the two lane marking are not consistent, then repeat step 1. Two lane marking $L_{1}$ and $L_{2}$ are considered consistent if the following \r\nconditions are met: \r\n\\[ \r\n\\left(|v_{1} \\cdot v_{2}| > \\theta_{1} \\right) \\textbf{ and }\r\n\\left(\\left|v_{1} \\cdot \\frac{P_{2} - P_{1}}{\\vectornorm{P_{2} - P_{1}}}\\right| \r\n> \\theta_{2} \\right) \\textbf{ and }\r\n\\left(\\left|v_{2} \\cdot \\frac{P_{2} - P_{1}}{\\vectornorm{P_{2} - P_{1}}}\\right| \r\n> \\theta_{3} \\right)\r\n\\] \r\nwhere $v_{1,2}$ are the normalized unit vector representing the line $L_{1,2}$ \r\nand $p_{1,2}$ is a point that is on the line $L_{1,2}$.\r\n\\item Estimate a line $l$ from the two lane marking found in step 2.\r\n\\item Using RANSAC find all the lane marking that are consistent with the line \r\n$l$ formed in the previous step. These consistent lane marking are the inliers \r\nof line $l$\r\n\\item Pick the set that contain the highest amount of inliers. \r\n\\item Remove the inliers found in step 5. \r\n\\item Repeat step 1 until no two lane marking that are consistent with each \r\nother can be found. \r\n\\end{enumerate}\r\n\r\nOnce the set of inliers $\\mathrm{I} =\\{\\mathrm{I_{1}}, \\mathrm{I_{2}}, . . \r\n., \\mathrm{I_{n}}\\}$ where $\\mathrm{I_{i}}$ are consistent lane \r\nmarking are found. The distance to the left and right boundary of the lane is \r\ncalculated as follow: \r\n\\begin{enumerate}\r\n \\item Split $\\mathrm{I}$ into $\\mathrm{I_{L}}$ and $\\mathrm{I_{R}}$ where $\\mathrm{I_{L}}$ contain inliers which has the closest lane marking on the left side of the origin. Similarly $\\mathrm{I_{R}}$ contain inliers which has the closest lane marking on the right side of the origin.\r\n \\item Find the distance from the origin to each inliers in $\\mathrm{I_{L}}$ and $\\mathrm{I_{R}}$\r\n \\item The minimum distance to the left and right inliers are assumed to be the distance to the left and right boundary of the lane. \r\n\\end{enumerate}", "meta": {"hexsha": "8cd51337079faef7fcc2cc68ae0978855c02d65a", "size": 14660, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "backup/ch3.tex", "max_stars_repo_name": "zkan/dissertation", "max_stars_repo_head_hexsha": "458c5fce241973008bdcc3958bdf962b9197e593", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "backup/ch3.tex", "max_issues_repo_name": "zkan/dissertation", "max_issues_repo_head_hexsha": "458c5fce241973008bdcc3958bdf962b9197e593", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "backup/ch3.tex", "max_forks_repo_name": "zkan/dissertation", "max_forks_repo_head_hexsha": "458c5fce241973008bdcc3958bdf962b9197e593", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.908496732, "max_line_length": 287, "alphanum_fraction": 0.7031377899, "num_tokens": 4251, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632979641571, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.34582063799634805}} {"text": "%% LaTeX2e Template by Stephen Iota (https://stepheniota.com/)\n%% last updated: May 2019\n\\documentclass[11pt]{article}\n\\usepackage[margin=2.5cm]{geometry}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath,amssymb,amsthm,physics}\n\\usepackage{mathtools} % for boxed answers in align environments\n\\usepackage{cancel}\n\\usepackage{graphicx}\n%\\usepackage[shortlabels]{enumitem} % change labels in enum/item environments\n\\usepackage[labelfont=bf,font=small]{caption}\n\\usepackage[dvipsnames]{xcolor}\n%\\usepackage[big]{titlesec} % [small,medium,big]\n\\usepackage{fancyhdr} %http://tug.ctan.org/tex-archive/macros/latex/contrib/fancyhdr/fancyhdr.pdf\n%\\usepackage[noadjust]{cite}\n%\\usepackage{lipsum}\n\\usepackage[\n\tcolorlinks=true,\n\tcitecolor=NavyBlue!90!black,\n\tlinkcolor=green!50!black,\n\turlcolor=RoyalBlue,\n\thypertexnames=false]{hyperref}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% My commands & environments %%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newcommand{\\email}[1]{\\texttt{\\href{mailto:#1}{#1}}}\n\\renewcommand{\\d}[1]{\\ensuremath{\\operatorname{d}\\!{#1}}}\n%\\newcommange{\\ave}[1]{\\ensuremath{\\langle {#1} \\rangle}}\n\n%\\newenvironment{question}[0]{\n%\\bigbreak\n%\\noindent\n%}\n\\theoremstyle{definition}\n\\newtheorem{question}{Part}[section]\n\\newtheorem*{solution}{Solution}\n%\\newenvironment{solution}[0]{\n%\\bigbreak\n%\\noindent\n%\\textbf{Solution:}\n%}\n\n%%%%%%%%%%%%%%%%%%\n%% Front Matter %%\n%%%%%%%%%%%%%%%%%%\n\\pagestyle{fancy}\n\\fancyhead[L]{\\footnotesize{\\leftmark}}\n\\fancyhead[C]{}\n\\fancyhead[R]{\\footnotesize{ IOTA \\textbf{\\thepage}}}\n\\fancyfoot[L,C,R]{}\n\\thispagestyle{plain} % no hf on first page\n%\\pagenumbering{gobble} % no page numbers\n%\\setcounter{section}{-1}\n\\graphicspath{{figures/}} % set directory for figures\n\\numberwithin{equation}{section}\n\\numberwithin{figure}{section}\n\n\n%%%%%%%%%%%%%\n%%% Title %%%\n%%%%%%%%%%%%%\n\\begin{document}\n\n\\begin{center}\n{\\LARGE \\textsc{Statistical Mechanics}: \\textbf{Final Exam}}\n\\end{center}\n\\bigbreak\n\n%%%%%%%%%%\n%% INFO %%\n%%%%%%%%%%\n\\begin{tabular}{rl}\n\\textsc{Name}:&\t\tStephen Iota (\\email{siota001@ucr.edu})\n\\\\\n\\textsc{Course}:&\t\tPhysics 133 (Spring 2019), Prof.~Kuhlman\n\\\\\n\\textsc{Date}:&\t\t\\today\n\\end{tabular}\n\\bigbreak\n\n\n\n%%%%%%%%%%%%%%\n%% PROBLEMS %%\n%%%%%%%%%%%%%%\n\n\\noindent\nSethna problems 2.1, 3.11, 5.2, 5.10, 7.3 and \\textbf{An unlikely event}. Final answers are \\boxed{\\text{boxed}}.\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% RANDOM WALKS IN GRADE SPACE %%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Random walks in grade space}\nConsider a multiple-choice exam with ten questions of ten points each. Each problem on the exam is equally difficult. Students score a mean of $70$ on the exam. Assume all students are identical, and each question is answered at random with a probability of $0.7$ of being correct.\n\n%%%%%%%%%%%%\n%% PART A %%\n%%%%%%%%%%%%\n\\begin{question}\n\\textbf{(a)}~What is the expected mean and standard deviation for the exam?\n\\end{question}\n\n\\begin{solution}\nLet $p_+ = 0.7$ be the probability of a correct answer and $1 - p_+ = p_-$ be the probability of an incorrect answer; let $s \\in \\mathbb{S}$ be an exam score of a student in the set $\\mathbb{S}$ of all possible student scores, and let $N$ be the number of questions on the exam. The mean score $\\expval{s}$ is given by\n\\begin{equation}\n\\expval{s_N} = \\sum_{n = 0}^N \\binom{N}{n} \\ (N-n)p_-\\cdot(N-n)p_+\\cdot10n \\ \\text{pts}\n\\end{equation}\nand the rms score $\\expval{s^2}$is\n\\begin{equation}\n\\expval{s_N^2} = \\sum_{n = 0}^N \\binom{N}{n} \\ (N-n)p_-\\cdot(N-n)p_+\\cdot10n \\ \\text{pts}^2.\n\\end{equation}\nStandard deviation $\\sigma$ is related to the mean and rms by\n\\begin{equation}\n\\sigma = \\sqrt{\\expval{s^2} - \\expval{s}^2}.\n\\end{equation}\n\nFirst lets consider $N = 1$ questions on the exam. The mean, rms and std dev in this case are\n\\begin{align}\n\\expval{s_1} &= p_+(10 \\ \\text{pts}) = 7 \\ \\text{pts}\n\\\\\n\\expval{s_1^2} &= p_+(10 \\ \\text{pts})^2 = 70 \\ \\text{pts}\n\\\\\n\\sigma &= \\sqrt{70 - 7^2} = \\sqrt{21} \\ \\text{pts}\n\\end{align}\nWe can use the properties of random walks to see the evolution of these parameters as the number of questions on the exam $N$ increases. The mean score $\\expval{s_N}$ will be $7N$ pts. The std dev scales by $\\sqrt{N}$. This means that $\\sigma_N = \\sqrt{21N}$ pts.\n\nFor an exam of 10 questions, we find a mean of 10 pts and std dev of $\\sqrt{210}$.\n\\begin{align}\n\\Aboxed{\\expval{s_{10}} &= 10 \\ \\text{pts}}\n\\\\\n\\Aboxed{\\expval{s_{10}^2} &= \\sqrt{210} \\ \\text{pts} \\approx 14.5 \\ \\text{pts}}\n\\end{align}\n\\end{solution}\n\n\n\\newpage\n%%%%%%%%%%%%\n%% PART B %%\n%%%%%%%%%%%%\n\\begin{question}\n\\textbf{(b)}~A typical exam with mean 70 might have an observed std dev of about 15. What physical interpretation do you make of the ratio of the random standard deviation and the observed one?\n\\end{question}\n\n\\begin{solution}\nFor a completely random model with 10 questions, the mean score is $\\expval{s} = 50$ pts and the std dev is $\\sigma_\\text{random} = \\sqrt{250}$.\n\\begin{equation}\n\\frac{\\sigma}{\\sigma_\\text{random}} = \\frac{\\sqrt{210}}{\\sqrt{250}} \\approx \\frac{14.5}{15.8} \\approx 0.92\n\\end{equation}\nStudents in our model have a smaller spread of scores than in a completely random model. In our model, we made our students ``smarter.'' In a real exam setting, students rarely randomly guess on all questions. They have a higher probability of getting each question correct, which is why our model is slightly more accurate than the random model.\n\\end{solution}\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%\n%% Maxwell Relations %%\n%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Maxwell relations}\n\nConsider the microcanonical formula for the equilibrium energy $E(S,V,N)$ of some general system. One knows that the second derivatives of energy are symmetric; at fixed $N$, we get the same answer whichever order we take partial derivatives with respect to $S$ and $V$.\n\n\\begin{question}\n\\textbf{(a)} Use this to show the Maxwell relation\n\\begin{equation}\n\\pdv{T}{V}\\eval_{S,N} = - \\pdv{P}{S}\\eval_{V,N}.\\label{eq:2.0}\n\\end{equation}\nGenerate two other similar formul{\\ae} by taking other second partial derivatives of $E$.\n\\end{question}\n\n\\begin{solution}\nConsider the fundamental thermodynamic relation\n\\begin{equation}\n\\d{E} = T\\d{S} - P\\d{V} + \\mu\\d{N}.\n\\end{equation}\nKeeping $N$ constant, take the following derivatives.\n\\begin{align}\n\\pdv{E}{V}\\eval_{S,N} &= -P\n\\\\\n\\pdv{S}\\eval_{V,N} \\pdv{E}{V}\\eval_{S,N} &= - \\pdv{P}{S}\\eval_{V,N}\\label{eq:2.1}\n\\end{align}\nSince the second derivative of energy is symmetric, we can switch the order of integration in the second step.\n\\begin{align}\n\\pdv{V}\\eval_{S,N} \\pdv{E}{S}\\eval_{V,N} &= \\pdv{T}{S}\\eval_{V,N}\\label{eq:2.2}\n\\end{align}\nFrom eq.~\\eqref{eq:2.1} and \\eqref{eq:2.2}, it follows that\n\\begin{equation}\n\\boxed{\\pdv{T}{V}\\eval_{S,N} = - \\pdv{P}{S}\\eval_{V,N}.}\n\\end{equation}\n\nUsing the same recipe, we can construct more Maxwell relations. Another relationship is given by taking partials of energy with respect to volume and number of particles.\n\\begin{align}\n\\pdv{E}{V}\\eval_{S,N} &= -P\n\\\\\n\\pdv{N}\\eval_{S,V} \\pdv{E}{V}\\eval_{S,N} &= -\\pdv{P}{N}\\eval_{S,V}\n\\\\\n\\pdv{V}\\eval_{S,N} \\pdv{E}{N}\\eval_{S,V} &= \\pdv{\\mu}{N}\\eval_{S,V}\n\\\\\n\\Aboxed{\\pdv{\\mu}{V}\\eval_{S,N} &= -\\pdv{P}{N}\\eval_{S,V}}\n\\end{align}\nAnother one by differentiating energy by entropy and number of particles.\n\\begin{align}\n\t\\pdv{E}{S}\\eval_{V,N} &= T\n\t\\\\\n\t\\pdv{N}\\eval_{S,V} \\pdv{E}{S}\\eval_{V,N} &= -\\pdv{T}{N}\\eval_{S,V}\n\t\\\\\n\t\\pdv{S}\\eval_{V,N} \\pdv{E}{N}\\eval_{S,V} &= \\pdv{\\mu}{S}\\eval_{V,N}\n\t\\\\\n\t\\Aboxed{\\pdv{T}{N}\\eval_{S,V} &= \\pdv{\\mu}{S}\\eval_{V,N}}\n\\end{align}\n\\end{solution}\n\n\\begin{question}\n\\textbf{(b)}~Statistical mechanics check of the Maxwell relation. Using\n\\begin{equation}\nS(E,V,N) = \\frac{5}{2} Nk_B + Nk_B\\log\\Bigg[\\frac{V}{Nh^3}\\bigg( \\frac{4\\pi m E}{3N} \\bigg)^{3/2}\\Bigg]\\label{eq:2b1}\n\\end{equation}\nderive formul{\\ae} for $E(S,V,N)$, $T(S,V,N) = (\\pdv*{E}{S})|_{V,N}$, and $P(S,V,N) = -(\\pdv*{E}{V})|_{S,N}$ for the ideal gas. Show explicitly that eq.~\\eqref{eq:2.0} is satisfied.\n\\end{question}\n\n\\begin{solution}\nTo find $E(S,V,N)$, rearrange eq.~\\eqref{eq:2b1}.\n\\begin{align}\n\\log\\Bigg[\\frac{V}{Nh^3}\\bigg( \\frac{4\\pi m E}{3N} \\bigg)^{3/2}\\Bigg] &= \\frac{S}{Nk_B} - \\frac{5}{2}\n\\\\\n\\frac{V}{Nh^3}\\bigg( \\frac{4\\pi m E}{3N} \\bigg)^{3/2} &= \\exp(\\frac{S}{Nk_B} - \\frac{5}{2})\n\\\\\n\\bigg( \\frac{4\\pi m E}{3N} \\bigg)^{3/2} &= \\frac{Nh^3}{V} \\exp(\\frac{S}{Nk_B} - \\frac{5}{2})\n\\\\\n\\frac{4\\pi m E}{3N} &= \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})\n\\\\\n\\Aboxed{E &= \\frac{3N}{4\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})}\\label{eq:2b2}\n\\end{align}\n\nTo find an expression for $T(S,V,N)$, take the derivative of eq.~\\eqref{eq:2b2} with respect to $S$, keeping $V,N$ constant.\n\\begin{align}\nT(S,V,N) &= \\pdv{E}{S}\\eval_{V,N}\n\\end{align}\n\\begin{align}\n\\begin{aligned}\n\\pdv{E}{S} &= \\pdv{S} \\frac{3N}{4\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\eval_{V,N}\n\\\\\n&= \\frac{3N}{4\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\pdv{S} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\eval_{V,N}\n\\\\\n&= \\frac{3N}{4\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\pdv{S} \\frac{2}{3}\\frac{S}{Nk_B} \\eval_{V,N}\n\\\\\n&= \\frac{1}{2k_B\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})\n\\end{aligned}\n\\end{align}\n\\begin{align}\n\\Aboxed{T(S,V,N) &= \\frac{1}{2k_B\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})}\\label{eq:2bT}\n\\end{align}\n\nNext, let's find an expression for $P(S,V,N)$.\n\\begin{equation}\nP(S,V,N) = -\\pdv{E}{V}\\eval_{S,N}\n\\end{equation}\n\\begin{align}\n\t\\begin{aligned}\n\\pdv{E}{V}\\eval_{S,N} &= - \\pdv{V} \\frac{3N}{4\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\eval_{S,N}\n\\\\\n&= - \\frac{3N}{4\\pi m} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\pdv{V} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\eval_{S,N}\n\\\\\n&= \\frac{1}{2}\\frac{N^2h^3}{\\pi m V^2} \\Big( \\frac{V}{Nh^3} \\Big)^{1/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})\n\\end{aligned}\n\\end{align}\n\\begin{equation}\n\\boxed{P(S,V,N) = \\frac{1}{2}\\frac{N^2h^3}{\\pi m V^2} \\Big( \\frac{V}{Nh^3} \\Big)^{1/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})}\\label{eq:2bP}\n\\end{equation}\n\nNow, we can show that eq.~\\eqref{eq:2.0} is valid. First take the derivative of eq.~\\eqref{eq:2bT} with respect to volume.\n\\begin{equation}\n\\begin{aligned}\n\\pdv{T}{V}\\eval_{S,N} &= \\pdv{V} \\frac{1}{2k_B\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\eval_{S,N}\n\\\\\n&= \\frac{1}{2k_B\\pi m} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\pdv{V} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3}\n\\\\\n&= - \\frac{1}{3} \\frac{1}{k_B\\pi m} \\Big(\\frac{V}{Nh^3} \\Big)^{1/3} \\Big( \\frac{Nh^3}{V^2} \\Big) \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\label{eq:2c}\n\\end{aligned}\n\\end{equation}\nNow, take the derivative of eq.~\\eqref{eq:2bT} with respect to $S$.\n\\begin{equation}\n\\begin{aligned}\n\\pdv{P}{S}\\eval_{V,N} &= \\pdv{S} \\frac{1}{2k_B\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\eval_{V,N}\n\\\\\n&= \\frac{1}{2k_B\\pi m} \\Big( \\frac{Nh^3}{V} \\Big) ^{2/3} \\pdv{S} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\eval_{V,N}\n\\\\\n&= \\frac{1}{3} \\frac{Nh^3}{\\pi m V^2k_B} \\Big( \\frac{V}{Nh^3} \\Big)^{1/3} \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3})\n\\\\\n&= \\frac{1}{3} \\frac{1}{k_B\\pi m} \\Big(\\frac{V}{Nh^3} \\Big)^{1/3} \\Big( \\frac{Nh^3}{V^2} \\Big) \\exp(\\frac{2}{3}\\frac{S}{Nk_B} - \\frac{5}{3}) \\label{eq:2d}\n\\end{aligned}\n\\end{equation}\n\nWe find that eq.~\\eqref{eq:2c} is equal to negative eq.~\\eqref{eq:2d}, thus proving the Maxwell relation eq.~\\eqref{eq:2.0}. $\\qedhere$\n\\end{solution}\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% Burning information and Maxwellian demons %%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Burning information and Maxwellian demons}\n\n\\begin{figure}[p]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{Final_Fig1}\n\\caption{Minimalist digital memory tape.\\label{fig:tape}}\n\\end{figure}\n\n\\begin{figure}[p]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{Final_Fig2}\n\\caption{Expanding piston.\\label{fig:piston}}\n\\end{figure}\n\n\\begin{figure}[p]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{Final_Fig3}\n\\caption{Information-burning engine.\\label{fig:engine}}\n\\end{figure}\n\n\\begin{figure}[p]\n\\centering\n\\includegraphics[width=0.4\\linewidth]{Final_Fig4}\n\\caption{Pistons resetting a \\textit{known} bit for free.\\label{fig:zero}}\n\\end{figure}\n\n\nIs there a minimum energy cost for taking a measurement? Let's try to find out. We start by addressing the connection between information entropy and thermodynamic entropy. Can we burn information as fuel?\n\nConsider a really frugal digital memory tape, with one atom used to store each bit (fig~\\ref{fig:tape}). The tape is a series of boxes, with each box containing one ideal gas atom. The box is split into two equal pieces by a removable central partition. If the atom is in the top half of the box, the tape reads one; else the tape reads zero. The side walls are frictionless pistons that may be used to push the atom around.\n\n\\emph{If} we know the atom position in the $n$th box, we can move the other side wall in, remove the partition, and gradually retract the piston to its original position (fig~\\ref{fig:piston})--destroying our information about where the atom is, but extracting useful work.\n\n\\begin{question}\n\\textbf{(a)~Burning the information.} Assuming the gas expands at a constant temperature $T$, how much work $\\int P\\d{V}$ is done by the atom as the piston retracts?\n\\end{question}\n\n\\begin{solution}\nThis is an isothermal process; $T$ and $N$ are constant. Work in thermodynamic processes is given by $\\int P\\d{V}$, where $P = Nk_BT/V$. Before the piston retracts, $P_iV_i$ is equal to $Nk_BT$. We make this substitution in the integral. Finally we integrate to find the magnitude of the result. The final answer is \\emph{negative}; work is being done \\emph{by} the environment \\emph{on} the atom and gas.\n\\begin{align}\n\\begin{aligned}\nW& = - \\int_{V_i}^{V_f} P \\d{V}\n= - P_iV_i \\int_{V_i}^{V_f} \\frac{\\d{V}}{V}\n\\\\\n& = \\boxed{P_iV_i \\log \\! \\Bigg[ \\frac{V_i}{V_f} \\Bigg]}\\label{eq:work}\n\\end{aligned}\n\\end{align}\n\\end{solution}\n\n\\newpage\n\\begin{question}\n\\textbf{(b)~Rewriting a bit.} Give a sequence of particle insertion, partition removal, and adiabatic side-wall motions that will reversibly convert a bit zero into a bit one, with no net work done on the system.\n\\end{question}\n\n\\begin{solution}\n~\n\\\\\n\\begin{enumerate}\n\\item Place the atom into the bottom half of the partitioned box. Our bit is now in state zero.\n\n\\item Adiabatically snap the top piston down to the position of the partition. Atom still bit zero. There is no time for heat transfer; entropy gain zero. This motion costs us work $W_1$ equal to what we calculated in part \\textbf{(a)} (eq.~\\ref{eq:work}).\n\n\\item Remove the partition from underneath the top piston and place it just above the bottom piston. The atom is now in the ``top'' part of the box (bit one).\n\n\\item Adiabatically snap the bottom piston down the same distance the top piston traveled; the volume of the box is now back to the initial value. The atom is in bit one state. This work done by the atom $W_2$ is equal to $-W_1$, thus we have successfully changed to atom from bit zero to bit one with zero net work done on the system.\n\\end{enumerate}\n\\end{solution}\n\n\\begin{question}\n\\textbf{(c)~Forgetting a bit.} Suppose the atom location in the $n$th box is initially known. What is the change in entropy, if the partition is removed and the available volume doubles? Give both the thermodynamic entropy and the information entropy\n\\end{question}\n\n\\begin{solution}\n\\textit{Thermodynamic entropy.} If we define thermodynamic entropy change to be the ratio of heat flow to temperature, we can argue $\\Delta S_\\text{thermo} = 0$ if we double the volume by adiabatically drawing the pistons back. In addition, it is difficult to define temperature for a single atom. However, we can still think of $\\Delta S_\\text{thermo}$ as entropy change of mixing.\n\nWe can write down the entropy of mixing as\n\\begin{align}\nS& = k_B \\log \\Omega \\label{eq:S},\n\\end{align}\nwhere $\\Omega$ is the configurational energy-shell volume. For $N$ atoms in a box, the number of configurations in a volume $V$ is given by $V^N/N!$. Thus for our atom in a box, $\\Omega_i = V$ and $\\Omega_f = 2\\Omega_i$\n\\begin{align}\n\\Aboxed{\\Delta S_\\text{thermo} & = S_f - S_i = k_B \\log \\frac{\\Omega_f}{\\Omega_i} = k_B \\log 2} \\label{eq:DSthermo}\n\\end{align}\n\n\\textit{Information entropy.} We can write down information entropy generally as\n\\begin{align}\nS_\\text{info} & = - k_S \\sum_k p_k \\log p_k = - \\sum_k p_k \\log_2 p_k\\label{eq:Sinfo},\n\\end{align}\nwhere $p_k$ is the probability of each possible state $k$ of a bit. Before the volume is doubled, we already know the position of the atom in the box. Suppose we know it is in a bit zero state. The initial probability of the atom being in bit zero state is one; probability of bit one state is zero. Once we remove the partition and double the volume, we lose information about the atom's state. Each state becomes equally probable. We use this change in probabilities to write down $\\Delta S_\\text{info}$.\\footnote{eqs.~\\eqref{eq:DSthermo} \\& \\eqref{eq:DSinfo} differ by nothing more than a normalization factor. We write thermodynamic entropy in units of energy per temperature, hence the Boltzmann factor $k_B$. When speaking about the entropy of bits, it makes more sense to present our results in base 2.}\n\\begin{align}\nS_i &= -k_S \\cdot(1) \\cdot \\log 1 = 0\n\\\\\nS_f &= -k_S \\cdot(2) \\cdot \\frac{1}{2} \\log \\frac{1}{2} = \\log_2{2}\n\\\\\n\\Aboxed{\n\\Delta S_\\text{info} &= S_f - S_i = k_S \\log{2} = 1}\\label{eq:DSinfo}\n\\end{align}\n\\end{solution}\n\n\\begin{question}\n\\textbf{(d)~Demonic states.} What prevents a Maxwellian demon from using an atom in an unknown state to extract work? The demon must first measure which side of the box the atom is on; this measurement needs no energy expenditure. (i) After the bit has been burned (see fig~\\ref{fig:engine}), is the demon in a known state?\\footnote{The demon can be thought of as a two-state system (e.g.~another partitioned box). If the demon starts in a known state, one can copy the state of the box into the demon with no cost in entropy or energy, by adiabatically turning on an appropriate coupling. The demon has now measured the atom position, and can extract work from the pistons.}\n(ii) What is its entropy? (iii) How much energy would it take to return the demon to its original state, at temperature T? (iv) Is the second law violated?\n\\end{question}\n\n\\begin{solution}\n~\\\\\n\\begin{enumerate}\n\\item[(i)] \\boxed{\\text{Yes,}} after the bit has been burned, the demon is in a known state. The information from the bit has been copied to the demon prior to the bit's burning-for-work.\n\n\\item[(ii)] Burning the bit, however, costs us entropy. Initially the bit was in a low entropy state. The cost of ``forgetting a bit'' is given by $\\boxed{k_B \\log{2}}$ (see eqs.~\\ref{eq:DSthermo} \\& \\ref{eq:DSinfo}).\n\n\\item[(iii)] It would take \\boxed{0} energy to return the demon to its original state, at temperature $T$. Its original state was a random state. We can wipe a bit by removing the partition and moving both pistons adiabatically in the same direction for a certain distance (see fig~\\ref{fig:zero}).\n\n\\item[(iv)] The Second Law of Thermodynamics states that the entropy of a closed system never decreases over time. This law is \\boxed{\\text{not violated.}} We can extract an unlimited amount of work with a demon by copying information from one bath to another. However, whenever we take energy from a bath, we have to pay for it with entropy. This is a fundamental truth about the Universe.\n\\end{enumerate}\n\\end{solution}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% ENTROPY INCREASES: DIFFUSION %%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Entropy increases: diffusion}\n\nEntropy technically does not increase for a closed system. However, in most of the coarse-grained theories, entropy does increase; when we integrate out degrees of freedom, we provide a means for the information about the initial conditions to be destroyed.\n\nLet $\\rho(x,t)$ obey the one-dimensional diffusion equation $\\pdv*{\\rho}{t} = D\\pdv*[2]{p}{x}$. Assume that the density $\\rho$ and all its gradients die away rapidly at $x = \\pm \\infty$.\\footnote{Also, assume $\\pdv*[n]{\\rho}{x} \\log{\\rho}$ goes to zero at $x = \\pm \\infty$, even though $\\log{\\rho}$ goes to $-\\infty$.}\n\n\\begin{question}\nDerive a formula for the time derivative of the entropy $S = -k_B \\int \\rho(x)\\log{\\rho(x)} \\d{x}$ and show that it is strictly increasing in time.\n\\end{question}\n\n\n\n\\begin{solution}\nWe start by taking the time derivative of the definition of entropy.\n\\begin{align}\nS &= -k_B \\int_{\\text{all space}} \\rho \\log{\\rho} \\d{x}\n\\\\\n\\pdv{S}{t} &= -k_B \\int \\dot{\\rho} \\log{\\rho} + \\dot{\\rho} \\d{x}\n\\end{align}\nSince $\\rho$ obeys the diffusion equation, we can make the following substitution.\n\\begin{align}\n\\dot{S} &= -Dk_b \\int \\rho'' \\log{\\rho} + \\rho'' \\d{x}\n\\\\\n\\dot{S} &= -Dk_B \\bigg[\\cancelto{0}{\\rho'\\eval_{-\\infty}^{\\infty}} + \\int \\rho'' \\log{\\rho} \\d{x}\\bigg]\n\\\\\n\\dot{S} &= -Dk_B \\int \\rho'' \\log{\\rho} \\d{x}\n\\end{align}\nThe $\\rho''$ term dies out when integrated over all space because we are assuming the probability density and all its gradients die out at the boundaries.\n\nNext, we will try to get the remaining integrand into a form that is always positive. We will use the \\textsc{di} method for integration by parts.\n\n\\begin{center}\n\\begin{tabular}{lrl}\n\t\t& \tD\t\t& \tI\n\\\\\n+\t\t&\t\t$\\log{\\rho}$\t\t&\t\t$\\rho''$\n\\\\\n-\t\t&\t\t$\\frac{1}{\\rho}\\rho'$\t\t&\t\t$\\rho'$\n\\end{tabular}\n\\end{center}\n\\begin{align}\n\\dot{S} &= -Dk_B \\int \\rho'' \\log{\\rho} \\d{x} = -Dk_B \\bigg[ \\cancelto{0}{\\rho'\\log{\\rho}\\eval_{-\\infty}^{\\infty}} - \\int \\frac{(\\rho')^2}{\\rho} \\d{x} \\bigg]\n\\end{align}\nAfter integrating by parts, our first term dies again because of our initial assumption regarding the probability density (see footnote). We are left with the following integrand.\n\\begin{equation}\n\\dot{S} = Dk_B \\int \\frac{(\\rho')^2}{\\rho} \\d{x}\\label{eq:4.0}\n\\end{equation}\nWe cannot evaluate this integral, but we don't need to. First, consider the $1/\\rho$ term. $\\rho(x,t) \\in [0,1]$ is a probability density, it cannot physically be negative. Next, consider the $(\\rho')^2$ term. The square of any number is always positive. So although we do not know what $\\rho'$ looks like, we only care about its square. Therefore, we know that the integrand in eq.~\\eqref{eq:4.0} will always be positive for all values of $x$ that we are integrating over.\n\nWe can confidently say that the time derivative of entropy is always increasing.\n\\begin{align}\n\\Aboxed{\n\\pdv{S}{t} &\\geq 0}\n\\end{align}\n\\end{solution}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% PHASE SPACE UNITS AND THE ZERO OF ENTROPY %%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{Phase-space units and the zero of entropy}\n\nIn classical mechanics, entropy goes to minus infinity at the temperature is lowered to zero. In quantum mechnics the entropy per particle goes to zero, because states are quantized and the ground state is the only one populated.\n\nThe classical phase-space shell volume $\\Omega(E)\\var E$ has units of ([momentum]$\\cdot$[distance])$^{3N}$. We can make it unitless with Planck's constant $h^{3N}$; if we measure phase-space volume in units of $h$ per dimension, $\\Omega(E)\\var E$ will be dimensionless. Of course, the correct dimension could be a constant times $h$, like $\\hbar \\ldots$\n\n\\begin{question}\n\\textbf{(a)}~\\textbf{Arbitrary zero of the classical entropy.} Show that the width of the energy shell $\\var E$ in the definition of $\\Omega(E)$ does not change the microcanonical entropy per particle\n\\begin{equation}\nS/N = k_B\\log{\\Omega(E})/N\\label{eq:5.0}\n\\end{equation}\nin a large system. Show that the choice of units in phase space does change the classical entropy per particle.\n\\end{question}\n\n\\begin{solution}\nThe definition of classical phase-space is\n\\begin{equation}\n\\Omega(E) = \\frac{1}{\\var E}\\int_{H < E < H+ \\var E } \\d{\\mathbb{P}} \\d{\\mathbb{Q}}\n\\end{equation}\nFor our ensemble, we assume particles are non-interacting, so we do not have to integrate over configurations $\\mathbb{Q}$. The integral over momenta $\\mathbb{P}$ is equal to the difference in volumes of $3N$ dimensional spheres of radii $\\sqrt{2m(E + \\delta E)}$ and $\\sqrt{2mE}$. When we take the logarithm of the different of phase-space volumes for each sphere, we find that the exponents are eliminated thanks to the nifty properties of logarithms, and the $\\var E$'s cancel out.\n\\begin{align}\n\\Omega(E) &= \\frac{1}{\\var E} \\frac{\\pi^{3N/2}}{(3N/2)!} \\Bigg[ (2m(E + \\delta E))^{3N/2} - (2mE)^{3N/2}\\Bigg]\n\\\\\nS &= k_B \\log(\\frac{1}{\\var E} \\frac{\\pi^{3N/2}}{(3N/2)!} \\Bigg[ (2m(E + \\delta E))^{3N/2} - (2mE)^{3N/2}\\Bigg])\n\\\\\n&= k_B \\Bigg[ \\log{\\frac{\\pi^{3N/2}}{(3N/2)!}} + 3N/2\\log( 2m(E + \\var E) - 2mE) - \\log(\\var E) \\Bigg]\n\\\\\nS/N &= k_B \\Bigg[ \\frac{1}{N} \\log{\\frac{\\pi^{3N/2}}{(3N/2)!}} + 3/2\\log{2m} + 3/2\\log{\\var E} -\\log{\\var E} \\Bigg]\n\\end{align}\nThe value of $1/2 \\log{\\var E}$ is very small, so in the end we have shown that $S/N$ is independent of $\\var E$.\n\nHowever, if we chose the normalize our phase-space volume with $h^{3N}$ or $\\hbar^{3N}$, we would get a different result. Therefore, choice of units in phase-space does matter.\n\\end{solution}\n\n\n\n\\begin{question}\n\\textbf{(b)}~\\textbf{Phase-space density of states for a particle in a one-dimensional box.} Show, or note, that the quantum momentum-space density of states for a free quantum particle in a one-dimensional box of length $L$ with periodic boundary conditions is $L/h$. Draw a picture of the classical phase space of this box $(p,x)$, and draw a rectangle of length $L$ for each quantum eigenstate. Is the phase-space area per eigenstate equal to $h$, as assumed?\n\\end{question}\n\n\\begin{solution}\nIn quantum mechanics, energy states are quantized. The energy eigenstates for a 1-$D$ particle in a box are $E_n = \\frac{n^2 h^2}{8m^2 L^2}$. Therefore, the momentum per eigenstate are $p_n = \\frac{nL}{2L}$. Taking into account the periodic boundary conditions (accounts for the factor of 2), we find that the phase space volume is indeed $L/h$.\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=.3\\linewidth]{Final_Fig5}\n\\caption{Phase-space volume of a 1-D particle in a box.}\n\\end{figure}\nThe phase-space volume per eigenstate is indeed equal to $h$.\n\\end{solution}\n\n\n\\begin{question}\n\\textbf{(c)}~\\textbf{Phase-space density of states for $N$ particles in a box.} Show that the density of states for $N$ free particles in a cubical box of volume $V$ with periodic boundary conditions is $V^N/h^{3N}$, and hence that the phase-space volume per state is $h^{3N}$.\n\\end{question}\n\n\\begin{solution}\nThe energy eigenstates are given by $E_{nx,ny,nz} = \\frac{\\hbar^2 \\vb{k}^2}{2m}$ and we find that each 3D wave vector is given by\n$k = \\frac{n_x \\pi}{L} + \\frac{n_y \\pi}{L} + \\frac{n_z \\pi}{L}$. Therefore the phase space volume for one particle is given by $V/h^3$. If we have $N$ particles in a box, each occupies a respective volume, so we take our answer and raise it to the power of $N$. The phase-space volume then turns out to be $V^N/h^{3N}$, and the volume per state is $h^{3N}$.\n\\end{solution}\n\n\n\\begin{question}\n\\textbf{(d)}~\\textbf{Phase-space density of states for a harmonic oscillator.} Consider a harmonic oscillator with Hamiltonian $\\mathcal{H} = p^2/2m + 1/2 mw^2q^2$. Draw a picture of the energy surface with energy $E$, and find the volume of phase space enclosed. What is the volume per energy state?\n\\end{question}\n\n\\begin{solution}\nFor a fixed energy, the phase-space volume turns out to be elliptical. The volume (or rather area) is given by $\\pi p x$.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=.3\\linewidth]{Final_Fig6}\n\t\\caption{Phase-space volume of a quantum harmonic oscillator with energy $E$.}\n\\end{figure}\n\n\\end{solution}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%\n%% AN UNLIKELY EVENT %%\n%%%%%%%%%%%%%%%%%%%%%%%\n\\newpage\n\\section{An unlikely event}\n\nOn June 12, 2019 at 6:59 pm, you are exiting the elevator, walking toward \\textsc{ms\\&e} to hand in your final take home exam for Physics 133. Just a few moments before 6:59 pm, a small meteor falls from the sky and crashes into a physics textbook on the desk of a philosopher trying to understand the meaning of entropy. The meteor is burning at such a high temperature that it instantly incinerates the book and continues burning through floor after floor of the building until coming to rest near a water line adjacent to a 50 year-old water fountain. The pipe becomes super heated, compressing the water in the plumbing and causing a catastrophic failure of the building’s pipes. The compression shock shakes the foundation of the building and, just as you exit the elevator, a micro-earthquake rumbles across the entire campus. Your exam slips from your hands and falls like an autumn leaf right between the crack that separates the elevator from the third floor's surface. The exam can't be recovered. Assume that the observation of this event is extremely unlikely. How much information is contained in such an event?\n\n\\begin{question}\n\\textbf{(a)} From Shannon, we know that the entropy is in fact a measure of \\textit{information}. Suppose that any event, including that described above, occurs with probability $p$. We can associate with the observation of this event a function that quantifies the amount of information, call it $I(p)$. Find the functional form of $I(p)$ such that the following conditions are met:\n\\begin{enumerate}\n\\item Information is \\textit{non-negative}.\n\\item If two events occur independently so that their joint probability is the product of event 1 and event 2, then their information is \\textit{additive}.\n\\item Information is a continuous function of the probability $p$.\n\\item There is \\textit{no information} content to an event that is always observed.\n\\end{enumerate}\n\\end{question}\n\n\\begin{solution}\nWe need a function that satisfies the following four criteria.\n\\begin{enumerate}\n\\item Information is non-negative.\n\\begin{align}\nI(p) & \\geq 0\n\\end{align}\n\n\\item For two independent events $i$ and $j$ where the joint probability that they both occur is $p_i \\cdot p_j$, the total information of the events is given by $I(p_i) + I(p_j)$.\n\\begin{align}\nI(p_i) + I(p_j) & = I(p_i + p_j)\n\\end{align}\n\n\\item Information is a continuous function of the probability $p$. A function $I$ is continuous if and only if the limit\n\\begin{equation}\n\\lim_{p \\to a} I(p) = I(a)\n\\end{equation}\nexists.\n\n\\item If an event $i$ is always observed, that event contains no information such that\n\\begin{align}\n\\text{for an event $i$, if $p_i = 1$, then}\\ I(p_i)& = 0.\n\\end{align}\n\\end{enumerate}\nA simple function that has these four properties is the logarithm function.\\footnote{Proofs left as an exercise for the reader. Or see \\url{https://en.wikipedia.org/wiki/Logarithm}.} Let\n\\begin{align}\n\\Aboxed{I(p) & = - \\log{p} = \\log{\\frac{1}{p}}}\\label{eq:info}\n\\end{align}\nwhere $p \\in [0,1]$ is the probability of a given event occurring.\n\\end{solution}\n\n\\begin{question}\n\\textbf{(b)} Find the value of $I(p)$ for the event described above (i.e.~the probability of observing such an event approaches zero). How much information is contained in an event as unlikely as this one?\n\\end{question}\n\n\\begin{solution}\nThe chances of such an event happening are \\textbf{unlikely}. Let the probability of the unlikely event occurring go to zero. The amount of information contained in the event is given by\n\\begin{equation}\n\\boxed{I(p) = \\lim_{p \\to 0} \\log\\frac{1}{p} = \\infty.}\n\\end{equation}\nThe amount of information contained in this event is infinite.\n\\end{solution}\n\n\\end{document}\n", "meta": {"hexsha": "78b6f3d4cab2051d7060d0dfe80eb53bb49f374a", "size": 32041, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "solved/P133_Final.tex", "max_stars_repo_name": "stepheniota/Physics133-UCR", "max_stars_repo_head_hexsha": "2018ae62e384205459175dbed525930e607f8956", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "solved/P133_Final.tex", "max_issues_repo_name": "stepheniota/Physics133-UCR", "max_issues_repo_head_hexsha": "2018ae62e384205459175dbed525930e607f8956", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "solved/P133_Final.tex", "max_forks_repo_name": "stepheniota/Physics133-UCR", "max_forks_repo_head_hexsha": "2018ae62e384205459175dbed525930e607f8956", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 44.439667129, "max_line_length": 1124, "alphanum_fraction": 0.6865266377, "num_tokens": 10388, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6370307944803832, "lm_q1q2_score": 0.34582062857361023}} {"text": "% Copyright (c) Microsoft Corporation. All rights reserved.\n% Licensed under the MIT License.\n\n%!TEX program=xelatex\n% This cheatsheet is based on the template\n% provided at https://gist.github.com/alexander-yakushev/c773543bf9a957749f79.\n\\documentclass[10pt,english,landscape]{article}\n\\usepackage{multicol}\n\\usepackage{calc}\n\\usepackage[landscape]{geometry}\n\\usepackage{color,graphicx,overpic}\n\\usepackage{amsmath}\n\n% Workaround for left brackets not rendering\n% https://tex.stackexchange.com/questions/520847/left-matrix-bracket-not-rendered-properly\n\n\\newbox\\matrixrepresentationright\n\\sbox\\matrixrepresentationright{\\parbox{\\textwidth}{\n \\footnotesize\n $\\qquad \\underrightarrow{n \\text{ columns}} \\\\\n \\rotatebox[origin=c]{270}{$\\underrightarrow{m \\text{ rows}}$}\n \\begin{bmatrix} \n a_{0,0} & \\cdots & a_{0,n-1} \\\\ \n \\vdots & \\ddots & \\vdots \\\\\n a_{m-1,0} & \\cdots & a_{m-1,n-1} \n \\end{bmatrix}$\n}}\n\n\\newbox\\vectorrepresentationright\n\\sbox\\vectorrepresentationright{\n \\footnotesize\n $\\begin{bmatrix}\n a_0 \\\\ \\vdots \\\\ a_{n-1}\n \\end{bmatrix}$\n}\n\n\\newbox\\matrixadditionleft\n\\sbox\\matrixadditionleft{\\parbox{\\textwidth}{\n \\footnotesize\n Addition \\linebreak\n $\\begin{bmatrix} \n a & b \\\\ \n c & d \n \\end{bmatrix} + \n \\begin{bmatrix}\n e & f \\\\\n g & h\n \\end{bmatrix}$ \n}}\n\n\\newbox\\matrixadditionright\n\\sbox\\matrixadditionright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} \n a + e & b + f \\\\\n c + g & d + h \n \\end{bmatrix} $\n}}\n\n\\newbox\\matrixscalarleft\n\\sbox\\matrixscalarleft{\\parbox{\\textwidth}{\n \\footnotesize\n Scalar product \\newline\n $ a \\cdot \\begin{bmatrix} \n b & c \\\\\n d & e\n \\end{bmatrix} $\n}}\n\n\\newbox\\matrixscalarright\n\\sbox\\matrixscalarright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix}\n a \\cdot b & a \\cdot c \\\\\n a \\cdot d & a \\cdot e\n \\end{bmatrix} $ \n}}\n\n\\newbox\\matrixmultiplicationleft\n\\sbox\\matrixmultiplicationleft{\\parbox{\\textwidth}{\n \\footnotesize\n Matrix product \\newline\n $ \\begin{bmatrix} \n a & b & c \\\\\n d & e & f\n \\end{bmatrix} \\cdot\n \\begin{bmatrix} \n x \\\\\n y \\\\\n z\n \\end{bmatrix} $\n}}\n\n\\newbox\\matrixmultiplicationright\n\\sbox\\matrixmultiplicationright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} \n a \\cdot x + b \\cdot y + c \\cdot z \\\\\n d \\cdot x + e \\cdot y + f \\cdot z \n \\end{bmatrix} $\n}}\n\n\\newbox\\matrixtransposeleft\n\\sbox\\matrixtransposeleft{\\parbox{\\textwidth}{\n \\footnotesize\n Transpose \\newline \n $ \\begin{bmatrix}\n a & b & c\\\\\n d & e & f \n \\end{bmatrix}^T $\n}}\n\n\\newbox\\matrixtransposeright\n\\sbox\\matrixtransposeright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix}\n a & d \\\\\n b & e \\\\\n c & f\n \\end{bmatrix} $ \n}}\n\n\\newbox\\matrixadjointleft\n\\sbox\\matrixadjointleft{\\parbox{\\textwidth}{\n \\footnotesize\n Adjoint \\newline \n $ \\begin{bmatrix}\n a & b & c \\\\\n d & e & f\n \\end{bmatrix}^\\dagger $ \n}}\n\n\\newbox\\matrixadjointright\n\\sbox\\matrixadjointright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix}\n \\overline{a} & \\overline{e} \\\\\n \\overline{b} & \\overline{d} \\\\\n \\overline{c} & \\overline{f}\n \\end{bmatrix} $ \n}}\n\n\\newbox\\outerproductleft\n\\sbox\\outerproductleft{\\parbox{\\textwidth}{\n \\footnotesize\n Outer product \\newline\n $ \\begin{bmatrix} \n a \\\\\n b\n \\end{bmatrix} $ and $\n \\begin{bmatrix} \n x \\\\ \n y \\\\\n z \n \\end{bmatrix} $\n}}\n\n\\newbox\\innerproductleft\n\\sbox\\innerproductleft{\\parbox{\\textwidth}{\n \\footnotesize\n Inner product \\newline\n $ \\left\\langle \n \\begin{bmatrix}\n a \\\\\n b\n \\end{bmatrix} ,\n \\begin{bmatrix} \n c \\\\\n d\n \\end{bmatrix} \n \\right\\rangle $ \n}}\n\n\\newbox\\innerproductright\n\\sbox\\innerproductright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix}\n a \\\\\n b\n \\end{bmatrix}^\\dagger\n \\begin{bmatrix}\n c \\\\\n d\n \\end{bmatrix} =\n \\begin{bmatrix} \n \\overline{a} & \\overline{b} \n \\end{bmatrix} \n \\begin{bmatrix}\n c \\\\\n d\n \\end{bmatrix} = \n \\overline{a} c + \\overline{b} d $\n}}\n\n\\newbox\\outerproductright\n\\sbox\\outerproductright{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix}\n a \\\\\n b\n \\end{bmatrix} \n \\begin{bmatrix}\n x \\\\\n y \\\\\n z\n \\end{bmatrix}^\\dagger =\n \\begin{bmatrix}\n a \\\\\n b \n \\end{bmatrix}\n \\begin{bmatrix}\n \\overline{x} & \\overline{y} & \\overline{z}\n \\end{bmatrix} = $ \\newline\n $ = \\begin{bmatrix} \n a \\cdot \\overline{x} & a \\cdot \\overline{y} & a \\cdot \\overline{z} \\\\\n b \\cdot \\overline{x} & b \\cdot \\overline{y} & b \\cdot \\overline{z} \\\\\n \\end{bmatrix} $ \n}}\n\n\\newbox\\xgate\n\\sbox\\xgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 0 & 1 \\\\ 1 & 0 \\end{bmatrix} $ \n}}\n\n\\newbox\\ygate\n\\sbox\\ygate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 0 & -i \\\\ i & 0 \\end{bmatrix} $\n}}\n\n\\newbox\\zgate\n\\sbox\\zgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 \\\\ 0 & -1 \\end{bmatrix} $\n}}\n\n\\newbox\\igate\n\\sbox\\igate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 \\\\ 0 & 1 \\end{bmatrix} $\n}}\n\n\\newbox\\hgate\n\\sbox\\hgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\frac{1}{\\sqrt{2}}\\begin{bmatrix} 1 & 1 \\\\ 1 & -1 \\end{bmatrix} $\n}}\n\n\\newbox\\sgate\n\\sbox\\sgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 \\\\ 0 & i \\end{bmatrix} $\n}}\n\n\\newbox\\tgate\n\\sbox\\tgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 \\\\ 0 & e^{i\\pi/4} \\end{bmatrix} $ \n}}\n\n\\newbox\\rxgate\n\\sbox\\rxgate{\\parbox{\\textwidth}{\n \\footnotesize\n $\\begin{bmatrix} \\cos\\frac{\\theta}{2} & -i\\sin\\frac{\\theta}{2} \\\\ -i\\sin\\frac{\\theta}{2} & \\cos\\frac{\\theta}{2} \\end{bmatrix} $ \n}}\n\n\\newbox\\rygate\n\\sbox\\rygate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} \\cos\\frac{\\theta}{2} & -\\sin\\frac{\\theta}{2} \\\\ \\sin\\frac{\\theta}{2} & \\cos\\frac{\\theta}{2} \\end{bmatrix} $\n}}\n\n\\newbox\\rzgate\n\\sbox\\rzgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} e^{-i\\theta/2} & 0 \\\\ 0 & e^{i\\theta/2} \\end{bmatrix} $ \n}}\n\n\n\\newbox\\rgate\n\\sbox\\rgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 \\\\ 0 & e^{i\\theta} \\end{bmatrix} $\n}}\n\n\\newbox\\cnotgate\n\\sbox\\cnotgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & \\mathbf{0} & \\mathbf{1} \\\\ 0 & 0 & \\mathbf{1} & \\mathbf{0} \\end{bmatrix} $\n}}\n\n\\newbox\\swapgate\n\\sbox\\swapgate{\\parbox{\\textwidth}{\n \\footnotesize\n $ \\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & \\mathbf{0} & \\mathbf{1} & 0 \\\\ 0 & \\mathbf{1} & \\mathbf{0} & 0 \\\\ 0 & 0 & 0 & 1 \\end{bmatrix} $\n}}\n\n\\newbox\\controlledugate\n\\sbox\\controlledugate{\\parbox{\\textwidth}{\n \\footnotesize\n $\\begin{bmatrix} 1 & 0 & 0 & 0 \\\\ 0 & 1 & 0 & 0 \\\\ 0 & 0 & a_{0,0} & a_{0,1} \\\\ 0 & 0 & a_{1,0} & a_{1,1} \\end{bmatrix} $\n}}\n\n\\newbox\\ccnotgate\n\\sbox\\ccnotgate{\\parbox{\\textwidth}{\n \\footnotesize\n $\\begin{bmatrix}\n 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 \\\\\n 0 & 0 &0 & 0 & 0 & 1 & 0 & 0 \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & \\mathbf{0} & \\mathbf{1} \\\\\n 0 & 0 & 0 & 0 & 0 & 0 & \\mathbf{1} & \\mathbf{0}\n\\end{bmatrix}$\n}}\n\n\\newbox\\braketcx\n\\sbox\\braketcx{\\parbox{\\textwidth}{\n \\footnotesize\n $ |00\\rangle\\langle00| + |01\\rangle\\langle01| + |11\\rangle\\langle10| + |10\\rangle\\langle11| \\newline \\text{or} \\newline |0\\rangle\\langle0| \\otimes I + |1\\rangle\\langle1| \\otimes X $\n}}\n\n\\newbox\\braketcu\n\\sbox\\braketcu{\\parbox{\\textwidth}{\n \\footnotesize\n $ |0\\rangle\\langle0| \\otimes I + |1\\rangle\\langle1| \\otimes U $\n}}\n\n\\newbox\\braketccx\n\\sbox\\braketccx{\\parbox{\\textwidth}{\n \\footnotesize\n $ |00\\rangle\\langle00| + |10\\rangle\\langle01| + a_{0,0}|10\\rangle\\langle10| + \\newline a_{0,1}|10\\rangle\\langle11| + a_{1,0}|10\\rangle\\langle11| + a_{1,1}|11\\rangle\\langle11| \\newline \\text{Or} \\newline |0\\rangle\\langle0| \\otimes I + |1\\rangle\\langle1| \\otimes U $\n}}\n\n\n\n% https://tex.stackexchange.com/a/192067\n\\usepackage{fontawesome}\n \\newfontfamily{\\FA}{[FontAwesome.otf]}\n\n% Known issue in {fontspec}: https://github.com/wspr/fontspec/issues/312#issuecomment-342125206\n% Since {fontspec} is included only when compiling with XeTeX, we guard\n% our fix accordingly.\n\\usepackage{ifxetex}\n \\ifxetex\n \\let\\latinencoding\\relax\n \\usepackage{fontspec}\n \\setmainfont{Segoe UI}\n \\setmonofont{Consolas}\n \\fi\n\n% \\usepackage[T1]{fontenc}\n% \\usepackage[bitstream-charter]{mathdesign}\n\\usepackage[utf8]{inputenc}\n\\usepackage{url}\n\\usepackage{amsfonts}\n\\usepackage{array,booktabs}\n\\usepackage{textcomp}\n\\usepackage[usenames,dvipsnames,table]{xcolor}\n\\usepackage[most]{tcolorbox}\n\\usepackage{menukeys}\n\\usepackage{tabularx}\n\\usepackage{multirow}\n\\usepackage{colortbl}\n\\usepackage{tikz}\n\\usepackage{environ}\n\\usepackage{braket}\n\n\\usetikzlibrary{calc}\n\\pgfdeclarelayer{background}\n\\pgfdeclarelayer{foreground}\n\\pgfsetlayers{background,main,foreground}\n\n\\geometry{top=1cm,left=1cm,right=1cm,bottom=1cm}\n\n\\pagestyle{empty} % Turn off header and footer\n\n% \\renewcommand\\rmdefault{phv} % Arial\n% \\renewcommand\\sfdefault{phv} % Arial\n\n% Redefine section commands to use less space\n\\makeatletter\n\\renewcommand{\\section}{\\@startsection{section}{1}{0mm}%\n {-1ex plus -.5ex minus -.2ex}%\n {0.5ex plus .2ex}%x\n {\\normalfont\\large\\bfseries}}\n\\renewcommand{\\subsection}{\\@startsection{subsection}{2}{0mm}%\n {-1explus -.5ex minus -.2ex}%\n {0.5ex plus .2ex}%\n {\\normalfont\\normalsize\\bfseries}}\n\\renewcommand{\\subsubsection}{\\@startsection{subsubsection}{3}{0mm}%\n {-1ex plus -.5ex minus -.2ex}%\n {1ex plus .2ex}%\n {\\normalfont\\small\\bfseries}}\n\\makeatother\n\n\\setcounter{secnumdepth}{0} % Don't print section numbers\n\\setlength{\\parindent}{0pt}\n\\setlength{\\parskip}{0pt plus 0.5ex}\n\n\\definecolor{TableHead}{rgb}{0.353, 0.329, 0.667}\n\\definecolor{TableRow}{rgb}{0.816, 0.812, 0.902}\n\n% Turn off warnings about slightly overfull boxes\n% https://tex.stackexchange.com/questions/50830/do-i-have-to-care-about-bad-boxes/50850\n\n\\hfuzz=10pt\n\n\\NewEnviron{keysref}[1]{\n % \\begin{center}\n \\smallskip\n \\begin{tikzpicture}\n \\rowcolors{1}{}{TableRow}\n\n \\node (tbl) [inner sep=0pt] {\n \\begin{tabular}{p{2.5cm}p{5.05cm}}\n \\rowcolor{TableHead}\n \\multicolumn{2}{l}{\\normalsize\\textbf{\\color{white}{#1}}}\\parbox{0pt}{\\rule{0pt}{0.3ex+\\baselineskip}}\\\\\n \\BODY\n \\arrayrulecolor{TableHead}\\specialrule{.17em}{0em}{.2em}\n \\end{tabular}};\n \\begin{pgfonlayer}{background}\n \\draw[rounded corners=2pt,top color=TableHead,bottom color=TableHead, draw=white]\n ($(tbl.north west)-(0,-0.05)$) rectangle ($(tbl.north east)-(0.0,0.15)$);\n \\draw[rounded corners=2pt,top color=TableHead,bottom color=TableHead, draw=white]\n ($(tbl.south west)-(0.0,-0.11)$) rectangle ($(tbl.south east)-(-0.0,-0.02)$);\n \\end{pgfonlayer}\n \\end{tikzpicture}\n % \\end{center}\n}\n\n\\NewEnviron{keysrefGates}[1]{\n % \\begin{center}\n \\smallskip\n \\begin{tikzpicture}\n \\rowcolors{1}{}{TableRow}\n\n \\node (tbl) [inner sep=0pt] {\n \\begin{tabular}{p{0.85cm}p{3.55cm}p{4.05cm}p{5.55cm}p{2.95cm}p{2.95cm}p{3.05cm}}\n \\rowcolor{TableHead}\n \\multicolumn{7}{l}{\\normalsize\\textbf{\\color{white}{#1}}}\\parbox{0pt}{\\rule{0pt}{0.3ex+\\baselineskip}}\\\\\n \\BODY\n \\arrayrulecolor{TableHead}\\specialrule{.17em}{0em}{.2em}\n \\end{tabular}};\n \\begin{pgfonlayer}{background}\n \\draw[rounded corners=2pt,top color=TableHead,bottom color=TableHead, draw=white]\n ($(tbl.north west)-(0,-0.05)$) rectangle ($(tbl.north east)-(0.0,0.15)$);\n \\draw[rounded corners=2pt,top color=TableHead,bottom color=TableHead, draw=white]\n ($(tbl.south west)-(0.0,-0.11)$) rectangle ($(tbl.south east)-(-0.0,-0.02)$);\n \\end{pgfonlayer}\n \\end{tikzpicture}\n % \\end{center}\n}\n\n\\NewEnviron{keysrefMultiGates}[1]{\n % \\begin{center}\n \\smallskip\n \\begin{tikzpicture}\n \\rowcolors{1}{}{TableRow}\n\n \\node (tbl) [inner sep=0pt] {\n \\begin{tabular}{p{1.55cm}p{3,75cm}p{5,35cm}p{6,05cm}p{2.95cm}p{3.75cm}}\n \\rowcolor{TableHead}\n \\multicolumn{6}{l}{\\normalsize\\textbf{\\color{white}{#1}}}\\parbox{0pt}{\\rule{0pt}{0.3ex+\\baselineskip}}\\\\\n \\BODY\n \\arrayrulecolor{TableHead}\\specialrule{.17em}{0em}{.2em}\n \\end{tabular}};\n \\begin{pgfonlayer}{background}\n \\draw[rounded corners=2pt,top color=TableHead,bottom color=TableHead, draw=white]\n ($(tbl.north west)-(0,-0.05)$) rectangle ($(tbl.north east)-(0.0,0.15)$);\n \\draw[rounded corners=2pt,top color=TableHead,bottom color=TableHead, draw=white]\n ($(tbl.south west)-(0.0,-0.11)$) rectangle ($(tbl.south east)-(-0.0,-0.02)$);\n \\end{pgfonlayer}\n \\end{tikzpicture}\n % \\end{center}\n}\n\n\n% https://tex.stackexchange.com/a/102523\n\\newcommand{\\forceindent}[1]{\\leavevmode{\\parindent=#1\\indent}}\n\n%% CUSTOM NOTATION %%\n\n\\newcommand{\\qs}{Q\\#}\n\\newcommand{\\unixlike}{\\hfill\\faApple\\faLinux}\n\\newcommand{\\ctrllike}{\\hfill\\faWindows\\faLinux}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\n\\begin{document}\n\n\\raggedright\\\n\n\\Large{\\qs~0.15 Language Quick Reference}\n\n\\footnotesize\n\\begin{multicols}{3}\n\n \\begin{keysref}{Primitive Types}\n 64-bit integers & \\texttt{Int} \\\\\n Double-precision \\newline floats & \\texttt{Double} \\\\\n Booleans & \\texttt{Bool} \\newline \n e.g.: \\texttt{true} or \\texttt{false} \\\\\n Qubits & \\texttt{Qubit} \\\\\n Pauli basis & \\texttt{Pauli} \\newline\n e.g.: \\texttt{PauliI}, \\texttt{PauliX}, \\texttt{PauliY}, or \\texttt{PauliZ} \\\\\n Measurement \\newline results & \\texttt{Result} \\newline\n e.g.: \\texttt{Zero} or \\texttt{One} \\\\\n Sequences of \\newline integers & \\texttt{Range} \\newline\n e.g.: \\texttt{1..10} or \\texttt{5..-1..0} \\\\\n Strings & \\texttt{String} \\newline \n e.g.: \\texttt{\"Hello Quantum!\"} \\\\\n \"Return no \\newline information\" type & \\texttt{Unit} \\newline e.g.: \\texttt{()} \\\\\n \\end{keysref}\n\n \\begin{keysref}{Derived Types}\n Arrays & \\texttt{\\emph{elementType}[]} \\\\\n Tuples & \\texttt{(\\emph{type0}, \\emph{type1}, ...)} \\newline\n e.g.: \\texttt{(Int, Qubit)} \\\\\n Functions & \\texttt{\\emph{input} -> \\emph{output}} \\newline\n e.g.: \\texttt{ArcCos : (Double) -> Double} \\\\\n Operations & \\texttt{\\emph{input} => \\emph{output} is \\emph{variants}} \\newline\n e.g.: \\texttt{H : (Qubit => Unit is Adj)} \\\\\n \\end{keysref}\n \n \\begin{keysref}{User-Defined Types}\n Declare UDT with \\newline anonymous items & \\texttt{newtype \\emph{Name} = (Type, Type);} \\newline\n e.g.: \\texttt{newtype \\emph{Pair} = (Int, Int);} \\\\\n Define UDT literal & \\texttt{\\emph{Name}(\\emph{baseTupleLiteral})} \\newline\n e.g.: \\texttt{let origin = Pair(0, 0);} \\\\\n Unwrap operator \\texttt{!} \\newline (convert UDT to \\newline underlying type) \n & \\texttt{\\emph{VarName}!} \\newline\n e.g.: \\texttt{let originTuple = origin!;} \\newline \\hphantom{..........}(now \\texttt{originTuple = (0, 0)}) \\\\\n Declare UDT with \\newline named items & \\texttt{newtype \\emph{Name} =} \\newline \\texttt{\\hphantom{....}(\\emph{Name1}: Type, \\emph{Name2}: Type);} \\newline\n e.g.: \\texttt{newtype Complex =} \\newline \\texttt{\\hphantom{....}(Re : Double, Im : Double);} \\\\\n Accessing named \\newline items of UDTs & \\texttt{\\emph{VarName}::\\emph{ItemName}} \\newline\n e.g.: \\texttt{complexVariable::Re} \\\\\n Update-and-\\newline reassign for named \\newline UDT items & \\texttt{set \\emph{VarName} w/= \\emph{ItemName} <- \\emph{val};} \\newline\n e.g.: \\texttt{mutable p = Complex(0., 0.);} \\newline\n \\texttt{\\hphantom{....}set p w/= Re <- 1.0;} \\\\\n \\end{keysref}\n \n \\begin{keysref}{Symbols and Variables}\n Declare immutable \\newline symbol & \\texttt{let \\emph{varName} = \\emph{value}} \\\\\n Declare mutable \\newline symbol (variable) & \\texttt{mutable \\emph{varName} = \\emph{initialValue}} \\\\\n Update mutable \\newline symbol (variable) & \\texttt{set \\emph{varName} = \\emph{newValue}} \\\\\n Apply-and-reassign & \\texttt{set \\emph{varName} \\emph{operator}= \\emph{expression}} \\newline\n e.g.: \\texttt{set counter += 1;} \\\\ \n \\end{keysref}\n\n \\columnbreak\n\n \\begin{keysref}{Functions and Operations}\n Define function \\newline (classical routine)\n & \\texttt{function \\emph{Name}(\\emph{in0} : \\emph{type0}, ...) \\newline\n : \\emph{returnType} \\{} \\newline\n \\texttt{\\hphantom{....}// \\emph{function body}} \\newline\n \\texttt{\\}} \\\\\n Call function & \\texttt{\\emph{Name}(\\emph{parameters})} \\newline\n e.g.: \\texttt{let two = Sqrt(4.0);} \\\\\n Define operation \\newline (quantum routine) \\newline with explicitly \\newline specified body, \\newline controlled and \\newline adjoint variants\n & \\texttt{operation \\emph{Name}(\\emph{in0} : \\emph{type0}, ...) \\newline \n : \\emph{returnType} \\{} \\newline\n \\texttt{\\hphantom{....}body \\{ ... \\}} \\newline\n \\texttt{\\hphantom{....}adjoint \\{ ... \\}} \\newline\n \\texttt{\\hphantom{....}controlled \\{ ... \\}} \\newline\n \\texttt{\\hphantom{....}adjoint controlled \\{ ... \\}} \\newline\n \\texttt{\\}} \\\\\n Define operation \\newline with automatically \\newline generated adjoint \\newline and controlled \\newline variants\n & \\texttt{operation \\emph{Name}(\\emph{in0} : \\emph{type0}, ...) \\newline \n : \\emph{returnType} is Adj + Ctl \\{} \\newline\n \\texttt{\\hphantom{....}... \\newline\n \\} } \\\\\n Call operation & \\texttt{\\emph{Name}(\\emph{parameters})} \\newline\n e.g.: \\texttt{Ry(0.5 * PI(), q);} \\\\\n Call adjoint \\newline operation & \\texttt{Adjoint \\emph{Name}(\\emph{parameters})} \\newline\n e.g.: \\texttt{Adjoint Ry(0.5 * PI(), q);} \\\\\n Call controlled \\newline operation & \\texttt{Controlled \\emph{Name}(\\emph{controlQubits}, \\newline \\hphantom{....}\\emph{parameters})} \\newline\n e.g.: \\texttt{Controlled Ry(controls, \\newline \\hphantom{....}(0.5 * PI(), target));} \\\\\n \\end{keysref}\n\n \\begin{keysref}{Control Flow}\n Iterate over \\newline a range of numbers\n & \\texttt{for \\emph{index} in \\emph{range} \\{ }\\newline \n \\texttt{\\hphantom{....}// Use integer \\emph{index}} \\newline\n \\texttt{\\hphantom{....}...} \\newline\n \\texttt{\\}} \\newline\n e.g.: \\texttt{for i in 0..N-1 \\{ ... \\}} \\\\\n While loop \\newline (within functions)\n & \\texttt{while (\\emph{condition}) \\{} \\newline\n \\texttt{\\hphantom{....}...} \\newline\n \\texttt{\\}} \\\\\n Iterate over \\newline an array & \\texttt{for \\emph{val} in \\emph{array} \\{ }\\newline \n \\texttt{\\hphantom{....}// Use value \\emph{val}} \\newline\n \\texttt{\\hphantom{....}...} \\newline\n \\texttt{\\}} \\newline\n e.g.: \\texttt{for q in register \\{ ... \\}} \\\\\n Repeat-until- \\newline success loop & \\texttt{repeat \\{ ... \\} \\newline until \\emph{condition} \\newline fixup \\{ ... \\}} \\\\\n Conditional \\newline statement & \\texttt{if \\emph{cond1} \\{ ... \\}} \\newline \n \\texttt{elif \\emph{cond2} \\{ ... \\}} \\newline\n \\texttt{else \\{ ... \\}} \\\\\n Ternary operator & \\texttt{\\emph{condition} ? \\emph{caseTrue} | \\emph{caseFalse} } \\\\\n Return a value & \\texttt{return \\emph{value}} \\\\\n Stop with an error & \\texttt{fail \"\\emph{Error message}\"} \\\\\n Conjugations \\newline ($ABA^\\dagger$ pattern)\n & \\texttt{within \\{ ... \\}} \\newline\n \\texttt{apply \\{ ... \\}} \\\\\n \\end{keysref}\n \n \\columnbreak\n\n \\begin{keysref}{Arrays}\n Allocate array & \\texttt{mutable \\emph{name} = new \\emph{Type}[\\emph{length}]} \\newline\n e.g.: \\texttt{mutable b = new Bool[2];} \\\\\n Get array length & \\texttt{Length(\\emph{name})} \\\\\n Access k-th element & \\texttt{\\emph{name}[k]} \\newline NB: indices are 0-based \\\\\n Assign k-th element \\newline (copy-and-update) \n & \\texttt{set \\emph{name} w/= \\emph{k} <- \\emph{value}} \\newline\n e.g.: \\texttt{set b w/= 0 <- true;} \\\\\n Array literal & \\texttt{[\\emph{value0}, \\emph{value1}, ...]} \\newline\n e.g.: \\texttt{let b = [true, false, true];} \\\\\n Array concatenation & \\texttt{\\emph{array1} + \\emph{array2}} \\newline \n e.g.: \\texttt{let t = [1, 2, 3] + [4, 5];} \\\\\n Slicing (subarray) & \\texttt{\\emph{name}[\\emph{sliceRange}]} \\newline\n e.g.: if \\texttt{t = [1, 2, 3, 4, 5]}, then \\newline\n \\texttt{\\hphantom{..}t[1 .. 3]\\hphantom{....}} is \\texttt{[2, 3, 4]} \\newline\n \\texttt{\\hphantom{..}t[3 ...]\\hphantom{.....}} is \\texttt{[4, 5]} \\newline\n \\texttt{\\hphantom{..}t[... 1]\\hphantom{.....}} is \\texttt{[1, 2]} \\newline\n \\texttt{\\hphantom{..}t[0 .. 2 ...]\\hphantom{}} is \\texttt{[1, 3, 5]} \\newline\n \\texttt{\\hphantom{..}t[...-1...]\\hphantom{..}} is \\texttt{[5, 4, 3, 2, 1]} \\\\\n \\end{keysref}\n\n \\begin{keysref}{Debugging (classical)}\n Print a string & \\texttt{Message(\"Hello Quantum!\")} \\\\\n Print an \\newline interpolated string \n & \\texttt{Message(\\$\"Value = \\{\\emph{val}\\}\")} \\\\\n \\end{keysref}\n \n \\section{Resources}\n \n \\begin{keysref}{Documentation}\n Quantum \\newline Development Kit & \\url{https://docs.microsoft.com/azure/quantum} \\\\\n QDK user guides & \\url{https://docs.microsoft.com/azure/quantum/user-guide} \\\\\n \\qs~Libraries \\newline Reference & \\url{https://docs.microsoft.com/qsharp/api} \\\\\n \\end{keysref}\n\n \\begin{keysref}{\\qs~Code Repositories}\n QDK Samples & \\url{https://github.com/microsoft/quantum} \\\\\n QDK Libraries & \\url{https://github.com/microsoft/QuantumLibraries} \\\\\n Quantum Katas \\newline (tutorials) & \\url{https://github.com/microsoft/QuantumKatas} \\\\\n \\qs~compiler and \\newline extensions & \\url{https://github.com/microsoft/qsharp-compiler} \\\\\n Simulation \\newline framework & \\url{https://github.com/microsoft/qsharp-runtime} \\\\\n Jupyter kernel and \\newline Python host & \\url{https://github.com/microsoft/iqsharp} \\\\\n Source code for \\newline the documentation & \\url{https://github.com/MicrosoftDocs/quantum-docs} \\\\\n \\end{keysref}\n\n \\columnbreak\n\n \\begin{keysref}{Qubit Allocation}\n Allocate a register \\newline of $N$ qubits \n & \\texttt{use \\emph{reg} = Qubit[\\emph{N}]; } \\newline\n \\texttt{//} Qubits in \\texttt{\\emph{reg}} start in $\\ket{0}$. \\newline\n \\texttt{...} \\newline\n \\texttt{//} Qubits must be returned to $\\ket{0}$. \\\\\n Allocate one qubit & \\texttt{use \\emph{one} = Qubit(); } \\newline\n \t\t\t\t\t\t \\texttt{...} \\\\\n Allocate a mix of \\newline qubit registers and \\newline individual qubits\n & \\texttt{use (\\emph{x}, \\emph{y}, ... ) = } \\newline \n \\texttt{\\hphantom{...}(Qubit[N], Qubit(), ... ); } \\newline\n \\texttt{...} \\\\\n \\end{keysref}\n\n \\begin{keysref}{Debugging (quantum)}\n Print amplitudes \\newline of wave function & \\texttt{DumpMachine(\"dump.txt\")} \\\\\n Assert that a qubit is in $\\ket{0}$ or $\\ket{1}$ state & \\texttt{AssertQubit(Zero, \\emph{zeroQubit})} \\newline\n \\texttt{AssertQubit(One, \\emph{oneQubit})} \\\\\n \\end{keysref}\n\n \n\n\n \n \\begin{keysref}{Measurements}\n Measure qubit in \\newline Pauli $Z$ basis & \\texttt{M(\\emph{oneQubit})} \\newline yields a \\texttt{Result} (\\texttt{Zero} or \\texttt{One}) \\\\\n Reset qubit to $\\ket{0}$ & \\texttt{Reset(\\emph{oneQubit})} \\\\\n Reset an array of \\newline qubits to $\\ket{0..0}$ & \\texttt{ResetAll(\\emph{register})} \\\\\n \\end{keysref}\n\n \n\n \\section{Working with \\qs~from command line}\n\n \\begin{keysref}{Command Line Basics}\n Change directory & \\texttt{cd \\emph{dirname}} \\\\\n Go to home & \\texttt{cd \\textasciitilde} \\\\\n Go up one directory & \\texttt{cd ..} \\\\\n Make new directory & \\texttt{mkdir \\emph{dirname}} \\\\\n Open current \\newline directory in VS Code & \\texttt{code .} \\\\\n \\end{keysref}\n\n \\begin{keysref}{Working with \\qs~Projects}\n Create new project & \\texttt{dotnet new console -lang Q\\# \\newline \n --output \\emph{project-dir}} \\\\\n Change directory to \\newline project directory & \\texttt{cd \\emph{project-dir}} \\\\\n Build project & \\texttt{dotnet build} \\\\\n Run all unit tests & \\texttt{dotnet test} \\\\\n \\end{keysref}\n\n \\columnbreak\n \n \\section{Math reference}\n\n \\begin{keysref}{Complex Arithmetic}\n $i^2 $ & $-1$ \\\\\n $ (a + bi) + (c + di) $ & $ (a + c) + (b + d)i $ \\\\\n $ (a + bi) (c + di) $ & $ a \\cdot c + a \\cdot di + b \\cdot ci + (b \\cdot d)i^2 =$ \\newline\n $ = (a \\cdot c - b \\cdot d) + (a \\cdot d + b \\cdot c)i $ \\\\\n\t Complex conjugate & $ \\overline{a + bi} = a - bi $ \\\\\n Division $ \\frac{a+bi}{c+di} $ & $ \\frac{a+bi}{c+di} \\cdot 1 = \\frac{a+bi}{c+di} \\cdot \\frac{c-di}{c-di} = \\frac{(a+bi)(c-di)}{c^2 + d^2} $ \\\\\n Modulus $|a + bi|$ & $ \\sqrt{a^2 + b^2} $ \\\\\n $e^{i\\theta} $ & $ \\cos \\theta + i \\sin \\theta $ \\\\ \n $ e^{a + bi} $ & $ e^a \\cdot e^{bi} = e^a \\cos b + i e^a \\sin b $ \\\\\n $ r^{a+bi} $ & $ r^a \\cdot r^{bi} = r^a \\cdot e^{bi\\ln r} = $ \\newline \n $ = r^a \\cos(b\\ln r) + i \\cdot r^a \\sin(b\\ln r) $ \\\\\n Polar form $re^{i\\theta}$ to \\newline Cartesian form $a + bi$ & $ a = r \\cos \\theta \\newline b = r \\sin \\theta $ \\\\\n Cartesian form $a + bi$ \\newline to polar form $re^{i\\theta}$ & $ r = \\sqrt{a^2 + b^2} \\newline \\theta = \\arctan(\\frac{b}{a}) $ \\\\\n \\end{keysref}\n\n \\begin{keysref}{Linear Algebra}\n\n $m \\times n$ matrix & \\usebox\\matrixrepresentationright \\\\\n\n Vector of size $n$ & \\usebox\\vectorrepresentationright \\\\\n \n \\usebox\\matrixadditionleft & \\usebox\\matrixadditionright \\\\\n \n \\usebox\\matrixscalarleft &\\usebox\\matrixscalarright \\\\\n \n \\usebox\\matrixmultiplicationleft & \\usebox\\matrixmultiplicationright \\\\\n\n \\usebox\\matrixtransposeleft & \\usebox\\matrixtransposeright \\\\ \n \n \\usebox\\matrixadjointleft & \\usebox\\matrixadjointright \\\\\n\n \\usebox\\innerproductleft & \\usebox\\innerproductright \\\\ \n \n \\usebox\\outerproductleft & \\usebox\\outerproductright \\\\\n \n \\end{keysref}\n \n \\newpage\n \\end{multicols}\n \\section{Gates reference}\n\n \\begin{keysrefGates}{Single Qubit gates}\n\n Gate & \n Matrix representation & \n Ket-bra representation & \n Applying to $ \\ket{\\psi} = \\alpha\\ket{0} + \\beta\\ket{1} $ & \n Applying to basis states: & $\\ket{0},\\: \\ket{1},\\: \\ket{+},\\: \\ket{-}\\:$ and & \n $\\ket{\\pm i} = \\frac{1}{\\sqrt2}(\\ket{0} \\pm i\\ket{1})$ \\\\\n \n X & \n \\usebox\\xgate & $ \\ket{0}\\bra{1} + \\ket{1}\\bra{0}$ & \n $ \\ket{\\psi} = \\alpha\\ket{1} + \\beta\\ket{0} $ & \n $ X\\ket{0} = \\ket{1} \\newline X\\ket{1} = \\ket{0} $ & \n $ X\\ket{+} = \\ket{+} \\newline X\\ket{-} = -\\ket{-} $ &\n $ X\\ket{i} = i\\ket{-i} \\newline X\\ket{-i} = -i\\ket{i} $ \\\\\n\n Y & \n \\usebox\\ygate & \n $ i(\\ket{1}\\bra{0} - \\ket{0}\\bra{1})$ & \n $ Y\\ket{\\psi} = i\\big(\\alpha\\ket{1} - \\beta\\ket{0}\\big)$ & \n $Y\\ket{0} = i\\ket{1} \\newline Y\\ket{1} = -i\\ket{0} $ & \n $ Y\\ket{+} = -i\\ket{-} \\newline Y\\ket{-} = i\\ket{+} $ & \n $ Y\\ket{i} = \\ket{i} \\newline Y\\ket{-i} = -\\ket{-i}$ \\\\\n\n Z & \n \\usebox\\zgate & \n $ \\ket{0}\\bra{0} - \\ket{1}\\bra{1} $ & \n $ Z\\ket{\\psi} = \\alpha\\ket{0} - \\beta\\ket{1} $ & \n $ Z\\ket{0} = \\ket{0} \\newline Z\\ket{1} = -\\ket{1} $ & \n $ Z\\ket{+} = \\ket{-} \\newline Z\\ket{-} = \\ket{+} $ & \n $ Z\\ket{i} = \\ket{-i} \\newline Z\\ket{-i} = \\ket{i} $ \\\\\n\n I & \n \\usebox\\igate & \n $ \\ket{0}\\bra{0} + \\ket{1}\\bra{1} $ & \n $ I\\ket{\\psi} = \\ket{\\psi} $ & & & \\\\\n\n H & \n \\usebox\\hgate & \n $ \\ket{0}\\bra{+} + \\ket{1}\\bra{-} $ & \n $ H\\ket{\\psi} = \\alpha\\ket{+} + \\beta\\ket{-} = \\frac{\\alpha + \\beta}{\\sqrt{2}}\\ket{0} + \\frac{\\alpha - \\beta}{\\sqrt{2}}\\ket{1} $ & \n $ H\\ket{0} = \\ket{+} \\newline H\\ket{1} = \\ket{-} $ & \n $ H\\ket{+} = \\ket{0} \\newline H\\ket{-} = \\ket{1} $ & \n $ H\\ket{i} = e^{i\\pi/4}\\ket{-i} \\newline H\\ket{-i} = e^{-i\\pi/4}\\ket{i} $ \\\\\n\n S & \n \\usebox\\sgate & \n $ \\ket{0}\\bra{0} + i\\ket{1}\\bra{1} $ & \n $ S\\ket{\\psi} = \\alpha\\ket{0} + i\\beta\\ket{1} $ & \n $ S\\ket{0} = \\ket{0} \\newline S\\ket{1} = i\\ket{1} $ & \n $ S\\ket{+} = \\ket{i} \\newline S\\ket{-} = \\ket{-i} $ & \n $ S\\ket{i} = \\ket{-} \\newline S\\ket{-i} = \\ket{+} $ \\\\\n\n T & \n \\usebox\\tgate & \n $ \\ket{0}\\bra{0} + e^{i\\pi/4}\\ket{1} \\bra{1} $ & \n $ T\\ket{\\psi} = \\alpha\\ket{0} + e^{i\\pi/4} \\beta \\ket{1} $ & \n $ T\\ket{0} = \\ket{0} $ &\n $ T\\ket{1} = e^{i\\pi/4}\\ket{1} $ & \\\\\n\n $ R_x(\\theta) $ & \n \\usebox\\rxgate & \n $ \\cos\\frac{\\theta}{2}\\ket{0}\\bra{0} -i\\sin\\frac{\\theta}{2}\\ket{1}\\bra{0} - \\newline \n -i\\sin\\frac{\\theta}{2}\\ket{0} \\bra{1} + \\cos\\frac{\\theta}{2}\\ket{1}\\bra{1} $ & \n $ R_x(\\theta)\\ket{\\psi} = \\ (\\alpha\\cos\\frac{\\theta}{2} - i\\beta\\sin\\frac{\\theta}{2})\\ket{0} + \\newline \n + (\\beta\\cos\\frac{\\theta}{2} - i\\alpha\\sin\\frac{\\theta}{2})\\ket{1} $ & \n $ R_x(\\theta)\\ket{0} = \\newline\n = \\cos\\frac{\\theta}{2}\\ket{0} - i\\sin\\frac{\\theta}{2}\\ket{1} $ & \n $ R_x(\\theta)\\ket{1} = \\newline\n = \\cos\\frac{\\theta}{2}\\ket{1} - i\\sin\\frac{\\theta}{2}\\ket{0} $ & \\\\ \n\n $ R_y(\\theta) $ & \n \\usebox\\rygate & \n $ \\cos\\frac{\\theta}{2}\\ket{0}\\bra{0} + \\sin\\frac{\\theta}{2} \\ket{1}\\bra{0} - \\newline \n -\\sin\\frac{\\theta}{2}\\ket{0}\\bra{1} + \\cos\\frac{\\theta}{2}\\ket{1}\\bra{1}$ & \n $ R_y(\\theta)\\ket{\\psi} = (\\alpha\\cos\\frac{\\theta}{2} - \\beta\\sin\\frac{\\theta}{2})\\ket{0} + \\newline \n + (\\beta\\cos\\frac{\\theta}{2} + \\alpha\\sin\\frac{\\theta}{2})\\ket{1} $ & \n $ R_y(\\theta)\\ket{0} = \\newline \n = \\cos\\frac{\\theta}{2}\\ket{0} + \\sin\\frac{\\theta}{2}\\ket{1} $ & \n $ R_y(\\theta)\\ket{1} = \\newline \n = \\cos\\frac{\\theta}{2}\\ket{1} - \\sin\\frac{\\theta}{2}\\ket{0} $ & \\\\\n\n $ R_z(\\theta) $ & \\usebox\\rzgate & \n $ e^{-i\\theta/2}\\ket{0}\\bra{0} + e^{i\\theta/2}\\ket{1} \\bra{1} $ & \n $ R_z(\\theta)\\ket{\\psi} = \\alpha e^{-i\\theta/2}\\ket{0} + \\beta e^{i\\theta/2}\\ket{1} $ & \n $ R_z(\\theta)\\ket{0} = e^{-i\\theta/2}\\ket{0} $ & \n $ R_z(\\theta)\\ket{1} = e^{i\\theta/2}\\ket{1} $ & \\\\\n\n $ R_1(\\theta) $ & \n \\usebox\\rgate & \n $ \\ket{0}\\bra{0} + e^{i\\theta}\\ket{1} \\bra{1} $ & \n $ R_1(\\theta)\\ket{\\psi} = \\alpha\\ket{0} + \\beta e^{i\\theta}\\ket{1} $ & \n $ R_1(\\theta)\\ket{0} = \\ket{0} $ & \n $ R_1(\\theta)\\ket{1} = e^{i\\theta}\\ket{1} $ & \\\\\n\n \\end{keysrefGates}\n \n \\begin{keysrefMultiGates}{Two-qubit gates}\n \n Gate &\n Matrix Representation & \n Ket-Bra Representation &\n Applying to $\\ket{\\psi} = \\alpha\\ket{00} + \\beta\\ket{01} + \\gamma\\ket{10} + \\delta\\ket{11}$ &\n Applying to basis states & \\\\\n \n CNOT &\n \\usebox\\cnotgate & \n \\usebox\\braketcx &\n $ \\text{CNOT}\\ket{\\psi} = \\alpha\\ket{00} + \\beta\\ket{01} + $ {\\boldmath $ \\delta\\ket{10} + \\gamma\\ket{11} $} &\n $ \\text{CNOT}\\ket{00} = \\ket{00} \\newline\n \\text{CNOT}\\ket{01} = \\ket{01} $ &\n $ \\text{CNOT}\\ket{10} = \\ket{11} \\newline\n \\text{CNOT}\\ket{11} = \\ket{10} $ \\\\\n \n SWAP &\n \\usebox\\swapgate & \n $ \\ket{00}\\bra{00} + \\ket{01}\\bra{10} + \\ket{10}\\bra{01} + \\ket{11}\\bra{11} $ &\n $ \\text{SWAP}\\ket{\\psi} = \\alpha\\ket{00} + $ {\\boldmath $ \\gamma\\ket{01} + \\beta\\ket{10} $} $+ \\delta\\ket{11}$ &\n $ \\text{SWAP}\\ket{00} = \\ket{00} \\newline\n \\text{SWAP}\\ket{01} = \\ket{10} $ &\n $\\text{SWAP}\\ket{10} = \\ket{01} \\newline\n \\text{SWAP}\\ket{11} = \\ket{11} $ \\\\\n \n Controlled U &\n \\usebox\\controlledugate & \n \\usebox\\braketcu & \n $ \\text{CU} \\ket{\\psi} = \\alpha\\ket{00} + \\beta\\ket{01} + \\newline \n + (\\gamma a_{0,0} + \\delta a_{0,1}) \\ket{10} + (\\gamma a_{1,0} + \\delta a_{1,1}) \\ket{11} $ & \n $ \\text{CU} \\ket{00} = \\ket{00} \\newline\n \\text{CU} \\ket{01} = \\ket{01} $ & \n $\\text{CU} \\ket{10} = a_{0,0} \\ket{10} + a_{1,0} \\ket{11} \\newline \n \\text{CU} \\ket{11} = a_{0,1} \\ket{10} + a_{1,1} \\ket{11} $ \\\\\n \\end{keysrefMultiGates}\n \n \\begin{keysrefMultiGates}{Toffoli (CCNOT) gate} \n \n Gate &\n Matrix Representation &\n Ket-Bra Representation &\n Applying to $\\ket{\\psi} = \\alpha\\ket{000} + \\beta\\ket{001} + \\gamma\\ket{010} + \\delta\\ket{011} + \\epsilon\\ket{100} + \\lambda\\ket{101} + \\eta\\ket{110} + \\kappa\\ket{111} $ &\n Applying to basis states & \\\\\n \n \n CCNOT &\n \\usebox\\ccnotgate & \n $ (I_2 - \\ket{11}\\bra{11}) \\otimes I_1 + \\ket{11}\\bra{11} \\otimes X $ & %$ \\ket{000}\\bra{000} + \\ket{001}\\bra{001} + \\ket{010}\\bra{010} + \\ket{011}\\bra{011} + \\ket{100}\\bra{100} + \\ket{101}\\bra{101} + \\ket{110}\\bra{111} + \\ket{111}\\bra{110} $ &\n $ \\text{CCNOT} \\ket{\\psi} = \\alpha\\ket{000} + \\beta\\ket{001} + \\gamma\\ket{010} + \\delta\\ket{011} + \\epsilon\\ket{100} + \\lambda\\ket{101} + $ \\boldmath$ \\kappa \\ket{110} + \\eta\\ket{111} $ &\n $ \\text{CCNOT} \\ket{000} = \\ket{000} \\newline \\text{CCNOT} \\ket{001} = \\ket{001} \n \\newline \\text{CCNOT} \\ket{010} = \\ket{010} \\newline \\text{CCNOT} \\ket{011} = \\ket{011} $ &\n $ \\text{CCNOT} \\ket{100} = \\ket{100} \\newline \\text{CCNOT} \\ket{101} = \\ket{101} \\newline \n \\text{CCNOT} \\ket{110} = \\ket{111} \\newline \\text{CCNOT} \\ket{111} = \\ket{110} $\\\\\n \n \\end{keysrefMultiGates}\n\n\\end{document} \n", "meta": {"hexsha": "08ebf7bffefaef35d81881d8983d430a6b2b8cb1", "size": 34886, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "quickref/qsharp-quick-reference.tex", "max_stars_repo_name": "raffaeleterribile/quantum-katas", "max_stars_repo_head_hexsha": "d0616b148fc3914c1fa5ed4140fa371846ebb284", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "quickref/qsharp-quick-reference.tex", "max_issues_repo_name": "raffaeleterribile/quantum-katas", "max_issues_repo_head_hexsha": "d0616b148fc3914c1fa5ed4140fa371846ebb284", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "quickref/qsharp-quick-reference.tex", "max_forks_repo_name": "raffaeleterribile/quantum-katas", "max_forks_repo_head_hexsha": "d0616b148fc3914c1fa5ed4140fa371846ebb284", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.1267759563, "max_line_length": 268, "alphanum_fraction": 0.5554377114, "num_tokens": 12568, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6370307944803831, "lm_q1q2_score": 0.3458206285736102}} {"text": "\\documentclass[../main.tex]{subfiles}\n\n\\begin{document}\n \n\\chapter{Learning Methods Survey} \\label{learning_methods_survey}\n\n\nIn this chapter, we outline a set of ideal characteristics and desired behavior of an ideal candidate classification algorithm, and conduct a survey of potential unsupervised learning methods. We then evaluate each of these surveyed methods against our selection criteria, and determine the optimal method with which to proceed.\n\n\\section{Evaluation Criteria}\n\nDespite being not entirely objective, existing classification heuristics have a range of desirable behavior that we would wish to replicate with our candidate clustering algorithm. Additionally, we would want to replicate this behavior while also maintaining objectivity and stability in our new heuristic by utilizing a highly nonparametric learning method.\n\nIn particular, we would be extremely interested in preserving the nested hierarchical clustering behavior of the current schemes. That is, to be able to classify a market into sectors, and in turn those sectors into subsectors. Furthermore, it would be desirable to be able to determine these nested sub-sectors in the context of the greater market, rather than in isolated analysis of a particular sector.\n\nAdditionally, we would also like to vary the number of resulting sectors of our algorithm while maintaining stability. That is, if we were to request two sectors from a heuristic that was initially resulting in four sectors, the two sectors would be some combination of the initial four sectors, as opposed to an entirely new segmentation profile. This behavior is reflective of the real world, where economic sectors often exhibit nesting, as opposed to independent clustering.\n\nFinally, as per RG-1 (see Section~\\ref{research_goals:specific_research_goals}), we are extremely motivated to design a heuristic that is either entirely non-parametric, or parameterized with highly objective, quantitatively derived criteria. In addition to preserving mathematical objectivity of our results, a nonparametric approach would ensure that no personal biases - either explicit or implicit - are introduced to the final learned sectors.\n\n\n\\section{Candidate Learning Methods}\n\nGiven the required behavior outlined above, we evaluated three major families of clustering algorithms. We empirically evaluate each clustering technique through the lens of the requirements outlined above.\n\n\n\\subsection{$K$-means Clustering}\n\n$K$-Means Clustering is a method of partitioning $n$-dimensional data into a set of $K$ distinct clusters. The basic algorithm is outlined below\\citeFormat{\\cite{Lloyd1982LeastPCM}}:\n\n\\begin{gather*}\n \\text{Let $C_1, C_2, \\ldots, C_K$} = \\text{Set of K possible clusters} \\\\\n \\text{Let $W(C_k)$} = \\text{Measure of pariwise difference of observations in a cluster} \\\\\n \\text{Let $x_{ij}$} = \\text{$j^\\text{th}$ feature in cluster $i$ with coordinates $x$} \\\\\n \\Rightarrow W(C_k) = \\frac{1}{|C_k|} \\sum_{i, i^\\prime \\in C_k} \\sum_{j = 1}^p (x_{ij} - x_{i^\\prime j})^2 \\\\\n \\\\\n \\Rightarrow \\text{$K$-Means Clusters}\n = \\underset{C_1, \\ldots, C_k}{\\text{minimize}} \\sum_{k=1}^K W(C_k)\n = \\underset{C_1, \\ldots, C_k}{\\text{minimize}} \\sum_{k=1}^K \\frac{1}{|C_k|} \\sum_{i, i^\\prime \\in C_k} \\sum_{j = 1}^p (x_{ij} - x_{i^\\prime j})^2\n\\end{gather*}\n\nNotice that in the algorithm outlined above, the $K$-means clustering process requires two sets of parameters at initialization. First, it requires the number of target clusters, $K$, as well as a set of random initializations for cluster centroids, $\\frac{1}{|C_k} \\sum_{i \\in C_k} x_{ij}$. This high level of parameterization, coupled with the clear lack of congruity of assignment across varying values of $K$ make this family of algorithms poorly suited to the task of sector classification, as per the constraints detailed above.\n\n\\subsection{Support Vector Classifier}\n\nThe support vector classifier is based on the notion of finding a set of hyperplanes in a higher dimensional feature space that optimally divides a set of data into classes. Data is mapped to a higher dimensional space to ensure orthogonal hyperplanes in the divisions of the clusters. The support vector classifier objective function is outlined below\\citeFormat{\\cite{Ben-Hur2001SupportClustering}}:\n\n\\begin{equation*}\n \\begin{aligned}\n & \\underset{R, a, \\boldsymbol{\\alpha}}{\\text{minimize}} & & R^2 - \\sum_i \\alpha_i (R^2 - ||x_i - a||^2) \\\\\n & \\text{subject to}\n & & \\alpha_i \\geq 0 \\\\\n & & & (R^2 - ||x_i - a||^2) = 0 \\; \\forall \\; i \\; \\text{(KKT Condition)}\n \\end{aligned}\n\\end{equation*}\n\nAs indicated by the literature, and through inspection of the objective function, it is clear that the support vector classifier is parameterized on the kernel used for optimization, as well as the specific loss function employed during model training. Furthermore, this model also optimizes to a fixed numbed of sectors, as opposed to a dynamic number. Therefore, this method too is inappropriate as per the evaluation criteria.\n\n\\subsection{Hierarchical Cluster Analysis} \\label{learning_methods_survey:hca}\n\nHierarchical clustering is a greedy algorithm which seeks to build clusters following either an agglomerative, or a divisive approach.\\citeFormat{\\cite{Ward1963HierarchicalFunction}} Agglomerative clustering is \\textit{bottom-up}, with each observation starting it its own cluster, whereas divisive is \\textit{top-down}, with all observations starting in one cluster and splits performed recursively at each level. The clusters output by this algorithm are determined by two model settings; the distance metric (i.e. the algorithm for computation of pairwise distance between observations), and the linkage method, which specifies the algorithm governing the dissimilarity of entire sets, as a function of the pairwise distances of observations in those sets.\n\nThis method has the distinct advantage of being entirely additively hierarchical, with groups being nested as described in the evaluation criteria. Furthermore, this method is entirely nonparametric, with the sole exception being the choice of linkage and distance metric. Additionally, it is extremely stable with varying sector counts. This is a direct result of the greedy nature of the algorithm, as it does not recompute the hierarchy each time a new cluster arity is extracted, but rather just changes the level of extraction from the same hierarchy.\n\nAs per the evaluation of the different families of learning methods detailed in this chapter, we selected Hierarchical Clustering to be the basis of our Learned Sectors classification heuristic.\n\n\n\\end{document}", "meta": {"hexsha": "ef8e0bf852541e12993a87f42c2ef08c95263e43", "size": 6687, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sections/05_learning_methods_survey.tex", "max_stars_repo_name": "rukmal/FE-800-Learned-Sectors-Report", "max_stars_repo_head_hexsha": "f60f09dc689c4adc172b576a261df5ed77518f8f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-02-14T08:39:24.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-14T08:39:24.000Z", "max_issues_repo_path": "sections/05_learning_methods_survey.tex", "max_issues_repo_name": "rukmal/MS-Thesis-Learned-Sectors-Report", "max_issues_repo_head_hexsha": "f60f09dc689c4adc172b576a261df5ed77518f8f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 39, "max_issues_repo_issues_event_min_datetime": "2019-05-13T22:30:23.000Z", "max_issues_repo_issues_event_max_datetime": "2019-05-17T03:37:55.000Z", "max_forks_repo_path": "sections/05_learning_methods_survey.tex", "max_forks_repo_name": "rukmal/MS-Thesis-Learned-Sectors-Report", "max_forks_repo_head_hexsha": "f60f09dc689c4adc172b576a261df5ed77518f8f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2019-06-01T06:32:20.000Z", "max_forks_repo_forks_event_max_datetime": "2019-06-01T06:32:20.000Z", "avg_line_length": 99.8059701493, "max_line_length": 759, "alphanum_fraction": 0.7803200239, "num_tokens": 1511, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.3458206210919017}} {"text": "\\section{Performance Analysis}\\label{sec:analysis}\n\nIn this section we analyze and discuss the overhead of the CCVPN design with respect\nto the additional processing time and state consumption needed to handle traffic.\n\n\\subsection{State Consumption}\nThe CCVPN design has an immediate impact on the FIB and PIT size of a gateway.\n(The content store size remains unaffected since only decapsulated content objects\nare ever cached.) Let $F_S$ be the total size of a standard forwarder\nFIB in terms of bytes and\n$N_F$ be the number of entries in the FIB. For simplicity, we will assume that\neach name prefix in the FIB has a constant size of $64$B. In practice we expect\nthis to be a comfortable upper bound. Thus, $F_S = N_Fs$, where $s$ is the size of\neach FIB entry. Here, $s$ includes a name prefix (of size $64$B) and a bit vector\nthat identifies the matching links for the interface. We assume that a gateway has\n$128$ links which, again, is a comfortable upper bound. Therefore, $F_S = 80N_F$B.\nNow consider the FIB size $F_G$ for a CCVPN gateway. Some entries in these FIBs will\npoint to ``private'' prefixes, i.e., other domains, and therefore have a larger size\nto account for the corresponding prefix and key material that must be stored.\nFor both public- and symmetric-key encryption, the key size is the same: $32$B \\cite{sodiumGithub}.\nTherefore, by taking into account two both the FIB entry prefix key, translation\nprefix, encryption key, and corresponding bit vector, the total size of one ``private''\nFIB entry will be $176$B, meaning that $F_G = 176N_F$B. By comparing $F_S$ to $F_G$, we\nsee that, in the worst case, the CCVPN FIB is at most $F_G/F_S = 176/80 = 2.2$ times larger\nthan the standard FIB. In practice, however, we expect this to be much smaller, since the\nfraction of public to private FIB entries in a gateway will be non-zero.\n\nWe will now apply the same analysis to the PIT size. A standard PIT entry includes a\ncomplete name and ingress bit vector. (They may also include the optional {\\tt KeyId}\nand {\\tt ContentId}, but since they are included in the gateway PIT as well we omit them\nfrom this analysis.) A gateway PIT entry will contain the same elements of a standard\nPIT entry but also a symmetric encryption key ($32$B), nonce ($12$B), and an\nencapsulation name ($64$B + $32$B).\nThe encapsulation name is the name of an encapsulated interest and includes an additional\n$32$B {\\tt PayloadID} segment to identify the encapsulated value in the payload. Let\n$P_S$ and $P_G$ be the sizes of the standard and gateway PIT, respectively, and let $N_P$\nbe the number of PIT entries in one such table. Based on the above discussion, and assuming\nagain that a name is at most $64$B, a standard PIT entry is of the size $80$B. In contrast,\na gateway PIT entry is of size $204$B. Therefore, in the worst case, the CCVPN PIT\nwill be at most $P_G / P_S = 204/80 = 2.55$B larger than the standard PIT. Assuming\na steady state size of approximately $1e^5$ entries \\cite{carofiglio2015pending},\nthis means that the PIT will be $20.4$MB, which is well within the capacity of\nmodern systems.\n\n\\subsection{Processing Overhead}\nIn terms of processing overhead, the gateway adds a number of new steps to the data\npath of a packet. The main computational burdens are packet encapsulation and decapsulation.\nIn the public-key variant of CCVPN, interests are processed using public-key encryption,\nwhereas content is always processed using symmetric-key encryption. Let $T_E^P(n)$ and $T_D^P(n)$\nbe the time to encrypt and decrypt $n$B of data using a suitable public-key encryption scheme.\nSimilarly, let $T_E^S(n)$ and $T_D^S(n)$ be the time to encrypt and decrypt $n$B of data\nusing a symmetric-key encryption scheme. Then, the latency in a single interest-content\nexchange is increased by $T = T_E^P(n_I) + T_D^P(n_I) + T_E^S(n_C) + T_D^S(n_C)$, where\n$n_I$ and $n_C$ are the original interest and content sizes, respectively. As a rough\nestimate, \\cite{benchmarks} lists the cost of AES-GCM to be $2.946\\mu s$ for setup\nfollowed by $102$MiB/second Intel Core 2 1.83 GHz processor under Windows Vista in\n32-bit mode (with AES ISA support). For packets that are at most $1500$B,\nthe total processing time is roughly $17\\mu s$. Moreover, The public-key encryption and\ndecryption operations will always be at least as expensive, so the total latency is\nincreased by at least $T = 4 \\times 17\\mu s = 68 \\mu s$. In comparison to the network\nlatency for a single packet this may not be noticable, but for a steady arrival state of\napproximatey $1e^5$, this would lead to an instable system that would quickly overflow.\n(This is because $65 \\mu s \\times 1e^5 = 6.8s$.) Therefore, there is an upper bound on\nthe number of private packets a gateway can process per second. This bound is entirely\ndependent on the system configuration and network conditions.\n\nAnother performance deficiency comes from the fact that gateways cannot process packets\nwithout allocating memory. Specifically, each packet requires either an encryption\nor decryption. However, this cannot be done entirely in-place, the gateway must allocate some amount\nof memory for every processed packet, e.g., to store the MAC tag, to account for\nciphertext expansion, or to apply padding, if desired. This overhead can outweigh\nthe cryptographic computations if the packet arrival rate is high enough. Therefore,\nwhen implementing CCVPN, special care must be taken to ensure that memory allocations\nare minimized or avoided where possible.\n", "meta": {"hexsha": "a6617079f76817a00694d6d084ee90ea977d27ee", "size": 5515, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/06-perf-analysis.tex", "max_stars_repo_name": "chris-wood/ccvpn", "max_stars_repo_head_hexsha": "61994629f6ebec54425ff3dfa7992a71ba3cc9a3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper/06-perf-analysis.tex", "max_issues_repo_name": "chris-wood/ccvpn", "max_issues_repo_head_hexsha": "61994629f6ebec54425ff3dfa7992a71ba3cc9a3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/06-perf-analysis.tex", "max_forks_repo_name": "chris-wood/ccvpn", "max_forks_repo_head_hexsha": "61994629f6ebec54425ff3dfa7992a71ba3cc9a3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 73.5333333333, "max_line_length": 100, "alphanum_fraction": 0.7747960109, "num_tokens": 1390, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6370307806984444, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3458206210919017}} {"text": "\\documentclass{article}\n\\usepackage{hyperref}\n\\usepackage{amsmath,amssymb}\n\\usepackage{graphicx}\n\\usepackage{caption}\n\\usepackage{subcaption}\n\\usepackage[section]{placeins}\n\\renewcommand{\\thesubsection}{\\thesection.\\alph{subsection}}\n\\usepackage{listings}\n\n\\title{\\bf{CSE397: Assignment \\#1}}\n\\author{Nicholas Malaya \\\\ Institute for Computational Engineering and Sciences \\\\ University of Texas at Austin} \\date{}\n\n\\begin{document}\n\\maketitle\n\n\\newpage\n\\section{Problem 1}\n\n\\begin{figure}[!htb]\n \\includegraphics[scale=.5]{plots/data.pdf}\n \\label{fig:data}\n \\caption{The data (after applying the filter) with and without\n normally distributed noise. } \n\\end{figure}\n\nAfter applying the new filter $k(x)$ and adding gaussian noise, the data\nused as input for the inverse problem is plotted in figure 1. \n\nNotice that this filter is significant: the raw data generated after\npassing through the filter (even without statistical noise) has lost\nseveral features of the underlying ``true'' signal. \n\n\\subsection{$T_{\\text{SVD}}$}\n\n\n\\begin{figure}[!htb]\n \\centering\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/tsvd0001.pdf}\n \\caption{$\\alpha=0.0001$}\n \\end{subfigure}%\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/tsvd001.pdf}\n \\caption{$\\alpha=0.001$}\n \\end{subfigure}\n \\centering\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/tsvd01.pdf}\n \\caption{$\\alpha=0.1$}\n \\end{subfigure}%\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/tsvd1.pdf}\n \\caption{$\\alpha=1.0$}\n \\end{subfigure}\n \\caption{$T_{\\text{SVD}}$ at varying values of $\\alpha$.}\n \\label{fig:svd}\n\\end{figure}\n\nFigure \\ref{fig:svd} plots the solution to the inverse problem using\nTruncated SVD at several different values of $\\alpha$, the\nregularization parameter. We can see that actually, the filter nicely\nregularizing the inverse problem even for small values of $\\alpha$. It\nis only for the largest value of $\\alpha$ that the parameter is obviously\ntoo large, at which point almost all the frequencies are filtered out,\nand the solution to the inverse problem is constant and zero. \n\nAs expected, none of the solutions capture any of the very fine (high\nfrequency) features of the original data. \n\n\\subsection{Tikhanov Filter}\n\n\n\\begin{figure}[!htb]\n \\centering\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/reconstruct0001.pdf}\n \\caption{$\\alpha=0.0001$}\n \\end{subfigure}%\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/reconstruct001.pdf}\n \\caption{$\\alpha=0.001$}\n \\end{subfigure}\n \\centering\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/reconstruct01.pdf}\n \\caption{$\\alpha=0.1$}\n \\end{subfigure}%\n \\begin{subfigure}[bh]{0.45\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/reconstruct1.pdf}\n \\caption{$\\alpha=1.0$}\n \\end{subfigure}\n \\caption{Solutions to the inverse problem using Tikhanov\n regularization at varying values of $\\alpha$.} \n \\label{fig:tik}\n\\end{figure}\n\nFigure \\ref{fig:tik} plots the solution to the inverse problem using\nTikhanov regularization at several different values of $\\alpha$, the\nregularization parameter. \n\nThe Tikhanov filter is more sensitive, with (a) and (b) (low values of\n$\\alpha]$) completely being dominated by noise. None of the underlying\nsignal is regained, as all the high frequency content blown up and\ndominating. \n\nFor higher levels of the regularization parameter, we regain a solution\nsimilar to in the TSVD cases, where we capture the underlying trend of\nthe data, but none of the sharp features. Also, the highest value of the\nfilter does not overwhelm the signal completely, as in the TSVD case,\nwhich is interesting. \n\n\n\\subsection{L-Curve}\n\n\\begin{figure}[!htb]\n \\includegraphics[scale=.6]{plots/L-curve.pdf}\n \\caption{The L-curve} \n \\label{fig:lcurve}\n\\end{figure}\n\nFigure \\ref{fig:lcurve} plots the results of the L-curve criterion. The\nred dot plotted is at 0.01. This is approximately the ``optimal'' value\nof regularization, which is generally consistent with the observations\nof the previous plots. \n\n\n\\subsection{Morozov's Discrepancy Criterion}\n\n\\begin{figure}[!htb]\n \\includegraphics[scale=.5]{plots/morozov.pdf}\n \\caption{Morozov discrepancy Criterion} \n \\label{fig:moro}\n\\end{figure}\n\nFigure \\ref{fig:moro} plots the results of Morozov discrepancy\ncriterion. I could not get the plots to draw a horizontal line, so that\nred dot instead dictates the limit of $\\delta$, the norm of the\nerror. This occurs at $\\approx 4e-2$. This is smaller than the value\nobtained using the L-curve, but not so much smaller that the value\nappears suspect. \n\n\n\\subsection{Actual Error}\n\n\\begin{figure}[!htb]\n \\includegraphics[scale=.5]{plots/true1d.pdf}\n \\caption{The actual error in the reconstruction as a function of\n alpha. } \n \\label{fig:actual}\n\\end{figure}\n\nFigure \\ref{fig:actual} plots actual error in the reconstruction,\ne.g. the norm of the difference between the reconstruction and the\nactual (true) image, as a function of alpha. You can see that this\nplot forms a rotated ``S'' curve, with the minimum occuring\napproximately at the red dot. The value plotted is 0.01,\nwhich actually agrees with the L-curve estimate. \n\nIn any case, the values arrived at using the L-curve, Morozov's\ndiscrepancy criterion and the actual error all agree within at least an\norder of magnitude. So we are in the right ``ballpark''. As an aside, I\nwas actually quite pleased with the result, I would not have expected\nthese criterion to so closely arrive at values that are reasonable\nguesses, even without knowledge of the true underlying signal. While\nthis is a toy problem, it does show that these criterion can be\neffective means of approximating $\\alpha$. Given the similarities\nbetween the TSVD and Tikhanov, I would not be surprised to find that\nthese heuristics can be more generally useful for other regularizations as well. \n\n\n\\newpage\n\\section{Problem 2}\n\n% \\begin{figure}[!htb]\n% \\includegraphics[scale=.5]{plots/longhorn.png}\n% \\label{fig:true}\n% \\caption{The true image.} \n% \\end{figure}\n\n% Figure \\ref{fig:true} plots the true image, that we will be attempting\n% to reconstruct through inversion. \n\n\\subsection{Memory Requirement}\n\nHow much memory would be required to form the entire matrix, $K$?\nWe have $N^2$ model parameters. K lives in $\\mathbb{R}^{N_2 * N_2}$. \nWith $N=128$, then \n\\begin{equation}\n\\frac{N^2 * N^2 * 8 \\text{Bytes}}{(1024)^3} = 2 \\text{GB}\n\\end{equation}\nSo while this is not inaccessible on modern HPC systems, it is certainly\nnot efficient. \n\n\\subsection{L-Curve}\n\n\\begin{figure}[!htb]\n \\includegraphics[scale=.5]{plots/L-curve2d.pdf}\n \\caption{The L-curve for the 2d problem.} \n \\label{fig:l2d}\n\\end{figure}\n\nFigure \\ref{fig:l2d} plots the L-curve. The red dot is approximately at\nthe location of the optimal value of $\\alpha$, which is found to be\n$\\approx 0.05 - 0.001$. \n\n\\subsection{Actual Error}\n\n\\begin{figure}[!htb]\n \\includegraphics[scale=.5]{plots/2d-true.pdf}\n \\caption{The actual error in the reconstruction as a function of\n alpha, for the 2d problem.} \n \\label{fig:2dt}\n\\end{figure}\n\nFigure \\ref{fig:2dt} plots the true error. Surprisingly, the error curve\nappears to flatten out at lower values of alpha, implying a constant\nerror. Clearly any values less than $1e-3$ appear to be prefered, as the\nerror increases at higher values of $\\alpha$. \n\nTo calculate the error, I only needed to solve the conjugate gradient\n(as before) and then calculate the norm of the misfit against $I$, the\ntrue image:\n\\begin{lstlisting}\n I_alpha = pcg(@(in)apply(in,K1,K2,N1,N2,alpha),K_Ibn(:),1e-6,1500);\n misfit(k) = norm((K1 * (K2 * reshape(I_alpha,N2,N1))')' - I); \n\\end{lstlisting}\n\\begin{figure}[!htb]\n \\centering\n \\begin{subfigure}[bh]{0.65\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/tikrecon0001.pdf}\n \\caption{$\\alpha=0.0001$}\n \\end{subfigure}%\n \\begin{subfigure}[bh]{0.65\\textwidth}\n \\includegraphics[width=\\textwidth]{plots/tikrecon001.pdf}\n \\caption{$\\alpha=0.001$}\n \\end{subfigure}\n \\caption{The reconstructed image.} \n\\label{fig:tik}\n\\end{figure}\n\nThe ``true'' image would imply then that the L-curve is too conservative\nan estimate, and smaller values of $\\alpha$ are acceptable. However, I\ndon't believe that. Figure 9 plots the Tikhanov\nreconstruction for $\\alpha = 0.001$ and $\\alpha = 0.0001$. Clearly the\ntrue image is recognizable in the right hand side, and not at all in the\nleft. So I believe that I either have a bug in my calculation of the\nactual error, or the L-curve is, somehow, a more reliable estimate of\n``good'' values of $\\alpha$, here. \n\n\\end{document}", "meta": {"hexsha": "d6a3d78accbaecf58595e021b7de7d4af3e0ba23", "size": 9214, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "inv_prob/ps1/report.tex", "max_stars_repo_name": "nicholasmalaya/paleologos", "max_stars_repo_head_hexsha": "11959056caa80d3c910759b714a0f8e42f986f0f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-04T17:49:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-04T17:49:42.000Z", "max_issues_repo_path": "inv_prob/ps1/report.tex", "max_issues_repo_name": "nicholasmalaya/paleologos", "max_issues_repo_head_hexsha": "11959056caa80d3c910759b714a0f8e42f986f0f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "inv_prob/ps1/report.tex", "max_forks_repo_name": "nicholasmalaya/paleologos", "max_forks_repo_head_hexsha": "11959056caa80d3c910759b714a0f8e42f986f0f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-01-04T16:08:18.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-16T19:34:24.000Z", "avg_line_length": 36.7091633466, "max_line_length": 121, "alphanum_fraction": 0.7100065118, "num_tokens": 2529, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.3458206210919017}} {"text": "% !TEX root = ../../../proposal.tex\n\n\\section{TLS}\\label{sec:subgroup_tls}\n\nTLS (Transport Layer Security) is a transport layer protocol designed to provide confidentiality,\nintegrity and (most commonly) one-side authentication for application sessions. \nIt is widely used to protect HTTP and mail protocols. \n\nA TLS client initiates a TLS handshake with the \\texttt{Client\\-Hello} message.\nThis message includes a list of supported cipher suites, and a client random\nnonce $r_c$. The server responds with a \\texttt{Server\\-Hello} message containing\nthe chosen cipher suite and server random nonce $r_s$, and a\n\\texttt{Certificate} message that includes the server's X.509 certificate. If\nthe server selects a cipher suite using ephemeral Diffie-Hellman key exchange,\nthe server additionally sends a \\texttt{Server\\-Key\\-Exchange} message containing\nthe server's choice of Diffie-Hellman parameters $p$ and $g$, the server's\nDiffie-Hellman public value $y_s = g^{x_s} \\bmod p$, a signature by the\nserver's private key over both the client and server nonces ($r_c$ and $r_s$),\nand the server's Diffie-Hellman parameters ($p$, $g$, and $y_s$). The client\nthen verifies the signature using the public key from the server's certificate,\nand responds with a \\texttt{Client\\-Key\\-Exchange} message containing the client's\nDiffie-Hellman public value $y_c = g^{x_c} \\bmod p$. The Diffie-Hellman shared\nsecret $Y = g^{x_s x_c} \\bmod p$ is used to derive encryption and MAC keys. The\nclient then sends \\texttt{Change\\-Cipher\\-Spec} and \\texttt{Finished} messages. The\n\\texttt{Finished} message contains a hash of the handshake transcript, and is\nencrypted and authenticated using the derived encryption and MAC keys. Upon\ndecrypting and authenticating this message, the server verifies that the hash\nof the transcript matches the expected hash. Provided the hash matches, the\nserver then sends its own \\texttt{Change\\-Cipher\\-Spec} and \\texttt{Finished}\nmessages, which the client then verifies. If either side fails to decrypt or\nauthenticate the \\texttt{Finished} messages, or if the transcript hashes do not\nmatch, the connection fails immediately~\\cite{rfc5246}.\n\nTLS also specifies a mode of using Diffie-Hellman with fixed parameters from\nthe server's certificate~\\cite{rfc3279}. This mode is not forward secret, was\nnever widely adopted, and has been removed from all modern browsers due to\ndangerous protocol flaws~\\cite{kci-tls-2015}. The only widely used form of\nDiffie-Hellman in TLS today is ephemeral Diffie-Hellman, described above.\n\n\\subsection{Small Subgroup Attacks in TLS}\n\\label{sec:tls-subgroup-attack}\n\n\\paragraph{Small subgroup confinement attacks}\nA malicious TLS server can perform a variant of the small subgroup attack\nagainst a client by selecting group parameters $g$ and $p$ such that $g$\ngenerates an insecure group order. TLS versions prior to 1.3 give the server\ncomplete liberty to choose the group, and they do not include any method for\nthe server to specify the desired group order $q$ to the client. This means a\nclient has no feasible way to validate that the group sent by the server has the\ndesired level of security or that a server's key exchange value is in\nthe correct group for a non-safe prime.\n\\looseness=-1\n\nSimilarly, a man in the middle with knowledge of the server's long-term private\nsigning key can use a small subgroup confinement attack to more easily\ncompromise perfect forward secrecy, without having to rewrite an entire\nconnection. The attack is similar to the those described by Bhargavan and\nDelignat-Lavaud~\\cite{bhargavan-channel-bindings-2015}. The attacker modifies the\nserver key exchange message, leaving the prime unchanged, but substituting a\ngenerator $g_i$ of a subgroup of small order $q_i$ for the group generator and \n$g_i$ for the server's key exchange value $y_s$. The attacker then forges a correct signature for the modified server key exchange message and passes it to the client. The client then\nresponds with a client key exchange message $y_c = g_i^{x_c} \\bmod p$, which\nthe man-in-the-middle leaves unchanged. The server's view of the shared secret\nis then $g_i^{x_c x_s} \\bmod p$, and the client's view of the shared secret is\n$g_i^{x_c} \\bmod p$. These views are identical when ${x_s} \\equiv 1 \\bmod q_i$,\nso this connection will succeed with probability $1/q_i$. For small enough\n$q_i$, this enables a man in the middle to use a compromised server signing key to\ndecrypt traffic from forward-secret ciphersuites with a reasonable probability\nof success, while only requiring tampering with a single handshake message,\nrather than having to actively rewrite the entire connection for the duration of the session.\n\nFurthermore, if the server uses a static Diffie-Hellman key exchange value,\nthen the attacker can perform a small subgroup key-recovery attack as the\nclient in order to learn the server's static exponent $x_s \\bmod q_i$ for the\nsmall subgroup. This enables the attacker to calculate a custom generator such\nthat the client and server views of the shared secret are always identical,\nraising the above attack to a 100\\% probability of success.\n\n\n\\paragraph{Small subgroup key recovery attacks} In TLS, the client must\nauthenticate the handshake before the server, by providing a valid\n\\texttt{Finished} message. This forces a small subgroup key recovery attack\nagainst TLS to be primarily online. To perform a Lim-Lee small subgroup key\nrecovery attack against a server static exponent, a malicious client initiates\na TLS handshake and sends a generator $g_i$ of a small subgroup of order $q_i$\nas its client key exchange message $y_c$. The server will calculate $Y_s =\ng_i^{x_s} \\bmod p$ as the shared secret. The server's view of the shared secret\nis confined to the subgroup of order $q_i$. However, since $g_i$ and $g$\ngenerate separate subgroups, the server's public value $y_s = g^x_s$ gives the\nattacker no information about the value of the shared secret $Y_s$. Instead,\nthe attacker must guess a value for $x_s \\bmod q_i$, and send the corresponding\nclient \\texttt{Finished} message. If the server continues the handshake, the\nattacker learns that the guess is correct. Therefore, assuming the server is\nreusing a static value for $x_s$, the attacker needs to perform at most $q_i$\nqueries to learn the server's secret $x_s \\bmod q_i$~\\cite{lim-1997}. This\nattack is feasible if $q_i$ is small enough and the server reuses\nDiffie-Hellman exponents for sufficiently many requests.\n\nThe attacker repeats this process for many different primes $q_i$, and uses the\nChinese remainder theorem to combine them modulo the product of the primes\n$q_i$. The attacker can also use the Pollard lambda algorithm to reconstruct\nany remaining bits of the exponent~\\cite{lim-1997}.\n\nWe note that the TLS False Start extension allows the\nserver to send application data before receiving the client's authentication~\\cite{rfc7918}.\nThe specification only allows this behavior for abbreviated handshakes, which\ndo not include a full key exchange. If a full key exchange were allowed, the\nfact that the server authenticates first would allow a malicious client to\nmount a mostly offline key recovery attack.\n%\\todo{Add discussion of the old unused TLS extension that makes this offline. \\ref{rfc7918}}\n\n\n\n\\subsection{OpenSSL}\n\nPrior to early 2015, OpenSSL defaulted to using static-ephemeral Diffie-Hellman\nvalues. Server applications generate a fresh Diffie-Hellman secret exponent on\nstartup, and reuse this exponent until they are restarted. A server would be\nvulnerable to small subgroup attacks if it chose a DSA prime, explicitly\nconfigured the \\texttt{dh->length} parameter to generate a short exponent, and\nfailed to set \\texttt{SSL\\_OP\\_SINGLE\\_DH\\_USE} to prevent repeated exponents.\nOpenSSL provides some test code for key generation which configures DSA group\nparameters, sets an exponent length to the group order, and correctly sets the\n\\texttt{SSL\\_OP\\_SINGLE\\_DH\\_USE} to generate new exponents on every\nconnection. We found this test code widely used across many applications. We\ndiscovered that Unbound, a DNS resolver, used the same parameters as the tests,\nbut without setting \\texttt{SSL\\_OP\\_SINGLE\\_DH\\_USE}, rendering them\nvulnerable to a key recovery attack. A number of other applications including\nLighttpd used the same or similar code with non-safe primes, but correctly set\n\\texttt{SSL\\_OP\\_SINGLE\\_DH\\_USE}.\n\nIn spring 2015, OpenSSL added explicit support for RFC~5114\ngroups~\\cite{openssl-changelog-102}, including the ability for servers to\nspecify a subgroup order in a set of Diffie-Hellman group parameters. When the\nsubgroup order is specified, the exponent length is automatically adjusted to\nmatch the subgroup size. However, the update did not contain code to validate\nsubgroup order for key exchange values, leaving OpenSSL users vulnerable to\nprecisely the key recovery attack outlined in\nSection~\\ref{sec:tls-subgroup-attack}.\n\nWe disclosed this vulnerability to OpenSSL in January 2016. The vulnerability\nwas patched by including code to validate subgroup order when a subgroup was\nspecified in a set of Diffie-Hellman parameters and setting\n\\texttt{SSL\\_OP\\_SINGLE\\_DH\\_USE} by default~\\cite{openssl-secadv-subgroup}.\nPrior to this patch, any code using OpenSSL for DSA-style Diffie-Hellman\nparameters was vulnerable to small subgroup attacks by default.\n\n\\ApplicationsTable\n\nExim~\\cite{exim}, a popular mail server that uses OpenSSL, provides a clear\nexample of the fragile situation created by this update. By default, Exim uses\nthe RFC~5114 Group 23 parameters with OpenSSL, does not set an exponent length,\nand does not set \\texttt{SSL\\_OP\\_SINGLE\\-\\_DH\\_USE}. In a blog post, an Exim\ndeveloper explains that because of ``numerous issues with automatic generation\nof DH parameters'', they added support for fixed groups specified in RFCs and picked\nGroup~23 as the default~\\cite{exim-blog}. Exim narrowly avoided being fully\nvulnerable to a key recovery attack by not including the size of the subgroup\ngenerated by $q$ in the Diffie-Hellman parameters that it passes to OpenSSL.\nHad this been included, OpenSSL would have automatically shortened the exponent\nlength, leaving the server fully vulnerable to a key recovery attack. For this\ngroup, an attacker can recover 130 bits of information about the secret\nexponent using $2^{33}$ online queries, but this does not allow the attacker to\nrecover the server's 2048-bit exponent modulo the correct 224-bit group order\n$q$ as the small subgroup orders $q_i$ are all relatively prime to $q$.\n\nWe looked at several other applications as well, but did not find them to be\nvulnerable to key recovery attacks (Table~\\ref{tab:common-applications}).\n\n\n% comprising 53.62\\% of all such servers~\\cite{esoft2016}.\n\n% Exim can be compiled either with GNU TLS or with OpenSSL. When compiled with OpenSSL,\n% it supports single use DH keys with an option that is off by default\n% (as of version 4.87, the latest at time of writing). In OpenSSL mode, it\n% includes checks for unsafe primes and incorrect generators, however it\n% ignores the generator check when the generator is not 2 or 5, due to a known\n% bug in OpenSSL. More significantly, it utilizes the DSA primes with small\n% subgroups. Their code comments note ``These have been thoroughly reviewed as\n% meeting certain eligibility criteria, which is more than can be said for primes\n% generated quickly''. However, they fail to note the difference in criteria for\n% the DSA primes as opposed to the others, resulting in the selection of vulnerable\n% groups. The subgroup size $q$ was not included in the Exim's default DSA parameter, saving \n% a default Exim installation from being vulnerable to the attack described \n% in Section~\\ref{sec:tls-subgroup-attack} due the fact that OpenSSL did not have the possibility \n% to match the exponent length to subgroup size. Said that, if the DSA parameter would have been generated using the \n% OpenSSL's genpkey utility~\\cite{genpkey} all the conditions to perform the Small subgroup key recovery attack\n% as per Section ~\\ref{sec:tls-subgroup-attack} would have been met.\n% In comparison Postfix an MX utilized for 32.80\\% of mail servers~\\cite{esoft2016}\n% suggests generating a group at install time but neglects to set the single use OpenSSL option\n% and includes two hardcoded groups by default.\n\n\\subsection{Other Implementations} \\label{subsec:nss}\n\nWe examined the source code of multiple TLS implementations\n(Table~\\ref{tab:tls-implementations}). Prior to January 2016, no TLS\nimplementations that we examined validated group order, even for the well-known\nDSA primes from RFC~5114, leaving them vulnerable to small subgroup confinement attacks.\n\nMost of the implementations we examined attempt to match exponent length to the\nperceived strength of the prime. For example, Mozilla Network Security Services\n(NSS), the TLS library used in the Firefox browser and some versions of\nChrome~\\cite{nss-overview,chrome-to-openssl}, uses NIST's ``comparable key strength'' recommendations\non key management~\\cite{sp800} to determine secret exponent lengths from the length of the prime.~\\cite{nss-line-of-code} Thus NSS uses 160-bit exponents with a 1024-bit prime, and 224-bit exponents with a 2048-bit prime.\n In fall 2015, NSS added an additional check to ensure that the\nshared secret $g^{x_ax_b} \\not \\equiv 1 \\bmod p$~\\cite{nss-code-secret-not-one}.\n\nSeveral implementations go to elaborate lengths to match exponent length to\nperceived prime strength. The Cryptlib library fits a quadratic curve to the\nsmall exponent attack cost table in the original van~Oorschot\npaper~\\cite{van1996diffie} and uses the fitted curve to determine safe key\nlengths~\\cite{cryptlib-fitted-curve}. The Crypto++ library uses an explicit\n``work factor'' calculation, evaluating the function $2.4 n^{1/3} (\\log\nn)^{2/3}$~\\cite{cryptoplusplus-work-factor}. Subgroup order and exponent\nlengths are set to twice the calculated work factor. The work factor\ncalculation is taken from a 1995 paper by Odlyzko on integer\nfactorization~\\cite{odlyzko-1995}. Botan, a C++ cryptography and TLS library,\nuses a similar work factor calculation, derived from RFC~3766~\\cite{rfc3766},\nwhich describes best practices as of 2004 for selecting public key strengths\nwhen exchanging symmetric keys. RFC~3766 uses a similar work factor algorithm\nto Odlyzko, intended to model the running time of the number-field\nsieve. Botan then doubles the length of the work factor to obtain subgroup and\nexponent lengths~\\cite{botan-double}.\n\n% Apple Safari on OS X and iOS performed no validation of\n% Diffie-Hellman prime lengths until July 2015, and supported connections where\n% the server offered a trivially-broken group using a 16-bit prime~\\cite{weakdh-ccs15}.\n\n\\subsection{Measurements}\n\\label{sec:tls-measurements}\n\nWe used ZMap~\\cite{zmap-2013} to probe the public IPv4 address space for hosts\nserving three TLS-based protocols: HTTPS, SMTP+STARTTLS, and POP3S\\@. To\ndetermine which primes servers were using, we sent a \\texttt{ClientHello}\nmessage containing only ephemeral Diffie-Hellman cipher suites. We combined\nthis data with scans from Censys~\\cite{censys} to determine the overall\npopulation. The results are summarized in Table~\\ref{tab:scandata}. \n\n\\ScanTable\n\n\\TLSHostValidationTable\n\nIn August 2016, we conducted additional scans of a random 1\\% sample\nof HTTPS hosts on the Internet. First, we checked for nontrivial\nsmall subgroup attack vulnerability. For servers that sent us a prime\n$p$ such that $p-1$ was divisible by 7, we attempted a handshake using\na client key exchange value of $g_7\\bmod p$, where $g_7$ is a\ngenerator of a subgroup of order $7$. (7 is the smallest prime factor\nof $p-1$ for Group 22.) When we send $g_7$, we expect to correctly\nguess the \\texttt{PreMasterSecret} and complete the handshake with one\nseventh of hosts that do not validate subgroup order. In our scan, we\nwere able to successfully complete a handshake with $1477$ of $10714$\nhosts that offered a prime such that $p-1$ was divisible by $7$,\nimplying that approximately 96\\% of these hosts fail to validate\nsubgroup order six months after OpenSSL pushed a patch adding group\norder validation for correctly configured groups.\n\nSecond, we measured how many hosts performed even the most basic\nvalidation of key exchange values. We attempted to connect to HTTPS hosts with\nthe client key exchange values of $y_c = 0 \\bmod p, 1 \\bmod p, -1 \\bmod p$. As\nTable~\\ref{tab:tlsvalidation} shows, we found that over 5\\% of hosts that\naccepted DHE ciphersuites accepted the key exchange value of $-1 \\bmod p$ and\nderived the \\texttt{PreMasterSecret} from it. These implementations are\nvulnerable to a trivial version of the small subgroup confinement attacks\ndescribed in Section~\\ref{sec:tls-subgroup-attack}, for \\emph{any} prime\nmodulus $p$. By examining the default web pages of many of these hosts,\nwe identified products from several notable companies including Microsoft,\nCisco, and VMWare. When we disclosed these findings, VMWare notified us that\nthey had already applied the fix in the latest version of their products;\nMicrosoft acknowledged the missing checks but chose not to include them since\nthey only use safe primes, and adding the checks may break functionality\nfor some clients that were sending unusual key exchange values; and Cisco\ninformed us that they would investigate the issue.\n\nOf 40.6\\,M total HTTPS hosts found in our scans, 10.8\\,M~(27\\%) supported\nephemeral Diffie-Hellman, of which 1.6\\,M~(4\\%) used a non-safe prime, and\n309\\,K~(0.8\\%) used a non-safe prime and reused exponents across multiple\nconnections, making them likely candidates for a small subgroup key recovery\nattack. We note that the numbers for hosts reusing exponents are an\nunderestimate, since we only mark hosts as such if we found them using the same\npublic Diffie-Hellman value across multiple connections, and some load\nbalancers that cycle among multiple values might have evaded detection.\n\nWhile 77\\%~of POP3S hosts and 39\\%~of SMTP servers used a non-safe prime, a\nmuch smaller number used a non-safe prime and reused exponents (<0.01\\% in both\nprotocols), suggesting that the popular implementations (Postfix and\nDovecot~\\cite{mail-2015}) that use these primes follow recommendations to use\nephemeral Diffie-Hellman values with DSA primes.\n\nTable~\\ref{tab:primes} shows nine groups that accounted for the majority of\nnon-safe primes used by hosts in the wild. Over 1.17\\,M hosts across all of our\nHTTPS scans negotiated Group~22 in a key exchange. To get a better picture of\nwhich implementations provide support for this group, we examined the default\nweb pages of these hosts to identify companies and products, which we show in\nTable~\\ref{tab:tls-group22-support}.\n\n\\TLSGroupSupport\n\nOf the the 307\\,K HTTPS hosts that both use non-safe primes and reuse\nexponents, 277\\,K~(90\\%) belong to hosts behind Amazon's Elastic Load\nBalancer~\\cite{amazon-elb}. These hosts use a 1024-bit prime with a 160-bit\nsubgroup. We set up our own load balancer instance and found that the\nimplementation failed to validate subgroup order. We were able to use a\nsmall-subgroup key recovery attack to compute 17 bits of our load balancer's\nprivate Diffie-Hellman exponent $x_s$ in only 3813 queries. We responsibly\ndisclosed this vulnerability to Amazon. Amazon informed us that they have\nremoved Diffie-Hellman from their recommended ELB security policy, and are\nencouraging customers to use the latest policy. In May 2016, we performed\nadditional scans and found that 88\\% of hosts using this prime no longer\nrepeated exponents. We give a partial factorization for $p-1$ in\nTable~\\ref{tab:group-order-factorization}; the next largest subgroups have 61\nand 89 bits and an offline attack against the remaining bits of a 160-bit\nexponent would take $2^{71}$ time. For more details on the computation, see\nSection~\\ref{sec:ecm}.\n\nSSLeay~\\cite{ssleay}, a predecessor for OpenSSL, includes several default\nDiffie-Hellman primes, including a 512-bit prime. We found that 717 SMTP\nservers used a version of the OpenSSL 512-bit prime with a single character\ndifference in the hexadecimal representation. The resulting modulus that these\nservers use for their Diffie-Hellman key exchange is no longer prime. We\ninclude the factorization of this modulus along with the factors of the\nresulting group order in Table~\\ref{tab:group-order-factorization}. The use of\na composite modulus further decreases the work required to perform a small\nsubgroup attack.\n\nAlthough TLS also includes static Diffie-Hellman cipher suites that require a\nDSS certificate, we did not include them in our study; no browser supports\nstatic Diffie-Hellman~\\cite{kci-tls-2015}, and Censys shows no hosts with DSS\ncertificates, with only 652 total hosts with non-RSA or ECDSA certificates.\n\n%In February 2016, we conducted an additional scan of a random 1\\% sample of the\n%Internet to check how many hosts performed even the most basic validation of\n%key exchange values. We attempted to connect to HTTPS hosts with a client\n%key exchange message of $y_c = 1$. We found that 322 of 589,241 hosts accepted\n%this key exchange value and derived the \\texttt{PreMasterSecret} from it. 248\n%of these hosts were Cisco devices, another 28 were VMware Horizon View servers,\n%and the remaining 46 devices did not fit into any clear category. These\n%implementations are vulnerable to a trivial version of the small subgroup\n%confinement attacks described in Section~\\ref{sec:tls-subgroup-attack}.\\todo{did we disclose this? we need to. also what about validation of values other than 0?}\n\n\n%\\TODO{classify implementations again and disclose to Cisco, etc}\n\n\\begin{landscape}\n\\TLSLibraryTable\n\\PrimesAllTLS\n\\end{landscape}\n\n\n\n\n", "meta": {"hexsha": "ee4ec5ff9e704fe40e2e2b794bda03e9e3b062ba", "size": 21823, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "papers/subgroup/paper/tls.tex", "max_stars_repo_name": "dadrian/dissertation", "max_stars_repo_head_hexsha": "5607114fb4340c5b6e944c73ed6019006d3ebec9", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "papers/subgroup/paper/tls.tex", "max_issues_repo_name": "dadrian/dissertation", "max_issues_repo_head_hexsha": "5607114fb4340c5b6e944c73ed6019006d3ebec9", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "papers/subgroup/paper/tls.tex", "max_forks_repo_name": "dadrian/dissertation", "max_forks_repo_head_hexsha": "5607114fb4340c5b6e944c73ed6019006d3ebec9", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 62.1737891738, "max_line_length": 222, "alphanum_fraction": 0.7940246529, "num_tokens": 5335, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.345799967212732}} {"text": "%%% Copyright 2017 International Business Machines Corporation\n\\documentclass[11pt]{article} % use larger type; default would be 10pt\n\n\\usepackage[utf8]{inputenc} % set input encoding (not needed with XeLaTeX)\n\\usepackage[T1]{fontenc}\n\n\n%%% PAGE DIMENSIONS\n\\usepackage[top=1.5in,bottom=1in,right=1in,left=1in,headheight=90pt,headsep=1cm]{geometry} % to change the page dimensions\n\\geometry{a4paper} % or letterpaper (US) or a5paper or....\n \\geometry{margin=0.7in} % for example, change the margins to 2 inches all round\n\n\\usepackage{graphicx} % support the \\includegraphics command and options\n\n \\usepackage[parfill]{parskip} % Activate to begin paragraphs with an empty line rather than an indent\n\n%%% PACKAGES\n\\usepackage{amsmath}\n\\usepackage{cases}\n\\usepackage{ulem}\n\\usepackage{listings}\n\n\n\\usepackage{fancyhdr} % This should be set AFTER setting up the page geometry\n\\pagestyle{fancy} % options: empty , plain , fancy\n\\renewcommand{\\headrulewidth}{0.4pt} % customise the layout...\n\\lhead{}\\chead{\\footnotesize \\textit{IBM}}\\rhead{}\n\\lfoot{}\\cfoot{\\thepage}\\rfoot{}\n\n%%% SECTION TITLE APPEARANCE\n\\usepackage{sectsty}\n\n\\newenvironment{aside}\n {\\begin{mdframed}[style=0,%\n leftline=false,rightline=false,leftmargin=2em,rightmargin=2em,%\n innerleftmargin=0pt,innerrightmargin=0pt,linewidth=0.75pt,%\n skipabove=7pt,skipbelow=7pt]\\small}\n {\\end{mdframed}}\n\n%COLOURING\n\\usepackage{color}\n\\newcommand{\\new}{\\textcolor{red}}\n%\\input{rgb}\n\n%%% ToC (table of contents) APPEARANCE\n\\usepackage[nottoc,notlof,notlot]{tocbibind} % Put the bibliography in the ToC\n\n\n\\title{The Iterative Reweighted Least Squares Method\\\\ for $\\ell_1$ Norm Minimisation}\n\\author{Cecilia Aas}\n\\date{October, 2015}\n\n\n\\def\\layersep{2.5cm}\n\n\\begin{document}\n\\lstset{language=C++,\n commentstyle=\\textcolor[rgb]{0.00,0.66,0.33},\n keywordstyle=\\textcolor[rgb]{0.00,0.00,1.00},\n basicstyle=\\footnotesize\\ttfamily,\n frame=lines,\n framexleftmargin=2mm,\n numbers=left,\n numberstyle=\\footnotesize,\n stepnumber=1,\n numbersep=1pt}\n\\maketitle\n\\section{Overview}\nThe algorithm aims to solve the problem\n\\begin{equation}\n\\label{eq:l1min}\n\\min\\left|\\left| \\uline{x} \\right| \\right|_1 \\text{ subject to } \\uuline{A}\\uline{x} = \\uline{y}\n\\end{equation}\nby finding the minimisers $\\left\\{x_0, x_1, \\cdots \\right\\}$ of the functional\n\\begin{equation}\nJ_{\\lambda}(x) = \\frac{1}{2} \\left| \\left|\\uuline{A} \\uline{x} - \\uline{y} \\right| \\right|_2^2 + \\lambda \\left| \\left| \\uline{x} \\right| \\right|_1 \\; .\n\\end{equation}\nfor decreasing $\\lambda$. The algorithm does this by tracing out the \\textit{homotopy path}, approaching the limit\n\\begin{equation}\n\\lim_{\\lambda \\rightarrow 0} J_{\\lambda}(x) = \\frac{1}{2} \\left| \\left|\\uuline{A} \\uline{x} - \\uline{y} \\right| \\right|_2^2 \\; .\n\\end{equation}\n\n\\section{Variables}\nWe have the following variables,\n\\begin{itemize}\n\\item $m \\times n$ sensing matrix, $\\uuline{A}$\n\\item signal vector of $m$ elements, $\\uline{y}$\n\\item solution vector of $n$ elements, $\\uline{x}$\n\\item weights vector of $n$ elements, $\\uline{w}$\n\\item the error, $\\epsilon$\n\\item iteration index $j$\n\\end{itemize}\n\n\\section{Algorithm}\n\nThe iterative reweighted least square algorithms requires initialisation of\n\\begin{itemize}\n\\item the value of $\\epsilon_0 = 0$\n\\item the weights vector to $\\uline{w} = \\uline{1}$ (i.e., a vector with all elements equal to one)\n\\end{itemize}\nIn each iterative step $j$, the iterative reweighted least square algorithms updates the solution approximation, $\\uline{x}^{(j)}$, and the weights, $\\uline{w}^{(j)}$, as follows.\n\\begin{itemize}\n\\item the approximate solution is updated\n\\begin{equation}\n\\uline{x}^{(j+1)} = \\uuline{D}_j^n \\uuline{A}^T\\left(\\uuline{A} \\uuline{D}^{-1} \\uuline{A}^T\\right) \\uline{y}\n\\end{equation}\nwhere the matrix $\\uuline{D}$ is a diagonal matrix containing the weights,\n\\begin{equation}\nD_{ii} = w_i^{(j)}\n\\end{equation}\n\\item the value of epsilon is updated\n\\begin{equation}\n\\epsilon_{j+1} = \\min\\left(\\epsilon_n, \\frac{r\\left(x^{(j+1)}\\right)_{K+1}}{N}\\right)\n\\end{equation}\nwhere the vector $\\uline{r}\\left(\\uline{x}\\right)$ is defined to contain the non-increasing arrangement of the absolute values of the elements in $\\uline{x}$; $r\\left(\\uline{x}\\right)_i$ is thus the $i^{\\text{th}}$ largest element of $\\uline{x}$; a vector $\\uline{x}$ is $k$-sparse if and only if $r\\left(\\uline{x}\\right)_{k+1} = 0$\n\\item the weights are updated\n\\begin{equation}\nw_j^{(j+1)} = \\frac{1}{\\sqrt{\\left(x_j^{(j+1)}\\right)^2 + \\epsilon_{j+1}^2}}\n\\end{equation}\n\\end{itemize}\n\n\nThe loop breaks either\n\\begin{itemize}\n\\item when reaching the maximum number of iterations, $N_{iter}$, or\n\\item when the infinity-norm of the residual vector (i.e., the lambda parameter in Eq.~(\\ref{eq:l1min})) becomes smaller than the pre-set tolerance,\n\\end{itemize}\nwhichever happens first.\n\\end{document}", "meta": {"hexsha": "0f63a02c6dc5b056a30577dc0a7ecd0b84001b6c", "size": 4916, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/algorithms/irls-method/src.tex", "max_stars_repo_name": "rayglover-ibm/sparse-solvers", "max_stars_repo_head_hexsha": "601d74c5230fcab4dc4f83fb90049515b3bf7d06", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2017-05-13T10:33:04.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-13T06:18:25.000Z", "max_issues_repo_path": "docs/algorithms/irls-method/src.tex", "max_issues_repo_name": "rayglover-ibm/sparse-solvers", "max_issues_repo_head_hexsha": "601d74c5230fcab4dc4f83fb90049515b3bf7d06", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2017-05-12T23:26:56.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-11T13:52:54.000Z", "max_forks_repo_path": "docs/algorithms/irls-method/src.tex", "max_forks_repo_name": "rayglover-ibm/sparse-solvers", "max_forks_repo_head_hexsha": "601d74c5230fcab4dc4f83fb90049515b3bf7d06", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2018-05-21T03:53:05.000Z", "max_forks_repo_forks_event_max_datetime": "2018-05-21T03:53:06.000Z", "avg_line_length": 38.1085271318, "max_line_length": 332, "alphanum_fraction": 0.7082994304, "num_tokens": 1628, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6757645944891559, "lm_q1q2_score": 0.345799963871278}} {"text": "\\chapter{Reinforcement Learning}\n\\label{ch:reinforcement_learning}\n\nThe standard way to solve Markov decision processes is through dynamic programming, which simply consists in solving the Bellman fixed-point equations discussed in the previous chapter. Following this approach, the problem of finding the optimal policy is transformed into the problem of finding the optimal value function. However, apart from the simplest cases where the MDP has a limited number of states and actions, dynamic programming becomes computationally infeasible. Moreover, this approach requires complete knowledge of the Markov transition kernel and of the reward function, which in many real-world applications might be unknown or too complex to use. \\glsfirst{RL} is a subfield of Machine Learning which aims to turn the infeasible dynamic programming methods into practical algorithms that can be applied to large-scale problems. \\gls{RL} algorithms are based on two key ideas: the first is to use samples to compactly represent the unknown dynamics of the controlled system. The second idea is to use powerful function approximation methods to compactly estimate value functions and policies in high-dimensional state and action spaces. In the following sections, we present the \\gls{RL} problem in more depth and discuss how it relates to the standard discrete-time stochastic optimal control theory and to the other subfields of machine learning, such as supervised learning. In particular, we will see how RL extends ideas from optimal control theory and stochastic approximation to address the broader goal of artificial intelligence. This quick overview of RL is propaedeutic to the following chapters, where we will present in more detail a particular class of algorithms, called policy gradient methods, which are well suited for continuous action spaces. For a more thorough presentation, the reader may consult \\cite{sutton1998introduction}, \\cite{szepesvari2010algorithms} or \\cite{wiering2012reinforcement}.\n\n\\section{The Reinforcement Learning Problem}\nReinforcement Learning (RL) is a general class of algorithms in the field of machine learning that allows an agent to learn how to behave in a stochastic and possibly unknown environment, where the only feedback consists of a scalar reward signal. In order to maximize the long-run reward, the agent must learn which actions are the most profitable by trial-and-error. Therefore, RL algorithms can be seen as computational methods to solve Markov decision processes by directly interacting with the environment, for which a model may or may not be available. Trial-and-error search and a delayed reward signal can be seen as the most characteristic features of reinforcement learning.\\\\\nCompared to supervised learning, one of the main branches of machine learning, the feedback the learner receives is much less. In supervised learning, the agent is provided with examples of the correct or expected behavior by a knowledgeable external supervisor and the agent's goal is to learn how to replicate these examples as well as possible and possibly generalize this knowledge to new examples. In reinforcement learning, the agent only receives a numerical reward that only gives a partial feedback of the goodness of actions taken. Therefore, this feedback system is evaluative rather than instructive and it is much more difficult for the agent to learn how to behave in uncharted territory without any external guidance.\\\\ \nThis particular framework generates some challenges that are not present in other kinds of learning. The first one is the trade-off between exploration and exploitation. In order to maximize his reward, an agent would greedily select actions that have already been tried in the past and found to be effective in producing rewards. However, to find these actions, the agent must also test actions that have not been chosen before in order to evaluate their potential. Clearly, this might result in worse performance in the short-term because the actions might be suboptimal. However, without trying them, the agent might not be able to find possible improvements. Thus, an agent must exploit what is known to obtain rewards but also needs to explore to select better actions in the future. A second challenge is the credit assignment problem. Since rewards might be delayed in time, it will be difficult for the agent to understand which actions are mostly responsible for the outcome. \n\n\\section{Model-Free RL Methods}\nIn Section \\ref{sec:policy_evaluation}, we discussed the policy iteration method for computing an optimal policy for an \\gls{MDP} in a finite state and action spaces. This algorithm belongs to the class of \\emph{model-based} methods, since it requires perfect knowledge of the Markov transition kernel and reward function, which consists of a model of the MDP. RL is primarily concerned with how to obtain an optimal policy when such a model is not available. In this section, we discuss some classes of \\emph{model-free} methods which do not rely on the MDP model. The lack of a model generates the need to sample the MDP to gather statistical knowledge about this unknown model. In the control setting, the goal is to approximate the optimal policy, which depends on the optimal value, which in turn depends on the model of the MDP, as shown in Figure \\ref{fig:control_dependences}. Indeed, we have already seen that a policy which is greedy with respect to the optimal action-value funtion $Q_*$, namely for which holds\n\\begin{equation}\n\t\\int_{\\A} \\pi_*(s,a) Q_*(s,a) da = \\sup_a Q_*(s,a)\n\\end{equation} \nis optimal. Therefore, we can derive the following three methodologies that differ in which part of the solution process is approximated\n\\begin{enumerate}[label={\\roman*)}]\n\t\\item \\emph{Model-approximation} algorithms approximate the MDP model and compute an estimate of the optimal policy by dynamic programming.\n\t\\item \\emph{Value-approximation} algorithms use samples to directly approximate $V_*$ or $Q_*$, from which an estimate of $\\pi_*$ can be derived by acting greedily.\n\t\\item \\emph{Policy-approximation} algorithms directly try to estimate the optimal policy.\n\\end{enumerate}\nIt should be noticed that these approaches are not mutually exclusive and can be combined to derive hybrid algorithms. In the following sections we discuss these three classes of algorithm in more detail, following closely \\cite{wiering2012reinforcement}.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\begin{tikzpicture}[node distance = 6em, auto, thick]\n\t\t\\node [block] (Model) {Model\\\\$\\calP$, $\\calR$};\n\t\t\\node [block, below of=Model] (Value) {Value functions\\\\$V_*$, $Q_*$};\n\t\t\\node [block, below of=Value] (Policy) {Policy\\\\$\\pi_*$};\t\t \n\t\t\\path [line] (Model) edge (Value);\n\t\t\\path [line] (Value) edge (Policy);\n\t\\end{tikzpicture}\n\t\\caption{Solution process for the control problem.}\n\t\\label{fig:control_dependences}\n\\end{figure}\n\n\n\\subsection{Model Approximation}\nModel-approximation algorithms approximate the MDP model and compute an optimal policy by dynamic programming, using the techniques discussed in the previous chapter. Since $\\S$, $\\A$ and $\\gamma$ are assumed to be known, these methods are based on learning an approximation of the Markov transition kernel $\\calP$ and the reward function $\\calR$. Thanks to the Markov property, these quantities only depend on the current state and action, so that their approximation corresponds to a density estimation problem and a regression problem respectively, which are fairly standard supervised learning problems. Learning the model may not be trivial, but it is in general easier than learning the value of a policy or optimizing the policy directly. The major drawback of model-based algorithms in continuous-state MDPs is that, even if the model is available, it is in general infeasible to compute the value functions by dynamic programming and to extract an optimal policy for all states by acting greedily. Alternatively, a transition model estimate may be used to generate sample trajectories from the MDP, which can then be used to estimate the value function or directly improve the policy. However, the value function and the policy estimated using these samples can only be as accurate as the learned model, so that in many cases it may be easier to directly approximate the value function or the policy using the methods described below. \n\n\\subsection{Value Approximation}\nValue-approximation algorithms use samples from the MDP to approximate $V_*$ or $Q_*$ directly and then derive an estimate of the optimal policy by acting greedily with respect to $Q_*$. Typically, when the state and action spaces are large, the value functions are estimated using a parametric function approximator, whose parameters are iteratively updated given the observed samples. Many reinforcement learning algorithms fall into this category and they can be distinguishes based on whether they are on-policy or off-policy and whether they update the value function estimates online or offline. \\emph{On-policy} algorithms approximate the state-value function $V_\\pi$ or the action-value function $Q_\\pi$ from samples of the MDP obtained by following the same policy $\\pi$ to be evaluated. Although the optimal policy $\\pi_*$ is initially unknown, such algorithms can eventually approximate the optimal value functions $V_*$ or $Q_*$ from which an approximation of the optimal policy can be derived. On the other hand, \\emph{off-policy} algorithms can learn the value of a policy different from the one use for obtaining the MDP samples. \\emph{Online} algorithms adapt their value approximation after each observed sample while \\emph{Offline} algorithms operate on batches of samples. Online algorithms typically require less computation per sample but their convergence is slower. \n%Online on-policy algorithms include temporal-difference (TD) algorithms, such as TD-learning, Sarsa. Offline on-policy algorithms include least-squares approaches, such least-squares temporal difference (LSTD), least-squares policy evaluation (LSPE) and least-squares policy iteration (LSPI). The most known model-free online off-policy algorithm is Q-learning.\n\n\\subsection{Policy Approximation}\nPolicy-approximation algorithms only store a policy and update this policy to maximize a given performance measure and eventually converge to the optimal policy. Since these algorithms only use an estimate of the optimal policy and do not rely on a value function approximation, they are also referred to as \\emph{direct policy-search} or \\emph{actor-only} algorithms. Algorithms that store both a policy and a value function are commonly known as \\emph{actor-critic} algorithms \\cite{sutton1999policy}, \\cite{konda1999actor}. Policy gradient algorithms, the most studied policy-approximation methods, will be discussed in much more detail in the next chapter. \n\n\n\n\n\n", "meta": {"hexsha": "5b215aea3d56613d1cda6b38a763c7355dbe15bb", "size": 10883, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Report/Chapters/3_Reinforcement_Learning.tex", "max_stars_repo_name": "AmineAboussalah/Thesis", "max_stars_repo_head_hexsha": "1a3ae97023acff1ee5e2d197a446734117a6fb99", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 80, "max_stars_repo_stars_event_min_datetime": "2016-06-13T15:20:29.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-24T23:47:13.000Z", "max_issues_repo_path": "Report/Chapters/3_Reinforcement_Learning.tex", "max_issues_repo_name": "pnecchi/Thesis", "max_issues_repo_head_hexsha": "1a3ae97023acff1ee5e2d197a446734117a6fb99", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Report/Chapters/3_Reinforcement_Learning.tex", "max_forks_repo_name": "pnecchi/Thesis", "max_forks_repo_head_hexsha": "1a3ae97023acff1ee5e2d197a446734117a6fb99", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 34, "max_forks_repo_forks_event_min_datetime": "2017-05-15T07:51:52.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-21T21:48:53.000Z", "avg_line_length": 209.2884615385, "max_line_length": 1937, "alphanum_fraction": 0.8070385004, "num_tokens": 2263, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5621765008857982, "lm_q1q2_score": 0.34578793835137167}} {"text": "% !TeX root = ../main.tex\n\\begin{landscape}\n\\section{Semantics Derivation Tree Example}\n\\label{sec:semanticsTree}\nThis is an example of how to create a derivation tree for a program written in HCL.\nIt is based on the following code snippet.\n\n\\begin{lstlisting}[language=HCL,label=lis:derivationCode,firstnumber=1]\nnum x\nx = 5\n\\end{lstlisting}\n\nUsing the abstract syntax described in section \\ref{sec:semantics}, the code can be written as the following big step semantic:\n\n\\begin{center}\n\t$env_V, env_F \\vdash \\langle num\\ x;\\ x = 5, env_V, env_F, sto \\rangle \\rightarrow_{stm} (sto', env_V', env_F')$\n\\end{center}\nUsing the Compositional Big Step Semantic rule, the following derivation is produced:\n\n\\begin{center}\n\t\\begin{math}\n\t\t\\cfrac\n\t\t{env_V, env_F \\vdash \\langle num\\ x, env_V, env_F, sto \\rangle \\rightarrow_{stm} (sto'', env_V'', env_F'')\\quad env_V'', env_F'' \\vdash \\langle x = 5, env_V'', env_F'', sto'' \\rangle \\rightarrow_{stm} (sto', env_V', env_F')}\n\t\t{env_V, env_F \\vdash \\langle num\\ x;\\ x = 5, env_V, env_F, sto \\rangle \\rightarrow_{stm} (sto', env_V', env_F')}\n\t\\end{math}\n\\end{center}\n\nTable \\ref{tbl:progstates} shows how the program state changes after each statement in the code is run.\n\n\\begin{table}[H]\n\t\\centering\n\t\\caption{Table of program states after each statement.}\n\t\\label{tbl:progstates}\n\t\\setlength\\extrarowheight{5pt}\n\t\\begin{tabular}{|c|c|c|}\n\t\t\\hline\n\t\t& Abstract program state & Concrete program state \\\\ \\hline\n\t\tInitial state & $(sto, env_V, env_F)$ & $(sto, env_V, env_F)$ \\\\ \\hline\n\t\tAfter first statement & $(sto'', env_V'', env_F'')$ & $(sto[l \\mapsto 0], env_V[x \\mapsto l], env_F)$ \\\\ \\hline\n\t\tAfter second statement & $(sto', env_V', env_F')$ & $(sto[l \\mapsto 5], env_V[x \\mapsto l], env_F)$ \\\\ \\hline\n\t\\end{tabular}\n\\end{table}\n\n\\end{landscape}", "meta": {"hexsha": "4a0d6f86a9b67153d9ab9b17a5834782f8295958", "size": 1871, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "report/Appendix/SemanticsTree.tex", "max_stars_repo_name": "C0DK/P3-HCL", "max_stars_repo_head_hexsha": "b6adbc46cc7347aacd45dce5fddffd6fee2d37ac", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2018-02-08T12:34:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-09T11:33:53.000Z", "max_issues_repo_path": "report/Appendix/SemanticsTree.tex", "max_issues_repo_name": "C0DK/P3-HCL", "max_issues_repo_head_hexsha": "b6adbc46cc7347aacd45dce5fddffd6fee2d37ac", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 10, "max_issues_repo_issues_event_min_datetime": "2018-02-17T14:30:17.000Z", "max_issues_repo_issues_event_max_datetime": "2018-05-13T17:37:09.000Z", "max_forks_repo_path": "report/Appendix/SemanticsTree.tex", "max_forks_repo_name": "C0DK/P3-HCL", "max_forks_repo_head_hexsha": "b6adbc46cc7347aacd45dce5fddffd6fee2d37ac", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-07-04T13:55:53.000Z", "max_forks_repo_forks_event_max_datetime": "2018-07-04T13:55:53.000Z", "avg_line_length": 42.5227272727, "max_line_length": 226, "alphanum_fraction": 0.6702298236, "num_tokens": 597, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.34578793835137167}} {"text": "\\documentclass{article}\n\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage[UKenglish]{babel}\n\\usepackage[backend=biber,sorting=none]{biblatex}\n\\usepackage{enumitem}\n\\usepackage{fancyhdr}\n\\usepackage[margin=1in]{geometry}\n\\usepackage{graphicx}\n\\usepackage[utf8]{inputenc}\n\\usepackage{listings}\n\\usepackage{mathtools}\n\\usepackage{tikz-cd}\n\\usepackage{csquotes}\n\n\\newcommand{\\F}{\\mathbb{F}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\A}{\\mathbb{A}}\n\\renewcommand{\\P}{\\mathbb{P}}\n\n\\newcommand{\\val}[1]{\\left. #1 \\right\\rvert}\n\\newcommand{\\rb}[1]{\\left( #1 \\right)}\n\\renewcommand{\\sb}[1]{\\left[ #1 \\right]}\n\\newcommand{\\cb}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\ab}[1]{\\left\\langle #1 \\right\\rangle}\n\\newcommand{\\abs}[1]{\\left\\lvert #1 \\right\\rvert}\n\n\\newcommand{\\twobyone}[2]{\\begin{pmatrix} #1 \\\\ #2 \\end{pmatrix}}\n\\newcommand{\\onebythree}[3]{\\begin{pmatrix} #1 & #2 & #3 \\end{pmatrix}}\n\\newcommand{\\threebyone}[3]{\\begin{pmatrix} #1 \\\\ #2 \\\\ #3 \\end{pmatrix}}\n\\newcommand{\\threebythree}[9]{\\begin{pmatrix} #1 & #2 & #3 \\\\ #4 & #5 & #6 \\\\ #7 & #8 & #9 \\end{pmatrix}}\n\n\\theoremstyle{definition}\n\\newtheorem*{definition}{Definition}\n\\newtheorem*{example}{Example}\n\\newtheorem*{remark}{Remark}\n\\newtheorem{proposition}{Proposition}[subsection]\n\\newtheorem{lemma}[proposition]{Lemma}\n\\newtheorem{theorem}[proposition]{Theorem}\n\\newtheorem{algorithm}[proposition]{Algorithm}\n\\newtheorem{conjecture}[proposition]{Conjecture}\n\n\\pagestyle{fancy}\n\\lhead{The Arithmetic of Elliptic Curves}\n\\rhead{David Kurniadi Angdinata}\n\n\\addbibresource{bibliography.bib}\n\n\\title{The Arithmetic of Elliptic Curves}\n\\author{David Kurniadi Angdinata}\n\\date{Monday, 2 July 2018 - Friday, 28 September 2018}\n\n\\begin{document}\n\n\\maketitle\n\n\\vspace{1cm}\n\n\\begin{center}\n\\includegraphics*[width=0.5\\textwidth]{img/EllipticCurveCatalog.png}\n\\end{center}\n\n\\vfill\n\n\\begin{center}\n\\emph{\"It is possible to write endlessly on elliptic curves. (This is not a threat.)\"\\cite{lang}} \\\\\n\\end{center}\n\n\\vspace{1cm}\n\n\\begin{abstract}\nIn the field of algebraic geometry, elliptic curves are deeply studied rich structures with far-fetching computational applications to classical number theory and contemporary cryptography. It is a fundamental tool in Wiles' proof of Fermat's last theorem, as well as the main object of discussion in the Birch and Swinnerton-Dyer conjecture, an open problem in number theory deemed worthy of being called one of the Millennium Prize Problems by the Clay Mathematics Institute. In this project, three of the four most fundamental theorems in the arithmetic of elliptic curves, namely the Hasse-Weil theorem, the Nagell-Lutz theorem, and the Mordell-Weil theorem, are proven in their respective special forms. Schoof's algorithm for counting rational points over Galois fields will also be briefly discussed, allowing for an application to integer factorisation and primality testing. An introductory section and a brief appendix on fields, varieties, curves, and groups are also included for completion.\n\\end{abstract}\n\n\\pagebreak\n\n\\tableofcontents\n\n\\pagebreak\n\n\\section*{Preface}\n\n\\addcontentsline{toc}{section}{Preface}\\par\n\nThe aim of this project is to provide a gentle introduction to the deep theory of \\emph{elliptic curves}. My approach for the report is to deliver information in the form of general propositions and fundamental theorems, providing adequate proofs whenever possible. Admittedly, almost all of the work here is unoriginal and should have been done elsewhere in a similar fashion, but I have tried to make the flow of information as coherent as possible, consulting various online resources as well as the two books \\cite{gtm} and \\cite{utm}.\n\nAs the theory of elliptic curves requires some technical background in algebraic geometry, which in turn requires prerequisites in commutative algebra, both of which I have zero background in, the report is rather superficial and may not provide as good of an insight to deeper theory. In addition to the first two years of undergraduate mathematics, the first three preliminary appendices A.1, A.2, and A.3 will provide all the required background to understand the first introductory section, while a few additional notions in appendix A.4 will allow the rest to be fully accessible to middle-year undergraduates such as myself.\n\nThe first section introduces elliptic curves while hiding away the definitions and results taken from algebraic geometry. For instance, the use of explicit \\emph{Weierstrass equations} and formulae for the \\emph{group law} is an active attempt to avoid bringing in the \\emph{Riemann-Roch theorem} in algebraic geometry to prove that they are related. As such, many proofs in this section are omitted, but are all given a direct reference.\n\nThe second section discusses elliptic curves over finite fields, which hinges on \\emph{Hasse's theorem} to explain \\emph{Schoof's algorithm} for counting rational points. This in turn paves the path for applications in the last section, as well as an alternative method of counting rational points in the following section. Proofs are mostly given in full, except for an overly lengthy but elementary proof based on induction.\n\nThe third section discusses elliptic curves over the rationals, which can be split into two parts due to the \\emph{fundamental theorem of finitely generated abelian groups}. The \\emph{Nagell-Lutz theorem} and \\emph{reduction modulo prime} are two related ways to compute the \\emph{torsion subgroup}, while \\emph{Mordell's theorem} proves that the fundamental theorem indeed holds and provides a semi-workable method to compute the \\emph{rank}. Again, proofs are mostly given in full, except for an assumption made in the last part of Mordell's theorem.\n\nThe last section touches on some applications to classical arithmetic, including integer factorisation and primality proving, as well as the basics of contemporary cryptography, which could potentially be explored into if time permitted. There was originally an intention to cover complex elliptic curves in here instead, leading to a brief exposition of Fermat's last theorem, but was omitted due to lack of time.\n\nThe appendices include the aforementioned preliminaries, as well as proofs of two algorithms and listings of code. The proofs are placed here as they are deemed less relevant and too lengthy to be included in the main text. All code under code listings are in the functional \\emph{Haskell} programming language compiled by the \\emph{Glassgow Haskell Compiler}, which is markedly different from many implementations publicly available.\n\nI will also present several remarks regarding the style of the report, which is an attempt to imitate typical lecture notes and textbooks. Definitions are all given in bold, while italics are reserved for less important terms that are undefined, all of which provided with adequate examples whenever possible. Theorems are important results and propositions those less important, while lemmas serve as intermediate checkpoints to theorems and propositions. All of these are provided immediate proofs after their statements, or postponed for later as \\emph{Proof of \\emph{statement}}. Remarks are largely irrelevant to the flow of the main discussion and can be disregarded, but are included for interesting points, which may include previously undefined terms.\n\nIn terms of notation, most of the symbols I have used are those found in books, such as $ \\F_p $, $ \\Z $, or $ \\Q $, and should be unambiguous. While there are unfortunate cases of equivalent symbols being used to mean completely different objects due to limitations of the English and Greek alphabet system, I have tried to minimise these or make them clear from context. An example being $ f, g, h $ being used for general functions, while homomorphisms are always denoted $ \\phi, \\psi, \\chi $ or their variants. In examples like the following,\n$$ f\\rb{x, y} = xg\\rb{y} + yh\\rb{x}, \\qquad g \\in F\\sb{y}, \\qquad h \\in F\\sb{x}, $$\n$ g $ and $ h $ are always existentially quantified, while $ f $ and $ F $ are initially fixed or explicitly universally quantified. A function $ f' $ will always be distinct, but possibly related, to the function $ f $, while differentiation with respect to a variable $ x $ will always be denoted $ d / dx $ or $ \\partial / \\partial x $. No distinction will be made between sums and \\emph{formal sums}, or derivatives and \\emph{formal derivatives}, as they are clear from the context of this report.\n\nI would like to thank my supervisor Prof Johannes Nicaise for his support and guidance throughout the duration of this project. I had many doubts and questions early on, all of which he clarified with great detail.\n\n\\pagebreak\n\n\\section{Introduction}\n\nInformally, an elliptic curve is a cubic curve with no cusps, self-intersections, or isolated points, whose solutions are confined to a region of space topologically equivalent to a torus. It can represented by a cubic equation in two variables, with its coefficients being elements of a specified field. Two elliptic curves over the field of real numbers are illustrated below.\n\n\\begin{center}\n\\includegraphics*[width=0.4\\textwidth]{img/ECClines-3.png}\n\\end{center}\n\n\\subsection{Definition}\n\nA formal definition is as follows.\n\n\\begin{definition}[Elliptic curve]\nAn \\textbf{elliptic curve} over a perfect field $ F $ is an ordered pair $ \\rb{E, \\mathcal{O}_E} $ such that $ E $ is a smooth projective plane curve of genus one over $ F $ and $ \\mathcal{O}_E \\in E $ is an $ F $-rational \\textbf{base point}.\n\\end{definition}\n\nThis definition uses several terms in other fields of mathematics, which are briefly covered in the appendices. In particular, one of the many characterisations of a perfect field is given in Appendix A.1, while several fundamental notions in projective and algebraic geometry such as projective planes and smoothness are laid out in Appendix A.2. Appendix A.3 defines a curve and the genus due to the \\emph{genus-degree formula}.\n\n\\begin{remark}\nThe genus in algebraic geometry is usually defined in general literature by the \\emph{Riemann-Roch theorem}, which does coincide with the topological definition.\n\\end{remark}\n\nAs the report is a gentle introduction to elliptic curves, further delving into the vast world of algebraic geometry will be avoided, and so explicit formulae will be provided whenever possible. To this end, the various definitions in the appendix can be summarised in the following proposition.\n\n\\begin{proposition}\n\\label{prop:appendices}\nLet $ \\rb{E, \\mathcal{O}_E} $ be an elliptic curve over a perfect field $ F $. Then:\n\\begin{enumerate}\n\\item $ I\\rb{E} = \\ab{e} $ for some homogeneous irreducible polynomial $ e $ of three variables,\n\\item any point $ P \\in E $ has multiplicity $ m_P\\rb{e} = 1 $, and\n\\item $ e $ have roots confined to a torus and is cubic.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\nThis follows directly from the appendices.\n\\end{proof}\n\nThus an elliptic curve can be fully defined in terms of its defining polynomial, which would need to satisfy certain conditions. As per the appendix, an abuse of notation will be used to denote an elliptic curve $ \\rb{E, \\mathcal{O}_E} $ over $ F $ \\emph{given} by a polynomial $ e $, namely\n$$ E : e\\rb{X, Y, Z} = 0 \\qquad \\iff \\qquad E : e\\rb{x, y} = 0, $$\nwhich are respectively the homogenised and dehomogenised forms of a polynomial that can be used interchangeably. For the rest of this section, let $ E : e\\rb{x, y} = 0 $ and $ E' : e'\\rb{x, y} = 0 $ be two elliptic curves over a perfect field $ F $ with algebraic closure $ K = \\overline{F} $. The notion of an isomorphism, as for any algebraic geometric structure, would be useful. This is captured in the following definition.\n\n\\begin{definition}[Isomorphism]\n$ \\rb{E, \\mathcal{O}_E} $ and $ \\rb{E', \\mathcal{O}_{E'}} $ are \\textbf{isomorphic}, denoted by $ \\rb{E, \\mathcal{O}_E} \\cong \\rb{E', \\mathcal{O}_{E'}} $, iff there is an isomorphism $ \\phi : E \\to E' $ such that $ \\phi\\rb{\\mathcal{O}_E} = \\mathcal{O}_{E'} $.\n\\end{definition}\n\n\\begin{remark}\nIsomorphism defines an equivalence relation of elliptic curves, such that two elliptic curves from an equivalence class are indistinguishable.\n\\end{remark}\n\nAgain, this abstract notion can be made explicit later by the defining polynomials of the elliptic curves.\n\n\\pagebreak\n\n\\subsection{Weierstrass equations}\n\nThe definition of an elliptic curve boils down to its defining polynomial, which will be made explicit in this subsection. A family of curves related to elliptic curves will be defined beforehand.\n\n\\begin{definition}[Weierstrass curve]\nA \\textbf{Weierstrass curve} is a projective plane curve $ W $ over $ F $ \\emph{given} by the \\textbf{Weierstrass equation}\n$$ W : w\\rb{x, y} = 0 : y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6, \\qquad a_i \\in F, $$\nwith associated quantities:\n$$ b_2 = a_1^2 + 4a_2, \\qquad b_4 = a_1a_3 + 2a_4, \\qquad b_6 = a_3^2 + 4a_6, \\qquad b_8 = a_1^2a_6 + 4a_2a_6 - a_1a_3a_4 + a_2a_3^2 - a_4^2, $$\n$$ c_4 = b_2^2 - 24b_4, \\qquad c_6 = 36b_2b_4 - b_2^3 - 216b_6, \\qquad \\Delta_W = 9b_2b_4b_6 - b_2^2b_8 - 8b_4^3 - 27b_6^2, \\qquad j_W = c_4^3 / \\Delta_W. $$\n\\end{definition}\n\nIt holds that $ 4b_8 = b_2b_6 - b_4^2 $ and $ 1728\\Delta_W = c_4^3 - c_6^2 $. It can be easily verified that Weierstrass curves, with the additional condition of smoothness, would almost satisfy Proposition \\ref{prop:appendices}. The only remaining requirement is having an additional base point in its definition, which can be easily fixed as follows.\n\n\\begin{definition}[Point at infinity]\nThe \\textbf{point at infinity} of $ E $ is the point $ \\mathcal{O} = \\sb{0, 1, 0} $.\n\\end{definition}\n\nIn contrast to general projective geometry, the \\emph{line at infinity} $ L : l\\rb{X, Y, Z} = Z = 0 $ intersects a Weierstrass curve only at $ \\mathcal{O} $, where $ X = Z = 0 $ and $ Y \\ne 0 $. As such, any other point would have $ Z \\ne 0 $ and can be treated as an affine point $ \\rb{a, b} $. Now since $ 0, 1 \\in F $, the point $ \\mathcal{O} $ is actually an $ F $-rational point, and can be paired with a smooth Weierstrass curve $ W $ to give an elliptic curve $ \\rb{W, \\mathcal{O}} $. Conversely, any elliptic curve can also be explicitly given by a smooth Weierstrass curve through an isomorphism as follows.\n\n\\begin{proposition}\n$ \\rb{E, \\mathcal{O}_E} \\cong \\rb{W, \\mathcal{O}} $ for some smooth Weierstrass curve $ W $ over $ F $.\n\\end{proposition}\n\n\\begin{proof}\nOmitted, see III.3.1a in \\cite{gtm}.\n\\end{proof}\n\nThere are even computerised algorithms to transform a general smooth projective plane cubic curve with a given arbitrary $ F $-rational flex point, or an elliptic curve, into a Weierstrass curve with the $ F $-rational point $ \\mathcal{O} $. The following algorithm summarises the process in \\cite{weierstrass} proven in the appendix.\n\n\\begin{algorithm}[Transformation of a cubic curve into Weierstrass form]\nInput: a cubic curve $ E $ over $ F $ with an $ F $-rational flex point $ P \\in E $. Output: $ E $ in Weierstrass form.\n\\begin{enumerate}\n\\item Get the unique tangent line $ L $ at $ P $.\n\\item Find the intersection $ L \\cap E $ to get a point $ Q \\in L \\setminus E $ distinct to $ P $.\n\\item Write down an invertible matrix $ M = \\onebythree{Q}{P}{R} $, where $ R \\in \\cb{\\sb{1, 0, 0}, \\sb{0, 1, 0}, \\sb{0, 0, 1}} $.\n\\item Transform $ \\sb{X, Y, Z} \\mapsto M\\sb{X, Y, Z}^T $ to get a scaled Weierstrass equation.\n\\item Rescale $ \\sb{X, Y, Z} \\mapsto \\sb{X, Y, cZ} $ for some $ c \\in K^* $ to get a Weierstrass equation.\n\\end{enumerate}\n\\end{algorithm}\n\nThe following example illustrates an implementation of the algorithm.\n\n\\begin{example}\nLet $ E : e\\rb{X, Y, Z} = 0 : X^3 + Y^3 = Z^3 $ be a smooth projective plane cubic curve over $ \\R $ with an $ \\R $-rational flex $ P = \\sb{1, -1, 0} \\in E $. Then the unique tangent at $ P $ is\n$$ L : \\twobyone{1}{1, 0, 0}\\val{\\dfrac{\\partial e}{\\partial X}}_P\\rb{X - 1} + \\twobyone{1}{0, 1, 0}\\val{\\dfrac{\\partial e}{\\partial Y}}_P\\rb{Y + 1} + \\twobyone{1}{0, 0, 1}\\val{\\dfrac{\\partial e}{\\partial Z}}_PZ = 3\\rb{X + Y} = 0, $$\nwhich intersects $ E $ at $ X^3 + \\rb{-X}^3 + Z^3 = 0 $, or $ Z = 0 $. Hence $ L \\cap E = \\cb{P} $ and let $ Q = \\sb{1, -1, 1} \\in L \\setminus E $. Then there is an invertible affine transformation matrix\n$$ M = \\threebythree{1}{1}{1}{-1}{-1}{0}{1}{0}{0} \\qquad \\implies \\qquad M^{-1} = \\threebythree{0}{0}{1}{0}{-1}{-1}{1}{1}{0}. $$\nsuch that the affine transformation $ \\sb{X, Y, Z} \\mapsto M\\sb{X, Y, Z}^{T} $ gives\n$$ \\rb{X + Y + Z}^3 + \\rb{-X - Y}^3 = X^3 \\qquad \\implies 3Y^2Z + 6XYZ + 3YZ^2 = X^3 - 3X^2Z - 3XZ^2 - Z^3. $$\nThus the affine transformation $ \\sb{X, Y, Z} \\mapsto \\sb{X, Y, \\tfrac{1}{3}Z} $ gives a Weierstrass curve\n$$ E : Y^2Z + 2XYZ + \\tfrac{1}{3}YZ^2 = X^3 - X^2Z - \\tfrac{1}{3}XZ^2 - \\tfrac{1}{27}Z^3. $$\n\\end{example}\n\nThis characterisation allows a smooth Weierstrass curve to act as an alternative definition for an elliptic curve, and will be done for ease of future discussions. For the rest of this subsection, let $ E $ and $ E' $ be respectively given by the two Weierstrass curves over $ F $\n$$ W : w\\rb{x, y} = 0 : y^2 + a_1xy + a_3y = x^3 + a_2x^2 + a_4x + a_6, \\qquad a_i \\in F, $$\n$$ W' : w'\\rb{x, y} = 0 : y^2 + a_1'xy + a_3'y = x^3 + a_2'x^2 + a_4'x + a_6', \\qquad a_i' \\in F, $$\nand write them interchangeably as an abuse of notation by\n$$ \\rb{E, \\mathcal{O}_E} = E : e\\rb{x, y} = 0 \\qquad \\iff \\qquad \\rb{W, \\mathcal{O}} = W : w\\rb{x, y} = 0, $$\n$$ \\rb{E', \\mathcal{O}_{E'}} = E' : e'\\rb{x, y} = 0 \\qquad \\iff \\qquad \\rb{W', \\mathcal{O}} = W' : w'\\rb{x, y} = 0. $$\nWith these explicit equations at hand, the abstract notion of isomorphism between elliptic curves can now be made explicit by considering affine transformations of these equations, which is given below.\n\n\\begin{proposition}\n\\label{prop:affine}\n$ E \\cong E' $ iff there is an affine transformation\n$$ \\rb{x, y} \\mapsto \\rb{u^2x + r, u^3y + u^2sx + t}, \\qquad u \\in K^*, \\qquad r, s, t \\in K $$\nfrom $ W $ to $ W' $.\n\\end{proposition}\n\n\\begin{proof}\nOmitted, see III.3.1b in \\cite{gtm}.\n\\end{proof}\n\n\\begin{remark}\nThis affine transformation also transforms the coefficients and quantities of $ W $ and $ W' $ by\n$$ a_1 \\mapsto \\dfrac{a_1 + 2s}{u}, \\qquad a_2 \\mapsto \\dfrac{a_2 - sa_1 + 3r - s^2}{u^2}, \\qquad a_3 \\mapsto \\dfrac{a_3 + ra_1 + 2t}{u^3}, $$\n$$ a_4 \\mapsto \\dfrac{a_4 - sa_3 + 2ra_2 - \\rb{t + rs}a_1 + 3r^2 - 2st}{u^4}, \\qquad a_6 \\mapsto \\dfrac{a_6 + ra_4 - ta_3 + r^2a_2 - rta_1 + r^3 - t^2}{u^6}, $$\n$$ b_2 \\mapsto \\dfrac{b_2 + 12r}{u^2}, \\qquad b_4 \\mapsto \\dfrac{b_4 + rb_2 + 6r^2}{u^4}, \\qquad b_6 \\mapsto \\dfrac{b_6 + 2rb_4 + r^2b_2 + 4r^3}{u^6}, $$\n$$ b_8 \\mapsto \\dfrac{b_8 + 3rb_6 + 3r^2b_4 + r^3b_2 + 3r^4}{u^8}, \\qquad c_4 \\mapsto \\dfrac{c_4}{u^4}, \\qquad c_6 \\mapsto \\dfrac{c_6}{u^6}, \\qquad \\Delta_W \\mapsto \\dfrac{\\Delta_W}{u^{12}}, \\qquad j_W \\mapsto j_W $$\nwhich can be tediously verified.\n\\end{remark}\n\nAgain, this will be treated as the definition of isomorphism between elliptic curves. Now in the original definition of a Weierstrass curve, it is given by a Weierstrass equation that is somewhat perverse. This \\emph{long} Weierstrass equation can in fact be greatly simplified, provided there are small restrictions on the characteristic of the underlying field.\n\n\\begin{proposition}\nIf $ char\\rb{F} \\ne 2 $, then\n$$ E : y^2 = x^3 + Ax^2 + Bx + C, \\qquad A, B, C \\in F. $$\nIf $ char\\rb{F} \\ne 3 $ as well, then\n$$ E : y^2 = x^3 + Ax + B, \\qquad A, B \\in F. $$\n\\end{proposition}\n\n\\begin{proof}\nLet $ char\\rb{F} \\ne 2 $, then the affine transformation $ \\rb{x, y} \\mapsto \\rb{x, y - \\tfrac{1}{2}\\rb{a_1x + a_3}} $ gives an isomorphism from $ E $ to the curve given by the \\emph{medium} Weierstrass equation\n$$ y^2 = x^3 + Ax^2 + Bx + C, \\qquad A = \\tfrac{1}{4}b_2, \\qquad B = \\tfrac{1}{2}b_4, \\qquad C = \\tfrac{1}{4}b_6. $$\nLet $ char\\rb{F} \\ne 3 $ as well, then the affine transformation $ \\rb{x, y} \\mapsto \\rb{x - \\tfrac{1}{12}b_2, y} $ gives an isomorphism from $ E $ to the curve given by the \\emph{short} Weierstrass equation\n$$ y^2 = x^3 + Ax + B, \\qquad A = -\\tfrac{1}{48}c_4, \\qquad B = -\\tfrac{1}{864}c_6. $$\n\\end{proof}\n\nMedium and short Weierstrass equations greatly reduce the tedium when manipulating them, since there is a symmetry to the equation itself, giving two $ y $ opposite in sign for each $ x $. As there are only two characteristics that do not permit the affine transformation to a short Weierstrass equation, they will be disregarded for ease of future discussions. Hence always assume that $ char\\rb{F} \\notin \\cb{2, 3} $ and write the Weierstrass equations of $ W $ and $ W' $ as\n$$ W : w\\rb{x, y} = 0 : y^2 = x^3 + Ax + B, \\qquad A, B \\in F, $$\n$$ W' : w'\\rb{x, y} = 0 : y^2 = x^3 + A'x + B', \\qquad A', B' \\in F. $$\nThe following example illustrates the affine transformation to a short Weierstrass equation.\n\n\\begin{example}\nLet\n$$ E : y^2 + 2xy + \\tfrac{1}{3}y = x^3 - x^2 - \\tfrac{1}{3}x - \\tfrac{1}{27} $$\nbe the Weierstrass curve over $ \\R $ from the example above. Since $ char\\rb{\\R} = 0 \\notin \\cb{2, 3} $, there is an affine transformation $ \\rb{x, y} \\mapsto \\rb{x, y - x - \\tfrac{1}{6}} $ such that\n$$ \\rb{y - x - \\tfrac{1}{6}}^2 + 2x\\rb{y - x - \\tfrac{1}{6}} + \\tfrac{1}{3}\\rb{y - x - \\tfrac{1}{6}} = x^3 - x^2 - \\tfrac{1}{3}x - \\tfrac{1}{27} \\qquad \\implies \\qquad y^2 = x^3 + \\tfrac{1}{108}, $$\nwhich is a short Weierstrass equation.\n\\end{example}\n\nAmong the quantities associated with Weierstrass curves, most are used in defining the simplified Weierstrass equations, while the last two, the discriminant $ \\Delta_W $ and the $ j $-invariant $ j_W $, encode various properties of the curve itself. As only short Weierstrass equations are considered, these two quantities can be restated in an equivalent form in terms of the new coefficients. The discriminant is redefined as follows.\n\n\\begin{definition}[Discriminant]\nThe \\textbf{discriminant} of $ W $ is\n$$ \\Delta_W = -16\\rb{4A^3 + 27B^2}. $$\n\\end{definition}\n\nThe discriminant is transformed as $ \\Delta_W \\mapsto \\Delta_W / u^{12} $ by the affine transformation in Proposition \\ref{prop:affine}. It encodes behaviours at the singularities of Weierstrass curves, and whether they exist. The following proposition allows for an easy method of checking the smoothness of a Weierstrass curve.\n\n\\begin{proposition}\n$ W $ is smooth iff $ \\Delta_W \\ne 0 $.\n\\end{proposition}\n\n\\begin{proof}\nAssume that $ W $ is not smooth and $ P = \\rb{a, b} \\in W $ is singular. Then\n$$ 0 = \\val{\\dfrac{\\partial w}{\\partial X}}_P = -3a^2 - A, \\qquad 0 = \\val{\\dfrac{\\partial w}{\\partial Y}}_P = 2b, \\qquad 0 = \\val{\\dfrac{\\partial w}{\\partial Z}}_P = b^2 - 2Aa - 3B. $$\nSince $ b = 0 $ and $ A = -3a^2 $, it holds that $ 0 = 2Aa + 3B = -6a^3 + 3B $, so $ B = 2a^3 $. Hence $ \\Delta_W = -16\\rb{4\\rb{-3a^2}^3 + 27\\rb{2a^3}^2} = 0 $. Conversely assume that $ \\Delta_W = -16\\rb{4A^3 + 27B^2} = 0 $, such that the discriminant of $ x^3 + Ax + B $ is $ -\\rb{4A^3 + 27B^2} = 0 $. Then there is a repeated root $ x = a \\in K $, so $ P = \\rb{a, 0} \\in W $ and\n$$ W : y^2 = \\rb{x - a}^2\\rb{x - a'}, \\qquad a' \\in K. $$\nThen\n$$ \\val{\\dfrac{\\partial w}{\\partial x}}_P = -2\\rb{a - a}\\rb{a - a'} - \\rb{a - a}^2 = 0, \\qquad \\val{\\dfrac{\\partial w}{\\partial y}}_P = 2\\rb{0} = 0. $$\nThus $ P $ is singular and $ W $ is not smooth.\n\\end{proof}\n\nHence $ W $ is eligible as an elliptic curve iff $ \\Delta_W \\ne 0 $, and by the proof above, iff $ x^3 + Ax + B $ has distinct factors. The following example illustrates the discriminant.\n\n\\begin{example}\nLet $ E $ be the Weierstrass curve over $ \\R $ from the example above. Then\n$$ \\Delta_E = -16\\rb{4\\rb{0}^3 + 27\\rb{\\tfrac{1}{108}}^2} = -\\tfrac{1}{27} < 0, $$\nso $ E $ is smooth. Thus $ E $ is an elliptic curve over $ \\R $.\n\\end{example}\n\nThe $ j $-invariant, defined only for smooth Weierstrass curves where $ \\Delta_W \\ne 0 $, is redefined as follows.\n\n\\begin{definition}[$ j $-invariant]\nThe \\textbf{$ j $-invariant} of $ W $ is\n$$ j_W = 1728\\rb{\\dfrac{4A^3}{4A^3 + 27B^2}}. $$\n\\end{definition}\n\nThe $ j $-invariant is transformed as $ j_W \\mapsto j_W $ by the affine transformation in Proposition \\ref{prop:affine}. It stays invariant between elliptic curves that are isomorphic, which gives its name. The following proposition allows for an alternative characterisation of an isomorphism.\n\n\\begin{proposition}\n$ E \\cong E' $ iff $ j_W = j_{W'} $.\n\\end{proposition}\n\n\\begin{proof}\nAssume that $ E \\cong E' $, then the affine transformation maps $ j_W $ to $ j_W $, so $ j_W = j_{W'} $. Conversely assume that $ j_W = j_{W'} $, so\n$$ 1728\\rb{\\dfrac{4A^3}{4A^3 + 27B^2}} = 1728\\rb{\\dfrac{4A'^3}{4A'^3 + 27B'^2}} \\qquad \\implies \\qquad A^3B'^2 = A'^3B^2. $$\nIf $ A = 0 $, then $ B \\ne 0 $ and $ A' = 0 $. Then there is an affine transformation\n$$ \\rb{x, y} \\mapsto \\rb{\\sqrt[3]{\\dfrac{B}{B'}}x, \\sqrt{\\dfrac{B}{B'}}y} \\qquad \\implies \\qquad \\dfrac{B}{B'}y^2 = \\dfrac{B}{B'}x^3 + A'\\sqrt[3]{\\dfrac{B}{B'}}x + B', $$\nsuch that $ y^2 = x^3 + B' = x^3 + A'x + B' $. If $ B = 0 $, then $ A \\ne 0 $ and $ B' = 0 $. Then there is also an affine transformation\n$$ \\rb{x, y} \\mapsto \\rb{\\sqrt{\\dfrac{A}{A'}}x, \\sqrt[4]{\\dfrac{A}{A'}}^3y} \\qquad \\implies \\qquad \\sqrt{\\dfrac{A}{A'}}^3y^2 = \\sqrt{\\dfrac{A}{A'}}^3x^3 + A'\\sqrt{\\dfrac{A}{A'}}x + B', $$\nsuch that $ y^2 = x^3 + A'x = x^3 + A'x + B' $. Otherwise $ A \\ne 0 $ and $ B \\ne 0 $, then there is an affine transformation from $ W $ to $ W' $ equal to the two affine transformations above. Thus $ E \\cong E' $.\n\\end{proof}\n\nWhile $ j $-invariant affine transformations preserve elliptic curves, this does not necessarily hold for their set of rational points. The following illustrates the $ j $-invariant.\n\n\\begin{example}\nLet $ E $ be the elliptic curve over $ \\R $ from the example above. Then\n$$ j_E = -1728\\rb{\\dfrac{4\\rb{0}^3}{4\\rb{0}^3 + 27\\rb{\\frac{1}{108}}^2}} = 0. $$\nHence $ E $ is isomorphic to any elliptic curve with zero $ j $-invariant. Now let $ E' : y^2 = x^3 + B $ for some $ B \\in \\R $ such that $ j_{E'} = 0 $, then there is an affine transformation\n$$ \\rb{x, y} \\mapsto \\rb{\\dfrac{1}{3\\sqrt[3]{2}^2B}x, \\dfrac{1}{2\\sqrt{3}^3B}y}. $$\nfrom $ E $ to $ E' $. Thus $ E \\cong E' $.\n\\end{example}\n\nThe definition and isomorphism classes of elliptic curves are now fully characterised.\n\n\\begin{remark}\nThere are alternate characterisations of elliptic curves by other families of curves, which will not be discussed here. One of these is the \\emph{Legendre form} of a Weierstrass curve, written as\n$$ E : y^2 = x\\rb{x - 1}\\rb{x - \\lambda}, \\qquad \\lambda \\in K \\setminus \\cb{0, 1}. $$\nThis is merely a transformation, but proves useful when studying elliptic curves over the reals.\n\\end{remark}\n\n\\pagebreak\n\n\\subsection{Group law}\n\nAn elliptic curve has an additional group theoretic property that makes it an \\emph{algebraic group}. This subsection provides a full definition of the additive group induced by an elliptic curve, as well as an attempt to prove that it is indeed one. The following lemma will be used in the definition of the addition operation.\n\n\\begin{lemma}\n\\label{lem:thirdpoint}\nLet $ P = \\sb{a, b, c} \\in E $ and $ Q = \\sb{a', b', c'} \\in E $ be points. Then:\n\\begin{enumerate}\n\\item if $ P \\ne Q $, there is a unique line joining $ P $ and $ Q $ given by\n$$ L : \\rb{bc' - b'c}X + \\rb{a'c - ac'}Y + \\rb{ab' - a'b}Z = 0, $$\n\\item if $ P = Q $, there is a unique tangent at $ P $ given by\n$$ L : \\rb{-3a^2 - Ac^2}X + 2bcY + \\rb{b^2 - 2Aac - 3Bc^2}Z = 0, $$\n\\item there is a unique third point $ R \\in E $ such that $ L $ intersects $ E $ at $ P $, $ Q $, and $ R $.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet $ L : l\\rb{X, Y, Z} = 0 $.\n\\begin{enumerate}\n\\item If $ P \\ne Q $, then\n$$ l\\rb{X, Y, Z} = \\threebyone{X}{Y}{Z} \\cdot \\rb{\\threebyone{a}{b}{c} \\times \\threebyone{a'}{b'}{c'}}. $$\n\\item If $ P = Q $, then\n$$ l\\rb{X, Y, Z} = \\twobyone{1}{1, 0, 0}\\val{\\dfrac{\\partial e}{\\partial X}}_P\\rb{X - a} + \\twobyone{1}{0, 1, 0}\\val{\\dfrac{\\partial e}{\\partial Y}}_P\\rb{Y - b} + \\twobyone{1}{0, 0, 1}\\val{\\dfrac{\\partial e}{\\partial Z}}_P\\rb{Z - c}. $$\n\\item Since $ \\deg\\rb{l} = 1 $ and $ \\deg\\rb{\\gcd\\rb{e, l}} = 0 $, B\\'ezout's theorem gives that $ L $ intersects $ E $ at three points up to multiplicity. Assume that $ P = \\sb{a, b, c} \\ne Q = \\sb{a', b', c'} $. If $ I_P\\rb{e, l} = 1 $ and $ I_Q\\rb{e, l} = 1 $, then there is a unique third point $ R \\in E $ such that $ R \\ne P, Q $ and $ I_R\\rb{e, l} = 1 $. Otherwise $ I_P\\rb{e, l} = 2 $ or $ I_Q\\rb{e, l} = 2 $, then there is also a unique third point $ R = P $ or $ R = Q $ respectively. Otherwise assume that $ P = Q = \\sb{a, b, c} $. Since $ \\cb{l} = T_P\\rb{l} \\in T_P\\rb{e} $, it holds that $ I_P\\rb{e, l} > m_P\\rb{e}m_P\\rb{l} = 1 $. If $ I_P\\rb{e, l} = 2 $, then there is a unique third point $ R \\in E $ such that $ R \\ne P $ and $ I_R\\rb{e, l} = 1 $. Otherwise $ I_P\\rb{e, l} = 3 $, then there is also a unique third point $ R = P $.\n\\end{enumerate}\n\\end{proof}\n\nThe following example illustrates the unique lines and tangents above.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + 2x + 1 $ be an elliptic curve over $ \\R $ with points $ P = \\rb{0, -1} \\in E $ and $ Q = \\rb{1, 2} \\in E $. Then the unique line joining $ P $ and $ Q $ is $ L : y = 3x - 1 $, while the tangent at $ P $ is $ L_P : y = -x - 1 $, and the tangent at $ Q $ is $ L_Q : y = \\tfrac{5}{4}x + \\tfrac{3}{4} $.\n\\end{example}\n\nInstead of defining the addition operation right away, it is clearer to define an intermediate operation with the above lemma as follows.\n\n\\begin{definition}[$ * $]\n$ * : E \\times E \\to E $ is defined by $ P * Q = R $, where $ R $ is the unique third point in Lemma \\ref{lem:thirdpoint}.\n\\end{definition}\n\nThe addition operation can then be defined immediately in terms of this intermediate operation, which are both symmetric and hence commutative.\n\n\\begin{definition}[$ + $]\n$ + : E \\times E \\to E $ is defined by $ P + Q = \\rb{P * Q} * \\mathcal{O} $.\n\\end{definition}\n\nThis definition is chosen carefully so as to make a group law possible. While it might be slightly convoluted, there is an easy geometrical interpretation. While $ P * Q \\in E $ is simply the unique third intersection point of two points $ P \\in E $ and $ Q \\in E $, reflecting it along the horizontal axis gives $ P + Q $. This motivates writing out several explicit formulae relating the affine coordinates of $ P $, $ Q $ and $ P + Q $, which will allow equation manipulations in later sections. The following algorithm summarises the explicit formulae for $ + $, which are proven in the appendix.\n\n\\begin{algorithm}[Group law explicit formulae]\nInput: points $ P, Q \\in E $. Output: $ P + Q $.\n$$ P + Q = \\begin{cases} R & P = \\rb{a, b}, \\ Q = \\rb{a', b'}, \\ a \\ne a' \\\\ S & P = Q = \\rb{a, b}, \\ b \\ne 0 \\\\ P & Q = \\mathcal{O} \\\\ \\mathcal{O} & P = Q = \\rb{a, 0} \\end{cases}, $$\nwhere\n$$ R = \\rb{\\dfrac{\\rb{A + aa'}\\rb{a + a'} + 2\\rb{B - bb'}}{\\rb{a - a'}^2}, \\dfrac{\\rb{Ab' - a'^2b}\\rb{3a + a'} + \\rb{a^2b' - Ab}\\rb{a + 3a'} - 4B\\rb{b - b'}}{\\rb{a - a'}^3}}, $$\n$$ S = \\rb{\\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4b^2}, \\dfrac{a^6 + 5Aa^4 + 20Ba^3 - 5A^2a^2 - 4ABa - A^3 - 8B^2}{8b^3}}. $$\n\\end{algorithm}\n\nThe first case is referred to as the \\emph{addition formula}, while the second case is referred to as the \\emph{duplication formula}. The last two cases allows the definition of a negation operation used for inverses in the group law. This is referred to as the \\emph{negation formula}, where $ -\\mathcal{O} = \\mathcal{O} $ and $ -\\rb{a, b} = \\rb{a, -b} $. Now the group law explicit formulae for characteristic two and three are more complicated and given in full under code listings in the appendix. The following example illustrates an implementation of the algorithm in the appendix.\n\n\\begin{example}\nLet $ E $ be the elliptic curve over $ \\R $ and let $ L $, $ L_P $, $ L_Q $ be the lines for the points $ P, Q \\in E $ from the example above. Then $ L $ intersects $ E $ at $ \\rb{3x - 1}^2 = x^3 + 2x + 1 $, or $ x\\rb{x - 1}\\rb{x - 8} = 0 $. Hence $ P * Q = \\rb{8, 23} $, so $ P + Q = \\rb{8, -23} $. Similarly $ L_P $ intersects $ E $ at $ \\rb{-x - 1}^2 = x^3 + 2x + 1 $, or $ x^2\\rb{x - 1} = 0 $, while $ L_Q $ intersects $ E $ at $ \\rb{\\tfrac{5}{4}x + \\tfrac{3}{4}}^2 = x^3 + 2x + 1 $, or $ \\rb{x - 1}^2\\rb{16x + 7} = 0 $. Thus $ P * P = \\rb{1, -2} $ and $ Q * Q = \\rb{-7 / 16, 13 / 64} $, so $ P + P = \\rb{1, 2} $ and $ Q + Q = \\rb{-7 / 16, -13 / 64} $.\n\\end{example}\n\nAn alternative formulation for $ + $ is such that three points $ P, Q, R \\in E $ are collinear iff\n$$ P + Q + R = P + \\rb{Q + R} = \\rb{P + Q} + R = \\mathcal{O}. $$\nThis formulation will help in proving that certain maps obey some property later, but also allows for a pictorial description for $ * $. As per the notation in the appendix: the first pane describes $ \\rb{*}_2 $; the second pane describes $ \\rb{*}_3 $; the third pane describes $ \\rb{*}_1 $ and $ \\rb{*}_5 $; the fourth pane describes $ \\rb{*}_4 $; the unillustrated line at infinity describes $ \\rb{*}_6 $.\n\\begin{center}\n\\includegraphics*[width=0.9\\textwidth]{img/ECClines.png}\n\\end{center}\n\n\\pagebreak\n\nThe group structure of an elliptic curve with respect to $ + $ can now be stated in the following theorem.\n\n\\begin{theorem}[Group law]\n$ \\rb{E, \\mathcal{O}, +} $ is an abelian group.\n\\end{theorem}\n\nAs full proofs for associativity such as in III.3.4 of \\cite{gtm} require further prerequisites on algebraic curves, particularly on \\emph{divisors} and \\emph{differentials}, only the sketch of an alternative geometric proof is given, of which the special case of nine pairwise distinct points is assumed.\n\n\\begin{proof}\nThe unique right identity is $ \\mathcal{O} \\in E $ and unique right inverses are given by the negation formula. Symmetry of $ + $ gives the unique identity, unique right inverses, and commutativity. Associativity of $ + $ can be checked with various methods, such as by tediously verifying cases of the explicit formulae in \\cite{grouplaw}. Alternatively, let $ P, Q, R \\in E $ be points, and let\n\\begin{itemize}\n\\item $ L_1 : l_1\\rb{X, Y, Z} = 0 $ be the line joining $ P $, $ Q $, and $ P * Q = -\\rb{P + Q} $,\n\\item $ L_2 : l_2\\rb{X, Y, Z} = 0 $ be the line joining $ Q $, $ R $, and $ Q * R = -\\rb{Q + R} $,\n\\item $ L_3 : l_3\\rb{X, Y, Z} = 0 $ be the line joining $ P + Q $, $ \\mathcal{O} $, and $ \\rb{P + Q} * \\mathcal{O} = -\\rb{P + Q} $,\n\\item $ L_4 : l_4\\rb{X, Y, Z} = 0 $ be the line joining $ Q + R $, $ \\mathcal{O} $, and $ \\rb{Q + R} * \\mathcal{O} = -\\rb{Q + R} $,\n\\item $ L_5 : l_5\\rb{X, Y, Z} = 0 $ be the line joining $ P + Q $, $ R $, and $ \\rb{P + Q} * R = -\\rb{\\rb{P + Q} + R} $, and\n\\item $ L_6 : l_6\\rb{X, Y, Z} = 0 $ be the line joining $ P $, $ Q + R $, and $ P * \\rb{Q + R} = -\\rb{P + \\rb{Q + R}} $,\n\\end{itemize}\nassuming that these points are pairwise distinct except for $ -\\rb{\\rb{P + Q} + R} $ and $ -\\rb{P + \\rb{Q + R}} $. Now let\n$$ C_1 : \\rb{l_1l_4l_5}\\rb{X, Y, Z} = 0, \\qquad C_2 : \\rb{l_2l_3l_6}\\rb{X, Y, Z} = 0, $$\nbe cubics such that\n$$ I = \\cb{\\mathcal{O}, P, Q, R, P + Q, Q + R, -\\rb{P + Q}, -\\rb{Q + R}} \\subseteq C_1 \\cap C_2. $$\nThen B\\'ezout's theorem gives that $ E $, $ C_1 $, and $ C_2 $ pairwise intersect at nine points up to multiplicity. Hence\n$$ E \\cap C_1 = I \\cup \\cb{-\\rb{\\rb{P + Q} + R}}, \\qquad E \\cap C_2 = I \\cup \\cb{-\\rb{P + \\rb{Q + R}}}, \\qquad C_1 \\cap C_2 = I \\cup \\cb{S}, $$\nfor some ninth point $ S \\in C_1 \\cap C_2 $. Since $ I \\subseteq E $, the Cayley-Bacharach theorem gives $ S \\in E $, so\n$$ -\\rb{\\rb{P + Q} + R} = S = -\\rb{P + \\rb{Q + R}}. $$\nThus $ \\rb{P + Q} + R = P + \\rb{Q + R} $.\n\\end{proof}\n\nWith an abelian group at hand, some group theoretic properties of an elliptic curve can be explored. In particular, restricting an elliptic curve onto its $ F $-rational points retain the group structure.\n\n\\begin{proposition}\n$ \\rb{E\\rb{F}, \\mathcal{O}, +} \\le \\rb{E, \\mathcal{O}, +} $.\n\\end{proposition}\n\n\\begin{proof}\nSince $ 0, 1 \\in F $, it holds that $ \\mathcal{O} \\in E\\rb{F} $. Let $ P, Q \\in E\\rb{F} $ be points, then the explicit formulae give $ -P, P + Q \\in E\\rb{F} $. Thus $ \\rb{E\\rb{F}, \\mathcal{O}, +} \\le \\rb{E, \\mathcal{O}, +} $.\n\\end{proof}\n\nAdditionally, the $ n $-torsion points of an elliptic curve also form a group, provided $ \\mathcal{O} $ is included. The following example illustrates the structure of the $ 2 $-torsion subgroup.\n\n\\begin{example}\nLet $ P = \\rb{a, b} \\in E\\sb{2} $, then $ b = -b = 0 $. Since $ x^3 + Ax + B = 0 $ has three distinct solutions, there are three distinct points $ P_1 = \\rb{a_1, 0} $, $ P_2 = \\rb{a_2, 0} $, and $ P_3 = \\rb{a_3, 0} $ in $ E\\sb{2} $. Thus $ \\rb{E\\sb{2}, \\mathcal{O}, +} = \\rb{\\cb{\\mathcal{O}, P_1, P_2, P_3}, \\mathcal{O}, +} \\cong \\rb{\\Z_2^2, 0, +} $.\n\\end{example}\n\nHence $ b = 0 $ iff $ ord\\rb{a, b} = 2 $.\n\n\\begin{remark}\nIn fact, the $ n $-torsion points of $ E $ form a subgroup $ E\\sb{n} $ of $ E $, such that $ \\rb{E\\sb{p}, \\mathcal{O}, +} \\cong \\rb{\\Z_p^2, 0, +} $ if $ char\\rb{F} \\nmid p $, and either $ \\rb{E\\sb{p^e}, \\mathcal{O}, +} \\cong \\rb{\\cb{0}, 0, +} $ or $ \\rb{E\\sb{p^e}, \\mathcal{O}, +} \\cong \\rb{\\Z_{p^e}, 0, +} $ for all $ e \\in \\Z_{> 0} $ if $ char\\rb{F} \\mid p $.\n\\end{remark}\n\n\\pagebreak\n\n\\subsection{Isogenies}\n\nPrior to this section, the only maps between elliptic curves that have been defined were affine transformations. Now that the group law is defined, group homomorphisms can also be considered. However, a slightly different approach to this will be taken with the following definition, noting that morphisms of curves are either constant or surjective.\n\n\\begin{definition}[Isogeny]\nAn \\textbf{isogeny} from $ E $ to $ E' $ is a surjective morphism $ \\phi : E \\to E' $ such that $ \\phi\\rb{\\mathcal{O}} = \\mathcal{O} $.\n\\end{definition}\n\nAs isomorphisms are defined as invertible morphisms that preserve the point at infinity, they are isogenies as well. Now despite the simple condition, isogenies are actually group homomorphisms, which also preserve the point at infinity. The following proposition then gives an equivalent definition.\n\n\\begin{proposition}\nLet $ \\phi : E \\to E' $ be an isogeny. Then $ \\phi $ is a group homomorphism.\n\\end{proposition}\n\n\\begin{proof}\nOmitted, see III.4.8 in \\cite{gtm}.\n\\end{proof}\n\nThe following is a typical example of an isogeny.\n\n\\begin{example}\nThe \\emph{multiplication by $ n $ map} $ \\sb{n} : E \\to E $ defined by $ \\sb{n}\\rb{P} = nP $ is an isogeny such that $ Ker\\rb{\\sb{n}} = E\\sb{n} $.\n\\end{example}\n\nLet $ \\phi : E \\to E' $ be an isogeny. While isomorphisms are easily characterised by $ j $-invariant affine transformations, the smaller restriction on isogenies allow for a wider range of coordinate transformations that still obey the group homomorphism property. In particular, rational functions that define isogenies can be characterised by the following lemma.\n\n\\begin{lemma}\nLet $ f \\in F\\rb{E} $ be a rational function. Then\n$$ f\\rb{x, y} = \\dfrac{f'\\rb{x} + f''\\rb{x}y}{f'''\\rb{x}}, \\qquad f', f'' \\in F\\sb{x}, \\qquad f''' \\in F\\sb{x} \\setminus \\cb{0}. $$\n\\end{lemma}\n\n\\begin{proof}\nLet $ f = g / h $ for some $ g \\in F\\sb{x, y} $ and some $ h \\in F\\sb{x, y} \\setminus \\cb{0} $. Then $ g\\rb{x, y} = \\sum_{i = 0}^n g_i\\rb{x}y^i $ for some $ g_i \\in F\\sb{x} $, some $ h_i \\in F\\sb{x} \\setminus \\cb{0} $, and some $ n, m \\in \\Z_{\\ge 0} $, so:\n\\begin{align*}\ng\\rb{x, y}\n& = \\sum_{i = 0}^n g_i\\rb{x}y^i\n= \\sum_{i = 0}^{n / 2} g_{2i}\\rb{x}y^{2i} + \\sum_{i = 0}^{n / 2} g_{2i + 1}\\rb{x}y^{2i + 1} \\\\\n& = \\sum_{i = 0}^{n / 2} g_{2i}\\rb{x}\\rb{x^3 + Ax + B}^i + \\sum_{i = 0}^{n / 2} g_{2i + 1}\\rb{x}\\rb{x^3 + Ax + B}^iy \\\\\n& = g'\\rb{x} + g''\\rb{x}y, \\qquad g', g'' \\in F\\sb{x}.\n\\end{align*}\nSimilarly $ h\\rb{x, y} = h'\\rb{x} + h''\\rb{x}y $ for some $ h', h'' \\in F\\sb{x} $. Thus\n\\begin{align*}\nf\\rb{x, y}\n& = \\dfrac{g\\rb{x, y}}{h\\rb{x, y}}\n= \\dfrac{g'\\rb{x} + g''\\rb{x}y}{h'\\rb{x} + h''\\rb{x}y}\n= \\dfrac{\\rb{g'\\rb{x} + g''\\rb{x}y}\\rb{h'\\rb{x} - h''\\rb{x}y}}{\\rb{h'\\rb{x} + h''\\rb{x}y}\\rb{h'\\rb{x} - h''\\rb{x}y}} \\\\\n& = \\dfrac{g'\\rb{x}h'\\rb{x} - g''\\rb{x}h''\\rb{x}y^2 - g'\\rb{x}h''\\rb{x}y + g''\\rb{x}h'\\rb{x}y}{h'\\rb{x}^2 - h''\\rb{x}^2y^2} \\\\\n& = \\dfrac{g'\\rb{x}h'\\rb{x} - g''\\rb{x}h''\\rb{x}\\rb{x^3 + Ax + B} - g'\\rb{x}h''\\rb{x}y + g''\\rb{x}h'\\rb{x}y}{h'\\rb{x}^2 - h''\\rb{x}^2\\rb{x^3 + Ax + B}} \\\\\n& = \\dfrac{f'\\rb{x} + f''\\rb{x}y}{f'''\\rb{x}}, \\qquad f, f' \\in F\\sb{x}, \\qquad f'' \\in F\\sb{x} \\setminus \\cb{0}.\n\\end{align*}\n\\end{proof}\n\n\\pagebreak\n\nAn entire isogeny can now be characterised similarly, noting the group homomorphism property. The following proposition gives the explicit \\emph{standard form} of an isogeny, defined in terms of its image.\n\n\\begin{proposition}\nLet $ P = \\rb{a, b} \\in E \\setminus Ker\\rb{\\phi} $ be a point. Then\n$$ \\phi\\rb{P} = \\rb{\\dfrac{r\\rb{a}}{s\\rb{a}}, \\dfrac{u\\rb{a}}{v\\rb{a}}b} \\qquad r, u \\in F\\sb{x}, \\qquad s, v \\in F\\sb{x} \\setminus \\cb{0}, $$\nsuch that $ \\gcd\\rb{r, s} = \\gcd\\rb{u, v} = 1 $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ \\phi = \\sb{\\phi_x, \\phi_y, \\phi_z} $ for some $ \\phi_x, \\phi_y, \\phi_z \\in F\\rb{E} $. Since $ \\phi\\rb{P} \\ne \\mathcal{O} $, it holds that $ \\phi_z\\rb{P} \\ne 0 $, so\n$$ \\phi\\rb{P} = \\sb{\\phi_x\\rb{P}, \\phi_y\\rb{P}, \\phi_z\\rb{P}} = \\rb{\\dfrac{\\phi_x\\rb{P}}{\\phi_z\\rb{P}}, \\dfrac{\\phi_y\\rb{P}}{\\phi_z\\rb{P}}}. $$\nThen $ \\phi_x\\rb{P} / \\phi_z\\rb{P}, \\phi_y\\rb{P} / \\phi_z\\rb{P} \\in F\\rb{E} $ are rational functions, so\n$$ \\dfrac{\\phi_x\\rb{P}}{\\phi_z\\rb{P}} = \\dfrac{\\psi\\rb{a} + \\psi'\\rb{a}b}{\\psi''\\rb{a}}, \\qquad \\dfrac{\\phi_y\\rb{P}}{\\phi_z\\rb{P}} = \\dfrac{\\chi\\rb{a} + \\chi'\\rb{a}b}{\\chi''\\rb{a}}, \\qquad \\psi, \\psi', \\chi, \\chi' \\in F\\sb{x}, \\qquad \\psi'', \\chi'' \\in F\\sb{x} \\setminus \\cb{0}. $$\nSince $ \\phi\\rb{-P} = -\\phi\\rb{P} $,\n$$ \\rb{\\dfrac{\\psi\\rb{a} + \\psi'\\rb{a}\\rb{-b}}{\\psi''\\rb{a}}, \\dfrac{\\chi\\rb{a} + \\chi'\\rb{a}\\rb{-b}}{\\chi''\\rb{a}}} = \\phi\\rb{-P} = -\\phi\\rb{P} = \\rb{\\dfrac{\\psi\\rb{a} + \\psi'\\rb{a}b}{\\psi''\\rb{a}}, -\\dfrac{\\chi\\rb{a} + \\chi'\\rb{a}b}{\\chi''\\rb{a}}}. $$\nHence $ \\psi'\\rb{a} = \\chi\\rb{a} = 0 $. Now let $ g = \\gcd\\rb{\\psi, \\psi''} $ and $ g' = \\gcd\\rb{\\chi', \\chi''} $. Thus let\n$$ r = \\dfrac{\\psi}{g}, u = \\dfrac{\\chi'}{g'} \\in F\\sb{x}, \\qquad s = \\dfrac{\\psi''}{g}, v = \\dfrac{\\chi''}{g'} \\in F\\sb{x} \\setminus \\cb{0}, $$\nsuch that $ \\gcd\\rb{r, s} = \\gcd\\rb{u, v} = 1 $.\n\\end{proof}\n\nIn the above proof, the assumption that a point $ P \\in E $ is not in the kernel allows for the isogeny to be scaled appropriately. If $ P $ is in the kernel, it would be mapped to the point at infinity, which would mean that $ \\phi_z $, and hence $ s $ or $ v $, is zero. With this in mind, an abuse of notation allows for the standard form to be written as\n$$ \\phi\\rb{x, y} = \\rb{\\dfrac{r\\rb{x}}{s\\rb{x}}, \\dfrac{u\\rb{x}}{v\\rb{x}}y}, \\qquad r, s, u, v \\in F\\sb{x}, \\qquad \\gcd\\rb{r, s} = \\gcd\\rb{u, v} = 1, $$\nremembering that $ \\phi\\rb{\\mathcal{O}} = \\mathcal{O} $, and $ \\phi\\rb{a, b} = \\mathcal{O} $ whenever $ s\\rb{a} = 0 $ or $ v\\rb{a} = 0 $ for any point $ \\rb{a, b} \\in E $. The following example rewrites the multiplication by two map with the familiar duplication formula.\n\n\\begin{example}\nBy the duplication formula,\n\\begin{align*}\n\\sb{2}\\rb{x, y}\n& = \\rb{\\dfrac{x^4 - 2Ax^2 - 8Bx + A^2}{4y^2}, \\dfrac{x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - A^3 - 8B^2}{8y^3}} \\\\\n& = \\rb{\\dfrac{x^4 - 2Ax^2 - 8Bx + A^2}{4\\rb{x^3 + Ax + B}}, \\dfrac{x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - A^3 - 8B^2}{8\\rb{x^3 + Ax + B}^2}y},\n\\end{align*}\nwhich is in standard form. Then $ \\sb{2}\\rb{a, b} = \\mathcal{O} $ iff $ b^2 = a^3 + Aa + B = 0 $ for any point $ \\rb{a, b} \\in E $.\n\\end{example}\n\nThere are also two useful notions of an isogeny, the first of which is its degree.\n\n\\begin{definition}[Isogeny degree]\nThe \\textbf{degree} of $ \\phi $ is $ \\deg\\rb{\\phi} = \\max\\cb{\\deg\\rb{r}, \\deg\\rb{s}} $.\n\\end{definition}\n\nThe degree of the constant morphism, while not an isogeny, is defined to be zero. The degrees of two trivial isogenies are given in the following example.\n\n\\begin{example}\nThe identity isogeny, or the multiplication by one map $ \\sb{1} $ has degree $ \\deg\\rb{\\sb{1}} = \\max\\cb{1, 1} = 1 $. Similarly, the multiplication by negative one map $ \\sb{-1} $ also has degree $ \\deg\\rb{\\sb{-1}} = 1 $.\n\\end{example}\n\n\\pagebreak\n\nThe second invariant notion of an isogeny is its separability.\n\n\\begin{definition}[Separable isogeny]\n$ \\phi $ is \\textbf{separable} iff $ d\\rb{r / s} / dx \\ne 0 $.\n\\end{definition}\n\n\\begin{remark}\nThe isogeny $ \\phi $ induces an injection $ \\phi^* : F\\rb{E'} \\to F\\rb{E} $ of function fields. Its separability is equivalently formulated as that of $ F\\rb{E} / \\phi^*F\\rb{E'} $, which reflects the definition of a separable extension.\n\\end{remark}\n\nThe following example of the multiplication by two map illustrates these two notions.\n\n\\begin{example}\n$ \\sb{2} $ has degree $ \\deg\\rb{\\sb{2}} = \\max\\cb{4, 3} = 4 $ and is separable since\n$$ \\dfrac{d}{dx}\\rb{\\dfrac{x^4 - 2Ax^2 - 8Bx + A^2}{4\\rb{x^3 + Ax + B}}} = \\dfrac{x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - A^3 - 8B^2}{4\\rb{x^3 + Ax + B}^2} \\ne 0. $$\n\\end{example}\n\nSeparability always holds in $ char\\rb{F} = 0 $, but there are inseparable isogenies in $ char\\rb{F} = p $ for some prime $ p \\in \\Z_{> 0} $. More on separable isogenies will be discussed in a later section. Now since $ + $ is a morphism, the set of all isogenies between $ E $ and $ E' $, together with the constant morphism, forms an abelian group $ Hom\\rb{E, E'} $ under the operation\n$$ \\rb{\\phi + \\psi}\\rb{P} = \\phi\\rb{P} + \\psi\\rb{P}. $$\nIsogenies in the group can also compose to form a ring when $ E = E' $ in the following definition.\n\n\\begin{definition}[Endomorphism]\n$ \\phi $ is an \\textbf{endomorphism} of $ E $ iff $ E = E' $. The \\textbf{endomorphism ring} $ End\\rb{E} $ of $ E $ is the ring of all endomorphisms of $ E $ with respect to $ + $ and $ \\circ $, where\n$$ \\rb{\\phi \\circ \\psi} = \\phi\\rb{\\psi\\rb{P}}. $$\n\\end{definition}\n\nThe following example gives an endomorphism of elliptic curves over fields of non-zero characteristic that is of particular interest.\n\n\\begin{example}\nLet $ F = \\F_p $ for some prime $ p \\in \\Z_{> 0} $. Then the \\emph{Frobenius endomorphism} $ Fr : E \\to E $ defined by $ Fr\\rb{x, y} = \\rb{x^p, y^p} $ is an inseparable endomorphism with degree $ \\deg\\rb{Fr} = p $.\n\\end{example}\n\nThe Frobenius endomorphism will be formally defined in a later section. On a final note, endomorphisms with inverses also form a multiplicative subgroup.\n\n\\begin{definition}[Automorphism]\n$ \\phi $ is an \\textbf{automorphism} of $ E $ iff it is an endomorphism and an isomorphism. The \\textbf{automorphism group} $ Aut\\rb{E} $ is the group of all automorphisms of $ E $.\n\\end{definition}\n\nUnlike the endomorphism ring, the automorphism group of an elliptic curve is easily characterised.\n\n\\begin{proposition}\n$$ Aut\\rb{E} \\cong \\begin{cases} \\Z_6 & j_E = 0 \\\\ \\Z_4 & j_E = 1728 \\\\ \\Z_2 & j_E \\notin \\cb{0, 1728} \\end{cases}. $$\n\\end{proposition}\n\n\\begin{proof}\nLet $ \\phi \\in Aut\\rb{E} $. Then $ \\phi $ induces a $ j $-invariant affine transformation\n$$ \\rb{x, y} \\mapsto \\rb{u^2x + r, u^3y + u^2sx + t}, \\qquad u \\in K^*, \\qquad r, s, t \\in K $$\nfrom $ W $ to itself. Since $ \\phi $ is an automorphism, it holds that $ r = s = t = 0 $, and $ A = A / u^4 $ and $ B = B / u^6 $. If $ j_E = 0 $, then $ A = 0 $ and $ B \\ne 0 $, so $ u^6 = 1 $. Hence $ u $ is a sixth root of unity and $ Aut\\rb{E} \\cong \\Z_6 $. If $ j_E = 1728 $, then $ A \\ne 0 $ and $ B = 0 $, so $ u^4 = 1 $. Hence $ u $ is a fourth root of unity and $ Aut\\rb{E} \\cong \\Z_4 $. Otherwise $ j_E \\notin \\cb{0, 1728} $, then $ A \\ne 0 $ and $ B \\ne 0 $, so $ u^6 = 1 $ and $ u^4 = 1 $. Hence $ u^2 = 1 $ and $ u $ is a second root of unity. Thus $ Aut\\rb{E} \\cong \\Z_2 $.\n\\end{proof}\n\n\\begin{remark}\nIf $ char\\rb{F} \\in \\cb{2, 3} $, then the above list of cases for $ Aut\\rb{E} $ with $ j_E = 0, 1728 $ is not exhaustive. In particular, if $ char\\rb{F} = 2 $, then $ Aut\\rb{E} \\cong \\Z_4 \\ltimes \\Z_3 $, otherwise $ char\\rb{F} = 3 $, then $ Aut\\rb{E} \\cong \\Z_3 \\ltimes Q_8 $.\n\\end{remark}\n\nThe above definitions are defined for $ E\\rb{F} $ as well, and are written $ Hom_F\\rb{E, E'} $, $ End_F\\rb{E} $, and $ Aut_F\\rb{E} $ respectively.\n\n\\pagebreak\n\n\\section{Elliptic curves over finite fields}\n\nWhen studying elliptic curves over a field or a family of fields, an important question would be to determine the set of solutions existing in that field. For instance, it is desirable to count the rational solutions in that field, which would have far fetching applications in number theory and cryptography. For finite fields, there is a finite process to compute the rational points that would always work. The following example illustrates a naive approach for this.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + x + 1 $ be an elliptic curve over $ \\F_5 $. Since there are five distinct values for $ x \\in \\F_5 = \\cb{0, 1, 2, 3, 4} $, computing $ x^3 + x + 1 $ for each value of $ x $ and checking if it is a quadratic residue $ y^2 $ in $ \\F_5 $ gives the following\n\\begin{itemize}\n\\item If $ x = 0 $, then $ y^2 = x^3 + x + 1 = 1 = 1^2 = 4^2 $, so $ y = 1 $ or $ y = 4 $.\n\\item If $ x = 1 $, then $ y^2 = x^3 + x + 1 = 3 $ is not a quadratic residue.\n\\item If $ x = 2 $, then $ y^2 = x^3 + x + 1 = 1 = 1^2 = 4^2 $, so $ y = 1 $ or $ y = 4 $.\n\\item If $ x = 3 $, then $ y^2 = x^3 + x + 1 = 1 = 1^2 = 4^2 $, so $ y = 1 $ or $ y = 4 $.\n\\item If $ x = 4 $, then $ y^2 = x^3 + x + 1 = 4 = 2^2 = 3^2 $, so $ y = 2 $ or $ y = 3 $.\n\\end{itemize}\nSince $ \\mathcal{O} \\in E\\rb{\\F_5} $, there are exactly nine $ \\F_5 $-rational points\n$$ E\\rb{\\F_5} = \\cb{\\mathcal{O}, \\rb{0, 1}, \\rb{0, 4}, \\rb{2, 1}, \\rb{2, 4}, \\rb{3, 1}, \\rb{3, 4}, \\rb{4, 2}, \\rb{4, 3}}. $$\nHence $ E\\rb{\\F_5} \\cong \\Z_3^2 $ or $ E\\rb{\\F_5} \\cong \\Z_9 $. Now Lagrange's theorem gives that $ ord\\rb{P} = 3 $ or $ ord\\rb{P} = 9 $ for any non-zero point $ P \\in E\\rb{\\F_5} $. Let $ P = \\rb{0, 1} \\in E\\rb{\\F_5} $. By the addition and duplication formulae, it holds that $ 3P = \\rb{2, 1} $ and $ 9P = \\mathcal{O} $, so it has order $ ord\\rb{P} = 9 $ and is a generator of $ E\\rb{\\F_5} $. Thus $ E\\rb{\\F_5} \\cong \\Z_9 $.\n\\end{example}\n\nThis finite process is straightforward in the sense that it always terminates. However, as it runs with an asymptotic time complexity of $ O\\rb{q} $ for a finite field $ \\F_q $, the approach becomes rather intractable for large prime powers $ q \\in \\Z_{> 0} $. This section will attempt to develop several techniques to compute $ E\\rb{\\F_q} $, or more specifically $ \\abs{E\\rb{\\F_q}} $, which will span the next few subsections. Now let $ E $ be an elliptic curve over the perfect field $ F = \\F_q = \\F_{p^e} $ for some prime $ p \\in \\Z_{> 0} \\setminus \\cb{2, 3} $ and some $ e \\in \\Z_{> 0} $, given by the Weierstrass curve\n$$ E : y^2 = x^3 + Ax + B, \\qquad A, B \\in F, $$\nwith the group of rational points $ E\\rb{F} = \\rb{E\\rb{F}, \\mathcal{O}, +} $. \n\n\\subsection{Hasse's theorem: inseparable isogenies}\n\nThe following theorem bounds the maximum cardinality of the group of rational points.\n\n\\begin{theorem}[Hasse]\n\\label{thm:hasse}\n$ \\abs{E\\rb{F}} = q - t + 1 $ for some \\emph{trace} $ t \\in \\Z $ such that $ \\abs{t} \\le 2\\sqrt{q} $.\n\\end{theorem}\n\n\\begin{remark}\nThis is a special case of the Hasse-Weil theorem, which states that $ \\abs{C\\rb{F}} = q - t + 1 $ for some $ \\abs{t} \\le 2g\\sqrt{q} $ for any projective algebraic curve $ C $ over $ F $ of genus $ g $.\n\\end{remark}\n\nProof of Hasse's theorem concerns the properties of separable and inseparable isogenies, which are given by separable and inseparable polynomials. The following lemma allows inseparable polynomials to be written in a reduced form.\n\n\\begin{lemma}\nLet $ f \\in F\\sb{x} $ be an inseparable polynomial. Then $ f\\rb{x} = g\\rb{x^p} $ for some $ g \\in F\\sb{x} $.\n\\end{lemma}\n\n\\begin{proof}\nLet $ f\\rb{x} = \\sum_{i = 0}^n a_ix^i = \\sum_{a_i \\ne 0} a_ix^{m_i} $ for some $ a_i \\in F $ and some $ n, m_i \\in \\Z_{> 0} $. Since $ f $ is separable, it holds that $ 0 = df / dx = \\sum_{a_i \\ne 0} m_ia_ix^{m_i - 1} $. Then $ m_ia_i = 0 $ for each $ a_i \\ne 0 $, so $ p \\mid m_i $ and $ m_i = pk $ for some $ k \\in \\Z_{\\ge 0} $. Thus $ f\\rb{x} = \\sum_{a_i \\ne 0} a_i\\rb{x^p}^k = g\\rb{x^p} $ for some $ g \\in F\\sb{x} $.\n\\end{proof}\n\nThe polynomial $ g $ would then be of a smaller degree than $ f $, which justifies why it is deemed as reduced. A similar argument allows inseparable isogenies to be reduced, so let $ E' $ be another elliptic curve over $ F $ given by the Weierstrass curve\n$$ E' : y^2 = x^3 + A'x + B', \\qquad A', B' \\in F, $$\nand let $ \\phi : E \\to E' $ be an isogeny. The following lemma again allows inseparable isogenies to be written in a reduced form.\n\n\\begin{lemma}\nLet $ \\phi $ be inseparable. Then\n$$ \\phi\\rb{x, y} = \\rb{\\dfrac{r'\\rb{x^p}}{s'\\rb{x^p}}, \\dfrac{u'\\rb{x^p}}{v'\\rb{x^p}}y^p}, \\qquad r', s', u', v' \\in F\\sb{x}. $$\n\\end{lemma}\n\n\\begin{proof}\nSince $ \\phi $ is inseparable,\n$$ 0 = \\dfrac{d}{dx}\\rb{\\dfrac{r}{s}} = \\dfrac{1}{s^2}\\rb{\\dfrac{dr}{dx}s - \\dfrac{ds}{dx}r} \\qquad \\implies \\qquad \\dfrac{dr}{dx}s = \\dfrac{ds}{dx}r. $$\nSince $ \\gcd\\rb{r, s} = 1 $, it holds that $ r \\mid dr / dx $. Since $ \\deg\\rb{dr / dx} < \\deg\\rb{r} $, it also holds that $ dr / dx = 0 $, so $ r $ is inseparable and $ r\\rb{x} = r'\\rb{x^p} $ for some $ r' \\in F\\sb{x} $. Similarly $ s $ is inseparable, so $ ds / dx = 0 $ and $ s\\rb{x} = s'\\rb{x^p} $ for some $ s' \\in F\\sb{x} $. Now\n$$ \\rb{\\dfrac{u}{v}y}^2 = \\rb{\\dfrac{r}{s}}^3 + A'\\dfrac{r}{s} + B' \\qquad \\implies \\qquad u^2s^3y^2 = v^2t, \\qquad t = r^3 + A'rs^2 + B's^3. $$\nThen $ dr / dx = 0 $ and $ ds / dx = 0 $ gives $ dt / dx = 0 $, which gives $ d\\rb{u^2y^2 / v^2} / dx = d\\rb{t / s^3} / dx = 0 $. Hence $ u\\rb{x}^2y^2 = y'\\rb{x^p} $ and $ v\\rb{x}^2 = v'\\rb{x^p} $ for some $ y', v' \\in F\\sb{x} $ similarly. Now since $ y^2 = x^3 + Ax + B $ has distinct factors, let $ y^2 = \\rb{x - \\alpha_1}\\rb{x - \\alpha_2}\\rb{x - \\alpha_3} $ for some $ \\alpha_i \\in K $. Then each $ \\rb{x - \\alpha_i} $ is a factor of $ y'\\rb{x^p} $, so $ \\rb{x^p - \\alpha_i^p} = \\rb{x - \\alpha_i}^p $ is also a factor of $ y'\\rb{x^p} $ and of $ u\\rb{x}y^2 $. Hence $ y'\\rb{x^p} = t'\\rb{x^p}\\rb{y^2}^p $ for some $ t' \\in F\\sb{x} $. Now any factor $ \\rb{x - \\alpha} $ of $ u\\rb{x} $ is such that $ \\rb{x - \\alpha}^p = \\rb{x^p - \\alpha^p} $ is a factor of $ t'\\rb{x^p} $. Since $ \\gcd\\rb{p, 2} = 1 $, it holds that $ \\rb{\\rb{x - \\alpha}^p}^2 $ is also a factor of $ t'\\rb{x^p} $, so $ t'\\rb{x^p} = u'\\rb{x^p}^2 $ for some $ u' \\in F\\sb{x} $. Thus $ u\\rb{x}^2y^2 = u'\\rb{x^p}^2\\rb{y^p}^2 $ and\n$$ \\phi\\rb{x, y} = \\rb{\\dfrac{r\\rb{x}}{s\\rb{x}}, \\dfrac{u\\rb{x}}{v\\rb{x}}y} = \\rb{\\dfrac{r'\\rb{x^p}}{s'\\rb{x^p}}, \\pm\\dfrac{u'\\rb{x^p}}{v'\\rb{x^p}}y^p}. $$\n\\end{proof}\n\nNow the above lemma might feel slightly arbitrary due to the presence of $ x^p $ and $ y^p $ in the isogeny. This brings the discussion to a particular endomorphism defined as follows, which would simplify the above expression.\n\n\\begin{definition}[Frobenius endomorphism]\nThe \\textbf{Frobenius endomorphism} $ Fr : E \\to E $ is defined by $ Fr\\rb{x, y} = \\rb{x^p, y^p} $ if $ e = 1 $. The $ q $-th power Frobenius endomorphism $ Fr_q : E \\to E $ is defined by $ Fr_q\\rb{x, y} = \\rb{x^q, y^q} $.\n\\end{definition}\n\nThe Frobenius endomorphism is also injective by virtue of the field characteristic, and hence bijective, which allows for an inverse isogeny to be easily defined.\n\n\\begin{remark}\nA remarkable equivalent characterisation of a perfect field is that the Frobenius endomorphism of a field with positive characteristic is an automorphism, which induces a similar property for isogenies defined over this field.\n\\end{remark}\n\nThe above lemmas for inseparable $ \\phi $ can now be written in terms of the Frobenius endomorphism $ \\phi = \\phi' \\circ Fr $, where\n$$ \\phi'\\rb{x, y} = \\rb{\\dfrac{r'\\rb{x}}{s'\\rb{x}}, \\dfrac{u'\\rb{x}}{v'\\rb{x}}y}, \\qquad r', s', u', v' \\in F\\sb{x}, $$\nwhich is a reduced standard form of an isogeny. In fact, any isogeny can be written as the composition of a Frobenius endomorphism. The following proposition summarises the above lemmas nicely.\n\n\\begin{proposition}\n$ \\phi = \\phi_s \\circ Fr^n $ for some separable isogeny $ \\phi_s : E \\to E' $ and some $ n \\in \\Z_{\\ge 0} $.\n\\end{proposition}\n\n\\begin{proof}\nIf $ \\phi $ is separable, then let $ \\phi_s = \\phi $ and $ n = 0 $. Otherwise $ \\phi = \\phi_1 \\circ Fr $ for some $ \\phi_1 : E \\to E' $. If $ \\phi_i : E \\to E' $ is inseparable, then $ \\phi_i = \\phi_{i + 1} \\circ Fr^i $ for some $ \\phi_{i + 1} : E \\to E' $. Since $ \\deg\\rb{\\phi} $ is finite, by induction, there is some $ n \\in \\Z_{\\ge 0} $ such that $ n \\le \\deg\\rb{\\phi} $ and $ \\phi_n : E \\to E' $ is separable. Thus let $ \\phi_s = \\phi_n $.\n\\end{proof}\n\n\\begin{remark}\nSince $ F $ is a perfect field, the isogeny $ \\phi $ can also be written as $ \\phi = Fr^n \\circ \\phi'_s $ for some separable isogeny $ \\phi'_s : E \\to E' $ such that $ \\deg\\rb{\\phi_s} = \\deg\\rb{\\phi'_s} $. If $ F $ is not a perfect field, the Frobenius endomorphism is not necessarily an automorphism, so $ Im\\rb{Fr} \\subseteq E $ and the domain of $ \\phi_s $ is only a subset of $ E $.\n\\end{remark}\n\nHence any isogeny can be decomposed as the unique composition of a separable isogeny and a Frobenius endomorphism, so $ \\phi $ will be written as\n$$ \\phi = \\phi_s \\circ Fr^n, \\qquad \\phi_s \\in F\\sb{E}, \\qquad n \\in \\Z_{\\ge 0}, $$\nwhere $ \\phi_s $ is a separable isogeny. Two additional notions of degree of an isogeny can then be defined as follows.\n\n\\begin{definition}[Separable degree]\nThe \\textbf{separable degree} of $ \\phi $ is $ \\deg_s\\rb{\\phi} = \\deg\\rb{\\phi_s} $. The \\textbf{inseparable degree} of $ \\phi $ is $ \\deg_i\\rb{\\phi} = p^n $.\n\\end{definition}\n\nIt is clear that the degree of an isogeny is related to these two degrees by\n$$ \\deg\\rb{\\phi} = \\deg_s\\rb{\\phi}\\deg_i\\rb{\\phi}. $$\nIf an isogeny is separable, its decomposition to a Frobenius endomorphism is trivial, so its separable degree is equal to its degree and its inseparable degree is one.\n\n\\begin{remark}\nAn inseparable isogeny does not necessarily have its inseparable degree equal to its degree and its separable degree equal to one. If this is the case, then the isogeny is \\emph{purely inseparable}. However, purely inseparable isogenies are not always inseparable, as with the case for degree one isogenies, which are isomorphisms, with all three degree equal to one.\n\\end{remark}\n\nThe following example illustrates the two additional notions of degree.\n\n\\begin{example}\n$ Fr $ has separable degree $ \\deg_s\\rb{Fr} = 1 $ and inseparable degree $ \\deg_i\\rb{Fr} = p $, while $ \\sb{2} $ has separable degree $ \\deg_s\\rb{\\sb{2}} = \\deg\\rb{\\sb{2}} = 4 $ and inseparable degree $ \\deg_i\\rb{\\sb{2}} = 1 $.\n\\end{example}\n\nThis digression leads to an important proposition relating the kernel and the separable degree of an isogeny as follows, which is crucial to the proof of Hasse's theorem.\n\n\\begin{proposition}\n$ \\abs{Ker\\rb{\\phi}} = \\deg_s\\rb{\\phi} $.\n\\end{proposition}\n\n\\begin{proof}\nLet\n$$ S_1 = \\cb{\\rb{a, 0} \\in E'} = E\\sb{2}, \\qquad S_2 = \\cb{\\rb{0, b} \\in E'}, \\qquad S_3 = \\cb{\\rb{a, b} \\in E' \\mid \\deg\\rb{r - as} < \\deg\\rb{\\phi_s}}, $$\n$$ S_4 = \\cb{\\rb{a, b} \\in E' \\ \\Bigg| \\ \\rb{\\dfrac{r}{s}}\\rb{a'} = a, \\ \\dfrac{d}{dx}\\rb{\\dfrac{r}{s}}\\rb{a'} = 0, \\ \\rb{a', b'} \\in E}, \\qquad S = S_1 \\cup S_2 \\cup S_3 \\cup S_4. $$\nThen $ \\abs{S_1} \\le 3 $ and $ \\abs{S_2} \\le 2 $ are finite. Since $ \\deg\\rb{\\phi_s} $ is finite, it holds that $ \\abs{S_3} \\le 2\\deg\\rb{\\phi_s} $ is also finite. Since $ \\phi_s $ is separable, it holds that $ d\\rb{r / s} / dx \\ne 0 $, so $ \\abs{S_4} \\le \\deg\\rb{r} $ is also finite. Hence $ S $ is finite and $ E' \\setminus S $ is non-empty. Now let $ P = \\rb{a, b} \\in E' \\setminus S $ and $ P' = \\rb{a', b'} \\in E $ be points, and let $ \\psi = r - as \\in K\\sb{x} $ be such that $ \\deg\\rb{\\psi} = \\deg\\rb{\\phi_s} $. Then $ \\phi_s\\rb{P'} = P $ iff $ \\rb{r / s}\\rb{a'} = a $ and $ \\rb{u / v}\\rb{a'}b' = b $. Since $ b \\ne 0 $ gives $ u\\rb{a'} \\ne 0 $, this also holds iff $ \\psi\\rb{a'} = r\\rb{a'} - as\\rb{a'} = 0 $ and $ b' = \\rb{v / u}\\rb{a'}b $. Hence $ \\abs{\\phi_s^{-1}\\rb{P}} $ is the number of distinct roots of $ \\psi $. Suppose for a contradiction that $ a' $ is a repeated root of $ \\psi $. Then\n$$ 0 = \\psi\\rb{a'} = r\\rb{a'} - as\\rb{a'}, \\qquad 0 = \\dfrac{d\\psi}{dx}\\rb{a'} = \\dfrac{dr}{dx}\\rb{a'} - a\\dfrac{ds}{dx}\\rb{a'}, $$\nsuch that\n$$ \\rb{\\dfrac{r}{s}}\\rb{a'} = a, \\qquad \\dfrac{dr}{dx}\\rb{a'}s\\rb{a'} = \\dfrac{ds}{dx}\\rb{a'}r\\rb{a'} \\qquad \\implies \\qquad \\dfrac{d}{dx}\\rb{\\dfrac{r}{s}}\\rb{a'} = 0, $$\nso $ P' \\in S_4 $, which is a contradiction. Hence $ \\psi $ splits over $ K $ and $ \\abs{\\phi_s^{-1}\\rb{P}} = \\deg\\rb{\\psi} $. Since $ \\chi : Ker\\rb{\\phi_s} \\to \\phi_s^{-1}\\rb{P} $ defined by $ \\chi\\rb{Q} = Q + P $ is a bijection, it holds that $ \\abs{Ker\\rb{\\phi_s}} = \\abs{\\phi_s^{-1}\\rb{P}} $. Since $ Fr $ is bijective, so are $ Fr^n $ and $ \\val{Fr^n}_{Ker\\rb{\\phi}} : Ker\\rb{\\phi} \\to Ker\\rb{\\phi_s} $, so $ \\abs{Ker\\rb{\\phi}} = \\abs{Ker\\rb{\\phi_s}} $. Thus\n$$ \\abs{Ker\\rb{\\phi}} = \\abs{Ker\\rb{\\phi_s}} = \\abs{\\phi_s^{-1}\\rb{P}} = \\deg\\rb{\\psi} = \\deg\\rb{\\phi_s} = \\deg_s\\rb{\\phi}. $$\n\\end{proof}\n\nMotivated by the endomorphism ring, composition of isogenies with appropriate domains can be seen as multiplication. In particular, their degrees multiply out naturally in the following lemma.\n\n\\begin{lemma}\nLet $ E'' $ be an elliptic curve over $ F $ such that $ \\psi : E' \\to E'' $ is an isogeny. Then\n$$ \\deg\\rb{\\psi \\circ \\phi} = \\deg\\rb{\\psi}\\deg\\rb{\\phi}, \\qquad \\deg_s\\rb{\\psi \\circ \\phi} = \\deg_s\\rb{\\psi}\\deg_s\\rb{\\phi}, \\qquad \\deg_i\\rb{\\psi \\circ \\phi} = \\deg_i\\rb{\\psi}\\deg_i\\rb{\\phi}. $$\n\\end{lemma}\n\n\\begin{proof}\nSince $ \\phi $ and $ \\psi $ are surjective, so is $ \\psi \\circ \\phi $, so the first isomorphism theorem gives\n$$ \\dfrac{E}{Ker\\rb{\\phi}} \\cong E', \\qquad \\dfrac{E'}{Ker\\rb{\\psi}} \\cong E'', \\qquad \\dfrac{E}{Ker\\rb{\\psi \\circ \\phi}} \\cong E'', $$\nsuch that\n$$ \\abs{Ker\\rb{\\psi \\circ \\phi}} = \\dfrac{\\abs{E}}{\\abs{E''}} = \\dfrac{\\abs{E'}\\abs{Ker\\rb{\\phi}}}{\\abs{E'} / \\abs{Ker\\rb{\\psi}}} = \\abs{Ker\\rb{\\psi}}\\abs{Ker\\rb{\\phi}}. $$\nHence $ \\deg_s\\rb{\\psi \\circ \\phi} = \\deg_s\\rb{\\psi}\\deg_s\\rb{\\phi} $. Now let $ \\psi = \\psi_s \\circ Fr^m $ and $ \\psi \\circ \\phi = \\chi_s \\circ Fr^k $ for some isogenies $ \\psi_s : E' \\to E'' $ and $ \\chi_s : E \\to E'' $ and some $ m, k \\in \\Z_{\\ge 0} $. Then\n$$ \\chi_s \\circ Fr^k = \\psi_s \\circ Fr^m \\circ \\phi_s \\circ Fr^n. $$\nSince $ \\deg_s\\rb{Fr} = 1 $, it holds that $ \\deg_s\\rb{Fr^m} = 1 $, so\n$$ \\deg_s\\rb{Fr^m \\circ \\phi_s} = \\deg_s\\rb{Fr^m}\\deg_s\\rb{\\phi_s} = \\deg_s\\rb{\\phi_s}. $$\nThen $ Fr^m \\circ \\phi_s = \\chi'_s \\circ Fr^m $ for some isogeny $ \\chi'_s : E \\to E' $ such that $ \\deg\\rb{\\phi_s} = \\deg\\rb{\\chi'_s} $, so\n$$ \\chi_s \\circ Fr^k = \\psi_s \\circ \\chi'_s \\circ Fr^{n + m}. $$\nSince $ \\psi_s \\circ \\chi'_s $ is separable, it holds that $ k = n + m $. Hence $ \\deg_i\\rb{\\psi \\circ \\phi} = \\deg_i\\rb{\\psi}\\deg_i\\rb{\\phi} $. Thus\n$$ \\deg\\rb{\\psi \\circ \\phi} = \\deg_s\\rb{\\psi}\\deg_i\\rb{\\psi}\\deg_s\\rb{\\phi}\\deg_i\\rb{\\phi} = \\deg\\rb{\\psi}\\deg\\rb{\\phi}. $$\n\\end{proof}\n\nThis paves the way to the proof of the following proposition on inseparable isogenies, which is also crucial to the proof of Hasse's theorem. Now let $ \\psi : E \\to E' $ be an isogeny.\n\n\\begin{proposition}\nLet $ \\phi $ and $ \\psi $ be inseparable, and let $ E'' $ and $ E''' $ be elliptic curves over $ F $ such that $ \\chi : E'' \\to E $ and $ \\chi' : E' \\to E''' $ are isogenies. Then $ \\phi \\circ \\chi $, $ \\chi' \\circ \\phi $, and $ \\phi - \\psi $ are inseparable.\n\\end{proposition}\n\n\\begin{proof}\nSince $ \\deg_i\\rb{\\phi \\circ \\chi} = \\deg_i\\rb{\\phi}\\deg_i\\rb{\\chi} > 1 $ and $ \\deg_i\\rb{\\chi' \\circ \\phi} = \\deg_i\\rb{\\phi}\\deg_i\\rb{\\chi} > 1 $, it holds that $ \\phi \\circ \\chi $ and $ \\chi' \\circ \\phi $ are inseparable. Now let $ \\phi = \\phi_s \\circ Fr^n $ and $ \\psi = \\psi_s \\circ Fr^m $ for some separable isogenies $ \\phi_s : E \\to E' $ and $ \\psi_s : E \\to E' $ and some $ n, m \\in \\Z_{> 0} $. Then\n$$ \\phi - \\psi = \\phi_s \\circ Fr^n - \\psi_s \\circ Fr^m = \\rb{\\phi_s \\circ Fr^{n - 1} - \\psi_s \\circ Fr^{m - 1}} \\circ Fr. $$\nThus $ \\phi - \\psi $ is inseparable.\n\\end{proof}\n\nHence adding a separable isogeny with an inseparable isogeny will give a separable isogeny. Returning to the initial motivation, letting $ E = E' = E'' = E''' $ in the above results implies that the set of all inseparable endomorphisms of $ E $ is an ideal of $ End\\rb{E} $.\n\n\\pagebreak\n\n\\subsection{Hasse's theorem: quadratic forms}\n\nNow the \\emph{degree map} $ \\deg : Hom\\rb{E, E'} \\to \\Z_{\\ge 0} $ has a particular property that allows a form of the Cauchy-Schwarz inequality to be defined on it. This property can be defined with the aid of the following notion.\n\n\\begin{definition}[Bilinear pairing]\nA pairing $ b : G \\times G \\to F $ of an group $ G $ is \\textbf{bilinear} iff $ b\\rb{x + y, z} = b\\rb{x, z} + b\\rb{y, z} $ and $ b\\rb{x, y + z} = b\\rb{x, y} + b\\rb{x, z} $ for any $ x, y, z \\in G $.\n\\end{definition}\n\nIn other words the pairing is linear in both components. A bilinear pairing can be defined in terms of the degree map, with the set of isogenies $ Hom\\rb{E, E'} $ as the abelian group, which has the following property.\n\n\\begin{definition}[Quadratic form]\nA \\textbf{quadratic form} is a map $ d : A \\to F $ of an abelian group $ A $ such that $ d\\rb{x} = d\\rb{-x} $ for any $ x \\in A $, and the \\textbf{associated pairing} $ b_d : A \\times A \\to F $ defined by\n$$ b_d\\rb{x, y} = \\tfrac{1}{2}\\rb{d\\rb{x + y} - d\\rb{x} - d\\rb{y}} $$\nis bilinear.\n\\end{definition}\n\nThe associated bilinear pairing is usually written $ \\ab{\\cdot, \\cdot} : A \\times A \\to F $ with context, and inherits all the definitions from linear algebra, such as the notions of being symmetric and positive definite.\n\n\\begin{remark}\nConversely, for any symmetric bilinear pairing $ \\ab{\\cdot, \\cdot} : A \\times A \\to F $, the map $ d : A \\to F $ defined by $ d\\rb{x} = \\ab{x, x} $ is a quadratic form, so notions related to symmetric bilinear pairings and quadratic forms are interchangeable, provided $ char\\rb{F} \\ne 2 $.\n\\end{remark}\n\nHence an aim would be to show that the degree map indeed is a positive definite quadratic form, as symmetry follows by definition. This could be done by proving a particular fundamental property that holds for all quadratic forms, which is given in the following theorem. Now denote $ -\\phi = \\sb{-1} \\circ \\phi $ and\n$$ \\phi + \\dots + \\phi = n\\phi = \\sb{n} \\circ \\phi, \\qquad \\psi + \\dots + \\psi = m\\psi = \\sb{m} \\circ \\psi, \\qquad n, m \\in \\Z, $$\nto ease the proofs below.\n\n\\begin{theorem}[Parallelogram law]\n$ \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\phi - \\psi} = 2\\deg\\rb{\\phi} + 2\\deg\\rb{\\psi} $.\n\\end{theorem}\n\n\\begin{proof}\nIf $ \\phi = 0 $ or $ \\psi = 0 $, then $ \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\phi - \\psi} = 2\\deg\\rb{\\phi} + 2\\deg\\rb{\\psi} $ holds. If $ \\phi = \\psi $ or $ \\phi = -\\psi $, then\n$$ \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\phi - \\psi} = \\deg\\rb{2\\phi} = \\deg\\rb{\\sb{2}}\\deg\\rb{\\phi} = 4\\deg\\rb{\\phi} = 2\\deg\\rb{\\phi} + 2\\deg\\rb{\\psi} $$\nalso holds. Otherwise let\n$$ \\phi\\rb{x, y} = \\rb{w_1, z_1}, \\qquad \\psi\\rb{x, y} = \\rb{w_2, z_2}, \\qquad \\rb{\\phi + \\psi}\\rb{x, y} = \\rb{w_3, z_3}, \\qquad \\rb{\\phi - \\psi}\\rb{x, y} = \\rb{w_4, z_4}, $$\nfor each $ w_i = r_i\\rb{x} / s_i\\rb{x} $ and $ z_i = u_i\\rb{x}y / v_i\\rb{x} $ for some homogeneous polynomials $ r_i, s_i, u_i, v_i \\in F\\sb{x} $ such that each $ \\gcd\\rb{r_i, s_i} = \\gcd\\rb{u_i, v_i} = 1 $ and\n$$ \\deg\\rb{\\phi} = \\deg\\rb{r_1} = \\deg\\rb{s_1}, \\qquad \\deg\\rb{\\psi} = \\deg\\rb{r_2} = \\deg\\rb{s_2}, $$\n$$ \\deg\\rb{\\phi + \\psi} = \\deg\\rb{r_3} = \\deg\\rb{s_3}, \\qquad \\deg\\rb{\\phi - \\psi} = \\deg\\rb{r_4} = \\deg\\rb{s_4}. $$\nBy the addition formula,\n$$ w_3 = \\dfrac{\\rb{A + w_1w_2}\\rb{w_1 + w_2} + 2\\rb{B - z_1z_2}}{\\rb{w_1 - w_2}^2}, \\qquad w_4 = \\dfrac{\\rb{A + w_1w_2}\\rb{w_1 + w_2} + 2\\rb{B + z_1z_2}}{\\rb{w_1 - w_2}^2}. $$\nAdding these two equations gives $ \\rb{w_3 + w_4}\\rb{w_1 - w_2}^2 = 2\\rb{A + w_1w_2}\\rb{w_1 + w_2} + 4B $, so\n$$ \\dfrac{r_3s_4 + r_4s_3}{s_3s_4} = \\dfrac{2\\rb{As_1s_2 + r_1r_2}\\rb{r_1s_2 + r_2s_1} + 4Bs_1^2s_2^2}{\\rb{r_1s_2 - r_2s_1}^2}. $$\nHence let\n$$ R = r_3s_4 + r_4s_3, \\qquad S = s_3s_4, \\qquad U = 2\\rb{As_1s_2 + r_1r_2}\\rb{r_1s_2 + r_2s_1} + 4Bs_1^2s_2^2, \\qquad V = \\rb{r_1s_2 - r_2s_1}^2. $$\nSimilarly multiplying these two equations gives\n\\begin{align*}\nw_3w_4\\rb{w_1 - w_2}^4\n& = \\rb{A + w_1w_2}^2\\rb{w_1 + w_2}^2 + 4B\\rb{A + w_1w_2}\\rb{w_1 + w_2} + 4B^2 - 4z_1^2z_2^2 \\\\\n& = \\rb{A^2 + 2Aw_1w_2 + w_1^2w_2^2}\\rb{w_1^2 + 2w_1w_2 + w_2^2} + 4B\\rb{Aw_1 + Aw_2 + w_1^2w_2 + w_1w_2^2} \\\\\n& \\qquad + 4B^2 - 4\\rb{w_1^3 + Aw_1 + B}\\rb{w_2^3 + Aw_2 + B} \\\\\n& = A^2w_1^2 - 2A^2w_1w_2 + A^2w_2^2 - 4Bw_1^3 + 4Bw_1^2w_2 + 4Bw_1w_2^2 - 4Bw_2^3 \\\\\n& \\qquad - 2Aw_1^3w_2 + 4Aw_1^2w_2^2 - 2Aw_1w_2^3 + w_1^4w_2^2 - 2w_1^3w_2^3 + w_1^2w_2^4 \\\\\n& = A^2\\rb{w_1 - w_2}^2 - 4Bw_1^2\\rb{w_1 - w_2} + 4Bw_2^2\\rb{w_1 - w_2} \\\\\n& \\qquad - 2Aw_1w_2\\rb{w_1 - w_2}^2 + w_1^2w_2^2\\rb{w_1 - w_2}^2 \\\\\n& = \\rb{A^2 - 2Aw_1w_2 + w_1^2w_2^2}\\rb{w_1 - w_2}^2 - 4B\\rb{w_1^2 - w_2^2}\\rb{w_1 - w_2} \\\\\n& = \\rb{A - w_1w_2}^2\\rb{w_1 - w_2}^2 - 4B\\rb{w_1 + w_2}\\rb{w_1 - w_2}^2,\n\\end{align*}\nsuch that $ w_3w_4\\rb{w_1 - w_2}^2 = \\rb{A - w_1w_2}^2 - 4B\\rb{w_1 + w_2} $, so\n$$ \\dfrac{r_3r_4}{s_3s_4} = \\dfrac{\\rb{As_1s_2 - r_1r_2}^2 - 4B\\rb{r_1s_2 + r_2s_1}s_1s_2}{\\rb{r_1s_2 - r_2s_1}^2}. $$\nHence also let\n$$ T = r_3r_4, \\qquad W = \\rb{As_1s_2 - r_1r_2}^2 - 4B\\rb{r_1s_2 + r_2s_1}s_1s_2, $$\nsuch that\n$$ \\deg\\rb{R} = \\deg\\rb{S} = \\deg\\rb{T} = \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\phi - \\psi}, $$\n$$ \\deg\\rb{U} = \\deg\\rb{V} = \\deg\\rb{W} = 2\\deg\\rb{\\phi} + 2\\deg\\rb{\\psi}. $$\nSuppose for a contradiction that $ \\gcd\\rb{R, S, T} \\ne 1 $, so $ g \\mid \\gcd\\rb{R, S, T} $ for some irreducible homogeneous polynomial $ g \\in F\\sb{x} $. If $ g \\mid r_3 $, then $ g \\nmid s_3 $, so $ g \\mid s_4 $ and $ g \\nmid r_4 $ gives $ g \\nmid r_3s_4 + r_4s_3 = R $. Otherwise $ g \\nmid r_3 $, then $ g \\mid r_4 $, so $ g \\nmid s_4 $ and $ g \\mid s_3 $ also gives $ g \\nmid r_3s_4 + r_4s_3 = R $, which is a contradiction. Hence $ \\gcd\\rb{R, S, T} = 1 $. Now let $ g' = \\gcd\\rb{U, V, W} $, so\n$$ U = g'U', \\qquad V = g'V', \\qquad W = g'W', \\qquad U', V', W' \\in F\\sb{x}, \\qquad \\gcd\\rb{U', V', W'} = 1, $$\nsuch that\n$$ \\deg\\rb{U'} = \\deg\\rb{V'} = \\deg\\rb{W'} = \\deg\\rb{U} - \\deg\\rb{g'}. $$\nCombining the two equations from adding and multiplying gives a ratio\n$$ \\sb{R, S, T} = \\sb{\\dfrac{R}{S}, 1, \\dfrac{T}{S}} = \\sb{\\dfrac{U}{V}, 1, \\dfrac{W}{V}} = \\sb{U, V, W} = \\sb{g'U', g'V', g'W'} = \\sb{U', V', W'}, $$\nsuch that $ R = U' $, $ T = W' $, and $ S = V' $. Hence\n$$ \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\phi - \\psi} = \\deg\\rb{R} = \\deg\\rb{U'} = \\deg\\rb{U} - \\deg\\rb{g'} \\le \\deg\\rb{U} = 2\\deg\\rb{\\phi} + 2\\deg\\rb{\\psi}. $$\nNow replacing $ \\rb{\\phi, \\psi} \\mapsto \\rb{\\phi + \\psi, \\psi + \\phi} $ gives the converse\n\\begin{align*}\n2\\deg\\rb{\\phi + \\psi} + 2\\deg\\rb{\\phi - \\psi}\n& \\ge \\deg\\rb{\\phi + \\psi + \\phi - \\psi} + \\deg\\rb{\\phi + \\psi - \\phi + \\psi} \\\\\n& = \\deg\\rb{\\sb{2}}\\deg\\rb{\\phi} + \\deg\\rb{\\sb{2}}\\deg\\rb{\\psi} \\\\\n& = 4\\deg\\rb{\\phi} + 4\\deg\\rb{\\psi},\n\\end{align*}\nThus $ \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\phi - \\psi} = 2\\deg\\rb{\\phi} + 2\\deg\\rb{\\psi} $.\n\\end{proof}\n\nAn application of the parallelogram law would be a simple inductive proof of the following lemma, which has many other proofs.\n\n\\begin{lemma}\nLet $ n \\in \\Z $. Then $ \\deg\\rb{\\sb{n}} = n^2 $.\n\\end{lemma}\n\n\\begin{proof}\n$ \\deg\\rb{\\sb{0}} = 0 $ and $ \\deg\\rb{\\sb{1}} = 1 $. Assume that $ \\deg\\rb{\\sb{m}} = m^2 $ for any $ m \\le n $ for some $ n \\in \\Z_{\\ge 0} $. Then\n$$ \\deg\\rb{\\sb{n + 1}} = 2\\deg\\rb{\\sb{n}} + 2\\deg\\rb{\\sb{1}} - \\deg\\rb{\\sb{n - 1}} = 2n^2 + 2 - \\rb{n - 1}^2 = n^2 + 2n + 1 = \\rb{n + 1}^2. $$\nHence $ \\deg\\rb{\\sb{n}} = n^2 $ for any $ n \\in \\Z_{\\ge 0} $ by induction. Similarly $ \\deg\\rb{\\sb{-n}} = \\deg\\rb{\\sb{-1}}\\deg\\rb{\\sb{n}} = \\deg\\rb{\\sb{n}} $ for any $ n \\in \\Z_{\\ge 0} $. Thus $ \\deg\\rb{\\sb{n}} = n^2 $ for any $ n \\in \\Z $.\n\\end{proof}\n\nIn fact, the above lemma can be generalised for arbitrary isogenies, as follows.\n\n\\begin{lemma}\nLet $ n, m \\in \\Z $. Then $ \\deg\\rb{n\\phi + m\\psi} = n^2\\deg\\rb{\\phi} + 2nm\\ab{\\phi, \\psi} + m^2\\deg\\rb{\\psi} $.\n\\end{lemma}\n\n\\begin{proof}\nSince $ \\ab{\\phi, \\phi} = \\tfrac{1}{2}\\rb{\\deg\\rb{2\\phi} - 2\\deg\\rb{\\phi}} = 2\\deg\\rb{\\phi} - \\deg\\rb{\\phi} = \\deg\\rb{\\phi} $, it holds that\n$$ \\deg\\rb{n\\phi + m\\psi} = \\ab{n\\phi + m\\psi, n\\phi + m\\psi} = n^2\\deg\\rb{\\phi} + 2nm\\ab{\\phi, \\psi} + m^2\\deg\\rb{\\psi}. $$\n\\end{proof}\n\nThe initial aim can then be proven in the following lemma.\n\n\\begin{lemma}\n$ \\deg : Hom\\rb{E, E'} \\to \\Z_{\\ge 0} $ is a positive definite quadratic form.\n\\end{lemma}\n\n\\begin{proof}\n$ \\deg\\rb{-\\phi} = \\deg\\rb{\\sb{-1}}\\deg\\rb{\\phi} = \\deg\\rb{\\phi} $. Let $ \\chi : E \\to E' $ be an isogeny. Since\n\\begin{align*}\n\\deg\\rb{\\phi + \\psi + \\chi}\n& = 2\\deg\\rb{\\phi + \\psi} + 2\\deg\\rb{\\chi} - \\deg\\rb{\\phi + \\psi - \\chi} \\\\\n& = 2\\deg\\rb{\\phi + \\psi} + 2\\deg\\rb{\\chi} - 2\\deg\\rb{\\phi - \\chi} - 2\\deg\\rb{\\psi} + \\deg\\rb{\\phi - \\psi - \\chi} \\\\\n& = 2\\deg\\rb{\\phi + \\psi} + 2\\deg\\rb{\\chi} - 2\\deg\\rb{\\phi - \\chi} - 2\\deg\\rb{\\psi} \\\\\n& \\qquad + 2\\deg\\rb{\\psi + \\chi} + 2\\deg\\rb{\\phi} - \\deg\\rb{\\phi + \\psi + \\chi},\n\\end{align*}\nit holds that $ \\deg\\rb{\\phi + \\psi + \\chi} = \\deg\\rb{\\phi + \\psi} + \\deg\\rb{\\chi} - \\deg\\rb{\\phi - \\chi} - \\deg\\rb{\\psi} + \\deg\\rb{\\psi + \\chi} + \\deg\\rb{\\phi} $.\nHence\n\\begin{align*}\n\\ab{\\phi + \\psi, \\chi}\n& = \\tfrac{1}{2}\\rb{\\deg\\rb{\\phi + \\psi + \\chi} - \\deg\\rb{\\phi + \\psi} - \\deg\\rb{\\chi}} \\\\\n& = \\tfrac{1}{2}\\rb{-\\deg\\rb{\\phi - \\chi} - \\deg\\rb{\\psi} + \\deg\\rb{\\psi + \\chi} + \\deg\\rb{\\phi}} \\\\\n& = \\tfrac{1}{2}\\rb{-2\\deg\\rb{\\phi} - 2\\deg\\rb{\\chi} + \\deg\\rb{\\phi + \\chi} - \\deg\\rb{\\psi} + \\deg\\rb{\\psi + \\chi} + \\deg\\rb{\\phi}} \\\\\n& = \\tfrac{1}{2}\\rb{\\deg\\rb{\\phi + \\chi} - \\deg\\rb{\\phi} - \\deg\\rb{\\chi} + \\deg\\rb{\\psi + \\chi} - \\deg\\rb{\\psi} - \\deg\\rb{\\chi}}\n= \\ab{\\phi, \\chi} + \\ab{\\psi, \\chi}.\n\\end{align*}\nSimilarly $ \\ab{\\phi, \\psi + \\chi} = \\ab{\\phi, \\psi} + \\ab{\\phi, \\chi} $ by symmetry. Thus since $ \\deg\\rb{\\phi} > 0 $ for any $ \\phi \\ne 0 $ and $ \\deg\\rb{0} = 0 $, it holds that $ \\deg $ is a positive definite quadratic form.\n\\end{proof}\n\nReplacing the degree map with any map satisfying the parallelogram law also gives a quadratic form. The following variant of the Cauchy-Schwarz inequality generalises to quadratic forms similarly.\n\n\\begin{theorem}[Cauchy-Schwarz]\n$ \\ab{\\phi, \\psi}^2 \\le \\deg\\rb{\\phi}\\deg\\rb{\\psi} $.\n\\end{theorem}\n\n\\begin{proof}\nLet $ n = -\\ab{\\phi, \\psi} $ and $ m = \\deg\\rb{\\phi} $. Then\n$$ 0 \\le \\ab{\\phi, \\psi}^2\\deg\\rb{\\phi} - 2\\ab{\\phi, \\psi}^2\\deg\\rb{\\phi} + \\deg\\rb{\\phi}^2\\deg\\rb{\\psi} = \\deg\\rb{\\phi}\\rb{\\deg\\rb{\\phi}\\deg\\rb{\\psi} - \\ab{\\phi, \\psi}^2}. $$\nThus $ \\ab{\\phi, \\psi}^2 \\le \\deg\\rb{\\phi}\\deg\\rb{\\psi} $.\n\\end{proof}\n\nHasse's theorem can finally be proven.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:hasse}]\nA point $ P = \\sb{a, b, c} \\in E\\rb{F} $ iff $ a^q = a $, $ b^q = b $, and $ c^q = c $ by Fermat's little theorem, or $ \\sb{a^q, b^q, c^q} = \\sb{a, b, c} $. This holds iff the $ q $-th power Frobenius endomorphism $ Fr_q : E \\to E $ is such that $ Fr_q\\rb{P} = P $, or $ P \\in Ker\\rb{Fr_q - \\sb{1}} $. Hence $ E\\rb{F} = Ker\\rb{Fr_q - \\sb{1}} $. Since $ \\sb{1} $ is separable and $ Fr_q $ is inseparable with degree $ \\deg\\rb{Fr_q} = \\deg_i\\rb{Fr_q} = q $, it holds that $ Fr_q - \\sb{1} $ is separable, so\n$$ Ker\\rb{Fr_q - \\sb{1}} = \\deg_s\\rb{Fr_q - \\sb{1}} = \\deg\\rb{Fr_q - \\sb{1}} = \\deg\\rb{Fr_q} - 2\\ab{Fr_q, \\sb{1}} + \\deg\\rb{\\sb{1}} = q - 2\\ab{Fr_q, \\sb{1}} + 1. $$\nThen let $ t = 2\\ab{Fr_q, \\sb{1}} $, so Cauchy-Schwarz gives $ t^2 = 4\\ab{Fr_q, 1}^2 \\le 4\\deg\\rb{Fr_q}\\deg\\rb{\\sb{1}} = 4q $. Thus $ \\abs{E\\rb{F}} = q - t + 1 $ for $ \\abs{t} \\le 2\\sqrt{q} $.\n\\end{proof}\n\n\\pagebreak\n\n\\subsection{Riemann hypothesis}\n\nHasse's theorem, or more accurately the Hasse-Weil theorem, is also sometimes referred to as the \\emph{Riemann hypothesis} for smooth projective algebraic curves over finite fields. It has an alternative formulation that makes it analogous to the famous classical Riemann hypothesis, an open problem in number theory deemed worthy of being called one of the Millennium Prize Problems by the Clay Mathematics Institute with a monetary prize of a million dollars. The conjecture revolves around zeroes of the following complex function.\n\n\\begin{definition}[Riemann zeta function]\nThe Riemann zeta function $ \\zeta : \\C \\to \\C $ is defined for any $ \\Re\\rb{s} > 1 $ as the power series $ \\zeta\\rb{s} = \\sum_{n = 1}^\\infty n^{-s} $, and extended to $ \\C $ by analytic continuation.\n\\end{definition}\n\nRiemann himself proved the analytic continuation, as well as a functional equation satisfied by the Riemann zeta function given by\n$$ \\xi\\rb{s} = \\xi\\rb{1 - s}, \\qquad \\xi\\rb{s} = \\tfrac{1}{2}\\sqrt{\\pi}^{-s}s\\rb{s - 1}\\Gamma\\rb{\\tfrac{1}{2}s}\\zeta\\rb{s}. $$\nThe conjecture is then formulated in \\cite{riemann} as follows.\n\n\\begin{conjecture}[Riemann]\nLet $ s \\in \\C $ be such that $ s \\notin -2\\Z_{> 0} $. If $ \\zeta\\rb{s} = 0 $, then $ \\Re\\rb{s} = \\tfrac{1}{2} $.\n\\end{conjecture}\n\nThe connection to this still open problem can be seen via a powerful theorem known as the \\emph{Weil conjectures}, proposed by Weil and proven in steps later by himself, Dwork, Deligne, Grothendieck, and many others. The so-called conjectures also involve a related zeta function encoding the number of rational points of a smooth projective algebraic varieties variety, which is defined as follows.\n\n\\begin{definition}[Local zeta function]\nThe \\textbf{local zeta function} of a projective algebraic variety $ V $ over $ F $ is the power series\n$$ Z_V\\rb{t} = \\exp\\rb{\\sum_{n = 1}^\\infty \\abs{V\\rb{F_n}}\\dfrac{t^n}{n}}, \\qquad \\abs{V\\rb{F_n}} = \\dfrac{1}{\\rb{n - 1}!}\\val{\\dfrac{d^n}{dt^n}\\ln\\rb{Z_V\\rb{t}}}_{t = 0}. $$\nwhere $ F_n = \\F_{q^n} $.\n\\end{definition}\n\nThe following example is a trivial application the local zeta function.\n\n\\begin{example}\nLet $ V\\rb{0} $ be the trivial projective algebraic variety over $ F $. Then $ \\abs{V\\rb{F_n}} = 1 $ for any $ n \\in \\Z_{> 0} $, so\n$$ Z_V\\rb{t} = \\exp\\rb{\\sum_{n = 1}^\\infty \\dfrac{t^n}{n}} = \\exp\\rb{\\ln\\rb{\\dfrac{1}{1 - t}}} = \\dfrac{1}{1 - t}. $$\n\\end{example}\n\nHis three conjectures are then formulated as follows, which are easily satisfied by the above example.\n\n\\begin{theorem}[Weil conjectures]\n\\label{thm:weil}\nLet $ V $ be a smooth projective algebraic variety over $ F $ of dimension $ n \\in \\Z_{\\ge 0} $.\n\\begin{itemize}\n\\item Rationality. $ Z_V\\rb{t} = P\\rb{t} / \\rb{1 - t}\\rb{1 - q^nt} \\in \\Q\\sb{t} $, where\n$$ P\\rb{t} = \\prod_{i = 1}^{2n - 1} P_i\\rb{t}^{\\rb{-1}^{i + 1}}, \\qquad P_i \\in \\Z\\sb{t}. $$\n\\item Functional equation. Let $ \\epsilon \\in \\Z $ be the \\emph{Euler characteristic} of $ V $. Then\n$$ Z_V\\rb{\\dfrac{1}{q^nt}} = \\pm \\sqrt{q}^{n\\epsilon} t^\\epsilon Z_V\\rb{t}. $$\n\\item Riemann hypothesis. Let $ S_i = \\cb{\\alpha \\in \\C \\mid \\abs{\\alpha} = \\sqrt{q}^i} $ and $ P_i $ be as per above. Then each\n$$ P_i\\rb{t} = \\prod_{\\alpha \\in S_i'} \\rb{1 - \\alpha t}, $$\nover some $ S_i' \\subseteq S_i $ such that $ P_i \\in \\Z\\rb{t} $.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{proof}\nOmitted, see \\cite{weil1}, \\cite{weil2}, and \\cite{weil3}.\n\\end{proof}\n\n\\pagebreak\n\n\\begin{remark}\nThere is a fourth Weil conjecture on \\emph{Betti numbers} that states if $ V $ is a \\emph{reduction modulo $ q $} of a smooth projective algebraic variety $ W $ over a number field, then $ \\deg\\rb{P_i} $ is the $ i^{th} $ topological Betti number of $ W $ for each $ P_i $.\n\\end{remark}\n\nIn the special case where $ V $ is a smooth projective algebraic curve $ C $ of genus $ g_C $, its dimension is $ n = 1 $ and its Euler characteristic is $ \\epsilon = 2 - 2g_C $, which greatly simplifies Theorem \\ref{thm:weil}. The following is a formulation for the elliptic curve $ E $ of genus one.\n\n\\begin{theorem}[Weil conjectures for elliptic curves]\n\\label{thm:weilcurve}\n$ Z_E $ satisfies the following properties.\n\\begin{itemize}\n\\item Rationality. $ Z_E\\rb{t} = P\\rb{t} / \\rb{1 - t}\\rb{1 - qt} \\in \\Q\\rb{t} $ for some $ P \\in \\Z\\rb{t} $.\n\\item Functional equation. $ Z_E\\rb{1 / qt} = \\pm Z_E\\rb{t} $.\n\\item Riemann hypothesis. $ P\\rb{t} = \\prod_\\alpha \\rb{1 - \\alpha t} $ for some $ \\alpha \\in \\C $ such that $ \\abs{\\alpha} = \\sqrt{q} $ and $ P \\in \\Z\\rb{t} $.\n\\end{itemize}\n\\end{theorem}\n\nAs full proofs of the Weil conjectures, even just for elliptic curves, requires further prerequisites on algebraic geometry, particularly on the \\emph{Tate module} and the \\emph{Weil pairing}, only the final part of the proof is given, of which the following lemma will be assumed.\n\n\\begin{lemma}\n$ \\abs{E\\rb{F_n}} = 1 + q^n - \\alpha^n - \\overline{\\alpha}^n $ for some $ \\alpha \\in \\C $ such that $ \\abs{\\alpha} = \\sqrt{q} $.\n\\end{lemma}\n\n\\begin{proof}\nOmitted, see V.2.3 in \\cite{gtm}.\n\\end{proof}\n\nLetting $ n = 1 $ in the above lemma for Theorem \\ref{thm:weilcurve} gives $ \\abs{\\abs{E\\rb{F}} - 1 - q} = \\abs{-\\alpha - \\overline{\\alpha}} \\le 2\\abs{\\alpha} = 2\\sqrt{q} $, which proves Hasse's theorem once again. The final part of the proof is as follows.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:weilcurve}]\nThe above lemma on the zeta function gives $ \\alpha \\in \\C $ such that $ \\abs{\\alpha} = \\sqrt{q} $, and\n$$ \\ln\\rb{Z_E\\rb{t}} = \\sum_{n = 1}^\\infty \\rb{1 + q^n - \\alpha^n - \\overline{\\alpha}^n}\\dfrac{t^n}{n} = -\\ln\\rb{1 - t} - \\ln\\rb{1 - qt} + \\ln\\rb{1 - \\alpha t} + \\ln\\rb{1 - \\overline{\\alpha} t}. $$\nThus\n$$ Z_E\\rb{t} = \\dfrac{\\rb{1 - \\alpha t}\\rb{1 - \\overline{\\alpha}t}}{\\rb{1 - t}\\rb{1 - qt}}, \\qquad \\abs{\\alpha} = \\sqrt{q}, $$\nwhich satisfies rationality and the Riemann hypothesis, and gives the functional equation\n$$ Z_E\\rb{\\dfrac{1}{qt}} = \\dfrac{\\rb{1 - \\dfrac{\\alpha}{qt}}\\rb{1 - \\dfrac{\\overline{\\alpha}}{qt}}}{\\rb{1 - \\dfrac{1}{qt}}\\rb{1 - \\dfrac{1}{t}}} = \\dfrac{qt^2 - \\rb{\\alpha + \\overline{\\alpha}}t + \\dfrac{\\alpha\\overline{\\alpha}}{q}}{\\rb{qt - 1}\\rb{t - 1}} = \\dfrac{\\rb{1 - \\alpha t}\\rb{1 - \\overline{\\alpha}t}}{\\rb{1 - t}\\rb{1 - qt}} = Z_E\\rb{t}. $$\n\\end{proof}\n\nBy the above proof, the connection to the classical Riemann hypothesis can then be seen as follows. An analogue of the Riemann zeta function can be defined for elliptic curves over $ F $ as $ \\zeta_E\\rb{s} = Z_E\\rb{q^{-s}} $. It then satisfies a similar functional equation,\n$$ \\zeta_E\\rb{s} = Z_E\\rb{q^{-s}} = Z_E\\rb{q^{s - 1}} = \\zeta_E\\rb{1 - s}. $$\nIf $ \\zeta_E\\rb{s} = 0 $, Theorem \\ref{thm:weilcurve} also gives\n$$ \\dfrac{\\rb{1 - \\alpha q^{-s}}\\rb{1 - \\overline{\\alpha}q^{-s}}}{\\rb{1 - q^{-s}}\\rb{1 - q^{1 - s}}} = 0, \\qquad \\abs{\\alpha} = \\sqrt{q}. $$\nHence $ 1 = \\alpha q^{-s} $ or $ 1 = \\overline{\\alpha} q^{-s} $, so $ q^{\\Re\\rb{s}} = \\abs{q^s} = \\sqrt{q} $. Thus $ \\Re\\rb{s} = \\tfrac{1}{2} $.\n\n\\begin{remark}\nThe Weil conjectures is a generalisation of Riemann hypothesis, which those for elliptic curves is in turn a special case of. In general, there are many zeta functions analogous to the Riemann zeta function. One such family of zeta functions is for a \\emph{finitely generated algebra} $ R $ over $ \\Z $, defined as\n$$ \\zeta_R\\rb{s} = \\prod_M \\dfrac{1}{1 - \\abs{R / M}^{-s}}, $$\nover all maximal ideals $ M \\subset R $.\n\\end{remark}\n\n\\pagebreak\n\n\\subsection{Schoof's algorithm}\n\nIn light of Hasse's theorem, there were improved algorithms to compute $ E\\rb{F} $ similar to the naive approach described in a previous subsection. Lagrange's theorem gives that $ ord\\rb{P} \\mid \\abs{E\\rb{F}} $ for any point $ P \\in E\\rb{F} $, the latter of which is bounded by Hasse's theorem. After obtaining a random point $ P \\in E\\rb{F} $ by inspection or otherwise, simply try all values of $ n \\in \\Z $ such that $ q - 2\\sqrt{q} + 1 \\le n \\le q + 2\\sqrt{q} + 1 $ to catch whenever $ nP = \\mathcal{O} $. If this $ n $ is unique, the point $ P $ is a generator of $ E\\rb{F} $ and hence $ \\abs{E\\rb{F}} = ord\\rb{P} = n $. Otherwise obtain a different random point $ P \\in E\\rb{F} $ and repeat. This process can be illustrated with a prior example.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + x + 1 $ be an elliptic curve over $ \\F_5 $ and $ P = \\rb{0, 1} \\in E\\rb{\\F_5} $ be a point. Hasse's theorem gives $ \\abs{E\\rb{\\F_5}} = 5 - t + 1 $ for some $ \\abs{t} \\le 2\\sqrt{5} $, so $ \\abs{E\\rb{\\F_5}} \\in \\cb{2, \\dots, 10} $. Then the addition formula gives only $ 9P = \\mathcal{O} $, so $ \\abs{E\\rb{\\F_5}} = 9 $.\n\\end{example}\n\nThere is then room for algorithms like \\emph{baby-step giant-step} that trades a space complexity of $ O\\rb{\\sqrt{q}} $ for a time complexity of also $ O\\rb{\\sqrt{q}} $, speeding up the computation further. However, discussions here will be on a different algorithm for computing $ \\abs{E\\rb{F}} $, which also builds upon Hasse's theorem. A high-level description of the \\emph{deterministic polynomial time} algorithm is as follows.\n\n\\begin{algorithm}[Schoof's algorithm]\nInput: an elliptic curve $ E $ over $ \\F_q $. Output: $ \\abs{E\\rb{\\F_q}} $.\n\\begin{enumerate}\n\\item Generate a set $ S $ of distinct primes excluding $ p $ with product $ N \\in \\Z_{> 0} $, such that $ N > 4\\sqrt{q} $.\n\\item Compute $ t \\mod n $ for each $ n \\in S $.\n\\item Obtain $ t \\mod N $ from each $ t \\mod n $.\n\\item Reduce $ t $ into a value between $ -2\\sqrt{q} $ and $ 2\\sqrt{q} $.\n\\item Calculate $ \\abs{E\\rb{F}} = q - t + 1 $.\n\\end{enumerate}\n\\end{algorithm}\n\nThe proof of this algorithm will be done in reverse. The first and last two steps will be made clear later, but several results will be proven for the second and third. In particular, the former generates a system of prime congruences for the latter, which in turn employs a classical theorem in number theory as follows.\n\n\\begin{theorem}[Chinese remainder]\nLet $ n_1, \\dots, n_k \\in \\Z_{> 1} $ be pairwise coprime with product $ N \\in \\Z_{> 0} $, and let $ t_1, \\dots, t_k \\in \\Z $. Then there is a unique $ t \\in \\Z_{\\ge 0} $ such that $ t < N $ and each $ t \\equiv t_i \\mod n_i $.\n\\end{theorem}\n\n\\begin{proof}\nLet $ k = 2 $. B\\'ezout's identity gives $ m_1n_1 + m_2n_2 = 1 $ for some $ m_i \\in \\Z $. Let $ t' = t_2m_1n_1 + t_1m_2n_2 $, so\n$$ t' = \\rb{t_2 - t_1}m_1n_1 + t_1\\rb{m_1n_1 + m_2n_2} \\equiv t_1 \\mod n_1, $$\n$$ t' = t_2\\rb{m_1n_1 + m_2n_2} - \\rb{t_2 - t_1}m_2n_2 \\equiv t_2 \\mod n_2. $$\nIf $ t'' \\in \\Z $ is such that $ t'' \\equiv t_1 \\mod n_1 $ and $ t'' \\equiv t_2 \\mod n_2 $, then $ t' \\equiv t'' \\mod n_1 $ and $ t' \\equiv t'' \\mod n_2 $, so $ n_1 \\mid t' - t'' $ and $ n_2 \\mid t' - t'' $. Then $ N = n_1n_2 \\mid t' - t'' $, so $ t' \\equiv t'' \\mod N $ and $ t' $ is unique up to congruences. Hence division gives a unique $ t \\in \\Z_{\\ge 0} $ such that $ t < N $ and $ t \\equiv t' \\mod N $. Now let $ k \\in \\Z_{\\ge 2} $ with product $ N_k \\in \\Z_{> 0} $ and assume that there is a unique $ t' \\in \\Z_{\\ge 0} $ such that $ t' < N_k $ and each $ t' \\equiv t_i \\mod n_i $. Since $ N_k $ and $ n_{k + 1} $ are coprime, the case $ k = 2 $ gives a unique $ t \\in \\Z_{\\ge 0} $ such that $ t < N_kn_{k + 1} $, and $ t \\equiv t' \\mod N_k $ and $ t \\equiv t_{k + 1} \\mod n_{k + 1} $. Thus the unique $ t \\in \\Z_{\\ge 0} $ holds by induction.\n\\end{proof}\n\n\\begin{remark}\nThe Chinese remainder theorem can be generalised to ideals $ I_i $ of arbitrary commutative unital rings $ R $, replacing the coprime condition with $ I_n + I_m = R $ for all $ n, m \\in \\Z $ and modulo with respect to $ I_i $.\n\\end{remark}\n\nA general process for computing this unique $ t \\in \\Z_{\\ge 0} $ can be inferred directly from the proof of the Chinese remainder theorem, using the extended Euclidean algorithm for B\\'ezout's identity, illustrated as follows.\n\n\\begin{example}\nLet\n$$ t \\equiv 1 \\mod 2, \\qquad t \\equiv 2 \\mod 3, \\qquad t \\equiv 3 \\mod 5 $$\nbe a system of congruences for $ t \\in \\Z_{\\ge 0} $. B\\'ezout's identity gives $ \\rb{-1}\\rb{2} + \\rb{1}\\rb{3} = 1 $, so let $ t' = 2\\rb{-1}\\rb{2} + 1\\rb{1}\\rb{3} = -1 $ be such that $ t' \\equiv 1 \\mod 2 $ and $ t' \\equiv 2 \\mod 3 $. Hence division gives $ t'' = 1\\rb{6} + \\rb{-1} = 5 < 6 $ such that $ t'' \\equiv t' \\mod 6 $. Similarly B\\'ezout's identity gives $ \\rb{1}\\rb{6} + \\rb{-1}\\rb{5} = 1 $, so let $ t''' = 3\\rb{1}\\rb{6} + 5\\rb{-1}\\rb{5} = -7 $ be such that $ t''' \\equiv 5 \\mod 6 $ and $ t''' \\equiv 3 \\mod 5 $. Thus division gives $ t = 1\\rb{30} + \\rb{-7} = 23 < 30 $ such that $ t \\equiv t''' \\mod 30 $ similarly.\n\\end{example}\n\nFor the rest of this section, let $ S $ be as in the first step of Schoof's algorithm and $ n \\in S $ be a prime. Now invoking the Chinese remainder theorem on the system of congruences $ t' \\equiv t \\mod n $ generated by the second step gives a unique $ t'' \\in \\Z_{\\ge 0} $ such that $ t'' < N $ and $ t'' \\equiv t' \\mod N $, as in the third step. The fourth step then ensures this $ t'' $ falls within the required bound using careful Euclidean division to give the trace $ t \\in \\Z $, of which the first step has made possible by forcing $ S $ to span the entire interval over which it could lie in. The fifth step is merely a simple application of Hasse's theorem. It only remains to understand the second step of Schoof's algorithm. This uses the properties of a general system of polynomials allowing for recursive operations, given in the following definition.\n\n\\begin{definition}[Division polynomial]\nThe \\textbf{$ n $-th division polynomial} $ \\psi_n \\in F\\sb{x, y} $ is defined for $ n \\in \\Z $ by\n\\begin{align*}\n\\psi_0\\rb{x, y} & = 0, \\\\\n\\psi_1\\rb{x, y} & = 1, \\\\\n\\psi_2\\rb{x, y} & = 2y, \\\\\n\\psi_3\\rb{x, y} & = 3x^4 + 6Ax^2 + 12Bx - A^2, \\\\\n\\psi_4\\rb{x, y} & = 4y\\rb{x^6 + 5Ax^4 + 20Bx^3 - 5A^2x^2 - 4ABx - A^3 - 8B^2},\n\\end{align*}\nrecursively defined for $ n > 4 $ by\n\\begin{align*}\n\\psi_{2m} & = \\tfrac{1}{2y}\\psi_m\\rb{\\psi_{m + 2}\\psi_{m - 1}^2 - \\psi_{m - 2}\\psi_{m + 1}^2}, \\\\\n\\psi_{2m + 1} & = \\psi_{m + 2}\\psi_m^3 - \\psi_{m - 1}\\psi_{m + 1}^3,\n\\end{align*}\nand for $ n < 0 $ by $ \\psi_{-n} = -\\psi_n $, with associated polynomials $ \\phi_n, \\omega_n \\in F\\sb{x, y} $ defined for $ n \\in \\Z_{\\ge 0} $ by\n\\begin{align*}\n\\phi_n & = x\\psi_n^2 - \\psi_{n + 1}\\psi_{n - 1}, \\\\\n\\omega_n & = \\tfrac{1}{4y}\\rb{\\psi_{n + 2}\\psi_{n - 1}^2 - \\psi_{n - 2}\\psi_{n + 1}^2}.\n\\end{align*}\n\\end{definition}\n\nIt holds that $ \\phi_{-n} = -\\phi_n $ and $ \\omega_{-n} = -\\omega_n $, and $ \\psi_{2m} = 2\\omega_n\\psi_n $. The following lemma allows certain division polynomials to be written solely in terms of $ x $.\n\n\\begin{lemma}\nIf $ n \\in \\Z $ is even, then\n$$ \\psi_n \\in 2y\\Z\\sb{x, A, B}, \\qquad \\phi_n \\in \\Z\\sb{x, A, B}, \\qquad \\omega_n \\in \\Z\\sb{x, A, B}, $$\notherwise $ n \\in \\Z $ is odd, then\n$$ \\psi_n \\in \\Z\\sb{x, A, B}, \\qquad \\phi_n \\in \\Z\\sb{x, A, B}, \\qquad \\omega_n \\in y\\Z\\sb{x, A, B}. $$\n\\end{lemma}\n\n\\begin{proof}\nLet $ Z = \\Z\\sb{x, A, B} $, then $ \\psi_0, \\psi_2, \\psi_4 \\in 2yZ $ and $ \\psi_1, \\psi_3 \\in Z $. Assume that $ \\psi_n \\in 2yZ $ for any even $ n \\in \\Z_{\\ge 0} $ and $ \\psi_n \\in Z $ for any odd $ n \\in \\Z_{\\ge 0} $ such that $ n < 2m $. If $ m $ is even, then\n$$ \\psi_m, \\psi_{m + 2}, \\psi_{m - 2} \\in 2yZ, \\qquad \\psi_{m - 1}, \\psi_{m + 1} \\in Z \\qquad \\implies \\qquad \\psi_{2m} \\in 2yZ, \\qquad \\psi_{2m + 1} \\in Z. $$\nOtherwise $ m $ is odd, then similarly\n$$ \\psi_{m - 1}, \\psi_{m + 1} \\in 2yZ, \\qquad \\psi_m, \\psi_{m + 2}, \\psi_{m - 2} \\in Z \\qquad \\implies \\qquad \\psi_{2m} \\in 2yZ, \\qquad \\psi_{2m + 1} \\in Z. $$\nHence $ \\psi_n \\in 2yZ $ for any even $ n \\in \\Z $ and $ \\psi_n \\in Z $ for any odd $ n \\in \\Z $. If $ n $ is even, then\n$$ \\psi_n^2 \\in y^2Z = Z, \\qquad \\psi_{n + 1}\\psi_{n - 1} \\in Z \\qquad \\implies \\qquad \\phi_n \\in Z. $$\nOtherwise $ n $ is odd, then similarly\n$$ \\psi_n^2 \\in Z, \\qquad \\psi_{n + 1}\\psi_{n + 1} \\in 4y^2Z = Z \\qquad \\implies \\qquad \\phi_n \\in Z. $$\nNow if $ n $ is even, then also\n$$ \\psi_{n + 2}, \\psi_{n - 2} \\in 2yZ, \\qquad \\psi_{n - 1}, \\psi_{n + 1} \\in Z \\qquad \\implies \\qquad \\omega_n \\in Z. $$\nOtherwise $ n $ is odd, then similarly also\n$$ \\psi_{n - 1}, \\psi_{n + 1} \\in 2yZ, \\qquad \\psi_{n + 2}, \\psi_{n - 2} \\in Z \\qquad \\implies \\qquad \\omega_n \\in yZ. $$\n\\end{proof}\n\nThe division polynomials $ \\phi_n\\rb{x, y} $, $ \\psi_n\\rb{x, y}^2 $, and $ \\omega_n\\rb{x, y}^2 $ can then be written as $ \\phi_n\\rb{x} $, $ \\psi_n\\rb{x}^2 $, and $ \\omega_n\\rb{x}^2 $ respectively as an abuse of notation without ambiguity. Now the familiar expression for $ \\psi_4 $ is that of the multiplication by two map, generalised as follows.\n\n\\begin{proposition}\nLet $ n \\in \\Z $. Then\n$$ \\sb{n}\\rb{x, y} = \\rb{x - \\dfrac{\\psi_{n + 1}\\rb{x, y}\\psi_{n - 1}\\rb{x, y}}{\\psi_n\\rb{x}^2}, \\dfrac{\\psi_{2n}\\rb{x, y}}{2\\psi_n\\rb{x, y}^4}} = \\rb{\\dfrac{\\phi_n\\rb{x}}{\\psi_n\\rb{x}^2}, \\dfrac{\\omega_n\\rb{x, y}}{\\psi_n\\rb{x, y}^3}}. $$\n\\end{proposition}\n\nThe proof of this proposition is through induction on $ n \\in \\Z_{\\ge 0} $ with base cases $ n \\in \\cb{0, \\dots, 4} $, while $ n \\in \\Z_{< 0} $ follows directly from the above observation. While it is completely elementary through the group law explicit formulae, it is extremely tedious and computational and hence are omitted altogether.\n\n\\begin{proof}\nOmitted, see III.E.3.7 in \\cite{gtm}.\n\\end{proof}\n\n\\begin{remark}\nThis proof can be approached via other ways, such as through properties of the \\emph{Weierstrass elliptic function} $ \\wp $ in 9.33 of \\cite{crypto}, which will not be discussed. The fact that $ \\gcd\\rb{\\phi_n, \\psi_n^2} = 1 $ and $ \\deg\\rb{\\phi_n} = n^2 $ also lends itself to another proof that $ \\deg\\rb{\\sb{n}} = n^2 $ for any $ n \\in \\Z $.\n\\end{remark}\n\nRelating this back to the standard form of isogenies, it holds that $ \\psi_n\\rb{a, b} = 0 $ iff $ \\sb{n}\\rb{P} = \\mathcal{O} $ for any point $ P = \\rb{a, b} \\in E $, which is the case whenever $ P \\in E\\sb{n} $. Group operations in $ End\\rb{E\\sb{n}} $ can be more easily done since the polynomials involved in the endomorphisms have bounded degrees in the \\emph{coordinate ring} $ F\\sb{x, y} / \\ab{y^2 - x^3 - Ax - B, \\psi_n} $, provided $ \\psi_n $ is already precomputed. Now arithmetic in $ End\\rb{E\\sb{n}} $ is motivated by the second step of Schoof's algorithm, where all congruences are modulo $ n $ and endomorphisms are computed modulo $ \\psi_n $. A \\emph{characteristic} equation that all endomorphisms satisfy will be given in the following lemma.\n\n\\begin{lemma}\nLet $ \\phi \\in End\\rb{E} $ be an endomorphism, and let $ d = \\deg\\rb{\\phi} $ and $ t = 2\\ab{\\phi, \\sb{1}} $. Then $ \\phi^2 - t\\phi + \\sb{d} = 0 $.\n\\end{lemma}\n\n\\begin{proof}\nLet $ n \\in \\cb{-1, 1} $. Since $ \\deg{\\phi + \\sb{n}} = \\deg\\rb{\\phi} + 2n\\ab{\\phi, \\sb{1}} + \\deg\\rb{\\sb{n}} = d + nt + 1 $,\nit holds that\n\\begin{align*}\n\\ab{\\phi^2, \\sb{1}}\n& = -\\tfrac{1}{2}\\rb{\\deg\\rb{\\phi^2 - \\sb{1}} - \\deg\\rb{\\phi^2} - \\deg\\rb{-\\sb{1}}} \\\\\n& = -\\tfrac{1}{2}\\rb{\\deg\\rb{\\phi - \\sb{1}}\\deg\\rb{\\phi + \\sb{1}} - \\deg\\rb{\\phi}^2 - 1} \\\\\n& = -\\tfrac{1}{2}\\rb{\\rb{d + t + 1}\\rb{d - t + 1} - d^2 - 1}\n= -\\tfrac{1}{2}\\rb{2d - t^2}.\n\\end{align*}\nSince also\n$$ \\ab{\\phi^2, \\phi} = \\tfrac{1}{2}\\rb{\\deg\\rb{\\phi^2 + \\phi} - \\deg\\rb{\\phi^2} - \\deg\\rb{\\phi}} = \\tfrac{1}{2}\\deg\\rb{\\phi}\\rb{\\deg\\rb{\\phi + \\sb{1}} - \\deg\\rb{\\phi} - \\sb{1}} = d\\ab{\\phi, \\sb{1}} = \\tfrac{1}{2}dt, $$\nit holds that\n\\begin{align*}\n\\deg\\rb{\\phi^2 - t\\phi + \\sb{d}}\n& = \\deg\\rb{\\phi^2} + \\deg\\rb{t\\phi} + \\deg\\rb{\\sb{d}} - 2\\ab{\\phi^2, t\\phi} + 2\\ab{\\phi^2, \\sb{d}} - 2\\ab{t\\phi, \\sb{d}} \\\\\n& = \\deg\\rb{\\phi}^2 + t^2\\deg\\rb{\\phi} + d^2 - 2t\\ab{\\phi^2, \\phi} + 2d\\ab{\\phi^2, \\sb{1}} - 2dt\\ab{\\phi, \\sb{1}} \\\\\n& = 2d^2 - 2t\\ab{\\phi^2, \\phi} + 2d\\ab{\\phi^2, \\sb{1}} \\\\\n& = 2d^2 - 2\\rb{\\tfrac{1}{2}dt^2} + 2d\\rb{-\\tfrac{1}{2}\\rb{2d - t^2}}\n= 0.\n\\end{align*}\nThus $ \\phi^2 - t\\phi + \\sb{d} = 0 $.\n\\end{proof}\n\nIn particular, the $ q $-th Frobenius endomorphism satisfies the characteristic equation, so it can be written as $ tFr_q = Fr_q^2 + \\sb{q} $. While it is possible to compute the right hand side directly and try all values of $ t $ until one satisfies the characteristic equation, the polynomials involved in $ Fr_q $ and $ Fr_q^2 $ will have rapidly increasing degrees, which is highly impractical for huge $ q $. The second step handles exactly this by reducing the equation in $ End\\rb{E} $ to one in $ End\\rb{E\\sb{n}} $ with affine points, as seen in the following lemma.\n\n\\begin{lemma}\nLet $ P = \\rb{a, b} \\in E\\sb{n} $ be a point. Then there are unique $ t_n \\in \\Z_{\\ge 0} $ and $ q_n \\in \\Z_{> 0} $ such that $ t_n \\equiv t, \\ q_n \\equiv q \\mod n $ with $ \\abs{t_n}, \\abs{q_n} < n $, and\n$$ t_n\\rb{a^q, b^q} = \\rb{a^{q^2}, b^{q^2}} + \\rb{a_q, b_q}, \\qquad q_nP = \\rb{a_q, b_q} \\in E\\sb{n}. $$\n\\end{lemma}\n\n\\begin{proof}\nSince $ Fr_q $ is injective, so is $ Fr_q^2 $, so $ Fr_q\\rb{P} = \\rb{a^q, b^q} $ and $ Fr_q^2\\rb{P} = \\rb{a^{q^2}, b^{q^2}} $. Hence\n$$ t\\rb{a^q, b^q} = \\rb{a^{q^2}, b^{q^2}} + q\\rb{a, b}. $$\nNow Lagrange's theorem gives that $ P \\in E\\sb{n} $ iff $ ord\\rb{P} = n $. Since $ q $ is prime and $ q \\ne n $, it holds that $ \\gcd\\rb{q, n} = 1 $, so $ qP \\ne \\mathcal{O} $. Then division gives a unique $ q_n \\in \\Z_{> 0} $ such that $ q_n \\equiv q \\mod n $ with $ \\abs{q_n} < n $. Hence $ q_nP = qP = \\rb{a_q, b_q} $ for some point $ \\rb{a_q, b_q} \\in E\\sb{n} $. Similarly division gives a unique $ t_n \\in \\Z_{\\ge 0} $ such that $ t_n \\equiv t \\mod n $ and $ \\abs{t_n} < n $. Since $ Fr_q $ has a trivial kernel and $ nFr_q\\rb{P} = Fr_q\\rb{nP} = Fr_q\\rb{\\mathcal{O}} = \\mathcal{O} $, it holds that $ ord\\rb{Fr_q\\rb{P}} = n = ord\\rb{P} $, so $ t_nFr_q\\rb{P} = tFr_q\\rb{P} $ similarly. Thus\n$$ t_n\\rb{a^q, b^q} = \\rb{a^{q^2}, b^{q^2}} + \\rb{a_q, b_q}. $$\n\\end{proof}\n\nHence it boils down to obtaining a suitable $ t_n \\in \\Z_{\\ge 0} $ satisfying\n$$ t_n\\rb{x^q, y^q} = \\rb{x^{q^2}, y^{q^2}} + q_n\\rb{x, y}, $$\nall of which can be computed as per usual, but in the coordinate ring $ F\\sb{x, y} / \\ab{y^2 - x^3 - Ax - B, \\psi_n} $. The following algorithm illustrates the process of computing this $ t_n $, with further details given in \\cite{crypto}.\n\n\\begin{algorithm}[Computation of the trace modulo prime]\nInput: an elliptic curve $ E $ over $ \\F_q $ and a prime $ n \\in S $. Output: $ t_n $. If $ n = 2 $, then\n$$ t_n = \\begin{cases} 0 & g \\ne 1 \\\\ 1 & g = 1 \\end{cases}, \\qquad g = \\gcd\\rb{x^q - x, x^3 + Ax + B}. $$\nOtherwise $ n > 2 $, then compute $ \\psi_n $ and $ q_n $, and reduce $ q_n $ into a value between $ -n / 2 $ and $ n / 2 $. Let\n$$ \\rb{x', y'} = \\rb{x^{q^2}, y^{q^2}} + q_n\\rb{x, y}, \\qquad \\rb{x'', y''} = \\rb{x^q, y^q}. $$\nIf $ x' = x_i $, where $ \\rb{x_i, y_i} = i\\rb{x'', y''} $ for some $ i \\in \\cb{1, \\dots, \\rb{n - 1} / 2} $, then\n$$ t_n = \\begin{cases} i & y' = y_i \\\\ -i & y' = -y_i \\end{cases}. $$\nOtherwise if $ q_n \\equiv r_n^2 \\mod n $ for some $ r_n \\in \\cb{1, \\dots, \\rb{n - 1} / 2} $, then let $ \\rb{x_r, y_r} = r_n\\rb{x, y} $ and\n$$ \\rb{\\dfrac{r\\rb{x}}{s\\rb{x}}, \\dfrac{u\\rb{x}}{v\\rb{x}}y} = \\rb{x'' - x_r, y'' - y_r}, \\qquad \\gcd\\rb{r, s} = \\gcd\\rb{u, v} = 1. $$\nIf $ \\gcd\\rb{r, \\psi_n} = 1 $, then\n$$ t_n = \\begin{cases} 2r_n & g' \\ne 1 \\\\ -2r_n & g' = 1 \\end{cases}, \\qquad g' = \\gcd\\rb{u, \\psi_n} $$\nOtherwise $ t_n = 0 $.\n\\end{algorithm}\n\nAn analysis of Schoof's algorithm shows that it has a time complexity of $ O\\rb{\\log^8\\rb{q}} $, which is asymptotically faster than that of the naive approach. Subsequently, there were refinements that restricted the primes in $ S $ into \\emph{Elkies primes} and \\emph{Atkin primes} rather than arbitrary small primes, and made use of \\emph{modular polynomials} rather than division polynomials. Now known as the \\emph{Schoof-Elkies-Atkin} algorithm, it has a time complexity of $ O\\rb{\\log^6\\rb{q}} $ and is widely used in practicality when the prime $ q $ in question is huge, seen in the \\emph{ellcard} command in the \\emph{PARI} programming language. In implementations when maximum efficiency is required, a probabilistic version is used, which allows even faster computations of many operations.\n\n\\pagebreak\n\n\\subsection{Point counting}\n\nAs per the aim of this section, Schoof's algorithm computes the number of rational points of elliptic curves over finite fields. Although computations are generally done by code due to routine tedium, the following simple example illustrates a possible execution process.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + 2x + 1 $ be an elliptic curve over $ \\F_{19} $, so let $ S = \\cb{2, 3, 5} $ be such that $ N = \\rb{2}\\rb{3}\\rb{5} = 30 > 20 = 4\\sqrt{25} > 4\\sqrt{19} $.\n\\begin{itemize}\n\\item Let $ n = 2 $. Then\n\\begin{align*}\nx^{19}\n& = x\\rb{-2x - 1}^6\n= 7x^7 + 2x^6 + 12x^5 + 8x^4 + 3x^3 + 12x^2 + x \\\\\n& = 7x\\rb{-2x - 1}^2 + 2\\rb{-2x - 1}^2 + 12x^2\\rb{-2x - 1} + 8x\\rb{-2x - 1} + 3\\rb{-2x - 1} + 12x^2 + x \\\\\n& = 4x^3 + x^2 + 2x + 18\n= 4\\rb{-2x - 1} + x^2 + 2x + 18\n= x^2 + 13x + 14,\n\\end{align*}\nso $ \\gcd\\rb{x^{19} - x, x^3 + 2x + 1} = \\gcd\\rb{x^2 + 12x + 14, x^3 + 2x + 1} = 1 $. Hence $ t_2 = 1 $.\n\\item Let $ n = 3 $. Then $ q_3 = 1 \\equiv 19 \\mod 3 $ such that $ -3 / 2 \\le 1 \\le 3 / 2 $, and $ \\psi_3\\rb{x} = 3x^4 + 12x^2 + 12x + 15 $. Since $ \\psi_3\\rb{8} = 3\\rb{8}^4 + 12\\rb{8}^2 + 12\\rb{8} + 15 = 0 $, it holds that $ \\rb{8, b} \\in E\\rb{\\F_{19}}\\sb{3} $ for some $ b \\in \\F_{19} $. Lagrange's theorem gives $ 3 \\mid \\abs{E\\rb{\\F_{19}}} $, so $ 19 - t + 1 \\equiv 0 \\mod 3 $ and $ t \\equiv 20 \\equiv 2 \\mod 3 $. Hence $ t_3 = 2 $.\n\\item Let $ n = 5 $. Then $ q_5 = -1 \\equiv 19 \\mod 5 $ such that $ -5 / 2 \\le 1 \\le 5 / 2 $, and\n\\begin{align*}\n\\psi_5\\rb{x}\n& = \\psi_4\\rb{x, y}\\psi_2\\rb{x, y}^3 - \\psi_1\\rb{x, y}\\psi_3\\rb{x, y}^3 \\\\\n& = 4y\\rb{x^6 + 10x^4 + x^3 + 18x^2 + 11x + 11 + 11}\\rb{2y}^3 - 1\\rb{3x^4 + 12x^2 + 12x + 15}^3 \\\\\n& = 13\\rb{x^3 + 2x + 1}^2\\rb{x^6 + 10x^4 + x^3 + 18x^2 + 11x + 3} \\\\\n& \\qquad + 11x^{12} + 18x^{10} + 18x^9 + 9x^8 + 11x^7 + 6x^6 + 12x^5 + 10x^4 + 18x^3 + 12x^2 + 13x + 7 \\\\\n& = 5x^{12} + 10x^{10} + 17x^8 + 5x^7 + x^6 + 9x^5 + 12x^4 + 2x^3 + 5x^2 + 8x + 8,\n\\end{align*}\nso let $ \\rb{x', y'} = \\rb{x^{361}, y^{361}} - \\rb{x, y} $ and $ \\rb{x'', y''} = \\rb{x^{19}, y^{19}} $. It can be tediously verified that $ x' \\ne x_1 $ but $ x' = x_2 $, where $ \\rb{x_i, y_i} = i\\rb{x'', y''} $, so $ t_n \\equiv 2 \\mod 5 $ or $ t_n \\equiv -2 \\mod 5 $. Another tedious verification gives $ y' = -y_2 $, so $ t_n \\equiv -2 \\equiv 3 \\mod 5 $. Hence $ t_5 = 3 $.\n\\end{itemize}\nThe Chinese remainder theorem from the example above gives $ t \\equiv 23 \\mod 30 $ such that $ 0 \\ge 23 < 30 = N $. Thus $ t = 23 - 30 = -7 $ is such that $ \\abs{-7} < 8 = 4\\sqrt{4} < 4\\sqrt{19} $ and $ \\abs{E\\rb{\\F_{19}}} = 19 - \\rb{-7} + 1 = 27 $.\n\\end{example}\n\nWhile just counting $ F $-rational points may have many practical applications, a subtler question would be characterising their group structure. This would be more than just Schoof's algorithm, but machinery from previous subsections can finally combine to give the following proposition.\n\n\\begin{proposition}\n$ E\\rb{F} \\cong \\Z_{n_1} $ or $ E\\rb{F} \\cong \\Z_{n_1} \\oplus \\Z_{n_2} $ for some $ n_1, n_2 \\in \\Z_{> 0} $ such that $ n_1 \\mid n_2 $.\n\\end{proposition}\n\n\\begin{proof}\nThe fundamental theorem of finite abelian groups gives\n$$ E\\rb{F} \\cong \\bigoplus_{i = 1}^m \\Z_{n_i}, \\qquad m \\in \\Z_{\\ge 0}, \\qquad n_i \\in \\Z_{> 0}, $$\nsuch that each $ n_i \\mid n_{i + 1} $. Let $ G_i = \\cb{x \\in \\Z_{n_i} \\mid ord\\rb{x} \\mid n_1} \\le \\Z_{n_i} $ be subgroups. Then each $ \\phi_i : \\Z_{n_1} \\to G_i $ defined by $ \\phi_i\\rb{x} = n_ix / n_1 $ is an isomorphism, so each $ \\abs{G_i} = \\abs{\\Z_{n_1}} = n_1 $. Hence\n$$ n_1^m = \\abs{\\bigoplus_{i = 1}^m G_i} = \\abs{E\\rb{F}\\sb{n_1}} \\le \\abs{E\\sb{n_1}} = \\abs{Ker\\rb{n_1}} = \\deg_s\\rb{\\sb{n_1}} \\le \\deg\\rb{\\sb{n_1}} = n_1^2. $$\nSince $ q \\notin \\cb{2, 3} $, it holds that $ \\abs{E\\rb{F}} = q - t + 1 \\ge q - 2\\sqrt{q} + 1 > 1 $. Thus $ \\abs{E\\rb{F}} \\ncong \\cb{0} $ and $ m \\in \\cb{1, 2} $.\n\\end{proof}\n\nBoth cases can arise from different elliptic curves and finite fields, as seen in the following example.\n\n\\begin{example}\n$ E\\rb{\\F_5} \\cong \\Z_9 $ in the above example, while $ E' : y^2 = x^3 + x $ over $ \\F_5 $ has $ E'\\rb{\\F_5} \\cong \\Z_2 \\oplus \\Z_2 $.\n\\end{example}\n\n\\begin{remark}\nIf $ q \\in \\cb{2, 3} $, then $ E\\rb{F} $ could be trivial, but the only examples with this property are $ E_2 : y^2 + y = x^3 + x + 1 $ and $ E_2' : y^2 + y = x^3 + x^2 + 1 $ over $ \\F_2 $ and $ E_3 : y^2 = x^3 - x - 1 $ over $ \\F_3 $, up to isomorphism.\n\\end{remark}\n\n\\pagebreak\n\n\\section{Elliptic curves over the rationals}\n\nAfter the discussion of elliptic curves over finite fields, the focus redirects to the field of rational numbers. Again, the question of computing the rational points arises again, with the unfortunate answer that it is not as straightforward as finite fields. Due to the countably infinite nature of the rationals, enumerating all possible rational solutions of all elliptic curves is not possible, so other techniques will be deployed. In particular, there will be an attempt to prove one of the most fundamental theorems of elliptic curves over the rationals, namely that the rational points form a finitely generated group. While finite groups arising from finite fields can be fully characterised by the fundamental theorem of finite abelian groups, finitely generated groups arising from the rationals can be fully characterised by the fundamental theorem of finitely generated abelian groups,\n$$ E\\rb{\\Q} \\cong \\Z^r \\oplus \\bigoplus_{i = 1}^m \\Z_{n_i}, \\qquad r, m \\in \\Z_{\\ge 0}, \\qquad n_i \\in \\Z_{> 1}, $$\nsuch that each $ n_i \\mid n_{i + 1} $, which is given in full in Appendix A.4. However, there are issues with computing $ r \\in \\Z_{\\ge 0} $, which will be discussed later. Now let $ E $ be an elliptic curve over the perfect field $ \\Q $, given by the Weierstrass curve\n$$ E : y^2 = x^3 + A'x + B', \\qquad A' = \\dfrac{p}{q}, B' = \\dfrac{p'}{q'} \\in \\Q, $$\nwith the group of rational points $ E\\rb{\\Q} = \\rb{E\\rb{\\Q}, \\mathcal{O}, +} $. Since there is a $ j $-invariant affine transformation $ \\rb{x, y} \\mapsto \\rb{q^{-2}q'^{-2}x, q^{-3}q'^{-3}y} $, there is an isomorphism from $ E $ to the curve given by the Weierstrass equation\n$$ \\rb{\\dfrac{1}{q^3q'^3}y}^2 = \\rb{\\dfrac{1}{q^2q'^2}x}^3 + \\dfrac{p}{q}\\rb{\\dfrac{1}{q^2q'^2}x} + \\dfrac{p'}{q'} \\qquad \\implies \\qquad y^2 = x^3 + pq^3q'^4x + p'q^6q'^5. $$\nHence for this section, assume without loss of generality that\n$$ E : y^2 = x^3 + Ax + B, \\qquad A, B \\in \\Z. $$\n\n\\subsection{Nagell-Lutz theorem}\n\nFor the following sections, let $ \\Delta'_E = \\tfrac{1}{16}\\Delta_E $ be the \\emph{reduced discriminant}. Then the following theorem characterises the affine coordinates of torsion points.\n\n\\begin{theorem}[Nagell-Lutz]\n\\label{thm:nagelllutz}\nLet $ P = \\rb{a, b} \\in E\\rb{\\Q} $ be a non-zero torsion point. Then:\n\\begin{enumerate}\n\\item $ a, b \\in \\Z $, and\n\\item $ b = 0 $ or $ b^2 \\mid \\Delta'_E $.\n\\end{enumerate}\n\\end{theorem}\n\nProof of the first part of the Nagell-Lutz theorem will be split into several definitions and lemmas, many of which follows from the properties of \\emph{$ p $-adic numbers}, which will not be discussed. Now let $ p \\in \\Z_{> 0} $ be a prime. A particular valuation in the construction of $ p $-adic numbers describing how a prime divides the numerator or denominator of a rational number is given in the following definition.\n\n\\begin{definition}[$ p $-adic valuation]\nThe \\textbf{$ p $-adic valuation} is a valuation $ v_p : \\Q \\to \\Z \\cup \\cb{\\infty} $ defined by\n$$ v_p\\rb{x} = \\begin{cases} \\max\\cb{v \\in \\Z_{\\ge 0} \\ \\Big| \\ x = \\dfrac{q}{r}p^v, \\ q \\in \\Z, \\ r \\in \\Z_{> 0}, \\ p \\nmid r} & x \\ne 0 \\\\ \\infty & x = 0 \\end{cases}. $$\n\\end{definition}\n\nHence any $ x \\in \\Q $ will be uniquely written as\n$$ x = \\dfrac{q}{r}p^v, \\qquad q \\in \\Z, \\qquad r \\in \\Z_{> 0}, $$\nsuch that $ p, q, r $ are pairwise coprime, where $ v = v_p\\rb{x} $. It is clear that $ v_p\\rb{q / r} $ is positive whenever $ p $ divides $ q $ and $ v_p\\rb{q / r} $ is negative whenever $ p $ divides $ r $, while $ v_p\\rb{q / r} $ is zero otherwise. This can be illustrated in the following example.\n\n\\begin{example}\n$$ v_5\\rb{\\dfrac{100}{10}} = v_5\\rb{\\dfrac{2}{1}5^1} = 1, \\qquad v_5\\rb{\\dfrac{10}{100}} = v_5\\rb{\\dfrac{1}{2}5^{-1}} = -1, \\qquad v_5\\rb{1} = v_5\\rb{5^0} = 0. $$\n\\end{example}\n\nThree properties given in the following lemma will come in handy when computing $ p $-adic valuations of sums and products.\n\n\\begin{lemma}\nLet $ x, y \\in \\Q $. Then:\n\\begin{enumerate}\n\\item $ v_p\\rb{1 / x} = -v_p\\rb{x} $,\n\\item $ v_p\\rb{xy} = v_p\\rb{x} + v_p\\rb{y} $, and\n\\item $ v_p\\rb{x + y} \\ge \\min\\cb{v_p\\rb{x}, v_p\\rb{y}} $, with equality if $ v_p\\rb{x} \\ne v_p\\rb{y} $.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nLet\n$$ x = \\dfrac{q}{r}p^v, \\qquad y = \\dfrac{q'}{r'}p^{v'}, \\qquad q, q' \\in \\Z, \\qquad r, r' \\in \\Z_{> 0}, $$\nsuch that $ p, q, r $ are pairwise coprime and $ p, q', r' $ are pairwise coprime, where $ v = v_p\\rb{x} $ and $ v' = v_p\\rb{y} $.\n\\begin{enumerate}\n\\item Since $ \\gcd\\rb{r, q} = 1 $,\n$$ v_p\\rb{\\dfrac{1}{x}} = v_p\\rb{\\dfrac{r}{q}p^{-v}} = -v = -v_p\\rb{x}. $$\n\\item Since $ \\gcd\\rb{p, qq'} = \\gcd\\rb{p, rr'} = 1 $,\n$$ v_p\\rb{xy} = v_p\\rb{\\dfrac{qq'}{rr'}p^{v + v'}} = v + v' = v_p\\rb{x} + v_p\\rb{y}. $$\n\\item Assume that $ v = v' $. Then\n$$ v_p\\rb{x + y} = v_p\\rb{\\dfrac{qr'p^v + q'rp^v}{rr'}} = v_p\\rb{\\dfrac{qr' + q'r}{rr'}p^v} \\ge v = \\min\\cb{v, v'} = \\min\\cb{v_p\\rb{x}, v_p\\rb{y}}. $$\nAssume otherwise that $ v > v' $. Since $ \\gcd\\rb{rr'} = \\gcd\\rb{p, qr'p^{v - v'} + q'r} = 1 $,\n$$ v_p\\rb{x + y} = v_p\\rb{\\dfrac{qr'p^v + q'rp^{v'}}{rr'}} = v_p\\rb{\\dfrac{qr'p^{v - v'} + q'r}{rr'}p^{v'}} = v' = \\min\\cb{v, v'} = \\min\\cb{v_p\\rb{x}, v_p\\rb{y}}. $$\nSimilarly, if $ v < v' $, then $ v_p\\rb{x + y} = \\min\\cb{v_p\\rb{x}, v_p\\rb{y}} $.\n\\end{enumerate}\n\\end{proof}\n\nThe following example illustrates the above lemma.\n\n\\begin{example}\n$$ v_5\\rb{\\dfrac{25}{5}} = v_5\\rb{5} = 1 = 2 - 1 = v_5\\rb{25} - v_5\\rb{5}, \\qquad v_2\\rb{8} = 3 > 2 = \\min\\cb{v_2\\rb{4}, v_2\\rb{4}}. $$\n\\end{example}\n\nWith this trick, a relation between the $ p $-adic valuated coordinates of any affine rational point in an elliptic curve can be seen in the following lemma.\n\n\\begin{lemma}\nLet $ P = \\rb{a, b} \\in E\\rb{\\Q} $ be a point. Then $ v_p\\rb{a} < 0 $ iff $ v_p\\rb{b} < 0 $, for which $ v_p\\rb{a} = -2v $ and $ v_p\\rb{b} = -3v $ for some $ v \\in \\Z_{> 0} $.\n\\end{lemma}\n\n\\begin{proof}\nAssume that $ v_p\\rb{a} < 0 $. Since $ A, B \\in \\Z $, it holds that $ v_p\\rb{A}, v_p\\rb{B} \\ge 0 $, so\n$$ 2v_p\\rb{b} = v_p\\rb{b^2} = v_p\\rb{a^3 + Aa + B} = \\min\\cb{3v_p\\rb{a}, v_p\\rb{A} + v_p\\rb{a}, v_p\\rb{B}} = 3v_p\\rb{a}. $$\nHence $ 2 \\mid v_p\\rb{a} $ and $ 3 \\mid v_p\\rb{b} $, so $ v_p\\rb{a} = -2v $ and $ v_p\\rb{b} = -3v $ for some $ v \\in \\Z_{> 0} $. Conversely assume that $ v_p\\rb{a} \\ge 0 $. Then $ 2v_p\\rb{b} \\ge \\min\\cb{3v_p\\rb{a}, v_p\\rb{A} + v_p\\rb{a}, v_p\\rb{B}} \\ge 0 $. Thus $ v_p\\rb{b} \\ge 0 $.\n\\end{proof}\n\nHence for any point $ P = \\rb{a, b} \\in E\\rb{\\Q} $,\n$$ a = \\dfrac{q}{d^2}, \\qquad b = \\dfrac{r}{d^3}, \\qquad q, r \\in \\Z, \\qquad d \\in \\Z_{> 0} $$\nsuch that $ \\gcd\\rb{q, d} = \\gcd\\rb{r, d} = 1 $. This fact will be proven explicitly here as it will be used several times in later subsections. Now a change of coordinates will be undertaken to ease discussions, namely\n$$ t = T = \\dfrac{X}{Y}, \\qquad s = S = \\dfrac{Z}{Y}, \\qquad \\sb{X, Y, Z} \\mapsto \\sb{T, 1, S} = \\rb{t, s}, $$\nwhich is an invertible projective transformation\n$$ E : Y^2Z = X^3 + AXZ^2 + BZ^3 \\qquad \\iff \\qquad E' : S = T^3 + ATS^2 + BS^3 : s = t^3 + Ats^2 + Bs^3. $$\nThis has the effect that\n$$ \\mathcal{O} \\mapsto \\rb{0, 0}, \\qquad \\rb{a, b} \\mapsto \\rb{\\dfrac{a}{b}, \\dfrac{1}{b}} $$\nfor any point $ \\rb{a, b} \\in E\\rb{\\Q} $ such that $ b \\ne 0 $, while the three $ 2 $-torsion points $ \\rb{a, 0} $ map to three points at infinity and can be disregarded for now. The modified group law is then given in the following lemma.\n\n\\begin{lemma}\nLet $ P = \\rb{a, b} \\in E'\\rb{\\Q} $ and $ Q = \\rb{a', b'} \\in E'\\rb{\\Q} $ be points such that $ P + Q = \\rb{a'', b''} \\in E'\\rb{\\Q} $. Then $ -P = \\rb{-a, -b} $ and\n$$ a'' = a + a' + \\dfrac{2A\\lambda\\mu + 3B\\lambda^2\\mu}{1 + A\\lambda^2 + B\\lambda^3}, \\qquad \\lambda = \\dfrac{a^2 + aa' + a'^2 + Ab'^2}{1 - Aa\\rb{b + b'} - B\\rb{b^2 + bb' + b'^2}}, \\qquad \\mu = b - \\lambda a. $$\n\\end{lemma}\n\n\\begin{proof}\nSince $ \\rb{a, b} \\mapsto \\rb{a / b, 1 / b} $, it holds that $ -\\rb{a, b} = \\rb{a, -b} \\mapsto \\rb{-a / b, -1 / b} $. Let $ P * Q = -\\rb{P + Q} = \\rb{-a'', -b''} $. If $ a \\ne a' $, then the line joining $ P $ and $ Q $ is\n$$ L : s = \\lambda_1 t + \\mu_1, \\qquad \\lambda_1 = \\dfrac{b - b'}{a - a'}, \\qquad \\mu_1 = b - \\lambda_1 a. $$\nOtherwise $ a = a' $, then the tangent at $ P $ is\n$$ L : s = \\lambda_2 t + \\mu_2, \\qquad \\lambda_2 = \\dfrac{3a^2 + Ab^2}{1 - 2Aab - 3Bb^2}, \\qquad \\mu_2 = b - \\lambda_2 a. $$\nSince\n\\begin{align*}\nb - b'\n& = a^3 + Aab^2 + Bb^3 - a'^3 - Aa'b'^2 - Bb'^3 \\\\\n& = a^3 - a'^3 + Aab^2 - Aab'^2 + Aab'^2 - Aa'b'^2 + Bb^3 - Bb'^3 \\\\\n& = \\rb{a - a'}\\rb{a^2 + aa' + a'^2} + Aa\\rb{b - b'}\\rb{b + b'} + Ab'^2\\rb{a - a'} + B\\rb{b - b'}\\rb{b^2 + bb' + b'^2},\n\\end{align*}\nit holds that\n$$ \\rb{b - b'}\\rb{1 - Aa\\rb{b + b'} - B\\rb{b^2 + bb' + b'^2}} = \\rb{a - a'}\\rb{a^2 + aa' + a'^2 + Ab'^2}, $$\nso $ \\rb{b - b'} / \\rb{a - a'} = \\lambda = \\lambda_1 = \\lambda_2 $ and $ \\mu = \\mu_1 = \\mu_2 $. Now $ L : s = \\lambda t + \\mu $ intersects $ E' $ at\n$$ \\rb{1 + A\\lambda^2 + B\\lambda^3}t^3 + \\rb{2A\\lambda\\mu + 3B\\lambda^2\\mu}t^2 + \\rb{A\\mu^2 + 3B\\lambda\\mu^2 - \\lambda}t - \\rb{\\mu - B\\mu^3} = 0. $$\nThus comparing coefficients gives $ -\\rb{2A\\lambda\\mu + 3B\\lambda^2\\mu} / \\rb{1 + A\\lambda^2 + B\\lambda^3} = a + a' - a'' $.\n\\end{proof}\n\nThe above proof is brief but can be verified manually. Now let\n$$ E\\rb{p^v} = \\cb{\\mathcal{O}} \\cup \\cb{\\rb{a, b} \\in E\\rb{\\Q} \\mid v_p\\rb{a} \\le -2v, \\ v_p\\rb{b} \\le -3v} $$\nbe a subset of $ E\\rb{\\Q} $. Rewriting coordinates accordingly gives $ v_p\\rb{a / b} \\ge v $ and $ v_p\\rb{1 / b} \\ge 3v $, so let\n$$ E'\\rb{p^v} = \\cb{\\rb{0, 0}} \\cup \\cb{\\rb{a, b} \\in E'\\rb{\\Q} \\mid v_p\\rb{a} \\ge v, \\ v_p\\rb{b} \\ge 3v} $$\nbe a subset of $ E'\\rb{\\Q} $ bijective to $ E\\rb{p^v} $. These two sets induce two decreasing sequences of subsets.\n\n\\begin{definition}[Filtration]\nA \\textbf{filtration} is a decreasing sequence of subsets $ S_i $ such that $ S_i \\supseteq S_j $ for any $ i \\le j $.\n\\end{definition}\n\nA simple rephrasal gives that $ E\\rb{p^v} $ and $ E'\\rb{p^v} $ \\emph{induce} two $ p $-adic filtrations\n$$ E\\rb{\\Q} \\supseteq E\\rb{p} \\supseteq E\\rb{p^2} \\supseteq E\\rb{p^3} \\supseteq \\dots \\supseteq \\cb{\\mathcal{O}}, \\qquad E'\\rb{\\Q} \\supseteq E'\\rb{p} \\supseteq E'\\rb{p^2} \\supseteq E'\\rb{p^3} \\supseteq \\dots \\supseteq \\cb{\\rb{0, 0}}. $$\nThe individual subsets in these filtrations are actually subgroups, giving a filtration of subgroups, which will be proven in the following lemma.\n\n\\begin{lemma}\nLet $ v \\in \\Z_{> 0} $ and $ P = \\rb{a, b} \\in E'\\rb{p^v} $ and $ Q = \\rb{a', b'} \\in E'\\rb{p^v} $ be points such that $ P + Q = \\rb{a'', b''} \\in E'\\rb{\\Q} $. Then $ -P, P + Q \\in E'\\rb{p^v} $ and $ v_p\\rb{a + a' + a''} \\ge 5v $.\n\\end{lemma}\n\n\\begin{proof}\nSince $ -P = \\rb{-a, -b} $, it holds that $ v_p\\rb{-a} = v_p\\rb{a} $, so $ -P \\in E'\\rb{p^v} $. Since $ A, B \\in \\Z $, it holds that $ v_p\\rb{A}, v_p\\rb{B} \\ge 0 $. Now the group law gives\n$$ a'' = a + a' + \\dfrac{2A\\lambda\\mu + 3B\\lambda^2\\mu}{1 + A\\lambda^2 + B\\lambda^3}, \\qquad \\lambda = \\dfrac{a^2 + aa' + a'^2 + Ab'^2}{1 - Aa\\rb{b + b'} - B\\rb{b^2 + bb' + b'^2}}, \\qquad \\mu = b - \\lambda a. $$\nThen\n\\begin{align*}\nv_p\\rb{a^2 + aa' + a'^2 + Ab'^2} & \\ge \\min\\cb{2v_p\\rb{a}, v_p\\rb{a} + v_p\\rb{a'}, 2v_p\\rb{a'}, v_p\\rb{A} + 2v_p\\rb{b'}} \\ge 2v, \\\\\nv_p\\rb{Aa\\rb{b + b'}} & \\ge \\min\\cb{v_p\\rb{a} + v_p\\rb{a'} + v_p\\rb{b}, v_p\\rb{a} + v_p\\rb{a'} + v_p\\rb{b'}} \\ge 5v, \\\\\nv_p\\rb{B\\rb{b^2 + bb' + b'^2}} & \\ge \\min\\cb{v_p\\rb{B} + 2v_p\\rb{b}, v_p\\rb{B} + v_p\\rb{b} + v_p\\rb{b'}, v_p\\rb{B} + 2v_p\\rb{b'}} \\ge 6v,\n\\end{align*}\nso $ v_p\\rb{\\lambda} \\ge 2v - \\min\\cb{0, 5v, 6v} = 2v $ and $ v_p\\rb{\\mu} \\ge \\min\\cb{3v, 3v} = 3v $. Hence\n\\begin{align*}\nv_p\\rb{2A\\lambda\\mu + 3B\\lambda^2\\mu} & \\ge \\min\\cb{v_p\\rb{2} + v_p\\rb{A} + v_p\\rb{\\lambda} + v_p\\rb{\\mu}, v_p\\rb{3} + v_p\\rb{B} + 2v_p\\rb{\\lambda} + v_p\\rb{\\mu}} \\ge 5v, \\\\\nv_p\\rb{1 + A\\lambda^2 + B\\lambda^3} & = \\min\\cb{v_p\\rb{1}, v_p\\rb{A} + 2v_p\\rb{\\lambda}, v_p\\rb{B} + 3v_p\\rb{\\lambda}} = 0,\n\\end{align*}\nso $ v_p\\rb{a''} \\ge \\min\\cb{a, a', 5v} \\ge v $. Thus $ P + Q \\in E'\\rb{p^v} $ and $ v_p\\rb{a + a' - a''} \\ge 5v $.\n\\end{proof}\n\nNote that the second part of the lemma proves something stronger, that the $ x $ coordinates of three collinear points add to give a large $ p $-adic valuation. Now let $ R = \\cb{x \\in \\Q \\mid v_p\\rb{x} \\ge 0} $ be a unique factorisation domain such that $ \\ab{p^v} = \\cb{x \\in \\Q \\mid v_p\\rb{x} \\ge v} \\subseteq R $ is a principal ideal. This also induces a filtration of subgroups\n$$ R \\ge \\ab{p} \\ge \\ab{p^2} \\ge \\ab{p^3} \\ge \\dots \\cb{0}. $$\nThen $ v_p\\rb{a + a' - a''} \\ge 5v $ from the previous lemma can be rephrased as $ a + a' - a'' \\in \\ab{p^{5v}} $, or even better as $ \\ab{p^{5v}} + a + a' = \\ab{p^{5v}} + a'' $. The following lemma attempts to makes use of this fact.\n\n\\begin{lemma}\nThere is an injective group homomorphism\n$$ \\phi : E\\rb{p^v} / E\\rb{p^{5v}} \\to \\ab{p^v} / \\ab{p^{5v}}, \\qquad \\phi\\rb{E\\rb{p^{5v}} + P} = \\begin{cases} \\ab{p^{5v}} + \\dfrac{a}{b} & P = \\rb{a, b} \\\\ \\ab{p^{5v}} & P = \\mathcal{O} \\end{cases}. $$\n\\end{lemma}\n\n\\begin{proof}\nLet $ \\psi : E\\rb{p^v} \\to \\ab{p^v} / \\ab{p^{5v}} $ be defined by\n$$ \\psi\\rb{P} = \\begin{cases} \\ab{p^{5v}} + \\dfrac{a}{b} & P = \\rb{a, b} \\\\ \\ab{p^{5v}} & P = \\mathcal{O} \\end{cases}, $$\nand let $ P, Q \\in E\\rb{p^v} $ be points. If $ P = \\mathcal{O} $, then\n$$ \\psi\\rb{P} + \\psi\\rb{Q} = \\ab{p^{5v}} + \\ab{p^{5v}} + \\dfrac{a'}{b'} = \\ab{p^{5v}} + \\dfrac{a'}{b'} = \\psi\\rb{Q} = \\psi\\rb{P + Q}, $$\nor similar for $ Q = \\mathcal{O} $. If $ P = \\rb{a, b} $ and $ Q = \\rb{a, -b} $, then\n$$ \\psi\\rb{P} + \\psi\\rb{Q} = \\ab{p^{5v}} + \\dfrac{a}{b} + \\ab{p^{5v}} - \\dfrac{a}{b} = \\ab{p^{5v}} = \\psi\\rb{\\mathcal{O}} = \\psi\\rb{P + Q}. $$\nOtherwise $ P = \\rb{a, b} $ and $ Q = \\rb{a', b'} $ such that $ P + Q = \\rb{a'', b''} $, then\n$$ \\psi\\rb{P} + \\psi\\rb{Q} = \\ab{p^{5v}} + \\dfrac{a}{b} + \\ab{p^{5v}} + \\dfrac{a'}{b'} = \\ab{p^{5v}} + \\dfrac{a}{b} + \\dfrac{a'}{b'} = \\ab{p^{5v}} + \\dfrac{a''}{b''} = \\psi\\rb{P + Q}. $$\nHence $ \\psi $ is a group homomorphism. Now $ \\mathcal{O} \\in Ker\\rb{\\psi} $, and $ \\rb{a, b} \\in Ker\\rb{\\psi} $ iff $ v_p\\rb{a / b} \\ge 5v $. This holds iff $ \\rb{a / b, 1 / b} \\in E'\\rb{p^{5v}} $ and $ \\rb{a, b} \\in E\\rb{p^{5v}} $, so $ Ker\\rb{\\psi} = E\\rb{p^{5v}} $. Thus the first isomorphism theorem gives a natural injective group homomorphism\n$$ \\phi : \\dfrac{E\\rb{p^v}}{E\\rb{p^{5v}}} \\to Im\\rb{\\psi} \\subseteq \\dfrac{\\ab{p^v}}{\\ab{p^{5v}}}. $$\n\\end{proof}\n\nNow the subgroup $ E\\rb{p} $ can be proven to be \\emph{torsion-free} with a proof by contradiction in the following lemma, from which the first part of the Nagell-Lutz theorem can be deduced.\n\n\\begin{lemma}\n$ E\\rb{p} $ has no non-zero torsion points.\n\\end{lemma}\n\n\\begin{proof}\nLet $ P = \\rb{a, b} \\in E\\rb{\\Q} $ be an $ n $-torsion point. Suppose for a contradiction that $ P \\in E\\rb{p} $, so $ v_p\\rb{a} = -2v $ for some $ v \\in \\Z_{> 0} $ and $ v_p\\rb{a / b} = v $. Then\n$$ \\ab{p^{5v}} = \\phi\\rb{E\\rb{p^{5v}}} = \\phi\\rb{E\\rb{p^{5v}} + nP} = n\\phi\\rb{E\\rb{p^{5v}} + P} = n\\rb{\\ab{p^{5v}} + \\dfrac{a}{b}} = \\ab{p^{5v}} + n\\dfrac{a}{b}, $$\nso $ n\\rb{a / b} \\in \\ab{p^{5v}} $. Assume that $ p \\nmid n $, so $ a / b \\in \\ab{p^{5v}} $ and $ v = v_p\\rb{a / b} \\ge 5v $, which is a contradiction. Hence $ P \\notin E\\rb{p} $. Otherwise assume that $ p \\mid n $, then $ n = mp $ for some $ m \\in \\Z_{> 0} $. Now let $ Q = mP = \\rb{a', b'} \\in E\\rb{\\Q} $ be a $ p $-torsion point. Since $ P \\in E\\rb{p} $, it holds that $ Q \\in E\\rb{p} $, so $ v_p\\rb{a'} = -2v' $ for some $ v' \\in \\Z_{> 0} $ and $ v_p\\rb{a' / b'} = v' $. Then\n$$ \\ab{p^{5v'}} = \\phi\\rb{E\\rb{p^{5v'}}} = \\phi\\rb{E\\rb{p^{5v'}} + pQ} = p\\phi\\rb{E\\rb{p^{5v'}} + Q} = p\\rb{\\ab{p^{5v'}} + \\dfrac{a'}{b'}} = \\ab{p^{5v}} + p\\dfrac{a'}{b'}, $$\nso $ p\\rb{a' / b'} \\in \\ab{p^{5v'}} $. Then $ 5v' \\le v_p\\rb{p\\rb{a' / b'}} = v_p\\rb{p} + v_p\\rb{a' / b'} = 1 + v' $, which is again a contradiction. Hence $ Q \\notin E\\rb{p} $ and $ P \\notin E\\rb{p} $. Thus $ E\\rb{p} $ has no non-zero torsion points.\n\\end{proof}\n\nBoth parts of the Nagell-Lutz theorem can finally be proven here, the second part a corollary of the first.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:nagelllutz}]\nLet $ P = \\rb{a, b} \\in E\\rb{\\Q} $ be a non-zero $ n $-torsion point.\n\\begin{enumerate}\n\\item Since $ P \\notin E\\rb{p} $ for any prime $ p \\in \\Z_{> 0} $, it holds that $ v_p\\rb{a} \\ge 0 $ and $ v_p\\rb{b} \\ge 0 $. Thus $ a, b \\in \\Z $.\n\\item Assume that $ b \\ne 0 $ and let $ 2P = \\rb{a', b'} \\in E\\rb{\\Q} $. By the duplication formula,\n$$ a' = \\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4b^2}. $$\nSince $ P $ and $ 2P $ are torsion points, it holds that $ a, b, a', b' \\in \\Z $, so $ b^2 \\mid a^4 - 2Aa^2 - 8Ba + A^2 $. Thus\n$$ b^2 \\mid \\rb{a^4 - 2Aa^2 - 8Ba + A^2}\\rb{3a^2 + 4A} - \\rb{a^3 + Aa + B}\\rb{3a^3 - 5Aa - 27B} = 4A^3 + 27B^2 = \\Delta'_E. $$\n\\end{enumerate}\n\\end{proof}\n\n\\pagebreak\n\n\\subsection{Torsion computation}\n\nAn application of the Nagell-Lutz theorem is as follows. Assuming the fundamental theorem of finite abelian groups,\n$$ E\\rb{\\Q} \\cong \\Z^r \\oplus E\\rb{\\Q}_{tors}, \\qquad r \\in \\Z_{\\ge 0}, $$\nthe Nagell-Lutz theorem can be used to compute the torsion subgroup $ E\\rb{\\Q}_{tors} $, since there are only finitely many torsion points $ \\rb{a, b} \\in E\\rb{\\Q} $ such that $ b^2 \\mid \\Delta'_E $. The following example illustrates the full computation of the torsion subgroup of an elliptic curve.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + 4 $ be an elliptic curve over $ \\Q $ and $ P = \\rb{a, b} \\in E\\rb{\\Q} $ be a torsion point. Then either $ b = 0 $ or $ b^2 \\mid 4\\rb{0}^3 + 27\\rb{4}^2 = 3\\rb{12}^2 $, so $ b \\in \\cb{0, \\pm 1, \\pm 2, \\pm 3, \\pm 4, \\pm 6, \\pm 12} $, of which only $ P_1 = \\rb{0, 2} \\in E\\rb{\\Q} $ and $ P_2 = \\rb{0, -2} \\in E\\rb{\\Q} $. Then\n$$ 2P_1 = \\rb{\\dfrac{0}{4\\rb{4}}, \\dfrac{2^2 - 3\\rb{4}}{2\\rb{2}}} = \\rb{0, -2} = P_2, $$\nso $ ord\\rb{P_1} = ord\\rb{P_2} = 3 $. Thus the torsion subgroup is $ E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}, P_1, P_2} \\cong \\Z_3 $.\n\\end{example}\n\nThe following algorithm summarises the process and code in the appendix.\n\n\\begin{algorithm}[Computation of the torsion subgroup]\nInput: an elliptic curve $ E $ over $ \\Q $. Output: $ E\\rb{\\Q}_{tors} $.\n\\begin{enumerate}\n\\item Calculate $ \\Delta'_E $ and get all non-negative $ b $ coordinates such that $ b^2 \\mid \\Delta'_E $.\n\\item Get all $ a $ coordinates for each non-negative $ b $ coordinate such that $ b^2 = a^3 + Aa + B $.\n\\item Add points $ \\rb{a, b} $ with itself repeatedly and stop at $ \\mathcal{O} $ or non-integer coordinates.\n\\item Negate each point $ \\rb{a, b} $ to $ \\rb{a, -b} $ and do the same.\n\\item Insert $ \\mathcal{O} $ into the list of all points that add to $ \\mathcal{O} $.\n\\end{enumerate}\n\\end{algorithm}\n\nThe torsion subgroups of the following examples of elliptic curves given by the Weierstrass equations $ y^2 = x^3 - px $ for $ p \\in \\Z_{> 0} $ can be computed similarly. This information will be used in a later subsection.\n\n\\begin{example}\nThe elliptic curves $ E : y^2 = x^3 - x $ has torsion subgroup $ E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}, \\rb{0, 0}, \\rb{1, 0}, \\rb{-1, 0}} \\cong \\Z_2^2 $, while the elliptic curves $ E : y^2 = x^3 - 5x $, $ E : y^2 = x^3 - 17x $, $ E : y^2 = x^3 - 226x $ all have torsion subgroup $ E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}, \\rb{0, 0}} \\cong \\Z_2 $.\n\\end{example}\n\nThe converse to the Nagell-Lutz theorem does not generally hold. It cannot be used to prove that a certain point is a torsion point, but it can be used to show the contrapositive, that a point is not a torsion point, by duplicating it until its coordinates are not integers. The following example illustrates this.\n\n\\begin{example}\nLet $ E : y^2 = x^3 - 4 $ be an elliptic curve over $ \\Q $ with torsion subgroup $ E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}} \\cong \\Z_1 $. Now let $ P = \\rb{2, 2} \\in E\\rb{\\Q} $ be a point. Then\n$$ 2P = \\rb{5, -11}, \\qquad 4P = \\rb{\\dfrac{785}{484}, -\\dfrac{5497}{10648}}. $$\nThus $ ord\\rb{P} $ is infinite.\n\\end{example}\n\nThese are several examples of different torsion subgroups. In fact, there are even elliptic curves with as large as $ 12 $-torsion elements, but there are strangely none with $ 11 $-torsion elements. The following difficult theorem was proven to be an exhaustive list of all possible torsion subgroups of all elliptic curves.\n\n\\begin{theorem}[Mazur]\n$ E\\rb{\\Q} $ is isomorphic to one of\n$$ \\Z_n, \\qquad n \\in \\cb{1, \\dots, 10, 12}, $$\n$$ \\Z_2 \\times \\Z_{2n}, \\qquad n \\in \\cb{1, \\dots, 4}. $$\n\\end{theorem}\n\n\\begin{proof}\nOmitted, see \\cite{mazur}.\n\\end{proof}\n\nAs such, the torsion subgroup of an elliptic curve $ E\\rb{\\Q} $ can be computed in a finite number of steps. However, computations may still be intensive if $ \\Delta'_E $ has many squared factors, as the computation involves solving a cubic equation. The next section provides an alternative method for this.\n\n\\pagebreak\n\n\\subsection{Reduction modulo prime}\n\nAnother method of computing the torsion subgroup is to reduce the elliptic curve over rationals into one over a finite field, by applying isomorphisms that simplify the Weierstrass equation, then applying a particular group homomorphism. The assumption of integer coefficients in a previous subsection makes the Weierstrass equation \\emph{integral}, but a further reduction can be done as follows.\n\n\\begin{definition}[Minimal]\nA Weierstrass equation is \\textbf{minimal} iff it is integral and $ g \\in \\cb{-1, 1} $ if $ g^4 \\mid A $ and $ g^6 \\mid B $.\n\\end{definition}\n\nA minimal Weierstrass equation is unique up to sign. The above definition reflects the minimality of the integer coefficients after $ j $-invariant affine transformations, which is illustrated in the following example.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + n^4x + n^6 $ be an elliptic curve over $ \\Q $ for some $ n \\in \\Q $. Since there is a $ j $-invariant affine transformation $ \\rb{x, y} \\mapsto \\rb{n^2x, n^3y} $, there is an isomorphism from $ E $ to the curve given by the Weierstrass equation $ y^2 = x^3 + x + 1 $, which is integral and minimal.\n\\end{example}\n\nMinimal Weierstrass equations can then be treated as if their coefficients are modulo a prime, which is stated formally as a map in the following definition.\n\n\\begin{definition}[Reduction map]\nThe \\textbf{reduction modulo $ p $ map} $ r_p : E\\rb{\\Q} \\to E_p\\rb{\\F_p} $ for some prime $ p \\in \\Z_{> 0} $ is defined by\n$$ E_p : y^2 = x^3 + \\tilde{A}x + \\tilde{B}, \\qquad r_p\\rb{P} = \\begin{cases} \\rb{\\tilde{a}, \\tilde{b}} & P = \\rb{a, b} \\\\ \\mathcal{O} & P = \\mathcal{O} \\end{cases}, $$\nwhere $ \\tilde{\\cdot} : \\Z \\to \\F_p $ denotes modulo $ p $.\n\\end{definition}\n\nThere is a minor hiccup with this definition, since $ E_p $ might not even define a smooth Weierstrass curve. However, since $ 0 \\ne \\Delta_E = p_1 \\dots p_n $ for some primes $ p_i \\in \\Z_{> 0} $ and $ \\Delta_{E_p} = 0 $ only if any $ p_i \\mid p $, this issue can be easily fixed by considering only the primes that are not $ p_i $, which is given in the following definition.\n\n\\begin{definition}[Good reduction]\nA prime $ p \\in \\Z_{> 0} $ is of \\textbf{good reduction} iff $ p \\nmid \\Delta_E $.\n\\end{definition}\n\nHence $ r_p $ has a well-defined codomain for infinitely many primes of good reduction, while those of bad reduction will not be considered. Additionally since the discriminant has a coefficient of $ 16 $, the prime $ 2 $ will always be considered one of bad reduction. Now $ r_p $ is also well-defined, which is immediate considering the following lemma.\n\n\\begin{lemma}\n\\label{lem:normalise}\nLet $ P = \\sb{a, b, c} \\in E\\rb{\\Q} $ be a point. Then $ P = \\sb{a', b', c'} $ for some $ a', b', c' \\in \\Z $ such that $ \\gcd\\rb{a', b', c'} = 1 $.\n\\end{lemma}\n\n\\begin{proof}\nIf $ c = 0 $, then $ P = \\mathcal{O} $, so $ \\gcd\\rb{0, 1, 0} = 1 $. Otherwise $ c \\ne 0 $, then $ P = \\rb{a / c, b / c} $. Then $ a / c = q / d^2 $ and $ b / c = r / d^3 $ for some $ q, r \\in \\Z $ and some $ d \\in \\Z_{> 0} $ such that $ \\gcd\\rb{q, d} = \\gcd\\rb{r, d} = 1 $. Thus $ P = \\rb{q / d^2, r / d^3} = \\sb{qd, r, d^3} $ is such that $ \\gcd\\rb{qd, r, d^3} = 1 $.\n\\end{proof}\n\nThis integral and minimal condition will also be defined as follows.\n\n\\begin{definition}[Normalised]\nA point $ P \\in E\\rb{\\Q} $ has \\textbf{normalised} coordinates iff it satisfies Lemma \\ref{lem:normalise}.\n\\end{definition}\n\nWith this representation, there must be one of $ a', b', c' $ coprime to $ p $ for any prime $ p \\in \\Z_{> 0} $ of good reduction, so $ r_p\\rb{P} = \\sb{\\tilde{a'}, \\tilde{b'}, \\tilde{c'}} \\in E_p\\rb{\\F_p} $ is well-defined. The normalised coordinates of any point is unique up to sign, which is illustrated with the following example.\n\n\\begin{example}\nLet $ P = \\rb{2 / 5, -1 / 3} \\in E\\rb{\\Q} $ be a point. Then\n$$ \\rb{\\dfrac{2}{5}, -\\dfrac{1}{3}} = \\sb{\\dfrac{2}{5}, -\\dfrac{1}{3}, 1} = \\sb{6, -5, 15}, \\sb{-6, 5, -15} $$\nare its normalised coordinates.\n\\end{example}\n\n\\pagebreak\n\nLet $ p \\in \\Z_{> 0} $ be a prime of good reduction. Then the following proposition characterises $ r_p $.\n\n\\begin{proposition}\n$ r_p : E\\rb{\\Q} \\to E_p\\rb{\\F_p} $ is a group homomorphism such that $ Ker\\rb{r_p} = E\\rb{p} $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ P, Q \\in E\\rb{\\Q} $ be points with normalised coordinates and\n$$ L : l\\rb{X, Y, Z} = kX + mY + nZ = 0 $$\nbe a line joining $ P $ and $ Q $ with coefficients in $ \\Q $ such that $ P, Q, -\\rb{P + Q} \\in E\\rb{\\Q} \\cap L $. Then normalising $ \\sb{l, m, n} $ similarly gives $ \\sb{l', m', n'} $ for some $ l', m', n '\\in \\Z $ such that $ \\gcd\\rb{l, m, n} = 1 $. Hence the line\n$$ L_p : l_p\\rb{X, Y, Z} = \\tilde{l}X + \\tilde{m}Y + \\tilde{n}Z = 0 $$\nwith coefficients in $ \\F_p $ is well-defined. Now let $ P = \\sb{a, b, c} $. Then\n$$ la + mb + nc = 0 \\qquad \\implies \\qquad \\tilde{l}\\tilde{a} + \\tilde{m}\\tilde{b} + \\tilde{n}\\tilde{c} = 0, $$\nso $ r_p\\rb{P} = \\rb{\\tilde{a}, \\tilde{b}, \\tilde{c}} \\in E_p\\rb{\\F_p} \\cap L_p $. Similarly $ r_p\\rb{Q} \\in E_p\\rb{\\F_p} \\cap L_p $. Since $ r_p\\rb{-\\mathcal{O}} = r_p\\rb{\\mathcal{O}} = -r_p\\rb{\\mathcal{O}} $ and\n$$ r_p\\rb{-\\rb{a, b}} = r_p\\rb{\\rb{a, -b}} = \\rb{\\tilde{a}, \\tilde{-b}} = \\rb{\\tilde{a}, -\\tilde{b}} = -\\rb{\\tilde{a}, \\tilde{b}} = -r_p\\rb{\\rb{a, b}} $$\nfor any point $ \\rb{a, b} \\in E\\rb{\\Q} $, similarly $ -r_p\\rb{P + Q} = r_p\\rb{-\\rb{P + Q}} \\in E_p\\rb{\\F_p} \\cap L_p $. Since $ \\gcd\\rb{e_p, l_p} = 1 $ where $ e_p\\rb{x, y} $ is the Weierstrass equation of $ E_p $, B\\'ezout's theorem gives that $ L_p $ intersects $ E_p\\rb{\\F_p} $ at three points up to multiplicity, so\n$$ E_p\\rb{\\F_p} \\cap L_p = \\cb{r_p\\rb{P}, r_p\\rb{Q}, -r_p\\rb{P + Q}}. $$\nHence $ r_p\\rb{P} + r_p\\rb{Q} = r_p\\rb{P + Q} $. Now let $ R = \\rb{a, b} \\in E\\rb{\\Q} $ be a point. Then $ a = q / d^2 $ and $ b = r / d^3 $ for some $ q, r \\in \\Z $ and some $ d \\in \\Z_{> 0} $ such that $ \\gcd\\rb{q, d} = \\gcd\\rb{r, d} = 1 $. Since $ R = \\sb{qd, r, d^3} $ has normalised coordinates, it holds that $ r_p\\rb{R} = \\sb{\\tilde{q}\\tilde{d}, \\tilde{r}, \\tilde{d}^3} \\in E_p\\rb{\\F_p} $. Then $ R \\in Ker\\rb{r_p} $ iff $ \\tilde{d}^3 = 0 $, or $ p \\mid d $. This holds iff $ v_p\\rb{a} \\le -2 $ and $ v_p\\rb{b} \\le -3 $, or $ R \\in E\\rb{p} $. Thus $ Ker\\rb{r_p} = E\\rb{p} $.\n\\end{proof}\n\nRestricting $ r_p $ into the torsion subgroup of its domain gives it a stronger property as follows.\n\n\\begin{theorem}[Reduction]\n$ E\\rb{\\Q}_{tors} \\cong G $ for some $ G \\le E_p\\rb{\\F_p} $.\n\\end{theorem}\n\n\\begin{proof}\nSince $ Ker\\rb{r_p} = E\\rb{p} $, it holds that $ v_p\\rb{a} \\le -2 $ and $ v_p\\rb{b} \\le -3 $ for any point $ P = \\rb{a, b} \\in Ker\\rb{r_p} $, so $ a, b \\notin \\Z $. Then the Nagell-Lutz theorem gives that $ ord\\rb{P} $ is infinite, so $ P \\notin E\\rb{\\Q}_{tors} $. Now let $ r_p' = \\val{r_p}_{E\\rb{\\Q}_{tors}} $ and $ G = Im\\rb{r_p'} $, so $ Ker\\rb{r_p'} = Ker\\rb{r_p} \\cap E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}} $. Thus the first isomorphism theorem gives $ G \\cong E\\rb{\\Q}_{tors} / Ker\\rb{r_p'} \\cong E\\rb{\\Q}_{tors} $.\n\\end{proof}\n\nLagrange's theorem then gives $ \\abs{E\\rb{\\Q}_{tors}} \\mid \\abs{E_p\\rb{\\F_p}} $, which enforces a restriction of the possible torsion subgroups. The following reignites a prior example, this time with the reduction theorem.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + 4 $ be an elliptic curve over $ \\Q $. Then $ \\Delta_E = -16\\rb{4\\rb{0}^3 + 27\\rb{4}^2} = -\\rb{2}^8\\rb{3}^3 $, so let $ p = 5 $ be a prime of good reduction. Then the previous section gives $ \\abs{E_5\\rb{\\F_5}} = 6 $. Since $ \\abs{E\\rb{\\Q}_{tors}} \\mid \\abs{E_5\\rb{\\F_5}} $, it holds that $ \\abs{E\\rb{\\Q}_{tors}} \\in \\cb{1, 2, 3, 6} $. Since $ ord\\rb{\\rb{0, 2}} = 3 $ and there are no points $ P \\in E\\rb{\\Q} $ such that $ ord\\rb{P} = 2 $, it holds that $ E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}, \\rb{0, 2}, \\rb{0, -2}} \\cong \\Z_3 $.\n\\end{example}\n\nWhile this might not seem much of a timesave, the following example begs to differ.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + 1680 $ be an elliptic curve over $ \\Q $. Then $ \\Delta'_E = 4\\rb{0}^3 + 27\\rb{1680}^2 = 3\\rb{5040}^2 = \\rb{2}^8\\rb{3}^5\\rb{5}^2\\rb{7}^2 $ and $ \\Delta_E = -\\rb{2}^{12}\\rb{3}^5\\rb{5}^2\\rb{7}^2 $, so $ p \\ge 11 $ are primes of good reduction. Now $ 5040 $ is a \\emph{colossally abundant number} with exactly $ 120 $ positive and negative divisors, so more than $ 120 $ values of $ b $ such that $ b^2 \\mid \\Delta'_E $ needs to be checked. Instead the previous section computes $ \\abs{E_{13}\\rb{\\F_{13}}} = 9 $ and $ \\abs{E_{19}\\rb{\\F_{19}}} = 28 $. Since $ \\abs{E\\rb{\\Q}_{tors}} \\mid \\abs{E_{13}\\rb{\\F_{13}}} $ and $ \\abs{E\\rb{\\Q}_{tors}} \\mid \\abs{E_{19}\\rb{\\F_{19}}} $, and $ \\gcd\\rb{9, 28} = 1 $, it holds that $ \\abs{E\\rb{\\Q}_{tors}} = 1 $. Thus $ E\\rb{\\Q}_{tors} = \\cb{\\mathcal{O}} $.\n\\end{example}\n\nCounting points over finite fields can generally be done very efficiently, so the reduction theorem allows for an immediate answer. In any case, computation of the torsion subgroup is relatively straightforward.\n\n\\pagebreak\n\n\\subsection{Mordell's theorem: descent}\n\nThe following theorem is one of the most fundamental theorems of elliptic curves over the rationals, as stated in a previous subsection.\n\n\\begin{theorem}[Mordell]\n\\label{thm:mordell}\nThe \\emph{Mordell-Weil group} $ E\\rb{\\Q} $ is finitely generated.\n\\end{theorem}\n\n\\begin{remark}\nThis is a special case of the \\emph{Mordell-Weil theorem}, which states that $ E\\rb{K} $ is finitely generated over any number field $ K $.\n\\end{remark}\n\nProof of Mordell's theorem will be split into two distinct steps. The first step of the proof develops some theory of a certain function that describes the size of points. The second step of the proof is a weak variant of the theorem stating that the index of a subgroup is finite. These two steps are then used in a variant of \\emph{Fermat's infinite descent}, which can be stated in full generalisation for arbitrary abelian groups as follows.\n\n\\begin{theorem}[Descent]\nLet $ G $ be an abelian group such that the index $ \\sb{G : 2G} $ is finite, and let $ h : G \\to \\R_{\\ge 0} $ be such that:\n\\begin{itemize}\n\\item the set $ \\cb{P \\in G \\mid h\\rb{P} \\le C_1} $ is finite for any $ C_1 \\in \\R_{\\ge 0} $,\n\\item for any $ Q \\in G $, there is a constant $ C_2 \\in \\R_{\\ge 0} $ such that $ h\\rb{P + Q} \\le 2h\\rb{P} + C_2 $ for any $ P \\in G $, and\n\\item there is a constant $ C_3 \\in \\R_{\\ge 0} $ such that $ h\\rb{2P} \\ge 4h\\rb{P} - C_3 $ for any $ P \\in G $.\n\\end{itemize}\nThen $ G $ is finitely generated.\n\\end{theorem}\n\n\\begin{proof}\nLet $ Q_1, \\dots, Q_n \\in G $ be representatives such that $ 2G + Q_i \\in G / 2G $ are distinct cosets. For any $ P \\in G $, the upper bound gives each $ h\\rb{P - Q_i} \\le 2h\\rb{P} + C_i $ for some $ C_i \\in \\R_{\\ge 0} $, so\n$$ h\\rb{P - Q_i} \\le 2h\\rb{P} + C, \\qquad i \\in \\cb{1, \\dots, n}, \\qquad C = \\max\\cb{C_i} \\in \\R_{\\ge 0}. $$\nFor any $ P \\in G $, the lower bound also gives\n$$ h\\rb{2P} \\ge 4h\\rb{P} - C', \\qquad C' \\in \\R_{\\ge 0}. $$\nThen there is a finite set\n$$ S = \\cb{P \\in G \\mid h\\rb{P} \\le C + C'}. $$\nNow let $ P \\in G $. Then $ 2G + P = 2G + Q_{i_0} $ for some $ i_0 \\in \\cb{1, \\dots, n} $, so $ P = 2P_0 + Q_{i_0} $ for some $ P_0 \\in G $. By induction, for any $ j \\in \\Z_{> 0} $, there is some $ i_j \\in \\cb{1, \\dots, n} $ such that $ 2G + P_{j - 1} = 2G + Q_{i_j} $, so\n$$ P_{j - 1} = 2P_j + Q_{i_j}, \\qquad P = 2^{j + 1}P_j + \\sum_{k = 0}^j 2^kQ_{i_k}, \\qquad P_j \\in G. $$\nNow for any $ j \\in \\Z_{> 0} $,\n$$ 4h\\rb{P_j} \\le h\\rb{2P_j} + C' = h\\rb{P_{j - 1} - Q_{i_j}} + C' \\le 2h\\rb{P_{j - 1}} + \\rb{C + C'}, $$\nso that\n$$ h\\rb{P_j} \\le \\tfrac{1}{2}h\\rb{P_{j - 1}} + \\tfrac{1}{4}\\rb{C + C'} = \\dfrac{3}{4}h\\rb{P_{j - 1}} - \\tfrac{1}{4}\\rb{h\\rb{P_{j - 1}} - \\rb{C + C'}}. $$\nIf $ h\\rb{P_{j - 1}} > C + C' $ for some $ j \\in \\Z_{> 0} $, then $ h\\rb{P_j} < \\dfrac{3}{4}h\\rb{P_{j - 1}} $, so $ h\\rb{P_m} \\le C + C' $ for some $ m \\in \\Z_{> 0} $ such that $ m \\ge j $ and $ P_m \\in S $. Otherwise $ h\\rb{P_{j - 1}} \\le C + C' $ for all $ j \\in \\Z_{> 0} $, so let $ m = 1 $ such that $ P_m \\in S $ as well. Hence\n$$ P = 2^{m + 1}P_m + \\sum_{k = 0}^m 2^kQ_{i_k} = \\sum_{S_i \\in S} n_iS_i + \\sum_{i = 1}^n m_iQ_i, \\qquad n_i, m_i \\in \\Z. $$\nThus $ G $ is finitely generated by $ S \\cup \\cb{Q_i} $.\n\\end{proof}\n\nMordell's theorem is simply an application of the general descent procedure.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:mordell}]\nThe three properties of the function $ h $ will be given in Propositions \\ref{prop:finiteness}, \\ref{prop:upperbound}, \\ref{prop:lowerbound} of the next section. The weak version of the theorem will be given in Theorem \\ref{thm:weakmordell} of the section after the next. Applying descent to $ G = E\\rb{\\Q} $ with $ h $ gives that $ E\\rb{\\Q} $ is finitely generated.\n\\end{proof}\n\nThe next subsections will be devoted to proving these claims.\n\n\\pagebreak\n\n\\subsection{Mordell's theorem: heights}\n\nThe function $ h $ can be defined as follows.\n\n\\begin{definition}[Height]\nThe \\textbf{height} of a point $ P \\in E\\rb{\\Q} $ is a function $ h\\rb{P} : E\\rb{\\Q} \\to \\R_{\\ge 0} $ defined by $ h\\rb{P} = \\log_2\\rb{H\\rb{P}} $, where\n$$ H\\rb{P} = \\begin{dcases} \\max\\cb{\\abs{p}, \\abs{q}} & P = \\rb{\\dfrac{p}{q}, y}, \\ \\gcd\\rb{p, q} = 1 \\\\ 1 & P = \\mathcal{O} \\end{dcases}. $$\n\\end{definition}\n\n\\begin{remark}\nThe above definition for heights is chosen due to its simplicity, and is not the \\emph{canonical height} in the general literature. The theory of height functions will not be discussed here.\n\\end{remark}\n\nThe three intended properties of height function will then be proven, the first of which states that there are a finite number of points less than a given height. This property is trivial and stated as follows.\n\n\\begin{proposition}\n\\label{prop:finiteness}\nThe set $ S = \\cb{P \\in E\\rb{\\Q} \\mid h\\rb{P} \\le C_1} $ is finite for any $ C_1 \\in \\R_{\\ge 0} $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ C_1 \\in \\R_{\\ge 0} $ and $ P \\in E\\rb{\\Q} $ be a point. If $ P = \\mathcal{O} $, then $ P \\in S $. Otherwise $ P = \\rb{p / q, y} $, then $ \\max\\cb{\\abs{p}, \\abs{q}} \\le 2^{C_1} $, so $ -2^{C_1} \\le p, q \\le 2^{C_1} $. Thus $ \\abs{S} \\le \\rb{2^{C_1 + 1} + 1}^2 + 1 $ is finite.\n\\end{proof}\n\nThe second property provides an upper bound for the height of added points. This is relatively easy and is stated in the following proposition.\n\n\\begin{proposition}\n\\label{prop:upperbound}\nLet $ Q \\in E\\rb{\\Q} $. Then there is a constant $ C_2 \\in \\R_{\\ge 0} $ such that $ h\\rb{P + Q} \\le 2h\\rb{P} + C_2 $ for any $ P \\in E\\rb{\\Q} $.\n\\end{proposition}\n\n\\begin{proof}\nIf $ P = \\mathcal{O} $ or $ Q = \\mathcal{O} $ or $ P + Q = \\mathcal{O} $, let $ C_2 = 2h\\rb{Q} $ such that $ h\\rb{P + Q} \\le 2h\\rb{P} + 2h\\rb{Q} $. Otherwise $ P = \\rb{a, b} $ and $ Q = \\rb{a', b'} $ for $ a \\ne a' $ or $ a = a' $ and $ b = b' \\ne 0 $. Assume that $ a = a' $ and $ b = b' \\ne 0 $, then let $ C_2 = h\\rb{2Q} $ such that $ h\\rb{P + Q} = h\\rb{2Q} \\le 2h\\rb{P} + h\\rb{2Q} $. Assume otherwise that $ a \\ne a' $, and let $ C_2 = \\log_2\\rb{\\max\\cb{K_3, K_2}} $, where\n$$ K_1 = \\sqrt{1 + \\abs{A} + \\abs{B}}, \\qquad K_2 = 1 + \\abs{a'}, \\qquad K_3 = \\rb{\\abs{A} + \\abs{a'}}K_2 + 2\\rb{\\abs{B} + \\abs{b'}K_1}. $$\nThen $ a = p / d^2 $ and $ b = q / d^3 $ for some $ p, q \\in \\Z $ and some $ d \\in \\Z_{> 0} $ such that $ \\gcd\\rb{p, d} = \\gcd\\rb{q, d} = 1 $, and $ q^2 = p^3 + Apd^4 + Bd^6 $. Since $ H\\rb{P} = \\max\\cb{\\abs{p}, \\abs{d}^2} $, it holds that $ \\abs{p}, \\abs{d}^2 \\le H\\rb{P} $, so $ \\abs{d} \\le \\sqrt{H\\rb{P}} $ and\n$$ \\abs{q}\\abs{d} = \\abs{\\sqrt{p^3 + Apd^4 + Bd^6}}\\abs{d} \\le \\sqrt{\\abs{p}^3\\abs{d}^2 + \\abs{A}\\abs{p}\\abs{d}^6 + \\abs{B}\\abs{d}^8} \\le K_1H\\rb{P}^2. $$\nNow let $ P + Q = \\rb{a'', b''} $. By the addition formula,\n$$ a'' = \\dfrac{\\rb{A + aa'}\\rb{a + a'} + 2\\rb{B - bb'}}{\\rb{a - a'}^2} = \\dfrac{\\rb{Ad^2 + a'p}\\rb{p + a'd^2} + 2\\rb{Bd^4 - b'qd}}{\\rb{p - a'd^2}^2}. $$\nThus\n\\begin{align*}\nh\\rb{P + Q}\n& \\le \\log_2\\rb{\\max\\cb{\\abs{\\rb{Ad^2 + a'p}\\rb{p + a'd^2} + 2\\rb{Bd^4 - b'qd}}, \\abs{\\rb{p - a'd^2}^2}}} \\\\\n& \\le \\log_2\\rb{\\max\\cb{\\rb{\\abs{A}\\abs{d}^2 + \\abs{a'}\\abs{p}}\\rb{\\abs{p} + \\abs{a'}\\abs{d}^2} + 2\\rb{\\abs{B}\\abs{d}^4 + \\abs{b'}\\abs{q}\\abs{d}}, \\rb{\\abs{p} + \\abs{a'}\\abs{d}^2}^2}} \\\\\n& \\le \\log_2\\rb{\\max\\cb{K_3H\\rb{P}^2, K_2H\\rb{P}^2}}\n= \\log_2\\rb{H\\rb{P}^2\\max\\cb{K_3, K_2}}\n= 2h\\rb{P} + C_2.\n\\end{align*}\n\\end{proof}\n\n\\pagebreak\n\nThe third property provides an lower bound for the height of doubled points. It is more difficult as it involves seemingly arbitrary identities, and is stated in the following proposition.\n\n\\begin{proposition}\n\\label{prop:lowerbound}\nThere is a constant $ C_3 \\in \\R_{\\ge 0} $ such that $ h\\rb{2P} \\ge 4h\\rb{P} - C_3 $ for any $ P \\in E\\rb{\\Q} $.\n\\end{proposition}\n\n\\begin{proof}\nIf $ P = \\mathcal{O} $, let $ C_3 = 0 $ such that $ h\\rb{2P} \\ge 4h\\rb{P} $. If $ P = \\rb{a, 0} $, let $ C_3 = 4h\\rb{P} $ such that $ h\\rb{2P} \\ge 0 $. Otherwise $ P = \\rb{a, b} $ for $ b \\ne 0 $. Let $ a = p / q $ for some $ p \\in \\Z $ and some $ q \\in \\Z^* $ such that $ \\gcd\\rb{p, q} = 1 $, and let\n\\begin{align*}\np' & = p^4 - 2Ap^2q^2 - 8Bpq^3 + A^2q^4, \\\\\nq' & = 4p^3q + 4Apq^3 + 4Bq^4, \\\\\n\\lambda & = 12p^2q + 16Aq^3, \\\\\n\\mu & = -3p^3 + 5Apq^2 + 27Bq^3, \\\\\n\\lambda' & = \\rb{16A^3 + 108B^2}p^3 - 4A^2Bp^2q + \\rb{12A^4 + 88AB^2}pq^2 + \\rb{12A^3B + 96B^3}q^3, \\\\\n\\mu' & = A^2Bp^3 + \\rb{5A^4 + 32AB^2}p^2q + \\rb{26A^3B + 192B^3}pq^2 - \\rb{3A^5 + 24A^2B^2}q^3, \\\\\nK_1 & = 4\\max\\cb{12, 16\\abs{A}}, \\\\\nK_2 & = 4\\max\\cb{3, 5\\abs{A}, 27\\abs{B}}, \\\\\nK_3 & = 4\\max\\cb{16\\abs{A}^3 + 108B^2, 4A^2\\abs{B}, 12A^4 + 88\\abs{A}B^2, 12\\abs{A}^3\\abs{B} + 96\\abs{B}^3}, \\\\\nK_4 & = 4\\max\\cb{A^2\\abs{B}, 5A^4 + 32\\abs{A}B^2, 26\\abs{A}^3\\abs{B} + 192\\abs{B}^3, 3\\abs{A}^5 + 24A^2B^2}.\n\\end{align*}\nThen it can be tediously verified that $ \\lambda p' + \\mu q' = 4\\Delta'_E q^7 $ and $ \\lambda' p' + \\mu' q' = 4\\Delta'_E p^7 $. Since $ \\abs{p}^2\\abs{q} $ and $ \\abs{p}\\abs{q}^2 $ are between $ \\abs{p}^3 $ and $ \\abs{q}^3 $, it holds that $ \\max\\cb{\\abs{p}^3, \\abs{p}^2\\abs{q}, \\abs{p}\\abs{q}^2, \\abs{q}^3} = \\max\\cb{\\abs{p}^3, \\abs{q}^3} $. Then it can also be verified that\n$$ \\abs{\\lambda} \\le K_1M, \\qquad \\abs{\\mu} \\le K_2M, \\qquad \\abs{\\lambda'} \\le K_3M, \\qquad \\abs{\\mu'} \\le K_4M, $$\nfor $ M = \\max\\cb{\\abs{p}^3, \\abs{q}^3} $, so let $ C_3 = \\log_2\\rb{2\\max\\cb{K_1, K_2, K_3, K_4}} $. Since\n\\begin{align*}\n4\\abs{\\Delta'_E}\\max\\cb{\\abs{p}^3, \\abs{q}^3}\\rb{\\max\\cb{\\abs{p}, \\abs{q}}}^4\n& = 4\\abs{\\Delta'_E}\\max\\cb{\\abs{q}^7, \\abs{p}^7}\n= \\max\\cb{\\abs{4\\Delta'_E q^7}, \\abs{4\\Delta'_E p^7}} \\\\\n& \\le \\max\\cb{\\abs{\\lambda}\\abs{p'} + \\abs{\\mu}\\abs{q'}, \\abs{\\lambda'}\\abs{p'} + \\abs{\\mu'}\\abs{q'}} \\\\\n& \\le 2\\max\\cb{\\abs{\\lambda}, \\abs{\\mu}, \\abs{\\lambda'}, \\abs{\\mu'}}\\max\\cb{\\abs{p'}, \\abs{q'}} \\\\\n& \\le 2M\\max\\cb{K_1, K_2, K_3, K_4}\\max\\cb{\\abs{p'}, \\abs{q'}},\n\\end{align*}\nit holds that\n$$ 4\\abs{\\Delta'_E}H\\rb{P}^4 = 4\\abs{\\Delta'_E}\\rb{\\max\\cb{\\abs{p}, \\abs{q}}}^4 \\le 2\\max\\cb{K_1, K_2, K_3, K_4}\\max\\cb{\\abs{p'}, \\abs{q'}}. $$\nNow let $ 2P = \\rb{a', b'} $. By the duplication formula,\n$$ a' = \\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4b^2} = \\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4a^3 + 4Aa + 4B} = \\dfrac{p'}{q'}. $$\nSince $ g = \\gcd\\rb{p', q'} \\mid \\gcd\\rb{4\\Delta'_E p^7, 4\\Delta'_E q^7} = 4\\Delta'_E $, it holds that $ 1 \\le \\abs{g} \\le 4\\abs{\\Delta'_E} $. Thus\n\\begin{align*}\nh\\rb{2P}\n& = \\log_2\\rb{\\max\\cb{\\abs{\\dfrac{p'}{g}}, \\abs{\\dfrac{q'}{g}}}}\n= \\log_2\\rb{\\dfrac{\\max\\cb{\\abs{p'}, \\abs{q'}}}{\\abs{g}}} \\\\\n& \\ge \\log_2\\rb{\\dfrac{\\max\\cb{\\abs{p'}, \\abs{q'}}}{4\\abs{\\Delta'_E}}}\n\\ge \\log_2\\rb{\\dfrac{H\\rb{P}^4}{2\\max\\cb{K_1, K_2, K_3, K_4}}}\n\\ge 4h\\rb{P} - C_3.\n\\end{align*}\n\\end{proof}\n\nThe properties of the height function $ h $ are now verified.\n\n\\pagebreak\n\n\\subsection{Mordell's theorem: weak Mordell}\n\nThe weak version of Mordell's theorem, restricted to $ \\Q $, states that the index of the normal subgroup $ 2E\\rb{\\Q} = \\cb{2P \\mid P \\in E\\rb{\\Q}} $ is finite.\n\n\\begin{theorem}[Weak Mordell]\n\\label{thm:weakmordell}\n$ \\abs{E\\rb{\\Q} : 2E\\rb{\\Q}} $ is finite.\n\\end{theorem}\n\nAs full proofs of the weak theorem, such as in VIII.1 of \\cite{gtm}, requires further prerequisites on algebraic number theory, particularly finiteness of the \\emph{ideal class group} of number fields, only an alternative proof is given, of which the special case of a rational $ 2 $-torsion point $ \\rb{a_0, 0} $ is assumed. Since there is a $ j $-invariant affine transformation $ \\rb{x, y} \\mapsto \\rb{x + a_0, y} $, there is an isomorphism from $ E $ to the curve given by the Weierstrass equation\n$$ y^2 = \\rb{x + a_0}^3 + A\\rb{x + a_0} + B \\qquad \\implies \\qquad y^2 = x^3 + 3a_0x^2 + \\rb{3a_0^2 + A}x. $$\nHence for this subsection and the next, assume without loss of generality that $ a_0 = 0 $ and\n$$ T = \\rb{a_0, 0} = \\rb{0, 0} \\in E : y^2 = x^3 + Ax^2 + Bx, \\qquad A, B \\in \\Z. $$\nThe modified discriminant and group law is then given in the following lemma.\n\n\\begin{lemma}\nThe following properties hold:\n\\begin{enumerate}\n\\item $ B \\ne 0 $ and $ A^2 - 4B \\ne 0 $.\n\\item Let $ P = \\rb{a, b} \\in E\\rb{\\Q} $ and $ Q = \\rb{a', b'} \\in E\\rb{\\Q} $ be points such that $ a \\ne a' $ and $ P + Q = \\rb{a'', b''} \\in E\\rb{\\Q} $. Then $ aa'a'' = \\mu^2 $ for some $ \\mu \\in \\Q $.\n\\item Let $ P = \\rb{a, b} \\in E\\rb{\\Q} $ be a point such that $ b \\ne 0 $. Then\n$$ 2P = \\rb{\\dfrac{\\rb{a^2 - B}^2}{4b^2}, \\dfrac{\\rb{a^2 - B}\\rb{a^4 + B^2 + 2Aa^3 + 2ABa + 6Ba^2}}{8b^3}} \\in E\\rb{\\Q}. $$\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nThe negation formula remains unmodified, so $ -\\rb{a, b} = \\rb{a, -b} $ for any point $ \\rb{a, b} \\in E $.\n\\begin{enumerate}\n\\item Since $ E $ is smooth and the discriminant is\n$$ \\Delta_E = 9\\rb{4A}\\rb{2B}\\rb{0} - \\tfrac{1}{4}\\rb{4A}^2\\rb{\\rb{4A}\\rb{0} - \\rb{2B}^2} - 8\\rb{2B}^3 - 27\\rb{0}^2 = 16B^2\\rb{A^2 - 4B}, $$\n$ 16B^2\\rb{A^2 - 4B} \\ne 0 $. Thus $ B \\ne 0 $ and $ A^2 - 4B \\ne 0 $.\n\\item The line joining $ P $ and $ Q $ is\n$$ L : y = \\lambda x + \\mu, \\qquad \\lambda = \\dfrac{b - b'}{a - a'}, \\qquad \\mu = \\dfrac{ab' - a'b}{a - a'}, $$\nwhich intersects $ E $ at $ x^3 - \\rb{\\lambda^2 - A}x^2 + \\rb{B - 2\\lambda\\mu}x - \\mu^2 = 0 $. Let $ P * Q = -\\rb{P + Q} = \\rb{a'', -b''} $. Thus comparing coefficients gives $ \\mu^2 = aa'a'' $.\n\\item The tangent at $ P $ is\n$$ L : y = \\lambda x + \\mu, \\qquad \\lambda = \\dfrac{3a^2 + 2Aa + B}{2b}, \\qquad \\mu = \\dfrac{b^2 - Aa^2 - 2Ba}{2b}, $$\nwhich intersects $ E $ at $ x^3 - \\rb{\\lambda^2 - A}x^2 + \\rb{B - 2\\lambda\\mu}x - \\mu^2 = 0 $. Let $ P * P = -2P = \\rb{a', -b'} $, so comparing coefficients gives $ \\lambda^2 - A = 2a + a' $. Thus\n$$ 2P = \\rb{\\lambda^2 - A - 2a, \\mu - \\lambda\\rb{\\lambda^2 - A - 2a}} \\in E\\rb{\\Q}. $$\n\\end{enumerate}\n\\end{proof}\n\nThe above proof is brief but can be verified manually. Let a related curve be\n$$ E' : y^2 = x^3 + A'x^2 + B'x, \\qquad A' = -2A, \\qquad B' = A^2 - 4B, $$\nsuch that $ T \\in E' $ and $ B' \\ne 0 $. Then $ A'^2 - 4B' = \\rb{-2A}^2 - 4\\rb{A^2 - 4B} = 16B $, and the group law is similar to that of $ E $ but with $ A' $ and $ B' $ instead of $ A $ and $ B $. Now let the two maps $ \\phi : E \\to E' $ and $ \\psi : E' \\to E $ be defined by\n$$ \\phi\\rb{P} = \\begin{dcases} \\rb{\\dfrac{b^2}{a^2}, \\dfrac{b\\rb{a^2 - B}}{a^2}} & P = \\rb{a, b} \\ne T \\\\ \\mathcal{O} & P \\in \\cb{\\mathcal{O}, T} \\end{dcases}, \\qquad \\psi\\rb{P} = \\begin{dcases} \\rb{\\dfrac{b^2}{4a^2}, \\dfrac{b\\rb{a^2 - B'}}{8a^2}} & P = \\rb{a, b} \\ne T \\\\ \\mathcal{O} & P \\in \\cb{\\mathcal{O}, T} \\end{dcases}. $$\nThese two maps are related in the obvious way, where one can be seen as the scaling of the other. They also relate the two elliptic curves, as seen in the following lemma.\n\n\\begin{lemma}\n$ \\phi : E \\to E' $ and $ \\psi : E' \\to E $ are isogenies such that $ \\psi \\circ \\phi = \\sb{2}_E $ and $ \\phi \\circ \\psi = \\sb{2}_{E'} $.\n\\end{lemma}\n\n\\begin{remark}\nPreserving the point at infinity induces a group homomorphism, but the full property can be tediously verified in III.4 of \\cite{utm} for each case of the group law.\n\\end{remark}\n\n\\begin{proof}\nFor any point $ P = \\rb{a, b} \\in E $,\n$$ \\rb{\\dfrac{b^2}{a^2}}^3 + A'\\rb{\\dfrac{b^2}{a^2}}^2 + B'\\dfrac{b^2}{a^2}\n= \\dfrac{b^2}{a^4}\\rb{\\dfrac{\\rb{b^2 - Aa^2}^2 - 4Ba^4}{a^2}}\n= \\dfrac{b^2}{a^4}\\rb{\\dfrac{\\rb{a^3 + Ba}^2 - 4Ba^4}{a^2}}\n= \\rb{\\dfrac{b\\rb{a^2 - B}}{a^2}}^2, $$\nso $ \\phi\\rb{P} \\in E' $. Since $ \\phi\\rb{T} = \\phi\\rb{\\mathcal{O}} = \\mathcal{O} $, it holds that $ \\phi $ is a well-defined non-constant morphism, and hence an isogeny. Since $ \\psi $ can be seen as applying $ \\chi \\circ \\phi $ to $ E $, where $ \\chi $ is the $ j $-invariant affine transformation $ \\rb{x, y} \\mapsto \\rb{x / 4, y / 8} $, it is also a well-defined non-constant morphism, and hence an isogeny. Now let $ P \\in E $. If $ P = \\mathcal{O} $ or $ P = \\rb{a, 0} $, then $ \\rb{\\psi \\circ \\phi}\\rb{P} = \\mathcal{O} = 2P $. Otherwise $ P = \\rb{a, b} $ such that $ a \\ne 0 $ and $ b \\ne 0 $, then\n\\begin{align*}\n\\rb{\\psi \\circ \\phi}\\rb{P}\n& = \\rb{\\dfrac{\\rb{b\\rb{a^2 - B} / a^2}^2}{4\\rb{b^2 / a^2}^2}, \\dfrac{\\rb{b\\rb{a^2 - B} / a^2}\\rb{\\rb{b^2 / a^2}^2 - B'}}{8\\rb{b^2 / a^2}^2}} \\\\\n& = \\rb{\\dfrac{\\rb{a^2 - B}^2}{4b^2}, \\dfrac{\\rb{a^2 - B}\\rb{b^4 - \\rb{A^2 - 4B}a^4}}{8b^3a^2}} \\\\\n& = \\rb{\\dfrac{\\rb{a^2 - B}^2}{4b^2}, \\dfrac{\\rb{a^2 - B}\\rb{a^4 + B^2 + 2Aa^3 + 2ABa + 6Ba^2}}{8b^3}} = 2P.\n\\end{align*}\nHence $ \\psi \\circ \\phi = \\sb{2}_E $. Similarly let $ P' \\in E' $. If $ P' = \\mathcal{O} $ or $ P' = \\rb{a, 0} $, then $ \\rb{\\phi \\circ \\psi}\\rb{P'} = \\mathcal{O} = 2P' $. Otherwise $ P' = \\rb{a, b} $ such that $ a \\ne 0 $ and $ b \\ne 0 $, then\n\\begin{align*}\n\\rb{\\phi \\circ \\psi}\\rb{P'}\n& = \\rb{\\dfrac{\\rb{b\\rb{a^2 - B'} / 8a^2}^2}{\\rb{b^2 / 4a^2}^2}, \\dfrac{\\rb{b\\rb{a^2 - B'} / 8a^2}\\rb{\\rb{b^2 / 4a^2}^2 - B}}{\\rb{b^2 / 4a^2}^2}} \\\\\n& = \\rb{\\dfrac{\\rb{a^2 - B'}^2}{4b^2}, \\dfrac{\\rb{a^2 - B'}\\rb{b^4 - 16\\rb{\\rb{A'^2 - 4B'} / 16}a^4}}{8b^3a^2}} \\\\\n& = \\rb{\\dfrac{\\rb{a^2 - B'}^2}{4b^2}, \\dfrac{\\rb{a^2 - B'}\\rb{a^4 + B'^2 + 2A'a^3 + 2A'B'a + 6B'a^2}}{8b^3}} = 2P'.\n\\end{align*}\nThus $ \\phi \\circ \\psi = \\sb{2}_{E'} $.\n\\end{proof}\n\nHence the multiplication by $ 2 $ map can be decomposed into two isogenies $ \\phi $ and $ \\psi $. As only the image of these isogenies will be used, their standard forms will not be used to prevent confusion.\n\n\\begin{remark}\nThese two isogenies are \\emph{dual isogenies} to each other. Any isogeny of degree $ n \\in \\Z_{> 0} $ has a dual isogeny, which composes with it to give two multiplication by $ n $ maps in their respective domains.\n\\end{remark}\n\n\\pagebreak\n\nThe image of the isogeny $ \\psi $ depends on whether $ B $ is a perfect square or whether $ x $ coordinates are in the normal subgroup $ \\rb{\\Q^*}^2 = \\cb{q^2 \\mid q \\in \\Q^*} $. In particular, the equation $ x^3 + A'x + B' = 0 $ with discriminant $ 16\\rb{A'^2 - 4B'} = 16B $ has two solutions in $ \\Q^* $ iff $ 16B \\in \\rb{\\Z^*}^2 $, or $ B \\in \\rb{\\Z^*}^2 $, stated as follows.\n\n\\begin{lemma}\nThe image $ Im\\rb{\\psi} $ is such that:\n\\begin{itemize}\n\\item $ \\mathcal{O} \\in Im\\rb{\\psi} $,\n\\item $ T \\in Im\\rb{\\psi} $ iff $ B \\in \\rb{\\Z^*}^2 $, and\n\\item $ \\rb{a, b} \\ne T \\in Im\\rb{\\psi} $ iff $ a \\in \\rb{\\Q^*}^2 $.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nSince $ \\psi\\rb{\\mathcal{O}} = \\mathcal{O} $, it holds that $ \\mathcal{O} \\in Im\\rb{\\psi} $. Now $ T \\in Im\\rb{\\psi} $ iff there is a point $ P = \\rb{a, b} \\in E'\\rb{\\Q} $ such that $ \\psi\\rb{P} = T $ and $ 0 = b^2 / 4a^2 $. This holds iff $ a \\in \\Q^* $ and $ b = 0 $, or $ B \\in \\rb{\\Z^*}^2 $. Now assume that $ P = \\rb{a, b} \\ne T \\in Im\\rb{\\psi} $. Then there is a point $ Q = \\rb{a', b'} \\in E'\\rb{\\Q} $ such that $ \\psi\\rb{Q} = P $, so $ a = b'^2 / 4a'^2 = \\rb{b' / 2a}^2 \\in \\rb{\\Q^*}^2 $. Conversely assume that $ P = \\rb{a, b} \\ne T \\in E\\rb{\\Q} $ and $ a \\in \\rb{\\Q^*}^2 $. Then $ a = c^2 $ for some $ c \\in \\Q^* $, so\n$$ b^2 = c^6 - \\dfrac{A'c^4}{2} + \\dfrac{A'^2 - 4B'}{16}c^2 \\qquad \\implies \\qquad B' = \\rb{2c^2 - \\dfrac{A'}{2} + \\dfrac{2b}{c}}\\rb{2c^2 - \\dfrac{A'}{2} - \\dfrac{2b}{c}}. $$\nNow let $ Q = \\rb{a', b'} $, where $ a' = 2c^2 - A' / 2 + 2b / c $ and $ b' = 2a'c $, such that $ B' = a'\\rb{a' - 4b / c} $. Then\n$$ a'^3 + A'a'^2 + B'a' = a'^3 + A'a'^2 + a'^2\\rb{a' - \\dfrac{2b}{c}} = 2a'^2\\rb{a' + \\dfrac{A'}{2} - \\dfrac{4b}{c}} = 4a'^2c^2 = b'^2, $$\nso $ Q \\in E'\\rb{\\Q} $, and\n$$ \\psi\\rb{Q} = \\rb{\\dfrac{b'^2}{4a'^2}, \\dfrac{b'\\rb{a'^2 - B'}}{8a'^2}} = \\rb{\\dfrac{4a'^2c^2}{4a'^2}, \\dfrac{2a'c\\rb{a'^2 - a'\\rb{a' - 4b / c}}}{8a'^2}} = \\rb{c^2, \\dfrac{c\\rb{4a'b / c}}{4a'}} = \\rb{a, b} = P. $$\nThus $ P \\in Im\\rb{\\psi} $.\n\\end{proof}\n\nThe image of the isogeny $ \\phi $ can be characterised analogously, and will not be explicitly stated here. Now let another map be defined as\n$$ \\alpha : E\\rb{\\Q} \\to \\Q^* / \\rb{\\Q^*}^2, \\qquad \\alpha\\rb{P} = \\begin{cases} \\rb{\\Q^*}^2a & P = \\rb{a, b} \\ne T \\\\ \\rb{\\Q^*}^2B & P = T \\\\ \\rb{\\Q^*}^2 & P = \\mathcal{O} \\end{cases}. $$\nThen $ \\psi $ and $ \\alpha $ induce an \\emph{exact sequence} $ E'\\rb{\\Q} \\xrightarrow{\\psi} E\\rb{\\Q} \\xrightarrow{\\alpha} \\Q^* / \\rb{\\Q^*}^2 $, which can be stated more concretely in the following lemma.\n\n\\begin{lemma}\n$ \\alpha : E\\rb{\\Q} \\to \\Q^* / \\rb{\\Q^*}^2 $ is a group homomorphism such that $ Im\\rb{\\psi} = Ker\\rb{\\alpha} $.\n\\end{lemma}\n\n\\begin{proof}\nLet $ P, Q \\in E\\rb{\\Q} $ be points. If $ P = \\mathcal{O} $,\n$$ \\alpha\\rb{P}\\alpha\\rb{Q} = \\rb{\\Q^*}^2\\alpha\\rb{Q} = \\alpha\\rb{Q} = \\alpha\\rb{P + Q}, $$\nor similar for $ Q = \\mathcal{O} $. If $ P = \\rb{a, b} $ and $ Q = \\rb{a', b'} $ such that $ a \\ne a' $ and $ P + Q = \\rb{a'', b''} \\in E\\rb{\\Q} $, then\n$$ \\alpha\\rb{P}\\alpha\\rb{Q} = \\rb{\\Q^*}^2a\\rb{\\Q^*}^2a' = \\rb{\\Q^*}^2aa' = \\rb{\\Q^*}^2\\dfrac{\\mu^2}{a''} = \\rb{\\Q^*}^2\\mu^2a'' = \\rb{\\Q^*}^2a'' = \\alpha\\rb{P + Q}. $$\nOtherwise $ P = \\rb{a, b} $ and $ Q = \\rb{a, b'} $, then\n$$ \\alpha\\rb{P}\\alpha\\rb{Q} = \\rb{\\Q^*}^2a\\rb{\\Q^*}^2a = \\rb{\\Q^*}^2a^2 = \\rb{\\Q^*}^2 = \\rb{\\Q^*}^2\\dfrac{\\rb{a^2 - B}^2}{4b^2} = \\alpha\\rb{P + Q}. $$\nHence $ \\alpha $ is a group homomorphism. Now $ \\mathcal{O} \\in Ker\\rb{\\alpha} $, the point $ T \\in Ker\\rb{\\alpha} $ iff $ B \\in \\rb{\\Q^*}^2 $, and a point $ \\rb{a, b} \\ne T \\in Ker\\rb{\\alpha} $ iff $ a \\in \\rb{\\Q^*}^2 $. Thus $ Im\\rb{\\psi} = Ker\\rb{\\alpha} $.\n\\end{proof}\n\nThe image of the group homomorphism $ \\alpha $ can again be characterised, as being contained in a finite subgroup of $ \\Q^* / \\rb{\\Q^*}^2 $. Now let $ S\\rb{B} $ be the set of primes $ p \\in \\Z_{> 0} $ such that $ p \\mid B $, and let\n$$ G\\rb{B} = \\cb{\\rb{\\Q^*}^2\\rb{\\prod_{p \\in S} p} \\ \\Bigg| \\ S \\subseteq S\\rb{B}} \\cup \\cb{\\rb{\\Q^*}^2\\rb{-\\prod_{p \\in S} p} \\ \\Bigg| \\ S \\subseteq S\\rb{B}}. $$\nRecalling the fact that for any point $ \\rb{a, b} \\in E\\rb{\\Q} $,\n$$ a = p / d^2, \\qquad b = q / d^3, \\qquad p, q \\in \\Z, \\qquad d \\in \\Z_{> 0}, $$ such that $ \\gcd\\rb{p, d} = \\gcd\\rb{q, d} = 1 $, the following lemma characterises $ \\alpha $.\n\n\\begin{lemma}\n$ G\\rb{B} $ is a group such that $ \\abs{G\\rb{B}} = 2^{\\abs{S\\rb{B}} + 1} $ and $ Im\\rb{\\alpha} \\le G\\rb{B} \\le \\Q^* / \\rb{\\Q^*}^2 $.\n\\end{lemma}\n\n\\begin{proof}\nSince $ \\emptyset \\subseteq S\\rb{B} $, it holds that $ \\rb{\\Q^*}^2 \\in G\\rb{B} $. Let $ a, b \\in G\\rb{B} $. Then\n$$ a = \\rb{\\Q^*}^2 j p_1 \\dots p_n p'_1 \\dots p'_{n'}, \\qquad b = \\rb{\\Q^*}^2 j' p_1 \\dots p_n p''_1 \\dots p''_{n''} $$\nfor some $ j, j' \\in \\cb{-1, 1} $ and some distinct primes $ p_i, p'_i, p''_i \\in S\\rb{B} $, so\n$$ \\dfrac{a}{b} = \\dfrac{\\rb{\\Q^*}^2 j p_1 \\dots p_n p'_1 \\dots p'_{n'}}{\\rb{\\Q^*}^2 j' p_1 \\dots p_n p''_1 \\dots p''_{n''}} = \\rb{\\Q^*}^2\\dfrac{j p'_1 \\dots p'_{n'}}{j' p''_1 \\dots p''_{n''}} = \\rb{\\Q^*}^2 j j' p'_1 \\dots p'_{n'} p''_1 \\dots p''_{n''} \\in G\\rb{B}. $$\nHence $ G\\rb{B} \\le \\Q^* / \\rb{\\Q^*}^2 $ and $ \\abs{G\\rb{B}} = 2^{\\abs{S\\rb{B}}} + 2^{\\abs{S\\rb{B}}} = 2^{\\abs{S\\rb{B}} + 1} $. Now let $ P \\in E\\rb{\\Q} $ be a point. If $ P = \\mathcal{O} $, then $ \\alpha\\rb{P} = \\rb{\\Q^*}^2 \\in G\\rb{B} $. If $ P = T $, then $ \\alpha\\rb{P} = \\rb{\\Q^*}^2B \\in G\\rb{B} $. Otherwise $ P = \\rb{a, b} \\ne T $, then $ a = r / d^2 $ and $ b = s / d^3 $ for some $ r, s \\in \\Z $ and some $ d \\in \\Z_{> 0} $ such that $ \\gcd\\rb{r, d} = \\gcd\\rb{s, d} = 1 $ and $ s^2 = r^3 + Ar^2d^2 + Brd^4 = r\\rb{r^2 + Ard^2 + Bd^4} $. Let $ g = \\gcd\\rb{r, r^2 + Ard^2 + Bd^4} $, then $ r = cg $ and $ r^2 + Ard^2 + Bd^4 = c'g $ for some $ c, c' \\in \\Z_{\\ge 0} $ such that $ \\gcd\\rb{c, c'} = 1 $. Since $ s^2 = \\rb{cg}\\rb{c'g} = cc'g^2 $, it holds that $ \\rb{s / g}^2 = cc' $, so $ c = k q_1^2 \\cdots q_m^2 $ for some $ k \\in \\cb{-1, 1} $ and some primes $ q_i \\in \\Z_{> 0} $. Since $ g \\mid r $ and $ g \\mid Bd^4 $, it also holds that $ g \\mid B $, so $ g = k' q'_1 \\dots q'_{m'} $ for some $ k' \\in \\cb{-1, 1} $ and some primes $ q'_i \\in \\Z_{> 0} $ such that $ q'_i \\mid B $, and hence $ q'_i \\in S\\rb{B} $. Hence\n$$ \\alpha\\rb{P} = \\rb{\\Q^*}^2a = \\rb{\\Q^*}^2\\dfrac{r}{d^2} = \\rb{\\Q^*}^2\\dfrac{k k' q_1^2 \\cdots q_m^2 q'_1 \\dots q'_{m'}}{d^2} = \\rb{\\Q^*}^2 k k' q'_1 \\dots q'_{m'} \\in G\\rb{B}. $$\nThus $ Im\\rb{\\alpha} \\le G\\rb{B} $.\n\\end{proof}\n\nA similar group homomorphism $ \\alpha' : E'\\rb{\\Q} \\to \\Q^* / \\rb{\\Q^*}^2 $ can again be characterised analogously, and will not be explicitly stated here. The weak theorem can then be proven here, for the special case of a rational $ 2 $-torsion point.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:weakmordell}]\nThe first isomorphism theorem with the preceding lemmas give two inclusions\n$$ \\dfrac{E\\rb{\\Q}}{Im\\rb{\\psi}} = \\dfrac{E\\rb{\\Q}}{Ker\\rb{\\alpha}} \\cong Im\\rb{\\alpha} \\le G\\rb{B}, \\qquad \\dfrac{E'\\rb{\\Q}}{Im\\rb{\\phi}} = \\dfrac{E'\\rb{\\Q}}{Ker\\rb{\\alpha'}} \\cong Im\\rb{\\alpha'} \\le G\\rb{B'}, $$\nwhich give finite indices\n$$ n = \\abs{E\\rb{\\Q} : Im\\rb{\\psi}} \\le \\abs{G\\rb{B}} = 2^{\\abs{S\\rb{B}} + 1}, \\qquad m = \\abs{E'\\rb{\\Q} : Im\\rb{\\phi}} \\le \\abs{G\\rb{B'}} = 2^{\\abs{S\\rb{B'}} + 1}. $$\nLet $ P_1, \\dots, P_n \\in E\\rb{\\Q} $ be representative points such that $ Im\\rb{\\psi} + P_i \\in E\\rb{\\Q} / Im\\rb{\\psi} $ are distinct cosets, and let $ Q_1, \\dots, Q_m \\in E'\\rb{\\Q} $ be representative points such that $ Im\\rb{\\phi} + Q_i \\in E'\\rb{\\Q} / Im\\rb{\\phi} $ are distinct cosets. Now let $ P \\in E\\rb{\\Q} $ be a point. Then $ Im\\rb{\\psi} + P = Im\\rb{\\psi} + P_j $ for some $ j \\in \\cb{1, \\dots, n} $, so $ P = \\psi\\rb{Q} + P_j $ for some $ Q \\in E'\\rb{\\Q} $ and $ \\psi\\rb{Q} \\in Im\\rb{\\psi} $. Similarly $ Im\\rb{\\phi} + Q = Im\\rb{\\phi} + Q_k $ for some $ k \\in \\cb{1, \\dots, m} $, so $ Q = \\phi\\rb{P'} + Q_k $ for some $ P' \\in E\\rb{\\Q} $ and $ \\phi\\rb{P'} \\in Im\\rb{\\phi} $. Hence\n$$ P = \\psi\\rb{Q} + P_j = \\psi\\rb{\\phi\\rb{P'} + Q_k} + P_j = \\psi\\rb{\\phi\\rb{P'}} + \\psi\\rb{Q_k} + P_j \\in 2E\\rb{\\Q} + \\psi\\rb{Q_k} + P_j, $$\nand $ \\psi\\rb{Q_k} + P_j \\in E\\rb{\\Q} $ represent all cosets in $ E\\rb{\\Q} / 2E\\rb{\\Q} $. Thus\n$$ \\abs{E\\rb{\\Q} : 2E\\rb{\\Q}} \\le nm = 2^{\\rb{\\abs{S\\rb{B}} + 1}\\rb{\\abs{S\\rb{B'}} + 1}} $$\nis finite.\n\\end{proof}\n\nThe proof of Mordell's theorem is now complete.\n\n\\pagebreak\n\n\\subsection{Rank computation}\n\nA direct application of Mordell's theorem would be the fundamental theorem of finite abelian groups,\n$$ E\\rb{\\Q} \\cong \\Z^r \\oplus \\bigoplus_{i = 1}^m \\Z_{n_i}, \\qquad r, m \\in \\Z_{\\ge 0}, \\qquad n_i \\in \\Z_{> 1}, $$\nsuch that each $ n_i \\mid n_{i + 1} $. Thus for any point $ P \\in E\\rb{\\Q} $,\n$$ P = \\sum_{i = 1}^r r_iP_i + \\sum_{i = 1}^m m_iQ_i, \\qquad r_i \\in \\Z, \\qquad m_i \\in \\Z_{n_i}, \\qquad P_i, Q_i \\in E\\rb{\\Q}. $$\nWhile the torsion subgroup can be easily computed, the rank $ r $ is generally difficult to compute, and can only be made slightly easier with Mordell's theorem. Noting that $ \\bigoplus_i \\rb{G_i / H_i} \\cong \\rb{\\bigoplus_i G_i} / \\rb{\\bigoplus_i H_i} $ for any groups $ G_i, H_i $, the following proposition gives a direct formula for the rank.\n\n\\begin{proposition}\nThe rank $ r = rk\\rb{E\\rb{\\Q}} $ is such that\n$$ 2^r = \\tfrac{1}{4}\\abs{Im\\rb{\\alpha}}\\abs{Im\\rb{\\alpha'}}. $$\n\\end{proposition}\n\n\\begin{proof}\nThe fundamental theorem of finite abelian groups gives\n$$ \\dfrac{E\\rb{\\Q}}{2E\\rb{\\Q}} \\cong \\dfrac{\\Z^r \\oplus \\bigoplus_{i = 1}^m \\Z_{n_i}}{r2\\Z \\oplus \\bigoplus_{i = 1}^m 2\\Z_{n_i}} \\cong r\\rb{\\dfrac{\\Z}{2\\Z}} \\oplus \\bigoplus_{i = 1}^m \\dfrac{\\Z_{n_i}}{2\\Z_{n_i}}. $$\nThen $ \\Z / 2\\Z \\cong \\Z_2 $. If $ n_i \\nmid 2 $, then $ 2^{-1} \\in \\Z_{n_i} $, so $ \\Z_{n_i} \\cong 2\\Z_{n_i} $ and $ \\Z_{n_i} / 2\\Z_{n_i} \\cong 0 $, otherwise $ n_i \\mid 2 $. Now $ P \\in E\\rb{\\Q}\\sb{2} $ iff $ 2P = 0 $, or each $ r_i = 0 $ and each $ 2m_i = 0 \\mod n_i $, which holds iff $ m_i = 0 $ or $ n_i \\mid 2 $, so $ E\\rb{\\Q}\\sb{2} = \\bigoplus_{n_i \\mid 2} \\Z_{n_i} $. Hence\n$$ \\dfrac{E\\rb{\\Q}}{2E\\rb{\\Q}} \\cong \\Z^r_2 \\oplus E\\rb{\\Q}\\sb{2} \\qquad \\implies \\qquad \\abs{E\\rb{\\Q} : 2E\\rb{\\Q}} = 2^r\\abs{E\\rb{\\Q}\\sb{2}}. $$\nNow let $ \\theta : E'\\rb{\\Q} \\to Im\\rb{\\psi} / 2E\\rb{\\Q} $ be a surjective group homomorphism defined by $ \\theta\\rb{P} = 2E\\rb{\\Q} + \\psi\\rb{P} $. Then $ P \\in Ker\\rb{\\theta} $ iff $ \\psi\\rb{P} \\in 2E\\rb{\\Q} $, or $ \\psi\\rb{P} = \\psi\\rb{\\phi\\rb{Q}} $ for some $ Q \\in E\\rb{\\Q} $. This holds iff $ \\psi\\rb{P - \\phi\\rb{Q}} = 0 $, or $ P - \\phi\\rb{Q} \\in Ker\\rb{\\psi} $ and $ P \\in Ker\\rb{\\psi} + Im\\rb{\\phi} $. Then the three isomorphism theorems with $ Ker\\rb{\\theta} = Ker\\rb{\\psi} + Im\\rb{\\phi} $ give\n$$ \\dfrac{Im\\rb{\\psi}}{2E\\rb{\\Q}} \\cong \\dfrac{E'\\rb{\\Q}}{Ker\\rb{\\psi} + Im\\rb{\\phi}} \\cong \\dfrac{\\dfrac{E'\\rb{\\Q}}{Im\\rb{\\phi}}}{\\dfrac{Ker\\rb{\\psi} + Im\\rb{\\phi}}{Im\\rb{\\phi}}} \\cong \\dfrac{\\dfrac{E'\\rb{\\Q}}{Im\\rb{\\phi}}}{\\dfrac{Ker\\rb{\\psi}}{Ker\\rb{\\psi} \\cap Im\\rb{\\phi}}}. $$\nHence\n$$ \\abs{E\\rb{\\Q} : 2E\\rb{\\Q}} = \\dfrac{\\abs{E\\rb{\\Q} : Im\\rb{\\psi}}\\abs{E'\\rb{\\Q} : Im\\rb{\\phi}}}{\\abs{Ker\\rb{\\psi} : Ker\\rb{\\psi} \\cap Im\\rb{\\phi}}} = \\dfrac{\\abs{Im\\rb{\\alpha}}\\abs{Im\\rb{\\alpha'}}}{\\abs{Ker\\rb{\\psi} : Ker\\rb{\\psi} \\cap Im\\rb{\\phi}}}. $$\nNow $ B' \\in \\rb{\\Z^*}^2 $ iff $ T \\in Im\\rb{\\phi} $ and the equation $ x^2 + Ax + B = 0 $ with discriminant $ 16\\rb{A^2 - 4B^2} = 16B' $ has solutions in $ \\Z^* $. Since $ Ker\\rb{\\psi} = \\cb{\\mathcal{O}, T} $ and $ \\mathcal{O} \\in Im\\rb{\\phi} $, this holds iff $ Ker\\rb{\\psi} \\cap Im\\rb{\\phi} = \\cb{\\mathcal{O}, T} $. Since $ \\mathcal{O}, T \\in E\\rb{\\Q}\\sb{2} $, this also holds iff $ \\rb{a, 0}, \\rb{a', 0} \\in E\\rb{\\Q}\\sb{2} $ for the solutions $ a, a' \\in \\Q^* $ of $ x^2 + Ax + B = 0 $. Hence\n$$ E\\rb{\\Q}\\sb{2} = \\begin{cases} \\cb{\\mathcal{O}, T, \\rb{a, 0}, \\rb{a', 0}} & B' \\in \\rb{\\Z^*}^2 \\\\ \\cb{\\mathcal{O}, T} & B' \\notin \\rb{\\Z^*}^2 \\end{cases}, \\qquad \\dfrac{Ker\\rb{\\psi}}{Ker\\rb{\\psi} \\cap Im\\rb{\\phi}} = \\begin{cases} \\cb{\\mathcal{O}} & B' \\in \\rb{\\Z^*}^2 \\\\ \\cb{\\mathcal{O}, T} & B' \\notin \\rb{\\Z^*}^2 \\end{cases}, $$\nso $ \\abs{Ker\\rb{\\psi} : Ker\\rb{\\psi} \\cap Im\\rb{\\phi}}\\abs{E\\rb{\\Q}\\sb{2}} = 4 $. Thus\n$$ 2^r = \\dfrac{\\abs{Im\\rb{\\alpha}}\\abs{Im\\rb{\\alpha'}}}{\\abs{Ker\\rb{\\psi} : Ker\\rb{\\psi} \\cap Im\\rb{\\phi}}\\abs{E\\rb{\\Q}\\sb{2}}} = \\tfrac{1}{4}\\abs{Im\\rb{\\alpha}}\\abs{Im\\rb{\\alpha'}}. $$\n\\end{proof}\n\nComputation of the rank simply reduces to determining images of $ \\alpha $ and $ \\alpha' $. This in turn can be rephrased as a question of Diophantine equations.\n\n\\begin{proposition}\nThe image $ Im\\rb{\\alpha} $ is such that\n$$ Im\\rb{\\alpha} = \\cb{\\rb{\\Q^*}^2\\beta \\ \\Big| \\ \\beta, B / \\beta \\in \\Z^*, \\ \\rb{X, Y, Z} \\in \\Z^3 \\setminus \\cb{\\rb{0, 0, 0}}, \\ Y^2 = \\beta X^4 + AX^2Z^2 + \\rb{B / \\beta}Z^4}. $$\n\\end{proposition}\n\n\\begin{proof}\nSince $ \\rb{\\Q^*}^2 \\in Im\\rb{\\alpha} $, there is a solution $ \\rb{X, Y, Z} = \\rb{1, 1, 0} $ for $ \\beta = 1 $. Since $ \\rb{\\Q^*}^2B \\in Im\\rb{\\alpha} $, there is also a solution $ \\rb{X, Y, Z} = \\rb{0, 1, 1} $ for $ \\beta = B $. Let $ P = \\rb{a, b} \\ne T \\in E\\rb{\\Q} $ such that $ \\rb{\\Q^*}^2a \\in Im\\rb{\\alpha} $. Then $ a = r / Z_0^2 $ and $ b = s / Z_0^3 $ for some $ r, s \\in \\Z $ and some $ Z_0 \\in \\Z_{> 0} $ such that $ \\gcd\\rb{r, Z_0} = \\gcd\\rb{s, Z_0} = 1 $. Now let $ r = X_0^2\\beta_0 $, where $ X_0 = p_1 \\dots p_n \\in \\Z_{> 0} $ for some primes $ p_i \\in \\Z_{> 0} $ and $ \\beta_0 = j q_1 \\dots q_n $ for some $ j \\in \\cb{-1, 1} $ and some distinct primes $ q_i \\in \\Z_{> 0} $. Since $ \\rb{\\Q^*}^2\\beta_0 = \\rb{\\Q^*}^2X_0^2\\beta_0 / Z_0^2 = \\rb{\\Q^*}^2\\beta_0 \\in G\\rb{B} $, each $ q_i \\mid B $, so $ \\beta_0 \\mid B $ and hence $ B / \\beta_0 \\in \\Z^* $. Then\n$$ \\rb{\\dfrac{s}{Z_0^3}}^2 = \\rb{\\dfrac{X_0^2\\beta_0}{Z_0^2}}^3 + A\\rb{\\dfrac{X_0^2\\beta_0}{Z_0^2}}^2 + B\\dfrac{X_0^2\\beta_0}{Z_0^2} \\qquad \\implies \\qquad s^2 = \\beta_0^2X_0^2\\rb{\\beta_0X_0^4 + AX_0^2Z_0^2 + \\rb{B / \\beta_0}Z_0^4}, $$\nso let $ Y_0 = s^2 / \\beta_0^2X_0^2 \\in \\Z $ such that $ Y_0^2 = \\beta_0X_0^4 + AX_0^2Z_0^2 + \\rb{B / \\beta_0}Z_0^4 $. Hence there is a non-zero solution $ \\rb{X, Y, Z} = \\rb{X_0, Y_0, Z_0} $ for $ \\beta = \\beta_0 $. Conversely let $ \\rb{X, Y, Z} = \\rb{X_0, Y_0, Z_0} $ be a non-zero solution for some $ \\beta = \\beta_0 \\in \\Z^* $, so $ Y_0^2 = \\beta_0 X_0^4 + AX_0^2Z_0^2 + \\rb{B / \\beta_0}Z_0^4 $. Then $ P = \\rb{\\beta_0 X_0^2 / Z_0^2, \\beta_0 X_0Y_0 / Z_0^3} $ is such that\n$$ \\rb{\\dfrac{\\beta_0 X_0Y_0}{Z_0^3}}^2 = \\dfrac{\\beta_0^2X_0^2\\rb{\\beta_0 X_0^4 + AX_0^2Z_0^2 + \\rb{B / \\beta_0}Z_0^4}}{Z_0^6} = \\rb{\\dfrac{\\beta_0 X_0^2}{Z_0^2}}^3 + A\\rb{\\dfrac{\\beta_0 X_0^2}{Z_0^2}}^2 + B\\dfrac{\\beta_0 X_0^2}{Z_0^2}, $$\nso $ P \\in E\\rb{\\Q} $ and $ \\alpha\\rb{P} = \\rb{\\Q^*}^2\\rb{\\beta_0 X_0^2 / Z_0^2} = \\rb{\\Q^*}^2\\beta_0 $. Thus any non-zero solution is in $ Im\\rb{\\alpha} $.\n\\end{proof}\n\nAgain, the image of $ \\alpha' $ is similar to that of $ \\alpha $ but with $ B' $ instead of $ B $. The following example illustrates the full computation of the rank of a simple elliptic curve.\n\n\\begin{example}\nLet $ E : y^2 = x^3 - x $ be an elliptic curve over $ \\Q $. Then $ \\beta \\in \\cb{\\pm 1} $. Since $ \\beta = 1 $ and $ \\beta = -1 = B $ have solutions, it holds that $ \\abs{Im\\rb{\\alpha}} = 2 $. Now $ E' : y^2 = x^3 + 4x $ gives $ \\beta \\in \\cb{\\pm 1, \\pm 2, \\pm 4} $. Since $ \\rb{\\Q^*}^2\\rb{\\pm 1} = \\rb{\\Q^*}^2\\rb{\\pm 4} $, the Diophantine equations to consider are:\n\\begin{enumerate}\n\\item $ \\beta = 1 $ gives $ Y^2 = X^4 + 4Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{0, 2, 1} $.\n\\item $ \\beta = 2 $ gives $ Y^2 = 2X^4 + 2Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{1, 2, 1} $.\n\\item $ \\beta = -1 $ gives $ Y^2 = -X^4 - 4Z^4 $, which has no solutions by sign disparity.\n\\item $ \\beta = -2 $ gives $ Y^2 = -2X^4 - 2Z^4 $, which has no solutions by sign disparity.\n\\end{enumerate}\nHence $ \\abs{Im\\rb{\\alpha'}} = 2 $ and $ 2^r = \\tfrac{1}{4}\\rb{2}\\rb{2} = 1 $. Thus $ rk\\rb{E\\rb{\\Q}} = 0 $ and $ E\\rb{\\Q} = E\\rb{\\Q}_{tors} \\cong \\Z_2 $.\n\\end{example}\n\nThe following algorithm summarises the process and code in the appendix.\n\n\\begin{algorithm}[Computation of the rank]\nInput: an elliptic curve $ E $ over $ \\Q $. Output: $ rk\\rb{E\\rb{\\Q}} $.\n\\begin{enumerate}\n\\item Get all positive $ \\beta $ such that $ \\beta \\mid B $ and free the squares from each $ \\beta $.\n\\item Print all Diophantine equations of the form $ Y^2 = \\beta X^4 + AX^2Z^2 + \\rb{B / \\beta}Z^4 $.\n\\item Write down the elliptic curve $ E' : y^2 = x^3 - 2Ax^2 + \\rb{A^2 - 4B}x $ and do the same.\n\\item Check if there are non-zero solutions to the systems of Diophantine equations.\n\\item Compute the rank with the formula $ rk\\rb{E\\rb{\\Q}} = \\log_2\\abs{Im\\rb{\\alpha}} + \\log_2\\abs{Im\\rb{\\alpha'}} - 2 $.\n\\end{enumerate}\n\\end{algorithm}\n\nUnfortunately, there are no known effective method for the second to last step. In contrast to attempting at a number theoretic algorithm like in \\cite{rank}, only ad-hoc congruences will be used to complete the computations in the following examples of elliptic curves given by the Weierstrass equations $ y^2 = x^3 - px $ for $ p \\in \\Z_{> 0} $. The following example is an elliptic curve of rank one.\n\n\\begin{example}\nLet $ E : y^2 = x^3 - 5x $ be an elliptic curve over $ \\Q $, which gives $ \\beta \\in \\cb{\\pm 1, \\pm 5} $. Since $ \\beta = 1 $ and $ \\beta = -5 $ have trivial solutions, the Diophantine equations to consider are $ Y^2 = -X^4 + 5Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{1, 2, 1} $, and $ Y^2 = 5X^4 - Z^4 $, which has a solution by symmetry. Hence $ \\abs{Im\\rb{\\alpha}} = 4 $. Now $ E' : y^2 = x^3 + 20x $ gives $ \\beta \\in \\cb{\\pm 1, \\pm 2, \\pm 5, \\pm 10} $. If $ \\beta < 0 $, there are no solutions by sign disparity. Since $ \\beta = 1 $ and $ \\beta = 5 $ have trivial solutions, the Diophantine equations to consider are $ Y^2 = 2X^4 + 10Z^4 $ and $ Y^2 = 10X^4 + 2Z^4 $. Since $ \\gcd\\rb{X_0, Y_0} = 1 $, if the first has a solution $ \\rb{X, Y, Z} = \\rb{X_0, Y_0, Z_0} $, then $ Y_0^2 \\equiv 2X_0^4 \\equiv 2 \\mod 5 $ gives no solutions for $ Y_0 $, so both equations have no solutions. Hence $ \\abs{Im\\rb{\\alpha'}} = 2 $. Thus $ rk\\rb{E\\rb{\\Q}} = \\log_2\\rb{4} + \\log_2\\rb{2} - 2 = 1 $ and $ E\\rb{\\Q} \\cong \\Z \\oplus E\\rb{\\Q}_{tors} \\cong \\Z \\times \\Z_2 $.\n\\end{example}\n\nThe following example is an elliptic curve of rank two.\n\n\\begin{example}\nLet $ E : y^2 = x^3 - 17x $ be an elliptic curve over $ \\Q $, which gives $ \\beta \\in \\cb{\\pm 1, \\pm 17} $. Since $ \\beta = 1 $ and $ \\beta = -17 $ have trivial solutions, the Diophantine equations to consider are $ Y^2 = -X^4 + 17Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{1, 4, 1} $, and $ Y^2 = 17X^4 - Z^4 $, which has a solution by symmetry. Hence $ \\abs{Im\\rb{\\alpha}} = 4 $. Now $ E' : y^2 = x^3 + 68x $ gives $ \\beta \\in \\cb{\\pm 1, \\pm 2, \\pm 17, \\pm 34} $. If $ \\beta < 0 $, there are no solutions by sign disparity. Since $ \\beta = 1 $ and $ \\beta = 17 $ have trivial solutions, the Diophantine equations to consider are $ Y^2 = 2X^4 + 34Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{1, 6, 1} $, and $ Y^2 = 34X^4 + 2Z^4 $, which has a solution by symmetry. Hence $ \\abs{Im\\rb{\\alpha'}} = 4 $. Thus $ rk\\rb{E\\rb{\\Q}} = \\log_2\\rb{4} + \\log_2\\rb{4} - 2 = 2 $ and $ E\\rb{\\Q} \\cong \\Z^2 \\oplus E\\rb{\\Q}_{tors} \\cong \\Z^2 \\times \\Z_2 $.\n\\end{example}\n\nThe following example is an elliptic curve of rank three.\n\n\\begin{example}\nLet $ E : y^2 = x^3 - 226x $ be an elliptic curve over $ \\Q $, which gives $ \\beta \\in \\cb{\\pm 1, \\pm 2, \\pm 113, \\pm 226} $. Since $ \\beta = 1 $ and $ \\beta = -226 $ have trivial solutions, the Diophantine equations to consider are $ Y^2 = -X^4 + 226Z^4 $, $ Y^2 = 2X^4 - 113Z^4 $, $ Y^2 = -2X^4 + 113Z^4 $, $ Y^2 = 113X^4 - 2Z^4 $, $ Y^2 = -113X^4 + 2Z^4 $, and $ Y^2 = 226X^4 - Z^4 $. The first three have solutions $ \\rb{X, Y, Z} = \\rb{1, 15, 1} $, $ \\rb{X, Y, Z} = \\rb{3, 7, 1} $, and $ \\rb{X, Y, Z} = \\rb{1, 9, 2} $ respectively, while the last three have solutions by symmetry. Hence $ \\abs{Im\\rb{\\alpha}} = 8 $. Now $ E' : y^2 = x^3 + 904x $ gives $ \\beta \\in \\cb{\\pm 1, \\pm 2, \\pm 113, \\pm 226} $. If $ \\beta < 0 $, there are no solutions by sign disparity. Since $ \\beta = 1 $ and $ \\beta = 226 $ have trivial solutions, the Diophantine equations to consider are $ Y^2 = 2X^4 + 452Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{1, 22, 2} $, and $ Y^2 = 113X^4 + 8Z^4 $, which has a solution $ \\rb{X, Y, Z} = \\rb{1, 11, 1} $. Hence $ \\abs{Im\\rb{\\alpha'}} = 4 $. Thus $ rk\\rb{E\\rb{\\Q}} = \\log_2\\rb{8} + \\log_2\\rb{4} - 2 = 3 $ and $ E\\rb{\\Q} \\cong \\Z^3 \\oplus E\\rb{\\Q}_{tors} \\cong \\Z^3 \\times \\Z_2 $.\n\\end{example}\n\nThe ranks of elliptic curves above are relatively small in value and easy to compute, but there are elliptic curves with larger rank values. The record as of 2018 in \\cite{rankhistory} for the elliptic curve with the largest rank was discovered by Elkies in 2006, and is given by the Weierstrass curve\n$$ E : y^2 + xy + y = x^3 - x^2 - 20067762415575526585033208209338542750930230312178956502x $$\n$$ + 34481611795030556467032985690390720374855944359319180361266008296291939448732243429, $$\nwhich is proven to have rank at least 28. There are also elliptic curves with relatively large ranks known exactly, the largest of which was also discovered by Elkies in 2009, and is given by the Weierstrass curve\n$$ E : y^2 + xy + y = x^3 - x^2 + 31368015812338065133318565292206590792820353345x $$\n$$ + 302038802698566087335643188429543498624522041683874493555186062568159847, $$\nwhich has rank 19. In fact, it is conjectured that the rank of an elliptic curve does not have an upper bound.\n\n\\begin{conjecture}\nThere are elliptic curves over $ \\Q $ of arbitrary large rank.\n\\end{conjecture}\n\nHowever, while they exist, elliptic curves of rank greater than one are rare. This notion of rarity is measured by the \\emph{average rank} of all elliptic curves, of which is conjectured to exist as a quantity.\n\n\\begin{conjecture}\nThe average rank of all elliptic curves over $ \\Q $ is $ \\tfrac{1}{2} $.\n\\end{conjecture}\n\nIn particular, rank zero constitute a half and rank one constitute the other half, while all higher ranks constitute zero percent, of all elliptic curves. While it has not been definitely proven, Bhargava and Shankar showed in \\cite{rankaverage} that the average rank of all elliptic curves is at most $ 7 / 6 $.\n\n\\pagebreak\n\n\\subsection{Birch and Swinnerton-Dyer conjecture}\n\nUltimately, the rank of an elliptic curve is not completely understood. It was greatly studied for decades, and had lead mathematicians to formalise one of the most influential conjectures in number theory, which is also deemed worthy of being called one of the Millennium Prize Problems. The problem, now commonly known as the \\emph{Birch and Swinnerton-Dyer conjecture}, relates the rank with Taylor expansion of a particular complex series. Letting $ t_p $ denote the trace in Hasse's theorem applied to $ E_p\\rb{\\F_p} $ for any prime $ p \\in \\Z_{> 0} $ of good reduction, the series can be given as follows.\n\n\\begin{definition}[Incomplete Hasse-Weil $ L $-series]\nThe \\textbf{incomplete Hasse-Weil $ L $-series} is defined for any $ \\Re\\rb{s} > 3 / 2 $ as the \\emph{Euler product}\n$$ L\\rb{E, s} = \\prod_p \\dfrac{1}{1 - t_pp^{-s} + p^{1 - 2s}} $$\nover all primes $ p \\in \\Z_{> 0} $ of good reduction, and extended to $ \\C $ by analytic continuation.\n\\end{definition}\n\nThis analytic continuation, as well as a functional equation similar to that of the Riemann zeta function, was originally known as the \\emph{Hasse-Weil conjecture}, but was subsequently implied by the \\emph{modularity theorem}.\n\n\\begin{remark}\nThe \\emph{complete Hasse-Weil $ L $-series} is defined over all primes $ p \\in \\Z_{> 0} $ as the Euler product\n$$ L^*\\rb{E, s} = \\prod_{p \\mid \\Delta_E} \\dfrac{1}{1 - t_pp^{-s}} \\prod_{p \\nmid \\Delta_E} \\dfrac{1}{1 - t_pp^{-s} + p^{1 - 2s}} = \\sum_{n = 1}^\\infty \\dfrac{a_n}{n^s}. $$\nNow $ E $ might be a singular cubic curve, so that $ t_p $ can be defined for primes of bad reduction as either $ t_p = \\pm 1 $ or $ t_p = 0 $, depending on whether $ E $ has \\emph{split} or \\emph{non-split multiplicative} reduction or \\emph{additive} reduction, which corresponds to whether $ E_p $ has a \\emph{node} or a \\emph{cusp} respectively.\n\\end{remark}\n\nDue to analyticity in $ \\C $, it makes sense to consider the Taylor expansion of $ L\\rb{E, s} $ given by\n$$ L\\rb{E, s} = \\sum_{i = 0}^\\infty c_i\\rb{s - s_0}^i, \\qquad s_0 \\in \\C, \\qquad c_i \\in \\C, $$\nas well its \\emph{order of vanishing} $ ord_{s = s_0} $ or order of zero at $ s_0 $, a value $ i $ such that $ c_i \\ne 0 $ but $ c_j = 0 $ for any $ j < i $. A different notion of rank can then be defined for $ E $, as follows.\n\n\\begin{definition}[Analytic rank]\nThe \\textbf{analytic rank} of $ E $ is $ rk_{an}\\rb{E\\rb{\\Q}} = ord_{s = 1}L\\rb{E, s} $.\n\\end{definition}\n\nThe conjecture then relates both notions of ranks as follows.\n\n\\begin{conjecture}[Birch and Swinnerton-Dyer] $ rk\\rb{E\\rb{\\Q}} = rk_{an}\\rb{E\\rb{\\Q}} $.\n\\end{conjecture}\n\n\\begin{remark}\nThere is also a refined version of the conjecture that involves the \\emph{Tate-Shafarevich group}, which is omitted for further discussion. Proving this strong version will then indirectly lead to efficient algorithms for rank computation.\n\\end{remark}\n\nA direct consequence of the conjecture is that $ E\\rb{\\Q} $ is infinite iff its ranks $ rk\\rb{E\\rb{\\Q}} $ and $ rk_{an}\\rb{E\\rb{\\Q}} $ are positive. This holds iff $ L\\rb{E, s} $ does not have a constant term, or iff $ L\\rb{E, 1} $ computes to give a value of $ 0 $. In other words, the finiteness of $ E\\rb{\\Q} $ holds iff $ L\\rb{E, 1} \\ne 0 $. Now the conjecture has been supported with much numerical evidence in \\cite{bsd0}, and can also be verified by prior examples with the \\emph{Sage} programming language as follows.\n\n\\begin{example}\nLet $ E : y^2 = x^3 - x $ be an elliptic curve over $ \\Q $. Then $ rk\\rb{E\\rb{\\Q}} = 0 $ and\n$$ L\\rb{E, s} \\approx 0.655514388573030 + 0.447208159472739s - 0.233131198781643s^2 + 0.0342258563577268s^3 + \\dots, $$\nHence $ L\\rb{E, 1} \\approx 0.655514388573030 \\ne 0 $. Now let $ E' : y^2 = x^3 - 5x $ be an elliptic curve over $ \\Q $. Then $ rk\\rb{E'\\rb{\\Q}} = 1 $ and\n$$ L\\rb{E', s} \\approx 0.000000000000000 + 2.22876814774675s - 2.06654309593994s^2 + 0.549852427979257s^3 + \\dots. $$\nThus $ L\\rb{E', 1} \\approx 0.000000000000000 = 0 $.\n\\end{example}\n\nHowever, only special cases of the conjecture have been proven to date. The first general result, proven by Coates and Wiles, states that an elliptic curve $ E $ with $ L\\rb{E, 1} \\ne 0 $ and \\emph{complex multiplication}, or when $ \\abs{End\\rb{E}} $ is strictly larger than $ \\Z $, has finite $ E\\rb{\\Q} $, and hence $ rk\\rb{E\\rb{\\Q}} = 0 $. A later result, proven by Gross and Zagier with \\emph{Heegner points}, states that a \\emph{modular} elliptic curve $ E $ with $ L\\rb{E, 1} = 0 $ and $ \\rb{d / ds}L\\rb{E, 1} \\ne 0 $, or equivalently $ rk_{an}\\rb{E\\rb{\\Q}} = 1 $, has a non-torsion rational point in $ E\\rb{\\Q} $, and hence $ rk\\rb{E\\rb{\\Q}} > 0 $. Subsequently, Kolyvagin extended this proof by showing that $ rk\\rb{E\\rb{\\Q}} = 1 $ must hold for this latter case, and that $ rk\\rb{E\\rb{\\Q}} = 0 $ if $ L\\rb{E, 1} \\ne 0 $ instead. With the modularity theorem proven by Breuil et al, it is now known that any elliptic curve over $ \\Q $ is modular, hence proving the following special case of the Birch and Swinnerton-Dyer conjecture.\n\n\\begin{theorem}[Breuil, Coates, Conrad, Diamond, Gross, Kolyvagin, Taylor, Wiles, Zagier]\n$ rk\\rb{E\\rb{\\Q}} = rk_{an}\\rb{E\\rb{\\Q}} $ for $ rk_{an}\\rb{E\\rb{\\Q}} \\in \\cb{0, 1} $.\n\\end{theorem}\n\n\\begin{proof}\nOmitted, see \\cite{bsd1}, \\cite{bsd2}, \\cite{bsd3}, and \\cite{modularity}.\n\\end{proof}\n\nThe very recent result due to Bhargava and Shankar in \\cite{rankaverage} also showed that a large proportion of all elliptic curves must have either rank zero or one, but the conjecture still remain unproven for elliptic curves with higher ranks. Now as a Millenium Prize Problem, the Birch and Swinnerton-Dyer conjecture has significant implications in number theory, particularly on finiteness of the Tate-Shafarevich group, but it also proves other more elementary results, one of which concerns integers with the following property.\n\n\\begin{definition}[Congruent number]\n$ n \\in \\Z_{> 0} $ is a \\textbf{congruent number} iff it is the area of some right triangle with sides in $ \\Q_{> 0} $.\n\\end{definition}\n\nCongruent numbers can be illustrated with the following example.\n\n\\begin{example}\n$ 5 = \\tfrac{1}{2}\\rb{3 / 2}\\rb{20 / 3} $ is a congruent number since it is the area of the right triangle with sides $ 3 / 2, 20 / 3, 41 / 6 \\in \\Q_{> 0} $, while $ 10 $ is not a congruent number.\n\\end{example}\n\nAn open problem is the classification of all congruent numbers, known as the \\emph{congruent number problem}, which boils down to obtaining simultaneous solutions for $ a^2 + b^2 = c^2 $ and $ 2n = ab $, for some $ n \\in \\Z_{> 0} $ and some $ a, b, c \\in \\Q_{> 0} $. Considering the non-zero inverse transformations\n$$ \\rb{x, y} = \\rb{\\dfrac{n\\rb{a + c}}{b}, \\dfrac{2n^2\\rb{a + c}}{b^2}}, \\qquad \\rb{a, b, c} = \\rb{\\dfrac{\\rb{x^2 - n^2}}{y}, \\dfrac{2nx}{y}, \\dfrac{\\rb{x^2 + n^2}}{y}}, $$\nthe system of equations can be transformed with a bijective correspondence to the Weierstrass equation $ y^2 = x^3 - n^2x $. Hence checking whether $ n $ is a congruent number is in turn equivalent to determining whether an affine rational point with non-zero coordinates exists in the elliptic curve $ E : y^2 = x^3 - n^2x $ over $ \\Q $. This prompts the following theorem that further classify the conditions for being a congruent number.\n\n\\begin{theorem}[Tunnell]\nLet $ n \\in \\Z_{> 0} $ be a square-free congruent number. If $ n $ is odd, then\n$$ 2\\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid n = 2x^2 + y^2 + 32z^2}} = \\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid n = 2x^2 + y^2 + 8z^2}}. $$\nOtherwise $ n $ is even, then\n$$ 2\\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid n = 2\\rb{4x^2 + y^2 + 32z^2}}} = \\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid n = 2\\rb{4x^2 + y^2 + 8z^2}}}. $$\n\\end{theorem}\n\n\\begin{proof}\nOmitted, see \\cite{tunnell}.\n\\end{proof}\n\nThe Birch and Swinnerton-Dyer conjecture, on the other hand, provides the converse to Tunnell's theorem, hence giving a single criterion for any congruent number that can be checked by enumerating the four sets involved. The following example illustrates the process, assuming the conjecture.\n\n\\begin{example}\nSince $ 5 $ is an odd square-free congruent number, it holds that\n$$ 2\\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid 5 = 2x^2 + y^2 + 32z^2}} = 0 = \\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid 5 = 2x^2 + y^2 + 8z^2}}. $$\nConversely, since $ 10 $ is an even square-free non-congruent number, it holds that\n$$ 2\\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid n = 2\\rb{4x^2 + y^2 + 32z^2}}} = 8 \\ne 4 = \\abs{\\cb{\\rb{x, y, z} \\in \\Z^3 \\mid n = 2\\rb{4x^2 + y^2 + 8z^2}}}. $$\n\\end{example}\n\nAs such, the conjecture, if proven even only for elliptic curves given by the Weierstrass equation $ y^2 = x^3 - n^2x $, would allow the congruence number problem to be fully resolved.\n\n\\pagebreak\n\n\\section{Applications}\n\nIn this section, two elementary applications of elliptic curves are briefly touched on, together with a brief history of their development prior to elliptic curves.\n\n\\subsection{Arithmetic}\n\nIn arithmetic, the widely known fact that $ \\Z $ is a unique factorisation domain acts as the modern rephrasing of the fundamental theorem of arithmetic, which states that any integer can be uniquely written as a product of primes up to rearrangement and sign. An classical problem in arithmetic is determining these primes, or more commonly known as integer factorisation, for a given huge integer $ N \\in \\Z_{> 0} $ with tens of digits. Clearly no work is required if $ N $ is prime, which can be verified for most cases by the following theorem.\n\n\\begin{theorem}[Fermat's little theorem]\nLet $ N $ be prime. Then $ a^{N - 1} \\equiv 1 \\mod N $ for any $ a \\in \\Z_{> 0} $ such that $ N \\nmid a $.\n\\end{theorem}\n\n\\begin{proof}\nSince $ N \\nmid a $, it holds that $ a \\not\\equiv 0 \\mod N $, so $ a \\in \\Z_N^* $. Lagrange's theorem gives $ ord\\rb{a} \\mid \\abs{\\Z_N^*} = N - 1 $, so $ a^{N - 1} = 1 $. Thus $ a^{N - 1} \\equiv 1 \\mod N $.\n\\end{proof}\n\nUnfortunately, its converse does not hold, due to the existence of \\emph{Carmichael numbers}, which are composite integers that seemingly satisfy Fermat's little theorem, the first of which being relatively small at $ 561 $. These integers can be avoided manually due to their relative rarity, but there are indeed more reliable tests to check if $ N $ is prime. Now simply assume that $ N $ is composite, with an unknown large prime factor $ p \\in \\Z_{> 0} $. On first sight, there is a simple naive approach to factorise $ N $ involving trial division, applying Euclidean division to every positive integer less than $ N $. A clear improvement can be made by only considering $ 2, 3 $ and integers of the form $ 6m \\pm 1 $, which is illustrated in the following example with a small value of $ N $.\n\n\\begin{example}\nLet $ N = 420 $ be composite. Trial division on $ \\cb{2, 3, 5, 7, \\dots} $ gives $ 420 = 2\\rb{210} + 0 $ and $ 210 = 2\\rb{105} + 0 $, but $ 105 = 2\\rb{52} + 1 $. Now $ 105 = 3\\rb{35} + 0 $, but $ 35 = 3\\rb{11} + 2 $. Then $ 35 = 7\\rb{5} + 0 $, but $ 7 = 1\\rb{5} + 2 $. Finally $ 7 = 1\\rb{7} + 0 $. Thus $ L' = \\cb{2, 2, 3, 5, 7} $ and $ N = \\rb{2}\\rb{2}\\rb{3}\\rb{5}\\rb{7} $.\n\\end{example}\n\nIt is immediately evident that this process of factorising $ N $ into $ p $ and $ N / p $ is laborious, and especially difficult if $ N / p $ is also prime and relatively similar in magnitude to $ p $, since every integer below $ p $ and $ N / p $ will be checked for failure. In fact, while multiplying $ p $ and $ N / p $ to produce $ N $ is relatively easy, there are no known efficient \\emph{non-quantum} polynomial time algorithms to deterministically do the reverse, of which is exactly the basis of modern cryptography. The fastest methods for integer factorisation involve either \\emph{sieves} or elliptic curves, the latter of which discussed here has motivations stemming from the classical \\emph{Pollard's $ p - 1 $ method}. The following definition, considering a fixed integer $ B \\in \\Z_{> 0} $, will be used.\n\n\\begin{definition}[$ B $-power smooth]\nLet $ n \\in \\Z_{> 0} $ be such that $ n = p_1^{e_1} \\dots p_n^{e_n} $ for some $ e_i \\in \\Z_{> 0} $ and some primes $ p_i \\in \\Z_{> 0} $. Then $ n $ is \\textbf{$ B $-power smooth} iff each $ p_i^{e_i} \\le B $.\n\\end{definition}\n\nThis definition is illustrated in the following example.\n\n\\begin{example}\n$ 420 $ is $ 7 $-smooth, and $ 2\\rb{420} $ is not $ 7 $-smooth but is $ 8 $-smooth.\n\\end{example}\n\n$ B $ is typically chosen to be fairly large, such that the composite integer $ p - 1 $ is likely to be $ B $-power smooth. Now let $ l_B = \\text{lcm}\\rb{2, \\dots, B} $, so that any prime below $ B $ divides $ l_B $, and so $ p - 1 \\mid l_B $. The algorithm again uses Fermat's little theorem to give $ a^{l_B} \\equiv 1 \\mod p $ for any $ a \\in \\Z_{> 0} $ such that $ p \\nmid a $. Hence $ p \\mid a^{l_B} - 1 $, so letting $ g_B = \\gcd\\rb{a^{l_B} - 1, N} $ gives $ p \\mid g_B \\mid N $. Obtaining $ g_B < N $ immediately implies that $ g_B $ is a proper divisor of $ N $, so the process can be repeated replacing $ N $ with $ g_B $. The algorithm is summarised as follows.\n\n\\begin{algorithm}[Pollard's $ p - 1 $ method]\nInput: an integer $ N \\in \\Z_{> 0} $. Output: a proper divisor of $ N $.\n\\begin{enumerate}\n\\item Choose a \\emph{smoothness bound} $ B \\in \\Z_{> 0} $.\n\\item Calculate $ l_B = \\text{lcm}\\rb{1, \\dots, B} $.\n\\item Compute $ g_B = \\gcd\\rb{a^{l_B} - 1, N} $ for some $ a \\in \\cb{2, \\dots, N - 1} $.\n\\item If $ g_B = 1 $, then choose a larger smoothness bound $ B $, else if $ g_B = N $, then choose a different $ a $.\n\\item Otherwise $ 1 < g_B < N $, then return $ g_B $.\n\\end{enumerate}\n\\end{algorithm}\n\n\\pagebreak\n\n\\begin{remark}\nIn practicality, this $ B $ typically fixed, and a different algorithm is forcibly switched into if this algorithm fails, which would mean that Pollard's $ p - 1 $ algorithm only serves to simplify the problem.\n\\end{remark}\n\nA simple observation picks out several possible concerns in the algorithm, most obviously being the question if the algorithm even terminates, which must be the case, as the smoothness bound $ B $ will eventually be large enough for $ l_B $ to exceed $ p - 1 $. Another issue might be the speed of computing $ l_B $ and $ g_B $, with the latter seemingly depending on the former. This is resolved by the realisation that $ l_B $ is arbitrarily defined as such to allow all factors below $ B $ to divide $ l_B $, so a purely multiplicative $ l_B = B! $ can be used. While $ g_B $ may require further exponentiation, this is not a concern as modular exponentiation with successive squaring to compute $ a^{l_B} $ is fast, and the result of this can be used in the computation instead of $ a^{l_B} - 1 $. In the highly unlikely chance that $ p \\mid a $, a fix to this issue could be first doing trial division up to a small integer $ n \\in \\Z_{> 0} $ to rule out any possibility that $ p < n $, and only considering the cases where $ a < n $. With these concerns in mind, Pollard's $ p - 1 $ algorithm can be illustrated with the following example involving a small $ N $, which on modern computers can actually be done instantly even with trial division.\n\n\\begin{example}\nLet $ N = 246082373 $. Then $ 2^{N - 1} \\equiv 114193013 \\ne 1 \\mod N $, so Fermat's little theorem gives that $ N $ is composite. Now let $ B = 7 $ and $ l_B = \\text{lcm}\\rb{1, \\dots, B} = 420 $, and let $ a = 2 $. Then $ a^{l_B} \\equiv 60592910 \\mod N $, so $ \\gcd\\rb{a^{l_B} - 1, N} = \\gcd\\rb{60592909, N} = 1 $. Hence choose $ B = 9 > 7 $ and $ l_B = \\text{lcm}\\rb{1, \\dots, B} = 2520 $, and let $ a = 2 $. Then $ a^{l_B} \\equiv 130940741 \\mod N $, so $ \\gcd\\rb{a^{l_B} - 1, N} = \\gcd\\rb{130940740, N} = 2521 < N $. Thus $ 2521 \\mid N $ and $ N = \\rb{2521}\\rb{97613} $, which are both prime.\n\\end{example}\n\nA final serious concern would be overall efficiency of the algorithm that is only reasonable with the initial supposition that $ p - 1 $ is $ B $-power smooth, which may not always be the case, inevitably forcing large values of $ B $. Despite so, it may still be desirable to check the $ B $-power smoothness of $ p + 1 $, or even $ p \\pm n $ for any small $ n \\in \\Z $, to see if the initially fixed $ B $ is sufficient to factorise $ N $. In this respect, Pollard's $ p - 1 $ algorithm has no way of allowing for other values of $ p \\pm n $ due to the restriction given by Fermat's little theorem. Now \\emph{Lenstra's elliptic curve factorisation method} takes this flaw into account by considering multiple random elliptic curves over finite fields. While the former considers the multiplicative group $ \\Z_p^* $ of order $ \\abs{\\Z_p^*} = p - 1 $ and checks if $ a \\in \\Z_p^* $ satisfies $ a^{l_B} = 1 $, the latter considers the elliptic curve $ E\\rb{\\F_p} $ of order $ \\abs{E\\rb{\\F_p}} = p - t + 1 $ and checks if $ P \\in E\\rb{\\F_p} $ satisfies $ l_BP = \\mathcal{O} $. Hasse's theorem gives $ \\abs{t} \\le 2\\sqrt{p} $, allowing $ \\abs{E\\rb{\\F_p}} $ to vary wildly within this interval and hence removing the aforementioned defect. The algorithm is stated as follows.\n\n\\begin{algorithm}[Lenstra's elliptic curve factorisation method]\nInput: an integer $ N \\in \\Z_{> 0} $. Output: a proper divisor of $ N $.\n\\begin{enumerate}\n\\item Set $ P = \\rb{a, b} $ and $ B = b^2 - a^3 - Aa $ for some $ A, a, b \\in \\cb{1, \\dots, N - 1} $.\n\\item If $ g = \\gcd\\rb{4A^3 + 27B^2, N} = N $, choose a different $ A $, else if $ 1 < g < N $, return $ g $, else let $ E : y^2 = x^3 + Ax + B $ be an elliptic curve over $ \\Q $, treated as over the ring $ \\Z_N^* $.\n\\item Choose a smoothness bound $ C $ and calculate $ l_C $ as per Pollard's $ p - 1 $ method.\n\\item Compute $ l_CP = \\rb{q / d^2, r / d^3} $ and $ g_C = \\gcd\\rb{d, N} $.\n\\item If $ g_C = 1 $, choose a larger $ C $ or a different $ A $, else if $ g_C = N $, choose a smaller $ C $, else return $ g_C $.\n\\end{enumerate}\n\\end{algorithm}\n\nA couple of remarks will be made on Lenstra's elliptic curve factorisation method, which has the first two steps markedly differently from Pollard's $ p - 1 $ method. The last three steps of the former mirror the latter, but it is worth noting that $ l_CP = \\mathcal{O} $ iff $ d = 0 $, which holds iff $ p \\mid d $, or $ p \\mid g_C = \\gcd\\rb{d, N} \\mid N $. While it is possible to directly obtain a random elliptic curve $ E : y^2 = x^3 + Ax + B $ over $ \\F_p $ by choosing some random $ A, B \\in \\Z $ and treating $ A, B $ as elements of $ \\F_p $, obtaining a point in $ E $ afterwards will involve the laborious process of finding a modular square root, and so the above approach is taken. Moreover, to avoid having $ P $ be a torsion point and hence resulting in $ N \\mid d $, a simple fix would be an additional condition to choose $ b^2 \\nmid 4A^3 + 27\\rb{A^3 + Aa} $ in this reverse approach, which by the contrapositive to the Nagell-Lutz theorem forces $ P $ to have infinite order. Again, computation of $ l_C $ will be done by successive duplication of points, but a concern arises from possibly having denominators $ d \\in \\Z_N $ being zero. Considering the cases when an inverse does not exist in $ \\Z_N^* $, or when $ \\gcd\\rb{d, N} > 1 $, either $ \\gcd\\rb{d, N} = N $, for which it is sufficient to simply choose a different curve $ E $, or $ 1 < \\gcd\\rb{d, N} < N $, for which $ \\gcd\\rb{d, N} $ can be returned as a proper divisor of $ N $. In practice, the affine coordinates are computed modulo $ N $, catching a division by zero error only when the extended Euclidean algorithm fails to give a modular inverse, then computing $ g_C $ separately. With these remarks at hand, Lenstra's elliptic curve factorisation method can be illustrated with the following example involving a small $ N $, which again can actually be done instantly with a modern computer.\n\n\\begin{example}\nLet $ N = 1715761513 $. Then $ 2^{N - 1} \\equiv 114094409 \\ne 1 \\mod N $, so Fermat's little theorem gives that $ N $ is composite. Let $ A = 1 $, $ a = 2 $, and $ b = 1 $, such that $ P = \\rb{2, 1} $ and $ B = 1^2 - 2^3 - 1\\rb{2} = -9 $. Then $ g = \\gcd\\rb{4A^3 + 27B^2, N} = 1 $, such that $ E : y^2 = x^3 + x - 9 $ is an elliptic curve over $ \\Q $. Now let $ C = 17 $ and $ l_C = \\text{lcm}\\rb{1, \\dots, C} = 12252240 $. Then $ l_CP = \\rb{1225303014, 142796033} $, so $ \\rb{d, N} = 1 $. Instead of choosing a larger $ C $, choose a different $ A \\in \\cb{2, \\dots, N} $ and recompute. Then $ A = 42 $ eventually returns $ \\rb{d, N} = 26927 $. Thus $ 26927 \\mid N $ and $ N = \\rb{26927}\\rb{63719} $, which are both prime.\n\\end{example}\n\nAs seen above, if choosing a different $ A $ does not work, a larger $ C $ could be tried, or even picking a different $ P = \\rb{a, b} $ if all else fails, a sign of high algorithmic flexibility. While it may involve more sophistication and possibly a larger overhead in maintaining the elliptic curve data structure than Pollard's $ p - 1 $ method, Lenstra's elliptic curve factorisation method \\emph{heuristically} runs at sub-exponential time complexity, due to a high probability that an integer is $ B $-power smooth within the interval in Hasse's theorem.\n\n\\begin{remark}\nUsing Pollard's $ p - 1 $ algorithm for this example will require $ B \\ge 13463 $, which will take a while.\n\\end{remark}\n\nWith the discussion on factorising the composite $ N $ completed, the initial question of checking if $ N $ is prime makes a comeback. After all, probabilistic factorisation algorithms may never terminate if $ N $ was never composite in the first place. Again, there is a trivial method to do this, which involves factoring $ N $ by trial division, Fermat's little theorem, or any other factorisation methods and hoping to result in a definite failure. While there were subsequent developments on fast non-deterministic tests, such as \\emph{Miller-Rabin}, that do not rely on factorisation but on Fermat's little theorem instead, the remaining discussion here focuses on one of the fastest \\emph{primality tests} that involves elliptic curves. In contrast to factorisation, currently doable for tens to hundreds of digits, the primality of integers up to tens of thousands of digits have been proven.\n\n\\begin{remark}\nThese forms of primality testing hold for arbitrary integers, but there are certain classes of huge integers that have been proven to be primes using specialised methods. For instance, the largest integer ever proven to be a prime is a \\emph{Mersenne prime} that has tens of millions of digits.\n\\end{remark}\n\nThe said primality test is also an elliptic curve version of another classical primality test, known as the \\emph{Pocklington-Lehmer primality test}, which has its basis on the following theorem.\n\n\\begin{theorem}[Pocklington-Lehmer]\n\\label{thm:pocklington}\nLet $ r \\in \\Z_{> 0} $ be such that $ r \\mid N - 1 $ and $ r \\ge \\sqrt{N} $. Then $ N $ is prime if\n$$ a_q^{N - 1} \\equiv 1 \\mod N, \\qquad \\gcd\\rb{a_q^{\\rb{N - 1} / q} - 1, N} = 1, \\qquad a_q \\in \\Z_{> 0}, $$\nfor any prime $ q \\in \\Z_{> 0} $ such that $ q \\mid r $.\n\\end{theorem}\n\n\\begin{proof}\nLet $ p \\in \\Z_{> 0} $ be a prime such that $ p \\mid N $, so $ p $ is a prime of good reduction, and let $ v = v_q\\rb{r} $ be the $ q $-adic valuation of $ r $. Now let $ a \\equiv a_q^{\\rb{N - 1} / q^v} \\mod p $ for some $ a \\in \\Z_p^* $. Then $ a^{q^v} = a_q^{N - 1} \\equiv 1 \\mod p $ and $ a^{q^{v - 1}} = a_q^{\\rb{N - 1} / q} \\not\\equiv 1 \\mod p $. Hence $ ord\\rb{a} = q^v $. Lagrange's theorem gives $ q^v \\mid \\abs{\\Z_p^*} = p - 1 $, so $ r \\mid p - 1 $ and $ p > r \\ge \\sqrt{N} $. Thus $ p = N $ and $ N $ is prime.\n\\end{proof}\n\nAgain, a simple observation points that integer factorisation is involved in finding $ r $, as well as its prime factors $ q $. Similar to Pollard's $ p - 1 $ method, a prior assumption is required for the algorithm to run at a reasonable time, namely that $ p - 1 $ has many small factors $ q $ that allow $ r $ to be generated quickly with trial division. It is also worth noting that finding $ a_q $ is equivalent to finding a generator of the group $ \\Z_p^* $, which might be laborious, but often letting $ a_q = 2 $ works. The test algorithm then simply follows from the theorem, but it is desirable to have a third party mechanism that checks validity of the primality proof, which will also help ensure functional correctness of implementations. In particular, the notion of a \\emph{primality certificate}, which is an ordered pair $ \\rb{r, c} $ as in Theorem \\ref{thm:pocklington}, where $ c $ is a list of ordered pairs $ \\rb{q, a_q} $, allows for a verification by simple modular exponentiation. The Pocklington-Lehmer primality test can be illustrated with a much larger value of $ N $ in the following example, owing to the fact that $ N - 1 $ has many small factors.\n\n\\begin{example}\nLet $ N = 9223372036854775783 $. Then $ N - 1 = 9223372036854775782 $. Trial division for primes less than $ 400000 $ gives $ r = \\rb{2}\\rb{3}^4\\rb{17}\\rb{23}\\rb{319279} \\mid N - 1 $, such that $ r = 20223770418 \\ge \\sqrt{N} $. Then\n$$ 2^{N - 1} \\equiv 3^{N - 1} \\equiv 1 \\mod N, \\qquad \\gcd\\rb{2^{\\rb{N - 1} / 3} - 1, N} = \\gcd\\rb{3^{\\rb{N - 1} / 2} - 1, N} = 1, $$\n$$ \\gcd\\rb{2^{\\rb{N - 1} / 17} - 1, N} = \\gcd\\rb{2^{\\rb{N - 1} / 23} - 1, N} = \\gcd\\rb{2^{\\rb{N - 1} / 319279} - 1, N} = 1. $$\nThus $ N $ is prime with primality certificate $ \\rb{r, \\sb{\\rb{2, 3}, \\rb{3, 2}, \\rb{17, 2}, \\rb{23, 2}, \\rb{319279, 2}}} $.\n\\end{example}\n\nAs for Lenstra's elliptic curve factorisation method, an elliptic curve analogue attempts to fix the $ p - 1 $ assumption by virtue of Hasse's theorem. By first choosing $ A, a, b \\in \\Z_N^* $, a valid elliptic curve over the ring $ \\Z_N^* $ is considered, giving the following analogous theorem.\n\n\\begin{theorem}[Goldwasser-Kilian]\nLet $ E : y^2 = x^3 + Ax + B $ be an elliptic curve over $ \\Q $ such that $ \\gcd\\rb{4A^3 + 27B^2, N} = 1 $, treated as over the ring $ \\Z_N^* $, and let $ m \\in \\Z_{> 0} $ be such that $ m > \\rb{\\sqrt[4]{N} + 1}^2 $. Then $ N $ is prime if\n$$ mP_q = \\mathcal{O}, \\qquad \\dfrac{m}{q}P_q \\ne \\mathcal{O}, \\qquad P_q \\in E\\rb{\\Q} \\setminus \\cb{\\mathcal{O}}, $$\nfor any distinct prime $ q \\in \\Z_{> 0} $ such that $ q \\mid m $.\n\\end{theorem}\n\n\\begin{proof}\nLet $ p \\in \\Z_{> 0} $ be a prime such that $ p \\mid N $, and let $ v = v_q\\rb{m} $ be the $ q $-adic valuation of $ m $. Now let $ r_p : E\\rb{\\Q} \\to E_p\\rb{\\F_p} $ be the reduction modulo $ p $ map, and let $ P_q' = \\rb{m / q^v}r_p\\rb{P_q} $. Then $ q^vP_q' = mr_p\\rb{P_q} = \\mathcal{O} $ and $ q^{v - 1}P_q' = \\rb{m / q}r_p\\rb{P_q} \\ne \\mathcal{O} $. Hence $ ord\\rb{P_q'} = q^v $. Lagrange's theorem gives $ q^v \\mid \\abs{E_p\\rb{\\F_p}} $, so $ m \\mid \\abs{E_p\\rb{\\F_p}} $. Now Hasse's theorem gives\n$$ \\rb{\\sqrt[4]{N} + 1}^2 < m \\le \\abs{E_p\\rb{\\F_p}} \\le p + 2\\sqrt{p} + 1 = \\rb{\\sqrt{p} + 1}^2. $$\nHence $ p > \\sqrt{N} $. Thus $ p = N $ and $ N $ is prime.\n\\end{proof}\n\nAs for the Pocklington-Lehmer primality test, an appropriate primality certificate, possibly with the form of an ordered triple $ \\rb{m, A, c} $ as in Theorem \\ref{thm:pocklington}, where $ c $ is a list of ordered pairs $ \\rb{q, P_q} $, will act as the output of a successful test. With all previous considerations, the Goldwasser-Kilian primality test can be illustrated with an even larger value of $ N $ in the following example.\n\n\\begin{example}\nLet $ N = 9223372036854775907 $. Instead let $ A = -2 $, $ a = 2 $, and $ b = 0 $, such that $ P = \\rb{2, 0} $ and $ B = 0^2 - 2^3 - -2\\rb{-2} = -4 $. Then $ \\gcd\\rb{4A^3 + 27B^2, N} = 1 $, such that $ E : y^2 = x^3 - 2x - 4 $ is an elliptic curve over $ \\Q $. Now let $ m = \\rb{2}\\rb{11}\\rb{13}\\rb{37}\\rb{269}\\rb{1327} = 3777382466 $ such that $ \\rb{\\sqrt[4]{N} + 1}^2 < 3037110719 < m $. Then $ ord\\rb{P} = 2 $, so $ mP = \\mathcal{O} $, and\n$$ \\dfrac{m}{2}P \\ne \\mathcal{O}, \\qquad \\dfrac{m}{11}P \\ne \\mathcal{O}, \\qquad \\dfrac{m}{13}P \\ne \\mathcal{O}, \\qquad \\dfrac{m}{37}P \\ne \\mathcal{O}, \\qquad \\dfrac{m}{269}P \\ne \\mathcal{O}, \\qquad \\dfrac{m}{1327}P \\ne \\mathcal{O}. $$\nThus $ N $ is prime with primality certificate $ \\rb{m, -2, \\sb{\\rb{2, P}, \\rb{11, P}, \\rb{13, P}, \\rb{37, P}, \\rb{269, P}, \\rb{1327, P}}} $.\n\\end{example}\n\nThe heart of the algorithm lies on finding a suitable elliptic curve $ E\\rb{\\Q} $, which in the above example is seemingly conjured from thin air. As for Lenstra's elliptic curve factorisation method, many random values of $ A $, $ a $, and $ b $ are generated, until their corresponding elliptic curve is found to have an order with enough small distinct prime factors to multiply and exceed $ \\rb{\\sqrt[4]{N} + 1}^2 $. This order $ E_p\\rb{F_p} $ is in turn efficiently computed through the Schoof-Elkies-Atkin algorithm.\n\n\\begin{remark}\nUsing the Pocklington-Lehmer primality test for this example will require $ r \\ge 273901883852669 $, which is a prime and will take forever, even with a supercomputer.\n\\end{remark}\n\nMost algorithms in this subsection are given in full under code listings in the appendix. Unfortunately, due to the difficulty in implementing Schoof's algorithm, the Goldwasser-Kilian primality test is omitted. Historically, Schoof's algorithm was seen as too cumbersome to implement, such that the theory of complex multiplication was utilised instead to construct an elliptic curve $ E $ with an easily computable order $ E_p\\rb{F_p} $. This became known as the \\emph{Atkin-Morain primality test}, which remained the fastest primality proving algorithm to date, despite the later advent of the much faster Schoof-Elkies-Atkin algorithm.\n\n\\pagebreak\n\n\\subsection{Cryptography}\n\n\\emph{Cryptography} is the study of computational techniques to allow for secure communication of information across public platforms in the presence of third party eavesdroppers. It is historically based on the computational intractability of certain mathematical problems, such as integer factorisation. As the difficult computations involved in elliptic curve point multiplication became more apparent, they saw great use in modern cryptography, of which a very brief introduction will be provided in this short subsection. With the standard notation of three parties, namely \\emph{Alice}, the sender, \\emph{Bob}, the receiver, and \\emph{Eve}, the eavesdropper, the standard process of secure communication is as follows.\n\n\\begin{algorithm}[Communication across a public domain]\nAlice sends a message to Bob.\n\\begin{enumerate}\n\\item Alice intends to send a \\emph{plaintext} message $ P $ to Bob, without Eve eavesdropping.\n\\item Alice \\emph{encrypts} $ P $ with an \\emph{encryption key} $ E $, which is owned only by Alice.\n\\item The \\emph{ciphertext} message $ C = E\\rb{P} $ is sent across a public domain, from Alice to Bob.\n\\item Bob \\emph{decrypts} $ C $ with a \\emph{decryption key} $ D $, which is owned only by Bob.\n\\item Bob recovers $ P = D\\rb{C} $ from Alice.\n\\end{enumerate}\n\\end{algorithm}\n\nIn this setup, it is clear that Bob should never leak $ D $ to Eve so as to keep $ P $ secret, and that $ D $ must be the left inverse of $ E $. In the case of \\emph{symmetric encryption}, the right inverse property also holds, allowing for the roles of $ E $ and $ D $ to swap. However, Alice and Bob would then need to secretly agree on $ E $ and $ D $ in advance, which may be infeasible if there are no means for prior contact. \\emph{Asymmetric encryption} solves this problem by introducing two pairs of inverse keys for Alice and Bob, such that they would publish their respective encryption keys $ E_A $ and $ E_B $, while hiding their respective decryption keys $ D_A $ and $ D_B $. Whenever Alice intends to send $ P $ to Bob, Alice would send $ C = E_B\\rb{P} $ across the public domain, so that only Bob could recover $ P = D_B\\rb{C} $, or vice versa. The modern \\emph{Rivest-Shamir-Adleman cryptosystem}, or more commonly known as the \\emph{RSA}, is a asymmetric encryption system that has its basis on the difficulty of integer factorisation, and is given in the following algorithm.\n\n\\begin{algorithm}[Rivest-Shamir-Adleman cryptosystem]\nOutputs a pair of keys.\n\\begin{enumerate}\n\\item Choose two distinct huge prime numbers $ p, q \\in \\Z_{> 0} $ of similar magnitude and compute $ n = pq $.\n\\item Compute $ \\lambda\\rb{n} = \\text{lcm}\\rb{p - 1, q - 1} $ and choose a small $ e \\in \\Z_{> 0} $ such that $ \\gcd\\rb{e, \\lambda\\rb{n}} = 1 $.\n\\item Compute the multiplicative inverse $ d $ of $ e $ modulo $ \\lambda\\rb{n} $.\n\\item Return the public encryption key $ \\rb{n, e} $ and function $ E\\rb{P} \\equiv P^e \\mod n $.\n\\item Return the private decryption key $ \\rb{n, d} $ and function $ D\\rb{C} \\equiv C^d \\mod n $.\n\\end{enumerate}\n\\end{algorithm}\n\nThe functional correctness of the algorithm trivially follows from Fermat's little theorem, and as such will not be discussed. Now $ e $ is typically chosen to be small enough for quick encryption computations, but not too small so as to be insecure. Moreover $ P $ is also assumed to have been translated into an integer beforehand with some form of \\emph{padded} cipher mechanism. The following example implements the RSA.\n\n\\begin{example}\nLet $ p = 2147483647 $ and $ q = 2147483659 $ be primes. Then $ n = pq = 4611686039902224373 $ and $ \\lambda\\rb{n} = \\text{lcm}\\rb{p - 1, q - 1} = 768614339267876178 $. Now choose a public encryption key $ n $ and $ e = 65537 $ such that $ \\gcd\\rb{e, \\lambda\\rb{n}} = 1 $. Thus the private decryption key is $ n $ and $ d = 73205833433176421 \\equiv e^{-1} \\mod l $.\n\\end{example}\n\nNow the availability of increasingly powerful techniques imposes a minimum bit size of $ p $ and $ q $ in order to make the RSA even remotely practical. Furthermore, the presence of specialised algorithms, such as Pollard's $ p - 1 $ method, forces a requirement on $ p - 1 $ and $ q - 1 $ to have large factors to prevent unprecedented \\emph{attacks} as well. To date, a few hundred bits of $ n $ is easily factorisable by a desktop computer within hours, if not seconds, with a suitably powerful algorithm. As such, the RSA generally chooses an $ n $ with thousands of bits, which is somewhat cumbersome for massive amounts of data, and would require larger chip sizes and power consumption in the process of encryption. On the other hand, this drawback is less evident in symmetric encryption systems, which are generally faster if $ E $ and $ D $ were somehow agreed by Alice and Bob in advance. This could be feasibly done without any prior contact with the following \\emph{key exchange protocol}.\n\n\\begin{algorithm}[Diffie-Hellman key exchange]\nOutputs a private symmetric key.\n\\begin{enumerate}\n\\item Alice and Bob agree on a huge prime $ p \\in \\Z_{> 0} $ and a small $ g \\in \\Z_{> 0} $ such that $ g < p $.\n\\item Alice chooses $ a \\in \\Z_{> 0} $ such that $ a < p $ and sends $ g_a \\equiv g^a \\mod p $ to Bob across the public domain.\n\\item Bob chooses $ b \\in \\Z_{> 0} $ such that $ b < p $ and sends $ g_b \\equiv g^b \\mod p $ to Alice across the public domain.\n\\item Alice computes $ s \\equiv g_b^a \\equiv g^{ab} \\mod p $ privately.\n\\item Bob computes $ s \\equiv g_a^b \\equiv g^{ab} \\mod p $ privately.\n\\end{enumerate}\n\\end{algorithm}\n\nThe algorithm simply uses basic properties of modular exponentiation and produces a private symmetric key $ s $ between Alice and Bob, which as aforementioned allows secure communication in a public domain. In this process, only $ p $, $ g $, $ g_a $, and $ g_b $ is immediately available to Eve, while $ a $, $ b $, and $ s $ are kept secret between Alice and Bob. The following example implements the Diffie-Hellman key exchange.\n\n\\begin{example}\nLet $ p = 2147483647 $ be prime and $ g = 65537 $. Alice chooses $ a = 16777259 $ and sends the public key $ g_a = 751856369 \\equiv g^a \\mod p $ to Bob, while Bob chooses $ b = 16777289 $ and sends the public key $ g_b = 1654172966 \\equiv g^b \\mod p $ to Alice. Thus the private symmetric key is $ s = 1288974049 \\equiv g^{ab} \\mod p $.\n\\end{example}\n\nReverse engineering these values as a third party attacker is known as the \\emph{discrete logarithm problem}, which is, analogous to integer factorisation, computationally infeasible provided $ p $ is chosen to be huge. While there is an impossibly slow, naive approach by simply trying $ g^n $ for each $ n \\in \\Z_{> 0} $ until $ g^n = g^a $ or $ g^n = g^b $ is obtained, there are no known algorithms efficient enough to crack the system.\n\n\\begin{remark}\nUnfortunately, the non-existence of efficient polynomial time algorithms for integer factorisation and the discrete logarithm problem is unproven. In fact, a \\emph{quantum computer} running \\emph{Shor's algorithm} would theoretically do both of these in polynomial time.\n\\end{remark}\n\nRephrasing the Diffie-Hellman key exchange in terms of finite cyclic groups, modulo operations involving $ p $ is equivalent to operating under the finite multiplicative cyclic group $ \\Z_p^* $, while $ g $ is simply any generator of $ \\Z_p^* $. A elliptic curve variant of Diffie-Hellman can then be stated easily as follows.\n\n\\begin{algorithm}[Elliptic curve Diffie-Hellman]\nOutputs a private symmetric key.\n\\begin{enumerate}\n\\item Alice and Bob agree on an elliptic curve $ E $ over $ \\F_p $ for some prime $ p \\in \\Z_{> 0} $ and a point $ P \\in E\\rb{\\F_p} $.\n\\item Alice chooses $ a \\in \\Z_p^* $ and sends $ P_a = aP $ to Bob across the public domain.\n\\item Bob chooses $ b \\in \\Z_p^* $ and sends $ P_b = bP $ to Alice across the public domain.\n\\item Alice computes $ S = a\\rb{P_b} = a\\rb{bP} = abP $ privately.\n\\item Bob computes $ S = b\\rb{P_a} = b\\rb{aP} = abP $ privately.\n\\end{enumerate}\n\\end{algorithm}\n\nThe prior example can then be illustrated with elliptic curve Diffie-Hellman as follows.\n\n\\begin{example}\nLet $ E : y^2 = x^3 + 65537x + 1 $ be an elliptic curve over $ \\F_p $ where $ p = 2147483647 $ and $ P = \\rb{0, 1} \\in E\\rb{\\F_p} $. Alice chooses $ a = 16777259 $ and sends the public key $ P_a = aP = \\rb{675295473, 1821381850} $ to Bob, while Bob chooses $ b = 16777289 $ and sends the public key $ P_b = bP = \\rb{294235749, 438747352} $ to Alice. Thus the private symmetric key is $ S = abP = \\rb{1210475635, 471187571} $.\n\\end{example}\n\nThe discrete logarithm problem on certain elliptic curves over finite fields is also significantly harder than that of generic finite fields, and as such a list of recommended elliptic curves for use in key exchanges was published publicly by the National Institute of Standards and Technology. These elliptic curves are used in several other cryptosystems, such as the underlying group of the \\emph{ElGamal encryption system}, or in other cryptographic contexts, such as \\emph{digital signatures}, but will not be discussed further. Most of these elliptic curve based systems require fewer bits than those based on traditional finite fields. An estimate places a key size of less than $ 2^9 $ bits having an equivalent security to a key size of $ 2^{12} $ bits for the RSA.\n\n\\begin{remark}\nUnfortunately, the discrete logarithm problem for elliptic curves is still susceptible to attacks from a quantum computer. The \\emph{supersingular isogeny key exchange} is an analogue of Diffie-Hellman that considers isogenies, and would in theory resist quantum attacks, but has a higher performance overhead.\n\\end{remark}\n\n\\pagebreak\n\n\\appendix\n\n\\section{Preliminaries}\n\n\\subsection{Rings and fields}\n\nLet $ R $ be a commutative unital ring and $ F \\subseteq K \\subseteq L $ be fields.\n\n\\begin{definition}[Automorphism]\nAn \\textbf{automorphism} of $ R $ is an isomorphism from $ R $ to itself, which are elements of the \\textbf{automorphism group} $ Aut\\rb{R} $ with respect to composition.\n\\end{definition}\n\n\\begin{example}\n$ id_R $ is an automorphism of $ R $.\n\\end{example}\n\n\\begin{definition}[Prime ideal]\nAn ideal $ I \\subset R $ is \\textbf{prime} iff $ ab \\in I $ implies $ a \\in I $ or $ b \\in I $ for any two elements $ a, b \\in I $.\n\\end{definition}\n\n\\begin{example}\nAn irreducible element $ r \\in R $ in a unique factorisation domain $ R $ generates a prime ideal $ \\ab{r} $.\n\\end{example}\n\n\\begin{definition}[Chain]\nA \\textbf{chain} of subsets in $ R $ of length $ n \\in \\Z_{\\ge 0} $ is a sequence of distinct subsets $ S_0 \\subset \\dots \\subset S_n \\subset R $.\n\\end{definition}\n\n\\begin{example}\n$ \\ab{0} \\subset \\ab{x_1} \\subset \\dots \\subset \\ab{x_1, \\dots, x_n} \\subset F\\sb{x_1, \\dots, x_n} $ is a chain of prime ideals of length $ n $.\n\\end{example}\n\n\\begin{definition}[Characteristic]\nThe \\textbf{characteristic} $ char\\rb{F} $ of $ F $ is the smallest $ n \\in \\Z_{> 0} $, if it exists, such that $ n \\cdot 1 = 1 + \\dots + 1 = 0 $. Otherwise $ char\\rb{F} $ is $ 0 $.\n\\end{definition}\n\n\\begin{example}\n$ char\\rb{\\C} = char\\rb{\\R} = char\\rb{\\Q} = 0 $ while $ char\\rb{\\F_{p^e}} = p $ for prime $ p \\in \\Z_{> 0} $ and $ e \\in \\Z_{\\ge 0} $.\n\\end{example}\n\n\\begin{definition}[Field extension]\n$ K $ is a \\textbf{field extension} of $ F $, denoted by $ K / F $, iff $ F $ is a subfield of $ K $.\n\\end{definition}\n\n\\begin{example}\n$ \\C / \\R $, $ \\R / \\Q $, $ \\C / \\Q $, and $ \\F_{p^e} / \\F_{p^{e'}} $ for prime $ p \\in \\Z_{> 0} $ and $ e \\mid e' $ are field extensions.\n\\end{example}\n\n\\begin{definition}[$ F $-homomorphism]\nAn \\textbf{$ F $-homomorphism} from $ K / F $ to another field extension $ K' / F $ is a field homomorphism $ \\phi : K \\to K' $ such that $ \\val{\\phi}_F = \\val{id}_F $. The definitions of \\textbf{$ F $-isomorphism} and \\textbf{$ F $-automorphism} extend naturally, with $ Aut_F\\rb{K} $ denoting the \\textbf{$ F $-automorphism group} of $ K $.\n\\end{definition}\n\n\\begin{example}\nComplex conjugation is an $ \\R $-automorphism of $ \\C $.\n\\end{example}\n\n\\begin{definition}[Finite extension]\n$ K / F $ is \\textbf{finite} iff the dimension $ dim_F K $ of $ K $ as a vector space over $ F $ is finite.\n\\end{definition}\n\n\\begin{example}\n$ \\C / \\R $ is a finite extension with $ \\sb{\\C : \\R} = 2 $ since $ \\cb{1, i} $ is a basis of $ \\C $ over $ \\R $.\n\\end{example}\n\n\\begin{definition}[Finitely generated]\n$ F\\rb{s_1, \\dots, s_n} $ is \\textbf{finitely generated} by $ s_1, \\dots, s_n \\in K $ over $ F $ iff $ F\\rb{S} $ is the smallest subfield of $ K $ containing $ s_1, \\dots, s_n $ and the elements of $ F $.\n\\end{definition}\n\n\\begin{example}\n$ \\Q\\rb{\\sqrt{2}, \\sqrt{3}} $ is finitely generated by $ \\cb{\\sqrt{2}, \\sqrt{3}} $ over $ \\Q $, by $ \\cb{\\sqrt{2}} $ over $ \\Q\\rb{\\sqrt{3}} $, and by $ \\cb{\\sqrt{3}} $ over $ \\Q\\rb{\\sqrt{2}} $.\n\\end{example}\n\n\\begin{definition}[Number field]\n$ K $ is a \\textbf{number field} iff $ K / F $ is finite and $ F = \\Q $.\n\\end{definition}\n\n\\begin{example}\n$ \\Q\\rb{\\sqrt{d}} $ for square-free $ d \\in \\Z $ are number fields with $ \\sb{\\Q\\rb{\\sqrt{d}} : \\Q} = 2 $.\n\\end{example}\n\n\\begin{definition}[Algebraic element]\n$ \\alpha \\in K $ is \\textbf{algebraic} over $ F $ iff it is a root of some non-zero polynomial in $ F\\sb{x} $. Otherwise $ \\alpha $ is \\textbf{transcendental} over $ F $.\n\\end{definition}\n\n\\begin{example}\n$ \\pi $ is transcendental over $ \\Q $ but algebraic over $ \\R $ since it is the root of $ x - \\pi $.\n\\end{example}\n\n\\begin{definition}[Minimal polynomial]\nThe \\textbf{minimal polynomial} $ m_\\alpha $ of $ \\alpha $ over $ F $ is the unique monic irreducible polynomial in $ F\\sb{x} $ with $ \\alpha $ as a root.\n\\end{definition}\n\n\\begin{example}\nThe minimal polynomial of $ \\sqrt{2} $ is $ x^2 - 2 $ over $ \\Q $ and is $ x - \\sqrt{2} $ over $ \\R $.\n\\end{example}\n\n\\begin{definition}[Algebraic extension]\n$ K / F $ is \\textbf{algebraic} iff any element in $ K $ is algebraic over $ F $. Otherwise $ K / F $ is \\textbf{transcendental}.\n\\end{definition}\n\n\\begin{example}\n$ \\C / \\Q $ is not an algebraic extension since $ \\pi $ is transcendental over $ \\Q $.\n\\end{example}\n\n\\begin{definition}[Algebraically closed]\n$ F $ is \\textbf{algebraically closed} iff any non-constant polynomial in $ F\\sb{x} $ has a root in $ F $.\n\\end{definition}\n\n\\begin{example}\n$ \\R $ is not algebraically closed since $ x^2 + 1 \\in \\R\\sb{x} $ has no roots in $ \\R $.\n\\end{example}\n\n\\begin{definition}[Algebraic closure]\nAn \\textbf{algebraic closure} of $ F $ is an algebraically closed algebraic extension of $ F $ that is unique up to $ F $-isomorphism.\n\\end{definition}\n\n\\begin{example}\n$ \\overline{\\R} = \\C $ while $ \\overline{\\Q} $ is the field of algebraic numbers.\n\\end{example}\n\nThe existence and uniqueness of algebraic closures can be proven from \\emph{Zorn's lemma}, which is equivalent to the \\emph{Axiom of choice}.\n\n\\begin{proposition}\nAn algebraic closure $ \\overline{F} $ of $ F $ exists and is unique up to $ F $-isomorphism.\n\\end{proposition}\n\n\\begin{definition}[Splits]\nA polynomial $ f\\rb{x} \\in F\\sb{x} $ of degree $ n > 0 $ \\textbf{splits} over $ K $ iff $ f\\rb{x} = c\\prod_{i = 0}^n \\rb{x - a_k} $ for some $ c \\in F $ and $ a_k \\in K $.\n\\end{definition}\n\n\\begin{example}\n$ x^2 - 2 $ splits over $ \\Q\\rb{\\sqrt{2}} $ but not over $ \\Q $ since $ x^2 - 2 = \\rb{x - \\sqrt{2}}\\rb{x + \\sqrt{2}} $ in $ \\Q\\rb{\\sqrt{2}} $.\n\\end{example}\n\n\\begin{definition}[Normal extension]\n$ K / F $ is \\textbf{normal} iff $ K / F $ is algebraic and any irreducible polynomial in $ F\\sb{x} $ with a root in $ K $ splits over $ F $.\n\\end{definition}\n\n\\begin{example}\n$ \\Q\\rb{\\sqrt{2}} / \\Q $ is a normal extension, while $ \\Q\\rb{\\sqrt[3]{2}} / \\Q $ is not a normal extension since $ f\\rb{x} = x^3 - 2 = \\rb{x - \\sqrt[3]{2}} \\rb{x^2 + \\sqrt[3]{2}x + \\sqrt[3]{4}} $ has a root $ x = \\sqrt[3]{2} $ but does not split over $ \\Q\\rb{\\sqrt[3]{2}} $.\n\\end{example}\n\n\\begin{definition}[Separable polynomial]\nAn polynomial $ f \\in F\\sb{x} $ is \\textbf{separable} iff $ df / dx \\ne 0 $.\n\\end{definition}\n\n\\begin{example}\n$ x^2 - 2 \\in \\Q\\sb{x} $ is a separable polynomial since $ d\\rb{x^2 - 2} / dx = 2x \\ne 0 $, while $ x^2 - y^2 \\in \\F_2\\rb{y^2} $ is an inseparable polynomial since $ d\\rb{x^2 - y^2} / d\\rb{y^2} = 0 $.\n\\end{example}\n\n\\begin{definition}[Separable extension]\n$ K / F $ is \\textbf{separable} iff $ K / F $ is algebraic and the minimal polynomial of any $ \\alpha \\in K $ is separable.\n\\end{definition}\n\n\\begin{example}\n$ \\Q\\rb{\\sqrt{2}} / \\Q $ is a separable extension, while $ \\F_2\\rb{y} / \\F_2\\rb{y^2} $ is an inseparable extension since the minimal polynomial of $ y $ over $ F_2\\rb{y^2} $ is $ x^2 - y^2 $, which is inseparable.\n\\end{example}\n\n\\begin{definition}[Galois extension]\n$ K / F $ is \\textbf{Galois} iff $ K / F $ is normal and separable.\n\\end{definition}\n\n\\begin{example}\n$ \\C / \\R $ and $ \\Q\\rb{\\sqrt{2}} / \\Q $ are Galois extensions.\n\\end{example}\n\n\\begin{definition}[Galois group]\n$ Aut_F\\rb{K} $ is the \\textbf{Galois group} $ Gal_F\\rb{K} $ of $ K $ over $ F $ iff $ K / F $ is Galois.\n\\end{definition}\n\n\\begin{example}\n$ Gal_\\R\\rb{\\C} = \\cb{id_\\R, \\phi} $ where $ \\phi $ is complex conjugation, while $ Gal_\\Q\\rb{\\Q\\sqrt{2}} = \\cb{id_\\Q, \\phi} $ where $ \\phi $ is the $ \\Q $-automorphism that swaps $ \\sqrt{2} $ and $ -\\sqrt{2} $.\n\\end{example}\n\n\\begin{definition}[Perfect field]\n$ F $ is \\textbf{perfect} iff the algebraic closure of $ F $ is Galois.\n\\end{definition}\n\n\\begin{example}\nExamples of perfect fields include any field of characteristic zero including $ \\Q $, $ \\R $, and $ \\C $, any finite field $ \\F_{p^e} $, and any algebraically closed field including $ \\overline{\\Q} $. Examples of imperfect fields include the field of rational functions $ \\F_p\\rb{y} $ of any finite field $ \\F_p $ since $ x^p - y \\in \\F_p\\rb{y} $ is irreducible but inseparable.\n\\end{example}\n\n\\pagebreak\n\n\\subsection{Algebraic varieties}\n\nLet $ F $ be a perfect field of $ char\\rb{F} \\notin \\cb{2, 3} $ with algebraic closure $ K = \\overline{F} $ and Galois group $ Gal_F\\rb{K} $.\n\n\\begin{definition}[Affine space]\nAn \\textbf{affine $ n $-space} over $ F $ is $ \\A^n = K^n $.\n\\end{definition}\n\n\\begin{definition}[Projective space]\nA \\textbf{projective $ n $-space} over $ F $ is $ \\P^n = \\rb{\\A^{n + 1} \\setminus \\cb{\\rb{0, \\dots, 0}}} / \\sim $, the set of equivalence classes of \\textbf{homogeneous coordinates} $ \\sb{p_0, \\dots, p_n} $, where $ \\rb{x_0, \\dots, x_n} \\sim \\rb{y_0, \\dots, y_n} $ iff each $ x_i = \\lambda y_i $ for some $ \\lambda \\in F^* $.\n\\end{definition}\n\n$ \\P^n $ can be considered a superset of $ n + 1 $ copies of $ \\A^n $ by the natural inclusions $ \\phi_i : \\A^n \\to \\P^n $ for each $ i \\in \\cb{0, \\dots, n} $ defined by\n$$ \\phi_i\\rb{x_1, \\dots, x_n} = \\sb{x_1, \\dots, x_i, 1, x_{i + 1}, \\dots, x_n}, $$\nso write $ \\A^n \\subseteq \\P^n $. Now let $ \\A^n $ be an affine $ n $-space over $ F $ and $ \\P^n $ be a projective $ n $-space over $ F $.\n\n\\begin{definition}[Rational point]\nThe set of \\textbf{$ F $-rational points} of $ \\A^n $ is $ \\A^n\\rb{F} = F^n $, and of $ \\P^n $ is\n$$ \\P^n\\rb{F} = \\cb{\\sb{p_0, \\dots, p_n} \\in \\P^n \\mid \\forall p_j \\ne 0, \\ \\forall p_i, \\ p_i / p_j \\in F}. $$\n\\end{definition}\n\n$ \\A^n $ can be equipped with $ Gal_F\\rb{K} $, such that $ \\A^n\\rb{F} = \\cb{a \\in \\A^n \\mid \\forall \\sigma \\in Gal_F\\rb{K}, \\ \\sigma\\rb{a} = a} $. This holds similarly in $ \\P^n $.\n\n\\begin{example}\n$ \\C^n $ is an affine $ n $-space over $ \\R $, with $ \\C^n\\rb{\\R} = \\R $. $ \\overline{\\F}^n $ is a projective $ n $-space over $ \\F^n $, with $ \\overline{\\F}^n\\rb{\\F^n} $ being the projective plane of order $ n $.\n\\end{example}\n\n\\begin{definition}[Homogeneous]\nA polynomial $ f \\in K\\sb{x_0, \\dots, x_n} $ is \\textbf{homogeneous} of degree $ d \\in \\Z_{\\ge 0} $ iff for any $ \\lambda \\in K $, it holds that $ f\\rb{\\lambda x_0, \\dots, \\lambda x_n} = \\lambda^d f\\rb{x_0, \\dots, x_n} $. An ideal $ I \\subseteq K\\sb{x_0, \\dots, x_n} $ is homogeneous iff $ I $ is generated by homogeneous polynomials in $ K\\sb{x_0, \\dots, x_n} $.\n\\end{definition}\n\nA homogeneous polynomial $ f^* \\in K\\sb{x_0, \\dots, x_n} $ can be \\emph{dehomogenised} into $ f \\in K\\sb{x_1, \\dots, x_n} $ by\n$$ f\\rb{x_1, \\dots, x_n} = f^*\\rb{1, x_1, \\dots, x_n}, $$\nwhile a non-homogeneous polynomial $ g \\in K\\sb{x_1, \\dots, x_n} $ can be \\emph{homogenised} into $ g^* \\in K\\sb{x_0, \\dots, x_n} $ by\n$$ g^*\\rb{x_0, \\dots, x_n} = x_0^d g\\rb{x_1 / x_0, \\dots, x_n / x_0}. $$\n\n\\begin{example}\n$ z^3 + wxy + 7w^3 \\in \\C\\sb{w, x, y, z} $ is homogeneous, which can be dehomogenised to $ z^3 + xy + 7 \\in \\C\\sb{x, y, z} $. Conversely $ x^3 + 3x^2y + z^7 \\in \\C\\sb{x, y, z} $ is non-homogeneous, which can be homogenised to $ w^4x^3 + 3w^4x^2y + z^7 \\in \\C\\sb{w, x, y, z} $. Thus $ \\ab{z^3 + wxy + 7w^3, w^4x^3 + 3w^4x^2y + z^7} \\subseteq \\C\\sb{w, x, y, z} $ is a homogeneous ideal.\n\\end{example}\n\nThe following definitions are simplified by considering only prime ideals in \\emph{Hilbert's Nullstellensatz}.\n\n\\begin{definition}[Algebraic variety]\nAn \\textbf{affine algebraic variety} of $ \\A^n $ over $ F $ is\n$$ A = \\cb{a \\in \\A^n \\mid \\forall f \\in I, \\ f\\rb{a} = 0} $$\nfor some finitely generated prime ideal $ I \\subseteq F\\sb{x_1, \\dots, x_n} $, denoted by $ A\\rb{I} $ and $ I\\rb{A} $ respectively. The set of $ F $-rational points of $ A $ is $ A\\rb{F} = A \\cap \\A^n\\rb{F} $. A \\textbf{projective algebraic variety} $ P $ of $ \\P^n $ over $ F $ and the set of $ F $-rational points of $ P $ are defined similarly but with homogeneous prime ideals.\n\\end{definition}\n\nSince $ I\\rb{A} $ can be finitely generated by $ f_1, \\dots, f_m \\in F\\sb{x_1, \\dots, x_n} $, it holds that $ A\\rb{f_1, \\dots, f_m}\\rb{F} $ is the set of solutions in $ F $ to the system of equations $ f_1\\rb{x_1, \\dots, x_n} = \\dots = f_m\\rb{x_1, \\dots, x_n} = 0 $. This holds similarly in $ \\P^n $.\n\n\\begin{example}\nLet $ \\ab{x^2 + y^2 - 1} \\subseteq \\R\\sb{x, y} $ be a finitely generated prime ideal. Thus $ A\\rb{x^2 + y^2 - 1} $ is an affine algebraic variety of $ \\C^2 $ over $ \\R $ and $ A\\rb{x^2 + y^2 - 1}\\rb{\\R} $ is the unit circle $ S^1 $. Homogenisation gives a finitely generated homogeneous ideal $ \\ab{x^2 + y^2 - w^2} \\subseteq \\R\\sb{w, x, y} $. Similarly $ P\\rb{x^2 + y^2 - w^2} $ is a projective algebraic variety of $ \\C^2 $ over $ \\R $ and $ P\\rb{x^2 + y^2 - w^2}\\rb{\\R} $ is the unit circle $ S^1 $.\n\\end{example}\n\nLet $ A $ be an affine algebraic variety of $ \\A^n $ over $ F $ and $ P $ be a projective algebraic variety of $ \\P^n $ over $ F $.\n\n\\begin{definition}[Dimension]\nThe \\textbf{dimension} $ dim\\rb{A} $ of $ A $ is the length of any longest chain of prime ideals in $ F\\sb{x_1, \\dots, x_n} / I\\rb{A} $. The dimension of $ P $ is $ dim\\rb{P} = dim\\rb{A\\rb{I\\rb{P}}} - 1 $ for any $ A\\rb{I\\rb{P}} \\subseteq \\A^n $.\n\\end{definition}\n\n\\begin{example}\nLet $ A\\rb{x - y} $ be an affine algebraic variety of $ \\C^3 $ over $ \\R $. Then a longest chain of prime ideals is $ \\ab{0} \\subset \\ab{y} \\subset \\ab{y, z} \\subset \\R\\sb{x, y, z} / \\ab{x - y} $, which has length two. Thus it has dimension $ dim\\rb{A\\rb{x - y}} = 2 $. A projective algebraic variety $ P\\rb{x - y} $ of $ \\C^2 $ over $ \\R $ has dimension $ dim\\rb{P\\rb{x - y}} = dim\\rb{A\\rb{x - y}} - 1 = 1 $.\n\\end{example}\n\nThe dimension of projective algebraic varieties can also be defined from \\emph{Krull's Hauptidealsatz}.\n\n\\begin{proposition}\n$ dim\\rb{P} = n - 1 $ iff $ I\\rb{P} $ is generated by a homogeneous irreducible polynomial in $ F\\sb{X_0, \\dots, X_n} $.\n\\end{proposition}\n\n\\begin{definition}[Smooth]\nA point $ a \\in A $ is \\textbf{singular} iff the Jacobian $ m \\times n $ matrix $ J $ defined by $ J_{ij} = \\partial f_i / \\partial x_j $ is such that $ rk\\rb{\\val{J}_a} < n - dim\\rb{A} $. $ A $ is \\textbf{smooth} if it has no singular points. This holds similarly for $ P $.\n\\end{definition}\n\n\\begin{example}\nLet $ A\\rb{x - y} $ be an affine algebraic variety of $ \\C^3 $ over $ \\R $. Then $ dim\\rb{A\\rb{x - y}} = 2 $, so a point $ a = \\rb{x, y, z} \\in A\\rb{x - y} $ is singular iff $ rk\\rb{\\val{J}_a} < 3 - 2 = 1 $, or\n$$ 0 = rk\\rb{\\val{J}_a} = rk\\onebythree{\\val{\\dfrac{\\partial \\rb{x - y}}{\\partial x}}_a}{\\val{\\dfrac{\\partial \\rb{x - y}}{\\partial y}}_a}{\\val{\\dfrac{\\partial \\rb{x - y}}{\\partial z}}_a} = rk\\onebythree{1}{-1}{0} = 1. $$\nThus there are no singular points and $ A\\rb{x - y} $ is smooth.\n\\end{example}\n\n\\begin{definition}[Function field]\nThe \\textbf{function field} of $ P $ is\n$$ F\\rb{P} = \\cb{f\\rb{x_0, \\dots, x_n} / g\\rb{x_0, \\dots, x_n} \\mid f, g \\in F\\sb{x_0, \\dots, x_n}, \\ \\deg\\rb{f} = \\deg\\rb{g}, \\ g \\notin I\\rb{P}} / \\sim, $$\nthe field of equivalence classes of \\textbf{rational functions} of homogeneous polynomials, where $ f / g \\sim f' / g' $ iff $ fg' - f'g \\in I\\rb{P} $.\n\\end{definition}\n\n\\begin{example}\nLet $ P\\rb{xy} $ be a projective algebraic variety of $ \\C $ over $ \\R $. Then $ x \\in \\R\\sb{x, y} $ and $ y \\in \\R\\sb{x, y} $ are homogeneous of degree one, and $ y \\notin I\\rb{P} $. Thus $ x / y \\in \\R\\rb{P} $.\n\\end{example}\n\nLet $ P' $ be a projective algebraic variety of $ \\P^m $ over $ F $.\n\n\\begin{definition}[Morphism]\nA \\textbf{morphism} from $ P $ to $ P' $ is an equivalence class of rational functions $ \\phi = \\sb{\\phi_0, \\dots, \\phi_m} : P \\to P' $ for some $ \\phi_i \\in F\\rb{P} $, such that for any $ p \\in P $, there is a rational function $ g \\in F\\rb{P} $ such that $ g\\phi_i\\rb{p} \\in P' $ for each $ \\phi_i $ and $ g\\phi_i\\rb{p} \\ne 0 $ for some $ \\phi_i $, where $ \\rb{\\phi_0, \\dots, \\phi_m} \\sim \\rb{\\psi_0, \\dots, \\psi_m} $ iff each $ \\phi_i = g'\\psi_i $ for some $ g' \\in F\\rb{P} $.\n\\end{definition}\n\n\\begin{example}\nLet $ P\\rb{x^2 + y^2 - w^2} $ be a projective algebraic variety of $ \\C^2 $ over $ \\R $ and $ P'\\rb{0} $ be a projective algebraic variety of $ \\C $ over $ \\R $, and let $ \\phi = \\sb{w + x, y} : P \\to P' $ be such that $ w + x, y \\in \\R\\rb{P} $. Let $ p = \\sb{w, x, y} \\in P $ be a point such that $ w + x \\ne 0 $ or $ y \\ne 0 $. Then $ w + x, y \\ne 0 $ are well-defined at $ p $. Now let $ p' = \\sb{w, x, y} \\in P $ be a point such that $ w + x = y = 0 $. Then $ \\rb{\\rb{w - x} / y}\\rb{w + x} = \\rb{w^2 - x^2} / y = y^2 / y = y $ and $ \\rb{\\rb{w - x} / y}y = w - x \\ne 0 $ are well-defined at $ p' $. Thus $ \\phi $ is a morphism.\n\\end{example}\n\nA standard result in algebraic geometry states that images of morphisms are projective algebraic varieties.\n\n\\begin{proposition}\nLet $ \\phi : P \\to P' $ be a morphism and $ dim\\rb{P} = dim\\rb{P'} = 1 $. Then $ \\phi $ is either constant or surjective.\n\\end{proposition}\n\n\\begin{definition}[Isomorphism]\nAn \\textbf{isomorphism} is a morphism $ \\phi : P \\to P' $ such that there is another morphism $ \\phi' : P' \\to P $ where $ \\phi' \\circ \\phi = id_P $ and $ \\phi \\circ \\phi' = id_{P'} $. $ P $ and $ P' $ are \\textbf{isomorphic}, denoted by $ P \\cong P' $, iff there is an isomorphism $ \\phi : P \\to P' $.\n\\end{definition}\n\n\\begin{example}\nLet $ P\\rb{x^2 + y^2 - w^2} $ be a projective algebraic variety of $ \\C^2 $ over $ \\R $ and $ P'\\rb{0} $ be a projective algebraic variety of $ \\C $ over $ \\R $ with a morphism $ \\phi : \\sb{w + x, y} : P \\to P' $. Then $ \\phi' = \\sb{x^2 + y^2, x^2 - y^2, 2xy} : P' \\to P $ is also a morphism such that $ \\phi \\circ \\phi' = \\sb{2x^2, 2xy} = \\sb{x, y} = id_{P'} $ and\n$$ \\phi' \\circ \\phi = \\sb{\\rb{w + x}^2 + y^2, \\rb{w + x}^2 - y^2, 2\\rb{w + x}y} = \\sb{2w\\rb{w + x}, 2x\\rb{w + x}, 2y\\rb{w + x}} = \\sb{w, x, y} = id_P. $$\nThus $ \\phi $ is an isomorphism and $ P \\cong P' $.\n\\end{example}\n\n\\pagebreak\n\n\\subsection{Algebraic curves}\n\nLet $ F $ be a perfect field of $ char\\rb{F} \\notin \\cb{2, 3} $ with algebraic closure $ K = \\overline{F} $ and $ V $ be a projective algebraic variety of $ \\P^n $ over $ F $.\n\n\\begin{definition}[Projective plane curve]\n$ V $ is a \\textbf{projective plane curve} iff $ dim\\rb{V} = 1 $ and $ n = 2 $.\n\\end{definition}\n\nSince a projective plane curve $ V $ is such that $ \\dim\\rb{V} = 1 = 2 - 1 $, it holds that $ I\\rb{V} $ is generated by some homogeneous irreducible polynomial $ f \\subseteq F\\sb{X, Y, Z} $. For ease of notation $ V $ will be written in the form $ V : f\\rb{X, Y, Z} = 0 $, or in its simpler dehomogeneous form $ V : f\\rb{x, y} = 0 $. Now let $ C : f\\rb{X, Y, Z} = 0 $ and $ C' : g\\rb{X, Y, Z} = 0 $ be two projective plane curves over $ F $ with a point $ P = \\sb{a, b, c} \\in C \\cap C' $.\n\n\\begin{definition}[Multiplicity]\nThe \\textbf{multiplicity} $ m_P\\rb{f} $ of $ C $ at $ P $ is the smallest $ m \\in \\Z_{> 0} $ such that\n$$ \\forall i, j, k \\in \\Z_{\\ge 0}, \\qquad i + j + k = n, \\qquad \\val{\\dfrac{\\partial^n f}{\\partial X^i \\partial Y^j \\partial Z^k}}_P = 0 $$\nfor any $ n \\in \\cb{0, \\dots, m - 1} $ but not $ n = m $.\n\\end{definition}\n\n$ P $ is singular iff $ rk\\rb{\\val{J}_P} < 1 $, or $ \\val{\\partial f / \\partial X}_P = \\val{\\partial f / \\partial Y}_P = \\val{\\partial f / \\partial Z}_P = 0 $, which holds iff $ m_P\\rb{f} > 1 $.\n\n\\begin{example}\nAssume $ char\\rb{F} = 0 $, and let $ f\\rb{X, Y, Z} = f\\rb{x, y} = \\rb{x^2 + y^2}^2 + 3x^2y - y^3 $ with $ P = \\sb{0, 0, 1} = \\rb{0, 0} $. Then\n$$ \\val{\\dfrac{\\partial f}{\\partial x}}_P = \\val{\\dfrac{\\partial f}{\\partial y}}_P = \\val{\\dfrac{\\partial^2 f}{\\partial x^2}}_P = \\val{\\dfrac{\\partial^2 f}{\\partial y^2}}_P = \\val{\\dfrac{\\partial^2 f}{\\partial x \\partial y}}_P = 0, \\qquad \\val{\\dfrac{\\partial^3 f}{\\partial y^3}}_P = -6 \\ne 0. $$\nThus the multiplicity of $ C $ at $ P $ is $ m_P\\rb{f} = 3 $ and $ P $ is singular.\n\\end{example}\n\n\\begin{definition}[Tangent]\nThe \\textbf{tangents} $ T_P\\rb{f} $ of $ C $ at $ P = \\sb{a, b, c} $ with multiplicity $ m = m_P\\rb{f} $ are the irreducible factors of the polynomial\n$$ t_P\\rb{f}\\rb{X, Y, Z} = \\sum_{i + j + k = m} \\twobyone{m}{i, j, k} \\val{\\dfrac{\\partial^m f}{\\partial X^i \\partial Y^j \\partial Z^k}}_P \\rb{X - a}^i \\rb{Y - b}^j \\rb{Z - c}^k. $$\n\\end{definition}\n\n\\begin{example}\nLet $ f\\rb{X, Y, Z} = f\\rb{x, y} = \\rb{x^2 + y^2}^2 + 3x^2y - y^3 $ with $ P = \\sb{0, 0, 1} = \\rb{0, 0} $ and $ m = m_P\\rb{f} = 3 $. Then\n\\begin{align*}\nt_P\\rb{f}\\rb{X, Y, Z} = t_P\\rb{f}\\rb{x, y}\n& = \\twobyone{3}{0}\\val{\\dfrac{\\partial^3 f}{\\partial x^3}}_Px^3 + \\twobyone{3}{1}\\val{\\dfrac{\\partial^3 f}{\\partial x^2 \\partial y}}_Px^2y + \\twobyone{3}{2}\\val{\\dfrac{\\partial^3 f}{\\partial x \\partial y^2}}_Pxy^2 + \\twobyone{3}{3}\\val{\\dfrac{\\partial^3 f}{\\partial y^3}}_Py^3 \\\\\n& = 18x^2y - 6y^3 = 6y\\rb{\\sqrt{3}x - y}\\rb{\\sqrt{3}x + y}.\n\\end{align*}\nThus the tangents of $ C $ at $ P $ are $ T_P\\rb{f} = \\cb{y, \\sqrt{3}x - y, \\sqrt{3}x + y} $.\n\\end{example}\n\n\\begin{definition}[Ordinary singularity]\nA singular point $ P \\in C $ is \\textbf{ordinary} iff $ t_P\\rb{f} $ has distinct factors.\n\\end{definition}\n\n\\begin{example}\nLet $ f\\rb{X, Y, Z} = f\\rb{x, y} = \\rb{x^2 + y^2}^2 + 3x^2y - y^3 $ with $ P = \\sb{0, 0, 1} = \\rb{0, 0} $. Then $ t_P\\rb{f} $ have distinct factors $ y $, $ \\sqrt{3}x - y $, and $ \\sqrt{3}x + y $. Thus $ P $ is ordinary.\n\\end{example}\n\n\\begin{definition}[Intersection number]\nThe \\textbf{intersection number} of $ C $ and $ C' $ at $ P $ if $ \\deg\\rb{\\gcd\\rb{f, g}} = 0 $ is $ I_P\\rb{f, g} $, where $ I_P : F\\sb{X, Y, Z} \\times F\\sb{X, Y, Z} \\to \\Z_{> 0} $ is defined for any $ f', g' \\in F\\sb{X, Y, Z} $ by:\n\\begin{itemize}\n\\item $ I_P\\rb{f', g'} = I_P\\rb{g', f'} $,\n\\item $ I_P\\rb{f', g'} = I_P\\rb{f', g' \\circ h} $ for any affine transformation $ h $,\n\\item $ I_P\\rb{f', g'} = I_P\\rb{f', g' + hf'} $ for any $ h \\in F\\sb{X, Y, Z} $,\n\\item $ I_P\\rb{f', hh'} = I_P\\rb{f', h} + I_P\\rb{f', h'} $ for any $ h, h' \\in F\\sb{X, Y, Z} $, and\n\\item $ I_P\\rb{f', g'} \\ge m_P\\rb{f'} m_P\\rb{g'} $, with equality iff $ T_P\\rb{f'} \\cap T_P\\rb{g'} = \\emptyset $.\n\\end{itemize}\n\\end{definition}\n\nSince $ T_P\\rb{X} = \\cb{X} $, $ T_P\\rb{Y} = \\cb{Y} $, and $ T_P\\rb{Z} = \\cb{Z} $ are all distinct, it holds that $ I_P\\rb{X, Y} = I_P\\rb{X, Z} = I_P\\rb{Y, Z} = 1 $.\n\n\\pagebreak\n\n\\begin{example}\nLet $ f\\rb{X, Y, Z} = f\\rb{x, y} = \\rb{x^2 + y^2}^2 + 3x^2y - y^3 $ and $ g\\rb{X, Y, Z} = g\\rb{x, y} = \\rb{x^2 + y^2}^3 - 4x^2y^2 $ with $ P = \\sb{0, 0, 1} = \\rb{0, 0} $. Then $ \\gcd\\rb{f, g} = 1 $, so $ \\deg\\rb{\\gcd\\rb{f, g}} = 0 $. Now let $ h\\rb{x, y} = 4x^2y + 4y^3 + 5x^2 - 3y^2 $, such that\n$$ g + \\rb{3y - x^2 - y^2}f = y^2h \\qquad \\text{and} \\qquad f + \\rb{y^2 - 2x^2y - y^3 - 3x^2}y = x^4. $$\nThen\n$$ I_P\\rb{f, g} = I_P\\rb{f, g + \\rb{3y - x^2 - y^2}f} = I_P\\rb{f, y^2h} = I_P\\rb{f, y^2} + I_P\\rb{f, h}. $$\nThe first term can be computed as\n$$ I_P\\rb{f, y^2} = 2I_P\\rb{f, y} = 2I_P\\rb{f + \\rb{y^2 - 2x^2y - y^3 - 3x^2}y, y} = 2I_P\\rb{x^4, y} = 8I_P\\rb{x, y} = 8. $$\nNow $ m_P\\rb{f} = 3 $ and $ T_P\\rb{f} = \\cb{y, \\sqrt{3}x - y, \\sqrt{3}x + y} $. Since $ C'' : h\\rb{x} = 0 $ is also a projective plane curve, its multiplicity at $ P $ can be computed to be $ m_P\\rb{h} = 2 $ and its tangents at $ P $ can also be computed to be $ T_P\\rb{h} = \\cb{\\sqrt{5}x - \\sqrt{3}y, \\sqrt{5}x + \\sqrt{3}y} $. Hence $ I_P\\rb{f, h} = m_P\\rb{f} m_P\\rb{h} = \\rb{3}\\rb{2} = 6 $. Thus the intersection number of $ C $ and $ C' $ at $ P $ is $ I_P\\rb{f, g} = 8 + 6 = 14 $.\n\\end{example}\n\n\\begin{definition}[Flex]\n$ P $ is a \\textbf{flex} iff $ I_P\\rb{f, g} > 2 $ is odd.\n\\end{definition}\n\n\\begin{example}\nLet $ f\\rb{X, Y, Z} = f\\rb{x, y} = y - x^3 $ with $ P = \\sb{0, 0, 1} = \\rb{0, 0} $. Then $ \\val{\\partial f / \\partial y}_P = 1 \\ne 0 $, so $ m_P\\rb{f} = 1 $. Since\n$$ g\\rb{X, Y, Z} = g\\rb{x, y} = t_P\\rb{f}\\rb{x, y} = \\val{\\partial f / \\partial x}_Px + \\val{\\partial f / \\partial y}_Py = y, $$\nit holds that $ \\gcd\\rb{f, g} = 1 $, so $ \\deg\\rb{\\gcd\\rb{f, g}} = 0 $. Hence\n$$ I_P\\rb{f, g} = I_P\\rb{f - y, y} = I_P\\rb{-x^3, y} = 3I_P\\rb{-x, y} = 3 > 2. $$\nThus $ P $ is a flex.\n\\end{example}\n\nThe following follows from the fundamental theorem of algebra on the \\emph{resultant} of $ f $ and $ g $.\n\n\\begin{theorem}[B\\'ezout]\n$ C $ intersects $ C' $ at $ \\rb{\\deg\\rb{f}}\\rb{\\deg\\rb{g}} $ points \\emph{up to multiplicity}, so\n$$ \\sum_{P \\in C \\cap C'} I_P\\rb{f, g} = \\rb{\\deg\\rb{f}}\\rb{\\deg\\rb{g}}. $$\n\\end{theorem}\n\nThe following follows from a \\emph{dimension counting} argument.\n\n\\begin{theorem}[Cayley-Bacharach]\nLet $ \\deg\\rb{f} = \\deg\\rb{g} = 3 $ such that $ C $ intersects $ C' $ at nine points up to multiplicity, and let $ C'' : h\\rb{X, Y, Z} = 0 $ be a cubic projective plane curve over $ F $ such that at least eight of these points are in $ C'' $. Then the ninth point is also in $ C'' $.\n\\end{theorem}\n\nThe following definition is the \\emph{genus-degree formula}, which is a corollary of the \\emph{adjunction formula} and the \\emph{Riemann-Roch theorem} for arbitrary curves and surfaces.\n\n\\begin{definition}[Degree]\nThe \\textbf{degree} of $ C $ is $ d_C = \\deg\\rb{f} $.\n\\end{definition}\n\n\\begin{definition}[Genus]\nThe \\textbf{genus} of $ C $ is\n$$ g_C = \\tfrac{1}{2}\\rb{d_C - 1}\\rb{d_C - 2} - \\tfrac{1}{2}\\sum_{P \\in C} m\\rb{m - 1}, $$\nover all ordinary singularities $ P \\in C $ with multiplicity $ m_P\\rb{f} = m $.\n\\end{definition}\n\nThe genus of $ C $ is $ g_C = \\tfrac{1}{2}\\rb{d_C - 1}\\rb{d_C - 2} $ if $ C $ is smooth.\n\n\\begin{example}\nThe line $ L : y = x $ is a smooth projective plane curve of degree one and genus zero. The unit circle $ S_1 : x^2 + y^2 = 1 $ is a smooth projective plane curve of degree two and genus zero. An elliptic curve $ E : y^2 = x^3 + Ax + B $ is a smooth projective plane curve of degree three and genus zero.\n\\end{example}\n\n\\pagebreak\n\n\\subsection{Groups}\n\nLet $ G $ be an additive abelian group, with multiplication $ \\cdot : \\Z \\times G \\to G $ defined by\n$$ nx = \\begin{dcases} x + \\dots + x & n > 0 \\\\ 0 & n = 0 \\\\ \\rb{-x} + \\dots + \\rb{-x} & n < 0 \\end{dcases}. $$\n\n\\begin{theorem}[Isomorphism theorems]\nThe following theorems hold:\n\\begin{enumerate}\n\\item Let $ H $ be a group and $ \\phi : G \\to H $ be a group homomorphism. Then:\n$$ Ker\\rb{\\phi} \\trianglelefteq G, \\qquad \\dfrac{G}{Ker\\rb{\\phi}} \\cong Im\\rb{\\phi}. $$\n\\item Let $ N \\trianglelefteq G $ and $ H \\le G $ be subgroups. Then:\n$$ N \\cap H \\trianglelefteq H, \\qquad \\dfrac{H}{N \\cap H} \\cong \\dfrac{N + H}{N}. $$\n\\item Let $ N \\trianglelefteq G $ and $ H \\trianglelefteq G $ be subgroups such that $ N \\le H $. Then:\n$$ \\dfrac{H}{N} \\trianglelefteq \\dfrac{G}{N}, \\qquad \\dfrac{G / N}{H / N} \\cong \\dfrac{G}{H}. $$\n\\end{enumerate}\n\\end{theorem}\n\nAll subgroups of $ G $ are normal, but the above theorems still hold if $ G $ is non-abelian.\n\n\\begin{definition}[Torsion element]\nAn \\textbf{$ n $-torsion element} is an element $ x \\in G $ such that $ n = ord\\rb{x} $ is finite.\n\\end{definition}\n\n\\begin{example}\n$ \\Z + p / q \\in \\Q / \\Z $ is a torsion element since $ ord\\rb{x} \\mid q $ is finite.\n\\end{example}\n\n\\begin{definition}[Torsion subgroup]\nThe \\textbf{$ n $-torsion subgroup} $ G\\sb{n} $ is the group of $ m $-torsion elements of $ G $ such that $ m \\mid n $. The \\textbf{torsion subgroup} $ G_{tors} $ of $ G $ is the group of $ m $-torsion elements of $ G $ for any $ m \\in \\Z_{\\ge 0} $.\n\\end{definition}\n\n\\begin{example}\n$ G = \\R / \\Z $ has torsion subgroup $ G_{tors} = \\Q / \\Z $ since any $ n $-torsion element $ \\Z + x \\in G $ is such that $ nx \\in \\Z $ and $ x \\in \\Q $.\n\\end{example}\n\n\\begin{definition}[Finitely generated]\n$ G $ is \\textbf{finitely generated} iff there are finitely many elements $ x_1, \\dots, x_n \\in G $ such that any element $ x \\in G $ is a sum\n$$ x = \\sum_{i = 1}^n m_ix_i, \\qquad m_i \\in \\Z. $$\n\\end{definition}\n\n\\begin{example}\n$ \\Z $ and $ \\Z_n $ are finitely generated abelian groups.\n\\end{example}\n\nThe \\emph{direct sum} $ \\oplus $ of finitely many abelian groups is equivalent to their direct product $ \\times $, thus $ \\Z^n = \\Z \\times \\dots \\times \\Z = \\Z \\oplus \\dots \\oplus \\Z $. Now let $ G $ be finitely generated.\n\n\\begin{theorem}[Fundamental theorem of finitely generated abelian groups]\n\\label{thm:ftfgag}\nThere are unique $ r, m \\in \\Z_{\\ge 0} $ and $ n_1, \\dots, n_m \\in \\Z_{> 1} $ such that\n$$ G \\cong r\\Z \\oplus \\bigoplus_{i = 1}^m \\Z_{n_i}, $$\nwith each $ n_i \\mid n_{i + 1} $.\n\\end{theorem}\n\n\\begin{definition}[Rank]\nThe \\textbf{rank} $ rk\\rb{G} $ of $ G $ is the unique $ r \\in \\Z_{\\ge 0} $ in Theorem \\ref{thm:ftfgag}.\n\\end{definition}\n\n\\begin{example}\nA finite abelian group $ G $ has rank $ rk\\rb{G} = 0 $ since $ G_{tors} = G $.\n\\end{example}\n\n\\pagebreak\n\n\\section{Algorithm proofs}\n\n\\subsection{Transformation of a cubic curve into Weierstrass form}\n\nLet\n$$ E : f\\rb{X, Y, Z} = a_1X^3 + a_2X^2Y + a_3XY^2 + a_4Y^3 + a_5X^2Z + a_6XYZ + a_7Y^2Z + a_8XZ^2 + a_9YZ^2 + a_{10}Z^3 = 0 $$\nfor some $ a_i \\in F $ be an elliptic curve over a perfect field $ F $, and let $ P = \\sb{a, b, c} \\in E $ be an $ F $-rational point. Then the unique tangent at $ P $ is\n\\begin{align*}\nL\n: & \\twobyone{1}{1, 0, 0}\\val{\\dfrac{\\partial f}{\\partial X}}_P\\rb{X - a} + \\twobyone{1}{0, 1, 0}\\val{\\dfrac{\\partial f}{\\partial Y}}_P\\rb{Y - b} + \\twobyone{1}{0, 0, 1}\\val{\\dfrac{\\partial f}{\\partial Z}}_P\\rb{Z - c} = 0 \\\\\n: & \\rb{3a_1a^2 + 2a_2ab + a_3b^2 + 2a_5ac + a_6bc + a_8c^2}\\rb{X - a} \\\\\n& + \\rb{a_2a^2 + 2a_3ab + 3a_4b^2 + a_6ac + 2a_7bc + a_9c^2}\\rb{Y - b} \\\\\n& + \\rb{a_5a^2 + a_6ab + a_7b^2 + 2a_8ac + 2a_9bc + 3a_{10}c^2}\\rb{Z - c} = 0.\n\\end{align*}\nIf $ P $ is not a flex, B\\'ezout's theorem gives that $ L $ intersects $ E $ at three points up to multiplicity, so $ E \\cap L = \\cb{P, P'} $ for some other $ F $-rational point $ P' \\in E $, and repeat inductively with $ P' $. This \\emph{chord-tangent} method eventually terminates until $ P' $ is a flex, so assume without loss of generality that $ P $ is a flex. Now let $ Q \\in L \\setminus E $ be a point distinct to $ P $ and define a matrix\n$$ M = \\onebythree{Q}{P}{R}, \\qquad R \\in \\cb{\\sb{1, 0, 0}^T, \\sb{0, 1, 0}^T, \\sb{0, 0, 1}^T}. $$\nSince $ P $ and $ Q $ are linearly independent, at least one of these is invertible. Then transforming $ \\sb{X, Y, Z} \\mapsto M\\sb{X, Y, Z}^T $ gives $ Q \\mapsto \\sb{1, 0, 0} $ and $ P \\mapsto \\sb{0, 1, 0} = \\mathcal{O} $, and the elliptic curve\n$$ E' : f'\\rb{X, Y, Z} = a_1'X^3 + a_2'X^2Y + a_3'XY^2 + a_4'Y^3 + a_5'X^2Z + a_6'XYZ + a_7'Y^2Z + a_8'XZ^2 + a_9'YZ^2 + a_{10}'Z^3 = 0. $$\nfor some $ a_i' \\in F $. Since $ \\mathcal{O} \\in E' $, it holds that $ f'\\rb{0, 1, 0} = a_4' = 0 $. Now the tangent at $ \\mathcal{O} $ is\n$$ L' : \\twobyone{1}{1, 0, 0}\\val{\\dfrac{\\partial f'}{\\partial X}}_\\mathcal{O}X + \\twobyone{1}{0, 1, 0}\\val{\\dfrac{\\partial f'}{\\partial Y}}_\\mathcal{O}\\rb{Y - 1} + \\twobyone{1}{0, 0, 1}\\val{\\dfrac{\\partial f'}{\\partial Z}}_\\mathcal{O}Z = 0 : a_3'X + a_7'Z = 0. $$\nSince $ L' : Z = 0 $, it holds that $ a_3' = 0 $ and $ a_7' \\ne 0 $. Then $ L' $ intersects $ E' $ at\n$$ a_1'X^3 + a_2'X^2Y + a_5'X^2\\rb{0} + a_6'XY\\rb{0} + a_7'Y^2\\rb{0} + a_8'X\\rb{0}^2 + a_9'Y\\rb{0}^2 + a_{10}'\\rb{0}^3 = X^2\\rb{a_1'X + a_2'Y} = 0. $$\nSince $ \\mathcal{O} $ is a flex, it holds that $ X = 0 $ is repeated three times, so $ a_1' \\ne 0 $ and $ a_2' = 0 $. Hence\n\\begin{align*}\nE'\n& : a_1'X^3 + a_5'X^2Z + a_6'XYZ + a_7'Y^2Z + a_8'XZ^2 + a_9'YZ^2 + a_{10}'Z^3 = 0 \\\\\n& : X^3 + \\dfrac{a_5'}{a_1'}X^2Z + \\dfrac{a_6'}{a_1'}XYZ + \\dfrac{a_7'}{a_1'}Y^2Z + \\dfrac{a_8'}{a_1'}XZ^2 + \\dfrac{a_9'}{a_1'}YZ^2 + \\dfrac{a_{10}'}{a_1'}Z^3 = 0.\n\\end{align*}\nThen rescaling $ \\sb{X, Y, Z} \\mapsto \\sb{X, Y, -\\rb{a_1' / a_7'}Z} $ gives\n\\begin{align*}\nE'\n& : X^3 - \\dfrac{a_5'}{a_7'}X^2Z - \\dfrac{a_6'}{a_7'}XYZ - Y^2Z + \\dfrac{a_8'a_1'}{a_7'^2}XZ^2 + \\dfrac{a_9'a_1'}{a_7'^2}YZ^2 - \\dfrac{a_{10}'a_1'^2}{a_7'^3}Z^3 = 0 \\\\\n& : Y^2Z + \\dfrac{a_5'}{a_7'}X^2Z + \\dfrac{a_6'}{a_7'}XYZ = X^3 + \\dfrac{a_8'a_1'}{a_7'^2}XZ^2 + \\dfrac{a_9'a_1'}{a_7'^2}YZ^2 - \\dfrac{a_{10}'a_1'^2}{a_7'^3}Z^3.\n\\end{align*}\nThus $ E' $ is a Weierstrass equation.\n\n\\pagebreak\n\n\\subsection{Group law explicit formulae}\n\nLet $ P, Q \\in E $ be points such that $ R = P * Q \\in E $. Since $ * $ is symmetric, it is commutative, so only the following six cases need to be considered.\n\\begin{enumerate}[label=$ \\rb{*}_\\arabic* $]\n\\item Assume that $ P = \\rb{a, b} $ and $ Q = \\rb{a, b'} $ for $ b \\ne b' $. Then the line joining $ P $ and $ Q $ is\n$$ L : \\rb{b - b'}X + \\rb{ab' - ab}Z = 0 : X = aZ, $$\nwhich intersects $ E $ at\n$$ Y^2Z = \\rb{aZ}^3 + A\\rb{aZ}Z^2 + BZ^3 \\qquad \\implies \\qquad Z\\rb{a^3Z^2 + AaZ^2 + BZ^2 - Y^2} = 0. $$\nIf $ Z \\ne 0 $, then this can be dehomogenised into\n$$ y^2 = a^3 + Aa + B, $$\nwhich has trivial solutions opposite in sign, so $ b' = -b $. Thus $ Z = 0 $ and $ R = \\mathcal{O} $.\n\\item Assume that $ P = \\rb{a, b} $ and $ Q = \\rb{a', b'} $ for $ a \\ne a' $, and let\n$$ \\lambda = \\dfrac{b - b'}{a - a'}, \\qquad \\mu = \\dfrac{ab' - a'b}{a - a'}. $$\nThen the line joining $ P $ and $ Q $ is\n$$ L : \\rb{b - b'}X + \\rb{a' - a}Y + \\rb{ab' - a'b}Z = 0 : Y = \\lambda X + \\mu Z, $$\nwhich intersects $ E $ at\n$$ \\rb{\\lambda X + \\mu Z}^2Z = X^3 + AXZ^2 + BZ^3 \\qquad \\implies \\qquad X^3 - \\lambda^2X^2Z + \\rb{A - 2\\lambda\\mu}XZ^2 - \\rb{\\mu^2 - B}Z^3 = 0. $$\nSince $ Z = 0 $ gives $ Y = 0 $, it holds that $ Z \\ne 0 $, and this can be dehomogenised into\n$$ x^3 - \\lambda^2x^2 + \\rb{A - 2\\lambda\\mu}x - \\rb{\\mu^2 - B} = 0. $$\nLet $ R = \\rb{a'', b''} $. Since\n$$ 0 = \\rb{x - a}\\rb{x - a'}\\rb{x - a''} = x^3 - \\rb{a + a' + a''}x^2 + \\rb{aa' + a'' + a'a''}x - aa'a'', $$\ncomparing coefficients gives\n$$ \\lambda^2 = a + a' + a'', \\qquad A - 2\\lambda\\mu = aa' + aa'' + a'a'', \\qquad \\mu^2 - B = aa'a''. $$\nThus $ R = \\rb{\\lambda^2 - a - a', \\lambda\\rb{\\lambda^2 - a - a'} + \\mu} $.\n\\item Assume that $ P = Q = \\rb{a, b} $ for $ b \\ne 0 $, and let\n$$ \\lambda = \\dfrac{3a^2 + A}{2b}, \\qquad \\mu = \\dfrac{2Aa + 3B - b^2}{2b}. $$\nThen the tangent at $ P $ is\n$$ L : \\rb{-3a^2 - A}X + 2bY + \\rb{b^2 - 2Aa - 3B}Z = 0 : Y = \\lambda X + \\mu Z, $$\nwhich intersects $ E $ by $ \\rb{*}_2 $ at\n$$ x^3 - \\lambda^2x^2 + \\rb{A - 2\\lambda\\mu}x - \\rb{\\mu^2 - B} = 0. $$\nLet $ R = \\rb{a', b'} $. Since\n$$ 0 = \\rb{x - a}^2\\rb{x - a'} = x^3 - \\rb{2a + a'}x^2 + \\rb{2aa' + a^2}x - a^2a', $$\ncomparing coefficients gives\n$$ \\lambda^2 = 2a + a', \\qquad A - 2\\lambda\\mu = 2aa' + a^2, \\qquad \\mu^2 - B = a^2a'. $$\nThus $ R = \\rb{\\lambda^2 - 2a, \\lambda\\rb{\\lambda^2 - 2a} + \\mu} $.\n\\item Assume that $ P = Q = \\rb{a, 0} $. Then since $ a\\rb{3a^2 + A} = 3\\rb{a^3} + Aa = 3\\rb{-Aa - B} + Aa = -2Aa - 3B $, the tangent at $ P $ is\n$$ L : \\rb{-3a^2 - A}X + \\rb{-2Aa - 3B}Z = 0 : X = aZ, $$\nwhich intersects $ E $ at $ \\rb{a, 0} $, $ \\rb{a, -0} = \\rb{a, 0} $, and $ \\mathcal{O} $ by $ \\rb{*}_1 $. Thus $ R = \\mathcal{O} $.\n\\item Assume that $ P = \\rb{a, b} $ and $ Q = \\mathcal{O} $. Then the line joining $ P $ and $ Q $ is\n$$ L : -X + aZ = 0 : X = aZ, $$\nwhich intersects $ E $ at $ \\rb{a, b} $, $ \\rb{a, -b} $, and $ \\mathcal{O} $ by $ \\rb{*}_1 $. Thus $ R = \\rb{a, -b} $.\n\\item Assume that $ P = Q = \\mathcal{O} $. Then the tangent at $ P $ is\n$$ L : Z = 0. $$\nThus $ R = \\mathcal{O} $.\n\\end{enumerate}\nSimilarly since $ + $ is symmetric, it is commutative, so only the following four cases need to be considered. If $ P = \\rb{a, b} $ and $ Q = \\rb{a', b'} $ for $ a \\ne a' $, then $ \\rb{*}_2 $ and $ \\rb{*}_5 $ give\n\\begin{align*}\nP + Q\n& = \\rb{\\lambda^2 - a - a', -\\rb{\\lambda\\rb{\\lambda^2 - a - a'} + \\mu}}, \\qquad \\lambda = \\dfrac{b - b'}{a - a'}, \\qquad \\mu = \\dfrac{ab' - a'b}{a - a'} \\\\\n& = \\rb{\\dfrac{\\rb{A + aa'}\\rb{a + a'} + 2\\rb{B - bb'}}{\\rb{a - a'}^2}, \\dfrac{a'b - ab' - \\rb{\\dfrac{\\rb{A + aa'}\\rb{a + a'} + 2\\rb{B - bb'}}{\\rb{a - a'}^2}}\\rb{b - b'}}{a - a'}} \\\\\n& = \\rb{\\dfrac{\\rb{A + aa'}\\rb{a + a'} + 2\\rb{B - bb'}}{\\rb{a - a'}^2}, \\dfrac{\\rb{Ab' - a'^2b}\\rb{3a + a'} + \\rb{a^2b' - Ab}\\rb{a + 3a'} - 4B\\rb{b - b'}}{\\rb{a - a'}^3}}.\n\\end{align*}\nIf $ P = Q = \\rb{a, b} $ for $ b \\ne 0 $, then $ \\rb{*}_3 $ and $ \\rb{*}_5 $ give\n\\begin{align*}\nP + Q\n& = \\rb{\\lambda^2 - 2a, -\\rb{\\lambda\\rb{\\lambda^2 - 2a} + \\mu}}, \\qquad \\lambda = \\dfrac{3a^2 + A}{2b}, \\qquad \\mu = \\dfrac{2Aa + 3B - b^2}{2b} \\\\\n& = \\rb{\\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4b^2}, \\dfrac{b^2 - 2Aa - 3B - \\rb{\\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4b^2}}\\rb{3a^2 + A}}{2b}} \\\\\n& = \\rb{\\dfrac{a^4 - 2Aa^2 - 8Ba + A^2}{4b^2}, \\dfrac{a^6 + 5Aa^4 + 20Ba^3 - 5A^2a^2 - 4ABa - A^3 - 8B^2}{8b^3}}.\n\\end{align*}\nIf $ P = \\rb{a, b} $ and $ Q = \\mathcal{O} $, then $ \\rb{*}_5 $ gives\n$$ P + Q = \\rb{\\rb{a, b} * \\mathcal{O}} * \\mathcal{O} = \\rb{a, -b} * \\mathcal{O} = \\rb{a, b} = P. $$\nOtherwise $ P + Q = \\mathcal{O} $ or $ P = Q = \\mathcal{O} $, then $ \\rb{*}_1 $, $ \\rb{*}_4 $, and $ \\rb{*}_6 $ give\n$$ P + Q = \\rb{P * Q} * \\mathcal{O} = \\mathcal{O} * \\mathcal{O} = \\mathcal{O}. $$\nThus the explicit formulae hold.\n\n\\pagebreak\n\n\\section{Code listings}\n\n\\subsection{Fields.hs}\n\nThis module includes basic types and instances for fields and prime subfields in Section 2.\n\n\\lstinputlisting{hs/Fields.hs}\n\n\\pagebreak\n\n\\subsection{WeierstrassEquations.hs}\n\nThis module includes data for Weierstrass curves in Section 1.2, as well as related quantities and transformations.\n\n\\lstinputlisting{hs/WeierstrassEquations.hs}\n\n\\pagebreak\n\n\\subsection{GroupLaw.hs}\n\nThis module includes data for the group law in Section 1.3, as well as those of points.\n\n\\lstinputlisting{hs/GroupLaw.hs}\n\n\\pagebreak\n\n\\subsection{Rationals.hs}\n\nThis module includes the algorithms in Section 3.2 and Section 3.7.\n\n\\lstinputlisting{hs/Rationals.hs}\n\n\\pagebreak\n\n\\subsection{Applications.hs}\n\nThis module includes the algorithms in Section 4.1 and Section 4.2.\n\n\\lstinputlisting{hs/Applications.hs}\n\n\\pagebreak\n\n\\subsection{Test.hs}\n\nThis module includes input of computations from previous modules.\n\n\\lstinputlisting{hs/Test.hs}\n\n\\pagebreak\n\n\\subsection{Output.txt}\n\nThis file includes output of computations from previous modules.\n\n\\lstinputlisting{hs/Output.txt}\n\n\\pagebreak\n\n\\nocite{*}\n\n\\printbibliography[heading=bibintoc]\n\n\\end{document}", "meta": {"hexsha": "23026d57af7a9fae07a277e67f7842b059b78ec9", "size": 250201, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "urop/report/EllipticCurves.tex", "max_stars_repo_name": "Multramate/EllipticCurves", "max_stars_repo_head_hexsha": "f46652e975c1d7af5977fc0f2b5b845972eec63d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-04-19T17:04:11.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-12T08:58:48.000Z", "max_issues_repo_path": "urop/report/EllipticCurves.tex", "max_issues_repo_name": "Multramate/EllipticCurves", "max_issues_repo_head_hexsha": "f46652e975c1d7af5977fc0f2b5b845972eec63d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "urop/report/EllipticCurves.tex", "max_forks_repo_name": "Multramate/EllipticCurves", "max_forks_repo_head_hexsha": "f46652e975c1d7af5977fc0f2b5b845972eec63d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 81.7917620137, "max_line_length": 1876, "alphanum_fraction": 0.6239383536, "num_tokens": 98905, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857981, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.34578793835137156}} {"text": "\n\n \\filetitle{xsf}{Power spectrum and spectral density of model variables}{model/xsf}\n\n\t\\paragraph{Syntax}\\label{syntax}\n\n\\begin{verbatim}\n[S,D,List] = xsf(M,Freq,...)\n[S,D,List,Freq] = xsf(M,NFreq,...)\n\\end{verbatim}\n\n\\paragraph{Input arguments}\\label{input-arguments}\n\n\\begin{itemize}\n\\item\n \\texttt{M} {[} model {]} - Model object.\n\\item\n \\texttt{Freq} {[} numeric {]} - Vector of frequencies at which the\n XSFs will be evaluated.\n\\item\n \\texttt{NFreq} {[} numeric {]} - Total number of requested\n frequencies; the frequencies will be evenly spread between 0 and\n \\texttt{pi}.\n\\end{itemize}\n\n\\paragraph{Output arguments}\\label{output-arguments}\n\n\\begin{itemize}\n\\item\n \\texttt{S} {[} namedmat \\textbar{} numeric {]} - Power spectrum\n matrices.\n\\item\n \\texttt{D} {[} namedmat \\textbar{} numeric {]} - Spectral density\n matrices.\n\\item\n \\texttt{List} {[} cellstr {]} - List of variable in order of\n appearance in rows and columns of \\texttt{S} and \\texttt{D}.\n\\item\n \\texttt{Freq} {[} numeric {]} - Vector of frequencies at which the\n XSFs has been evaluated.\n\\end{itemize}\n\n\\paragraph{Options}\\label{options}\n\n\\begin{itemize}\n\\item\n \\texttt{'applyTo='} {[} cellstr \\textbar{} char \\textbar{}\n \\emph{\\texttt{Inf}} {]} - List of variables to which the option\n \\texttt{'filter='} will be applied; \\texttt{Inf} means all variables.\n\\item\n \\texttt{'filter='} {[} char \\textbar{} \\emph{empty} {]} - Linear\n filter that is applied to variables specified by `applyto'.\n\\item\n \\texttt{'nFreq='} {[} numeric \\textbar{} \\emph{\\texttt{256}} {]} -\n Number of equally spaced frequencies over which the `filter' is\n numerically integrated.\n\\item\n \\texttt{'matrixFmt='} {[} \\emph{\\texttt{'namedmat'}} \\textbar{}\n \\texttt{'plain'} {]} - Return matrices \\texttt{S} and \\texttt{D} as\n either \\href{namedmat/Contents}{\\texttt{namedmat}} objects\n (i.e.~matrices with named rows and columns) or plain numeric arrays.\n\\item\n \\texttt{'progress='} {[} \\texttt{true} \\textbar{}\n \\emph{\\texttt{false}} {]} - Display progress bar on in the command\n window.\n\\item\n \\texttt{'select='} {[} \\emph{\\texttt{@all}} \\textbar{} char \\textbar{}\n cellstr {]} - Return XSF for selected variables only; \\texttt{@all}\n means all variables.\n\\end{itemize}\n\n\\paragraph{Description}\\label{description}\n\n\\paragraph{Example}\\label{example}\n\n\n", "meta": {"hexsha": "a07a0819a99e8cef9c49024af5ad48045eab84f8", "size": 2331, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "-help/model/xsf.tex", "max_stars_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_stars_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-12-06T13:38:38.000Z", "max_stars_repo_stars_event_max_datetime": "2017-12-06T13:38:38.000Z", "max_issues_repo_path": "-help/model/xsf.tex", "max_issues_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_issues_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2017-03-28T08:13:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-02T10:40:25.000Z", "max_forks_repo_path": "-help/model/xsf.tex", "max_forks_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_forks_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-17T07:06:39.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-17T07:06:39.000Z", "avg_line_length": 30.2727272727, "max_line_length": 86, "alphanum_fraction": 0.6846846847, "num_tokens": 735, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5195213368305399, "lm_q2_score": 0.665410558746814, "lm_q1q2_score": 0.3456949830213013}} {"text": "\n\\documentclass{article}\n\\usepackage{graphicx}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath, amssymb, latexsym}\n\\usepackage{neuralnetwork}\n\\usepackage{multicol}\n\\usepackage{expl3}\n\n\\usepackage{pgfplots}\n\\usepackage{algorithm}\n\\usepackage[noend]{algpseudocode}\n\\usepackage{tikz}\n\\usepackage{nicefrac}\n\\pgfplotsset{every axis legend/.append style={\nat={(0,0)},\nanchor=north east}}\n\\usetikzlibrary{shapes,positioning,intersections,quotes}\n\\usetikzlibrary{arrows.meta,\n bending,\n intersections,\n quotes,\n shapes.geometric}\n \n\\definecolor{darkgreen}{rgb}{0.0, 0.6, 0.0}\n\\definecolor{darkred}{rgb}{0.7, 0.0, 0.0}\n\\makeatletter\n\\def\\BState{\\State\\hskip-\\ALG@thistlm}\n\\makeatother\n\\title{Week 13}\n\\begin{document}\n\\pagenumbering{gobble}\n\\maketitle\n\\newpage\n\\pagenumbering{arabic}\n\n\\section*{Unsupervised learning}\n\n\\begin{itemize}\n \\item Try to figure out the structure of the data.\n \\item The clustering algorithm organizes data based on data characteristics.\n \\item Market segmentation is categorizing clients into different market categories.\n \\item Social network analysis.\n \\item Computer clusters and data centers are organized for network structure and location.\n \\item Understanding galaxy creation through astronomical data analysis.\n\\end{itemize}\n\n\\section*{K-means algorithm}\n\n\\begin{itemize}\n \\item Would you like an algorithm to automatically arrange data into coherent clusters?\n \\item By far the most used clustering algorithm is K-means.\n\\end{itemize}\n\n~\\\\\nAlgorithm overview:\n\n\\begin{enumerate}\n \\item Assign k locations at random as cluster centroids.\n \\item Go through each example and assign each point to one of the k clusters based on which center it is closest to.\n\n\\includegraphics[width=\\textwidth]{resources/kclusters_1}\n\n \\item Move to the average of the similarly allocated data-points for each centroid.\n\n\\includegraphics[width=\\textwidth]{resources/kclusters_2}\n\n \\item Repeat 2) and 3) until convergence.\n\\end{enumerate}\n\n\\section*{K-means for non-separated clusters}\n\n\n\\begin{itemize}\n \\item So far, we've looked at K-means, which has well-defined clusters.\n \\item However, K-means is frequently used on datasets with poorly defined clusters.\n \\item As an example, consider t-shirt sizes. How large do you make them if you want three sizes (S,M,L)?\n \\item As a result, three clusters are formed, even if they are not actually there.\n \\item This is an example of market segmentation; create items that are tailored to the demands of your subpopulations.\n\\end{itemize}\n\n\\includegraphics[width=0.5\\textwidth]{resources/t_shirt}\n\n\\section*{K means optimization objective}\n\n\\begin{itemize}\n \\item K-means, like the supervised learning functions we've examined, has an optimization goal.\n \\item While K-means is running, we keep track of two sets of variables.\n \\item $c^i$ is the index of clusters ${1,2, ..., K}$ to which $x^i$ is currently assigned.\n \\item $\\mu_k$, is the cluster associated with centroid $k$.\n \\item $\\mu_c^i$, is the cluster centroid of the cluster to which example $x^i$ has been assigned to.\n \\item We may write the optimization objective using this notation:\n\\end{itemize}\n\n$$J(c^{(1)}, ..., c^{(m)}, \\mu_1, ...,\\mu_K)=\\frac{1}{m}\\sum_{i=1}^{m}||x^{(i)}-\\mu_{c^{(i)}}||^2$$\n\n~\\\\\ni.e. squared distances between training example $x^i$ and the cluster centroid to which $x^i$ has been assigned to.\n\n\\includegraphics[width=0.5\\textwidth]{resources/cost_cluster}\n\n~\\\\\nWhen we look at the k-means method:\n\n\\begin{itemize}\n \\item The cluster assigned step is minimizing $J(...)$ with respect to $c_1, c_2 ... c_i$ i.e. find the centroid closest to each example. Doesn't change the centroids themselves.\n \\item The move centroid step. We can show this step is choosing the values of $\\mu$ which minimizes $J(...)$ with respect to $\\mu$.\n \\item So, we're partitioning the algorithm into two parts: First part minimizes the $c$ variables. Second part minimizes the $J$ variables.\n\\end{itemize}\n\n\\section*{Random initialization}\nDepending on the starting setting, K means might converge to different solutions.\n\n\\includegraphics[width=\\textwidth]{resources/optimum_cluster}\n\n\\begin{itemize}\n \\item Randomly initialize K-means.\n \\item For each n (e.g. 100) random initialization run K-means.\n \\item Then compute the distortion on the set of cluster assignments and centroids at convergent.\n \\item End with n ways of cluster the data.\n \\item Pick the clustering which gave the lowest distortion.\n\\end{itemize}\n\n\\section*{Elbow method}\n\n\\begin{itemize}\n \\item How do we choose the number of clusters K?\n \\item Vary K and compute cost function at a range of K values.\n \\item $J(...)$'s minimum value should decrease as K rises (i.e. you decrease the granularity so centroids can better optimize).\n \\item Look for the \"elbow\" on the graph ($K$ vs $J()$).\n\\end{itemize}\n\n\\includegraphics[width=0.47\\textwidth]{resources/elbow}\n\n\\end{document}\n", "meta": {"hexsha": "4eb33aa9a6c1c4422d0e3197a4841ecd5f474f28", "size": 4991, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "slides/week_13.tex", "max_stars_repo_name": "djeada/Stanford-Machine-Learning", "max_stars_repo_head_hexsha": "e6ef77939b7c581aebb5e9454669ad2dbb4f98f0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "slides/week_13.tex", "max_issues_repo_name": "djeada/Stanford-Machine-Learning", "max_issues_repo_head_hexsha": "e6ef77939b7c581aebb5e9454669ad2dbb4f98f0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "slides/week_13.tex", "max_forks_repo_name": "djeada/Stanford-Machine-Learning", "max_forks_repo_head_hexsha": "e6ef77939b7c581aebb5e9454669ad2dbb4f98f0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.6985294118, "max_line_length": 180, "alphanum_fraction": 0.7387297135, "num_tokens": 1340, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6859494485880928, "lm_q1q2_score": 0.3456541598145602}} {"text": "% arara: makechapters: {items: [syllabus, dtt, pi, inductive, identity, equivalences, contractible, fundamental, hierarchy, funext, pullback, univalence]}\n\n\\documentclass[11pt]{memoir} %[ebook,10pt,oneside]\n \n\\usepackage{hott}\n\n\\title{Introduction to homotopy type theory}\n\\author{Egbert Rijke}\n\\date{Carnegie Mellon University\\\\Pittsburgh PA\\\\Summer 2019}%\\\\Version: \\today}\n%\\address{Carnegie Mellon University}\n%\\email{erijke@andrew.cmu.edu}\n\n\\pretitle{\\begin{center}\\textsc\\bgroup\\LARGE}\n\\posttitle{\\egroup\\end{center}\\vspace{2cm}}\n\\preauthor{\\begin{center}\\textsc\\bgroup\\Large}\\postauthor{\\egroup\\end{center}\\vfill}\n\\predate{\\begin{center}\\textsc\\bgroup}{\\postdate{\\egroup\\end{center}}\n\n\n % The following is to avoid overfull hboxes in the table of contents.\n% https://tex.stackexchange.com/questions/49887/overfull-hbox-warning-for-toc-entries-when-using-memoir-documentclass\n%\\renewcommand*{\\cftdotsep}{1}\n\\setpnumwidth{2em}\n\\setrmarg{3em}\n\\setlength{\\cftchapternumwidth}{2em}\n\\setlength{\\cftsectionindent}{2em}\n\\setlength{\\cftsectionnumwidth}{2em}\n\\setlength{\\cftsubsectionindent}{4em}\n\\setlength{\\cftsubsectionnumwidth}{3em}\n\n% We number sections independently of chapters, and subsections will be\n% numbered too.\n\n\\counterwithout{section}{chapter}\n\\settocdepth{subsection}\n\n% We set up the exercise environment, which produces list environment in a new\n% unnumbered subsection that also gets mentioned in the table of contents.\n\n\\newlist{exenum}{enumerate}{1}\n\\setlist[exenum]{noitemsep,label=\\thesection.\\arabic*}\n %,ref=\\thechapter.\\arabic*}\n\n\\crefname{exenumi}{Exercise}{Exercises}\n \n\\newlist{subexenum}{enumerate}{1}\n\\setlist[subexenum]{noitemsep,label=(\\alph*),ref=\\theexenumi.\\alph*}\n\\crefname{subexenumi}{Exercise}{Exercises}\n \n\\newenvironment{exercises}\n{%\n\\subsection*{Exercises}%\n\\addcontentsline{toc}{subsection}{Exercises}%\n\\sectionmark{Exercises}%\n\\begin{exenum}}\n{%\n\\end{exenum}}\n \n\\addbibresource{bibliography.bib}\n\n\\makeindex\n\n\\begin{document}\n\n\\begin{titlingpage}\n \\maketitle \n\\end{titlingpage}\n\n\\mbox{}\n\\vfill\n\\doclicenseThis\\thispagestyle{empty}\n\n\\cleardoublepage\n\n\\frontmatter\n\n\\tableofcontents\n\n%\\include{intro}\n\n\\chapter{Introduction}\n\nTo include introduction:\n\\begin{enumerate}\n\\item What are types in mathematics. Dependent types and dependent functions are everywhere in mathematics.\n\\item Why univalent foundations. Why should homotopy be in the foundation of mathematics\n\\item Constructive nature of homotopy type theory. Discuss differences with set theory.\n\\item What this course is about\n\\item How to use this book\n\\item Formal type theory versus informal type theory\n\\item Mention the formalization\n\\end{enumerate}\n\n\\begin{rmk}\n One difference between set theory and type theory is that every well-formed term is specified along with its type and with its context. One way of looking at this is that there are three sorts in type theory: contexts, types, and terms. On the other hand, there is only one sort in set theory: sets. Sets are governed by the elementhood relation: the formula $x\\in y$ is a well-formed formula of set theory for any two sets $x$ and $y$. In particular, for a given set $x$, the formula $x\\in y$ can be true for many sets $y$, which is very different to the situation in type theory, where every term is assigned a unique type.\n \n Another important difference between set theory and type theory is that set theory is formulated in the language of first order logic, whereas type theory is its own deductive system, not making use of any ambient logic. We will see in the present chapter and in the next few chapters what this deductive system looks like.\n\\end{rmk}\n\n\\mainmatter\n\n\\renewcommand{\\thechapter}{\\Roman{chapter}}\n\\setsecnumdepth{subsection}\n\n\\chapter{Martin-L\\\"of's dependent type theory}\n\\input{dtt}\n\\input{pi}\n\\input{nat}\n\\input{inductive}\n\\input{identity}\n\\input{universes-relations}\n\\input{number-theory}\n\\input{finite-types}\n\n\\chapter{Basic concepts of type theory}\n\\input{equivalences}\n\\input{contractible}\n\\input{fundamental}\n\\input{hierarchy}\n\n\\chapter{Univalent mathematics}\n\\input{funext}\n\\input{univalence}\n\\input{circle}\n\n\\chapter{Homotopy pullbacks and pushouts}\n\\input{pullback}\n\\input{pushout}\n\\input{cubical}\n\\input{descent}\n\\input{id-pushout}\n\n\\chapter{The homotopy image of a map}\n\\input{sequences}\n\\input{image}\n\\input{replacement}\n\\input{sets}\n\\section{The classifying type of a group}\n\\input{truncation}\n\\section{Truncations}\n\n\\chapter{Synthetic homotopy theory}\n\\input{homotopy_groups}\n\\input{les}\n\\input{connected}\n\\input{smash}\n\\input{blakers-massey}\n\n%\\include{open_problems}\n\n\\backmatter\n\n\\printbibliography\n\n\\printindex\n\n\\end{document}\n", "meta": {"hexsha": "a5adbc6770439c3241a6b03878f6fa1aa72e0084", "size": 4655, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Book/hott-intro.tex", "max_stars_repo_name": "tadejpetric/HoTT-Intro", "max_stars_repo_head_hexsha": "f4228d6ecfc6cdb119c6e8b0e711fea05b98b2d5", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Book/hott-intro.tex", "max_issues_repo_name": "tadejpetric/HoTT-Intro", "max_issues_repo_head_hexsha": "f4228d6ecfc6cdb119c6e8b0e711fea05b98b2d5", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Book/hott-intro.tex", "max_forks_repo_name": "tadejpetric/HoTT-Intro", "max_forks_repo_head_hexsha": "f4228d6ecfc6cdb119c6e8b0e711fea05b98b2d5", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.6496815287, "max_line_length": 627, "alphanum_fraction": 0.7767991407, "num_tokens": 1315, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583270090337583, "lm_q2_score": 0.6187804337438502, "lm_q1q2_score": 0.3454818288208155}} {"text": "\\subsubsection{Total number of messages sent}\\label{subsubsec:ld2krmessages}\n\nThis is an indication of the energy efficiency of the entire network.\n\nThe maximum number of copies is the dominant factor (\\(85.18\\%\\)), as in the\nhigh density case. In the previous case the size of the hear window was the\nsecond most important factor. Here, instead, we have the broadcast radius\n(\\(6.57\\%\\)) and its combination with the maximum number of copies. The\nunexplained variation is extremely low \\(0.31\\%\\).\n\nFrom \\figref{fig:ldperfmessagesR} we notice that increasing the broadcast radius\nis useful to reduce the total number of messages sent when the maximum number of\ncopies is low. This can be explained by the fact that with an higher value for\n\\(m\\) means that nearly all the users in the network will relay the message\nafter they have successfully heard it. So, if \\(m\\) is high, there is not much\nto do to reduce the total number of messages sent.\n\n\\begin{figure}[htb]\n\t\\centering\n\t\\includegraphics[width=0.6\\textwidth]{img/ld/messages-R-perfplot}\n\t\\caption{Performance plot for the total number of messages sent. The\n\tbroadcast radius can be used to reduce the total number of messages sent\n\twhen the maximum number of copies is not\n\thigh}\\label{fig:ldperfmessagesR}\n\\end{figure}\n", "meta": {"hexsha": "5c08cfa651fa3f739aafabb610861e0525d2d726", "size": 1280, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/chapters/scenarios/low-density/2kr/messages.tex", "max_stars_repo_name": "SpeedJack/pecsn", "max_stars_repo_head_hexsha": "40c757cddec978e06de766c9dff00abf57ccd6b3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/chapters/scenarios/low-density/2kr/messages.tex", "max_issues_repo_name": "SpeedJack/pecsn", "max_issues_repo_head_hexsha": "40c757cddec978e06de766c9dff00abf57ccd6b3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/chapters/scenarios/low-density/2kr/messages.tex", "max_forks_repo_name": "SpeedJack/pecsn", "max_forks_repo_head_hexsha": "40c757cddec978e06de766c9dff00abf57ccd6b3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.2307692308, "max_line_length": 80, "alphanum_fraction": 0.778125, "num_tokens": 317, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34548181972574316}} {"text": "\\documentclass{dw2014}\n\n\\usepackage{mathptmx}\n\\usepackage{helvet}\n\\usepackage{courier}\n\\usepackage{graphicx}\n\n\\title{Identification of human control during perturbed walking}\n\n\\author{Jason K. Moore, Sandra K. Hnat, Antonie J. van den Bogert\\\\\n Human Motion and Control Laboratory, Cleveland State University, Cleveland, Ohio, USA\\\\\n {\\it j.k.moore19@csuohio.edu, s.hnat@vikes.csuohio.edu, a.vandenbogert@csuohio.edu}}\n\n\\date{}\n\n\\begin{document}\n\n\\maketitle\n\n\\section*{Introduction}\n%\nRecent research and commercial activity have shown that gait-related powered\nprosthetics will play an important role in assisting humans with disabilities.\nThese devices include a variety of sensors and actuators than can be coupled to\na control system to provide acutated gait assistance. However, the available\nlightweight, lower extremity exoskeletons lack gait that resembles an\nable-bodied human. To improve the gait of powered prosthetics, our intent is to\nidentify a simple, linear controller from a large set of data collected from\nable-bodied subjects being perturbed by random longitudinal forces.\n%\n\\section*{Methods}\nThe example data shown here was collected from a single subject (age: 29, mass:\n63 kg, height: 172 cm) walking on an instrumented treadmill (V-Gait, Motek\nMedical). The subject was longitudinally perturbed using random white noise\nwith 10\\% std around a nominal 1.2 m/s belt speed. Data was recorded for 8\nminutes at 100 Hz, which included approximately 500 steps of walking. Ankle\nplantarflexion, knee flexion, and hip flexion angles, rates, and moments were\ncomputed using 2D inverse dynamics.\n%\n\\begin{figure}[hbt]\n \\begin{center}\n \\includegraphics[width=\\columnwidth]{fig/gains.pdf}\n \\caption{Gait phase percent scheduled gains for right (blue) and left (red) legs.}\n \\label{fig:gains}\n \\end{center}\n\\end{figure}\n\nJoint angle, rate, and torque time series were sectioned into steps based on\nthe right foot's heel strike. Subsequently, 20 evenly spaced data points from\neach series were interpolated along the gait cycle. We assume a simple\nscheduled proportional derivative controller that generates the joint torques\ngiven the joint angles and rates that fits the following form.\n%\n\\begin{equation}\n \\mathbf{m}(t) = \\mathbf{m}^*(\\varphi(t)) -\n \\mathbf{K}(\\varphi(t))\\mathbf{s}(t)\n\\end{equation}\n%\nwhere $t$ is an instance of time, $\\varphi(t)$ is the phase in the right leg\ngait cycle, $\\mathbf{m}(t)$ is a vector of joint torques,\n$\\mathbf{m}^*(\\varphi(t))$ is a vector of the nominal joint torques,\n$\\mathbf{K}(\\varphi(t))$ is a gain matrix scheduled with respect to gait\nphase which multiplies the vector of joint angles and rates, $\\mathbf{s}(t)$.\nThis equation is linear in the gains and the reference torques. Given\nsufficient joint angle, rate, and torque measurements, the reference torques\nand the gains can be solved for using linear least squares.\n%\n\\section*{Results}\n%\nHere we present an example result from a controller structure which is limited\nto joint torque generation only from error in the sensors from the same joint.\nFigure \\ref{fig:gains} shows the estimates of the scheduled gains with respect\nto the percent gait cycle in each leg. Figure \\ref{fig:fit} demonstrates an\nexample prediction of the measured ankle plantarflexion torque in the right leg\nby the identified control model.\n%\n\\begin{figure}[b]\n \\begin{center}\n \\includegraphics[width=\\columnwidth]{fig/fit.pdf}\n \\caption{Predicted torque compared to independent validation data.}\n \\label{fig:fit}\n \\end{center}\n\\end{figure}\n%\n\\section*{Discussion}\n%\nWe are able to identify a simple linear controller that exhibits larger gains\nin the stance phase than in the swing phase. Additionally, similar gain\npatterns in the right and left legs are observed that use both positive and\nnegative feedback. The controller is capable of predicting the measured joint\ntorques with greater than 65\\% VAF in all joints. Results and conclusions from\na larger sample of subjects and conditions will be presented at the conference.\n\n%\\section*{Acknowledgments}\n%\n%This research was funded by the Ohio's Wright Center for Sensor Systems\n%Engineering and the Parker Hannifin Corporation.\n\\end{document}\n", "meta": {"hexsha": "c30c8428287eaca2172fe9b4e8c9c0be4d8d7fa2", "size": 4211, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "abstract/moore.tex", "max_stars_repo_name": "moorepants/DW2014", "max_stars_repo_head_hexsha": "b588f837ecaaa098dea5db6a72903632edd352aa", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "abstract/moore.tex", "max_issues_repo_name": "moorepants/DW2014", "max_issues_repo_head_hexsha": "b588f837ecaaa098dea5db6a72903632edd352aa", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "abstract/moore.tex", "max_forks_repo_name": "moorepants/DW2014", "max_forks_repo_head_hexsha": "b588f837ecaaa098dea5db6a72903632edd352aa", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.5353535354, "max_line_length": 89, "alphanum_fraction": 0.7791498456, "num_tokens": 1060, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.34548181972574304}} {"text": "%!TEX root = ../thesis.tex\n%*******************************************************************************\n%****************************** Second Chapter *********************************\n%*******************************************************************************\n\n\\chapter{The Rotating Sun}\n\n\\section{Differential Rotation}\nThe sun is known to rotate differentially, unlike the earth. This means that angular rate of rotation of a point in the sun about its spin axis is depends on depth and latitude, $\\Omega = \\Omega(r,\\theta)$. Splitting of p-mode\\footnote{pressure waves are also called p-modes} frequencies due to differential rotation is well understood and has a long history of inversion analysis \\cite{ritzwoller,lavely92,schou98}.\nThis $\\Omega$ is generally taken to be symmetric about the equitorial plane \\cite{ritzwoller}.\n\nAs a result of Alfven's freezing theorem\\footnote{Alfven's freezing theorem states that in a perfectly conducting plasma, magnetic flux through \\textit{every} surface is conserved as it gets dragged along with the ambient flow. See \\cite{goedbloed2004} for proof. This theorem implies that magnetic field lines are $frozen$ in the plasma as it flows.}, differential rotation is responsible for winding the solar magnetic field around its spin axis in an axisymmetric fashion. Thus we'll be investigating the effects of an axis symmetric magnetic field on the spectrum in this work, although the method we develop can very well accommodate non-axisymmetric fields too.\n\\section{Detection from frequency spectrum}\nBecause of axis symmetry of differential rotation, the flow profile is given as,\n\\begin{equation}\n\\vrot = \\sum_{s = 1,3,5,...}^{\\infty} -w_s^0(r) \\partial_{\\theta} Y_s^0 \\ev{\\phi}\n\\end{equation}\n\nNote that $w_1^0$ is responsible for shell like (pure) rotation as it couples with $\\partial_{\\theta} Y_1^0 \\sim \\sin\\theta$. Hence $w_3^0$ onwards components of flow are responsible for the differential part of the rotation. The fact that the azimuthal index for the $\\vrot$ profile is $0$ means that it has no azimuthal dependence. Finding frequency splittings due to differential rotation is a problem in DPT or QDPT depending on whether we're using the isolated multiplet approximation or not. The isolated multiplet approximation is the assumption that there exists negligible cross coupling between different $\\mode{n}{l}$ modes. \nBelow we outline the QDPT approach to the problem because when applied to a single multiplet $_n S_l$ it reduces to the DPT approach. It has been argued via analysis differential rotation coupling of \\mode{n}{1} and \\mode{n}{3} multiplets that eigenfrequency correction to DPT via QDPT from cross coupling between these two is negligible ($\\sim 1 \\mu Hz$) \\cite{lavely92}. We'll show in Chapter 5 (figure \\ref{fig:split_dr}) that for $l\\sim 100$ modes, a frequency corrections of upto $600 nHz$ frequencies are obtained when QDPT is used.\n\n%which is close to the correction obtained due presence of realistically strong magnetic fields too, and hence cannot be ignored in an analysis which accounts for both differential rotation and magnetic fields.\n\n\\subsection{QDPT Analysis}\nThe pertubation operator \\dLd for a differential rotaional flow is given by \n\\begin{equation}\n\\dLd = -2i \\omega \\rho \\vrot \\cdot \\grad\n\\end{equation}\nwhere $\\omega$ is the reference frequency in the problem, and $\\rho$ is the static background density profile \\cite{ritzwoller}.\nThe supermatrix element $Z_{k' k}$ for a pertubation \\dLd is given by\n\\begin{equation}\nZ_{k' k} = \\Lamdr_{k'k} - \\delta_{k'k} (\\omref^2 - \\omega_k^2)\n\\end{equation}\nwhere $\\Lamdr$ is the coupling matrix element $\\Lamdr = \\inner{\\xiv_{k'}}{\\dLd \\xiv_{k}}$ .\nCoupling matrix element is given by\n\\begin{equation}\n\\Lamdr_{k'k} = 8 \\pi \\omref \\gam{l'}\\gam{l} \\oddsum{s} \\gam{s} \\wigred{-m}{0}{m} \\ints dr \nr^2 w_s^0(r) T_s(r)\n\\end{equation}\nwhere the sensitivity kernel $T_s$ is given by\n\\begin{dmath}\nT_s(r) = (1-(-1)^{s+l+l'}) \\om{l'}{0} \\om{l}{0} \\wigred{-1}{0}{1} r^{-1}\n \\enc{U'V+V'U-U'U-\\frac{1}{2}\\enc{l'(l'+1) + l(l+1) - s(s+1) V'V}}\n\\end{dmath}\nwhere the rounded brackets represent Wigner 3j symbols \\cite{ritzwoller}, and $\\gam{l}$ and $\\Omega_{l}^{N}$ are constants defined in Appendix \\ref{app_conventions}.\n\\subsection{Selecition rules in mode coupling}\\label{sec:selec_rules}\nThis matrix element enforces the following selection rules for inter-mode interaction which derive from the properties of Wigner 3j symbols \\cite{lavely92}.\n\\begin{enumerate}\n\\item $m'=m$\n\\item $l'+l+s = \\text{odd}$\n\\item $|l'-l| \\leq s \\leq l'+l$\n\\end{enumerate}\nIt should be noted here that even though only sum over odd s is considered, the expression for $T_s$ is general and holds for all $s$. This has been verified independently using the Mathematica packaged developed for the sake of this work \\cite{GSH_repo}. Hence, as far as self coupling is concerned ($l'=l$), $T_s$ vanishes for even $s$. This means that even if $\\Omega(\\theta,\\phi)$ had a component which is antisymmetric about the equatorial plane, DPT analysis would not reveal any signature of that in the frequency splittings.\n\n\\subsubsection{Form of supermatrix $Z_{k'k}$}\n\nIt is clear from selection rule (1) in \\ref{sec:selec_rules} that $Z_{k'k}$ is going to be a sparse matrix consisting of a diagonal and a number of sub-diagonals (proportional to number of multiplets being considered). Also, as $s$ is always odd, $l-l'$ has to be even for non-zero coupling as consequence of selection rule (2). This influences our choice of modes whose inter-coupling will be studied hereafter. Figure (\\ref{fig:coup_mat}) is a visual representation of a typical supermatrix consisting of three multiplets. The logarithm scale demonstrates the order of magnitude difference between self-coupling (main diagonal elements), and cross-coupling (subdiagonal elements). Also notice that largest elements (yellow) are in the first and third section in the main diagonal. These are large because these frequencies are placed away from $\\omref$ which is taken to be mean of the three mode frequencies. The relative weakness of the sub-diagonal terms compared to the main-diagonals is because $l-l'\\geq 2$ forces $s\\geq 2$ which makes the largest component of differential rotation, i.e. $w_1^0$, unable to couple these modes; $w_3^0$ and $w_5^0$ are one and two orders of magnitude smaller than $w_1^0$ respectively.\n\n\\begin{figure}[h]\n\\includegraphics[scale=0.6,center]{Chapter2/figs/coup_mat}\n\\caption{Visualisation of $\\log_{10}|Z_{k'k}|$ in $\\mu Hz^2$ for the multiplets $\\mode{0}{11}$, $\\mode{0}{13}$, and $\\mode{0}{15}$ with frequencies $603.69 \\mu Hz$, $641.84\\mu Hz$, and $677.55 \\mu Hz$ respectively. Numbers on the X and Y axes represent cumulative $m$ of all three multiplets. White spaces correspond to $0$ value.} \n\\label{fig:coup_mat}\n\\end{figure}", "meta": {"hexsha": "c74f22f3a8fffcbeadc64de1a41447e4e3f662b9", "size": 6849, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Chapter2/chapter2.tex", "max_stars_repo_name": "tuneerch/masters_thesis", "max_stars_repo_head_hexsha": "487646d71dc5f1f2bfbb8e29ee4878d14825c344", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Chapter2/chapter2.tex", "max_issues_repo_name": "tuneerch/masters_thesis", "max_issues_repo_head_hexsha": "487646d71dc5f1f2bfbb8e29ee4878d14825c344", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter2/chapter2.tex", "max_forks_repo_name": "tuneerch/masters_thesis", "max_forks_repo_head_hexsha": "487646d71dc5f1f2bfbb8e29ee4878d14825c344", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 108.7142857143, "max_line_length": 1226, "alphanum_fraction": 0.734413783, "num_tokens": 1816, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34548181972574304}} {"text": "\\documentclass[a4paper,12pt]{article}\n\n% Font\n\\usepackage[T1]{fontenc}\n\\usepackage{gentium}\n\n% Math packages\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{bm}\n\n% Define symbol shortcuts\n\\newcommand{\\cc}{\\mathcal{C}}\n\\newcommand{\\dd}{\\mathcal{D}}\n\\newcommand{\\hh}{\\mathcal{H}}\n\\newcommand{\\xx}{{\\bm x}}\n\\newcommand{\\yy}{{\\bm y}}\n\n% Math environment\n\\newtheorem*{thm}{Theorem}\n\n% Better list management:\n% - vertical spacing in lists\n% - items in lists start with dash not bullet point.\n\\usepackage{enumitem}\n\\setlist{label=\\textemdash,\n itemsep=0pt, topsep=3pt, partopsep=0pt} \n\n% Include graphics\n\\usepackage{graphicx}\n\\usepackage{subcaption}\n\n% Page format \n\\usepackage[top=2cm,left=2cm,right=2cm,bottom=2cm]{geometry}\n\\usepackage{float}\n\n\\begin{document}\n%%% HEADER\n\\raisebox{0.6in}[0in]{\\makebox[\\textwidth][r]{\\it Unproofed version }}\n\\vspace{-0.7in}\n\n\\begin{center}\n\\bf\\large MA2823 : Foundations of Machine Learning \\\\\nChapter 5 : Linear Regression\n\\end{center}\n\n\\noindent\nLecturer : Chlo\\'e-Agathe Azencott \n\\hfill\nScribes : Emilie Leblanc, Antoine Salon, Hippolyte Jacomet \n\n\\noindent\n\\rule{\\textwidth}{1pt}\n\n\\medskip\n\n%%% NOTES START HERE\nIn this chapter we will see how to :\n\\begin{itemize}\n\\item define parametric methods ;\n\\item define the maximum likelihood estimator and compute it\nfor Bernoulli, multinomial and Gaussian densities ;\n\\item define the Bayes estimator and compute it for normal\npriors ;\n\\item compute the maximum likelihood estimator / least-square\nfit solution for linear regression ;\n\\item compute the maximum likelihood estimator for logistic\nregression.\n\\end{itemize}\n\n\\section{Parametric methods}\n\nIn this chapter we would like to define different ways of building estimators that are part of a statistical model, and how to assess their performances.\n\n\\paragraph{Definitions and notations} Let us consider a training set of $n$ $p$-dimensional vectors denoted by $\\mathcal{X}=\\{x^i\\}_{i=1,\\dots,n}$, and let's assume that each of these observation follows a law of probability given a parameter $\\theta$. \n\n\\[\n\\forall i \\in [1,\\dots,n],\\ \\xx^{i}\\sim{}p(\\xx|\\theta)\n\\]\\\\\nThe {\\em parametric estimation} consists in {\\em assuming} a form for $p(\\xx|\\theta)$ and then {\\em estimating} the parameter $\\theta$ using our training set $\\mathcal{X}$. \\\\\nFor example, if we assume that the observations of our training set follow a Gaussian distribution $p(x_{j}|\\theta_j)\\sim{}\\mathcal{N}(\\mu_j,{\\sigma_j}^2)$ our goal would be to estimate the parameter $\\theta=\\{\\mu_1,\\sigma_1,\\dots,\\mu_p,\\sigma_p\\}$. \\\\\n\\\\\nTo achieve that goal, the observations of the training set $\\xx^i$ are usually assumed to be {\\em independent and identically distributed} (i.i.d.). \\\\\n\nIn the following sections we sill study two different methods to estimate the parameters : the {\\em maximum likelihood estimation} and the {\\em Bayes estimation}.\n\n\n\\subsection{Maximum likelihood estimation}\n\nThis first method relies on the idea that $\\theta$ should maximize the likelihood for $\\mathcal{X}$ to be drawn from the law of probability we chose in our model.\n\n\\paragraph{Likelihood and log-likelihood} We call {\\em likelihood of} $\\theta$ given $\\mathcal{X}$ the function : $\\ell(\\cdot{}|\\mathcal{X}) : \\theta \\mapsto p(\\mathcal{X}|\\theta).$\\\\\nIn the case of an i.i.d. sample $\\mathcal{X}$, this becomes :\n\n\\[\\ell(\\theta|\\mathcal{X})=p(\\mathcal{X}|\\theta)=p(\\xx^1|\\theta)p(\\xx^2|\\theta)\\dots{}p(\\xx^p|\\theta)\\] \\\\\nIt is often useful to also consider the {\\em log-likelihood}, simply defined by $\\mathcal{L} (\\cdot{}|\\mathcal{X}) : \\theta \\mapsto \\log \\ell(\\theta|\\mathcal{X})$, which in our previously mentioned case becomes :\n\\[\\mathcal{L}(\\theta|\\mathcal{X})=\\log{p(\\xx^1|\\theta)}+\\dots{}+\\log{p(\\xx^n|\\theta)}\\]\n\\\\\nGiven these two functions, we will then try to find the estimator that maximizes them, called the {\\em maximum likelihood estimator} or MLE, and defined by :\n\n\\[\\hat{\\theta}=\\arg\\underset{\\theta}\\max \\ \\ell(\\theta|\\mathcal{X})=\\arg\\underset{\\theta}\\max \\ \\mathcal{L}(\\theta|\\mathcal{X})\\]\\\\\n\n\nLet us now look at these MLE for Bernoulli, Multinomial and Normal laws.\n\n\\subsubsection{Bernoulli density}\n\nSuch a law corresponds to observations that reflect two states of either failure or success (think of a coin toss for example), and thus let us consider a data set $\\mathcal{X}=\\{x^i\\}_{i=1,\\dots,n}$ with $x^i\\in\\{0,1\\}$ $\\forall i\\in[1,\\dots,n]$.\\\\\n\\\\\nThe Bernoulli density is given by $P(X=x|p_0)={p_0}^{x}(1-p_0)^{(1-x)}$, let us try to find the MLE $\\ \\hat{p_0}$ of the parameter $p_0$.\\\\\n\\\\\nWe first compute the log-likelihood :\n\n\\[\\mathcal{L}(p_0|\\mathcal{X})=\\log{P(\\mathcal{X}|p_0)}=\\sum_{i=1}^n\\left(x^i\\log p_0 + (1-x^i)\\log{(1-p_0)}\\right)\\]\nThis is a concave function which we can easily maximize by setting its gradient to 0 :\n\n\\[\\frac{\\sum_{i=1}^nx^i}{\\hat{p_0}}-\\frac{n}{(1-\\hat{p_0})}+\\frac{\\sum_{i=1}^nx^i}{(1-\\hat{p_0})}= 0\n\\]\nWhich yields the MLE of $p_0$ :\n\n\\[\\hat{p_0}=\\frac{1}{n}\\sum_{i=1}^nx^i\\]\n\n\n\\subsubsection{Multinomial density}\n\nLet us consider $K$ mutually exclusive and exhaustive classes. The probability for each class to occur is $p_k$, with $\\sum_{k=1}^Kp_k=1$. We represent these classes with $K$ indicator variables $x_1, x_2, \\dots, x_K$ knowing \n\\[\n x_k= \n\\begin{cases}\n 1& \\text{if the outcome is class } k\\\\\n 0 & \\text{otherwise}\n\\end{cases}\n\\]\nThe joint probability distribution is given by :\n\\[P(x_1,x_2,\\dots,x_K)=\\displaystyle\\prod_{k=1}^{K}p_k^{x_k}\\]\\\\\n\\\\\nLet us compute the MLE $\\bm{\\hat{p}}=(\\hat{p_1},\\dots,\\hat{p_K})$. The observations are i.i.d. thus the log-likelihood looks dramatically like :\n\\\\\n\\[\\mathcal{L}(\\bm p|\\mathcal{X})=\\sum_{i=1}^n\\sum_{k=1}^K x_k^i\\log p_k\n\\]\\\\\nAnd we now have to maximize it under the constraint $\\displaystyle\\sum_{k=1}^Kp_k=1$.\\\\\nWe must call on to a Lagrange multiplier $\\lambda$ and define the Lagrangian function \n\\[\\mathcal{F} : (\\bm p, \\lambda) \\mapsto \\mathcal{L}(\\bm p|\\mathcal{X})-\\lambda \\left(\\sum_{k=1}^Kp_k - 1\\right)\\]\nSetting its gradient to zero gently yields a system of equations from which we are happy to learn that $\\lambda=n$ and consequently that\n\n\\[\\hat{p}_k=\\dfrac{1}{n}\\displaystyle\\sum_{i=1}^nx_k^i\\ \\ \\ \\ \\ \\ \\ \\ \\forall k \\in [1,\\dots,K]\\]\n\n\n\\subsubsection{Gaussian distribution}\n\nLet us now assume that our observations $x^i$ are i.i.d. following a Gaussian distribution, i.e. : \n\n$\\forall i \\in [1,\\dots,n]$,\n\\[x^i\\sim\\mathcal{N}(\\mu,\\sigma^2)\\]\n\n\\[p(x^i|\\mu,\\theta)=\\dfrac{1}{\\sqrt[]{2\\pi}\\sigma}\\exp\\left[-\\frac{(x^i-\\mu)^2}{2\\sigma^2}\\right]\\]\n\\\\\n\nSetting the gradient of the log-likelihood to zero yields the estimators :\n\n\\[ \\hat{\\mu} = \\frac{1}{n} \\sum_{i=1}^nx^i\\]\n\\[ \\hat{\\sigma}^2 = \\frac{1}{n} \\sum_{i=1}^n (x^i-\\hat{\\mu})^2\\]\n\n\\subsection{Bias-variance trade-off}\n\nWe can assess the performance of the estimator (be it obtain through MLE or another method) by computing the {\\em mean squared error} or MLE, defined by :\n\n\\[\\mbox{MSE}(\\hat{\\theta}) = \\mathbb{E}[(\\hat{\\theta}-\\theta_0)^2]\\]\n\n\\paragraph{Bias-variance trade-off} Let us try to link this MSE to two previously known notions that help characterize our estimator, the variance and the bias. As a reminder, the {\\em bias} of an estimator is the difference between this estimator's expected value and the true value of the parameter being estimated : $\\mbox{Bias}(\\hat{\\theta})=\\mathbb{E}[\\hat{\\theta}]-\\theta_0$. \n\n\\begin{align*}\n\\mbox{MSE}(\\hat{\\theta}) &= \\mathbb{E}[(\\hat{\\theta}-\\theta_0)^2]\\\\ \n&= \\mathbb{E}\\left[(\\hat{\\theta}-\\mathbb{E}[\\hat{\\theta}])^2+2\\hat{\\theta}\\mathbb{E}[\\hat{\\theta}]-\\mathbb{E}[\\hat{\\theta}]^2-2\\hat{\\theta}\\theta_0+\\theta_0^2\\right]\\\\\n&=\\mbox{Var}(\\hat{\\theta})+\\mathbb{E}[\\hat{\\theta}]^2-2\\mathbb{E}[\\hat{\\theta}]\\theta_0+\\theta_0^2\\\\\n&=\\mbox{Var}(\\hat{\\theta})+\\mbox{Bias}^2(\\hat{\\theta})\n\\end{align*}\\\\\nWe see that the MSE is a balance between these two positive expressions, hence the idea of a {\\em bias-variance trade-off}. We understand that a biased estimator may achieve better MSE than an\nunbiased one, as illustrated below :\n\n\\begin{center}\n\\includegraphics[width=100mm]{biasvariance.png}\n\\end{center}\n\n\\subsection{Bayes estimator}\nFrom now on let us study another way of building an estimator, using Bayes rule which in its general form is :\n\\[P(C|\\xx) = \\dfrac{P(C)p(\\xx|C)}{p(\\xx)}\\]\\\\\nHere we choose to treat $\\theta$ as a random variable with a given prior distribution $p(\\theta)$. Considering an observation-set $\\mathcal{X}$, the Bayes rule yields :\n\n\\[p(\\theta|\\mathcal{X})=\\frac{p(x|\\theta)p(\\theta)}{p(\\mathcal{X})}\\]\\\\\n\\\\\nWe define the {\\em Bayes estimate} as the conditional expected value of $\\theta$ given our data set $\\mathcal{X}$ :\n\\[\\hat{\\theta}_{\\mbox{Bayes}}=\\mathbb{E}[\\theta|\\mathcal{X}]=\\int{\\!\\theta p(\\theta|\\mathcal{X})d\\theta}\\]\\\\\n\\\\\nHere's a reminder of the estimates we have previously seen :\\\\\n\n{\\em Maximum a posteriori} estimate :\n\\[\\hat{\\theta}_{\\mbox{MAP}}=\\arg\\underset{\\theta}\\max \\ p(\\theta|\\mathcal{X})\\]\n\n{\\em Maximum likelihood} estimate :\n\\[\\hat{\\theta}_{\\mbox{MLE}}=\\arg\\underset{\\theta}\\max \\ p(\\mathcal{X}|\\theta)\\]\\\\\n\n\n\\subsubsection{Bayes estimator for normal priors}\n\nLet us consider $n$ i.i.d. data points $x^i$ following a normal law, $x^i\\sim\\mathcal{N}(\\theta,\\sigma^2)$ and let's assume that $\\theta$ is of Gaussian prior distribution : $\\theta\\sim\\mathcal{N}(\\mu,\\sigma_0^2)$.\\\\\n\\\\\nWe know that the MLE of $\\theta$ is its sample mean : $\\hat{\\theta}_{\\mbox{MLE}} = \\frac{1}{n} \\sum_{i=1}^nx^i$ and we would like to compare it to its Bayes estimator.\\\\\n\\\\\nAfter a long and troublesome computation, a quick look at $p(\\theta|\\mathcal{X})$ shows us that it follows a normal distribution with mean $m$ and variance $s^2$ :\n\n\\[m = \\frac{n\\hat{\\theta}_{\\mbox{MLE}}\\sigma^2+\\mu\\sigma_0^2}{n\\sigma^2+\\sigma_0^2}\n\\]\n\n\\[s^2 = \\frac{\\sigma^2\\sigma_0^2}{n\\sigma^2+\\sigma_0^2}\n\\]\\\\\n\\\\\nKnowing that :\n\\[\\mathbb{E}[\\theta|\\mathcal{X}]=m\\]\nWe can conclude with :\n\\[\\hat{\\theta}_{\\mbox{Bayes}}=\\frac{\\frac{n}{\\sigma_0^2}}{\\frac{n}{\\sigma_0^2}+\\frac{1}{\\sigma^2}}\\hat{\\theta}_{\\mbox{MLE}}+\\frac{\\frac{1}{\\sigma^2}}{\\frac{n}{\\sigma_0^2}+\\frac{1}{\\sigma^2}}\\mu\n\\]\\\\\n\\\\\nIt is interesting to notice that when n increases, $\\hat{\\theta}_{\\mbox{Bayes}}$ gets closer to the sample average (uses\ninformation from the sample) and When $\\sigma$ is small, $\\hat{\\theta}_{\\mbox{Bayes}}$ gets closer to $\\mu$ (little uncertainty\nabout the prior). \n\n\\section{Linear regression}\n\nThe goal of a {\\em linear regression} is to approximate the law followed by our data \\(\\textit{y}\\) by a linear combination of observed variables \\(\\xx\\) such as :\n\\[ f(\\xx|\\beta)= \\displaystyle\\sum_{j=1}^{p} \\beta_j*x_j + \\beta_0\\]\nwith \\(\\xx^i \\in \\mathbb{R}^p\\) and \\(\\textit{y}^i \\in \\mathbb{R}\\) and noting \\(\\dd = {\\{\\xx^i,\\textit{y}^i\\}}_{\\textit{i}=1,..,\\textit{n}}\\). \n\\begin{center}\n\\includegraphics[width=100mm]{linear_regression_example.png}\n\\end{center}\n\nWe assume that the error \\(\\epsilon\\) we have while doing a regression is \\textbf{Gaussian distributed}. Therefore, with \\(y = g(\\xx) + \\epsilon \\), \\(f(\\xx|\\beta)\\) being \\(g\\)'s estimator, we have : \n\\begin{center}\n\\(\\epsilon = y - g(\\xx)\\) and \\(\\epsilon\\sim{}\\mathcal{N}(0,{\\sigma}^2)\\)\n\\end{center}\n\nWe therefore have our \\(\\dd = {\\{\\xx^i,\\textit{y}^i\\}}_{\\textit{i}=1,..,\\textit{n}}\\) observations (the blue dots), the linear regression obtained \\(\\mathbb{E}(y|x) = \\beta*x + \\beta_0\\) and, for each new \\(x^*\\) point, we can compute the hypothetical \\(y^* = \\mathbb{E}(y|x^*)\\) value and the corresponding error \\(\\epsilon^* \\) and confidence interval given by \\(p(y|x^*)\\) since : \n\\[p(y|\\xx)\\sim{}\\mathcal{N}(f(\\xx|\\beta),{\\sigma}^2)\\]\n\\begin{center}\n\\includegraphics[width=100mm]{LR_2.png}\n\\end{center}\n\n\\subsection{Maximum likelihood estimation under Gaussian noise}\n\nWe are now going to use the \\textit{maximum likelihood estimation} in order to find the best linear regression possible to fit the data and predict new points. \\\\\n\\\\\nUnder Gaussian noise, considering i.i.d. observations and with \\(p(y|\\xx)\\sim{}\\mathcal{N}(f(\\xx|\\beta),{\\sigma}^2)\\), the log-likelihood defined earlier becomes : \n\\begin{align*}\n\\mathcal{L}(\\beta|\\dd) &= \\log\\displaystyle\\prod_{i=1}^{n} p(y^i|\\xx^i) + \\log\\displaystyle\\prod_{i=1}^{n} p(\\xx^i) \\\\ &= \\log \\left(\\displaystyle\\prod_{i=1}^{n} \\frac{1}{\\sqrt{2\\pi}\\sigma}*exp\\left[ - \\frac{(y^i - f(\\xx^i|\\beta))^2}{2\\sigma^2} \\right] \\right ) + Cte \\\\\n& = Cte - \\frac{1}{2\\sigma^2} \\displaystyle\\sum_{i=1}^{n} (y^i - f(\\xx^i|\\beta))^2\n\\end{align*}\nSince \\(\\log\\displaystyle\\prod_{i=1}^{n} p(\\xx^i)\\) is independent of \\(\\beta\\), its value is added to a term \\(Cte\\) along with others. \\\\\nAssuming Gaussian error, maximizing the likelihood \\(\\mathcal{L}(\\beta|\\dd)\\) is therefore equivalent to minimizing the sum of squared residuals \\(\\displaystyle\\sum_{i=1}^{n} (y^i - f(\\xx^i|\\beta))^2\\). \nThis expression of the residual sum of squares can also be written in matrix form : \n\\begin{align*}\nRSS(\\beta) &= \\displaystyle\\sum_{i=1}^{n} (y^i - f(\\xx^i))^2 \\\\ &= \\displaystyle\\sum_{i=1}^{n} \\left( y^i - \\beta_0 - \\displaystyle\\sum_{j=1}^{p} x_j^i \\beta_j \\right)^2 \\\\\n& = (y - \\textbf{\\textit{X}} \\beta)^T * (y - \\textbf{\\textit{X}} \\beta)\n\\end{align*}\nwith \\(\\textbf{\\textit{X}} = \\left( \\begin{array}{ccccc}\n1 & x_1^1 & x_2^1 & ... & x_p^1 \\\\\n1 & x_1^2 & x_2^2 & ... & x_p^2 \\\\\n. & . & . & ... & .\\\\\n. & . & . & ... & .\\\\\n. & . & . & ... & .\\\\\n1 & x_1^n & x_2^n & ... & x_p^n \\end{array} \\right) \\). \nHere, we added a vector \\(\\left( \\begin{array}{c}\n1 \\\\\n1 \\\\\n.\\\\\n. \\\\\n. \\\\\n1 \\end{array} \\right)\\) to \\({\\{\\xx^i\\}}_{\\textit{i}=1,..,\\textit{n}}\\) in \\(\\textbf{\\textit{X}}\\)'s expression in order to compute the scalar \\(\\beta_0\\). \\\\\n\\\\ \nHistorically, the use of the residual sum of squares minimization can be attributed to Carl Friedrich Gauss (to predict the location of Ceres) and Adrien Marie Legendre. \n\\\\\n\\\\\n\\textbf{In this method, under which conditions is \\(\\beta\\)'s estimator unique ? }\n\\\\\n\n\\begin{itemize}\n\\item \nIn order to minimize \\(RSS(\\beta)\\), \\(\\textbf{\\textit{X}}\\) must have a full column rank, hence \\(\\textbf{\\textit{X}}^T\\textbf{\\textit{X}}\\) \\textit{invertible}. In this case, the \\(\\beta\\) minimizing \\(RSS(\\beta)\\) is : \n\\[\\hat{\\beta} = (\\textbf{\\textit{X}}^T\\textbf{\\textit{X}})^{-1} \\textbf{\\textit{X}}^T y\\]\n\\item \nIf \\(\\textbf{\\textit{X}}^T\\textbf{\\textit{X}}\\) is not invertible (or \\textit{rank-deficient}), we can use its pseudo-inverse. However, when doing so, we must keep in mind that the solution is \\textit{not} unique. \\\\ A pseudo-inverse of\\textbf{ A} is a matrix \\textbf{G} such as \\textbf{AGA = A}.\n\\end{itemize}\n \n\n\n\\subsection{Gauss-Markov Theorem}\n\n\\textbf{Gauss-Markov Theorem :} Under the assumption that \\(\\epsilon\\sim{}\\mathcal{N}(0,{\\sigma}^2)\\), the least-squares estimator of \\(\\beta\\) is its best linear unbiased estimator. This Best Linear Unbiased Estimator is unique if \\(\\textbf{\\textit{X}}^T\\textbf{\\textit{X}}\\) is invertible. We call it the \"\\textit{BLUE}\". \n\\\\\n\\\\\nIndeed, with \\(\\hat{\\beta} = (\\textbf{\\textit{X}}^T\\textbf{\\textit{X}})^{-1} \\textbf{\\textit{X}}^T y\\), we can show that (demonstration of the Gauss-Markov Theorem) : \n\\begin{center}\n\\(\\forall \\beta^*\\) unbiased estimator of \\(\\beta\\), \\(\\mathbb{V}ar(\\beta^*) \\geq \\mathbb{V}ar(\\hat{\\beta})\\) and when \\(\\mathbb{V}ar(\\beta^*) = \\mathbb{V}ar(\\hat{\\beta}) => \\beta^* = \\hat{\\beta} \\)\n\\end{center}\n\\vspace{1cm}\n\n\\textbf{Proof of the Gauss-Markov theorem :} \n\\\\ We have \\(\\hat{\\beta} = (\\textbf{\\textit{X}}^T\\textbf{\\textit{X}})^{-1} \\textbf{\\textit{X}}^T y\\) \\textit{BLUE} estimator and let \\(\\beta^* = A y\\) be another linear estimator of \\(\\beta\\). As we're restricting to unbiased estimators, minimum mean squared error implies minimum variance. The goal is therefore to show that such an estimator has a variance no smaller than that of \\(\\hat{\\beta}\\). \nWe have \\(A = (\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D\\) where \\(D\\) is a non-zero matrix. \n\\\\ In order to have \\(\\beta^*\\) unbiased, we compute: \n\\begin{align*}\n\\mathbb{E}[\\beta^*] &= \\mathbb{E}[Ay] \\\\ &= \\mathbb{E}[((\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D)(\\textit{\\textbf{X}}\\beta + \\epsilon)] \\\\\n& = ((\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D)\\textit{\\textbf{X}}\\beta + ((\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D)\\mathbb{E}[\\epsilon] \\\\ &= ((\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D)\\textit{\\textbf{X}}\\beta \\\\ &= (\\textit{\\textbf{I}} + D\\textit{\\textbf{X}})\\beta\n\\end{align*}Since \\(\\mathbb{E}(\\epsilon) = 0\\). Therefore, in order to have an unbiased \\(\\beta^*\\) estimator, we must have \\(DX = 0\\). \\\\\nNow, we calculate and compare the variances :\n\\begin{align*}\n\\mathbb{V}ar(\\hat{\\beta}) &= \\mathbb{E}[((\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T \\epsilon \\epsilon^T \\textit{\\textbf{X}} (\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1}] \\\\ &= (\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T * \\sigma \\textit{\\textbf{I}} * \\textit{\\textbf{X}} (\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\\\\n& = \\sigma^2 (\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \n\\end{align*}\n\\begin{align*}\n\\mathbb{V}ar(\\beta^*) &= \\mathbb{V}ar(Ay) \\\\ &= A\\mathbb{V}ar(y)A^T \\\\\n& = \\sigma^2*AA^T \\\\ &= \\sigma^2*((\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D)(\\textit{\\textbf{X}}^T \\textit{\\textbf{X}})^{-1} \\textit{\\textbf{X}}^T + D)^T \\\\ &= \\sigma^2[(\\textit{\\textbf{X}}^T\\textit{\\textbf{X}})^{-1} + (\\textit{\\textbf{X}}^T\\textit{\\textbf{X}})^{-1}(D\\textit{\\textbf{X}})^T + D(\\textit{\\textbf{X}}(\\textit{\\textbf{X}}^T\\textit{\\textbf{X}})^{-1} + DD^T] \\\\ &= \\sigma^2\\textit{\\textbf{X}}^T\\textit{\\textbf{X}})^{-1} + \\sigma^2 DD^T \\\\ &= \\mathbb{V}ar(\\hat{\\beta}) + \\sigma^2 DD^T\n\\end{align*}since \\(DX = 0\\). \\\\\nTherefore \\(\\mathbb{V}ar(\\beta^*) = \\sigma^2 D D^T + \\mathbb{V}ar(\\hat{\\beta})\\) and since \\(\\sigma^2 D D^T \\) is positive semi-defined and minimal for \\(D = 0\\), \\(\\mathbb{V}ar(\\beta^*)\\) exceeds \\(\\mathbb{V}ar(\\hat{\\beta})\\). \n\n\n\\subsection{Interpretation with correlated or uncorrelated variables}\n\nThe interpretation of the coefficients of a linear regression can be tricky. Indeed, once we have found \\(\\beta_0,...,\\beta_p\\) such as : \n\\[f(\\textbf{\\textit{X}}) = \\beta_0 + \\beta_1 x_1 + ... + \\beta_p x_p\\]\nwe first have to study the correlation of the variables. \n\\begin{itemize}\n\\item If the variables are \\textit{decorrelated}, each coefficient can be estimated separately and the interpretation is thus easy : \"A change in 1 in \\(x_j\\) is associated with a change of \\(\\beta_j\\) in Y, while everything else stays the same.\"\n\\end{itemize}\n\\begin{itemize}\n\\item This interpretation is no longer true if the variables are \\textit{correlated}. The correlations between variables cause problems : the variance of all coefficients tend to increase and the interpretation is much harder (when \\(x_j\\) changes, so does everything else). \n\\end{itemize}\n\n\\section{Logistic regression}\n\n\\paragraph{What about classification ?} \nThere are different conceptual issues with linear regression that make it unsuitable for classification. The main one is that we can't model the probability \\(P(y=1|\\xx)\\) as a linear function of $\\xx$ because it must be between 0 and 1. Moreover, this cannot model \"diminishing returns\" where the impact of a change in $\\xx$ is not constant along the probability range. Indeed, if \\(P(y=1|\\xx)\\) is close to 0 or +1, $\\xx$ must change a lot for y to change and this is not the case when \\(P(y=1|\\xx)\\) is close to 0.5. \\\\\nHence we use a \\textbf{logit transformation} through a \\textit{logistic regression}.\\\\\n\n\\begin{figure}[H]\n\\centering\n\\begin{minipage}[b!]{0.4\\textwidth}\n\\includegraphics[width=70mm]{graphe1.png}\n\\end{minipage}\n\\hfill\n\\begin{minipage}[b!]{0.4\\textwidth}\n\\includegraphics[width=70mm]{graphe2.png}\n\\end{minipage}\n\\end{figure}\n\\begin{center}\n\\(\\log \\dfrac{P(y=1|\\xx)}{1 - P(y=1|\\xx)} = \\beta^T\\xx + \\beta_0\\) with \\(\\beta = (\\beta_0, ..., \\beta_p)\\)\\\\\n\\end{center}\n\n\\paragraph{Maximum likelihood estimation of logistic regression coefficients}\n\nAs usual, let's compute the log likelihood for this logistic regression, knowing our data \\(\\dd = {\\{\\xx^i,\\textit{y}^i\\}}_{\\textit{i}=1,..,\\textit{n}}\\) :\\\\\n\\\\\n\\[\\mathcal{L}(\\beta|\\dd) = \\displaystyle\\sum_{i=1}^{n} log P(y^i|\\xx^i) + Cte = \\displaystyle\\sum_{i=1}^{n} (y^i \\log g^i + (1-y^i) \\log(1-g^i))\\]\n\\begin{center}\nwith \\(g = P(y=1|\\xx) = \\frac{1}{1+e^{-\\beta^T\\xx}}\\) \n\\end{center}\nWe find the maximum of the log-likelihood by calculating its gradient \\(\\nabla_{\\beta}\\mathcal{L}\\): \n\n\\[\\nabla_{\\beta}g^i = \\xx^i g^i (1-g^i)\\]\n\n\\[\\nabla_{\\beta}\\mathcal{L} = \\displaystyle\\sum_{i=1}^{n} (y^i-g^i)\\xx^i\\]\\\\\n\\\\\nAnd solving for zero in the gradient formula :\n\n\\[\\displaystyle\\sum_{i=1}^{n} \\left( y^i - \\frac{1}{1 + e^{-\\beta^T\\xx^i}} \\right) = 0\\]\\\\\n\\\\\nHowever, this expression cannot be solved analytically. \\\\ Since $\\mathcal{L}$ is concave, hence there is no local minima, we can use the gradient ascent method.\n\\\\\n\\paragraph{Gradient ascent method} For a function J concave in $\\beta$ :\n\n\\begin{itemize}\n\\item Update rule : \\(\\beta^{(t+1)} \\leftarrow \\beta^{(t)} + \\eta \\nabla_\\beta J(\\beta^{(t)})\\)\n\\item Iterate until change is inferior to a chosen margin $\\epsilon$\n\\item $\\eta$ is the learning rate\n\\end{itemize}\n\n\\begin{center}\n\\includegraphics[width=100mm]{gradient.png}\n\\end{center}\nOther methods remain possible such as the Newton method, conjugate gradient ascent or IRLS.\n\n\n\\section{Summary}\n\nThe main points to keep in mind from this lesson are the following : \n\\begin{itemize}\n\\item MAP estimate : \\(\\hat{\\theta}_{\\mbox{MAP}}=\\arg\\underset{\\theta}\\max \\ p(\\theta|\\mathcal{X})\\)\n\\item MLE : \\(\\hat{\\theta}_{\\mbox{MLE}}=\\arg\\underset{\\theta}\\max \\ p(\\mathcal{X}|\\theta)\\)\n\\item Bayes estimate : \\(\\hat{\\theta}_{\\mbox{Bayes}}=\\mathbb{E}[\\theta|\\mathcal{X}]=\\int{\\!\\theta p(\\theta|\\mathcal{X})d\\theta}\\)\n\\item Assuming a Gaussian error, maximizing the likelihood (MLE) is equivalent to minimizing the RSS (residual sum of squares). \n\\item Linear regression MLE : \\(\\hat{\\beta} = (\\textbf{\\textit{X}}^T\\textbf{\\textit{X}})^{-1} \\textbf{\\textit{X}}^T y\\) \n\\item Logistic regression : to solve with gradient ascent. \n \\end{itemize}\n\\end{document}\n", "meta": {"hexsha": "aa6fca5b7507e9701f313bb2ae9a1dd320c226f1", "size": 22340, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "lecture_notes/chap05_linreg.tex", "max_stars_repo_name": "taeraa/ma2823_2016", "max_stars_repo_head_hexsha": "b91642e8b3cac059f91d6785f9da6a436b382db8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 19, "max_stars_repo_stars_event_min_datetime": "2016-09-13T16:35:40.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-05T02:16:53.000Z", "max_issues_repo_path": "lecture_notes/chap05_linreg.tex", "max_issues_repo_name": "taeraa/ma2823_2016", "max_issues_repo_head_hexsha": "b91642e8b3cac059f91d6785f9da6a436b382db8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2016-10-03T10:42:43.000Z", "max_issues_repo_issues_event_max_datetime": "2016-10-03T10:43:16.000Z", "max_forks_repo_path": "lecture_notes/chap05_linreg.tex", "max_forks_repo_name": "taeraa/ma2823_2016", "max_forks_repo_head_hexsha": "b91642e8b3cac059f91d6785f9da6a436b382db8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 62, "max_forks_repo_forks_event_min_datetime": "2016-09-07T06:25:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-30T13:21:05.000Z", "avg_line_length": 55.0246305419, "max_line_length": 527, "alphanum_fraction": 0.6714861235, "num_tokens": 7754, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353744, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.34548181972574304}} {"text": "\\documentclass[runningheads]{llncs}\n\\usepackage[hyphens]{url}\n\\usepackage{graphicx}\n\\usepackage{amsmath}\n\n\n\\begin{document}\n\n\\title{Genetic Model Design for 2021 Huawei Delivery Optimization Competition}\n\n\\author{Zhenshuo Chen\\inst{1}\\orcidID{0000-0003-2091-4160} \\and\n Guowen Liu\\inst{2}\\orcidID{0000-0002-8375-5729}}\n\n\\institute{Dublin City University, Dublin, Ireland \\and\n Trinity College Dublin, Dublin, Ireland}\n\n\\maketitle\n\n\n\\begin{abstract}\nWe design a genetic model for 2021 Huawei Delivery Optimization Competition,\nwhich aims to minimize multi-vehicle transportation costs within the constraints of vehicle specifications.\nWe use the one-dimensional array as genotype to represent the transportation routes of multiple vehicles.\nThis representation allows us to directly use existing genetic algorithms for multi-vehicle route optimization.\n\n\\keywords{Genetic Algorithm \\and Multi-vehicle Route Optimization.}\n\\end{abstract}\n\n\n\\section{INTRODUCTION}\nThe competition task is to takes care of deliveries of consumables from a depot to customers, as shown in Fig.~\\ref{fig:overview}.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.8\\linewidth]{figures/overview.png}}\n \\caption{The overview}\n \\label{fig:overview}\n\\end{figure}\n\n\\begin{itemize}\n \\item\n The depot has a given set of home vehicles, each with a certain capacity for carrying consumables.\n\n \\item\n Customers are represented as nodes in a graph, in which edges have an associated transportation cost equal to the distance between the nodes.\n\n \\item\n Each customer node has an associated demand for a consumable.\n\\end{itemize}\n\nThe target is to minimize the total transportation cost by determining a set of routes that meet the following requirements:\n\n\\begin{itemize}\n \\item Each vehicle has one route that starts and finishes at the depot.\n\n \\item Each customer is visited exactly once.\n\n \\item The total demand along each route is less than the vehicle capacity.\n\\end{itemize}\n\nIn a route, the depot is represented by node $1$ and customer numbers start from node $2$.\nFor example, the route $[1, 2, 3, 4, 1]$ means a vehicle starts from the depot ($1$), visits customer nodes $2$, $3$, $4$ and returns to the depot ($1$).\n\n\n\\section{MODEL DESIGN}\nWe used the genetic algorithm as our main process.\nIt simulates the process of natural selection which means those species who can adapt to changes in their environment are able to survive, reproduce and go to next generation as Fig.~\\ref{fig:genetic}.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=\\linewidth]{figures/genetic.jpg}}\n \\caption{A genetic model}\n \\label{fig:genetic}\n\\end{figure}\n\n\\subsection{Individual}\nFor a task containing $N$ customers and $M$ vehicles, we used a permutation of $N + M - 1$ consecutive numbers (0-based) to represent a transportation plan, consisting of both customers and depots.\n$M - 1$ depots can split the customers into $M$ segments. Each segment represents a route.\nFor instance, if a task contains eight customers and three vehicles, a transportation plan can be generated as Fig.~\\ref{fig:individual}.\nIn the example individual, $0, \\ldots, 7$ are customers and $8, 9$ are the depot.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=\\linewidth]{figures/individual.png}}\n \\caption{Using an individual to represent multi-vehicle routes}\n \\label{fig:individual}\n\\end{figure}\n\nThis representation can produce empty routes such as the individual $[3,0,1,2]$ for a task containing three customers and two vehicles.\nThe depot $3$ is in the first position, so the routes are $[1, 1]$ and $[1, 2, 3, 4, 1]$, transformed from an empty segment and $[0,1,2]$.\n\n\\subsection{Mutation}\nIn a mutation operation, we randomly swapped the order of two nodes as Fig.~\\ref{fig:mutation}.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.8\\linewidth]{figures/mutation.png}}\n \\caption{Mutation}\n \\label{fig:mutation}\n\\end{figure}\n\n\\subsection{Crossover}\nIn a crossover operation, we executed ordered crossover on two parents to produce children, as shown in Fig.~\\ref{fig:crossover}.\n\n\\begin{figure}[htbp]\n \\centerline{\\includegraphics[width=0.8\\linewidth]{figures/crossover.png}}\n \\caption{Ordered crossover}\n \\label{fig:crossover}\n\\end{figure}\n\n\\subsection{Selection}\nWe use tournament selection, keeping transportation plans with the highest fitness from randomly chosen candidates.\n\nThe fitness is the cost of a transportation plan.\nFor overloaded plans, a large value is added to fitness as a penalty. This value increases as overload weight increases, see Equation.~\\ref{eq:penalty}.\n\n\\begin{equation}\n Penalty = \\sum_{i}\\sum_{j}c_{i, j} + 2 \\cdot W\n \\label{eq:penalty}\n\\end{equation}\n\n$c_{i, j}$ is the cost between the node $i$ and $j$, and $W$ is overloaded weight of the transportation plan.\n\n\n\\section{IMPLEMENTATION}\nOur implementation is based on \\verb|DEAP| framework \\cite{deap}.\nThe source code is available on the GitHub \\footnote{\\url{https://github.com/Zhuagenborn/Huawei-Delivery-Optimization}}.\n\n\n\\bibliographystyle{splncs04}\n\\bibliography{huawei-delivery-optimization}\n\n\\end{document}", "meta": {"hexsha": "68ff354d26ddb7a67cb2011abc6d004099f2f909", "size": 5175, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper-tex/huawei-delivery-optimization.tex", "max_stars_repo_name": "Zhuagenborn/Huawei-Delivery-Optimization", "max_stars_repo_head_hexsha": "af7ccf077eadda1e40af88a555598e844d16f5db", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-20T15:44:36.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T15:44:36.000Z", "max_issues_repo_path": "paper-tex/huawei-delivery-optimization.tex", "max_issues_repo_name": "Zhuagenborn/Huawei-Delivery-Optimization", "max_issues_repo_head_hexsha": "af7ccf077eadda1e40af88a555598e844d16f5db", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper-tex/huawei-delivery-optimization.tex", "max_forks_repo_name": "Zhuagenborn/Huawei-Delivery-Optimization", "max_forks_repo_head_hexsha": "af7ccf077eadda1e40af88a555598e844d16f5db", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.1162790698, "max_line_length": 201, "alphanum_fraction": 0.7605797101, "num_tokens": 1325, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804196836383, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.3454818118755473}} {"text": "\\subsection{Multi-Agent Proximal Policy Optimization}\n\\label{subsec:41}\n\nSince the naive application of standard reinforcement learning algorithms performs poorly in multi-agent settings, our goal is to derive an algorithm that can operate under the following constraints:\n\\begin{itemize}\n\\item{The learned policies can only use local information (i.e. own observations) at execution time.}\n\\item{There is no particular communication method between the agents, since it can restrict the scalability of the solution.}\n\\item{We do not assume a differential model of the environment dynamics.}\n\\end{itemize}\nFulfilling the above desiderata would provide a general-purpose multi-agent learning algorithm that could be applied not just to cooperative games with explicit communication channels, but also to competitive games and games involving only physical interactions between agents.\n\nWe accomplish our goal by adopting the framework of centralized training with decentralized execution in which agent policies are composed of two separate networks with different parameters: a policy network (actor) that generates an action distribution and a critic network (critic) that predicts discounted future returns. The critic uses extra information to ease training process whilst actors take actions based solely on their own local observations. This is still a valid approach as long as the information used during the training of the agents are not used at test/execution time.\n\n% For figures use\n%\n\\begin{figure}[h!]\n%\\sidecaption\n% Use the relevant command for your figure-insertion program\n% to insert the figure file.\n% For example, with the graphicx style use\n\\includegraphics[scale=.65]{images/MAPPO}\n%\n% If no graphics program available, insert a blank space i.e. use\n%\\picplace{5cm}{2cm} % Give the correct figure height and width in cm\n%\n\\caption{Overview of our multi-agent decentralized actor, centralized critic approach. The centralized action-value function (critic) for each agent takes as input the actions and observations of all agents and predicts the discounted future returns (Q-values) for the agent while the actor produces an action distribution based solely on the agent's observations and actions.}\n\\label{fig:MAPPO} % Give a unique label\n\\end{figure}\n\nAs illustrated in Fig. \\ref{fig:MAPPO}, our centralized action-value function (critic) for each agent takes as input the actions and observations of all agents and predicts the discounted future returns for the agent. Additional information can also be included in the inputs. Since each action-value function is learned separately, each agent can have arbitrary rewards including conflicting rewards in competitive scenarios.\n\nFurthermore, we extend the idea of \\cite{schulman2017proximal} by adopting the proximal policy optimization algorithm (PPO) as the main policy optimization algorithm instead of deep deterministic policy gradient (DDPG) as presented in the paper. PPO is a family of policy optimization methods that use multiple epochs of stochastic gradient ascent to perform each policy update. It performs comparably to or better than DDPG while being much simpler to implement and tune.\n\n\\subsection{Cooperation and Competition}\n\\label{subsec:42}\nAs we seek the emergence of coordination strategies between the components of the micro-grid, the agents controlling the micro-grid components are trained using \\textit{self-play} in an \\textit{autocurricula} configuration. This acts as a natural curriculum as these agents are always in a mixed cooperative and competitive setting. For example, the energy loads (i.e. production machines) of the micro-grid are always in competitive settings. If there is not enough electrical energy available in the grids, then some production machines have to turn themselves off so that the production can be carried on by the other machines. In the same direction, cooperation can take place if, for example, the energy storage components need to be switched to the discharging state in order to increase the electrical energy available on the energy bus and thus enable the production machines to perform additional production tasks.\n\nAdopting such a configuration also offers many advantages. If a new successful strategy is adopted by an agent, it implicitly changes the task distribution for the competing agents which are then forced to develop a better strategy. These evolutionary arms races create an implicit autocurriculum in which competing agents constantly create new tasks for each other and thus discover new powerful and robust strategies and counter-strategies. In addition, the multi-agent autocurriculum is open-ended. That means that the training will only stabilizes once equilibrium between the different agents is founded which can only be reached if a suitable coordination strategy between the agents has emerged. Due to the exploration factors, many coordination strategies can also emerge at different stages of the training.\n\nIn order to create a competitive setting, competing agents are rewarded with competitive rewards. If an agent wins the game, it receives a positive reward and all other competing agents are penalized by an opposite negative reward. If nobody wins the game, then all competing agent are penalized. In a cooperative setting, agents are given a team based reward; meaning that all members of the team receive a positive reward, if a team member wins the game. To confine agent behavior to a reasonable space, we also introduced an environmental-based penalization.\n\nThe success of agents in the autocurricula setting requires the agents to occasionally solve the task (win the game) by random actions. The probability of this happening in most games is minuscule as they require as a prerequisite some fundamental skills like the ability to properly control the operational state machine of the grid component. For example the agent must first quickly learn how to bring a production machine in the $execute$-state in order to start a production task or how to turn on/off the storage component without suspending/halting the production. To overcome this problem, we use simple dense rewards at each step so that the agents can first learn basic motor skills that increase the likelihood that random actions of the agent will result in a relatively small positive reward. We refer to this reward as the \\textit{exploration reward}. The exploration reward is then gradually annealed to zero, in favor of the \\textit{competition/cooperation reward}, to allow the agents to learn coordination strategies for the remaining training time. This is achieved using a linear annealing factor $\\alpha$. We also introduce an \\textit{environmental reward} which aims to penalize action sequences leading to dangerous outcomes (for example system configurations violating the constraints formulated in Eq. \\ref{eq:constraint_energy_balance} and Eq. \\ref{eq:constraint_load_profile}) and a \\textit{productivity reward} which encourage production machines to execute production tasks more frequently. \n\nSo, at time-step $t$, if the exploration reward is $r_{explo}$, the competition reward is $r_{comp}$, the environmental reward $r_{env}$, the the productivity reward $r_{prod}$ and T is the termination time-step, then the total agent reward is:\n\\begin{equation}\n\\label{eq:total_reward}\t\n\tr_t = \\alpha \\cdot r_{explo} + (1 - \\alpha) \\cdot (r_{comp} + r_{env} + r_{prod})\n\\end{equation} \n\nWith this formulation, the agents are trained initially on the dense reward for only about 10-20\\% of the training epochs in order to gain some basic skills first. \n", "meta": {"hexsha": "f50fbfe84855314c6ac6a69011bb657ced44a7db", "size": 7623, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "author/approach.tex", "max_stars_repo_name": "jupiterbak/Artificial-Intelligence-in-Industry-4.0", "max_stars_repo_head_hexsha": "7ddeb55de44c4e50b195edf7a75aa4afb99fcd9e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-06-09T11:05:49.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-09T11:05:49.000Z", "max_issues_repo_path": "author/approach.tex", "max_issues_repo_name": "jupiterbak/Artificial-Intelligence-in-Industry-4.0", "max_issues_repo_head_hexsha": "7ddeb55de44c4e50b195edf7a75aa4afb99fcd9e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "author/approach.tex", "max_forks_repo_name": "jupiterbak/Artificial-Intelligence-in-Industry-4.0", "max_forks_repo_head_hexsha": "7ddeb55de44c4e50b195edf7a75aa4afb99fcd9e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 149.4705882353, "max_line_length": 1520, "alphanum_fraction": 0.8126721763, "num_tokens": 1518, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7248702761768248, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.34545842328678156}} {"text": "\\section{Results}\n\\label{results}\n\nOur Spark program is written in Scala using Hadoop distribution $2.7.2$, and run on a Intel (R) Xeon machine with two physical CPUs. Each CPU has $16$ E$5-2650$ @ $2.00$GHz cores, giving us a total $32$ cores. The machine also has $32$GB of RAM, and runs Ubuntu $16.04$. \n\n\\subsection{Input description}\n\nAs discussed earlier, our input graph $G$ is an undirected, unweighted graph with $679965$ edges and $899162$ vertices. Each vertex and edge is represented using 8 bytes and so the total amount of memory required by our graph is about $20$MB. \n\nOur input graph also consists of $199004$ connected components, with the maximum component size at $145$ and the minimum at $2$. The distribution of component sizes is shown in Fig. \\ref{fig:vertices-components-all}, where the numbers of components with sizes above, say $40$, were dwarfed by the components with smaller sizes. Fig. \\ref{fig:vertices-components-40} (resp. Fig. \\ref{fig:vertices-components-20}) is an alternative figure that reveals the distribution of larger (resp. smaller) components. \n\nTo investigate the scalability of our code for increasing input, we replicate the graph by doubling it (labeled graph $G_2$) then quadrupling it (labeled graph $G_4$), whilst preserving the same structure as far as connected components are concerned.\n\n\\subsection{Failures and connectivity loss analysis}\n\nFor each of the four scenarios described in Sec. \\ref{methods}, we vary the number of threads and partitions, compute the percentage of loss, and plot this value against the number of nodes removed from the graph. In all the tables and figures below, $R$ corresponds to random failures, $D$ to degree based failures, $BC$ to betweenness centrality (load) failures, and $C$ to cascading failures. Fig. \\ref{fig:loss-100} and \\ref{fig:loss-1000} present a refined view demonstrating the progression of loss for the first 100 and 1000 failures on transmission and distribution nodes. Fig. \\ref{fig:loss-all} shows the loss for the entire failures. The connectivity loss is faint and proportional to the number of failures in the random case. It is worse for the cascading scenario, whereas the connectivity losses associated with the load-based versus degree based scenarios are more or less comparable. These findings are confirmed in Table~\\ref{resilience}, which presents, for each scenario, the percentage of failed nodes effecting in $60\\%$ and $80\\%$ (nearly total) connectivity loss. Some of these figures, particularly, the relatively high percentages required in each of the $BC$, $D$, and $R$ scenarios that precede total failure, can be interpreted to say that the Lebanese grid is highly redundant, and thus somehow resilient, thanks to its decomposition into numerous components and its reliance on local diesel generators that alleviate the effects of blackouts and failures. \n\n\n\\begin{table}[t]\n\\centering\n\\caption{Percentage of vertices to be removed to reach $60\\%$ and $80\\%$ losses}\n\\label{tab:threshold-percentage}\n{\\small\n\\begin{tabular}{||c||c|c|c|c||}\n\\hline\n\\textbf{Loss}\t&\\cellcolor{black!10}BC&\t\\cellcolor{black!10}C & \\cellcolor{black!10} R & \\cellcolor{black!10} D \\\\ \\hline \\hline\n60\\% &\t\t11.6\\%&\t4.3\\%\t&59.8\\%&\t7.7\\%\t \\\\ \\hline\t\t\n80\\%\t&\t\t27.8\\%&\t14.6\\%\t&79.7\\%&\t29.5\\%\t \\\\ \\hline\t\t\n\\end{tabular}\n}\n\\label{resilience}\n\\end{table}\n\n\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/loss-100-crop.pdf}\n\\caption{Loss percentage: first 100 attacks}\n\\label{fig:loss-100}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/loss-1000-crop.pdf}\n\\caption{Loss percentage: first 1000 attacks}\n\\label{fig:loss-1000}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/loss-all-crop.pdf}\n\\caption{Loss percentage: overall attacks}\n\\label{fig:loss-all}\n\\end{figure}\n\n\\subsection{Run-time and parallel efficiency analysis}\n\\secondedited{\nThe run-time for each pair of ({\\it thread},{\\it partition}) values using our input graphs are shown in the tables~\\ref{tab:graph1}, \\ref{tab:graph2}, and \\ref{tab:graph4}, for $G$, $G_2$, and $G_4$ respectively. Moreover, the corresponding performance efficiency plots are shown in Fig. \\ref{fig:effbc1}, \\ref{fig:effbc2}, \\ref{fig:effbc4}, \\ref{fig:effc1}, \\ref{fig:effc2}, \\ref{fig:effc4}, \\ref{fig:effd1}, \\ref{fig:effd2}, \\ref{fig:effd4}, \\ref{fig:effr1}, \\ref{fig:effr2}, and \\ref{fig:effr4}. \n}\n%\n\\edited{\nHere, parallel efficiency is defined as $\\frac{T_s}{p\\cdot T_p}$, where $T_s$ denotes the serial run-time and $T_p$ denotes the parallel run-time given $p$ parallel processes.}\n%\n\\secondedited{\nThe timings shown in the tables correspond to the parallel phase of the algorithm. The sequential run-time was omitted because it was extremely negligible compared to the parallel part that is of higher order (in contrast to linear running time in steps 7 and 8).\n%\nBy default, Spark sets the partition size at 64 MB. This is too large for our given graph. As a result, we choose to override the default value by specifying the number of partitions at compile time. When this is done, Spark re-adjusts the size of each partition based on their total number as well as on the size of the input graph. The several connected components in each graph get mapped onto all partitions such that the number of vertices in each partition is balanced across all partitions. We experiment with a number of partitions ranging from $4$, $8$, $16$, and $32$ in order to explore the effect that the number of partitions have on run-time and parallel efficiency. From Fig. \\ref{fig:vertices-components-all} and \\ref{fig:vertices-components-40}, we gather that there would be enough connected components in each partition to engage all threads assigned to the partition. Also, the fact that many connected components have sizes greater than, say, 40, ensures that the distributed work is more or less balanced.}\n\nAs expected, for all three graphs, the fastest and yet the worst parallel efficiency correspond to the R (random failures) scenario that does not rely on any centrality measure computation. In contrast, the highest run-times and hence best efficiency are for the C (cascading scenario) that updates the betweenness centrality of all the nodes following each node removal. It is clear that the more work entailed by a certain scenario, the better it will be for the parallel program to compensate for the overheads associated with the partitioning and scheduling operations. Reading across the three tables for the same partition size and same scenario, our tool shows improved scalability as the input size grows. \n\nFor each given graph and for each scenario employed, we notice that the actual run-time has opposing trends that depend on the number of partitions. For all three tables, there is a cut-off value for the number of partitions, before which run-time continues to improve, and after which it starts to deteriorate. For graph $G$, the cut-off number is at $8$, for graph $G_2$ it is somewhere between $8$ and $16$, and for $G_4$, it is at $16$. We justify the improvement in run-time as we move closer to the cut-off threshold as follows. With a higher number of partitions, one would expect that the multiple threads will be spread about, sharing the work but on data that is more split into distinct regions of main memory. As such, we have reduced contention over the shared address space, as well as scheduling costs associated with managing threads on one single partition. We now argue that creating more partitions beyond the cut-off number drastically affects the performance and introduces a huge overhead -- particularly when the number of threads is low. For instance, in Table II, if we consider only one thread of the BC scenario, it takes 161 (resp. 423) seconds in case of 4 (resp. 32) partitions. We attribute this overhead to the shuffling and repartition operations taking place at the end of each stage that assigns one or more threads from one partition to another. In that phase, some transformations (e.g., \\texttt{groupByKey}, \\texttt{reduceByKey}, \\texttt{join} operations) require shuffling and repartitioning of data, for instance, to group all the items with the same keys in the same partition. \n\nWith that said, we observe that efficiency actually improves for larger partitions. This isn't to be construed, however, to mean that the parallelisation has improved in any way, but rather that the rate of deterioration in the performance for a smaller number of threads (particularly, the case of one thread) is higher when the number of partitions exceeds the cut-off number, resulting in a higher-speedup as the number of threads grows larger. \n\n%Our own Spark implementation of contingency analysis executes in real-time for the entire power grid, achieving a speed-up of ? on ? processors. We conclude with a spatial understanding of the hotspots on Lebanese soil where energy centers can be exposed and are at risk, using a spatial correlation supported by a binary search tree. The amenability of our work to big data processing makes it extendable to larger networks of networks, towards a fuller understanding of resilence at many vital levels beyond the power grid. Examples are as communications network, Internet networks, transportation networks, hospitals and medical centers networks, to name a few.\n\n\\secondedited{\nMoreover, we notice that the parallel efficiency in some cases is above one (e.g., in case of betweenness centrality 32 partitions and 2 threads). This may due to the effect of the garbage collectors. In case of one thread and several partitions, the threshold of the garbage collector would be reached ahead of time. Whereas in case of more threads/processes, less data needs to be freed.\n%\nAdditionally, another cause behind super-linear speedup can be attributed to the ``caching effect'', which results from the varying speeds in accessing different levels of the memory hierarchy on which the input graph and intermediate data are stored. As the number of threads increases, the data assigned to each thread decreases, rendering it into the smaller cache levels which are faster to access. As a result, the reduction in run-time is not solely explained by the increase in the number of working threads but also in the reduction of the time spent on I/O, which causes the theoretical estimate for parallel speedup to go beyond $p$ (given $p$ threads), or equivalently, for parallel efficiency to go beyond 1.}\n\n%\\begin{figure}\n%\\centering\n%\\includegraphics[scale=0.35]{bench/generated/frequencyall-crop.pdf}\n%\\caption{Components size -- overall distribution}\n%\\label{fig:cs-overall}\n%\\end{figure}\n\n%\\begin{figure}\n%\\centering\n%\\includegraphics[scale=0.35]{bench/generated/frequency-selected-crop.pdf}\n%\\caption{Components size -- cropped distribution}\n%\\label{fig:cs-selected}\n%\\end{figure}\n\n\\begin{table*}[t]\n\\begin{minipage}[b]{\\textwidth}\n\\caption{Run-time \\edited{(in seconds)} for graph $G$}\n\\label{tab:graph1}\n{\\small\n\\centering\n\\begin{tabular}{||c||c|c|c|c||c|c|c|c||c|c|c|c||c|c|c|c|}\n\\hline\n\\textbf{Threads}\t&\\cellcolor{black!10}BC-4&\t\\cellcolor{black!10}BC-8\t&\\cellcolor{black!10}BC-16\t&\\cellcolor{black!10}BC-32&\t\\cellcolor{black!10}C-4\t&\\cellcolor{black!10}C-8&\t\\cellcolor{black!10}C-16&\t\\cellcolor{black!10}C-32&\t\\cellcolor{black!10}D-4&\t\\cellcolor{black!10}D-8\t&\\cellcolor{black!10}D-16&\\cellcolor{black!10}\tD-32&\t\\cellcolor{black!10}R-4\t&\\cellcolor{black!10}R-8&\t\\cellcolor{black!10}R-16&\t\\cellcolor{black!10}R-32 \\\\ \\hline \\hline\n64&\t\t27\t&17\t&22\t&41\t&53\t&34\t&32\t&52\t&26\t&15\t&20\t&41\t&23\t&17\t&20\t&39\t\t\t\\\\ \\hline\t\t\t\t\t\n32&\t\t24\t&17\t&19\t&37\t&52\t&33\t&31\t&49\t&20\t&16\t&18\t&36\t&20\t&19\t&19\t&36\t\\\\ \\hline\n16&\t\t24\t&18\t&22\t&39\t&53\t&35\t&37\t&54\t&21\t&16\t&19\t&37\t&19\t&16\t&20&37\t\\\\ \\hline\n8\t&\t25\t&25\t&30\t&54\t&55\t&50\t&56\t&80\t&21\t&21\t&26\t&51\t&20\t&20\t&26\t&50\t\\\\ \\hline\n4\t&\t41\t&41\t&53\t&97\t&87\t&86\t&98\t&138\t&33\t&33\t&45\t&89\t&30\t&31\t&44\t&90\t\\\\ \\hline\n2&\t\t82\t&84\t&108\t&206\t&165\t&165\t&189\t&287\t&65\t&66\t&92\t&188\t&55\t&62\t&87\t&186\t\\\\ \\hline\n1&\t\t161\t&167\t&219\t&423\t&312\t&320\t&369\t&543\t&124\t&127\t&187\t&384\t&104\t&114\t&174\t&344\t\\\\ \\hline\n\\end{tabular}\n}\n\\end{minipage}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\\begin{minipage}[b]{\\textwidth}\n\\caption{Run-time \\edited{(in seconds)} for graph $G_2$}\n\\label{tab:graph2}\n{\\small\n\\begin{tabular}{||c||c|c|c|c||c|c|c|c||c|c|c|c||c|c|c|c|}\n\\hline\n\\textbf{Threads}\t&\\cellcolor{black!10}BC-4&\t\\cellcolor{black!10}BC-8\t&\\cellcolor{black!10}BC-16\t&\\cellcolor{black!10}BC-32&\t\\cellcolor{black!10}C-4\t&\\cellcolor{black!10}C-8&\t\\cellcolor{black!10}C-16&\t\\cellcolor{black!10}C-32&\t\\cellcolor{black!10}D-4&\t\\cellcolor{black!10}D-8\t&\\cellcolor{black!10}D-16&\\cellcolor{black!10}\tD-32&\t\\cellcolor{black!10}R-4\t&\\cellcolor{black!10}R-8&\t\\cellcolor{black!10}R-16&\t\\cellcolor{black!10}R-32 \\\\ \\hline \\hline\n64\t\t&127\t&71\t&48\t&55\t&119\t&64\t&60\t&76\t&108\t&58\t&45\t&53\t&45\t&32\t&31\t&50\t\t \\\\ \\hline\t\t\n32\t\t&120\t&59\t&45\t&49\t&116\t&68\t&56\t&71\t&108\t&57\t&42\t&47\t&40\t&29\t&29\t&45\t\t \\\\ \\hline\t\t\t\n16\t\t&128\t&66\t&39\t&52\t&115\t&66\t&65\t&79\t&112\t&57\t&36\t&50\t&41\t&33\t&30\t&46 \\\\ \\hline\t\t\t\t\n8\t\t&123\t&66\t&56\t&77\t&120\t&97\t&98\t&121\t&106\t&54\t&46\t&68\t&40\t&34\t&39\t&63\t \\\\ \\hline\t\t\t\n4\t\t&164\t&104\t&101\t&142\t&210\t&191\t&185\t&223\t&134\t&83\t&80\t&119\t&60\t&57\t&69\t&113\t \\\\ \\hline\t\t\t\n2\t\t&263\t&203\t&207\t&292\t&416\t&353\t&366\t&462\t&209\t&159\t&153\t&239\t&111\t&110\t&131\t&224\t \\\\ \\hline\t\t\n1\t\t&452&357\t&371\t&525\t&761\t&804\t&793\t&969\t&355\t&281\t&286\t&434\t&196\t&201\t&230\t&402 \\\\ \\hline\t\t\n\\end{tabular}\n}\n\\end{minipage}\n\\end{table*}\n\n\\begin{table*}[t]\n\\begin{minipage}[b]{\\textwidth}\n\\caption{Run-time \\edited{(in seconds)} for graph $G_4$}\n\\label{tab:graph4}\n{\\small\n\\begin{tabular}{||c||c|c|c|c||c|c|c|c||c|c|c|c||c|c|c|c|}\n\\hline\n\\textbf{Threads}\t&\\cellcolor{black!10}BC-4&\t\\cellcolor{black!10}BC-8\t&\\cellcolor{black!10}BC-16\t&\\cellcolor{black!10}BC-32&\t\\cellcolor{black!10}C-4\t&\\cellcolor{black!10}C-8&\t\\cellcolor{black!10}C-16&\t\\cellcolor{black!10}C-32&\t\\cellcolor{black!10}D-4&\t\\cellcolor{black!10}D-8\t&\\cellcolor{black!10}D-16&\\cellcolor{black!10}\tD-32&\t\\cellcolor{black!10}R-4\t&\\cellcolor{black!10}R-8&\t\\cellcolor{black!10}R-16&\t\\cellcolor{black!10}R-32 \\\\ \\hline \\hline\n64 &\t\t525&\t317\t&171&\t117\t&411\t&179&\t127\t&123\t&430\t&220\t&134\t&104&\t133&\t79&\t70\t&87\t \\\\ \\hline\t\t\n32\t&\t491&\t274\t&153\t&99\t&387\t&202\t&123\t&121 &449\t&243&\t149&\t100\t&122\t&101&\t64&\t75\\\\ \\hline\n16\t&\t473\t&311\t&133\t&96\t&401\t&184\t&127\t&137 &457\t&236\t&113\t&77\t&130\t&86\t&62\t&78 \\\\ \\hline\n8\t&\t557&\t250\t&132\t&126\t&387\t&234\t&204\t&212 &452\t&222\t&113\t&105\t&119&\t93\t&75\t&93 \\\\ \\hline\n4\t\t&750&\t333\t&221\t&234\t&612\t&431\t&374\t&398 &601\t&274 &168\t&190\t&151\t&126&\t120\t&163 \\\\ \\hline\n2\t\t&1041 &526 &430 &467\t&1042\t&791\t&736\t&793 &888\t&436\t&344\t&395\t&273\t&235\t&237\t&337 \\\\ \\hline\n1\t\t&1626&902&879 &1021 &2156 &1784 &1577 &1754 &1584 &905 &730 &783& 533 &458 &456 &647 \\\\ \\hline\n\\end{tabular}\n}\n\\end{minipage}\n\\end{table*}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-bc-1-crop.pdf}\n\\caption{Efficiency for the load based (BC) scenario and graph $G$}\n\\label{fig:effbc1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-c-1-crop.pdf}\n\\caption{Efficiency for the cascading (C) scenario and graph $G$}\n\\label{fig:effc1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-d-1-crop.pdf}\n\\caption{Efficiency for the degree-based (D) scenario and graph $G$}\n\\label{fig:effd1}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-r-1-crop.pdf}\n\\caption{Efficiency for the random (R) scenario and graph $G$}\n\\label{fig:effr1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-bc-2-crop.pdf}\n\\caption{Efficiency for the load based (BC) scenario and graph $G_2$}\n\\label{fig:effbc2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-c-2-crop.pdf}\n\\caption{Efficiency for the cascading (C) scenario and graph $G_2$}\n\\label{fig:effc2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-d-2-crop.pdf}\n\\caption{Efficiency for the degree-based (D) scenario and graph $G_2$}\n\\label{fig:effd2}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-r-2-crop.pdf}\n\\caption{Efficiency for the random (R) scenario and graph $G_2$}\n\\label{fig:effr2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-bc-4-crop.pdf}\n\\caption{Efficiency for the load based (BC) scenario and graph $G_4$}\n\\label{fig:effbc4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-c-4-crop.pdf}\n\\caption{Efficiency for the cascading (C) scenario and graph $G_4$}\n\\label{fig:effc4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-d-4-crop.pdf}\n\\caption{Efficiency for the degree-based (D) scenario and graph $G_4$}\n\\label{fig:effd4}\n\\end{figure}\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.35]{bench/generated/efficiency-r-4-crop.pdf}\n\\caption{Efficiency for the random (R) scenario and graph $G_4$}\n\\label{fig:effr4}\n\\end{figure}\n\n\n\n\\subsection{Spatial analysis}\nThe spatial correlation analysis shown in Fig. \\ref{fig:correlation} reveals a long range correlated case with correlation decaying slowly with distance, particularly, in the linear regime where $C(r) \\propto r^{-\\gamma} $, $\\gamma = 1.13$, which is in agreement with the literature results in \\cite{DaqingAl14}. Overload failures usually propagate through collective interactions among system components. Our results reveal that high failures in critical nodes has impact that propagates across long path lengths on Lebanese soil.\n\n\\begin{figure}\n\\centering\n\\includegraphics[scale=0.65]{bench/fixed/correlation-eps-converted-to.pdf}\n\\caption{Spatial correlation between nodes in the cascading scenario on a logarithmic scale.}\n\\label{fig:correlation}\n\\end{figure}\n\n\n\n\n", "meta": {"hexsha": "85671cf2949510f61c397e701567ea5dcf6bf8c7", "size": 17996, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/journal-tcss/results.tex", "max_stars_repo_name": "okm02/Power-grid-analysis", "max_stars_repo_head_hexsha": "1c24a2c8bcdedd04d3e63f2db7abfa6ab135a107", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tex/journal-tcss/results.tex", "max_issues_repo_name": "okm02/Power-grid-analysis", "max_issues_repo_head_hexsha": "1c24a2c8bcdedd04d3e63f2db7abfa6ab135a107", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/journal-tcss/results.tex", "max_forks_repo_name": "okm02/Power-grid-analysis", "max_forks_repo_head_hexsha": "1c24a2c8bcdedd04d3e63f2db7abfa6ab135a107", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 68.6870229008, "max_line_length": 1619, "alphanum_fraction": 0.7438319627, "num_tokens": 5743, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795672, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.34533680814633055}} {"text": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% # TO DO LIST\n% \n% * read again and put all the citations needed when something is stated \n% without demonstration (a lot of times...)\n% * find a reference for the duality theory (other that Bourbaki ?)\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\documentclass[a4paper,11pt]{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage[english]{babel}\n\\usepackage{amsmath,amssymb,amsthm,amsopn}\n\\usepackage{mathrsfs}\n%\\usepackage{stmaryrd} % to get <--| \\mapsfrom\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{tikz}\n\\usepackage{array}\n%\\usepackage[top=1cm,bottom=1cm]{geometry}\n%\\usepackage{listings}\n%\\usepackage{xcolor}\n\\usepackage{bm}\n\\usepackage{bbm}\n\n% Création des labels Théorème, Lemme, etc...\n\n\\newtheoremstyle{break}%\n{}{}%\n{\\itshape}{}%\n{\\bfseries}{}% % Note that final punctuation is omitted.\n{\\newline}{}\n\n\\theoremstyle{break}\n\\newtheorem{thm}{Theorem}[section]\n\\newtheorem{lm}[thm]{Lemma}\n\\newtheorem{prop}[thm]{Proposition}\n\\newtheorem{cor}[thm]{Corollary}\n\n\\theoremstyle{definition}\n\\newtheorem{defi}[thm]{Definition}\n\\newtheorem{ex}[thm]{Example}\n\n\\theoremstyle{remark}\n%\\newtheorem{rem}[thm]{Remarque}\n\n% Raccourcis pour les opérateurs mathématiques (les espaces avant-après sont\n% modifiés pour mieux rentrer dans les codes mathématiques usuels)\n% \\deg already exists !!\n\\DeclareMathOperator{\\Ker}{Ker}\n\\DeclareMathOperator{\\Id}{Id}\n\\DeclareMathOperator{\\Img}{Im}\n\\DeclareMathOperator{\\Card}{Card}\n\\DeclareMathOperator{\\Vect}{Vect}\n\\DeclareMathOperator{\\rev}{rev}\n\\DeclareMathOperator{\\rem}{rem}\n\n\\DeclareBoldMathCommand{\\bxi}{\\xi}\n\\DeclareBoldMathCommand{\\bupsilon}{\\upsilon}\n\\DeclareBoldMathCommand{\\bzeta}{\\zeta}\n\n% Nouvelles commandes\n\\newcommand{\\ps}[2]{\\langle#1,#2\\rangle}\n\\newcommand{\\ent}[2]{[\\![#1,#2]\\!]}\n\\newcommand{\\psdot}{\\ps{\\cdot}{\\cdot}}\n\\newcommand{\\remt}{\\rem^t}\n\n% opening\n\\title{Internship report}\n\\author{Édouard \\textsc{Rousseau}\\\\edouard.rousseau@u-psud.fr\\\\ \\\\Internship \nsupervised by Luca \\textsc{De \nFeo}}\n\n\\usepackage{tikz-cd}\n\\usetikzlibrary{matrix, calc, arrows}\n\n\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\n\n This internship took place during July and August 2016, the aim was to study\n the algorithms presented in~\\cite{DeDoSc14}, and to implement them in\n Nemo~\\cite{Nemo}, a computer algebra package for the Julia~\\cite{Julia} programming\n language. These algorithms were designed to deal with the algebraic closure \n of finite fields, and often have a quasi-linear cost, whereas traditional \n algorithms have quadratic cost. This paper intends to compare the Julia \n and the C implementations of the code, in terms of speed, but also genericity \n and ease of writing and reading such code. As we will see, writing code in \n Julia is really easy, and the tools available to develop the code are \n sharp, but the speed of the implementation in Julia does not match the speed \n of the C implementation.\n\n\\end{abstract}\n\n\\tableofcontents\n\n\\clearpage\n\n\\section{Julia}\n\\subsection{Overview of Julia's caracteristics}\nJulia is a free and open-source, high-level programming language developed since \n2012, with dynamic\ntype system and high-performance. It is a compiled language, with a\n\\emph{just-in-time} (jit) compilation. This means that when one writes a\nfunction, e.g. \n\\begin{verbatim}\njulia> function myFunction(n)\n return n^3+n+2\n end\n\\end{verbatim}\nthe function will be compiled during the very first call. After this first\ncall, it will be much faster to run the compiled function\n\\begin{verbatim}\njulia> myFunction(BigInt(2)^300)\n\\end{verbatim}\nthan the expression below.\n\\begin{verbatim}\njulia> (BigInt(2)^300)^3 + BigInt(2)^300 + 2\n\\end{verbatim}\nIn order to optimize the code, the compiler creates a new function for each\ntype of input that one can think of when calling myFunction. For example, in the\ncode \nbelow, Julia will compile myFunction three times, which will result in\nthree functions, each one of them optimized for \\texttt{Int64} (the default type\nof $17$), \\texttt{Float64}, or \\texttt{BigInt}.\n\\begin{verbatim}\njulia> myFunction(17)\n4932\njulia> myFunction(17.)\n4932.0\njulia> myFunction(BigInt(17))\n4932\n\\end{verbatim}\nJulia is a very easy-to-learn language, the syntax is intuitive for someone who\nalready worked with high-level langages, like Python.\nThe code written in \\href{https://github.com/edouardRousseau/FastArithmetic.jl}{FastArithmetic.jl} is\nvery generic, thanks to the type system of Julia, and works for fields\n$\\mathbb{F}_{p^n}$ with a small $p$ as well as large $p$. More importantly, the\n\\emph{jit} compiler creates a function for each kind of $p$. Indeed the elements\nof $\\mathbb{F}_{p^n}$ have type \\texttt{fq\\_nmod} for $p$ small or \\texttt{fq}\nfor $p$ large, so one can write only one function and have two compiled\nfunctions: one for each type, which will be optimized by the compiler for this \ntype.\n\n\\subsection{Nemo}\n\nNemo is a computer algebra package of Julia, it can be installed with \n\\begin{verbatim}\njulia> Pkg.add(\"Nemo\")\njulia> using Nemo\n\\end{verbatim}\nand tested with the following.\n\\begin{verbatim}\njulia> Pkg.test(``Nemo'')\n\\end{verbatim}\nNemo is based on C/C++ libraries such as Flint, Antic, Pari, etc. and is also\nwritten in Julia. It provides a lot of features, and all the work of this\ninternship is based on Nemo. Indeed, we work with Nemo finite fields and Nemo\npolynomials over these fields. \\texttt{fq}, \\texttt{fq\\_nmod}, and\n\\texttt{fq\\_nmod\\_poly} (the type of the polynomials over $\\mathbb{F}_{p^n}$) \nare\nall Nemo types. The interested reader can learn more in Nemo's manual, which is very well\ndocumented.\n\n\\section{A bit of theory}\n\\subsection{General background}\nIn most of the computer algebra systems, like Sage, Magma, Flint, etc., it is \npossible to work with finite\nfields and their algebraic closure. Let $p$ be a prime number and\n$k=\\mathbb{F}_p$ the field with $p$ elements, the algebraic closure of $k$ is\ninfinite and is $\\bar k = \\cup_{i\\geq 0} \\mathbb{F}_{p^i}$, dealing with $\\bar\nk$ means being able to compute the finite fields $\\mathbb{F}_{p^i}$, and also\nbeing able to embed small fields into large ones, or project elements of large\nfields into smaller ones (when possible).\n\nThese algorithms often rely on linear algebra, and the cost is at least \nquadratic in the degree of the extension used. The algorithms presented\nin~\\cite{DeDoSc14} rely on polynomial arithmetic, and have a quasi-linear cost for \nmost of the algorithms needed to\ncompute in the algebraic closure of a finite field. In this section, we will\nintroduce (briefly) the tools used in the algorithms, and explain how they work.\nThe most important result is the \\emph{transposition principle}, an\nalgorithmic proposition that guarantees that with any linear algorithm\nperforming a matrix-vector product $v\\mapsto Mv$, one can \\emph{transpose} it to\nobtain another algorithm, which has essentially the same cost, performing the \ntransposed matrix-vector product $v\\mapsto\nM^tv$. But we first need to talk about \\emph{duality}, because the transposition\nprinciple also changes the used basis. \n\\subsection{Trace and duality}\nLet $k$ be a finite field, and $E,F$ be two $k$-vector spaces with the same\nfinite dimension $\\dim E = \\dim F < \\infty$. Let also\n$\\ps{\\cdot}{\\cdot}:E\\times F\\rightarrow k$ be a non-degenerate bilinear form. \nThen,\nfor any basis $\\bxi=(\\xi_i)_i$ of $E$, there exists a unique basis\n$\\bxi^*=(\\xi_j^*)_j$ of $F$\nsuch that $\\forall (i,j),\\ps{\\xi_i}{\\xi_j^*}=\\delta_{i,j}$ where $\\delta_{i,j}$ \nis\nthe \\textsc{Kronecker} symbol and $\\delta_{i,j}=1$ if $i=j$ and 0 otherwise.\nThe basis $\\bxi^*$ is called the \\emph{dual basis} of $\\bxi$.\nSimilarly, let $E',F'$ be two other $k$-vector spaces with $\\dim E' = \\dim F' <\n\\infty$ and $\\ps{\\cdot}{\\cdot}':E'\\times F'\\rightarrow k$ another \nnon-degenerate\nbilinear form. Then, for any linear map $u:E\\rightarrow E'$, there exists a\nunique map $u^t:F'\\rightarrow F$ such that $\\forall (a,b)\\in E\\times\nF',\\ps{u(a)}{b}'=\\ps{a}{u^t(b)}$. This map is called the \\emph{dual} map of $u$,\nwith respect to $\\ps{\\cdot}{\\cdot}$ and $\\ps{\\cdot}{\\cdot}'$.\n\nIt is now time to speak about the \\emph{trace form}, it will be our candidate to \nbe \n$\\ps{\\cdot}{\\cdot}$, the non-degenerate bilinear form. Let $P$ be an irreducible\npolynomial of $k[x]$ of degree $m$, let $K=k[x]/(P)$ be a finite\nextension of $k$, and $a\\in K$. We denote by $\\mu_a$ the application of\nmultiplication-by-$a$, $\\mu_a:K\\rightarrow K, b\\mapsto ab$. $K$ is a $k$-vector\nspace of dimension $m$ and $\\mu_a$ is an endomorphism of $K$. For all \n$a,b\\in K$, we also denote by $\\tau_P(ab)$ the trace of the endomorphism \n$\\mu_{ab}$, and we define\n$\\ps{a}{b}_P=\\tau_P(ab)$. This defines a non-degenerate bilinear form on \n$K\\times\nK$. Thus, if $\\bxi$ is a basis of $K$, one can obtain a dual basis $\\bxi^*$ with\nrespect to the bilinear form $\\ps{\\cdot}{\\cdot}_P$. Hence, if $a\\in K$ has the\ncoordinates $(a_i)_i$ in the basis $\\bxi^*$, these coordinates are given by\n$a_i=\\ps{\\xi_i}{a}$, so $a=\\sum a_i\\xi_i^*=\\sum\\ps{\\xi_i}{a}\\xi_i^*$. This\nformula will be important for the change-of-basis algorithms.\n\nLet now $Q$ be an irreducible polynomial of $k[y]$ of degree $n$, with $n$\ncoprime to $m$, then $L=k[x,y]/(P,Q)\\cong \\mathbb{F}_{p^{mn}}$ is also a finite \nextension of $k$, of\ndegree $mn$. Since $L$ is a $k$-vector space (of dimension $mn$), it is again\npossible to construct the trace $\\tau_{P,Q}$ of an element of $L$, and we define \nthe same way\na non-degenerate bilinear form $\\psdot_{P,Q}$ on $L\\times L$. If\n$\\bxi=(x^i)_{0\\leq i\\leq m-1}$ and $\\bupsilon=(y^j)_{0\\leq j \\leq n-1}$ are the\nmonomial bases of respectively $k[x]/(P)$ and $k[y]/(Q)$, then\n$\\bxi\\otimes\\bupsilon=(x^iy^j)_{0\\leq i \\leq m-1,0\\leq j \\leq n-1}$ is the\ncannonical monomial basis of $L$. But, if $\\bxi^*$ and $\\bupsilon^*$ are\nrespectively the dual basis of $\\bxi$ with respect to $\\tau_P$ and the dual\nbasis of $\\bupsilon$ with respect to $\\tau_Q$, $\\bxi\\otimes\\bupsilon^*$, \n$\\bxi^*\\otimes\\bupsilon$, and $\\bxi^*\\otimes\\bupsilon^*$, are also bases of $L$.\nWhat's more, $\\bxi\\otimes\\bupsilon$ and $\\bxi^*\\otimes\\bupsilon^*$ are dual\nwith respect to $\\tau_{P,Q}$, and so are $\\bxi\\otimes\\bupsilon^*$ and\n$\\bxi^*\\otimes\\bupsilon$.\n\nConstructing\n$\\mathbb{F}_{p^{mn}}$ this way leads to a bivariate reprentation of the\nelements. We prefer a univariate representation, because the algorithms we have\nin this case are more efficient, even for simple operations such as \nmultiplication. Hence, we can represent $\\mathbb{F}_{p^{mn}}$ by $k[z]/(R)$\nwhere $R=P\\odot Q$ is the composed product of $P$ and $Q$. It means that if\n$(p_i)_{0\\leq i \\leq m-1}$ and $(q_j)_{0\\leq j\\leq n-1}$ are respectively the\nroots of $P$ and $Q$ in a algebraic closure of $k$, $R$ is the polynomial whose\nroots are the products $p_iq_j$. So we have embeddings $\\varphi_x$, $\\varphi_y$\nand an isomorphism $\\Phi$ of the form:\n\\[\n\\begin{array}{crcl}\n \\varphi_x: & k[x]/(P) & \\to & k[z]/(R),\\\\[2mm]\n \\varphi_y: & k[y]/(Q) & \\to & k[z]/(R),\\text{ and}\\\\[2mm]\n \\Phi:& A=k[x,y]/(P,Q) & \\to & k[z]/(R) \\\\\n & xy & \\mapsto & z.\n\\end{array}\n\\]\n\nWe will provide algorithms for $\\varphi_x$, $\\varphi_y$ and $\\Phi$ in Section~\\ref{sec-algo}.\n\n\\subsection{Transposition principle}\nLet, as in the previous section, $\\bxi$ be a basis of $E$, and $\\bxi^*$ the\nbasis of $F$ which is dual to $\\bxi$. Let also $\\bupsilon$ and $\\bupsilon^*$ be, respectively,\na\nbasis of $E'$ and its dual basis of $F'$. If $M$ is the matrix of a linear map\n$u:E\\rightarrow E'$ in the\nbases $(\\bxi,\\bupsilon)$, then the matrix of the linear map $u^t:F'\\rightarrow \nF$ in the bases\n$(\\bupsilon^*,\\bxi^*)$ is $M^t$. Given an algorithm to compute a linear map\n$u:E\\rightarrow E'$ in the bases $(\\bxi,\\bupsilon)$, the transposition principle\nsays that one can transpose it to obtain a new algorithm computing the dual map \n$u^t:F'\\rightarrow F$ in the bases\n$(\\bupsilon^*,\\bxi^*)$. The costs of the two algorithms only differ by a\nconstant. Roughly speaking, transposing an algorithm consists on transposing all\nthe subroutines and inverse their orders.\n\n\\begin{ex}\nLet $A$ be an algorithm taking $x_1$ and $x_2$ as input, and computing \n$y_1=ax_1+bx_2$ and $y_2=cx_1+dx_2$. $A$ is linear and we obtain its \ntransposition $A^t$ by reversing all the arrows and exchanging $\\times$ and $+$ \nin the graph representing $A$. We obtain an algorithm that takes $y_1$ and \n$y_2$ as input and compute $x_1=ay_1+cy_2$ and $x_2=by_1+dy_2$.\n\n\\begin{figure}[!h]\n\\begin{tikzpicture}\n \\matrix[matrix of math nodes,column sep={60pt,between origins},row\n sep={60pt,between origins},nodes={asymmetrical rectangle}] (s)\n {\n &|[name=x1]| x_1 &|[name=1]| \\times &|[name=2]| + & |[name=y1]| y_1\\\\\n %\n &|[name=x2]| x_2 &|[name=3]| \\times &|[name=4]| + &|[name=y2]| y_2 \\\\\n };\n \\draw[->,>=latex] (1) edge node[auto] {\\(a\\)} (2)\n \t\t\t(3) edge node[auto] {\\(d\\)} (4)\n (1) edge (4)\n (3) edge (2)\n (x1) edge (1)\n (x2) edge (3)\n (2) edge (y1)\n (4) edge (y2)\n ;\n \\draw (1.5,0.3) node {$b$}\n (1.5,-0.3) node {$c$}\n ;\n\t \\draw[->,>=latex,gray,dotted,thick] (1,-1.5) -- (1,-2.5);\n\t \\draw[gray] (3,-2) node {transposition principle};\n\\end{tikzpicture}\n\n\\begin{tikzpicture}\n \\matrix[matrix of math nodes,column sep={60pt,between origins},row\n sep={60pt,between origins},nodes={asymmetrical rectangle}] (s)\n {\n &|[name=x1]| x_1 &|[name=1]| + &|[name=2]| \\times & |[name=y1]| y_1\\\\\n %\n &|[name=x2]| x_2 &|[name=3]| + &|[name=4]| \\times &|[name=y2]| y_2 \\\\\n };\n \\draw[->,>=latex] (2) edge node[auto] {\\(a\\)} (1)\n \t\t(4) edge node[auto] {\\(d\\)} (3)\n (4) edge (1)\n (2) edge (3)\n (1) edge (x1)\n (3) edge (x2)\n (y1) edge (2)\n (y2) edge (4)\n ;\n \\draw (1.5,0.3) node {$b$}\n (1.5,-0.3) node {$c$};\n\\end{tikzpicture}\n\\caption{Transposition of a simple arithmetic circuit}\n\\end{figure}\n\\end{ex}\n\nLet us introduce some operations and their transposes. Let $k$ be a field and \n$k[x]_m$ the vector space of polynomials of degree at most $m$, the reversal \noperator of $k[x]_m$ is the map such that\n\n\\[ \n\\begin{array}{rccc}\n \\textbf{rev}(\\cdot,m): & k[x]_m & \\to & k[x]_m \\\\\n & Q & \\mapsto & x^mQ(1/x)\n\\end{array}.\n\\]\nWe can see that \\textbf{rev} is its own transpose, indeed \\textbf{rev} is a \nlinear map and its matrix in the cannocial basis of $k[x]_m$ is symmetric: it \nis the matrix with $1$'s on the anti-diagonal. Let now $P\\in k[x]$ be a monic \npolynomial of degree $m$ and $n\\in\\mathbb{N}$ an integer, we denote by \n$\\textbf{rem}(\\cdot,P,n)$ the remainder by $P$ with length $n$, defined by \n\\[ \n\\begin{array}{rccc}\n \\textbf{rem}(\\cdot,P,n): & k[x]_n & \\to & k[x]_{m-1} \\\\\n & Q & \\mapsto & Q\\mod P\n\\end{array}.\n\\]\n\nAs explained in~\\cite{BoLeSc03}, $\\textbf{rem}^t(\\cdot,P,n)$ is the linear \nsequence extension: it takes as input the $m$ first terms of a linear \nrecurring sequence and computes the $n+1$ first terms using $P$ as the minimal \npolynomial of the sequence.\n\n\\subsection{The algorithms}\n\\label{sec-algo}\nAll the algorithms described here, and their complexity analysis, have been\ndiscussed in~\\cite{DeDoSc14}, that is why we will not show all the pseudo-codes but\nrather try to explain how they work and on which result they are based. The \ncomplexities will also be given without any proofs. \n\nAs we saw in the previous section, transposing an algorithm results in a change\nof basis. Thus, it is important to be able to go from a basis to another. Here\nagain $k$ is a finite field, $P$ an irreducible monic polynomial of $k[x]$, \nand\n$K=k[x]/(P)$ is a finite extension of $k$. We will be considering two bases of\n$K$: the monomial basis $\\bxi=(x^i)_{0\\leq i \\leq m-1}$ (where $m$ is the\ndegree of $P$), and its dual basis $\\bxi^*$ with respect to $\\psdot_P$.\n\n\\paragraph{Change of basis.}The two\nfirst algorithms allow us to juggle between the monomial and the dual basis. They are\nboth based on a lemma which describes the generating series of the traces\n$\\tau_P(ax^i)_{0\\leq i \\leq m-1}$, where $a\\in K$. \n\\begin{lm}\n \\label{lm-basis}\n Let $a\\in K$ be an element of $K$. In the formal power series ring $k[[x]]$,\n we have:\n \\[\n \\sum_{i\\geq 0}\\tau_P(ax^i)x^i=\\frac{\\rev(P'a\\bmod P,m)}{\\rev(P,m+1)}.\n \\]\n\\end{lm}\nGoing from the monomial basis to the dual basis can be done by computing\n$\\rev(P'a\\bmod P,m)\\mod x^m$, $(\\rev(P,m+1)\\bmod x^m)^{-1}$, and the product of\nthem. We obtain, by Lemma~\\ref{lm-basis}:\n\\[\n \\frac{\\rev(P'a\\bmod P,m)}{\\rev(P,m+1)}\\mod x^m=\\sum_{i=0}^{m-1}\\tau_P(ax^i)x^i.\n\\]\nSince we said that\n$(\\tau_P(ax^i))_{0\\leq i \\leq m-1}$ are the coordinates of $a$ in the dual basis\n$\\bxi^*$, taking the coefficient of the polynomial\n$\\sum_{i=0}^{m-1}\\tau_P(ax^i)x^i$ gives a solution to go from the monomial basis\nto the dual basis, that can be written as follows.\n\\newline\n\n\\texttt{monomialToDual}(a, P)\\\\\n\\textbf{Input} $\\textbf{a}=(a_i)_{0\\leq i\\leq m-1}\\in k^m$, $P$ an irreducible\nmonic polynomial of degree~$m$.\\\\\n\\textbf{Output} $(\\tau_P(ax^i))_{0\\leq i \\leq m-1}$, with $a=\\sum_{0\\leq i\n\\leq m-1}a_ix^i$\n\\begin{enumerate}\n \\item $T=1/\\rev(P,m+1) \\mod x^m$\n \\item $b=\\rev(P'a=\\sum_{0\\leq i\\leq m-1}a_ix^i \\mod P,m)T\\mod x^m$\n \\item \\textbf{return} (coefficient($b,x^i))_{0\\leq i \\leq m-1}$\n\\end{enumerate}\n\nObtaining the coefficients in the monomial basis $\\bxi$\nknowing the coefficients in the dual basis $\\bxi^*$ is done using the same\nequality, but starting from the other half. So, computing $\\sum_{0\\leq i \\leq\nm-1}\\tau_P(ax^i)x^i \\mod x^m$, $\\rev(P,m+1)$, and multiplying them gives, by\nLemma~\\ref{lm-basis} again: \n\\[\n\\rev(P,m+1)\\sum_{0\\leq i \\leq m-1}\\tau_P(ax^i)x^i \\mod x^m = \\rev(P'a\\mod P,m) \n\\mod x^m.\n\\]\nThen, applying $\\rev(\\cdot,m)$ and multiplying by $(P')^{-1}\\mod P$ gives $a\\mod\nP$, so taking the coefficients of $a$ gives the coefficients in the monomial\nbasis $\\bxi$.\n\\newline\n\n\\texttt{dualToMonomial}(b, P)\\\\\n\\textbf{Input} $\\textbf{b}=(b_i)_{0\\leq i\\leq m-1}\\in k^m$, $P$ an irreducible\nmonic polynomial of degree $m$.\\\\\n\\textbf{Output} $(a_i)_{0\\leq i \\leq m-1}$ such that\n$\\tau_P(\\sum_{0\\leq i \\leq m-1}ax^{i+j})=b_j$ for all $j$\n\\begin{enumerate}\n \\item $S=1/P \\mod P$\n \\item $b=\\rev(P,m+1)\\sum_{0\\leq i\\leq m-1}b_ix^i\\mod x^m$\n \\item $c=\\rev(b,m)$\n \\item $d=cS \\mod P$\n \\item \\textbf{return} (coefficient($d,x^i))_{0\\leq i \\leq m-1}$\n\\end{enumerate}\n\nAs we see, even if the lemma holds in $k[[x]]$, we just need to work $\\mod x^m$\nso the inverses can be obtained with fast algorithms for \\textsc{Euclide}\ndivision. \\texttt{monomialToDual} uses $O(M(m))$ operations in $k$, where \n$M(m)$ is the number of operations in $A$ of a multiplication of polynomials of \ndegree at most $m$ on any ring $A$. The cost of \\texttt{dualToMonomial} is \n$O(M(m)\\log(m))$, but can be reduced to $O(M(m))$ with precomputations. $M$ is \na function that is quasi-linear, so the change-of-basis algorithms are \nquasi-linaear too. \n\n\\paragraph{Embedding and computing R.} Let $Q$ be an irreducible monic\npolynomial of $k[y]$ of degree $n$, with $n$ coprime to $m$. Let also $R=P\\odot\nQ$ be the composed product of $P$ and $Q$. We have the extensions $k[x]/(P)$,\n$k[y]/(Q)$, $k[x,y]/(P,Q)$, and $k[z]/(R)$ which respectively have the traces\n$\\tau_P$, $\\tau_Q$, $\\tau_{P,Q}$, $\\tau_R$ and the associated bilinear forms.\nThese extensions are respectively endowed with the monomial bases $\\bxi$,\n$\\bupsilon$, $\\bxi\\otimes\\bupsilon$ and $\\bzeta$. Finally, we denote by\n$\\bxi^*$, $\\bupsilon^*$ and $\\bzeta^*$ the dual bases of $\\bxi$, $\\bupsilon$ and\n$\\bzeta$ with respect to $\\psdot_P$, $\\psdot_Q$ and $\\psdot_R$. In this paragraph, we will\ngive algorithms to compute the embeddings $\\varphi_x$, $\\varphi_y$, their\nsections, and to compute the polynomial $R$. The algorithms follow from the next\nlemma.\n\\begin{lm}\n \\label{lm-trac}\n Let $b$ be in $k[x]/(P)$ and $c$ in $k[y]/(Q)$. Then we have\n $\\tau_R(\\Phi(bc))=\\tau_{P,Q}(bc)=\\tau_P(b)\\tau_Q(c)$.\n\\end{lm}\nThis lemma will permit us to compute the restriction of $\\Phi$ to the set \n\\[\n \\Pi=\\left\\{ bc \\;|\\; b\\in k[x]/(P), c\\in k[y]/(Q) \\right\\}\\subset\n k[x,y]/(P,Q).\n\\]\nIndeed, if we have the coordinates $(\\tau_P(bx^i))$ and $(\\tau_Q(cy^j))$ of $b$ and\n$c$ in the dual bases, we can extend them up to $mn-1$ using $\\remt$, then, by Lemma~\\ref{lm-trac}, we have\n$(\\tau_P(bx^i)\\tau_Q(cy^i))_{0\\leq i \\leq mn-1}=(\\tau_R(\\Phi(bc)z^i))_{0\\leq i\n\\leq mn-1}$ which gives us the coordinates of $\\Phi(bc)$ in the dual basis\n$\\bzeta^*$. So we can obtain $\\varphi_x$ and $\\varphi_y$ by applying the last\nformula with respectively $c=1$ and $b=1$. This can be written as follows.\n\\newline\n\n\\texttt{embed}(b, c, r)\\\\\n\\textbf{Input} $\\textbf{b}=(b_i)_{0\\leq i\\leq m-1}\\in k^m$,\n$\\textbf{c}=(c_i)_{0\\leq i\\leq n-1}\\in k^n$, an optional integer $r\\geq mn$ set\nto $r=mn$ by defaut\\\\\n\\textbf{Output} $\\textbf{a}=(a_i)_{0\\leq i\\leq r-1}\\in k^r$\n\\begin{enumerate}\n \\item $(t_i)_{0\\leq i \\leq r-1}=\\remt(\\textbf{b},P,r)$\n \\item $(u_i)_{0\\leq i \\leq r-1}=\\remt(\\textbf{c},P,r)$\n \\item \\textbf{return} $(t_iu_i)_{0\\leq i \\leq r-1}$\n\\end{enumerate}\n\\texttt{embed} uses $O(r(M(m)/m + M(n)/n))$ operations in $k$.\nIf $u_P$ is the vector of the coordinates of $1\\in k[x]/(P)$ in the dual basis\n$\\bxi^*$ and $u_Q$ the vector defined similarly, we can compute\n$\\varphi_x=\\texttt{embed}(\\cdot,u_Q)$ and $\\varphi_y=\\texttt{embed}(u_P,\\cdot)$.\nSince $R$ is the minimal polynomial of the sequence\n$(\\tau_R(z^i))=\\texttt{embed}(u_P,u_Q)$, we can obtain $R$ by computing the\n\\textsc{Berlekamp-Massey} algorithm (which computes minimal polynomials of linear\nsequences).\n\nAs we have just seen, the theory of duality gives us a very simple algorithm for\n\\texttt{embed}, but together with the transposition principle, it will also\nprove useful to find algorithms for the section of $\\varphi_x$, and, as we will\nsee later, for $\\Phi^{-1}$. Let $c\\in k[y]/(Q)$ such that $\\tau_Q(c)=1$ (for\nexample $c=\\upsilon^*_0$ is convenient), $\\varepsilon:k[x]/(P)\\to k[z]/(R),\nb\\mapsto \\Phi(bc)$, and $\\varepsilon^t:k[z]/(R)\\to k[x](P)$ its dual map with\nrespect to $\\psdot_P$ and $\\psdot_R$. Then, for any $b$, $b'$ in $k[x]/(P)$, we\nhave\n\\[\n\\begin{array}{rcl}\n \\ps{b}{b'}_P &=& \\tau_P(bb') = \\tau_P(bb')\\tau_Q(c) = \\tau_R(\\Phi(bb'c))\\\\\n &=& \\ps{\\varepsilon(b)}{\\Phi(b')}_R = \\ps{b}{\\varepsilon^t(\\Phi(b'))}_P.\n\\end{array}\n\\]\nSince $\\psdot_P$ is non-degenerate, this means that \n$b'=\\varepsilon^t(\\Phi(b'))=\\varepsilon^t(\\varphi_x(b'))$ for all $b'\\in \nk[x]/(P)$. In other words, $\\varepsilon^t$ is an inverse of $\\varphi_x$ on its \nimage, also called a section of $\\varphi_x$. Remarking that $\\varepsilon$ is \nnothing else that $\\texttt{embed}(\\cdot,c)$, where $c=(1,0,\\dots,0)$, we have \nan algorithm for $\\varepsilon$, and, by transposing this algorithm, we have an \nalgorithm for $\\varepsilon^t$, which is the section of $\\varphi_x$ we were \nsearching. This algorithm uses $O(nM(m)+nM(n))$ operations in $k$.\n\n\\paragraph{Isomorphism.} We will apply the same kind of strategy to deduce an \nalgorithm for $\\Phi^{-1}$ from an algorithm for $\\Phi$. First, recall that \n$\\bxi\\otimes\\bupsilon$, $\\bxi^*\\otimes\\bupsilon^*$, $\\bxi^*\\otimes\\bupsilon$ \nand $\\bxi\\otimes\\bupsilon^*$ are all bases of $k[x,y]/(P,Q)$ and that the \nsecond is dual to the first and the third to the fourth, with respect to \n$\\psdot_{P,Q}$. Let $\\Phi^t$ be the dual map to $\\Phi$ with respect to \n$\\psdot_{P,Q}$ and $\\psdot_R$. For any $b$ and $b'$ in $k[x,y]/(P,Q)$, we have\n\\[\n\\begin{array}[here]{rcl}\n \\ps{b}{b'}_{P,Q} &=& \\tau_{P,Q}(bb') = \\tau_R(\\Phi(bb')) \\\\\n &=& \\ps{\\Phi(b)}{\\Phi(b')}_R = \\ps{b}{\\Phi^t(\\Phi(b'))}_{P,Q}.\n\\end{array}\n\\]\nSo, here again, thanks to the non-degeneracy of $\\psdot_{P,Q}$, we have \n$\\Phi^t=\\Phi^{-1}$. Hence, if we have an algorithm computing $\\Phi$ in \nsome bases $\\textbf{a}$ and $\\textbf{b}$, we can deduce, using the \ntransposition principle, an algorithm computing $\\Phi^{-1}$ in the dual bases \n$\\textbf{b}^*$ and $\\textbf{a}^*$. We have two algorithms for computing $\\Phi$, \nthe first is better in cases where $m$ is small compared to $n$, and uses the \nlinearity of $\\Phi$ and the work already done. The idea is to express $b\\in \nk[x,y]/(P,Q)$ in the form $b=\\sum_i^{m-1}b_ix^i$ with $b_i\\in k[y]/(Q)$ for all \n$0\\leq i \\leq m-1$. Since the $b_ix^i$ are all in $\\Pi$, we can apply \n\\texttt{embed} and sum the result to obtain $\\Phi(b)=\\sum_i^{m-1}\\Phi(b_ix^i)$. \nTo apply this strategy, the $b_i$ must be expressed in the dual basis \n$\\bupsilon^*$ of $k[y]/(Q)$, because \\texttt{embed} takes its input in the dual \nbases. In the end, exploiting the fact that the coordinates of the $x^i$ in the \ndual basis $\\bxi^*$ are $(\\tau_P(x^{i+j}))_{0\\leq j\\leq m-1}$ ($u_P$ shifted \n$i$ times), we have an algorithm to compute $\\Phi$ in the basis \n$\\bxi\\otimes\\bupsilon^*$ and giving the result in the basis $\\bzeta^*$. As we \nnoticed, transposing this algorithm will create a new algorithm computing \n$\\Phi^{-1}$ and expressed in the bases $\\bzeta$ and $\\bxi^*\\otimes\\bupsilon$. \nThese algorithms both have a quadratic cost in the degree $m$: $O(m^2M(n))$.\n\nThe second way of computing $\\Phi$ does not depend on previous algorithms, but \nwe will still be able to deduce an algorithm for $\\Phi^{-1}$ from it using the \ntransposition principle. It is best suited when dealing with $m$ and $n$ of the \nsame magnitude and it uses baby steps / giant steps: a technique consisting of \nsacrificing memory to gain speed. Recall that $\\Phi(xy)=z$, then, if $b\\in \nk[x,y]/(P,Q)$, we can write\n\\begin{eqnarray*}\nb&=&\\sum_{i=0}^{m-1}\\sum_{j=0}^{n-1} b_{i,j}x^i y^j\n=~\\sum_{i=0}^{m-1}\\sum_{j=0}^{n-1} b_{i,j}x^i y^i y^{j-i}\\\\\n&=&\\sum_{h=-m+1}^{n-1}\\sum_{i=0}^{m-1} b_{i,i+h}(xy)^i y^h\n=~\\frac{1}{y^{m-1}} \\sum_{h=0}^{m+n-2} c_h(xy) y^h,\n\\end{eqnarray*}\nwith $c_h(xy)=\\sum_{0 \\leq i \\leq m-1} b_{i,i+h-m+1} (xy)^i$, and where \nundefined indices $b_{i,i+h-m+1}$ equal zero. So $\\Phi(xy)=z$ is known, but \nalso $T=\\Phi(y)$, because $y\\in\\Pi$ so we can use $\\texttt{embed}$ to compute \nit. Then, we have \n\\[\na = \\frac{1}{T^{m-1}}\\tilde a \\mod R \\text{ with } \\tilde a = \\sum_{h=0}^{m+n-2} \nc_h(z) T^h.\n\\]\nThis is the equality that will be exploited to compute $\\Phi$, and, thanks to \nbaby steps / giant steps techniques, this will be reduced to matrix \nmultiplication (but the coefficients of those matrices are polynomials). This \ntechnique gives an algorithm expressed in the bases $\\bxi\\otimes\\bupsilon$ and \n$\\bzeta$. Details can be found in~\\cite{DeDoSc14}. As we said earlier, we \nimmediatly deduce an algorithm for $\\Phi^{-1}$ using the \ntransposition principle, this last algorithm will be expressed in the bases \n$\\bzeta^*$ and $\\bxi^*\\otimes\\bupsilon^*$. Both these algorithms use \n$O(M(mn)n^{1/2}+M(m)n^{(\\omega+1)/2})$ operations in $k$, where $\\omega$ is a \nconstant such that one can multiply two matrices of size $n\\times n$ over any \nring $A$ in $O(n^\\omega)$ operations in $A$ (we know that $2 < \\omega \\leq \n2.38$).\n\n\n\\section{Julia/Nemo in practice}\nSince Julia is easy-to-write, we hope it is also\neasy-to-read. If not, it is probably due to the way FastArithmetic.jl is written,\nrather than to Julia itself. The code can be found at\n\\url{https://github.com/erou/FastArithmetic.jl}. What's more, it is really\neasy to install personal packages, one just has to clone the repository\ninside Julia, using\n\\begin{verbatim}\njulia> Pkg.clone(``https://github.com/edouardRousseau/FastArithmetic.jl'') \n\\end{verbatim}\nand the package can be easily tested by running the following.\n\\begin{verbatim}\njulia> Pkg.test(``FastArithmetic'') \n\\end{verbatim}\nThe code is very likely to change, but it is possible to update the package as\nfollows.\n\\begin{verbatim}\njulia> Pkg.update()\n\\end{verbatim}\n\\subsection{Benchmarks}\nWe will now look at the speed of the code in Julia, and compare it with\nthe speed in C. All the benchmarks in Julia were realised with the\n\\texttt{@benchmark} macro available in the BenchmarkTools package. This macro\ncan be used to benchmark any function with \n\n\\begin{verbatim}\njulia>@benchmark 1+1\nBenchmarkTools.Trial: \n samples: 10000\n evals/sample: 1000\n time tolerance: 5.00%\n memory tolerance: 1.00%\n memory estimate: 0.00 bytes\n allocs estimate: 0\n minimum time: 3.00 ns (0.00% GC)\n median time: 3.00 ns (0.00% GC)\n mean time: 3.04 ns (0.00% GC)\n maximum time: 10.00 ns (0.00% GC)\n\\end{verbatim}\nwhich runs the desired function and measures the time it takes. It is possible\nto choose all the parameters (number of samples, number of evaluation\nper sample,\\dots) of the benchmark and one can use different\nestimators. For the following benchmarks, we use the median time, because it\nis more stable than the mean time and still gives an idea of the average time\nthe function will take. The stability we are talking about is linked with the\nmemory management in Julia. Indeed, in Julia, we do not have to concern about\nmemory, and the variables we assign are cleared by the Garbage Collector\n(GC). The time Julia passes in the GC is not very clear nor predictible, and the\noperations in the GC take a significant time. That is why the time taken by a\nfunction can change between two samples, depending on the time spent in the GC.\nBy default, before every benchmark, the garbage is automaticaly collected, but\nthere is still a bit of instability. We benchmark our functions in the case\n$p=5$, as in~\\cite{DeDoSc14}, because no difference is seen between different\nvalues of $p$.\nSince all the functions written in FastArithmetic.jl depend on at least one\npolynomial $P$, sometimes also $Q$ and $R=P\\odot Q$, we let $m=\\deg (P)$ grow\nfrom $1$ to $200$ and benchmark the functions for all these values of $m$. We\nchoose $\\deg (Q)=m+1$, and both $P$ and $Q$ are irreducible polynomials.\nIrreducible polynomials of degree $1$ to $201$ were precomputed, by choosing\nrandom elements in $\\mathbb{F}_{p}[X]$, in order to\nalways use the same list and to be sure that we are not using special\npolynomials that could be speeding up the functions (like sparse\npolynomials). When only one field is involved in an operation (such as in \nmodular multiplication or in change of basis) the field used is \n$\\mathbb{F}_{p^{m(m+1)}}\\cong k[z]/(R)$.\n\nThe benchmarks of the C code were realized thanks to a python \nscript written by the authors of the code, that can be found at \n\\url{https://github.com/defeo/ff\\_compositum}. The benchmarks work the same but not \nall the values of $m$ were tested, so we just chose to compare the results for \ncertain values of $m$ tested (we also could have change the script\\dots).\n\nIn the tables, the speed of\nthe C code is set to $1.00$, and smaller is better. One of the functions \nbenchmarked in the C code, the modular multiplication \\texttt{mulmod}, was \nalready implemented in Nemo, so we start to compare the results. We see no\ndifference in terms of speed and it was predictible: the Julia\nfunction is calling C code.\n\n\\begin{center}\n \\begin{tabular}[here]{c}\n \\texttt{mulmod} \\\\\n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $0.99$\\\\\n $51$ & $1.04$\\\\\n $104$ & $1.03$\\\\\n $140$ & $1.05$\\\\\n $187$ & $1.07$\\\\\n \\end{tabular}\n \\end{tabular}\n\\end{center}\nThe speed of \\texttt{monomialToDual} and of \\texttt{dualToMonomial} are the \nsame, either in C or Julia, so we show the results only for the first.\nAll the Julia functions differ from the C ones by a\nconstant factor, which depends on the functions but is around $8$.\nSurprisingly, \\texttt{inversePhi1} is three times faster than its transposition\n\\texttt{phi1}, and \\texttt{inversePhi2} is slower than \\texttt{phi2} for small\nvalues of $m$ but equals its speed at $m\\approx 100$ and is two times faster for $m\\geq170$.\nIn Julia, the cross point between \\texttt{phi1} and \\texttt{phi2} is earlyer than\nin C, it occures at $m=30$.\n\nWe did not mention the time to compute $R$ in these tables, the reason \nis because the implementation of the \\textsc{Berlekamp-Massey} algorithm used \nin it is naive, so the algorithm \\texttt{computeR} can be improved easily by \nimproving \\textsc{Berlekamp-Massey}.\n\n\\begin{center}\n\\begin{tabular}[here]{ccc}\n \\texttt{monomialToDual} & \\texttt{monomialToDual\\_pre} & \\texttt{mulModT} \\\\\n\n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $10.50$\\\\\n $51$ & $10.92$\\\\\n $104$ & $10.26$\\\\\n $140$ & $10.43$\\\\\n $187$ & $11.03$\\\\\n\n \\end{tabular}\n\n &\n\n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $3.13$\\\\\n $51$ & $2.99$\\\\\n $104$ & $1.96$\\\\\n $140$ & $2.06$\\\\\n $187$ & $3.06$\\\\\n\n \\end{tabular}\n\n\n &\n \n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $19.92$\\\\\n $51$ & $15.62$\\\\\n $104$ & $16.23$\\\\\n $140$ & $17.73$\\\\\n $187$ & $17.97$\\\\\n\n \\end{tabular}\n\n\\end{tabular}\n\\end{center}\n \n\n\\begin{center}\n\\begin{tabular}[here]{cccc}\n \\texttt{embed} & \\texttt{project} & \\texttt{phi1} & \\texttt{phi2} \\\\\n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $5.29$\\\\\n $51$ & $4.51$\\\\\n $104$ & $3.85$\\\\\n $140$ & $6.78$\\\\\n $187$ & $5.42$\\\\\n\n \\end{tabular}\n\n &\n\n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $5.73$\\\\\n $51$ & $3.54$\\\\\n $104$ & $3.00$\\\\\n $140$ & $2.65$\\\\\n $187$ & $6.44$\\\\\n\n \\end{tabular}\n\n &\n\n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $5.39$\\\\\n $51$ & $6.29$\\\\\n $104$ & $5.26$\\\\\n $140$ & $4.91$\\\\\n $187$ & $5.74$\\\\\n\n \\end{tabular}\n\n &\n \n \\begin{tabular}[here]{cc}\n $m$ & Julia \\\\\n $10$ & $1.90$\\\\\n $51$ & $3.40$\\\\\n $104$ & $5.78$\\\\\n $140$ & $8.28$\\\\\n $187$ & $9.94$\\\\\n\n \\end{tabular}\n\\end{tabular}\n\\end{center}\n\n\\section{Conclusion}\nAs we see, we are not able to reach C speed, and the Julia code is slower by a\nconstant factor, approximately $8$. Part of this factor is due to\nprecomputations in C that are not done in Julia. With further investigations and\nunderstanding of the Julia langage, we could probably diminish this constant.\nAnyway, Julia provides a much more generic code and we could have benchmarked\nthe code for $p=2^{127}-1$ without changing a single line. On top of that, Julia\nprovides a variety of tools that makes life easier when coding : the benchmark\nparckage, the package manager, a high-level syntax\\dots It will be up to reader\nto decide if the price of the constant factor $8$ is too much.\n\nAny error in this report or in the code of FastArithmetic.jl can be signaled by\ne-mail (edouard.rousseau@u-psud.fr). Finally, this work would not have been done\nwithout the help of Luca \\textsc{De Feo}, who took the time to answer all the\nnaive questions he received.\n\n\n\\clearpage\n\\bibliographystyle{unsrt}\n\\bibliography{biblio}\n\\end{document}\n", "meta": {"hexsha": "ba2d6539e948bfda3241a9d967a574351e7cb0c0", "size": 34784, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "report/report.tex", "max_stars_repo_name": "artichaut/fastarithmetic", "max_stars_repo_head_hexsha": "b0484612cec0b9e0d442452fcb17f3569673fb38", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "report/report.tex", "max_issues_repo_name": "artichaut/fastarithmetic", "max_issues_repo_head_hexsha": "b0484612cec0b9e0d442452fcb17f3569673fb38", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "report/report.tex", "max_forks_repo_name": "artichaut/fastarithmetic", "max_forks_repo_head_hexsha": "b0484612cec0b9e0d442452fcb17f3569673fb38", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.6274509804, "max_line_length": 107, "alphanum_fraction": 0.6795078197, "num_tokens": 11711, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795672, "lm_q2_score": 0.6548947155710234, "lm_q1q2_score": 0.3453368045945626}} {"text": "%#########################################################\n\\chapter{Diffusion Modeling to Study Microarchitecture in Human Skeletal Muscles}\n\\label{ch: DiffusionExp}\n%#########################################################\nThe resolution of the MR images is not sufficient to perform direct measurements of the muscle fiber geometry and other physical parameters such as membrane permeability at the level of individual fiber. \nWhile typical resolution of the MR images is $\\approx \\SI{e-3}{\\m}$ histology studies showed that cross-sectional liner dimensions are in the range of $10^{-5} - \\SI{e-4}{\\m}$ \\cite{TAYLOR200335}. \nIn recent years a number of diffusion models were developed in the effort to relate macroscopic signal measured in MRI experiments to the microscopic parameters of muscle tissue. \n%-new paragraph-%\n\n%-new paragraph-%\nThis chapter presents work on application of two diffusion models to interpret disuse atrophy and age related changes in human lower leg muscle.\n\n%=========================================================\n\\section{Monitoring Changes in the Medial Gastrocnemius in Disuse Atrophy Induced by Unilateral Limb Suspension}\n\\label{sec: DTI ULLS}\n%=========================================================\nThe feasibility of diffusion tensor imaging (DTI) of skeletal muscle including muscle fiber tracking for extraction of fiber lengths and pennation angles has been established, and the technique has been applied to monitor muscle injury, aging, gender related differences, and compartmental syndrome~\\cite{RND1, RND2, RND3, RND4, RND5, RND6}. \nAdditionally, some studies have explored changes in muscle DTI with atrophy; e.g., arising from age-related atrophy in human skeletal muscle~\\cite{RND7}, from denervation and from Achilles tenotomy-induced atrophy in rodent models~\\cite{RND8, RND9}.\nFurther, DTI combined with diffusion models is increasingly being used to explore tissue microstructural parameters such as fiber diameter, permeability or intracellular volume fraction~\\cite{RND10}.\nWhile most modeling studies have focused on the brain, a few recent studies have extended modeling efforts to the diffusion in skeletal muscle~\\cite{RND11, RND12, RND13}. \nRecent studies have combined combine magnetic resonance imaging deformation analyses and diffusion tensor imaging tractography in the medial gastrocnemius~\\cite{RNSS4,RNCS4}. \nKarakuzu~et~al. showed that sub-maximal plantar flexion activity at 15\\% Maximum voluntary contraction (MVIC) causes heterogeneous length changes along the fascicles of human medial gastrocnemius (MG) muscle~\\cite{RNCS4}.\nThe heterogeneity of fascicle strains was explained on the basis of epimuscular myofascial force transmission. \n%-new paragraph-%\n\n%-new paragraph-%\nIt is well established that disuse (e.g., by chronic unloading) leads to skeletal muscle atrophy that is accompanied by a significant loss of muscle force~\\cite{RNS1}.\nThe Unilateral Limb Suspension (ULLS) is a validated model to study the effects of chronic unloading~\\cite{RNS8}.\nUnilateral limb suspension results in both loss of muscle mass (muscle fiber atrophy) as well as a decrease in muscle force~\\cite{RNS8}.\nPrior studies of induced immobilization indicate that muscle remodeling with inactivity is a fast process that occurs after about a week of unloading~\\cite{RNS8, RNS5}.\nThese studies documented the decrease in both fiber length and pennation angle, a decrease in Focal Adhesion Kinase content ($-20\\%$) and activity ($-30\\%$), associated with a $50\\%$ fall in muscle protein synthesis and a $5\\%$ decrease in quadriceps muscle anatomical cross-sectional area (ACSA)~\\cite{RNS8, RNS5}.\nDTI with its ability to probe at the microstructural level is ideally suited to investigate potential muscle remodeling that occurs with chronic unloading.\nHowever, no studies to date have investigated unloading induced changes in skeletal muscle using DTI in human subjects.\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Methods}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n%---------------------------------------------------------\n\\subsubsection{\\textit{In-vivo} experiments}\n%---------------------------------------------------------\nThe study was carried out under the approval of the Medical Research Ethics Board of UC~San~Diego and conformed to all standards for the use of human subjects in research as outlined in the Declaration of Helsinki on the use of human subjects in research.\nIRB approval was obtained from the Medical Research Ethics Board of UC~San~Diego and all subjects were recruited after obtaining written informed consent. A total of 7 normal healthy young subjects (2 females, $29.1 \\pm 5.7$ years, body mass $75.4 \\pm \\SI{22.7}{\\kilogram}$, height $168.1 \\pm \\SI{7.4}{\\centi\\meter}$) were recruited for this study.\nThe criterion for inclusion was that subjects should be moderately physically active. Subjects participating in competitive sports as well as those with any surgical procedures performed on the lower leg were excluded.\n%---------------------------------------------------------\n\\subsubsection{Study design} \n%---------------------------------------------------------\nThe effect of chronic unloading on the force production capability and diffusion tensor indices of the MG muscle were assessed by comparing the baseline (pre) to immediately after four weeks of limb suspension (post).\nDuring the four week suspension period, subject compliance to the protocol was monitored at two weeks to check for muscle atrophy (MRI morphological scan) and loss of force production.\nIn addition, compliance was also monitored by a wireless activity tracker that was integrated into the crutches; the subject was not informed of the tracker to ensure that it was not removed or tampered with to simulate crutch usage.\nAfter the four week suspension, subjects were required to attend structured physical rehabilitation sessions.\nMRI morphological scans were performed at the end of the rehabilitation period (four weeks) to confirm that the muscle had recovered to baseline status.\n%---------------------------------------------------------\n\\subsubsection{Unilateral Limb Suspension (ULLS)}\n%---------------------------------------------------------\nThe ULLS model is an established model of inducing controlled atrophy~\\cite{RNS19} and was used in the current study to induce muscle atrophy on the non-dominant leg with four weeks of chronic unloading.\nThe dominant leg was self-identified by subjects as the one they preferentially used to regain balance from a jostle.\nThe non-dominant leg was the left leg for all subjects in this study.\nThe ULLS protocol allowed the subjects a reasonable amount of freedom to carry out their daily activities including driving since the dominant leg (right in this study) was not unloaded.\nA crutch was used to prevent the foot (of the left leg) from touching the ground.\nThe right foot was raised with a $\\SI{5}{\\centi\\meter}$ sole on the shoe to further minimize accidental loading of the foot.\n%---------------------------------------------------------\n\\subsubsection{MR imaging} \n%---------------------------------------------------------\nSubjects were positioned supine in a $\\SI{3}{\\tesla}$ whole-body scanner (GE Medical Systems, WI, USA) and only the limb selected for suspension was scanned (pre- and post-suspension).\nA custom-built receive-only phased array coil with a large FOV was used to image approximately $\\SI{30}{\\centi\\meter}$ of the lower leg without moving the subject or coil; the intent was to cover the medial gastrocnemius muscle from its origin to insertion in two acquisitions without having to reposition the subject in the coil.\nThe coil moved relative to the magnet between acquisitions, the subject did not move with respect to the coil.\nOne anatomical slice was common between the two sets of acquisitions and the two sets were reoriented by a transformation obtained from the rigid alignment of the slice common to both sets.\nThe subject's leg was fully extended while the foot was maintained in a fixed position at $\\approx \\SI{15}{\\degree}$ in plantarflexion; the foot rested on an adjustable wedge to maintain the angle constant for the scan duration. This foot position was used to ensure that there was no passive tension in the lower leg muscles.\n%-new paragraph-%\n\n%-new paragraph-%\nThe MRI pulse sequence used in the image acquisition was a fat suppressed single shot EPI spin-echo sequence with monopolar diffusion gradients (diffusion gradient duration, $\\delta = \\SI{12}{\\milli\\second}$, and diffusion time $\\Delta = \\SI{20}{\\milli\\second}$). \nSpectral-spatial water selective excitation was used to suppress fat.\nThe total number of slices (acquired in two multi-slice 2D sets) required to cover the calf muscles ranged from 32 to 40 slices.\n$B_0$ shimming in the selected region of interest was performed with vendor supplied automated second order shims.\nThirty-two non-collinear gradient directions with a \\mbox{\\textit{b-}value} of $\\SI{400}{\\second \\per \\milli\\meter^2}$ were used to map the direction dependent diffusion.\nImaging parameters: echo time (TE) $\\SI{52.7}{\\milli\\second}$, repetition time (TR) $\\SI{7000}{\\milli\\second}$ ms with 4 signal averages, field of view (FOV) $240 \\times 240 \\; \\SI{}{\\milli\\meter^2}$, slice thickness $\\SI{5}{\\milli\\meter}$, no gap and acquisition matrix $80 \\times 80$; no parallel imaging was employed. The $80 \\times 80$ matrix was reconstructed to $128 \\times 128$ matrix by partial fourier with homodyne reconstruction.\nThe homodyne algorithm pre-weights the \\mbox{\\textit{k-}space} data so that when the real part of the image is extracted, it corresponds to uniform weighting in \\mbox{\\textit{k-}space}~\\cite{RND20}.\nThe $128 \\times 128$ matrix is then extrapolated to a $256 \\times 256$ matrix yielding a voxel resolution of $\\SI{0.94}{\\milli\\meter} \\times \\SI{0.94}{\\milli\\meter} \\times \\SI{5.00}{\\milli\\meter}$.\nIn addition to the diffusion weighted images, a morphological volume was acquired with a fat saturated fast gradient echo (FGRE) sequence for two echo times with the following parameters: $\\mathrm{TE_1}/\\mathrm{TE_2}$/TR: $\\SI{3.2}{\\milli\\second}/\\SI{5.6}{\\milli\\second}/\\SI{350}{\\milli\\second}$ and a flip angle (FA) of $\\SI{20}{\\degree}$; all geometric parameters were the same as in the DTI scans.\nThe acquired morphological volume was used to generate phase maps to correct $B_0$ distortions in the echo planar images.\nThe total scan time including the morphological images for each session was 17 minutes. \n%---------------------------------------------------------\n\\subsubsection{Force measurements}\n%---------------------------------------------------------\nIsometric MVIC of the plantarflexor muscles was determined for each subject prior to MR imaging. For this purpose, the ankle was fixed in a neutral position ($\\SI{90}{\\degree}$ angle between the axis of the foot and the shank).\nTo estimate the maximum force acting along the Triceps Surae tendon, the force recorded by the force transducer was divided by the Achilles tendon moment arm corresponding to $\\SI{90}{\\degree}$ angle between the axis of the foot and the shank detailed earlier~\\cite{RNS20}.\nIn brief, a sagittal MR image of the lower leg and foot was used to identify the joint (ankle) center of rotation as well as Achilles tendon line of action (the latter marked as a straight line along the center of the tendon). \nThe perpendicular distance of the joint center to the line of action was measured as the Achilles tendon moment arm~\\cite{RNS20}.\nThe measured muscle force is that generated by the gastrocnemius (lateral and medial) and the soleus muscles.\n%---------------------------------------------------------\n\\subsubsection{Muscle volume measurements}\n%---------------------------------------------------------\nThe manual contouring was performed using parametric Bezier curves every fifth slice using OsiriX~\\cite{RND22}.\nAutomated interpolation was performed for the intervening slices; all automated contours were examined and edited if needed.\n%---------------------------------------------------------\n\\subsubsection{Image preprocessing and DTI indices computation}\n%---------------------------------------------------------\nFigure~\\ref{fig: DTI-flowchart} is the flowchart of the sequence of image processing steps~\\cite{DTI}.\n%*********************************************************\n\\begin{figure}[!h]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures/DTI_flowchart.pdf}\n\\caption[The image processing flowchart showing the sequence of steps leading to the extraction of the diffusion tensor indices]{The image processing flowchart showing the sequence of steps leading to the extraction of the diffusion tensor indices.}\n\\label{fig: DTI-flowchart}\n\\end{figure}\n%*********************************************************\nAll diffusion weighted image volumes were registered to the baseline ($b=0$) image to correct for eddy currents and motion related artifacts using the \\textit{eddy} algorithm from FSL~\\cite{RND23}. \nThe phase images of the dual echo (FGRE) volumes were used to correct for $B_0$ field inhomogeneities using the \\textit{fugue} algorithm from FSL~\\cite{RND23}.\nDiffusion weighted images were then denoised using a Joint Rician Linear Minimum Mean Square Error (LMMSE) estimator~\\cite{RND24}. \nThe noise in magnitude MR images is best modeled by a Rician distribution and the LMMSE estimator has been shown to out-perform other Rician denoising techniques~\\cite{RND25}.\nThe tensor was calculated from the distortion corrected and denoised data using a Gaussian model of diffusion.\nThe tensor was then diagonalized to yield the eigenvalues followed by the calculation of the fractional anisotropy (FA), apparent diffusion coefficient (ADC) and coefficient of planarity (CP) maps. \nThe contour of the MG traced in the FGRE volume slices was transferred to the DTI indices map.\nAverage values of the DTI indices were obtained from a $5 \\times 5$ region of interest (ROI) automatically placed at the centroid of the MG mask at each anatomical slice (in most subjects, 40 axial slices were required to cover the MG); this placement ensured that the ROI was not at the edges which would bias estimates due to partial volume effects. \nA manual check was also made to ensure that the ROI was not in the region of a blood vessel, fascia or contained artifacts; such ROIs were manually adjusted. \nThough the ROI was small at 25 pixels~/~slice, the total number of pixels across all slices was $\\approx 900 (25 \\times 36)$. \n%---------------------------------------------------------\n\\subsubsection{Simulation}\n%---------------------------------------------------------\nThe model used here closely follows that proposed originally by Karampinos~et~al.~\\cite{RND12}. \nSimulations were carried out by varying the fit parameters ($\\alpha$, $d$, $\\tau_{\\mathrm{in}}$, $\\nu_{\\mathrm{in}}$, $\\nu_{\\mathrm{col}}$) to compute DTI eigenvalues. \nThe five parameter-space was searched to find the optimal combination of the five parameters that minimized the normalized root mean square difference (NRMSE) between the eigenvalues computed from the simulation and those determined from experiments (Equation~\\ref{eq: KargerRMSE}).\n%.........................................................\n\\begin{equation}\\label{eq: KargerRMSE}\n\\mathrm{NRMSE} = \\sqrt{\\dfrac{\\sum\\limits_{i=1}^3 \\left( \\lambda_{i,\\mathrm{expr}}-\\lambda_{i, \\mathrm{sim}} \\right)^2}{\\sum\\limits_{i=1}^3 \\lambda_{i,\\mathrm{expr}}^2}}\n\\end{equation}\n%.........................................................\nwhere $\\lambda_{i,\\mathrm{expr}}$ is the experimentally determined eigenvalues while $\\lambda_{i,\\mathrm{sim}}$ is the eigenvalue from the model simulation. \nSince there are five variables that are fit to the model, the search space was limited to physiologically relevant ranges in order to prevent the search from minimizing at non-physiological values, these ranges were taken from~\\cite{RND12}. \nThe simulations were carried out with the values for intra-cellular diffusion coefficient ($D_{\\mathrm{in}}$), extracellular matrix diffusion coefficient ($D_{\\mathrm{ex}}$), intra-cellular $\\mathrm{T_2}$ ($\\mathrm{T}_{\\mathrm{2,in}}$), and extra-cellular $\\mathrm{T_2}$ ($\\mathrm{T}_{\\mathrm{2,ex}}$) taken from the literature~\\cite{RND12, RND26}.\nIt should be noted here that Karampinos~et~al. fitted $D_{\\mathrm{in}}$ and $D_{\\mathrm{ex}}$ to the diffusion model, but these variables were kept fixed in the current paper at the values obtained for young normal subjects in the former paper~\\cite{RND12}. \nThis was done to limit the number of model variables in order to increase the robustness of the fit.\nFurther, as $D_{\\mathrm{in}}$ is determined by the various intracellular membrane and protein structure; it is not anticipated to change with atrophy. \n$D_{\\mathrm{ex}}$, the diffusion in the extracellular compartment is determined by the collagen fraction in the endomysium and the diffusion coefficient of water.\nBased on the findings from imaging that \\% connective tissue fraction does not change with atrophy (unreported study on the same subjects), it is also anticipated that \\% collagen fraction in the endomysium may not change significantly with atrophy. \nThe experimental values of the eigenvalues ($\\lambda_{i,\\mathrm{expr}}$) used in obtaining the optimal solution from the simulation were the mean values of all the subjects for pre- and post-suspension. \nThe optimization was performed in Wolfram Research, Inc., Mathematica, (Version 11.3, Champaign, IL). \nThe conjugate gradient search algorithm was used to find the parameters that minimize NRMSE. \nAs this method only guarantees a local minimum, the global minimum was obtained by a search performed over all combinations of the five-parameter space~\\cite{BiComp}; the range and step size of the search space are provided in Table~\\ref{tab: Karger2}.\n%=========================================================\n\\begin{table}[!htb]\n\\vspace{+0.2cm}\n\\caption[Tissue microstructural parameters from bicompartmental diffusion model]{Tissue microstructural parameters from bicompartmental diffusion model.}\n\\label{tab: Karger2}\n\\begin{center}\n\\begin{tabular}{@{}llllll@{}}\n\\toprule[1pt]\\midrule[0.3pt]\n & & Pre-ULLS & Post-ULLS & Range & Step size \\\\ \\midrule\n$\\alpha$ & & 0.55 & 0.95 & 0.05 - 0.95 & 0.05 \\\\[6pt]\n$d$ & $[\\SI{}{\\micro\\meter}]$ & 82 & 65 & 50 - 110 & 5 \\\\[6pt]\n$\\tau_{\\mathrm{in}}$ & $\\left[\\SI{}{\\second}\\right]$ & 0.8 & 2.6 & 0.8 - 2.6 & 0.2 \\\\[6pt]\n$\\nu_{\\mathrm{in}}$ & &0.95 & 0.89 & 0.89 - 0.95 & 0.0025 \\\\[6pt]\n$\\nu_{\\mathrm{col}}$ & &0.5 & 0.6 & 0.5 - 0.6 & 0.0025 \\\\ \\midrule[0.3pt]\\toprule[1pt]\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n\\end{table}\n%=========================================================\n%---------------------------------------------------------\n\\subsubsection{Statistical analysis} \n%---------------------------------------------------------\nThe outcome variables of the analysis are the diffusion tensor indices ($\\lambda_1$, $\\lambda_2$, $\\lambda_3$, FA and CP). \nThe assumption of normality of data was tested using both the Shapiro-Wilk test and by visual inspection of quantile-quantile plots. \nSince deviations from normality were only minor and no non-parametric alternative to factorial ANOVA exists, differences between pre- and post-ULLS values were assessed using two-way repeated measures ANOVAs. \nData are reported as mean $\\pm$ standard deviation. \nFor all tests, the level of significance was set at 0.05. \nThe statistical analyses were carried out using SPSS for Mac OSX (SPSS 23.0, IBM Inc, Armonk, NY).\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Bicompartmental Diffusion Model with Exchange}\n\\label{subsec: bicompart}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nThe multi-compartment diffusion model used here is originally proposed by K\\\"arger~\\cite{KARGER19881} and extended by Karampinos~\\cite{RND12}. \nMuscle fibers are approximated with infinite cylinders of an elliptical cross-section and ratio between major axis $\\alpha = d_s/d_l$ as shown in Figure~\\ref{fig: KargerModel}. \n%*********************************************************\n\\begin{figure}[!htb]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[scale=.3]{Figures/FibersModel.pdf}\n\\caption[Schematic of muscle fibers and endomaysium matrix]{Schematic of muscle fibers and endomaysium matrix.}\n\\label{fig: KargerModel}\n\\end{figure}\n%*********************************************************\nMuscle fiber diameter $d$ defined as the mean of the short axis and long axis lengths.\nThe model allows exchange between two compartments: the intracellular space (inside the muscle fiber) and the extracellular space (collagenous intramuscular connective tissue consisting of endomysium and perimysium).\nThe volumes of the two compartments are denoted as $\\nu_{\\mathrm{in}}$ (intramyocellular volume fraction) and $\\nu_{\\mathrm{ex}}$ (extracellular volume fraction), with $\\nu_{\\mathrm{in}} + \\nu_{\\mathrm{ex}} = 1$. \nFurther, the collagen fraction in the extracellular matrix is represented by $\\nu_{\\mathrm{coll}}$. \nThe last parameter of the fit is the intracellular residence time $(\\tau_{\\mathrm{in}})$, which is related to the permeability ($\\kappa$) of the sarcolemma (i.e. the cell membrane of muscle fibers) through the relationship: $\\tau_{\\mathrm{in}} = d/2\\kappa$. \nTotal MR signal $S$ which is a function of time $t$ and \\mbox{\\textit{b-}value} is a sum of signals from intracellular and extracellular compartments:\n%.........................................................\n\\begin{equation}\\label{eq: Karger Signal}\nS(t,b) = S_{\\mathrm{in}}(t,b)+S_{\\mathrm{ex}}(t,b)\n\\end{equation}\n%.........................................................\nTime evolution of the signals $S_\\mathrm{in}$ and $S_\\mathrm{ex}$ can be found from the system of ordinary differential equations:\n%.........................................................\n\\begin{equation}\\label{eq: Karger ODE}\n{\\setstretch{1.0}\n\\begin{cases}\n\\dfrac{dS_{\\mathrm{in}}}{dt}=-q^2D^{\\mathrm{app}}_{\\mathrm{in}}S_{\\mathrm{in}}-\\dfrac{1}{\\tau_{\\mathrm{in}}}S_{\\mathrm{in}}+\\dfrac{1}{\\tau_{\\mathrm{ex}}}S_{\\mathrm{ex}}-\\dfrac{1}{\\mathrm{T}_{2,\\mathrm{in}}}S_{\\mathrm{in}}\\\\[20pt]\n\\dfrac{dS_{\\mathrm{ex}}}{dt}=-q^2D^{\\mathrm{app}}_{\\mathrm{ex}}S_{\\mathrm{ex}}-\\dfrac{1}{\\tau_{\\mathrm{ex}}}S_{\\mathrm{ex}}+\\dfrac{1}{\\tau_{\\mathrm{in}}}S_{\\mathrm{in}}-\\dfrac{1}{\\mathrm{T}_{2,\\mathrm{ex}}}S_{\\mathrm{ex}}\n\\end{cases}\n}\n\\end{equation}\n%.........................................................\nwhere $q$ is a product of gyromagnetic ratio $\\gamma$, gradient $G$ of duration $\\delta$. \nUsing $q$ the \\mbox{\\textit{b-}value} can be expressed as $b=q^2(\\Delta-\\delta/3)$, $\\Delta$ is time separation between diffusion gradients. \nWith initial conditions:\n%.........................................................\n\\begin{equation}\\label{eq: Karger IC}\n{\\setstretch{1.0}\n\\begin{aligned}\n& S_{\\mathrm{in}}(0,b)=\\nu_{\\mathrm{in}}\\\\[10pt]\n& S_{\\mathrm{ex}}(0,b)=\\nu_{\\mathrm{ex}}\\\\[10pt]\n& \\dfrac{S_{\\mathrm{in}}(0,b)}{\\tau_{\\mathrm{in}}}=\\dfrac{S_{\\mathrm{ex}}(0,b)}{\\tau_{\\mathrm{ex}}}\n\\end{aligned}\n}\n\\end{equation}\n%.........................................................\nsystem of Equations~\\ref{eq: Karger ODE} has the solution:\n%.........................................................\n\\begin{equation}\\label{eq: Karger Solution}\nS(t,b)=\\nu'_{\\mathrm{in}} e^{-q^2tD'_{\\mathrm{in}}}+\\nu'_{\\mathrm{ex}} e^{-q^2tD'_{\\mathrm{ex}}}\n\\end{equation}\n%.........................................................\nwith modified volumes and diffusion coefficients:\n%.........................................................\n\\begin{equation}\\label{eq: Karger v modified}\n\\begin{array}{l}\n\\nu'_{\\mathrm{ex}} = \\dfrac{1}{D'_{\\mathrm{ex}}-D'_{\\mathrm{in}}} \\left[\\nu_{\\mathrm{in}}\\left( D^{\\mathrm{app}}_{\\mathrm{in}} + \\dfrac{1}{q^2 \\mathrm{T}_{2,\\mathrm{in}}} \\right) + \\nu_{\\mathrm{in}}\\left( D^{\\mathrm{app}}_{\\mathrm{in}} + \\dfrac{1}{q^2 \\mathrm{T}_{2,\\mathrm{in}}} \\right) - D'_{\\mathrm{in}} \\right]\\\\[10pt]\n\\nu'_{\\mathrm{in}} = 1 - \\nu'_{\\mathrm{ex}}\n\\end{array}\n\\end{equation}\n%.........................................................\n%.........................................................\n\\begin{equation}\\label{eq: Karger ADC}\n\\begin{split}\nD'_\\mathrm{in,ex} &= \\frac{1}{2}\\left( D^{\\mathrm{app}}_\\mathrm{in} + D^{\\mathrm{app}}_\\mathrm{ex} + \\frac{1}{q^2}\\left(\\frac{1}{\\tau_{\\mathrm{in}}} + \\frac{1}{\\tau_{\\mathrm{ex}}} + \\frac{1}{\\mathrm{T}_{2,\\mathrm{in}}} + \\frac{1}{\\mathrm{T}_{2,\\mathrm{ex}}} \\right) \\right) \\mp \\\\ &\\mp \\frac{1}{2}\\sqrt{\\left( D^{\\mathrm{app}}_\\mathrm{in} - D^{\\mathrm{app}}_\\mathrm{ex} + \\frac{1}{q^2}\\left(\\frac{1}{\\tau_{\\mathrm{in}}} - \\frac{1}{\\tau_{\\mathrm{ex}}} - \\frac{1}{\\mathrm{T}_{2,\\mathrm{in}}} + \\frac{1}{\\mathrm{T}_{2,\\mathrm{ex}}} \\right) \\right) + \\frac{4}{q^4\\tau_{\\mathrm{in}}\\tau_{\\mathrm{ex}}}}\\\\\n\\end{split}\n\\end{equation}\n%.........................................................\n\\\\\nIntracellular apparent diffusion coefficient $D^{\\mathrm{app}}_\\mathrm{in}$ is found by investigating signal attenuation due to diffusion inside two impermeable walls separated by distance $d$ with gradient $G$ applied perpendicular to these walls. \nSignal $S$ defined as ratio between $\\bar{M}$ and $M_{0}$ expressed through conditional probability $P(z_2|z_1, \\Delta)$:\n%.........................................................\n\\begin{equation}\\label{eq: Karger AppEx}\nS (t,b)= \\dfrac{1}{d}\\iint dz_1 dz_2 \\cos \\left[ \\gamma \\delta G (z_2-z_1)\\right]P(z_2|z_1, \\Delta)\n\\end{equation}\n%.........................................................\nfrom Fick's law (Equation~\\ref{eq:probability_diffusion}) with initial conditions imposed:\n%.........................................................\n\\begin{equation}\\label{eq: probability initial conditions}\n\\begin{array}{c}\n\tP\\left( z,0\\right) = \\delta\\left( z-z_1\\right)\\\\[10pt]\n\t\\left. \\dfrac{\\partial P}{\\partial z}\\right\\vert_{z=\\pm d/2} = 0\n\\end{array}\n\\end{equation}\n%.........................................................\nusing Equation~\\ref{eq: Diffusion from bvalue} intracellular apparent diffusion coefficient is expressed:\n%.........................................................\n\\begin{equation}\\label{eq: Karger AppEx}\n\\begin{split}\n\tD^{\\mathrm{app}}_{\\mathrm{in}} (D_{\\mathrm{in}},\\Delta, b, d) &= -\\dfrac{1}{b} \\ln \\left[ \\frac{2 - 2 \\cos(qd)}{(qd)^2} + \\vphantom{ \\exp \\left( - \\frac{n^2 \\pi^2 D_{\\mathrm{in}} \\Delta}{d^2}\\right)\\frac{1-(-1)^n \\cos{(qd)}}{\\left( \\left(qd\\right)^2 - \\left( n \\pi\\right)^2\\right)^2}} \\right. \\\\\n &+ \\left. 4(qd)^2 \\sum^{\\infty}_{n=1} \\exp \\left( - \\frac{n^2 \\pi^2 D_{\\mathrm{in}} \\Delta}{d^2}\\right)\\frac{1-(-1)^n \\cos{(qd)}}{\\left( \\left(qd\\right)^2 - \\left( n \\pi\\right)^2\\right)^2} \\right]\n\\end{split}\n\\end{equation}\n%.........................................................\nExtracellular apparent diffusion coefficient as a function of water diffusion coefficient $(D_{\\mathrm{w}})$, orientation parameter $(\\zeta)$, extracellular $(\\nu_{\\mathrm{ex}})$ and collagen $(\\nu_{\\mathrm{col}})$ volume fractions was defined in~\\cite{RND12}:\n%.........................................................\n\\begin{equation}\\label{eq: Karger AppEx}\nD^{\\mathrm{app}}_{\\mathrm{ex}}=D_{\\mathrm{w}}(1-\\nu_{\\mathrm{col}})^{2/3}\\nu_{\\mathrm{ex}}^\\zeta\n\\end{equation}\n%.........................................................\nwhere $\\zeta$ can take three values: 0 when considering diffusion parallel to the longitudinal axes of cylinder, $\\alpha$ when parallel to the major axis in the cylinder cross-section and $1/\\alpha$ when parallel to the minor axis.\nFinally the diffusion eigenvalue from the model can then be calculated as following:\n%.........................................................\n\\begin{equation}\\label{eq: Karger AppEx}\n\\lambda_i = -\\frac{1}{b} \\mathrm{ln}\\left( \\frac{S_i(q,\\Delta)}{S_i(q,0)}\\right)\n\\end{equation}\n%.........................................................\n\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Results}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nMuscle force decreased significantly post-suspension (change of $-32.6 \\pm 24.7\\%$; $p=0.013$); this is the average over the seven subjects. \nThe average change (over all subjects) in the volume of the MG, LG, and SOL muscles after the 4-week suspension was $-9.6 \\pm 4.5\\%$ ($p=0.001$), $-11.1 \\pm 7.4\\%$ ($p=0.008$), and $-7.4 \\pm 5.9\\%$ ($p=0.006$) respectively (significance of pre- and post- suspension volume changes indicated for each muscle).\nWhile the DTI measurements~/~modeling are focused on the MG, the measured isometric plantarflexion force is generated by the Triceps Surae muscles and thus the volume change of all three muscles are included here. \nTypical maps of the eigenvalues, ADC, FA, and CP at one anatomical location of the lower leg in one subject pre- and post-suspension are shown in Figure~\\ref{fig: KargerEV}. \nThe two left columns are pre-suspension and the two right columns are corresponding images from the same subjects post-suspension; \nan anatomical level close top row shows the maps pre-suspension and the bottom the maps at a corresponding anatomic location post-suspension. \nThe maps are (top to bottom, column 1 and 3): ADC, $\\lambda_1, \\lambda_2, \\lambda_3$; (top to bottom, column 2 and 4): FA, CP, lead eigenvector, mask of MG muscle with ROI superposed on the eigenvalue weighted eigenvector color map. \nEigenvalues are in units of $\\SI{}{\\micro\\meter^2 / \\milli\\second}$, the FA and CP are unitless. $\\mathrm{ev}_1$ is the colormap of the eigenvector corresponding to the lead eigenvalue, $\\lambda_1$. \nThe colormap is the projection of the primary eigenvector following the convention: Left $\\rightarrow$ Right ($x$-projection): red, Anterior $\\rightarrow$ Posterior ($y$-projection): green, Superior $\\rightarrow$ Inferior ($z$-projection): blue. \nThe predominantly blue hue confirms the superior-inferior direction of the muscle fiber. \nThe same anatomic location is shown pre- and post-suspension; \nthe atrophy of the muscles is clearly seen comparing the pre- to the post-suspension images. \nThe effectiveness of the fat suppression can be seen by the absence of fat artifacts (shifted subcutaneous fat) in the image.\n%*********************************************************\n\\begin{figure}[!htb]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[width=\\textwidth]{Figures/DTI_2compart.pdf}\n\\caption[The parametric maps extracted from the diffusion tensor]{The parametric maps extracted from the diffusion tensor.}\n\\label{fig: KargerEV}\n\\end{figure}\n%*********************************************************\nVisual assessment of the parametric DTI maps confirms the image quality (SNR) and lack of image artifacts (good fat suppression as well as small image distortions).\n%-new paragraph-%\n\n%-new paragraph-%\nThe DTI indices evaluated pre- and post-suspension in the medial gastrocnemius are summarized in Table~\\ref{tab: Karger1};\nthe values listed here are the averages for all subjects. Three eigenvalues and apparent diffusion coefficient decreased significantly post-suspension ($\\lambda_1$: $p = 0.025$, $\\lambda_2$: $p = 0.035$, $\\lambda_3$: $p = 0.049$, ADC: $p = 0.029$) while FA increased and CP decreased with suspension, although the latter indices two did not show significant changes ($p = 0.239$ for FA and $p=0.763$ for CP). \nThe maximum decrease was in the secondary eigenvalue that in turn resulted in a smaller difference between the secondary and tertiary eigenvalue post suspension (reflected also as a decrease in CP).\n%=========================================================\n\\begin{table}[!htb]\n\\vspace{+0.2cm}\n\\caption[Diffusion tensor indices pre- and post-suspension]{Diffusion tensor indices pre- and post-Unilateral Limb Suspension (ULLS).}\n\\label{tab: Karger1}\n\\begin{center}\n\\begin{threeparttable}\n\\begin{tabular}{@{}llll@{}}\n\\toprule[1pt]\\midrule[0.3pt]\n & & Pre-ULLS & Post-ULLS \\\\ \\midrule\n$\\lambda_1$\\tnote{$\\dagger$} & $\\left[\\SI{}{\\micro\\meter^2 / \\milli\\second}\\right]$ & 2.06 $\\pm$ 0.11 & 1.91 $\\pm$ 0.15 \\\\[6pt]\n$\\lambda_2$\\tnote{$\\dagger$} & $\\left[\\SI{}{\\micro\\meter^2 / \\milli\\second}\\right]$ & 1.44 $\\pm$ 0.08 & 1.31 $\\pm$ 0.11 \\\\[6pt]\n$\\lambda_3$\\tnote{$\\dagger$}& $\\left[\\SI{}{\\micro\\meter^2 / \\milli\\second}\\right]$ & 1.30 $\\pm$ 0.11 & 1.17 $\\pm$ 0.10 \\\\[6pt]\nADC\\tnote{$\\dagger$} \t&\t& 1.60 $\\pm$ 0.07 & 1.47 $\\pm$ 0.11 \\\\[6pt]\nFA \t\t\t\t\t&\t& 0.25 $\\pm$ 0.04 & 0.27 $\\pm$ 0.04 \\\\ \\midrule[0.3pt]\\bottomrule[1pt]\n\\end{tabular}\n\\begin{tablenotes}[flushleft]\\footnotesize\n\\item[$\\dagger$] significant difference between pre- and post-suspension\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\vspace{-0.2cm}\n\\end{table}\n%=========================================================\nIn addition to the average over all subjects listed in Table~\\ref{tab: Karger1}, the DTI indices (eigenvalues, ADC, FA and CP) pre- and post-suspension are also plotted for each subject individually in Figure~\\ref{fig: KargerPlots}. \nThis was done to confirm the consistency of DTI changes with suspension across the seven subjects and hence the validity of the data.\nPre- and post- values for each subject are connected for ease of visualization of the changes in the indices with suspension. \nFor most of the DTI indices the majority of subjects show the same direction of changes: e.g., six out of seven subjects show a decrease in $\\lambda_1, \\lambda_2$, ADC, and an increase in FA, while five out of seven subjects show a decrease in $\\lambda_3$ and in CP.\n%*********************************************************\n\\begin{figure}[!htb]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[width=\\textwidth]{Figures/KargerPlots.pdf}\n\\caption[Individual plots of the DTI indices pre- and post-suspension]{Individual plots of the DTI indices pre- and post-suspension.}\n\\label{fig: KargerPlots}\n\\end{figure}\n%*********************************************************\nTable~\\ref{tab: Karger2} lists the model parameters obtained by the search of the parameter space for the values of ($\\alpha$, $d$, $\\tau_{\\mathrm{in}}$, $\\nu_{\\mathrm{in}}$, $\\nu_{\\mathrm{col}}$) that yielded the lowest value of the error index~\\cite{RND12}. \nThe increase in the ellipticity index with disuse indicates that the fiber becomes more circular, while the decrease in muscle fiber diameter is as anticipated with atrophy. \n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Discussion}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nWhile it is known that the quadriceps muscles show greater atrophy than the medial gastrocnemius in aging and are a better correlate to physical function~\\cite{RND27}, the motivation for the focus on the MG in the current study is based on the ease of performing functional MRI studies on the MG for correlation to structural MRI studies~\\cite{RNSS4, RNCS4, Malis:2018fr, RNS16}. \nFurther, calf circumference (and by extension calf muscle mass as well) has been shown in a large-scale study to provide information on muscle related disability and physical function~\\cite{RND30}.\n%-new paragraph-%\n\n%-new paragraph-%\nMuscle force loss was approximately greater by a factor of 3 compared to reduction in muscle mass. \nPrior work on gravitational unloading effects on fast and slow rat hind-limb muscles data indicate that predominantly slow muscles are more responsive to unloading than predominantly fast muscles~\\cite{RND31}.\nSince the soleus has a higher proportion of slow twitch muscles compared to the gastrocnemius muscles, it would be anticipated that the soleus would show the highest atrophy (volume decrease). \nHowever, comparing \\% volume changes in the three muscles in the current study, larger changes were seen in the gastrocnemius muscles. \nIt should be noted though that the soleus muscle had the highest changes in terms of absolute volume changes comparing pre- and post-suspension muscles.\n%-new paragraph-%\n\n%-new paragraph-%\nComparing the values of DTI indices in the current study with that reported by Karampinos~et~al.~\\cite{RND12}, there are considerable differences in FA and in CP between the two studies.\nThe reasons for these differences may arise from differences in the measurement methodology.\nA simulated echo EPI diffusion weighted sequence was used by Karampinos~et~al.~\\cite{RND12} in contrast to the current study that uses a spin echo EPI diffusion weighted sequence.\nIt has been shown earlier that the simulated echo is better for diffusion imaging of muscle than a spin echo~\\cite{RND32}. \nHowever, the sequence used in~\\cite{RND12} has a TE of $\\SI{52}{\\milli\\second}$~\\cite{RND12} which is really long for a stimulated echo that has half the signal of a spin echo with a similar TE.\nThis raises the issue that the SNR of the sequence used in~\\cite{RND12} may not have been high and low SNR has been shown to bias the estimation of DTI indices~\\cite{RND33}.\nFurther, values reported for the eigenvalues and FA for the MG are in good agreement between the current study with another recent rigorous study that evaluated DTI indices for voxels with SNR $> 20$~\\cite{RND33}.\n%-new paragraph-%\n\n%-new paragraph-%\nPrevious studies on denervation induced atrophy on rodent models showed a reduction in $\\lambda_2$ and $\\lambda_3$, an increase in fractional anisotropy while $\\lambda_1$ was unchanged~\\cite{RND8}. \nIt is well accepted that DTI measurements in muscle reflect water diffusion in the intracellular space, thus the authors of this paper attributed the decrease in the secondary and tertiary eigenvalues to a decrease in the diameter of myofibers (i.e., myofiber atrophy). \nIn another study, the triceps surae was assessed with DTI after Achilles tenotomy in rats~\\cite{RND9}. \nBoth muscle atrophy as well as a decrease in plantarflexion force was seen post-tenotomy. \nSignificant decreases in $\\lambda_1$, $\\lambda_2$, and $\\lambda_3$ and an increase in FA were seen between the control and treatment sides at four weeks post-tenotomy~\\cite{RND9}. \nThis is in contrast to denervation induced atrophy where no changes were seen in $\\lambda_1$.\nIn the present study, decreases in all three eigenvalues were significant. \nThe eigenvector corresponding to the lead eigenvalue is in the direction of the muscle fiber and thus $\\lambda_1$ represents diffusion along the long axis of the muscle fiber. \nSince the muscle fiber is orders of magnitude greater than the diffusion distances, changes in muscle fiber length are not expected to cause changes in $\\lambda_1$. \nFurther, changes in fiber cross-section diameters should also not affect $\\lambda_1$ as the diffusion direction is along the long axis of the muscle fiber. \nThus, the decrease in $\\lambda_1$ is difficult to explain. \nPossibly, the reduced diffusivity along the fibers' long axis is related to a loss in myosin content~\\cite{RNS10} and reduction in the packing density of actin filament proteins~\\cite{RND35} that result in structurally weakened sarcomeres. \nWhile the underlying physiological mechanisms are unclear, presented observations of a decrease in $\\lambda_1$ is similar to that seen in the tenotomy induced atrophy~\\cite{RND9}.\nThe decrease in $\\lambda_2$ and $\\lambda_3$ can potentially be related to a change in muscle fiber diameter and is similar to that seen in denervation induced atrophy~\\cite{RND8}. \nIt should also be noted that the largest decrease was seen in $\\lambda_2$. \nIf the secondary and tertiary eigenvalues reflect the long axis and short axis diameters respectively of the muscle fiber cross-section, then the muscle fiber is more circular post-suspension. \n%-new paragraph-%\n\n%-new paragraph-%\nIn this context, Karampinos~et~al.~\\cite{RND12} advanced the hypothesis that the elliptic fiber cross-section arises from a response to the mechanical stimulus as the fiber is strained more along one direction in the muscle fiber cross-section than in the orthogonal direction. \nWhen the mechanical stimulus is reduced or removed as with ULLS induced disuse, the preferential deformation along one axis in the fiber cross-section is removed.\nThis may well result in the fiber cross-section becoming more circular in the absence of a mechanical stimulus.\nAs opposed to the progressive circularity accompanying disuse-atrophy~\\cite{RND36},\nADC decreased significantly with suspension and may potentially arise from a combination of a decrease in inherent intracellular diffusivity (decrease in all three eigenvalues) and a decrease in muscle fiber size (decrease in secondary and tertiary eigenvalues).\nIn the context of a reduction of a mechanical stimulus with suspension, the discussion of findings from related studies using strain~/~strain rate tensor imaging is worthwhile~\\cite{RNSS4, RNCS4, Malis:2018fr, RNS16}. \nStrain and strain rate tensor imaging evaluate tissue deformation and the tensor provides information of the strain (or strain rate) in three orthogonal directions. \nThe primary, secondary, and tertiary eigenvectors of the strain rate tensor correspond approximately to the muscle fiber direction and to two orthogonal directions in the muscle fiber cross-section, respectively. \nWhen the muscle fiber contracts, a negative strain will be seen along the muscle fiber direction and accompanying this, a positive strain from radial expansion will be seen in the fiber cross-section.\nHowever, this positive strain is not symmetric in the fiber cross-section, with much larger expansion along one direction than the orthogonal direction; this asymmetry has been reported in a number of studies~\\cite{Malis:2018fr, RNS16, RNS31}.\n%-new paragraph-%\n\n%-new paragraph-%\nA recent study on strain rate tensor imaging after unilateral limb suspension reported that the fiber cross-section strain rate asymmetry decreased post-suspension~\\cite{Malis:2018fr}.\nThis decrease in strain rate asymmetry may also be tied to the structural response to the lack of mechanical stimulus.\nThis potentially may imply that the long axis of the elliptical cross-section (proportional to the secondary eigenvalue of the diffusion tensor) is a consequence of the external mechanical stimulus since the radial expansion is preferentially along this direction resulting in an elongation of the fiber cross-section in one direction. Once this stimulus is removed (as in limb suspension) the elongated axes is no longer preferentially stretched, and may potentially become less elliptical (more circular).\nThis is reflected in a lower CP value post-suspension. \nAnother potential explanation for the observed structural changes may be the altered muscle fiber-extracellular matrix interactions from limb suspension~\\cite{RNSS4, RNCS4}. \n%-new paragraph-%\n\n%-new paragraph-%\nThe values of $\\alpha$ reported in the literature for the vastus lateralis muscle are in the range of $0.4$ to $0.68$~\\cite{RND38, RND39}; the current value of $0.55$ for MG ellipticity in the pre-suspension case appears to be reasonable~\\cite{RND38, RND39}.\nKarampinos~et~al. report the value of $\\alpha$ from the fit to the diffusion model with a range of $0.6 - 0.9$; the lower end of this range is close to the current paper.\nThe MG muscle fiber diameter extracted from the model in the current study ($\\SI{82}{\\micro\\meter}$) is in agreement with values reported in the literature~\\cite{RND38} and also with the range ($70 - \\SI{110}{\\micro\\meter}$) determined for the MG in the earlier diffusion modeling study by Karampinos~et~al.~\\cite{RND12}.\nFurther, the focus of the modeling was on determining if meaningful changes in microstructure with suspension can be extracted from the experimentally determined diffusion eigenvalues.\nWith suspension, the ratio of the short axis to long axis muscle fiber length increases which implies that the fiber becomes more circular.\nIn the light of the hypothesis that the elliptical muscle fiber shape arises from the mechanical stimulus, the modeling result of a more circular muscle fiber can be explained by the lack of mechanical stimulus during the 4-week suspension.\nThe diameter of the muscle fiber decreases; this finding is consistent with muscle fiber atrophy. \nIn fitting both pre- and post-suspension DTI data, the search algorithm converged on the upper or lower limits for $\\tau_{\\mathrm{in}}, \\nu_{\\mathrm{in}}$ and $\\nu_{\\mathrm{col}}$.\nThus, the best-fit values for these parameters may be in error. \nThe NRMSE was quite flat with respect to these variables; this was noted for $\\tau_{\\mathrm{in}}$ by Karampinos~et~al.~\\cite{RND12}, and thus, fitting the available DTI data to this model does not allow one to draw conclusions regarding $\\tau_{\\mathrm{in}}, \\nu_{\\mathrm{in}}$ and $\\nu_{\\mathrm{col}}$.\n%-new paragraph-%\n\n%-new paragraph-%\nThere are limitations to the current study: the study population is small but since it is a longitudinal study (pre- and post-ULLS), statistical significance was reached for the changes in the eigenvalues. \nA one-point diffusion measurement is not ideal for extracting tissue parameters through modeling; future work using multi-point DTI data (obtained for example, by varying diffusion time and~/~or \\mbox{\\textit{b-}value}s) will allow more robust modeling to extract accurate values for these microstructural features. \nFurther, multi-point DTI data may allow greater flexibility to extend to single subject rather than limit to cohort modeling as in the current study.\nIt should also be noted that if the fit was performed independently on each subject, it would have allowed testing of the significant changes in the microstructural parameters with limb suspension.\nMeasurements on the contralateral leg would have provided information on the effects of a change in mechanical loading of the unsuspended leg.\nHowever, the current study was part of a larger protocol that besides DTI, included quantification of fat, connective tissue as well as functional measurements and acquiring data on the contralateral leg would have prolonged the scanning time excessively.\nFurther, imaging both legs at the same time was restricted by the customized coil that accommodated only one leg.\nHowever, the customized coil provided higher SNR than vendor coils and allowed the acquisition of the entire lower leg without repositioning the subject.\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Conclusion}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nThe current study shows that the DTI eigenvalues decrease with suspension induced disuse atrophy. \nExperimentally, the secondary eigenvalue showed the largest decrease with suspension.\nThis could potentially be related to a hypothesis that attributes the elliptical shape of the muscle fiber cross-section as arising from a response to an external load~\\cite{RND12}. \nWhile it is still a conjecture, the asymmetry of deformation in the fiber cross-section may result in one axis becoming longer than the orthogonal one and then unloading conditions can cause this shape asymmetry to be reduced. \nAs the secondary and tertiary eigenvalues are related to the fiber cross-section diameters, this results in larger changes in the secondary eigenvalue as the fiber reduces the most in diameter along this direction with unloading. \nAn exploratory modeling of the DTI data to extract microstructural parameters showed that it is feasible and the disuse atrophy related changes in muscle ellipticity, mean diameter, residence time, intracellular volume and collagen volume extracted from the model were physiologically reasonable.\n%=========================================================\n\\section{Random Permeable Barrier Modeling of Age Induced Changes in the Time Dependent Diffusion Eigenvalues}\n\\label{sec: STEAM RPBM}\n%=========================================================\nMuscle mass loss have been reported in the aging muscle and this is in part, responsible for functional loss. \nHowever, non-invasive monitoring of microstructural changes as in muscle fiber diameter with age has not been reported. \nCommonly employed diffusion weighted acquisition protocols collect data at a single diffusion time. \nThe time dependence of the diffusion tensor eigenvalues can provide additional information to improve quantification of tissue microstructure. \nA diffusion model is required to make inferences about the microstructure from the time dependent eigenvalue data.\nRandom permeability barrier model (RPBM) was recently introduced and applied to retrieve microscopic parameters such as membrane permeability and fiber diameter in human skeletal muscles~\\cite{NovikovRPBM, RND13}. \n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Random Permeable Barrier Model}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nThe model treats muscle as a volume with randomly oriented infinite flat semipermeable membranes. \nFor each membrane permeability ($\\kappa$) relates the difference in concentrations on both sides of the membrane and the flux through it which provides the following boundary conditions:\n%.........................................................\n\\begin{equation}\\label{eq: RPBM bc}\nD_0\\mathbf{n}\\left.\\frac{\\partial{c}}{\\partial r} \\right\\vert_{\\pm} = \\kappa \\left[ c|_+ - c|_- \\right]\n\\end{equation}\n%.........................................................\nRPBM model is characterized by three parameters: the free diffusion coefficient ($D_0$), membrane surface to volume ratio ($S/V$) and permeability ($\\kappa$). Time-dependent diffusion coefficient is found by solving the diffusion equation (Equation~\\ref{eq:Fick2}) and then averaging the result over disorder using real-space renormalization group~\\cite{NovikovRPBM}.\n%.........................................................\n\\begin{equation}\\label{eq: RPBM}\nD(t) = \\frac{D_0}{2\\pi t}\\int\\limits^{\\infty}_{-\\infty}\\dfrac{d\\omega}{(-i\\omega)^2}\\frac{e^{-i\\omega t}}{1 + \\zeta +2z_{\\omega}(1-z_{\\omega}) \\left[\\sqrt{1+\\zeta/\\left(1-z_{\\omega}\\right)}-1 \\right] }\n\\end{equation}\n%.........................................................\nwhere $z_\\omega = i \\sqrt{i \\omega t}$ and $\\zeta=(S/V)D_0/4\\kappa$ is an effective volume fraction occupied by membranes.\nThe integral~(\\ref{eq: RPBM bc}) can be integrated numerically as outlined in reference~\\cite{RND13}.\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Materials and Methods}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nImage acquisition protocol used a custom-built STEAM-EPI DTI sequence (Figure~\\ref{fig: STEAM_GE}) discussed in more details in section~\\ref{sec: EPI}. \nA water selective SPSP RF-pulse was used for fat suppression. \nSix non-collinear gradient directions with a nominal \\mbox{\\textit{b-}value} of $\\SI{400}{\\second \\per\\milli\\meter^2}$ were used to map the direction dependent diffusion at ten values of the diffusion time~$\\Delta$ ($\\SI{20}{\\milli\\second}$ to $\\SI{600}{\\milli\\second}$). \nImaging parameters were; echo time (TE):~$\\SI{32}{\\milli\\second}$, repetition time (TR):~$\\SI{4000}{\\milli\\second}$, signal averages (NEX):~$2$ , acquisition matrix size:~$80~\\times~80$, field of view (FOV):~$200 \\times 200 \\; \\SI{}{\\milli\\meter^2}$, 3 slices of $\\SI{5}{\\milli\\meter}$ thickness. \nDiffusion data were pre-processed for eddy current mis-registration and denoised by applying Joint-Rican LMMSE filter~\\cite{RND24} prior to computing the diffusion tensor and the diffusion eigenvalues. \nThe corrected full \\textit{b-}matrix that accounted for the diffusion, imaging and crusher gradients was calculated as outlined in subsection~\\ref{subsection: STEAM b value}. For the nominal $b = 0$ images, the \\mbox{\\textit{b-}value} varied from $\\SI{1.7}{\\second \\per \\milli\\meter^2}$ at $\\Delta = \\SI{20}{\\milli\\second}$ to $\\SI{22}{\\second\\per\\milli\\meter^2}$ at $\\Delta = \\SI{600}{\\milli\\second}$ and for the nominal $b = \\SI{400}{\\second\\per\\milli\\meter^2}$ images, the \\mbox{\\textit{b-}value} varied from $\\SI{371}{\\second\\per\\milli\\meter^2}$ at $\\Delta = \\SI{20}{\\milli\\second}$ to $\\SI{547}{\\second\\per\\milli\\meter^2}$ at $\\SI{600}{\\milli\\second}$, emphasizing the need for full \\textit{b-}matrix calculation. \nPrior to human scans a set of images was obtained in a water phantom following the same imaging protocol as described above. \nFractional anisotropy (FA) maps were calculated from the acquired data to validate diffusion tensor calculations at long diffusion times. \nFA maps side by side with the corresponding magnitude images at nominal \\mbox{\\textit{b-}value} for diffusion times 20 and $\\SI{600}{\\milli\\second}$ are show in the Figure~\\ref{fig:STEAM phantom}. \n%*********************************************************\n\\begin{figure}[!htb]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures/STEAM_Phantom.pdf}\n\\caption[Baseline magnitude images and FA colormaps of water phantom acquired with STEAM-EPI DTI image sequence at two diffusion times]{Baseline magnitude images and FA colormaps of water phantom acquired with STEAM-EPI DTI image sequence at two mixing times: 20 and $\\SI{590}{\\milli\\second}$.}\n\\label{fig:STEAM phantom}\n\\end{figure}\n%*********************************************************\n%-new paragraph-%\n\n%-new paragraph-%\nAll human imaging studies were performed after IRB approval on a $\\SI{3}{\\tesla}$ scanner (GE Medical Systems, WI, USA) on seven young ($31 \\pm 8$ years) and six senior ($75 \\pm 5$ years) subjects. \nThe RPBM model was fitted by fixing the free diffusion coefficient $D_0$ at the long diffusion time limit of the primary eigenvalue ($\\lambda_1$) to extract the following parameters: the membrane permeability ($\\kappa$), and the membrane surface to volume ratio ($S/V$). \nThe myofiber size (a) is derived from the surface to volume ratio as: $4V/S$. \nThe RPBM fits were made to the time dependence of the average of $\\lambda_2$, and $\\lambda_3$ ($D_{\\perp}$); the values of $D_{\\perp}$ were the average over the medial gastrocnemius (MG) muscle segmented from all three~slices.\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Results}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nFigure~\\ref{fig:STEAM_EV} shows parametric $D_\\parallel$ and $D_\\perp$ maps for a young and for a senior subject at two diffusion times.\nExcellent fat suppression was seen in all the STEAM DTI data. \nTypical eigenvalue images ($D_\\parallel \\equiv \\lambda_1$) and ($D_\\perp \\equiv (\\lambda_2 + \\lambda_3)/2$) at two values of mixing time ($20$ and $\\SI{590}{\\milli\\second}$) for a young (left panel) and a senior (right panel) subject. \nThe decrease in SNR with diffusion time can be visualized by the noisier images of the bottom panel. \n%*********************************************************\n\\begin{figure}[!htb]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures/STEAM_EV.pdf}\n\\caption[Diffusion tensor eigenvalues grayscale maps at two values of mixing time for a young and a senior subject]{Diffusion tensor eigenvalues $D_\\parallel$ and $D_\\perp$ grayscale maps at two values of mixing time (20 and $\\SI{600}{\\milli\\second}$) for a young (left) and a senior (right) subject. The decrease in SNR with TM can be visualized by the noisier images of the bottom panel.}\n\\label{fig:STEAM_EV}\n\\end{figure}\n%*********************************************************\n%=========================================================\n\\begin{table}[!htb]\n\\vspace{+0.2cm}\n\\begin{center}\n\\caption[Diffusion eigenvalues at different mixing times]{Diffusion eigenvalues ($\\lambda_1, \\lambda_2, \\lambda_3$) at different mixing times.}\n\\label{tab:RPBM}\n\\begin{tabular}{@{}ccccccccc@{}}\n\\toprule[1pt]\\midrule[0.3pt]\n\\multirow{2}{*}{$\\mathrm{TM}$ {[}ms{]}} & \\multicolumn{2}{c}{$\\lambda_1 \\quad [\\SI{}{\\micro\\meter^2\\per\\milli\\second}]$} & & \\multicolumn{2}{c}{$\\lambda_2 \\quad [\\SI{}{\\micro\\meter^2\\per\\milli\\second}]$} & & \\multicolumn{2}{c}{$\\lambda_3 \\quad [\\SI{}{\\micro\\meter^2\\per\\milli\\second}]$} \\\\ \\cmidrule(lr){2-3} \\cmidrule(lr){5-6} \\cmidrule(lr){8-9} \n & Young & Senior & & Young & Senior & & Young & Senior \\\\ \\cmidrule(){1-9}\n20 & 2.8 $\\pm$ 0.2 & 2.8 $\\pm$ 0.2 & & 2.4 $\\pm$ 0.1 & 2.3 $\\pm$ 0.3 & & 1.2 $\\pm$ 0.1 & 1.2 $\\pm$ 0.1 \\\\\n30 & 2.6 $\\pm$ 0.2 & 2.7 $\\pm$ 0.2 & & 2.0 $\\pm$ 0.1 & 2.1 $\\pm$ 0.3 & & 1.2 $\\pm$ 0.1 & 1.3 $\\pm$ 0.1 \\\\\n60 & 2.4 $\\pm$ 0.2 & 2.5 $\\pm$ 0.2 & & 1.9 $\\pm$ 0.2 & 1.8 $\\pm$ 0.2 & & 0.8 $\\pm$ 0.1 & 1.0 $\\pm$ 0.2 \\\\\n90 & 2.4 $\\pm$ 0.3 & 2.4 $\\pm$ 0.2 & & 1.7 $\\pm$ 0.2 & 1.7 $\\pm$ 0.2 & & 0.8 $\\pm$ 0.1 & 0.9 $\\pm$ 0.1 \\\\\n190 & 2.3 $\\pm$ 0.3 & 2.3 $\\pm$ 0.2 & & 1.7 $\\pm$ 0.1 & 1.7 $\\pm$ 0.2 & & 0.4 $\\pm$ 0.1 & 0.6 $\\pm$ 0.1 \\\\\n290 & 2.2 $\\pm$ 0.2 & 2.2 $\\pm$ 0.2 & & 1.8 $\\pm$ 0.1 & 1.7 $\\pm$ 0.1 & & 0.2 $\\pm$ 0.1 & 0.4 $\\pm$ 0.1 \\\\\n340 & 2.2 $\\pm$ 0.1 & 2.2 $\\pm$ 0.2 & & 1.2 $\\pm$ 0.1 & 1.3 $\\pm$ 0.1 & & 0.7 $\\pm$ 0.1 & 0.8 $\\pm$ 0.1 \\\\\n390 & 2.2 $\\pm$ 0.1 & 2.2 $\\pm$ 0.3 & & 1.3 $\\pm$ 0.2 & 1.3 $\\pm$ 0.2 & & 0.7 $\\pm$ 0.1 & 0.7 $\\pm$ 0.1 \\\\\n490 & 2.1 $\\pm$ 0.1 & 2.2 $\\pm$ 0.2 & & 1.3 $\\pm$ 0.2 & 1.3 $\\pm$ 0.1 & & 0.6 $\\pm$ 0.1 & 0.7 $\\pm$ 0.1 \\\\\n590 & 2.1 $\\pm$ 0.2 & 2.2 $\\pm$ 0.2 & & 1.3 $\\pm$ 0.2 & 1.3 $\\pm$ 0.1 & & 0.6 $\\pm$ 0.1 & 0.7 $\\pm$ 0.1 \\\\ \\midrule[0.3pt]\\bottomrule[1pt]\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n\\end{table}\n%=========================================================\nTable~\\ref{tab:RPBM} summarizes the eigenvalues for the MG muscle in the young and senior cohort at the ten diffusion times. \nLarger changes with diffusion time are seen in $\\lambda_2$, and $\\lambda_3$ (compared~to~$\\lambda_1$). \nThe RPBM fits to the time dependence of $D_\\perp$ for a subject from the young cohort and from the senior cohort are shown for the Figure~\\ref{fig:RPBM fit}. The large variation of $D_\\perp$ with TM is in contrast to the smaller variation of $D_\\parallel$ with TM.\nTable~\\ref{tab:RPBM2} is a list of model derived muscle microstructure parameters; volume fraction decreased with age while diffusion time and residence time in a cell increased with age while other model parameters such as the fiber size and membrane permeability increased with age but did not reach significance.\n%*********************************************************\n\\begin{figure}[!htb]\n\\vspace{+0.2cm}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures/RPBM_fit.pdf}\n\\caption[RPBM model fit to the time dependence of diffusion indices for a young and senior subjects]{RPBM model fit to the time dependence of diffusion indices for a young and senior subjects.}\n\\label{fig:RPBM fit}\n\\end{figure}\n%*********************************************************\n%=========================================================\n\\begin{table}[!htb]\n\\vspace{+0.2cm}\n\\begin{center}\n\\caption[Parameters of muscle tissue extracted from the RPBM model]{Parameters of muscle tissue extracted from the RPBM model.}\n\\label{tab:RPBM2}\n\\begin{tabular}{@{}lrr@{}}\n\\toprule[1pt]\\midrule[0.3pt]\n \t\t\t\t\t\t\t\t& \\multicolumn{1}{c}{Young} & \\multicolumn{1}{c}{Senior} \\\\ \\cmidrule(){1-3}\n$D_0 \\; \\left[\\SI{}{\\micro\\meter^2\\per\\milli\\second}\\right]$ & \\multirow{2}{*}{2.20 $\\pm$ 0.16} & \\multirow{2}{*}{2.18 $\\pm$ 0.24} \\\\\nfree diffusion \t\t\t\t\t\t\t\t& \t& \\\\[6pt]\n$\\zeta$ \t\t\t\t\t\t\t\t& \\multirow{2}{*}{3.26 $\\pm$ 1.62} & \\multirow{2}{*}{2.79 $\\pm$ 0.63} \\\\\nvolume fraction \t\t\t\t\t\t\t\t& \t& \\\\[6pt]\n$S/V \\; \\left[\\SI{}{\\micro\\meter^{-1}}\\right]$ & \\multirow{2}{*}{0.14 $\\pm$ 0.03} & \\multirow{2}{*}{0.13 $\\pm$ 0.02} \\\\\nsurface to volume ratio \t\t\t\t\t\t\t\t& \t& \\\\[6pt]\n$\\kappa \\; \\left[\\SI{}{\\micro\\meter / \\milli\\second}\\right]$\t\t& \\multirow{2}{*}{0.025 $\\pm$ 0.005}& \\multirow{2}{*}{0.027 $\\pm$ 0.009} \\\\\npermeability \t\t\t\t\t\t\t\t& \t& \\\\[6pt]\n$a \\; \\left[\\SI{}{\\micro\\meter}\\right]$ \t& \\multirow{2}{*}{28.63 $\\pm$ 4.94} & \\multirow{2}{*}{32.00 $\\pm$ 3.72} \\\\\nmyofiber diameter \t\t\t\t\t\t\t\t& \t& \\\\[6pt]\n$\\tau_{R} \\; \\left[\\SI{}{\\milli\\second}\\right]$ \t\t\t\t\t\t\t\t& \\multirow{2}{*}{552.2 $\\pm$ 122.1}& \\multirow{2}{*}{650.6 $\\pm$ 159.2} \\\\\nresidence time \t\t\t\t\t\t\t\t& \t& \\\\[6pt]\n$\\tau_{D} \\; \\left[\\SI{}{\\milli\\second}\\right]$ \t\t\t\t\t\t\t\t& \\multirow{2}{*}{194.5 $\\pm$ 71.9} & \\multirow{2}{*}{238.1 $\\pm$ 50.7} \\\\\ndiffusion time \t\t\t\t\t\t\t\t& \t& \\\\ \\midrule[0.3pt]\\bottomrule[1pt]\n\\end{tabular}\n\\end{center}\n\\vspace{-0.2cm}\n\\end{table}\n%=========================================================\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Discussion} \n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nThe values of the model derived parameters are in general agreement with that reported for the MG in an earlier study. \nThe RPBM model yields a lower volume fraction for the aging muscle which is a measure of the membrane's ability to hinder diffusion. \nIt is possible that aging muscle may have compromised sarcolemma integrity that makes it more permeable and thus poses less of a barrier to diffusion. \nBased on the fact that muscle atrophies with age a decrease in fiber diameter with age is expected. \nHowever fiber diameters derived from the RPBM model did not show anticipated changes. \nA sarcolemma with compromised integrity (as seen here in aging muscle) may potentially affect lateral transmission of force, the latter is mediated by proteins in the sarcolemma.\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\subsection{Conclusions} \n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nThe study has demonstrated the potential of diffusion modeling to extract muscle tissue parameters from time dependent $D_\\perp$ derived from DTI data and its potential application it to study age related skeletal muscle remodeling.\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\n\\section{Acknowledgments}\n%~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\nSection~\\ref{sec: DTI ULLS} is a reprint of material, with additional details provided in subsection~\\ref{subsec: bicompart}, as it appears in: V.~Malis, U.~Sinha, R.~Csapo, M.~Narici, E.~Smitaman, and S.~Sinha, ``Diffusion tensor imaging and diffusion modeling: Application to monitoring changes in the medial gastrocnemius in disuse atrophy induced by unilateral limb suspension,'' \\emph{J. Magn. Reson. Imaging}, vol. 49, no. 6, pp. 1655-1664, Dec. 2018.\nThe author of the dissertation was the primary author of this paper.\n%-new paragraph-%\n\n%-new paragraph-%\nSection~\\ref{sec: STEAM RPBM} is a reprint of material, with additional details, as it appears in: V.~Malis, S.~Sinha, E.~Smitaman, and U.~Sinha, ``Skeletal Muscle Diffusion Modeling to Identify Age Related Remodeling of Muscle Microstructure,'' \\emph{Proceedings of the International Society of Magnetic Resonance in Medicine}, Sydney, 2020.\nThe author of the dissertation was the primary author of this abstract.", "meta": {"hexsha": "3b3b7434facb881329cbe1b636ffb812cba15b05", "size": 63813, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapter6.tex", "max_stars_repo_name": "vmalis/PhDissertation", "max_stars_repo_head_hexsha": "7c6a343f902eb7a76d3f0ceca9aeb54def160c59", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapter6.tex", "max_issues_repo_name": "vmalis/PhDissertation", "max_issues_repo_head_hexsha": "7c6a343f902eb7a76d3f0ceca9aeb54def160c59", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapter6.tex", "max_forks_repo_name": "vmalis/PhDissertation", "max_forks_repo_head_hexsha": "7c6a343f902eb7a76d3f0ceca9aeb54def160c59", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 104.7832512315, "max_line_length": 720, "alphanum_fraction": 0.6765235924, "num_tokens": 16367, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6548947155710233, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34533680459456245}} {"text": "There are many types of neural networks and each of them perform differently according to the task. Considering the low vocabulary of our task and the large amount of data, we'll be looking to search for patterns through the identification of templates with neural networks. Therefore, CNN comes immediately to mind. But many works suggests the use of recurrent neural networks. Therefore we'll focus on those two types of networks that seems to be the most promising. Finally, MLPs are usually mentionned but not for their performances, mostly because they are easy to implement and it's always a nice comparison.\n\n\\vspace{5mm}\n\nKeyword spotting task are extremely fit to CNNs according to many research \\cite{CNN}, \\cite{init}. A DNN cannot take advantage of the topology of the signal, however CNNs are extremely good at acoustic modeling which permits to give a good representation of a speech spectrum. The differences of frequencies between each speaker and also the timing differences between each examples are known to be easily handled by CNNs.\n\n\\vspace{5mm}\n\nRNNs and especially LSTMs are also fit to this task because of their ability to take into account time sequences through feedback connections. They also benefits from the combination of lstm layers with deep or convolutionnal layers \\cite{graves}. But one of the main problem encountered with RNNs, is that they are slow for many reasons (no parrellelization, slow learning, ...). Therefore the trend is to move away from recurrent structures.\n\n\n\n\\section{MLP}\n\nNeural networks are used to approximate functions which solving a non linear optimization problem that tries to find the minimum \"distance\" between the approximation and the real function. To do so, training algorithms try to find the set of weights that minimize the \"distance\" represented by the loss function.\n\n\\vspace{5mm}\n\nMLPs are one of the most simple form yet usually reliable of neural networks. They are composed of one input layer, one output layer and at least one hidden layer. They are feed forward that means that the flow of information goes in only one direction : from input layer to output layer.\n\n\n\\section{CNN}\nCNNs are an evolution of MLPs. Indeed, usually they have an MLP base that is situated at the end of the network but they also have some convolutions layers. the role of those convolution layers will be to split the input in order to find interesting information in a small subset of it. This will make CNN be able space dependecies in 2-d data. In addition of that, CNNs are usually more effective than MLPs in term of computation. \n\n\\section{LSTM}\n\nLSTM structures (Long short term memory) are a chain like network where you gives the output of a block as an input of the next one. The idea being similar to recurrent neural networks, it's about creating an artificial memory of how inputs evoluate over time. It is extremely interesting for tasks with temporal dependencies such as speech recognition. The improvement over the RNN structures is the forget gate that allows to manipulate weights of recurrents connections in order to face the vanishing gradient problem. Therefore \n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=1\\textwidth]{chapters/pictures/lstm_unit.png}\n \\caption{Inside of a LSTM unit}\n \\label{fig:mlp}\n\\end{figure}\n", "meta": {"hexsha": "995dc1ca754746200da35d297183ef662e7efafc", "size": 3324, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "report/chapters/3-Literrature_review.tex", "max_stars_repo_name": "TMarquet/speech_recognition", "max_stars_repo_head_hexsha": "06cf66b89e706306d676e127126d525bd18588f0", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "report/chapters/3-Literrature_review.tex", "max_issues_repo_name": "TMarquet/speech_recognition", "max_issues_repo_head_hexsha": "06cf66b89e706306d676e127126d525bd18588f0", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "report/chapters/3-Literrature_review.tex", "max_forks_repo_name": "TMarquet/speech_recognition", "max_forks_repo_head_hexsha": "06cf66b89e706306d676e127126d525bd18588f0", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 94.9714285714, "max_line_length": 614, "alphanum_fraction": 0.7999398315, "num_tokens": 695, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.640635854839898, "lm_q2_score": 0.5389832206876841, "lm_q1q2_score": 0.34529197632961595}} {"text": "\\documentclass[a4paper]{article}\n\n% Language and Font Encoding\n\\usepackage[english]{babel}\n%\\usepackage[utf8x]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage{listings}\n\\usepackage{float}\n\\usepackage{caption} % For captions\n\\usepackage{fancyhdr} % For header and footer\n\\usepackage{rotating}\n\\usepackage{amssymb}\n%\\usepackage{varwidth}\n\\usepackage[colorlinks=true, allcolors=blue]{hyperref}\n\n% Page size and margins\n\\usepackage[a4paper,top=3cm,bottom=2cm,left=2cm,right=2cm,marginparwidth=1.75cm]{geometry}\n\n% Useful Packages\n\\usepackage{amsmath}\n\\usepackage{graphicx}\n\\usepackage{amsthm}\n%\\usepackage{bbm}\n\\usepackage{siunitx}\n\\usepackage{cite}\n\\DeclareMathOperator*{\\argmax}{argmax}\n\\DeclareMathOperator*{\\argmin}{argmin}\n\\DeclareMathOperator*{\\softmax}{softmax}\n\n\\newcommand{\\squeezeup}{\\vspace{-2.5mm}}\n\\newcommand{\\norm}[1]{\\left\\lVert#1\\right\\rVert}\n\n\n\\pagestyle{fancy}\n\\lhead{Appendix: Topology}\n\\rhead{}\n\n\\begin{document}\n\n\\title{Optimization on Manifolds, Appendix: Topology}\n\\author{Ibrahim Akbar}\n\\maketitle\n\n\\section{Definitions}\nThis serves as a light introduction into the topic of topology, but is not sufficient to have a complete or deep understanding. For further information, I suggest reading James Munkres' \\textit{Topology} text for a thorough introduction.\\\\\n\n\\noindent Just as an n-dimensional vector space is an abstraction of $\\mathbb{R}^{n}$, a topology on a set $X$ is an abstraction of open sets in $\\mathbb{R}^{n}$.\\\\\n\n\\noindent A \\textbf{topology} on a set $X$ is a collection $\\mathcal{T}$ of open subsets of $X$ such that:\n\n\\begin{enumerate}\n\\item $X,\\{0\\}\\in\\mathcal{T}$\n\\item Given a subcollection $\\mathcal{S}$ of $\\mathcal{T}$, $\\cup_{i}S_{i}\\in\\mathcal{T}$\n\\item Given a finite subcollection $\\mathcal{S}$ of $\\mathcal{T}$, $\\cap_{i}S_{i}\\in\\mathcal{T}$\n\\end{enumerate}\n\n\\noindent Thus a topology is the couple $(X,\\mathcal{T})$, but when notation allows it will be abbreviated to $X$. Currently, the definition for topology is not well-defined as we do not know what \"open\" means.\\\\\n\n\\noindent Let $X$ be a topological space. Let $A$ be a subset of $X$. A \\textbf{neighborhood} of a point $x$ on $X$ is a subset $\\mathcal{V}$ that includes an open set, $\\mathcal{U}$ containing $x$.\n$$\nx\\in\\mathcal{U}\\subset\\mathcal{V}\\subseteq X\n$$\n\n\\noindent A \\textbf{limit point} or \\textbf{accumulation point} of a subset $A$ of $X$ is a point $x$ of $X$ such that every neighborhood of $x$ intersects $A$ in some point \\textit{other than} $x$. Note that it is important to distinguish between intersecting at $x$ and somewhere else. If the restriction of intersecting at points other than were to be removed $x$ would be a \\textbf{point of closure}. Thus every point of closure is a limit point but not necessarily every limit point is a point of closure.\\\\\n\n\\noindent A subset is \\textbf{closed} if and only if it contains all it's limit points. An open set is naturally one that does not contain all it's limit points.\\\\\n\n\\noindent A sequence of points $\\{x_{k}\\}_{k=1,2,\\ldots}$ of $X$ \\textbf{converges} to a point $x\\in X$ if for every neighborhood $\\mathcal{U}$ of $x$ there exists a positive integer $K$ such that $x_{k}$ belongs to $\\mathcal{U}$ for all $k\\geq K$.\\\\\n\n\\noindent Since topology needs to satisfy relatively few axioms it is natural that properties that may hold for $\\mathbb{R}^{n}$ may not hold for $X$. As an example a \\textbf{singleton} which is a set containing only one element may not be closed in the topological sense. Take for example the topology of $[-1,1]$. The open sets are $[-1, a)$ for $a > 0$,$(b,a)$ for $a < 0, b > 0$, and $(b, 1]$ for $b < 0$ \\textit{(Why?)}. To avoid such situations \\textbf{separation axioms} have been introduced to make a distinction between topologies.\\\\\n\n\\noindent A topological space $X$ is $T_{1}$; \\textbf{accessible} or \\textbf{Fr\\'{e}chet}, if for any distinct points $x$ and $y$ of $X$ there is an open set containing $x$ and not $y$. (Every singleton is closed.)\\\\\n\n\\noindent A topological space $X$ is $T_{2}$, \\textbf{Hausdorff}, if any two distinct points of $X$ have disjoint neighborhoods. A Hausdorff topology means that any sequence of points on $X$ converges to at most one point of $X$.\\textit{(Why?)}\\\\\n\n\\noindent Let $\\mathcal{T}_{1}$ and $\\mathcal{T}_{2}$ be topological spaces of $X$. If $\\mathcal{T}_{1}\\subseteq\\mathcal{T}_{2}$ then $\\mathcal{T}_{2}$ is said to be \\textbf{finer}.\\\\\n\n\\newpage\n\\noindent A \\textbf{base} or \\textbf{basis} for a topology on set $X$ is a collection $\\mathcal{B}$ of subsets of $X$ such that\n\\begin{enumerate}\n\\item each $x\\in X$ belongs to at least one element in $\\mathcal{B}$\n\\item if $x\\in(B_{1}\\cap B_{2})$ with $B_{1},B_{2}\\in\\mathcal{B}$, then there exists $B_{3}\\in\\mathcal{B}$ such that $x\\in B_{3}\\subseteq B_{1}\\cap B_{2}$.\n\\end{enumerate}\n\n\\noindent If $\\mathcal{B}$ is a base for topology $\\mathcal{T}$ then $\\cup_{i}\\mathcal{B} = \\mathcal{T}$ and a topology is denoted as \\textbf{second-countable} if it's base is finite.\\\\\n\n\\noindent If $X$ and $Y$ are topological spaces then the \\textbf{product topology} $X\\times Y$ has the base $\\mathcal{B}$ which is the collection of all sets of the form $U\\times V$ where $U$ is an open set of $X$ and $V$ is an open set of $Y$.\\\\\n\n\\noindent If $Y$ is a subset of a topological space $(X,\\mathcal{T})$. Then $\\mathcal{T}_{Y} = \\{Y\\cap U\\mid U\\in\\mathcal{T}\\}$ is a topology on $Y$ called the \\textbf{subspace topology}.\\\\\n\n\\noindent A collection $\\mathcal{A}$ of subsets of $X$ \\textbf{cover} or is a \\textbf{covering} of $X$ if the union of the elements of $\\mathcal{A}$ is equal to $X$.\\\\\n\n\\noindent \\textbf{Heine-Borel Theorem}: A subset of $\\mathbb{R}^{n}$ with the subspace topology is compact if and only if it is closed and bounded.\\\\ \n\n\\end{document}\n\n", "meta": {"hexsha": "c6b669774dcccf3e5e19be5d0a27db50c7f413f3", "size": 5775, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "files/notes/tex/opti-manifold/append/opti-manifold-append-topo.tex", "max_stars_repo_name": "ibrahimkakbar/ibrahimkakbar.github.io", "max_stars_repo_head_hexsha": "5e6b0ea67f5e5f8f3a7bb4394095ea7f7992673a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "files/notes/tex/opti-manifold/append/opti-manifold-append-topo.tex", "max_issues_repo_name": "ibrahimkakbar/ibrahimkakbar.github.io", "max_issues_repo_head_hexsha": "5e6b0ea67f5e5f8f3a7bb4394095ea7f7992673a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "files/notes/tex/opti-manifold/append/opti-manifold-append-topo.tex", "max_forks_repo_name": "ibrahimkakbar/ibrahimkakbar.github.io", "max_forks_repo_head_hexsha": "5e6b0ea67f5e5f8f3a7bb4394095ea7f7992673a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.5360824742, "max_line_length": 543, "alphanum_fraction": 0.7168831169, "num_tokens": 1822, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832206876841, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.34529196893354036}} {"text": "\\subsection{Affine coordinate systems}\\label{subsec:affine_coordinate_system}\n\n\\begin{remark}\\label{rem:affine_coordinate_system_concept}\n Most humans possess a strong intuition for visual information like drawings or diagrams. A paper or a painting is only a medium for communicating information and emotions. \\Fullref{def:euclidean_plane/figures} contains some highlighted curves that our mind maps to abstract geometric figures, without considering the size limitations of the page, the precision of the drawings or the thickness of the lines.\n\n \\begin{figure}[b]\n \\centering\n \\text{\\todo{Add diagram}}\\iffalse\\begin{mplibcode}\n u := 1cm;\n\n beginfig(1);\n draw (0, -1) * u -- (3, 0) * u;\n draw (-1, 2) * u -- (3, 1) * u -- (1, 3) * u -- cycle;\n draw fullcircle scaled 1.5u shifted ((0, 0.5) * u);\n endfig;\n \\end{mplibcode}\\fi\n \\caption{A triangle, a circle and a line in the Euclidean plane.}\\label{def:euclidean_plane/figures}\n \\end{figure}\n\n Our goal is to map these visualizations to the concept of vector spaces. Formalisms at the level of formal \\hyperref[def:first_order_language]{logic} will not be stated because we only want to sketch some high-level concepts. We only give definitions that are strictly necessary, plane geometry itself is described in \\fullref{subsec:analytic_geometry_in_the_plane}. We will proceed as follows:\n\n \\begin{itemize}\n \\item Define an affine plane in \\fullref{def:affine_plane} with auxiliary definitions.\n \\item Describe the Euclidean plane \\( A_2 \\) in \\fullref{def:euclidean_plane} as a very special affine plane.\n \\item Give additional definitions for the Euclidean plane in \\fullref{def:euclidean_plane_auxiliary_definitions}.\n \\item Define the set \\( F_2 \\) of free vectors over \\( A_2 \\) in \\fullref{def:euclidean_plane_free_vector}.\n \\item Show that \\( F_2 \\) is a two-dimensional vector space over \\( \\BbbR \\) in \\fullref{thm:euclidean_plane_factorization}.\n \\item Define coordinate systems that give explicit isomorphisms between \\( A_2 \\), \\( F_2 \\) and \\( \\BbbR^2 \\) in \\fullref{def:euclidean_plane_coordinate_system}.\n \\item Generalize these notions in \\fullref{rem:coordinate_systems}\n \\end{itemize}\n\\end{remark}\n\n\\begin{definition}\\label{def:affine_plane}\\mcite[1]{Hartshorne1967}\n An \\term{affine plane} consists of\n \\begin{itemize}\n \\item a set \\( X \\), whose elements are called \\term{points},\n \\item a family of subsets of \\( X \\), whose members are called \\term{lines}\n \\end{itemize}\n with the additional relations\n \\begin{itemize}\n \\item a \\term{parallel} relation \\( l \\parallel g \\) for lines that holds if either \\( l = g \\) or if they have no points in common,\n \\item a \\term{collinearity} relation for a set \\( B \\) of points that holds if \\( B \\) is a subset of some line,\n \\end{itemize}\n such that\n \\begin{thmenum}\n \\thmitem[def:affine_plane/A1]{A1} Given two distinct points, there exists only one line that contains both.\n \\thmitem[def:affine_plane/A2]{A2} Given a line \\( l \\) and a point \\( P \\not\\in l \\), there exists exactly one line \\( g \\parallel l \\) that contains \\( P \\).\n \\thmitem[def:affine_plane/A3]{A3} There exist three non-collinear points.\n \\end{thmenum}\n\\end{definition}\n\n\\begin{definition}\\label{def:euclidean_plane}\n The \\term{Euclidean plane} \\( A_2 \\) is a formalization of a straight infinite surface. An axiomatic definition can be found in \\cite{nLab:euclidean_geometry}. We will use that\n \\begin{itemize}\n \\item The Euclidean plane \\( A_2 \\) is an \\hyperref[def:affine_plane]{affine plane}\n \\item \\( A_2 \\) is a \\hyperref[def:complete_metric_space]{complete metric space} with distance \\( \\op{dist} \\).\n \\item There is a \\term{betweenness} relation for points that says if the point \\( R \\) is \\term{between} \\( P \\) and \\( Q \\).\n \\end{itemize}\n\n \\begin{figure}\n \\centering\n \\text{\\todo{Add diagram}}\\iffalse\\begin{mplibcode}\n input metapost/plotting;\n\n u := 1.5cm;\n\n beginfig(1);\n path l, g, h, P, Q, R;\n l = (0, -1) * u -- (3, 0) * u;\n draw l;\n label.top(\"$l$\", midpoint of l);\n\n g = (0, -2) * u -- (3, -1) * u;\n draw g;\n label.bot(\"$g$\", midpoint of g);\n\n h = (0, 0) * u -- (3, -2) * u;\n draw h;\n label.urt(\"$h$\", midpoint of h);\n\n P = dot shifted point 0.2 of h;\n fill P;\n label.llft(\"$P$\", midpoint of P);\n\n Q = dot shifted point 0.8 of h;\n fill Q;\n label.llft(\"$Q$\", midpoint of Q);\n\n R = dot shifted point 0.4 of h;\n fill R;\n label.llft(\"$R$\", midpoint of R);\n endfig;\n \\end{mplibcode}\\fi\n \\caption{Three lines and three points in the Euclidean plane. The lines \\( l \\) and \\( g \\) are collinear, while the point \\( R \\) is between \\( P \\) and \\( Q \\)}\\label{def:affine_plane/figure}\n \\end{figure}\n\\end{definition}\n\n\\begin{definition}\\label{def:euclidean_plane_auxiliary_definitions}\n We will also need the following definitions:\n \\begin{thmenum}\n \\thmitem{def:affine_plane/half_plane} Every line \\( l \\) gives rise to two (closed) \\term{half-planes} \\( H^+ \\) and \\( H^- \\) as follows:\n \\begin{itemize}\n \\item \\( H^+ \\cap H^- = l \\)\n \\item \\( H^+ \\cup H^- = A_2 \\)\n \\item If \\( P \\in H^+ \\setminus l \\) and \\( Q \\in H^- \\setminus l \\), then there is a point \\( R \\in l \\) between \\( P \\) and \\( Q \\)\n \\end{itemize}\n\n Note that the superscripts \\( + \\) and \\( - \\) are only for distinguishing between the two half-planes and are not assigned based on some property of the half-planes. See \\fullref{def:half_space} for a definition of a half-plane that actually has a concept of signs.\n\n \\begin{figure}\n \\centering\n \\text{\\todo{Add diagram}}\\iffalse\\begin{mplibcode}\n input metapost/plotting;\n\n u := 1cm;\n\n beginfig(1);\n input hatching;\n\n path l, Hp, Hm;\n l = (0, -1) * u -- (3, 0) * u;\n draw l;\n\n Hp = l -- (3, 0.5) * u -- (0, 0.5) * u -- cycle;\n hatchfill Hp withcolor (45, 1mm, -.5bp);\n label.ulft(\"$H^+$\", startpoint of l);\n\n Hm = l -- (3, -1.5) * u -- (0, -1.5) * u -- cycle;\n hatchfill Hm withcolor (135, 1mm, -.5bp);\n label.lrt(\"$H^-$\", endpoint of l);\n endfig;\n \\end{mplibcode}\\fi\n\n \\caption{Differently hatched half-planes in the Euclidean plane.}\\label{def:affine_plane/bound_vector/half_plane}\n \\end{figure}\n\n \\thmitem{def:affine_plane/ray} Every line \\( l \\) and every point \\( R \\) give rise to two (closed) \\term{rays} \\( l^+ \\) and \\( l^- \\) as follows:\n \\begin{itemize}\n \\item \\( l^+ \\cap l^- = \\{ R \\} \\) are disjoint\n \\item \\( l^+ \\cup l^- = l \\)\n \\item If \\( P \\in l^+ \\setminus \\{ R \\} \\) and \\( Q \\in l^- \\setminus \\{ R \\} \\), then \\( R \\) is between \\( P \\) and \\( Q \\)\n \\end{itemize}\n\n The rays \\( l^+ \\) and \\( l^- \\) are called \\term{opposite} of each other.\n\n We say that \\( R \\) is the \\term{vertex} of \\( l^+ \\) and \\( l^- \\).\n\n See \\fullref{def:geometric_ray} for a definition of a ray that actually has a concept of signs.\n\n \\begin{figure}\n \\centering\n \\text{\\todo{Add diagram}}\\iffalse\\begin{mplibcode}\n input metapost/plotting;\n\n u := 1cm;\n\n beginfig(1);\n path l, R;\n\n l = (0, -1) * u -- (3, 0) * u;\n drawdblarrow l;\n label.lft(\"$l^-$\", startpoint of l);\n label.rt(\"$l^+$\", endpoint of l);\n\n R = dot shifted midpoint of l;\n fill R;\n label.bot(\"$R$\", midpoint of R);\n endfig;\n \\end{mplibcode}\\fi\n\n \\caption{Opposite rays in the Euclidean plane.}\\label{def:affine_plane/day/figure}\n \\end{figure}\n\n \\thmitem{def:affine_plane/rays_unidirectional} Two rays are said to be \\term{unidirectional} if there exists a line distinct from the lines containing the rays, such that both rays are contained in the same half-plane with respect to the line.\n\n \\thmitem{def:affine_plane/bound_vector} An ordered pair \\( \\vect{PQ} \\) of points is called a \\term{bound vector}. The point \\( P \\) is called the \\term{beginning} of \\( \\vect{PQ} \\) and \\( Q \\) is called the \\term{end} of \\( \\vect{PQ} \\).\n\n \\begin{figure}\n \\centering\n \\text{\\todo{Add diagram}}\\iffalse\\begin{mplibcode}\n input metapost/plotting;\n\n u := 0.75cm;\n\n beginfig(1);\n path P, Q, R, PQ, PR;\n\n PQ = (0, -1) * u -- (3, 0) * u;\n drawarrow PQ;\n label.bot(\"$\\vect{PQ}$\", midpoint of PQ);\n\n P = dot shifted startpoint of PQ;\n fill P;\n label.bot(\"$P$\", midpoint of P);\n\n Q = dot shifted endpoint of PQ;\n label.bot(\"$Q$\", midpoint of Q);\n\n PR = (0, -1) * u -- (-2, 0.5) * u;\n drawarrow PR;\n label.llft(\"$\\vect{PR}$\", midpoint of PR);\n\n R = dot shifted endpoint of PR;\n label.llft(\"$R$\", midpoint of R);\n endfig;\n \\end{mplibcode}\\fi\n\n \\caption{Bound vectors in the Euclidean plane can be regarded as oriented line segment.}\\label{def:affine_plane/bound_vector/figure}\n \\end{figure}\n \\end{thmenum}\n\\end{definition}\n\n\\begin{definition}\\label{def:euclidean_plane_free_vector}\n We say that the bound vectors \\( \\vect{P_1 Q_1} \\) and \\( \\vect{P_2 Q_2} \\) in \\( A_2 \\) are \\term{congruent} if \\( \\op{dist}(P_1, Q_1) = \\op{dist}(P_2, Q_2) \\) and if the rays \\( r_i, i = 1, 2 \\) beginning at \\( P_i \\) and containing \\( Q_i \\), are unidirectional.\n\n We define \\term{free vectors} as \\hyperref[thm:equivalence_partition]{equivalence classes} of bound vectors by this congruence relation. We denote the corresponding equivalence partition by \\( F_2 \\).\n\\end{definition}\n\n\\begin{theorem}\\label{thm:euclidean_plane_factorization}\n The set \\( F_2 \\) of free vectors over \\( A_2 \\) is a two-dimensional \\hyperref[def:vector_space]{vector space} over \\( \\BbbR \\) with the following operations:\n \\begin{thmenum}\n \\thmitem{thm:euclidean_plane_factorization/sum} We define the \\term{sum} of the cosets \\( [\\vect{PQ}] \\) and \\( [\\vect{QR}] \\) as the coset \\( [\\vect{PR}] \\).\n\n \\thmitem{thm:euclidean_plane_factorization/scalar_product} We define the \\term{scalar multiplication} of \\( \\lambda \\in \\BbbR \\) with the coset \\( [\\vect{PQ}] \\) to be the coset \\( [\\vect{PR}] \\), where \\( \\vect{PR} \\) is the unique vector that is unidirectional with \\( \\vect{PQ} \\) and \\( \\op{dist}(P, R) = \\lambda \\op{dist}(P, Q) \\).\n \\end{thmenum}\n\\end{theorem}\n\\begin{proof}\n Proving the well-definedness of the operations and verifying that \\( F_2 \\) is a two-dimensional vector space requires a lot of work and the proof is skipped.\n\\end{proof}\n\n\\begin{definition}\\label{def:euclidean_plane_coordinate_system}\n Just because \\fullref{thm:euclidean_plane_factorization} states that the set \\( F_2 \\) of free vectors is a vector space does not mean that we can work with it as with \\( \\BbbR^2 \\). \\Fullref{thm:finite_dimensional_spaces_are_isomorphic} says that \\( F_2 \\) is isomorphic to \\( \\BbbR^2 \\), however the proof requires the \\hyperref[def:zfc/choice]{axiom of choice}. The concrete way to select a basis in \\( F_2 \\) is through coordinate systems.\n\n Somewhat confusingly, we define coordinate systems over \\( A_2 \\) rather than over \\( F_2 \\), but this will, soon be justified.\n\n A \\term{coordinate system} \\( Oxy \\) in \\( A_2 \\) is a choice of\n \\begin{thmenum}\n \\thmitem{def:euclidean_plane_coordinate_system/origin} A point \\( O \\in A_2 \\), called the \\term{origin} of the coordinate system.\n \\thmitem{def:euclidean_plane_coordinate_system/basis} An \\hyperref[def:partially_ordered_set]{ordered} \\hyperref[def:hamel_basis]{basis} \\( (x, y) \\) of \\( F_2 \\), called the \\term{basis} of \\( Oxy \\).\n \\end{thmenum}\n\n What we achieve through the choice of \\( O \\) is that, for each point \\( P \\in A_2 \\), we select the bound vector \\( \\vect{OP} \\in V_2 \\), called the \\term{radius vector} of \\( P \\). This injects \\( A_2 \\) into \\( V_2 \\), however if we take the free vector \\( [\\vect{OP}] \\), we instead obtain a bijection between \\( A_2 \\) and \\( F_2 \\).\n\n Now that we have a correspondence between \\( A_2 \\) and \\( F_2 \\), coordinates for the point \\( P \\) are defined simply as the \\hyperref[def:module_basis_projection]{coordinates} of \\( [\\vect{OP}] \\) with respect to the basis \\( (x, y) \\).\n\n Thus, the pair \\( (A_2, Oxy) \\) has an explicit isomorphism with \\( \\BbbR^2 \\).\n\n The \\term{coordinate axis} of \\( x \\) is the unique \\hyperref[def:affine_plane/ray]{ray} starting at \\( O \\) and containing the end of \\( x \\). It is called the \\term{abscissa}. The coordinate axis of \\( y \\) is called the \\term{ordinate}.\n\\end{definition}\n\n\\begin{remark}\\label{rem:coordinate_systems}\n We sketched how to embed mental images of planes into \\( \\BbbR^2 \\), however in mathematics we are often interested in the opposite: given a set of points in \\( \\BbbR^2 \\), visualize them on a screen or paper and then absorb the the resulting image in our brain.\n\n This is one of the most powerful constructions in mathematics, yet it is, so intuitive that it is not really given a lot of attention, at least until generalizations are required. Given any vector space \\( V \\) in the sense of \\fullref{def:vector_space}, we want a way to assign a pair of numbers to each vector in \\( V \\). This is only possible if \\( \\dim V = 2 \\), however we can generalize this to tuples of coordinates via bases - see \\fullref{def:hamel_basis}. This well for finitely dimensional vector spaces, however we need to generalize these notion for infinitely dimensional vector spaces and general modules over \\hyperref[def:module]{rings}. This allows us to generalize coordinates further to manifolds - see \\fullref{def:topological_manifold}.\n\n See \\fullref{subsec:vector_space_geometry} for immediate generalizations of the concepts introduced here.\n\\end{remark}\n", "meta": {"hexsha": "744a36d759047625427553c4e9dc4268353a140f", "size": 13802, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/affine_coordinate_systems.tex", "max_stars_repo_name": "v--/anthology", "max_stars_repo_head_hexsha": "89a91b5182f187bc1aa37a2054762dd0078a7b56", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/affine_coordinate_systems.tex", "max_issues_repo_name": "v--/anthology", "max_issues_repo_head_hexsha": "89a91b5182f187bc1aa37a2054762dd0078a7b56", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/affine_coordinate_systems.tex", "max_forks_repo_name": "v--/anthology", "max_forks_repo_head_hexsha": "89a91b5182f187bc1aa37a2054762dd0078a7b56", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 53.496124031, "max_line_length": 760, "alphanum_fraction": 0.6567888712, "num_tokens": 4180, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832058771036, "lm_q2_score": 0.6406358411176238, "lm_q1q2_score": 0.3452919594453517}} {"text": "\\subsection{Gradient computation, case of NewtonEuler with quaternion}\n\nIn the section, $q$ is the quaternion of the dynamical system.\n\n\\begin{figure}[h]\n \\centering\n \n %\\input{./Figures/NewtonEulerImpact.pstex_t}\n \\input{./Figures/NewtonEulerImpact.pdf_t}\n \n \\caption{Impact of one DS.}\n \\label{figCase}\n\\end{figure}\n\nThe normal vector $N$ is view as a constant.\n\\[~\\tilde h(q)=P_c(\\frac{q}{\\|q\\|})\\]\n\\[^t \\nabla h(q)(\\delta q) = \\lim _{e \\to 0}\\frac{(\\tilde h (q+e\\delta q)-\\tilde h (q)).N}{e} \\]\n\n$\\nabla _q h$ consist in computing $P_c(\\frac{q+\\delta q}{\\|q+\\delta q\\|})-P_c(q)$.\n\\[GP(q)=qG_0P_0~^cq\\]\n\\[GP(\\frac{q+\\delta q}{\\|q+\\delta q\\|})=(q+\\delta q)G_0P_0~^c(q+\\delta q)\\frac{1}{\\|q+\\delta q\\|^2}\\]\n\\[=(q+\\delta q)~^cqGP(q)q~^c(q+\\delta q)\\frac{1}{\\|q+\\delta q\\|^2}\\]\n\\[=((1,0,0,0)+\\delta q~^cq)GP(q)((1,0,0,0)+q~^c\\delta q)\\frac{1}{\\|q+\\delta q\\|^2}\\]\n\\[=GP(q)+\\delta q~^cqGP(q) + GP(q)q~^c\\delta q+0(\\delta q)^2\\frac{1}{\\|q+\\delta q\\|^2}\\]\nSo, because G is independant of $q$:\n\\[P(\\frac{q+\\delta q}{\\|q+\\delta q\\|})-P(q)=qGP(\\frac{q+\\delta q}{\\|q+\\delta q\\|})-GP(q)=\\delta q~^cqGP(q) + GP(q)q~^c\\delta q+0(\\delta q)^2 + GP(q)\\frac{1}{\\|q+\\delta q\\|^2}\\]\nFor the directional derivation, we chose $\\delta q = \\epsilon * (1,0,0,0)$. using a equivalent to $\\frac{1}{1+\\epsilon}$\n\\[\\lim_{\\epsilon \\to 0}\\frac{P(\\frac{q+\\delta q}{\\|q+\\delta q\\|})-P(q)}{\\epsilon}=~^cqGP(q) + GP(q)q-2q_iGP(q)\\]\nFor the directional derivation, we chose $\\delta q = \\epsilon * (0,1,0,0)=\\epsilon * e_i$\n\\[\\lim_{\\epsilon \\to 0}\\frac{P(\\frac{q+\\delta q}{\\|q+\\delta q\\|})-P(q)}{\\epsilon}=e_i~^cqGP(q) - GP(q)qe_i-2q_iGP(q)\\]\nApplication to the NewtonEulerRImpact:\n\\[H:\\mathbb{R}^7 \\to \\mathbb{R}\\]\n\\[\\nabla _q H \\in \\mathcal{M}^{1,7}\\]\n\\[\\nabla _q H =\\left(\\begin{array}{c} N_x\\\\N_y\\\\N_z\\\\\n(~^cqGP(q) + GP(q)q-2q_0GP(q)).N\\\\\n(e_2~^cqGP(q) - GP(q)qe_2-2q_1GP(q)).N\\\\\n(e_3~^cqGP(q) - GP(q)qe_3-2q_2GP(q)).N\\\\\n(e_4~^cqGP(q) - GP(q)qe_4-2q_3GP(q)).N\\\\\n\\end{array}\\right)\\]\n\\subsection{Ball case}\nIt is the case where $GP=-N$:\nfor $e2$:\n\\[(0,1,0,0).(q_0,-\\underline p).(0,-N)=\\]\n\\[\\left(\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right).\\underline p,\\left(\\begin{array}{c}q_0\\\\0\\\\0\\end{array}\\right) -\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*\\underline p \\right).(0,-N)=\\]\n\\[\\left(?, -\\underline p_x~N-\\left(\\left(\\begin{array}{c}q_0\\\\0\\\\0\\end{array}\\right)- \\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*\\underline p \\right)*N\\right)=\\]\nand:\n\\[(0,-N).(q_0,\\underline p).(0,1,0,0)=\\]\n\\[(N.\\underline p,-q_0N-N*\\underline p).(0,1,0,0)=\\]\n\\[\\left(?,(N.\\underline p)\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right) + \\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*(q_0N+N*\\underline p)\\right)=\\]\n\\[\\left(?,(N.\\underline p)\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)+q_0 \\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*N+\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*(N*\\underline p)\\right)\\]\nsub then and get the resulting vector.N:\n\\[\\left[ -\\underline p_x~N -N.\\underline p~\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)+()*N-\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*(N*\\underline p)\\right].N=\\]\n\\[-\\underline p_x-N_xN.\\underline p+0-(\\left(\\begin{array}{c}1\\\\0\\\\0\\end{array}\\right)*(N*\\underline p)).N=\\]\n using $a*(b*c)=b(a.c)-c(a.b)$ leads to\n \\[-q_1-N_xN.\\underline p-(q_1~N-N_x~\\underline p).N=\\]\n\\[-q_1-N_xN.\\underline p-q_1+N_xN.\\underline p=-2q_1\\]\nfor $e1=(1,0,0,0)$:\n\\[(q_0,-\\underline p).(0,-N)=(?,-q_0N+\\underline p*N)\\]\n\\[(0,-N).(q_0,\\underline p)=(?,-q_0N-\\underline p*N)\\]\nSo\n\\[\\nabla _q H =\\left(\\begin{array}{c} N_x\\\\N_y\\\\N_z\\\\\n0\\\\\n0\\\\\n0\\\\\n0\\\\\n\\end{array}\\right)\\]\n\n\\subsection{Case FC3D: using the local frame and momentum}\n\n\\[\\left(\\begin{array}{c}m \\dot V\\\\I \\dot \\Omega + \\Omega I \\Omega \\end{array}\\right)= \\left(\\begin{array}{c}Fect+R\\\\Mext _{R_{obj}} + (R*PG) _{R_{obj}} \\end{array}\\right)\\]\n with * vectoriel product, $R$ reaction in the globla frame. $P$ the point of contact.\n $r$ is the reaction in the local frame. $M_{R_{obj}toR_{abs}}=M_{R_{abs}toR_{obj}}^t r=R$ with:\n \\[M_{R_{C}toR_{abs}}=\\left(\\begin{array}{ccc} nx&t_1x&t_2x \\\\ny&t_1y&t_2y\\\\nz&t_1z&t_2z \\end{array}\\right)\\]\n we have :\n \\[\\left(\\begin{array}{c}R\\\\(R*PG) _{R_{obj}}\\end{array}\\right)=\\left(\\begin{array}{c} I_3\\\\M_{R_{abs}toR_{obj}}N_{PG}\\end{array}\\right).R\\]\n \\[=\\left(\\begin{array}{c} I_3\\\\M_{R_{abs}toR_{obj}}N_{PG}\\end{array}\\right).M_{R_{obj}toR_{abs}}r\\]\n \\[ N_{PG}=\\left(\\begin{array}{ccc} 0&PG_z&-PG_y\\\\-PG_z&0&PG_x\\\\PG_y&-PG_X&0\\end{array}\\right)\\]\n that is:\n \n\\[\\left(\\begin{array}{c}m \\dot V\\\\I \\dot \\Omega + \\Omega I \\Omega \\end{array}\\right)=\n\\left(\\begin{array}{c} M_{R_{C}toR_{abs}} \\\\\n M_{R_{abs}toR_{obj}}N_{PG}M_{R_{C}toR_{abs}}\n\\end{array}\\right) r\\]\nSo $jachqt=MN$\n\n\\subsection{Case FC3D: using the local frame local velocities}\n\\begin{figure}[h!]\n \\centering\n \\scalebox{0.6}{\n \\input{./Figures/SolideContact.pdf_t}\n % \\input{./Figures/SolideContact.pdf_t}\n }\n \\caption{Two objects colliding.}\n \\label{figCase}\n\\end{figure}\n\n\nWe are looking for an operator named $CT$ such that:\n\n\\[V_C=\\left(\\begin{array}{c} V_N \\\\ V_T \\\\ V_S \\end{array}\\right)_{R_{C}}=CT \\left(\\begin{array}{c} V_{G1}~_{R_{abs}} \\\\ \\Omega_1~_{R_{obj1}} \\\\ V_{G2}~_{R_{abs}}\\\\ \\Omega_2~_{R_{obj2}} \\end{array}\\right)\\]\n\n\\[V_c=V_{G1}~_{R_{abs}} + w_1 * G_1P~_{R_{abs}} -(V_{G2}~_{R_{abs}} + w_2 * G_1P~_{R_{abs}})\\]\nwhere $w_1$ and $w_2$ are given in $R_{abs}$. We note $M_{R_{obj1}toR_{abs}}$ the matrice converting the object 1 coordinate to the absolute coordinate. We note $N_{GP}$ the matrice such that $w_1*G_1P~_{R_{abs}} = N_{GC} w_1$. Endly, we note $M_{R_{abs}toR_C}$ converting the absolute coordinate to the $R_C$ frame.\nwe get:\n\\[CT= M_{R_{abs}toR_C} \\left(\\begin{array}{cccc} I_3 & N_{G_1C}M_{R_{obj1}toR_{abs}} & -I_3 & -N_{G_2C}M_{R_{obj2}toR_{abs}} \\end{array}\\right)\\]\n\n\\subsubsection{Expression of $M_{R_{obj1}toR_{abs}}$}\nUsing quaternion, we get :\n\\begin{equation}\n \\label{eq:newton_Mobjtoabs}\nM_{R_{obj1}toR_{abs}} = \\left(\\begin{array}{ccc} q \\left(\\begin{array}{c}1\\\\0\\\\0 \\end{array}\\right)~^cq & q \\left(\\begin{array}{c} 0\\\\1\\\\0 \\end{array}\\right)~ ^cq & q \\left(\\begin{array}{c} 0\\\\0\\\\1 \\end{array}\\right)~ ^cq \\end{array}\\right)\n\\end{equation}\n\n\\subsubsection{Expression of $N_1$}\n\\[N_{GC}=\\left(\\begin{array}{ccc} 0&G_1C_z&-G_1C_y\\\\-G_1C_z&0&G_1C_x\\\\G_1C_y&-G_1C_X&0\\end{array}\\right)\\]\n\n%%% Local Variables: \n%%% mode: latex\n%%% TeX-master: \"DevNotes\"\n%%% End: \n", "meta": {"hexsha": "da7702b4541619915497af573995242be78c25a9", "size": 6422, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/sphinx/devel_guide/notes/NewtonImpactJacobian.tex", "max_stars_repo_name": "ljktest/siconos", "max_stars_repo_head_hexsha": "85b60e62beca46e6bf06bfbd65670089e86607c7", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 137, "max_stars_repo_stars_event_min_datetime": "2015-06-16T15:55:28.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-26T06:01:59.000Z", "max_issues_repo_path": "docs/sphinx/devel_guide/notes/NewtonImpactJacobian.tex", "max_issues_repo_name": "ljktest/siconos", "max_issues_repo_head_hexsha": "85b60e62beca46e6bf06bfbd65670089e86607c7", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 381, "max_issues_repo_issues_event_min_datetime": "2015-09-22T15:31:08.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-14T09:05:23.000Z", "max_forks_repo_path": "docs/sphinx/devel_guide/notes/NewtonImpactJacobian.tex", "max_forks_repo_name": "ljktest/siconos", "max_forks_repo_head_hexsha": "85b60e62beca46e6bf06bfbd65670089e86607c7", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 30, "max_forks_repo_forks_event_min_datetime": "2015-08-06T22:57:51.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-02T20:30:20.000Z", "avg_line_length": 53.0743801653, "max_line_length": 316, "alphanum_fraction": 0.6222360635, "num_tokens": 2843, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185943925708561, "lm_q2_score": 0.48047867804790706, "lm_q1q2_score": 0.3452692837950837}} {"text": "\\documentclass[11pt]{article}\n\\usepackage{fullpage}\n\\usepackage{amsmath, amssymb, bigints, graphicx}\n\n\\title{Classical Lamination Theory \\\\\n { \\normalsize ~ \\\\ A note in support of \\textit{Fiber Orientation Tools} \\\\\n \\texttt{http://github.com/charlestucker3/Fiber-Orientation-Tools}}}\n\n\\author{Charles L.~Tucker III \\\\\n Department of Mechanical Science and Engineering \\\\\n University of Illinois at Urbana-Champaign \\\\\n 1206 W.~Green St. \\\\\n Urbana, IL 61801 \\\\\n }\n\\include{defs} % Macro definitions\n\n\\begin{document}\n\\maketitle\n\nThe function \\texttt{Clayer2laminate} in \\emph{Fiber Orientation Tools} uses classical lamination theory to find the stiffness properties of a composite where the fiber orientation, fiber volume fraction, and/or the fiber length distribution vary across the thickness. Such structures are typical for injection molded composites. \n\nThis note explains the nomenclature used by \\texttt{Clayer2laminate} and summarizes the underlying theory. We develop the laminate properties in a notation that is common in the structural mechanics of laminated composite materials\\footnote{E.g., R. M. Jones, \\emph{Mechanics of Composite Materials}. Taylor \\& Francis, Philadelphia, 2nd edition, 1999.}. Equation numbers with a period, e.g., Eqn.~(8.2), and section numbers refer to items from \\emph{Fundamentals of Fiber Orientation}, C. L. Tucker, Hanser, Munich, 2022. \n\n\\subsection*{Coordinate System}\n\nIn the language of laminated composites, a \\emph{laminate} is a plate-like structure consisting of multiple discrete layers, or \\emph{lamina}, bonded together. Choose $(x,y,z)$ coordinates so that $z = 0$ is the midplane of the laminate, and let the total thickness of the laminate be $H$. \n\n\\subsection*{Strains and Curvatures}\n\nLet the components of the displacement vector be $(u, v, w)$. The components of the strain tensor follow from the usual definitions, e.g., Eqns.~(8.2) and (8.3):\n\\begin{equation}\n \\varepsilon_{xx} = \\frac{\\partial u}{\\partial x} \\qquad \\qquad\n \\varepsilon_{yy} = \\frac{\\partial v}{\\partial y} \\qquad \\qquad\n \\gamma_{xy} = \\frac{\\partial u}{\\partial y} + \\frac{\\partial v}{\\partial x} \n \\label{epsilonDef}\n\\end{equation}\nNote that we are using engineering shear strain here, rather than tensor shear strain. \n\nLet $w_0$ be the $z$-direction displacement of the laminate midplane, $z = 0$. The midplane curvatures of the laminate are then described by the second derivatives of $w_0$:\n\\begin{equation}\n \\kappa_{xx} = -\\frac{\\partial^2 w_0}{\\partial x^2} \\qquad \\qquad\n \\kappa_{yy} = -\\frac{\\partial^2 w_0}{\\partial y^2} \\qquad \\qquad\n \\kappa_{xy} = -2\\frac{\\partial ^2 w_0}{\\partial x \\partial y}\n \\label{kappaDef}\n\\end{equation}\nThe inclusion of a factor of 2 in $\\kappa_{xy}$ is common, and parallels the use of the engineering shear strain. These two choices are a departure from the notation in \\emph{Fundamentals of Fiber Orientation}, where all tensors are contracted using a consistent procedure, but they are standard in laminate analysis. \n\n\n\\subsection*{Lamina Stress-Strain Equations}\n\nThe next task is to find the stress-strain relationships for a lamina in plane stress. In our case we know (or can calculate) the 3-D stiffness tensor $\\Cfour$ for each lamina, so we will start with that. Using $[\\Cfour]$ to represent the $6 \\times 6$ matrix that is the contracted form of this tensor, the lamina stress-strain equation is\n\\begin{equation}\n \\left\\{ \\begin{array}{c}\n \\sigma_{xx} \\\\ \\sigma_{yy} \\\\ \\sigma_{zz} \\\\ \\sigma_{yz} \\\\ \\sigma_{zx} \\\\ \\sigma_{xy}\n \\end{array} \\right\\}\n =\n \\left[ \\begin{array}{cccccc}\n \\Cfour_{11} & \\Cfour_{12} & \\Cfour_{13} & \\Cfour_{14} & \\Cfour_{15} & \\Cfour_{16} \\\\\n \\Cfour_{21} & \\Cfour_{22} & \\Cfour_{23} & \\Cfour_{24} & \\Cfour_{25} & \\Cfour_{26} \\\\\n \\Cfour_{31} & \\Cfour_{32} & \\Cfour_{33} & \\Cfour_{34} & \\Cfour_{35} & \\Cfour_{36} \\\\\n \\Cfour_{41} & \\Cfour_{42} & \\Cfour_{43} & \\Cfour_{44} & \\Cfour_{45} & \\Cfour_{46} \\\\\n \\Cfour_{51} & \\Cfour_{52} & \\Cfour_{53} & \\Cfour_{54} & \\Cfour_{55} & \\Cfour_{56} \\\\\n \\Cfour_{61} & \\Cfour_{62} & \\Cfour_{63} & \\Cfour_{64} & \\Cfour_{65} & \\Cfour_{66} \n \\end{array} \\right] \n \\left\\{ \\begin{array}{c}\n \\varepsilon_{xx} \\\\ \\varepsilon_{yy} \\\\ \\varepsilon_{zz} \\\\ \\gamma_{yz} \\\\ \\gamma_{zx} \\\\ \\gamma_{xy}\n \\end{array} \\right\\}\n \\label{stiffnessC}\n \\end{equation}\n\nThe laminate is assumed to be thin relative to its in-plane dimensions, so each lamina is in a state of \\emph{plane stress}: $\\sigma_{zz} = \\sigma_{xz} = \\sigma_{yz} = 0$. We would like to specialize Eqn.~(\\ref{stiffnessC}) for plane stress. To do this, take the matrix inverse of $[ \\Cfour ]$ to find $[ \\tilde{\\Sfour} ] = [ \\Cfour ]^{-1}$. In many books $[ \\tilde{\\Sfour} ]$ is called the compliance matrix. This converts Eqn.~(\\ref{stiffnessC}) to a strain-stress relationship:\n\\begin{equation}\n \\left\\{ \\begin{array}{c}\n \\varepsilon_{xx} \\\\ \\varepsilon_{yy} \\\\ \\varepsilon_{zz} \\\\ \\gamma_{yz} \\\\ \\gamma_{zx} \\\\ \\gamma_{xy}\n \\end{array} \\right\\}\n =\n \\left[ \\begin{array}{cccccc}\n \\tilde{\\Sfour}_{11} & \\tilde{\\Sfour}_{12} & \\tilde{\\Sfour}_{13} & \\tilde{\\Sfour}_{14} & \\tilde{\\Sfour}_{15} & \\tilde{\\Sfour}_{16} \\\\\n \\tilde{\\Sfour}_{21} & \\tilde{\\Sfour}_{22} & \\tilde{\\Sfour}_{23} & \\tilde{\\Sfour}_{24} & \\tilde{\\Sfour}_{25} & \\tilde{\\Sfour}_{26} \\\\\n \\tilde{\\Sfour}_{31} & \\tilde{\\Sfour}_{32} & \\tilde{\\Sfour}_{33} & \\tilde{\\Sfour}_{34} & \\tilde{\\Sfour}_{35} & \\tilde{\\Sfour}_{36} \\\\\n \\tilde{\\Sfour}_{41} & \\tilde{\\Sfour}_{42} & \\tilde{\\Sfour}_{43} & \\tilde{\\Sfour}_{44} & \\tilde{\\Sfour}_{45} & \\tilde{\\Sfour}_{46} \\\\\n \\tilde{\\Sfour}_{51} & \\tilde{\\Sfour}_{52} & \\tilde{\\Sfour}_{53} & \\tilde{\\Sfour}_{54} & \\tilde{\\Sfour}_{55} & \\tilde{\\Sfour}_{56} \\\\\n \\tilde{\\Sfour}_{61} & \\tilde{\\Sfour}_{62} & \\tilde{\\Sfour}_{63} & \\tilde{\\Sfour}_{64} & \\tilde{\\Sfour}_{65} & \\tilde{\\Sfour}_{66} \n \\end{array} \\right] \n \\left\\{ \\begin{array}{c}\n \\sigma_{xx} \\\\ \\sigma_{yy} \\\\ \\sigma_{zz} \\\\ \\sigma_{yz} \\\\ \\sigma_{zx} \\\\ \\sigma_{xy}\n \\end{array} \\right\\}\n \\label{3Dcompliance}\n \\end{equation}\nNow we can write the in-plane strains as functions of the in-plane stresses by selecting the first, second, and sixth lines of Eqn.~(\\ref{3Dcompliance}) and setting $\\sigma_{zz}$, $\\sigma_{xz}$, and $\\sigma_{yz}$ to zero:\n\\begin{equation}\n \\left\\{ \\begin{array}{c}\n \\varepsilon_{xx} \\\\ \\varepsilon_{yy} \\\\ \\gamma_{xy}\n \\end{array} \\right\\}\n =\n \\left[ \\begin{array}{ccc}\n \\tilde{\\Sfour}_{11} & \\tilde{\\Sfour}_{12} & \\tilde{\\Sfour}_{16} \\\\\n \\tilde{\\Sfour}_{21} & \\tilde{\\Sfour}_{22} & \\tilde{\\Sfour}_{26} \\\\\n \\tilde{\\Sfour}_{61} & \\tilde{\\Sfour}_{62} & \\tilde{\\Sfour}_{66} \n \\end{array} \\right] \n \\left\\{ \\begin{array}{c}\n \\sigma_{xx} \\\\ \\sigma_{yy} \\\\ \\sigma_{xy}\n \\end{array} \\right\\}\n \\label{2Dcompliance}\n \\end{equation}\n This matrix simply uses selected components from the full $[ \\tilde{\\Sfour} ]$ matrix. \n \n To obtain stress-strain relationships we invert the matrix in Eqn.~(\\ref{2Dcompliance}) to find the \\emph{plane-stress lamina stiffness matrix} $[Q]$,\n \\begin{equation}\n \\left[ \\begin{array}{ccc}\n Q_{11} & Q_{12} & Q_{16} \\\\\n Q_{21} & Q_{22} & Q_{26} \\\\\n Q_{61} & Q_{62} & Q_{66} \n \\end{array} \\right] \n =\n \\left[ \\begin{array}{ccc}\n \\tilde{\\Sfour}_{11} & \\tilde{\\Sfour}_{12} & \\tilde{\\Sfour}_{16} \\\\\n \\tilde{\\Sfour}_{21} & \\tilde{\\Sfour}_{22} & \\tilde{\\Sfour}_{26} \\\\\n \\tilde{\\Sfour}_{61} & \\tilde{\\Sfour}_{62} & \\tilde{\\Sfour}_{66} \n \\end{array} \\right] ^{-1}\n \\label{Qdef}\n \\end{equation}\n Now the lamina stress-strain relationship for plane stress is\n \\begin{equation}\n \\left\\{ \\begin{array}{c}\n \\sigma_{xx} \\\\ \\sigma_{yy} \\\\ \\sigma_{xy}\n \\end{array} \\right\\}\n =\n \\left[ \\begin{array}{ccc}\n Q_{11} & Q_{12} & Q_{16} \\\\\n Q_{21} & Q_{22} & Q_{26} \\\\\n Q_{61} & Q_{62} & Q_{66} \n \\end{array} \\right] \n \\left\\{ \\begin{array}{c}\n \\varepsilon_{xx} \\\\ \\varepsilon_{yy} \\\\ \\gamma_{xy}\n \\end{array} \\right\\}\n \\label{laminaQ}\n \\end{equation}\nThis numbering of the entries in $[Q]$ is standard. \n\nThis route to obtain $[Q]$ is different from the ones usually used in books on composite mechanics, but it starts from the lamina information that is available when analyzing discontinuous fiber composites, which is the lamina stiffness tensor $\\mathbf{\\Cfour}$. In practice all of these calculations are carried out numerically.\n\n\\subsection*{Laminate Loads and Moments}\n\nThe standard way to describe the in-plane loads in a laminate is to define a vector of forces per unit width $\\{ N \\}$ whose components are\n\\begin{equation}\n \\left\\{ \\begin{array}{c}\n N_x \\\\ N_y \\\\ N_{xy}\n \\end{array} \\right\\}\n = \\bigintss_{-H/2}^{H/2} \n \\left\\{ \\begin{array}{c}\n \\sigma_{xx} \\\\ \\sigma_{yy} \\\\ \\sigma_{xy}\n \\end{array} \\right\\}\n \\ud z\n \\label{Ndef}\n\\end{equation}\nFigure~\\ref{Nfig} sketches these loads. \nNote that $N_x$ is the $x$-direction force per unit width in the $y$ direction, acting on the entire thickness of the laminate. The stress $\\sigma_{xx}$ will vary across the laminate thickness because the stiffness varies with $z$, and possibly because the laminate is being bent. The average value of $\\sigma_{xx}$ across the thickness of the laminate is $N_x / H$. \n\n\\begin{figure}[tb]\n \\centering\n \\includegraphics[width = 0.48\\textwidth, trim = 10 00 10 10, clip = true, keepaspectratio = true]{Nlaminate.png}\n \\includegraphics[width = 0.50\\textwidth, trim = 10 15 10 10, clip = true, keepaspectratio = true]{Mlaminate.png}\n \\caption{The in-plane loads per unit width (left) and moments per unit width (right) on a laminate.}\n \\label{Nfig}\n\\end{figure}\n\n\nSimilarly, the vector of moments per unit width $\\{M\\}$ is defined as\n\\begin{equation}\n \\left\\{ \\begin{array}{c}\n M_x \\\\ M_y \\\\ M_{xy}\n \\end{array} \\right\\}\n = \\bigintss_{-H/2}^{H/2} \n \\left\\{ \\begin{array}{c}\n \\sigma_{xx} \\\\ \\sigma_{yy} \\\\ \\sigma_{xy}\n \\end{array} \\right\\}\n z \\ud z\n \\label{Mdef}\n\\end{equation}\nThese are also sketched in Fig.~\\ref{Nfig}, where you can see that $M_x$ and $M_y$ are bending moments, while $M_{xy}$ is a twisting moment in the $x$--$y$ coordinate system. \n\n\n\n\\subsection*{Kirchoff-Love Hypothesis}\n\nWe would like to relate $\\{ N \\}$ and $\\{ M \\}$ to the deformation of the laminate. To do this, classical lamination theory uses the Kirchoff-Love hypothesis. This states that any material line initially normal to the $x$--$y$ plane remains straight and normal during deformation. This is the laminated-plate analogy of slender beam theory. However, while both a plate and a beam can bend in response to $M_x$, a plate can also bend in response to $M_y$ and twist in response to $M_{xy}$. \n\nWith this hypothesis, the in-plane strains at any height $z$ are related to the strains on the midplane, which are $\\varepsilon^0_{xx}$, $\\varepsilon^0_{yy}$, and $\\gamma^0_{xy}$, and to the midplane curvatures by\n\\begin{equation}\n\\renewcommand\\arraystretch{1.2}\n \\left\\{ \\begin{array}{c}\n \\varepsilon_{xx} \\\\ \\varepsilon_{yy} \\\\ \\gamma_{xy}\n \\end{array} \\right\\}\n =\n \\left\\{ \\begin{array}{c}\n \\varepsilon^0_{xx} \\\\ \\varepsilon^0_{yy} \\\\ \\gamma^0_{xy}\n \\end{array} \\right\\}\n + z\n \\left\\{ \\begin{array}{c}\n \\kappa_{xx} \\\\ \\kappa_{yy} \\\\ \\kappa_{xy}\n \\end{array} \\right\\}\n \\label{Kirchoff}\n\\end{equation}\nIn the absence of curvature, every layer in a laminate has the same in-plane strain. With curvature, strain increases linearly with $z$, in proportion to the corresponding component of curvature. For example, a positive $\\kappa_{xx}$ stretches the layers above the midplane, and compresses the layers below the midplane. \n\n\\subsection*{Laminate Stiffness Equations}\n\nEquations~(\\ref{laminaQ}--\\ref{Kirchoff}) can be combined to write an overall relationship between the forces and moments on the laminate and its midplane strains and curvatures. That equation is\n\\begin{equation}\n\\renewcommand\\arraystretch{1.2}\n \\left\\{ \\begin{array}{c}\n N_x \\\\ N_y \\\\ N_{xy} \\\\ M_x \\\\ M_y \\\\ M_{xy}\n \\end{array} \\right\\}\n =\n\\renewcommand\\arraystretch{2}\n \\left[ \\begin{array}{c|c}\n A & B \\\\\n \\hline\n B & D\n \\end{array} \\right]\n\\renewcommand\\arraystretch{1.2}\n \\left\\{ \\begin{array}{c}\n \\varepsilon^0_{xx} \\\\ \\varepsilon^0_{yy} \\\\ \\gamma^0_{xy} \\\\ \\kappa_{xx} \\\\ \\kappa_{yy} \\\\ \\kappa_{xy}\n \\end{array} \\right\\}\n \\label{laminateStiffness} \n\\end{equation}\nThe $6 \\times 6$ matrix on the right-hand side of this equation contains several $3 \\times 3$ sub-matrices, which are defined as\n\\begin{align}\n [A] &= \\int_{-H/2}^{H/2} \\: [Q(z)] \\: \\ud z \\label{Amatrix} \\\\\n [B] &= \\int_{-H/2}^{H/2} \\: [Q(z)] \\: z \\ud z \\label{Bmatrix} \\\\\n [D] &= \\int_{-H/2}^{H/2}\\: [Q(z)] \\: z^2 \\ud z \\label{Dmatrix}\n\\end{align}\nThe plane-strain stiffness matrix $[Q]$ is a function of $z$ because different layers have different fiber orientation. We can see that $[A]$ relates the midplane strains to the in-plane loads $\\{ N \\}$, while $[D]$ relates the the midplane curvatures to the moments $\\{ M \\}$. The $[B]$ matrix describes coupling, either between curvature and in-plane loads or between in-plane strains and moments. If the laminate is symmetric about the midplane, i.e., if $[Q(z)] = [Q(-z)]$, then $[B] = 0$ and there is no coupling. \n\nTo compute these matrices in a practical situation (and in \\texttt{Clayer2laminate}), divide the laminate into $n$ discrete layers, and treat each layer as having a uniform $[Q]$. Number the layers $i = 1$ to $n$ from the bottom of the laminate to the top, and let $[Q_i]$ be the plane-strain stiffness matrix for layer $i$. Define $z_i$ such that layer $i$ extends from $z_{i-1}$ to $z_{i}$, with $z_0 = -H/2$ and $z_n = H/2$. Now Eqns.~(\\ref{Amatrix}--\\ref{Dmatrix}) can be integrated analytically for each layer, and the matrices computed as\n\\begin{align}\n [A] &= \\sum_{i=1}^n \\: [Q_i] \\: \\left( z_i - z_{i-1} \\right) \\label{Asum} \\\\\n [B] &= \\sum_{i=1}^n \\: [Q_i] \\: \\left( \\frac{z^2_i - z^2_{i-1}}{2} \\right) \\label{Bsum} \\\\\n [D] &= \\sum_{i=1}^n \\: [Q_i] \\: \\left( \\frac{z^3_i - z^3_{i-1}}{3} \\right) \\label{Dsum}\n\\end{align}\n$[A]$, $[B]$ and $[D]$ fully describe the elastic behavior of the laminate, and they are returned by \\texttt{Clayer2laminate}. \n\n\\subsection*{Extracting Tensile and Flexural Moduli}\n\nAs discussed in Section~8.1.5, any engineering constant like Young's modulus is defined in terms of a mechanical test where loads are applied and deformations are measured. For the tensile modulus in the $x$ direction, a stress $\\sigma_{xx}$ is applied, with all other stresses equal to zero, and the modulus is the ratio of the applied stress to $\\varepsilon_{xx}$. For a laminate we'll used the midplane strain, and also require that there be no moments applied. Then, the $x$-direction tensile modulus is\n\\begin{equation}\n E^t_{xx} = \\frac{\\sigma_{xx}}{\\varepsilon^0_{xx}} = \\frac{N_{x}/H}{\\varepsilon^0_{xx}} \n \\label{EtensileDef}\n \\end{equation}\nWe have used a superscript $t$ to indicate that this is a tensile modulus, and the second expression replaces the average stress $\\sigma_{xx}$ with the load per unit width $N_x$. A similar expression gives the $y$-direction tensile modulus.\n\nTo find the flexural modulus in the $x$ direction, one subjects the sample to a bending test where a moment $M_x$ is applied, with no other loads or moments, and the curvature $\\kappa_{xx}$ is determined by measuring the deflection. If this test were performed on a homogeneous beam of width $b$ and thickness $H$, the moment-curvature relationship would be\n\\begin{equation}\n M_x b = E^f_{xx} I \\, \\kappa_{xx} = \\frac{E^f_{xx} b H^3}{12} \\kappa_{xx}\n \\label{momentCurvature}\n\\end{equation}\nHere $I = bH^3/12$ is the moment of inertia of the cross-section, and the superscript $f$ on $E$ indicates a flexural modulus. Note that $M_x$ is a moment per unit width, so the total applied moment is $M_x b$. We can cancel $b$ from both sides and rearrange this equation to get the effective flexural modulus,\n\\begin{equation}\n E^f_{xx} = \\frac{12 M_x}{H^3 \\kappa_{xx}}\n \\label{EflexDef}\n\\end{equation}\nAnalogous equations apply in the $y$ direction. \n\nTo apply these results to a specific laminate, we need to find the midplane strain and the midplane curvature for the two load cases. To do this numerically, invert the $6 \\times 6$ matrix from the right-hand side of Eqn.~(\\ref{laminateStiffness}). The resulting matrix seems to have no official title, so we'll call it the laminate compliance matrix $[S^\\mathrm{lam}]$:\n\\begin{equation}\n [S^\\mathrm{lam}] \n = \n \\left[ \\begin{array}{c|c}\n A & B \\\\\n \\hline\n B & D\n \\end{array} \\right] ^{-1}\n \\label{SlamDef}\n\\end{equation}\nNow the strain-stress relationship for the laminate is\n\\begin{equation}\n\\renewcommand\\arraystretch{1.2}\n \\left\\{ \\begin{array}{c}\n \\varepsilon^0_{xx} \\\\ \\varepsilon^0_{yy} \\\\ \\gamma^0_{xy} \\\\ \\kappa_{xx} \\\\ \\kappa_{yy} \\\\ \\kappa_{xy}\n \\end{array} \\right\\}\n = [S^\\mathrm{lam}] \n\\renewcommand\\arraystretch{1.2}\n \\left\\{ \\begin{array}{c}\n N_x \\\\ N_y \\\\ N_{xy} \\\\ M_x \\\\ M_y \\\\ M_{xy}\n \\end{array} \\right\\}\n \\label{laminateCompliance} \n\\end{equation}\nFor a tensile test, with all loads and moments zero except $N_x$, the midplane strain is $\\varepsilon^0_{xx} = S^\\mathrm{lam}_{11} N_x$. Using this in Eqn.~(\\ref{EtensileDef}), we can calculate the laminate's tensile modulus in the $x$ direction as\n\\begin{equation}\n E^t_{xx} = \\frac{1}{S^\\mathrm{lam}_{11} H} \n \\label{EtxxFinal}\n \\end{equation}\n Similarly, the tensile modulus in the $y$ direction is\n\\begin{equation}\n E^t_{yy} = \\frac{1}{S^\\mathrm{lam}_{22} H} \n \\label{EtyyFinal}\n \\end{equation}\n For a bending test in the $x$ direction, the moment and curvature are related by $\\kappa_{xx} = S^\\mathrm{lam}_{44} M_x$. Substituting this into Eqn.~(\\ref{EflexDef}), we obtain an equation for the flexural modulus of the laminate in the $x$ direction,\n \\begin{equation}\n E^f_{xx} = \\frac{12}{S^\\mathrm{lam}_{44} H^3}\n \\label{EfxxFinal}\n\\end{equation}\nThe $y$-direction flexural modulus is found similarly,\n \\begin{equation}\n E^f_{yy} = \\frac{12}{S^\\mathrm{lam}_{55} H^3}\n \\label{EfyyFinal}\n\\end{equation}\n Equations~(\\ref{EtxxFinal}--\\ref{EfyyFinal}) are implemented in \\texttt{Clayer2laminate}. \n For an example calculation, see \\break \\texttt{LaminatedPlateProperties.mlx}.\n\n\n\\end{document}", "meta": {"hexsha": "9619e00fba0d34f55612e3a08402a9bcb46547a8", "size": 19201, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Documentation/LaTeX Source Files/LaminateTheory.tex", "max_stars_repo_name": "charlestucker3/Fiber-Orientation-Tools", "max_stars_repo_head_hexsha": "4047e06f2bf88f349be494c7078a7101e61f57b8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-01-19T20:38:27.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-19T20:38:27.000Z", "max_issues_repo_path": "Documentation/LaTeX Source Files/LaminateTheory.tex", "max_issues_repo_name": "charlestucker3/Fiber-Orientation-Tools", "max_issues_repo_head_hexsha": "4047e06f2bf88f349be494c7078a7101e61f57b8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Documentation/LaTeX Source Files/LaminateTheory.tex", "max_forks_repo_name": "charlestucker3/Fiber-Orientation-Tools", "max_forks_repo_head_hexsha": "4047e06f2bf88f349be494c7078a7101e61f57b8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.3450479233, "max_line_length": 547, "alphanum_fraction": 0.649497422, "num_tokens": 6357, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6959583250334526, "lm_q2_score": 0.49609382947091946, "lm_q1q2_score": 0.3452606306180124}} {"text": "\\pagenumbering{arabic}\n\\setcounter{page}{1}\n\n\\chapter{Introduction}\n\n\\section{Magnetohydrodynamics}\n\n\n\n\\subsection{Navier-Stokes}\n\n\\subsection{Maxwell}\n\n\\section{A model problem}\n\nstationary incompressible and resistive magnetohydrodynamics (MHD) system:\n\\begin{subequations}\n\\label{eq:mhd}\n\\begin{alignat}2\n\\label{eq:mhd1} - \\nu \\, \\Delta\\uu{u} + (\\uu{u} \\cdot \\nabla)\n\\uu{u}+\\nabla p - \\kappa\\,\n(\\nabla\\times\\uu{b})\\times\\uu{b} &= \\uu{f} & \\qquad &\\mbox{in $\\Omega$},\\\\[.1cm]\n\\label{eq:mhd2}\n\\nabla\\cdot\\uu{u} &= 0 & \\qquad &\\mbox{in $\\Omega$},\\\\[.1cm]\n\\label{eq:mhd3}\n\\kappa\\nu_m \\, \\nabla\\times( \\nabla\\times \\uu{b})\n+ \\nabla r\n- \\kappa \\, \\nabla\\times(\\uu{u}\\times \\uu{b}) &= \\uu{g} & \\qquad &\\mbox{in $\\Omega$},\\\\[.1cm]\n\\label{eq:mhd4} \\nabla\\cdot\\uu{b} &= 0 & \\qquad &\\mbox{in\n$\\Omega$}.\n\\end{alignat}\n\\end{subequations}\n\n\\begin{subequations}\n\\label{eq:bc}\n\\begin{alignat}2\n\\label{eq:bc1} \\uu{u} &= \\uu{u_D} & \\qquad &\\mbox{on $\\partial\\Omega$},\\\\[.1cm]\n\\label{eq:bc2}\n \\uu{n}\\times\\uu{b} &= \\uu{n} \\times \\uu{b_D} & \\qquad &\\mbox{on $\\partial\\Omega$},\\\\[.1cm]\n\\label{eq:bc3} r &=0 &\\qquad &\\mbox{on $\\partial\\Omega$},\n\\end{alignat}\n\\end{subequations}\n\n\\section{Finite element methods}\n${\\mathcal P}_{k}$: the space of polynomials of total degree at most $k$\n$ \\uu{R}_k$: the space of homogeneous vector polynomials of total degree $k$ that are orthogonal to the position vector $\\uu{x}$\n\n\\section{Iterative Methods}\n\n\\section{Objectives and contributions}\n\n\n%{\\bf [ changed order of equations to $(u,p,b,r)$] }\n% Here, $\\Omega$ is a bounded simply-connected Lipschitz polyhedron in~$\\mathbb{R}^3$, with a connected boundary~$\\partial\\Omega$. The unknowns are the velocity~$\\uu{u}$, the hydrodynamic pressure~$p$, the magnetic field $\\uu{b}$, and the Lagrange multiplier $r$ associated with the divergence constraint on the magnetic field. The functions $\\uu{f}$ and $\\uu{g}$ represent\n% external force terms.\n\n% The equations \\eqref{eq:mhd} are characterized by three dimensionless parameters: the hydrodynamic Reynolds number ${\\rm Re}=\\nu^{-1}$, the magnetic Reynolds number ${\\rm Rm}~=~\\nu_m^{-1}$, and the coupling number~$\\kappa$. For further discussion of these parameters and their typical values, we refer the reader to~\\cite{ArmeroSimo96, Gerbeau2006, Roberts67}. We consider the following homogeneous and inhomogeneous Dirichlet boundary conditions:\n% with $\\uu{n}$ being the unit outward normal on $\\partial\\Omega$. By taking the divergence of the magnetostatic equation~(\\ref{eq:mhd3}), we obtain the Poisson problem\n% \\begin{equation}\n% \\label{eq:zero-r} \\Delta r =\\nabla \\cdot \\uu{g} \\quad \\mbox{in\n% $\\Omega$}, \\qquad r=0 \\quad\\mbox{on $\\partial\\Omega$}.\n% \\end{equation}\n% Since $\\uu{g}$ is divergence-free in physically relevant applications, the multiplier $r$ is typically zero and its primary purpose is to ensure stability; see also~\\cite{Greif10}.\n\n% Our goal in this paper is to derive a scalable numerical solution procedure for solving \\eqref{eq:mhd}-\\eqref{eq:bc}. We first present in Section~\\ref{sec:discretization} the finite element discretization and the structure of the matrices that arise throughout the nonlinear iterations. In Section~\\ref{sec:preconditioning} we introduce the proposed preconditioners for the indefinite linear systems that arise; our ideas are based on combining preconditioners for the incompressible Navier-Stokes and Maxwell sub-systems appearing in the MHD system~\\eqref{eq:mhd1}--\\eqref{eq:mhd4} while taking into account the presence of coupling terms. Spectral analysis is performed in Section~\\ref{sec:mhd_eigenvalue}. {\\bf [DS: More precise and quote earlier papers]}. In Section~\\ref{sec:decoupling} we consider circumstances where the incompressible Navier-Stokes and the Maxwell systems can be decoupled; in such cases the preconditioning approach may be significantly simplified. In Section~\\ref{sec:numerical_results_mhd_\n% solver} we provide preliminary results in two dimensions that show that our numerical solution techniques are feasible and reasonably scalable. Finally, we offer some concluding remarks in Section~\\ref{sec:conclusions}.\n", "meta": {"hexsha": "c95acc223c8c0cdee95d92644bd331a021641fff", "size": 4132, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "MHD/THESISnew/Intro/Intro.tex", "max_stars_repo_name": "wathen/PhD", "max_stars_repo_head_hexsha": "35524f40028541a4d611d8c78574e4cf9ddc3278", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-10-25T13:30:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-08-10T21:27:30.000Z", "max_issues_repo_path": "MHD/THESISnew/Intro/Intro.tex", "max_issues_repo_name": "wathen/PhD", "max_issues_repo_head_hexsha": "35524f40028541a4d611d8c78574e4cf9ddc3278", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "MHD/THESISnew/Intro/Intro.tex", "max_forks_repo_name": "wathen/PhD", "max_forks_repo_head_hexsha": "35524f40028541a4d611d8c78574e4cf9ddc3278", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-28T16:12:13.000Z", "max_forks_repo_forks_event_max_datetime": "2020-01-13T13:59:44.000Z", "avg_line_length": 61.671641791, "max_line_length": 1020, "alphanum_fraction": 0.7347531462, "num_tokens": 1280, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6224593312018545, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34513531479044884}} {"text": "\\input{docs/preamble}\n\\title{The Speed of Light}\n\\author{Max Bigras and David Frawley}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{The speed of light}\n\\subsection{Light speed in air}\nUsing a pulse modulated diode laser and fast photodiode detector we measured the time lag between the laser pulse and detection. We adjusted the distance the laser pulse traveled in air using mirrors. By plotting distance vs. time, shown in Figure \\ref{air}, we determined $c^{\\mathrm{air}}_{\\mathrm{exp}} = 3.02 \\times 10^{8} \\pm 0.03 $ m/s, which agrees with the accepted value, $c^{\\mathrm{air}} = 2.99792458 \\times 10^{8}$ m/s.\n\\begin{figure}[H]\n \\includegraphics[totalheight=0.6\\textwidth]{figs/air_final}\n \\caption{Determining the speed of light in air}\n \\label{air}\n\\end{figure}\n\n\\subsection{Light speed in an optical fiber}\nUsing a pulse modulated diode laser and fast photodiode detector we measured the time lag between the laser pulse and detection. We adjusted the distance the laser pulse traveled in an optical fiber by using different lengths of fiber. By plotting distance vs. time, shown in Figure \\ref{fiber}, we determined $c^{\\mathrm{fiber}}_{\\mathrm{exp}} = 1.85 \\times 10^{8} \\pm 0.03 $ m/s. The refractive index for a material is given by the equation\n\\begin{equation}\nn^{\\mathrm{material}} = c^{\\mathrm{vacuum}}/c^{\\mathrm{material}}\n\\label{n}\n\\end{equation}\n\nApplying Equation \\ref{n} and using $c^{\\mathrm{air}}_{\\mathrm{exp}} \\approx c^{\\mathrm{vacuum}}$ we determined $n^{\\mathrm{fiber}}_{\\mathrm{exp}} = 1.63 \\pm 0.01$ which agrees with the accepted value for the core of a optical fiber $n^{\\mathrm{fiber}} \\approx 1.62$ \\cite{wiki}.\n\n\\begin{figure}[H]\n \\includegraphics[totalheight=0.6\\textwidth]{figs/fiber_final}\n \\caption{Determining the speed of light in optical fiber}\n \\label{fiber}\n\\end{figure}\n\\subsection{Light speed in a coaxial cable}\nWe propagated electromagnetic waves in a RG-6/U coaxial cable terminated by a resistor box. Depending on the resistance of the box we observed the signal at the end of the cable be: reflected, inverted and absorbed. We measured the speed of an electromagnetic wave in the cable, $c^{\\mathrm{cable}} = 2.32 \\times 10^{8}$. We determined the velocity factor = 0.79, which agrees with the accepted velocity factor for a RG-6/U coaxial cable = 0.82 \\cite{cable}.\n\n\\begin{thebibliography}{99}\n\n\\bibitem{manual} Physics Dept., ``The Speed of Light'', Quantum Lab,\n California Polytechnic University, (2013).\n\n\\bibitem{wiki} ``Optical fiber.'' Wikipedia: The Free Encyclopedia. Wikimedia Foundation, Inc., 05 December 2013. Web.\n\n\\bibitem{cable} C. R. Parashare, and R. F. Bradley, ``75 $\\Omega$ Transmission System'', National Radio Astronomy Observatory Technology Center, Virginia, 2006.\n\n\\end{thebibliography}\n\n\\end{document}\n", "meta": {"hexsha": "15cf8db9d4d49da994feb97b07ec6a0b16fada1d", "size": 2786, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "senior_project/report/speed_of_light/report/lab_report.tex", "max_stars_repo_name": "mbigras/physics_projects", "max_stars_repo_head_hexsha": "7dd29707b3ac8adea7ed8b63786245e34097345e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2016-12-05T23:34:30.000Z", "max_stars_repo_stars_event_max_datetime": "2016-12-11T19:45:07.000Z", "max_issues_repo_path": "senior_project/report/speed_of_light/report/lab_report.tex", "max_issues_repo_name": "mbigras/physics_projects", "max_issues_repo_head_hexsha": "7dd29707b3ac8adea7ed8b63786245e34097345e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "senior_project/report/speed_of_light/report/lab_report.tex", "max_forks_repo_name": "mbigras/physics_projects", "max_forks_repo_head_hexsha": "7dd29707b3ac8adea7ed8b63786245e34097345e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.2765957447, "max_line_length": 458, "alphanum_fraction": 0.7465900933, "num_tokens": 794, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6224593241981982, "lm_q1q2_score": 0.3451353109071283}} {"text": "\\documentclass[10pt]{article}\n\\usepackage{multicol, caption}\n\\usepackage[margin=1in]{geometry}\n\\usepackage{graphicx}\n\\newenvironment{Figure}\n {\\par\\medskip\\noindent\\minipage{\\linewidth}}\n {\\endminipage\\par\\medskip}\n\\graphicspath{ {images/} }\n\\title{\\textbf{Real-time Mesh Deformation using Volumetric Graph Laplacian}}\n\\author{Elias Eberhardsson}\n\\date{}\n\\begin{document}\n\\maketitle\n\n\\textit{\nThis paper describes the implementation and testing of a mesh deformation method using volumetric graph laplacian for use in real-time. First a graph of nodes in a crystal structure inside the 3D mesh is constructed. The nodes are connected together and linked with the mesh. These nodes and their connections are used to preserve volumetric detail when deforming vertices contained in the mesh. The laplacian on vertices based on this graph is how volumetric details are preserved, coupled with the laplacian on the mesh, which preserves outer details. And such the deformed locations are calculated. This method should ideally be used together with quadratic minimization, but due to the time constraint this was left out in favour of a simple interpolation of new vertex position and the two laplacian values.\n}\n\\begin{multicols}{2}\n\\section{Introduction}\nMesh deformation has been interesting to computer scientists for as long as we have been able to draw vertices on the screen. Much research has been done and many many methods have been created and put forth through scientific publications. \n\nThere are many applications for mesh deformation: animation, modelling and simulations are some of the more prominent ones.\n\nThe method showcased in this paper aims to in real-time deform a mesh while balancing volumetric and surface details and such prevent self-intersection and unnatural looking bends and twists. \nThe method builds on the Volumetric Graph Laplacian (VGL), used in [Zhou et al. 2005], which owes itself to differential domain techniques and graph theory. This represents detail as points differences from origo to be balanced between.\n\\section{Related work}\n\nThis project is an attempt at implementing the method described in the paper Large Mesh Deformation Using the Volumetric Graph Laplacian [Zhou et al. 2005] which was an attempt at much better preserve detail and volume when doing deformations such as bending and twisting parts of a mesh. This method used quadratic minimization on energy functions to as closely as possible simulate a real deformation.\n\nSupposed to be used but cut due to time constraints are WIRE deformation [Singh and Fiume 1998] where a method of connecting vertices in a mesh on a curve and deforming with some constraints to how the curve may bend. This combined with volumetric graph laplacian is what is described in [Zhou et al. 2005].\n\nAs mentioned in the introduction this is also closely related to differential domain techniques [Yu et al. 2004] and spectral graph theory [Chung 1997].\n\n\\section{Deformation on Volumetric Graphs}\n\nGiven a mesh \\textit{M = (V, K, N)}, where \\textit{ \\[ V = \\{ p_i \\in R^3|1\\leq i \\leq n \\} \\] } are the vertex positions as points in 3D-space and \\textit{K} is the connective data to each of the adjacent volumetric nodes in the 3-dimensional array that is used to store them: \\textit{ \\[ K = \\{ I_i \\in R^3|1\\leq i \\leq m \\} \\] } And \\textit{N} is similarily the conncetive data to adjacent vertices in the mesh: \\textit{ \\[ N = \\{ N_i \\in R|1\\leq i \\leq k \\} \\] } Using this as a starting point we can lay out the details of the deformation method.\n\\subsection{Volumetric graph}\n\\begin{Figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{graph1.png}\n\t\\captionof{figure}{Graph structure.}\n\\end{Figure}\n\\begin{Figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{graph2.png}\n\t\\captionof{figure}{Final volume graph.}\n\\end{Figure}\nTo construct the graph that is used to represent volume in the deformation we build two 3-dimensional fields of nodes. The distance, \\textit{d}, between each node is set to the average edge-distance in the mesh. The fields are offset by half of \\textit{d} in each direction to create a structually sound, crystal-like, volume, see figure 2.\n \nWhen this is done all nodes which are not contained within the mesh are discarded. Finally each node in the resulting graph are connected to each of its neighbors and any mesh-vertices within \\textit{d\\textbackslash2} is connected and that vertex is also given information on which connections that have been made, see figure 3.\n\nThe method used to determine whether a node is contained within the mesh is to calculate triangle-ray intersections from the node in an arbitrary direction. Given an even number of intersections the node is outside of the mesh, and given an uneven number, the node is contained within the mesh.\n\n\\subsection{Laplacian of graph}\nThe laplacian of a graph is the same as the laplacian operator on manifolds [Chung 1997] and computes the difference of a point \\textit{p} and all of its adjacencies. Given a weight \\textit{w} for each neighbor that satisfies \\[ \\sum_{j\\in N_i \\cup K_i} w_{ij} = 1\\] we linearly compute the difference of the graph adjacencies: \\[ \\bigtriangledown_G(p_i) = p_i - \\sum_{j\\in K_i}w_{ij}p_j \\] Similarly we do this for mesh adjacencies: \\[ \\bigtriangledown_M(p_i) = p_i - \\sum_{j\\in N_i}w_{ij}p_j \\] Here $\\bigtriangledown$ is the laplacian operator and \\textit{G} and \\textit{M} is the volumetric graph and the 3D mesh respectivly.\n\nThis gives us two terms, one which encodes volumetric detail and one that contains surface detail.\n\\subsection{Deformation}\nGiven a point \\textit{q\\textsubscript{i}} that is the target deformation point for \\textit{p\\textsubscript{i}} we need to find the point \\textit{p`\\textsubscript{i}} that balances volume and detail but still deforms \\textit{p\\textsubscript{i}} satisfactorily. \n\nWe have the three terms a, b, and c: \n\\[ a = \\bigtriangledown_G(h_{i}) - \\bigtriangledown_G(p_i)\\]\n\\[ b = h_i - q_i\\]\n\\[ c = \\bigtriangledown_M(h_{i}) - \\bigtriangledown_M(p_i)\\]\nwhere \\[ h_i = p_i + \\gamma*(p_i - q_i)\\]\n$\\gamma \\in \\{ 0..1 \\}$ gives $h_i$ an approximate value of a deformed location. $\\gamma$ is set to 0.6 in our implementation.\n\nThe final deformed location $p`_i$ is calculated:\n\\[ p`_i = a*(1 - \\alpha - \\beta) + b*\\alpha + c*\\beta \\]\n$\\beta$ balances between surface and volume details. $\\beta = n/N$ works well where n is the number of vertices on the mesh and N is the count of the volumetric nodes contained in the graph. \n\nExperimenting with $\\alpha$ is needed to achive good results based on the mesh that is to be deformed, but 0.4 seems like a good number, as used in [Zhout et al. 2005].\n\n\\subsection{Propagation}\nWhen a new position has been calculated we use it to get the relative translation needed to be done. $T = p`_i - p_i$. The translation \\textit{T} is propagated throughout the mesh using a simple propagation scheme. So that each of $p_i$'s neighbor $n$ is translated $T*(1/||p_n - p_i||)$.\n\n\\section{Results}\n\nThe results were subpar if you were to compare this to any proper implementation using energy functions and quadratic minimization. Also a better way to propagate would have been great to get good looking reults. \n\n\\begin{Figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{mondef.png}\n\t\\captionof{figure}{Face deformation.}\n\\end{Figure}\n\nSmall deformations work fine, moving parts of a face as an example looks fine, as shown in figure 4. But trying to bend bigger shapes do not come out good without much tweaking to the balancing and changes to the weight used in propagation, shown in figure 5 and 6.\n\n\\begin{Figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{korv1.png}\n\t\\captionof{figure}{Pre deformed.}\n\\end{Figure}\n\\begin{Figure}\n\t\\centering\n\t\\includegraphics[width=\\linewidth]{korv2.png}\n\t\\captionof{figure}{Post deformed.}\n\\end{Figure}\n\n\\section{Conclusion and Reflections}\n\nOur implementation of these deformation algorithms are subpar at best. We underestimated the work and planning needed to properly implement and use VGL [Zhou et al. 2005], completely missing the need for WIRE deformation to do large deformations the first ten times I read the paper.\n\nWe had intended to use quadratic minimization on energy functions, as laid out in [Zhou et al. 2005], but we did not understand the bounds used nor had we enough time to implement or get used to any library needed for such computation. And as such our simple interpolation does prove decent at preserving volume and detail in small deformation and does prevent self-intersection. But it fails at any large deformation, or any greater deformations tried, which results in odd looking results.\n\nInstead of simply a translation to be propagated, it should be a transformation-matrix, otherwise proper turning and twisting would have a hard time looking proper.\n\nIf I got to do this all over again, I would plan in much greater detail all parts of the implementation, choice of datastructures used and all things that actually needed to be understood.\n\n\\begin{thebibliography}{7}\n\\bibitem{zhou} \nKun Zhou, Jin Huang, John Snyder, Xinguo Liu, Hunun Bao, Baining Guo and Heung-Yeung Shun. 2005.\n\\textit{Large Mesh Deformation Using the Volumetric Graph Laplacian}. \nACM Trans. Graphics (Proc. ACM SIGGRAPH), vol. 24, no. 3, pp. 496-503.\n \n\\bibitem{moller} \nTomas M{\\\"o}ller. 1997.\n\\textit{A Fast Triangle-Triangle Intersection Test}.\nJournal of Graphics Tools, 2(2):25-30.\n \n\\bibitem{moller2}\nTomas M{\\\"o}ller and Ben Trumbore. 1997.\n\\textit{Fast, Minimum Storage Ray/Triangle Intersection}.\nJournal of Graphics Tools, 2(1):21-28.\n\n\\bibitem{chung} \nChung F.R.K. 1997.\n\\textit{Spectral Graph Theory}.\nCBMS 92, AMS.\n\n\\bibitem{gingh} \nKaran Singh and Eugene Fiume. 1998.\n\\textit{Wires: A Gerometric Deformation Technique}.\nSIGGRAPH 98 Conference Proceedings, pages 405-414.\n\\end{thebibliography}\n\n\n\n\\end{multicols}\n\\end{document}\n", "meta": {"hexsha": "481545216a66acb999f9d30b94c8a97807e0dff7", "size": 9896, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper-eliebe-5/paper-eliebe-5.tex", "max_stars_repo_name": "eliebe-5/vgl-deform", "max_stars_repo_head_hexsha": "d82e12587cc9aa5618ecf110b50b9902fbed6300", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper-eliebe-5/paper-eliebe-5.tex", "max_issues_repo_name": "eliebe-5/vgl-deform", "max_issues_repo_head_hexsha": "d82e12587cc9aa5618ecf110b50b9902fbed6300", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper-eliebe-5/paper-eliebe-5.tex", "max_forks_repo_name": "eliebe-5/vgl-deform", "max_forks_repo_head_hexsha": "d82e12587cc9aa5618ecf110b50b9902fbed6300", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 70.1843971631, "max_line_length": 812, "alphanum_fraction": 0.7747574778, "num_tokens": 2577, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3451353070238076}} {"text": "\\documentclass[leqno,11pt,a4paper]{article}\n\\usepackage{amsmath,amsthm}\n\n\\textheight=9.0truein\n\\hoffset=-0.4truein\n\\textwidth=6.1truein\n\\voffset=-0.5truein\n\n\\input{header.tex}\n\\input{defs_AHM.tex}\n\n%\\usepackage{mathabx} %package utilisé pour définir \"\\lll\" comme le symbole d'absolue continuité de 2 mesures\n\\usepackage[toc,page]{appendix} %package utilisé pour l'appendice voir : https://texfaq.org/FAQ-appendix\n\\usepackage{tikz}\n%\\usetikzlibrary{shapes}\n%\"Ce document a été généré avec la version de TikZ \\pgfversion .\" : à mettre après \"begin{document}\" si affichage souhaité...\n\\newtheorem{rem}{Remark}\n\\setcounter{secnumdepth}{3} %permet de numéroter les sous-sous-sections\n\n\\begin{document}\n\\title{A quantitative Mac Diarmid's inequality for geometrically ergodic Markov chains}\n%\\title{Density estimation for RWRE : a Bayesian non-parametric approach}\n%\\author{A. Havet, M. Lerasle, E. Moulines and E. Vernet}\n\\author{D. Belomestny, E. Moulines and S. Samsonov}\n\\maketitle\n\n\\begin{abstract}\n We state and prove a quantitative version of the bounded difference inequality for geometrically ergodic Markov chains.\n Our proof uses the same martingale decomposition as \\cite{MR3407208} but, compared to this paper, the exact coupling argument is modified to fill a gap between the strongly aperiodic case and the general aperiodic case.\n\\end{abstract}\n\n\n\\noindent\n{\\small {\\bf Keywords:} Concentration inequalities ; Markov chains ; Geometric ergodicity ; Coupling.}\n\n\\noindent\n{\\small {\\bf AMS MSC 2010:} 60J05; 60E15.}\n\n\\section{Introduction}\nThe purpose of this note is to establish a quantitative version of Mc Diarmid's inequality for geometrically ergodic Markov chains.\nLet $X_0,\\ldots,X_{n-1}$ denote independent random variables taking values in a measurable space $(\\sX,\\cX)$ and $c=(c_0,\\ldots,c_{n-1})$ denote a vector of non-negative real numbers. A function $f:\\sX^n\\to\\Rbb$ satisfies the bounded difference inequality if for all $x=(x_0,\\ldots,x_{n-1})$ and\n$y=(y_0,\\ldots,y_{n-1})\\in \\sX^{n}$, we have\n\\begin{equation}\\label{def:BDP}\n|f(x)-f(y)|\\leqslant \\sum_{i=0}^{n-1}c_i\\indiacc{x_i\\ne y_i}\\eqsp.\n\\end{equation}\nThe bounded difference inequality, first established in \\cite{MR1036755}, shows that for all $t > 0$,\n\\[\n\\Pbb\\big(f(X_0,\\ldots,X_{n-1})-\\Ebb[f(X_0,\\ldots,X_{n-1})]>t\\big)\\leqslant \\rme^{-2t^2/\\|c\\|^2}\\eqsp,\n\\]\nwhere $\\|c\\|^2=\\sum_{i=0}^{n-1}c_i^2$.\nSeveral attempts have been made to extend this result to Markov chains.\nIn \\cite{MR2424985}, the concentration of particular functionals of the form $f(x_0,\\ldots,x_{n-1})=\\sup_{g\\in \\cF}\\sum_{i=0}^{n-1}g(x_i)$, for centered functions $g$ in a class $\\cF$ is established.\nThe concentration of general functionals (satisfying \\eqref{def:BDP}) of geometrically ergodic Markov chains was established in \\cite{MR3407208}, where it is also proved that geometric ergodicity is a necessary assumption. However, the result in \\cite{MR3407208} is not quantitative.\nIt states that for all geometrically recurrent set $C$, there exists a constant $\\beta$, depending on $C$ such that for all $x \\in C$ and $t > 0$,\n\\begin{equation}\n\\label{eq:mcdiarmid-markov}\n\\Pbb_x\\big(f(X_0,\\ldots,X_{n-1})-\\Ebb_x[f(X_0,\\ldots,X_{n-1})]>t\\big)\\leqslant \\rme^{-\\beta t^2/\\|c\\|^2}\\eqsp,\n\\end{equation}\nwhere for any $x\\in\\Xsigma$, $\\Pbb_x$ is the distribution of the Markov chain $\\{X_k\\}_{k=0}^\\infty$ starting from $x$ (see the precise definition below).\nIn many applications, it is necessary to get the explicit dependence of the constant $\\beta$ as a function of the set $C$.\nIn particular, this problem arises when establishing posterior concentration rates of Bayesian non-parametric estimators; see for example \\cite{rousseau:2016,ghosal:vandervaart:2017} for recent accounts on this theory.\nTo extend these results to Markovian settings, the result of \\cite{MR3407208} cannot be applied directly and a quantitative version of \\eqref{eq:mcdiarmid-markov} is required, where the dependence of $\\beta$ on constants characterizing the mixing of the Markov chain is needed; see for example \\cite{vernet:2015,lecorff:lerasle:vernet:2018}.\n\n\nA quantitative version of Mc Diarmid's inequality for Markov chains was established in \\cite{paulin:2015}, where the constant $\\beta$ depends here explicitly on the mixing time of the chain.\nThe existence of finite mixing times requires \\emph{uniform} ergodicity of the chain, see for example \\cite[Section 3.3]{MR2095565}, an assumption that typically fails when the chain takes value in general state spaces.\nIn this note, we prove an extension of Mc Diarmid's inequality to geometrically ergodic Markov chains.\nOur proof is based on \\cite{MR3407208},\nbut avoids the use of \\cite[Lemma~6]{MR3407208} which requires the construction of an exact coupling.\nExact coupling can actually be built in the strongly aperiodic case but there is a gap in the general aperiodic case.\n\nThe remaining of the paper is decomposed as follows, Section~\\ref{sec:Setting} introduces formally the notations and the assumptions of the main result, which is stated and proved in Section~\\ref{sec:Main}.\n\n\n\\section{Notations and assumptions}\\label{sec:Setting}\nLet $(\\sX,\\cX)$ be a measurable space.\nWe denote by $\\tvdistsym$ the total variation distance between probability measures.\nFor any sequence $x=\\sequence{x}[n][\\Nbb]$ and any non-negative integers $a$ and $b$, with $a\\leqslant b$, let $\\chunk[a]{x}[b]=(x_a,x_{a+1},\\ldots,x_b)$.\nFor any $n\\geqslant 0$ and any vector $c=\\chunk[0]{c}[n-1]\\in \\R^n$, let $\\|c\\|$ denote the Euclidean norm of $c$ and $\\|c\\|_\\infty=\\max_{0\\leqslant i\\leqslant n-1}|c_i|$ denote its sup-norm.\n\n\nWe denote by $(\\sX^{\\Zbb_+},\\cX^{\\otimes\\Zbb_+}, (\\cF_k)_{k\\geqslant 0})$ the canonical filtered space,\n$\\{X_n\\}_{n=0}^\\infty$ the canonical process and $\\theta: \\sX^{\\Zbb_+}\\to\\sX^{\\Zbb_+}$ the shift operator on the canonical space defined, for any $x=(x_{n})_{n\\geqslant 0}\\in \\sX^{\\Zbb_+}$ by $\\theta(x)\\in \\sX^{\\Zbb_+}$, where, for any $n\\geqslant 0$, $\\theta(x)_n=x_{n+1}$.\nSet $\\theta_1=\\theta$ and for $n\\in \\Nbb^*$, define inductively, $\\theta_n=\\theta_{n-1} \\circ \\theta$. We also need to define $\\theta_\\infty$. To this aim, fix an arbitrary $x^*\\in \\Xset$, we define $\\theta_\\infty:\\Xset^\\Nbb \\to \\Xset^\\Nbb$ such that for $z=\\sequence{z}[k][\\Nbb] \\in \\Xset^\\Nbb$, $\\theta_\\infty z\\in \\Xset^\\Nbb$ is the constant sequence $(\\theta_\\infty z)_k=x^*$ for all $k \\in \\Nbb$.\n\n\nLet $P$ be a Markov kernel on $\\sX \\times \\cX$.\nFor any probability measure $\\xi$ on $(\\Xset,\\Xsigma)$, denote by $\\Pbb_{\\xi}$ the unique probability under which $(X_n)_{n\\geqslant 0}$ is a Markov chain with Markov kernel $P$ and initial distribution $\\xi$ and let $\\Ebb_{\\xi}$ denote the expectation under the distribution $\\Pbb_{\\xi}$.\nRecall that $\\cF_n$ denotes the $\\sigma$-algebra generated by $X_0, \\ldots,X_n$.\nFor any $x\\in \\sX$, let $\\delta_x$ denote the Dirac mass at point $x$.\nWith some abuse of notation, we also denote $\\Pbb_x$ (resp. $\\Ebb_x$) instead of $\\Pbb_{\\delta_x}$ (resp. $\\Ebb_{\\delta_x}$).\n\nFor any $\\cB\\in \\cX$ and any integer $i\\geqslant 0$, let\n\\[\n\\tau_{\\cB}^i=\\inf\\{n\\geqslant i: X_n\\in \\cB\\}=i+\\tau_{\\cB}^0\\circ \\theta^i\n\\qquad \\text{and} \\qquad\n\\sigma_{\\cB}=\\tau_{\\cB}^1=1+\\tau_{\\cB}^0\\circ \\theta\\eqsp.\n\\]\nFor $c = \\chunk[0]{c}[n-1] \\in \\R_+^n$, we denote by $\\functionboundeddiff{\\sX^n}{c}$ the set of measurable functions $f:\\sX^n\\to\\R$ such that for all $x= (x_0,\\dots,x_{n-1})$ and $y= (y_0,\\dots,y_{n-1})$, $|f(x)-f(y)|\\leqslant \\sum_{i=0}^{n-1} c_i\\indiacc{x_i\\ne y_i}$\nThe main result is established under the following conditions.\n\\begin{itemize}\n\\item[{\\bf H1}] The Markov kernel $P$ is irreducible and aperiodic, with unique invariant probability $\\pi$.\n\\item[{\\bf H2 (drift condition)}] There exist a measurable function $V: X \\rightarrow [1; +\\infty)$, real numbers $\\lambda \\in (0,1)$, $b > 0$ and a non-empty set $\\cC \\in \\cX$ such that for any $x \\in X$,\n\\[\nPV(x) \\leq \\lambda V(x) + b\\indiacc{x \\in \\cC}\n\\]\n%\\item[{\\bf H2}] There exist a non-empty set $\\cC \\in \\cX$ and two real numbers $u>1$ and $M>0$ such that\n% \\[\\sup_{x\\in \\cC}\\Ebb_x[u^{\\sigma_{\\cC}}]\\leqslant M\\eqsp.\\]\n%\\item[{\\bf H3}] There exist $r\\in (0,1)$ and $L\\geqslant 1$ such that, for any $x$ in the set $\\cC$ of {\\bf H2} and any $n\\geqslant 0$,\n%\\[\n%\\tvdist{\\delta_xP^n}{\\pi} \\leqslant Lr^n\\eqsp,\n%\\]\n%where $\\pi$ is the unique invariant measure granted in {\\bf H1}.\n\\item[{\\bf H3 (V-ergodicity)}] There exist $r \\in (0,1)$, $L \\geqslant 1$ and function $V: X \\rightarrow \\mathbb{R}$ such that for any $x \\in X$ and any $n \\geqslant 0$, it holds\n\\[\nd_V(\\delta_xP^n,\\pi) \\leqslant L V(x) r^n\\eqsp,\n\\]\n\\end{itemize}\nwhere $\\pi$ is the unique invariant measure from {\\bf H1}.\n\\par\nWhen the Markov kernel $P$ is uniformly ergodic, then {\\bf H3} holds with $V(x) = 1$. \n\\par\nNote that \\cite[Proposition 14.3.1]{douc:moulines:priouret:soulier:2018} implies that if Markov kernel $P$ satisfies the drift condition {\\bf H2} with the set $\\cC \\in \\cX$, $\\cC = \\{x: V(x) < d\\}$, then it holds for any $x \\in \\cC$ that\n\\begin{equation}\n\\label{eq:exp_moment_hit}\n\\sup_{x\\in \\cC}\\Ebb_x[\\lambda^{-\\sigma_{\\cC}}]\\leqslant M\\eqsp\n\\end{equation}\nwith some $M$ depending on $V(x)$ and $d$.\n\\par\nThe following Lemma is a coupling result that replaces \\cite[Lemma~6]{MR3407208}. It is instrumental in the sequel.\n\\begin{lemma}\\label{lem:DMPS18}\n For any probability measures $\\xi$ and $\\xi'$ on $(\\sX,\\cX)$, any $n\\geqslant 1$, any $c\\in \\R_+^n$ and any $h\\in \\bB\\Dbb(\\sX^n,c)$,\n\\begin{equation*}\\label{eq:Step1}\n |\\Ebb_\\xi[h(\\chunk[0]{X}[n-1])]-\\Ebb_{\\xi'}[h(\\chunk[0]{X}[n-1])]|\\leqslant 2\\sum_{i=0}^{n-1}c_i \\tvdist{\\xi P^i}{\\xi'P^i} \\leqslant 2\\sum_{i=0}^{n-1}c_i \\vfuncdist{\\xi P^i}{\\xi'P^i} \\eqsp.\n\\end{equation*}\n\\end{lemma}\n\\begin{remark}\n It is possible to avoid the factor $2$ in \\eqref{eq:Step1} under additional technical conditions, for example, when there exists a maximal coupling for $(\\Pbb_\\xi,\\Pbb_{\\xi'})$, see \\cite[Lemma 23.2.1]{douc:moulines:priouret:soulier:2018}.\n\\end{remark}\n\\begin{proof}\nFix an arbitrary\n$x^* \\in \\Xset$. For $i \\in \\{1,\\ldots,n-1\\}$, we set $\\bar{h}_i(\\chunk[i]{x}[n-1])=h(x^*,\\dots,x^*,\\chunk[i]{x}[n-1])$. By convention,\nwe set $\\bar{h}_n$ the constant function $\\bar{h}_n=h(x^*,\\ldots,x^*)$ and $\\bar{h}_0=h$. With these\nnotations, we have the decomposition\n\\[\nh(\\chunk{x})=\\sum_{i=0}^{n-1} \\{\\bar{h}_i(\\chunk[i]{x}[n-1]) - \\bar{h}_{i+1}(\\chunk[i+1]{x}[n-1]) \\} +\\bar{h}_n \\eqsp.\n\\]\nFor all $i \\in \\{0,\\ldots,n-1\\}$ and all $x_i \\in \\Xset$, let\n\\begin{align}\n\\bar{w}_i(x_i)&=\\int \\lrcb{\\bar{h}_i(\\chunk[i]{x}[n-1])-\\bar{h}_{i+1}(\\chunk[i+1]{x}[n-1])} \\prod_{\\ell=i+1}^{n-1} P(x_{\\ell-1}, \\rmd x_{\\ell})\\eqsp, \\nonumber \\\\\n&=\\int \\lrcb{h(x^*,\\ldots,x^*,\\chunk[i]{x})-h(x^*,\\ldots,x^*,\\chunk[i+1]{x})} \\prod_{\\ell=i+1}^{n-1} P(x_{\\ell-1}, \\rmd x_{\\ell}) \\eqsp. \\label{eq:delta:diarmid:fond}\n\\end{align}\nIt is easily seen that $\\CPE{\\{\\bar{h}_i(\\chunk[i]{X}[n-1])-\\bar{h}_{i+1}(\\chunk[i+1]{X}[n-1])\\}}{\\cF_{i}}=\\bar{w}_i(X_i)$, $\\Pbb_\\xi-\\as$, which implies that\n\\[\n\\Ebb_\\xi \\lrb{h(\\chunk{X})}=\\sum_{i=0}^{n-1} \\xi P^i \\bar{w}_i + \\bar{h}_n \\eqsp.\n\\]\nSince $h \\in\\functionboundeddiff{\\Xset^n}{c}$, \\eqref{eq:delta:diarmid:fond} shows that $\\supnorm{\\bar{w}_i} \\leq c_i$.\nTherefore,\n\\[\n|\\Ebb_\\xi \\lrb{h(X^{n-1})}-\\Ebb_{\\xi'} \\lrb{h(X^{n-1})} |\\\\\n\\leq \\sum_{i=0}^{n-1} |\\xi P^i \\bar{w}_i-\\xi' P^i \\bar{w}_i| \\leq 2 \\sum_{i=0}^{n-1} c_{i}\\tvdist{\\xi P^{i}}{\\xi' P^{i}} \\eqsp.\n\\]\nThe result now follows from the trivial bound $\\tvdist{\\xi}{\\eta} \\leq \\vfuncdist{\\xi}{\\eta}$ for arbitrary measures $\\xi,\\eta$ given that $V(x) \\geq 1$ for any $x \\in X$.\n\\end{proof}\n\n\\section{Main result}\\label{sec:Main}\nThe main result of this paper is the following quantitative version of Mac Diarmid's inequality for $V-$ergodic Markov chains under certain drift condition.\n%\\begin{theorem}\\label{thm:ConcMarkQuant}\n %Assume {\\bf H1}, {\\bf H2}, {\\bf H3}. Let $n\\geqslant 1$, $c\\in \\R^n$ and $f\\in \\functionboundeddiff{\\sX^n}{c}$. Then, for all $x\\in \\cC$ and $t > 0$,\n %\\[\n%\\Pbb_x\\big(f( \\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]>t\\big)\\leqslant \\exp\\bigg(-\\frac{\\beta t^2}{\\|c\\|^2}\\bigg)\\eqsp,\n% \\]\n%where $\\beta$ is given by\n %\\[\n %\\beta=\\frac{(1-r\\vee u^{-1/4})^2}{16L}\\bigg(\\frac5{\\log u}+4ML\\bigg)^{-1}\\eqsp.\n %\\]\n%\\end{theorem}\n\\begin{theorem}\\label{thm:ConcMarkVErgodic}\n Assume {\\bf H1}, {\\bf H2} and {\\bf H3} holds with drift function $V(x)$. Let $M$ be a constant from \\ref{eq:exp_moment_hit}, $\\cC = \\{V(x) < d\\}$, $n\\geqslant 1$, $c\\in \\R^n$ and $f\\in \\functionboundeddiff{\\sX^n}{c}$. Then, for all $x\\in \\cC$ and $t > 0$,\n \\[\n\\Pbb_x\\big(f( \\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]>t\\big)\\leqslant \\exp\\bigg(-\\frac{\\beta t^2}{\\|c\\|^2}\\bigg)\\eqsp,\n \\]\nwhere $\\beta$ is given by\n \\[\n \\beta=\\frac{(1-r\\vee \\lambda^{1/4})^2}{16Ld}\\bigg(-\\frac5{\\log \\lambda}+4MLd\\bigg)^{-1}\\eqsp.\n \\]\n\\end{theorem}\n\n\\begin{proof}[Proof of \\Cref{thm:ConcMarkVErgodic}]\nFix $c\\in \\Rbb^n$, $x\\in\\sX$ and $f\\in \\functionboundeddiff{\\sX^n}{c}$. Note that drift condition {\\bf H3} implies that\n\\[\n\\sup_{x\\in \\cC}\\Ebb_x[\\lambda^{-\\sigma_{\\cC}}]\\leqslant M\\eqsp\n\\]\nFollowing \\cite{MR3407208}, we decompose $f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]$ into martingale increments by conditioning to the stopping times $\\tau_{\\cC}^i$, $i=0,\\ldots, n-1$.\nFor any integer $i\\in [0,n-1]$, define\n\\[\nG_i=\\Ebb_x\\big[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^i}\\big]\\eqsp.\n\\]\nAs $\\tau_{\\cC}^0=0$ $\\Pbb_x$-a.s., it holds $\\Ebb_x[f(\\chunk[0]{X}[n-1])]=\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^0}]=G_0$.\nMoreover, as $\\tau_{\\cC}^{n-1}\\geqslant n-1$, it also holds $G_{n-1}=\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^{n-1}}]=f(\\chunk[0]{X}[n-1])$.\nTherefore, the difference $f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]$ is decomposed into a sum of the martingale increments $G_{i+1}-G_i$ as follows\n\\begin{equation}\\label{eq:LinkWithG}\n f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]=G_{n-1}-G_0=\\sum_{i=0}^{n-2}(G_{i+1}-G_i)\\eqsp.\n\\end{equation}\nThe proof is now decomposed into three facts that aim at bounding the Laplace transform of $f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]$.\n\n\\noindent\n{\\bf Fact 1.} \\emph{For any $i\\in \\{1,\\ldots,n-1\\}$,}\n\\begin{equation}\\label{eq:Fact1}\n G_i-G_{i-1}=(G_i-G_{i-1})\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\eqsp.\n\\end{equation}\n\\begin{proof}[Proof of Fact 1.] By definition $\\tau_{\\cC}^{i-1}\\geqslant i-1$ and $\\tau_{\\cC}^{i-1}> i-1$ if and only if $\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}$.\n Therefore,\n \\[\n G_i-G_{i-1}=(G_i-G_{i-1})\\left(\\indiacc{\\tau_{\\cC}^{i-1}=i-1}+\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}}\\right)\\eqsp.\n \\]\nTo prove that $(G_i-G_{i-1})\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}}=0$, we decompose according to the values of $\\tau_{\\cC}^{i}$:\n \\[\n(G_i-G_{i-1})\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}}=\\sum_{j\\geqslant i} (G_i-G_{i-1})\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}=j}\\eqsp.\n \\]\nNow, remark that, for any $i\\geqslant 0$,\n\\begin{align}\n G_i\\indiacc{\\tau_{\\cC}^{i}=j}=\n\\begin{cases}\\label{eq:Gitaui=j}\n \\Ebb_x\\big[f(\\chunk[0]{X}[n-1])|\\cF_j\\big]&\\text{ if } j\\leqslant n-2\\eqsp,\\\\\n f(\\chunk[0]{X}[n-1])&\\text{ if } j\\geqslant n-1\\eqsp.\n\\end{cases}\n\\end{align}\nThen, for any $j\\geqslant i$,\n\\[\n G_i\\indiacc{\\tau_{\\cC}^{i}=j}\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}}= G_{i-1}\\indiacc{\\tau_{\\cC}^{i-1}=j}\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}}=G_{i-1}\\indiacc{\\tau_{\\cC}^{i}=j}\\indiacc{\\tau_{\\cC}^{i-1}=\\tau_{\\cC}^{i}}\\eqsp.\n\\]\nThis proves Fact 1.\n\\end{proof}\n\nFact 2. bounds the increments $G_i-G_{i-1}$. The proof relies on the following lemma which is a consequence of the coupling result Lemma~\\ref{lem:DMPS18}.\nDefine $g_{n-1}=g_{n-1,\\pi}=f$ and, for any $i\\in [0,n-2]$, let $g_i$ and $g_{i,\\pi}$ denote the functions defined for any $\\chunk[0]{x}[i]\\in\\sX^{i+1}$ by \n\\begin{equation}\n\\label{eq:definition-g-i}\ng_i(\\chunk[0]{x}[i])=\\Ebb_{x_i}[f(\\chunk[0]{x}[i],\\chunk[1]{X}[n-1-i])],\\qquad g_{i,\\pi}(\\chunk[0]{x}[i])=\\Ebb_{\\pi}[f(\\chunk[0]{x}[i],\\chunk[1]{X}[n-1-i])]\\eqsp.\n\\end{equation}\n\n\\begin{lemma}\n\\label{lem:gi-gipi}\n Assume {\\bf H1}, {\\bf H2}, {\\bf H3} and let $\\cC = \\{x:V(x) < d\\}$. \n For any $i\\in \\{0,\\ldots,n-1\\}$ and $(\\chunk[0]{x}[i-1],x_i)$ in $\\sX^i\\times\\cC$,\n\\begin{equation}\\label{eq:gi-gipi}\n|g_i(\\chunk[0]{x}[i])-g_{i,\\pi}(\\chunk[0]{x}[i])|\\leqslant 2LV(x_i)\\sum_{j=i+1}^{n-1}c_jr^{j-i} \\leqslant 2Ld \\sum_{j=i+1}^{n-1}c_jr^{j-i}\\eqsp.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n Fix $i\\in\\{0,\\ldots,n-1\\}$ and $\\chunk[0]{x}[i]\\in \\sX^{i+1}$.\nAs $f\\in\\functionboundeddiff{\\sX^n}{c}$, the function $\\tilde{f}_i:\\chunk[1]{y}[n-1-i]\\in \\sX^{n-1-i}\\mapsto f(\\chunk[0]{x}[i],\\chunk[1]{y}[n-1-i])\\in \\R$ satisfies\n\\[\n|\\tilde{f}_i(\\chunk[1]{y}[n-1-i])-\\tilde{f}_i(\\chunk[1]{z}[n-1-i])|\\leqslant \\sum_{k=1}^{n-1-i}c_{i+k}\\indiacc{y_k\\ne z_k}\\eqsp.\n\\]\nHence, $\\tilde{f}_i\\in\\bB\\Dbb(\\sX^{n-1-i},c_{i+1:n-1})$. Applying Lemma~\\ref{lem:DMPS18} to the function $h=\\tilde{f}_i$ yields\n\\begin{align*}\n |g_i(\\chunk[0]{x}[i])-g_{i,\\pi}(\\chunk[0]{x}[i])|&=|\\Ebb_{x_i}[f(\\chunk[0]{x}[i],\\chunk[1]{X}[n-1-i])]-\\Ebb_{\\pi}[f(\\chunk[0]{x}[i],\\chunk[1]{X}[n-1-i])]|\\\\\n &=|\\Ebb_{x_i}[\\tilde{f}_i(\\chunk[1]{X}[n-1-i])]-\\Ebb_{\\pi}[\\tilde{f}_i(\\chunk[1]{X}[n-1-i])]| \\leqslant 2\\sum_{j=i+1}^{n-1}c_j \\vfuncdist{\\delta_{x_i}P^{j-i}}{\\pi} \\eqsp.\n\\end{align*}\nNow inequality \\eqref{eq:gi-gipi} follows from {\\bf H3}.\n\\end{proof}\n\n\n\n\n\n\\noindent\n{\\bf Fact 2.} \\emph{Let $\\rho$ such that $r\\leqslant \\rho<1$ and $i\\in\\{1,\\ldots,n-1\\}$.\nThen,}\n\\begin{align}\n\\label{eq:Gi-Gi-1} |G_i-G_{i-1}|&\\leqslant C_1\\|c\\|_{\\infty}\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\sigma_{\\cC}\\circ\\theta^{i-1}\\eqsp,\\\\\n\\label{eq:Gi-Gi-1Sq} |G_i-G_{i-1}|^2&\\leqslant C_2\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\frac1{\\rho^{2\\sigma_{\\cC}\\circ\\theta^{i-1}}}\\sum_{k=i}^{n-1}c_k^2\\rho^{k-i}\\eqsp.\n\\end{align}\n\\emph{where, $C_1=5Ld/(1-r)$ and $C_2=16L^2d^2/(1-\\rho)$.}\n\\begin{proof}[Proof of Fact 2.]\nFor any integer $i\\in\\{1,\\ldots,n\\}$, let\n\\[\nG_{i,1}=\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^{i-1}}]\\indiacc{\\tau_{\\cC}^{i-1}=i-1},\\qquad G_{i,2}=\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^{i}}]\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\eqsp.\n\\]\nFrom Fact 1., $G_i-G_{i-1}=G_{i,2}-G_{i,1}$.\nBy Markov's property, for any $i\\in \\{0,\\ldots,n-1\\}$ and $x\\in \\sX$,\n\\[\n\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_i]=g_i(X_{0:i}),\\qquad \\Pbb_{x}-\\text{a.s.}\\eqsp.\n\\]\nNow, let $R_{i,1}=g_{i-1}(\\chunk[0]{X}[i-1])\\indiacc{\\tau_{\\cC}^{i-1}=i-1}-g_{i-1,\\pi}(\\chunk[0]{X}[i-1])\\indiacc{\\tau_{\\cC}^{i-1}=i-1}$.\nWe have\n\\begin{align}\n\\notag G_{i,1}&=\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^{i-1}}]\\indiacc{\\tau_{\\cC}^{i-1}=i-1}=\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{i-1}]\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\\\\n\\label{eq:Gi1} &=g_{i-1}(\\chunk[0]{X}[i-1])\\indiacc{\\tau_{\\cC}^{i-1}=i-1}=g_{i-1,\\pi}(\\chunk[0]{X}[i-1])\\indiacc{\\tau_{\\cC}^{i-1}=i-1}+R_{i,1}\\eqsp.\n\\end{align}\nMoreover, as $\\tau_{\\cC}^i\\geqslant i$, by \\eqref{eq:Gitaui=j},\n\\begin{align}\n\\notag G_{i,2}&=\\sum_{j\\geqslant i}\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{\\tau_{\\cC}^{i}}]\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\indiacc{\\tau_{\\cC}^i=j}\\\\\n%\\notag &=\\sum_{j=i}^{n-2}\\Ebb_x[f(\\chunk[0]{X}[n-1])|\\cF_{j}]\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\indiacc{\\tau_{\\cC}^i=j}+f(\\chunk[0]{X}[n-1])\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\indiacc{\\tau_{\\cC}^i\\geqslant n-1}\\\\\n\\label{eq:Gi2} &=\\sum_{j=i}^{n-2}g_j(\\chunk[0]{X}[j])\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}+f(\\chunk[0]{X}[n-1])\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i\\geqslant n-1}\\eqsp.\n\\end{align}\nLet $R_{i,2}=\\sum_{j=i}^{n-2}(g_j(\\chunk[0]{X}[j])-g_{j,\\pi}(\\chunk[0]{X}[j]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}$.\nFrom \\eqref{eq:Gi1} and \\eqref{eq:Gi2},\n\\begin{align}\n\\label{eq:Gi1-Gi2} |G_{i,2}-G_{i,1}|=|&R_{i,2}-R_{i,1}+\\sum_{j=i}^{n-2}(g_{j,\\pi}(\\chunk[0]{X}[j])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}\\\\\n\\notag &+(f(\\chunk[0]{X}[n-1])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i\\geqslant n-1}|\n% \\\\\n% \\leqslant 4&\\indiacc{\\tau_{\\cC}^{i-1}=i-1}L\\sum_{j=i+1}^{n-1}c_jr^{j-i}+\\sum_{j=i}^{n-2}|(g_{j,\\pi}(\\chunk[0]{X}[j])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}|\\\\\n% &+|(f(\\chunk[0]{X}[n-1])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i\\geqslant n-1}|\n\\eqsp.\n\\end{align}\nWe bound separately all the terms in this decomposition.\nFirst, as $\\pi$ is invariant and $f\\in \\functionboundeddiff{\\sX^n}{c}$, for any $j\\in \\{i+1,\\ldots,n-1\\}$ and any $\\chunk[0]{x}[j]\\in \\sX^{j+1}$,\n\\[\n|g_{j,\\pi}(\\chunk[0]{x}[j])-g_{i-1,\\pi}(\\chunk[0]{x}[i-1])|=\\Ebb_{\\pi}[f(\\chunk[0]{x}[j],\\chunk[j+1]{X}[n-1])-f(\\chunk[0]{x}[i-1],\\chunk[i]{X}[n-1])]\\leqslant \\sum_{k=i}^jc_k\\eqsp.\n\\]\nHence,\n\\begin{equation}\n\\label{eq:Bound:Easyterms}\n\\begin{aligned} \\sum_{j=i}^{n-2}|(g_{j,\\pi}(\\chunk[0]{X}[j])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))|\\indiacc{\\tau_{\\cC}^i=j}&\\leqslant \\sum_{j=i}^{n-2}\\indiacc{\\tau_{\\cC}^i=j}\\sum_{k=i}^jc_k=\\indiacc{\\tau_{\\cC}^i\\leqslant n-2}\\sum_{k=i}^{\\tau_{\\cC}^i}c_k\\eqsp,\\\\\n|f(\\chunk[0]{X}[n-1])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1])|\\indiacc{\\tau_{\\cC}^i\\geqslant n-1}&\\leqslant \\indiacc{\\tau_{\\cC}^i\\geqslant n-1}\\sum_{k=i}^{n-1}c_k\\eqsp.\n\\end{aligned}\n\\end{equation}\n\n\n\n %where $|R_{i,1}|$ and $|R_{i,2}|$ are respectively bounded in \\eqref{eq:Bound:R1} and \\eqref{eq:Bound:R2}.\n%\\anici{Selon moi, ta majoration de $|R_{i,1}|$ et $|R_{i,2}|$ par le même terme n'était pas juste : j'ai commenté dans le fichier source (au dessus de cette anotation) et propose donc}\n%\\begin{align*}\n% |G_{i,2}-G_{i,1}|=|&R_{i,2}-R_{i,1}+\\sum_{j=i}^{n-2}(g_{j,\\pi}(\\chunk[0]{X}[j])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}\\\\\n% &+(f(\\chunk[0]{X}[n-1])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i\\geqslant n-1}|\\\\\n% \\leqslant |& R_{i,1}| + |R_{i,2}|+\\sum_{j=i}^{n-2}|(g_{j,\\pi}(\\chunk[0]{X}[j])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}|\\\\\n% &+|(f(\\chunk[0]{X}[n-1])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i\\geqslant n-1}|\\\\\n% \\leqslant 2&\\indiacc{\\tau_{\\cC}^{i-1}=i-1}L \\left( \\sum_{j=i+1}^{n-1}c_jr^{j-i} + \\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i} \\right)\\\\\n% & + \\sum_{j=i}^{n-2}|(g_{j,\\pi}(\\chunk[0]{X}[j])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i=j}|\\\\\n% &+|(f(\\chunk[0]{X}[n-1])-g_{i-1,\\pi}(\\chunk[0]{X}[i-1]))\\indiacc{\\tau_{\\cC}^{i-1}=i-1,\\tau_{\\cC}^i\\geqslant n-1}|\\eqsp.\n%\\end{align*}\nTo bound $|R_{i,1}|$ and $|R_{i,2}|$ in \\eqref{eq:Gi1-Gi2}, we use Lemma~\\ref{lem:gi-gipi}.\nFirst, \\eqref{eq:gi-gipi} directly yields\n\\begin{equation}\\label{eq:Bound:R1}\n|R_{i,1}|\\leqslant 2\\indiacc{\\tau_{\\cC}^{i-1}=i-1}Ld\\sum_{j=i+1}^{n-1}c_jr^{j-i}\\eqsp.\n\\end{equation}\nMoreover, as $\\{\\tau_{\\cC}^i=j\\}\\subset \\{X_j\\in \\cC\\}$, \\eqref{eq:gi-gipi} also yields\n\\[\n(g_j(\\chunk[0]{X}[j])-g_{j,\\pi}(\\chunk[0]{X}[j]))\\indiacc{\\tau_{\\cC}^i=j}\\leqslant 2L\\sum_{k=j+1}^{n-1}c_kr^{k-j}\\indiacc{\\tau_{\\cC}^i=j}\\leqslant 2Ld\\indiacc{\\tau_{\\cC}^i=j}\\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i}\\eqsp.\n\\]\nTherefore,\n\\begin{equation}\\label{eq:Bound:R2}\n|R_{i,2}|\\leqslant 2Ld\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i}\\eqsp.\n\\end{equation}\nPlugging \\eqref{eq:Bound:Easyterms}, \\eqref{eq:Bound:R1} and \\eqref{eq:Bound:R2} in \\eqref{eq:Gi1-Gi2} yields\n\\begin{align}\n\\label{eq:Gi1-Gi22} |G_{i,2}-G_{i,1}|\\leqslant &2Ld\\bigg(\\sum_{j=i+1}^{n-1}c_jr^{j-i}+\\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i}+\\frac1{2Ld}\\sum_{k=i}^{\\tau_{\\cC}^i\\wedge (n-1)}c_k\\bigg)\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\eqsp.\n\\end{align}\nBoth \\eqref{eq:Gi-Gi-1} and \\eqref{eq:Gi-Gi-1Sq} follow from \\eqref{eq:Gi1-Gi22} by bounding separately the $3$ terms in the right-hand side of this inequality.\nLet us first establish \\eqref{eq:Gi-Gi-1}.\nSince $r<1$,\n\\[\n\\sum_{j=i+1}^{n-1}c_jr^{j-i}\\leqslant \\frac{\\|c\\|_\\infty r}{1-r},\\qquad \\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i}\\leqslant \\frac{\\|c\\|_\\infty r}{1-r}\\eqsp.\n\\]\n%\\anici{Dans toutes les équations suivantes, je pense qu'il y avait une erreur dans l'indice du $\\theta$ car par définition $\\tau_{\\cC}^i = i + \\tau_{\\cC}^0 \\circ \\theta^i$ et $\\sigma_{\\cC}=1+\\tau_{\\cC}^0 \\circ \\theta$ d'où $1-i+\\tau_{\\cC}^i\\wedge (n-1) \\leqslant 1-i+\\tau_{\\cC}^i = 1 - i + i +\\tau_{\\cC}^0 \\circ\\theta \\circ \\theta^{i-1}=\\sigma_{\\cC}\\circ\\theta^{i-1}$. Cela est par ailleurs cohérent avec \\eqref{eq:Gi-Gi-1} que l'on souhaite montrer. J'ai donc modifié en conséquence.}\nMoreover,\n\\[\n\\sum_{k=i}^{\\tau_{\\cC}^i\\wedge (n-1)}c_k\\leqslant \\|c\\|_{\\infty}[1-i+\\tau_{\\cC}^i\\wedge (n-1) ]\\leqslant \\|c\\|_{\\infty}[1+\\tau_{\\cC}^0\\circ \\theta^i]=\\|c\\|_{\\infty}\\sigma_{\\cC}\\circ\\theta^{i-1}\\eqsp.\n\\]\nAs $r<1\\leqslant \\sigma_{\\cC}\\circ\\theta^{i-1}$, plugging these upper bounds in \\eqref{eq:Gi1-Gi22} shows\n\\[\n|G_{i}-G_{i-1}|=|G_{i,2}-G_{i,1}|\\leqslant \\frac{5Ld\\|c\\|_{\\infty}}{1-r}\\sigma_{\\cC}\\circ\\theta^{i-1}\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\eqsp.\n\\]\nThis proves \\eqref{eq:Gi-Gi-1}.\nWe use slightly different controls to prove \\eqref{eq:Gi-Gi-1Sq} from \\eqref{eq:Gi1-Gi22}.\nAs $r\\leqslant \\rho<1$, $\\rho^{-\\sigma_{\\cC}\\circ\\theta^{i-1}}\\geqslant 1$, and\n\\begin{align}\n\\label{eq:Step1Var} \\sum_{j=i+1}^{n-1}c_jr^{j-i}\\leqslant \\sum_{j=i}^{n-1}c_j\\rho^{j-i}\\leqslant \\rho^{-\\sigma_{\\cC}\\circ\\theta^{i-1}}\\sum_{j=i}^{n-1}c_j\\rho^{j-i}\\eqsp.\n\\end{align}\n%\\anici{Ci après, je pense qu'il y avait également la même erreur d'indice pour $\\theta$ et j'ai modifié en conséquence.}\nMoreover,\n\\begin{align*}\n \\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i}\\leqslant\\rho^{i-\\tau_{\\cC}^i}\\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_k\\rho^{k-i}\\eqsp.\n\\end{align*}\nAs $\\tau_{\\cC}^i\\geqslant i$ and $i-\\tau_{\\cC}^i=1-\\sigma_{\\cC}\\circ\\theta^{i-1}$,\n\\begin{equation}\\label{eq:Step2Var}\n\\sum_{k=\\tau_{\\cC}^i+1}^{n-1}c_kr^{k-\\tau_{\\cC}^i}\\leqslant \\rho^{1-\\sigma_{\\cC}\\circ\\theta^{i-1}} \\sum_{j=\\tau^i_{\\cC}+1}^{n-1}c_j\\rho^{j-i}\\leqslant \\rho^{-\\sigma_{\\cC}\\circ\\theta^{i-1}}\\sum_{j=\\tau^i_{\\cC}+1}^{n-1}c_j\\rho^{j-i}\\eqsp.\n\\end{equation}\nIn addition,\n\\begin{align}\\label{eq:Step3Var}\n \\sum_{k=i}^{\\tau_{\\cC}^i\\wedge (n-1)}c_k\\leqslant &\\sum_{k=i}^{\\tau_{\\cC}^i\\wedge (n-1)}c_k\\rho^{k-\\tau_{\\cC}^i}=\\sum_{k=i}^{\\tau_{\\cC}^i\\wedge (n-1)}c_k\\rho^{k-i-\\sigma_{\\cC}\\circ\\theta^{i-1}+1}\\leqslant \\rho^{-\\sigma_{\\cC}\\circ\\theta^{i-1}}\\sum_{k=i}^{\\tau_{\\cC}^i\\wedge (n-1)}c_k\\rho^{k-i}\\eqsp.\n\\end{align}\n%\\anici{Je ne sais pas si c'est important quantitativement mais dans les deux majorations précédentes, on se débarasses du $\\tau_{\\cC}^i$ assez \"brutalement\" alors que si on fait comme dans mon manuscrit de thèse, on parvient dans l'inégalité suivante non pas à $25L^2$ mais $16 L^2$ (en supposant bien évidemment comme ici que $L\\geq 1$).}\nPlugging \\eqref{eq:Step1Var}, \\eqref{eq:Step2Var} and \\eqref{eq:Step3Var} in \\eqref{eq:Gi1-Gi22} and applying Cauchy-Schwarz inequality shows\n\\begin{align*}\n |G_{i}-G_{i-1}|^2=&|G_{i,2}-G_{i,1}|^2\\leqslant16L^2d^2\\rho^{-2\\sigma_{\\cC}\\circ\\theta^{i-1}}\\bigg(\\sum_{k=i}^{n-1}c_k\\rho^{k-i}\\bigg)^2\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\\\\n \\leqslant &\\frac{16L^2d^2}{1-\\rho}\\rho^{-2\\sigma_{\\cC}\\circ\\theta^{i-1}}\\sum_{k=i}^{n-1}c_k^2\\rho^{k-i}\\indiacc{\\tau_{\\cC}^{i-1}=i-1}\\eqsp.\n\\end{align*}\nThis proves \\eqref{eq:Gi-Gi-1Sq} and thus {\\bf Fact 2.}\n\\end{proof}\n\n\n\\noindent\n{\\bf Fact 3.} \\emph{ Assume {\\bf H1}, {\\bf H2}, {\\bf H3}. Let $\\cC = \\{x: V(x) < d\\}$. Then for any $x\\in \\cC$,}\n\\begin{align}\n\\Ebb_x\\bigg[\\rme^{f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]}\\bigg]\\leqslant \\rme^{C_3\\|c\\|^2}\\eqsp.\n\\end{align}\n\\emph{where $C_3=\\fracaa{4Ld\\left(-\\fracaa{5}{\\log \\lambda}+4MLd\\right)}{(1-r\\vee \\lambda^{1/4})^2}$.}\n\\begin{proof}[Proof of Fact 3.]\nFor any $t\\in \\Rbb$, $\\rme^t\\leqslant 1+t+t^2\\rme^{|t|}$. Hence, as $\\Ebb_{x}[G_{i+1}-G_i|\\cF_{\\tau_{\\cC}^{i}}]=0$, for any $i\\geqslant 0$, we have\n\\begin{align*}\n \\Ebb_x[\\rme^{G_{i+1}-G_i}|\\cF_{\\tau_{\\cC}^{i}}]\\leqslant 1+\\Ebb_x[(G_{i+1}-G_i)^2\\rme^{|G_{i+1}-G_i|}|\\cF_{\\tau_{\\cC}^{i}}]\\eqsp.\n\\end{align*}\nBy Fact 2.,\n\\begin{align*}\n \\Ebb_x[\\rme^{G_{i+1}-G_i}|\\cF_{\\tau_{\\cC}^{i}}]&\\leqslant 1+C_2\\sum_{k=i+1}^{n-1}c_k^2\\rho^{k-i-1}\\indiacc{\\tau_{\\cC}^{i}=i}\\Ebb_x[\\rho^{-2\\sigma_{\\cC}\\circ\\theta^{i}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}\\circ\\theta^{i}}|\\cF_{\\tau_{\\cC}^{i}}]\\eqsp.\n \\end{align*}\n Now by Markov's property,\n %\n\\begin{align*}\n\\indiacc{\\tau_{\\cC}^{i}=i} \\Ebb_x[\\rho^{-2\\sigma_{\\cC}\\circ\\theta^{i}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}\\circ\\theta^{i}}|\\cF_{\\tau_{\\cC}^{i}}]&=\\indiacc{\\tau_{\\cC}^{i}=i} \\Ebb_x[\\rho^{-2\\sigma_{\\cC}\\circ\\theta^{i}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}\\circ\\theta^{i}}|\\cF_{i}]\\\\\n&=\\indiacc{\\tau_{\\cC}^{i}=i}\\Ebb_{X_i}[\\rho^{-2\\sigma_{\\cC}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}}]\\eqsp.\n\\end{align*}\nHence,\n \\begin{align*}\n \\Ebb_x[\\rme^{G_{i+1}-G_i}|\\cF_{\\tau_{\\cC}^{i}}]\n &=1+C_2\\sum_{k=i+1}^{n-1}c_k^2\\rho^{k-i-1}\\indiacc{\\tau_{\\cC}^{i}=i}\\Ebb_{X_i}[\\rho^{-2\\sigma_{\\cC}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}}]\\eqsp.\n \\end{align*}\nLet $\\rho=r\\vee u^{-1/4}$, $\\varepsilon=\\log u/(2C_1)$ and assume first that $\\|c\\|_{\\infty}\\leqslant \\varepsilon$. By {\\bf H$2$},\n\\[\n\\indiacc{\\tau_{\\cC}^{i}=i}\\Ebb_{X_i}[\\rho^{-2\\sigma_{\\cC}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}}]\\leqslant \\indiacc{\\tau_{\\cC}^{i}=i}\\sup_{x\\in\\cC}\\Ebb_x[\\rho^{-2\\sigma_{\\cC}}\\rme^{C_1\\|c\\|_{\\infty}\\sigma_{\\cC}}]\\leqslant\\sup_{x\\in\\cC}\\Ebb_x[u^{\\sigma_{\\cC}}]\\leqslant M\\enspace.\n\\]\nHence,\n\\[\n \\Ebb_x[\\rme^{G_{i+1}-G_i}|\\cF_{\\tau_{\\cC}^{i}}]\\leqslant 1+C_2M\\sum_{k=i+1}^{n-1}c_k^2\\rho^{k-i-1}\\leqslant \\rme^{C_2M\\sum_{k=i+1}^{n-1}c_k^2\\rho^{k-i-1}}\\eqsp.\n\\]\nBy recurrence, it follows that\n\\begin{align*}\n\\Ebb_x\\bigg[\\rme^{f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]}\\bigg]&\\leqslant \\rme^{C_2M\\sum_{i=0}^{n-2}\\sum_{k=i+1}^{n-1}c_k^2\\rho^{k-i-1}}\\\\\n&=\\rme^{C_2M\\sum_{k=1}^{n-1}c_k^2\\sum_{i=0}^{k-1}\\rho^{k-i-1}}\\leqslant \\rme^{\\frac{C_2M}{1-\\rho}\\|c\\|^2} \\eqsp.\n\\end{align*}\nFix $\\tilde{x}$ in $\\sX$ and let $\\tilde{f} : \\sX^n \\rightarrow \\Rbb$ be defined, for any $x_{0:n-1}$ in $\\sX^n$, by\n\\[\n\\tilde{f}(\\chunk[0]{X}[n-1])\n=\nf(x_0\\mathbbm{1}_{\\{c_{0}\\leq\\varepsilon\\}} + \\tilde{x} \\mathbbm{1}_{\\{c_{0}>\\varepsilon\\}},\\dots, x_{n-1}\\mathbbm{1}_{\\{c_{n-1}\\leq\\varepsilon\\}} + \\tilde{x} \\mathbbm{1}_{\\{c_{n-1}>\\varepsilon\\}})\n\\eqsp .\n\\]\nAs $f$ belongs to $\\mathbb{BD}\\left(\\sX^n,c\\right)$, $\\tilde{f}$ belongs to $ \\mathbb{BD}\\left(\\sX^n,\\tilde{c}\\right)$, where\n\\[\n\\tilde{c}\n=\n\\left( c_0\\mathbbm{1}_{\\{c_{0}\\leq\\varepsilon\\}},\\dots,c_{n-1}\\mathbbm{1}_{\\{c_{n-1}\\leq\\varepsilon\\}}\\right)\n\\eqsp .\n\\]\nSince $\\|\\tilde{c}\\|_{\\infty} < \\varepsilon$ and $\\|\\tilde{c}\\|\\leqslant \\|c\\|$, $\\tilde{f}$ satisfies\n\\begin{align}\\label{ineq:max-gamma-leq-varepsilon-ftilde}\n\\Ebb_{x}\\left[\\rme^{\\tilde{f}(\\chunk[0]{X}[n-1])-\\Ebb_{x}[\\tilde{f}(\\chunk[0]{X}[n-1])]}\\right]\n& \\leqslant\n\\rme^{\\frac{MC_2}{1-\\rho}\n\\| \\tilde{c}\\|^2} \\leqslant\n\\rme^{\\frac{MC_2}{1-\\rho}\n\\| c\\|^2}\n\\eqsp .\n\\end{align}\nFurthermore, by definition of $\\tilde{f}$ and since $f$ is in $\\mathbb{BD}(\\sX^n,c)$, for any $x\\in\\sX^n$,\n\\begin{align}\\label{ineq:difference-f-tildef}\n|f(x)-\\tilde{f}(x)|\n& =\n%|f(x_0,\\dots,x_{n-1})-f(x_0\\mathbbm{1}_{\\{\\gamma_{0}\\leq\\varepsilon\\}} + \\tilde{x} \\mathbbm{1}_{\\{\\gamma_{0}>\\varepsilon\\}},\\dots, x_{n-1}\\mathbbm{1}_{\\{\\gamma_{n-1}\\leq\\varepsilon\\}} + \\tilde{x} \\mathbbm{1}_{\\{\\gamma_{n-1}>\\varepsilon\\}})|\\\\\n%& \\leq\n\\sum_{i=0}^{n-1} c_i \\mathbbm{1}_{\\{c_i>\\varepsilon\\}}\n\\leq\n\\sum_{i=0}^{n-1} c_i \\frac{c_i}{\\varepsilon}\n\\leq\n\\frac{\\|c\\|^2}{\\varepsilon}.\n\\end{align}\nThis implies\n\\[\n\\Ebb_x\\bigg[\\rme^{f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]}\\bigg]\\leqslant \\rme^{\\frac{2\\|c\\|^2}{\\varepsilon}}\\Ebb_{x}\\left[\\rme^{\\tilde{f}(\\chunk[0]{X}[n-1])-\\Ebb_{x}[\\tilde{f}(\\chunk[0]{X}[n-1])]}\\right]\\leqslant \\rme^{\\bigg(\\frac2{\\varepsilon}+\\frac{MC_2}{1-\\rho}\\bigg)\n\\| c\\|^2}\n\\eqsp .\n\\]\nThis shows {\\bf Fact 3} since\n\\[\n\\frac2{\\varepsilon}+\\frac{MC_2}{1-\\rho}\\leqslant \\frac{4Ld}{(1-r\\vee \\lambda^{1/4})^2}\\bigg(-\\frac5{\\log \\lambda}+4MLd\\bigg)\\eqsp.\n\\]\n\\end{proof}\n\n\\noindent\n{\\bf Fact 3} proves that there exists a constant $C=2C_3$ such that, for any $c\\in \\Rbb^n$, $f\\in \\functionboundeddiff{\\sX^n}{c}$ and $x\\in \\cC$,\n\\begin{equation}\\label{ControlLaplace}\n \\Ebb_x\\bigg[\\rme^{f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]}\\bigg]\\leqslant \\rme^{C\\| c\\|^2/2}\n\\eqsp .\n\\end{equation}\nLet $f\\in \\functionboundeddiff{\\sX^n}{c}$ and $x\\in \\cC$. \nFor any $s>0$, $sf\\in \\functionboundeddiff{\\sX^n}{c}$.\nHence, from \\eqref{ControlLaplace}, for any $s,t>0$,\n\\begin{align*}\n \\Pbb\\big(f(\\chunk[0]{X}[n-1])-\\Ebb_x[f(\\chunk[0]{X}[n-1])]>t\\big)&\\leqslant \\rme^{-st+\\log\\Ebb_x\\big[\\rme^{sf(\\chunk[0]{X}[n-1])-\\Ebb_x[sf(\\chunk[0]{X}[n-1])]}\\big]}\\\\\n &\\leqslant \\rme^{-st+s^2C\\| c\\|^2/2}\\enspace.\n\\end{align*}\nChoosing $s=t/(C\\|c\\|^2)$ proves Theorem~\\ref{thm:ConcMarkQuant} with\n\\[\n\\beta=\\frac1{2C}=\\frac1{4C_3}=\\frac{(1-r\\vee \\lambda^{1/4})^2}{16Ld}\\bigg(-\\frac5{\\log \\lambda}+4MLd\\bigg)^{-1}\\eqsp.\n\\]\n\\end{proof}\n\n\\section{Applications to Unadjusted Langevin Algorithm}\\label{sec:ULA}\nWe illustrate the applicability of the results by the example on one of the MCMC methods. Suppose that we aim at sampling from the distribution with density $\\pi(x) = \\frac{e^{-U(x)}}{\\int\\limits_{\\mathbb{R}^d} e^{-U(y)}\\,dy}$. One of the popular algorithms for solving this problem is the Unadjusted Langevin Algorithm, which suggests constructing the Markov Chain\n$$\nX_{k+1} = h_{k+1}(X_k) + \\sqrt{2\\gamma_{k+1}}Z_{k+1}, h_{k+1}(X_k) := X_{k} - \\gamma_{k+1}\\nabla U(X_{k}) \n$$\nwith some non-increasing sequence $\\gamma_k \\geq 0$, where $Z_k \\sim \\mathcal{N}(0,I)$ are i.i.d. standard normal $d$-dimensional random variables. Let us denote ULA kernel with parameter $\\gamma$ as $P_{\\gamma}$, that is,\n\\begin{equation}\n\\label{eq:ula_kernel}\nP_{\\gamma}(x,dy) = \\frac{1}{(4\\pi\\gamma)^{\\frac{d}{2}}}\\exp{\\left(-\\frac{\\|y-x+\\gamma\\nabla U(x)\\|^2}{4\\gamma}\\right)}\\,dy\n\\end{equation}\nLet us also introduce the multiple steps kernel\n$$\nQ^{p,n} = P_{\\gamma_p}\\ldots P_{\\gamma_n}, Q^{n} := Q^{1,n}\n$$\nStandard assumptions on the sequence $\\{\\gamma_k\\}$ are $\\sum\\limits_{k=0}^{\\infty}\\gamma_k = \\infty$ and $\\sum\\limits_{k=0}^{\\infty}\\gamma_k^2 < \\infty$. Thus it is impossible to analyze properties of additive functionals of $\\{X_k\\}$ by standard tools, since the assumptions {\\bf H2} and {\\bf H3} fails in case $\\gamma_k \\rightarrow 0$. Yet we might establish some alternative bounds on the speed of convergence to stationary distribution. We need to impose condition\n\\par\n[{\\bf C1}] Potential $U(x)$ satisfies for all $x,y \\in \\mathbb{R}^d$ the condition\n$$\n\\|U(x) - U(y)\\| \\leq L\\|x-y\\|\n$$\nThis condition obviously implies\n$$\n\\|h_{k}(X) - h_k(Y)\\| \\leq (1+L\\gamma_k)\\|X-Y\\|\n$$\nHence, by \\cite[Theorem~19]{durmus:moulines:2018}, we might obtain the following minorization condition\n\\begin{lemma} Assume that the condition {\\bf C1} holds. Then for all $x,y \\in \\mathbb{R}^d, n\\geq 1$ it holds\n$$\n\\|\\delta_xQ^n - \\delta_yQ^n\\|_{TV} \\leq \\mathbb{I}(x \\neq y)\\left(1 - 2\\Phi\\left(-\\frac{\\|x-y\\|}{2\\sqrt{b_n}}\\right)\\right)\n$$\nwhere $b_n := \\sum\\nolimits_{i=1}^{k}\\frac{\\gamma_i}{\\prod_{j=1}^{i}(1+L\\gamma_j)^2}$\n\\end{lemma}\nNow we aim at establishing Foster-Lyapunov drift conditions for the kernel $P_{\\gamma_n}$.We will say that function $V(x): \\mathbb{R}^d \\rightarrow [1,+\\infty)$ satisfies a Foster-Lyapunov drift condition for the Markov kernel $P_\\gamma$ if. there are such $\\overline{\\gamma} > 0, \\lambda \\in [0,1), C > 0$ such that for any $\\gamma \\in (0, \\overline{\\gamma}]$ it holds\n\\begin{equation}\n\\label{eq:drift}\nP_{\\gamma}V \\leq \\lambda^{\\gamma}V + \\gamma C\n\\end{equation}\nGiven that this condition holds, we may prove drift condition for. the kernel $Q^n$.\n\n\\begin{lemma}\n\\label{lem:drift_prod}\n Assume that $P_{\\gamma}$ satisfies condition \\ref{eq:drift} for any $0 < \\gamma < \\overline{\\gamma}$. Then it holds\n$$\nQ^{n,p}V \\leq \\lambda^{\\Gamma_{p,n}}V + C\\frac{\\lambda^{\\Gamma_{1,p}} - \\lambda^{\\Gamma_{1,n}}}{\\log{\\frac{1}{\\lambda}}}\n$$ \n\\end{lemma}\n\\begin{proof} For notation simplicity let us consider only $p=1$. Then by induction\n$$\nQ^{n}V \\leq \\lambda^{\\Gamma_{1,n}}V + C \\sum\\limits_{k=1}^{n}\\gamma_k \\lambda^{\\Gamma_{1,k-1}} \\leq \\lambda^{\\Gamma_{1,n}}V + C \\int\\limits_{0}^{\\Gamma_{1,n}}\\lambda^{x}\\,dx \\leq\n$$\n$$\n\\leq \\lambda^{\\Gamma_{1,n}}V + C\\frac{1-\\lambda^{\\Gamma_{1,n}}}{\\log{\\frac{1}{\\lambda}}}\n$$\n\\end{proof}\nLet us check that ULA kernel \\ref{} satisfies the drift condition \\label{eq:drift}. We will need additional assumption\n\\par\n[{\\bf C2}] There exists such constants $K_1 > 0$ and $m > 0$, such that for any $x \\notin B(0,K_1)$ it holds\n$$\n\\langle \\nabla U(x), x \\rangle \\geq \\frac{m}{2}\\|x\\|^2\n$$\nAnother popular condition on the potential is strong convexity outside some euclidean ball:\n\\par\n[{\\bf C3}] There exist constants $K_1 > 0$ and $m > 0$ such that for any $x \\notin B(0,K_1)$ and $y \\in \\mathbb{R}^d$, it holds $\\langle D^2U(x)y,y\\rangle \\geq m\\|y\\|^2$. Moreover, there exists $M \\geq 0$ such that for any $x \\in \\mathbb{R}^d$, $\\|D^3U(x)\\| \\leq M$.\n\\par\nYet it is possible to show that assumption {\\bf C3} and {\\bf C1} imply {\\bf C2} (see \\cite[Lemma~14]{brosse:2018:diffusion}), so in the sequel we are going to work with the condition {\\bf C2}. It allows us to prove the following result\n\n\\begin{lemma} Assume that the potential $U(x)$ satisfies conditions {\\bf C1} and {\\bf C2} and without loss of generality consider $\\nabla U(0) = 0$. Then the kernel \\ref{} satisfies drift condition \\ref{eq:drift} for any $0 < \\gamma < \\overline{\\gamma} = \\frac{m}{4L^2}$ with drift function $V(x) = 1 + \\|x\\|^2$, constants $\\lambda = \\exp{-\\left(\\frac{m}{2}\\right)}$, $C = 3K_1^2 + 2d + C_1$ for some absolute constant $C_1 > 0$.\n\\end{lemma}\n\n\\begin{proof} Consider $V(x) = 1 + \\|x\\|^2$, then\n$$\nP_{\\gamma}V(x) = \\int\\limits_{\\mathbb{R}^d}V(y)P_{\\gamma}(x,dy) = \\int\\limits_{\\mathbb{R}^d}(1+\\|y\\|^2)\\frac{1}{(4\\pi\\gamma)^{\\frac{d}{2}}}\\exp{\\left(-\\frac{\\|y-x+\\gamma\\nabla U(x)\\|^2}{4\\gamma}\\right)}\\,dy = \n$$\n\n$$\n=1 + \\frac{1}{(4\\pi\\gamma)^{\\frac{d}{2}}}\\int\\limits_{\\mathbb{R}^d}\\|z + x - \\gamma \\nabla U(x)\\|^2 \\exp{\\left(-\\frac{\\|z\\|^2}{4\\gamma}\\right)}\\,dz\n$$\nLet us first consider the case $x \\notin B(0,K_1)$. Note that $\\|z+x-\\gamma \\nabla U(x)\\|^2 = \\|z\\|^2 + 2 \\langle z, x - \\gamma \\nabla U(x) \\rangle + \\gamma^2 \\|x - \\gamma \\nabla U(x)\\|^2$ and the linear term vanishes after integration, moreover, for $Z \\sim \\mathcal{N}(0,2\\gamma I_d$, it holds $\\mathbb{E}\\|Z\\|^2 = 2\\gamma d$. It remains to notice that due to {\\bf C1} and {\\bf C2}\n$$\n\\|x - \\gamma \\nabla U(x) \\|^2 = \\|x\\|^2 - 2\\gamma \\langle \\nabla U(x), x\\rangle + \\gamma^2 \\|\\nabla U(x)\\|^2 \\leq\n$$\n$$\n \\leq (1-\\gamma m)\\|x\\|^2 + 2\\gamma^2 L^2 \\|x\\|^2\n$$\nThus, plugging everything into expression for $P_{\\gamma}V$ and using $\\gamma < \\frac{m}{4L}$, we obtain\n$$\nP_{\\gamma}V(x) \\leq (1-\\gamma m + 2\\gamma^2 L^2)V(x) + 2\\gamma d + (\\gamma m - 2\\gamma^2 L^2) \\leq \\exp^{-\\frac{\\gamma m}{2}}V(x) + 2\\gamma d\n$$\nNow let $x \\in B(0,K_1)$. Then simply using $\\|x-\\gamma \\nabla U(x)\\|^2 \\leq 2(1+L\\gamma)^2\\|x\\|^2$, we obtain\n$$\nP_{\\gamma}V(x) \\leq (1-\\gamma m + 2\\gamma^2 L^2)V(x) + \\gamma\\left((m - 2\\gamma L^2)(1+\\|x\\|^2) + 2d + 2(1+L\\gamma)^2\\|x\\|^d\\right) \\leq \n$$\n\n$$\n\\leq \\exp^{-\\frac{\\gamma m}{2}}V(x) + \\gamma(3K_1^2 + 2d + C_1)\n$$\nfor some absolute constant $C_1$, which does not depend on $\\gamma, d, x$.\n\\end{proof}\n\nNow the geometric ergodicity result readily follows from \\cite[Theorem~19.4.1]{douc:moulines:priouret:soulier:2018}, namely, the following lemma holds:\n\n\\begin{lemma} Let the potential $U(x), x \\in \\mathbb{R}^D$ satisfy conditions {\\bf C1} and {\\bf C2}. Then for $0 < \\gamma < \\overline{\\gamma}=\\frac{m}{4L^2}$, for any $x \\in X$ it holds\n$$\nd_V(\\delta_xQ^n_\\gamma,\\pi_\\gamma) \\leq C\\rho^n\\left(V(x) + \\pi_\\gamma(V)\\right)\n$$ \nwith $V(x) = 1 + \\|x\\|^2$ and constants\n$$\nb = \\gamma(3K_1^2 + 2D + C_1); \\quad \\overline{b} = b\\exp^{-\\frac{m\\gamma}{2}} + d; \\quad \\varepsilon = 2\\Phi\\left(-\\frac{\\sqrt{d}(1+L\\gamma)}{2\\sqrt{\\gamma}}\\right); \\quad\n$$\n\n$$\nC = \\left(1 + \\exp^{-\\frac{m\\gamma}{2}}\\right)\\left(1+\\frac{\\overline{b}}{(1-\\varepsilon)(1-\\exp^{-\\frac{m\\gamma}{2}} - \\frac{2b}{1+d})}\\right);\n$$\n\n$$\n\\rho = \\exp^{-\\gamma\\left(\\frac{m}{2} - 2\\frac{3K_1^2 + 2D + C_1}{d}\\right)\\frac{\\log{(1-\\varepsilon)}}{\\log{(1-\\varepsilon)} + \\log{(\\exp^{-\\frac{m\\gamma}{2}} + \\frac{2b}{d+1})}}}\n$$\nfor some $C_1 > 0$, which does not depend on $d, \\gamma$ and $n$.\n\\end{lemma}\n\n\\begin{proof} Note that the condition {\\bf C1} implies that the Markov kernel $Q^n_\\gamma$ satisfies $(1,\\varepsilon)$-Doeblin condition with $\\varepsilon = 2\\Phi\\left(-\\frac{\\sqrt{d}(1+L\\gamma)}{2\\sqrt{\\gamma}}\\right)$. Together with drift condition {\\bf C2} it allows to apply \\cite[Theorem~19.4.1]{douc:moulines:priouret:soulier:2018} with appropriate constants. \n\\end{proof}\n\n\\bibliographystyle{plain}\n\\bibliography{BiblioSource}\n\n\n\\end{document} ", "meta": {"hexsha": "69d1ea0393b1a89933fde1e20d6a3eaf0de381aa", "size": 40673, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Note_exponential_moment/MarkovCon.tex", "max_stars_repo_name": "svsamsonov/VR-MCMC", "max_stars_repo_head_hexsha": "335ffb0835917b1cdce633877099c8452e3aeb78", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Note_exponential_moment/MarkovCon.tex", "max_issues_repo_name": "svsamsonov/VR-MCMC", "max_issues_repo_head_hexsha": "335ffb0835917b1cdce633877099c8452e3aeb78", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Note_exponential_moment/MarkovCon.tex", "max_forks_repo_name": "svsamsonov/VR-MCMC", "max_forks_repo_head_hexsha": "335ffb0835917b1cdce633877099c8452e3aeb78", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 64.7659235669, "max_line_length": 486, "alphanum_fraction": 0.6356059302, "num_tokens": 18385, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.6791787121629465, "lm_q1q2_score": 0.344895008002215}} {"text": "\\section{Jastrow Factors}\n\\label{sec:jastrow}\n\nJastrow factors are among the simplest and most effective ways of including\ndynamical correlation in the trial many body wavefunction. The resulting many body\nwavefunction is expressed as the product of an antisymmetric (in the case\nof Fermions) or symmetric (for Bosons) part and a correlating Jastrow factor\nlike so:\n\\begin{equation}\n\\Psi(\\vec{R}) = \\mathcal{A}(\\vec{R}) \\exp\\left[J(\\vec{R})\\right]\n\\end{equation}\n\nIn this section we will detail the types and forms of Jastrow factor used \nin QMCPACK. Note that each type of Jastrow factor needs to be specified using\nits own individual \\texttt{jastrow} XML element. For this reason, we have repeated the\nspecification of the \\texttt{jastrow} tag in each section, with specialization for the\noptions available for that given type of Jastrow.\n\n\\subsection{One-body Jastrow functions}\n\\label{sec:onebodyjastrow}\nThe one-body Jastrow factor is a form that allows for the direct inclusion\nof correlations between particles that are included in the wavefunction with\nparticles that are not explicitly part of it. The most common example of\nthis are correlations between electrons and ions. \n\nThe Jastrow function is specified within a \\texttt{wavefunction} element\nand must contain one or more \\texttt{correlation} elements specifying\nadditional parameters as well as the actual coefficients. Section\n\\ref{sec:1bjsplineexamples} gives examples of the typical nesting of\n\\texttt{jastrow}, \\texttt{correlation}, and \\texttt{coefficient} elements.\n\n\\subsubsection{Input Specification}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{l c c c l }\n\\hline\n\\multicolumn{5}{l}{Jastrow element} \\\\\n\\hline\n\\bfseries name & \\bfseries datatype & \\bfseries values & \\bfseries defaults & \\bfseries description \\\\\n\\hline\nname & text & & (required) & Unique name for this Jastrow function \\\\\ntype & text & One-body & (required) & Define a one-body function \\\\ \nfunction & text & Bspline & (required) & BSpline Jastrow \\\\\n & text & pade2 & & Pade form \\\\\n & text & \\ldots & & \\ldots \\\\\nsource & text & name & (required) & Name of attribute of classical particle set \\\\ \nprint & text & yes / no & yes & Jastrow factor printed in external file?\\\\\n \\hline\n\\multicolumn{5}{l}{elements}\\\\ \\hline\n& Correlation & & & \\\\ \\hline\n\\multicolumn{5}{l}{Contents}\\\\ \\hline\n& (None) & & & \\\\ \\hline\n\\end{tabular}\n%\\end{tabular*}\n\\end{center}\n\\end{table}\n\nTo be more concrete, the one-body Jastrow factors used to describe correlations\nbetween electrons and ions take the form below\n\\begin{equation}\nJ1=\\sum_I^{ion0}\\sum_i^e u_{ab}(|r_i-R_I|)\n\\end{equation}\nwhere I runs over all of the ions in the calculation, i runs over the electrons\nand $u_{ab}$ describes the functional form of the correlation between them.\nMany different forms of $u_{ab}$ are implemented in QMCPACK. We will detail \ntwo of the most common ones below.\n\\include{jastrow_one_body_spline}\n\\include{jastrow_one_body_pade}\n\n\\subsection{Two-body Jastrow functions}\nThe two-body Jastrow factor is a form that allows for the explicit inclusion\nof dynamic correlation between two particles included in the wavefunction. It\nis almost always given in a spin dependent form so as to satisfy the Kato cusp\ncondition between electrons of different spins\\cite{kato}.\n\n The two body Jastrow function is specified within a \\texttt{wavefunction} element\nand must contain one or more correlation elements specifying additional parameters\nas well as the actual coefficients. Section \\ref{sec:2bjsplineexamples} gives \nexamples of the typical nesting of \\texttt{jastrow}, \\texttt{correlation} and\n\\texttt{coefficient} elements.\n\n\\subsubsection{Input Specification}\n\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{l c c c l }\n\\hline\n\\multicolumn{5}{l}{Jastrow element} \\\\\n\\hline\n\\bfseries name & \\bfseries datatype & \\bfseries values & \\bfseries defaults & \\bfseries description \\\\\n\\hline\nname & text & & (required) & Unique name for this Jastrow function \\\\\ntype & text & Two-body & (required) & Define a one-body function \\\\ \nfunction & text & Bspline & (required) & BSpline Jastrow \\\\\nprint & text & yes / no & yes & Jastrow factor printed in external file?\\\\\n \\hline\n\\multicolumn{5}{l}{elements}\\\\ \\hline\n& Correlation & & & \\\\ \\hline\n\\multicolumn{5}{l}{Contents}\\\\ \\hline\n& (None) & & & \\\\ \\hline\n\\end{tabular}\n%\\end{tabular*}\n\\end{center}\n\\end{table}\n\nThe two-body Jastrow factors used to describe correlations between electrons take the form\n\\begin{equation}\nJ2=\\sum_i^{e}\\sum_{j>i}^{e} u_{ab}(|r_i-r_j|)\n\\end{equation}\n\nThe most commonly used form of two body Jastrow factor supported by the code is a splined\nJastrow factor, with many similarities to the one body spline Jastrow.\n\n\\include{jastrow_two_body_spline}\n\n\n\\subsection{Long-ranged Jastrow factors}\nWhile short-ranged Jastrow factors capture the majority of the benefit \nfor minimizing the total energy and the energy variance, long-ranged \nJastrow factors are important to accurately reproduce the short-ranged \n(long wavelength) behavior of quantities such as the static structure \nfactor, and are therefore essential for modern accurate finite size \ncorrections in periodic systems.\n\nBelow two types of long-ranged Jastrow factors are described. The \nfirst (the k-space Jastrow) is simply an expansion of the one and/or \ntwo body correlation functions in plane waves, with the coefficients \ncomprising the optimizable parameters. The second type have few \nvariational parameters and use the optimized breakup method of Natoli \nand Ceperley\\cite{Natoli1995} (the Yukawa and Gaskell RPA Jastrows).\n\n\n\\subsubsection{Long-ranged Jastrow: k-space Jastrow}\nThe k-space Jastrow introduces explicit long-ranged dependence commensurate with the periodic supercell. This Jastrow is to be used in periodic boundary conditions only. \n\nThe input for the k-space Jastrow fuses both one and two-body forms into a single element and so they are discussed together here. The one- and two-body terms in the k-Space Jastrow have the form:\n\\begin{align}\n J_1 &= \\sum_{G\\ne 0}b_G\\rho_G^I\\rho_{-G} \\\\\n J_2 &= \\sum_{G\\ne 0}a_G\\rho_G\\rho_{-G}\n\\end{align}\nHere $\\rho_G$ is the Fourier transform of the instantaneous electron density:\n\\begin{align}\n \\rho_G=\\sum_{n\\in electrons}e^{iG\\cdot r_n}\n\\end{align}\nand $\\rho_G^I$ has the same form, but for the fixed ions. In both cases the coefficients are restricted to be real, though in general the coefficients for the one-body term need not be. See section \\ref{sec:feature_kspace_jastrow} for more detail.\n\nInput for the k-space Jastrow follows the familar nesting of \\texttt{jastrow-correlation-coefficients} elements, with attributes unique to the k-space Jastrow at the \\texttt{correlation} input level.\n\n\\FloatBarrier\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabularx}{\\textwidth}{l l l l l l }\n\\hline\n\\multicolumn{6}{l}{\\texttt{jastrow type=kSpace} element} \\\\\n\\hline\n\\multicolumn{2}{l}{parent elements:} & \\multicolumn{4}{l}{\\texttt{wavefunction}}\\\\\n\\multicolumn{2}{l}{child elements:} & \\multicolumn{4}{l}{\\texttt{correlation}}\\\\\n\\multicolumn{2}{l}{attributes} & \\multicolumn{4}{l}{}\\\\\n & \\bfseries name & \\bfseries datatype & \\bfseries values & \\bfseries default & \\bfseries description \\\\\n & \\texttt{type}$^r$ & text & \\textbf{kSpace} & & Must be kSpace \\\\\n & \\texttt{name}$^r$ & text & \\textit{anything} & 0 & Unique name for Jastrow \\\\\n & \\texttt{source}$^r$ & text & \\texttt{particleset.name} & & Ion particleset name\\\\\n \\hline\n\\end{tabularx}\n\\end{center}\n\\end{table}\n\\FloatBarrier\n\n\\FloatBarrier\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabularx}{\\textwidth}{l l l l l l }\n\\hline\n\\multicolumn{6}{l}{\\texttt{correlation} element} \\\\\n\\hline\n\\multicolumn{2}{l}{parent elements:} & \\multicolumn{4}{l}{\\texttt{jastrow type=kSpace}}\\\\\n\\multicolumn{2}{l}{child elements:} & \\multicolumn{4}{l}{\\texttt{coefficients}}\\\\\n\\multicolumn{2}{l}{attributes} & \\multicolumn{4}{l}{}\\\\\n & \\bfseries name & \\bfseries datatype & \\bfseries values & \\bfseries default & \\bfseries description \\\\\n & \\texttt{type}$^r$ & text & \\textbf{One-Body},\\textbf{Two-Body} & & Must be One-Body/Two-Body \\\\\n & \\texttt{kc}$^r$ & real & kc$\\ge$ 0 & 0.0 & k-space cutoff in a.u. \\\\\n & \\texttt{symmetry}$^o$ & text & crystal,isotropic,none & crystal & Symmetry of coefficients\\\\\n & \\texttt{spinDependent}$^o$ & boolean & yes,no & no & \\textit{No current function} \\\\\n \\hline\n\\end{tabularx}\n\\end{center}\n\\end{table}\n\\FloatBarrier\n\n\\FloatBarrier\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabularx}{\\textwidth}{l l l l l l }\n\\hline\n\\multicolumn{6}{l}{\\texttt{coefficients} element} \\\\\n\\hline\n\\multicolumn{2}{l}{parent elements:} & \\multicolumn{4}{l}{\\texttt{correlation}}\\\\\n\\multicolumn{2}{l}{child elements:} & \\multicolumn{4}{l}{\\textit{None}}\\\\\n\\multicolumn{2}{l}{attributes} & \\multicolumn{4}{l}{}\\\\\n & \\bfseries name & \\bfseries datatype & \\bfseries values & \\bfseries default & \\bfseries description \\\\\n & \\texttt{id}$^r$ & text & \\textit{anything} & cG1/cG2 & Label for coeffs \\\\\n & \\texttt{type}$^r$ & text & \\texttt{Array} & 0 & Must be Array \\\\\n\\multicolumn{2}{l}{body text} & \\multicolumn{4}{l}{}\\\\\n & & \\multicolumn{4}{l}{The body text is a list of real values for the parameters.} \\\\\n \\hline\n\\end{tabularx}\n\\end{center}\n\\end{table}\n\\FloatBarrier\n\n\nAdditional information:\n\\begin{itemize}\n \\item{It is normal to provide no coefficients as an initial guess. The number of coefficients will be automatically calculated according to the k-space cutoff + symmetry and set to zero. }\n \\item{Providing an incorrect number of parameters also results in all parameters being set to zero.}\n \\item{There is currently no way to turn optimization on/off for the k-space Jastrow. The coefficients are always optimized.}\n \\item{Spin dependence is currently not implemented for this Jastrow.}\n \\item{\\texttt{kc}: Parameters with G vectors magnitudes less than \\texttt{kc} are included in the Jastrow. If \\texttt{kc} is zero, it is the same as excluding the k-space term.}\n \\item{\\texttt{symmetry=crystal}: Impose crystal symmetry on coefficients according to the structure factor.}\n \\item{\\texttt{symmetry=isotropic}: Impose spherical symmetry on coefficients according to G-vector magnitude.}\n \\item{\\texttt{symmetry=none}: Impose no symmetry on the coefficients.}\n\\end{itemize}\n\n\n\\begin{lstlisting}[style=XML,caption=k-space Jastrow with one- and two-body terms.]\n \n \n \n \n \n \n \n \n \n \n\\end{lstlisting}\n\n\\begin{lstlisting}[style=XML,caption=k-space Jastrow with one-body term only.]\n \n \n \n \n \n \n\\end{lstlisting}\n\n\\begin{lstlisting}[style=XML,caption=k-space Jastrow with two-body term only.]\n \n \n \n \n \n \n\\end{lstlisting}\n\n\n\n\n\\subsubsection{Long-ranged Jastrows: Gaskell RPA and Yukawa forms}\n\\label{sec:twobodyjastrowlr}\n\\textbf{NOTE: The Yukawa and RPA Jastrows do not work at present \nand are currently being revived. Please contact the developers if \nyou are interested in using them.} \n\nThe exact Jastrow correlation functions contain terms which have a \nform similar to the Coulomb pair potential. In periodic systems \nthe Coulomb potential is replaced by an Ewald summation of the \nbare potential over all periodic image cells. This sum is often \nhandled by the optimized breakup method\\cite{Natoli1995} and this \nsame approach is applied to the long-ranged Jastrow factors in QMCPACK.\n\nThere are two main long-ranged Jastrow factors of this type \nimplemented in QMCPACK: the Gaskell RPA\\cite{Gaskell1961,Gaskell1962} \nform and the Yukawa\\cite{Ceperley1978} form. Both of these forms \nwere used by Ceperley in early studies of the electron gas\\cite{Ceperley1978}, \nbut they are also appropriate starting points for general solids. \n\nThe Yukawa form is defined in real space. It's long-range form is \nformally defined as\n\\begin{align}\n u_Y^{PBC}(r) = \\sum_{L\\ne 0}\\sum_{i2k_F \\\\\n \\frac{3k}{4k_F}-\\frac{1}{2}\\left(\\frac{k}{2k_F}\\right)^3 & k<2k_F\n \\end{cases}\n \\right.\n\\]\nWhen written in atomic units, RPA Jastrow implemented in QMCPACK has the \nform\n\\begin{align}\n u_{RPA}(k) = \\frac{1}{2N_e}\\left(-\\frac{1}{S_0(k)}+\\left(\\frac{1}{S_0(k)^2}+\\frac{12}{r_s^3k^4}\\right)^{1/2}\\right)\n\\end{align}\nHere ``$r_s$'' is again a variational parameter and $k_F\\equiv(\\tfrac{9\\pi}{4r_s^3})^{1/3}$.\n\nFor both the Yukawa and Gaskell RPA Jastrows, the default value for $r_s$ is $r_s=(\\tfrac{3\\Omega}{4\\pi N_e})^{1/3}$.\n\n\n\\FloatBarrier\n\\begin{table}[h]\n\\begin{center}\n\\begin{tabularx}{\\textwidth}{l l l l l l }\n\\hline\n\\multicolumn{6}{l}{\\texttt{jastrow type=Two-Body function=rpa/yukawa} element} \\\\\n\\hline\n\\multicolumn{2}{l}{parent elements:} & \\multicolumn{4}{l}{\\texttt{wavefunction}}\\\\\n\\multicolumn{2}{l}{child elements:} & \\multicolumn{4}{l}{\\texttt{correlation}}\\\\\n\\multicolumn{2}{l}{attributes} & \\multicolumn{4}{l}{}\\\\\n & \\bfseries name & \\bfseries datatype & \\bfseries values & \\bfseries default & \\bfseries description \\\\\n & \\texttt{type}$^r$ & text & \\textbf{Two-Body} & & Must be Two-Body \\\\\n & \\texttt{function}$^r$ & text & \\textbf{rpa}/\\textbf{yukawa} & & Must be rpa or yukawa \\\\\n & \\texttt{name}$^r$ & text & \\textit{anything} & RPA\\_Jee & Unique name for Jastrow \\\\\n & \\texttt{longrange}$^o$ & boolean & yes/no & yes & Use long-range part \\\\\n & \\texttt{shortrange}$^o$ & boolean & yes/no & yes & Use short-range part \\\\\n\\multicolumn{2}{l}{parameters} & \\multicolumn{4}{l}{}\\\\\n & \\texttt{rs}$^o$ & rs & $r_s>0$ & $\\tfrac{3\\Omega}{4\\pi N_e}$ & Avg. elec-elec distance \\\\\n & \\texttt{kc}$^o$ & kc & $k_c>0$ & $2\\left(\\tfrac{9\\pi}{4}\\right)^{1/3}\\tfrac{4\\pi N_e}{3\\Omega}$ & K-space cutoff\\\\\n \\hline\n\\end{tabularx}\n\\end{center}\n\\end{table}\n\\FloatBarrier\n\n\n\\begin{lstlisting}[style=XML,caption=Two body RPA Jastrow with long- and short-ranged parts.]\n\n\n\\end{lstlisting}\n\n\n\n% J1 RPA (intended for electron-proton system)\n% source = particleset.name\n% function = RPA\n% name = anything [Jep]\n% rs = >0 [-1]\n% kc = >0 [-1]\n% \n% J2 RPA\n% attributes\n% function = yukawa or rpa\n% name = anything [RPA_Jee]\n% longrange = yes/no [yes]\n% shortrange = yes/no [yes]\n% parameters\n% rs = >0 [-1] 3\\Omega/4\\pi N_e\n% kc = >0 [-1] 2 (9\\pi/4)^1/3 * 4\\pi N_e/3\\Omega\n\n\n\n\\subsection{Three-body Jastrow functions}\nExplicit three-body correlations can be included in the wavefunction via the three-body Jastrow factor.\nThe three-body electron-electron-ion correlation function ($u_{\\sigma\\sigma'I}$) currently used in \\qmcpack is identical to the one proposed in \\cite{Drummond2004}:\n\\begin{eqnarray}\nu_{\\sigma\\sigma'I}(r_{\\sigma I},r_{\\sigma'I},r_{\\sigma\\sigma'}) &= \\sum_{\\ell=0}^{M_{eI}}\\sum_{m=0}^{M_{eI}}\\sum_{n=0}^{M_{ee}}\\gamma_{\\ell mn} r_{\\sigma I}^\\ell r_{\\sigma'I}^m r_{\\sigma\\sigma'}^n \\\\\n &\\times \\left(r_{\\sigma I}-\\frac{r_c}{2}\\right)^3 \\Theta\\left(r_{\\sigma I}-\\frac{r_c}{2}\\right) \\nonumber \\\\\n &\\times \\left(r_{\\sigma' I}-\\frac{r_c}{2}\\right)^3 \\Theta\\left(r_{\\sigma' I}-\\frac{r_c}{2}\\right) \\nonumber\n\\end{eqnarray}\nHere $M_{eI}$ and $M_{ee}$ are the maximum polynomial orders of the\nelectron-ion and electron-electron distances, respectively,\n$\\{\\gamma_{\\ell mn}\\}$ are the optimizable parameters (modulo\nconstraints), $r_c$ is a cutoff radius, and $r_{ab}$ are the distances\nbetween electrons or ions $a$ and $b$. i.e. The correlation function\nis only a function of the interparticle distances and not a more\ncomplex function of the particle positions, $\\mathbf{r}$. As indicated by the\n$\\Theta$ functions, correlations are set to zero beyond a distance of\n$r_c/2$ in either of the electron-ion distances and the largest\nmeaningful electron-electron distance is $r_c$. This is the\nhighest-order Jastrow correlation function currently implemented.\n\n\nToday, solid state applications of \\qmcpack usually utilize one and\ntwo-body B-spline Jastrow functions, with calculations on heavier\nelements often also using the three-body term described above.\n\n\\paragraph{Example use case}\nHere is an example of H2O molecule. After optimizing one and two body Jastrow factors, add the following block in the wavefunction.\nThe coefficients will be filled zero automatically if not given.\n\\begin{lstlisting}[style=XML]\n\n \n \n \n \n \n \n \n \n \n \n \n \n\n\\end{lstlisting}\n", "meta": {"hexsha": "73e111e686a3a3773abc3592b3f74f2e5f702f02", "size": 19499, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "manual/jastrow.tex", "max_stars_repo_name": "bwvdg/qmcpack", "max_stars_repo_head_hexsha": "cd09fc54b36de2579c9802f5e64b7ec15506f3c3", "max_stars_repo_licenses": ["NCSA"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "manual/jastrow.tex", "max_issues_repo_name": "bwvdg/qmcpack", "max_issues_repo_head_hexsha": "cd09fc54b36de2579c9802f5e64b7ec15506f3c3", "max_issues_repo_licenses": ["NCSA"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-04-10T15:33:28.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-10T15:35:59.000Z", "max_forks_repo_path": "manual/jastrow.tex", "max_forks_repo_name": "bwvdg/qmcpack", "max_forks_repo_head_hexsha": "cd09fc54b36de2579c9802f5e64b7ec15506f3c3", "max_forks_repo_licenses": ["NCSA"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 46.7601918465, "max_line_length": 248, "alphanum_fraction": 0.6841376481, "num_tokens": 5896, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.34489500140698126}} {"text": "\\section{Preliminaries} \\label{gua:sec:prelim}\\label{gua:sec:definitions}\nMany definitions intersect with those defined in previous chapters,\nbut to keep the chapter self-contained we define them here.\n\nNotation:\n$\\bbB = \\{\\true,\\false\\}$ is the set of Boolean values,\n$\\bbN$ is the set of natural numbers (excluding $0$),\n$\\bbN_0 = \\bbN\\cup\\{0\\}$,\n$[k]$ is the set $\\{i \\in \\bbN \\| i \\leq k\\}$\nand $[0..k]$ is the set $[k] \\cup \\{0\\}$ for $k \\in \\bbN$.\nFor a sequence $x=x_1x_2\\ldots$ denote the $i$-$j$-subsequence as $x\\slice{i}{j}$,\ni.e., ${x\\slice{i}{j}}=x_i \\ldots x_j$.\n\n\n\\subsection{System Model} \\label{gua:sec:model}\n\nWe consider systems $A {\\parallel} B^n$, usually written $\\largesys$, \nconsisting of\none copy of a process template $A$ and $n$ copies of a process template $B$,\nin an interleaving parallel composition.%\n%% AK: moved to a separate note, to be able to explain why we _cannot_ generalize for 1-conj\n%\\footnote{As shown in \\cite{Emerson00}, cutoffs for this case generalize to cutoffs \n% for systems of the form $A^m {\\parallel} B^n$, and further \n% to systems with an arbitrary number of process templates \n% $U_1^{n_1} {\\parallel} \\ldots {\\parallel} U_m^{n_m}$.} \nWe distinguish objects that belong to different templates by indexing them with\nthe template. E.g., for process template $U \\in \\{A,B\\}$, $Q_U$ is the set of\nstates of $U$. For this section, fix two disjoint finite sets $Q_A$, $Q_B$ as\nsets of states of process templates $A$ and $B$, and a positive integer $n$.\n\n\\parbf{Processes} A \\emph{process template} \n is a transition system\n $U=(\\stateset, \\init, \\inputs, \\delta)$ with \n\t\\begin{itemize}\n\t\\item $\\stateset$ is a finite set of states including the\n initial state $\\init$,\n\t\\item $\\inputs$ is a finite input alphabet,\n\t\\item $\\delta: \\stateset \\times \\inputs \\times \\mP(Q_A \\cupdot Q_B) \\times \\stateset$ is a guarded transition relation.\n\t\\end{itemize}\nA process template is \\emph{closed} if $\\inputs = \\emptyset$, and otherwise \\emph{open}.\n\nBy $\\transition{q_i}{q_j}{e:g}$\nwe denote a process transition from $q_i$ to $q_j$\nfor input $e \\in \\Sigma$ and guarded by guard $g \\in \\mP(Q_A \\cupdot Q_B)$.\nWe skip the input $e$ and guard $g$\nif they are not important or can be inferred from the context.\n\nWe define the size $\\card{U}$ of a process template $U \\in \\{A,B\\}$ as $\\card{\\stateset_U}$. A copy of a template $U$ will be called a \\emph{$U$-process}.\nDifferent $B$-processes are distinguished by subscript, i.e., for $i \\in [1..n]$, $B_i$ is the $i$th copy of $B$, and $\\state_{B_i}$ is a state of $B_i$. A state of the $A$-process is denoted by $q_A$. \n\nFor the rest of this subsection, fix templates $A$ and $B$. We assume that $\\inputs_A \\cap \\inputs_B = \\emptyset$. We will also write $p$ for a process in $\\{ A, B_1, \\ldots, B_n\\}$, unless $p$ is specified explicitly.\nWe often denote the set $\\{B_1,...,B_n\\}$ as $\\mB$.\n\n\n\\parbf{Disjunctive and conjunctive systems}\nIn a system $\\largesys$,\nconsider the global state $s = (\\state_A,\\state_{B_1},\\ldots,\\state_{B_n})$ and\nglobal input $e=(\\localin_A,\\localin_{B_1},\\ldots,\\localin_{B_n})$.\nWe write $s(p)$ for $q_p$, and $e(p)$ for $\\sigma_p$.\nA local transition $(\\state_p,\\localin_p,g,\\state_p') \\in \\delta_U$ of a process $p$ is \\emph{enabled for $s$ and $e$}\nif the \\emph{guard} $g$ is satisfied by the state $s$ wrt.\\ the process $p$, written $(s,p) \\models g$ (defined below).\nThe semantics of $(s,p) \\models g$ differs for disjunctive and conjunctive systems:\n%\n\\begin{align*}\n\\text{In disjunctive systems: } & (s,p) \\models g \\text{~~~iff~~~} \n\\exists p' \\in \\{A,B_1,\\ldots,B_n\\} \\setminus \\{p\\}:\\ \\ \\state_{p'} \\in g. \\\\\n\\text{In conjunctive systems: } & (s,p) \\models g \\text{~~~iff~~~} \n\\forall p' \\in \\{A,B_1,\\ldots,B_n\\} \\setminus \\{p\\}:\\ \\ \\state_{p'} \\in g.\n\\end{align*}\n\nNote that we check containment in the guard (disjunctively or conjunctively) \nonly for local states of processes \\emph{different from} $p$. A process is \\emph{enabled} for $s$ and $e$ if at least one of its transitions is enabled for $s$ and $e$, otherwise it is \\emph{disabled}.\n\nLike Emerson and Kahlon~\\cite{Emerson00}, \nwe assume that in conjunctive systems $\\init_A$ and $\\init_B$ are contained in all guards,\ni.e., they act as neutral states.\nFurthermore, we call a conjunctive system \\emph{$1$-conjunctive} if every guard is of the form $(Q_A \\cupdot Q_B) \\setminus \\{q\\}$ for some $q \\in Q_A\\cupdot Q_B$.\n\nThen, \\largesys is defined as the transition \nsystem $(S,\\init_S,\\globIn,\\delta)$ with \n\\begin{itemize}\n\\item set of global states $S = \\stateset_A \\times \\stateset_B^{n}$, \n\\item global initial state $\\init_S = (\\initstate_A,\\initstate_B,\\ldots,\\initstate_B)$, \n\\item set of global inputs $\\globIn = (\\inputs_A) \\times (\\inputs_B)^{n}$,\n\\item and global transition relation $\\delta \\subseteq S \\times \\globIn \\times S$ with $(s,e,s') \\in \\delta$ iff \n\\begin{enumerate}[label=\\roman*)] \n \\item $s=(\\state_A,\\state_{B_1},\\ldots,\\state_{B_n})$, \n \\item $e=(\\localin_A, \\localin_{B_1},\\ldots,\\localin_{B_n})$, and \n \\item $s'$ is obtained from $s$ by replacing one local state $\\state_p$ with a new local state $\\state_p'$, where $p$ is a $U$-process with local transition $(\\state_{p},\\localin_{p},g,\\state_p') \\in \\delta_U$ and $(s,p) \\models g$. \n Thus, we consider so-called interleaved systems,\n where in each step exactly one process transits.\n\\end{enumerate}\n\\end{itemize}\nWe say that a system $\\largesys$ is \\emph{of type} $(A,B)$. It is called a\n\\emph{conjunctive system} if guards are interpreted conjunctively, and a\n\\emph{disjunctive system} if guards are interpreted disjunctively. \nA system is \\emph{closed} if all of its templates are closed.\n\n\n\\parbf{Runs} \nA \\emph{configuration} of a system is a triple $(s,e,p)$, where $s \\in S$, $e \n\\in \\globIn$, and $p$ is either a system process, or the special symbol $\\bot$.\n A \\emph{path} of a system is a configuration sequence \n$x = (s_1,e_1,p_1),(s_2,e_2,p_2),\\ldots$ such that, for all $\\time < |x|$, there is a \ntransition $(s_\\time,e_\\time,s_{\\time+1}) \\in \\delta$ based on a local \ntransition of process $p_\\time$. We say that process \n$p_\\time$ \\emph{moves} at \\emph{moment} $\\time$. \nConfiguration $(s,e,\\bot)$ appears\n iff all processes are disabled for $s$ and $e$.\nAlso, for every $p$ and $\\time < |x|$: \neither $e_{\\time+1}(p) = e_\\time(p)$ or process $p$ moves at moment $\\time$. \nThat is, the environment keeps the input to each process unchanged until \nthe process can read it.\\footnote{By only considering inputs that are actually processed, we \napproximate an \naction-based semantics. Paths that do not fulfill this requirement are not \nvery interesting, since the environment can violate any interesting \nspecification that involves input signals by manipulating them when the \ncorresponding process is not allowed to move.} \n\nA system \\emph{run} is a maximal path starting in the initial state. Runs are either infinite, or they end in a configuration $(s,e,\\bot)$. We say that a run is \\emph{initializing} if every \n%$B$-process \nprocess\nthat moves infinitely often also visits \n%$\\initstate_B$ \nits $\\initstate$ \ninfinitely often.\n\nGiven a system path $x = (s_1,e_1,p_1),(s_2,e_2,p_2),\\ldots$ and a process $p$, the \\emph{local path} of $p$ in $x$ is the projection $x(p) = (s_1(p),e_1(p)),(s_2(p),e_2(p)),\\ldots$ of $x$ onto local states and inputs of $p$.\nSimilarly, we define the projection on two processes $p_1,p_2$ denoted by $x(p_1,p_2)$.\n\n%The \\emph{destuttering} $\\destutter(x)$\\ak{make it work with inf runs} of a (local) path \\sj{local path not defined} $x=x_0,x_1,\\ldots$ is obtained by removing stuttering steps from the sequence, i.e., $\\destutter(x)$ is the maximal subsequence $x'$ of $x$ such that for every $\\time$ we have $x'_\\time \\neq x'_{\\time+1}$. Two (local) paths $x$ and $y$ are \\emph{stutter-equivalent}, written $x \\simeq y$, if $\\destutter(x)=\\destutter(y)$. Define an extension of $\\destutter$ to sets of paths in the obvious way. Then two systems $S_1, S_2$ are \\emph{stutter-equivalent}, written $S_1 \\simeq S_2$, if $\\destutter(X_1) = \\destutter(X_2)$, where $X_i$ is the set of all infinite runs of system $S_i$.\n\n\\parbf{Deadlocks and fairness}\nA run is \\emph{globally deadlocked} if it is finite.\nAn infinite run is \\emph{locally deadlocked} for process $p$ if there exists $\\time$ such that $p$ is disabled for all $s_{\\time'},e_{\\time'}$ with $\\time'\\ge \\time$. A run is \\emph{deadlocked} if it is locally or globally deadlocked.\nA system \\emph{has a (local/global) deadlock} if it has a (locally/globally) deadlocked run. Note that the absence of local deadlocks for all $p$ implies the absence of global deadlocks, but not the other way around.\n\nA run $(s_1,e_1,p_1), (s_2,e_2,p_2),...$ is \\emph{unconditionally-fair} if every process moves infinitely often. \nA run is \\emph{strong-fair} if it is infinite and, for every process $p$, if $p$ is enabled infinitely often, then $p$ moves infinitely often.\n%\\sj{weak fairness needed?} Finally, $x$ is \\emph{weak-fair} if it is infinite and for every process $p$, if there exists $t$ such that $p$ is enabled for every $s_{\\time'}, e_{\\time'}$ with $\\time' \\ge \\time$, then $p$ moves infinitely often.\nWe will discuss the role of deadlocks and fairness in synthesis in Section~\\ref{gua:sec:paramsynt}.\n\n\\begin{remark}[$A^m {\\parallel} B^n$]\nOne usually starts with studying parameterized systems of the form $A^n$ (having one process template),\nthen proceeds to systems of the form $A^m {\\parallel} B^n$ (having two templates)\nand $U_1^{n_1} {\\parallel} \\ldots {\\parallel} U_m^{n_m}$ (having an arbitrary fixed number of templates).\nOur work studies systems $A {\\parallel} B^n$,\nwhich have one $A$-process and a parameterized number of $B$-processes,\nbecause the results for such systems can be generalized to systems $U_1^{n_1} {\\parallel} \\ldots {\\parallel} U_m^{n_m}$\n(see~\\cite{Emerson00} for details).\nThis generalization works for our results as well,\nexcept for the cutoffs for deadlock detection that are restricted to 1-conjunctive systems of the form $A\\parallel B^n$\n(Section~\\ref{gua:sec:cutoffs}).\n\\end{remark}\n\n\\subsection{Specifications}\n\\label{gua:sec:semantics}\nFix templates $(A,B)$.\nWe consider formulas in $\\LTLmX$---$\\LTL$ without the next-time operator $\\nextt$---%\nthat are prefixed by path quantifiers $\\E$ or $\\A$\n(for LTL and path quantifiers see Section~\\ref{defs:ctlstar}).\nLet $h(A,B_{i_1},\\ldots,B_{i_k})$ be an $\\LTLmX$ formula over atomic propositions from $Q_A \\cup \\Sigma_A$ and indexed propositions from $(Q_B \\cup \\Sigma_B) \\times \\{i_1,\\ldots,i_k\\}$.\nFor a system $\\largesys$ with $n \\geq k$ and every $i_j \\in [1..n]$,\nsatisfaction of $\\A h(A,B_{i_1},\\ldots,B_{i_k})$ and $\\E h(A,B_{i_1},\\ldots,B_{i_k})$ is defined in the usual way.\n\n\n\\parbf{Parameterized specifications} \\label{gua:sec:parameterized}\nA \\emph{parameterized specification} is a temporal logic formula\nwith indexed atomic propositions and quantification over indices. \nWe consider formulas of the forms\n$\\forall{i_1,\\ldots,i_k.} \\A h(A,B_{i_1},\\ldots,B_{i_k})$ and\\\\ \n$\\forall{i_1,\\ldots,i_k.} \\E h(A,B_{i_1},\\ldots,B_{i_k})$. \nFor a given $n \\geq k$, \n$$\n\\largesys \\models \\forall{i_1,{\\ldots},i_k.} \\A h(A,B_{i_1},{\\ldots},B_{i_k})\n$$\n~iff~\n$$\n\\largesys \\models \\!\\!\\!\\!\\!\\!\\!\\!\\bigwedge_{j_1 \\neq {\\ldots} \\neq j_k \\in [1..n]}\\!\\!\\!\\!\\!\\!\\!\\!\\A h(A,B_{j_1},{\\ldots},B_{j_k}).\n$$ \nBy symmetry of guarded systems (see~\\cite{Emerson00}),\nthe second formula is equivalent to\n$\\largesys \\models \\A h(A,B_1,\\ldots,B_k)$. \nThe formula $\\A h(A,B_1,\\ldots,B_k)$ is denoted by $\\A h(A,B^{(k)})$, \nand we often use it instead of the original $\\forall{i_1,\\ldots,i_k.} \\A h(A,B_{i_1},...,B_{i_k})$.\nFor formulas with the path quantifier $\\E$,\nsatisfaction is defined analogously\nand is equivalent to satisfaction of $\\E h(A,B^{(k)})$.\n\\begin{example}\nConsider the formula\n$$\n\\forall{i_1,i_2}.\\A \\big(\\G (r_{i_1} \\impl \\F g_{i_1}) \\land \\G \\neg (g_{i_1} \\land g_{i_2})\\big).\n$$\nBy our definition, its satisfaction by a system $(A,B)^{(1,3)}$ means\n\\begin{align*}\n(A,B)^{(1,3)} \\models\n\\A \\left(\n\\begin{aligned}\n&\\G(r_1 \\impl \\F g_1) \\land \\G(r_2 \\impl \\F g_2) \\land \\G(r_3 \\impl \\F g_3) \\land \\\\\n&\\G \\neg (g_1 \\land g_2) \\land \\G \\neg (g_1 \\land g_3) \\land \\G \\neg (g_2 \\land g_3)\n\\end{aligned}\n\\right),\n\\end{align*}\nwhere $g_1$ and $r_1$ refer to the propositions $g$ and $r$ of the process $B_1$,\n$g_2$ and $r_2$ belong to $B_2$, and so on.\nBy symmetry, the latter satisfaction is equivalent to\n\\begin{align*}\n(A,B)^{(1,3)} \\models\n\\A \\left(\n\\begin{aligned}\n&\\G(r_1 \\impl \\F g_1) \\land \\G(r_2 \\impl \\F g_2) \\land \\\\\n&\\G \\neg (g_1 \\land g_2)\n\\end{aligned}\n\\right).\n\\end{align*}\nNote that this formula talks about processes $B_1$ and $B_2$, but does not mention $B_3$.\n\\end{example}\n\n\n\\parbf{Specification of fairness and local deadlocks}\nIt is often convenient to express fairness assumptions and local deadlocks \nas parameterized specifications.\nTo this end,\ndefine auxiliary atomic propositions $\\sched_p$ and $\\enabled_p$ for every process $p$ of system $(A,B)^{(1,n)}$. At moment $\\time$ of a given run $(s_1,e_1,p_1),(s_2,e_2,p_2), \\ldots$, let $\\sched_p$ be true whenever $p_\\time = p$, and let $\\enabled_p$ be true if $p$ is enabled for $s_\\time, e_\\time$. Note that we only allow the use of these propositions to define fairness, but not in general specifications.\nThen, an infinite run is \n\\begin{itemize}\n\\item \\emph{local-deadlock-free} if it satisfies $\\forall{p}. \\GF \\enabled_p$, abbreviated as $\\spec_{\\neg dead}$,\n\\item \\emph{strong-fair} if it satisfies $\\forall{p}. \\GF \\enabled_p \\impl \\GF \\sched_p$, abbreviated as $\\spec_{strong}$, and \n\\item \\emph{unconditionally-fair} if it satisfies $\\forall{p}. \\GF \\sched_p$, abbreviated as $\\spec_{uncond}$.\n%\\item \\sj{needed?:}\\emph{weak-fair} if it satisfies $\\forall{p}. \\A \\spec_{weak}$, where $\\spec_{weak} = \\FG \\enabled_p \\impl \\GF \\sched_p$.\n\\end{itemize}\n\nIf $f \\in \\{strong, uncond\\}$ is a fairness notion and \n$\\A h(A,B^{(k)})$\na specification, then we write\n$\\A_{f} h(A,B^{(k)})$ for $\\A (\\spec_{f} \n\\rightarrow h(A,B^{(k)}))$.\nSimilarly, we write $\\E_{f} h(A,B^{(k)})$ for $\\E (\\spec_{f} \\land h(A,B^{(k)}))$.\n\n\n\n\\subsection{Model Checking and Synthesis Problems}\n\\label{gua:sec:nonparameterized_synthesis}\n%\nGiven a system $\\largesys$ and a specification $\\A h(A,B^{(k)})$, where $n \\ge k$. Then:\n\\begin{itemize}\n\\item the \\emph{model checking problem} is to decide whether $\\largesys \\models \\A h(A,B^{(k)})$,\n\\item the \\emph{deadlock detection problem} is to decide whether $\\largesys$\n does not have global nor local deadlocks,\n%\\item the \\emph{deadlock detection problem} is to decide whether all runs of $\\largesys$\n%are infinite and $\\largesys \\models \\A \\spec_{\\neg dead}$, \n%i.e., there are no local deadlocks,\n\\item the \\emph{parameterized model checking problem} (PMCP) is to decide whether $\\forall m \\ge n:\\ (A,B)^{(1,m)} \\models \\A h(A,B^{(k)})$, and \n%\\item the \\emph{parameterized deadlock detection problem} is to decide whether for all $m \\ge n$, all runs of $(A,B)^{(1,m)}$ are infinite and $(A,B)^{(1,m)} \\models \\A \\spec_{\\neg dead}$.\n\\item the \\emph{parameterized deadlock detection problem} is to decide whether, \n for all $m \\ge n$, $(A,B)^{(1,m)}$ does not have global nor local deadlocks.\n\\end{itemize}\nFor a given number $n \\in \\bbN$ and specification $\\A h(A,B^{(k)})$ with $n \\ge k$,\n\\begin{itemize}\n\\item the \\emph{template synthesis problem} is to find process templates $A,B$ such that\n$\\largesys \\models \\A h(A,B^{(k)})$ and $\\largesys$ does not have global deadlocks\\footnote{\\label{footnote:local-deadlocks}Here we do not explicitly mention local deadlocks because they can be specified as a part of $\\A h(A,B^{(k)})$.}\n\\item \nthe \\emph{bounded template synthesis problem} for a pair of bounds $(\\bound_A,\\bound_B) \\in \\bbN \\times \\bbN$ \nis to solve the template synthesis problem with\n$\\card{A} \\leq \\bound_A$ and $\\card{B} \\leq \\bound_B$.\n\\item the \\emph{parameterized template synthesis problem} is to find process templates $A,B$ such that $\\forall m \\ge n:\\ (A,B)^{(1,m)} \\models \\A h(A,B^{(k)})$ and $(A,B)^{(1,m)}$ does not have global deadlocks\\footnoteref{footnote:local-deadlocks}.\n\\end{itemize}\nSimilarly, we define problems for specifications having $\\E$ instead of $\\A$.\nThe definitions can be flavored with different notions of fairness.\n", "meta": {"hexsha": "4e9e1b2324ffe0f7589909970b0e3321103f80bb", "size": 16385, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "thesis/guarded-systems/prelim.tex", "max_stars_repo_name": "5nizza/phd-thesis", "max_stars_repo_head_hexsha": "74a7a4c6ed06aa2894d2ba05f417f5f812730b78", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "thesis/guarded-systems/prelim.tex", "max_issues_repo_name": "5nizza/phd-thesis", "max_issues_repo_head_hexsha": "74a7a4c6ed06aa2894d2ba05f417f5f812730b78", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "thesis/guarded-systems/prelim.tex", "max_forks_repo_name": "5nizza/phd-thesis", "max_forks_repo_head_hexsha": "74a7a4c6ed06aa2894d2ba05f417f5f812730b78", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 60.6851851852, "max_line_length": 699, "alphanum_fraction": 0.6956972841, "num_tokens": 5379, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.679178686187839, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3448949948117473}} {"text": "\\documentclass[a4paper]{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{tikz}\n\\usetikzlibrary{quantikz}\n\\usepackage{physics}\n\\usepackage{pgfplots}\n\\usepgfplotslibrary{units}\n\\usetikzlibrary{patterns}\n\\usepackage{siunitx}\n\\usetikzlibrary{angles, quotes}\n\\usepackage{amssymb}\n\\usetikzlibrary{angles, arrows.meta, quotes}\n\\pgfplotsset{compat=1.17}\n\n\\usepackage{minted}\n\\usepackage[most]{tcolorbox}\n\\definecolor{lightgreen}{rgb}{0.56, 0.93, 0.56}\n\\definecolor{moonstoneblue}{rgb}{0.45, 0.66, 0.76}\n\n\\newtcblisting{myminted}{%\n listing engine=minted,\n minted language=c,\n listing only,\n breakable,\n enhanced,\n minted options = {\n linenos,\n breaklines=true,\n breakbefore=.,\n fontsize=\\footnotesize,\n numbersep=2mm\n },\n overlay={%\n \\begin{tcbclipinterior}\n \\fill[gray!25] (frame.south west) rectangle ([xshift=4mm]frame.north west);\n \\end{tcbclipinterior}\n }\n}\n\\begin{document}\n\t\\title{\n\n\t\\vspace{1cm}\n\t\\Huge Grover's Algorithm\n\t}\n\n\t\\vspace{1cm}\n\n\t% if you are the only author, you might use the following\n\t% \\author{Name of student}\n\n\t% Insert here your name and correct mail address\n\t\\author{\\Large {Rahul Goel} \\Large \\\\ \\ \\ 2019111034\n\t\\date{November 2020}\n\t\\vspace{0.5cm}}\n\n\t% name of the course and module\n\t\\date{\n\t\\large Algorithm Analysis And Design \\\\\n\t\\vspace{1cm}\n\t}\n\n\t\\maketitle\n\t\\setlength{\\parindent}{0pt}\n\n\\vspace{8cm}\n\\begin{abstract}\n\\begin{center}\n\\normalsize\nThis document can be used as an introductory tutorial to Grover's Algorithm. It also includes certain peculiar observations that help to deepen the understanding about the differences between Quantum Computing and Classical Computing.\n\\end{center}\n\\end{abstract}\n\t\\newpage\n\t\\tableofcontents\n\t\\newpage\n\n\n\n\\maketitle\n\n\\textbf{Note - It has been assumed that $N = 2 ^ n$ throughout this sheet.}\n\n\\section{The Problem - Needle in a Haystack}\n\\indent The problem that Grover's Algorithm solves is the unstructured search problem. In simple words, among $N$ given entities, our task is to find one unique entity that satisfies a particular property.\n\nIn most cases, the situation is random and the only way a classical computer knows how to solve it is by brute force. This can take $O(N / 2)$ on average and up to $O(N)$ time in the worst case.\n\nThe problem of unstructured search is considered to be a hard problem for a classical computer as $N$ gets bigger in size. Grover's Algorithm provides a quadratic speedup to this and reduces the number of queries (time taken) to $O(\\sqrt{N})$. In fact, it has been proven that for a quantum computer, this is the best possible time complexity.\n\nNote - The problem gets easier and the complexity gets reduced by a constant factor when the entities that satisfy the property are more in number.\n\n\\section{How the Algorithm works}\n\\subsection{Intution 1}\nInitially, we start with $n$ qubits in such a way that in their superposition, all the $N$ states have equal amplitude (each has an equal probability of popping up when we measure the state).\n$$\\ket{\\psi_{initial}} = \\sum_{0}^{N-1} {\\alpha_i}\\ket{i} \\text{ where } \\alpha_i = \\frac{1} {\\sqrt{N}}$$\n\nLet $\\ket{j}$ be our needle in the haystack.\nNow, by means of certain quantum gates, the amplitude $\\alpha_j$ is first negated i.e. it becomes $-\\alpha_j$. And then every single $\\alpha_i$ is flipped about the mean of all amplitudes. As a result, $\\alpha_j$ stands out more than every other $\\alpha_i$.\n\\\\[3cm]\n\\indent This has been explained by means of the following diagrams (Not to scale).\nInitially the amplitudes of all the states are equal and the sum of their squares is equal to $1$.\n\n\\begin{tikzpicture}\n\\begin{axis}[ybar interval, ymax=0.8,ymin=-0.5, minor y tick num = 1, xmax = 8]\n\\addplot coordinates { (0, 0.353) (1, 0.353) (2, 0.353) (3, 0.353) (4, 0.353) (5, 0.353) (6, 0.353) (7,0.353)};\n\\draw [dashed] (0,0.353) -- (7, 0.353);\n\n\\end{axis}\n\\end{tikzpicture}\n\nThen a quantum circuit is applied to the state which flips the amplitude of the desired state.\n\n\\begin{tikzpicture}\n\\begin{axis}[ybar interval, ymax=0.8,ymin=-0.5, minor y tick num = 1, xmax = 8]\n\\addplot coordinates { (0, 0.353) (1, 0.353) (2, -0.353) (3, 0.353) (4, 0.353) (5, 0.353) (6, 0.353) (7,0.353)};\n\\draw [dashed] (0,0.3) -- (7, 0.3);\n\\end{axis}\n\\end{tikzpicture}\n\nNow the inversion of one of the amplitudes results in a slight decrement of the mean of all the amplitudes. If we now flip the amplitudes about their mean, we land up in the following state.\n\n\\begin{tikzpicture}\n\\begin{axis}[ybar interval, ymax=0.8,ymin=-0.5, minor y tick num = 1, xmax = 8]\n\\addplot coordinates { (0, 0.25) (1, 0.25) (2, 0.65) (3, 0.25) (4, 0.25) (5, 0.25) (6, 0.25) (7,0.25)};\n\\draw [dashed] (0,0.3) -- (7, 0.3);\n\\end{axis}\n\\end{tikzpicture}\n\nThe amplitude of the needle in the haystack has increased and hence if we measure the state now, the chance of getting the correct value is more than it was previously. However, the sum of probabilities of getting other states is still greater than the probability of getting the answer.\\\\[4pt]\n\nHence, we repeat the negation and inversion steps a number of times so that when we measure the state, we can be pretty sure that the outcome is the answer (the desirable state).\\\\[4pt]\n\nInitially $\\alpha_j = \\dfrac{1}{\\sqrt{N}}$. As we run iterations of the algorithm, $\\alpha_j$ increases and $\\alpha_i$, $\\forall$ $i \\neq j$, decreases, all in such a way that $\\sum_0 ^{N-1} {\\alpha_i}^2 = 1$.\nIf we want the probability of getting $\\ket{j}$ to be at least half, we need to pump up the magnitude $\\alpha_j$ to $\\dfrac{1}{\\sqrt{2}}$. At the point in time when $\\alpha_j = \\dfrac{1}{\\sqrt{2}}$, $\\alpha_i \\approx \\dfrac{1}{\\sqrt{2N}}$ $\\forall$ $i \\neq j$. Because the amplitude of the other states decreases continuously, we can say that throughout our iterations $\\alpha_i \\geq \\dfrac{1}{\\sqrt{2N}}$. Since in one iteration, the amplitude $\\alpha_j$ increases by at least by $2 \\alpha_i$ $\\left( i \\neq j\\right)$, the increment is at least $\\sqrt{\\dfrac{2}{N}}$. Therefore the number of iterations required is at least $\\dfrac{1 / \\sqrt{2}}{\\sqrt{2 / N}}$ $=$ $\\sqrt{N}$.\\\\\nWhen the number of favourable states is $M$, the number of iterations comes out to be $\\sqrt{\\dfrac{N}{M}}$.\n\n\\subsection{Intution 2}\n\nLet $\\ket{a}$ represent our desirable state i.e. $\\ket{j}$. And $\\ket{b}$ represent the superposition of all the remaining states with equal amplitudes i.e. $\\ket{b} = \\sum_{i \\neq j} \\dfrac{1}{\\sqrt{N - 1}} \\ket{i}$. It can be easily noticed that $\\ket{a}$ and $\\ket{b}$ are orthogonal to each other. Let $\\psi_0$ be our initial state which is $\\sum_{0}^{N - 1} \\dfrac{1}{\\sqrt{N}} \\ket{i}$. Now initially $\\ket{\\psi_0}$ lies really close to $\\ket{b}$ and it is the task of Grover's Algorithm to move it closer to $\\ket{a}$.\\\\[4pt]\n\n\\begin{tikzpicture}[\n > = Straight Barb,\nphasor/.style = {very thick,-{Stealth}},\nangles/.style = {draw, <->, angle eccentricity=1,\n right, angle radius=15mm}\n ]\n% coordinates\n \\draw[->, thick] (-0.5,0) -- (4,0) coordinate (x) node[below left] {$\\ket{b}$};\n \\draw[->, thick] (0,-0.5) -- (0,4) node[below left] (y) {$\\ket{a}$};\n% phasors\n \\draw[phasor] (0,0) -- (15:4) coordinate (i) node[right] {$\\psi_0$};% used polar coordinates\n% angles drawn by pic\n\\coordinate (X) at (0,0);\n\\draw\n pic[\"$\\theta$\",angles] {angle=x--X--i}\n ;\n\\end{tikzpicture}\n\nThe amplitude negation step leads to reflection of $\\ket{\\psi_0}$ about $\\ket{b}$. This is true because $\\ket{b}$ represents the superposition of all the undesirable states of equal amplitude. So the projection of $\\ket{\\psi_0}$ on $\\ket{b}$ will remain as it is while the projection of $\\ket{\\psi_0}$ perpendicular to $\\ket{b}$ will get negated. This is nothing but reflection about $\\ket{b}$.\n\n\\begin{tikzpicture}[\n > = Straight Barb,\nphasor/.style = {very thick,-{Stealth}},\nangles/.style = {draw, <->, angle eccentricity=1,\n right, angle radius=15mm}\n ]\n% coordinates\n \\draw[->, thick] (-0.5,0) -- (4,0) coordinate (x) node[below left] {$\\ket{b}$};\n \\draw[->, thick] (0,-0.5) -- (0,4) node[below left] (y) {$\\ket{a}$};\n% phasors\n \\draw[phasor, dotted] (0,0) -- (15:4) coordinate (i) node[right] {$\\psi_0$};% used polar coordinates\n \\draw[phasor] (0,0) -- (-15:4) coordinate (j) node[right] {$\\psi$};% used polar coordinates\n% angles drawn by pic\n\\coordinate (X) at (0,0);\n\\draw\n pic[\"$\\theta$\",angles] {angle=x--X--i}\n pic[\"$\\theta$\",angles] {angle=j--X--x}\n ;\n\\end{tikzpicture}\n\nNow, we need to do inversion about the mean which is equivalent to reflection about the the equal amplitude state $\\sum_{0}^{N - 1} \\dfrac{1}{\\sqrt{N}} \\ket{i}$.\n\n\\begin{tikzpicture}[\n > = Straight Barb,\nphasor/.style = {very thick,-{Stealth}},\nangles/.style = {draw, <->, angle eccentricity=1,\n right, angle radius=15mm}\n ]\n% coordinates\n \\draw[->, thick] (-0.5,0) -- (4,0) coordinate (x) node[below left] {$\\ket{b}$};\n \\draw[->, thick] (0,-0.5) -- (0,4) node[below left] (y) {$\\ket{a}$};\n% phasors\n \\draw[phasor, dotted] (0,0) -- (15:4) coordinate (i) node[right] {$\\psi_0$};% used polar coordinates\n \\draw[phasor, dotted] (0,0) -- (-15:4) coordinate (j) node[right] {$\\psi$};% used polar coordinates\n \\draw[phasor] (0,0) -- (45:4) coordinate (k) node[right] {$\\psi$};% used polar coordinates\n% angles drawn by pic\n\\coordinate (X) at (0,0);\n\\draw\n pic[\"$\\theta$\",angles] {angle=x--X--i}\n pic[\"$\\theta$\",angles] {angle=j--X--x}\n pic[\"\",angles] {angle=x--X--k}\n;\n\\end{tikzpicture}\n\nSo, after one iteration we have moved closer to the required state. We need to carry these iterations certain number of times so that when we measure a state, it collapses to the desirable state with high probability.\n\nEvery time the inversion about mean is done using the equal amplitude state only. It happens to be our initial state too but this is true only for the first iteration and not for further iterations.\n\n\\section{How to Implement the circuit?}\n\nNow that we know what to do, we have to question how to do it.\n\\subsection{Negation of Amplitude}\nOur first task is to negate the amplitude of the desirable quantum state.\nThe situation is like this that we have been given a quantum black box that outputs 1 for the desirable states and 0 for the undesirable states.\n\nInsert diagram here.\n\nIn other words, given the superposition $\\ket{x}\\ket{0}$, the circuit spits out the superposition $\\ket{x}\\ket{f(x)}$. We know the quantum circuits are designed in such a way that the answer register gets XORed with the output $f(x)$. So if the input superposition is $\\ket{x}\\ket{y}$, then the output superposition will be $\\ket{x}\\ket{y \\oplus{f(x)}}$.\n\nWith this knowledge, we can apply neat little hack and trick our black box into doing the task for us. We will initialise the answer register to $\\ket{-}$. Since\n\n\\begin{align*}\n\\ket{-} &= \\dfrac{1}{\\sqrt{2}} ( \\ket{0} - \\ket{1} )\\\\\n\\ket{- \\oplus {f(x)}} &= \\dfrac{1}{\\sqrt{2}} \\left( \\ket{0 \\oplus {f(x)}} - \\ket{0 \\oplus {f(x)}} \\right)\\\\\n\\ket{- \\oplus {f(x)}} &= \\dfrac{1}{\\sqrt{2}} \\left( \\ket{{f(x)}} - \\ket{\\overline{f(x)}} \\right)\\\\\n\\ket{- \\oplus {f(x)}} &= \\dfrac{1}{\\sqrt{2}} \\left( -1 \\right) ^ {f(x)} \\left( \\ket{0} - \\ket{1} \\right)\\\\\n\\ket{- \\oplus {f(x)}} &= \\left( -1 \\right) ^ {f(x)} \\ket{-}\\\\\n\\end{align*}\n\nTherefore, the entire transition will be\n$$\\ket{x}\\ket{-} \\longrightarrow \\left( -1 \\right) ^ {f(x)} \\ket{x} \\ket{-}$$\n\nNow, since it is a superposition that is being looked at as a whole, the term $\\left( -1 \\right) ^ {f(x)}$ can be associated with either $\\ket{x}$ or $\\ket{-}$.\nPutting $\\ket{x} = \\sum_0^{N-1}\\alpha_i\\ket{i}$, we get the output from the black box as $\\sum_0^{N-1}\\alpha_i\\ket{i} \\left( -1 \\right) ^ {f(x)}$. Now since $f(x) = 1$ for $\\ket{j}$ and $0$ for all others, the amplitude is negated only for $\\ket{j}$. Thus we can do the amplitude negation easily.\n\\subsection{Inversion about mean}\nInversion about mean is the same as reflection about the uniform state i.e. reflection about the state $\\sum_0 ^{N - 1} \\dfrac{1}{\\sqrt{N}}$. Now for reflection about the uniform state, we can first map the uniform state to the all-zero state using a Hadamard Gate and then reflect about the all-zero state and then map the state back to the uniform state using another Hadamard Gate (Since Hadamard Gate is it's own inverse). So now we need a method to reflect about the all-zero state ($\\ket{00\\dots0}$ or simply $\\ket{0}$). Consider the following matrix :-\n\n\\[\nA = \\begin{bmatrix}\n 1 & 0 & \\dots & 0 \\\\\n 0 & -1 & \\dots & 0 \\\\\n \\vdots & \\vdots & \\ddots & \\vdots \\\\\n 0 & 0 & \\dots & -1\n \\end{bmatrix}\n\\]\n\nThis matrix maps negates all the states except for the state $\\ket{0}$. This is the same as reflection about $\\ket{0}$. This matrix is also a unitary matrix because $A A^{\\dagger} = I$ i.e. the conjugate transpose is its inverse. So we can conclude that the sequence of matrices $H_N A H_N$ (where $H_N$ stands for the unitary matrix corresponding to Hadamard Gate applied to n qubits.) does a reflection about the mean. And since all the matrices are unitary, they can be implemented using quantum gates.\n\nSo when we apply the operations of amplitude negation and inversion about mean repeatedly for around $\\sqrt{N}$ times, the probability of getting the correct answer exceeds half and we are highly likely to get the current answer. We can perform this several times to be sure of the answer since the method is probabilistic.\n\n\\section{Qiskit Code}\n\\textbf{Note - Please refer to the Jupyter Notebook for the output of the program and the explanatory comments and further experimentation.}\n\n\\begin{tcolorbox}[\n enhanced,\n attach boxed title to top left={xshift=6mm,yshift=-3mm},\n title=Simulation Code,\n boxed title style={size=small,sharp corners},\n sharp corners,\n]\n\\begin{minted}{python}\nimport matplotlib.pyplot as plt\nimport numpy as np\n\nfrom qiskit import IBMQ, Aer, QuantumCircuit,\n ClassicalRegister, QuantumRegister, execute\nfrom qiskit.providers.ibmq import least_busy\nfrom qiskit.quantum_info import Statevector\nfrom qiskit.visualization import plot_histogram\nfrom qiskit.extensions import UnitaryGate\nfrom math import sqrt\n\nn = 5\nN = 2 ** n\n\ndef initialize(qc, qubits):\n for q in qubits:\n qc.h(q)\n return qc\n\ndef get_oracle_matrix(N, values):\n oracle_matrix = np.identity(N)\n for value in values:\n oracle_matrix[value][value] = -1\n return oracle_matrix\n\ndef get_diffusion_matrix(N):\n diffusion_matrix = np.zeros((N, N), dtype = float)\n diffusion_matrix.fill(2 / N)\n diffusion_matrix -= np.identity(N)\n return diffusion_matrix\n\n\\end{minted}\n\\end{tcolorbox}\n\\begin{tcolorbox}[\n enhanced,\n attach boxed title to top left={xshift=6mm,yshift=-3mm},\n title=Simulation Code,\n boxed title style={size=small,sharp corners},\n sharp corners,\n]\n\\begin{minted}{python}\n\noracle_matrix = get_oracle_matrix(N, values = [2])\noracle_unitary_gate = UnitaryGate(oracle_matrix)\n\ndiffusion_matrix = get_diffusion_matrix(N)\ndiffusion_unitary_gate = UnitaryGate(diffusion_matrix)\n\nqc = QuantumCircuit(n)\nqc = initialize(qc, [x for x in range(n)])\n\nfor i in range(5):\n qc.unitary(oracle_matrix, [x for x in range(n)])\n qc.unitary(diffusion_matrix, [x for x in range(n)])\nqc.draw()\n\nqc.measure_all()\n\nqasm_simulator = Aer.get_backend('qasm_simulator')\nshots = 1024\nresults = execute(qc, backend=qasm_simulator,\n shots=shots).result()\nanswer = results.get_counts()\nplot_histogram(answer, figsize = (15, 12))\n\\end{minted}\n\\end{tcolorbox}\n\n\\section{Findings}\nWhile it was mostly studying stuff discovered by others, I did observe some things on my own that most probably have already been observed.\n\n\\subsection{Going beyond $\\sqrt{N}$ queries}\n\nIn classical computers, it just doesn't hurt to go a few extra steps (since they don't harm the overall time complexity) just to be sure of our result.\nI tried this out on the implementation that I wrote and the results surprised me. After exceeding the required number of queries the amplitude of the desired state starts to decrease again and reaches a certain value and increases again and oscillates like this. Also, the period of this oscillation varies along with the number of qubits we are dealing with. When the number of qubits is small, the value decreases quite suddenly and even deviating by 1 query can alter the probability of the outcome easily. However, when the number of qubits is large this is smoothened out, and a query or two give or take doesn't affect the outcome by much.\n\nAll in all, minimum $\\sqrt{N}$ queries are required and should be sufficient to give the answer. One should not go beyond this limit. Rather, one should re-conduct the experiment from the initial state to verify the answer.\n\nSo, what may be happening has been attempted to be explained with the following diagrams.\\\\\nAfter $\\sqrt{N}$ iterations, the state $\\psi$ is quite close to the desirable state. The probability of collapsing to the desirable state is as high as $80\\% - 90\\%$ (usual number as seen from the simulator code).\n\n\\begin{tikzpicture}[\n > = Straight Barb,\nphasor/.style = {very thick,-{Stealth}},\nangles/.style = {draw, <->, angle eccentricity=1,\n right, angle radius=15mm}\n ]\n% coordinates\n \\draw[->, thick] (-4,0) -- (4,0) coordinate (x) node[below left] {$\\ket{b}$};\n \\draw[->, thick] (0,-1) -- (0,4) node[below left] (y) {$\\ket{a}$};\n% phasors\n \\draw[phasor, dotted] (0,0) -- (15:4) coordinate (i) node[right] {$\\psi_0$};% used polar coordinates\n \\draw[phasor] (0,0) -- (80:4) coordinate (k) node[right] {$\\psi$};% used polar coordinates\n% angles drawn by pic\n\\coordinate (X) at (0,0);\n\\draw\n pic[\"\",angles] {angle=x--X--i}\n pic[\"$\\theta$\",angles] {angle=x--X--k}\n;\n\\end{tikzpicture}\n\nNow, when we do amplitude negation by reflecting about $\\ket{b}$, we get state as shown.\n\n\\begin{tikzpicture}[\n > = Straight Barb,\nphasor/.style = {very thick,-{Stealth}},\nangles/.style = {draw, <->, angle eccentricity=1,\n right, angle radius=15mm}\n ]\n% coordinates\n \\draw[->, thick] (-4,0) -- (4,0) coordinate (x) node[below left] {$\\ket{b}$};\n \\draw[->, thick] (0,-4) -- (0,4) node[below left] (y) {$\\ket{a}$};\n% phasors\n \\draw[phasor, dotted] (0,0) -- (15:4) coordinate (i) node[right] {$\\psi_0$};% used polar coordinates\n \\draw[phasor] (0,0) -- (-80:4) coordinate (k) node[right] {$\\psi$};% used polar coordinates\n% angles drawn by pic\n\\coordinate (X) at (0,0);\n\\draw\n pic[\"\",angles] {angle=x--X--i}\n pic[\"$\\theta$\",angles] {angle=k--X--x}\n;\n\\end{tikzpicture}\n\nAnd when we do the inversion about mean, we get the state as shown.\n\n\\begin{tikzpicture}[\n > = Straight Barb,\nphasor/.style = {very thick,-{Stealth}},\nangles/.style = {draw, <->, angle eccentricity=1,\n right, angle radius=15mm}\n ]\n% coordinates\n \\draw[->, thick] (-4,0) -- (4,0) coordinate (x) node[below left] {$\\ket{b}$};\n \\draw[->, thick] (0,-1) -- (0,4) node[below left] (y) {$\\ket{a}$};\n% phasors\n \\draw[phasor, dotted] (0,0) -- (15:4) coordinate (i) node[right] {$\\psi_0$};% used polar coordinates\n \\draw[phasor] (0,0) -- (105:4) coordinate (k) node[right] {$\\psi$};% used polar coordinates\n% angles drawn by pic\n\\coordinate (X) at (0,0);\n\\draw\n pic[\"\",angles] {angle=x--X--i}\n pic[\"$\\theta$\",angles] {angle=x--X--k}\n;\n\\end{tikzpicture}\n\nSo exceeding one query does not make much difference, but when we do this a few number of times, $\\psi$ keeps on deviating from the required state. And then after some number of such iterations, it again starts to return to the position. And this \"some number of iterations\" is less than $\\sqrt{N}$ because since the angle made from $\\ket{b}$ is large, the iterations of the algorithm make huge changes in the states and thus requires fewer steps.\n\n\\textbf{Note - This was observed when there was only one desirable state. With multiple desirable states a similar pattern was followed but not accurately.}\n\n\\subsection{Assumption about the given black box}\nIt is assumed that a quantum circuit has already been implemented that identifies the answer that we use as a black box. Well, if such a black box has been implemented, the maker of the black box would already know about the desired state.\\\\\nI think that maybe in future a situation may arise when the implemented black box is provided by nature and the algorithm may be used to find the desirable state.\\\\\n\n\\section{Conclusion}\n\nRegardless of the black-box property mentioned above, this amazing algorithm does give an idea about the power of Quantum Computing. It gives a quadratic speed up in searching which can be used in database searching. The quadratic speed up can be used to solve some NP-Complete Problems by doing exhaustive searching over the set of all possible solutions (though not for large $N$ as exponential term still dominates).\n\n\\end{document}.\n", "meta": {"hexsha": "f9628cba3e0435e374dd017ebd707cd437f56a8c", "size": 21177, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Sheets/GroversAlgorithm/GroversAlgorithm.tex", "max_stars_repo_name": "ishaanshah/Cuantum-Qomputing", "max_stars_repo_head_hexsha": "79035133f148f37d049f3e009408b17262115efd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-12-04T12:45:43.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-04T12:45:43.000Z", "max_issues_repo_path": "Sheets/GroversAlgorithm/GroversAlgorithm.tex", "max_issues_repo_name": "ishaanshah/Cuantum-Qomputing", "max_issues_repo_head_hexsha": "79035133f148f37d049f3e009408b17262115efd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Sheets/GroversAlgorithm/GroversAlgorithm.tex", "max_forks_repo_name": "ishaanshah/Cuantum-Qomputing", "max_forks_repo_head_hexsha": "79035133f148f37d049f3e009408b17262115efd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-08-24T04:34:35.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-24T04:34:35.000Z", "avg_line_length": 50.6626794258, "max_line_length": 679, "alphanum_fraction": 0.6866411673, "num_tokens": 6276, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.6688802735722128, "lm_q1q2_score": 0.34488799028809203}} {"text": "\\chapter{Background}\n\\label{chap:background}\nImage classification refers to a task that requires from method to determine the categories to which picture belongs. Such challenges that utilize standard datasets like ImageNet and PASCAL VOC limit this problem to a question if objects of particular classes presented on the image \\cite{Russakovsky2015ImageNet, Everingham2010PASCAL-VOC}. However, real-world settings might require extending this question to more general one \\cite{Wang2016CNN-RNN:Classification}. For instance, in this study, the search extended beyond merely objects to include other image properties including color, weather, action, and sceneries. %In other words, image classification system should be able to find a correct placement of a picture in a multi-dimensional space of categories.\n\n% \\section{Image classification}\n% definition of image classification\n\n\\section{Deep convolutional neural networks}\n\nA standard neural network consists of computational units called neurons. Neurons are connected to each other in particular way to form a network. Neurons in such networks are organized in groups or ``layers'' for computational efficiency reasons since it allows to apply vector operations \\cite{cs231n-nn1}. Each input connection to a neuron has dedicated weight. Non-liner \\textit{activation function} use these input weights to determine when to activate particular neuron and send a signal further \\cite{Schmidhuber2015DeepOverview}. Activation function can be different, however recent years ReLU function is mostly used in modern DNN due to its computational efficiency and good output results \\cite{relu, Krizhevsky2012ImageNetDNN}. The ``depth'' of the neural network is identified as an amount of layers except input one \\cite{Schmidhuber2015DeepOverview}.\n\nThe process of \\textit{training} the neural network is optimization of each neuron input weights in such way that the particular input to a network will produce desired output. This optimization is achieved by first introducing loss function which represents how ``far'' given output is from the desired one. After that, on each iteration of training, by calculating gradients of a loss function on given weights, the system determines how it should adjust weights to decrease the computed loss \\cite{cs231n-opt1}. Modern DNNs use backpropagation approach of gradient calculation due to its efficiency \\cite{cs231n-opt2}. \n\nConvolutional Neural Networks (CNN) have the same core ideas as any other neural networks except that CNN architectures assume that the inputs are images \\cite{cs231n-conv}. Unlike conventional neural networks, instead of processing images pixel-by-pixel CNN is searching for patterns in image patches of different sizes. Each convolution can also be perceived as a feature extractor, such as the one that detects particular edge angle or color. Such convolution functions are also organized in layers where output features from one layer can be an input data for next convolutional filters. Therefore, an end system consists of filters cascade, where, similar to human vision, low-level features, like edges, are recognized on the fist levels, and higher levels trigger on more complex features like different shapes and patterns~\\cite{Zeiler2014VisualizingNets}\n\nThe main difference between machine learning approach and deep convolutional neural networks is that the former one has an additional step of supervised feature extraction. In contrast, in deep learning approach feature extraction happens automatically by the internal network itself. Therefore DNN can be considered as a ``black box'' system. For instance, in the case of image classification, such system will have raw picture pixels as an input and classified category numbers as output.\n\nTo improve the generalizability of results and to be able to compare networks with different hyperparameters the common approach is to use dataset splitting where an image collection is partitioned into three parts \\cite{cs231n-cls}. The first part, \\textit{called training} set, is employed directly for training, while the second part, called \\textit{validation set}, is used to compare different architectures and control overfitting. Overfitting is a state of the system when it optimizes to a noise in a training data instead of searching the underlying patterns \\cite{cs231n-nn1}. Such optimization can potentially reduce the generalizability of the system and should be avoided \\cite{cs231n-nn1}. The usual way to control overfitting of the model is to check system performance on a validation set periodically during the training process \\cite{cs231n-nn3}. If the loss computed on a training set gets smaller with more iterations, but validation loss increases or stays the same, then it is an evidence that the model is starting to overfit on the training set \\cite{cs231n-nn3}. To be able to monitor the generalizability of the method, validation set should not contain any images from the training set. The third part of the dataset, which is called \\textit{test set}, is used for a final evaluation of the method. The separation between validation and test sets is important for the similar reason as with separation between training and validation sets. When different network implementations are compared using the validation set, and the best one is selected, hyperparameters can overfit on the validation set similar to the state when network is overfitting on the training set \\cite{cs231n-cls}. Meaning that chosen parameters will be optimized for the particular images in the validation set. Therefore, it is important to test the final system on a separate set of pictures, which CNN has not seen at any time during the training process. According to Andrew Ng, The common ratio between these tree sets is 60\\%/20\\%/20\\%, where the training set gets 60\\% of images while validation and test sets receive 20\\% each \\cite{Ng2016NutsLearning}.\n% train, val, test split\n% different layers\n% overfitting\n% backpropagation\n\n\\section{Multi-label image classification}\nOpposite to a single-label image classification, in multi-label case picture can belong to more than one category. Several labels can both describe all objects presented on the picture and characterize the objects or picture itself.\n\nThere are two group of multi-label image classification methods:\n\\begin{itemize}\n \\item That process parts of the images and use CNN to find a single label for each of the part \\cite{Wei2016HCP, Ren2016, Yang2015},\n \\item That train CNN on the whole picture, similarly to single label methods, but use different loss functions that take several labels into account \\cite{Gong2013DeepRanking}.\n\\end{itemize}\n\nAn example of the first type of network is described by Wei et al. in their paper ``HCP: A Flexible CNN Framework for Multi-Label Image Classification'' \\cite{Wei2016HCP}. The core idea of the proposed method is to reuse already developed CNNs that were trained to classify single-labeled pictures in a multi-label scenario. The authors proposed deep CNN infrastructure called ``Hypotheses-CNN-Pooling'' (HCP) to achieve this. Its architecture is shown on Figure \\ref{fig:hcp-net}.\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=\\textwidth]{hcp-net}\n \\caption{Hypotheses-CNN-Pooling infrastructure \\cite{Wei2016HCP}}\n \\label{fig:hcp-net}\n\\end{figure}\n\nThe whole classification pipeline proposed by the authors consists of the following steps:\n\\begin{enumerate}\n \\item Applying state of the art object detection technique (for example BING \\cite{Cheng2014} or EdgeBoxes \\cite{Zitnick2014}) to get the set of hypothesis,\n \\item Using proposed hypothesis selection method reduce the number of initial hypotheses,\n \\item Use pre-trained CNN to assign single label for each hypothesis,\n \\item Use cross-hypothesis max-pooling to create the vector of labels found on the whole picture.\n\\end{enumerate}\n\nAuthors claim of getting up to 90.9\\% of Mean Average Precision (which is defined in PASCAL Challenge \\cite{Everingham2010PASCAL-VOC}) on PASCAL VOC 2012 image dataset.\n\nYang et al. proposed similar infrastructure, but extended it with additional large-margin nearest neighbor (LMNN) CNN that incorporates knowledge of ground truth bounding boxes of the training set to improve results \\cite{Yang2015}. This extra layer tries to find similar objects in the test image to the ones extracted from the training set (ground-truth object set) independently from the other layer that employs more usual CNN. Results from the both layers are taken into account in the final prediction vector.\n\nThe method described by Girshik et al. in ``Rich feature hierarchies for accurate object detection and semantic segmentation'' paper also highly rely on the ground-truth bounding box during the training phase \\cite{Girshick2014}. In particular, generated hypothesis are ranged based on the overlap with ground-truth box.\n\nOquab et al. proposed method that also use bounding box information to produce labels for training images \\cite{Oquab2014TransferringMidLevel}. Specifically, they propose to generate around 500 square patches from the picture and put a single label on each patch dependently on the overlap with the ground truth bounding box. Patches which intersect with two or more objects are filtered out, and the once which do not intersect with any objects are labeled as a background. Example training data generated from the image and ground-truth information is shown on Figure \\ref{fig:patches-example}.\n\n\\begin{figure}[H]\n \\centering\n \\includegraphics[scale=0.35]{patches-example}\n \\caption{Training data generation example using patching method \\cite{Oquab2014TransferringMidLevel}}\n \\label{fig:patches-example}\n\\end{figure}\n\nCurrently, the first type of multi-label classification methods with image partitioning show improved performance compared to the ones that use the whole pictures \\cite{Wei2016HCP, Ren2016, Yang2015}. However, such methods require having either ground truth bounding box encoded in metadata in addition to labels which identifies the location of all objects on the image \\cite{Chen2015ContextualizingClassification, Dong2013Subcategory-AwareClassification}, or pre-trained single-label classifier for the same set of categories \\cite{Wei2016HCP}. In addition, this approach might be more suitable for searching for objects in the picture, but not perform as good to assign labels describing the whole image since it processes only part of a picture at a time. For instance, the dataset used in this study had label \\textit{alone} that separates images with single objects, or \\textit{landscapes} which characterize the whole picture. These kinds of categories might be challenging to classify using method that splits an image.\n\nIn contrast to previous papers, Gong et al. applied CNN to the whole picture without partitioning \\cite{Gong2013DeepRanking}. The main contribution of this study was not to produce new multi-label classification method, but to compare different loss functions in multi-label classification context. Results showed that weighted approximated-ranking loss (WARP) improved the classification performance of the categories with a small number of pictures which can be an essential factor for unbalanced datasets.\n\nWang et al. in their research were also trying to improve multi-label image classification performance for realistic image collections \\cite{Wang2016CNN-RNN:Classification}. Authors point out on issues connected with methods that transforming multi-label task into a single-label one. Researchers identify the lack of modeling semantic dependencies between categories in such methods. The main reason for this is that such approaches treat image categories completely independently, while some of the labels might occur together more often than others. To solve this problem, the authors propose to utilize Recurrent Neural Network (RNN) in addition to CNN, which can model semantic relevance between images.\n\nMost of the presented methods use so-called transfer learning technique in different ways to improve their results and to utilize the work from the other studies to their benefit. Brief description and motivation behind this approach presented in the next section.\n\n% \\section{Caffe framework}\n\n% pipeline of computations going through layers\n% sigmoid cross entropy\n\n\\section{Transfer learning}\n\\label{sec:transfer-learning}\n\nUsually, the usage of deep learning approach adds additional requirements: to have enough processing power and big enough image dataset. One of the most common methods to reduce these constraints called \\textit{transfer learning}. The main principal behind it is to reuse ``knowledge'' of a pre-trained neural network on one task and transfer this knowledge to another one \\cite{Pan2010TransferLearningSurvey, Oquab2014TransferringMidLevel}. This idea appeared when people noticed that first layers of most CNNs learn to trigger on the same features like corners, edges, color conjunctions, therefore there is no need to discover them in each network \\cite{Zeiler2014VisualizingNets}.\n\nThere are two types of transfer learning \\cite{Yosinski2014HowTransferable}:\n\\begin{itemize}\n \\item With frozen layers, when layers copied from the pre-trained network do not change their weights during the backpropagation phase, in other words, their learning rates are set to zero,\n \\item Fine-tuning, when copied layers only start with pre-trained weights, but during the training phase adjust just their values like all other layers.\n\\end{itemize}\n\nOquab et al. describe in detail the usage of transfer learning for multi-label classification. In particular, they show how pre-trained network on one category set can be used to find labels for more general categories (for example source categories might include different dog breeds while target set may contain only general label ``dog'') or even entirely new ones \\cite{Oquab2014TransferringMidLevel}. Insights provided in this research will be applied in the project since target NTB dataset also has a different set of categories compared to the ones available in the standard image collections.\n\nRecent studies also suggest that the usage of a pre-trained model can improve the generalizability of the final image classification system in contrast to the system trained on the target dataset from scratch \\cite{Oquab2014TransferringMidLevel, Yosinski2014HowTransferable}. The transfer learning method employed in this project, which is described in the next chapter, allowed to reduce dataset size and performance requirements needed for successful system training. \n\n\\section{Real-world datasets}\nSome datasets like PASCAL VOC or NUS WIDE are created from realistic pictures that contain several objects or people in different combination and interactions between them \\cite{Everingham2010PASCAL-VOC, Chua2009NUS-WIDE}. Creators of such datasets, as well as other researchers, argue that real-world images are more likely to have multi-label nature by having both multiple entities in the picture as well as such category set that can describe image or object in several prospectives \\cite{Everingham2010PASCAL-VOC, Chua2009NUS-WIDE, Wang2016CNN-RNN:Classification, Dong2013Subcategory-AwareClassification}. Wang et al. poit out that in addition to objects, such images can contain labels that represent scenes, actions, and parts of some entities \\cite{Wang2016CNN-RNN:Classification}. Authors also identify the challenge of small objects on such images that can be ignored by classification methods, especially when full pictures are processed in a neural network.\n\nExcept for the picture collection itself, dataset should also contain metadata for image and a defined set of categories. While datasets described earlier consist of realistic pictures, the set of categories and therefore images were designed and moderated for the purpose of further training and comparison of image classification systems. Therefore, there is still a question on how different state of the art methods can be applied to datasets that were created for other reasons and how their metadata utilization can be maximized. In addition to realistic pictures, such datasets can have additional challenges connected with a unique set of labels, a less balanced size of categories, a big number of both systematic and random errors.\n\n\\subsection{Current state of the NTB dataset}\nAs it was mentioned earlier, NTB provided their labeled image collection for the purpose of these project. Currently, NTB dataset contains around one million of manually annotated pictures and more than ten millions of images without labels. Therefore, the fist usage of the automatic image classification system for the company is to propagate annotations on other images from the collection. NTB has its unique set of categories organized in a tree structure. The company uses these tags to provide a solution for pictures search to their customers. Therefore the second application of the system would be to incorporate an image classification in this search solution. The dataset is specific to Norway and to particular topics including sports, finances, politics, and media. More in-depth analysis of the NTB dataset will be presented in Section \\ref{sec:dataset-analysis}.\n\n% own set of categories, have to trafsfer them as many as possible", "meta": {"hexsha": "781a192c9b822c5ab54b84a3bd319d5a81058c10", "size": 17415, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "background.tex", "max_stars_repo_name": "uvNikita/master-thesis", "max_stars_repo_head_hexsha": "56b22f7b8a645ee3b144860bccf8c2c8551b287b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2019-02-27T13:25:51.000Z", "max_stars_repo_stars_event_max_datetime": "2019-06-04T03:49:18.000Z", "max_issues_repo_path": "background.tex", "max_issues_repo_name": "uvNikita/master-thesis", "max_issues_repo_head_hexsha": "56b22f7b8a645ee3b144860bccf8c2c8551b287b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "background.tex", "max_forks_repo_name": "uvNikita/master-thesis", "max_forks_repo_head_hexsha": "56b22f7b8a645ee3b144860bccf8c2c8551b287b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 172.4257425743, "max_line_length": 2161, "alphanum_fraction": 0.8164226242, "num_tokens": 3576, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5156199157230157, "lm_q1q2_score": 0.34488799028809203}} {"text": "%\n% CMPT 454: Database Systems II - A Course Overview\n% Section: Query Optimization\n%\n% Author: Jeffrey Leung\n%\n\n\\section{Query Optimization}\n\t\\label{sec:query-optimization}\n\\begin{easylist}\n\n& Notation: Given a schema $R$:\n\t&& The size in pages is denoted as $|R|$\n\t&& The number of records per page is denoted as $p_R$\n\t&& The size of the buffer is $B$\n\n& Clustered indices are only applicable for terms which form a prefix of the search key\n\t&& E.g. The index $$ can be used for the search key $A = 1, B = 3$\n\n\\clearpage\n\\end{easylist}\n\\subsection{Optimizing Equality Joins}\n\t\\label{subsec:optimizing-equality-joins}\n\\begin{easylist}\n\n& \\textbf{Join:} Database operation where a single set of rows are returned, created from two relations\n\n& E.g. \\textrm{SELECT * FROM Sailors S, Reserves R WHERE R.id = S.id}\n\t\n& \\textbf{Simple Nested Loops (SNL) join:} Equality join implementation where nested loops compare equality values using 1-page buffers\n\t&& Consists of multiple nested loops, each of which loops through a table\n\t\t&&& \\textbf{Outer table:} Table in an SNL join which is scanned outside the other nested loop(s)\n\t\t&&& \\textbf{Inner table:} table in an SNL join which is scanned inside the other nested loop(s)\n\t&& Runtime:\n\t\t\\begin{align*}\n\t\t\t|Outer| + |Outer| \\cdot |Inner|\n\t\t\\end{align*}\n\t\t&&& $|Outer|$ is the cost of loading each value from the outer table into memory\n\t\t&&& $|Outer| * |Inner|$ is the cost of loading each value from the inner table into memory, and comparing with an outer table value\n\t\t&&& To optimize, smaller tables should be implemented as the outer\n\t\t\t\n& \\textbf{Block Nested Loops (BNL) join:} Equality join implementation where nested loops compare equality values using multi-page buffers\n\t&& \\textbf{Block:} Fixed-size set of pages representing the size of a buffer\n\t&& Runtime:\n\t\t\\begin{align*}\n\t\t\t|Outer| + |Outer\\ Blocks| \\cdot |Inner| \\\\\n\t\t\t= |Outer| + \\left\\lceil \\frac{|Outer|}{B-2} \\right\\rceil \\cdot |Inner|\n\t\t\\end{align*}\n\t\t&&& Two buffer pages are used; one as the output buffer and one as the input buffer for the inner table\n\t\t&&& To optimize, smaller tables should be implemented as the outer\n\n& \\textbf{Index Nested Loops (INL) join:} Equality join implementation where nested loops compare equality values, and inner loops use an index to search matching values from the outer loop\n\t&& Runtime:\n\t\t\\begin{align*}\n\t\t\t|Outer| + |Outer| \\cdot p_{outer} \\cdot \\textrm{probing costs}\n\t\t\\end{align*}\n\t\t&&& $|Outer| \\cdot p_{outer}$ represents the number of records in table $R$\n\t\t&&& \\textbf{Probing cost:} I/O cost to search for an entry and record using an index\n\t\t\t&&&& Probing cost of data entries in a hash index: 1.2\n\t\t\t&&&& Probing cost of data entries in a B+ tree: 3-4\n\t\t\t&&&& Probing cost of data records when the search key is a candidate key: 1\n\t\t\t&&&& Probing cost of data records when the search key is not a candidate key, in a clustered index: 1\n\t\t\t&&&& Probing cost of data records when the search key is not a candidate key, in an unclustered index: $\\frac{\\textrm{records in inner}}{\\textrm{records in outer}}$\n\t\t\t\t\n\t&& Equality with multiple attributes: Index must apply to one or more of the attributes\n\t&& Inequality: Index must be clustered B+ tree\n\n& \\textbf{Sort-merge-join (SMJ):} Equality join implementation where tables are sorted externally, written back to the disk, merged by equality, then sent to output\n\t&& Runtime:\n\t\t&&& Sorting:\n\t\t\\begin{align*}\n\t\t\t 2 \\cdot |T_1| \\cdot (\\left\\lceil \\log_{B-1} (\\frac{|T_1|}{B}) \\right\\rceil + 1) \\\\\n\t\t\t+ 2 \\cdot |T_2| \\cdot (\\left\\lceil \\log_{B-1} (\\frac{|T_2|}{B}) \\right\\rceil + 1)\n\t\t\\end{align*}\n\t\twhere $T_1$, $T_2$ are tables and $B$ is the buffer size\n\t\t&&& Merging:\n\t\t\\begin{align*}\n\t\t\t|T_1| + |T_2|\n\t\t\\end{align*}\n\t\t&&& Optimization 1: Find the tables which have the fewest maximum count of any repeating value, then use those those tables as the outer loop (to fit as much outer loop data as possible into the input buffer)\n\t\t&&& Optimization 2: Join the runs of the sorted tables after only the first pass of the external sort\n\t\t\t&&&& When joining, each run is allocated a single buffer\n\t\t\t&&&& Only possible if $B \\geq \\sqrt{\\textrm{max} \\{ |T_1|, |T_2| \\} }$\n\t\t\t&&&& Runtime:\n\t\t\t\\end{easylist}\n\t\t\t\\begin{align*}\n\t\t\t\t& \\textrm{(Cost of pass 0)} + \\textrm{(Merging all runs)} \\\\\n\t\t\t\t& = 2 (|R|+|S|) + (|R|+|S|) \\\\\n\t\t\t\t& = 3 (|R|+|S|)\n\t\t\t\\end{align*}\n\t\t\t\\begin{easylist}\n\t\t\t\t\n\t&& Equality with multiple attributes: Sort based on all joined attributes\n\t&& Inequality: Not applicable\n\t\t\t\n\n& \\textbf{In-memory probing by hash table:} CPU runtime optimization which hashes database table pages into buckets\n\t&& Reduces excess page retrieval and optimizes access speed to specific pages\n\n& \\textbf{Hash-join (HJ):} Equality join implementation where tables are partitioned into hash buckets, then the values in each hash bucket is joined with the matching values in the hash bucket from the other table\n\t&& Hash bucket notation:\n\t\t&&& $T_i$ represents the contents of hash bucket $i$ created by partitioning $T$\n\t\t&&& Assuming partitions are roughly equally sized: $|T_i| = \\frac{|T|}{B-1}$\n\t&& Tables can be hash-joined in one scan if and only if $B-2 \\geq \\textrm{min} \\{ |R_i|, |S_i| \\}$ for all $R_i$, $S_i$ given tables $R$, $S$\n\t\t&&& Implies $B \\geq \\sqrt{\\textrm{min} \\{ |R_i|, |S_i| \\} }$\n\t\t&&& If tables cannot be hash-joined in one scan, recursively apply hash-join to the bucket(s) which are too large to fit in memory\n\t&& Runtime (assuming $R_i$ or $S_i$ can fit in memory):\n\t\\end{easylist}\n\t\\begin{align*}\n\t\t& \\textrm{Cost of partitioning both relations - read and write} + \\textrm{Cost of joining relations} \\\\\n\t\t& = 2( |R|+|S|) + (|R|+|S|) \\\\\n\t\t& = 3( |R|+|S|)\n\t\\end{align*}\n\t\\begin{easylist}\n\t&& Parallelizable\n\t&& Sensitive to data skew\n\t\n\t&& Equality with multiple attributes: Hash based on all joined attributes\n\t&& Inequality: Not applicable\n\n\\clearpage\n\\end{easylist}\n\\subsection{Optimizing Selections}\n\t\\label{subsec:optimizing-selections}\n\\begin{easylist}\n\n& \\textbf{Selection:} Database operation where a subset of rows are returned from a relation\n\n& Notation:\n\t&& \\textbf{Reduction factor (RF):} Percent of data that a table is reduced by through a selection operation\n\n& Size of result:\n\\begin{align*}\n\t|T| \\cdot \\textrm{RF}\n\\end{align*}\n\n& Runtime:\n\t&& No index or not clustered on the attributes: $|T|$\n\t&& Clustered on the attributes: $|T| \\cdot \\textrm{RF}$\n\n& Page I/O:\n\t&& No index or not clustered on the attributes: $|T| \\cdot \\textrm{RF} \\cdot p_T$\n\t\t&&& Optimization: Collect and sort record IDs before retrieving them\n\t&& Clustered on the attributes: $|T| \\cdot \\textrm{RF}$\n\t\t&&& Clustered records are more likely to be on the same page\n\n& Converted to conjunctive normal form (CNF) before processing\n\n& Optimizing with multiple indices:\n\t&& Approach 1: Retrieve a set of records using the most selective/restrictive index (minimizing returned values), then apply remaining terms\n\t&& Approach 2: Retrieve sets of record IDs using all indices, intersect them, sort them, retrieve the records, then apply remaining terms\n\n\\clearpage\n\\end{easylist}\n\\subsection{Optimizing Projections}\n\t\\label{subsec:optimizing-projections}\n\\begin{easylist}\n\n& \\textbf{Projection:} Database operation where a subset of columns are returned from a relation\n\t&& Often includes a $\\textrm{DISTINCT}$ operation\n\n& \\textbf{Standard projection:} Projection implementation using a modified version of the external sort Pass 0\n\t&& When writing during external sort Pass 0, drop unnecessary fields\n\t&& During external sort merging of runs, drop duplicate rows\n\n\t&& Page I/O:\n\t\\end{easylist}\n\t\\begin{align*}\n\t& \\textrm{(Reading all records)} + \\textrm{(Writing partial columns)} \\\\\n\t& + \\textrm{(Reading the same data)} \\\\\n\t& + \\textrm{(Merging non-duplicate rows)} \\\\\n\t= & |T| + |T| \\cdot \\textrm{(\\% of data projected)} * 2 \\\\\n\t & + |T| \\cdot \\textrm{(\\% of data projected)} \\cdot \\textrm{(\\% of distinct data merged)}\n\t\\end{align*}\n\t\\begin{easylist}\n\n& \\textbf{Hash projection:} Projection implementation where hashing is used to eliminate duplicates\n\t&& Process: Use a hash function on all wanted attributes to partition the table into $B-1$ partitions (while discarding unprojected fields), use a new hash function on all fields of each partition to discard duplicates\n\n& Index terms:\n\t&& \\textbf{Index-only projection:} Projection operation where all projected attributes are in the index's search key\n\t\t&&& Requires sorting afterwards if the projected attributes are not a prefix of the search key\n\t\t&&& Requires duplicate removal\n\t&& \\textbf{Index-only, no sort needed projection:} Projection operation where all projected attributes constitute a prefix of the index's search key\n\t\t&&& Requires duplicate removal\n\n\\clearpage\n\\end{easylist}\n\\subsection{Optimizing Unions}\n\t\\label{subsec:optimizing-unions}\n\\begin{easylist}\n\n& \\textbf{Union:} Database operation where two relations with matching columns are combined\n\t&& Includes removal of duplicates\n\n& \\textbf{Hash-based union:} Union implementation which utilizes two hash functions\n\t&& Process:\n\t\t&&& Hash both relations into partitions\n\t\t&&& Re-hash a partition from the first relation, into new partitions\n\t\t&&& Re-hash the matching partition from the second relation, into the same re-partitions (avoiding adding duplicates)\n\n\\end{easylist}\n\\subsection{Optimizing Differences}\n\t\\label{subsec:optimizing-differences}\n\\begin{easylist}\n\n& \\textbf{Difference:} Database operation between two relations where any row from the first relation which exists in the second relation is removed\n\n& \\textbf{Hash-based difference:} Difference implementation which utilizes two hash functions\n\t&& Process:\n\t\t&&& Hash both relations into partitions\n\t\t&&& Re-hash a partition from the first relation, into new partitions\n\t\t&&& For each value in the second relation's matching partition:\n\t\t\t&&&& Re-hash it to find the corresponding partition\n\t\t\t&&&& Remove it from the partition if it exists\n\n\\end{easylist}\n\\subsection{Optimizing Aggregation}\n\t\\label{subsec:optimizing-aggregation}\n\\begin{easylist}\n\n& \\textbf{Aggregation:} Database operation where numerical values in rows are combined\n\n& If the index is clustered based on the attribute to be grouped, then no sorting will be required\n\n& \\textbf{Index-only aggregation:} Aggregation case where the search key contains all attributes to be searched and grouped, therefore no record access is required (only entries)\n\n\\end{easylist}\n\\subsection{Combined Operations}\n\t\\label{subsec:combined-operations}\n\\begin{easylist}\n\n& If grouping by a specific attribute:\n\t&& Ensure the relation is clustered/sorted on the attribute (if not, sort it externally)\n\t&& Use the relation with the attribute as the outer loop to preserve order \n\n\\end{easylist}\n\\clearpage\n\n\n\n\n\n\n\n\n\n\n\n\n\n", "meta": {"hexsha": "493f0bbc6b466785e606ba5210924aee3b9cae1e", "size": 10813, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "cmpt-454-database-systems-ii/tex/query-optimization.tex", "max_stars_repo_name": "AmirNaghibi/notes", "max_stars_repo_head_hexsha": "c4640bbcb65c94b8756ccc3e4c1bbc7d5c3f8e92", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 25, "max_stars_repo_stars_event_min_datetime": "2019-08-11T08:45:10.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-09T02:37:39.000Z", "max_issues_repo_path": "cmpt-454-database-systems-ii/tex/query-optimization.tex", "max_issues_repo_name": "AmirNaghibi/notes", "max_issues_repo_head_hexsha": "c4640bbcb65c94b8756ccc3e4c1bbc7d5c3f8e92", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "cmpt-454-database-systems-ii/tex/query-optimization.tex", "max_forks_repo_name": "AmirNaghibi/notes", "max_forks_repo_head_hexsha": "c4640bbcb65c94b8756ccc3e4c1bbc7d5c3f8e92", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2020-11-18T09:17:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-27T21:44:56.000Z", "avg_line_length": 42.9087301587, "max_line_length": 219, "alphanum_fraction": 0.7217238509, "num_tokens": 3066, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6688802735722128, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.344887990288092}} {"text": "\n\\section{The Inverse Method}\n\n\\subsection{Requirements}\n\n\\begin{enumerate} \n\\item Typechecker\n \\begin{itemize} \n \\item Internal api, e.g. ``\\%doublecheck''\n \\item External api, for Deepak say. Perhaps support explicit sharing of terms.\n \\end{itemize} \n\\item Term Reconstruction\n\\item Logic programming engine (Bottom up)\n\\item M2 Checker (\\%mode,\\%worlds,\\%terminates,\\%covers,\\%total)\n\\item logic programming engine (Top down)\n\\item Meta-theorem prover (produce .prf files from .thm files)\n\\end{enumerate} \n\n\\subsection{The Subformula Property}\n\n\\begin{Theorem} The cut rule is admissible in LF \\end{Theorem} \n\\XXX{Needs reference}\n\nIt is a property of cut-free sequent calculi that all propositions occurring\nin a derivation are \\emph{subformulas} of the endsequent in the following sense.\n\nWe define \\emph{signed subformulas} of a formula $F$ with judgments:\n\n\\begin{align*} \n\\PSubform{A} & & \\mbox{$A$ is a positive subformula of the initial goal.}\\\\\n\\NSubform{A} & & \\mbox{$A$ is a negative subformula of the initial goal.}\n\\end{align*} \n\nwhere \n$$\\Psi ::= \\cdot \\Spb \\Psi, v:A \\Spb \\Psi, x:A$$\n\nHere $v$ ranges over arbitrary terms and $x$ ranges over\nfixed parameters.\n\n\nThese rules are defined as a forward logic program by the following inference\nrules:\n\n\\bigskip \n\\begin{tabular}{cc}\n\\infer{\\NSubform{A}}{\\PSubform{(A\\LFArrow B)}} &\n\\infer{\\NSubform{B}}{\\PSubform{(A\\LFArrow B)}} \\\\\n\\infer{\\PSubform{A}}{\\NSubform{(A\\LFArrow B)}} &\n\\infer{\\PSubform{B}}{\\NSubform{(A\\LFArrow B)}} \\\\\n\\infer{\\PSubform[\\Psi,x:A]{B}}{\\PSubform{(\\PiTyp{x}{A}{B})}} &\n\\infer{\\NSubform[\\Psi,u:A]{B}}{\\NSubform{(\\PiTyp{u}{A}{B})}} \\\\\n\\end{tabular} \n\\bigskip \n\nNote that there are no rules for families of kind $\\Type$.\n\nTo find the subformulas of a given goal formula $G$, we run\nthe above rules as a forward logic program to saturation on $\\PSubform{G}$. \n\n\\subsection{Admissible Substitutions}\n\nExplicit substitutions are an integral part of the theory and \nimplementation of our development. They are necessary because\nwhen logic variables arise during proof search, it is not always\npossible to apply a substitution immediately. While contexts \ndo the necessary bookkeeping of tracking the types of bound variables,\nsubstitutions track variable (term) instantiations.\nThe language of substitutions is\n\n$$\\sigma ::= \\circ \\Spb \\sigma,(M/u) \\Spb \\sigma, (y//x)$$ \n\nwhere $M$ ranges over arbitrary terms, and $y$ ranges over\nparameters.\n\nAn invariant(*) of admissible substitutions is that if $\\AdmSub{\\sigma}{\\Psi}$ then\nthe domain of $\\sigma$ is exactly the free variables of $\\Psi$. (The codomain\nneed not mention all of $\\Sigma$.) Similarly, whenever we apply \n$A[\\Theta]$ it will be the case that $\\Theta$ substitutes exactly for the\nfree variables of $A$. \n\nThe judgment on admissible substitutions will be \n$$\\AdmSub{\\sigma}{\\Psi}$$\nread ``$\\sigma$ is an admissible substitution from $\\Psi$ to $\\Sigma$''.\nand meaning that $\\sigma$ substitutes for all the free variables in\n$\\Psi$, but can mention variables in $\\Sigma$. \n\n\\bigskip \n\n\\infer{\\AdmSub{\\cdot}{\\cdot}}{} \n\\bigskip \n\n\\infer{\\AdmSub{(\\sigma,M/u)}{\\Psi,u:A}}{\\AdmSub{\\sigma}{\\Psi} & \\CheckTy[\\Sigma]{M}{A[\\sigma]}} \n\\smallskip \nwhen $u\\not\\in\\Psi$\n\\bigskip \n\n\\infer{\\AdmSub{(\\sigma,y//x)}{\\Psi,x:A}}{\\AdmSub{\\sigma}{\\Psi} & \\SynthTy[\\Sigma]{y}{A'} & A' = \\HerSub{A}} \n\\smallskip \nwhen $x\\not\\in\\Psi$\n\\bigskip \n\n\\subsection{Forward Sequents}\n\nFor the inverse method, we think of the usual sequent calculus rules, which work bottom-up, \nbackwards to work top-down. \n\nThe judgment \n\n$$\\Sequent{\\Sigma}{A_1[\\tau_1],\\ldots,A_n[\\tau_n]}{C[\\sigma]}$$\n\nwhere for all $i$, \n\n\\begin{align*} \n &\\AdmSub{\\tau_i}{\\Psi_i}\\\\\n &\\NSubform[\\Psi_i]{A_i}\\\\\n &\\AdmSub{\\sigma}{\\Psi}\\\\\n &\\PSubform[\\Psi]{C}\n\\end{align*} \n\ncan be read as (modulo constraints on substitutions)\n``Given the parameters $\\Sigma$ and hypotheses and derived formulas \n$A_i$ under admissible substitutions $\\tau_i$, the goal $C$ under admissible\nsubstitution $\\sigma$ is derivable.''\n\n\\subsection{Subsumption}\n\n$$\n(\\Sequent{\\Sigma}{A_1[\\tau_1],\\ldots,A_n[\\tau_n]}{C[\\sigma]}) \\leq \n(\\Sequent{\\Sigma'}{A_1[\\tau_1'],\\ldots,A_k[\\tau_k']}{C[\\sigma']})\n$$\n\nif there exists an admissible substitution $\\AdmSub[\\Sigma']{\\Theta}{\\Sigma}$\nsuch that \n$\\tau_1' = \\tau_1 \\Comp \\Theta, \\ldots, \\tau_k' = \\tau_k \\Comp \\Theta, \\sigma' = \\sigma \\Comp \\Theta$\n\nfor some $n\\leq k$. \n\n\\subsection{Unification}\n\n\\subsection{Most General Unifiers}\nIn the pattern fragment of LF, most general unifiers exist. Therefore the\nfunction $\\mgu(x,y)$ is well defined. Keeping our invariant (*) in mind, \nwe will write $\\mgu = (\\Theta_1,\\Theta_2)$ when the domain of the intended\nterms to which we apply the substitutions have different sets of free variables.\nThat is, if the most general unifier of $A_1,A_2$ is $\\Theta$ and \n$fv(A_1) \\neq fv(A_2)$ then \n\\begin{align*} \n \\Theta_1 &= \\mathbf{filter}\\ (\\lambda(x,\\_).\\ \\mathbf{mem}(fv(A_1,x),x))\\ \\Theta \\\\ \n \\Theta_2 &= \\mathbf{filter}\\ (\\lambda(x,\\_).\\ \\mathbf{mem}(fv(A_2,x),x))\\ \\Theta\n\\end{align*} \n\n\n\\subsection{Forward Sequent Rules, Pattern Fragment}\n\n\\infer[\\textbf{Init}]{\\Sequent{\\Sigma}{P_1[\\Theta_1]}{P_2[\\Theta_2]}}{}\nwhen\n\\begin{align*} \n &\\NSubform[\\Psi_1]{P_1}\\\\\n &\\PSubform[\\Psi_2]{P_2}\\\\\n &\\MGU{(\\Theta_1,\\Theta_2)}{\\NSubform[\\Psi_1]{P_1}}{\\PSubform[\\Psi_2]{P_2}}\\\\\n &\\AdmSub{\\Theta_1}{\\Psi_1}\\\\\n &\\AdmSub{\\Theta_2}{\\Psi_2}\n\\end{align*} \n\n\\infer[\\textbf{Contract}]{\\Sequent{\\Sigma'}{\\Gamma[\\Theta],A[\\tau_1] \\Comp \\Theta}{C[\\sigma \\Comp \\Theta]}}\n {\\Sequent{\\Sigma}{\\Gamma,A[\\tau_1],A[\\tau_2]}{C[\\sigma]}}\nwhen\n\\begin{align*} \n &\\AdmSub{\\tau_1}{\\Psi}\\\\\n &\\AdmSub{\\tau_2}{\\Psi}\\\\ \n &\\MGU[\\Sigma']{\\Theta}{\\tau_1}{\\tau_2}\n\\end{align*} \n\n\\infer[\\textbf{ArrowR1}]{\\Sequent{\\Sigma'}{\\Gamma[\\Theta]}{(A\\LFArrow B)[\\tau \\Comp \\Theta]}}\n {\\Sequent{\\Sigma}{\\Gamma,A[\\tau]}{B[\\sigma]}}\nwhen \n\\begin{align*} \n &\\MGU[\\Sigma']{\\Theta}{\\tau}{\\sigma} \\\\\n &\\AdmSub{\\Theta}{\\Psi} \\\\\n &\\AdmSub{\\sigma}{\\Psi}\\\\\n &\\PSubform{A \\LFArrow B}\n\\end{align*} \n\n\\infer[\\textbf{ArrowR2}]{\\Sequent{\\Sigma}{\\Gamma}{(A\\LFArrow B)[\\sigma]}}\n {\\Sequent{\\Sigma}{\\Gamma}{B[\\sigma]}}\nwhen \n\\begin{align*} \n &\\AdmSub{\\sigma}{\\Psi}\\\\\n &\\PSubform{A \\LFArrow B}\n\\end{align*} \n\n\\infer[\\textbf{ArrowL}]{\\Sequent{\\Sigma}{\\Gamma_1[\\Theta_1],\\Gamma_2[\\Theta_2],(A\\LFArrow B)[\\tau_1 \\Comp \\Theta_1]}{C[\\sigma \\Comp \\tau_2]}}\n {\\Sequent{\\Sigma_1}{\\Gamma_1}{A[\\tau_1]} & \\Sequent{\\Sigma_2}{\\Gamma_2,B[\\tau_2]}{C[\\sigma]}}\nwhen\n\\begin{align*} \n &\\MGU{(\\Theta_1,\\Theta_2)}{\\AdmSub{\\Theta_1}{\\Sigma_1}}{\\AdmSub{\\Theta_2}{\\Sigma_2}}\\\\\n &\\AdmSub{\\tau_1}{\\Psi}\\\\\n &\\AdmSub{\\tau_2}{\\Psi}\\\\\n &\\NSubform{A \\LFArrow B}\n\\end{align*} \n\n\\infer[\\textbf{PiL}]{\\Sequent{\\Sigma}{\\Gamma, (\\PiTyp{u}{A}{B})[\\tau]}{C[\\sigma]}}\n {\\Sequent{\\Sigma}{\\Gamma, B[\\tau,M/u]}{C[\\sigma]}}\nwhen \n\\begin{align*} \n \\NSubform{\\PiTyp{u}{A}{B}}\n\\end{align*} \n\n\\infer[\\textbf{PiR}]{\\Sequent{\\Sigma}{\\Gamma}{(\\PiTyp{x}{A}{B})[\\sigma]}}\n {\\Sequent{\\Sigma,y:A'}{\\Gamma}{B[\\sigma,y//x]}}\n\nwhen \n\\begin{align*} \n &y\\not\\in \\Gamma \\\\\n &A' = A[\\sigma] \\\\\n &\\PSubform{\\PiTyp{x}{A}{B}}\n\\end{align*} \n\n%% \\subsection{Junk}\n\n\n%% \\newcommand{\\init}{(\\Gamma_0^-,C_0^+)}\n\n%% \\infer[$Atom$]{P \\vdash P}{}\n%% \\bigskip \n%% when $P \\leq^+ \\init$ and $P \\leq^- \\init$\n%% \\bigskip \n\n%% \\infer[$Contract$]{\\Gamma,A \\vdash B}{\\Gamma,A,A\\vdash B}\n%% \\bigskip \n\n%% \\infer[\\to$R1$]{\\Gamma\\vdash A \\LFArrow B}{\\Gamma,A \\vdash B & (A\\LFArrow B \\leq^+ \\init)}\n%% \\bigskip \n\n%% \\infer[\\LFArrow$R2$]{\\Gamma\\vdash A \\LFArrow B}{\\Gamma \\vdash B & A \\not\\in\\Gamma & (A\\LFArrow B \\leq^+ \\init)}\n%% \\bigskip \n\n%% \\infer[\\LFArrow$L$]{\\Gamma_1,\\Gamma_2,A \\LFArrow B\\vdash C}{\\Gamma_1,\\vdash A,& \\Gamma_2,A \\vdash B & (A\\LFArrow B \\leq^- \\init)}\n%% \\bigskip \n\n%% \\subsection{The First Order Case}\n\n%% \\newcommand{\\seq}[3][\\Sigma]{#1 ; #2 \\vdash #3}\n%% \\newcommand{\\unif}[3][\\Sigma]{#1 ; #2 = #3}\n%% \\newcommand{\\renaming}[1]{#1\\ \\mathsf{renaming}}\n\n%% General Form:\n\n%% $$\\Sigma; \\Gamma[\\tau] \\vdash C[\\sigma]$$\n\n%% which is shorthand for\n\n%% $$\\Sigma; A_1[\\tau_1],\\ldots,A_n[\\tau_n] \\vdash C[\\sigma]$$\n\n%% \\bigskip \n\n%% \\infer[$Atom$]{\\seq{P_1^-[\\rho\\circ\\tau]}{P_2^+[\\tau]}} {}\n%% \\bigskip \n%% when $\\unif{P_1^-[\\rho\\circ\\tau]}{P_2^+[\\tau]}$ and $\\renaming{\\rho}$\n%% \\bigskip \n\n%% \\infer[\\Pi$-R$]{\\seq{\\Gamma}{}}{}\n%% \\bigskip \n\n", "meta": {"hexsha": "0d9c32d9a048329353b08e955edd9904c6a7471e", "size": 8425, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/inverse/tex/inverse.tex", "max_stars_repo_name": "kryptine/twelf", "max_stars_repo_head_hexsha": "1edad1846921cc962138cd4a5a703d3b1e880af2", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 61, "max_stars_repo_stars_event_min_datetime": "2015-01-24T18:10:58.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-25T12:41:05.000Z", "max_issues_repo_path": "src/inverse/tex/inverse.tex", "max_issues_repo_name": "kryptine/twelf", "max_issues_repo_head_hexsha": "1edad1846921cc962138cd4a5a703d3b1e880af2", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-02-27T22:17:51.000Z", "max_issues_repo_issues_event_max_datetime": "2021-02-27T22:17:51.000Z", "max_forks_repo_path": "src/inverse/tex/inverse.tex", "max_forks_repo_name": "kryptine/twelf", "max_forks_repo_head_hexsha": "1edad1846921cc962138cd4a5a703d3b1e880af2", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 10, "max_forks_repo_forks_event_min_datetime": "2016-05-06T01:32:34.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T19:33:29.000Z", "avg_line_length": 31.7924528302, "max_line_length": 141, "alphanum_fraction": 0.6490207715, "num_tokens": 3037, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.5156199157230156, "lm_q1q2_score": 0.3448879766744842}} {"text": "\\documentclass[journal=jpcbfk,manuscript=article,layout=singlecolumn,articletitle=true]{achemso}\n\\setkeys{acs}{articletitle = true}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% Place any additional packages needed here. Only include packages\n%% which are essential, to avoid problems later.\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage{graphicx,bm,lineno,mathpazo} \n\\usepackage{amssymb,amsfonts,amsmath,mathtools}\n\\usepackage{algorithm,algpseudocode,booktabs,dcolumn}\n\\captionsetup{labelfont=bf}\n\\usepackage[input-decimal-markers={.},separate-uncertainty=true,table-auto-round,scientific-notation = false,zero-decimal-to-integer=true]{siunitx}\n\\DeclareMathOperator*{\\tr}{tr}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\usepackage[caption=false,font=normalsize,position=top]{subfig}\n\\usepackage[kerning,spacing,tracking]{microtype}\n\\microtypecontext{spacing=nonfrench}\n\\usepackage{indentfirst}\n\\overfullrule=5pt\n\\usepackage[normalem]{ulem} \n\\usepackage{color}\n\\usepackage{empheq}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% New command\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\newcommand{\\boxedeq}[2]{\\begin{empheq}[box={\\fboxsep=6pt\\fbox}]{align}\\label{#1}#2\\end{empheq}}\n\n\\title{EM: From Langevin-Dynamics}\n\\author{Yi-Tsao Chen}\n\\date{December 2019}\n\n\\begin{document}\n \n\\maketitle\n\n\\section{Langevin-Dynamics}\nGiven a potential mean force $V(x)$ and a diffusion coefficient $D$, we simulate a trajectory, $X(t)$, from\n\\begin{equation}\n\\label{eq:ODLD}\ndx_t=D F(x_t)dt + \\sqrt{2D}d{W}_t,\n\\end{equation}\n\n\\section{Formulate the problem as Hidden Markov Model}\nWe denote a particular configuration as $(\\textbf{x}, \\textbf{y})$:\n\\begin{equation}\n(\\textbf{x}, \\textbf{y})=(x_0, x_1, \\cdots, x_T, y_0, y_1, \\cdots, y_T)\n\\end{equation}\nThe joint probability is:\n\\begin{equation}\np(\\textbf{x}, \\textbf{y})= p(x_0)\\prod_{t=0}^{T-1} p(x_{t+1}|x_t) \\prod_{t=0}^{T}p(y_t|x_t)\n\\end{equation}\n\n\\begin{figure}[h]\n\\includegraphics{figures/graphical.pdf}\n\\caption[Graphical Model]{Graphical Model}\n\\label{domaindecomp}\n\\end{figure}\n\n\\section{Inference}\n\\subsection{state representation}\nWe represent the state at time $t$ as a multinomial random varialbe $x_{t}$, with components $x_t^i$ for $i=0,...,M$. Thus $x_t^i$ is equal to one for a particular value of $i$ and is equal to zero for $j \\neq i$.\n\\begin{equation}\nx_t = \n\\begin{bmatrix}\nx_t^0 & x_t^1 & x_t^2 &\\cdots & x_t^M\n\\end{bmatrix}^T\n\\end{equation}\nFor example, if the state is in the second state, \n\\begin{equation}\nx_t = \n\\begin{bmatrix}\n0 & 0 & 1 & \\cdots & 0\n\\end{bmatrix}^T\n\\end{equation}\n\\subsection{alpha-beta}\nWe focus on a particular state node $x_t$ and ask to calculate its posterior probability, $p(x_t| \\textbf{y})$. And by Bayes' theorem,\n\\begin{equation}\np(x_t | \\textbf{y}) = \\frac{p(\\textbf{y}|x_t) p(x_t)}{p(\\textbf{y})}\n\\end{equation}\nThen, conditioning on $x_t$ and use conditional independence:\n\\begin{equation}\np(x_t | \\textbf{y}) = \\frac{p(y_0,\\cdots,y_t|x_t) p(y_{t+1},\\cdots,y_T|x_t) p(x_t)}{p(\\textbf{y})}\n\\end{equation}\nRegroup the terms\n\\begin{equation}\n\\begin{split}\np(x_t | \\textbf{y}) &= \\frac{p(y_0,\\cdots,y_t,x_t) p(y_{t+1},\\cdots,y_T|x_t)}{p(\\textbf{y})}\\\\\n&=\\frac{\\alpha(x_t)\\beta(x_t)}{p(\\textbf{y})}\n\\end{split}\n\\end{equation}\nwhere \n\\begin{equation}\n\\alpha(x_t) \\triangleq p(y_0,\\cdots,y_t,x_t)\n\\end{equation}\nis the probability of emitting a partial sequence of outputs $y_0,\\cdots,y_t$ and ending up in state $x_t$, and\n\\begin{figure}[h]\n\\includegraphics{figures/alpha_example.pdf}\n\\caption[Alpha]{The illustration of $\\alpha(x_2)=p(y_0,y_1,y_2,x_2)$}\n\\label{alpha}\n\\end{figure}\n\\begin{equation}\n\\beta(x_t) \\triangleq p(y_{t+1},\\cdots,y_T|x_t)\n\\end{equation}\nis the probability of emitting a partial sequence of outputs $y_{t+1},\\cdots,y_T$ given that the system starts in state $x_t$. \n\\begin{figure}[h]\n\\includegraphics{figures/beta_example.pdf}\n\\caption[Beta]{The illustration of $\\beta(x_2)=p(y_3,y_4|x_2)$}\n\\label{beta}\n\\end{figure}\n\n\\begin{figure}[b]\n\\includegraphics{figures/graphical_example.pdf}\n\\caption[Graphical Model]{The example of forward-backward}\n\\label{domaindecomp}\n\\end{figure}\n\n\\subsection{Likelihood function}\nThe sum of $p(x_t|\\textbf{y})$ over the possible values of $x_t$ must equal one.\n\\begin{equation}\n1 = \\sum_{x_t} p(x_t|\\textbf{y}) = \\frac{\\sum_{x_t} \\alpha(x_t)\\beta(x_t)}{p(\\textbf{y})}\n\\end{equation}\nSo, the likelihood function $p(\\textbf{y})$ can be written as\n\\begin{equation}\np(\\textbf{y})=\\sum_{x_t} \\alpha(x_t)\\beta(x_t)\n\\end{equation}\n\n\\subsection{Posterior probability $p(x_t | \\textbf{y})$}\nWe denote the posterior probablity $p(x_t | \\textbf{y})$ as $\\gamma(x_t)$\n\\begin{equation}\n\\gamma(x_t) \\triangleq \\frac{\\alpha(x_t)\\beta(x_t)}{p(\\textbf{y})}\n\\end{equation}\n\n\\subsection{Forward algorithm}\n\\begin{equation}\n\\alpha(x_{t+1}) = \\sum_{x_t} \\alpha(x_t) a_{x_t,x_{t+1}} p(y_{t+1}|x_{t+1})\n\\end{equation}\nwhere $a_{x_t,x_{t+1}}$ is the transition probability from $x_t$ to $x_{t+1}$. The definition of alpha at the first time step yields:\n\\begin{equation}\n\\alpha(x_0) = p(y_0, x_0) = p(y_0|x_0)p(x_0) = p(y_0|x_0) \\pi_{x_0} \n\\end{equation}\nand these values are used to initialize the recursion.\n\n\\subsection{Backward algorithm}\n\\begin{equation}\n\\beta(x_t) = \\sum_{x_{t+1}} \\beta(x_{t+1}) a_{x_t,x_{t+1}} p(y_{t+1}|x_{t+1})\n\\end{equation}\nThe beta recursion is a backwards recursion; that is, we start at the final time step $T$ and proceed backwards to the initial time step.\n\nSome remarks:\n\\begin{enumerate}\n\\item $\\beta(x_T)$ is unhelpful because it makes reference to a non-existent $y_{T+1}$\n\\item If we assign $\\beta(x_T)$ to be a vector of ones, i.e., $\\beta(x_T)=\\textbf{1}$, we can compute $\\beta(x_{T-1})$ correctly\n\\begin{equation}\n\\begin{split}\n\\beta(x_{T-1}) &= p(y_T|x_{T-1}) \\\\\n&= \\sum_{x_T} \\beta(x_T) a_{x_{T-1}, x_T} p(y_T|x_T) \\\\\n&= \\sum_{x_T} a_{x_{T-1}, x_T} p(y_T|x_T) \n\\end{split}\n\\end{equation}\n\\end{enumerate}\n\n\\subsection{Computing $p(\\textbf{y})$ by a single forward pass for the alphas}\n\\begin{equation}\n\\begin{split}\np(\\textbf{y}) &= \\sum_{i} \\alpha(x_T^i) \\beta(x_T^i) \\\\\n&= \\sum_{i} \\alpha(x_T^i) \\\\\n&= \\sum_{i} p(y_0,y_1,\\cdots,y_T,x_T^i)\n\\end{split}\n\\end{equation}\n\\end{enumerate}\n\n\n\\section{Expectation maximization}\nPreparing an initial potential mean force $V_0(x)$ and a diffusion coefficient $D_0$, and try to using the framework in the 2013 JPCB paper to learn $V(x)$ and $D$ back.\n\n\\section{Question}\n\\begin{enumerate}\n\\item If we ignore the information of photons, what is the observed data, $Y(t)$?\n\\item What is the likelihood functional?\n\\begin{equation}\n\\label{eq:PathP}\n\\begin{split}\n\\mathcal{P}(Y(t) ; F(x),D) &= \\mathcal{P}(Y(t) ; \\theta)= \\mathcal{L}[\\theta]\\\\\n&=\\langle \\alpha_{t_0} |e^{-\\bm{H}\\Delta t_1}\\bm{y}_1e^{-\\bm{H}\\Delta t_2}\\bm{y}_2 \\ldots e^{-\\bm{H}\\Delta t_{N_{\\mathrm{P}}}}\\bm{y}_{N_{\\mathrm{P}}} | \\beta_{t_{\\mathrm{exp}}} \\rangle\n\\end{split}\n\\end{equation}\n\\end{enumerate}\n\n\\section{Appendix A: Bayes' theorem}\n\\begin{equation}\np(A|B) = \\frac{p(B|A)p(A)}{p(B)}\n\\end{equation}\n\n\\end{document}\n", "meta": {"hexsha": "83426360984ae6dcb39705c9991fc24906b4332d", "size": 7077, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "old/em_0.tex", "max_stars_repo_name": "yizaochen/em_theory", "max_stars_repo_head_hexsha": "a9260f17ff59d7a265dd9e629607376b8d909ae6", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "old/em_0.tex", "max_issues_repo_name": "yizaochen/em_theory", "max_issues_repo_head_hexsha": "a9260f17ff59d7a265dd9e629607376b8d909ae6", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "old/em_0.tex", "max_forks_repo_name": "yizaochen/em_theory", "max_forks_repo_head_hexsha": "a9260f17ff59d7a265dd9e629607376b8d909ae6", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.2923076923, "max_line_length": 213, "alphanum_fraction": 0.6850360322, "num_tokens": 2512, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5506073655352404, "lm_q2_score": 0.6261241702517975, "lm_q1q2_score": 0.3447485798802805}} {"text": "% arara: pdflatex\n% arara: bibtex\n% arara: pdflatex\n% arara: pdflatex\n\n\\documentclass[a4paper,11pt]{article}\n\\pdfoutput=1 % if your are submitting a pdflatex (i.e. if you have\n % images in pdf, png or jpg format)\n\n\\usepackage[a4paper,\ntotal={170mm,257mm},\nleft=20mm,\ntop=20mm,]{geometry}\n\n\\usepackage[dvipsnames]{xcolor}\n\\usepackage{longtable}\n\\usepackage{authblk}\n\\usepackage{hyperref}\n\n\\usepackage{titlesec}\n\\titleformat{\\section}[block]{\\color{blue}\\Large\\bfseries\\filcenter}{}{1em}{}\n\\titleformat{\\subsection}[hang]{\\bfseries}{}{1em}{}\n\n\n\\title{\\textcolor{red}{\\texttt{\\Huge TpMp}} - topology mapping}\n\n\n\\author{Andrey Pikelner}\n\\affil{II.~Institut f\\\"ur Theoretische Physik, Universit\\\"at Hamburg,\\\\\nLuruper Chaussee 149, 22761 Hamburg, Germany}\n\n\n% \\emailAdd{andrey.pikelner@desy.de}\n\n\n% Mathematica commands in blue\n\\newcommand{\\mma}[1]{\\textcolor{BlueGreen}{\\texttt{#1}}}\n% Color tt\n\\newcommand{\\ctt}[1]{\\textcolor{OliveGreen}{\\texttt{#1}}}\n\n\\setcounter{secnumdepth}{0}\n\\begin{document} \n\\maketitle\n\n\\abstract{We present program for diagram generation and mapping on\n topology working with \\texttt{Mathematica} interface. Available from\n \\url{https://github.com/apik/TpMp}}\n\n\n\\section*{Design ideas}\n\n\\subsection*{QGRAF input conventions}\n\\begin{itemize}\n\\item We use all particles as internal in QGRAF and having momentum\n $p_1,\\dots,p_n$. Field out kept blank.\n\\begin{verbatim}\n in = q, Q ;\n out = ;\n\\end{verbatim}\n\n\\item All fields in model file must contain field \\ctt{type} with\n possible values \\ctt{F,M,S,C,A} in addition to\n comutativity flag $\\pm$.\n\\begin{verbatim}\n [ phi, phi, + ; type='S']\n\\end{verbatim}\n\n\\item We set internal momentum to be \\ctt{k}\n\\end{itemize}\n\n\n\\subsection*{DB storage}\n\\label{sec:dbstore}\n\n\n\\section*{Main commands available in package }\n\n\\subsection*{Loading diagrams}\n\\begingroup\n\\renewcommand\\arraystretch{2}\n\\begin{longtable}{p{4cm}|p{11cm}}\n \\mma{LoadQGRAF[\"qlist\"]}\n & Load \\texttt{QGRAF}\\cite{Nogueira:1991ex} output \\texttt{qlist.yaml} in YAML format produced with the help of\n yaml.sty style file and produce SQLite 3 DB with generated\n diagrams and name \\texttt{qlist.sqlite3}.\\\\\n \\mma{nDB=LoadDB[\"db\"]}\n & Load SQLite 3 DB from file \\texttt{db.sqlite3} with diagrams and\n return numeric descriptor \\texttt{nDB} of open DB.\\\\\n \\mma{GetDia[nDia,nDB]}\n & Retrive diagram with number \\texttt{nDia} from DB with descriptor\n \\texttt{nDB}.\\\\\n\\end{longtable}\n\\endgroup\n\nWhen DB loaded all commands need DB decriptor to be specified. If DB\ndescriptor not specified explicitly - all commands applyed to first\nopen DB.\n\n\\subsection*{Selecting diagrams}\n\n\\begingroup\n\\renewcommand\\arraystretch{2}\n\\begin{longtable}{p{4cm}|p{11cm}}\n \\mma{WithField[\"f\"]}\n & Select diagrams with field named \"f\".\\\\\n \\mma{WithFieldType[T]}\n & Select diagrams with field type \\texttt{T}, where \\texttt{T} is\n one of: \\texttt{F} - for Dirac fermion, \\texttt{M} - for Majorana\n fermion, \\texttt{S} - for scalar, \\texttt{C} - for ghost and\n \\texttt{A} - for auxiliary field type.\\\\\n\\end{longtable}\n\\endgroup\n\n\\subsection*{Plotting diagrams}\n\n\n\\section{Mapping}\n\\subsection{\\boldmath $U$ and $F$ polynomials}\n\nIf we define $P_i=-q_j^2+m_j^2$, than:\n\\begin{equation}\n \\label{eq:MQJdefMass}\n \\sum\\limits_{j=1}^{n}x_j(-q_j^2+m_j^2)=-\\sum\\limits_{r=1}^{l}\\sum\\limits_{s=1}^{l}k_rM_{rs}k_s+\\sum\\limits_{r=1}^{l}2k_r\\cdot\n Q_r+J\n\\end{equation}\nTerm $J$ contain all dependence on external momentum products and\nmasses and can be decomposed:\n\n\\begin{equation}\n \\label{eq:Jdecompose}\n J=J^{p}+J^{m} = J^{p}(p_i^2,p_i\\cdot p_j)+\\sum\\limits_{j=1}^{n}x_jm_j^2\n\\end{equation}\n\nIf only information about momentum distribution needed we can neglect\nmasses and use $P$ polynomial instead of $F$\n\n\\begin{equation}\n \\label{eq:MQJdefNoMass}\n -\\sum\\limits_{j=1}^{n}x_jq_j^2=-\\sum\\limits_{r=1}^{l}\\sum\\limits_{s=1}^{l}k_rM_{rs}k_s+\\sum\\limits_{r=1}^{l}2k_r\\cdot\n Q_r+J^{p}\n\\end{equation}\n\nAnd Feynman graph polymnomials defined as\n\\begin{equation}\n \\label{eq:UFMdef}\n U=\\det(M),\\quad P=\\det(M)(J^p+QM^{-1}Q),\\quad F=\\det(M)(J+QM^{-1}Q).\n\\end{equation}\n\n$U$ polynomial determine internal structure of diagram, $P$ polynomial\ndetermines external momenta distribution and kinematic of the diagram\nand $F$-polynomial determines how masses are distributed between lines.\n\n\n\\begin{verbatim}\n<|-x[1] -> \n <|{} -> <|{1} -> {{{}, T12[{1, k1, m1}]}}|>|>, \n\n-x[1] - x[2] -> \n <|{{1, 1}} -> \n <|{1, 1} -> {{{{1, 1} -> -p1^2}, T12[{1, k1, m1}, {2, k1 + p1, m2}]}}, \n {0, 1} -> {{{{1, 1} -> -p1^2}, T02[{2, k1 + p1, m2}, {1, k1, 0}]}}, \n {1, 0} -> {{{{1, 1} -> -p1^2}, T10[{1, k1, m1}, {2, k1 + p1, 0}]}}, \n {0, 0} -> {{{{1, 1} -> -p1^2}, T00[{1, k1, 0}, {2, k1 + p1, 0}]}}|>|>|>\n\\end{verbatim}\n\n\\subsection{External momentum flow}\n\\label{sec:extflow}\n\nIt is possible to map diagram with $p_1,\\dots,p_n$ external momentum on topology\nwith $p_1,\\dots,p_k,k\\{p2,p3,...\\}}.\n\n\\subsection{Internal structure}\n\nFor set of diagrams it is possible to define ``projections'' i.e.\ndiagrams with contraints on kinematical variables: masses and external\nmomentums.\n\nProjections created with the help of command \\texttt{Project[\"pr\",subs]} where\nparameters \\texttt{\"pr\"} - unique name of projection and \\texttt{subs}\n- substitution rules.\n\n% \\appendix\n% \\section{Additions}\n% \\acknowledgments\n\n\\bibliographystyle{plain}\n\\bibliography{TpMp} \n\n\\end{document}\n\n%%% Local Variables:\n%%% mode: latex\n%%% TeX-master: t\n%%% End:\n", "meta": {"hexsha": "818f987dcfbcbb8583bd76e320810cd2bd155c08", "size": 5764, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/TpMp.tex", "max_stars_repo_name": "apik/TpMp", "max_stars_repo_head_hexsha": "840416d0eededfc3b0cfbe21f3c2af3a438be080", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/TpMp.tex", "max_issues_repo_name": "apik/TpMp", "max_issues_repo_head_hexsha": "840416d0eededfc3b0cfbe21f3c2af3a438be080", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/TpMp.tex", "max_forks_repo_name": "apik/TpMp", "max_forks_repo_head_hexsha": "840416d0eededfc3b0cfbe21f3c2af3a438be080", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.1111111111, "max_line_length": 127, "alphanum_fraction": 0.6835530881, "num_tokens": 2012, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251064863697, "lm_q2_score": 0.5350984286266115, "lm_q1q2_score": 0.3447238421626679}} {"text": "\\documentclass[11pt]{scrartcl} % Font size\n\\input{structure.tex} % Include the file specifying the document structure and custom commands\n\n%----------------------------------------------------------------------------------------\n%\tTITLE SECTION\n%----------------------------------------------------------------------------------------\n\n\\title{\n\t\\normalfont\\normalsize\n\t\\textsc{Harvard Privacy Tools Project}\\\\ % Your university, school and/or department name(s)\n\t\\vspace{25pt} % Whitespace\n\t\\rule{\\linewidth}{0.5pt}\\\\ % Thin top horizontal rule\n\t\\vspace{20pt} % Whitespace\n\t{\\huge Generalized Resize Notes}\\\\ % The assignment title\n\t\\vspace{12pt} % Whitespace\n\t\\rule{\\linewidth}{2pt}\\\\ % Thick bottom horizontal rule\n\t\\vspace{12pt} % Whitespace\n}\n\n\\author{} % Your name\n\n\\date{} % Today's date (\\today) or a custom date\n\n\\begin{document}\n\n\\maketitle\n\n\\textbf{This is a work in progress.}\n\n\\section{Goals}\nThe generalized resize component is a means of jointly achieving a few different goals:\n\\begin{enumerate}\n \\item guarantee known $n$,\n \\item give users flexibility in how they trade off between bias and privacy usage, and\n \\item allow for a combination of c-stability and privacy amplification from subsampling.\n\\end{enumerate}\n\n\\section{Algorithm Statement}\n\\label{sec:algorithm_statement}\nThe function will take the following inputs:\n\\begin{enumerate}\n \\item $X$: The private underlying data.\n \\item $\\tilde{n}$: The size of the private underlying data.\n \\item $n$: The desired size of the new data.\n \\item $p$: The proportion of the underlying data that can be used to construct the new data. Can be $> 1$.\n \\item $...$: Various arguments explaining imputation rules (not of interest for this doc)\n\\end{enumerate}\n\nLet $sample(Y, m)$ be a function that samples $m$ elements from data set $Y$ without replacement. \nLet $Aug(Y, m, \\hdots)$ be a function that imputes new elements independent of the data (using imputation parameters given by $\\hdots$) for a data set $Y$ until it is of size $m$. \nThe algorithm will look something like the following:\n\\begin{algorithm}[H]\n \\caption{Generalized Resize: resize(X, n, p, neighboring, ...)}\n \\label{alg:gen_resize}\n \\begin{algorithmic}[1]\n \\State $c \\gets \\lceil p \\rceil$ \\Comment{sets c-stability property}\n \\State $s \\gets p/c$ \\Comment{sets subsampled\\_proportion property}\n \\State $X_c \\gets \\bigcup_{i=1}^{c}X$ \\Comment{create new database of size $c\\tilde{n}$, composed of $c$ copies of $X$}\n \\If{neighboring == ``replace one''}\n \\State $m \\gets \\lfloor sc\\tilde{n} \\rfloor$ \\Comment{number of records that can be filled using subsampled private data}\n \\ElsIf{neighboring == ``add/remove one''}\n \\State $m \\gets Binomial(c\\tilde{n}, s)$ \\Comment{number of records that can be filled using subsampled private data}\n \\EndIf\n \\State $(\\epsilon', \\delta') \\gets \\left(\\log\\left(1+s\\left(e^{c\\epsilon}-1\\right) \\right), s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)\\delta \\right)$ \\Comment{privacy amplification via subsampling}\n \\State $X' \\gets sample\\left( X_c, \\max(m, n), neighboring \\right) \\bigcup \\left( Aug(\\varnothing, \\max(0, n - m), \\hdots) \\right)$\n \\\\ \\Return $(X', \\epsilon', \\delta')$\n \\end{algorithmic}\n\\end{algorithm}\n\nThe $\\epsilon', \\delta'$ terms come from first applying the group privacy definition with group size $c$ to the \ndatabase $X_c$ to get $\\left(c\\epsilon, \\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)\\delta \\right)$ \\cite{Vad17}\nand then applying privacy amplification by subsampling results from Theorems 8 and 9 of \\cite{BBG18}. \nNote that, for the ``replace one'' definition, we could be using $\\frac{m}{c\\tilde{n}}$ instead of $s$ in the privacy calculation. Using $s$ gives us a very slightly \nworse privacy guarantee (the only difference is the $\\lfloor \\cdot \\rfloor$ we used to get $m$), but is nice for \nconsistency between the methods and not having to keep track of $m$ as an extra property.\n\n\\section{Functional Privacy Parameters}\nWe established in section~\\ref{sec:algorithm_statement} that a user asking for an $(\\epsilon, \\delta)$-DP guarantee will get an \n$(\\epsilon', \\delta')$-DP guarantee with respect to the original private data. What we'd really like, however, is for the user to ask for an $(\\epsilon, \\delta)$-DP \ngurantee and have the library come up with what we will call a \\emph{functional $(\\epsilon'', \\delta'')$} that will ensure $(\\epsilon, \\delta)$-DP \non the original data. Any components that operate on the resized data will use the \\emph{functional $(\\epsilon'', \\delta'')$} internally instead of \nthe parameters passed by the user.\n\n\\begin{theorem}\n A mechanism that respects \n \\[ \\left( \\frac{1}{c}\\log\\left(\\frac{e^{\\epsilon'}-1}{s} + 1 \\right), \\frac{\\delta'}{s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)} \\right)\\text{-DP} \\] \n on the resized data respects $(\\epsilon, \\delta)$-DP on the true private data.\n \\begin{proof}\n We know that an $(\\epsilon, \\delta)$-DP on the resized data corresponds to a \n \\[ (\\epsilon', \\delta') = \\left(\\log\\left(1+s\\left(e^{c\\epsilon}-1\\right) \\right), s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)\\delta \\right) \\]\n guarantee on the private data. We just need to invert the function to find $(\\epsilon, \\delta)$ in terms of $(\\epsilon', \\delta')$. \\newline \n\n Let's start with $\\epsilon, \\epsilon'$:\n \\begin{align*}\n \\epsilon' &= \\log\\left(1+s\\left(e^{c\\epsilon}-1\\right)\\right) \\\\\n e^{\\epsilon'} &= 1+s\\left(e^{c\\epsilon}-1\\right) \\\\\n \\frac{e^{\\epsilon'}-1}{s} &= e^{c\\epsilon}-1 \\\\\n \\frac{1}{c}\\log\\left(\\frac{e^{\\epsilon'}-1}{s} + 1 \\right) &= \\epsilon.\n \\end{align*}\n\n We carry out a similar calculation for $\\delta, \\delta'$:\n \\begin{align*}\n \\delta' &= s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)\\delta \\\\\n \\frac{\\delta'}{s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)} &= \\delta.\n \\end{align*}\n \\end{proof} \n So, in order for a mechanism to respect $(\\epsilon, \\delta)$-DP on the original data, it must respect \n $\\left( \\frac{1}{c}\\log\\left(\\frac{e^{\\epsilon}-1}{s} + 1 \\right), \\frac{\\delta}{s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)} \\right)$-DP \n on the resized data.\n\\end{theorem}\n\nWe now present an mini-algorithm for finding the \\emph{functional} $(\\epsilon, \\delta)$.\n\\begin{algorithm}\n \\caption{Finding Functional $(\\epsilon_f, \\delta_f)$: get\\_func\\_priv($p, \\epsilon, \\delta$)}\n \\label{alg:finding_functional_privacy}\n \\begin{algorithmic}[1]\n \\State $c \\gets \\lceil p \\rceil$ \\Comment{sets c-stability property}\n \\State $s \\gets p/c$ \\Comment{sets subsampled\\_proportion property}\n \\State $\\epsilon_{f} \\gets \\frac{1}{c}\\log\\left(\\frac{e^{\\epsilon}-1}{s} + 1\\right)$\n \\State $\\delta_{f} \\gets \\frac{\\delta}{s\\left(\\sum_{i=1}^{c-1}e^{i \\epsilon}\\right)}$\n \\\\ \\Return $(\\epsilon_{f}, \\delta_{f})$ \n \\end{algorithmic}\n\\end{algorithm}\n\n\\section{Examples} \\label{sec:examples}\nLet $X$ be such that $\\tilde{n} = \\vert X \\vert = 100$. We can look at a few examples of calls to the generalized resize function (which will return $X'$) and check the behavior.\n\\begin{enumerate}\n \\item resize(X, 150, 1, ...): $X'$ will be made up of the 100 true elements of $X$ and 50 imputed values. The functional privacy parameters are identical to the ones the user provides. \n \\item resize(X, 100, 0.75, ...): $X'$ will be made up of 75 true elements of $X$ and 25 imputed values. The functional privacy parameters will benefit (lower noise) from amplification via subsampling.\n \\item resize(X, 90, 1.5, ...): $X'$ will be a random sample of $X \\bigcup X$ of size 90. \n The functional privacy parameters will lead to greater noise than what the user provides, as they have to take into account the new c-stability of $2$. \n This example is illustrative in that is shows that the functional privacy usage is affected only by $p$ -- it has nothing to do with the relative sizes of the $X$ and $X'$.\n\\end{enumerate}\n\n\\bibliographystyle{alpha}\n\\bibliography{generalized_resize}\n\n\\end{document}\n", "meta": {"hexsha": "8f56c84eee3c8ac3f2e5eb9cfce73234d732cf2d", "size": 8203, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "whitepapers/data_processing/generalized_resize.tex", "max_stars_repo_name": "confiare/SN-Core", "max_stars_repo_head_hexsha": "40604b288b407b154541740290db9eeac990b4ae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 53, "max_stars_repo_stars_event_min_datetime": "2021-02-18T07:02:53.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T22:10:13.000Z", "max_issues_repo_path": "whitepapers/data_processing/generalized_resize.tex", "max_issues_repo_name": "confiare/SN-Core", "max_issues_repo_head_hexsha": "40604b288b407b154541740290db9eeac990b4ae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 34, "max_issues_repo_issues_event_min_datetime": "2020-10-22T13:56:57.000Z", "max_issues_repo_issues_event_max_datetime": "2021-01-11T13:44:20.000Z", "max_forks_repo_path": "whitepapers/data_processing/generalized_resize.tex", "max_forks_repo_name": "confiare/SN-Core", "max_forks_repo_head_hexsha": "40604b288b407b154541740290db9eeac990b4ae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2020-10-22T13:29:54.000Z", "max_forks_repo_forks_event_max_datetime": "2020-12-08T15:57:02.000Z", "avg_line_length": 59.0143884892, "max_line_length": 207, "alphanum_fraction": 0.6592709984, "num_tokens": 2378, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.6893056167854461, "lm_q2_score": 0.5, "lm_q1q2_score": 0.34465280839272305}} {"text": "\\documentclass[12pt]{article}\n\\usepackage{geometry}\n\\geometry{left=1in,right=0.75in,top=1in,bottom=1in}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Replace ABCDEF in the next line with your chosen problem\n% and replace 1111111 with your Team Control Number\n\\newcommand{\\Problem}{E}\n\\newcommand{\\Team}{2102362}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\usepackage{newtxtext}\n\\usepackage{amsmath,amssymb,amsthm}\n\\usepackage{indentfirst}\n\\usepackage{gensymb}\n\\usepackage{setspace}\n\\usepackage{subfigure}\n\\usepackage{listings}\n\\usepackage{bm}\n\\usepackage{float}\n\\usepackage{newtxmath} % must come after amsXXX\n\\usepackage{textcomp}\n\\usepackage{graphicx}\n\\usepackage{cite}\n\\usepackage{xcolor}\n\\usepackage{fancyhdr}\n\\usepackage{booktabs}\n\\usepackage{multirow}\n\\usepackage{makecell}\n\\usepackage{epstopdf}\n\\usepackage{hyperref}\n\\usepackage{titlesec}\n\\usepackage{titletoc}\n\\usepackage{appendix}\n\\usepackage{url}\n\\usepackage[final]{pdfpages}\n\\linespread{1}\n\\setlength{\\parskip}{0.5\\baselineskip}\n\\setlength{\\parindent}{2em}\n\\lhead{Team \\Team}\n\\rhead{}\n\\cfoot{}\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{definition}{Definition}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\setlength{\\headheight}{15pt}\n\n\\begin{document}\n\\graphicspath{{.}} % Place your graphic files in the same directory as your main document\n\\DeclareGraphicsExtensions{.pdf, .jpg, .tif, .png}\n\\thispagestyle{empty}\n\\vspace*{-16ex}\n\\centerline{\\begin{tabular}{*3{c}}\n \\parbox[t]{0.3\\linewidth}{\\begin{center}\\textbf{Problem Chosen}\\\\ \\Large \\textcolor{black}{\\Problem}\\end{center}}\n & \\parbox[t]{0.3\\linewidth}{\\begin{center}\\textbf{2021\\\\ MCM/ICM\\\\ Summary Sheet}\\end{center}}\n & \\parbox[t]{0.3\\linewidth}{\\begin{center}\\textbf{Team Control Number}\\\\ \\Large \\textcolor{black}{\\Team}\\end{center}} \\\\\n \\hline\n \\end{tabular}}\n%%%%%%%%%%% Begin Summary %%%%%%%%%%%\n\n\\begin{center}\n {\\Large \\quad Prior Factors of a Food System: Money or Balance? \\newline Case Studies on China, USA, and Ethiopia }\n\n \\vspace{1em}\n {\\large \\textbf{Summary}}\n\\end{center}\n\\vspace{-0.5em}\n\n\\textbf{Food system} is an integrated concept reflecting human activities in the production and acquisition of food. A food system is strongly associated with other systems such as economy system, ecology system, and society system. Nowadays, the stability of a food system are of greater and greater significance for building a harmonious society and promoting national development. However, at present our global food system works mainly based on the control of market, where people in the system are blindly in pursuit of high efficiency and high profitability, causing problems including the imbalance of distribution and environmental degradation. The aim of this report is to \\textbf{build a re-optimization model} to evaluate the result of reordering the significance of different \\textbf{indicators}. These indicators are efficiency, profitability, equity, and sustainability. We also establish a \\textbf{predictive model} to reflect the change of the food system along with time variation. Besides, we do case studies and discuss the completeness of the model. \n\nFor the re-optimization model, we collect data of 18 indices in 3 countries from 2000 to 2020 in total, and use these indices to build four models for the four indicators. In this part, we first adopt the \\textbf{logistic model} to predict the population change. Then we build an differential equation for finding efficiency, and use \\textbf{Runge-Kutta method} to get the numerical solution. According to the results, we build a model for evaluating equity. Next, in the model for evaluating profitability, we use \\textbf{linear programming} to ensure that the data we use is within a reasonable range. Finally, we use \\textbf{linear} and \\textbf{nonlinear fitting} method to evaluate sustainability. \n\nFor the prediction model, we normalize all the data we derived, and then obtain the values of weight through \\textbf{analytic hierarchy process (AHP)}. We assess the validity of the weights to prove the correctness of comparison matrix. We also provide calculations of the final prioritized food system index $PFSI$ by doing a special \\textbf{weighted summation}. The result of $PFSI$ shows the food system evaluation results under a certain degree of emphasis for the four indicators. If $PFSI$ is closer to 0, then the focus of the food system we analyze is more on efficiency and profitability. If the result is closer to 0, then the focus is more on sustainability and equity.\n\nFor the discussion section, we first use the data we collected for the prediction of the food system after re-optimization. Then, we discuss the benefits and costs for changing the order of the four indicators. We apply our model into \\textbf{China}, \\textbf{USA}, and \\textbf{Ethiopia} for case studies and analyze the different consequence between developing countries and developed countries. Moreover, we pay attention to the scalability and adaptability of the model. Finally, we discuss the strengths and potential improvements of the food system model.\n\n\n\n\\vspace{2em}\n\\textbf{Keywords}: Food system, Analytic hierarchy process (AHP), Logistic model, Linear Programming, Runge-Kutta method, Nonlinear fitting, Linear fitting\n\n% to here\n%%%%%%%%%%% End Summary %%%%%%%%%%%\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\clearpage\n\\pagestyle{fancy}\n% Uncomment the next line to generate a Table of Contents\n\n\\newpage\n\\setcounter{page}{1}\n\\rhead{Page \\thepage\\ }\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\titlecontents{section}[0cm]{\\fontsize{12pt}{\\baselineskip}\\selectfont}{\\hspace*{3em}\\contentslabel{2em}\\ }%\n{}{\\titlerule*[0.5pc]{$\\cdot$}\\contentspage\\hspace*{1cm}\\vspace*{-0.3em}}%\n\n\\titlecontents{subsection}[1cm]{\\fontsize{10pt}{\\baselineskip}\\selectfont}{\\hspace*{3em}\\contentslabel{2em}\\ }%\n{}{\\titlerule*[0.5pc]{$\\cdot$}\\contentspage\\hspace*{1cm}\\vspace*{-0.5em}}%\n\n\\titlecontents{subsubsection}[2cm]{\\fontsize{10pt}{\\baselineskip}\\selectfont}{\\hspace*{3em}\\contentslabel{2em}\\ }%\n{}{\\titlerule*[0.5pc]{$\\cdot$}\\contentspage\\hspace*{1cm}\\vspace*{-0.6em}}%\n\n\\tableofcontents\n\n\n\\newpage\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction}\n\\vspace{-1em}\n\\subsection{Background}\n\n\nThe food system is conceived as a set of human activities from food production to processing, distribution, and consumption \\cite{cite:Prior}. Its complexity shows up in its manifold drivers, feedback, outcomes, and relations with other human systems, and its concepts are constantly enriched with human's increasing attention to its connotation. From a macroscopic perspective, the current analysis of food systems is strongly connected to the topic of \\textbf{food security}, which refers to a goal that everyone in the world can get access to sufficient and nutritious food to satisfy the diet needs \\cite{cite:Concept}. For example, a scientific group for the 2021 United Nations food systems summit suggests that the ability of the food system to enable every human to be well-nourished should be the core of the overall action tracks, which can be recognized as a pursuit for sustainability and equity \\cite{cite:Summit}.\n\n\\vspace{-1em}\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 0.5\\textwidth]{figure/FS.pdf}\n \\vspace{-1em}\n \\caption {Sustainable food systems \\cite{cite:fig}}\n\\end{figure}\n\n\\vspace{-1em}\nHowever, due to the leading rule of the market, at present the prior objective of the global food system is maximum efficiency along with the highest profit \\cite{cite:Market}. The market control of trade and prices and regional inherent inequities make the basic demands for food by many kinds of vulnerable groups hard to be satisfied. As a result, distribution problems such as food waste and food shortage have emerged. For example, in 2016 about 10.7\\% people in the world suffer from malnutrition problems, but at the same time about 1/3 of all the food produced is wasted \\cite{cite:USDA}. Furthermore, negative interactions that are exerted between the food system and other correlated systems like environment system and economic system make the dilemma of the food system harder to solve. \n\n\\vspace{-1em}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Restatement of the Problem}\n\nTo ensure a more stable food system with a more comprehensive selection of different objectives, an optimization model must be established to improve the current conditions of the food system. As a consultant team for the International Comestibles Management (ICM) Committee, we are going to implement the following tasks:\n\n\\begin{itemize}\n \\item Find the factors related to the food system, determine the objectives of our model, and choose representative indicators.\n \\item According to the indicators we fix, build a comprehensive evaluation model for the current food system.\n \\item Re-optimize the food system given a more preferable objective, and compare the ideal food system with the current one, and then analyze the feasibility of improving the food system.\n \\item Discuss the trade-offs of optimization, including both benefits and costs of the change.\n \\item Apply the model to countries with different levels of development and compare the results.\n \\item Analyze the extendability and adaptability of the food system model. \n\\end{itemize}\n\n\\vspace{-1em}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Our Work}\nOur efforts can be summarized in the following flow chart.\n\\begin{figure}[H]\n \\centering\n \\includegraphics[width = 1\\textwidth]{figure/procedure.pdf}\n \\caption{Structure of the model and paper}\n \\label{procedure}\n\\end{figure}\n\n\\section{Assumptions and Justifications}\n\nWe make the following assumptions in the modelling process. \n\n\\noindent \\textbf{Assumption 1.} \\textit{Country is the unit of a food system.}\n\nAccording to the instructions of the question of comparison between developed and developing countries, we assume that our evaluation for food systems is conducted in terms of country, and the model only synthesizes the consideration of food system for profitability, efficiency, equity, and sustainability.\n\n\\noindent \\textbf{Assumption 2.} \\textit{Imports and exports among countries are neglected.}\n\nWe assume that we do not consider the import and export of food among countries. In addition, we assume that food is distributed equally within the country we study. Since based on the data provided by FAO almost all the major countries in the world have the self-sufficiency rate of food larger than 0.8 \\cite{cite:Self_support}, we regard each country we analyze as an independent place where people produce and market all by themselves. \n\n\\noindent \\textbf{Assumption 3.} \\textit{All the data is reliable and precise.}\n\nWe assume that the data we collected from databases are reliable. They are from major websites including FAO \\cite{cite:FAO_data}, World bank \\cite{cite:Worldbank}, Statista \\cite{cite:Statista}, and Worldometers \\cite{cite:Worldometers}.\n\n\\noindent \\textbf{Assumption 4.} \\textit{Unpredictable natural disasters and other sudden changes will not happen.}\n\nUnforeseen natural hazard such as drought and flood disasters will affect the food system as well as its indices dramatically, so we assume that within the predicted time there are no sudden far-reaching changes.\n\n\\noindent \\textbf{Assumption 5.} \\textit{The change in the population of a country is continuous.}\n\nSince indices of the model is based on the population in a country, we assume that there are not mass migrations or wars that could affect the population in the country. \n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Model Preparation}\n\\subsection{Notation}\n\nIn this paper, we define some symbols and parameters, and their notations are shown in Table \\ref{tab:notation}.\n\n\n\\begin{table}[!h]\n \\centering\n \\caption{\\label{tab:notation} Notations}\n \\vspace{1em}\n \\begin{tabular}{lll}\n \\toprule\n Symbols & Description & Units \\\\\n \\midrule\n $t$ & time & year \\\\\n $PM$ & Policy motivation, an adjustable constant & 1 \\\\\n $UR_{2020}$ & Undernourishment rate in 2020 & \\% \\\\\n $OR_{2020}$ & Obesity rate in 2020 & \\% \\\\\n $p(t)$ & Population & person\\\\\n $p_{2020}$ & Population in 2020 & person \\\\\n $S_m(t)$ & Target total yield in year $t$ & ton \\\\\n $S(t)$ & Total yield & ton \\\\\n $S_{2020}$ & Total yield in 2020 & ton \\\\\n $s_1(t)$ &Target total yield of cereal in year $t$& ton\\\\\n $s_2(t)$ &Target total yield of fruit in year $t$& ton\\\\\n $s_3(t)$ &Target total yield of oil crops in year $t$& ton\\\\\n $s_4(t)$ &Target total yield of vegetables in year $t$& ton\\\\\n $s_5(t)$ &Target total yield of dairy in year $t$& ton\\\\\n $s_6(t)$ &Target total yield of meat in year $t$& ton \\\\\n $PR_i(t)$ &The corresponding average prices for six different foods in year $t$ & \\$/ton \\\\\n $IF$ &Average inflation rate of the studied country from 2000 to 2020 & \\%\\\\\n $IN(t)$ &The annual income of the Food sales in year $t$ & 100000 \\$ \\\\\n $GHG(t)$ & Total greenhouse gas emissions from agriculture in year $t$ & CO$_2$ equivalent \\\\\n $r_1$ & Weight of efficiency & 1 \\\\\n $r_2$ & Weight of equity & 1 \\\\\n $r_3$ & Weight of profitability & 1 \\\\\n $r_4$ & Weight of sustainability & 1 \\\\\n $rs$ & Relative significance of \\textit{EQ} (or \\textit{SU}) with respect to \\textit{EF} (or \\textit{PF}) & 1\\\\\n $EF(t)$ & Score of efficiency in the food system & 1 \\\\\n $EQ(t)$ & Score of equity in the food system & 1 \\\\\n $PF(t)$ & Score of profitability in the food system & 1 \\\\\n $SU(t)$ & Score of sustainability in the food system & 1 \\\\\n $PFSI(t)$ & Prioritized food system index & 1 \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Model Design}\n\nTo complete our model for the food system, we use $EF$, $EQ$, $PR$, and $SU$ to denote the score of efficiency, score of equity, score of profitability, and score of sustainability respectively. Then the final food system index $PFSI$ can be derived by a special weighted average process. \n\n\\subsection{Preparation: Classic Population Logistic Prediction Model}\n\nTo predict the population in the future for each year, we searched for possible population forecasting models in the database. We choose \\textbf{logistic model} to complete the prediction \\cite{cite:logi}, since compared with \\textbf{Malthus model} we should consider constraints on natural resources, environmental capacity and policies to give a changing population growth rate $r$. We use \\textbf{linear least square method} to obtain the initial value $r_0$ and maximum population $P_{max}$. The solution procedure is shown below.\n\n\\begin{enumerate}\n \\item Collect population data of China, USA, and Ethiopia from 2000 to 2020.\n \\item Suppose the population growth rate $r$ is a function varying with population $P$: $r(P) = r_0 - P$. The equation of the logistic model is given by $$\\frac{\\mathrm{d} P}{\\mathrm{d} t} = r_0 (1-\\frac{P(t)}{P_{max}})P(t).$$\n \\item Get the difference equation by backward difference:\n \n $$\\frac{P(k)-P(k-1)}{P(k)} = r_0 - s\\cdot P(k),\\text{ }k = 2, 3, \\cdots, 21.$$\n \n \\item Fit the parameters $r_0$ and $P_{max}$ by linear least square method. The result of China and USA are shown in Table \\ref{tab:pplt} below. \n \n \\begin{table}[H]\n \\centering\n \\caption{\\label{tab:pplt} Values of $r_0$ and $P_{max}$}\n \\vspace{1em}\n \\begin{tabular}{ccc}\n \\toprule\n & $r_0$ & $P_{max}$ \\\\\n \\midrule\n China & -0.0219 & $1.83\\times 10^9$ \\\\ \n USA & -0.0332 & $0.408\\times 10^9$ \\\\\n Ethiopia & -0.0381 & 0.328\\times 10^9\\\\\n \\bottomrule\n \\end{tabular}\n \\end{table}\n \n \\item Use the separation of variables method for differential equation to derive the analytical solution \n $$P(t) = \\frac{P_{max}}{1+(\\frac{P_{max}}{P_{2020}}-1)\\cdot e^{r_0(t-2020)}}.$$\n\\end{enumerate}\n\nThe result of our fitting for the population in China and USA is shown in Figure \\ref{fig:USA_pplt} and \\ref{fig:China_pplt}.\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/USA/USA_pplt.eps}\n \\caption{Logistic Fit of Population of USA.\\label{fig:USA_pplt}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/China/China_pplt.eps}\n \\caption{Logistic Fit of Population of USA.\\label{fig:China_pplt}}\n \\end{minipage}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Model for evaluating \\textit{EF}} \\label{sec:EF}\n\nStockholm Environment Institute defines the efficiency in the food system as ``the ratio of outputs to inputs'' \\cite{cite:Efficiency_1}. However, to estimate factors of input and output of different kinds of food requires a huge set of data from measurements and experiments, and from all the literature we read no one even try to explore the overall efficiency of a complex food system. We finally select the sum of the annual food yield of different food as an indicator for efficiency. The selection of the indicator refers to work by \\cite{cite:Esti_efficiency}. \n\nIn order to calculate the value of efficiency in the food system we choose, we do the following steps:\n\n\\begin{itemize}\n \\item \\textbf{Find the total yield.} Based on the food pyramid provided by a Harvard research \\cite{cite:Food_pyramid}, we divide food into six groups: cereal, fruit, oil crops, vegetable, dairy, and meat. The total food yield is given by Formula \\eqref{eq:yield} below.\n \n \\begin{align}\n \\label{eq:yield} S = \\sum^6_{i=1}s_i, \\text{ where }s_i\\text{ is the yield of the i-th food.}\n \\end{align}\n \n \\item \\textbf{Determine the target yield in year $t$.} Consider the malnutrition rate and obesity rate among adults, the value of the target yield in the i-th year $S_m(t)$ can be obtained by Formula \\eqref{eq:target} below.\n \n \\begin{align}\n \\label{eq:target} S_m(t) = \\frac{S_{2020}/(1+OR_{2020}-UR_{2020})}{p_{2020}}\\cdot p(t).\n \\end{align}\n \n \\item \\textbf{Establish the equation for yield $S(t)$.} In the equation we introduce a coefficient $PM$, called policy motivation, to reflect the regulation of priorities. First, our aim is to ensure that the larger $PM$ is, the greater the efficiency $EF$ changes. Second, we anticipate that if the current food yield $S$ is larger than the target yield $S_m$, $S(t)$ will decrease with respect to time $t$, and if the current $S$ is less than the target yield $S_m$, $S(t)$ will increase with respect to time $t$. Third, the first two changes should be applied to food yield $S(t)$ itself. Then we obtain Equation \\eqref{eq:logi}, which is similar to the equation of logistic model, but \\textbf{the denominator in the middle entry is now a variable, instead of a constant}: \n \n \n \\begin{equation}\n \\left\\{\n \\begin{aligned}\n \\label{eq:logi} & \\frac{\\mathrm{d} S}{\\mathrm{d} t} = PM\\cdot (1-\\frac{S(t)}{S_m(t)})\\cdot S(t) \\\\\n & S_{\\text{initial}} = S_{2020} \\\\ \n \\end{aligned}\n \\right.\n \\end{equation}\n\n \\item \\textbf{Solve the differential equation.} In order to obtain the numerical solution of this differential equation, we adopt \\textbf{Runge-Kutta method} for the initial-value problem \\cite{cite:Runge}. First, we separate the interval $[2020, 2050]$ (indicating the year from 2020 to 2050 as our evaluation range) into $N$ sub-intervals $[t_n, t_{n+1}]$ ($n = 0, 1, \\cdots, N-1$). Then, by the mean value theorem we have \n \n \\begin{align}\n S(t_{n+1}) - S(t_n) = \\int^{t_{n+1}}_{t_n} f(t,S(t)) \\mathrm{d}t = (t_{n+1}-t_{n}) f(\\xi, S(\\xi)), \\text{where }\\xi\\in [t_n, t_{n+1}] \\text{\\cite{cite:Runge}}.\n \\end{align}\n \n Finally, by approximation we get \n \n \\begin{align}\n S_{n+1} -S_n = (t_{n+1}-t_n) \\sum^m_{i = 1} c_i f(\\xi_i, y(\\xi_i)), \\text{where we choose }m = 4, c_1 = c_4 =1/6, \n \\end{align}\n \n and $c_2 = c_3 = 1/3$.\n \n \\item \\textbf{Normalize the result.} After obtaining results of the total yield, we rate the values by making a \\textbf{min-max normalization}. The form is shown in Formula \\eqref{eq:Normal}.\n \n \\begin{align}\n \\label{eq:Normal} EF(t) = \\frac{S(t)-S_{\\min}}{S_{\\max}-S_{\\min}}\n \\end{align}\n\\end{itemize}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Model for evaluating \\textit{EQ}}\n\nAccording to the literature found on the database \\cite{cite:Equity_def, cite:Equity_def2}, equity problems in the food system includes racism, gap between the rich and the poor, gender inequality, etc. In our model for the food system, since we take each country as a unit, we are aimed at regarding a country as a whole and finding the difference between countries. \n\nTherefore, for all studied objects, we assume that categories in the dietary structure are totally included by the types of agricultural products produced, which is actually mostly the case. Therefore we claim that countries we selected can be \\textbf{self-sufficient} in food categories. We then come up with a method making full use of the outcomes of Subsection \\ref{sec:EF}. \n\nWe use \\textbf{the ratio of supply and demand $S(t)/S_m(t)$} to denote the level of equity. The interpretation comes from the fact that if the current total yield is close to the target yield then everyone in the country has a chance to have the necessary amount of food, neither too excessive (which implies that there are some people in the country possess the food amount more than they need) nor too insufficient (which implies that there are some people in the country possess the food amount less than they need). Finally, we normalize the result in the same procedure as what is shown in Subsection \\ref{sec:EF}.\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Model for evaluating \\textit{PF}} \\label{sec:pf}\n\nThe profitability in the food system can be defined as the \\textbf{total income} by selling all the food in terms of their different prices. Besides, the price by prediction should have limits in the scope of value to prevent price mutations in a certain year. We assume that people are rational enough to always pursue the maximum profit. Therefore, we use \\textbf{linear programming} to solve this problem. \n\nFormula \\eqref{eq:profit} shows the calculation form. For the constraints on linear programming problems, we stipulate the yields of each food \\textbf{within the yields interval for recent five years}, so that we can ensure the continuity of the yields. For the future price of food, we apply \\textbf{linear fit} to each kind of food to predict.\n\n\n\\begin{gather}\n% \n% \\centering\n\\max \\quad \\sum_{i=1}^{6} (s_i \\frac{PR_i}{IF}), \\notag\\\\\n\\mbox{s.t.} \\quad \n\\sum_{i=1}^{6} s_i =S(t),\\label{eq:profit} \\\\\n\\min \\{ \\frac{s_k(t-j)}{S(t-j)} \\} \\leq \\frac{s_k(t)}{S(t)} \\leq \\max \\{ \\frac{s_k(t-j)}{S(t-j)} \\}, j=1,2,3,\\ldots,5, \\quad k = 1,2,3,\\ldots,6. \\notag\n\\end{gather}\n% &\\min \\{ \\frac{s_2(t-j)}{S(t-j)} \\} \\leq \\frac{s_2(t)}{S(t)} \\leq \\max \\{ \\frac{s_2(t-j)}{S(t-j)} \\},&j=1,2,3\\ldots,5 \\\\\n% &\\min \\{ \\frac{s_3(t-j)}{S(t-j)} \\} \\leq \\frac{s_3(t)}{S(t)} \\leq \\max \\{ \\frac{s_3(t-j)}{S(t-j)} \\},&j=1,2,3\\ldots,5 \\\\\n% &\\min \\{ \\frac{s_4(t-j)}{S(t-j)} \\} \\leq \\frac{s_4(t)}{S(t)} \\leq \\max \\{ \\frac{s_4(t-j)}{S(t-j)} \\},&j=1,2,3\\ldots,5 \\\\\n% &\\min \\{ \\frac{s_5(t-j)}{S(t-j)} \\} \\leq \\frac{s_5(t)}{S(t)} \\leq \\max \\{ \\frac{s_5(t-j)}{S(t-j)} \\},&j=1,2,3\\ldots,5 \\\\\n% &\\min \\{ \\frac{s_6(t-j)}{S(t-j)} \\} \\leq \\frac{s_6(t)}{S(t)} \\leq \\max \\{ \\frac{s_6(t-j)}{S(t-j)} \\},&j=1,2,3\\ldots,5 \\\\\n\n\nFinally, we normalize the result in the same procedure as what is shown in Subsection \\ref{sec:EF}.\n\n\n\\subsection{Model for evaluating \\textit{SU}}\n\nAccording to Holden et al. \\cite{cite:Sus1}, the excessive consumption of fossil fuel is one of the most severe influence current food systems have on the natural environment, and causing the food system unsustainable. Hence, we choose the total emissions from agriculture in one country as an indicator of the sustainability. Considering the fact that the larger food production is, the more emissions from agriculture will be, we take the quotient, i.e., the emissions per unit of the yield, to reflect the performance of sustainability. We should also take the policy motivation into account:\n\n\\begin{itemize}\n \\item Before the policy change, we do \\textbf{liner fittings} for the data of greenhouse gas emissions and the data of total yield per year we collected respectively. Figure \\ref{fig:lf_E_yield_GHG} shows the fitting performance of greenhouse gas emission and annual yield. The coefficients of determination of linear fit of them are $0.9706$ and $0.9603$ respectively, which are very close to the ideal value $1$. Equations \\eqref{eq:lf_E_GDG} and \\eqref{eq:lf_E_yield} show the fitting results.\n \\begin{figure}[htbp]\n \\centering\n \t\\subfigure[Linear Fit of the Annual Yield of Ethiopia.]{\\includegraphics[width=0.48\\textwidth]{figure/model/Ethiopia/Ethiopia_yield_fit.eps}} \n \t\\subfigure[Linear Fit of the Greenhouse Gas Emmision of Ethiopia.]{\\includegraphics[width=0.48\\textwidth]{figure/model/Ethiopia/Ethiopia_GHG_fit.eps}} \n \\caption{Sensitivity Test for Policy Motivation \\textit{PM}.\\label{fig:lf_E_yield_GHG}}\n \t\\vspace{0.2in}\n \\end{figure}\n \\begin{gather}\n GHG(t) = a\\cdot t - b, \\text{where } a = 2576, b = 5.049\\times10^6 \\label{eq:lf_E_GDG}\\\\\n S(t) = c\\cdot t - d, \\text{where } c = 1.228\\times10^6, d = 2.443\\times10^9 \\label{eq:lf_E_yield}\n \\end{gather}\n Then we have the expression below:\n \n $$\\frac{GHG(t)}{S(t)} = \\frac{a t-b}{c t-d}.$$\n \\item \\textbf{Modify the equation.} After the policy change, we take $PM$ into account, and update the expression as below (suppose the policy change occurs in 2020):\n \n $$\\frac{GHG(t)}{S(t)} = \\frac{a (t-2020) + (2020 a -b)(1-PM)}{c (t-2020) + (2020 c - b)(1-PM)},$$\n \n from where we should notice that the new function has \\textbf{the same asymptotic line as the older one} (which is $y = a/c$ as $x\\to \\infty$), and the two functions \\textbf{intersect at \\textit{t = 2020}}. The new function also indicates that \\textbf{when \\textit{PM} = 0 the two functions are equal}, and given a stronger policy motivation $PM$ the new function will drop more steeply.\n\\end{itemize}\n\nAfter we predict the value of sustainability by year, we normalize the result in the same procedure as what is shown in Subsection \\ref{sec:EF}.\n\n\\subsection{Model for evaluating \\textit{PFSI}}\nWe obtain the prioritized food system index $PFSI$ by making weighted summation of $EF$, $EQ$, $PR$, and $SU$. Values of weight are obtained through \\textbf{Analytic Hierarchy Process (AHP)}. To reasonably indicate the main focus of the food system, we define Formula \\eqref{eq:PFSI} below to find the value of $PFSI$:\n\n\\begin{align}\n \\label{eq:PFSI} PFSI(t) = (1-EF(t))\\cdot r_1 + (1-PR(t))\\cdot r_3 + EQ(t)\\cdot r_2 + SU(t)\\cdot r_4\n\\end{align}\n\nIt is suggested that \\textbf{when \\textit{PFSI} is closer to 0 the focus of the food system is more on \\textit{EF} and \\textit{PR}}, and \\textbf{when \\textit{PFSI} is closer to 1 the focus of the food system is more on \\textit{SU} and \\textit{EQ}}.\n\n%图？\n\nThe steps of calculating weights by \\textbf{AHP} are shown below:\n\\begin{itemize}\n \\item We first determine the relative significance of four factors (\\textit{EF}, \\textit{PF}, \\textit{EQ}, and \\textit{SU}). Specifically, \\textit{EF} and \\textit{PF} are considered to have the same significance, and \\textit{EQ} and \\textit{SU} are considered to have the same significance. We set the the significance of \\textit{EQ} (or \\textit{SU}) with respect to \\textit{EF} (or \\textit{PF}) to be $rs$. Then, the comparison matrix $A$ (Equ. \\eqref{eq:comM}) is set up.\n \\begin{equation}\n A=\\left[\\begin{array}{cccc}\n 1 & 1 & 1/rs & 1/rs \\\\\n 1 & 1 & 1/rs & 1/rs \\\\\n rs & rs & 1 & 1 \\\\\n rs & rs & 1 & 1\n \\end{array}\\right]\n \\label{eq:comM}\n \\end{equation}\n In the case study (Sec. \\ref{sec:case}), we set $rs = 7$. However, the absolute value of $rs$ does not affect the general shape of the final result, which will be verified in sensitivity analysis (Sec. \\ref{sec:sense}).\n \\item Calculated the eigenvalues of matrix $A$, we can further determine the weights of \\textit{EF}, \\textit{PF}, \\textit{EQ}, and \\textit{SU} by normalize the eigenvector of the maximum eigenvalue. Namely, the weight vector can be calculated through Equ. \\eqref{eq:AHPweight}.\n \\begin{equation}\n \\mathbf{w} = \\frac{\\mathbf{v}}{\\|\\mathbf{v}\\|_1} = \\frac{\\mathbf{v}}{\\sum_{i=1}^4|v_i|}\n \\label{eq:AHPweight}\n \\end{equation}\n \\item To prevent the possible conflict caused by arbitrarily set significance, we are supposed to assess the validity of the weights. We define the consistency index $CI$ as\n \\begin{equation}\n CI = \\frac{\\lambda - 4}{4-1} = \\frac{\\lambda - 4}{3},\n \\end{equation}\n where $\\lambda$ is the maximum eigenvalue of comparison matrix $A$. Then, the consistency ratio $CR$ can be calculated through\n \\begin{equation}\n CR = \\frac{CI}{RI},\n \\end{equation}\n where $RI$ is random consistency index. Here, we have 4 factors in total, so $RI=0.9$. In general, if $CR<0.1$, the weights obtained from AHP is valid.\n\\end{itemize}\n\n\\section{Discussion}\n\\subsection{Task 1: Re-Optimization for the Food System}\n\nSince there are interactions among the model for efficiency, profitability, sustainability, and equity, after the optimization for equity and sustainability we need to look at the consequence of the changes of their weight. We consider the situation in different countries. \n\n\\begin{itemize}\n \\item \\textbf{For a country in which the undernourishment rate is less than the obesity rate, we claim that the total food production amount is enough to satisfy the need of all the population.} The country produces food surpluses. As a result, under the influence of policy motivation, the country will tend to produce exactly the amount of food they need, so that the total yield of food \\textbf{will decrease at first}, causing a decline of $EF$. When the intermediate rate state is reached, the prediction of the level of efficiency will depend on the population change as well. In the countries where the population is growing continuously, the annual food yield should try to catch up with the population, and accordingly \\textbf{the total yield and $EF$ will increase since then}. In the countries where the population is declining, the change will be in the opposite case.\n \n Besides, the value of the level of \\textbf{equity ($S/S_m$) will decrease to 1} at first since the food yield is large. According to the differential equation, we can see that at this time the growth rate of the yield is 0. Hence, the value of $S/S_m$ will go on decreasing until less than 1. However, the slope of $S$ will be higher until it surpasses the population growth rate, so that \\textbf{in the end $S/S_m$ will approach to 1}, implying that the amount of food per person is more equitable. \n \n As for the profitability, since the prices of each kind of food is not fixed, these prices are predicted by linear fitting. However, one thing is known that we use the yields interval for recent five years for the linear programming, so that we may infer from the expression that \\textbf{the level of profitability will change smoothly}. Another supporting argument is that we aim to find the maximum income for each year so that the \\textbf{food system still seek for the maximum benefits even sustainability is considered now}.\n \n For the prediction of the change of sustainability, we can find that the \\textbf{greenhouse gas emissions keep increasing in developing countries} like China, and remain nearly constant in developed countries like the USA. Then after exerting policy motivation, \\textbf{the value of $GHG(t)/S(t)$ will decrease faster than the case before re-optimization}. However, for our model in the end the level of sustainability will be the same as before because the two functions have the same asymptotic line. It reflects that this model loses some accuracy for a very long term prediction.\n \n \n \\item \\textbf{For a country in which the undernourishment rate is higher than the obesity rate}, the total food production amount is not enough. In this case, the total yield will be raised to meet the need for food by residents. If the population growth rate is larger than \\textbf{$PM$, $S/S_m$ will be firstly less than 1 then approach 1}. In the end, the population reaches its limit, and correspondingly the yield will approach approximately a constant.\n \n On the contrary, \\textbf{if the population growth rate is less than $PM$}, then $S/S_m$ will decline due to the fact that the yield does not catch up with the rate of population growth. When the population growth rate begins decreasing, $S/S_m$ will rise up again until leveling off to 1.\n \n The analysis of profitability and sustainability is the same as above.\n \n\\end{itemize}\n\n\n\\textbf{We define 0.5 as the balance value for efficiency, profitability, equity, and sustainability.} It means that we regard them as equally important. According to our model, the time that a food system needs to implement should be counted from the occurrence of policy change to the moment when $PFSI$ reaches 0.5 from either side for the first time. The interpretation is that when $PFSI > 0.5$ indicators including sustainability and equity become more important than efficiency and profitability. \n\n%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Task 2: Benefits and Costs}\n\nWe only consider the case of changing the priorities of a food system from efficiency and profitability to sustainability and equity. The opposite direction will give the contrary result. \n\n\\subsubsection{Potential benefits}\n\n\\begin{itemize}\n \\item \\textbf{Distribution become more reasonable within developed countries, and will first receive a drop in some developing countries, but later will recover.} When we change the prioritization of food system to sustainability and equity, it is obvious that \\textbf{ developed countries will tend to distribute its food more fairly} among its residents, because the equity is always approaching 1. However, for some developing countries, Ethiopia, for example, firstly received a drop in equity. This can result from the fact that the production rate fails to exceed the population rate at first. But later, equity of developing countries will approach 1 again as the production rate increases and the population growth rate decreases.\n \\item \\textbf{Under the motivation of policy, sustainability for both developed countries and developing countries become stronger.} As it will also \\textbf{limits the increase of emission amount of greenhouse gas} if the total yields grow. Such effect becomes \\tetxbf{mostly apparent in the developed country}, as the obesity rate is much larger than the undernourishment rate, and the yield generally decreases. While in some developing country, which has a large undernourishment rate, yield growth is also limited with stronger sustainability, so that the atmosphere and environment are protected, and used more efficiently.\n\\end{itemize}\n\n\\subsubsection{Potential costs}\n\\begin{itemize}\n \\item \\textbf{Developing countries will become tougher when cooperating with the increase of the population} For some developing countries, whose obesity rate is slightly larger than the undernourishment rate, they will first drop the yields to fit the current population and then try to keep pace with the population growth. Comparing with such a \"following\" strategy, if countries try to increase their yields at first to cope with the future population, they will become more prepared.\n \n \\item \\textbf{Profit from the food industry will always decrease regardless of developed and developing countries.} In the later case study section, we illustrate the change of profitability, and these charts show that all developed and developing countries received profit loss. On one hand, this can result from some yield drop in some developed countries. On the other hand, some developing countries have a relatively large inflation rate, causing the actual profit to become less valuable. \n\\end{itemize}\n\n\\subsubsection{Prediction on the occurrence of the benefits and costs}\nHere, we firstly define \\textbf{the occurrence of the previously mentioned benefits and costs as the time when equity is within the interval [0.95,1.05] and is continuously monotonically increasing/ monotonically decreasing to approach 1}. With that definition and referring to the findings in section \\ref{sec:case}, we can find that China's occurrence time happened in the approximately year 2045 and that Ethiopia's occurrence time happened in the approximately year 2025, while America's occurrence time happened in the approximately year 2051. Therefore, with these experiment result, we can qualitatively conclude that \\textbf{the occurrence time of developed countries is later than the developing countries under the same policy motivation (PM)}.\n\n%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Task 3: Case Studies}\\label{sec:case}\n\nInstead, we need to utilize the prediction result of food prices and greenhouse gas emissions. Here we only show the plot of the scores for each indicator by using the method mentioned before. \n\n\\subsubsection{China: Developing Country with $UR_{2020}$ relatively less than $OR_{2020}$}\nChina is the representative developing country whose economic situation is very close to some developed countries. Specifically, the undernourishment rate of China is much smaller than the obesity rate in year 2020. This fact reflected a high possibility that the food China produced is more than the demand of people. However, as a developing country, its population is still in the quickly rising period. Thus, \\textbf{the current supply of food can not satisfy the demand of its future maximum population}. To satisfy the food demand of current population, \\textbf{the yield amount will firstly receive a slight drop until year 2035. But after that, the yield will continuously increase to follow the increase of population and corresponding demand.} Figure \\ref{fig:China_yield} shows the predicted situation with policy motivation. This result is obtained from the model for evaluating \\textit{EF} with \\textit{PM}$=0.1$. Meanwhile, the equity of food will be improved due to a better arrangement of supply. As the annual yield is controlled by the policy, the supply is gradually close to the demand (Fig. \\ref{fig:China_equity}), which ensures the domestic equity of food system. As shown in the Figure \\ref{fig:China_yield} and \\ref{fig:China_equity}, the effect of the policy will reach the maximum at approximately year \\textbf{2035}, namely, 15 years after the implementation of policy. After about \\textbf{25 years} of continuously policy motivation, the re-optimized food system is finally set up. \\textbf{At that time, the ratio of supply demand is extremely close to 1 and the yield is almost constant.}\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/China/China_yield.eps}\n \\caption{Annual Yield of China with Policy Interference.\\label{fig:China_yield}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/China/China_equity.eps}\n \\caption{Ratio of Supply and Demand of Food of China with Policy Interference.\\label{fig:China_equity}}\n \\end{minipage}\n\\end{figure}\n\nBesides, the policy will have a \\textbf{negative effect on the income} of the people served in the food system and a \\textbf{positive effect on the sustainability} of the food system. Due to the effect of inflation, the short-term decrease of yield will result in a \\textbf{continuous attenuation of the income}. Using the model stated in Section \\ref{sec:pf}, we can predict the income (Fig. \\ref{fig:China_income}). Furthermore, the policy will also \\textbf{push the development of science} in order to decrease the labor cost. This in turn improves the production method such that less greenhouse gas emission is necessary for the same yield. Red line in figure \\ref{fig:China_GHG} shows the \\textbf{accelerated decreasing trend greenhouse gas emission} with policy motivation. \n\\begin{figure}[H]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/China/China_profit.eps}\n \\caption{Income of Food System of China with Policy Interference.\\label{fig:China_income}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/China/China_sustainability.eps}\n \\caption{Greenhouse Gas Emission per yield of the Food System of China with Policy Interference.\\label{fig:China_GHG}}\n \\end{minipage}\n\\end{figure}\n\nFinally, with the economic, social, scientific, and environmental factors taken into account, the food system in China will prioritize equity and sustainability \\textbf{after the 15 years} of implementation of the policy. With \\textbf{10 more years} of development, the benefits and the costs of the new food system will be manifest. Figure \\ref{fig:China_score} shows the change of evaluated \\textit{PFSI} with \\textit{EF}, \\textit{PF}, \\textit{EQ}, and \\textit{SU} marked. Figure \\ref{fig:China_radar} shows the standardized score of \\textit{EF}, \\textit{PF}, \\textit{EQ}, and \\textit{SU} in 2021, 2035 (15 years after policy's implementation), 2045 (25 years after policy's implementation).\n\\begin{figure}[!htb]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width = 0.8\\textwidth]{figure/model/China/China_score.eps}\n \\caption{\\textit{PFSI} of China with All Four Factors Specified from 2015 to 2045.\\label{fig:China_score}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width = 0.8\\textwidth]{figure/radar/China_radar.pdf}\n \\caption{The Change of Four Factors of China's Food System.\\label{fig:China_radar}}\n \\end{minipage}\n\\end{figure}\n\n\\subsubsection{Ethiopia: Developing Country with $UR_{2020}$ larger than $OR_{2020}$} \nEthiopia is a typical developing country with a high undernourishment rate and low obesity rate. Therefore, the current food supply of Ethiopia is far away from the demand of its possible maximum population. As we change the priority of the food system by introducing the policy motivation, the annual yield will \\textbf{quickly increase to satisfy the demand of the population} (Fig. \\ref{fig:Ethiopia_yield}). As for the equity, policy motivation \\textbf{inhibits increasing speed of equity} to prevent it exceeds the ideal value $1$. Similar to Ethiopia, the income will be affected by the policy so that it \\textbf{decreases quickly at the beginning of the policy's implementation}. However, due to the large inflation rate, the \\textbf{decreasing speed is much larger} than in China. Besides, the sustainability of Ethiopia will also be improved with the policy motivation. The biggest difference between China and Ethiopia is the value of policy motivation \\textit{PM}. To obtain the same effect as the food system of China, Ethiopia needs to \\textbf{implement a much stronger policy regulation}. Through our estimation, the value of policy motivation of Ethiopia should be approximately 10 times that of China to get the same effect. In this case study, we choose to use policy motivation \\textit{PM}$=1$.\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/Ethiopia/Ethiopia_yield.eps}\n \\caption{Annual Yield of Ethiopia with Policy Interference.\\label{fig:Ethiopia_yield}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/Ethiopia/Ethiopia_equity.eps}\n \\caption{Ratio of Supply and Demand of Food of Ethiopia with Policy Interference.\\label{fig:Ethiopia_equity}}\n \\end{minipage}\n\\end{figure}\n\nFinally, we calculate the line of \\textit{PFSI}. After changing the weight of efficiency, profitability, equity, and sustainability, a gap appears in the year 2020. Different from China, the growth rate of Ethiopia's \\textit{PFSI} is not that big. Instead, \\textit{PFSI} \\textbf{increases gradually in the beginning} and \\textbf{is stable in year 2030}. Namely, only \\textbf{10 years of policy motivation} can bring the underdeveloped food system of Ethiopia back to the right track, the balanced mode. Figure \\ref{fig:Ethiopia_radar} shows the standardized score of \\textit{EF}, \\textit{PF}, \\textit{EQ}, and \\textit{SU} in 2021, 2035, and 2045.\n\\begin{figure}[!htb]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width = 0.8\\textwidth]{figure/model/Ethiopia/Ethiopia_score.eps}\n \\caption{\\textit{PFSI} of Ethiopia with All Four Factors Specified from 2015 to 2045.\\label{fig:Ethiopia_score}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width = 0.8\\textwidth]{figure/radar/Ethiopia_radar.pdf}\n \\caption{The Change of Four Factors of Ethiopia's Food System.\\label{fig:Ethiopia_radar}}\n \\end{minipage}\n\\end{figure}\n\n\\subsubsection{The USA: Developed Country with $UR_{2020}$ much less than $OR_{2020}$}\nThe USA is a developed country where the undernourishment rate is very small while the obesity rate is very large. \\textbf{Therefore, the current food supply of the USA even exceeds the demand of its possible maximum population.} After we change the priority to sustainability and equity, \\textbf{the total yield amount (Fig. \\ref{fig:USA_yield}) will generally receive a drop to satisfy the demand of population}, even in the year 2050, there will be a historically lowest point for the yields. Moreover, as the equity index($S/S_m$) (Fig. \\ref{fig:USA_equity}) is high at the present yield speed (about 1.4 in 2020), under the control of $PM$(policy motivation, set as 0.1), the equity will drop to 0.98 at the lowest level in the approximately year 2050 and then recover to 1 gradually so that it can compensate the increasing food demand caused by population growth. We can conclude that dropping the unnecessary yield of food can improve the sustainability of the food system, and increase equity in the long run.\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/USA/USA_yield.eps}\n \\caption{Annual Yield of USA with Policy Interference.\\label{fig:USA_yield}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.9\\textwidth]{figure/model/USA/USA_equity.eps}\n \\caption{Ratio of Supply and Demand of Food of USA with Policy Interference.\\label{fig:USA_equity}}\n \\end{minipage}\n\\end{figure}\n\nSimilar to China, policy motivation will have a \\textbf{negative effect on the income} of the people served in the food system due to the effect of inflation and price change. This is reasonable after taking the decrease in food yield into consideration. Besides, the ratio $GHG(t)/S(t)$ is also continuously drop with time. With the motivation of policy ($PM$=0.1), \\textbf{the drops will be more quickly} comparing with the slope of $PM=0$.\n\nFinally, we can calculate the line of $PFSI$ (figure\\ref{fig:USA_score}). After changing the weight of equity, sustainability, profitability, and efficiency, a sudden drop happens in the year 2020. Then, \\textbf{$PFSI$ will continuously grow} with its slope continuously drop. In approximately \\textbf{the year 2027}, the scoring of $PFSI$ will exceed 0.5, which is the time the food system is optimized for equity and sustainability. In approximately \\textbf{the year 2040},\nefficiency received its historical highest point, and the differential of $PFSI$ received a sudden drop, which is the time the benefits and cost become manifest. Figure\\ref{fig:USA_score} shows the standardized score of $EF$, $PF$, $EQ$, and $SU$ in 2021,2035, and 2045.\n\n\n\\begin{figure}[!htb]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width = 0.8\\textwidth]{figure/model/USA/USA_score.eps}\n \\caption{\\textit{PFSI} of USA with All Four Factors Specified from 2015 to 2045.\\label{fig:USA_score}}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width = 0.8\\textwidth]{figure/radar/USA_radar.pdf}\n \\caption{The Change of Four Factors of USA's Food System.\\label{fig:USA_radar}}\n \\end{minipage}\n\\end{figure}\n\n\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%\n\\subsection{Task 4: Scalability and Adaptability of the Model}\n\n\\subsubsection{Scalability}\n\nFood systems with different scales are under consideration. After analysis, we make the following conclusions.\n\n\\begin{itemize}\n \\item \\textbf{The larger the food system is, the better our optimization will be}. Given a larger food system (major economies or unions of nations), the production structure of the food system will fit better with the common dietary structure for the residents. The reason is that we do not consider the food trade as well as other food communications between countries, and a larger scale can provide a more self-sufficient system.\n \\item \\textbf{Food systems of countries suit best}. Since the model is designed for countries, we consider the yearly inflation rate of a country in the model of profitability. The data processing may be revised if given other systems than countries.\n\\end{itemize}\n\n\\subsubsection{Adaptability}\n\nFood systems with different properties are under consideration. Adaptability reflects the consequence when we change the object of the food system model. We have the following results.\n\n\\begin{itemize}\n \\item \\textbf{The model fits with countries with different $PM$}. For big countries like China and the USA, there is a large population along with strong scientific and technological support. However, for countries which have a smaller population and weaker policy support, the intensity of policy motivation needs to be modified to reflect the difference.\n \\item \\textbf{The model reflects the global polarization between rich countries and poor countries}. In Section \\ref{sec:EF}, we include the undernourishment rate and the obesity rate into our evaluation model. With regard to Formula \\eqref{eq:logi}, if $S_{2020}$ of a country is less than $S_m$, namely the average undernourishment rate is larger than the average obesity rate, then $1-S/S_m$ will be larger than 0. It indicates that the formula adapts to countries with different conditions of rich and poor.\n \\item \\textbf{The model is suitable for the comparison among countries with similar production capacity for food}. Since the costs in the production of food are neglected, the closer the production capacity among countries, the more comparable the result of profitability will be.\n\\end{itemize}\n\n\n\\section{Sensitivity Analysis}\\label{sec:sense}\nIn our model, there are two variables whose value is set by us without derivation. They are the policy motivation \\textit{PM} and the relative significance $rs$. \\textit{PM} is set to $0.1$ in the cases of China and the USA and is set to $0.5$ in the cases of Ethiopia. $rs$ is set to $7$ in calculating weights of \\textit{EF}, \\textit{PF}, \\textit{EQ}, and \\textit{SU}. In this part, we will argue that the results we get from the model are not sensitive to the change of the values of \\textit{PM} and $rs$.\n\n\\subsection{\\textit{PM} Fluctuation Test.}\nPolicy motivation $PM$ represents the strength of policy in encouraging the balanced development of the food system. It is first used to calculate the annual yield and equity index in our model for a prioritized food system. We re-calculate the annual yield and equity index with a different value of \\textit{PM}, ranging from $0.04$ to $0.4$. The results are shown in Figure \\ref{fig:sense_pm}.\n\\begin{figure}[!htb]\n \\centering\n\t\\subfigure[Re-calculating of Annual Yield of USA with Different Values of \\textit{PM}.]{\\includegraphics[width=0.48\\textwidth]{figure/sense/sense_pm_yield.pdf}} \n\t\\subfigure[Re-calculating of Equity Index of USA with Different Values of \\textit{PM}.]{\\includegraphics[width=0.48\\textwidth]{figure/sense/sense_pm_equity.pdf}} \n \\caption{Sensitivity Test for Policy Motivation \\textit{PM}.\\label{fig:sense_pm}}\n\t%\\vspace{0.2in}\n\\end{figure}\n\nAlthough the value of \\textit{PM} varies dramatically, the general shape of the annual yield does not change a lot. Annual yield always has a trend of decreasing to a minimum and increasing to a fixed asymptote. it is the same for the equity index (ratio of supply and demand). No matter how $PM$ swings, it will approach 1 as the year goes to infinity. Thus, we can conclude that the selection of the value \\textit{PM} is not a significant issue in our model, since the annual yield and equity index is convergent with respect to \\textit{PM}.\n\n\\subsection{\\textbf{\\textit{rs}} Fluctuation Test.}\n$rs$ represents the relative significance of \\textit{EQ} (or \\textit{SU}) with respect to \\textit{EF} (or \\textit{PF}). It is used to calculate the weights to get the final score of \\textit{PFSI}. To verify that the change of this value dose not significantly incluence the final score, we test different values of $rs$, ranging from $1$ to $8$. Figure \\ref{fig:sense_rs} shows the result of the recalculated \\textit{PFSI}. \n\nAlthough the change of $rs$ brings a difference to the weights, the final score does not change significantly. Meanwhile, its trend is generally the same. Namely, \\textit{PFSI} will increases faster in the beginning, and the speed of increasing decreases at approximately the year 2040. This observation confirms that $PFSI$ is not sensitive to the value of $rs$ in our model.\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width = 0.6\\textwidth]{figure/sense/sense_AHP.eps}\n \\caption{Sensitivity Test for $rs$ by Re-calculating the Corresponding \\textit{PFSI}.\\label{fig:sense_rs}}\n\\end{figure}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Strengths and Possible Improvements}\n\\subsection{Strength}\n\n\\begin{itemize}\n \\item \\textbf{Practicability}: To make our model conform to reality, we divide the food into six groups: cereal. fruit, oil crops, vegetable, dairy, and meat. The model considers all the components of the dietary pyramid.\n \\item \\textbf{Globality}: We have a macroscopic view of the current food system in the world without taking internal effects into account.\n \\item \\textbf{Flexibility}: By introducing a variable parameter named policy motivation ($PM$), we can adjust the priorities of efficiency, profitability, sustainability, and equity accordingly. Assigning $PM$ with different possible values (implying that the intensity of policy motivation may vary), our prediction result can be varied to be more flexible.\n\\end{itemize}\n\n\\subsection{Possible Improvements}\n\n\\begin{itemize}\n \\item \\textbf{We do not consider imports and exports}. If for a particular country there is a huge difference between the total amount of food imports and exports, the deviation of our model with reality will increase. \n \\item \\textbf{Indicators for our model could be more comprehensive}. In the model of equity, we can add another parameter depicting the matching degree of the dietary structure and the production structure, so that differences in the food distribution within countries can be considered. In addition, in the model of sustainability, more indicators except for greenhouse gas emissions may be added to improve the completeness of the model. The cost of food production may also be included to perfect the model. \n\\end{itemize}\n\n\n\\section{Conclusions}\n\nIn this paper, we first adopt the logistic model to predict the population growth from 2020 to 2050. Next, we build four models for evaluating efficiency, profitability, equity, and sustainability respectively. We use differential equations, linear programming, and linear fitting to help to find the solutions of our model. Then we propose an index called Prioritized Food System Index ($PFSI$) to denote the emphasis of the food system. In the discussion part, we analyze the benefits and costs for re-optimizing the food system, do case studies on China, the USA, and Ethiopia, and discuss the scalability and adaptability of the model. Finally, we conduct a sensitivity analysis to show the influence of policy motivation on the food system index and discuss strengths as well as improvements for our work.\n\n\n\\bibliographystyle{IEEEtran}\n\\bibliography{mybib}\n\n% \\newpage\n\\begin{appendices}\n\\section{Supporting Figures}\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/USA/USA_profit.eps}\n \\caption{Income of Food System of USA with Policy Interference.}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/USA/USA_sustainability.eps}\n \\caption{Greenhouse Gas Emission per yield of the Food System of USA with Policy Interference.}\n \\end{minipage}\n\\end{figure}\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/Ethiopia/Ethiopia_profit.eps}\n \\caption{Income of Food System of Ethiopia with Policy Interference.}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/Ethiopia/Ethiopia_sustainability.eps}\n \\caption{Greenhouse Gas Emission per yield of the Food System of Ethiopia with Policy Interference.}\n \\end{minipage}\n\\end{figure}\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/USA/USA_GHG_fit.eps}\n \\caption{Linear Fit of the Greenhouse Gas Emission of USA.}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/USA/USA_yield_fit.eps}\n \\caption{Linear Fit of the Annual Yield of USA.}\n \\end{minipage}\n\\end{figure}\n\\begin{figure}[htbp]\n \\centering\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/China/China_GHG_fit.eps}\n \\caption{Linear Fit of the Greenhouse Gas Emission of China.}\n \\end{minipage}\n \\begin{minipage}[t]{0.48\\textwidth}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{figure/model/China/China_yield_fit.eps}\n \\caption{Linear Fit of the Annual Yield of China.}\n \\end{minipage}\n\\end{figure}\n\\begin{figure}\n \\centering\n \\includegraphics[width = 0.5\\textwidth]{figure/model/Ethiopia/Ethiopia_pplt.eps}\n \\caption{Logistic Fit of Population of Ethiopia.}\n\\end{figure}\n% \\section{Code Example}\n% % \\lstset{\n% % basicstyle = \\sffamily, % 基本代码风格\n% % keywordstyle = \\bfseries, % 关键字风格\n% % commentstyle = \\rmfamily\\itshape, % 注释的风格，斜体\n% % stringstyle = \\ttfamily, % 字符串风格\n% % flexiblecolumns, % 别问为什么，加上这个\n% % numbers = left, % 行号的位置在左边\n% % showspaces = false, % 是否显示空格，显示了有点乱，所以不现实了\n% % % numberstyle = \\zihao{-5}\\ttfamily, % 行号的样式，小五号，tt等宽字体\n% % showstringspaces = false,\n% % captionpos = t, % 这段代码的名字所呈现的位置，t指的是top上面\n% % frame = lrtb, % 显示边框\n% % }\n\n% \\vspace{-2em}\n% \\lstset{frame=tb,\n% language=Python,\n% aboveskip=3mm,\n% belowskip=3mm,\n% showstringspaces=false,\n% columns=flexible,\n% basicstyle={\\small\\ttfamily},\n% numbers=none,\n% numberstyle=\\tiny\\color{gray},\n% keywordstyle=\\color{blue},\n% % commentstyle=\\color{dkgreen},\n% % stringstyle=\\color{mauve},\n% breaklines=true,\n% breakatwhitespace=true,\n% tabsize=3\n% }\n% % \\lstdefinestyle{Python}{\n% % language = Python, % 语言选Python\n% % basicstyle = \\tt\n% % % numberstyle = \\zihao{-5}\\ttfamily,\n% % keywordstyle = \\color{blue},\n% % % keywordstyle = [2] \\color{teal},\n% % stringstyle = \\color{magenta},\n% % commentstyle = \\color{red}\\ttfamily,\n% % breaklines = true, % 自动换行，建议不要写太长的行\n% % columns = fixed, % 如果不加这一句，字间距就不固定，很丑，必须加\n% % basewidth = 0.5em,\n% % }\n% % \\lstinputlisting[title = Grey Forecast]{code/GreyPre.py}\n% \\lstinputlisting[title = Moving Entropy Weight Method]{code/EWM.py}\n% \\lstinputlisting[title = Grey Forecast]{code/GreyPre.py}\n\n\n\\end{appendices}\n\\end{document}\n\n", "meta": {"hexsha": "2762d2f72de74fd6f521dc6ead1d433241fbe05e", "size": 61769, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "main.tex", "max_stars_repo_name": "zyxum/2021-ICM", "max_stars_repo_head_hexsha": "5533ff4f0606a1ad20a57a8e964ced6a4c026c18", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "main.tex", "max_issues_repo_name": "zyxum/2021-ICM", "max_issues_repo_head_hexsha": "5533ff4f0606a1ad20a57a8e964ced6a4c026c18", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "main.tex", "max_forks_repo_name": "zyxum/2021-ICM", "max_forks_repo_head_hexsha": "5533ff4f0606a1ad20a57a8e964ced6a4c026c18", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 77.4047619048, "max_line_length": 1618, "alphanum_fraction": 0.7302206608, "num_tokens": 16261, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.523420348936324, "lm_q2_score": 0.658417487156366, "lm_q1q2_score": 0.3446291108731628}} {"text": "% 19DistributionsaFoliations.tex\n% Fund Science! & Help Ernest finish his Physics Research! : quantum super-A-polynomials - a thesis by Ernest Yeung\n% \n% http://igg.me/at/ernestyalumni2014 \n% \n% Facebook : ernestyalumni \n% github : ernestyalumni \n% gmail : ernestyalumni \n% google : ernestyalumni \n% linkedin : ernestyalumni \n% tumblr : ernestyalumni \n% twitter : ernestyalumni \n% youtube : ernestyalumni \n% indiegogo : ernestyalumni \n%\n% Ernest Yeung was supported by Mr. and Mrs. C.W. Yeung, Prof. Robert A. Rosenstone, Michael Drown, Arvid Kingl, Mr. and Mrs. Valerie Cheng, and the Foundation for Polish Sciences, Warsaw University. \n%\n%These notes are open-source, governed by the Creative Common license. Use of these notes is governed by the Caltech Honor Code: ``No member of the Caltech community shall take unfair advantage of any other member of the Caltech community.'' \\\\\n%\n\n\\subsection*{Distributions and Involutivity}\n\n\\textbf{distribution on $M$ of rank $k$} is rank-$k$ subbundle of $TM$, \\textbf{smooth distribution} if it's smooth subbundle \\\\\nOften rank-$k$ distribution described by specifying $\\forall \\, p \\in M$ linear subspace $D_p \\subseteq T_pM$ of $\\text{dim}D_p = k$, \\\\\n\\phantom{\\quad \\quad \\,} $D = \\bigcup_{p \\in M} D_p$\n\nLemma 10.32, local frame criterion for subbundles, that $D$ smooth distribution iff $\\forall \\, p \\in M$, $\\exists \\, $ open $U \\ni p$ on which $\\exists \\, $ smooth vector fields $X_1 \\dots X_k : U \\to TM$ s.t. $\\left. X_1 \\right|_q \\dots \\left. X_k \\right|_q$ is basis for $D_q$ $\\forall \\, q \\in U$\n\n\\subsubsection*{Integral Manifolds and Involutivity}\n\nSuppose smooth distribution $D \\subseteq TM$ \\\\\n\\textbf{integral manifold of $D$} : immersed submanifold $N \\neq \\emptyset$, $N \\subseteq M$ if $T_pN = D_p$ $\\forall \\, p \\in N$\n\n\\textbf{Example 19.1 (Distributions and Integral Manifolds)}\n\n\\begin{enumerate}\n\\item[(a)] \n\\item[(b)]\n\\item[(c)]\n\\item[(d)]\n\\end{enumerate}\n\n$D$ \\textbf{involutive} if $\\forall \\, $ pair of smooth local sections of $D$ (i.e. smooth vector fields $X,Y$ defined on open subset of $M$ s.t. $X_p, Y_p \\in D_p$ \\, $\\forall \\, p$) \\\\\n\n\\textbf{integrable} : smooth distribution $D$ on $M$ integrable if $\\forall \\, p \\in M$, $p$ in integral manifold of $D$, i.e. \\\\\n\\phantom{\\quad \\quad \\, } $T_p M = D_p$\n\n\\begin{proposition}[19.3] $\\forall \\, $ integrable distribution is involutive. \\end{proposition}\n\n\\begin{proof} Let $D \\subseteq TM$ is integrable distribution. \\\\\nsuppose smooth local sections of $D$, $X,Y$ on some open $U\\subseteq M$. \\\\\n$\\forall \\, p \\in U$, let $N$ integral manifold of $D$, $N \\ni p$ \\\\\n$X,Y$ sections of $D$, so $X,Y$ tangent to $N$ \\\\\nBy Corollary 8.32, $[X,Y]$ also tangent to $N$, so $[X,Y]_p \\in D_p$\n\\end{proof}\n\n\n\n\\subsubsection*{Involutivity and Differential Forms}\n\n\\begin{lemma}[19.5] (\\textbf{1-form Criterion for Smooth Distributions}) Suppose smooth $n$-dim. manifold $M$, distribution $D \\subseteq TM$, rank $k$ \\\\ \n$D$ smooth iff $\\forall \\, p \\in M$, $\\exists \\, $ neighborhood $U$ on which $\\exists \\, $ smooth 1-forms $\\omega^1 \\dots \\omega^{n-k}$ s.t. $\\forall \\, q \\in U$, \n\\begin{equation}\n D_q = \\text{ker} \\left. \\omega^1 \\right|_q \\bigcap \\dots \\bigcap \\left. \\text{ker} \\omega^{n-k} \\right|_q \\quad \\quad \\quad \\, (19.1)\n\\end{equation}\n\\end{lemma}\n \\begin{proof}\nBy Prop. 10.15, complete forms $\\omega^1 \\dots \\omega^{n-k}$ to smooth coframe $(\\omega^1 \\dots \\omega^n)$ \\quad \\, $\\forall \\, p$ \\\\\nif $(E_1 \\dots E_n)$ dual frame, easy to sheet that $D$ locally spanned by $E_{n-k+1 }, \\dots , E_n$, so smooth by local frame criterion. \n\nConverse, suppose $D$ smooth. \\\\\n$\\forall \\, $ open $U \\ni p \\in M$, $\\exists \\, $ smooth vector fields $Y_1 \\dots Y_k$ spanning $D$. \\\\\nBy Prop. 16.5, complete $Y_1 \\dots Y_k$ to smooth local frame $(Y_1 \\dots Y_n)$ for $M$ in open $U\\ni p$ \\\\\nwith dual coframe $(\\epsilon^1 \\dots \\epsilon^n)$, it follows easily that $D$ characterized locally by $D_q = \\text{ker} \\left. \\epsilon^{k+1} \\right|_q \\bigcap \\dots \\bigcap \\text{ker} \\left. \\epsilon^n \\right|_q$ \n\n\\end{proof}\n\n\nif $D$ rank-$k$ distribution on smooth $n$-manifold $M$, any \\\\\n\\phantom{if $D$ } $n-k$ linearly independent 1-forms $\\omega^1 \\dots \\omega^{n-k}$ on open $U\\subseteq M$ s.t. (19.1) \n\\[\nD_q = \\left. \\text{ker}\\omega^1 \\right|_q \\bigcap \\dots \\bigcap \\left. \\text{ker}\\omega^{n-k} \\right|_q = \\lbrace X | X=X^iX_i, \\, i =1 \\dots k, \\, \\omega^1(X) = 0\\rbrace \\bigcap \\dots \\bigcap \\lbrace X | \\omega^{n-k}(X) = 0\\rbrace\n\\]\n$\\forall \\, q \\in U$ are \\textbf{local defining forms} for $D$\n\n\n\n\\begin{proposition}[19.8] \\textbf{(Local Coframe Criterion for Involutivity)}\n Let $D$ smooth distribution of rank $k$ on smooth $n$-manifold $M$ \\\\\nlet $\\omega^1 \\dots \\omega^{n-k}$ smooth defining forms for $D$ on open $U \\subseteq M$. \n\nThe following are equivalent:\n\\begin{enumerate}\n\\item[(a)] $D$ is involutive on $U$ \n\\item[(b)] $d\\omega^1 \\dots d\\omega^{n-k}$ annihilate $D$\n\\item[(c)] $\\exists \\, $ smooth 1-forms $\\lbrace \\alpha^i_j | i, j =1 \\dots n-k \\rbrace$ s.t. \n\\[\nd\\omega^i = \\sum_{j=1}^{n-k} \\omega^j \\wedge \\alpha^i_j \\quad \\quad \\, \\forall \\, i = 1 \\dots n-k\n\\]\n\\end{enumerate}\n\\end{proposition}\n\n\\exercisehead{19.9} Prove the preceding proposition, 19.8. \n\n\\begin{proof}\n(a) $\\Longrightarrow$ (b) \n\nOn open $U\\subseteq M$, $\\forall \\, q \\in U$, $\\omega^i$ smooth defining form for $D$, $i=1\\dots k$, and $\\omega^i(X) = 0$ \\, $\\forall \\, X \\in D_q$ \\\\\n\\phantom{ On open } Then $d\\omega^i$ also annihilates $D$ on $U$ (Thm. 19.7 1-form Criterion for Involutivity (19.3) ) \\\\\n$d\\omega^1 \\dots d\\omega^{n-k}$ annihilate $D$\n\n(b) $\\Longrightarrow $ (c)\n\n$d\\omega^i \\in \\Omega^2_q(M)$, $\\forall \\, q \\in U$ \\\\\nBy Lemma 19.6, smooth $p$-form $\\eta$ on $U$ annihilates $D$ iff $\\eta$ ofform $\\eta = \\sum_{i=1}^{n-k} \\omega^i\\wedge \\beta^i$, for $(p-1)$ forms $\\beta^1 \\dots \\beta^{n-k}$ on $U$ \\\\\n$d\\omega^i$ annihilates $D$ \\\\\n\\phantom{\\quad \\, } $\\Longrightarrow d\\omega^i = \\sum_{j=1}^{n-k} \\omega^j \\wedge \\beta_j^i \\quad \\quad \\, \\beta^i_{ \\, j} $ smooth 1-forms on $U$, $i,j=1\\dots n-k$\n\n(c) $\\Longrightarrow $ (a) \n\nUse Thm. 19.7 Proof\n\\[\n\\omega^i([X,Y]) = X(\\omega^i(Y)) - Y(\\omega^i(X)) - d\\omega^i(X,Y) = 0 -0 - d\\omega^i(X,Y)\n\\]\n\\[\nd\\omega^i(X,Y) - \\sum_{j=1}^{n-k} \\omega^j \\wedge \\alpha^i_{ \\, j}(X,Y) = \\sum_{j=1}^{n-k} \\omega^j(X) \\alpha^i_{ \\, j }(Y) - \\alpha^i_{ \\, j}\\omega^j(Y) = 0 - 0 = 0 \n\\]\nwhere I used this local formula:\n\\[\n(\\alpha \\wedge \\beta)_p(v,w) = \\alpha_p(v) \\beta_p(w) - \\alpha_p(w) \\beta_p(v)\n\\]\n$\\omega^i([X,Y]) = 0$ so $[X,Y] \\in \\text{ker}\\omega^i$ \\quad \\, $\\forall \\, i = 1 \\dots n-k$\n\n\\end{proof}\n\n\n\n\\subsection*{Problems}\n\n\\problemhead{19-3}\n\nLet $\\omega \\in \\Omega^1(M)$ \\\\\nintegrating factor $\\mu $ for $\\omega \\equiv \\mu \\in C^{\\infty}(M)$, $\\mu > 0$, and $\\mu \\omega $ exact on $U$, i.e. $\\mu \\omega = df$, for some $f \\in \\mathcal{C}^{\\infty}(M)$\n\n\\begin{enumerate}\n\\item[(a)] If $\\omega \\neq 0$ on $U$, \\\\\nSuppose $\\omega$ admits an integrating factor $\\mu$. \n\n\\[\nd\\omega \\wedge \\omega = d\\left( \\frac{df}{\\mu} \\right) \\wedge \\frac{df}{\\mu} = \\left( \\frac{d^2 f}{ \\mu} + -\\frac{df}{\\mu^2} \\frac{ \\partial \\mu }{ \\partial x^i } dx^i \\wedge df \\right) \\wedge \\frac{df}{\\mu} = 0 \n\\]\nas $d^2f =0$ and $df\\wedge df =0$\n\nIf $d\\omega \\wedge \\omega =0$, consider $\\mu \\in \\mathcal{C}^{\\infty}(M)$ s.t. $\\mu >0$ (i.e. positive) on open $U\\subseteq M$ (build it up with partitions of unity if need to). \n\nNow, using the formula for exterior differentiation, \n\\[\nd(\\mu \\omega) = d\\mu \\wedge \\omega + (-1)^0 \\mu d\\omega \n\\]\nso that \n\\[\nd (\\mu \\omega) \\wedge \\omega = d\\mu \\wedge \\omega \\wedge \\omega + \\mu d\\omega \\wedge \\omega = 0 + \\mu d\\omega \\wedge \\omega = 0 + 0 = 0 \n\\]\n\n$\\omega$ nonzero, so $d(\\mu \\omega) =0$. EY : 20150221 I'm not sure about this statement. Surely, locally,\n\n\\[\nd(\\mu \\omega) \\wedge \\omega = \\frac{1}{2} ( d(\\mu \\omega) )_{ij} \\omega_k dx^i \\wedge dx^j \\wedge dx^k = d(\\mu \\omega)_{ \\underline{I}} \\omega_k dx^{\\underline{I}} \\wedge dx^k\n\\]\nwith $\\underline{I} = (i_1,i_2)$ and $i_1 < i_2$. \n\nBy considering every $k \\neq \\underline{I}$, then I think one can conclude, component by component, that $d(\\mu \\omega) =0$. \n\nThen, consider a compact submanifold $B$, $\\text{dim}{B} =3$ that is a submersion of $U$. Then use Stoke's theorem in the following:\n\n\\[\n\\int_B d(\\mu \\omega) = \\int_{\\partial B} \\mu \\omega = 0 \\Longrightarrow \\mu \\omega = df\n\\]\n\nSo \n\\[\n\\boxed{ \\begin{gathered} \\text{ If } \\omega \\neq 0 \\text{ on } U \\\\\n \\omega \\text{ admits an integrating factor $\\mu$ } \\text{ iff } d\\omega \\wedge \\omega =0 \\end{gathered} }\n\\]\n\nEY 20150221 : I didn't use Frobenius' theorem for the converse. Should I have?\n\\item[(b)] If $\\text{dim}{M}=2$, $d\\omega \\wedge \\omega =0$ (immediately) \\\\\nThen $\\omega $ admits an integrating factor by the above solution.\n\\end{enumerate}\n", "meta": {"hexsha": "567af46d98cfd842f617558a655e899ed6bba31a", "size": 9603, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "LeeJM/19DistributionsaFoliations.tex", "max_stars_repo_name": "wacfeldwang333/mathphysics", "max_stars_repo_head_hexsha": "59eb794dfa46e2b80e43df0440bb8ec3c472d973", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 50, "max_stars_repo_stars_event_min_datetime": "2017-01-10T14:24:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T11:19:23.000Z", "max_issues_repo_path": "LeeJM/19DistributionsaFoliations.tex", "max_issues_repo_name": "wacfeldwang333/mathphysics", "max_issues_repo_head_hexsha": "59eb794dfa46e2b80e43df0440bb8ec3c472d973", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2017-09-29T09:29:53.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-12T03:12:29.000Z", "max_forks_repo_path": "LeeJM/19DistributionsaFoliations.tex", "max_forks_repo_name": "wacfeldwang333/mathphysics", "max_forks_repo_head_hexsha": "59eb794dfa46e2b80e43df0440bb8ec3c472d973", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 25, "max_forks_repo_forks_event_min_datetime": "2018-01-21T05:33:31.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-03T20:15:13.000Z", "avg_line_length": 50.5421052632, "max_line_length": 300, "alphanum_fraction": 0.585129647, "num_tokens": 3259, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011686727231, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34455971863092255}} {"text": "\\documentclass[main.tex]{subfiles}\n\\begin{document}\n\n% \\section*{Wed Nov 06 2019}\n\n\\subsection{Non-isothermal winds}\n\nNow, we consider the possibility that our winds are \\emph{not isothermal}.\nThis will change the structure of the wind, by the introduction of an additional pressure gradient.\n\nIt will change the speed of sound and thus the Mach number.\n\nIt is useful to define the energy per unit mass \\(e\\): \n%\n\\begin{align}\n e(r) = \\frac{v^2(r)}{2} - \\frac{GM}{r} + \\frac{\\gamma }{\\gamma -1} \\frac{ \\mathcal{R} T}{\\mu }\n\\,,\n\\end{align}\n%\nwhere \\(\\gamma / (\\gamma -1) = 5/2\\) for a monoatomic gas, which has \\(\\gamma = 3/2\\).\n\nIn the lower boundary of the wind the velocity is much less than the escape velocity (\\(v \\ll v _{\\text{esc}}\\)), and also the thermal velocity of the particles is not enough for them to escape the gravitational well: \\(\\mathcal{R} T / \\mu \\ll v _{\\text{esc}}\\) at the surface of the star, while far from the star we have \\(v \\gg v _{\\text{esc}}\\).\n\nIf we have an isothemal wind, some energy must be added in order to prevent the adiabatic cooling of the gas, lift it from the potential well, and to increase its kinetic energy.\n\nIf a force is applied it increases the momentum, but the heat transmission \\(q\\) also appears in the momentum equation, which is \\(\\Delta e = \\int f + q \\dd{r} \\).\nHeat transmission changes the pressure profile, which affects the momentum, even though \\(q\\) does not appear explicitly in the momentum equation.\n\nWe define the total heat deposition \\(Q\\) and the total work done by the force \\(W\\):\n%\n\\begin{align}\nQ(r) = \\int_{r_0 }^{r} q(\\widetilde{r}) \\dd{\\widetilde{r}}\n\\qquad \\text{and} \\qquad\nW(r) = \\int_{r_0 }^{r} f(\\widetilde{r}) \\dd{\\widetilde{r}}\n\\,,\n\\end{align}\n%\nand we will have \\(e( \\infty ) - e(r_0 ) = Q( \\infty ) + W( \\infty )\\). \n\nThe most general momentum equation is given by: \n%\n\\begin{align}\n \\frac{1}{v} \\dv{v}{r} = \\qty(2 \\frac{c_s^2}{r} - \\frac{GM}{r^2} + f -(\\gamma - 1) q) / \\qty(v^2- c_s^2)\n\\,,\n\\end{align}\n%\nwhere we introduce the adiabatic speed of sound \\(c_s = \\sqrt{\\gamma a^2} \\), \\(a\\) being the \\emph{isothermal} speed of sound.\nIn general \\(- (\\gamma - 1)<0\\), therefore if we add heat this is equivalent to pushing \\emph{inward}.\n\nIf either \\(f\\) or \\(q\\) depend on the velocity gradient \\(\\dv*{v}{r}\\) then the sonic point can \\emph{decouple} from the critical point.\n\nThere are cases in which we have multiple critical points (specifically, multiple zeros of the denominator).\n\nThe momentum equation plus the energy equation \n%\n\\begin{align}\n \\dv{}{r} \\qty(\\frac{v^2}{2} + \\frac{5}{2} \\frac{RT}{\\mu } - \\frac{GM}{r}) = f(r) + q(r)\n\\,,\n\\end{align}\n%\ncan be solved numerically, and if we impose smooth passage through the critical point this yields the mass loss rate.\n\nQualitatively, the results are the same as in the isothermal case.\nAdding either momentum or energy to the subsonic region of the wind increases the bottom-of-the-corona velocity and the mass loss rate. Doing it in the supersonic region has no effect.\n\nThis ends our general introduction to stellar winds.\n\n\\subsection{Exercises}\n\nNow we will do a couple of exercises to get familiar with the theory. \n\n\\begin{greenbox} \\textbf{Exercise}\n\nThe wind is isothermal.\nThe solar wind has a mean coronal temperature of \\SI{1.5e6}{K} and a mass loss rate of \\num{2e-14} solar masses per year.\nThe bottom of the corona is at \\(r_0 \\approx 1.003 R_{\\odot}\\), where the density is \\(\\rho (r_0 ) = \\SI{e-14}{g\\per\\centi\\metre\\cubed}\\).\n\nCalculate the potential energy, the kinetic energy and the enthalpy of the gas at \\(r_0 \\).\n\nCalculate the same quantities at the critical point. Which of these energies has absorbed the largest fraction of the energy input?\n\\end{greenbox}\n\n\\begin{bluebox}\nWe can use the continuity equation \\(\\dot{M} = 4 \\pi \\rho_0 r_0^2 v_0 \\) to get \n%\n\\begin{align}\n v_0 = \\frac{\\dot{M}}{4 \\pi r_0^2 \\rho_0 }\n \\approx \\SI{21}{m/s}\n\\,,\n\\end{align}\n%\nand with this we can calculate \n%\n\\begin{align}\n e(r_0 ) = - \\frac{GM}{r_0 } + \\frac{1}{2} v_0^2 + \\frac{5}{2} \\frac{RT}{\\mu }\n\\,.\n\\end{align}\n\nWe get: \n%\n\\begin{align}\n E _{\\text{kin, 0}} = \\frac{v_0^2}{2} \\approx \\SI{212}{J/kg}\n\\,,\n\\end{align}\n%\nwhile for the gravitational energy we'd need the mass of the star. Assuming it is equal to the solar mass, we find \n%\n\\begin{align}\n E _{\\text{grav, 0}} = -\\frac{GM}{r_0 } \\approx \\SI{-1.9e11}{J/kg}\n\\,.\n\\end{align}\n%\n\nThe mean molecular weight of the gas for the Sun is something like \\(\\mu = 0.62\\) (we count electrons in it).\nThen we get \n%\n\\begin{align}\n E _{\\text{chem, 0}} = E _{\\text{\\text{chem, crit}}} = \\frac{5}{2} \\frac{RT}{\\mu } \\approx \\SI{5.0e7}{J/kg} \n\\,.\n\\end{align}\n%\n\nThe enthalpy is the same everywhere in the flow, since the flow is isothermal.\n\nThe values at the critical radius are calculated with the same formula. The velocity will be the speed of sound \\(a = \\sqrt{RT / \\mu } \\approx \\SI{4.5e3}{m/s} \\).\n\nThen we find: \n%\n\\begin{align}\n E _{\\text{kin, crit}} = \\frac{a^2}{2} \\approx \\SI{1.0e7}{J/kg} \\approx \\num{5e4} \\times E _{\\text{kin, 0}} \n\\,.\n\\end{align}\n%\n\nThe critical radius is given by \\(r_c = GM / (2 a^2)\\): so, we get \n%\n\\begin{align}\n E _{\\text{grav, crit}} = - \\frac{GM}{r_{c}} = -2 a^2 \\approx - \\SI{4.0e7}{J/kg} \\approx \\num{2.1e-4} \\times E _{\\text{grav, 0}}\n\\,.\n\\end{align}\n%\n\nQualitatively, at the corona we have \n%\n\\begin{align}\n \\abs{ E _{\\text{grav, 0}}} \\gg E _{\\text{chem, 0}} \\gg E _{\\text{kin, 0}}\n\\,,\n\\end{align}\n%\nwhile at the critical point they are similar, and specifically \n%\n\\begin{align}\n E _{\\text{chem, crit}} \\gtrsim \\abs{E _{\\text{grav, crit}}} \\gtrsim E _{\\text{kin, crit}}\n\\,.\n\\end{align}\n%\n\n\\end{bluebox}\n\n\\begin{greenbox}\n \\textbf{Exercise}\n\n A star with \\(T _{\\text{eff}} = \\SI{3200}{K}\\), \\(R_{*} = 30 R_{\\odot}\\), \\(L_{*} = 85 L_{\\odot}\\) and \\(M_{*} = 6 M_{\\odot}\\) has an isothermal corona of \\(T = \\SI{e6}{K}\\) with a density at the lower boundary of \\SI{e-13}{g \\per\\centi\\metre\\cubed}.\n\n Calculate the energy per unit mass at the bottom of the corona at \\(r_0 = R_{*}\\).\n\n Calculate the location of the critical point, \\(r_c\\), and the mass loss rate.\n\n Calculate the energy per gram gained by the wind between \\(r_0 \\) and \\(r\\). What fraction of the stellar luminosity is used to drive the wind up to the critical point?\n\\end{greenbox}\n\n\\begin{bluebox}\n The energy per unit mass at the bottom of the corona is given by \n %\n \\begin{align}\n e(r_0 ) = - \\frac{GM}{r_0 } + \\frac{1}{2} v_0^2 + \\frac{5}{2} \\frac{RT}{\\mu }\n \\,,\n \\end{align}\n\n where we have everything but \\(v_0 \\): we can calculate it by imposing that the solution passes through the critical point, the formula we found for it is \n %\n \\begin{align}\n v_0 = a \\qty(\\frac{r_c}{r_0 })^2 \\exp(- 2 \\frac{r_c}{r_0 } + \\frac{3}{2}) \\approx \\SI{61}{km/s}\n \\,.\n \\end{align}\n %\n \n Using this figure, we can find \n %\n \\begin{align}\n e(r_0 ) &\\approx \\SI{-2.76e9}{m^2/s^2} \\\\ \n e(r_c ) &\\approx \\SI{1.34e10}{m^2/s^2}\n \\,,\n \\end{align}\n %\n where we assumed \\(\\mu = \\num{.62}\\) since the matter is ionized, and computed the energy density for both the bottom of the corona and the critical point \\(r_c = GM/2a^2 \\approx \\SI{2.97e10}{m} \\approx 1.4 R_{*}\\).\n\n The isothermal speed of sound is given by \\(a = \\sqrt{RT/\\mu } \\approx \\SI{116}{km/s}\\). \n\n The mass loss rate can be computed at the bottom of the corona: it is given by \n %\n \\begin{align}\n \\dot{M} = 4 \\pi r_0 ^2 v_0 \\rho_0 \\approx \\SI{3.34e16}{kg/s}\n \\,,\n \\end{align}\n %\n therefore we can compute the power needed to accelerate the wind: \n %\n \\begin{align}\n P = \\dot{M} \\qty(e(r_c) - e(r_0 )) \\approx \\SI{5.40e26}{J/s} \\approx \\SI{1.66}{\\percent} L_{*}\n \\,,\n \\end{align}\n %\n \n %\n% but we cannot use this formula since we do not have \\(v_0 \\) nor \\(\\dot{M}\\). However, we can approximate the density profile as an exponential, applying the formula \n% %\n% \\begin{subequations}\n% \\begin{align}\n% \\dot{M} &= 4 \\pi r_c^2 a \\rho_{c} \\\\\n% &= 4 \\pi r_c^2 a \\rho_0 \\exp(- \\frac{r_c - r_0 }{H_0 } \\frac{r_0 }{r_c}) \n% \\,,\n% \\end{align}\n% \\end{subequations}\n% %\n% where we have: the length scale \\(H_0 = RT r_0^2 / (GM \\mu )\\), the critical velocity \\(a = \\sqrt{RT/\\mu }\\), and the critical radius \\(r_c = GM/(2a^2)\\).\n\n% Plugging these in, we find: \n% %\n% \\begin{subequations}\n% \\begin{align}\n% \\dot{M} &= 4 \\pi \\qty(\\frac{GM \\mu }{2 RT})^2 \\sqrt{\\frac{RT}{\\mu }} \\rho_0 \n% \\exp(\\frac{\\mu GM}{RT} \\qty(\\frac{2RT}{GM \\mu } - \\frac{1}{r_0 })) \\\\\n% &= \\pi (GM)^2 \\qty(\\frac{\\mu }{RT})^{3/2} \\rho_0 \n% \\exp(2 - \\frac{\\mu GM}{RT r_0 })\n% \\,.\n% \\end{align}\n% \\end{subequations}\n% %\n% \\todo[inline]{One temperature should probably be the effective temperature, but I do not really know what that means. }\n\n% Then, the mass loss rate can be calculated, since we have all of these quantities. It comes out to be barely anything, since \\(\\mu GM / RT r_0 \\) is very large and we have a negative exponential of it\\dots\n\n\\end{bluebox}\n\n% Lamers-Cassinelli: chap 4, section 3: multiple critical points.\n\n% Next Tuesday, the 12th, after the lecture, we will likely move to the DFA in Via Marzolo until 14.30.\n\n\\section{Wind types}\n\n\\paragraph{Coronal winds}\nThese are driven by gas pressure due to high temperature.\nStars with a convection zone right under the photosphere can have coronas of a few million Kelvin degrees. \nThese are rather well described as isothermal, however the temperature decreases slowly because of conduction.\n\n\\paragraph{Dust driven winds}\nFor these winds, the driving mechanism is the radiation pressure on the dust grains, which are continuum absorbers. The gas component is dragged along by momentum transfer. \nThis happens for cool stars, whose envelopes have temperatures of less than a thousand degrees Kelvin, in which the dust grains can actually be formed. \nThese can be well modelled as winds with a force like \\(f = [r \\geq r_d] A / r^2\\), where the radius \\(r_d\\) is the one after which the temperature is low enough for the grains to form.\n\n\\paragraph{Line-driven winds}\nThese are winds of hot stars, driven by radiation pressure on spectral lines of abundant ions which have many in the UV and far UV.\nThe pressure depends on the Doppler effect: the ions can absorb different wavelengths of light at different radii because of it.\nThe force depends on the velocity gradient: it cannot be modelled with what we discussed so far.\n\n\\paragraph{Pulsation driven winds}\nStars such as Miras and those in the Asymptotic Giant Branch may pulsate: the atmosphere is tossed up and then falls back. Because of the low gravity, they fall slowly and are hit by an outmoving layer before they have completed their fall. \nThis means that for each pulsation cycle they get a ``kick''. \n\nThis can be a very efficient driving method if we account for dust formation. The shockfront from the pulsation cycle basically travels outwards to infinity.\n\n\\paragraph{Sound wave driven winds}\nThey are modelled in a way that is similar to coronal winds. \nThe resulting amplitudes are usually small. \n\n\\paragraph{Alfvén Wave Driven winds}\nDue to magnetic fields: if the points at which the field lines make contact with the surface move, than a magnetic wave travels forward with a speed \\(v_A = B / \\sqrt{4 \\pi \\rho } \\gg a\\). \nThis can be very efficient, and result in high wind speeds. \nIt is relevant for stars which do not have strong radiation pressure (ie not more than a thousand times more luminous than the Sun). \n\n\\paragraph{Magnetic rotating winds}\nMaterial is ``flung out'' moving along magnetic field lines. \nThe terminal velocity is heavily dependent on the strength of the field.\n\n\\section{Line driven winds}\n\nNow, we introduce the next topic.\nWe deal with hot, luminous stars: at the top left of the HR diagram.\n\n[Picture from the slides]\n\nWe talk of line-driven winds for hot stars.\nThese are winds driven by \\emph{spectral lines}. \nThe blackbody emission for these stars is mainly at high frequencies, like the UV.\n\nOne may ask: hydrogen is much more abundant than heavier elements like carbon, nitrogen\\dots why do we see the spectral lines for these heavier elements?\n\nThis is because hydrogen is completely ionized at these temperatures, and helium is also.\n\nThe strongest lines are far in the UV, where the stellar flux is low: only few atoms are hit, the rest of the gas is dragged along.\n\nA peculiar characteristic of the spectrum is the so-called P-Cygni profile.\n\nNow, an overview of the formation of spectral lines:\nthere are 5 processes \\begin{enumerate}\n \\item Line scattering: if it comes from the ground state of the atom, it is called \\emph{resonance scattering}, which is the main phenomenon.\n \\item Emission by recombination: the ion recombines to an excited state.\n \\item Emissional from collisional or photo-excitation. A photon is absorbed, it excites the atom which then descends to a lower energy level.\n \\item Pure absorption and then de-excitation of an already excited atom.\n \\item Masering by stimulated emission: it can happen in a very narrow set of circumstances: an excited atom is hit by a photon which has exactly the same energy as the one between the atom's state and the ground state, so the atom is deexcited and now there are two photons.\n This happens when there are many excited atoms, and when the velocity gradient is very small --- otherwise, the photons are Doppler-shifted out of the right frequency.\n\\end{enumerate}\n\n\n\\end{document}", "meta": {"hexsha": "da193ab535cda447608790147b6cc5317d5ac4c3", "size": 13442, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ap_first_semester/advanced_astrophysics/06nov.tex", "max_stars_repo_name": "jacopok/notes", "max_stars_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-10-10T13:10:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-13T14:52:50.000Z", "max_issues_repo_path": "ap_first_semester/advanced_astrophysics/06nov.tex", "max_issues_repo_name": "jacopok/notes", "max_issues_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "ap_first_semester/advanced_astrophysics/06nov.tex", "max_forks_repo_name": "jacopok/notes", "max_forks_repo_head_hexsha": "805ebe1be49bbd14c6b46b24055f9fc7d1cd2586", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2019-10-03T16:20:19.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-06T16:11:07.000Z", "avg_line_length": 42.5379746835, "max_line_length": 348, "alphanum_fraction": 0.6961017706, "num_tokens": 4130, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.34455971010999603}} {"text": "\\documentclass{article}\n\n\\usepackage{luatexja}\n\n\\usepackage{url}\n\\usepackage{amssymb,amsmath,stmaryrd,bcprules,xspace}\n\\usepackage{etoolbox,xcolor,here}\n\n\\usepackage[top=0pt,bottom=0pt,left=2cm,right=2cm]{geometry}\n\n\\input{macro.tex}\n\n\\author{nymphium}\n\\title{\\lambdaeff}\n\n\\begin{document}\n\\thispagestyle{empty}\n\\maketitle\n\n\\section*{Syntax}\n\\begin{figure}[ht]\n \\[\n \\begin{array}{rrl}\n x & \\in & \\variable{Variables}\\\\\n \\variable{eff} & \\in & \\variable{Effects}\\\\\n &&\\\\\n v & ::= & x \\mid h \\mid \\lambda x. e \\mid \\variable{eff}\\\\\n e & ::= & v \\mid e\\ e \\mid \\letT{x}{e}{e}\\\\\n &\\mid & \\instT \\mid \\withhandle{v\\ }{e} \\\\\n &\\mid & \\perform{\\variable{e\\ }{e}}\\\\\n h & ::= & \\handlerK v\\ \\left(\\keyword{val} x \\rightarrow e\\right) \\left(\\left(x, k\\right)\\rightarrow e\\right)\\\\\n &&\\\\\n F & ::= & e\\ \\square \\mid \\square\\ v \\mid \\letT{x}{\\square}{e} \\\\\n &\\mid & \\withhandle{v\\ }{\\square} \\mid \\perform{\\square\\ }{e} \\mid \\perform{\\variable{v\\ }}{\\square}\\\\\n s &::= & \\left[\\right] \\mid F :: s\n \\end{array}\n \\]\n \\caption{the syntax of \\lambdaeff}\n \\label{fig:lambdaeff}\n\\end{figure}\n\n\\section*{Semantics}\n\\newcommand{\\flatfn}[1]{%\n \\variable{flatfn}\\ #1\n}\n\n\\begin{figure}[ht]\n \\begin{flalign*}\n \\flatfn{\\left[\\right]} &= \\lambda x. x\\\\\n \\flatfn{\\B{F :: s}} &= \\lambda x. \\flatfn{s}\\ \\B{F\\left[x\\right]}\n \\end{flalign*}\n\n \\caption{utility function \\variable{flatfn}}\n\\end{figure}\n\n\\begin{figure}[ht]\n \\newcommand{\\defspace}{\\vskip.5\\baselineskip}\n\n \\centering\n\n \\infax[Push]{\\model{F\\left[e\\right]}{s}{es} \\mapsto \\model{e}{F :: s}{es}}%\n \\defspace\n\n \\infax[Pop]{\\model{v}{F :: s}{es} \\mapsto \\model{F\\left[v\\right]}{s}{es}}%\n \\defspace\n\n \\infax[Result]{\\model{v}{[]}{es} \\mapsto \\model{v}{[]}{es}}%\n \\defspace\n\n \\infax[Apply]{\\model{\\lambda x. e}{\\left(\\square\\ v\\right) :: s}{es} \\mapsto \\model{e\\left[x = v\\right]}{s}{es}}%\n \\defspace\n\n \\infax[Instanciate]{\\model{\\instT}{s}{es} \\mapsto \\model{\\variable{eff}}{s}{es}}%\n \\defspace\n\n \\infax[Rethrow]{%\n \\model{\\perform{\\variable{eff}}{v}}{F :: s}{es}%\n \\mapsto%\n \\model{\\perform{\\variable{eff}{v}}}{s}{F :: es}}%\n \\defspace\n\n \\infax[Handle\\textsubscript{eff}]{%\n \\begin{array}{l}\n \\model{\\perform{\\variable{eff}}{v}}{F :: s}{es}%\n \\mapsto%\n \\model{e_\\variable{eff}\\left[x = v, k = \\flatfn{es}\\right]}{F :: s}{\\left[\\right]}\\\\%\n \\begin{array}{ll}\n \\mathrm{where} & F = \\withhandle{h\\ }{\\square} \\\\\n & h = \\handlerK \\variable{eff} \\left(\\valK x \\rightarrow e_v\\right) \\left(\\left(x, k\\right) \\rightarrow e_\\variable{eff}\\right)\n \\end{array}\n \\end{array}}%\n \\defspace\n\n \\infax[Handle\\textsubscript{v}]{%\n \\model{v}{F :: s}{es}%\n \\mapsto%\n \\model{e_v[x = v]}{s}{es}\\\\\n \\begin{array}{ll}\n \\mathrm{where} & F = \\withhandle{h\\ }{\\square} \\\\\n & h = \\handlerK \\variable{eff} \\left(\\valK x \\rightarrow e_v\\right) \\left(\\left(x, k\\right) \\rightarrow e_\\variable{eff}\\right)\n \\end{array}\n }\n \\defspace\n\n \\infax[Leak]{%\n \\model{\\perform{\\variable{eff}}{v}}{\\left[\\right]}{es}%\n \\mapsto%\n \\textsl{abort}}%\n\\caption{the semantics of \\lambdaeff}\n\\end{figure}\n\\end{document}\n", "meta": {"hexsha": "91ec848ea503e5fcd4c501b8eff9677f939f1e88", "size": 3442, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "text/main.tex", "max_stars_repo_name": "Nymphium/lambdaeff", "max_stars_repo_head_hexsha": "afaa9736c6151a969884c3bc413c0b5f03e3ff57", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-10-01T11:26:04.000Z", "max_stars_repo_stars_event_max_datetime": "2020-10-01T11:26:04.000Z", "max_issues_repo_path": "text/main.tex", "max_issues_repo_name": "Nymphium/lambdaeff", "max_issues_repo_head_hexsha": "afaa9736c6151a969884c3bc413c0b5f03e3ff57", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "text/main.tex", "max_forks_repo_name": "Nymphium/lambdaeff", "max_forks_repo_head_hexsha": "afaa9736c6151a969884c3bc413c0b5f03e3ff57", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.7321428571, "max_line_length": 154, "alphanum_fraction": 0.5435793144, "num_tokens": 1214, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34455971010999603}} {"text": "\\documentclass[12pt,preprint]{aastex}\n\n\\pdfoutput=1\n\n\\usepackage{color,hyperref}\n\\definecolor{linkcolor}{rgb}{0,0,0.5}\n\\hypersetup{colorlinks=true,linkcolor=linkcolor,citecolor=linkcolor,\n filecolor=linkcolor,urlcolor=linkcolor}\n\\usepackage{url}\n\\usepackage{amssymb,amsmath}\n\\usepackage{subfigure}\n\\usepackage{booktabs}\n\n\\usepackage{natbib}\n\\bibliographystyle{apj}\n\n\n\\newcommand{\\project}[1]{\\textsl{#1}}\n\\newcommand{\\kepler}{\\project{Kepler}}\n\\newcommand{\\terra}{\\project{TERRA}}\n\\newcommand{\\license}{MIT License}\n\n\\newcommand{\\paper}{\\textsl{Article}}\n\n\\newcommand{\\foreign}[1]{\\emph{#1}}\n\\newcommand{\\etal}{\\foreign{et\\,al.}}\n\\newcommand{\\etc}{\\foreign{etc.}}\n\\newcommand{\\True}{\\foreign{True}}\n\\newcommand{\\Truth}{\\foreign{Truth}}\n\n\\newcommand{\\figref}[1]{\\ref{fig:#1}}\n\\newcommand{\\Fig}[1]{\\figurename~\\figref{#1}}\n\\newcommand{\\fig}[1]{\\Fig{#1}}\n\\newcommand{\\figlabel}[1]{\\label{fig:#1}}\n\\newcommand{\\Tab}[1]{Table~\\ref{tab:#1}}\n\\newcommand{\\tab}[1]{\\Tab{#1}}\n\\newcommand{\\tablabel}[1]{\\label{tab:#1}}\n\\newcommand{\\Eq}[1]{Equation~(\\ref{eq:#1})}\n\\newcommand{\\eq}[1]{\\Eq{#1}}\n\\newcommand{\\eqalt}[1]{Equation~\\ref{eq:#1}}\n\\newcommand{\\eqlabel}[1]{\\label{eq:#1}}\n\\newcommand{\\sectionname}{Section}\n\\newcommand{\\Sect}[1]{\\sectionname~\\ref{sect:#1}}\n\\newcommand{\\sect}[1]{\\Sect{#1}}\n\\newcommand{\\sectalt}[1]{\\ref{sect:#1}}\n\\newcommand{\\App}[1]{Appendix~\\ref{sect:#1}}\n\\newcommand{\\app}[1]{\\App{#1}}\n\\newcommand{\\sectlabel}[1]{\\label{sect:#1}}\n\n\\newcommand{\\dd}{\\ensuremath{\\,\\mathrm{d}}}\n\\newcommand{\\bvec}[1]{\\ensuremath{\\boldsymbol{#1}}}\n\\newcommand{\\densityunit}{{\\ensuremath{\\mathrm{nat}^{-2}}}}\n\n% TO DOS\n\\newcommand{\\todo}[3]{{\\color{#2} \\emph{#1} TODO: #3}}\n\\newcommand{\\dfmtodo}[1]{\\todo{DFM}{red}{#1}}\n\\newcommand{\\hoggtodo}[1]{\\todo{HOGG}{blue}{#1}}\n\n% Response to referee\n\\definecolor{mygreen}{rgb}{0, 0.50196, 0}\n\\newcommand{\\response}[1]{#1}\n% \\newcommand{\\response}[1]{{\\color{mygreen} {\\bf #1}}}\n\n% Document specific variables.\n\\newcommand{\\rate}{\\ensuremath{\\lambda}}\n\\newcommand{\\intrate}{\\ensuremath{\\Lambda}}\n\n\\begin{document}\n\n\\title{%\n Note about the Poisson Process for population inference\n}\n\n\\newcommand{\\uw}{2}\n\\newcommand{\\nyu}{3}\n\\author{%\n Daniel~Foreman-Mackey\\altaffilmark{1,\\uw,\\nyu}\n}\n\\altaffiltext{1} {Sagan Fellow; \\url{danfm@nyu.edu}}\n\\altaffiltext{\\uw} {Department of Astronomy, University of Washington,\n Seattle, WA, 98195, USA}\n\\altaffiltext{\\nyu} {Center for Cosmology and Particle Physics,\n Department of Physics, New York University,\n 4 Washington Place, New York, NY, 10003, USA}\n\n\\section{The Poisson Process}\n\nGiven a rate density function $\\rate(\\bvec{w})$, the Poisson probability of\nobserving $N$ events in some volume $\\bvec{w} \\in \\bvec{W}$ is\n\\begin{eqnarray}\\eqlabel{count}\np(N) &=& \\frac{\\intrate^N\\,e^{-\\intrate}}{N!}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\intrate &=& \\int_{\\bvec{W}} \\rate(\\bvec{w}) \\dd\\bvec{w}\n\\end{eqnarray}\nis the total rate in the volume.\n\nUnder this same model, the probability of an event at $\\bvec{w}_n$ is\n\\begin{eqnarray}\np(\\bvec{w}_n) &=& \\frac{\\rate(\\bvec{w}_n)}{\\intrate} \\quad.\n\\end{eqnarray}\nThen, once we have $N$ independent events $\\{\\bvec{w}_n\\}_{n=1}^N$, the\nprobability across the full sample is\n\\begin{eqnarray}\\eqlabel{ind}\np(\\{\\bvec{w}_n\\}) &=& N!\\,\\prod_{n=1}^N\\frac{\\rate(\\bvec{w}_n)}{\\intrate}\n\\end{eqnarray}\nwhere the $N!$ combinatoric term comes from the exchangeability of the\nsamples and it is important for the next step.\n\nCombining \\eq{count} and \\eq{ind}, we can derive the standard expression for\nthe Poisson Process likelihood function\n\\begin{eqnarray}\np(N,\\,\\{\\bvec{w}_n\\}) &=& p(N)\\,p(\\{\\bvec{w}_n\\}) \\\\\n&=& \\frac{\\intrate^N\\,e^{-\\intrate}}{N!} \\,\n N!\\,\\prod_{n=1}^N\\frac{\\rate(\\bvec{w}_n)}{\\intrate} \\\\\n&=& e^{-\\intrate} \\, \\prod_{n=1}^N\\rate(\\bvec{w}_n) \\quad.\n\\end{eqnarray}\n\n\\section{Mixture of Poisson Processes}\n\nLet's say that each point $n$ in our sample is generated from one of $K$\ndistinct populations with underlying rate densities $\\rate_k(\\bvec{w})$.\nIn this case, the joint likelihood across the full sample can be written as\n\\begin{eqnarray}\np(N,\\,\\{\\bvec{w}_n\\}\\,|\\,\\{q_n\\}) &=&\n \\exp\\left(-\\sum_{k=1}^K\\intrate_k\\right) \\,\n \\prod_{n=1}^N\\prod_{k=1}^K\n \\left[\\rate_k(\\bvec{w}_n)\\right]^{\\bvec{1}(q_n = k)}\n\\end{eqnarray}\nwhere $q_n$ is the class of object $n$ and $\\bvec{1}(\\cdot)$ is the indicator\nfunction that equals one if the argument is true and zero otherwise.\nThis equation can be easily derived by splitting the sample into the $K$\nsubsamples, writing the Poisson Process likelihood for each subsample, and\ntaking the product.\n\nNow, we will relax our assumptions and allow for the fact that we don't know\nthe class memberships $\\{q_n\\}$ \\foreign{a priori}.\nInstead, we can choose a prior over class memberships\n\\begin{eqnarray}\np(q_n) &=& \\prod_{k=1}^K {Q_k}^{\\bvec{1}(q_n = k)}\n\\end{eqnarray}\nwhere $\\sum_k Q_k = 1$, and marginalize over choice of class for each sample.\nDoing this, we find\n\\begin{eqnarray}\np(N,\\,\\{\\bvec{w}_n\\}) &=&\n \\exp\\left(-\\sum_{k=1}^K\\intrate_k\\right) \\,\n \\prod_{n=1}^N\\sum_{k=1}^K\n Q_k\\,\\rate_k(\\bvec{w}_n)\n\\end{eqnarray}\n\n\n% \\begin{figure}[p]\n% \\begin{center}\n% \\includegraphics[width=\\textwidth]{figures/smooth/results.pdf}\n% \\end{center}\n% \\caption{%\n% {\\bf Simulated data}.\n% Inferences about the rate density based on the simulated catalog \\modela.\n% \\emph{Center:} the points with error bars show the exoplanet candidates in the\n% simulated incomplete catalog, the contours show the survey completeness\n% function (\\citealt{petigura}), and the grayscale shows the median posterior\n% occurrence surface.\n% \\emph{Top and left:} the red dashed line shows the true distribution that was\n% used to generate the catalog, the points with error bars show the results of\n% the inverse-detection-efficiency procedure, and the histograms are posterior\n% samples from the marginalized rate density as inferred by our method.\n% \\figlabel{smooth-results}}\n% \\end{figure}\n\n\\end{document}\n", "meta": {"hexsha": "d649cba7c1d3cb80a77804aad83a6ae4ca1ea7dc", "size": 6081, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "document/poisson.tex", "max_stars_repo_name": "dfm/exopop2", "max_stars_repo_head_hexsha": "fb67b4bab39a7c77811548f365690da159fb88d7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "document/poisson.tex", "max_issues_repo_name": "dfm/exopop2", "max_issues_repo_head_hexsha": "fb67b4bab39a7c77811548f365690da159fb88d7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "document/poisson.tex", "max_forks_repo_name": "dfm/exopop2", "max_forks_repo_head_hexsha": "fb67b4bab39a7c77811548f365690da159fb88d7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 35.3546511628, "max_line_length": 80, "alphanum_fraction": 0.6883736228, "num_tokens": 2073, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34455971010999603}} {"text": "\\chapter{Introduction} \\label{sec:intro}\nPositioning assets, such as facilities and equipment, within a pre-defined region, such as a plot of land or a building, in a fashion that is tailoured towards a criteria of optimality for a specific problem is one endeavour that has multiple applications in different fields, primarily due to the benefits it provides. Finding the best possible asset positioning can result in improved operations efficiency, better productivity \\cite{El-Baz2004}, and even decreases in expenses \\cite{Sahin2011}. As a matter of fact, due to the benefits of asset positioning, \\$300 billion dollars have been spent each year on just determining suboptimal locations of buildings and facilities in the United States alone \\cite{Asl2015a}. This is further proof of the importance of asset positioning. One entertaining example of said application is showcased by Barriga et al. (2014). In their paper, the authors developed a genetic algorithm that optimized placement of buildings in a StarCraft match. The algorithm produced building placements that allowed the defending player's base to better survive base assaults from the opposing player \\cite{Barriga2014}. Developing an open-plan office layout is another application of asset positioning. Chen et al. (2020) also developed a genetic algorithm that generates an open-office layout where the space utilization is maximized as possible \\cite{Chen2020}. This task of arranging assets within a given space according to some criteria has a formal term, which is the \"facility layout problem\", often abbreviated as \"FLP\". We will be discussing facility layout problems in more detail in this chapter.\n\nFLP is a field that has been researched as early as 1957 (with Koopsman T.C., and Beckman, M. being the first to model the problem) \\cite{Kusiak1987}, and there is still active research around it to this day. This research paper is one of the testaments to that. In this research, we will be solving the classical facility layout problem using a recent optimization algorithm called the Grey Wolf Optimization (GWO) algorithm. The specific type of FLP that we will solve is called the unequal area static FLP. The categorization will be discussed later in this paper. The proposed algorithm will then be compared to a genetic algorithm using experimental data used in other related papers.\n\n\\section{Facility Layout Problem}\nThe problem of arranging a set of facilities and/or machines in a pre-determined area, or a set of possible locations (such as in the work of Farmakis, P., and Chassiakos, A. \\cite{Farmakis2018}) is called the facility layout problem (FLP). The facilities and/or machines are arranged in such a way that the resulting layout is in line with some criteria or objectives and under certain constraints. These constraints, which must not be violated, include shape, size, orientation, pick-up/drop-off points \\cite{Hosseini-Nasab2018}, and usable area \\cite{Fernando2015}. Facilities and/or machines must also not overlap. Solutions that satisfy the aforementioned conditions are called feasible solutions \\cite{Meller1996}.\n\nGenerally, the facility layout problem is considered to be an \\textbf{NP-Hard} problem \\cite{Drira2007}. Hosseini-Nasab, H., Fereidouni, S., and Fatemi, S. have noted in their systematic review of FLP that most researches dealing with the facility layout problem model their problems either as a quadratic assignment problem (QAP) or a mixed integer programming problem \\cite{Hosseini-Nasab2018}. According to Drira, A., Pierreval, H., and Hajri-Gabouj, S., the former is sometimes used in discrete FLP formulations, while the latter is often used in continuous formulations \\cite{Drira2007}. Discrete and continuous FLP formulations will be discussed later. \\textbf{Quadratic assignment problems} deal with placing $n$ facilities in $n$ locations in such a way that minimizes the assignment cost. The assignment cost is the sum of all facility pairs's flow rate between each other multiplied by their flow rate \\cite{QAPDefinition}. This assignment cost is commonly seen in many FLP researches, as we will discuss later. QAP is also known to be an NP-Hard problem \\cite{Garey1979}. It should be noted though that \\textit{some} instances of QAP are easy to solve \\cite{Feizollahi2015}. The other modeling framework, \\textbf{mixed integer programming}, can solve problems with both discrete decisions and continuous variables. An example of such problem is the assignment problem \\cite{Richards2005}, which the FLP can be classified under. In this formulation, a set of integer and real-valued integers are being optimized based on an objective function that is being minimized or maximized, while satisfying constraints which are linear equations or inequalities \\cite{Wolsey2008}. Mixed integer programming, when in the context of optimization, is also known to be NP-Hard \\cite{Richards2005}. These two formulations being known to be generally NP-Hard proves that FLP is indeed generally NP-Hard.\n\nThe fact that FLP is an NP-Hard problem has resulted in many research works that utilize heuristics (such as simulated annealing and genetic algorithms). Note that there are also works that utilize exact methods, which seek to find the \\textit{optimal} solution for a problem. However, the NP-Hard nature of the FLP prevents them from finding the solution in large problems within reasonable time \\cite{Asl2015}.\n\n\\subsection{The Basic Mathematical Model}\nEach problem instances of the facility layout problem naturally will have their own mathematical models tailor-fit for their problem instance. Nevertheless, based on our observations and from readings, most of those models are derivatives of or use (such as in \\cite{Garcia-Hernandez2013}, \\cite{Lin2019}, and \\cite{Navarro2016}) what will be calling a basic minimization function, which is defined as:\n\n$$\n\\text{min} F = \\sum_{i=1}^{n}\\sum_{j=1}^{n}c_{ij}f_{ij}d_{ij}\n$$\n\nwhere $N$ is the number of facilities, $c_{ij}$ is the cost of handling materials between locations $i$ and $j$, $f_{ij}$ is the flow rate between $i$ and $j$, and $d_{ij}$ is the distance between the centroids of $i$ and $j$. The distance function may differ from work to work. For example, Liu, J., et. al. uses the Manhattan distance in their work \\cite{Liu2018}, while in the work of Ripon, K. S. N., et. al., Euclidean distance was used \\cite{Ripon2013}. In works that derive from this formula, such as in \\cite{Farmakis2018}, \\cite{Solimanpur2008}, and \\cite{Peng2018}, it was observed that $d_{ij}$, or a similar variable or expression, is commonly present in the work's objective function while $c_{ij}$ and $f_{ij}$ \\textit{may} be present and/or the work uses more or fewer variables.\n\nDrira, A., Pierreval, H., and Hajri-Gabouj, S. note the same observation but showcase a slightly differing formula in their 2007 survey of facility layout problems. Unlike the basic minimization formula above, their formula has $f_{ij}$ and $c_{ij}$ combined. They also note that the function above is typically used in continuous formulations of the FLP. The discrete formulation uses a similar function, but ensures that a facility is only in one location, a location only contains one facility, and makes sure that only pairs of locations that contain facilities contribute to the fitness value of a solution. (The descriptions and the differences of the discrete and continuous formulations are discussed in the next section.) Additionally, they mention that the function is also subject to the following constraints: (1) facilities must obviously not overlap with one another, and (2) the total area used by the facilities must be equal to or less than the allotted area \\cite{Drira2007}. These constraints have been observed to be generally in many FLP works.\n\n\\subsection{Discrete vs Continuous Formulations}\nSolving instances of the facility layout problem requires determining the form of the solution. The form is highly dependent on the problem being solved. Some problems may require a solution that assigns assets to pre-existing locations, while others may require more flexibility. Facility layout problems may be categorized based on the characteristics of these solutions, or formally known as formulations: discrete, and continuous.\n\nIn a discrete formulation, the region where the facilities will be laid out are divided into equal rectangular blocks of the same shape and size, or have pre-determined possible facility locations \\cite{Drira2007}. Each facility will be given a number of blocks, or be assigned to one facility location, respectively. This formulation, however, does not suit well when the facilities require exact positions and it cannot model facility attributes such as orientation. In problems that have such requirements, a continuous formulation is more appropriate \\cite{Hosseini-Nasab2018}. Facilities in a continuous formulation are usually located by either their centroid coordinates, half length, and half width, or by their bottom-left coordinates, length, and width \\cite{Drira2007}. This allows for the formulation's flexibility compared to its discrete counterpart. However, this does provide challenges towards ensuring that no two facilities overlap with one another. Discrete formulations do not need to consider this problem due to their inherent characteristics.\n\n\\subsection{Static vs Dynamic Facility Layout Problems}\nAnother categorization for facility layout problems is based on whether the layouts will change over time. There are situations where a regular change of layout over some periods of time is necessitated. The layout of facilities in a construction is one example. As the construction of a building moves to from phase to another, the layout of facilities within the construction site change to better fit the needs of the current phase of construction \\cite{Farmakis2018}. A similar need is the motivation behind changing layouts in manufactories. Product demand variations, and even a change in product design can incline a factory's management to reorganize facilities in the building to be more efficient in response to the changes \\cite{Pourhassan2017}. There are two categories for the aformentioned criteria. These are: (1) static, and (2) dynamic. We will refer to these categories as \\textbf{\"period-based layout categories\"} in this paper.\n\nThe survey of Hoisseini-Nasab et al. (2018) showed that the most common period-based categorization in literature is the static facility layout problem \\cite{Hosseini-Nasab2018}. This is likely due to the fact that static facility layout problems are easier to solve than dynamic facility layout problems. Though, it is also possible that many problems just happen to not require consideration of variable changes over time. The \\textbf{static facility layout problem}, abbreviated as SFLP, is a type of FLP where variables to be considered such as material handling costs do not change for a considerable amount of time \\cite{Perez-Gosende2020}. For this type of problems, only a single layout is generated since no changes are made in the considered variables over time.\n\nHowever, some industries will find SFLPs inadequate for their needs. There are companies that require adaptability to changes to, for example, product demands. For cases like this, the other category, dynamic facility layout problems are more appropriate \\cite{DerakhshanAsl2017}. In the dynamic facility layout problem, abbreviated to DFLP, the variables to be considered change over time, unlike in SFLP. The cost of rearranging facilities are also considered in the problem \\cite{Hosseini2016}. The solutions for DFLPs are also divided into time periods, where each period has a different layout. This period may equate to years, seasons, months, or weeks \\cite{DerakhshanAsl2017}. DFLPs can also be viewed as extensions of SFLP, since each layout in a period can be viewed as a solution to an SFLP with that period's variables into consideration but with rearrangement costs considered. While most research today is focused on SFLPs, Hosseini-Nasab et al. (2018) recommends that research should deal with DFLPs more these days due to rapid scientific developments, and product changes \\cite{Hosseini-Nasab2018}.\n\n\\subsection{Other FLP Classifications}\nFacility layout problems can also be categorized based on different characteristics. FLPs can be divided by the area of their facilities. The facilities may have the same areas, referred to as equal areas, or have different areas, referred this time to as unequal areas \\cite{DerakhshanAsl2017}. They can also be divided based on the possible arrangements of facilities. Some problems may have facilities located only in a single pre-defined row (single-row), or they may be placed anywhere in the region (open field) \\cite{Drira2007}. There are multiple classifications for FLP and discussing them in this chapter would take long and dislocate the focus of this paper. Due to that, we would like to refer the reader to the papers of Drira et al. (2007) \\cite{Drira2007} and Hosseini-Nasab et al. (2018) \\cite{Hosseini-Nasab2018} for more information on FLP classifications.\n\n\\section{The Grey Wolf Optimization Algorithm}\nIn this paper, we will be using the Grey Wolf Optimization algorithm to solve the unequal-area static facility layout problem. As such, we will be introducing the algorithm here for us to gain a better understanding of the algorithm.\n\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics{./images/chap01-introduction/wolves.jpg}\n\t\\caption{The Grey Wolf Optimization algorithm was inspired from the behaviour of grey wolves. Pictured are white wolves, different from grey wolves, but why pass up the opportunity to add a meme in a research paper? Profeshonal.}\n\t\\label{wolves-meme}\n\\end{figure}\n\nThe Grey Wolf Optimization algorithm, abbreviated as GWO, was first conceived by Mirjalili et al. (2014) in 2014. The optimization algorithm is inspired from the hunting and social behaviour of grey wolves. There is a hierarchy in packs of wolves. Each category in the hierarchy have specific responsibilities. There are four categories: alpha ($\\alpha$), beta ($\\beta$), delta ($\\delta$), and omega ($\\omega$). Alpha wolves are responsible for making major decisions for the pack. Every wolf must follow the alpha. However, sometimes the alpha follows other wolves. The second in line is the beta, which ensures the discipline of the pack and advises the alpha. They also command other wolves and reinforces the alpha's commands. The lowest in the hierarchy are the omegas. They must follow the orders of the other wolves, and are the last to eat. Despite their low status, they are still crucial in the pack as their absence causes the pack to face internal fighting and problems. If a wolf is not an alpha, beta, nor omega, they are considered to a delta, the third category in the hierarchy. Deltas may act as scouts, sentinels, elders, hunters, or caretakers. They are also at a category higher than the omegas \\cite{Mirjalili2014} \\cite{Gupta2018}. As an interesting side note, the inspiration for Grey Wolf Optimization initially came from The Grey, a movie where survivors of a plane crash must survive, but a pack of grey wolves surround them \\cite{TheGreyPlotSummary2011}.\n\n\\subsection{Mathematical Model}\nThe mathematical model assumes the existence of a \"pack of wolves\". The number of wolves in this pack can be determined by the researcher. Each wolf of this a solution to the problem. We will be delving into the model more in this section, discussing about the model of leadership hierarchy, encircling, and hunting behaviour of grey wolves. The prey being hunted in this scenario is the best solution for a given problem \\cite{Mirjalili2014}.\n\n\\subsubsection{Leadership Hierarchy}\nSolutions are assigned to a certain hierarchy in the mathematical model of GWO. The fittest solution is considered the alpha ($\\alpha$), while the second and third fittest are considered to be the beta (\\beta) and delta ($\\delta$) solutions. The rest of the solutions are referred to as the omega solutions. The leading wolves guide the omegas towards the prey throughout the search process \\cite{Gupta2018}.\n\n\\subsubsection{Encircling the Prey}\nPrey encirclement, which is one of the first steps when grey wolves hunt for their prey, can be modeled with the following:\n\n\\begin{align*}\n\t\\vec{X}(t + 1) &= \\vec{X_{p}}(t)\n\t\t\t\t\t - \\vec{A} \\cdot \\vec{D} \\\\\n\t\\vec{D} &= \\left | \\vec{C} \\cdot \\vec{X_{p}}(t)\n\t\t\t\t\t - \\vec{X}(t) \\right | \\\\\n\t\\vec{A} &= 2 \\cdot \\vec{a} \\cdot \\vec{r_{1}}\n\t\t\t\t\t - \\vec{a} \\\\\n\t\\vec{C} &= 2 \\cdot \\vec{r_{2}}\n\\end{align*}\n\nwhere $\\vec{X}(t)$ and $\\vec{X}(t + 1)$ are the positions of the wolf at the iteration $t$ and $t + 1$ respectively, $\\vec{X_{p}}(t)$ represents the location of the prey at the iteration $t$, $\\vec{D}$ is the difference vector, $\\vec{A}$ and $\\vec{C}$ are coefficient vectors, and $\\vec{r_{1}}$ and $\\vec{r_{2}}$ are uniformly random vectors with the range $\\left[ 0, 1 \\right]$. $\\vec{a}$ is vector that linearly decreases from $2$ to $0$ over the course of iterations \\cite{Mirjalili2014}. The original paper on GWO does not specify but Gupta, S. and Deep, K. provided the following equation to specify the decrease of $\\vec{a}$ from $2$ to $0$ \\cite{Gupta2018}:\n\n$$\n\\vec{a} = 2 - 2 \\cdot \\left( \\frac{t}{\\text{maximum number of iterations}} \\right)\n$$\n\nNote that the multiplication of vectors in the equations above is a component-wise multiplication, and not a dot product \\cite{Mirjalili2020-MathModel}.\n\n\\subsubsection{Hunting}\nWe typically do not know the position of the prey in an abstract search space. As such, it is presumed that the $\\alpha$, $\\beta$, and $\\delta$ solutions have the best idea so far of the position of the prey \\cite{Mirjalili2014}. Each wolf updates their positions based on the following equations.\n\n\\begin{align}\n\t\\vec{D}_{\\alpha} &= \\left | \\left ( \\vec{C}_{1} \\cdot \\vec{X}_{\\alpha} \\right ) - \\vec{X} \\right | \\\\\n\t\\vec{D}_{\\beta} &= \\left | \\left ( \\vec{C}_{2} \\cdot \\vec{X}_{\\beta} \\right ) - \\vec{X} \\right | \\\\\n\t\\vec{D}_{\\delta} &= \\left | \\left ( \\vec{C}_{3} \\cdot \\vec{X}_{\\delta} \\right ) - \\vec{X} \\right | \\\\\n\t\\vec{X_{1}^{'}} &= \\vec{X_{\\alpha}}(t) - \\vec{A_{\\alpha}} \\cdot \\vec{D_{\\alpha}} \\\\\n\t\\vec{X_{2}^{'}} &= \\vec{X_{\\beta}}(t) - \\vec{A_{\\beta}} \\cdot \\vec{D_{\\beta}} \\\\\n\t\\vec{X_{3}^{'}} &= \\vec{X_{\\delta}}(t) - \\vec{A_{\\delta}} \\cdot \\vec{D_{\\delta}} \\\\\n\t\\vec{X}(t + 1) &= \\frac{\\vec{X_{1}^{'}} + \\vec{X_{2}^{'}} + \\vec{X_{3}^{'}}}{3}\n\\end{align}\n\nwhere $\\vec{X_{\\alpha}}$, $\\vec{X_{\\beta}}$, and $\\vec{X_{\\delta}}$ represent the $\\alpha$, $\\beta$, and $\\delta$ solutions \\cite{Gupta2018}.\n\n\\subsubsection{Exploration and Exploitation}\nThe exploration phase of metaheuristics is modeled by the search phase, while exploitation is modeled by the attack phase. When $\\left| \\vec{A} \\right| < 1$, or $\\vec{C} < 1$, GWO is undergoing exploitation of the search space. Exploitation can be viewed as the wolves approaching towards the prey. On the other hand, when $\\left| \\vec{A} \\right| > 1$, or $\\vec{C} > 1$, the algorithm is in the search phase, where the wolves can be viewed as searching for the prey. In the search process, as the number of iterations $t$ reach the maximun possible number, the algorithm tends to focus more on exploitation than exploration. $\\vec{A}$ and $\\vec{a}$ eventually approach $0$, leaving $\\vec{C}$ the sole vector to eventually influence the search exploration. At this point, the algorithm will intensify towards exploitation. ", "meta": {"hexsha": "34e7dbf24ad38ad4867bc9c47b9b7c396c8690e3", "size": 19632, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapter1_intro.tex", "max_stars_repo_name": "seanballais/undergraduate-thesis-manuscript", "max_stars_repo_head_hexsha": "d2ae4c524b93ed1cb7a5fb6eedcfd3db3f90799d", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapter1_intro.tex", "max_issues_repo_name": "seanballais/undergraduate-thesis-manuscript", "max_issues_repo_head_hexsha": "d2ae4c524b93ed1cb7a5fb6eedcfd3db3f90799d", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapter1_intro.tex", "max_forks_repo_name": "seanballais/undergraduate-thesis-manuscript", "max_forks_repo_head_hexsha": "d2ae4c524b93ed1cb7a5fb6eedcfd3db3f90799d", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 208.8510638298, "max_line_length": 1898, "alphanum_fraction": 0.7763854931, "num_tokens": 4700, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.588889130767832, "lm_q2_score": 0.5851011542032312, "lm_q1q2_score": 0.34455971010999603}} {"text": "\\chapter{Basics on classification}\\label{ch:2}\n\n\\begin{remark}{Outline}\nIn this chapter, we introduce classification models from a machine learning perspective. \nFirst, in Section~\\ref{sec:2:classification}, we give a self contained introduction to \nclassification, including the most-common algorithms, and the main applications of \nclassification models. Then, in Section~\\ref{sec:2:measures}, we present the different evaluation \nmeasures that are normally used for analyzing the performance of classification methods.\n\\end{remark}\n\n\\section{Introduction}\n\\label{sec:2:classification}\n\nIn machine learning, classification refers to the attempt of identifying to which of a set of \nclasses a new example belongs, based on learning from examples whose class membership is known. \nA classification task begins with a training set in which the class of a set of examples is known. \nFor example, a classification model that predicts credit card fraud is developed by analyzing \nmany observed credit transactions over a period of time. The class in this case is a variable which \nindicates for each example whether or not the transaction was or not a fraud. Also, the features, \nare the transaction attributes like place, amount and time of the transaction.\n\nThen, during the training process, a classification algorithm finds the patterns and relationships \nbetween the values of the features and the values of the target class. Different algorithms use \ndifferent methods and techniques to estimate these relationships. Afterwards, these relationships \nare summarized in a model that is able to make predictions on new sets of data.\n\nIn general, there are two types of classification models: binary and multi-class. In binary \nclassification problems, the objective is to classify examples between two classes, usually \nreferred to as the negative and positive classes. On the other hand, multi-class problems are not \nbound to two classes but instead aim to classify examples among a number of classes. In this work we \nfocus on binary classification problems.\n\nBinary classification algorithms are widely used across a variety of domains. For example in the \nmedical field, models have been used for making predictions about tumors, probability \nof a disease, probability of selecting the right drug for a particular patient, and estimating the \nprobability of relapsing, among others \\citep{Herland2014}. In the financial sector, classification \nmodels have been successfully applied for fraud detection, credit scoring, portfolio management and \nalgorithmic trading. Also, in marketing, several models are being currently used for churn modeling, \ncustomer targeting, behavior prediction and direct marketing \\citep{Baesens2014}. Additionally, \nclassification algorithms are used in many other emerging applications such as terrorism \nprevention, malware detection, computer security, energy consumption prediction, spam \nclassification, and others \\citep{Kriegel2007}.\n\nFormally, a binary classification algorithm deals with the problem\tof predicting the class $y_i$ \nof a set $\\mathcal{S}$ of examples or instances $i$, given their $k$ features \\mbox{$\\mathbf{x}_i \n\\in \\mathbb{R}^k$}. The objective is to construct a function $f(\\cal{S})$ that makes a prediction \n$c_i$ of the class of each example $i$ from $\\cal{S}$ using its feature vector $\\mathbf{x}_i$, \nwhere $\\vert \\mathcal{S} \\vert= N$. Moreover, some algorithms allow to not only estimate the \nprediction, but also its confidence, in the form of the probability $\\hat p_i$ of belonging to the \npositive class, i.e. $c_i = 1$. The way for finding from $\\hat p_i$ to $c_i$ is simply by defining \na probability threshold $t$, and applying the following formula\n\\begin{equation}\\label{eq_pred}\n c_i = \n \\begin{cases}\n \\phantom{-}0 \\phantom{-} \\mbox{if} \\phantom{-} \\hat p_i \\le t\\\\\n \\phantom{-}1 \\phantom{-}\\mbox{otherwise,}\n \\end{cases}\n\\end{equation}\nUsually $t=\\frac{1}{2}$ \\citep{Hastie2009}. However, if $t \\ne \\frac{1}{2}$, the \nfunction that generates the predicted class labels $\\mathbf{c}=[c_i]$ is denoted as \n$f^t$.\n\n\\begin{figure}\n\t\\centering\n\t\\input{figures/src/ch2_fig_classification_process}\n \\caption{Classification process}\n \\label{fig:2:1}\n\\end{figure}\n\nIn \\figurename{ \\ref{fig:2:1}}, the process of training and prediction in a classification \nalgorithm are summarized. First, during the training phase, using a training set \n$\\mathcal{S}_{train}$, an algorithm is trained to predict $\\mathbf{y}$, where\n$\\mathbf{y}=[y_i]$. Then the algorithm is used to estimate the classes \n$\\mathbf{c}$ of a set of testing examples $\\mathcal{S}_{test}$.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics{ch2_fig1b}\n\\caption{Example of a classification algorithm. Using a set of examples from two classes, a \n\tclassification algorithm is learned in order to separate between the positives and the negatives. \n}\n\\label{fig:2:2}\n\\end{figure} \n\nThere exists several algorithms that can be used for classification tasks. In general a \nclassification algorithm is learned with the objective of finding patterns that separate between \nthe different classes \\citep{Hastie2009}. In order to clarify this intuition, in \\figurename{ \n\\ref{fig:2:2}} an example of a classification algorithm is shown. Let us consider a set of \nexamples, where the red points represent the positive examples and the blue ones the negative \nexamples. The objective of a classifier is to find the best way to separate between the positive and \nnegative examples. Then, the output of a classifier learned using the set of training examples is \nshown as the dashed black line. It is observed that this classifier is able to separate almost all \nthe examples using a linear classifier. However, not all examples are correctly classified. In \nparticular, there are four negative examples that were predicted as positive, and five positive \nexamples that were predicted as negative. In the next section, we present the standard methods for \nevaluating the performance of a classification algorithm.\n\n\n\\section{Traditional evaluation measures}\n\\label{sec:2:measures}\n\nWhen evaluating the performance of a classification algorithm, the first thing to do is to check \nthe number of examples that were misclassified, since the true class of the examples \nis known. Therefore, evaluating the error of a model is as simple as counting the number of times \nan example is misclassified divided by the number of examples\n\\begin{equation}\\label{eqn:ch2:error}\nErr(f({\\cal S})) = 1 -\\frac{1}{N} \\sum_{i=1}^N \\mathbf{1}_{y_i}(c_i),\n\\end{equation}\nwhere $\\mathbf{1}_q(z)$ is an indicator function that is calculated as:\n\\begin{equation}\n \\mathbf{1}_q(z) = \n \\begin{cases}\n \\phantom{-}1 \\phantom{-} \\mbox{if} \\phantom{-} z = q\\\\\n \\phantom{-}0 \\phantom{-} \\mbox{if} \\phantom{-} z \\neq q.\n \\end{cases}\n\\end{equation}\nMoreover the accuracy is defined as the percentage of times the algorithm \nmade the correct prediction\n\\begin{equation}\\label{eqn:2:accuracy}\nAcc(f({\\cal S})) = 1- Err(f({\\cal S})).\n\\end{equation}\n\nHowever, just knowing these statistics is not enough to make decisions, as in many applications \nit is important to know where the errors are coming from. In particular, the misclassified examples \nmay belong only to one class, which may give interesting insights about the problem. A way to \nobserve the different errors is by looking at the confusion matrix, as shown in \n\\mbox{\\tablename{~\\ref{tab:2:1}}}. Afterwards, using the cost matrix several statistics are \nextracted. In particular:\n \\begin{flalign}\n &Recall = \\frac{TP}{TP+FN} &\\\\\n &Precision = \\frac{TP}{TP+FP}& \\\\\n &F_1Score = 2\\frac{Precision \\cdot Recall}{Precision + Recall}&\n \\end{flalign}\n \n\t\\begin{table}[!t]\n\t\t\\centering\n\t\t\\footnotesize\n \\begin{tabular}{c|c|c}\n \\multicolumn{3}{c}{}\\\\\n\t\t\t\\multicolumn{1}{c|}{} & Actual Positive& Actual Negative \\\\\n\t\t\t\\multicolumn{1}{c|}{} & $y_i=1$& $y_i=0$ \\\\\n\t\t\t\\hline\n\t\t\tPredicted Positive \t\t& \\multirow{ 2}{*}{True Positive ($TP$)} & \\multirow{ \n\t\t\t2}{*}{False Positive ($FP$)} \\\\\n\t\t\t$c_i=1$ & &\\\\\n\t\t\t\\hline\n\t\t\tPredicted Negative \t& \\multirow{ 2}{*}{False Negative ($FN$)} & \\multirow{ \n\t\t\t2}{*}{True Negative ($TN$)} \\\\\n\t\t\t$c_i=0$ & &\\\\\n\t\t\\end{tabular}\n\t\t\\caption{Classification confusion matrix}\n\t\t\\label{tab:2:1}\n \\end{table} \n \n \\newpage\n As an illustrative example, the different statistics are calculated for the example presented \n in Section~\\ref{sec:2:classification}. First, the confusion matrix is calculated as \n follows:\n \\begin{center}\n \\footnotesize\n \\begin{tabular}{c|c|c}\n \\multicolumn{1}{c|}{} & Actual Positive& Actual Negative \\\\\n \\multicolumn{1}{c|}{} & $y_i=1$& $y_i=0$ \\\\\n \\hline\n Predicted Positive & \\multirow{ 2}{*}{36} & \\multirow{ \n 2}{*}{4} \\\\\n $c_i=1$ & &\\\\\n \\hline\n Predicted Negative & \\multirow{ 2}{*}{5} & \\multirow{ \n 2}{*}{68} \\\\\n $c_i=0$ & &\\\\\n \\end{tabular}\n \\end{center}\n Then using the confusion matrix, the different statistics are calculated as: Error = \n11.11\\%, Recall = 87.8\\%, Precision = 90\\% and $F_1Score$ = 88.8\\%.\n\t\nThere are, however, several instances that are misclassified, that is because the simple linear \nclassifier that was used in this example may not be good enough to separate between the positive \nand negative classes. In order to make a comparison, using the same example, a new \nalgorithm is learned. This time the algorithm made the correct prediction more often as shown in \n\\figurename{ \\ref{fig:2:3}}. Afterwards, the confusion matrix is calculated as follows:\n\n\\begin{figure}[t!]\n \\centering\n \\includegraphics{ch2_fig2}\n \\caption{Example of a classification algorithm. Using a set of examples from two classes, a \n classification algorithm is learned in order to separate between the positives and the negatives. \n}\n \\label{fig:2:3}\n\\end{figure}\n\n\\begin{center}\n\t\t\\footnotesize\n \\begin{tabular}{c|c|c}\n\t\t\t\\multicolumn{1}{c|}{} & Actual Positive& Actual Negative \\\\\n\t\t\t\\multicolumn{1}{c|}{} & $y_i=1$& $y_i=0$ \\\\\n\t\t\t\\hline\n\t\t\tPredicted Positive \t\t& \\multirow{ 2}{*}{37} & \\multirow{ \n\t\t\t2}{*}{2} \\\\\n\t\t\t$c_i=1$ & &\\\\\n\t\t\t\\hline\n\t\t\tPredicted Negative \t& \\multirow{ 2}{*}{4} & \\multirow{ \n\t\t\t2}{*}{70} \\\\\n\t\t\t$c_i=0$ & &\\\\\n\t\t\\end{tabular}\n\\end{center}\nThen the different statistics are calculated as: Error = 5.3\\%, Recall = 90.2\\%, Precision = \n94.9\\% and $F_1Score$ = 92.5\\%.\nIt is observed that in this case the FP are reduced more than the FN, this leads to a higher \nincrease in precision than in recall. There is not a single rule regarding which one is more \nimportant than the other, it depends on the application. For example in applications with a high \nfalse negative cost such as failing to identify a tumor in a medical exam, the recall should be the \npriority, even if that implies having a significant number of false positives. On the other hand, \nIn applications such as spam detection, predicting a normal email as spam it may have a large \nimpact on the customer, therefore, in this example is better to allow some false negatives and \nfocus on the false positives.\n\nIt is not always straightforward to define the right tradeoff between false positives \nand false negatives. The best approximation to solve that, is to focus on the actual costs incurred \nby the different decisions. This is usually solved using cost-sensitive classification methods. \n\n\\subsection{Brier score}\n\\label{sec:2:brier}\n\nTraditional evaluation measures of binary classification problems, such as Accuracy and \n$F_1Score$, provide a way to analyze the performance of a model. However, when using the classifier \noutput as a basis for decision making, there is a need of a measure that takes into account not \nonly the misclassification of a classifier predicted class $c$, but also the quality of the \nestimated probabilities $\\mathbf{\\hat p}$ \\citep{cohen2004}. The most appropriate is the Brier \nscore \\citep{brier1950}. The Brier score belongs to the class of so-called proper scores which are \nused in evaluating the subjective probability assessment of the prediction \\citep{DeGroot1983}. The \nBrier score is the average squared difference between the estimated probability and the true class \nlabel. It is defined as:\n\\begin{equation}\n BS(f(\\mathcal{S})) = \\frac{1}{N} \\sum_{i=1}^{N} (\\hat p_i - y_i)^2.\n\\end{equation}\n\nThe main justification of this score is based on decision theoretic considerations, in the sense \nthat, a forecaster should pay a price proportional to the confidence with which it asserts its \ndecision.\n\n\n\\subsection*{}\n", "meta": {"hexsha": "ce19b02c27161585902893c6ced7dfaa4cd3bffc", "size": 12553, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/chapters/chapter02.tex", "max_stars_repo_name": "albahnsen/phd-thesis", "max_stars_repo_head_hexsha": "8aedb00cba939b6f8a2f891f453a37206db1f635", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2015-10-07T13:31:49.000Z", "max_stars_repo_stars_event_max_datetime": "2019-08-09T12:02:27.000Z", "max_issues_repo_path": "tex/chapters/chapter02.tex", "max_issues_repo_name": "albahnsen/phd-thesis", "max_issues_repo_head_hexsha": "8aedb00cba939b6f8a2f891f453a37206db1f635", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/chapters/chapter02.tex", "max_forks_repo_name": "albahnsen/phd-thesis", "max_forks_repo_head_hexsha": "8aedb00cba939b6f8a2f891f453a37206db1f635", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2017-01-25T17:16:32.000Z", "max_forks_repo_forks_event_max_datetime": "2020-02-11T13:47:21.000Z", "avg_line_length": 51.4467213115, "max_line_length": 101, "alphanum_fraction": 0.7463554529, "num_tokens": 3296, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5851011542032312, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34455971010999603}} {"text": "\\renewcommand*\\chappic{img/intro.pdf}\n\\renewcommand*\\chapquote{}\n\\renewcommand*\\chapquotesrc{}\n%\n\\chapter{Introduction}\n\\label{ch:intro}\n\\section{Overview}\n\\label{sec:intro-overview}\n%\nHash functions are used as cryptographic primitives in many applications and protocols.\nThey take an arbitrary input message and provide a hash value. Input message and hash value\nare considered as byte strings in a particular encoding.\nThe hash value is of fixed length and satisfies several properties which make it useful\nin a variety of applications.\n\nIn this thesis, we consider the hash algorithms MD4 and SHA-256.\nOur goal is to find hash collisions using differential cryptanalysis.\nWe define differences between two messages and determine actual bits\nsuch that the two messages result in the same hash value. \\\\\nThis whole equation system will be modelled as a satisfiability problem.\nA SAT solver reports satisfiability if and only if the particular\ndifferences can be resolved and an actual hash collision is found.\nWe introduce a bit condition notation which allows us to visualize\nsuch differential states. Verification is done by several SAT solvers\nand we compare their runtime. Because the Boolean functions modelled in\na CNF have a major influence on the runtime, we investigate several\napproaches and compare them.\n\nBased on experience with these kind of problems with previous heuristic search tools\nwe aim to apply best practices to a satisfiability setting.\nWe will discuss, which SAT techniques lead to best performance characteristics\nfor our MD4 and SHA-256 testcases.\n\n\\section{Thesis Outline}\n\\label{sec:intro-outline}\n%\nThis thesis is organized as follows:\n\n\\begin{description}\n\\item[In Chapter~\\ref{ch:intro},] we briefly introduce basic subjects of this\n thesis. We explain our high-level goal involving hash functions and SAT solvers.\n\n\\item[In Chapter~\\ref{ch:hash},] we introduce the MD4 and SHA-256 hash functions.\n Certain design decisions imply certain properties which can be used in differential\n cryptanalysis. We discuss those decisions in this chapter after a formal definition\n of the function itself. Beginning with this chapter we develop a theoretical notion\n of our tools.\n\n\\item[In Chapter~\\ref{ch:dc},] we discuss approaches of differential cryptanalysis.\n We start with work done by Wang, et al. and followingly introduce differential\n notation to simplify representation of differential states. This way we can easily dump\n hash collisions.\n\n\\item[In Chapter~\\ref{ch:sat},] we discuss SAT solving techniques. We discuss how\n the problem needs to be encoded and give a brief overview over used SAT solvers.\n This includes a customized lingeling version by Armin Biere for our purposes.\n\n\\item[In Chapter~\\ref{ch:features},] we define SAT features which help us to\n classify SAT problems. This is a small subproject we did to look at properties\n of resulting DIMACS CNF files.\n\n\\item[In Chapter~\\ref{ch:enc},] we discuss how we represent a problem (i.e. the hash\n function and a differential characteristic) as SAT~problem. This ultimatively\n allows us to solve the problem using a SAT solver.\n\n\\item[In Chapter~\\ref{ch:results},] we present the result of our work.\n Runtimes are the main part of this chapter, but also results of Chapter~\\ref{ch:features}\n are presented.\n\n\\item[In Chapter~\\ref{ch:summary},] we conclude and discuss future work based on our results.\n\\end{description}\n", "meta": {"hexsha": "eda0fb23ce93353c2c8eeeceb92f00087a0671b0", "size": 3434, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "intro.tex", "max_stars_repo_name": "prokls/master_iaik", "max_stars_repo_head_hexsha": "30703774eaac6a514add2424f5c408a549cd4db1", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2016-05-13T21:45:11.000Z", "max_stars_repo_stars_event_max_datetime": "2016-05-13T21:45:11.000Z", "max_issues_repo_path": "intro.tex", "max_issues_repo_name": "prokls/master_iaik", "max_issues_repo_head_hexsha": "30703774eaac6a514add2424f5c408a549cd4db1", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "intro.tex", "max_forks_repo_name": "prokls/master_iaik", "max_forks_repo_head_hexsha": "30703774eaac6a514add2424f5c408a549cd4db1", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.6944444444, "max_line_length": 93, "alphanum_fraction": 0.795573675, "num_tokens": 775, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143433998, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3445189346395196}} {"text": "\\documentclass[letterpaper,11pt]{article}\n\\usepackage{ltcadiz}\n\n%%side margin is 1.0in + oddside, so we're creating a 1 inch margin\n\\oddsidemargin 0.0in \n\\topmargin 0.0in\n\\textwidth 6.5in\n\n\\usepackage{fullpage}\n\n\\title{Z Examples Found on the Internet}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{Introduction}\n\n\\subsection{Z section heading}\n\n \\begin{zsection}\n \\SECTION gearsystem \\parents standard\\_toolkit\n \\end{zsection}\n\n\\subsection{Free types paragraph}\n\n \\begin{zed}\n gearing ::= Single | Hub \\ldata \\nat \\rdata\n | Derailleurs \\ldata \\nat \\cross \\nat \\rdata\n \\end{zed}\n\n\\subsection{Horizontal definition paragraph}\n\n \\begin{zed}\n Wellgeared == [ gears : gearing | gears \\in \\ran Derailleurs ]\n \\end{zed}\n\n\\subsection{Generic axiomatic description paragraph}\n\n \\begin{gendef}[X,Y]\n First : X \\cross Y \\fun X \\\\\n Second : X \\cross Y \\fun Y\n \\where\n \\forall x : X; y : Y @ \\\\\n \\t1 First~(x,y) = x \\land \\\\\n \\t1 Second~(x,y) = y\n \\end{gendef}\n\n\\subsection{Schema definition paragraph}\n\n \\begin{schema}{Gear}\n Wellgeared \\\\\n ingear : \\nat \\cross \\nat\n \\where\n \\exists r, s : \\nat | gears = Derailleurs (r,s) @\\\\\n \\t1 First~ingear < r \\land Second~ingear < s\n \\end{schema}\n\n\\subsection{Generic schema definition paragraph}\n\n \\begin{schema}{Bicycle}[G]\n gears:\\power G\n \\end{schema}\n\n\\subsection{Axiomatic description paragraph}\n\n \\begin{axdef}\n numgears : gearing \\fun \\nat\n \\where\n \\forall g : gearing @ \\\\\n \\t1 (g = Single \\implies numgears~g = 1) \\land \\\\\n \\t1 (\\exists x : \\nat @ g = Hub~x \\implies numgears~g = x) \\land \\\\\n \\t1 (\\exists r, s :\\nat @ g = Derailleurs~(r, s) \\implies numgears~g = r*s)\n \\end{axdef}\n\n\\subsection{Conjecture paragraph}\n\n \\begin{theorem}{Lemma1}\n \\thrm numgears (Derailleurs (3,7)) = 21\n \\end{theorem}\n\n\n\\end{document}\n", "meta": {"hexsha": "3244020adce9d04b5d522dc55e027b66f1f918e0", "size": 1879, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "z/z_example.tex", "max_stars_repo_name": "CraigKelly/latex-samples", "max_stars_repo_head_hexsha": "48330f4b91ce2bf721441f27f2f0950cabcde1e3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2018-08-09T08:11:58.000Z", "max_stars_repo_stars_event_max_datetime": "2018-08-09T08:11:58.000Z", "max_issues_repo_path": "z/z_example.tex", "max_issues_repo_name": "CraigKelly/latex-samples", "max_issues_repo_head_hexsha": "48330f4b91ce2bf721441f27f2f0950cabcde1e3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "z/z_example.tex", "max_forks_repo_name": "CraigKelly/latex-samples", "max_forks_repo_head_hexsha": "48330f4b91ce2bf721441f27f2f0950cabcde1e3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-08-09T08:11:59.000Z", "max_forks_repo_forks_event_max_datetime": "2018-08-09T08:11:59.000Z", "avg_line_length": 22.369047619, "max_line_length": 79, "alphanum_fraction": 0.6391697712, "num_tokens": 616, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540697, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.3445189262749576}} {"text": "\n% This LaTeX was auto-generated from an M-file by MATLAB.\n% To make changes, update the M-file and republish this document.\n\n\n\n \n \n\n\\subsection*{conT.m} \n\n\\begin{par}\n\\textbf{Summary:} Test derivatives of controller functions. It is assumed that the controller function computes the mean and the variance of the control signal for a Gaussian distributed input $x\\sim\\mathcal N(m,s)$\n\\end{par} \\vspace{1em}\n\\begin{verbatim}function [dd dy dh] = conT(deriv, policy, m, s, delta)\\end{verbatim}\n\\begin{par}\n\\textbf{Input arguments:}\n\\end{par} \\vspace{1em}\n\\begin{verbatim}deriv desired derivative. options:\n (i) 'dMdm' - derivative of the mean of the predicted control\n wrt the mean of the input distribution\n (ii) 'dMds' - derivative of the mean of the predicted control\n wrt the variance of the input distribution\n (iii) 'dMdp' - derivative of the mean of the predicted control\n wrt the controller parameters\n (iv) 'dSdm' - derivative of the variance of the predicted control\n wrt the mean of the input distribution\n (v) 'dSds' - derivative of the variance of the predicted control\n wrt the variance of the input distribution\n (vi) 'dSdp' - derivative of the variance of the predicted control\n wrt the controller parameters\n (vii) 'dCdm' - derivative of inv(s)*(covariance of the input and the\n predicted control) wrt the mean of the input distribution\n (viii) 'dCds' - derivative of inv(s)*(covariance of the input and the\n predicted control) wrt the variance of the input distribution\n (ix) 'dCdp' - derivative of inv(s)*(covariance of the input and the\n predicted control) wrt the controller parameters\npolicy policy structure\n .fcn function handle to policy\n .\\ensuremath{<}\\ensuremath{>} other fields that are passed on to the policy\nm mean of the input distribution\ns covariance of the input distribution\ndelta (optional) finite difference parameter. Default: 1e-4\\end{verbatim}\n\\begin{par}\n\\textbf{Output arguments:}\n\\end{par} \\vspace{1em}\n\\begin{verbatim}dd relative error of analytical vs. finite difference gradient\ndy analytical gradient\ndh finite difference gradient\\end{verbatim}\n\\begin{par}\nCopyright (C) 2008-2013 by Marc Deisenroth, Andrew McHutchon, Joe Hall, and Carl Edward Rasmussen.\n\\end{par} \\vspace{1em}\n\\begin{par}\nLast modified: 2013-05-30\n\\end{par} \\vspace{1em}\n\n\\begin{lstlisting}\nfunction [dd dy dh] = conT(deriv, policy, m, s, delta)\n\\end{lstlisting}\n\n\n\\subsection*{Code} \n\n\n\\begin{lstlisting}\nif nargin < 5; delta = 1e-4; end % default value\n\n% if no input arguments, create random policy parameters and check derivatives\nif nargin == 0\n D = 2;\n d = 3;\n policy.w = randn(D,d);\n policy.b = randn(D,1);\n m = randn(d,1);\n s = randn(d); s = s*s';\n policy.maxU = '2 3';\n policy.fcn = @(policy, m, s)conCat(@conLin, @gSat, policy, m, s);\nend\n\nD = length(policy.maxU);\nd = length(m);\n\nswitch deriv\n case 'dMdm'\n [dd dy dh] = checkgrad(@conT0, m, delta, policy, s);\n\n case 'dSdm'\n [dd dy dh] = checkgrad(@conT1, m, delta, policy, s);\n\n case 'dCdm'\n [dd dy dh] = checkgrad(@conT2, m, delta, policy, s);\n\n case 'dMds'\n [dd dy dh] = checkgrad(@conT3, s(tril(ones(d))==1), delta, policy, m);\n\n case 'dSds'\n [dd dy dh] = checkgrad(@conT4, s(tril(ones(d))==1), delta, policy, m);\n\n case 'dCds'\n [dd dy dh] = checkgrad(@conT5, s(tril(ones(d))==1), delta, policy, m);\n\n case 'dMdp'\n [dd dy dh] = checkgrad(@conT6, policy.p, delta, policy, m, s);\n\n case 'dSdp'\n [dd dy dh] = checkgrad(@conT7, policy.p, delta, policy, m, s);\n\n case 'dCdp'\n [dd dy dh] = checkgrad(@conT8, policy.p, delta, policy, m, s);\n\nend\n\\end{lstlisting}\n\n\\begin{lstlisting}\nfunction [f, df] = conT0(m, policy, s) % dMdm\nif nargout < 2\n M = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm] = policy.fcn(policy, m, s);\n df = dMdm;\nend\nf = M;\n\nfunction [f, df] = conT1(m, policy, s) % dSdm\nif nargout < 2\n [M, S] = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm] = policy.fcn(policy, m, s);\n df = dSdm;\nend\nf = S;\n\nfunction [f, df] = conT2(m, policy, s) % dCdm\nif nargout < 2\n [M, S, C] = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm] = policy.fcn(policy, m, s);\n df = dCdm;\nend\nf = C;\n\nfunction [f, df] = conT3(s, policy, m) % dMds\nd = length(m);\nv(tril(ones(d))==1) = s; s = reshape(v,d,d); s = s+s'-diag(diag(s));\nif nargout < 2\n M = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm, dMds] = policy.fcn(policy, m, s);\n dd = length(M); dMds = reshape(dMds,dd,d,d); df = zeros(dd,d*(d+1)/2);\n for i=1:dd;\n dMdsi(:,:) = dMds(i,:,:); dMdsi = dMdsi + dMdsi'-diag(diag(dMdsi));\n df(i,:) = dMdsi(tril(ones(d))==1);\n end\nend\nf = M;\n\nfunction [f, df] = conT4(s, policy, m) % dSds\nd = length(m);\nv(tril(ones(d))==1) = s; s = reshape(v,d,d); s = s+s'-diag(diag(s));\nif nargout < 2\n [M, S] = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm, dMds, dSds] = policy.fcn(policy, m, s);\n dd = length(M); dSds = reshape(dSds,dd,dd,d,d); df = zeros(dd,dd,d*(d+1)/2);\n for i=1:dd; for j=1:dd\n dSdsi(:,:) = dSds(i,j,:,:); dSdsi = dSdsi+dSdsi'-diag(diag(dSdsi));\n df(i,j,:) = dSdsi(tril(ones(d))==1);\n end; end\nend\nf = S;\n\nfunction [f, df] = conT5(s, policy, m) % dCds\nd = length(m);\nv(tril(ones(d))==1) = s; s = reshape(v,d,d); s = s+s'-diag(diag(s));\nif nargout < 2\n [M, S, C] = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm, dMds, dSds, dCds] = policy.fcn(policy, m, s);\n dd = length(M); dCds = reshape(dCds,d,dd,d,d); df = zeros(d,dd,d*(d+1)/2);\n for i=1:d; for j=1:dd\n dCdsi = squeeze(dCds(i,j,:,:)); dCdsi = dCdsi+dCdsi'-diag(diag(dCdsi));\n df(i,j,:) = dCdsi(tril(ones(d))==1);\n end; end\nend\nf = C;\n\nfunction [f, df] = conT6(p, policy, m, s) % dMdp\npolicy.p = p;\nif nargout < 2\n M = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm, dMds, dSds, dCds, dMdp] = policy.fcn(policy, m, s);\n df = dMdp;\nend\nf = M;\n\nfunction [f, df] = conT7(p, policy, m, s)\npolicy.p = p;\nif nargout < 2\n [M, S] = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm, dMds, dSds, dCds, dMdp, dSdp] = policy.fcn(policy, m, s);\n df = dSdp;\nend\nf = S;\n\nfunction [f, df] = conT8(p, policy, m, s)\npolicy.p = p;\nif nargout < 2\n [M, S, C] = policy.fcn(policy, m, s);\nelse\n [M, S, C, dMdm, dSdm, dCdm, dMds, dSds, dCds, dMdp, dSdp, dCdp] = ...\n policy.fcn(policy, m, s);\n df = dCdp;\nend\nf = C;\n\\end{lstlisting}\n", "meta": {"hexsha": "aa1e8533d28b64823978eccb71de295ab6313611", "size": 6919, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/tex/conT.tex", "max_stars_repo_name": "SJTUGuofei/pilco-matlab", "max_stars_repo_head_hexsha": "a0b48b7831911837d060617903c76c22e4180d0b", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 53, "max_stars_repo_stars_event_min_datetime": "2016-12-17T15:15:48.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-09T16:59:27.000Z", "max_issues_repo_path": "doc/tex/conT.tex", "max_issues_repo_name": "sahandrez/quad_pilco", "max_issues_repo_head_hexsha": "2c99152e3a910d147cd0a52822da306063e6a834", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-04-24T11:02:23.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-24T11:09:45.000Z", "max_forks_repo_path": "doc/tex/conT.tex", "max_forks_repo_name": "sahandrez/quad_pilco", "max_forks_repo_head_hexsha": "2c99152e3a910d147cd0a52822da306063e6a834", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 36, "max_forks_repo_forks_event_min_datetime": "2017-04-19T06:55:25.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-19T10:19:12.000Z", "avg_line_length": 31.7385321101, "max_line_length": 215, "alphanum_fraction": 0.5852001734, "num_tokens": 2445, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3445189262749576}} {"text": "\\chapter{Discussion}\\label{c4}\n\nAs mentioned earlier, kNN compares the test sample to k neighbouring training samples to determine the class. This means all training samples have to be stored in memory \\cite{Gutie} which in turn could slow down the computer, causing the classifier to take a longer time to produce results. Being a non-parametric technique \\cite{16N} unlike DT and RF, kNN is prone to the curse of dimensionality \\cite{Gutie}, which happens when the dimensions or number of features increases \\cite{Distr}. This might explain the lower performance metric scores in comparison. There are big differences between the optimal k values for some turbines as shown in \\autoref{f3} which could be due to the data for each turbine having different distributions and characteristics.\n\nOverall, using a single multiclass-multilabel classifier with imbalanced training data produced better scores, which could be due to presence of correlations between the different turbine categories used as labels that are generalised better using this approach \\cite{110}.\n\nAs the labels `gearbox' and `electrical system' are in the top three out of 14 labels used causing longest downtimes, they should have more samples classed as `faulty' and `X hours before fault' compared to other categories, which therefore should result in better performance as the classifier would have learned the characteristics of different samples belonging to the same classes. This was not the case, however, looking at the confusion matrices in \\autoref{a4}. The classifier tends to misclassify `faulty' and `X hours before fault' classes as `normal' at a higher rate than the accuracy for these classes. This is still the case after removing the `curtailment' class, which saw an improvement in the accuracy of the `faulty' class. The accuracies of `X hours before fault' classes could potentially be increased by reducing the 6-hour intervals used and the maximum of 48 hours before a fault. The analysis should be repeated by reducing 48 hours to a smaller timescale, such as 12 hours, or by combining all samples that fall under this with `faulty' points to make the classification binary. Although this is likely to improve the performance, the classifier will not be able to give an indication of the timescale before a potential fault, therefore making it more difficult to decide on the appropriate action to be taken to avoid catastrophic failure in the turbine.\n\nBased on the feature importance in \\autoref{t6}, active power SCADA fields were least influential in predicting faults for both labels. The labelled power curves for `electrical system' in \\autoref{f5} below shows no clear relationship between fault points and the power curve shape. There are also many overlapping `normal', `faulty' and `X hours before fault' points even after filtration of curtailment and anomalies, which could explain why this feature was less important.\n\n\\begin{figure}\n \\centering\n \\begin{subfigure}[t]{.5\\textwidth}\n \\centering\n \\includegraphics[height=5cm]{../images/f5a}\n \\caption{\\label{f5a}}\n \\end{subfigure}%\n \\begin{subfigure}[t]{.5\\textwidth}\n \\centering\n \\includegraphics[height=5cm]{../images/f5b}\n \\caption{\\label{f5b}}\n \\end{subfigure}\n \\begin{subfigure}[t]{.5\\textwidth}\n \\centering\n \\includegraphics[height=5cm]{../images/f5c}\n \\caption{\\label{f5c}}\n \\end{subfigure}\n \\caption{\\label{f5}Labelled power curve for turbine 1 with turbine category 10 (`electrical system') through the two stages of filtering out anomalous and curtailment points labelled as `normal'. The original power curve is shown in \\ref{f5a}. The first stage involves a filter based on a pitch angle threshold, which produces \\ref{f5b}. The second stage involves several additional filters to produce the final power curve \\ref{f5c}.}\n\\end{figure}\n\nThe reactive power and generator speed played a bigger role in the classification for `electrical system', which makes sense considering the reactive power is produced as a result of impedance in the current due to electromagnetic fields produced by generators and transformers \\cite{React}. It is likely that the features used in classification for this label are unsuitable. A fault in the electrical system would be reflected in voltages, currents, frequencies \\cite{Overb} and temperature of power switchboards and cables. Electrical system faults could also be caused by environmental conditions such as lightning strikes and contact of wires with wildlife \\cite{Overb}. If there are such conditions recorded as environmental downtime categories, these should be accounted for when analysing faults in the electrical system.\n\nThe `gearbox' label was also found to perform poorly for these classes, despite having a higher mean F1 score than `electrical faults', which could be due to feature selection as well. Statistics from the National Renewable Energy Laboratory's gearbox failure database indicate that most faults are caused by bearings, gears and other components including filtration and lubrication systems \\cite{Stati15}. These are mostly due to wear, fatigue and cracks \\cite{Sheng11} and may be detected with higher accuracy if the features include quantities such as torque, oil pressure and gearbox temperature.\n\nThe SCADA data provided for this project only had 17 SCADA fields, of which 10 are used as features. This did not include voltages, currents, frequencies, torques or temperature readings, but the SCADA system for the turbine model used does measure these parameters. When a more complete SCADA data is available, the evaluation should be done by increasing the number of features to include these fields. The role of environmental conditions on failures could explain why wind speed and direction were very influential in detecting faults in the two labels analysed, although further in-depth analysis is required to verify this.\n\nBalancing the training dataset improved the classification accuracy of `X hours before fault' classes slightly. An overall improved model may be developed by oversampling only these classes for training, but there is a trade-off between this and the training time and computational resources required.\n\nThe dataset could have incorrect readings in the SCADA fields caused by broken or unresponsive sensors which are not detected as unusual when the downtime data is used in labelling. This was why a power threshold before cut-in speed was applied to remove redundant data points, by visually inspecting the power curve of the turbine as seen in \\autoref{f1}. The dataset should be manually inspected for all other features using curves such as pitch versus power and power versus rotor speed, to see if there are any other incorrect values previously undetected which may affect the classifier's accuracy in detecting faults. The rows of data corresponding to these values should then be excluded from the training data.\n\n\\section{Future work}\n\nIn addition to possible areas for future work discussed above, the following were identified.\n\nWhen more data is available, the analysis should be repeated using historic datasets spanning the life of the turbine. Historic data would have recorded the different states a turbine has experienced over its life, and therefore when a classifier is trained on this, it could detect future turbine states easier. However, this will mean the training data will be bigger, which in turn causes longer training time and more computing resources to be used. Another area of work is to test the performance of a classifier using different lengths of datasets for training while keeping the testing set and hyperparameter settings constant. This will allow for the most appropriate length of dataset for training to be determined based on the resources available to produce satisfactory results in terms of training time and classification accuracy.\n\nAfter a classifier has been trained and used in practice, its performance over time should be monitored. If the performance is found to diminish over time or after a major component replacement, the classifier should be retrained using recent data. As the classification makes a distinction between the different faults, the ability to alert relevant maintenance professionals for a specific fault automatically is possible.\n\nInstead of using turbine categories in the downtime data, which is supervised, for labelling, the alarm logs, which are unsupervised, can instead be used to compare the results. The number of alarm logs for the turbines used, however, is 480, compared to 23 turbine categories. This will mean the number of labels will be much higher, which will cause longer computational time. A solution to reduce this is to group similar alarms into one class. Another approach would be to use a single label with each alarm as a separate class, but there is likely to be overlap between classes when fault prediction is also included. If actual failure records are available, a comparison can be made between the predicted classes, actual classes, and actual failures that have occurred and their costs.\n\nFurther optimisation of hyperparameters is possible, such as finding the optimal number of estimators for RF. Each optimisation takes time and is limited by the specifications of the computer used, which is why this was not carried out in this project. Detailed analysis done on the results using RF for two labels above should be repeated for each label and classifier used for fair comparisons to be made.\n\nThe methodology could also be tested for wind turbines of different models in different sites. Provided these turbines have similar SCADA data with downtime records, only slight modifications to the codes, such as the data source, field names, and number of features, would be required in order to be used on other turbine models.\n\nIn industry, a cost function analysis needs to be done prior to implementing this fault detection method. It is defined as the cost of a false alarm (false positive) or failing to detect a fault in advance (false negative). False positives and false negatives both incur charges. The first is due to transporting labour and equipment to site, which could be expensive especially for sites in harsh environments, such as offshore wind farms. The second would cause unscheduled downtime, the loss of revenue due to no power generation and replacement of turbine components due to irreversible damage. This cost should then be compared to the cost of alternatively using a condition monitoring system, and the overall cost of running the wind farm or wind turbine. The analysis should give an indication on which performance metric is more important; if the cost of false negatives is more, attention should be paid to the recall score, while the precision is more important if false positives cost more \\cite{deRu15}.\n", "meta": {"hexsha": "5c87833c31727cb40c6d7bb6d53c8b08e88a48fd", "size": 10869, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/discussion.tex", "max_stars_repo_name": "nmstreethran/WindTurbineClassification", "max_stars_repo_head_hexsha": "b0ea6de909ccd5bb425cee291ca3c252c11df4eb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 34, "max_stars_repo_stars_event_min_datetime": "2019-03-01T21:24:40.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T09:59:17.000Z", "max_issues_repo_path": "docs/discussion.tex", "max_issues_repo_name": "nmstreethran/WindTurbineClassification", "max_issues_repo_head_hexsha": "b0ea6de909ccd5bb425cee291ca3c252c11df4eb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/discussion.tex", "max_forks_repo_name": "nmstreethran/WindTurbineClassification", "max_forks_repo_head_hexsha": "b0ea6de909ccd5bb425cee291ca3c252c11df4eb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2019-03-01T21:24:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-26T15:04:14.000Z", "avg_line_length": 194.0892857143, "max_line_length": 1381, "alphanum_fraction": 0.8094580918, "num_tokens": 2179, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.3445189262749576}} {"text": "\\documentclass[]{aastex62}\n\\usepackage{amssymb,amsmath}\n\n \n\\newcommand{\\vk}{von K\\'{a}rm\\'{a}n}\n\\newcommand{\\vdag}{(v)^\\dagger}\n\\newcommand\\aastex{AAS\\TeX}\n\\newcommand\\latex{La\\TeX}\n\n\n\\def\\eq#1{\\begin{equation} #1 \\end{equation}}\n\\def\\mic {\\hbox{$\\mu\\mathrm{m}$}}\n\\def\\atm {\\hbox{\\rm{ATM}}}\n\\def\\dC {\\hbox{$^{o}C$}}\n\\graphicspath{{./}{figures/}}\n\n%% Reintroduced the \\received and \\accepted commands from AASTeX v5.2\n%\\received{May 7, 2019}\n%\\revised{May 8, 2019}\n%\\accepted{--}\n%% Command to document which AAS Journal the manuscript was submitted to.\n%% Adds \"Submitted to \" the arguement.\n%\\submitjournal{Icarus}\n\n\\shorttitle{Variation of CubeSat Temperature along its Orbit}\n\\shortauthors{}\n\n\\begin{document}\n\n\\title{A Simple Model for the Variation of CubeSat Temperature along its Orbit} \n \n%\\email{ivezic@uw.edu}\n\n\\author[0000-0001-5250-2633]{\\v{Z}eljko Ivezi\\'{c}}\n\\affiliation{Department of Astronomy, University of Washington, 3910 15th Avenue NE, Seattle, \n WA 98195, USA; e-mail: ivezic@uw.edu}\n\n\\begin{abstract}\nWe discuss a simple model for the variation of CubeSat temperature along its orbit. First \nwe consider an analytic solution for the satellite temperature variation with time when subjected to \na bistable heat source: two segments with piece-wise constant power, such as orbits with an eclipsed \nsegment when the Sun is not directly visible. The model assumes that at a given time a single \ntemperature applies to the entire satellite body. Discussion is focused on CubeSat satellites in \nlow-Earth orbits, the uncertainties in predicted temperatures due to uncertain input parameters, and \nit emphasizes the importance of the satellite thermal inertia in setting the amplitude of the temperature \nvariation along the orbit. This simplified ``spherical cow'' model is suitable for studying the relative \neffects of surfaces with different emissivities, the effects of small changes in the solar flux between \nJune and December, the impact of thermal inertia, and as a ``sanity check'' for the results obtained \nwith numerical thermal models that utilize detailed geometrical and thermal descriptions of all satellite \ncomponents. We also developed a numerical model with arbitrary time dependence of the heating power, including its \ndependence on the satellite temperature, and validated it using analytic solution for a bistable heat \nsource. Analysis of a typical 2U CubeSat shows that low temperatures are more worrisome than\nhigh temperatures, and that low temperatures can be mitigated by active temperature control such\nas releasing heat when the satellite is in eclipse using electrical energy stored in batteries that\nare charged during non-eclipsed portion of the orbit. \n\\end{abstract}\n\n\\keywords{Satellites --- CubeSat --- Radiative transfer -- Analytic solution}\n\n\n\\section{Introduction} \n \nOver the last two decades, the concept of a small modular satellite named CubeSat\\footnote{For more \ndetails and references, see https://en.wikipedia.org/wiki/CubeSat} has become\nextremely popular -- to date about two thousand satellites were launched or are in preparation \nstage\\footnote{See https://en.wikipedia.org/wiki/List\\_of\\_CubeSats}. CubeSat projects have both\nscientific and educational values, and are suported by many universities as well as space organizations\nsuch as NASA and ESO. \n\nWhen designing a satellite, it is crucial to establish that the temperature variation for each component \nwill be within its operating range. In practice, professional engineering tools (such as Ansys framework)\nand numerical analysis are used to analyze complex systems. Nevertheless, approximately correct temperature estimates \nand physical insight can be derived even using simple analytic models. Such models can be used as\na ``sanity check'' for the results obtained with complex thermal models that include \nnumerous input parameters, and for fast input parameter exploration (e.g., runtime for a numerical \ntransient model using the Ansys code can be several hours). Simplified models are also useful as an\neducational tool and they help develop deeper physical understanding than numerical simulations. \n\nWhile there is relatively abundant literature on CubeSat thermal modeling, it appears that a \ncompact reference appropriate for undergraduate and graduate students entering this field,\nand supported by easy to use open source code, has not been published yet. This paper is\nattempting to fill this gap and it is a result of our work with two CubeSat projects: the SOC-i satellite project\\footnote{https://www.aa.washington.edu/news/article/2019-02-11/cubesat-team}\nat the University of Washington and the Perun satellite project\\footnote{https://perun-i.hr/en/} in Croatia. \n\nA simple model for the satellite temperature variation with time can be derived by assuming that a single\ntemperature applies to the entire satellite body at any given time. An implication of this assumption is \nthat the thermal resistivity across the surface is vanishing and thus the entire surface can reach the\nsame temperature very quickly (under a minute or less). This model permits an analytic solution \nwhen subjected to a bistable heat source with two segments that each have constant input power \n(the sum of input heating flux and internal power dissipation). A bistable heat source is a good \napproximation for a satellite orbit with an eclipsed segment when the Sun is not directly visible. \n\nThe governing equations for such a simple model with bistable heat source are presented in the \nfollowing Section, and numerical results are discussed in Section 3. In Section 4 we discuss\nactive temperature control and summarize our conclusions in Section 5. \n\n\n\\section{Simple single-temperature thermal model} \n\nConsider a satellite with a given geometry and assume a uniform temperature over its surface\nat any given time, $T(t)$. The temperature variation with time depends on the difference between\nheat source, $Q_{in}$, and heat sink, $Q_{out}$, \n\\eq{\n\\label{eq:dTdt}\n m C {dT \\over dt} = Q_{in} - Q_{out}, \n}\nwhere $m$ is the satellite mass and $C$ is the material heat capacity. The heat capacity for \nmost common materials used in satellites is listed in Table~\\ref{tab:inputsMatProp}. \nThe $mC$ product is often called the thermal inertia. Heat sources and sinks are measured in \nWatts (W = Js$^{-1}$). \n\n\\begin{table}[ht!]\n\t\\centering\n\t\\caption{Common material properties (Gilmore 2002). }\n\t\\label{tab:inputsMatProp}\n\t\\begin{tabular}{r|r|r|r} % \n\t\t\\hline\n \t Material & density & specific heat & thermal conductivity \\\\\n\t\t\\hline\n \\phantom{x} & $\\rho$ (kg\\,m$^{-3}$) & $C$ (J\\,kg$^{-1}$\\,K$^{-1}$) & $k$ (W\\,m$^{-1}$\\,K$^{-1}$) \\\\\n Aluminum & 2,710 & 768--921 & 120--205 \\\\ \n Solar cells & 2,285 & 300--700 & 60--100 \\\\\n \t\t\\hline\n\t\\end{tabular} \n % $^a$ $\\rho$ is mass density, $C$ is heat capacity (specific heat), and $k$ is thermal conductivity. \n\\end{table}\n \n\\subsection{Radiative heat sink} \n\nAssuming that the satellite is in vacuum, the heat sink is due to radiative losses\n\\eq{\n\\label{eq:Qout}\n Q_{out} = A_{tot} \\epsilon_T \\, \\sigma T^4 \n}\nwhere $A_{tot}$ is the satellite total surface area (e.g., for a spherical satellite $A_{tot} = 4\\pi R^2$,\nwhere $R$ is the satellite radius), $\\sigma=5.67\\times10^{-8}$ Wm$^{-2}$K$^{-4}$ is the Stefan-Boltzmann \nconstant, and $\\epsilon_T$ is the wavelength-averaged surface emissivity over the thermal flux distribution. \nThe surface emissivity $\\epsilon_T$ is approximately equal to the emissivity at the wavelength of the peak emission. From Wien's \nlaw, this wavelength is equal to (3000 K$\\mic$)/T and thus for $T \\approx 300$ K, $\\epsilon_T$ is approximately \nequal to the material emissivity around 10 $\\mic$. Typical values of $\\epsilon_T$ for materials used in \nsatellite industry are in the range 0.8-0.9; values of $\\epsilon_T$ for common materials are listed in \nTable~\\ref{tab:inputsAbsEmiss}. \n\nThe energy spent on battery charging (the conversion of incoming solar flux to chemical energy) is \nalso a heat sink. However, essentially all of that energy is returned back as a heat source at a later time. \nThe treatment of these effects is discussed separately further below (see \\S~\\ref{sec:batteries}). \n\n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Surface optical absorptivity and infrared emissivity (Gilmore 2002). }\n\t\\label{tab:inputsAbsEmiss}\n\t\\begin{tabular}{r|r|r} % \n\t\t\\hline\n \t Surface & $\\alpha_S$ & $\\epsilon_T$ \\\\\n\t\t\\hline\n Black anodized aluminum & 0.86 & 0.86 \\\\ \n Blue anodized aluminum & 0.67 & 0.86 \\\\ \n Yellow anodized aluminum & 0.47 & 0.86 \\\\ \n Solar panels & 0.92 & 0.85 \\\\\n Black plastic & 0.95 & 0.87 \\\\ \n Catalac White Paint & 0.24 & 0.90 \\\\ \n Dupont Silver Paint & 0.43 & 0.49 \\\\\n Buffed Aluminum & 0.16 & 0.03 \\\\ \n Buffed Copper & 0.30 & 0.03 \\\\ \n Polished stainless steel & 0.42 & 0.11 \\\\ \n Gold coating & 0.19 & 0.02 \\\\\n Kapton foil & 0.11 & 0.33 \\\\\n\t\t\\hline\n\t\\end{tabular} \n\\end{table}\n\n\n\n\\subsection{Heat sources} \n\nThe heat sources can include direct solar radiation, $Q_{sun}$, solar radiation reflected from Earth, \n$Q_{ref}$, and thermal infrared emission from Earth, $Q_{IR}$. When the satellite is exposed to direct \nsunlight, the maximum possible heat source corresponds to \n\\eq{\n\\label{eq:QinSun} \n Q_{in}^{sun} = Q_{sun} + Q_{ref} + Q_{IR} \n} \nwhile the minimum possible heating corresponds to \n\\eq{\n\\label{eq:QinEclipse} \n Q_{in}^{eclipse} = Q_{IR},\n} \nwhen the satellite is in Earth's shadow (eclipsed by Earth). For illustration, see figure~\\ref{fig:JacquesFig22}. \n\n%The two extremes, ``hot'' and ``cold'', should not be confused with ``hot'' and ``cold'' scenarios\n% that maximize/minimize the satellite temperature over an orbital cycle. \n \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.65\\textwidth, keepaspectratio]{figures/Jacques_heatSources.png} \n\\caption{An illustration of the satellite heat balance (here 1U CubeSat satellite is shown). \nThe heat sources include direct solar radiation, solar radiation reflected from Earth (albedo), \ninfrared radiation emitted by Earth, and internally dissipated power. Conversion of input\nradiation to chemical energy in batteries is not shown. The heat sink is thermal infrared radiation \nemitted by the satellite and energy for battery charging. Credit: Figure 2.2 from the master thesis \nby Lionel Jacques (2009, University of Liege). \n\\label{fig:JacquesFig22}}\n\\end{figure}\n\n\n\\subsubsection{Solar radiation} \n\nThe time-averaged solar flux is about $F_{sun}$=1372 Wm$^{-2}$ and its spectral energy distribution peaks \nat wavelenghts of about 0.5 $\\mic$ (yellow light; the Sun's surface temperature is about 5,800 K). Since \nEarth's orbit is not circular, the solar flux varies from 1322 Wm$^{-2}$ in June to 1422 Wm$^{-2}$ in December, \nor by about 4\\% around its mean value (see Table~\\ref{tab:inputsEnvParam}). The absorbed energy due to \nsolar flux is then \n\\eq{\n\\label{eq:Qsun}\n Q_{sun} = A_S \\, \\alpha_S \\, F_{sun} = \\eta_S \\, A_{tot} \\, \\alpha_S \\, F_{sun} \n} \nwhere $A_S$ is the satellite's mean projected surface area towards the Sun (for sphere, $A_S = \\pi R^2$ and\n$\\eta_S=1/4$), and $\\alpha_S$ is the wavelength-averaged surface absorptivity over the solar flux distribution. \nFollowing Kirchhoff's law, absorptivity $\\alpha_S$ is approximately equal to the emissivity at 0.5 $\\mic$, the wavelength \nof the peak of the solar spectral energy distribution\\footnote{The following convention is used in ESA and NASA \nliterature: $\\epsilon$ is the mean emissivity (and absorptivity) in the infrared wavelength range (5--35 \\mic), \nand $\\alpha$ is the mean absorptivity (and emissivity) in the optical wavelength range (0.3--2.4 \\mic).}. The low \nvalues of $\\alpha_S$ imply high reflectivity and ``shiny'' surfaces. The values of $\\alpha_S$ for common \nmaterials are listed in Table~\\ref{tab:inputsAbsEmiss}. \n \n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{The range of input enviromental parameters. }\n\t\\label{tab:inputsEnvParam}\n\t\\begin{tabular}{r|r|r|r|r} % \n\t\t\\hline\n \t Quantity & max & min & mean & unit \\\\\n\t\t\\hline\n Solar flux & 1422 & 1322 & 1372 & Wm$^{-2}$ \\\\\n Earth albedo & 35 & 25 & 30 & \\% \\\\ \n Earth IR flux & 260 & 220 & 240 & Wm$^{-2}$ \\\\\n \t\t\\hline\n\t\\end{tabular} \n\\end{table}\n\n\\subsubsection{Solar radiation reflected from Earth} \n\n \nThe fraction of solar flux reflected by Earth back towards the satellite is typically $\\rho_E=0.3$, and it \nvaries in the range $\\rho_E=0.2-0.4$ across Earths' surface and oceans. The reflected flux depends on the\n\"Sun-Earth-satellite\" angle, $\\theta$, and it is maximized when the satellite is at the subsolar point. \nGilmore (2002) gives an approximate formula for the variation of reflected light with $\\theta$ as \n\\eq{\n f(\\theta) = \\left[ \\cos(0.9\\theta)\\right]^{1.5},\n}\nthat can be used to derive mean correction, $f_{alb}$, for a given orbit. For example, for a polar orbit passing\nthrough subsolar point, $f_{alb}=0.62$ for the non-eclipsed part of the orbit, and for a polar orbit perpendicular \nto it (with $\\theta=90$ deg. and no eclipsed part), $f_{alb}=0.06$. \n \nThe absorbed energy from the reflected solar radiation is then\n\\eq{\n\\label{eq:Qref}\nQ_{ref} = f_E \\, A_E \\, f_{alb} \\, \\rho_E \\, \\alpha_S \\, F_{sun} = f_E \\, \\eta_E \\, A_{tot} \\, f_{alb} \\, \\rho_E \\, \\alpha_S \\, F_{sun} \n}\nwhere $A_E$ is the satellite's effective projected surface area towards Earth. The $f_E$ factor \naccounts for the fact that Earth fills less than 2$\\pi$ srad (``half the sky'') as viewed from the satellite, and is defined as \n\\eq{\n f_E = \\left( R_E \\over R_E + h \\right)^2, \n}\nwhere $R_E=6,378$ km is the Earth's mean radius and $h$ is the satellite's altitude (typically, $h=550$ km \nfor a low-Earth orbit, giving $f_E \\sim 0.85$). The ratio $\\eta_E=A_E/A_{tot}$ needs to account for the fact\nthat incoming radiation from Earth is not plane-parallel as is the case for direct solar radiation. The computation\nof $\\eta_E$ is based on the concept of radiative viewing factors and it is discussed in more detail in Appendix A. \nThe resulting $\\eta_E$ for spherical and CubeSat satellites are further discussed in \\S~\\ref{sec:effA}. \n\nTherefore, \n\\eq{\n Q_{ref} = f_E \\, {\\eta_E \\over \\eta_S} \\, f_{alb} \\, \\rho_E \\, Q_{sun}. \n}\n\n\n\\subsubsection{Infrared radiation emitted by Earth} \n\nThe thermal infrared flux emitted by Earth is about $F_{IR}$=240 Wm$^{-2}$ on average (see Table~\\ref{tab:inputsEnvParam}), \nand it is equal to one quarter (the ratio $A_S/A_{tot}$ for Earth) of the absorbed solar radiation (for $\\rho_E=0.3$, 70\\% of \n$F_{sun}$ is absorbed by Earth). Since Earth's equilibrium temperature of $\\approx$300 K, the spectral energy \ndistribution of this radiation peaks at about 10 $\\mic$. Therefore, \n\\eq{\n Q_{IR} = f_E \\, A_E \\, \\alpha_{IR} \\, F_{IR} = f_E \\, \\eta_E \\, A_{tot} \\, \\alpha_{IR} \\, F_{IR} \n}\nwhere $\\alpha_{IR}$ is the wavelength-averaged surface absorptivity over the Earth's thermal flux distribution.\nGiven Kirchhoff's law and the fact that the satellite and Earth's temperatures are similar, \n$\\alpha_{IR} \\approx \\epsilon_T$. Due to varying emission properties of Earth's surface (oceans, continents,\nclouds), $F_{IR}$ can vary by about $\\pm$10\\% along the satellite's orbit. \n\n\n\n\\subsection{Internal power dissipation \\label{sec:batteries}} \n\nA fraction of absorbed optical flux (the sum of $Q_{sun}$ and $Q_{ref}$) is often used to charge \non-board batteries. Up to about 30\\% of absorbed flux can be thus converted into chemical energy.\nThis energy conversion is also a heat sink. However, essentially all of that energy is returned back \nas a heat source at a later time (except for a small fraction needed to power on-board computer and\nto emit communication signal back to Earth), motivating a separate treatment. \n\nThe energy stored in batteries can be dissipated in various ways, including at a constant rate and \nin short bursts. Here it will be assumed that batteries are charged using 30\\% of absorbed flux \n(solar cell efficiency $\\eta_{cell} = 0.3$) during non-eclipsed portion of the orbit, and that this \naccumulated energy is dissipated at a constant rate during the entire orbit. If fraction $\\eta_P$ \nof the orbital period $P$ is spent in Earth's shadow, then eqs.~\\ref{eq:QinSun} and \\ref{eq:QinEclipse} \nhave to be modified as \n\\eq{\n Q_{in}^{sun} = (1 - \\eta_P * \\eta_{cell}) * (Q_{sun} + Q_{ref}) + Q_{IR} \n} \nand \n\\eq{\n\\label{eq:Qeclipse}\n Q_{in}^{eclipse} = Q_{IR} + \\eta_{cell} \\, (1 - \\eta_P) \\, (Q_{sun} + Q_{ref}), \n} \nwhere the second term in eq.~\\ref{eq:Qeclipse} is the battery power internally dissipated as heat\nat a constant rate during the entire orbit,\n\\eq{\n Q_{dissip} = \\eta_{cell} \\, (1 - \\eta_P) \\, (Q_{sun} + Q_{ref}).\n} \nOf course, eqs.~\\ref{eq:QinSun} and \\ref{eq:QinEclipse} are recovered when $\\eta_{cell} = 0$. Finally,\nit is good to emphasize that $\\eta_{cell}$ represents the fraction of all absorbed radiation that\nwas converted to battery charge. For example, if the cells occupy 2/3 of all external surfaces, and \nthe cell conversion efficiency is 30\\%, then $\\eta_{cell}$= 0.2. For randomized orientiations, it's only\n``effective'' quantities that count in the model considered here; however, when a specific satellite orientation\nis known, one could incorporate information about where exactly the solar cell panels are positioned, too. \n\n\n\\subsection{Effective surface areas $A_S$ and $A_E$ for 1U and 2U CubeSat satellites \\label{sec:effA}} \n\nThree surface areas matter for heat balance: \n\\begin{itemize}\n\\item The total surface area, $A_{tot}$, that controls infrared radiation emitted by the satellite.\n\\item The satellite projected surface area as viewed from the direction of incoming solar radiation, $A_S=\\eta_SA_{tot}$.\n For example, $\\eta_S= 1/4$ in case of spherical satellites. \n\\item The projected surface area as viewed from Earth, with satellite in zenith, $A_E=\\eta_EA_{tot}$. In more\n detail, the computation of $A_E$ is a bit more complicated than in case of $A_S$ because Earth is much closer\n than the Sun and it fills a much larger solid angle on the sky (e.g., in case of a spherical satellite very \n close to Earth, $\\eta_S = 1/4$ and $\\eta_E = 1/2$, while asymptotically $\\eta_E = 1/4$ when the satellite\n is much further away; for an orbit altitude of 550 km, $\\eta_E =0.36$). \n\\end{itemize}\n\nIn case of non-spherical satellites, the satellite orientation matters. For a given orientation and CubeSat\nsatellites, $\\eta_S$ and $\\eta_E$ can be computed by adding values for six individual sides, which are \ncomputed as discussed in Appendix A. When averaged over plausible orientations and orbits, \nfor 1U and 2U CubeSat geometries $\\eta_S = 0.21$ and $\\eta_E = 0.36$, with a plausible uncertainty due to\nactual orbit specifics of the order 10\\%. \nGiven that equilibrium temperature is proportional to \n$\\eta^{1/4}$ (see eq.~\\ref{eq:Teq} below), the implied temperature uncertainty due to 10\\% uncertainties \nin $\\eta$ factors is about 2.5\\%, or about 7 \\dC\\ assuming a typical temperature of 273 K. \n% The actual variation of a 2U CubeSat temperature along the orbit will have a much smaller amplitude \n% due to finite thermal inertia (discussed in detail further below and illustrated in Figure~\\ref{fig:Tt}). \n\nFor an orbit altitude of 550 km, for a spherical satellite $\\eta_S = 0.25$ and $\\eta_E =0.36$, while\nfor a 2U CubeSat $\\eta_S \\approx 0.21$ and $\\eta_E \\approx 0.36$. Therefore, given everything else same, \nthe equilibrium temperature for the spherical satellite will be slightly higher (typically of the order \n10 $^\\circ$C) than for the CubeSat because of 20\\% higher absorbed direct solar radiation. \n\n\n\\subsection{Typical numerical values of heat sources and sinks for 2U CubeSat} \n\nA ``randomly oriented'' 2U CubeSat is used for numerical analysis and illustration, with $A_{tot}=0.1$ m$^2$, $\\eta_S=0.21$,\nand $\\eta_E=0.36$. Numerical input assumptions include mean environmental parameters from \nTable~\\ref{tab:inputsEnvParam}, aluminum heat capacity $C=921$ J\\,kg$^{-1}$\\,K$^{-1}$, satellite \nmass $m=2.0$ kg, surfaces with $\\alpha_S=0.86$ and $\\epsilon_T=0.86$ (black anodized aluminum), \n$\\eta_{cell}=0.2$, and a Sun-synchronous orbit with $h=550$ km (assumed orbital period of 90 minutes), \nwith $\\eta_P=0.33$ and $f_{alb}=0.62$. Note that these parameters do {\\bf not} correspond to any particular satellite. \n\nWith these input parameters, $Q_{in}^{sun}=$40.1 W and $Q_{in}^{eclipse}$=11.1 W, with absorbed direct\nsolar radiation $Q_{sun}$=29.5 W, and with dissipated thermal power contributing a constant rate of\n$Q_{dissip}=$4.8 W. Absorbed direct solar radiation is about five times as large as absorbed\nreflected solar radiation, and larger by a similar factor than absorbed Earth's infrared emission. \nThe total battery energy charged and then dissipated during one orbital period is 7.2 Wh. \n\n\n\\subsection{High and low equilibrium temperatures} \n\nThe equilibrium temperature can be computed by assuming that the satellite is exposed to a constant \nheat source for an infinitely long time and thus $dT/dt=0$. It then follows from eqs.~\\ref{eq:dTdt} and \n\\ref{eq:Qout} that\n\\eq{\n\\label{eq:Teq}\n T_{eq} = \\left( {Q_{in} \\over A_{tot} \\, \\epsilon_T \\, \\sigma} \\right)^{1/4}. \n}\nNote that the equilibrium temperature does {\\bf not} depend on satellite's thermal inertia (the product of \nmass and heat capacity). Because for a given geometry and orientation $Q_{in}$ is proportional to the\ntotal area $A_{tot}$, the equilibrium temperature does {\\bf not} depend on satellite's size either. \n\nAssuming $Q_{in}^{sun}=$40.1 W and $Q_{in}^{eclipse}$=11.1 W, the corresponding equilibrium temperatures\nare $T_{eq}^{sun}$ = 301.1 K (27.9 $^\\circ$C) and $T_{eq}^{eclipse}$ = 218.6 K ($-$54.6 $^\\circ$C).\nThis temperature range is {\\bf much larger} than satellite temperature variation expected for oscillatory \nheating in a typical orbit, as discussed next. \n\n\n\\subsection{Analytic solution for the temperature's return to its equilibrium value} \n\n \n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/analyticUp.png}\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/analyticDown.png}\n\\caption{The analytic solution for dimensionless temperature parameter $\\tau(t) = T(t)/T_{eq}$\nas a function of dimensionless time parameter $t/t_o$ for initial conditions with $\\tau_o<1$\n(left) and $\\tau_o>1$ (right). Note that in both cases $\\tau(t)$ asymptotically approaches unity,\nthat is, $T(t)$ asymptotically approaches $T_{eq}$. \n\\label{fig:analytic}}\n\\end{figure}\n\n\n\nEquation~\\ref{eq:dTdt} is typically solved using numerical integration. When the heat source is constant\nin time, the solution can be obtained analytically. Equation~\\ref{eq:dTdt} can be recast using \neqs.~\\ref{eq:Qout} and \\ref{eq:Teq} as \n\\eq{\n\\label{eq:dtaudx}\n {d\\tau \\over dx } = 1 - \\tau^4, \n}\nwhere $\\tau=T/T_{eq}$, $x=t/t_o$ and the time scale $t_o$ is given by \n\\eq{\n\\label{eq:t0}\n t_o = { C m \\over A_{tot} \\, \\epsilon_T \\, \\sigma \\, T_{eq}^3}. \n}\nThe derivative $d\\tau/ dx$ is positive when starting temperature $T_o=T(t=0)$ is\n$T_o < T_{eq}$. Thus, when $Q_{in} = Q_{in}^{sun}$ and $T_{eq} = T_{eq}^{sun}$, the temperature\nwill be increasing with time, while for $Q_{in} = Q_{in}^{eclipse}$ and $T_{eq} = T_{eq}^{eclipse}$\nthe derivative is negative and the temperature decreases with time.\n\nThe simplified dimensionless differential equation \\ref{eq:dtaudx} admits an implicit analytic \nsolution:\n\\eq{\n\\label{eq:analytic}\n{t \\over t_o} = {1\\over 2}\\left[\\arctan(\\tau) - \\arctan(\\tau_o)\\right] + {1\\over 4}\\left[\\ln\\left({\\tau+1\\over\\tau_o+1}\\right) - \\ln\\left({\\tau-1\\over\\tau_o-1}\\right) \\right], \n}\nwhere $\\tau_o = T_o/T_{eq}$ is the initial condition. When $t \\gg t_o$, $\\tau$ asymptotically approaches unity. \nFigure~\\ref{fig:analytic} illustrates solutions given by eq.~\\ref{eq:analytic} for both $\\tau_o<1$ and $\\tau_o>1$. \n\n\n\n\\subsection{Temperature variation for a bistable heat source} \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.31\\textwidth, keepaspectratio]{figures/3tempsVStime_DefaultsMassVariationMsmall.png}\n\\includegraphics[width=0.31\\textwidth, keepaspectratio]{figures/3tempsVStime_DefaultsMassVariationMdefault.png}\n\\includegraphics[width=0.31\\textwidth, keepaspectratio]{figures/3tempsVStime_DefaultsMassVariationMlarge.png}\n\\caption{The satellite orbital temperature variation as a function of the surface emissivity properties \nand thermal inertia. The satellite total surface area is 0.1 m$^2$ (similar to 2U CubeSat), with \neffective absorptive surfaces corresponding to a spherical satellite. Three different types of anodized \naluminum surfaces are modeled: black with $\\alpha_S, \\epsilon_T$ = (0.86, 0.86), blue: (0.67, 0.87) \nand yellow: (0.47, 0.87). The thermal inertia is controlled by the satellite mass; left: low (0.05 kg), \nmiddle: medium (2.2 kg), right: high (10 kg). The temperature variation is compared to a typical battery operating \ntemperature range (the blue horizontal band). \n\\label{fig:Tt}}\n\\end{figure}\n\n\nNow consider a bistable heat source, such as a satellite in an orbit and the heat source periodically\nswitching between $Q_{in}^{sun}$ and $Q_{in}^{eclipse}$. Because corresponding equilibrium temperatures \n$T_{eq}^{sun}$ and $T_{eq}^{eclipse}$ are different, the implied time scales for the temperature's return to its\nequilibrium value, $t_o$ given by eq.~\\ref{eq:t0}, will be different, too. Eq. ~\\ref{eq:t0} implies that\nthe ratio of time scales in eclipse and when exposed to sunlight is \n\\eq{\n { t_o^{eclipse} \\over t_o^{sun} }= \\left(Q_{in}^{sun} \\over Q_{in}^{eclipse} \\right)^{3/4} \\approx 3. \n}\n\nFor a bistable heat source, the temperature at the end of the rising phase must be equal to the\ntemperature at the start of cooling phase and vice versa. As a result of this condition, the \ntemperature will oscillate between two extremes, $T_{min}$ and $T_{max}$ with $T_{min} \\ge T_{eq}^{eclipse}$ \nand $T_{max} \\le T_{eq}^{sun}$. Eq.~\\ref{eq:analytic} appears too cumbersome to derive closed-form\nanalytic solutions for $T_{min}$ and $T_{max}$; in practice, $T_{min}$ and $T_{max}$ are easily determined\nnumerically (see Appendix B). \n\nWhen thermal inertia is vanishing, the temperature will return to its equilibrium values essentially\ninstantaneously and most of the time the satellite temperature will be either $T_{eq}^{eclipse}$ or $T_{eq}^{sun}$. \nOn the other hand, for infinitely large thermal inertia the temperature will assume an equilibrium\nvalue that corresponds to the heat source averaged over the satellite orbit. For example, if the satellite \nspends one third of the orbital period in eclipse, then\n\\eq{\n\\label{eq:TeqAve} \n T_{eq}^{ave} = \\left[ {1\\over 3} \\left(T_{eq}^{eclipse}\\right)^4 + {2\\over 3} \\left(T_{eq}^{sun}\\right)^4 \\right]^{1/4}. \n}\nWith $T_{eq}^{sun} = 301.1$ K and $T_{eq}^{eclipse} = 218.6$ K, $T_{eq}^{ave}$ = 281.1 K (7.9 $^\\circ$C). \nWith thermal inertia corresponding to heat capacity for aluminum ($C=921$ J\\,kg$^{-1}$\\,K$^{-1}$, see\nTable~\\ref{tab:inputsMatProp}), and satellite mass $m=2.0$ kg, the actual temperature extremes are $T_{min}$ = 272.4 K \n($-$0.8 $^\\circ$C) and $T_{max}$ = 289.1 K (16.0 $^\\circ$C). Note that the $T_{max} - T_{min}$ difference is about\nfive times smaller than the $T_{eq}^{sun} - T_{eq}^{eclipse}$ difference. The variation of these extreme orbital \ntemperatures on various input parameters is discussed next. \n\n\n\\section{Numerical Examples} \n\nThis section explores the impact of variations in input parameters on the mean satellite temperature and the \namplitude of temperature variations. The concept of hot and cold cases is also discussed. Note that numerical\nvalues of various parameters were chosen to be similar to 2U CubeSat parameters; however, they do {\\bf not}\ncorrespond to any particular satellite. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodizedMassVariation.png}\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodizedFsunVariation.png}\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodizedChargingVariation.png}\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodizedOrbitVariation.png}\n\\caption{The impact of thermal inertia (top left), solar flux variation (top right), solar cell efficiency (bottom left)\nand the eclipse duration (bottom right) on satellite orbital temperature variation (all for black anodized aluminum \nsurface with $\\alpha_S, \\epsilon_T$ = 0.86, 0.86). In the bottom left panel, it is assumed that a fraction of \nabsorbed solar radiation is used to charge batteries, and it is then dissipated as heat at a constant rate\nthroughout the orbit. In the bottom right panel, the no eclipse case corresponds to a polar orbit whose normal\nvector points to the Sun. The effective albedo is varied from 0.06 times its maximum value, as expected for such\nan orbit, to 0.6 times its maximum value, as expected for an orbit that includes subsolar point. The temperature variation \nis compared to a typical battery operating temperature range (0--40 $^\\circ$C, the blue horizontal band). \n\\label{fig:Tt2}}\n\\end{figure}\n\n\n\\newpage\n\\subsection{The impact of thermal inertia on the amplitude of temperature variation} \n\nFigure~\\ref{fig:Tt} shows the satellite orbital temperature variation as a function of the surface emissivity\nproperties and thermal inertia. The temperature variation is computed using analytic solution given by\neq.~\\ref{eq:analytic} and emissivity properties corresponding to three different types of anodized aluminum \nsurfaces (see figure caption). It is assumed that the orbital period is 90 min, with the eclipse portion lasting \n30 min. The aluminium heat capacity is assumed and the thermal inertia is controlled by the satellite mass. \n\nFor low thermal inertia (left panel), the temperature displays large variation, drops quickly to the cold \nequilibrium temperature and rises back even faster to the hot equilibrium temperature. For very high\nthermal inertia (right panel) the temperature varies by only a few degrees around the value given by \neq.~\\ref{eq:TeqAve} (7.9 $^\\circ$C for black anodized Al surface). In the most realistic case shown in the \nmiddle panel, the temperature variation amplitude is 10--17 degrees, depending on the surface properties. \n\nThis behavior is similar to potatoes taken from a hot oven: small satellites would cool faster \nthan their scaled-up larger versions. However, here the difference in behavior is due to different \nthermal inertia for satellites that look identical from the outside (same size, shape and surface properties). \nInstead of small and large potatoes, a better analogy is solid and hollow potatoes of the same size.\n\nIt is important to recognize that some components within the satellite could achieve temperatures \nhigher than $T_{max}$ (components close to the locations of internal power dissipation) but {\\bf never\nlower} than $T_{min}$ (assuming steady-state after many cycles). To obtain temperature variation for \nindividual components, a professional tool (e.g., Thermal Desktop, Ansys) and detailed numerical \ncomputations need to be employed. Nevertheless, the essential impact of thermal inertia on the \namplitude of temperature variation will remain. Perhaps the most important conclusion of this \nsimplified analysis that pertains to detailed numerical modeling is that {\\bf a full transient model\nmust be employed to assess the temperature variation} between $T_{min}$ and $T_{max}$. The \nextreme equilibrium temperatures, $T_{eq}^{sun}$ and $T_{eq}^{eclipse}$ are {\\bf not} representative of the\nactual temperature variation experienced by the satellite. \n\nIn practice, the uncertainty in thermal inertia is much smaller than discussed in figure~\\ref{fig:Tt}.\nThe top left panel in figure~\\ref{fig:Tt2} shows that varying thermal inertia by $\\pm$30\\% \naround its mean value changes temperature predictions by about 5 $^\\circ$C. \n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodizedShotVScold.png} \n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodized2UareaVariation.png} \n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodized2UhotVScold.png} \n\\includegraphics[width=0.45\\textwidth, keepaspectratio]{figures/3tempsVStime_BlackAnodized2UhotVScoldExtreme.png} \n\n\\caption{The impact of choosing extreme values of environmental conditions (top left), varying geometry expressed\nthrough 20\\% variation of the absorptive surface area (top right), extreme values of environmental conditions when \nassuming 2U CubeSat satellite geometry with randomized orientation (bottom left, $\\eta_S=0.21$ and $\\eta_E=0.36$), \nand with orientation that maximizes the temperature range between these so-called ``hot'' and ``cold'' cases (bottom\nright, hot: $\\eta_S=0.30$ and $\\eta_E=0.38$; cold: $\\eta_S=0.10$ and $\\eta_E=0.34$). The temperature variation is \ncompared to a typical battery operating temperature range (the blue horizontal band). \n\\label{fig:Tt3}}\n\\end{figure}\n\n\n\n\\subsection{The impact of variable solar flux on predicted satellite temperature} \n\nDue to Earth's elliptical orbit around the Sun, the solar flux at Earth's location varies by about 3.6\\%\naround its mean value, between its maximum at winter solstice and its minimum at summer solstice\n(see Table~\\ref{tab:inputsEnvParam}). The top right panel in figure~\\ref{fig:Tt2} shows that this \nvariation changes the minimum and maximum temperatures by about 3-4 degrees. \n\n \n\\subsection{The impact of battery charging on the amplitude of temperature variation} \n\nThe energy spent to charge on-board batteries is converted to chemical energy and subtracted\nfrom heat balance (see eq.~\\ref{eq:dTdt}). It is returned to heat balance in the form of resistive\nheat dissipation, at a constant rate as assumed here. Because the charging does not happen \nduring the eclipsed portion of the orbit, this dissipation effectively ``flattens'' the heat source \nvariation and decreases the amplitude of temperature variation. The bottom left panel in \nfigure~\\ref{fig:Tt2} shows that the conversion of 20\\% of incoming solar flux to battery charge\nand release as heat can decrease the amplitude of temperature variation by about 5 degrees\ncompared to no-battery case. \n \n\n\\subsection{The impact of eclipse duration on the amplitude of temperature variation} \n\nGiven a fixed orbital period, the shorter is the eclipse the higher is the total accumulated\nenergy. The bottom right panel in figure~\\ref{fig:Tt2} compares two polar orbits, one whose \nnormal vector points to the Sun, with no eclipse, and another one that includes subsolar point\nand has one third of orbital period spent in eclipse. The impact on mean temperature is about \n10-15 degrees. Uncertainties in effective albedo contribute to the uncertainty in predicted\ntemperatures; when\nalbedo is varied from 0.06 times its maximum value, as expected for the first orbit with\nno eclipse, to 0.6 times its maximum value, the temperature is raised by another 10 degrees.\n\n\n\\subsection{The concept of hot and cold cases} \n\nDue to uncertainties in input parameters, including environmental, orbital and satellite parameters,\nengineering pre-launch analysis often focuses on the most extreme scenarios that predict the coldest \nand the hottest satellite temperatures. \n \nThe top left panel in figure~\\ref{fig:Tt3} compares the cold and hot cases for a spherical satellite, \nwith the extreme values of environmental parameters taken from Table~\\ref{tab:inputsEnvParam}. \nThe predicted temperature extremes differ by about 25 degrees.\n \nThe top right panel in figure~\\ref{fig:Tt3} explores the impact of uncertainties in orbital parameters \nand satellite orientation by varying $\\eta_E$ by 20\\% around its mean value (about twice as much \nas typical uncertainties for a 2U CubeSat). The predicted temperature extremes differ by about \n10 degrees.\n\nThe bottom left panel is analogous to the top left panel, except that typical values of $\\eta_S$\nand $\\eta_E$ for 2U CubeSat are used instead of values for a spherical satellite. \nNote that these parameters do {\\bf not} correspond to any particular satellite. \nAs expected, the predicted temperatures are about 8 $^\\circ$C higher for the spherical satellite \nbecause of higher absorbed direct solar radiation. \n\nThe bottom right panel in figure~\\ref{fig:Tt3} pushes the comparison of hot and cold cases for \n2U CubeSat to its extreme. It is assumed that for hot case the satellite orientation is actively \ncontrolled so that during non-eclipsed portion its maximum possible projected area is always \npointing towards the Sun, while during the eclipse it's pointed towards Earth (see figure caption). \nFor cold case, the projected areas towards the Sun and Earth are minimized. The resulting\ntemperature extremes differ by as much as 80 degrees. It is noteworthy that it is possible\nto reverse this scenario. If the satellite orientation is such that the projected areas are\nminimized for hot case, and maximized for cold case, the impact of environmental parameters \ncan be reversed and hot case can be made colder than cold case. In other words, {\\bf the satellite \norientation can be more important than the variation of environmental parameters.} \n\n\n\n\\section{Active temperature control \\label{sec:active}} \n\nGiven the allowed operating temperature ranges for satellite components (the most stringent requirement\ncomes from batteries, chosen here as 0--40 $^\\circ$C for illustration), these results imply that \nlow temperatures will be more worrisome than high temperatures. Motivated by this finding, we \nexplored a model for active temperature control.\n\nAs a concrete satellite example, we used SOC-i CubeSat developed at the University of Washington. \nAdopted parameter values are discussed and listed in Appendix D. \n\n\n\\subsection{A toy model for active temperature control } \n\nWe developed a toy model for active temperature control that assumes an additional internal power \ndissipation whenever the satellite temperature drops below a pre-defined threshold. Analytic solution\ngiven by eq.~\\ref{eq:analytic} is not applicable any more because the time dependence of heat source\nis now an unspecified function and numerical integration is used to obtain the solution\\footnote{Python\ncode and Jupyter notebooks are publicly available at https://github.com/ivezic/CubeSats}. Analytic \nsolution given by eq.~\\ref{eq:analytic} was used to validate the numerical solution code in case of bistable \nheat source. \n\nWe investigated cold case and three levels of power (2 W, 5, W, 10 W) that is applied whenever the temperature \ndrops below 273 K (0 $^\\circ$C). Results are shown in figure~\\ref{fig:SOCi2}. \nAdditional power can raise the satellite temperature by 5 to 11 degrees. The consumed power ranges\nfrom 1.9 Wh to 4.3 Wh, and it is under the total available battery power (5.6 Wh for cold case and\n$\\eta_{cell}=0.2$; for hot, extreme case, it could be boosted to 18 Wh with $\\eta_{cell}=0.3$). \nThese results show that such an approach is a viable method for mitigating low temperatures.\n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.3\\textwidth, keepaspectratio]{figures/TempsPlotCompare_SOCi-cold-heated2.png} \n\\includegraphics[width=0.3\\textwidth, keepaspectratio]{figures/TempsPlotCompare_SOCi-cold-heated5.png} \n\\includegraphics[width=0.3\\textwidth, keepaspectratio]{figures/TempsPlotCompare_SOCi-cold-heated10.png} \n\\caption{Illustration of the impact of active thermal control. A toy model assumes that whenever the satellite temperature\ndrops below zero $^\\circ$C (273 K), an additional heating source with power of 2 W (left), 5 W (middle), or 10 W (right) contributes\nto the heat balance. The blue lines correspond to the blue line in the left panel in figure~\\ref{fig:SOCi1} (cold case) and the \nred dashed line is the corresponding temperature prediction with this additional heating power. The consumed power is about \n1.9 Wh, 3.4 Wh and 4.3 Wh, respectively (the total available battery power is 5.6 Wh). \n\\label{fig:SOCi2}}\n\\end{figure}\n\n\n\n\n\n\\newpage\n\\section{Conclusions} \n\nWhen a body assumed to have a uniform temperature field is subjected to a bistable heat source,\nthere exists an analytic solution for the temperature variation with time. This simplified model is \nsuitable for addressing a variety of satellite thermal analysis problems: studying the relative effects \nof surfaces with different emissivities, the effects of small changes in the solar flux between June \nand December, the impact of thermal inertia on predicted amplitude of temperature variation, and \nas a ``sanity check'' for the results obtained with numerical thermal models that utilize detailed \ngeometrical and thermal descriptions of all satellite components. \n\nBrief examples of such studies are presented here. The most notable conclusions for further,\nmore detailed studies with numerical tools, include:\n\\begin{itemize}\n\\item The mean satellite temperature depends on the extreme values of steady-state equilibrium \n temperatures for eclipsed and non-eclipsed parts of the orbit, and the duration of the eclipse\n relative to the orbital period (see eq.~\\ref{eq:TeqAve}). \n\\item The amplitude of temperature variation around the mean temperature is by and large \n controlled by the satellite thermal inertia (see figure~\\ref{fig:Tt}).\n\\item Although one might naively think that the satellite temperature is lower during the eclipse\n than when the satellite is exposed to direct sunlight, the satellite temperature range is {\\bf identical} \n for these two orbital phases because of cyclic boundary condition (unless the thermal inertia \n is unrealistically low). \n\\item Satellites with non-spherical geometry can be modeled within the same framework with \n judiciously chosen effective surface areas, parametrized with $\\eta_S$ and $\\eta_E$\n (see eq.~\\ref{eq:Qsun} and Appendix A). \n\\item For non-spherical satellites, such as 2U CubeSat, the satellite orientation can have a\n significant impact on the predicted temperatures; indeed, with active 2U CubeSat orientation \n control, the impact of environmental variations could be mitigated entirely (i.e., ``hot case'' \n achieving lower temperatures than ``cold case''). \n\\item {\\bf Model uncertainties, including uncertainties in input parameters, result in uncertainties\n of predicted temperatures of at least 10 $^\\circ$C!} \n\\end{itemize} \n\nSince it appeared that low temperatures will be more concerning than high temperatures,\nwe also explored a toy model for active temperature control that assumes an additional internal \npower dissipation whenever the satellite temperature drops below a pre-defined threshold. \nWe found out that such an approach is a viable method for mitigating low temperatures.\n\nThe latex source for this document, and the supporting python code for evaluating analytic and\nnumerical models and producing all the plots presented here, are publicly available\\footnote{https://github.com/ivezic/CubeSats}. \n\n\n\\vskip 0.2in \n%\\newpage \n\\leftline{\\bf Acknowledgments} \nAn initial version of the code for obtaining a numerical solution of the differential equation \\ref{eq:dTdt} was \ncontributed by Haley Stewart (University of Washington). I thank the University of Washington SOC-i team, in \nparticular Boone Tate, Henry Brown and Charlie Kelly, for access to technical parameters describing their \n2U CubeSat named SOC-i. Without seeing their enthusiasm, I would have never written this paper. \n \n\n%\\bibliographystyle{aasjournal}\n%\\bibliography{ref}{}\n%\\input{appendix}\n\n\\vskip 0.2in \n\\leftline{\\bf References}\nGilmore, D. 2002, ``Spacecraft Thermal Control Handbook: Fundamental Technologies'', 2nd ed. Aerospace Press \n\nJacques, L. 2009, ``Thermal Design of the Oufti-1 Nanosatellite'', Master Thesis, University of Liege\n\n\n%\\newpage\n\\appendix{}\n\n\\vskip 0.2in\n\\leftline{\\bf A. Effective area for the absorption of radiation from Earth}\n\n\\vskip 0.1in\n Effective area for the absorption of radiation from Earth can be computed using geometric radiative viewing factors\n(and remembering that the factor $f_E$ is explicitly included in eq.~\\ref{eq:Qref}). \n\nFor a spherical satellite, \n\\eq{\n \\eta_E = {1\\over 2} \\left(1 - \\sqrt{1- f_E} \\right) \\, f_E^{-1},\n} \nwhere $f_E=[R_E/(R_E+h)]^2$, with Earth's radius $R_E=6,378$ km and $h$ is the satellite's altitude. As $h$\nincreases, $\\eta_E$ for sphere varies from 1/2 to 1/4; for $h=550$ km, $\\eta_E=0.36$.\n\nFor CubeSat satellites, $\\eta_E$ can be obtained as the sum of values for all 6 sides because the viewing\nfactors are additive. For a flat surface whose normal is at angle $\\beta$ relative to Earth's surface, \n\\eq{\n \\eta_E = \\cos(\\beta), \n} \nfor $|\\beta|\\le\\arccos(\\sqrt{f_E})$, and otherwise\n\\eq{\n \\eta_E = \\pi^{-1} \\left[ \\left( \\cos(\\beta)\\arccos(y) - x \\,z \\, \\sin(\\beta) \\right) + f_E^{-1} \\, \\arctan(x^{-1} \\, y \\, \\sin(\\beta)) \\right] \n} \nwhere $x=\\sqrt{f_E^{-1}-1}$, $y=-x\\tan(\\beta)^{-1}$ and $z=\\sqrt{1-y^2}$. The same expressions can\nbe used to compute $\\eta_S$ for CubeSat satellites by setting $h$ to a very large value. \n\nFor randomly oriented CubeSat satellites, $\\eta_S \\approx 0.21$ and $\\eta_E \\approx 0.36$ for both 1U and 2U versions,\nwith a scatter of about 10\\% around these mean values for realistic orientations. For 2U CubeSat\nwith $h=550$ km, the possible ranges are $\\eta_S = 0.10 - 0.30$ and $\\eta_E = 0.34 - 0.38$. \n\n\\vskip 0.2in\n\\leftline{\\bf B. A method for enforcing cyclic boundary condition for equation~\\ref{eq:dTdt}}\n\n\\vskip 0.1in\nGiven the orbital period and eclipse duration, two cooling time scales $t_o$ (see eq.~\\ref{eq:t0}) and two \nsteady-state equilibrium temperatures $T_{eq}^{sun}$ and $T_{eq}^{eclipse}$, there are four unknowns to \nbe solved for: $\\tau_o^C$, $\\tau_f^C$, $\\tau_o^H$, and $\\tau_f^H$, where $\\tau=T/T_{eq}$, subscripts \n$o$ and $f$ correspond to the initial and final values, and superscripts $C$ and $H$ correspond to \neclipsed and non-eclipsed parts of the orbit. \n \nTwo equations come from the cyclic boundary condition that the final temperature for the $C$ phase\nmust be equal to the initial temperature for the $H$ phase, and vice versa\n\\eq{\n\\label{eq:taueqs}\n \\tau_o^H = C_1 \\,\\tau_f^C \\,\\, \\,\\, {\\rm and} \\,\\, \\,\\, \\tau_f^H = C_1 \\, \\tau_o^C,\n} \nwhere $C_1 = T_{eq}^{eclipse}/T_{eq}^{sun} \\le 1$. The remaining two equations come\nfrom applying eq.~\\ref{eq:analytic} to $C$ and $H$ phases, where the left side is known and the right\nhand side involves $\\tau_o^C$ and $\\tau_f^C$, and $\\tau_o^H$ and $\\tau_f^H$, respectively. \n\nAfter substituting eqs.~\\ref{eq:taueqs} into two eqs.~\\ref{eq:analytic}, the resulting system of \ntwo equations with two unknowns is easily solved numerically. In case of numerical solution\nfor an arbitrary time dependence of the heating source, the cyclic boundary condition is satisfied\nusing iterations (usually only a few iterations are sufficient). \n\n%\\vskip 0.2in\n\\newpage\n\\leftline{\\bf C. Model parameters for the University of Washington SOC-i CubeSat}\n\nThe University of Washington SOC-i\\footnote{A nod toward the Pacific Northwest salmon.} \n(the Satellite for Optimal Control and Imaging) project\\footnote{https://www.aa.washington.edu/news/article/2019-02-11/cubesat-team}\nhas a specific mission to demonstrate the ability to satisfy two constraints with its orientation\ncontrol and imaging systems. It was selected by NASA CubeSat Launch Initiative for launch in\n2022 or 2023. We use it here as a specific example to quantitatively demonstrate the impact\nof active temperature control on the satellite's minimum temperature. \n\n\\vskip 0.1in \n\\leftline{\\bf Input satellite parameters}\n\nWe used the SOC-i CAD model\\footnote{I am grateful to Boone Tate for extracting the model parameters.} \ndeveloped for structural analysis to extract information about\nexternal satellite surfaces and their material properties. We adopted the following description \nof external SOC-i surfaces: \n\\begin{itemize}\n\\item Top: 60\\% solar panel, 40\\% aluminum frame\n\\item Bottom: 38\\% aluminum frame, 62\\% PCB (printed circuit board). \n\\item Sides (2): 64\\% solar panels, 19\\% aluminum panels (outside), 17\\% aluminum frame rails\n\\item Sides (2): 57\\% solar panels, 26\\% aluminum panels (outside), 17\\% aluminum frame rails\n\\end{itemize}\n\nWith the values of absorptivity and emissivity listed in Table~\\ref{tab:inputsAbsEmiss}, we obtained\ntheir surface-weighted values $\\alpha=0.83$ and $\\epsilon=0.79$ (54\\% of external surface area\nis covered by solar panels). In addition, we assumed that the satellite mass is $m=2.6$ kg and \nadopted specific heat corresponding to aluminum ($C=768$ J\\,kg$^{-1}$\\,K$^{-1}$), yielding a\nthermal intertia of $mC = 2.00$ kJ\\,K$^{-1}$ (we note that a more accurate value of thermal inertia \ncan be obtained by summing the $mC$ product for all individual structural components\nin the SOC-i CAD model). \n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Surface optical absorptivity and infrared emissivity for SOC-i surface materials. }\n\t\\label{tab:inputsAbsEmiss}\n\t\\begin{tabular}{r|r|r|r|r|r} % \n\t\t\\hline\n \t Part & Surface & $\\alpha$ & $\\epsilon$ & $k$ (W\\,m$^{-1}$\\,K$^{-1}$) & $C$ (J\\,kg$^{-1}$\\,K$^{-1}$) \\\\\n\t \t\\hline\n Solar panels & GaAs, with AR coating & 0.92 & 0.85 & 60.6 & 324 \\\\ \n Al panels, outside & 7075 Al, Kapton & 0.87 & 0.81 & 121.2 & 801 \\\\ \n Al frame rails & 5052 Al, hard anodized & 0.86 & 0.86 & 138.5 & 768 \\\\ \n Al frame & 5052 Al, alodine & 0.08 & 0.15 & 138.5 & 768 \\\\\n PCBs & FR4 & 0.81 & 0.90 & 18.0 & 1544 \\\\ \n \t\t\\hline \n\t\\end{tabular} \n\\end{table}\n \n\n\\vskip 0.1in \n\\leftline{\\bf Assumptions for orbital parameters}\n\nIt is already known that SOC-i will have a nearly-polar sun-synchronous \norbit\\footnote{See https://en.wikipedia.org/wiki/Sun-synchronous\\_orbit} \nwith an altitude of $h=550$ km and orbital inclination of 97.7 degrees. A satellite in sun-synchronous \norbit passes over any given point of the planet's surface at the same local mean solar time because the \norbit precesses through one complete revolution each year (that is, the orbit always maintains the same \nrelationship with the Sun). \n\nThe eclipse duration for sun-synchronous orbits depends on their right ascension of the \nascending node (RAAN), which will not be known until the launch date (RAAN is determined\nby the exact launch time). The orbital period for sun-synchronous orbit with an altitude of \n$h=550$ km is 96 mins, and the maximum eclipse duration is 36 mins. When the orbital \nplane is perpendicular to incoming solar radiation, there is no eclipse (the satellite is following \nthe terminator line at all times). \n\n\n\n\\vskip 0.1in \n\\leftline{\\bf Hot and cold cases} \n\nWe define ``hot'' and ``cold'' cases by first adopting \nthe extreme values of environmental parameters from Table~\\ref{tab:inputsEnvParam}. \nIn addition, we make an assumption that the orientation of SOC-i's sun-synchronous orbit\nresults in an eclipse with maximum duration (36 min) for cold case, and no eclipse at all for \nhot case. \n\nThe intensity of solar radiation reflected from Earth varies along the orbit. For a polar \norbit passing through subsolar point, $f_{alb}=0.62$ for the non-eclipsed part of the orbit, while \nfor a polar orbit aligned with the terminator (with no eclipse), $f_{alb}=0.06$. Therefore, \nwe adopt $f_{alb}=0.62$ for hot case and $f_{alb}=0.06$ for cold case (note that reflected solar \nradiation contributes more flux for cold case, when not in eclipse). \n \nWith these assumptions, we compute incoming heating flux.\nFor cold case, the only heating flux during the eclipsed portion of the orbit is IR flux from Earth. \nThe variation of flux between hot and cold cases for three main heat sources is summarized in \nTable~\\ref{tab:inputflux}. Note that reflected solar flux is smaller for hot case but this difference\nis compensated by the absence of eclipsed orbital portion in hot case. \n\nTable~\\ref{tab:inputflux} lists absorbed flux per unit area, assuming SOC-i's effective \nabsorption coefficient ($\\alpha$). The listed value are also appropriate for detailed \nAnsys-based modeling, but need to be corrected for $\\alpha$ of each surface material.\nThe actual absorbed power (absorbed energy per unit time) depends on \nthe values of $\\eta_S$ and $\\eta_E$, which in turn depend on orientation. We make additional \nassumptions about satellite orientation, as discussed next. \n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Absorbed flux ($\\alpha=0.83$) for hot and cold cases (in Wm$^{-2}$). }\n\t\\label{tab:inputflux}\n\t\\begin{tabular}{r|r|r|r} % \n\t\t\\hline\n \t Quantity & hot case & cold case & ratio hot/cold \\\\\n\t\t\\hline\n Direct solar flux & 1181 & 1098 & 1.08 \\\\\n Reflected solar flux & 21.0 & 144.2 & 0.15 \\\\ \n Earth IR flux & 174.6 & 147.8 & 1.18 \\\\\n\t\t\\hline\n\t\\end{tabular} \n\\end{table}\n\n\n\n\\vskip 0.1in \n\\leftline{\\bf Assumptions for satellite orientation}\n\nThe satellite orientation determines effective surface areas for the absorption of radiation from \nthe Sun and Earth. For convenience, these surface areas are expressed relative to the total surface area, $A_{tot}$\n(=0.1 m$^2$ for SOC-i), using $\\eta$ factors ($\\eta_S$ and $\\eta_E$, respectively). \nWe used results discussed in Section 2.4 to adopt the following values for SOC-i.\n\nWhen averaged over plausible orientations and orbits, $\\eta_S = 0.21$ and $\\eta_E = 0.36$, \nwith an uncertainty due to actual orbit specifics of the order 10\\%. The limits of \npossible ranges are $\\eta_S = 0.10 - 0.30$ and $\\eta_E = 0.34 - 0.38$. The limits for \n$\\eta_S$ reflect the range of projected area towards plane-parallel rays for 2U CubeSat geometry, \nwith the minimum value corresponding to one small side oriented perpendicularly to the incoming \nsolar radiation. For $\\eta_E$, the variation is much smaller because typically all six sides can \n``see'' Earth's surface\\footnote{We note that even for spherical geometry $\\eta_S$ and $\\eta_E$\nare generally different: $\\eta_S=1/4$, while $\\eta_E$ decreases from 1/2 to 1/4 as the orbit\naltitude varies from zero to infinity.}. \n \nWe do not adopt specific satellite orientation for hot and cold cases but instead explore two \noptions in each case. First, we adopt averaged orientations for both hot and cold cases,\nwith $\\eta_S = 0.21$ and $\\eta_E = 0.36$ corresponding to 2U CubeSat values. As the second \nassumption, we consider the following extreme cases: $\\eta_S = 0.30$ and $\\eta_E = 0.34$ for \nhot case, and $\\eta_S = 0.10$ and $\\eta_E = 0.38$ for cold case. \n \nThe second set of values assumes that the satellite orientation is actively controlled. For hot\ncase, the maximum possible projected satellite area for plane-parallel rays is always pointing \ntowards the Sun ($\\eta_S = 0.30$). For cold case and during non-eclipsed portion, the smallest \nsatellite side is always pointing towards the Sun ($\\eta_S = 0.10$). The adopted values of \n$\\eta_E$ are its extreme values. \n\n \n\\vskip 0.1in \n\\leftline{\\bf Assumptions for internal power dissipation}\n\n\nA fraction of absorbed optical flux (the sum of direct solar flux and reflected solar flux) is often \nused to charge on-board batteries. We assume that 20\\% of absorbed flux\\footnote{Here, $\\eta_{cell}$ \nrepresents the fraction of all absorbed radiation that was converted to battery charge. For example, if \nthe cells occupy 2/3 of all external surfaces, and the cell conversion efficiency is 30\\%, then $\\eta_{cell}$= 0.2. \nFor randomized orientiations, it's only ``effective'' quantities that count in the model considered here; \nhowever, when a specific satellite orientation is known, one could incorporate information about where \nexactly the solar cell panels are positioned, too.} is converted into chemical\nenergy ($\\eta_{cell}=0.2$). This energy is returned back at a constant rate as internal heat dissipation. \n\nIn hot case, satellite is always exposed to the Sun and there is {\\bf no net effect} within the context\nof single-temperature model considered here. In reality, and in detailed Ansys models, this\ninternal heat dissipation can modify the temperature distribution within the satellite (areas closer\nto the heater will have elevated temperature). In cold case, the effect of internal heat dissipation\nis to {\\bf minimize} the amplitude of temperature variation, or equivalently, to {\\bf raise the minimum\ntemperature} (at the end of eclipsed portion). \n \nThese assumptions complete the specification of SOC-i hot and cold thermal models. \n\n\\begin{table}[t]\n\t\\centering\n\t\\caption{Absorbed power (in Watt) and equilibrium temperatures for hot and cold SOC-i cases. }\n\t\\label{tab:powertemp}\n\t\\begin{tabular}{r|r|r|r|r} % \n\t\t\\hline\n \t Quantity & hot, random & hot, extreme & cold, random & cold, extreme \\\\\n\t\t\\hline\n Absorbed direct solar & 24.8 & 36.6 & 23.1 & 11.0 \\\\\n Absorbed Earth albedo & 0.8 & 0.8 & 5.2 & 4.9 \\\\ \n Absorbed Earth IR & 6.3 & 6.6 & 5.3 & 5.0 \\\\\n Internal dissipation & 5.1 & 7.5 & 3.5 & 2.0 \\\\\n\t\t\\hline \n Total input in eclipse & --- & --- & 8.9 & 7.0 \\\\\n Total input in sun & 31.8 & 44.0 & 31.4 & 19.7 \\\\\n\t\t\\hline\n Equilibrium T in eclipse & --- & --- & 211 & 199 \\\\\n Equilibrium T in sun & 290 & 315 & 289 & 257 \\\\\n \t\t\\hline\n $T_{min}$ (K) & 290 & 315 & 259 & 235 \\\\\n $T_{max}$ (K) & 290 & 315 & 274 & 244 \\\\\n\t\t\\hline\n $T_{min}$ ($^\\circ$C) & 16.7 & 41.5 & {\\bf $-$14.2 } & {\\bf $-$37.6} \\\\\n $T_{max}$ ($^\\circ$C) & {\\bf 16.7 } & {\\bf 41.5} & 1.3 & $-$28.9 \\\\\n\t\t\\hline\n\t\\end{tabular} \n\\end{table}\n\n \n\n\n\\vskip 0.1in \n\\leftline{\\bf Predicted absorbed power and equilibrium temperatures for hot and cold cases} \n\n\nGiven all the input assumptions described above, it is straightforward to solve the\ngoverning equation with direct numerical integration. Table~\\ref{tab:powertemp} lists predicted \nabsorbed power for all four modeled cases. We note that the total energy stored in batteries, and \ndissipated as heat at a constant rate, ranges from 3.2 Wh for cold, extreme case to 12 Wh for hot, \nextreme case. \n\n\\end{document}\n ", "meta": {"hexsha": "e3e97a14c008d261d9fc06f62d6e662490f2bce5", "size": 60498, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/cubsats.tex", "max_stars_repo_name": "ivezic/CubeSats", "max_stars_repo_head_hexsha": "b392124c8481b10721008184b884fe488801b0bf", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper/cubsats.tex", "max_issues_repo_name": "ivezic/CubeSats", "max_issues_repo_head_hexsha": "b392124c8481b10721008184b884fe488801b0bf", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/cubsats.tex", "max_forks_repo_name": "ivezic/CubeSats", "max_forks_repo_head_hexsha": "b392124c8481b10721008184b884fe488801b0bf", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.6627218935, "max_line_length": 191, "alphanum_fraction": 0.7235776389, "num_tokens": 16180, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3445189262749576}} {"text": "\\documentclass[pdftex,12pt,a4paper]{article}\n\n\\usepackage[pdftex]{graphicx}\n\n\\newcommand{\\HRule}{\\rule{\\linewidth}{0.5mm}}\n\n\\begin{document}\n\t\\input{./title.tex}\n\t\n\t\\tableofcontents\n\t\\newpage\n\t\n\t\\section{Problem Statement}\n\t\tUsing the movielens ml-100k data set, build a system that predicts for an unknown user (whose simple demographics are\navailable):\n\t\t\\begin{itemize}\n\t\t\t\\item Whether the user is likely to see a given movie.\n\t\t\t\\item And if the answer to the above is Yes then the star rating (on a scale of 1 to 5) that the user will most likely give to the movie.\n\t\t\\end{itemize}\n\t\n\t\\section{Challenges}\n\t\tToday's world is full of choices across all product categories which makes it important for consumers to always make a better choice. Recommender systems were created to solve this need and have revolutionalized e-commerce and advertising. Movie recommendation systems study a user's movie viewing and rating history and intelligently predict the genre of movies that a particular user might prefer and the rating he might give to a movie. This provides a powerful tool to both movie viewers and distributers. In a movie recommender system there can be several challenges:\n\t\t\\begin{itemize}\n\t\t\t\\item If a movie has very few ratings, it is difficult to predict recommendations on it.\n\t\t\t\\item Users have varying rating patterns. Some users give an average rating to almost all movies while some always give extreme ratings. Still others show a spread from 1 to 5. Some users give higher ratings to some directors and stars.\n\t\t\t\\item There can be attributes which are not given in the existing data set but which contribute to a user's rating.\n\t\t\\end{itemize}\n\t\n\t\\section{Related Work}\n\t\tEarlier works~\\cite{Schein2002,MarlinThesis2004} have implemented several techniques like K-Nearest Neighbour classifiers, Naive Bayes classifiers and K-Medians clustering to make recommendations on this same dataset and to compare the results to Probabilistic and Collaborative prediction techniques. The most effective techniques for movie recommendation on this dataset have been Collaborative Filtering~\\cite{Sarwar2001,Agarwal2009} and other matrix factorization methods~\\cite{CoTriFactor2009}. MovieLens itself uses collaborative filtering to make recommendations~\\cite{Herlocker1999}.\n\t\n\t\\section{DataSet Review}\n\t\tThe data set consists of entries for 943 users, 1,682 movies and 100,000 ratings. Demographic data of the users has also been provided and includes age, gender, occupation and zipcode. The genre of the movies and some other information about the movies is also provided.\n\t\n\t\\section{Methodology}\n\t\tWe are using Collaborative Filtering to provide movie recommendations. The rating $r$ which we predict for a movie $i$ by a user $u$ will be an aggregation of ratings given by top N users who are most similar to user $u$ and rated movie $i$.\n\t\t\\begin{center}\n\t\t\t$r_{u,i} = aggr_{u^\\prime \\in U} r_{u^\\prime, i}$\n\t\t\\end{center}\n\t\twhere $U$ is the set of all users except $u$.\\\\\\\\\n\t\tFor finding the most similar N users, we will use either the Pearson correlation similarity between two users x, y which is:\n\t\t\\begin{center}\n\t\t\t$simil(x,y) = \\frac{\\sum\\limits_{i \\in I_{xy}}(r_{x,i}-\\bar{r_x})(r_{y,i}-\\bar{r_y})}{\\sqrt{\\sum\\limits_{i \\in I_{xy}}(r_{x,i}-\\bar{r_x})^2\\sum\\limits_{i \\in I_{xy}}(r_{y,i}-\\bar{r_y})^2}}$\n\t\t\\end{center}\n\t\tor the cosine similarity which is:\n\t\t\\begin{center}\n\t\t\t$simil(x,y) = \\frac{\\sum\\limits_{i \\in I_{xy}}r_{x,i}r_{y,i}}{\\sqrt{\\sum\\limits_{i \\in I_{xy}}r_{x,i}^2}\\sqrt{\\sum\\limits_{i \\in I_{xy}}r_{y,i}^2}}$\n\t\t\\end{center}\n\t\twhere $I_{xy}$ is the set of movies rated by both users x and y.\\\\\\\\\n\t\tWhile both of these are memory based approaches, we might also go for model based approaches. The similarity functions can be used to find similarity between both users and movies. The final recommeder system will handle 2 tasks:\n\t\t\\begin{itemize}\n\t\t\t\\item Recommendation: Will the user like a movie? This can either be given as a confidence measure or as a simple yes or no answer by keeping a threshold on the confidence measure. For this, we will first find out some N similar users to our new user using user attributes if the user is completely new or using user attributes and past history if he has rated some movies. Based on this we will recommend a movie.\n\t\t\t\\item Prediction: If the user will view a movie, what rating will he give to it? This will be a number from 1 to 5. This will also take into account ratings the user has given to similar movies.\n\t\t\\end{itemize}\n\n\t\\bibliographystyle{amsplain}\n\t\\bibliography{references}\n\\end{document}", "meta": {"hexsha": "c86ea53f92cadbd9c8f608d510fdda1a8c540f4f", "size": 4597, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Reports/Initial Report/Group_11_Project_Proposal.tex", "max_stars_repo_name": "kracwarlock/Movie-Recommender-and-Score-Prediction-System", "max_stars_repo_head_hexsha": "4298a8e3a8e621fff76f7c751e5ec64f8f6ad67b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2016-02-23T23:17:06.000Z", "max_stars_repo_stars_event_max_datetime": "2019-03-17T10:55:10.000Z", "max_issues_repo_path": "Reports/Initial Report/Group_11_Project_Proposal.tex", "max_issues_repo_name": "kracwarlock/Movie-Recommender-and-Score-Prediction-System", "max_issues_repo_head_hexsha": "4298a8e3a8e621fff76f7c751e5ec64f8f6ad67b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Reports/Initial Report/Group_11_Project_Proposal.tex", "max_forks_repo_name": "kracwarlock/Movie-Recommender-and-Score-Prediction-System", "max_forks_repo_head_hexsha": "4298a8e3a8e621fff76f7c751e5ec64f8f6ad67b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2015-09-27T18:04:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-08-29T03:23:51.000Z", "avg_line_length": 79.2586206897, "max_line_length": 593, "alphanum_fraction": 0.7624537742, "num_tokens": 1223, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030761371503, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.3445189176770466}} {"text": "% A '%' character causes TeX to ignore all remaining text on the line,\n% and is used for comments like this one.\n% All LaTex files require the commands:\n% \\documentclass (with argument)\n% \\begin{document} command\n% \\end{document} command\n\n% specifies the document class. We usually use article but there are others. \n\\documentclass{article} \n\n% these are standard packages used for the math symbols\n\\usepackage{amsmath,amssymb,amsthm, graphicx, color, gensymb,mathpazo, enumitem}\n\n% These commands below is to make sure the numbering of these are consistent with theorem\n% If you are not sure what something means, delete them, build a new file and see the\n% difference between the files. You can ignore this part for now.\n\\newtheorem{theorem}{Theorem}[section]\n\\newtheorem{conjecture}[theorem]{Conjecture}\n\\newtheorem{observation}[theorem]{Observation}\n\\newtheorem{definition}[theorem]{Definition}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{lemma}[theorem]{Lemma}\n\\newtheorem{example}[theorem]{Example}\n\\newtheorem{remark}[theorem]{Remark}\n\\newtheorem{notation}[theorem]{Notation}\n\n\n\\begin{document}\n\n% Title of my document\n\\title{\\Large Infinite Stream of Electrons}\n\n% The author command places text right after title\n\\author{Prannaya Gupta}\n\\date{\\today}\n\\maketitle\n\n\\section{Question}\nLet's say you have devised a system called the \"Stop-Time-inator\" that has allowed you to momentarily stop time. In that moment, no effect of force is seen but the force you yourself apply, and your physical body does not age. You thus stop time and open up a dimension, let's just call this the Cartesian Dimension, where it's just a single infinite plane with no gravitational force present. You enter it, starting at a certain origin which is assumed to be the centre of this dimension. From here, you can also see four paths around you, representing the positive and negative x-axis and positive and negative y-axis of the Cartesian Coordinate plane. Now, in stopped time, you get to work.\n\\\\\n\\\\\nYou bring along a device called the \"Charged-Particle-inator\" which releases a charged particle of charge q, set by the user. Now, you place your hand on the knob and cover the dial so that you cannot see the charge q that you set. You turn the knob and release it. Let's say this charge is denoted as a value $q_o$.\n\\\\\n\\\\\nNow, you decide to do very interesting thing: you use the python random module to generate a random integer between 2 and a super large number. You label this value as k in the interpreter, but do not check what k is. Now you load this values of k into a \"Move-inator\", which allows you to go to a specific x and y with respect to the assumed origin. What this \"Move-inator\" now does is it sends the \"Charged-Particle-inator\" to points ($k^n$, $k^n$) for all n from 0 to infinity (let's just assume that the \"Move-inator\" is able to do this within the stopped time) and releases one charge there.\n\\\\\n\\\\\nNow it returns, and you haven't died (consider yourself lucky). Now you have set the Move-inator to go to the points ($-k^n$, $-k^n$) for all n from 0 to infinity (again, let's just assume that the \"Move-inator\" is able to do this within the stopped time), and to each n, it swaps another charge knob that toggles the charge between positive and negative while keeping the magnitude intact (i.e. at ($-k^0$, $-k^0$), a negative charge of $q_o$ is released, while at (-k, -k), a positive charge of $q_o$ is released and so on). \n\\\\\n\\\\\nNow you use another device you have created called the \"Uncharge-inator\" and use it to completely discharge yourself, and then another called the \"Charge-inator\" to charge yourself to the same $q_o$ which you still do not know. Now you relocate to the assumed origin and stop the the \"Stop-Time-inator\". What is the acceleration that you, a 60-year old esteemed physicist who has spent years devising this complex system and has a mass of m, experience?\n\n\\begin{enumerate}[label=\\uppercase\\alph*)]\n \\item $\\frac{q_o}{4\\pi\\epsilon_0} \\frac{k^2}{m(k^4 - 1)}$, facing north-east\n \\item \\textbf{$\\frac{q_o}{4\\pi\\epsilon_0} \\frac{k^4}{m(k^4 - 1)}$, facing south-west}\n \\item $\\frac{q_o}{4\\pi\\epsilon_0} \\frac{k^4}{m(k^4 - 1)}$, facing north-east\n \\item $\\frac{q_o}{4\\pi\\epsilon_0} \\frac{k^2}{m(k^4 - 1)}$, facing south-west\n\\end{enumerate}\n\n\\newpage\n\\section{Solution}\n\nTo calculate this, you must use the formula \n\n\\begin{align*}\nF &= \\frac{1}{4\\pi\\epsilon_0} \\frac{Qq}{r^2}\n\\end{align*}\n\\\\\nHere, we know that Q and q are both equal to $q_o$, which you already know as a constant, but the direction may change based on the charge of the charges dropped by the \"Charged-Particle-inator\". Thus, you extract it as \n\\\\\n\\begin{align*}\nF &= \\frac{q_o}{4\\pi\\epsilon_0} \\frac{1}{r^2}\n\\end{align*}\n\\\\\nTo calculate $r^2$, what you do is you calculate based on the points as given. From this we have $r^2 = k^{2n} + k^{2n} = 2k^{2n}$.\n\\\\\\\\\nWe thus derive a series representing the force exhibited for the positive direction (i.e. x and y are both positive) \n\\\\\n\\begin{align*}\nF_1 &= \\frac{q_o}{4\\pi\\epsilon_0}(\\frac{1}{2k^0} + \\frac{1}{2k^2} + \\frac{1}{2k^4} + ...)\n\\end{align*}\n\\\\\nwhich faces south-west.\n\\\\\n\\\\\nMeanwhile, we can also derive a similar series representing the force exhibited for the positive direction (i.e. x and y are both positive) \n\\\\\n\\begin{align*}\nF_2 &= \\frac{q_o}{4\\pi\\epsilon_0}(\\frac{1}{2k^0} - \\frac{1}{2k^2} + \\frac{1}{2k^4} - ...) \n\\end{align*}\n\\\\\nwhich faces south-west as well.\n\\\\\n\\\\\nWe add these two forces up and we get\n\\\\\n\\begin{align*}\nF_{net} &= \\frac{q_o}{4\\pi\\epsilon_0}(\\frac{1}{k^0} + \\frac{1}{k^4} + \\frac{1}{k^8} + ...)\n\\end{align*}\n\\\\\n which faces south-west.\n\\\\\n\\\\\nNow, I would like to divert to derive an expression for\n\\\\\n\\begin{align*}\nS &= \\frac{1}{k^0} + \\frac{1}{k^4} + \\frac{1}{k^8} + ...\n\\end{align*}\n\\\\\n\\\\\nTo do this, we derive \n\\\\\n\\begin{align*}\nk^4S &= k^4 + \\frac{1}{k^0} + \\frac{1}{k^4} + .... \\\\\n&= k^4 + S\n\\end{align*}\n\\\\\n\\\\\nFrom this, we get \n\\\\\n\\begin{align*}\nS &= \\frac{k^4}{k4 - 1}\n\\end{align*}\n\\\\\n\\\\\nNow substituting back in, we get \n\\\\\n\\begin{align*}\nF_{net} &= \\frac{q_o}{4\\pi\\epsilon_0} \\frac{k^4}{k4 - 1}\n\\end{align*}\n\\\\\n\\\\\nSince $F_{net} = ma$\n\\\\\n$\\therefore a &= \\frac{q_o}{4\\pi\\epsilon_0} \\frac{k^4}{m(k^4 - 1)}$, facing south-west.\n\n\\end{document}", "meta": {"hexsha": "dc717a1735a6ee23673ff31dfb5527b37cc80cc2", "size": 6273, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Y3S2/Electrostatics/Inator/Inator.tex", "max_stars_repo_name": "ThePyProgrammer/peerwise", "max_stars_repo_head_hexsha": "1da806a6b238ce13ae9c7328953b2497d2da9e25", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Y3S2/Electrostatics/Inator/Inator.tex", "max_issues_repo_name": "ThePyProgrammer/peerwise", "max_issues_repo_head_hexsha": "1da806a6b238ce13ae9c7328953b2497d2da9e25", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Y3S2/Electrostatics/Inator/Inator.tex", "max_forks_repo_name": "ThePyProgrammer/peerwise", "max_forks_repo_head_hexsha": "1da806a6b238ce13ae9c7328953b2497d2da9e25", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.7883211679, "max_line_length": 693, "alphanum_fraction": 0.7122588873, "num_tokens": 1907, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953797290153, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.34443738628506126}} {"text": "\\documentclass[a4paper]{article}\n\n\\input{temp}\n\n\\begin{document}\n\n\\title{Model Theory Example Sheet 3}\n\n\\author{Xiaojia Rao}\n\n\\maketitle\n\n\\setcounter{secnumdepth}{0}\n\n\\section{Question 4}\n\nSuppose $a \\not\\in acl(A)$. Then there exists a model $\\mathcal{M} \\preccurlyeq U$ which contains $A$ but not $a$. But $a \\in acl(A,b)$ for any realization $b$ of $\\phi(z)$, so $\\mathcal{M}$ cannot contain any of those $b$, i.e. $\\mathcal{M}$ does not contain any realization of $\\phi(z)$. But it does contain $A$, so the sentence $\\exists z \\phi(z)$ should be true in $\\mathcal{M}$ since it is true in $U$ and by elementarity. Contradiction.\n\nFor types it's similar: similarly we have a model $\\mathcal{M}$ containing $A$ that does not contain $a$ and has no realization of $p(z)$. Note that by elementarity $p(z)$ is still finitely consistent in $\\mathcal{M}$, so $p(z)$ must be an infinite type.\n\n\\section{Question 7}\n\nSuppose $C \\cap acl(A) = \\phi$. Then for each $c \\in C$, consider its type over $A$, $p_c(x):=tp(c/A)$: for $\\phi(x) \\in p_c(x)$, we have $|\\phi(U)| \\geq \\omega$. So by the same argument from lecture we know $|p_c(U)| = |U|$.\\\\\nWe continue mimicking the proof: let $\\mathcal{M} \\supset A$ be a model, then $p_c(U) \\setminus \\mathcal{M} \\neq \\phi$ for each $c \\in C$. Pick $b_c \\in p_c(U) \\setminus \\mathcal{M}$, then by homogeneity\n\n\\end{document}\n", "meta": {"hexsha": "7d8dd4fa6fc9197a8f86f62232643c1091f80eda", "size": 1360, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Notes/Model Theory ES3.tex", "max_stars_repo_name": "raoxiaojia/raoxiaojia.github.io", "max_stars_repo_head_hexsha": "d20c23a64794b500f2e0356fd01017ee31830fa2", "max_stars_repo_licenses": ["CC-BY-3.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-01-25T17:34:25.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-25T17:34:25.000Z", "max_issues_repo_path": "Notes/Model Theory ES3.tex", "max_issues_repo_name": "raoxiaojia/raoxiaojia.github.io", "max_issues_repo_head_hexsha": "d20c23a64794b500f2e0356fd01017ee31830fa2", "max_issues_repo_licenses": ["CC-BY-3.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Notes/Model Theory ES3.tex", "max_forks_repo_name": "raoxiaojia/raoxiaojia.github.io", "max_forks_repo_head_hexsha": "d20c23a64794b500f2e0356fd01017ee31830fa2", "max_forks_repo_licenses": ["CC-BY-3.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 50.3703703704, "max_line_length": 442, "alphanum_fraction": 0.6801470588, "num_tokens": 450, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3444373776110126}} {"text": "\\chapter{Microbial Network Modelling}\n\\pagenumbering{arabic} \\setcounter{page}{21}\n\nAs stated in the previous chapters, the microbes form associations, which can be studied using the principles of network science. There is a long chain of essential process from the abundance matrix of sequences to community network models. This chapter reviews the strategies involved while making microbial co-occurrence networks.\n\n\\begin{figure}[!hb]\n \\centering\n \\includegraphics[width=12cm, height=6cm] {../figures/Figure6.png}\n \\caption{Graphical representation of the pairwise interactions}\n \\label{fig:figure6}\n\\end{figure}\n\n\\section{Pairwise Interactions}\nThe associations are primarily classified into three categories,i.e., positive interaction, negative interaction, of neutral interaction. Figure \\ref{fig:figure6} shows the graphical summary of the interaction types \\cite{faust_2012_microbial}. The most basic interaction is \\emph{mutualism}, which is a win-win scenario for both members of the pair. These members benefit from each other under this association (e.g. co-operative and symbiotic pairs). Opposite to that, the \\emph{Competition} association is the lose-lose situation as both the members compete for resources. \\emph{Predation} \\& \\emph{Parasitism} can have win-lose effects on either member as if one is the predator/parasite; the other has to be the prey/host. Following that, in a situation where a member is neutral, and the other member in the pair is being harmed, this kind of association is known as \\emph{Amensalism} or \\emph{Commensalism} if other benefits. Co-occurrence networks derived using pairwise associations are perceived as simple networks and made using similarity-based inferences as discussed in the previous chapters \\cite{faust_2012_microbial}. Initially, all the normally derived permutation-combinations are used to calculate the mathematical measures and then the significant ones are retained to form co-occurrence networks [Figure \\ref{fig:figure7}].\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=12cm, height=7.5cm] {../figures/Figure7.png}\n \\caption{Steps involved for designing the pairwise-interaction network}\n \\label{fig:figure7}\n\\end{figure}\n\n\\section{Complex Interactions and Dynamic Networks}\nThe cases may arise when more than one microbial interaction is under study. The associations involve more than two members undergoing either positive, negative or neutral associations. The rules of pairwise interaction still apply; however, the number of contributing members changes in order to map the complex interactions methods like multiple-regression and associative rule mining is used\\cite{faust_2012_microbial} . The multiple-regression predicts the dependent variable/interaction based on the independent variables/group of pairwise interactions. The networks generated are called the directed hypergraphs, which have hyperedges \\cite{faust_2012_microbial}. Figure \\ref{fig:figure8} shows a typical complex co-occurrence network made using associative rule mining. The generalized Lotka-Volterra equations used to simulate a dynamic model of a microbial community. As input, they need the growth rates $b_{i}$ and interaction strengths $a_{ij}$ of the different community members, which can either be obtained experimentally or estimated from time-series data. The abundance of taxa $i$ at time $t$ are denoted by $x_{i}(t)$\n\n$$\\frac{dx_{i}(t)}{dt} = x_{i}(t)(b_{i} + \\sum^{N}_{j=1}a_{ij}x_{j}(t))$$\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=12cm, height=8cm] {../figures/Figure8.png}\n \\caption[A typical directed hypergraph]{A typical directed hypergraph made using the OTUs. Hyper-edges connects more than two nodes}\n \\label{fig:figure8}\n\\end{figure}\n\n\\subsection*{Pitfalls}\nThe networks rec-construction is not independent of compositional bias. Arising from the sampling, the measure of the relative abundance can heavily impact the downstream analysis \\cite{faust_2012_microbial}. Other than that, microbial datasets are full of zeros that could arise because either sampling is inappropriate or the member is missing. Even if the construction is successful, an inference could be error-prone, as it is difficult to segregate between the interaction.", "meta": {"hexsha": "59364fea01121a61160994da229a6734a73e005d", "size": 4241, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "literatureReview/tex/ch7.tex", "max_stars_repo_name": "spriyansh/metagenomics-CSG-2021", "max_stars_repo_head_hexsha": "bfa93af6bb8631ee0e19ac6005db72bbf18f0fae", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "literatureReview/tex/ch7.tex", "max_issues_repo_name": "spriyansh/metagenomics-CSG-2021", "max_issues_repo_head_hexsha": "bfa93af6bb8631ee0e19ac6005db72bbf18f0fae", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "literatureReview/tex/ch7.tex", "max_forks_repo_name": "spriyansh/metagenomics-CSG-2021", "max_forks_repo_head_hexsha": "bfa93af6bb8631ee0e19ac6005db72bbf18f0fae", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 117.8055555556, "max_line_length": 1345, "alphanum_fraction": 0.8033482669, "num_tokens": 978, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5774953651858117, "lm_q1q2_score": 0.34443737761101256}} {"text": "\\section{Inference}\n\\label{inf}\n\nTRILL systems can answer many different queries. To do so, it exploits an algorithm called \\emph{tableau} algorithm, which is able to collect explanations. In the following you can find an example that shows how the tableau works. In section~\\ref{sec:trillq} we will see how queries can be asked with TRILL systems.\n\nConsider a simple knowledge base inspired by the film ``The Godfather'' containing the following axioms:\n\\begin{align}\n&tom : Cat\\\\\n&(donVito, tom) : hasPet\\\\\n&Cat \\sqsubseteq Pet\\\\\n&\\exists hasAnimal.Pet \\sqsubseteq NatureLover\\\\\n&NatureLover \\sqsubseteq GoodPerson\\\\\n&hasPet \\sqsubseteq hasAnimal\n\\end{align}\n\nThe axioms are telling what is known about the domain: (1) Tom is an individual of the domain, and he is a Cat; (2) donVito (Vito Corleone) has tom as his pet; (3) all cats are also pets; (4) everyone having at least one animal which is a pet is a nature lover; (5) nature lovers are good people; and (6) if one has a pet, she/he also has an animal.\n\nThis KB can be defined by the following TRILL syntax axioms:\n\\begin{verbatim}\nclassAssertion(cat, tom).\npropertyAssertion(hasPet, donVito, tom).\nsubClassOf(cat, pet).\nsubClassOf(someValuesFrom(hasAnimal, pet), natureLover).\nsubClassOf(natureLover,goodPerson).\nsubPropertyOf(hasPet,hasAnimal).\n\\end{verbatim}\nYou can run this example \\href{http://trill-sw.eu/example/trill/donVito.pl}{here}.\n\nThe first two axioms are assertional axioms (hence they constitute the ABox), the other four axioms define the TBox. Axiom 1 is called class assertion, 2 is called property assertion, 3,4,5 are called class subsumption axioms, and axiom 6 is called property subsumption axiom.\n\nTo check, for example, whether don Vito Corleone is a good person, the tableau algorithm builds a graph, called the \\emph{tableau}. The initial tableau contains information from the ABox plus the negation of the query, as depicted in Figure\\ref{fig:tab1}. This last axiom is added since the underlying proof mechanism uses refutation. In logic, working by refutation means assuming the opposite of the query one wants to prove. Then, if this assumption leads to a contradiction, this means that the axioms of the ontology allows to prove that the query is true, and thus that its opposite is false. In practice, working by refutation means that the graph must assume that the posed query be false, the tableau algorithm expands all the known axioms (including the negation of the query) and looks for contradictions present in the final graph. The presence of a contradiction in a node proves that the query is true because the graph depicts at least one way to contradict the negation of that query, and thus it depicts at least one way to prove that the opposite of the query contradicts what is defined by the ontology. \n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.5\\linewidth]{img/tab1}\n\t\\caption{Initial tableau}\n\t\\label{fig:tab1}\n\\end{figure}\n\nThis means that if the opposite of the query is (artificially) added to the knowledge base as a new axiom, this ontology will contain at least two pieces of information one contradicting the other.\n\nThe tableau has one node for each individual: tom is labelled as cat, $donVito$ is labelled as not a good person (the negation of the query), and the edge between them is labelled as $hasPet$ because the individuals are connected by this property (Figure~\\ref{fig:tab1}).\n\nAt this point, the graph of Figure~\\ref{fig:tab1} is expanded using the axioms of the ontology to check the truth of the query and to build the justifications. Therefore, the tableau algorithm takes e.g. the axiom 3, ``cats are pets'', and adds to the node for tom also the label $pet$ since he is a cat. This new information is true and its justification is given directly by the set of axioms {1,3}: axiom 3 because since $tom$ is a $cat$ (axiom 1) he is also a $pet$. The same operation can be done for the edge (relationship) between tom and $donVito$, which can be labelled also as $hasAnimal$ because of axioms~2 and~6.\n\nAt this point, the calculus can deduce that $donVito$ belongs to the class \\linebreak $\\exists hasAnimal.Pet$ because $donVito$ is connected with $tom$, which is a $pet$ (axioms {1,3}), via property $hasAnimal$ (axioms {2,6}). Therefore, $donVito$’s node is labelled also as $\\exists hasAnimal.Pet$ with a justification given by the union of the axioms associated with the used axioms, therefore its justification is given by the set of the involved axioms {1,2,3,6}. Then, the tableau graph is further expanded by adding the class $NatureLover$ to $donVito$’s node using axiom 4 and finally, by adding also the class $GoodPerson$ using label $NatureLover$ (axioms {1,2,3,4,6}) and axiom 5, creating as justification the set of axioms {1,2,3,4,5,6}.\n\nThe final graph is shown in Figure~\\ref{fig:tab2}.\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.5\\linewidth]{img/tab2}\n\t\\caption{Final tableau}\n\t\\label{fig:tab2}\n\\end{figure}\nThe expanded graph contains now a contradiction, i.e., $donVito$ is labelled as $GoodPerson$ and as not a $GoodPerson$ (i.e., $\\neg GoodPerson$), therefore, by refutation, the query ``Is don Vito Corleone a good person?'' is true, with justification given by the axioms {1,2,3,4,5,6}, that are the axioms of the KB necessary to deduce this information.\n\nFrom this example, it would be clear why the use of probabilistic information is useful. Indeed, don Vito Corleone is hardly classifiable as a good person. This is because not all people who are nature lovers are also good, and therefore, one could say that axiom 5 is true with probability 0.4. It would also be arguable that everyone who has animals is also a nature lover, making probabilistic also this axiom. For a formal description of how the probability of the query is computed see the Appendix~\\ref{app:inf}.", "meta": {"hexsha": "766cdb661e3e9bb3656b6a54452a4d8805fdc6e3", "size": 5848, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/inference.tex", "max_stars_repo_name": "rzese/trill", "max_stars_repo_head_hexsha": "145781966a4203e43c3cfe18e62435557e764297", "max_stars_repo_licenses": ["Artistic-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2016-02-02T07:36:34.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-17T01:23:49.000Z", "max_issues_repo_path": "doc/inference.tex", "max_issues_repo_name": "rzese/trill", "max_issues_repo_head_hexsha": "145781966a4203e43c3cfe18e62435557e764297", "max_issues_repo_licenses": ["Artistic-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2019-10-19T12:53:55.000Z", "max_issues_repo_issues_event_max_datetime": "2019-10-29T15:14:51.000Z", "max_forks_repo_path": "doc/inference.tex", "max_forks_repo_name": "rzese/trill", "max_forks_repo_head_hexsha": "145781966a4203e43c3cfe18e62435557e764297", "max_forks_repo_licenses": ["Artistic-2.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2016-04-05T10:34:18.000Z", "max_forks_repo_forks_event_max_datetime": "2019-10-30T10:46:54.000Z", "avg_line_length": 102.5964912281, "max_line_length": 1123, "alphanum_fraction": 0.7742818057, "num_tokens": 1519, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858117, "lm_q2_score": 0.5964331462646255, "lm_q1q2_score": 0.34443737761101256}} {"text": "\\documentclass{amsart}\r\n\r\n\\usepackage{amsmath}\r\n\\usepackage[framemethod=TikZ]{mdframed}\r\n\\usepackage{xparse}\r\n\\ExplSyntaxOn\r\n\\NewDocumentCommand{\\cfracdots}{ }\r\n {\r\n \\rule{0pt}{1.5\\baselineskip}\r\n \\raisebox{.5\\baselineskip}{\\enspace$\\ddots$\\enspace}\r\n }\r\n\\NewDocumentCommand{\\cfraccdots}{}{\\cdots}\r\n\\NewDocumentCommand{\\cfracddots}{}{\\ddots}\r\n\\NewDocumentCommand{\\cfracldots}{}{\\ldots}\r\n\r\n\\NewDocumentCommand{\\xcontfrac}{ s O{c} >{\\SplitArgument{1}{;}}m }\r\n { \r\n \\IfBooleanTF{#1}\r\n { \\cfrac_inline:nn #3 }\r\n { \\cfrac_map:nnn { #2 } #3 }\r\n }\r\n\r\n\\cs_new:Npn \\cfrac_inline:nn #1 #2\r\n {\r\n \\IfNoValueTF { #2 }\r\n {\r\n \\tl_use:N \\c_cfrac_message_tl\r\n \\xcontfrac*{;#1}\r\n }\r\n {\r\n \\group_begin:\r\n \\cs_set_eq:NN \\cfracdots \\dots\r\n [\\, \\tl_if_empty:nTF { #1 } { 0 } { #1 } ; #2 \\,]\r\n \\group_end:\r\n }\r\n }\r\n\r\n\\tl_const:Nn \\c_cfrac_lbrace_tl { \\if_true: { \\else: } \\fi: }\r\n\\tl_const:Nn \\c_cfrac_rbrace_tl { \\if_false: { \\else: } \\fi: }\r\n\\tl_const:Nn \\c_cfrac_strut_tl { \\vrule width 0pt depth .3\\baselineskip }\r\n\\tl_new:N \\l_cfrac_left_tl\r\n\\tl_new:N \\l_cfrac_right_tl\r\n\\msg_new:nnn { cfrac } { wrong-syntax }\r\n {\r\n Wrong~syntax~for~\\token_to_str:N \\xcontfrac,~\r\n assuming~0~in~the~integer~part,~on~line~\\msg_line_number:.\r\n }\r\n\r\n\\cs_new:Npn \\cfrac_map:nnn #1 #2 #3\r\n {\r\n \\tl_clear:N \\l_cfrac_left_tl \\tl_clear:N \\l_cfrac_right_tl\r\n \\IfNoValueTF { #3 }\r\n { \r\n \\msg_warning:nn { cfrac } { wrong-syntax }\r\n \\xcontfrac[#1]{;#2}\r\n }\r\n {\r\n \\tl_if_empty:nTF { #2 }\r\n { \\cfrac_map_aux:nn { #1 } { \\exp_not:N \\use_none:n , #3 } }\r\n { \\cfrac_map_aux:nn { #1 } { #2 , #3 } }\r\n }\r\n }\r\n\\cs_new:Npn \\cfrac_map_aux:nn #1 #2\r\n {\r\n \\clist_map_inline:nn { #2 }\r\n {\r\n \\tl_put_right:Nn \\l_cfrac_left_tl { \\cfrac_begin:nn { #1 } { ##1 } }\r\n \\tl_put_right:Nn \\l_cfrac_right_tl { \\exp_not:N \\c_cfrac_rbrace_tl }\r\n }\r\n \\tl_set:Nx \\l_cfrac_left_tl\r\n { \\l_cfrac_left_tl \\c_cfrac_strut_tl \\l_cfrac_right_tl }\r\n \\tl_set:Nx \\l_cfrac_left_tl { \\l_cfrac_left_tl }\r\n \\exp_after:wN \\use_none:nnnnnn \\l_cfrac_left_tl\r\n }\r\n\\cs_new:Npn \\cfrac_begin:nn #1 #2\r\n {\r\n \\exp_not:n\r\n { + \\exp_not:N \\cfrac[#1] { 1 } \\c_cfrac_lbrace_tl \\exp_not:N \\mathstrut #2 }\r\n }\r\n\\ExplSyntaxOff\r\n\r\n\\usepackage{listings}\r\n\\usepackage{color}\r\n\r\n\\definecolor{mygreen}{rgb}{0,0.6,0}\r\n\\definecolor{mygray}{rgb}{0.5,0.5,0.5}\r\n\\definecolor{mymauve}{rgb}{0.58,0,0.82}\r\n\r\n\\lstset{ \r\n backgroundcolor=\\color{white}, % choose the background color; you must add \\usepackage{color} or \\usepackage{xcolor}; should come as last argument\r\n basicstyle=\\footnotesize, % the size of the fonts that are used for the code\r\n breakatwhitespace=false, % sets if automatic breaks should only happen at whitespace\r\n breaklines=true, % sets automatic line breaking\r\n captionpos=b, % sets the caption-position to bottom\r\n commentstyle=\\color{mygreen}, % comment style\r\n deletekeywords={...}, % if you want to delete keywords from the given language\r\n escapeinside={\\%*}{*)}, % if you want to add LaTeX within your code\r\n extendedchars=true, % lets you use non-ASCII characters; for 8-bits encodings only, does not work with UTF-8\r\n firstnumber=1000, % start line enumeration with line 1000\r\n frame=single,\t % adds a frame around the code\r\n keepspaces=true, % keeps spaces in text, useful for keeping indentation of code (possibly needs columns=flexible)\r\n keywordstyle=\\color{blue}, % keyword style\r\n language=Octave, % the language of the code\r\n morekeywords={*,...}, % if you want to add more keywords to the set\r\n numbers=none, % where to put the line-numbers; possible values are (none, left, right)\r\n numbersep=5pt, % how far the line-numbers are from the code\r\n numberstyle=\\tiny\\color{mygray}, % the style that is used for the line-numbers\r\n rulecolor=\\color{black}, % if not set, the frame-color may be changed on line-breaks within not-black text (e.g. comments (green here))\r\n showspaces=false, % show spaces everywhere adding particular underscores; it overrides 'showstringspaces'\r\n showstringspaces=false, % underline spaces within strings only\r\n showtabs=false, % show tabs within strings adding particular underscores\r\n stepnumber=2, % the step between two line-numbers. If it's 1, each line will be numbered\r\n stringstyle=\\color{mymauve}, % string literal style\r\n tabsize=2,\t % sets default tabsize to 2 spaces\r\n title=\\lstname % show the filename of files included with \\lstinputlisting; also try caption instead of title\r\n}\r\n\r\n\\usepackage{tikz}\r\n\\usetikzlibrary{calc}\r\n\\newcommand{\\tikzmark}[1]{\\tikz[overlay,remember picture] \\node (#1) {};}\r\n\r\n\r\n\\newmdtheoremenv{theorem}{Theorem}\r\n\\newtheorem{example}{Example}\r\n\r\n\\newcounter{theo}[section]\\setcounter{theo}{0}\r\n\\newenvironment{theo}[2][]{%\r\n \\refstepcounter{theo}\r\n \\ifstrempty{#1}%\r\n% if condition (without title)\r\n{\\mdfsetup{%\r\n frametitle={%\r\n \\tikz[baseline=(current bounding box.east),outer sep=0pt]\r\n \\node[anchor=east,rectangle,fill=blue!20]\r\n {\\strut Theorem~\\thetheo};}\r\n }%\r\n% else condition (with title)\r\n}{\\mdfsetup{%\r\n frametitle={%\r\n \\tikz[baseline=(current bounding box.east),outer sep=0pt]\r\n \\node[anchor=east,rectangle,fill=blue!20]\r\n {\\strut Theorem~\\thetheo:~#1};}%\r\n }%\r\n}%\r\n% Both conditions\r\n\\mdfsetup{%\r\n innertopmargin=10pt,linecolor=blue!20,%\r\n linewidth=2pt,topline=true,%\r\n frametitleaboveskip=\\dimexpr-\\ht\\strutbox\\relax%\r\n}\r\n\\begin{mdframed}[]\\relax}{%\r\n\\end{mdframed}}\r\n\r\n\\newcounter{lem}[section]\\setcounter{lem}{0}\r\n\\renewcommand{\\thelem}{\\arabic{section}.\\arabic{lem}}\r\n\\newenvironment{lem}[2][]{%\r\n\\refstepcounter{lem}%\r\n\\ifstrempty{#1}%\r\n{\\mdfsetup{%\r\nframetitle={%\r\n\\tikz[baseline=(current bounding box.east),outer sep=0pt]\r\n\\node[anchor=east,rectangle,fill=green!20]\r\n{\\strut ~\\thelem};}}\r\n}%\r\n{\\mdfsetup{%\r\nframetitle={%\r\n\\tikz[baseline=(current bounding box.east),outer sep=0pt]\r\n\\node[anchor=east,rectangle,fill=green!20]\r\n{\\strut ~#1};}}%\r\n}%\r\n\\mdfsetup{innertopmargin=10pt,linecolor=green!20,%\r\nlinewidth=2pt,topline=true,%\r\nframetitleaboveskip=\\dimexpr-\\ht\\strutbox\\relax\r\n}\r\n\\begin{mdframed}[]\\relax%\r\n\\label{#2}}\r\n{\\end{mdframed}}\r\n\r\n\\begin{document}\r\n\\title{Euclid's extended algorithm and applications}\r\n\\author{Franklin}\r\n\\maketitle\r\n\r\nThe foundation of the algorithm is the following theorem\r\n\r\n\\begin{theo}[Euclidean division theorem]{}\\label{euclid}\r\nIf $a,b$ are integers, then there exist integers $q,r$, such that \r\n\\begin{align*}\r\na&=bq+r\\\\\r\n0&\\leq r<|b|\r\n\\end{align*}\r\n\\end{theo}\r\n\r\n\\begin{lem}[Extended Euclidean Algorithm]{}\r\n\\noindent\\textbf{Input:} Two integers $a$ and $b$. The input could also be two polynomials, or in general anything that satisfies Theorem \\ref{euclid}.\r\n\r\n\\noindent\\textbf{Algorithm:}\r\n\\begin{itemize}\r\n\\item \\textbf{Step 1:} If $b>a$, then swap $a$ and $b$. In other words, assume that $a\\geq b$.\r\n\\item \\textbf{Step 2:} Call $r_{0}=a$ and $r_1=b$ and set $i=0$.\r\n\\item \\textbf{Step 3:} While $r_{i+1}\\neq 0$, divide $r_{i}$ by $r_{i+1}$ as in Theorem \\ref{euclid} $$r_{i}=r_{i+1}q_{i}+r_{i+2}$$ and increment $i$.\r\n\\end{itemize}\r\n\r\n\\noindent\\textbf{Output:}\r\nThe algorithm outputs all the equations obtained in the loop of Step 2.\r\nThese are \r\n\\begin{align*}\r\nr_{0}&=r_1q_0+r_2\\\\\r\nr_1&=r_2q_1+r_3\\\\\r\nr_2&=r_3q_2+r_4\\\\\r\n\\vdots&\\phantom{{}={}}\\vdots\\\\\r\nr_{n-1}&=r_{n}q_{n-1}+r_{n+1}\\\\\r\nr_{n}&=r_{n+1}q_{n} + 0\r\n\\end{align*}\r\n\\end{lem}\r\n\r\n\r\nThe algorithm always ends when the last remainder computed is $0$.\r\n\r\n\\section*{Information that can be read from the output of Euclid's algorithm}\r\n\\begin{itemize}\r\n\\item The $\\gcd(a,b)$ satisfies $$\\gcd(a,b)=r_{n+1}$$This is, the greatest common divisor of $a$ and $b$ is the last number used as divisor when the algorithm terminated.\r\n\\item Bezout's equation expressing $\\gcd(a,b)$ as a combination $$ax+by=\\gcd(a,b)$$ for some integers $x$ and $y$.\r\n\\end{itemize}\r\n\r\n\\section*{Bezout's equation}\r\n\r\nTo produce Bezout's equation for $a,b$, from the output of Euclid's extended algorithm we first write the output by solving in each equation for the remainder \r\n\\begin{align*}\r\nr_{0}-r_1q_0&=\\tikzmark{a}r_2\\\\\r\nr_1-\\tikzmark{b}r_2q_1&=\\tikzmark{c}r_3\\\\\r\nr_2-\\tikzmark{d}r_3q_2&=r_4\\\\\r\n\\vdots&\\phantom{{}={}}\\vdots\\\\\r\nr_{n-2}-r_{n-1}q_{n-2}+&=\\tikzmark{e}r_{n}\\\\\r\nr_{n-1}-\\tikzmark{f}r_{n}q_{n-1}+&=r_{n+1}\\\\\r\nr_{n}&=r_{n+1}q_{n} + 0\r\n\\begin{tikzpicture}[overlay,remember picture,out=225,out=335,distance=0.6cm]\r\n \\draw[->,red] (a.south) to (b.north);\r\n\t\t\\draw[->,red] (c.south) to (d.north);\r\n\t\t\\draw[->,red] (e.south) to (f.north);\r\n \\end{tikzpicture}\r\n\\end{align*}\r\n\r\nSubstituting all of these equations into the second to last one, we get $\\gcd(a,b)$ expressed in terms of $r_0$ and $r_1$, which are $a$ and $b$, as wanted.\r\n\r\n\\section*{Multiplicative inverse in modular arithmetic}\r\nIn the case that $\\gcd(a,b)=1$, Bezout's equation takes the form $$ax+by=1$$\r\nIf we reduce this equation modulo $b$, the term $bx$ becomes $0$, for being a multiple of $b$. We obtain $$ax\\equiv 1\\pmod{b}$$\r\nTherefore, $x$, or rather its remainder modulo $b$, is the multiplicative inverse of $a$, modulo $b$.\r\n\r\n\\section*{Numeric example} Let us compute the $\\gcd$ of $2519$ and $377$.\r\n\r\nFirst, we perform Euclid's Extended Algorithm.\r\n\r\n\\begin{align*}\r\n2519&=\\tikzmark{a}377\\cdot6+\\tikzmark{c}257\\\\\r\n&\\\\\r\n\\tikzmark{b}377&=\\tikzmark{d}257\\cdot1+\\tikzmark{f}120\\\\\r\n&\\\\\r\n\\tikzmark{e}257&=\\tikzmark{g}120\\cdot2+\\tikzmark{h}17\\\\\r\n&\\\\\r\n\\tikzmark{i}120&=\\tikzmark{j}17\\cdot7+\\tikzmark{k}\\boxed{1}\\\\\r\n&\\\\\r\n\\tikzmark{l}17&=\\tikzmark{m}\\boxed{1}\\cdot 17+0\r\n\\begin{tikzpicture}[overlay,remember picture,out=225,out=335,distance=0.6cm]\r\n \\draw[->,red] (a.south) to (b.north);\r\n\t\t\\draw[->,red] (c.south) to (d.north);\r\n\t\t\\draw[->,red] (d.south) to (e.north);\r\n\t\t\\draw[->,red] (f.south) to (g.north);\r\n\t\t\\draw[->,red] (g.south) to (i.north);\r\n\t\t\\draw[->,red] (h.south) to (j.north);\r\n\t\t\\draw[->,red] (j.south) to (l.north);\r\n\t\t\\draw[->,red] (k.south) to (m.north);\r\n \\end{tikzpicture}\r\n\\end{align*}\r\n\r\nThe last divisor used is the $\\gcd$. So, $\\gcd(2519,377)=1$.\r\n\r\nNow, to form Bezout's equation, let's solve for the remainders in all these equations and substitute each into the next one until we get to the second-to-last equation. We don't want to carry out any of the arithmetic operations, while we are doing the substitutions, at least not the ones with the numbers $2519$ and $377$.\r\n\r\n\\begin{align*}\r\n2519-\\tikzmark{a}377\\cdot6&=\\tikzmark{c}257\\\\\r\n&\\\\\r\n\\tikzmark{b}377-\\tikzmark{d}257\\cdot1&=\\tikzmark{f}120\\\\\r\n&\\\\\r\n\\tikzmark{e}257-\\tikzmark{g}120\\cdot2&=\\tikzmark{h}17\\\\\r\n&\\\\\r\n\\tikzmark{i}120-\\tikzmark{j}17\\cdot7&=\\tikzmark{k}\\boxed{1}\r\n\\begin{tikzpicture}[overlay,remember picture,out=225,out=335,distance=0.6cm]\r\n \\draw[->,red] (c.south) to (d.north);\r\n\t\t\\draw[->,red] (f.south) to (g.north);\r\n\t\t\\draw[->,red] (h.south) to (j.north);\r\n \\end{tikzpicture}\r\n\\end{align*}\r\n\r\nSubstituting the third (second-to-last) equation into the last one to eliminate the remainder $17$, we get \r\n\r\n$$120-(257-120\\cdot 2)\\cdot 7=\\boxed{1}$$\r\n\r\nNow we can use the second equation to eliminate from this one all occurrences of the remainder $120$. We get \r\n\r\n$$(377-257\\cdot1)-(257-(377-257\\cdot1)\\cdot 2)\\cdot 7=\\boxed{1}$$\r\n\r\nNow we use the first equation to eliminate from this one all occurences of the remainder $257$. We get\r\n\r\n$$(377-(2519-377\\cdot 6)\\cdot1)-((2519-377\\cdot 6)-(377-(2519-377\\cdot 6)\\cdot1)\\cdot 2)\\cdot 7=\\boxed{1}$$\r\n\r\nFinally, we gather together all terms that are multiplied by $2519$ and all that are multiplied by $377$. We get\r\n\r\n$$2519\\cdot (-22)+377\\cdot (147) =\\boxed{1}$$\r\n\r\nThe two factors $-22$ and $147$ in Bezout's equation are not unique. We could, for example add and subtract a multiple of $2519\\cdot 377$ and get \r\n\r\n\\begin{align*}\\boxed{1}&=2519\\cdot (-22)+2519\\cdot377\\cdot k-2519\\cdot377\\cdot k+377\\cdot (147)\\\\\r\n&=2519\\cdot (377\\cdot k-22)+377\\cdot (147-2519\\cdot k)\r\n\\end{align*}\r\n\r\nSo, the factors $377\\cdot k-22$ and $147-2519\\cdot k$ also work.\r\n\r\nIn the process we also got that $$377\\cdot 147\\equiv1\\pmod{2519}$$\r\nand that $$2519\\cdot (-22)\\equiv1\\pmod{377}$$\r\n\r\n\\section*{Implementation}\r\n\r\nIt is important to practice the computations above a few times, by hand. \r\nTo verify your computations we can use a computer.\r\n\r\nThe following is a function in Python that inputs $a$ and $b$ and returns a triple $d,x,y$ such that $$ax+by=d$$\r\n\r\n\\begin{lstlisting}[language=Python]\r\ndef xgcd(a, b):\r\n \"\"\"return (d, x, y) such that a*x + b*y = d = gcd(a, b)\"\"\"\r\n x0, x1, y0, y1 = 0, 1, 1, 0\r\n while a != 0:\r\n (q, a), b = divmod(b, a), a\r\n y0, y1 = y1, y0 - q * y1\r\n x0, x1 = x1, x0 - q * x1\r\n return b, x0, y0\r\n\\end{lstlisting}\r\n\r\n\\noindent\\textbf{Note:} To copy this code into a Python interpreter, remember that in Python the indentation of the lines is important. So, give $4$ spaces to indent each indented line.\r\n\r\nExecuting \\lstinline{xgcd(2519,377)} we get \\lstinline{(1,-22,147)}. So, the example should be OK (or both the example and the Python code are wrong).\r\n\r\n\\section*{Continued fractions}\r\n\r\nA byproduct of the output of the Extended Euclid's Algorithm for the input $a,b$, with $a\\geq b$, is a \\emph{continued fraction} expansion of the rational number $\\frac{a}{b}$. With the notation above, we get that \r\n\r\n$$\\frac{a}{b}=\\frac{r_0}{r_1}=\\xcontfrac{q_0;q_1,q_2,\\cfracdots,q_n}$$\r\n\r\n\r\n\r\n\r\n\\end{document}", "meta": {"hexsha": "beef9eadf459f1691e0d59c6b0a6167f2d34ccbe", "size": 13750, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "assets/files/ExtendedEuclid.tex", "max_stars_repo_name": "franklinvp/franklinvp.github.io", "max_stars_repo_head_hexsha": "fea6ac7ad8df6a00eb1b5fd04f54eab5ba756006", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "assets/files/ExtendedEuclid.tex", "max_issues_repo_name": "franklinvp/franklinvp.github.io", "max_issues_repo_head_hexsha": "fea6ac7ad8df6a00eb1b5fd04f54eab5ba756006", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "assets/files/ExtendedEuclid.tex", "max_forks_repo_name": "franklinvp/franklinvp.github.io", "max_forks_repo_head_hexsha": "fea6ac7ad8df6a00eb1b5fd04f54eab5ba756006", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.8418079096, "max_line_length": 325, "alphanum_fraction": 0.6521454545, "num_tokens": 4730, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.546738151984614, "lm_q2_score": 0.6297746074044135, "lm_q1q2_score": 0.3443218050191248}} {"text": "\\chapterquote{%\nTime and Space\\dots It is not nature which imposes them upon us, it is we who impose them upon nature because we find them convenient.}%\n{-- Henri Poincar\\'e, \\textit{The Value of Science} (1905)}\n\nIn this chapter\\footnote{The present chapter was based on the articles \\cite{dester2018} and \\cite{dester2021part}.}, we include the spatial position of the nodes in the analysis.\n%\nTherefore, the transmission success probability $p_s$ can no longer be an arbitrary function of the traffic, because it needs to include the spatial positions into its model.\n%\nThus, we need a more specific model for $p_s$, preferably one that entails analytical tractability.\n%\nIn the following section, we propose a model inspired in Proposition~\\ref{prop:PPP_ps} and other models adopted throughout the literature.\n\nAnother important concept we deal with in this chapter is the \\textit{stability} of the queued packets. When a queue is not stable the number of queued packets tends to infinity \\textit{almost surely} as time tends to infinity.\n%\nWe are very interested in stable queues, because one of the important metrics, namely the delay, tends to infinity in unstable systems, and that is not desirable.\n\nThe additional notations used in this chapter are summarized in Table~\\ref{tab:symbols}.\n%\n\\begin{table}[hbt]\n \\centering\n \\caption{Notations and symbols used in this chapter}\n \\label{tab:symbols}\n \\setlength{\\tabcolsep}{3pt}\n \\begin{tabular}{l l}\n \\hline\n \\hline\n \\textbf{Symbol} & \\textbf{Definition/explanation} \\\\\n \\hline\n $\\alpha\\in(2,\\infty)$\t& path loss exponent \\\\\n $\\delta\\in(0,1)$\t\t& $\\triangleq 2/\\alpha$ \\\\\n $N \\in \\N^*$ & number of user classes \\\\\n $\\cal{C}$\t\t\t\t& $\\triangleq \\{1,2,\\dots,N\\}$, set of classes \\\\\n $n\\in\\cal{C}$\t\t\t& refers to the $n$th user class \\\\\n $p_n\\in(0,1)$\t\t\t& medium access probability \\\\\n \t$a_n\\in(0,1)$\t\t\t& packet arrival rate per time slot \\\\\n $\\bm{a}\\in(0,1)^N$\t\t& $=(a_1,a_2,\\dots,a_N)$ \\\\\n $p_{s,n}\\in(0,1)$\t\t& transmission success probability \\\\\n $\\theta_n \\in\\R_+$\t\t& SIR threshold for successful communication \\\\\n $\\overline{R}_n\\in\\R_+$\t& average transmission distance \\\\\n $D_n\\in(1,\\infty)$\t\t& average packet transmission delay \\\\\n $\\bm{D}\\in(1,\\infty)^N$ & $=(D_1,D_2,\\dots,D_N)$ \\\\\n $P_n\\in\\R_+$\t\t\t& transmission power \\\\\n $\\bm{P}\\in\\R_+^N$ \t\t& $=(P_1,P_2,\\dots,P_N)$ \\\\\n $\\Phi_\\TX^{(n)}$\t & Poisson point process for the transmitters \\\\\n $\\lambda_n\\in\\R_+$\t\t& density of $\\Phi_\\TX^{(n)}$ \\\\\n $\\psi_n\\in\\R_+$\t\t\t& $\\triangleq 4\\,\\overline{R}_n^2 \\,\\theta_n^{\\delta} \\,\\pi\\delta/\\sin(\\pi \\delta)$ \\\\\n \\hline\n \\hline\n \\end{tabular}\n\\end{table}\n\n\\newpage\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % \n% % % % % % % % % % % % % % % % % % % % % % % % % % \n\\section{Multi-class High-mobility Bipolar Networks} \\label{sec:N-class}\n\n\\subsection{Main Contributions}\nThe main contributions of the work presented in this section when compared to the literature, particularly the paper by Stamatiou and Haenggi \\cite{stamatiou2010random}, are twofold.\n%\nFirstly, we have extended the analysis of stability (Definition~\\ref{def:stability}) and delay in random-access wireless networks to the case of a network with an arbitrary number $N$ of user classes. As far as we know, this is the first time in the literature the stability region is found in closed form for a network with $N > 3$ different user classes sharing the same channel.\n%\nSecondly, we have expanded the analysis to show that the channel sharing mechanism in the investigated scenario can be seen as a process of partitioning a fixed and well-defined quantity into portions, each portion allotted to each user class, the size of which varying in accordance with the user class parameters.\n\nMore specifically, the novelty of the results on this section are summarized below:\n\\begin{itemize}[itemsep=3pt,parsep=3pt,topsep=3pt,partopsep=3pt]\n \\item We propose a tractable scenario to study the performance and stability of a Poisson network with an arbitrary number $N$ of classes of users sharing the same channel;\n \\item a simple and elegant expression relating mean delays, arrival rates, user densities, mean link distances, and bit rates of all $N$ classes is derived for the case of a stable network. This expression clearly shows that each class of user takes a well-defined portion of the available finite resource in the RF channel (Proposition~\\ref{prop:identity_1});\n \\item a closed form solution (Definition~\\ref{def:closed_form}) to the fixed-point system of equations that determine the stationary transmission success probabilities for $N$ user classes is found;\n \\item an intuitive equation is presented relating link quality, packet arrival rate, density of users, and stationary mean delay (Proposition 1);\n \\item we prove the necessary and sufficient conditions that determine whether a given network is stable (Theorems \\ref{TH:NEC_SUFF} and \\ref{TH:STABILITY});\n \\item we establish a simple necessary condition for stability that does not depend on the transmit powers (Corollary~\\ref{cor:stab});\n \\item the optimum transmit powers per user class that achieve the optimum stationary mean delays for each user class (Proposition~\\ref{prop:opt}) are derived;\n \\item the optimum packet arrival rates per user class that achieve the maximum channel throughput per unit of area (Proposition~\\ref{prop:eps}) are derived;\n \\item we conclude that depending on the channel and user classes, the best strategy to maximize channel throughput is to share the channel, instead of using one single class per channel.\n\\end{itemize}\n\n% This section is organized as follows: Section \\ref{sec:SysMod} describes the model used throughout the chapter and provides some important results from the literature to be used in the following sections;\n% Section \\ref{sec:N-users} presents the main results of the paper, i.e., necessary and sufficient conditions for stability when we have $N$ interacting user classes, and shows a simple expression for the stationary mean delay and the packet success probability; Section \\ref{sec:application} applies the obtained results in two general scenarios: one scenario optimizes the transmission power of different user classes with different delay requirements sharing the same channel and the other optimizes the throughput per unit of area; Section \\ref{sec:conclusion} concludes the paper.\n\n\n\\subsection{System Model}\n\nWe consider a stationary high-mobility Poisson network with density $\\lambda$ on $\\R^2$ and with $N$ classes of users that share the same radio frequency channel as defined in the general network model of Chapter~\\ref{cap:P2_00}.\n%\nTime is slotted, $\\T=\\N$, and for each time slot $t \\in \\T$ and each user class $n \\in \\cal{C} \\triangleq \\{1,2,\\dots,N\\}$, we have a homogeneous Poisson point process (PPP) denoted by $\\Phi_\\TX^{(n)}(t)\\stackrel{*}{=} \\{ X_{i\\,n}(t) \\}_{i\\in\\N}$ of density $\\lambda_n$ on $\\R^2$, which represents the position of the sources. These PPPs are independent from each other and from the past.\n%\nFurthermore,\n\\begin{align*}\n \\lambda = \\sum_{n\\in\\cal{C}} \\lambda_n,\n \\quad\\text{then,}\\quad\n \\Phi_\\TX(t) = \\sum_{n\\in\\cal{C}} \\Phi_\\TX^{(n)}(t) \\quad\\text{for every $t\\in\\T$}.\n\\end{align*}\n\n\\begin{remark}\n Using the general network model of Chapter~\\ref{cap:P2_00}, the set of transmitters $\\cal{N}_\\TX = \\N\\times\\cal{C}$. Thus, each transmitter should be indexed by a number and a class. However, for ease of notation and since the network is stationary across the users within a class, we shall omit the number and only refer to the class for parameters and metrics.\n %\n For example, the transmission success probability $p_{s,n}$ refers to $p_{s,\\N\\times\\{n\\}}$, i.e., it refers to the set of all users from $n$th class (or a typical user of that class).\n\\end{remark}\n\n\\begin{figure}[H]\n \\centering\n \\if\\printfig1\n \\includegraphics[width=0.8\\textwidth]{Figures/Ch7_BipolarQueuedNetwork.pdf}\n \\else\n \\includegraphics[draft,width=0.8\\textwidth]{Figures/Ch7_BipolarQueuedNetwork.pdf}\n \\fi\n \\caption{Example of a bipolar high-mobility random network with $N=3$ user classes (one for each color). The queues represent the transmitters, and the potential receivers are represented by circular shapes of the corresponding color. The purpose of the arrows is to remember that the nodes are moving. Each transmitter communicates with the closest potential receiver, as it is shown by the dashed lines. Unconnected circles represent inactive receivers. The quantities $\\lambda$, $a$, $p_s$ and $\\phi$ are related to density of users, rate of arrival of packets, rate of service of packets and link quality, respectively.}\n \\label{fig:BipolarNetwork}\n\\end{figure}\n\nEach transmitter of user class $n$ transmits with power $P_n$ constant over time, and the transmitted signal is subjected to Rayleigh short-term fading and power law path loss function $\\ell(r) = r^{-\\alpha}, r>0$, where $\\alpha>2$ is the path loss exponent.\n\n% For each time slot the position $X_{i\\,n}(t)$ of the $i$th transmitter is reallocated following the high-mobility random walk model \\cite{baccelli2010stochastic}.\n%\n% The $i$th transmitter of user class $n$ communicates with a receiver located at $Y_{i\\,n}(t)$. Thus, the distance between the $i$th transmitter of class $n$ and its destination is given by $R_{i\\,n}(t) = || X_{i\\,n}(t) - Y_{i\\,n}(t) ||$.\n%\nWe further assume that each transmitter is associated with a ``son'' PPP that models the locations of its potential receivers. The receiver associated with the $i$th transmitter of class $n$ is chosen as the closest point in the respective son PPP as illustrated in Figure~\\ref{fig:BipolarNetwork}.\n\nAs a consequence, the link distances $\\{ R_{i\\,n}(t) \\}_t$ are iid Rayleigh random variables\\footnote{The iid random variables for the link separation distance are of grave importance for the theoretical model. Otherwise, there would exist unstable queues and, consequently, the queueing network would be unstable.} \\cite[Eq.~(2.35)]{kingman1992poisson}.\n%\nRayleigh distributed link separation distance has been used in several other works investigating similar scenarios (see \\cite{haenggi2013diversity}).\n%\nWe denote the mean transmission distance $\\E[R_{i\\,n}(t)]$ simply by $\\overline{R}_n$ because we have stationarity across time and across users of a given class. Then, the density of $R_{i\\,n}$ can be expressed as\n\\begin{align*}\n f_{R_n}(r) = \\frac{\\pi\\,r}{2 \\overline{R}_n^2} \\exp\\!\\left[ - \\frac{\\pi\\,r^2}{4\\overline{R}_n^2}\\right], \\qquad r\\in\\R_+.\n\\end{align*}\n\nThe occupation of the buffer at each transmitter is represented by its queue length $\\{ Q_{i\\,n}(t) \\}_t$ of infinite capacity.\n%\nThe packet arrival probability at each queue is denoted by $a_n$ and the medium access probability by $p_n$.\n%\nWithin each slot, the first event to take place for each transmitter with a non-empty queue is the medium access decision with probability $p_n$. If it is granted access and the signal to interference ratio (SIR) %\\footnote{We assume thermal noise is negligible; refer to \\cite{haenggi2012stochastic} for further details.}\nis greater than a threshold $\\theta_n>0$, a packet is successfully transmitted and leaves the queue. Then, we have the arrival of the next packet with probability $a_n$. The last event to take place is the displacement of the transmitters and destinations.\n\nThe queue lengths of the $i$th transmitter, user class $n$ are Markov chains represented by\n\\begin{equation*}\n\tQ_{i\\,n}(t+1) = (Q_{i\\,n}(t) - B_{i\\,n}(t))_+ + E_{i\\,n}(t), \\quad t \\in \\N,\n\\end{equation*}\nwhere $(\\cdot)_+ \\triangleq \\max\\{\\cdot,0\\}$, $\\{E_{i\\,n}(t)\\}_t$ are iid Bernoulli random variables of parameter $a_n$, i.e., $E_{i\\,n}(t)\\sim\\mathscr{B}(a_n)$ and represents the arrival process,\n$\n\tB_{i\\,n}(t) = e_{i\\,n}(t)\\,\\ind\\{\\text{SIR}_{i\\,n}(t)>\\theta_{n}\\}\n$\nrepresents the departure process, where $\\{e_{i\\,n}(t)\\}_t$ are iid Bernoulli random variables of parameter $p_n$, i.e., $e_{i\\,n}(t)\\sim\\mathscr{B}(p_n)$, and the constant $\\theta_n > 0$ represents the SIR threshold for successful communication.\n\n\\begin{remark}\n In view of the model of Chapter~\\ref{cap:P2_00}, we have that the arrival point process $A_{i\\,(n)}$ is a Bernoulli point process of parameter $a_n$, the access point process $T_{i\\,(n)}$ is a Bernoulli point process of parameter $p_n$ and the transmissions times $T_{i\\,(n)}^*$ is a thinning of $T_{i\\,(n)}$ conditioned to the corresponding queue being non-empty, and the success probability model $\\mathscr{S}(f) = \\ind\\{f(0) > 1\\}$, where $f(t') = \\SIR_{i\\,n}(t-t')/\\theta_n$.\n\\end{remark}\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %\n\\subsection{Analysis and Results}\n\\label{ssec:N-users}\n\nFrom the stationarity across users of a class and ergodicity regarding time (if the system is stable), the transmission success probability $p_{s,n}$ is the limiting probability of a successful transmission from a typical user of class $n$, i.e., $$p_{s,n} = \\lim_{t\\to\\infty} \\P(\\mathrm{SIR}_{i,n}(t) > \\theta_n).$$\n\n\\begin{note}\n Remember that, \\textit{à priori}, we do not need to take $t\\to\\infty$, any $t$ would suffice in an ergodic process.\n %\n However, as discussed in Section~\\ref{sec:poisson_network} we would have to start the system at $-\\infty$ or start the system distributed according to the stationary distribution.\n\\end{note}\n\nThe stationary mean delay $D_n$ to transmit packets of class $n$ is defined as the limiting ($t\\to\\infty$) expected time a packet spends in the buffer and the server.\n%\nSee Theorem~\\ref{th:little}.\n% For each time slot and for each class we have an independent homogeneous PPP, which is stationary and isotropic (invariant to translation and rotation, respectively).\n\n% The following results from the literature are used in many proofs throughout the chapter. In a wireless network, let us assume that \\textit{(i)} the separation distance between a given pair TX - RX is equal to $r$, \\textit{(ii)} the positions of the interferers (users who will transmit packets in a given time slot) follow a PPP of density $\\lambda_\\mathrm{eff}$, and \\textit{(iii)} every transmitter has the same transmit power. Then, the probability of a successful transmission between TX and RX is given by \\cite[Sec.~III.A]{haenggi2009stochastic}\n% \\begin{align}\n% \t\\P(\\mathrm{SIR} > \\theta) &= \\E[\\euler^{-\\theta r^\\alpha I}] \\nonumber\\\\\n% \t&= \\exp\\left( - \\pi\\,\\Gamma(1+\\delta) \\Gamma(1-\\delta)\\,\\theta^{\\delta}\\,r^2\\,\\lambda_\\mathrm{eff} \\right),\n% \\end{align}\n% where $\\delta \\triangleq 2/\\alpha$ and $I \\triangleq \\sum_{X\\in\\Phi} ||X||^{-\\alpha}$ is the interference received by RX normalized by the transmit power and $\\Phi$ is a PPP of density $\\lambda_\\mathrm{eff}$, which is the effective density of active sources.\n%\n\n% As described in the previous section, we consider a network with $N$ classes of users.\n%\nThe following proposition presents the stationary success probability and mean delay when transmitting a packet in a stable network. The results that guarantee stability are presented later in the sequence, in Theorem~\\ref{TH:NEC_SUFF}.\n\n\\begin{proposition} \\label{prop:psk}\n\tIf the network is stable, then the stationary success probability and mean delay for a typical user of class $n \\in \\cal{C}$ are given by\n\t\\begin{align}\n \tp_{s,n} &= \\left( 1 + \\dfrac{\\psi_n}{P_n^\\delta}\\, \n \\dfrac{\\sum_j P_j^\\delta\\,a_j\\lambda_j}\n {1 - \\sum_j \\psi_j\\,a_j\\lambda_j} \\right)^{-1},\\label{eq:psn}\\\\ \n D_n &= \\dfrac{1 - a_n}{p_n\\,p_{s,n} - a_n},\t\\label{eq:Dn}\n \\end{align}\n where the sums are taken over the set of user classes $\\cal{C}$, ${\\delta \\triangleq 2/\\alpha}$, and \n \\begin{equation}\\label{eq:DefPhi}\n \\psi_n\n \\triangleq 4\\,\\Gamma(1+\\delta) \\Gamma(1-\\delta) \\overline{R}_n^2 \\theta_n^\\delta\n = 4\\,\\overline{R}_n^2\\,\\theta_n^{\\delta}\\, \\frac{\\pi\\delta}{\\sin(\\pi \\delta)}.\n \\end{equation}\n\\end{proposition}\n\n\\begin{proof}\n First, we need to calculate the transmission success probability for a given traffic and a given link distance.\n %\n Using the reasoning of Proposition~\\ref{prop:PPP_ps} for the case of several classes of users, we have\n \\begin{align} \\label{eq:P_SIR}\n \t\\P(\\text{SIR}_{i\\,n}(t)>\\theta_n \\mid R_{i\\,n}) &= \\E\\!\\left[\\exp\\left(-\\frac{\\theta\\,R_{i\\,n}^\\alpha}{P_n} \\sum_{k\\in\\cal{C}} P_k\\,I_k\\right)\\right]\\nonumber\\\\\n \t&= \\prod_{k\\in\\cal{C}} \\E\\!\\left[\\exp\\left(-\\theta\\,R_{i\\,n}^\\alpha\\frac{P_k}{P_n} I_k\\right)\\right] \\nonumber\\\\\n &= \\prod_{k\\in\\cal{C}} \\exp\\!\\left( - \\pi R_{i\\,n}^2\\, \\theta^{\\delta}\\,\\frac{\\pi\\delta}{\\sin(\\pi\\delta)} \\frac{P_k^\\delta}{P_n^\\delta}\\lambda_\\mathrm{eff}^{(k)}(t)\\right) \\nonumber\\\\\n \t&= \\exp\\!\\left( - \\pi R_{i\\,n}^2\\,\\theta^{\\delta}\\,\\frac{\\pi\\delta}{\\sin(\\pi\\delta)} \\,\\sum_{k\\in\\cal{C}}\\frac{P_k^\\delta}{P_n^\\delta}\\lambda_\\mathrm{eff}^{(k)}(t)\\right),\n \\end{align}\n where $I_n$ is the interference from the $n$th class normalized by the transmit power, and $\\lambda_{\\mathrm{eff}}^{(n)}$ is the density of the thinned Poisson point process $\\Phi_\\TX^{(n)}$ by the nodes that are transmitting.\n %\n It is assumed that $\\{I_n\\}_{n \\in \\cal{C}}$ is iid.\n\n As $t\\to\\infty$, the effective PPP density of active sources $\\lambda_\\mathrm{eff}^{(n)}$ for each user class $n\\in\\cal{C}$ converges (by hypothesis) to $\\lambda_n\\,p_n\\,\\rho_n$, where $\\rho_n = a_n/(p_n\\,p_{s,n})$ is the load of the queue (or the probability of having a non-empty queue), which is the ratio between the arrival rate and the service rate of packets. Thus, $\\lambda_\\mathrm{eff}^{(n)} = \\lambda_n\\,a_n/p_{s,n}$.\n\n Then, to calculate the transmission success probability $p_{s,n}$, we use \\eqref{eq:P_SIR}. Thus, by deconditioning the transmission success probability on $R_{i\\,n}$, we take into account that $R_{i\\,n}$ is Rayleigh distributed, that is\n \\begin{align} \\label{eq:aux_psk}\n \tp_{s,n} &= \\lim_{t\\to\\infty}\\P(\\text{SIR}_{i\\,n}(t)>\\theta_n) \\nonumber\\\\\n \t&= \\int_0^\\infty \\lim_{t\\to\\infty} \\P(\\text{SIR}_{i\\,n}(t)>\\theta_n\n \t\\mid R_{i\\,n}(t) = r)\\,f_{R_n}(r)\\,\\mathrm{d}r \\nonumber\\\\\n &= \\int_0^\\infty \\frac{\\pi r}{2 \\overline{R}_n^2} \\exp\\!\\left[ - \\frac{\\pi\\,r^2}{4\\overline{R}_n^2} \\left(1+\\psi_n\\sum_{k\\in\\cal{C}}\\frac{P_k^\\delta}{P_n^\\delta}\\lambda_\\mathrm{eff}^{(k)}\\right)\\right]\\mathrm{d}r \\nonumber\\\\\n &= \\left( 1 + \\dfrac{\\psi_n}{P_n^\\delta}\n \t\t\t\\sum_{k\\in\\cal{C}} P_k^\\delta \\dfrac{a_k\\lambda_k}{p_{s,k}} \\right)^{-1}.\n \\end{align}\n This expression can be rearranged as\n \\begin{equation} \\label{eq:psk_equivalence}\n \t\\dfrac{P_n^\\delta}{\\psi_n} \\left( \\dfrac{1-p_{s,n}}{p_{s,n}}\\right) = \n \\sum_{k\\in\\cal{C}} P_k^\\delta \\dfrac{a_k\\lambda_k}{p_{s,k}}.\n \\end{equation}\n Note that the right-hand side of \\eqref{eq:psk_equivalence} does not depend on $n$. Then, for all $j\\in\\cal{C}$, we have%\n \\begin{equation} \\label{eq:Pi_Pk}\n \t\\dfrac{P_j^\\delta}{\\psi_j} \\left( \\dfrac{1-p_{s,j}}{p_{s,j}}\\right) = \n \\dfrac{P_n^\\delta}{\\psi_n} \\left( \\dfrac{1-p_{s,n}}{p_{s,n}}\\right).\n \\end{equation}\n For each $j$, we can solve the above equation for $p_{s,j}$ and plug it into the sum of \\eqref{eq:aux_psk}. Then, we can solve it for $p_{s,n}$, which ends the proof for the $p_{s,n}$.\n \n The buffer (plus server) is a discrete time Geo/Geo/1 queue (Definition~\\ref{def:geo/geo/1}) at stationary \\cite{stamatiou2010random} and the equation for the delay $D_n$ comes from Theorem~\\ref{th:geo/geo/1}.\n\\end{proof}\n\nThe following theorem shows the conditions for which the network is stable, i.e., it presents the region formed by all arrival rates $\\bm{a}$ that make the system stable.\n\n\\begin{theorem} \\label{TH:NEC_SUFF}\n\tA necessary and sufficient condition for the system network to be stable is that $\\bm{a} \\in \\bigcup_{\\nu\\in\\cal{V}} \\cal{E}_\\nu$, where $\\cal{V}$ is the space of all bijective functions from $\\cal{C}$ to $\\cal{C}$ and $\\cal{E}_\\nu$ is defined below with the convention $\\sum_{k=1}^0 \\cdot = 0$.\n\t\\begin{align} \\label{eq:stab_long}\n \t\\cal{E}_{\\nu} \\triangleq \\Bigg\\{ \\bm{a} \\in [0,1)^N \\Bigm\\vert~ \n \t& 0 \\le \n \\dfrac{\\psi_{\\nu(n)}}{P_{\\nu(n)}^\\delta}\\dfrac{a_{\\nu(n)}}{p_{\\nu(n)}-a_{\\nu(n)}} \\nonumber\\\\\n & < \\dfrac{ 1 - \\sum_{k=1}^{n-1} \\psi_{\\nu(k)} a_{\\nu(k)} \\lambda_{\\nu(k)} }\n {\\sum_{k=1}^{n-1} P_{\\nu(k)}^{\\delta} a_{\\nu(k)}\\lambda_{\\nu(k)} +\n \\sum_{k=n}^N P_{\\nu(k)}^\\delta p_{\\nu(k)}\\lambda_{\\nu(k)}},\n \\quad n \\in \\cal{C} \\Bigg\\}.\n \\end{align}\n\\end{theorem}\n%\n\\begin{proof}\n Using stochastic dominance through dominant networks \\cite[Section~2.1.2]{kompella2014stable}, it is possible to derive necessary and sufficient conditions for stability.\n %\n A dominant network behaves exactly the same as the original network, except that all user classes in a subset $\\cal{D}$ of $\\cal{C}$ transmit dummy packets.\n %\n Thus, the dominant network have more or the same number of buffered packets as the original network \\textit{almost surely}.\n %\n Then, if the dominant network is stable, the original network is stable as well. On the other hand, if the queues of the user classes in $\\cal{D}$ are not empty in the original network, then this system behaves exactly like the dominant network, i.e, both systems are \\emph{indistinguishable}, see Definition~\\ref{def:indistinguishable} and \\cite[Section~3.2]{szpankowski1994stability}.\n %\n Therefore, if the dominant network is unstable, then the original network is unstable as well.\n %\n In order to have necessary and sufficient conditions, we must perform this verification for all $\\cal{D} \\subset \\cal{C}$.\n \n Let us start with $\\cal{D}=\\cal{C}$, i.e., all users transmit dummy packets. For each step of the verification, we remove the stable user class from the set $\\cal{D}$. This procedure repeats until the set $\\cal{D}$ becomes empty. In order to attain stability of the dominant network we must have an arrival rate smaller than the service rate (Theorem~\\ref{th:loynes}). Thus, a sufficient condition for the first user class stability is, for any queue $i$ of this class (by symmetry),\n \\begin{equation*}\n \ta_1 < p_1\\,\\P(\\widetilde{\\text{SIR}}_{i,1} > \\theta_1) = p_1 \\left( 1 + \\dfrac{\\psi_1}{P_1^\\delta} \\sum_{k=1}^N P_k^\\delta\\,p_k\\lambda_k \\right)^{-1},\n \\end{equation*}\n where $\\widetilde{\\text{SIR}}$ represents the signal-interference ratio in the dominant network and the second equality comes from the same procedure to obtain \\eqref{eq:aux_psk} with $\\lambda_\\mathrm{eff}^{(n)} = \\lambda_n$ (all users are active, since every TX transmits dummy packets).\n %\n This guarantees stability for the first user class. Let us remove it from the set $\\cal{D}$. Then, we calculate the stationary success probability of the first user class $\\widetilde{p}^{(1)}_{s,1}$ for this dominant network. At steady state, we have\n \\begin{equation*}\n \t\\widetilde{p}^{(1)}_{s,1} = \\left( 1 + \\dfrac{\\psi_1}{P_1^\\delta} \\left( P_1^\\delta\\,p_1\\lambda_1\\dfrac{a_1}{p_1\\widetilde{p}^{(1)}_{s,1}} + \\sum_{k=2}^N P_k^\\delta\\,p_k\\lambda_k \\right) \\right)^{-1},\n \\end{equation*}\n which can be solved for $\\widetilde{p}^{(1)}_{s,1}$,\n \\begin{equation*}\n \t\\widetilde{p}^{(1)}_{s,1} = \\dfrac{1 - \\psi_1\\,\\lambda_1\\,a_1}{1 + \\frac{\\psi_1}{P_1^\\delta} \\sum_{k=2}^N P_k^\\delta\\,p_k\\lambda_k}.\n \\end{equation*}\n The next step is to verify the conditions of stability for the second user class, when the first user class is at steady state. After that, we remove the second user class from the set $\\cal{D}$ and calculate the stationary success probability of the two stable user classes in the dominant network. We repeat these steps until we remove all user classes, \\textit{i.e}, $\\cal{D} = \\{\\}$. We show this by induction; we suppose stability of the user classes $1,2,\\dots,j-1$. Let $\\cal{D} = \\{j,j+1,\\dots\\,N\\}$; the $j$th user class is stable, given that all the user classes in $\\cal{C}\\setminus\\cal{D}$ are stable, when\n \\begin{align} \\label{eq:aux_aj}\n \ta_j \n \t&< p_j\\,\\P(\\widetilde{\\text{SIR}}_{i,j} > \\theta_j) \\nonumber\\\\\n \t&= p_j \\left( 1+\\dfrac{\\psi_j}{P_j^\\delta} \n \\left( \\sum_{k=1}^{j-1} P_k^\\delta\\,\\lambda_k\\,\\dfrac{a_k}\n {\\widetilde{p}^{(j)}_{s,k}} + \\sum_{k=j}^N P_k^\\delta\\,p_k\\lambda_k \\right) \\right)^{-1},\n \\end{align}\n where $\\widetilde{p}^{(j)}_{s,k}$ is the $k$th user class success probability ($1 \\leq k < j$) at steady state in the dominant network at the $j$th step. To calculate this probability, we must solve the following system of equations. For $k \\in \\{1,2,\\dots,j-1\\}$\n \\begin{equation*}\n \t\\widetilde{p}^{(j)}_{s,k} = \\left( 1 + \\dfrac{\\psi_k}{P_k^\\delta}\n \\left( \\sum_{\\ell = 1}^{j-1} P_\\ell^\\delta\\,\\lambda_\\ell\\,\n \\dfrac{a_\\ell}{\\widetilde{p}^{(j)}_{s,\\ell}} + \n \\sum_{\\ell = j}^{N} P_\\ell^\\delta\\,p_\\ell\\lambda_\\ell \\right) \\right)^{-1}.\n \\end{equation*}\n Using an analogous approach as the one presented in the proof of Proposition~\\ref{prop:psk}, we have that for $k \\in \\{1,2,\\dots,j-1\\}$,\n \\begin{equation*}\n \t\\widetilde{p}^{(j)}_{s,k} =\n \\left( 1 + \\dfrac{\\psi_k}{P_k^\\delta} \n \\dfrac{ \\sum_{\\ell=1}^{j-1} P_\\ell^\\delta\\,a_{\\ell}\\,\\lambda_\\ell +\n \\sum_{\\ell=j}^{N} P_\\ell^\\delta\\,p_\\ell\\lambda_\\ell }\n { 1 - \\sum_{\\ell=1}^{j-1} \\psi_{\\ell}\\,a_{\\ell}\\,\\lambda_\\ell} \n \\right)^{-1}.\n \\end{equation*}\n Comparing the last two equations, it is easy to see that\n \\begin{align*}\n \t\\sum_{\\ell = 1}^{j-1} P_\\ell^\\delta\\,\\lambda_\\ell\\,\n &\\dfrac{a_\\ell}{\\widetilde{p}^{(j)}_{s,\\ell}} + \n \\sum_{\\ell = j}^{N} P_\\ell^\\delta\\,p_\\ell\\,\\lambda_\\ell = \\dfrac{ \\sum_{\\ell=1}^{j-1} P_\\ell^\\delta\\,a_{\\ell}\\,\\lambda_\\ell +\n \\sum_{\\ell=j}^{N} P_\\ell^\\delta\\,p_\\ell\\lambda_\\ell }\n { 1 - \\sum_{\\ell=1}^{j-1} \\psi_{\\ell}\\,a_{\\ell}\\,\\lambda_\\ell } .\n \\end{align*} \n Finally, we can use this result to rewrite \\eqref{eq:aux_aj} as, for all $j \\in \\cal{C}$,\n \\begin{equation*}\n 0 \\le\n \t\\dfrac{\\psi_j}{P_j^\\delta}\\,\\dfrac{a_j}{p_j-a_j} <\n \\dfrac{1 - \\sum_{k=1}^{j-1} \\psi_k\\,a_k\\,\\lambda_k}\n {\\sum_{k=1}^{j-1} P_k^\\delta\\,a_k\\lambda_k +\n \\sum_{k=j}^N P_k^\\delta\\,p_k\\lambda_k}.\n \\end{equation*}\n This concludes the proof since the extension for the other partitions of $\\cal{C}$ is analogous.\n\\end{proof}\n\nTheorem~\\ref{TH:NEC_SUFF} requires verifying $N\\!\\times\\!N!$ inequalities, whereas the following theorem is equivalent and it involves only $N$ inequalities.\n%\nThus, Theorem~\\ref{TH:STABILITY} presents a simpler form of verifying the conditions for stability.\n% Also, it relates (in the proof) the stability condition with the stationary mean delays $\\bm{D}$.\nHowever, we cannot prove \\ref{TH:STABILITY} without \\ref{TH:NEC_SUFF}.\n\n\\begin{theorem}[\\textbf{Network Stability}] \\label{TH:STABILITY}\n\tThe system network is stable if and only of $\\bm{a}\\in\\cal{R}$,\n \\begin{align*}\n \t\\cal{R}\n \\triangleq \\bigg\\{ \\bm{a}\\in[0,1)^N \\bigm\\vert ~ a_n < p_n, \\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} < \n \\dfrac{1 - \\sum_{k} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k} P_k^{\\delta}\\,a_k \\lambda_k} \n \\quad\\forall n \\in \\cal{C} \\bigg\\}.\n % &= \\bigg\\{ \\bm{a}\\in[0,1)^N \\bigm\\vert ~ a_n < p_n, \\\\\n % &\\hspace{2mm} \\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} < \n % \\dfrac{1 - \\sum_{k \\neq n} \\psi_k\\,a_k\\,\\lambda_k}\n % {P_n^\\delta\\,p_n\\lambda_n + \\sum_{k \\neq n} P_k^{\\delta}\\,a_k\\,\\lambda_k}\n % ~\\forall n \\in \\cal{C} \\bigg\\}.\n \\end{align*}\n\\end{theorem}\n\n\\begin{proof}\n\tThe proof consists of showing that the set $\\cal{R}$ is equal to the set defined in Theorem~\\ref{TH:NEC_SUFF}.\n %\n First, let us prove that $\\bigcup_{\\nu\\in\\cal{V}} \\cal{E}_\\nu \\subset \\cal{R}$.\n %\n For that we suppose $\\bm{a} \\in \\cal{E}_\\nu$ and we show $\\bm{a} \\in \\cal{R}$ for all $\\nu\\in\\cal{V}$ by induction.\n %\n For simplicity of exposition let us take $\\cal{E}_\\nu$ with $\\nu: n \\longmapsto n$, $n\\in\\cal{C}$.\n %\n We assume the inequality\n \\begin{align}\n \\dfrac{\\psi_{N-j}}{P_{N-j}^\\delta}\\,\\dfrac{a_{N-j}}{p_{N-j}-a_{N-j}} \n & < \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k}{\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k}. \\label{eq:th2_basecase}\n \\end{align}\n %\n is true for all $j\\in\\{0,\\dots,m-1\\}$ and we prove that it is also true for $j=m$.\n %\n First, we have to prove the base case $m = 1$.\n %\n Since $\\bm{a}\\in \\cal{E}_\\nu$, then for $j=0$ ($n=N$)\n %\n \\begin{align*}\n \t\\dfrac{\\psi_N}{P_N^\\delta}\\,\\dfrac{a_N}{p_N-a_N} \n \t& < \\dfrac{1 - \\sum_{k=1}^{N-1} \\psi_k\\,a_k \\lambda_k}{\\sum_{k=1}^{N-1} P_k^\\delta\\,a_k\\lambda_k + P_N^\\delta\\,p_N\\lambda_N} \\\\\n & = \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k + \\psi_N\\,a_N \\lambda_N}{\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k + P_N^\\delta\\,(p_N-a_N)\\lambda_N}.\n \\end{align*}\n %\n Then, using simple manipulations, we can show that the above inequality is equivalent to\n %\n \\begin{align*}\n \t\\dfrac{\\psi_N}{P_N^\\delta}\\,\\dfrac{a_N}{p_N-a_N} \n \t& < \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k}{\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k}.\n \\end{align*}\n %\n Thus, the base case $m = 1$ is true.\n %\n Now, for $j=m$ and $\\bm{a}\\in \\cal{E}_\\nu$, we know that\n \\begin{align*}\n \\dfrac{\\psi_{N-m}}{P_{N-m}^\\delta}\\,\\dfrac{a_{N-m}}{p_{N-m}-a_{N-m}}\n & < \\dfrac{1 - \\sum_{k=1}^{N-m-1} \\psi_k\\,a_k \\lambda_k}\n \t{\\sum_{k=1}^{N-m-1} P_k^\\delta\\,a_k\\lambda_k \n \t + \\sum_{k=N-m}^{N} P_k^\\delta\\,p_k\\lambda_k} \\\\\n & = \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k \n + \\sum_{k=N-m}^{N} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k \n + \\sum_{k=N-m}^{N} P_k^\\delta\\,(p_k-a_k)\\lambda_k}.\n \\end{align*}\n Through simple manipulations we can show that the above inequality is equivalent to\n \\begin{align*}\n & \\dfrac{\\psi_{N-m}}{P_{N-m}^\\delta}\\,\\dfrac{a_{N-m}}{p_{N-m}-a_{N-m}} < \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k \n + \\sum_{k=N-m+1}^{N} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k \n + \\sum_{k=N-m+1}^{N} P_k^\\delta\\,(p_k-a_k)\\lambda_k}.\n \\end{align*}\n Now, we only need to verify that\n \\begin{align*}\n &\\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k \n + \\sum_{k=N-m+1}^{N} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k \n + \\sum_{k=N-m+1}^{N} P_k^\\delta\\,(p_k-a_k)\\lambda_k} < \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k}.\n \\end{align*}\n Again, simple manipulations lead to the equivalent inequality\n \\begin{align*}\n &\\dfrac{\\sum_{k=N-m+1}^{N} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k=N-m+1}^{N} P_k^\\delta\\,(p_k-a_k)\\lambda_k}\n < \\dfrac{1 - \\sum_{k=1}^{N} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k=1}^{N} P_k^\\delta\\,a_k\\lambda_k},\n \\end{align*}\n which is true from the base case. This can be seen by multiplying \\eqref{eq:th2_basecase} by $P^\\delta_{N-j}(p_{N-j}-a_{N-j})$ at both sides of the inequality and summing over $j\\in\\{0,\\dots,m-1\\}$. Thus, $\\cal{E}_\\nu \\subset \\cal{R}$ for the mapping $\\nu: n \\longmapsto n$. The extension for another instances of $\\nu\\in\\cal{V}$ is analogous. This concludes the proof that $\\bigcup_{\\nu\\in\\cal{V}} \\cal{E}_\\nu \\subset \\cal{R}$.\n %\n \n However, we still need to prove the converse, that is ${\\cal{R} \\subset \\bigcup_{\\nu\\in\\cal{V}}} \\cal{E}_\\nu$.\n %\n Note that the set of arrival rates that makes the system stable in Theorem~\\ref{TH:NEC_SUFF} requires that at least one $a_n$ ($n\\in\\cal{C}$) satisfies\n \\begin{equation} \\label{eq:aux_stab_first}\n \t\\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} <\n \\dfrac{1}{\\sum_{k=1}^N P_k^\\delta\\,p_k \\lambda_k}.\n \\end{equation}\n \tLet us show that $\\cal{R}$ requires the same restriction by contradiction. Suppose that there exist $\\bm{a}\\in\\cal{R}$ such that\n %\n \\begin{equation} \\label{eq:aux_contr0}\n \\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} \\ge\n \\dfrac{1}{\\sum_{k=1}^N P_k^\\delta\\,p_k \\lambda_k}\n \\quad \\forall n \\in \\cal{C}.\n \\end{equation}\n %\n Multiplying \\eqref{eq:aux_contr0} by $P_n^\\delta(p_n-a_n)\\lambda_n > 0$ at both sides and summing over $n\\in\\cal{C}$ we have\n %\n \\begin{align*}\n \\sum_{n=1}^N \\psi_n a_n \\lambda_n \n \\ge \\frac{\\sum_{n=1}^N P_n^\\delta (p_n-a_n) \\lambda_n}\n {\\sum_{k=1}^N P_k^\\delta\\,p_k \\lambda_k},\n \\end{align*}\n %\n which is equivalent to\n %\n \\begin{align} \\label{eq:aux_case0}\n \\left(\\sum_{n=1}^N \\psi_n a_n \\lambda_n \\right)\n \\left(\\sum_{k=1}^N P_k^\\delta\\,p_k \\lambda_k \\right)\n \\ge \\sum_{n=1}^N P_n^\\delta (p_n-a_n) \\lambda_n.\n \\end{align}\n %\n Since $\\bm{a}\\in\\cal{R}$, then\n \\begin{align} \\label{eq:aux_theo2}\n \\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} < \n \\dfrac{1 - \\sum_{k} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k} P_k^{\\delta}\\,a_k \\lambda_k} \\quad \\forall n\\in\\cal{C}.\n \\end{align}\n %\n Again, multiplying \\eqref{eq:aux_theo2} by $P_n^\\delta(p_n-a_n)\\lambda_n > 0$ at both sides, summing over $n\\in\\cal{C}$ and performing some manipulations we have\n %\n \\begin{align} \\label{eq:aux_theo2_case0}\n &\\left(\\sum_{n=1}^N \\psi_n a_n \\lambda_n \\right)\n \\left(\\sum_{k=1}^N P_k^\\delta\\,a_k \\lambda_k \\right) < \\left(1 - \\sum_{k=1}^N \\psi_k\\,a_k \\lambda_k \\right)\\left(\\sum_{n=1}^N P_n^\\delta (p_n-a_n) \\lambda_n\\right).\n \\end{align}\n %\n Then, through some manipulations on \\eqref{eq:aux_case0} and \\eqref{eq:aux_theo2_case0}, we have\n %\n \\begin{align*}\n 0 &\\le \\sum_{n=1}^N P_n^\\delta a_n \\lambda_n - \\left(1 - \\sum_{k=1}^N \\psi_k\\,a_k \\lambda_k \\right)\\left(\\sum_{n=1}^N P_n^\\delta p_n \\lambda_n\\right) < 0,\n \\end{align*}\n which clearly is a contradiction, since $\\cal{R}$ is a non-empty set. Thus, there exists at least one $a_n$, $n\\in\\cal{C}$ that satisfies \\eqref{eq:aux_stab_first}.\n %\n For simplicity of exposition, let us suppose that the arrival rate $a_n$ that satisfies this restriction is from the first user class ($n=1$). The next step is to show that as in the set $\\bigcup_{\\nu\\in\\cal{V}}\\cal{E}_\\nu$, the set $\\cal{R}$ also requires that we have at least one $a_n$, aside from $a_1$, that satisfies\n %\n \\begin{equation*}\n \t\\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} <\n \\dfrac{1-\\psi_1\\,\\lambda_1\\,a_1}{P_1^\\delta\\,a_1\\lambda_1+\n \\sum_{k=2}^N P_k^\\delta\\,p_k\\lambda_k}.\n \\end{equation*}\n %\n We can also prove this by contradiction and then, for simplicity of exposition, suppose that $a_2$ is the one that satisfies this restriction. We repeat this procedure until we reach all user classes. Let us show the $j$th step for completeness, $j\\in\\cal{C}$. Suppose that for all $n \\in \\{j,j+1,\\dots,N\\}$,\n %\n \\begin{align} \\label{eq:aux_contr_j}\n \\dfrac{\\psi_n}{P_n^\\delta}\\dfrac{a_n}{p_n-a_n}\n \\ge \\dfrac{ 1 - \\sum_{k=1}^{j-1} \\psi_k a_k \\lambda_k }\n {\\sum_{k=1}^{j-1} P_k^{\\delta} a_k\\lambda_k +\n \\sum_{k=j}^N P_k^\\delta p_k\\lambda_k}.\n \\end{align}\n Multiplying \\eqref{eq:aux_contr_j} by $P_n^\\delta(p_n-a_n)\\lambda_n > 0$ at both sides, summing over $n\\in\\{j,j+1,\\dots,N\\}$ and manipulating we have\n %\n \\begin{align} \\label{eq:aux_case_j}\n &\\left(\\sum_{n=j}^N \\psi_n a_n \\lambda_n \\right)\\!\n \\left(\\sum_{k=1}^N P_k^\\delta\\,a_k \\lambda_k + \\sum_{k=j}^N P_k^\\delta\\,(p_k-a_k) \\lambda_k \\right) \\ge \\left( 1 - \\sum_{k=1}^{j-1} \\psi_k a_k \\lambda_k \\right)\\!\n \\left( \\sum_{n=j}^N P_n^\\delta (p_n-a_n) \\lambda_n \\right).\n \\end{align}\n %\n Once again, multiplying \\eqref{eq:aux_theo2} by $P_n^\\delta(p_n-a_n)\\lambda_n > 0$ at both sides, summing over $n\\in\\{j,j+1,\\dots,N\\}$ and manipulating we have\n %\n \\begin{align} \\label{eq:aux_theo2_case_j}\n &\\left(\\sum_{n=j}^N \\psi_n a_n \\lambda_n \\right)\n \\left(\\sum_{k=1}^N P_k^\\delta\\,a_k \\lambda_k \\right) < \\left(1 - \\sum_{k=1}^N \\psi_k\\,a_k \\lambda_k \\right)\\left(\\sum_{n=j}^N P_n^\\delta (p_n-a_n) \\lambda_n\\right).\n \\end{align}\n %\n Then, through some manipulations on \\eqref{eq:aux_case_j} and \\eqref{eq:aux_theo2_case_j}, we have\n %\n \\begin{align*}\n 0 &\\le \\left(1 - \\sum_{k=1}^N \\psi_k\\,a_k \\lambda_k \\right)\\left(\\sum_{n=j}^N P_n^\\delta (p_n-a_n) \\lambda_n\\right) - \\left(\\sum_{n=j}^N \\psi_n a_n \\lambda_n \\right)\n \\left(\\sum_{k=1}^N P_k^\\delta\\,a_k \\lambda_k \\right) < 0.\n \\end{align*}\n %\n As expected, we have a contradiction. Then, we must have at least one $a_n$, $n\\in{\\{j,j+1,\\dots,N\\}}$ that satisfies\n %\n \\begin{equation} \\label{eq:aux_region_j}\n \t\\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} <\n \\dfrac{1-\\sum_{k=1}^{j-1} \\psi_k\\,a_k\\,\\lambda_k}\n {\\sum_{k=1}^{j-1} P_k^\\delta\\,a_k\\lambda_k+\n \\sum_{k=j}^N P_k^\\delta\\,p_k\\lambda_k}.\n \\end{equation}\n %\n We assume that this is satisfied by the $j$th class and in this case, $\\cal{R} \\subset \\cal{E}_\\nu$ for $\\nu: n \\longmapsto n$.\n %\n Without the assumption of the ordering in which \\eqref{eq:aux_region_j} is satisfied, we conclude that \\eqref{eq:aux_region_j} must hold for at least one permutation of $\\cal{C}$. This region is exactly $\\bigcup_{\\nu\\in\\cal{V}} \\cal{E}_\\nu$. Therefore, ${\\cal{R} \\subset \\bigcup_{\\nu\\in\\cal{V}} \\cal{E}_\\nu}$.\n %\n Finally, ${\\cal{R} = \\bigcup_{\\nu\\in\\cal{V}} \\cal{E}_\\nu}$.\n\\end{proof}\n\n\\begin{remark}\n The convoluted conditions of Theorem~\\ref{TH:NEC_SUFF} are extensively reduced in Theorem~\\ref{TH:STABILITY}, for which we can see that as the density of users $\\lambda_n$ increases or the quantity $\\psi_n$ (which is inversely related to link quality, see \\eqref{eq:DefPhi}) increases for some $n\\in\\cal{C}$, then the stability region $\\cal{R}$ decreases for all user classes. On the other hand, if the transmission power $P_n$ increases for some $n\\in\\cal{C}$, then the stability region $\\cal{R}$ increases for the $n$th class and decreases for all other classes. Surprisingly, when the access probability $p_n$ varies, the only affected class (regarding stability region) is the $n$th class, as long $p_n>a_n$.\n\\end{remark}\n\nFigure~\\ref{fig:ThStab} shows an example of stability region for $N=3$ classes, where we have used only three non-linear inequalities instead of 18.\n%\nThe following corollary establishes a simple result on stability, which is used in Section~\\ref{sec:application} to propose and solve optimization problems regarding delay and throughput.\n\n\\begin{corollary} \\label{cor:stab}\n\tThere exists a vector of transmit powers ${\\bm{P} \\in \\R_+^N}$ such that the network is stable if and only if $\\bm{a}\\in\\cal{S}_0$, where\n \\begin{equation*}\n \t\\cal{S}_0 \\triangleq \\left\\lbrace \\bm{a}\\in[0,1)^N \\bigm\\vert 0 \\le \\sum_{n\\in\\cal{C}} \\dfrac{\\psi_n \\lambda_n}{\\frac{1}{a_n}-\\frac{1}{p_n}} < 1 \\right\\rbrace.\n \\end{equation*}\n\\end{corollary}\n\\begin{proof}\n First, let us show that $\\cal{R} \\subset \\cal{S}_0$ for all ${\\bm{P} \\in \\R_+^N}$.\n %\n If $\\bm{a}\\in\\cal{R}$, then for all $n\\in\\cal{C}$\n \\begin{align*}\n \\dfrac{\\psi_n}{P_n^\\delta}\\,\\dfrac{a_n}{p_n-a_n} < \n \\dfrac{1 - \\sum_{k} \\psi_k\\,a_k \\lambda_k}\n {\\sum_{k} P_k^{\\delta}\\,a_k \\lambda_k}.\n \\end{align*}\n %\n Multiplying both sides of the above equation by $P_n a_n \\lambda_n$ and summing over all $n\\in\\cal{C}$ results in \\vspace{-5mm}\n \\begin{align*}\n \\sum_{n\\in\\cal{C}} \\psi_n \\lambda_n\\,\\dfrac{p_n\\,a_n}{p_n-a_n} < 1\n \\end{align*}\n after some manipulations. Thus, $a\\in\\cal{S}_0$.\n \n Now, let us show that $\\cal{S}_0 \\subset \\cal{R}$ for some $\\bm{P}\\in\\R_+^N$.\n %\n In particular, let us choose $P_n = \\psi_n a_n/(p_n-a_n)$, $n\\in\\cal{C}$. Then, the inequalities that describe the region $\\cal{R}$ can be rewritten as one unique inequality\n \\begin{align*}\n 1 < \\frac{1 - \\sum_{k} \\psi_k a_k \\lambda_k}\n {\\sum_k \\psi_k a_k^2 \\lambda_k / (p_k-a_k)}\n \\end{align*}\n that does not depend on $n$ anymore. It is easy to show that this inequality is the same as the one that defines the region $\\cal{S}_0$. Thus, if $\\bm{a}\\in\\cal{S}_0$, then $\\bm{a}\\in\\cal{R}$ for that choice of $\\bm{P}$ (or any scalar multiple). This ends the proof.\n\\end{proof}\n\nFigure~\\ref{fig:CorStab} shows the region of arrival rates, according to Corollary~\\ref{cor:stab}, for which it is possible to find transmit powers that make the network stable.\n%\nOn the other hand, out of this region, the system is always unstable. It is worth mentioning that $\\psi_n$, defined in \\eqref{eq:DefPhi}, is related to the quality of the link between receiver and transmitter for class $n\\in\\cal{C}$; the larger is the value of $\\psi_n$, the poorer is the quality of the link.\n%\nAlso, the stability region $\\cal{R}$ showed in Fig.~\\ref{fig:ThStab} is contained in $\\cal{S}_0$. This is expected since we used the same parameters for both sets and Corollary~\\ref{cor:stab} considers the best case scenario, where we can choose the transmit powers $\\bm{P}$ for each $\\bm{a}$.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[t]{.45\\textwidth}\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_theorem_stab.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_theorem_stab.pdf}\n \\fi\n \\caption{Stability region $\\cal{R}$ according to Theorem~\\ref{TH:STABILITY} for $p_1 = 1/3$, $p_2 = 2/3$, $p_3 = 1$, $\\psi_1\\lambda_1=1$, $\\psi_2\\lambda_2=2$, $\\psi_3\\lambda_3=3$, $\\psi_1/P_1 = 1/3$, $\\psi_2/P_2 = 1/2$, $\\psi_3/P_3= 1$.}\n\\label{fig:ThStab}\n\\end{subfigure}%\n\\begin{subfigure}{.05\\textwidth}\n\\hspace{.05\\textwidth}\n\\end{subfigure}%\n\\begin{subfigure}[t]{.45\\textwidth}\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_corollary_stab.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_corollary_stab.pdf}\n \\fi\n \\caption{Maximum stability region $\\cal{S}_0$ according to Corollary~\\ref{cor:stab} for $p_1 = 1/3$, $p_2 = 2/3$, $p_3 = 1$, $\\psi_1\\lambda_1=1$, $\\psi_2\\lambda_2=2$, $\\psi_3\\lambda_3=3$.}\n\\label{fig:CorStab}\n\\end{subfigure}\n\\caption{}\n\\end{figure}\n\nFrom now on, we assume that whenever there is a packet in the buffer, the corresponding transmitter attempts to transmit, i.e., the medium access probability $p_n=1$ for all $n\\in\\cal{C}$.\n%\nWe discuss in Section~\\ref{sec:high-mobility} the validity of this assumption and the high-mobility assumption. The motivation is that, when the access probability of all classes is equal to one, we maximize the stability region $\\cal{R}$.\n%\nThis is easy to see with the inequalities of Theorem~\\ref{TH:STABILITY}, where the right-hand side does not depend on $p_n$ and the left-hand side decreases monotonically with $p_n$. Thus, the stability region is maximized when $p_n=1$ for all $n\\in\\cal{C}$. The same occurs in Corollary~\\ref{cor:stab}.\n%\nThis result is surprising and might be explained by the fact that we have independence between adjacent time slots and, therefore, for each time slot there is a new scenario (a new effective PPP).\n%\nThen, it makes sense to always try retransmission. This approach also minimizes the mean delay according to \\eqref{eq:Dn}, since the success probability $p_{s,n}$ in \\eqref{eq:psn} does not depend on the access probability in a stable network.\n\nUsing Proposition \\ref{prop:psk}, Proposition~\\ref{prop:identity_1} is introduced, which presents an equation that relates all performance parameters independently of the transmission powers. Also, the conditions for stability are extensively simplified, see Corollary~\\ref{cor:stab}.\n%\n\\begin{proposition} \\label{prop:identity_1}\n\tIf the network is stable and $p_n=1$ for all ${n\\in\\cal{C}}$, then the following identities hold (at stationary state):\n \\begin{equation}\\label{eq:identity_1}\n \t\\sum_{n\\in\\cal{C}} \\psi_n\\,\\lambda_n\\,\\dfrac{D_n}{D_n-1}\\,\\dfrac{a_n}{1-a_n} = 1,\n \\end{equation}\n and\n \\begin{equation*}\n \t\\frac{\\psi_j}{P_j^\\delta}\n \\left( \\frac{D_j}{D_j-1}\\,\\frac{1}{1-a_j} - 1\\right) =\n \\frac{\\psi_k}{P_k^\\delta} \\left( \\frac{D_k}{D_k-1}\\,\n \\frac{1}{1-a_k} - 1 \\right)\n \\quad \\forall\\,j,k\\in\\cal{C}.\n \\end{equation*}\n\\end{proposition}\n\n\\begin{proof}\n\tWe start with the terms of the sum,\n\t\\begin{align*}\n\t\t\\psi_n\\,\\lambda_n\\,\\dfrac{D_n}{D_n-1}\\,\\dfrac{a_n}{1-a_n}\n &\\stackrel{\\text{(i)}}{=} \\psi_n\\lambda_n\\dfrac{a_n}{1-p_{s,n}}\\\\ \n\t\t&= P_n^\\delta \\dfrac{\\lambda_n\\,a_n}{p_{s,n}} \\left( \n \\dfrac{\\psi_n}{P_n^\\delta} \\dfrac{p_{s,n}}{1-p_{s,n}} \\right)\\\\\n &\\stackrel{\\text{(ii)}}{=} \\frac{ P_n^\\delta\\frac{\\lambda_n\\,a_n}{p_{s,n}} }\n { \\sum_j P_j^\\delta\\frac{\\lambda_j\\,a_j}{p_{s,j}} },\n\t\\end{align*}\n where (i) comes from \\eqref{eq:Dn} with $p_n=1$ and (ii) comes from \\eqref{eq:psk_equivalence}. Summing over $\\cal{C}$ ends the proof of the first identity.\n %\n For the second relation of Proposition~\\ref{prop:identity_1}, we use \\eqref{eq:Dn} once again to find\n \\begin{equation*}\n \t\\dfrac{\\psi_n}{P_n^\\delta} \\left( \\dfrac{D_n}{D_n-1}\\,\\dfrac{1}{1-a_n} - 1\\right) = \\dfrac{\\psi_n}{P_n^\\delta} \\dfrac{p_{s,n}}{1-p_{s,n}}.\n \\end{equation*}\n Comparing this expression with \\eqref{eq:Pi_Pk} ends the proof.\n\\end{proof}\n\nProposition~\\ref{prop:identity_1} is an elegant form to see that a channel is a limited resource regarding traffic intensity and delay. Let us rewrite the identity \\eqref{eq:identity_1} in terms of physical parameters,\n\\begin{equation} \\label{eq:physical}\n\t\\sum_{n=1}^N 4\\,\\lambda_n\\,\\overline{R}_n^2\\,\\theta_n^{2/\\alpha}\\,\\dfrac{D_n}{D_n-1}\\,\\dfrac{a_n}{1-a_n} = \\dfrac{\\sin(2\\pi/\\alpha)}{2\\pi/\\alpha}.\n\\end{equation}\n%\nNote that $\\frac{a_n}{1-a_n}$ and $\\frac{\\sin(2\\pi/\\alpha)}{2\\pi/\\alpha}$ are monotonic increasing functions and $\\frac{D_n}{D_n-1}$ is a monotonic decreasing function. The right hand-side of \\eqref{eq:physical} can be seen as the amount of resource available to all users of the channel.\n%\nLarger $\\alpha$ results in higher $\\frac{\\sin(2\\pi/\\alpha)}{2\\pi/\\alpha}$, meaning that a larger amount of resource is available to users. This can be explained by recalling that a larger path loss exponent leads to stronger isolation among links sharing the channel and, consequently, more users can be accommodated in the network.\n%\nTherefore, the larger the path loss exponent $\\alpha$, the larger (smaller) the terms $\\lambda_n$, $\\overline{R}_n$, $\\theta_n$, $a_n$ ($D_n$) can be. The identity \\eqref{eq:physical} also tells us that the $n$th class of user takes a well-defined portion of the amount of resource available in the network, which is given by the $n$th term in the summation. \n%\nThis means that the values of $\\lambda_n$, $\\overline{R}_n$, $\\theta_n$, $a_n$ and $D_n$ for a given class $n$ can be adjusted, while keeping the quantity $\\lambda_n\\,\\overline{R}_n^2\\,\\theta_n^{2/\\alpha}\\,\\frac{D_n}{D_n-1}\\,\\frac{a_n}{1-a_n}$ unchanged.\n%\nFor instance, we can make a direct exchange between decreasing the delay $D_n$ and decreasing the arrival rate of packets $a_n$ (by controlling the ratio of transmit power levels), such that the term $\\frac{D_n}{D_n-1}\\,\\frac{a_n}{1-a_n}$ remains constant; or else, increase the arrival rate of packets and decrease the density of users, such that the term $\\lambda_n\\,\\frac{a_n}{1-a_n}$ remains constant. \n%\nTherefore, Proposition \\ref{prop:identity_1} reveals, through a simple expression, the interplay among traffic intensity, mean delay, density of users, link distance, and outage probability, when the network is stable.\n\n\\begin{remark}\n Corollary~\\ref{cor:stab} and Proposition~\\ref{prop:identity_1} are the simplest and the most meaningful results of the present chapter, as they translate the behavior of the network in simple equations, which do not directly depend on the transmission powers.\n\\end{remark}\n\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[t]{.45\\textwidth}\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_Opt_Delay_equal_a.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_Opt_Delay_equal_a.pdf}\n \\fi\n \\caption{Optimum delays}\n\\label{fig:Opt_Delay_equal}\n\\end{subfigure}%\n\\begin{subfigure}{.05\\textwidth}\n\\hspace{.05\\textwidth}\n\\end{subfigure}%\n\\begin{subfigure}[t]{.45\\textwidth}\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_Opt_Power_equal_a.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_Opt_Power_equal_a.pdf}\n \\fi\n \\caption{Optimum transmit powers}\n\\label{fig:Opt_Power_equal}\n\\end{subfigure}\n\\caption{These figures represent the optimization of a 3-class network with the following parameters: $a_1=a_2=a_3=a$, $\\psi_1\\,\\lambda_1 = 0.1$, $\\psi_2\\,\\lambda_2 = 0.3$, $\\psi_3\\,\\lambda_3 = 0.6$ and $c_1 = \\frac{10}{16},~ c_2 = \\frac{5}{16},~ c_3 = \\frac{1}{16}$.}\n\\label{fig:opt}\n\\end{figure}\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %\n\\subsection{Interpretation and Application}\n\\label{sec:application}\n\nIn this section, we solve two optimization problems using the proposed formulation applied to scenarios of different classes of terminals sharing a radio channel.\n\n% -------------------------------- %\n\\subsubsection{Delay Optimization}\n\\label{ssec:opt_delay}\n\nLet us consider the scenario with $N$ classes sharing a channel. Each class may represent a particular user application, with each application having a different delay requirement in the network.\n%\nLet us suppose we are interested in adjusting the transmit power of each user class, such that the weighted average delay among all classes is minimized.\n%\nThis problem is addressed as follows. For fixed arrival rates of vector $\\bm{a}$ that satisfies Corollary~\\ref{cor:stab}, i.e., for $\\bm{a}\\in\\cal{S}_0$, let us minimize the delays $\\bm{D}$ by changing the ratio between the transmit powers $\\bm{P}$.\n%\nEach user class requires a different response time, then we weight the optimization problem with the vector $(c_1,c_2,\\dots,c_N) \\in \\R_+^N$. The larger the coefficient of a class, the smaller the resulting mean delay to deliver packets for that class. Then, we have\n\\begin{equation} \\label{eq:opt_prob}\n\t\\min_{\\bm{P}\\in\\R_+^N} \\sum_{n\\in\\cal{C}} c_n D_n\n = \\min_{\\bm{P}\\in\\R_+^N} \\sum_{n\\in\\cal{C}} \\frac{c_n\\,(1-a_n)}\n \t{\\left( 1 + \\frac{\\psi_n}{P_n^\\delta}\n \t\\frac{\\sum_j P_j^\\delta\\,a_j\\lambda_j}\n \t{1 - \\sum_j \\psi_j\\,a_j\\lambda_j} \\right)^{\\hspace{-1mm}-1}\\!\\!\\! - a_n},\n\\end{equation}\nwhere $D_n$ is given by Proposition~\\ref{prop:psk}. Note that as thermal noise is not considered in our model, we have a degree of freedom for the optimum solution $\\bm{P}^*$, which agrees with the formulation in \\eqref{eq:opt_prob}.\n\n\\begin{proposition} \\label{prop:opt}\n\tThe minimum of the optimization problem \\eqref{eq:opt_prob} is attained by\n \\begin{equation} \\label{eq:opt_D}\n {P_n^*}^\\delta\n \t= \\dfrac{\\beta}{\\lambda_n a_n} \\left( \\frac{a_n\\,\\cal{A}_n}\n {1 - \\sum_{k} \\cal{A}_k }\n + \\frac{\\sqrt{c_n\\,\\cal{A}_n}}\n { \\sum_{k} \\sqrt{c_k\\,\\cal{A}_k } } \\right), \\quad n\\in\\cal{C},\n \\end{equation}\n where $\\beta$ is any positive real constant, $\\cal{A}_n \\triangleq \\psi_n\\,\\lambda_n\\,\\frac{a_n}{1-a_n}$ and the sums are over $\\cal{C}$.\n\\end{proposition}\n\n\\begin{proof}\n\tSince we have one degree of freedom for the solution $\\bm{P}^*$, let us set $\\sum_j P_j^\\delta\\,a_j\\lambda_j = 1 - \\sum_j \\psi_j\\,a_j\\lambda_j$ to extensively simplify the algebraic manipulations. Then, we use the Karush-Kuhn-Tucker conditions \\cite[Section~3.3.1]{bertsekas1999nonlinear} in the \\emph{Lagrangian} function\n \\begin{equation*}\n \\begin{split}\n \t\\cal{L}(\\bm{P},\\mu) \n \t &= \\sum_{n\\in\\cal{C}} \\frac{c_n (1-a_n)}\n \t {\\left( 1 + \\frac{\\psi_n}{P_n^\\delta}\\right)^{-1}\\! - a_n} + \\mu\\left[ \\sum_{j\\in\\cal{C}} P_j^\\delta\\,a_j\\lambda_j \n \t- \\left(1 - \\sum_{j\\in\\cal{C}} \\psi_j\\,a_j\\lambda_j\\right) \\right],\n \\end{split}\n \\end{equation*}\n where $\\mu\\in\\R$ is the \\emph{Lagrange} multiplier.\n %\n The objective function is strictly convex (the Hessian is a diagonal matrix with positive eigenvalues) and the feasible region is a hyperplane, therefore the solution is the global optimum.\n %\n Now, we return to the original problem that does not have the artificial constraint. Thus, we multiply the solution by an arbitrary constant $\\beta>0$ to obtain the general solution.\n\\end{proof}\n\nIt is interesting to note that ${\\sum_k\\cal{A}_k<1}$ by Corollary~\\ref{cor:stab}. Therefore, ${P_n^*}^\\delta$ is always a positive quantity.\n%\nAlso, if $c_n = \\cal{A}_n$ for all $n\\in\\cal{C}$, then the optimum delays are all equal and given by ${D_1 = D_2 = \\cdots = D_N = \\left( 1 - \\sum_k \\cal{A}_k \\right)^{-1}}$. Thus, we can always choose transmit powers, such that we have the same mean delay for all classes!\n\n\nAs an example, let us consider a 3-class network, where Class 1 has a more restrictive delay requirement than Class 2, which is more restrictive than Class 3. We consider that all classes have the same arrival rate of packets, i.e., $a_1=a_2=a_3=a$. Figure~\\ref{fig:Opt_Delay_equal} shows the expected waiting time of a packet before a successful transmission, which is $D_n^*-1$, ($n = 1,2,3$) since a transmission takes exactly one time slot. As expected, the optimization resulted in monotonic increasing functions and $D_1^* < D_2^* < D_3^*$ for all $a$.\n\n\nFigure~\\ref{fig:Opt_Power_equal} shows the normalized\\footnote{Whenever we refer to normalized $b_n {P_n^*}^\\delta$, it means that we choose $\\beta$ in Proposition~\\ref{prop:opt} such that $\\sum_k b_k {P_k^*}^\\delta = 1$ and $b_n$ is any term that depends on $n$, for example $b_n = \\lambda_n$ or $b_n = a_n \\lambda_n$.} transmit powers per unit of area $\\lambda_n {P_n^*}^\\delta$ as a function of $a$. In this case, we do not have a clear hierarchy among the transmit powers, as it depends on the traffic intensity. For $n\\in\\cal{C}$, if the network is close to saturation, i.e., $\\sum_k\\cal{A}_k$ tends to 1, then the normalized $\\lambda_n {P_n^*}^\\delta$ approaches $\\cal{A}_n/\\sum_k\\cal{A}_k$ and, at first order, it does not depend on the coefficients $c_1, c_2, \\dots, c_N$. On the other hand, if the network is at low traffic, i.e., $\\sum_k\\cal{A}_k$ tends to 0, then the normalized $a_n \\lambda_n {P_n^*}^\\delta$ approaches $\\sqrt{c_n\\cal{A}_n}/\\sum_k\\sqrt{c_k\\cal{A}_k}$.\n\n% -------------------------------- %\n\\subsubsection{Throughput Optimization}\n\nNow, let us maximize the total throughput of the channel per unit of area with the constraint that the system is stable.\n%\nSince each transmitter performs retransmissions until the packet is correctly received by the intended receiver, then all packets are successfully transmitted (eventually) in a stable system. Thus, the throughput per transmitter is given by the packet arrival rate $a_n$.\n%\nThe density of users per unit of area is given by $\\lambda_n$, then the throughput of the $n$th user class per unit of area is simply $\\mathscr{T}_n = \\lambda_n\\,a_n$ and the throughput per unit of area of the entire system $\\mathscr{T}$ is the sum of the throughput for all classes $n\\in\\cal{C}$.\n\nUsing Corollary~\\ref{cor:stab}, we can formulate the optimization problem as\n${\\displaystyle\\max_{\\bm{a}\\in\\cal{S}_0}} \\sum_{n} \\lambda_n\\,a_n$.\n%\nHowever, the strict inequality in the region $\\cal{S}_0$ of Corollary~\\ref{cor:stab} results in an optimization problem that is not well-posed. In this case, if the optimum solution lies in the boundary of the feasible region, then the solution does not exist.\n%\nTo circumvent this problem, we propose a new region $\\cal{S}_\\epsilon\\subset\\cal{S}_0$ by adding an arbitrarily small parameter $\\epsilon \\in (0,1)$ in the inequality, i.e., the new region is given by\n\\begin{equation*}\n\t\\cal{S}_{\\epsilon} = \\left\\lbrace \\bm{a}\\in[0,1)^N \\bigm\\vert \\sum_{n\\in\\cal{C}} \\psi_n\\,\\lambda_n\\,\\dfrac{a_n}{1-a_n} \\le 1-\\epsilon \\right\\rbrace.\n\\end{equation*}\n%\nAs the parameter $\\epsilon$ increases, the system becomes less sensitive to perturbations\\footnote{When the system parameters suffer a sufficiently small change, the system remains stable.}.\nNow, the optimization problem is posed as\n\\begin{equation} \\label{eq:opt_thr}\n\t\\max_{\\bm{a}\\in\\cal{S}_\\epsilon} \\mathscr{T} = \\max_{\\bm{a}\\in\\cal{S}_\\epsilon} \\sum_{n\\in\\cal{C}} \\lambda_n\\,a_n.\n\\end{equation}\n%\nThe following proposition presents the solution to the optimization, i.e., the optimum arrival rates $\\bm{a}^*$ that maximize the throughput per unit of area and maintain the system stable.\n\n\\begin{proposition} \\label{prop:eps}\n\tIf \n \\begin{equation} \\label{eq:opt_req}\n\t\t\\sum_{k\\in\\cal{C}}\\lambda_k\\psi_k\\left(\\sqrt{\\frac{\\max_n \\psi_n}{\\psi_k}}-1\\right)<1-\\epsilon,\n\t\\end{equation} then the solution of \\eqref{eq:opt_thr} is attained by\n \\begin{equation}\n \ta_n^* = 1 - \\frac{\\sum_k \\lambda_k \\sqrt{\\psi_n\\psi_k}}{1-\\epsilon+\\sum_k\\lambda_k\\psi_k}, \\quad n\\in\\cal{C},\n \\end{equation}\n where the sums are over $\\cal{C}$. If the inequality \\eqref{eq:opt_req} is not satisfied, then the $m$th class is excluded, where $m = \\arg\\max_n \\psi_n$, and the inequality is checked again.\n\\end{proposition}\n\n\\begin{proof}\n\tIt is a direct application of the Karush-Kuhn-Tucker conditions \\cite[Section~3.3.1]{bertsekas1999nonlinear} in the \\emph{Lagrangian} function\n \\begin{align*}\n \t\\cal{L}(\\bm{a},\\mu) = \\sum_{n\\in\\cal{C}} \\lambda_n a_n + \\mu \\left[ \\sum_{k\\in\\cal{C}}\\psi_k\\lambda_k\\frac{a_k}{1-a_k} - (1-\\epsilon)\\right],\n \\end{align*}\n where $\\mu\\in\\R$ is the Lagrange multiplier associated with the constraint of stability. Equation~\\eqref{eq:opt_req} guarantees that the solution $\\bm{a}^*\\in[0,1)^N$.\n %\n The objective function is convex (affine function) and the region $\\cal{S}_\\epsilon$ is strictly convex because the Hessian of the function that defines the region is a diagonal matrix with negative eigenvalues. Therefore the presented solution is the global optimum and it is unique.\n\\end{proof}\n\nIn the optimization \\eqref{eq:opt_thr}, we still have freedom to choose the transmit powers $\\bm{P}$, as long the network remains stable. The best way of choosing $\\bm{P}$ is by minimizing the delays, which we have already done in Subsection~\\ref{ssec:opt_delay}, Proposition~\\ref{prop:opt}, where the arrival rates $\\bm{a} \\in \\cal{S}_\\epsilon \\subset \\cal{S}_0$ are fixed. When the optimization is performed in this sequence (maximization of throughput, then minimization of delay), we have the optimum throughput (per unit of area) and the optimum delays for the optimum configuration of arrival rates.\n%%\nLater in this section, we illustrate this procedure with a numerical example.\n\nIn order to solve the optimization problem \\eqref{eq:opt_thr} we did not have to handle with the transmit powers $\\bm{P}$ directly, which would make the solution and the problem formulation more cumbersome. This shows the usefulness of Corollary~\\ref{cor:stab}.\n\n\\begin{figure}\n\\centering\n\\begin{subfigure}[t]{.45\\textwidth}\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_Opt_a_eps.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_Opt_a_eps.pdf}\n \\fi\n \\caption{Optimum arrival rates}%\n \\label{fig:Opt_a_eps}\n\\end{subfigure}%\n\\begin{subfigure}{.05\\textwidth}\n \\hspace{.05\\textwidth}\n\\end{subfigure}%\n\\begin{subfigure}[t]{.45\\textwidth}%%\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_Opt_lam_a_eps.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_Opt_lam_a_eps.pdf}\n \\fi\n \\caption{Optimum throughput. The dashed curve corresponds to a scenario with only the best performing class}\n \\label{fig:Opt_lam_a_eps}\n\\end{subfigure}\n%\n\\begin{subfigure}[t]{.45\\textwidth}%%%\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_Opt_D_eps.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_Opt_D_eps.pdf}\n \\fi\n \\caption{Optimum delays}\n \\label{fig:Opt_D_eps}\n\\end{subfigure}%\n\\begin{subfigure}{.05\\textwidth}\n \\hspace{.05\\textwidth}\n\\end{subfigure}%\n\\begin{subfigure}[t]{.45\\textwidth}%%%%\n \\centering\n \\if\\printfig1\n \\includegraphics[width=\\textwidth]{Figures/Ch7_Opt_P_eps.pdf}\n \\else\n \\includegraphics[draft,width=\\textwidth]{Figures/Ch7_Opt_P_eps.pdf}\n \\fi\n \\caption{Optimum transmit powers}\n \\label{fig:Opt_P_eps}\n\\end{subfigure}\n\\caption{These figures represent the optimization of the throughput and mean delay of a 3-class network with the parameters given in Table~\\ref{tab:param}.}\n\\label{fig:opt_eps}\n\\end{figure}\n\nLet us illustrate the throughput optimization problem with a system for which the parameters are shown in Table~\\ref{tab:param}. Figure~\\ref{fig:Opt_a_eps} shows the optimum arrival rates $\\bm{a^*}$ that maximizes the throughput per unit of area.\n%\n\\begin{table}[hbt]\n \\centering\n \\caption{Network parameters for Fig.~\\ref{fig:opt_eps}}\n \\begin{tabular}{l l}\n \\hline\n \\hline\n \\textbf{Parameters} & \\textbf{Values} \\\\\n \\hline\n $(\\lambda_1,\\lambda_2,\\lambda_3)$\t& $=(1,2,3)$ \\\\\n $(\\psi_1,\\psi_2,\\psi_3)$\t\t\t& $=(0.3,0.5,0.4)$ \\\\\n\t\t$(c_1,c_2,c_3)$\t\t\t\t\t\t& $=(\\frac{1}{3},\\frac{1}{3},\\frac{1}{3})$ \\\\\n \\hline\n \\hline\n \\end{tabular}\n \\label{tab:param}\n% \\vspace*{-\\baselineskip}\n\\end{table}\n%\nIt is quite interesting that the optimum solution is not (necessarily) exclusively activating the class with the best link quality (i.e., the class with the smallest $\\psi$, which is Class 1 in this example). In Fig.~\\ref{fig:Opt_lam_a_eps} it is shown the optimum throughput for each class and the total throughput of the system.\n%\nFor comparison, we plotted a dashed curve representing the total throughput if we only use the best performing user class, regarding throughput.\nThe dashed curve is below the optimum total throughput for all $\\epsilon$. Therefore, the best solution is always a combination of all user classes, as long as \\eqref{eq:opt_req} is satisfied.\n%\nOn the other hand, if this equation is not satisfied, it means that there is at least one user class with a bad link quality, such that it is better (regarding throughput efficiency) to reallocate this user class to another channel.\n\nNow that we have, for each $\\epsilon$, the arrival rate configuration $\\bm{a}^*(\\epsilon)$ which gives the maximum throughput, we can use Proposition~\\ref{prop:opt} to find the best configuration of transmit powers $\\bm{P}^*(\\epsilon)$ that minimizes the sum of the mean delays for each optimum configuration of arrival rates $\\bm{a}^*(\\epsilon)$.\n%\nFigure~\\ref{fig:Opt_D_eps} shows the result of this optimization, which is a direct application of \\eqref{eq:opt_D}.\n%\nIt is worth noting that as we increase $\\epsilon$ the system is farther from instability, which corresponds to having a smaller delay to transmit packets, as we can see in Fig.~\\ref{fig:Opt_D_eps}, and a smaller throughput, as shown in Fig.~\\ref{fig:Opt_lam_a_eps}.\n%\nFigure~\\ref{fig:Opt_P_eps} shows the optimum distribution of power per unit of area required by each user class. Notice that the first user class, which has the best link quality, uses the smallest power per unit of area. However, this behavior is more intricate; it also depends on the density of users of the corresponding class. Notice, for example, the inversion between user classes 2 and 3 as we increase $\\epsilon$ in Fig.~\\ref{fig:Opt_P_eps}.\n\nAnother interesting and direct result from Corollary~\\ref{cor:stab} is to provide an upper bound for the total throughput per unit of area $\\mathscr{T}$ in a stable system, which is given by $1/(\\min_n\\psi_n)$.\n%\nThis is not a tight bound, however it is interesting on its own, due to its simplicity and the fact that it does not depend on the density of users. The proof follows\n\\begin{align}\n\t\\mathscr{T} = \\sum_{k\\in\\cal{C}} a_k\\lambda_k &\\le \\frac{1}{\\min_{n}\\psi_n} \\sum_{k\\in\\cal{C}} \\psi_k\\,a_k\\,\\lambda_k \\nonumber\\\\\n &< \\frac{1}{\\min_{n}\\psi_n} \\sum_{k\\in\\cal{C}}\\psi_k\\,\\lambda_k\\,\\frac{a_k}{1-a_k} \\nonumber\\\\\n &< \\left(\\textstyle\\min_{n}\\psi_n \\right)^{-1},\n\\end{align}\nwhere the last inequality comes from Corollary~\\ref{cor:stab}.\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % \n% % % % % % % % % % % % % % % % % % % % % % % % % % \n\\section{On the High-mobility Assumption} \\label{sec:high-mobility}\n\nIn this section, we address the high-mobility assumption, which may not be realistic in real wireless networks, since the mobility of transmitters does not change drastically between adjacent time slots. Therefore, the independence assumption would not hold.\n%\nNevertheless, in a stable wireless network that has a small packet arrival rate $a$ per user or a small access probability $p$, the correlation might be sufficiently small such that the independence (high-mobility) assumption is reasonable. In \\cite{haenggi2013diversity} the authors show that if the access probability $p$ is sufficiently small, then the independence assumption provides a good approximation.\n\nWhen the packet arrival rate $a$ or the access probability $p$ are small, the typical user sees a significantly different PPP of transmitters for each time slot, which justifies the independence (high-mobility) assumption. We verified this claim through simulations and the result is shown in Fig.~\\ref{fig:high-mobility}, where it is used one user class with ${\\lambda c \\overline{R}^2 = \\pi/4}$, ${\\alpha = 3}$ and ${\\theta = 1}$.\n%\n\\begin{figure}\n \\centering\n \\if\\printfig1\n \\includegraphics[width=0.55\\textwidth]{Figures/Ch7_high-mobility.pdf}\n \\else\n \\includegraphics[draft,width=0.75\\textwidth]{Figures/Ch7_high-mobility.pdf}\n \\fi\n \\caption{Queue load $\\rho$ as a function of the access probability $p$. Simulation results with a static network are presented in marks and the theoretical results with the high-mobility assumption are presented in curves.}\n \\label{fig:high-mobility}\n\\end{figure}\n\nThe mean load of the queues $\\rho$, which is equivalent to the percentage of queues with packets to transmit, are plotted as a function of the access probability $p$ for several values of arrival rate $a$. As expected, for small values of $a$ or $p$, the theoretical model presents good estimations of the average queue load.\n%\nIt is important to emphasize that we did not plot the mean delay $D$, because in a static PPP there might exist a set of unstable users, whose queues and delays tend to infinity. This would raise the average delay to infinity too. Then, we chose to plot the mean load $\\rho$, which is equal to 1 for unstable users and does not tend to infinity opposed to the mean delay $D$.\n\nTo establish Proposition~\\ref{prop:identity_1}, we supposed that the access probability $p$ is equal to one for all users. In the context of high-mobility, this approach makes sense, since the typical user sees a different interference scenario for each time slot. Thus, it is reasonable to attempt a retransmission every time slot until the packet is successfully transmitted. This also minimizes the mean delay $D$, which is in accordance with \\eqref{eq:Dn}, as the transmission success probability $p_{s,n}$ does not depend on the access probability $p_n$ in a stable network.\n\nThere is another scenario, which does not require high-mobility to achieve spatial independence between adjacent time slots. This scenario is a network that uses the frequency-hopping scheme over a set of channels \\cite{tse2005fundamentals}. For each time slot, there is a different PPP pattern, since the transmitting nodes select with equal probability one channel to transmit. Thus, the spatial correlation between time slots decreases with the number of channels available for selection.\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % \n% % % % % % % % % % % % % % % % % % % % % % % % % % \n\\section{Bandwidth Partitioning on High-mobility Bipolar Networks}\n\\label{sec:bandwidth}\n\nIn this section, we consider the model presented in Section~\\ref{sec:N-class} with only two classes of users, i.e., $N=2$.\n%\nHowever, differently from our previous models, the available frequency band is partitioned among users of one of the classes, such that users of this class can access only a fraction of the original bandwidth available to the network at a time.\n%\nThe other class continues using the whole frequency band available.\n%\nThe motivation for such bandwidth partitioning is to reduce the interference among users, leading to a higher network capacity.\n\nThe work presented in this section generalizes the paper by Jindal et al. in \\cite{jindal2008bandwidth} in the sense that it (\\textit{i}) considers two user classes and (\\textit{ii}) investigates the effects of bandwidth partitioning on the mean \\textit{queuing} delay.\n%\nBy considering two classes of users, we are able to study the performance of a network in which users of one class are allowed to transmit on a fraction of the channel accessed by users of the other class. Also, by studying the delay, we are able to have a better understanding of the effects of bandwidth partitioning on network performance.\n%\n% The present work also extends the study presented in \\cite{dester2018} by using bandwidth partitioning on one of the user classes. \n% More specifically, we recover some results from \\cite{jindal2008bandwidth} and \\cite{dester2018} by setting the arrival rate of Class 2 $a_2 = 0$ and by setting the number of band partitions $M=1$, respectively.\n\nWe used the wireless network formulation proposed in Section~\\ref{sec:N-class} to achieve a tractable framework for the analysis of the bandwidth partitioning.\n\nThe main contributions presented in this section can be summarized as follows:\n\\begin{itemize}\n \\item We have found a pair of simple equations that relate the main parameters of the network and guarantees network stability (Theorem~\\ref{th:identity});\n \\item A transcendental equation to find the optimum Class 1 spectral efficiency is derived, along with the optimum number of partitions (Theorem~\\ref{th:optimum_eta});\n \\item The first-order expansion of the optimum spectral efficiency around a small use of the channel by Class 2 (Eq.~\\eqref{eq:eta_expansion});\n \\item We show that the bandwidth partitioning strategy is more effective when the path loss exponent is not small and the performance requirements of Class 2, the one whose users access the whole bandwidth, is not high in respect to the maximum performance attainable (Fig.~\\ref{fig:a_ratio}).\n\\end{itemize}\n\n% The paper is organized as follows. In Section~\\ref{sec:sysmod_M} the system model is presented and in Section~\\ref{sec:stab} we derive the stability conditions and expressions for stationary transmission success probability and mean delay. Using these expressions, we present in Section~\\ref{sec:opt} the optimization problem involving bandwidth partitioning and proves the existence and uniqueness of the solution. Section~\\ref{sec:num} provides some numerical examples of bandwidth optimization. %, also including a case involving cellular and D2D.\n% Section~\\ref{sec:conc} concludes the paper.\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %\n\\subsection{System Model} \\label{sec:sysmod_M}\n\nWe consider the network model presented in Section~\\ref{sec:N-class} with two user classes (namely Class 1 and Class 2), and only one and important difference that users of Class 1 access a fraction of the bandwidth during each transmission.\n\n%The high-mobility model, which is used in several works (see, for instance \\cite{jindal2008bandwidth}, \\cite{baccelli2010stochastic}, \\cite{stamatiou2010random}, \\cite{dester2018}), assumes that users occupy a different point in space for each time slot. \n%The communication protocol is the slotted Aloha, i.e., for each time slot, if a source (transmitter) has packets to transmit, it will transmit one packet with a fixed probability, independently from other sources and the past. The position of source $i \\in \\N$ of class $n\\in\\{1,2\\}$ at time $t\\in\\N$ is denoted by $X_{i,n}(t) \\in \\R^2$. We assume that $\\{X_{i,n}(t)\\}_i\\subset \\R^2$ is a marked homogeneous Poisson point process (PPP) of density $\\lambda_n$. These PPPs are independent across classes and time slots. The transmit power of a source of class $n$ is denoted by $P_n$, assumed to be constant over time slots and the same for all sources within a class.\n%\n% Let $Y_{i,n}(t)\\in\\R^2$ be the position of the destination terminal with which the $i$th source of class $n$ communicates. The distribution of $Y_{i,n}(t)$ is such that the location of the destination terminal is at a random distance $R_{i,n}(t) = ||X_{i,n}(t) - Y_{i,n}(t)||$ from the corresponding transmitter in a uniformly random direction, where $||\\cdot||$ is the euclidean norm. We assume that $R_{i,n}(t)$ is iid across time slots and users, and it follows a Rayleigh distribution of mean $\\overline{R_n}$.\n%\n% This assumption contributes to the tractability of the model and it was also used in \\cite{dester2018, lin2014spectrum, di2014stochastic}. In this context, a reasonable assumption is to consider an interference-limited network, i.e., the noise is negligible.\n% It should be noted that the destination terminals are not part of the Poisson Point Processes that model the positions of the sources.\n\n% Each source has a buffer of infinite capacity for arriving packets. The total number of packets in the $i$th source of class $n$ at time $t$ is denoted by the queue length $Q_{i,n}(t)$. If the queue of a given source is not empty, then the source tries to transmit a packet with probability $p_n$ (medium access probability) following the first-come-first-serve\n% (FCFS) discipline and the transmission takes exactly one time slot.\n% %\n% Thus, each queue length $Q_{i,n}(t)$ forms a Markov chain (for more details, see \\cite{dester2018}).\n% %\n% The packet is successfully transmitted when the signal-to-interference ratio (SIR) of the received signal is greater than a threshold $\\theta_n$ (capture effect). In this case, the receiver sends an acknowledge through an error-free channel and the packet leaves the queue. Packets arrive at the queue according to an iid Bernoulli distribution of parameter $a_n$.\n% %\n% Chronologically, within each time slot, we have the transmission of packets, then the arrival of packets, and, at the end of the time slot, the displacement of the sources occurs to form a new and independent PPP realization.\n\n% The SIR experienced by the typical user from class $n$ is given by $\\mathrm{SIR}_{i,n}= P_n h_{i,n} R_{i,n}^{-\\alpha}/I$, where $\\alpha>2$ is the path-loss exponent, the Rayleigh fading effect is represented by the fading coefficient $h_{i,n}$, which is an iid (with respect to time and users) exponentially distributed random variable of unit mean and remains constant during the time slot, and $I$ is the aggregate interference from other sources. It is known that $\\mathrm{SIR}_{i,n}$ is an iid random variable with respect to time \\cite{baccelli2010stochastic}.\n\nMore specifically, the frequency band of bandwidth $B_0$ is shared by users of both classes. For Class 1 transmissions, this frequency band is divided into $M$ partitions (sub-bands) of bandwidth $B_0/M$, and each Class 1 user transmits over one randomly selected sub-band. Therefore, the density of sources for each partition becomes $\\lambda_1/M$. On the other hand, users of Class 2 transmit over the whole available bandwidth $B_0$.\n%\nNote that front-end receiver filters of the first class destinations will capture $1/M$ of the power from transmissions of users of the second class and do not capture anything from other partitions of the first class. On the other hand, the second class destinations capture all the power coming from sources of both classes.\n\nThe transmission rate $R_T$ for the first-class users is fixed and, as in \\cite{jindal2008bandwidth}, the SIR threshold\\footnote{The transmission successfully happens if and only if the SIR is greater than the threshold.} $\\theta_1$ comes from the following equation: ${R_T = (B_0/M)\\,\\ln(1+\\theta_1)}$. Then, ${\\theta_1 = \\euler^{M\\eta_0}-1}$, where ${\\eta_0 \\triangleq R_T/B_0}$ is the spectral efficiency without partitioning the frequency band. Note that the transmission rate for the first class of users remains unchanged with bandwidth partitioning.\n\n% Both user classes use the same bandwidth and the goal is to find the optimum number of partitions $M$ for the first user class.\n% Table~\\ref{tab:symbols} summarizes the notation.\n\n% \\begin{table}[hbt]\n% \\centering\n% \\caption{Symbols and Definitions}\n% \\begin{tabular}{ll}\n% \\toprule\n% \\textbf{Symbol} & \\textbf{Definitions} \\\\\n% \\midrule\n% $\\alpha\\in(2,\\infty)$\t& Path loss exponent \\\\\n% $\\delta\\in(0,1)$\t\t& $\\triangleq 2/\\alpha$ \\\\\n% \t\t$M \\in \\N$\t\t\t\t& Number of partitions \\\\\n% $\\eta_0 \\in \\R_+$\t\t& Spectral efficiency without bandwidth partitioning \\\\\n% $\\eta \\in \\R_+$\t\t& $\\triangleq M \\eta_0 $, spectral efficiency of Class 1 \\\\\n% $p_n\\in[0,1]$ \t\t& Medium access probability \\\\\n% \t$a_n\\in[0,1]$\t\t\t& Packet arrival rate \\\\\n% $p_{s,n}\\in[0,1]$\t\t& Stationary transmission success probability \\\\\n% $\\theta_n\\in \\R_+$\t\t& SIR threshold for successful transmission \\\\\n% $D_n\\in(1,\\infty)$\t\t& Average packet transmission delay \\\\\n% $\\overline{R}_n\\in \\R_+$& Mean source-destination separation distance \\\\\n% $P_n\\in \\R_+$\t\t\t& Transmit power \\\\\n% $\\lambda_n\\in \\R_+$\t\t& User density of class $n$\\\\\n% $\\psi_n\\in \\R_+$ \t\t& $\\triangleq 4\\,\\Gamma(1+\\delta)\\,\n% \t\t\t\t\t\t \\Gamma(1-\\delta)\\,\\overline{R}_n^2\\,\n% \\theta_n^{\\delta}$~~(link quality)\\\\\n% % $||\\cdot||$\t\t\t& Euclidean norm \\\\\n% % $\\ind_A(x)$\t\t\t\t& indicator function\\\\\n% $\\Psi_n\\in \\R_+$ ~& $\\triangleq \\psi_n\\lambda_n \\left(\\frac{D_n}{D_n-1}\\right) \\left(\\frac{a_n}{1-a_n}\\right)$~~(resource utilization)\\\\\n% \\bottomrule\n% \\end{tabular}\n% \\label{tab:symbols}\n% % \\vspace*{-\\baselineskip}\n% \\end{table}\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %\n\\subsection{Stability Conditions and Stationary Analysis}\n\\label{sec:stab}\n\nIn this section, we derive conditions for stability and carry a stationary analysis of the network, which leads to expressions relating network parameters and performance metrics. These expressions are used in Section \\ref{sec:opt} to formulate an optimization problem to maximize the network performance when bandwidth partitioning is employed. \n\n% In order to derive the stability conditions, we first determine the successful transmission probability of a typical user from class $n\\in\\{1,2\\}$, given the \\textit{effective density of active sources} $\\lambda^\\mathrm{eff}_k(t)$ for each class $k\\in\\{1,2\\}$ in a time slot $t\\in\\Z_+$.\n% %\n% The effective density of active sources of a class is the PPP density of the sources with non-empty queues and allowed by the medium access control technique to transmit a packet.\n% %\n% This access control is modeled in this work by the medium access probability $p_n$. One can calculate the probability of successful transmission by deconditioning the general expression for the successful probability given in \\cite[Eq.~(9)]{haenggi2009stochastic} for the distance $R$, when $R$ follows a Rayleigh distribution. This was done in \\cite[Proposition~1]{dester2018}. Then, the successful transmission probability is given by\n\nFrom the proof of Proposition~\\ref{prop:psk} in Eq.~\\eqref{eq:P_SIR}, for two classes of users, we can write that%\n\\begin{equation} \\label{eq:P_SIR_M}\n\t\\P(\\mathrm{SIR}_{i,n}(t) > \\theta_n) \n \t= \\nu_n\\!\\left( P_1^\\delta\\,\\lambda_\\mathrm{eff}^{(1)}(t)\n \t+ P_2^\\delta\\,\\lambda_\\mathrm{eff}^{(2)}(t) \\right), \\quad t\\in\\N,\n\\end{equation}\nwhere the function ${\\nu_n: \\R_+ \\longrightarrow \\R_+}$ is defined as\n\\begin{equation}\\label{defNu}\n\\nu_n(x) \\triangleq \\left(1+\\frac{\\psi_n}{P_n^\\delta}x\\right)^{-1}, \\quad x\\in\\R_+, n\\in\\{1,2\\}.\n\\end{equation}\n\nA necessary and sufficient condition for the stability of the buffers for both user classes is given by the following proposition.\n%\n% A system is stable whenever the Markov chain describing the system admits a proper limit distribution when $t\\to\\infty$ \\cite{szpankowski1994stability}.\n%\nStability region refers to the set of arrival rates for which the system is stable.\n\n\\begin{proposition} \\label{prop:stability}\nThe system network is stable if and only if $(a_1,a_2) \\in \\mathcal{E}_1 \\cup \\mathcal{E}_2$, where\n\\begin{align*}\n\t\\mathcal{E}_1\n \t&= \\left\\lbrace (a_1,a_2)\\in [0,1]^2 \\mid\n \t\ta_1 < p_1\\,\\nu_1\\!\\left( P_1^\\delta\\tfrac{\\lambda_1}{M}p_1\n \t+\\tfrac{P_2^\\delta}{M^\\delta}\\lambda_2 p_2\\right), \n a_2 < p_2\\,\\nu_2\\!\\left( P_1^\\delta\\lambda_1\n \t\t\\tfrac{a_1}{\\widetilde{p}_{s,1}} + P_2^\\delta\\lambda_2 p_2 \\right) \\right\\rbrace,\\\\\n\t\\mathcal{E}_2\n \t&= \\left\\lbrace (a_1,a_2)\\in [0,1]^2 \\mid\n \t\ta_2 < p_2\\,\\nu_2\\!\\left( P_1^\\delta\\lambda_1 p_1\n \t+P_2^\\delta\\lambda_2p_2\\right), \n a_1 < p_1\\,\\nu_1\\!\\left(P_1^\\delta\n \t\t\\tfrac{\\lambda_1}{M}p_1+\\tfrac{P_2^\\delta}{M^\\delta}\n \\lambda_2\\tfrac{a_2}{\\widetilde{p}_{s,2}} \\right)\n\t\t\t\\right\\rbrace,\n\\end{align*}\nwhere\n\\begin{equation*}\n\t\\widetilde{p}_{s,1} = \\frac{1-\\psi_1\\tfrac{\\lambda_1}{M}a_1}\n \t{1+\\psi_1(\\tfrac{P_2}{M P_1})^\\delta\\lambda_2p_2}, \\qquad\n \\widetilde{p}_{s,2} = \\frac{1-\\psi_2\\lambda_2 a_2}\n \t{1+\\psi_2(\\tfrac{P_1}{P_2})^\\delta\\lambda_1 p_1}.\n\\end{equation*}\n\\end{proposition}\n%\n\\begin{proof}\n\tSufficient conditions for the network stability are obtained by using the concept of dominant network \\cite[Section~2.1.2]{kompella2014stable}, which corresponds to a network almost identical to the original one, differing only on the fact that users of some classes of the dominant network transmit dummy packets when their queues are empty. If the dominant network is stable, then the original network is stable as well.\n\t%\n Let us analyze a dominant network where both classes transmit dummy packets with the correspondent medium access probability $p_1, p_2$. In this case, the effective density of active sources for the $n$th traffic class is $\\lambda_\\mathrm{eff}^{(n)} = \\lambda_n\\,p_n$ for all $t$, since we are assuming that sources transmit dummy packets when their queues are empty.\n %\n Note that we have partitioned the frequency band of the first class of users into $M$ sub-bands, i.e., the destinations of packets of this user class only receive signals within their bandwidth, i.e., signals from $1/M$ of the sources of the first user class and $1/M$ of the power of signals the sources of the second class of users.\n\n% In general, a sufficient condition for stability\\footnote{The system is stable if the expected rate of incoming packets is smaller than the expected number of packets leaving the queue per time slot \\cite{loynes1962stability}.} for both classes are \n\tA sufficient condition for stability is\n \\begin{equation}\n a_n < p_n \\, P(\\mathrm{SIR}_{i,n} > \\theta_n), \\quad n\\in\\{1,2\\},\n \\end{equation}\nby Loynes' theorem (Theorem~\\ref{th:loynes}). Then, using Eq.~\\eqref{eq:P_SIR_M}, sufficient conditions for stability for the first and the second classes are \n \\begin{equation} \\label{eq:stabClass1}\n \ta_1\\theta_k)$, $k\\in\\{1,2\\}$, is the Class $k$ coverage probability in the dominant network. Therefore, the stationary effective density of active sources from the first class is $\\lambda_1^\\mathrm{eff} = \\lambda_1\\,p_1\\,\\rho_1$. From Eq.~\\eqref{eq:P_SIR_M}, we have the following fixed-point equation\n \\begin{equation*}\n \t\\widetilde{p}_{s,1}\n \t= p_1\\,\\nu_1\\!\\left[ P_1^\\delta\\tfrac{\\lambda_1}{M}\\tfrac{a_1}{\\widetilde{p}_{s,1}}\n \t+ \\left(\\tfrac{P_2}{M}\\right)^\\delta\\!\\lambda_2 p_2 \\right],\n \\end{equation*}\n which is easily solvable for $\\widetilde{p}_{s,1}$. Then, we can use this value to derive a weaker stability condition for the second user class, i.e., $a_2 1$ and \n\\begin{equation} \\label{eq:defPsi}\n\\Psi_n \\triangleq \\psi_n\\lambda_n \\left(\\frac{D_n}{D_n-1}\\right) \\left(\\frac{a_n}{1-a_n}\\right) \\ge 0, \\qquad n\\in\\{1,2\\}.\n\\end{equation} \n% with $D_n\\in(1,\\infty)$, $n\\in\\{1,2\\}$.\n\\end{theorem}\n\n\\begin{proof}\n\tIf the system is stable, then the equations can be verified through Proposition~\\ref{prop:stationary} and some manipulations. On the other hand, to prove that a set of parameters that satisfy the equations of the theorem implies stability, one can follow these steps: plug Eq.~\\eqref{eq:theor02} into the inequalities of Proposition~\\ref{prop:stability}; then use Eq.~\\eqref{eq:theor01} and the fact that $\\Psi_n > \\psi_n\\lambda_n\\frac{a_n}{1-a_n}$ (since $\\frac{D_n}{D_n-1}>1$) to show that the inequalities of Proposition~\\ref{prop:stability} are satisfied. Then, the system is stable.\n\\end{proof}\n\\begin{remark} \\label{rmk:Th.Stb.}\n\tFrom Theorem~\\ref{th:identity},\n\t$\\frac{\\Psi_1}{M} + \\Psi_2 \\le 1$.\n\tThis follows directly from equations \\eqref{eq:theor01} and \\eqref{eq:defPsi}.\n\t%\n\tFurthermore, for fixed $M$, the quantity $\\Psi_1$ decreases if $\\Psi_2$ increases. For example, let the parameters $M,\\psi_1,\\psi_2,a_1,a_2,D_1$, and $D_2$ be fixed; if we wish to increase the density of users $\\lambda_1$ of Class 1 and maintain the aforementioned parameters constant, we necessarily need to decrease the density $\\lambda_2$ of Class 2.\n\t%\n\tNote also that $\\Psi_n$ increases with $a_n$ and decreases with $D_n$, such that $\\Psi_n$ can be seen as a comprehensive measure of the performance of users of class $n \\in \\{1,2\\}$.\n\\end{remark}\n\n\\begin{remark} \\label{rmk:FreePowers}\nIn a network where we can freely choose the transmit power ratio $P_1/P_2$, at first we only need to satisfy Eq.~\\eqref{eq:theor01} when specifying system parameters ($M,\\psi_1,\\psi_2,\\lambda_1$,$\\lambda_2$) and performance parameters ($a_1,a_2,D_1$,$D_2$). After that, we use Eq.~\\eqref{eq:theor02} to specify the transmit power levels $P_1$ and $P_2$.\n\\end{remark}\n\nTheorem~\\ref{th:identity} considers $p_1=p_2=1$, which minimizes the mean delay for both classes (see Proposition~\\ref{prop:stationary}) and maximizes the stability region (see Remark~\\ref{rem:stab}). Also, if $M=1$, we recover the result in Proposition~\\ref{prop:identity_1}, i.e., $\\sum_n \\Psi_n = 1$.\n\nIt is worth noticing the simple form through which the performance parameters $a_1,a_2,D_1$, and $D_2$ of both user classes and system stability are related.\n% \\textcolor{blue}{If the system does not have constraints regarding the ratio of the transmit powers, we may only use equation \\eqref{eq:theor01} of Theorem~\\ref{th:identity}, since it is always possible to satisfy equation \\eqref{eq:theor02} in this case. Precisamos explicar melhor essa ultima sentenca}\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %\n\\subsection{Optimum Bandwidth Partition}\n\\label{sec:opt}\nLet us suppose we want to maximize the performance of Class 1 users when bandwidth partitioning is employed. More specifically, for given fixed arrival rate $a_2$ and required mean delay $D_2$ of the second user class (i.e., fixed $\\Psi_2$), we are interested in maximizing the quantity $\\frac{\\Psi_1}{\\psi_1\\lambda_1}=\\frac{D_1}{D_1-1}\\frac{a_1}{1-a_1}$, when $p_1=p_2=1$ (see Remark~\\ref{rem:stab}), by adjusting the number of partitions $M$. Note that, if we set a maximum tolerable mean delay $D_1$, this choice of optimization leads to the maximum arrival rate $a_1$ admissible. On the other hand, if we fix $a_1$, this choice of optimization leads to the minimum $D_1$ achievable. From Eq. \\eqref{eq:theor01} of Theorem~\\ref{th:identity} and recalling that $\\psi_1 = 4\\,\\Gamma(1+\\delta) \\Gamma(1-\\delta) \\overline{R}_1^2 \\theta_1^\\delta$, where $\\theta_1 = \\euler^{M\\eta_0} - 1$, one can show with simple manipulations of the ratio $\\frac{\\Psi_1}{\\psi_1\\lambda_1}$ that maximizing the quantity $\\frac{D_1}{D_1-1}\\frac{a_1}{1-a_1}$ is equivalent to the following optimization problem:\n%\n\\begin{equation} \\label{eq:opt_M}\n\tM^* = \\argmax_{M\\in\\N} \\dfrac{M\\,(\\euler^{M\\eta_0}-1)^{-\\delta}}\n \t{1+(M^{1-\\delta}-1)\\,\\Psi_2}.\n\\end{equation}\n\n% Our interest lies in finding the optimum number of bandwidth partitions $M^*$ for the first user class.\nAfter finding the optimum number of partitions $M^*$, it is necessary to adjust the transmit power levels to satisfy Theorem~\\ref{th:identity}. In this sense, we are also choosing the optimum transmit power ratio $P_1/P_2$ (see Remark~\\ref{rmk:FreePowers}).\n\nLet us now relax the constraint that $M$ must be integer to find a closed form equation for the optimization problem \\eqref{eq:opt_M}, analogously to \\cite{jindal2008bandwidth}. Note that after the bandwidth partition, the spectral efficiency is given by $\\eta = M\\eta_0$. Now, without the constraint that $M$ must be integer, we have a new goal, which is to find the optimum spectral efficiency $\\eta^*$. An equivalent relaxed problem of \\eqref{eq:opt_M} is\n\\begin{equation} \\label{eq:opt_eta}\n\t\\eta^* = \\argmin_{\\eta\\in\\R_+} \\left(\\euler^\\eta-1\\right)^\\delta\\,\n \\left( \\dfrac{1}{\\eta} + \\dfrac{\\beta}{\\eta^\\delta}\\right),\n\\end{equation}\nwhere \n\\begin{equation}\n\\beta \\triangleq \\frac{\\Psi_2}{1-\\Psi_2}\\eta_0^{-(1-\\delta)} \\ge 0. \n\\end{equation}\nThis new formulation can be obtained by taking the multiplicative inverse of the objective function in \\eqref{eq:opt_M}, followed by some manipulations. The following theorem guarantees the existence and uniqueness of the optimum spectral efficiency and shows how to find it.\n\n\\begin{theorem} \\label{th:optimum_eta}\n\tThe optimum spectral efficiency $\\eta^*$ is given by the unique positive solution of the following equation\n\\begin{equation}\\label{eq:Theor02}\n% \t\\dfrac{h(\\eta^*) - \\delta}{1 - h(\\eta^*)} =\n% \\delta\\,\\beta\\,{\\eta^*}^{1-\\delta},\n\t\\big(1-h(\\eta^*)\\big) \\big(1+\\delta\\,\\beta\\,{\\eta^*}^{1-\\delta}\\big) = 1-\\delta,\n\\end{equation}\n\twhere $h(\\eta) \\triangleq (1-\\euler^{-\\eta})/\\eta$. Furthermore, $\\eta^*$ decreases monotonically with respect to $\\beta$.\n\\end{theorem}\n\n\\begin{proof}\n\tLet us prove that the objective function \\eqref{eq:opt_eta} is strictly convex in the region of interest. Since the sum of two strictly convex functions is strictly convex, then it is enough to show that ${[(\\euler^\\eta-1)/\\eta]^\\delta}$ and ${(\\euler^\\eta-1)^\\delta/\\eta}$ are convex with respect to $\\eta > 0$. Throughout the proof we need the inequalities\n \\begin{equation} \\label{eq:aux_ineq}\n \t0 < \\euler^{-\\eta} < h(\\eta)^2 < h(\\eta) < 1,\n \\end{equation}\n which are valid when $\\eta > 0$, and are easily proved using series expansion. First, let us show that\n $\n \t{\\frac{\\partial^2}{\\partial \\eta^2} \\left(\\frac{\\euler^\\eta-1}\n {\\eta}\\right)^\\delta > 0}.\n $\n For $\\eta > 0$ and $\\delta > 0$ and after taking the derivatives with respect to $\\eta$, this inequality can be written as\n\t$\n\t{\\delta\\left( 1 - h(\\eta) \\right)^2 > \\euler^{-\\eta} - h(\\eta)^2},\n\t$\nwhich is true by \\eqref{eq:aux_ineq}. Therefore, the function ${[(\\euler^\\eta-1)/\\eta]^\\delta}$ is strictly convex, since its second derivative is positive. Now, let us show that\n\t$\n\t\t\\frac{\\partial^2}{\\partial \\eta^2} \n \\frac{(\\euler^\\eta-1)^\\delta}{\\eta} > 0.\n\t$\n\tAgain, for $\\eta > 0$ and $\\delta > 0$, this inequality can be written as\n\\begin{equation}\\label{eq:der2eq}\n\t\\delta^2 - \\left(2\\,h(\\eta) +\n \\euler^{-\\eta} \\right)\\,\\delta + 2\\,h(\\eta)^2 > 0.\n\\end{equation}\n\tFrom \\eqref{eq:aux_ineq}, the inequality \\eqref{eq:der2eq} is satisfied if $\\delta \\notin [1,2]$. Since $\\alpha >2$, then $\\delta\\in(0,1)$, and this is enough to prove strict convexity for the function ${(\\euler^\\eta-1)^\\delta/\\eta}$ as well. Therefore, the function to be optimized is strictly convex and differentiable, i.e., if there exists a point where the derivative vanishes, then this point is unique and it is the global minimum.\n It is easy to verify that this strictly convex function is arbitrarily large when $\\eta$ approaches 0 or $\\infty$, then the global minimum exists and $\\eta^*\\in(0,\\infty)$. Now, manipulating the equation \n\\begin{equation*}\n\\frac{\\partial}{\\partial\\eta} \\left[ \\left( 1 + \\beta\\,\\eta^{1-\\delta}\\right) \\frac{\\left(\\euler^\\eta-1\\right)^\\delta}{\\eta}\\right] = 0\n\\end{equation*}\nwe obtain \\eqref{eq:Theor02}, concluding the proof.\nThe proof that $\\eta^*$ decreases monotonically with respect to $\\beta$ is immediate, since $h$ is a monotonically decreasing function.\n\\end{proof}\n\nFigure~\\ref{fig:optimum_partition} shows the optimum spectral efficiency $\\eta^*$ of the first class as a function of $\\beta$ for some values of the path loss exponent $\\alpha$.\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[]{./Figures/Ch7_optimum_partition.pdf}\t\n\t\\caption{Optimum spectral efficiency $\\eta^*$ of the first class as a function of the parameter $\\beta$, which is an increasing function of the parameter $\\Psi_2$ of the second user class.}\n\t\\label{fig:optimum_partition}\n\\end{figure}\nRecall that $\\beta$ is an increasing function of $\\Psi_2$, which, in turn, is a comprehensive measure of the performance requirement of Class 2 users. As expected from the previous discussion, $\\eta^*$ decreases as the performance requirement of the second class becomes more stringent, i.e., when $\\Psi_2$ increases. This behavior of $\\eta^*$ can be explained by recalling that higher spectral efficiency requires higher SIR threshold $\\theta_1$, causing stronger interference to users of the second class.\n%\nThis increased interference reduces the maximum admissible rate $a_2$ and increases the minimum achievable delay $D_2$. Since $\\Psi_2$ is kept fixed in this optimization problem, $\\eta^*$ must be limited. In fact, from Eq. \\eqref{eq:theor01} we can see that if $\\Psi_2$ increases, then $\\Psi_1$ must be reduced in order to keep the system stable.\n%\nThe propagation environment also affects the optimum spectrum efficiency. Environments with larger path loss exponent $\\alpha$ reduce the interference among users, improving the channel quality and allowing for the use of higher spectral efficiency transmission schemes.\n\nFinally, note that without the second user class ($\\Psi_2 = 0$), we have $\\beta=0$ and the equation for $\\eta^*$ in Theorem~\\ref{th:optimum_eta} has a closed form solution, which is\n\\begin{equation}\\label{eq:opt_eff}\n\t\\eta^*(\\beta=0) = \\tfrac{\\alpha}{2} +\n W_0\\big(-\\tfrac{\\alpha}{2}\\,\\euler^{-\\alpha/2} \\big),\n\\end{equation}\nwhere $W_0$ is the principal branch of the Lambert-$W$ function, which is defined as the solution on $[-1,\\infty)$ of the equation $W_0(x)\\,\\euler^{W_0(x)} = x$, for $x \\ge -1/\\euler$.\n%\nThe above result is consistent with \\cite[Theorem~2]{jindal2008bandwidth}, where the optimization is over the traffic density achievable and the system consists of a single user class. Expression \\eqref{eq:opt_eff} was also obtained by Haenggi in \\cite{Haenggi_Lambert}.\n\nAs discussed in \\cite{jindal2008bandwidth}, the quantity $\\delta \\eta^*(0) \\to 1$ as $\\delta \\to 0$ (i.e. $\\alpha\\to\\infty$), which means that $\\eta^*(0)$ grows asymptotically as $\\alpha/2$ when the path loss exponent $\\alpha$ tends to infinity.\n%\nHowever, for a given $\\beta>0$, the same does not hold true. We can show that $\\delta \\eta^*(\\beta) \\to 0$ as $\\delta \\to 0$.\n\nIn fact, we can be more precise and show through \\eqref{eq:Theor02} that if $\\beta>0$, then $\\sqrt{\\delta} \\eta^* \\to 1/\\sqrt{\\beta}$ as $\\delta\\to 0$, which means that $\\eta^*$ grows asymptotically as $\\sqrt{\\alpha/2\\beta}$ when the path loss exponent $\\alpha$ tends to infinity.\n%\nThus, by introducing interference from another class in the system ($\\beta>0$, i.e. $\\Psi_2>0$), we change the asymptotic behavior of the optimal spectral efficiency from linear growth to squared root growth with respect to the path loss exponent $\\alpha$.\n\nThis asymptotic result can be seen in Figure~\\ref{fig:optimum_eta_delta}, where we plot the optimal spectral efficiency $\\eta^*$ adjusted by the multiplicative factor $\\sqrt{\\beta\\delta}$, which corresponds to the reciprocal of the asymptote as $\\delta$ tends to $0$ (that is why all curves converge to $1$ at $\\delta = 0$).\n%\nAnother simple asymptote of $\\eta^*$ is given by $2(1-\\delta)/(1+\\beta)$ when $\\delta$ tends to $1$ (i.e. $\\alpha\\to2$).\n\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[]{./Figures/Ch7_optimum_eta_delta.pdf}\t\n\t\\caption{Optimum spectral efficiency $\\eta^*$ of the first class adjusted by the multiplicative factor $\\sqrt{\\beta\\delta}$ as a function of the parameter $\\delta = 2/\\alpha$ for several values of $\\beta$.}\n\t\\label{fig:optimum_eta_delta}\n\\end{figure}\n\nAlso, we can obtain a generalization of \\eqref{eq:opt_eff} when the performance requirements of Class 2 is small (i.e., $\\Psi_2 \\approx 0$), which is equivalent to $\\beta \\approx 0$. Then, we have the following first order expansion for the optimum spectral efficiency\n\\begin{align} \\label{eq:eta_expansion}\n \\frac{\\eta^*\\!(\\beta)}{\\eta^*\\!(0)} = 1 - \\frac{(1-\\delta)\\eta^*\\!(0)^{1-\\delta}}{\\eta^*\\!(0)+1-\\frac{1}{\\delta}} \\beta + \\cal{O}(\\beta^2),\n\\end{align}\nas $\\beta$ tends to zero, $\\eta^*\\!(0)$ is given by Eq.~\\eqref{eq:opt_eff} and $\\cal{O}$ is the big O notation as defined in Definition~\\ref{def:landau}.\n% \\footnote{\n% We say $f(x) = \\cal{O}(g(x))$ as $x\\to a$, if $\\limsup\\limits_{x\\to a} \\frac{|f(x)|}{g(x)} < \\infty$.\n% }.\n%\nThe above expression is found using implicit differentiation on \\eqref{eq:opt_eta} and some manipulations.\n\nWe can verify through \\eqref{eq:eta_expansion} that the optimal spectral efficiency has a steeper (relative) decay for larger values of the path loss exponent $\\alpha$, when $\\alpha > 2.48$.\n%\nThis means that the presence of interference from another class has a larger relative impact on the optimal spectral efficiency when $\\alpha$ is large.\n\n% \\begin{figure}[!t]\n% \t\\centering\n% \t\\input{./Plots/optimum_partition_norm.tex}\t\n% \t\\caption{Optimum spectral efficiency $\\eta^*$ of the first class as a function of the parameter $\\beta$, which is an increasing function of the parameter $\\Psi_2$ of the second user class.}\n% \t\\label{fig:optimum_partition_norm}\n% \\end{figure}\n\n% % % % % % % % % % % % % % % % % % % % % % % % % % % % % %\n\\subsection{Applications}\n\\label{sec:num}\n\nIn this section, we illustrate the analytical results through some numerical examples. Let the first and second user classes represent the opportunist and the main users of a given bandwidth $B_0$, respectively.\n%\nThe opportunist users access the same frequency band as the main users and we perform bandwidth partitioning for the opportunistic access in order to improve the performance of the opportunist users for a given performance of the main users, which is equivalent to fixing the quantity $\\Psi_2$.\n\nFrom now on, we shall use the numerical values in Table~\\ref{tab:user_classes} for the numerical examples.\n%\n\\begin{table}[!ht]\n \\centering\n \\caption{Parameter values.}\n \\begin{tabular}{ c|l }\n \\hline\n Parameters & Values \\\\\n \\hline\\hline\n $\\lambda_1$ & $\\SI{0.005}{/m^2}$ \\\\\n $\\lambda_2$ & $\\SI{0.002}{/m^2}$ \\\\\n $\\overline{R}_1$ & $\\SI{10}{m}$ \\\\\n $\\overline{R}_2$ & $\\SI{20}{m}$ \\\\\n $\\eta_0$ & $\\SI{0.2}{bits/s/\\hertz}$ \\\\\n $p_1=p_2$ & $1$ \\\\\n \\hline\n \\end{tabular}\n \\label{tab:user_classes}\n\\end{table}\n%\nWe begin our analysis by comparing the stability regions with and without bandwidth partition. \n% Let the system parameters be $\\lambda_1 = \\SI{0.005}{/m^2}$, $\\lambda_2 = \\SI{0.002}{/m^2}$, $\\overline{R}_1 = \\SI{10}{m}$, $\\overline{R}_2 = \\SI{20}{m}$ and the (initial) spectral efficiency $\\eta_0 = 0.2\\,\\ln(2)\\,\\si{nats/s/\\hertz}$ for both classes.\n%\nOur approach to determine the stability regions is the following: we begin by using Theorem~\\ref{th:optimum_eta} to find the optimum spectral efficiency $\\eta^*$ of the first class for a given value of ${\\Psi_2\\in(0,1)}$ (see Remark~\\ref{rmk:Th.Stb.}); then, we choose the number of sub-bands $M$ as the closest integer%\n\\footnote{This is a functional thumb rule to find the optimum number of partitions $M^*$. However, the ideal approach is to verify through the objective function if the best is to round up or round down.\nIn our plots, we used the thumb rule, since there is no visual difference.}\nto $\\eta^*/\\eta_0$ (it is worth remembering that $\\Psi_2$ is fixed, so the performance requirements of Class 2 are still satisfied); finally, we use Theorem~\\ref{th:identity} along with the fact that $p_1=p_2=1$ maximizes the stability region to obtain the union of the stability regions for all the transmit power ratios $P_1/P_2$ and for all medium access probabilities $p_1$ and $p_2$. The results are shown in Fig.~\\ref{fig:optimum_stab_region} for $\\alpha = 2.5$ and $4.5$.\n%\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[]{./Figures/Ch7_optimum_stab_region.pdf}%\n\t\\caption{Stability regions of the network for different values of path loss exponent $\\alpha$. The arrival rates $a_1$ and $a_2$ are the opportunist and main users, respectively; full and dashed curves correspond to the boundaries of the stability region with and without bandwidth partition, respectively. Parameter values are shown in Table~\\ref{tab:user_classes}.}\n\t\\label{fig:optimum_stab_region}\n\\end{figure}\n%\nWe can see that the optimization of the spectral efficiency of the first user class (through bandwidth partitioning) expands the stability region. Note that when the path loss exponent $\\alpha$ is small (close to 2), the optimization does not improve significantly the system performance.\n%\nAnother observation is that if the traffic from main users (Class 2) is high, i.e., it is responsible for the majority of the interference, then it makes practically no difference to perform bandwidth partitioning in the opportunist class (Class 1). On the other hand, as the traffic from opportunist users increases, the impact of the optimization also increases\n% \\textcolor{red}{(esse melhor desempenho no canto direito inferior da região ($a_1$ alto e $a_2$ baixo) tem a ver apenas com a baixa interferência causada pelos usuários 2? Acho que poderíamos explorar esse melhor desempenho nessa região, pois é a região de interesse da ideia da partição.)}.\n\nLet us now consider another example to analyze the effects of bandwidth partitioning on the mean delay.\n% For this, we set the initial spectral efficiency for the opportunist class $\\eta_0 = 0.2\\,\\ln(2)\\,\\si{nats/s/\\hertz}$, the access probabilities $p_1=p_2=1$.\n%\nWe then determine the mean delay $D_1$ of the opportunist class as a function of its arrival rate of packets $a_1$, for $\\Psi_2 = 0.25, \\, 0.5, \\, 0.75$. Note that $\\Psi_2$ can be seen here as a measure of the performance required by the main class (see Remark~\\ref{rmk:Th.Stb.}). The procedure to find $M^*$ is analogous to the one used in the last example. The results are shown in Fig.~\\ref{fig:optimum_delay}.\n%\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[]{./Figures/Ch7_optimum_delay.pdf}%\n\t\\caption{Opportunist (first class) mean delay $D_1$ as a function of opportunist arrival rate $a_1$, for different values of the main user performance requirement $\\Psi_2$. Full and dashed curves correspond to the cases with and without bandwidth partition, respectively. Parameter values are shown in Table~\\ref{tab:user_classes}.}\n\t\\label{fig:optimum_delay}\n\\end{figure}\n%\nWhen the performance requirements of the main users are modest (small $\\Psi_2$), bandwidth partitioning allows for a more efficient use of the channel by opportunist users, i.e., a larger arrival rate $a_1$ is possible, or a smaller mean delay is achieved. On the other hand, and as expected from the previous analysis, we can also see that when the requirements of main users become more stringent (larger $\\Psi_2$), then less noticeable is the improvement from the bandwidth partitioning.\n%\n\\begin{figure}[!t]\n\t\\centering\n\t\\includegraphics[width=0.7\\columnwidth]{./Figures/Ch7_a_ratio.pdf}%\n\t\\caption{Maximum throughput relative improvement of Class 1 after band partition. Parameter values are shown in Table~\\ref{tab:user_classes}.}\n\t\\label{fig:a_ratio}\n\\end{figure}\n\n% \\begin{figure}[!t]\n% \t\\centering\n% \t\\input{./Plots/a_ratio_v2.tex}\t\n% \t\\caption{Maximum throughput improvement of Class 1 after band partition. Parameter values are shown in Table~\\ref{tab:user_classes}. \\textcolor{red}{Daria para colocar ``curvas de nível'', relativas a alguns valores de ganho?}}\n% \t\\label{fig:a_ratio_v2}\n% \\end{figure}\n\nFigure~\\ref{fig:a_ratio} shows the relative improvement of the maximum throughput after band partition of Class 1 relative to the maximum throughput before band partition, for all possible values of $\\Psi_2$ and $\\delta$. For example, in the yellow region, we have almost doubled the throughput ($\\approx 100\\%$ improvement) after implementing the optimum number of band partitions. As we have previously seen through figures \\ref{fig:optimum_stab_region} and \\ref{fig:optimum_delay}, the bandwidth partitioning strategy of the first user class improves significantly the performance of Class 1 users as the Class 2 performance requirements $\\Psi_2$ is small and the path loss exponent $\\alpha$ is large (which is equivalent to small $\\delta = 2/\\alpha$). On the other hand, when the path loss exponent $\\alpha$ is small or the performance requirements of the main (Class 2) users $\\Psi_2$ is stringent, there is almost no gain in performing bandwidth partitioning of the opportunistic class (Class 1).\n\n% % % % % % % % % % % % % % \n\\section{Summary} \\label{sec:summ_P2_03}\n\nIn this chapter, we derived necessary and sufficient conditions for stability in a network with $N$ user classes; we also provided simple closed form expressions for the packet success probability and mean delay. The advantage of using this model as a base to model other network effects is its analytic tractability.\n%\nAs an example, we were able to derive simple conditions to verify the stability of an interference-limited network with undetermined transmit powers using Corollary~\\ref{cor:stab}.\n%\nWe also solved (analytically and in closed form) two optimization problems regarding the minimization of the delays in a network and maximization of the total throughput per unit of area.\n%\nAn interesting insight from the optimization problems is that the best solution to maximize the throughput of a channel is not necessarily using solely the user class with the best link quality, i.e., a mix with other user classes may result in better use of the channel.\n% \n% All in all, this paper provides a simple way to evaluate the existing trade-offs involved in the design of wireless networks when different classes of nodes co-exist.\n\nFurther, we showed that, under certain conditions, bandwidth partitioning can significantly improve the performance of the network (under stability condition) by means of higher allowable arrival rates or lower mean delays. This is the first time the problem of bandwidth partitioning is treated analytically for two interacting user classes, for which arriving packets are queued.", "meta": {"hexsha": "8e4f7f4e6004c0c0be8aadf8c5b36d8261e0a7bb", "size": 114407, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "P2_03.tex", "max_stars_repo_name": "pliniodester/PhD_thesis", "max_stars_repo_head_hexsha": "65b00b31ebdac9dda68b1e83e937fcaed92ce354", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "P2_03.tex", "max_issues_repo_name": "pliniodester/PhD_thesis", "max_issues_repo_head_hexsha": "65b00b31ebdac9dda68b1e83e937fcaed92ce354", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "P2_03.tex", "max_forks_repo_name": "pliniodester/PhD_thesis", "max_forks_repo_head_hexsha": "65b00b31ebdac9dda68b1e83e937fcaed92ce354", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 79.7261324042, "max_line_length": 1089, "alphanum_fraction": 0.7009798351, "num_tokens": 35275, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.3443218050191247}} {"text": "\\documentclass[12pt, a4paper]{scrartcl}\n\\usepackage[utf8]{inputenc}\n\\usepackage{graphicx}\n\\usepackage{amsmath, amsthm, amssymb, textcomp}\n\\usepackage{setspace}\n\\usepackage{paralist}\n\\usepackage{graphicx}\n\\usepackage{caption}\n\\graphicspath{{WSK_im/}} %Graphic is in a folder named WSK_im in the currend directory\n\\usepackage{float}\n\\usepackage{authblk}\n\\renewcommand\\Authfont{\\fontsize{12}{14.4}\\selectfont}\n\\title{Bayesian probability theory - Lesson 8:\\\\\nContinuous probability distributions and invariance}\n\n\\author{Wolfgang von der Linden, Gerhard Dorn, Johanna Moser}\n\\date{Transcript}\n\n\\begin{document}\n\\setlength{\\parindent}{0pt}\n\\maketitle\n\\onehalfspacing\n\nWelcome to unit 8 of the course on Bayesian probability theory. My name\nis Wolfgang von der Linden and I will enable you to help Captain Bayes and\nher crew to deal with \\textbf{continuous variables} and to learn how to\\textit{assign prior probabilites}.\n\\begin{itemize}\\item We will learn how to work with \\textbf{probability densities} instead of probability mass functions.\n\\item We will learn how to use the \\textbf{transformation law} to locate the lighthouse and the principle of \\textbf{transformation invariance} will allow us to derive \\textbf{Jeffrey’s prior}\n\\item We will study the probability distribution of the first digits of mathematical tables.\n\\end{itemize}\n\n\n\\section*{Continuous variables}\nUp to this point we have actually only introduced the rules of probability\ntheory for \\textbf{discrete variables}. But you certainly realized that \\textbf{continuous\nvariables} have also crept into the problems of the last unit, for example in the star\npath problem or the fishing ground problem. We even mentioned probability\ndensities. \\\\\nThe values of the parameters in the continuous case can take on\n\\textit{any real number} in a given interval or even on the entire real axis.\nNow it’s time to extend the rules of probability theory to continuous variables.\\\\\n\nLet’s start with the following consideration: Assume we are interested in the height of trees. Typically it ranges between 0 and and a maximum height $h_{max}$. To get started we discretize the possible heights into $N$ intervals of equal length. For simplicity, we assume that all heights are equally probable. Therefore,\nthe probability that the height $l$ of a randomly selected tree lies in\nthe interval $I_i$ is the same for all intervals.\n\\[P(\\Delta I_i)=\\frac 1N = \\frac{\\Delta l}{l_{max}}\\]%\n\nTwo things are important to note here:\\\\\na) If we increase the number of intervals the probability goes to zero. So\nthe \\textit{probability for a continuous variable to have a precise value is zero} and\nhence it makes no sense to speak of probabilities or to use a probability mass\nfunction to assign probabilities to certain variables.\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 1.} Which of the following statements are true?\\\\\na) The height of trees shall be uniform in the interval [0,30]. Then, the probability for a tree to have a height of 20 meters is $\\frac{1}{30}$.\\\\\nb) The height of trees shall be uniform in the interval [0,30]. Then, the probability to have a tree of 15 to 17 meters is $\\frac{1}{15}$.\\\\\nc) The height of trees shall be uniform in the interval [0,30]. Then, the probability to have a tree of 14 to 16 meters is $\\frac{1}{15}$.\\\\\nd) Assuming a unifor mistribution with the probability given for an interval $P(\\Delta I)=A$, the probability for $P(\\Delta I/2)=A/2$.\n}}\n\\\\\n\nb) We see that the probability is \\textit{proportional to the interval size} $\\Delta l$ .\\\\\nBoth points stay valid in the case of non-equidistant intervals and also if the height\ndistribution of trees is not uniform.\\\\\n\nIn general, we find for the probability, that the height of a tree lies in a certain interval, is \\textit{proportional} to the length of this interval. The proportionality function is the \\textbf{probability density} $p(l)$.\n It is the probability to find a tree with a length in the interval $[l,l+\\Delta l]$ divided by the interval length.\n \\begin{equation*}\\boxed{p(l)=\\frac{P(\\Delta I_l)}{\\Delta l}\n}\\end{equation*}\\\\\nThe probability for the height of a tree to fall into an interval can also be obtained\nform the difference\n\\[P(\\Delta I_l)=P(\\text{height}\\leq l+\\Delta l)-P(\\text{height} \\leq l)\\]\n\nWe introduce a special function $P(\\text{height}\\leq l)=F(l)$ for these probabilities, which are called \\textbf{cumulative distribution functions}. %\nThe reason for this term becomes clear when we rewrite $p(l)$ in\nterms of the cdf:\\\\\n\\[p(l)=\\frac{F(l+\\Delta l)-F(l)}{\\Delta l}\\]\n\nFor infinitesimal interval length the pdf turns into the \\textit{derivative} of the cdf\nand in turn the cdf is the integral of the pdf.\\\\\n \\begin{equation*}\\boxed{F(l)=\\int_0^lp(l')\\text{d}l'\n }\\end{equation*}\\\\\n\nThat explains why it is called cumulative. We also see that the normalization\nimplies that the cdf approaches 1 when the length approaches the upper\nlimit of the the range of this random variable.\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 2.} Which of the following statements are true?\\\\\na) If the pdf is Gaussian, then the cdf is also Gaussian.\\\\\nb) If the pmf is given, then the pdf can be obtained by the derivative.\\\\\nc) If I want to know the probability that a tree is at max 20 meters high, I can use the cdf $F(20)$.\\\\\nd) The cdf has the following form: $F(x)\\propto x^2$. Then the pdf reads $p(x)\\propto x$.\n}}\n\\\\\n\nTo keep the notation clear, we denote pdfs with the lowercase letter $p$ and\nstick with the uppercase letter $P$ for the probability mass functions.\\\\\n\n\\section*{Calculations with continuous variables}\nThe sum and product rule are the same for continuous and discrete variables.\nBut the normalization and the marginalization rule are slightly different, as we do have integrals instead of sums. %8_1\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_1.png}\n\\end{figure}\nHere, the difference between the four marginalization formulas is that capital letters stand for discrete lower case letters for continuous variables. Similarly, the mean value of functions $f(x)$ of continuous random variables x are also converted to integrals. \\\\\n\\begin{equation*}\\boxed{\\langle f(X)\\rangle = \\sum_i f(X_i)P(X_i)\\qquad \\langle f(x)\\rangle = \\int \\text{d}x f(x)p(x)\n}\\end{equation*}\\\\\nHere, the probability mass function is replaced by the pdf.\\\\\n\n\nNow there is a specialty of continuous variables: You can introduce \\textbf{transformations} like $l\\mapsto m(l)=l^3$, which for a spherical tree would be proportional\nto the mass and if one is interested in the mass then the corresponding pdf would be $p(m)$. Then a question occurs naturally: \\textit{If we know the pdf $p(l)$ for the length what is the pdf $p(m)$ for the mass?}\\\\\n\nThe probability, that the height of a tree lies in the interval $[l,l+\\Delta l]$, is given by d$P$.\nThe mass of these trees lies in the interval $[m,m+\\Delta m]$. Since they are the same trees, we can express dP in terms of the mass and finally, due to the unique relation ``$m$ is a unique function of $l$ and $l$ is a unique function of $m$'' (i.e. bijection), we obtain the very important transformation law.\\\\\n\\begin{equation*}\\boxed{p(m)=p(l(m))\\left|\\frac{\\text{d}l(m)}{\\text{d}m}\\right|\n}\\end{equation*}\\\\\n\nFor the case of the uniform height distribution from above, $p(l)$ is constant, we\nobtain the pdf for the mass. \\[p(m)=\\frac{1}{l_{\\text{max}}}\\frac{\\text{d}(m^{1/3})}{\\text{d}m}=\\frac{1}{3l_{\\text{max}}}m^{-2/3}\\]\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 3.} Which of the following transformations are correct?\\\\\na) $p(x)=\\frac 1x$ and $y(x)=\\frac 1x$. Then, the pdf of $y$ is $p(y)=\\frac{1}{y^3}$\\\\\nb) $p(x)=\\frac 1x$ and $y(x)=\\frac 1x$. Then, the pdf of $y$ is $p(y)=\\frac{1}{y^2}$\\\\\nc) $p(x)=x^2$ and $y(x)=\\tan(x)$. Then, the pdf of $y$ is $p(y)=\\arctan(y)^2\\frac{1}{1+y^2}$\n}}\n\\\\\n\n%8_2\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_2.png}\n\\end{figure}\nSo instead of being uniform, the pdf for the mass is sharply peaked at zero.\nThat brings us to a very fundamental question: \\textit{How do we describe ignorant pdfs?}\\\\\n\nIn the discrete case the law of indifference implied equal probability but in the\ncase of continuous variables ignorance does not necessarily imply a uniform pdf.\nAs a further example, let us consider a non-negative quantity $x$ that is known\nto have a uniform pdf in the interval $[0,x_{\\text{max}}]$. But we are interested in the\npdf of a variable $y=\\exp(x)$. The desired pdf is proportional\nto $\\frac{1}{y}$. So we have found, if a variable is \\textit{linear on a logarithmic scale $\\log(y)=kx+d$}, \nthen it's pdf has a \\textit{$\\frac 1y$ dependance on the linear scale}.\\\\\n\nFinally, we want to consider the transformation law for the \\textbf{multivariate case}.\nLet $\\vec{x}$ be a set of variables with $N$ elements and $\\vec{y}$ a new set of variables with the invertible transformations: \n$\\vec{y}(\\vec{x})$ is a unique function of $\\vec{x}$ and $\\vec{x}(\\vec{y})$ in turn is a unique function of $\\vec{y}$. Then the pdfs are related by the following expression, where the last factor is the \\textbf{Jacobi-determinant}.\\\\\n\\begin{equation*}\\boxed{p(\\vec{y})=p(\\vec{x}(\\vec{y}))\\left|\\frac{\\text{d}x_i}{\\text{d}y_j}\\right|\n}\\end{equation*}\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 4.} We look at the polar transformation of a uniform pdf:\\\\\nGiven is a uniform pdf of $\\vec{x}=(x_1,x_2), \\vec{y}=(r,\\phi)$ with $x_1=r\\cos(\\phi),x_2=r\\sin(\\phi)$. Which of the following statements are true?\\\\\na) The pdf of $\\vec{y}$ is given by $p(r,\\phi)\\propto r$\\\\\nb) If $p(\\vec{x})=p(x_1,x_2)\\propto x_1+x_2$, then $p(\\vec{y})=p(r,\\phi)=r^2(\\cos(\\phi)+\\sin(\\phi))$\\\\\nc) The determinant of the Jacobi Matrix is given by $r^2(\\sin^2(\\phi)+\\cos^2(\\phi)=r^2$\\\\\nd) The Jacobi matrix is given by $\\left( {\\begin{array}{cc}\n \\cos(\\phi) & -r\\sin(\\phi)\\\\\n \\sin(\\phi) & r\\cos(\\phi) \\\\\n \\end{array} } \\right)$\n}}\n\\\\\n\n\\section*{Bertrand paradox - ignorant priors}\nThe crew of Captain Bayes was pondering about the so-called \\textbf{Bertrand paradox} which is defined as follows:\\\\\n\\textit{Given a circle of radius R and a random set of lines that cross the circle,\nwhat is the probability that a random line has a distance from the center that is less than half the radius?}\\\\%8_3\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_3.png}\n\\end{figure}\n\nThe line cuts the circle at two points $P_1$ and $P_2$.\nWe can define these endpoints by the angles $\\varphi_1$ and $\\varphi_2$ formed by the \\textit{radial lines to the endpoints with the x-axis}. These are two independent variables that uniquely define the line.\\\\%8_4\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{8_4.png}\n\\end{figure}\n\nThe line is also uniquely defined by the midpoint $P$ of the two endpoints $P_1$ and $P_2$. \nThis involves again two\nindependent variables, namely the $x$ and $y$ coordinate of the point. Finally, for solving the problem, it is enough to know the distance $r$ of the line from the center of the circle. \\\\%8_5\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.4\\textwidth]{8_5.png}\n\\end{figure}\n\nNow think of how you would simulate the problem on the computer.\nBernoulli came up with the idea to choose the points $P_1$ and $P_2$ at random\non the circle. That is equivalent to choosing the two angles $\\varphi_1$ and $\\varphi_2$\nat random. That in turn implies that the \\textit{pdf of the intermediate angle is\nuniform.}\\\\\n\nIn this variable, the distance of the line to the center is given by $R\\cos(\\varphi/2)$\n and the\ncondition, that the line has a distance less than $R$ half from the center, is given by \\[R\\cos(\\varphi/2)\\leq R/2\\]\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 5.} What is the range of $\\varphi\\in(0,\\pi]$ fulfilling the inequality $R\\cos(\\varphi/2)\\leq R/2$?\\\\\na) $\\varphi \\in [\\pi/4,\\pi/3]$\\\\\nb) $\\varphi \\in [0,\\pi/2]$\\\\\nc) $\\varphi \\in [0,2\\pi/3]$\\\\\nd) $\\varphi \\in [2\\pi/3,\\pi]$\\\\\n}}\n\\\\\n\nwhich implies $\\varphi < \\frac{\\pi}{6}$. Hence the desired probability is $\\frac 13$.\\\\%8_6\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_6.png}\n\\end{figure}\n\nAnother way of generating the line at random was proposed in the adventure\nby Pascal. She generates the midpoint at random, which means the points\nare \\textit{uniformly distributed within the circle}. Then the desired probability is\nthe probability that the point lands in the circle of radius $\\frac R2$. This is a nice\nillustration of the transformation law for two variables.\\\\\nInstead of the Cartesian coordinates $x$ and $y$ we use spherical coordinates $r$ and $\\varphi$. The transformation law yields for the correctly normalized pdf the following expression. %8_7\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_7.png}\n\\end{figure}\nThe desired result requires that $r<\\frac R2$ and the corresponding probability gives $\\frac 14$.\n\nFinally, Laplace suggested to generate the radial distance uniformly between 0 and $R$ resulting in a constant pdf. In that case the probability is $0.5$. \nThe question is now, \\textit{which algorithm describes the physical experiment?}\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 6.} Guess who is right! What is the probability for a free day (= that the line does not cut the inner circle)? Do you know why?\\\\\na) Laplace: It is a 50\\% chance to have a free day.\\\\\nb) Pascal: It is a 75\\% chance to have a free day.\\\\\nc) Bernoulli: It is a 67\\% chance to have a free day.\n}}\n\\\\\n\nThis question is already on the mind of scientists for generations and still is.\nWe can phrase this problem more generally: If we know nothing about the\npdf apart from the definition of the setup - so there are no experimental\ndata or additional theoretical constraints - what is the correct way to find an\nun-informative prior? So how do we assign probabilities in the continuous way?\nSuch an object is also called an \\textbf{ignorant prior}.\\\\\n\nFor ignorant priors Ed Jaynes came up with the following idea. \\textit{If there is a\ncontinuous and invertible map $\\vec{T}(\\vec{x})$ from one set of variables $\\vec{x}$ to a\nnew set $\\vec{y}$, then the corresponding pdfs should be the same, i.e $p(\\vec{y})=p(\\vec{x})$.} In the case of the die it\nmeans that putting the pips on different sites of the cube should not modify the probability mass function.\nIn other words, \\textit{the probability mass function should be invariant against\npermuting the indices}, which results in the well-known uniform distribution.\nSince the mapping in the continuous case is assumed to be continuous as\nwell, we can introduce a parameter$\\epsilon$ such that for $\\epsilon\\rightarrow 0$ the mapping is the\nidentity. \\[\\lim_{\\epsilon\\rightarrow 0}\\vec{T}_{\\epsilon}(\\vec{x})=\\vec{x}\\]\n\nUsing the transformation law from above and the invariance properties of\nthe pdf, Jaynes came up with the following equation for the pdf:%8_8\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_8.png}\n\\end{figure}\n\nLet’s illustrate this approach for a \\textbf{scale variable} $y=\\alpha x$. A scale variable $x$ is a\n\\textit{non-negative quantity for which the pdf should be scale invariant}. In this case, the infinitesimal transformation has the following form:\\[T_{\\epsilon}=(1+\\epsilon)x\\]\nJaynes’ equation can be solved by elementary tools\nand the result is \n\\begin{equation*}\\boxed{p(x)=\\frac 1x\n}\\end{equation*}\\\\\nNote that this is a so-called \\textbf{improper pdf}, as it \\textit{cannot be normalized}. In the\nframe of Bayesian probability theory, such a prior is not forbidden as long as\nthe resulting posterior is normalizable.\nThis scale-invariant pdf is often called \\textbf{Jeffreys’ prior}, although it is only a\nspecial application of Jeffreys’ idea.\\\\\n\nThe uninformative prior pdf of parameters a depends of course on the\nmeaning of the parameters. In many cases these are uniquely determined\nby the likelihood function $p(d|a)$. In that case, a pdf that is invariant against reparameterization is given by $p$ of a is given by the determinant of g (see picture).%8_9\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_9.png}\n\\end{figure}\n\nA typical example of the scale variable is the standard deviation $\\sigma$ that enters\na Gaussian likelihood. This is a nice and informative exercise for the application of Jeffrey’s prior. The result is the same as that obtained by Jaynes’\napproach $p(\\sigma)=\\frac{1}{\\sigma}$\\\\\n\nEd Jaynes applied the principle of transformation invariance to the Bertrand paradox.\nHe used the following transformations under which the problem is invariant:%\n\\begin{enumerate}\n\t\\item Translation of the center of the circle\n\t\\item Rescaling of the radius\n\t\\item Rotation of the circle about the center\n\\end{enumerate} \nThe solution of his equation yields the same result that Laplace proposed :\\\\\n\\[p(r)=\\text{const}=\\frac 1R\\]\n\nThere is another way to approach the Bertrand paradox. Since the lines\nare drawn randomly and without knowing the position of the circle, we can\nequally well draw the line first and then randomly place the circle on the\nplane. The circle is uniquely defined by its center and therefore this is equivalent to randomly choosing the coordinates $x$ and $y$ of the center. If we define\nthe direction of the line as the $x$-axis, then the $x$-coordinate of the center is\nirrelevant. Consequently, the distance of the line from the center of the circle is given by the absolute value of $y$. As the coordinates of the center are\nchosen at random, the distance is a uniform random variable corroborating\nJaynes' result.\\\\%\n\n\\section*{Maximum Entropy}\nWe have encountered the \\textbf{maximum entropy principle} already in the discrete case. It\nwas introduced to obtain the probability mass function in the case of testable\ninformation given for instance in the form of mean values. Maximum entropy can also\nbe invoked for pdfs given testable information for instance in the following form:\\[\\int \\text{d}x f_i(x)p(x)=\\mu_i\\]\n\nThe only complication in the continuous case is that the entropy is not entirely what you would expect.\\\\\n\\begin{equation*}\\boxed{S(p)=-\\int dx p(x)ln\\left(\\frac{p(x)}{p_0(x)}\\right)\n}\\end{equation*}\\\\\n\nThe function $p_0(x)$ is sometimes called the \\textbf{default model}, because the MaxEnt\nsolution without any constraints apart from normalization yields $p(x)=p_0(x)$.\\\\\nThe deeper reason why we have a default model in the continuous case and\nnot in the discrete case is that the transition by discretization from the\ncontinuous to the discrete case is not unique.\nThe default model plays the role of an ignorant prior.\nOtherwise, the implementation of testable information in the maximum entropy procedure is the same as in the discrete case.\nOn passing we want to discuss a simple application from statistical physics.\nWe want to infer the pdf $p(x)$ of particles at the height $x$ above the earth.\nThe mean potential energy of these particles is proportional to the mean\nheight.\\[E=mg\\int x p(x)\\text{d}x \\propto \\langle x\\rangle\\]\n\nIn a canonical ensemble the mean energy is given by $k_BT$\nSo the constraint has the following form: \\[ \\langle x \\rangle = \\int xp(x)\\text{d}x = \\frac{k_BT}{mg}\\]\n\nMaximizing the entropy yields the \\textbf{Barometric formula} $P\\propto \\exp(-\\alpha x)$.\nNote that $x$ is \\textit{not\na scale variable here as there is an upper limit for it}. Therefore, a uniform prior\nis adequate, which is corroborated by the experimental observation for free\nparticles without potential energy.\\\\\n\nIn closing, it should be mentioned that this form of entropy is also called\n \\textbf{cross-entropy} or in other context the \\textbf{Kullback-Leibler distance}.\n \\[d_{KL}(p,q)=\\int_{-\\infty}^{+\\infty}p(x)log\\left(\\frac{p(x)}{q(x)}\\right)\\text{d}x\\]\n It can be\nconsidered as distance between distributions $p(x)$ and $p_0(x)$, and it plays\nan important role in machine learning.\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 7.} Which of the following statements is true?\\\\\na) We need a default model $p_0$ in the entropy $S$ in the continuous case, since the mapping from the continuous to the discrete case is not unique.\\\\\nb) The height variable $x$ when deriving the barometric formula is uniform since the earth has a finite atmosphere.\\\\\nc) We need a default model $p_0$ of the entropy $S$ in the continuous case, since variables are scale invariant.\n}}\n\\\\\n\n\n\\section*{The lighthouse problem}\nThe lighthouse problem that Lyra addressed in the adventure is another\nexample of the \\textbf{transformation of coordinates}. Let $x_0$ and $y_0$ be the coordinates\nof the lighthouse and let the coast line define the $x$-axis with the origin of\nthe coordinate system being on the coast line.\nThe lighthouse emits light signals that are uniformly distributed in the rotation angle $\\varphi$ about the lighthouse axis. We are interested in the pdf for a\nlight signal being detected at point $\\boldsymbol{x}$ on the coast line.\nThe following relation exists between the coastline position x and the\nangle $\\varphi$, and the transformation law yields the desired pdf for $\\boldsymbol{x}$ from the uniform pdf for\nthe angle.\\\\ %8_10\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_10.png}\n\\end{figure}\n\n$p(\\boldsymbol{x}|x_0,y_0)$ is the probability to find a light signal at position $\\boldsymbol{x}$ given the parameters\n$x_0,y_0$ and therefore it represents the likelihood of the problem.\nWe assume for simplicity that the data points $\\boldsymbol{x}_i$ are uncorrelated. Bayes\ntheorem then yields for the desired pdf for the position of the lighthouse the following form.\n\\[p(x_0,y_0|\\boldsymbol{x})=\\frac 1Z \\prod_i p(\\boldsymbol{x}_i|x_0,y_0)p(x_0,y_0)\\]\nThe prior probability depends on our knowledge about lighthouses.\n\\textit{In the Pluto notebook you can gain experience with this problem and see that\nit is indeed possible to locate the lighthouse on the basis of the distribution\nof the light flashes on the coastline.}\n\n\\section*{Important distributions}\nThere are many important distributions for continuous variables. Here we\nonly want to mention the most popular ones.\nWe have repeatedly used Bayes’ theorem to compute the posterior pdf for a\nparameter a given some data d.\nLet’s first ignore the prior $p(a)$, then the posterior is proportional to the likelihood $p(a|\\boldsymbol{d})\\propto p(\\boldsymbol{d}|a)$.\nIn case of a counting experiment with a Poisson likelihood the parameter\n$a$ corresponds to the mean $\\mu$ and the data are the counts $\\boldsymbol{N}$. Then the\nposterior is of the following form:\n\\begin{equation*}\\boxed{p(\\mu|\\boldsymbol{N})\\propto e^{-\\mu}\\mu^\\boldsymbol{N}\n}\\end{equation*}\\\\\nThis function is a \\textbf{Gamma-distribution}, that is defined for the range $x\\in [0,\\infty)$\nand with the correct normalization it reads.%8_11\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_11.png}\n\\end{figure}\nIt has a \\textit{power-law dependence for small x} and \\textit{decays exponentially for large\nx}. Mean and variance are given by the following expressions.\n\\begin{equation*}\\boxed{\\langle x\\rangle = \\frac{\\alpha}{\\beta} \\qquad \\text{var}(x)=\\frac{\\alpha}{\\beta^2}\n}\\end{equation*}\\\\\nThey can easily be derived from the Gamma function which is part of\nthe normalization.\n\\[\\Gamma(\\alpha):=\\int_0^{\\infty}t^{\\alpha-1}e^{-t}\\text{d}t\\]\nThe Gamma function has a very convenient recursion relation\nand for an integer argument it is related to the factorial: \\[\\Gamma(N)=(N-1)! \\quad N\\in\\mathbb{N}\\]\n\nMoreover, we want to mention that $\\Gamma(\\frac 12)=\\sqrt{\\pi}$.\nSpecial cases of the Gamma distribution are the $\\mathbf{\\chi^2}=\\Gamma(\\frac n2, \\frac 12)$ \\textbf{distribution} \nand the \\textbf{exponential distribution} $\\Gamma(1, \\beta)$.\\\\%\n\nSo far we have only discussed the likelihood which lead to the Gamma distribution. If we include an arbitrary prior, then the posterior will be different.\nFor that reason it is common practice to also use a Gamma distribution for\nthe prior. This is not a significant restriction, as prior knowledge is typically\nvague and the Gamma distribution is very flexible and can cover a wide range\nof prior knowledge. Such a so-called \\textbf{conjugate prior} has the advantage that\nthe posterior is still a Gamma distribution and mean and covariance can be\ncomputed analytically.\\\\\n\n\nNow we turn to a Bernoulli experiment with sample size $N$. Here, the parameter a\ncorresponds to the intrinsic probability $q$ and the data is the number $K$ of\noccurrences of the outcome of interest, so $p(a|d)=p(q|K,N)$. Then the posterior is of the following form:\n\\begin{equation*}\\boxed{p(q|K,N))\\propto q^K(1-q)^{N-K}\n}\\end{equation*}\\\\\nThis function is a \\textbf{Beta-distribution}, which has the range$x\\in [0,1]$ and the\nnormalized distribution reads.%8_12\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_12.png}\n\\end{figure}\nMean and variance\ncan be determined by means of the beta function.\nIt also enters the normalization of the beta distribution.\nIn case of a Bernoulli experiment the conjugate prior is therefore a Beta\ndistribution.\\\\\n\nFinally, we want to mention the \\textbf{Gaussian distribution}, which we have already\nencountered several times. The Gaussian is of central importance for the\ndescription of experimental noise and the range of the random variable in\nthe one-dimensional case is the entire real axis. The normalized Gaussian has a mean and a variance which is given by the following expressions.\\\\%8_13\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_13.png}\n\\end{figure}\n\\textit{See the Pluto notebook about popular distributions and study the possible\nshapes that emerge by varying the parameters.}\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 8.} Match the distributions: \\{Gauss, Gamma, Beta\\}\\\\\nThe domain of the .............. distribution is $x\\in[0,1]$, as it describes the probability of a probability.\\\\\nThe ......... distribution is the conjugate prior of a Poisson Likelihood.\\\\\nThe .......... distribution is parametrized by mean and variance.\n}}\n\\\\\n\n\\section*{Benford's law}\nLyra in the adventure also mentioned a very interesting discovery that \\textit{the\nleading digits of large numbers form a pattern} no matter what is considered: %8_14\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_14.png}\n\\end{figure}\nFor the amount of collected shells, bottles, coloured stones per day, the leading numbers all behave the same and\nthey are \\textbf{not} uniformly distributed. This is in contradiction to the principle\nof indifference.\\\\\nThis phenomenon has to do with \\textbf{scale invariances}. We have seen before that\nthe pdf of scale invariant quantities $x$ is given by Jeffrey’s prior $p(x)\\propto \\frac 12$.\nPut differently the pdf for log($x$) is uniform, so for $y=\\log_{10}(x)\\rightarrow p(y)= \\text{const}$.\nIf you recall the intervals on logarithmic axes you may have noticed that the\ninterval sizes \\textit{decrease} from 1 to 9. %\nLet’s consider numbers x between 1 and 9.99999$\\ldots$\n\nThe leading digit corresponds to the interval $x \\in [1,2)$.\nThen the probability is $P_1\\propto \\int_1^2\\frac{\\text{d}x}{x}\\ln(2)-\\ln(1)$.\nIn general, the probability that the leading digit has the value $I$ is proportional to $\\log(I+1)-\\log(I)$.\nWith the correct normalization for all possible digits 1 up to 9 we have the probability that the leading digit is $I$ given by $\\log_{10}(I+1)-\\log_{10}(I)$\nThis distribution is depicted in the figure.\\\\%8_15\n \\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.75\\textwidth]{8_15.png}\n\\end{figure}\n\nThe same result is obtained if the numbers are in the interval 10 to 99.9999\nor a 100 to 999.999 and so on.\nActually, the same law holds true also for \\textit{counting experiments when the\nnumbers vary over many orders of magnitude}. Also in this case the probability mass function is \\textit{uniform on a logarithmic scale} or \\textbf{log-uniform} as it is\nsometimes called.\\\\\n\n\\fbox{\\parbox{\\linewidth}{\\textbf{Question 9.} Which of the following variables could be scale invariant?\\\\\na) Rain drops per day/km$^2$\\\\\nb) The results of repeated dice throws, using a 100 faced die\\\\\nc) The number of sand grains in your shoes\n}}\n\\\\\n\n\\textit{You will find a Pluto notebook to perform random experiments with log-uniform distributions.}\\\\\n\n\nThis concludes lesson 8. We have learned about continuous variables and\nhow to assign prior probabilities using the principle of transformation invariance or the maximum entropy principle. We have discussed Bertrand's paradox and know now why Benford’s law\ndescribes the leading digit of counting lists and measurement tables.\nCheck out the interactive Pluto notebooks to solve the lighthouse problem,\nto do your own Benford computer experiments, and to experiment with\nvarious popular distributions. Feel free to ask questions in the forum and feel encouraged to test your knowledge in the quiz.\n\n\\vspace{2cm}\n\\begin{minipage}[t]{1\\textwidth}\n\t\\raggedleft\n\t\\centering\n\t\\includegraphics[width = 0.20\\textwidth]{CC-BY_icon}\n\t\\vspace{0.2cm}\n\t\n\t\\centering\n\t{\\large ITPCP, TU Graz} \\\\\n\thttps://creativecommons.org/licenses/by/4.0/legalcode\n\\end{minipage}\n\\end{document}", "meta": {"hexsha": "03137b9a2a26341bc768e63d2f923fac0eac39c6", "size": 29202, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "transcripts/tex-files/Lection_8.tex", "max_stars_repo_name": "moserjo/bayes-imoox", "max_stars_repo_head_hexsha": "144bde64410b7e740efa28c82b5e7dfe66885141", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 9, "max_stars_repo_stars_event_min_datetime": "2021-03-19T04:40:18.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-30T14:12:16.000Z", "max_issues_repo_path": "transcripts/tex-files/Lection_8.tex", "max_issues_repo_name": "moserjo/bayes-imoox", "max_issues_repo_head_hexsha": "144bde64410b7e740efa28c82b5e7dfe66885141", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2021-02-20T10:51:38.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-26T13:57:05.000Z", "max_forks_repo_path": "transcripts/tex-files/Lection_8.tex", "max_forks_repo_name": "moserjo/bayes-imoox", "max_forks_repo_head_hexsha": "144bde64410b7e740efa28c82b5e7dfe66885141", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2021-02-19T18:14:35.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-27T07:07:35.000Z", "avg_line_length": 57.4842519685, "max_line_length": 322, "alphanum_fraction": 0.7527224163, "num_tokens": 8051, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6297746074044134, "lm_q1q2_score": 0.3443218050191247}} {"text": "\\documentclass[10pt,fleqn]{article}\n\\usepackage{amsmath}\n\\usepackage{cancel}\n\\usepackage[letterpaper, landscape, margin=1in]{geometry}\n\n\\setlength{\\parindent}{0pt}\n\n\n\\begin{document}\n\n\\section{Isotropic visco-acoustic constant density second order self-adjoint system}\nJohn Washbourne, Ken Bube\\\\\nSeptember 20, 2013\n\n\n\\section{Introduction}\nThis note shows the derivation of \\textit{time update equations} and the linearization \nfor the isotropic constant density self-adjoint system. We implement attenuation with a \nmonochromatic approximation to Maxwell bodies, and use this attenuation model to \nimplement zero outgoing absorbing boundary conditions on the exterior of the modeling \ndomain.\n\\vspace{10pt}\n\nThe time update equations are used to advance solutions in time, expressing the pressure\nwavefield at time $p_{(t + \\Delta)}$ as a function of $p_{(t - \\Delta)}$ and $p_{(t)}$.\n\n\\subsection{Symbols}\n\\begin{center}\n\t\\begin{tabular}{ll} \\\\[-10pt]\n\t\t$\\partial_t$ & $\\displaystyle \\frac{\\partial }{\\partial t}$ \\\\[10pt]\n\t\t$\\nabla p $ & Laplacian: $ \\left(\n\t\t\t\\displaystyle \\frac{\\partial^2 }{\\partial x^2} +\n\t\t\t\\displaystyle \\frac{\\partial^2 }{\\partial y^2} +\n\t\t\t\\displaystyle \\frac{\\partial^2 }{\\partial z^2} \\right) p $ \\\\[15pt]\n\t\t$\\Delta$ & Temporal sampling rate \\\\[10pt]\n\t\t$\\omega$ & reference frequency for attenuation \\\\[10pt]\n\t\t$Q$ & attenuation at frequency $\\omega$ \\\\[10pt]\n\t\t$P$ & Pressure wavefields \\\\[10pt]\n\t\t$S(x,y,z,t)$ & Pressure source term \\\\[10pt]\n\t\t$b$ & buoyancy = $\\displaystyle 1/\\rho $ (reciprocal density) \\\\[10pt]\n\t\t$ \\{\\ V_p\\ \\} $ & Material parameters \\\\[10pt]\n\t\\end{tabular}\n\\end{center}\n\n\\newpage\n\\subsection{Modeling system}\nEquation \\ref{eq:system} shows the modeling system with absorbing boundaries implemented \nusing amplitude only (dissipation only, no dispersion) Q. \\vspace{10pt}\n\nWe apply the time derivative in the term $\\displaystyle \\frac{\\omega}{Q} \\partial_t p$ \nusing a backward one-sided numerical difference. We tested both forward one-sided and \ncentered difference alternatives and found them to be less stable.\n\n\\begin{equation}\n\t\\frac{1}{v^2} \\left( \\partial_t^2 p + \\frac{\\omega}{Q} \\partial_t p \\right) = \\nabla p + S\n\t\\label{eq:system}\n\\end{equation}\n\n\\section{Time update equations}\n\n\\subsection{Time update numerical difference formulas, first and second order}\n\\begin{equation}\n\t\\partial_t p = \\frac{1}{\\Delta} \\left[ p_{(t)} - p_{(t - \\Delta)} \\right]\n\t\\label{eq:diff1}\n\\end{equation}\n\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\partial_t^2 p & = \\frac{1}{\\Delta^2} \\left[ p_{(t+\\Delta)} - 2 p_{(t)} + p_{(t - \\Delta)} \\right] \\\\[10pt]\n\t\tp_{(t+\\Delta)} & = \\Delta^2 \\partial_t^2 p + 2 p_{(t)} - p_{(t - \\Delta)}\n\t\\end{aligned}\n\t\\label{eq:diff2}\n\\end{equation}\n\n\\subsection{Rearrange equation \\ref{eq:system} for $\\partial_t^2 p$}\n\\begin{equation}\n\t\\partial_t^2 p = v^2 \\left( \\nabla p + S \\right) - \\frac{\\omega}{Q} \\partial_t p\n\\end{equation}\n\n\\subsection{Apply equations \\ref{eq:diff1} and \\ref{eq:diff2}, and rearrange}\n\\begin{equation}\n\tp_{(t+\\Delta)} = \\Delta^2 v^2 \\left( \\nabla p + S \\right) - \\Delta \\frac{\\omega}{Q} \\left[ p_{(t)} - p_{(t - \\Delta)} \\right]\n\t+ 2 p_{(t)} - p_{(t - \\Delta)}\n\\end{equation}\n\n\\newpage\n\\section{Linearization and Born modeling equation}\n\n\\subsection{Nonlinear modeling equation}\n\\begin{equation}\n\t\\frac{1}{v^2} \\left( \\partial_t^2 p + \\frac{\\omega}{Q} \\partial_t p \\right) = \\nabla p + S\n\\end{equation}\n\n\\subsection{Taylor expand $\\displaystyle \\frac{1}{v^2} \\rightarrow \\left( \\frac{1}{v_0^2} - \\frac{2}{v_0^3} \\delta v \\right) $\n\tand replace $\\displaystyle p \\rightarrow \\displaystyle (p_0 + \\delta p)$}\n\\begin{equation}\n\t\\left( \\frac{1}{v_0^2} - \\frac{2}{v_0^3} \\delta v \\right) \\left( \\partial_t^2 p_0 + \\partial_t^2 \\delta p +\n\t\\frac{\\omega}{Q} \\partial_t p_0 + \\frac{\\omega}{Q} \\partial_t \\delta p \\right) =\n\t\\nabla p_0 + \\nabla \\delta p + S\n\\end{equation}\n\n\\subsection{Expand}\n\\begin{equation}\n\t\\frac{1}{v_0^2} \\partial_t^2 p_0\n\t+ \\frac{1}{v_0^2} \\partial_t^2 \\delta p\n\t+ \\frac{1}{v_0^2} \\frac{\\omega}{Q} \\partial_t p_0\n\t+ \\frac{1}{v_0^2} \\frac{\\omega}{Q} \\partial_t \\delta p\n\t- \\frac{2}{v_0^3} \\delta v \\partial_t^2 p_0\n\t- \\frac{2}{v_0^3} \\delta v \\partial_t^2 \\delta p\n\t- \\frac{2}{v_0^3} \\delta v \\frac{\\omega}{Q} \\partial_t p_0\n\t- \\frac{2}{v_0^3} \\delta v \\frac{\\omega}{Q} \\partial_t \\delta p\n\t= \\nabla p_0 + \\nabla \\delta p + S\n\\end{equation}\n\n\\subsection{Cancel reference terms}\n\\begin{equation}\n\t\\cancel{ \\frac{1}{v_0^2} \\partial_t^2 p_0 }\n\t+ \\frac{1}{v_0^2} \\partial_t^2 \\delta p\n\t+ \\cancel{ \\frac{1}{v_0^2} \\frac{\\omega}{Q} \\partial_t p_0 }\n\t+ \\frac{1}{v_0^2} \\frac{\\omega}{Q} \\partial_t \\delta p\n\t- \\frac{2}{v_0^3} \\delta v \\partial_t^2 p_0\n\t- \\frac{2}{v_0^3} \\delta v \\partial_t^2 \\delta p\n\t- \\frac{2}{v_0^3} \\delta v \\frac{\\omega}{Q} \\partial_t p_0\n\t- \\frac{2}{v_0^3} \\delta v \\frac{\\omega}{Q} \\partial_t \\delta p\n\t= \\cancel{ \\nabla p_0 } + \\nabla \\delta p + \\cancel{ S }\n\\end{equation}\n\n\\subsection{Zero terms second order in perturbations $\\delta p,\\delta v$ }\n\\begin{equation}\n\t\\frac{1}{v_0^2} \\partial_t^2 \\delta p\n\t+ \\frac{1}{v_0^2} \\frac{\\omega}{Q} \\partial_t \\delta p\n\t- \\frac{2}{v_0^3} \\delta v \\partial_t^2 p_0\n\t- \\cancel{ \\frac{2}{v_0^3} \\delta v \\partial_t^2 \\delta p }\n\t- \\frac{2}{v_0^3} \\delta v \\frac{\\omega}{Q} \\partial_t p_0\n\t- \\cancel{ \\frac{2}{v_0^3} \\delta v \\frac{\\omega}{Q} \\partial_t \\delta p }\n\t= \\nabla \\delta p\n\\end{equation}\n\n\\subsection{Rearrange for the Born modeling equation}\n\\begin{equation}\n\t\\begin{aligned}\n\t\t\\frac{1}{v_0^2} \\left( \\partial_t^2 \\delta p + \\frac{\\omega}{Q} \\partial_t \\delta p \\right)\n\t\t & = \\nabla \\delta p + \\frac{2}{v_0^3} \\delta v \\left( \\partial_t^2 p_0 + \\frac{\\omega}{Q} \\partial_t p_0 \\right) \\\\[10pt]\n\t\t & = \\nabla \\delta p + \\frac{2}{v_0^3} \\delta v \\left( v_0^2 \\nabla p_0 + v_0^2 S \\right) \\\\[10pt]\n\t\\end{aligned}\n\\end{equation}\n\nNote: it is procedurally simplest to serialize the quantity \n$\\displaystyle \\left( v_0^2 \\nabla p_0 + v_0^2 S \\right)$ when we perform nonlinear \nforward modeling, and use that field as the Born source for the linearized forward and \nadjoint operators.\n\\end{document}\n", "meta": {"hexsha": "4f69247b822816e25ad90459fa3023b566b32191", "size": 6436, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/latex_notes/IsotropicAcousticQ.tex", "max_stars_repo_name": "ChevronETC/WaveFD", "max_stars_repo_head_hexsha": "555cf24fc70835e789a8126a1d9866e25de4a8d3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2020-10-28T16:59:56.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-25T21:55:17.000Z", "max_issues_repo_path": "docs/latex_notes/IsotropicAcousticQ.tex", "max_issues_repo_name": "ChevronETC/WaveFD", "max_issues_repo_head_hexsha": "555cf24fc70835e789a8126a1d9866e25de4a8d3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 20, "max_issues_repo_issues_event_min_datetime": "2020-10-06T03:03:51.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-09T23:29:11.000Z", "max_forks_repo_path": "docs/latex_notes/IsotropicAcousticQ.tex", "max_forks_repo_name": "ChevronETC/WaveFD", "max_forks_repo_head_hexsha": "555cf24fc70835e789a8126a1d9866e25de4a8d3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-10-26T15:19:51.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-26T15:19:51.000Z", "avg_line_length": 42.0653594771, "max_line_length": 126, "alphanum_fraction": 0.6390615289, "num_tokens": 2298, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381372136563, "lm_q2_score": 0.6297746074044135, "lm_q1q2_score": 0.34432179571675076}} {"text": "%!TEX root = paper.tex\n%\n% Riemann\n%\n\n\\subsection{Riemann: A Community-Driven Collection of Approximate Riemann\nSolvers}\\label{sec:riemann}\n\nThe methods implemented in \\clawpack, and all modern Godunov-type\nmethods for hyperbolic PDEs, are based on the solution of Riemann\nproblems as discussed in \\cref{sec:hyp}. Whereas most existing codes\nfor hyperbolic PDEs use Riemann solvers to compute fluxes, \\clawpack\nRiemann solvers instead compute the waves (or discontinuities) that\nmake up the Riemann solution. In the unsplit algorithm, \\clawpack\nalso makes use of {\\em transverse} Riemann solvers, responsible for\ncomputing transport between cells that are only corner (in 2d) or edge\n(in 3d) adjacent.\n\nFor nonlinear systems, the exact solution of the Riemann problem is\ncomputationally costly and may involve both discontinuities (shocks and contact\nwaves) and rarefactions. It is almost always preferable to employ inexact\nRiemann solvers that approximate the solution using discontinuities only, with\nan appropriate entropy condition. The solvers available in \\clawpack are\nall approximate solvers, although one could easily implement their own exact\nsolver and make it available in the format needed by \\clawpack routines.\n\nA common feature in all packages in the \\clawpack suite is the\nuse of a standard interface for Fortran Riemann solver routines. This ensures that new\nsolvers or solver improvements developed for one package can immediately\nbe used by all packages. To further facilitate this sharing and to avoid\nduplication, Riemann solvers are (with rare exceptions) not maintained under\nthe other packages but are collected in a single repository named\n\\texttt{riemann}.\nUsers who develop new solvers for \\clawpack are encouraged to submit them to the\nRiemann repository.\n\nIn the Fortran-based packages (Classic, AMRClaw, and GeoClaw) the Riemann solver\nis selected at compile-time by modifying a problem-specific Makefile. In\n\\pyclaw, the Riemann solver to be used is selected at run-time. This is made\npossible by compiling all of the Riemann solvers (when \\pyclaw is installed) and\ngenerating Python wrappers with \\texttt{f2py}. For \\pyclaw, \\texttt{riemann}\nalso\nprovides metadata (such as the number of equations, the number of waves,\nand the names of the conserved quantities) for each solver so that setup is\nmade more transparent.\n", "meta": {"hexsha": "5fe7a055a7d36c28a1f60d7bf5321887c0faa877", "size": 2361, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "papers/clawpack-5x/riemann.tex", "max_stars_repo_name": "kbarnhart/doc", "max_stars_repo_head_hexsha": "ede1e86c883ad9336f6334c967a89212d2b9cdc7", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2019-01-08T12:17:16.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-15T21:17:44.000Z", "max_issues_repo_path": "papers/clawpack-5x/riemann.tex", "max_issues_repo_name": "kbarnhart/doc", "max_issues_repo_head_hexsha": "ede1e86c883ad9336f6334c967a89212d2b9cdc7", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 76, "max_issues_repo_issues_event_min_datetime": "2015-01-31T20:22:01.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-05T16:55:53.000Z", "max_forks_repo_path": "papers/clawpack-5x/riemann.tex", "max_forks_repo_name": "kbarnhart/doc", "max_forks_repo_head_hexsha": "ede1e86c883ad9336f6334c967a89212d2b9cdc7", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 12, "max_forks_repo_forks_event_min_datetime": "2015-03-01T08:26:32.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-06T02:59:13.000Z", "avg_line_length": 51.3260869565, "max_line_length": 87, "alphanum_fraction": 0.8085556967, "num_tokens": 591, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5736784074525096, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.3443151021119291}} {"text": "\\section{Results}\n\\label{Resul}\n\nThe following Section summarises the work by presenting the obtained results. Comments on the goodness of such results are also present, together with possible future developments. \nThe popularity distribution computed through Eq. (\\ref{Popc}) shows a gaussian-like distribution centred in 0 (Fig. \\ref{PopDist}), meaning that on average comments are neutral, with lateral tails of comments with strong polarity, both positive and negative. Recommendations play a small role, because of some outliers causing the normalised recommendation count to be quite a small number. It is fundamental for this distribution to be unbiased as it will be used to estimate controversy. \n\n\\begin{figure}[b]\n\\centering\n \\begin{subfigure}{0.5\\tw}\n \\centering\n \\includegraphics[height=4.5cm]{Pictures/PopularityDist.pdf}\n \\caption{Popularity distribution}\n \\label{PopDist}\n \\end{subfigure}%\n\\hfill\n \\begin{subfigure}{0.5\\tw}\n \\centering\n \\includegraphics[height=4.5cm]{Pictures/articleScoreDist.pdf}\n \\caption{Article score distribution.}\n \\label{AScore}\n \\end{subfigure}\n\\caption{Preliminary distributions.}\n\\end{figure}\n\nThe article score distribution shows a similar behaviour; it is reported in Fig. \\ref{AScore} even though it will not be included in further analyses. Again, the article score calculated with Eq. \\ref{ASeq} shows a quite symmetrical distribution, meaning that the estimator is consistent and the articles are equally distributed among the polarity spectrum.\n\n\\subsection{Strategy 1 (linear)}\n\\label{Strategy1}\nThe first strategy to estimate controversy in news articles (Eq. \\ref{Str1}) was to multiply the number of comments related to each article with the IQR of the popularity distribution for such comments (labelled as \\textit{debate}). This calculation lead to an index of controversy peaked near 0. Furthermore, the controversy distribution looks very similar to the comment number distribution (see Fig. \\ref{Str1Dist}). This suggests that popularity and controversy are strongly correlated, while debate plays an insignificant role in the compute. \n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat1Dist.pdf}\n\\caption{Popularity, Debate and Controversy distributions for Strategy 1}\n\\label{Str1Dist}\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat1Corr.pdf}\n\\caption{Correlation plots for Strategy 1.}\n\\label{Str1Corr}\n\\end{figure}\n\nTo corroborate this hypothesis, the correlation scatter plots for the three variables were built, and Fig. \\ref{Str1Corr} shows a very strong linear correlation between popularity and controversy. This correlation rows against the thesis of the project, which is trying to differentiate the so-called \"hot topics\" from the most controversial ones. For this reason, another strategy was implemented to see if some other definition of controversy can detach more efficiently from the concept of popularity.\n\nTo go deeper into the analysis, the controversy scores for each articles were averaged among articles of the same section. This grouping was made to see if any article section is more controversial than others. Unfortunately, Strategy 1 did not prove to be useful in this case, as the most controversial topics are also the most popular one. Figure \\ref{RG1} shows how, for each section, the controversy and the popularity score look very similar to each other (after proper normalisation).\n\n\\begin{figure}[htb]\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat1SN.pdf}\n\\caption{Most controversial sections for Strategy 1}\n\\label{RG1}\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat1ND.pdf}\n\\caption{Most controversial desks for Strategy 1}\n\\label{RG1nd}\n\\end{figure}\n\nAfter that, articles were also grouped by the \"new desk\" index, to see if there is any category that stands out as the most controversial. Again, results (Fig. \\ref{RG1nd}) indicate a high similarity between controversy and popularity, with \"National\" desk as the most popular desk.\n\n%\\begin{figure}[hbt]\n\n%\\end{figure}\n\n\\subsection{Strategy 2 (log)}\n\nStrategy 2 consisted on applying the natural logarithm to the number of comments under each article, in order to dampen the strong dependencies between the former and controversy. Thanks to this new strategy, results change significantly, and they are reported in the same format as Section \\ref{Strategy1}.\nFirst of all, the popularity distribution (hence the controversy index) are represented in their new shapes, while the debate distribution obviously does not change (Fig. \\ref{Str2Dist}).\nBy looking at the correlation plots, it can be seen that the strong linear correlation between the two distributions has been removed (Fig. \\ref{Str2Corr}). The problem is now that distributions are not smooth for small values of popularity, because of the logarithm. However, the distribution of controversy looks smoother and more balanced compared to the one adopted in Strategy 1.\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat2Dist.pdf}\n\\caption{Popularity, Debate and Controversy distributions for Strategy 2.}\n\\label{Str2Dist}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat2Corr.pdf}\n\\caption{Correlation plots for Strategy 2.}\n\\label{Str2Corr}\n\\end{figure}\n\n\nIf we look at the averages on categories (Fig. \\ref{RG2} and \\ref{RG2nd}), it can be seen that popularity and controversy are now two very different quantities. The bar plots both for section name and desk show different behaviour between the two data series. However, the problem with Strategy 2 is that the categories labelled as \"highly controversial\" are quite unexpected, such as Cycling and Entertainment. On the other hand, controversial categories in Strategy 1 were Politics, Editorials and Middle East (categories one would expect to be controversial).\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat2SN.pdf}\n\\caption{Most controversial sections for Strategy 2}\n\\label{RG2}\n\\end{figure}\n\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\tw]{Pictures/Strat2ND.pdf}\n\\caption{Most controversial desks for Strategy 2}\n\\label{RG2nd}\n\\end{figure}\n\n\n\\subsection{Keyword extraction}\nLastly, a feature extraction was performed to check which words were common in comments related to the most controversial articles (top 10\\%). A corpus vectorisation was performed via a {\\tt TfidfVectorizer}, and then a $\\chi2$ analysis was invoked to select words with higher frequency. Since TF-IDF tends to boost the score of words which are very uncommon in the corpus, among the \"top controversial words\" there were many nonsensical words that occurred only once in the whole corpus, probably originated by typos of the comment authors. For this reason, only the words with a match in the AFINN dictionary were kept. Figure \\ref{TopWords} depicts the words selected by the $\\chi^2$ analysis. An ample fraction of the word list is made of polarised words, indicating that polarity is at the basis of controversy.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\linewidth]{Pictures/TopWs.png}\n\\caption{Most controversial words in the comment dataset.}\n\\label{TopWords}\n\\end{figure}\n\n\\section{Conclusions}\n\nThe aim of this project was to define controversy in news articles by analysing the comments below the articles. Two different strategies were adopted, each one with its pros and cons.\nStrategy 1 defined controversy through a linear relationship; it was the one who guaranteed the results with most common sense (Politics and Middle East were among the top categories), but it failed to detach from popularity.\nOn the other hand, strategy 2 guaranteed a significant shift from popularity and removed linear correlations, but the results leave us with some doubts.\nIn any case, it was proved that in either case polarity in the comments is the key to analyse controversy. \nSome future developments to this work could be:\n\\begin{itemize}\n\\item Perform aspect-based sentiment analysis to go deeper into the comments, and analyse which aspects within a given comment generated controversy\n\\item Using topic modelling, extract the article section name from the comments, since nearly half of the articles in the dataset were provided with \"Unknown\" section name\n\\item Demonstrate the robustness of this controversy estimator by changing the lexicon (for example, WordNet could be used instead of AFINN) and look for similar behaviour of the controversy distribution\n\\item Find another strategy to remove correlation between popularity and controversy without the need to apply a logarithm\n\\item Train a classifier able to predict controversy in other datasets of news and comments\n\\end{itemize}", "meta": {"hexsha": "de83e5ead316f582e995674a5082c012ab6b4acf", "size": 8731, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Report/Results.tex", "max_stars_repo_name": "andreasala98/NYMines", "max_stars_repo_head_hexsha": "3e9eb3da22f815d3179f10ff98b9887a2a4f1360", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Report/Results.tex", "max_issues_repo_name": "andreasala98/NYMines", "max_issues_repo_head_hexsha": "3e9eb3da22f815d3179f10ff98b9887a2a4f1360", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Report/Results.tex", "max_forks_repo_name": "andreasala98/NYMines", "max_forks_repo_head_hexsha": "3e9eb3da22f815d3179f10ff98b9887a2a4f1360", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 69.848, "max_line_length": 816, "alphanum_fraction": 0.7991066315, "num_tokens": 1959, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6001883449573376, "lm_q2_score": 0.5736784074525098, "lm_q1q2_score": 0.344315093906683}} {"text": "\\documentclass[a4paper,12pt]{article}\n\n% Font\n\\usepackage[T1]{fontenc}\n\\usepackage{gentium}\n\n% Math packages\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage{bm}\n\n% Define symbol shortcuts\n\\newcommand{\\cc}{\\mathcal{C}}\n\\newcommand{\\dd}{\\mathcal{D}}\n\\newcommand{\\hh}{\\mathcal{H}}\n\\newcommand{\\xx}{{\\bm x}}\n\\newcommand{\\yy}{{\\bm y}}\n\n% Math environment\n\\newtheorem*{thm}{Theorem}\n\n% Better list management:\n% - vertical spacing in lists\n% - items in lists start with dash not bullet point.\n\\usepackage{enumitem}\n\\setlist{label=\\textemdash,\n itemsep=0pt, topsep=3pt, partopsep=0pt}\n\n% Include graphics\n\\usepackage{graphicx}\n\\usepackage{subcaption}\n\n% Page format \n\\usepackage[top=2cm,left=2cm,right=2cm,bottom=2cm]{geometry}\n\n\\begin{document}\n%%% HEADER\n\\raisebox{0.6in}[0in]{\\makebox[\\textwidth][r]{\\it Unproofed version }}\n\\vspace{-0.7in}\n\n\\begin{center}\n\\bf\\large MA2823: Foundations of Machine Learning \\\\\nChapter 6: Regularized Linear Regression\n\\end{center}\n\n\\noindent\nLecturer: Chlo\\'e-Agathe Azencott \n\\hfill\nScribe: Adrien Galamez \\\\\n\\null \\hfill Paul Magon de la Villehuchet\n\n\n\\noindent\n\\rule{\\textwidth}{1pt}\n\n\\medskip\n\nIn this chapter we will see:\n\\begin{itemize}\n\\item what is regularization;\n\\item how to use regularization as a means to control model complexity;\n\\item several forms of regularizing linear regression models.\n\\end{itemize}\n\n\\section{Regression setting}\n\\subsection{Large $p$, small $n$}\nThis section is a reminder of the context in which we work. We consider a dataset that has $p$ features and $n$ samples. Thus, we're working with a data matrix $X$ that have $n$ rows and $p$ columns. The outcome vector $y$ describes the quantity we want to predict. The goal of linear regression model is to approximate $y$ as a linear combination of $X$.\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[scale = 0.3]{figures/RegressionSettings.png}}\n\\caption{Figure illustrating the dimensions of the data matrix and the outcome vector.}\n\\label{DataMatrix}\n\\end{figure}\n\nOn the Figure \\ref{DataMatrix}, we can see that the data matrix is taller than it is wide ($p < n$). This is a common shape. But, conversed settings can also appear. \n\nThis configuration is called large $p$, small $n$. The data matrix is wider than it is tall. This is the kind of settings we have in genetics and neuroimaging: we have many more descriptors than we have data points. For instance, in genetics, a data set can gather informations for thousands of genes as this information is easy to access. Yet, as each row holds data gathered on a patient, we have at best hundreds of patients. Thus, we have a wide but not tall data matrix. In neuroimaging, brain images are big objets, typically thousands pixels or voxels (3D images). Yet, as brainscans are costly to obtain, we repeat the process on thousands of patient only. Thus, we have more features than observations. In this configuration, regularization is usefull.\n\n\\subsection{Linear regression}\n\\subsubsection{Pro and cons of least-squares fit}\nLinear regression is aimed at approximating $y$ as a linear combination of $X$. In the previous chapter, we said that, if we're doing a least-squares fit (which is equivalent to Maximum Likelihood estimation under the assumption of Gaussian noise), we obtain this solution\n\\begin{equation}\n\\hat{\\beta} = \\arg\\min_{\\beta} (y-X\\beta)^{T} (y-X\\beta) = (X^T X)^{-1} X^T y.\n\\label{LR}\n\\end{equation}\nThis solution is uniquely defined when $X^T X$ is invertible, hence when $X$ has a full column rank. As $X$ is an $n \\times p$ matrix, if $p$ is larger than $n$ (as in the large $p$, small $n$ configuration), $X$ can not be inverted.\n\nThis fit has several advantages.\n\\begin{itemize}\n\\item The predicted vector $\\hat{\\beta}$ is unbiased ($\\mathbb{E}[\\hat{\\beta}]= \\beta$).\n\\item If we're restricting to unbiased estimators, minimum mean squared error implies minimum variance.\n\\item This fit gives an explicit solution (Equation \\ref{LR}).\n\\item If $n \\gg p$, the computational time is linear in the number of samples. Indeed, the computational time is $O(\\underbrace{np^2}_{\\text{compute } X^T X}+\\underbrace{p^3}_{\\text{invert } X^T X})$. \n\\end{itemize}\n\nYet, this fit has also some drawbacks.\n\\begin{itemize}\n\\item Correlated variable leads to high variance of the estimator.\n\\item Prediction error increases linearly as a function of $p$.\n\\item The solution is hard to interpret when $p$ is large as shown in the section \\ref{LargeP}.\n\\end{itemize}\n\n\\subsubsection{When $p$ in larger than $n$}\n\\label{LargeP}\nWhen $X^T X$ is not invertible, we still have ways to find $\\hat{\\beta}$. In this case, we can use the pseudo-inverse of $X$. We can also use numerical methods to solve a linear system of $p$ equations such as gradient descent (aimed at minimizing a convexe function), Gaussian elimination or LU decomposition.\nOn the Figure \\ref{LRPseudoInverse}, we can compare the predicted coefficient (thanks to a linear regression using pseudo-inverse) and the true coefficient when $p = 1000$ and $n = 10$. So, we're in the case of the large $p$, small $n$ configuration. In fact, the outcome vector $y$ was created by a linear combination of the data-matrix $X$ using the weights shown on the left hand graph. \n\nThe objectif was to retrieve those weights using a linear regression on the vectors $X$ and $y$. $10$ causal features were highlighted in orange on both graphs. A good approximation would be to use only those $10$ causal features. Nevertheless, as the right hand graph shows, the linear regression gives coefficients that have approximately the same weight. The causal features are lost entirely in the noise created by the others coefficients. Information has been lost. Regularization is a way of addressing this issue.\n\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[scale = 0.3]{Figures/linear_regression_pseudoinverse.png}}\n\\caption{Example of weights obtained from linear regression when $X$ is not invertible, thus using the pseudo-inverse. On the left, the graph is showing the true coefficients of the vector $\\beta$. On the right, the graph presents the predicted coefficients of the vector $\\hat{\\beta}$ obtained from linear regression.}\n\\label{LRPseudoInverse}\n\\end{figure}\n\n\\section{Regularization}\nThe Figure \\ref{LRPseudoInverse} showed that the linear regression used all the features available to predict the outcome vector $y$. Yet, all the weights are very small. Thus, the solution is hard to interpret. Usually, we prefer having a small subset of features with strong weights. This is one of the advantages of regularization. Moreover, the more we have variables in our model, the more complex our model is. And, the more complex our model is, the more chances we have to overfit our data. Thus, we want to find a way to simplify our model.\n\n So, instead of minimizing only the sum of squared error, the idea of regularization is to minimize\n \\[ \\text{Sum of squared error} + \\lambda \\text{ prenalty on model complexity}.\\]\nThus, when using regularization, the estimator is biased (if $\\lambda \\not = 0$). Yet, because the model will be less complex, we'll have a smaller variance. We're willing to accept a biased estimator in exchange for smaller variance. So, there's a tradeoff to make between bias and variance. $\\lambda$ can be set by cross-validation. \\label{CVlambda}\n\nThis method is also called \\emph{shrinkage} in the context of linear regression. Indeed, this is going to shrink the weights of the model. Thus, the final model is simpler.\n\nThe following sections are presenting examples of regularization technique.\n\n\\subsection{Ridge regression}\n\\subsubsection{Ridge estimator}\nInstead of minimizing our sum of squared error, ridge regression is aimed at minimizing $||y - X\\beta||_2^2 + \\lambda ||\\beta||_2^2$. Thus, the ridge estimator is given by\n\\[ \\hat{\\beta}_\\text{ridge} = \\arg \\min_\\beta ||y - X\\beta||_2^2 + \\lambda ||\\beta||_2^2. \\]\n\n\\begin{thm} The ridge regression estimator is given by\n\\[ \\hat{\\beta}_\\text{ridge} = (X^T X+ \\lambda I)^{-1} X^T y. \\]\n\\end{thm}\n\n\\begin{proof}\nWe consider the function $f$ defined by\n\\[ f(\\beta) = ||y - X\\beta||_2^2 + \\lambda ||\\beta||_2^2.\\]\nIn order to minimize this function, we take its gradient.\n\\[ \\nabla_{\\beta}f(\\beta) = -2X^T (y-X\\beta) + 2\\lambda \\beta.\\]\nThen, $\\beta_\\text{ridge}$ is defined as $\\nabla_{\\beta}f(\\beta_{\\text{ridge}}) = 0$. Thus,\n\\[ (X^TX+\\lambda I)\\beta_\\text{ridge} = X^T y.\\]\nIf $\\lambda > 0$, $(X^T X + \\lambda I)$ is invertible, then\n\\[ \\hat{\\beta}_\\text{ridge} = (X^T X+ \\lambda I)^{-1} X^T y. \\]\n\\end{proof}\n\n\\subsubsection{Solution path}\n\\label{SectionSolutionPathRR}\nWe've said in section \\ref{CVlambda} that we can find the optimal value of $\\lambda$ by cross-validation. We can compute solution paths (Figure \\ref{SolutionPathRR}) that is a plot showing how the feature coefficient evolves when we decrease the value of $\\lambda$. Indeed, when $\\lambda = 0$, there's no regularization anymore. And, if $\\lambda \\gg 1$, we only want to minimize the model complexity. So, all coefficient are then equal to $0$. Then, on the Figure \\ref{SolutionPathRR}, on the left hand, we start with the model in which all coefficients are equal to zero and we end, on the right hand, with the linear regression without any regularization model.\n\nThe vertical red line shows the value of $\\lambda$ that was obtained by cross-validation. The goal is to choose the value of $\\lambda$ that gives the best generalization on another set of data.\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.7\\textwidth]{figures/solution_path_ridge_regression.png}}\n\\caption{Ridge regression solution path}\n\\label{SolutionPathRR}\n\\end{figure}\n\n\\subsubsection{Standardization}\nThe goal of this section is to describe what will happen to our model if we multiply our features by a constant.\n\\begin{enumerate}\n\\item \\textbf{Standard linear regression}~: Without any regularization, we multiply, in the data matrix $X$, the column $j$ by a real $c \\not = 0$. So, \n\\[\\forall i \\in [|0;n-1|], x_{ij} \\to c x_{ij}.\\]\nThen, the corresponding weight $\\beta_j$ is going to be divided by $c$.\n\\[ \\beta_j \\to \\frac{1}{c} \\beta_j.\\]\nThus, the weight is different but the solution is similar.\n\\item \\textbf{Ridge regression}~: When we multiply, in the data matrix $X$, the column $j$ by a real $c \\not = 0$, we can't know what will happen because of the penalization term $\\lambda \\beta_j^2$. So, it's important to use standardized feature \\emph{before} regularizing linear regression.\n\\end{enumerate}\n\\subsubsection{Advantages and drawbacks}\nFinally,\n\\begin{enumerate}\n\\item \\textbf{Advantages}~: Ridge regression has several advantages\n\\begin{itemize}\n\\item Correlated variables get similar weights.\n\\item Identical variables get identical weights.\n\\item An analytical solution is provided\n\\end{itemize}\n\\item \\textbf{Drawbacks}~: And it also have drawbacks\n\\begin{itemize}\n\\item Ridge regression shrinks coefficients but does \\emph{not} result in a sparse model. A model is sparse when many coefficient get a weight of $0$. Then, when a model is sparse, many coefficient can be eliminated from the model.\n\\end{itemize} \n\\end{enumerate}\n\n\\subsection{Lasso regression}\n\\subsubsection{Lasso estimator}\nInstead of using the $L^2$ penalty, like the Ridge estimator, the Lasso estimator is based on a $L^1$ penalty. Hence we have to minimize : $||y - X\\beta||_2^2 + \\lambda ||\\beta||_1$.\n\\begin{equation}\n\\hat{\\beta}_\\text{lasso} = \\arg \\min_\\beta ||y - X\\beta||_2^2 + \\lambda ||\\beta||_1.\n\\label{DefLagrange}\n\\end{equation}\n\nUnlike the Ridge estimator, there is no explicit solution. We can however, transform this problem into a quadratic problem.\n\n\\begin{thm} There is a bijection between $\\lambda$ and $t$ such as ${\\beta}_\\text{lasso}$ is solution of the quadratic problem : \n\\begin{equation}\n\\hat{\\beta}_\\text{lasso} = \\arg \\min_\\beta ||y - X\\beta||_2^2 \\quad s.t \\quad ||\\beta||_1\\leq t.\n\\label{DefQuadratic}\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nWe consider the functions $f$ and $g$ defined by\n\\[ f(\\beta) = ||y - X\\beta||_2^2.\\]\n\\[ g(\\beta) = ||\\beta||_1 - t.\\]\nFrom (\\ref{DefQuadratic}) to (\\ref{DefLagrange}). Let $H$ be the feasible region, i.e, $\\{ \\beta$ $, g(\\beta) \\leq 0\\}$\n\\[ \\hat{\\beta}_\\text{0} = \\arg \\min_{\\beta \\in H} ||y - X\\beta||_2^2 \\quad.\\]\nTwo cases are possible:\\\\\n\\begin{itemize}\n\\item Case 1 : the unconstrained minimum lies in the feasible region. Problem is solved.\n\\item Case 2 : the unconstrained minimum does not lie in the feasible region. \\\\\nIn this case, the solution is the intersection of the iso-contours of $f$ with the boundary of $H$, as shown in the next figure.\n\\end{itemize}\n~\\\\\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.7\\textwidth]{figures/sol_lasso.png}}\n\\caption{Solution of Lasso-estimator}\n\\label{SolutionLasso}\n\\end{figure}\n~\\\\\nFurthermore, we know that the gradient of $g(\\beta)$ is orthogonal to the boundary of $H$.\nThe gradient of $f(\\beta)$ is also orthogonal to the iso-contours of $f$.\\\\\nHence, at $\\beta_{0}$, the gradient of $f$ and $g$ are parallel.\\\\\nWe also know that $\\nabla_{\\beta}g(\\beta)$ points towards the unconstrained minimum of $f$ while $\\nabla_{\\beta}f(\\beta)$ points in the opposite direction.\\\\\n~\\\\\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.7\\textwidth]{figures/sol_lasso2.png}}\n\\caption{Gradients of $f$ and $g$ and solution of Lasso-estimator}\n\\label{SolutionLasso2}\n\\end{figure}\n~\\\\\nWe can conclude from previous results that $\\nabla_{\\beta}f(\\beta) = -\\lambda \\nabla_{\\beta}g(\\beta)$.\\\\\nThe Lagrangian of the Lasso-problem : $L(\\beta) = f(\\beta) + \\lambda g(\\beta)$ is minimized at $\\beta_{0}$. \\\\\n\nFrom (\\ref{DefLagrange}) to (\\ref{DefQuadratic}). Let us fix $\\lambda$ and define $\\beta_* = \\arg \\min_\\beta f(\\beta) + \\lambda ||\\beta||_1^2$. Let us define $t = ||\\beta_*||_1^2.$\\\\ \nThen, for any $\\beta, f(\\beta_*)+ \\lambda ||\\beta_*||_1^2 \\leq f(\\beta)+\\lambda||\\beta||_1^2$. Hence $f(\\beta_*)\\leq f(\\beta)+\\lambda(||\\beta||_1^2 - t).$\\\\\nUnder the constraint that $||\\beta||_1^2 \\leq t, \\lambda(||\\beta||_1^2 - t) \\leq 0$. Hence $f(\\beta_*) \\leq f(\\beta)$, i.e. $\\beta_* = \\beta_{lasso}$\\\\\n\nThus, the two problems are equivalent.\n\\end{proof}\n\\subsubsection{Solution path}\nAs we did before in Section \\ref{SectionSolutionPathRR} for the Ridge Regression estimator, we can also compute solution paths for the Lasso estimator (Figure \\ref{SolutionPathLasso}). On the left hand of the graph, we start with high values of $\\lambda$ that give very simple models. We move towards lower values of $\\lambda$ that gives more complex models. Yet, there's one difference with the ridge regression solution path. In fact, we see here that our features ``enter the model one by one'': their weight is equal to $0$ for all values of $\\lambda$ lower to a certain $\\lambda_0$. For instance, here, when the model complexity increases, features ``enter'' the model in this order: \\texttt{lcavol}, \\texttt{lweight}, \\texttt{svi}, \\ldots, \\texttt{gleason}. Thus, at every stage, we have a \\emph{sparse model}. For example, on the vertical red line, we have a model with only $5$ parameters that are not null (\\texttt{lcavol}, \\texttt{svi}, \\texttt{lweight}, \\texttt{pgg45}, \\texttt{lbph}).\n\n\\subsubsection{Forward stepwise regression}\nWe just saw that, in the lasso regularization, the features are ``entering'' the model one by one. Then, we can also get the lasso solution by using forward stepwise regression: the model is built sequentially by adding one variable at a time. Then, this a greedy method.\n\\begin{itemize}\n\\item We start with the intercept only.\n\\item At each step, the variable that most improves the fit is added. All variables are tested and the one that gives the better fit is kept.\n\\item We stop when $||\\beta ||_1 \\leq t$.\n\\end{itemize}\n\n\\subsubsection{Least Angle Regression}\nThe computation of the lasso solution is a quadratic problem, and can be tackled by standard algorithms. But, the least angle regression algorithm is a better approach. In the previous section, when a variable and its weight was added in the model, and this weight was never updated. The least angle regression procedure follow the same general scheme, but doesn't add a feature fully into the model. Here, at each step, ``only as much of a variable as needed'' is added. In fact, the coefficient of that feature is increased only until that feature is no longer the one most correlated with the residual $r$. This algorithms can be described as follow\n\\begin{enumerate}\n\\item Standardize the features to have mean zero and unit norm. Start with the residual $r = y - \\bar{y}$ and $\\beta_1, \\ldots, \\beta_p = 0$. The residual is ``what is left to explain from our model''. The perfect model would have a residual that is equal to $0$.\n\\item Find the one feature that explains $r$ the most. In other words, find the feature $x_j$ most correlated with $r$.\n\\item Update the weight $\\beta_j$ until $x_j$ is no longer the one most correlated with $r$: move $\\beta_j$ from $0$ towards in the direction of the sign of its correlation with $y$. In fact, move $\\beta_j$ from $0$ towards its least-squares coefficient $\\langle x_j, r \\rangle$ until some other competitor $x_k$ has as much correlation with the current residual as does $x_j$. So\n\\[ \\beta_j + \\alpha \\frac{1}{\\sum_{i=1}^n (x_j^i)^2} \\sum_{i=1}^n x_j^i r^i \\to \\beta_j,\\]\n\\[ (y - \\bar{y}) - \\beta_j x_j \\to r,\\]\nwhere $\\alpha$ is called the step size.\n\\item Move $\\beta_j$ and $\\beta_k$ in the direction defined by their joint least squares coefficient of the current residual on $\\langle x_j, x_k \\rangle$ until some other competitor $x_l$ has a much correlation with the current residual $r = (y-\\bar{y} - \\beta_j x_j - \\beta_k x_k)$.\n\\item If a non-zero coefficient hits zero, drop its variable from the active set of variables and recompute the current joint least squares direction.\n\\item Continue in this way until all $p$ predictors have been entered.\n\\end{enumerate}\n\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.7\\textwidth]{figures/solution_path_lasso.png}}\n\\caption{Ridge regression solution path}\n\\label{SolutionPathLasso}\n\\end{figure}\n\n\\subsection{Elastic Net}\n\\subsubsection{Elastic Net estimator}\nThe Elastic Net estimator is a convex combination of Lasso and Ridge. It is defined by :\n\\[ \\hat{\\beta}_\\text{enet} = \\arg \\min_\\beta ||y - X\\beta||_2^2 + \\lambda(\\alpha||\\beta||_2^2 + (1 - \\alpha)||\\beta||_1) \\]\nThe main advantage is to combine both :\n\\begin{itemize}\n\\item It selects variables like the Lasso-estimator. \n\\item It shrinks together correlated and identical variables like the Ridge-estimator\n\\end{itemize}\nOf course, this cames with the cost of having two parameters ($\\lambda$, $\\alpha$) to set instead of one.\n\n\\subsubsection{Comparison of Elastic Net-estimator and Lasso-estimator}\nThe Figure \\ref{Lassoenet} represents results of both Lasso-estimator and Elastic Net-estimator on Leukemia data. We can see that Elastic Net results in more non-zero coefficients than Lasso, but with smaller amplitudes.\n\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.9\\textwidth]{figures/lassoenet.png}}\n\\caption{Comparison of weights on Lasso-estimator and Elastic Net estimator}\n\\label{Lassoenet}\n\\end{figure}\n\n\\subsection{$L^q$ regression}\n\\subsubsection{$L^q$ estimator}\nThe $L^q$ estimator is a generalization of the Ridge and Lasso estimator : the penalty is based on the $L^q$ norm of $\\beta$.\n\\[||\\beta||_q^q = \\sum_{i = 1}^{n}{\\beta_{i}^q}\\]\nThe $L^q$ estimator is then defined by :\n\\[ \\hat{\\beta}_\\text{$L^q$} = \\arg \\min_\\beta ||y - X\\beta||_2^2 + \\lambda ||\\beta||_q^q. \\]\nWe have the same theorem to transform this problem into a quadratic problem with constraints.\\\\\n\n\\begin{thm} There is a bijection between $\\lambda$ and $t$ such that ${\\beta}_\\text{$L^q$}$ is solution of the quadratic problem : \n\\[ \\hat{\\beta}_\\text{$L^q$} = \\arg \\min_\\beta ||y - X\\beta||_2^2 \\quad s.t \\quad ||\\beta||_q^q\\leq t. \\]\n\\end{thm}\n\n\\begin{proof}\nThe proof follow the same steps than the previous one.\n\\end{proof}\n\nThe Figure \\ref{Bound2} represents the feasible sets for different values of $q$.\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.7\\textwidth]{figures/bound.png}}\n\\caption{Boundaries of the feasible set with different values of q}\n\\label{Bound2}\n\\end{figure}\n\n\\subsubsection{Comparison between Lasso and Ridge}\nBecause the feasible depends on the value of $q$, the Ridge $(q = 2)$ and Lasso $(q = 1)$ estimator will be different. The feasible set will be a square in one case and a circle in the other. This helps understand why the $L^1$ norm gives sparse models. Indeed, the $L^1$ ball has corners. This means that when we're moving from the unconstrained minimum towards the ball, we're more likely to ``hit'' a corner of the ball. And, on a corner of the ball, only one feature has a weight non equal to zero.\n\n\\begin{figure}[!h]\n\\centerline{\n\\includegraphics[width = 0.7\\textwidth]{figures/comparison.png}}\n\\caption{Comparison of the Lasso-estimator and the Ridge-estimator}\n\\label{Comparison}\n\\end{figure}\n\n\\subsection{Structured Regularization}\n\\subsubsection{Group Lasso}\nThe \\emph{group Lasso} is a variant of Lasso. All $p$ variables are partitioned in $K$ predefined groups of variables that are known to work together and expected to work together and thus be all active or inactive together. For instance, genes belonging to the same biological pathway can form a group.\n\nLet $X_k$ be the features belonging to group $k$. Let $p_k$ be the size of group $k$. We have : $\\sum_{k = 1}^K{p_k} = p$. We can define the Group Lasso-estimator:\n\\[ \\hat{\\beta}_\\text{grlasso} = \\arg \\min_\\beta ||y - \\sum_{k = 1}^K{X_k\\beta_k}||_2^2 + \\lambda \\sum_{k = 1}^K{\\sqrt{p_k}||\\beta_k||_2}\\]\n\n\\subsubsection{Other structured penalties}\nOther regularizations exist :\n\\begin{itemize}\n\\item Overlapping groups\n\\item Graphs\n\\item Trees\n\\item Multiple related Tasks\n\\end{itemize}\n\n\\section{Conclusion}\nWe've replaced the minimization of the sum of squared errors (that gave us the solution of the linear regression problem) by minimizing the sum of squared errors \\emph{plus} a regularizer that penalizes the model complexity. We've seen 3 major forms of regularizers:\n\\begin{itemize}\n\\item Ridge regression ($L^2$ norm)\n\t\\begin{itemize}\n\t\\item gives similar weights to similar variables,\n\t\\item doesn't really give sparse solution,\n\t\\item but has the advantage of having an explicit solution.\n\t\\end{itemize}\n\\item Lasso regression ($L^1$ norm)\n\t\\begin{itemize}\n\t\\item randomly picks one of several correlated variables,\n\t\\item gives sparse solution,\n\t\\item can be implemented by the Least Angle Regression algorithm.\n\t\\end{itemize}\n\\item Elastic net (combining $L^1$ and $L^2$ norm)\n\t\\begin{itemize}\n\t\\item selects variables like the lasso,\n\t\\item shrinks together the coefficients of correlated variables.\n\t\\end{itemize}\n\\end{itemize}\n\n\\end{document}", "meta": {"hexsha": "6bb877238b6eecd02f462f193e3a67c6221a42b8", "size": 22906, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "lecture_notes/chap06_regularization.tex", "max_stars_repo_name": "taeraa/ma2823_2016", "max_stars_repo_head_hexsha": "b91642e8b3cac059f91d6785f9da6a436b382db8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 19, "max_stars_repo_stars_event_min_datetime": "2016-09-13T16:35:40.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-05T02:16:53.000Z", "max_issues_repo_path": "lecture_notes/chap06_regularization.tex", "max_issues_repo_name": "taeraa/ma2823_2016", "max_issues_repo_head_hexsha": "b91642e8b3cac059f91d6785f9da6a436b382db8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2016-10-03T10:42:43.000Z", "max_issues_repo_issues_event_max_datetime": "2016-10-03T10:43:16.000Z", "max_forks_repo_path": "lecture_notes/chap06_regularization.tex", "max_forks_repo_name": "taeraa/ma2823_2016", "max_forks_repo_head_hexsha": "b91642e8b3cac059f91d6785f9da6a436b382db8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 62, "max_forks_repo_forks_event_min_datetime": "2016-09-07T06:25:40.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-30T13:21:05.000Z", "avg_line_length": 61.9081081081, "max_line_length": 996, "alphanum_fraction": 0.740679298, "num_tokens": 6441, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.34415215672215116}} {"text": "%\\section{Discussion}\n\n\n\\subsection{Comparison of beading schemes}\nWe can see from \\cref{TEST_naive_accuracy}(top) and~\\ref{over_underfill} that the uniform technique causes a lot of overfills and underfills: on average \\revise{approximately}{} \\revise{\\SI{1}{\\percent}}{\\SI{1.6}{\\percent}} of the total target area is covered by underfill and likewise for overfill.\nTo our knowledge, the uniform beading scheme, as well as the outer beading scheme, is of little use to FDM printers.\n\nThe constant bead count scheme effectively deals with underfills, but generates orders of magnitude more overfills compared to the other schemes. \nAlso, the scheme comes at the cost of greatly varying bead widths and an average bead width that is not close to the preferred bead width.\nNote that most overfill areas occur near regions of alternating bead width. \n\\revise{}{While the scheme results in short toolpaths, as indicated by the idealized print time, it also results in a wide range of bead widths, which cause the back pressure compensation print time to be very large.\nSee \\cref{statisticsfig}.}\nFor an input outline shape which contains both very small and very large features, the constant bead count scheme produces bead widths which can fall outside of the range of manufacturable bead widths.\nMoreover the centrality marking is not robust against small perturbations in the outline; adding a small chamfer in a corner causes the unmarked ST to be very small at that location, which results in tiny bead widths.\n\\revise{}{See top right of \\cref{TEST_Constant_accuracy}.}\n\nIn \\cref{TEST_Center_accuracy} we can see that\nthe centered beading scheme effectively deals with \\revise{both }{}overfill \\revise{and underfill }{}and produces desired bead widths in all locations, except for the extrusion paths in the center, where the bead widths \\revise{are within a factor 2 off from the desired bead width.}{range between $0.25 w^*$ and $1.8w^*$.}\n\\revise{}{However, it does produce some narrow underfill regions.}\n\\revise{%gs\nAccording to \\cref{over_underfill} the overfill and underfill for the centered, the evenly distributed and the inward distributed scheme are all approximately \\SI{0.2}{\\percent}, which is a considerable improvement over the uniform technique.\n}{%sg\nCompared to the uniform technique the centered technique increases the (open) path count, but considerably reduces over- and underfill and decimates the number of toolpath angles below \\SI{45}{\\degree}.\nSee \\cref{statisticsfig}.%afs\n}%zgds\n\nHowever, according to \\cref{widthHistogram} the centered scheme exhibits a wider range of bead widths than the distributed schemes:\nthe standard deviation of the bead widths in the centered scheme is approximately \\revise{\\SI{39}{\\micro\\meter}}{\\SI{53}{\\micro\\meter}}, while that of the distributed schemes is approximately \\revise{\\SI{14}{\\micro\\meter}}{\\SI{23}{\\micro\\meter}}.%_\n\\revise{}{\\footnote{\\revise{}{Although the standard deviation $\\sigma$ of the inward distributed scheme is slightly higher than that of the evenly distributed scheme, the mean absolute deviation is lower (i.e. \\SI{9}{\\micro\\meter} versus \\SI{11}{\\micro\\meter}), because its distribution is more peaked.}}}\n\\revise{}{Moreover, because the quantization operator rounds to the nearest number of beads, in the worst case where we switch from a single to two beads the widths switch from $0.75w^*$ to $1.5w^*$, which is a considerably smaller range than in the centered scheme.}\nWe therefore conclude that the distributed schemes \\revise{result in bead widths closer to the preferred widths}{exhibit a lower bead width variation and lower (open) path count} compared to the centered scheme.\n%This is desirable for the manufacturability of the beads and can therefore have a positive effect on the mechanical properties and surface quality of the 3D prints. \n\n\\revise{It is hard to visually identify the difference between the evenly and the inward distributed scheme in \\cref{visualized_accuracy}, because that particular example shape does not have wide features.}\n{\\Cref{distributed_comparison,TEST_Distributed_accuracy,TEST_InwardDistributed_accuracy} show that in the inward distributed scheme the outer toolpaths have the preferred width more often than in the evenly distributed scheme, which means that the outline accuracy of the inward distributed beading is less affected by inaccuracies in the adaptive width control system.}\n%{\\Cref{TEST_InwardDistributed_accuracy} shows that in the inward distributed scheme the outer beads are more often equal to the preferred width compared to the evenly distributed scheme in \\cref{TEST_Distributed_accuracy}, but that effect is more pronounced for wider geometry such as in \\cref{distributed_comparison}.}\n%While the difference between the evenly and inward distributed scheme in \\cref{visualized_accuracy} results only in the outer bead being the preferred width in some locations, the effect of the inward distributed scheme is more pronounced in larger geometry, as can be seen in \\cref{distributed_comparison}.}\n\\revise{However, \\cref{distributed_comparison} and \\cref{widthIndexedHistogram} confirm that the outer toolpaths have the preferred bead width more often.}{}% _\nFurthermore, \\revise{from \\cref{smoothness} }{}we find that compared to evenly distributed, the inward distributed scheme produces \\revise{smoother toolpaths overall}{less corners with angles above \\SI{130}{\\degree} and less overfill, because the area of influence that bead count transitions have is limited in the inward distributed scheme}.\nThus the inward distributed scheme prevents over- and underfill, generates smooth toolpaths with more homogeneous width and affects smaller more centered parts of the print than the other schemes\\revise{}{, while incurring little to no extra print time}. \n\n\n\n\\revise{}\n{\n\\subsection{Limitations}\n% design considerations\nBecause the performance of the various toolpathing techniques depends on the geometry of a model, they have ramifications for the practice of design for additive manufacturing.\nBecause the naive method produces under- or overfill for parts of an outline with a constant diameter $d \\neq 2 i w^*$ it is best practice to design a model such that horizontal cross-sections have a feature diameter of an even integer multiple $i$ of the bead width.\nFor the center\\revise{ed} scheme and for the current state of the art one should only avoid parts for which $(2i + 1.8) w^* < d < (2i + 0.25) w^*$ in order to avoid underfill.\nFor the distributed schemes however, there is no diameter at which the framework produces under- or overfill for a part with a constant diameter $d$.\nThe design consideration therefore reduces to limiting the diameter of your parts to be within the range $[w_\\text{min}, \\infty)$,\nwhere $w_\\text{min}$ is a configurable parameter when using the widening meta-scheme.\n\n\n% single line segments & continuity\nThe default limit bisector angle $\\alpha_\\text{max} = \\SI{135}{\\degree}$ ensures that we don't employ transitioning in shallow wedge regions, which would result in a lot of short odd single bead polylines, which would break up the semi-continuous nature of polygonal extrusion paths;\n$\\alpha_\\text{max} = \\SI{135}{\\degree}$ corresponds to $w^* / \\cos \\nicefrac12 \\alpha_\\text{max} \\approx \\SI{0.4}{\\milli\\meter}$ long segments\nand under-/overfill areas of $\\nicefrac14 (w^*)^2 \\left( \\tan ( \\alpha / 2) - \\alpha / 2 \\right) \\approx \\SI{0.05}{\\milli\\meter\\squared}$.\nHowever, future work might be aimed at reducing under-/overfill in regions with a low bisector angle without the introduction of short single polyline extrusion segments.\nIf the over-/underfill problem is also solved for non-significant regions we might be able to increase $\\alpha_\\text{max}$ and reduce the discontinuity introduced by short extrusion segments.\n\n% single beads\nAnother limitation of our method is that in a location $v$ with locally maximal $R(v) \\approx (i + \\nicefrac12) w^*$ the odd bead count will result in a single polyline extrusion segment consisting of only a single point.\nThis can be viewed in the bottom right of \\cref{TEST_InwardDistributed_accuracy} for example.\nIn order to print such a dot, we make it into a \\SI{10}{\\micro\\meter} long extrusion segment, with an altered width such that the total volume remains correct.\nA more principled way of dealing with such a situation remains future work.\n\n% emulation troubles\nFinally it should be noted that although our framework can accurately emulate the constant bead count approach by \\citeauthor{Ding2016a}, its emulation of the centered approach by \\citeauthor{Jin2017JMS} is imperfect.\nThe transitions resulting from out framework introduce sharper corners and there is more width variation in those corners.\nWhereas the width of the connecting segment in the approach by \\citeauthor{Jin2017JMS} is the preferred width $w^*$, the bead widths closer to the center resulting from our framework will be twice the local radius, which is larger than $w^*$.\nHowever, this inflated bead width variation is expected to have an insignificant impact on the measured bead width variation.\n}\n\n\n\n\\input{10_applications}\n\n\n\n\n\n\n\\revise{\n\\subsection{Discussion on implications}\nNote that the current industry standard in FDM printing employs little to no bead width variation.\nProperly performing bead width variation calls for adaptations and developments in printers and firmware.\nIn the beading schemes we set a transition length of $t(n) = w^*$.\nThat will demand changes in cross-sectional area of the bead up to \\SI{200}{\\percent} over a small distance that is comparable to the nozzle size, which is challenging for some hardware systems.\nVarying the movement speed can be utilized to change the cross-sectional area, but this approach is limited, since the movement speed is constrained by acceleration considerations near bends in the toolpath~\\cite{Ertay2018,Kuipers2018}.\nOur schemes require a more accurate control of the volumetric flow rate in \\si{\\milli\\meter\\cubed\\per\\second}.\nUsing a filament feeder directly mounted on the print head (a.k.a. direct drive) can control the flow more dynamicaly then FDM printers where the material is fed through a Bowden tube from a feeder mounted on the frame.\nStill direct drive printers require some control system in order to accurately change the volumetric flow rate such as pressure advance algorithms~\\cite{tronvoll2019investigating}.\nYet inaccuracies in direct drive systems employing advance algorithms might arise due to the changes in back-pressure required by changing bead size.\nWe expect that developments in printing hardware and firmware will address these challenges in the future.\n\nAnother limiting factor for adopting adaptive bead width is the format of G-code which stores machine instructions.\nG-code does not support moves with varying cross-sectional area.\nA typical extrusion move \\lstinline!G1 X$x$ Y$y$ E$v$! only specifies the total amount of volume $v$ to be extruded in the move, not how that total amount should be distributed along the extrusion move.\nA workaround is to approximate a variable width extrusion segment by smaller segments with constant width.\nHowever, this introduces errors nevertheless.\nIdeally the G-code language would be expanded in some way to allow for extrusion segments with varying cross-sectional area.\n}{}\n\n% Taking a broader perspective, we note that our proposed inward distributed scheme is a pragmatic solution.\n% Rather than deriving some optimal beading scheme from a clear specification of the objective, we propose some arbitrary inward distributed beading scheme and show that it is better than the other beading schemes.\n% An optimal beading scheme can be derived if the objective is formalized terms of a unambiguous fitness function, but that would depend on the specific hardware setup and application for which toolpaths are generated.\n% This manuscript is therefore limited to showing the flexibility and versatility of the framework, rather than deriving an optimal beading scheme.\n\n\n\n\n", "meta": {"hexsha": "9702ff8602a12f61074c20b824542721c021126d", "size": 12057, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "8_discussion.tex", "max_stars_repo_name": "BagelOrb/variable_width_paper", "max_stars_repo_head_hexsha": "8b8b7a2b9b913e56267a578d2a15ed7e97fa3503", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-03-23T10:22:42.000Z", "max_stars_repo_stars_event_max_datetime": "2021-03-23T10:22:42.000Z", "max_issues_repo_path": "8_discussion.tex", "max_issues_repo_name": "BagelOrb/variable_width_paper", "max_issues_repo_head_hexsha": "8b8b7a2b9b913e56267a578d2a15ed7e97fa3503", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "8_discussion.tex", "max_forks_repo_name": "BagelOrb/variable_width_paper", "max_forks_repo_head_hexsha": "8b8b7a2b9b913e56267a578d2a15ed7e97fa3503", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 105.7631578947, "max_line_length": 370, "alphanum_fraction": 0.7987890852, "num_tokens": 2676, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.6039318337259584, "lm_q1q2_score": 0.34415214789850984}} {"text": "\\subsection{Fishing Mortality in Stock Synthesis}\n\nThe implementation and reporting of fishing mortality rate, $F$, in SS3 has some aspects that can be confusing. This description provides an overview of the ways in which $F$ is calculated, used, and reported. \n\n\\myparagraph{Rationale}\nFishery management systems expect to have a measure of annual fishing mortality that describes the intensity of the fishery such that an optimal level of $F$ can be articulated and accountability measures can be invoked if $F$ is too high, e.g., overfishing. This concept is simple and straightforward if the model is a simple biomass dynamics such that a single annual $F$ value operates on the entirety of a non-age structured population. It also is simple for age-structured models that have a single fishing fleet and knife-edge selectivity beginning at some specified age.\n\nThe simplicity of $F$ disappears quickly as models invoke a variety of realistic complexities such as: allowing the $F$ to differ among ages or to be based on size; using a collection of fleets with different $F$ levels and different age patterns for $F$; spreading the population across areas and allowing different fleets with different $F$ among the areas. An unambiguous measure of annual fishing intensity that represents the cumulative effect of all that complexity has not been defined. This problem has not been solved with SS, but some logical alternatives have been made available.\n\n\\myparagraph{Nomenclature}\nThe nomenclature below ignores sex, morphs and areas for simplicity. The quantities associated with $F$ calculations are defined as:\n\n$f$ is fleet.\n\n$t$ is a time step; continuous across years $y$ and seasons $s$; equivalent to year if only 1 season.\n\n$a$ is age.\n\n$C_{t,f}$ is fleet-specific catch in a time step.\n\n$B_{t,f}$ is fleet specific available biomass, e.g., total biomass filtered by fleet-specific age selectivity, $s_{t,f,a}$.\n\n$s_{t,f,a}$ is age-specific selectivity for a fleet. If selectivity is length-specific, then age-specific selectivity is calculated as the dot product across length bins of length selectivity and the normal (or lognormal) distribution of length-at-age. If selectivity is both length- and age-based, which is an entirely normal concept in SS, then age selectivity due to length selectivity is calculated first, then multiplied by the direct age selectivity. This compound age selectivity is used in the mortality calculations and is reported as asel2 in report file. See appendix to \\citet{methotstock2013} for more detail on this.\n\n$F_{t,f}'$ is the apical fishing mortality for a fleet. This means that it is the rate for the age that has selectivity equal to 1.0. If your model is using $F'$s as parameters, then the parameter values are for $F'$.\n\n$F_{t,f,a}$ is age and fleet-specific fishing mortality rate equal to $F_{t,f}' * s_{t,f,a}$. Note that it is possible for no age to have a selectivity equal to 1.0. In this case, $F'$ is still the rate for the hypothetical age that has selectivity equal to 1.0. The reported $F'$ values are not rescaled to be an $F$ for the age with peak selectivity. Users need to take this into account if they are comparing reported $F'$ values to reported vector of $F_{t,f,a}$ values.\n\n$\\text{ann}F_y$ is a measure of the total fishing intensity for a year, based on one of several user-specified options (see below).\n\n$F\\text{std}_y$ is a standardized measure of the total fishing intensity for a year and is reported in the derived quantities, so variance is calculated for this quantity. See below for how it relates to $annF$.\n\nTerminology and reporting of $\\text{ann}F$ and $F\\text{std}$ has been slightly revised for clarity in 3.30.15.00 and the description here follows the new conventions.\n\n\\myparagraph{$F$ Calculation}\nSS3 allows for three approaches to estimate the $F'$ that will match the input values for retained catch. Note that SS3 is calculating the $F'$ to match the retained catch conditional on the fraction of total catch that is retained, e.g., the total catch is partitioned into retained and discarded portions.\n\n\\begin{enumerate}\n\t\\item Pope’s method decays the numbers-at-age to the middle of the season, calculates a harvest rate for each fleet, $H_{t,f}$, that is the ratio of $C_{t,f}$ to $B_{t,f}$, then decays the survivors to the end of the season. the total mortality, $Z_{t,a}$, from the ratio of survivors to initial numbers, is then calculated. The $Z$ is subsequently used for in-season interpolation to get expected values for observations.\n\t\n\t\\item $F$ as parameters uses the standard Baranov catch equation and lets ADMB find the $F'$ parameter values that produce the lowest negative log-likelihood, which includes fit to the input catch data. $F$ as parameters method tends to work better than Pope’s or hybrid in high $F$ situations because it allows for some lack of fit to catch levels in early iterations and can later improve this fit as it closes in on the best solution.\n\t\n\t\\item Hybrid $F$ starts by calculating a harvest rate, $H$, using Pope’s, then converts these $H$ values, which have units of fractional harvest rate, into an approximate of $F'$ in exponential units, tuning these $F'$ values over a few iterations to get a better match to each fleet’s catch.\n\\end{enumerate}\n\nItems to note:\n\\begin{itemize}\n\t\\item SS3 includes a permutation on the $F$ as parameters method. In the first few phases, SS3 uses hybrid, then between phases it converts these directly calculated $F'$ values into parameters and proceeds in subsequent phases and MCMC to use the parameter approach. This variation on the parameter method is the recommend approach in high $F$ situations.\n\t\n\t\\item With Pope’s method, the $H$ values are fraction caught, so duration of the season does not matter. Parameter and hybrid treat $F'$ identically and multiply the $F'$ values by season duration (which has units of fraction of a year) as it is used. Each of the $F$ methods ends up with a $Z_{t,f}$ that is used for in-season interpolation.\n\\end{itemize}\n\n\\myparagraph{Relative $F$ and $F$mult}\nThe $F'$ is fleet-specific, so it is useful to have a concept of relative $F$, $\\text{rel}F_f$, among fleets. In SS, $\\text{rel}F_f= F_{t,f}'/\\sum_{f}^{}F_{t,f}'$ for a single time period $t$. In the benchmark and forecast routines, SS3 can calculate $\\text{rel}F_f$ using $F_{t,f}'$ over a range of years, or the user can input custom $\\text{rel}F$ values for benchmark and forecast in the forecast.ss file. Note that in a multi-season model setup, $\\text{rel}F_f$ is implemented as $\\text{rel}F_{s,f}$ where $s$ is the season. These get multiplied by season duration as they are used.\n\nIn the benchmark section of the code, SS3 searches for an $F$mult to achieve various management reference points (often referred to as benchmarks). In this search, SS3 calculates a benchmark $F$ as $F_{ben,f}' = F\\text{mult} * \\text{rel}F_f$, then calculates equilibrium yield and spawning biomass per recruit (SPR). SS3 searches for the $F$mult that satisfies the search conditions, first for user-specified SPR, then for user-specified spawning biomass at a management target (B\\textsubscript{TGT} or $F_{0.1}$), then for MSY. The resultant benchmark quantities are reported in the derived quantities, but $F$mult and $F_{ben,f}'$ are only reported in the Forecast\\_report.sso file. SS3 stores the benchmark $F$mult values so that user can invoke them for the forecast.\n\n\\myparagraph{Annual $F$}\nThe $\\text{ann}F$ is a single annual value across all fleets and areas according to F\\_report\\_units, which is specified by users in the starter file. If there are many fleets, across several areas and with very different selectivity patterns, $\\text{ann}F$ can have a complicated relationship to apical $F$. The F\\_report\\_units specification in the starter.ss file, see example line below, allows user to calculate it using $F'$ directly, use exploitation rate, or be derived from $Z$-at-age.\n\nExample $F$ reporting unit specification in the starter.ss file:\n\n\\begin{center}\n\t\\begin{longtable}{p{2cm} p{12cm}}\n\t\t\\hline\n\t\t5 & \\# F\\_report\\_units:\\Tstrut\\\\\n\t\t & 0 = skip; \\\\\n\t\t & 1 = exploitation(Bio); \\\\\n\t\t & 2 = exploitation(Num); \\\\ \n\t\t & 3 = sum(Frates); \\\\\n\t\t & 4 = true F for range of ages; \\\\\n\t\t & 5 = unweighted avg. F for range of ages. \\Bstrut\\\\\n\t\t\\hline\n\t\t3 7 & \\# min and max age over which average F will be calculated \\Tstrut\\Bstrut\\\\\n\t\t\\hline\n\t\\end{longtable}\n\\end{center}\n\nFor options 4 and 5 of F\\_report\\_units, the $F$ is calculated as $Z-M$ where $Z$ is calculated as $ln(N_{t+1,a+1}/N_{t,a})$, thus $Z$ subsumes the effect of $F$.\n\nThe ann$F$ is calculated for each year of the estimated time series and of the forecast. Additionally, an ann$F$ is calculated in the benchmark calculations to provide equilibrium values that have the same units as ann$F$ from the time series. In versions previous to 3.30.15, it was labeled inaccurately as $F$std in the output, not ann$F$. For example, in the Management Quantities section of derived quantities prior to 3.30.15, there is a quantity labeled Fstd\\_Btgt. This is more accurately labeled as the annual $F$ associated with the biomass target, ann\\_F\\_Btgt, in 3.30.15.\n\n\\myparagraph{$F$std}\n$F$std is a single annual value based on ann$F$ and the relationship to ann$F$ is specified by F\\_report\\_basis in the starter.ss file. The benchmark ann$F$ may be used to rescale the time series of ann$F$s to become a time series of standardized values representing the intensity of fishing, $F$std. The report basis is selected in the starter file as:\n\n\\begin{center}\n\t\\begin{longtable}{p{2cm} p{12cm}}\n\t\t%\\multicolumn{2}{l}{The starter file line:}\\\\\n\t\t\\hline\n\t\t0 & \\# F\\_report\\_basis: \\Tstrut\\\\\n\t\t& 0 = raw F report; \\\\\n\t\t& 1 = F / F\\textsubscript{SPR}; \\\\ \n\t\t& 2 = F / F\\textsubscript{MSY}; \\\\\n\t\t& 3 = F / F\\textsubscript{BTGT}.\\Bstrut\\\\\n\t\t\\hline\n\t\\end{longtable}\n\\end{center}\n\nFor example, if user selects option 1, $F$ / $F_\\text{SPR}$, the time series of ann$F$ will be divided by each value by the ann$F$ calculated in benchmark.\n\n\\myparagraph{Units for Stock Synthesis inputs related to $F$}\nBelow is a list of items to consider in terms of units for $F$ in SS:\n\\begin{itemize}\n\t\\item If F\\_ballpark is specified in the control.ss file, its units are the same as ann$F$, so is not fleet-specific.\n\t\n\t\\item $F$ as parameter values has units of fleet-specific apical $F'$.\n\t\n\t\\item In the forecast.ss file there is an option to input a vector of rel$F$ values. These are dimensionless and will be rescaled to sum to 1.0.\n\t\n\t\\item In the forecast.ss file there is an option to specify an $F$ scalar for the forecast. The units of $F$ scalar are the same as the $F$mult values calculated in benchmark. There are a full set of options for forecast $F$ scalar that can be selected in the forecast file \n\t%(-1 = none; 0 = simple; 1 = F\\textsubscript{SPR}; 2 = F\\textsubscript{MSY} 3 = F\\textsubscript{BTGT} or F\\textsubscript{0.1}; 4 = Ave F (uses first-last relative F years); and 5 = input annual F scalar). \n\tIf the forecast $F$ scalar is set as $F_\\text{SPR}$, then SS3 will use SPR\\_Fmult calculated in benchmark and reported in Forecast-report.sso. If user selects the option to input an annual $F$ scalar, option 5, then the value is input on a following line. Whichever method the user selects for forecast $F$ scalar ($F$mult), SS3 will start the forecast by creating a fleet-specific vector of apical $F$ values from $F$mult*rel$F_f$.\n\t\n\t\\item Also in the forecast.ss file, the last section of inputs allows for input of time and fleet specific apical $F_{t,f}'$ values that override the basic forecast $F$ specification described above.\n\\end{itemize}", "meta": {"hexsha": "72a0bdfbefe01c765d79b48f7228051e9e4a2234", "size": 11721, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "_f_mortality.tex", "max_stars_repo_name": "nmfs-stock-synthesis/ss-documentation", "max_stars_repo_head_hexsha": "941401b1f21582ad9d17b3b261761b307d6ea669", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-03-05T22:03:14.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-05T15:23:29.000Z", "max_issues_repo_path": "_f_mortality.tex", "max_issues_repo_name": "nmfs-stock-synthesis/ss-documentation", "max_issues_repo_head_hexsha": "941401b1f21582ad9d17b3b261761b307d6ea669", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 70, "max_issues_repo_issues_event_min_datetime": "2021-03-02T22:47:17.000Z", "max_issues_repo_issues_event_max_datetime": "2021-12-30T17:51:05.000Z", "max_forks_repo_path": "_f_mortality.tex", "max_forks_repo_name": "nmfs-stock-synthesis/ss-documentation", "max_forks_repo_head_hexsha": "941401b1f21582ad9d17b3b261761b307d6ea669", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 101.9217391304, "max_line_length": 772, "alphanum_fraction": 0.7536899582, "num_tokens": 3057, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.34411648543097656}} {"text": "% Chapter 2\n\n\\chapter{Conclusion} % Main chapter title\n\n\\label{Chapter6} % For referencing the chapter elsewhere, use \\ref{Chapter2} \n\n\nWe have reviewed some of the algorithms used for sampling a uniform spanning tree. The main impetus for going into the details of \\citet{harvey2016generating} is due to the fact that it uses the Sherman-Morrison-Woodbury identity for updating the laplacian pseudoinverse. Recently \\citet{DBLP:journals/corr/abs-2004-12739} also used the same identity for showing that \\textbf{REACHABILITY} problem is in \\texttt{DynFO} + \\texttt{Mod} $2(\\leq, +, \\times)$. Hence we explored the possibility of using the same framework for sampling spanning trees in the dynamic setting. But it turns out that there are a lot of subtleties in coming up with a proper formulation. \n\n%----------------------------------------------------------------------------------------\n\n\n%----------------------------------------------------------------------------------------\n\n\n\n\n\n", "meta": {"hexsha": "6bfdcda11b11cbc89c0fa32207f7f7fd46caee67", "size": 987, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "thesis/Chapters/Chapter6.tex", "max_stars_repo_name": "severus-tux/masters-thesis", "max_stars_repo_head_hexsha": "c6d3856cccda06735a01699c91ad923590f08ab7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "thesis/Chapters/Chapter6.tex", "max_issues_repo_name": "severus-tux/masters-thesis", "max_issues_repo_head_hexsha": "c6d3856cccda06735a01699c91ad923590f08ab7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "thesis/Chapters/Chapter6.tex", "max_forks_repo_name": "severus-tux/masters-thesis", "max_forks_repo_head_hexsha": "c6d3856cccda06735a01699c91ad923590f08ab7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-06-14T15:34:34.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-14T15:34:34.000Z", "avg_line_length": 51.9473684211, "max_line_length": 662, "alphanum_fraction": 0.6382978723, "num_tokens": 213, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.3440433010583621}} {"text": "\\documentclass[12]{scrartcl}\n\\usepackage{amssymb,amsmath,gensymb,dsfont,calc,multicol,fullpage}\n\\makeatletter\n\\newcommand\\Aboxed[1]{\n \\@Aboxed#1\\ENDDNE}\n\\def\\@Aboxed#1\\ENDDNE{%\n &\n \\settowidth\\@tempdima{$\\displaystyle#1{}$}\n \\setlength\\@tempdima{\\@tempdima+\\fboxsep+\\fboxrule}\n \\kern-\\@tempdima\n \\boxed{#1#2}\n}\n\\makeatother\n\n\\begin{document}\n\n\\title{Homework 1, Section 5.2: 21, 22, 23}\n\\author{Alex Gordon}\n\\date{\\today}\n\\maketitle\n\\section*{Homework}\n\\subsection*{21. A)}\nFalse. If a matrix is not invertible then its determinant is zero. By the invertible Matrix Theorem, then \n\\subsection*{21. B)}\nFalse. This contradicts the properties of a determinant (see theorem 3).\n\\subsection*{21. C)}\nTrue. This contradicts the properties of a determinant (see theorem 3).\n\\subsection*{21. D)}\nFalse. Example 4 proves this wrong. \n\n\\subsection*{22. A)}\nFalse. When A is a 3 x 3 matrix, det A turns out to be the volume of the parallelepiped determined by the columns of $a_1, a_2, a_3$\n\\subsection*{22. B)}\nFalse. This contradicts the properties of a determinant (see theorem 3).\n\\subsection*{22. C)}\nTrue. In general, the algebraic multiplicity of an eigenvalue is its multiplicity as a root of the characteristic equation. \n\\subsection*{22. D)}\nFalse. The slightly incorrect warning on page 277 shows this. \n\n\\subsection*{23.}\nIf $A = QR$, with Q invertible, and if $A_1 = RQ$ then $A_1 = Q^{-1}QRQ = Q^{-1} AQ$ which shows that $A_1$ is similar to A\n\n\n\n\\end{document}", "meta": {"hexsha": "2eb6bd7672a4c3263023022fa7310bda3494d80d", "size": 1480, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "LinearAlgebra/Homework28.tex", "max_stars_repo_name": "alexggordon/latex", "max_stars_repo_head_hexsha": "7dd945f33490e6585e26cff39d9cf6ad8f582a0e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "LinearAlgebra/Homework28.tex", "max_issues_repo_name": "alexggordon/latex", "max_issues_repo_head_hexsha": "7dd945f33490e6585e26cff39d9cf6ad8f582a0e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "LinearAlgebra/Homework28.tex", "max_forks_repo_name": "alexggordon/latex", "max_forks_repo_head_hexsha": "7dd945f33490e6585e26cff39d9cf6ad8f582a0e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.8888888889, "max_line_length": 132, "alphanum_fraction": 0.722972973, "num_tokens": 495, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6723316991792861, "lm_q1q2_score": 0.3440432943390066}} {"text": "\\documentclass[12pt]{article}\n\\input{physics1}\n\\newcommand{\\ap}{\\mathrm{ap}}\n\\newcommand{\\peri}{\\mathrm{peri}}\n\\begin{document}\n\n\\section*{NYU Physics I---Problem Set 12}\n\nDue Thursday 2018 November 29 at the beginning of lecture.\n\n\\paragraph{\\problemname~\\theproblem:}\\refstepcounter{problem}%\nWhat is the most expensive ingredient of a typical, traditional\nThanksgiving dinner \\emph{by weight} (that is, in dollars per ounce or\nper pound). Start with the turkey and show your work (that is,\ncompare some ingredients). Don't forget the trace (that is, small\nin quantity) ingredients!\nWhat is the relevance of all this to world\nhistory? Keep it traditional---traditional food with traditional\ningredients, like you could have cooked in 1850.\nYou might want to discuss with someone who cooked\na Thanksgiving dinner (or did the shopping for it).\n\n\\paragraph{\\problemname~\\theproblem:}\\refstepcounter{problem}%\nIf all goes well in class on 2018-11-20, we will get a quadratic equation\nfor the radii $r_\\ap$ and $r_\\peri$ of aphelion and perihelion. The argument\ngoes like this: The total energy of an orbit can be written in terms of the\nangular momentum and the radial velocity\n\\begin{eqnarray}\nE & = & \\frac{1}{2}\\,m\\,v^2 - \\frac{G\\,M\\,m}{r}\n\\\\\nE & = & \\frac{1}{2}\\,m\\,v_r^2 + \\frac{1}{2}\\,m\\,v_\\perp^2 - \\frac{G\\,M\\,m}{r}\n\\\\\nE & = & \\frac{1}{2}\\,m\\,v_r^2 + \\frac{L^2}{2\\,m\\,r^2} - \\frac{G\\,M\\,m}{r}\n\\label{foo}\n\\end{eqnarray}\nwhere $E$ is the total energy of the orbit, $v_r$ is the radial component of the velocity,\nand $L$ is the angular momentum of the orbit.\nThe radial velocity $v_r$ is exactly zero at aphelion and perihelion.\nSo set it to zero in equation~(\\ref{foo}),\nand solve the resulting quadratic equation! It will give two answers,\nwhich are $r_\\ap$ and $r_\\peri$.\nUse the definition of eccentricity\n\\begin{equation}\ne \\equiv \\frac{r_\\ap - r_\\peri}{r_\\ap + r_\\peri}\n\\end{equation}\nto figure out\nthe relationship between eccentricity $e$ of an elliptical orbit and\nthe total energy $E$ and the angular momentum $L$.\n\n\\paragraph{\\problemname~\\theproblem:}\\refstepcounter{problem}%\n\\textsl{(a)}\nSketch orbits of fixed semi-major axis but increasing\neccentricity, from a circular orbit, to one that is close to radial\n(eccentricity close to unity). Make sure you show the location of\nthe point around which the object is orbiting!\n\n\\textsl{(b)}\nWhat is the transfer time for a radial plunge orbit\nfrom the radius of the Moon's orbit down to the surface of the Earth?\nUse the period of the Moon's orbit, the relevant Kepler's law, and\nthe properties of the unit-eccentricity and circular orbits.\n\n\\textsl{(c)}\nLook up the timeline of the Apollo~11 mission, especially\nthe return to Earth. Do you see any issues there? What's your best\nexplanation of what happened?\n\n\\paragraph{\\problemname~\\theproblem:}\\refstepcounter{problem}%\n\\textsl{(a)}\nHow fast do you have to move with respect to the Earth's surface to\nescape Earth's gravity? That is, what is escape velocity from the Earth.\nCalculate it yourself in terms of the radius $R$ of the Earth and the\nvalue of $g$ at the surface. Then give it also in $\\mps$.\n\n\\textsl{(b)} A spaceship of mass $m$ resting on the surface of the\nEarth is bound to the Earth but also to the Sun. If we make\nthe naive (and close to correct) assumption that these energies just\nadd, what is the total binding energy of the spaceship in the Solar\nSystem? This calculation can be confusing, because although you can assume\nthe spaceship is stationary with respect to the Earth (so there is\nonly gravitational potential energy with respect to the Earth), the\nspaceship is moving fast relative to the Sun (so there is both\ngravitational potential and kinetic energy with respect to the\nSun). The best way to do the calculation is to just pick the Newtonian reference\nframe centered on the Sun, and compute the kinetic and potential\nenergies in that frame. Now what is the escape velocity from the Solar\nSystem? Give your answer in $\\mps$.\n\n\\textsl{(c)} Look up the derivation of how a rocket accelerates. You\nshould be able to find a rocket equation that relates the initial mass\nof the rocket+fuel, the final mass of the rocket after the fuel is\nspent, the speed at which the rocket ejects exhaust, and the final\nspeed of the rocket. (Hint: The equation is exponential in a mass\nratio.) If the rocket can eject exhaust at 10 times the speed of sound\nin air at STP (and that's optimistic!), what is the ratio of initial\nmass to final mass of a rocket that will leave the Solar System? What\nis the maximum fraction of the spaceship initial mass that can be used\nfor payload---that is, for cabin, crew, and cargo?\n\n\\textsl{(d)} Now imagine the spaceship is going to another planet just\nlike the Earth. What fraction of the spaceship can be used for\nnon-fuel payload in this case? The point is that it takes just as much\nvelocity change to slow down at the end of the journey as it took to\ntake off at the beginning, and that the end-of-flight fuel is part of\nthe cargo that the ship has to take with it at launch. You should get\nthat the payload fraction is (something like) the square of what you got in\nthe previous part.\n\n\\paragraph{Extra Problem (will not be graded for credit):}%\nWhat do you think the above problem means for interstellar travel?\n\n\\paragraph{Extra Problem (will not be graded for credit):}%\nMake a spreadsheet integration that integrates a test-particle\n(low-mass) orbit in the central force law, and show that you do indeed\nget an elliptical orbit. You want to use a time step that is $<0.001$\nof the period and go for $>1000$ timesteps if you want the integration\nto look good! This integration is harder than other integrations you\nhave done in this class, because you have to project the force onto\nthe $x$ and $y$ directions correctly. That is, you have to keep track\nof both $x$ and $y$ positions, velocities, and accelerations. Do the\nproblem in the two-dimensional plane of the orbit. If you want\nfeedback or get stuck, bring your intermediate work to Prof~Hogg for\ndiscussion.\n\n\\end{document}\n", "meta": {"hexsha": "cfdbdfd2a60a07000b0695b38a029beee0215819", "size": 6060, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/physics1_ps12.tex", "max_stars_repo_name": "davidwhogg/Physics1", "max_stars_repo_head_hexsha": "6723ce2a5088f17b13d3cd6b64c24f67b70e3bda", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-11-13T03:48:56.000Z", "max_stars_repo_stars_event_max_datetime": "2017-11-13T03:48:56.000Z", "max_issues_repo_path": "tex/physics1_ps12.tex", "max_issues_repo_name": "davidwhogg/Physics1", "max_issues_repo_head_hexsha": "6723ce2a5088f17b13d3cd6b64c24f67b70e3bda", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 29, "max_issues_repo_issues_event_min_datetime": "2016-10-07T19:48:57.000Z", "max_issues_repo_issues_event_max_datetime": "2019-01-29T22:47:25.000Z", "max_forks_repo_path": "tex/physics1_ps12.tex", "max_forks_repo_name": "davidwhogg/Physics1", "max_forks_repo_head_hexsha": "6723ce2a5088f17b13d3cd6b64c24f67b70e3bda", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.0952380952, "max_line_length": 90, "alphanum_fraction": 0.7630363036, "num_tokens": 1601, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.672331699179286, "lm_q1q2_score": 0.34404329433900654}} {"text": "\\documentclass[twocolumn,longbibliography]{quantumarticle-customized}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{graphicx}\n\\usepackage[pdfpagelabels,pdftex,bookmarks,breaklinks]{hyperref}\n\\usepackage[all]{hypcap}\n\\hypersetup{colorlinks,citecolor=blue,urlcolor=blue,linkcolor=blue}\n\n\\title{Factoring with $n+2$ clean qubits and $n-1$ dirty qubits}\n\\author{Craig Gidney}\n\\affiliation{Google, Santa Barbara, CA 93117, USA}\n\\email{craiggidney@google.com}\n\n\\def\\sectionautorefname{Section}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nWe present reversible classical circuits for performing various arithmetic operations aided by dirty ancillae (i.e. extra bits/qubits in an unknown state that must be restored before the circuit ends).\nWe improve the number of clean qubits needed to factor an $n$-bit number with Shor's algorithm \\cite{Shor1999} from $1.5n + O(1)$ \\cite{zalka2006} to $n+2$, assisted by $n-1$ dirty qubits, without increasing the asymptotic size or depth of the circuit.\n\\end{abstract}\n\n\\section{Introduction} \\label{sec:introduction}\n\nWhen constructing quantum circuits, or classical reversible circuits, an important resource is the number of available ancillae.\nAn ancilla is an extra bit or qubit that is available for use by a circuit as temporary workspace.\nAncillae may be initialized to a known computational basis state (``clean bit\"), or be given to the circuit in an unknown and potentially entangled state that must be restored before the circuit finishes (``dirty bit\").\nClean bits are more valuable, allowing for simpler and more compact circuit constructions, but dirty bits are more plentiful, since any temporarily unused bit is a borrowable dirty bit.\n\nOne part of a circuit can borrow dirty bits from another part of the same circuit, so circuit constructions that require only dirty bits are easier to apply under tight space constraints, or on circuit topologies where other ancillae are too far away to be acquired quickly.\nWhen attempting to reduce the number of bits or qubits required by a circuit, replacing constructions that use clean ancillae with ones that use dirty ancillae is a useful intermediate goal.\n\nIt is important to note that, pragmatically speaking, it is far more important to, for example, achieve low T gate counts under plausible architectural constraints than to reduce the number of clean qubits required by a circuit.\nOur goal in this paper is not to come up with an implementation of Shor's algorithm optimized in the ways that matter for plausible future quantum computer architectures.\nOur goal is to explore the consequences of picking a metric, trying to optimize it, and seeing what ideas fall out.\nThe circuit constructions we present will not be optimized to achieve good constant factors on circuit depth or gate count (though their asymptotics are fine).\nAnd they will ignore machine architecture; they assume all-to-all connectivity between qubits.\nAnd the {\\em total} number of qubits we use ($2n+1$) is higher than previous work.\nBut interesting ideas don't always come from thinking directly about a problem; there is much to be found by placing artificial goals and seeing where they lead.\nThat exploration is our intent.\nIn this paper, we reduce the number of clean qubits required to factor an $n$-bit number with Shor's algorithm from $1.5n + O(1)$ clean qubits \\cite{zalka2006} to $n+2$ clean qubits assisted by $n-1$ dirty qubits.\nWe do so without increasing the asymptotic circuit depth or size.\n\nOur paper is structured as follows.\n\\autoref{sec:introduction} introduces and describes the conventions our circuit constructions and circuit diagrams will follow.\nIn \\autoref{sec:construct} we describe all the circuit constructions we use to reduce the period finding step of Shor's algorithm into constant-sized gates, while tracking the number of required dirty ancillae.\nThen, in \\autoref{sec:costs}, we discuss the novelty and comparative costs of the presented circuit constructions.\nFinally, \\autoref{sec:conclusion} concludes with a discussion of possible future improvements.\n\nAll constructions in this paper use a two's-complement representation of integers.\nWhen a result would exceed the size of a register, it wraps (i.e. all the non-modular arithmetic constructions we discuss actually perform arithmetic modulo $2^n$, where $n$ is the size of the target register).\n\nAll diagrams order qubits from least significant bit (LSB) at the top to most significant bit (MSB) at the bottom.\n\nAll our circuit diagrams annotate operations with the number of clean and/or dirty ancillae they need.\nFor example, a dashed line from an operation down to a triangle inscribed with ``3 dirty\" means that the operation needs 3 dirty ancilla.\nIf there are 3 unused wires that the operation can borrow, then the triangle will be green and a note of ``(satisfied)\" will be written underneath.\nIf there are not enough unused wires shown in the diagram, the triangle will be yellow.\n\nThe ancillae counts shown in diagrams and discussed in the text are not optimal; they are entirely based on what the constructions we discuss in this paper achieve.\nNearly all the circuit constructions we present are classical reversible circuits (i.e. they do not use any quantum operations), so tricks such as the ancilla bootstrap shown in \\autoref{fig:bootstrap-ancilla} could cut the ancilla count in several places.\nHowever, none of these improvements decrease the ancillae required at any crucial bottleneck, and so they do not reduce the number of qubits required by our overall construction of Shor's algorithm.\n\nTo avoid ambiguity, circuit diagrams will divide multi-register operations into separate parts.\nFor example, for an addition operation that adds a register $x$ into a register $y$ (i.e. performs $|x\\rangle |y\\rangle \\rightarrow |x\\rangle |y + x \\pmod{2^n} \\rangle$), we will place a light-gray box with the text ``Input A\" over the wires corresponding to $x$ and a white box with the text ``+A\" over the wires corresponding to $y$.\nThe ``A\" symbol refers to the value of the input register, in the computational basis, at the time of the operation.\nDiagrams will often use a sequence of operations each with their own ``Input A\" box specific to that operation.\nThis reuse of ``A\" does not indicate any relation between those input values.\nThe use of ``A\" is merely a convention for indicating how the input-indicating box and the effect-indicating box are supposed to combine to form a single operation.\n\nIn the case of operations parametrized by compile-time constants, diagrams will use the letters ``K\" and/or ``R\".\nFor example, a box showing ``+K (mod R)\" refers to the modular-offset operation $|x\\rangle \\rightarrow |x+K \\pmod{R}\\rangle$.\nWe use $R$ to refer to the modulus in modular-arithmetic operations, and to the value that is factored by Shor's algorithm.\nIn several constructions we assume that $R$ is odd.\nRemoving factors of 2 from a factoring problem is trivial, so this assumption does not decrease the generality of the overall construction.\n\nBecause adding a constant into a register is more expensive than adding one register into another, we avoid ambiguity between the two operations by always referring to the operation that adds a constant as an ``offset operation\".\nWe will refer to the operation that adds one register into another as just ``addition\" or, if needed for clarity, ``enregistered addition\".\n\nWhen an operation is controlled by a wire bundle, it is conditioned on all wires in the bundle.\nWhen a single-qubit gate is applied to a wire bundle, the gate applies separately to every wire.\nFor example, when a NOT gate controlled by one wire bundle is applied to another wire bundle, every wire in the latter is toggled when and only when every wire in the former is on.\n\nFor completeness, even when the period finding reduction doesn't require a controlled version of an operation, we nevertheless provide a controllable construction that scales linearly with the number of controls.\n\nFor brevity, we take for granted that each circuit's inverse operation is decomposed into the same operations as the original operation but run in reverse order and with each sub-operation inverted.\nA decrement is a reversed increment, a multiply-add is a reversed multiply-subtract, a modular division is a reversed modular multiplication, a modular halving is a reversed modular double, and so forth.\n\nWe do not attempt to define or implement reasonable behavior for modular arithmetic circuits applied to out of range values (ones equal to or larger than the modulus).\nThis {\\em includes input values}.\nFor example, if our modular addition construction ($x \\mathrel{{+}{=}} y \\pmod{R}$) is applied to a $y \\geq R$, then we don't guarantee that $y \\mod R$ will be added into $x$, that the operation will commute with other additions, that optimizations won't change the behavior, or that anything sensible at all will happen.\nIf the precondition $y < R$ is violated, what occurs is undefined behavior \\cite{c11}.\n\n\n\\section{Constructions} \\label{sec:construct}\n\nOur reduction, from the period finding step of Shor's algorithm down to constant-sized gates, uses many circuit constructions.\n\\autoref{fig:dependencies} shows an overview of the various operations we will use, and the path through them that we will take.\n\n\\begin{figure}\n \\centering\n \\includegraphics[height=17.3cm]{assets/dependencies.png}\n \\caption{\n Graph of the transitive reduction of dependencies between constructions in our paper.\n Edge labels indicate which constructions use a dependency more than a constant number of times.\n }\n \\label{fig:dependencies}\n\\end{figure}\n\n\\subsection{Period Finding for Modular Exponentiation}\n\n\\begin{figure}\n \\centering\n \\makebox[\\linewidth]{\n \\includegraphics[width=\\linewidth]{assets/shor-period-finding.png}\n }\n \\caption{\n\tHigh-level definition of the period finding circuit \\cite{Shor1999} used by Shor's algorithm.\n\t$R$ is the modulus and the number to be factored, $B$ is a randomly chosen base, $n$ is the number of bits needed to store $R$, and $p \\in \\Theta(n)$ controls the precision of the phase estimation step.\n The triangles indicate how many ancillae are needed ``behind the scenes\", by our constructions, to perform an operation.\n\tRecovering the period requires classical post-processing of the sampled output.\n }\n \\label{fig:period-finding}\n\\end{figure}\n\nA high-level view of the circuit for period finding applied to modular exponentiation, the core quantum subroutine of Shor's quantum factoring algorithm \\cite{Shor1999}, is shown in \\autoref{fig:period-finding}.\n\nBefore the quantum circuit is constructed, a base $B$ is chosen at random.\nThe randomly chosen $B$ must be co-prime to $R$.\nWhen a $B$ that is not co-prime to $R$ is accidentally chosen, the lucky victim can instead factor $R$ by recursively factoring $R_1 = \\gcd(B, R)$ and $R_2 = R / \\gcd(B, R)$.\n\nThe circuit begins by preparing a uniform superposition $|\\psi_0\\rangle = \\sqrt{2^{-n}} \\sum_{k=0}^{2^n-1} |k\\rangle$, then uses the $\\times B^A {\\pmod R}$ operation to separate that superposition into equivalence classes modulo the unknown period $l$ of this operation.\nThat is to say, w.l.o.g. the state ends up equal to $|\\psi_{1,x}\\rangle = \\sqrt{l 2^{-n}} \\sum_{k=0}^{\\approx (2^n/l)-1} |l k + x \\rangle$ for some $x$.\nThe circuit then applies an inverse Fourier transform to the state.\nFourier transforming a uniform signal with period $l$ produces a spectrum with peaks near $N \\cdot 0/l$, $N \\cdot 1/l$, $N \\cdot 2/l$, ..., $N \\cdot (p-1)/l$.\nShor's algorithm recovers the period $l$ by sampling values $s_i$ from this spectrum, then using a continued fractions algorithm to compute the denominator of the fraction (with denominator below $R$) that is closest to $s_i/N$.\n\nBecause period-finding measures all qubits immediately after performing a QFT, most of the transformed qubits can be measured earlier than shown.\nIn fact, each qubit can be measured so early that the next qubit needed for the QFT does not even need to be initialized yet!\nOnly one of the qubits in the phase-estimation register needs to be present at a time, and so the phase register can be reduced to a single repeatedly-used qubit \\cite{zalka1998, mosca1999, parker2000, beauregard2003}.\n\\autoref{fig:period-finding-solo-phase-qubit} shows a period-finding circuit with this property.\nIt uses (a) controlled modular multiplication, (b) measurement, (c) X-axis rotations classically parametrized by previous measurements, and (d) qubit resets.\nThe only non-trivial operation is (a), the controlled modular multiplication of an $n$-qubit register.\n\nWe perform modular multiplication with modular scaled-addition operations and an ancilla register as in \\cite{beauregard2003}.\nHowever, to allow our ancilla register to be mostly dirty, we extend the operation so that it has a well defined effect on the second register: multiplying by the inverse factor.\nWe will refer to this combined operation as a ``bimultiply\".\n\nThe bimultiplications we perform throughout the algorithm do change the value of the ancilla register, but after the usual end of the circuit we can undo the damage.\nThe key insight is that the work register is initialized to $|1\\rangle$ and gets multiplied by constants inverse to the ones trashing the dirty ancilla register.\nInstead of discarding the work register after performing phase estimation, as is normally done, we have a use for it.\nWe measure the work register to recover the fixup factor needed to restore the dirty register's original value.\n\\autoref{fig:period-finding-solo-phase-qubit-explicit-dirty-register} shows this construction.\n\nBecause our modular circuit constructions would have undefined behavior if any register had a value equal to or larger than $R$, we need the ancilla register to contain a value less than $R$.\nWe ensure this by requiring that the ancilla register's MSB be $|0\\rangle$ (i.e. clean) and that the registers be as small as possible (i.e. have size $n = \\lceil \\lg_2(R) \\rceil$).\n\n\\begin{figure*}\n \\centering\n \\makebox[\\linewidth]{\n \\includegraphics[width=\\linewidth]{assets/shor-period-finding-solo-phase-qubit.png}\n }\n \\caption{\n\tPeriod finding with a single phase-estimation qubit \\cite{beauregard2003}.\n\tThe small oplus' ({\\tiny $\\oplus$}) are ``X-axis controls\".\n\tAn X-axis control is equivalent to a normal control, but with a Hadamard gate applied before and after.\n\tIt conditions on the state $\\frac{1}{\\sqrt 2}|0\\rangle - \\frac{1}{\\sqrt 2}|1\\rangle$ instead of on the state $|1\\rangle$.\n }\n \\label{fig:period-finding-solo-phase-qubit}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\makebox[\\linewidth]{\n \\includegraphics[width=\\linewidth]{assets/shor-period-finding-solo-phase-qubit-double-register.png}\n }\n \\caption{\n\tPeriod finding with a single phase-estimation qubit and paired inverse multiplications (``bimultiplications\").\n\tUses $O(n^3 \\lg n)$ gates, $O(n^3)$ depth, and no additional qubits beyond those shown.\n }\n \\label{fig:period-finding-solo-phase-qubit-explicit-dirty-register}\n\\end{figure*}\n\n\n\\subsection{Modular Bimultiplication}\n\nAs in \\cite{beauregard2003}, we perform controlled modular multiplication with a second register and modular scaled-addition operations.\nHowever, to allow the second register to be dirty, we use an additional scale-add as well as a negation operation as in \\cite{zalka2006}.\nSee \\autoref{fig:controlled-modular-multiply} for the circuit diagram.\n\nTo show that the circuit works, suppose that the two registers start in the state $(x, y)$, working modulo $R$.\nWe scale-add the first register times $K$ into the second register to produce the state $(x, y+xK)$.\nThen a scale-subtract times $K^{-1}$ out of the first register puts the system into the state $(x-yK^{-1}-xKK^{-1}, y+xK)$, which is just $(-yK^{-1}, y+xK)$.\nNext, we cancel the $y$ term in the second register by scale-adding the first register times $K$ into it again, leaving $(-yK^{-1}, y+xK-yK^{-1} \\cdot K)$ which is simply $(-yK^{-1}, xK)$.\nFinally, we swap the terms and negate the second term to get $(xK, yK^{-1})$ as desired.\n\nIn the case where $K$ has no multiplicative inverse modulo $R$, this construction will not work (it would define an invalid irreversible operation).\nHowever, that would mean $K$ is a factor of $R$; a case that can be checked for and handled classically before bothering to factor $R$ with a quantum computer.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/controlled-modular-multiply.png}\n \\caption{\n Controlled modular bimultiplication, with a constant multiplier $K$, using three modular scaled-additions and a swap.\n $K$ must have a multiplicative inverse modulo $R$.\n Uses no ancillae, $O(c + n^2 \\lg n)$ gates, and $O(c + n^2)$ depth where $c$ is the number of controls and $n$ is the register size.\n The negation operation requires 2 dirty ancillae, but notice there are free wires in the other register that can be borrowed, so the overall construction requires no additional dirty ancilla (for non-trivial $n$).\n }\n \\label{fig:controlled-modular-multiply}\n\\end{figure}\n\n\n\\subsection{Modular Scaled-Addition}\n\nTo perform modular scale-add operations, we use a shift-and-add approach similar to \\cite{beauregard2003}.\nWe right-shift (i.e. divide by $2 {\\pmod R}$) the target register $n-1$ times, then begin iteratively left-shifting (i.e. multiplying by $2 {\\pmod R}$) and adding $K$ into the target.\nWe condition the first modular addition on the most significant bit of the input, the second addition on the next most significant bit, and so forth.\n(The more significant bits go first because their effects must be hit by more left-shifts.)\nSee \\autoref{fig:controlled-modular-scale-accumulate}.\n\nThe conditional offset and modular doubling operations need a dirty bit, but for non-trivial $n$ there are more than enough unused bits available to borrow, so the circuit as a whole doesn't require any dirty bits.\n\nNote that the modular doublings (and halvings) require $R$ to be odd, since otherwise the operation would be irreversible.\nHowever, given that in the intended use case $R$ is a number to be factored, it is reasonable to require callers to have factored out multiples of two beforehand.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/controlled-modular-multiply-accumulate.png}\n \\caption{\n Reducing a controlled modular scale-add, with a constant multiplier $K$, into the modular equivalent of shift-and-add.\n Requires the modulus $R$ to be odd.\n Uses no ancillae, $O(c + n^2 \\lg n)$ gates, and $O(c + n^2)$ depth where $c$ is the number of controls and $n$ is the register size.\n }\n \\label{fig:controlled-modular-scale-accumulate}\n\\end{figure}\n\n\n\\subsection{Modular Doubling}\n\nTo multiply a register by 2 modulo an odd $R$, we note that the permutation this operation performs is to take the values from $[(R+1)/2, R)$, the top-half of the valid range, and interleave them between the values from $[0, (R+1)/2)$, the bottom-half of the valid range.\n\nWe can perform this interleaving by moving the valid-top-half up until it starts at $2^{n-1}$, i.e. aligns with the MSB boundary.\nA left-rotate of the register bits then moves the MSB to the LSB, converting the alignment into interleaving.\n\nBecause it's expensive to perform an offset that affects values above $R/2$ while leaving values below $R/2$ alone, the circuit shown in \\autoref{fig:modular-double} uses a different alignment strategy.\nIt starts by offset-ing the whole range down by $\\lceil R/2 \\rceil$, which wraps the valid-bottom-half of the input around to the top of the register's range.\nThis offset also aligns what was originally the valid-top-half against the bottom of the register range.\nThen the circuit uses a controlled offset to move just the top half of the register (including the invalid range above $R$) up by $\\lceil R/2 \\rceil$, re-wrapping what was originally the valid-bottom-half past the top and around to the middle, aligning it with the MSB boundary.\nThe circuit then toggles the MSB, fixing the fact that, although the valid-top-half is aligned with 0 and the valid-bottom-half is aligned with $2^{n-1}$, that's exactly the reverse of what we want.\nWith the desired alignment achieved, the circuit interleaves the two halves with a left-rotate.\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/controlled-modular-double.png}\n \\caption{\n Controlled modular doubling.\n $R$ must be odd.\n Uses 1 dirty ancilla, $O(c + n \\lg n)$ gates, and $O(c + n)$ depth, where $c$ is the number of controls and $n$ is the register size.\n }\n \\label{fig:modular-double}\n\\end{figure}\n\n\n\\subsection{Pivot-Flips} \\label{sec:pivot-flips}\n\nWe will implement both modular negation and modular offset/addition in terms of a non-standard operation we call a ``pivot-flip\".\nA pivot-flip is an operation that reverses the order of states less than a given pivot value, without affecting other states.\nFor example, a pivot-flip with the pivot equal to 4 would swap $|0\\rangle$ and $|3\\rangle$, swap $|1\\rangle$ and $|2\\rangle$, and leave all other states untouched.\n\nThe exact permutation performed by a pivot-flip with pivot equal to $K$ is:\n\n$$\\text{PivotFlip}_K = \\sum_{i=0}^{K-1} |K-i-1\\rangle \\langle i| + \\sum_{i=K}^{N-1} |i\\rangle \\langle i|$$\n\nTo perform a pivot-flip efficiently, we use the fact that $x \\rightarrow \\lnot(x - K)$ nearly does what is required: it flips the range below $K$ but unfortunately also flips the range above-and-including $K$.\nIt's a ``bi-flip\".\n\nThe bi-flip operation is its own inverse, so it can be ``toggle-controlled''.\nIf we apply two bi-flips controlled by the same control qubit, then toggling the control qubit determines whether or not a bi-flip is applied to the target register.\nWhen the control qubit stays off, neither bi-flip fires and nothing happens to the target.\nWhen the control qubit stays on, both bi-flips fire and they undo each other.\nBut if the control qubit is toggled after the first bi-flip but before the second, then exactly one of them will fire and the target register ends up bi-flipped.\n\nAnother useful property of a bi-flip is that it doesn't move any states across the pivot.\nA bi-flip at $K$ preserves $x n$ is the size of the target register.\n }\n \\label{fig:controlled-addition}\n\\end{figure}\n\n\n\\subsection{Increment}\n\nA register can be incremented by subtracting both $x$ and $\\neg x = -x-1$ from it, for any $x$.\nWhen $n$ dirty bits are available, $x$ can come from a register defined by those $n$ arbitrary bits, as shown in \\autoref{fig:increment-many-dirty}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/increment-many-dirty.png}\n \\caption{ Subtracting $x$ and $-x-1$ from a register increments it. Requires $O(n)$ depth and size, and $n$ dirty ancillae.}\n \\label{fig:increment-many-dirty}\n\\end{figure}\n\nTo improve from $n$ dirty bits to the single dirty bit used by the circuit in \\autoref{fig:controlled-increment-odd}, we break the register into two halves.\nA high-half that is incremented only if all of the low bits are on, and a low-half that is unconditionally incremented.\n\nIf the increment is uncontrolled, the low-half can be incremented with the double-subtraction trick by borrowing the high-half.\nWhen there are controls, we instead increment the low-half using a commutator-control construction where one operation subtracts the ancilla register out of the target and the other applies a NOT gate to every input wire and target wire.\nThis works because $\\lnot (\\lnot T + \\lnot K) - K = \\lnot (-T - 1 - K - 1) - K = (T+2+K) - 1 - K = T + 1$.\n\nThe high-half is trickier to deal with.\nWe want to borrow the low-half for the double-subtraction trick, but the low-half is being used as a control and so can't be borrowed.\nTo work around not being able to operate on the borrowed low-half bits while using them as a control, we use more commutator control tricks and some knowledge of what state the low-half must be in if operations it is controlling are firing.\n\nWe add and subtract the low-half out of the high-half, but frame the addition with NOT gates controlled by all bits in the low half.\nWhen any of the bits in the low-half are off, the NOT gates don't fire and the addition and subtraction will cancel each other.\nWhen all of the bits in the low-half are on, i.e. when the low-half is storing the two's complement representation of -1, the NOT gates do fire.\nThis inverts the addition into a subtraction, and the low-half (which is storing -1) is subtracted out of the high-half twice.\nTherefore the high-half was incremented by 2.\nTo halve the +2 into a +1, we prepend a dirty LSB onto the target register.\n\nRecall that, earlier in the paper, we used increments and decrements to implement addition where the target was larger than the input.\nTo avoid an expensive cyclic dependency in our increment construction, the additions we use must not be larger-target additions.\n(Also, when the two registers are not the same size, they define arithmetic modulo different powers of 2.\nThis breaks properties such as $T + \\lnot K = T - K - 1$.)\n\nBecause we can only perform additions and subtractions with input and target registers of the same size, the construction described so far in this section only works for odd-sized registers.\nTo handle even-sized registers, we decompose the increment into a controlled-increment and a NOT gate.\nWe use the LSB as a control determining whether the (odd-sized) rest of the register is incremented, and then toggle the LSB.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/controlled-increment-odd.png}\n \\caption{\n Odd-sized controlled increment.\n For the even-sized case, separate the LSB from the rest of the register, increment the rest of the register using the LSB as an extra control, then toggle the LSB.\n Uses $O(c+n)$ gates, $O(c+n)$ depth, and 1 dirty ancilla.\n }\n \\label{fig:controlled-increment-odd}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/ancilla-bootstrap.png}\n \\caption{ Bootstrapping a dirty ancilla out of an increment gate using quantum operations.\n The $i^{\\text{frac}}$ gate is a ``phase gradient\" operation that phases each computational basis state $|v\\rangle$ by an amount proportional to $v/2^d$, where $d$ is the size of the register.\n In this case each state is phased by $e^{i \\frac{\\pi}{2} v/2^d}$.\n The phase gradient is implemented by a column of $Z^{2^{-k}}$ gates.}\n \\label{fig:bootstrap-ancilla}\n\\end{figure}\n\n\n\\subsection{Bit Swaps, Rotations, and Reversals}\n\nBit permuting operations can usually be emulated by re-labelling qubits, so they are easy to overlook in circuits.\nBut some of our circuit diagrams have used controlled bit rotations which require actual gates.\nWe provide the relevant constructions in \\autoref{fig:bit-rotate} and \\autoref{fig:bit-reverse}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/controlled-bit-rotate.png}\n \\caption{\n A controlled bit rotation / bit swap is three controlled bit-reverses.\n Uses no ancillae, $O(c + n)$ gates, and $O(c + n)$ depth.\n }\n \\label{fig:bit-rotate}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/controlled-reverse.png}\n \\caption{\n Inline controlled bit order reversal on an odd-sized register of size $n=2k+1$, and on an even-sized register of size $n=2k+2$.\n Each XOR operation is a series of independent CNOTs (note that the inputs have opposite endian-ness to the outputs).\n Uses no ancillae, $O(c + n)$ gates, and $O(c + n)$ depth.\n }\n \\label{fig:bit-reverse}\n\\end{figure}\n\n\n\\subsection{Multi-Nots}\n\nSeveral of our constructions have used CNOTs applied to wire bundles, with many controls and many targets.\nA naive approach to implementing these operations would be to apply a separate NOT, each controlled by every control, to every target.\nBut this would use $O(n \\cdot c)$ gates, which is not linear in the number of controls.\n\nTo avoid paying the overhead of $c$ controls for every target, we toggle {\\em one} target conditioned on all of the controls.\nWe then use toggle-controlling to spread the toggling effect to all of the other targets.\n\nTo efficiently reduce the single remaining CNOT with $c$ controls into constant-sized Toffoli gates, we take advantage of the many available dirty ancilla.\nSee \\autoref{fig:multi-not} and \\autoref{fig:cnot-reduction}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/multi-cnot-to-single-cnots.png}\n \\caption{\n Reducing a many-control many-target CNOT into one many-control single-target CNOT and many single-target single-control CNOTs.\n Uses no ancillae, $O(c + n)$ gates, and $O(c + n)$ depth where $n$ is the number of targets and $c$ is the number of controls.\n The depth can be reduced to $O(c + \\lg n)$ by spreading the toggling effect more intelligently.\n }\n \\label{fig:multi-not}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{assets/cnot-reduction.png}\n \\caption{\n When $c-2$ dirty ancillae are available, a controlled-not with $c$ controls can be reduced into $4c - 8$ Toffoli gates \\cite{barenco1995}.\n It is possible to reduce the number of ancilla needed to just one, by using twice as many Toffoli gates, but that construction isn't needed in the context of period finding because there are always enough unused qubits to borrow.\n }\n \\label{fig:cnot-reduction}\n\\end{figure}\n\n\\section{Overview and Improvements} \\label{sec:costs}\n\n\\begin{figure*}\n \\centering\n \\makebox[\\linewidth]{\n \\begin{tabular}{ l | c c c c c l }\n & Year & Depth & Gates & Clean Qubits & Total Qubits \\\\\n \\hline\n Shor \\cite{Shor1999} & 1994 & $\\Theta(n M(n))$ & $\\Theta(n M(n))$ & $\\Theta(n)$ & $\\Theta(n)$ \\\\\n Beckman et al. \\cite{beckman1996} & 1996 & $\\Theta(n^3)$ & $\\Theta(n^3)$ & $5n + 1$ & $5n + 1$ \\\\\n Veldral et al. \\cite{vedral1996} & 1996 & $\\Theta(n^3)$ & $\\Theta(n^3)$ & $4n + 3$ & $4n + 3$ \\\\\n Beauregard \\cite{beauregard2003} & 2003 & $\\Theta(n^3 \\lg \\frac{1}{\\varepsilon})$ & $\\Theta(n^3 \\lg \\frac{n}{\\varepsilon} \\lg \\frac{1}{\\varepsilon})$ & $2n+3$ & $2n+3$ \\\\\n Takahashi et al. \\cite{takahashi2006} & 2006 & $\\Theta(n^3 \\lg \\frac{1}{\\varepsilon})$ & $\\Theta(n^3 \\lg \\frac{n}{\\varepsilon} \\lg \\frac{1}{\\varepsilon})$ & $2n+2$ & $2n+2$ \\\\\n Zalka \\cite{zalka2006} & 2006 & $\\Theta(n^3 \\lg \\frac{1}{\\varepsilon})$ & $\\Theta(n^3 \\lg \\frac{n}{\\varepsilon} \\lg \\frac{1}{\\varepsilon})$ & $1.5n+O(1)$ & $1.5n+O(1)$ \\\\\n H\\\"{a}ner et al. \\cite{haner2016} & 2016 & $\\Theta(n^3)$ & $\\Theta(n^3 \\lg n)$ & $2n+2$ & $2n+2$ \\\\\n (ours) & 2017 & $\\Theta(n^3)$ & $\\Theta(n^3 \\lg n)$ & $n+2$ & $2n+1$ \\\\\n \\end{tabular}\n }\n \\caption{\n Space-efficient constructions of Shor's algorithm over time.\n The table only includes papers that presented explicit circuit constructions and improved the minimum number of qubits required to perform the algorithm (or else the time complexity at a slightly larger qubit count).\n $M(n)$ is the classical time-complexity of multiplication, which is known to be asymptotically at most $n \\cdot (\\lg n) \\cdot 2^{O(\\lg^* n)}$ \\cite{furer2007}.\n $\\varepsilon$ is the maximum error when synthesizing the circuit out of a fixed set of universal gates, which is asymptotically relevant for algorithms that use Draper addition \\cite{draper2000} (i.e. applying phase gradients in frequency space) instead of Toffoli-based addition/offset constructions.\n }\n \\label{fig:table-over-time}\n\\end{figure*}\n\n\nRecall that \\autoref{fig:dependencies} shows a dependency graph of the constructions discussed in this paper.\nOur asymptotic costs are dominated by performing $O(n)$ modular multiplications, each of which uses $O(n)$ modular additions and offsets, each of which uses $O(n \\lg n)$ constant-sized classical gates \\cite{haner2016} and $O(n)$ depth.\nThe total cost of the period-finding step in Shor's algorithm, using our construction, is $O(n^3 \\lg n)$ gates and $O(n^3)$ depth.\n\nIn \\autoref{fig:table-over-time} we show how the number of qubits needed for Shor's algorithm has improved over time.\nOur main improvements over previous arithmetic constructions are 1) the use of pivot-flips for modular addition, 2) the use of dirty bimultiplication for modular multiplication, and 3) the $O(n)$ incrementer requiring only a single dirty ancilla.\n\nPrevious modular addition constructions worked by temporarily storing an is-wraparound-needed comparison in a clean ancilla \\cite{takahashi2006, haner2016}.\nPivot flips also require ancillae, but the ancillae can be dirty and, in the context of Shor's algorithm, there are always qubits available to borrow whenever a pivot-flip is needed.\nThis improvement ends up saving a qubit, reducing the total number of qubits we would have needed from $2n+2$ as in \\cite{haner2016} to $2n+1$.\n\nPrevious modular multiplication constructions did not work with a dirty ancilla register.\nThey either required a clean ancilla register because they leaked the ancilla register into the work register \\cite{haner2016}, or else a trashable ancilla register because there was no way to undo the damage being done by the multiplications \\cite{zalka2006}.\nThe leakage problem is fixed by adding an extra scale-add operation and a negation operation \\cite{zalka2006}.\nWe fixed the trashing problem by measuring the work register at the end of the circuit, and using its value to drive a clean-up multiplication that restores the ancilla register.\nThis improvement allows $n-1$ of the qubits in the ancilla register to be dirty.\nThe MSB has to stay clean to ensure the register's value is less than the modulus being factored.\n\nPrevious published incrementers (not counting an unpublished version of our construction being cited by \\cite{haner2016}) required either $O(n^2)$ gates or $\\omega(1)$ ancillae \\cite{draper2000, barenco1995}.\nOur classical incrementer construction uses $O(n)$ gates and a single dirty ancilla.\n\nNote that, for classical reversible computation, 1 dirty ancilla is the minimum possible for an incrementer.\nAn increment operation on $n$ bits is equivalent to performing a state-permutation that uses $2^n-1$ swaps to sweep the state $|2^n-1\\rangle$ from the top of the state-space to the bottom of the state-space one step at a time.\nNote that $2^n-1$ is odd for non-trivial $n$, and therefore the parity of the state permutation performed by an increment operation is odd.\nHowever, the parity of the permutation performed by any classical gate that doesn't cover the entire circuit is even.\nSince odd-parity permutations can't be implemented by composing even-parity permutations, it is impossible to compose smaller operations into an increment operation that covers every bit.\nAn uncovered bit (i.e. a dirty ancilla) must be present.\n\nWhen quantum operations are available, the parity barrier can be bypassed and incrementing can be performed without any ancilla.\nThe asymptotic cost is slightly worse due to the cost of synthesizing tiny rotations, and the constant factors hidden by the asymptotic notation are also worse.\nSee \\autoref{fig:bootstrap-ancilla} for the construction.\n\nWe used \\href{https://github.com/Strilanc/Quirk}{Quirk} \\cite{quirk2016} to explore \\cite{victor2013}, check, and refine individual constructions.\nThe full construction down to Toffoli gates was tested in \\href{https://github.com/ProjectQ-Framework/ProjectQ}{ProjectQ} \\cite{projq2016}.\nThe python source code for our test implementation, and an issue tracker for submitting errata, can be found online at \\href{https://github.com/Strilanc/PaperImpl-2017-DirtyPeriodFinding}{https://github.com/Strilanc/PaperImpl-2017-DirtyPeriodFinding}.\n\nAlthough the asymptotic costs of our constructions match previous work, the constant factors are significantly worse.\nFor example, our equivalent of a controlled 32-qubit multiplication uses roughly 1.3 million Toffoli gates (the specific number depends on the factor to multiply by, the modulus, whether or not a number of optimizations are used, and whether or not those optimizations apply to a given case).\nThis cost is over an order of magnitude worse than previous work \\cite{haner2016}.\nThe reason for the increased cost is that using dirty ancillae results in more uncomputation work, in repeating some operations twice conditionally to do them once unconditionally, and in less efficient constructions in general.\nThese inefficiencies stack multiplicatively when one construction uses another as a subroutine.\n\n\n\\section{Future Work and Conclusion} \\label{sec:conclusion}\n\nIn this paper we described how to perform various arithmetic operations using only dirty ancilla.\nWe also described various techniques we used to find the constructions: pivot flipping, toggle controlling, commutator controlling, and exploration by directly manipulating circuit diagrams.\nWe showed how these constructions reduce the number of clean qubits required to perform Shor's algorithm from $1.5n+O(1)$ to $n+2$.\n\nOf the $2n+1$ qubits our period-finding construction requires, $1$ is used for the phase estimation qubit, $n$ are used to store the work register, and $n$ are ancillae used to implement modular multiplication in terms of scaled modular addition.\nExcept for multiplication, all our arithmetic constructions use two or fewer dirty ancillae.\nSince we reduce modular scale-addition into operations that have a $\\Theta(n)$ surplus of unused qubits, in context there are always more than enough dirty ancillae available to implement the simpler arithmetic inline.\nBecause there is slack in the simpler arithmetic constructions, improving the overall number of qubits used by Shor's algorithm requires only that we improve the first step of reducing modular multiplication into some other operation.\n\nOne way we could reduce modular multiplication into better operations is by knowing the factorization $p \\cdot q$ of the modulus $R$, with $p$ and $q$ each having size $\\approx n/2$.\nThe Chinese remainder theorem guarantees that $x \\mod R$ could be uniquely represented as the pair of half-sized values $(x \\mod p, x \\mod q)$.\nMultiplications of the half-sized values could use and reuse the same half-sized ancilla register, for a total of three half-sized registers (one for $x \\mod p$, one for $x \\mod q$, and one hopefully-dirty ancilla register).\nDespite the large reduction in the number of ancillae available to the underlying arithmetic operations, each operation would still have more than enough ancillae and contribute no additional ancillae to the overall circuit.\n(Note: because the initial value $x_0 = 1$ is trivial, and the final value is measured or discarded, it's not necessary to implement circuits that translate between the Chinese-remainder representation and the usual 2s-complement representation.)\nOf course, since we use Shor's algorithm to compute the factorization of $R$, using the factorization of $R$ to optimize Shor's algorithm would be paradoxical.\nThe $1.5n$ achieved in \\cite{zalka2006} was done via this kind of method, except that instead of factoring $R$ Zalka factored the value to multiply by and the factoring was done modulo $R$ (by stopping a generalized gcd at a midway point) instead of in $\\mathbb{Z}$.\n\nA second way to reduce modular multiplication into (slightly) smaller operations could be with commutator control.\nNote that the modular Fourier transform inverts the effect of modular multiplication: $QFT_{R} \\cdot (\\times K \\,\\text{mod}\\, R) \\cdot QFT_R^{-1} = (\\times K^{-1} \\,\\text{mod}\\, R)$.\nThis has two useful high-level effects.\nFirst, by applying the bimultiplication gate used in this paper twice, but inverting the effect on the ancilla register for the second application, we can build a proper modular multiplication gate that only affects a single register.\nSecond, we can use commutator control to move controls from the modular multiplication gate onto modular Fourier transform gates.\nAssuming controlled modular QFTs can be performed with fewer ancillae than controlled modular multiplication, we could save a qubit when performing the modular multiplication by borrowing the control qubit.\n\nPossible future improvements aside, in this paper we showed that the number of clean qubits sufficient to perform Shor's algorithm is no more than $n + O(1)$.\nWe consider this to be a step towards a construction for Shor's algorithm that uses only $n + O(1)$ qubits total.\nWe do note, however, that these improvements are of more theoretical than practical interest.\nIn practice, on error corrected quantum computers, the important cost to optimize is the number of T gates.\nIn that sense, the main {\\em practical} contribution of this paper is the improved increment circuit and the demonstration of constructing circuits under tight space constraints.\n\n\n\\section{Acknowledgements}\n\nWe thank Matthew Neeley, Dave Bacon, and Austin Fowler for comments on earlier versions of this paper, from which it greatly benefited.\n\n\n\\bibliographystyle{plain}\n\\bibliography{citations}\n\n\\end{document}\n", "meta": {"hexsha": "3a6087d1b0819c259f55781cca19925cb33e75e8", "size": 51348, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/paper.tex", "max_stars_repo_name": "Strilanc/PaperImpl-PeriodFindCutQubits", "max_stars_repo_head_hexsha": "402d216e5c1641f878153dc604792f529bcb79f4", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2017-09-05T13:37:05.000Z", "max_stars_repo_stars_event_max_datetime": "2021-04-01T01:10:35.000Z", "max_issues_repo_path": "doc/paper.tex", "max_issues_repo_name": "Strilanc/PaperImpl-PeriodFindCutQubits", "max_issues_repo_head_hexsha": "402d216e5c1641f878153dc604792f529bcb79f4", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-09-16T21:11:25.000Z", "max_issues_repo_issues_event_max_datetime": "2017-09-19T22:20:58.000Z", "max_forks_repo_path": "doc/paper.tex", "max_forks_repo_name": "Strilanc/PaperImpl-PeriodFindCutQubits", "max_forks_repo_head_hexsha": "402d216e5c1641f878153dc604792f529bcb79f4", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2017-09-27T20:07:34.000Z", "max_forks_repo_forks_event_max_datetime": "2017-09-27T20:07:34.000Z", "avg_line_length": 71.9159663866, "max_line_length": 335, "alphanum_fraction": 0.7616654982, "num_tokens": 12857, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34404329433900654}} {"text": "\\chapter[Ionisation]{Ionisation}\n \n\n The class G4eIonisation calculates the continuous energy loss\n due to ionisation and\n simulates the 'discrete' part of the ionisation , the Moller and Bhabha\nscattering or delta ray production by $e^-/e^+$ . The formulae used here\n are the same than those in GEANT3 (\\cite{eion.geant3}).\n\n\\section{Method}\n\nLet: \\[\\frac{d\\sigma(E,T)}{dT}\\]\nbe the differential cross-section\nfor the ejection of an electron with kinetic energy $T$ by an incident\n${e^{\\pm}}$ of total energy $E$ moving in a medium of density $\\rho$.\n Let us denote the value of the {\\em kinetic energy cut-off} or\n {\\em production threshold for $\\delta$-rays} by $T_{cut}$.\nBelow this threshold the soft\nelectrons ejected are simulated as continuous energy loss by the incident\n$e^+/e^-$, and above it they are explicitly generated.\n\nThe mean value of the energy lost by the incident ${\\rm e^{\\pm}}$ to\nthe soft $\\delta$-rays is:\n\\begin{equation}\nE_{soft}(E,T_{cut})\n= \\int_{0}^{T_{cut}} \\frac{d \\sigma (E,T)}{dT} T \\: dT \\label{eion.a}\n\\end{equation}\nwhereas the total cross-section for the ejection of\nan electron of energy \\linebreak $T > T_{cut}$ is:\n\\begin{equation}\n\\sigma (E,T_{cut})\n= \\int_{T_{cut}}^{T_{Max}}\\frac{d \\sigma (E,T)} {dT} \\: dT \\label{eion.b}\n\\end{equation}\nwhere $T_{Max}$ is the maximum energy transferable to the free electron:\n\\begin{equation}\n\\label{eion.c}\nT_{Max} = \\left\\{ \\begin{array}{ll}\n E-m & {for \\hspace{.2cm} e^+} \\\\\n (E-m)/2 & {for \\hspace{.2cm} e^- ,} \\\\\n \\end{array} \\right .\n\\end{equation}\nwhere $m$ is the electron mass.\nThe method of calculation of the continuous energy loss\nand the total cross-section are\nexplained below. \n\n\\section{Continuous energy loss}\nThe integration of \\ref{eion.a} leads to the Berger-Seltzer\nformulae ( see e.g in \n \\cite{eion.messel}) :\n\n\\begin{equation}\n\\label{eion.d}\n\\frac {dE}{dx} = \\frac{2 \\pi r_0^ 2 mn }{\\beta^2}\n \\left [\\ln \\frac{2(\\tau + 2)} {(I/m)^2}+ F^{\\pm} (\\tau , \\Delta )\n- \\delta \\right ],\n\\end{equation}\n where\n\n\\[\n\\begin{array}{ll}\n\\gamma & \\frac{E}{m} \\\\\n\\beta^2 & 1-\\frac{1}{\\gamma^2} \\\\\n\\tau & \\gamma-1 \\\\\n\\tau_c & \\frac{T_{cut}}{m} \\\\\n T_{cut} & \\mbox{energy cut for} \\: e^{\\pm} \\\\\n\\tau_{max} & \\mbox{maximum possible energy transfer in $e^-$ mass:\n $\\tau$ for $e^+$, $\\tau/2$ for $e^-$} \\\\\n\\Delta & \\min(\\tau_c,\\tau_{max}) \\\\\nn & \\mbox{electron density of the medium} \\\\\nI & \\mbox{average mean ionisation energy} \\\\\n\\delta & \\mbox{density effect correction}.\n\\end{array}\n\\]\n\nThe functions $ F^{\\pm}$ are given by\n\\begin{equation}\n\\label{eion.e}\nF^+ (\\tau,\\Delta) =\\ln(\\tau\\Delta ) -\n\\frac{\\Delta^2}{\\tau}\\left[\\tau + 2 \\Delta -\n\\frac{3\\Delta^2 y } {2} -\\left(\\Delta - \\frac{\\Delta^3 }{3} \\right) y^2 \n- \\left (\\frac{\\Delta^2}{2} - \\tau\n \\frac{\\Delta^3}{3} + \\frac{\\Delta^4 } {4} \\right)\n y^3 \\right] \n\\end{equation}\n\n\\begin{equation}\n\\label{eion.f}\nF^- (\\tau,\\Delta ) = -1 -\\beta^2 +\\ln \\left [(\\tau - \\Delta)\n\\Delta \\right ] + \\frac{\\tau}{\\tau -\\Delta}+\\frac{\\left [\n\\frac{\\Delta^2}{2} + ( 2\\tau +1) \\ln\n\\left (1- \\frac{\\Delta}{\\tau} \\right ) \\right ]}{\\gamma^2}\n\\end{equation}\n\nwhere $ y = 1/(\\gamma+1) $.\n\nThe density effect correction is calculated according to the\n formalism of Sternheimer ( \\cite{eion.sternheimer}). \n\n\\section{Total cross-sections}\nThe integration of formula \\ref{eion.b} gives the total cross-section\n (\\cite{eion.messel}) for M\\\"{o}ller scattering ($e^- e^-$):\n\\begin{equation}\n\\label{eion.g}\n\\sigma ( Z,E,T_{cut} ) =\\frac {2 \\pi r_0^2 mZ}{\\beta^2(E-m)}\n \\left[\\frac{(\\gamma-1)} {\\gamma^2}\\left(\\frac{1}{x}-1\\right)\n +\\frac{1}{x}-\\frac{1}{1-x}-\\frac{2\\gamma-1}{\\gamma^2}\\ln\n\\frac{1-x}{x}\\right]\n\\end{equation}\nand for Bhabha scattering ($e^+ e^-$) :\n\\begin{equation}\n\\label{eion.h}\n\\sigma (Z,E,T_{cut}) =\\frac{ 2 \\pi r^2_0 mZ }\n {(E-m)}\\left [\\frac {1 }{\\beta^2} \\left(\\frac{1}{x}-1\\right)\n + B_1 x + B_2 (1-x) -\n \\frac {B_3 } {2} ( 1-x^2 ) +\\frac{B_4}{3}(1-x^3)\\right]\n\\end{equation}\nwhere\n\n\\[\n\\begin{array}{ll}\n\\gamma = \\frac{E}{m} & \\beta^2 = 1-\\frac{1}{\\gamma^2} \\\\ [.2cm]\n x =\\frac {T_{cut}}{E-m} & \\gamma=\\frac{1}{\\gamma + 1} \\\\ [.2cm]\n B_1=2-y^2 & B_2=(1-2y)(3+y^2) \\\\ [.2cm]\nB_3=(1-2y)^2+(1-2y)^3 & B_4=(1-2y)^3\n\\end{array}\n\\]\n\nThe formulae [\\ref{eion.g}] and [\\ref{eion.h}]\ngive the total cross-section of the scattering\nabove the threshold energies\n\n\\begin{equation}\nT_{\\rm Moller}^{\\rm thr} =2T_{cut} \\mbox{\\hspace{2cm}and\\hspace{2cm}}\nT_{\\rm Bhabha}^{\\rm thr} = T_{cut}\n\\end{equation}\nThe interaction length for the production of $\\delta$-rays is calculated\nduring initialisation .\n\n\\section{Simulation of the $\\delta -ray$ production}\n\\subsection{Differential cross section}\n\nThe differential cross-section of the $\\delta$-ray production can\nbe written as in equation \\ref{eion.i} , \\ref{eion.j}\n (\\cite{eion.messel}). For the\nelectron/electron (M\\\"{o}ller) scattering we have:\n\\begin{equation}\n\\label{eion.i}\n\\frac{d\\sigma }{d \\epsilon }=\\frac{2 \\pi Z r^2 _0 m }{\\beta^2 (E-m)}\n\\left[ \\frac{(\\gamma -1 )^2} {\\gamma^2 }+\\frac{1}{\\epsilon}\n\\left(\\frac{1}{\\epsilon}-\\frac{2 \\gamma -1 } {\\gamma^2 } \\right) +\n\\frac{1}{1- \\epsilon}\\left(\\frac{1} {1- \\epsilon} \\frac{2 \\gamma - 1}\n{\\gamma^2 }\\right) \\right]\n\\end{equation}\nand for the positron-electron (Bhabha) scattering:\n\\begin{equation}\n\\label{eion.j}\n\\frac{d \\sigma}{d \\epsilon}=\\frac{2 \\pi Z r^2_0 m }{(E-m)}\\left[\n\\frac{1} {\\beta^2 \\epsilon^2}-\\frac{B_1}{\\epsilon}+B_2 - B_3 \\epsilon\n+ B_4 \\epsilon^2\\right]\n \\end{equation}\n\nwhere\n\n\\[\n\\begin{array}{lcllcl}\nZ & = & \\mbox{atomic number of the medium} &\nE & = & \\mbox{energy of the incident particle} \\\\\nM & = & \\mbox{rest mass of the incident particle} &\n\\gamma & = & \\frac{E}{M} \\\\ [.2cm]\n\\beta^2 & = & 1- \\frac{1} {\\gamma^2 } &\n y & = & \\frac{1} {\\gamma + 1} \\\\ [.2cm]\n B_1 & = & 2-y^2 &\n B_2 & = & (1-2y)(3+y^2 ) \\\\\n B_3 & = & (1-2y)^2+(1-2y)^3 &\n B_4 & = & (1-2y)^3 \\\\\n\\epsilon & = & \\frac{T} {E-m}\n\\end{array}\n\\]\n\nwith $T$ the kinematic energy of the scattered electron (of the lower\nenergy in the case of $e^+ e^-$ scattering).\n\nThe kinematical limits for the variable $\\epsilon$ are:\n\n\\[\n\\epsilon_0 = \\frac{T_{cut}}{E-m} \\leq \\epsilon \\leq \\frac{1}{2}\n\\mbox{\\hspace{.2cm} for $e^- e^-$} \\hspace{2cm}\n\\epsilon_0 = \\frac{T_{cut}}{E-m} \\leq \\epsilon \\leq 1\n\\mbox{\\hspace{.2cm} for $e^+ e^-$}\n\\]\n\n\\subsection{Sampling}\n\nApart from the normalisation, the cross-section can be written as\n\\[\n\\frac{d\\sigma}{d\\epsilon}=f(\\epsilon) g(\\epsilon),\n\\]\nwhere, for $e^- e^-$ scattering,\n\\begin{eqnarray*}\nf(\\epsilon)&=&\\frac{1}{\\epsilon^2} \\frac{\\epsilon_0 }{1- 2\\epsilon_0} \\\\\ng(\\epsilon)&=&\\frac{4}{9\\gamma^2 - 10 \\gamma + 5}\\left[(\\gamma -1)^2\n\\epsilon^2 - (2 \\gamma^2 +2\\gamma -1) \\frac{\\epsilon} {1- \\epsilon }+\n\\frac{\\gamma^2}{(1- \\epsilon )^2 }\\right]\n\\end{eqnarray*}\nand for $e^+ e^-$ scattering\n\\begin{eqnarray*}\n f(\\epsilon)&=&\\frac{1}{\\epsilon^2} \\frac{\\epsilon_0}{1- \\epsilon_0 } \\\\\n g(\\epsilon)&=&\\frac{B_0 -B_1 \\epsilon +B_2 \\epsilon^2\n -B_3 \\epsilon^3 +B_4 \\epsilon ^4}{B_ 0-B_1\\epsilon_0\n+B_2\\epsilon^2_0\n -B_3 \\epsilon^3_0 +B_4 \\epsilon^4_0 }\n\\end{eqnarray*}\nHere $ B_0=\\gamma^2/(\\gamma^2-1)$ and\nall the other quantities have been defined above.\n\nThe variable $\\epsilon$ is sampled by:\n\\begin{enumerate}\n\\item sample $\\epsilon$ from $f(\\epsilon)$\n\\item calculate the rejection function $g(\\epsilon)$ and accept the\nsampled $\\epsilon$ with a probability of $g(\\epsilon)$.\n\\end{enumerate}\n\nAfter the successful sampling of $\\epsilon$, the direction\n of the scattered electron is generated with respect to the direction of the\nincident particle. The azimuthal angle $\\phi$ is generated isotropically;\nthe polar angle\n$\\theta$ is calculated from the energy momentum conservation.\nThis information\nis used to calculate the energy and momentum of both scattered\nparticles and to transform them into the {\\em global} coordinate system.\n\n\\section{Status of this document}\n\n 9.10.98 created by L. Urb\\'an.\n\n\\begin{thebibliography}{99}\n\n\\bibitem[GEANT3]{eion.geant3}\n GEANT3 manual ,CERN Program Library Long Writeup W5013 (October 1994).\n\\bibitem[Mess70]{eion.messel}\n H.Messel and D.F.Crawford. Pergamon Press,Oxford,1970.\n\\bibitem[Ster71]{eion.sternheimer}\n R.M.Sternheimer. Phys.Rev. B3 (1971) 3681.\n\n\\end{thebibliography}\n\n", "meta": {"hexsha": "6e029cbbef6d346586d02b7c41c5a2df22216e82", "size": 8648, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "geant4/electromagnetic/standard/eion.tex", "max_stars_repo_name": "berghaus/cernlib-docs", "max_stars_repo_head_hexsha": "76048db0ca60708a16661e8494e1fcaa76a83db7", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-07-24T12:30:01.000Z", "max_stars_repo_stars_event_max_datetime": "2019-07-24T12:30:01.000Z", "max_issues_repo_path": "geant4/electromagnetic/standard/eion.tex", "max_issues_repo_name": "berghaus/cernlib-docs", "max_issues_repo_head_hexsha": "76048db0ca60708a16661e8494e1fcaa76a83db7", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "geant4/electromagnetic/standard/eion.tex", "max_forks_repo_name": "berghaus/cernlib-docs", "max_forks_repo_head_hexsha": "76048db0ca60708a16661e8494e1fcaa76a83db7", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 34.8709677419, "max_line_length": 76, "alphanum_fraction": 0.6187557817, "num_tokens": 3112, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.34404329433900654}} {"text": "\\documentclass[a4paper,11pt]{article}\n\\usepackage{setspace}\n\\usepackage[utf8]{inputenc}\n%\\usepackage{subfigure}\n\n\\pagestyle{plain}\n\n\\usepackage{amssymb, bm, blkarray, multicol}\n\\usepackage{listings,xcolor,lmodern}\n\\usepackage{graphicx}\n\\usepackage{enumitem}\n\\usepackage{tikz}\n\\usepackage{booktabs}\n\\usepackage{amsmath}\n\\usepackage{algorithm}\n\\usepackage{algpseudocode}\n\\usepackage[nottoc]{tocbibind}\n\\usepackage{upgreek}\n\n\\usepackage{mathtools}\n\\mathtoolsset{showonlyrefs}\n\n\\usepackage{hyperref}\n\\usepackage{url}\n\\usepackage{nicefrac}\n\\usepackage{microtype}\n\n\\usepackage{latexsym}\n% \\usepackage{a4wide}\n\n\\newtheorem{theorem}{THEOREM}\n\\newtheorem{lemma}[theorem]{LEMMA}\n\\newtheorem{corollary}[theorem]{COROLLARY}\n\\newtheorem{proposition}[theorem]{PROPOSITION}\n\\newtheorem{remark}[theorem]{REMARK}\n\\newtheorem{definition}[theorem]{DEFINITION}\n\\newtheorem{fact}[theorem]{FACT}\n\n\\newtheorem{problem}[theorem]{PROBLEM}\n\\newtheorem{exercise}[theorem]{EXERCISE}\n\\def \\set#1{\\{#1\\} }\n\n\\lstset{\n basicstyle=\\ttfamily,\n columns=fullflexible,\n frame=single,\n breaklines=true,\n postbreak=\\mbox{\\textcolor{red}{$\\hookrightarrow$}\\space},\n}\n\n\\newcommand{\\E}{\\mathbb{E}}\n\\newcommand{\\var}{\\mathrm{Var}}\n\\newcommand{\\cov}{\\mathrm{Cov}}\n\\newcommand{\\KL}{\\mathrm{KL}}\n\\newcommand{\\neighb}{\\text{ne}}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\\newcommand{\\Perp}{\\mathrel{\\text{\\scalebox{1.07}{$\\perp\\mkern-10mu\\perp$}}}}\n\n\\usetikzlibrary{shapes.geometric}\n\n\\begin{document}\nExisting tests test moments of the marginal distributions of $\\theta$ and sometimes the evaluations of $p(y|\\theta), p(\\theta)$. What test functions should we choose for the MMD tests?\n\n\\section{MMD-SC}\n\nLet $q_{\\text{SC}}$ denote the transition kernel for Algorithm \\ref{alg:sc-sampler}. Then $q_{\\text{SC}}(\\theta', y' | \\theta, y) = q(\\theta' | y') p(y' | \\theta) q(\\theta, y)$ and \n\\begin{align*}\n q_{\\text{SC}}(\\theta', y') &= \\int q_{\\text{SC}}(\\theta', y' | \\theta, y) q(\\theta, y) d\\theta dy \\\\\n &= \\int q_{\\text{SC}}(\\theta' | y') p(y' | \\theta) q_{\\text{SC}}(\\theta, y) d\\theta dy \\\\\n &= q_{\\text{SC}}(\\theta' | y') \\int p(y' | \\theta) q_{\\text{SC}}(\\theta) d\\theta \\\\\n\\end{align*}\nIf the marginals are correct, $q_{\\text{SC}}(\\theta) = p(\\theta)$ and\n\\begin{align*}\n q_{\\text{SC}}(\\theta', y') = q_{\\text{SC}}(\\theta' | y') \\int p(y' | \\theta) q_{\\text{SC}}(\\theta) d\\theta = q_{\\text{SC}}(\\theta' | y') p(y')\n\\end{align*}\n$q_{\\text{SC}}(\\theta' | y') = p(\\theta'|y') \\rightarrow q_{\\text{SC}}(\\theta' | y') p(y') = p(\\theta', y') \\rightarrow q_{\\text{SC}}(\\theta) = p(\\theta) $, but the other direction does not necessarily hold. Equality of the marginal distributions of $\\theta$ is not enough to show equality of the joints.\n\nHowever, $q_{\\text{SC}}(\\theta) = p(\\theta)$ does imply $q_{\\text{SC}}(y) = p(y)$, so\n\\begin{align*}\n q_{\\text{SC}}(\\theta' | y') &= \\frac{q_{\\text{SC}}(y' | \\theta') q_{\\text{SC}}(\\theta')}{q_{\\text{SC}}(y')} \\\\\n &= \\frac{q_{\\text{SC}}(y' | \\theta') p(\\theta')}{p(y')} \\\\\n\\end{align*}\n\nThus, if $q_{\\text{SC}}(y' | \\theta') = p(y' | \\theta')$, then $q_{\\text{SC}}(\\theta' | y') = p(\\theta' | y')$ and the joint distributions are the same. \n\nIs testing the evaluation of the likelihood enough? Example where it is not:\n\\begin{align*}\n \\theta &\\sim \\mathcal{U}(\\{0, 1\\}) \\\\\n y &\\sim \\mathcal{U}(\\{0, 1\\}) \\\\\n p(y | \\theta) &= 0.25\n\\end{align*}\nvs\n\\begin{align*}\n \\theta &\\sim \\mathcal{U}(\\{0, 1\\}) \\\\\n q(y=i|\\theta) &= \\begin{cases}\n 1 & \\theta = i \\\\\n 0 & \\text{otherwise}\n \\end{cases} \\quad i \\in \\{0, 1\\}\n\\end{align*}\nThe evaluations of $p(y | \\theta)$ and the marginals are the same, but the joint distributions differ; the first model gives $(i, j) \\quad \\forall i, j \\in \\{0,1\\}$, while the second gives $(i, i)$. This example relies on the independence of $\\theta$ and $y$. \n\nCan we come up with other examples? How might these arise in MCMC?\n\n\\section{MMD-BC}\nLet $q_{\\text{BC}}$ denote the joint distribution from Algorithm \\ref{alg:bc-sampler}.\n\\begin{align*}\n q_{\\text{BC}}(\\theta', y') &= q_{\\text{BC}}(\\theta'|y')q(y') = q_{\\text{BC}}(\\theta'|y')p(y') \\\\\n\\end{align*}\nBy the same logic as for the MMD-SC test, if $q_{\\text{BC}}(y' | \\theta') = p(y' | \\theta')$, then $q_{\\text{BC}}(\\theta' | y') = p(\\theta' | y')$ and the joint distributions are the same.\n\\newpage \n\n\\begin{algorithm}\n \\centering\n \\caption{Marginal-conditional (MC) joint simulator}\\label{alg:mc-sampler}\n \\begin{algorithmic}[1]\n \\State \\text{Initialize} $\\mathbf{g}_{MC} \\in \\mathbb{R}_{N\\times |\\mathbf{g}|}$\n \\For{$n = 1, \\ldots, N$}\n \\State $\\mathbf{\\Theta}_{n} \\sim P(\\mathbf{\\Theta})$ \n \\State $\\mathbf{Y}_{n} \\sim P(\\mathbf{Y}|\\mathbf{\\Theta}_{n})$ \n \\State $\\mathbf{g}_{MC}[n, :] = \\mathbf{g}(\\mathbf{\\Theta}_{n}, \\mathbf{Y}_{n})$ \n \\EndFor \n \\State \\textbf{return} $\\mathbf{g}_{MC}$\n \\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}\n \\centering\n \\caption{Successive-conditional (SC) joint simulator}\\label{alg:sc-sampler}\n \\begin{algorithmic}[1]\n \\State \\text{Initialize} $\\mathbf{g}_{SC} \\in \\mathbb{R}_{N\\times |\\mathbf{g}|}$\n \\State $\\mathbf{\\Theta}_{0} \\sim P(\\mathbf{\\Theta})$ \n \\For{$n = 1, \\ldots, N$}\n \\State $\\mathbf{Y}_{n} \\sim P(\\mathbf{Y}|\\mathbf{\\Theta}_{n-1})$ \n \\State $\\mathbf{\\Theta}_{n} \\sim \\text{PosteriorSampler}(\\mathbf{\\Theta}_{n-1}, \\mathbf{Y}_{n})$ \n \\State $\\mathbf{g}_{SC}[n, :] = \\mathbf{g}(\\mathbf{\\Theta}_{n}, \\mathbf{Y}_{n})$ \n \\EndFor \n \\State \\textbf{return} $\\mathbf{g}_{SC}$\n \\end{algorithmic}\n\\end{algorithm}\n\n\\begin{algorithm}[H]\n \\centering\n \\caption{Backward-conditional (BC) joint simulator}\\label{alg:bc-sampler}\n \\begin{algorithmic}[1]\n \\State \\text{Initialize} $\\mathbf{g}_{BC} \\in \\mathbb{R}_{N\\times |\\mathbf{g}|}$\n \\For{$n = 1, \\ldots, N_{BC}$}\n \\State $\\mathbf{\\Theta}_{0} \\sim P(\\mathbf{\\Theta})$ \n \\State $\\mathbf{Y}_{n} \\sim P(\\mathbf{Y}|\\mathbf{\\Theta}_{0})$ \n \\For{$m = 1, \\ldots, M$} \n \\State $\\mathbf{\\Theta}_{n} \\sim \\text{PosteriorSampler}(\\mathbf{\\Theta}_{n}, \\mathbf{Y}_{n})$\n \\EndFor\n \\State $\\mathbf{g}_{BC}[n, :] = \\mathbf{g}(\\mathbf{\\Theta}_{n}, \\mathbf{Y}_{n})$ \n \\EndFor \n \\State \\textbf{return} $\\mathbf{g}_{BC}$\n \\end{algorithmic}\n\\end{algorithm}\n\n\\end{document}", "meta": {"hexsha": "0dfa54c7455f59ffb77c49e139d03a864ab969e9", "size": 6621, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "notes.tex", "max_stars_repo_name": "awjiang/mcmc-joint-tests", "max_stars_repo_head_hexsha": "ce94b9070f8976ad2dc0731e46a5125db48c66b1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notes.tex", "max_issues_repo_name": "awjiang/mcmc-joint-tests", "max_issues_repo_head_hexsha": "ce94b9070f8976ad2dc0731e46a5125db48c66b1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notes.tex", "max_forks_repo_name": "awjiang/mcmc-joint-tests", "max_forks_repo_head_hexsha": "ce94b9070f8976ad2dc0731e46a5125db48c66b1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.641509434, "max_line_length": 304, "alphanum_fraction": 0.6076121432, "num_tokens": 2324, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041654, "lm_q2_score": 0.6723316926137811, "lm_q1q2_score": 0.3440432909793286}} {"text": "\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% First Chapter %\n% Problem Formulation %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\chapter{Posture Generation: Problem Formulation}\n\\label{cha:posture_generation_problem_formulation}\n\n%\\nomenclature[z-PG]{PG}{Posture Generation}\n%\\nomenclature[z-IK]{IK}{Inverse Kinematics}\n%\\nomenclature[z-DoF]{DoF}{degrees of freedom}\n%\\nomenclature[z-w.r.t]{w.r.t}{with respect to}\n%\\nomenclature[x-I]{$\\mathbb{I}_n$}{Matrix identity of dimension $n$}\n%\\nomenclature[x-w]{$\\wedge$}{cross product}\n%\\nomenclature[a-F]{F}{a frame}\n%\\nomenclature[a-W]{W}{the world frame}\n%\\nomenclature[a-w]{w}{a wrench or a force}\n%\\nomenclature[a-f]{f}{a force resultant}\n%\\nomenclature[a-m]{m}{a force moment}\n\n\n\\graphicspath{{Chapter2-PG/Figs/}}\n\n%{{{List of contributions\n%\\section{List of contributions}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% SECTION LIST OF CONTRIBUTIONS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%\\begin{itemize}\n %\\item Generalities, introduction\n %\\item Presentation of the existing methods\n %\\item From Inverse Kinematics to Generalized IK/posture Generation/pose estimation (addition of articular limits, forces, stability etc).\n %\\item Topology of the parametrization space (Free-flyer, q, f, other)\n %\\item Formulation as a nonlinear constrained optimization problem\n %\\item Adrien \\& Karim's formulations\n %\\item Formulation of several types of cost/constraints\n %\\begin{itemize}\n %\\item Contact with planar surface\n %\\item Collision avoidance\n %\\item Auto-Collision avoidance\n %\\item Static equilibrium: Newton/CoM projection\n %\\item Forces in friction cones\n %\\item Articular limits\n %\\item Torque limits\n %\\item Torque minimization\n %\\item Goal Posture\n %\\end{itemize}\n %\\item Reasons why it is not enough and why we needed a new PG\n %\\begin{itemize}\n %\\item Having an easier way to formulate problems\n %\\item Avoid having to de some gymnastic to remain on manifolds\n %\\item Automatic variable management\n %\\item Robustness\n %\\end{itemize}\n %\\item Utilization of posture generation in planning\n%\\end{itemize}\n%}}}\n\n\nIn this Chapter, we present in detail the formulation of a posture generation problem.\nWe present the algorithms used to compute the kinematics of a robot and its derivatives as well as the joint torques.\nThen we formulate some classical functions that are often used in posture generation: joint limits, contact constraints, collision avoidance, stability, torque limits and friction cones.\nIn~\\Figref{fig:PG}, we illustrate those constraints with the result of a posture generation problem where the HRP-4 robot must climb on a stack of cubes while being statically stable, respecting its joint and torque limits, the contact forces must remain within their respective friction cones and the robot must avoid auto-collisions and non-desired contacts (i.e. collisions) with the environment.\n\n%\\begin{figure}[ht]\n %\\centering\n %\\includegraphics[width=0.7\\textwidth]{PG.pdf}\n %\\caption{HRP4 on a stack of cubes. The color of each box corresponds to the color of the item it depicts.}\n%\\label{fig:PG}\n%\\end{figure}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SECTION PROBLEM FORMULATION %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\section{Kinematics Formulation}\n%\\label{sec:kinematics_formulation}\n\n\n\\section{Forward Kinematics}\n\\label{sec:forward_kinematics}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION FORWARD KINEMATICS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn this section, we present a formulation of robotic systems that allows specifying most of the typical constraints encountered in robotics problems.\n\nWe consider a robotic system made of $n_B$ bodies and $n_J(=n_B+1)$ joints.\nThe global structure of the robot is described by an ordered graph called multibody graph.\nThe base body (World) has index $0$ and each of the remaining bodies get a different positive integer index.\nWe denote the body of index $i$, $B_i$.\n$B_0$ refers to the World.\nEach body $B_i$ has its reference frame $F_i$ attached to it.\n$F_0$ denotes the World frame.\nBodies are linked together by joints that also are indexed by positive integers, we denote the joint of index $i$, $J_i$, and the body that comes after it is $B_i$.\nEach joint defines the relation between its predecessor and successor bodies.\nFor joint $J_i$, they are respectively denoted $pred(i)$ and $succ(i)$, and $B_{pred(i)}$ is called the parent body of $B_{succ(i)}$.\nWe denote $\\lambda(j)$ the index of the parent body of $B_j$.\nThe number of degrees of freedom of $J_i$ is denoted $dof^J_i$ and the number of degrees of freedom of the whole robot is denoted $dof$.\n\\Figref{fig:mbg} illustrates this numbering system for a simple robot with 4 joints and 5 bodies (including the basis)\n\nThe geometric relations between bodies and joints are described through transformations between their reference frames.\nWe use transformations as described in the Spatial Vector Algebra chapter of `Rigid Body Dynamics Algorithm' by Roy Featherstone~\\cite{featherstone:book:2007}.\nMotion vectors (vectors describing motion quantities as positions, velocities and accelerations) and their force counterpart are defined in~\\cite{featherstone:book:2007}.\n\nFor any 3D vector $v\\in\\mathbb{R}^3$, $\\hat{v}$ denotes the $3\\times 3$ skew-symmetric matrix such that $\\hat{v}u = v\\wedge u$.\nWhere $\\wedge$ denotes the cross product operator.\n\nLet A and B be Cartesian frames with origins O and P respectively.\nLet $\\mathbf{t}$ be the coordinate vector expressing $\\overrightarrow{OP}$ in A.\nAnd $\\mathbf{R}$ be the rotation matrix that transforms 3D vectors from A to B coordinates.\nThe transformation from A to B for a motion vector is defined by:\n\\begin{equation}\n {}^B X_A =\n \\begin{bmatrix}\n \\mathbf{R} & \\mathbf{0} \\\\\n -\\mathbf{R}\\hat{\\mathbf{t}} & \\mathbf{R} \\\\\n \\end{bmatrix}\n\\end{equation}\nIts inverse is:\n\\begin{equation}\n {{}^B X_A}^{-1} = {}^A X_B =\n \\begin{bmatrix}\n \\mathbf{R}^T & \\mathbf{0} \\\\\n \\hat{\\mathbf{t}}\\mathbf{R}^T & \\mathbf{R}^T \\\\\n \\end{bmatrix}\n\\end{equation}\nThe transformation from A to B for a force vector is defined by:\n\\begin{equation}\n {}^B X_A^* =\n \\begin{bmatrix}\n \\mathbf{R} & -\\mathbf{R}\\hat{\\mathbf{t}} \\\\\n \\mathbf{0} & \\mathbf{R} \\\\\n \\end{bmatrix}\n\\end{equation}\nIts inverse is:\n\\begin{equation}\n {}^B X_A^{-*} = {}^A X_B^* =\n \\begin{bmatrix}\n \\mathbf{R}^T & \\hat{\\mathbf{t}}\\mathbf{R}^T \\\\\n \\mathbf{0} & \\mathbf{R}^T \\\\\n \\end{bmatrix}\n\\end{equation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.7\\textwidth]{mbg.pdf}\n \\caption{MultiBody graph}\n\\label{fig:mbg}\n\\end{figure}\n\n%Each joint $J_i$ is defined in the reference frame of its predecessor body by a static transformation $X^x_i = \\{\\mathbf{R}^x_i, \\mathbf{t}^x_i\\}$ from the base of the body to the base of the joint.\nEach joint $J_i$ is defined by a static transformation $X^x_i = \\{\\mathbf{R}^x_i, \\mathbf{t}^x_i\\}$ between the reference frame of its predecessor body and its own reference frame.\nEach joint $J_i$ is associated with a motion subspace which representation matrix is denoted $S_i$.\nEach column of $S_i$ described a degree of freedom of $J_i$ its upper part for the rotations and lower for translations (see Section~\\ref{sec:joints_formulations}).\n\\begin{equation}\n S_i =\n \\begin{bmatrix}\n S^R_{i,0} & \\cdots &\n S^R_{i,j} & \\cdots &\n S^R_{i,dof} \\\\\n S^t_{i,0} & \\cdots &\n S^t_{i,j} & \\cdots &\n S^t_{i,dof}\n \\end{bmatrix}\n\\end{equation}\n\nFor a given joint configuration $\\mathbf{q}$, the transformation due to the joint $J_i$ current configuration from its reference frame to the reference frame of its successor body is denoted \\\\$X^J_i (\\mathbf{q}) = \\{\\mathbf{R}^J_i (\\mathbf{q}), \\mathbf{t}^J_i (\\mathbf{q})\\}$.\n\nThe transformation between $B_{\\lambda(i)}$ and $B_i$ is denoted $X^{PtS}_i (\\mathbf{q}) = \\{\\mathbf{R}^{PtS}_i, \\mathbf{t}^{PtS}_i\\}$ (PtS stands for `Parent to Son') can then be computed as:\n\\begin{equation}\n {X}^{PtS}_i (\\mathbf{q}) = {}^{i}X_{\\lambda (i)} (\\mathbf{q}) = X^J_i (\\mathbf{q}) X^x_i\n \\label{eq:PtS}\n\\end{equation}\n\nLet $\\kappa (i) =\\{0, i_1, i_2 \\ldots i\\}$ be the list of indexes of successive joints going from $B_0$ to $B_i$.\nIt can easily be computed by adding iteratively the parent of the current body:\n\n\\begin{algorithm}\n \\caption{Joint Path to $B_i$}\n\\label{alg:JP}\n\\begin{algorithmic}\n \\State{$j \\leftarrow i$, $\\kappa(i)=[i]$}\n \\While{$j \\neq 0$}\n \\State{$j \\leftarrow \\lambda(j)$}\n \\State{$\\kappa(i) \\leftarrow [\\kappa(i),\\ j]$}\n \\EndWhile{}\n\\end{algorithmic}\n\\end{algorithm}\n\nThe transformation from the World base to $B_i$ is denoted \\\\ ${}^i X_0 (\\mathbf{q}) = \\{{}^i \\mathbf{R}_0 (\\mathbf{q}), {}^i \\mathbf{t}_0 (\\mathbf{q})\\}$.\nThe formula~\\Eqref{eq:PtS} can be used iteratively on all bodies of the robot to obtain the expression of ${}^i X_0 (\\mathbf{q})$.\n\nWe obtain the full expression of ${}^i X_0$ as:\n\\begin{equation}\n {}^i X_0 (\\mathbf{q}) = \\prod_{j\\in\\kappa (i)}X^J_j (\\mathbf{q}) X^x_j = \\prod_{j\\in\\kappa (i)}\\ {}^j X_{\\lambda (j)}\n = {}^i X_{\\kappa (1)}\\ ^{\\kappa (1)}X_{\\kappa (2)} \\dots ^{\\kappa (\\text{end}-1)}X_{W}\n\\end{equation}\n\nWhich can be computed recursively by a Forward Kinematics algorithm:\n\n\\begin{algorithm}\n \\caption{Forward Kinematics}\n\\label{alg:FK}\n\\begin{algorithmic}\n \\For{$i = 0:n_J$}\n \\If{$\\lambda (i) \\neq -1$} ${}^i X_0 = {}^i X_{\\lambda (i)}\\ ^{\\lambda (i)}X_0$\n \\Else$\\ {}^i X_0 = X^{PtS}_i$\n \\EndIf{}\n \\EndFor{}\n\\end{algorithmic}\n\\end{algorithm}\n\nIn the following section, we provide some detailed description of how to compute $X_J (\\mathbf{q})$ for a variety of useful joints.\nUsing the joint descriptions and the Forward Kinematics algorithm, we are able to explicit a relation between $\\mathbf{q}$ the joint parameters of the robot and the 3D position and orientation of any geometric quantity defined in the reference frame of a body of the robot.\nGiven a transformation ${}^p X_i$ defined in the frame of $B_i$, its value in the world frame is given by ${}^p X_0 (\\mathbf{q}) = {}^p X_i\\ {}^i X_0 (\\mathbf{q})$\n\n\n\n\\section{Joints formulations}\n\\label{sec:joints_formulations}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION JOINTS FORMULATIONS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe entire geometry of our system is described by the list of static transformations $X^x_j$ and of joint transformations $X^J_j (q)$.\nIn this section, we explicit the descriptions and formulations of common joints type.\n\nLet us consider a joint $J$ that governs the transformation between two frames $F_1=\\{O_1, x_1, y_1, z_1\\}$ and $F_2=\\{O_2, x_2, y_2, z_2\\}$.\nThe most common type of joint encountered in robotics (and even virtual avatars) systems is the revolute joint, that allows a rotation around a fixed axis.\nIf $J$ is a revolute joint around the axis $(O_1,z_1)$ with parameter $q$, its motion subspace, rotation, and translation are as follows:\n\\begin{table} [H]\n\\centering\n\\begin{tabular}{cccc}\n \\toprule\n Joint type & $S$ & $Rotation$ & $translation$ \\\\\n \\midrule\n Revolute $(O_1,z_1)$\n &\n $\\begin{bmatrix}\n 0 \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 0 \\\\ 0\n \\end{bmatrix}$\n &\n $\\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos (q) & \\sin (q) \\\\\n 0 & -\\sin (q) & \\cos (q) \\\\\n \\end{bmatrix}$\n &\n ${\\bf 0}_{3\\times1}$\n \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nSimilar formulas can be devised for rotations around any other axis, provided that $R$ describes the rotation of angle $q$ around that axis.\n\nIn the case of a prismatic joint, all rotations are blocked, and only one translation along a given axis is allowed.\nA prismatic joint along $x_1$ is described by the following formulas:\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{cccc}\n \\toprule\n Joint type & $S$ & $Rotation$ & $translation$ \\\\\n \\midrule\n Prismatic $(x_1)$\n &\n $\\begin{bmatrix}\n 0 \\\\ 0 \\\\ 0 \\\\ 1 \\\\ 0 \\\\ 0\n \\end{bmatrix}$\n &\n ${\\bf 1}_{3\\times3} $\n &\n $\\begin{bmatrix}\n q \\\\ 0 \\\\ 0\n \\end{bmatrix}$\n \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nPlanar joints are also frequently used in robotics.\nA planar joint describes a plane sliding on another plane. Assuming that the normal to both planes is $z_1 = z_2$, this type of joint allows free relative rotation of $F_2$ around $z_1$ and relative translations along $x_1$ and $y_1$.\nWe denote $q = \\{q_1, q_2, q_3\\}$ the joint parameters, $q_1$ corresponding to the rotation and $q_2,\\ q_3$ to the translations.\nWe get:\n\\begin{table}[H]\n\\centering\n\\begin{tabular}{cccc}\n \\toprule\n Joint type & $S$ & $Rotation$ & $translation$ \\\\\n \\midrule\n Planar $(z_1)$\n &\n $\\begin{bmatrix}\n 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0\n \\end{bmatrix}$\n &\n $\\begin{bmatrix}\n \\cos(q_1) & \\sin(q_1) & 0 \\\\\n -\\sin(q_1) & \\cos(q_1) & 0 \\\\\n 0 & 0 & 1 \\\\\n \\end{bmatrix}$\n &\n $\\begin{bmatrix}\n \\cos(q_1)q_2 - \\sin(q_1)q_3 \\\\ \\sin(q_1)q_2 + \\cos(q_1)q_3 \\\\ 0\n \\end{bmatrix}$\n \\\\\n \\bottomrule\n\\end{tabular}\n\\end{table}\n\nA spherical joint blocks all translations and allows all rotations.\nThis joint must be parametrized by a 3D rotation.\nThe space of 3D rotations $SO(3)$ can be represented in many different ways.\nThe simplest and most intuitive way to parametrize $SO(3)$ is to use Euler Angles.\nIt comes down to decomposing the 3D rotation into a succession of three 1D rotations around different axes.\nFor example, the roll, pitch, yaw is a succession of a rotation of $F_1$ around its $x$ axis, followed by a rotation around the $y$ axis of the resulting frame, finally a rotation around the $z$ axis of the resulting frame of the latter rotation.\nThe rotation matrix for such a convention is given by:\n\\begin{equation}\n {\\bf R} =\n \\begin{bmatrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos(q_3) & \\sin(q_3) \\\\\n 0 & -\\sin(q_3) & \\cos(q_3) \\\\\n \\end{bmatrix}\n \\cdot\n \\begin{bmatrix}\n \\cos(q_2) & 0 & -\\sin(q_2) \\\\\n 0 & 1 & 0 \\\\\n \\sin(q_2) & 0 & \\cos(q_2) \\\\\n \\end{bmatrix}\n \\cdot\n \\begin{bmatrix}\n \\cos(q_1) & \\sin(q_1) & 0 \\\\\n -\\sin(q_1) & \\cos(q_1) & 0 \\\\\n 0 & 0 & 1\n \\end{bmatrix}\n\\end{equation}\n\nEuler Angle formulations have the advantage to be simple and intuitive.\nThere are many other possible choices of the rotation order and conventions.\nBut they all suffer from the so-called gimbal-lock problem or more generally from singularities, which happens when two of the three rotation axes become aligned.\nIn such a configuration, the only rotations possible are one rotation around the two aligned axis and one rotation around the third axis.\nThus, one degree of freedom is lost.\nThose singularities are prohibitive for the use of that type of formulation in a posture generation.\nIn~\\cite{grassia1998}, Grassia states that any attempt to parametrize the entire set of 3D rotations by an open subset of Euclidean space (as do Euler angles) will suffer from gimbal lock.\nNote that this singularity is only due to the user's choice of parametrization, it is not intrinsic to the manifold $SO(3)$.\n%One can prove that there is no smooth mapping between $SO(3)$ and $\\mathbb{R}^3$ that is free of singularities.\nIt is possible to parametrize $SO(3)$ without having to face singularities by parametrizing it over another non-Euclidean manifold.\nThe most common ones are the set of unit quaternion embedded in $\\mathbb{R}^4$ and the set of rotation matrices embedded in $\\mathbb{R}^{3\\times 3}$.\n%To avoid singularities, one must use a higher dimension parametrization, such as the most common $SO(3)$ representation in robotics that are the unit quaternions and the rotation matrices.\n%The unit quaternions space is the subset of $\\mathbb{R}^4$ where all elements have unit norm.\nWith the unit quaternion parametrization, a variable on $SO(3)$ is represented by 4 parameters $q = [q_w, q_x, q_y, q_z]$, and it is necessary to ensure that the quaternion is of norm 1, $\\{q\\in\\mathbb{R}^4:||q||=1\\}$.\nSimilarly, if a variable is parametrized by a rotation matrix, then the matrix $M$ representing it has 9 parameters and M must be orthogonal and have determinant 1: $\\{M\\in\\mathbb{R}^{3\\times 3}:M^T M = \\mathbf{1}_3\\ \\text{and}\\ \\det (M) = 1\\}$.\nSimilar issues can be found with the parametrization of other non-Euclidean manifolds, like $S^2$ for example.\n\nA quaternion $q = [q_w, q_x, q_y, q_z]$ is a unit quaternion iff $q_w^2+q_x^2+q_y^2+q_z^2 = 1$.\nIt represents a rotation of angle $\\theta$ around an axis ${\\bf u}$ such that:\n\\begin{align}\n q_w &= \\cos(\\theta/2) \\\\\n q_x &= \\sin(\\theta/2)u_x \\\\\n q_y &= \\sin(\\theta/2)u_y \\\\\n q_z &= \\sin(\\theta/2)u_z \\\\\n\\end{align}\nThe rotation matrix associated with this quaternion is:\n\\begin{equation}\n {\\bf R} = 2 \\begin{bmatrix}\n \\frac{1}{2} - {q_y}^2 - {q_z}^2 &\tq_x q_y - q_z q_w &\tq_x q_z + q_y q_w \\\\\n q_x q_y + q_z q_w\t& \\frac{1}{2} - {q_x}^2 - {q_z}^2 &\tq_y q_z - q_x q_w \\\\\n q_x q_z - q_y q_w &\tq_y q_z + q_x q_w\t& \\frac{1}{2} - {q_x}^2 - {q_y}^2 \\\\\n \\end{bmatrix}\n\\end{equation}\n\nThat formulation does not suffer from singularities, but it requires to maintain 4 parameters for a 3D rotation.\nAnd those 4 parameters must satisfy the unit norm constraint which in turn would become an additional constraint in the optimization formulation.\nGiven a parameter set $q = \\{ q_w, q_x, q_y, q_z\\}$, we get the following table.\n\n\\begin{table}[ht]\n \\centering\n \\begin{tabular}{cccc}\n \\toprule\n Joint type & $S$ & $Rotation$ & $translation$ \\\\\n \\midrule\n Spherical\n &\n $\\begin{bmatrix}\n 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0\n \\end{bmatrix}$\n &\n $2 \\begin{bmatrix}\n \\frac{1}{2} - {q_y}^2 - {q_z}^2 &\tq_x q_y - q_z q_w &\tq_x q_z + q_y q_w \\\\\n q_x q_y + q_z q_w\t& \\frac{1}{2} - {q_x}^2 - {q_z}^2 &\tq_y q_z - q_x q_w \\\\\n q_x q_z - q_y q_w &\tq_y q_z + q_x q_w\t& \\frac{1}{2} - {q_x}^2 - {q_y}^2 \\\\\n \\end{bmatrix}$\n &\n $\\begin{bmatrix}\n 0 \\\\ 0 \\\\ 0\n \\end{bmatrix}$\n \\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\nFinally, a free joint allows free motion of its successor body with respect to its predecessor body.\nIt can be viewed as a combination of a spherical joint and 3 perpendicular prismatic joints.\nGiven a parameter set $q = \\{ q_w, q_x, q_y, q_z, t_x, t_y, t_z\\}$, we get the following table.\n\n\\begin{tabular}{cccc}\n \\toprule\n Joint type & $S$ & $Rotation$ & $translation$ \\\\\n \\midrule\n Spherical\n &\n $\\mathbf{1}_6$\n &\n $2 \\begin{bmatrix}\n \\frac{1}{2} - {q_y}^2 - {q_z}^2 &\tq_x q_y - q_z q_w &\tq_x q_z + q_y q_w \\\\\n q_x q_y + q_z q_w\t& \\frac{1}{2} - {q_x}^2 - {q_z}^2 &\tq_y q_z - q_x q_w \\\\\n q_x q_z - q_y q_w &\tq_y q_z + q_x q_w\t& \\frac{1}{2} - {q_x}^2 - {q_y}^2 \\\\\n \\end{bmatrix}$\n &\n ${\\bf R}^{-1}\\begin{bmatrix}\n t_x \\\\ t_y \\\\ t_z\n \\end{bmatrix}$\n \\\\\n \\bottomrule\n\\end{tabular}\n\n\n\n\\section{Jacobian computation}\n\\label{sec:jacobian_computation}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION JACOBIAN COMPUTATION %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn order to solve our problem with a gradient-based nonlinear optimization algorithm, it is useful to compute the derivatives of every function used as constraint or cost with respect to any variable of the problem.\nThe transformations ${}^i X_0$ are used in many functions, therefore, having an efficient algorithm to compute their derivatives and the derivatives of any transformation defined in $B_i$ is necessary.\n\nGiven a static transformation ${}^p X_i$ defined in body $B_i$.\nIts expression in the world frame is ${}^p X_0 = {}^p X_i\\ {}^i X_0$ and its expression in the frame of $B_j$ is ${}^p X_j = {}^p X_i\\ {}^i X_0\\ {{}^j X_i}^{-1}$.\n\nWe denote $\\text{Jac}_i$ the Jacobian of body $i$, and $\\text{Jac}_i(X)$ the jacobian of the frame defined by $X$ in the referential of body $i$.\n$q_i$ is the part of $q$ that corresponds to the degrees of freedom of joint $J_i$.\nWe denote $\\text{Jac}_i.\\text{cols}(j)$ the columns of $\\text{Jac}_i$ associated with joint $J_j$.\nThe jacobian of the frame defined by ${}^p X_i$ in $B_i$ with respect to $q_j$ is given by\n\\begin{equation}\n\\label{partial_jacobian}\n \\text{Jac}_i({}^p X_i).\\text{cols}(j) = {}^p X_j\\ S_i\n\\end{equation}\n\nThe complete Jacobian of a body $\\text{Jac}_i$ is a $6\\times \\text{dof}$ matrix that can be computed by using the formula~\\ref{partial_jacobian} on every index $j$ in $\\kappa(i)$ and filling the rest of $\\text{Jac}_i$ with zeros.\n\nThe algorithm that we use to compute $\\text{Jac}_i({}{}^p X_i)$ writes as follows:\n\n\\begin{algorithm}\n \\caption{Jacobian Computation}\n\\label{alg:jacobian_computation}\n\\begin{algorithmic}\n \\State{$\\text{Jac}_i({}^p X_i) = {\\bf 0}_{6\\times\\text{dof}}$}\n \\State{${}^p X_0 = {}^p X_i\\ {({}^i X_0)}^{-1}$}\n \\For{$j = 0:\\text{size}(\\kappa(i))$}\n \\State{$k \\leftarrow \\kappa(j)$}\n \\State{${}^p X_k = {}^p X_0\\ {({}^k X_0)}^{-1}$}\n \\State{$\\text{Jac}_i({}^p X_i).\\text{cols}(k) = {}^p X_k\\ S_k$}\n \\EndFor{}\n\\end{algorithmic}\n\\end{algorithm}\n\nWe write the jacobian of each body at its origin as follows:\n\\begin{equation}\n \\mathbf{Jac}^0_i =\n \\begin{bmatrix}\n \\frac{\\partial {}^i\\mathbf{R}_0}{\\partial q_0} & \\cdots &\n \\frac{\\partial {}^i\\mathbf{R}_0}{\\partial q_j} & \\cdots &\n \\frac{\\partial {}^i\\mathbf{R}_0}{\\partial q_{dof}} \\\\\n \\frac{\\partial {}^i\\mathbf{t}_0}{\\partial q_0} & \\cdots &\n \\frac{\\partial {}^i\\mathbf{t}_0}{\\partial q_j} & \\cdots &\n \\frac{\\partial {}^i\\mathbf{t}_0}{\\partial q_{dof}}\n \\end{bmatrix}\n=\n \\begin{bmatrix}\n \\omega_{i,0} & \\cdots &\n \\omega_{i,j} & \\cdots &\n \\omega_{i,dof} \\\\\n v_{i,0} & \\cdots &\n v_{i,j} & \\cdots &\n v_{i,dof}\n \\end{bmatrix}\n\\end{equation}\n\n\n\n%\\section{Geometric constraints}\n%\\label{sec:geometric_constraints}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SECTION GEOMETRIC CONSTRAINTS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Joint Limits}\n\\label{sec:joint_limits}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION JOINT LIMITS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nMost robotic joints have limits which define the range of value that can be accessed by the joint variables.\nThe joint limits for 1D joints like revolute and prismatic joints are trivial to formulate: We denote $q^-$ and $q^+$ the lower and upper values accessible and add a boundary constraint to the optimization problem:\n\\begin{equation}\n\\label{eq:joint_limits}\n \\boxed{q^- \\leq q \\leq q^+}\n\\end{equation}\n\nMost joints are easy to limit because their variables are independent.\nLimiting the movements of a spherical joint, and by extension, of a free joint, is more complicated.\nIn humanoid robotics, spherical joints can be used to model the shoulder or hip joint of the robot.\nA common approach to limit shoulder joint, inspired from the biomechanics field, considers the spherical motion (or swing) and the axial motion (or twist) separately as shown in~\\Figref{fig:ballAndSocket}.\n\n\\begin{figure}[htb]\n \\centering\n \\includegraphics[width=0.7\\linewidth]{ballAndSocket.pdf}\n \\caption{Swing and Twist in ball and socket joint}\n\\label{fig:ballAndSocket}\n\\end{figure}\n\nThe spherical motion can be parametrized by a vector of the 3D unit sphere $S^2$ and constrained to lay within a limit cone, the axial motion can be parametrized in $\\mathbb{R}$ and limited by equation~\\Eqref{eq:joint_limits}.\nThis type of formulation is presented in~\\cite{baerlocher}.\n\n\n\n\\section{Contact constraints}\n\\label{sec:contact_constraints}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION CONTACT CONSTRAINTS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nMost of the tasks the robots achieve (grasping, manipulation, walking, etc.) are made by making and breaking contacts.\nA contact can be defined between 2 surfaces: one on the robot and the other on the entity (that can be another robot, object or the environment) to contact with.\nThe most usual types of contact constraint encountered are the planar contact and the fixed contact.\nA planar contact constraint is used when a planar surface of a robot is put in contact with a planar surface of the environment.\nWe denote $F_1 = \\{O_1, \\vec{x_1}, \\vec{y_1}, \\vec{z_1}\\}$ a frame defined on $S_1$, the surface of the first body involved in the contact, such that the 3D point $O_1$ is on $S_1$ and the vector $\\vec{z_1}$ is normal to $S_1$ and pointing toward the inside of the body.\n$F2 = \\{O_2, \\vec{x_2}, \\vec{y_2}, \\vec{z_2}\\}$ is a frame on $S_2$, the surface of the second body involved, such that $O_2$ is on $S_2$ and the vector $\\vec{z_2}$ is normal to $S_2$ and pointing away from the body.\n\nConstraining $S_1$ and $S_2$ to be coplanar boils down to aligning $\\vec{z_1}$ with $\\vec{z_2}$ and to ensure that the projection of the distance between $O_1$ and $O_2$ along $\\vec{z_1}$ is null.\nNote that we avoid using the dot product of two vectors that are meant to be aligned e.g. $\\vec{z_1}\\cdot\\vec{z_2} = 1$ because when that constraint is satisfied, its gradient is zero, which implies that in the optimization context it is unqualified.\nThat is why we prefer imposing orthogonality constraints.\nThis translates into adding the following set of constraints to our problem:\n\n\\begin{equation}\n\\label{eq:coplanarity}\n\\boxed{\\left\\{\n \\begin{array}{l}\n \\overrightarrow{O_1O_2} \\cdot \\vec{z_1} = 0\\\\\n \\vec{x_1}\\cdot\\vec{z_2} = 0\\\\\n \\vec{y_1}\\cdot\\vec{z_2} = 0\\\\\n \\vec{z_1}\\cdot\\vec{z_2} \\geq 0\n \\end{array}\n \\right.}\n\\end{equation}\n\nThis set of constraints leaves free the displacements of $F_2$ along $\\vec{x_1}$ and $\\vec{y_1}$ as well as its rotation around $\\vec{z_1}$.\nWe call this a floating planar contact, the optimization algorithm will be able to choose the location of $F_2$ in the plane $\\{O_1, \\vec{x_1}, \\vec{y_1}\\}$.\nThis contact has 3 degrees of freedom.\n\n\\begin{figure}[htpb]\n \\centering\n \\includegraphics[width=0.8\\linewidth]{contactConstraint.pdf}\n \\caption{Floating and Fixed Contacts}\n\\label{fig:contactConstraint}\n\\end{figure}\n\nIf we constrain the location of $F_2$ in $\\{O_1, \\vec{x_1}, \\vec{y_1}\\}$ such that $O_1$ and $O_2$ are superimposed and $\\vec{x_1}$, $\\vec{y_1}$, $\\vec{z_1}$ are aligned with respectively $\\vec{x_2}$, $\\vec{y_2}$, $\\vec{z_2}$, we obtain a fixed contact with zero degrees of freedom.\nThis translates into adding the following set of constraints to our problem:\n\n\\begin{equation}\n\\label{eq:fixed_contact}\n\\boxed{\\left\\{\n \\begin{array}{l}\n \\overrightarrow{O_1O_2} = \\vec{0}\\\\\n \\vec{x_1}\\cdot\\vec{z_2} = 0\\\\\n \\vec{y_1}\\cdot\\vec{z_2} = 0\\\\\n \\vec{x_1}\\cdot\\vec{y_2} = 0\\\\\n \\vec{z_1}\\cdot\\vec{z_2} \\geq 0\\\\\n \\vec{x_1}\\cdot\\vec{x_2} \\geq 0\\\\\n \\end{array}\n \\right.}\n\\end{equation}\n\nWe illustrate those two types of contacts in~\\Figref{fig:contactConstraint}\n\n\n\n\\section{Collision avoidance}\n\\label{sec:collision_avoidance}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION COLLISION AVOIDANCE %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn order to avoid unwanted collisions between bodies of robots, for two bodies $B_1$ and $B_2$, we want to define a continuously differentiable function $d_{\\{B_1, B_2\\}}(q)$ that has the properties of a pseudo-distance:\n\\begin{itemize}\n \\item $d_{\\{B_1, B_2\\}}(q) > 0$ when the bodies are not touching each other\n \\item $d_{\\{B_1, B_2\\}}(q) = 0$ when the bodies are in collision without interpenetration\n \\item $d_{\\{B_1, B_2\\}}(q) < 0$ when the bodies are in collision with interpenetration\n\\end{itemize}\n\nUsing the cartesian distance between the exact surfaces of $B_1$ and $B_2$ might result in a discontinuous gradient of $d_{\\{B_1, B_2\\}}$ if the surfaces of $B_1$ and $B_2$ are not convex.\nA conservative approach is to associate to each body, a strictly convex bounding volume and to compute the distance between those volumes.\n\\cite{escande:humanoids:2007} and~\\cite{escande:itro:2014} proposes a method to generate a strictly convex Sphere-Torus-Patch Bounding Volumes (STP-BV) that guarantees the gradient continuity of the proximity distance.\nThe distance between the STP-BV of $B_1$ and $B_2$ computed by an enhanced GJK~\\cite{gilbert-1988a} collision-detection algorithm is a continuously differentiable pseudo-distance.\nThus, we can use this function in our optimization algorithm to ensure that the distance between the bodies is greater than a safety distance $\\epsilon_{12}$:\n\n\\begin{equation}\n \\boxed{d_{\\{B_1, B_2\\}}(q) \\geq \\epsilon_{12}}\n\\end{equation}\n\nThis function can be used to avoid collisions between a robot and the environment as well as auto collisions between bodies of the same robot.\nWe denote $Coll$ the list of triplets $\\{B_i, B_j, \\epsilon_{ij}\\}$ defining each collision that we want to avoid.\n\nThen the set of constraints to add to our problem is:\n\n\\begin{equation}\n \\boxed{\\forall \\{B_i, B_j, \\epsilon_{ij}\\} \\in Coll,\\ d_{\\{B_i, B_j\\}}(q) \\geq \\epsilon_{ij}}\n\\end{equation}\n\nIn many cases, it is possible to avoid the collision between two bodies of a robot by modifying the joint limits and reducing them to a span where the collision of interest cannot happen.\nThat approach is conservative and ad-hoc but can save some precious computation time.\n\n\n%\\section{Static constraints}\n%\\label{sec:static_constraints}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SECTION STATIC CONSTRAINTS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{External Forces}\n\\label{sec:external_forces}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION EXTERNAL FORCES %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nFor a robot to contact with the real world (or another robot), its geometric description is not enough.\nThe robot is subject to forces applied on its bodies as a reaction to that it applies, and which can be generated by contacts with the environment or with another actor (human, another robot, manipulated object...), by the effect of physical forces fields like gravitation or magnetism, or by contacts between two bodies of the robot.\nOur posture generator must take those `External forces' into account, to be able to estimate the stability of the robot and compute the internal torques generated in the joints, or to be able to generate desired force-driven posture for given tasks when needed.\n\nAn external force applied on a rigid body can be represented by a \\emph{wrench}, from screw theory, and is composed of a resultant part $f$ (sometimes called force) and a moment part (sometimes called couple).\nLet $w$ be a wrench, $w|_F^O$ is the expression of $w$ calculated at the point $O$ expressed in the frame $F$.\nWe denote $\\vec{f}$ the resultant part of $w$, and $\\vec{f}|_F$ the expression of $\\vec{f}$ in $F$.\n$\\vec{m}$ is the moment part of $w$ and $\\vec{m}|_F^O$ the expression of $\\vec{m}$ in $F$ calculated at the point $O$.\n\n\\begin{equation}\n w|_F^O = \\left\\{ \\begin{array}{r}\n \\vec{m}\\\\\n \\vec{f}\\\\\n \\end{array} \\right\\}^O_F\n = \\left\\{ \\begin{array}{r}\n \\vec{m}|_F^O\\\\\n \\vec{f}|_F\\\\\n \\end{array}\\right\\}\n\\end{equation}\n\nThe expression of the moment part on a different point $P$ is given by the Varignon formula:\n\n\\begin{equation}\n \\vec{m}|_F^P = \\vec{m}|_F^O + \\overrightarrow{PO} \\wedge \\vec{f}|_F\n\\end{equation}\n\nThe resultant part is invariant with respect to the point at which the wrench is calculated.\n\nWe drop the frame subscript when the choice of the frame does not matter and all quantities are computed in the same frame.\n\n\n\n\\section{Static stability}\n\\label{sec:static_stability}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION STATIC STABILITY %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWe denote $g$ the acceleration of gravity on earth $g = 9.81 m.s^{-2}$.\nThe wrench associated with the action of gravity on a body of mass $M$ which center of mass is denoted $G$ with $\\vec{z}$ the upward vertical vector in the world frame $F_0$ is:\n\\begin{equation}\n w_g|^G_{F_0} = \\left\\{ \\begin{array}{r}\n \\vec{0} \\\\\n -Mg\\vec{z} \\\\\n \\end{array}\\right\\}^G_{F_0}\n\\end{equation}\n\nA solid is statically stable if it satisfies the Euler-Newton Equation.\nWe consider a robot on which $m$ external wrench $w_i = \\left\\{ \\begin{array}{r}\n \\vec{m_i}\\\\\n \\vec{f_i}\\\\\n\\end{array} \\right\\}^{P_i}$ are applied.\nWe denote $P$ the application point at which the equation and all its terms are calculated:\n\\begin{equation}\n \\sum\\limits_i w_i|^P + w_g|^P = 0\n\\end{equation}\nwhich is equivalent to:\n\\begin{equation}\n\\left\\{\n\\begin{array}{r}\n \\sum\\limits_i \\vec{m_i}|^P + \\overrightarrow{GP}\\wedge Mg\\vec{z} = 0 \\\\\n \\sum\\limits_i \\vec{f_i} - Mg\\vec{z} = 0 \\\\\n\\end{array}\n\\right.\n\\end{equation}\n\nThis equation can be simplified by applying it at the center of mass of the body as:\n\\begin{equation}\n \\left\\{\n \\begin{array}{r}\n \\sum\\limits_i \\vec{m_i}|^G = 0 \\\\\n \\sum\\limits_i \\vec{f_i} - Mg\\vec{z} = 0 \\\\\n \\end{array}\n \\right.\n\\label{eq:stability}\n\\end{equation}\n\nSatisfying equation~\\Eqref{eq:stability} ensures the static equilibrium of a rigid body.\nIf the robot's actuators are powerful enough to maintain its posture under any external perturbation, namely, when they can generate infinite or at least large enough torques, then the robot can be approximated as a rigid body and satisfying equation~\\Eqref{eq:stability} is enough to ensure its stability.\n%In some cases, an articulated robot is considered as a rigid body and this equation alone can be used to ensure its stability.\n%It is only valid if the robot can generate infinite torques in its articulations, or at least if we have some guarantee that the robot is able to generate large enough torques.\nOtherwise, it is necessary to verify that the robot's actuators can generate large enough torques to maintain that posture.\nThe details concerning the torque computations are discussed in Section~\\ref{sec:torque_limits}\n\nEquation~\\Eqref{eq:stability} can be used in an optimization problem.\nWe consider that each wrench $w_i$ applied on the system is defined by the position of its application point $P_i$ and the values $m_i$ and $f_i$ that represent the moment and resultant of $w_i$ at $P_i$.\n$P_i$ depends on $q$ the joint parameter of the robot.\n$m_i$ and $f_i$ are new variables that need to be added to the problem.\nIn summary, $w_i$ depends on $q$, $m_i$ and $f_i$.\nWe denote $f$ the concatenation of all the variables $m_i$ and $f_i$.\n\n\\begin{equation}\n \\boxed{s(q,f) = \\left\\{\n \\begin{array}{r}\n \\sum\\limits_i \\vec{m_i} + \\overrightarrow{P_i G}\\wedge \\vec{f_i} \\\\\n \\sum\\limits_i \\vec{f_i} - Mg\\vec{z} \\\\\n \\end{array}\n \\right\\}\n = 0}\n\\end{equation}\n\nThe optimization problem~\\Eqref{eq:optim_form_PG} becomes (we denote m the dimension of the force variables):\n\n\\begin{equation}\n\\label{eq:optim_form_PG_with_stab}\n \\left\\{\n \\begin{array}{l}\n \\min\\limits_{q\\in\\mathcal{C}, f\\in \\mathbb{R}^m}{f(q)}\\\\\n \\text{ s.t. }\n \\left\\{\n \\begin{array}{l}\n s(q,f) = 0\\\\\n c_i(q) = 0,\\ \\forall i\\in{E}\\\\\n c_i(q) \\geq 0,\\ \\forall i\\in{I}\\\\\n \\end{array}\n \\right.\n \\end{array}\n \\right.\n\\end{equation}\n\nThe derivation of the static stability constraint is straightforward.\nAll the terms of equation~\\Eqref{eq:stability} are components of wrenches.\nA wrench is completely defined by the frame in which it is expressed and its values of resultant and moment in that frame.\nDeriving the stability condition boils down to deriving each term w.r.t its components values and w.r.t the transformation of its frame.\n\n\\begin{equation}\n\\left\\{\n\\begin{array}{r}\n \\partial\\left(\\sum\\limits_i m_i|^G\\right) = \\sum\\limits_i \\partial(m_i|^G) \\\\\n \\partial\\left(\\sum\\limits_i f_i\\right) = \\sum\\limits_i \\partial(f_i) \\\\\n\\end{array}\n\\right.\n\\label{eq:derivation_stability}\n\\end{equation}\n\nWe will explicit a method to automatically compute those derivatives in a later chapter.\n\n\n\n\\section{Center of mass projection}\n\\label{sec:center_of_mass_projection}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION CENTER OF MASS PROJECTION %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nWhen all the wrenches applied to the body are due to unilateral punctual contacts on the same horizontal plane $H = \\{O, \\vec{x}, \\vec{y}\\}$, the stability criterion~\\Eqref{eq:stability} can be simplified.\nThe wrench $w_i$ generated by a unilateral punctual contact is a pure force resultant, its moment part is zero on the contact point.\n\n\\begin{equation*}\n \\left. w_i \\right|^{P_i} =\n \\left\\{\n \\begin{array}{r}\n \\vec{0}\\\\\n \\vec{f_i}\\\\\n \\end{array} \\right\\}^{P_i}\n\\end{equation*}\n\nEquation~\\Eqref{eq:stability} becomes:\n\n\\begin{equation}\n\\left\\{\n\\begin{array}{r}\n \\sum\\limits_i \\overrightarrow{OP_i}\\wedge \\vec{f_i} - \\overrightarrow{OG} \\wedge Mg\\vec{z} = 0 \\\\\n \\sum\\limits_i \\vec{f_i} - Mg\\vec{z} = 0 \\\\\n\\end{array}\n\\right.\n\\end{equation}\n\nWe can write $\\overrightarrow{OG} = \\overrightarrow{OG_p} + z_G\\vec{z}$ with $G_P$ the projection of $G$ on $H$. Replacing in the moment equation gives:\n\n\\begin{equation}\n \\sum\\limits_i \\overrightarrow{OP_i}\\wedge \\vec{f_i} - \\sum\\limits_i\\overrightarrow{OG_P} \\wedge \\vec{f_i} = 0\n\\label{eq:projCoM}\n\\end{equation}\n\nWith $f_i = f_i^x\\vec{x} + f_i^y\\vec{y} + f_i^z\\vec{z}$, $G$ and $P_i$ can be written as $\\overrightarrow{OG_P} = G_x \\vec{x} + G_y\\vec{y}$ and $\\overrightarrow{OP_i} = P_{ix} \\vec{x} + P_{iy} \\vec{y}$. The two first lines of equation~\\Eqref{eq:projCoM} give:\n\n\\begin{align}\n\\sum\\limits_i \\left\\{\\begin{array}{r} P_{iy}f_i^z\\\\-P_{ix}f_i^z\\end{array}\\right\\}\n= \\left\\{\\begin{array}{r} G_{y}\\\\-G_{x}\\end{array}\\right\\} \\sum\\limits_i f_i^z\\\\\n \\overrightarrow{OG_P} = \\frac{\\sum\\limits_i \\overrightarrow{OP_i} f_i^z}{\\sum\\limits_i f_i^z}\n\\end{align}\n\nSince all the contacts are unilateral, all the $f_i^z$ must be positive.\nFor any set of $f_i^z\\geq0$, $G_P$ is a barycenter with positive coefficients of the $P_i$.\nAny point $G_P$ that is included in the convex hull of all the $P_i$ is a solution.\n\nThus, we have the property: a rigid body that has all its contacts with the environment being punctual, unilateral and all lying on the same horizontal plane $H$, is said stable if and only if the projection of its center of mass on $H$ is inside the convex hull of all its contact points.\n\n\n\n\\section{Torque limits}\n\\label{sec:torque_limits}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION TORQUE LIMITS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn general, satisfying equation~\\Eqref{eq:stability} is not enough to ensure that a robot can be statically stable.\nThe joint torques that are required to hold the posture must be within the physical capabilities of the robot, namely, its torque limits.\nWe denote $\\tau_i^-$ and $\\tau_i^+$ the minimal and maximal torques that can be generated by the robot's actuators on joint $J_i$.\nIn some cases, the torque limits can depend on the joint parameters $\\tau_i^-(q)$ and $\\tau_i^+(q)$.\nFor example, it is the case with the Atlas robot that is hydraulically actuated.\n\nFeatherstone~\\cite{featherstone:book:2007} proposes a recursive algorithm to compute the torques, accelerations, and velocities generated in a multi-articulated system by a set of external forces called the Inverse Dynamics Algorithm.\nFor the purpose of generating statically stable postures, the velocities and accelerations are not needed.\nThus, we devise a specialized algorithm to fit our needs and call it the Inverse Static Algorithm.\n\nWe denote $a_g$ the gravity acceleration vector with $\\vec{a_c}$ its rotation part and $\\vec{a_f}$ its translation part:\n\\begin{equation}\n a_g = \\left\\{ \\begin{array}{r}\n \\vec{a_c} \\\\\n \\vec{a_f} \\\\\n \\end{array} \\right\\}\n = \\left\\{ \\begin{array}{r}\n \\vec{0} \\\\\n g\\vec{z}\\\\\n \\end{array} \\right\\}\n\\end{equation}\n\nThe algorithm first computes the generalized forces $f^G_i$ applied to each body.\nIt is the sum of the action of gravity and of the external forces applied on a body calculated at the origin of the world frame, expressed in the world frame.\nThen, the generalized forces are used to compute the torques.\nWe denote $\\mathbf{I}_i$ the inertia matrix of body $i$.\n\n\\begin{algorithm}\n \\caption{Inverse Static Algorithm}\n\\label{alg:IS}\n\\begin{algorithmic}\n \\For{$i = 0:n_B$}\n \\State$f^G_i = \\mathbf{I}_i {}^i\\mathbf{X}_0 a_g - {{}^i\\mathbf{X}_0}^*f_i^{ext}$\n \\EndFor{}\n \\For{$i = n_J-1:0$}\n \\State$\\tau_i = {f^G_i}^T S_i$\n \\If{$pred(i) \\neq -1$}\n \\State$f^G_{pred(i)} += {\\mathbf{X}^{PtS}_i(\\mathbf{q})}^{-*} f^G_i$\n \\EndIf{}\n \\EndFor{}\n\\end{algorithmic}\n\\end{algorithm}\n\nWe can write the torques as a function of the joint parameters and the external forces $\\tau(\\mathbf{q},\\mathbf{f})$.\nThe torque limit constraint writes as:\n\\begin{equation}\n \\boxed{\\tau^- \\leq \\tau(\\mathbf{q},\\mathbf{f}) \\leq \\tau^+}\n\\end{equation}\n\nWe will detail the derivation of this constraint in Section~\\ref{sec:torque_derivation}.\n\n\n\\section{Contact Forces and Friction Cones}\n\\label{sec:contact_forces_and_friction_cones}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SUBSECTION CONTACT FORCES AND FRICTION CONES %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%Here we use the same frames and notations as introduced in Section~\\ref{sec:contact_constraints}.\n\nThe contacts involved in a posture generation problem can be separated into two types:\n\\begin{itemize}\n \\item Geometric Contacts, and\n \\item Stability Contacts.\n\\end{itemize}\nThe Geometric Contact is a contact where the position of contact is reached with theoretically zero force, i.e. that contact does not support any forces (this is a necessary step to generate a sequence of quasi-static transitions).\nPhysically, that correspond to the transition state between a configuration without contact and a configuration with contact on which forces are applied.\nWe defined the Geometric Contacts in Section~\\ref{sec:contact_constraints}.\nThe Stability Contact is a Geometric Contact that bears contact forces.\n\n%In planning, contacts are added or removed one by one between successive postures.\n%In order to add or remove a stability contact, it is necessary to go through a geometric contact (that guarantees the continuity on contact forces).\n%First a geometric contact configuration is found, then on the next posture, that geometric contact is fixed ond forces are added on it, it becomes a stability contact.\n%Similarly, to remove a stability contact, we first look for a posture with a fixed geometric contact in place of the stability contact to remove.\n%And on the next posture, the contact can be removed.\n\nWe denote $w_{1\\rightarrow 2}$ the force applied by body $B_1$ on body $B_2$.\nThen the force applied by $B_2$ on $B_1$ is $w_{2\\rightarrow 1} = -w_{1\\rightarrow 2}$.\n%These interaction forces must be taken into account in the stability and torque computations of each robot involved in the contact.\n\nThe interaction force resulting from a punctual contact (see~\\Figref{fig:frictionCone}) on a point $P$ can be modeled as a pure force resultant along the $z_1$ direction $\\vec{f_n} = f_z \\vec{z_1}$ and the tangential efforts due to the friction in that contact can be modeled as $\\vec{f_t} = f_x \\vec{x_1} + f_y \\vec{y_1}$.\n%A punctual contact cannot carry moments on its contact point.\n\n\\begin{equation}\n\\label{eq:punctual_force}\n\\left. w_{2\\rightarrow 1}\\right|^P = \\left\\{\n \\begin{array}{l}\n \\vec{0} \\\\\n \\overrightarrow{f_{2\\rightarrow 1}} = \\vec{f_t} + \\vec{f_n} \\\\\n \\end{array}\n \\right\\}^P\n\\end{equation}\n\n\\begin{figure}[htpb]\n \\centering\n \\includegraphics[width=0.6\\linewidth]{frictionCone.pdf}\n \\caption{Punctual unilateral stability contact with friction}\n\\label{fig:frictionCone}\n\\end{figure}\n\nThis formulation of the interaction force defines a bilateral contact, in the sense that the force can be in any direction, $B_1$ can push as well as pull on $B_2$.\n\nTo model a unilateral contact, we must constrain the normal component of $f_{2\\rightarrow 1}$ to be oriented toward the inside of $B_1$.\nThis means that only pushing actions can be generated, not pulling ones.\nThis translates into:\n\\begin{equation}\n \\overrightarrow{f_{2\\rightarrow 1}}\\cdot \\vec{z_1} = f_z \\geq 0\n\\end{equation}\n\nFurthermore, to avoid slippage, the Coulomb friction law must be respected for each contact force $\\vec{f}$.\n%This law states that the contact force resultant must lay inside a friction cone of angle $\\mu$, the friction coefficient.\nWhich translates into the following equation, with $\\vec{f_n}$ and $f_t$, respectively the normal and tangential parts of $\\vec{f}$ and $\\mu$, the friction coefficient:\n\n\\begin{equation}\n \\mu\\|\\vec{f_n}\\| \\geq \\|\\vec{f_t}\\|\n\\end{equation}\n\nGiven the decomposition of $f_{2\\rightarrow 1}$ in $F_1$, $f_{2\\rightarrow 1} = f_x \\vec{x_1} + f_y \\vec{y_1} + f_z \\vec{z_1}$, for any punctual contact in a posture generation problem, we can add the following set of constraint to our optimization problem:\n\\begin{equation}\n \\label{eq:unilateralContact}\n \\left\\{\n \\begin{array}{l}\n f_z \\geq 0 \\\\\n \\mu^2 f_z^2 - f_x^2 +f_y^2 \\geq 0\n \\end{array}\n \\right.\n\\end{equation}\n\nWhen it comes to planar contacts on surface $S$ with $\\vec{n}$ the outbound normal to $S$, the interaction force can have components of forces and moments in all directions.\nThe forces components intrinsic to the planar contact model are a resultant part aligned with $\\vec{n}$ $\\vec{f_n} = f_z \\vec{z_1}$ and a moment part tangential to $S$: $\\vec{m_t} = m_x \\vec{x_1} + m_y \\vec{y_1}$.\nThe forces due to friction are a tangential friction resultant part $\\vec{f_t} = f_x \\vec{x_1} + f_y \\vec{y_1}$ and a normal friction moment $\\vec{m_n} = m_z \\vec{z_1}$.\n\nThis type of force can be modeled by a set of unilateral punctual efforts applied on each vertex of a polygon describing the contact area.\nAnd ensuring that each of them lay in their respective friction cone, thus satisfying the equation~\\Eqref{eq:unilateralContact}.\nAs depicted in~\\Figref{fig:planarContact}\n\n\\begin{figure}[htpb]\n \\centering\n \\setlength{\\fboxsep}{0pt}%\n \\setlength{\\fboxrule}{1pt}%\n \\fbox{\\includegraphics[width=0.6\\linewidth]{planarContact.pdf}}\n \\caption{Modeling of a planar contact between the foot on HRP-4 and the ground.}\n\\label{fig:planarContact}\n\\end{figure}\n\n\n\\section{Cost Functions}\n\\label{sec:cost_functions}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SECTION COST FUNCTIONS %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn addition to constraints, it is often useful to add a cost function to our optimization problem.\nThe submanifold of feasible configurations $\\mathcal{C}_F$ can contain an infinity of solutions and even some continuous solution areas in which all points are solutions.\nThe cost function helps to choose the `best' candidate solution.\nVarious types of cost functions can be chosen, for example, we can minimize the distance to a reference posture $q_R$:\n\n\\begin{equation*}\n f_\\text{posture}(\\mathbf{q}) = {\\|\\mathbf{q}-q_R\\|}^2\n\\end{equation*}\n\nThe effect of that type of cost function is illustrated in~\\Figref{fig:cost}.\nOn both images, the HRP-2Kai robot is stable, respects its joints and torques limits.\nThe only difference is that the right one minimizes the distance to a reference posture (standing straight with bent knees) while the left result does not use a cost function.\n\n\\begin{figure}[htpb]\n \\centering\n \\setlength{\\fboxsep}{0pt}%\n \\setlength{\\fboxrule}{1pt}%\n \\fbox{\\includegraphics[width=0.5\\linewidth]{cost.pdf}}\n \\caption{Effect of cost function. Left: no cost function. Right: Distance to a half-sitting predefined posture.}\n\\label{fig:cost}\n\\end{figure}\n\nOne can also want to minimize the sum of norms of the contact forces:\n\\begin{equation*}\n f_\\text{forces}(\\mathbf{f}) = \\sum\\limits_i {\\|f_i(\\mathbf{f})\\|}^2\n\\end{equation*}\nor the torques in the robot's joints:\n\\begin{equation*}\n f_\\text{torques}(\\mathbf{q},\\mathbf{f}) = \\sum\\limits_i {\\|\\tau_i(\\mathbf{q},\\mathbf{f})\\|}^2\n\\end{equation*}\nSome more custom cost functions can also be considered, for example, we may want a point $P_i$ on body $B_i$ with to be as far as possible in a direction $\\vec{d}$\n\\begin{equation*}\n f_\\text{point} (\\mathbf{q}) = -{\\overrightarrow{O_0 P_i}}\\cdot{\\vec{d}}\n\\end{equation*}\n\nAny positively weighted combination of cost function can be used, in which case it is important to choose the weights $p_i$ carefully to scale all the costs so that they all can influence the result and none is completely dominated by another.\n\\begin{equation}\n f_\\text{cost}(\\mathbf{q},\\mathbf{f}) = \\sum\\limits_i{p_i f_i(\\mathbf{q},\\mathbf{f})} = p_0 f_\\text{posture}(\\mathbf{q}) + p_1 f_\\text{forces}(\\mathbf{f}) + p_2 f_\\text{torques}(\\mathbf{q},\\mathbf{f}) + \\cdots\n\\end{equation}\n\n\n\n\\section{Conclusion}\n\\label{sec:Ch1_Conclusion}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SECTION CONCLUSION %\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn this Chapter, we have seen how to formulate a robotics problem with several different tasks and objectives as an optimization problem.\n\nWe denote $\\mathcal{T}_i$ the additional tasks added to the problem, which is described by the set of equalities $g_i(\\mathbf{q},\\mathbf{f}) = 0$ and inequalities $h_i(\\mathbf{q},\\mathbf{f}) \\geq 0$.\nThe contact tasks are included in those and the equations describing them must encompass the geometric contact constraint equation like~\\Eqref{eq:coplanarity} as well as the unilateral and friction constraints equations~\\Eqref{eq:unilateralContact} in the case of a unilateral contact.\n\nA typical robotics problem can be written as a combination of all those costs and constraints:\n\\begin{align}\n\\minimize_{\\mathbf{q}, \\mathbf{f}} & \\quad f_\\text{cost}(\\mathbf{q},\\mathbf{f}) \\nonumber\\\\\n\\text{s.t.}&\n\\left\\{\n\\begin{array}{lr}\nq^- \\le \\mathbf{q} \\le q^+\\\\\ns(\\mathbf{q},\\mathbf{f}) = 0 \\\\\n\\tau^- \\le \\tau(\\mathbf{f},\\mathbf{q}) \\le \\tau^+\\\\\n\\forall \\{B_i, B_j, \\epsilon_{ij}\\} \\in Coll,\\ d_{\\{B_i, B_j\\}}(\\mathbf{q}) > \\epsilon_{ij}\\\\\ng_i(\\mathbf{q},\\mathbf{f}) = 0\\ \\ \\forall\\mathcal{T}_i,\\\\\nh_i(\\mathbf{q},\\mathbf{f}) \\geq 0\\ \\ \\forall\\mathcal{T}_i.\n\\end{array}\\right.\n\\label{eq:PG}\n\\end{align}\n\nIn the following chapter, we present an extension to the contact constraint formulation that allows generating non-inclusive contacts, an algorithm to compute the exact derivatives of the torques in robot's joints, and our endeavor to apply a different optimization approach to solving posture generation problems.\n", "meta": {"hexsha": "343675c6dd142142c4d7ba707de7a5d9f93618aa", "size": 51900, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Chapter2-PG/chapter2.tex", "max_stars_repo_name": "stanislas-brossette/phd-thesis", "max_stars_repo_head_hexsha": "7f4d2d46dfdd1f59ac29770585e8cee6dc4f2668", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "Chapter2-PG/chapter2.tex", "max_issues_repo_name": "stanislas-brossette/phd-thesis", "max_issues_repo_head_hexsha": "7f4d2d46dfdd1f59ac29770585e8cee6dc4f2668", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Chapter2-PG/chapter2.tex", "max_forks_repo_name": "stanislas-brossette/phd-thesis", "max_forks_repo_head_hexsha": "7f4d2d46dfdd1f59ac29770585e8cee6dc4f2668", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 48.1894150418, "max_line_length": 399, "alphanum_fraction": 0.6678034682, "num_tokens": 15265, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6825737214979745, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3439531101038274}} {"text": "\\documentclass[11pt]{article}\n\\usepackage{graphicx}\n\\usepackage{amssymb}\n\\usepackage{multicol}\n\\usepackage{epstopdf}\n\\DeclareGraphicsRule{.tif}{png}{.png}{`convert #1 `dirname #1`/`basename #1 .tif`.png}\n\n\\textwidth = 6.5 in\n\\textheight = 9 in\n\\oddsidemargin = 0.0 in\n\\evensidemargin = 0.0 in\n\\topmargin = 0.0 in\n\\headheight = 0.0 in\n\\headsep = 0.0 in\n\\parskip = 0.2in\n\\parindent = 0.0in\n\n\\newtheorem{theorem}{Theorem}\n\\newtheorem{corollary}[theorem]{Corollary}\n\\newtheorem{definition}{Definition}\n\n\\title{{\\tt kuramoto\\underline{ }1d.m} - Kuramoto-Sivashinsky Solvers}\n\\author{Jeff Borggaard}\n\\begin{document}\n\\maketitle\n\n\\section*{Summary}\nThis software (and the similar {\\tt kuramoto\\underline{ }1da.m}) solves the \nKuramoto-Sivashinsky equation using $C^1$ Hermite cubic elements in a \nperiodic domain.\n\n\\section*{The Equations}\nThere are two popular forms of the Kuramoto-Sivashinsky equations. The first\nmimics the form originally derived by Kuramoto \\cite{kuramoto}.\n%\n\\begin{equation}\n\\label{eq:ks1}\n d\n\\end{equation}\n\nThe second has the form of the derivative of equation (\\ref{eq:ks1}), namely\n%\n\\begin{displaymath}\n w_t + w_{xx} + w_{xxxx} + 2 w w_x = 0.\n\\end{displaymath}\n%\nNote that if we scale the spatial variable so that the computational\ndomain occurs over the unit interval: $\\bar{x} = x/L$, and define\n$\\epsilon = 1/L^2$, then we have the following version of the Kuramoto-Sivashinsky equation\n%\n\\begin{equation}\n\\label{eq:ks2}\n w_t + \\epsilon w_{\\bar{x}\\bar{x}} + \\epsilon^2 w_{\\bar{x}\\bar{x}\\bar{x}\\bar{x}} + 2 \\epsilon w w_{\\bar{x}} = 0\n\\end{equation}\n%\nwith $t>0$, $\\bar{x} \\in (0,1)$, $w(0,t) = w(1,t)$ and $w_{\\bar{x}}(0,t) = \nw_{\\bar{x}}(1,t)$. The initial condition is scaled so that \n$w(\\bar{x},0) = w_0(\\bar{x})$.\n\nA nice property of solutions to equation (\\ref{eq:ks2}) is that the integral\n%\n\\begin{equation}\n\\label{eq:w_ave}\n \\int_0^1 w(\\bar{x},t) \\ d\\bar{x}\n\\end{equation}\n%\nis independent of time. In other words, the average of the solutions is\ndetermined from the average of the initial conditions. This imposes a \ngreat deal of structure on the solutions as will be seen in the numerical\ntests.\n\n\n\\section*{Parameters}\nInput parameters\n%\n\\begin{itemize}\n \\item {\\tt epsilon} - \\newline\n This parameter has the effect of changing the length of the periodic\n domain. See the scaling of the equations above.\n\\end{itemize}\n\nInternal parameters\n%\n\\begin{itemize}\n \\item {\\tt n\\underline{ }gauss} - \\newline\n The number of Gauss point used in element integration\n \\item {\\tt n\\underline{ }nodes} - \\newline\n The number of nodes (including the shared endpoints) used in the \n spatial discretization. The number of Hermite elements is equal to\n {\\tt n\\underline{ }nodes-1}.\n \\item {\\tt t\\underline{ }initial}, {\\tt t\\underline{ }step}, \n {\\tt t\\underline{ }save}, and {\\tt t\\underline{ }final} - \\newline\n Time integration parameters. The solution is saved at multiples of\n {\\tt t\\underline{ }save}, so {\\tt t\\underline{ }save}/{\\tt t\\underline{ }step}\n should be a positive integer.\n \\item {\\tt resid\\underline{ }tol} - \\newline\n Residual tolerance for Newton solve at each timestep.\n \\item {\\tt step\\underline{ }tol} - \\newline\n Step tolerance for Newton solve at each timestep.\n \\item {\\tt max\\underline{ }iterations} - \\newline\n Number of Newton iterations attempted to achieve one of the convergence\n tolerances in each timestep.\n\\end{itemize}\n\n\\section*{Dynamical Systems Analysis}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lll}\n\\hline\nlength $L$ & \\multicolumn{1}{c}{$\\epsilon$} & solution behavior \\\\\n\\hline\n12.5664 & 0.00633257 & bifurcation \\\\\n12.8767 & 0.00603102 & heteroclinic bifurcation \\\\\n13.1403 & 0.00579148 & hopf bifurcation \\\\\n402.2590 & 0.00000618 & ``chaos'' \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\end{document}", "meta": {"hexsha": "4a415ed8eabddf974da033ef20d89a3403142210", "size": 3821, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "SplitDMD_KS-master/FEM code/kuramoto_1d.tex", "max_stars_repo_name": "jovanzigic/SplitDMD_KS", "max_stars_repo_head_hexsha": "ddfe2eb81ffcda58837e3e7fe63476ff578b8bf4", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "SplitDMD_KS-master/FEM code/kuramoto_1d.tex", "max_issues_repo_name": "jovanzigic/SplitDMD_KS", "max_issues_repo_head_hexsha": "ddfe2eb81ffcda58837e3e7fe63476ff578b8bf4", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "SplitDMD_KS-master/FEM code/kuramoto_1d.tex", "max_forks_repo_name": "jovanzigic/SplitDMD_KS", "max_forks_repo_head_hexsha": "ddfe2eb81ffcda58837e3e7fe63476ff578b8bf4", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.3196721311, "max_line_length": 112, "alphanum_fraction": 0.7110704004, "num_tokens": 1264, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961013, "lm_q2_score": 0.6076631698328917, "lm_q1q2_score": 0.3439486172814331}} {"text": "\n\n \\filetitle{srf}{Shock (impulse) response function}{SVAR/srf}\n\n\t\\paragraph{Syntax}\\label{syntax}\n\n\\begin{verbatim}\n[Resp,Cum] = srf(V,NPer)\n[Resp,Cum] = srf(V,Range)\n\\end{verbatim}\n\n\\paragraph{Input arguments}\\label{input-arguments}\n\n\\begin{itemize}\n\\item\n \\texttt{V} {[} SVAR {]} - SVAR object for which the impulse response\n function will be computed.\n\\item\n \\texttt{NPer} {[} numeric {]} - Number of periods.\n\\item\n \\texttt{Range} {[} numeric {]} - Date range.\n\\end{itemize}\n\n\\paragraph{Output arguments}\\label{output-arguments}\n\n\\begin{itemize}\n\\item\n \\texttt{Resp} {[} tseries \\textbar{} struct {]} - Shock response\n functions.\n\\item\n \\texttt{Cum} {[} tseries \\textbar{} struct {]} - Cumulative shock\n response functions.\n\\end{itemize}\n\n\\paragraph{Options}\\label{options}\n\n\\begin{itemize}\n\\item\n \\texttt{'presample='} {[} \\texttt{true} \\textbar{}\n \\emph{\\texttt{false}} {]} - Include zeros for pre-sample initial\n conditions in the output data.\n\\item\n \\texttt{'select='} {[} cellstr \\textbar{} char \\textbar{} logical\n \\textbar{} numeric \\textbar{} \\emph{\\texttt{Inf}} {]} - Selection of\n shocks to which the responses will be simulated.\n\\end{itemize}\n\n\\paragraph{Description}\\label{description}\n\n\\paragraph{Example}\\label{example}\n\n\n", "meta": {"hexsha": "8c49f6d7789dc88e78110a168716c3879187fcb2", "size": 1258, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "-help/SVAR/srf.tex", "max_stars_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_stars_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-12-06T13:38:38.000Z", "max_stars_repo_stars_event_max_datetime": "2017-12-06T13:38:38.000Z", "max_issues_repo_path": "-help/SVAR/srf.tex", "max_issues_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_issues_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2017-03-28T08:13:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-02T10:40:25.000Z", "max_forks_repo_path": "-help/SVAR/srf.tex", "max_forks_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_forks_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-17T07:06:39.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-17T07:06:39.000Z", "avg_line_length": 23.7358490566, "max_line_length": 70, "alphanum_fraction": 0.6939586645, "num_tokens": 394, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.3439486172814331}} {"text": "\\section*{Results}\nWhile the prospect of private property drives cooperation,\\footnote{I don't know how this assertion finds an operationalization in the model $\\rightarrow$ maybe rephrase or even drop} cooperation is itself sensitive to property violation. While small quantities of property violation introduce some amount of randomness that enhances cooperation (see Figure \\ref{fig:phase_transition}a), we focus on the limit amount of property violation beyond which cooperation transitions to defection.\\footnote{as said above the overall rational is that private property enforcement is costly for society, and hence there is an incentive to find the maximum bearable level of private property.}\\\\\n\nOur model operationalizes the action of property violation as a uniform random variable [$p(S \\geqslant s) = 1$], with sure strategy update ($r=0$),\\footnote{for strategy update we resort to the update method proposed by Helbing and Yu \\cite{helbing2009outbreak}, which in our noiseless case is equivalent to the Fermi update with Fermi temperature equal to 1.} no random strategy reset ($q=0$) and no random migration to an empty site ($m=0$). And as players choose their best shot in the success-driven migration step, we can say that they are fully rational agents (in their migration range), which choice to expel another individual results only from odds of property enforcement.\\footnote{These parameters settings ensure the simplest possible model that guaranties that the observed stylized facts are genuine to the property violation phenomenon.} \\\\\n\n\\subsection*{Population density, Migration Range and Cooperation Level} \nThe outcome of the property violation game is nevertheless constrained by the initial conditions, such as population density and migration range: Increased population density reduces the opportunities to find an empty location, and thus, mechanically increases the odds of property violation. The migration range on the one hand increases the chances to find an empty sweet spot in case of success-driven or forced migrations. On the other hand, it also increases opportunities for property violators. Migration influences the capability by defectors to penetrate clusters of cooperators. However, expelled cooperators can move far enough to make the cluster move away from the defector. Hence, larger migration ranges increase tolerance to property violation (see Figure \\ref{fig:phase_transition}b), but their beneficial effects are limited for highly densely populations. On the contrary, populations with very short migration ranges -- here, embodied by unitary migration distance ($M=1$) -- can hardly stand any property violation. \n\n%Nevertheless, the intensity of property violation is determinant in the ability by defectors to break clusters of cooperators %(see Figure \\ref{fig:viz} and online visualization tool).\n\n\\subsection*{Optimizing for Property Violation is Managing at the Edge of Collapse}\nProperty enforcement is costly, and thus, there is an incentive to limit enforcement to the strict minimum, which will ensure high cooperation (which is equivalent to decide how much property crime a society can stand). For large enough migration ranges ($M\\geqslant 3$), a little property violation ($s < 1 \\%$), brings the necessary randomness that reduces friction and enhances cooperation, similarly to previous results obtained with other types of noise involved in success-driven migration games \\cite{helbing2009outbreak}. For larger property violation $s > 1\\%$, cooperation decreases linearly according to $ c = 1 - 2.2\\cdot s$, until a critical phase transition is reached (see Figure \\ref{fig:phase_transition}a), which itself depends on the population density and the migration range as reported on Figure \\ref{fig:phase_transition}b. There is no hard limit, but rather a transition from sure striving society with high cooperation to a sure collapse. The region in between is large compared to the absolute value of the upper limit: For instance, for $M=5$ and $d=0.5$, the phase transition spans from $s = 3.7\\%$ to $s= 4.2\\%$, with an equal chance to survive or collapse for property violation $s=4\\%$. Hence, the ``danger zone\" spans over more than 10\\% of the limit value, and the decreasing probability of striving in the critical phase transition region, makes it very hard to determine whether the course of a society will end up positively or negatively.\\footnote{We have verified that the outcome is independent from the initial grid organization, by running multiple simulations from the same initial grid.}\n\n\\subsection*{Decoupling for Sustainable Cooperation}\nThe sudden phase transition from high cooperation to collapse suggests a ``grain of sand\" factor deciding whether a society operating at the limit level of property violation will strive or collapse.\\footnote{Again, it's important to stress that it is natural for a society to operate close to the limit, since private property enforcement is costly, and even sometimes socially undesired.}\\\\\n\nTo better determine this ``grain of sand\", we resort to simple of economics implied by the property game and opportunities evolve over time: To maximize their payoff, individuals are endowed with three basic rules, \n\n\\begin{itemize}\n \\item {\\bf strategy update rule:} the player updates its prisoner's dilemma strategy (i.e., cooperate or defect), as a function of her neighbors strategies,\n \\item {\\bf success driven migration:} the player migrates to the best empty site within her migration area determined by the Moore distance $M$,\n \\item {\\bf property violation:} the player attempts to expel with probability $s$ another player from her location within their migration area determined by the Moore distance $M$.\n\\end{itemize}\n\nThe three strategies are opportunity based, deterministic for the first two (i.e., if players are presented with better options they will update their strategy or migrate) and random for property violation, reflecting the unsure nature of taking the property of someone else. While decisions are idiosyncratic at the individual level, at the aggregate level, they reveal the expected utility $U$ carried by each rule, which can be formulated as \n\n\\begin{equation}\nU_{rule} = p_{rule} \\times u_{rule},\n\\label{U}\n\\end{equation}\n\nwith $p_{rule}$ the probability to execute one of the 3 {\\it rules} defined above (i.e., $rule = \\{$update, migration, expel$\\}$, and $u_{rule}$ the payoff increase from executing a {\\it rule}. For prisoner's dilemma strategy update and success-driven migration $p_{rule} = (1~|~u_{rule} > 0)$ (reflecting the deterministic decision to execute these rules) and for property violation $p_{rule} = (s~|~u_{rule} > 0)$, reflecting the unsure nature of attempting to expel another individual. For each rule, $U_{rule}(t)$ provides a direct measure -- at the aggregate level -- of opportunity associated with each rule and how it evolves. Similarly, one can define an expected mobility distance,\n\n\\begin{equation}\nN_{rule} = p_{rule} \\times n_{rule},\n\\end{equation}\n\nwhere $p_{rule}$ is similar as in (\\ref{U}) and $n_{rule}$ is the migration distance. $N_{rule}$ shows the expected travel distance for success-driven and property violation migration rules.\\\\\n\nStarting from a population randomly scattered across the grid with an even number of cooperators and defectors, $U$ and $N$ undergo significant changes over time, with subtle yet determinant differences between a striving and a collapsing society (see Figure \\ref{fig:tseries} for an illustration of a successful and a failed outcome with similar parameters $d=0.5$, $M=5$ and $s=0.037$; This particular case illustrates similar stylized facts obtained for different values of $d$, $M$ and $s$). In both the successful and the failed dynamics, {\\it success-driven} migration provides the highest expected payoff increase ($U_{migration} > U_{update} > U_{expel}$) in the early iterations $t < 1.5\\times10^{5}$ (over $20\\times10^6$ MCS iterations). After $t > 1.5\\times10^{5}$, a significant change occurs: In striving societies (Figure \\ref{fig:tseries}b) a switch of expected payoff occurs, which becomes : $U_{expell} > U_{update} > U_{migration}$, while in the failed society case (Figure \\ref{fig:tseries}b), strategy update provides the highest expected increased payoff, and the expected increased payoff from success-driven and property violation migrations converge.\\footnote{ nb: It is important to note that expected payoff is not necessarily increasing overall the entire population: players expelled from their site suffer a negative expected payoff increase (not shown), as well as neighbors around the focal player who changes prisoner's dilemma strategy to defect, or a defector migrating or performing a property violation.} At the time of the switch, we observe an inflection point of the cooperation level (purple dashed circle on Figure \\ref{fig:tseries}b): After this point, cooperation still increases in the failed case, but at much lower pace,\\footnote{Mostly driven by strategy updates, which provide the highest expected payoff increase? These updates are at first of the type : $D \\rightarrow C$ and then, after the peak rather $C \\rightarrow D$. The intuition is that, somewhen migration (whether success driven or property violation) does bring enough payoff increase, compared to strategy update, therefore something weird occurs which is that cooperation increases (more or larger clusters?), which in turn prepare for an invasion of defectors!} on the contrary to striving societies for which we observe a smooth continuous increase of cooperation, up to saturation and stabilization at a high level ($c > 0.8$). In the failed scenario illustrated in Figure \\ref{fig:tseries}, the peak of cooperation is reached at $t \\approx 5\\times10^5$, and collapse occurs after roughly$10^6$ iterations, nearly $850,000$ iterations after the switching point!\\\\\n\nWhile it is hard to observe the detailed switching mechanisms occurring at the transition point,\\footnote{This may be done in the future.} we find that the expected mobility (the probability to remains high in the failed scenario in comparison with the successful case (Figures \\ref{fig:tseries}c and \\ref{fig:tseries}d).\\footnote{It is important to note that mobility by property violation and consecutively by individuals forced to move, is not nearly as large as the mobility induced by success-driven actions (c.f., Figures \\ref{fig:tseries}b Inset 2).} Also, we find that in the successful scenario, a strong decoupling between expected payoff increase occurs (Figures \\ref{fig:tseries}a inset 1), while for the collapsing scenario, dependence between expected payoff remain high and get even reinforced (Figures \\ref{fig:tseries}b inset 2).\\footnote{The correlogram before the switch also shows high dependence (not shown here), which is natural somehow, as the system gets organized {\\bf [This correlogram (pre-switching) should be shown too, at least in the supplementary materials. In the failing scenario, it shows that property violation may influence ]}} Moreover, in the latter case, success-driven migration actions strongly drive private property violations (green correlogram) and strategy updates (red correlogram).\\footnote{The correlation peak occurs for a negative lag (roughly 15-20 bins {\\bf [I must check the size of a bin in terms of iterations]}) when considering the influence of private property violations and strategy updates on migration} Dependence between property violation and strategy update is also high (blue correlogram), but with no lag thus one cannot say whether $E \\rightarrow U$ or $U \\rightarrow E$. One can infer however that they are both influenced by success-driven migration (see red and green correlograms).\\footnote{It is unclear however what makes success-driven migration the driver of other actions. Something special must occur before or at the switching point.}\\\\\n\nPast the very first iterations where success-driven migration is the predominant way to increase payoff (in order to build and reach clusters of cooperators, regardless of whether the moving player is a cooperator or a defector), our findings draw two completely different stories leading to either collapse or well-established cooperative societies. In the latter scenario, expected payoff from success-driven migration rapidly decays to a level comparable to the expected payoff derived from strategy update and property violation. Additionally expected payoff increase from the three rules get highly independent: A spatio-temporal decoupling occurs and players no longer undergo long-range migration, and while property violation brings the highest expected payoff increase, it is tackled by some kind of locally established self-organization \\footnote{(which remains to be explained/documented in further details)}. On the contrary, the collapse scenario is driven by long-range migrations,\\footnote{We need perhaps more insights on the nature of these migrations, e.g., do they spread defectors or rather cooperators? Does this evolve over time?} which in turn trigger strategy updates and property violation.\\footnote{These long-range spatio-temporal dependences look like systemic risk to me, which may appear in a seamless, subtle and hardly noticeable way!} \\footnote{It would be interesting to check the dynamics with a residual level of property violation, e.g., $s=0.01$, and with a super high level of $s$, to see if these extreme cases magnify differences between success and failure.}\n\n\\subsection*{Enduring and Adaptive Societies Facing Property Violation}\nOne striking observation is that even though the system switches early on, and determines the final outcome, changes may remain unnoticed for a while, as cooperation increases for 4 times more iterations (since $t=0$) until collapse actually occurs.\\footnote{In some simulations, the inflection point in the increase of cooperation is hardly noticed (e.g. Figure \\ref{fig:adaptation})}\\\\\n\nThus, on the one hand it may take time to realize that a society is running towards the edge of collapse, and on the other hand, it may take time to take corrective action, such as taking actions to reduce private property violations.\\footnote{As a result of Figure \\ref{fig:phase_transition}b, one could also think of increasing migration ranges (which would require massive and long-term infrastructure investments) and decreasing population density, which concretely in our world would mostly require reducing population and would be ethically questionable. Here, we limit our study to private property violations as the main control parameter.}\\\\\n\nBudgets allocated to property right enforcement may vary over time, in particular, a typical situation is that property crime is low and therefore there is a reduced sense of urgency and budgets might be reduced as a consequence. Also on the long term, institutions concerned with property enforcement may experience downturns of efficiency (e.g., through some kind of corruption).\\\\\n\nHere, we ask two questions: \n\n\\begin{enumerate}\n \\item {\\bf Resistance:} Given that a cooperative society is in the process of establishing \\footnote{I am currently running simulations for this.} or has stabilized at a high level of cooperation (given a level of property violation), how much more property violation and for how long can this society stand a higher level of property violation?\n \\item {\\bf Adaptation:} Given that a society is on the path to collapse (as shown e.g., on Figure \\ref{fig:tseries}b), how early and by how much property violation shall be reduced to ensure it will strive by reaching a stable high level of cooperation?\n\\end{enumerate}\n\n\\paragraph{Resistance:} We find that when a society has stabilized at a high cooperation level ($s=4\\%$), increasing a property violation a little ($0.05 \\leqslant s \\leqslant 0.06$) increases the chance that cooperation will collapse suddenly yet on the long run ( iterations > $3 \\times 10^5$ after change of property violation level). For higher levels of property violation ($s>0.06$), the cooperation curves decreases in a more or less parabolic shape, which seems to be invariant in the limit of $s \\rightarrow \\infty$, and it looks like that the zero-cooperation level can be hit at earliest around $3 \\times 10^5$ after the increase of property violation (see Figure \\ref{fig:resistance}).\n\n\\paragraph{Adaptation:} The tipping point ($t \\approx 10^5$) occurs way before the edge of actual collapse ($t \\approx 10^{5.7}$), and thus, a very natural question is whether it is possible to take corrective actions to prevent the collapse of cooperation once the tipping point has passed and has \nbeen detected. We find that unless property violation can be reduced drastically (e.g., 75\\%, c.f., Figure \\ref{fig:adaptation}d), action must be taken at latest when cooperation is at its maximum to ensure resilience (see Figure \\ref{fig:adaptation}c when property violation is halved). For more realistic property violation reduction (10\\% to 25\\%, c.f. Figures \\ref{fig:adaptation}a and\\ref{fig:adaptation}b), the policy maker must intervene as the cooperation still increases. We could of course imagine more complicated and progressive property violation mitigation schemes (which is actually more relevant for drastic actions to reduce of property violation that may take time to take effect), but the idea is that in order to avoid collapse, action must be taken {\\bf before} the effects of property violation get actually widely noticed.\\footnote{with more simulations, we could actually compute thoroughly the odds of survival given timing and effort deployed to reduce property violation, but I think what is presented here is enough to demonstrate that it is super important to avoid doing what politicians do -- or are forced to do by the ``real-politics\" democratic mechanisms, which is taking action only when the problem get sufficiently noticeable by the population.}\\\\\n\n", "meta": {"hexsha": "a56223a7a5d329dace58cbf52581f9706adf0c89", "size": 18010, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "manuscript/sections/results.tex", "max_stars_repo_name": "wazaahhh/pgames", "max_stars_repo_head_hexsha": "acf6fbb86d689ee307b6b2f807bc29fb6a818535", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "manuscript/sections/results.tex", "max_issues_repo_name": "wazaahhh/pgames", "max_issues_repo_head_hexsha": "acf6fbb86d689ee307b6b2f807bc29fb6a818535", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2015-11-06T18:21:13.000Z", "max_issues_repo_issues_event_max_datetime": "2015-11-06T20:28:50.000Z", "max_forks_repo_path": "manuscript/sections/results.tex", "max_forks_repo_name": "wazaahhh/pgames", "max_forks_repo_head_hexsha": "acf6fbb86d689ee307b6b2f807bc29fb6a818535", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-02-01T15:55:10.000Z", "max_forks_repo_forks_event_max_datetime": "2022-02-01T15:55:10.000Z", "avg_line_length": 277.0769230769, "max_line_length": 2681, "alphanum_fraction": 0.7962798445, "num_tokens": 3837, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6619228758499941, "lm_q2_score": 0.519521321952093, "lm_q1q2_score": 0.34388304749192006}} {"text": "% !Mode:: \"TeX:UTF-8\"\n% !TEX program = xelatex\n\\section{Summary}\nIf English is not your first language then writing the mathematical paper is doubly difficult, since you cannot guarantee the proper use of words even though with a large scientific and technical vocabulary. Therefore, Halmos comes up with three specific advice on how to use scientific and technical terms carefully,\n\\begin{enumerate}[label={\\Alph*.}]\n\t\\item Avoid using technical terms as much as possible, especially creating new ones;\n\t\\item Be careful when thinking about the new terms you have to create by looking up the dictionary to make it as appropriate as possible;\n\t\\item Correctly and consistently use the old terms without awkwardness.\n\\end{enumerate}\n\nIn addition to the terminology, you should also pay attention to the language itself. Science and technology English is generally not emotional, and its expression is relatively straightforward, which helps to make the readers easy to understand without creating too much imagination irrelevant. That is, the vocabulary using in science and technology English requires specific and sTable. Therefore, you need to accumulate basic mathematical vocabulary, mathematical symbols with their pronunciation, common phrases and language structures to better write rational mathematics papers.\n\n\n\\subsection{Basic Vocabulary}\nBasic vocabulary is an important prerequisite because you have to grapple with a large vocabulary as you try to express your thoughts. There is a large mathematical vocabulary, but the spelling is relatively simple compared to other disciplines, and as long as you master a certain amount of mathematical vocabulary, writing an uncomplicated mathematics paper will not be too difficult.\n\nPlease see Table~2.1 to Table~2.13 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the basic vocabulary in different fields of mathematics.\n\n\n\\subsection{Symbol Overview}\nMathematical language is the universal language, therefore, when using English to write technical papers or using spoken English to express certain symbols in scientific communication, you should understand the English expression of the common mathematical symbols. In this way, you can do more with less.\n\nPlease see Table~2.14 to Table~2.17 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the common symbols in different fields of mathematics.\n\n\n\\subsection{Common Phrases}\nThe common mathematical phrases found in professional English textbooks or theoretically papers are very practical and require proper collection and organization because proficiently using these phrases can be beneficial to your writing.\n\nPlease see Table~2.18 to Table~2.20 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the common phrases.\n\n\n\\subsection{Language Structures}\nSince mathematics writing has certain rules and formats in language structures and styles, there are many fixed modes in mathematics writing, which include some common sentence patterns, modified vocabulary, verbs in mathematical operations and transition statements.\n\\begin{itemize}[label={\\dag}]\n\t\\item Please see Table~2.21 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the common sentence patterns;\n\t\\item Please see Table~2.22 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the modified vocabulary;\n\t\\item Please see Table~2.23 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the verbs used in mathematical operations;\n\t\\item Please see Table~2.24 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the common verbs used in mathematics writing;\n\t\\item Please see Section~2.4.5 in \\emph{Mathematical Writing in English}\\cite{2013数学之英文写作} about the transition statements.\n\\end{itemize}\n\n\n\n\\nocite{2013数学之英文写作,Nicholas1998Handbook}\n\\bibliography{ref}\n", "meta": {"hexsha": "a4ebc91721c3d910be15c55713658253226a50a5", "size": 3844, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "MA320/sections/3.tex", "max_stars_repo_name": "iydon/homework", "max_stars_repo_head_hexsha": "253d4746528ef62d33eba1de0b90dcb17ec587ed", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 8, "max_stars_repo_stars_event_min_datetime": "2019-10-20T08:18:54.000Z", "max_stars_repo_stars_event_max_datetime": "2021-07-11T12:14:56.000Z", "max_issues_repo_path": "MA320/sections/3.tex", "max_issues_repo_name": "iydon/homework", "max_issues_repo_head_hexsha": "253d4746528ef62d33eba1de0b90dcb17ec587ed", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2022-01-13T03:04:10.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-12T00:49:10.000Z", "max_forks_repo_path": "MA320/sections/3.tex", "max_forks_repo_name": "iydon/homework", "max_forks_repo_head_hexsha": "253d4746528ef62d33eba1de0b90dcb17ec587ed", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-11-02T05:46:01.000Z", "max_forks_repo_forks_event_max_datetime": "2020-03-12T23:11:28.000Z", "avg_line_length": 83.5652173913, "max_line_length": 585, "alphanum_fraction": 0.8163371488, "num_tokens": 858, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.519521321952093, "lm_q2_score": 0.6619228625116081, "lm_q1q2_score": 0.34388304056234414}} {"text": "\\section{Data}\n\\label{sec:data}\n\n\\newcommand{\\ncomoving}{10,000}\n\\newcommand{\\nfollow}{800}\n\\newcommand{\\nfollowme}{363}\n\\newcommand{\\ohpercent}{80}\n\\newcommand{\\mypercent}{80}\n\\newcommand{\\nconfirmed}{500}\n\\newcommand{\\pairprob}{99.99\\%}\n\\newcommand{\\badpairprob}{50\\%}\n\nHere we use comoving pairs and groups of stars identified in the TGAS\ncatalogue by \\citet{Oh2016}.\nThese stars do not have precise absolute ages: their absolute ages are\ninferred by fitting isochrones to their photometric colours and parallaxes.\nHowever their relative ages are very precise---they should be identical to\nwithin a few million years as we assume that they formed at the same time from\nthe same molecular cloud.\nWe use the fact that these stars are coeval to test gyrochronology by\npredicting their ages from their rotation periods.\nWe use the gyrochronal ages predicted for each star in the pair to infer the\nintrinsic scatter the gyrochronology relations.\n\nThese comoving pairs were identified in the TGAS catalogue by identifying\nstars with similar positions and proper motions by \\citet{Oh2016}.\nThe probability that the stars were comoving was compared with the probability\nthat the stars were drawn from an isotropic, random distribution of\nvelocities.\nIn this way, \\citet{Oh2016} identified \\ncomoving\\ comoving pair candidates.\n\nRadial velocity (RV) follow-up of \\nfollow\\ comoving pair candidates was\nperformed in February 2017.\n\\ohpercent\\% of these candidates were confirmed to have RVs consistent with\nbeing a comoving pair.\nOver seven nights, from the 26$^{th}$ of June to the 2${^nd}$ of July, we\nobtained RVs of \\nfollowme\\ further comoving pair candidates which fall in\nthe \\kepler\\ field of view.\nSince a small number of stars in the original \\citet{Oh2016} catalogue fall in\nthe \\kepler\\ field, we used a slightly different sample with less stringent\nfalse positive rates.\nThe original catalogue contains stars with a formal probability of being\ncomoving of \\pairprob.\nWe used a catalogue of lower probability stars, specifically with a\n\\badpairprob\\ formal probability of being a true comoving pair and selected\nonly stars within 10 parsecs of each other in order to reduce the false\npositive rate.\n\\badpairprob\\ may seem like a low probability, however with follow-up RV\nobservations we were able to identify many of the false positives in this\nlist.\nWe found \\mypercent\\% to be true comoving pairs based on their RVs.\nWe targeted stars in the \\kepler\\ field as these targets have ultra-precise\ntime-series photometry, from which we can measure rotation periods.\nWe selected the \\nconfirmed\\ confirmed comoving pairs to perform further\nanalysis.\n", "meta": {"hexsha": "9f2b4770a1d6271ba68048f6ab3a18a0e3848ce7", "size": 2652, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/data.tex", "max_stars_repo_name": "RuthAngus/teacups", "max_stars_repo_head_hexsha": "299863eb2e579cb71efce3769cf13b38eb460fd5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper/data.tex", "max_issues_repo_name": "RuthAngus/teacups", "max_issues_repo_head_hexsha": "299863eb2e579cb71efce3769cf13b38eb460fd5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/data.tex", "max_forks_repo_name": "RuthAngus/teacups", "max_forks_repo_head_hexsha": "299863eb2e579cb71efce3769cf13b38eb460fd5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 47.3571428571, "max_line_length": 78, "alphanum_fraction": 0.8009049774, "num_tokens": 644, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6334102636778401, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3438551753353145}} {"text": "\\section{Introduction}\nThe idea of quantum computation was first proposed by Richard Feynman in 1981 to simulate quantum systems that are too hard to simulate using conventional classical digital computers.\n\nIn 1994, Peter Shor found an efficient quantum algorithm for factoring large numbers, which caused great intrest in quantum computation for its implications for cryptanalysis.\n\nThen, it was again Peter Shor who found quantum error-correcting codes and fault-tolerant methods for executing a quantum computation reliably using noisy hardware. It makes it possible for quantum computing to be scaled up to large devices that solve very hard problems in principal.\n\nWe often use NISQ to describe the current state of quantum computation. It stands for \\textit{Noisy Intermediate-Scale Quantum}. \"Intermediate-Scale\" conveys that today's device with more than 50 well-controlled qubits cannot be simulated by brute force using the most powerful currently existing classical computer.\n\"noisy\" means it is still not error corrected, which limits its computational power.\n\n\n\nAlthough physical realization of universal quantum computation is still in a primitive stage, there has been a lot of work in software field like IBM's QisKit, Google's Cirq and Microsoft's Q$\\sharp$.\nThey provide software tools to describe and simulate quantum algorithms and give access of cloud quantum processors or simulators to wide community.\n\nThere are three main models of quantum computing: \\textit{Quantum Turing Machine}, \\textit{Quantum $\\lambda$-Calculus} and \\textit{Quantum Circuit}, among which the third one is the most practical.\nMost quantum programming languages are \\textit{quantum-circuit description languages}, which means they are used to describe the architecture of quantum circuits.\nThe current quantum programming languages can be categorized into two categories according to their styles: functional quantum programming languages and imperative quantum programming languages.\n\\begin{multicols}{2}\n \\begin{center}\n Functional:\n \\end{center}\n \\begin{itemize}\n \\item Qwire\n \\item QML\n \\item Quipper\n \\item QuaFL\n \\item Silq\n \\end{itemize}\n\n \\columnbreak\n\n \\begin{center}\n Imperative:\n \\end{center}\n \\begin{itemize}\n \\item QASM\n \\item QCL\n \\item ScaffCC\n \\item Qiskit\n \\item Quil\n \\end{itemize}\n\\end{multicols}\n\nAdvanced quantum programming languages can use more powerful abstract constructs and type systems to make it easier for programmers to write correct quantum programs.\nIn particular, we design a simple quantum programming language named $\\lambda_Q$, which means \\textbf{$\\boldsymbol{\\lambda}$-calculus with quantum circuit}.\nIts syntax consists of a traditional part, which is a simple $\\lambda$-calculus, and a quantum part, whose syntax is based on Qwire~\\cite{qwire}, a functional quantum programming language with linear type system.\nWhat's more, we implement a compiler from $\\lambda_Q$ to QASM (Quantum Assembly Language) ~\\cite{qasm}, an imperative quantum programming language with low-level instruction sets.\nThe output QASM program can run on the IBM cloud quantum machine.\n\nThe main feature of $\\lambda_Q$ is that the syntax for traditional computation and quantum computation are separated.\nThey communicate with each other via some specific operations : quantum circuit can be \\textit{abstracted} or \\textit{lifted (measured)} into traditional term, and traditional term for quantum circuit can be applied to quantum bits.\nThus, the syntax of quantum circuit can use \\textit{linear type system} to guarantee that the Quantum Non-cloning Theorem is not violated, meanwhile the $\\lambda$-calculus of the traditional part makes it easier to write quantum programs.\nWe will explain it in detail in Section \\ref{spec}.\n\nThe overall structure of the compiler can be visualized in Figure \\ref{compiler}.\nThe frontend is implemented using \\texttt{Haskell}, and the backend is implemented using \\texttt{C}.\nThe code can be found in \\url{https://github.com/thwfhk/lambdaQ}.\nWe will discuss the implementation of $\\lambda_Q$ compiler in detail in Section \\ref{front} and \\ref{back}.\n\n\n\\begin{center}\n \\begin{figure}\n \\label{compiler}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{images/overview.png}\n \\caption{Structure of $\\lambda_Q$ compiler.}\n \\end{figure}\n\\end{center}", "meta": {"hexsha": "6f44d3ea7351f9c97f0c083b6b53ff8b1efd0987", "size": 4353, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "report/motivation.tex", "max_stars_repo_name": "thwfhk/lambdaQ", "max_stars_repo_head_hexsha": "834c0a42e234f486ac0f7b55f76e096d70cf7262", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2021-04-10T08:42:04.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-09T07:09:53.000Z", "max_issues_repo_path": "report/motivation.tex", "max_issues_repo_name": "thwfhk/lambdaQ", "max_issues_repo_head_hexsha": "834c0a42e234f486ac0f7b55f76e096d70cf7262", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "report/motivation.tex", "max_forks_repo_name": "thwfhk/lambdaQ", "max_forks_repo_head_hexsha": "834c0a42e234f486ac0f7b55f76e096d70cf7262", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-05-09T06:13:15.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-09T06:13:15.000Z", "avg_line_length": 63.0869565217, "max_line_length": 316, "alphanum_fraction": 0.7898001378, "num_tokens": 973, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3438551753353145}} {"text": "\n\n%############################################################################################\n%############################################################################################\n\\section{Fixed Model-Driven MU-MIMO Evaluation}\n\\label{sec:model}\nIn order to understand how different environments and operational frequencies will effect the performance of a \\ac{MU-MIMO} system, we first turn to modern statistical MIMO channel models \\cite{liu2012cost}.\nSince this statistical model requires tuning for different environments and frequencies, we compare the results using two published parametrizations for 300~MHz \\cite{zhu2013cost} and 5.8~GHz\\cite{poutanen2011cost}. These results provide the theoretical motivation for over-the-air experiments to explore common application scenarios for UHF and 2.4/5.8~GHz WiFi.\n\n\\subsection{UHF vs. 2.4/5 GHz: Channel Models}\n\\label{sec:specDiff}\n\n\tSpectrum differences between 2.4/5~GHz WiFi and sub-gigahertz frequencies are essentially due to the different manifestations of Doppler effects given each band's wavelength.\n\tDoppler effects are a result of transmitter, receiver, and client movement with respect to a transmission's wavelength. \n\tBecause sub-gigahertz wavelengths are 2-4 times longer than 2.4/5~GHz, environmental variation will affect sub-gigahertz transmissions 2-4 times less (without considering multi-path effects). \n\n\tFig.~\\ref{fig:doppler} shows the theoretical, freespace 50\\% coherence time for various sub-gigahertz and 2.4/5~GHz frequencies \\cite{rappaport1996wireless}.\n\tThe 50\\% coherence time is expected length of time that the channel characteristics will vary at most 50\\% given some velocity (effectively channel variation).\n\n\tThe coherence time difference between 2.4/5~GHz WiFi and sub-gigahertz frequencies is between 1-2 \\textit{orders of magnitude}.\n\tThis channel characterization does not consider many real world effects such as multi-path or fading but provides a coarse characterization of the key differences in the two bands.\n\n\\begin{figure}[htbp!]\n\t\\centering\n \t\\includegraphics[width=0.7\\linewidth]{figs/doppler} \n \t\\caption{50\\% coherence time for various sub-gigahertz and 2.4/5~GHz WiFi frequencies.}\n\t\\label{fig:doppler}\n\\end{figure}\n\n\tFor a more realistic characterization of the spectrum differences, we employ the COST 2100 MIMO channel model, a flexible channel model that is well suited for \\ac{MU-MIMO} scenarios \\cite{liu2012cost}.\n\tThis channel model is tuned with parameters that are extracted from empirical measurements and thus does consider real-world channel effects such as fading, multi-path, and \\ac{NLoS} transmissions.\n\tParametrized realizations of the COST 2100 model have been created for 300~MHz \\cite{zhu2013cost} and 5~GHz \\cite{poutanen2011cost} bands.\n\tUsing these models, we generate 15,000 channel snapshots at a simulated rate of 100 snapshots per second to characterize the variation of channel state over time and the separability of individual users. \n\tSpecifically we explore the temporal correlation and receiver separability (shown in Fig.~\\ref{fig:corr}) of the generated matrices.\n\n\\begin{figure}[ht]\n\t\\centering\n\t\\subfigure[Temporal correlation between channel snapshots from 0 to 10 seconds apart. Higher time correlation allows for more robust MU-MIMO performance. $T_{0.9}$ is 50~ms and 4~s for 5 GHz and 300 MHz respectively.] {\n\t\\includegraphics[width=0.48\\linewidth]{figs/theo_tempDiff}\n\t\t\\label{fig:theo_tempCorr}\n\t}\n\t\\subfigure[CDF of model-generated Demmel condition number. Left is better for \\ac{MU-MIMO}] {\n\t\t\\includegraphics[width=0.48\\linewidth]{figs/dcond_cdf} \n\t\t\\label{fig:spatCorr}\n\t}\n \t\\caption{Temporal correlation and channel condition of 300~MHz and 5~GHz 2x2 MU-MIMO channels generated by COST 2100 MIMO channel model. \\label{fig:corr}} \n\\end{figure}\n \n\tThe correlation coefficient $\\rho$ presented in Equation~\\ref{eq_corr_coeff} is calculated for all combinations of transmit antenna, receive antenna, and starting time sample.\n\tWe show the magnitude of the temporal correlation coefficient in Fig.~\\ref{fig:theo_tempCorr} for our generated channels. \n\tLower temporal correlation results in less robust MU-MIMO transmissions because the measured channel state has a high probability of being stale.\n\tAs seen in Fig.~\\ref{fig:theo_tempCorr}, the temporal correlation of 5~GHz WiFi almost immediately drops to below 0.9 ($T_{0.9}$) a point when when re-sounding the channel is strongly suggested \\cite{breit2009coherence}. \n\nAccording to the channel models, the approximate re-sounding time for 5~GHz is 50~ms and 300~MHz is approximately 4.5~s (almost two orders of magnitude longer). \nThis result is similar to what we expect from Doppler effects of the different frequency bands (Fig.~\\ref{fig:doppler}) and is similar to our indoor temporal characterization in \\S~\\ref{sec:indoor}.\n\n\\subsection{Demmel Condition Number}\n\\label{sec:demmel}\n\nUser separability refers to how well a multi-antenna transmitter can serve a set of users in parallel. \nThe Demmel condition number is a modified matrix condition number that directly predicts the efficacy of an adaptive MIMO or MU-MIMO transmission for a particular channel realization \\cite{zhong2011distribution}.\n\nThe Demmel condition number is computed using the eigenvalues $\\lambda_k$ of $HH^{\\dagger}$ as:\n\\begin{align}\nd\\triangleq\\frac{\\sum^{n}_{k=1}{\\lambda_k}}{\\lambda_{n}}\n\\label{eq:dcond}\n\\end{align}\nwhere $\\lambda_1 > \\lambda_2 > \\hdots > \\lambda_n$.\nThis ratio represents how well a matrix can be inverted, a key component of many adaptive MU-MIMO techniques such as Zero-Forcing Beamforming \\cite{goldsmith2006zf} and MMSE \\cite{tse2005fundamentals}.\nSpecifically, the higher the condition number, the more numerically unstable the inverse and thus the more inter-user interference during MU-MIMO transmissions reducing received SINR. \nThe condition number ranges from 1 to infinity for well to ill-conditioned matrices, respectively. \n\nThis method of calculating the condition number is less forgiving than the traditional singular value ratio. \nThe singular values of $H$ are the square root of the eigenvalues of $HH^{\\dagger}$.\nThus, instead of $\\sigma_k/\\sigma_n$, the Demmel condition number is equivalent to $\\sum{\\sigma^2}/\\sigma_n^2$ meaning that channel matrices with low singular values (resulting in inaccurate inversion) are even further ``penalized.``\nThis modification to the condition number better predicts MU-MIMO performance, in fact, it is consistent and accurate enough to be used for determining parameters such as supported modulation rate and user selection \\cite{zhong2011distribution}.\n\n\nThe COST channel models show a significant difference between the 5~GHz WiFi and UHF bands. \nThe CDF shown in Fig.~\\ref{fig:spatCorr} depicts how almost all of the generated 5~GHz channel matrices have a Demmel condition number less than 10 while UHF's channel condition varies far more and is significantly worse.\nThis results in an increased ability for a \\ac{MU-MIMO} transmitter to invert the channel matrix and send orthogonal streams to each intended user. \n\n\nThus, existing MIMO channel models show that while the UHF channel is more temporally stable over time, its ill-conditioned channel matrices can result in lower served SINR due to inter-user interference. \nHowever, the available parametrizations of the COST model are for indoor 5~GHz and outdoor UHF scenarios. \n%We show in \\S~\\ref{sec:expDrivenEval} how restricting these bands to these transmission environments does not tell the full story.\n\n\\section{Fixed Indoor MU-MIMO Channel Characterization}\n\\label{sec:mumimo_channels} \n\n%############################################################################################\n%############################################################################################\n%\\subsection{Experiment-Driven Fixed Wireless Evaluation}\n%\\label{sec:expDrivenEval}\nThe models analyzed in Section~\\ref{sec:model} are parametrized for particular environments, frequency bands, and topologies. While they suggest that the performance of \\ac{MU-MIMO} beamforming in UHF bands may be advantageous, it it difficult to directly predict or simulate UHF performance using these models as they were not validated for application scenarios such as indoor or urban outdoor, nor the UHF frequency band.\n\nIn order to address uncertainty in these models for our target application (indoor and outdoor \\ac{WLAN}), we perform a set of experiments utilizing our custom \\ac{SDR} radio platform that allows us to measure the performance of a \\ac{MU-MIMO} transmission over a diverse set of carrier frequencies and characterize the wireless \\ac{MU-MIMO} channel for important temporal and spatial correlation properties.\n\nWe perform over-the-air beam-forming transmissions in a densely packed, challenging office scenario with multiple subscriber nodes and demonstrate not only the ability to simultaneously beamform to distinct users in relatively close proximity, but also the relative improvement that shifting to UHF frequencies provides.\n\nFinally, we perform two sets of experiments with a customized MAC and PHY designed to gather dense, wideband, over-the-air channel estimates in realistic indoor and outdoor \\ac{WLAN} scenarios with multiple subscriber nodes. Using this data, we then demonstrate that the spatial correlation for outdoor users remains similar to that of 2.4~GHz WiFi, thus incurring no beamforming ``penalty'' for utilizing a frequency band with superior propagation and temporal correlation.\n\n\\subsection{Indoor MU-MIMO Transmissions}\n\\label{sec_static_indoor_exp}\n\n\\label{sec:indoor}\n\\textbf{Experimental Setup.}\nFirst, we evaluate the performance of UHF \\ac{MU-MIMO} in an indoor \\ac{NLoS} office environment.\nExperiments were conducted during the work day with people walking through the halls in the environment depicted in Fig.~\\ref{fig:indoorExp}.\n\n\n\\begin{figure*}[t!]\n\t\\centering\n \t\\includegraphics[width=1\\linewidth]{figs/indoor_wl_all} \n \t\\caption{Indoor fixed MU-MIMO achievable rate.\\label{fig:indoor_cap_all}}\n\\end{figure*}\n\n\n\\begin{figure}[th]\n\\vspace{-5mm}\n\t\\centering\n \t\\includegraphics[width=0.7\\linewidth]{figs/indoorExp} \n \t\\caption{Indoor fixed WURC-based experimental test setup.\n\t\\label{fig:indoorExp}}\n\\end{figure}\n\n\nThe \\hspace{0.01pt} transmitting \\hspace{0.01pt} array\\hspace{0.01pt} was \\hspace{0.01pt} placed \\hspace{0.01pt} on \\hspace{0.01pt} a \\hspace{0.01pt} third \\hspace{0.01pt} floor \\hspace{0.01pt} walkway bridge and 6 separate receivers in two adjacent offices within the adjoining hallway.\nNote that the to-scale depiction in Fig.~\\ref{fig:indoorExp} shows the relative co-location of all receiving nodes with respect to the distance from the transmitter to simulate a densely packed office environment.\nThis represents a realistic, challenging case for indoor stationary MU-MIMO transmissions due to the co-located receivers.\n\nTo encompass a wide range of user grouping conditions, every possible combination of transmit and receive antennas are considered. Sixty transmissions are performed for each topology. The center frequencies for each frequency band (\\textit{i.e.} channel) were chosen so that transmissions did not encounter interference from other equipment. Specifically, the UHF channel was first directly scanned for existing DTV or microphone transmissions and an experimental license was obtained to operate equipment on that channel. The channels selected for 2.4 and 5.8 GHz are not currently supported by the regulatory domain where these experiments were performed, thus ensuring minimal ISM-band interference.\n\tUsing the measurement technique specified in Section~\\ref{sec_static_miso_chan_est}, every possible topology's MU-MIMO capacity is measured for each frequency band and shown in Fig.~\\ref{fig:indoor_cap_all}.\n\n\n\n\\begin{figure}[th]\n\t\t\\centering \n\t\\includegraphics[width=0.7\\linewidth]{figs/meas_indoor_dcond} \n \t\\caption{Demmel condition number measured for the indoor environment. Left is better for \\ac{MU-MIMO}.\n\t\\label{fig:indoor_demmel}}\n\\end{figure}\n\n\n \n\\subsubsection{Fixed Wireless Measured Achievable Sum-Rate Capacity.}\n\nBased on the channel models and accompanying analysis presented in Section~\\ref{sec:specDiff}, we expect that the increased spatial correlation of UHF channels will not \\ allow \\ for \\ MU-MIMO \\ transmissions\\ to \\ accurately separate nearby users.\nHowever, we find that UHF \\ac{MU-MIMO} transmissions can actually achieve a sum capacity similar to that of 2.4~GHz WiFi transmissions (always between 1-2~b/s/Hz above of below the 2.4~GHz band).\n\nIn fact, we find that majority of the intuition and channel models surrounding UHF MU-MIMO are not specific to the frequency band itself but rather generalized characteristics of MU-MIMO transmissions.\nFor example, the available MU-MIMO channel models characterize \\textit{indoor} WiFi and \\textit{outdoor} UHF channel environments where, regardless of frequency band, we expect increased difficulty in user separability in outdoor environments. \nNote the channel condition of the different transmission bands in the NLOS environment in Figure~\\ref{fig:indoor_demmel} are similar in contrast to Fig.~\\ref{fig:spatCorr}. \nEven though the wavelength of UHF is longer resulting in better propagation through materials, the UHF-band transmission still experiences enough multi-path to successfully beamform to multiple users in parallel.\n\n\n\nAdditionally, the results shown in Figure~\\ref{fig:indoor_cap_all} show a known trend of achievable capacity for MU-MIMO transmissions where the MU-MIMO gain plateaus as the available \\ac{DoF}\\footnote{\\ac{DoF} here refers to how many more transmit antennas there are than receive antennas in a MIMO transmission.} are reached.\nThe consistently worse performance of 5.8~GHz is explained by the high attenuation experienced by that frequency band in \\ac{NLoS} conditions combined with its sensitivity to environmental variation. \n\nNote that UHF MU-MIMO consistently outperforms 2.4~GHz transmissions except for in the 2x2 transmission scenario. \nBecause the sum transmit power emanating from the array is held constant regardless of the number of transmit antennas in use, the performance differential is solely a result of channel state, specifically it is an indicator of temporal channel correlation due to the WARPLab measurement platform.\n\n\n\tAs discussed in Section~\\ref{sec_warplab_timing}, the latency in the WARPLab platform is due to the rate at which the host PC can download and upload samples to each of the WARP boards over Ethernet. \n\tIn our system, we benchmark a read/write rate of approximately 2.5~ms per buffer and the closed loop beamforming method employed requires between 10 to 20~ms to complete depending on the number of transmit and receive antennas (the difference between a 2x2 and 4x4 transmission scenario).\n\n\n\\subsection{Fixed Wireless Measured Temporal Correlation.}\n\\label{sec_fixed_temporal}\n\nTo gain additional insight into the measured capacity results and to infer real world performance from \nour MU-MIMO transmissions, we also consider the channel correlation measured during each experiment.\n\nFor each topology, we consider each of the 60 MU-MIMO transmissions and their channel matrices.\nWe calculate channel correlation between varying times during the experiment to measure the rate of change of the channel information with respect to time. \nThese calculations are an average over all topologies (all combinations of transmit and receive antennas).\n\n\n\\begin{figure}[th]\n\t\\centering\n \t \\includegraphics[width=0.7\\linewidth]{figs/measCorrWl} \n \t \\caption{Measured Temporal Channel Correlation, depicting Beacon Interval, and $T_{0.9}$. WARPLab latency is 10-20~ms depending number of transmit and receive antennas.\n\t\\label{fig:meas_corr}}\n\\end{figure}\n\n\nFig.~\\ref{fig:meas_corr} shows how the channels decorrelate over the course of one measured second in time.\nThis is effectively an indicator of how long a transmitter has after measuring the channel matrix and before actually transmitting parallel streams using that measurement. \nA coherence time of $T_{0.9}$ represents when the probability of the channel being too stale to successfully beamform over is high.\n\nFirst, note that the WARPLab latency range of 10 to 20~ms is approximately $T_{0.9}$ for the \nthe two WiFi frequency bands.\nThis indicates why only the 2x2 transmission scenario has the 2.4~GHz transmitter outperform UHF MU-MIMO; the latency between the sounding and transmission phase was the lowest and just at the $T_{0.9}$ limit.\n\nWhile the 2.4/5~GHz frequencies both drop significantly within 100~ms, UHF remains above the $T_{0.9}$ threshold for the maximum one measured second difference between channel matrices. \nWhile these correlation values are not asymptotic and will eventually degrade, the performance of 2.4 and 5.8~GHz is sufficently low for stationary devices \\cite{breit2009coherence}. \n\nThe 802.11 infrastructure \\ac{BSS} beacon rate\\footnote{The IEEE Std 802.11-2012 Target Beacon Transmission Time (TBTT) is configured in ``Time Units'' defined as 1 TU = 1024 $\\mu s$, and is often set to 100~TU.} (102.4~ms \\cite{std11_2012}) is greater than the interval that 2.4/5~GHz MU-MIMO channels decorrelate. \nHowever, the stability of the UHF channel implies that a UHF MU-MIMO system could use periodic protocol packets for exchanging channel state information. \n\nFinally, the channel correlation result shown in Fig.~\\ref{fig:meas_corr} effectively scales the MU-MIMO achievable rate shown in Fig.~\\ref{fig:indoorExp}.\nThe rate at which the 2.4/5~GHz channel decorrelates necessitates channel sounding on a per packet basis adding considerable overhead to MU-MIMO transmissions. \nHowever, the temporal stability of the UHF MU-MIMO channel allows a transmitter to significantly reduce this overhead intensive sounding process and thus significantly increase the potential MU-MIMO gains. \n\n%############################################################################################\n%############################################################################################\n\n\\begin{figure}[p]\n\t\\centering\n\t\\subfigure[Experimental setup for outdoor channel sounding.] {\n\t\\centering\n\t\\includegraphics[width=0.48\\linewidth]{figs/meas/outdoorExp}\n\t\t\\label{fig_fix_outdoor_diagram}\n\t}\n\t\\subfigure[Example of outdoor STA with dual antennas.] {\n\t\t\\includegraphics[width=0.48\\linewidth]{figs/meas/ryan_tripod_platform} \n\t\t\\label{fig_fix_outdoor_sta}\n\t}\n \t\\caption{Fixed outdoor multi-band measurement setup \\label{fig_fixed_outdoor}} \n\\end{figure}\n\n\\section{Fixed Outdoor MU-MIMO Channel Characterization}\n\n%\\rgnote{TODO: I'm missing autocorrelation results from this experiment. Need to add.}\n\n%\\begin{figure}[th!]\n%\\centering\n %\\includegraphics[width=0.5\\linewidth]{figs/meas/outdoorExp} \n %\\caption{}\\label{fig:outdoor}\n%\\end{figure}\n\n\n\n\n\tFinally, using the experimental framework developed in Section~\\ref{sec_static_miso_chan_est}, we perform outdoor channel sounding experiments to directly compare the performance and stability of UHF \\ac{MU-MIMO} channels.\n\tTo that end, we setup an experimental network of a collection of nodes located outdoors being served by our array from a third floor balcony. \n\tAlthough the UHF transmitter is capable of transmitting much further distances, we limited the scale of the topology as shown in Figure~\\ref{fig_fix_outdoor_diagram} to ensure a fair comparison between UHF and 2.4/5~GHz bands. \n\tThe locations of the nodes were chosen such that the transmissions from the UHF and 2.4/5~GHz bands would reach the receivers (the UHF band transmitters can easily transmit further than 50~m).\n\tHowever, even by reducing the receiver distance to what is shown in Figure~\\ref{fig_fix_outdoor_diagram}, the 5~GHz band transmissions did not reliably reach the receiving nodes.\n\tSince this severely limited the number of measured channel matrices at 5~GHz, we restrict our outdoor comparison to the UHF and 2.4~GHz bands.\n\n\tJust as we evaluated temporal correlation in the multi-path rich, indoor transmission environment, we seek to similarly characterize the most detrimental aspect of the outdoor MU-MIMO channel: receiver separability. \n\tIll-conditioned channel matrices, as discussed in \\S~\\ref{sec:specDiff}, have a detrimental effect on an MU-MIMO enabled transmitter's ability to separate multiple users. \n\n\tIn the previous section, we found that while temporal stability of UHF was greater than that of 2.4/5 GHz, spatial correlation did not suffer as the UHF MU-MIMO transmissions were able to separate the co-located receivers.\n\tHowever, in an open, outdoor \\ac{LoS} environment, we find that both the UHF and 2.4~GHz bands exhibit the same Demmel condition number. \n\tAdditionally, the CDF of the Demmel condition number closely matches the COST UHF channel condition shown in Fig.~\\ref{fig:spatCorr}.\n\tThis suggests that the comparison shown in Fig.~\\ref{fig:spatCorr} is not a result of the frequency band itself, but rather the wholly different channel environments in which the model was parametrized.\n\n\\begin{figure}[t]\n\\centering\n \\includegraphics[width=0.7\\linewidth]{figs/measSpatCorr} \n \\caption{Measured Demmel Condition Number of the outdoor MU-MIMO channel.}\\label{fig:measSpat}\n\t\t\\vspace{-5mm}\n\\end{figure}\n\n%% UHF-band MUMIMO Conclusions\n%\\subsection{Discussion}\n\t\t%\\rgnote{what does this mean for system design? the implications are very important}\n", "meta": {"hexsha": "c90335c0829ae0a040587fc7b197cbf898554ef6", "size": 21547, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sec/uhf_mumimo.tex", "max_stars_repo_name": "RyanEGuerra/ryan_guerra_phd_thesis", "max_stars_repo_head_hexsha": "acf1ebafee00a8e4375008e60e35da8affc97d9b", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "sec/uhf_mumimo.tex", "max_issues_repo_name": "RyanEGuerra/ryan_guerra_phd_thesis", "max_issues_repo_head_hexsha": "acf1ebafee00a8e4375008e60e35da8affc97d9b", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "sec/uhf_mumimo.tex", "max_forks_repo_name": "RyanEGuerra/ryan_guerra_phd_thesis", "max_forks_repo_head_hexsha": "acf1ebafee00a8e4375008e60e35da8affc97d9b", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 82.5555555556, "max_line_length": 702, "alphanum_fraction": 0.7777880911, "num_tokens": 5043, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5428632831725052, "lm_q2_score": 0.6334102567576901, "lm_q1q2_score": 0.34385517157861917}} {"text": "%---------------------------------------------------------------------\n\\section{Summary and Conclusions}\n\\label{summary}\n\nWe have used leading order matrix elements to separate expected single\ntop quark signals from background. This analysis used lepton+jets\nevents selected from nearly 1~fb$^{-1}$ of Run~II~ data. The\ntwo-dimensional discriminant output from the analysis is used to\nmeasure the single top cross section. We obtain the following result:\n$$\n\\sigma\\left({\\ppbar}{\\rargap}tb+tqb+X\\right)\n= 4.6^{+1.8}_{-1.5}~{\\rm pb}\n$$\n\nThis result has a p-value of $0.22\\%$, corresponding to a $2.9\\sigma$\nGaussian equivalent significance, meaning it is highly unlikely to be\nan upward fluctuation of the background that has produced this\nmeasured cross section.\n", "meta": {"hexsha": "15e36c75645dae1c29ab3efd87f5e2b606e72211", "size": 763, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "MEnote/Summary.tex", "max_stars_repo_name": "tgadf/thesis", "max_stars_repo_head_hexsha": "19d4a6bc7f7ac8660fce582322703d50e0d6bd31", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "MEnote/Summary.tex", "max_issues_repo_name": "tgadf/thesis", "max_issues_repo_head_hexsha": "19d4a6bc7f7ac8660fce582322703d50e0d6bd31", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "MEnote/Summary.tex", "max_forks_repo_name": "tgadf/thesis", "max_forks_repo_head_hexsha": "19d4a6bc7f7ac8660fce582322703d50e0d6bd31", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.1578947368, "max_line_length": 70, "alphanum_fraction": 0.6959370904, "num_tokens": 190, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.785308580887758, "lm_q2_score": 0.4378234991142019, "lm_q1q2_score": 0.34382655076868646}} {"text": "\\documentclass{article}\n\\usepackage[T1]{fontenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{lmodern}\n\\usepackage{textcomp}\n\\usepackage{lastpage}\n\n\\input{common_symbols_and_format.tex}\n\\usepackage{tocloft}\n\\renewcommand{\\cfttoctitlefont}{\\Large\\bfseries}\n\n\\begin{document}\n\\normalsize\n\\logo\n\\rulename{Difference Regression}\n\\tblofcontents\n\\ruledescription{This trading rules regresses the 1-day price changes seen historical against the prior day's \\% change of the research series.}\n\\ruleparameters{Difference Coefficient}{0.1}{Multiplier for regression.}{$k_1$}{Constant Coefficient}{0.1}{Initial constant for regression.}{$k_2$}\n\\stoptable\n\\section{Equation}\nBelow is the equation which governs how this specific trading rule calculates a trading position.\n\\begin{equation}\n\\position_\\currenttime = k_1*(\\frac{R_t}{P_{t-1}}-1)+k_2\n\\end{equation}\n\\\\\nwhere:\n\n$\\position_\\currenttime$ is the portfolio allocation at time $\\currenttime$\n\n$R_t$: Research value at time $\\currenttime$\n\n$P_{t-1}$: Price value at time $\\currenttime -1$ \n\n$k_1$: Difference coefficient\n\n$k_2$: Constant coefficient\n\n\\keyterms\n\\furtherlinks\n\\end{document}", "meta": {"hexsha": "6d3b22fef9cfffe362dc45921e70f57c578162a2", "size": 1129, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/strategies/tex/DifferenceRegression.tex", "max_stars_repo_name": "parthgajjar4/infertrade", "max_stars_repo_head_hexsha": "2eebf2286f5cc669759de632970e4f8f8a40f232", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 34, "max_stars_repo_stars_event_min_datetime": "2021-03-25T13:32:54.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-06T23:03:01.000Z", "max_issues_repo_path": "docs/strategies/tex/DifferenceRegression.tex", "max_issues_repo_name": "parthgajjar4/infertrade", "max_issues_repo_head_hexsha": "2eebf2286f5cc669759de632970e4f8f8a40f232", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 137, "max_issues_repo_issues_event_min_datetime": "2021-03-25T10:59:46.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-28T19:36:30.000Z", "max_forks_repo_path": "docs/strategies/tex/DifferenceRegression.tex", "max_forks_repo_name": "parthgajjar4/infertrade", "max_forks_repo_head_hexsha": "2eebf2286f5cc669759de632970e4f8f8a40f232", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 28, "max_forks_repo_forks_event_min_datetime": "2021-03-26T14:26:04.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-10T18:21:14.000Z", "avg_line_length": 28.225, "max_line_length": 147, "alphanum_fraction": 0.7794508415, "num_tokens": 323, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6113819874558604, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3437045864125405}} {"text": "\\documentclass[runningheads]{llncs}\n\\usepackage{graphicx}\n\\usepackage{times}\n\\usepackage{float}\n\\usepackage{latexsym}\n\\usepackage[T1]{fontenc}\n\\usepackage[utf8]{inputenc}\n\\usepackage{microtype}\n\\usepackage[utf8]{inputenc}\n\\usepackage[ruled,vlined]{algorithm2e}\n\\graphicspath{ {./images/} }\n\\begin{document}\n\\title{Neural Network Pruning as a N Player NIM Game}\n\\author{Daniel Campos\\orcidID{0000-0002-5138-8426} }\n\\authorrunning{D. Campos et al.}\n\\institute{University of Illinois at Urbana-Champaign, Urbana, Il, 61801, USA}\n\\maketitle % typeset the header of the contribution\n%\n\\begin{abstract}\nUnstructured pruning has become one of the most popular methods for Neural Network model pruning since it provides an intuitive method of extracting a sub network from the original Neural Network which matches the pruning goals. Pruning approaches have been used to reduce model size, increase adversarial robustness, increase stability and generizability, and increase model inference speed but each of these goals works independently. By framing Network Pruning as a $N$ player NIM game we provide a framework which can be used to combine pruning strategies depending on needs. In our work, each pruning method takes the role of a NIM game player where they remove weights from a network in an iterative fashion. We find that 2 player methods combining $L1$ and Random pruning outperform existing method across datasets and models. We believe this formulation of network pruning provides a comprehensive framework which can be used to prune networks for multiple goals. \n\\keywords{Neural Network \\and Subtraction Game \\and Neural Network Pruning.}\n\\end{abstract}\n\\section{Introduction}\nNeural Networks have become popular choices for complex computation tasks like image recognition \\cite{Howard2017MobileNetsEC}, image generation \\cite{Goodfellow2014GenerativeAN}, speech processing \\cite{Zhao2017RecurrentCN}, and question answering \\cite{Seo2017BidirectionalAF}. These networks have grown to hundred billions of parameters \\cite{Brown2020LanguageMA} which require specialized hardware like clusters of GPUs and FPGAs in order to infer on unseen data. Recent work has shown that larger networks can learn quicker \\cite{Li2020TrainLT}, are more accurate and are more sample efficient \\cite{Kaplan2020ScalingLF}. Seeking to allow the improvements that large models have brought to be used on smaller devices and in a more energy efficient way researchers have produced many methods which produce smaller networks that approximate, match, or exceed the original network performance. \\\\ \nModel distillation, quantization, and pruning have emerged as successful methods to produce smaller networks from the original over-parametized network. In model distillation \\cite{Ba2014DoDN} the original network is used to train a smaller network to mimic the behavior of the large network. In model quantization \\cite{Han2016DeepCC} models are made smaller by reducing the numbers of bits that represent each weight. Model Pruning \\cite{LeCun1989OptimalBD} has focused on finding sub-networks in the original network by structured and unstructured pruning. In structured pruning, successful sub-networks are found by removing neurons \\cite{Wang2019StructuredPF} or larger network specific structures like attention heads \\cite{Voita2019AnalyzingMS}. In unstructured pruning the successful sub-networks are found by setting individual weights to zero \\cite{Kwon2019StructuredCB}. \\\\\nUsing network pruning, The Lottery Ticket Hypothesis \\cite{Frankle2019TheLT} proved the concept that in large neural networks there exists a sub network which can match the accuracy of the full network despite its smaller size. Building on the notion that there are many sub networks in an overparamertized network we formulate network pruning as a combinatorial problem. Given an initial structure $S$ and a target network size $\\epsilon$ the goal is to find the sub network $s_m$ of size $\\epsilon$ which maximizes the optimization metric. Sub networks can have many different optimization goals such as: accuracy, size, adversarial robustness, computation speed, explainability, etc. Given that networks commonly have millions of parameters and the network can continually be updated by retraining even a greedy strategy for optimal model selection would require a millions of combinations. Instead, using network pruning we can can produce sub network $s_a$ with minimal compute and using a targeted pruning strategy we only remove neurons that minimize our optimization goal. \\\\\nBelieving that the future of network pruning is likely to focus on multi metric optimization we formulate network pruning as a $N$ player game of NIM. In a game of NIM \\cite{Dass2006SecretsBT} players remove items from a variety of bins in alternating turns. Using this strategy we are able to combine different pruning strategies to produce networks that maximize sparsity and performance and outperform traditional iterative pruning mechanisms. \n\\section{Related Work}\nNeural network compression is an area that has attracted the attention of researchers for the last few decades. Methods for producing smaller networks that approximate original network performance include: distillation, quantization, structured and unstructured pruning. While each of these methods can compress models substantially on their own many researchers have found that some combination of these methods can produce the smallest models with the highest performance \\cite{Polino2018ModelCV}, \\cite{Sanh2020MovementPA}. \\\\\nModel distillation \\cite{Ba2014DoDN} addresses compressing models by first training a large network and calling it a teacher. Then using this teacher model a smaller student model learns to approximate what the teacher model would do. This framework is quite popular because it can leverage existing large models easily and the student model can be designed to fit the application requirements in terms of speed and model size. Distillation has been one of the most common methods of deployment of large scale language models where student models like DistilBERT \\cite{Sanh2019DistilBERTAD} can approximate full model performance at a fraction of the size. \\\\\nModel quantization \\cite{Gong2014CompressingDC} \\cite{Han2016DeepCC} addresses compressing models by reducing the number of bits that are require to represent parameters in a model. In simple implementations this means changing representation of weights from Float32 to float16(effectively cutting model size in half). Complex implementations tune networks to find the smallest amount of bits that can be used for weights, biases, and gradient updates using values as low as 1 bit \\cite{Courbariaux2016BinarizedNN}. Quantization is particularly effective because it both leverages that networks are defaulted to a level of precision which is too high and by decreasing the size of representations networks are forced to share weights making networks more robust. \\\\\nModel pruning \\cite{LeCun1989OptimalBD} addresses model compression by decreasing the connection in a network. The goal of network pruning is to produce a sub network of the original network which optimizes some network property(accuracy, speed, robustness) while preserving the original network function. Network pruning has been show to produce a similar effect to random noise injection \\cite{Bartoldson2019TheGT} and this noise can be used to make the network more efficient. Bartoldson et al., showed that network pruning is not just used for decreasing size but can be used to increase the generalization of the network. The formulate that pruning methods generally generally focus on how important a prune target is currently to the network but believe that algorithms should also should be designed to consider the damage that removal has on a network. \\\\\nAs mentioned earlier, there is structured pruning and unstructured pruning. In structured pruning, the structure of the network is altered by removal of entire neurons, layers, or portions neural network. This method has proven especially successful in language model compression where despite having dozens of attention heads \\cite{Vaswani2017AttentionIA} few heads do most of the work and the rest can be removed \\cite{Michel2019AreSH}. In unstructured pruning the network structure is altered by removal of individual weights. Unstructured pruning when paired with optimization engines can produce networks that are smaller, more accurate, and run faster than the original network. \\\\\nLiu et al., \\cite{Liu2019RethinkingTV} proposed that network pruning is doing something similair to neural architecture search to find an optimal architecture given a set of optimization metrics. Recently, researchers have explored pruning methods that simultaneously remove and reintroduce weights \\cite{Jia2020StochasticMP} which helps keep a pruned network from being stuck in a local minimum. Within unstructured pruning methods can be local or global. Global pruning pools together all parameters across layers while local performs each pruning operation focused on individual layers. \\\\\nThe use of Game Theory has proven successful for model compresion. KDGAN \\cite{Wang2018KDGANKD} explores formulation of distillation as a 3 player game which\nimproves model performance, train time, and training efficient. Many more\nmethods \\cite{Guo2018SparseDW} \\cite{Dhillon2018StochasticAP}, \\cite{Sehwag2020OnPA}, \\cite{Xie2020BlindAP}, explore the use of Adversarial methods in model\npruning with a goal of making NN more robust to adversarial inputs. HYDRA \\cite{Sehwag2020HYDRAPA} prunes models for adversarial robustness by formulating pruning as a empirical risk minimization (ERM) problem removing neurons that maximize risk.\n\\section{Pruning as an N Player Game}\nWe propose the formulation of pruning as an $N$ player game of NIM as a way to allow multi objective network pruning. NIM is traditionally a combinatorial game where two players take turns removing objects from several heaps. NIM is the most well known example of an impartial game because every player has the same amount of moves in a game. In our implementation, the game is played on unstructured local pruning where each player is able to remove a single weights from the layer of the neural network which is currently being played. This game is an extensive subtraction game since it happen over many rounds of play and players make moves by removing parts of a structure. \\\\\nIn this game, pruning methods are implemented as a collection of $P$ players in which each player $p \\in P$ has a unique strategy which does does not change based on other players nor any prior round. Prior to playing their turn, each player ranks the existing weights based on how each weight maximizes their optimization metric. When its a players turn to play they simply select the first non zero weight in their ranking. At the beginning of each round the order in which the players $p \\in P$ execute their prune is selected at random to ensure that one method is not favored over others. Since the outcome of the game is independent of players performance we will not discuss their reward and utility functions further. \\\\\nThe utility in this implementation of pruning is the ability to easily combine strategies and ratio of strategies depending on targeted applications. In single objective pruning there is only one player who plays the game alone. As more optimization goals get introduced the amount of players grow. In a complex game there may be multiple players with the same strategy which gives the game organizer(the network pruner) the ability to set importance of pruning optimization metrics. \n\\section{Experiment}\nThe structure of our experiments is straightforward: train an initial baseline method and prune this networks iteratively using each of the candidate methodologies. To evaluate the effect of our formulation we select 3 popular image recognition frameworks which vary in structure, size and recency: VGG-16 \\cite{Simonyan2015VeryDC}, ResNet50 \\cite{He2016DeepRL}, DPN92 \\cite{Chen2017DualPN}. We train each network on the CIFAR-10 \\cite{Krizhevsky2009LearningML} and CIFAR-100 \\cite{CIFAR-10} dataset which are common benchmarks for evaluating image systems. The CIFAR-10 dataset consists of 60,000 32x32 color images where each image belongs to one of ten classes. The CIFAR-100 dataset is just like CIFAR-10 but it has 100 classes instead of 10 and each class includes 600 examples instead of 6,000.\n\\subsection{Generating Baselines}\nEach of our model baselines are trained in an identical fashion. For each combination of model architecture and dataset each model is trained with a batch size of 128, an initial learning rate of 0.1 and trained for 1000 epochs. Models are trained using 2 Nvidia 2080 Ti GPUs with a Intel Core™ i9-10900X CPU and 128GB of RAM. Training time ranges from 1 hour for VGG-16 to 1 day for DPN-92. During this training regime we evaluate the model on the validation portion of the dataset at the end of each epoch and the final candidate baseline is the checkpoint that maximised validation accuracy. Numerical details on baseline models can be found in Table \\ref{tab:tab1}.\n\\subsection{Pruning}\nOur pruning experiments are focused on Iterative Pruning as it has become one of the most popular methods. In these experiments we only prune the weights of convolutions layers and we prune each layer independently. While other research has shown that optimal pruning of a network has different sparsity at different layers \\cite{Blalock2020WhatIT} we keep a uniform pruning for ease of experimentation. By focusing only on pruning convolutional layers we are not exploring every possible sub network but as shown in Table \\ref{tab:tab1}, convolutional weights make up the majority of each network. The process is formalized in Algorithm 1 where $NN$ is a neural network, $lambda_i$ is the $i$th prunable layer in a $NN$, $\\pi$ represents what percentage of weights need to be trimmed at each time step and $M$ represents a sorting function which maximizes the pruning optimization metric, $P$ represents the players playing our NIM game where each player $p_i$ has a strategy $s_i$ which is based on a $m_i$, $s_j$ represents sparsity at epoch j and $s_{target}$ represents the target network sparsity. \\\\\nIn our experiments we set $s_{target} = 0.95$, $\\pi=0.05$ and we evaluate model performance after each epoch. Our iterative pruning has three steps: stabilization, pruning, and fine tuning. In stabilization the network is trained with a learning rate of $0.1$ for 2 epochs. In network pruning we gradually remove $\\pi\\%$ of weights and retrain the network with one pass the entire training data with a learning rate of 0.01. This stage lasts 19 epochs as $s_j = s_{j-1} + \\pi$ takes 19 steps to reach $s_{target}$. In fine tuning we set the learning rate to 0.001 and train the network for 10 epochs on the full dataset to ensure the network has reached optimal performance and is stable. \\\\\nIn the network pruning step, we prune each layer independently and implement it as a NIM game as shown in Algorithm 2. In this NIM game there are some set of players $N$ which each has a strategy for network pruning $s_i$. For each round, player order is randomly selected and each players removes a single weight from the layer. A NIM game is played for each $\\lambda_i \\in NN$ and if the network sparsity is under the $s_{current sparsity}$ then another round of NIM is played on each layer that is still under the target. Since target sparsity is a global parameter some layer sparsity will have lower and higher sparsity than the target. If a layer is over pruned in one epoch it will not be pruned until its sparsity is lower than the target. Each player $n_i \\in N$ is defined by what pruning strategy is used and while our experiments focus on 2 and 3 player games $N$ could be much bigger. We will expand on network pruning as an $N$ payer game shortly.\n\\subsection{Pruning Methods}\nUsing the methodology previously described we explore the effect of. To explore how NIM can be used in Network pruning we formulate 8 games with a mixture and strategies. Each method is implement as $1-n$ player game where each player has a strategy $s_i$ which is represented via a sorting method $M_i$ which they use to sort the weights $W$ of the given layer they are pruning. Each of these ranking methods ranks weights of 0 as lowest and then ranks all existing weights to maximize their desired outcome. Details on each pruning method can be founding in the appendix.\n\\section{Results}\nIn our experiments our main goal was to see if combining pruning strategies could lead to improved models across sparcity. If we focus initially on performance on high sparcity networks(90\\% +) we find that L1 + Random pruning out performs other methods on five out of our six baselines. We believe this is because the combination of L1 and random pruning simultaneously keeps the most important sub networks while introducing generalization as the network cannot only rely on the strongest weights. We also note that across all experiments we see a large variation in optimal pruning mechanism given a target sparcity. We believe this further underscores the utility of the NIM game because it allows researchers to choose the most optimal game formulation depending on desired results. We also note that we see a wider spread in model accuracy on pruning of models targeting CIFAR-10 vs CIFAR-100. We believe this is because the dataset has few classes which results in more independent class specific sub networks which our various methods effect differently. \\\\\nOne of the surprises in our experiments was the consistent high performance of Random pruning even at high sparcity. We believe this is the case because even at high scarcities since the datasets are easier the networks are still highly over parametized. We reason that as we scale the difficulty of the target task random pruning alone would start to cause complete model failure. Since the focus of our work is on the formulation of pruning as a $N$ player game we will not further analyze our results but the full results can be found in the appendix. \n\\section{Conclusion and Future Work}\nThe addition of game theory to AI has proven to be a robust framework which leads to new discoveries and methods. We connect this literature to Neural Network pruning to build a framework which can optimize networks to a variety of goals. Using this framework we then show that the combination of strategies can be used to surpass single strategy pruning mechanism across a wide and representative set of models. By using a game of NIM to prune a network we have created a framework which can be used to create pruning strategies which outperforms traditional iterative pruning with limited computational overhead. While our method shows promise on CIFAR-10 and CIFAR-100 we believe that larger and more complex datasets are required to explore the effects of our pruning strategies further. \\\\\nGoing forward we many areas we wish to continue to explore. First off, we wish to explore what other games can be used to model Neural Networks pruning. While we show that NIM can be used successfully to improve pruning we believe that there may be other game formulations which are a better fit. Second, we wish to explore how NIM pruning can be used in the pruning of large language models. These models have scaled to billions of parameters and are used for a breadth of tasks so multi goal pruning will likely prove beneficial. Third, we want to explore network performance with the addition of new pruning goals. We believe the addition of FLOP optimization, Adversarial Robustness, and class aware pruning can all be introduced to NIM pruning to produce networks that maximize many goals and are small. Finally, in future work we seek to improve our framework for pruning to allow for global weight pruning. By moving to global weight pruning the game can expand to have strategies which optimize to network depth and breath, weight distribution, and take into account the varying sensitivity to pruning each layer may have. \n\\bibliographystyle{splncs04}\n\\bibliography{references}\n\\section{Appendix}\n\\subsection{Baseline Statistics}\n\\begin{tiny}\n\\begin{table}\n\\begin{tabular}{|l|l|l|l|l|}\n\\hline\nModel& Data & Param & Conv& Acc\\\\ \\hline\nVGG-16 & CIFAR-10 & 14.73 & 14.71 & 86.85 \\\\ \\hline\nVGG-16 & CIFAR-100 & 14.77 & 14.71 & 58.43 \\\\ \\hline\nRESNET50 & CIFAR-10 & 23.52 & 20.68 & 86.89 \\\\ \\hline\nRESNET50 & CIFAR-100 & 23.71 & 20.68 & 62.74 \\\\\\hline\nDPN-92 & CIFAR-10 & 34.24 & 29.88 & 88.61 \\\\\\hline\nDPN-92 & CIFAR-100 & 34.47 & 29.88 & 66.13 \\\\ \\hline \n\\end{tabular}\n\\caption{Baseline models accuracy, characteristics, and parameter size. Parameters are in millions and Conv represents the parameters in convolutional layers. }\n\\label{tab:tab1}\n\\end{table}\n\\end{tiny}\n\\subsection{L1 Pruning Mechanism}\nWe implement L1 as a single player game where player $n_1$ sorts weights by ranking those closest to the L1-Norm as lowest. The L1-norm is the sum of the magnitudes of the vectors in a space. L1 pruning is thought to be a successful because weights with a small l1 are likely to produce small activation's and thus not influence the end result much.In most existing pruning implementation L1 pruning usually produces the best results. \n\\subsubsection{Random Pruning Mechanism}\nWe implement Random as a single player game where player $n_1$ sorts weights at random. Random pruning has shown to be highly effective as it acts as a regularizer in the network forcing many sub networks to learn effective representations as the network cannot rely on only one part of the network.\n\\subsubsection{Magnitude Pruning Mechanism}\nMagnitude pruning is a common strategy in which all weights closest to zero are removed. We implement this strategy as a two player game where the first player does positive magnitude pruning and the second does negative magnitude pruning. The positive pruning agent, $n_1$, sorts weights by ranking $w_j \\in W$ where $w_j >= 0$ which removes smallest positive weights. The negative pruning agent, $n_2$, behaves the same as positive but removes the smallest negative weights.\n\\subsection{L1 + Random Pruning Mechanism}\nWe implement L1 + Random as a two player game where the first player, $n_1$, plays an L1 strategy and the second player, $n_2$, plays a random strategy. This strategies goal is to remove unimportant weights and force the existing network to learn a sparse generalizable representation.\n\\subsection{L1 + Magnitude Pruning Mechanism}\nWe implement L1 + Magnitude as a three player game where the first player, $n_1$, plays an L1 strategy, the second player, $n_2$, plays a positive magnitude strategy and the third player, $n_3$, plays a negative magnitude strategy.\n\\subsection{Random + Magnitude Pruning Mechanism}\nWe implement Random + Magnitude as a three player game where the first player, $n_1$, plays an random strategy, the second player, $n_2$, plays a positive magnitude strategy and the third player, $n_3$, plays a negative magnitude strategy.\n\\begin{tiny}\n\\begin{algorithm}[]\n\\label{algo:iterprune}\n\\SetAlgoLined\n\\KwResult{Sparse Neural Network}\n\\SetKwFunction{FPrune}{Prune}\n\\SetKwFunction{FMain}{IterativePrune}\n\\SetKwFunction{FTrain}{Train}\n\\SetKwFunction{FEval}{Evaluate}\n\\SetKwFunction{FNim}{NimGame}\n\\SetKwFunction{Fc}{isConvLayer}\n\\SetKwFunction{Fs}{GetNetworksparsity}\n\\SetKwFunction{Fss}{GetLayersparsity}\n \\SetKwProg{Fn}{Function}{:}{}\n \\Fn{\\FMain{$NN, \\pi, s_{target}, P$}}{\n pruningepochs = $\\frac{s_{target}}{\\pi}$\\;\n \\FEval($NN$)\\;\n \\For{i in range(0,2)}{\n \\FTrain{$NN$}\\;\n \\FEval($NN$)\\;\n }\n $s_{current target} = 0$\\;\n \\For{i in range(0,pruningepochs)}{\n $s_{current target} = s_{current target} + \\pi$ \\;\n $s_{current} =$ \\Fs{$NN$}\\;\n \\While{$s_{current} < s_{current target}$}{\n \\For{$\\lambda \\in NN$}{\n \\If{\\Fc{$\\lambda$}} {\n \\If{\\Fss{$\\lambda $} $ < s_{current target}$} {\n $\\lambda = $ \\FNim{$\\lambda , P$}\\;\n }\n }\n $s_{current} =$ \\Fs{$NN$}\\;\n }\n }\n \\FEval($NN$)\\;\n \\FTrain{$NN$}\\;\n \\FEval($NN$)\\;\n }\n \\For{i in range(0,10)}{\n \\FTrain{$NN$}\\;\n \\FEval($NN$)\\;\n }\n}\n\\caption{Iterative Network Pruning}\n\\end{algorithm}\n\\end{tiny}\n\\begin{algorithm}[]\n\\label{algo:nim}\n\\SetAlgoLined\n\\KwResult{pruned layer of a NN}\n\\SetKwFunction{FPrune}{Prune}\n\\SetKwFunction{FMain}{NimGame}\n\\SetKwFunction{Fs}{getStrategy}\n\\S\n \\SetKwProg{Fn}{Function}{:}{}\n \\Fn{\\FMain{$\\lambda_i, P$}}{\n \\For{p $\\in$ P}{$\\lambda_i$=\n \\FPrune{$\\lambda_i$,\\Fs{$p$}}\\; \n }\n}\n\\caption{Iterative Network Pruning}\n\\end{algorithm}\n\\begin{algorithm}[]\n\\label{algo:prune}\n\\SetAlgoLined\n\\KwResult{Pruned Neural Network Layer}\n\\SetKwFunction{FMain}{Prune}\n \\SetKwProg{Fn}{Function}{:}{}\n \\Fn{\\FMain{$\\lambda_i$, $M$}}{\n idx-sort = Sort weights $w_j \\in \\lambda_i$ using by $M$ \\CommentSty{//produce a sorted lookup index of weights in layer based on a sorting function $M$}\\;\n j = 0 \\; \n \\While{$\\lambda_i$[idx-sort[j]] == 0} {\n j = j + 1 \\CommentSty{//Find first non pruned weight}\n }\n $\\lambda_i$[idx-sort[j]] = 0 \\CommentSty{Prune weight by setting to 0} \\; \n \\KwRet $\\lambda_i$\n}\n\\caption{Prune Function sorts the weights in layer given using the optimization goal and sets the least important non zero weigth to 0 }\n\\end{algorithm}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm, height=6cm]{Paper/images/resnet50cifar10.png}\n\\caption{RESNET50 on CIFAR10}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm, height=6cm]{Paper/images/resnet50cifar100.png}\n\\caption{RESNET50 on CIFAR100}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm, height=6cm]{Paper/images/dpn92cifar10.png}\n\\caption{DPN92 on CIFAR10}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm, height=6cm]{Paper/images/dpn92cifar100.png}\n\\caption{DPN92 on CIFAR100}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm, height=6cm]{Paper/images/vgg16cifar10.png}\n\\caption{VGG16 on CIFAR10}\n\\end{figure}\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=6cm, height=6cm]{Paper/images/vgg16cifar100.png}\n\\caption{VGG16 on CIFAR100}\n\\end{figure}\n\\begin{table}[h]\n\\begin{tabular}{|l|r|r|r|}\n\\hline\n prune\\_method & sparsity & accuracy \\\\ \\hline\n Baseline & 0.000000 & 66.13 \\\\ \\hline\n MAGNITUDE & 0.925926 & 74.10 \\\\ \\hline\n L1+RANDOM & 0.925926 & 74.82 \\\\ \\hline\n Negative & 0.925926 & 74.18 \\\\ \\hline\n Positive & 0.925926 & 74.35 \\\\ \\hline\n L1 & 0.925926 & 74.82 \\\\ \\hline\n RANDOM & 0.925926 & 74.21 \\\\ \\hline\n MAGNITUDE+RANDOM & 0.925926 & 73.91 \\\\ \\hline\n MAGNITUDE+L1 & 0.925926 & 74.79 \\\\ \\hline\n\\end{tabular}\n\\label{tab:cifar100-DPN92}\n\\caption{DPN-92 on CIFAR 100. }\n\\end{table}\n\n\n\\begin{table}[h]\n\\begin{tabular}{|l|r|r|r|}\n\\hline\n prune\\_method & sparsity & accuracy \\\\ \\hline\n Baseline & 0.000000 & 88.61 \\\\ \\hline\n L1 & 0.925926 & 95.45 \\\\ \\hline\n Positive & 0.925926 & 95.25 \\\\ \\hline\n Negative & 0.925926 & 93.85 \\\\ \\hline\n MAGNITUDE+RANDOM & 0.925926 & 94.32 \\\\ \\hline\n MAGNITUDE & 0.925926 & 92.78 \\\\ \\hline\n RANDOM & 0.925926 & 95.17 \\\\ \\hline\n L1+RANDOM & 0.925926 & 95.16 \\\\ \\hline\n MAGNITUDE+L1 & 0.925926 & 94.30 \\\\ \\hline \n\\end{tabular}\n\\label{tab:cifar10-DPN92}\n\\caption{DPN-92 on CIFAR 10. Unlike all other experiments the most successful mechanism is a one player game.}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{tabular}{|l|r|r|r|}\n\\hline\n prune\\_method & sparsity & accuracy \\\\ \\hline \n Baseline & 0.000000 & 86.85 \\\\ \\hline\n Positive & 0.961008 & 90.81 \\\\ \\hline\n MAGNITUDE & 0.985989 & 90.16 \\\\ \\hline\n Negative & 0.961008 & 88.92 \\\\ \\hline\n RANDOM & 0.975409 & 90.27 \\\\ \\hline\n MAGNITUDE+RANDOM & 0.994909 & 90.80 \\\\ \\hline\n L1+RANDOM & 0.981350 & 91.10 \\\\ \\hline\n L1 & 0.975409 & 90.37 \\\\ \\hline\n MAGNITUDE+L1 & 0.921536 & 90.04 \\\\ \\hline\n\\end{tabular}\n\\label{tab:cifar10-VGG16}\n\\caption{VGG16 on CIFAR 10. While all methods outperform initial baseline the combination of L1+Random outperforms all other methods.}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{tabular}{|l|r|r|r|}\n\\hline\n prune\\_method & sparsity & accuracy \\\\ \\hline \n Baseline & 0.000000 & 58.43 \\\\ \\hline\n Positive & 0.961008 & 67.27 \\\\ \\hline\n RANDOM & 0.975409 & 67.61 \\\\ \\hline\n MAGNITUDE+L1 & 0.921484 & 65.71 \\\\ \\hline\n MAGNITUDE & 0.985989 & 66.66 \\\\ \\hline\n L1+RANDOM & 0.981350 & 67.98 \\\\ \\hline\n Negative & 0.961008 & 64.96 \\\\ \\hline\n MAGNITUDE+RANDOM & 0.994906 & 67.39 \\\\ \\hline\n L1 & 0.975409 & 67.44 \\\\ \\hline\n\\end{tabular}\n\\label{tab:cifar100-VGG16}\n\\caption{VGG16 on CIFAR 100. While all methods outperform initial baseline the combination of L1+Random outperforms all other methods.}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{tabular}{|l|r|r|r|}\n\\hline\n prune\\_method & sparsity & accuracy \\\\ \\hline\n Baseline & 0.000000 & 62.74 \\\\ \\hline\n RANDOM & 0.925926 & 73.23 \\\\ \\hline\n Positive & 0.925926 & 73.06 \\\\ \\hline\n MAGNITUDE & 0.925926 & 72.00 \\\\ \\hline\n MAGNITUDE+RANDOM & 0.925926 & 71.54 \\\\ \\hline\n MAGNITUDE+L1 & 0.925926 & 71.63 \\\\ \\hline\n L1+RANDOM & 0.925926 & 73.35 \\\\ \\hline\n Negative & 0.925926 & 72.31 \\\\ \\hline\n L1 & 0.925926 & 72.75 \\\\ \\hline\n\\end{tabular}\n\\label{tab:cifar100-RESNET50}\n\\caption{RESNET50 on CIFAR 100. While all methods outperform initial baseline the combination of L1+Random outperforms all other methods.}\n\\end{table}\n\n\\begin{table}[h]\n\\begin{tabular}{|l|r|r|r|}\n\\hline\n prune\\_method & sparsity & accuracy \\\\ \\hline\n Baseline & 0.000000 & 86.89 \\\\ \\hline\n Positive & 0.925926 & 91.69 \\\\ \\hline\n Negative & 0.925926 & 91.68 \\\\ \\hline\n MAGNITUDE+RANDOM & 0.925926 & 92.09 \\\\ \\hline\n MAGNITUDE & 0.925926 & 90.78 \\\\ \\hline\n L1+RANDOM & 0.925926 & 92.36 \\\\ \\hline\n L1 & 0.925926 & 91.92 \\\\ \\hline\n RANDOM & 0.925926 & 92.20 \\\\ \\hline\n MAGNITUDE+L1 & 0.925926 & 91.49 \\\\ \\hline\n\\end{tabular}\n\\label{tab:cifar10-RESNET50}\n\\caption{RESNET50 on CIFAR 10. While all methods outperform initial baseline the combination of L1+Random outperforms all other methods.}\n\\end{table}\n\\end{document}\n", "meta": {"hexsha": "7ddbf45e08bd747c593e2e271b17c5e0103a4229", "size": 31178, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Paper/other_main.tex", "max_stars_repo_name": "spacemanidol/PriceOfAnarchyPruning", "max_stars_repo_head_hexsha": "ea20c251a1eca04abb566d5968c3e41204dc3fc9", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2020-12-02T09:02:34.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-02T09:02:34.000Z", "max_issues_repo_path": "Paper/other_main.tex", "max_issues_repo_name": "spacemanidol/PriceOfAnarchyPruning", "max_issues_repo_head_hexsha": "ea20c251a1eca04abb566d5968c3e41204dc3fc9", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-12-03T08:39:42.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-03T08:39:42.000Z", "max_forks_repo_path": "Paper/other_main.tex", "max_forks_repo_name": "spacemanidol/GameTheoreticPruning", "max_forks_repo_head_hexsha": "ea20c251a1eca04abb566d5968c3e41204dc3fc9", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 100.5741935484, "max_line_length": 1131, "alphanum_fraction": 0.7508499583, "num_tokens": 7977, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6113819732941511, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.3437045784511603}} {"text": "\\chapter{Proposed Translator}\n\\label{ch:proposed-translator}\n\nThis chapter focuses on proposing a solution to ODMTP.\nFirst formalizing the solution. And then modelling it\nwith elements of software engineering such as use cases\nand requirements.\n\n\\section{Translator Formalization}\nAs a solution to the previous chapter, this section focuses on\nproposing a application $f : S' \\rightarrow M$ such that\napplied on a schema, results in a domain model based on\nplain objects.\n\nLets define $f(S') = \\begin{bmatrix}f'(s_1')\\\\ f'(s_2')\\\\ \\vdots\\\\ f'(s_n')\\end{bmatrix}$ and\n$f'(s_i') = \\begin{bmatrix}f''(e_1')\\\\ f''(e_2')\\\\ \\vdots\\\\ f''(e_n')\\end{bmatrix}$. Then $f''(e_i')$\nis the application that maps a triple expression $e'$ from $N \\times T_{g} \\times \\{(1,1),(0,\\infty)\\}$\nto $N \\times T_g$. To find such a function we will use the knowledge that we already have.\nWe know that $p$ has a direct mapping as it belongs to $N$, $T_g$ maps to $T_g$ if\nthe cardinality value is $(1,1)$ or $(1, \\infty)$. And the cardinality is aggregated to the type\nso its not needed to map it. Then we define the application $f''(e')$ as $f:(p,t,c) \\in N \\times T_g \\times \\{(1,1), (0,\\infty)\\} \\rightarrow (n,t)\\in N \\times T_g$\nand therefore,\n\n\\begin{equation}\\label{eq:transformation}\nf''(e_i')\n\\begin{cases}\n (p,Proy_{t_g}lst) & if \\; c=(1,1) \\\\\n (p,List[Proy_{t_g}lst]) & if \\; c=(0,\\infty)\n\\end{cases}.\n\\end{equation}\n\nThis application's function is to transform a triple expression\ninto an annotated type property. Where the $Proy_{tg}lst$ represents\nthe projection of the generic type from the abstraction of languages\nof representation of plain objects on to the language specific type.\n\\cref{fig:lst-diagram} illustrates how the same input can lead to\nmultiple types due to the specific translators, that perform the \n$Proy_{tg}lst$ operation.\n\n\\begin{figure}\n \\includegraphics[scale=0.8]{images/lsc-diagram.pdf}\n \\centering\n\t\\caption[Different target types generated by specific translators]{Different target types generated by specific translators.}\n \\label{fig:lst-diagram}\n\\end{figure}\n\n\\section{Translator modelling}\nAt this point we already have an abstraction of our system ready.\nWe know that you must implement the transformation function previously explained.\nNow we will lower our abstraction one level.\nFor this we will model our system by means of software engineering techniques\nsuch as use cases, requirements or class diagrams. First, we will use the use case\nmethod to find the necessary functionality of our system.\n\n\\begin{figure}[h!]\n \\includegraphics[scale=0.8]{images/trans-use-case.pdf}\n \\centering\n \\caption[Translator use cases]{Translator use cases.}\n \\label{fig:trans-use-case}\n\\end{figure}\n\n\\cref{fig:trans-use-case} shows us that in our system we will, of course, have the functionality to\ntranslate ShEx schemas to domain models based on plain objects.\nBut we can also see that the entry may be wrong.\nThis implies that there must be some kind of input validation.\nIn addition, an error management system is also necessary.\n\\cref{fig:trans-flow} illustrates a high level view of the flowchart that the translator follows.\n\n\\begin{figure}\n \\includegraphics[width=\\textwidth]{images/diagrama-flujo-traductor.pdf}\n \\centering\n \\caption[Translator high level flowchart]{Translator high level flowchart.}\n \\label{fig:trans-flow}\n\\end{figure}\n\nAlso from the developed use cases we can extract the list of functional and\nnon-functional requirements (external interfaces) that our system must support.\n\n\\begin{figure}[h!]\n \\includegraphics[width=\\textwidth]{images/trans-reqf.pdf}\n \\centering\n \\caption[Translator functional requirements]{Translator functional requirements.}\n \\label{fig:trans-reqf}\n\\end{figure}\n\n\\begin{figure}[h!]\n \\includegraphics[width=\\textwidth]{images/trans-reqnf.pdf}\n \\centering\n \\caption[Translator non functional requirements]{Translator non functional requirements.}\n \\label{fig:trans-reqnf}\n\\end{figure}\n\nThus, for the previous use cases and requirements the implementation abstract diagram will be.\n\n\\begin{figure}[h!]\n \\includegraphics[width=\\textwidth]{images/trans-diagram.pdf}\n \\centering\n \\caption[Translator component and class diagrams]{Translator component and class diagrams.}\n \\label{fig:trans-diag}\n\\end{figure}", "meta": {"hexsha": "a155989829c95dd4cf752664522e8e8dbd5aeef3", "size": 4339, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapters/part_2/translator-analysis.tex", "max_stars_repo_name": "thewilly/shex-lite-book", "max_stars_repo_head_hexsha": "ce22ebcf6cd1ce5bafd79398ee87e50dcb18c41d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapters/part_2/translator-analysis.tex", "max_issues_repo_name": "thewilly/shex-lite-book", "max_issues_repo_head_hexsha": "ce22ebcf6cd1ce5bafd79398ee87e50dcb18c41d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapters/part_2/translator-analysis.tex", "max_forks_repo_name": "thewilly/shex-lite-book", "max_forks_repo_head_hexsha": "ce22ebcf6cd1ce5bafd79398ee87e50dcb18c41d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 43.39, "max_line_length": 164, "alphanum_fraction": 0.7483291081, "num_tokens": 1153, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7577943712746406, "lm_q2_score": 0.45326184801538616, "lm_q1q2_score": 0.34347927713960125}} {"text": "\\documentclass[oneside,9pt]{article}\n\\usepackage[a4paper,width=170mm,top=18mm,bottom=22mm,includeheadfoot]{geometry}\n\\pdfoutput=1\n\\usepackage{multicol}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{caption}\n\\usepackage{hyperref}\n\\usepackage{mathtools}\n\\usepackage[linesnumbered,algoruled,boxed,lined]{algorithm2e}\n\\usepackage{graphicx}\n\\usepackage{float}\n\n\\hypersetup{\n colorlinks=true,\n urlcolor=cyan\n}\n\\hypersetup{final}\n\n% paper version\n\\newcommand{\\PaperVersionNumber}{0.1 - 2018-02-28}\n\n\\providecommand{\\keywords}[1]{\\textbf{\\textit{Keywords---}} #1}\n\n% list of elements\n\\newcommand{\\lista}[3][_]{ #2#10, \\dots, #2#1{#3} }\n% set\n\\newcommand{\\setd}[1]{ \\mathit{#1} }\n\\newcommand{\\setl}[2]{ \\left\\{ \\lista{#1}{#2} \\right\\} }\n% vector\n\\newcommand{\\vecd}[1]{ \\textbf{#1} }\n\\newcommand{\\vecl}[2]{ (\\lista{#1}{#2}) }\n% poly\n\\newcommand{\\poly}[2]{ p_{\\textbf{#1}}(#2) }\n\\newcommand{\\polysum}[3][x]{ #2_0 + #2_1 #1 + #2_2 #1^2 + \\dots + #2_{#3} #1^{#3} }\n% big O\n\\newcommand{\\bigo}[1]{ \\mathcal{O}(#1) }\n\n% evaluated point\n\\newcommand{\\ele}{s}\n% linear codes \\ec{n}{k}\n\\newcommand{\\ec}[2]{ \\mathcal{C}(#1, #2) }\n% finite field \\gf{q} or \\gf[n]{p}\n\\newcommand{\\gf}[2][]{ \\mathbb{F}_{{#2}^{#1}} }\n% ring nf4\n\\newcommand{\\ringn}[1][n]{ \\mathbb{R}_{F_4}(#1) }\n% fft\n\\newcommand{\\fft}[1]{ \\text{FFT}(#1) }\n\\newcommand{\\fftl}{N}\n% coef\n\\newcommand{\\ith}[1]{#1^\\text{th}}\n% set of points\n\\newcommand{\\sete}{\\setd{S}}\n% complexity\n\\newcommand{\\an}[1][N]{\\mathcal{A}(#1)}\n\\newcommand{\\mn}[1][N]{\\mathcal{M}(#1)}\n% F4\n\\newcommand{\\ffour}{65537}\n\\newcommand{\\ffourMinusOne}{65536}\n% operator\n\\newcommand{\\op}{\\diamond}\n\n\\DeclarePairedDelimiter{\\ceil}{\\lceil}{\\rceil}\n\\newcommand*\\eg{e.g.\\@\\xspace}\n\\newcommand*\\Eg{e.g.\\@\\xspace}\n\\newcommand*\\ie{i.e.\\@\\xspace}\n\\newcommand*\\samethanks[1][\\value{footnote}]{\\footnotemark[#1]}\n\n%\\renewcommand{\\itemhook}{\\setlength{\\topsep}{0pt} \\setlength{\\itemsep}{0pt}\\setlength{\\leftmargin}{15pt}}\n\n%% Authors\n\\author{ Vianney Rancurel\\thanks{Scality in San Francisco, CA, 94104.}\\\\ \\texttt{vianney.rancurel@scality.com}\n \\and L\\^am Pham-Sy\\thanks{Scality France.}\\\\ \\texttt{lam.pham-sy@scality.com} \n \\and Sylvain Laperche\\samethanks[3]\\\\ \\texttt{sylvain.laperche@scality.com}\n}\n\n\\title{QuadIron: A library for Number Theoretic Transform-based Erasure Codes\\thanks{Manuscript received Januray 12, 2020.}}\n\n\\begin{document}\n\\maketitle\n\n\\begin{abstract}\nReed-Solomon codes can be seen as polynomial operations which can be implemented by number theoretic transforms and accelerated by different fast Fourier transform algorithms according to the characteristics of the fields. We present a library, QuadIron, which implements FFTs in binary, prime, and extension fields, as well as a set of optimizations to make them practical for a very large number of symbols. These FFT-based erasure codes could be useful for applications like decentralized storage over Internet, or real-time video high-quality streams.\n\\end{abstract}\n\n\n\\keywords{Number Theoretic Transforms, Discrete Fourier Transforms, Finite Fields, Erasure Codes, Digital Storage, Distributed storage, Decentralized Storage, Decentralized Cloud Storage, Real-time video, high-quality streaming}\n\n\n\\section{Introduction}\n\\label{sec:introduction}\n\nA {$\\ec{n}{k}$} erasure code is defined by $n=k+m$, $k$ being the number of data fragments, $m$ being the number of desired erasure fragments. In an application it is required to transmit the $n$ fragments. A Maximum Distance Separable (MDS) code guarantees that any $k$ fragments can be used to decode a file. Erasure codes can be either systematic or non-systematic. Systematic codes generate $n-k$ erasure fragments and therefore maintain $k$ data fragments. Non-systematic codes generate $n$ erasure fragments. In the case of systematic codes, we try to retrieve primarily the $k$ data fragments if possible because there is nothing to decode. A decoding is necessary only if one or more data fragments are missing. In the case of non-systematic codes, we need to decode $k$ fragments. Erasure codes can also be compared by their sensitivity to the rate $r=k/n$, which may or may not impact the encoding and decoding speed. Another comparison criterion is the support of adaptive rate: does the erasure code allows to change $k$ and $m$ dynamically, without having to regenerate the whole set of erasure fragments. Another critical property is called the 'confidentiality' \\cite{li_confidentiality} which is determined if an attacker can partially decode the data if he obtains less than k fragments. Finally, we can also compare erasure code according to their repair bandwidth, i.e.\\ the number of fragments required to repair a fragment. To sum up, here is a list of codes' properties that are of interest for us:\n\\begin{itemize}\n\\item MDS/non-MDS\n\\item Systematic/Non-systematic\n\\item Encoding/Decoding speed according to various $n$\n\\item Encoding/Decoding speed predictivity and stability acc/ to $n$\n\\item Rate sensitivity\n\\item Adaptive vs non-adaptive rate\n\\item Confidentiality\n\\item Repair bandwidth\n\\end{itemize}\n\nReed-Solomon (RS) codes are MDS codes constructed from Vandermonde or Cauchy matrices \\cite{reed_sol} that are systematic and support adaptive rates. The RS encoding process is traditionally performed by multiplying a $[n \\times k]$ matrix by a $k$ data vector in $\\gf{2^w}$, leading to a high complexity of $\\bigo{k \\times n}$. The topic of optimizing those codes has been widely discussed. While such optimizations reduce the overall complexity, most of them rely solely on hardware optimizations, therefore their real complexity in ''big O'' notation has not been widely studied.\n% * 2018-03-02T18:03:30.509Z:\n% \n% Question on notation: I am assuming that an n*k matrix is a different expression than k x n. If not, make them the same.\n% \n% ^.\n\nLow-Density-Parity-Check (LDPC) codes are also an important class of erasure codes and are constructed over sparse parity-check matrices. Although initially used in networking applications, some researchers recently showed that it is possible to use them in distributed storage scenarios \\cite{plank_ldpc}. Those codes, which even though require to store $n=k+m$ fragments (like MDS codes), need to retrieve $[f \\times k]$ fragments to recover the data (instead of only $k$ for MDS codes), $f$ being called the overhead or the inefficiency. The study \\cite{plank_ldpc} shows 3 types of LDPC codes where $f$ oscillates between $1.10$ and $1.30$ for $n \\lessapprox 100$, and shows that you need a larger $n$, e.g.\\ $n \\gtrapprox 1000$ in order to have an $f$ approaching $1.0$. As shown by \\cite{plank_ldpc}, those codes are more sensible to network latency because of the extra fragments, due to the overhead, that they need to retrieve, so in cases where latency can be important RS codes seems more interesting than LDPC codes. More recently hybrid-LDPC schemes such as \\cite{wirehair} have reduced the overhead to $[k + f]$ with a very small f. Also, \\cite{park_ldpc} have shown that it is possible to design LDPC codes which beat RS codes when taking into account the repair bandwidth, because RS codes always need to retrieve $k$ fragments to be able to repair the data, while it is possible to design LDPC codes that require less than $k$ fragments for the repair process. However:\n\\begin{itemize}\n\\item LDPC are not MDS: it is always possible to find a pattern (e.g.\\ stopping sets) that cannot decode (e.g. having only k fragments out of n).\n\\item You can always find/design an LDPC code optimized for few properties (i.e.\\ tailored for a specific use case) that beats other codes on those few properties, but there is no silver bullet: it will be sub-optimal for the other properties (its a trade-off, e.g.\\ good for large $n$ and with an optimal repair bandwidth, but not good for small $n$ and cannot support adaptive rate): these cannot be used in a generic library.\n\\item Designing a good LDPC code is some kind of black art that requires a lot of fine tuning and experimentation. Ultimately an LDPC code optimal for all the interesting properties for a given use case could exist but would be very complex and/or would only be available in a commercial library.\n\\end{itemize}\n\nRecently some other types of codes, called Locally-Repairable-Codes (LRC) \\cite{xor_elephant} \\cite{erasure_azure}, have tackled the repair bandwidth issue of the RS codes. They combine multiple layers of RS: the local codes and the global codes. However those codes are not MDS and they require an higher storage overhead than MDS codes.\n\nFast Fourier transform (FFT) based RS codes remain relatively simple, and can be used to perform encoding on finite fields with clearer and lower announced complexities therefore having a good set of desirable properties:\n\\begin{itemize}\n\\item They are MDS\n\\item Support both systematic or non-systematic codes\n\\item Fast for a wide range of $n$\n\\item Rate insensitive (for non-systematic codes).\n\\item Confidential (for systematic codes).\n\\end{itemize}\nWe focus our research on optimizing their encoding and decoding speed.\n% * 2018-03-02T18:07:38.602Z:\n% \n% > Recently, thanks to fast Fourier transform\n% Are you announcing your findings here? If so: \"We have found that using FFT...\"\n% \n% ^.\n\nSince the computational complexities of FFT operations mainly depend on the chosen finite field, we investigate two types of finite fields: prime finite fields and binary extension finite fields. For each type, there are different approaches to accelerate the FFT operations.\n\n%\\hfill The authors\n \n%\\hfill April 26, 2018\n\n\\section{Reed-Solomon codes}\n%%%%%%%%%%%%%%%%%%%%%%%%\n\nA Reed-Solomon (RS) code is an MDS code $\\ec{n}{k}$ defined over a finite field $\\gf{q}$ where $q$ is a prime power $q=p^w$. A code word composed of $n$ symbols is generated from a message of length $k$. The message is recovered from a subset of at least $k$ elements of the code word.\n% * 2018-03-02T18:23:50.138Z:\n% \n% > code word\n% Changing this to code word from \"codeword\" globally. If you know better (i.e., if \"codeword\" is a widely accepted neologism), you can roll it back.\n% \n% ^.\n\n\\subsection{Encoding}\nAs Reed and Solomon described in their foundational paper\\cite{reed_sol}, a code word is encoded from a message using polynomial evaluation. \n\nA message $\\vecd{m} = \\vecl{m}{k} \\in \\gf{q}^k$ can be represented as a polynomial $\\poly{m}{x}$ of degree $k-1$:\n\n\\begin{equation}\\label{eq:poly} \n\t\\poly{m}{x} = \\polysum{m}{k-1}\n\\end{equation}\n\nThe code word $\\vecd{c} = \\vecl{c}{n-1} \\in \\gf{q}^n$ of the message $\\vecd{m}$ is obtained by evaluating $\\poly{m}{x}$ at a given but arbitrary set of $n$ \\emph different points $\\sete = \\setl{\\ele}{n-1}$ of the field $\\gf{q}$, called the set of evaluation points. Concretely,\n\n\\begin{equation}\\label{eq:enc_def}\n\\begin{split}\n\tc_0 &= \\polysum[\\ele_0]{m}{k-1},\\\\\n\tc_1 &= \\polysum[\\ele_1]{m}{k-1},\\\\\n\t& \\dots \\\\\n\tc_{n-1} &= \\polysum[\\ele_{n-1}]{m}{k-1}\\\\\n\\end{split}\n\\end{equation}\n\nEquation (\\ref{eq:enc_def}) can be represented as a multiplication of the message $\\vecd{m}$ and matrix $G$:\n\n\\begin{equation}\n\\vecd{c} = \\vecd{m} \\times G\n\\end{equation}\n where $G$ is the generator matrix of the RS codes:\n\n\\begin{equation}\\label{eq:gen_mat}\nG = \\begin{bmatrix}\n1 & 1 & \\dots & 1 \\\\\n\\ele_0 & \\ele_1 & \\dots & \\ele_{n-1} \\\\\n\\ele_0^2 & \\ele_1^2 & \\dots & \\ele_{n-1}^2 \\\\\n\\vdots & \\vdots & & \\vdots \\\\\n\\ele_0^{k-1} & \\ele_1^{k-1} & \\dots & \\ele_{n-1} ^{k-1}\n\\end{bmatrix}\n\\end{equation}\n\nSince $\\ele_i \\neq \\ele_j$ when $i \\neq j$, $G$ is actually a Vandermonde matrix \\cite{book_ecc}.\n% * 2018-03-02T18:31:07.849Z:\n% \n% > for\n% do you mean \"when\" or \"because\"?\n% \n% ^.\n\nStraightforwardly, the encoding process has a computational complexity of $\\bigo{k \\times n}$ for performing a multiplication of a vector $\\vecd{m}$ and a matrix $G$.\n\nThis complexity can be reduced by using a Discrete Fourier Transform (DFT) on a finite field that evaluates polynomial $\\poly{m}{x}$ at the set $\\sete$ of evaluating points. Currently, two main Fast Fourier Transform (FFT) techniques apply for prime field $\\gf{p}$ \\cite{cooley_tukey,fnt_lacan,fnt_ida}, and for binary extension field $\\gf{2^w}$ \\cite{fft_add_wang_zhu,fft_add_cantor,fft_add_gao_mateer,fft_add_chung}. We summarize these techniques in Section \\ref{sec:fft}. Briefly, thanks to the FFT techniques, the encoding complexity drops to $\\bigo{n \\times \\log{n}}$.\n\n%Briefly, the codeword $\\vecd{c}$ can be calculated by one FFT operation on message $\\vecd{m}$:\n%\\begin{equation}\\label{eq:fft_1}\n%\\vecd{c} = \\fft{\\vecd{m}}\n%\\end{equation}\n\n\\subsection{Decoding}\n\nA message $\\vecd{m}$ can be decoded from a code word $\\vecd{c}$ that could miss at most $n-k$ elements. Indeed, given a set of $k$ coefficients of $\\vecd{c}$, the polynomial $\\poly{m}{x}$ in Equation (\\ref{eq:poly}) can be regenerated using Lagrange interpolation because its degree is at most $k-1$. Let $\\setl{\\tilde{c}}{k-1}$ denote the set of $k$ known coefficients corresponding to $k$ evaluation points $\\setl{\\tilde{\\ele}}{k-1}$. The polynomial $\\poly{m}{x}$ is thus interpolated:\n\n\\begin{equation}\n\\label{eq:lagrange}\n\\poly{m}{x} = \\sum_{i=0}^{k-1} \\big( \\tilde{c}_i \\times \\sum_{0 \\leq j \\leq k-1, j \\neq i} \\frac{x - \\tilde{\\ele}_j}{ \\tilde{\\ele}_i - \\tilde{\\ele}_j } \\big)\n\\end{equation}\n\nCoefficients of the message $\\vecd{m}$ are actually coefficients of $\\poly{m}{x}$.\n\nOn prime field, equation \\ref{eq:lagrange} can be simplified as proposed in \\cite{fnt_lacan}. Note that degree of $\\poly{m}{x}$ is at most $(k-1)$, the polynomial is derived as:\n\n\\begin{equation}\n\\poly{m}{x} = \\Big( A(x) \\times \\sum_{i=0}^{k-1} \\frac{n_i}{x - \\tilde{\\ele}_i} \\Big) \\text{ mod } x^k\n\\end{equation}\nwhere\n\\[\n\\begin{split}\nA(x) &= \\prod_{i=0}^{k-1}(x - \\tilde{\\ele}_i), \\\\\nn_i &= \\frac{ \\tilde{c}_i }{ A'( \\tilde{\\ele}_i) }\n\\end{split}\n\\]\n\nUsing Taylor series, we have\n\n\\[\n\\frac{1}{ \\tilde{\\ele}_i - x } = \\sum_j \\frac{ x^j }{ {\\tilde{\\ele}_i}^{j+1} }\n\\]\n\nAnd by defining the following polynomial:\n\\[\nN(x) := \\sum_{i = 0}^{k-1} \\frac{n_i}{\\tilde{\\ele}_i} \\times x^{z_i}\n\\]\nwhere its exponents $\\{z_i, i = 0, \\dots, k-1 \\}$ are determined by the evaluation set $S$ and the $\\ith{n}$-root of unity $r$\n\\[\n\\tilde{\\ele}_i = r^{z_i}, \\ i = 0, \\dots, k-1\n\\]\n \n We obtain a simpler formula:\n\n\\begin{equation}\n\\poly{m}{x} = \\Big( A(x) \\times B(x) \\Big) \\text{ mod } x^k\n\\end{equation}\nwhere $B(x) := \\sum_{i=0}^{k-1} N(r^{-i}) x^i $ whose coefficients are obtained by evaluating $N(x)$ at $k$ points $\\{ r^{-i} \\}$ that can be efficiently performed using the inverse FFT algorithm.\n\nThanks to the convolution theorem, we can use FFT techniques to accelerate the multiplication of two polynomials $A(x), B(x)$:\n\\begin{equation}\nA(x) \\times B(x) = iFFT_{2k} ( FFT_{2k}(A) \\cdot FFT_{2k}(B) )\n\\end{equation}\nwhere $FFT_{2k}, iFFT_{2k}$ are FFT and inverse FFT of length $2 \\times k$.\n\nNote that $A(x)$ depends only on evaluation points, hence $A(x), A'(x), FFT_{2k}(A)$ can be calculated once.\n\nThe decoding computation is composed of:\n(1) one inverse FFT of length $n$ to compute $N(x)$;\n(2) one FFT and one inverse FFT of length $2k$ to compute $A(x) \\times B(x)$.\nIt leads to a decoding complexity of $\\bigo{ n \\log n + 4 k \\log{2k}}$.\n\nCompared to \\cite{fnt_lacan}, our computations are based on the fact that the degree of $\\poly{m}{x}$ is smaller than $k$. Hence we use $2k$-length FFT instead of $2n$-length FFT.\n\n%%%%%%%%% Section %%%%%%%%%%\n\\section{How can FFT reduce Encoding Complexity?}\\label{sec:fft}\n%%%%%%%%%%%%%%%%%%%%%%%%\n\nIn this section, we summarize two FFT techniques on the field $\\gf{q}$ that can the reduce computational complexity of the encoding process. Their common methodology is to split an FFT operation on a vector $\\vecd{m}$ into two FFT operations on a smaller set $\\vecd{m}_1$ and $\\vecd{m}_2$. \nLet $\\fftl$ denote the length of input vector $\\vecd{m}$ that is also represented as a polynomial $\\poly{m}{x}$\n\n\\begin{equation}\\label{eq:poly_fft} \n\t\\poly{m}{x} = \\polysum{m}{\\fftl-1}\n\\end{equation}\n\nGiven a set $\\sete = \\setl{\\ele}{\\fftl-1}$ of different elements of $\\gf{q}$, FFT transforms the input vector to an output vector $\\vecd{c}$ whose coefficients are values of $\\poly{m}{x}$ evaluated at points of $\\sete$.\n\n\\begin{equation}\\label{eq:enc_fft} \n\t\\vecd{c} = \\big( \\poly{m}{\\ele_0}, \\poly{m}{\\ele_1}, \\dots, \\poly{m}{\\ele_{\\fftl-1}} \\big)\n\\end{equation}\n\nThis is similar to the described RS encoding process, except that the input and output vectors are of the same length $\\fftl$.\n\nThe key factor that allows splitting FFTs into smaller transformations is choosing the set $\\sete$. The two methods of constructing such a set are described here.\n% * 2018-03-02T18:43:34.525Z:\n% \n% > The key factor that allows splitting FFT into smaller ones is choosing the set $\\sete$.\n% Smaller whats? \n% \n% ^.\n\n\\subsection{$\\sete$ as multiplicative group}\n\nThe set $\\sete$ is constructed as a multiplicative group whose generator is the $\\ith{\\fftl}$ root of unity of the field $\\gf{q}$, i.e.\n\n\\begin{equation}\n\\sete = \\left\\{ \\lista[^]{\\alpha}{\\fftl-1} \\right\\}\n\\end{equation}\nwhere $\\alpha$ is the $\\ith{\\fftl}$ root of unity of the field $\\gf{q}$. We call this technique \\textit{multiplicative FFT}.\n\nThe FFT technique applied for the constructed set $\\sete$ was first introduced in \\cite{cooley_tukey}. Supposing that $\\fftl = \\fftl_1 \\times \\fftl_2$, the FFT operation on $\\vecd{m}$ is split into two FFT operations on two vectors $\\vecd{m}_1$ and $\\vecd{m}_2$ of length $\\fftl_1$ and $\\fftl_2$ respectively. This results in a computational complexity $\\bigo{\\fftl(\\fftl_1+\\fftl_2)}$. If $\\fftl$ is highly composite, this technique reduces the computation time to $\\bigo{\\fftl\\log{\\fftl}}$.\n\nTo obtain such advantages, however, this technique requires the satisfaction of two conditions:\n\\begin{enumerate}\n\\item $\\fftl$ is a divisor of $q-1$ (for the existing $\\ith{\\fftl}$ root of unity of the field $\\gf{q}$),\n\\item $\\fftl$ is highly composite, e.g. $\\fftl = 2^v$ ideally\n\\end{enumerate}\n\nA simple solution to satisfying these conditions is to use the field $\\gf{q=p^w}$ where $p$ is prime and $p-1$ is highly composite. Fermat numbers, i.e. $F_i = 2^{2^i} + 1$ for $i \\leq 4$, are perfectly suitable for that. This technique, also called Fermat Number Transform (FNT) based erasure codes, was introduced and analyzed in \\cite{fnt_lacan, fnt_ida}.\n\nThis technique, however, does not efficiently apply to the binary extension field $\\gf[w]{2}$ because divisors of $(2^w-1)$ are not highly composite. However, $\\gf[w]{2}$ is perfectly suitable for most practical applications because each element can be expressed by $n$ bits. The following FFT technique focuses on this field.\n\n\\subsection{$\\sete$ as additive group}\n\nThis technique focuses on the binary extension field $\\gf[w]{2}$. Introduced in\\cite{fft_add_wang_zhu, fft_add_cantor}, it was recently improved in \\cite{fft_add_gao_mateer} by using Taylor extension algorithms. A further improvement is shown in \\cite{fft_add_chung}.\n\n%We describe briefly the technique introduced in \\cite{fft_add_gao_mateer}. \nThe FFT length is necessarily a power of 2, i.e. $\\fftl = 2^m$ with $m \\leq w$.\nLet $\\lista{\\beta}{m-1}$ be $m$ linearly independent elements of $\\gf[n]{2}$. The set $\\sete$ is chosen as a subspace spanned by $\\beta_i$ over $\\gf{2}$, i.e.\n\n\\begin{equation}\n\\begin{split}\n\\ele_i = i_0 \\beta_0 + i_1 \\beta_1 + i_2 \\beta_2 + \\dots + i_{m-1} \\beta_{m-1},\\\\\n\\text{ for } 0 \\leq i \\leq 2^m-1\\\\\n\\end{split}\n\\end{equation}\n\nwhere $i = \\polysum[2]{i}{m-1}$ with $i_j \\in \\gf{2}$.\nThe FFT operation on a vector $\\vecd{m}$ is re-expressed as two FFTs on two vectors of half length.\n\n\\subsection{Computational complexity}\n\nLet $\\an[\\fftl]$ denote the number of additions or subtractions in the field $\\gf{q}$ needed for an FFT operation with an input vector of length $\\fftl$ (assuming that an addition requires the same amount of effort to implement as a subtraction).\n\nLet $\\mn[\\fftl]$ denote the number of multiplications required in the field $\\gf{q}$ needed for an FFT operation with an input vector of length $\\fftl$.\n\nThe computational complexities of these algorithms are shown in Table \\ref{tab:compare}, which shows that, with suitable parametrization of the FFT length and the field $\\gf{q}$, multiplicative FFT is faster than all available additive FFTs.\n\n\\begin{table*}[!ht]\n \\caption{Computational complexity of FFT techniques}\n \\label{tab:compare}\n \\begin{center}\n \\begin{tabular}{|c | c | c | c|} \n \\hline\n Name\n \t& Restriction\n\t& Addition $\\an[\\fftl]$\n\t& Multiplication $\\mn[\\fftl]$\n\t\\\\ \\hline\n Multiplicative FFT \n \t& $\\fftl = 2^v$ and $(q-1) \\% \\fftl = 0$ \n \t& $\\fftl \\log_2(\\fftl)$ \n\t& $\\frac{1}{2} \\fftl \\log_2(\\fftl) - \\fftl + 1$\n\t\\\\ \\hline\n Additive FFT \\cite{fft_add_gao_mateer}\n \t& $\\fftl = 2^v$ and $q=2^w$\n\t& $4 \\fftl (\\log_2(\\fftl))^2 + \\frac{3}{4} \\fftl \\log_2(\\fftl) - \\frac{1}{2} \\fftl$ \n\t& $2 \\fftl \\log_2(\\fftl) - 2 \\fftl + 1$\n\t\\\\ \\hline \n Additive FFT \\cite{fft_add_gao_mateer}\n \t& $\\fftl = 2^{2^v}$ and $q=2^w$\n\t& $\\fftl \\log_2(\\fftl) + \\frac{1}{2} \\fftl \\log_2(\\fftl) \\log_2\\log_2(\\fftl)$ \n\t& $\\frac{1}{2} \\fftl \\log_2(\\fftl)$\n\t\\\\ \\hline \n Additive FFT \\cite{fft_add_chung}\n \t& $\\fftl = 2^{v}$ and $q=2^w$\n\t& $\\fftl \\log_2(\\fftl) $ \n\t& $\\frac{1}{2} \\fftl \\log_2(\\fftl)$\n\t\\\\ \\hline \n \\end{tabular}\n \\end{center}\n\\end{table*}\n\n%%%%%%%%% Section %%%%%%%%%%\n\\section{The QuadIron library}\n%%%%%%%%%%%%%%%%%%%%%%%%\n\nThe QuadIron library is a C++ library, written in C++14, that provides a streaming API to use the different flavors of NTT-based erasure codes.\n\nThe library focuses primarily on high fragmentation, i.e. $n \\gg k$.\n\nIt includes general modular arithmetic routines, algorithms for manipulating rings of integers modulo \\textit{n}, finite fields (including binary, prime, and non-binary extension fields), polynomial operations, different flavors of discrete Fourier transforms, and forward error correction (FEC) algorithms.\n% * 2018-03-02T19:25:31.132Z:\n% \n% > (non-binary)\n% Parens why?\n% \n% ^.\n% * 2018-03-02T19:24:18.536Z:\n% \n% > n\n% italic?\n% \n% ^.\n\nThe library also includes an abstraction for writing systematic and non-systematic codes, although for applications requiring high fragmentation, systematic codes are not especially useful.\n\n\\subsection{Optimizations}\n\nThe QuadIron library's code relies heavily on templates to generate specifically optimized code at compile time. We use this template to allow a selection of numeric types matching the size of the processor's registers to optimize performance.\n\n\\subsection{Vectorization} \\label{sec:vectorization}\n\nWe focus here on vectorizing multiplicative FFTs which are based on prime (non-binary) field $\\gf{q}$ operations.\n\nWe observe that there are three operations that are costly in multiplicative FFT transformation: addition, subtraction and Hadamard multiplication. All of them are element-wise operations where the inputs are two vectors $\\vec{a}, \\vec{b}$ and the output is a vector $\\vec{c}$. All of these vectors are of the same length $n$.\n\n\\begin{equation}\n\\begin{split}\n\\vec{a} = (a_1, a_2, \\dots, a_n) \\\\\n\\vec{b} = (b_1, b_2, \\dots, b_n) \\\\\n\\vec{c} = (c_1, c_2, \\dots, c_n)\n\\end{split}\n\\end{equation}\n\n\\begin{itemize}\n\\item Addition\n$$\nc_i = (a_i + b_i) \\% q, \\text{ for } i = 1, \\dots, n\n$$\n\\item Subtraction\n$$\nc_i = (a_i - b_i) \\% q, \\text{ for } i = 1, \\dots, n\n$$\n\n\\item Hadamard multiplication\n$$\nc_i = (a_i * b_i) \\% q, \\text{ for } i = 1, \\dots, n\n$$\n\n\\end{itemize}\n\nFor simplicity, we express these operations by using a super-operator $\\oplus$ as below:\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{operator.pdf}\n\\caption{Element-wise operations $\\vec{c} = \\vec{a} \\oplus \\vec{b}$}\n\\label{fig:operator}\n\\end{figure}\n\nWe accelerate these operations in two ways: horizontal and vertical vectorizations. Note that the number of operations performed in parallel depends on bit-size of the elements and the bit-size of the vector registers. For example, if elements are 32-bit integers and registers are 128-bit wide, then there are 128/32 = 4 operations that can be performed in parallel.\n\nWithout loss of generality, we assume that 4 element-wise operations can be performed in parallel.\n\n\\subsubsection{Horizontal vectorization}\n\nEach vector is defined as an $n$-length array of integers. Four operations applied on four consecutive elements of $\\vec{a}$ and $\\vec{b}$ are performed in parallel. Therefore, operations on $\\vec{a}$ and $\\vec{b}$ corresponds to $\\frac{n}{4}$ vector operations.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{vec_horizontal.pdf}\n\\caption{Horizontal vectorization}\n\\label{fig:vec_horizontal}\n\\end{figure}\n\n\n\\subsubsection{Vertical vectorization}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.5\\columnwidth]{vec_buffers.pdf}\n\\caption{Vector of buffers}\n\\label{fig:vec_buffers}\n\\end{figure}\n\n\\begin{figure*}[!ht]\n\\includegraphics[width=\\textwidth]{vec_vertical.pdf}\n\\caption{Vertical vectorization}\n\\label{fig:vec_vertical}\n\\end{figure*}\n\nEach vector is defined as an $n$-length array of buffers. Each element of these vectors is a buffer composed of integers, see Figure \\ref{fig:vec_buffers}.\n\nAn operation $c_i = a_i \\oplus b_i$ is equivalent to $s$ operations performed on $s$ elements each of $a_i$ and $b_i$. \nWe accelerate operations performed on each pair of buffers $a_i$ and $b_i$, i.e.\\ in vertical way as shown in Figure \\ref{fig:vec_vertical}. Four operations applied on four consecutive elements of $a_i$ and $b_i$ are performed in parallel. Therefore, $s$ operations on $a_i$ and $b_i$ corresponds to $\\frac{s}{4}$ vector operations.\n\n\\subsection{Vectorized modular operations in $\\gf{q}$ where $q$ is a Fermat number}\n\nIn the previous section, we described the two methods to transform element-wise operations to packed vector operations. Each vector operation performs several element-wise operations in parallel. In this section, we describe how the parallel operations are performed on packed vectors.\n\nNote that we focus on vectorizing modular arithmetic of finite field $\\gf{q}$ where $q=2^{2^w}+1$, a Fermat number.\nThis case is very interesting since it allows performing multiplicative FFT of low computational complexity (see the 1st row of Table \\ref{tab:compare}) \\cite{fnt_lacan}\n\nLet $\\vec{q}$ denote a packed vector whose elements are $q$.\nLet $\\vec{h}$ denote a packed vector whose elements are $q-1$.\nLet $\\vec{u}, \\vec{v}$ denote two input packed vectors. \n\nLet $vector_(\\vec{u}, \\vec{v})$ denote an instruction that applies an operation on two input packed vectors $\\vec{u}, \\vec{v}$.\n\n\\begin{itemize}\n\\item \\texttt{vector\\_add}$(\\vec{u}, \\vec{v})$ does element-wise addition\n\\item \\texttt{vector\\_sub}$(\\vec{u}, \\vec{v})$ does element-wise subtraction\n\\item \\texttt{vector\\_mullo}$(\\vec{u}, \\vec{v})$ does element-wise multiplication\n\\item \\texttt{vector\\_or}$(\\vec{u}, \\vec{v})$ compute bitwise OR operation\n\\item \\texttt{vector\\_and}$(\\vec{u}, \\vec{v})$ compute bitwise AND operation\n\\item \\texttt{vector\\_andnot}$(\\vec{u}, \\vec{v})$ compute bitwise NOT of $\\vec{u}$ then AND with $\\vec{v}$\n\\item \\texttt{vector\\_cmpgt}$(\\vec{u}, \\vec{v})$ returns $((u_i > v_i) ? 1 : 0, i=1, \\dots, n)$\n\\item \\texttt{vector\\_cmpeq}$(\\vec{u}, \\vec{v})$ returns $((u_i == v_i) ? 1 : 0, i=1, \\dots, n)$\n\\item \\texttt{vector\\_rshift}$(\\vec{u})$ returns $(x[i] \\gg m, i=1, \\dots, n)$\n\\end{itemize}\n\n\\subsubsection{Addition}\n\nThe pseudocode of addition is as below. This algorithm works as $0 \\leq u_i + v_i < 2q$, hence\n\\begin{equation}\nu_i + v_i = \n\\begin{cases}\n\tu_i + v_i, & \\text{if}\\ u_i + v_i < q \\\\\n\tu_i + v_i - q, & \\text{otherwise}\n\\end{cases}\n\\end{equation}\n\n\\begin{algorithm}[]\n \\KwData{two vectors $\\vec{u}, \\vec{v}$}\n \\KwResult{Element-wise addition modulo $q$ of $\\vec{u}, \\vec{v}$}\n \\tcc{compute element-wise addition}\n $\\vec{r} := \\texttt{vector\\_add}(\\vec{u}, \\vec{v})$\\;\n \\tcc{subtract to $q$}\n $\\vec{m} := \\texttt{vector\\_sub}(\\vec{r}, \\vec{q})$\\;\n \\tcc{If $r_i \\ge q $, result is $m_i$. Otherwise, i.e. $r_i < q$, result is $r_i$}\n \\KwRet{$\\texttt{vector\\_min}(\\vec{r}, \\vec{m})$}\n \\caption{Addition}\n\\end{algorithm}\n\n\\subsubsection{Subtraction}\nThe pseudocode of subtraction is as below. This algorithm works as \n\\begin{equation}\nu_i - v_i = \n\\begin{cases}\n\tu_i - v_i, & \\text{if}\\ u_i \\geq v_i \\\\\n\tq + u_i - v_i, & \\text{otherwise}\n\\end{cases}\n\\end{equation}\n\n\\begin{algorithm}[]\n \\KwData{two vectors $\\vec{u}, \\vec{v}$}\n \\KwResult{Element-wise subtraction modulo $q$ of $\\vec{u}, \\vec{v}$}\n \\tcc{compute element-wise subtraction}\n $\\vec{r} := \\texttt{vector\\_sub}(\\vec{u}, \\vec{v})$\\;\n \\tcc{add by $q$}\n $\\vec{m} := \\texttt{vector\\_add}(\\vec{r}, \\vec{q})$\\;\n \\tcc{If $r_i \\ge 0 $, result is $r_i$. Otherwise, i.e. $r_i < 0$, result is $m_i$}\n \\KwRet{$\\texttt{vector\\_min}(\\vec{r}, \\vec{m})$}\n \\caption{Subtraction}\n\\end{algorithm}\n\n\\subsubsection{Hadamard multiplication}\n\nNote, we need to check and to perform extra operations for the case both elements $u_i, v_i$ are equal to $(q-1)$. However, this case occurs with a low probability. Hence, we implement two multiplication operations:\n\\begin{itemize}\n\t\\item Simple multiplication of $u, v$ whose pair elements $u_i, v_i$ are not both equal to $(q-1)$.\n\t\\item General multiplication of $u, v$ where the special case will be handled.\n\\end{itemize}\n\nObviously the simple multiplication is faster than the general multiplication. And it?s useful in performing FFT techniques in which a term of multiplications (twiddle factors) is generally known at compile-time. \n\nThe general multiplication consists in two steps:\n\\begin{itemize}\n\\item Step 1: compute element-wise multiplication.\n\t\\texttt{vector\\_mullo}$(\\vec{u}, \\vec{v})$ does element-wise multiplication. We should handle the overflow case where\n\t\\begin{itemize}\n\t\t\\item $u_i = v_i = (q-1)$\n\t\t\\item bit-size of elements smaller than $2^{w+1}$\n\t\\end{itemize}\n\tIndeed, $u_i * v_i = 2^{2^{w+1}}$ is out-of-range of defined integer for elements.\n\tNote that in the overflow case the product result is one as $(u_i * v_i) \\% q = (q-1)^2 \\% q = 1$.\n\\item Step 2: perform a modulo on the multiplication result.\n\\end{itemize}\nLet $r := (u[i]* v[i])$. Let $r_{\\text{hi}}$ and $r_{\\text{lo}}$ denote the high and low $2^w$-bit values of $r$.\nWe express $r$ in the following way thanks to $q = 2^{2^{w} + 1}$\n\n\\begin{equation}\n\\begin{split}\nr &= r_{\\text{hi}} * (q-1) + r_{\\text{lo}} \\\\\n &= r_{\\text{hi}} * q + (r_{\\text{lo}} - r_{\\text{hi}}) \\\\\n\\Rightarrow r \\% q &= (r_{\\text{lo}} - r_{\\text{hi}}) \\% q\n\\end{split}\n\\end{equation}\n\n\\begin{algorithm}[]\n \\KwData{two vectors $\\vec{u}, \\vec{v}$ whose elements $u_i, v_i$ are not both equal to $(q-1)$}\n \\KwResult{$\\texttt{simp\\_mul}(\\vec{u}, \\vec{v})$: simple multiplication modulo $q$ of $\\vec{u}, \\vec{v}$}\n \\tcc{compute element-wise addition}\n $\\vec{s} := \\texttt{vector\\_mullo}(\\vec{u}, \\vec{v})$\\;\n \\tcc{get low part of $s$}\n $\\vec{l} := \\texttt{vector\\_blend}(0, \\vec{s},0x55)$\\;\n \\tcc{get high part of $s$}\n $\\vec{h} := \\texttt{vector\\_blend}(0, \\texttt{vector\\_rshift}(\\vec{s}, 2), 0x55)$\\;\n \\tcc{result is returned by subtracting the low part by the high part}\n \\KwRet{$subtraction(\\vec{l}, \\vec{h})$}\n \\caption{Simple multiplication}\n\\end{algorithm}\n\n\\begin{algorithm}[]\n \\KwData{two vectors $\\vec{u}, \\vec{v}$}\n \\KwResult{General multiplication modulo $q$ of $\\vec{u}, \\vec{v}$}\n \\tcc{perform a simple multiplication}\n $\\vec{s} := \\texttt{simp\\_mul}(\\vec{u}, \\vec{v})$\\;\n \\tcc{create a mask checking whether both of elements $u_i, v_i$ are equal to $q-1$}\n $\\vec{m} := \\texttt{vector\\_and}(\\texttt{vector\\_cmpeq}(u, h), \\texttt{vector\\_cmpeq}(v, h))$\\;\n \\tcc{return simple multiplication if no pair of elements $u_i, v_i$ are equal to $q-1$}\n \\lIf{$\\texttt{is\\_zero}(\\vec{m})$} { \\Return $\\vec{s}$ }\n \\tcc{create a vector whose element is 1 according to the mask, otherwise zero}\n $\\vec{d} := \\texttt{vector\\_and}(1, \\vec{m})$\\;\n \\tcc{set specified element $s_i=1$ for $u_i= v_i=q-1$ and return it}\n \\KwRet{$\\texttt{vector\\_or}(\\vec{s}, \\vec{d})$}\n \\caption{General multiplication}\n\\end{algorithm}\n\n\\subsection{Testing}\n\nThe QuadIron library contains a test suite (to ensure correctness) and a benchmark suite (to measure performance).\nThese suites are integrated in our continuous integration system and run regularly, allowing early detection of performance regressions and bugs.\n\n\\subsection{Documentation}\n\nThe library comes with a Doxygen-generated documentation that associates the implementation with corresponding formulas and references.\n\n\\subsection{Contribution process}\n\nThe implementation is open source (licensed under the 3-Clause BSD License) and open to external contributions. The code can be found on Github, on the scality/quadiron repository.\n\n%\\href{https://github.com/scality/quadiron}{quadiron repository}.\n\n\\section{Performance}\n\nEncoding performance is expressed by the encoding speed metric that is the ratio of encoded data size to encoding time.\n\nIn our measurements, for systematic codes whose encoding performs a matrix multiplication, the encoding speed is given by:\n\n\\begin{equation}\\label{eq:enc_metric_def_sys}\n\\text{Encoding speed} = \\frac{ n - k }{ k } \\times \\frac{ \\text{file size} }{ \\text{encoding time} }\n\\end{equation}\n\nFor other codes, i.e. FFT-based codes, the encoding speed is given by:\n\n\\begin{equation}\\label{eq:enc_metric_def}\n\\text{Encoding speed} = \\frac{ n }{ k } \\times \\frac{ \\text{file size} }{ \\text{encoding time} }\n\\end{equation}\n\nWe use the two different formulas that expresses the computational cost of the corresponding encoder.\n\nDecoding performance is expressed by the decoding speed metric that is the ratio of information data size to decoding time, i.e.\n\n\\begin{equation}\\label{eq:dec_metric_def}\n\\text{Decoding speed} = \\frac{ \\text{file size} }{ \\text{decoding time} }\n\\end{equation}\n\nSimulated files are composed of $k$ fragments each of a fixed $50$-KiB size. Encoding speed is therefore linear to $\\frac{n}{\\text{encoding time}}$. With our optimization resulting in an encoding complexity of $\\bigo{\\fftl \\log{k}}$, the encoding speed is thus linear to $\\frac{1}{\\log{k}}$:\n\\begin{equation}\\label{eq:enc_metric_on_k}\n\\text{Encoding speed} \\sim \\frac{ 1 }{ \\log{k} }\n\\end{equation}\nWe measured performance of RS codes $\\ec{n}{k}$ over $\\gf{2^{2^4}+1}$ on a Intel Core i5 processor rated at 2.7 GHz, running Mac OS X v.10.12.5 with 64-bit compilation, L2 cache (per core) of 256 KB and L3 cache of 3 MB. All tests were on a single core.\n% * 2018-03-02T20:12:27.248Z:\n% \n% > operating at 2.7 GHz \n% I added \"operating at\" You need to say either \"operating at 2.7 GHZ\" (meaning we tested or otherwise verified the clock speed) or \"rated at 2.7 GHZ\" (meaning \"that's what it said on the box.\")\n% \n% ^.\n\n\\subsection{Vertical vs. horizontal vectorizations}\n\nWe have two methods of vectorizations as described in section \\ref{sec:vectorization}. Theoretically, the two optimization methods result in similar improvements. However, in practice the vertical vectorization enhances significantly the performance as it benefits from the cache memory.\n\nWe measure encoding performance of RS($k=8, n=128$) and RS($k=8, n=1024$) by using the two methods. With horizontal vectorization using 128-bit vector, encoding speed of the two codes are roughly 90 MiB/s and 99 MiB/s. These encoding speeds can be improved significantly by using vertical vectorization using also a 128-bit vector. The packet size $s$, i.e.\\ number of elements in each buffers, impact on the encoding speed is shown in Figure \\ref{fig:fnt_vertical_vec_pkt_size}. By choosing a \"good\" packet size, e.g. $s=1024$, these RS codes can be encoded at very high speeds, such as 1424 MiB/s and 1294 MiB/s.\n\nIn both curves of figure \\ref{fig:fnt_vertical_vec_pkt_size} there are three parts. The first part that increases quickly for $s$ from 16 to 256 represents the advantages of L2 cache on small size. The second part on larger packet sizes corresponds to the use of the L3 cache. Finally, when the packet size is too large to fit into the L3 cache, the performance decreases dramatically.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{fnt_vertical_vec_pkt_size.pdf}\n\\caption{Vertical vectorization: impact of packet size $s$ on encoding speed of RS codes over $\\gf{2^{2^4}+1}$: RS($k=8, n=128$) and RS($k=8, n=1024$).}\n\\label{fig:fnt_vertical_vec_pkt_size}\n\\end{figure}\n\n\n\\subsection{Impact of data length $k$}\nMindful of hyper-fragmentation, we measured the following parameters:\n\n\\begin{itemize}\n\\item $n \\in \\left\\{ 256, 1024, 2048 \\right\\}$\n\\item $k \\in \\left\\{ 8, 16, 32 \\right\\}$\n\\end{itemize}\n\nFigure \\ref{fig:fnt_k_n} shows the encoding speed for the three cases of code length $n$ over different number of data fragments $k$. Each curve represents the encoding speed for a given $n$. We observe that for each code length $n$ the speed is linear to $\\frac{1}{\\log{k}}$. The gap between the curves shows that the advantages of cache memory is more important for short code length than long ones. For short code length, encoding process can benefit from the different cache memories L2 and L3, while long code lengths can only exploit the L3 cache.\n% * 2018-03-02T20:19:37.872Z:\n% \n% Is there a unit? \n% \n% ^.\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{fnt_k_n.pdf}\n\\caption{Encoding speed of RS codes over $\\gf{2^{2^4}+1}$. Each curve represents a fixed $n$.}\n\\label{fig:fnt_k_n}\n\\end{figure}\n\nFigure \\ref{fig:dec_fnt_k_n} shows the decoding speed.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{dec_fnt_k_n.pdf}\n\\caption{Decoding speed of RS codes over $\\gf{2^{2^4}+1}$. Each curve represents a fixed $n$.}\n\\label{fig:dec_fnt_k_n}\n\\end{figure}\n\n\\section{Applications}\n\n\\subsection{Decentralized Storage over the Internet}\n\nAs drive density continues to be driven higher in keeping with Moore's Law, the price of storage continues to fall. This makes extra data copies cheaper. For example, generating and spreading hundreds of fragments from a file makes it possible to reconstruct the data while having only a fraction of the total data available. In this context, it is possible and interesting to develop a reliable decentralized cloud storage system that allows to exploit unused storage on the heterogeneous edge nodes (smartphones, laptops, servers, \\dots) of the network (see figure \\ref{fig:network}).\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{network.png}\n\\caption{A decentralized storage system based on heterogeneous nodes}\n\\label{fig:network}\n\\end{figure}\n\nWe have to compensate the unreliability of the underlying devices (which can go off-line anytime for any duration of time) if we want to have a reliable and resilient storage. Erasure coding is an excellent solution to increase durability while keeping the storage overhead low. Existing practical systems, such as Storj, are going toward encoding schemes like $\\ec{30}{10}$, $\\ec{60}{20}$ or $\\ec{120}{40}$.\n\nThe systematic property is interesting but not absolutely necessary in a decentralized storage application: Indeed a decentralized storage software may choose to place the plain text data fragments on the peers having the best quality of service. But in an environment like the Internet where it is difficult to evaluate the quality of service of peers, or when the quality of service is very variable, the probability of reaching a peer that contains a plain text data fragment is getting lower as k gets bigger (see figure \\ref{fig:systematic}). Therefore the systematic property becomes useless. Moreover non-systematic code are rate insensitive.\n\n\\begin{figure}[!h]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{systematic.png}\n\\caption{Evolution of the reachability of the whole set of data fragments in function of k}\n\\label{fig:systematic}\n\\end{figure}\n\nThe adaptive rate property may also be useful in a decentralized storage application because it is not practically possible to delete and rewrite the erasure code fragments which are already in place. However if the initial durability chosen is already very high, the need for re-dimensioning the code is not critical.\n\nThe repair bandwidth is indeed a problem for RS codes but it can be mitigated in multiple ways in a decentralized storage scenario:\n\\begin{itemize}\n\\item Generally we cut a file in sub-blocks of fixed size called stripes. Since the stripes are spread in a very high number of peers, the loss of a peer will affect only one stripe with a very high probability.\n\\item It is possible to detect a missing fragment at the time of the read and repair it immediately after (read-repair).\n\\item Because of the chosen very high durability, a pro-active repair mechanism (called scrubbing) is not absolutely critical and can be scheduled rarely. \n\\end{itemize}\n\nFinally one can argue that both the systematic property and the optimization of the repair bandwidth at the level of a stripe is still advantageous in order to support get\\_range() operations: the ability to read a portion of a stripe without decoding or by minimizing the decoding. The get\\_range() properties might indeed be of interest in the case of a local distributed file system or a local big data cluster, but not in a decentralized storage system which targets primarily the cloud storage use case (for archival purpose) where people generally wants the files stored and retrieved in their totality.\n\nWe do compare our erasure codes with other libraries (ISA-L, Leopard, Wirehair) for k up to 1024. The ISA-L library \\cite{isal} is a performance-oriented open-source library developed by Intel that implements a systematic RS code defined over $\\gf{2^8}$ (which limits the code length to at most 255). Leopard \\cite{leopard} is a performance-oriented RS codes using additive FFT techniques. Wirehair \\cite{wirehair} is a performance-oriented hybrid LDPC library that implements a systematic code defined over $\\gf{2^8}$ that also leverage SIMD instructions.\n\nFigures \\ref{fig:comp_enc_rate_1_4}, \\ref{fig:comp_enc_rate_1_3}, \\ref{fig:comp_enc_rate_1_2}, \\ref{fig:comp_enc_rate_2_3} show encoding speeds of these codes for different coding rates 1/4, 1/3, 1/2 and 2/3 respectively. We observe that for small $k$, ISA-L codes are very good. But for $k > 24$, our codes outperform ISA-L codes thanks to FFT operations. \n%Figure \\ref{fig:isal_vs_fnt_vs_leo_enc} shows encoding speeds of these codes for different number of data fragments. We observe that for small $k$, ISA-L codes are very good. But for $k > 24$, our codes outperform ISA-L codes thanks to FFT operations. \n\nLDPC encoding speed worsen as $k$ grows. Note the graphs shows the pure decoding speed and does not show the additional time needed to retrieve the additional fragments due to the overhead (see \\cite{plank_ldpc} for a detailed study on this topic).\n\nWirehair is also less good for higher code rates.\n\nFigures \\ref{fig:comp_dec_rate_1_4}, \\ref{fig:comp_dec_rate_1_3}, \\ref{fig:comp_dec_rate_1_2}, \\ref{fig:comp_dec_rate_2_3} show decoding speeds of these codes for different coding rates 1/4, 1/3, 1/2 and 2/3 respectively. We observe that average decoding time of FNT codes is about only 3.5 times longer than encoding time, compared to 8 times mentioned in \\cite{fnt_lacan}. However our FNT decode speed is still slow compared to others. We plan to implement the approach defined in \\cite{fnt_ida}.\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Enc_Rate1-4.pdf}\n\\caption{Encoding speed of RS codes of rate 1/4: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_enc_rate_1_4}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Enc_Rate1-3.pdf}\n\\caption{Encoding speed of RS codes of rate 1/3: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_enc_rate_1_3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Enc_Rate1-2.pdf}\n\\caption{Encoding speed of RS codes of rate 1/2: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_enc_rate_1_2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Enc_Rate2-3.pdf}\n\\caption{Encoding speed of RS codes of rate 2/3: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_enc_rate_2_3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Dec_Rate1-4.pdf}\n\\caption{Decoding speed of RS codes of rate 1/4: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_dec_rate_1_4}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Dec_Rate1-3.pdf}\n\\caption{Decoding speed of RS codes of rate 1/3: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_dec_rate_1_3}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Dec_Rate1-2.pdf}\n\\caption{Decoding speed of RS codes of rate 1/2: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_dec_rate_1_2}\n\\end{figure}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=0.7\\columnwidth]{Dec_Rate2-3.pdf}\n\\caption{Decoding speed of RS codes of rate 2/3: comparison our FNT codes vs. ISA-L vs. Leopard vs. WireHair}\n\\label{fig:comp_dec_rate_2_3}\n\\end{figure}\n\nAs wee see in practical decentralized storage applications like Storj the typical code rate is 1/3 or 1/4 which means relatively high durability and relatively low number of fragments to retrieve which is good for latency. LDPC such as Wirehair show to be very good, although non MDS, we show that FNT has a relatively good performance but is still inferior to LDPC, we think we can improve FNT perf in improving the FFT method (we currently implement a bit-reverse method that requires copying data and we could replace by other more efficient methods). We also think that accelerating additive FFT such as what has been done in Leopard \\cite{leopard} is very promising.]\n\nAnother interesting property for this application is the confidentiality of the erasure code, even though the client decides to encrypt or not the files beforehand, using a non-systematic FNT augments the security.\n\n\\subsection{Real-Time Video Encoding High-Quality Streaming}\n\nThere are many types of video streams. For high quality ingress of a broadcast system, the master stream is typically broken up into pieces and encoded to lower quality with a few seconds of delay.\n\nThose real-time video high quality streams with 30\\% interleaving (code rate 2/3) have many pieces, and are generally encoded using $\\bigo{N^2}$ streaming convolutional codes e.g. like CauchyCaterpillar \\cite{cauchycaterpillar} which operates on $\\gf{2^8}$ and is limited to 2MB/s. To go faster the technique generally used is interleaving block codes (which is how video FEC is usually done \\cite{al_fec}). The overhead from using something fast like a XOR based LDPC on $\\gf{2^2}$ is highly inefficient. Our MDS NTT approach is much better because we have no overhead and we can keep up with the speed.\n\nIn this type of application there is a need for a systematic code. As wee see in \\cite{leopard}, \\cite{fft_add_chung} and \\cite{fnt_lacan} we can implement systematic NTT based codes with lesser but similar performance.\n\n\\section{Conclusion}\n\nNTT codes offer superior encoding performance compared to matrix-based RS erasure codes for applications requiring $n \\gtrapprox 24$ symbols, and are simpler than LDPC codes, while supporting all the desirable properties: Fast, MDS, systematic or non-systematic, confidential (for systematic codes). As we have seen the optimization of repair bandwidth as offered by LRC codes is not critical for a decentralized cloud storage application. The most important property for us remains the MDS property as a rock solid contract: being sure than if k fragments are available then the data is recoverable. Also, as we seen, FNT aka multiplicative FFT codes offer not necessarily always faster but more predictive performance than additive FFT codes for various $n$. \n\nThose codes may have other potential applications such as high-quality video streaming but a lot of experiments and analysis have yet to be conducted.\n\n\\section{Future Work}\nThe next steps for the QuadIron library are in order of priority:\n\\begin{itemize}\n\\item Optimize the multiplicative FFT decoding, which is for now relatively slow (\\cite{fnt_ida} and \\cite{gao_decode} has shown a way to optimize). It can be possible for special values of k and m, e.g. $k \\% m = 0$ or $m \\% k = 0$.\n\\item Optimize additive FFTs (for now we don't have any HW acceleration).\n\\item Implement additive FFT based systematic codes.\n\\item Design and implement NTT based adaptive codes for multiplicative FFTs.\n\\item Design and implement NTT based adaptive codes for additive FFTs.\n\\item Implement Frobenius multiplicative FFT \\cite{frobenius_fft}.\n\\item Implement Frobenius additive FFT \\cite{frob_additive}.\n\\end{itemize}\n\n\\bibliographystyle{unsrt}\n\\bibliography{bib}\n\n\n\\section*{Authors}\n\n\\begin{description}\n\n\\item[Vianney Rancurel] is a computer professional with more than 20 years of experience designing and developing computer products. He currently leads the Scality research team which is responsible for feasibility studies, prototypes and patents/publications. Previously he worked in the messaging and telco industries on both software and hardware technologies at Bizanga, Borea Technologies, and Freescale. He was also a professor at the EPITA engineering school and was responsible for its System Research Lab. His specialties are system software programming, distributed systems, embedded systems and security. Vianney has a degree of Systems \\& Networks Engineering from EPITA.\n\n% if you will not have a photo at all:\n\\item[Lam Pham-Sy] is a research engineer working on information theory and computer science. His main research focuses on different families of forward erasure correcting codes such as Reed-Solomon codes, Low-Density Parity-Check codes, Locally Repairable codes etc. Their application covers from digital communication to data storage. He did his PhD program in a collaboration between CEA-Leti and Eutelsat S.A. on the subject of forward erasure codes for satellite communications. Afterwards, he continued his researches at ETIS laboratory and at Orange Labs. Currently he works at Scality S.A. as a research engineer whose research topics include application of erasure codes in distributed storage systems, finite field arithmetics.\n\n\\item[Sylvain Laperche] is a code craftsman. With a background in biotech engineering,he learnt how to hack bacteria before learning how to hack a computer. That changed when it studied bioinformatics, and since then he honed and applied its skill on a wide set of problematics: genome sequencing, complex embedded systems, climate modelling at European scale, mass-scale geolocation for telco industries. Its steps led him to work on distributed storage systems and he currently works as an R\\&D engineer at Scality.\nSylvain Laperche has an Engineer's degree in Hardware, Circuit Design and Embedded Systems from ISIMA.\n\n\\end{description}\n\n\\end{document}\n\n\n", "meta": {"hexsha": "99c730dff77d1c2e55a9d42074f8cdd0370eea35", "size": 51480, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/quadiron.tex", "max_stars_repo_name": "scality/quadiron", "max_stars_repo_head_hexsha": "c062e908ba39a3989acad3234044ec11a5891cbf", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 21, "max_stars_repo_stars_event_min_datetime": "2018-09-20T05:42:44.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-22T14:53:04.000Z", "max_issues_repo_path": "paper/quadiron.tex", "max_issues_repo_name": "scality/quadiron", "max_issues_repo_head_hexsha": "c062e908ba39a3989acad3234044ec11a5891cbf", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 67, "max_issues_repo_issues_event_min_datetime": "2018-07-26T14:44:11.000Z", "max_issues_repo_issues_event_max_datetime": "2020-04-16T16:58:23.000Z", "max_forks_repo_path": "paper/quadiron.tex", "max_forks_repo_name": "scality/quadiron", "max_forks_repo_head_hexsha": "c062e908ba39a3989acad3234044ec11a5891cbf", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2018-10-07T00:41:23.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-05T19:25:50.000Z", "avg_line_length": 59.5144508671, "max_line_length": 1520, "alphanum_fraction": 0.7387917638, "num_tokens": 14883, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6513548646660543, "lm_q1q2_score": 0.3434701827220722}} {"text": "\\documentclass[12pt]{article}\n\\usepackage{amsmath}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{xcolor}\n\\usepackage{subcaption}\n\\usepackage{siunitx}\n\n\\addtolength{\\oddsidemargin}{-.875in}\n\\addtolength{\\evensidemargin}{-.875in}\n\\addtolength{\\textwidth}{1.75in}\n\n\\addtolength{\\topmargin}{-.875in}\n\\addtolength{\\textheight}{1.75in}\n\n\\title{Model Predictive Control}\n\\date{2020}\n\\author{sentry5588, MIT License}\n\n\\begin{document}\n\\pagecolor{lightgray}\n\\maketitle\n\\section{Introduction}\nThis note documents the Model Predictive Control (MPC) method for the two\nwheel balancing robot. Many people has built similar self-balancing robots.\nAlmost all of them uses PID control as the control strategy. For position\nestimation, some uses Kalman filters, and others uses complementary filters.\n\nThe purpose of using MPC in this robot is not trying to invent a new MPC\ntechnique, which usually the case for research papers.\nBut rather I intend to 1) practice MPC and 2) test how well MPC behaves\ncompares to other control schemes.\n\n\\section{Robot Coordinates}\nAs shown in Figure~\\ref{fig_coordinates}, $\\theta_k$ and $\\omega_k$ denotes \nthe angular position at time step $k$, respectively.\nCounter-clockwise rotation is positive. Angular acceleration is denoted\nby $\\dot{\\omega}_k$.\n$u_k$ is the horizontal force\nwith positive to the right. Robot specifications can be found in \nTable~\\ref{tab_robot_specification}\n\n\n\\begin{figure}\n\n\\begin{subfigure}{0.3\\textwidth}\n\\includegraphics[width=\\linewidth]{./figures/coordinates.png}\n\\caption{Robot Coordinates} \\label{fig_coordinates}\n\\end{subfigure}\n\\hspace*{\\fill} % separation between the subfigures\n\\begin{subfigure}{0.3\\textwidth}\n\\includegraphics[width=\\linewidth]{./figures/one_d_rotation.png}\n\\caption{1D Rotation} \\label{fig_one_d_rotation}\n\\end{subfigure}\n\\hspace*{\\fill} % separation between the subfigures\n\\begin{subfigure}{0.3\\textwidth}\n\\includegraphics[width=\\linewidth]{./figures/free_body_diagram.png}\n\\caption{1D Free Body Diagram} \\label{fig_1d_free_body_diagram}\n\\end{subfigure}\n\\caption{Two Wheel Balancing Robot} \\label{fig_robots}\n\\end{figure}\n\n\n\\begin{table}\n \\centering\n \\begin{tabular}{l|c|c}\n \\hline\n\tSpecification & Notation & Value \\\\ \\hline\n Center of Gravity (CoG) & $h$ & ?? 0.2 m \\\\ \n\tMass & $m$ & ?? 1.1 kg \\\\ \n\tMoment of Inertia & I & ?? 0.8 \\si{\\kilogram\\cdot\\meter^2} \\\\ \\hline\n \\end{tabular}\n \\caption{Robot specification} \n \\label{tab_robot_specification}\n\\end{table}\n\n\n\\section{Problem Formulation}\nThe model is a 1-input-2-output model.\n\\begin{align}\n\\label{equ_orig_nonlinear_dynamics}\nx_{k+1} & = f(x_k, u_k) \\\\\ny_k & = C(x_k)x_k\n\\end{align}\n\n$x_k=[\\theta_k\\;\\;\\omega_k\\;\\;\\dot{\\omega}_k]^T$ is the system state. \nLinearize Equ~\\ref{equ_orig_nonlinear_dynamics}, we have\n\\begin{align}\nx_{k+1} & = A(x_k)x_k + B(x_k)u_k \\\\\ny_k & = C(x_k)x_k\n\\end{align}\n\nwhere $A(x_k)$ and $B(x_k)$ are Jacobians and given by\n\\begin{align}\nA(x_k) = \\frac{\\partial f}{\\partial x_k},\nB(x_k) = \\frac{\\partial f}{\\partial u_k}\n\\end{align}\n\nFigure~\\ref{fig_1d_free_body_diagram} is the 1-D free body diagram.\nIn the next subsection, I will develop the nonlinear continuous-time\ndynamics.\n\n\\subsection{Nonlinear Continuous-Time Model}\n\\subsubsection{Derived from Newton's 2nd law}\nSumming the forces in figure~\\ref{fig_1d_free_body_diagram} in the horizontal\ndirection we have\n\\begin{align}\nF_x=m\\ddot{p}_x\n\\end{align}\nwhere $p_x$ is the horizontal velocity of the CoG.\nSumming the forces in figure~\\ref{fig_1d_free_body_diagram} in the vertical\ndirection we have\n\\begin{align}\nmg-F_z = m\\ddot{p}_z\n\\label{equ_vertical_newton}\n\\end{align}\nwhere $p_x$ is the horizontal velocity of the CoG.\nSumming the torques in figure~\\ref{fig_1d_free_body_diagram} around the\ncenter of gravity we have\n\\begin{align}\nF_x h\\cos(\\theta) + F_z h\\sin(\\theta) + mg\\cdot 0= \\ddot{\\theta}I\n\\label{equ_rotation_newton}\n\\end{align}\n\\textcolor{blue}{\nThe contact point between the robot and the ground has $0$ vertical velocity.\nIt's vertical velocity is a combination of the rod rotation around \nCoG, $\\dot{\\theta}h\\sin(\\theta)$,\nand vertical velocity of CoG, $\\dot{p}_z$. Therefore\n\\begin{align}\n-\\dot{\\theta}h\\sin(\\theta) + \\dot{p}_z = 0\n\\label{equ_velocity_equation}\n\\end{align}\nTake time derivative of~(\\ref{equ_velocity_equation})\n\\begin{align}\n-\\ddot{\\theta}h\\sin(\\theta) - \\dot{\\theta}^2h\\cos(\\theta) + \\ddot{p}_z = 0\\\\\n\\ddot{p}_z = \\ddot{\\theta}h\\sin(\\theta) + \\dot{\\theta}^2h\\cos(\\theta)\n\\label{equ_ddot_p_z}\n\\end{align}}\nSubstitute $\\ddot{p}_z$ in~(\\ref{equ_vertical_newton}) \nwith~(\\ref{equ_ddot_p_z})\n\\begin{align}\nmg-F_z = m(\\ddot{\\theta}h\\sin(\\theta) + \\dot{\\theta}^2h\\cos(\\theta)) \\\\\nmg-F_z = m\\ddot{\\theta}h\\sin(\\theta) + m\\dot{\\theta}^2h\\cos(\\theta) \\\\\nF_z = mg - m\\ddot{\\theta}h\\sin(\\theta) - m\\dot{\\theta}^2h\\cos(\\theta)\n\\end{align}\nSubstitute $F_z$ with above equation in~(\\ref{equ_rotation_newton})\n\\begin{align}\nF_xh\\cos(\\theta) + (mg - m\\ddot{\\theta}h\\sin(\\theta) \n- m\\dot{\\theta}^2h\\cos(\\theta))h\\sin(\\theta) + mg\\cdot 0= \\ddot{\\theta}I\\\\\nF_xh\\cos(\\theta) + mgh\\sin(\\theta) - m\\ddot{\\theta}h^2\\sin^2(\\theta) \n- m\\dot{\\theta}^2h^2\\cos(\\theta)\\sin(\\theta) = \\ddot{\\theta}I \\\\\n(I+mh^2\\sin^2(\\theta))\\ddot{\\theta}+mh^2\\cos(\\theta)\\sin(\\theta)\\dot{\\theta}^2\n=F_xh\\cos(\\theta) + mgh\\sin(\\theta)\n\\end{align}\n\nSo the dynamics can be written as \n\\begin{align}\nF_x &= m\\ddot{p}_x \\\\\n(I+mh^2\\sin^2(\\theta))\\ddot{\\theta}+mh^2\\cos(\\theta)\\sin(\\theta)\\dot{\\theta}^2\n&= F_xh\\cos(\\theta) + mgh\\sin(\\theta)\n\\label{equ_nonlinear_CT_dynamics}\n\\end{align}\n\n\\textbf{Check dynamics at special points \nfor~(\\ref{equ_nonlinear_CT_dynamics})}\n\nWhen $\\theta=0$, i.e. the vertical up position,\n(\\ref{equ_nonlinear_CT_dynamics}) becomes\n\\begin{align}\nI\\ddot{\\theta} = F_xh\n\\end{align}\nWhen $\\theta=\\pi/2$, i.e. the horizontal position pointing to the left\n(assume single contact point with the ground)\n(\\ref{equ_nonlinear_CT_dynamics}) becomes\n\\begin{align}\n(I+mh^2)\\ddot{\\theta} = mgh\n\\end{align}\nWhen $\\theta=-\\pi/2$, i.e. the horizontal position pointing to the right\n(assume single contact point with the ground)\n(\\ref{equ_nonlinear_CT_dynamics}) becomes\n\\begin{align}\n(I+mh^2)\\ddot{\\theta} = -mgh\n\\end{align}\nWhen $\\theta=\\pi/4$, i.e. the horizontal position pointing to the left\n(assume single contact point with the ground)\n(\\ref{equ_nonlinear_CT_dynamics}) becomes\n\\begin{align}\n(I+\\frac{1}{2}mh^2)\\ddot{\\theta}+\\frac{1}{2}mh^2\\dot{\\theta}^2\n&= \\frac{\\sqrt{2}}{2}F_xh + \\frac{\\sqrt{2}}{2}mgh\n\\end{align}\n\n\\subsubsection{Derived from Lagrangian mechanics}\nThe derivation follows~\\cite{peacock_2007_mit_lagrange}.\nThe Lagrangian is $L=KE-PE$ where $KE$ and $PE$ are the\nkinematic energy and potential energy, respectively.\n$W$ is the virtual work.\nThe equation of motion can be determined by applying\nLagrange mechanics in two generalized coordinate $p_x$ and $\\theta$\n\\begin{align}\n\\label{equ_lagrangian_original}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\bigg(\n\\frac{\\partial L}{\\partial \\dot{p}_x}\\bigg)\n-\\frac{\\partial L}{\\partial p_x}=\n\\frac{\\partial W}{\\partial p_x}, \\quad\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\bigg(\n\\frac{\\partial L}{\\partial \\dot{\\theta}}\\bigg)\n-\\frac{\\partial L}{\\partial \\theta}=\n\\frac{\\partial W}{\\partial \\theta}\n\\end{align}\nThe contact point displacement relative to the center of gravity is\n$h\\theta \\cos(\\theta)$. Therefore the absolute velocity of\nthe contact point is $p_x + h\\theta \\cos(\\theta)$.\nThe kinematic energy, potential energy and virtual work are\n\\begin{align}\nKE = \\frac{1}{2}m\\dot{p}_x^2\n+\\frac{1}{2}I\\dot{\\theta}^2,\\quad\nPE = mgh\\cos(\\theta), \\quad\nW = F_x (p_x+ h\\theta \\cos(\\theta)) + F_x h \\theta \\cos(\\theta)\n= F_x p_x + 2F_x h \\theta \\cos(\\theta)\n\\label{equ_lagrangian_energy_work}\n\\end{align}\nSubstitute~(\\ref{equ_lagrangian_original})\nwith~(\\ref{equ_lagrangian_energy_work}), in $p_x$ direction we have\n\\begin{align}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\\bigg(\n\\frac{\\partial}{\\partial \\dot{p}_x}\\bigg(\n\\frac{1}{2}m\\dot{p}_x^2 + \\frac{1}{2}I\\dot{\\theta}^2\\bigg)\\bigg) \\quad &\\\\\n-\\frac{\\partial}{\\partial p_x}\\bigg(\n\\frac{1}{2}m\\dot{p}_x^2+\\frac{1}{2}I\\dot{\\theta}^2\n-mgh\\cos(\\theta)\\bigg)=&\n\\frac{\\partial}{\\partial p_x}F_x p_x\n+\\frac{\\partial}{\\partial p_x} 2F_x h \\theta \\cos(\\theta) \\\\\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\n(m\\dot{p}_x + 0)\n-(0+0-0)=&\nF_x + 0\\\\\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\nm\\dot{p}_x = & F_x \\\\\nm\\ddot{p}_x =& F_x\n\\end{align}\nSubstitute~(\\ref{equ_lagrangian_original})\nwith~(\\ref{equ_lagrangian_energy_work}), in $\\theta$ direction we have\n\\begin{align}\n&\\frac{\\mathrm{d}}{\\mathrm{d}t}\\bigg(\n\\frac{\\partial}{\\partial\\dot{\\theta}}\\bigg(\n\\frac{1}{2}m\\dot{p}_x^2+\\frac{1}{2}I\\dot{\\theta}^2\n-mgh\\cos(\\theta)\\bigg)\\bigg)\n-\\frac{\\partial}{\\partial \\theta}\\bigg(\n\\frac{1}{2}m\\dot{p}_x^2+\\frac{1}{2}I\\dot{\\theta}^2\n-mgh\\cos(\\theta)\\bigg) \\\\\n=&\\frac{\\partial}{\\partial \\theta}F_x p_x\n+\\frac{\\partial}{\\partial \\theta} 2F_x h \\theta \\cos(\\theta) \n\\end{align}\n\\begin{align}\n\\frac{\\mathrm{d}}{\\mathrm{d}t}\n(0+I\\dot{\\theta}-0)\n-(0+0+mgh\\sin(\\theta))=&-2 F_x h\\theta \\sin(\\theta)\n+2F_x h\\cos(\\theta)\\\\\n(0+I\\ddot{\\theta}-0)\n-(0+0+mgh\\sin(\\theta))=& -2 F_x h\\theta \\sin(\\theta)\n+2F_x h\\cos(\\theta) \\\\\nI\\ddot{\\theta}-mgh\\sin(\\theta)=&-2 F_x h\\theta \\sin(\\theta)\n+2F_x h\\cos(\\theta) \\\\\nI\\ddot{\\theta}\n=& 2 F_x h\\cos(\\theta)+mgh\\sin(\\theta)\n-2 F_x h\\theta \\sin(\\theta)\n\\end{align}\n\n\n\\subsection{Model Linearization and discretization}\nI follow~\\cite{zhakatayev_2017_successive_linearize_MPC}\nto linearize and discretize the nonlinear continuous-time dynamics\nin~\\cite{???}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\bibliographystyle{apalike}\n\\bibliography{MPC}{}\n\n\n\\end{document}\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n%\n", "meta": {"hexsha": "88e187a2c48201f6917b3ab37ad7a56fc9ad3311", "size": 9602, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "control/tex/MPC.tex", "max_stars_repo_name": "sentry5588/two_wheeler", "max_stars_repo_head_hexsha": "ba62de060af3bb47052e21157bb440ba55d3a7a5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2019-04-18T23:29:09.000Z", "max_stars_repo_stars_event_max_datetime": "2019-10-30T23:05:04.000Z", "max_issues_repo_path": "control/tex/MPC.tex", "max_issues_repo_name": "sentry5588/two_wheeler", "max_issues_repo_head_hexsha": "ba62de060af3bb47052e21157bb440ba55d3a7a5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2019-04-04T14:43:45.000Z", "max_issues_repo_issues_event_max_datetime": "2019-06-11T19:27:44.000Z", "max_forks_repo_path": "control/tex/MPC.tex", "max_forks_repo_name": "sentry5588/two_wheeler", "max_forks_repo_head_hexsha": "ba62de060af3bb47052e21157bb440ba55d3a7a5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 32.32996633, "max_line_length": 78, "alphanum_fraction": 0.7114142887, "num_tokens": 3381, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.34347018272207214}} {"text": "\\section{Wavelet transforms}\n\n\\begin{frame} \\frametitle{Wavelet transforms for EXAFS analysis}\n\n\\begin{cenpage}{105mm}\n\n The Fourier transform is fundamental to understanding EXAFS.\n\n Wavelet transforms extend Fourier transforms, mixing $k$ and $R$.\n\n They have been used by a handful of groups.\n\n\n\n\\end{cenpage}\n\n\\begin{columns}\n \\begin{column}[T]{70mm}\n\n {\\onslide+<2-> \\includegraphics[width=70mm]{figs/wavelets/wavelet_composite_mag} }\n\n \\end{column}\n\n\\begin{column}[T]{52mm}\n\n {\\onslide+<2-> \\includegraphics[width=52mm]{figs/wavelets/wavelet_surface} }\n\n \\vmm\n\n Wavelet transform for FeO data.\n\n \\vmm \\vmm\n\n{\\hspace{-15mm}{\n {\\onslide+<2->\n \\begin{minipage}{60mm}\n {\\tiny{\n This is using the ``Continuous Cauchy'' wavelet transform, as\n described by Munoz, Argoul, and Farges, {\\emph{Am Mineralogist}},\n 2003\n\n Others (Funke {\\emph{et al}}, Penfold, {\\emph{et al}}) have used Morlet\n wavelets.\n }}\n \\end{minipage}\n}\n}}\n\n\n \\end{column}\n\n\\end{columns}\n\\end{frame}\n", "meta": {"hexsha": "ceffb80be1edf6506d9af23e757faa821e12d219", "size": 1047, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "slides/wavelets.tex", "max_stars_repo_name": "newville/xafsfun", "max_stars_repo_head_hexsha": "525b0b8fb6ec61396dc7dd2950a3e2a3ab6c17d1", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "slides/wavelets.tex", "max_issues_repo_name": "newville/xafsfun", "max_issues_repo_head_hexsha": "525b0b8fb6ec61396dc7dd2950a3e2a3ab6c17d1", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "slides/wavelets.tex", "max_forks_repo_name": "newville/xafsfun", "max_forks_repo_head_hexsha": "525b0b8fb6ec61396dc7dd2950a3e2a3ab6c17d1", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 19.3888888889, "max_line_length": 88, "alphanum_fraction": 0.6561604585, "num_tokens": 330, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6513548646660542, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34347018272207214}} {"text": "\\section{Fl---The Meta Language of Voss}\n\nIn this section\\footnote{This section is to a large extent modeled\nafter Chapter 1 in the HOL System DESCRIPTION from Cambridge University.\nIn particular, many of the early examples are taken from this source.}\nwe provide an introduction to the functional language FL.\n\nSimilar to many theorem provers (e.g., the HOL%\n\\index{HOL}%\n{}\nsystem~\\cite{HOLsystem,HOLbook}) the\nVoss command language for the verification system is a general\npurpose programming language%\n\\index{programming language}%\n{}.\nIn fact, the FL language shows a strong degree of influence from\nthe version of ML%\n\\index{ML}%\n{} used in the HOL-88%\n\\index{HOL-88}%\n{} system.\nHowever, there are several differences: many syntactic but some\nmore fundamental.\nIn particular, the functional language used in Voss has lazy evaluation\nsemantics.\nIn other words, no expression is evaluated until it is\nabsolutely needed.\nSimilarly, no expression is evaluated more than once.\nAnother difference is that Boolean functions are first-class\nobjects and can be created, evaluated, compared and printed out.\nFor efficiency reasons these Boolean functions are represented as\nordered binary decision diagrams%\n\\index{ordered binary decision diagrams}%\n{}. \n\nFl is an interactive language.\nAt top-level one can:\n\\begin{enumerate}\n\\item define functions (possibly of arity 0),\n\\item define new concrete types,\n\\item define aliases for types,\n\\item modify the parser, and\n\\item evaluate expressions.\n\\end{enumerate}\nIn this section we introduce the language by several examples.\n\n\\subsection{Invoking FL}\n\nIf the Voss system is installed on your system\nand you have the suitable search path set up, it suffices\nto type \\verb@fl@ to get a stand-alone version of FL. In this manual, we\nhave used Voss 1.8b throughout. In other words, typing \\verb@fl@ yielded:\n\n\\begin{hol}\n% fl\n /\\verb@\\@\n / \\verb@\\@/\\verb@\\@\n /\\verb@\\@ / \\verb@\\@\n/ Voss 1.8b\\verb@\\@\nVOSS-LIBRARY-DIRECTORY = /isd/local/generic/lib/vosslib\n\n:\n\\end{hol}\nNote that the VOSS-LIBRARY-DIRECTORY%\n\\index{VOSS-LIBRARY-DIRECTORY}%\n{} is installation dependent.\nWe return to this below and in Section~\\ref{vossrc}.\n\nThe fl program can take a number of arguments.\n\\index{command-line arguments}\nIn particular, \n\\begin{description}\n\\item[-d]\nTurn debugging mode%\n\\index{debugging mode}%\n{} on.\nIn this mode a function call trace%\n\\index{call trace}%\n{} will be generated\nwhen a failure that is not caught by an exception handler occurs.\nThis feature is not as useful as in many other languages, but\nis retained for backwards compatibility.\n\\item[-D]\nTurn debugging mode off.\nUnless the DEBUG-ON flag is set in the .vossrc file, the default\nis for debugging mode to be turned off.\n\\item[-f n]\nStart FL by first reading in the content of the file named n.\n\\item[-h]\nA flag that is needed when fl is used as a subroutine in the\nHOL%\n\\index{HOL}%\n{} theorem prover in the HOL-Voss%\n\\index{HOL-Voss}%\n{}\nsystem~\\cite{JoyceSegerDAC,SegerJoyceTwoLevel}.\nThis flag should not be used when FL is used stand-alone.\n\\item[-I dir]\nSet the default search directory%\n\\index{search directory}%\n{} to dir.\n\\item[-s i]\nSet the initial OBDD table size to be $2^i$, where $i$ can range\nfrom 12 to 24.\nNote that each OBDD node in this table requires 16 bytes, and thus\nthe memory requirement increases very rapidly with increasing table size.\nNote that FL automatically grows the OBDD table if it needs to, and\nthe only reason using this flag is to reduce the time it takes\nto grow the tables.\nThe flag is kept for historical reasons, but with the modern\ndynamical re-ordering version of the OBDD table, this user setable\nconstant is rarely needed.\n\\index{OBDD!table size}\n\\index{dynamic variable re-ordering}\n\\item[-o]\nBefore reaching the fl command prompt, print out all defaults\nread from the .vossrc%\n\\index{.vossrc}%\n{} file.\nThis is useful both for finding out the names of most (but unfortunately\nnot all) options%\n\\index{options}%\n{} as well as checking that a particular option\nhas the correct value.\n\\item[-r i]\nInitialize the random number%\n\\index{random number}%\n{} generator with the seed i.\nThis allows the {\\tt rvariable}\n\\index{rvariable}%\n{} command to create new sets\nof random variable values.\nSee the rvariable command description in Section~\\ref{rvariable} for\nmore details.\n\\item[-v fn]\nStore the variable ordering obtained by dynamic variable re-ordering%\n\\index{variable re-ordering}%\n{}\nin the file fn.\n\\index{dynamic variable re-ordering}\n\\end{description}\n\n\\subsection{Expressions}\n\\index{expressions}\nThe Fl prompt%\n\\index{prompt}%\n{} is : so lines beginning with this contain\nthe user's input; all other lines are output of the system.\n\\begin{hol}\n: 2+3;\nit::int\n5\n\\end{hol}\nHere we simply evaluated the expression 2+3 and FL reduced\nit to normal form; in this case computed the result 5. Note that\nfl does only support integers%\n\\index{integers}%\n{} as number types.\nThe integers are represented in arbitrary precision, and grow as needed.\n\nNote also that Voss stores the result of the most recent expression\nin a variable called {\\tt it}.\n\\index{it}%\nThus, continuing the example by evaluating the expression {\\tt it;} yields:\n\\begin{hol}\n: it;\nit::int\n5\n\\end{hol}\n\n\\subsection{Declarations}\n\nThe declaration%\n\\index{declaration}%\n{} let x = e binds a computation of e to\nthe variable x.\nNote that it does not evaluate e (since the language is fully lazy%\n\\index{lazy}%\n{}).\nOnly if x is printed or used in some other expression that is evaluated\nwill it be evaluated.\nAlso, once e is evaluated, x will refer to the result of the evaluation\nrather than the computation.\nHence, the expression e is evaluated at most once, but it may\nnot be evaluated at all.\n\\begin{hol}\n: let x = 3+3;\nx::int\n\\end{hol}\nNote that when expressions are bound to variables, the system simply\nprints out the inferred type%\n\\index{inferred type}%\n{} of the expression. We will return to\nthe typing scheme in FL later.\nFor now, it suffices to say that FL tries to find as general type as\npossible that is consistent with the type of the expression.\n\nA declaration%\n\\index{declaration!local}%\n{} can be made local to the evaluation of an expression\ne by evaluating the expression decl in e.\nFor example:\n\\begin{hol}\n: let y = let x = 4 in x-5;\ny::int\n\\end{hol}\nwould bind the expression 4 to x only inside the expression\nbound to y. Thus, we get:\n\\begin{hol}\n: let x = 2;\nx::int\n: let y = let x = 4 in x-5;\ny::int\n: x;\n2\n: y;\n-1\n\\end{hol}\nFL is lexically scoped%\n\\index{scoping}%\n{}, and thus\nthe binding in effect at the time of definition is the one used.\nIn other words, if we write:\n\\begin{hol}\n: let x = 2;\nx::int\n: let y = x*5;\ny::int\n: let x = 12;\nx::int\n\\end{hol}\nand we then evaluate y we will get 10\nrather than 60.\n\nContrary to ML, at the top level FL 1.8b does not allow\nsimultaneous bindings%\n\\index{simultaneous bindings}%\n{}.\nHence, if we would like to bind the expressions 2 and 4-5 to the\nvariables x and y respectively, we would have to write:\n\\begin{hol}\n: let x = 2;\nx::int\n: let y = 4-5;\ny::int\n\\end{hol}\nHowever, inside local bindings, a special keyword {\\tt val}\n\\index{val}%\n{}\nis available to take a complicated object apart automatically.\nFor example, if e is an expression of type (int\\verb@#@bool) then\nwe could write:\n\\begin{hol}\n: let top\\_level q =\n\tval (i,b) = q in\n\ti < 3 => b | F;\ntop\\_level::(int\\verb@#@bool)->bool\n\\end{hol}\nIn general, the expression to the right of the val keyword\ncan be an arbitrary complex pattern similar to the patterns%\n\\index{patterns}%\n{}\nallowed in function definitions and lambda expressions.\nFor more details, see the section on pattern matching on page~\\ref{val}.\n\n\\subsection{Functions}\n\nTo define a function%\n\\index{function}%\n{} f with formal parameter x and body e one performs\nthe declaration: let f x = e.\nTo apply the function f to an actual parameter e one evaluates\nthe expression f e.\n\\begin{hol}\n: let f x = x+2;\nf::(int) -> (int)\n: f 4;\n6\n\\end{hol}\nNote that the type inferred for f is essentially ``a function taking\nan int as argument and returning an int''.\nApplications binds more tightly than anything else in FL; thus for example:\nf 3 * 4 would be evaluated as: ((f 3)*4) and thus yield 20.\n\nFunctions of several arguments can also be defined:\n\\begin{hol}\n: let add x y = x+2*y;\nadd::(int) -> ((int) -> (int))\n: add 1 4;\n9\n: let f = add 1;\nf::(int) -> (int)\n: f 4;\n9\n\\end{hol}\nApplications associate%\n\\index{association rules}%\n{} to the left so add 3 4 means (add 3) 4. In the\nexpression add 3, the\nfunction add is partially applied to 3; the resulting value is the\nfunction of type int\\funarrow int which\nadds 3 to twice its argument. Thus add takes its arguments one at a time.\nWe could have made add take a single argument of the cartesian product%\n\\index{cartesian product}%\n{}\ntype (int\\verb@#@int):\n\\begin{hol}\n: let add (x,y) = x+y;\nadd::((int # int)) -> (int)\n: add (3,4);\n7\n: add 3;\n===Type mismatch: (int # int) and int\n#### Run-time error\n---- Type error\n\\end{hol}\nAs well as taking structured arguments (e.g. (3,4)) functions may also return\nstructured results:\n\\begin{hol}\n: let manhat\\_dist (x1,y1) (x2,y2) = (x2-x1, y2-y1);\nmanhat\\_dist::((int # int)) -> (((int # int)) -> ((int # int)))\n: manhat\\_dist (1,1) (3,5);\n(2,4)\n\\end{hol}\n\nTrying to print%\n\\index{print}%\n{} a function with insufficient number of actual arguments\nyield a dash for the function and the type of the expression is printed out.\nFor example:\n\\begin{hol}\n: (5, manhat\\_dist (1,2));\n(5,-) ::(int # ((int # int)) -> ((int # int)))\n\\end{hol}\nThe only exception to this rule is for concrete types\nfor which the user has installed a printing function%\n\\index{printing function}%\n{}.\nFor more details of concrete types, see page~\\ref{concreteTypes}.\n\n\\subsection{Recursion}\n\nThe following is an attempt to define the factorial function:\n\\begin{hol}\n: let fact n = n=0 => 1 | n*fact (n-1);\n#### Run-time error\n---- Undefined variable (fact)\n\\end{hol}\nThe problem is that any free variables in the body of a function have\nthe bindings they had\njust before the function was declared; fact is such a free variable\nin the body of the declaration\nabove, and since it is not defined before its own declaration, an error\nresults. To make things\nclear consider:\n\\begin{hol}\n: let f n = n+1;\nf::(int) -> (int)\n: let f n = n=0 => 1 | n*f (n-1);\nf::(int) -> (int)\n: f 3;\n9\n\\end{hol}\nHere 3 results in the evaluation of 3*(f 2), but now the first f is\nused so f 2 evaluates to 2+1=3.\nTo make a function declaration hold within its own body, letrec%\n\\index{letrec}%\n{}\ninstead of let must be used.\nThe correct recursive%\n\\index{recursive}%\n{} definition of the factorial function is thus:\n\\begin{hol}\n: letrec fact n = n=0 => 1 | n*fact (n-1);\nfact::(int) -> (int)\n: fact 5;\n120\n\\end{hol}\n\nIt should be pointed out that FL currently does not allow direct definition\nof mutually recursive functions%\n\\index{mutually recursive functions}%\n{}.\nFor an example on how this limitation can be dealt with, see\nthe subsection on concrete type decalartions.\n\n\\subsection{Tuples}\n\nIf $e\\_1, e\\_2, \\ldots , e\\_n$ have types $t\\_1, t\\_2, \\ldots , t\\_n$, then\nthe FL expression $(e\\_1, e\\_2, \\ldots , e\\_n)$ have type\n$t\\_1$\\verb@#@$t\\_2$\\verb@#@$\\ldots$\\verb@#@$t\\_n$.\nThe standard functions on tuples are fst%\n\\index{fst}%\n{} (first), snd%\n\\index{snd}%\n{} (second), and the infix\noperation , (pair%\n\\index{pair}%\n{}).\n\\begin{hol}\n: let q = ((1,2),3);\nq::((int # int) # int)\n: let qq = (1,2,3);\nqq::(int # (int # int))\n: q;\n((1,2),3)\n: qq;\n(1,2,3)\n: let qqq = (1,\"abc\");\nqqq::(int # string)\n: qqq;\n(1,\"abc\")\n\\end{hol}\n\n\\subsection{Lists}\n\nIf $e\\_1, e\\_2, \\ldots , e\\_n$ have type $t$, then the FL expression\n$[e\\_1, e\\_2, \\ldots , e\\_n]$ has type ($t$ list%\n\\index{list}%\n{}).\nThe standard functions on lists are hd%\n\\index{hd}%\n{} (head%\n\\index{head}%\n{}), tl%\n\\index{tl}%\n{} (tail%\n\\index{tail}%\n{}), []%\n\\index{[]}%\n{} (the empty list%\n\\index{empty list}%\n{}), and the infix operation : (cons%\n\\index{cons}%\n{}).\nNote that all elements of a list must have the same type (compare\nthis with a tuple where the size is determined but each member of\nthe tuple can have different type).\n\\begin{hol}\n: let l = [1,2,3,3,2,1,2];\nl::(int list)\n: hd l;\n1\n: tl l;\n[2,3,3,2,1,2]\n: 0:l;\n[0,1,2,3,3,2,1,2]\n: letrec (len [] = 0) /\\verb!\\! (len (a:rest) = 1+len rest);\nlen::((* list)) -> (int)\n: len l;\n7\n\\end{hol}\n\n\\subsection{Strings}\n\nA sequence of characters enclosed between \\verb@\"@ or \\verb@`@ is a string%\n\\index{string}%\n{}.\nThe standard functions on strings are \\verb@^@ (catenation%\n\\index{catenation}%\n{}), explode%\n\\index{explode}%\n{}\n(make string into list of strings) and\nimplode%\n\\index{implode}%\n{} (make list of strings into single string).\nThere are also int2str%\n\\index{int2str}%\n{} and bool2str%\n\\index{bool2str}%\n{} functions that create a string\nfrom an integer or an object of type boolean.\nWe will return to these later.\n\\begin{hol}\n: let q = \"abc and \\_12!@@#\";\nq::string\n: let qq = `qw\"q qw`;\nqq::string\n: q^qq;\n\"abc and \\_12!@@#qw\"q qw\"\n: explode q;\n[\"a\",\"b\",\"c\",\" \",\"a\",\"n\",\"d\",\" \",\"\\_\",\"1\",\"2\",\"!\",\"@\",\"@\",\"#\"]\n: implode [\"1\", \"2\"];\n\"12\"\n: int2str (1-34);\n\"-33\"\n: bool2str ((variable \"a\") AND (variable \"b\"));\n\"a&b\"\n\\end{hol}\n\n\\subsection{Polymorphism}\n\nThe list processing functions hd, tl, etc. can be used on all types of lists.\n\\begin{hol}\n: hd [1,2,3];\n1\n: hd [\"abc\", \"edf\"];\n\"abc\"\n: (hd [\"a\", \"b\"]), hd [4,2,1];\n(\"a\",4)\n: let q = [T,T,F];\nq::(bool list)\n: hd q;\nT\n\\end{hol}\nThus hd has several types; for example, it is used above with types\n(int list) \\funarrow int,\n(string list) \\funarrow string, and (bool list) \\funarrow bool.\nIn fact if ty is any type then hd has the type (ty list) \\funarrow ty.\nFunctions, like hd, with many types are called polymorphic%\n\\index{polymorphic}%\n{},\nand FL uses type variables%\n\\index{type variables}%\n{} *, **, ***, etc. to represent their types.\n\\begin{hol}\n: let f x = hd x;\nf::((* list)) -> (*)\n: letrec map fn [] = []\n /\\verb!\\! map fn (h:rest) = (fn h) : (map fn rest);\nmap::((*) -> (**)) -> (((* list)) -> ((** list)))\n: letrec fact n = n=0 => 1 | n*fact (n-1);\nfact::(int) -> (int)\n: map fact [1,2,3,4,5,6,7];\n[1,2,6,24,120,720,5040]\n\\end{hol}\nThe FL function map%\n\\index{map}%\n{} takes a function f (with argument type * and result\ntype **), and a list l\n(of elements of type *), and returns the list obtained by applying\nf to each element of l\n(which is a list of elements of type **). Map can be used at any instance\nof its type: above, both *\nand ** were instantiated to int; below, * is instantiated to (int list)\nand ** to bool. Notice that the\ninstance need not be specified; it is determined by the type checker.\n\\begin{hol}\n: let eq1 x = x=1;\neq1::(int) -> (bool)\n: map eq1 [1,2,3,4,12,2,1,2];\n[T,F,F,F,F,F,T,F]\n\\end{hol}\n\nIt should be pointed out that FL has a polymorphic type system that\nis sightly different from\nstandard ML's. In particular, only ``top-level'' user-defined functions\ncan be polymorphic. In\nother words, the following works as we would expect.\n\\begin{hol}\n: let null l = l = [];\nnull::((* list)) -> (bool)\n: let f x y = null x OR null y;\nf::((* list)) -> (((** list)) -> (bool))\n: f [1,2,3] [\"abc\", \"cdef\"];\nF\n\\end{hol}\nHowever, if we use the same declaration inside the expression, it must\nbe monomorphic%\n\\index{monomorphic}%\n{}. In other\nwords, the following example fails.\n\\begin{hol}\n: let f x y =\n let null l = l = [] in\n null x OR null y;\nf::((* list)) -> (((* list)) -> (bool))\n: f [1,2,3] [\"abc\", \"cdef\"];\n===Type mismatch: int and string\n#### Run-time error\n---- Type error\n\\end{hol}\nIn this respect, FL is similar to the functional language called\nMiranda\\footnote{Miranda is a trademark of\nResearch Software Ltd.}~\\cite{Miranda}.\n\n\\subsection{Type Annotations}\nSometimes it is useful to inform the type inference mechanism\nof FL what type is expected.\nIn particular, when using overloaded%\n\\index{overloaded}%\n{} operators or functions\nit is quite common to have to explicitly type arguments.\nIn FL a variable or expression can be annotated%\n\\index{type annotation}%\n{} with its expected\ntype by enclosing it in curly braces and decorate the expression\nwith a type expression.\nIn particular, if we would like to define a function ihd that\nreturn the head of a list, but that only can be used on integer lists,\nwe could define ihd as follows:\n\\begin{hol}\nlet ihd \\{l::int list\\} = hd l;\n\\end{hol}\nSimilarly, if we have overloaded the operator + to either\noperate over strings or integers, we can extend this opverloading\nwith yet another function by declaring\n\\begin{hol}\noverload + \\{+::int->int->int\\} \\{+::string->string->string\\} foo;\n\\end{hol}\nIf we now were to write a function that adds three integer numbers\nwe would have to decorate the types so that the overload resolution\ncan succeed. For example, we could write:\n\\begin{hol}\nlet f \\{x::int\\} \\{y::int\\} \\{z::int\\} = x+y+z;\n\\end{hol}\nor we could write\n\\begin{hol}\nlet f x y z = \\{x+\\{y::int\\} ::int\\}+z;\n\\end{hol}\nalthough the former is easier to read and more general.\n\n\\subsection{Lambda Expressions}\n\nThe expression \\verb@\\@x.e evaluates to a function with formal parameter x\nand body e. Thus the\ndeclaration let f x = e is equivalent to let f = \\verb@\\@x.e. The character\n\\verb@\\@ is our representation of lambda,\nand expressions like \\verb@\\@x.e are called lambda-expressions%\n\\index{lambda-expressions}%\n{}.\n\\begin{hol}\n: \\verb!\\!x.x+1;\n- ::(int) -> (int)\n\n: let q = \\verb!\\!x.x+1;\nq::(int) -> (int)\n: q 1;\n2\n: map (\\verb!\\!x.x*x) [1,2,3,4,5];\n[1,4,9,16,25]\n\\end{hol}\n\n\\subsection{Failures}\n\nSome standard functions fail at run-time on certain arguments, printing\nout a string (which is usually the function name) to identify the sort\nof failure. A failure%\n\\index{failure}%\n{} with string \"t\" may also be\ngenerated explicitly by evaluating the expression error \"t\" (or more\ngenerally error%\n\\index{error}%\n{} e where e has type string).\n\\begin{hol}\n: hd(tl [2]);\nFailure: ---- Cannot compute hd of the empty list\n\n: 1/0;\nFailure: ---- Division by zero\n\n\n: error \"My message\";\nFailure: ---- My message\n\\end{hol}\n\nA failure can be trapped%\n\\index{trap a failure}%\n{} by catch%\n\\index{catch}%\n{}; the value of the expression $e\\_1$ catch\n$e\\_2$ is that of $e\\_1$, unless\n$e\\_1$ causes a failure, in which case it is the value of $e\\_2$. One important\nproperty of catch is that it is\n(very) strict in its first argument. In other words, (hd ($e\\_1$ catch\n$e\\_2$)) will completely evaluate $e\\_1$\neven though only the first element in the list may be needed. In view\nof FL's lazy semantics%\n\\index{lazy semantics}%\n{}, the\nuse of catch should be very carefully considered. In particular, the\nbindings of catch is dynamic\nrather than static so the user beware!\n\n\\subsection{Boolean Expressions}\n\nAll Boolean expressions%\n\\index{Boolean expression}%\n{} in FL are maintained as ordered binary decision\ndiagrams. Hence, it\nis very easy to compare complex Boolean expressions and to combine\nthem in different ways.\nBoolean variables%\n\\index{Boolean variable}%\n{} are created by variable%\n\\index{variable}%\n{} s, where s is of type string.\nThe system uses name\nequivalence, and thus\n\\begin{hol}\n: let v = variable \"v\";\nv::bool\n: v=v;\nT\n: variable \"v\" = variable\"v\";\nT\n\\end{hol}\n\nThe constants true%\n\\index{true}%\n{} and false%\n\\index{false}%\n{} are denoted T and F respectively. The\nstandard boolean functions\nare available, i.e., AND%\n\\index{AND}%\n{}, OR%\n\\index{OR}%\n{}, NOT%\n\\index{NOT}%\n{}, XOR%\n\\index{XOR}%\n{}, and =%\n\\index{=}%\n{} are all defined for objects of type Boolean..\nFurthermore, there\nis a special identity%\n\\index{identity}%\n{} operator ==%\n\\index{==}%\n{} that return true or false depending\non whether the two arguments\nrepresent the same Boolean function or not.\n\nNote that the variable ordering%\n\\index{variable ordering}%\n{} in the OBDD representation is defined\nby the order in which each variable function call {\\em gets evaluated}.\nSince FL is a fully lazy language%\n\\index{lazy language}%\n{}, and thus the order in which expressions\nare evaluated is often difficult to predict, it is strongly recommended\nthat each variable declaration is forced to be evaluated before it\nis being used.\nAlternatively, one can request that FL re-orders the variables\nby evaluating the function var\\_order%\n\\index{var\\_order}%\n{} and give as argument a list\nof variable names.\nFL will then apply the dynamic variable re-ordering mechanism and\nenforce that the variables mentioned in the list will be the first\narguments and that they will occur in exactly this order.\n\\begin{hol}\n: let a = variable \"a\";\na::bool\n: let b = variable \"b\";\nb::bool\n: a AND b;\na&b\n: a OR b;\na + b\n: NOT a AND NOT b AND T;\na'&b'\n: a = b;\na&b + a'&b'\n: a == b;\nF\n: (a=b) == (a AND b OR NOT a AND NOT b);\nT\n\\end{hol}\n\nThe default style for printing Boolean expressions%\n\\index{printing Boolean expressions}%\n{} is as a sum-of-products%\n\\index{sum-of-products}%\n{}.\nSince this may require printing an extremely large expression, there\nis a user-setable limit on how many products that will be printed\nand the maximum size of a product.\nFor more details how to modify these two parameters, see Section~\\ref{vossrc}.\n\n\\subsection{Quantifiers}\n\nThere are several ways of using quantification%\n\\index{quantification}%\n{}.\nThe ``traditional'' !x. e%\n\\index{$!x.\\,e$}%\n{} (for all x) and ?x. e%\n\\index{$?x. e$}%\n{}\n(there is an x) can be used as long as the type of x and e is bool.\nIn addition, you can also quantify away a variable in an expression\nby quant\\_forall%\n\\index{quant\\_forall}%\n{} v e or quant\\_thereis%\n\\index{quant\\_thereis}%\n{} v e.\n\\begin{hol}\n: !a. ?b. (a XOR b);\nT\n: let a = variable \"a\"; let b = variable \"b\"; let c = variable \"c\";\na::bool\nb::bool\nc::bool\n: : : a AND b OR c;\na&b + c\n: quant\\_forall a (a AND b OR c);\nc\n: quant\\_thereis (a OR c) (a AND b OR c);\nT\n\\end{hol}\nIn fact, quant\\_forall and quant\\_thereis quantifies away all variables\nin the first Boolean expression.\nFor example:\n\\begin{hol}\n: let v s = variable s;\nv::(string) -> (bool)\n: let a = v \"a\"; let b = v \"b\"; let c = v \"c\"; let d = v \"d\";\na::bool\nb::bool\nc::bool\nd::bool\n: : : : let ex = (a AND NOT b);\nex::bool\n: ex;\na&b'\n: let nex = ex AND (a=c) AND (b=d);\nnex::bool\n: quant\\_thereis (a AND b) nex;\nc&d'\n\\end{hol}\nNote that the actual Boolean expression used as first argument is irrelevant.\nThe only important fact is on what variables the expression depends.\n\n\\subsection{Dependencies}\n\nSometimes it is useful to find out which Boolean variables a\nBoolean function actually depends on.\nThe built-in function depends%\n\\index{depends}%\n{} takes a list of elements of type\nbool and return the union of the variables these functions depend on.\nFor example:\n\\begin{hol}\n: let v s = variable s;\nv::(string) -> (bool)\n: let a = v \"a\"; let b = v \"b\"; let c = v \"c\"; let d = v \"d\";\na::bool\nb::bool\nc::bool\nd::bool\n: : : : let ex1 = (a=c) AND d;\nex1::bool\n: let ex2 = a = b;\nex2::bool\n: depends [ex1];\n[\"a\",\"c\",\"d\"]\n: depends [ex1,ex2];\n[\"a\",\"b\",\"c\",\"d\"]\n\\end{hol}\n\nNote that the order of the variables in the list returned by depends\nis the variable order of the OBDD representation.\n\n\\subsection{Substitutions}\n\nGiven a Boolean function represented as an OBDD, it is convenient to\nbe able to apply the function to some arguments.\nThis can be accomplished by the substitute%\n\\index{substitute}%\n{} command that takes a\nlist of (variable name, expression) and an expression in which the\nsimultaneous substitution%\n\\index{substitution}%\n{} is to be made.\nFor example,:\n\\begin{hol}\n: let v s = variable s;\nv::(string) -> (bool)\n: let a = v \"a\"; let b = v \"b\"; let c = v \"c\"; let d = v \"d\";\na::bool\nb::bool\nc::bool\nd::bool\n: : : : let ex = (a AND NOT b);\nex::bool\n: ex;\na&b'\n: substitute [(\"a\", c), (\"b\", d)] ex;\nc&d'\n\\end{hol}\n\nIt should be pointed out that there are no restrictions on the expressions\nin the substitutions.\nIn particular, it is possible to ``swap'' variables.\nWe illustrate this by continuing the example above:\n\\begin{hol}\n: ex;\na&b'\n: substitute [(\"a\", b), (\"b\", a)] ex;\na'&b\n\\end{hol}\n\n\\subsection{Type Abbreviations}\n\\index{type abbreviations}\n\nTypes can be given names:\n\\begin{hol}\n: new\\_type\\_abbrev pair = int#int;\n: let p = (1,2);\np::(int # int)\n\\end{hol}\nHowever, as can be seen from the example, the system does not\nmake any distinction between the new type name\nand the actual type. It is purely a short hand that is useful when\ndefining concrete types below.\n\n\\subsection{Concrete Types}\n\nNew types (rather than mere abbreviations) can also be defined.\nConcrete types%\n\\index{concrete type}%\n{} are types\ndefined by a set of constructors%\n\\index{type constructor}%\n{} which can be used to create objects\nof that type and also (in\npatterns) to decompose objects of that type. For example, to define\na type card one could use the\nconstruct type:\n\\begin{hol}\n: lettype card = king | queen | jack | other int;\nother::(int) -> (card)\njack::card\nqueen::card\nking::card\n\\end{hol}\nSuch a declaration declares king, queen, jack and other as constructors\nand gives them\nvalues. The value of a 0-ary constructor such as king is the constant\nvalue king. The value of a\nconstructor such as other is a function that given an integer value\nn produces other(n).\n\\begin{hol}\n: king;\n- ::card\n\n: other 9;\n- ::card\n\\end{hol}\n\nNote that there is no print routine%\n\\index{print routine}%\n{} for concrete types. If a print\nroutine is desired, one has to\ndefine it.\nTo define functions that take their argument from a concrete type,\nwe introduce the idea of pattern matching%\n\\index{pattern matching}%\n{}.\nIn particular\n\\begin{hol}\nlet f pat1 = e1\n /\\verb@\\@ f pat2 = e2\n /\\verb@\\@ ...\n /\\verb@\\@ f patn = en;\n\\end{hol}\ndenotes a function that given a value v selects the first pattern that\nmatches $v$, say pati, binds the variables of pati to\nthe corresponding components of the value $v$ and then evaluates the\nexpression ei.\nWe could for example define a print function for the cards in the following way:\n\\begin{hol}\n: let pr\\_card king = \"K\"\n /\\verb@\\@ pr\\_card queen = \"Q\"\n /\\verb@\\@ pr\\_card jack = \"J\"\n /\\verb@\\@ pr\\_card (other n) = int2str n;\npr\\_card::(card) -> (string)\n: pr\\_card king;\n\"K\"\n: pr\\_card queen;\n\"Q\"\n: pr\\_card jack;\n\"J\"\n: pr\\_card (other 5);\n\"5\"\n\\end{hol}\nIf we now issue the top-level command \n\\begin{hol}\n: install\\_print\\_function pr\\_card;\nInstalled pretty printer for objects of type: card\n\\end{hol}\n\\index{install\\_print\\_function}\nevery time we evaluate an expression of type card this routine would\nbe called and the string printed out on standard out.\n\nMutually recursive types%\n\\index{mutually recursive types}%\n{} can also be defined. To do so, simply list\nthe type names on the left\nhand side of the equality sign and list the type expressions on the\nright hand side. For example:\n\\begin{hol}\n: lettype IExpr, BExpr = Ivar string | Plus IExpr IExpr | ITE BExpr IExpr IExpr,\n And BExpr BExpr | GEQ IExpr IExpr;\nGEQ::IExpr->IExpr->BExpr\nAnd::BExpr->BExpr->BExpr\nITE::BExpr->IExpr->IExpr->IExpr\nPlus::IExpr->IExpr->IExpr\nIvar::string->IExpr\n\\end{hol}\ndefines two mutually recursive concrete data types for integer expressions\nand Boolean expressions (very simple versions!).\n\nCurrently, FL does not provide any direct way of defining\nmutually recursive functions%\n\\index{mutually recursive functions}%\n{}.\nThe easiest work-around is to pass the later defined functions as\nparameters to the earlier function.\nAfter all functions have been defined, one can re-define the early ones.\nTo illustrate the approach, consider writing functions that converts\nobjects of type IExpr and BExpr to strings.\nOne possible solution is as follows:\n\\begin{hol}\n: let prIExpr prBexpr expr =\n letrec prIExpr (Ivar s) = s\n /\\verb@\\@ prIExpr (Plus a b) = (prIExpr a)^\" + \"^(prIExpr b)\n /\\verb@\\@ prIExpr (ITE c t e) = \"if \"^(prBexpr c)^\" then \"^\n (prIExpr t)^\" else \"^(prIExpr e) in\n prIExpr expr;\nprIExpr::(BExpr->string)->IExpr->string\n\n: letrec prBExpr (And a b) = (prBExpr a) ^ \" AND \" ^ (prBExpr b)\n /\\verb@\\@ prBExpr (GEQ a b) = (prIExpr prBExpr a)^\" >= \"^\n\t\t\t (prIExpr prBExpr b);\nprBExpr::BExpr->string\n\n: let prIExpr e = prIExpr prBExpr e;\nprIExpr::IExpr->string\n\\end{hol}\nNote that we simply pass prBExpr as an argument to the initial definition\nof prIExpr.\n\n\\subsection{Abstract Types}\n\nIn FL one can also hide the definitions of types, type constructors,\nand functions. By\nenclosing a sequence of type declarations and function definitions\nwithin begin\\_abstype%\n\\index{begin\\_abstype}%\n{}\nend\\_abstype%\n\\index{end\\_abstype}%\n{} elist, only the constructors and/or functions mentioned\nin the elist will be visible\nand accessible for other functions and definitions. Thus, one can protect\na concrete type and only\nmake some abstract constructor functions available. To illustrate the\nconcept, consider defining a\nconcrete type called theorem. The only way we would like the user to\nbe able to create a new theorem is to give a Boolean expression that\ndenotes a tautology (something always true).\nFirst we define the expression type.\n\\begin{hol}\n: lettype expr = Forall string expr |\n Thereis string expr |\n Var string |\n True |\n False |\n And expr expr |\n Or expr expr |\n Not expr;\nNot::(expr) -> (expr)\nOr::(expr) -> ((expr) -> (expr))\nAnd::(expr) -> ((expr) -> (expr))\nFalse::expr\nTrue::expr\nVar::(string) -> (expr)\nThereis::(string) -> ((expr) -> (expr))\nForall::(string) -> ((expr) -> (expr))\n\\end{hol}\n\nWe then define the concrete type theorem and the constructor function\nis\\_taut. Note that we\nalso define a couple of help functions. However, only the is\\_taut function\nis exported out of the\nabstract type%\n\\index{abstract type}%\n{}, and thus is the only way of creating a theorem.\n\\begin{hol}\n: begin\\_abstype;\n: lettype theorem = Thm expr;\nThm::(expr) -> (theorem)\n: letrec assoc x l = l = [] => error \"assoc\" |\n let h = hd l in\n (fst h) = x => (snd h) | assoc x (tl l);\nassoc::(*) -> ((((* # **) list)) -> (**))\n: letrec (eval (Forall s e) al = !x.(eval e ((s,x):al))) /\\verb@\\@\n (eval (Thereis s e) al = ?x.(eval e ((s,x):al))) /\\verb@\\@\n (eval (Var s) al = (assoc s al) catch\n (error \"Free variable\")) /\\verb@\\@\n (eval True al = T) /\\verb@\\@\n (eval False al = F) /\\verb@\\@\n (eval (And e1 e2) al = (eval e1 al) AND (eval e2 al))/\\verb@\\@\n (eval (Or e1 e2) al = (eval e1 al) OR (eval e2 al)) /\\verb@\\@\n (eval (Not e) al = NOT (eval e al));\neval::(expr) -> ((((string # bool) list)) -> (bool))\n: let is\\_taut e = ((eval e []) == T) => Thm e |\n error \"Not a tautology\";\nis\\_taut::(expr) -> (theorem)\nend\\_abstype is\\_taut;\n\\end{hol}\n\nWe can now use this very safe theorem system, since we can only generate\ntheorems that are\ntautologies. For example\n\\begin{hol}\n: let e = (Forall \"a\" (Thereis \"b\" (And (Var \"a\") (Var \"b\"))));\ne::expr\n: is\\_taut e;\nFailure: ---- Not a tautology\n\n\n: let f = (Forall \"a\" (Thereis \"b\" (Or (Var \"a\") (Var \"b\"))));\nf::expr\n: is\\_taut f;\n- ::theorem\n\\end{hol}\n\n\\subsection{Infix Operators}\n\nIn order to make the FL code more readable, it is possible to declare\na function to be infix%\n\\index{infix}%\n{} (associating from the left), infixr%\n\\index{infixr}%\n{}\n(associating from the right), nonfix%\n\\index{nonfix}%\n{} (no fixity at all), prefix\n\\index{prefix}%\n{} (prefix operator with tighter binding than ``normal'' function definitions),\npostfix%\n\\index{postfix}%\n{}, or\nof a binder%\n\\index{binder}%\n{} type.\nFor the infix and infixr directives, the precedence can be given as\na number from 1 to 9, where a higher number binds tighter.\nSimilarly, prefix also takes a precedence number, but only 0 or 1.\nNote that prefix and postfix functions bind higher than any infix function.\nBeware that the fixity declaration modifies the parser and\nthus remains in effect whether\nthe function is exported out of an abstract data type or note.\nAs an illustration of this idea, consider the following example:\n\\begin{hol}\n: lettype expr = Var int |\n Mult expr expr |\n Plus expr expr |\n Negate expr;\nNegate::expr->expr\nPlus::expr->expr->expr\nMult::expr->expr->expr\nVar::int->expr\n: letrec eval (Var i) = i\n /\\verb@\\@ eval (Mult e1 e2) = (eval e1) * (eval e2)\n /\\verb@\\@ eval (Plus e1 e2) = (eval e1) + (eval e2)\n /\\verb@\\@ eval (Negate e1) = 0-(eval e1);\neval::expr->int\n: let ** a b = Mult a b;\n**::expr->expr->expr\n: let ++ a b = Plus a b;\n++::expr->expr->expr\n: infix 4 **;\n: infix 3 ++;\n: let ' i = Var i;\n'::int->expr\n: prefix 0 ';\n: let q = '1 ++ Negate '2 ** Negate '4;\nq::expr\n: eval q;\nit::int\n9\n\\end{hol}\n\nThe next example illustrates how postfix declarations can make the\ncode more readable.\n\\begin{hol}\n: let ns i = 1000*i;\nns::(int) -> (int)\n: postfix ns;\n: let to a b = (a,b);\nto::(*) -> ((**) -> ((* # **)))\n: infix 3 to;\n: 2 ns to 4 ns;\n(2000,4000)\n\\end{hol}\n\nOur final example deals with the binder declaration.\nThe command binder takes a function and makes it into a binder, i.e., an\nobject that introduces a new bound variable in an expression.\nNote that the type of the function declared to be a binder must be\n(*\\funarrow **)\\funarrow string\\funarrow **, since the first argument of\na binder function will be a lambda expression and the second argument will\nbe a string with the name of the bound variable.\nThus, if a function f has been declared as a binder, then\nf x.E will be parsed as f (\\protect\\verb!\\! x.E) \"x\".\n\n\\subsection{Overloading}\n\nFL supports a limited amount of user defined overloading%\n\\index{overloading}%\n{} of functions\nand operators.\nHowever, in order to avoid an exponential type inference algorithm, the\noverloaded operators must be resolved from the types of their arguments only. \nTo illustrate the construct, consider the following example:\n\\begin{hol}\n: let pplus (a,b) (c,d) = (a+c, b+d);\npplus::(int#int)->(int#int)->(int#int)\n: let pprod (a,b) (c,d) = (a*c, b*d);\npprod::(int#int)->(int#int)->(int#int)\n: overload + + pplus;\n: overload * * pprod;\n: (1,2)+(3,4)*(9,3*2);\nit::int#int\n(28,26)\n\\end{hol}\nHere we overloaded the symbols + and *.\nNote that we essentially added new meanings to + and * since we included\nthe (built-in) versions as possible candidates.\n\nThe one area where overloaded operators are somewhat cumbersome to use is\nwhen defining new functions.\nHere it is often necessary to decorate the arguments to the functions\nin order to make the resolution function succeed.\n\\begin{hol}\n: let f x y z = x*(y+z);\n===Type error around line 1\nCannot resolve overloading of `+'\nBoth `pplus' and `+' matches the type\n: let f \\{x::int\\} \\{y::int\\} \\{z::int\\} = x*(y+z);\nf::int->int->int->int\n\\end{hol}\nFinally, overloaded operators and functions can of course also\nbe declared infix, binders, or postfix as any other function or operator.\n\n\\subsection{Quotation of Expressions}\n\\index{quotation of expression}\nSometimes it is convenient to retain the actual text that was used\nto denote an expression.\nBy enclosing the expression in \\verb@``@ and \\verb@''@, FL will return\na pair consisting of the text of the expression as well as the expression\nitself.\n\\begin{hol}\n: `` 1+3*4 '';\nit::string#int\n(\" 1+3*4 \",13)\n\\end{hol}\nNote that it is advisable to include an extra space after the opening quotes\nand before the closing quotes since the parser otherwise is likely to\nview the quotes as being part of an operator or variable.\n\n\\subsection{Circuit Models}\n\nSince the main use of the FL system, and its historical root, is related\nto hardware verification, there are a number of built-in\nfunctions specifically tailored towards hardware modeling\nand symbolic trajectory evaluation in particular. \nInternally, a circuit is represented by a list of nodes (names),\nand a next state function%\n\\index{next state function}%\n{}.\nThe next state function is mapping the current state of the circuit\n(including the current values on the inputs) to a new state\nof the circuit.\nSince the circuit representation is intended for trajectory evaluation,\nthe value domain for each node in the circuit is $\\{0,1,\\X,\\top\\}$\nand thus the next state function consists of quaternary%\n\\index{quaternary}%\n{} (four-valued)\nextensions of the usual Boolean function.\nThe type of such circuit is fsm%\n\\index{fsm}%\n{} (for finite state machine%\n\\index{finite state machine}%\n{}) and, by\ndefault, it does not have a print function (since the machines\nare usually much too large to be meaningful anyway to the user.\nHowever, for the curious reader, the command print\\_fsm%\n\\index{print\\_fsm}%\n{} will print out\na pretty complete version of the state machine.\nNote that for efficiency reasons, the next state function also\ncontains delay and fan-in and fan-out information to aid in the\nefficient simulation.\n\nIn general, there are two ways of creating a fsm object:\n\\begin{enumerate}\n\\item\nLoading a pre-compiled version of a circuit in .exe format.\n\\item\nConverting a FL structure into an fsm.\n\\end{enumerate}\n\n\\subsubsection{Loading a .exe File}\n\nIf the original circuit was described in Berkeley .sim format or\nas a SILOS II gate list, there are programs distributed with the\nsystem that can be used to compile an fsm model directly from\nthese formats.\nThe common format for such pre-compiled circuit model is a (binary)\nfile with a .exe%\n\\index{.exe file}%\n{} suffix.\n%In Section~\\ref{CompanionTools} we discuss in more detail how\n%to create such .exe file.\nLoading in the .exe file and making it an fsm object simply\ninvolves calling the load\\_exe function.\n\nFor example, if there is a full\\_adder.exe file in the current\nworking directory, the following command would create an fsm object\nof the circuit.\n\\begin{hol}\n: let ckt1 = load\\_exe \"full\\_adder.exe\";\nckt1::fsm\n\\end{hol}\nNote that there is no theoretical limit on the number of circuit that can\nbe loaded into the FL system at any particular point in time.\nHowever, since fsm models are often quite large, it is generally\nadvisable not to load more models than absolutely necessary.\n%(In most cases, only one model will be used, but there are times\n%when several models can be useful. We will return to this later\n%in Section~\\ref{ModelChecking}.)\n\n\\subsubsection{Creating an fsm Object Inside FL}\n\\label{makeSTEsection}\n\n{\\bf Warning:} This addition is fairly young and has thus not\nbeen extensively tested. Also, it leaves quite a bit to\nbe desired in terms of not being very ``clean and simple''.\nFor a reader that is more interested in using the Voss system rather\nthan writing a new conversion program from some other netlist format, this\nsection can be skipped.\n\nThe main routine for creating an fsm object is make\\_fsm%\n\\index{make\\_fsm}%\n{}.\nThe type of make\\_fsm is (Set)\\funarrow (fsm)\\footnote{Currently, when\nFL is invoked, the type of make\\_fsm is actually (*)\\funarrow (fsm), but\nthis is a bug that will be removed in the next release.}\nBefore going into how an object of type Set%\n\\index{Set}%\n{} can be constructed, we\nneed to make a small digression and introduce the quaternary logic type.\nFor historical reasons, the name of the quaternary objects is tern%\n\\index{tern}%\n{},\nand tern is defined as the concrete type:\n\\begin{hol}\n: lettype tern = One |\n Zero |\n X |\n Z |\n Val string |\n And tern tern |\n Or tern tern |\n Not tern |\n Glb tern tern |\n Lub tern tern;\nLub::tern->tern->tern\nGlb::tern->tern->tern\nNot::tern->tern\nOr::tern->tern->tern\nAnd::tern->tern->tern\nVal::string->tern\nZ::tern\nX::tern\nZero::tern\nOne::tern\n\\end{hol}\nwhere One%\n\\index{One}%\n{}, Zero%\n\\index{Zero}%\n{}, X%\n\\index{X}%\n{}, And%\n\\index{And}%\n{}, Or%\n\\index{Or}%\n{}, and Not%\n\\index{Not}%\n{} are the obvious functions.\nGlb%\n\\index{Glb}%\n{} stands for the greatest lower bound and Lub%\n\\index{Lub}%\n{} for the least upper bound.\nZ%\n\\index{Z}%\n{} is used to represent the top value%\n\\index{top value}%\n{}.\nFinally, Val%\n\\index{Val}%\n{} s is used to refer to the value on node s.\nThus, as a concrete example, the next state function of a node \"o\" may\nbe described as (Not (And (Val \"in1\") (Val \"in2\"))).\n\nThere are four constructor functions for an object of type Set:\n\\begin{enumerate}\n\\item\nEmpty%\n\\index{Empty}%\n{},\n\\item\nElement%\n\\index{Element}%\n{} (string\\verb@#@((tern\\verb@#@tern) list)),\n\\item\nUnion%\n\\index{Union}%\n{} Set Set, and\n\\item\nSequential%\n\\index{Sequential}%\n{} Set Set.\n\\end{enumerate}\nThe Empty is introduced to make writing functions that generate\nobjects of type Set easier.\nElement is the constructor that actually introduces new nodes and\nalso give driver functions for the node.\nIn general, Element n gvl, will create a node named n.\nThis node will at least have (more about this later) the drivers\nlisted in the gvl list.\nA driver is simply a pair of quaternary expressions:\nthe first object of the pair is a guard, and the second part\nis the value being driven when the guard is true.\nFor a combinational node, the driver list is a single pair whose\nguard is One and whose value component is the next state function\nof the gate.\nFor more complex nodes, like register nodes or bus nodes, the guard-value\nlist is often much longer.\n\nThe Union construct is used to gather a collection of these Element\ndeclarations so that an fsm can eventually be made.\nFinally, the Sequential constructor takes two objects S1 and S2 of\ntype Set and merges them into a single object of type Set.\nIn that sense, Sequential behaves exactly like Union.\nThe difference is how the two deal with nodes defined in both set S1 and set S2\nand for which both nodes may have at least one of their guards enabled\nat the same time.\nUnion will find the greatest lower bound of the values being driven\nat the same time whereas Sequential will assume that the driver in the\nsecond set overrides the driver in the first.\nTo illustrate the difference, consider the following example:\n\\begin{hol}\n: let a = Element (\"in1\", [(One, One)]);\na::Set\n: let b = Element (\"in1\", [(One, Zero)]);\nb::Set\n: let ex1 = Union a b;\nex1::Set\n: let ex2 = Sequential a b;\nex2::Set\n: let m1 = make\\_fsm ex1;\nm1::fsm\n: let m2 = make\\_fsm ex2;\nm2::fsm\n: m1;\n\\end{hol}\nNow m1 and m2 will both be fsm models with a single node called in1.\nHowever, in m1, the next state function of in1 will be $\\X$, since that\nis the most we can say from the inconsistent driver commands given\nin the example.\nOn the other hand, the next state function of the in1 node in m2 will be Zero.\n\nFor an example of using this facility, the directory HDL in the demo\ndistribution illustrates how one can go about defining a new description\nlanguage in order to create an fsm object.\n\n\\subsection{Queries to the Circuit Model}\n\nThere are five built-in functions that are useful in dealing with\nan fsm model: nodes%\n\\index{nodes}%\n{}, fanin%\n\\index{fanin}%\n{}, fanout%\n\\index{fanout}%\n{}, is\\_stable%\n\\index{is\\_stable}%\n{}, and get\\_node\\_val%\n\\index{get\\_node\\_val}%\n{}.\nNodes takes an fsm model and returns a {\\em list of lists} of circuit nodes.\nThe reason for the double listing is that a node may have more than\none name (only from .exe file translations).\nFanin takes an fsm model and a name of a node and returns the\nlist of node names that the next state function depend on.\nFanout works in a similar way, but for the fanout nodes.\nIs\\_stable returns T (true) if the circuit is currently stable.\nIt is mainly used when the simulation is aborted at some\nparticular time with the \"-m\" option to STE.\nFinally, get\\_node\\_val takes an fsm object and the name of a node\nand returns the encoded version of the current value of the node.\nThe encoding used is: $\\X = (T,T)$, $1=(T,F)$, $0=(F,T)$, and $\\top=(F,F)$.\nOf course, the two Boolean values are often Boolean functions over some\nvariables.\nThe main use of the get\\_node\\_val function is when the simulation\nis aborted for some reason.\nGet\\_node\\_val can then be used to probe the current state of the system.\nFor example, assuming there is a full\\_adder.exe file in the current\nworking directory, we would get:\n\\begin{hol}\n: let ckt = load\\_exe \"full\\_adder.exe\";\nckt::fsm\n: nodes ckt;\n..[[\"cout\"],[\"t4\"],[\"t3\"],[\"t2\"],[\"result\"],[\"cin\"],[\"t1\"],[\"b\"],[\"a\"]]\n: fanin ckt \"cout\";\n[\"t2\",\"t3\",\"t4\"]\n: fanout ckt \"t3\";\n[\"cout\"]\n: fanout ckt \"t4\";\n[\"cout\"]\n: fanout ckt \"a\";\n[\"t3\",\"t2\",\"t1\"]\n: get\\_node\\_val ckt \"t2\";\n..(T,T)\n\\end{hol}\n\nIf the user also loads in the \"default.fl\" standard library, there is\nthe very useful function {\\em excitation}%\n\\index{excitation}\nthat also takes an fsm model\nand a node name, but that returns the next state function for binary inputs.\nNote, however, that the current version of the excitation function\nonly works correctly for unit delay nodes.\n\\begin{hol}\n: load \"defaults.fl\";\n-Loading file defaults.fl\nT\n\n: excitation ckt \"cout\";\n....\nTrace started for node: cout \n Current value:X\n.Time: 1\n.Trace: Node cout at time 1: t4' + t3' + t2'\nTime: 2\n\nTrace ended for node: cout \n\"t4' + t3' + t2'\"\n\\end{hol}\nWe will return to this in Section~\\ref{vossrc}.\n\n\\subsection{Symbolic Trajectory Evaluation}\n\nThere is actually only one built-in command for symbolic trajectory%\n\\index{symbolic trajectory evaluation}%\n{}\nevaluation called STE%\n\\index{STE}%\n{}.\nIn general, STE determines, through symbolic trajectory evaluation, whether\nan antecedent/consequent pair hold in for some circuit.\nSTE will return a Boolean function that gives the condition for the\nverification to succeed.\nFor most applications the desired return value is T.\n\nIn general STE is invoked as\n\\begin{hol}\nSTE options fsm weak\\_list ant\\_list cons\\_list trace\\_list\n\\end{hol}\nwhere options%\n\\index{STE options}%\n{} is a string that can contain a combinations of the\nfollowing flags:\n\\begin{description}\n\\item[-a] Abort%\n\\index{abort STE}%\n{} the verification at the first antecedent or consequent failure.\nIf the verification is aborted, STE will return a Boolean function that\ngives {\\em the condition for this failure to manifest itself}.\nNote that this is contrary to STE's usual behavior which is to return\nthe Boolean function that gives the conditions for the verification to\nsucceed.\n\\item [-m n]\nAbort the verification after reaching time n.\n\\item [-e]\nEvaluate evey node in the circuit whether the node is in the\nfan-in tree of some node that is checked or traced or not.\nIn other words, with the -e option, the simulator will compute\nthe value on every node. Normally, a node whose value\ncannot be observed (directly or indirectly) is simply kept at X.\nOf course, the STE command can run significantly slower with the\n-e option, so beware!\n\\item [-i]\nAllow antecedent failures%\n\\index{antecedent failures}%\n{}.\nIn other words, compute a straight implication.\nThe normal behavior of the verification process is to disallow\nantecedent failures.\nThus the default verification condition is both to check that every\ntrajectory the circuit can go thorough that is consistent with the\nantecedent is also consistent with the consequent, and that there\nis at least one (real) circuit trajectory that is consistent\nwith the antecedent.\n\\item [-n]\nAllow over-constrained values, i.e., allow nodes to take\non the top value in the value lattice.\nNormally, over-constrained values are viewed as failures.\n\\item[-w]\nDo not print out warning messages%\n\\index{warning messages}%\n{}.\n\\item[-t s]\nIn addition to printing out trace messages on stderr, also send\nthe trace events in Postscript%\n\\index{Postscript waveform}%\n{} format to the file s.\nBy previewing or printing out the file the user gets a waveform diagram for\nthe traced signals.\n\\item[-T s]\nSame as -t, but generate Postscript code in landscape mode.\n\\end{description}\n\nThe second argument to STE must be an object of the fsm type representing\na circuit that is to be simulated.\n\nThe weak\\_list is a list of 4-tuples.\nEach 4-tuple is of the form (g,n,f,t), where $n$ is a node name, $f$ and $t$\ndenote start and stop times, and $g$ is a Boolean function\ndenoting the domain for which this node should be weakened%\n\\index{weakened}%\n{}.\nFor assignments making the guard true, the next state function of the\ncircuit node is set to $X$.\nThis is often useful when verifying a subsystem of a larger system\nthat has shared input signals.\nBy weakening%\n\\index{weakening}%\n{} the model, unnecessary computations does not have\nto be performed.\nNote that weakening a model is a ``safe'' operation in that\nthe monotonicity of the circuit models guarantee that if we\ncan verify something in the weakened model, then that same property\nis guaranteed to hold in the original model.\n\nThe ant\\_list and cons\\_list are both lists of five-tuples.\nEach five-tuple%\n\\index{five-tuple}%\n{} is of the form $(g,n,v,s,t)$, where $g$ is a Boolean function\ndenoting the domain for which this assertion/check should be carried out,\n$n$ is the name of a node, $v$ is the value to be asserted/checked, and\n$s$ and $t$ denote the start and stop times for this assertion/check\nrespectively.\n\nFinally, the last argument to STE is a list of triples.\nEach triple is of the form $(n,s,t)$, where $n$ is a name of a node\nto be traced%\n\\index{traced}%\n{} and $s$ and $t$ are the start and stop times for this\ntrace respectively.\n\nOf course, in practice, it would be quite tedious to have to\nwrite all specifications in terms of lists of five-tuples.\nConsequently, a small language (actually a small set of useful functions)\nhas been defined in the library file \"verification.fl\".\nThese functions make it much easier to write specification.\nHowever, it should be remembered that when the verification\nis actually performed, all these higher level constructs gets translated\ndown to the two lists of five-tuples.\n\nFor more details on how to use the STE function, we refer the reader\nto Section~\\ref{tutorial}.\n", "meta": {"hexsha": "24ec134bb1252f526176c88ee4c53b9181852e92", "size": 49967, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/new_doc/03_fl.tex", "max_stars_repo_name": "mfernan2/VossII", "max_stars_repo_head_hexsha": "e1c1142c9c94255995c13efe2e1b7af4ae03f2e2", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/new_doc/03_fl.tex", "max_issues_repo_name": "mfernan2/VossII", "max_issues_repo_head_hexsha": "e1c1142c9c94255995c13efe2e1b7af4ae03f2e2", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/new_doc/03_fl.tex", "max_forks_repo_name": "mfernan2/VossII", "max_forks_repo_head_hexsha": "e1c1142c9c94255995c13efe2e1b7af4ae03f2e2", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 29.8667065152, "max_line_length": 80, "alphanum_fraction": 0.7087277603, "num_tokens": 14097, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6513548646660542, "lm_q1q2_score": 0.34347018272207214}} {"text": "\\documentclass[xcolor=dvipsnames]{beamer}\n\n\\usepackage[utf8]{inputenc}\n\\usepackage{default}\n\\usepackage{beamerthemesplit}\n\\usepackage{tikz}\n\\usetikzlibrary{positioning,shapes,arrows}\n\\usetheme{Berkeley}\n\\usecolortheme[named=Black]{structure}\n\n\\title[]{Floating Base Rigid Body Dynamics Estimation}\n\\author[]{Francesco Nori, Jorhabib Eljaik}\n\\date{\\today}\n\n\n\\begin{document}\n% \\maketitle\n\n\\begin{frame}\n \\titlepage\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Table of contents}\n \\tableofcontents\n\\end{frame}\n\n\\section{Introduction}\n\\begin{frame}\n \\frametitle{Sensor fusion on the iCub}\n\\begin{figure} \n \\centering \n\t \\includegraphics[height=0.55\\hsize]{images/png/FT.png} \n\t \\includegraphics[height=0.55\\hsize]{images/png/acc.png} \n\t \\includegraphics[height=0.55\\hsize]{images/png/gyroAcc.png} \n\\end{figure}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Sensor fusion on the iCub}\n\\begin{figure} \n \\centering \n\t \\includegraphics[height=0.55\\hsize]{images/pdf/rigidBody.pdf} \n\\end{figure}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Sensor fusion on the iCub}\n\\begin{figure} \n \\centering \n\t \\includegraphics[height=0.55\\hsize]{images/pdf/foot.pdf} \n\\end{figure}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Sensor fusion on the iCub}\n \\begin{eqnarray*}\nm \\dot v^B + \\omega^B \\times (m v^B) & = & f^B_1 + ... + f^B_n + mg^B \\\\\nI^B \\dot \\omega^B + \\omega^B \\times (I^B \\omega^B) & = & \\mu^B_1 + ... + \\mu^B_n\n\\end{eqnarray*}\n\\begin{itemize}\n\\item $I^B$ : inertia in the body reference frame\n\\item $m$ : mass of the rigid body\n\\item $f^B_i$ : i-th force expressed in the body reference frame\n\\item $\\mu^B_i$ : i-th torque expressed in the body reference frame\n\\item $\\omega^B$: angular velocity expressed in the body reference frame\n\\item $v^B$ : linear velocity in the body reference frame\n\\item $q$ : quaternion representing the rigid body orientation\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Sensor fusion on the iCub}\n \\begin{eqnarray*}\nm \\textcolor{red}{\\dot v^B} + \\omega^B \\times (m v^B) & = & \\textcolor{red}{f^B_1 + ... + f^B_n} + mg^B \\\\\nI^B \\dot \\omega^B + \\omega^B \\times (I^B \\omega^B) & = & \\textcolor{red}{\\mu^B_1 + ... + \\mu^B_n}\n\\end{eqnarray*}\n\\begin{itemize}\n\\item $I^B$ : inertia in the body reference frame\n\\item $m$ : mass of the rigid body\n\\item \\textcolor{red}{$f^B_i$} : i-th force expressed in the body reference frame\n\\item \\textcolor{red}{$\\mu^B_i$} : i-th torque expressed in the body reference frame\n\\item \\textcolor{green}{$\\omega^B$}: angular velocity expressed in the body reference frame\n\\item \\textcolor{red}{$\\dot v^B$} : linear acceleration in the body reference frame\n\\item $q$ : quaternion representing the rigid body orientation\n\\end{itemize}\n\\end{frame}\n\n\\section{Bayesian Network for a Floating Base Rigid Body System}\n\\begin{frame}\n \\frametitle{Bayesian Network for a Floating Base Rigid Body System}\n \n \\centering\n \\begin{tikzpicture}[\n node distance=1cm and 0cm,\n mynode/.style={draw,circle,text width=1cm,align=center}\n ]\n \\node[mynode] (x) {$x$};\n \\node[mynode,below=of x] (y) {$y$};\n \\path (x) edge[-latex] (y);\n \n \\node[right=of x, right=0.5cm of x]\n {\n $x = \\left[ v^B \\quad \\omega^B \\quad f_o \\quad \\mu_o \\quad f_c \\quad \\mu_c \\right]^T$ \n };\n \n \\node[right=of y, right=0.5cm of y]\n {\n $y = \\left[ f_o \\quad \\mu_o \\quad f_c \\quad \\mu_c \\quad \\dot{v}^B \\right]^T$\n };\n \\end{tikzpicture}\n\n\\end{frame}\n\n\n\\section{The Extended Kalman Filter}\n\\begin{frame}\n \\frametitle{The Extended Kalman Filter}\n Given a nonlinear system:\n \\begin{align}\n y_k &= h(x) + \\rho_m \\\\\n x_k &= f(x_{k-1}) + \\rho_p\n \\end{align}\n \n Where $y$, the measurement vector and $x$, the state vector have gaussian uncertainties.\n\n After linearizing our dynamical system about an estimate of the current state ${x}_{k|k-1}$, the corresponding Extended Kalman Filter can be obtained following a \\textbf{prediction} and \\textbf{update} steps.\n \\begin{alertblock}{Prediction}\n $\\mathbf{x}_{k|k-1} = f(\\mathbf{x}_{k-1|k-1})$\n \\end{alertblock}\n \n \\begin{alertblock}{\\textbf{Update}}\n $\\mathbf{x}_{k|k} = x_{k|k-1} + K_k(y_k - h(\\mathbf{x}_{k|k-1}))$\n \\end{alertblock}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{The Extended Kalman Filter}\n Linearized measurement model: $y = Cx + e$ where:\\\\\n y = \\begin{pmatrix}\n 0& 0& \\mathbf{I}& 0& 0& 0\\\\\n 0& 0& 0& \\mathbf{I}& 0& 0\\\\\n 0& 0& 0& 0& \\mathbf{I}& 0\\\\\n 0& 0& 0& 0& 0& \\mathbf{I}\\\\ \n S(\\bar{\\omega}_B)& -S(\\bar{V}_B)& \\mathbf{I}& 0& \\mathbf{I}& 0\n \\end{pmatrix} x \n+ \\begin{pmatrix}\n 0\\\\\n 0\\\\\n 0\\\\\n 0\\\\\n S(\\bar{V}^B)\\bar{\\omega}^B\n \\end{pmatrix}\n\\end{frame}\n\n\n\\begin{frame}\n \\frametitle{Extended Kalman Filter results}\n Linear Velocities smoothing\n\\begin{figure} \n \\centering \n\t \\includegraphics[height=0.70\\hsize]{images/png/linVel.png} \n\\end{figure}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Extended Kalman Filter results}\n Angular velocities smoothing\n\\begin{figure} \n \\centering \n\t \\includegraphics[height=0.70\\hsize]{images/png/angVel.png} \n\\end{figure}\n\\end{frame}\n\n\n% \\section{Update Step of the Kalman Filter as a Bayesian Network}\n% \\begin{frame}\n% \\frametitle{Update Step of the Kalman Filter as a Bayesian Network}\n% \\textbf{Update Step} $\\leftrightarrow $ \\textbf{Conditional probability of} $\\mathbf{x}_{k|k-1}$ \\textbf{given} $\\mathbf{y}_k$ \\\\\n% in other words, we can also obtain $\\mathbf{x}_{k|k}$ as:\n% \\begin{equation}\n% \\mathbf{x}_{k|k} = E[x_k \\; \\vert \\; y_k, x_{k|k-1}]\n% \\end{equation}\n% Where:\n% \\begin{align}\n% \\mathbf{y}_k \\vert_{x_{k|k-1}} &\\backsim \\mathcal{N}(C\\mathbf{x}, R) \\\\\n% \\mathbf{x} &\\backsim \\mathcal{N}(x_{k|k-1},P_{k|k-1})\n% \\end{align}\n% \n% And $\\mathbf{y}$ has been linearized as: $y=Cx + e$\n% \\end{frame}\n\n\n% \\section{Solving a Linear System with a Bayesian Network}\n% \\begin{frame}\n% \\frametitle{Solving a Linear System with a Bayesian Network}\n% Given a linear system $\\mathbf{Ax} = \\mathbf{b}$, the computation of the solution vector $\\mathbf{x^*}$ is identical to the inference of the vector of marginal means $\\mu = \\{\\mu_1, ... , \\mu_n\\}$ over the graph $\\mathcal{G}$ with the associated Joint Gaussian Probability Density Function\n% \n% \\begin{equation}\n% p(\\mathbf{x}) \\backsim \\mathcal{N}(\\mu \\triangleq \\mathbf{A}^{-1}\\mathbf{b}, \\mathbf{A}^{-1})\n% \\end{equation}\n% \n% The mean of this probability distribution is the target solution. The solution of the system implies inferring the marginal densities:\n% \n% \\begin{equation}\n% p(x_i) \\backsim \\mathcal{N}(\\mu_i = \\{\\mathbf{A^{-1}}\\mathbf{b}\\}_i,P^{-1}_i = \\{\\mathbf{A}^{-1}_{ii})\n% \\end{equation}\n% \n% Where $\\mu_i$ and $P_i$ are the marginal mean and inverse variance.\n% \n% % Reference: Gaussian Belief Propagation Solver for Systems of Linear Equations. Ori Shental 1 , Paul H. Siegel and Jack K. Wolf, Danny Bickson and Danny Dolev\n% \\end{frame}\n\n\\end{document}\n", "meta": {"hexsha": "e37dcd9cd0a08dc677d53a47cecfd05402033877", "size": 6900, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ICRA2015/VVV14Pres/EKFforDynamicalSystems.tex", "max_stars_repo_name": "jeljaik/extended-kalman-filter", "max_stars_repo_head_hexsha": "b8d2d510291bc67b503eb799f18fe459dfe6a4f7", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 35, "max_stars_repo_stars_event_min_datetime": "2015-04-10T13:04:27.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-17T09:42:04.000Z", "max_issues_repo_path": "ICRA2015/VVV14Pres/EKFforDynamicalSystems.tex", "max_issues_repo_name": "jeljaik/extended-kalman-filter", "max_issues_repo_head_hexsha": "b8d2d510291bc67b503eb799f18fe459dfe6a4f7", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 32, "max_issues_repo_issues_event_min_datetime": "2015-02-06T14:59:44.000Z", "max_issues_repo_issues_event_max_datetime": "2018-11-19T17:41:01.000Z", "max_forks_repo_path": "ICRA2015/VVV14Pres/EKFforDynamicalSystems.tex", "max_forks_repo_name": "jeljaik/extended-kalman-filter", "max_forks_repo_head_hexsha": "b8d2d510291bc67b503eb799f18fe459dfe6a4f7", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 36, "max_forks_repo_forks_event_min_datetime": "2015-09-09T10:18:47.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-25T09:51:19.000Z", "avg_line_length": 32.0930232558, "max_line_length": 293, "alphanum_fraction": 0.6695652174, "num_tokens": 2459, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795671, "lm_q2_score": 0.6513548511303336, "lm_q1q2_score": 0.3434701755844631}} {"text": "\\section{Debugging}%\n\\label{sec:debugging}%\n\\subsection{Net Weights}\n\\label{subsec:net_weights}\n%\nIn an attempt to better understand the source of this unexpected behavior,\nmultiplicative weights (referred to as `nw' for `network weights' in\nFig.~\\ref{fig:bad_convergence}) were introduced to scale the individual\ncontribution from each of the `learned' functions $S$, $T$, and $Q$,\nexplicitly:\n%\n\\begin{align}\n S &\\rightarrow \\alpha_{S}\\, S \\\\\n T &\\rightarrow \\alpha_{T}\\, T \\\\\n Q &\\rightarrow \\alpha_{Q}\\, Q.\n\\end{align}\n%\nBy varying each of these weights individually allows us to selectively `tune'\nhow much each of these functions contribute when running inference on the\ntrained model.\n%\nNote that in the limit $\\alpha_{S}, \\alpha_{T}, \\alpha_{Q} \\rightarrow 0$, we\nrecover generic HMC, and as can be seen in Fig.~\\ref{fig:bad_convergence},\nthe error in the average plaquette $\\delta_{\\phi_{P}} \\simeq 0$, as expected.\n%\nAs an additional sanity check, we looked at how the error in the average\nplaquette behaves for different values of the weights $\\alpha_{i}$ ($i = S, T,\nQ$).\n%\nExplicitly, beginning with $\\vec{\\alpha} \\equiv [\\alpha_{S}, \\alpha_{Q},\n\\alpha_{T}] = [0, 0, 0]$, we increase each of the weights one by one and compute\nthe average value of the plaquette difference.\n%\nFor example, the blue line (Transformation $(Q)$ function) in\nFig.~\\ref{fig:plaq_diff_vs_net_weights} was obtained by keeping both\n$\\alpha_{S}$ and $\\alpha_{T}$ fixed and $0$ and varying $\\alpha_{Q} \\in\n[0.1, 0.25, 0.5, 0.75, 1.0, 1.5, 2.0, 5.0]$, and similarly for $\\alpha_{S}$ and\n$\\alpha_{Q}$.\n%\n\\begin{figure}\n \\centering\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\caption{$N_{LF} = 10$}\n \\includegraphics[width=\\textwidth]{new_figures/plaq_error/plaq_diff_vs_net_weights_lf10.pdf}\n \\end{subfigure}\n % \\vspace{2pt}\n % \\hfill\n % \\centering\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\caption{$N_{LF} = 12$}\n \\includegraphics[width=\\textwidth]{new_figures/plaq_error/plaq_diff_vs_net_weights_lf12.pdf}\n \\end{subfigure}\n % \\vspace{2pt}\n \\begin{subfigure}[b]{0.48\\textwidth}\n \\caption{$N_{LF} = 16$}\n \\includegraphics[width=\\textwidth]{new_figures/plaq_error/plaq_diff_vs_net_weights_lf16.pdf}\n \\end{subfigure}\n \\caption{Plaquette difference $\\delta_{\\phi_{P}}$ for different values of the\n net weights $\\vec{\\alpha} \\equiv [\\alpha_{S}, \\alpha_{T}, \\alpha_{Q}]$.\n Note that $\\delta_{\\phi_{P}} \\rightarrow 0$ as $\\vec{\\alpha} \\rightarrow\n [0, 0, 0]$, as expected.}%\n\\label{fig:plaq_diff_vs_net_weights}\n\\end{figure}\n%\nThese results seem to indicate that each of the individual functions contribute\nseparately to the error, with the scaling ($S$) and translation ($T$) functions\nhaving the largest effect.\n% }}}\n\\subsection{Updates (11/11/2019)}\n\\begin{todolist}\n \\item[\\done] Ensure reproducibility across training/inference runs when\n using same input parameters.\n \\begin{itemize}\n \\item Essential for debugging the bias in the average plaquette.\n \\item Somewhat tricky problem due to the fact that tensorflow \n has two distinct methods of specifying a seed: graph-level and\n operation-level. Additionally, when using horovod for distributed\n training across multiple ranks, we must ensure that each rank gets a\n different seed otherwise they will all be training identical copies of\n the model.\n \\end{itemize}\n\n \\item[\\done] Implement reversibility checker that ensures that the L2HMC\n dynamics (i.e.\\ the augmented HMC sampler) is reversible.\n \\begin{itemize}\n \\item Starting with $\\xi = (x, v, d)$, run the dynamics in the forward\n direction to get $\\xi^{\\prime}$. If $\\mathbf{L}_{\\theta}\n \\xi = \\xi^{\\prime}$, and $\\mathbf{F} \\xi = \\mathbf{F} (x, v, d) = (x,\n v, -d)$, then a complete (invertible) update step can be written as\n $\\mathbf{FL}_{\\theta}\\xi = \\xi^{\\prime}$.\n %\n \\item If our sampler is reversible, running the dynamics backwards on\n $\\xi^{\\prime}$ should return us to the original state $\\xi$. \n %\n \\begin{equation}\n \\mathbf{FL}_{\\theta}\\mathbf{FL}_{\\theta} \\xi = \\xi\n \\end{equation}\n \\end{itemize}\n\n \\item[\\done] Try increasing floating point precision (\\texttt{tf.float32 -->\n tf.float64}) (\\textbf{\\textcolor{red}{Error still present.}}) \n\n \\item[\\done] Try anti-symmetric Gaussian Mixture Model and see if the trained\n model is an accurate representation of the target distribution (e.g.\\ by\n looking at the locations of the means).\n % \\href{https://l2hmc.slack.com/archives/CK0SMC6NS/p1567623563026900}{(link\n % to post)}.\n\n \\item[\\done] At James' suggestion, we decided to look at the\n kinetic/potential energies and the Hamiltonian at the beginning and end of\n each trajectory (\\textbf{\\textcolor{red}{Solved! Issue was being caused by\n unexpected resampling of the momentum}}).\n % Fig~\\ref{fig:potential_energy},~\\ref{fig:kinetic_energy},~\\ref{fig:hamiltonian}).\n \\end{todolist}\n\\section{GMM Results}%\n\\label{sec:gmm_results}\n\\begin{figure}[htpb] \n \\centering \n \\includegraphics[width=0.5\\textwidth]{updates/single_chain}\n \\caption{Inference run shown for a single chain using the L2HMC\n sampler trained on this gaussian mixture model.}%\n \\label{fig:gmm_single_chain}\n\\end{figure}\n\n\\begin{figure}[htpb] \n \\centering \n \\begin{subfigure}[t]{0.48\\textwidth}\n \\caption{L2HMC samples}\n \\includegraphics[width=\\textwidth]{updates/means_hist_observed}\n \\end{subfigure}\n % \\hfill\n \\begin{subfigure}[t]{0.48\\textwidth}\n \\caption{true target distribution}\n \\includegraphics[width=\\textwidth]{updates/means_hist_true}\n \\end{subfigure}\n \\caption{Histograms of $\\langle x\\rangle$ and $\\langle y\\rangle$ in this\n two-dimensional target space.}\n\\end{figure}\n% }}}\n", "meta": {"hexsha": "9812510b831096aead1754c5194e8ad1de051bb0", "size": 5801, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/debugging/debugging.tex", "max_stars_repo_name": "saforem2/l2hmc-qcd", "max_stars_repo_head_hexsha": "b5fe06243fae663607b6c88e71373b68b19558fc", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 32, "max_stars_repo_stars_event_min_datetime": "2019-04-18T18:50:28.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T18:30:48.000Z", "max_issues_repo_path": "doc/debugging/debugging.tex", "max_issues_repo_name": "saforem2/l2hmc-qcd", "max_issues_repo_head_hexsha": "b5fe06243fae663607b6c88e71373b68b19558fc", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 21, "max_issues_repo_issues_event_min_datetime": "2019-09-09T21:10:48.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-26T17:43:51.000Z", "max_forks_repo_path": "doc/debugging/debugging.tex", "max_forks_repo_name": "saforem2/l2hmc-qcd", "max_forks_repo_head_hexsha": "b5fe06243fae663607b6c88e71373b68b19558fc", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2020-10-31T02:25:04.000Z", "max_forks_repo_forks_event_max_datetime": "2021-05-25T00:49:14.000Z", "avg_line_length": 41.4357142857, "max_line_length": 96, "alphanum_fraction": 0.7021203241, "num_tokens": 1737, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160666, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34342015362247647}} {"text": "\\chapter{Long-term surface water change detection}\n\\label{ch5}\n\n\\begin{abstract}\n\tMethods to detect long-term surface water changes are discussed. A simple method based on reflectance percentile composites and linear regression is presented. As an example, Siling Lake in the Tibetan Plateau is used as a study area. The advantages and drawbacks of the method are discussed.\n\t\n\t\\begin{center}\n\t\t\\begin{tikzpicture}[every node/.style={inner sep=0,outer sep=0}]\n\t\t\\node[draw=none,shade,blur shadow={shadow blur steps=5}] {\n\t\t\t\\includegraphics[width=3in]{01.4-change-detection/figures/change-detection-dubai}\n\t\t}; \n\t\t\\end{tikzpicture}\n\t\\end{center}\n\t\n\t\\textbf{Keywords:} $M_3$, reflectance percentiles, NDWI, linear regression, long-term surface water changes.\n\t\n\\end{abstract}\n\n{\n\t\\setlength{\\parindent}{0pt}This chapter is based on \\bibentry{donchyts2016earth}.\n}\n\n\n%% Start the actual chapter on a new page.\n\\newpage\n\n\\section{Introduction}\n\\dropcap{A}{s} we have seen in the previous three chapters, accurate surface water detection is a complex task and may require many steps. However, different land use changes may occur at different temporal scales. If we are interested only in long-term (inter-annual) land-use changes, more generic statistical methods can be applied to detect these changes. With sufficient observations available and under the assumption that statistics of reflectance values for different years remain the same, we can use relatively simple methods, when compared to those discussed in the Chapter \\ref{ch4}. \n\nThe first step that should be done when analyzing optical passive sensor satellite imagery is the elimination of clouds and cloud shadow effects. Cleaning images from cloud, cloud shadows, and snow can be a challenging and time-consuming process, especially when images partially covered by clouds need to be considered. Many methods were developed to do this properly, as has been discussed in chapters \\ref{ch2}, \\ref{ch3}, and \\ref{ch4}. These methods can be used to construct completely cloud-free images, or to extract information required to analyze temporal changes in different land cover classes. \n\nAs an alternative to this, we can generate inter-annual composites directly from the top-of-atmosphere or surface reflectance values. In recent years, this approach was used in \\citet{Hansen2013} to study forest cover changes by employing a combination of metrics, such as: \n\\begin{enumerate*}[label=(\\emph{\\alph*})]\n\t\\item selected percentile values\n\t\\item mean reflectance values for selected percentile ranges\n\t\\item slope of linear regression of band reflectance values versus image date\n\\end{enumerate*}. Furthermore, they make use of supervised classification methods based on decision trees to relate these metrics to homogeneously varying subsets of data.\n\nHere, a similar approach to estimate long-term surface water changes is introduced. However, instead of developing multiple metrics like in \\citet{Hansen2013}, a single combined metric employing percentiles is used to estimate average cloud-free reflectance values and the slope of linear regression to identify pixels where long-term changes have been occurring. \n\nThe resulting method is also more resource efficient because applying linear regression after averaging is cheaper to compute. The cost of applying linear regression directly to reflectance values is $O(n^3)$, and the cost to calculate the average value for every pixel is $O(n)$, where n is the number of analyzed images. The $O$ notation is frequently used in computer science to indicate scalability of the algorithm with the growth of the dataset. By employing a two-step approach, surface water changes can be estimated in almost $O(n)$ time, making it attractive to be applied across multiple sensors, assuming similarity in reflectance values among different satellite sensors for some spectral bands. For surface water change studies, the most used spectral index is \\gls{NDWI}. The index uses near-infrared and green bands. These bands have similar spatial and spectral resolution across multiple Landsat missions and seem to be in agreement with regards to spectral response and spectral resolution, except for Landsat 8. For Landsat 8, the OLI sensor covers the thinner spectral range for the near-infrared band when compared to TM and ETM+ sensors used by Landsat 4, 5, and 7 \\citet{roy2016characterization, angal2014cross}. A detailed comparison of spectral responses between Landsat 5 and 7 can be found in \\citet{teillet2001radiometric} and between Landsat 7 and 8 in \\citet{flood2014continuity}. Additionally, the literature says that data produced from Landsat was cross-calibrated to ensure consistency and continuity of values between different missions \\citet{mishra2014radiometric}. This makes combining images across different Landsat sensors, and the use of statistical methods to process their values, very promising.\n\n\\begin{figure}[H]\n\t\\includegraphics[width=1.0\\textwidth,left]{01.4-change-detection/figures/aral}\n\t\\caption{Aral Sea, Uzbekistan. False-color reflectance percentile composite (swir1, nir, green, 20\\%). The red circle indicates the location where reflectance and \\gls{NDWI} values shown on charts were plotted. The composite images and the time series were computed from 4325 Landsat scenes between 1984 and 2017. \\opt{source}{Source: \\url{https://code.earthengine.google.com/c1ccc09c6b5e7d73bd3a3601f8edfaf3}}}\n\t\\label{fig:ch4-aral-sea}\n\\end{figure}\n\nSurface water is very dynamic and may change on a daily, monthly, and annual basis. To estimate long-term, inter-annual surface water changes from raw satellite images, we have to exclude effects caused by both clouds and by intra-annual or seasonal variations, assuming these changes remain the same across years. Examples of intra-annual changes can be seasonal changes in surface water due to natural processes, a regular water cycle, changes of surface water levels caused by tidal effects in coastal areas, and variability in surface water levels due to man-made activities, such as the operation of reservoirs, which may also cause surface water changes in the downstream rivers. Long-term surface water changes may be due to natural changes of water boundaries, such as erosion or accretion, natural changes caused by climate changes, or manmade changes in rivers or coastal zones (embankments, land reclamation, construction of reservoirs). \n\nIn reality, a combination of these surface water changes may occur and interfere with each other, making the task of surface water change detection extremely difficult.\n\nTo demonstrate the variability of reflectance values in different bands for an area where both long and short-term changes took place, a long-term time series in the middle of the Aral Sea is obtained for pixels which have dried up during the past few decades, as shown in Figure \\ref{fig:ch4-aral-sea}. This area has faced dramatic changes regarding surface water and, as can be seen from the reflectance values, this has influenced both reflectance and corresponding \\gls{NDWI} value changes. While it would be possible to detect the trend of \\gls{NDWI} values from the raw reflectance value time series, doing this may be less efficient, mainly because \\gls{NDWI} values for many pixels corresponding to clouds may look very similar to the values corresponding to water. Additionally, it would require significantly more computing resources.\n\n\\section{Long-term surface water change detection}\n\nTo address long-term surface water changes, we can first try to exclude short-term variability of reflectance values by computing average images for a long time intervals, excluding effects of clouds and cloud shadows in this way. By estimating this type of composite images for multiple time intervals, followed by the application of spectral water indices, such as \\gls{NDWI}, we should be able to reconstruct long-term variability of the surface water changes, which will be reflected in the variability of the \\gls{NDWI} values. One of the important questions arising using this method is how to select the best percentile, allowing to minimize the effect of clouds and cloud shadows, and at the same time, avoiding the appearance of cloud shadows in the images. \n\nThe method will work only if a sufficient number of observations is available. A large number of images is required to ensure that the distribution of values, used to compute reflectance percentile composites, converges to the actual distribution representing different land or atmosphere values for a given location is based on similar distributions. Ideally, the resulting percentile should look the same for two areas when no long-term surface water changes take place. \n\nBecause surface water has a very distinctive spectral signature, with very low reflectance values in almost all bands, the presence or absence of water in a given pixel will be reflected in the final distribution represented as a probability distribution function (PDF) or a cumulative distribution function (CDF) for a given location.\n\n\\afterpage{%\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width=1.0\\textwidth,left]{01.4-change-detection/figures/Tibet-lake-changes}\n\t\t\\caption{Siling Lake, Tibetan Plateau, China. Two reflectance composite images at the top-left show how the lake looked like in 80's (left) and how it looks like today (right). The blue color indicates submerged land due to climate changes. \\opt{source}{Source: \\url{https://code.earthengine.google.com/87d0057d8b3d28817b1da7cfc1bd6b6e}}}\n\t\t\\label{fig:ch4-tibet-lake}\n\t\\end{figure}\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width=1.0\\textwidth,left]{01.4-change-detection/figures/Tibet-time-series}\n\t\t\\caption{Estimating surface Siling Lake, Tibetan Plateau, China. The actual TOA reflectance values (left-top), annually-averaged (20\\%) reflectance values (left-middle) and \\gls{NDWI} values, estimated from annually-averaged reflectance values (left-bottom). The three charts on the left indicate values and distributions of the sun parameters for the images used to compute these values.}\n\t\t\\label{fig:ch4-tibet-lake-time-series}\n\t\\end{figure}\n\t\n\t\\clearpage\n}\n\nTo demonstrate the main steps used in the methods, surface water changes which have occurred around the Siling Lake are analyzed - one of the lakes of Tibetan Plateau, China. In the last few decades, an enormous area of new surface water has been created due to climate changes. Figure \\ref{fig:ch4-tibet-lake} shows the final estimate for the submerged land, equal to $22.5 km^2$. \n\nA straightforward and efficient way to estimate long-term surface water changes may be to use a two-step approach. First, the intensity percentile composites are computed for all temporal intervals $T$ used for the change detection analysis:\n\n\\begin{equation}\nF(\\rho) = \\int_{-\\infty}^{\\rho^i} P(\\rho)d\\rho\n\\end{equation}\n\nwhere $F$ - cumulative distribution function, the superscript index $i$ denotes the $i^{th}$ percentile, $\\rho$ - reflectance values for a given band.\n\nThen, for every percentile reflectance value $\\rho^i$, the spectral index is computed as an image:\n\n\\begin{equation}\nI = \\frac{\\rho_{green}^i-\\rho_{nir}^i}{\\rho_{green}^i+\\rho_{nir}^i} \\\\\n\\end{equation}\n\nAfter that, the linear regression is performed on a set of all index values for all intervals $T$:\n\n\\begin{equation}\nI = \\beta_0 + \\beta_1 T + \\epsilon\n\\end{equation}\n\nThe resulting slope of the linear regression $\\beta_1$ is then analyzed to identify pixels, where significant surface water changes took place. \n\nUsually, the percentile is determined empirically and corresponds to low reflectance values, but not too low, to avoid confusing water with cloud shadows, which can also be very dark for green and nir (or swir) bands used to estimate spectral indices. For those pixels, where the slope $\\beta_1$ of the linear regression is significant. The spectral index values $I$ are also tested to exclude false positive changes, detected for images where most images correspond to water or land:\n\n\\begin{equation}\n\\begin{split}\n\\begin{aligned}\nI &= min\\left(I, I_{min}^{water}\\right) \\\\\nI &= max\\left(I, I_{max}^{land}\\right) \n\\end{aligned}\n\\end{split}\n\\label{eq:water-land-suppression}\n\\end{equation}\n\nwhere $I_{min}^{water}$ and $I_{max}^{land}$ indicate minimum and maximum values of the spectral index to be considered as water and land correspondingly. The equation \\ref{eq:water-land-suppression} is applied for every interval used during temporal averaging.\n\nThe first equation of the \\ref{eq:water-land-suppression} allows to filter pixels, where the slope of the regression is significant, with all values still belonging to water. The second equation of the \\ref{eq:water-land-suppression} does the same to eliminate locations where all of the temporal intensity values belong to the land.\n\nAdditionally, \\gls{NDVI} vegetation index is used similarly to remove locations, where, for example, deforestation took place. In this case, \\gls{NDWI} values may also change significantly in time, but the actual changes do not correspond to surface water changes.\n\n\\afterpage{%\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width=1.0\\textwidth,left]{01.4-change-detection/figures/images_real}\n\t\t\\caption{Examples of false-color (swir1, nir, green) images from 1998 acquired by Landsat TM sensor for Siling Lake, Tibetan Plateau, China. \\opt{source}{Source: \\url{https://code.earthengine.google.com/882ab2929f536676d56bc79cbc895ded}}\n\t\t}\n\t\t\\label{fig:ch4-tibet-lake-false}\n\t\\end{figure}\n\t\n\t\\begin{figure}\n\t\t\\includegraphics[width=1.0\\textwidth,left]{01.4-change-detection/figures/images_percentiles}\n\t\t\\caption{False-color (swir1, nir, green) reflectance percentile images estimated from 751 images acquired by multiple NASA Landsat missions during 1984-2017, Siling Lake, Tibetan Plateau, China}\n\t\t\\label{fig:ch4-tibet-lake-percentiles}\n\t\\end{figure}\n\t\\clearpage\n}\n\nThe actual number of images available need to be adjusted based on the cloud frequency present for a given area. For the area around this lake, the average cloud frequency is 13\\%, with relatively low intra-annual variation, as estimated by \\citet{wilson2016remotely} from MODIS images. However, for some areas on Earth, mainly near the equator, 95\\% of the images may be covered by clouds. This means, that to apply this method, many more images need to be analyzed. In this case, only long-term surface water changes may be estimated using passive optical sensors. \n\n% t-test using regression \\url{https://en.wikipedia.org/wiki/Student%27s_t-test#Slope_of_a_regression_line}\n% non-parametric tests \\url{ftp://cran.r-project.org/pub/R/web/packages/trend/vignettes/trend.pdf}\n\n% CDF sun parameters mismatch, topographic errors \\url{https://code.earthengine.google.com/c9ed4d65ee90db7d4944e4c257969e1b}\n% \\url{https://code.earthengine.google.com/5ddaeea0946642afe23d7557c3c0c657}\n\n\\section{Topographic noise for inconsistent image collections}\nEven though the method described in Chapter \\ref{ch5} allows detection of surface water changes at high accuracy, a substantial commission error may occur in mountain regions. These errors are caused mainly by combined errors of elevation and azimuth, causing differences in hill shadows in composite images, used to detect surface water changes. For global analysis, most of these errors were eliminated by masking-out the final surface water changes images with the topographic index derived from \\gls{HAND}. The topographic mask, constructed by HAND < 40m, was used to filter out these errors. The \\gls{HAND} dataset used for the processing was generated from 30m \\gls{SRTM} and some auxiliary 90m datasets to cover areas where it was not acquired (>60 degree north latitude) \\citet{donchyts2016hand}. However, for very accurate surface water change estimates this may also result in omission errors in the areas when DEM values are incorrect. For the time being, no free high-resolution DEM exists to be used as a topographic index (via \\gls{HAND}). Therefore, the Aqua Monitor website does not perform this correction during on-the-fly estimation of surface water changes. This may change in future versions, when a more appropriate method is introduced.\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=1\\textwidth]{01.8-aqua-monitor/figures/problem1}\n\t\\caption{False-positive composites (swir1, nir, green) and surface water changes commission errors due to inconsistent sun parameters in the samples used to compute changes in the hilly areas (2000-2015). Top: upstream of the Lake Mead (\\href{http://aqua-monitor.appspot.com/?from=2000&to=2013&view=36.103119715292486,-113.93795967102045,13z&max_doy=365}{36.10, -113.90}); bottom: river near Yogongxiang, China (\\href{http://aqua-monitor.appspot.com/?mode=dynamic&from=2000&to=2013&view=30.269128374988604,94.89677429199217,12z&max_doy=365&averaging_months1=36&averaging_months2=36}{30.27, 94.90}).}\n\t\\label{fig:am-topographic-errors}\n\\end{figure}\n\nExamples of these errors are demonstrated in Figure \\ref{fig:am-topographic-errors}. \n\nThe same errors were reported in the \\citet{Hansen2013} dataset, which uses similar methods to estimate surface water changes (interval mean percentile composites versus percentile composites in Aqua Monitor).\n\nA time consuming approach to eliminate these errors could be to perform topographic correction on every image before passing them to the change detection algorithm. However, this may significantly increase the amount of resources needed, making this type of analysis hardly achievable today.\n\n\n\\section{Conclusions and Discussion}\n\nThe method presented in this chapter showed surprisingly good performance for most of the places on Earth and was used to analyze global surface water changes over the last thirty years. Despite its simplicity, the method can very accurately detect long-term surface changes. Additionally, it is easy to implement using the Google Earth Engine parallel processing platform.\n\nAnother advantage of the method is that it is less resource intensive when compared to a more streamlined approach to detecting surface water changes. Hence it can be used for large planetary-scale studies and for a wide range of users, which is demonstrated in Chapter \\ref{ch6}. For example, all images are first classified e.g. using supervised machine learning methods, and then the changes are computed from the resulting thematic maps.\n\nHowever, it may be difficult to reliably detect surface water changes when:\n\n\\begin{enumerate}[label=(\\alph*)]\n\t\\item \\label{enum:changes-low-count} the number of images used to compute percentiles is low\n\t\\item \\label{enum:changes-complex-trend} surface water changes follow a more complex pattern than a one directional trend\n\t\\item \\label{enum:changes-multple-classes} multiple land-use changes are present, represented by low reflectance values but irrelevant to surface water changes\n\t\\item \\label{enum:changes-topo} complex topographic conditions combined with unequally distributed sun parameters in different time intervals\n\\end{enumerate}\n\nSome of these issues can be easily addressed, for example, by changing empirically chosen percentiles to ensure one captures pixels free from cloud and snow effects.\n\nAdditionally, image samples used to compute percentiles can also be selected to only represent certain seasons, where some of the effects are less present, filtering image collections by day-of-year or by sun parameters, to avoid the most evident topographic effects. \n\nThe methods presented here work best when reflectance values, representing complex land-use and atmospheric changes, are similarly distributed. While for surface water this is frequently true, it may be less trivial under very complex land-use change \\ref{enum:changes-multple-classes}. To detect these kind of changes automatically, additional inference steps may be required, for example, where different patterns can be recognized in the distributions such as those presented in Chapter \\ref{ch4}, Figure \\ref{fig:prob-sampled-water-cloud-snow}.\n\nWhile linear regression may be a good choice to detect monotonic long-term changes, surface water may follow much more complex trends, requiring the use of more complex methods to capture them. For example, logistic regression can be used instead of linear regression to detect abrupt changes such as construction or decommissioning of reservoirs, or to analyze autocorrelation to detect recurring surface water changes.\n\nWith the increasing number of satellite images available, the use of statistical methods may be a powerful instrument, providing an easy way for change or anomaly detection before the use of more complex methods.\n\n\n", "meta": {"hexsha": "a289d442fc57093e16aaeef9862e012c538b5415", "size": 20662, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "01.4-change-detection/change-detection.tex", "max_stars_repo_name": "songyingxu/thesis", "max_stars_repo_head_hexsha": "b4f027f6d5362c61a2ca5b76d99b6563466610cd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 18, "max_stars_repo_stars_event_min_datetime": "2018-06-22T20:41:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-19T11:57:16.000Z", "max_issues_repo_path": "01.4-change-detection/change-detection.tex", "max_issues_repo_name": "Yorklii/thesis", "max_issues_repo_head_hexsha": "b4f027f6d5362c61a2ca5b76d99b6563466610cd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "01.4-change-detection/change-detection.tex", "max_forks_repo_name": "Yorklii/thesis", "max_forks_repo_head_hexsha": "b4f027f6d5362c61a2ca5b76d99b6563466610cd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 11, "max_forks_repo_forks_event_min_datetime": "2019-05-02T05:50:04.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-13T14:45:59.000Z", "avg_line_length": 108.7473684211, "max_line_length": 1741, "alphanum_fraction": 0.8010841158, "num_tokens": 4725, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878414043814, "lm_q2_score": 0.5583269943353744, "lm_q1q2_score": 0.34342014574354174}} {"text": "\\section{Essential ML ideas}\n\\smallskip \\hrule height 2pt \\smallskip\n\n\\begin{itemize}\n\t\\item Never ever \\underline{ever} touch the test set\n\t\\item You know you are overfitting when there is a big test between train and test results. E.g. metric of percent wrong. \n\t\\item Need to be comfortable taking a hit on fitting accuracy if you can get a benefit on the result.\n\t\\item Bias vs variance trade-off. \n\t\tHigh bias when the model is too simple \\& doesn't fit the data well. \n\t\tHigh variance is when small changes to the data set lead to large solution changes. \n\t\\item If features are non discriminative in the beginning, they don't work for any classifier. % week 4 reminder\n\t\\item Your feature vector often has a smaller dimension that the feature space. % week 4, Friday. \n\t\tIf you have too long of a feature vector, you may get overfitting. \n\t\\item You need to prevent the optimizer from getting an easy way out. % week 7 audio\n\t\\item Whenever we are building a discriminative classifier, \n \t\twe should not expect it to reason about things it has never seen before\n \t\tYou can't classify on something that has only seen + points. \n \t\tThe classifier needs to see some + and some - or it will fail miserably. % week 8 audio\n\t\\item We should always avoid making hard decisions early. % week 8 audo. \n\t\t E.g. don't put a lot of trust in classifiers early on in bagging. \n\\end{itemize}\n\nYou can do $l_2$ normalization for a feature vector to get a unit vector: % week 6 audio. \n\tConvert $x$ to $\\hat{x}$ so that if you form $||\\hat{x}||_2^2 = 1$\n\tCan also do $l_1$ \n", "meta": {"hexsha": "610c53efe0d1a20057485fd557a215f391a518f2", "size": 1591, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/essential_ideas.tex", "max_stars_repo_name": "JanetMatsen/Machine-Learning", "max_stars_repo_head_hexsha": "12e1f701eb7de89b97d5caffe86b0267731e4cb5", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 25, "max_stars_repo_stars_event_min_datetime": "2016-02-07T23:35:57.000Z", "max_stars_repo_stars_event_max_datetime": "2021-01-26T05:13:33.000Z", "max_issues_repo_path": "tex/essential_ideas.tex", "max_issues_repo_name": "JanetMatsen/Machine-Learning", "max_issues_repo_head_hexsha": "12e1f701eb7de89b97d5caffe86b0267731e4cb5", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/essential_ideas.tex", "max_forks_repo_name": "JanetMatsen/Machine-Learning", "max_forks_repo_head_hexsha": "12e1f701eb7de89b97d5caffe86b0267731e4cb5", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 9, "max_forks_repo_forks_event_min_datetime": "2016-08-29T00:15:49.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-06T22:36:19.000Z", "avg_line_length": 61.1923076923, "max_line_length": 124, "alphanum_fraction": 0.7297297297, "num_tokens": 427, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982647, "lm_q2_score": 0.6370307806984444, "lm_q1q2_score": 0.34334891129283307}} {"text": "In this section, we discuss dynamic steady-state power and temperature analysis\nunder process variation. Unlike transient analysis, this analysis cannot be done\none time step at a time, since the repetitive workload needs to be taken into\naccount at once in order to calculate the corresponding dynamic steady state.\nThus, similarly to the contrast between \\sref{transient-analysis} and\n\\sref{dynamic-steady-analysis}, the solutions in \\sref{chaos-transient-analysis}\nand this section differ. However, they still rely on the same methodology\noutlined in \\sref{chaos-uncertainty-analysis} and shown in\n\\fref{chaos-overview}.\n\n\\subsection{\\problemtitle}\n\nThe system model is the same as the one in \\sref{chaos-transient-analysis}. The\nonly difference is that it is more convenient to define the power model as\nfollows:\n\\begin{equation} \\elab{chaos-power-model-bulk}\n \\mp = \\f(\\vu, \\mq).\n\\end{equation}\nThe function $\\f: \\real^\\nu \\times \\real^{\\np \\times \\ns} \\to \\real^{\\np \\times\n\\ns}$ is supposed to return the periodic power profile that corresponds to the\nperiodic workload being analyzed.\n\n\\inputalgorithm{chaos-dynamic-steady-solution-iterative}\nThe solution to probabilistic dynamic steady-state analysis is based on the\ndeterministic one presented in \\sref{dynamic-steady-solution}. The power,\ntemperature, and other vectors that appear in \\sref{dynamic-steady-solution}\nbecome stochastic in the present context, which also concerns the boundary\ncondition given in \\eref{dynamic-steady-condition}. The pseudocode for a\nprocedure that delivers \\mq for a fixed \\vu is listed in\n\\aref{dream-dynamic-steady-solution}. The algorithm does not take account of the\ninterdependence between power and temperature; however, this interdependence can\nbe addressed via one of the techniques presented in \\sref{power-temperature}. In\nthis section, we use the iterative approach illustrated in\n\\aref{dream-dynamic-steady-solution-iterative}. For clarity, this algorithm is\nrewritten here as shown in \\aref{chaos-dynamic-steady-solution-iterative}; the\nmain difference is that the calculation of power on line~3 is now based on\n\\eref{chaos-power-model-bulk}.\n\n\\begin{remark}\nThe application of the linear approximation described in\n\\sref{power-temperature} is problematic in this case. This technique is suitable\nwhen the only varying parameter is temperature, and all other parameters have\nnominal values. In that case, it is relatively easy to decide on a\nrepresentative temperature range and apply a curve-fitting procedure. In this\ncase, however, the power model has multiple parameters that range far from their\nnominal values.\n\\end{remark}\n\nTo recapitulate, the quantity of interest \\g in \\fref{chaos-overview} is the\ndynamic steady-state power and temperature profiles, which are denoted by \\mp\nand \\mq, respectively. The calculation of this quantity is shown in\n\\aref{chaos-dynamic-steady-solution-iterative}.\n\n\\subsection{Surrogate Construction}\n\nAt Stage~3 in \\fref{chaos-overview}, the procedure delineated in\n\\sref{chaos-construction} is applied to\n\\aref{chaos-dynamic-steady-solution-iterative} from Stage~1 with respect to the\noutput of Stage~2. Concretely, \\aref{chaos-dynamic-steady-solution-iterative} is\nutilized inside \\aref{chaos-construction} via Algorithm~G. In this case,\nAlgorithm~G calls \\aref{chaos-dynamic-steady-solution-iterative} and returns \\mp\nor \\mq or both, depending on what is actually needed for the subsequent\ncalculations, in a suitable format.\n\nSuppose, for instance, that the designer is interested in analyzing solely the\ndynamic steady-state temperature profile \\mq. Following\n\\rref{chaos-multidimensional-output} concerning vector-valued quantities, \\g is\ntreated as an $\\np \\ns$-element row vector, in which case each coefficient\n$\\hat{\\g}_{\\vi}$ in \\eref{chaos-expansion} is also such a vector. The projection\nmatrix, which is defined in \\eref{chaos-projection-matrix}, and\n\\aref{chaos-construction} should be reinterpreted accordingly: \\vg is an $\\nq\n\\times \\np \\ns$ matrix whose row~$j$ is \\mq computed at point~$j$ of the\nquadrature in \\eref{chaos-coefficient} and reshaped into a row vector.\nSimilarly, $\\hat{\\vg}$ should be understood as an $\\nc \\times \\np \\ns$ matrix\nwhose row~$i$ is coefficient~$i$ of the expansion in \\eref{chaos-expansion}.\nRecall that a certain ordering is assumed to be imposed on quadrature points and\npolynomial terms.\n\nThe constructed \\ac{PC} expansion can now be post-processed as required, which\nis already the topic of Stage~4 in \\fref{chaos-overview}; see\n\\sref{chaos-processing}.\n", "meta": {"hexsha": "54b1e7d6be8202d91625f443d91fcd938f669bb4", "size": 4554, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "include/uncertainty/process/development/dynamic-steady-analysis.tex", "max_stars_repo_name": "IvanUkhov/thesis", "max_stars_repo_head_hexsha": "95a7e2ee7664b94156906322610555e36e53cfe0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "include/uncertainty/process/development/dynamic-steady-analysis.tex", "max_issues_repo_name": "IvanUkhov/thesis", "max_issues_repo_head_hexsha": "95a7e2ee7664b94156906322610555e36e53cfe0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "include/uncertainty/process/development/dynamic-steady-analysis.tex", "max_forks_repo_name": "IvanUkhov/thesis", "max_forks_repo_head_hexsha": "95a7e2ee7664b94156906322610555e36e53cfe0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 55.5365853659, "max_line_length": 80, "alphanum_fraction": 0.7940272288, "num_tokens": 1165, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6757646010190476, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3431612818573753}} {"text": "\\documentclass[11pt]{article}\n\\usepackage{geometry} \n\\geometry{a4paper,left=2.5cm,right=2.5cm,top=2.5cm,bottom=2.5cm}\n\\usepackage{natbib}\n\\usepackage{color}\n\\definecolor{mygreen}{RGB}{28,172,0} % color values Red, Green, Blue\n\\definecolor{mylilas}{RGB}{170,55,241}\n\\usepackage{epsfig}\n\\usepackage{amssymb,amsmath}\n\\usepackage{enumerate}\n\\usepackage{enumitem}\n\\usepackage[utf8]{inputenc}\n\\usepackage{hyperref}\n\\usepackage{mathtools}\n\n\n\\newcommand{\\ssd}{\\text{ssd}}\n\\newcommand{\\sS}{\\mathsf{S}}\n\\newcommand{\\tot}{\\text{tot}}\n\n\\begin{document}\n\n\n\\input{symbols}\n\n\\section*{One-Layer Rotating Shallow Water Model}\n\n\\subsection*{Classical Form}\n\nThe one-layer, two-dimensional rotating shallow water model\non a rotating $f$-plane can be written as,\n\\begin{align} \n\\frac{\\partial u}{\\partial t} + \\left(\\bf{u} \\cdot \\nabla\\right) u - f v \n&= - g \\frac{\\partial h}{\\partial x},\\\\\n\\frac{\\partial v}{\\partial t} + \\left(\\bf{u} \\cdot \\nabla\\right) v + f u \n&= - g \\frac{\\partial h}{\\partial y},\\\\\n\\frac{\\partial h}{\\partial t} + \\nabla \\cdot \\left( h \\bf{u} \\right) &= 0.\n\\end{align}\nThis describes the motion of a pancake like fluid in that it is thin in the vertical\nand much longer in the horizontal. It contains pressure forces due to the free surface\nand a Coriolis pseudo-force because of the rotating frame of reference.\n\nThe fluid moves as columns that can be\ntranslated in the horizontal and stretched/contracted in the vertical. If the \nheight of a column changes then the voriticty must change, as can be reflected \nin the fact that, in the absence of forcing and dissipation, \nPotential Vorticity is conserved following the motion,\n$$\n\\frac{D}{Dt} \\left( \n\\frac{ \\frac{\\partial v}{\\partial x} - \\frac{\\partial u}{\\partial y} + f }{h} \\right) = 0.\n$$\n\n\\subsection*{Conservation Form}\n\nThere are a variety of forms, these equations can be written. \nOne of them is conservation where the three fields that have time \nderivatives are $U = hu$, $V = hv$ and $h$. \n\\begin{align} \n\\frac{\\partial U}{\\partial t} \n+ \\frac{\\partial}{\\partial x}\\left( \\frac{U^2}{h} + \\frac{g h^2}{2} \\right)\n+ \\frac{\\partial}{\\partial y}\\left( \\frac{U V}{h} \\right) \n- f V \n&= 0,\\\\\n\\frac{\\partial V}{\\partial t} \n+ \\frac{\\partial}{\\partial x}\\left( \\frac{U V}{h} \\right) \n+ \\frac{\\partial}{\\partial y}\\left( \\frac{V^2}{h} + \\frac{g h^2}{2} \\right)\n+ f U \n&= 0,\\\\\n\\frac{\\partial h}{\\partial t} + \\nabla \\cdot \\left( h \\bf{u} \\right) &= 0.\n\\end{align}\n\nConservation form is attractive because it can help to ensure that, using a clever numerical\nscheme, some quantitites are conserved. Note that if you have topography then there\nare source terms that appear on the right-hand side and this causes some problems.\n\n\\subsection*{Vorticity-Bernoulli Form}\n\nA third form arises from rewriting the nonlinear \nacceleration terms as a gradient and a cross product term. Using a vector identity,\nit can be shown that the above system is mathematically equivalent to the following,\n\\begin{align} \n\\frac{\\partial u}{\\partial t} - q h v \n&= - \\frac{\\partial B}{\\partial x},\\\\\n\\frac{\\partial v}{\\partial t} + q h u \n&= - \\frac{\\partial B}{\\partial y},\\\\\n\\frac{\\partial h}{\\partial t} + \\nabla \\left( h \\bf{u} \\right) &= 0.\n\\end{align}\nNote that above we have defined the vorticity and the Bernoulli function,\n\\begin{align}\nq &= \\frac{\\zeta + f}{h} \n = \\frac{\\frac{\\partial v}{\\partial x} - \\frac{\\partial u}{\\partial y} + f}{h},\\\\\nB & = g h + \\frac12 \\left( u^2 + v^2 \\right).\n\\end{align}\n\nBefore we look at solving this complicated set of equations \n\n\\subsection*{ $1 \\frac12$ Dimensional Limit}\n\nWe assume that none of the variables depend on one horizontal direction, say $y$. \nBut, it is very important to realize that the velocity in that direction is not necessarily zero.\nIndeed, if you have flow in the $x$-direction, the Coriolis force will deflect it to the right,\nwhich will then generate a flow that is perpendicular. This will continue and often give rise\nto inerital oscialltions in the horizontal. So we can have motion in either direction but\nthe motion only changes with respect to $x$. \n\nIf we simply the governing equations we get\n\\begin{align} \n\\frac{\\partial u}{\\partial t} - q h v \n&= - \\frac{\\partial B}{\\partial x},\\\\\n\\frac{\\partial v}{\\partial t} + q h u \n&= 0,\\\\\n\\frac{\\partial h}{\\partial t} + \\frac{\\partial}{\\partial x} \\left( h u \\right) &= 0.\n\\end{align}\nwhere the vorticity and the Bernoulli function simplify to,\n\\begin{align}\nq & = \\frac{\\zeta + f}{h} = \\frac{\\frac{\\partial v}{\\partial x} + f}{h},\\\\\nB & = g h^2 + \\frac12 \\left( u^2 + v^2 \\right).\n\\end{align}\n\n\\subsection*{Conserved Quantities}\n\nIn the purely conservative or nondissipative limit, \nthere are three quantities that are exactly conserved. \n\n1) Mass:\n$$\nM = \\int_D h \\, dA\n$$\n\n2) Total Energy: sum of potential and kinetic energies\n$$\nE = \\frac12 \\int_D \\left(g h^2 + h( u^2 + v^2)\\right) \\, dA\n$$\n\n3) Potenal Enstrophy: \n$$\nQ = \\frac12 \\int_D h q^2 \\, dA\n$$\n\nIn a numerical model we cannot expect these to be conserved \nbut we would like them to be close to be conserved. \nIf they are very badly conserved than this could reflect that\nthe numerical scheme is behaving badly.\nHowever, just because these are conserved that does not \nguarantee that the solution is correct. \nBut they are usually good indicators as to how our method is \ndoing in the conservative limit. \n\nOf course when nonconservative forces are introduced things will change.\nOne might argue that since the world is non-dissipative then we don't\nneed to worry about conserving these. However, it is desirable to\nknow that basis of your model is well behaved and therefore why\nwe should worry about conserved quantities.\n\n\\end{document}", "meta": {"hexsha": "bde87055db77967afe77f52508d4db1ff7c9d5a4", "size": 5699, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/sw_intro.tex", "max_stars_repo_name": "nishantsule/PyRsw", "max_stars_repo_head_hexsha": "753788608a0d227b5c8dc8b863d85bfc3a907310", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 16, "max_stars_repo_stars_event_min_datetime": "2015-11-10T21:45:20.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-13T08:47:38.000Z", "max_issues_repo_path": "docs/sw_intro.tex", "max_issues_repo_name": "nishantsule/PyRsw", "max_issues_repo_head_hexsha": "753788608a0d227b5c8dc8b863d85bfc3a907310", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2015-11-26T16:14:28.000Z", "max_issues_repo_issues_event_max_datetime": "2018-01-09T04:49:37.000Z", "max_forks_repo_path": "docs/sw_intro.tex", "max_forks_repo_name": "nishantsule/PyRsw", "max_forks_repo_head_hexsha": "753788608a0d227b5c8dc8b863d85bfc3a907310", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2015-11-10T10:30:06.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-21T18:21:13.000Z", "avg_line_length": 36.7677419355, "max_line_length": 97, "alphanum_fraction": 0.7085453588, "num_tokens": 1757, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.6757646010190476, "lm_q1q2_score": 0.3431612818573753}} {"text": "%-----------------------------------------------------------------------------------------------\n\\section{Introduction}\n\n Polytropic processes---\\emph{poly}: many, \\emph{tropic}: forms,\n ways~\\cite{2020-NaaktgeborenC-engrXiv}---are a versatile class of equilibrium thermodynamics\n processes with many applications in engineering\n thermodynamics~\\cite{2002-MoranMJ+ShapiroHN-LTC, 2013-CengelYA+BolesMA-AMGH,\n 2015-KroosKA+PotterMC-Cengage, 1986-JonesJB+HawkinsGA-Wiley}, given by the following\n relationship\n %\n \\begin{equation}\n Pv^n = \\mathsf{c} = \\mbox{const.},\n \\label{eq:polytropic}\n \\end{equation}\n %\n \\noindent where $P$ is the system pressure, $v$ the system specific volume, and $n$ is the\n polytropic exponent.\n\n Polytropic processes are said to find support in\n measurements~\\cite{2013-CengelYA+BolesMA-AMGH, 2002-MoranMJ+ShapiroHN-LTC,\n 1985-WylenG-Wiley}; however, theory~\\cite{2012-ChristiansJ-IntJMechEngEduc,\n 2020-NaaktgeborenC-engrXiv} predicts a fairly restrictive set of conditions---on the process\n boundary and internal conditions and on the underlying substance model---are required for a\n process to follow Eq.~(\\ref{eq:polytropic}) \\emph{exactly}---see, for instance, Theorem~3 of\n reference~\\cite{2020-NaaktgeborenC-engrXiv}.\n\n Thus, in non-ideal settings, polytropic processes either (i)~hold approximately in a\n \\emph{local} neighborhood of a given state, for constant-$n$, or (ii)~the polytropic\n exponent must be generalized into a function of a state property, such as temperature, or\n even further generalized.\n\n Reference~\\cite{2020-NaaktgeborenC-engrXiv} brings useful definitions for discussions like\n this, such as that of a \\emph{logical process}, and also those of \\emph{exact} and of\n \\emph{local} polytropic processes.\n\n In \\emph{exact polytropic process}, Eq.~(\\ref{eq:polytropic}) holds exactly with a constant,\n unique polytropic exponent, for the entire duration of the \\emph{logical process}. They are\n shown to be able to represent any straight line segment process in $\\log P \\times \\log v$\n coordinates~\\cite{2020-NaaktgeborenC-engrXiv}.\n\n Moreover, process whose representations in $\\log P \\times \\log v$ coordinates are curved,\n are shown to be able to be approximated by a finite number of \\emph{local polytropic\n processes} within finite error intervals~\\cite{2020-NaaktgeborenC-engrXiv}. Observations\n like these attest the flexibility of processes based on the polytropic relationship, thus\n justifying the flexibility encoded in their name.\n\n Despite their enormous flexibility, the defining concept of polytropic processes can be\n further generalized as to allow for increased flexibility, so that some processes families\n with curved representation in $\\log P \\times \\log v$ coordinates might have an exact\n representation in such further generalized process relation concept. The introduction of\n such generalizations can be of theoretical and of applied interest.\n\n Therefore, this work proposes one such generalization, named \\emph{polyekthetic processes},\n that form a \\emph{larger process set} of which the polytropic process set is a proper\n subset, so that any polytropic process is a particular case of a polyekthetic process, but\n not necessarily the opposite.\n\n The origin (etymology) for the proposed `polyekthetic' term is also given, and an\n application yielding exact polyekthetic processes (but not exactly polytropic ones) is\n provided.\n\n%-----------------------------------------------------------------------------------------------\n\n", "meta": {"hexsha": "507d288fcf5cfbfd5039d55d0bcbfee47eddb8b8", "size": 3939, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "01-01-Introduction.tex", "max_stars_repo_name": "cnaak/man-Polyekthetic", "max_stars_repo_head_hexsha": "0918d9d87e9e8841126059ce3346f15bd6826999", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "01-01-Introduction.tex", "max_issues_repo_name": "cnaak/man-Polyekthetic", "max_issues_repo_head_hexsha": "0918d9d87e9e8841126059ce3346f15bd6826999", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "01-01-Introduction.tex", "max_forks_repo_name": "cnaak/man-Polyekthetic", "max_forks_repo_head_hexsha": "0918d9d87e9e8841126059ce3346f15bd6826999", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 61.546875, "max_line_length": 96, "alphanum_fraction": 0.6636202082, "num_tokens": 1058, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.6757645944891558, "lm_q1q2_score": 0.3431612785414188}} {"text": "\n\n \\filetitle{rngcmp}{Compare two IRIS date ranges}{dates/rngcmp}\n\n\t\\paragraph{Syntax}\\label{syntax}\n\n\\begin{verbatim}\nFlag = rngcmp(R1,R2)\n\\end{verbatim}\n\n\\paragraph{Input arguments}\\label{input-arguments}\n\n\\begin{itemize}\n\\itemsep1pt\\parskip0pt\\parsep0pt\n\\item\n \\texttt{R1}, \\texttt{R2} {[} numeric {]} - Two IRIS date ranges that\n will be compared.\n\\end{itemize}\n\n\\paragraph{Output arguments}\\label{output-arguments}\n\n\\begin{itemize}\n\\itemsep1pt\\parskip0pt\\parsep0pt\n\\item\n \\texttt{Flag} {[} \\texttt{true} \\textbar{} \\texttt{false} {]} - True\n if the two date ranges are the same.\n\\end{itemize}\n\n\\paragraph{Description}\\label{description}\n\nAn IRIS date range is distinct from a vector of dates in that only the\nfirst and the last dates matter. Often, date ranges are context\nsensitive. In that case, you can use \\texttt{-Inf} for the start date\n(meaning the earliest possible date in the given context) and\n\\texttt{Inf} for the end date (meaning the latest possible date in the\ngiven context), or simply \\texttt{Inf} for the whole range (meaning from\nthe earliest possible date to the latest possible date in the given\ncontext).\n\n\\paragraph{Example}\\label{example}\n\n\\begin{verbatim}\nr1 = qq(2010,1):qq(2020,4);\nr2 = [qq(2010,1),qq(2020,4)];\n\nrngcmp(r1,r2)\nans =\n 1\n\\end{verbatim}\n\n\n", "meta": {"hexsha": "0772ee22a693d3995bf3ab4aeb2bf817219299af", "size": 1295, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "-help/dates/rngcmp.tex", "max_stars_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_stars_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2017-12-06T13:38:38.000Z", "max_stars_repo_stars_event_max_datetime": "2017-12-06T13:38:38.000Z", "max_issues_repo_path": "-help/dates/rngcmp.tex", "max_issues_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_issues_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2017-03-28T08:13:20.000Z", "max_issues_repo_issues_event_max_datetime": "2020-09-02T10:40:25.000Z", "max_forks_repo_path": "-help/dates/rngcmp.tex", "max_forks_repo_name": "OGResearch/IRIS-Toolbox-For-Octave", "max_forks_repo_head_hexsha": "682ea1960229dc701e446137623b120688953cef", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2022-01-17T07:06:39.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-17T07:06:39.000Z", "avg_line_length": 24.9038461538, "max_line_length": 72, "alphanum_fraction": 0.7374517375, "num_tokens": 403, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.6654105653819836, "lm_q1q2_score": 0.3430989396434626}} {"text": "% vim: set spell spelllang=en\n\\documentclass[10pt]{article}\n\\usepackage[pdftex]{graphicx}\n\\usepackage{amssymb, amsmath}\n\\usepackage{url}\n\n\\usepackage{fullpage}\n\\setlength{\\parindent}{0pt}\n\\setlength{\\parskip}{\\baselineskip}\n\n\\title{Implementation of the Density Evolution method with\nFokker-Planck for L-NLIF Model in Python}\n\\author{Valentin Haenel \\\\\nvalentin.haenel@gmx.de \\\\\nBCCN Berlin}\n\n\\begin{document} \n\n\\maketitle\n\n\\section{Introduction}\n\nThis report is a description of the work done by Valentin Haenel\nfor a Lab Rotation within the Bernstein Center for Computational\nNeuroscience in Berlin. The rotation was completed under the\nsupervision of Gabriel and Bartosz Telenczuk. \n\nThroughout the project I have implemented a variant of the Leaky\nIntegrate and Fire Neuron (LIF), and a method to fit the parameters of this\nneuron using a model of the input and extracellularly recorded spikes.\nPrimarily this involved solving a partial differential equation and constructing\na maximum likelihood objective function. \n\nThis project is motivated by the fact that recording intracellularly from awake\nbehaving animals is usually not possible, and extracellularly recorded spike\ntrains are much more common in this setting. The model implemented allows us to\ngain at least some partial insight into the subthreshold voltage of such\nneurons. A further motivation comes from the fact, that algorithms described are\nnot publicly available in the form of an easy to use software package. \n\n\\section{Model}\nThe original model is a L-NLIF model, which consists of a Linear\n(L) Filter, followed by a probabilistic or Noisy (N) form of Leaky\nIntegrate and Fire spike generation (LIF) (Paninski et\nal.\\cite{PaninskiPillowSimoncelli}). For sake of completeness the\nmodel is briefly described here.\n\nThe evolution of the Voltage $V$ is given by:\n\\begin{equation}\n dV = (-g(V(t) -V_{leak}) +I_{stim}(t) + I_{hist}(t)) dt + W_{t}\n\\end{equation}\nWhere $g$ is the leak conductance $V_{leak}$ is the leak reversal\npotential $I_{stim}$ is the convolution of linear filter with the\ninput signal, $I_{hist}$ is the spike current history and $W_t$ is\nstandard Gaussian white noise. More precisely:\n\\begin{equation}\n I_{stim}(t) = \\vec{k} \\cdot \\vec{x}(t) \n\\end{equation}\n\\begin{equation}\n I_{hist}(t) = \\sum_{j=0}^{i-1}h(t-t_j)\n\\end{equation}\nWhere $k$ is a linear filter and $h$ is a fixed postspike current waveform. \n\n\\section{Algorithms}\n\n\\subsection{PDE Solver}\n\n\\subsubsection{The Problem}\nLet \n\\begin{equation}\n P(V,t) \\equiv P(V(t) \\cap V(s) < V_{th} \\forall s < t)\n\\end{equation}\nThis refers to the probability of $V(t)$, the membrane potential,\nbeing less than $V_{th}$ the firing threshold until time $t$. \n\nAs stated in \\cite{PaninskiPillowSimoncelli} the method of density\nevolution requires us to solve the\nfollowing Fokker-Planck drift diffusion equation numerically.\n\\begin{equation}\n \\frac{\\partial P(V,t)}{\\partial t} =\n \\frac{\\sigma^2}{2} \\frac{\\partial^2 P(V,t) } {\\partial V^2} +\n g\\frac{\\partial[(V-V_{rest})P(V,t)]}{\\partial V}\n \\label{fokkerplanck}\n\\end{equation}\n\nWe will be solving the evolution of the probability density for a\ngiven inter-spike interval (ISI). And therefore we will constrain the range\nof $V$ and $t$.\n\\begin{equation}\n P(V_{th},t) = 0\n \\label{boundarytop}\n\\end{equation}\n\\begin{equation}\n P(V,0) = \\delta(V-V_{reset})\n \\label{boundaryleft}\n\\end{equation}\n\nCondition (\\ref{boundarytop}) ensures that the probability of obtaining a spike\nduring an ISI is zero. Condition (\\ref{boundaryleft}) ensures\nthat at the beginning of an ISI, i.e. at the time of the last spike,\nthe probability of the neuron being at reset potential is exactly one.\nThereby we have obtained the constraints for the top and the left hand\nside of our solution grid, however, when solving numerically we also\nneed a lower bound on the voltage $V_{lb}$, so this becomes an\nadditional boundary condition, for the bottom:\n\\begin{equation}\n P(V_{lb},t) = 0 \n \\label{boundarybottom}\n\\end{equation}\n\nIdeally $V_{lb} = -\\infty $.\n\n$V_{rest}$ is defined as the stationary point of the noiseless\nsubthreshold dynamics:\n\\begin{equation}\n V_{rest}(t) = V_{leak} + \\frac{1}{g}(\\vec{k} \\cdot \\vec{x}(t) +\n \\sum_{j=0}^{i-1}h(t-t_j))\n\\end{equation}\n\n% This was just wasting space\n%\n%We can rewrite (\\ref{fokkerplanck}) as:\n%\\begin{equation}\n% \\frac{\\sigma^2}{2} \\frac{\\partial^2 P(V,t) } {\\partial V^2} +\n% g(V-V_{rest})\\frac{\\partial P(V,t)}{\\partial V} +\n% gP(V,t) -\n% \\frac{\\partial P(V,t)}{\\partial t} = \n% 0\n%\\end{equation}\n%Since we know that:\n%\\begin{equation}\n% \\frac{\\partial[(V-V_{rest})P(V,t)]}{\\partial V} =\n% \\frac{\\partial (V-V_{rest})}{\\partial V} P +\n% \\frac{\\partial P}{\\partial V} (V-V_{rest})\n%\\end{equation}\n%and\n%\\begin{equation}\n% \\frac{\\partial (V-V_{rest})}{\\partial V} = 1\n%\\end{equation}\n\n\\subsubsection{Finite Differencing}\n\nIn order to solve (\\ref{fokkerplanck}) numerically we discretized time\nand potential. We adopt the notation that Potential is discretized\ninto $W$ intervals of length $w$ and indexed\nby $\\nu= 0,1, \\dots W $. Time is discretized into $U$ intervals of\nlength $u$ and indexed by: $\\tau= 0,1, \\dots U $.\nWe adopt the notation: $P_{\\nu,\\tau} = P(\\nu w,\\tau u)$\n\nA forward finite differencing scheme would like:\n\\begin{eqnarray}\n \\hat{P}(v,t) =& P_{\\nu,\\tau}\\\\\n%\n\\frac{\\partial \\hat{P}(v,t)}{\\partial t} =& \\frac{P_{\\nu,\\tau +1 } -\n P_{\\nu,\\tau}}{u} \\\\\n%\n \\frac{\\partial \\hat{P}(v,t)}{\\partial V} =& \n \\frac{P_{\\nu +1,\\tau } -\n P_{\\nu - 1,\\tau } }\n {2w} \\\\ \n%\n \\frac{\\partial^2 \\hat{P}(v,t)}{\\partial V^2} =& \n \\frac{P_{\\nu+1,\\tau} - 2 P_{\\nu,\\tau} + P_{\\nu-1,\\tau}}\n {w^2} \n\\end{eqnarray}\n\nHowever as suggested in \\cite{PaninskiHaithSzirtes} we will use the\ncomputationally more stable Crank-Nicolson scheme \\cite{press}. \nWe rewrite the derivatives using the new scheme which is centered around $t +\nu/2$.\n\n\\begin{eqnarray}\n P_{CN}(v,t) =& \\frac{P_{\\nu,\\tau} + P_{\\nu,\\tau + 1}}{2} \\\\\n %\n \\frac{\\partial P_{CN}(v,t)}{\\partial t} =& \\frac{P_{\\nu,\\tau +1 } -\n P_{\\nu,\\tau}}{u} \\\\\n %\n \\frac{\\partial P_{CN}(v,t)}{\\partial V} =&\n \\frac{P_{\\nu +1,\\tau } + P_{\\nu +1,\\tau +1 } -\n P_{\\nu - 1,\\tau } - P_{\\nu -1,\\tau +1}} \n {4w} \\\\\n %\n \\frac{\\partial^2 P_{CN}(v,t)}{\\partial V^2} =&\n \\frac{P_{\\nu+1,\\tau} - 2 P_{\\nu,\\tau} + P_{\\nu-1,\\tau} +\n P_{\\nu+1,\\tau+1} - 2 P_{\\nu,\\tau+1} + P_{\\nu-1,\\tau+1}}\n {2w^2} \n\\end{eqnarray}\n\n\\subsubsection{Tridiagonal Equations}\n\nIf we now let:\n\\begin{align*}\na &= \\frac{\\sigma^2}{2} \\\\\nb &= g(V - V_{rest}) \\\\\nc &= g \\\\\n\\end{align*}\nand multiply throughout with $4w^2u$\nwe may rearrange all the $P_{*,\\tau+1} $ terms on the left hand side:\n\\begin{multline}\n \\overbrace{-(2au+bwu)}^{A_\\nu} P_{\\nu+1,\\tau+1} + \n \\overbrace{(4au - 2cw^2u + 4w^2)}^{B_\\nu} P_{\\nu,\\tau+1}\n \\overbrace{-(2au-bwu)}^{C_\\nu} P_{\\nu-1,\\tau+1}\n = \\\\\n \\underbrace{(2au+bwu) P_{\\nu+1,\\tau} + \n (-4au +2cw^2u + 4w^2) P_{\\nu,\\tau} + \n (2au-bwu) P_{\\nu-1,\\tau}}_{D_{\\nu}}\n \\label{rearrange}\n\\end{multline}\nFor each $ \\nu = 1 , \\dots , W-1 $\n\nUsing (\\ref{rearrange}) and incorporating the boundary conditions\n\\ref{boundarytop} and \\ref{boundaryleft} we have obtain a complete system \nof $W-1$ linear equations which we may now rewrite in the following\ntridiagonal matrix notation.\n\n\\begin{equation}\n\\underbrace{\n\\begin{pmatrix}\n & B_1 & A_1 & 0 & \\cdots \\\\\n & C_2 & B_2 & A_2 & 0 \\\\\n & & \\ddots & \\ddots & \\ddots \\\\\n & \\cdots & 0 & C_{W-1}& B_{W-1} \\\\\n\\end{pmatrix}}_{\\Lambda}\n\\underbrace{\n\\begin{pmatrix}\n P_{1,\\tau+1} \\\\\n P_{2,\\tau+1} \\\\\n \\vdots \\\\\n P_{W-1,\\tau+1} \\\\\n\\end{pmatrix}}_{\\chi}\n=\n\\underbrace{\n\\begin{pmatrix}\n D_1 \\\\\n D_2 \\\\\n \\vdots \\\\\n D_{W-1} \\\\\n\\end{pmatrix}}_{\\beta}\n\\end{equation}\n\nWe can then use a tridiagonal matrix algorithm to solve $\\Lambda \\chi\n= \\beta$. This allows us to iteratively obtain the density evolution $P(V,t)$ of the\nmembrane potential for a given ISI. Where the initial $\\beta$ for an ISI is given by\n(\\ref{boundaryleft}) and the halting criterion is the time of the next\nspike. So for the $i$th spike we obtain values for $V$ in the range\n$[V_{lb}, V_{th}]$ and for $t$ in the range $[t_{i-1},t_{i}]$. An\nexample can be seen in figure \\ref{P_vt}.\n\n\n\\subsection{First Passage Time}\n\nThe maximum likelihood optimizer proposed in\n\\cite{PaninskiPillowSimoncelli} however uses the first passage time\ndensity at a given time point which is defined as:\n\n\\begin{equation}\n \\mathrm{FPT}(t) = -\\frac{\\partial}{\\partial t} \\int P(V,t)dV \n \\label{eq:fpt}\n\\end{equation}\n\nand this can be easily computed from a $P(V,t)$ matrix by a finite differences\n/ sums scheme.\n\n\\subsection{Maximum Likelihood Estimator}\n\nPutting together the PDE Solver and method for computing the First\nPassage Time \nThe final likelihood is then simply\n\n\\begin{equation}\n L_{\\vec{x},t_{i}} = \\prod_{i} \\mathrm{FPT}(t_{i})\n\\end{equation}\n\nThis is the product of the first passage time at the times of\nspiking for all spikes.\n\nThe pseudocode for the objective function can be described as follows:\n\n{\\tt\nfor each spike interval do:\n\\begin{enumerate}\n \\item compute $P(V,t)$ as matrix\n \\item compute FPT$(t)$ as vector\n\\end{enumerate}\nreturn the product of the last scalars in the FPT vectors.}\n\nThis objective function could be presented to a standard numerical\noptimizer, for example the Nelder-Mead Downhill simplex \\cite{press},\nthis needs no gradients and is fairly robust but slow.\n\n\\subsection{Monte Carlo Method}\n\nTo double check the results of the PDE Solver and the resulting First\nPassage Time, we additionally implemented some basic Monte Carlo\nsimulations. The basic idea is to simulate a few thousand traces of the neuron\nup to the first spike, with noise. The resulting potential traces, and spike\ntimes were used to obtain both $P(V,t)$ and $fpt$. The density evolution is\ncompared qualitatively to the density evolution technique as can bee seen in:\n\\ref{P_vt}. The FPT is compared using a Kolmogorov-Smirnov (K-S)\ntest\\cite{press}.\n\n\n\\section{Implementation}\n\nThis section describes the concrete implementation\n\n\\subsection{Availability and Design}\n\nThe algorithm has been implemented in Python, and the full source code\nhas been made available within a revision control system under a free\nsoftware license at: http://github.com/esc/molif/tree/master. Given\nthat the Python Language was named after the British Comedians Monty\nPython we decided to call the software {\\it Meaning Of LIF},\n(abbreviated to {\\it molif}) which is a reference to the Monty Python\nmovie {\\it Meaning of Life}. \n\nThe software is structured into a series of modules contained within\nthe package {\\tt molif} \n\n\\begin{itemize}\n \\item {\\tt model} implementation of the neuron model\n \\item {\\tt density} implementation of the PDE solver \n \\item {\\tt likelihood} connects the optimizer and the PDE solver\n \\item {\\tt montecarlo} implementation of Monte Carlo simulator\n \\item {\\tt plotting} methods used to generate plots\n \\item {\\tt util} methods to determine timing\n\\end{itemize}\n\nThe software uses the additional packages numpy, scipy for vector\nbased and scientific computing, and pylab/matplotlib for plotting the\nresults. We used the modules {\\it scipy.sparse} and {\\it\nscipy.linsolve} for computing the solution to tridiagonal equations, and the\nmethod {\\it optimize.fmin}, which implements the Nelder-Mead Downhill\nSimplex, for minimizing the negative likelihood.\n\n\n\\subsection{Limitations and Future Work}\n\nThe PDE solver uses sparse matrix algorithms to store and solve the\ntridiagonal matrix. Ideally we would like to use the routine $TRIDAG$\ndescribed in \\cite{press}, however an open source implementation in\npython was not available, although feasible. This improvement could\npossibly speed up the computation of $P(V,t)$.\n\nThere is a serious bug, whereby the value of sigma,\nmust be within a specific range of $0.1 - 0.05$, otherwise strange\noscillatory instabilities occur. Among the possible causes, are a\nnumerical instability in the finite differencing scheme, a bug in our\ncode, a bug in the solver for the linear equations or the\ndiscretization of $V$ and $t$ being to large. \n\nCurrently the gradient of the objective function isn't implemented,\nand any gradient based optimizer would need to compute their\napproximation from the function values. \n\nWe used a Kolmogorov-Smirnov (K-S) test to check that the first\npassage time computed by the PDE-solver and the Monte Carlo method\ncome from the same distribution. However the test currently returns a\nnegative result, although at first glance the distributions look very\nsimilar. \n\nLastly we should mention that the objective function isn't fully\noperational yet. A simple test is executed as follows: first we\ngenerate some spikes with given parameters, and then we attempt to fit\nparameters to these spikes. If the code operates correctly the\noptimizer should not deviate away from the initial parameters vector.\nHowever currently this is not the case and the optimizer finds a\nminimum at a different position in parameter space.\n\n\\subsection{Results}\n\nThis section presents some plots.\n\nFigure \\ref{P_vt} shows a comparison of PDE Solver (top) and Monte Carlo\n(bottom) method when computing the density evolution. As expected the two plots\nlook the same, and corresponds very nicely to what we would expect from the\niterative solution to a Fokker-Planck equation. The drift term is responsible for\nthe shift of the mean, and the diffusion is responsible for the flattening. \n\nFigure \\ref{fpt} shows the comparison of the PDE Solver and Monte\nCarlo method when computing the first passage time. Unfortunately a\nK-S test allows us to reject the hypothesis that these two come from\nthe same distribution, even though they look very much alike at a\nfirst glance.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.60]{P_vt}\n\\caption{Density Evolution}\n\\label{P_vt}\n\\end{figure}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.60]{fpt.png}\n\\caption{First Passage Time Comparison}\n\\label{fpt}\n\\end{figure}\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.60]{traces.png}\n\\caption{Voltage Traces for 5000 Monte Carlo simulations }\n\\label{traces}\n\\end{figure}\n\n\\section{Conclusion}\n\nNot all of the initial Project Goals were reached to my full\nsatisfaction. We did implement an Integrate and Fire\nneuron, and a method to fit the parameters, however we did not get as\nfar as actually fitting real experimental data. None the less \nuseful code has been produced and solid numerical techniques (Solving\nPartial Differential Equations) were\nlearnt, and I consider the lab rotation to be successful. \n\n\\bibliography{documentation}{ \\bibliographystyle{abbrv} }\n\n\\end{document}\n", "meta": {"hexsha": "9198874bdb67cd7fc71c6272c58e64a4797681dd", "size": 14877, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "documentation/documentation.tex", "max_stars_repo_name": "tibrewalabhay/molif", "max_stars_repo_head_hexsha": "fd7e6478ffd81f44f17b3e32a286f9c773e03b0c", "max_stars_repo_licenses": ["WTFPL"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2016-05-08T19:56:45.000Z", "max_stars_repo_stars_event_max_datetime": "2016-05-08T19:56:45.000Z", "max_issues_repo_path": "documentation/documentation.tex", "max_issues_repo_name": "tibrewalabhay/molif", "max_issues_repo_head_hexsha": "fd7e6478ffd81f44f17b3e32a286f9c773e03b0c", "max_issues_repo_licenses": ["WTFPL"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-10-02T13:05:23.000Z", "max_issues_repo_issues_event_max_datetime": "2020-10-02T13:05:23.000Z", "max_forks_repo_path": "documentation/documentation.tex", "max_forks_repo_name": "tibrewalabhay/molif", "max_forks_repo_head_hexsha": "fd7e6478ffd81f44f17b3e32a286f9c773e03b0c", "max_forks_repo_licenses": ["WTFPL"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-10-02T12:57:43.000Z", "max_forks_repo_forks_event_max_datetime": "2020-10-02T12:57:43.000Z", "avg_line_length": 35.4214285714, "max_line_length": 84, "alphanum_fraction": 0.7188277206, "num_tokens": 4369, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6187804337438501, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.34309547480636743}} {"text": "\\section*{Results}\n\nIn order to demonstrate the efficacy of our approach, we experiment with D-NeRF's synthetic\ndataset which contains 6 different scenes.\n\n\\subsection*{Quantitative Results}\n\nOur method is able to slightly outperform D-NeRF on their synthetic dataset. This is likely because D-NeRF can predict non-realistic movement, such as jumping from frame-to-frame. In practice, D-NeRF does not do this, but the movement is not bound to have any smoothness properties, so it may not resemble real movement, by coming to full stops or suddenly accelerating. In contrast, our method enforces that movement is fluid, and thus is better able to reproduce missing frames. This only leads to a small improvement in PSNR, because the dataset contains relatively simple movement, but we expect that in longer sequences this difference would become more apparent.\n\n\\subsection*{Qualitative Results}\n\nThe difference between the architectures can be observed in the difference of flow between scenes. It's clear from Figure~\\ref{fig:dnerf_cmp} that our method more accurately captures squashing and stretching of movement, as the top of the ball is not moving but the bottom of the ball is falling. In contrast, D-NeRF contains approximately equal flow for the entire ball. Since there is no forced prior on movement given by D-NeRF, there is also flow on rigid components.\n\nThe difference between the two is also more clearly seen in videos of reconstruction. $C^0$-NeRF visibly has the effect of 'tweening between views, slowing into stops, while D-NeRF has more abrupt starts and stops. While it's difficult to characterize the plausibility of this movement, future work may seek to characterize smoothness of the velocity, as well as smoothness of acceleration and use that as a way to characterize movement.\n\n\\begin{table*}[t]\n \\centering\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\textbf{PSNR$^\\uparrow$} & Bouncing Balls & Hellwarrior & Hook & Jumping Jacks \\\\\n \\hline\n D-NeRF & 22.978 & 32.712 & 26.757 & 26.278 \\\\\n \\hline\n $C^0$-NeRF & 24.251 & 33.504 & 27.705 & 27.432 \\\\\n \\hline\n & Lego & Mutant & Standup & T-Rex \\\\\n \\hline\n D-NeRF & 22.679 & 27.408 & 29.435 & 24.494 \\\\\n \\hline\n $C^0$-NeRF & 22.945 & 28.486 & 31.050 & 25.421 \\\\\n \\hline\n \\end{tabular}\n \\centering\n \\begin{tabular}{|c|c|c|c|c|}\n \\hline\n \\textbf{MS-SSIM$^\\uparrow$} & Bouncing Balls & Hellwarrior & Hook & Jumping Jacks \\\\\n \\hline\n D-NeRF & 0.917 & 0.949 & 0.961 & 0.931 \\\\\n \\hline\n $C^0$-NeRF & 0.971 & 0.966 & 0.976 & 0.978 \\\\\n \\hline\n & Lego & Mutant & Standup & T-Rex \\\\\n \\hline\n D-NeRF & 0.912 & 0.960 & 0.975 & 0.957 \\\\\n \\hline\n $C^0$-NeRF & 0.935 & 0.980 & 0.988 & 0.972 \\\\\n \\hline\n \\end{tabular}\n \\caption{\n Our method is able to recover movement with slightly improved accuracy in dynamic scenes as compared to D-NeRF~\\cite{pumarola2020dnerf}. Despite, or maybe because of, the forced prior of continuous movement, we are able to learn a smooth interpolation through each frame. In our training regime which randomly samples all frames, we reconstruct each with much lower variance as compared to D-NeRF.\n }\n\\end{table*}\n\n\\begin{figure*}\n \\includegraphics[width=\\textwidth]{dnerf_compare}\n \\caption{\n \\label{fig:dnerf_cmp}\n Results comparing movement in D-NeRF~\\cite{pumarola2020dnerf} versus our work for a small timestep. There is substantial qualitative difference in predicted flow, since D-NeRF cannot guarantee the initial frame has no movement. There is also a significant difference in rigidity, which is not easily explained. We assume that it may be significantly easier to model low movement with splines, thus there is less of a need for rigidity gating.\n }\n\\end{figure*}\n", "meta": {"hexsha": "e1f4b7d8050f456f946fecb5ce5459e2e45413d5", "size": 3796, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "c0_paper/results.tex", "max_stars_repo_name": "princeton-computational-imaging/nerf_atlas", "max_stars_repo_head_hexsha": "f66ba284ea440cd816b303cdb7312288901da97e", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 14, "max_stars_repo_stars_event_min_datetime": "2021-05-17T13:17:02.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-05T00:44:44.000Z", "max_issues_repo_path": "c0_paper/results.tex", "max_issues_repo_name": "princeton-computational-imaging/nerf_atlas", "max_issues_repo_head_hexsha": "f66ba284ea440cd816b303cdb7312288901da97e", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2021-09-07T08:31:18.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-28T06:55:40.000Z", "max_forks_repo_path": "c0_paper/results.tex", "max_forks_repo_name": "princeton-computational-imaging/nerf_atlas", "max_forks_repo_head_hexsha": "f66ba284ea440cd816b303cdb7312288901da97e", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2021-05-16T01:06:03.000Z", "max_forks_repo_forks_event_max_datetime": "2021-12-26T01:48:09.000Z", "avg_line_length": 62.2295081967, "max_line_length": 668, "alphanum_fraction": 0.7241833509, "num_tokens": 1071, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6187804337438501, "lm_q1q2_score": 0.34309547480636743}} {"text": "The multi-model concept was introduced in \\mf using GWF Model Exchange objects to support a tight coupling between any two groundwater flow models at the matrix level. This concept has proven to be successful and valued by the modeling community, but the implementation has had its drawbacks too. The information on the spatial discretization at the interface between models as it is available in the Model Exchange object is limited, designed to carry out the basic conductance calculation between connected cells but not sufficient for more advanced discretization schemes. For example, the XT3D option in the NPF package enables simulation of fully three-dimensional (3D) anisotropy by taking into account the full, three-dimensional conductivity tensor \\cite{modflow6xt3d}. In doing so, it requires data not only from any pair of cells between which a flow needs to be calculated but also from their neighboring cells. These data are not available in the Model Exchange object and the XT3D option could not be applied across the model interface, at least not without a significant restructuring of the code.\n\nThe extension of \\mf with the transport model (GWT) comes with the need for a more generic coupling of sub-models too. Here the XT3D option is used in a similar fashion for the dispersion calculation, and the TVD (total variation diminishing) scheme in the advective transport calculation has a computational stencil (i.e. the group of cells required to determine the flux through a particular cell face) that requires information from neighboring cells as well. Finally, the GWF Model Exchange object merely replicates logic already present in the standard GWF packages, e.g. the standard NPF conductance calculation, the Newton-Raphson formulation, or the cell rewetting algorithm, and is not easily adapted when additions or modifications to those packages are made. We introduce a generalized method for coupling of Numerical Models that addresses these challenges and extends \\mf with the capability to couple GWT models.\n\nThe generalized coupling uses the concept of an Interface Model and is described in more detail below. In essence, this Interface Model uses the object-oriented paradigm in \\mf to mimic a regular GWF or GWT model by means of type extension. This way it can use the existing packages and their algorithms in those models to calculate coefficients for the matrix and right-hand-side vector in the Numerical Solution, a responsibility it takes over from the Model Exchange object on which it relies. Its grid is constructed around the interface defined by the Model Exchange with a sufficiently large extent to perform the calculations. Note that the Interface Model is never being solved and neither its configuration data nor its solution vector have any independent meaning: they are merely an image of those parts of the actual Numerical Models that contribute to the interface grid. As a matter of fact, and this is probably the most important point to make here, this generalized coupling with the Interface Model does not introduce any new model functionality. It even avoids the need for the alternative formulations currently present in the GWF Model Exchange. However, it does enable the user to apply the well-tested concepts of existing \\mf packages not only on a model’s interior domain, but also across the interface between connected models.\n \n\\subsection{The Interface Model}\nGeneralized coupling is based on the Interface Model to provide the coefficients for the linear system in Numerical Solution. In order to utilize the routines in the existing flow and transport packages to calculate these coefficients, the grid for the Interface Model is constructed from the individual model grids and the coupling information specified by the user and its package data is kept synchronized with the model data during each step of the simulation. Because the Model Exchange makes it possible to connect model grids of different type (DIS, DISV, and DISU), the resulting interface grid is always unstructured, i.e. of type DISU.\n\nConnecting models such that the dynamics at the interface can be described properly by an Interface Model does impose a few conditions on the configuration. For more advanced functionality such as XT3D, it is required that the relative position of the model grids is specified. They should also share the cell faces over which they are connected, such that the resulting grid for the interface can be specified as a valid DISU discretization. Another consequence is that a certain compatibility is required with respect to the active processes and configuration in the individual models: enabling advection (ADV) or dispersion (DSP), for example, only in one of two connected models introduces ambiguity in how to construct the Interface Model for the transport process. Similar arguments hold when two groundwater models are connected and only one of them is configured with a buoyancy package (BUY). Such model setups are currently not supported.\n\n\\begin{figure}[!t]\n\t\\begin{center}\n\t\\includegraphics[width=0.5\\textwidth]{./Figures/InterfaceModel/mt3dms-p10-modelgrid.png}\n\t\\caption[A plan view of the model grid from MT3DMS problem 10]{A plan view of the model grid for the simulation as defined in MT3DMS problem 10. The green dashed rectangle shows the location of the interface that is used for decomposing this case into a system of coupled GWF and GWT models}\n\t\\label{fig:gwtgwt-fullgrid}\n\t\\end{center}\n\\end{figure}\n\nTo illustrate the concept of this new coupling, a well known, single-model case study (problem 10 in the MT3DMS manual \\cite{zheng1999mt3dms}) is implemented as a coupled system where the connection between the GWT models is handled by the Interface Model framework. This study is part of the official set of examples that comes with every \\mf software release\\footnote{https://modflow6-examples.readthedocs.io/en/master/}. The coefficients for the flow coupling in this case are calculated with the basic GWF Model Exchange module and the following discussion will focus on the GWT-GWT exchange. Figure~\\ref{fig:gwtgwt-fullgrid} shows a plan view of the grid and the location of the interface between the submodels. Note that changing only the composition of the grid and leaving the model configuration the same, does not affect the outcome of the simulation\\footnote{At the highest level of detail, the outcome does change because the numbering of the grid nodes is different in both scenarios leading to a reordered matrix system to be solved. As expected, these differences turn out to be well within the configured (IMS) solver tolerance.} and therefore makes for a very suitable test case of this coupling.\n\n\\begin{figure}[!ht]\n\t\\begin{center}\n\t\\includegraphics[width=0.8\\textwidth]{./Figures/InterfaceModel/gwt-ifmod-grid.png}\n\t\\caption[A top view of the Interface Model grid]{A top view of the grid for the Interface Model belonging to the inner GWT model in the example described in the text. The blue cells are located in the outer model’s grid, the red cells are internal. Note that deeper layers have an identical horizontal structure.}\n\t\\label{fig:gwtgwt-interface-grid}\n\t\\end{center}\n\\end{figure}\n\nFigure~\\ref{fig:gwtgwt-interface-grid} shows a plan view of the grid that is reconstructed as part of the Interface Model for the inner GWT model. The red band with cells belong to the inner model’s grid and the blue cells are part of the outer, coarser grid. Because two Interface Models are constructed for every Exchange, (one for each of the models in it) a similar though not identical picture can be drawn for the outer GWT model. However, the extent of this grid is not necessarily equal on both sides of the exchange. The task of the Interface Model is twofold. First of all, it is designed to calculate the coefficients for the linear system for fluxes through the faces directly at the interface. Additionally, it can provide coefficients for those fluxes that would be affected by cells on the other side of the interface just as if the simulation would have been set up as a single model. It is the requirement to support the latter that causes the asymmetry. This becomes clearer when looking at Figure~\\ref{fig:gwtgwt-stencils} which shows the XT3D computational stencils and how they determine what the required extent of the interface grid is. To determine the flux through a face at the model interface, the cells on either side and their immediate neighbors are sufficient (the circle). To correctly calculate the flux through cell faces not directly at the interface but influenced by the connection to another model, more additional levels of connectivity are required, depending on the exact size of the computation stencil (the diamond).\n\n\\begin{figure}[!ht]\n\t\\begin{center}\n\t\\includegraphics[width=0.8\\textwidth]{./Figures/InterfaceModel/gwt-ifmod-stencils.png}\n\t\\caption[Stencil size and the extent of the Interface Model grid]{Top view when zoomed in to the South-West corner of the interface grid. The dashed line surrounds the grid cells participating in the calculation of the dispersive XT3D flux directly at the interface (the white circle) and through a face in the model’s interior (the white diamond) which is nonetheless affected by the connection to the outer GWT model. The extension of the computational stencil in the vertical direction is omitted but straightforward.}\n\t\\label{fig:gwtgwt-stencils}\n\t\\end{center}\n\\end{figure}\n\nThis case study is used to demonstrate the concept of the generalized coupling method with the Interface Model. It serves as an excellent test case because the outcome should be identical to the known results, at least within the configured tolerance of the Numerical Solution. However, from a modeling perspective it would make more sense to redesign the model grid of the problem. With the new capability to use the multi-model approach also for problems containing solute transport, there is no longer a need to refine the study area by using varying column and row width as was done in the original MT3DMS problem. Both the study area and the surrounding domain can be configured as coupled models with a regularly structured (DIS) grid. The granularity of the inner grid can be independently refined to improve the resolution of the results without the need to set up the entire problem based on an unstructured (DISU) grid.\n\nFinally, such a setup would allow to apply the Interface Model in another powerful way. A refinement of the grid is known to produce inaccuracies at the interface, as discussed in the introduction of \\cite{modflow6xt3d}. This can be remedied by enabling the XT3D option in the GWF NPF package. The new coupling mechanism based on the Interface Model makes it now possible to enable this option \\emph{only} at the interface, which will mitigate the distorting effects of such a grid refiniment on the calculated flow field and still avoids the computational overhead of calculating the XT3D terms on the entire domain.\n\n", "meta": {"hexsha": "c1e33587524276b4a85fd540ec388106b25be10c", "size": 11031, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/SuppTechInfo/gen-coupling.tex", "max_stars_repo_name": "kzeiler/modflow6", "max_stars_repo_head_hexsha": "a185d95b91985e965f8a04ae353305dff19b9637", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/SuppTechInfo/gen-coupling.tex", "max_issues_repo_name": "kzeiler/modflow6", "max_issues_repo_head_hexsha": "a185d95b91985e965f8a04ae353305dff19b9637", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/SuppTechInfo/gen-coupling.tex", "max_forks_repo_name": "kzeiler/modflow6", "max_forks_repo_head_hexsha": "a185d95b91985e965f8a04ae353305dff19b9637", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 250.7045454545, "max_line_length": 1559, "alphanum_fraction": 0.812347022, "num_tokens": 2264, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7025300573952054, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.3430337617223085}} {"text": "\\section{Joint SED modeling of Photometry and Spectra} \\label{sec:methods}\n\\subsection{Stellar Population Synthesis Modeling} \\label{sec:sps} \nPROVABGS will provide galaxy properties inferred from joint SED modeling of\nDESI photometry and spectra. \nFor the SED modeling, we use a state-of-the-art stellar population synthesis\n(SPS) model that uses a non-parametric SFH with a starburst, a non-parametric\nZH that varies with time, and a flexible dust attenuation prescription. \n\n% describe SFH prescription\nThe form of the SFH is one of the most important factors in the accuracy of an\nSPS model.\nIn general, the form of the SFH requires balancing between being flexible enough\nto describe the wide range of SFHs in observations while not being too flexible\nthat it can describe any SFH at the expense of constraining power. \nIf the model SFH is not flexible enough to describe actual SFHs of galaxies,\nthen unbiased galaxy properties cannot be inferred using the SPS model. \nFor instance, most SPS models~(\\emph{e.g.} CIGALE,~\\citealt{serra2011,\nboquien2019}; BAGPIPES,~\\citealt{carnall2017}) use parametric SFH such as the\nexponentially declining $\\tau$-model.\nSuch functional forms, however, produce biased estimates of galaxy properties\n(\\emph{e.g.} $M_*$ and SFR) when used to fit mock observations of simulated \ngalaxies~\\citep{simha2014, pacifici2015, ciesla2017, carnall2018}.\nOn the other hand, many non-parametric forms of the SFH are overly flexible\nand allow unphysical SFHs~\\citep{leja2019}, which unncessarily increases \nparameter degeneracies and discards constraining power. \n\nIn our SPS model, we use a non-parametric SFH with two components: one based on\nnon-negative matrix factorization (NMF) basis functions and a starburst component.\nFor the first component, SFH is a linear combination of four NMF SFH bases:\n\\begin{equation} \\label{eq:nmf} \n {\\rm SFH}^{\\rm NMF} (t, t_{\\rm age}) = \\sum\\limits_{i=1}^{4} \\beta_i\n \\frac{s_i^{\\rm SFH}(t)}{\\int\\limits_0^{t_{\\rm age}} s_i^{\\rm SFH}(t) \\,\n {\\rm d}t}. \n\\end{equation} \n$\\{s^{\\rm SFH}_i\\}$ are the NMF basis functions and $\\{\\beta_i\\}$ are the\ncoefficients. \nThe integral in the denominator normalizes the NMF basis functions to unity. \nWe constrain $\\sum_i \\beta_i = 1$, so the total SFH of the component over the\nage of the galaxy ($t_{\\rm age})$ is normalized to unity.\n$\\{s^{\\rm SFH}_i\\}$ are derived from the Illustris cosmological hydrodynamic\nsimulation~\\citep{vogelsberger2014, genel2014, nelson2015}.\nWe compile, rebin, and smooth the SFHs of Illustris galaxies and then perform\nnon-negative matrix factorization~\\citep{lee1999,cichocki2009, fevotte2011} on\nthem to derive $\\{s^{\\rm SFH}_i\\}$. \nWe find that 4 components is sufficient to accurately reconstruct the SFHs\nfrom Illustris. \nWe present the NMF SFH bases as a function of lookback time in\nleft panel of Figure~\\ref{fig:nmf}.\n\\edits{\n By using NMF instead of \\emph{e.g.} Principal Component Analysis (PCA), we\n ensure that all of the SFH bases are non-negative and, thus, physically\n meaningful. \n}\nFor further details on the derivation of the NMF bases, we refer readers to\nAppendix~\\ref{sec:nmf}. \nAssuming that the SFHs of Illustris galaxies resemble the SFHs of actual\nobserved galaxies, our NMF form provides a compact and flexible representation\nof the SFHs. \n\nThe NMF basis functions are derived from smooth SFHs, which means that it does\nnot include any stochasticity. \nHowever, observations and high resolution zoom-in hydrodyanmical simulations\nboth find significant stochasticity in galaxy SFHs~\\citep{sparre2017,\ncaplar2019, hahn2019b, iyer2020}. \nTo include some stochasticity in our SPS model, we include a starburst\ncomponent that consists of a SSP. \nThus, for the total SFH, we use\n\\begin{equation} \\label{eq:sfh}\n {\\rm SFH} (t, t_{\\rm age}) = (1 - f_{\\rm burst})~{\\rm SFH}^{\\rm NMF} (t,\n t_{\\rm age}) + f_{\\rm burst}~\\delta_{\\rm D}(t - t_{\\rm burst}).\n\\end{equation}\n$f_{\\rm burst}$ is the fraction of total stellar mass formed during the\nstarburst; $t_{\\rm burst}$ is the time at which the starburst occurs; \n$\\delta_{\\rm D}$ is the Dirac delta function.\nIn total we use 6 free parameters in our SFH: 4 NMF basis coefficients \n($\\beta_i$), $f_{\\rm burst}$, and $t_{\\rm burst}$. \n\n% describe ZH \nAnother key part of an SPS model is the chemical enrichment history, or ZH. \nCurrent SPS models mostly assume a\n\\edits{\n flat ZH, constant metallicity over time~\\citep{carnall2017, leja2019}.\n} \nSince galaxies do not have constant metallicities throughout their history,\nthis assumption can significantly bias the inferred galaxy\nproperties~\\citep{thorne2021}. \nInstead, we take a similar approach to the SFH and use NMF basis functions for\nour ZH:\n\\begin{equation}\n {\\rm ZH}(t) = \\sum\\limits_{i=1}^2 \\gamma_i s_i^{\\rm ZH}(t).\n\\end{equation} \n$\\{s_i^{\\rm ZH}(t)\\}$ are the ZH NMF basis functions and $\\{\\gamma_i\\}$ are the\ncoefficients. \n$\\{s_i^{\\rm ZH}(t)\\}$ are fit using the ZHs of simulated galaxies from\nIllustris in the same fashion as the SFH. \nIn the right panel of Figure~\\ref{fig:nmf}, we present the ZH NMF bases as a\nfunction of lookback time. \nWe use two NMF components, so our ZH prescription has 2 free parameters. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{figs/nmf_bases.pdf} \n \\caption{\n Non-negative matrix factorization basis functions for the SFH (left)\n and ZH (right) used in the non-parametric SFH and ZH prescriptions of\n our SPS model. \n These basis functions are derived from the SFHs and ZHs of simulated\n galaxies in the Illustris cosmological hydrodynamic simulations. \n With the NMF basis functions, we can reproduce the wide range of SFHs\n and ZHs of Illustris galaxies (Appendix~\\ref{sec:nmf}). \n }\n \\label{fig:nmf}\n\\end{center}\n\\end{figure}\n\nWe use the SFH and ZH above to model the unattenuated rest-frame luminosity as\na linear combination of multiple SSPs, evaluated at logarithmically-spaced\nlookback time bins.\nWe use a fixed log-binning with the bin egdes starting with $(0, 10^{6.05}{\\rm\nyr})$, $(10^{6.05}, 10^{6.15}{\\rm yr})$, and continuing on with bins of width\n0.1 dex.\nThe binning is truncated at the age of the model galaxy. \nFor a $z=0$ galaxy, this binning produces 43 $\\tlb$ bins.\nWe use log-spaced $\\tlb$ bins because it better reproduces galaxy luminosities\nevaluated with much higher resolution $\\tlb$ binning than linearly-spacing, for\nthe same number of bins. \n\\edits{At each of the 43 $\\tlb$ bin $i$,}\nwe evaluate the luminosity of a SSP with ${\\rm ZH}(t_i)$, where $t_i$ is the\ncenter of $\\tlb$ bin, and total stellar mass calculated by resampling the SFH\nin Eq.~\\ref{eq:sfh}. \n\\edits{\n We use \\fsps~to evaluate the SSP luminosities and use the MIST isochrones, the\n combination of MILES and BaSeL spectral libraries, and the \\cite{chabrier2003}\n IMF (same as in Section~\\ref{sec:sed}). \n}\nSince we use MIST isochrones, we impose a minimum and maximum limit to ${\\rm\nZH}$ based on its coverage: $4.49\\times10^{-5}$ and $4.49\\times10^{-2}$,\nrespectively.\n\\edits{\n These metallicity values are in units of absolute metallicity and can be\n converted to solar metallicity using $Z_\\odot = 0.019$. \n}\nWe note that our stellar metallicity range is significantly broader than\nprevious studies for additional flexibility~\\citep[\\emph{e.g.}][]{carnall2017,\nleja2017, tacchella2021}. \n\\edits{\n Since we model galaxies solely as a linear combination of SSPs, we do not\n model nebular emission. \n We, therefore, exclude emission lines in our SED modeling by masking the\n wavelength ranges of emission lines.\n}\n\nBefore we combine the SSP luminosities, we apply dust attenuation.\nWe use a two component \\cite{charlot2000} dust attenuation model with birth\ncloud (BC) and diffuse-dust (ISM) components. \nThe BC component represents the extra dust attenuation of young stars that are\nembedded in modecular clouds and HII regions. \nFor SSPs younger than $t_i < 100{\\rm Myr}$, we apply the\nfollowing BC dust attenuation: \n\\begin{equation}\n L_i(\\lambda) = L_i^{\\rm unatten.}(\\lambda) \\exp\\left[-\\tau_{\\rm BC} \\left(\n \\frac{\\lambda}{5500\\AA} \\right)^{-0.7} \\right].\n\\end{equation}\n$\\tau_{\\rm BC}$ is the BC optical depth that determines the strength of the BC\nattenuation. \nAfterwards, {\\em all} SSPs are attenuated by the diffuse dust using the\n\\cite{kriek2013} attenuation curve parameterization: \n\\begin{equation}\n L_i(\\lambda) = L_i^{\\rm unatten.}(\\lambda) \\exp\\left[-\\tau_{\\rm ISM} \\left(\n \\frac{\\lambda}{5500\\AA} \\right)^{n_{\\rm dust}} \\left(k_{\\rm Cal}(\\lambda) +\n D(\\lambda) \\right) \\right].\n\\end{equation}\n$\\tau_{\\rm ISM}$ is the diffuse dust optical depth.\n$n_{\\rm dust}$ is the \\cite{calzetti2001} dust index, which determines the\nslope of the attenuation curve. \n$k_{\\rm Cal}(\\lambda)$ is the \\cite{calzetti2001} attenuation curve and\n$D(\\lambda)$ is the UV dust bump, parameterized using a Lorentzian-like Drude \nprofile:\n\\begin{equation}\n D(\\lambda) = \\frac{E_b(\\lambda~\\Delta \\lambda)^2}{(\\lambda^2 -\n \\lambda_0^2)^2 + (\\lambda~\\Delta \\lambda)^2}\n\\end{equation}\nwhere $\\lambda_0 = 2175 \\AA$, $\\Delta \\lambda = 350\\AA$, and \n$E_b=0.85 - 1.9\\,n_{\\rm dust}$ are the central wavelength, full width at half\nmaximum, and strength of the bump, respectively. \nOnce dust attenuation is applied to the SSPs, we sum them up to get the\nrest-frame luminosity of the galaxy. \nIn total, our SPS model has 12 free parameters: $M_*$, 4 SFH basis\ncoefficients, $f_{\\rm burst}$, $t_{\\rm burst}$, 2 ZH basis coefficients,\n$\\tau_{\\rm BC}$, $\\tau_{\\rm ISM}$, and $n_{\\rm dust}$. \n\n%description of our speculator SED model \\citep{alsing2019}, which is based on FSPS. We use Chabrier IMF \\ch{do we need to justify htis?}. \n\nIn practice, evaluating each SSP using \\fsps~requires \\ch{X} seconds. \nFor each model evaluation, we evaluate $\\sim 43$ SSPs in each of the log-spaced\n$\\tlb$ bins. \nThough this is not a prohibitive computational cost on its own, sampling a\nhigh dimensional parameter space for inference requires $>100,000$ evaluations\n--- \\emph{i.e.} \\ch{$>100$} CPU hours \\emph{per galaxy}. \nFor the >10 million BGS galaxies, this would require \\ch{\\emph{a billion}}\nCPU hours. \nInstead, we use an emulator for the model luminosity, which uses a PCA neural\nnetwork (NN) following the approach of \\cite{alsing2019}. \n\n%The NN provides a flexible and accurate mapping between the SPS model\n%parameters and PCA coefficients --- \\emph{i.e.} the NN predicts PCA\n%coefficients for a given set of SPS parameters. \n%Then the linear combination of the predicted coefficients and PCA basis\n%functions give us the emulated model luminosity. \n%The PCA basis functions and NN are trained using 1,000,000 SPS parameters and\n%model luminosity pairs, $\\{(\\theta, L(\\lambda;\\theta))\\}$. \n\n\\edits{\n To construct our emulator, we first generate $N_{\\rm model} = 1,000,000$\n model luminosities, $L(\\lambda;\\theta)$, from unique SPS parameters,\n $\\theta$, sampled from the prior (Section~\\ref{sec:infer},\n Table~\\ref{tab:params}).\n We then split the model luminosities into four wavelength bins: 2000 - 3600,\n 3600 - 5500, 5500 - 7410, and 7410 - $60000\\AA$ with $N_{\\rm spec}$ = 127, 2109,\n 2113, and 549 resolution elements, respectively.\n For each wavelength bin, a PCA is done in the $N_{\\rm spec}$-dimensional\n space to yield PCA basis functions, or eigenspectra. \n We represent the model luminosity using the first $N_{\\rm basis}$ = 50, 50, 50, and 30\n eigenspectra and their corresponding PCA coefficients. \n A NN is then trained on the set of $N_{\\rm model}$ models to\n derive a mapping from the 12 SPS parameters to the $N_{\\rm basis}$ PCA\n coefficients for each wavelength bin. \n}\n\n\\edits{\n Once trained, our emulator works as follows.\n For a given set of SPS parameters, the NN for each wavelength bin predicts\n PCA coefficients. \n The coefficients are then linearly combined with the eigenspectra to\n predict the model luminosity in the wavelength bin. \n The luminosity in all four wavelength bins are concatenated to produce the\n full model luminosity. \n}\nThroughout the wavelength range relevant for BGS, $3000 < \\lambda < 9800\\AA$,\nwe achieve $< 1\\%$ accurate with the emulator. \nFor details on the training, validation, and performance of our PCA NN\nemulator, we refer readers to Kwon \\etal~(in prep.). \n\nFrom the rest-frame luminosity, we obtain the observed-frame, redshifted, flux\nin the same way as Eq.~\\ref{eq:sed}.\nIn our case, redshift is not a free parameter since we will have high quality\nspectroscopic redshifts for every DESI BGS galaxy.\nBGS redshifts will have small redshift error, $\\sigma_z < 0.0005 (1+z)$\n(150 km/s), and <5\\% catastrophic failures, $\\Delta z/(1+z) < 0.003$ (<1000\nkm/s).\nTo model DESI photometry, we convolve the model flux with the LS broadband\nfilters as in Eq.~\\ref{eq:photo}.\nTo model DESI spectra, we first apply Gaussian velocity dispersion. \nIn this work, we keep velocity dispersion fixed at 0 km/s but in practice\nvelocity dispersion can be set as a free parameter. \nThen the broadened flux is resampled into the DESI wavelength binning. \nSince DESI spectra do not necessarily include all the light of a galaxy, we\ninclude a nuisance parameter $f_{\\rm fiber}$, a normalization factor on the\nspectra to account for fiber aperture effects. \nFinally, the SPS model photometry and spectrum can be directly compared to\nobservations.\n\n\\begin{table} \n\\caption{Parameters of the PROVABGS SPS model and their priors used for joint\n SED modeling of DESI photometry and spectroscopy.} \n\\begin{center}\n \\begin{tabular}{ccc} \\toprule\n name & description & prior \\\\[3pt]\n \\hline \n $\\log M_*$ & log galaxy stellar mass & uniform over [7, 12.5] \\\\\n $\\beta_1, \\beta_2, \\beta_3, \\beta_4$ & NMF basis coefficients for SFH & Dirichlet prior \\\\\n $f_{\\rm burst}$ & fraction of total stellar mass formed in starburst event & uniform over [0, 1] \\\\\n $t_{\\rm burst}$ & time of starburst event & uniform over [10Myr, 13.2Gyr] \\\\\n $\\gamma_1, \\gamma_2$ & NMF basis coefficients for ZH & log uniform over\n [$4.5\\times10^{-5}, 1.5\\times10^{-2}$] \\\\\n $\\tau_{\\rm BC}$ & Birth cloud optical depth & uniform over [$0, 3$] \\\\\n $\\tau_{\\rm ISM}$ & diffuse-dust optical depth & uniform over [$0, 3$] \\\\\n $n_{\\rm dust}$ & \\cite{calzetti2001} dust index & unifrom over[$-2, 1$]\\\\\n $f_{\\rm fiber}$ & spectrum fiber-aperture effect normalization &\n Gaussian $\\mathcal{N}(\\hat{f}^{\\rm fiber}_r, \\frac{f^{\\rm fiber}_r}{f_r} \\sigma_r)$\\\\\n \\hline \n\\end{tabular} \\label{tab:params}\n\\end{center}\n\\end{table}\n\n\\begin{figure}\n\\begin{center}\n \\includegraphics[width=0.9\\textwidth]{figs/mcmc_posterior_demo.pdf}\n \\caption{\n \\emph{Top}: \n Posterior probability distribution of our 12 SPS model parameters\n derived from joint SED modeling of the mock DESI photometry and\n spectrum.\n The contours mark the 68 and 95\\% percentiles.\n We use a Gaussian likelihood and the prior specified in\n Table~\\ref{tab:params} to evaluate the posterior and sample the\n distribution using ensemble slice MCMC. \n \\emph{With our Bayesian SED modeling approach, we capture the significiant\n parameter degeneracies and multimodality of the posterior\n distribution.}\\\\\n \\emph{Bottom}: \n We compare the best-fit model observables (orange) to the mock\n observations (black). \n We find excellent agreement for both the LS photometry (left) and the\n DESI spectrum (right). \n } \\label{fig:posterior}\n\\end{center}\n\\end{figure}\n\n\n\\subsection{Bayesian Parameter Inference} \\label{sec:infer} \nUsing the SPS model above, we perform Bayesian parameter inference to derive\nposterior probability distributions of the SPS parameters from photometry and\nspectroscopy. \nFrom Bayes rule, we write down the posterior as\n\\begin{equation} \\label{eq:bayes}\n p(\\theta\\given {\\bf X}) \\propto p(\\theta)~p({\\bf X} \\given \\theta)\n\\end{equation}\nwhere ${\\bf X}$ is the photometry or spectrum and $\\theta$ is the set of SPS\nparameters. \n$p({\\bf X} \\given \\theta)$ is the likelihood, which we calculate separately for\nthe photometry\n\\begin{equation}\n \\mathcal{L}^{\\rm photo} \\propto \\exp\\left[-\\frac{1}{2} \\left(\\frac{X^{\\rm photo} -\n m^{\\rm photo}(\\theta)}{\\sigma^{\\rm photo}}\\right)\\right]\n\\end{equation}\nand for the spectrum\n\\begin{equation}\n \\mathcal{L}^{\\rm spec} \\propto \\exp\\left[-\\frac{1}{2} \\left(\\frac{X^{\\rm spec} -\n m^{\\rm spec}(\\theta)}{\\sigma^{\\rm spec}} \\right)^2\\right].\n\\end{equation}\n$m^{\\rm photo}$ and $m^{\\rm spec}$ represent SPS model for photometry and\nspectroscopy. \n$\\sigma^{\\rm photo}$ and $\\sigma^{\\rm spec}$ respresent the uncertainties on\nthe measured photometry and spectrum. \nIn calculating $\\mathcal{L}^{\\rm spec}$, we exclude wavelength ranges of width\n40\\AA~surrounding the OII, H$\\beta$, OIII, and H$\\alpha$ emission lines since\nour SED model does not model gas emissions.\nWe consider the photometry indepedent from the spectrum so we combine the\nlikelihoods when jointly modeling the spectrophotometry: \n\\begin{equation}\n \\log \\mathcal{L} \\approx \\log \\mathcal{L}^{\\rm photo} + \\log\n \\mathcal{L}^{\\rm spec}.\n\\end{equation}\n$p(\\theta)$ in Eq.~\\ref{eq:bayes} is the prior on the SPS parameters. \nFor most of our parameters, we use uninformative uniform priors with\nconservatively chosen ranges that are listed in Table~\\ref{tab:params}. \nHowever, for the priors of $\\{\\beta_1, \\beta_2, \\beta_3, \\beta_4 \\}$, the NMF coefficients\nfor the SFH, we use a Dirichlet distribution to main the normalization of the SFH in Eq.~\\ref{eq:nmf}. \nWith a Dirichlet distribution, $\\beta_i$ are within $0 < \\beta_i < 1$ and\nsatisfy the constraint $\\sum_i \\beta_i = 1$. \n\nNow that we can evaluate the posterior at given $\\theta$, we estimate the\nposterior distributions using Markov Chain Monte Carlo (MCMC) sampling. \nWe use the \\cite{karamanis2020} ensemble slice sampling MCMC algorithm with the\n{\\sc zeus} Python\npackage\\footnote{\\href{https://zeus-mcmc.readthedocs.io/}{https://zeus-mcmc.readthedocs.io/}}. \nEnsemble slice sampling is an extension of standard slice sampling that does\nnot requires specifying the initial length scale or any further hand-tuning.\nIt generally converges faster than other MCMC algorithms (\\emph{e.g.}\nMetropolis) and generates chains with significantly lower autocorrelation.\n\nWhen we sample the posterior, we do not directly sample our 12 dimensional\nSPS parameter space because we use a Dirichlet prior on the SFH NMF\ncoefficients. \nDirichlet distributions are difficult to directly sample so we instead use the\n\\cite{betancourt2012} sampling method, which transforms an $N$ dimensional\nDirichlet distribution into an easier to sample $N-1$ dimensional space.\nHence, we sample the posterior in the transformed 11 dimensional space. \nGiven this dimensionality, we run our MCMC sampling with 30 walkers.\nOverall, we find that the sampling converges after 2,500 iterations with a 500\niteration burn in. \nDeriving the posterior distribution from a joint SED modeling of photometry and\nspectra, with the emulator, takes \\ch{0.5 CPU hours}. \nIn principle, since our emulator uses a PCA NN, we can further expedite our\nparemeter inference using more efficient sampling methods that exploit gradient\ninformation, such as Hamiltonian Monte Carlo. \nWe will explore further speed ups to our SED modeling in future works. \n\nIn Figure~\\ref{fig:posterior} we present the posterior distribution of our 12\nSPS model parameters for an arbitrarily chosen \\lgal~mock observation. \nWe mark the 68 and 95 percentiles of the distribution with the contours. \nThe posterior distribution reveal there are significant degeneracies between\nSPS parameters: \\emph{e.g.} $\\beta_2^{\\rm SFH}$ and $f_{\\rm burst}$. \nFurthermore, the distribution is multimodal (see $f_{\\rm burst}$ panels). \nWith our Bayesian SED modeling, we are able to capture such complexities in the\nposterior that would be lost with point estimates or maximum likelihood\napproaches.\nIn the bottom panels, we compare our SPS model evaluated at the best-fit\nparameters (orange) with the \\lgal~mock observations (black). \nOn the left, we compare the $g$, $r$, $z$ band magnitudes; on the right, we\ncompare spectra. \nWe find excellent agreement between the best-fit SPS model and mock\nobservations.\nThe entire PROVABGS SED modeling pipeline, including the neural emulators and\nparameter inference framework, is publicly available at\n\\href{https://github.com/changhoonhahn/provabgs/}{https://github.com/changhoonhahn/provabgs/}. \n", "meta": {"hexsha": "63aae0deee6cd3f03c56ff0b2752c25abe3f4c98", "size": 20758, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/paper/methods.tex", "max_stars_repo_name": "changhoonhahn/gqp_mc", "max_stars_repo_head_hexsha": "abdfadac2e7d7e12c1642743e2b3c6a6a6fe370b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2019-12-18T20:51:45.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-11T05:59:24.000Z", "max_issues_repo_path": "doc/paper/methods.tex", "max_issues_repo_name": "changhoonhahn/gqp_mc", "max_issues_repo_head_hexsha": "abdfadac2e7d7e12c1642743e2b3c6a6a6fe370b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 44, "max_issues_repo_issues_event_min_datetime": "2020-02-20T06:02:00.000Z", "max_issues_repo_issues_event_max_datetime": "2021-04-13T20:00:50.000Z", "max_forks_repo_path": "doc/paper/methods.tex", "max_forks_repo_name": "changhoonhahn/gqp_mc", "max_forks_repo_head_hexsha": "abdfadac2e7d7e12c1642743e2b3c6a6a6fe370b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2019-10-04T22:25:44.000Z", "max_forks_repo_forks_event_max_datetime": "2020-07-20T02:05:03.000Z", "avg_line_length": 52.1557788945, "max_line_length": 139, "alphanum_fraction": 0.7343674728, "num_tokens": 5939, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6859494421679929, "lm_q2_score": 0.5, "lm_q1q2_score": 0.34297472108399646}} {"text": "\\pdfoutput=1\n\n%\\documentclass[preprint]{article}\n\\documentclass[10pt]{amsart}\n\\usepackage{stmaryrd}\n\n\\usepackage{fullpage}\n\\usepackage[colorlinks=true]{hyperref}\n\\usepackage{amsmath,amssymb,amsfonts,amsthm}\n\\usepackage{array} \n\\usepackage{mathtools}\n%\\usepackage{pdfpages}\n\\usepackage{bm}\n\\usepackage{bbm}\n\\usepackage{tikz}\n\\usepackage[normalem]{ulem}\n\\usepackage{hhline}\n\\usepackage{graphicx}\n\\usepackage{subfig}\n\\usepackage{color}\n\\usepackage{doi}\n\n%% ====================================== graphics\n\n\\usepackage{pgfplots}\n\\usepackage{pgfplotstable}\n\\definecolor{markercolor}{RGB}{124.9, 255, 160.65}\n\\pgfplotsset{\ncompat=1.3,\nwidth=10cm,\ntick label style={font=\\small},\nlabel style={font=\\small},\nlegend style={font=\\small}\n}\n\n\\usetikzlibrary{calc}\n\\usetikzlibrary{intersections} \n\n%%% START MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\\newcommand{\\logLogSlopeTriangle}[5]\n{\n % #1. Relative offset in x direction.\n % #2. Width in x direction, so xA-xB.\n % #3. Relative offset in y direction.\n % #4. Slope d(y)/d(log10(x)).\n % #5. Plot options.\n\n \\pgfplotsextra\n {\n \\pgfkeysgetvalue{/pgfplots/xmin}{\\xmin}\n \\pgfkeysgetvalue{/pgfplots/xmax}{\\xmax}\n \\pgfkeysgetvalue{/pgfplots/ymin}{\\ymin}\n \\pgfkeysgetvalue{/pgfplots/ymax}{\\ymax}\n\n % Calculate auxilliary quantities, in relative sense.\n \\pgfmathsetmacro{\\xArel}{#1}\n \\pgfmathsetmacro{\\yArel}{#3}\n \\pgfmathsetmacro{\\xBrel}{#1-#2}\n \\pgfmathsetmacro{\\yBrel}{\\yArel}\n \\pgfmathsetmacro{\\xCrel}{\\xArel}\n\n \\pgfmathsetmacro{\\lnxB}{\\xmin*(1-(#1-#2))+\\xmax*(#1-#2)} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnxA}{\\xmin*(1-#1)+\\xmax*#1} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnyA}{\\ymin*(1-#3)+\\ymax*#3} % in [ymin,ymax].\n \\pgfmathsetmacro{\\lnyC}{\\lnyA+#4*(\\lnxA-\\lnxB)}\n \\pgfmathsetmacro{\\yCrel}{\\lnyC-\\ymin)/(\\ymax-\\ymin)} % THE IMPROVED EXPRESSION WITHOUT 'DIMENSION TOO LARGE' ERROR.\n\n % Define coordinates for \\draw. MIND THE 'rel axis cs' as opposed to the 'axis cs'.\n \\coordinate (A) at (rel axis cs:\\xArel,\\yArel);\n \\coordinate (B) at (rel axis cs:\\xBrel,\\yBrel);\n \\coordinate (C) at (rel axis cs:\\xCrel,\\yCrel);\n\n % Draw slope triangle.\n \\draw[#5] (A)-- node[pos=0.5,anchor=north] {}\n (B)-- \n (C)-- node[pos=0.5,anchor=west] {#4}\n cycle;\n }\n}\n%%% END MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\n\\newcommand{\\logLogSlopeTriangleNeg}[5]\n{\n % #1. Relative offset in x direction.\n % #2. Width in x direction, so xA-xB.\n % #3. Relative offset in y direction.\n % #4. Slope d(y)/d(log10(x)).\n % #5. Plot options.\n\n \\pgfplotsextra\n {\n \\pgfkeysgetvalue{/pgfplots/xmin}{\\xmin}\n \\pgfkeysgetvalue{/pgfplots/xmax}{\\xmax}\n \\pgfkeysgetvalue{/pgfplots/ymin}{\\ymin}\n \\pgfkeysgetvalue{/pgfplots/ymax}{\\ymax}\n\n % Calculate auxilliary quantities, in relative sense.\n \\pgfmathsetmacro{\\xArel}{#1}\n \\pgfmathsetmacro{\\yArel}{#3}\n \\pgfmathsetmacro{\\xBrel}{#1-#2}\n \\pgfmathsetmacro{\\yBrel}{\\yArel}\n \\pgfmathsetmacro{\\xCrel}{\\xArel}\n\n \\pgfmathsetmacro{\\lnxB}{\\xmin*(1-(#1-#2))+\\xmax*(#1-#2)} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnxA}{\\xmin*(1-#1)+\\xmax*#1} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnyA}{\\ymin*(1-#3)+\\ymax*#3} % in [ymin,ymax].\n \\pgfmathsetmacro{\\lnyC}{\\lnyA+#4*(\\lnxA-\\lnxB)}\n \\pgfmathsetmacro{\\yCrel}{\\lnyC-\\ymin)/(\\ymax-\\ymin)} % THE IMPROVED EXPRESSION WITHOUT 'DIMENSION TOO LARGE' ERROR.\n\n % Define coordinates for \\draw. MIND THE 'rel axis cs' as opposed to the 'axis cs'.\n \\coordinate (A) at (rel axis cs:\\xArel,\\yArel);\n \\coordinate (B) at (rel axis cs:\\xBrel,\\yBrel);\n \\coordinate (C) at (rel axis cs:\\xCrel,\\yCrel);\n\n % Draw slope triangle.\n \\draw[#5] (A)-- node[pos=.5,anchor=south] {}\n (B)-- \n (C)-- node[pos=0.5,anchor=west] {#4}\n cycle;\n }\n}\n%%% END MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\n%%% START MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\\newcommand{\\logLogSlopeTriangleFlipNeg}[5]\n{\n % #1. Relative offset in x direction.\n % #2. Width in x direction, so xA-xB.\n % #3. Relative offset in y direction.\n % #4. Slope d(y)/d(log10(x)).\n % #5. Plot options.\n\n \\pgfplotsextra\n {\n \\pgfkeysgetvalue{/pgfplots/xmin}{\\xmin}\n \\pgfkeysgetvalue{/pgfplots/xmax}{\\xmax}\n \\pgfkeysgetvalue{/pgfplots/ymin}{\\ymin}\n \\pgfkeysgetvalue{/pgfplots/ymax}{\\ymax}\n\n % Calculate auxilliary quantities, in relative sense.\n %\\pgfmathsetmacro{\\xArel}{#1}\n %\\pgfmathsetmacro{\\yArel}{#3}\n \\pgfmathsetmacro{\\xBrel}{#1-#2}\n \\pgfmathsetmacro{\\yBrel}{#3}\n \\pgfmathsetmacro{\\xCrel}{#1}\n\n \\pgfmathsetmacro{\\lnxB}{\\xmin*(1-(#1-#2))+\\xmax*(#1-#2)} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnxA}{\\xmin*(1-#1)+\\xmax*#1} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnyA}{\\ymin*(1-#3)+\\ymax*#3} % in [ymin,ymax].\n \\pgfmathsetmacro{\\lnyC}{\\lnyA+#4*(\\lnxA-\\lnxB)}\n \\pgfmathsetmacro{\\yCrel}{\\lnyC-\\ymin)/(\\ymax-\\ymin)} % THE IMPROVED EXPRESSION WITHOUT 'DIMENSION TOO LARGE' ERROR.\n\n\t\\pgfmathsetmacro{\\xArel}{\\xBrel}\n \\pgfmathsetmacro{\\yArel}{\\yCrel}\n\n % Define coordinates for \\draw. MIND THE 'rel axis cs' as opposed to the 'axis cs'.\n \\coordinate (A) at (rel axis cs:\\xArel,\\yArel);\n \\coordinate (B) at (rel axis cs:\\xBrel,\\yBrel);\n \\coordinate (C) at (rel axis cs:\\xCrel,\\yCrel);\n\n % Draw slope triangle.\n \\draw[#5] (A)-- node[pos=0.5,anchor=east] {#4}\n (B)-- \n (C)-- node[pos=0.5,anchor=north] {1}\n cycle;\n }\n}\n%%% END MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\n\n%%% START MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\\newcommand{\\logLogSlopeTriangleFlip}[5]\n{\n % #1. Relative offset in x direction.\n % #2. Width in x direction, so xA-xB.\n % #3. Relative offset in y direction.\n % #4. Slope d(y)/d(log10(x)).\n % #5. Plot options.\n\n \\pgfplotsextra\n {\n \\pgfkeysgetvalue{/pgfplots/xmin}{\\xmin}\n \\pgfkeysgetvalue{/pgfplots/xmax}{\\xmax}\n \\pgfkeysgetvalue{/pgfplots/ymin}{\\ymin}\n \\pgfkeysgetvalue{/pgfplots/ymax}{\\ymax}\n\n % Calculate auxilliary quantities, in relative sense.\n %\\pgfmathsetmacro{\\xArel}{#1}\n %\\pgfmathsetmacro{\\yArel}{#3}\n \\pgfmathsetmacro{\\xBrel}{#1-#2}\n \\pgfmathsetmacro{\\yBrel}{#3}\n \\pgfmathsetmacro{\\xCrel}{#1}\n\n \\pgfmathsetmacro{\\lnxB}{\\xmin*(1-(#1-#2))+\\xmax*(#1-#2)} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnxA}{\\xmin*(1-#1)+\\xmax*#1} % in [xmin,xmax].\n \\pgfmathsetmacro{\\lnyA}{\\ymin*(1-#3)+\\ymax*#3} % in [ymin,ymax].\n \\pgfmathsetmacro{\\lnyC}{\\lnyA+#4*(\\lnxA-\\lnxB)}\n \\pgfmathsetmacro{\\yCrel}{\\lnyC-\\ymin)/(\\ymax-\\ymin)} % THE IMPROVED EXPRESSION WITHOUT 'DIMENSION TOO LARGE' ERROR.\n\n\t\\pgfmathsetmacro{\\xArel}{\\xBrel}\n \\pgfmathsetmacro{\\yArel}{\\yCrel}\n\n % Define coordinates for \\draw. MIND THE 'rel axis cs' as opposed to the 'axis cs'.\n \\coordinate (A) at (rel axis cs:\\xArel,\\yArel);\n \\coordinate (B) at (rel axis cs:\\xBrel,\\yBrel);\n \\coordinate (C) at (rel axis cs:\\xCrel,\\yCrel);\n\n % Draw slope triangle.\n \\draw[#5] (A)-- node[pos=0.5,anchor=east] {#4}\n (B)-- \n (C)-- node[pos=0.5,anchor=south] {}\n cycle;\n }\n}\n%%% END MACRO FOR ANNOTATION OF TRIANGLE WITH SLOPE %%%.\n\n\\theoremstyle{definition}\n\\newtheorem{definition}{Definition}\n\\theoremstyle{lemma}\n\\newtheorem{lemma}{Lemma}\n\\newtheorem*{remark}{Remark}\n\\theoremstyle{theorem}\n\\newtheorem{theorem}{Theorem}\n\\theoremstyle{assumption}\n\\newtheorem{assumption}{Assumption}\n\n%\\DeclareMathOperator{\\diag}{diag}\n\\DeclareMathOperator{\\diam}{diam}\n\n\\renewcommand{\\topfraction}{0.85}\n\\renewcommand{\\textfraction}{0.1}\n\\renewcommand{\\floatpagefraction}{0.75}\n\n\n\\newcommand{\\bbm}[1]{\\mathbbm{#1}}\n\\newcommand{\\bs}[1]{\\boldsymbol{#1}}\n\\newcommand{\\equaldef}{\\stackrel{\\mathrm{def}}{=}}\n\n\n\\newcommand{\\mb}[1]{\\mathbf{#1}}\n\\newcommand{\\mbb}[1]{\\mathbb{#1}}\n\\newcommand{\\mc}[1]{\\mathcal{#1}}\n\n\\renewcommand{\\hat}{\\widehat}\n\\renewcommand{\\tilde}{\\widetilde}\n\\newcommand{\\td}[2]{\\frac{{\\rm d}#1}{{\\rm d}{\\rm #2}}}\n\\newcommand{\\pd}[2]{\\frac{\\partial#1}{\\partial#2}}\n\\newcommand{\\pdn}[3]{\\frac{\\partial^{#3}#1}{\\partial#2^{#3}}}\n\\newcommand{\\snor}[1]{\\left| #1 \\right|}\n\\newcommand{\\nor}[1]{\\left\\| #1 \\right\\|}\n\\newcommand{\\LRp}[1]{\\left( #1 \\right)}\n\\newcommand{\\LRs}[1]{\\left[ #1 \\right]}\n\\newcommand{\\LRa}[1]{\\left\\langle #1 \\right\\rangle}\n\\newcommand{\\LRb}[1]{\\left| #1 \\right|}\n\\newcommand{\\LRc}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\LRceil}[1]{\\left\\lceil #1 \\right\\rceil}\n\\newcommand{\\LRl}[1]{\\left. \\LRp{#1} \\right|}\n\\newcommand{\\jump}[1] {\\ensuremath{\\llbracket#1\\rrbracket}}\n\\newcommand{\\avg}[1] {\\ensuremath{\\LRc{\\!\\{#1\\}\\!}}}\n\\newcommand{\\Grad} {\\ensuremath{\\nabla}}\n\\newcommand{\\note}[1]{{\\color{blue}{#1}}}\n\\newcommand{\\lcwnote}[1]{{\\color{magenta}{#1}}}\n\\renewcommand{\\d}{\\partial}\n\\newcommand{\\diag}[1]{{\\rm diag}\\LRp{#1}}\n\n\n\n\\newcommand{\\LK}{L^2\\LRp{D^k}}\n\\newcommand{\\LdK}{L^2\\LRp{\\partial D^k}}\n\\newcommand{\\Dhat}{\\widehat{D}}\n\\newcommand{\\Lhat}{L^2\\LRp{\\Dhat}}\n\n\n\\newcommand*\\diff[1]{\\mathop{}\\!{\\mathrm{d}#1}} % d in integrand\n\n\\date{}\n\\author{Jesse Chan, Lucas C. Wilcox}\n\\title{On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes}% and GPU acceleration}\n\\graphicspath{{./figs/}}\n\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nWe construct entropy conservative and entropy stable high order accurate discontinuous Galerkin (DG) discretizations for time-dependent nonlinear hyperbolic conservation laws on curvilinear meshes by extending the methodology of \\cite{chan2017discretely}. The resulting schemes preserve a semi-discrete quadrature approximation of a continuous global entropy inequality. The proof requires the satisfaction of a discrete geometric conservation law, which we enforce through a modification of an approach introduced in \\cite{kopriva2006metric}. We extend the construction of entropy conservative and entropy stable DG schemes to the case when high order accurate curvilinear mass matrices are approximated using low-storage weight-adjusted approximations \\cite{chan2016weight1, chan2016weight2}. We present numerical experiments which verify theoretical results for the compressible Euler equations on triangular and tetrahedral meshes. %in two and three dimensions.\n\\end{abstract}\n\n%\\tableofcontents\n\n\\section{Introduction}\n\nHigh order discontinuous Galerkin (DG) methods are attractive for the simulation of time-dependent wave propagation due to their low numerical dispersion and dissipation \\cite{hu1999analysis, ainsworth2004dispersive} and ability to handle unstructured meshes and complex geometries. These same properties make them attractive for the resolution of transient waves and vortices in compressible flow \\cite{wang2013high}. However, whereas the construction of stable DG methods for wave propagation is relatively well-established, it is not possible to extend the same formulations directly from linear wave problems to the nonlinear conservation laws which govern compressible fluid flow. \n\nThe low numerical dissipation of high order DG methods combined with the lack of inherently stable formulations for nonlinear conservation laws has given high order discretizations the reputation of being non-robust and highly sensitive to instabilities and under-resolved features \\cite{wang2013high}. This instability is addressed in practice by adding additional stabilization through limiting, filtering, or artificial viscosity \\cite{persson2006sub, krivodonova2007limiters, barter2010shock, guermond2011entropy}. However, these approaches are typically ad-hoc, and do not guarantee the stability of the resulting scheme. Moreover, high order accuracy can be lost if the stabilization is too strong. \n\nThe lack of inherently stable formulations was addressed for high order nodal DG methods on quadrilateral and hexahedral meshes in \\cite{fisher2013high, carpenter2014entropy}. The resulting schemes ensure that that the numerical solution satisfies a semi-discrete version of an entropy inequality, independently of discrete effects such as under-integration. This results in significantly more robust high order simulations, where the numerical solution does not blow up even in the presence of under-resolved features such as shock discontinuities or turbulence. High order entropy stable schemes have since been extended to staggered grid and non-conforming \\cite{parsani2016entropy, friedrich2017entropy} tensor product elements, as well as to simplicial meshes \\cite{chen2017entropy, chan2017discretely, crean2018entropy}. \n\nEntropy stable methods have largely relied on a finite difference summation-by-parts (SBP) framework. The summation-by-parts property holds for spectral element DG methods (DG-SEM), which assume a polynomial basis which collocates the solution at Gauss--Legendre--Lobatto (GLL) quadrature nodes on tensor product elements. The collocation approach requires the number of quadrature nodes to be identical to the number of basis functions. Because analogous quadrature rules on the triangle and tetrahedron must contain more nodes than the dimension of the underlying polynomial space \\cite{helenbrook2009existence, hicken2016multidimensional}, GLL-like collocation cannot be replicated on simplicial elements. It is still possible to construct entropy stable schemes on simplicial elements within the SBP framework \\cite{chen2017entropy, crean2018entropy}. However, the resulting SBP operators are not ``modal'', in the sense that the matrices are not associated with an underlying basis or approximation space. \n\nEntropy stable schemes were extended to modal discretizations on affine meshes in \\cite{chan2017discretely}, allowing for the use of arbitrary basis functions and over-integrated quadrature rules. In this work, we show how to extend the construction of modal high order entropy stable schemes to curved meshes. As noted in \\cite{crean2018entropy}, this requires the use of a split formulation for the geometric terms involved in differentiation, as well as the satisfaction of a discrete geometric conservation law (GCL) \\cite{thomas1979geometric, kopriva2006metric}. We also show how to extend entropy stable schemes to accomodate weight-adjusted mass matrices, which provide low storage approximations of inverse weighted mass matrices appearing for curved meshes \\cite{chan2016weight2}. These weight-adjusted approximations can be applied more efficiently than inverse weighted mass matrices on many-core architectures such as Graphics Processing Units (GPUs) \\cite{chan2017weight}. However, additional steps are required to ensure entropy stability and the conservation of mass, momentum, and energy under weight-adjusted mass matrices.\n\nThe paper is organized as follows: Section~\\ref{sec:1} reviews the derivation of entropy inequalities for systems of nonlinear conservation laws, and describes why this does not hold under a high order DG discretization. Section~\\ref{sec:2} describes the construction of entropy stable DG methods for curved meshes using weighted mass matrices, and Section~\\ref{sec:3} describes how to extend this to the case of weight-adjusted mass matrices. Section~\\ref{sec:4} describes how to enforce the geometric conservation law using a modification of the approach described in \\cite{kopriva2006metric}, and Section~\\ref{sec:num} concludes by presenting two and three-dimensional numerical experiments which verify the accuracy and stability of the presented schemes. \n\n\\section{Systems of nonlinear conservation laws}\n\\label{sec:1}\nThis work addresses high order accurate schemes for the following system of $n$ nonlinear conservation laws in $d$ dimensions \n\\begin{equation}\n \\pd{\\bm{u}}{t} + \\sum_{j=1}^d\\pd{\\bm{f}_j(\\bm{u})}{x_j} = 0, \\qquad \\bm{u} : \\mathbb{R}^d \\times [0,\\infty) \\rightarrow \\mathbb{R}^n, \\qquad \\bm{f}_j : \\mathbb{R}^n\\rightarrow \\mathbb{R}^n,\n\\label{eq:nonlineqs}\n\\end{equation}\nwhere $\\bm{u}(\\bm{x},t)$ denotes the \\emph{conservative variables} for this system. %We assume the system is well-posed under appropriate boundary conditions.\nWe are interested in nonlinear conservation laws for which an entropy function $U(\\bm{u})$ exists, where $U(\\bm{u})$ is convex with respect to the conservative variables $\\bm{u}$. If this function exists, then it is possible to define \\emph{entropy variables} $\\bm{v}(\\bm{u}) = \\pd{U}{\\bm{u}}$. These functions symmetrize the system of nonlinear conservation laws (\\ref{eq:nonlineqs}) \\cite{hughes1986new}. \n\nIt can be shown (see, for example, \\cite{mock1980systems}) that symmetrization is equivalent to the existence of entropy flux functions $F_j(\\bm{u})$ and entropy potentials $\\psi_j$ such that\n\\[\n\\bm{v}^T \\pd{\\bm{f}_j}{\\bm{u}} = \\pd{F_j(\\bm{u})}{\\bm{u}}^T, \\qquad \\psi_j(\\bm{v}) = \\bm{v}^T\\bm{f}_j(\\bm{u}(\\bm{v})) - F_j(\\bm{u}(\\bm{v})), \\qquad \\psi_j'(\\bm{v}) = \\bm{f}_j(\\bm{u}(\\bm{v})).\n\\]\nSmooth solutions of (\\ref{eq:nonlineqs}) can be shown to satisfy a conservation of entropy by multiplying (\\ref{eq:nonlineqs}) by $\\bm{v}(\\bm{u})$. Using the definition of the entropy variables, entropy flux, and the chain rule yields\n\\begin{equation}\n\\bm{v}^T\\pd{\\bm{f}_j(\\bm{u})}{x_j} = \\pd{U(\\bm{u})}{\\bm{u}}^T\\pd{\\bm{f}_j(\\bm{u})}{\\bm{u}}\\pd{\\bm{u}}{x_j} = \\pd{F_j(\\bm{u})}{x_j}, \\qquad \\pd{U(\\bm{u})}{t} + \\sum_{j=1}^d \\pd{F_j(\\bm{u})}{x_j} = 0.\n\\label{eq:chainrule}\n\\end{equation}\n%and that\n%\\begin{equation*}\n%\n%%\\label{eq:entropyeqstrong}\n%\\end{equation*}\nLet $\\Omega \\subset \\mathbb{R}^d$ be a closed domain with boundary $\\partial \\Omega$. Integrating over $\\Omega$ and using Gauss' theorem on the spatial derivative yields\n\\begin{equation}\n \\int_{\\Omega}\\pd{U(\\bm{u})}{t}\\diff{\\bm{x}} + \\int_{\\partial \\Omega} \\sum_{j=1}^d \\LRp{\\bm{v}{(\\bm{u})}^T\\bm{f}_j(\\bm{u}) - \\psi_j\\LRp{\\bm{v}(\\bm{u})}}n_j \\diff{\\bm{x}} = 0,\n\\label{eq:entropyeq}\n\\end{equation}\nwhere $\\bm{n} = \\LRp{n_1,\\ldots,n_d}^T$ denotes the unit outward normal vector on $\\partial \\Omega$. \n\nGeneral solutions (including non-smooth solutions such as shocks) satisfy an entropy \\emph{inequality}\n\\begin{equation}\n \\int_{\\Omega}\\pd{U(\\bm{u})}{t}\\diff{\\bm{x}} + \\int_{\\partial \\Omega} \\sum_{j=1}^d \\LRp{\\bm{v}{(\\bm{u})}^T\\bm{f}_j(\\bm{u}) - \\psi_j\\LRp{\\bm{v}(\\bm{u})}}n_j \\diff{\\bm{x}} \\leq 0,\n\\label{eq:entropyineq}\n\\end{equation}\nwhich results from considering solutions of an appropriate viscous form of the equations (\\ref{eq:nonlineqs}) and taking the limit as viscosity vanishes. In this work, schemes which satisfy a discrete form of (\\ref{eq:entropyineq}) will be constructed by first enforcing a discrete version of entropy conservation (\\ref{eq:entropyeq}), then adding an appropriate numerical dissipation which will enforce the entropy inequality (\\ref{eq:entropyineq}). \n\n\n\\subsection{Standard DG formulations for nonlinear conservation laws}\n\nWe begin by reviewing the construction of standard high order accurate DG formulations for (\\ref{eq:nonlineqs}). \n\n\\subsubsection{Mathematical notation}\n\nLet the domain $\\Omega \\subset \\mathbb{R}^d$ be decomposed into elements (subdomains) $D^k$, and let $\\hat{D}$ denote a $d$-dimensional reference element with boundary $\\partial \\hat{D}$. Let $\\hat{\\bm{x}} = \\LRc{\\hat{x}_1,\\ldots,\\hat{x}_d}$ denote coordinates on $\\hat{D}$, and let $\\hat{n}_i$ denote and the $i$th component of the unit normal vector on $\\partial \\hat{D}$. We assume that $\\hat{n}_i$ is constant; i.e., that the faces of the reference element are planar (this assumption holds for all commonly used reference elements \\cite{chan2015gpu}). \n\nWe will assume that each physical element $D^k$ is the image of $\\hat{D}$ under some smoothly differentiable mapping $\\bm{\\Phi}_k(\\hat{\\bm{x}})$ such that\n\\[\n\\bm{x} = \\bm{\\Phi}_k(\\hat{\\bm{x}}), \\qquad \\bm{x}\\in D^k.\n\\]\nThis also implies that integrals over physical elements can be mapped back to the reference element as follows\n\\[\n\\int_{D^k} u \\diff{\\bm{x}} = \\int_{\\hat{D}} u J^k\\diff{\\hat{\\bm{x}}}, \n\\]\nwhere $J^k$ denotes the determinant of the Jacobian of $\\bm{\\Phi}_k$. Integrals over physical faces of $D^k$ can similarly be mapped back to reference faces.\n\n\nWe define an approximation space using degree $N$ polynomials on the reference element. For example, on a $d$-dimensional reference simplex, the natural polynomial space are total degree $N$ polynomials \n\\[\nP^N\\LRp{\\widehat{D}} = \\LRc{\\hat{x}_1^{i_1}\\ldots\\hat{x}_d^{i_d}, \\quad \\hat{\\bm{x}} \\in \\widehat{D}, \\quad 0\\leq \\sum_{k=1}^d i_k \\leq N}.\n\\]\nOther element types possess different natural polynomial spaces \\cite{chan2015gpu}, but typically contain the space of total degree $N$ polynomials. This work is directly applicable to other elements and spaces as well. We denote the dimension of the approximation space $P^N$ as $N_p = {\\rm dim}\\LRp{P^N\\LRp{\\widehat{D}}}$. We also define trace spaces for each face of the reference element. Let $\\hat{f}$ be a face of the reference element $\\hat{D}$. The trace space over $\\hat{f}$ is defined as the space of traces of functions in $P^N\\LRp{\\hat{D}}$\n\\[\nP^N_f \\LRp{\\hat{f}} = \\LRc{ \\left.u\\right|_{\\hat{f}}, \\quad u \\in P^N\\LRp{\\hat{D}}}, \\qquad \\hat{f}\\in \\partial\\hat{D}.\n\\]\nWe denote the dimension of the trace space as ${\\rm dim}\\LRp{P^N_f\\LRp{\\hat{f}}} = N^f_p$.\n\nWe next define the $L^2$ norm and inner products over the reference element $\\hat{D}$ and the surface of the reference element $\\partial \\hat{D}$ as\n\\[\n \\LRp{\\bm{u},\\bm{v}}_{\\hat{D}} = \\int_{\\widehat{D}} \\bm{u}\\cdot\\bm{v} \\diff{\\hat{\\bm{x}}}, \\qquad \\nor{\\bm{u}}^2_{\\hat{D}} = (\\bm{u},\\bm{u})_{\\hat{D}}, \\qquad \\LRa{\\bm{u},\\bm{v}}_{\\partial \\hat{D}} = \\int_{\\partial \\hat{D}} \\bm{u} \\cdot \\bm{v} \\diff{\\hat{\\bm{x}}}.\n\\]\nWe also introduce the continuous $L^2$ projection operator $\\Pi_N$ and lifting operator $L$. For $u \\in L^2\\LRp{\\widehat{D}}$, the $L^2$ projection $\\Pi_N u$ is defined through\n\\begin{equation}\n\\int_{\\widehat{D}} \\Pi_N u v \\diff{\\hat{\\bm{x}}} = \\int_{\\widehat{D}} u v \\diff{\\hat{\\bm{x}}}, \\qquad \\forall v\\in P^N\\LRp{\\hat{D}}.\n\\label{eq:l2proj}\n\\end{equation}\nLikewise, for a boundary function $u \\in L^2\\LRp{\\partial \\hat{D}}$, the lifting operator $L$ \\cite{hesthaven2007nodal, di2011mathematical} is defined through \n\\begin{equation}\n\\LRp{L u,v}_{\\hat{D}} = \\LRa{u,v}_{\\partial \\hat{D}}, \\qquad \\forall v \\in P^N\\LRp{\\hat{D}}.\n\\label{eq:lift}\n\\end{equation}\n\nFinally, we introduce $L^2, L^{\\infty}$ Sobolev norms and spaces, which will be utilized for error estimates. The $L^2$ space is defined as the space of functions with finite $L^2$ norm. The Lebesgue $L^\\infty$ norm and the associated $L^\\infty$ space over a general domain $\\Omega$ are \n\\begin{align*}\n\\nor{u}_{L^{\\infty}\\LRp{\\Omega}} &= \\inf\\LRc{C \\geq 0: \\LRb{u\\LRp{\\bm{x}}} \\leq C \\quad \\forall \\bm{x}\\in \\Omega}, \\qquad\nL^{\\infty}\\LRp{\\Omega} = \\LRc{u: \\Omega\\rightarrow \\mathbb{R}, \\quad \\nor{u}_{L^{\\infty}\\LRp{\\Omega}} < \\infty}.\n\\end{align*}\nThe $L^2$ and $L^{\\infty}$ Sobolev norms of degree $s$ are then defined as\n\\begin{align*}\n%\\LRb{u}_{W^{s,p}\\LRp{\\Omega}} &= \\LRp{\\sum_{\\LRb{\\alpha}= s} \\nor{ D^{\\alpha} u}_{L^p\\LRp{\\Omega}}^p}^{1/p}, \\qquad \\LRb{u}_{W^{s,\\infty}\\LRp{\\Omega}} = \\max_{\\LRb{\\alpha}= s} \\nor{D^{\\alpha}u}_{L^{\\infty}\\LRp{\\Omega}}\\\\\n\\nor{u}_{W^{s,2}\\LRp{\\Omega}}^2 &= {\\sum_{\\LRb{\\alpha}\\leq s} \\nor{ D^{\\alpha} u}_{L^2\\LRp{\\Omega}}^2}, \\qquad \\nor{u}_{W^{s,\\infty}\\LRp{\\Omega}} = \\max_{\\LRb{\\alpha}\\leq s} \\nor{D^{\\alpha}u}_{L^{\\infty}\\LRp{\\Omega}},\n\\end{align*}\nrespectively. Here $\\alpha = \\LRc{\\alpha_1,\\ldots,\\alpha_d}$ is a multi-index\nof order $\\LRb{\\alpha} = \\alpha_1 + \\cdots + \\alpha_d$ such that\n\\[\nD^{\\alpha}u = \\pd{^{\\alpha_1}}{x_1^{\\alpha_1}}\\cdots\\pd{^{\\alpha_d}}{x_d^{\\alpha_d}} u.\n\\]\nThe Sobolev spaces $W^{s,2}$ and $W^{s,\\infty}$ are defined as the spaces of functions with finite $L^2$ and $L^\\infty$ Sobolev norms of degree $s$, respectively.\n\n\n\n\\subsubsection{Discontinuous Galerkin formulations and the $L^2$ projection}\n\nDiscontinuous Galerkin methods have been widely applied to systems of nonlinear conservation laws (\\ref{eq:nonlineqs}) \\cite{cockburn1989tvb, cockburn1998runge, cockburn2001devising}. The development of new discontinuous Galerkin methods for nonlinear conservation laws has focused heavily on the choice of numerical flux \\cite{qiu2006numerical} or the development of slope limiters \\cite{krivodonova2007limiters, zhang2012maximum} and artificial viscosity strategies \\cite{persson2006sub, barter2010shock, klockner2011viscous}. However, the treatment of the underlying volume discretization remains relatively unchanged between each of these approaches. \n\nIgnoring terms involving filters, limiters, or artificial viscosity, a semi-discrete ``weak'' DG formulation for (\\ref{eq:nonlineqs}) can be given locally over an element $D^k$: find $\\bm{u}\\in \\LRp{P^N\\LRp{D^k} \\times [0,\\infty)}^n$ such that\n\\begin{align}\n\\int_{D^k} \\LRp{\\pd{\\bm{u}}{t}\\cdot \\bm{v} - \\sum_{j=1}^d\\bm{f}_j(\\bm{u}) \\cdot \\pd{\\bm{v}}{x_i}} \\diff{\\bm{x}} \n+ \\sum_{j=1}^d \\int_{\\partial D^k} \\LRp{\\bm{f}^*_j\\LRp{\\bm{u}^+,\\bm{u}} }\\cdot \\bm{v} n_j \\diff{\\bm{x}} = 0, \\qquad \\forall \\bm{v}\\in \\LRp{P^N\\LRp{D^k}}^n,\n\\label{eq:weakdg}\n\\end{align}\nwhere the numerical flux $\\bm{f}^*$ is a function of the solution $\\bm{u}$ on both $D^k$ and neighboring elements. \n\nUnfortunately, solutions to (\\ref{eq:weakdg}) do not (in general) obey a discrete version of the entropy inequality (\\ref{eq:entropyineq}). Since (\\ref{eq:entropyineq}) is a generalized statement of energy stability, the lack of a discrete entropy inequality implies that the discrete solution can blow up in finite time. The reason for this is due to the fact that, in practice, the integrals in (\\ref{eq:weakdg}) are not computed exactly and are instead approximated using polynomially exact quadratures. This is compounded by the fact that the nonlinear flux function $\\bm{f}_j\\LRp{\\bm{u}}$ is often rational and impossible to integrate exactly using polynomial quadratures. \n\nWhile this paper focuses on curved meshes, the inexactness of quadrature leads to the loss of the chain rule and thus the loss of entropy conservation or entropy dissipation (\\ref{eq:entropyineq}) even on affine meshes. We can rewrite (\\ref{eq:weakdg}) in a strong form using a discrete quadrature-based $L^2$ projection. For polynomial approximation spaces on affine meshes, $\\pd{\\bm{v}}{x_i}$ is polynomial. \nThen, mapping (\\ref{eq:weakdg}) back to the reference element $\\hat{D}$ and using the $L^2$ projection and (\\ref{eq:l2proj}), we have that\n\\[\n\\int_{D^k} \\bm{f}_j(\\bm{u}) \\cdot \\pd{\\bm{v}}{x_i} \\diff{\\bm{x}} = \\int_{\\hat{D}} \\Pi_N \\bm{f}_j(\\bm{u}) \\cdot \\pd{\\bm{v}}{x_i} J^k\\diff{\\bm{x}}.\n\\]\nThus, integrating by parts (\\ref{eq:weakdg}) recovers a ``strong'' DG formulation involving the projection operator\n\\begin{align}\n&\\int_{D^k} \\LRp{\\pd{\\bm{u}}{t} - \\sum_{j=1}^d \\pd{\\Pi_N \\bm{f}_j(\\bm{u})}{x_j}} \\cdot \\bm{v} \\diff{\\bm{x}} \\nonumber\\\\\n&+ \\sum_{j=1}^d \\int_{\\partial D^k} \\LRp{\\bm{f}^*_j\\LRp{\\bm{u}^+,\\bm{u}} - \\Pi_N\\bm{f}_j(\\bm{u})}\\cdot \\bm{v} n_j \\diff{\\bm{x}} = 0, \\qquad \\forall \\bm{v}\\in \\LRp{P^N\\LRp{D^k}}^n. \n\\label{eq:strongdg}\n\\end{align}\nFrom this, we see that our discrete scheme does not differentiate the nonlinear flux function $\\bm{f}_j\\LRp{\\bm{u}}$ exactly, but instead differentiates the projection of $\\Pi_N \\bm{f}_j\\LRp{\\bm{u}}$ onto polynomials of degree $N$. Because the $L^2$ projection operator is introduced, the chain rule no longer holds at the discrete level and step (\\ref{eq:chainrule}) of the proof of entropy conservation is no longer valid. Thus, ensuring discrete entropy stability will require a discrete formulation of the system of nonlinear conservation laws (\\ref{eq:nonlineqs}) from which we can prove a discrete entropy inequality without relying on the chain rule. \n\n%We note that it is not strictly necessary to introduce the projection operator; however, constructing a ``strong'' DG formulation without a projection operator requires an analytical expression for derivatives of the nonlinear flux function $\\bm{f}_j\\LRp{\\bm{u}}$, which can be complex or expensive to compute. Moreover, because polynomial quadrature rules are typically used in practice to evaluate integrals, it is not possible to use integration by parts when the nonlinear flux function is rational due to errors in approximating integrals. Introducing the $L^2$ projection operator resolves both issues, as the integrands involved in integration by parts are polynomials of degree $2N-1$, for which efficient quadratures exist. \n\n\\section{Discretely entropy stable DG methods on curved meshes}\n\\label{sec:2}\nWe will first show how to construct discretely entropy stable high order accurate DG methods on curvilinear meshes, but will present this using a matrix formulation as opposed to a continuous formulation. This is to ensure that the effects of discretization, nonlinear, and quadrature are accounted for in the proof of semi-discrete entropy stability. We first introduce quadrature-based matrices, which we will then use to construct discretely entropy stable DG formulations.\n\n\n\\subsection{Basis and quadrature rules}\n\n{We now introduce quadrature-based matrices for the $d$-dimensional reference element $\\widehat{D}$, which we will use to construct matrix-vector formulations of DG methods. Assuming $u(\\hat{\\bm{x}}) \\in P^N\\LRp{\\widehat{D}}$, it can be represented in terms of the vector of coefficients $\\bm{u}$ using some polynomial basis $\\phi_i$ of degree $N$ and dimension $N_p$ \n\\[\n u(\\hat{\\bm{x}}) = \\sum_{j=1}^{N_p}\\bm{u}_j \\phi_j(\\widehat{\\bm{x}}), \\qquad P^N\\LRp{\\widehat{D}} = {\\rm span}\\LRc{\\phi_i(\\widehat{x})}_{i=1}^{N_p}.\n\\]\n\nWe construct quadrature-based matrices based on $\\phi_i$ and appropriate volume and surface quadrature rules. The volume and surface quadrature rules are given by points and positive weights $\\LRc{(\\hat{\\bm{x}}_i, \\hat{w}_i)}_{i=1}^{N_q}$ and $\\LRc{(\\hat{\\bm{x}}^f_i, \\hat{w}^f_i)}_{i=1}^{N^f_q}$, respectively. We make the following assumptions on the strength of these quadratures: %\\note{finish! Note that we want quadratures such that discrete IBP holds. For example, GLL or $2N-1$ vol and $2N$ surface. }\n\\begin{assumption}[Integration by parts under quadrature]\n The volume quadrature rule $\\LRc{(\\hat{\\bm{x}}_i, \\hat{w}_i)}_{i=1}^{N_q}$ is exact for polynomials of degree $2N-1$. Additionally, \nfor any $u, v \\in P^N\\LRp{\\hat{D}}$, integration by parts \n\\[\n \\LRp{\\pd{u}{\\hat{x}_i},v}_{\\hat{D}} = \\LRa{u,v\\hat{n}_i}_{\\partial \\hat{D}} - \\LRp{u,\\pd{v}{\\hat{x}_i}}_{\\hat{D}}\n\\]\nholds when the volume and surface integrals are approximated using quadrature.\n\\label{ass:quad}\n\\end{assumption}\nAssumption~\\ref{ass:quad} holds, for example, for any surface quadrature rule which is exact for degree $2N$ polynomials on the boundary of the reference element $\\partial \\hat{D}$. \n\n\n\\subsection{Reference element matrices}\n\\label{sec:matrix}\n\nLet $\\bm{W}, \\bm{W}_f$ denote diagonal matrices whose entries are volume and surface quadrature weights, respectively. The surface quadrature weights are given by quadrature weights on reference faces, which are mapped to faces of the reference element. We define the volume and surface quadrature interpolation matrices $\\bm{V}_q$ and $\\bm{V}_f$ such that\n\\begin{align}\n\\LRp{\\bm{V}_q}_{ij} &= \\phi_j(\\hat{\\bm{x}}_i), \\qquad 1 \\leq j \\leq N_p, \\qquad 1 \\leq i \\leq N_q, \\nonumber\\\\\n\\LRp{\\bm{V}_f}_{ij} &= \\phi_j(\\hat{\\bm{x}}^f_i), \\qquad 1 \\leq j \\leq N_p, \\qquad 1 \\leq i \\leq N^f_q,\\label{eq:qinterp}\n\\end{align}\nwhich map coefficients $\\bm{u}$ to evaluations of $u$ at volume and surface quadrature points. \n\nNext, let ${\\bm{D}}_i$ denote the differentiation matrix with respect to the $i$th coordinate, defined implicitly through the relations\n\\[\nu(\\hat{\\bm{x}}) = \\sum_{j=1}^{N_p} \\bm{u}_j \\phi_j(\\hat{\\bm{x}}), \\qquad \\pd{u}{\\hat{\\bm{x}}_i} = \\sum_{j=1}^{N_p} \\LRp{{\\bm{D}}_i \\bm{u}}_j\\phi_j(\\hat{\\bm{x}}).\n\\]\nThe matrix ${\\bm{D}}_i$ maps basis coefficients of some polynomial $u \\in P^N\\LRp{\\hat{D}}$ to coefficients of its $i$th derivative with respect to the reference coordinate $\\hat{\\bm{x}}$, and is sometimes referred to as a ``modal'' differentiation matrix (with respect to a general non-nodal ``modal'' basis \\cite{hicken2016multidimensional}). \n\nUsing the volume quadrature interpolation matrix $\\bm{V}_q$, we can compute a quadrature-based mass matrix $\\bm{M}$ by evaluating $L^2$ inner products of different basis functions using quadrature\n\\[\n \\bm{M} = \\bm{V}_q^T\\bm{W}\\bm{V}_q, \\qquad \\bm{M}_{ij} = \\sum_{k=1}^{N_q} \\hat{w}_k \\phi_j(\\hat{\\bm{x}}_k)\\phi_i(\\hat{\\bm{x}}_k) \\approx \\int_{\\hat{D}}\\phi_j\\phi_i \\diff{\\hat{\\bm{x}}} = \\LRp{\\phi_j,\\phi_i}_{\\hat{D}}.\n\\]\nThe approximation in the formula for the mass matrix becomes an equality if the volume quadrature rule is exact for polynomials of degree $2N$. The mass matrix is symmetric and positive definite under Assumption~\\ref{ass:quad}; however, we do not make any distinctions between diagonal and dense (lumped) mass matrices in this work. \n\n\nThe mass matrix appears in the discretization of $L^2$ projection (\\ref{eq:l2proj}) and lift operators (\\ref{eq:lift}) using quadrature. The result are quadrature-based $L^2$ projection and lift operators $\\bm{P}_q, \\bm{L}_q$, \n\\begin{equation}\n\\bm{P}_q = \\bm{M}^{-1}\\bm{V}_q^T\\bm{W}, \\qquad \\bm{L}_q = \\bm{M}^{-1}\\bm{V}_f^T \\bm{W}_f,\n\\label{eq:projlift}\n\\end{equation}\nwhich are discretizations of the continuous $L^2$ projection operator $\\Pi_N$ and continuous lift operator $L$. The matrix $\\bm{P}_q$ maps a function (in terms of its evaluation at quadrature points) to coefficients of the $L^2$ projection in the basis $\\phi_j(x)$, while the matrix $\\bm{L}_q$ ``lifts'' a function (evaluated at surface quadrature points) from the boundary of an element to coefficients of a basis defined in the interior of the element. \n\nFinally, we introduce quadrature-based operators $\\bm{D}_N^i$ which will be used to construct discretizations of our nonlinear conservation laws. This operator was introduced in \\cite{chan2017discretely} as a ``decoupled summation-by-parts'' operator\n\\begin{equation}\n\\bm{D}_N^i =\\LRs{\n\\begin{array}{cc}\n\\bm{V}_q\\bm{D}_i \\bm{P}_q - \\frac{1}{2}\\bm{V}_q\\bm{L}_q \\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f}\\bm{V}_f\\bm{P}_q & \\frac{1}{2}\\bm{V}_q\\bm{L}_q \\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f}\\\\\n- \\frac{1}{2}\\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f} \\bm{V}_f\\bm{P}_q & \\frac{1}{2}\\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f}\n%=======\n% \\bm{V}_q\\bm{D}_i \\bm{P}_q - \\frac{1}{2}\\bm{V}_q\\bm{L}_q \\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f}\\bm{V}_f\\bm{P}_q & \\frac{1}{2}\\bm{V}_q\\bm{L}_q \\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f}\\\\\n% - \\frac{1}{2}\\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f} \\bm{V}_f\\bm{P}_q & \\frac{1}{2}\\diag{\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f}\n%>>>>>>> cd72d858ec79933e123c7bf1361e488006d5956d\n\\end{array}\n}\n\\label{eq:decoupledsbp}\n\\end{equation}\nwhere $\\hat{\\bm{n}}_i$ is the vector containing values of the $i$th component of the unit normal on the surface of the reference element $\\hat{D}$, and $\\bm{\\hat{J}}_f$ is the vector containing values of the face Jacobian factor $\\hat{J}_f$ which result from mapping a face of $\\hat{D}$ to a reference face. Here $\\hat{\\bm{n}}_i\\circ\\bm{\\hat{J}}_f$ is the Hadamard product (i.e., the entrywise product) of the vectors $\\hat{\\bm{n}}_i$ and $\\bm{\\hat{J}}_f$. When combined with projection and lifting matrices, $\\bm{D}_N^i$ produces a high order approximation of non-conservative products. Let $\\bm{f},\\bm{g}$ denote vectors containing the evaluation of functions $f(\\bm{x}),g(\\bm{x})$ at both volume and surface quadrature points\n\\[\n\\LRs{\\begin{array}{cc}\\bm{P}_q & \\bm{L}_q\\end{array}} \\diag{\\bm{f}}\\bm{D}_N^i \\bm{g} \\approx f\\pd{g}{\\hat{x}_i}.\n\\]\nIt was shown in \\cite{chan2017discretely} that the matrix $\\bm{D}_N^i$ satisfies several key properties. First, it can be observed that $\\bm{D}_N^i\\bm{1} = 0$, where $\\bm{1}$ is the vector of all ones. Second, $\\bm{D}_N^i$ satisfies a summation-by-parts property. Let $\\bm{Q}_N^i$ be the scaling of $\\bm{D}_N^i$ by the diagonal matrix of volume and surface quadrature weights \n\\[\n\\bm{Q}_N^i = \\bm{W}_N \\bm{D}_N^i, \\qquad \\bm{W}_N = \\LRp{\\begin{array}{cc}\n\\bm{W} &\\\\\n& \\bm{W}_f \n\\end{array}}.\n\\]\nThen, $\\bm{Q}_N^i$ satisfies the following discrete analogue of integration by parts \n\\begin{equation}\n\\bm{Q}_N^i + \\LRp{\\bm{Q}_N^i}^T = \\bm{B}^i_N, \\qquad \\bm{B}_N = \\LRp{\\begin{array}{cc}\n\\bm{0}&\\\\\n & \\bm{W}_f \\diag{\\hat{\\bm{n}}_i\\circ\\hat{\\bm{J}}_f}\n\\end{array}}.\n\\label{eq:sbpprop1}\n\\end{equation}\nThe matrix $\\bm{D}_N^i$ reduces to polynomial differentiation when applied to polynomials, in the sense that\n\\begin{align}\n\\bm{D}_N^i \\LRs{\\begin{array}{c}\n\\bm{V}_q\\\\\n\\bm{V}_f\n\\end{array}} = \\LRs{\\begin{array}{c}\n\\bm{V}_q\\bm{D}_i\\\\\n\\bm{0}\n\\end{array}}.\n\\label{eq:dnvqvf}\n\\end{align}\n\n%, and $\\hat{J}_f$ denotes the Jacobian of the mapping from a face of $\\hat{D}$ to a reference face (for example, the faces of a triangle are mapped back to the bi-unit interval $[-1,1]$). \n\n\\subsection{Matrices on curved physical elements}\n\\label{sec:curv}\n\nThe key difference between curvilinear and affine meshes is that geometric terms now vary spatially over each element. In practice, derivatives are computed over the reference element and mapped to the physical element $D^k$ through a change of variables formula\n\\[\nJ^k \\pd{u}{x_i} = \\sum_{j=1}^d G^k_{ij}\\pd{u}{\\hat{x}_j}, \\qquad G^k_{ij} = J^k\\pd{\\hat{x}_j}{x_i},\n\\]\nwhere we have defined the elements of the matrix $\\bm{G}^k$ as the derivatives of the reference coordinates $\\hat{x}_j$ with respect to the physical coordinates $x_i$ on $D^k$ times the Jacobian of the transformation from reference to physical coordinates $J^k$. We denote evaluations of $G^k_{ij}$ at both volume and surface quadrature points as the vector $\\bm{G}^k_{ij}$.\n\nWe assume in this work that the mesh is stationary. It can be shown at the continuous level that, for any differentiable and invertible mapping, the quantity $\\bm{G}^k$ satisfies a geometric conservation law (GCL) \\cite{kopriva2006metric, thomas1979geometric}\n\\begin{equation}\n\\sum_{j=1}^d\\pd{}{\\hat{x}_j}G^k_{ij} = 0,\n\\label{eq:gcl}\n\\end{equation}\nor that $\\hat{\\Grad}\\cdot \\bm{G}^k = 0$. Using (\\ref{eq:gcl}), the scaled physical derivative $J^k\\pd{u}{x_i}$ can be computed via\n\\begin{equation}\nJ^k\\pd{u}{x_i} = \\frac{1}{2}\\sum_{j=1}^d \\LRp{G^k_{ij}\\pd{u}{\\hat{x}_j} + \\pd{\\LRp{G^k_{ij}u}}{\\hat{x}_j}}.\n\\label{eq:splitderiv}\n\\end{equation}\n\nWe will require the following assumptions on the mesh, as well as the geometric terms and outward normal vectors:\n\\begin{assumption}[Mesh assumptions]\nWe assume that the mesh is quasi-uniform. The mesh is also assumed to be watertight, such that normals are consistent across neighboring elements as follows: for a shared face $f$ between $D^k$ and $D^{k,+}$, the scaled outward normal vectors for each element are equal and opposite at all points such that \n\\begin{align}\n\\bm{n}J^k_f = -\\bm{n}^+J^{k,+}_f.\n\\label{eq:normalsign}\n\\end{align}\nWe also assume that the scaled matrix of geometric terms transforms scaled reference normal vectors to scaled physical normals, such that\n\\begin{align}\n\\sum_{j=1}^d G^k_{ij} \\hat{n}_j\\hat{J}_f = n_i J^k_f, \\qquad \\LRp{\\sum_{j=1}^d \\LRp{\\bm{G}^k_{ij}}_f \\circ\\LRp{\\hat{\\bm{n}}_j\\circ\\hat{\\bm{J}}_f}}\n= \\LRp{\\bm{n}_i\\circ\\bm{J}^k_f},\n\\label{eq:normalconsistency}\n\\end{align}\nwhere $\\bm{n}_j$ and $\\bm{J}^k_f$ are vectors containing evaluations of the physical unit normals and face Jacobian factors for $D^k$ at surface quadrature points, respectively.\nLikewise, $\\LRp{\\bm{G}^k_{ij}}_f$ is a vector containing evaluations of $G^k_{ij}$ at the surface quadrature points.\n\\label{ass:norm}\n\\end{assumption}\nThe properties (\\ref{eq:normalsign}) and (\\ref{eq:normalconsistency}) hold at the continuous level for a watertight mesh \\cite{ciarlet1978finite}, and thus at all points where the geometric terms are computed exactly. However, we will also consider cases where the geometric terms $G^k_{ij}$ are modified to enforce a discrete form of (\\ref{eq:gcl}); in these situations, it will be important to ensure that (\\ref{eq:normalconsistency}) holds after such modifications. \n\nSimilar to what is done to stabilize finite difference discretizations~\\cite{nordstrom2006cfdf, gassner2016split}, we define physical differentiation matrices based on the approximation of (\\ref{eq:splitderiv}). Define $\\bm{D}^i_k$ as\n\\[\n \\bm{D}^i_k = \\frac12 \\sum_{j=1}^d \\left(\\diag{\\bm{G}^k_{ij}}\\bm{D}^j_N + \\bm{D}^j_N\\diag{\\bm{G}^k_{ij}}\\right).\n\\]\nUsing properties of the Hadamard product \\cite{horn2012matrix}, we can rewrite $\\bm{D}^i_k$ as \n\\begin{equation}\n\\bm{D}^i_k = \\sum_{j=1}^d \\LRp{\\bm{D}^j_N \\circ \\avg{\\bm{G}^k_{ij}}}, \\qquad \\avg{\\bm{G}^k_{ij}}_{mn} = \\frac{1}{2}\\LRp{\\LRp{\\bm{G}^k_{ij}}_m + \\LRp{\\bm{G}^k_{ij}}_n},\n\\label{eq:dik}\n\\end{equation}\nwhere $\\avg{\\bm{G}^k_{ij}}$ denotes the matrix of averages between each of the entries of $\\bm{G}^k_{ij}$. From Assumption~\\ref{ass:norm} and (\\ref{eq:dik}), it is straightforward to show that (because $\\avg{\\bm{G}^k_{ij}}$ is symmetric) $\\bm{Q}^i_k = \\bm{W}_N\\bm{D}^i_k$ also satisfies a summation-by-parts property\n\\begin{equation}\n\\bm{Q}^i_k + \\LRp{\\bm{Q}^i_k}^T = \\bm{B}^i_k, \\qquad \\bm{B}^i_k = \n\\LRp{\\begin{array}{cc}\n\\bm{0}&\\\\\n& \\bm{W}_f \\diag{\\bm{n}_i\\circ\\bm{J}^k_f}\n\\end{array}}.\n\\label{eq:sbpk}\n\\end{equation}\n\nCurvilinear mappings also imply that integrals over each physical element $D^k$ are no longer simple scalings of integrals over $\\hat{D}$. The $L^2$ projection of $u\\in L^2\\LRp{D^k}$ over a curvilinear element $D^k$ is defined through \n\\begin{equation}\n\\LRp{\\Pi^k_N u,v}_{D^k} = \\LRp{u,v}_{D^k}, \\qquad \\forall v\\in P^N\\LRp{\\hat{D}}.\n\\label{eq:l2curv}\n\\end{equation}\nMapping integrals to the reference element $\\hat{D}$ yields\n\\begin{equation}\n\\LRp{\\Pi^k_N u,v J^k}_{\\hat{D}} = \\LRp{u,vJ^k}_{\\hat{D}}, \\qquad \\forall v\\in P^N\\LRp{\\hat{D}}.\n\\label{eq:l2curvmap}\n\\end{equation}\nFor affine elements, $J^k$ is constant and can be cancelled. Thus, the $L^2$ projection over affine elements is equivalent to simply taking the $L^2$ projection of a function over the reference element. However, for curved elements, $J$ acts as a spatially varying weight within the $L^2$ inner product. \n\nDiscretizing (\\ref{eq:l2curvmap}) requires a weighted mass matrix. We define a curved mass matrix over an element $D^k$ by weighting the discrete $L^2$ norm with values of $J$ at quadrature points\n\\begin{equation}\n\\bm{M}^k = \\bm{V}_q^T \\bm{W}\\diag{\\bm{J}^k}\\bm{V}_q,\n\\label{eq:curvedmass}\n\\end{equation}\nwhere $\\bm{J}^k$ is a vector containing evaluation of the physical Jacobian factors for $D^k$ at volume quadrature points.\nThen, curvilinear $L^2$ projection and lift matrices can be defined in a manner analogous to (\\ref{eq:projlift})\n\\begin{equation}\n\\bm{P}^k_q = \\LRp{\\bm{M}^k}^{-1}\\bm{V}_q^T\\bm{W}\\diag{\\bm{J}^k}, \\qquad \\bm{L}^k_q = \\LRp{\\bm{M}^k}^{-1}\\bm{V}_f^T\\bm{W}_f\\diag{\\bm{J}^k_f}.\n\\label{eq:projliftcurved}\n\\end{equation}\nThese matrices are distinct from element to element, reflecting the fact that problem (\\ref{eq:l2curvmap}) is distinct from element to element. \n\n\\subsection{A discretely entropy stable DG formulation on curved meshes}\n\nGiven the matrices in Section~\\ref{sec:curv}, we can now define a local entropy stable DG formulation on an element $D^k$. Here, we seek an approximation solution $\\bm{u}_N(\\bm{x},t)$ to (\\ref{eq:nonlineqs}), which is represented using vector-valued coefficients $\\bm{u}_h(t)$ such that\n\\[\n\\bm{u}_N(\\bm{x},t) = \\sum_{j=1}^{N_p} \\LRp{\\bm{u}_h(t)}_j \\phi_j(\\bm{x}), \\qquad \\LRp{\\bm{u}_h(t)}_j \\in \\mathbb{R}^n.\n\\]\nSince the coefficients are vector valued, we assume that all matrices act component-wise on $\\bm{u}_h$ in the Kronecker product sense. \n\nWe first define the numerical fluxes $\\bm{f}_{i,S}\\LRp{\\bm{u}_L,\\bm{u}_R}$ as the bivariate function of ``left'' and ``right'' conservative variable states $\\bm{u}_L, \\bm{u}_R$. % Such a numerical flux is referred to as entropy conservative (or entropy stable) if it satisfies the following conditions:\n\\begin{definition}\nThe numerical flux $\\bm{f}_{i,S}\\LRp{\\bm{u}_L,\\bm{u}_R}$ is entropy conservative (or entropy stable) if it satisfies the following conditions:\n\\begin{enumerate}\n\\item $\\bm{f}_{i,S}\\LRp{\\bm{u}_L,\\bm{u}_R} = \\bm{f}_{i,S}\\LRp{\\bm{u}_R,\\bm{u}_L}$ (symmetry).\n\\item $\\bm{f}_{i,S}\\LRp{\\bm{u},\\bm{u}} = \\bm{f}_i\\LRp{\\bm{u}}$ (consistency).\n\\item $\\bm{f}_{i,S}$ is referred to as entropy conservative if it satisfies conditions 1, 2, and \n\\[\n \\LRp{\\bm{v}_L-\\bm{v}_R}^T\\bm{f}_{i,S}\\LRp{\\bm{u}_L,\\bm{u}_R} = \\psi_i\\LRp{\\bm{u}_L}-\\psi_i\\LRp{\\bm{u}_R}.\n\\]\n%\\item $\\bm{f}_{i,S}$ is referred to as entropy stable if it satisfies conditions 1, 2, and \n%\\[\n%\\LRp{\\bm{v}_L-\\bm{v}_R}^T\\bm{f}_{i,S}\\LRp{\\bm{u}_L,\\bm{u}_R} \\leq \\psi_i\\LRp{\\bm{u}_L}-\\psi_i\\LRp{\\bm{u}_R}.\n%\\]\n\\end{enumerate}\n\\label{def:entropyflux}\n\\end{definition}\n\nWe now introduce the $L^2$ projection of the entropy variables $\\bm{v}_h$ and the entropy-projected conservative variables $\\tilde{\\bm{u}}$ \n\\begin{align}\n\\bm{u}_q = \\bm{V}_q \\bm{u}_h, \\qquad \\bm{v}_h = \\bm{P}^k_q \\bm{v}\\LRp{\\bm{u}_q}, \\qquad \n\\tilde{\\bm{v}} = \\LRs{\\begin{array}{c}\n\\tilde{\\bm{v}}_q\\\\\n\\tilde{\\bm{v}}_f\n\\end{array}} = \\LRs{\\begin{array}{c}\n\\bm{V}_q\\\\\n\\bm{V}_f\n\\end{array}}\\bm{v}_h, \\qquad \\tilde{\\bm{u}} = \\LRs{\\begin{array}{c}\n\\tilde{\\bm{u}}_q\\\\\n\\tilde{\\bm{u}}_f\n\\end{array}} = \\bm{u}\\LRp{\\tilde{\\bm{v}}}.\n\\label{eq:evars1}\n\\end{align}\nIn (\\ref{eq:evars1}), the entropy-projected conservative variables $\\tilde{\\bm{u}}$ denote the evaluation of the conservative variables in terms of the projected entropy variables at volume and face quadrature points. We note that, under an appropriate choice of quadrature on quadrilaterals and hexahedra, this approach is equivalent to the approach taken in \\cite{parsani2016entropy}, where the entropy variables are evaluated at Gauss nodes, then interpolated to a different set of nodes and used to compute the nonlinear fluxes. \n\nWe now introduce a semi-discrete DG formulation for $\\bm{u}_h$\n\\begin{align}\n&\\td{\\bm{u}_h}{t} + \\LRs{\\begin{array}{cc}\\bm{P}^k_q & \\bm{L}^k_q\\end{array}}\n \\sum_{j=1}^d \\LRp{2\\bm{D}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\bm{L}^k_q \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)} = 0,\\label{eq:dgform1}\\\\\n &\\LRp{\\bm{F}_{j,S}}_{mn} = \\bm{f}_{j,S}\\LRp{\\LRp{\\tilde{\\bm{u}}}_m,\\LRp{\\tilde{\\bm{u}}}_n}, \\qquad 1 \\leq m,n \\leq N_q + N^f_q,\\nonumber\\\\\n &\\bm{f}_j^* = \\bm{f}_{j,S}(\\tilde{\\bm{u}}_f^+,\\tilde{\\bm{u}}_f) \\text{ on interior faces},\\nonumber\n\\end{align}\nwhere $\\tilde{\\bm{u}}^+$ denotes the values of the entropy-projected conservative variables on the neighboring element across each face of $D^k$, and $\\bm{f}_j^*$ on the boundary denotes the $j$th component of some numerical flux through which boundary conditions are imposed. Note that the face/surface Jacobian factors $\\bm{J}^k_f$ are incorporated into the definition of $\\bm{L}^k_q$. \n\nDefine the diagonal boundary quadrature matrix $\\bm{W}_{\\partial \\Omega}$ such that\n\\[\n\\LRp{\\bm{W}_{\\partial \\Omega} }_{ii} = \\begin{cases}\n \\bm{W}_f, & \\text{if $\\hat{\\bm{x}}^f_i$ is on the $\\partial \\Omega$}\\\\\n0, & \\text{otherwise}. \n\\end{cases}\n\\]\nWe have the following semi-discrete statement of entropy conservation:\n\\begin{theorem}\n Let $\\bm{f}_{i,S}$ be an entropy conservative flux from Definition~\\ref{def:entropyflux} and assume that $\\bm{Q}^j_k\\bm{1} = 0$ for $j = 1,\\ldots,d$ over each element $D^k$. Then, (\\ref{eq:dgform1}) is entropy conservative in the sense that\n\\[\n \\sum_k \\bm{1}^T\\diag{\\bm{J}^k}\\bm{W}\\td{U(\\bm{u}_q)}{t} = \\sum_k \\sum_{j=1}^d \\bm{1}^T\\diag{\\bm{n}_j\\circ\\bm{J}^k_f}\\bm{W}_{\\partial \\Omega} \\LRp{\\psi_j\\LRp{\\tilde{\\bm{u}}_f}-\\tilde{\\bm{v}}_f^T\\bm{f}_j^*}.\n\\]\n\\label{thm:stab1}\n\\end{theorem}\n\\begin{proof}\nUnder the assumption that $\\bm{Q}^j_k\\bm{1} = 0$ for $i = 1,\\ldots,d$ over each element and (\\ref{eq:sbpk}), the proof of entropy conservation is identical to that of \\cite{chan2017discretely}. \n%\\note{Expand out! Do spatial term only, punt for time term. }\n%To prove entropy conservation, we multiply on both sides by $\\bm{M}_k$ \n%\\begin{align}\n%\\bm{M}_k\\td{\\bm{u}_h}{t} + \\LRs{\\begin{array}{c}\\bm{V}_q \\\\ \\bm{V}_f\\end{array}}^T\n%\\sum_{j=1}^d \\LRp{\\bm{Q}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\bm{V}_f^T \\bm{W}_f\\diag{\\bm{n}_j\\bm{J}^k_f}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)} = 0.\n%\\end{align}\n%We now test on both sides by $\\bm{v}_h^T$ and note that $\\bm{v}_h^T\\bm{M} = \\bm{v}(\\bm{u}_q)^T \\bm{W} \\bm{J}^k \\bm{V}_q$. For the time term, we assuming continuity in time and use the chain rule\n%\\[\n%\\bm{v}(\\bm{u}_q)^T \\bm{W} \\bm{J}^k \\td{\\bm{V}_q\\bm{u}_h}{t} = \\bm{1}^T \\bm{W} \\bm{J}^k \\bm{v}(\\bm{u}_q)^T\\td{\\bm{u}_q}{t} = \n%\\bm{1}^T \\bm{W} \\bm{J}^k \\td{U(\\bm{u}_q)}{t},\n%\\]\n%where we have used that $\\bm{W}$ is diagonal, that $\\bm{v}(\\bm{u}) = \\pd{U(\\bm{u})}{\\bm{u}}$, and the chain rule in time. The right hand side terms\n%\\[\n%\\sum_{j=1}^d \\LRp{\\bm{Q}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\bm{V}_f^T \\bm{W}_f\\diag{\\bm{n}_j\\bm{J}^k_f}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)} = 0.\n%\\]\n%\\note{Finish proof}\n\\end{proof}\nAn entropy stable scheme can be constructed by adding an entropy-dissipating penalty term, such as a Lax-Friedrichs penalization or the matrix dissipation terms introduced in \\cite{chandrashekar2013kinetic, winters2017uniquely}. For example, Lax-Friedrichs penalization can be incorporated by replacing the flux term with\n\\[\n\\bm{L}^k_q \\diag{\\bm{n}_j} \\LRp{\\bm{f}_j^*-\\bm{f}(\\bm{u})} \\Longrightarrow \\bm{L}^k_q\\LRp{\\diag{\\bm{n}_j} \\LRp{\\bm{f}_j^*- \\bm{f}(\\bm{u})} - \\frac{\\lambda}{2}\\jump{\\tilde{\\bm{u}}_f}},\n\\]\nwhere $\\lambda$ is an estimate of the maximum eigenvalue of $\\pd{\\bm{f}(\\bm{u})}{\\bm{u}}$ \\cite{chen2017entropy, chan2017discretely}.\n\n%\\note{maybe add more here?}\n\n\\section{Discretely stable and low storage DG methods on curved meshes}\n\\label{sec:3}\nA disadvantage of the formulation (\\ref{eq:dgform1}) is high storage costs, especially at high orders of approximation. While the matrices $\\bm{Q}^i_k$ can be applied to a vector without needing to explicitly store the matrix, the projection and lifting matrices (\\ref{eq:projliftcurved}) differ from element to element, necessitating either explicit pre-computation and storage or the assembly and inversion of a weighted mass matrix for each right hand side evaluation. The latter option is computationally expensive, while the former option increases storage costs. This increase in storage can result in suboptimal performance on modern computational architectures \\cite{chan2017weight}, due to the increasing cost of memory operations and data movement compared to arithmetic operations. \n\nIn this section, we present a discretely entropy stable scheme which avoids this high storage cost through the use of a low-storage weight-adjusted approximation to the inverse of a weighted mass matrix. To ensure a discrete entropy conservation or a discrete entropy inequality, we also modify the formulation (\\ref{eq:dgform1}) to take into account the use of a weight-adjusted mass matrix.\n\n\\subsection{A weight-adjusted approximation to the curvilinear mass matrix}\n\n%Recall that, by mapping back to the reference element $\\hat{D}$, the $L^2$ projection $\\Pi_N^k$ on a curved element $D^k$ is defined as\n%\\begin{equation}\n%\\LRp{J \\Pi_N^k u,v}_{\\hat{D}} = \\LRp{uJ,v}_{\\hat{D}}, \\qquad \\forall v\\in P^N\\LRp{\\hat{D}}.\n%\\label{eq:l2projcurv}\n%\\end{equation}\nThe presence of the weighted $L^2$ inner product $\\LRp{J^k \\Pi_N^k u,v}_{\\hat{D}}$ in (\\ref{eq:l2curvmap}) results in the presence of a weighted mass matrix. Because the weight $J^k$ varies spatially over each element, the inverse of a weighted mass matrix is no longer a scaling of the inverse reference mass matrix. The motivation for the weight-adjusted mass matrix is to replace the inversion of weighted mass matrices over each element with the application of inverse reference mass matrices and quadrature-based operations involving the spatially varying weights $J^k$ \\cite{chan2016weight1, chan2016weight2}. \n\nTo define a weight-adjusted approximation to the curvilinear $L^2$ inner product, we first define the operator $T_{w}^{-1}: L^2\\rightarrow P^N$ as follows\n\\begin{equation}\n \\LRp{wT_{w}^{-1} u,v}_{\\hat{D}} = \\LRp{ u,v}_{\\hat{D}}, \\qquad \\forall v\\in P^N\\LRp{\\hat{D}}.\n\\label{eq:wadgTw}\n\\end{equation}\nRoughly speaking, $T_{w} u$ approximates $u/w$. Thus, taking $w = 1/J^k$ provides an approximation of the curvilinear $L^2$ inner product\n\\begin{equation*}\n\\LRp{J^k u,v}_{\\hat{D}} \\approx \\LRp{T_{1/J^k}^{-1} u,v}_{\\hat{D}}.\n\\end{equation*}\n\nComputing $T_{1/J^k}^{-1}u$ requires solving (\\ref{eq:wadgTw}). Let $u \\in P^N\\LRp{\\hat{D}}$, and let $\\bm{u}_J$ denote coefficients for the polynomial $T_{1/J^k}^{-1}u$. This results in the following matrix system\n\\[\n\\bm{M}_{1/J^k}\\bm{u}_{J} = \\bm{M}\\bm{u}, \\qquad {\\bm{M}_{1/J^k}} = \\bm{V}_q^T \\bm{W}\\diag{1/\\bm{J}^k}\\bm{V}_q,%\\LRp{\\frac{1}{J^k}\\phi_i,\\phi_j}_{\\hat{D}},\n\\]\nwhich implies that, when restricted to polynomials, the matrix form of $T_{1/J^k}^{-1}$ is $\\bm{M}_{1/J^k}^{-1}\\bm{M}$. Then, the weight-adjusted mass matrix is the Gram matrix with respect to the weight-adjusted inner product $\\LRp{T_{1/J^k}^{-1} u,v}_{\\hat{D}}$, such that\n\\[\n\\bm{M}^k \\approx \\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}, \\qquad \\LRp{\\bm{M}^k}^{-1} \\approx \\bm{M}^{-1}\\bm{M}_{1/J^k}\\bm{M}^{-1}.\n\\]\nThe inverse of the weight-adjusted mass matrix can be applied in a matrix-free fashion by using quadrature to form $\\bm{M}_{1/J^k} $. This requires storage of the inverse reference mass matrix and the values of $\\bm{J}^k$ at quadrature points. Assuming that the number of quadrature points scales as $O(N^d)$ in $d$ dimensions, this yields a storage cost of $O(N^d)$ per-element compared to an $O(N^{2d})$ per element storage cost required for the storage of inverse weighted mass matrices $\\LRp{\\bm{M}^k}^{-1}$. This application of the weight-adjusted mass matrix is typically applied using the $L^2$ projection matrix $\\bm{P}_q$ as follows\n\\[\n\\bm{M}^{-1}\\bm{M}_{1/J^k}\\bm{M}^{-1} = \\bm{P}_q \\diag{1/\\bm{J}^k} \\bm{V}_q \\bm{M}^{-1}.\n\\]\nWhen evaluating the right hand side of a semi-discrete formulation such as (\\ref{eq:dgform1}), the inverse mass matrix is typically merged into operations on the right hand side, such that the main work in applying the weight-adjusted mass matrix consists of applying the interpolation matrix $\\bm{V}_q$, scaling by pointwise values of $1/\\bm{J}^k$ at quadrature points, and multiplying by the $L^2$ projection matrix $\\bm{P}_q$.\n\n\n\\subsection{A discretely entropy stable low storage DG formulation on curved meshes}\n\nGiven the weight-adjusted inverse mass matrix, we can also define a weight-adjusted version of the $L^2$ projection over a curved element $D^k$. We refer to this operator as $\\tilde{\\Pi}^k_N: L^2\\rightarrow P^N$, which satisfies\n\\[\n\\LRp{T^{-1}_{1/J^k}\\tilde{\\Pi}^k_N u,v}_{\\hat{D}} = \\LRp{uJ^k,v}_{\\hat{D}}, \\qquad \\forall v\\in P^N\\LRp{\\hat{D}}.\n\\]\nIt was shown in \\cite{chan2016weight2} that $\\tilde{\\Pi}^k_N $ is given explicitly by\n\\begin{equation}\n\\tilde{\\Pi}^k_N u = \\Pi_N\\LRp{\\frac{1}{J^k}\\Pi_N\\LRp{uJ^k}},\n\\label{eq:wadgprojop}\n\\end{equation}\nwhere $\\Pi_N$ is the $L^2$ projection operator on the reference element $\\hat{D}$. We can discretize $\\tilde{\\Pi}^k_N$ using quadrature to yield a weight-adjusted projection matrix $\\tilde{\\bm{P}}^k_q$ \n\\begin{align}\n\\tilde{\\bm{P}}^k_q &= \\bm{M}^{-1}\\bm{M}_{1/\\bm{J}^k}\\bm{M}^{-1}\\bm{V}_q^T\\bm{W}\\diag{\\bm{J}^k} = \\bm{M}^{-1}\\bm{V}_q^T\\bm{W}\\diag{1/\\bm{J}^k} \\bm{V}_q\\bm{P}_q\\diag{\\bm{J}^k} \\nonumber\\\\\n&= \\bm{P}_q \\diag{{1}/{\\bm{J}^k}} \\bm{V}_q\\bm{P}_q \\diag{\\bm{J}^k}.\n\\label{eq:wadgproj}\n\\end{align}\nWe can similarly define a weight-adjusted lifting matrix $\\tilde{\\bm{L}}_q$ by replacing the weighted mass matrix in (\\ref{eq:projliftcurved}) with the weight-adjusted mass matrix\n\\begin{align}\n\\tilde{\\bm{L}}^k_q &= \\bm{M}^{-1}\\bm{M}_{1/\\bm{J}^k}\\bm{M}^{-1}\\bm{V}_f^T\\bm{W}_f\\diag{\\bm{J}^k_f} = \\bm{M}^{-1}\\bm{V}_q^T\\bm{W}\\diag{1/\\bm{J}^k} \\bm{V}_q\\bm{L}_q\\diag{\\bm{J}^k_f} \\nonumber\\\\\n&= \\bm{P}_q \\diag{{1}/{\\bm{J}^k}} \\bm{V}_q\\bm{L}_q \\diag{\\bm{J}^k_f}.\n\\label{eq:wadglift}\n\\end{align}\n\nWe can now introduce the weight-adjusted projection of the entropy variables $\\bm{v}_h$ and the corresponding entropy-projected conservative variables $\\tilde{\\bm{u}}$ \n\\begin{align}\n\\bm{u}_q = \\bm{V}_q \\bm{u}_h, \\qquad \\bm{v}_h = \\tilde{\\bm{P}}^k_q \\bm{v}\\LRp{\\bm{u}_q}, \\qquad \n\\tilde{\\bm{v}} = \\LRs{\\begin{array}{c}\n\\bm{V}_q\\\\\n\\bm{V}_f\n\\end{array}}\\bm{v}_h, \\qquad \\tilde{\\bm{u}} = \\LRs{\\begin{array}{c}\n\\tilde{\\bm{u}}_q\\\\\n\\tilde{\\bm{u}}_f\n\\end{array}} = \\bm{u}\\LRp{\\tilde{\\bm{v}}}.\n\\label{eq:evars2}\n\\end{align}\nA semi-discrete DG formulation for $\\bm{u}_h$ can be constructed using the variables defined in (\\ref{eq:evars2})\n\\begin{align}\n&\\td{\\bm{u}_h}{t} + \\LRs{\\begin{array}{cc}\\tilde{\\bm{P}}^k_q & \\tilde{\\bm{L}}^k_q\\end{array}}\n \\sum_{j=1}^d \\LRp{2\\bm{D}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\tilde{\\bm{L}}^k_q \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)} = 0,\\label{eq:dgform2}\\\\\n &\\LRp{\\bm{F}_{j,S}}_{mn} = \\bm{f}_{j,S}\\LRp{\\LRp{\\tilde{\\bm{u}}}_m,\\LRp{\\tilde{\\bm{u}}}_n}, \\qquad 1 \\leq m,n \\leq N_q + N^f_q,\\nonumber\\\\\n &\\bm{f}_j^* = \\bm{f}_{j,S}(\\tilde{\\bm{u}}_f^+,\\tilde{\\bm{u}}_f) \\text{ on interior faces}.\\nonumber\n\\end{align}\nSince the weight-adjusted mass matrix inverse is low-storage, and since the matrices $\\bm{D}^j_k$ in (\\ref{eq:dik}) can be assembled from reference matrices $\\bm{D}^i_N$ and the values of geometric terms at quadrature points, the overall scheme requires only $O(N^d)$ storage per element. We can additionally show that formulation (\\ref{eq:dgform2}) is entropy conservative in the same sense as Theorem~\\ref{thm:stab1}: \n\\begin{theorem}\n Let $\\bm{f}_{i,S}$ be an entropy conservative flux from Definition~\\ref{def:entropyflux} and assume that $\\bm{Q}^j_k\\bm{1} = 0$ for $j = 1,\\ldots,d$ over each element $D^k$. Then, (\\ref{eq:dgform2}) is entropy conservative in the sense that\n\\[\n \\sum_k \\bm{1}^T\\diag{\\bm{J}^k}\\bm{W}\\td{U(\\bm{u}_q)}{t} = \\sum_k \\sum_{j=1}^d \\bm{1}^T\\diag{\\bm{n}_j\\circ\\bm{J}^k_f}\\bm{W}_{\\partial \\Omega} \\LRp{\\psi_j\\LRp{\\tilde{\\bm{u}}_f}-\\tilde{\\bm{v}}_f^T\\bm{f}_j^*}.\n\\]\n\\label{thm:stab2}\n\\end{theorem}\n\\begin{proof}\nThe proof is similar to that of Theorem~\\ref{thm:stab1}. \nFirst, recall from the definitions of the weight-adjusted projection matrix~(\\ref{eq:wadgproj}) and weight-adjusted lift matrix~(\\ref{eq:wadglift}) that\n\\begin{align}\n\\tilde{\\bm{P}}^k_q &= \\bm{M}^{-1}\\bm{M}_{1/J^k} \\bm{M}^{-1} \\bm{V}_q^T \\bm{W} \\diag{\\bm{J}^k }\\label{eq:projidentity},\\\\\n\\tilde{\\bm{L}}^k_q &= \\bm{M}^{-1}\\bm{M}_{1/J^k} \\bm{M}^{-1} \\bm{V}_f^T \\bm{W}_f\\diag{\\bm{J}^k_f}\\label{eq:liftidentity}.\n\\end{align}\nThen, multiplying by the weight-adjusted mass matrix $\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}$ on both sides of (\\ref{eq:dgform2}) and using (\\ref{eq:projidentity}), (\\ref{eq:liftidentity}) yields the weak form of (\\ref{eq:dgform2})\n\\begin{equation}\n\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t} + \\LRs{\\begin{array}{cc}{\\bm{V}}_q^T & {\\bm{V}}_f^T\\end{array}}\n\\sum_{j=1}^d \\LRp{2\\bm{Q}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d {\\bm{V}}_f^T\\bm{W}_f \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)} = 0.\n\\label{eq:dgform2weak}\n\\end{equation}\nTesting with the weight-adjusted projection of the entropy variables $\\bm{v}_h = \\tilde{\\bm{P}}^k_q \\bm{v}\\LRp{\\bm{u}_q}$ and using (\\ref{eq:projidentity}) then yields for the time term\n\\begin{align*}\n\\LRp{\\tilde{\\bm{P}}^k_q \\bm{v}\\LRp{\\bm{u}_q}}^T\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t} &= \\bm{v}\\LRp{\\bm{u}_q}^T\\bm{W}\\diag{\\bm{J}^k}\\bm{V}_q \\bm{M}^{-1}\\bm{M}_{1/J^k} \\bm{M}^{-1} \\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t}\\\\\n &= \\bm{v}\\LRp{\\bm{u}_q}^T\\bm{W}\\diag{\\bm{J}^k} \\td{\\bm{V}_q\\bm{u}_h}{t} = \\bm{1}^T\\bm{W}\\diag{\\bm{J}^k} \\left({\\rm diag}\\LRp{\\bm{v}\\LRp{\\bm{u}_q}}\\td{\\bm{u}_q}{t}\\right) \\\\\n &= \\bm{1}^T\\bm{W}\\diag{\\bm{J}^k} \\td{U(\\bm{u}_q)}{t}.\n\\end{align*}\nThe remainder of the proof is identical to that of Theorem~\\ref{thm:stab1} and \\cite{chan2017discretely}.\n\\end{proof}\n\n\\subsection{Analysis of weight-adjusted projection} \n\nThe construction of the discretely entropy stable weight-adjusted DG formulation (\\ref{eq:dgform2}) replaces the $L^2$ projection operator $\\Pi_N^k$ with the weight-adjusted projection operator $\\tilde{\\Pi}^k_N$.\nWhile this preserves entropy stability, it is unclear whether $\\tilde{\\Pi}^k_N$ is high order accurate. In this section, we prove that the weight-adjusted projection is high order accurate due to the fact that, for a fixed geometric mapping and sufficiently regular $u$, the difference between the $L^2$ and weight-adjusted projection is $\\nor{\\Pi_N^k u - \\tilde{\\Pi}^k_N u}_{L^2\\LRp{\\Omega}} = O(h^{N+2})$. Because the approximation error for the $L^2$ projection is $O(h^{N+1})$ for sufficiently regular $u$, the difference between the $L^2$ and weight-adjusted projection converges faster than the $L^2$ best approximation error. Consequentially, solutions computed using the $L^2$ and weight-adjusted projection are typically indistinguishable for a fixed geometric mapping \\cite{chan2018multi}. \n\nWe first note that $\\tilde{\\Pi}^k_N = \\Pi_N\\LRp{\\frac{1}{J^k}\\Pi_N\\LRp{uJ^k}}$ is self-adjoint with respect to the $J$-weighted $L^2$ inner product\n\\begin{equation}\n \\LRp{J^k \\tilde{\\Pi}^k_N u, v}_{\\hat{D}} = \\LRp{\\Pi_N\\LRp{\\frac{1}{J^k}\\Pi_N\\LRp{uJ^k}}, vJ^k}_{\\hat{D}} = \\LRp{uJ^k, \\Pi_N\\LRp{\\frac{1}{J^k}\\Pi_N\\LRp{vJ^k}}}_{\\hat{D}} = \\LRp{uJ^k, \\tilde{\\Pi}^k_N v}_{\\hat{D}}.\n\\label{eq:PNsym}\n\\end{equation}\nFurthermore, using that the operator $T_{1/J^k}^{-1}$ is self-adjoint for $v \\in P^N\\LRp{\\hat{D}}$ with respect to the $L^2$ inner product \\cite{chan2016weight1}, we find that a projection-like property holds for the weight-adjusted $L^2$ inner product\n\\begin{equation}\n \\LRp{T_{1/J^k}^{-1} \\tilde{\\Pi}^k_N u,v}_{\\hat{D}} = \\LRp{ \\frac{1}{J^k}\\Pi_N(uJ^k),T_{1/J^k}^{-1}v}_{\\hat{D}} = \\LRp{\\Pi_N(u J^k),v}_{\\hat{D}} = \\LRp{u J^k,v}_{\\hat{D}}, \\qquad \\forall v\\in P^N{\\hat{D}}.\n\\label{eq:PNproj}\n\\end{equation}\n\nTo prove $\\nor{\\Pi_N^k u - \\tilde{\\Pi}^k_N u}_{L^2\\LRp{\\hat{D}}} = O(h^{N+2})$, we use a generalized inverse inequality and results from \\cite{chan2016weight1,chan2016weight2}. We first introduce a modification of Theorem 3.1 in \\cite{warburton2013low, chan2016weight1}\n\\begin{theorem}%[Theorem 3.1 in \\cite{warburton2013low, chan2016weight1}]\nLet $D^k$ be a quasi-regular element with representative size $h = \\diam\\LRp{D^k}$, and let $\\hat{D}$ be the reference element. For $N \\geq 0$, $w\\in W^{N+1,\\infty}\\LRp{D^k}$, and $u\\in W^{r,2}\\LRp{D^k}$, \n\\begin{align*}\n%\\nor{u - \\frac{1}{w} \\Pi_N\\LRp{{u}{w}}}_{L^2\\LRp{D^k}} &\\leq C h^{N+1}\\nor{\\frac{1}{\\sqrt{J^k}}}_{L^{\\infty}\\LRp{D^k}}\\nor{\\frac{\\sqrt{J^k}}{w}}_{L^{\\infty}\\LRp{D^k}} \\nor{w}_{W^{N+1,\\infty}\\LRp{D^k}} \\nor{u}_{W^{N+1,2}\\LRp{D^k}},\\\\\n\\nor{u - \\frac{1}{w} \\Pi_N\\LRp{{u}{w}}}_{L^2\\LRp{\\hat{D}}} &\\leq C h^{\\min\\LRp{r,N+1}} \\nor{\\frac{1}{\\sqrt{J^k}}}_{L^{\\infty}\\LRp{D^k}}\\nor{\\frac{1}{w}}_{L^{\\infty}\\LRp{D^k}} \\nor{w}_{W^{N+1,\\infty}\\LRp{D^k}} \\nor{u}_{W^{r,2}\\LRp{D^k}}.\n\\end{align*}\n\\label{thm:wproj}\n\\end{theorem}\nThe proof is a straightforward modification of the proofs presented in \\cite{warburton2013low, chan2016weight1} accounting for reduced regularity of $u$ when $r < (N+1)$. The next result we need is a generalized inverse inequality. \n\\begin{lemma}\n\\label{lemma:sobolev}\nLet $v \\in P^N\\LRp{\\hat{D}}$, and let $h = \\diam\\LRp{D^k}$. Then,\n\\[\n \\nor{v}_{W^{N+1,2}\\LRp{D^k}} \\leq C_{N} h^{-N} \\nor{\\sqrt{J^k}}_{L^{\\infty}} \\nor{\\frac{1}{\\sqrt{J^k}}}_{L^{\\infty}} \\nor{\\frac{1}{J^k}\\bm{G}^k}_{W^{N,\\infty}\\LRp{D^k}} \\nor{v}_{L^2\\LRp{D^k}}.\n\\]\nwhere $C_{N}$ depends on $N$, but is independent of $h$.\n\\end{lemma}\n\\begin{proof}\n By applying Fa\\`{a} di Bruno's formula, we can express the degree $(N+1)$ Sobolev norm of $v$ on $D^k$ in terms of derivatives of $v$ on the reference element $\\hat{D}$. Noting that all $(N+1)$ derivatives of $v$ disappear for $v\\in P^N\\LRp{\\hat{D}}$ allows us to bound the degree $(N+1)$ Sobolev norm of $v$ by its degree $N$ Sobolev norm\n\\[\n \\nor{v}_{W^{N+1,2}\\LRp{D^k}} \\leq C_N \\nor{\\frac{1}{J^k}\\bm{G}^k}_{W^{N,\\infty}\\LRp{D^k}} \\nor{v}_{W^{N,2}\\LRp{D^k}},\n\\]\nwhere $\\bm{G}^k$ is the matrix of scaled geometric terms for $D^k$. Then, a scaling argument \\cite{ciarlet1978finite, brenner2007mathematical} yields\n\\begin{align}\n\\nor{v}_{W^{N,2}\\LRp{D^k}} \\leq C_1 h^{-N} \\nor{\\sqrt{J^k}}_{L^{\\infty}} \\nor{v}_{W^{N,2}\\LRp{\\hat{D}}}. \n\\end{align}\nThe quantity $\\nor{v}_{W^{N,2}\\LRp{\\hat{D}}}$ can be bounded by noting that $v\\in P^N\\LRp{\\hat{D}}$. Since $P^N\\LRp{\\hat{D}}$ is finite-dimensional, the Sobolev norm can be bounded from above by the $L^2$ norm of $\\hat{D}$ with a constant $C_{2}$ depending on $N, d$. By another scaling argument, we have \n\\begin{align}\n\\nor{v}_{W^{N+1,2}\\LRp{\\hat{D}}} \\leq C_{2} \\nor{v}_{L^2\\LRp{\\hat{D}}} \\leq \\nor{\\frac{1}{\\sqrt{J^k}}}_{L^{\\infty}} C_{2} \\nor{v}_{L^2\\LRp{D^k}}.\n\\end{align}\n%where $\\bm{M}$ is the $L^2$ mass matrix over $D^k$ and $\\bm{K}_N$ is the Gram matrix corresponding to the Sobolev inner product on the reference element $W^{N+1,2}\\LRp{\\hat{D}}$. The constant $C_N$ depends on the largest generalized eigenvalue of \\ref{eq:eig}, which in turn depends on the order $N$ and dimension $d$. \n\\end{proof}\n\nWe can now prove that $\\tilde{\\Pi}^k_N u$ is superconvergent to the curvilinear $L^2$ projection $\\Pi_N^k u$: \n\\begin{theorem}\nLet $u \\in W^{r,2}\\LRp{D^k}$. The difference between the $L^2$ projection $\\Pi^k_Nu$ and the weight-adjusted projection $\\tilde{\\Pi}^k_N u$ is\n\\[\n\\nor{\\Pi_N^k u - \\tilde{\\Pi}^k_N u}_{L^2\\LRp{D^k}} \\leq C_N C_J h^{\\min\\LRp{r,N+1}+1}\\nor{u}_{W^{N+1,2}\\LRp{D^k}},\n\\]\nwhere $C_N$ is a mesh-independent constant which depends on $N, d$ and $C_J$ is\n\\[\n C_J = \\nor{J^k}_{L^{\\infty}\\LRp{D^k}}^{1.5} \\nor{\\frac{1}{J^k}}_{L^{\\infty}\\LRp{D^k}}^{1.5} \\nor{\\frac{1}{J^k}}_{W^{N+1,\\infty}\\LRp{D^k}}\\nor{J^k}_{W^{N+1,\\infty}\\LRp{D^k}}\\nor{\\frac{1}{J^k}\\bm{G}^k}_{W^{N,\\infty}\\LRp{D^k}}.\n\\]\n\\label{thm:superconverge}\n\\end{theorem}\n\\begin{proof}\nWe can rewrite the norm of the difference between the weight-adjusted and $L^2$ projections\n\\[\n\\nor{\\Pi_N^k u - \\tilde{\\Pi}^k_N u}_{L^2\\LRp{D^k}}^2 = \\LRp{\\Pi_N^k u - \\tilde{\\Pi}^k_N u,vJ^k}_{\\hat{D}}, \\qquad v = \\Pi_N^k u - \\tilde{\\Pi}^k_N u.\n\\]\nBecause $v \\in P^N\\LRp{\\hat{D}}$, we can also evaluate the squared error as\n\\begin{align*}\n\\nor{\\Pi_N^k u - \\tilde{\\Pi}^k_N u}_{L^2\\LRp{D^k}}^2 &= \\LRb{\\LRp{\\Pi_N^k u,vJ^k}_{\\hat{D}} - \\LRp{\\tilde{\\Pi}^k_N u,vJ^k}_{\\hat{D}}} = \\LRb{\\LRp{u,vJ^k}_{\\hat{D}} - \\LRp{\\tilde{\\Pi}^k_N u,vJ^k}_{\\hat{D}}} \\\\\n%&\\leq \\LRp{J^k,1}_{\\hat{D}} \\LRb{\\LRp{u-\\tilde{\\Pi}^k_Nu,v}_{\\hat{D}}} = \\LRb{D^k}\\LRb{\\LRp{u-\\tilde{\\Pi}^k_Nu,v}_{\\hat{D}}},\n&\\leq \\nor{J^k}_{L^{\\infty}\\LRp{D^k}} \\LRb{\\LRp{u-\\tilde{\\Pi}^k_Nu,v}_{\\hat{D}}}.% = \\LRb{D^k}\\LRb{\\LRp{u-\\tilde{\\Pi}^k_Nu,v}_{\\hat{D}}},\n\\end{align*}\n%where we have applied the Cauchy-Schwarz inequality in the last steps. \nWe can then note that $\\tilde{\\Pi}^k_Nu = \\Pi_N\\LRp{\\frac{1}{J^k} \\Pi_N\\LRp{uJ^k}}$ to show that\n\\begin{align*}\n{\\LRp{u-\\tilde{\\Pi}^k_Nu,v}_{\\hat{D}}} &= {\\LRp{uJ^k,\\frac{v}{J^k}}_{\\hat{D}}-\\LRp{\\Pi_N\\LRp{\\frac{1}{J^k} \\Pi_N\\LRp{uJ^k}},v}_{\\hat{D}}} \\\\\n&= \\LRp{uJ^k,\\frac{v}{J^k}}_{\\hat{D}}-\\LRp{\\Pi_N\\LRp{uJ^k},\\frac{v}{J^k}}_{\\hat{D}}. %+ \\LRp{\\Pi_N\\LRp{uJ^k},\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}} - \\LRp{\\Pi_N\\LRp{uJ^k},\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}}\\\\\n%&= \\LRp{uJ^k,\\frac{v}{J^k}}_{\\hat{D}}-\\LRp{\\Pi_N\\LRp{uJ^k},\\frac{v}{J^k}}_{\\hat{D}} + \\LRp{\\Pi_N\\LRp{uJ^k},\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}} - \\LRp{\\Pi_N\\LRp{uJ^k},\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}}\n\\end{align*}\nAdding and subtracting $\\LRp{\\Pi_N\\LRp{uJ^k},\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}}$ and using Theorem~\\ref{thm:wproj} (noting that $v\\in W^{N+1,2}\\LRp{D^k}$ since it is polynomial) gives\n\\begin{align*}\n\\nor{\\Pi^k_Nu-\\tilde{\\Pi}^k_Nu}_{L^2\\LRp{D^k}}^2 &\\leq \\nor{J^k}_{L^{\\infty}\\LRp{D^k}}\\LRb{\\LRp{u-\\tilde{\\Pi}^k_Nu,v}_{\\hat{D}}}\\\\\n&= \\nor{J^k}_{L^{\\infty}\\LRp{D^k}}\\LRb{\\LRp{uJ^k - \\Pi_N\\LRp{uJ^k},\\frac{v}{J^k}-\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}}} \\\\\n&\\leq \\nor{J^k}_{L^{\\infty}\\LRp{D^k}}\\nor{uJ^k - \\Pi_N\\LRp{uJ^k}}_{\\hat{D}}\\nor{\\frac{v}{J^k}-\\Pi_N\\LRp{\\frac{v}{J^k}}}_{\\hat{D}}\\\\\n&\\leq C h^{\\min\\LRp{r,N+1}+N+1} \\tilde{C}_J \\nor{u}_{W^{r,2}\\LRp{D^k}} \\nor{v}_{W^{N+1,2}\\LRp{D^k}},\n\\end{align*}\nwhere \n\\[\n\\tilde{C}_J = \\nor{J^k}_{L^{\\infty}\\LRp{D^k}}\\nor{\\frac{1}{J^k}}_{L^{\\infty}\\LRp{D^k}} \\nor{J^k}_{W^{N+1,\\infty}\\LRp{D^k}}\\nor{\\frac{1}{J^k}}_{W^{N+1,\\infty}\\LRp{D^k}}.\n\\]\nApplying Lemma~\\ref{lemma:sobolev} then yields\n\\begin{align*}\n\\nor{\\Pi^k_Nu-\\tilde{\\Pi}^k_Nu}_{L^2\\LRp{D^k}}^2 &\\leq C_N h^{\\min\\LRp{r,N+1}+1} C_J \\nor{u}_{W^{N+1,2}\\LRp{D^k}} \\nor{v}_{L^2\\LRp{D^k}}.\n\\end{align*}\nDividing through by $\\nor{v}_{L^2\\LRp{D^k}} = \\nor{\\Pi^k_N u - \\tilde{\\Pi}^k_N u}_{L^2\\LRp{D^k}}$ gives the desired result. % $C_J = \\nor{J}_{L^{\\infty}} \\nor{\\frac{1}{J}}^2_{L^\\infty} \\nor{J}^2_{W^{N+1,\\infty}\\LRp{\\hat{D}}}$.\n\\end{proof}\nTheorem~\\ref{thm:wproj} can be used to show that the $L^2$ error between $\\Pi^k_N u, \\tilde{\\Pi}^k_N u$ and $u \\in W^{r,2}\\LRp{D^k}$ is $O(h^r)$. Theorem~\\ref{thm:superconverge} demonstrates that the $L^2$ difference between $\\Pi^k_N u, \\tilde{\\Pi}^k_N u$ is $O(h^{r+1})$, or at least one order higher than the approximation error. We note that optimal convergence of the weight-adjusted projection requires that the geometric mapping $\\bm{\\Phi}_k$ is asymptotically affine (i.e., the Sobolev norm of $J^k, \\bm{G}^k$ does not grow under mesh refinement), which is ensured under nested mesh refinement and appropriate curvilinear blending strategies \\cite{lenoir1986optimal, warburton2013low, chan2016weight2}. \n%We note that, while Theorem~\\ref{thm:superconverge} requires sufficient smoothness of $u$, numerical experiments in Section~\\ref{sec:num} demonstrate that even for discontinuous functions, the $L^2$ and weight-adjusted projections are indistinguishable from each other on a curved element. \n\n\\subsubsection{Local and global conservation}\n\\label{sec:conservation}\nWe next address local conservation of the weight-adjusted scheme (\\ref{eq:dgform2}), which is also referred to as primary conservation \\cite{fisher2013discretely, fisher2013high, carpenter2014entropy, friedrich2017entropy}. We begin by noting that (\\ref{eq:dgform2}) is locally conservative with respect to the weight-adjusted inner product. Testing (\\ref{eq:dgform2weak}) with $\\bm{1}$ yields\n\\[\n\\bm{1}^T\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t} + \n\\sum_{j=1}^d \\bm{1}^T \\LRp{2\\bm{Q}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\bm{1}^T\\bm{W}_f \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)} = 0.\n\\]\nLocal conservation can be shown by applying the SBP property (\\ref{eq:sbpk}) and noting that $\\LRp{\\bm{B}^i_k \\circ \\bm{F}_{j,S}}\\bm{1} = \\bm{W}_f\\diag{\\bm{n}_j \\circ \\bm{J}^k_f}\\bm{f}_j(\\bm{u})$ (using the consistency of $\\bm{f}_S$ and diagonal nature of $\\bm{B}^i_k$) yields\n\\[\n\\bm{1}^T\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t} + \n\\sum_{j=1}^d \\bm{1}^T \\LRp{\\LRp{\\bm{Q}^j_k-\\LRp{\\bm{Q}^j_k}^T} \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\bm{1}^T\\bm{W}_f \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^*} = 0.\n\\]\nSince $\\LRp{\\bm{Q}^j_k-\\LRp{\\bm{Q}^j_k}^T} $ is skew-symmetric and $\\bm{F}_{j,S}$ is symmetric, the Hadamard product of these two matrices is skew-symmetric. As a result, $\\bm{1}^T \\LRp{\\LRp{\\bm{Q}^j_k-\\LRp{\\bm{Q}^j_k}^T} \\circ \\bm{F}_{j,S}}\\bm{1} = 0$ and \n\\begin{equation}\n\\bm{1}^T\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t} + \\sum_{j=1}^d \\bm{1}^T\\bm{W}_f \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^*} = 0.\n\\label{eq:localcons}\n\\end{equation}\nGlobal conservation is shown by summing (\\ref{eq:localcons}) over all elements $D^k$. Because the flux $\\bm{f}_j^*$ is single-valued on each face and the outward normal $\\bm{n}_j$ changes sign on adjacent elements, the contributions $\\bm{1}^T\\bm{W}_f \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^*}$ cancel on all interior interfaces.\\footnote{Conservation still holds when incorporating entropy dissipation through penalty terms involving jumps. This is because the definition of the jump changes sign on adjacent elements, such that jump contributions cancel when summing over all elements.} On periodic meshes, this yields conservation with respect to the weight-adjusted mass matrix\n\\[\n\\sum_k \\bm{1}^T\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\td{\\bm{u}_h}{t} = 0.\n\\]\nHowever, while the weight-adjusted approximation of the mass matrix is high order accurate and efficient, it does not preserve the average over a physical element, which is equivalent to the $J$-weighted average over the reference element. This is due to the fact that, in general,\n\\begin{equation}\n\\int_{\\hat{D}}u J^k \\diff{\\hat{\\bm{x}}} - \\int_{\\hat{D}} T_{1/J^k}^{-1}u \\diff{\\hat{\\bm{x}}} \\approx \\bm{1}^T\\bm{M}_{J^k}\\bm{u} - \\bm{1}^T\\bm{M}\\bm{M}_{1/J^k}^{-1}\\bm{M}\\bm{u} \\neq 0. \n\\label{eq:conserr}\n\\end{equation}\nResults in \\cite{chan2016weight1} show that the difference between the true mean and weight-adjusted mean in (\\ref{eq:conserr}) converges extremely fast at a rate of $O(h^{2N+2})$. However, for systems of conservation laws, it is often desired that the local element average is preserved exactly up to machine precision. We present two simpler approaches to ensuring local conservation in this section. \n\nThe simplest way to ensure local conservation is to approximate $J$ using a degree $N$ polynomial and utilize a sufficiently accurate quadrature. Then, we have the following lemma:\n\\begin{lemma}\nLet $J^k \\in P^N$, and let integrals be computed using quadrature which is exact for degree $2N$ polynomials. Then, \n\\[\n\\bm{1}^T\\bm{M}\\bm{M}^{-1}_{1/J^k}\\bm{M}\\bm{u} = \\int_{\\hat{D}}T^{-1}_{1/J^k} u \\diff{\\hat{\\bm{x}}}= \\int_{\\hat{D}} u J^k\\diff{\\hat{\\bm{x}}}.\n\\]\n\\label{eq:conscorrect1}\n\\end{lemma}\n\\begin{proof}\nThe proof relies on (\\ref{eq:wadgTw}), which states that $\\LRp{\\frac{1}{J^k} T^{-1}_{1/J^k}u,v}_{\\hat{D}} = \\LRp{u,v}_{\\hat{D}}$ for all $v\\in P^N$. If $J^k$ is a polynomial of degree $N$, then taking $v = 1$ yields\n\\[\n\\LRp{T^{-1}_{1/J^k}u,1}_{\\hat{D}} = \\LRp{\\frac{1}{J^k}T^{-1}_{1/J^k}u,{J^k}}_{\\hat{D}}= \\LRp{u,{J^k}}_{\\hat{D}} = \\int_{\\hat{D}} uJ^k\\diff{\\hat{\\bm{x}}}. \n\\]\nAdditionally, the proof still holds if integrals are approximated using a quadrature rule which exactly integrates $uJ^k \\in P^{2N}$.\n\\end{proof}\nWe note that, for isoparametric curved elements, $J^k \\not\\in P^N$ in general (in 2D, $J^k \\in P^{2N-2}$, while in 3D, $J^k \\in P^{3N-3}$ \\cite{johnen2013geometrical}). Thus, to ensure local conservation, we will approximate $J^k$ using a degree $N$ polynomial (for example, the interpolant or $L^2$ projection onto $P^N$). We note that this approximation is required only in the weight-adjusted mass matrix, and does not modify the scaled geometric terms $G^k_{ij}$. \n\nThe second approach relies on a simple correction which restores exact conservation of the true mean. In \\cite{chan2016weight2}, it was shown that a rank one correction of the weight-adjusted mass matrix inverse preserves the mean exactly. However, this requires the use of the Sherman--Morrison formula to compute the inverse of a rank one matrix update, which can be cumbersome to incorporate. We present a simpler explicit correction formula which does not involve matrices. Let $u_J$ and $u_{\\rm WADG}$ be defined through \n%coefficients \n%\\[\n%\\bm{u} = \\bm{M}^{-1}\\bm{M}_{1/J}\\bm{M}^{-1} \\LRp{\\bm{V}_q^T\\bm{W}\\bm{f}},\n%\\]\n%where $\\bm{f}$ are the values of $f$ at quadrature points on $\\hat{D}$. \n\\begin{align*}\n\\LRp{u_J J,v}_{\\hat{D}} &= \\LRp{f,v}_{\\hat{D}}, \\qquad \\forall v \\in P^N\\LRp{\\hat{D}},\\\\\n\\LRp{T_{1/J}^{-1}u_{\\rm WADG},v}_{\\hat{D}} &= \\LRp{f,v}_{\\hat{D}}, \\qquad \\forall v \\in P^N\\LRp{\\hat{D}},\n\\end{align*}\nwhere $u_J$ corresponds to the inversion of the weighted mass matrix and $u_{\\rm WADG}$ corresponds to the inversion of a weight-adjusted mass matrix. For example, if $f = u J$ for some function $u(\\bm{x})$, then $u_J = \\Pi^k_N u$ and $u_{\\rm WADG} = \\tilde{\\Pi}^k_N u$. To ensure both local and global conservation, we require that the weighted average of $u_{\\rm WADG}$ is the same as the weighted average of $u_J$. Let the conservative weight-adjusted $\\tilde{u}_{\\rm WADG}$ be defined as\n\\begin{equation}\n \\tilde{u}_{\\rm WADG} = u_{\\rm WADG} + \\frac{\\int_{\\hat{D}} \\left(f - J u_{\\rm WADG}\\right)\\diff{\\hat{\\bm{x}}}}{\\int_{\\hat{D}} J \\diff{\\hat{\\bm{x}}}}. \n \\label{eq:conscorrect}\n\\end{equation}\nTaking the weighted integral of $\\tilde{u}_{\\rm WADG}$ yields\n\\[\n \\int_{\\hat{D}}\\tilde{u}_{\\rm WADG}J \\diff{\\hat{\\bm{x}}} = \\int_{\\hat{D}} f \\diff{\\hat{\\bm{x}}}.\n\\]\nIn other words, (\\ref{eq:conscorrect}) ensures local conservation by correcting the weighted average of $u_{\\rm WADG}$ to match that of $u_J$. Applying this correction to the right hand side of (\\ref{eq:dgform2}) then yields a scheme which locally and globally conserves mean values of the conservative variables.\n\nThis approach is applicable to an arbitrary weight, and can be generalized to matrix-valued weights as well \\cite{chan2017weight}. Moreover, using Theorem 6 in \\cite{chan2016weight1}, one can show that the $L^2$ norm of the difference $\\tilde{u}_{\\rm WADG} - u_{\\rm WADG}$ is $O(h^{2N+1})$, and does not affect high order accuracy. \n\n\\section{Enforcing the discrete geometric conservation law}\n\\label{sec:4}\nAn important aspect of Theorem~\\ref{thm:stab1} is the assumption that $\\bm{Q}^j_k\\bm{1} = 0$ for $j = 1,\\ldots,d$. However, this is not always guaranteed to hold for $\\bm{Q}^j_k$ as defined through (\\ref{eq:dik}). In this section, we discuss methods of constructing the geometric terms $G^k_{ij}$ for curvilinear meshes in a way that ensures $\\bm{Q}^j_k\\bm{1} = 0$. \n\nFrom (\\ref{eq:dik}), the condition $\\bm{Q}^j_k\\bm{1} = 0$ is equivalent to\n\\begin{align*}\n\\bm{Q}^j_k\\bm{1} = \\bm{W}_N\\sum_{j=1}^d\\bm{D}^j_N\\circ \\avg{\\bm{G}_{ij}^k}\\bm{1} &= \\frac{1}{2}\\bm{W}_N\\sum_{j=1}^d \\LRp{ \\diag{\\bm{G}_{ij}^k}\\bm{D}^j_N \\bm{1} + \\bm{D}^j_N\\diag{\\bm{G}_{ij}^k}\\bm{1}} \\\\\n&= \\frac{1}{2}\\bm{W}_N\\sum_{j=1}^d \\bm{D}^j_N \\bm{G}_{ij}^k = 0,\n\\end{align*}\nwhere we have used that $\\bm{D}^j_N \\bm{1} = 0$ to eliminate the first term. Since $\\bm{W}_N$ is a diagonal matrix with positive entries, $\\bm{Q}^j_k\\bm{1} = 0$ is equivalent to ensuring that a discrete version of the GCL (\\ref{eq:gcl}) holds\n\\begin{equation}\n\\sum_{j=1}^d \\bm{D}^j_N \\bm{G}_{ij}^k = 0.\n\\label{eq:dgcl}\n\\end{equation}\nThis condition is required to ensure that free-stream preservation holds at the discrete level. In other words, we wish to ensure that the semi-discrete scheme preserves (for $\\bm{u}$ constant) \n\\[\n\\pd{\\bm{u}}{t} + \\Grad \\cdot \\bm{f}(\\bm{u}) = \\pd{\\bm{u}}{t} = 0.\n\\]\nFor isoparametric geometric mappings (where the degree of the mapping matches the degree of the polynomial approximation) in two dimensions, the GCL is naturally enforced by the ``cross-product'' form, noting that the scaled metric terms $G^k_{ij}$ are exactly polynomials of degree $N$. As a result, computing the metric terms exactly automatically enforces that both the continuous GCL (\\ref{eq:gcl}) and the discrete GCL (\\ref{eq:dgcl}) are satisfied. However, the discrete GCL is not always maintained at the discrete level in 3D. \n\nIn three dimensions, geometric terms are typically computed in ``cross-product'' form\n\\begin{align}\n\\LRs{\\begin{array}{c}\nG^k_{1i}\\\\\nG^k_{2i}\\\\\nG^k_{3i}\\end{array}} &= \\pd{\\bm{x}}{\\hat{x}_j}\\times \\pd{\\bm{x}}{\\hat{x}_k}, \\qquad (i,j,k) = (1,2,3), \\text{ cyclic}.\n\\end{align}\nNote the abuse of notation here and in the sequel, the superscript $k$ refers to the element number and the subscript $k$ to the cyclic index. \nThis formula can be used to compute the geometric terms exactly at volume and surface quadrature points. However, because $\\pd{\\bm{x}}{\\hat{x}_j},\\pd{\\bm{x}}{\\hat{x}_k} \\in P^{N-1}$, the geometric terms $G^k_{ij}$ are polynomials of degree $P^{2N-2}$. The discrete GCL condition holds only if $G^k_{ij} \\in P^{2N-2}$ are differentiated exactly; however, because applying $\\bm{D}^j_N$ involves the $L^2$ projection, and because $G^k_{ij}$ and its $L^2$ projection onto degree $N$ polynomials can differ, the discrete GCL (\\ref{eq:dgcl}) does not hold in general \\cite{kopriva2006metric}. \n\nThis can be remedied by using an alternative form of the geometric terms, which ensures that (\\ref{eq:dgcl}) is satisfied a-priori \\cite{thomas1979geometric, visbal2002use, kopriva2006metric}. The geometric terms $G^k_{ij}$ can also be computed using a ``conservative curl'' form, where $G^k_{ij}$ are the image of the curl applied to some quantity\n%We use the invariant curl form, which is given as\n%\\begin{align}\n%J^k\\bm{G}^k_{nj} = -\\frac{1}{2}\\bm{e}_j\\cdot \\hat{\\Grad} \\times \\LRp{x_l\\hat{\\Grad}x_m - x_m\\hat{\\Grad}x_l}, \\qquad j = 1,2,3, \\quad (n,m,l) = (1,2,3), \\text{ cyclic},\n%\\label{eq:conscurl}\n%\\end{align}\n%where $\\bm{e}_j$ is the $j$th canonical vector and $\\hat{\\Grad} \\times $ denotes the curl with respect to the reference element coordinates. \n% Expanding and simplifying allows (\\ref{eq:conscurl}) to be rewritten \n as follows: % \\cite{hindenlang2012explicit}:\n\\begin{align}\n\\LRs{\\begin{array}{c}\nG^k_{1i}\\\\\nG^k_{2i}\\\\\nG^k_{3i}\\end{array}} =\n\\LRs{\\begin{array}{c}\n {\\left( - \\hat{\\nabla} \\times \\left(x_3 \\hat{\\nabla} x_2\\right)\\right)}_i\\\\\n {\\left(\\phantom{-}\\hat{\\nabla} \\times \\left(x_3 \\hat{\\nabla} x_1\\right)\\right)}_i\\\\\n {\\left(\\phantom{-}\\hat{\\nabla} \\times \\left(x_1 \\hat{\\nabla} x_2\\right)\\right)}_i\\end{array}},\n\\label{eq:conscurl}\n\\end{align}\nwhere the subscript $i$ denotes the $i$th component of the vector quantity.\nFrom (\\ref{eq:conscurl}), it can be observed that, because the divergence of a curl vanishes, the continuous GCL condition (\\ref{eq:gcl}) holds. \n%\\begin{align}\n%\\hat{\\Grad}\\cdot \\LRs{\\begin{array}{c}\n%J^k\\bm{G}^k_{1i}\\\\\n%J^k\\bm{G}^k_{2i}\\\\\n%J^k\\bm{G}^k_{3i}\\end{array}} = \\sum_{j=1}^d \\pd{}{\\hat{x}_j}J^k\\bm{G}^k_{ij} = 0.\n%\\end{align}\nThe central idea of \\cite{visbal2002use, kopriva2006metric} is to use (\\ref{eq:conscurl}), but to interpolate before applying the curl %This approximates the geometric terms via\n\\begin{align}\n\\LRs{\\begin{array}{c}\nG^k_{1i}\\\\\nG^k_{2i}\\\\\nG^k_{3i}\\end{array}} =\n\\LRs{\\begin{array}{c}\n {\\left( - \\hat{\\nabla} \\times I_N\\left(x_3 \\hat{\\nabla} x_2\\right)\\right)}_i\\\\\n {\\left(\\phantom{-}\\hat{\\nabla} \\times I_N\\left(x_3 \\hat{\\nabla} x_1\\right)\\right)}_i\\\\\n {\\left(\\phantom{-}\\hat{\\nabla} \\times I_N\\left(x_1 \\hat{\\nabla} x_2\\right)\\right)}_i\\end{array}},\n\\label{eq:iconscurl}\n\\end{align}\nwhere $I_N$ denotes the degree $N$ polynomial interpolation operator. Since the geometric terms are still computed by applying a curl, the continuous GCL condition (\\ref{eq:gcl}) is still satisfied. We shall also show that this approximation also satisfies the discrete GCL condition. \n\nWe adopt a slight modification of (\\ref{eq:iconscurl}) in this work which is tailored towards triangular and tetrahedral elements. Because the geometric terms are computed by applying the curl, the geometric terms are approximated as degree $(N-1)$ polynomials rather than degree $N$ polynomials, which can reduce accuracy. Instead, we approximate geometric terms by using the interpolation operator $I_{N+1}$ onto degree $(N+1)$ polynomials, then interpolating back to degree $N$ polynomials \n\\begin{align}\n\\LRs{\\begin{array}{c}\nG^k_{1i}\\\\\nG^k_{2i}\\\\\nG^k_{3i}\\end{array}} =\n\\LRs{\\begin{array}{c}\n I_N{\\left( - \\hat{\\nabla} \\times I_{N+1}\\left(x_3 \\hat{\\nabla} x_2\\right)\\right)}_i\\\\\n I_N{\\left(\\phantom{-}\\hat{\\nabla} \\times I_{N+1}\\left(x_3 \\hat{\\nabla} x_1\\right)\\right)}_i\\\\\n I_N{\\left(\\phantom{-}\\hat{\\nabla} \\times I_{N+1}\\left(x_1 \\hat{\\nabla} x_2\\right)\\right)}_i\\end{array}}.\n\\label{eq:iconscurl2}\n\\end{align}\nFor any $u \\in P^N\\LRp{\\hat{D}}$, $\\pd{u}{\\hat{x}_i} \\in P^{N-1}\\LRp{\\hat{D}}$ for $i = 1,2,3$. Thus, the interpolation to degree $N$ polynomials is exact, since the derivatives of a degree $(N+1)$ polynomial are degree $N$ on triangles and tetrahedra. % the approximation generated by (\\ref{eq:iconscurl2}) is a degree $N$ polynomial. \n\n\\begin{remark}\nThe accuracy of (\\ref{eq:iconscurl2}) depends on the choice of interpolation points. It is well known that interpolating at equispaced points can result in inaccurate polynomial approximations. One can determine good interpolation point sets by optimizing over some measure of interpolation quality (such as the Lebesgue constant), and in practice, sets of interpolation points are pre-computed for some polynomial degrees $N = 1,\\ldots, N_{\\max}$ on the reference element and stored \\cite{chen1996optimal, hesthaven1998electrostatics}, or explicitly computed as the image of equispaced points under an appropriately defined mapping \\cite{blyth2006lobatto, warburton2006explicit, chan2015comparison}. \n\\end{remark}\n\nTo prove that the construction (\\ref{eq:iconscurl2}) satisfies Assumption~\\ref{ass:norm}, we must assume that the interpolation points for a degree $N$ element include an appropriate number of points on each face. We note that these assumptions exclude interpolation points which lie purely in the interior of an element, such as those introduced in \\cite{williams2014symmetric, witherden2015identification}. We can now show that the geometric terms satisfy all conditions necessary to guarantee entropy stability:\n\\begin{theorem}\n%<<<<<<< HEAD\nLet the mesh consist of triangles or tetrahedral elements which satisfy Assumption~\\ref{ass:norm}, and let the interpolation points which define the degree $N$ interpolation operator be distributed such that $N^f_p$ points lie on each face. Then, the approximate geometric terms $\\tilde{J^k\\bm{G}^k_{ij}}$ and approximate scaled normals $\\tilde{n_iJ^k_f}$ computed using (\\ref{eq:iconscurl2}) and (\\ref{eq:normalconsistency})) satisfy both the discrete GCL condition (\\ref{eq:dgcl}) and Assumption~\\ref{ass:norm}. Additionally, the error in the approximation satisfies\n%=======\n%Let the mesh satisfy Assumption~\\ref{ass:norm}, and let the interpolation points which define the interpolation operator $I_N$ be distributed such that $N^f_p$ points lie on each face. Then, the approximate geometric terms $\\tilde{G^k_{ij}}$ and approximate scaled normals $\\tilde{n_iJ^k_f}$ computed using (\\ref{eq:iconscurl2}) and (\\ref{eq:normalconsistency})) satisfy both the discrete GCL condition (\\ref{eq:dgcl}) and Assumption~\\ref{ass:norm}. Additionally, the error in the approximation satisfies\n%>>>>>>> 97186f466cf2432ab515d0a2d65cf4dd49645850\n\\[\n%\\sqrt{\\sum_{k}\\sum_{ij=1}^d\\nor{{J^k\\bm{G}^k_{ij}}-\\tilde{J^k\\bm{G}^k_{ij}}}_{L^2\\LRp{\\hat{D}}}^2} \\leq C_N h^{N+2} \\nor{\\cdot}_{W^{N+2,2}\\LRp{D^k}}.\n\\nor{{G^k_{ij}}-\\tilde{G^k_{ij}}}_{L^2\\LRp{\\Omega}}^2 \\leq C_N \\LRb{\\Omega} h^{N+2} \\sqrt{\\sum_{i=1}^d \\sum_k \\nor{\\bm{r}_i}^2_{W^{N+2,2}\\LRp{D^k}}}.\n\\]\n\\label{thm:koprivagcl}\n\\end{theorem}\n\\begin{proof}\n The satisfaction of (\\ref{eq:dgcl}) relies on the fact that $\\tilde{G^k_{ij}}$ is a degree $N$ polynomial and is equal to its own $L^2$ projection. Let $\\tilde{\\bm{G}^k_{ij}}$ denote the polynomial coefficients of $\\tilde{G^k_{ij}}$. \nThen, applying $\\bm{D}^j_N$ to evaluations of $\\tilde{G^k_{ij}}$ at volume and surface quadrature points and using (\\ref{eq:dnvqvf}), we have\n\\[\n\\sum_{j=1}^d\\bm{D}^j_N \\LRs{\n\\begin{array}{c}\n\\bm{V}_q\\\\\n\\bm{V}_f\n\\end{array}}\n\\tilde{\\bm{G}^k_{ij}} = \n\\sum_{j=1}^d \\LRs{\\begin{array}{c}\n\\bm{V}_q\\bm{D}_i\\tilde{\\bm{G}^k_{ij}} \\\\\n0\n\\end{array}}.\n\\]\nThe entries of $\\bm{V}_q\\bm{D}_i\\tilde{\\bm{G}^k_{ij}}$ correspond to values of the derivatives of $\\tilde{G^k_{ij}}$ evaluated at quadrature points. Since \n\\[\n\\sum_{j=1}^d\\pd{}{\\hat{x}_j}\\tilde{G^k_{ij}} = 0\n\\]\nby construction using (\\ref{eq:iconscurl2}), $\\sum_{j=1,\\ldots,d} \\bm{V}_q\\bm{D}_i\\tilde{\\bm{G}^k_{ij}} = \\bm{0}$ as well. \n\nEquation (\\ref{eq:normalconsistency}) of Assumption~\\ref{ass:norm} is satisfied by directly constructing the scaled normals $\\tilde{\\bm{n}}J^k_f$ using values of $\\tilde{G^k_{ij}}$ at quadrature points. We must now prove that the construction of $\\tilde{\\bm{n}}J^k_f$ implies that Equation (\\ref{eq:normalsign}) holds. This is not immediately clear; since the normals are constructed from the approximate geometric terms and the formula (\\ref{eq:normalconsistency}), it is not guaranteed that $\\tilde{\\bm{n}}^+J^{k,+}_f = -\\tilde{\\bm{n}}J^k_f$ will hold across a shared face. However, the scaled normal vectors involve only nodal values on the shared face because the normals are computed in terms of the tangential reference derivatives \\cite{kopriva2006metric}. Thus, assuming a watertight mesh, the interpolation nodes on two neighboring elements will coincide for a shared face $f$, such that the trace of $\\tilde{G^k_{ij}}$ from either neighboring element will be the same lower-dimensional polynomial on $f$. This is sufficient to ensure that the scaled normal vectors $\\tilde{\\bm{n}}^+J^{k,+}_f, \\tilde{\\bm{n}}J^k_f$ will be equal and opposite. \n\nThe local $L^2$ error $\\nor{{G^k_{ij}}-\\tilde{G^k_{ij}}}_{L^2\\LRp{D^k}}$ can be bounded by noting that, since the error $G^k_{ij}-\\tilde{G^k_{ij}}$ consists of linear combinations of derivatives of the interpolation error $\\bm{r}_i-I_{N+1}\\bm{r}_i$, it can be bounded by the $H^1$-seminorm of the latter quantity\n\\begin{align*}\n\\nor{{G^k_{ij}}-\\tilde{G^k_{ij}}}_{L^2\\LRp{D^k}} &\\leq %\\nor{\\sqrt{J^k}}_{L^2\\LRp{\\hat{D}}} \\nor{\\hat{\\Grad}\\times \\LRp{\\bm{r}_i-I_{N+1}\\bm{r}_i}}_{L^2\\LRp{\\hat{D}}}\\\\\n%\\leq \n \\sum_{i=1}^d C_1 \\nor{\\sqrt{J^k}}_{L^2\\LRp{\\hat{D}}} \\LRb{\\LRp{\\bm{r}_i-I_{N+1}\\bm{r}_i}}_{H^1\\LRp{\\hat{D}}}\\\\\n &\\leq \\sum_{i=1}^d \\tilde{C}_N \\nor{\\sqrt{J^k}}_{L^2\\LRp{\\hat{D}}} \\LRb{\\bm{r}_i}_{W^{N+2,2}\\LRp{\\hat{D}}},\n\\end{align*}\nwhere we have used the Bramble-Hilbert lemma \\cite{brenner2007mathematical} on the reference element in the last step. Since it is applied on the reference element $\\hat{D}$ rather than the physical element $D^k$, the constant $\\tilde{C}_N$ depends on the reference element and order of approximation, but not the mesh size $h$. A scaling argument for quasi-uniform meshes then yields that\n\\begin{align}\n\\LRb{\\bm{r}_i}_{W^{N+2,2}\\LRp{\\hat{D}}} \\leq C_2 h^{N+2}\\nor{\\bm{r}_i}_{W^{N+2,2}\\LRp{D^k}}.\n\\label{eq:localgeoerr}\n\\end{align}\nThe global estimate results from squaring (\\ref{eq:localgeoerr}), summing over all elements and using $\\nor{\\sqrt{J^k}}^2_{L^2\\LRp{\\hat{D}}} = \\LRb{D^k}$.\n%\\[\n%%\\sqrt{\\sum_k \\nor{{J^k\\bm{G}^k_{ij}}-\\tilde{J^k\\bm{G}^k_{ij}}}_{L^2\\LRp{D^k}}^2 }\n%\\nor{{J^k\\bm{G}^k_{ij}}-\\tilde{J^k\\bm{G}^k_{ij}}}_{L^2\\LRp{\\Omega}}^2 \\leq C_N \\LRb{\\Omega} h^{N+2} \\sqrt{\\sum_k \\nor{\\bm{r}}_2_{W^{N+2,2}\\LRp{D^k}}}.\n%\\]\n\\end{proof}\n\n%<<<<<<< HEAD\n\\begin{remark}\nIt should be pointed out that this approach does not work on hexahedral elements. This is due to the fact that the natural polynomial space on hexahedral elements is the tensor product space $Q^N\\LRp{\\widehat{D}}$\n\\[\nQ^N\\LRp{\\widehat{D}} = \\LRc{\\hat{x}_1^{i_1}\\ldots\\hat{x}_d^{i_d}, \\quad \\hat{\\bm{x}} \\in \\widehat{D}, \\quad 0 \\leq i_k \\leq N}.\n\\]\nFor $u\\in Q^{N+1}\\LRp{\\hat{D}}$, $\\pd{u}{\\hat{x}_i}$ is at most degree $N$ in the coordinate $\\hat{x}_i$, but can remain a polynomial of degree $N+1$ in all other coordinates. Thus, interpolating from degree $N+1$ to degree $N$ polynomials in (\\ref{eq:iconscurl2}) introduces aliasing errors and is no longer exact. The result of (\\ref{eq:iconscurl2}) is no longer the image of a curl, and thus does not satisfy the discrete GCL by construction. \n\\end{remark}\n\n\n%We briefly outline how to compute $\\tilde{J^k\\bm{G}^k_{ij}}$ in three dimensions. Let $\\LRc{\\hat{\\bm{x}}^N_i}_{i=1}^{N_p}$ denote the set of degree $N$ interpolation points, and let $\\ell^N_i(\\hat{\\bm{x}})$ denote the $i$th degree $N$ Lagrange basis function on the reference element. We define interpolation matrices $\\bm{V}_N^{N+1}$ and $\\bm{V}^N_{N+1}$ between degree $N$ and $N+1$ polynomials such that \n%=======\nWe briefly outline how to compute $\\tilde{G^k_{ij}}$ in three dimensions. Let $\\LRc{\\hat{\\bm{x}}^N_i}_{i=1}^{N_p}$ denote the set of degree $N$ interpolation points, and let $\\ell^N_i(\\hat{\\bm{x}})$ denote the $i$th degree $N$ Lagrange basis function on the reference element. We define interpolation matrices $\\bm{V}_N^{N+1}$ and $\\bm{V}^N_{N+1}$ between degree $N$ and $N+1$ polynomials such that \n%>>>>>>> 97186f466cf2432ab515d0a2d65cf4dd49645850\n\\begin{align}\n\\LRp{\\bm{V}_N^{N+1}}_{ij} &= \\ell^N_j(\\hat{\\bm{x}}^{N+1}_i) , \\qquad 1\\leq i \\leq N_p, \\qquad \\qquad 1\\leq i \\leq (N+1)_p\\\\\n\\LRp{\\bm{V}^N_{N+1}}_{ij} &= \\ell^{N+1}_j(\\hat{\\bm{x}}^{N}_i) , \\qquad 1\\leq i \\leq (N+1)_p, \\qquad \\qquad 1\\leq i \\leq N_p, \\nonumber\n\\end{align}\nwhere $N_p, (N+1)_p$ denotes the number of interpolation points for degree $N$ and $N+1$ polynomials, respectively. Next, let $\\bm{x}_1,\\bm{x}_2,\\bm{x}_3$ denote vectors containing $x_1,x_2,x_3$ coordinates of degree $N$ interpolation points on a curved physical element $D^k$, and let $\\tilde{\\bm{x}}_1,\\tilde{\\bm{x}}_2,\\tilde{\\bm{x}}_3$ denote their evaluation at degree $(N+1)$ interpolation points\n\\begin{align}\n\\tilde{\\bm{x}}_1 = \\bm{V}_N^{N+1}\\bm{x}_1, \\qquad \\tilde{\\bm{x}}_2 = \\bm{V}_N^{N+1}\\bm{x}_2, \\qquad \\tilde{\\bm{x}}_3 = \\bm{V}_N^{N+1}\\bm{x}_3.\n\\end{align}\nLet $\\tilde{\\bm{D}}^{N+1}_i$ denote the nodal differentiation matrix of degree $N+1$ with respect to the $i$th coordinate direction. The geometric factors are computed as follows:\n\\begin{align}\n\\bm{G^k}_{11} &= \\phantom{-}\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_3 \\LRp{\\LRp{ \\tilde{\\bm{D}}^{N+1}_2 \\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_3} - \\tilde{\\bm{D}}^{N+1}_2 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_3\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_3}} \\\\\n\\bm{G^k}_{12} &= \\phantom{-}\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_1 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_3\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_3} - \\tilde{\\bm{D}}^{N+1}_3 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_1\\tilde{\\bm{x}}_2 }\\circ\\tilde{\\bm{x}}_3}} \\nonumber\\\\\n\\bm{G^k}_{13} &= \\phantom{-}\\bm{V}_{N+1}^N{\\LRp{\\tilde{\\bm{D}}^{N+1}_2 \\LRp{\\LRp{ \\tilde{\\bm{D}}^{N+1}_1\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_3} - \\tilde{\\bm{D}}^{N+1}_1 \\LRp{\\LRp{ \\tilde{\\bm{D}}^{N+1}_2\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_3}}} \\nonumber\\\\\n%\n\\bm{G^k}_{21} &= -\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_3 \\LRp{\\LRp{ \\tilde{\\bm{D}}^{N+1}_2 \\tilde{\\bm{x}}_1} \\circ\\tilde{\\bm{x}}_3} - \\tilde{\\bm{D}}^{N+1}_2 \\LRp{\\LRp{ \\tilde{\\bm{D}}^{N+1}_3\\tilde{\\bm{x}}_1} \\circ\\tilde{\\bm{x}}_3}} \\nonumber \\\\\n\\bm{G^k}_{22} &= -\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_1 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_3\\tilde{\\bm{x}}_1} \\circ\\tilde{\\bm{x}}_3} - \\tilde{\\bm{D}}^{N+1}_3 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_1\\tilde{\\bm{x}}_1 }\\circ\\tilde{\\bm{x}}_3}} \\nonumber\\\\\n\\bm{G^k}_{23} &= -\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_2 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_1\\tilde{\\bm{x}}_1} \\circ\\tilde{\\bm{x}}_3} - \\tilde{\\bm{D}}^{N+1}_1 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_2\\tilde{\\bm{x}}_1 }\\circ\\tilde{\\bm{x}}_3}} \\nonumber\\\\\n%\n\\bm{G^k}_{31} &= -\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_3 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_2 \\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_1} - \\tilde{\\bm{D}}^{N+1}_2 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_3\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_1}} \\nonumber\\\\\n\\bm{G^k}_{32} &= -\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_1 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_3\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_1} - \\tilde{\\bm{D}}^{N+1}_3 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_1\\tilde{\\bm{x}}_2 }\\circ\\tilde{\\bm{x}}_1}} \\nonumber\\\\\n\\bm{G^k}_{33} &= -\\bm{V}_{N+1}^N\\LRp{\\tilde{\\bm{D}}^{N+1}_2 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_1\\tilde{\\bm{x}}_2} \\circ\\tilde{\\bm{x}}_1} - \\tilde{\\bm{D}}^{N+1}_1 \\LRp{ \\LRp{\\tilde{\\bm{D}}^{N+1}_2\\tilde{\\bm{x}}_2 }\\circ\\tilde{\\bm{x}}_1}}. \\nonumber\n\\end{align}\n\\begin{remark}\n\nWe note that the discrete GCL (\\ref{eq:dgcl}) can also be enforced directly through a local constrained minimization problem \\cite{fernandez2016simultaneous, crean2018entropy}, which yields a solution in terms of a pseudo-inverse. However, we have not found a straightforward way to simultaneously enforce both the discrete GCL condition (\\ref{eq:dgcl}) and Assumption~\\ref{ass:norm} using this approach.\n\\end{remark}\n\n\n\\section{Numerical experiments}\n\\label{sec:num}\n\nIn this section, we present numerical results which verify the theoretical results in this work. We first verify that the weight-adjusted $L^2$ projection $\\tilde{\\Pi}^k_N$ and the GCL-satisfying geometric factors $G^k_{ij}$ obey the error estimates in Theorem~\\ref{thm:superconverge} and Theorem~\\ref{thm:koprivagcl}. \n\nNext, we verify the semi-discrete entropy conservation, primary conservation, and accuracy of the proposed high order accurate methods for the compressible Euler equations on curved meshes in two and three dimensions. For all curved meshes, we utilize the low storage weight-adjusted formulation (\\ref{eq:dgform2}). \n\nIn choosing the timestep $dt$, we follow \\cite{chan2015gpu} and set \n\\begin{equation}\ndt = \\min_{k} {\\rm CFL} \\frac{\\nor{J}_{L^{\\infty}\\LRp{D^k}}}{C_N\\nor{J^f}_{L^{\\infty}\\LRp{\\partial D^k}}},\n\\end{equation}\nwhere $C_N$ is the $O(N^2)$ order-dependent constant in the surface trace inequality \\cite{chan2015gpu} and ${\\rm CFL}$ is a user-chosen constant. All numerical experiments in this work utilize the five-stage fourth order low storage Runge--Kutta (LSRK-45) time-stepper \\cite{carpenter1994fourth}. \n\n\n\\subsection{Accuracy of weight-adjusted projection and geometric terms}\n\nIn this section, we verify Theorems~\\ref{thm:superconverge} and \\ref{thm:koprivagcl} concerning the accuracy of the weight-adjusted projection and modified construction of geometric terms satisfying the discrete geometric conservation law. Figure~\\ref{fig:superconverge} shows $L^2$ errors for both the standard $L^2$ projection (\\ref{eq:l2curv}) and the weight-adjusted projection (\\ref{eq:wadgproj}) for a series of warped meshes of degree $N = 4$. Errors are estimated using degree $2N+1$ quadratures for triangles and tetrahedra \\cite{xiao2010quadrature}. We compute $L^2$ errors for both smooth and discontinuous functions\n\\[\nf(\\bm{x}) = e^{x_1+x_2}\\sin\\LRp{{\\pi x_1}}\\sin\\LRp{{\\pi x_2}}, \\qquad g(\\bm{x}) = f(\\bm{x}) + H(x_1+x_2-\\sin(\\pi x_1)),\n\\]\nwhere $H$ is the Heaviside function. For the smooth function $f(\\bm{x})$, we observe that the difference between the $L^2$ and weight-adjusted projections indeed converges at a rate of $O\\LRp{h^{N+2}}$ as predicted by Theorem~\\ref{fig:superconverge}, such that the $L^2$ errors for each projection appear identical. The $L^2$ errors for the $L^2$ and weight-adjusted projections of the discontinuous function $g(\\bm{x})$ are also virtually identical. However, the difference beteween the $L^2$ and weight-adjusted projection converges faster than estimated by Theorem~\\ref{fig:superconverge}, with the $L^2$ error converging as $O(h^{1/2})$ and the difference converging as $O(h^{2+1/2})$. \n\n%, despite the fact that Theorem~\\ref{thm:superconverge} only holds for functions with sufficient regularity. Moreover, the difference between the two projections appears to converge even more rapidly for the discontinuous function $g(\\bm{x})$, with the $L^2$ error converging as $O(h^{1/2})$ and the difference converging as $O(h^{2+1/2})$. \n\n\\begin{figure}\n\\centering\n\\subfloat[Warped curvilinear mesh]{\\includegraphics[width=.285\\textwidth]{figs/wadgCurved.png}}\n\\subfloat[$f(\\bm{x}) = e^{x_1+x_2}\\sin\\LRp{{\\pi x_1}}\\sin\\LRp{{\\pi x_2}}$]{\n\\begin{tikzpicture}\n\\begin{loglogaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.37\\textwidth, \n xlabel={Mesh size $h$},\n ylabel={$L^2$ error}, \n ymin=1e-11, ymax=1e-1, \n grid style=dashed,\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.5,0.0268241)(0.25,0.00198388)(0.125,7.29929e-05)(0.0625,2.43094e-06)(0.03125,7.71592e-08)};\n\\logLogSlopeTriangleFlip{0.3}{0.15}{0.48}{5}{blue}\n\\addplot[color=red,mark=x,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.5,0.0269391)(0.25,0.00197894)(0.125,7.29563e-05)(0.0625,2.43063e-06)(0.03125,7.71567e-08)};\n\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.5,0.00237193)(0.25,5.41179e-05)(0.125,7.6934e-07)(0.0625,1.18362e-08)(0.03125,1.84801e-10)};\n\\logLogSlopeTriangleFlip{0.3}{0.15}{0.225}{6}{black}\n\n\\legend{$L^2$ projection,Weight-adjusted,Difference}\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{loglogaxis}\n\\end{tikzpicture}\n}\n\\subfloat[$g(\\bm{x}) = f(\\bm{x}) + H(x_1+x_2-\\sin(\\pi x_1))$]{\n\\begin{tikzpicture}\n\\begin{loglogaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.37\\textwidth,\n xlabel={Mesh size $h$},\n% ymin=1e-7, ymax=1e1,\n% ylabel={$L^2$ error}, \n grid style=dashed,\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.5,0.323258)(0.25,0.227822)(0.125,0.160387)(0.0625,0.114753)(0.03125,0.0792918)};\n\\logLogSlopeTriangleFlip{0.4}{0.25}{0.875}{.5}{blue}\n\\addplot[color=red,mark=x,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.5,0.324117)(0.25,0.227927)(0.125,0.160411)(0.0625,0.114756)(0.03125,0.0792924)};\n\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.5,0.00466574)(0.25,0.000804201)(0.125,0.000182886)(0.0625,2.96701e-05)(0.03125,5.01885e-06)};\n\\logLogSlopeTriangleFlip{0.3}{0.15}{0.18}{2.5}{black}\n\n%\\legend{$L^2$ projection,Weight-adjusted, Difference}\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{loglogaxis}\n\\end{tikzpicture}\n}\n\\caption{$L^2$ errors in approximating both smooth and discontinuous functions using $L^2$ and weight-adjusted projections on a curved mesh. The approximation order is $N=4$, and a degree $2N$ quadrature rule is used to compute integrals over the reference triangle. }\n\\label{fig:superconverge}\n\\end{figure}\n\nWe next compare the approximation of the geometric factors on a curved three-dimensional mesh. We generate a sequence of quasi-uniform unstructured tetrahedral meshes using GMSH \\cite{geuzaine2009gmsh} and construct a curvilinear mesh from the distorted coordinates $\\tilde{\\bm{x}} = \\bm{x} + \\frac{1}{8}\\cos\\LRp{\\frac{\\pi}{2}x_1}\\cos\\LRp{\\frac{\\pi}{2}x_2}\\cos\\LRp{\\frac{\\pi}{2}x_3}$.\nWe then compute the $L^2$ error in approximating geometric terms for each element $D^k$ by computing $G^k_{ij}-\\tilde{G^k_{ij}}$ at quadrature points. \n%\\[\n%\\sqrt{\\sum_{k}\\sum_{ij=1}^d \\nor{G^k_{ij} - \\tilde{G^k_{ij}}}^2_{L^2\\LRp{\\hat{D}}}}. \n%\\]\nWe estimate the mesh size as $h = \\max_k \\nor{J^k/J^k_f}_{L^{\\infty}}$, since $J^k = O(h^d)$ and $J^k_f = O(h^{d-1})$ in $d$ dimensions \\cite{chan2015gpu}. Figure~\\ref{fig:geomerr} shows errors for an $N=3$ and $N=4$ mesh. We refer to the construction of approximate geometric terms $\\tilde{G^k_{ij}}$ introduced in \\cite{kopriva2006metric, hindenlang2012explicit} as ``Geo-$N$'', since the interpolation is performed using a degree $N$ interpolation operator. We refer to the construction of $\\tilde{G^k_{ij}}$ in (\\ref{eq:iconscurl2}) and Theorem~\\ref{thm:koprivagcl} as ``Geo-$(N+1)$'', since the main interpolation step is performed on degree $(N+1)$ polynomials instead. It can be observed that the Geo-N scheme converges at a rate of $O(h^{N+1})$, while the Geo-(N+1) scheme converges at a rate of $O(h^{N+2})$. We note that the error in both the Geo-$N$ and Geo-$(N+1)$ approximations of $\\tilde{G^k_{ij}}$ converge at the same rate or faster than the best approximation error. \n%\\note{Optimal rates are observed for $N=5$, geometry order 3 by Hindenlang, but this is be because $G^k_{ij} \\in Q^{2N_{\\rm geo}-2} = Q^4$, so it should not be necessary to enforce the GCL in that situation. Note sensitivity of nonlinear solvers to geometry error \\cite{toulorge2016optimizing}}.\n\n\\begin{figure}\n\\centering\n\\subfloat[$N=3$]{\n\\begin{tikzpicture}\n\\begin{loglogaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.45\\textwidth, \n xlabel={Mesh size $h$},\n ylabel={$L^2$ error}, \n% ymin=1e-10, ymax=1e-1, \n grid style=dashed,\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.67574,0.00231891)(0.374388,7.21634e-05)(0.16654,3.2502e-06)(0.0867431,1.31183e-07)};\n\\addplot[color=magenta,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.67574,0.00231891)(0.374388,0.000121059)(0.16654,2.10855e-06)(0.0867431,8.08278e-08)};\n\\addplot[color=red,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.67574,0.00635039)(0.374388,0.000542466)(0.16654,4.85003e-05)(0.0867431,4.03792e-06)};\n\\addplot[color=black,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.67574,0.00952559)(0.374388,0.000897556)(0.16654,3.51441e-05)(0.0867431,2.58651e-06)};\n\n%coordinates{(0.5,0.00237193)(0.25,5.41179e-05)(0.125,7.6934e-07)(0.0625,1.18362e-08)(0.03125,1.84801e-10)};\n%\\logLogSlopeTriangleFlip{0.3}{0.15}{0.225}{6}{black}\n\n\\legend{Geo-$(N+1)$, $h^{N+2}$, Geo-$N$, $h^{N+1}$}\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{loglogaxis}\n\\end{tikzpicture}\n}\n\\subfloat[$N=4$]{\n\\begin{tikzpicture}\n\\begin{loglogaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.45\\textwidth,\n xlabel={Mesh size $h$},\n% ymin=1e-10, ymax=1e-1, \n% ymin=1e-7, ymax=1e1,\n% ylabel={$L^2$ error}, \n grid style=dashed,\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.673351,0.000766195)(0.373777,1.44835e-05)(0.166655,2.91207e-07)(0.0867501,5.69685e-09)};\n\\addplot[color=magenta,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.673351,0.000957743)(0.373777,2.80204e-05)(0.166655,2.20143e-07)(0.0867501,4.37946e-09)};\n\\addplot[color=red,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.673351,0.00231677)(0.373777,0.000127702)(0.166655,5.3742e-06)(0.0867501,2.11781e-07)};\n\\addplot[color=black,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.673351,0.00405434)(0.373777,0.000213685)(0.166655,3.7653e-06)(0.0867501,1.43901e-07)};\n\n%coordinates{(0.5,0.00237193)(0.25,5.41179e-05)(0.125,7.6934e-07)(0.0625,1.18362e-08)(0.03125,1.84801e-10)};\n%\\logLogSlopeTriangleFlip{0.3}{0.15}{0.225}{6}{black}\n\n\\legend{Geo-$(N+1)$, $h^{N+2}$, Geo-$N$, $h^{N+1}$}\n\\end{loglogaxis}\n\\end{tikzpicture}\n}\n\\caption{$L^2$ errors in the approximation of metric terms $G^k_{ij}$ with metric terms $\\tilde{G^k_{ij}}$ satisfying the discrete GCL condition (\\ref{eq:dgcl}) and Assumption~\\ref{ass:norm}.}\n\\label{fig:geomerr}\n\\end{figure}\n\n% N = 3\n\\subsection{Two-dimensional compressible Euler equations}\n\nThe compressible Euler equations in two dimensions are given as follows:\n\\begin{align}\n\\pd{\\rho}{t} + \\pd{\\LRp{\\rho u}}{x_1} + \\pd{\\LRp{\\rho v}}{x_2} &= 0,\\\\\n\\pd{\\rho u}{t} + \\pd{\\LRp{\\rho u^2 + p }}{x_1} + \\pd{\\LRp{\\rho uv}}{x_2} &= 0,\\nonumber\\\\\n\\pd{\\rho v}{t} + \\pd{\\LRp{\\rho uv}}{x_1} + \\pd{\\LRp{\\rho v^2 + p }}{x_2} &= 0,\\nonumber\\\\\n\\pd{E}{t} + \\pd{\\LRp{u(E+p)}}{x_1} + \\pd{\\LRp{v(E+p)}}{x_2}&= 0.\\nonumber\n\\end{align}\nIn two dimensions, the pressure is $p = (\\gamma-1)\\LRp{E - \\frac{1}{2}\\rho (u^2+v^2)}$, and the specific internal energy is $\\rho e = E - \\frac{1}{2}\\rho (u^2+v^2)$. \n\nThe choice of convex entropy for the Euler equations is non-unique \\cite{harten1983symmetric}. However, a unique entropy can be chosen by restricting to choices of entropy variables which symmetrize the viscous heat conduction term in the compressible Navier-Stokes equations \\cite{hughes1986new}. This leads to $U(\\bm{u})$ of the form\n\\begin{equation}\nU(\\bm{u}) = -\\frac{\\rho s}{\\gamma-1},\n\\label{eq:entropy2d}\n\\end{equation}\nwhere $s = \\log\\LRp{\\frac{p}{\\rho^\\gamma}}$ is the physical specific entropy. The entropy variables in two dimensions are \n\\begin{align}\nv_1 = \\frac{\\rho e (\\gamma + 1 - s) - E}{\\rho e}, \\qquad v_2 = \\frac{\\rho u}{\\rho e}, \\qquad v_3 = \\frac{\\rho v}{\\rho e}, \\qquad v_4 = -\\frac{\\rho}{\\rho e}.\n\\end{align}\nThe conservation variables in terms of the entropy variables are given by\n\\begin{equation}\n\\rho = -(\\rho e) v_4, \\qquad \\rho u = (\\rho e) v_2, \\qquad \\rho v = (\\rho e) v_3, \\qquad E = (\\rho e)\\LRp{1 - \\frac{{v_2^2+v_3^2}}{2 v_4}},\n\\end{equation}\nwhere $\\rho e$ and $s$ in terms of the entropy variables are \n\\begin{equation}\n\\rho e = \\LRp{\\frac{(\\gamma-1)}{\\LRp{-v_4}^{\\gamma}}}^{1/(\\gamma-1)}e^{\\frac{-s}{\\gamma-1}}, \\qquad s = \\gamma - v_1 + \\frac{{v_2^2+v_3^2}}{2v_4}.\n\\end{equation}\nThe entropy conservative numerical fluxes for the two-dimensional compressible Euler equations are given by Chandrashekar \\cite{chandrashekar2013kinetic}\n\\begin{align}\n&f^1_{1,S}(\\bm{u}_L,\\bm{u}_R) = \\avg{\\rho}^{\\log} \\avg{u},& &f^1_{2,S}(\\bm{u}_L,\\bm{u}_R) = \\avg{\\rho}^{\\log} \\avg{v},&\\\\\n&f^2_{1,S}(\\bm{u}_L,\\bm{u}_R) = f^1_{1,S} \\avg{u} + p_{\\rm avg},& &f^2_{2,S}(\\bm{u}_L,\\bm{u}_R) = f^1_{2,S} \\avg{u},&\\nonumber\\\\\n&f^3_{1,S}(\\bm{u}_L,\\bm{u}_R) = f^2_{2,S},& &f^3_{2,S}(\\bm{u}_L,\\bm{u}_R) = f^1_{2,S} \\avg{v} + p_{\\rm avg},&\\nonumber\\\\\n&f^4_{1,S}(\\bm{u}_L,\\bm{u}_R) = \\LRp{E_{\\rm avg} + p_{\\rm avg}}\\avg{u},& &f^4_{2,S}(\\bm{u}_L,\\bm{u}_R) = \\LRp{E_{\\rm avg} + p_{\\rm avg} }\\avg{v},& \\nonumber\n\\end{align}\nwhere we have defined the auxiliary quantities \n\\begin{gather}\np_{\\rm avg} = \\frac{\\avg{\\rho}}{2\\avg{\\beta}}, \\qquad E_{\\rm avg} = \\frac{\\avg{\\rho}^{\\log}}{2\\avg{\\beta}^{\\log}\\LRp{\\gamma -1}} + \\frac{\\nor{\\bm{u}}^2_{\\rm avg}}{2}, \\\\\n \\nor{\\bm{u}}^2_{\\rm avg} = 2(\\avg{u}^2 + \\avg{v}^2) - \\LRp{\\avg{u^2} +\\avg{v^2}} \\nonumber. \n\\end{gather}\n\n\\subsubsection{Entropy conservation}\n\nWe begin by testing the propagation of a shock on a two-dimensional curved mesh using a discontinuous profile on the domain $\\Omega = [0,20] \\times [-5,5]$. We set the initial velocities to be zero, and initialize the density and pressure as a discontinuous square pulse as in \\cite{chan2017discretely}\n\\begin{equation}\n\\rho(\\bm{x},t) = \\begin{cases}\n3 & \\LRb{x_1} < 1/2 \\text{ and } \\LRb{x_2} < 1/2\\\\\n2 & \\text{otherwise},\n\\end{cases} \\qquad \nu(\\bm{x},t) = v(\\bm{x},t) = 0, \\qquad\np(\\bm{x},t) = \\rho^\\gamma.\n\\label{eq:discontin}\n\\end{equation}\nTo test the scheme (\\ref{eq:dgform2}), we utilize an entropy conservative flux and run it on a uniform triangular mesh with a curvilinear warping shown in Figure~\\ref{fig:warp2d}. Theorem~\\ref{thm:stab2} ensures that, under an entropy conservative flux, (\\ref{eq:dgform2}) is semi-discretely entropy conservative. This does not hold at the fully discrete level; however, it is possible to verify that (\\ref{eq:dgform2}) is entropy stable using other approaches. \n\nFirst, can examine the entropy RHS, which we define as the right hand side of (\\ref{eq:dgform2}) tested with $\\bm{v}_h$ \n\\begin{equation}\n \\text{entropy RHS}= -\\sum_{j=1}^d\\LRp{ \\tilde{\\bm{v}}^T\\LRp{2\\bm{Q}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\tilde{\\bm{v}}_f^T\\bm{W}_f^k \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)}}. \n\\label{eq:entropyrhs}\n\\end{equation}\n%\\lcwnote{I put a 2 in the previous equation to match \\eqref{eq:dgform2weak}. Do we also want minus signs since these terms are moved to the RHS?} JC: thanks! sure - added a zero, though the term should be zero so the sign shouldn't matter\nFor positive density and pressure, (\\ref{eq:entropyrhs}) should be zero to machine precision. We can also track the change in entropy $\\Delta U = \\LRb{U(\\bm{u}(\\bm{x},t))-U(\\bm{u}(\\bm{x},0))}$, which should converge to $0$ as the timestep $dt$ approaches zero. Furthermore, the rate of convergence should match the order of the time-stepper used \\cite{gassner2016well, chan2017discretely}. \n\n\\begin{figure}\n\\centering\n\\subfloat[Uniform mesh]{\\includegraphics[width=.35\\textwidth]{figs/vortexMeshUniform.png}}\n\\hspace{1em}\n\\subfloat[Warped mesh]{\\includegraphics[width=.35\\textwidth]{figs/vortexMeshCurved.png}}\n\\caption{2D curved meshes used for testing entropy conservation and primary conservation.}\n\\label{fig:warp2d}\n\\end{figure}\n\nFigure~\\ref{fig:dSconverge} shows the evolution of entropy $U(\\bm{u})$ over time $[0,T]$, where the final time $T = 2$. We compare when the entropy-projected conservative variables $\\tilde{\\bm{u}}$ are computed using the standard $L^2$ projection on the reference element, and when $\\tilde{\\bm{u}}$ are computed using the weight-adjusted projection. For the standard $L^2$ projection, the change in entropy does not decrease as $dt$ decreases (for a fixed mesh and order $N$). When $\\tilde{\\bm{u}}$ is defined using the weight-adjusted projection, the entropy decreases as $dt$ decreases. Moreover, the rate of convergence is approximately $O(dt^{4.675})$, which is slightly higher than the expected rate of $O(dt^4)$ when using LSRK-45. Regardless of whether the standard and weight-adjusted projection was used, the entropy RHS (\\ref{eq:entropyrhs}) is $O(10^{-14})$, indicating that the proposed scheme is implemented correctly. \n\n\\begin{figure}\n\\centering\n\\subfloat[With weight-adjusted projection]{\n\\begin{tikzpicture}\n\\begin{semilogyaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.45\\textwidth, \n xlabel={Time $t$},\n ylabel={Change in entropy $\\Delta U(\\bm{u})$}, \n ymin=1e-9, ymax=5e-4, \n grid style=dashed,\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.025641,0)(0.128205,1.81455e-05)(0.230769,1.84993e-05)(0.333333,1.81016e-05)(0.435897,2.38212e-05)(0.538462,3.43804e-05)(0.641026,3.75077e-05)(0.74359,3.80509e-05)(0.846154,4.02406e-05)(0.948718,4.57864e-05)(1.05128,5.21023e-05)(1.15385,5.52339e-05)(1.25641,5.93653e-05)(1.35897,6.53204e-05)(1.46154,7.10508e-05)(1.5641,7.58692e-05)(1.66667,8.21148e-05)(1.76923,8.86933e-05)(1.87179,9.52306e-05)(1.97436,0.000102832)};\n%\\addplot[color=blue,dashed,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.025641,9.95731e-16)(0.128205,4.11303e-15)(0.230769,1.09496e-14)(0.333333,1.86934e-14)(0.435897,1.8624e-14)(0.538462,4.82253e-14)(0.641026,2.80886e-14)(0.74359,3.34073e-14)(0.846154,3.41116e-14)(0.948718,1.12826e-14)(1.05128,1.02106e-14)(1.15385,7.27543e-15)(1.25641,3.21965e-15)(1.35897,2.70617e-15)(1.46154,2.28116e-15)(1.5641,1.01134e-14)(1.66667,9.05179e-15)(1.76923,8.96852e-15)(1.87179,2.69368e-14)(1.97436,1.20043e-15)};\n\\addplot[color=red,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.0129032,0)(0.116129,9.54018e-07)(0.219355,8.64409e-07)(0.322581,7.28292e-07)(0.425806,1.00547e-06)(0.529032,1.54507e-06)(0.632258,1.58376e-06)(0.735484,1.48926e-06)(0.83871,1.52599e-06)(0.941935,1.77263e-06)(1.04516,2.04306e-06)(1.14839,2.10785e-06)(1.25161,2.25231e-06)(1.35484,2.49471e-06)(1.45806,2.7075e-06)(1.56129,2.86391e-06)(1.66452,3.11015e-06)(1.76774,3.35711e-06)(1.87097,3.6045e-06)(1.97419,3.88568e-06)};\n%\\addplot[color=red,dotted,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.0129032,1.00787e-15)(0.116129,2.48065e-16)(0.219355,8.43769e-15)(0.322581,1.07483e-14)(0.425806,6.18255e-15)(0.529032,4.95159e-14)(0.632258,1.19835e-14)(0.735484,3.68837e-14)(0.83871,2.35957e-14)(0.941935,1.19904e-14)(1.04516,1.73785e-14)(1.14839,7.88952e-15)(1.25161,2.44249e-14)(1.35484,6.38031e-15)(1.45806,1.78781e-14)(1.56129,5.72459e-16)(1.66452,4.02456e-15)(1.76774,1.7316e-14)(1.87097,2.50425e-14)(1.97419,3.55375e-14)};\n\\addplot[color=black,mark=pentagon*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.00647249,0)(0.110032,5.38994e-08)(0.213592,4.43651e-08)(0.317152,3.18878e-08)(0.420712,4.65967e-08)(0.524272,7.64772e-08)(0.627832,7.36288e-08)(0.731392,6.33475e-08)(0.834951,6.22881e-08)(0.938511,7.43794e-08)(1.04207,8.71374e-08)(1.14563,8.67872e-08)(1.24919,9.19998e-08)(1.35275,1.02781e-07)(1.45631,1.11204e-07)(1.55987,1.16121e-07)(1.66343,1.26634e-07)(1.76699,1.36562e-07)(1.87055,1.46567e-07)(1.97411,1.57581e-07)};\n%\\addplot[color=black,dashdotted,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.00647249,2.19226e-16)(0.110032,5.88071e-15)(0.213592,1.34059e-14)(0.317152,2.30337e-14)(0.420712,8.23647e-15)(0.524272,2.68709e-14)(0.627832,2.53304e-14)(0.731392,2.95527e-14)(0.834951,2.5948e-14)(0.938511,4.56579e-15)(1.04207,2.28428e-14)(1.14563,8.22259e-15)(1.24919,1.51094e-14)(1.35275,7.86871e-15)(1.45631,1.11508e-14)(1.55987,1.57721e-14)(1.66343,6.95624e-15)(1.76699,2.07681e-14)(1.87055,2.72421e-14)(1.97411,1.7028e-14)};\n\n% % N = 4, K= 8, dt = .25\n % N = 4, K= 8, dt = .125\n % N = 4, K= 8, dt = .0625\n\n\\legend{${\\rm CFL} = .25$,${\\rm CFL} = .125$,${\\rm CFL} = .0625$ }\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{semilogyaxis}\n\\end{tikzpicture}\n}\n\\subfloat[Without weight-adjusted projection]{\n\\begin{tikzpicture}\n\\begin{semilogyaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.45\\textwidth,\n xlabel={Time $t$},\n% ymin=1e-10, ymax=1e-1, \n% ymin=1e-7, ymax=1e1,\n ymin=1e-9, ymax=5e-4, \n% ylabel={$L^2$ error}, \n grid style=dashed,\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.025641,0)(0.128205,4.45681e-05)(0.230769,6.82313e-05)(0.333333,0.000131517)(0.435897,0.000108761)(0.538462,8.51562e-05)(0.641026,2.94791e-05)(0.74359,3.62904e-05)(0.846154,2.97564e-05)(0.948718,9.87886e-05)(1.05128,0.000136806)(1.15385,7.64601e-05)(1.25641,0.000111044)(1.35897,5.72761e-05)(1.46154,4.27e-05)(1.5641,5.03819e-05)(1.66667,4.99789e-05)(1.76923,5.07438e-05)(1.87179,4.26361e-05)(1.97436,2.48694e-05)};\n%\\addplot[color=blue,dashed,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.025641,5.11743e-16)(0.128205,1.16547e-14)(0.230769,7.91034e-15)(0.333333,1.28231e-14)(0.435897,4.79131e-15)(0.538462,3.44134e-14)(0.641026,6.92502e-15)(0.74359,2.28047e-14)(0.846154,2.15314e-14)(0.948718,4.36456e-15)(1.05128,2.09416e-14)(1.15385,4.86763e-15)(1.25641,3.59088e-15)(1.35897,3.43475e-16)(1.46154,1.03632e-14)(1.5641,1.03598e-14)(1.66667,1.83881e-16)(1.76923,5.94663e-15)(1.87179,3.18252e-14)(1.97436,1.88495e-14)};\n\\addplot[color=red,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.0129032,0)(0.116129,6.48664e-05)(0.219355,8.24474e-05)(0.322581,0.000152373)(0.425806,0.000138342)(0.529032,0.000126709)(0.632258,1.85501e-05)(0.735484,6.98555e-05)(0.83871,7.53129e-05)(0.941935,0.00013915)(1.04516,0.000196476)(1.14839,0.000133645)(1.25161,0.000173751)(1.35484,0.000126668)(1.45806,0.000116008)(1.56129,0.000127919)(1.66452,0.0001343)(1.76774,0.000140992)(1.87097,0.00013971)(1.97419,0.0001291)};\n%\\addplot[color=red,dotted,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.0129032,3.00107e-16)(0.116129,6.93196e-15)(0.219355,1.00198e-14)(0.322581,7.79932e-15)(0.425806,2.48759e-15)(0.529032,3.46181e-14)(0.632258,1.7205e-14)(0.735484,2.30718e-14)(0.83871,3.0146e-14)(0.941935,9.4369e-15)(1.04516,1.07969e-14)(1.14839,5.59275e-15)(1.25161,5.34295e-15)(1.35484,5.7801e-15)(1.45806,8.74648e-15)(1.56129,1.24137e-14)(1.66452,2.95597e-15)(1.76774,2.64441e-14)(1.87097,2.58023e-14)(1.97419,6.984e-15)};\n\\addplot[color=black,mark=pentagon*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.00647249,0)(0.110032,6.52844e-05)(0.213592,8.08628e-05)(0.317152,0.000153077)(0.420712,0.000141625)(0.524272,0.000130883)(0.627832,2.58046e-05)(0.731392,6.81127e-05)(0.834951,7.92926e-05)(0.938511,0.000137768)(1.04207,0.000201777)(1.14563,0.000136705)(1.24919,0.000177364)(1.35275,0.000131153)(1.45631,0.000119851)(1.55987,0.000131686)(1.66343,0.000138676)(1.76699,0.00014539)(1.87055,0.000144608)(1.97411,0.000134142)};\n%\\addplot[color=black,dashdotted,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.00647249,7.31403e-16)(0.110032,1.05055e-14)(0.213592,2.58127e-15)(0.317152,1.04916e-14)(0.420712,5.12437e-15)(0.524272,4.40203e-14)(0.627832,1.147e-14)(0.731392,2.81025e-14)(0.834951,3.01946e-14)(0.938511,4.51028e-15)(1.04207,1.096e-14)(1.14563,6.11317e-15)(1.24919,3.69843e-15)(1.35275,6.47399e-15)(1.45631,9.41608e-15)(1.55987,1.58901e-15)(1.66343,6.39766e-15)(1.76699,1.56819e-14)(1.87055,2.11949e-14)(1.97411,1.12063e-14)};\n\n%\\legend{Geo-$(N+1)$, $h^{N+2}$, Geo-$N$, $h^{N+1}$}\n\\legend{${\\rm CFL} = .25$,${\\rm CFL} = .125$,${\\rm CFL} = .0625$ }\n\\end{semilogyaxis}\n\\end{tikzpicture}\n}\n\\caption{Change in entropy under an entropy conservative formulation with $N=4$. In both cases, the magnitude of the entropy RHS (\\ref{eq:entropyrhs}) is $O\\LRp{10^{-14}}$. }\n\\label{fig:dSconverge}\n\\end{figure}\n\nThese results suggest that the weight-adjusted projection is necessary to produce an entropy conservative scheme on curvilinear meshes. However, entropy conservative schemes result in spurious oscillations and lower convergence rates \\cite{chan2017discretely}. In practice, dissipative interface terms are added to produce entropy stable schemes. In the presence of interface dissipation, the evolution of entropy over time, $L^2$ errors for smooth solutions, and the qualitative behavior of the solution are very similar with or without the weight-adjusted projection. This may reflect the fact that aliasing-driven instabilities arise from the spatial discretization, and the entropy RHS (\\ref{eq:entropyrhs}) is machine precision zero with or without the weight-adjusted projection. In contrast, the presence of the weight-adjusted projection affects only the time-derivative on the left-hand side, and may not play as large a role in suppressing aliasing instabilities. \n\n\n\\subsubsection{Local and global conservation}\n\\label{sec:conservationcheck}\nNext, we check that primary conservation is maintained numerically on curved meshes. We follow \\cite{friedrich2017entropy} and examine the semi-discrete evolution of the average of $\\bm{u}$ over the domain $\\Omega$\n\\begin{align}\n \\pd{}{t}\\int_{\\Omega} \\bm{u} \\diff{\\bm{x}} = \\sum_k \\int_{D^k} \\pd{\\bm{u}}{t} \\diff{\\bm{x}} = \\sum_k \\int_{\\hat{D}} \\pd{\\bm{u}}{t} J^k \\diff{\\hat{\\bm{x}}} = \\sum_k \\bm{1}^T \\bm{W}\\diag{\\bm{J}^k} \\bm{V}_q \\td{\\bm{u}_h}{t}. \n\\label{eq:conssemidiscrete}\n\\end{align}\nThe quantity (\\ref{eq:conssemidiscrete}) is computed using quadrature by multiplying the semi-discrete system (\\ref{eq:dgform2}) by $\\bm{1}^T\\bm{W}\\diag{\\bm{J}^k} \\bm{V}_q$\n\\begin{align}\n&\\sum_k \\bm{1}^T \\bm{W}\\diag{\\bm{J}^k} \\bm{V}_q \\td{\\bm{u}_h}{t}= \\nonumber\\\\\n& \\underbrace{\\bm{1}^T\\bm{W}\\diag{\\bm{J}^k}\\bm{V}_q\\LRp{\n \\LRs{\\begin{array}{cc}\\tilde{\\bm{P}}^k_q & \\tilde{\\bm{L}}^k_q\\end{array}} \\sum_{j=1}^d \\LRp{2\\bm{D}^j_k \\circ \\bm{F}_{j,S}}\\bm{1} + \\sum_{j=1}^d \\tilde{\\bm{L}}^k_q \\diag{\\bm{n}_j}\\LRp{\\bm{f}_j^* - \\bm{f}_j(\\tilde{\\bm{u}}_f)}}}_{\\text{conservation residual}}.\n \\label{eq:consres}\n\\end{align}\n\nWe consider three different cases:\n\\begin{enumerate}\n\\item \\textbf{Unmodified:} the weight-adjusted DG scheme (\\ref{eq:dgform2}) is used without any special modifications,\n\\item \\textbf{Polynomial approximation:} the weight-adjusted DG scheme (\\ref{eq:dgform2}) utilizes the $L^2$ projection of $J^k$ onto $P^N(\\hat{D})$ in the weight-adjusted mass matrix (such that Lemma~\\ref{eq:conscorrect1} holds), \n\\item \\textbf{Conservative correction:} the conservation correction (\\ref{eq:conscorrect}) is applied to the right hand side of the scheme (\\ref{eq:dgform2}). \n\\end{enumerate}\nAll results use a Lax-Friedrichs flux, a CFL of $1/2$, $N=4$, and the curved mesh warping shown in Figure~\\ref{fig:warp2d}. We use volume quadrature rules from \\cite{xiao2010quadrature} which are exact for degree $2N+1$ polynomials, and use $(N+1)$-node 1D Gauss--Legendre quadrature rules on the faces. \n\nFigure~\\ref{subfig:cons1} shows computed conservation residuals from (\\ref{eq:consres}) on the curved mesh in Figure~\\ref{fig:warp2d} for the discontinuous pulse initial condition (\\ref{eq:discontin}). The conservation residual oscillates around $O(10^{-6})$ for the unmodified scheme (\\ref{eq:dgform2}). Modifying the scheme using either the ``Polynomial approximation'' or ``Conservative correction'' approaches reduces the conservation error to $O(10^{-12})$. %The magnitude of the conservation residual is due to roundoff effects. Repeating the same experiments on a smaller mesh of $8\\times 8$ curved elements reduces the conservation residual to $O(10^{-13})$, while using a larger mesh of $32\\times 32$ curved elements results in a conservation residual of $O(10^{-11})$. \nWe have also studied the accuracy of each case in by comparing $L^2$ errors on a curved mesh for a smooth vortex solution at time $T=5$. We observe that in all cases, the $L^2$ errors are virtually identical. For $N=4$ on an $8\\times 8$ mesh, the $L^2$ errors for each case differ only in the 5th digit, while for the $16\\times 16$ and $32\\times 32$ meshes, they differ only in the 8th digit. This confirms that both approaches outlined in Section~\\ref{sec:conservation} restore primary conservation with no perceivable effect on accuracy. \n\n%\\note{finish}\n\n\\begin{figure}\n\\centering\n\\subfloat[ Conservation residual over time (unmodified)]{\n\\begin{tikzpicture}\n\\begin{semilogyaxis}[\n legend cell align=left,\n legend style={legend pos=north east, font=\\tiny},\n width=.45\\textwidth, \n xlabel={Time},\n ylabel={Conservation residual}, \n ymin=1e-8, ymax=1e-5, \n grid style=dashed,\n% ytick={1e-12,1e-10,1e-8,1e-6,1e-4,1e-2 }\n] \n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.00642674,9.9266e-07)(0.404884,4.12992e-07)(0.803342,5.14326e-07)(1.2018,1.86091e-07)(1.60026,1.48526e-07)(1.99871,2.64763e-07)(2.39717,5.64483e-07)(2.79563,2.71857e-07)(3.19409,2.62073e-07)(3.59254,2.40249e-07)(3.991,2.80807e-07)(4.38946,3.23725e-07)(4.78792,3.9606e-07)(5.18638,3.46717e-07)(5.58483,2.17698e-07)(5.98329,1.99781e-07)(6.38175,9.91165e-08)(6.78021,2.63862e-07)(7.17866,3.13545e-07)(7.57712,4.01892e-07)(7.97558,2.37462e-07)(8.37404,3.91475e-07)(8.77249,7.92905e-07)(9.17095,2.75479e-07)(9.56941,4.42523e-07)(9.96787,6.28215e-07)};\n\n%\\addplot[color=red,mark=x,dashed,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.00642674,3.34034e-12)(0.404884,3.31953e-12)(0.803342,3.47682e-12)(1.2018,3.48782e-12)(1.60026,3.69874e-12)(1.99871,3.55961e-12)(2.39717,3.69915e-12)(2.79563,3.67725e-12)(3.19409,3.84246e-12)(3.59254,3.84801e-12)(3.991,3.9449e-12)(4.38946,3.95075e-12)(4.78792,4.05986e-12)(5.18638,4.07415e-12)(5.58483,4.18282e-12)(5.98329,4.34447e-12)(6.38175,4.3659e-12)(6.78021,4.34614e-12)(7.17866,4.46538e-12)(7.57712,4.37571e-12)(7.97558,4.35258e-12)(8.37404,4.16896e-12)(8.77249,4.22267e-12)(9.17095,4.14187e-12)(9.56941,4.10735e-12)(9.96787,3.90628e-12)};\n%\n%\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.00642674,3.28579e-12)(0.404884,3.32419e-12)(0.803342,3.37232e-12)(1.2018,3.50534e-12)(1.60026,3.64225e-12)(1.99871,3.47872e-12)(2.39717,3.62415e-12)(2.79563,3.68648e-12)(3.19409,3.79388e-12)(3.59254,3.84432e-12)(3.991,3.97812e-12)(4.38946,3.96804e-12)(4.78792,4.0684e-12)(5.18638,4.04986e-12)(5.58483,4.22413e-12)(5.98329,4.25356e-12)(6.38175,4.39801e-12)(6.78021,4.37474e-12)(7.17866,4.39737e-12)(7.57712,4.38242e-12)(7.97558,4.31792e-12)(8.37404,4.28398e-12)(8.77249,4.21688e-12)(9.17095,4.10925e-12)(9.56941,4.17163e-12)(9.96787,3.96996e-12)};\n\\legend{Unmodified}\n%\\legend{Unmodified, Polynomial approx., Conservative correction }\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{semilogyaxis}\n\\end{tikzpicture}\n\\label{subfig:cons1}}\n\\subfloat[Conservation residual over time (modified)]{\n\\begin{tikzpicture}\n\\begin{semilogyaxis}[\n legend cell align=left,\n legend style={legend pos=north east, font=\\tiny},\n width=.45\\textwidth, \n xlabel={Time},\n ylabel={Conservation residual}, \n ymin=.5e-12, ymax=2e-11, \n grid style=dashed,\n ytick={1e-12, 1e-11},\n yticklabels={{$ 10^{-12}$},{$ 10^{-11}$}}\n] \n\n%\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\n%coordinates{(0.00642674,9.9266e-07)(0.404884,4.12992e-07)(0.803342,5.14326e-07)(1.2018,1.86091e-07)(1.60026,1.48526e-07)(1.99871,2.64763e-07)(2.39717,5.64483e-07)(2.79563,2.71857e-07)(3.19409,2.62073e-07)(3.59254,2.40249e-07)(3.991,2.80807e-07)(4.38946,3.23725e-07)(4.78792,3.9606e-07)(5.18638,3.46717e-07)(5.58483,2.17698e-07)(5.98329,1.99781e-07)(6.38175,9.91165e-08)(6.78021,2.63862e-07)(7.17866,3.13545e-07)(7.57712,4.01892e-07)(7.97558,2.37462e-07)(8.37404,3.91475e-07)(8.77249,7.92905e-07)(9.17095,2.75479e-07)(9.56941,4.42523e-07)(9.96787,6.28215e-07)};\n\n\\addplot[color=red,mark=x,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.00642674,3.34034e-12)(0.404884,3.31953e-12)(0.803342,3.47682e-12)(1.2018,3.48782e-12)(1.60026,3.69874e-12)(1.99871,3.55961e-12)(2.39717,3.69915e-12)(2.79563,3.67725e-12)(3.19409,3.84246e-12)(3.59254,3.84801e-12)(3.991,3.9449e-12)(4.38946,3.95075e-12)(4.78792,4.05986e-12)(5.18638,4.07415e-12)(5.58483,4.18282e-12)(5.98329,4.34447e-12)(6.38175,4.3659e-12)(6.78021,4.34614e-12)(7.17866,4.46538e-12)(7.57712,4.37571e-12)(7.97558,4.35258e-12)(8.37404,4.16896e-12)(8.77249,4.22267e-12)(9.17095,4.14187e-12)(9.56941,4.10735e-12)(9.96787,3.90628e-12)};\n\n\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.00642674,3.28579e-12)(0.404884,3.32419e-12)(0.803342,3.37232e-12)(1.2018,3.50534e-12)(1.60026,3.64225e-12)(1.99871,3.47872e-12)(2.39717,3.62415e-12)(2.79563,3.68648e-12)(3.19409,3.79388e-12)(3.59254,3.84432e-12)(3.991,3.97812e-12)(4.38946,3.96804e-12)(4.78792,4.0684e-12)(5.18638,4.04986e-12)(5.58483,4.22413e-12)(5.98329,4.25356e-12)(6.38175,4.39801e-12)(6.78021,4.37474e-12)(7.17866,4.39737e-12)(7.57712,4.38242e-12)(7.97558,4.31792e-12)(8.37404,4.28398e-12)(8.77249,4.21688e-12)(9.17095,4.10925e-12)(9.56941,4.17163e-12)(9.96787,3.96996e-12)};\n\\legend{Polynomial approx., Conservative correction }\n%\\legend{Unmodified, Polynomial approx., Conservative correction }\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{semilogyaxis}\n\\end{tikzpicture}\n\\label{subfig:cons2}}\n%\\subfloat[$L^2$ errors at time $T=5$]{\n%\\begin{tikzpicture}\n%\\begin{loglogaxis}[\n% legend cell align=left,\n% legend style={legend pos=south east, font=\\tiny},\n% width=.4\\textwidth, \n% xlabel={Time},\n% ylabel={Conservation residual}, \n% ymin=1e-2, ymax=2, \n% grid style=dashed,\n% ytick={1e-13,1e-11,1e-9,1e-7,1e-5,1e-3,1e-1 }\n%] \n%\n%\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\n%% 1.315466363259058 0.263564848187665 0.018309692309614\n%coordinates{(1.25,1.315466)(0.625,0.263565)(0.3125,0.0183097)};\n%\n%\\addplot[color=red,mark=x,dashed,semithick, mark options={solid,fill=markercolor}]\n%%1.315440699275237 0.263564850056684 0.018309677771005\n%coordinates{(1.25,1.31544)(0.625,0.263565)(0.3125,0.0183097)};\n%\n%\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\n%% 1.315467094406914\n%coordinates{(1.25,1.315467)(0.625,0.263565)(0.3125,0.0183097)};\n%\n%\\legend{Unmodified, Polynomial approx., Conservative correction }\n%\n%\\end{loglogaxis}\n%\\end{tikzpicture}\n%\\label{subfig:cons2}}\n%\\caption{Conservation residuals for a discontinuous solution and $L^2$ errors for a vortex solution using the unmodified WADG formulation (\\ref{eq:dgform2}) and two different techniques for restoring local conservation. Both experiments use a Lax-Friedrichs flux, a CFL of $1/2$, $N=4$, and the curved mesh warping shown in Figure~\\ref{fig:warp2d}. }\n\\caption{Conservation residuals for a discontinuous initial condition using the unmodified WADG formulation (\\ref{eq:dgform2}) and two different techniques for restoring local conservation. Both experiments use a Lax-Friedrichs flux, a CFL of $1/2$, $N=4$, and the curved mesh warping shown in Figure~\\ref{fig:warp2d}. }\n\\label{fig:cons}\n\\end{figure}\n\n\n\\subsubsection{Accuracy and convergence}\n\n\\begin{figure}\n\\centering\n\\subfloat[Affine mesh]{\\includegraphics[width=.4\\textwidth]{figs/mesh2d_affine_converge2.png}}\n\\hspace{2em}\n\\subfloat[Curved mesh]{\\includegraphics[width=.4\\textwidth]{figs/mesh2d_curved_converge2.png}}\n\\caption{Example of affine and warped meshes (corresponding to $h = 1$) used in convergence studies.}\n\\label{fig:warp2dconverge}\n\\end{figure}\n\nFinally, we test the accuracy of the proposed schemes for smooth solutions on curved meshes in two dimensions. \nWe use the isentropic vortex problem \\cite{shu1998essentially}, which has an analytical solution \n\\begin{align}\n\\rho(\\bm{x},t) &= \\LRp{1 - \\frac{\\frac{1}{2}(\\gamma-1)(\\beta e^{1-r(\\bm{x},t)^2})^2}{8\\gamma \\pi^2}}^{\\frac{1}{\\gamma-1}}, \\qquad p = \\rho^{\\gamma},\\\\\nu(\\bm{x},t) &= 1 - \\frac{\\beta}{2\\pi} e^{1-r(\\bm{x},t)^2}(x_2-c_2), \\qquad v(\\bm{x},t) = \\frac{\\beta}{2\\pi} e^{1-r(\\bm{x},t)^2}(x_2-c_2),\\nonumber\n\\end{align}\nwhere $u, v$ are the $x_1$ and $x_2$ velocity and $r(\\bm{x},t) = \\sqrt{(x_1-c_1-t)^2 + (x_2-c_2)^2}$. Here, we take $c_1 = 5, c_2 = 0$ and $\\beta = 5$. \nThe solution is computed on a periodic rectangular domain $[0, 20] \\times [-5,5]$ at final time $T=5$. Quasi-uniform triangular meshes are generated using \\textsc{GMSH} \\cite{geuzaine2009gmsh}, and a curvilinear warping is applied to the mesh to test the effect of non-affine mappings. This warping is shown in Figure~\\ref{fig:warp2dconverge} and is defined by mapping nodal positions on each triangle to warped nodal positions $(\\tilde{x_1},\\tilde{x_2})$ via\n\\begin{align*}\n\\tilde{x_1} &= x_1 + \\sin\\LRp{\\pi x_1 / 20}\\sin\\LRp{2\\pi (x_2+5)/10}\\\\\n\\tilde{x_2} &= x_2 - \\frac{1}{2}\\sin\\LRp{2 \\pi x_1 / 20}\\sin\\LRp{\\pi (x_2+5)/10}.\n\\end{align*} \nTo ensure primary conservation, we use the ``Polynomial Approximation'' strategy described in Section~\\ref{sec:conservation} and compute the inverse of the weight-adjusted mass matrix using the degree $N$ $L^2$ projection $\\hat{\\Pi}_N J^k$ instead of $J^k$. The computed $L^2$ errors are shown in Figure~\\ref{fig:converge2d}. We observe optimal $O(h^{N+1})$ rates of convergence for $N=2, N= 4$. For degree $N=3$, the rate of convergence is slightly higher than $O(h^{4})$ and may indicate that the mesh is not yet sufficiently fine for the $L^2$ error to show the asymptotic convergence rate. \n\n\\begin{figure}\n\\centering\n\\begin{tikzpicture}\n\\begin{loglogaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.45\\textwidth, \n xlabel={Mesh size $h$},\n ylabel={$L^2$ error}, \n ymin=5e-6, ymax=2, \n xmin=1e-1, xmax=2.5, \n grid style=dashed,\n legend entries={Affine,Curved}\n] \n\\addlegendimage{no markers,black}\n\\addlegendimage{no markers,dashed,black}\n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,1.06717)(1,0.149639)(0.5,0.025693)(0.25,0.00342827)} [yshift=4pt] node[left, pos=1.05, color=blue] {$N = 2$};\n\\addplot[color=blue,mark=*,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,1.37709)(1,0.219501)(0.5,0.0385896)(0.25,0.0039906)};\n\\logLogSlopeTriangleFlip{0.45}{0.15}{0.575}{3}{blue}\n\n\n\\addplot[color=red,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.50782)(1,0.0516053)(0.5,0.00249425)(0.25,0.000110618)}[yshift=8pt] node[left, pos=1.05, color=red] {$N = 3$};\n\\addplot[color=red,mark=square*,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.535036)(1,0.0657418)(0.5,0.00501536)(0.25,0.000243005)};\n\\logLogSlopeTriangleFlip{0.45}{0.15}{0.34}{4}{red}\n\n\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.235374)(1,0.0125714)(0.5,0.000321559)(0.25,1.06097e-05)} [yshift=8pt] node[left, pos=1.05, color=black] {$N = 4$};\n\\addplot[color=black,mark=triangle*,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.297758)(1,0.0207604)(0.5,0.000816006)(0.25,2.36102e-05)};\n\\logLogSlopeTriangle{0.45}{0.15}{0.05}{5}{black}\n\n%\\legend{$L^2$ projection,Weight-adjusted,Difference}\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{loglogaxis}\n\\end{tikzpicture}\n\\caption{Convergence of $L^2$ errors for the 2D isentropic vortex problem on affine and curved meshes.}\n\\label{fig:converge2d}\n\\end{figure}\n\n%The mesh is constructed by first building a mesh of uniform quadrilateral elements and subdividing each quadrilateral into two uniform triangles. We estimate the $L^2$ errors and their rates of convergence, which are shown in Figure~\\ref{fig:converge2d}. We observe $L^2$ optimal $O(h^{N+1})$ rates of convergence for $N = 1,\\ldots, 3$, while for $N = 4$ we observe a rate of convergence between $O(h^{N+1})$ and $O(h^{N+1/2})$. We note that the rate of $O(h^{N+1/2})$ is the theoretically proven rate of convergence for upwind DG methods on general meshes applied to linear hyperbolic problems \\cite{johnson1986analysis,cockburn2008optimal}. We note that this observed rate does not change significantly if the time-step is halved (improving from $4.785$ to $4.8$), suggesting that this slight degradation in convergence rate is not due to temporal errors. \n\n\\subsection{Three dimensional compressible Euler equations}\n\nIn three dimensions, the compressible Euler equations are given by\n\\begin{align}\n\\pd{}{t}\\LRp{\\begin{array}{c}\n\\rho\\\\\n\\rho u\\\\\n\\rho v\\\\\n\\rho w\\\\\nE\n\\end{array}} +\n\\pd{}{x_1}\\LRp{\\begin{array}{c}\n\\rho u\\\\\n\\rho u^2+p\\\\\n\\rho uv\\\\\n\\rho uw\\\\\nu(E+p)\n\\end{array}} + \\pd{}{x_2}\\LRp{\\begin{array}{c}\n\\rho v\\\\\n\\rho uv\\\\\n\\rho v^2+p\\\\\n\\rho vw\\\\\nv(E+p)\n\\end{array}} + \\pd{}{x_3}\\LRp{\\begin{array}{c}\n\\rho w\\\\\n\\rho uw\\\\\n\\rho vw\\\\\n\\rho w^2+p\\\\\nw(E+p)\n\\end{array}} &= 0,\n\\label{eq:euler3d}\n\\end{align}\nwhere the pressure $p$ and specific internal energy $\\rho e$ are defined \n\\begin{align}\np = (\\gamma-1)\\LRp{E - \\frac{1}{2}\\rho (u^2+v^2+w^2)}, \\qquad \\rho e = E - \\frac{1}{2}\\rho (u^2+v^2+w^2). \n\\label{eq:pressure3d}\n\\end{align}\nThe formula for the entropy $U(\\bm{u})$ in three dimensions is the same as the two-dimensional formula (\\ref{eq:entropy2d}). \n The entropy variables in three dimensions are \n\\begin{align}\nv_1 = \\frac{\\rho e (\\gamma + 1 - s) - E}{\\rho e}, \\qquad v_2 = \\frac{\\rho u}{\\rho e}, \\qquad v_3 = \\frac{\\rho v}{\\rho e}, \\qquad v_4 = \\frac{\\rho w}{\\rho e}, \\qquad v_5 = -\\frac{\\rho}{\\rho e}.\n\\end{align}\nThe conservation variables in terms of the entropy variables are given by\n\\begin{equation}\n\\rho = -(\\rho e) v_5, \\qquad \\rho u = (\\rho e) v_2, \\qquad \\rho v = (\\rho e) v_3, \\qquad \\rho w = (\\rho e) v_4, \\qquad E = (\\rho e)\\LRp{1 - \\frac{{v_2^2+v_3^2+v_4^2}}{2 v_5}},\n\\end{equation}\nwhere $\\rho e$ and $s$ in terms of the entropy variables are \n\\begin{equation}\n\\rho e = \\LRp{\\frac{(\\gamma-1)}{\\LRp{-v_5}^{\\gamma}}}^{1/(\\gamma-1)}e^{\\frac{-s}{\\gamma-1}}, \\qquad s = \\gamma - v_1 + \\frac{{v_2^2+v_3^2+v_4^2}}{2v_5}.\n\\end{equation}\nA set of entropy conservative numerical fluxes for the three-dimensional compressible Euler equations can be written as\n\\begin{gather}\n\\bm{f}_{1,S} = \\LRp{\\begin{array}{c}\n\\avg{\\rho}^{\\log}\\avg{u}\\\\\n\\avg{\\rho}^{\\log}\\avg{u}^2 + p_{\\rm avg}\\\\\n\\avg{\\rho}^{\\log}\\avg{u}\\avg{v}\\\\\n\\avg{\\rho}^{\\log}\\avg{u}\\avg{w}\\\\\n(E_{\\rm avg}+ p_{\\rm avg})\\avg{u}\\\\\n\\end{array}}, \n\\qquad \n\\bm{f}_{2,S} = \\LRp{\\begin{array}{c}\n\\avg{\\rho}^{\\log}\\avg{v}\\\\\n\\avg{\\rho}^{\\log}\\avg{u}\\avg{v}\\\\\n\\avg{\\rho}^{\\log}\\avg{v}^2 + p_{\\rm avg}\\\\\n\\avg{\\rho}^{\\log}\\avg{v}\\avg{w}\\\\\n(E_{\\rm avg}+ p_{\\rm avg})\\avg{v}\\\\\n\\end{array}},\\\\\n\\bm{f}_{3,S} = \\LRp{\\begin{array}{c}\n\\avg{\\rho}^{\\log}\\avg{w}\\\\\n\\avg{\\rho}^{\\log}\\avg{u}\\avg{w}\\\\\n\\avg{\\rho}^{\\log}\\avg{v}\\avg{w}\\\\\n\\avg{\\rho}^{\\log}\\avg{w}^2 + p_{\\rm avg}\\\\\n(E_{\\rm avg}+ p_{\\rm avg})\\avg{w}\\\\\n\\end{array}}.\\nonumber\n%&f^1_{1,S}(\\bm{u}_L,\\bm{u}_R) = \\avg{\\rho}^{\\log} \\avg{u},& &f^1_{2,S}(\\bm{u}_L,\\bm{u}_R) = \\avg{\\rho}^{\\log} \\avg{v},&\\\\\n%&f^2_{1,S}(\\bm{u}_L,\\bm{u}_R) = f^1_{1,S} \\avg{u} + p_{\\rm avg},& &f^2_{2,S}(\\bm{u}_L,\\bm{u}_R) = f^1_{2,S} \\avg{u},&\\nonumber\\\\\n%&f^3_{1,S}(\\bm{u}_L,\\bm{u}_R) = f^2_{2,S},& &f^3_{2,S}(\\bm{u}_L,\\bm{u}_R) = f^1_{2,S} \\avg{v} + p_{\\rm avg},&\\nonumber\\\\\n%&f^4_{1,S}(\\bm{u}_L,\\bm{u}_R) = \\LRp{\\frac{p_{\\rm avg}^{\\log}}{\\gamma -1} + p_{\\rm avg} + \\frac{\\nor{\\bm{u}}^2_{\\rm avg}}{2}}\\avg{u},& &f^4_{2,S}(\\bm{u}_L,\\bm{u}_R) = \\LRp{\\frac{p_{\\rm avg}^{\\log}}{\\gamma -1} + p_{\\rm avg} + \\frac{\\nor{\\bm{u}}^2_{\\rm avg}}{2}}\\avg{v},& \\nonumber\n\\end{gather}\nwhere we have defined the auxiliary quantities\n\\begin{align}\np_{\\rm avg} &= \\frac{\\avg{\\rho}}{2\\avg{\\beta}}, \\qquad E_{\\rm avg} = \\frac{\\avg{\\rho}^{\\log}}{2(\\gamma-1)\\avg{\\beta}^{\\log}} + \\frac{1}{2}\\avg{\\rho}^{\\log}\\nor{\\bm{u}}^2_{\\rm avg}\\\\\n\\nor{\\bm{u}}^2_{\\rm avg} &= 2(\\avg{u}^2 + \\avg{v}^2 + \\avg{w}^2) - \\LRp{\\avg{u^2} +\\avg{v^2} + \\avg{w^2}}.\\nonumber\n\\end{align}\n\n\\subsubsection{Accuracy and convergence}\n\nAs before, we test the accuracy of the proposed scheme using an isentropic vortex solution adapted to three dimensions. We take the solution to be the extruded 2D vortex propagating in the $x_2$ direction, whose analytic expression is derived from \\cite{williams2013nodal}\n\\begin{align*}\n\\rho(\\bm{x},t) &= \\LRp{1-\\frac{(\\gamma-1)}{2}\\Pi^2}^{\\frac{1}{\\gamma-1}}\\\\\n\\bm{u}(\\bm{x},t) &= \\Pi \\bm{r}, \\\\\nE(\\bm{x},t) &= \\frac{p_0}{\\gamma-1}\\LRp{1-\\frac{\\gamma-1}{2}\\Pi^2}^{\\frac{\\gamma}{\\gamma-1}} + \\frac{\\rho}{2}\\LRb{\\bm{u}}.\n\\end{align*}\nwhere $\\bm{u} = (u,v,w)^T$ is the velocity vector and % $\\Pi$, $\\bm{r}$ are defined as\n\\[\n\\Pi = \\Pi_{\\max}e^{\\frac{1-\\bm{r}^T\\bm{r}}{2}}, \\qquad \\bm{r} = \\begin{pmatrix}\n-(x_2-c_2-t)\\\\\nx_1-c_1\\\\\n0\n\\end{pmatrix}.\n\\]\nIn this problem, we take $c_1 = c_2 = 5$, $p_0 = {1}/{\\gamma}$, and $\\Pi_{\\max} = 0.4$. The problem is solved on the domain $[0,10]\\times [0,20]\\times [0,10]$. \n%\\begin{figure}\n%\\centering\n%\\subfloat[$h = 1$]{\\includegraphics[width=.4\\textwidth]{figs/periodicCube2.png}}\n%\\hspace{2em}\n%\\subfloat[$h = 1/2$]{\\includegraphics[width=.4\\textwidth]{figs/periodicCube3.png}}\n%\\caption{Unstructured affine tetrahedral meshes used for convergence tests.}\n%\\label{fig:mesh3d}\n%\\end{figure}\nWe use \\textsc{GMSH} to construct three unstructured meshes consisting of 1354, 9543, and 72923 affine tetrahedra, corresponding to $h = 2$, $h = 1$, and $h = 1/2$ (shown in Figure~\\ref{subfig:mesh3d}). We apply a curvilinear warping by again mapping nodal positions each tetrahedron to warped nodal positions $(\\tilde{x_1},\\tilde{x_2},\\tilde{x_3})$ via\n\\begin{align*}\n\\tilde{x_1} &= x_1 + \\frac{1}{2}\\sin\\LRp{\\pi \\frac{x_1}{10}}\\sin\\LRp{2\\pi \\frac{x_2}{20}}\\sin\\LRp{\\pi \\frac{x_3}{10}},\\\\\n\\tilde{x_2} &= x_2 - \\sin\\LRp{2 \\pi \\frac{x_1}{10}}\\sin\\LRp{\\pi \\frac{x_2}{10}}\\sin\\LRp{2\\pi \\frac{x_3}{10}},\\\\\n\\tilde{x_3} &= x_3 + \\frac{1}{2}\\sin\\LRp{\\pi \\frac{x_1}{10}}\\sin\\LRp{2\\pi \\frac{x_2}{10}}\\sin\\LRp{\\pi \\frac{x_3}{10}}.\n\\end{align*} \nThe geometric terms are approximated using (\\ref{eq:iconscurl2}) to ensure the satisfaction of the discrete GCL. \n\n\\begin{figure}\n\\centering\n\\subfloat[Mesh for $h = 1/2$]{\\raisebox{2em}{\\includegraphics[width=.4\\textwidth]{figs/periodicCube3.png}}\\label{subfig:mesh3d}}\n\\subfloat[$L^2$ errors]{\\begin{tikzpicture}\n\\begin{loglogaxis}[\n legend cell align=left,\n legend style={legend pos=south east, font=\\tiny},\n width=.45\\textwidth, \n xlabel={Mesh size $h$},\n ylabel={$L^2$ error}, \n ymin=1e-4, ymax=2, \n xmin=2.5e-1, xmax=2.5, \n grid style=dashed,\n legend entries={Affine,Curved}\n] \n\\addlegendimage{no markers,black}\n\\addlegendimage{no markers,dashed,black}\n\n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,1.05519)(1,0.143515)(0.5,0.0212682)}[yshift=4pt] node[left, pos=1.05, color=blue] {$N = 2$};\n\\addplot[color=blue,mark=*,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,1.20915)(1,0.2069)(0.5,0.0284505)};\n\\logLogSlopeTriangleFlip{0.45}{0.15}{0.6}{3}{blue}\n\n\\addplot[color=red,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.339318)(1,0.0314342)(0.5,0.00197699)}[yshift=8pt] node[left, pos=1.05, color=red] {$N = 3$};\n\\addplot[color=red,mark=square*,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.464613)(1,0.0513369)(0.5,0.00334595)};\n\\logLogSlopeTriangleFlip{0.45}{0.15}{0.375}{4}{red}\n\n\\addplot[color=black,mark=triangle*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.122229)(1,0.00488434)(0.5,0.000192453)}[yshift=8pt] node[left, pos=1.05, color=black] {$N = 4$};\n\\addplot[color=black,mark=triangle*,dashed,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(2,0.184547)(1,0.0104361)(.5,0.0003960681)};\n\\logLogSlopeTriangle{0.45}{0.15}{0.04}{5}{black}\n\n%\\legend{$L^2$ projection,Weight-adjusted,Difference}\n%\\legend{Uniform, Optimal, Smoothed}\n\\end{loglogaxis}\n\\end{tikzpicture}}\n\\caption{Convergence of $L^2$ errors for the 3D isentropic vortex problem on unstructured affine and curved meshes, with optimal $O(h^{N+1})$ rates of convergence shown for reference.}\n\\label{fig:converge3d}\n\\end{figure}\n\nFigure~\\ref{fig:converge3d} shows $L^2$ errors for $N = 2,3,4$ on both affine and curved meshes. In both cases, we observe optimal $O(h^{N+1})$ asymptotic rates of convergence for $N = 2,3$, while for $N = 4$ we observe a rate which is slightly higher than $O(h^{N+1/2})$ but not quite $O(h^{N+1})$. This may be due to the fact that the time-stepper is $4$th order while the spatial discretization is $5$th order. \n\nWe also compared approximations of geometric factors using degree $N$ polynomials via (\\ref{eq:iconscurl2}) and degree $N-1$ polynomials via (\\ref{eq:iconscurl}). For the isentropic vortex problem on warped meshes, the approximation of the geometric factors does not impact accuracy significantly. Utilizing (\\ref{eq:iconscurl}) and approximating geometric factors with degree $N-1$ polynomials only changed the error in the $5$th significant digit, and resulted in the same asymptotic rates of convergence on curved 3D meshes. Future work will explore the effect of a more accurate geometric approximation on curved boundaries where a solid wall boundary condition is applied, where the approximation of geometry has been shown to have a more significant effect \\cite{toulorge2016optimizing}. \n\n%\\note{Isentropic vortex on curved and non-curved meshes}\n\n\\subsubsection{Inviscid Taylor--Green vortex}\n\nOur last numerical experiment investigates the behavior of entropy stable DG schemes for the inviscid Taylor--Green vortex \\cite{ae1937mechanism, gassner2016split, crean2018entropy}. The domain is the periodic box $[-\\pi,\\pi]^3$, and the initial conditions are given as\n\\begin{align*}\n\\rho &= 1\\\\\nu &= \\sin(x_1)\\cos(x_2)\\cos(x_3),\\\\\nv &= -\\cos(x_1)\\sin(x_2)\\cos(x_3),\\\\ \nw &= 0,\\\\\np &= \\frac{100}{\\gamma} + \\frac{1}{16} \\LRp{\\cos(2x_1) + \\cos(2x_2)}\\LRp{2+\\cos(2x_3)}.\n\\end{align*}\nThe Taylor--Green vortex is used to study the transition and decay of turbulence \\cite{debonis2013solutions}. In the absence of viscosity, the Taylor--Green vortex develops smaller and smaller scales, implying that for sufficiently large times, the solution contains under-resolved features. We study the evolution of the kinetic energy $\\kappa(t)$ \n\\[\n\\kappa(t) =\\frac{1}{\\LRb{\\Omega}} \\int_{\\Omega} \\rho \\bm{u}\\cdot\\bm{u} \\diff{\\bm{x}},\n\\]\nas well as the kinetic energy dissipation rate $-\\pd{\\kappa}{t}$, which we approximate by differencing $\\kappa(t)$. Figure~\\ref{fig:tg} shows the evolution of $\\kappa(t)$ over time for both affine and curvilinear meshes with $h = \\pi/8$ and $N = 3$. We also use a curved coarse mesh with element size $h = \\pi$ test the convergence of the average entropy for a non-dissipative entropy conservative formulation as the timestep decreases. In both cases, the mesh is defined by constructing nodal positions $\\tilde{\\bm{u}}$ from warpings of affine nodal positions $\\bm{x}$ as follows \n\\[\n\\tilde{\\bm{x}} = \\bm{x} + .125 \\sin(x_1)\\sin(x_2)\\sin(x_3).\n\\]\nThe kinetic energy is plotted at 100 equally spaced times between $[0,20]$. The simulation is stable and does not blow up despite the lack of filtering, limiting, or artificial viscosity. The results are qualitatively similar to those reported in the literature, with the kinetic energy dissipation rate increasing around $t = 4$ and peaking with a value of roughly $0.014$ before $t = 9$ \\cite{debonis2013solutions, gassner2016split}. We also observe that, for an entropy conservative formulation, the average entropy appears to converge at a rate of $O(dt^{5})$ for a 4th order time-stepping method. The same phenomena was also observed in \\cite{chan2017discretely, crean2018entropy}. \n\n\\begin{figure}\n\\centering\n\\subfloat[KE dissipation rate for $N=3$, $h = \\pi/8$]{\n\\begin{tikzpicture}\n\\begin{axis}[\n scaled ticks=false, \n tick label style={/pgf/number format/fixed},\n\tlegend cell align=left,\n\tlegend style={font=\\tiny},\n\twidth=.475\\textwidth,\n xlabel={Time $t$},\n ylabel={$-\\pd{\\kappa}{t}$},\n% xmin=.005, xmax=1,\n% ymin=1e-10, ymax=1e-1,\nymin=-.0025, ymax=.017,\n legend pos=north east,\n xmajorgrids=true,\n ymajorgrids=true,\n grid style=dashed,\n ytick={0, .005, .01, .015},\n yticklabels={0, .005, .01, .015} \n] \n\\addplot[color=blue,semithick, mark options={fill=markercolor}]\ncoordinates{(0,-0)(0.18796,-8.45875e-06)(0.37592,-9.02267e-06)(0.563879,-1.63536e-05)(0.751839,-1.40979e-05)(0.939799,-2.48123e-05)(1.12776,-2.65041e-05)(1.31572,-2.76319e-05)(1.50368,-2.53763e-05)(1.69164,-2.0301e-05)(1.8796,-2.31206e-05)(2.06756,-1.69175e-05)(2.25552,-1.3534e-05)(2.44348,2.25567e-06)(2.63144,1.63536e-05)(2.8194,2.98876e-05)(3.00736,4.90608e-05)(3.19532,7.38731e-05)(3.38328,0.000117859)(3.57124,0.000183837)(3.7592,0.000306771)(3.94716,0.000514292)(4.13512,0.000819935)(4.32308,0.00122765)(4.51104,0.00175378)(4.699,0.00240454)(4.88696,0.00307729)(5.07491,0.00363332)(5.26288,0.00394291)(5.45083,0.00414817)(5.6388,0.00436754)(5.82676,0.00455927)(6.01471,0.00484235)(6.20267,0.00527544)(6.39063,0.00577169)(6.57859,0.0063638)(6.76655,0.00709971)(6.95451,0.00787679)(7.14247,0.00860142)(7.33043,0.0093097)(7.51839,0.0100575)(7.70635,0.0109236)(7.89431,0.011972)(8.08227,0.0129661)(8.27023,0.013675)(8.45819,0.0141244)(8.64615,0.0142863)(8.83411,0.0141814)(9.02207,0.0139541)(9.21003,0.0136913)(9.39799,0.0134325)(9.58595,0.0133152)(9.77391,0.0132047)(9.96187,0.0130733)(10.1498,0.0127473)(10.3378,0.0122686)(10.5258,0.0118191)(10.7137,0.0114588)(10.9017,0.011108)(11.0896,0.0107578)(11.2776,0.0104415)(11.4656,0.0101274)(11.6535,0.00979918)(11.8415,0.00946253)(12.0294,0.00913827)(12.2174,0.00885124)(12.4053,0.00859409)(12.5933,0.00836514)(12.7813,0.00814465)(12.9692,0.00790724)(13.1572,0.00766025)(13.3451,0.00740818)(13.5331,0.0071578)(13.7211,0.00691137)(13.909,0.00667057)(14.097,0.0064264)(14.2849,0.00616305)(14.4729,0.0058918)(14.6609,0.00565045)(14.8488,0.00543559)(15.0368,0.00524499)(15.2247,0.00506736)(15.4127,0.00490326)(15.6007,0.00475551)(15.7886,0.00460156)(15.9766,0.00443803)(16.1645,0.00427562)(16.3525,0.00412054)(16.5405,0.00397618)(16.7284,0.00384422)(16.9164,0.00371339)(17.1043,0.00357805)(17.2923,0.00344835)(17.4803,0.00333275)(17.6682,0.00323237)(17.8562,0.00313369)(18.0441,0.00303105)(18.2321,0.00292842)(18.4201,0.00282973)(18.608,0.00273556)(18.796,0.00264082)(18.9839,0.00254834)(19.1719,0.00246319)(19.3599,0.0023848)(19.5478,0.00231431)(19.7358,0.00225059)(19.9237,0.00218969)};\n\n\\addplot[color=red, dashed,semithick, mark options={fill=markercolor}]\ncoordinates{(0,5.36239e-07)(0.18796,-9.65229e-06)(0.37592,-1.01885e-05)(0.563879,-1.98408e-05)(0.751839,-1.76959e-05)(0.939799,-1.93046e-05)(1.12776,-2.14495e-05)(1.31572,-1.93046e-05)(1.50368,-2.4667e-05)(1.69164,-1.66234e-05)(1.8796,-8.57982e-06)(2.06756,-0)(2.25552,1.17972e-05)(2.44348,1.76959e-05)(2.63144,3.70005e-05)(2.8194,6.43486e-05)(3.00736,0.000107784)(3.19532,0.000173741)(3.38328,0.000268656)(3.57124,0.000432745)(3.7592,0.000693357)(3.94716,0.00106122)(4.13512,0.00156796)(4.32308,0.00222485)(4.51104,0.00299597)(4.699,0.00373597)(4.88696,0.00432584)(5.07491,0.00468404)(5.26288,0.00495645)(5.45083,0.00521599)(5.6388,0.00551361)(5.82676,0.0059426)(6.01471,0.00650672)(6.20267,0.00717005)(6.39063,0.00793258)(6.57859,0.00878681)(6.76655,0.0095858)(6.95451,0.0103457)(7.14247,0.0110685)(7.33043,0.0116551)(7.51839,0.0124011)(7.70635,0.013015)(7.89431,0.0133341)(8.08227,0.0135652)(8.27023,0.0136097)(8.45819,0.0135175)(8.64615,0.0134918)(8.83411,0.0133647)(9.02207,0.0131925)(9.21003,0.0130242)(9.39799,0.012703)(9.58595,0.0123292)(9.77391,0.011971)(9.96187,0.0115586)(10.1498,0.0111345)(10.3378,0.0107687)(10.5258,0.0103671)(10.7137,0.0100089)(10.9017,0.00969788)(11.0896,0.00941206)(11.2776,0.00916861)(11.4656,0.00893803)(11.6535,0.00868707)(11.8415,0.00840018)(12.0294,0.00807897)(12.2174,0.00776742)(12.4053,0.00747785)(12.5933,0.00721187)(12.7813,0.00696306)(12.9692,0.00670834)(13.1572,0.00645792)(13.3451,0.00619624)(13.5331,0.00595922)(13.7211,0.00574312)(13.909,0.00552219)(14.097,0.00531841)(14.2849,0.0051318)(14.4729,0.0049527)(14.6609,0.00476984)(14.8488,0.00459717)(15.0368,0.00443791)(15.2247,0.00428401)(15.4127,0.00412689)(15.6007,0.00396924)(15.7886,0.00382231)(15.9766,0.00368503)(16.1645,0.00355633)(16.3525,0.00343783)(16.5405,0.00332253)(16.7284,0.00320563)(16.9164,0.00309088)(17.1043,0.00298524)(17.2923,0.00288282)(17.4803,0.00277932)(17.6682,0.00268602)(17.8562,0.00260719)(18.0441,0.00253641)(18.2321,0.00245865)(18.4201,0.00237071)(18.608,0.0022833)(18.796,0.00220501)(18.9839,0.00213584)(19.1719,0.00207203)(19.3599,0.00201358)(19.5478,0.00196102)(19.7358,0.0019074)(19.9237,0.00185056)};\n\n\\legend{Affine, Curved}\n\\end{axis}\\end{tikzpicture}\n}\n%\\hspace{1em}\n\\subfloat[Entropy conservative convergence of $\\int_{\\Omega}U(\\bm{u})$ ]{\n\\begin{tikzpicture}\n\\begin{loglogaxis}[\n scaled ticks=false, \n tick label style={/pgf/number format/fixed},\n\tlegend cell align=left,\n\tlegend style={font=\\tiny},\n\twidth=.475\\textwidth,\n xlabel={CFL},\n ylabel={Average entropy},\n% xmin=.005, xmax=1,\n%ymin=-.0025, ymax=.017,\n legend pos=south east,\n xmajorgrids=true,\n ymajorgrids=true,\n grid style=dashed,\n% ytick={0, .005, .01, .015},\n% yticklabels={0, .005, .01, .015} \n] \n\\addplot[color=blue,mark=*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.25,6.22584e-05)(0.125,2.42033e-05)(0.0625,2.70793e-06)(0.03125,9.23535e-08)};\n\n\\addplot[color=red,dashed,mark=square*,semithick, mark options={solid,fill=markercolor}]\ncoordinates{(0.25,5.32822e-05)(0.125,1.5792e-05)(0.0625,9.458e-07)(0.03125,3.05056e-08)};\n\n\\logLogSlopeTriangleFlip{0.275}{0.15}{0.3}{5}{black}\n%\\addplot[color=red, dashed,semithick, mark options={fill=markercolor}]\n\n\\legend{Affine, Curved} %Affine\n\\end{loglogaxis}\n\\end{tikzpicture}\n}\n\\caption{Evolution of the kinetic energy dissipation rate over time on affine and curvilinear meshes, as well as dependence of average entropy over the domain $\\int_{\\Omega} U(\\bm{u})$ at time $T = 20$ for an entropy conservative formulation. }\n\\label{fig:tg}\n\\end{figure}\n\n%\\note{Add Taylor--Green vortex}\n\n\\section{Conclusions}\n\nThis work describes how to extend entropy conservative and entropy stable DG ``modal'' discretizations to curvilinear meshes using both weighted and weight-adjusted mass matrices. Assuming that the geometric terms satisfy a discrete geometric conservation law, the presented schemes allow for the use of over-integration while satisfying a semi-discrete entropy equality or inequality on curved meshes. Numerical results show the presented schemes achieve optimal rates of convergence for smooth solutions on both two and three dimensional affine and curvilinear meshes while remaining robust in the presence of under-resolved solutions such as shocks and turbulence. \n\nSeveral outstanding computational questions remain to be answered. First, the use of weight-adjusted mass matrices is motivated by the low storage requirements and their efficient application on GPUs. However, to guarantee discrete entropy stability, the computation of a weight-adjusted projection is necessary, which adds additional computational cost compared to the affine case. Furthermore, numerical results suggest that, while computing the entropy-projected conservative variables $\\tilde{\\bm{u}}$ using the weight-adjusted projection is necessary to ensure a semi-discrete conservation of entropy, the difference between using a weight-adjusted projection and regular projection may be negligible in practice. The necessity of the weight-adjusted projection in computing $\\tilde{\\bm{u}}$ should be examined further. Secondly, on triangular and tetrahedral meshes, the proposed schemes involve more computational work than under-integrated collocation-style SBP schemes \\cite{hicken2016multidimensional, chen2017entropy, crean2018entropy}. A careful computational comparison of the presented schemes with existing methods should be done to weigh the benefits of improved accuracy with additional computational costs. \n\nFinally, we note that while this work has focused on triangular and tetrahedral elements, the approaches outlined here are applicable to more arbitrary pairings of approximations spaces and quadratures. For example, similar techniques can be used to construct entropy stable B-spline or Galerkin difference discretizations on curved meshes \\cite{banks2016galerkin, chan2018multi}. \n\n\\section{Acknowledgements}\n\nThe authors thank Mark H.\\ Carpenter and David C.\\ Del Rey Fernandez for informative discussions. Jesse Chan is supported by NSF DMS-1719818. \n\n\\bibliographystyle{unsrturl}\n\\bibliography{dg}\n\n\n\\end{document}\n\n\n\n\n", "meta": {"hexsha": "97f7f5acf06f50052a7768b87eb9fdc33e3eb90b", "size": 158066, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/main.tex", "max_stars_repo_name": "ntan15/scratch_wadge", "max_stars_repo_head_hexsha": "0657069749b9507062c1f7e875c6545076fb85c8", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/main.tex", "max_issues_repo_name": "ntan15/scratch_wadge", "max_issues_repo_head_hexsha": "0657069749b9507062c1f7e875c6545076fb85c8", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/main.tex", "max_forks_repo_name": "ntan15/scratch_wadge", "max_forks_repo_head_hexsha": "0657069749b9507062c1f7e875c6545076fb85c8", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 80.6459183673, "max_line_length": 2134, "alphanum_fraction": 0.6944820518, "num_tokens": 59412, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5350984286266115, "lm_q2_score": 0.640635847978761, "lm_q1q2_score": 0.3428032355753118}} {"text": "%!TEX root=report.tex\n\\section{Data}\n\nThe GRACE dataset can be downloaded from GRGS \\cite{GRACE-data-source}.\nThis report is based on the Equivalent Water Height (denoted EWH) dataset from release 2 in the GRGS format with a 10-day interval.\n\nThis dataset contains quite a few text files, each containing information in both its filename and its content.\nThe filenames have the format:\n\n\\begin{lstlisting}\ngrid.water.10day_model_minus_RL02MF.19202_19211.txt\ngrid.water.10day_model_minus_RL02MF.19212_19221.txt\n...\n\\end{lstlisting}\n\nIn the filename, the last two numbers (e.g. \\texttt{19202\\_19211}) are important.\nThe first number denotes the start date and the second number the end date for the file content.\nEach number specifies the number of days since ``1950-01-01'' \\cite{GRACE-data-format-dates}.\n\nThe actual content of each text file should be read as a ``Space Separated Values'' format.\nWhen this is done one will have a $6480 \\times 10$ matrix.\nThis matrix can then be \\texttt{reshape}'ed row-wise into a $180 \\times 360$ matrix.\nThe result is a matrix with decreasing latitude on the rows and increasing longitude on the columns \\cite{GRACE-data-format-grids}.\n\n\\subsection{Data example}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures/data-example-world}\n\t\\caption{Plot of the data, 10 day period starting on 3 April 2009}\n\t\\label{fig:data-example-world}\n\\end{figure}\n\n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=\\textwidth]{figures/data-example-scatter}\n\t\\caption{Plot of EWH at 63.5 N 49.5 W, west coast of Greenland.}\n\t\\label{fig:data-example-scatter}\n\\end{figure}\n\nFrom Figure \\ref{fig:data-example-world} local mass losses at Greenland and the South Pole are seen.\nIn Figure \\ref{fig:data-example-scatter}, the mass loss at Greenland is seen with a yearly periodic trend. For both locations, the ice melting may be caused by ice melting.\nHowever it is also possible that these changes are caused by post glacial rebound, in partially because no glacial isostatic adjustment (GIA) have been performed on this version of the GRACE dataset.\nIn fact it is quite complicated to do sufficient adjustments to conclusively say something about ice melting \\cite{NASA-GIA-incomplete}.\nWhen that is said, attempts to analyze the GRACE data with the prospect of saying something about the ice melting patterns, will still be made.\n\nFrom Figure \\ref{fig:data-example-world} a mass increment in South America can also been seen, this is caused by the rain season.\n", "meta": {"hexsha": "a39063ea1bf6af21907c4247b46a4db93f7b2a86", "size": 2506, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Rapport/data.tex", "max_stars_repo_name": "AndreasMadsen/grace", "max_stars_repo_head_hexsha": "bf472d30a2fac76145d3f68e819c92da4a1970ba", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2016-05-17T22:52:19.000Z", "max_stars_repo_stars_event_max_datetime": "2016-05-17T22:52:19.000Z", "max_issues_repo_path": "Rapport/data.tex", "max_issues_repo_name": "AndreasMadsen/grace", "max_issues_repo_head_hexsha": "bf472d30a2fac76145d3f68e819c92da4a1970ba", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Rapport/data.tex", "max_forks_repo_name": "AndreasMadsen/grace", "max_forks_repo_head_hexsha": "bf472d30a2fac76145d3f68e819c92da4a1970ba", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.2083333333, "max_line_length": 199, "alphanum_fraction": 0.7849162011, "num_tokens": 626, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5234203638047913, "lm_q2_score": 0.6548947357776795, "lm_q1q2_score": 0.34278524085459566}} {"text": "%% This is emulateapj reformatting of the AASTEX sample document\n%%\n%\\documentclass[trackchanges]{aastex6}\n%\\documentclass[twocolumn,twocolappendix]{aastex6}\n%\\usepackage{amssymb,amsmath}\t% Advanced maths commands\n%\\usepackage{graphicx}\n%\\usepackage{todonotes}\n%\\usepackage{xcolor}\n%\\usepackage{hyperref}\n%\\newcommand\\earl[1]{\\textcolor{red}{ #1}}\n\n%\\newcommand\\earl[1]{\\textcolor{red}{ [#1]}}\n%usepackage{aas_macros}\n\n%\\newcommand\\abs[1]{\\left|#1\\right|}\n\n%\\newcommand{\\vdag}{(v)^\\dagger}\n%\\newcommand{\\myemail}{angelou@mps.mpg.de}\n\n\n%% You can insert a short comment on the title page using the command below.\n\n%\\slugcomment{Not to appear in Nonlearned J., 45.}\n\n%% If you wish, you may supply running head information, although\n%% this information may be modified by the editorial offices.\n%% The left head contains a list of authors,\n%% usually a maximum of three (otherwise use et al.). The right\n%% head is a modified title of up to roughly 44 characters.\n%% Running heads will not print in the manuscript style.\n\n%\\shorttitle{Statistical Properties of the MS}\n%\\shortauthors{Angelou \\& Bellinger et al.}\n\n%% This is the end of the preamble. Indicate the beginning of the\n%% paper itself with \\begin{document}.\n\n%\\begin{document}\n\n%% LaTeX will automatically break titles if they run longer than\n%% one line. However, you may use \\\\ to force a line break if\n%% you desire.\n\n%\\title{\n\\chapter{On the Statistical Properties of the Lower Main Sequence}\n\\label{chap:statistical}\n\nThe contents of this chapter were authored \n%as ``\\emph{On The Statistical Properties of the Lower Main Sequence}'' \nby G.~C.~Angelou, E.~P.~Bellinger, S.~Hekker, and S.~Basu and published in April of 2017 in \\emph{The Astrophysical Journal}, 839 (2), 116.\\footnote{Contribution statement: The work and writing of this chapter were done in equal parts between G.~C.~Angelou and myself, under the supervision of S.~Hekker and S.~Basu.} \n\\nocite{2017apj...839..116a} \n\n%% Use \\author, \\affil, and the \\and command to format\n%% author and affiliation information.\n%% Note that \\email has replaced the old \\authoremail command\n%% from AASTeX v4.0. You can use \\email to mark an email address\n%% anywhere in the paper, not just in the front matter.\n%% As in the title, use \\\\ to force line breaks.\n\n%\\author{George C. Angelou \\altaffilmark{1,2}, Earl P. Bellinger\\altaffilmark{1,2,3,4}, Saskia Hekker\\altaffilmark{1,2} and Sarbani Basu\\altaffilmark{4}}\n%\\affil{\\altaffilmark{1} Max-Planck-Institut f\\\"{u}r Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077 G\\\"{o}ttingen, Germany\\\\\n%\\altaffilmark{2} Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark \\\\\n%\\altaffilmark{3} Institut f\\\"{u}r Informatik, Georg-August-Universit\\\"{a}t G\\\"{o}ttingen, Goldschmidtstrasse 7, 37077 G\\\"{o}ttingen, Germany \\\\\n%\\altaffilmark{4} Department of Astronomy, Yale University, New Haven, CT 06520, USA}\n\n\n%% Mark off your abstract in the ``abstract'' environment. In the manuscript\n%% style, abstract will output a Received/Accepted line after the\n%% title and affiliation information. No date will appear since the author\n%% does not have this information. The dates will be filled in by the\n%% editorial office after submission.\n\n\\section*{Chapter Summary}\nAstronomy is in an era where all-sky surveys are mapping the Galaxy. \nThe plethora of photometric, spectroscopic, asteroseismic and astrometric data allows us to characterize the comprising stars in detail. \nHere we quantify to what extent precise stellar observations reveal information about the properties of a star, including properties that are unobserved, or even unobservable. \nWe analyze the diagnostic potential of classical and asteroseismic observations for inferring stellar parameters such as age, mass and radius from evolutionary tracks of solar-like oscillators on the lower main sequence. \nWe perform rank correlation tests in order to determine the capacity of each observable quantity to probe structural components of stars and infer their evolutionary histories. We also analyze the principal components of classic and asteroseismic observables to highlight the degree of redundancy present in the measured quantities and demonstrate the extent to which information of the model parameters can be extracted.\nWe perform multiple regression using combinations of observable quantities in a grid of evolutionary simulations and appraise the predictive utility of each combination in determining the properties of stars.\nWe identify the combinations that are useful and provide limits to where each type of observable quantity can reveal information about a star. We investigate the accuracy with which targets in the upcoming TESS and PLATO missions can be characterized. We demonstrate that the combination of observations from GAIA and PLATO will allow us to tightly constrain stellar masses, ages and radii with machine learning for the purposes of galactic and planetary studies.\n%\\end{abstract}\n\n%% Keywords should appear after the \\end{abstract} command. The uncommented\n%% example has been keyed in ApJ style. See the instructions to authors\n%% for the journal to which you are submitting your paper to determine\n%% what keyword punctuation is appropriate.\n\n%% Authors who wish to have the most important objects in their paper\n%% linked in the electronic edition to a data center may do so in the\n%% subject header. Objects should be in the appropriate \"individual\"\n%% headers (e.g. quasars: individual, stars: individual, etc.) with the\n%% additional provision that the total number of headers, including each\n%% individual object, not exceed six. The \\objectname{} macro, and its\n%% alias \\object{}, is used to mark each object. The macro takes the object\n%% name as its primary argument. This name will appear in the paper\n%% and serve as the link's anchor in the electronic edition if the name\n%% is recognized by the data centers. The macro also takes an optional\n%% argument in parentheses in cases where the data center identification\n%% differs from what is to be printed in the paper.\n\n%\\keywords{methods: statistical --- stars: abundances --- stars: fundamental parameters --- stars: low-mass --- stars: oscillations --- stars: solar-type}\n\n%% From the front matter, we move on to the body of the paper.\n%% In the first two sections, notice the use of the natbib \\citep\n%% and \\citet commands to identify citations. The citations are\n%% tied to the reference list via symbolic KEYs. The KEY corresponds\n%% to the KEY in the \\bibitem in the reference list below. We have\n%% chosen the first three characters of the first author's name plus\n%% the last two numeral of the year of publication as our KEY for\n%% each reference.\n\n\\section{Introduction} \n\nThe main sequence is generally considered the most well-understood phase of stellar evolution. \nOur Sun is a main-sequence star, and its proximity provides a wealth of constraints to the physics that may occur in low-mass counterparts during this phase \\citep[e.g.,][]{2015SSRv..196...49B,2016lrsp...13....2b}. \nCore-hydrogen burning stars are long-lived and hence numerous: indeed, the majority of the stars for which we can resolve parallaxes reside on the main sequence \\citep{2016arXiv160904172G}. \nAdditionally, many stars of this type display stochastic or ``solar-like'' oscillations that serve to reveal the stellar interior (see, for example, \\citealt{2013ARA&A..51..353C} for a review on solar-like oscillators). \nMain-sequence stars are important astrophysical laboratories for testing theories of stellar physics, structure, and evolution; and are a testbed for general physical theories such as nuclear fusion, diffusion, and convection \\citep[e.g.,][]{1994MNRAS.269.1137B,1990ARAA..28..263S}. \n\nDespite all of this, however, the ages of main-sequence stars remain uncertain to at least $10\\%$. This uncertainty stems not only from observational imprecision, but also from the inability of observations to fully constrain stellar parameters. \nRecently, \\citetalias{2016apj...830...31b} showed that even for stellar models without observational uncertainties, some model attributes of stars---such as their initial helium abundance or efficiency of convection---could not be fully resolved via global information that can be gleaned from their surfaces. \n\n\nIt is well-known that different observable quantities of stars constrain different model properties. For example, in the now-famous Christensen-Dalsgaard diagram (C--D diagram, the so-called ``asteroseismic HR diagram''), in which the large frequency separation is plotted against the small frequency separation (Appendix \\ref{sec:sdefs}), the large frequency separation covaries with the mass of the star and the small frequency separation covaries with its core-hydrogen abundance. Hence, observing one of these quantities sheds light on its unobservable counterpart.\nHowever, to date, a systematic investigation of the extent to which each observable quantity constrains each model property has not been performed.\n\nThe equations dictating stellar structure and evolution, and the corresponding microphysics that these equations respond to, give rise to emergent behaviors that are difficult to characterize through examination of the constituting ingredients themselves. To elucidate these opaque relationships, we seek to determine the extent to which observable stellar properties are capable of constraining the internal structures, chemical mixtures, and evolutionary histories of stars. Here we employ the methodology of exploratory data science, a statistical philosophy by which underlying structure in data---simulated or otherwise---can be unearthed.\n\n\n\nBA1 used machine learning to build a statistical description of main-sequence stellar evolution. They trained a random forest (RF) of decision trees to learn the relationships that exist between model input parameters and their resultant observable quantities. The technique was developed with particular focus on the determination of stellar ages. Ages are essential for understanding stellar evolution, characterizing extrasolar planetary systems and advancing models of galactic chemical evolution. \nNotably, the RF developed by BA1 was able to accurately predict stellar properties such as radii and luminosities using other information collected from the stars in their sample. This illustrates that there is redundant information in the stellar quantities, and that there exist model covariances between these quantities that can be characterized and exploited. \n\n\n\nThe philosophy employed in BA1 is a departure from the standard practice of stellar model fitting. Ordinarily, stellar parameters of observed stars are sought via $\\chi^2$-minimization.\nThe difference in approaches give rise to two points that motivate this paper:\n\\begin{enumerate}\n\n \\item Methods based on $\\chi^2$-minimization assume that each bit of observed information contributes to the objective of constraining the model properties of a star in an exact proportion to how precisely it has been measured. However, two quantities may be measured independently with no measured covariance, and yet still provide redundant information about the star. The result of such a minimization procedure will therefore be a model that is biased towards that redundant information. The RF developed in BA1, on the other hand, uses the process of statistical bagging to mitigate over-fitting of the data (see also \\citealt{hastie2005elements}). Here we demonstrate the degree to which the observables carry redundant information about the star. \n\n \\item The %blind\n optimization searches of iterative model finding procedures provide solutions but do not indicate the elements that were important in doing so. The use of regression requires that the observables correlate with those model parameters that we wish to infer. We therefore identify to what extent each observable constrains each model property, and how well the observables must be measured to achieve a desired precision from the regression. \n \n\\end{enumerate}\n\n\nThe method developed in BA1 makes use of an artificial intelligence strategy known as supervised learning. The RF that they train seeks relations in evolutionary simulations that enable model properties to be inferred as precisely as possible. Although the RF performs the analysis quickly, precisely, and automatically; supervised machine learning strategies do not provide much insight into how the end result is obtained. The algorithm essentially produces a formula for inferring stellar properties from observations, but one that is too complex for people to use analytically by hand. \n\nHere we incorporate a complementary strategy. We use the counterpart of supervised learning---\\emph{unsupervised learning}---to explicitly uncover the relations between observable properties of stars and their model parameters. Hence, BA1 is of a strictly practical nature: stellar parameters can be inferred rapidly without regard for the how or why; and this paper is aimed to further an understanding of the processes actually involved in such a deduction. \n\nIn this study we draw heavily from the work presented in BA1.\nOur analysis initially focuses on elucidating the inherent statistical properties of the grid of stellar models used to train the BA1 RF. \nWe determine the relationships and covariances between a chosen subset of stellar parameters and asteroseismic quantities (see Table~\\ref{tab:parmdefs}). \nWe carry out simultaneous rank correlation tests on the chosen parameters and identify the necessary, dispensable, and irrelevant information for determining each stellar property.\nThen, using principal component analysis %(PCA)\nwe reduce the dimensionality of the observable quantities and identify to what extent they reveal information of the model parameters.\nWe subsequently shift the focus of our analysis to how the grid properties are used by the RF and how the choices in the parameters impact on the precision of the regression. \nWe train RFs using all combinations of observable quantities in our dataset. The purpose of this is two-fold: first, it is often the case that we wish to quickly characterize a star from a few easily observed quantities---the Hertzsprung-Russell (HR) diagram serves as the classic example. Training and scoring all possible RF combinations provides a means to \\emph{quantify} the utility and predictive power of classical and asteroseismic parameters for inferring stellar properties. Secondly, it provides insight into the relationships determined by machine learning algorithms. \nFinally, we identify the observational accuracy required to satisfactorily constrain key stellar parameters. We investigate the observable quantities independently as well as consider the measurements expected from the upcoming TESS and PLATO missions.\n\n\n\\section{Stellar Models and Parameters}\n\\begin{table}\n\\centering\n{\n\\renewcommand{\\arraystretch}{1.2}\n\\centering\n\\begin{tabular}{lll} \n\\hline \\hline \n\\textbf{Qty} & \\textbf{Definition} & \\textbf{Unit} \\\\ \n\\hline\n\\multicolumn{3}{l}{Model Input Parameters} \\\\ \n$M$ & Initial mass & M$_{\\odot}$ \\\\\n$Y_0$ & Initial helium mass fraction & \\\\\n$Z_0$ & Initial metal mass fraction & \\\\\n$\\alpha_{\\text{MLT}}$ & Mixing length parameter & \\\\\n$\\alpha_{\\text{ov}}$ & Overshoot parameter & \\\\ \n$D$ & Diffusion efficiency factor & \\\\[8pt] \n%\\hline\n\\multicolumn{3}{l}{Stellar Attributes} \\\\\n$\\tau$ & Age & yr \\\\\n$\\tau_{\\text{MS}}$ & Normalized main-sequence lifetime & \\\\\nM$_{\\text{cc}}$ & Convective core mass & M$_{\\odot}$ \\\\\nX$_{\\text{surf}}$ & Surface hydrogen mass fraction & \\\\\n$Y_{\\text{surf}}$ & Surface helium mass fraction & \\\\\nX$_c$ & Central hydrogen mass fraction & \\\\\n$L$ & Luminosity & L$_{\\odot}$ \\\\ \n$R$ & Radius & R$_{\\odot}$ \\\\[8pt]\n%X & Hydrogen mass fraction & \\\\ \n%Y & Surface helium mass fraction & \\\\[8pt] \n%\\hline\n\\multicolumn{3}{l}{Classical Observables} \\\\\n$[\\text{Fe/H}]$ & Surface metallicity & \\\\\n$\\log{} g$ & Logarithmic surface gravity & \\\\ %cm s$^{-2}$ \\\\ \n$T_{\\text{eff}}$ & Effective temperature & K \\\\[8pt]\n\n%\\hline\n\\multicolumn{3}{l}{Asteroseismic Observables} \\\\\n$\\nu_{\\max}$ & Frequency of maximum oscillation power & $\\mu$Hz \\\\\n$\\langle\\Delta\\nu_0\\rangle$ & Large frequency separation ($\\ell=0$) & $\\mu$Hz \\\\\n$\\langle\\delta\\nu_{02}\\rangle$ & Small frequency separation ($\\ell=0,2$) & $\\mu$Hz \\\\\n$\\langle\\delta\\nu_{13}\\rangle$ & Small frequency separation ($\\ell=1,3$) & $\\mu$Hz \\\\\n$\\langle r_{02}\\rangle$ & Frequency separation ratio ($\\ell=0,2$) & \\\\\n$\\langle r_{13}\\rangle$ & Frequency separation ratio ($\\ell=1,3$) & \\\\\n$\\langle r_{01}\\rangle$ & Frequency average ratio ($\\ell=0,1$) & \\\\\n$\\langle r_{10}\\rangle$ & Frequency average ratio ($\\ell=1,0$) & \\\\[8pt]\\hline \n\\end{tabular}\n}\n\\caption{Definitions of the quantities analyzed in this study separated into four parts: model input parameters, stellar attributes, classical observables, and asteroseismic observables. \nAsteroseismic definitions are in Appendix \\ref{sec:sdefs}. \nAngled parenthesis indicate the quantity is a calculated weighted median.} \n\\label{tab:parmdefs} \n\\end{table} \n\n\n\nWe used \\emph{Modules for Experiments in Stellar Astrophysics} \\citep[MESA,][]{2011apjs..192....3p} to generate a grid of stellar evolutionary sequences initially for the purpose of training a random forest. The tracks are varied in initial mass $M$, helium $Y_0$, metallicity $Z_0$, mixing length parameter $\\alpha_{\\text{MLT}}$, overshoot coefficient $\\alpha_{\\text{ov}}$, and atomic diffusion multiplication factor $D$ (see BA1 Section~2.1 for details).\nInitial model parameters were chosen in a quasi-random fashion from the parameter ranges listed in Table~\\ref{tab:prange}.\nIn total $5325$ evolutionary tracks were evolved from ZAMS to either an age of ${\\tau=15}$~Gyr or until terminal-age main sequence (TAMS), which we define as having a fractional core-hydrogen abundance $X_{\\text{c}}$ below $10^{-3}$. We conduct our analysis on a subset of stellar models chosen from each sequence so not to bias our statistics towards longer lived stars or numerically challenging evolutionary tracks. \nDetails of the choice of input physics, grid generation strategy, and model selection procedure are further outlined in BA1. \nIn addition to computing the stellar structure we post process each model with the ADIPLS pulsation package \\citep{2008Ap&SS.316..113C}. P-mode oscillations up to spherical degree ${\\ell=3}$ below the acoustic cut-off frequency are computed, and from these, frequency separations and separation ratios calculated (see Appendix \\ref{sec:sdefs} for mathematical definitions). \n\n\nThere are many quantities that could be included in the current analysis. \nThe $25$ parameters we have selected to investigate are listed in Table~\\ref{tab:parmdefs}. \nThey comprise key asteroseismic and structural quantities and reflect our focus on characterizing the relationships between observable quantities (observables hereinafter) and those variables that allow us to generate detailed stellar models. \n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{llll}\n\\hline \\hline\nParameter\t&\tMin Value\t&\tMax Value\t&\tVariation \t\\\\ \n\\hline \nMass\t&\t0.7\t&\t1.6\t&\tlinear\\\\\n$Y_0$\t&\t0.22\t&\t0.34\t&\tlinear\\\\\n$Z_0$\t&\t$10^{-5}$\t&\t$10^{-1}$\t&\tlogarithmic\\\\\n$\\alpha_{\\text{MLT}}$\t&\t1.5\t&\t2.5\t&\tlinear\\\\\n$\\alpha_{\\text{ov}}$\t&\t$10^{-4}$\t&\t1\t&\tlogarithmic\\\\\nD\t&\t$10^{-6}$\t&\t$10^{2}$\t&\tlogarithmic\t\\\\\n\\hline\n\\end{tabular}\n\\caption{Ranges and sampling strategy for the initial model parameters in the BA1 grid.}\n\\label{tab:prange}\n\\end{table}\n\nWe consider two parameters not included in the RF training data. \nBA1 elected to omit the frequency of maximum oscillation power, $\\nu_{\\max}$ (Equation~\\ref{equ:nmax}), in their regression model\\footnote{$\\nu_{\\max}$ does have some role in the algorithm developed by BA1, as it is responsible for the location of the Gaussian envelope used to weight and derive averaged/median frequency separations.}. This quantity displays a strong correlation with ${\\langle\\Delta\\nu_0\\rangle}$ (see Figure~\\ref{fig:filt_corr} or \\citealt{2009A&A...506..465H,2009MNRAS.400L..80S}) and thus offers very little additional information when frequencies are known. \nWe include it in the current analysis because $\\nu_{\\max}$ is the simplest global asteroseismic parameter to extract from time-series observations, and because recent work by Theme{\\ss}l~et~al.~(private communication) indicates that the $\\nu_{\\max}$ scaling relation more accurately reproduces stellar parameters in well-constrained binary systems than the ${\\langle\\Delta\\nu_0\\rangle}$ relation (Equation~\\ref{equ:dnu}). This is despite the fact that ${\\langle\\Delta\\nu_0\\rangle}$ can be measured more precisely and that the relation can be corrected for temperature and metallicity dependencies (Equation~\\ref{eq:corrfunc2}) to yield greater accuracy \\citep{2016MNRAS.460.4277G, 2016ApJ...822...15S}. \n\nTo complement $\\tau$, we have also added normalized main-sequence age, $\\tau_{\\text{MS}}$, which describes how parameters change as a function of stellar evolution. \nMany low-mass stars in the grid do not reach the terminal-age main sequence (TAMS) before their evolution is stopped. \nTheir main-sequence lifetime is estimated by linearly extrapolating the rate at which the central hydrogen is depleted,\n\\begin{equation}\n\\tau_{\\text{TAMS}} = \\frac{\\tau_{\\text{last}}}{1-(X_{\\text{c, last}}/X_{\\text{c, init}})}\n\\end{equation}\nwhere $\\tau_{\\text{TAMS}}$ is the TAMS age, $\\tau_{\\text{last}}$ is the age of the last model in the track, $X_{\\text{c, last}}$ is the corresponding core-hydrogen abundance for that model and $X_{\\text{c, init}}$ is the core-hydrogen abundance of the initial model in that track. \nFor the longest-lived stars we find such an extrapolation is within about $25\\%$ of the true TAMS age. The uncertainty in the extrapolation for these stars stems from the fact we only capture the hydrogen depletion in the early part of the main sequence i.e., when ${X_{\\text{c, last}} > 0.3}$. Estimating the TAMS age in this manner, however, will not impact our conclusions.\nLarge discrepancies are limited to a small number of tracks (192) and differences between the true and extrapolated ages are reduced as ${X_{\\text{c, last}} \\to 0}$. \nMain sequence lifetime provides insight into the general correlations that develop as a function of main-sequence stellar evolution. Thus it is the monotonicity of $\\tau_{\\text{MS}}$ within a given track that is key. The stellar age parameter, on the other hand, is useful for exploring correlations across the whole parameter space. \n\n\n\\needspace{5\\baselineskip}\n\\section{Rank Correlation Test}\n\\label{sec:RCT}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.6\\linewidth]{HRD6.png}\n \\caption[Hertzsprung-Russell diagram for the grid of models]{Hertzsprung-Russell diagram for those tracks in the truncated grid (see text for details). Here each model is coloured by stellar radius.}\n \\label{fig:HRDRad}\n\\end{figure}\n\n\n\n\\begin{figure*}\n \\centering\n \\includegraphics[trim={1.5cm 0 2cm 1cm},clip, width=\\textwidth]{filter-corr-spearman2.pdf}\n \\caption[Rank correlation diagram]{Spearman rank correlation matrix comprising various stellar and asteroseismic parameters. The quantities are as described in Table~\\ref{tab:parmdefs}\n with model input parameters marked in green.\n The size and the color of each circle both indicate the magnitude of the Spearman coefficient with red and blue denoting negative and positive correlations respectively. The presence of a cross indicates that \nthe two parameters have failed our significance test; i.e., the correlation is indistinguishable from nil. \nThe variables are ordered according to their correlation with the first eigensolution of the correlation matrix$^2$.}\n \\label{fig:filt_corr}\n\\end{figure*}\n\n%No support for footnote in caption. This footnote corresponds to the $^2$ in figure \\ref{fig:filt_corr} above\n\\footnotetext{As principal component analysis is the eigensolution of the correlation (or covariance) matrix, the first eigenvalue indicates the maximum variance in the variables that can be accounted for by a linear model with a single underlying `factor.'\nOrdering the parameters in this way demonstrates the direction of the first principal component (PC$_1$) vector. Figure~\\ref{fig:filt_corr} thus offers a visual representation of principal component analysis which we employ in Section~\\ref{sec:PCA}.}\n\nWe begin our analysis with a rank correlation test, the purpose of which being to understand the \nstatistical properties of the collective lower main sequence.\nThis is distinct from typical analyses that focus on the evolutionary properties within \nindividual stellar tracks or chemically homogeneous isochrones. By identifying correlations present across the entire parameter space we reveal exploitable relationships available to model fitting and regression methods. \n\n\nSince many quantities (see Table~\\ref{tab:parmdefs}) are known to vary in a highly non-linear fashion, we opt to study \\emph{rank} statistics. In particular, we replace each quantity by its rank, i.e., an integer representing how big or small a particular quantity is compared to the other models; and calculate Spearman's correlation coefficient $\\rho$ between all variables. We further calculate the significance of these correlations (p-values) using the Spearman $\\rho$ test. We adopt a conservative significance cut-off of ${\\alpha = 10^{-5}}$ and use the Bonferroni correction to account for the fact that we are making multiple ($625$) comparisons \\citep[e.g.,][]{doi:10.1080/01621459.1955.10501294}. \n\nThis analysis allows us to determine whether quantities vary monotonically in the same direction (${\\rho \\approx 1}$), i.e.~both increasing or both decreasing; monotonically apart (${\\rho \\approx -1}$), i.e.~one increases while the other decreases; or neither (${\\rho \\approx 0}$)\\footnote{Spearman's~$\\rho$ is equivalent to Pearson's~$r$ on ranked quantities. We note also that ${\\rho = 0}$ does not necessarily indicate a relationship does not exist; simply that the relationship is not monotonic. A parabolic function for example would result in ${\\rho = 0}$.}. When $\\abs{\\rho}$ is nearly one, the information from one parameter can be used to determine information about the other. Therefore, this is a valuable tool for probing the relationships that exist in and across evolutionary tracks and determining which model properties can be inferred from which observable quantities. \n\nIn the current analysis, we are strictly interested in the relationships expected from the observational data. We apply cuts to the grid computed by BA1 as it spans a wide parameter range\\footnote{When training a RF for the purposes of characterizing stellar systems, sampling the parameter space well beyond the expected ranges of each quantity is prudent. RFs do not extrapolate---doing so would be undesirable anyway---so characterizing a star requires that all of its observations are firmly within the boundaries of the grid used to train the RF. Doing this furthermore avoids pre-conceived biases in the analysis: it allows the observations to dictate the interesting regions of the parameter space rather than limiting the ranges to the values we \\emph{expect} the parameters to take.}. The full set of tracks in the BA1 grid includes models with temperatures exceeding the limit in which solar-like oscillations are thought to develop (${ T_{\\text{eff}} \\approx 6700}$~K, i.e., the approximate surface temperature beyond which the stellar envelopes are radiative rather than convective). \nEvolutionary tracks in the training grid with more than half of the constituent models having $T{_{\\text{eff}} > 6700}$~K are excluded from the rank correlation analysis.\nNote that the grid will still contain models with $T{_{\\text{eff}} > 6700}$~K if more than half the models in a track display temperatures below this cutoff; there is some chance we may observe such stars. \nLikewise, we omit tracks where high atomic-diffusion rates significantly drain metals from the surface, i.e., tracks where more than half the models display surface-hydrogen mass fractions ${> 0.95}$. The dearth of stars observed at zero metallicity indicates that there are some physical processes not included in our models (e.g., radiative levitation or turbulent diffusion) which inhibit the unabated flow of metals from the stellar surface.\nThis is a common result in models of high-mass stars that include gravitational settling and therefore the process is \\emph{ordinarily} suppressed once ${M \\gtrsim 1.1\\;M_{\\odot}}$. Metal depletion may also arise in cases when settling is made to operate extremely efficiently. \nThe removal of these sequences reduces the BA1 training set from $5325$ to $2010$ evolutionary tracks (truncated grid hereinafter) for the current analysis. \nIn Figure~\\ref{fig:HRDRad} we plot the truncated grid in the HR diagram and color the models according to radius. \n\n\n\nFigure~\\ref{fig:filt_corr} shows the results of the correlation analysis for the truncated grid.\nWe defer correlation analysis on the full grid of models to Appendix \\ref{sec:fullcorr}. \nCare is needed when interpreting Figure~\\ref{fig:filt_corr}. \nFirst, it is important to remember that correlation is not transitive\\footnote{This is irrespective of whether one is using Pearson's~$r$, Spearman's~$\\rho$ or Kendall's~$\\tau$.} \\citep{lang}, i.e.,\n\\begin{equation}\n\\corr(X,Y) \\wedge \\corr(Y,Z) \\not \\Rightarrow \\corr(X,Z)\n\\end{equation}\neven when the correlations are due to causative relationships \\citep{stav}.\nIn fact one can only draw inference on the direction of ${\\corr(X,Z)}$ in cases when \n\\begin{equation}\n\\rho_{X,Y}^2 + \\rho_{Y,Z}^2 > 1\n\\end{equation}\n(transitive criterion hereinafter).\n\nSecond, recall that these correlations hold only for the main sequence. During the main sequence there is generally a positive correlation between, say, $L$ and $T_{\\text{eff}}$. \nThis relationship will change as the stars evolve further beyond the main-sequence turnoff. \n\nThird, save for correlations with $\\tau_{\\text{MS}}$, the relationships presented here do not describe how parameters correlate internally throughout an evolutionary track. Rather, they describe how they correlate across \\emph{all} tracks. For example, as a star ascends the main sequence, luminosity increases and therefore one may expect a strong positive correlation between $\\tau$ and $L$. The fact that we report a negative correlation is because higher-mass stars are shorter lived -- thus high $L$ corresponds to a lower $\\tau$ when the whole parameter space is considered. This correlation is in fact stronger in the analysis of the complete grid used in BA1 which we report in Appendix \\ref{sec:fullcorr}, as our grid truncation preferentially selects against higher-mass stars. \nFurthermore we note that some initial model variables ($M$, $Y_0$, $Z_0$, $\\alpha_{\\text{MLT}}$, $\\alpha_{\\text{ov}}$ and $D$; all indicated in green) correlate with other parameters. This would not be the case if we reported correlations within tracks, as these parameters do not change within a given track. \n\nIt should be noted that there is some bias present in the grid as the low-mass stars are not computed to the end of their main-sequence lifetime. The strengths of some correlations would change had we considered evolution beyond the age of the Universe. \n\\subsection{Interpreting the Correlations}\nHaving set the general context in which to interpret Figure~\\ref{fig:filt_corr}, we highlight some statistical features of the lower main sequence that can be extracted: \n\n\\begin{itemize}\n\\item Most pairs of parameters with ${| \\rho | \\approx 1}$ correspond to well known main-sequence and/or asteroseismic relations. Pairs displaying strong correlations include:\n\n%\\begin{math}\n%\\langle\\Delta\\nu_0\\rangle - \\log{} g; \\qquad \\langle\\Delta\\nu_0\\rangle - \\nu_{\\max}; \\qquad \\log{} g - \\nu_{\\max}; \\\\\n%\\langle\\Delta\\nu_0\\rangle - R; \\qquad \\qquad \\log{} g - R; \\qquad \\qquad M - R; \\\\\n% L - R; \\qquad \\quad \\qquad\\langle\\delta\\nu_{02}\\rangle - X_c.\n%\\end{math}\n\\begin{tabular}{lll}\n $\\langle\\Delta\\nu_0\\rangle - \\log{} g$; \n& $\\langle\\Delta\\nu_0\\rangle - \\nu_{\\max}$; \n& $\\log{} g - \\nu_{\\max}$; \\\\\n $\\langle\\Delta\\nu_0\\rangle - R$; \n& $\\log{} g - R$; \n& $M - R$; \\\\\n $L - R$; \n& $\\langle\\delta\\nu_{02}\\rangle - X_c$. \n&\n\\end{tabular}\n\n\\item Figure~\\ref{fig:HRDRad} illustrates why $T_{\\text{eff}}$ and its correlations with $R$ and $L$ are weaker than those listed above. \nMany of the tracks evolve past the main-sequence turn off before exhausting their core-hydrogen abundance. The change in morphology of the HR diagram and resultant increase in radius impacts on the monotonicity of the respective correlations. \n\n \n\n\\item The mass of the convective core (M$_{\\text{cc}}$) displays a moderate negative correlation with age whereas it barely registers a relationship with $\\tau_{\\text{MS}}$. \nIt is the higher-mass and hence shorter-lived stars that preferentially develop convective cores. A negative correlation with age is therefore according to expectations. \nIn stars that burn hydrogen radiatively no correlation will develop between M$_{\\text{cc}}$ and $\\tau_{\\text{MS}}$. \nIn those stars that burn convectively, the size of the convective core will grow but then recede as the CNO-burning region becomes more centrally condensed. \nThese two factors lead to an (essentially) null result between M$_{\\text{cc}}$ and $\\tau_{\\text{MS}}$. \n\n\n\\item The correlations between $\\tau$ and the ratios ${\\langle r_{02}\\rangle}$ and ${\\langle r_{13}\\rangle}$ are stronger than the correlation between $\\tau$ and X$_c$. \nThe grid comprises large ranges in mass and metallicity and hence stars at different ages can possess the same X$_c$, thereby weakening the strength of that correlation.\nConversely, as one might expect, $\\tau_{\\text{MS}}$ exhibits a stronger relationship with X$_c$ than the ratios. \n\n\\item The small frequency separations and the asteroseismic frequency ratios strongly correlate with both $\\tau$ and X$_c$. The large frequency separation, however, demonstrates a much stronger correlation with X$_c$ than it does with $\\tau$. The rate at which stars burn their central fuel will largely depend on their mass, thus the \nmodels can attain the same density (which is proportional to the large frequency separation) at a range of ages. Both $\\tau_{\\text{MS}}$ and X$_c$ are evolutionary variables and display the expected correlations with ${\\langle\\Delta\\nu_0\\rangle}$. \n\n\\item We lack the necessary information to constrain some of the initial model variables. Indeed ${[\\text{Fe/H}]}$ provides some constraints on the diffusion efficiency factor $D$, but there is much degeneracy: a model can attain the same surface Y starting with a low ${[\\text{Fe/H}]}$ and low diffusion rate as a track with a high ${[\\text{Fe/H}]}$ and high diffusion rate. It is possible that fitting for the base of the convective envelope through seismic analysis of the acoustic glitch signal \\citep{2014ApJ...782...18M, 2014ApJ...794..114V} could help further constrain these parameters. \n\n\n\\end{itemize}\n\nFigure~\\ref{fig:filt_corr} immediately reveals information about the relationships utilized in the machine learning algorithms. \nFor those parameter pairs that failed the significance test, neither is likely to feature in the regression model that predicts the other, except in a circumstance where a subset of models exhibit a trend that is absent from the general case of all the models being considered together.\nConversely, where possible, the regressor will attempt to draw on information from pairs that display the strongest correlations.\nQuantities such as radius illustrate that there is indeed redundant information in independently measured parameters.\nThis is useful as the observables measured, and their corresponding accuracy, will vary from survey to survey. \nIf a key piece of datum is missing or unreliable, a new regression model can be trained using an appropriate substituted quantity in its place. \nThis requires that the redundant information in the observables are treated correctly, if however they are not, then they will lead to biases in model finding procedures. We explore this point further in the next section. \n\n\n\\section{Principal Component Analysis}\n\\label{sec:PCA}\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.8\\linewidth]{PCA_CUMUL3.pdf}\n \\caption[Explained variance of principal components]{{The explained variance (Ve$_{\\,\\text{pca}}$) and cumulative Ve$_{\\,\\text{pca}}$ of the principal components comprising the \\emph{observable} quantities listed in Table~\\ref{tab:parmdefs}. The figure demonstrates that $98\\%$ of the variance in the $11$ observational parameters can be explained by four independent components and $99.2\\%$ of the variance explained when a fifth component is considered. The Ve$_{\\,\\text{pca}}$ of each component is also presented in the second column of Table~\\ref{tab:PCAEV}. }}\n \\label{fig:GCA-pca}\n\\end{figure}\n\n\n\\begin{figure} \\centering\n\\includegraphics[width=0.7\\textwidth]{barchart2a.pdf}%\n\\caption[Correlation between principal components and stellar observables]{ Pearson correlation strength between the first five principal components and the stellar observables. Quantities are ordered according to their correlation strength with the first principal component. Strong correlations indicate that much of the variance of the quantity is captured by the given PC. Note that the ordinate axes in this figure are on different scales. }\n\\label{fig:GCA-pcabar}\n\\end{figure}\n\n\\begin{figure} \\centering\n\\includegraphics[width=0.7\\textwidth]{barchart3a.pdf}%\n\\caption[Correlation between principal components and model parameters]{ Pearson correlation strength between the first five principal components and the model parameters (\\emph{cf.} Figure~\\ref{fig:GCA-pcabar}). }\n\\label{fig:GCA-pcabarb}\n\\end{figure}\n\n\n\n\\iffalse\n\\begin{figure} \\centering\n%\\gridline{\\fig{barchart2a.pdf}{0.48\\textwidth}{\\qquad \\qquad (a)}\n% \\fig{barchart3a.pdf}{0.48\\textwidth}{\\qquad \\qquad (b)}}\n\\includegraphics[width=0.5\\textwidth]{barchart2a.pdf}%\n\\includegraphics[width=0.5\\textwidth]{barchart3a.pdf}%\n\\caption{ Pearson correlation strength between the first five principal components and (left) the stellar observables and (right) the model parameters. Quantities are ordered according to their correlation strength with the first principal component. Strong correlations indicate that much of the variance of the quantity is captured by the given PC. Note that the ordinate axes in this figure are on different scales. }\n\\label{fig:GCA-pcabar}\n\\end{figure}\n\n\n\\afterpage{\n\\clearpage\n\\begin{landscape}\n\\begin{figure} \\centering\n%\\gridline{\\fig{barchart2a.pdf}{0.48\\textwidth}{\\qquad \\qquad (a)}\n% \\fig{barchart3a.pdf}{0.48\\textwidth}{\\qquad \\qquad (b)}}\n\\includegraphics[height=\\textwidth,angle=-90,origin=c]{barchart2a.pdf}\\\\\n\\includegraphics[height=\\textwidth,angle=-90,origin=c]{barchart3a.pdf}%\n\\caption{ Pearson correlation strength between the first five principal components and (left) the stellar observables and (right) the model parameters. Quantities are ordered according to their correlation strength with the first principal component. Strong correlations indicate that much of the variance of the quantity is captured by the given PC. Note that the ordinate axes in this figure are on different scales. }\n\\label{fig:GCA-pcabar}\n\\end{figure}\n\\end{landscape}\n}\n\\fi\n\n\n\\iffalse\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figs/GCA/barchart2a.pdf}\n \\caption{Pearson correlation strength between the stellar observables and first five principal components. Parameters are ordered according to their correlation strength with the first principal component. Strong correlations indicate that much of the variance of the quantity is captured by the given PC. Note that the ordinate axes in this figure are on different scales. }\n \\label{fig:GCA-pcabar2}\n\\end{figure}\n\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\columnwidth]{figs/GCA/barchart3a.pdf}\n \\caption{Pearson correlation strength between the model parameters and first five principal components. Parameters are ordered according to their correlation strength with the first principal component. Strong correlations indicate a significant degree of linearity exists between the quantity and the observables. Note that the ordinate axes in this figure are on different scales.} \n \\label{fig:GCA-pcabar3}\n\\end{figure}\n\n\\fi\n\nPast studies, particularly \\citet{1994ApJ...427.1013B}, have argued that redundancies and covariances in the stellar observables should be taken into account during any model fitting procedure. \nThey demonstrated one particular method (singular value decomposition, SVD hereinafter) of avoiding such biases. In the previous section we identified correlations present in the lower main sequence. Here we demonstrate the degree of redundant information contained in the observables by applying dimensionality reduction. \nWe perform principal component analysis (PCA) in order to discover latent structure in observable stellar quantities such that they may be related more directly---and without redundancy---to parameters of stellar modelling. \nThrough the principal components (PCs) we quantify the extent to which the observables capture information of the model parameters.\n\n\nA natural strategy for dealing with high-dimensional data is to reduce the dimensionality in search of \\emph{latent variables}; i.e., hidden variables that are more useful than the original quantities under consideration. \nPrincipal component analysis (PCA) is a technique to transform data into a sequence of orthogonal, and hence independent, linear combinations of the variables.\nEach successive component is constructed to maximize the residual variance from the original data whilst remaining orthogonal to the previous components.\nIt is a linear transformation in which the change of basis captures the variance contained in original data. \nIf parameters in the data are highly correlated, then PCA can potentially produce a lower-dimensional representation without significant loss of the information. \nThe method can therefore introduce a new set of variables capable of revealing the underlying structure of an originally high-dimensional space.\n\n \n\nPCA belongs to a family of matrix decomposition techniques that also include methods such as non-negative matrix factorization and independent components analysis as well as variations such as sparse PCA and kernel PCA. \nIt has previously been employed in an astrophysical context \\citep{2008ApJ...686.1349B, 1987ASSL..131.....M} along with SVD \\citep{1994ApJ...427.1013B, 2009ApJ...699..373M} to handle correlated errors in observational data. \nThe PCs in this work are calculated from the eigensolution of the correlation matrix, the results of which are not scale invariant. \nWe note that PCA can be interpreted as the singular value decomposition of a data matrix in cases where the columns have first been centered by their means. \nThus SVD analysis\\footnote{This method is in fact more numerically stable but more computationally expensive for extracting PCs.} is an alternative method for extracting the PCs (see also Appendix \\ref{sec:lambdaa}). \nWe indeed compare both methods as a check on our methodology and find that the magnitude of PC scores are identical although the direction (sign) of the vector may differ on occasion.\n\n\n\n\n\n\\subsection{Explained Variance of the Principal Components}\n\\label{sec:ev}\nWe perform PCA on $11$ classical and asteroseismic observables listed in Table~\\ref{tab:parmdefs}.\nThe chosen parameters reflect the quantities typically extracted\\footnote{Radius and luminosity are in some cases observable, but not ubiquitously available in the pre-GAIA era. We concede that the inclusion of ${\\ell=3}$ modes is an optimistic assumption.} from stars in the \\emph{Kepler} \\citep{2004SPIE.5487.1491K,2010Sci...327..977B} field.\nOur analysis focuses on the truncated grid of models\\footnote{To extract a robust interpretation of the PCs we consider different subsets of the BA1 grid (see Appendix \\ref{sec:fullPCA}).} (see Section~\\ref{sec:RCT}).\nThe truncated grid reduces our matrix to size ${128640 \\times 11}$ on which we perform the PCA (there are $340,800$ models in the full BA1 grid). \n\nThe PCs throughout this analysis are calculated from the eigendecomposition of observables in the correlation matrix. Here we wish to explain the variance in the data values rather than their rankings. \n We employ Pearson's~$r$ in the computation of the correlation matrix for the PCA analysis rather than Spearman's~$\\rho$. This allows us to transform freely back and forth between the original data space and the space of Pearson PCs. \n \nA given data matrix $\\mathbf{X}$ (grid) is of size ${n \\times p}$ where $n$ is the number of models and $p$ is the number of observable parameters. \nEach entry $x_{np}$ in $\\mathbf{X}$ is centered and scaled such that \n\\begin{equation}\n\\bar{x}_{np} = (x_{np} -\\hat{x_n})/\\sigma_{x_n}\n\\end{equation}\nwhere $\\bar{x}_{np}$ is the centered and scaled value, $x_{np}$ is the original entry, $\\hat{x}_n$ is the mean of the particular parameter and $\\sigma_{x_n}$ is its standard deviation. \nWith all variables having zero mean and unit variance ($\\mathbf{\\bar{X}}$), our analysis is equivalent to performing eigendecomposition on the covariance matrix\\footnote{We are essentially performing the eigendecompostion of the normalized covariance matrix.}. \nWe compute $\\boldsymbol\\Sigma$, the matrix of Pearson's~$r$ coefficients, between all entries in $\\mathbf{\\bar{X}}$; and compute the eigenvalues and eigenvectors of $\\boldsymbol\\Sigma$ to determine the PCs.\nThe eigenvalues, $\\lambda_i$, of $\\boldsymbol\\Sigma$ indicate the absolute variance explained by the eigenvectors. We use this to compute the fraction of variance explained by the eigenvector in the dataset, Ve$_{\\, \\text{pca}}$, such that:\n\\begin{equation}\n\\text{Ve}_{\\, \\text{pca}} \\ (\\text{PC}_i)=\\frac{\\lambda_i}{\\sum_{i=1}^p \\lambda_i}, \n\\label{eqn:pcaev}\n\\end{equation}\nwhere the number of observables in the data matrix, $p$, is equivalent to the number of principal components we extract.\n\n\n\n\nThe Ve$_{\\,\\text{pca}}$ and the cumulative explained variance of the PCs are reported in Figure~\\ref{fig:GCA-pca} (see also the second column in Table~\\ref{tab:PCAEV}).\nRemarkably, we find that $99.2\\%$ of the variance in our 11-dimensional observable space can be explained by a space of five components.\nHence, observable stellar quantities are clearly highly redundant in what they reveal, as only five dimensions contain original information about the star. \n\n\nFurther insight into the PCs can be gained through correlation analysis between the transformed data (i.e., data matrix projected onto the new PC features) and the original data matrix of observables.\nAny observable that correlates with a PC contributes to the linear combination of parameters that comprise that PC -- the PC is capturing part of the variance in that observable/dimension. \nMultiple parameters that simultaneously have a large fraction of their variance explained by the same PC, must therefore carry redundant information about the star\\footnote{The correlation analysis is in general similar to reporting the PC loadings. \nIn PCA loadings are the elements of the eigenvector scaled by the square roots of the respective eigenvalues.\nThe elements of the eigenvector are coefficients that indicate the weighting of the original data parameters that combine to form that PC. \nAs we have centred and scaled the data before performing the PCA, the correlation coefficients are equivalent to the loadings.}.\nIn Figure~\\ref{fig:GCA-pcabar} we quantify, through Pearson's~$r$ coefficient, the extent to which each observable correlates with the first five PCs.\nThe parameters in the top panel of Figure~\\ref{fig:GCA-pcabar} are ordered by their correlation with the first principal component. \nPC$_1$ accounts for a significant fraction of the variance in the observables (Ve${_{\\, \\text{pca}}=42.36\\%}$). \nThe top panel of Figure~ \\ref{fig:GCA-pcabar} reveals that this component correlates very strongly \n(${r > 0.85}$) with \n$\\nu_{\\max}$, \n${\\langle\\Delta\\nu_0\\rangle}$,\n${\\langle\\delta\\nu_{02}\\rangle}$, \n${\\langle\\delta\\nu_{13}\\rangle}$,\nand ${\\log{} g}$. \nThe strong correlations imply that the basis vector captures most of the variance across the five parameters simultaneously and points to a common latent variable. \n\n\n\\subsection{Interpreting the Principal Components}\n\\label{sec:intPC}\n\n\nIn Figure~ \\ref{fig:GCA-pcabar} and Figure~\\ref{fig:GCA-pcabarb} we plot the results of correlation analysis between all parameters in the grid and the transformed observables (PCs). \nThe figures offer a quantitative overview of the PCs allowing us to identify what interpretable features the PCs have captured.\nWe have seen that Figure~\\ref{fig:GCA-pcabar} demonstrates the extent to which each observable correlates with the first five PCs, similarly Figure~\\ref{fig:GCA-pcabarb} demonstrates how the principal components correlate with the model parameters. The corresponding correlation coefficients between the parameters and \\emph{all} PCs are listed in Tables \\ref{tab:ocoefs} \\& \\ref{tab:mcoefs}.\n\nAny interpretation of the PCs based on Figures \\ref{fig:GCA-pcabar} and \\ref{fig:GCA-pcabarb} are only valid for the truncated grid of models to which this PCA has been applied. For results on other sub grids we refer the reader to Appendices \\ref{sec:fullPCA} and \\ref{sec:PCAg}. \nWe draw upon the figures for generality in the discussion section (Section~\\ref{sec:disc}).\n\n\nInformation about \\emph{direct} correlations between parameters can be extracted from PCA which further helps with interpreting the underlying features.\nAny two parameters that correlate with a given principal component and meet the transitive criterion will be positively correlated if they both have the same sign with respect to the PC, and negatively correlated if their signs differ. \n\nAs is often the case with PCA, the first few principal components can be interpreted as describing the large-scale physical behavior of the system.\nWe interpret that the underlying feature that PC$_1$ captures is straightforwardly the stellar radius. \nThis is the physical property that has the greatest impact on the observables.\nFrom PC$_1$ in Figures \\ref{fig:GCA-pcabar} and \\ref{fig:GCA-pcabarb} we can infer (from the transitive criterion) that as a star evolves along the main sequence, i.e., $\\tau_{\\text{MS}}$ increases or $X_c$ decreases, radius (and for the most part L) will increase. \nThe consequence of increasing radius being $\\nu_{\\max}$, \n${\\langle\\Delta\\nu_0\\rangle}$,\n${\\langle\\delta\\nu_{02}\\rangle}$, \n${\\langle\\delta\\nu_{13}\\rangle}$,\n ${\\log{} g}$ all decrease and thus their variance is explained by PC$_1$. \nWe note that this PC also correlates with $M$ as stars with larger $M$ will have larger radii. \n\n\nPC$_2$ can be interpreted as a `core-surface' feature.\nPC$_2$ correlates strongly with different combinations of seismic ratios and small frequency separations. % across the four grids.\nWith strong weightings from the core it is no surprise that PC$_2$ features a moderate-to-strong correlation with $\\tau$. \nThis direction of maximal variance comprises information from all the observables and correlates with (mostly) all the dependent model variables further suggesting some form of time evolution. \nThe information from the surface is provided by $T_{\\text{eff}}$. \nThere is a degree to which the variance in $T_{\\text{eff}}$ is captured by the time-evolutionary aspect of this component. \nHowever PC$_2$ also displays a moderate correlation with the time-independent $Z_0$ and thus there is a second aspect to PC$_2$.\n$Z_0$ dictates the temperature at the surface through opacities and nuclear burning in the core.\n\nPC$_3$ appears to have the role of capturing the more extreme models in the grid. \nIn the truncated grid the correlations with $[\\text{Fe/H}]$ and $T_{\\text{eff}}$ suggest that the focus of this PC to account for the variance in the observations imparted by low-metallicity models. \n\nPC$_4$ appears to be a secondary `core-surface' feature much like PC$_2$. \nIt uses surface information, in this case ${[\\text{Fe/H}]}$, in conjunction with some information from the core in the form of the ${\\langle r_{02}\\rangle}$ and ${\\langle r_{13}\\rangle}$ ratios. \n\nPC$_5$ encapsulates the mixing processes that impact upon the surface abundances of the star but it is only required to explain a small fraction of the total variance in the data. \n\n\n\n\n\n\n\\subsection{Inferring Stellar Parameters} \n\\label{sec:ISP}\n\n\n\nThe dimensionality reduction achieved by the PCA quantifies the degree of redundancy in the stellar observables alluded to by Figure~\\ref{fig:filt_corr}. \nHowever, we also wish to quantify the extent to which the observed stellar properties constrain the internal structures and chemical mixtures of the star, i.e., the model properties. \n\n\nIn our application of RF regression the machine tries to fit for each model parameter, the success of which we can appraise (see Section~\\ref{sec:qu}). \nHere we conduct a more fundamental evaluation: how well can we capture the variance in the model parameters simply by explaining the variance in the observed data? \nIn other words: having removed the redundancies, to what extent is information of the model parameters encoded in the observables? We hence devise a score, $\\Lambda$, such that:\n\\begin{equation}\n \\Lambda(X) = \\ \\sum^{p}_{i=1} r(X, PC_i)^2\n\\end{equation}\nwhere $X$ is the parameter of interest, $p$ is the number of PCs ($11$ in our case) and ${r(X, PC_i)}$ is the Pearson coefficient between the parameter and the PC.\nAs we centred and scaled our data before computing the correlation matrix and extracting the PCs, the $\\Lambda(X)$ score is equivalent to summing the square of the PC loadings. The square of each loading indicates the variation in an observable that is explained by the component. A useful property of having scaled our data is that ${\\Lambda(X) =1}$ for each of our observables. We demonstrate these properties further in Appendix \\ref{sec:lambdaa}.\n\nIn Figure~\\ref{fig:GCA-pcabarb} we projected the parameter space of our model quantities onto the PC space. Whilst these are not the optimum vectors to explain our model parameters, that is not their purpose; we instead wish to determine what we can learn about the model quantities by understanding the observables. \nAs the square of the correlation coefficients (loadings) will indicate the fraction of explained variance for the parameter by a given PC, determining the \n$\\Lambda(X)$ score for the model parameters gives an indication of the extent the model data are retrievable from the observables. \n\nIn Table~\\ref{tab:corrL} we list the $\\Lambda$ score for each of the model parameters in Table~\\ref{tab:parmdefs}. Parameters with larger $\\Lambda$ scores have much of their variance captured by the linear models used to explain the observables. We expect to be able to infer parameters such as $R$, $L$ and $\\tau_{\\text{MS}}$ with a great deal of confidence through regression. \nParameters with intermediate values of $\\Lambda$ ($\\tau$, M$_{cc}$) we can expect to recover with some success by employing more sophisticated modelling, however, it is not clear that there is enough information contained in the observables to always do so. In cases with the lowest values of $\\Lambda$, such as the initial model parameters $\\alpha_{\\text{MLT}}$, $Y_0$ and $\\alpha_{\\text{ov}}$, explaining the variance in the observables does not explain the variance in the model parameters. New observables that provide independent information about the star are required to recover these parameters with higher confidence. Fitting the acoustic glitch for example may (eventually) provide constraints on the degree of convective envelope overshoot or atomic diffusion \\citep{2017ApJ...837...47V}. \n\n\n\n\\begin{table}\n \\centering\n \\begin{tabular}{ccc}\n \\hline \\hline\n \nParameter & & $\\Lambda_{\\text{param}}$ \\\\ \\hline \nR\t&\t&\t0.97\t\\\\\nL\t&\t&\t0.96\t\\\\\n$X_c$\t&\t&\t0.94\t\\\\\n$\\tau_{\\text{MS}}$\t&\t&\t0.93\t\\\\\nM\t&\t&\t0.91\t\\\\\n$\\tau$\t&\t&\t0.79\t\\\\\n$Z_0$\t&\t&\t0.73\t\\\\\nM$_{cc}$\t&\t&\t0.61\t\\\\\n$Y_{\\text{surf}}$\t&\t&\t0.50\t\\\\\nX$_{\\text{surf}}$\t&\t&\t0.48\t\\\\\n$\\alpha_{\\text{MLT}}$\t&\t&\t0.38\t\\\\\n$Y_0$\t&\t&\t0.31\t\\\\\nD\t&\t&\t0.29\t\\\\\n$\\alpha_{\\text{ov}}$\t&\t&\t0.08\t\\\\\n \\hline\n \\end{tabular}\n \\caption{The $\\Lambda$ score is a sum of the squares of $r_{\\text{PC, param}}$. Any parameter with high $\\Lambda$ is explained well by a linear model and can be confidently inferred. We have insufficient information to constrain those parameters with the lowest $\\Lambda$.}\n \\label{tab:corrL}\n\\end{table}\n\n\n\n\\section{Quantifying the Utility of Stellar Observables} \n\\label{sec:qu}\n\nThere is certainly value and a degree of intuition in dimensionality reduction.\nPCA has demonstrated the significant information redundancy in our data. \nIt has also allowed us to identify information from the model parameters manifested in the observables, and indicated to what extent those parameters can be extracted. \nWe now turn to another strategy of exploratory data science, which is to let machine learning algorithms fit complicated models to the data. As we shift our focus from \\emph{what} information is present to \\emph{how} it can be exploited, we transition from unsupervised to supervised learning methods. \n\n\nIn the PCA we determined orthogonal vectors that are the best fit to the observables. \nHere we utilize a RF to perform non-parametric, multiple regression in order to create the best functions capable of inferring each stellar parameter. With this particular form of supervised learning \nthe relationships between observables and model parameters remain hidden. Though some insight into the regression function can be gained through examination of the feature importances, the tree topology makes further interpretation difficult. \nWe thus seek to elucidate the RF's decision making processes by appraising\nhow well different combinations of parameters can predict the quantities in Table~\\ref{tab:parmdefs}. \n \n This approach not only illustrates the RF's ability to recover missing observational data, say for a rapid stellar evolution calculation, but also systematically \\emph{quantifies} the usefulness of each parameter in predicting all other quantities in the limit of perfect information. It is analogous to a seismic inversion in that it demonstrates the inherent uncertainty with which information can be reconstructed from the available observations. \nWhereas PCA serves to remove the redundant stellar information in the parameters, the analysis here is designed to highlight them. \n\nUsing the full grid of BA1 models, we perform multiple regression on every unique combination of observables in Table~\\ref{tab:parmdefs}. We omit those combinations that contain the quantity we are training for and include models with $R$ and $L$ as observables, resulting in the calculation of $49,153$ RFs. \n\n\nWe divide the full grid into a testing (${\\approx 15,000}$ models) and training set as per the method ascribed in Appendix D of BA1 so not to over-estimate the performance of the regression. \nWe perform two-fold cross-validation on each RF and, as in BA1, measure their success on the test data with an explained variance score, V$_{\\text{e}}$:\n\\begin{equation}\n\\label{eqn:ve}\n \\text{V}_{\\text{e}} = 1 - \\frac{\\text{Var}\\{ y - \\hat y \\}}{\\text{Var}\\{ y \\}}.\n\\end{equation}\nHere $y$ is the \\emph{true} value we want to predict, $\\hat{y}$ is the predicted value from the random forest, and Var is the variance. This score tells us the extent to which the regressor has reduced the variance in the parameter it is predicting with a score of one implying that the model predicts all values with zero error. This is a different but equivalent definition by which to measure the same quantity in Equation~(\\ref{eqn:pcaev}). We have adopted the same notation as BA1 for evaluating the RF which we use to distinguish from the definition used in the PCA (Section~\\ref{sec:PCA}). We also provide a measure of the `typical' error in the predictions, ${\\mu (\\epsilon)}$, which is calculated by averaging the absolute difference ($\\epsilon$) between the predicted and true values for each parameter. More formally: \n\\begin{equation}\n\\label{eqn:mu}\n \\mu (\\epsilon) = \\frac{1}{n} \\sum^n_{i=1} | \\hat y_i - y_i |,\n\\end{equation}\nwhere $n$ is the number of models in the test data. Through ${\\mu (\\epsilon)}$ we provide an indicative error associated with the regression model, over the whole parameter space, and in units of the quantity of interest. \n\nThe best combinations of parameters for inferring each quantity of interest are listed in Table~\\ref{tab:regparms}. \nWe present combinations of up to five parameters after which there is negligible improvement to the predictions. \nWe mark with a dash the occasions where the regressor is unable to produce a positive $V_e$ score. \nIt is important to remember that while a score of one implies a perfect predictor, any ${V_e < 1}$ implies there is still \\emph{some} error in the model. We thus opt for truncation rather than rounding when listing the scores. \nPredictions of the seismic quantities are omitted here. They strongly co-vary and are easily recovered when other seismic parameters are known; they are discussed separately in Section~\\ref{sec:seispr}. \nTheir strong covariances also mean that many of the ratios and separations used in the regression models are interchangeable (e.g., ${\\langle r_{02}\\rangle}$ for \n${\\langle r_{13}\\rangle}$ or ${\\langle r_{01}\\rangle}$ for ${\\langle r_{10}\\rangle}$) resulting in negligible differences to our two scores. \n\n%\\clearpage\n\\afterpage{\n\\clearpage% To flush out all floats, might not be what you want\n\\begin{landscape}\n\\thispagestyle{lscape}\n\\pagestyle{lscape}\n%\\vspace*{5.5cm}\n%\\hspace*{-1.25cm}\n\\begin{table} \\scriptsize\n%\\begin{rotatetable}\n%\\begin{table}\n%\\tablewidth{700pt}\n%\\tabletypesize{\\scriptsize}\n%\\tabletypesize{\\tiny}\n\\caption{The best combinations of observables for constraining the non-seismic parameters in Table~\\ref{tab:parmdefs}. For each combination we provide the ${V_e}$ score (Equation~\\ref{eqn:ve}) and ${\\mu (\\epsilon)}$ score (Equation~\\ref{eqn:mu}, given in the units indicated by the predicted quantity column).\\label{tab:regparms}}\n\\iffalse\n\\tablehead{\n \\colhead{Predicted} &\n \\multicolumn{3}{c}{One Parameter} &\n \\multicolumn{4}{c}{Two Parameters} &\n \\multicolumn{4}{c}{Three Parameters} &\n \\multicolumn{4}{c}{Four Parameters} &\n \\multicolumn{4}{c}{Five Parameters} \\\\ \n \\colhead{Quantity} & \n \\colhead{Observable} & \n \\colhead{$V_e$} & \n \\colhead{$\\mu (\\epsilon)$} & \n \\multicolumn{2}{c}{Observables} &\n \\colhead{$V_e$} & \n \\colhead{$\\mu (\\epsilon)$} & \n \\multicolumn{2}{c}{Observables} &\n \\colhead{$V_e$} & \n \\colhead{$\\mu (\\epsilon)$} & \n \\multicolumn{2}{c}{Observables} &\n \\colhead{$V_e$} & \n \\colhead{$\\mu (\\epsilon)$} & \n \\multicolumn{2}{c}{Observables} &\n \\colhead{$V_e$} & \n \\colhead{$\\mu (\\epsilon)$} \n }\n\\startdata\n\\fi\n\\hspace*{-1.25cm}\n\\begin{tabular}{ l | l l l | l l l l | l l l l | l l l l | l l l l }\nPredicted &\n\\multicolumn{3}{c|}{One Parameter} &\n\\multicolumn{4}{c|}{Two Parameters} &\n\\multicolumn{4}{c|}{Three Parameters} &\n\\multicolumn{4}{c|}{Four Parameters} &\n\\multicolumn{4}{c}{Five Parameters} \\\\ \\cline{2-20}\nQuantity & \nObservable & \n$V_e$ & \n$\\mu (\\epsilon)$ & \n\\multicolumn{2}{c}{Observables} &\n$V_e$ & \n$\\mu (\\epsilon)$ & \n \\multicolumn{2}{c}{Observables} &\n$V_e$ & \n$\\mu (\\epsilon)$ & \n \\multicolumn{2}{c}{Observables} &\n$V_e$ & \n$\\mu (\\epsilon)$ & \n \\multicolumn{2}{c}{Observables} &\n$V_e$ & \n$\\mu (\\epsilon)$ \\\\ \\hline\\hline\n $R/R_\\odot$ & $\\langle\\Delta\\nu_0\\rangle$ & 0.955 & 0.046 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$ && 0.985 & 0.027 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, && 0.999 & 0.009 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, && 0.999 & 0.008 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, $T_{\\text{eff}}$, && 0.999 & 0.008\\\\\n & & & & & & & & $T_{\\text{eff}}$ & & & & $T_{\\text{eff}}$, $\\log{} g$ && & & $\\log{} g$, $\\langle r_{10}\\rangle$ & & &\\\\[3pt]\n $\\log{} g$ & $\\langle\\Delta\\nu_0\\rangle$ & 0.86 & 0.046 & $T_{\\text{eff}}$, $\\nu_{\\max}$ && 0.999 & 0.004 & $T_{\\text{eff}}$, $\\nu_{\\max}$, && 0.999 & 0.003 & $T_{\\text{eff}}$, $\\nu_{\\max}$, && 0.999 & 0.002 & $T_{\\text{eff}}$, $\\nu_{\\max}$, $[\\text{Fe/H}]$, && 0.999 & 0.002\\\\\n & & & & & & & & $[\\text{Fe/H}]$ & & & & $[\\text{Fe/H}]$, $\\langle r_{13}\\rangle$ & & & & $\\langle r_{02}\\rangle$, $\\langle r_{13}\\rangle$ & & &\\\\[3pt]\n $L/L_\\odot$ & $T_{\\text{eff}}$ & 0.739 & 1.583 & $T_{\\text{eff}}$, $\\langle\\Delta\\nu_0\\rangle$ && 0.993 & 0.254 & $T_{\\text{eff}}$, $\\langle\\Delta\\nu_0\\rangle$, &&0.999 & 0.13 & $T_{\\text{eff}}$, $\\langle\\Delta\\nu_0\\rangle$, && 0.999 & 0.136 & $T_{\\text{eff}}$, $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, && 0.999 & 0.135\\\\\n & & & & & & & & $\\nu_{\\max}$ & & & & $\\nu_{\\max}$, $\\langle r_{10}\\rangle$ && & & $\\log{} g$, $\\langle r_{10}\\rangle$ & & &\\\\[3pt] \n $T_{\\text{eff}}$/K & $[\\text{Fe/H}]$ & 0.298 & 1216 & $\\log{} g$, $\\nu_{\\max}$ && 0.989 & 104 & $\\log{} g$, $\\nu_{\\max}$, && 0.991 & 95 & $\\log{} g$, $\\nu_{\\max}$, && 0.992 & 96 & $\\log{} g$, $\\nu_{\\max}$, $\\langle r_{01}\\rangle$, && 0.992 & 96\\\\\n & & & & & & & & $\\langle r_{01}\\rangle$ & & & & $\\langle r_{01}\\rangle$, $\\langle\\delta\\nu_{13}\\rangle$ && & & $\\langle\\Delta\\nu_0\\rangle$, $\\langle\\delta\\nu_{13}\\rangle$ & & &\\\\[3pt]\n $Z_0$ & $[\\text{Fe/H}]$ & 0.927 & 0.003 & $[\\text{Fe/H}]$, $\\langle\\delta\\nu_{02}\\rangle$ && 0.96 & 0.002 & $[\\text{Fe/H}]$, $T_{\\text{eff}}$, && 0.982 & 0.001 & $[\\text{Fe/H}]$, $T_{\\text{eff}}$, && 0.986 & 0.001 & $[\\text{Fe/H}]$, $T_{\\text{eff}}$, $\\langle\\Delta\\nu_0\\rangle$, && 0.987 & 0.001\\\\\n & & & & & & & & $\\langle\\Delta\\nu_0\\rangle$ & & & & $\\langle\\Delta\\nu_0\\rangle$, $\\langle r_{13}\\rangle$ && & & $\\langle r_{01}\\rangle$, $\\langle r_{13}\\rangle$ & & &\\\\[3pt]\n $M/M_\\odot$ & $\\langle\\Delta\\nu_0\\rangle$ & 0.348 & 0.157 & $\\langle\\Delta\\nu_0\\rangle$, $\\log{} g$ && 0.857 & 0.072 & $\\langle\\Delta\\nu_0\\rangle$, $T_{\\text{eff}}$, && 0.982 & 0.022 & $\\langle\\Delta\\nu_0\\rangle$, $\\log{} g$, && 0.986 & 0.02 & $\\langle\\Delta\\nu_0\\rangle$, $\\log{} g$, $\\nu_{\\max}$, && 0.982 & 0.024\\\\\n & & & & & & & & $\\nu_{\\max}$ & & & & $\\nu_{\\max}$, $T_{\\text{eff}}$ && & & $T_{\\text{eff}}$, $\\langle r_{10}\\rangle$ & & &\\\\[3pt] \n $\\tau_{\\text{MS}}$ & $\\langle\\delta\\nu_{02}\\rangle$ & 0.543 & 0.147 & $\\langle r_{02}\\rangle$, $\\langle r_{01}\\rangle$ && 0.846 & 0.077 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, && 0.957 & 0.038 & $\\langle r_{02}\\rangle$, $\\nu_{\\max}$, && 0.977 & 0.025 & $\\langle r_{02}\\rangle$, $\\nu_{\\max}$, $\\langle r_{10}\\rangle$, && 0.981 & 0.021\\\\\n & & & & & & & & $\\langle r_{13}\\rangle$ & & & & $\\langle r_{10}\\rangle$, $T_{\\text{eff}}$ && & & $T_{\\text{eff}}$, $[\\text{Fe/H}]$ & & &\\\\[3pt] \n X$_c$ & $\\langle\\delta\\nu_{02}\\rangle$ & 0.508 & 0.113 & $\\nu_{\\max}$, $\\langle r_{13}\\rangle$ && 0.842 & 0.062 & $\\nu_{\\max}$, $\\langle r_{13}\\rangle$, && 0.958 & 0.031 & $\\nu_{\\max}$, $\\langle r_{13}\\rangle$,& & 0.978 & 0.023 & $\\nu_{\\max}$, $\\langle r_{13}\\rangle$, $\\langle\\Delta\\nu_0\\rangle$, && 0.979 & 0.022\\\\\n & & & & & & & & $\\langle\\Delta\\nu_0\\rangle$ & & & & $\\langle\\Delta\\nu_0\\rangle$, $\\langle r_{10}\\rangle$ && & & $\\log{} g$, $\\langle r_{10}\\rangle$ & & &\\\\[3pt]\n $\\tau$ (Gyr)& $\\langle r_{02}\\rangle$ & 0.645 & 0.995 & $\\langle r_{02}\\rangle$, $\\nu_{\\max}$ && 0.844 & 0.642 & $\\langle r_{13}\\rangle$, $\\nu_{\\max}$, && 0.907 & 0.468 & $\\langle r_{02}\\rangle$, $T_{\\text{eff}}$, && 0.931 & 0.332 & $\\langle r_{02}\\rangle$, $\\nu_{\\max}$, $\\langle r_{01}\\rangle$, && 0.943 & 0.282\\\\\n & & & & & & & & $\\langle r_{10}\\rangle$ & & & & $\\langle r_{01}\\rangle$, $\\langle\\Delta\\nu_0\\rangle$ && & & $T_{\\text{eff}}$, $[\\text{Fe/H}]$ & & &\\\\[3pt] \n X$_{\\text{surf}}$ & $[\\text{Fe/H}]$ & 0.655 & 0.051 & $[\\text{Fe/H}]$, $\\log{} g$ && 0.772 & 0.041 & $[\\text{Fe/H}]$, $\\langle\\Delta\\nu_0\\rangle$, && 0.895 & 0.027 & $[\\text{Fe/H}]$, $\\langle\\Delta\\nu_0\\rangle$, && 0.928 & 0.024 & $[\\text{Fe/H}]$, $\\langle\\Delta\\nu_0\\rangle$, $T_{\\text{eff}}$, && 0.936 & 0.022\\\\\n & & & & & & & & $\\nu_{\\max}$ & & & & $T_{\\text{eff}}$, $\\langle r_{02}\\rangle$ && & & $\\langle r_{02}\\rangle$, $\\nu_{\\max}$ & & &\\\\[3pt] \n $M_{\\text{cc}}/M_\\odot$ & --- & --- & --- & $\\langle r_{13}\\rangle$, $\\langle\\delta\\nu_{02}\\rangle$ && 0.679 & 0.015 & $\\langle r_{13}\\rangle$, $\\nu_{\\max}$, & &0.862 & 0.009 & $\\langle r_{13}\\rangle$, $\\nu_{\\max}$, && 0.908 & 0.007 & $\\langle r_{13}\\rangle$, $\\nu_{\\max}$, $\\langle r_{10}\\rangle$, && 0.928 & 0.006\\\\\n & & & & & & & & $\\langle r_{10}\\rangle$ & & & & $\\langle r_{10}\\rangle$, $T_{\\text{eff}}$ && & & $T_{\\text{eff}}$, $[\\text{Fe/H}]$ & & &\\\\[3pt]\n $Y_{\\text{surf}}$ & $[\\text{Fe/H}]$ & 0.597 & 0.052 & $[\\text{Fe/H}]$, $\\log{} g$ && 0.736 & 0.041 & $[\\text{Fe/H}]$, $\\langle\\Delta\\nu_0\\rangle$, && 0.887 & 0.025 & $[\\text{Fe/H}]$, $\\langle\\Delta\\nu_0\\rangle$, && 0.916 & 0.024 & $[\\text{Fe/H}]$, $\\langle\\Delta\\nu_0\\rangle$, $\\langle r_{02}\\rangle$, && 0.927 & 0.022\\\\\n & & & & & & & & $\\nu_{\\max}$ & & & & $\\langle r_{02}\\rangle$, $T_{\\text{eff}}$ && & & $T_{\\text{eff}}$, $\\nu_{\\max}$ & & &\\\\[3pt] \n $Y_0$ & --- & --- & --- & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$ && 0.077 & 0.027 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, && 0.471 & 0.02 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, && 0.536 & 0.019 & $\\langle\\Delta\\nu_0\\rangle$, $\\nu_{\\max}$, $[\\text{Fe/H}]$, && 0.625 & 0.017\\\\\n & & & & & & & & $[\\text{Fe/H}]$ & & & & $[\\text{Fe/H}]$, $\\log{} g$ && & & $\\log{} g$, $\\langle\\delta\\nu_{13}\\rangle$ & & &\\\\[3pt] \n $\\alpha_{\\text{ov}}$ & --- & --- & --- & $\\langle r_{13}\\rangle$, $\\langle r_{02}\\rangle$ && 0.231 & 0.089 & $\\langle r_{13}\\rangle$, $\\langle r_{10}\\rangle$, && 0.44 & 0.075 & $\\langle r_{13}\\rangle$, $\\langle r_{10}\\rangle$, && 0.524 & 0.068 & $\\langle r_{13}\\rangle$, $\\langle r_{10}\\rangle$, $\\nu_{\\max}$, && 0.55 & 0.067\\\\\n & & & & & & & & $\\nu_{\\max}$ & & & & $\\nu_{\\max}$, $T_{\\text{eff}}$ && & & $T_{\\text{eff}}$, $[\\text{Fe/H}]$ & & &\\\\[3pt] \n D & --- & --- & --- & $[\\text{Fe/H}]$, $\\langle\\delta\\nu_{02}\\rangle$ && 0.022 & 5.393 & $[\\text{Fe/H}]$, $T_{\\text{eff}}$, && 0.295 & 4.483 & $[\\text{Fe/H}]$, $T_{\\text{eff}}$, && 0.446 & 3.706 & $[\\text{Fe/H}]$, $T_{\\text{eff}}$, $\\langle r_{02}\\rangle$, && 0.519 & 3.333\\\\\n & & & & & & & & $\\langle\\Delta\\nu_0\\rangle$ & & & & $\\langle r_{02}\\rangle$, $\\langle\\Delta\\nu_0\\rangle$ && & & $\\log{} g$, $\\langle r_{10}\\rangle$ & & &\\\\[3pt] \n $[\\text{Fe/H}]$ &--- & --- & --- & $\\nu_{\\max}$, $\\log{} g$ && 0.179 & 2.777 & $\\nu_{\\max}$, $\\log{} g$, && 0.273 & 2.439 & $\\nu_{\\max}$, $\\log{} g$, && 0.309 & 2.312 & $\\nu_{\\max}$, $\\log{} g$, $\\langle r_{02}\\rangle$, && 0.312 & 2.277\\\\\n & & & & & & & & $\\langle r_{02}\\rangle$ & & & & $\\langle r_{02}\\rangle$, $\\langle r_{10}\\rangle$ && & & $\\langle r_{01}\\rangle$, $\\langle r_{13}\\rangle$ & & &\\\\[3pt] \n $\\alpha_{\\text{MLT}}$ & --- & --- & --- & --- --- && --- & --- & $T_{\\text{eff}}$, $\\nu_{\\max}$, && 0.069 & 0.234 & $T_{\\text{eff}}$, $\\nu_{\\max}$, && 0.201 & 0.211 & $T_{\\text{eff}}$, $\\nu_{\\max}$, $\\langle r_{01}\\rangle$, && 0.229 & 0.207\\\\\n & & & & & & & & $\\langle\\delta\\nu_{13}\\rangle$ & & & & $[\\text{Fe/H}]$, $\\langle r_{02}\\rangle$ && & & $[\\text{Fe/H}]$, $\\langle r_{13}\\rangle$ & & &\\\\[3pt] \n%\\enddata\n \\end{tabular} \n \\end{table}\n%\\end{rotatetable}\n%\\end{figure}\n\\end{landscape}\n}\n%\\clearpage\n\nMany of the RFs we trained do not provide a satisfactory regression model for the quantity we are training for. Below we provide a deeper analysis for some of the more interesting results, focusing primarily on the predictions of ages and surface abundances. \n\n\\subsection{Ages}\n\\label{sec:sages}\nThe current exercise allows us to evaluate the theoretical limit in which parameter pairs, such as those used in the C--D diagram, can constrain stellar ages. \nRecall that there are six initial model parameters varied simultaneously in the BA1 grid. \nDescribing a six dimensional parameter space with two quantities invariably leads to degenerate solutions for age and necessarily high uncertainties.\nThe parameter pairs that offer similarly the best constraints on $\\tau$ are listed in Table~\\ref{tab:cd}. The combination of ${\\langle r_{02}\\rangle}$ and $\\nu_{\\max}$ marginally provide the best probe, explaining the largest fraction of the variance and inferring ages with uncertainty ${\\mu (\\epsilon) = \\pm 642}$~Myr. \n\n\\begin{table}\n\\centering\n\\caption{The best two-parameter combinations of observables for constraining stellar age. Below the dividing horizontal line we include the best spectroscopic pair for comparison as well as ${\\log{} g}$ -- ${\\langle\\Delta\\nu_0\\rangle}$ to highlight the necessity of the small frequency separation in determining stellar ages. The BA1 grid is varied in six dimensions and with such a high-dimensional parameter space the quantities in the C--D diagram (fifth row) constrain age with `typical' uncertainty of $701$~Myr.}\n \\begin{tabular}{cccc}\n \\hline \\hline\n\\multicolumn{2}{c}{Parameters} & $V_e$ & $\\mu (\\epsilon)$ [Gyr] \\\\ \\hline \n$\\langle r_{02}\\rangle$ & $\\nu_{\\max}$ &0.844 & 0.642\\\\\n$\\langle r_{02}\\rangle$ & $\\log{} g$ &0.833 & 0.683\\\\\n$\\langle r_{13}\\rangle$ &$\\nu_{\\max}$ &0.827 & 0.711\\\\\n$\\langle r_{02}\\rangle$ & $\\langle\\Delta\\nu_0\\rangle$ &0.825 & 0.694\\\\\n$\\langle\\Delta\\nu_0\\rangle$ &$\\langle\\delta\\nu_{02}\\rangle$ &0.824 & 0.701\\\\\n$\\langle r_{02}\\rangle$ & $\\langle\\delta\\nu_{02}\\rangle$ &0.821 & 0.701\\\\\nPC$_2$ & PC$_8$ & 0.788 & 0.767 \\\\\nPC$_2$ & PC$_4$ & 0.776 & 0.762 \\\\\n\\hline\n $\\log{} g$ & $\\langle\\Delta\\nu_0\\rangle$ & 0.481 & 1.29 \\\\\n $\\log{} g$ & $T_{\\text{eff}}$ & 0.321 & 1.53 \\\\\n \\hline\n \\end{tabular}\n \\label{tab:cd}\n\\end{table}\n\nThis is in comparison to ${\\mu (\\epsilon) = \\pm 701}$~Myr for ${\\langle\\Delta\\nu_0\\rangle}$ and ${\\langle\\delta\\nu_{02}\\rangle}$ as per the C--D diagram. \nIn Table~\\ref{tab:cd} we also include results from regression calculated with the PCs and find they perform comparably well. The results here omit any uncertainty stemming from the surface effect suggesting that the ${\\langle r_{02}\\rangle}$ and $\\nu_{\\max}$ pair are indeed the preferable choice. \n\n\n\nIt is clear from Tables~\\ref{tab:regparms} and \\ref{tab:cd} how important the small frequency separation and frequency ratios are for the determination of stellar ages on the MS. \nIf we limit the combinations to the classical observables, we find that ${\\log{} g}$ and $T_{\\text{eff}}$ can explain just $32.1\\%$ of the variance in $\\tau$ with uncertainty ${\\mu (\\epsilon) = \\pm 1.5}$~Gyr across the whole grid. The introduction of the large separation offers little improvement. The parameter pair ${\\log{} g}$ and ${\\langle\\Delta\\nu_0\\rangle}$ explain $48.1\\%$ of the variance with ${\\mu (\\epsilon) = \\pm 1.29}$~Gyr.\nIf we permit the RF to draw upon five observables for its regression model, some of the degeneracy in $\\tau$ is lifted. The last column in Table~\\ref{tab:regparms}\nindicates that the RF can reduce the average uncertainty in predicting $\\tau$ such that ${\\mu (\\epsilon) =\\pm 282}$~Myr.\n\n\\subsection{Abundances}\n\\begin{figure}\n \\centering\n \\includegraphics[width=0.9\\linewidth]{hehist2.pdf}\n \\caption[Distributions of initial and surface helium abundances in the generated stellar models]{Distributions of $Y_{\\text{surf}}$ and $Y_0$ in the BA1 grid.}\n \\label{fig:Hehist}\n\\end{figure}\nThe small frequency separations and separation ratios are integral for the determination of ages. \nHowever, the feature importances in BA1 (their Figure~5) indicate that the RF relies predominately on $T_{\\text{eff}}$ and $[\\text{Fe/H}]$ to infer other model parameters. \nTable~\\ref{tab:regparms} confirms how important measuring $[\\text{Fe/H}]$ is for characterizing stars. This quantity is preferentially selected in the many RFs and their regression models, whilst $[\\text{Fe/H}]$ itself cannot be determined from the other observables with any degree of confidence. \n$[\\text{Fe/H}]$ is an indispensable piece of independent information.\n\nAccurate determination of $[\\text{Fe/H}]$ is paramount for inferring many of the current-age stellar attributes. $[\\text{Fe/H}]$ also features prominently in the retrodiction of the initial model parameters but these quantities are characterized by large uncertainties. Foremost, we have no observable that satisfactorily constrains diffusion; $D$ demonstrates an average uncertainty spanning three orders of magnitude. This in turn introduces uncertainty in retrodicting the initial metal content. \n\nPredictions for $Z_0$ at first glace appear to be robust; we report $V_e$ and ${\\mu (\\epsilon) = \\pm 0.001}$. However we contend that a reported error of ${\\mu (\\epsilon) = \\pm 0.001}$ is not all that insightful given that the grid is sampled down to ${Z_0 = 10^{-5}}$. $Z_0$ is sampled logarithmically and takes a small (linear) range in values. In such cases a relative error is a more useful measure of performance than an absolute difference. \n\n\nIn Table~\\ref{tab:relabunds} we devise a series of measures that better appraise the performance of the RF in predicting abundances. We report the average absolute difference as per Table~\\ref{tab:regparms} [${\\mu (\\epsilon)}$], the maximum absolute difference [${\\max(\\epsilon)}$] and the median absolute difference [${\\tilde{\\epsilon}}$]. We also consider the average relative error [${\\mu (\\eta)}$], the maximum relative error [${\\max(\\eta)}$] and median relative error [$\\tilde{\\eta}$], where the relative error is a percentage defined as\n\\begin{equation}\n \\eta= \\frac{| \\hat y_i - y_i |}{|y_i|} \\cdot 100.\n\\end{equation}\n\n\n\\begin{table}\n\\centering\n\\caption{Different measures of uncertainty in predicting stellar abundances with the RF. See text for definitions and motivations.}\n \\begin{tabular}{lccc}\n \\hline \\hline\n Error Measure& $Y_{\\text{surf}}$ & $Y_0$ & $Z_0$ \\\\ \\hline\n $\\mu (\\epsilon)$ & 0.02 &0.017 & 0.001 \\\\\n Max($\\epsilon$) & 0.25 &0.09 & 0.037\\\\ \n$\\tilde {\\epsilon}$ & 0.016 &0.02 &0.00019\\\\\n$\\mu (\\eta)$ [\\%] & $10^{13}$ &8.92& 124.5 \\\\ \n Max($\\eta$) [\\%] & $10^{14}$ &40.34 & 9052 \\\\\n$\\tilde {\\eta}$ [\\%] & 10.88 &7.68 & 13.5 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:relabunds}\n\\end{table}\n\n\nWe find ${\\mu (\\eta) = 125\\%}$ in the retrodiction of metallicity. We attribute the seemingly large uncertainty to the bias imparted by extreme models that have undergone significant diffusion -- we report a maximum relative error of $9000\\%$. With less sensitivity to the outlying metal-depleted models, the median relative uncertainty, ${\\tilde{\\eta} = 13.5\\%}$, offers the most appropriate measure of error in the regression. \nLikewise, the extreme ${\\mu (\\eta)}$ and $\\max(\\eta)$ scores for $Y_{\\text{surf}}$ also stem from models with high diffusion leading to very small non-zero abundances by which we normalize. \n\nIt is interesting to compare the regressor's ability to infer $Y_{\\text{surf}}$ and $Y_0$ abundances. We find that $Y_{\\text{surf}}$ can be well fit (${V_e = 0.927}$) with ${\\mu (\\epsilon) = \\pm 0.022}$. In contrast, the initial abundance, $Y_0$, cannot be confidently retrodicted (${V_e=0.625}$) yet results in a smaller average error [${\\mu (\\epsilon) = \\pm 0.017}$]. \nThis initially surprising result can be understood through examination of the respective parameter distributions in the BA1 grid (Figure~\\ref{fig:Hehist}).\nThe grid is uniformly sampled in initial helium with ${Y_0 \\in [0.22, 0.34]}$. \nAtomic diffusion acts to drain helium from the surface layers and in fact, in some models, completely depletes this species from the envelope. \nThe surface helium abundance of a stellar model can thus attain values in the larger range\n${Y_{\\text{surf}} \\in [0.0, 0.34]}$. %As discussed in Bellinger et al. (submitted) \nIn a uniform distribution, such as we have for $Y_0$, the largest theoretical uncertainty is \n\\begin{equation}\n \\max \\left( \\frac{\\sigma^2(Y_0)}{Y_0} \\right) = \\frac{|b-a|}{|a|} \\cdot 100 = 54.51\\%,\n\\end{equation}\nwhere $a$ and $b$ are the respective minimum and maximum values in our parameter range. \nThis means that if the regressor was unable to explain \\emph{any} of the variance in this quantity and was randomly choosing $Y_0$ values from the initial distribution, the worst relative uncertainty we would expect is $54.51\\%$. The fact that we do go someway to predicting this quantity results in ${\\mu (\\eta) \\approx 8\\% }$ and more accurate inferences than for $Y_{\\text{surf}}$. \n\n\n\n\n\n\n\n\\subsection{Other Results}\nWe mention briefly other interesting results from the approximately $50,000$ RFs not necessarily reported in Table~\\ref{tab:regparms}.\nStellar masses can be accurately inferred from spectroscopic measurements. The combination of ${\\log{} g}$, $T_{\\text{eff}}$ and $[\\text{Fe/H}]$ constrains mass equally well as the pair ${\\langle\\Delta\\nu_0\\rangle}$ -- ${\\log{} g}$. Both combinations explain \n $86\\%$ of the variance in mass with ${\\mu (\\epsilon) = \\pm 0.07 \\; M_{\\odot}}$. With six degrees of freedom in the BA1 grid, we cannot determine mass to an accuracy better than ${\\mu (\\epsilon) = \\pm 0.02\\;M_{\\odot}}$. \n Whilst all observables correlate with $M$, they do not contain sufficient information to separate out the redundant structures that are possible by tweaking the other initial model parameters. We in fact find no improvement in our regression for $M$ beyond three parameters\\footnote{Numerics accounts for the differences in the third decimal place for scores in Table~\\ref{tab:regparms}.}. \n\nIf required, the RF can determine $T_{\\text{eff}}$ with high accuracy. \nAlthough this is almost certainly always an input for the RF, with two or more observables $T_{\\text{eff}}$ can be determined with ${\\mu (\\epsilon) \\approx 100}$~K -- an uncertainty comparable to typical spectroscopic errors. If one of $L$ or $R$ are provided as an input to the RF, a factor of two reduction in the uncertainty is achieved with ${\\mu (\\epsilon) \\lesssim 50}$~K. Furthermore, our testing of the RF (not included here) indicates that if both $L$ and $R$ are provided as observables the Stefan-Boltzmann law is recovered with ${\\mu (\\epsilon) = 4}$~K. \n \n\n\n\\subsection{Seismic Quantities} \\label{sec:seispr}\nWe did not include the predictions for the seismic parameters in Table~\\ref{tab:regparms} as they often carry redundant information. Indeed we accomplish little \nby reporting how the different combinations of ratios and separations can be used to recover each other. \nWe thus opt to analyze the seismic parameters separately, where we can employ discretion to present useful comparisons and highlight noteworthy results. \n\n\\subsubsection*{The large frequency separation -- $\\langle\\Delta\\nu_0\\rangle$}\n\n\n\nIn lieu of a direct measurement, ${\\langle\\Delta\\nu_0\\rangle}$ can be estimated from stellar models via an asteroseismic scaling relation (Equation~\\ref{equ:dnu}). Alternatively, it may be inferred from the observables through an empirical power law that relates ${\\langle\\Delta\\nu_0\\rangle}$ to $\\nu_{\\max}$ \\citep{2009A&A...506..465H,2009MNRAS.400L..80S}. \nThe power law estimates ${\\langle\\Delta\\nu_0\\rangle}$ within $15\\%$ of its measured value \\citep{2009MNRAS.400L..80S}.\nWe compare the RF's ability to likewise predict ${\\langle\\Delta\\nu_0\\rangle}$ from $\\nu_{\\max}$ in Table~\\ref{tab:dnu}. We also consider two and three parameter combinations for inferring ${\\langle\\Delta\\nu_0\\rangle}$ with the requirement that they do not comprise the remaining seismic observables. \n\n\\begin{table}\n\\centering\n\\caption{Combinations of observables that best constrain $\\langle\\Delta\\nu_0\\rangle$.}\n \\label{tab:dnu}\n \\begin{tabular}{ccccccc}\n \\hline \\hline\n\\multicolumn{3}{c}{Parameters} && $V_e$ & $\\mu (\\epsilon)$ & $\\mu (\\eta)$ \\\\ \n&&&&&[$\\mu$Hz] & [\\%] \\\\ \\hline\n$\\nu_{\\max}$ & & &&0.930 & 7.815 & 6.11 \\\\ \n$T_{\\text{eff}}$ & $\\nu_{\\max}$& &&0.990 & 3.09 & 2.46 \\\\\n$\\log{} g$ &$\\nu_{\\max}$ & &&0.990 & 2.95 & 2.34 \\\\\n$\\log{} g$ & $T_{\\text{eff}}$& & &0.990 & 2.92 & 2.31\\\\\n$[\\text{Fe/H}]$ & $\\nu_{\\max}$ & &&0.991 & 2.81 & 2.24\\\\\n$T_{\\text{eff}}$ & $[\\text{Fe/H}]$ & $\\nu_{\\max}$ && 0.995 & 1.67 & 2.13 \\\\\n$\\log{} g$ & $[\\text{Fe/H}]$ & $\\nu_{\\max}$ && 0.995 & 1.65 & 2.11\\\\\n\\hline\n \\end{tabular}\n\\end{table}\n\nWe find that the RF predicts ${\\langle\\Delta\\nu_0\\rangle}$ from $\\nu_{\\max}$ with ${\\mu (\\eta) \\approx 6\\%}$. These results are based on error free information (cross-validation hence no measurement noise) and the inclusion of $\\nu_{\\max}$ from a scaling law. \nIn order to conduct a more faithful comparison with \\citet{2009MNRAS.400L..80S}, we analyze the same data used in the derivation of their power law. \nTheir Table~1 is a compilation of $\\nu_{\\max}$ and ${\\langle\\Delta\\nu_0\\rangle}$ values from the literature. The data are predominately from radial velocity studies and measured with less precision than we have come to expect from \\emph{Kepler} timeseries; they provide a robust test of the RF. \nWe feed the RF the quoted $\\nu_{\\max}$ measurements and predict associated ${\\langle\\Delta\\nu_0\\rangle}$ values. We compare our predictions to the ${\\langle\\Delta\\nu_0\\rangle}$ values from the literature which are used to calculate corresponding \n$\\epsilon$ and $\\eta$ scores. Our results are presented in Table~\\ref{tab:stello}. \nWe omit entries from the \\citet{2009MNRAS.400L..80S} dataset that are outside the parameter ranges of our training grid.\nFor the remaining $17$ stars we find ${\\mu (\\eta) \\approx 8\\%}$ which is comparable to ${\\mu (\\eta) \\approx 6\\%}$ accuracy achieved from cross-validation test (approximately $15,000$ stars). \n\n\\begin{table}\n\\centering\n\\caption{Predictions of ${\\langle\\Delta\\nu_0\\rangle}$ for stars listed in \\citet{2009MNRAS.400L..80S}. Results pertain to a random forest trained with $\\nu_{\\max}$ as the only input. Predictions are compared to literature values from the sources listed in Table~1 of \\citet{2009MNRAS.400L..80S}. The RF performs as well as the power-law relation ($10$-$15\\%$) even on data measured with less precision than stars observed by \\emph{Kepler}.}\n\\label{tab:stello}\n\\begin{tabular}{llllll}\n\\hline \\hline\nStar & $\\nu_{\\max}$ & $\\langle\\Delta\\nu_0\\rangle_{\\text{lit}}$ & $\\langle\\Delta\\nu_0\\rangle_{\\text{pred}}$ & $\\epsilon$ & $\\eta$ \\\\ \n& ($\\mu$Hz) & ($\\mu$Hz) &($\\mu$Hz) &($\\mu$Hz) & (\\%) \\\\ \\hline\n$\\tau\\;$Cet &4500 & 170 & 171 & 1 & 1 \\\\\n$\\alpha\\;$Cen B &4100 & 161 & 184 & 22& 14 \\\\\nSun &3100 & 135 & 138 & 3 & 2 \\\\\n$\\iota\\;$Hor &2700 & 120 & 136 & 16&14 \\\\\n$\\gamma\\;$Pav &2600 & 120 & 122 & 1 & 1 \\\\\n$\\alpha\\;$Cen A &2400 & 106 & 124 & 18 & 17 \\\\\nHD 175726 &2000 & 97 & 100 & 3 & 3 \\\\\n$\\mu\\;$Ara &2000 & 90 & 100 & 10 & 11 \\\\\nHD 181906 &1900 & 88 & 97 & 10 & 11 \\\\\nHD 49933 &1760 & 86 & 101 & 15 & 18 \\\\\nHD 181420 &1500 & 75 & 76 & 1 & 1 \\\\\n$\\beta\\;$Vir &1400 & 72 & 77 & 5 & 8 \\\\\n$\\mu\\;$Her &1200 & 57 & 63 & 7 & 12 \\\\\n$\\beta\\;$Hyi &1000 & 57 & 57 & 0 & 0 \\\\\nProcyon &1000 & 55 & 57 & 2 & 4 \\\\\n$\\eta\\;$Boo &750 & 40 & 45 & 5 & 13 \\\\\n$\\nu\\;$Ind &320 & 25 & 23 & 3 & 10 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\nThe last column in Table~\\ref{tab:stello} indicates that the accuracy from the RF is similar to that of the power law. In addition, we find that parameterizing the RF regression as a function of two observables reduces the uncertainty by a factor of $2$--$3$ (Table~\\ref{tab:dnu}).\nThis hints that the inclusion of a temperature or metallicity dependence may also improve the fit offered by the power law\\footnote{Symbolic regression will help determine whether, in this case, the fitting by the RF has a sensible functional form that can be straightforwardly expressed by two independent variables. This result seems reasonable as the additional information is likely providing a better handle on the stellar mass.}. \n \nAnalysis of recent \\emph{Kepler} data yields a similar result. In Figure~\\ref{fig:chap} we present the percentage error in our predictions of $467$ stars measured by \\emph{Kepler} as reported in Table~1 of \\citet{2014ApJS..210....1C}. We analyze stars for which $\\nu_{\\max}$, ${\\langle\\Delta\\nu_0\\rangle}$ have been measured from the oscillation spectra along with $T_{\\text{eff}}$ as determined by \\citet{2012ApJS..199...30P} based on Sloan Digital Sky Survey (SDSS) photometry. Results from the \\emph{Kepler} sample confirm that predictions for ${\\langle\\Delta\\nu_0\\rangle}$ are improved with the inclusion of $T_{\\text{eff}}$ (lavender distribution). The blue distribution indicates that ${\\langle\\Delta\\nu_0\\rangle}$ is systematically overestimated when the RF only has access to information from $\\nu_{\\max}$ -- a bias that may very well be present in the power-law fit. With the inclusion of $T_{\\text{eff}}$ our predictions become more accurate and precise with the bias from the single parameter function mitigated. We do not quite reproduce the accuracy achieved in the cross validation (Table~\\ref{tab:dnu}) using error free information. Unsurprisingly, measurement uncertainty, which we do not consider here, does not permit the accuracy attained in the ideal case.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{chaplin.pdf}\n\\caption[Relative error in predictions for $\\Delta\\nu$]{Relative error (\\%) in the predictions for ${\\langle\\Delta\\nu_0\\rangle}$ for $467$ stars reported in \\citet{2014ApJS..210....1C}. The blue colored distribution indicates the error in the predictions from the random forest using $\\nu_{\\max}$ as the only input observation whilst the distribution marked in lavender are the results from providing $\\nu_{\\max}$ and $T_{\\text{eff}}$. In the calculations we employ the effective temperatures determined from \\citet{2012ApJS..199...30P} based on SDSS photometry.} \n\\label{fig:chap}\n\\end{figure}\n\n\n\n\\subsubsection*{The frequency of maximum oscillation power -- $\\nu_{\\max}$} \n\n\nCurrently we are unable to predict the frequency of maximum oscillation power from first principles. \\citet{1991ApJ...368..599B} and \\citet{1995A&A...293...87K} showed that this quantity does scale with the acoustic cut-off frequency and can thus be estimated via the Equation~(\\ref{equ:nmax}) scaling relation. It is therefore expected that Table~\\ref{tab:nmx} indicates that $\\nu_{\\max}$ is best inferred from ${\\log{} g}$ and $T_{\\text{eff}}$. These are the two observables that correlate strongest those parameters used to calculate $\\nu_{\\max}$ in the training grid. \n\n\\begin{table}\n\\centering\n\\caption{Combinations of observables that best constrain $\\nu_{\\max}$.}\n \\begin{tabular}{ccccc}\n\\hline \\hline\n\\multicolumn{2}{c}{Parameters} && $V_e$ &$\\mu (\\epsilon)$ [$\\mu$Hz] \\\\ \\hline \n$\\langle\\Delta\\nu_0\\rangle$ &&&0.923 & 7.88 \\\\\n$\\log{} g$ & $[\\text{Fe/H}]$ && 0.888 & 9.99 \\\\\n$\\log{} g$ & $\\langle\\Delta\\nu_0\\rangle$ &&0.954 & 5.38\\\\\n$T_{\\text{eff}}$ & $\\langle r_{10}\\rangle$ & &0.960 & 5.11\\\\\n$[\\text{Fe/H}]$ &$\\langle\\Delta\\nu_0\\rangle$ & &0.992 & 2.90\\\\\n$T_{\\text{eff}}$ & $\\langle\\Delta\\nu_0\\rangle$ & &0.992 & 2.84\\\\\n$\\log{} g$ & $T_{\\text{eff}}$ &&0.999 & 0.83\\\\\n\\hline\n \\end{tabular}\n \\label{tab:nmx}\n\\end{table}\n\n\n\\subsubsection*{The small frequency separation -- $\\langle\\delta\\nu_{02}\\rangle$}\nThe small frequency separation is an indispensable piece of independent information for determining stellar age. In the asymptotic limit \\citep{1980ApJS...43..469T}\n\\begin{equation}\n\\langle\\delta\\nu_{13}\\rangle = \\frac{5}{3} \\langle\\delta\\nu_{02}\\rangle \n\\end{equation}\nand as Table~\\ref{tab:d02} demonstrates, the RF recovers ${\\langle\\delta\\nu_{02}\\rangle}$ in the unlikely case that it is not extracted but ${\\langle\\delta\\nu_{13}\\rangle}$ is.\nIf we disregard combinations that include the seismic ratios, which also contain information of the local small frequency separation, we lack sufficient information to satisfactorily constrain ${\\langle\\delta\\nu_{02}\\rangle}$. \nClearly much of the evolutionary aspect of this quantity can be explained though parameters \nthat correlate with main-sequence lifetime e.g., ${\\log{} g}$, ${\\langle\\Delta\\nu_0\\rangle}$, $\\nu_{\\max}$ and $T_{\\text{eff}}$. However the associated errors of ${\\mu (\\epsilon) > 1.0 \\; \\mu}$Hz can correspond to large age uncertainties for main sequence stars (${\\eta > 10\\%}$). \n\n\n\n\n\\begin{table}\n\\centering\n\\caption{Combinations of observables, without the asteroseismic ratios, that best constrain $\\langle\\delta\\nu_{02}\\rangle$.}\n \\begin{tabular}{cccccc}\n \\hline\n\\multicolumn{3}{c}{Parameters} && $V_e$ & $\\mu (\\epsilon)$ [$\\mu$Hz] \\\\ \\hline \\hline\n$\\langle\\delta\\nu_{13}\\rangle$ & & &&0.944 & 0.66 \\\\ \n$\\log{} g$ & & &&0.542 & 2.08 \\\\ \n$\\langle\\delta\\nu_{13}\\rangle$ & $\\langle r_{10}\\rangle$ & &&0.987 & 0.320 \\\\ \n$T_{\\text{eff}}$ & $\\nu_{\\max}$& &&0.776 & 1.40\\\\\n$\\log{} g$ & $T_{\\text{eff}}$& & &0.775 & 1.40\\\\\n$\\log{} g$ &$\\nu_{\\max}$ & &&0.772 & 1.41\\\\\n$\\log{} g$ &$\\langle\\Delta\\nu_0\\rangle$ & &&0.723 & 1.54\\\\\n$T_{\\text{eff}}$ & $\\langle\\Delta\\nu_0\\rangle$& &&0.720 & 1.58\\\\\n$\\log{} g$ & $[\\text{Fe/H}]$ & && 0.720 & 1.59 \\\\\n$\\log{} g$ & $[\\text{Fe/H}]$ & $\\langle\\Delta\\nu_0\\rangle$ && 0.861 & 1.06 \\\\\n$\\log{} g$ & $\\nu_{\\max}$ & $\\langle\\Delta\\nu_0\\rangle$ && 0.860 & 1.09 \\\\\n\\hline\n \\end{tabular}\n \\label{tab:d02}\n\\end{table}\n\n\n\n\n%\\section{Quantifying the Required Measurement Accuracy of Stellar Observables}\n\\section[Quantifying the Required Measurement Accuracy]{Quantifying the Required Measurement Accuracy of Stellar Observables}\n\\label{sec:accu}\nIn the previous section we used RF regression to appraise how well combinations of observables constrain other stellar parameters. The ${\\approx 50,000}$ RFs were \nevaluated using cross-validation. The tests are a pure measure of the regressor's performance as we have error-free information that we attempt to reproduce (withheld models).\n As we have already alluded to, like all procedures that seek to infer stellar parameters, we must also consider the consequences of measurement uncertainty in our method. \n\n \nMeasurement uncertainty will impact the RF results in a manner that is different to model finding algorithms. Consider an iterative model finding procedure in which we seek an optimum model for a set of observations. We can typically expect $T_{\\text{eff}}$ as a constraint with an associated uncertainty of ${\\sigma = 100}$~K. The minimization algorithm will identify a set of candidate models, many with quite different structures. Hence the uncertainty in $T_{\\text{eff}}$ will impact %on the outcome of\nall stellar quantities simultaneously. The RF, on the other hand,\n builds a statistical description of stellar evolution by calculating a regression model for each individual parameter from the training data. \n The BA1 method requires that each input observable is perturbed with random Gaussian noise according to its measurement uncertainty. Monte Carlo perturbations are performed $10,000$ times and each instantiation evaluated by the RF to yield individual density distributions for each stellar parameter. Thus the uncertainty in $T_{\\text{eff}}$, or any observable for that matter, will only impact on the predictions \n of each parameter in proportion to the degree to which it features in that parameter's regression model. \n\n \n The methodology, combined with the speed of the RF, provides a tractable means to \n asses how the individual measurement uncertainty of an observable will impact upon each predicted stellar quantity. We hence determine how accurately the observables must be measured in order to achieve a desired precision from the RF.\n \n \n \nWe train a RF on the observables listed in Table~\\ref{tab:sun}.\nWe take the (approximate) solar value of each observable as our measurement and consider\n`observational uncertainties' ($\\sigma$) within the ranges specified in Table~\\ref{tab:sun}. \nWe first perturb the measurement values with Gaussian noise assuming the minimum $\\sigma$ values listed. We produce $10,000$ instantiations for that set of $\\sigma$ values, ensuring each perturbed observable remains within the limits of our training grid.\nWe evaluate stellar parameters and determine detailed distributions for that set of uncertainties. We repeat the process increasing the $\\sigma$ for a single observable always keeping the $\\sigma$ values of the other observables at their minimum. \nWe draw $50$ $\\sigma$ values for each observable sampling their specified ranges evenly. We produce probability density distributions for $250$ sets of $\\sigma$ values, the results of which are summarized in Figure~\\ref{fig:uncert1}. \n\n\n \n \\begin{table}\n \\centering\n \\caption{Central values and uncertainty ranges used for predicting the Sun in Figure~\\ref{fig:uncert1}.}\n \\begin{tabular}{lccc}\n \\hline \\hline\nQuantity & Value & Min($\\sigma$) & Max($\\sigma$) \\\\ \\hline \n$T_{\\text{eff}}$ (K) & 5777 & 10 & 500\\\\\n$\\log{} g$ & 4.43812 & 0.00013 & 1.0\\\\\n$[\\text{Fe/H}]$ & 0.0 & 0.05 & 0.2 \\\\\n%$\\nu_{\\max}$ & 3050 & 30 \\\\\n$\\langle\\Delta\\nu_0\\rangle$ ($\\mu$Hz) & 136.0 & 0.5 & 10\\\\\n$\\langle\\delta\\nu_{02}\\rangle$ ($\\mu$Hz)& 9.0 & 0.5 & 5 \\\\\n\\hline\n \\end{tabular}\n \\label{tab:sun}\n\\end{table}\n\n \nIn Figure~\\ref{fig:uncert1} we plot the median value (solid line) and the $68\\%$ confidence interval (shaded region)\n for $M$, $\\tau$, $L$ and $R$ as a function of the uncertainty applied to each observable. The figure is organised such that each row (and color) corresponds to the observable that has had its uncertainty increased and each column corresponds to the model parameter of interest. In this Figure, the left axis indicates the predicted value from the RF and the right axis indicates the relative error with reference to the true values of the Sun. The horizontal dotted grey lines mark the reference value in each case whilst the dotted vertical lines indicate a typical uncertainty for the perturbed observable. \n\nThe particular RF we have trained does not significantly rely on $T_{\\text{eff}}$ in its regression model for $M$, $\\tau$ or $R$. As the radius is supplemented by the seismic quantities, any uncertainty in $T_{\\text{eff}}$ is propagated as uncertainty in the luminosity. We find a typical uncertainty of $100$~K corresponds to an error of ${\\pm 0.2}$ $L$/L$_{\\odot}$ at the $68\\%$ confidence level.\n\nThe inference on solar mass is affected once ${\\delta \\, \\log{} g > 0.03}$. \nHowever, even at unreasonably large values of ${\\delta \\, \\log{} g = 1}$, the uncertainties for mass and age remained relatively constrained by additional seismic information. We find \nthat $L$ and $R$ are far more reliant on ${\\log{} g}$ in their regression function with uncertainties in these quantities growing significantly once ${\\delta \\, \\log{} g > 0.1}$.\n\nThe feature importances in BA1 indicate that ${[\\text{Fe/H}]}$ is used most often by the RF in crafting its decision rules. The four stellar parameters we investigate here indeed all rely on information from ${[\\text{Fe/H}]}$, however, they are supplemented by seismic information which helps to constrain the uncertainty in their predictions. It is the model parameters such as the mixing length, degree of overshoot and initial metallicty that become much less certain as we increase ${\\sigma([\\text{Fe/H}])}$ (not shown here). \n\n \n\nThe seismic diagnostics are very sensitive to the stellar structure, and hence also those parameters we use to characterize a star ($M$, $\\tau$, $L$ and $R$). We have seen how reliant the RF is on the seismic diagnostics in the regression models, allowing us to still predict the structural properties with relatively good precision in the face of large spectroscopic uncertainties. Without accurate measurement of ${\\langle\\Delta\\nu_0\\rangle}$ the uncertainty in structure parameters increase significantly. Whilst the uncertainty in ${\\langle\\delta\\nu_{02}\\rangle}$ does introduce some small uncertainty in $M$, $L$ and $R$, as expected, its accuracy significantly impacts upon our ability constrain stellar age. \n\n\n\n\n\n%\\afterpage{\n%\\clearpage\n%\\cleartoleftpage%\\cleardoublepage\n %\\clearpage% flush all other floats\n %\\ifodd\\value{page}\n %\\else% uncomment this else to get odd/even instead of even/odd\n % \\expandafter\\afterpage% put it on the next page if this one is odd\n %\\fi\n\\begin{landscape}\n%\\pagestyle{lscape}\n\\begin{figure}\n \\centering\\vspace*{-0.75cm}\n \\includegraphics[width=0.9\\linewidth,keepaspectratio]{sununcert.pdf}\n \\caption[Impact of uncertainties on predictions of mass, age, luminosity and radius]{(Caption on other page.) \\label{fig:uncert1}}\n %\\includegraphics[width=0.95\\textheight,height=0.95\\textwidth,keepaspectratio, angle=90]{stest.pdf}\n %\\caption{(Continued on the following page.) \\label{fig:uncert1}}\n %\\caption{Predictions for the solar mass, age, luminosity and radius as a function of the uncertainties applied to key observables. In each panel we have perturbed the quantity on the abscissa in isolation, centred around the measured value listed in Table~\\ref{tab:sun} and with the uncertainties in the ranges specified therein. We indicate the median predicted value (solid line) and the 68\\% confidence interval (shaded region). The dotted horizontal lines mark the zero point or true value in each panel and the vertical line indicates a typical observational uncertainty for the perturbed quantity.}\n %Predictions for the mass, age, luminosity and radius of the Sun each as a function of the uncertainty applied to a key observable.} \n\\end{figure}\n\\end{landscape}\n%}\n\\begin{figure}\n \\contcaption{Predictions for the solar mass, age, luminosity and radius as a function of the uncertainties applied to key observables. In each panel we have perturbed the quantity on the abscissa in isolation, centred around the measured value listed in Table~\\ref{tab:sun} and with the uncertainties in the ranges specified therein. We indicate the median predicted value (solid line) and the $68\\%$ confidence interval (shaded region). The dotted horizontal lines mark the zero point or true value in each panel and the vertical line indicates a typical observational uncertainty for the perturbed quantity.}% Continued caption\n\\end{figure}\n%\\end{landscape}\n%}\n\n\n\n\\section{Discussion}\n\\label{sec:disc}\n\n\nAdvances in stellar evolution theory are usually sought through refinement of the standard canonical model. In this classical approach, observations reveal behaviour that cannot be explained by the current stellar theory, a model is constructed, analysis of the resultant predictions are carried out and conclusions on the efficacy of that model drawn. In this study we adopted a complementary approach: an exploratory based method whereby we performed statistical analysis of models covering a large range of known physics. Rather than first develop a new model to evaluate, we explored the current paradigm to quantify existing relationships and draw new conclusions. \n\nSome of the techniques employed in this analysis are over $100$ years old and in many areas of research are powerful standalone tools. They have rarely featured in the field of stellar modelling. Here we comment briefly on the timing of our manuscript which we attribute to two main factors: the advent of supervised machine learning techniques and modern computing resources. \n\nRandom forests are an integral part of the present analysis and are a modern technology.\nThey help place the use of statistical methods in stellar evolution in a wider practical context. Elucidating both the relationships found by RF and the exploitable information inherent in the model data provided motivation for the use of techniques such as PCA and correlation analysis. The RF further facilitated the application of these methods due to the requirement that the models be cast into a comprehensive evolutionary matrix; something that is not strictly necessary for grid based searches. \n\nOur approach shares similarities to that taken by \\citet{1994ApJ...427.1013B} although we differ in methodology. Since their work, we have seen the necessary increase in computing power and the success of the \\emph{Kepler} and CoRoT space missions. The statistical analysis here requires a well sampled grid of stellar models both with structure and oscillations computed. It cost a week of modern supercomputing time to generate the matrix upon which these operations are performed. Evaluating and training approximately $50,000$ RFs itself is also a computationally expensive endeavour.\n\n\n\\subsection{Features of the Dataset}\nIt is not clear \\emph{a priori} through inspection of the equations of stellar structure, if and how any two emergent quantities of the models co-vary. \nThere are, of course, combinations of parameters whose covariances are well-founded in stellar theory, but there exist quantities whose diagnostic power remain underutilized and could in fact offer additional insight into the underlying models.\nBringing such relationships to light over the collective lower main sequence is a key aim of our statistical investigation. \nThe correlations in the truncated grid (Figure~\\ref{fig:filt_corr}) and full BA1 grid (\\ref{fig:corr}) reveal the relationships that can be utilized to constrain each of the quantities listed in Table~\\ref{tab:parmdefs}.\nMany of the model properties that we wish to infer correlate with several observables simultaneously. This indicates that the observables carry redundant information about the star. \nIn addition, observables co-vary amongst themselves. During iterative model searches some of the covariances, such as between the seismic ratios, are taken into account. However, for example, it is possible to obtain independent measurements of $\\nu_{\\max}$, ${\\langle\\Delta\\nu_0\\rangle}$, and ${\\log{} g}$. Treating these as independent degrees of freedom without considering model covariances then biases the fit towards the parameters to which these quantities pertain and can result in a solution that is overfit. \n\nWe determined the degree of degeneracy in the observables through PCA dimensionality reduction. \nAs mentioned previously, RF regression falls under the umbrella of \\emph{supervised} learning, whereas PCA is a form of \\emph{unsupervised} learning. \nThe difference is that in supervised learning, there is a correct answer that the algorithm is trying to understand how to reproduce.\nIn the case of unsupervised learning, the machine attempts to directly infer properties of data without any help from the supervisor. Hence, regression and classification analyses are forms of supervised learning, whereas cluster and factor analyses are examples of unsupervised learning.\nIn the case of supervised learning there is a clear measure of success in the resultant model. \nThere is a desired output that the inputs try to match. \nThe efficacy can be quantified and evaluated via, say, cross-validation or information-theoretic metrics. \nUnsupervised learning methods simply try to identify features and in the case of PCA these features are not necessarily interpretable. \n\nThe PCA in Section~\\ref{sec:PCA} focused on the truncated grid. It comprises $11$ stellar observables of all which carry information on the model properties to varying degrees. \n We found that $99.2\\%$ of the variance in the observables could be explained by five components with\nnearly $98\\%$ of the data are explained by four components. It could be argued that PC$_5$ explains noise rather than features, however, we found that PC$_5$ displays distinct enough correlations (i.e., with near surface physics) that it warrants inclusion in our analysis. \nThe clear dimensionality reduction, from $11$ observables to five PCs, highlights the value in performing PCA: had we found comparable contributions from each component, we would have instead confirmed a clear dominance from higher order relations and an inadequacy of an approach based on linear analysis. \n\n\nOur primary goal in Section~\\ref{sec:PCA} was to reduce the dimensionality of the observables.\nWe initially considered regions of the parameter space where observations have shown stars to occupy. \nFollowing on from the rank correlation tests in Section~\\ref{sec:RCT} we applied PCA to a truncated version of the BA1 grid. \nHowever, the results of the PCA depend on the properties of the data and will change depending on features such as the parameter ranges and number of models in the grid. \nFor example performing PCA on the full set of evolutionary tracks ($340,800$ models) demands that components are dedicated to explaining variance in (wider) unobserved regions of the parameter space. \nIn order to demonstrate that our interpretations of the PCs are robust, we repeated the PCA on four different subsets of the BA1 grid. \n We made cuts to the mass and metallicity ranges on the training data the results of which are included in Appendix \\ref{sec:PCAg} by means of qualitative correlation plots. \n \nThe PCs of the respective grids explain a similar percentage of the variance in each grid: PC$_1$ accounts for approximately $40\\%$ of the variance, PC$_2$ approximately $35\\%$ etc., with more than $75\\%$ of the variance in the observables explained by the first two PCs. \nWe interpret this result as the PCA capturing essentially the same five inherent `features' in the observables. \nIt follows that the choices in grid size and parameter range have only a small effect on the explained variances.\nAnalysis of all four grids helps further illustrate that there is redundant information carried in some observables, particularly the seismic separations and ratios. \nVarying the parameter ranges changes the correlations between the PCs and observables (loadings) yet the PCs still explain a similar percentage of the variance in each case. \nDue to the information redundancies the PCs can be constructed such that same features are captured with different linear combinations of the observables.\nHow exactly a PC is constructed in a particular grid will depend on the amount of variance in the observables imparted by the chosen parameter ranges. \n\n\nWith respect to the independent model parameters, it is no surprise that in general PC$_1$ is strongly correlated with the stellar mass ($M$) and and PC$_2$ with initial metallicity ($Z_0$). \nThese are the principal determinants of stellar evolution in that order and both impact upon the stellar structure independently.\nIn the two grids where we have cut the mass and metallicity ranges we find that the loading of $T_{\\text{eff}}$ is larger in PC$_1$.\nThis is because in the more solar-like tracks $T_{\\text{eff}}$ is a strongly monotonic function of evolution.\nThe surface aspect of PC$_2$ is then supplemented with some information from ${\\log{} g}$ and ${[\\text{Fe/H}]}$.\n\nReducing the dimensionality of the observables and relating them back to the model parameters without redundancy aided with the interpretation of the PCs. \nWhilst it is useful to have the observables so succinctly described, it does not provide insight into the model parameters we wish to infer. \nWe thus condensed the information from the correlation plots into a $\\Lambda$ score which \nis the sum of the square of the correlation coefficients between the model parameters and the PCs (determined for the observables). Squaring the correlation coefficients is equivalent to the squaring the PC loadings of the centered and scaled observables. \nThe score is a means to quantify the extent to which information from the model parameters, dependent and independent, are encoded in the observables.\nWe calculated $\\Lambda$ scores for all four grids upon which PCA was performed (Appendix \\ref{sec:lambdaa}) and indeed found mostly consistent results. We note some differences arise in the initial model parameters such as $\\alpha_{\\text{MLT}}$ and $\\alpha_{\\text{OV}}$ which reflect their underlying \ndistributions from the choices in grid truncation. The above analyses can be applied to any combination of observables and model parameters to gauge their utility.\n\n\n\n\\subsection{Exploiting the Inherent Relationships}\n\nUnderstanding the inherent properties of the collective lower main sequence is the first step in elucidating the BA1 RF regression. The statistical analysis quantified what information was present in the training data for the RF to exploit. We illustrated why the available data permit BA1 to predict parameters such as $M$, $R$ and $L$ with such high precision and why initial model parameters such as $D$ and $\\alpha_{\\text{MLT}}$ remain uncertain in comparison. Whilst Section~\\ref{sec:RCT} and Section~\\ref{sec:PCA} demonstrated the breadth of information available to the RF, in Section~\\ref{sec:qu} we determined how the information could best be used. \n\nRFs are amongst the most powerful tools in mathematics for non-linear regression. \nThe BA1 RF uses the observables, creating a set of decision rules that reduce the variance in the parameter it is fitting.\nWhilst feature importances provide some insight into this process as a whole it does not provide specific details for the individual parameters. \nBy performing non-parametric multiple regression with every combination of observable in our grid, \nwe demonstrated how the correlations in Figure~\\ref{fig:filt_corr} could best be exploited and best combined to reveal the most information about each stellar quantity.\nTwo of the observables, ${[\\text{Fe/H}]}$ and ${\\langle\\delta\\nu_{02}\\rangle}$ (or as a ratio), are of vital importance in model fitting procedures as they provide indispensable pieces of independent information that cannot be inferred from other quantities. \n\nWe in effect invert the observations for the model parameters based on functions learnt from the training data. \nThus we can determine the relative importance of each observable for inferring the model parameters. We, in addition, provide a precision with which we can determine each model parameter \\emph{directly} from the information contained in the observables. \nThe attainable precision is a function of the number of initial model parameters that are varied and the model degeneracy in the data. \nFor example, with perfect information from the observables, the six dimensions in the BA1 grid limits our inference on mass to \n${\\mu (\\epsilon) =0.02 \\ \\Mo}$. \n\n\nMany of the Tables in Section~\\ref{sec:qu} demonstrated an important property of the RF. \nIn the case of missing or unreliable measurements of an observable, the RF can draw upon information redundancies in the data to determine new regression rules for the model parameters.\nIn principle, such redundancies can lead to biases and overfitting in iterative model finding methods.\nDuring such search procedures the best fitting stellar model is the one that best matches all of the observations but each observation only bares on some parts of the model, and observations can contain redundant information.\n%In practise, procedures that match oscillation modes will be dominated by the precision of the individual frequencies and the accurate constraints they place on the stellar interior.\n\n\nThrough statistical bagging and multiple regression the RF is less likely to overfit. %avoids the problem of overfitting. \nThese underlying methodologies are the reason why in Section~\\ref{sec:accu} many of the parameters we inferred remained well constrained despite large uncertainties in some of the observables. \nIn statistical bagging different subsets of the training grid are sent to different nodes. Each node will use information theory to create a set of decision trees to explain the parameter of interest. The nodes will differ in their rules and choice of parameters. Thus the uncertainty in an observable will only impact on the parameter we infer to the extent to which the observable is used in the rules. \nTake the example from Figure~\\ref{fig:uncert1} where with a ${5\\;\\mu}$Hz uncertainty in ${\\langle\\Delta\\nu_0\\rangle}$ the RF still predicts the solar properties albeit with slightly less confidence.\nThe other observables help constrain the predictions. \n\nPart of the analysis in Section~\\ref{sec:qu} demonstrated the best possible (average) precision in which we can hope to infer stellar parameters. Our error analysis in Section~\\ref{sec:accu} is an extension of this. Rather than assume perfect information we determined \n the measurement accuracy required of the observables to attain a desired precision from the RF.\nOur analysis focused on the Sun and is indicative of solar-like analogues.\nIn Table~\\ref{tab:relabunds} we saw some of the large uncertainties associated with retrodicting abundances in low-metallicity stars. We have greater degeneracy with the efficiency of diffusion and the initial abundances. These large error scores by no means indicate that the RF is incapable of characterizing low-metallicity stars. Rather it is an honest appraisal of stellar uncertainties when we do not make assumptions of the initial abundance say through a dY/dZ chemical evolution ``law\" or a fixed diffusion efficiency. Our error analysis here does not take into account covariances and was designed to investigate the impact on an observable-by-observable basis.\nA more detailed error analysis and the associated issues at low metallicity form the focus of a forthcoming paper. \n\n\n\\subsection{Implications for the TESS and PLATO missions}\n\nThe NASA TESS mission \\citep{2015JATIS...1a4003R} and ESA's PLATO \\citep{2014ExA....38..249R} herald a new age for the space-based photometry and the detection of planetary transits. Due to launch in 2018 and 2025 respectively, their common primary science mission is to identify terrestrial planets around bright stars. The pre-selection of bright targets will ensure that the stellar hosts can be further analyzed with spectroscopy and it is expected that many of the planet candidates will be suitable for atmospheric follow-up (ideally) with the James Webb Space Telescope. As was the case with the \\emph{Kepler} and \\emph{CoRoT} missions, the photometric time-series observations will prove useful to asteroseismology. In the case of PLATO the study of the stellar structure through asteroseismology is a key science goal in the mission design \\citep{2014ExA....38..249R}. \n\nTESS will monitor photometric variations of ${> 10^5}$ low-mass main-sequence stars. Under its `step and stare' pointing strategy, fields will be monitored for periods ranging from one month to one year depending primarily on their ecliptic latitude. With its two minute and $30$ minute cadences, TESS will be able to detect small rocky planets around solar like stars at $\\le$ 7th magnitude. It is expected to detect of the order $1,700$ planets with sub-Neptune masses \\citep{2016ApJ...830..138C} and will identify many more larger planets around dimmer targets. The asteroseismic potential of TESS has been rigorously investigated by \\citet{2016ApJ...830..138C}. Their analysis of the expected TESS photometry indicates the presence of an oscillation power excess \nin low-mass main-sequence stars when there is no systematic noise present in the data. With an expected systematic noise level of $60$ ppm hr$^{1/2}$ from the mission, their analysis indicates a detectable power-excess in F-dwarfs as well as sub giants and red giants -- this owing to the higher luminosity and hence larger mode amplitudes in these stars. For a majority of stars the $27$ day pointing is insufficient to extract detailed asteroseismic diagnostics such as mode frequencies or separations. Rather, the seismic information will be limited to the determination of $\\nu_{\\max}$ in stars where the power-excess is detected. As a consequence, masses and radii for the TESS targets are to be determined using a combination of GAIA data, the $\\nu_{\\max}$ -- ${\\langle\\Delta\\nu_0\\rangle}$ power law \\citep{2009A&A...506..465H,2009MNRAS.400L..80S} , asteroseismic scaling relations and grid-based searches. \n \n\nThe number of small planet detections from the PLATO mission is expected to eclipse the number found by \\emph{Kepler}\nand TESS by up to three orders of magnitude. In addition, the PLATO pointing strategy will allow for the measurement of oscillation frequencies in $> 80,000$ dwarf and subgiant stars with magnitudes less than $11$. In total the mission will provide approximately one million light curves for stars with brightness $\\le$ 13th magnitude \\citep{2014ExA....38..249R}. In many stars modes up to spherical degree ${\\ell =3}$ will be detected with typical frequency uncertainties in the range $0.1$ -- $0.3\\;\\mu$Hz. The second major science goal of PLATO is to \nprobe stellar structure and evolution by asteroseismology and provide support to exoplanet science through determining\n\\begin{itemize}\n\\item stellar masses with an accuracy of better than $10\\%$,\n\\item stellar radii accurate to $1$--$2\\%$, and\n\\item ages of solar-like stars accurate to $10\\%$.\n\\end{itemize}\n\nHere we treat the `Sun as a star' in order to quantify how well we can characterize target systems observed by the \nupcoming space missions and to determine the prospect of meeting the accuracy requirements. \nIn Table~\\ref{tab:tpl} we indicate the observables the missions are likely to provide. \nWe degrade the corresponding solar data according to the expected uncertainty from the respective measurements. \nAs GAIA is complete down to 20th magnitude we have assumed that distances and hence luminosities will be available for all targets in these missions. We consider data for TESS targets assuming both $60$~ppm~hr$^{1/2}$ and no systematic noise in the photometry. Thus in the case of the latter we anticipate that an oscillation power excess can be extracted for a solar-like star and $\\nu_{\\max}$ determined. The large and small frequency separations for the PLATO data are determined by degrading a subset of solar frequencies using the method described in BA1. We take a conservative approach in this calculation and assume that the ${\\ell=3}$ modes are not extracted.\n\n\n\n\n\\afterpage{\n\\cleardoublepage\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\textwidth,height=1.02\\textheight, keepaspectratio ]{tess_plato3.pdf}\n \\caption[Recovering solar parameters using observations expected for targets from TESS and PLATO]{(Caption on other page.) \\label{fig:TessPlato}}\n\\end{figure}\n\\begin{landscape}\n\\begin{figure}\n \\contcaption{Predictions for the `Sun as a star' using observations expected for targets from TESS (assuming two different systematic noise levels) and PLATO space missions. In each panel we list the median with uncertainties ($84\\%$--$50\\%$ confidence intervals and $50\\%$--$16\\%$ confidence intervals) for the quantities as well as the relative error in our prediction.} \n\\end{figure}\n%\\begin{deluxetable*}{lccccccccc}\n\\begin{table} %\\scriptsize\n\\centering\n%\\tablewidth{0pt}\n\\caption{Solar data degraded to the level expected for sun-like stars in: the TESS catalogue assuming systematic noise of $60$~ppm~hr$^{1/2}$ from the mission, TESS assuming no systematic noise and from PLATO. For each set of observables we include the feature importances from the random forest used in characterizing the `Sun as a star.' Note that in the case of the expected PLATO data we have perturbed a subset of frequencies according to their distance from $\\nu_{\\max}$. The numbers reported for the separations and ratios are thus the respective means and standard deviations of $10,000$ perturbations to the data which we evaluate to determine our parameter distributions. \\label{tab:tpl}} \n%\\hspace*{-1.1cm}\n\\begin{tabular}{l|ccc|ccc|ccc}\n\\multicolumn{1}{c}{} &\n\\multicolumn{3}{c}{TESS ($60$~ppm~hr$^{1/2}$)} &\n\\multicolumn{3}{|c|}{TESS ($0$~ppm~hr$^{1/2}$)} &\n\\multicolumn{3}{c}{PLATO} \\\\\\hline\nParameter &\nValue &\nUncertainty & \nImportance &\nValue &\nUncertainty &\nImportance &\nValue &\nUncertainty &\nImportance \\\\\\hline\n$T_{\\text{eff}}$ (K) & 5778 &100 &29.3\\% & 5778 &100 &26.7\\%& 5778 &100 &16.2\\%\\\\\n$[\\text{Fe/H}]$ &-0.014 &0.021 &34.3\\% &-0.014 &0.021 &33.4\\% &-0.014 &0.021 &27.9\\%\\\\\n$\\log{} g$ &4.43 &0.07 &18.5\\% &4.43 &0.07 &12.4\\% &4.43 &0.07 &8.8\\\\\n$L \\ (L/\\text{L}_{\\odot})$ &0.98 &0.04 &18.0\\% &0.98 &0.04 &16.7\\% &0.98 &0.04 &7.8\\%\\\\\n$\\nu_{\\max} \\ (\\mu \\text{Hz})$ & -- & -- & -- &3093 &100 &10.8\\% & -- &-- & --\\\\\n$\\langle\\Delta\\nu_0\\rangle \\ (\\mu \\text{Hz})$ & -- & -- & -- & -- & -- & -- &134.81 &0.05 &6.4\\%\\\\\n$\\langle\\delta\\nu_{02}\\rangle \\ (\\mu \\text{Hz})$ & -- & -- & -- & -- & -- & -- &9.02 &0.15 &7.1\\% \\\\\n$\\langle r_{01}\\rangle$ & -- & -- & -- & -- & -- & -- &0.0226 &0.0005 &7.4\\%\\\\\n$\\langle r_{10}\\rangle$ & -- & -- & -- & -- & -- & -- &0.0227 &0.0005 &7.3\\% \\\\\n$\\langle r_{02}\\rangle$& -- & -- & -- & -- & -- & -- &0.0668 &0.0011 & 11.1\\%\n\\end{tabular}\n\\end{table}\n%\\enddata\n%\\end{deluxetable*}\n\\end{landscape}\n%\\newpage\n%\\afterpage{\n}\n%}\n\n\nFigure~\\ref{fig:TessPlato} shows our predictions for masses, radii, ages, initial helium and metallicity for a `Sun-as-a-star' exercise. In each panel we indicate the median of the probability density distribution and the corresponding uncertainty from the $16\\%$ and $84\\%$ confidence intervals for the parameter we are predicting. In addition we determine the relative error which we define as ${\\epsilon = 100 \\cdot \\sigma/\\mu}$ where $\\mu$ is the mean and $\\sigma$ is the standard deviation of the distributions. In Appendix \\ref{sec:PSM} we further demonstrate the impact of the measurement uncertainty on the prediction of each quantity as per Figure~\\ref{fig:uncert1}.\n\n\nAlthough we can expect accurate mass determinations for targets in both missions, the supplementary seismic data from PLATO allows us to improve the precision with which we determine mass by approximately a factor of two. This is despite the fact the RF has identified a less-likely but not impossible (slightly) younger, higher-mass, higher-metallicity solution from the PLATO data (we find bimodalities for most quantities predicted with the PLATO observables). In the case of TESS, the absence of the large frequency separation leads to greater uncertainty. One of the methods discussed by \\citet{2016ApJ...830..138C} for the mass determination of TESS targets is to use the power law linking $\\nu_{\\max}$ to ${\\langle\\Delta\\nu_0\\rangle}$ (which has been shown to be accurate to $10$--$15\\%$) and apply the asteroseismic scaling laws (Equations \\ref{equ:nmax} and \\ref{equ:dnu}). In Section~\\ref{sec:seispr} we demonstrated that the random forest exploits further information from temperature or metallicity measurements to improve the accuracy of the $\\nu_{\\max}$ -- ${\\langle\\Delta\\nu_0\\rangle}$ relation. Thus we expect the accuracy with which we predict mass from TESS data to represent an upper limit to that attainable by applying the power-law and scaling relations. \n\nThe assumption of GAIA distances and hence stellar luminosities ensure that radii can be determined for targets in both missions; the seismology is essentially redundant for the inference of the stellar radius. We note that the relative error for PLATO in our `Sun-as-a-star' test is a factor of two higher than the $1$--$2\\%$ expected by the consortium. This is a consequence of having identified bimodal solutions. Their target accuracy can likely be met if the uncertainties in the measurements are further reduced and a unimodal solution found. \n\nThe analysis in Section~\\ref{sec:seispr} has highlighted the necessity of the small frequency separation in order to \ntightly constrain the ages of field stars. The predictions for age in Figure~\\ref{fig:TessPlato} are therefore as expected. \nThe inclusion of oscillation frequencies and determination of the small frequency separation (and ratios) from PLATO data result in age uncertainties for solar-like stars to within the $10\\%$ level. Without information from the core, ages for TESS targets remain largely unconstrained and consistent with the accuracy typically expected when dating field stars spectroscopically. \n\n\n\n\n\n\\section{Conclusions}\nIn this work we examined the processes that allow random forest regression to rapidly and accurately infer stellar parameters \\citep{2016apj...830...31b}. We shed light on the inherent properties of the model training data that the algorithm can exploit. \n\n\\begin{itemize}\n \n \\item We demonstrated that there is a large amount of information redundancy in the stellar parameters which is integral to the efficacy of the random forest algorithm. Through statistical bagging, the random forest creates sets of decision rules using different combinations of observables to infer a given quantity. The methodology results in robust predictions and includes the ability to compensate for data that are missing or unreliable. \n \n \\item We illustrated the behaviour of parameters across the collective lower main sequence with the relationships that arise (e.g., age -- luminosity) different to those that develop internally along an evolutionary track. This is the inherent information the random forest draws upon in its regression.\n \n \\item We found the parameter pairs that exhibit the strongest correlations correspond to well known asteroseismic and main-sequence relations. \n \n \n\n\n \\item The random forest works well in cases when there is sufficient information and sufficient redundancy.\n Through principal component analysis we quantified the degree of degeneracy in the observables. \n Our analysis demonstrated that $99.2\\%$ of the variance in the $11$ stellar observables could be explained by five principal components.\n \n \n \\item The observables we have considered only carry five pieces of independent information.\n During iterative model searches it is common that independently determined parameters such as $\\nu_{\\max}$, ${\\langle\\Delta\\nu_0\\rangle}$, and ${\\log{} g}$ are treated as independent degrees of freedom. The composition of the principal components indicate that by not considering their model covariances, any fit is biased towards the common stellar information to which these parameters pertain.\n \n\n \\item We devised a score which allows us to rank the degree to which model parameters can be inferred from the observables. Radius, luminosity, and main-sequence lifetime can be extracted with confidence, however, the initial model parameters such as $\\alpha_{\\text{MLT}}$, $Y_0$ and $\\alpha_{\\text{ov}}$ are not sufficiently constrained by the observables and cannot be inferred directly from the data. Our analysis can be extended in a straightforward manner to model parameters and observables not considered here. \n \n \n \\item Having elucidated the statistical properties of the training data, we sought to better understand how the random forest uses the data in its decision making rules.\nBy performing non-parametric multiple regression with every combination of observable in our grid we determined:\n\\begin{enumerate}\n \\item which observables are the most important/useful for each model parameter,\n \\item the minimum set of observables that satisfactorily constrain each model parameter, and\n \\item the precision with which we can determine each model parameter \\emph{directly} from the information contained in the observables. \n\\end{enumerate}\n\n\n\n\\item We examined the quantities on a parameter by parameter basis and here highlight the results for mass and age. In a grid of stellar evolution models varied in six initial parameters we find that the average error in predicting mass across the grid is ${\\pm 0.02 \\ \\Mo}$ and ${\\pm 282}$ Myr for age. The average error in age increases by a factor of three when we are limited to information from only two observables such as in the Christensen-Dalsgaard diagram. Three parameters are sufficient for constraining mass whereas we require five observables to determine age. \n\n\n\\item We determined whether the random forest could reproduce the well-known power law that relates ${\\langle\\Delta\\nu_0\\rangle}$ to $\\nu_{\\max}$ and found that additional information from $T_{\\text{eff}}$ or $[\\text{Fe/H}]$ reduces the average error in the relation by a factor of two. \n\n\\item We investigated the measurement accuracy required of the observables to attain a desired precision from the random forest. \nThe processes of statistical bagging and multiple regression help mitigate the impact of large spectroscopic errors as the random draws upon complementary seismic information when devising its decision rules. \nThe results confirm that $[\\text{Fe/H}]$ and ${\\langle\\delta\\nu_{02}\\rangle}$ are indispensable independent pieces of information for model fitting algorithms.\n\n\\item Finally, we determined the accuracy and precision with which we can expect to characterize solar-like stars observed by the upcoming TESS and PLATO space missions. In both cases masses can be accurately inferred and measurements from GAIA will ensure that radii are well constrained. Oscillation frequencies will not be detectable in most low-mass main sequence stars observed by TESS. In contrast, the availability of the small frequency separation for PLATO targets will permit accurately determined stellar ages. \n\\end{itemize}\n\n\n\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%% Acknowledgements %%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\acknowledgments \n\\paragraph*{Acknowledgements}\n\\noindent The research leading to the presented results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no 338251 (StellarAges). E.B.\\ undertook this research in the context of the International Max Planck Research School for Solar System Research. S.B.\\ acknowledges partial support from NSF grant AST-1514676 and NASA grant NNX13AE70G. We thank Alexey Mints and the anonymous referee for their useful comments and discussions which helped improve this manuscript. \n\n\n%\\software \n\\paragraph*{Software} \n\\noindent Stellar models were calculated with \\emph{Modules for Experiments in Stellar Astrophysics} r8118 \\citep[MESA,][]{2011apjs..192....3p} and stellar oscillations with the ADIPLS pulsation package 0.2 \\citep{2008Ap&SS.316..113C}. \nAnalysis in this manuscript was performed with python 3.5.1 libraries scikit-learn 0.17.1 \\citep{scikit-learn}, NumPy 1.11.0 \\citep{van2011numpy}, matplotlib 1.5.1 \\citep{Hunter:2007}, biokit 0.3.2 \\citep{biokit} and pandas 0.19.0 \\citep{mckinney2010data} as well as R 3.3.2 \\citep{R} and the R libraries magicaxis 2.0.0 \\citep{magicaxis}, RColorBrewer 1.1-2 \\citep{RColorBrewer}, parallelMap 1.3 \\citep{parallelMap}, data.table 1.9.6 \\citep{data.table}, ggplot2 2.1.0 \\citep{ggplot2}, GGally 1.2.0 \\citep{GGally}, scales 0.4.0 \\citep{scales} and Corrplot 0.77. \n\n%\\bibliographystyle{aasjournal.bst}\n%\\bibliography{statistical_relations}\n\n\n\n\n\n\n\n\n\n\\section{Appendix}\n\n\n\\subsection{Seismic Definitions} \n\\label{sec:sdefs}\nWe denote any frequency separation $S$ as the difference between a frequency $\\nu$ of spherical degree $\\ell$ and radial order $n$ and another frequency: \n\\begin{equation} \n S_{(\\ell_1, \\ell_2)}(n_1, n_2) \\equiv \\nu_{\\ell_1}(n_1) - \\nu_{\\ell_2}(n_2).\n\\end{equation}\nThe large-frequency separation is defined as\n\\begin{equation} \n \\Delta\\nu_\\ell(n) \\equiv S_{(\\ell, \\ell)}(n, n-1)\n\\end{equation}\nand the small-frequency separation is\n\\begin{equation}\n \\delta\\nu_{(\\ell, \\ell+2)}(n) \\equiv S_{(\\ell, \\ell+2)}(n, n-1).\n\\end{equation}\n\\citet{2003A&A...411..215R} have demonstrated that taking the ratio of the \\emph{local} large and small-frequency separations reduces the systematic offset introduced from improper modelling of the near-surface super-adiabatic region. This ratio is defined as: \n\\begin{equation} %\\label{eqn:LSratio}\n \\mathrm{r}_{(\\ell,\\ell +2)}(n) \\equiv \\frac{\\delta\\nu_{(\\ell, \\ell+2)}(n)}{\\Delta\\nu_{(1-\\ell)}(n+\\ell)}.\n\\end{equation}\nIn addition, it was shown that the frequency-dependent offset can be somewhat mitigated by constructing ratios from five-point frequency separations and the \\emph{local} large separation:\n\\begin{equation} \n \\mathrm{r}_{(\\ell, 1-\\ell)}(n) \\equiv \\frac{\\mathrm{dd}_{(\\ell,1-\\ell)}(n)}{\\Delta\\nu_{(1-\\ell)}(n+\\ell)} %\\label{eqn:rnl}\n\\end{equation}\nwhere the five point separations are defined as:\n\\begin{align} \n \\mathrm{dd}_{0,1} \\equiv \\frac{1}{8} \\big[&\\nu_0(n-1) - 4\\nu_1(n-1) \\notag\\\\\n &+6\\nu_0(n) - 4\\nu_1(n) + \\nu_0(n+1)\\big]\\\\ \n \\mathrm{dd}_{1,0} \\equiv -\\frac{1}{8} \\big[&\\nu_1(n-1) - 4\\nu_0(n) \\notag\\\\\n &+6\\nu_1(n) - 4\\nu_0(n+1) + \\nu_1(n+1)\\big]. \\label{eqn:dlast}\n\\end{align}\nWe calculate dozens of oscillation frequencies per star with the mode sets available dependent on the internal structure of an individual model. We thus determine a single representative value \nby following the prescription of \\citet{2012A&A...537A..30M}. In order to mimic how the oscillation spectra would appear in an observational data, we weight all frequencies by their position in a Gaussian envelope with full-width at half-maximum of ${0.66\\cdot\\nu_{\\max}{}^{0.88}}$ and centered at the predicted frequency of maximum oscillation power $\\nu_{\\max}$. We then calculate the weighted median of each variable, which we denote with angled parentheses (e.g.\\ ${\\langle r_{1,0}\\rangle}$).\n\n\n\n\n\n\\subsection{Asteroseismic Scaling Relations}\n\n\n\\begin{equation}\n\\nu_{\\max} \\approx \\frac{ M/M_{\\odot}(T_{\\text{eff}}/T_{\\text{eff},\\odot})^{3.5}}{L/L_{\\odot}} \\nu_{\\max,\\odot} \\: \n\\label{equ:nmax}\n\\end{equation}\n\n\\begin{equation}\n\\Delta\\nu \\approx \\frac{(M/M_{\\odot})^{0.5}(T_{\\text{eff}}/T_{\\text{eff},\\odot})^{3}}{(L/L_{\\odot})^{0.75}}\n\\Delta\\nu_{\\odot} \\: \n\\label{equ:dnu}\n\\end{equation}\n\n\\citet{2016MNRAS.460.4277G} have shown that a metallicity-dependent correction is required for the Equation~(\\ref{equ:dnu}) scaling relation.\nThe ${\\Delta\\nu_{\\odot}}$ term can be replaced with a more appropriate reference value which can be calcuated according to:\n\n\n\\begin{equation}\n\\Delta\\nu_{\\text{ref}}= A \\cdot e^{\\lambda T_{\\text{eff}}/10^4K} \\cdot (\\cos(\\omega \\cdot T_{\\text{eff}}/10^4K+\\phi))+B,\n\\label{eq:corrfunc2}\n\\end{equation}\nand where the unkown terms are listed in Table~\\ref{tab:pars2}. \n\n \n\\begin{table}\n\t\\centering\n\t\\caption{Parameters of the correction function.}\n\t\\label{tab:pars2}\n\t\\begin{tabular}{lc} % four columns, alignment for each\n\t\t\\hline\n\t\tA & 0.64$\\cdot$[Fe/H] + 1.78 $\\mu Hz$ \\\\\n\t\t$\\lambda$ & $-$0.55$\\cdot$[Fe/H] + 1.23 \\\\\n\t\t$\\omega$ & 22.21 rad/K \\\\\n\t\t$\\phi$ & 0.48$\\cdot$[Fe/H] + 0.12 \\\\\n\t\tB & 0.66$\\cdot$[Fe/H] + 134.92 $\\mu Hz$ \\\\\n\t\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\n\n\\subsection{Correlation Plot} \n\\label{sec:fullcorr}\n\n\nThe full BA1 grid introduces some biases in our correlation analysis, particularly from tracks with calculated with high-mass and/or high-diffusion. Correlation analysis with all models included are presented in Figure~\\ref{fig:corr}. \n\\begin{figure*}\n \\centering\n \\includegraphics[trim={1cm 0 2cm 1cm},clip,\n width=\\textwidth]{corr-spearman.pdf}\n \\caption[]{Spearman rank correlation matrix comprising various stellar and asteroseismic parameters. The quantities are as described in Table~\\ref{tab:parmdefs}\n with model input parameters marked in purple above. The complete grid of models are considered here. The size and color of each circle indicates the sign and magnitude of the Spearman rank coefficient, $\\rho$, between two variables. All correlations are significant excepting the entries indicated with a cross. The variables are ordered by the first principal component of the correlation matrix. }\n \\label{fig:corr}\n\\end{figure*}\n\nA major difference that arises between Figure~\\ref{fig:filt_corr} and Figure~\\ref{fig:corr} is in the ordering of variables. Recall that we report the quantities according to the first principal component of the correlation matrix. Different combinations of variables are required to maximise the variance of each principal component in the new parameter space. Although the PCA analysis in Figures \\ref{fig:corr-pcaobs} and Figures \\ref{fig:corr-pcamods} rely on Pearson rather than Spearman correlations, they do demonstrate the difference in the composition of the PCs in each grid. \n\nWe also find differences in the correlations that pertain to current surface abundance parameters. Consider the pair M -- $Y_{\\text{surf}}$. In Figure~\\ref{fig:corr} we find a small but non-negligible negative correlation. The reason being that higher mass tracks diffuse the helium from their surface more efficiently than low-mass stars. Without the influence of these stars in our sample, our significance test yields a null correlation in Figure~\\ref{fig:filt_corr}; the expected result from a quasi-random distribution of initial abundances. \n\nTwo interesting features emanating from our grid selection relates to the parameter pairs ${\\langle\\delta\\nu_{02}\\rangle}$ -- $T_{\\text{eff}}$ and ${\\langle r_{02}\\rangle}$ -- ${\\log{} g}$. \nWe find a null correlation between ${\\langle\\delta\\nu_{02}\\rangle}$ -- $T_{\\text{eff}}$ in truncated grid however this emerges as a small positive correlation when the full grid is considered. \nIn Section~\\ref{sec:sages} we discussed the redundancy in the C--D diagram when projecting stellar models varied in six dimensions into a two-dimensional parameter space. Thus the null correlation \narising from the truncated grid reflects the fact there many combinations of (primarily) mass and metallicity and hence temperature at a given age. \nThe full grid, however, consists of a large number of hot short-lived stars that impart a noticeable trend. \n\nA similar argument applies to ${\\langle r_{02}\\rangle}$ -- ${\\log{} g}$. There are a great number of combinations of ${\\langle\\Delta\\nu_0\\rangle}$ and\n${\\langle\\delta\\nu_{02}\\rangle}$ for a given ${\\langle r_{02}\\rangle}$ thus in the truncated grid no correlation with ${\\log{} g}$ is registered. \nOnce again the number of massive short-lived stars bias this previous null correlation. \n\n\nFinally we note two minor results. Some pairs of parameters in the truncated grid which report null correlations in Figure~\\ref{fig:filt_corr}, show very weak correlations in Figure~\\ref{fig:corr}. \nWe refer to L -- $\\alpha_{\\text{ov}}$ and $\\alpha_{\\text{MLT}}$ -- ${\\langle\\delta\\nu_{02}\\rangle}$ as cases in point. The correlations remain very weak in the current analysis and the larger sample size has introduced a minor trend that in this case passes our conservative significance criterion. We note also that most variables display a much stronger correlation with age in the full grid. \n \n\n\n\n\n\n\n\\subsection{Principal Component Analysis Explained Variance} \n\\label{sec:fullPCA}\n%\\iffalse\n\nThe PCs and their correlations will change depending on the number of dimensions included in the grid and the range of values each parameter takes; the PCs identify vectors of maximal variance. \nOur aim is to determine whether the PCs capture fundamental features ubiquitously encoded in the observables. \nThus, we wish to investigate the information inherent to the dimensions and mitigate the impact of parameter ranges on our PCs. \nIn order to provide a more robust interpretation we have calculated the PCs and their correlations with four different considerations given to the BA1 grid:\n\\begin{description}\n \\item[\\textbf{Grid A}] The full BA1 training grid;\n \\item[\\textbf{Grid B}] The truncated grid;\n \\item[\\textbf{Grid C}] A grid where more than half the models in each track possess metallicities of [Fe/H] $> -2$; and\n \\item[\\textbf{Grid D}] A grid with masses limited to ${M < 1.2}$~M$_{\\odot}$.\n\\end{description}\nQualitative correlations between the stellar parameters and the PCs in each grid are presented in Figures \\ref{fig:corr-pcaobs} and Figures \\ref{fig:corr-pcamods}.\n\n\n%\\begin{deluxetable*}{ccccccc}\n\\begin{table}\n\\centering\n%\\tablewidth{0pt}\n\\caption{Percentage of the variance explained by each principal component. We report the explained variance percentages for the complete grid of training models (Grid A) and for the truncated set (Grid B, see Section~\\ref{sec:RCT}) that better encompasses the observational parameter space. In each case we consider the grid with and without the inclusion of $\\nu_{\\max}$ which is estimated using the \\citet{1995A&A...293...87K} scaling relations rather than calculated from first principle equations. We also consider the explained variances when limits are placed on the metallicity (Grid C) and mass (Grid D) ranges of the models. These grids are used in Section~\\ref{sec:disc} to help interpret the PCs. \\label{tab:PCAEV}} \n\\iffalse\n\\tablehead{\n \\colhead{} &\n \\multicolumn{4}{c}{$\\nu_{\\max}$ Included} &\n \\multicolumn{2}{c}{$\\nu_{\\max}$ Excluded}\\\\\n \\colhead{Component} &\n \\colhead{Grid A} &\n \\colhead{Grid B} &\n \\colhead{Grid C} &\n \\colhead{Grid D} &\n \\colhead{Grid A} &\n \\colhead{Grid B} \n}\n\\startdata\n\\fi\n\\begin{tabular}{c|cccc|cc}\n \\multicolumn{1}{c}{} &\n \\multicolumn{4}{c|}{$\\nu_{\\max}$ Included} &\n \\multicolumn{2}{c}{$\\nu_{\\max}$ Excluded}\\\\\\hline\n Component &\n Grid A &\n Grid B &\n Grid C &\n Grid D &\n Grid A &\n Grid B \\\\\\hline\nPC$_1$ \t&\t41.79\t&\t42.36\t& 42.49 & 42.74 &\t40.89\t&\t41.47\t\\\\\nPC$_2$ \t&\t36.12\t&\t34.18\t& 37.49 & 35.89 &\t36.52\t&\t33.65\t\\\\\nPC$_3$\t&\t9.17\t&\t11.65\t& 9.39 & 10.25 &\t8.99\t&\t12.21\t\\\\\nPC$_4$\t&\t7.69\t&\t9.79\t& 7.69 & 6.89 &\t8.27\t&\t10.58\t\\\\\nPC$_5$ \t&\t4.23\t&\t1.23\t& 2.14 & 3.36 &\t4.55\t&\t1.36\t\\\\\nPC$_6$\t&\t0.54\t&\t0.48\t&0.41 & 0.53 &\t0.48\t&\t0.51\t\\\\\nPC$_7$\t&\t0.25\t&\t0.18\t&0.24 &0.18 &\t0.16\t&\t0.12\t\\\\\nPC$_8$ \t&\t0.12\t&\t0.08\t&0.09 &0.10 &\t0.10\t&\t0.09\t\\\\\nPC$_9$\t&\t0.05\t&\t0.03\t&0.04 &0.04 &\t0.03\t&\t0.02\t\\\\\nPC$_{10}$ \t&\t0.02\t&\t0.01 &0.01 &0.01\t&\t0.01\t&\t0.00\t\\\\\nPC$_{11}$\t&\t0.01\t&\t0.00 &0.00 &0.00 &\t--\t &\t--\t\n%\\enddata\n% \\end{deluxetable*} \n\\end{tabular}\n\\end{table}\n\n\\subsection{PCA Correlation Analysis} \\label{sec:ccoefs}\nFigures \\ref{fig:GCA-pcabar} and \\ref{fig:GCA-pcabarb} demonstrate the correlation strengths between our stellar parameters and the first five PCs. In Tables \\ref{tab:ocoefs} and \\ref{tab:ocoefs} we list the coefficients between all parameters and all PCs. The table is useful for determining whether the transitive criterion applies to parameters within a given PC. It also aids in the calculation of the $\\Lambda$ scores in Section~\\ref{sec:ISP}.\n\n\n\\afterpage{\n\\clearpage\n\\begin{landscape}\n\\begin{table}\n\\centering\n\\caption{Pearson's~$r$ coefficients between the principal components and observables in the truncated grid.}\n\\label{tab:ocoefs}\n\\begin{tabular}{c|ccccccccccc}\n& $\\log{} g$ &$T_{\\text{eff}}$ & $[\\text{Fe/H}]$ & $\\langle\\Delta\\nu_0\\rangle$ & $\\langle\\delta\\nu_{02}\\rangle$ & $\\langle r_{02}\\rangle$ & $\\langle r_{01}\\rangle$ & $\\langle\\delta\\nu_{13}\\rangle$ & $\\langle r_{13}\\rangle$ & $\\langle r_{10}\\rangle$ & $\\nu_{\\max}$ \\\\ \\hline \\hline\nPC$_1$ & 0.93 & -0.20 & -0.35 & 0.92 & 0.93 & 0.38 & -0.07 & 0.95 & 0.32 & -0.07 & 0.87 \\\\\nPC$_2$ & -0.30 & 0.73 & -0.29 & -0.33 & 0.34 & 0.85 & 0.81 & 0.22 & 0.76 & 0.81 & -0.42 \\\\\nPC$_3$ & 0.00 & -0.60 & 0.63 & 0.04 & 0.06 & 0.15 & 0.45 & -0.13 & -0.17 & 0.45 & 0.22 \\\\\nPC$_4$ & 0.08 & 0.08 & -0.60 & 0.19 & -0.08 & -0.30 & 0.37 & -0.13 & -0.52 & 0.37 & 0.10 \\\\\nPC$_5$ & 0.14 & 0.25 & 0.20 & 0.06 & -0.02 & -0.05 & 0.03 & 0.00 & -0.07 & 0.03 & 0.01 \\\\\nPC$_6$ & 0.11 & -0.01 & -0.04 & 0.01 & -0.11 & 0.09 & 0.00 & -0.12 & 0.06 & 0.00 & 0.04 \\\\\nPC$_7$ & -0.09 & 0.03 & 0.01 & 0.05 & -0.03 & -0.01 & 0.00 & -0.01 & 0.03 & 0.00 & 0.09 \\\\\nPC$_8$ & 0.02 & -0.02 & 0.00 & 0.00 & -0.04 & -0.05 & 0.02 & 0.03 & 0.05 & 0.02 & -0.01 \\\\\nPC$_9$ & 0.01 & 0.01 & 0.00 & -0.04 & 0.02 & -0.02 & 0.00 & -0.02 & 0.01 & 0.00 & 0.03 \\\\\nPC$_{10}$ & 0.00 & 0.00 & 0.00 & 0.02 & 0.02 & -0.01 & 0.00 & -0.02 & 0.01 & 0.00 & -0.01 \\\\\nPC$_{11}$ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.01 & 0.00 & 0.00 & -0.01 & 0.00 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\n\n\n\\begin{table} %\\scriptsize\n\\centering\n\\caption{Pearson's~$r$ coefficients between the principal components and model parameters in the truncated grid.}\n\\label{tab:mcoefs}\n\\hspace*{-0.5cm}\n\\begin{tabular}{c|cccccccccccccc}\n &$M$ & $Y$ & $Z$ & $\\alpha_{\\text{MLT}}$ & $\\alpha_{\\text{ov}}$ & $D$ & $\\tau$ & $\\tau_{\\text{MS}}$ & $X_c$ & $M_{\\text{cc}}$ & $X_{\\text{surf}}$ & $Y_{\\text{surf}}$ & $R$ & $L$ \\\\ \\hline \\hline\nPC$_1$ & -0.67 & -0.08 & -0.30 & 0.06 & -0.05 & -0.12 & -0.19 & -0.72 & 0.77 & -0.45 & -0.04 & 0.16 & -0.86 & -0.69 \\\\\nPC$_2$ & 0.29 & 0.10 & -0.41 & -0.15 & -0.04 & -0.17 & -0.56 & -0.26 & 0.14 & 0.17 & 0.05 & 0.11 & 0.30 & 0.55 \\\\\nPC$_3$ & 0.13 & 0.04 & 0.48 & -0.10 & -0.07 & -0.03 & -0.01 & -0.04 & 0.14 & -0.11 & -0.33 & 0.17 & 0.09 & -0.04 \\\\\nPC$_4$ & -0.51 & -0.17 & -0.40 & 0.03 & -0.09 & -0.08 & 0.51 & 0.48 & -0.49 & -0.34 & 0.29 & -0.17 & -0.23 & -0.13 \\\\\nPC$_5$ & -0.02 & 0.17 & -0.09 & 0.44 & -0.16 & -0.43 & 0.13 & 0.13 & -0.17 & -0.29 & -0.50 & 0.59 & -0.14 & -0.03 \\\\\nPC$_6$ & -0.02 & -0.20 & 0.08 & -0.01 & -0.15 & -0.10 & 0.28 & 0.07 & -0.02 & -0.25 & 0.05 & -0.10 & -0.14 & -0.03 \\\\\nPC$_7$ & 0.09 & 0.05 & -0.05 & 0.14 & 0.11 & -0.15 & -0.10 & 0.03 & 0.10 & 0.30 & -0.14 & 0.17 & 0.18 & 0.38 \\\\\nPC$_8$ & -0.11 & 0.08 & -0.06 & -0.25 & 0.05 & -0.07 & 0.03 & -0.03 & -0.02 & -0.12 & -0.05 & 0.08 & -0.07 & -0.09 \\\\\nPC$_9$ & 0.21 & -0.35 & 0.22 & 0.20 & -0.06 & -0.08 & -0.22 & -0.12 & 0.15 & 0.04 & -0.01 & -0.10 & 0.06 & 0.07 \\\\\nPC$_{10}$ & -0.17 & 0.26 & -0.14 & -0.15 & 0.02 & 0.06 & 0.17 & 0.12 & -0.10 & -0.01 & 0.01 & 0.06 & -0.09 & -0.04 \\\\\nPC$_{11}$ & -0.01 & -0.01 & -0.01 & 0.01 & 0.01 & 0.00 & 0.00 & 0.00 & 0.00 & -0.02 & 0.01 & -0.01 & 0.00 & -0.01 \\\\ \\hline\n\\end{tabular} \n\\end{table}\n\\end{landscape}\n}\n\n\n\\subsection{PC correlations with different grids}\n\\label{sec:PCAg}\nIn Section~\\ref{sec:intPC} we presented the correlation strengths between the PCs and observables (Figure~\\ref{fig:GCA-pcabar}) and the PCs and the model parameters (\\ref{fig:GCA-pcabarb}). \n Here we perform the same analysis with the different subsets of the BA1 grid described in Appendix \\ref{sec:fullPCA}. In order to compare the results for each grid, in Figures \\ref{fig:corr-pcaobs} and \\ref{fig:corr-pcamods} we employ a correlation plot rather than the quantitative bar chart used in Section~\\ref{sec:intPC}. This allows an inspection of the qualitative behaviour of the PCs in each case. We find a similar explained variance from the corresponding PCs in each grid. This suggests that the PCs capture essentially the same inherent features in model data and that the PCs are not due to the number of models in our analysis or the chosen parameter ranges.\n\n\\afterpage{\n\\clearpage\n\\begin{landscape}\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{PCAObs4.pdf}\n \\caption[]{Pearson correlation matrices relating the principal components back to the stellar observables in each of the four grids described in Appendix \\ref{sec:fullPCA}. \\label{fig:corr-pcaobs}}\n\\end{figure}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{PCAMod4.pdf}\n \\caption[]{Pearson correlation matrices relating the principal components back to the model quantities in each of the four grids described in Appendix \\ref{sec:fullPCA}. \\label{fig:corr-pcamods}} \n\\end{figure}\n\\end{landscape}\n}\n\n\\subsection{$\\Lambda$ Analysis}\n\\label{sec:lambdaa}\n\nThe data matrix of observables $\\mathbf{X}$ is size ${n \\times p}$ where n is the number of training models and p the number of parameters.\nWe centre and scale the entries according to the mean and standard deviation of each parameter.\nThe resultant matrix, $\\mathbf{\\bar{X}}$, therefore has the property that for each parameter, $p$, ${\\mu(p) =0}$ and ${\\sigma(p) =1}$. \nWe compute the correlation matrix, $R$, for the matrix $\\mathbf{\\bar{X}}$ :\n\\begin{eqnarray}\n \\mathbf{R} &= \\text{Corr}(\\mathbf{\\bar{X}}) \\\\[3pt] \\nonumber\n &= \\mathbf{\\bar{X}}\\mathbf{\\bar{X}^\\top}. \n\\end{eqnarray}\nAs the correlation and covariance matrices are symmetric we calculate the eigendecomposition of R such that:\n\\begin{equation}\n\\mathbf{R}=\\mathbf{VLV^\\top},\n\\end{equation} \nwhere \\textbf{V} a matrix of eigenvector columns and \\textbf{L} a diagonal matrix of eigenvalues.\nThe eigenvectors specify the principal axes of the data and the eigenvalues indicate the amount of variance there is in the data in the direction of the corresponding eigenvector.\nWe can define the projection matrix \\textbf{P} such that we project/transform our data into the new space \n\\begin{equation}\n\\mathbf{P} = \\mathbf{\\bar{X}} \\mathbf{V}.\n\\end{equation}\n\n\nThe correlation matrix is a special case of the covariance matrix in that the former is normalised.\nFor generality let us consider the covariance matrix, such that the original data matrix was centred but not scaled ($\\mathbf{\\hat{X}}$), then\n\\begin{eqnarray}\n \\mathbf{C} &= \\text{Cov}(\\mathbf{\\hat{X}}) \\\\[3pt] \\nonumber\n &= \\frac{1}{n-1}\\mathbf{\\hat{X}}\\mathbf{\\hat{X}^\\top}\\\\[3pt] \\nonumber\n &= \\mathbf{V} \\mathbf{L} \\mathbf{V^\\top},\n\\end{eqnarray}\nwhere we divide by (n-1) to unbias to covariance (the covariance entries will have different scales).\n\nAlternatively and equivalently, we may extract our PCs through SVD of $\\mathbf{\\hat{X}}$ such that: \n\\begin{equation}\n \\mathbf{\\hat{X}} = \\mathbf{U} \\mathbf{\\Sigma} \\mathbf{ V^\\top}\n\\end{equation}\nwhere \\textbf{U} is the left matrix of singular orthogonal vectors with dimensions ${n \\times n}$,\n$\\mathbf{\\Sigma}$ is a diagonal matrix of singular values with dimensions $n \\times p$, \nand $\\mathbf{V^{\\top}}$ is the right matrix of singular orthogonal vectors with diemsnions ${p \\times p}$. \nThe diagonal elements of $\\mathbf{\\Sigma}$ assign a relative importance to each vector whereas the vectors of \\textbf{V} are the principal directions/axes.\nAs the matricies \\textbf{U} and \\textbf{V} comprise orthogonal components they have the property\n\\begin{eqnarray}\n\\label{eqn:ident}\n\\mathbf{U^{\\top}U}=\\mathbf I_{n \\times n} \\\\ \\nonumber\n\\mathbf{V^{\\top}V}=\\mathbf I_{p \\times p}.\n\\end{eqnarray}\nWe note also that \n\\begin{align}\n \\left(\\mathbf {A \\cdot B \\cdot C }\\right)^\\top &= \\mathbf{C^\\top \\cdot B^\\top \\cdot A^\\top} \\\\\n \\implies (\\mathbf U\\mathbf \\Sigma\\mathbf V^\\top)^\\top &= (\\mathbf V\\mathbf \\Sigma\\mathbf U^\\top)\n\\end{align}\nas $\\mathbf{\\Sigma}$ is a diagonal matrix.\n\n\nWe can reconstruct the eigendecomposition of the covariance matrix from the SVD: \n\\begin{eqnarray}\n \\frac{1}{n-1}\\mathbf{\\hat{X}}\\mathbf{\\hat{X}}^\\top &= \\frac{1}{n-1} (\\mathbf U\\mathbf \\Sigma\\mathbf V^\\top)(\\mathbf U\\mathbf \\Sigma\\mathbf V^\\top)^\\top\\\\[3pt] \\nonumber\n&= \\frac{1}{n-1}(\\mathbf U\\mathbf \\Sigma\\mathbf V^\\top)(\\mathbf V\\mathbf \\Sigma\\mathbf U^\\top)\n\\end{eqnarray}\nand from our identities in Equation~(\\ref{eqn:ident})\n\\begin{equation}\n \\frac{1}{n-1}\\mathbf{\\hat{X}}\\mathbf{\\hat{X}}^\\top=\\mathbf U \\frac{\\mathbf \\Sigma^2}{n-1} \\mathbf U^\\top.\n\\end{equation}\nWe therefore find that the square roots of the eigenvalues of $\\mathbf{C}$ are the singular values of $\\mathbf{\\bar{X}}$ and that the vectors in the right singular matrix, \\textbf{V}, are the principal directions/axes. The projection matrix can be calculated from the SVD such that \n\\begin{align}\n\\mathbf{P} &= \\mathbf{\\hat{X}} \\mathbf{V} \\\\[3pt] \\nonumber\n &= \\mathbf U \\mathbf \\Sigma \\mathbf V^\\top \\mathbf{V} \\\\[3pt] \\nonumber\n &= \\mathbf U \\mathbf \\Sigma.\n\\end{align}\n\nThe PCA loadings are the columns of \\textbf{L} which implies that \n\\begin{equation}\n\\mathbf{L}=\\mathbf{V}\\frac{\\mathbf \\Sigma}{\\sqrt{n-1}}.\n\\end{equation}\nWe can see that the loadings are the eigenvectors scaled by the square roots of the respective eigenvalues.\nWith these definitions we can compute the cross-covariance matrix between original variables and the standardized projection matrix. \nTo calculate the standardized PC scores for \\textbf{P} we require each column of \\textbf{U} to have unit variance. As $\\mathbf{\\Sigma}$ is diagonal it is simply a scaling matrix and can be dropped here yielding: \n\\begin{align}\n\\frac{1}{n-1}\\mathbf{X}^\\top(\\sqrt{n-1}\\mathbf{U}) &=\\frac{1}{\\sqrt{n-1}}\\mathbf{V}\\mathbf{\\Sigma}\\mathbf{U}^\\top\\mathbf{U} \\\\&=\\frac{1}{\\sqrt{n-1}}\\mathbf{V}\\mathbf{\\Sigma}\n\\\\&=\\mathbf{L}.\n\\end{align}\nWe find that the covariance matrix between the standardized PCs and original variables is in fact given by the loadings. \nIn Section~\\ref{sec:ev} we computed the \\emph{correlations} between the observables and their PCs rather than the covariances, requiring that the observables are normalized by their standard deviation. As we centred and scaled our data prior to performing the PCA, their values are unity and our correlation analysis is therefore equivalent to reporting the loadings. \n\nThe correlation analysis allowed us to project the model data onto the PC space and determine the `equivalent' loadings for each parameter. Through the $\\lambda$ score we can therefore determine to what extent the variance in the model data is captured by the PCs. In Table~\\ref{tab:corrLFull} we compare the results of the analysis for each grid. We find similar results for most parameters \nwith differences in some of the initial model parameters due to their underlying distributions as a result of the grid truncations.\n\n\n\n\n\\subsection{Impact of Uncertainties for Upcoming Photometric Space Missions} \nBelow we demonstrate the impact of measurement uncertainty on the prediction of parameters from the upcoming TESS (Figure~\\ref{fig:uncerttess}) and PLATO (Figure~\\ref{fig:uncertplato}) space missions. We produce probability density distributions for $250$ sets of $\\sigma$ values for each parameter we predict. The ranges for each parameter from which we draw our $\\sigma$ values are listed in Table~\\ref{tab:sunstar}. We restrict out observables to those we are likely to possess from the respective missions. \nIn each figure we plot the median value (solid line) and the $68\\%$ confidence interval (shaded region).\n\n\\afterpage{\n\\begin{table}\n \\centering\n \\caption{The $\\Lambda$ score is a sum of the squares of ${r(X, PC_i)}$ indicating the variance explained for a given parameter. These scores are by definition unity for our observables.}\n \\label{tab:corrLFull}\n \\begin{tabular}{ccccc}\n \\hline \\hline\n & \\multicolumn{4}{c}{$\\Lambda_{\\text{param}}$} \\\\\nParameter & Grid A & Grid B & Grid C & Grid D \\\\ \\hline \nR\t&\t0.97\t&\t0.97\t&\t0.98\t&\t0.97\t\\\\\nL\t&\t0.93\t&\t0.96\t&\t0.93\t&\t0.95\t\\\\\n$X_c$\t&\t0.93\t&\t0.94\t&\t0.93\t&\t0.94\t\\\\\n$\\tau_{\\text{MS}}$\t&\t0.93\t&\t0.93\t&\t0.93\t&\t0.94\t\\\\\nM\t&\t0.91\t&\t0.91\t&\t0.92\t&\t0.88\t\\\\\n$\\tau$\t&\t0.74\t&\t0.79\t&\t0.78\t&\t0.76\t\\\\\n$Z_0$\t&\t0.76\t&\t0.73\t&\t0.78\t&\t0.80\t\\\\\nM$_{cc}$\t&\t0.58\t&\t0.61\t&\t0.68\t&\t0.41\t\\\\\n$Y_{\\text{surf}}$\t&\t0.48\t&\t0.50\t&\t0.55\t&\t0.54\t\\\\\nX$_{\\text{surf}}$\t&\t0.50\t&\t0.48\t&\t0.53\t&\t0.55\t\\\\\n$\\alpha_{\\text{MLT}}$\t&\t0.02\t&\t0.38\t&\t0.04\t&\t0.06\t\\\\\n$Y_0$\t&\t0.10\t&\t0.31\t&\t0.27\t&\t0.09\t\\\\\nD\t&\t0.13\t&\t0.29\t&\t0.22\t&\t0.21\t\\\\\n$\\alpha_{\\text{ov}}$\t&\t0.10\t&\t0.08\t&\t0.11\t&\t0.12\t\\\\\n\n \\hline\n \\end{tabular}\n\\end{table}\n\n \\begin{table}\n \\centering\n \\caption{Central solar values and uncertainty ranges used for predictions in Figures \\ref{fig:uncerttess} and \\ref{fig:uncertplato}.}\n \\begin{tabular}{lcccccc}\n \\hline \\hline\n \\multicolumn{1}{c}{} & \\multicolumn{3}{c}{TESS}& \\multicolumn{3}{c}{PLATO} \\\\\nQuantity & Value & Min($\\sigma$) & Max($\\sigma$) & Value & Min($\\sigma$) & Max($\\sigma$)\\\\ \\hline \n$T_{\\text{eff}}$ (K) & 5777 & 10 & 500& 5777 & 10 & 500\\\\\n$\\log{} g$ & 4.44 & 0.0001 & 1.0& 4.44 & 0.0001 & 1.0\\\\\n$[\\text{Fe/H}]$ & 0.0 & 0.05 & 0.5 & 0.0 & 0.05 & 0.5\\\\\n$L$ & 1.0 & 0.001 & 10 & 1.0 & 0.001 & 10 \\\\\n$\\nu_{\\max}$ & 3050 & 10 & 500 & -- & -- & --\\\\\n$\\langle\\Delta\\nu_0\\rangle$ ($\\mu$Hz) & -- & -- & --& 136.0 & 0.5 & 50\\\\\n$\\langle\\delta\\nu_{02}\\rangle$ ($\\mu$Hz)& -- & -- & -- & 9.0 & 0.5 & 5 \\\\\n\\hline\n \\end{tabular}\n \\label{tab:sunstar}\n\\end{table}\n}\n\n\\clearpage\n\n\\label{sec:PSM}\n\n\n\n\n\n\\afterpage{\n%\\clearpage\n%\\cleartoleftpage%\\cleardoublepage\n %\\clearpage% flush all other floats\n %\\ifodd\\value{page}\n %\\else% uncomment this else to get odd/even instead of even/odd\n % \\expandafter\\afterpage% put it on the next page if this one is odd\n %\\fi\n\\begin{landscape}\n%\\pagestyle{lscape}\n\\begin{figure}\\vspace*{-0.75cm}\n \\centering\n \\includegraphics[width=0.9\\linewidth,keepaspectratio]{tess2.pdf}\n \\caption[]{(Caption on other page.)\\label{fig:uncerttess}}\n\\end{figure}\n\\end{landscape}\n\n\\begin{figure}\n \\contcaption{Predictions for the solar mass, age, luminosity and radius as a function of the uncertainties applied to key observables. In each panel we have perturbed the quantity on the abscissa in isolation, centred around the measured value listed in Table~\\ref{tab:sunstar} and with the uncertainties in the ranges specified therein. We indicate the median predicted value (solid line) and the $68\\%$ confidence interval (shaded region). Here the observables comprise those expected from the TESS space mission assuming that the p-mode power excess can be extracted.} \n\\end{figure}\n%\\end{landscape}\n%}\n\n\n\n\n%\\afterpage{\n\n\\afterpage{\n%\\clearpage\n%\\cleartoleftpage%\\cleardoublepage\n %\\clearpage% flush all other floats\n %\\ifodd\\value{page}\n %\\else% uncomment this else to get odd/even instead of even/odd\n % \\expandafter\\afterpage% put it on the next page if this one is odd\n %\\fi\n\\begin{landscape}\n%\\cleartoleftpage%\\cleardoublepage\n%\\begin{landscape}\n%\\pagestyle{lscape}\n\\begin{figure}\\vspace*{-0.75cm}\n \\centering\n \\includegraphics[width=0.9\\linewidth,keepaspectratio]{plato2.pdf}\n \\caption[]{The same as Figure~\\ref{fig:uncerttess}, but for PLATO. \\label{fig:uncertplato}} %(Caption on other page.) \\label{fig:uncertplato}}\n\\end{figure}\n\\end{landscape}\n}\n%\\begin{figure}\n %\\contcaption{}{Predictions for the solar mass, age, luminosity and radius as a function of the uncertainties applied to key observables. In each panel we have perturbed the quantity on the abscissa in isolation, centred around the measured value listed in Table~\\ref{tab:sunstar} and with the uncertainties in the ranges specified therein. We indicate the median predicted value (solid line) and the $68\\%$ confidence interval (shaded region). Here the observables comprise those expected from the PLATO space mission.} \n%\\end{figure}\n}\n\\clearpage\n\n\n\n\n\n\\iffalse\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.95\\textheight,height=0.95\\textwidth,keepaspectratio, angle=90]{tess2.pdf}\n \\caption{Predictions for the solar mass, age, luminosity and radius as a function of the uncertainties applied to key observables. In each panel we have perturbed the quantity on the abscissa in isolation, centred around the measured value listed in Table~\\ref{tab:sunstar} and with the uncertainties in the ranges specified therein. We indicate the median predicted value (solid line) and the $68\\%$ confidence interval (shaded region). Here the observables comprise those expected from the TESS space mission assuming that the p-mode power excess can be extracted.} \n \\label{fig:uncerttess}\n\\end{figure*}\n\n\\begin{figure*}\n \\centering\n \\includegraphics[width=0.95\\textheight,height=0.95\\textwidth,keepaspectratio, angle=90]{plato2.pdf}\n \\caption{Predictions for the solar mass, age, luminosity and radius as a function of the uncertainties applied to key observables. In each panel we have perturbed the quantity on the abscissa in isolation, centred around the measured value listed in Table~\\ref{tab:sunstar} and with the uncertainties in the ranges specified therein. We indicate the median predicted value (solid line) and the $68\\%$ confidence interval (shaded region). Here the observables comprise those expected from the PLATO space mission.} \n \\label{fig:uncertplato}\n\\end{figure*}\n\\fi\n\n%\\listofchanges\n%\\end{document}\n\n\n", "meta": {"hexsha": "845e9033da5830f718c17a5b11b3bbaaa5fb81ef", "size": 166086, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/ch3_stats/statistical_relations.tex", "max_stars_repo_name": "earlbellinger/Ph.D.-Thesis", "max_stars_repo_head_hexsha": "101c3f822b505c9e62beeb41ca30fe46c279ad79", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "tex/ch3_stats/statistical_relations.tex", "max_issues_repo_name": "earlbellinger/Ph.D.-Thesis", "max_issues_repo_head_hexsha": "101c3f822b505c9e62beeb41ca30fe46c279ad79", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/ch3_stats/statistical_relations.tex", "max_forks_repo_name": "earlbellinger/Ph.D.-Thesis", "max_forks_repo_head_hexsha": "101c3f822b505c9e62beeb41ca30fe46c279ad79", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 86.8650627615, "max_line_length": 1284, "alphanum_fraction": 0.7299531568, "num_tokens": 45787, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5234203638047913, "lm_q2_score": 0.6548947223065755, "lm_q1q2_score": 0.34278523380354553}} {"text": "\\title{Tendon Robot Statics Model Variants}\n\\author{John Till}\n\\date{}\n\n\\documentclass[12pt]{article}\n\n\\usepackage[a4paper, margin=0.75in]{geometry}\n\\usepackage[colorlinks=true,urlcolor=blue]{hyperref}\n\\usepackage{amsmath,amssymb}\n\\usepackage{graphicx}\n\n\\usepackage{xcolor}\n\\definecolor{OffWhite}{rgb}{0.93,0.93,0.93}\n\\definecolor{QtCommentColor}{rgb}{0,0.5,0}\n\\definecolor{QtKeywordColor}{rgb}{0.5,0.5,0}\n\\definecolor{QtPurpleColor}{rgb}{0.5,0,0.5}\n\\definecolor{QtGlobal}{rgb}{0.808,0.361,0}\n\\definecolor{QtFunctionColor}{rgb}{0,0.404,0.486}\n\n\\usepackage[T1]{fontenc} %for upquotes in listings\n\\usepackage{textcomp} %for upquotes in listings\n\\usepackage{listings}\n\\lstset{\n\t\tlanguage=C++,\n\t\tescapeinside={!-}{-!},\n\t\tupquote=true,\n\t\t%\n\t\totherkeywords={Vector3d, DiagonalMatrix, VectorXd, Matrix3d, Map, MatrixXd, Vector6d, Vector4d,\n\t\t Matrix6d, Eigen, Upper, std, fstream, Matrix3Xd,\n\t\t\t\t\t\t\t\t\tUnitX, pow, inverse, transposeMultiply, segment, data, UnitZ, cross, hat_squared,\n\t\t\t\t\t\t\t\t\that_postmultiply, hat_premultiply, normalized,\n\t\t\t\t\t\t\t\t\tZero, Identity, UnitY, cosseratRodOde, ode4, cols, row, main, shootingFunction,\n\t\t\t\t\t\t\t\t\tblock, rotation_error, solveLevenbergMarquardt, cos, sin, toDenseMatrix,\n\t\t\t\t\t\t\t\t\tselfadjointView, llt, close, kirchhoffTendonRobotOde, cwiseProduct, cwiseMin,\n\t\t\t\t\t\t\t\t\tcwiseMax, getRouting},\n morekeywords=[2]{Vector3d, DiagonalMatrix, VectorXd, Matrix3d, Map, MatrixXd, Vector6d, Vector4d,\n\t\t Matrix6d, Eigen, Upper, std, fstream, Matrix3Xd},\n\t\tmorekeywords=[3]{UnitX, UnitZ, pow, inverse, transposeMultiply, segment, data, cross, hat_squared,\n\t\t hat_postmultiply, hat_premultiply, normalized,\n\t\t Zero, Identity, UnitY, cosseratRodOde, ode4, cols, row, main, shootingFunction,\n\t\t\t\t\t\t\t\t\t\t block, rotation_error, solveLevenbergMarquardt, cos, sin, toDenseMatrix,\n\t\t\t\t\t\t\t\t\t\t selfadjointView, llt, close, kirchhoffTendonRobotOde, cwiseProduct, cwiseMin,\n\t\t\t\t\t\t\t\t\t\t cwiseMax, getRouting},\n %\n\t\tframe = single,\n\t\trulecolor=\\color{black},\n tabsize=4, % tab space width\n showstringspaces=false, % don't mark spaces in strings\n\t\t%\n\t\tbasicstyle=\\footnotesize,%\\color{QtIdentifier},\n\t\tbackgroundcolor=\\color{OffWhite},\n commentstyle=\\color{QtCommentColor}, % comment color\n keywordstyle=\\color{QtKeywordColor}, % keyword color\n\t\tkeywordstyle=[2]{\\color{QtPurpleColor}},\n\t\tkeywordstyle=[3]{\\color{QtFunctionColor}},\n stringstyle=\\color{QtCommentColor} % string color\n}\n\n\\begin{document}\n\n\\makeatletter\n\\renewcommand{\\@maketitle}{\n\\newpage\n\\null\n\\vskip 2em\n\\begin{center}\n{\\LARGE \\@title \\par}\n\\end{center}\n\\par\n} \\makeatother\n\n\\maketitle\n\nIn the last example we implemented a relatively simple tendon robot model where the backbone was modeled as a Cosserat rod and the tendons were routed in parallel with the backbone. There is a plethora of potential modifications to the model, and three potential variations are considered in different sections here.\n\n\\section{Kirchhoff Rod Backbone}\n\nThere are significant simplifications to the model if we neglect the shear and extension modes as described in \\href{https://ieeexplore.ieee.org/document/5957337}{``Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading''}. This results in $\\boldsymbol{v} = \\boldsymbol{e}_3$, which is the elastic rod problem as studied by Gustav Kirchhoff prior to the Cosserat brothers. The result for the tendon robot is that one only needs to solve a 3x3 linear system for $\\dot{\\boldsymbol{u}}$ instead of the 6x6 linear system considered previously for $\\dot{\\boldsymbol{v}}$ and $\\dot{\\boldsymbol{u}}$.\n\nThe state vector has $\\boldsymbol{n}^b$ instead of $\\boldsymbol{v}$, and the ODE function is changed accordingly:\n\\begin{lstlisting}\nvoid kirchhoffTendonRobotOde(VectorXd& y_s_out, VectorXd& y){\n //Unpack state vector\n Matrix3d R = Map(&y[3]);\n Vector3d nb = Map(&y[12]);\n Vector3d u = Map(&y[15]);\n\\end{lstlisting}\nOf course calculations involving shear or extension are modified so that $\\boldsymbol{v} = \\boldsymbol{e}_3$. The ODEs are also changed so that there is only a 3x3 linear system as in Equation (17):\n\\begin{lstlisting}\n //Pack state vector derivative\n Map p_s(&y_s_out[0]);\n Map R_s(&y_s_out[3]);\n Map nb_s(&y_s_out[12]);\n Map u_s(&y_s_out[15]);\n\n //ODEs\n p_s = R.col(2);\n R_s = hat_postmultiply(R,u);\n u_s = H.inverse()*(-u.cross(Kbt*u) - Vector3d::UnitZ().cross(nb) - b);\n nb_s = -u.cross(nb) - G*u_s - a - transposeMultiply(R,rho*area*g);\n}\n\\end{lstlisting}\nAn inverse is used instead of a linear solver because Eigen has specialized methods to take efficient inverses of small fixed-size matrices.\nWe also make some minor changes to the objective function since $\\boldsymbol{n}^b$ is a state variable instead of $\\boldsymbol{v}$:\n\\begin{lstlisting}\ny0 << p0, Map(Matrix3d(R0).data(), 9), guess;\n\n//Numerically integrate the Cosserat rod equations\nY = ode4(y0, L);\n\n//Find the internal forces in the backbone prior to the final plate\nVector3d nb = Y.block<3,1>(12,Y.cols()-1);\nVector3d uL = Y.block<3,1>(15,Y.cols()-1);\nVector3d mb = Kbt*uL;\n\\end{lstlisting}\nIf we rendered the solution, we would see no visible difference compared the Cosserat model.\n\n\\section{Non-parallel Tendon Routing}\n\nThe examples so far have assumed the tendons are routed parallel with the backbone. However, the model in\n\\href{https://ieeexplore.ieee.org/document/5957337}{``Statics and Dynamics of Continuum Robots With General Tendon Routing and External Loading''} is formulated for any general routing path, and helical routing strategies are demonstrated. The Cosserat backbone model is altered to have a helical routing strategy here.\n\nThere are a multitude of options for how to represent the tendon routing functions. Lambda functions might be the most straightforward way, and if we were willing to implement our own numerical integration function, the most efficient way would be to precompute all the values at grid points. The approach here is somewhere in between; we'll calculate all the vectors at a given point on the backbone:\n\\begin{lstlisting}\n//Helical routing\nconst double offset = 0.01506;\nvoid getRouting(double arclength, Vector3d* r, Vector3d* r_s, Vector3d* r_ss){\n const double f = 2*pi/L; //tendons make a full revolution\n\n double c = offset*cos(f*arclength);\n double s = offset*sin(f*arclength);\n double fc = f*c;\n double fs = f*s;\n double ffc = f*fc;\n double ffs = f*fs;\n\n r[0] = Vector3d(c, s, 0);\n r[1] = Vector3d(-s, c, 0);\n r[2] = Vector3d(-c, -s, 0);\n r[3] = Vector3d(s, -c, 0);\n\n r_s[0] = Vector3d(-fs, fc, 0);\n r_s[1] = Vector3d(-fc, -fs, 0);\n r_s[2] = Vector3d(fs, -fc, 0);\n r_s[3] = Vector3d(fc, fs, 0);\n\n r_ss[0] = Vector3d(-ffc, -ffs, 0);\n r_ss[1] = Vector3d(ffs, -ffc, 0);\n r_ss[2] = Vector3d(ffc, ffs, 0);\n r_ss[3] = Vector3d(-ffs, ffc, 0);\n}\n\\end{lstlisting}\nThe tendons are still separated by $90^\\circ$, but they are routed in helices around the backbone with a frequency ``f''. There are three arrays passed in having four vectors each. ``r'', ``r\\_s'', and ``r\\_ss'' are set based on the arc length. The ODE includes the derivatives of $\\boldsymbol{r}$:\n\\begin{lstlisting}\nVector3d r[num_tendons], r_s[num_tendons], r_ss[num_tendons];\ngetRouting(s, r, r_s, r_ss);\n\nfor(int i = 0; i < num_tendons; i++){\n Vector3d ri = r[i];\n Vector3d ri_s = r_s[i];\n Vector3d ri_ss = r_ss[i];\n\n Vector3d pb_si = u.cross(ri) + ri_s + v;\n double pb_s_norm = pb_si.!-\\textcolor{QtFunctionColor}{norm}-!();\n Matrix3d A_i = -hat_squared(pb_si)*(tau(i)/pow(pb_s_norm,3));\n Matrix3d G_i = -hat_postmultiply(A_i,ri);\n Vector3d a_i = A_i*(u.cross(pb_si + ri_s) + ri_ss);\n\\end{lstlisting}\nThe tendon routing vectors are calculated for the current arclength ``s'' before setting up the linear system. Over in the objective function, we also need to use the tendon routing vectors at the tip when finding the equilibrium error:\n\\begin{lstlisting}\n//Find the equilibrium error at the tip, considering tendon forces\nVector3d force_error = -nb;\nVector3d moment_error = -mb;\nVector3d r[num_tendons], r_s[num_tendons], r_ss[num_tendons];\ngetRouting(L, r, r_s, r_ss);\nfor(int i = 0; i < num_tendons; i++){\n Vector3d pb_si = uL.cross(r[i]) + r_s[i] + vL;\n Vector3d Fb_i = -tau(i)*pb_si.normalized();\n force_error += Fb_i;\n moment_error += r[i].cross(Fb_i);\n}\n\\end{lstlisting}\nThen we can solve the problem and visualize the effects of helical routing:\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.8\\textwidth]{fig/Helical.jpg}\n\\end{figure}\n\nWe could also implement a converging tendon strategy like in the paper \\href{https://ieeexplore.ieee.org/document/8606257}{``Continuum Robot Stiffness Under External Loads and Prescribed Tendon Displacements''}. We revise the tendon function:\n\\begin{lstlisting}\n//Converging routing\nconst double offset = 0.01506;\nconst double routing_slope = -offset/L;\nvoid getRouting(double s, Vector3d* r, Vector3d* r_s, Vector3d* r_ss){\n double b = (1 - s/L)*offset;\n\n r[0] = Vector3d(b, 0, 0);\n r[1] = Vector3d(0, b, 0);\n r[2] = Vector3d(-b, 0, 0);\n r[3] = Vector3d(0, -b, 0);\n\n r_s[0] = Vector3d(routing_slope, 0, 0);\n r_s[1] = Vector3d(0, routing_slope, 0);\n r_s[2] = Vector3d(-routing_slope, 0, 0);\n r_s[3] = Vector3d(0, -routing_slope, 0);\n\n r_ss[0] = r_ss[1] = r_ss[2] = r_ss[3] = Vector3d(0, 0, 0);\n}\n\\end{lstlisting}\nThen solving with converging routing yields the following solution:\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.8\\textwidth]{fig/Converging.jpg}\n\\end{figure}\n\n\\section{Tendon Displacement Control}\n\nIn the examples so far, the forward kinematics input has been the tendon tensions. However, most actuator control position instead of force. This scenario is also considered in the paper \\href{https://ieeexplore.ieee.org/document/8606257}{``Continuum Robot Stiffness Under External Loads and Prescribed Tendon Displacements''}. There are additional unknowns since the tendon tensions are not given, and additional constraints since the backbone must take a shape so that the tendon lengths are correct. For example, increasing the distance $q_i$ the tendon extends below the baseplate will cause the backbone to bend so that the arc length of the tendon above the baseplate is reduced, and the total tendon length remains constant (assuming an inextensible tendon).\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.5\\textwidth]{fig/DisplacementControl.jpg}\n\\end{figure}\n\nWe declare additional independent parameters for the displacement control:\n\\begin{lstlisting}\nconst VectorXd q = Vector4d(5e-3, 0, -5e-3, 0);\nconst VectorXd l_star = Vector4d(L,L,L,L);\n\\end{lstlisting}\nThe length a tendon extends behind the baseplate is $q_i$, and the total length of a tendon before stretching is $l_i^*$. We're careful when setting the actuator displacements because it is especially easy to have an \\emph{ill-posed} problem. For instance, pulling all the tendons back by half the length of the backbone is simply not realistic, and the solver would probably fail to find a solution.\nThe tendon tensions are now dependent variables, so they are declared as static:\n\\begin{lstlisting}\nstatic VectorXd tau;\n\\end{lstlisting}\nThese tensions will be guessed, so we change the guess initialization in the main function:\n\\begin{lstlisting}\nVectorXd init_guess = VectorXd::Zero(6 + num_tendons); //nb, u, and tau\n\\end{lstlisting}\nand we unpack the last elements of the guess as the tendon tensions:\n\\begin{lstlisting}\ntau = guess.segment(6);\n\\end{lstlisting}\nBased on how the backbone deforms, each tendon will have some arc length it is routed along the backbone, with errors resulting from incorrect lengths. Calculating the errors requires modifying the ODE function to also integrate the tendon arc lengths. We extend the state vector so that the last entries are the tendon arc lengths: \n\\begin{lstlisting}\nMap pb_s_norm(&y_s_out[18], num_tendons);\n\\end{lstlisting}\nNow the change in tendon arc length with respect to the change in reference arc length is stored as part of the state vector derivative:\n\\begin{lstlisting}\npb_s_norm(i) = pb_si.!-\\textcolor{QtFunctionColor}{norm}-!();\nMatrix3d A_i = -hat_squared(pb_si)*(tau(i)/pow(pb_s_norm(i),3));\n\\end{lstlisting}\nWe modify the initial conditions to include the arc lengths of the tendons behind the base plate:\n\\begin{lstlisting}\nVectorXd y0(18+num_tendons);\ny0 << p0, Map(Matrix3d(R0).data(),9), v0, u0, q;\n\\end{lstlisting}\nThen we have some error in the integrated lengths of the tendons versus the known lengths specified as design parameters:\n\\begin{lstlisting}\n//Find the length violation error\nVectorXd integrated_lengths = Y.block(18,Y.cols()-1);\nVectorXd length_error = integrated_lengths - l_star;\n\nVectorXd distal_error(6 + num_tendons);\ndistal_error << force_error, moment_error, length_error;\n\\end{lstlisting}\n\nNow if we run the program, the robot is solved using tendon displacement as the forward kinematics input. However, our approach currently assumes the tendons are inextensible and capable of supporting compression. We address the former first by adding compliance as an independent variable:\n\\begin{lstlisting}\nconst VectorXd C = Vector4d(1e-4,1e-4,1e-4,1e-4);\n\\end{lstlisting}\nThen the length error is revised to include the tendon stretch, which is calculated from Hooke's law:\n\\begin{lstlisting}\nVectorXd stretch = l_star.cwiseProduct( C.cwiseProduct(tau) );\nVectorXd length_error = integrated_lengths - (l_star + stretch);\n\\end{lstlisting}\nNow the tendons will stretch under tension. Unfortunately, they will also contract under compression. \nTo disallow compression, we include the effects of slack. Since slack and tension are mutually exclusive effects, we use a single variable to represent them both when we unpack the guess:\n\\begin{lstlisting}\ntau = guess.segment(6).cwiseMax(0);\nVectorXd slack = -(guess.segment(6).cwiseMin(0));\n\\end{lstlisting}\nDepending on the sign, we have either tension or slack. Hopefully the shooting method optimization space is smooth around the transition; it seems to work well. Then the length error is revised to include slack:\n\\begin{lstlisting}\nVectorXd length_error = integrated_lengths + slack - (l_star + stretch);\n\\end{lstlisting}\nNow we have a model with displacement controlled tendons which stretch under tension and go slack when necessary.\n\n\\end{document}", "meta": {"hexsha": "2ada42bb8a1adb928a3d620df736d0e694c4e618", "size": 14628, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "01_Statics/06_Tendon_Robot_Variants/LaTeX/TendonRobotVariants.tex", "max_stars_repo_name": "matthiastummers/ContinuumRobotExamples", "max_stars_repo_head_hexsha": "2575fa71075c8867f7be78c112695de2f9b6c403", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 73, "max_stars_repo_stars_event_min_datetime": "2019-03-21T04:59:34.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T18:44:50.000Z", "max_issues_repo_path": "01_Statics/06_Tendon_Robot_Variants/LaTeX/TendonRobotVariants.tex", "max_issues_repo_name": "matthiastummers/ContinuumRobotExamples", "max_issues_repo_head_hexsha": "2575fa71075c8867f7be78c112695de2f9b6c403", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2019-04-23T18:22:39.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-29T07:05:29.000Z", "max_forks_repo_path": "01_Statics/06_Tendon_Robot_Variants/LaTeX/TendonRobotVariants.tex", "max_forks_repo_name": "matthiastummers/ContinuumRobotExamples", "max_forks_repo_head_hexsha": "2575fa71075c8867f7be78c112695de2f9b6c403", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 28, "max_forks_repo_forks_event_min_datetime": "2019-04-02T04:01:39.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-31T06:59:02.000Z", "avg_line_length": 50.7916666667, "max_line_length": 765, "alphanum_fraction": 0.7407027618, "num_tokens": 4166, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3427852275917898}} {"text": "\\documentclass[p.tex]{subfiles}\n\\begin{document}\n\\section{Background}\\label{sec:background}\n\n\\subsection{Target Language}\\label{sec:language}\n\n\\begin{figure}\n\\begin{spreadlines}{5pt}\n\\begin{gather*}\n \\inference[\\textsc{E-Assert}]{%\n \\eeval{b}{\\sigma}{\\btrue}\n }{%\n \\ceval{\\cassert{b}}{\\sigma}{\\sigma}\n }\n \\qquad\n \\inference[\\textsc{E-SumLeft}]{%\n \\ceval{c_0}{\\sigma_0}{\\sigma_1}\n }{%\n \\ceval{\\csum{c_0}{c_1}}{\\sigma_0}{\\sigma_1}\n }\n \\qquad\n \\inference[\\textsc{E-SumRight}]{%\n \\ceval{c_1}{\\sigma_0}{\\sigma_1}\n }{%\n \\ceval{\\csum{c_0}{c_1}}{\\sigma_0}{\\sigma_1}\n }\n \\\\\n \\inference[\\textsc{E-Assign}]{%\n }{%\n \\ceval{\\cassign{x}{a}}{\\sigma}{\\subst{x}{a}{(\\sigma)}}\n }\n \\qquad\n \\inference[\\textsc{E-Sequence}]{%\n \\ceval{c_0}{\\sigma_0}{\\sigma_1}&\n \\ceval{c_1}{\\sigma_1}{\\sigma_2}&\n }{%\n \\ceval{\\cseq{c_0}{c_1}}{\\sigma_0}{\\sigma_2}\n }\n \\\\\n \\inference[\\textsc{E-Skip}]{%\n }{%\n \\ceval{\\cskip}{\\sigma}{\\sigma}\n }\n \\qquad\n \\inference[\\textsc{E-Loop}]{%\n \\ceval{c}{\\sigma_0}{\\sigma_1}&\n \\ceval{\\cloop{c}}{\\sigma_1}{\\sigma_2}\n }{%\n \\ceval{\\cloop{c}}{\\sigma_0}{\\sigma_2}\n }\n \\qquad\n \\inference[\\textsc{E-Break}]{%\n }{%\n \\ceval{\\cloop{c}}{\\sigma}{\\sigma}\n }\n\\end{gather*}\n\\end{spreadlines}\n \\caption{%\n The formal semantics of the target language $\\lang$, defined in\n terms of the constituent imperative commands in \\com.\n Metavariables $\\sigma, \\sigma_0, \\ldots$ denote program states in\n \\state, metavariable $x$ denotes a variable in \\var, metavariables\n $c, c_0, \\ldots$ denote commands in \\com, $a$ denotes an\n integer-valued expression in $\\aexpr$, and $b$ denotes a\n boolean-valued expression in $\\bexpr$.\n }\\label{fig:semantics}\n\\end{figure}\n\nIn this section we present the imperative language, $\\lang$, that we will extend\nand analyze for the remainder of this paper.\n$\\lang$ includes\nassignments, branches, loops, and assertions.\n%\n$\\lang$ makes use of expressions with boolean and integer\nvalue, denoted $\\bexpr$ and $\\aexpr$ respectively.\n%\nWe denote the space of program variables as \\var.\n%\nEach program in $\\lang$ is an imperative command.\nWe define the space of imperative commands inductively:\n\\begin{align*}\n \\com ::=&~\\cskip\\\\\n |&~\\cassign{\\var}{\\aexpr}\\\\\n |&~\\cassert{\\bexpr}\\\\\n |&~\\cseq{\\com}{\\com}\\\\\n |&~\\csum{\\com}{\\com}\\\\\n |&~\\cloop{\\com}\n\\end{align*}\n\nThe command $\\cseq{c_0}{c_1}$ is the command which first executes\n$c_0$ and then executes $c_1$ and the command $\\cskip$ is a no-op. The\ncommand $\\cassert{b}$ is a static claim that the boolean expression\n$b$ evaluates to true in every possible run of the command. We define\nthe formal semantics of the space of commands in \\cref{fig:semantics}.\n%\nThe program semantics are defined with reference to members of the\nspace of program states, which we denote $\\state$. A state\nis a map from a variables to integer values:\n%\n\\[\\state ::= \\var \\rightarrow \\mathbb{Z}\\]\n\n\\subsection{Hoare Logic}\n\\begin{figure}\n\\begin{spreadlines}{5pt}\n\\begin{gather*}\n \\inference[\\textsc{Skip}]{%\n }{%\n \\hoare{P}{\\cskip}{P}\n }\n \\qquad\n \\inference[\\textsc{Assign}]{%\n }{%\n \\hoare{\\subst{x}{a}{P}}{\\cassign{x}{a}}{P}\n }\n \\qquad\n \\inference[\\textsc{Assert}]{%\n }{%\n \\hoare{P}{\\cassert{b}}{P \\land b}\n }\n \\\\\n \\inference[\\textsc{Seq}]{%\n \\hoare{P}{c_0}{R}&\n \\hoare{R}{c_1}{Q}\n }{%\n \\hoare{P}{\\cseq{c_0}{c_1}}{Q}\n }\n \\qquad\n \\inference[\\textsc{Sum}]{%\n \\hoare{P}{c_0}{Q}&\n \\hoare{P}{c_1}{Q}\n }{%\n \\hoare{P}{\\csum{c_0}{c_1}}{Q}\n }\n \\\\\n \\inference[\\textsc{Loop}]{%\n \\hoare{P}{c}{P}\n }{%\n \\hoare{P}{\\cloop{c}}{P}\n }\n \\qquad\n \\inference[\\textsc{Cons}]{%\n P \\Rightarrow P'&\n \\hoare{P'}{c}{Q'}&\n Q' \\Rightarrow Q\n }{%\n \\hoare{P}{c}{Q}\n }\n \\end{gather*}\n\\end{spreadlines}\n\\caption{%\n Hoare Logic judgements over $\\lang$.\n Metavariables $P, Q, R\\ldots$ represent program\n assertions. Other metavariables are consistent with\n \\cref{fig:semantics}\n}\\label{fig:base-proof-system}\n\\end{figure}\n\nHoare Logic is a framework for reasoning about imperative commands by\nrelating them to program \\emph{assertions}.\nAn assertion is a proposition over a program state. Formally, we\ndefine the space of assertions as functions from program states to\npropositions.\n\\[ \\textsf{Assert} ::= \\state \\rightarrow \\textsf{Prop} \\]\nTypically, a Hoare logic system provides a set of judgements over\n\\emph{Hoare Triples}.\nIf $P, Q \\in \\textsf{Assert}$ are assertions and $c \\in \\com$ is a\ncommand, then \\hoare{P}{c}{Q} is a Hoare Triple.\n%\nInformally, this triple can be interpreted as stating \\textbf{(1)} if\nthe assertion $P$ holds over some program state $\\sigma$ and\n\\textbf{(2)} the command $c$ takes $\\sigma$ to $\\sigma'$, then\n\\textbf{(3)} the assertion $Q$ must hold over $\\sigma'$.\n%\nThis definition can be formalized as follows:\n\\[\n \\hoare{P}{c}{Q} ::= \\forall \\sigma,\n \\sigma'.~\\ceval{c_0}{\\sigma}{\\sigma'} \\Rightarrow P~\\sigma \\Rightarrow Q~\\sigma'\n\\]\nBy establishing a set of sound rules, a Hoare logic system provides a\ncomposable framework for structuring program proofs.\nWe provide a set of typical Hoare logic rules for $\\lang$ in\n\\Cref{fig:base-proof-system}.\n\n\n\n\\end{document}\n", "meta": {"hexsha": "8c1f6e4a2b2ec9114524b5f530cb5a5be536328b", "size": 5215, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/background.tex", "max_stars_repo_name": "DAHeath/self-relational", "max_stars_repo_head_hexsha": "f230a31945e35119c7bf571d40e5bfc991e89ce1", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper/background.tex", "max_issues_repo_name": "DAHeath/self-relational", "max_issues_repo_head_hexsha": "f230a31945e35119c7bf571d40e5bfc991e89ce1", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/background.tex", "max_forks_repo_name": "DAHeath/self-relational", "max_forks_repo_head_hexsha": "f230a31945e35119c7bf571d40e5bfc991e89ce1", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 27.8877005348, "max_line_length": 82, "alphanum_fraction": 0.6504314477, "num_tokens": 1846, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6548947290421275, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.3427852275917898}} {"text": "\\chapter{\\label{ch1-intro}Introduction} \n\n\\minitoc\n\n\\section{Cherenkov Radiation}\n\n\\begin{figure}\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\textwidth]{dipole_slow}\n \\caption{$v < \\frac{c}{n}$}\n \\label{fig:dipole_slow}\n \\end{subfigure}\n \\hfill\n \\begin{subfigure}[b]{0.49\\textwidth}\n \\includegraphics[width=\\textwidth]{dipole_fast}\n \\caption{$v \\ge \\frac{c}{n}$}\n \\label{fig:dipole_fast}\n \\end{subfigure}\n \\caption[Polarisation produced in a dielectric medium due to the presence of a charged particle.]{Polarisation produced in a dielectric medium due to the presence of a charged particle, for the cases of a non-relativistic (a) and relativistic particle (b). Where $v$ is the speed of the particle and $c/n$ is the speed light travels in a medium with refractive index $n$.}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=0.5\\textwidth]{cherenkov_geom} \n\t\\caption[Geometry of the wavefronts involved in Cherenkov radiation production.]{Geometry of the wavefronts involved in Cherenkov radiation production. The particle travels at a greater speed than the wavefronts propagate.}\n\t\\label{fig:cherenkov_geom}\n\\end{figure}\n\nWhen a charged particle moves slowly through a dielectric medium, the electric field of the particle distorts the nearby atoms. Momentarily, these atoms are transformed into elementary dipoles where the charged particles that constitute the atom are aligned with respect to the electric field of the travelling particle (Figure~\\ref{fig:dipole_slow}). Due to the complete symmetry of this polarisation around the travelling particle, no net field is produced by the dielectric medium. However, if instead the velocity of the charged particle is faster than the speed light travels in that medium, an asymmetry along the particle trajectory is formed in the polarisation of the surrounding atoms (Figure~\\ref{fig:dipole_fast}), resulting in a net dipole field. As the particle continues through the medium, elements of the polarised medium will release a brief burst of electromagnetic radiation. Generally these electromagnetic waves interfere destructively, except in the forward direction along the particle's trajectory in an opening angle $\\theta$. Although the full characterisation of this relativistic effect is complex, a simple consideration of the geometry involved, shown in Figure~\\ref{fig:cherenkov_geom}, can be used to describe $\\theta$ \\cite{Jelley1958a}. In a time $\\Delta t$ a particle travels a distance $\\beta c \\Delta t$ where $\\beta = \\frac{v}{c}$, while the emitted light will travel a distance $\\frac{c}{n} \\Delta t$ in a medium with refractive index $n$. This results in the relation:\n\\begin{equation} \\label{eq:cherenkov_angle}\n\\cos \\theta = \\frac{c}{vn}.\n\\end{equation}\nThe blue light emitted in this constrained opening angle, via this phenomena, is known as Cherenkov radiation.\n\n\\section{Atmospheric Cherenkov Showers} \\label{section:cherenkov_shower_intro}\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\textwidth]{cascade} \n\t\\caption[Production of an extended electromagnetic particle cascade.]{Production of a extended electromagnetic particle cascade, demonstrating the different components and interactions.}\n\t\\label{fig:cascade}\n\\end{figure}\n\nThe Earth's atmosphere is effectively opaque to photons with energy above \\SI{10}{eV} \\cite{Weekes2003}. To conduct astronomical observations at higher energies, one must usually leave the Earth's atmosphere, as was done by the Fermi Gamma-ray Space Telescope. However, at energies above \\SI{\\ge 10}{GeV}, a ``gamma-ray window'' in the atmosphere exists where the pursuit of gamma-ray observations can be performed using the Cherenkov radiation produced by the cascade of particles resulting from the interaction between the gamma ray and the atmosphere.\n\nTwo electromagnetic interactions are responsible for the creation of this cascade:\n\\begin{description}\n\\item [Pair Production] The conversion of a photon into an electron-positron pair in the presence of an atom (such as an atmospheric particle). The energy of the photon must exceed the sum of the rest masses of an electron and positron (\\SI{1.022}{MeV}). The electron-positron pair share the energy of the progenitor photon, and continue on a similar trajectory. This is the dominating interaction process for photons above \\SI{\\ge 10}{MeV} \\cite{Weekes2003}.\n\\item [Bremsstrahlung Radiation] The emission of a photon due to the interaction of a charged particle with the electric field of an atom (such as an atmospheric particle). This process allows further gamma rays to be produced.\n\\end{description}\nThe interplay between these two processes, occurring after each traversal of a radiation length, produces an extensive cascade of energetic electromagnetic particles. This is illustrated in Figure~\\ref{fig:cascade}. The charged particles produced in this cascade are responsible for the generation of the Cherenkov light. This cascade is often known as a ``Cherenkov shower''.\n\nThis cascade continues until the ionisation energy losses are equal to the radiation losses. The number of remaining particles after this point, known as the ``shower maximum'', begins to diminish. For a \\SI{1}{TeV} shower, this occurs at \\SI{\\sim 8.4}{km} altitude~\\cite{Weekes2003}. The produced Cherenkov light is collimated along the progenitor gamma ray trajectory, and produces a pool of blue light on the ground, with a radius of \\SI{\\sim 120}{m}~\\cite{Hillas1996a}. If the direction of the Cherenkov shower is extrapolated back to the cosmic sphere, the location of the source that produced the gamma ray can be inferred. Although the amount of energy that goes into Cherenkov photon production is a tiny fraction of the total energy, the atmosphere acts as a consistent calorimeter, therefore allowing an accurate reconstruction of the progenitor's energy from the amount of Cherenkov photons produced. \n\nA further characteristic of the Cherenkov shower is the time profile. The total Cherenkov light observed from the entire shower has a duration of \\SI{\\sim 5}{ns}. Therefore, despite the abundance of showers in the sky, and the visible wavelength of the Cherenkov light, they are imperceivable by the human eye. Furthermore, due to the faster-than-light velocities of the particles inside the cascade, the last Cherenkov photons produced at the end of the shower reach the ground before the first Cherenkov photons produced at the start of the shower. With different sections of the showers arriving at different times, the Cherenkov shower measurements display a time gradient across the image. \n\n\\section{Imaging Atmospheric Cherenkov Telescopes}\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\textwidth]{nsb} \n\t\\caption[Comparison of Cherenkov and NSB spectrum.]{Comparison of Cherenkov and NSB spectrum. The Cherenkov spectrum shown is expected at an altitude of \\SI{2200}{m}. The NSB spectrum shown was measured at La Palma \\cite{Bouvier2013}.}\n\t\\label{fig:nsb}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[b]{0.35\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{hess}\n\t\t\\caption{H.E.S.S.}\n\t\t\\label{fig:hess}\n\t\\end{subfigure}\n\t~\n\t\\begin{subfigure}[b]{0.35\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{magic}\n\t\t\\caption{MAGIC}\n\t\t\\label{fig:magic}\n\t\\end{subfigure}\n\t~\n\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{veritas}\n\t\t\\caption{VERITAS}\n\t\t\\label{fig:veritas}\n\t\\end{subfigure}\n\t\\caption{Images of modern IACTs.}\n\t\\label{fig:iacts}\n\\end{figure}\n\nA primary issue in \\gls{vhe} astronomy is the low flux (${\\sim} 0.2$ per \\si{m \\squared} per year \\cite{Franco2016}), requiring a collection area that is not feasible for space telescopes. If instead the Cherenkov showers are used to detect the gamma rays, large arrays of optical telescopes can be built to provide stereoscopic imaging of the Cherenkov showers. These telescopes are known as \\glspl{iact}. The multiple stereoscopic views of individual showers provided by arrays of \\glspl{iact} allow accurate reconstruction of the properties of the shower, such as direction and energy. The topic of reconstruction is discussed in Chapter~\\ref{ch6-reduction}.\n\nAs the \\gls{iact} technique involves imaging the Cherenkov showers, which are much larger than typical astronomy targets, \\glspl{iact} do not require the resolving power of typical optical telescopes. Instead, the priorities of an \\gls{iact} optical system are to maximise: \n\\begin{itemize}\n\\item Mirror collection area, such that more photons can be collected. This enables fainter showers to be detected, thereby lowering the energy threshold.\n\\item \\gls{fov}, which improves the surveying capabilities and eases the study of extended sources.\n\\end{itemize}\nFurthermore, the large collection area provided by the light pool of the Cherenkov shower enables a modest telescope to still make a large amount of gamma ray detections, enabling this technique to be viable despite the small flux.\n\nTwo major background components need to be accounted for in \\glspl{iact}:\n\\begin{description}\n\\item [Cosmic Ray Background] Protons (and heavier hadronic nuclei) are also capable of producing Cherenkov showers that are not entirely dissimilar to electromagnetic showers. As these particles are charged, they have been deflected by interstellar magnetic fields on their journey from their source, and therefore cannot be used to reconstruct the location of its source on the sky. These showers provide an isotropic background whose rate is $10^3$~-~$10^4$ times as numerous than the shower rate received from the discreet gamma-ray sources. However, a hadronic shower exhibits a morphology that is broader and less symmetric than that obtained from gamma-ray showers. Additionally, distinct features such as ``muon rings'', produced by highly penetrating muons reaching low altitudes such that the full Cherenkov cone is visible in a single telescope, accompany hadronic showers. Parametrisations of the Cherenkov shower image therefore enable the discrimination between the hadronic and gamma-induced showers (see Chapter~\\ref{ch6-reduction}).\n\\item [Night Sky Background] Due to the optical sensitivity of the cameras used by \\glspl{iact}, the measurements taken are susceptible to starlight, moonlight, and artificial light pollution. To minimise this interference, \\glspl{iact} are built in locations with minimal light pollution. One such location is La Palma, for which the \\gls{nsb} spectrum, compared to the expected Cherenkov spectrum at an altitude of \\SI{2200}{m}, is displayed in Figure~\\ref{fig:nsb}. This background is also excluded from signal readout in three ways. Firstly, smart trigger logic and strict thresholds (such as the one described in Chapter~\\ref{ch2-mechanics}) prevent triggering on \\gls{nsb} photons. Secondly, unbiased charge extraction techniques (described in Chapter~\\ref{ch6-reduction}) exclude this noise from the signal.\n\\end{description}\n\nThe application of the \\gls{iact} technique was first attempted in the 1960s. The first large optical reflector built with the purpose of gamma-ray astronomy was the Whipple 10 m telescope in southern Arizona, 1968. At first, gamma-ray astronomy suffered from a number of unsubstantial claims of transient signals from a variety of pulsars and binaries, but these signals had marginal statistical significance \\cite[][p.~9]{Weekes2003}. It wasn't until 20 years later, after further refinement of the technique, that the Crab Nebula was detected by Whipple in 1989, thus reigniting interest in gamma-ray astronomy.\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\textwidth]{sensitivity} \n\t\\caption[Differential sensitivity of CTA.]{Differential sensitivity of CTA predicted by Monte Carlo simulations, compared to the performance of other gamma-ray instruments. Obtained from the CTA website \\cite{cta-performance}. ``LAT'' refers to the Fermi Large Area Telescope, a gamma-ray space telescope. HAWC is the High-Altitude Water Cherenkov array. The differential sensitivity has been defined as the minimum flux needed by CTA to obtain a 5-standard-deviation detection of a point-like source.}\n\t\\label{fig:sensitivity}\n\\end{figure}\n\nModern \\glspl{iact} include \\gls{magic}, \\gls{veritas}, and the most recent, \\gls{hess} (Figure~\\ref{fig:iacts}). All three of these telescope systems operate with the advantage of stereoscopic collaboration. \n\n\\section{The Cherenkov Telescope Array}\n\nIn order to improve on the current \\glspl{iact}, an array of ${\\sim} 100$ telescopes was proposed, called the \\gls{cta}. This array will have \\cite{Acharya2013}:\n\\begin{itemize}\n\\item an improved sensitivity of 10 times over previous \\glspl{iact} ,\n\\item an observable gamma-ray energy range of \\SI{20}{GeV} to \\SI{300}{TeV},\n\\item a large (\\SI{\\sim 8}{\\degree}) field of view for surveys,\n\\item improved angular and energy resolution,\n\\item and will be the first \\gls{iact} to operate as an open observatory.\n\\end{itemize}\n\nFigure~\\ref{fig:sensitivity} highlights the improvements in sensitivity \\gls{cta} aims to provide, relative to existing gamma-ray instruments. Non-\\gls{iact} instruments are included in the comparison, such as the Fermi LAT (Large Area Telescope) and \\gls{hawc}. Fermi LAT is the principle instrument on the Fermi Gamma-Ray Satellite, providing direct detection of gamma rays in the energy range of \\SIrange{0.02}{300}{GeV} \\cite{Atwood2009}. \\gls{hawc} is a facility which utilises the water Cherenkov technique, an alternative to \\glspl{iact}, to detect the particle showers produced by gamma rays within the energy range of \\SIrange{0.1}{100}{TeV} \\cite{Abeysekara2018}.\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\textwidth]{sensitivity_tel} \n\t\\caption[Differential sensitivity of the different CTA telescope types.]{Contribution of each telescope type within \\gls{cta} to the total differential sensitivity, obtained from \\textcite{Marano2014}.}\n\t\\label{fig:sensitivity_tel}\n\\end{figure}\n\n\\gls{cta} will consist of three different sized telescopes:\n\\begin{itemize}\n\\item The \\gls{lst}, with a mirror diameter of about \\SI{23}{m} to enable the collection of as many photons as possible from the low energy showers (\\SIrange{20}{150}{GeV}). Only a few \\glspl{lst} are needed, as these low-energy showers are relatively frequent.\n\\item The \\gls{mst}, covering the mid-range energies between \\SIrange{0.1}{10}{TeV}, with mirror diameters of \\SI{12}{m}. \n\\item The \\gls{sst}, measuring the high energies of \\SIrange{1}{300}{TeV}, with mirror diameters of around \\SI{4}{m}. Only a small mirror area is necessary as the showers at these energies are very bright. However, due to the rarity of higher energy showers, many \\glspl{sst} need to be spread over an area of several square kilometres, to increase the chance of a detection \\cite{Acharya2013}.\n\\end{itemize}\nThe contributions of each telescope type to the sensitivity of \\gls{cta} is shown in Figure~\\ref{fig:sensitivity_tel}.\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\textwidth]{cta_south} \n\t\\caption[The southern-hemisphere Cherenkov Telescope Array.]{Computer-rendered graphic of the southern hemisphere site for CTA, obtained from the CTA website \\cite{cta-south}.}\n\t\\label{fig:cta_south}\n\\end{figure}\n\n\\gls{cta} will be hosted at two different sites. A northern hemisphere site for extragalactic observations will be built at La Palma, and is planned to contain 4 \\glspl{lst} and 16 \\glspl{mst}. As this site will focus on the energy range from \\SI{20}{GeV} to \\SI{20}{TeV}, no \\glspl{sst} are included on the northern site. A southern hemisphere site will provide observations of the galactic plane, spanning the full energy range of \\gls{cta}. Planned to be built nearby the Paranal Observatory in the Atacama Desert in Chile, the southern array is intended to feature 4 \\glspl{lst}, 15 \\glspl{mst}, and 70 \\glspl{sst}, spread over \\SI{4}{km \\squared}. A visualisation of the \\gls{cta} southern array is shown in Figure~\\ref{fig:cta_south}.\n\n\\section{Small-Sized Telescopes}\n\n\\begin{figure}\n\t\\centering\\includegraphics[width=\\textwidth]{ssts} \n\t\\caption[The three SST designs.]{Computer-rendered graphics of the three SST designs: GCT, ASTRI and SST-1M. Obtained from the CTA website \\cite{cta-sst}.}\n\t\\label{fig:ssts}\n\\end{figure}\n\nThree designs for an \\gls{sst} have been proposed:\n\\begin{itemize}\n\\item The SST-1M design, a single-mirror Davies-Cotton telescope developed in collaboration between the Czech Republic, Ireland, Poland, Switzerland and Ukraine \\cite{cta-sst}. The prototype structure was installed at the Institute of Nuclear Physics in Kraków, Poland in November 2013.\n\\item The \\gls{astri} design features a dual-mirror Schwarzschild-Couder telescope structure. \\gls{astri} is predominantly developed by Italy, however contributions were provided from Brazil and South Africa \\cite{cta-sst}. The \\gls{astri} prototype completed construction on Mt. Etna, Italy in 2014.\n\\item The \\gls{gct} design also features a dual-mirror Schwarzschild-Couder telescope structure. \\gls{gct} is being developed through collaboration between Australia, France, Germany, Japan, the Netherlands and the United Kingdom \\cite{cta-sst}. The prototype telescope structure was inaugurated at the Observatoire de Paris-Meudon, France in November 2015. \\gls{gct} is the telescope which I have been associated with during my DPhil study. The focus of this thesis is on the cameras designed for the \\gls{gct}.\n\\end{itemize}\nPictures of all three telescopes are shown in Figure~\\ref{fig:ssts}.\n\nThe Schwarzschild-Couder optical design was first proposed by German astrophysicist Karl Schwarzschild to eliminate optical aberrations across the \\gls{fov} \\cite{Scharzschild1905}. This optical design has since gone through many iterations, however it was never utilised for a reflector telescope due to the complexity and cost required to construct the mirrors \\cite{Giro2017}. However, interest in this optical design was recently reawaken by \\textcite{Vassiliev2007}, especially as it enables the utilisation of novel compact photosensors. Due to the adoption of similar Schwarzschild-Couder optics, the telescopes of \\gls{astri} and \\gls{gct} have been specifically designed to accommodate the cameras from both dual-mirror \\glspl{sst}. This increases the possibilities for the final \\gls{sst} design for \\gls{cta}.\n\n\\section{Science with the SSTs}\n\nAs shown in Figures~\\ref{fig:sensitivity}~and~\\ref{fig:sensitivity_tel}, the \\glspl{sst} are responsible for exploring beyond the current energy frontier in gamma-ray astronomy. Within 50 hours of observation time, the \\glspl{sst} will be able provide the same sensitivity as five years of observations with the \\gls{hawc} observatory \\cite{Consortium2018}. This enables \\gls{cta} to provide insights into the most energetic processes in the universe, and address prevalent topics of debate in \\gls{vhe} astronomy and particle physics.\n\nThe high energy science cases of \\gls{cta} are mostly concerned with the acceleration mechanisms that produce high energy cosmic rays. This has been an active topic of discussion in the past 100 years since their initial detection. It is therefore hoped that the \\gls{cta} \\glspl{sst} can provide new insight into these mechanisms. The different investigations related to this topic can be loosely consolidated into the following categories:\n\\begin{description}\n\\item [Supernova Remnants] It is known that the galactic population of \\glspl{snr} plays an important role in the acceleration of cosmic rays to high energies. The detection of \\si{TeV} photons from \\glspl{snr} (suggesting an efficient acceleration mechanism), and the description of the diffusive shock acceleration mechanism, both corroborate with the detection of high-energy cosmic rays in the Earth's atmosphere \\cite{Cristofari2017}. However, the detection of \\si{TeV} photons from \\glspl{snr} could instead be explained by the inverse-Compton scattering between accelerated electrons and the ambient photon background. Therefore, the debate between a leptonic or hadronic origin is still ongoing \\cite{Acharya2013}. While studies of individual \\glspl{snr} have improved our understanding of the acceleration mechanisms, a population wide study may help constrain the parameters involved \\cite{Cristofari2017}. The probe into higher energies with the \\glspl{sst} will provide further information about the spectral energy distribution of the currently known \\glspl{snr}, and the enhanced sensitivity of \\gls{cta} will increase the population of \\glspl{snr} known to emit at these energies.\n\\item [Origin of Cosmic Rays] Another important question regarding the locally measured flux of high-energy cosmic rays is their origin \\cite{Bigongiari2016}. As just described, \\glspl{snr} appear to be a dominant source for these particles, but are they the only major contributor to the galactic cosmic rays? Expanding on the discovered \\gls{vhe} galactic source population is the key to answering this question.\n\\item [Pevatrons] A further capability of \\gls{cta} (provided by the \\glspl{sst}) is the detection of extreme accelerators that power particles up to the \\si{PeV} scale. As a result of the acceleration of hadronic cosmic rays to these energies, gamma rays with energies of \\SI{100}{TeV} should be detectable from the accelerator. However, as the cross-section for inverse-Compton electron-photon interactions decreases very quickly above a few tens of \\si{TeV} \\cite{Consortium2018}, the absence of \\SI{100}{TeV} gamma rays from these accelerators would suggest a leptonic origin. The identification of even one Pevatron accelerator would therefore provide a huge breakthrough in the investigations into the origins of \\gls{vhe} gamma rays.\n\\end{description}\n\n\\section{Thesis Outline}\n\nThe focus of this thesis is on the camera for the \\gls{gct} \\gls{sst}. I begin with a detailed description of the camera's components and waveform readout in Chapter~\\ref{ch2-mechanics}. Specific focus is given to the Silicon Photomultiplier (SiPM), the photosensor utilised in the latest camera implementation. This photosensor exhibits high photoelectron counting resolution for a large dynamic range, but suffers from a phenomena known as optical crosstalk, which degrades the ability to resolve the number of incident photons detected with the photosensor.\n\nI provide background on the \\gls{cta} system architecture in Chapter~\\ref{ch3-architecture} in order to introduce the performance requirements the camera must meet. The most important requirement in the context of this thesis is the Charge Resolution, a measure of how accurately the signal contained in waveform can be resolved.\n\nChapter~\\ref{ch4-software} outlines the software packages relevant to the processing pipeline for the Cherenkov shower data recorded by the camera, many of which I have been actively involved in developing. This includes the low-level \\cpp~libraries used to read out and calibrate the waveforms from the camera, the Python packages responsible for reconstructing the Cherenkov shower parameters from the waveforms, and finally the science tools developed to analyse the reduced photon event lists provided by the \\gls{cta} Observatory to the public.\n\nThe calibration procedures I have developed for the camera are described in Chapter~\\ref{ch5-calibration}. These procedures remove the artefacts that are particular to the camera's electronics, transforming the waveform readout into a state ready for common signal processing techniques. The calibration procedures for the photosensor are also covered, which act to unify the illumination response between the different pixels in the camera.\n\nChapter~\\ref{ch6-reduction} describes the signal processing techniques commonly utilised by \\glspl{iact} to extract the Cherenkov shower signal from the waveforms. The \\textit{cross-correlation} approach I have developed and adopted for signal extraction is also described. Within this chapter I continue on to describe how the information extracted from the waveform is used to reconstruct the properties of the Cherenkov shower progenitor. \n\nI assess the cumulative performance of the camera (resulting from its components, calibration, and waveform processing) in Chapter~\\ref{ch7-performance}. The requirements introduced in Chapter~\\ref{ch3-architecture} are used as the benchmarks to assess the performance against, as the requirements must be met for the camera to be accepted as a contribution to \\gls{cta}. I also highlight the significance of the optical crosstalk on the Charge Resolution, and explore the impact of future improvements to the photosensor.\n\nIn Chapter~\\ref{ch8-onsky} I report on the on-telescope campaigns performed with an earlier iteration of the \\gls{gct} camera, demonstrating the application of Cherenkov shower parametrisation techniques introduced in Chapter~\\ref{ch6-reduction}.\n", "meta": {"hexsha": "0c3405c0369ba77a929346dac5246d4af16223d0", "size": 24889, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "thesis/text/ch1-intro.tex", "max_stars_repo_name": "watsonjj/Thesis", "max_stars_repo_head_hexsha": "6b0a0869b9a9a8abbc2ab74b367f7a9d7d741644", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-11-16T14:30:06.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-16T14:30:06.000Z", "max_issues_repo_path": "thesis/text/ch1-intro.tex", "max_issues_repo_name": "watsonjj/Thesis", "max_issues_repo_head_hexsha": "6b0a0869b9a9a8abbc2ab74b367f7a9d7d741644", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "thesis/text/ch1-intro.tex", "max_forks_repo_name": "watsonjj/Thesis", "max_forks_repo_head_hexsha": "6b0a0869b9a9a8abbc2ab74b367f7a9d7d741644", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 130.9947368421, "max_line_length": 1509, "alphanum_fraction": 0.797500904, "num_tokens": 5967, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6224593171945417, "lm_q2_score": 0.5506073655352404, "lm_q1q2_score": 0.34273068479335117}} {"text": "\\section{(Initial) Causal Graph Construction}\n\\label{sec:graph-construction}\n\nConstruction of an initial causal graph typically proceeds as follows.\nAt a high level, we\n\\begin{enumerate}\n \\item adopt a population perspective,\n \\item brainstorm all variables that we think affect the system that generates our observations,\n \\item remove any variables that could cause bias in our causal inferences,\n \\item connect all variables in our graph according to our a-priori beliefs about causal relations amongst them\n \\item consider how the graph structure may differ between individuals and subgroups within the population.\n\\end{enumerate}\nThe following paragraphs describe these steps in detail.\n\nTo begin, we adopt the position of a researcher concerned about population level relationships.\nThis means we will think through what is a likely generative model for all individuals.\nLater in this section, we will devote time to thinking about how subgroups and individual heterogeneity may affect our causal graphs.\n\nNow, we add our first variable(s) to our graph, the outcome variable(s) of interest in our problem.\nNote that we should consider relations and dependencies between these outcomes, and we should draw these onto our graphs.\nSuch inter-outcome dependencies may be of great relevance or even focus.\nRecall, for example, the case of activity-based modellers that was mentioned in Section \\ref{sec:choice-graphs}.\nSimilarly, medical researchers with data on multiple health measures or companies with multiple business metrics may all be interested in how the outcomes cause each other.\n\nAfter adding our outcomes to our graph, we list all the variables we believe to cause them.\nWe refer to these influencing variables as our initial explanatory variables.\n\nNext, we iterate through these initial explanatory variables.\nFor each current explanatory variable in the iteration, we think of variables that may modify the effect of the current explanatory variable on the outcome(s) of interest.\nWe refer to these variables as effect modifiers\\footnote{Note, effect modifiers and confounders are easily confused. Both variables cause the outcome. The difference is that effect modifiers do not cause the explanatory / treatment variables. Confounders do. For discussion and classification of the different types of effect modification, based on one's causal graph, see \\citet{vanderweele_2007_four}}.\nNote that some effect modifiers may be a part of our list of initial explanatory variables.\nFor any effect modifiers that we think of, outside of the list of initial explanatory variables, we add them to our causal graph.\n\nOverall, modifiers are important because our treatment effects systematically vary with them.\nAccordingly, if we better understand when our treatments will be effective, then we can better target them.\n% For instance, the effect of a transit voucher on increasing an individual's probability of using transit is likely modified by the recipients income.\n% We expect a smaller treatment effect on wealthy individual's than on individual's with low income\nFor instance, imagine that a region-wide lockdown reduces the 14-day rolling average of new COVID-19 cases by X\\% (on average).\nOf course, we know that a lockdown's effectiveness is modified by the percentage of workers who must continue going out to work.\nIf most residents in an area are essential workers, then a lockdown will be less effective there, as compared with other locales.\nWe might wish to target other interventions for that region, as a replacement or supplement for the lockdown.\nTargeting aside, knowledge of modifiers is also crucial to generalizing treatment effect inferences from one population to another.\nTo credibly transport our inferences, we must know what variables cause the treatment effects to differ between populations, and we must know how the distributions of those variables differs across populations \\citep{pearl_2014_external}.\nIn general, see \\citet{zheng_2018_automated} for a thorough introduction to moderation, its differences from other variables in one's causal graph, and for instructions on how to find moderators through data analysis alone.\n\nAfter adding explanatory and effect modifying variables to the graph, we turn our attention to mediating variables.\nA mediating variable is one through which an explanatory variable influences our outcome(s) of interest.\nSuch variables have multiple uses.\nUnder certain instances of confounding, mediators enable the ``front-door'' criterion to identify one's causal effect \\citep{glynn_2018_front, bellemare_2019_paper, gupta_2020_estimating}.\nSimilarly, subject to particular causal assumptions, mediating variables permit inference on long-term outcomes of a selected intervention, given only its short-term proxies \\citep{athey_2019_estimating, yang_2020_targeting}.\n\nTo find these mediators, we again iterate through each explanatory variable.\nOn each iteration, we brainstorm variables along paths of influence from our explanatory variable to our outcome.\nFor instance, consider how the presence of a bike lane influences bicycle mode choice.\nWe hypothesize that an individual's subjective perception of safety is the primary (or sole) mediator through which bicycle lane presence influences mode choice.\nAccordingly, we add subjective perception of safety to our causal graph for travel mode choice.\n\nAfter considering the variables above, we turn our attention to variables that complicate our analyses.\nTo begin with, we think of confounding variables.\nThe process is similar to how we generated effect modifying variables.\nWe iterate through each of the explanatory, mediating, and effect modifying variables, thinking specifically of any variables that both cause the current variable in the iteration and cause the outcome variable(s).\nWe call these variables, which cause our outcome and current variables in the iteration, confounding variables \\citep{elwert_2013_graphical, greenland_1999_confounding}.\nAs an example, consider a person's attitude towards environmental conservation.\nThis attitude may cause both that individual's observed distance to their workplace (another explanatory variable) and that individual's choice of travel mode.\nBoth in this example and in general, we should add such confounding variables to our causal graph.\n\nNext, we consider the effects of selection.\nAs noted by \\citet{greenland_2020_causal}, all datasets have a causal graph that implicitly conditions on a selection node.\nI.e., we only analyze data that has been selected to be a part of our dataset.\nWe should therefore consider how all of the other nodes in our causal graph relate to the selection node.\nIn particular, will we suffer any selection bias due to the outcomes influencing whether an observation is selected for inclusion in our dataset?\nSelection bias, if present, can cause our estimated causal effects to differ greatly from their population counterparts.\nThis stems from systematic differences between the observations that have been selected into our dataset and the observations in our population of interest.\nFor more details, see \\citet{heckman_1979_sample} and \\citet{hernan_2004_structural} as canonical references.\n\nAnother universally implied yet only implicitly described element of one's causal graph is the prior data and code that led to one's dataset \\citep[Pg.7]{greenland_2020_causal}.\nPresumably, prior data and potentially code-enabled-analysis influenced the sample design that led to your dataset.\nPerhaps some data transformations and code to implement those transformations was used to convert a raw dataset into the dataset being used for causal inference.\nAnd at all times, one uses computer programs to compute your reported results.\nIn each case, the prior data is variable that influences your current data, and your code is computational (sub)graph that is implicit in your causal graph.\nThese elements should perhaps be made explicit, and their influence on your causal effect estimates should definitely be assessed and reported.\n\nNext, we should explicitly consider the role of time, even in research that may be cross-sectional due to the data that is available to us or due to the problem itself.\nIn reality, how do we think our system evolves over time?\nIf we consider multiple observations of a given decision maker, how does that decision maker's observed variables at time $t$ partially cause future variables important to the context or outcome(s) for that decision maker at time $t' > t$?\nHow do the actions of a decision maker $i$ at time $t$ partially cause the future context or outcomes of a decision maker $j$?\nWe should add explicit nodes to our graph, subscripted or denoted by time, to show the cross-time causal relationships in our system.\nFor in-depth discussion of time-related causal inference topics, see papers such as \\citet{gill_2001_causal}, \\citet{eichler_2007_granger}, and \\citet{peters_2013_causal}.\nPlease note that the literature on this topic is vast, and the cited authors are not at all exhaustive or representative of all papers in this space.\nInterested readers are encouraged to perform further literature searches on their own.\n\nSimilarly, we will frequently want to consider the role of space.\nIn the context of choice modelling, this includes questions such as ``how does a decision maker's existence in a particular geographical area shape their choices?''\nFor example, consider multinational corporations where the business operates differently across state borders.\nHere, the borders associated with space directly causes a difference in the causal graph of how these businesses deliver their goods and services to customers.\nIn other instances, one can consider space as a noisy proxy for unobserved confounders, such as cultural attitudes of a region's inhabitants \\citep{paciorek_2010_importance}.\nRooted in the causal problem of unobserved confounding, this results in known statistical issues for choice modellers such as spatial correlation of model residuals \\citep{fleming_2004_techniques}.\nFortunately, some progress has been made in dealing with such problems.\nFor general discussion of recent techniques in causal inference for spatial-causal modelling to deal with such issues, see \\citet{osama_2019_inferring}.\n\nAt this point, we have added to our causal graph all the\noutcome, explanatory, effect modifying, mediating, confounding, selection, data/code, time-indexed, and space-indexed variables that we believe are relevant for our problem.\nHowever, many of these variables may be disconnected nodes, i.e., singletons in the graph.\nWe now focus on pruning nodes from this graph, before drawing our final hypothesized connections.\nIn particular, we focus on pruning ``post-outcome'' variables that are not part of the causal graph for future time periods or other observations.\nThe reason for this is that conditioning on such post-outcome variables would bias our causal effect estimates.\nOur pruning exception is the selection variable which we have no choice but to condition on.\nIt may unfortunately be a post-outcome.\n\nTo remove the problematic variables, we iterate through each of the non-outcome variables in our graph, and we assess whether each variable is actually a result of the outcome (perhaps in combination with other variables in our graph).\nThese post-outcome variables temporally follow the outcome variable(s) but do not cause variables in the causal graph for other observations.\nWe remove all such post-outcome variables from our graph.\n\nNow is a good time to step back and consider what other researchers have thought about our problem.\nSpecifically, we should conduct a literature review to see how other researchers have conceptualized the topic that we are working on.\nHave they included variables that we have not?\nWere those variables related our outcomes of interest?\nIf so, should we add these variables to our causal graph? How should these variables enter our graph?\nDo the included variables of other researchers suggest the existence of confounders in their work that we should include in our graph?\nHave other researchers ascribed differing roles to our graph's current variables than we have?\nFor example, have other researchers judged a variable to be a confounder, when we solely thought of the variable as an effect modifier?\nAs we answer these questions, we should critically examine the evidence for these alternative decisions to see if we should also reconsider how we're judging our variables.\n\nFinally, we need to connect the variables in our graph.\n\\begin{enumerate}\n \\item Draw direct arrows from our explanatory variables, confounders, and effect modifiers to the outcomes.\n \\item Draw arrows from the explanatory variables to the mediators, and then draw arrows from the mediators to the outcomes.\n \\item Draw arrows from the confounders to the explanatory variables and mediators that they may cause.\n \\item Draw arrows from the variables in time $t$ to the variables that they cause in time $t+1$.\n \\item Draw arrows from the variables that cause one's location in space to spatial variable nodes, and draw arrows to variables that are caused by one's location in space.\n \\item Draw arrows from all other nodes in one's graph to the selection node, based on which variables cause inclusion in one's dataset.\n\\end{enumerate}\nAfter drawing in all arrows, we should now have a fully connected causal graph.\nPause.\nLook over the graph to ensure there are no remaining singletons, and ensure that we have not drawn any spurious connections.\nThen, stop briefly to celebrate.\nDrawing a project's first causal graph is hard work!\n\nAfter celebrating, take a moment to pursue the following graph editing exercises.\nFirst, think about how the graph might differ across sub-populations.\nWhat sub-populations, if any, exist in your population of interest?\nAre there any causal relationships that should, or should not, not exist for a given sub-population?\nFor instance, are the outcomes in some sub-populations independent of a given explanatory variable?\nCan you think of any inverted causal relationships that are specific to this sub-population?\n(I.e., for a given sub-population, does $B \\rightarrow A$ instead of $A \\rightarrow B$?)\nAs explained in \\citet[Sec. 4]{druzdzel_2003_combining},\nadd these sub-populations to one's initial causal graph via a ``selection node,''\nor if this is not clear enough,\ndraw modified causal graphs for each sub-population of interest.\nNow, one can actually relax.\nThis concludes the ``purely mental'' drafting of one's causal graph.\nIn the next section, we'll look at testing this graph against data, and making any edits deemed empirically necessary.\n", "meta": {"hexsha": "5af0d12ef67c70da0f91df97e7fdc25c9b2027af", "size": 14739, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "article/sections/_4_graph_construction.tex", "max_stars_repo_name": "hassanobeid1994/tr_b_causal_2020", "max_stars_repo_head_hexsha": "1ffaeb7dcefccf5e1f24c459e9a2f140b2a052a5", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "article/sections/_4_graph_construction.tex", "max_issues_repo_name": "hassanobeid1994/tr_b_causal_2020", "max_issues_repo_head_hexsha": "1ffaeb7dcefccf5e1f24c459e9a2f140b2a052a5", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 89, "max_issues_repo_issues_event_min_datetime": "2020-02-10T02:52:11.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-23T03:50:27.000Z", "max_forks_repo_path": "article/sections/_4_graph_construction.tex", "max_forks_repo_name": "hassan-obeid/tr_b_causal_2020", "max_forks_repo_head_hexsha": "1ffaeb7dcefccf5e1f24c459e9a2f140b2a052a5", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 96.3333333333, "max_line_length": 404, "alphanum_fraction": 0.8146414275, "num_tokens": 2931, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.550607350786733, "lm_q2_score": 0.6224593312018546, "lm_q1q2_score": 0.34273068332553475}} {"text": "\\chapter{DNA Models}\n\\label{chap:dna}\n\n\n\\section{3SPN.2C}\n\\label{sec:dna_3spn2c}\n\nThe series of the 3SPN.x DNA models have been developed by de Pablo's\ngroup~\\cite{Freeman2011, Hinckley2013, Freeman2014}. The 3SPN.2C model is the\none for modeling sequence-dependent curvature of double-stranded DNA\n(dsDNA)~\\cite{Freeman2014}. Particularly, the model has been well-tuned to\nreproduce both mechanical and geometrical properties, such as persistent length\nand major/minor groove widths~\\cite{Hinckley2013, Freeman2014}. \n\n\\subsection{Topology}\n\\label{subsec:dna_3spn2c_top}\n\nIn this model, each nucleotide is represented by three CG particles, P\n(phosphate), S (sugar), and B (base), as shown in \\autoref{fig:dna_3spn2c_top}.\nB has four types: A (adenine), C (cytosine), G (guanine), and T (thymine). The\nCG particles are put at the center-of-mass of each chemical moiety.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.3\\textwidth]{figures/DNA_3spn2c_top.png}\n \\caption{Topology of the 3SPN.2C DNA model: each nucleotide is represented by\n 3 sites corresponding to phosphate (P), deoxyribose sugar (S), and\n nitrogenous base (B). A typical nucleotide ``residue'' is enclosed by the\n dashed line.}\n \\label{fig:dna_3spn2c_top}\n\\end{figure}\n\n\n\\begin{table}[ht]\n \\centering\n \\begin{tabular}{lc}\n \\toprule\n Particle Type & Mass (amu) \\\\\n \\midrule\n P & 94.97 \\\\\n S & 83.11 \\\\\n A & 134.1 \\\\\n C & 110.1 \\\\\n G & 150.1 \\\\\n T & 125.1 \\\\\n \\bottomrule\n \\end{tabular}\n \\caption{Mass of the 3SPN.2C DNA particles.}\n \\label{tab:dna_3spn2c_top_mass}\n\\end{table}\n\n\\subsection{Potentials}\n\\label{subsec:dna_3spn2c_potential}\n\nThe 3SPN.2C potentials can be devided into two parts, as usual, bonded, and\nnonbonded:\n\\begin{displaymath}\n U = U_b + U_{nb}.\n\\end{displaymath}\n\nThe bonded potentials include the following terms:\n\\begin{itemize}\n\\item bonds ($U_{bond}$),\n\\item angles ($U_{ang}$),\n\\item dihedral angles ($U_{dih}$).\n\\end{itemize}\n\\begin{equation}\n \\label{eq:dna_3spn2c_local}\n U_b = U_{bond} + U_{ang} + U_{dih}.\n\\end{equation}\n\nThe nonbonded potentials include these terms:\n\\begin{itemize}\n\\item base-base interactions:\n \\begin{itemize}\n \\item base stacking ($U_{bstk}$),\n \\item base pairing ($U_{bp}$),\n \\item cross base stacking ($U_{cstk}$),\n \\end{itemize}\n\\item excluded volume interactions ($U_{exv}$),\n\\item electrostatic interactions ($U_{ele}$).\n\\end{itemize}\n\\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal}\n U_{nb} = U_{bstk} + U_{bp} + U_{cstk} + U_{exv} + U_{ele}.\n\\end{equation}\n\n\n\\subsubsection{Bond}\n\\label{sec:dna_3spn2c_potential_bond}\n\n\\begin{smallpage}{3SPN.2C bond potential}\n \\begin{equation}\n \\label{eq:dna_3spn2c_local_bond}\n U_{bond} = \\sum_{i}^{bonds} k_b (r_i - r_{i,0})^2 + 100 k_b (r_i - r_{i,0})^4.\n \\end{equation}\n \\tcblower\n \\begin{itemize}\n \\item Bonds involved:\n \\begin{itemize}\n \\item P-S\n \\item S-B\n \\item S-P\n \\end{itemize}\n \\item $r_{i, 0}$: based on the structure of B-form DNA.\n \\item $k_b = 0.6\\ \\mathrm{kJ/mol/\\angstrom^2}$.\n \\end{itemize}\n\\end{smallpage}\n\n\n\n\\subsubsection{Angle}\n\\label{sec:dna_3spn2c_potential_angle}\n\n\\begin{smallpage}{3SPN.2C bond angle potential}\n \\begin{equation}\n \\label{eq:dna_3spn2c_local_angle}\n U_{ang} = \\sum_{i}^{angles} k_\\theta (\\theta_i - \\theta_{i,0})^2.\n \\end{equation}\n \\tcblower\n \\begin{itemize}\n \\item Angles involved:\n \\begin{itemize}\n \\item P-S-P\n \\item S-P-S\n \\item P-S-B\n \\item B-S-P\n \\end{itemize}\n \\item $\\theta_{i, 0}$: based on the structure of B-form DNA.\n \\item $k_\\theta$: see the table below: (unit: $\\mathrm{kJ/mol/rad^2}$)\n \\end{itemize}\n \\begin{center}\n \\begin{tabular}{l|cl|cl|cl|cl}\n \\toprule\n \\emph{PSP} & all & \\multicolumn{7}{l}{300}\\\\\n \\midrule\n \\multirow{4}{*}{\\emph{SPS}} &\n AA & 355 & AT & 147 & AC & 464 & AG & 368 \\\\\n & TA & 230 & TT & 355 & TC & 442 & TG & 273 \\\\\n & CA & 273 & CT & 368 & CC & 165 & CG & 478 \\\\\n & GA & 442 & GT & 464 & GC & 228 & GG & 165 \\\\\n \\midrule\n \\multirow{4}{*}{\\emph{PSB}} &\n AA & 460 & TA & 120 & CA & 206 & GA & 383 \\\\\n & AT & 370 & TT & 460 & CT & 358 & GT & 442 \\\\\n & AC & 442 & TC & 383 & CC & 278 & GC & 336 \\\\\n & AG & 358 & TG & 206 & CG & 278 & GG & 278 \\\\\n \\midrule\n \\multirow{4}{*}{\\emph{BSP}} &\n AA & 460 & AT & 370 & AC & 442 & AG & 358 \\\\\n & TA & 120 & TT & 460 & TC & 383 & TG & 206 \\\\\n & CA & 206 & CT & 358 & CC & 278 & CG & 278 \\\\\n & GA & 383 & GT & 442 & GC & 336 & GG & 278 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{center}\n\\end{smallpage}\n\n\\subsubsection{Dihedral Angle}\n\\label{sec:dna_3spn2c_potential_dihedral_angle}\n\nThe dihedral angle potential in the 3SPN.2C model contains two parts:\n\\begin{enumerate}\n\\item A strong Gaussian type potential applied on the backbone dihedrals\n ($U_{dih, Gaussian}$);\n\\item A weak periodic potential applied on all dihedral angles ($U_{dih, periodic}$).\n\\end{enumerate}\n\\begin{equation}\n \\label{eq:dna_3spn2c_local_dihedral}\n U_{dih} = U_{dih, Gaussian} + U_{dih, periodic}.\n\\end{equation}\n\n\\begin{smallpage}{3SPN.2C dihedral angle potential (Gaussian)}\n \\begin{equation}\n \\label{eq:dna_3spn2c_local_dihedral_Gaussian}\n U_{dih, Gaussian} = \\sum_{i}^{dihedrals} -k_{ \\phi, Gaussian } \\exp\\big( \\frac{-(\\phi_i - \\phi_{i,0})^2}{2\\sigma_{\\phi}^2} \\big).\n \\end{equation}\n \\tcblower\n \\begin{itemize}\n \\item Dihedral angles involved:\n \\begin{itemize}\n \\item P-S-P-S\n \\item S-P-S-P\n \\end{itemize}\n \\item $\\phi_{i, 0}$: based on the structure of B-form DNA.\n \\item $k_{\\phi, Gaussian} = 7.0\\ \\mathrm{kJ/mol}$.\n \\item $\\sigma_\\phi = 0.3$.\n \\end{itemize}\n\\end{smallpage}\n\n\\begin{smallpage}{3SPN.2C dihedral angle potential (periodic)}\n \\begin{equation}\n \\label{eq:dna_3spn2c_local_dihedral_periodic}\n U_{dih, periodic} = \\sum_{i}^{dihedrals} k_{\\phi, periodic} \\big[ 1+\\cos(\\phi_i - \\phi_{ i,0 }) \\big].\n \\end{equation}\n \\tcblower\n \\begin{itemize}\n \\item Dihedral angles involved:\n \\begin{itemize}\n \\item P-S-P-S\n \\item S-P-S-P\n \\item S-P-S-B\n \\item B-S-P-S\n \\end{itemize}\n \\item $\\phi_{i, 0}$: based on the structure of B-form DNA.\n \\item $k_{\\phi, periodic} = 2.0\\ \\mathrm{kJ/mol}$.\n \\end{itemize}\n\\end{smallpage}\n\n\n\\subsubsection{Base Stacking}\n\\label{sec:dna_3spn2c_potential_bstk}\n\nBefore we introduce the nonbonded terms, we would like to first show the\ntopology of the double stranded DNA and clarify the CG particles involved in the\nbase-base interaction terms (see \\autoref{fig:DNA_3spn2c_nonbonded_all}).\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.45\\textwidth]{figures/DNA_3spn2c_nonbonded_all.png}\n \\caption{The base-related nonbonded interactions among DNA CG particles. Here\n we introduce local indices (circled numbers) for easy understanding. The\n base-stacking interactinos are calculated for the neighboring bases\n ($U_{bstk}$). The base-pairing interactions are considered for Watson-Crick\n pairs, namely, the A-T and G-C pairs ($U_{bp}$). Whereas the cross-stacking\n terms are only considered for bases next to the pairsing bases ($U_{cstk}$).}\n \\label{fig:DNA_3spn2c_nonbonded_all}\n\\end{figure}\n\nWe would like to also define several common functions used by the\nbase-interaction potentials:\n\\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal_base_rep}\n U_m^{rep}(\\epsilon_{ij}, \\alpha_{ij}, r_{ij}) =\n \\begin{cases}\n \\epsilon_{ij} \\Big( 1-e^{\\big(-\\alpha_{ij}(r_{ij}-r_{ij,0})\\big)} \\Big)^2 & r_{ij} < r_{ij, 0}, \\\\[.5em]\n 0 & r_{ij} \\ge r_{ij, 0}.\n \\end{cases}\n\\end{equation}\n\n\\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal_base_attr}\n U_m^{attr}(\\epsilon_{ij}, \\alpha_{ij}, r_{ij}) =\n \\begin{cases}\n -\\epsilon_{ij} & r_{ij} < r_{ij, 0}, \\\\[.5em]\n \\epsilon_{ij} \\Big(1-e^{\\big(-\\alpha_{ij}(r_{ij}-r_{ij,0})\\big)} \\Big)^2 - \\epsilon_{ij} & r_{ij} \\ge r_{ij, 0}.\n \\end{cases}\n\\end{equation}\n\nIn ~\\autoref{eq:dna_3spn2c_nonlocal_base_rep} and\n\\autoref{eq:dna_3spn2c_nonlocal_base_attr}, $r_{ij}$ is the distance between\nparticle $i$ and $j$, whereas $\\epsilon_{ij}$ and $\\alpha_{ij}$ are sequence\ndependent parameters.\n\n\\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal_base_angle_mod}\n f(K, \\Delta \\theta) =\n \\begin{cases}\n 1 & \\displaystyle -\\frac{\\pi}{2K} < \\Delta \\theta < \\frac{\\pi}{2K}, \\\\[.7em]\n 1 - \\cos^2(K\\Delta\\theta) & \\displaystyle -\\frac{\\pi}{K} < \\Delta \\theta < -\\frac{\\pi}{2K} \\textrm{ or } \\frac{\\pi}{2K} < \\Delta \\theta < \\frac{\\pi}{K}, \\\\[.7em]\n 0 & \\displaystyle \\Delta \\theta < -\\frac{\\pi}{K} \\textrm{ or } \\Delta \\theta > \\frac{\\pi}{K}. \\\\\n \\end{cases}\n\\end{equation}\n\nThe $\\theta$ in \\autoref{eq:dna_3spn2c_nonlocal_base_angle_mod} has\ndifferent definitions in $U_{bstk}$, $U_{bp}$, and $U_{cstk}$.\n\nFor the base-stacking interactions, $\\theta_{BS}$ is the angle formed by\n\\circled{1}-\\circled{2}-\\circled{6}\n(\\autoref{fig:DNA_3spn2c_nonbonded_bstk}).\n\nPractically, we treat the $U_{bstk}$ as a ``local'' potential, because all the\ninteractions can be determined by the topology, regardless of the conformation.\nBesides, the order of magnitude of computational cost of $U_{bstk}$ is only\n$\\mathcal{O}(n)$, where $n$ is the number of basepairs in DNA.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.2\\textwidth]{figures/DNA_3spn2c_nonbonded_bstk.png}\n \\caption{Definition of $\\theta_{BS}$ in the base-stacking interactions.}\n \\label{fig:DNA_3spn2c_nonbonded_bstk}\n\\end{figure}\n\n\\begin{smallpage}{3SPN.2C base-stacking potential}\n \\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal_base_stacking}\n U_{bstk} = \\sum^{n_{bstk}} U_m^{rep}(\\epsilon_{BS}, \\alpha_{BS}, r_{ij}) +\n f(K_{BS}, \\Delta\\theta_{BS})\n U_m^{attr} (\\epsilon_{BS}, \\alpha_{BS}, r_{ij}).\n \\end{equation}\n \\tcblower\n \\begin{itemize}\n \\item Particles involved (see \\autoref{fig:DNA_3spn2c_nonbonded_bstk}):\n \\begin{itemize}\n \\item $r_{ij}$: \\circled{2}-\\circled{6}\n \\item $\\theta_{BS}$: \\circled{1}-\\circled{2}-\\circled{6}\n \\end{itemize}\n \\item $\\alpha_{BS} = 3.0$.\n \\item $K_{BS} = 6.0$.\n \\item $\\epsilon_{BS}$, $r_{ij, 0}$, and $\\theta_{BS, 0}$: see the table below:\n \\end{itemize}\n \\begin{center}\n \\begin{footnotesize}\n \\begin{tabular}{ll|rrrr|rrrr|rrrr}\n \\toprule\n & & \\multicolumn{12}{c}{\\circled{6}}\\\\\n & & A & T & G & C & A & T & G & C & A & T & G & C \\\\\n \\midrule\n & & \\multicolumn{4}{c|}{$\\epsilon_{BS}$ (kJ/mol)} & \\multicolumn{4}{c|}{$r_{ij, 0}$ (\\angstrom)} & \\multicolumn{4}{c}{$\\theta_{BS, 0}$ ($^\\circ$)}\\\\\n \\multirow{4}{*}{\\circled{2}}\n & A & 13.82 & 15.05 & 13.32 & 15.82 & 3.58 & 3.56 & 3.85 & 3.45 & 100.13 & 90.48 & 104.39 & 93.23 \\\\\n & T & 9.15 & 12.44 & 9.58 & 13.11 & 4.15 & 3.93 & 4.32 & 3.87 & 102.59 & 93.32 & 103.70 & 94.55 \\\\\n & G & 13.76 & 14.59 & 14.77 & 15.17 & 3.51 & 3.47 & 3.67 & 3.42 & 95.45 & 87.63 & 106.36 & 83.12 \\\\\n & C & 9.25 & 12.42 & 8.83 & 14.01 & 4.15 & 3.99 & 4.34 & 3.84 & 102.69 & 96.05 & 100.46 & 100.68 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{footnotesize}\n \\end{center}\n\\end{smallpage}\n\n\n\n\\subsubsection{Base Pairing and Cross-Stacking}\n\\label{sec:dna_3spn2c_potential_bp_cstk}\n\nIt is convenient to consider the $U_{bp}$ and $U_{cstk}$ potentials at the same\ntime, since $U_{cstk}$ only applies to bases next to the pairsing bases. There\nare also several angles and dihedral angles involved in the calculation of these\npotentials (\\autoref{fig:DNA_3spn2c_nonbonded_bp_cstk}).\n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{figures/DNA_3spn2c_nonbonded_bp_cstk.png}\n \\caption{Angles and dihedral angles involved in the $U_{bp}$ (red terms) and\n $U_{cstk}$ (blue terms) calculations.}\n \\label{fig:DNA_3spn2c_nonbonded_bp_cstk}\n\\end{figure}\n\nNote that the base pairsing interactions are considered when two bases satisfy\nthe following conditions:\n\\begin{itemize}\n\\item Base types can form Watson-Crick pairs: A-T, T-A, C-G, or G-C;\n\\item Base indices $i$ and $j$ are ``nonlocal'':\n \\begin{itemize}\n \\item $i$ and $j$ are in different strands;\n \\item $i$ and $j$ are in the same strand, but $| i - j | > 10$.\n \\end{itemize}\n\\end{itemize}\n\n\n\\begin{smallpage}{3SPN.2C base-pairing and cross-stacking potential}\n \\vspace{-1em}\n \\begin{align}\n U_{bp} &= \\sum^{n_{bp}} U_m^{rep}(\\epsilon_{BP}, \\alpha_{BP}, r_{ij}) \\nonumber \\\\\n &+ \\frac{1}{2} \\big( 1+\\cos(\\Delta \\phi_1) \\big)\n f(K_{BP}, \\Delta\\theta_{1})\n f(K_{BP}, \\Delta\\theta_{2})\n U_m^{attr} (\\epsilon_{BP}, \\alpha_{BP}, r_{ij}).\n \\label{eq:dna_3spn2c_nonlocal_base_pairing} \\\\[.5em]\n U_{cstk} &= \\sum^{n_{cstk}} f(K_{BP}, \\Delta\\theta_{3})\n f(K_{CS}, \\Delta\\theta_{CS})\n U_m^{attr} (\\epsilon_{CS}, \\alpha_{CS}, r_{kl}).\n \\label{eq:dna_3spn2c_nonlocal_cross_stacking}\n \\end{align}\n \\tcblower\n \\begin{itemize}\n \\item Particles involved (see \\autoref{fig:DNA_3spn2c_nonbonded_bp_cstk}):\n \\begin{itemize}\n \\item $r_{ij}$: \\circled{2}-\\circled{4}\n \\item $r_{kl}$: \\circled{2}-\\circled{5} and \\circled{4}-\\circled{6}, if available\n \\item $\\theta_{1}$: \\circled{1}-\\circled{2}-\\circled{4}\n \\item $\\theta_{2}$: \\circled{3}-\\circled{4}-\\circled{2}\n \\item $\\theta_{3}$: angle between vectors conecting \\circled{1}-\\circled{2} and \\circled{3}-\\circled{4}\n \\item $\\phi_{1}$: dihedral angle \\circled{1}-\\circled{2}-\\circled{4}-\\circled{3}\n \\item $\\theta_{CS}$: \\circled{1}-\\circled{2}-\\circled{5} or \\circled{3}-\\circled{4}-\\circled{6}\n \\end{itemize}\n \\item $\\alpha_{BP} = 2.0$.\n \\item $K_{BP} = 12.0$.\n \\item $\\alpha_{CS} = 4.0$.\n \\item $K_{CS} = 8.0$.\n \\item All the other parameters are listed in the table below:\n \\end{itemize}\n \\begin{center}\n \\begin{footnotesize}\n \\begin{tabular}{l|rrrrrr}\n \\toprule\n \\circled{2}-\\circled{4} & $\\theta_{1, 0}$ ($^\\circ$) & $\\theta_{2, 0}$ ($^\\circ$) & $\\theta_{3, 0}$ ($^\\circ$) & $\\phi_{1, 0}$ ($^\\circ$) & $r_{ij, 0}$ ($\\angstrom$) & $\\epsilon_{BP}$ (kJ/mol) \\\\\n \\midrule\n A-T & 153.17 & 133.51 & 110.92 & -38.18 & 5.82 & 14.41 \\\\\n T-A & 133.51 & 153.17 & 110.92 & -38.18 & 5.82 & 14.41 \\\\\n G-C & 159.50 & 138.08 & 120.45 & -35.75 & 5.52 & 18.24 \\\\\n C-G & 138.08 & 159.50 & 120.45 & -35.75 & 5.52 & 18.24 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{footnotesize}\n \\end{center}\n\n \\begin{center}\n \\begin{footnotesize}\n \\begin{tabular}{ll|rrrr|rrrr|rrrr}\n \\toprule\n & & \\multicolumn{12}{c}{\\circled{5}}\\\\\n & & A & T & G & C & A & T & G & C & A & T & G & C \\\\\n \\midrule\n & & \\multicolumn{4}{c|}{$\\epsilon_{CS}$ (kJ/mol)} & \\multicolumn{4}{c|}{$r_{kl, 0}$ (\\angstrom)} & \\multicolumn{4}{c}{$\\theta_{CS, 0}$ ($^\\circ$)}\\\\\n \\multirow{4}{*}{\\circled{2}}\n & A & 1.882 & 2.388 & 2.439 & 1.680 & 6.42 & 6.77 & 6.27 & 6.84 & 154.04 & 158.77 & 153.88 & 157.69 \\\\\n & T & 2.388 & 1.882 & 2.187 & 2.566 & 6.77 & 7.21 & 6.53 & 7.08 & 148.62 & 155.05 & 147.54 & 153.61 \\\\\n & G & 2.439 & 2.187 & 3.250 & 0.972 & 6.27 & 6.53 & 5.74 & 6.86 & 153.91 & 155.72 & 151.84 & 157.80 \\\\\n & C & 1.680 & 2.566 & 0.972 & 4.135 & 6.84 & 7.08 & 6.86 & 6.79 & 152.04 & 157.72 & 151.65 & 154.49 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{footnotesize}\n \\end{center}\n\n \\begin{center}\n \\begin{footnotesize}\n \\begin{tabular}{ll|rrrr|rrrr|rrrr}\n \\toprule\n & & \\multicolumn{12}{c}{\\circled{6}}\\\\\n & & A & T & G & C & A & T & G & C & A & T & G & C \\\\\n \\midrule\n & & \\multicolumn{4}{c|}{$\\epsilon_{CS}$ (kJ/mol)} & \\multicolumn{4}{c|}{$r_{kl, 0}$ (\\angstrom)} & \\multicolumn{4}{c}{$\\theta_{CS, 0}$ ($^\\circ$)}\\\\\n \\multirow{4}{*}{\\circled{4}}\n & A & 1.882 & 2.388 & 2.566 & 2.187 & 5.58 & 6.14 & 5.63 & 6.18 & 116.34 & 119.61 & 115.19 & 120.92 \\\\\n & T & 2.388 & 1.882 & 1.680 & 2.439 & 6.14 & 6.80 & 6.07 & 6.64 & 107.40 & 110.76 & 106.33 & 111.57 \\\\\n & G & 2.566 & 1.680 & 4.135 & 0.972 & 5.63 & 6.07 & 5.87 & 5.66 & 121.61 & 124.92 & 120.52 & 124.88 \\\\\n & C & 2.187 & 2.439 & 0.972 & 3.250 & 6.18 & 6.64 & 5.66 & 6.80 & 112.45 & 115.43 & 110.51 & 115.80 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{footnotesize}\n \\end{center}\n\\end{smallpage}\n\n\n\n\n\\subsubsection{Excluded-Volume Interactions}\n\\label{sec:dna_3spn2c_potential_exv}\n\n\n\\begin{smallpage}{3SPN.2C excluded volume interaction}\n \\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal_exv}\n U_{exv} = \\sum_{i \\sigma_{ij}.\n \\end{cases}\n \\end{equation}\n \\tcblower\n \\begin{itemize}\n \\item $i, j$ are all pairs of CG particles that are \\emph{not} in the\n following list:\n \\begin{itemize}\n \\item bonded (involved in bonds, angles, or dihedrals);\n \\item particles in the neighboring nucleotides;\n \\item Watson-Crick base pairs.\n \\end{itemize}\n \\item $\\epsilon_{r} = 1.0$ kJ/mol.\n \\item $\\displaystyle \\sigma_{ij} = \\frac{1}{2}\\big( \\sigma_i + \\sigma_j\n \\big)$, where $\\sigma_i$ ($\\sigma_j$) is listed below:\n \\end{itemize}\n \\begin{center}\n \\begin{tabular}{lc}\n \\toprule\n Particle Type & $\\sigma$ ($\\angstrom$) \\\\\n \\midrule\n P & 4.5 \\\\\n S & 6.2 \\\\\n A & 5.4 \\\\\n C & 6.4 \\\\\n G & 4.9 \\\\\n T & 7.1 \\\\\n \\bottomrule\n \\end{tabular}\n \\end{center}\n\\end{smallpage}\n\n\n\n\\subsubsection{Electrostatic Interactions}\n\\label{sec:dna_3spn2c_potential_ele}\n\n\n\\begin{smallpage}{3SPN.2C electrostatic interaction}\n \\begin{equation}\n \\label{eq:dna_3spn2c_nonlocal_ele}\n U_{ele} = \\sum_{i 30 GeV and > 70 GeV respectively. A cut point of BDT (Boost Decision Tree) is set for enhancing the ratio of VBF process over ggF and \\ztt +jets processes, and VBF region is therefore separated into VBF\\_0 (BDT > -0.1) and VBF\\_1 (BDT(VBF) < -0.1) regions. The boost category in SR selects the events that failed the VBF requirements, that is, the events from other three main Higgs production modes. Two boost categories are defined to target topologies where the Higgs boson is recoiling from a jet. The boost tight category is defined by requiring $\\Delta R<= 1.5$ and $p_T(H) >$140GeV. Events failing this criteria are assigned to the boost loose category ($\\Delta R > 1.5$ and $p_T(H) < $140GeV).\n\nControl region for \\ztt (ZCR) selects the events in window 60 GeV < \\MMC < 110 GeV, which is used for extracting the \\ztt normalisation in the fit (ZCR 0), as well as for estimating the \\phistarCP shape uncertainty in \\ztt events (ZCR 1). The preselection, VBF and boost regions for ZCR share the same requirements except with the \\MMC window. \n%\n\\begin{figure}[h!]\n\t\\begin{center}\n\t\t\\begin{tabular}{ccc}\n\t\t\t\\subfigure[hh]{\\includegraphics[width=.3\\textwidth]{plot_ditau_mmc_mlm_m_hh_preselection.pdf}} \\subfigure[lh]{\\includegraphics[width=.3\\textwidth]{plot_ditau_mmc_mlm_m_l1p_preselection.pdf}}\n\t\t\\end{tabular}\n\t\\end{center}\n\t\\caption{MMC distributions in the preselection region in the hadhad and lephad channels.}\n\t\\label{fig:selection:MMC}\n\\end{figure}\n%\n", "meta": {"hexsha": "203506caba78d58073ca60d056fbf97edd48564e", "size": 2788, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "eventselection.tex", "max_stars_repo_name": "beatrice-pan/ResearchProposal_TongPan", "max_stars_repo_head_hexsha": "29b94ac443c04e26301dd351c857ce1abe9780fa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "eventselection.tex", "max_issues_repo_name": "beatrice-pan/ResearchProposal_TongPan", "max_issues_repo_head_hexsha": "29b94ac443c04e26301dd351c857ce1abe9780fa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "eventselection.tex", "max_forks_repo_name": "beatrice-pan/ResearchProposal_TongPan", "max_forks_repo_head_hexsha": "29b94ac443c04e26301dd351c857ce1abe9780fa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 126.7272727273, "max_line_length": 997, "alphanum_fraction": 0.775107604, "num_tokens": 753, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.7185943805178139, "lm_q2_score": 0.47657965106367595, "lm_q1q2_score": 0.34246745912349813}} {"text": "\\documentclass{beamer}\n\\usepackage{blindtext}\n\\usepackage{graphicx}\n\n\\title[Short Title]{Longer Full Sample Title}\n\\subtitle{Sample Subtitle}\n\\author{Ben Morcos}\n\\date{October 15, 2018}\n\n\\usetheme{SimplePurple}\n\\begin{document}\n\n\t\\frame {\n\t\t\\titlepage\n\t}\n\n\t\\section{First Section}\n\n\t\\frame {\n\t\t\\frametitle{Mathmode}\n\t\tThis is the quadratic equation:\n\t\t\\begin{align}\n\t\t\t\\frac{-b \\pm \\sqrt{b^2 - c}}{2a}\n\t\t\\end{align}\n\t}\n\t\\frame{\n\t\t\\frametitle{Some bullet points}\n\t\t\\framesubtitle{And a list!}\n\t\t\\begin{itemize}\n\t\t\t\\item 1\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item 1.1\n\t\t\t\\end{itemize}\n\t\t\t\\item 2\n\t\t\t\\item 3\n\t\t\\end{itemize}\n\n\t\tText between? \\\\\n\n\t\tWhy not.\n\n\t\t\\begin{enumerate}\n\t\t\t\\item A thing\n\t\t\t\\begin{enumerate}\n\t\t\t\t\\item A related thing\n\t\t\t\t\\item \\alert{An important thing}\n\t\t\t\\end{enumerate}\n\t\t\t\\item Another thing\n\t\t\\end{enumerate}\n\t}\n\n\t\\section{Another Section}\n\n\t\\frame{\n\t \\frametitle{Paragraph Content}\n\t \\framesubtitle{Latin is cool}\n\t \\blindtext\n\t}\n\n\t\\frame{\n\t\t\\frametitle{Figures}\n\t\t\\framesubtitle{With caption}\n\t\t\\begin{figure}\n\t\t\t\\centering\n\t \t\\includegraphics[height=0.6\\textheight]{example-image}\n\t \t\\vspace{-10pt}\n\t \t\\caption{Built in examples are \\alert{\\textbf{genius}}}\n\t \\end{figure}\n\t}\n\\end{document}\n", "meta": {"hexsha": "8163731b891b4c0c732f34283c2c4f28597b0e72", "size": 1229, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sample.tex", "max_stars_repo_name": "bmorcos/beamer-SimplePurple", "max_stars_repo_head_hexsha": "a5f3f30c7d7299e8e1da45e3b6673ec4f38d64dd", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "sample.tex", "max_issues_repo_name": "bmorcos/beamer-SimplePurple", "max_issues_repo_head_hexsha": "a5f3f30c7d7299e8e1da45e3b6673ec4f38d64dd", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "sample.tex", "max_forks_repo_name": "bmorcos/beamer-SimplePurple", "max_forks_repo_head_hexsha": "a5f3f30c7d7299e8e1da45e3b6673ec4f38d64dd", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.3098591549, "max_line_length": 61, "alphanum_fraction": 0.6647681041, "num_tokens": 444, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34232597557068944}} {"text": "\\section{The output of the simulator}\n\\label{sec:mskoutput}\n\nAny report produced by the simulator contains the following elements:\n\\begin{itemize}\n\\item General information about the experiment\n\\item Statistics for each selected performance measure\n\\end{itemize}\n\n\\subsection{The contents of a report}\n\nGeneral information includes the names of the parameter files, the\ndate at which the experiment started, the CPU time required to carry\nout the complete simulation, the sample size, etc. This information\nis presented in the form of one (key: value) pair per line.\nThe main part of the report contains statistics for performance\nmeasures.\n\nSeveral quantities are computed during the simulation: event\ncounts, average times, and integrals. Each time a call exits\nthe system, counters are updated to keep track of various quantities.\nAll these random variables can be regrouped into a random vector\n$\\boldX=(X_0,\\ldots,X_{d-1})$ we will call\nan \\emph{observation}. Using some experimental techniques\npresented in section~\\ref{sec:mskexp}, the simulator can obtain $n$\ncopies of $\\boldX$ which is called a \\emph{sample}. Statistics are\ncomputed by applying some functions on this sample\n$\\boldX_0,\\ldots,\\boldX_{n-1}$, where $\\boldX_r=(X_{0, r}, \\ldots,\nX_{d-1, r})$ is the $r$th observation.\nLet\n\\[\\bar{\\boldX}_n=(\\bar X_{0, n}, \\ldots, \\bar X_{d-1, n})=\\frac1n\\sum_{r=0}^{n-1}\\boldX_r\\]\nthe \\emph{average} of $\\boldX_r$,\nwhich is used to estimate the expected value of $\\boldX$, denoted as\n\\[\\E[\\boldX]=(\\E[X_0], \\ldots, \\E[X_{d-1}]).\\]\nThe vector\n$\\bar{\\boldX}_n$ is an unbiased estimator of $\\E[\\boldX]$ if the\nobservations are independent and identically distributed (i.i.d.).\nWe are also interested in functions $g(\\boldX)$ such as ratios.\nThe function of averages $g(\\bar{\\boldX}_n)$ is used to estimate the\nfunction\nof expectations $g(\\E[\\boldX])$. This estimator is biased, unless\n$g(\\boldX)$ is a linear function of $\\boldX$.\nSubsection~\\ref{javadoc:umontreal.iro.lecuyer.contactcenters.app.PerformanceMeasureType}\nexplains how these quantities are regrouped in matrices for easier reporting.\n\nFor each performance measure, i.e., element $\\E[X_j]$ or function\n$g(\\E[\\boldX])$, the simulator outputs the following\nstatistics (all statistics are of course undefined when $n=0$):\n\\begin{description}\n\\item[Minimum] The minimal value of $X_j$ among all observations. No\n minimum is available for functions of multiple averages.\n\\item[Maximum] The maximal value of $X_j$ among all observations. No\n maximum is available for functions of multiple averages.\n\\item[Average] The average $\\bar X_{j, n}$, or function of averages\n $g(\\bar{\\boldX}_n)$, of the\n observations.\n\\item[Standard deviation] The sample standard deviation of the observations,\n i.e., $\\sqrt{\\Var[X_j]}$ or $\\sqrt{n\\Var[g(\\bar{\\boldX}_n)]}$. This\n corresponds to the asymptotic standard deviation in the case of a\n function of several averages.\n The value is undefined if $n<2$.\n\\item[Confidence interval] An interval $[a, b]$ containing $\\E[X_j]$ (or\n $g(\\E[\\boldX])$) with probability $1-\\alpha$, where $1-\\alpha$\n is a \\emph{confidence level} that can be adjusted via a simulation\n parameter.\n The interval is computed using the normality assumption.\n The value is undefined if $n<2$.\n\\end{description}\nUsing the element \\texttt{printed\\-Stat\\-Params} in the parameters of\nthe report, %(see\n%section~\\ref{javadoc:umontreal.iro.lecuyer.contactcenters.app.PrintedStatParams})\none can decide which performance measures appear in the report, and\ndetermine if all the statistics or only the averages are needed.\n\nWe now explain the last two statistics in more details.\nLet\n\\[\n\\boldS_n = \\frac1{n-1}\\sum_{r=0}^{n-1} (\\boldX_r -\n\\bar{\\boldX}_n)^\\tr(\\boldX_r - \\bar{\\boldX}_n),\n\\]\nbe the \\emph{sample covariance} of the $\\boldX_r$'s,\nwhich is used to estimate the covariance matrix\n\\[\\boldSigma = \\E[(\\boldX - \\E[\\boldX])^\\tr(\\boldX - \\E[\\boldX])].\\]\nThe simulator computes only parts of this sample covariance matrix, in\nparticular\nelements $(j, j)$ estimating $\\Var[X_j]=n\\Var[\\bar X_{j, n}]$,\nwhich can be used to estimate the error on $\\bar X_{j,\n n}$.\n% To estimate the error on $\\bar X_{j, n}$ for $j=0,\\ldots,d-1$, the\n% simulator computes the \\emph{sample variance}\n% \\[S_{X_j, n}^2=\\frac1{n-1}\\sum_{r=0}^{n-1} (X_{j, r} - \\bar X_{j,\n% n})^2\\]\n% which estimates the variance of the random variate $X_j$, defined as\n% \\[\\Var(X_j)=\\E[X_j^2] - \\E[X_j]^2.\\]\n\nThe sample variance is used to compute a confidence interval on the\ntrue mean $\\E[X_j]$ for any $j\\in\\{0, \\ldots, d-1\\}$. Assuming that\n$X_{j, r}$ follows the normal distribution,\n\\[\\sqrt{n}(\\bar X_{j, n} - \\E[X_j])/S_{X_j, n}\\]\nfollows the Student-$t$ distribution with $n-1$ degrees of freedom.\nHere, $S_{X_j, n}$ is the sample standard deviation of $X_j$.\nIf the desired probability that this (random)\ninterval covers the true mean $\\E[X_j]$ (a constant) is $1-\\alpha$,\nthe interval is given by $\\bar X_{j, n}\\pm t_{n-1, 1-\\alpha/2}S_{X_j,\n n}/\\sqrt{n}$, where $t_{n-1,1-\\alpha/2}$ is the inverse of the\nStudent-$t$ distribution function with $n-1$ degrees of freedom,\nevaluated at $1-\\alpha/2$.\n\nConfidence intervals on functions of means are computed\nusing the delta theorem \\cite{tSER80a}.\nHere, we explain the special case of ratios used by the simulator.\nOne can refer to \\cite{tSER80a,sLEC06a,iBUI05b} for the general case.\nLet $(X,Y)$ be a random vector for which the simulator can\ngenerate a sample $((X_0, Y_0), \\ldots, (X_{n-1}, Y_{n-1}))$. Let\n$\\bar X_n$ be the average for $X$ and $\\bar Y_n$ the average for $Y$;\nthese quantities estimate $\\mu_1=\\E[X]$, and $\\mu_2=\\E[Y]$,\nrespectively.\nThen, the function $\\bar\\nu_n=\\bar X_n/\\bar Y_n$\nestimates the ratio of means $\\nu=\\mu_1/\\mu_2$. By a Taylor expansion\nof the ratio of averages,\n%if $(\\bar X_n, \\bar Y_n)$ is a multinormal vector,\nthe asymptotic variance of $\\bar X_n/\\bar Y_n$, i.e., the variance\nwhen $n$ is large, is given by $\\sigma^2/n$, where\n\\[\\sigma^2=(\\Var[X] + \\nu^2\\Var[Y] - 2\\nu\\Cov[X, Y])/\\mu_2^2.\\]\nThe variance\n$\\sigma^2$ can be estimated by using sample means, variances and covariance.\n%as follows:\n%\\[\\hat\\sigma_n^2=(S_{X,n}^2+\\bar\\nu_n^2S_{Y,n}^2\n%-2\\bar\\nu_nS_{XY, n})/\\bar Y_n^2\\]\n%where $S_{X, n}$ estimates the variance of $X$, $S_{Y, n}$ estimates\n%the variance of $Y$, and $S_{XY, n}$ estimates the covariance of $X$\n%and $Y$.\n%\\[S_{XY, n}=\\frac1{n-1}\\sum_{r=0}^{n-1} (X_r - \\bar X_n)(Y_r - \\bar\n%Y_n).\\]\n\nAssuming that $(\\bar X_n, \\bar Y_n)$ follows the multinormal\ndistribution,\nthe confidence interval on the ratio of expectations with confidence\nlevel $1-\\alpha$ is given by $\\bar\\nu_n\\pm\nz_{1-\\alpha/2}\\sigma_n/\\sqrt{n}$, where\n$\\Phi(z_{1-\\alpha/2})=1-\\alpha/2$, $\\Phi(x)$ being the distribution\nfunction of a standard normal variable.\n\nNote that each confidence interval is computed for a single mean or\nratio of means, independently of other performance measures of the\nsystem. As a result, if $d>1$ output values are analyzed\nsimultaneously, the confidence level of the $d$ intervals\nis $1-d\\alpha<1-\\alpha$.\nThe confidence level for individual performance measures must then be\nhigher to get the same overall confidence level.\n\nThe value of $0/0$ is usually undefined and assigned the NaN (Not a Number) flag.\nHowever, in some ratios, $0/0$ can have some meaning.\nIn our simulator, $0/0$ is defined as 0 for most performance measures except service level.\nFor example, if there is no arrival, it is sensible to set the\nabandonment ratio and the average waiting time to 0, and\nthe service level to 1.\n\nFor expectations of ratios, $0/0$ observations are not collected,\nbecause fixing an arbitrary value would result in biased estimators. As a\nresult, the average is made on less observations, and the average is\nNaN if all observations are rejected.\n\n\\subsection{The format of the report}\n\\label{sec:reportformat}\n\nAt this moment, four file formats are supported for reports:\nXML, plain text, \\LaTeX, and\nMicrosoft Excel.\nThe first format is intended to be readable by programs while\nthe last three formats are human-readable.\n\n\\subsubsection{Program-readable format}\n\nThe XML format is intended to be parsed by Java programs\nusing the ContactCenters library.\nIt could also be parsed and processed by any other program\ncompatible with XML.\nProduced XML output files have root element\n\\texttt{Contact\\-Center\\-Sim\\-Results}, in\nnamespace URI\n\\path{http://www.iro.umontreal.ca/lecuyer/contactcenters/app}.\nThe XML schema for output files can be found in the \\texttt{schemas}\nsubdirectory of ContactCenters, and HTML documentation is available\nin \\path{doc/schemas}.\n%The binary format, using Java\n%serialization, may be faster to read, but the XML format may be\n%more easily\n%reusable by other programs not using ContactCenters, even not using\n%Java.\n\nAlternatively, the program can export to a XML file\nand compress the file using GZip to save\ndisk space.\nThis can be done by giving a file name with the \\texttt{.xml.gz}\nextension rather than \\texttt{.xml}.\n\n\\subsubsection{Plain text}\n\nWhen exporting to plain text, the simulator uses the\nplatform-default character encoding and line separator.\nAs a result, the created text file can be opened in any text editor\nsuch as Notepad, GNU Emacs, etc.\nAfter the general information, the report contains a table of summary\nresults, e.g., the performance measures concerning every call type,\nagent group, and period. Then, for each group of performance\nmeasures, a table of detailed results appear in the report.\nNote that the formatting of numbers is locale-specific. For example,\nif the current locale is set to French, the decimal separator\nis the comma while the separator is a period for the US locale.\n\nThe \\LaTeX\\ output, which is also plain text but with formatting\ninstructions,\nis intended to be processed by \\LaTeX\\ to generate printable tables of\nresults.\n\n\\subsubsection{Microsoft Excel}\n\nThe Microsoft Excel format is used to transfer results to spreadsheets\nfor further analysis and reporting.\nOne is not restricted to Microsoft Excel since\nmany other spreadsheets, e.g., OpenOffice.org and KOffice,\ncan read and write Excel files.\n\nThe Excel report is divided in at most three sheets.\nThe first sheet provides summary information only: the general\ninformation, and summary statistics, i.e., statistics for aggregate\nperformance measures.\nThe latter are split in two groups: source-related (or call-related)\nstatistics and destination-related (agent-related and waiting\nqueue-related) statistics.\nThe second sheet provides a detailed report for all time-aggregate\nperformance measures. This includes, e.g., the service level for\neach individual call type but not for each period.\nThe last sheet contains a detailed report for all performance\nmeasures, including statistics for individual periods.\n\n\\subsubsection{Localized format for reports}\n\nSome aspects of the reports produced by \\texttt{mskcallcentersim}\ndepend on the host environment of the JVM.\nThese aspects include the character encoding of the report,\nthe line delimiters, strings describing types of performance measures,\nand the format of the numbers.\nThe last two elements are influenced by the locale of the virtual\nmachine which executes the simulator, which corresponds to the default\nlocale of the host environment.\nAt this moment, only English and French are supported as languages for\nreports. If the current locale corresponds to another language,\nall text in the report will be in English, with locale-specific\nformatting for numbers.\n\nSome of these aspects can be customized using OS-specific options.\nFor example, calling \\texttt{LC\\_ALL=en\\_US.UTF-8 mskcallcentersim}\nlaunches the simulator with the US English locale, and UTF-8 encoding;\nthis is the default for most UNIX/Linux distributions.\nHowever, on other operating systems, such as Microsoft's Windows,\nthere is no built-in way to alter the default locale for a given\nprogram without changing the system-wide regional settings.\nHowever, Java properties can be changed to alter the default locale,\nand other parameters, in a platform-independent way.\nTable~\\ref{tab:repjavaprop} lists such properties.\nThe properties can be modified through the \\texttt{-D} JVM option.\nFor example, setting the \\texttt{CCJVMOPT} environment variable\nto \\texttt{-Duser.language=en} sets the language of reports to\nEnglish.\n\n\\begin{table}\n\\caption{Most common Java properties affecting reporting}\n\\label{tab:repjavaprop}\n\n\\centering\\begin{tabular}{|l|ll|}\\hline\nProperty & Action & Sample value \\\\ \\hline\n\\texttt{user.language} & Language of strings in reports & \\texttt{en}\n\\\\\n\\texttt{user.region} & Region affecting number and date format &\n\\texttt{US} \\\\\n\\texttt{file.encoding} & Character encoding for reports &\n\\texttt{UTF-8} \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n", "meta": {"hexsha": "168b17ea1889433be19dba461863415235399d73", "size": 12799, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/msk/mskoutput.tex", "max_stars_repo_name": "umontreal-simul/contactcenters", "max_stars_repo_head_hexsha": "f5ddb4a0a4b30dbf436ac36e6d97facce2f3576d", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/msk/mskoutput.tex", "max_issues_repo_name": "umontreal-simul/contactcenters", "max_issues_repo_head_hexsha": "f5ddb4a0a4b30dbf436ac36e6d97facce2f3576d", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/msk/mskoutput.tex", "max_forks_repo_name": "umontreal-simul/contactcenters", "max_forks_repo_head_hexsha": "f5ddb4a0a4b30dbf436ac36e6d97facce2f3576d", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.2261484099, "max_line_length": 91, "alphanum_fraction": 0.7541214157, "num_tokens": 3485, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5467381519846138, "lm_q2_score": 0.6261241772283034, "lm_q1q2_score": 0.34232597557068944}} {"text": "% Created 2021-07-06 Tue 08:01\n% Intended LaTeX compiler: pdflatex\n\\documentclass[presentation,aspectratio=169]{beamer}\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage{graphicx}\n\\usepackage{grffile}\n\\usepackage{longtable}\n\\usepackage{wrapfig}\n\\usepackage{rotating}\n\\usepackage[normalem]{ulem}\n\\usepackage{amsmath}\n\\usepackage{textcomp}\n\\usepackage{amssymb}\n\\usepackage{capt-of}\n\\usepackage{hyperref}\n\\usepackage{khpreamble}\n\\usepackage{amssymb}\n\\DeclareMathOperator{\\shift}{q}\n\\DeclareMathOperator{\\diff}{p}\n\\usetheme{default}\n\\author{Kjartan Halvorsen}\n\\date{2019-09-05}\n\\title{Computerized Control - analysis of discrete-time systems}\n\\hypersetup{\n pdfauthor={Kjartan Halvorsen},\n pdftitle={Computerized Control - analysis of discrete-time systems},\n pdfkeywords={},\n pdfsubject={},\n pdfcreator={Emacs 26.3 (Org mode 9.4.6)}, \n pdflang={English}}\n\\begin{document}\n\n\\maketitle\n\n\\section{Intro}\n\\label{sec:org06d2c71}\n\\begin{frame}[label={sec:org53fc2d8}]{Result from quizz}\n\\begin{itemize}\n\\item Very good results!\n\\item Nyquist plot + criterion\n\\item Robustness\n\\end{itemize}\n\\end{frame}\n\\section{Pole placement}\n\\label{sec:orgc540c2c}\n\\begin{frame}[label={sec:org88d54f5}]{Pole-placement and time-response}\nPair the pole-placement with the correct time-response (from HW1)\n\\begin{columns}\n\\begin{column}{0.4\\columnwidth}\n\\begin{center}\n\\includegraphics[width=\\linewidth]{../../figures/pzmap-apollo}\n\\end{center}\n\\end{column}\n\\begin{column}{0.6\\columnwidth}\n\\begin{center}\n\\includegraphics[width=\\linewidth]{../../figures/step-response-apollo}\n\\end{center}\n\\end{column}\n\\end{columns}\n\\end{frame}\n\n\\begin{frame}[label={sec:org75a559f}]{Mapping of poles from continuous time to discrete time}\n\\begin{center}\n\\begin{tabular}{ll}\nContinuous time & Discrete time\\\\\n\\hline\n\\(Y(s) \\triangleq \\laplace{y(t)}\\) & \\(Y(z) \\triangleq \\ztrf{y(kh)}\\)\\\\\n\\(Y(s) = G(s)U(s) = \\frac{b}{s+a}U(s)\\) & \\(Y(z) = H(z)U(z) = \\frac{\\beta}{z+\\alpha}U(z)\\)\\\\\nPole of the system: \\(s+a=0 \\; \\Rightarrow \\; s = -a\\) & Pole of the system: \\(z+\\alpha = 0 \\; \\Rightarrow \\; z = -\\alpha\\)\\\\\n\\includegraphics[width=0.22\\linewidth]{../../figures/cont-stable} & \\includegraphics[width=0.22\\linewidth]{../../figures/discrete-stable}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\nThe \\alert{s-domain} of continuous-time systems is related to the \\alert{z-domain} of discrete-time systems as \\[z = \\mathrm{e}^{sh}\\]\n\\end{frame}\n\n\\begin{frame}[label={sec:org4177f6b}]{Mapping of poles from continuous time to discrete time}\nDo excercise on paper!\n\\end{frame}\n\n\\section{Root locus}\n\\label{sec:org1da210f}\n\n\n\\begin{frame}[label={sec:org2d36761}]{Root locus: A brief review}\n\\begin{LaTeX}\n\\begin{center}\n \\begin{tikzpicture}[node distance=22mm, block/.style={rectangle, draw, minimum width=15mm}, sumnode/.style={circle, draw, inner sep=2pt}]\n\n \\node[coordinate] (input) {};\n \\node[sumnode, right of=input, node distance=16mm] (sum) {\\tiny $\\Sigma$};\n \\node[block, right of=sum, node distance=20mm] (plant) {$H(z)=\\frac{B(z)}{A(z)}$};\n \\node[block, below of=plant, node distance=12mm] (controller) {$F(z)=K\\frac{D(z)}{C(z)}$};\n \\node[coordinate, right of=plant, node distance=30mm] (output) {};\n\n \\draw[->] (input) -- node[above, pos=0.3] {} (sum);\n \\draw[->] (sum) -- node[above] {} (plant);\n \\draw[->] (plant) -- node[coordinate] (measure) {} node[above, near end] {} (output);\n \\draw[->] (measure) |- (controller);\n \\draw[->] (controller) -| (sum);\n \\end{tikzpicture}\n\\end{center}\n\\end{LaTeX}\n\n\\begin{itemize}\n\\item The loop pulse-transfer function (loop gain) becomes \\(L(z) = H(z)F(z) = K\\frac{\\overbrace{B(z)D(z)}^{Q(z)}}{\\underbrace{A(z)C(z)}_{P(z)}} = K \\frac{Q(z)}{P(z)}\\).\n\\item The roots of \\(Q(z)\\) are called the \\alert{open loop zeros}.\n\\item The roots of \\(P(z)\\) are called the \\alert{open loop poles}.\n\\item The characteristic equation for the closed-loop system is \\[ 1 + K\\frac{Q(z)}{P(z)} = 0 \\quad \\Leftrightarrow \\quad P(z) + KQ(z) = 0\\]\n\\end{itemize}\n\\end{frame}\n\n\n\\begin{frame}[label={sec:org0e0a1d3}]{Root locus: Definition}\nLet\n\\[\\begin{cases} P(z)&=z^n+a_1z^{n-1}+\\dots+a_n = (z-p_1)(z-p_2)\\cdots(z-p_n)\\\\ \nQ(z)&=z^m+b_1 z^{m-1}+\\dots+b_m=(s-q_1)(z-q_2)\\cdots(z-q_m) \\end{cases},\\ \\ \\ n\\ge m \\]\n\nThe root locus shows how the \\alert{solution} to the characteristic equation\n\\begin{equation}\n\\label{eq:P(z)+KQ(z)=0}\nP(z)+K\\cdot Q(z)=0,\\ \\ \\ 0\\le K<\\infty\n\\end{equation}\ndepend on the parameter \\(K\\). The root locus consists of the set of all points in the complex plane that are solutions to \\eqref{eq:P(z)+KQ(z)=0} for some non-negative value of \\(K\\).\n\\end{frame}\n\n\\begin{frame}[label={sec:org1e38a16}]{Root locus: Characteristics}\n\\begin{description}\n\\item[{Start points}] The \\(n\\) roots of \\(P(z)\\), marked by crosses\n\\item[{End points}] The \\(m\\) roots of \\(Q(z)\\), marked by circles\n\\item[{Asymptotes}] Number equal to the \\emph{pole excess} \\(n-m\\)\n\\item[{Real axis}] Some segments of the real axis belong to the root locus\n\\end{description}\n\\end{frame}\n\n\\begin{frame}[label={sec:org4d84931}]{Root locus: Direction of the asymptotes}\nThe characteristic equation \\(P(z)+K Q(z)=0\\) can be written \\(\\frac{P(z)}{Q(z)} = -K\\) and for large \\(z\\) it can be approximated as \n\\[ \\frac{z^n}{z^m} = -K \\quad \\Leftrightarrow \\quad z^{n-m} = -K.\\]\n\nTaking the argument of both sides of the equation gives \n\\((n-m)\\arg z = \\pi + k2\\pi, \\; k \\in \\mathbb{Z}\\)\nSo, the \\alert{directions} of the asymptotes are given by the expression\n\\[ \\theta_k = \\arg z = \\frac{(2k+1)\\pi}{n-m}, \\; k \\in \\mathbb{Z} \\]\n\\end{frame}\n\n\\begin{frame}[label={sec:orgd3d871c}]{Root locus: The asymptotes' intersection with the real axis}\n\\[ z_{ip} = \\frac{ \\sum_{i=0}^n p_i - \\sum_{i=0}^m q_i}{n-m}, \\]\nwhere \\(\\{p_i\\}\\) are the starting points (open-loop poles) and \\(\\{q_i\\}\\) are the end points (open-loop zeros). \n\\end{frame}\n\n\\begin{frame}[label={sec:org7ea8274}]{Root locus exerise: Pair the pulse-trf fcn and root locus}\n\\begin{columns}\n\\begin{column}{0.35\\columnwidth}\n\\begin{LaTeX}\n\\small\n\\begin{align*}\n G_1(z) &= K\\frac{(z+2.9)(z+0.2)}{(z-1)^2(z-0.3)}\\\\[3mm]\n G_2(z) &= K\\frac{(z-0.5)(z+0.4)}{(z-1)(z-0.3)(z-0.1)}\\\\[3mm]\n G_3(z) &= K\\frac{(z-0.5)(z+0.8)}{(z-1)^2(z-0.3)}\\\\[3mm]\n G_4(z) &= K \\frac{z-0.6}{(z-1)(z-0.3)}\n\\end{align*}\n\\end{LaTeX}\n\\end{column}\n\n\\begin{column}{0.65\\columnwidth}\n\\begin{center}\n\\includegraphics[width=1.04\\linewidth]{../../matlab/rlocus_2x2-crop}\n\\end{center}\n\\end{column}\n\\end{columns}\n\\end{frame}\n\n\n\\begin{frame}[label={sec:orgaba0adc}]{Draw a root locus}\nLevel control in a hydro power plant dam\n\n\\begin{center}\n\\small\n\\def\\svgwidth{0.5\\linewidth}\n\\input{hydroplant.pdf_tex}\n\\end{center}\n\nDiscrete-time model: \\(y(k+1) - y(k) = \\frac{h}{A} u(k) + \\frac{h}{A}v(k)\\), where \\(y(k)\\) is the deviation in water level from a standard level, \\(u(k)\\) is the (negative) deviation in flow through the dam ports and \\(v(k)\\) is a deviation in other flows (disturbance). \n\\end{frame}\n\n\\begin{frame}[label={sec:org0713314}]{What happens if the poles are \\alert{on the} unit circle?}\nSay, in \\(z = \\mathrm{e}^{\\pm i \\omega_0}\\)\n\\begin{center}\n\\includegraphics[width=0.3\\linewidth]{../../figures/rlocusA.png}\n\\end{center}\n\n\\[H_c(z) = \\frac{k z}{(z-\\mathrm{e}^{i \\omega_0})(z-\\mathrm{e}^{-i \\omega_0})} \\overbrace{+ \\cdots}^{\\text{stable term}}\\].\n\\end{frame}\n\n\\begin{frame}[label={sec:orgb1ee06f}]{What happens if the poles are \\alert{on the} unit circle?}\nSay, in \\(z = \\mathrm{e}^{\\pm i \\omega_0}\\)\n\\begin{center}\n\\includegraphics[width=0.3\\linewidth]{../../figures/rlocusA.png}\n\\includegraphics[height=0.34\\textheight]{../../figures/zgrid-crop}\\\\\n\\end{center}\n\n\\begin{align*}\nH_c(z) &= \\frac{k z}{(z-\\mathrm{e}^{i \\omega_0})(z-\\mathrm{e}^{-i \\omega_0})} + \\cdots\\\\\n &= \\frac{k z}{z^2 -2\\cos\\omega_0 z + 1} + \\cdots\n\\end{align*}\n\nIf \\(\\omega_0 = \\frac{\\pi}{6}\\) and the sampling period is \\unit{0.4}{\\second}, what is the \\alert{frequency} (in \\unit{}{\\radian\\per\\second} and in Hz) of the oscillations in the pulse response?\n\\end{frame}\n\n\\section{Bode diagrams and Nyquist plots}\n\\label{sec:org25d4b12}\n\\begin{frame}[label={sec:org291269e}]{Bode diagram and Nyquist plots}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgfc87a48}]{Sine in --- sine out}\n\\begin{LaTeX}\n\\begin{center}\n \\begin{tikzpicture}[node distance=22mm, block/.style={rectangle, draw, minimum width=15mm}, sumnode/.style={circle, draw, inner sep=2pt}]\n\n \\node[coordinate] (input) {};\n \\node[block, right of=sum, node distance=25mm] (plant) {$H(z)$};\n \\node[coordinate, right of=plant, node distance=40mm] (output) {};\n\n \\draw[->] (input) -- node[above, pos=0.3] {$u(kh) = \\sin(\\omega kh)$} (plant);\n \\draw[->] (plant) -- node [above, pos=1.1] {$y(kh) = |H(\\mathrm{e}^{i\\omega h})|\\sin(\\omega kh + \\arg H(\\mathrm{e}^{i\\omega h}))$} (output);\n \\end{tikzpicture}\n\\end{center}\n\\end{LaTeX}\n\\end{frame}\n\\begin{frame}[label={sec:orgceaa23e}]{Sine in --- sine out}\n\\begin{LaTeX}\n\\begin{center}\n \\begin{tikzpicture}[node distance=22mm, block/.style={rectangle, draw, minimum width=15mm}, sumnode/.style={circle, draw, inner sep=2pt}]\n\n \\node[coordinate] (input) {};\n \\node[block, right of=sum, node distance=25mm] (plant) {$H(z)$};\n \\node[coordinate, right of=plant, node distance=40mm] (output) {};\n\n \\draw[->] (input) -- node[above, pos=0.3] {$u(kh) = \\sin(\\omega kh)$} (plant);\n \\draw[->] (plant) -- node [above, pos=1.1] {$y(kh) = |H(\\mathrm{e}^{i\\omega h})|\\sin(\\omega kh + \\arg H(\\mathrm{e}^{i\\omega h}))$} (output);\n \\end{tikzpicture}\n\\end{center}\n\\end{LaTeX}\n\\alert{Prove it!} Some hints:\n\\begin{itemize}\n\\item Write \\(\\sin(\\omega kh) = \\mathrm{Im}\\{\\mathrm{e}^{i\\omega kh}\\}\\).\n\\item Use \\(H(z)U(z) \\stackrel{\\mathcal{Z}}{\\leftrightarrow} h(k)\\ast u(k)\\), and write out the discrete-time convolution \\(h \\ast u = \\sum_{n=-\\infty}^\\infty h(n)u(k-n) = \\sum_{n=0}^\\infty h(n)u(k-n)\\)\n\\item Try to rewrite to obtain as a factor \\(\\sum_{n=0}^\\infty h(n) \\mathrm{e}^{-i\\omega nh} = H(\\mathrm{e}^{i\\omega h})\\).\n\\end{itemize}\n\\end{frame}\n\n\n\\section{Relative stability}\n\\label{sec:org218756b}\n\n\\begin{frame}[label={sec:orgfe84823}]{The Nyquist plot}\nExample of a \\alert{Nyquist plot} or \\alert{frequency curve}.\n\\begin{LaTeX}\n\\begin{center}\n \\begin{tikzpicture}\n \\node at (0,0) {\\includegraphics[width=0.4\\linewidth]{../../figures/fig3-3.png}};\n \\node[pin=120:{$H(z)$ evaluated at $z=\\mathrm{e}^{i\\omega h}$, $0<\\omega<\\frac{\\pi}{h}$}] at (-1,1) {};\n \\node[pin=40:{$G(s)$ evaluated at $s=i\\omega$, $0<\\omega<\\infty$}] at (-0.98,0.4) {};\n \\end{tikzpicture}\n\\end{center}\n\\end{LaTeX}\n\n\n\nThe system \\(G(s) = \\frac{1}{s^2 + 1.4s + 1}\\) is sampled with ZOH-sampling (\\(h=\\unit{0.4}{\\second}\\)) to get \\(H(z) = \\frac{0.066z + 0.055}{z^2 - 1.45 z + 0.571}\\)\n\\end{frame}\n\n\\begin{frame}[label={sec:orgbad54b5}]{Simplified Nyquist criterion}\n\\begin{center}\n\\includegraphics[width=0.4\\linewidth]{../../figures/fig3-3.png}\n\\end{center}\n\nConsider the loop pulse-transfer function \\(L(z)\\) of a closed-loop system. If \\(L(z)\\) is stable (no poles outside the unit circle), then the closed-loop system with characteristic equation \\(1 + L(z) = 0\\) will be stable iff \\(L(z)\\) evaluated on the unit circle (i.e, the Nyquist plot of \\(L\\)) has the point \\alert{-1 to the left}.\n\\end{frame}\n\n\\begin{frame}[label={sec:org75821a8}]{Stability margins - phase margin}\n\\begin{center}\n\\includegraphics[width=0.38\\linewidth]{../../figures/implane-nyquist-margins}\n\\end{center}\n\\begin{itemize}\n\\item Cross-over frequency: The frequency \\(\\omega_c\\) for which \\(|L(\\mathrm{e}^{i\\omega h})| = 1\\).\n\\item Phase margin: The angle \\(\\varphi_m\\) to the negative real axis for the point where the Nyquist curve intersects the unit circle. \\[\\varphi_m = \\arg L(\\mathrm{e}^{i\\omega_c h}) - (-180\\degree) = \\arg L(\\mathrm{e}^{i\\omega_c h}) + 180\\degree\\]\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgb16d58a}]{Stability margins - gain margin}\n\\begin{center}\n\\includegraphics[width=0.34\\linewidth]{../../figures/implane-nyquist-margins}\n\\end{center}\n\\begin{itemize}\n\\item phase-cross-over frequency: The frequency \\(\\omega_p\\) for which \\(\\arg L(\\mathrm{e}^{i\\omega h}) = -180\\degree\\).\n\\item Gain margin: The gain \\(K=A\\) that would make the Nyquist curve of \\(K L(\\mathrm{e}^{i\\omega h})\\) go through the point \\(-1 + i0\\). This means that \\[ |L(\\mathrm{e}^{i\\omega_p h})| = \\frac{1}{A}. \\]\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[label={sec:orge210cb8}]{The relationship between Bode diagrams and frequency curves (Nyquist plots)}\nThey are both showing the value of a pulse-transfer function \\(H(z)\\) evaluated for \\(z = \\mathrm{e}^{i\\omega h}\\), \\(0<\\omega \\le \\frac{\\pi}{h}\\).\n\n\\begin{center}\n\\includegraphics[width=\\linewidth]{../../figures/fig3-3-4.png}\n\\end{center}\n\\end{frame}\n\\end{document}", "meta": {"hexsha": "b0eb3bf682f881313462642ae602eadd42a67ec6", "size": 12715, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "discrete-time-systems/slides/discrete-time-sys-analysis.tex", "max_stars_repo_name": "kjartan-at-tec/mr2007-computerized-control", "max_stars_repo_head_hexsha": "16e35f5007f53870eaf344eea1165507505ab4aa", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2020-11-07T05:20:37.000Z", "max_stars_repo_stars_event_max_datetime": "2020-12-22T09:46:13.000Z", "max_issues_repo_path": "discrete-time-systems/slides/discrete-time-sys-analysis.tex", "max_issues_repo_name": "kjartan-at-tec/mr2007-computerized-control", "max_issues_repo_head_hexsha": "16e35f5007f53870eaf344eea1165507505ab4aa", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2020-06-12T20:44:41.000Z", "max_issues_repo_issues_event_max_datetime": "2020-06-12T20:49:00.000Z", "max_forks_repo_path": "discrete-time-systems/slides/discrete-time-sys-analysis.tex", "max_forks_repo_name": "kjartan-at-tec/mr2007-computerized-control", "max_forks_repo_head_hexsha": "16e35f5007f53870eaf344eea1165507505ab4aa", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-03-14T03:55:27.000Z", "max_forks_repo_forks_event_max_datetime": "2021-03-14T03:55:27.000Z", "avg_line_length": 41.0161290323, "max_line_length": 335, "alphanum_fraction": 0.6675580024, "num_tokens": 4594, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6261241772283034, "lm_q2_score": 0.5467381519846138, "lm_q1q2_score": 0.34232597557068944}} {"text": "%!TEX root = /home/renaud/Documents/EPL/tfe/latex/tfe.tex\n\\chapter{From clusters to compartments: the method} \\label{chap:method}\nIn this chapter, we formalize the method used to go from the numerical implementation of the particle trajectories to the clustering allowing to delineate the compartments in a box model. In other words, it is explained how to use a clustering algorithm on an advection-diffusion problem, and then how to choose which communities found by the stability method to use as compartments in a box model. \n\nThe underpinning idea of the method is that communities found on the dynamic of the flow could be relevant compartments for the box model. The idea makes sense although it is far from being obvious that the communities are indeed the compartments that we seek. In a first instance, we should only check that, at least in some cases, community detection leads to relevant partitions. In the next chapter, a test problem is build for which we know in advances what compartments to expect, and the method is tested on that problem.\n%Then we apply the method on the less obvious, though still very naive overturner model of the Atlantic ocean.\n\n\\paragraph{Remark} \\label{remark:straightboundaries} In the context of interpreting the communities as compartments for a box model, we should require that the communities have vertical and horizontal boundaries. For example, when dealing with marine problems, the goal of a box model is to provide a simple and intuitive description of the problem. Complex shaped compartments are neither simple nor intuitive for marine models, hence the requirement.\n\n\\input{inputs/method/generalmethod}\n% \\input{inputs/method/overturner_clustering}\n", "meta": {"hexsha": "381ba7a49f02d9ad179f455c61cabd31f03cb119", "size": 1708, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "inputs/method/method.tex", "max_stars_repo_name": "dufaysr/tfe", "max_stars_repo_head_hexsha": "75c6191e1533da84233d4a38dea3cc3f3884a286", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "inputs/method/method.tex", "max_issues_repo_name": "dufaysr/tfe", "max_issues_repo_head_hexsha": "75c6191e1533da84233d4a38dea3cc3f3884a286", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "inputs/method/method.tex", "max_forks_repo_name": "dufaysr/tfe", "max_forks_repo_head_hexsha": "75c6191e1533da84233d4a38dea3cc3f3884a286", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 142.3333333333, "max_line_length": 528, "alphanum_fraction": 0.8120608899, "num_tokens": 346, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5813031051514763, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3423230803052945}} {"text": "%\\chapter{Advanced Techniques V: Ferns-based miniKanren}\\label{fernschapter}\n\\chapter{Techniques IV: Ferns}\\label{fernschapter}\n\nIn this chapter we provide a bottom-avoiding generalization of core\nminiKanren using \\emph{ferns} \\cite{ferns81}, a shareable data\nstructure designed to avoid divergence.\n\nThe chapter is organized as follows. Section~\\ref{fernsintro}\nintroduces the ferns data structure and shows examples of familiar\nrecursive functions using ferns. Section~\\ref{Sharing} describes the\npromotion algorithm \\cite{Friedman79b} that characterizes the\nnecessary sharing properties of ferns. Section~\\ref{lp-system}\ndefines bottom-avoiding logic programming goal constructors,\ncorresponding to core miniKanren with non-interleaving search.\nChapter~\\ref{fernsimpl} presents a complete \\emph{shallow\n embedding} \\cite{Boulton92tassel.experience} of the ferns data\nstructure and related operators.\n\n\\section{Introduction to Ferns}\\label{fernsintro}\n\nFerns are constructed with \\scheme|cons| and \\scheme|frons|,\noriginally called \\textbf{frons} \\cite{DFried80}, and accessed by\n\\scheme|fcar| and \\scheme|fcdr|, generalizations of \\scheme|car| and\n\\scheme|cdr|, respectively. Ferns built with \\scheme|frons| are like\nstreams in that the \\emph{evaluation} of elements is delayed,\npermitting unbounded data structures. In contrast to streams, the\n\\emph{ordering} of elements is also delayed: convergent values form\nthe prefix in some unspecified order, while divergent values form the\nsuffix.\n\nWe begin with several examples that illustrate the\nproperties of ferns, showing their similarities to\nand differences from traditional lists and streams. Later, we include\nexamples that show that a natural recursive style can be used when\nprogramming with ferns and point out the advantages ferns afford the\nuser.\n\n\\subsection{Two Simple Programs}\n\nConvergent elements of a fern form its prefix in some unspecified\norder. For example, evaluating the expression\n\n\\schemedisplayspace\n\\schemeinput{fernscode/intro-1}\n\n\\noindent \nprints either \\schemeresult|010| or \\schemeresult|101|,\ndemonstrating that the order of values within a fern is not specified in\nadvance but remains consistent once determined, while\n\n\\schemedisplayspace\n\\schemeinput{fernscode/intro-2}\n\n\\noindent returns \\mbox{\\schemeresult|(720 . 120)|}, demonstrating\nthat accessing a fern avoids divergence as much as possible.\n(\\scheme|bottom| is any expression whose evaluation diverges.) In the\nlatter example, each fern contains only one convergent value; taking\nthe \\scheme|fcdr| of \\scheme|s1| or the \\scheme|fcadr| of \\scheme|s2|\nresults in divergence.\n\nFerns are \\emph{shareable} data structures; sharing, combined with\ndelayed ordering of values, can result in surprising behavior. For\nexample, consider these expressions:\n\n\\schemedisplayspace\n\\schemeinput{fernscode/not-so-weird-sharing}\n\n\\noindent and\n\n\n\\schemedisplayspace\n\\schemeinput{fernscode/weird-sharing-1}\n\n\\noindent\nThe first expression must evaluate to \\mbox{\\schemeresult|(1 2\n 2)|}. The second expression may also return this value---as\nexpected, the car of \\scheme|b| would then be equal to the cadr of\n\\scheme|a|. The second expression might instead return\n\\mbox{\\schemeresult|(2 1 2)|} however; in this case, the car of \\scheme|b|\nwould be equal to the car of \\scheme|a| rather than to its cadr.\nSection~\\ref{Sharing} discusses sharing in detail.\n\n\\subsection{Recursion}\\label{fernsRecursiveExamples}\n\nWe now present examples of the use of ferns in simple recursive\nfunctions. Consider the definition of \\scheme|ints-bottom|\\footnote{\\scheme|timed-lambda| is identical to\n \\scheme|lambda|, except it creates preemptible procedures. \n(See Appendix~\\ref{nestable-engines}.)}.\n\n\\schemedisplayspace\n\\schemeinput{fernscode/ints}\n\n\\noindent Then \\mbox{\\scheme|(fcaddr (ints-bottom 0))|} \ncould return any non-negative integer, whereas a\nstream version would return \\schemeresult|2|.\n\n%But wait---there's more!\n\n% \\schemedisplayspace\n%\\schemeinput{fernscode/code/ints-bottom}\n\nThere is a tight relationship between ferns and lists, since every\ncons pair is a fern. The empty\nfern is also represented by \\scheme|`()|, and\n\\mbox{\\scheme|(pair? (frons e1 e2))|} returns \\schemeresult|#t| for\nall \\scheme|e1| and \\scheme|e2|. After replacing the list constructor\n\\scheme|cons| with the fern constructor \\scheme|frons|, many recursive\nfunctions operating on lists avoid divergence. For example,\n\\scheme|map-bottom| is defined\nby replacing \\scheme|cons| with \\scheme|frons|, \\scheme|car| with\n\\scheme|fcar|, and \\scheme|cdr| with \\scheme|fcdr| in the definition of\n\\scheme|map|, and can map a function over an unbounded fern:\n% inlined this from code/map-example.ss\nthe value of \\mbox{\\scheme|(fcaddr (map-bottom add1 (ints-bottom 0)))|} can be any positive integer.\n\n\n%The similarities between ferns and the more traditional list and\n%stream data structures make it possible to write bottom-avoiding\n%functions in the natural recursive style---indeed, this ability to\n%define functions in a familiar style is one of the advantages of using\n%ferns.\n\nFerns work especially well with \\emph{annihilators}. True values are\nannihilators for \\scheme|or-bottom|\n\n\\schemedisplayspace\n\\schemeinput{fernscode/or-fn}\n\n\\noindent which searches in a fern for a true convergent value\nand avoids divergence if it finds one:\n\\mbox{\\scheme|(or-bottom (fern bottom (odd? 1) (! 5) bottom (odd? 0)))|}\nreturns some true value, where \\scheme|fern| is defined as follows.\n\n\\schemedisplayspace\n\\schemeinput{fernscode/fern}\n\n% We get additional benefit when annihilators are common. Scheme's\n% \\scheme|or|, for example, uses any true value as an annihilator. We\n% can define \\scheme|or-bottom|, similar to \\scheme|or|, that takes any\n% number of expressions and returns a true value nondeterministically.\n% Thus, \\scheme|(or-bottom (odd? 0) bottom (! 5) (odd? 1) bottom)|\n% returns either \\scheme|#t| or \\scheme|120|, both of which\n% are true values in Scheme. \\scheme|or-bottom| is defined as a macro\n% that wraps its arguments in \\scheme|fern| before passing them to a\n% recursive auxiliary for two reasons: first, to make the auxiliary\n% lazy, and second, to allow the auxiliary to choose convergent\n% arguments before divergent ones. This strategy will be used again in\n% Section~\\ref{mplus-bottom-and-bind-bottom} to define \\scheme|mplus-bottom|.\n\n%\\schemedisplayspace\n%\\schemeinput{fernscode/code/or-fn}\n\n\\noindent\nLet us define \\scheme|append-bottom| for ferns. \n\n\\schemedisplayspace\n\\schemeinput{fernscode/append}\n\n\\noindent\nTo observe the behavior of \\scheme|append-bottom|, we define \\scheme|take-bottom| whose\nfirst argument is either \\scheme|#f| (all results) or $n > 0$ (no more than $n$ results).\n\n\\schemedisplayspace\n\\schemeinput{fernscode/take}\n\n\\noindent \nWhen determining the $n$th value, it is necessary to avoid taking the\n\\scheme|fcdr| after the $n$th value is determined, since it is that \\scheme|fcdr|\nthat might not terminate and we already have $n$ results.\n\nThe definition of \\scheme|append-bottom| appears to work as expected:\n\n\\wspace\n\\noindent\n\\scheme|(take-bottom 2 (append-bottom (fern 1) (fern bottom 2)))| \\schemeresult|=> (1 2)|.\n\\wspace\n\n\\noindent Moving \\scheme|bottom| from the second argument to the\nfirst, however, reveals a problem:\n\n\\wspace\n\\noindent\n\\scheme|(take-bottom 2 (append-bottom (fern bottom 1) (fern 2)))| \\schemeresult|=> bottom|.\n\\wspace\n\n\\noindent\nEven though the result of the call to\n\\scheme|append-bottom| should contain two convergent elements, taking the first\ntwo elements of that result diverges. This is because the\ndefinition of \\scheme|append-bottom| requires that \\scheme|s1| be completely\nexhausted before any elements from \\scheme|s2| can appear in the result.\nIf one of the elements of \\scheme|s1| is \\scheme|bottom|, then no\nelement from \\scheme|s2| will ever appear. The same is true if\n\\scheme|s1| contains an unbounded number of convergent elements: since\n\\scheme|s1| is never null, the result will never contain elements from\n\\scheme|s2|. With the definition of \\scheme|mplus-bottom| in\nSection~\\ref{mplus-bottom-and-bind-bottom}, it becomes clear that the solution to these problems is to\ninterleave the elements from \\scheme|s1| and \\scheme|s2| in the\nresulting fern as in the next example.\n\nFunctional programs often share rather than copy data, and ferns are\ndesigned to encourage this programming style. Consider a procedure to\ncompute the Cartesian product of two ferns:\n\n\\schemedisplayspace\n\\schemeinput{fernscode/cartesian}\n\n\\schemeinput{fernscode/cartesian-example}\\schemeresult|~> `((a . x) (a . y) (b . x) (a . z) (b . y) (b . z))|\n\\belowcodeskip \\medskipamount\n\\medskip\n\n\\noindent\nwhere \\schemeresult|~>| indicates \\emph{one} of the possible values.\nThis definition ensures that the resulting fern shares\nelements with the ferns passed as arguments. Many references to a\nparticular element may be made without repeating computations, hence the\nexpression\n\n\\wspace\n\n\\noindent\n\\scheme|(take-bottom 2 (Cartesian-product-bottom (fern (begin (display #t) 5)) (fern 'a bottom 'b)))|\n\n\\tspace\n\n\\noindent\n\\schemeresult|~> `((5 . a) (5 . b))|\n\\belowcodeskip \\medskipamount\n\\medskip\n\n\\noindent prints \\schemeresult|#t| \\emph{exactly once}.\n(There are more examples of the use of ferns in~\\citet{Johnson-83}, \\citet{Filman-Friedman-84}, and~\\citet{Jeschke-PHD-95}.)\n\nIn the next section we look at how the sharing properties of ferns are\nmaintained alongside bottom-avoidance.\n\n\\section{Sharing and Promotion}\\label{Sharing}\n\nIn this section, we provide examples and a high-level description of\nthe \\emph{promotion algorithm} of Friedman and Wise~\\cite{Friedman79b}. \nThe values in a fern are computed and\n\\emph{promoted} across the fern while ensuring that the correct values\nare available from each subfern, \\scheme|bottom|'s are avoided, and\nnon-\\scheme|bottom| values are computed only once.\n% A proper fern is either the empty list or a pair whose cdr is another fern. \nFerns have structure, and there may be references to more than one\nsubfern of a particular fern. Consider the example expression\n\n\\schemedisplayspace\n\\schemeinput{fernscode/sharing-1}\n\n\\begin{schemeresponse}~> `((6 120 720) (6 120 720) (6 720) (720))\n\\end{schemeresponse}\n\n\\noindent assuming \\scheme|list| evaluates its arguments\nleft-to-right. Importantly,\naccessing \\scheme|delta| cannot retrieve values in the prefix of the\nenclosing fern \\scheme|alpha|. We now describe in detail how the\nresult of \\mbox{\\scheme|(take-bottom 3 alpha)|} is determined along with\nthe necessary changes to the fern data structure during this\nprocess. Whenever we encounter a choice, we shall assume a choice\nconsistent with the value returned in the example.\n\nDuring the first access of \\scheme|alpha| the cdrs are evaluated, as\nindicated by the arrows in Figure~\\ref{fig:solid}a.\nFigure~\\ref{fig:solid}b depicts the data structure after\n\\mbox{\\scheme|(fcar alpha)|} is evaluated. We assume that, of the\npossible values for \\mbox{\\scheme|(fcar alpha)|}, namely\n\\scheme|bottom| (which is never chosen), \\mbox{\\scheme|(! 5)|},\n\\mbox{\\scheme|(! 3)|}, and \\mbox{\\scheme|(! 6)|}, the value of\n\\mbox{\\scheme|(! 3)|} is chosen and promoted. Since the value\nof \\mbox{\\scheme|(! 3)|} might be a value for \\mbox{\\scheme|(fcar beta)|}\n and \\mbox{\\scheme|(fcar gamma)|}, we replace the cars of all\nthree pairs with the value of \\mbox{\\scheme|(! 3)|}, which\nis \\schemeresult|6|. We replace the cdrs of \\scheme|alpha| and\n\\scheme|beta| with new frons pairs containing \\scheme|bottom| and\n\\mbox{\\scheme|(! 5)|}, which were not chosen. The new frons pairs are\nlinked together, and linked at the end to the old cdr of\n\\scheme|gamma|. Thus \\scheme|alpha|, \\scheme|beta|, and \\scheme|gamma|\nbecome a fern with \\schemeresult|6| in the car and a fern of the\nrest of their original possible values in their cdrs. As a result of\nthe promotion, $\\alpha$, $\\beta$, and $\\gamma$ become cons pairs,\nrepresented in the figures by rectangles.\n\nFigure~\\ref{fig:solid}c depicts the data structure after\n\\mbox{\\scheme|(fcadr alpha)|} is evaluated. This time,\n\\mbox{\\scheme|(! 5)|} is chosen from \\scheme|bottom|, \\mbox{\\scheme|(! 5)|}, \nand \\mbox{\\scheme|(! 6)|}. Since the value of \\mbox{\\scheme|(! 5)|} is\nalso a possible value for \\mbox{\\scheme|(fcadr beta)|}, we replace\nthe cadrs of both \\scheme|alpha| and \\scheme|beta| with the value of\n\\mbox{\\scheme|(! 5)|}, which is \\schemeresult|120|, and replace the cddr of\n\\scheme|alpha| with a frons pair containing the \\scheme|bottom| that\nwas not chosen and a pointer to \\scheme|delta|. The cddr of \\scheme|beta| points to \\scheme|delta|; no\nnew fern with remaining possible values is needed because the value\nchosen for \\mbox{\\scheme|(fcadr beta)|} was the first value\navailable. As before, the pairs containing values become cons pairs.\n\n% \\renewcommand{\\figurename}{Program}\n\n% \\begin{figure}[H]\n% \\begin{schemeregion}\n% \\schemeinput{fernscode/code/sharing-1}\n% \\end{schemeregion}\n% \\caption{Program that constructs and accesses fern $\\alpha$.\\label{fig:code/sharing-1}}\n% \\end{figure}\n\n\\renewcommand{\\figurename}{Figure}\n\\addtocounter{figure}{0}\n\n% \\newcommand{\\ahwd}{4}\n% \\newcommand{\\ahht}{3}\n% \\newcommand{\\dahwd}{5}\n% \\newcommand{\\hahht}{2}\n% \\newcommand{\\thahht}{1}\n% \\newcommand{\\dahht}{6}\n% \\newcommand{\\namegap}{8}\n% \\newcommand{\\ftxht}{12}\n% \\newcommand{\\ctxht}{10}\n% \\newcommand{\\ebxwd}{3}\n% \\newcommand{\\qbxwd}{10}\n% \\newcommand{\\hbxwd}{20}\n% \\newcommand{\\sbxwd}{6}\n% \\newcommand{\\fbxwd}{40}\n\n\\newcommand{\\ahwd}{4}\n\\newcommand{\\ahht}{3}\n\\newcommand{\\dahwd}{5}\n\\newcommand{\\hahht}{2}\n\\newcommand{\\thahht}{1}\n\\newcommand{\\dahht}{6}\n\\newcommand{\\namegap}{8}\n\\newcommand{\\ftxht}{10}\n\\newcommand{\\ctxht}{8}\n\\newcommand{\\ebxwd}{2}\n\\newcommand{\\qbxwd}{7}\n\\newcommand{\\hbxwd}{14}\n\\newcommand{\\sbxwd}{4}\n\\newcommand{\\fbxwd}{28}\n\n\n\\Define\\namebox(1) {\n \\Move(0,\\hbxwd) \n \\Move(0,\\namegap) \n \\Text(--#1--) \n \\Move(0,-\\namegap) \n \\Move(0,-\\hbxwd)\n}\n\n\\Define\\namefig(3) { % name, half width, height\n \\Move(#2,-#3)\n \\Move(0,-\\namegap)\n \\Text(--$\\mbox{(#1)}$--)\n \\Move(0,\\namegap)\n \\Move(-#2,#3)\n}\n\n\\newcommand{\\boxtext}{Z}\n\n\\Define\\fronsbox {\n \\Line(0,\\hbxwd) \n \\MarkLoc(p1)\n \\Move(\\qbxwd,\\qbxwd)\n \\MarkLoc(q1)\n \\Move(\\hbxwd,-\\hbxwd)\n \\MarkLoc(q2)\n \\Move(\\qbxwd,\\qbxwd)\n \\MarkLoc(p2) \n \\Curve(p1,q1,q2,p2) \n \\Line(0,-\\hbxwd)\n \\MarkLoc(p1)\n \\Move(-\\qbxwd,-\\qbxwd)\n \\MarkLoc(q1)\n \\Move(-\\hbxwd,\\hbxwd)\n \\MarkLoc(q2)\n \\Move(-\\qbxwd,-\\qbxwd)\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n \\Move(\\qbxwd,\\ftxht)\n \\Text(--\\boxtext--)\n \\Move(\\qbxwd,-\\ftxht)\n \\Line(0,\\hbxwd)\n \\Move(\\qbxwd,-\\qbxwd)\n}\n\n\\Define\\arrowhead {\n \\Move(-\\ahwd,-\\ahht) \\Line(\\ahwd,\\ahht)\n \\Move(-\\ahwd,\\ahht) \\Line(\\ahwd,-\\ahht)\n \\Move(1,0)\n}\n\n\\Define\\rightdotted {\n \\Do(0,\\qbxwd){\\Line(1,0)\\Move(2,0)}\n \\arrowhead\n \\Move(0,-\\qbxwd)\n}\n\n\\Define\\rightsolid {\n \\Do(0,\\qbxwd){\\Line(3,0)}\n \\arrowhead\n \\Move(0,-\\qbxwd)\n}\n\n\\Define\\boxnil {\n \\Move(-\\qbxwd,-\\qbxwd)\n \\Line(\\hbxwd,\\hbxwd)\n \\Move(0,-\\qbxwd)\n}\n\n\\Define\\dottednil {\n \\Move(-\\qbxwd,-\\qbxwd)\n \\Do(0,\\sbxwd){\\Line(1,1)\\Move(2,2)}\n \\Move(0,-\\qbxwd)\n}\n\n\\Define\\blankbox {\n \\Move(\\hbxwd,0)\n \\Move(\\qbxwd,0)\n \\Do(0,\\qbxwd){\\Move(3,0)}\n \\Move(1,0)\n}\n\n\\Define\\consbox {\n \\Line(0,\\hbxwd) \n \\Line(\\fbxwd,0)\n \\Line(0,-\\hbxwd)\n \\Line(-\\fbxwd,0)\n \\Move(\\qbxwd,\\ctxht)\n \\Text(--\\boxtext--)\n \\Move(\\qbxwd,-\\ctxht)\n \\Line(0,\\hbxwd)\n \\Move(\\qbxwd,-\\qbxwd)\n}\n\n\\Define\\downhead {\n \\Move(-\\ahht,\\ahwd) \\Line(\\ahht,-\\ahwd)\n \\Move(\\ahht,\\ahwd) \\Line(-\\ahht,-\\ahwd)\n \\Move(0,-1)\n}\n\n\\Define\\downsolid {\n \\Line(0,-\\hbxwd)\n \\Line(0,-\\qbxwd)\n \\downhead\n \\Move(\\qbxwd,0)\n \\Move(-\\fbxwd,-\\hbxwd)\n}\n\n\\Define\\longdownsolid {\n \\Line(0,-\\hbxwd)\n \\Line(0,-\\qbxwd)\n \\Line(0,-\\ebxwd)\n \\downhead\n \\Move(\\qbxwd,\\ebxwd)\n \\Move(-\\fbxwd,-\\hbxwd)\n}\n\n\\Define\\blankleft {\n \\Move(-\\hbxwd,0)\n \\Move(-\\qbxwd,0)\n \\Do(0,\\qbxwd){\\Move(-3,0)}\n \\Move(-1,0)\n}\n\n\\Define\\blankup {\n \\Move(0,\\fbxwd)\n \\Move(0,1)\n}\n\n\\Define\\rightuphead {\n \\Move(-\\dahwd,-\\hahht) \\Line(\\dahwd,\\hahht)\n \\Move(0,-\\dahht) \\Line(0,\\dahht)\n}\n\n\\Define\\tiltedrightuphead {\n \\Move(-\\dahwd,-\\thahht) \\Line(\\dahwd,\\thahht)\n \\Move(-1,-\\dahht) \\Line(1,\\dahht)\n}\n\n\\Define\\rightupdoublesolid {\n \\MarkLoc(p1)\n \\Move(\\fbxwd,0)\n \\MarkLoc(q1)\n \\Move(\\fbxwd,0)\n \\MarkLoc(q2)\n \\blankup\n \\Move(\\hbxwd,-\\qbxwd)\n \\Move(-1,-1)\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n \\rightuphead\n \\Move(1,1)\n}\n\n\\Define\\rightupdoublelonger {\n \\MarkLoc(p1)\n \\Move(\\fbxwd,0)\n \\MarkLoc(q1)\n \\Move(\\fbxwd,0)\n \\MarkLoc(q2)\n \\blankup\n \\Move(\\hbxwd,-\\qbxwd)\n \\Move(4,4)\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n \\rightuphead\n \\Move(-4,-4)\n}\n\n\\Define\\blankdown {\n \\Move(0,-\\fbxwd)\n}\n\n\\Define\\rightuptriplesolid {\n \\MarkLoc(p1)\n \\blankbox\n \\blankbox\n \\MarkLoc(q1)\n \\MarkLoc(q2)\n \\Do(0,\\qbxwd){\\Move(3,0)}\n \\Move(1,0)\n \\blankup\n \\Move(0,\\hbxwd)\n \\Move(0,\\qbxwd)\n \\Move(-3,0)\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n% \\rightuphead\n \\Move(3,0)\n \\blankleft \\blankleft \\blankdown\n}\n\n\n\\vspace{20pt}\n\n\\Define\\edown {\n% \\Line(0,-\\hbxwd)\n% \\Line(0,-\\qbxwd)\n \\Do(0,\\qbxwd){\\Line(0,-2)}\n \\downhead\n \\Move(0,-1)\n}\n\n\\Define\\unedown {\n \\Move(0,1)\n \\Move(0,1)\n \\Do(0,\\qbxwd){\\Move(0,2)}\n% \\Move(0,\\qbxwd)\n% \\Move(0,\\hbxwd)\n}\n\n\\newcommand\\egap{3}\n\n\\Define\\lengine {\n \\edown\n \\Move(\\egap,0)\n \\Line(-\\hbxwd,0)\n \\Line(0,-\\hbxwd)\n \\Line(\\qbxwd,0)\n \\Move(0,\\qbxwd)\n \\Text(--\\boxtext--)\n \\Move(0,-\\qbxwd)\n \\Line(\\qbxwd,0)\n \\Line(0,\\hbxwd)\n \\Move(-\\egap,0)\n \\unedown\n}\n\n\\Define\\rengine {\n \\edown\n \\Move(-\\egap,0)\n \\Line(\\hbxwd,0)\n \\Line(0,-\\hbxwd)\n \\Line(-\\qbxwd,0)\n \\Move(0,\\qbxwd)\n \\Text(--\\boxtext--)\n \\Move(0,-\\qbxwd)\n \\Line(-\\qbxwd,0)\n \\Line(0,\\hbxwd)\n \\Move(\\egap,0)\n \\unedown\n}\n\n\\Define\\nodownrengine {\n \\Move(0,-\\hbxwd)\n \\Move(0,-\\qbxwd)\n \\Move(0,-2)\n \\Move(-\\egap,0)\n \\Line(\\hbxwd,0)\n \\Line(0,-\\hbxwd)\n \\Line(-\\qbxwd,0)\n \\Move(0,\\qbxwd)\n \\Text(--\\boxtext--)\n \\Move(0,-\\qbxwd)\n \\Line(-\\qbxwd,0)\n \\Line(0,\\hbxwd)\n \\Move(\\egap,0)\n \\Move(0,2)\n \\Move(0,\\qbxwd)\n \\Move(0,\\hbxwd)\n}\n\n\\Define\\efronsbox {\n \\Line(0,\\hbxwd) \n \\MarkLoc(p1)\n \\Move(\\qbxwd,\\qbxwd)\n \\MarkLoc(q1)\n \\Move(\\hbxwd,-\\hbxwd)\n \\MarkLoc(q2)\n \\Move(\\qbxwd,\\qbxwd)\n \\MarkLoc(p2) \n \\Curve(p1,q1,q2,p2) \n \\Line(0,-\\hbxwd)\n \\MarkLoc(p1)\n \\Move(-\\qbxwd,-\\qbxwd)\n \\MarkLoc(q1)\n \\Move(-\\hbxwd,\\hbxwd)\n \\MarkLoc(q2)\n \\Move(-\\qbxwd,-\\qbxwd)\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n \\Move(\\qbxwd, \\qbxwd)\n \\lengine\n \\Move(\\qbxwd, -\\qbxwd)\n \\Line(0,\\hbxwd)\n \\Move(\\qbxwd,-\\qbxwd)\n}\n\n\\Define\\rightunright{\n \\Do(0,\\qbxwd){\\Line(3,0)}\n \\arrowhead\n \\Move(-1,0)\n \\Do(0,\\qbxwd){\\Move(-3,0)}\n}\n\n%%\\enlargethispage{20pt}\n\n\\renewcommand\\egap{7}\n\n\\begin{figure}[h]\n\n \\wspace\n\n \\wspace \n\n \\wspace\n \n\\begin{schemeregion}\n\\begin{picture}(160,60)(0,-60)\n\\Draw\\PenSize(1pt)\n\\namefig(a,80,45)\n\\namebox(\\scheme|alpha|)\n\\renewcommand{\\boxtext}{\\scheme|bottom|}\n\\efronsbox\\rightsolid\n\\namebox(\\scheme|beta|)\n\\renewcommand{\\boxtext}{\\scheme|! 5|}\n\\efronsbox\\rightsolid\n\\namebox(\\scheme|gamma|)\n\\renewcommand{\\boxtext}{\\scheme|! 3|}\n\\efronsbox\\rightsolid\n\\namebox(\\scheme|delta|)\n\\renewcommand{\\boxtext}{\\scheme|! 6|}\n\\efronsbox\\boxnil\n\\EndDraw\n\\end{picture}\n\\end{schemeregion}\n\n\\begin{schemeregion}\n\\begin{picture}(160,0)(-180,-70)\n\\Draw\\PenSize(1pt)\n\\namefig(b,80,45)\n\\namebox(\\scheme|alpha|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\longdownsolid\n\\renewcommand{\\boxtext}{\\scheme|bottom|}\n\\efronsbox\\rightsolid\n\\renewcommand{\\boxtext}{\\scheme|! 5|}\n\\efronsbox\\rightupdoublesolid\n\\blankleft \\blankleft \\Move(-1,0)\n\\namebox(\\scheme|beta|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\longdownsolid\n\\blankup \\blankbox\n\\namebox(\\scheme|gamma|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\rightsolid\n\\namebox(\\scheme|delta|)\n\\renewcommand{\\boxtext}{\\scheme|! 6|}\n\\efronsbox\\boxnil\n\\EndDraw\n\\end{picture}\n\\end{schemeregion}\n\n\\begin{schemeregion}\n\\begin{picture}(160,110)(0,-90)\n\\Draw\\PenSize(1pt)\n\\namefig(c,80,95)\n\\namebox(\\scheme|alpha|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\downsolid\n\\renewcommand{\\boxtext}{\\schemeresult|onetwenty|}\n\\consbox\\longdownsolid\n\\renewcommand{\\boxtext}{\\scheme|bottom|}\n\\efronsbox\\rightuptriplesolid\n\\renewcommand{\\boxtext}{\\schemeresult|onetwenty|}\n\\consbox\\rightupdoublesolid\n\\blankleft \\blankleft\n\\namebox(\\scheme|beta|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\downsolid\n\\blankup \\blankbox\n\\namebox(\\scheme|gamma|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\rightsolid\n\\namebox(\\scheme|delta|)\n\\renewcommand{\\boxtext}{\\scheme|! 6|}\n\\efronsbox\\boxnil\n\\EndDraw\n\\end{picture}\n\\end{schemeregion}\n\n\\begin{schemeregion}\n\\begin{picture}(160,0)(-180,-100)\n\\Draw\\PenSize(1pt)\n\\namefig(d,80,95)\n\\namebox(\\scheme|alpha|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\downsolid\n\\renewcommand{\\boxtext}{\\schemeresult|onetwenty|}\n\\consbox\\downsolid\n%\\renewcommand{\\boxtext}{\\schemeresult|720|}\n\\renewcommand{\\boxtext}{\\schemeresult|seventwenty|}\n\\consbox\\longdownsolid\n\\renewcommand{\\boxtext}{\\scheme|bottom|}\n\\efronsbox\\boxnil\n\\blankup\n\\Move(\\hbxwd,\\hbxwd)\n\\Move(\\qbxwd,\\qbxwd)\n\\Move(-3,1)\n\\renewcommand{\\boxtext}{\\schemeresult|onetwenty|}\n\\consbox\\rightupdoublesolid\n\\blankleft \\blankleft\n\\namebox(\\scheme|beta|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\downsolid\n\\blankup \\blankbox\n\\namebox(\\scheme|gamma|)\n\\renewcommand{\\boxtext}{\\schemeresult|six|}\n\\consbox\\rightsolid\n\\namebox(\\scheme|delta|)\n\\renewcommand{\\boxtext}{\\schemeresult|seventwenty|}\n\\consbox\\boxnil\n\\EndDraw\n\\end{picture}\n\\end{schemeregion}\n\n\\vspace{20pt}\n\n\\caption{Fern $\\alpha$ immediately after evaluation of cdrs, but before any cars have finished evaluation (a) and after the values, $6$ (b), $120$ (c), and $720$ (d) have been promoted.\\label{fig:solid}}\n\\end{figure}\n\n\\renewcommand\\egap{3}\n\nFigure~\\ref{fig:solid}d depicts the data structure after\n\\mbox{\\scheme|(fcaddr alpha)|} is evaluated. Of \\scheme|bottom| and\n\\mbox{\\scheme|(! 6)|}, it comes as no surprise that \\mbox{\\scheme|(!\n 6)|} is chosen. Since the value of \\mbox{\\scheme|(! 6)|}, which is\n\\schemeresult|720|, is also a possible value for \\mbox{\\scheme|(fcar\n delta)|} (and in fact the only one), we update the car of\n\\scheme|delta| and the car of the cddr of \\mbox{\\scheme|alpha|} with\n\\schemeresult|720|. The cdr of \\scheme|delta| remains as the empty\nlist, and the cdr of the cddr of \\mbox{\\scheme|alpha|} becomes a new\nfrons pair containing \\scheme|bottom|. The cdr of the new frons pair\nis the empty list copied from the cdr of \\scheme|delta|. The\nremaining values are obvious given the final state of the data\nstructure. No further manipulation of the data structure is necessary\nto evaluate the three remaining calls to \\scheme|take-bottom|.\n\nIn Figure~\\ref{fig:solid}d each of the ferns\n\\scheme|alpha|, \\scheme|beta|, \\scheme|gamma|, and \\scheme|delta|\ncontains some permutation of its original possible values, and\n\\scheme|bottom| has been pushed to the end of\n\\scheme|alpha|. Furthermore, if there are no shared references to\n\\scheme|beta|, \\scheme|gamma|, and \\scheme|delta|, the number of\naccessible pairs is linear in the length of the fern. If there are\nreferences to subferns, for a fern of size $n$, the worst case is\n$(n^2+n)/2$. But, as these shared references vanish, so do the\nadditional cons pairs.\n\nIf \\scheme|list| evaluated from right-to-left instead of evaluating\nfrom left-to-right, the example expression would return\n\\mbox{\\scheme|`((720 6 120) (720 6 120) (720 6) (720))|}. Each list\nwould be independent of the others and the last pair of\n\\scheme|alpha| would be a frons pair with \\scheme|bottom| in the car\nand the empty list in the cdr. This demonstrates that if there is\nsharing of these lists, the lists contain four pairs, three pairs, two\npairs, and one pair, respectively. If the example expression just\nreturned \\scheme|alpha|, then only four pairs would be accessible.\n\n%\\caption{Final state of fern $\\alpha$ after promotion of $120$ (top) and $720$ (bottom). \\label{fig:p3}}\n\n%\\begin{figure}[H]\n%\\end{figure}\n\n\n\n\n%\\begin{figure}[H]\n%\\end{figure}\n%%%%%\n\n%\\vspace{10pt}\n\n% In our implementation, it is the requesting of a value from a fern that\n% provokes evaluation of the potential values. Therefore changing the\n% order of requests affects how the data structure is manipulated.\n% Program~\\ref{fig:code/sharing-2} is a variation of\n% Program~\\ref{fig:code/sharing-1}, with the same initial data structure\n% but with the \\scheme|display| lines in reverse order; this code might,\n% on a particular run, print:\n% \\begin{schemeresponse}\n% (720)\n% (720 6)\n% (720 6 120)\n% (720 6 120)\n% \\end{schemeresponse}\n\n% Figures~\\ref{fig:dotted}, \\ref{fig:singles} and \\ref{fig:bottom}\n% depict some milestones in the evolution of the fern $\\alpha$ during\n% the evaluation of Program~\\ref{fig:code/sharing-2}.\n% Figure~\\ref{fig:dotted} is still applicable because\n% Programs~\\ref{fig:code/sharing-1} and \\ref{fig:code/sharing-2} differ\n% only in the bodies of their innermost \\scheme|let| expressions.\n% The bottom part of Figure~\\ref{fig:bottom} shows that if there are references to\n% \\scheme|beta|, \\scheme|gamma|, and \\scheme|delta|, then there are\n% respectively three additional cons pairs if $\\beta$ is referenced;\n% two additional cons pairs if $\\gamma$ is referenced; and one additional \n% cons pair if $\\delta$ is referenced. Thus, for a fern\n% of size $n$, the worst case is $(n^2+n)/2$. But, as these sharings\n% vanish, the number of additional cons pairs approaches zero.\n\n% is not applicable,\n% since Program~\\ref{fig:code/sharing-2} evaluates \\mbox{\\scheme|(fcar\n% delta)|} first, and the links between previous pairs remain\n% dotted. Figure~\\ref{fig:singles} depicts the changes to $\\alpha$ as\n% \\mbox{\\scheme|(take-bottom 1 delta)|}, \\mbox{\\scheme|(take-bottom 2 gamma)|}, and\n% \\mbox{\\scheme|(take-bottom 3 beta)|} are evaluated: in each case the first\n% frons pair of the respective subfern contains the only potential\n% value, so that frons pair is simply turned into a cons pair.\n\n% \\addtocounter{figure}{-4}\n% \\renewcommand{\\figurename}{Program}\n\n% \\begin{figure}[H]\n% \\begin{schemeregion}\n% \\schemeinput{fernscode/code/sharing-2}\n% \\end{schemeregion}\n% \\caption{Variant of Program~\\ref{fig:code/sharing-1} with \\textit{displays} reversed.\\label{fig:code/sharing-2}}\n% \\end{figure}\n\n% \\renewcommand{\\figurename}{Figure}\n% \\addtocounter{figure}{3}\n\n% \\begin{figure}[H]\n% \\begin{schemeregion}\n% \\begin{picture}(160,40)(0,-8)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% \\fronsbox\\rightdotted\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|(!! 5)|}\n% \\fronsbox\\rightdotted\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|(!! 3)|}\n% \\fronsbox\\rightdotted\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n% \\caption{Fern $\\alpha$ after promotion of the first value, 720. (See Program~\\ref{fig:code/sharing-2}.)\\label{fig:singles}}\n% \\end{figure}\n\n\n% \\begin{schemeregion}\n% \\begin{picture}(160,40)(0,-10)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% \\fronsbox\\rightdotted\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|(!! 5)|}\n% \\fronsbox\\rightdotted\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n\n% \\begin{schemeregion}\n% \\begin{picture}(160,40)(0,-10)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% \\fronsbox\\rightdotted\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|120|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n\n% Figure~\\ref{fig:singles} shows that \\mbox{\\scheme|(take-bottom 1 delta)|},\n% which is the same as \\mbox{\\scheme|(fcar delta)|}, evaluates only\n% \\mbox{\\scheme|(! 7)|}, accounting for \\scheme|720| in\n% $\\delta$. Once we have a cons pair in a fern, its car will generally\n% dominate any frons pairs that refer to it directly or indirectly, so\n% we end up in turn with \\mbox{\\scheme|(fcar gamma)|},\n% \\mbox{\\scheme|(fcar beta)|}, and \\mbox{\\scheme|(fcar alpha)|} also\n% being \\scheme|720|. The values in the other pairs dominate for\n% similar reasons, but then it is for the \\scheme|fcadr| and finally for\n% the \\scheme|fcaddr|. These lead to the before and after figures\n% for the last promotion that corresponds to taking \\mbox{\\scheme|(fcaddr alpha)|}\n% as shown in the top diagram of Figure~\\ref{fig:bottom}.\n\n \\Define\\rightdoubleupsolid {\n \\MarkLoc(p1)\n \\blankbox\n \\MarkLoc(q1)\n \\MarkLoc(q2)\n \\Do(0,\\qbxwd){\\Move(3,0)}\n \\Move(1,-\\qbxwd)\n \\blankup\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n \\tiltedrightuphead\n \\blankleft\n }\n\n \\Define\\righttripleupsolid {\n \\MarkLoc(p1)\n \\blankbox\n \\blankbox\n \\blankup\n \\MarkLoc(q1)\n \\MarkLoc(q2)\n \\Do(0,\\qbxwd){\\Move(3,0)}\n \\Move(1,-\\qbxwd)\n \\blankup\n \\MarkLoc(p2)\n \\Curve(p1,q1,q2,p2)\n \\tiltedrightuphead\n \\blankleft\n \\blankleft\n}\n\n%\\begin{figure}[H]\n% \\begin{schemeregion}\n% \\begin{picture}(160,80)(0,-50)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\schemeresult|120|}\n% \\consbox\\longdownsolid\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% \\fronsbox\\rightdoubleupsolid\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|120|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n\n% \\begin{schemeregion}\n% \\begin{picture}(160,120)(0,-90)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\schemeresult|120|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\longdownsolid\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% \\fronsbox\\righttripleupsolid\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|120|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\rightsolid\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n\n%%%% penultimate promotion.\n% \\begin{figure}[H]\n% \\begin{schemeregion}\n% \\begin{picture}(160,160)(0,-130)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% %\\consbox\\downsolid\n% %\\renewcommand{\\boxtext}{\\schemeresult|120|}\n% \\consbox\\longdownsolid\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% %\\fronsbox\\boxnil\\blankup\\blankup\n% \\fronsbox\\rightsolid\\blankup\\blankup\n% %\\Move(\\hbxwd,-\\qbxwd)\n% \\Move(3,0)\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|720|}\n% %%%\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\schemeresult|120|}\n% \\consbox\\boxnil\\blankup\\blankup\n% \\Move(\\hbxwd,-\\qbxwd)\\Move(3,0)\n% %\\consbox\\downsolid\n% %%%\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|720|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\boxnil\\blankup\n% \\Move(\\hbxwd,-\\qbxwd)\\Move(3,0)\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n\n% \\begin{schemeregion}\n% \\begin{picture}(160,160)(0,-130)\n% \\Draw\\PenSize(1pt)\n% \\namebox(\\scheme|alpha|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\schemeresult|120|}\n% \\consbox\\longdownsolid\n% \\renewcommand{\\boxtext}{\\scheme|bottom|}\n% \\fronsbox\\boxnil\\blankup\\blankup\\blankup\n% \\Move(\\hbxwd,-\\qbxwd)\\Move(3,0)\n% \\namebox(\\scheme|beta|)\n% \\renewcommand{\\boxtext}{\\scheme|720|}\n%%%\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\schemeresult|120|}\n% \\consbox\\boxnil\\blankup\\blankup\n% \\Move(\\hbxwd,-\\qbxwd)\\Move(3,0)\n% %\\consbox\\downsolid\n% %%%\n% \\namebox(\\scheme|gamma|)\n% \\renewcommand{\\boxtext}{\\scheme|720|}\n% \\consbox\\downsolid\n% \\renewcommand{\\boxtext}{\\scheme|6|}\n% \\consbox\\boxnil\\blankup\n% \\Move(\\hbxwd,-\\qbxwd)\\Move(3,0)\n% \\namebox(\\scheme|delta|)\n% \\renewcommand{\\boxtext}{\\schemeresult|720|}\n% \\consbox\\boxnil\n% \\EndDraw\n% \\end{picture}\n% \\end{schemeregion}\n% \\caption{Final promotion within fern $\\alpha$ from Program~\\ref{fig:code/sharing-2}.\\label{fig:bottom}}\n% \\end{figure}\n\nThe example presented in this section provides a direct view of\npromotion. When a fern is accessed by multiple computations, the\npromotion algorithm must be able to handle various issues such as\nmultiple values becoming available for promotion at once. The code\npresented in Chapter~\\ref{fernsimpl} handles these details.\n\nWe are now ready to consider a ferns-based implementation of\nminiKanren.\n\n\\section{Ferns-based miniKanren}\\label{lp-system}\n\nIn this section we describe a simple bottom-avoiding logic programming\nlanguage, which corresponds to core miniKanren with non-interleaving\nsearch.\n% In this section, we use the task of logic programming as an extended\n% example of the use of ferns in avoiding bottom while maintaining a\n% natural recursive style. \n%We compare two sets of goal constructors, one\n%using streams and the other using ferns. \nWe begin by describing and\nimplementing operators \\scheme|mplus-bottom| and \\scheme|bind-bottom| over ferns,\nand go on to implement goal constructors in terms of these\noperators. The fern-based goal constructors are shown to be more\ngeneral than the standard stream-based ones presented in Chapter~\\ref{mkimplchapter}\\footnote{See~\\citet{Wand04relatingmodels} for a historical account of logic combinators.}.\n\n\\subsection{\\protect\\scheme|mplus-bottom| and \\protect\\scheme|bind-bottom|}\\label{mplus-bottom-and-bind-bottom}\n\nSince we are developing goal constructors in Scheme, a call-by-value language,\nwe make \\scheme|mplus-bottom| itself lazy to avoid diverging when one or\nmore of its arguments diverge. This is accomplished by defining \\scheme|mplus-bottom| as a\nmacro that wraps its two arguments in \\scheme|fern| before passing\nthem to \\scheme|mplus-fn-bottom|. In addition, \\scheme|mplus-bottom| must interleave\nelements from both of its arguments so that a fern of unbounded length\nin the first argument will not cause the second argument to be\nignored.\n\n%% TODO: bind should be lazy!\n%% Bind should not need to be lazy, since its arguments are streams\n%% and functions and we know all the places it is called.\n\n%% TODO: examples of why we need frons?\n\n% The \\scheme|mplus-func-bottom| procedure does exactly that, returning a fern\n% that is guaranteed to contain all the convergent elements of both\n% \\scheme|s1| and \\scheme|s2|, even if both contain divergent elements\n% or are infinite in length.\n\n% \\noindent Because \\scheme|mplus-func-bottom| immediately returns a fern,\n% \\mbox{\\scheme|(mplus-func-bottom s1 s2) converges|} for all ferns \\scheme|s1| and\n% \\scheme|s2|. In addition, since the results are interleaved, divergent\n% values are pushed to the end of the resulting fern. Thus,\n\n% \\belowcodeskip 0pt\n% \\schemeinput{fernscode/code/mplus-example-1}\\schemeresult|=> `(5 6)|\n% \\belowcodeskip \\medskipamount\n\n% \\medskip\n\n% \\noindent \\scheme|mplus-func-bottom| even works on infinite ferns, interleaving\n% them:\n\n% \\belowcodeskip 0pt\n% \\schemeinput{fernscode/code/mplus-example-2}\\schemeresult|=> `(1 200 201 2)|\n% \\belowcodeskip \\medskipamount\n\n% \\medskip\n\n% Unfortunately, this definition of \\scheme|mplus-func-bottom| can handle only one\n% type of divergence: while \\mbox{\\scheme|(mplus-func-bottom s1 s2) converges|} when\n% \\mbox{\\scheme|s1| \\scheme|converges|} and \\mbox{\\scheme|s2| \\scheme|converges|}, \n% \\mbox{\\scheme|(mplus-func-bottom s1 bottom) => bottom|}. \n% Since \\scheme|s1| may have non-\\scheme|bottom|\n% values, we would prefer that \\scheme|mplus-func-bottom| avoid this type of\n% divergence by waiting to evaluate its arguments until necessary. We can\n% accomplish this by writing \\scheme|mplus-func-bottom| as a macro that creates a\n% two-value fern of its arguments before passing that fern to the\n% \\scheme|mplus-fn-bottom| procedure. $\n\n\\schemedisplayspace\n\\schemeinput{fernscode/mplus-fn}\n\n\\schemeinput{fernscode/bind}\n\n% We use a fern constructor to make \\scheme|mplus-bottom| lazy: if one of the ferns in the \n% argument to \\scheme|mplus-fn-bottom| is divergent, it can select the other\n% one. For example, consider \\mbox{\\scheme|(fcar (mplus-bottom bottom (fern 5)))|},\n% which evaluates to \\schemeresult|5|. \n\\noindent\n\\scheme|bind-bottom| avoids the same types of\ndivergence as \\scheme|map-bottom| described in Section~\\ref{fernsRecursiveExamples} but\nuses \\scheme|mplus-bottom| to merge the results of the calls to \\scheme|f|.\nThus, \\mbox{\\scheme|(bind-bottom (ints-bottom 0) ints-bottom)|} is an unbounded fern of\nintegers; for every (nonnegative) integer $n$, it contains the\nintegers starting from $n$ and therefore every nonnegative integer $n$\nis contained $n+1$ times. The interleaving leads to\nduplicates in the following example:\n\n\\wspace\n\\noindent\\scheme|(take-bottom 13 (bind-bottom (ints-bottom 0) ints-bottom))| \\schemeresult|~> `(0 1 2 1 3 4 5 6 7 8 9 2 10)|.\n\\wspace\n\n\\noindent\nThe addition of \\scheme|unit-bottom| and \\scheme|mzero-bottom| rounds out the set of\noperators we need to implement a minimal miniKanren-like language.\n\n\\schemedisplayspace\n\\schemeinput{fernscode/mzerounit}\n\n\\noindent Using these definitions, we can run programs that require\nmultiple unbounded ferns, such as this program inspired by Seres and\nSpivey~\\cite{CombinatorsforLP} that searches for a pair $a$ and $b$ of\ndivisors of $9$ by enumerating the integers from $2$ in a fern of\npossible values for $a$ and similarly for $b$:\n\n\\schemedisplayspace\n\\belowcodeskip 0pt\n\\schemeinput{fernscode/spivey}\\schemeresult|=> `(3 3)|.\n\\belowcodeskip \\medskipamount\n\\medskip\n\n\\noindent Using streams instead of ferns in this example, which would be\nlike nesting ``for'' loops, would result in divergence since $2$ does\nnot evenly divide $9$.\n\n\\subsection{Goal Constructors}\n\n%\\enlargethispage{30pt}\nWe are now ready to define three goal constructors:\n\\scheme|==-bottom|, which unifies terms; \\scheme|disj-bottom|, which\nperforms disjunction over goals; and \\scheme|conj-bottom|, which\nperforms conjunction over goals\\footnote{\\scheme|disj-bottom| is just a simplified version of \\scheme|conde|, while \\scheme|conj-bottom| is just a simplified version of \\scheme|exist|.}. These goal constructors are required\nto terminate, and they always return a goal. A \\emph{goal} is a\nprocedure that takes a substitution and returns a fern of\nsubstitutions (rather than a stream of substitutions, as in\nChapter~\\ref{mkimplchapter}).\n\n\\schemedisplayspace\n\\schemeinput{fernscode/mk-without-run}\n\nA logic program evaluates to a goal; to obtain answers, this goal is\napplied to the empty substitution. The result is a fern of\nsubstitutions representing answers. We define \\scheme|run-bottom| in terms\nof \\scheme|take-bottom|, described in Section \\ref{fernsRecursiveExamples}, to\nobtain a list of answers from the fern of substitutions\n\n\\schemedisplayspace\n\\schemeinput{fernscode/mk-run}\n\\noindent where \\scheme|n| is a non-negative integer (or \\scheme|#f|) and \n\\scheme|g| is a goal. \n% (See Section~\\ref{lp-helpers} for the rest of the definitions used in this section.)\n\nGiven two logic variables \\scheme|x| and \\scheme|y|, here are\nsome simple logic programs that produce the same answers using both\nfern-based and stream-based goal constructors.\n\n\\medskip\n\n\\indent \\scheme|(run-bottom #f (==-bottom 1 x))| \\schemeresult|=> `($\\{x/1\\}$)| \\\\\n\\indent \\scheme|(run-bottom 1 (conj-bottom (==-bottom y 3) (==-bottom x y)))| %$\n\\schemeresult|=> `($\\{x/3, y/3\\}$)| \\\\\n\\indent \\scheme|(run-bottom 1 (disj-bottom (==-bottom x y) (==-bottom y 3)))| %$\n\\schemeresult|=> `($\\{x/y\\}$)| \\\\\n \\indent \\scheme|(run-bottom 5 (disj-bottom (==-bottom x y) (==-bottom y 3)))| %$\n\\schemeresult|=> `($\\{x/y\\}$ $\\{y/3\\}$)| \\\\\n \\indent \\scheme|(run-bottom 1 (conj-bottom (==-bottom x 5) (conj-bottom (==-bottom x y) (==-bottom y 4))))|\n\\schemeresult|=> `()| \\\\\n\\indent \\scheme|(run-bottom #f (conj-bottom (==-bottom x 5) (disj-bottom (==-bottom x 5) (==-bottom x 6))))|\n\\schemeresult|=> `($\\{x/5\\}$)|\n\n\\medskip\n\n\\noindent It is not difficult, however, to find examples of logic\nprograms that diverge when using stream-based goal constructors but\nconverge using fern-based constructors:\n\n\\medskip\n\n\\scheme|(run-bottom 1 (disj-bottom bottom (==-bottom x 3)))| \\schemeresult|=> `($\\{x/3\\}$)| \\\\\n\\indent \\scheme|(run-bottom 1 (disj-bottom (==-bottom bottom x) (==-bottom x 5)))| \\schemeresult|=> `($\\{x/5\\}$)|\n\n\\medskip\n\n\\noindent and given idempotent substitutions \\cite{lloyd:lp}, the\nfern-based operators can even avoid some circularity-based divergence\nwithout the occurs-check, while stream-based operators cannot:\n\n\\wspace\n\n\\scheme|(run-bottom 1 (disj-bottom (==-bottom (list x) x) (==-bottom x 6)))| \\schemeresult|=> `($\\{x/6\\}$)|\n\n\\wspace\n\nThere are functions that represent relations. The relation\n\\scheme|always-five-bottom| associates 5 with its argument an unbounded number of times:\n\n\\schemedisplayspace\n\\schemeinput{fernscode/always-five}\n\n\\noindent Because both stream and fern constructors\ndo not evaluate their arguments,\nwe may safely evaluate the goal \\mbox{\\scheme|(always-five-bottom x)|}, %$\nobtaining an unbounded collection of answers. Using\n\\scheme|run-bottom|, we can ask for a finite number of these answers. Because\nthe ordering of streams is determined at construction time, however, the\nstream-based operators cannot even determine the first answer in that\ncollection. This is because the definition of \\scheme|always-five-bottom| is left recursive. The fern-based operators, however, compute as many answers \nas desired:\n\n\\medskip\n\n\\mbox{\\scheme|(run-bottom 4 (always-five-bottom x))| \\schemeresult|=> `($\\{x/5\\}$ $\\{x/5\\}$ $\\{x/5\\}$ $\\{x/5\\}$)|}.\n\n\\medskip\n\n\n% \\scheme|(take-bottom #f ((==-bottom 1 x) empty-s)) =>| \\schemeresult|`($\\{x/1\\}$)|\n\n% \\scheme|(take-bottom #f ((conj-bottom (==-bottom 1 x) (==-bottom 2 x)) empty-s) =>|\n% \\schemeresult|`()|\n\n% \\scheme|(take-bottom #f ((disj-bottom (==-bottom 2 x) (==-bottom 3 x)) empty-s)) =>| \\schemeresult|`($\\{x/2\\}$ $\\{x/3\\}$)|\n\n% \\noindent With \\scheme|cons| instead of \\scheme|frons|, the following examples diverge:\n\n% \\scheme|(take-bottom 1 ((disj-bottom bottom (==-bottom 4 x)) empty-s)) =>| \\schemeresult|`($\\{x/4\\}$)|\n\n% \\scheme|(take-bottom 1 ((disj-bottom (==-bottom bottom x) (==-bottom 5 x)) empty-s)) =>| \\schemeresult|`($\\{x/5\\}$)|\n\n% \\noindent Without the occurs check, the following example also diverges\n% with \\scheme|cons| instead of \\scheme|frons|:\n\n% \\scheme|(take-bottom 1 ((disj-bottom (==-bottom `(,x) x) (==-bottom x 6)) empty-s)) =>|\n% \\schemeresult|`($\\{x/6\\}$)|\n\n% \\section{Helper Functions}\\label{lp-helpers}\n\n% To complete the implementation of the bottom-avoiding logic\n% programming operators presented in Section~\\ref{lp-system}, we provide\n% a logic variable constructor \\scheme|make-var|, a unification\n% algorithm \\scheme|unify|, and substitution helpers \\scheme|empty-s|,\n% \\scheme|ext-s|, and \\scheme|walk|. We represent logic variables by\n% R$^6$RS~\\cite{r6rs} records (syntactic layer); defining the record\n% type \\scheme|var| creates the constructor \\scheme|make-var|\n% automatically. We represent substitutions as association lists, and\n% use the triangular substitution model~\\cite{FBaade01}.\n\n% \\schemedisplayspace\n% \\schemeinput{fernscode/subst}\n", "meta": {"hexsha": "636263e7c5ef8a9b0e0a783bdc90d3035baece66", "size": 43870, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "ferns.tex", "max_stars_repo_name": "holtzermann17/dissertation-single-spaced", "max_stars_repo_head_hexsha": "aca0e56a33916596c98709308342d9ccabd4718b", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 50, "max_stars_repo_stars_event_min_datetime": "2015-01-11T21:22:55.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-10T12:49:11.000Z", "max_issues_repo_path": "ferns.tex", "max_issues_repo_name": "holtzermann17/dissertation-single-spaced", "max_issues_repo_head_hexsha": "aca0e56a33916596c98709308342d9ccabd4718b", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-08-08T18:10:18.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-09T02:33:25.000Z", "max_forks_repo_path": "ferns.tex", "max_forks_repo_name": "holtzermann17/dissertation-single-spaced", "max_forks_repo_head_hexsha": "aca0e56a33916596c98709308342d9ccabd4718b", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 4, "max_forks_repo_forks_event_min_datetime": "2017-07-29T13:58:01.000Z", "max_forks_repo_forks_event_max_datetime": "2018-09-14T05:01:31.000Z", "avg_line_length": 32.7143922446, "max_line_length": 223, "alphanum_fraction": 0.7229541828, "num_tokens": 14376, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5888891451980403, "lm_q2_score": 0.5813030906443133, "lm_q1q2_score": 0.34232308015050855}} {"text": "\\chapter{``Zfh'' and ``Zfhmin'' Standard Extensions for Half-Precision Floating-Point,\n Version 1.0}\n\nThis chapter describes the Zfh standard extension for 16-bit half-precision\nbinary floating-point instructions compliant with the IEEE 754-2008 arithmetic\nstandard.\nThe Zfh extension depends on the single-precision floating-point extension, F.\nThe NaN-boxing scheme described in Section~\\ref{nanboxing} is extended to\nallow a half-precision value to be NaN-boxed inside a single-precision value\n(which may be recursively NaN-boxed inside a double- or quad-precision value\nwhen the D or Q extension is present).\n\n\\begin{commentary}\nThis extension primarily provides instructions that consume half-precision\noperands and produce half-precision results.\nHowever, it is also common to compute on half-precision data using higher\nintermediate precision.\nAlthough this extension provides explicit conversion instructions that suffice\nto implement that pattern, future extensions might further accelerate such\ncomputation with additional instructions that implicitly widen their\noperands---e.g., half$\\times$half$+$single$\\rightarrow$single---or implicitly\nnarrow their results---e.g., half$+$single$\\rightarrow$half.\n\\end{commentary}\n\n\\section{Half-Precision Load and Store Instructions}\n\nNew 16-bit variants of LOAD-FP and STORE-FP instructions are added,\nencoded with a new value for the funct3 width field.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{M@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{imm[11:0]} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{width} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n12 & 5 & 3 & 5 & 7 \\\\\noffset[11:0] & base & H & dest & LOAD-FP \\\\\n\\end{tabular}\n\\end{center}\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{O@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{imm[11:5]} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{width} &\n\\multicolumn{1}{c|}{imm[4:0]} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n7 & 5 & 5 & 3 & 5 & 7 \\\\\noffset[11:5] & src & base & H & offset[4:0] & STORE-FP \\\\\n\\end{tabular}\n\\end{center}\n\nFLH and FSH are only guaranteed to execute atomically if the effective address\nis naturally aligned.\n\nFLH and FSH do not modify the bits being transferred; in particular, the\npayloads of non-canonical NaNs are preserved.\nFLH NaN-boxes the result written to {\\em rd}, whereas FSH ignores all but\nthe lower 16 bits in {\\em rs2}.\n\n\\section{Half-Precision Computational Instructions}\n\nA new supported format is added to the format field of most\ninstructions, as shown in Table~\\ref{tab:fpextfmth}.\n\n\\begin{table}[htp]\n\\begin{center}\n\\begin{tabular}{|c|c|l|}\n\\hline\n{\\em fmt} field &\nMnemonic &\nMeaning \\\\\n\\hline\n00 & S & 32-bit single-precision \\\\\n01 & D & 64-bit double-precision \\\\\n10 & H & 16-bit half-precision \\\\\n11 & Q & 128-bit quad-precision \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Format field encoding.}\n\\label{tab:fpextfmth}\n\\end{table}\n\nThe half-precision floating-point computational instructions are\ndefined analogously to their single-precision counterparts, but operate on\nhalf-precision operands and produce half-precision results.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{R@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFADD/FSUB & H & src2 & src1 & RM & dest & OP-FP \\\\\nFMUL/FDIV & H & src2 & src1 & RM & dest & OP-FP \\\\\nFMIN-MAX & H & src2 & src1 & MIN/MAX & dest & OP-FP \\\\\nFSQRT & H & 0 & src & RM & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{R@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{rs3} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nsrc3 & H & src2 & src1 & RM & dest & F[N]MADD/F[N]MSUB \\\\\n\\end{tabular}\n\\end{center}\n\n\\section{Half-Precision Conversion and Move Instructions}\n\nNew floating-point-to-integer and integer-to-floating-point conversion\ninstructions are added. These instructions are defined analogously to the\nsingle-precision-to-integer and integer-to-single-precision conversion\ninstructions. FCVT.W.H or FCVT.L.H converts a half-precision floating-point\nnumber to a signed 32-bit or 64-bit integer, respectively. FCVT.H.W or\nFCVT.H.L converts a 32-bit or 64-bit signed integer, respectively, into a\nhalf-precision floating-point number. FCVT.WU.H, FCVT.LU.H, FCVT.H.WU, and\nFCVT.H.LU variants convert to or from unsigned integer values. FCVT.L[U].H and\nFCVT.H.L[U] are RV64-only instructions.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{R@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFCVT.{\\em int}.H & H & W[U]/L[U] & src & RM & dest & OP-FP \\\\\nFCVT.H.{\\em int} & H & W[U]/L[U] & src & RM & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\nNew floating-point-to-floating-point conversion instructions are added. These\ninstructions are defined analogously to the double-precision\nfloating-point-to-floating-point conversion instructions.\nFCVT.S.H or FCVT.H.S converts a half-precision floating-point number to\na single-precision floating-point number, or vice-versa, respectively.\nIf the D extension is present, FCVT.D.H or FCVT.H.D converts a half-precision\nfloating-point number to a double-precision floating-point number, or\nvice-versa, respectively.\nIf the Q extension is present, FCVT.Q.H or FCVT.H.Q converts a half-precision\nfloating-point number to a quad-precision floating-point number, or\nvice-versa, respectively.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{R@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFCVT.S.H & S & H & src & RM & dest & OP-FP \\\\\nFCVT.H.S & H & S & src & RM & dest & OP-FP \\\\\nFCVT.D.H & D & H & src & RM & dest & OP-FP \\\\\nFCVT.H.D & H & D & src & RM & dest & OP-FP \\\\\nFCVT.Q.H & Q & H & src & RM & dest & OP-FP \\\\\nFCVT.H.Q & H & Q & src & RM & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\nFloating-point to floating-point sign-injection instructions, FSGNJ.H,\nFSGNJN.H, and FSGNJX.H are defined analogously to the single-precision\nsign-injection instruction.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{R@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFSGNJ & H & src2 & src1 & J[N]/JX & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\n\nInstructions are provided to move bit patterns between the floating-point and\ninteger registers.\nFMV.X.H moves the half-precision value in floating-point register {\\em rs1} to\na representation in IEEE 754-2008 standard encoding in integer register {\\em\nrd}, filling the upper XLEN-16 bits with copies of the floating-point number's\nsign bit.\n\nFMV.H.X moves the half-precision value encoded in IEEE 754-2008 standard\nencoding from the lower 16 bits of integer register {\\em rs1} to the\nfloating-point register {\\em rd}, NaN-boxing the result.\n\nFMV.X.H and FMV.H.X do not modify the bits being transferred; in particular,\nthe payloads of non-canonical NaNs are preserved.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{R@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFMV.X.H & H & 0 & src & 000 & dest & OP-FP \\\\\nFMV.H.X & H & 0 & src & 000 & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\n\\section{Half-Precision Floating-Point Compare Instructions}\n\nThe half-precision floating-point compare instructions are\ndefined analogously to their single-precision counterparts, but operate on\nhalf-precision operands.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{S@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFCMP & H & src2 & src1 & EQ/LT/LE & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\n\\section{Half-Precision Floating-Point Classify Instruction}\n\nThe half-precision floating-point classify instruction, FCLASS.H, is\ndefined analogously to its single-precision counterpart, but operates on\nhalf-precision operands.\n\n\\vspace{-0.2in}\n\\begin{center}\n\\begin{tabular}{S@{}F@{}R@{}R@{}F@{}R@{}O}\n\\\\\n\\instbitrange{31}{27} &\n\\instbitrange{26}{25} &\n\\instbitrange{24}{20} &\n\\instbitrange{19}{15} &\n\\instbitrange{14}{12} &\n\\instbitrange{11}{7} &\n\\instbitrange{6}{0} \\\\\n\\hline\n\\multicolumn{1}{|c|}{funct5} &\n\\multicolumn{1}{c|}{fmt} &\n\\multicolumn{1}{c|}{rs2} &\n\\multicolumn{1}{c|}{rs1} &\n\\multicolumn{1}{c|}{rm} &\n\\multicolumn{1}{c|}{rd} &\n\\multicolumn{1}{c|}{opcode} \\\\\n\\hline\n5 & 2 & 5 & 5 & 3 & 5 & 7 \\\\\nFCLASS & H & 0 & src & 001 & dest & OP-FP \\\\\n\\end{tabular}\n\\end{center}\n\n\\section{``Zfhmin'' Standard Extension for Minimal Half-Precision Floating-Point Support}\n\nThis section describes the Zfhmin standard extension, which provides minimal\nsupport for 16-bit half-precision binary floating-point instructions.\nThe Zfhmin extension is a subset of the Zfh extension, consisting only\nof data transfer and conversion instructions.\nLike Zfh, the Zfhmin extension depends on the single-precision floating-point\nextension, F.\nThe expectation is that Zfhmin software primarily uses the half-precision\nformat for storage, performing most computation in higher precision.\n\nThe Zfhmin extension includes the following instructions from the Zfh\nextension: FLH, FSH, FMV.X.H, FMV.H.X, FCVT.S.H, and FCVT.H.S.\nIf the D extension is present, the FCVT.D.H and FCVT.H.D instructions are\nalso included.\nIf the Q extension is present, the FCVT.Q.H and FCVT.H.Q instructions are\nadditionally included.\n\n\\begin{commentary}\nZfhmin does not include the FSGNJ.H instruction, because it suffices to\ninstead use the FSGNJ.S instruction to move half-precision values between\nfloating-point registers.\n\\end{commentary}\n\n\\begin{commentary}\nHalf-precision addition, subtraction, multiplication, division, and\nsquare-root operations can be faithfully emulated by converting the\nhalf-precision operands to single-precision, performing the operation\nusing single-precision arithmetic, then converting back to\nhalf-precision~\\cite{roux:hal-01091186}.\nPerforming half-precision fused multiply-addition using this method incurs\na 1-ulp error on some inputs for the RNE and RMM rounding modes.\n\nConversion from 8- or 16-bit integers to half-precision can be emulated by\nfirst converting to single-precision, then converting to half-precision.\nConversion from 32-bit integer can be emulated by first converting to\ndouble-precision.\nIf the D extension is not present and a 1-ulp error under RNE or RMM is\ntolerable, 32-bit integers can be first converted to single-precision instead.\nThe same remark applies to conversions from 64-bit integers without the Q\nextension.\n\\end{commentary}\n", "meta": {"hexsha": "e215ed415b6c22cdf620a03899e0c851269b0f02", "size": 13192, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/zfh.tex", "max_stars_repo_name": "frantony/riscv-isa-manual", "max_stars_repo_head_hexsha": "25d7fc01be062b052759627333a2cb3241730910", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/zfh.tex", "max_issues_repo_name": "frantony/riscv-isa-manual", "max_issues_repo_head_hexsha": "25d7fc01be062b052759627333a2cb3241730910", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "src/zfh.tex", "max_forks_repo_name": "frantony/riscv-isa-manual", "max_forks_repo_head_hexsha": "25d7fc01be062b052759627333a2cb3241730910", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.6354916067, "max_line_length": 89, "alphanum_fraction": 0.6976955731, "num_tokens": 4533, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443134, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.34232307176218396}} {"text": "\\documentclass[11pt, oneside]{article} \t% use \"amsart\" instead of \"article\" for AMSLaTeX format\n\\usepackage{geometry} \t\t% See geometry.pdf to learn the layout options. There are lots.\n\\geometry{letterpaper} \t\t% ... or a4paper or a5paper or ... \n%\\geometry{landscape} \t\t% Activate for rotated page geometry\n%\\usepackage[parfill]{parskip} \t\t% Activate to begin paragraphs with an empty line rather than an indent\n\\usepackage{graphicx}\t\t\t\t% Use pdf, png, jpg, or eps§ with pdflatex; use eps in DVI mode\n\t\t\t\t\t\t\t\t% TeX will automatically convert eps --> pdf in pdflatex\t\t\n\\usepackage{amssymb}\n\n%SetFonts\n\n%SetFonts\n\n%\\newcommand{\\vectr}[1]{\\mathbf{#1}}\n\\newcommand{\\vectr}[1]{\\hbox{\\boldmath$#1$}}\n\\newcommand{\\tensor}[1]{\\hbox{\\boldmath$#1$}}\n\n\n\\title{Turbulent Inflow Tool}\n\\author{Jay and Peter}\n%\\date{}\t\t\t\t\t\t\t% Activate to display a given date or no date\n\n\\begin{document}\n\\maketitle\n\n\\section{Abstract}\n\nThis tool helps you to create all necessary input to utilize the turbulent inflow boundary condition in OpenFOAM.\n\n\\section{Theory}\n\nWait for Jay's write-up.\n\n\\subsection{Uniform model}\n\n\\begin{equation}\n u = \\bar\\phi\n \\label{Eq1}\n\\end{equation}\n\n\\begin{equation}\n \\vectr{u} = u\\,\\vectr{N}\n \\label{Eq2}\n\\end{equation}\n\n\\subsection{Exponential model}\n\n\\begin{equation}\n \\phi = \\bar\\phi \\left( \\frac{\\vectr{n} \\cdot \\vectr{x} }{d_0} \\right)^\\alpha\n \\label{Eq11}\n\\end{equation}\n\n\\begin{equation}\n \\ell_{u,y} = R_{uy:ux} \\, \\ell_{u,x} \\qquad \\ell_{u,z} = R_{uz:ux} \\, \\ell_{u,x}\n \\label{Eq12}\n\\end{equation}\n\\begin{equation}\n \\ell_{v,y} = R_{vy:vx} \\, \\ell_{v,x} \\qquad \\ell_{v,z} = R_{vz:vx} \\, \\ell_{v,x}\n \\label{Eq13}\n\\end{equation}\n\\begin{equation}\n \\ell_{w,y} = R_{wy:wx} \\, \\ell_{w,x} \\qquad \\ell_{w,z} = R_{wz:wx} \\, \\ell_{w,x}\n \\label{Eq14}\n\\end{equation}\n\n\n\n\\end{document} ", "meta": {"hexsha": "92e3d9e6e1fddeb5dadfe7bdfa880f6a13b24b9c", "size": 1873, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/help/theory.tex", "max_stars_repo_name": "moreff/TurbulenceInflowTool", "max_stars_repo_head_hexsha": "c2885a3a8c93736f6af93bc51caf33fb0e6506c7", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": 21, "max_stars_repo_stars_event_min_datetime": "2020-04-05T22:06:54.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-29T13:28:43.000Z", "max_issues_repo_path": "docs/help/theory.tex", "max_issues_repo_name": "moreff/TurbulenceInflowTool", "max_issues_repo_head_hexsha": "c2885a3a8c93736f6af93bc51caf33fb0e6506c7", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-06-08T13:32:50.000Z", "max_issues_repo_issues_event_max_datetime": "2020-07-14T08:45:08.000Z", "max_forks_repo_path": "docs/help/theory.tex", "max_forks_repo_name": "moreff/TurbulenceInflowTool", "max_forks_repo_head_hexsha": "c2885a3a8c93736f6af93bc51caf33fb0e6506c7", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": 11, "max_forks_repo_forks_event_min_datetime": "2020-07-01T13:27:56.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-24T04:33:07.000Z", "avg_line_length": 27.5441176471, "max_line_length": 113, "alphanum_fraction": 0.6486919381, "num_tokens": 656, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5813030906443133, "lm_q2_score": 0.588889130767832, "lm_q1q2_score": 0.3423230717621839}} {"text": "\\documentclass[10pt]{book}\n\\usepackage[german]{babel}\n\\usepackage[utf8]{inputenc}\n\\usepackage{fancyhdr}\n\\usepackage{amsmath}\n\\usepackage{calc}\n\\usepackage{listings}\n\\usepackage{hyperref}\n\\usepackage{tikz-network}\n\\usetikzlibrary{decorations.pathreplacing,calc}\n\\usepackage{multicol}\n\n\\newcommand{\\ueberschrift}{How does a neural net, work?}\n\n\\title{\\ueberschrift}\n\\author{Philipp Zettl}\n\n\\begin{document}\n \\pagestyle{fancy}\n \\maketitle\n\n At some point one can write a preamble to explain what a NN is, what to use it for\n and why it's actually better for us to use them then old numerical methods like the euler method to\n calculate the root(s) of a function\n\n \\chapter{Neural Networks}\n Neural networks are a widely used technique to solve complex problems in a very simplified mathematical\n way. There are many different implementations of neural networks, but they all share the same core idea.\n\n \\textbf{F}eed \\textbf{f}orward \\textbf{n}eural \\textbf{n}etworks, a type or neural network, are a great entry point to start discovering\n neural networks. The following chapter will introduce the idea of general neural networks and use this motivation\n to discover the first type of neural networks.\n \\section{Motivation}\n Unlike us humans computers have rather a hard time recognizing handwritten characters, digits or even full words.\n Humans taught themselves over millions of years of evolution how to recognize things.\\\\\n Since handwriting is mostly unique to a person one will have a hard time implementing an algorithm to detect digits from handwritten text,\n using an enormous amount of \\lstinline{if/else} conditions will be the least problem, but figuring out how to provide a stable algorithmically solution\n to this problem seems impossible.\n Nowadays we can use neural networks to mock this behavior and create models to do that task for us.\n In the following chapter I will describe how to create such model as well as how to train it using real live\n examples.\n\n To train a NN one uses a so called \\dq Training-Set\\dq of data to \\dq teach\\dq the network what prediction to give for a given input. In some way we can assume\n achieving similar results using equal training data sets.\\\\\n But wait, how does a neural net, work?\\\\ \n Now let us construct a basic neural network.\n \\section{General NN}\n To define the structure of Neural Networks we need to define a bunch of\n technical terms. In biology we differentiate between different types of neurons.\n Perceptrons, a type of neuron, are basically binary decision makers having \\(n\\) input values and a single output value.\n We can illustrate them using the following notation.\n \\begin{figure}[h]\n \\begin{center}\n \\begin{tikzpicture}\n \\Vertex[x=0,y=0,label=$x_1$]{A}\n \\Vertex[x=0,y=-1,label=$x_2$]{B}\n \\Vertex[x=0,y=-2,label=$x_3$]{C}\n \\Vertex[x=2,y=-1,label=$p_1$]{D}\n \\Vertex[x=4,y=-1,label=$o_1$]{F}\n \\Edge(A)(D)\n \\Edge(B)(D)\n \\Edge(C)(D)\n \\Edge(D)(F)\n \\end{tikzpicture}\n \\end{center}\n \\caption{Simple illustration of a single perceptron.\\label{fig:Perceptron}}\n \\end{figure}\n This model was introduced in 1958 by Frank Rosenblatt. He proposed a simple, yet complicate,\n rule to compute the output value of a perceptron. He introduced weights \\(w_i\\) to express the importance\n of an input value \\(x_i\\) to compute the output value \\(o_i\\).\n \\begin{align}\n o_i =\n \\left\\{\n \\begin{matrix}\n 0 & \\text{if } \\sum_j w_ix_i \\leq t\\\\\n 1 & \\text{if } \\sum_j w_ix_i > t\n \\end{matrix}\n \\right.\n \\label{eq:perceptron}\n \\end{align}\n with \\(t\\) a threshold.\n Obviously a single perceptron is not even close to real human decision making and building\n a neural net out of a single perceptron seems simply too unflexible and stiff.\n But it works great to illustrate the idea behind a complex model without using too complicated notations,\n we can imagine a perceptron being a single element of a complex model to make a more detailed decision.\\\\\n Hence a NN in the from\\\\\n \\begin{figure}[h]\n \\begin{center}\n \\begin{tikzpicture}\n \\Vertex[x=-4,y=-0.5,Pseudo,label=$i_1$]{i1}\n \\Vertex[x=-4,y=-1.5,Pseudo,label=$i_2$]{i2}\n\n \\Vertex[x=-2,y=0]{A}\n \\Vertex[x=-2,y=-1]{B}\n \\Vertex[x=-2,y=-2]{C}\n\n \\Vertex[x=0,y=1]{a}\n \\Vertex[x=0,y=0]{b}\n \\Vertex[x=0,y=-1]{c}\n \\Vertex[x=0,y=-2]{d}\n \\Vertex[x=0,y=-3]{e}\n\n \\Vertex[x=2,y=-1]{D}\n \\Vertex[x=4,y=-1,Pseudo,label=$o_1$]{F}\n\n \\Edge(i1)(A)\n \\Edge(i1)(B)\n \\Edge(i1)(C)\n \\Edge(i2)(A)\n \\Edge(i2)(B)\n \\Edge(i2)(C)\n\n \\Edge(A)(a)\n \\Edge(B)(a)\n \\Edge(C)(a)\n\n \\Edge(A)(b)\n \\Edge(B)(b)\n \\Edge(C)(b)\n\n \\Edge(A)(c)\n \\Edge(B)(c)\n \\Edge(C)(c)\n\n \\Edge(A)(d)\n \\Edge(B)(d)\n \\Edge(C)(d)\n\n \\Edge(A)(e)\n \\Edge(B)(e)\n \\Edge(C)(e)\n\n \\Edge(a)(D)\n \\Edge(b)(D)\n \\Edge(c)(D)\n \\Edge(d)(D)\n \\Edge(e)(D)\n \\Edge(D)(F)\n \\end{tikzpicture}\n \\end{center}\n \\caption{Simple illustration of a NN using multiple perceptrons.\\label{fig:simpleNN}}\n \\end{figure}\\\\\n will perform a more detailed analysis then (Figure \\ref{fig:Perceptron}).\\\\\n A network consists of the input values \\(i_j\\), several columns of perceptrons – we will call them from now on layers – and the output \\(o_j\\).\n The above illustrated network contains 3 layers of perceptrons. The first layer is making three simple decisions out of the input data,\n then forwards these decisions to the second layer of perceptrons, which will make 5 decisions out of each of the previous 3 decisions by weighting their\n outputs. Before we can analyse the above displayed network in more detail we need to introduce a good notation, how to call a single perceptron.\n Say we have \\(l > 0\\) layers in our network and use \\(n\\) input values \\(i_1, i_2, ..., i_n\\) to predict \\(m\\) output values \\(o_1, o_2, ..., o_m\\).\n Then we call the \\(i^{\\text{th}}\\) perceptron of the \\(l^{\\text{th}}\\) layer\n \\begin{align}\n p_i^l\n \\end{align}\n \n Now let's take a more detailed look at our network and calculate the amount of decisions we actually perform.\\\\\n We have two input values \\(i_1\\) and \\(i_2\\), based on each we perform 3 decisions, so after processing the first layer \\(l=1\\) of perceptrons we made\n \\begin{align}\n n_l\\cdot \\underbrace{n_{l-1}}_{=n} = 3\\cdot 2 = 6\n \\end{align}\n decisions. Moving on to the second layer we again count the number of perceptrons in the current layer \\(n_2 = 5\\) and\n multiply it with the amount of perceptrons in the previous layer\n \\begin{align}\n \\Rightarrow n_2 \\cdot n_1 = 5 \\cdot 3 = 15\n \\end{align}\n And for the third layer \\(l=3\\) with one perceptron we perform \\(5\\) decision.\\\\\n We also need to include the last layer into the calculation with a single perceptron.\n Adding those values together \\(6+15+5+1 = 27\\), we actually perform 27 decisions for a single output.\n\n This makes us assume that increasing the amount of layers and perceptrons per layer will increase the complexity\n behind a decision and therefor a big neural network the way to compute complex and sophisticated decisions.\n\n In \\eqref{eq:perceptron} we introduced a way how to calculate the output of a perceptron, we will change this notation\n a bit to achieve a way to calculate the outcome of a whole NN, not just a single perceptron of it.\n The previous notation of \\(\\sum_j w_jx_j > t\\) can be written as a dot product \\(w \\cdot x\\), where \\(w\\) and \\(x\\) are \n vectors whose components are the weights and input values respectively. The second change is to move the \n threshold from the right side of the inequality. But before we do that we rename it to its widely used name \\dq Bias\\dq, \\(b = -t\\)\n using these changes we can write a perceptron now using\n \\begin{align}\n o_i =\n \\left\\{\n \\begin{matrix}\n 0 & \\text{if } w\\cdot x + b \\leq 0\\\\\n 1 & \\text{if } w\\cdot x + b > 0\n \\end{matrix}\n \\right.\n \\label{eq:firstNN}\n \\end{align}\n Think of the bias as how easy it is to get 1 out of the perceptron. Or in a more biological\n way, the bias is a measure of how easy it is to get the perceptron to \\textit{fire}.\\newline\n Previous descriptions of perceptrons were based on the idea that a perceptron implements a method of weighting\n input values to make decisions. A more CS way to use perceptrons would be to compute elementary logical functions\n such as \\lstinline{AND, OR, NOR, XOR, ...} So we can for example build a network having the weights \\(-2\\) and the bias of \\(3\\), like this \n \\begin{figure}[h]\n \\begin{center}\n \\begin{tikzpicture}\n \\Vertex[x=0,y=0,label=$x_1$]{A}\n \\Vertex[x=0,y=-2,label=$x_2$]{B}\n \\Vertex[x=2,y=-1,label=$p_1$]{D}\n \\Vertex[x=4,y=-1,label=$o_1$]{F}\n \\Edge[label=$-2$](A)(D)\n \\Edge[label=$-2$](B)(D)\n \\Edge(D)(F)\n \\end{tikzpicture}\n \\end{center}\n \\caption{Logical \\lstinline{NAND} gate.\\label{fig:NANDGate}}\n \\end{figure}\n The above displayed NN \\ref{fig:NANDGate} implements a so called \\lstinline{NAND}-Gate.\n An input of \\(x = \\begin{pmatrix}0, 0\\end{pmatrix}^T\\) results in \\((-2)\\cdot 0 + (-2) \\cdot 0 + 3 = 3 \\Rightarrow o_1 = 1\\).\n Similar calculations reveal that the inputs \\(\\begin{pmatrix}0, 1\\end{pmatrix}^T, \\begin{pmatrix}1, 0\\end{pmatrix}^T\\) produce the output\n \\(1\\), but \\(\\begin{pmatrix}1, 1\\end{pmatrix}^T\\) results in \\((-2)\\cdot 1 + (-2)\\cdot 1 + 3 = -1\\) therefor \\(o_1 = 0\\).\n To make it even more obvious lets take a look at a simple truth table from the first semester, some of you might even know that stuff since their born, anyway.\n \\begin{center}\n \\begin{tabular}{c|c|c|c}\n A & B & A\\&B & !(A\\&B)\\\\\n \\hline\n 1 & 0 & 0 & 1\\\\\n 1 & 1 & 1 & 0\\\\\n 0 & 0 & 0 & 1\\\\\n 0 & 1 & 0 & 1\\\\\n \\end{tabular}\n \\end{center}\n as we can see the above mentioned results of our NN are equal to a regular \\lstinline{NAND} operation.\n\n This \\lstinline{NAND} example shows that we can use perceptrons to compute simple logical functions. In fact we can use \n perceptrons to compute \\textbf{any} logical function. But the reason for this is not the flexibility or any other aspect of neural networks\n or perceptrons per se, this is due to the universal definition of logical operations.\n Since we can recreate these universal definitions it follows that a perceptron is a similar universal definition of an operation.\n This fact is actually quite disappointing because it seems it's merely possible achieve something more complex then combinations of \\lstinline{NAND}\n gates and other logical operations. But on the other hand it might be reassuring to know that NNs are as powerful as modern computers.\n\n But don't take this introduction too serious, keep in mind we did not cover learning algorithms yet. Which open up\n a whole new world. We will use these learning algorithms to adjust weights and biases automatically within a network of\n artificial neurons. There are several methods of training all respond to external stimuli and do not require manual adjustments by\n a programmer. These algorithms allow us to use artificial neurons in a way which is radically different from\n logical gates. Instead of explicitly laying out a logical function our networks can simply learn to solve problems, which are way beyond\n simple conventional logical gates.\n \n \\section{Activated Neurons}\n Writing a learning algorithm sounds scary on first glance, but how can one achieve such behavior for a neural network?\n Suppose we have a network of perceptrons as previously described which we want to use to solve a problem. For example, the inputs\n to the network might be raw pixels from a scanned, handwritten image of a digit. And we'd like our network to learn it's weights and biases so that the\n output correctly classifies the digit within the image. To illustrate how learning might work we make small adjustments to the weights (or biases) of the network.\n We want those changes in the weights to be small to cause only small changes in the corresponding output from the network. Those changes will be called\n \\(\\Delta w, \\Delta b\\) and \\(\\Delta o\\). Suppose we input an image of a 9 and the network classifies this input as a 8, we then could figure out how to make a small\n change to the weights or biases so the network gets a bit closer to classify the image as a 9. We then repeat this process over and over again. The network is learning.\n\n Unfortunately our current definition of neural networks is not able to achieve this behavior using the perceptron model. Doing small adjustments on their weights and values\n will cause big changes in the networks output. This is due to the restriction in the binary output of perceptrons. Sometimes a change in a single weight flips a perceptrons output\n and the results in a complete different output of the network. So while the expected 9 might be classified correctly the behavior of the network in respect to other inputs\n might vary drastically. That makes it difficult to see how to gradually modify the weights and biases so the network achieves expected results.\n\n We simply overcome this issue with introducing a new type of neuron, the sigmoid neuron. Sigmoids are similar to perceptrons, with the small difference that small changes in their weights and biases cause only\n small changes in the networks output. That's a necessary feature to allow networks of sigmoids to learn. We depict sigmoids the same way as perceptrons. \n\n \\begin{figure}[h]\n \\begin{center}\n \\begin{tikzpicture}\n \\Vertex[x=0,y=0,label=$x_1$]{A}\n \\Vertex[x=0,y=-1,label=$x_2$]{B}\n \\Vertex[x=0,y=-2,label=$x_3$]{C}\n \\Vertex[x=2,y=-1,label=$p_1$]{D}\n \\Vertex[x=4,y=-1,label=$o_1$]{F}\n \\Edge(A)(D)\n \\Edge(B)(D)\n \\Edge(C)(D)\n \\Edge(D)(F)\n \\end{tikzpicture}\n \\end{center}\n \\caption{Illustration of a Sigmoid Neuron.\\label{fig:Sigmoid}}\n \\end{figure}\n Just like the perceptron the sigmoid neuron gets \\(n\\) input values \\(x_i, i=1, ...,n\\) but instead of forcing those values to be binary,\n they can be any number between \\(0\\) and \\(0\\), so \\(x_i \\in [0, 1]\\) as well as the perceptron the sigmoid neuron has a weight \\(w\\) for each input and overall biases \\(b\\). But it's output is\n again not binary. Instead it's \\(\\sigma(w \\cdot x + b)\\), where \\(\\sigma\\) is the \\textit{sigmoid} function (also called logistical function),\n \\begin{multicols}{2}\n \\includegraphics[width=.45\\textwidth]{sigmoid.eps}\n \n \\columnbreak\n \\vspace*{.6cm}\n\n \\begin{align}\n \\sigma(z) = \\frac{1}{1+e^{-z}}\n \\end{align}\n\n \\end{multicols}\n to be more precise the output of a sigmoid neuron is\n \\begin{align}\n \\frac{1}{1+e^{-w \\cdot x - b}}\n \\end{align}\n To understand the similarity between the previously discussed perceptron neuron and the sigmoid neuron, suppose\n \\(z = w \\cdot x + b\\) is a large positive number. Then \\(e^{-z} \\approx 0\\) so \\(\\sigma(z) \\approx 1\\). Or just\n \\dq Is \\(z\\) large and positive, the output is approximately 1\\dq, just like for the perceptron. So \\(\\sigma\\) is a way\n to enforce values being \\(\\in [0, 1]\\). One can achieve the behavior of perceptrons by redefining \\(\\sigma\\) as in \\eqref{eq:firstNN}.\n So Sigmoid Neurons are \\dq just\\dq smoothened perceptrons. But exactly this smoothening is the crucial fact which allows learning.\n This smoothness means that small changes in weights \\(\\Delta w\\) and biases \\(\\Delta b\\) will produce a small change in the output \\(\\Delta o\\).\n Hence we achieve the same behavior of perceptrons with the additional effect of being able to predict the change in output which will occur while\n modifying a weight or bias within the network.\n In fact, calculus tells us how to calculate that change as\n \\begin{align}\n \\Delta o \\approx \\sum \\limits_{j=1}^n \\frac{\\partial o}{\\partial w_j} \\Delta w_j + \\frac{\\partial o}{\\partial b} \\Delta b\n \\end{align}\n where \\(\\frac{\\partial o}{\\partial w_j}\\) denote partial derivatives of the output with respect to \\(w_j\\) and \\(b\\) respectively.\n This might sound overwhelming in the beginning and partial derivatives aren't that easy to understand. But what we can see from here\n is that \\(\\Delta o\\) is a linear function, since it's a combination of the linear function \\(\\Delta w\\) and \\(\\Delta b\\). Therefor\n a small change in weights or biases will cause only a small change in the output. So while sigmoid neurons have much of the same quality\n as perceptrons, they make it easier to figure out which weights and biases to adjust in order to achieve a small change of the output\n into a desired direction.\\newline\n \n \\section{Summary}\n We've discussed two types of neurons in this chapter. The perceptron which is ideal to reconstruct logical\n functions and the sigmoid neuron, which has exactly the same features as the perceptron\n with the addition that we can perform small adjustments to modify the used weights and biases in order\n to \\dq train \\dq the model. This brings one big conclusion. When we talk about neurons we can assume these given values:\n \\begin{enumerate}\n \\item \\(n:=\\) Number input elements\n \\item \\(m:=\\) Number output elements\n \\item \\(n_l:=\\) Number desired \\dq layers\\dq of neurons\n \\item Type of neurons in layer \\(l\\)\n \\end{enumerate}\n Depending on the type of neuron, the behavior differs. But what is the actual difference between the sigmoid neuron and the perceptron?\n The answer is quite easy. The main and mostly only difference is their activation function.\n \\begin{figure}[h]\n \\includegraphics[width=\\textwidth]{activation_functions.eps}\n \\end{figure}\n Both clamp the values \\(o \\in [0,1]\\), but the smoothness of the sigmoid function allows learning behavior.\n \n \\subsection{Important things to remember}\n Imagine having a big network of perceptrons. Multiplying every weight and bias within this network with a constant \\(c > 0\\) will \\underline{\\textbf{not}} change the overall network output.\n\n \\section{Network architecture}\n Over the next section we will introduce a neural network that can do a pretty good job classifying handwritten digits. It helps to first introduce some terminology\n that lets us name different parts of the network.\n Suppose we have the previously mentioned network\n\n \\begin{figure}[h]\n \\begin{center}\n \\begin{tikzpicture}\n \\Vertex[x=-4,y=-0.5,Pseudo,label=$i_1$]{i1}\n \\Vertex[x=-4,y=-1.5,Pseudo,label=$i_2$]{i2}\n\n \\Vertex[x=-4,y=-4,Pseudo]{B0}\n\n \\Vertex[x=-2,y=0]{A}\n \\Vertex[x=-2,y=-1]{B}\n \\Vertex[x=-2,y=-2]{C}\n\n\n \\Vertex[x=-2,y=-4,Pseudo]{B1}\n\n \\Vertex[x=0,y=1]{a}\n \\Vertex[x=0,y=0]{b}\n \\Vertex[x=0,y=-1]{c}\n \\Vertex[x=0,y=-2]{d}\n \\Vertex[x=0,y=-3]{e}\n\n\n \\Vertex[x=0,y=-4,Pseudo]{B2}\n\n \\Vertex[x=2,y=-1]{D}\n \\Vertex[x=2,y=-4,Pseudo]{B3}\n \\Vertex[x=4,y=-1,Pseudo,label=$o_1$]{F}\n \\Vertex[x=4,y=-4,Pseudo]{B4}\n\n \\Edge(i1)(A)\n \\Edge(i1)(B)\n \\Edge(i1)(C)\n \\Edge(i2)(A)\n \\Edge(i2)(B)\n \\Edge(i2)(C)\n\n \\Edge(A)(a)\n \\Edge(B)(a)\n \\Edge(C)(a)\n\n \\Edge(A)(b)\n \\Edge(B)(b)\n \\Edge(C)(b)\n\n \\Edge(A)(c)\n \\Edge(B)(c)\n \\Edge(C)(c)\n\n \\Edge(A)(d)\n \\Edge(B)(d)\n \\Edge(C)(d)\n\n \\Edge(A)(e)\n \\Edge(B)(e)\n \\Edge(C)(e)\n\n \\Edge(a)(D)\n \\Edge(b)(D)\n \\Edge(c)(D)\n \\Edge(d)(D)\n \\Edge(e)(D)\n \\Edge(D)(F)\n \\draw[-,decorate,decoration={brace,mirror}] (B0)-- node[below]{Input layer} (B0);\n \\draw[-,decorate,decoration={brace,mirror}] (B1)-- node[below]{Hidden layers} (B3);\n \\draw[-,decorate,decoration={brace,mirror}] (B4)-- node[below]{Output layer} (B4);\n \\end{tikzpicture}\n \\end{center}\n \\caption{Simple illustration of a NN.\\label{fig:NN}}\n \\end{figure}\n The most left layer is called the input layer and the neurons within this layer are called \\textit{input neurons}.\n The outermost right layer is called the output layer, and as you can guess it's neurons are called \\textit{output neurons}, in this case a single neuron.\n The layers in between are called \\textit{hidden} layers since their neurons are neither input nor output neurons. The term \\dq hidden\\dq might sound magical or\n deeply philosophical the first time you hear about it, but it's nothing else then another word for \\dq not an input/output\\dq.\n This might be somewhat confusion, and I just mention it for historical reason, some people call these networks\n MLPs, multiple layer perceptron networks, despite the fact that some of them hold sigmoid neurons, this should be mentioned\n so you're aware of that term.\\newline\n \\textbf{Important to note, I might call sigmoid neurons every now and then activated neurons, this is due to the nature of sigmoid neurons which use the sigmoid function as activation function.\n Whenever this activation function differs I will explicitly name this activation function. Other then that activated neurons are sigmoid neurons.}\n \n \\subsection{Input and output layers}\n The design for input and output layers is straight forward. Depending on the problem we want to solve we\n figure out the number of input values. For the example of classifying handwritten digits we have a dataset of 64x64 px images\n therefor we have \\(n = 64 \\cdot 64 = 4096\\) input parameters, which need to be clamped into \\([0,1]\\). To normalize the data we can use\n \\begin{align}\n \\tilde{x_i} = \\frac{x_i - \\min(x)}{\\max(x)-\\min(x)}\n \\end{align}\n The output layer will contain \\(m\\) neurons, to simplify for now we want to classify if a picture shows a 9 or not, so a single neuron with values\n \\(o \\leq 0.5\\) indicating it is \\underline{not} a 9 and \\(o > 0.5\\) that the input image is a 9.\n\n \\subsection{Hidden Layers}\n Albeit the design of input- and output layers is straight forward, it is kind of an art to architect hidden layers.\n To be more precise it is impossible to state a rule of thumb for hidden layer architectures. Instead researchers have found\n several heuristics to describe models which help solving different problems.\n As an example for the use of these heuristics they can be used to determine how to trade off hidden layers against the time required training the model.\n We'll meet several such design heuristics later.\n \n \\subsection{Prospects}\n For now we learned about networks of layers of neurons where the output of one layer is used as the input of the next layer.\n These networks are called \\textbf{Feed Forward Neural Networks} (FFNNs). The net works in a single operation forward feeding the input\n values to gain output values. No loops are involved. Actually it would be very hard to achieve loops with this design, so we don't allow them.\n However there are other designs of neural networks which allow these kind of designs.\n\n These models are called \\textbf{Recurrent Neural Networks} (RNNs).\n The idea in these models is to have neurons which fire for some limited duration of time, before becoming quiescent. That firing can stimulate other neurons, which may fire a little while later, also for a limited duration. That causes still more neurons to fire, and so over time we get a cascade of neurons firing. Loops don't cause problems in such a model, since a neuron's output only affects its input at some later time, not instantaneously.\n Recurrent neural nets have been less influential than feedforward networks, in part because the learning algorithms for recurrent nets are (at least to date) less powerful. But recurrent networks are still extremely interesting. They're much closer in spirit to how our brains work than feedforward networks. And it's possible that recurrent networks can solve important problems which can only be solved with great difficulty by feedforward networks.\n However, to limit our scope, we're going to concentrate for now on the more widely-used feedforward networks.\n\n \\chapter{NN to classifying handwritten digits}\n After this long and very dry introduction to neural networks, lets get our hands dirty and build something fun!\n In the following chapter we will implement the example we previously used to motivate the idea behind NNs. Classifying \n handwritten digits.\n \\section{Problem description}\n We can split the problem of recognizing handwritten digits into two sub-problems.\n First we want to split an image of multiple\n handwritten digits into separate images, each containing a single digit.\n\n \\begin{multicols}{2}\n So for example we want this number\\\\\n \\begin{center}\n \\includegraphics{digits-complete.eps}\n \\end{center}\n \\columnbreak\n\n To be split into 3 seperate images like\\\\\n \\begin{center}\n \\includegraphics{digits-split.eps}\n \\end{center}\n \\end{multicols}\n A human solves this quite easily, but we want to solve this using the computer.\n Once the image has been segmented, the program then needs to classify each digit individually.\n So for example we'd like the program to recognize the first digit in the image above to be 4.\n\n For now we focus on writing a program to solve the second problem. There are many approaches to solve\n the segmentation problem, but after we discussed the classification you will have a good idea how to tackle\n the segmentation. One method can be to generate many \\dq trial\\dq \\; segmentations and use an individually trained\n classifier to score each segmentation. A trial gets a higher score when the classifier is more certain about the classification\n and a lower score whenever it is uncertain about the classification. The idea is that if the classifier has trouble classifying then\n the segmentation is not good enough. So instead of worrying about the first problem in the beginning, we assume for now that we have\n in fact already segmented training sets and can come back to the segmentation model later.\n\n To solve the more interesting problem, to namely classify handwritten digits we will use a three-layer neural network.\n It consists of one input layer having \\(784\\) input neurons (one for each pixel of a 28x28 px image), a hidden layer with \\(15\\) neurons and an output layer with \\(10\\) output neurons (one for each digit 0-9 ;-) )\n\n The input pixels are greyscale, with values of \\(0.0\\) representing white and \\(1.0\\) black. Values in between are different shades of grey.\n\n You might wonder why the output layer consists of 10 neurons and not the minimum amount required being 4. In fact using 4 neurons can be a lightweight solution\n here since we can encode the output as a binary value since \\lstinline{111}\\({}_2=7< n_o <13=\\) \\lstinline{1111}\\({}_2\\). The ultimate justification is empirical\n so let's try out both approaches later on and compare the results, but it will turn out that the network with 10 output values will achieve better results then the \n network using 4 output values. That leaves us with the big question: Does the number of output values influence the general behavior of the network?\n Is there maybe some heuristic which gives us a direction to chose the number of output values?\n And can this heuristic possibly tell us upfront which solution fits better?\n\n Many resources online and books start now to describe how NNs work by using a explicit example. Most of them start splitting the image of a\n digit into several parts and justify the way how the neurons \\dq learn\\dq their weights/biases by using these segments and saying \\dq This weight is for the top arch of this digit...\\dq but\n that's just a useless simplification. Imagine that in fact weights represent features of the images fed into the network during the training period. But the odds that you will chose the right weight\n which actually describes this particular feature is very low. Imagine weights and biases as feature parts of the fed data. This can be a line, a cloud of pixels anything within the data.\n But their explanation motivates the decision to chose 10 output neurons, by claiming\n \\begin{quote}\n If the combination of hidden neurons is responsible to classify multiple digits it will get very hard for the output neuron to figure out\n which digit do chose from, since \\lstinline{1000} and \\lstinline{1001} aren't that different. But \\lstinline{0000000001} and \\lstinline{0000000010} do very much.\n \\end{quote}\n This description works as an heuristic and is widely used. Although I disagree with the detailed description it is a nice simplification to remember.\n\n Note that in case you want to achieve the same results as the 10 output neuron model, you can add an additional layer to the end of your model having 4 output neurons.\n You can use the perceptron neurons for that and manually precalculate the weights for it.\n\n \\section{Gradient decent}\n After we defined the design of our network, how can we train it?\\\\\n Hold your horses, first we need a dataset which we want to use to train our model. For this purpose we use\n the dataset provided by MNIST for \\href{http://yann.lecun.com/exdb/mnist/}{handwritten digits}.\n The MNIST dataset comes in split into two parts, training with 60k images of handwritten digits by 25 individuals. The images are greyscaled and 28 by 28 pixel in size.\n The second part is a test set containing 10k images with the same features.\n We will use the test set to validate the quality of our network. To make it a good test set it has been acquired by\n 250 individuals. This helps building confidence in the current state during the training phase.\n \n What we'd like is an algorithm which lets us find the weights and biases so that the output approximates \\(o(x)\\) for all training inputs \\(x\\). To quantify how well\n we're achieving this goal at the moment we define the \\textit{cost}-function:\n \\begin{align}\n C(w,b) = \\frac{1}{2n}\\sum \\limits_x^n || o(x) - a ||^2\n \\label{eq:cost}\n \\end{align}\n \\(w\\) denotes the collection of all weights in the network, \\(b\\) all the biases, \\(n\\) the total number of training inputs, \\(a\\) is a vector of outputs from the network when \\(x\\) is input (so the current estimation)\n and the sum is over all training inputs \\(x\\). We call \\(C\\) the quadratic cost function or mean square error (MSE). Nice to see is that for \\(o(x) \\approx a\\) we get \\(C\\approx 0\\) and for big differences between \\(o(x)\\) and \\(a\\)\n we get big values for \\(C\\). This is nice! Whenever the model is good the cost is significantly lower then whenever we do a bad prediction.\n So the aim of training is to minimize \\(C\\), in other words we want the weights and biases that produce the least possible \\(C\\).\n We do that using the \\textbf{gradient decent} algorithm.\n\n Why introduce the quadratic cost? After all, aren't we primarily interested in the number of images correctly classified by the network? Why not try to maximize that number directly, rather than minimizing a proxy measure like the quadratic cost? The problem with that is that the number of images correctly classified is not a smooth function of the weights and biases in the network. For the most part, making small changes to the weights and biases won't cause any change at all in the number of training images classified correctly. That makes it difficult to figure out how to change the weights and biases to get improved performance. If we instead use a smooth cost function like the quadratic cost it turns out to be easy to figure out how to make small changes in the weights and biases so as to get an improvement in the cost. That's why we focus first on minimizing the quadratic cost, and only after that will we examine the classification accuracy.\n\n It turns out that we can understand a tremendous amount by ignoring most of that structure, and just concentrating on the minimization aspect. So for now we're going to forget all about the specific form of the cost function, the connection to neural networks, and so on. Instead, we're going to imagine that we've simply been given a function of many variables and we want to minimize that function. We're going to develop a technique called gradient descent which can be used to solve such minimization problems. Then we'll come back to the specific function we want to minimize for neural networks.\n\n Gradient decent is an algorithm which you can imagine the following.\n You have a plane with minima and maxima in it, some places are higher some are lower, now you place a ball on that plane and gradient decent will find for you the minima of your plane. It uses\n derivatives of the function on which to calculate the extreme on to ensure convergence into the direction of the minima.\n\n So how can we apply it to neural networks? The idea is to use it to fit the models weights \\(w_k\\) and biases \\(b_l\\) during the training phase and therefor to minimize the cost function \\(C\\).\n Writing the gradient decent update rule in terms of components we get\n \\begin{align}\n w_k' &= w_k - \\eta \\frac{\\partial C}{\\partial w_k}\\\\\n b_l' &= b_l - \\eta \\frac{\\partial C}{\\partial b_l}\n \\label{eq:gradient-decent}\n \\end{align}\n By repeatedly applying this update rule we can \"roll down the hill\", and hopefully find a minimum of the cost function. In other words, this is a rule which can be used to learn in a neural network.\\newline\n There are several challenges applying the gradient decent rule. In following chapters we will take a look at some of them in depth.\n But for now I just want to mention one problem. To understand the problem we need to take a look at \\eqref{eq:cost} again.\n To compute the proper gradient of \\(C\\), \\(\\nabla C\\), we need to compute the gradients \\(\\nabla C_x\\), separately the cost for each input \\(x\\). This will become a big problem\n for big training sets. Hence we need to use a different approach to calculate \\(\\nabla C\\).\n\n The solution for this problem is simple, we use statistical tools to calculate the gradient, to be precise we use the stochastic gradient decent.\n The idea is to make a proper estimation of \\(\\nabla C\\) by computing \\(\\nabla C_x\\) for a small, random, subset of the training set.\n By averaging over the small subset we can achieve a pretty well result for the true gradient \\(\\nabla C\\), which helps speeding up the computation of the gradient as well as the time spent to train the model.\n\n To give a proper explanation, stochastic gradient decent is using a small subset of the training set with \\(m\\) randomly chosen elements \\(X_1, X_2, ..., X_m\\), we refer to them as a mini-batch. For a sample size\n \\(m\\), which needs to be large enough, but not too large, we expect that the average value for \\(\\nabla C_{X_j}\\) will be roughly equal to the over all average of \\(\\nabla C_x\\), or\n \\begin{align}\n \\underbrace{\\frac{\\sum_{j=1}^m \\nabla C_x}{m}}_{\\text{Gradient of mini batch}}\n \\approx \\underbrace{\\frac{\\sum_x \\nabla C_x}{n}}_{\\text{Gradient of complete training set}}\n = \\underbrace{\\nabla C}_{\\text{overall gradient}}\n \\end{align}\n Hence we can use the gradient of the mini-batch to estimate the overall gradient!\n Applied to the cost calculation for weights and biases \\eqref{eq:gradient-decent} we get\n \n \\begin{align}\n w_k' &= w_k - \\frac{\\eta}{m} \\sum_j \\frac{\\partial C_{X_j}}{\\partial w_k}\\\\\n b_l' &= b_l - \\frac{\\eta}{m} \\sum_j \\frac{\\partial C_{X_j}}{\\partial b_l}\n \\end{align}\n where the sums are over all the training examples \\(X_j\\) of the mini-batch. Then we pick out the next \\(m\\)\n elements for the next mini-batch. And so on, until we exhausted the whole dataset, which then completes an \\textit{epoch}.\n At that point we start over with a new training epoch.\\newline\n It's worth noting that the implementation of \\eqref{eq:cost} might vary. We scaled the cost function using the factor \\(\\frac{1}{n}\\) people sometimes\n omit this scaling and compute the sum of cost of the current mini-batch. In a similar way the equations \\eqref{eq:gradient-decent} sometimes omit the term \\(\\frac{1}{m}\\).\n Conceptually this makes little difference in the result, since it's equivalent of scaling the learning rate \\(\\eta\\).\n\n \\section{Implementation}\n Alright, lets get them hands dirty and implement a program which learns how to recognize handwritten digits using gradient decision and the MNIST dataset.\n \n Download the repository\n\n \\begin{lstlisting}[basicstyle=\\tiny]{language=bash}\ngit clone https://github.com/MichalDanielDobrzanski/DeepLearningPython35\n \\end{lstlisting}\n which contains python 3 compatible code for a neural network as well as the training dataset by MNIST.\n Initially I stated that we split the training set into 60k training and 10k testing sets, but that's the official\n dataset form. We'll do it a bit different and keep the test set as is and split the training set into 50k training and 10k\n validation sets. We won't use the validation set for now, but later on it will come in handy.\n\n Check out the code in the repository, II will only explain certain features of it, so it's quite necessary that you understand how to read python code.\n The heavy lifting in form of mathematical operations is mostly lifted by the library \\lstinline{numpy}. But no worries, I will explain implementation parts of the code whenever it's necessary and\n will provide the mathematical representation in those cases to enable the reader to implement the example in any programming language.\n\n So without any further ado, here goes nothing!\n\n For now the main focus is on \\lstinline{network.py}. Let's run the statistic gradient decent training algorithm.\n In order to do so, include the \\lstinline{network} and \\lstinline{mnist_loader} scripts into a python shell.\n Afterwards use \\lstinline{lnist_loader.load_data_wrapper()} method to \n\\begin{lstlisting}{language=Python}\n>>> import network\n>>> import mnist_loader\n>>> training_data, validation_data, test_data \\\n = mnist_loader.load_data_wrapper()\n>>> nn=network.Network([784,30,10])\n>>> nn.SGD(training_data, 30, 10, 3.0, test_data=test_data)\n\\end{lstlisting}\n%http://neuralnetworksanddeeplearning.com/chap1.html#MathJax-Span-788\n \n \\section{Backpropagation}\n \\subsection{Notation}\n\n Das Gewicht \\(w\\), welches die Verbindung zwischen dem \\(k\\)-ten Neuron des \\(l-1\\)-ten Layers und dem \\(j\\)-ten Neuron des \\(l\\)-ten Layers\n wird geschrieben als\n\n\n\\end{document}\n", "meta": {"hexsha": "a017f79135399788114777f2461aede6046b06af", "size": 40567, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "include/math/ds/docs/Backpropagation.tex", "max_stars_repo_name": "philsupertramp/game-math", "max_stars_repo_head_hexsha": "56526b673bfc9c71515c8402b5ffa237f1819a39", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "include/math/ds/docs/Backpropagation.tex", "max_issues_repo_name": "philsupertramp/game-math", "max_issues_repo_head_hexsha": "56526b673bfc9c71515c8402b5ffa237f1819a39", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 34, "max_issues_repo_issues_event_min_datetime": "2020-11-27T13:33:04.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-06T13:40:39.000Z", "max_forks_repo_path": "include/math/ds/docs/Backpropagation.tex", "max_forks_repo_name": "philsupertramp/game-math", "max_forks_repo_head_hexsha": "56526b673bfc9c71515c8402b5ffa237f1819a39", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 66.9422442244, "max_line_length": 965, "alphanum_fraction": 0.689082259, "num_tokens": 9935, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5888891163376235, "lm_q2_score": 0.5813030906443134, "lm_q1q2_score": 0.3423230633738592}} {"text": "\\documentclass[12pt, fullpage,letterpaper]{article}\n\n\\usepackage[margin=1in]{geometry}\n\\usepackage{url}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{xspace}\n\\usepackage{graphicx}\n\\usepackage{hyperref}\n\\usepackage{cancel}\n%\\usepackage{graphicx}\n%\\usepackage{subfig}\n\n\\newcommand{\\semester}{Spring 2022}\n\\newcommand{\\assignmentId}{4}\n\\newcommand{\\releaseDate}{31 March, 2022}\n\\newcommand{\\dueDate}{11:59pm, 15 April, 2022}\n\\newcommand\\independent{\\protect\\mathpalette{\\protect\\independenT}{\\perp}}\n\\def\\independenT#1#2{\\mathrel{\\rlap{$#1#2$}\\mkern2mu{#1#2}}}\n\n\\newcommand{\\bx}{{\\bf x}}\n\\newcommand{\\bw}{{\\bf w}}\n\n\\title{CS 6190: Probabilistic Machine Learning \\semester}\n\\author{Homework \\assignmentId}\n\\date{Handed out: \\releaseDate\\\\\n Due: \\dueDate}\n\n\\begin{document}\n\\maketitle\n\n\\input{emacscomm}\n\\footnotesize\n\t\\begin{itemize}\n\t\t\\item You are welcome to talk to other members of the class about\n\t\tthe homework. I am more concerned that you understand the\n\t\tunderlying concepts. However, you should write down your own\n\t\tsolution. Please keep the class collaboration policy in mind.\n\t\t\n\t\t\\item Feel free discuss the homework with the instructor or the TAs.\n\t\t\n\t\t\\item Your written solutions should be brief and clear. You need to\n\t\tshow your work, not just the final answer, but you do \\emph{not}\n\t\tneed to write it in gory detail. Your assignment should be {\\bf no\n\t\t\tmore than 10 pages}. Every extra page will cost a point.\n\t\t\n\t\t\\item Handwritten solutions will not be accepted.\n\t\t\n\t\t\\item The homework is due by \\textbf{midnight of the due date}. Please submit\n\t\tthe homework on Canvas.\n\t\\end{itemize}\n\n%GMM\n%Bayes logistic regression, VB, Laplace, Laplace Hession, \n%LDA \n\n\\section*{Practice [100 points + 100 bonus]}\t\n\\label{sec:q1}\n\\begin{enumerate}\n\t\\item~[20 points] Suppose we have a scalar distribution, \n\t\\[\n\tp(z) \\propto \\exp(-z^2)\\sigma(10z + 3).\n\t\\]\n\t\\begin{enumerate}\n\t\t\\item~[3 points] Although the normalization constant is not analytical, we can use Gauss-Hermite quadrature to calculate an accurate approximation. Please base on the example in ``data/example-code/gmq\\_example.py'', calculate the numerical approximation of the normalization constant, and report its value. With the normalization constant, please draw the density curve of $p(z)$, in the range $z \\in [-5, 5]$. \n\t\t\\item~[5 points] Implement the Laplace approximation of $p(z)$, and report the mean and variance of your Gaussian distribution. Draw the density of your Laplace approximation in the same plot as in (a). \n\t\t\\item~[10 points] Use the local variational inference method and EM-style updates as we discussed in the class (for logistic regression) to implement the variational approximation to $p(z)$. Report the form of your approximate distribution, and draw its density in the same plot as above.\n\t\t\\item~[2 points] By comparing the ``ground-truth'' (from (a)) and the approximations (from (b,c)), what do you observe and conclude?\n\t\\end{enumerate}\n\n\t\\item~[50 points] Let us work on a real-world dataset we have met before. Please download the data from the folder``data/bank-note''. The features and labels are listed in the file ``data-desc.txt''. The training data are stored in the file ``train.csv'', consisting of $872$ examples. The test data are stored in ``test.csv'', and comprise of $500$ examples. In both the training and testing datasets, feature values and labels are separated by commas. We assign the feature weight vector $\\w$ a standard normal prior $\\N(\\0, \\I)$. \n\t\\begin{enumerate}\n\t\t\\item~[7 points] Implement the standard Laplace approximation to the posterior distribution of the feature weight vector. Report your approximate posterior. Now, use Gauss-Hermite quadrature to implement the calculation of the predictive distribution. Please be careful: \\textbf{you need to do a proper variable transformation in the integral before applying the Gauss-Hermite quadrature because you integrate with a Gaussian like $\\N(x|\\mu, \\sigma^2)$ rather than $\\exp(-x^2)$!} Now we test the performance with two measures. First, we calculate the inner-product between the posterior mean of the weight vector and the feature vector of each test example, and throw the inner-product into the sigmoid function to calculate the probability that the test example is positive. If the probability is no less than 0.5, we classify the example to be positive (\\ie 1) otherwise we classify the example to be negative (\\ie 0). Report the prediction accuracy. Second, we calculate the average predictive likelihood of the test samples, namely we evaluate the predictive density value of each test sample based on the predictive distribution and then take an average. Note that in Bayesian learning, the predictive likelihood includes all the information of the (approximate) posterior distribution, hence is more preferred in the evaluation. \n\t\t\\item~[3 points] Implement Laplace approximation with the diagonal Hessian. Report the approximate posterior distribution of the feature weights, the prediction accuracy and average predictive likelihood. \n\t\t\\item~[20 points] Implement variational logistic regression we introduced in the class. Use EM-style updates. Report the variational posterior of the feature weight vector you obtained (\\ie a multivariate Gaussian). Report the prediction accuracy and average predictive likelihood. \n\t\t\\item~[15 points] Implement variational logistic regression we introduced in the class. But this time, you will use the fully factorized posterior, $q(\\w)= \\prod_i q(w_i)$ where $w_i$ is $i$-th element in the weight vector $\\w$. In the E step, please use the standard mean-field update to alternatively optimize each $q(w_i)$ given all the others fixed. Report your variational posterior (\\ie diagonal Gaussian), the prediction accuracy and average predictive likelihood on the test data. \n\t\t\n\t\t\\item~[5 points] Compare the results of the above four approximations. What do you observe and conclude?\n\t\\end{enumerate}\n\n\t\\item~[30 points] Gaussian Mixture Model (GMM). Please download the data ``data/faithful/faithful.txt'' from Canvas. Each row is a sample, including 2 features. Please normalize the features in each column to be in [-1, 1]. Specifically, denote the column by $\\x$; then we compute for each $x_i \\leftarrow (x_i - \\mathrm{mean}(\\x))/(\\max(\\x) - \\min(\\x))$. \n\t\\begin{enumerate}\n\t\t\\item~[20 points] Implement EM algorithm for GMM. Set the number of clusters to $2$. Initialize the cluster centers to be [-1, 1] and [1,-1], and the covariance matrix to be $0.1\\cdot\\I$ for both clusters. Run your EM algorithm for $100$ iterations. For iteration $1$, $2$, $5$, $100$, please draw the figures showing the corresponding cluster centers and memberships. Specifically, for each figure, first draw the scatter plots of the data points and your cluster centers. Each data point is assigned to the cluster that has a great posterior probability to include that data point. Please draw the cluster memberships with different colors. \n\t\t\\item~[7 points] Now initialize the cluster centers to be [-1, -1] and [1, 1] and covariance matrix to be $0.5 \\cdot \\I$ for both clusters. Run your EM algorithm for $100$ iterations. Draw the figures showing the cluster centers and memberships for iteration $1$, $2$, $5$, $100$. \n\t\t\\item~[3 points] Compare the results in (a) and (b), what do you observe and conclude?\n\t\\end{enumerate}\n\n\t\\item~[100 points][\\textbf{Bonus}] Latent Dirichlet Allocation (LDA). Please download the pre-processed corpus from ``data/lda''. From ``ap.txt'', you can see the original Associated Press corpus. ``ap.dat'' are the processed data which you will work on. Each row in ``ap.dat'' represents a document. In total, we have $2,246$ documents. The first number in each row is the number of words in the document. The following are a list of \\textbf{word-id}:\\textbf{count} items. Word-id starts with 0. The corresponding word list is given in ``vocab.txt''. The first row correspond to Word-id 0, second, Word-id 1, and continue. \n\t\n\t\\begin{enumerate}\n\t\t\\item~[70 points] Implement the mean-field variational EM algorithm for LDA inference as we discussed in the class. Following the orignal LDA paper ( \\url{http://www.cs.columbia.edu/~blei/papers/BleiNgJordan2003.pdf}) to implement the perplexity calculation on test documents (Sec. 7.1). Please randomly select $10\\%$ documents as the test set, and run your LDA inference algorithm on the remaining $90\\%$ documents. Vary the number of topics from \\{5, 10, 15, 20, 50, 100, 200\\}. Run your algorithm until convergence or $500$ iterations have achieved. Draw a figure to show how the perplexity vary along with the growth of the topic numbers. What do you observe and conclude?\n\t\t\\item~[30 points] Set the number of topics to $20$ and run your variational inference algorithm. Examine the top 15 words (\\ie with the largest probability) in each learned topic distribution. List a few topics which you think is semantically meaningful and explain why. \n\t\\end{enumerate}\n\\end{enumerate}\n\n\n\\end{document}\n", "meta": {"hexsha": "7660dfae55fd4fa257142a8a2a2abd46708fc1df", "size": 9038, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "assignments/a4/hw4.tex", "max_stars_repo_name": "tgautam03/CS6190-ProbabilisticML", "max_stars_repo_head_hexsha": "f000f571d1068ab640a360b490a40f0f15d8502b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-08T06:17:05.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-08T06:17:05.000Z", "max_issues_repo_path": "assignments/a4/hw4.tex", "max_issues_repo_name": "tgautam03/CS6190-ProbabilisticML", "max_issues_repo_head_hexsha": "f000f571d1068ab640a360b490a40f0f15d8502b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "assignments/a4/hw4.tex", "max_forks_repo_name": "tgautam03/CS6190-ProbabilisticML", "max_forks_repo_head_hexsha": "f000f571d1068ab640a360b490a40f0f15d8502b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 92.2244897959, "max_line_length": 1339, "alphanum_fraction": 0.7621155123, "num_tokens": 2283, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166195971441, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.3422771423746722}} {"text": "\\documentclass{article}\n\\usepackage{fullpage}\n\\usepackage{nopageno}\n\\usepackage{amsmath}\n\\allowdisplaybreaks\n\n\\newcommand{\\abs}[1]{\\left\\lvert #1 \\right\\rvert}\n\n\\begin{document}\n\\title{Notes}\n\\date{February 3, 2014}\n\\maketitle\n\\section*{last time}\n\\section*{lesson 7}\n\\subsection*{step 1}\nsolve pde for fundamental solutions\n\\begin{align*}\n u&=T(t)X(x)\\\\\n \\frac{T'(t)}{\\alpha^2T(t)}&=\\frac{X'(x)}{X(x)}=\\text{separation constant}\\\\\n \\intertext{phyysical considerationsindicatethat solutionsdeca to 0. simplify by assigning sep constant $\\leq0$}\\\\\n \\frac{T'(t)}{\\alpha^2T(t)}&=\\frac{X''(x)}{X(x)}=-\\lambda^2\\leq0 &T(x)X(x)=e^{-\\alpha^2\\lambda^2t}\\\\\n\\end{align*}\n\\subsection*{step 2}\nsolve pde and bc. solve $X''(x)+\\lambda^2X(x)=0 with X(0)=0$.\n\n\\subsubsection*{case $\\lambda=0$}\n\\begin{align*}\n X''(x)&=0\\\\\n X&=ax+b\\\\\n X(0)&=0 \\text{ gives} b=0\\\\\n X'(1)+hX^{(1)}&=0\\\\\n \\intertext{only trivial solution$X=0$}\n X&=ax\n \\intertext{ohter BC gives $a(h+1)=0$ give $a=0$}\n\\end{align*}\n\\subsubsection*{case $\\lambda>0$}\n\\begin{align*}\n X(x)&=a\\sin(\\lambda x)+b\\cos(\\lambda x)\\\\\n X(0)&=0\\to b=0\\\\\n &=a\\sin(\\lambda x)\\\\\n X'(1)+hX(1)&=0\\\\\n \\lambda\\cos(\\lambda)+h\\sin(\\lambda)&=0\\\\\n X'(x)&=a\\lambda\\cos(\\lambda x)\\\\\n \\intertext{this is on page 52}\n a\\lambda\\cos(\\lambda)+ha\\sin(\\lambda&=0\\\\\n a(\\lambda\\cos(\\lambda)+h\\sin(\\lambda))&=0\\\\\n\\end{align*}\nquestion: can $\\cos(\\lambda)=0$? no! then $\\sin(\\lambda)=0$. but $\\cos^2+\\sin^2=1$. Sothe equation becomes $\\tan\\lambda)=-\\frac{\\lambda}{h}$. And from the graph we see that there are infinitely many positively roots $0<\\lambda<\\lambda_2<\\lambda_3<\\dots$.\n\\begin{align*}\n \\frac{\\pi}{2}&<\\lambda_1<\\pi\\\\\n \\frac{3\\pi}{2}&<\\lambda_2<2\\pi\\\\\n n\\pi-\\pi/2& 0 \\\\\n 0 & \\op{B} = 0\n \\end{cases}\n\\end{align}\n$\\cond AB$ semantically asks \\enquote{assuming $B$ to happen with certainty, what is the probability that $A$ will happen}.\n$A$ and $B$ are independent if and only if $\\cond AB = \\op{A}$. Or in general $A_1, A_2, \\ldots, A_n \\in \\mathcal{A}$ is independent if $\\forall\\: 1 \\leq i_1 < i_2 < \\ldots < i_k \\leq n$\n\n\\[\n \\op{A_{i_1} \\cap A_{i_2} \\cap \\ldots \\cap A_{i_k}}\n = \\op{A_{i_1}} \\op{A_{i_2}} \\cdot \\ldots \\cdot \\op{A_{i_k}}\n\\]\n\n\\subsection{What is marginal/joint distribution? What's their relation?}\n%\nLet $A$ and $B$ be two events occuring with \\key{marginal} probability $\\op{A}$ and $\\op{B}$. The \\key{joint probability} is intuitively defined as probability that $A$ \\emph{and} $B$ will happen. Formally $\\op{A, B}$, defined as the intersection $\\op{A \\cap B}$.\n\n\\begin{align*}\n \\text{marginal probability: } && p_{X,Y}(x, y) &= \\os{X=x, Y=y} \\\\\n \\text{joint probability: } && p_X(x_1, x_2, \\ldots, x_n) &= \\os{X = (x_1, x_2, \\ldots, x_n)}\n\\end{align*}\n\nThen the following relation holds:\n\\[\n \\cond AB = \\begin{cases}\n \\frac{\\op{A \\cap B}}{\\op{B}} & \\text{if } \\op{B} > 0 \\\\\n 0 & \\text{if } \\op{B} = 0\n \\end{cases}\n\\]\n\nBayes' Theorem is given by\n\\[\n \\cond AB = \\frac{\\cond BA \\cdot \\op A}{\\op B}\n\\]\n\n\\subsection{Prove $\\mathbb{E}(X\\cdot Y) = \\mathbb{E}(X) \\cdot \\mathbb{E}(Y)$ for independent variables in the discrete case}\n\n\\begin{align*}\n \\Ex(X) &= \\sum_x x \\cdot p_X(x) \\\\\n \\Ex(Y) &= \\sum_y y \\cdot p_Y(y) \\\\\n \\Ex(X \\cdot Y) &= \\sum_{x,y} xy \\cdot p_{X,Y}(x,y) \\\\\n &= \\sum_x \\sum_y xy \\cdot p_X(x) \\cdot p_Y(y) \\\\\n &= \\left(\\sum_x x \\cdot p_X(x)\\right) \\cdot \\left(\\sum_y y \\cdot p_Y(y)\\right) \\\\\n &= \\Ex(X) \\cdot \\Ex(Y)\n\\end{align*}\n\n\\subsection{How is covariance and variance defined?}\n\nThe expected value $\\Ex(X)$ of a random variable $X$ is also known as mean $\\mu$.\n\nCovariance is a measure of how much two random variables change together. If the greater values of one variable primarily correspond with the greater values of the other variable (and vice versa for small values), the covariance is positive. If they are linearly disproportional, it is negative.\n\n\\[ \\sigma(X, Y) = \\Ex((X - \\Ex(X))(Y - \\Ex(Y))) = \\Ex(XY) - \\Ex(X) \\Ex(Y) \\]\n\nVariance is a special case of covariance with only one parameter. Variance is the covariance of a variable with itself: $\\sigma^2(X) = \\sigma(X, X)$. It measures how far a set of numbers is spread out. A variance of zero indicates that all the values are identical. Variance is always non-negative.\n\n\\[ \\sigma^2(X) = \\Va(X) = \\Ex((X - \\Ex(X))^2) \\]\n\\begin{align*}\n \\text{discrete:} && \\Va(X) &= \\sum_{i=1}^n (p_i \\cdot x_i^2) - \\mu^2 \\\\\n \\text{continuous:} && \\Va(X) &= \\int x^2 p(x) \\dt x - \\mu^2\n\\end{align*}\n\nCovariance is symmetrical: $\\sigma(X, Y) = \\sigma(Y, X)$. If $X$ and $Y$ are independent, then $\\sigma(X, Y) = 0$.\n\n\\subsection{Derive Bienaymé's equation for $\\Va(\\overline{X_n})$}\n\nBienaymé's equation states that\n%\n\\[\n \\Va\\left(\\sum_{i=1}^n X_i\\right) = \\sum_{i=1}^n \\Va(X_i)\n\\]\n%\n\\dots which makes variance preferrable in many applications. Furthermore it implies that \n%\n\\begin{align*}\n \\overline{X_n} &= \\frac{X_1 + \\ldots + X_n}{n} \\\\\n% \\Ex(\\overline{X_n}) &= \\frac{\\Ex(X_1) + \\ldots + \\Ex(X_n)}{n} = \\mu \\\\\n \\Va\\left(\\overline{X_n}\\right) &= \\Va\\left(\\frac1n \\sum_{i=1}^n X_i\\right) \\\\\n &= \\frac1{n^2} \\sum_{i=1}^n \\Va(X_i) \\\\\n &= \\frac1{n^2} n \\Va(X) \\\\\n &= \\frac1n \\Va(X) \\\\\n &= \\frac{\\sigma^2}{n}\n\\end{align*}\n\n\\subsection{Distinguish between convergence \\enquote{almost surely} and \\enquote{in probability}}\n\nLet $\\left(X_n\\right)_{n \\in \\mathbb{N}}$ be a sequence of random variables.\n\n\\begin{align*}\n \\lim_{n \\rightarrow \\infty} X_n = X \\text{ almost surely if }\n & \\mathbb{P}[\\exists \\lim X_n \\land \\lim X_n = X] = 1 \\\\\n \\lim_{n \\rightarrow \\infty} X_n = X \\text{ in probability if }\n & \\forall\\: a > 0, \\lim_{n \\rightarrow \\infty} \\mathbb{P}[\\left|X_n - X\\right| \\geq a] = 0\n\\end{align*}\n\nIntuitively almost surely means convergence is given until infinity. In probability means $n$ converges until a fixed constant (not necessarily infinity).\n\n\\subsection{In our notation, what is the difference between $x$, $X$, $X_n$ and $\\mathcal{X}$?}\n% source: 10\n\n\\begin{description}\n \\item[$x$] \\hfill{} \\\\\n A specific value (realization) of a random value. For example we iterate over all values of the discrete domain. The $x$ is typically used as iterator $x \\in X$.\n\n \\item[$X$] \\hfill{} \\\\\n $X$ denotes a random variable used in an equation. Often $Y$ accompanies $X$ as second random variable to discuss joint probability.\n\n \\item[$X_n$] \\hfill{} \\\\\n $X_n$ denotes a sequence of random variables.\n\n \\item[$\\mathcal{X}$] \\hfill{} \\\\\n $\\mathcal{X}$ is used with two different meanings:\n \\begin{itemize}\n \\item In the context of Markov chains or information channels, we used $\\mathcal{X}$ as the given input values for the Markov process.\n \\item In the context of probability theory, $\\mathcal{X}$ denotes the domain of a random variable $X$ or the sequence $X_n$ (if they all use the same domain).\n \\item In the context of codes, $\\mathcal{X}$ denotes the alphabet before encoding.\n \\end{itemize}\n\\end{description}\n\n\\subsection{Define and prove Markov's inequality}\n\n\\begin{mdframed}\n Let $X \\geq 0$ be a random variable with $0 < \\Ex(X) < \\infty$, then \n \\[ \\os{X > a \\cdot \\Ex(X)} \\leq \\frac1a \\quad\\forall\\: a > 0 \\]\n\\end{mdframed}\n\n\\begin{align*}\n A &= [X \\geq a \\Ex(X)] \\\\\n &= \\set{w \\in \\Omega: X(w) \\geq a \\cdot \\Ex(X)} \\\\\n X &\\geq X \\ind_A \\\\\n &\\geq \\underbrace{a \\Ex(X)}_{\\text{constant } c} \\cdot \\ind_A \\Leftrightarrow \\\\\n X(w) &\\geq X(w) \\ind_A(w) \\\\\n &\\geq c \\cdot \\ind_A(w) \\Rightarrow \\\\\n \\Ex(X) &\\geq \\Ex(c \\cdot \\ind_A) \\\\\n &= a\\Ex(X) \\os A \\Rightarrow \\\\\n \\frac{\\Ex(X)}{\\Ex(X)} &\\geq a \\os A \\Rightarrow \\\\\n \\os A &\\leq \\frac 1a\n\\end{align*}\n\n\\subsection{Define and prove Chebyshev's inequality}\n\n\\begin{mdframed}\n Let $X$ be a random variable with finite $\\Ex(X)$ and $\\Va(X)$, then $\\forall\\: a > 0$\n \\[ \\os{\\card{X - \\Ex(X)} \\geq a} \\leq \\frac{\\Va(X)}{a^2} \\]\n\\end{mdframed}\n\n\\begin{align*}\n Y &= (X - \\Ex(X))^2 \\geq 0 \\\\\n \\Ex(Y) &= \\Va(X) \\\\\n \\os{\\card{X - \\Ex(X)} \\geq a}\n &= \\os{Y \\geq a^2} \\\\\n &= \\os{Y \\geq \\frac{a^2}{\\Va(X)} \\Ex(Y)} \\\\\n &\\leq \\frac{1}{\\frac{a^2}{\\Va(X)}} && \\text{[apply Markov inequality]} \\\\\n & = \\frac{\\Va(X)}{a^2}\n\\end{align*}\n\n\n\n\\subsection{Distinguish between the Weak and Strong Law of Large Numbers}\n\nThe Weak Law of Large Numbers derives from Chebyshev's inequality and states,\n\n\\begin{mdframed}\n Given $X_n$ as sequence of iid random variables, $\\mu = \\Ex(X_n)$ and $\\sigma^2 = \\Va(X_n) < \\infty$, then\n \\[ \\overline{X_n} \\converges \\mu \\text{ in probability} \\]\n\\end{mdframed}\n\nThe Strong Law of Large Numbers states,\n\n\\begin{mdframed}\n Given $X_n$ as sequence of iid random variables then there exists a finite random variable $Y$ such that\n \\[ \\overline{X_n} \\converges Y \\text{ almost surely} \\]\n\\end{mdframed}\n\nThe Strong Law of Large Numbers can be equivalently defined as $\\Ex(X_n)$ being finite and in this case $Y = \\Ex(X_n)$ almost surely.\n\n\\subsection{Prove the Weak Law of Large Numbers}\n% source: Patrick\n\n\\begin{align*}\n \\os{\\card{\\overline{X_n} - \\mu} \\geq a} &\\leq \\frac{\\Va(\\overline{X_n})}{a^2} && \\text{[Chebyshev inequality must hold]} \\\\\n &= \\frac{\\sigma^2}{na^2} \\\\\n &\\converges 0\n\\end{align*}\n\n\\section{Entropy}\n\n\\subsection{Give Hartley's definition of information value}\n%\n\\begin{quote}\n The answer to a question that can assume the two values \\enquote{yes} and \\enquote{no} (without taking into account the meaning of the question) contains one unit of information.\n\\end{quote}\n\nLet $U_N$ be a set of $N$ elements of uniform probability; hence the information amount to identify an element is\n\\[ \\op[H]{U_N} = \\log_2(N) \\in \\mathbb{R} \\]\n\nIt holds that\n\\begin{itemize}\n \\item $\\op[H]{U_2} = 1$\n \\item $\\op[H]{U_N} \\leq \\op[H]{U_{N+1}}$\n \\item $\\op[H]{U_{M\\cdot N}} = \\op[H]{U_M} + \\op[H]{U_N}$\n\\end{itemize}\n\n\\subsection{Derive Shannon's entropy definition from Hartley's information value}\n\nAs can be seen by the third corollary of Hartley, we can group information into (not necessarily equally large) groups of values:\n\n\\[ U_N = U_{N_1} \\dotcup U_{N_2} \\dotcup \\ldots \\dotcup U_{N_n} \\]\n\nIf we want to identify the value in those groups, we need to ask two questions:\n\\begin{enumerate}\n \\item In which group?\n \\item Which element in the group?\n\\end{enumerate}\n\nIf the value can be found in group $k$, then $\\log_2{N_k}$ questions are needed to identify the value.\nThe average number of questions for the second question is:\n\n\\[ H_2 = \\sum_{k=1}^n \\frac{N_k}{N} \\log_2{N_k} \\]\n\nWhat about $H_1$?\n\n\\begin{align*}\n \\op[H]{U_N} &= H_1 + H_2 \\\\\n \\log_2{N} &= H_1 + \\sum_{k=1}^n \\frac{N_k}{N} \\log_2{N_k} \\\\\n H_1 &= \\log_2{N} - \\sum_{k=1}^n \\frac{N_k}{N} \\log_2{N_k} \\\\\n &= \\sum_{k=1}^n \\frac{N_k}{N} \\log_2{N} - \\sum_{k=1}^n \\frac{N_k}{N} \\log_2{N_k} \\\\\n &= -\\sum_{k=1}^n \\frac{N_k}{N} \\log_2{\\frac{N_k}{N}} \\\\\n &= -\\sum_{k=1}^n p_k \\log_2{p_k} && \\text{with } p_k = \\frac{N_k}{N}\n\\end{align*}\n\n\\subsection{Give Shannon's definition of \\key{entropy} and provide examples for small/large entropy}\n\n\\key{Entropy} is a measure of uncertainty of a random variable. Let $X$ be a discrete random variable with alphabet $\\mathcal{X}$ and probability mass function $p_X(x) = \\os{X = x}$ for $x \\in \\mathcal{X}$. The entropy $\\op[H]{X}$ of a discrete random variable $X$ is defined by\n\\[ \\op[H]{X} = -\\sum_{x \\in \\mathcal{X}} p_X(x) \\log_2{p_X(x)} = -\\Ex(\\log_2 p_X(x)) \\]\n$\\op[H]{X} \\geq 0$ holds for all $X$. Furthermore we define\n\\[\n 0 \\log 0 = 0 \\quad \\forall\\, b \\geq 0 \\qquad\n a \\log \\frac a0 = \\infty \\quad \\forall\\, a > 0\n\\]\n\nFor example a uniform distribution of two discrete values gives\n\\[ \\op[H]{X} = - \\left(\\frac12 \\log_2{\\frac12} + \\frac12 \\log_2{\\frac12} \\right) = 1 \\]\nUniform distribution always provides a \\key{large} value of entropy. Certain events provide a \\key{small} value. For example consider two events, where only event $A$ always happens:\n\\[ \\op[H]{X} = - \\left(0\\cdot\\log_2{0} + 1\\cdot\\log_2{1}\\right) = \\left(1\\cdot0\\right) = 0 \\]\n\nThe following holds:\n\\begin{itemize}\n \\item $\\op[H]{X}$ depends only on the probabilities; not on the specific interpretation.\n \\[ f: \\mathcal{X} \\rightarrow \\mathcal{X}' \\land X' = f(X) \\Rightarrow \\op[H]{X'} = \\op[H]{X} \\]\n \\item $p_0 \\mapsto \\op[H]{p_0, 1 - p_0}$ is continuous. The maximum is attained at $\\frac12$:\n \\[ \\op[H]{p_0, 1 - p_0} = -p_0 \\log_2(p_0) - (1 - p_0) \\log_2(1 - p_0) \\]\n \\[\n -\\frac12 \\log_2\\left(\\frac12\\right)\n - \\left(1 - \\frac12\\right) \\log_2 \\left(1 - \\frac12\\right)\n = +\\frac12 - \\left(-\\frac12\\right)\n = 1\n \\]\n \\item For some fixed $n$,\n \\[ \\op[H]{p_1, \\ldots, p_n} \\leq \\op[H]{\\frac1n, \\ldots, \\frac1n} = \\log_2(n) \\]\n\\end{itemize}\n\n\\subsection{What is joint entropy? What is the entropy of conditional distribution aka. conditional entropy?}\n\nGiven a pair of discrete random variable ($X, Y$). The joint entropy of $X$ and $Y$ is defined as:\n\\begin{align*}\n \\op[H]{X, Y} &= -\\sum_{x \\in \\mathcal{X}} \\sum_{y \\in \\mathcal{Y}} p(x, y) \\log_2 p(x, y) \\\\\n &= -\\Ex\\left(\\log_2{p(X, Y)}\\right)\n\\end{align*}\n\nEntropy of conditional distribution:\n\\begin{align*}\n \\cond[H]YX &= \\sum_{x \\in \\mathcal X} p_X(x) \\cond[H]{Y}{X=x} \\\\\n &= \\sum_{x \\in \\mathcal X} \\op{X=x} \\left(-\\sum_{y \\in \\mathcal Y} \\cond{Y=y}{X=x} \\log_2 \\cond{Y=y}{X=x} \\right) \\\\\n &= -\\sum_{x \\in \\mathcal X} \\sum_{y \\in \\mathcal Y} \\op{X=x} \\cond{Y=y}{X=x} \\log_2 \\cond{Y=y}{X=x}\n & \\left[\\op x \\cdot \\cond yx = \\op{y,x}\\right] \\\\\n &= -\\sum_{x \\in \\mathcal X} \\sum_{y \\in \\mathcal Y} \\op{Y=y,X=x} \\log_2 \\frac{\\op{Y=y,X=x}}{\\op{Y=y}} \\\\\n &= \\sum_{x \\in \\mathcal X} \\sum_{y \\in \\mathcal Y} \\op{Y=y,X=x} \\log_2 \\frac{\\op{Y=y}}{\\op{Y=y,X=x}} \\\\\n &= \\sum_{x \\in \\mathcal X} \\sum_{y \\in \\mathcal Y} p_{X,Y}(x,y) \\log_2 \\frac{p_Y(y)}{p_{X,Y}(x,y)}\n\\end{align*}\n\n\\subsection{Prove that $\\cond[H] YX = \\op[H]{X, Y} - \\op[H]{X}$}\n\n\\begin{align*}\n \\cond[H] YX &= -\\sum_x \\sum_y p_X(x) p_{X,Y}(y\\,|\\,x) \\log_2 p_{X,Y}(y,x) \\\\\n &= -\\sum_x \\sum_y p_{X,Y}(x,y) \\left[\\log_2 p_{X,Y}(x,y) - \\log_2 p_X(x)\\right] \\\\\n &= \\op[H]{X,Y} + \\sum_x \\log_2 p_X(x) \\sum_y p_{X,Y}(x,y) \\\\\n &= \\op[H]{X,Y} - \\op[H]{X}\n\\end{align*}\n\n\\subsection{What is the Kullback-Leibler distance? Are the parameters symmetrical?}\n\nLet $p$ and $q$ be two probability distributions on the finite set $\\mathcal{X}$.\n$X$ is a random variable with distance probability $p$.\nThe \\key{relative entropy} or \\key{Kullback-Leibler distance} of $p$ with respect to $q$ is\n\\[ \\kld{p}{q} = \\sum_x p(x) \\log_2 \\frac{p(x)}{q(x)} = \\Ex\\left(\\log_2 \\frac{p(x)}{q(x)}\\right) \\]\n\n\\begin{itemize}\n \\item Be aware that for $p(x) > 0$ and $q(x) = 0$: $\\kld pq = \\infty$.\n \\item $\\kld pq $ is always non-negative.\n \\item From $\\kld pq = 0$ it follows that $p(x) = q(x) \\: \\forall x \\in \\mathcal{X}$.\n \\item $\\kld pq \\neq \\kld qp$\n \\item Intuitively it is a measure of the information lost when $q$ is used to approximate $p$.\n\\end{itemize}\n\n\\subsection{What is mutual information? What is conditional mutual information? What is the chain rule?}\n\nLet $X$ and $Y$ be a pair of random variables. \\key{Conditional mutual information} is defined as,\n\n\\begin{align*}\n \\op[I]{X, Y} &= \\kld{p_{(X,Y)}}{p_X \\bigotimes p_Y} \\\\\n &= \\sum_{x,y} p(x,y) \\log_2 \\frac{p(x,y)}{p_X(x)\\cdot p_Y(y)} \\\\\n &= \\op[H]{X} - \\cond[H] XY = \\op[H]{Y} - \\cond[H] YX\n\\end{align*}\n\nIntuitively, mutual information measures the information that X and Y share. It measures how much knowing one of these variables reduces uncertainty about the other.\n\nSpecifically it holds that $\\op[I]{X, X} = \\op[H]{X}$. The \\key{chain rule} states that\n\\[\n \\op[H]{X_1, \\dotsc, X_n} = \\sum_{k=1}^n \\cond[H]{X_k}{X_{k-1}, \\dotsc, X_{1}}\n\\]\n\nThe chain rule of mutual information is defined as,\n\\[\n \\op[I]{(X_1, \\dotsc, X_k); Y} = \\sum_{k=1}^n \\cond[I]{X_k; Y}{X_{k-1},\\dotsc,X_{1}}\n\\]\n\n\\key{Conditional mutual information} of $X$ and $Y$ given $Z$:\n\n\\begin{align*}\n \\cond[I]{X;Y}{Z} &= \\sum_z p_Z(z) \\cond[I]{X;Y}{Z=z} \\\\\n &= \\begin{cases}\n \\cond[H]YZ - \\cond[H]Y{X,Z} \\\\\n \\cond[H]XZ - \\cond[H]X{Y,Z}\n \\end{cases}\n\\end{align*}\n\n\\subsection{What is Jensen's inequality?}\n%\nGiven a continuous, convex function $f: I \\rightarrow \\mathbb{R}$ (with $I$ as open interval) and $X$ as random variable,\n\\[ \\Ex(f(X)) \\geq f(\\Ex(X)) \\]\n\n\\subsection{What is information inequality?}\n\nLet $p(x)$ and $q(x)$ be a pair of probability distributions with $x \\in \\mathcal{X}$, then $\\kld{p}{q} \\geq 0$.\n\nEquality holds if and only if $p(x) = q(x) \\quad\\fall x$.\n\nCorollaries:\n\\begin{itemize}\n \\item $\\op[I]{X;Y} \\geq 0$\n \\item $\\cond[H]XY \\leq \\op[H]{X}$\n \\item $\\cond[I]{X;Y}{Z} \\geq 0$\n \\item $\\op[H]{X_1, \\dotsc, X_n} \\leq \\sum_{k=1}^n \\op[H]{X_k}$\n\\end{itemize}\n\n\\subsection{What is the log-sum inequality?}\n\nGiven $a_1, \\dotsc, a_n$ and $b_1, \\dotsc, b_n \\geq 0$, then\n\\[\n \\sum_{i=1}^n a_i \\log{\\frac{a_i}{b_i}}\n \\geq \\left(\\sum_{i=1}^n a_i\\right) \\log{\\frac{\\sum_{i=1}^n a_i}{\\sum_{i=1}^n b_i}}\n\\]\n\nGiven $p^{(1)}(\\cdot), p^{(2)}(\\cdot)$ and $q^{(1)}(\\cdot), q^{(2)}(\\cdot)$\nas probability densities on $\\mathcal{X}$ with $0 \\leq \\lambda \\leq 1$\n\\begin{multline*}\n \\Rightarrow\n \\kld{\\lambda \\cdot p^{(1)} + (1 - \\lambda) p^{(2)}}%\n {\\lambda \\cdot q^{(1)} + (1 - \\lambda) \\cdot q^{(2)}} \\\\\n \\leq\n \\lambda \\kld{p^{(1)}}{q^{(1)}} +\n (1 - \\lambda) \\kld{p^{(2)}}{q^{(2)}}\n\\end{multline*}\n\n\\subsection{What is a Markov chain?}\n%\nA \\key{Markov chain} is a random process satisfying the Markov property which means transitions happen memoryless.\nGiven a state space, transitions can happen between any two states with associated probabilities. Transition probabilities are stored in the so-called \\enquote{stochastic matrix}.\n\nHence a Markov chain is a sequence of random variables $X_1, X_2, X_3, \\dotsc$ with the Markov property, namely that, given the present state, the future and past states are independent. Formally $\\cond{X_{n+1}}{X_1 = x_1, X_2 = x_2, \\dotsc, X_n = x_n} = \\cond{X_{n+1} = x}{X_n = x_n}$.\n\nA Markov chain is ergodic iff every transition probability is greater zero. A Markov triple is a Markov chain of three states.\n\n\\subsection{Define Data Processing inequality}\n\nLet $X \\rightarrow Y \\rightarrow Z$ be a Markov triple \\emph{in that order} meaning $X$ and $Z$ are independent but conditional on $Y$. Hence\n\\[ \\cons{X=x, Z=z}{Y=y} = \\cons{X=x}{Y=y} \\cdot \\cons{Z=z}{Y=y} \\]\n\nIt holds that\n\\[ \\op[I]{X;Y} \\geq \\op[I]{X;Z} \\]\n\nThe information that $Y$ knows about $X$ cannot be increased by manipulating (processing) $Y$ deterministically or at random.\n\n\\subsection{Prove Data Processing inequality}\n\nThe chain rule tells\n\\[ \\op[I]{(Y, Z); X} = \\op[I]{Y;X} + \\cond[I]{Z;X}{Y} \\]\n\nSo,\n\\begin{align*}\n \\op[I]{X;Y,Z} &= \\op[I]{X;Y,Z} \\\\\n \\op[I]{X;Y} + \\underbrace{\\cond[I]{X;Z}{Y}}_{\\geq0} &= \\op[I]{X;Z} + \\underbrace{\\cond[I]{X;Y}{Z}}_{=0} \\\\\n \\op[I]{X;Y} &\\geq \\op[I]{X;Z}\n\\end{align*}\n\n\\subsection{Define and prove Fano's inequality}\n% source: 10, Felix\n\n\\begin{mdframed}\n Define a Markov triple $X \\rightarrow Y \\rightarrow \\hat X$\n with $\\mathbb{P}_{\\text{err}}$ as the error that $X \\neq \\hat X$.\n \\[\n H(\\Perr) + \\Perr \\log{\\card{\\mathcal{X}}}\n \\geq \\cond[H]{X}{\\hat X}\n \\geq \\cond[H]XY\n \\]\n\n Equivalently\n \\[ 1 + \\Perr \\log{\\card{\\mathcal{X}}} \\geq \\cond[H]XY \\]\n \\[ \\Perr \\geq \\frac{\\cond[H]XY - 1}{\\log_2\\card{\\mathcal{X}}} \\]\n\\end{mdframed}\n\n\\[\n E = \\begin{cases}\n 1 & \\text{if } \\hat X \\neq X \\\\\n 0 & \\text{if } \\hat X = X\n \\end{cases}\n\\]\n\n\\begin{align*}\n \\cond[H]{E,X}{\\hat X} &= \\cond[H]{E,X}{\\hat X} \\\\\n \\underbrace{\\cond[H]{E}{\\hat X}}_{\\leq \\op[H]{\\Perr}} + \\underbrace{\\cond[H]{X}{E,\\hat X}}_{\\leq \\Perr \\log\\card{\\mathcal{X}}} &= \\cond[H]{X}{\\hat X} + \\underbrace{\\cond[H]{E}{X,\\hat X}}_{=0} \\\\\n\\end{align*}\n\n$\\cond[H]{X}{E,\\hat X}$ is bounded by\n\\begin{align*}\n \\cond[H]{X}{E,\\hat X} &= \\os{E = 0}\\cond[H]{X}{\\hat X, E = 0} + \\os{E = 1}\\cond[H]{X}{\\hat X, E = 1} \\\\\n &\\leq (1 - \\Perr) \\cdot 0 + \\Perr \\log \\card{\\mathcal{X}}\n\\end{align*}\n\nHence, we obtain Fano's inequality. Intuitively it relates the probability of error in guessing the random variable $X$ to its conditional entropy $\\cond[H]XY$. Given $X$ as a function of $Y$. Iff $X$ can estimate $Y$ with zero probability of error, then $\\cond[H]XY = 0$. Fano's inequality uses this fact: Estimate $X$ with a low probability of error if the conditional entropy $\\cond[H]XY$ is small.\n\n\\section{Asymptotic entropy}\n\n\\subsection{What does iid mean?}\n\niid is a property of a set of random variables. A set of random variables can be \\enquote{independent, identically distributed} meaning that they all utilize the same probability distribution, but are independent in every possible form.\n\n\\subsection{What is a stochastic process?}\n\nA stochastic process (in discrete time) is a sequence of random variables\n\\[ \\left(X_n\\right)_{n\\geq1} \\]\n\n\\subsection{What is asymptotic entropy?}\n\nGiven $X_n$ as discrete random variables and its values in some finite/countable set $\\mathcal{X}$.\nThe \\key{asymptotic entropy} or \\key{asymptotic rate} $h$ of the stochastic process is defined as\n\\[ h = \\lim_{n \\rightarrow \\infty} \\frac1n \\op[H]{X_1,\\dotsc,X_n} \\]\n\nFurthermore\n\\[ h' = \\lim_{n \\rightarrow \\infty} \\cond[H]{X_n}{X_{n-1},\\dotsc,X_1} \\]\nif the limit exists. If $h'$ exists, then $h$ exists and $h = h'$.\n\nConvergence is interpreted as\n\\[\n \\lim_{n\\rightarrow\\infty} a_n = a\n \\Leftrightarrow\n \\fall \\varepsilon > 0\n \\ex N_\\varepsilon:\n \\fall n > N_\\varepsilon:\n \\card{a_n - a} < \\varepsilon\n\\]\n\n\\subsection{What is a stationary distribution?}\n\nA distribution $\\left(X_n\\right)_{n\\geq1}$ is stationary if $\\fall n, l \\in \\mathbb{N}:$ $(X_1,\\dotsc,X_n)$ and $(X_{l+1},\\dotsc,X_{l+n})$ have the same joint distribution.\n\n\\[ \\fall x_1, \\dotsc, x_n \\in \\mathcal{X}: \\os{X_1=x_1,\\dotsc,X_n=x_n} = \\os{X_{l+1}=x_1,\\dotsc,X_{l+n}=x_n} \\]\n\nIf $\\left(X_n\\right)_{n\\geq1}$ is stationary, $h'$ exists.\n\n\\subsection{What is time homogeneity?}\n\nA Markov chain is called time-homogeneous if\n\\[ p_n(y\\,|\\,x) = p_m(y\\,|\\,x) \\qquad \\fall m,n \\]\n\nHence, transition probability do not depend on time.\n\n\\subsection{Prove that at least one stationary distribution exists for every stochastic matrix.}\n\nFormally, a distribution at time $n$ is given as\n\\[\n \\os{X_n=y}\n = \\sum_{x\\in\\mathcal{X}} \\cons{X_n=y}{X_0=x} \\cdot \\os{X_0=x}\n = \\sum_x p_{x,y}^{(n)} \\mu_x\n = \\left(\\mu P^n\\right)_x\n\\]\n\nNow for an arbitrary stochastic matrix $P = \\left(p_{x,y}\\right)_{x,y \\in \\mathcal{X}}$ at least one stationary distribution $\\nu$ exists.\n\nConsider an initial distribution $\\mu = \\mu_0$.\n\\[ \\left( \\mu + \\mu P + \\mu P^2 + \\dotsb + \\mu P^{n-1} \\right) \\frac1n = \\mu_n \\]\n$\\mu_n$ is a probability vector on $\\mathcal{X}$ for various $n$. A subsequence $\\left(\\mu_{n_k}\\right)_{k \\in \\mathbb{N}}$ exists which converges to some vector $\\nu$.\n\n\\begin{align*}\n \\sum_{x \\in \\mathcal{X}} \\nu_X &= \\sum_{x \\in \\mathcal{X}} \\lim \\mu_{n_k} \\\\\n &= \\lim \\sum_{x \\in \\mathcal{X}} \\mu_{n_k} && \\text{[because $\\mathcal{X}$ is finite]} \\\\\n &= 1 \\\\\n \\mu_n^P - \\mu_n &= \\frac1n \\left(\\mu P + \\mu P^2 + \\dotsb + \\mu P^n\\right) \\\\\n &= \\frac1n \\left(\\mu + \\mu P + \\dotsb + \\mu P^{n+1}\\right) \\\\\n &= \\frac1n \\underbrace{\\left(\\mu - \\mu P^n\\right)}_{\\text{bounded by } 1} \\\\\n &\\converges 0\n\\end{align*}\n\\[\n \\begin{array}{ccccc}\n \\mu_{n_k} P & - & \\mu_{n_k} & \\rightarrow & 0 \\\\\n \\downarrow & & \\downarrow & & \\\\\n \\nu P & - & \\nu & = & 0\n \\end{array}\n\\]\n\n\\subsection{What does irreducibility of $(\\mathcal{X}, P)$ mean?}\n\n$(\\mathcal{X}, P)$ is called irreducible\\footnote{for finite $\\mathcal{X}$ it is called ergodic} if $\\fall x,y \\in \\mathcal{X}: \\exists n = n_{x,y}: p_{x,y}^n > 0$. Graph theoretically irreducibility means strongly connectedness.\n\nIf $(\\mathcal{X}, P)$ is finite and irreducible then there is a unique stationary distribution $\\mu$ and\n\\[\n \\nu_x = \\frac{1}{\\Ex_X(t^x)}\n\\]\nwhere $\\Ex_X(t^x)$ denotes the expected return time to $x$ with $t_{(w)}^x = \\inf\\set{n \\geq 1: X_n^{(w)} = x}$.\n$t^x$ is almost surely finite: $\\os{t^x < \\infty} = 1$ and $\\Ex_X(t^x) < \\infty$.\n\n\\subsection{How can we compute the stationary distribution for a given irreducible distribution?}\n\n\\subsection{Discuss a random walk.}\n\n\\subsection{Under which circumstances does a unique stationary initial distribution exist? Prove it.}\n\nA unique stationary initial distribution $\\nu$ exists, if $(\\mathcal{X}, P)$ is irreducible.\n\nProof: missing\n% TODO: lecture of 2014.05.07\n\n\\section{Asymptotic equipartition}\n\n\\subsection{What is asymptotic equipartition?}\n\nSuppose that $(X_n)_{n \\geq 1}$ is an $\\mathcal{X}$-valued stochastic process with entropy rate $h$. Then the process is said to have asymptotic equipartition property (AEP), if\n\\[ -\\frac1n \\log_2 p_n(X_1, \\dotsc, X_n) \\converges h \\text{ almost surely} \\]\n(Sometimes convergence in probability suffices)\n\n\\subsection{Under which conditions does the asymptotic equipartition property hold? What is it good for?}\n\n\\[ -\\frac1n \\log_2 p_n(X_1, \\dotsc, X_n) \\converges h \\text{ almost surely} \\]\nmust hold. Furthermore the AEP is good for data compression.\n\n\\subsection{What is a typical set?}\n\nThe typical set is defined as\n\\[\n A_\\varepsilon^{(n)}\n = \\left\\{\n (x_1, \\dotsc, x_n) \\in \\mathcal{X}^n:\n \\left|-\\frac1n \\log_2 p_n(x_1,\\dotsc,x_n) - h\\right| < \\varepsilon\n \\right\\}\n\\]\nwhere $\\varepsilon > 0$. The following theorems hold for typical sets\n\\begin{itemize}\n \\item $2^{-n(h+\\varepsilon)} < p_n(x_1,\\dotsc,x_n) < 2^{-n(h - \\varepsilon)} \\quad\\fall (x_1,\\dotsc,x_n) \\in A_\\varepsilon^{(n)}$\n \\item $\\os{(X_1,\\dotsc,X_n) \\in A_\\varepsilon^{(n)}} > 1 - \\varepsilon \\quad \\fall n \\geq N_\\varepsilon$\n \\item $(1 - \\varepsilon) 2^{n(h-\\varepsilon)} \\leq \\card{A_\\varepsilon^{(n)}} \\leq 2^{n(h + \\varepsilon)}$\n\\end{itemize}\nThose theorems follow from the definition of typical sets.\n\n\\section{Coding and compression}\n\n\\subsection{How is expected and average code length defined?}\n\n\\begin{description}\n \\item[\\key{Expected code length}] \\hfill{} \\\\\n $\\sum_{(x_1,\\dotsc,x_n)} l(C(x_1,\\dotsc,x_n)) p_n(x_1,\\dotsc,x_n)$\n \\item[\\key{Average code length}] \\hfill{} \\\\\n $\\frac1n \\Ex\\left(l(x_1,\\dotsc,x_n)\\right) \\leq h + \\varepsilon' \\qquad\\fall n > N_\\varepsilon$\n\\end{description}\n\n\\subsection{What does the Ergodic theorem state?}\n\nFor finite irreducible Markov chains $(\\mathcal{X}, P)$ it holds that for any $f: \\mathcal{X} \\rightarrow \\mathbb{R}$\n\\begin{align*}\n \\frac1n\\left(f(X_0) + f(X_1) + \\dotsb + f(X_{n-1})\\right)\n &= \\sum_{x \\in \\mathcal{X}} f(X) \\nu_x \\text{ almost surely} && \\text{ [discrete case]} \\\\\n &= \\int_\\mathcal{X} f \\dt\\nu && \\text{ [continuous case]} \\\\\n\\end{align*}\n\n\\section{Codes}\n\n\\subsection{Define the properties non-singularity, unique decodability and prefix-freedom.}\n\n\\begin{description}\n \\item[\\key{nonsingular}]\n $C: \\mathcal{X} \\rightarrow \\Sigma^+$ is injective\n \\item[\\key{uniquely decodable}]\n $C: \\mathcal{X}^* \\rightarrow \\Sigma^*$ is injective\n \\item[\\key{instantaneous or prefix-free}]\n $\\fall x,y \\in \\mathcal{X}: C(x)$ is not a prefix of $C(y)$\n\\end{description}\n\n\\subsection{What is the Theorem Kraft Inequality?}\n\nLet $D = \\card{\\Sigma}$ and $C: \\mathcal{X} \\rightarrow \\Sigma$ is a prefix-free code. It holds that\n\\[\n \\sum_{x \\in \\mathcal{X}} D^{-l(C(x))} \\leq 1\n\\]\n\nConversely if $l_X \\in \\mathbb{N}$ ($x \\in \\mathcal{X}$) are such that $\\sum_{x \\in \\mathcal{X}} D^{-l_x} \\leq 1$\nthen there is a prefix-free code $C$ with $C(L(x)) = L_x \\fall x \\in \\mathcal{X}$.\n\n\\subsection{Define a general integer optimization problem.}\n\n\\subsection{What are Lagrange multipliers and what are they used for?}\n\n\\subsection{Which theorem regarding code length holds for every prefix-free code?}\n\nFor every prefix-free code $C: \\mathcal{X} \\rightarrow \\Sigma^+$ the expected code length satisfies $L_C \\geq H_D(X)$.\nIf equality holds, then $p(x) = D^{-l(C(x))}$. Remember that $D = \\card{\\Sigma}$.\n\nIt follows that if $p$ is D-adic\\footnote{meaning $p(x) \\in \\set{D^{-n}: n \\in \\mathbb{N}}$} then the minimum value $H_D(X)$ is attained. We can set $l_X = -\\log_D p(x)$ satisfy Kraft equality.\n\nOtherwise minimise $\\log_D{B} + \\frac{1}{\\log_2 D} \\kld pr$.\n\n\\subsection{Which theorem holds for optimal code lengths?}\n\nOptimal code lengths $l_x^*$ with $x \\in \\mathcal{X}$ are such that the expected code length satisfies $H_D(X) \\leq L^* < H_D(X) + 1$.\n\n\\subsection{How does the Huffman algorithm work?}\n\n\\subsection{Prove that Huffman codes are optimal.}\n\n\\subsection{Which codes are canonical? Are Huffman codes canonical?}\n\nCanonical codes satisfy the following properties:\n\n\\begin{enumerate}\n \\item if $p(x) > p(y)$ then $l(C(x)) \\leq l(C(y))$\n \\item If $v,w \\in C(\\mathcal{X})$ are longest code words, then $l(v) = l(w)$\n \\item $C$ can be modified into another optimal prefix-free code $C'$. $C'$ satisfies:\n Let $v, w$ be the least likely symbols\n \\begin{itemize}\n \\item $v$ and $w$ are siblings in the coding tree\n \\item $\\fall z \\in \\mathcal{X} \\setminus \\set{x,y}: p(z) \\geq p(x) \\geq p(y)$\n \\end{itemize}\n\\end{enumerate}\n\nHuffman codes are canonical.\n\n\\section{Information channels}\n\n\\subsection{Define discrete channels. What distinguishes a memoryless channel and the $n$-th extension of $\\mathcal{C}$ without feedback?}\n\nInput alphabet $\\mathcal X$ \\\\\nOutput alphabet $\\mathcal Y$ \\\\\nProbability transition matrix $\\cond yx$\n\nThe channel is \\key{memoryless} iff the probability distribution of the output depends only on the current input and is conditionally independent of previous channel inputs or outputs.\n\n%A \\key{discrete memoryless channel} $\\mathcal{C} = (\\mathcal{X}, P, Y)$ with $P = (p(y\\,|\\,x))_{x \\in \\mathcal{X}}$ with $p(y\\,|\\,x)$ is the probability that the outcome is $y$ given that the input is $x$.\n\nThe \\key{n-th extension of $\\mathcal{C}$ without feedback} is the channel\n\\[ \\mathcal{C}^n = (\\mathcal{X}^n, P_n, Y^n) \\]\n\nWithout feedback means than input symbol are conditionally independent of previous output values.\nIn this case the transition function satisfies\n\\[ \\cond{y^n}{x^n} = \\prod_{i=1}^n \\cond{y_i}{x_i} \\]\n\n\\subsection{How is capacity of a channel defined?}\n\n\\[ \\Capacity(\\mathcal C) = \\max_{p(x)}\\set{\\op[I]{X;Y}: p_X \\in M(\\mathcal{X})} \\]\nwhere $M(\\mathcal{X}) = \\set{p(\\cdot) \\text{ probability distributions on } \\mathcal{X}}$\ngiven $P = (p(y\\,|\\,x))_{x\\in\\mathcal{X}}$ and $p_X$. This defines \\key{information channel capacity}.\n\nFrom Shannon's second theorem it follows that information channel capacity equals to operational channel capacity.\n\n\\subsection{Give examples for channels and their capacity}\n\n\\begin{itemize}\n \\item Noise-less binary channel\n \\item Channel with non-overlapping outputs\n \\item Noisy typewriter\n \\item Binary symmetric channel\n \\item Binary erasure channel\n\\end{itemize}\n\n\\paragraph{A noise-less binary channel}\ntransmits both bits always without error.\nIf we want to compute the capacity\n\\begin{align*}\n C &= \\max\\left\\{\\op[I]{X;Y}\\right\\} \\\\\n &= \\op[H]{X} - \\cond[H]XY \\\\\n &= -2\\left(\\frac12 \\log_2 \\frac12\\right) + \\left(2 \\cdot 0 \\log_2 0 + 2 \\cdot 1 \\log_2 1\\right) \\\\\n &= 1 + 0 = 1\n\\end{align*}\n\n\\paragraph{A noisy typewriter} transmits a letter of the latin alphabet correctly with probability $\\frac12$. With probability $\\frac12$ the next letter of the alphabet will be transmitted instead.\n\n\\[ \\Capacity(\\mathcal C) = \\max\\left\\{\\op[I]{X;Y}\\right\\} = \\op[H]{X} - \\cond[H]{X}{Y} \\]\n\n\\begin{align*}\n \\cond[H]{X}{Y} &= \\sum_{y \\in \\mathcal Y} \\sum_{x \\in \\mathcal X} \\op{X=x,Y=y} \\log_2 \\frac{\\op{Y=y}}{\\cond{X=x}{Y=y}} \\\\\n &= \\sum_{y \\in \\mathcal Y} \\sum_{x \\in \\mathcal X} \\frac1{2\\cdot 26} \\log_2 \\frac{\\frac{1}{26}}{\\frac{1}{52}} \\\\\n &= \\sum_{y \\in \\mathcal Y} \\sum_{x \\in \\mathcal X} \\frac{1}{2\\cdot 26} \\log_2 2 \\\\\n &= 26 \\cdot 2 \\cdot \\frac{1}{2\\cdot 26} \\cdot 1 \\\\\n &= 1\n\\end{align*}\n\nor\n\n\\begin{align*}\n \\cond[H]{X}{Y} &= \\sum_{y \\in \\mathcal Y} \\op{Y=y} \\cond[H]{X}{Y=y} \\\\\n &= \\sum_{y \\in \\mathcal Y} \\frac1{26} \\cdot 1 \\\\\n &= 26 \\cdot \\frac1{26} \\\\\n &= 1\n\\end{align*}\n\nwith\n\n\\begin{align*}\n H(X) &= -\\sum_{x \\in \\mathcal X} \\op{X=x} \\log_2 \\op{X=x} \\\\\n &= -\\sum_{x \\in \\mathcal X} \\frac{1}{26} \\log_2 \\frac{1}{26} \\\\\n &= -26 \\cdot \\frac{1}{26} \\log_2 \\frac{1}{26} \\\\\n &= \\log_2 \\frac1{26}\n\\end{align*}\n\n\\paragraph{A binary symmetric channel} transmits a bit correctly\nwith probability $(1-p)$. The wrong bit is sent with probability $p$.\n\n\\begin{align*}\n \\Capacity(\\mathcal{C}) &= \\op[H]{X} - \\sum_{y \\in \\mathcal Y} \\op{Y=y} \\cond[H]{X}{Y=y} \\\\\n &= \\op[H]{\\frac12, \\frac12} - \\op[H]{p, 1-p} \\\\\n &= 1 - \\op[H]{p}\n\\end{align*}\n\n\\paragraph{A binary erasure channel} transmits a bit correctly with probability\n$1 - \\alpha$. With probability $\\alpha$ a bit will be transmitted as detectable\nerror bit.\n\n\\begin{align*}\n \\Capacity(\\mathcal C) &= \\max_{p(x)}\\left(\\op[H]{X} - \\sum_{y \\in \\mathcal Y} \\op{Y=y} \\cond[H]{X}{Y=y}\\right) \\\\\n &= \\max_{p(x)}\\left(\\op[H]{X} - \\alpha \\op[H]{X}\\right) \\\\\n &= \\max_{\\op[H]{X}}\\left(\\op[H]{X} - \\alpha \\op[H]{X}\\right) \\\\\n &= \\left(1 - \\alpha \\cdot 1\\right) \\\\\n &= 1 - \\alpha\n\\end{align*}\n\n\\subsection{When is $P$ of an information channel called (weakly) symmetric? Which associated theorem exists?}\n\n$P = (p(y\\,|\\,x))_{x \\in \\mathcal{X}, y \\in \\mathcal{Y}}$ is called \\key{symmetric} iff\n\\begin{enumerate}\n \\item rows are permutations of each other\n \\item columns are permutations of each other\n\\end{enumerate}\n\n$P = (p(y\\,|\\,x))_{x \\in \\mathcal{X}, y \\in \\mathcal{Y}}$ is called \\key{weakly symmetric} if\n\\begin{enumerate}\n \\item rows are permutations of each other\n \\item all columns yield the same sum\n \\[ \\exists c: \\sum_x p(y\\,|\\,x) = c \\qquad \\fall y \\in Y \\]\n\\end{enumerate}\n\nThe theorem states,\n\\begin{quote}\n If $P$ is weakly symmetric, then $\\Capacity(\\mathcal{C}) = \\log_2\\card{Y} - \\op[H]{p(\\cdot\\,|\\,x)}$ and the maximum is achieved when $X$ is uniform on $\\mathcal{X}$.\n\\end{quote}\n\n\\subsection{Prove $\\Capacity(\\mathcal{C}^n) = n \\cdot \\Capacity(\\mathcal{C})$}\n\n\\subsection{What is a $(M, n)$ code? What is the average and maximum error of it?}\n\nAn $(M, n)$ code for the channel $C = (\\mathcal{X}, P, Y)$ consists of the following:\n\\begin{enumerate}\n \\item An input set $W$ with $\\card{W} = M$ or wlog. $W = \\set{1,\\dotsc,M}$\n \\item The codebook $B$ (set of codewords)\n \\item A decoding function $g: Y^n \\rightarrow \\hat W$\n\\end{enumerate}\n\nThe average error is defined as\n\\begin{align*}\n \\lambda_w^{(n)} &= \\cons{g(Y^{(n)}) \\neq w}{X^{(n)} = x^{(n)}(w)} \\\\\n &= \\sum_{y \\in Y} p(y \\,|\\, x(w))\n\\end{align*}\n\nThe maximum probability of the error is $\\lambda^{(n)} = \\max_w \\lambda_w^{(n)}$.\n\n\\subsection{What is the rate of a code?}\n\n\\[ R = \\frac{\\log_2 M}{n} \\]\n\n%A rate $R$ is said to be \\key{achievable} if there exists a sequence of $\\left(\\lceil 2^{nR} \\rceil, n\\right)$ codes such that the maximal probability of error $\\lambda^{(n)}$ converges towards $0$ as $n \\to \\infty$.\n\n\\subsection{What is an achievable rate? What is an achievable capacity?}\n\nA rate $R > 0$ is achievable if there is a sequence of $(M_n, n)$ codes such that\n\\[ R_n = \\frac{\\log_2 M_n}{n} \\converges R \\]\nand\n\\[ \\lambda^{(n)} \\converges 0 \\]\n\nThe achievable capacity $R^*$ of a channel is the supremum of all achievable rates:\n\\[\n \\fall \\varepsilon > 0 \\qquad\n \\exists (M,n) \\text{ code with rate } R > R^* - \\varepsilon \\land \\lambda^{(n)} < \\varepsilon\n\\]\n\n\\subsection{What is the set of jointly typical sequences?}\n\nIntuitively, we decode a channel output $Y^n$ as the $i$-th index iff the codeword $X^n(i)$ is in the jointly typical set of the received signal $Y^n$.\n\nThe set $A_\\varepsilon^{(n)}$ of \\key{jointly typical} sequences $\\set{(x^n, y^n)}$ with respect to the distribution $\\op{x,y}$ is the set of $n$-sequences with empirical entropies $\\varepsilon$-close to the true entropies:\n\\[\n\tA_\\varepsilon^{(n)} = \\set{\n\t\t(x^n, y^n) \\in \\mathcal X^n \\times \\mathcal Y^n:\n\t\t\\card{-\\frac1{n} \\log_2 \\op{x^n} - \\op[H]{X}} < \\varepsilon\n\t}\n\\]\n\nThe following statements hold for jointly typical sequences:\n\\begin{enumerate}\n \\item $\\op{(X^{(n)}, Y^{(n)}) \\in A_\\varepsilon^{(n)}} \\to 1$ as $n \\to \\infty$\n \\item $(1 - \\varepsilon) 2^{n(\\op[H]{X,Y} - \\varepsilon)} \\leq \\card{A_n^\\varepsilon} \\leq \\varepsilon 2^{n(\\op[H]{X,Y} - \\varepsilon)}$\n \\item $(X'_1, \\ldots, X'_n), (Y'_1, \\ldots, Y'_n)$ are independent. Then\n \\[\n\t\t\\op{(X'_1, \\ldots, X'_n, Y'_1, \\ldots, Y'_n) \\in A_\\varepsilon^{(n)}} =\n\t\t\\begin{cases}\n\t\t \\leq 2^{-n(\\op[I]{X,Y} - 3\\varepsilon)} & \\text{else} \\\\\n\t\t \\geq (1 - \\varepsilon) 2^{-n (\\op[I]{X,Y} + 3\\varepsilon)} & n \\geq N_\\varepsilon\n\t\t\\end{cases}\n \\]\n\\end{enumerate}\n\n\\subsection{Define Shannon's 2nd theorem and give an overview over the proof.}\n\nAlso called ``Channel coding theorem''.\n\n\\begin{quote}\n For a discrete memoryless channel, all rates below capacity C are achievable.\n Specifically for every rate $R < C$, there exists a sequence of $(2^{nR}, n)$ codes with maximum probability of error $\\lambda^n \\to 0$.\n\n Conversely any sequence of $(2^{nR}, n)$ codes with $\\lambda^{(n)} \\to 0$ must have $R \\leq C$.\n\\end{quote}\n\n\\[ R^* = \\Capacity(\\mathcal{C}) \\]\n\n\\section{List of theorems}\n\nSee also\n\\begin{itemize}\n \\item Markov inequality\n \\item Chebyshev inequality\n \\item Weak and Strong Law of Large Numbers\n\\end{itemize}\n\n\\begin{thm}\n If $X,Y$ are independent random variables with finite $\\Ex(X)$ and $\\Ex(Y)$, then\n \\[ \\Ex(X\\cdot Y) = \\Ex(X) \\cdot \\Ex(Y) \\]\n\\end{thm}\n\n\\begin{thm}\n If $X_n \\rightarrow X$ almost surely then $X_n \\rightarrow X$ in probability.\n\\end{thm}\n\n\\begin{thm}\n $X_n \\rightarrow X$ almost surely $\\Leftrightarrow$ $U_k \\rightarrow 0$ in probability with\n $ U_k = \\sup\\set{\\card{X_n - X}: n \\geq k} $\n\\end{thm}\n\n\\begin{thm}\n If $\\lim X_n = X$ almost surely and $\\lim X_n = X'$ almost surely, then $X = X'$ almost surely.\n If $\\lim X_n = X$ in probability and $\\lim X_n = X'$ in probability, then $X = X'$ almost surely.\n\\end{thm}\n\n\\begin{thm}\n The function $N \\mapsto H(U_N) = \\log_2{N}$ is the \\emph{unique} function satisfying the associated 3~axioms to Hartley formula:\n \\begin{itemize}\n \\itemsep0pt\n \\item $H(U_2) = 1$\n \\item $H(U_N) \\leq H(U_{N+1})$ (\\enquote{monotonicity})\n \\item $H(U_{M\\cdot N}) = H(U_M) + H(U_N)$\n \\end{itemize}\n It is also the \\emph{unique} function satisfying\n \\begin{itemize}\n \\itemsep0pt\n \\item $H(U_{N+1}) - H(U_N) \\xrightarrow{N \\rightarrow \\infty} 0$\n \\end{itemize}\n\\end{thm}\n\n\\begin{thm}\n \\begin{align*}\n & \\cond[H] YX = \\op[H]{X,Y} - \\op[H]{X} \\\\\n \\Leftrightarrow{ } & \\op[H]{X,Y} = \\op[H]{X} + \\cond[H] YX\n \\end{align*}\n\\end{thm}\n\n\\end{document}\n", "meta": {"hexsha": "9cb6b9a698463d85295f10f865e3139ee6ea0059", "size": 44757, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "pub/dsit_elaboration.tex", "max_stars_repo_name": "meisterluk/tug_lp", "max_stars_repo_head_hexsha": "eaf7e0a9bfaa91400248f7231c6891531ee71275", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "pub/dsit_elaboration.tex", "max_issues_repo_name": "meisterluk/tug_lp", "max_issues_repo_head_hexsha": "eaf7e0a9bfaa91400248f7231c6891531ee71275", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "pub/dsit_elaboration.tex", "max_forks_repo_name": "meisterluk/tug_lp", "max_forks_repo_head_hexsha": "eaf7e0a9bfaa91400248f7231c6891531ee71275", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 42.3434247871, "max_line_length": 401, "alphanum_fraction": 0.6346269857, "num_tokens": 15647, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5117166047041652, "lm_q2_score": 0.6688802537704063, "lm_q1q2_score": 0.34227713241305274}} {"text": "\n\\subsection{Recap of Hidden Markov Models (HMMs)}\n\nWe don’t see state\n\nEach state produces a visible output. this output is drawn from a distribution for each state.\n\nWe observe a sequence of outputs, not states.\n\n", "meta": {"hexsha": "1d9e50fb166c1edc56946863f33e01a131de90f8", "size": 215, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/pug/theory/statistics/markovHMMEstimation/01-01-HMM.tex", "max_stars_repo_name": "adamdboult/nodeHomePage", "max_stars_repo_head_hexsha": "266bfc6865bb8f6b1530499dde3aa6206bb09b93", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "src/pug/theory/statistics/markovHMMEstimation/01-01-HMM.tex", "max_issues_repo_name": "adamdboult/nodeHomePage", "max_issues_repo_head_hexsha": "266bfc6865bb8f6b1530499dde3aa6206bb09b93", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2021-03-03T12:36:56.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-01T22:16:09.000Z", "max_forks_repo_path": "src/pug/theory/statistics/markovHMMEstimation/01-01-HMM.tex", "max_forks_repo_name": "adamdboult/nodeHomePage", "max_forks_repo_head_hexsha": "266bfc6865bb8f6b1530499dde3aa6206bb09b93", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 21.5, "max_line_length": 94, "alphanum_fraction": 0.7720930233, "num_tokens": 50, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.668880247169804, "lm_q2_score": 0.5117166047041654, "lm_q1q2_score": 0.342277129035415}} {"text": "% appendices are not necessary and should aid understanding, not reproduce results from elsewhere\n\n\\chapter{Stability of degenerate internal squeezing}\n\\label{app:dIS_stability}\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.95\\textwidth]{dIS_stability.pdf}\n \\caption{Degenerate internal squeezing's stability for lossless (left panel) and lossy (right panel) cases. The different colours indicate different poles found numerically.\n % The squeezer parameter is normalised to the appropriate threshold in each case.\n A positive imaginary part of a pole indicates instability.\n Both cases are stable below singularity threshold. I use the parameters in Table~\\ref{tab:dIS_parameters}.}\n \\label{fig:dIS_stability}\n\\end{figure}\n\nI determine the stability of degenerate internal squeezing via the poles of its noise and signal responses. These responses are related fractions of polynomials.\nThe denominator of the noise response squared is $\\Omega^4 q(\\Omega,\\chi)$ where $q$ is a polynomial in $\\Omega,\\chi$~\\cite{korobkoQuantumExpanderGravitationalwave2019}.\nSince the zero $\\Omega=0$ comes from the horizontally free-falling mass assumption, the remaining zeros of the noise denominator are the zeros of $q$ (which are shared with the signal response~\\cite{korobkoQuantumExpanderGravitationalwave2019}). In the complex $\\Omega$ plane, if any of these poles~\\footnote{I check that the numerator is not also zero at that point.} have a positive imaginary part, then the system is unstable\\jam{(check this, IFT suggests other sign?)}~\\cite{nise_2019}. %~\\footnote{This result is often expressed instead with respect to the Laplace Transform variable $s=i\\Omega$, where a positive real part of $s$ indicates instability~\\cite{}.}. %This condition is equivalent to others such as\\jam{(... the Nyquist criterion?)}.\nAs shown in Fig.~\\ref{fig:dIS_stability}, therefore, degenerate internal squeezing is stable in the lossless case below threshold and the lossy case below the singularity threshold determined in Section~\\ref{sec:singularity_threshold}.\n\n\\chapter{Comparison of abstract mode structures}\n\\label{app:mode_structure}\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[angle=-90,width=1\\textwidth]{all_mode_structures.pdf}\n \\caption{Abstract mode diagrams of the different configurations considered in this thesis. The modes and coupling rates are explained in Sections~\\ref{sec:Hamiltonian_modelling}, \\ref{sec:dIS_results}, \\ref{sec:sWLC}, and~\\ref{sec:modal_equivalence}. Whenever the arm mode $\\hat a$ is shown, it is implicitly connected to the test mass mechanical mode $\\hat x$ and the gravitational wave signal $h(t)$. Losses are not shown. The photodetector shows the possibility for idler (optical or mechanical) or signal readout.}\n \\label{fig:mode_diagram}\n\\end{figure}\n\nIn Fig.~\\ref{fig:mode_diagram}, I compare the abstract mode structure of each of the configurations in this thesis. The parallels between the OPOs and the internal squeezing configurations can be seen. Nondegenerate internal squeezing and stable optomechanical filtering are modally equivalent but are optomechanical and all-optical, respectively, which means that their performance might be different given the different losses they encounter. When idler readout is used, although the internal mode structure is the same as signal readout, the photodetector ``sees'' the structure differently because it measures a different mode. %Idler readout of the mechanical idler mode is more difficult but potentially possible~\\cite{liEnhancingInterferometerSensitivity2021}.\n\n\n\\chapter{Singularity threshold for degenerate internal squeezing}\n\\label{app:dIS_singularity_thr}\n\n% \\renewcommand{\\thefigure}{A\\arabic{figure}}\n% \\setcounter{figure}{0}\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=0.8\\textwidth]{dIS_threshold_traj_compressed.pdf}\n \\caption{Degenerate internal squeezing trajectories of the singularities of the anti-squeezed quadrature of the noise and the extrema of the squeezed quadrature in $(\\Omega, \\chi)$ space as the arm loss is changed from $T_{l,a}\\in(0,1)$. I assume that the extrema are minima because of the shape of the noise response in Fig.~\\ref{fig:dIS_sensitivity}.\n The singularities and minima both achieve the same (a) lossless and (d) high arm loss limits, where neither reaches (d) because of limited numerical sampling. However, the singularities and minima diverge at high arm losses (e.g.\\ $T_{l,a}>0.1$) and merge with their counterparts moving in from infinity at different squeezer parameters shown at (b) and (c), respectively. I use the parameters in Table~\\ref{tab:signal_RO_parameters} and zero signal loss. %\\jam{(rest of explanation moved into text, does it still make sense?)}\n }\n \\label{fig:dIS_threshold_traj} % frequency scale is not logarithmic\n\\end{figure}\n\n\\newpage\nThe singularity threshold, see Section~\\ref{sec:singularity_threshold}, for degenerate internal squeezing shows that maximising the anti-squeezed quadrature is not the same as minimising the squeezed quadrature.\nAs shown in Fig.~\\ref{fig:dIS_threshold_traj}, in the lossless case, the singularities $(\\Omega,\\chi)$ are at $(0,\\infty)$ and $(\\omega_s,\\gamma^b_\\text{tot})$ which recovers threshold from Section~\\ref{sec:dIS_results}. %$(\\Omega_\\text{thr},\\chi_\\text{thr})\\xrightarrow[\\gamma_a\\rightarrow0]{}(\\omega_s,\\gamma^b_\\mathrm{tot})$\nAs the arm loss $\\gamma_a$ is increased from zero, the singularities move and merge at the $\\Omega=0$ axis when $\\gamma_a=\\omega_s$, and then the remaining singularity converges to the degenerate OPO threshold $(\\Omega_\\text{thr},\\chi_\\text{thr})\\xrightarrow[\\gamma_a\\rightarrow\\infty]{}(0,\\gamma^b_\\mathrm{tot})$ in the high arm loss limit as expected.\n% \\subsubsection{Disagreement between squeezed and anti-squeezed quadratures}\n% Degenerate internal squeezing - different notions of threshold\n%shouldn't this be in the dIS sub-chapter? no.\n% make clear that knowing threshold just gives the physical bounds of the parameter space, it doesn't correspond to the optimal squeezing (see results section) for sensitivity improvement -- just as minimising N does not necessarily maximise SNR if signal is not considered. \n%\\jam{(One explanation for the difference between position of the singularity of the anti-squeezed quadrature and the minimum of the squeezed quadrature is that the expression for the sloshing frequency in Korobko et al, 2019 is not valid at large arm losses. If this is so, then what does this fixed--sloshing frequency Hamiltonian correspond to? And for that system, why does the supposedly Gaussian squeezing not maximise antisqz at the frequency that it minimises squeezing? --> Look at pump phase and/or covariance matrix)}\n% plot: both quadratures comparison (black-green plot -- what is this?)\n% \\begin{figure}\n% \\centering\n% \\includegraphics[width=\\textwidth]{dIS_threshold_quadratures.pdf}\n% \\caption{\\jam{(Purpose: show problem with singularity threshold. Check if this section is staying in before fixing plot. Cut this plot even if moving to an appendix since the trajectories show this already.)}\\jam{(Normalise legend to threshold)} Degenerate internal squeeing noise quadratures (anti-squeezed in the top panel and squeezed in the bottom panel) and the difference between singularity threshold (that maximises the anti-squeezing peak) and the squeezing parameter that minimises the squeezing peak. This difference is only significant in the high arm loss regime that is unlike\\jam{(word choice?)} future detectors.\\jam{(Answer why squeezing is so small -- see Bram's question (it is just down to the parameters chosen, sloshing frequency and bandwidth, being poorly suited to dIS))}}\n% \\label{fig:dIS_on_threshold}\n% \\end{figure}\nHowever, where the anti-squeezed quadrature is divergent does not necessarily correspond to where the squeezed quadrature has the minimum value. This is unlike the degenerate OPO in Eq.~\\ref{eq:dOPO_fixed_phase} where the squeezed quadrature is minimised on threshold.\nIf the minima~\\footnote{Since the zeros of the squeezed quadrature are not robust to losses, as shown for the OPO in Fig.~\\ref{fig:dOPO_variances}, I consider the minima instead.} of the squeezed quadrature were used to define threshold, then their trajectories in (real) $(\\Omega,\\chi)$ space would be as shown in Fig.~\\ref{fig:dIS_threshold_traj}. Although they achieve the same limits, these trajectories are not the same as the singularity trajectories.\nThis does not violate the Heisenberg Uncertainty Principle because the losses increase the uncertainty product. %, i.e.\\ the anti-squeezed variance is increased more than the squeezed variance is decreased. %~\\footnote{The squeezing remains Gaussian~\\cite{}, i.e.\\ the squeezed noise ellipse's area is increased but remains an ellipse with semi-axes that represent the standard deviations of the Gaussian noise in each quadrature.\\jam{(check this)}}. \nMoreover, the difference between the minima and singularities is only significant with high arm losses that are far above the realistic loss $100~\\text{ppm}$ expected for future gravitational-wave detectors (e.g.\\ the squeezing curves only diverge by more than 0.1~dB around $T_{l,a}=0.1$).\n% e.g.\\\\jam{in Fig.~\\ref{fig:dIS_on_threshold} (quantify without plot)} uses $T_{l,a}=800000~\\text{ppm}$ compared to $100~\\text{ppm}$ predicted for future gravitational-wave detectors.\nTherefore, this difference is not of concern for future work involving singularity threshold.\n% because of this and that I am studying nondegenerate squeezing, this problem is not of concern for using singularity threshold in nondegenerate internal squeezing\\jam{(what about combined readout?)}.\n% Although I do not understand why this difference occurs\\jam{(check this)},\nFor completeness, two possible explanations for this behaviour are that it comes from the different DC behaviour of the limiting degenerate OPO's quadratures in Fig.~\\ref{fig:dOPO_variances} or that the approximation to the sloshing frequency in Section~\\ref{sec:nIS_model} breaks down in the high arm loss limit~\\footnote{In which case, this model would not represent the physical system in that limit.}~\\cite{korobkoQuantumExpanderGravitationalwave2019}. This might be understood better if the singularity threshold is verified against a pump-depletion model in future work.\n\n% This is shown\\jam{in Fig.~\\ref{fig:dIS_on_threshold} (explain without plot)} where the minimum squeezing occurs at $76\\%$\\jam{(check this)} singularity threshold and the relative difference is only large (above a percent threshold) in the high arm loss limit\\jam{(quantify this)}.\n% The extrema, of which there are a maximum of two at any point, start at the same points, but have different trajectories, e.g.\\ the first point moving initially into the $\\Omega>\\omega_s$ region instead, and merge at the different point on the y-axis, but converge to the same OPO limit (although the numerical sampling does not show it here). This difference does not violate the Heisenberg Uncertainty Principle\\jam{(but why does it occur?)}.\n% Singularity threshold uses singularities of the anti-squeezed quadrature rather than zeroes of the squeezed quadrature because the former are robust to losses, as shown for the OPO in Fig.~\\ref{fig:dOPO_variances}. \n% Since this problem\\jam{(is it even a problem?)} is not relevant to nondegenerate internal squeezing, ...\n\n\n\n\\begin{comment}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Appendix: Nondegenerate internal squeezing, 2x2 matrix model}\n\n\\jam{(For reference only, delete this from the thesis after checking that all information is in the nIS model given)}\n\nThis derivation is based on the lossless model in Ref.~\\cite{liBroadbandSensitivityImprovement2020} with the differential mode $\\hat{a}$, the signal mode in the signal-recycling cavity $\\hat{b}$, and the idler mode in the signal-recycling cavity $\\hat{c}$.\\jam{Approximations have already been made: single mode and semi-classical pump}. In that model, the only source of vacuum is into the $\\hat{b}$ mode from the readout -- i.e. behind the signal-recycling mirror.\nTo that model, I add in optical loss: (1) intra-cavity to each of the three modes $\\hat{a}, \\hat{b}, \\hat{c}$ through loss ports\\jam{(explain loss port, give a picture)} with transmissivity $T_{l,a}, T_{l,b}, T_{l,c}$, respectively; and (2) at the photodetector, through a beamsplitter with reflectivity $R_{PD}$\\jam{shown in fig}.\nTo study the low-frequency quantum noise, I introduce radiation pressure effects through coupling the gravitational-wave signal $h(t)$ to the end test-mass mirror motion $\\hat{x}$ with associated momentum $\\hat{p}$.\nFollowing the advice\\jam{(word choice)} in Ref.~\\cite{liBroadbandSensitivityImprovement2020}, I also couple the $\\hat{c}$ mode to a back-action evading mechanical mode $\\hat{y}$ with momentum $\\hat{q}$ and negative effective mass $-\\mu$. This is done to retain the PT-symmetry of the system with the P symmetry now exchanging $\\hat{a}\\leftrightarrow\\hat{c}$ and $\\hat{x}\\leftrightarrow\\hat{y}$.\n\nThe full Hamiltonian of the system is given by: $$\\hat{H}=\\hat{H}_0+\\hat{H}_\\mathrm{I}+\\hat{H}_{\\mathrm{GW}}+\\hat{H}_{\\mathrm{BAE}}+\\hat{H}_{\\gamma_R}+\\hat{H}_{\\gamma_a}+\\hat{H}_{\\gamma_b}+\\hat{H}_{\\gamma_c}.$$ Where each term describes:\n\\begin{itemize}\n\\item $\\hat{H}_0=\\hbar\\omega_0(\\hat{a}^\\dagger\\hat{a}+\\frac{1}{2})+\\hbar\\omega_0(\\hat{b}^\\dagger\\hat{b}+\\frac{1}{2})+\\hbar\\omega_\\mathrm{idler}(\\hat{c}^\\dagger\\hat{c}+\\frac{1}{2})$, the uncoupled behaviour of the optical modes, in the Interaction Picture, the model can ignore this behaviour (moving it from the operators onto the states)\\jam{(is RWA also used?)} % and uses a rotating frame at the carrier frequency in order to ignore the evolution of the optical modes from $\\hat{H}_0$\\jam{(be careful, are we? -- also, is RWA used?)}\n\\item $\\hat{H}_\\mathrm{I}=i\\hbar\\omega_s(\\hat{a}\\hat{b}^\\dag-\\hat{a}^\\dag\\hat{b})+i\\hbar\\chi(\\hat{b}^\\dag\\hat{c}^\\dag-\\hat{b}\\hat{c})$, the interaction of the three optical modes $\\hat{a}, \\hat{b}, \\hat{c}$, where a semi-classical approximation has been taken to the pump field\\jam{(spell out what this means for $\\chi$)} and a single-mode approximation to each of the cavity modes which are assumed to be on resonance\\jam{(this needs more discussion, do more in deg. int. sqz. section)}\n\\item $\\hat{H}_\\mathrm{GW}=-\\alpha (\\hat{x}-L_\\mathrm{arm}h)(\\frac{\\hat{a}+\\hat{a}^\\dag}{\\sqrt{2}})+\\frac{1}{2\\mu}\\hat{p}^2$,\\jam{($\\alpha$ is $\\alpha_\\mathrm{GW}$, not the alpha in Li)} the coupling to the gravitational-wave signal $h(t)$\\jam{(Mention that authors disagree on the exact form of the RP term -- there are many equivalent methods to couple the GW signal, mirror mode, and optical mode~\\cite{})}\n\\item $\\hat{H}_{\\mathrm{BAE}}=-\\alpha \\hat{y}(\\frac{\\hat{c}+\\hat{c}^\\dag}{\\sqrt{2}})-\\frac{1}{2\\mu}\\hat{q}^2$, the PT-symmetry enabling, back-action evasion\\jam{(clarify, what is BAE?)} mode\n\\item $\\hat{H}_{\\gamma_R}$\\jam{(formula, state but just use Langevin terms)}, the readout of the $\\hat{b}$ mode\n\\item $\\hat{H}_{\\gamma_i}$ for $i=a,b,c$, the intra-cavity loss ports, these and $\\hat{H}_{\\gamma_R}$ give the standard\\jam{(word choice)} Langevin terms in the Heisenberg-Langevin equations of motion\\jam{(cite Gardiner and Collete?)}.\n\\end{itemize}\n\nFrom this Hamiltonian $\\hat{H}$, I find the Heisenberg-Langevin equations of motion in the Interaction Picture (separating away the evolution with respect to $\\hat{H}_0$):\n$$\\begin{cases}\n\\dot{\\hat{a}}=-\\omega_s\\hat{b} - \\gamma_a \\hat{a} + \\sqrt{2\\gamma_a}\\hat{n}^L_a+\\frac{i}{\\hbar}\\alpha(\\hat{x}-L_\\mathrm{arm}h)\\frac{1}{\\sqrt{2}}\\\\\n\\dot{\\hat{b}}=\\omega_s\\hat{a} + \\chi\\hat{c}^\\dagger - \\gamma^b_\\mathrm{tot} \\hat{b} + \\sqrt{2\\gamma_R}\\hat{B}_\\mathrm{in} + \\sqrt{2\\gamma_b}\\hat{n}^L_b\\\\\n\\dot{\\hat{c}}=\\chi\\hat{b}^\\dagger - \\gamma_c \\hat{c} + \\sqrt{2\\gamma_c}\\hat{n}^L_c + \\frac{i}{\\hbar}\\alpha \\hat{y}\\frac{1}{\\sqrt{2}}\\\\\n\\dot{\\hat{x}}=\\frac{1}{\\mu}\\hat{p},\\quad \\dot{\\hat{p}}=\\alpha(\\frac{\\hat{a}+\\hat{a}^\\dag}{\\sqrt{2}})\\\\\n\\dot{\\hat{y}}=-\\frac{1}{\\mu}\\hat{q},\\quad \\dot{\\hat{q}}=\\alpha(\\frac{\\hat{c}+\\hat{c}^\\dag}{\\sqrt{2}})\n\\end{cases}$$\nI separate out the fluctuating part of each mode $\\hat{a}(t)=\\langle\\hat{a}\\rangle+\\delta\\hat{a}(t)$ from its time average (or large classical motion) $\\langle\\hat{a}\\rangle$. This does not change the form of the equations.\\jam{(Why? This is not clear, the LHS does not change but the time-averages remain)} \n% solving the equations\nIn Fourier space, these equations become (with the simplified notation $\\tilde{\\delta\\hat{Q}}(\\Omega)\\mapsto\\hat{Q}(\\Omega)$ and sign convention $\\partial_t\\mapsto-i\\Omega$):\n\\begin{equation}\n\\begin{cases}\n\\label{eq:nIS-2}\n-i\\Omega\\hat{a}(\\Omega)=-\\omega_s\\hat{b}(\\Omega) - \\gamma_a \\hat{a}(\\Omega) + \\sqrt{2\\gamma_a}\\hat{n}^L_a(\\Omega)+i(\\frac{1}{-\\Omega^2}\\rho_\\mathrm{RP}(\\frac{\\hat{a}(\\Omega)+\\hat{a}^\\dag(-\\Omega)}{\\sqrt{2}})-\\beta\\tilde{h}(\\Omega))\\\\\n-i\\Omega\\hat{b}(\\Omega)=\\omega_s\\hat{a}(\\Omega) + \\chi\\hat{c}^\\dagger(-\\Omega) - \\gamma^b_\\mathrm{tot} \\hat{b}(\\Omega) + \\sqrt{2\\gamma_R}\\hat{B}_\\mathrm{in}(\\Omega) + \\sqrt{2\\gamma_b}\\hat{n}^L_b(\\Omega)\\\\\n-i\\Omega\\hat{c}(\\Omega)=\\chi\\hat{b}^\\dagger(-\\Omega) - \\gamma_c \\hat{c}(\\Omega) + \\sqrt{2\\gamma_c}\\hat{n}^L_c(\\Omega) + \\frac{-i}{-\\Omega^2}\\rho_\\mathrm{BAE}(\\frac{\\hat{c}(\\Omega)+\\hat{c}^\\dag(-\\Omega)}{\\sqrt{2}}).\n\\end{cases}\n\\end{equation}\n\nWhere $\\beta=\\frac{\\alpha L_\\mathrm{arm}}{\\sqrt{2}\\hbar}$ and $\\rho_\\mathrm{RP}=\\rho_\\mathrm{BAE}=\\frac{\\alpha^2}{\\sqrt{2}\\hbar\\mu}$ where the radiation pressure (RP) and back-action evasion (BAE) effects have been separated even though the coupling constant is the same, for PT-symmetry.\n\nSolving these Equations~\\ref{eq:nIS-2} for simultaneous solutions of the $\\hat{Q}(\\Omega)$ and $\\hat{Q}^\\dag(-\\Omega)$ fields is easier when expressed in matrix form. Let $\\vec{\\hat{Q}}(\\Omega)=[\\hat{Q}(\\Omega),\\hat{Q}^\\dag(-\\Omega)]^T$, then the above equations can be re-written as:\n$$\\begin{cases}\n\\label{eq:nIS-3}\n((\\gamma_a-i\\Omega)\\mathrm{I}+\\frac{i\\rho_\\mathrm{RP}}{\\Omega^2 \\sqrt{2}}\\begin{bsmallmatrix}\n1 & 1 \\\\ \n-1 & -1\n\\end{bsmallmatrix})\\vec{\\hat{a}}(\\Omega)=-\\omega_s\\vec{\\hat{b}}(\\Omega) + \\sqrt{2\\gamma_a}\\vec{\\hat{n}}^L_a(\\Omega)-i\\beta\\begin{bsmallmatrix}\n1 & 0 \\\\ \n0 & -1\n\\end{bsmallmatrix}\\vec{\\tilde{h}}(\\Omega)\\\\\n(\\gamma^b_\\mathrm{tot}-i\\Omega)\\vec{\\hat{b}}(\\Omega)=\\omega_s\\vec{\\hat{a}}(\\Omega) + \\chi\\begin{bsmallmatrix}\n0 & 1 \\\\ \n1 & 0\n\\end{bsmallmatrix}\\vec{\\hat{c}}(\\Omega) + \\sqrt{2\\gamma_R}\\vec{\\hat{B}}_\\mathrm{in}(\\Omega) + \\sqrt{2\\gamma_b}\\vec{\\hat{n}}^L_b(\\Omega)\\\\\n((\\gamma_c-i\\Omega)\\mathrm{I}-\\frac{i\\rho_\\mathrm{BAE}}{\\Omega^2\\sqrt{2}}\\begin{bsmallmatrix}\n1 & 1 \\\\ \n-1 & -1\n\\end{bsmallmatrix})\\vec{\\hat{c}}(\\Omega)=\\chi\\begin{bsmallmatrix}\n0 & 1 \\\\ \n1 & 0\n\\end{bsmallmatrix}\\vec{\\hat{b}}(\\Omega) + \\sqrt{2\\gamma_c}\\vec{\\hat{n}}^L_c(\\Omega).\n\\end{cases}$$\n\nWhere $\\mathrm{I}$ is the $2\\times2$ identity matrix. Solving for $\\vec{\\hat{b}}(\\Omega)$, these give\n\\begin{align}\n\\vec{\\hat{b}}(\\Omega)=\\mathrm{M}_b^{-1}( &\\omega_s\\mathrm{M}_a^{-1}(\\sqrt{2\\gamma_a}\\vec{\\hat{n}}^L_a(\\Omega)-i\\beta\\begin{bsmallmatrix}\n1 & 0 \\\\ \n0 & -1\n\\end{bsmallmatrix}\\vec{\\tilde{h}}(\\Omega)) + \\chi\\begin{bsmallmatrix}\n0 & 1 \\\\ \n1 & 0\n\\end{bsmallmatrix}\\mathrm{M}_c^{-1}\\sqrt{2\\gamma_c}\\vec{\\hat{n}}^L_c(\\Omega)\\\\&+ \\sqrt{2\\gamma_R}\\vec{\\hat{B}}_\\mathrm{in}(\\Omega) + \\sqrt{2\\gamma_b}\\vec{\\hat{n}}^L_b(\\Omega)).\n\\end{align}\nWhere the matrices $\\mathrm{M}_i,\\; i=a,b,c$ are given by:\n\\begin{equation}\n\\begin{cases}\n\\mathrm{M}_a = (\\gamma_a-i\\Omega)\\mathrm{I}+\\frac{i\\rho_\\mathrm{RP}}{\\Omega^2 \\sqrt{2}}\\begin{bsmallmatrix}\n1 & 1 \\\\ \n-1 & -1\n\\end{bsmallmatrix}\\\\\n\\mathrm{M}_b = (\\gamma^b_\\mathrm{tot}-i\\Omega)\\mathrm{I} + \\omega_s^2 \\mathrm{M}_a^{-1} - \\chi^2\\begin{bsmallmatrix}\n0 & 1 \\\\ \n1 & 0\n\\end{bsmallmatrix} \\mathrm{M}_c^{-1} \\begin{bsmallmatrix}\n0 & 1 \\\\ \n1 & 0\n\\end{bsmallmatrix}\\\\\n\\mathrm{M}_c = (\\gamma_c-i\\Omega)\\mathrm{I}-\\frac{i\\rho_\\mathrm{BAE}}{\\Omega^2\\sqrt{2}}\\begin{bsmallmatrix}\n1 & 1 \\\\ \n-1 & -1\n\\end{bsmallmatrix}.\n\\end{cases}\n\\end{equation}\n\nHaving found the light inside the cavities, I want to find the light at the photodetector. The light right outside and travelling away from the signal-recycling mirror is given by the Input/Output (I/O) relation: $\\hat{B}_\\mathrm{out}=\\hat{B}_\\mathrm{in}-\\sqrt{2\\gamma_R}\\hat{b}$. Using the beamsplitter model of detection loss at the photodiode, for a beamsplitter of reflectivity $R_\\mathrm{PD}$, the quadratures of light at the photodetector are given by: $\\vec{\\hat{X}}_\\mathrm{PD}=\\sqrt{R_\\mathrm{PD}} \\vec{\\hat{X}}_\\mathrm{PD}^L + \\sqrt{1-R_\\mathrm{PD}} \\vec{\\hat{X}}_{B_\\mathrm{out}}$. Where $\\vec{\\hat{X}}_Q=\\Gamma \\vec{\\hat{Q}}=[\\hat{X}_{1,Q},\\hat{X}_{2,Q}]^T$ for $\\hat{X}_{i,Q}$ the ith quadrature of $Q$ and $\\Gamma=\\frac{1}{\\sqrt{2}}\\begin{bsmallmatrix}\n1 & 1 \\\\ \n-i & i\\end{bsmallmatrix}$ the quadrature matrix.\nPutting everything together, I find the quadratures of the light at the photodetector to be given by:\n\\begin{align}\n\\vec{\\hat{X}}_\\mathrm{PD}(\\Omega)&=\n% \\sqrt{R_\\mathrm{PD}} \\vec{\\hat{X}}_\\mathrm{PD}^L(\\Omega) + \\sqrt{1-R_\\mathrm{PD}} \\Gamma(\\hat{B}_\\mathrm{in}(\\Omega)-\\sqrt{2\\gamma_R}\\hat{b}(\\Omega))\\\\\n% &= \\sqrt{R_\\mathrm{PD}} \\vec{\\hat{X}}_\\mathrm{PD}^L(\\Omega) + \\sqrt{1-R_\\mathrm{PD}} \\Gamma(\\mathrm{I}-2\\gamma_R\\mathrm{M}_b^{-1})\\Gamma^{-1}\\vec{\\hat{X}}_{B_\\mathrm{in}}(\\Omega)\\\\\n% &-\\sqrt{1-R_\\mathrm{PD}} \\Gamma\\sqrt{2\\gamma_R}\\mathrm{M}_b^{-1}\n% (\\omega_s\\mathrm{M}_a^{-1}\\sqrt{2\\gamma_a}\\vec{\\hat{n}}^L_a(\\Omega)\n% -\\omega_s\\mathrm{M}_a^{-1}i\\beta\\begin{bsmallmatrix}\n% 1 & 0 \\\\ \n% 0 & -1\n% \\end{bsmallmatrix}\\vec{\\tilde{h}}(\\Omega) \\\\& \\hspace{5cm}+ \\chi\\begin{bsmallmatrix}\n% 0 & 1 \\\\ \n% 1 & 0\n% \\end{bsmallmatrix}\\mathrm{M}_c^{-1}\\sqrt{2\\gamma_c}\\vec{\\hat{n}}^L_c(\\Omega) + \\sqrt{2\\gamma_b}\\vec{\\hat{n}}^L_b(\\Omega))\\\\&=\n\\mathrm{R_{in}}\\vec{\\hat{X}}_{B_\\mathrm{in}}(\\Omega)\n+ \\mathrm{R}^L_a\\vec{\\hat{X}}^L_a(\\Omega)\n+ \\mathrm{R}^L_b\\vec{\\hat{X}}^L_b(\\Omega)\n+ \\mathrm{R}^L_c\\vec{\\hat{X}}^L_c(\\Omega)\n+ \\mathrm{R}^L_\\mathrm{PD}\\vec{\\hat{X}}_\\mathrm{PD}^L(\\Omega)\n+ \\mathrm{T}\\vec{\\tilde{h}}(\\Omega).\n\\end{align}\n\nWhere $\\vec{\\hat{X}}_{B_\\mathrm{in}}(\\Omega)$ is the quadrature vector of the vacuum from the main vacuum port behind the signal-recycling mirror, $\\vec{\\hat{X}}^L_a(\\Omega)$ is the vacuum from the $\\hat{a}$ intra-cavity loss, etc.. And where the transfer function matrices $\\mathrm{R}_i, \\mathrm{T}$ for the noises and signal, respectively, are given by the following (where the frequency dependence is inside each $\\mathrm{M}_i^{-1}$):\n\\begin{equation}\n\\begin{cases}\n\\mathrm{R_{in}}=\\sqrt{1-R_\\mathrm{PD}} \\Gamma(\\mathrm{I}-2\\gamma_R\\mathrm{M}_b^{-1})\\Gamma^{-1}\\\\\n\\mathrm{R}^L_a=-\\sqrt{1-R_\\mathrm{PD}} 2\\sqrt{\\gamma_R\\gamma_c}\\omega_s\\Gamma\\mathrm{M}_b^{-1}\\mathrm{M}_a^{-1}\\Gamma^{-1}\\\\\n\\mathrm{R}^L_b=-\\sqrt{1-R_\\mathrm{PD}} 2\\sqrt{\\gamma_R\\gamma_b}\\Gamma\\mathrm{M}_b^{-1}\\Gamma^{-1}\\\\\n\\mathrm{R}^L_c=-\\sqrt{1-R_\\mathrm{PD}} 2\\sqrt{\\gamma_R\\gamma_c}\\chi\\Gamma\\mathrm{M}_b^{-1}\\begin{bsmallmatrix}\n0 & 1 \\\\ \n1 & 0\n\\end{bsmallmatrix}\\mathrm{M}_c^{-1}\\Gamma^{-1}\\\\\n\\mathrm{R}^L_\\mathrm{PD}=\\sqrt{R_\\mathrm{PD}}\\\\\n\\mathrm{T}=\\sqrt{1-R_\\mathrm{PD}} \\sqrt{2\\gamma_R}i\\omega_s\\beta \\Gamma\\mathrm{M}_b^{-1}\\mathrm{M}_a^{-1}\\begin{bsmallmatrix}\n1 & 0 \\\\ \n0 & -1\n\\end{bsmallmatrix}.\n\\end{cases}\n\\end{equation} \n\n\n\\jam{(set up the combined noise, spectral densities matrix Sx)}\n\n\\begin{equation}\n\\mathrm{S}_X(\\Omega)2\\pi\\delta(\\Omega-\\Omega')=\\ev{\\vec{\\hat{X}}_\\mathrm{PD}(\\Omega)\\cdot\\vec{\\hat{X}}_\\mathrm{PD}(\\Omega')^\\dag}\n\\end{equation}\n\n\\jam{(explain uncorrelated vacuum assumption -- i and j are not indices of vectors)}\n\n\\begin{equation}\n\\ev{\\vec{\\hat{X}}_{(i)}(\\Omega)\\cdot\\vec{\\hat{X}}_{(j)}(\\Omega')^\\dag}=\\delta_{i,j}\\mathrm{I}\\,2\\pi\\delta(\\Omega-\\Omega'),\\quad \\mathrm{S_{vac}}=\\mathrm{I}=\\begin{bsmallmatrix}\n1 & & & \\\\ \n & 1 & & \\\\\n & & 1 & \\\\ \n & & & 1\n\\end{bsmallmatrix}\n\\end{equation}\n\n\\jam{(versus single-sided power spectral density of single function, not a vector: $A\\circ B=\\frac{1}{2}(A\\cdot B+B\\cdot A)$, check factors of two again --> single sided, vac=1, quadratures normalised as 1/rt2, quadrature is Hermitian and therefore commutes with its dagger)}\n\n\n\\begin{equation}\n\\mathcal{S}_X(\\Omega)2\\pi\\delta(\\Omega-\\Omega')=(\\text{S}_X)_{2,2}(\\Omega)2\\pi\\delta(\\Omega-\\Omega')=\\ev{\\hat{X}_{\\mathrm{PD},2}(\\Omega)\\circ\\hat{X}_{\\mathrm{PD},2}(\\Omega')^\\dag},\\quad \\mathcal{S}_\\mathrm{vac}=1\n\\end{equation}\n\n\\begin{equation}\n\\mathcal{T}(\\Omega)=(\\mathrm{T}.\\begin{bsmallmatrix}\n1\\\\ \n1\n\\end{bsmallmatrix})_2\n\\end{equation}\n\nI define the sensitivity of the detector to be the noise-to-signal ratio, i.e. the ratio of the noise and signal transfer functions (or, the signal-normalised noise).\nAssuming that we measure the second quadrature, e.g. via balanced homodyne readout\\jam{(address optimisation of quadrature angle, might not be 2 just because signal is only there. Also, need to describe homodyne readout somewhere)}, this gives:\n$$S_h(\\Omega)=\\frac{(\\mathrm{S}_X(\\Omega))_{2,2}}{\\abs{\\mathcal{T}(\\Omega)}^2}.$$\n\\end{comment}\n\n", "meta": {"hexsha": "528c2a3493ea9fe63ff98afa4d2f6e0413741923", "size": 25111, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "thesis/appendixA.tex", "max_stars_repo_name": "daccordeon/nondegDog", "max_stars_repo_head_hexsha": "570598b59ac8c70dee6387088698ed768a2d1247", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-02-22T05:25:14.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-24T23:42:29.000Z", "max_issues_repo_path": "thesis/appendixA.tex", "max_issues_repo_name": "daccordeon/nondegDog", "max_issues_repo_head_hexsha": "570598b59ac8c70dee6387088698ed768a2d1247", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "thesis/appendixA.tex", "max_forks_repo_name": "daccordeon/nondegDog", "max_forks_repo_head_hexsha": "570598b59ac8c70dee6387088698ed768a2d1247", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 97.3294573643, "max_line_length": 801, "alphanum_fraction": 0.7284855243, "num_tokens": 7766, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926666143434, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.3422622228836804}} {"text": "\\documentclass{article}\n\\usepackage{amsmath}\n\\usepackage{amsfonts} \n\\usepackage{graphicx}\n\\usepackage{biblatex} \n\\usepackage{authblk}\n\\usepackage{mathtools}\n\\usepackage{xurl} %see https://tex.stackexchange.com/questions/23394/url-linebreak-in-footnote for why we use xurl to get line breaks instead of regular url\n\\usepackage[hidelinks]{hyperref}\n\\usepackage{listings}\n\\usepackage{cancel}\n\\usepackage{enumitem}\n\\usepackage[bb=boondox]{mathalfa}\n\\addbibresource{three_books.bib}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\DeclareMathOperator*{\\argmin}{arg\\,min}\n\\usepackage{fancyhdr}\n\\usepackage{textcomp}\n\n\n\\setlist[description]{leftmargin=2cm,labelindent=1cm}\n\n\\lstdefinestyle{mystyle}{\n % backgroundcolor=\\color{backcolour}, \n % commentstyle=\\color{codegreen},\n % keywordstyle=\\color{magenta},\n % numberstyle=\\tiny\\color{codegray},\n % stringstyle=\\color{codepurple},\n % basicstyle=\\ttfamily\\footnotesize,\n % breakatwhitespace=false, \n % breaklines=true, \n captionpos=b, \n keepspaces=true, \n % numbers=left, \n % numbersep=5pt, \n % showspaces=false, \n % showstringspaces=false,\n % showtabs=false, \n tabsize=2,\n frame=single,\n}\n\\lstset{style=mystyle}\n\n\\title{Blueprints for building reinforcement learning algorithms in customer-facing applications}\n\\author[1]{Douglas Mason}\n\\affil[1]{Koyote Science, LLC \\footnote{\\href{http://www.koyotescience.com}{\\texttt{http://www.koyotescience.com}}}}\n\\date{November 2021}\n\n\\begin{document}\n\n\\maketitle\n\n\\tableofcontents\n\n\\pagestyle{fancy}\n% \\fancyhf{}\n\\cfoot{\\\\\n\\\\\n\\includegraphics[scale=0.15,valign=c]{koyote_science_logo.png}\n}\n\\lhead{Page \\thepage}\n\\rhead{Section \\thesection}\n\n\n\\section{Introduction}\nThis document outlines primary considerations for designing production-worthy reinforcement learning algorithms in customer-facing settings. We will (1) establish the notation, (2) discuss how policies are evaluated, (3) show how policies can be defined by evaluation functions alone, (4) demonstrate how we train an evaluation for our policy, even from customer interactions driven by another policy, known as off-policy training, (5) show how we can train the policy directly from customer interactions, and (6) demonstrate how we can combine both approaches for the current state-of-the-art in the field. We also (7) provide practical considerations under the hood for building our policies and evaluation functions, as well as (8) additional resources to help the reader dive more deeply into the field, as well as discover implementations that can be used right now. This document attempts to be brief but mathematically thorough. Please write to the author at \\href{mailto:douglas@koyotescience.com}{douglas@koyotescience.com} for questions or feedback. We provide a quick schematic of the algorithms we will cover in Figure \\ref{fig:rl_schematic}.\n\\begin{figure}\\label{fig:rl_schematic}\n \\begin{center}\n \\includegraphics[width=0.75\\linewidth]{rl_schematic}\n \\caption{Schematic for the reinforcement learning approaches covered in this document.\\footnotemark}\n \\end{center}\n\\end{figure}\n\\footnotetext{ Source: \\url{https://towardsdatascience.com/an-overview-of-classic-reinforcement-learning-algorithms-part-1-f79c8b87e5af}}\n\nAt a high-level overview, reinforcement learning algorithms consist of up to three primary components:\n\\begin{enumerate}\n\\item A \\textit{policy evaluator}, which can be used to define the policy on its own. This is often referred to as the state- or action-value function.\n\\item A \\textit{policy function}, which defines the policy on its own, and can use a policy evaluator to improve its performance\n\\item A \\textit{rollout planning} system for using the policy evaluator and policy function to fine-tune performance\n\\end{enumerate}\nThis documents covers the first two components, and lightly touches on the third, since it is the riskiest to implement in a production system, and requires the ability to predict future states of the system, known as the \\textit{environment model}. Moreover, the policy evaluator is a development of the multi-armed bandit formalism, allowing it to act as a stepping stone between optimization, such as hyperparameter tuning, and sequential planning that uses an explicit policy function.\n\n\\section{Notation and policy evaluation}\n\nWhen designing products that take users through a sequence of actions to get to a desired goal, it’s a natural instinct to look at the field of reinforcement learning to optimize your design. In this formulation, we have a state vector $\\mathbf{s}$ (e.g., the current page a user is on) which may also incorporate a context vector $\\mathbf{c}$ (e.g., user demographics), an action vector $\\mathbf{a}$ that represents a decision that the system can take (e.g., presenting different options along the way), and a reward for taking an action at a given state $r(\\mathbf{s},\\mathbf{a})$ (e.g., whether a user signs up at the end of a sign-up flow). Note that we can also define the reward without considering the action, $r(\\mathbf{s})$, which is identical except that it is defined by the state you end up in to get the reward rather than the state and action you take to get it. \n\nWe are interested in learning a policy $\\pi(\\mathbf{a}| \\mathbf{s})\\in [0,1]$ that gives us the probability of choosing a given action $\\mathbf{a}$ with a given state $\\mathbf{s}$. Note that when implemented, such a function depends on both $\\mathbf{a}$ and $\\mathbf{s}$. The policy is normalized so that its returns for all available actions at a given state add up to one, i.e., \\begin{equation}\\label{eq:normalization}\n\\sum_\\mathbf{a}\\pi(\\mathbf{a}|\\mathbf{s})=1\n\\end{equation} When learning this policy, our goal is to maximize the discounted sum of future rewards $G_t(\\boldsymbol{\\tau})$\\footnote{The letter $G$ is chosen for historical reasons, while $J(\\pi_{\\boldsymbol{\\theta}})=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}[G_0({\\boldsymbol{\\tau}})]$ is used later in Equation \\ref{eq:J}} from time step $t$ through the problem horizon $H$, as we step through a given trajectory of states $\\boldsymbol{\\tau}$ which we write out as\n\\begin{equation}\n\\boldsymbol{\\tau}=(\\mathbf{s}_0,\\mathbf{a}_0,\\mathbf{s}_1,\\mathbf{a}_1,\\dots,\\mathbf{s}_H,\\mathbf{a}_H)\n\\end{equation} and where the discount factor $\\gamma\\in(0,1]$. We write this quantity out as\n\\begin{equation}\n G_t(\\boldsymbol{\\tau})=\\sum_{t'=t}^{H}r(\\mathbf{s}_{t'},\\mathbf{a}_{t'})\\gamma^{t'-t}\n\\end{equation}\nWe can accomplish this by learning a Q-value or action-value function for each state and action, which returns the expected discounted sum of future rewards assuming we follow a policy $\\pi$ starting from state $\\mathbf{s}$ and action $\\mathbf{a}$, and can be written as\n\\begin{equation}\n\\begin{split}\nQ_\\pi(\\mathbf{s},\\mathbf{a})&=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi}[G_t({\\boldsymbol{\\tau}})|\\mathbf{s}_t=\\mathbf{s},\\mathbf{a}_t=\\mathbf{a}]\\\\&=\\sum_{t'=t}^{H}\\pi(\\mathbf{a}_{t'}|\\mathbf{s}_{t'})r(\\mathbf{s}_{t'},\\mathbf{a}_{t'})\\gamma^{t'-t}\n\\end{split}\n\\end{equation} Other formulations may work with a state-value function \\begin{equation}\\begin{split}V_{\\pi}(\\mathbf{s})&=\\sum_{\\mathbf{a}\\sim\\mathcal{A}}\\pi(\\mathbf{a}|\\mathbf{s})Q_\\pi(\\mathbf{s},\\mathbf{a})\\\\&=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi}[G_t({\\boldsymbol{\\tau}})|\\mathbf{s}_t=\\mathbf{s}]\\\\&=\\sum_{t'=t}^{H}\\sum_{\\mathbf{a}\\sim\\mathcal{A}}\\pi(\\mathbf{a}_{t'}|\\mathbf{s}_{t'})r(\\mathbf{s}_{t'},\\mathbf{a}_{t'})\\gamma^{t'-t}\\end{split}\\end{equation}which only depends on the state.\n\n\\section{Deriving the policy from the evaluation}\nThe policy can be determined by the $Q$-value using a greedy strategy\n\\begin{equation}\\label{eq:greedy_policy}\n \\pi(\\mathbf{a}|\\mathbf{s})=\\mathbb{1}\\left[\\mathbf{a}=\\argmax_\\mathbf{a}Q(\\mathbf{s},\\mathbf{a}))\\right]\n\\end{equation}where $\\mathbb{1}[\\cdot]$ returns a 1 when the argument is true and 0 otherwise. This policy is deterministic, but we can make a stochastic policy using a variety of other strategies. In $\\epsilon$-greedy, the greedy action is chosen with a fixed probability (say, 75\\%), and other actions are chosen at random for the remaining 25\\% of the time. Another strategy is to use the softmax function\n\\begin{equation}\n \\pi(\\mathbf{a}|\\mathbf{s})=\\frac{\\exp(\\beta Q(\\mathbf{s},\\mathbf{a}))}{\\sum_\\mathbf{a}\\exp(\\beta Q(\\mathbf{s},\\mathbf{a}))}\n\\end{equation}which becomes the greedy policy as $\\beta\\rightarrow\\infty$. And yet another strategy is to perform gradient ascent against the $Q$-value (when such gradients are available) to maximize it over the action space, which is used in the \"deterministic policy gradient\" algorithm\\cite{DDPG}.\n\nIf we use the state-value instead of the action-value, then we need to know how actions lead to state transitions, which is known as environment model $p(\\mathbf{s}_{t+1}|\\mathbf{s},\\mathbf{a}$), and is distinguished from the models used for policy function or state- and action-value function approximation. How do we learn the environment model? Given transitions observed by the environment, $p_\\text{obs.}(\\mathbf{s}'|\\mathbf{s},\\mathbf{a})$ which sum to one over all possible states $\\mathbf{s}'$, we can sample transitions using any policy $\\pi$ by storing or learning a function approximator to \\begin{equation}p(\\mathbf{s}'|\\mathbf{s},\\mathbf{a})=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi}\\left[\\text{IPW}(\\mathbf{a}_t|\\mathbf{s}_t)p_\\text{obs.}(\\mathbf{s}_{t+1}|\\mathbf{s_t},\\mathbf{a}_t)\\mathbb{1}\\left[\\mathbf{s}_t=\\mathbf{s},\\mathbf{a}_t=\\mathbf{a},\\mathbf{s}_{t+1}=\\mathbf{s}'\\right]\\right]\\end{equation}where the \\textit{inverse probability weighting}\n\\begin{equation}\\text{IPW}(\\mathbf{a}|\\mathbf{s})=\\frac{1}{\\pi(\\mathbf{a}|\\mathbf{s})}\\end{equation}is the inverse likelihood of choosing the given action with the policy used to sample trajectories $\\pi$. The IPW allows us to rectify the bias in the data distribution determined by the policy and used to train our environment model. Sometimes the word \"propensity\" is used instead, as a rarer synonym for \"probability\" to help distinguish it from other probabilities. While IPW will be used later in this document, the topic of environment \"model-based\" reinforcement learning \\cite{model_based_rl_survey} will remain outside our scope.\n\nAnother strategy makes use of Bayesian models that possess the ability to sample their parameters according to their epistemic uncertainty \\cite{GP_BLR}, and is called Thompson sampling \\cite{thompson_sampling}. In this strategy, the model parameters are sampled, the $Q$-value is calculated from this sampled model for each available action, and the strategy chooses the action with the highest score. This approach has been shown to optimize the exploration-exploitation trade-off in bandits\\cite{thompson_sampling,bootstrap_DQN}. The use of Bootstrap Thompson Sampling (BTS), which approximates the Bayesian distribution by training an ensemble of non-Bayesian models on different bootstrap samples of the training data, was popularized by the algorithm group at Facebook for its engineering benefits, among them that prediction only requires one non-Bayesian model from the ensemble at a time, and debugging is straightforward\\cite{bootstrap_facebook}. Further developments, such as Randomized Least Squares Value Iteration, use epistemic uncertainty sampling even when determining the next-state maximizing action in the $Q$-learning phase\\cite{RLSVI} to encourage exploration.\n\nLastly, a recent and amazingly-successful strategy, called Monte Carlo Tree Search (MCTS), performs simulations during each planning stage, using a simplified policy and an environment model, where the simplified policy is guided by state- and/or value-functions that are also trained. These strategies have been very successful in perfect-information, adversarial games, but they are outside the scope of this overview. Read Section 8.11 of \\cite{sutton_barto_rl} as well as \\cite{go1} and \\cite{go2} to learn more.\n\n\\section{Training the state- and value-function policy evaluators\\label{sec:value_training}}\n\nWith this notation secured, we can finally bring in the \\textbf{Bellman equation}, which is a major foundation of both reinforcement learning and dynamic programming. It states that the best choices we make now can be determined greedily by assuming we continue to make the best possible choices later, and we can write it out mathematically as\n\\begin{equation}\\label{eq:bellman}\nQ^\\ast(\\mathbf{s},\\mathbf{a})=r(\\mathbf{s},\\mathbf{a})+\\gamma \\argmax_{a'}Q^\\ast(\\mathbf{s}',\\mathbf{a}')\n\\end{equation}where we assume that action $\\mathbf{a}$ leads us from state $\\mathbf{s}$ to state $\\mathbf{s}'$. We denote $Q^\\ast$ as the optimal Q-value (which can then be used to define the optimal policy $\\pi^\\ast$), and any function approximation for the $Q$-value, using model parameters $\\boldsymbol{\\phi}$, as $Q_{\\boldsymbol{\\phi}}$ with the associated policy $\\pi_{\\boldsymbol{\\phi}}$.\n\n\\subsection{Elaboration on the Bellman equation}\n\nWe can update the $Q$-value using \\textbf{temporal-difference} (TD) update rules that use the fact that we can always improve our estimates of the $Q$-value by iterating over our trajectories, at each time step updating our function approximator of the $Q$-value to predict targets that increment against the old estimate. The incremental value, or target, is supplied by the difference between the reward returned by the environment for the current state and action pair ($\\mathbf{s}$, $\\mathbf{a}$) and the reward for the current time step predicted by our function approximator. We then multiply it by a learning rate $\\alpha\\in(0,1]$ in the explicit update rules, or use the learning rate in the computed policy gradient update rules (more on this later). \n\nSince the approximator covers the discounted sum of future rewards $G_t$, rather than specific rewards at each time step, we use the Bellman equation to predict the reward for the current time step as the temporal difference $G_{t+1}-G_t$ predicted by the function approximator. We can also look at higher-order approxiamtions using more than one time step, and define these quantities as \\begin{equation}\\begin{split}\nG_t^{(1)}=G_{t:t+1}^\\text{(state)}&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma V (\\mathbf{s}_{t+1})\\\\G_t^{(2)}=G_{t:t+2}^\\text{(state)}&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma r(\\mathbf{s}_{t+1},\\mathbf{a}_{t+1})+\\gamma^2 V (\\mathbf{s}_{t+2})\\\\G_t^{(n)}=G_{t:t+n}^\\text{(state)}&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma r(\\mathbf{s}_{t+1},\\mathbf{a}_{t+1})+\\cdots\\\\&\\nobreakspace\\nobreakspace\\nobreakspace\\nobreakspace+\\gamma^{n-1}r(\\mathbf{s}_{t+n-1},\\mathbf{a}_{t+n-1})+\\gamma^n V (\\mathbf{s}_{t+n})\\end{split}\\end{equation} where we have dropped the explicit policy markers for clarity. The analogous expression using the action-value function is \\begin{equation}\\begin{split}G_{t:t+n}^\\text{(action)}&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma r(\\mathbf{s}_{t+1},\\mathbf{a}_{t+1})+\\cdots\\\\&\\nobreakspace\\nobreakspace\\nobreakspace\\nobreakspace+\\gamma^{n-1}r(\\mathbf{s}_{t+n-1},\\mathbf{a}_{t+n-1})+\\gamma^n Q (\\mathbf{s}_{t+n},\\mathbf{a}_{t+n})\\end{split}\\end{equation} where the use of the action- or state-value versions is implied by context. The analagous expression when using expectations is \\begin{equation}\\begin{split}\nG_{t:t+n}^{(\\text{expected})}&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma r(\\mathbf{s}_{t+1},\\mathbf{a}_{t+1})+\\cdots\\\\&\\nobreakspace\\nobreakspace\\nobreakspace\\nobreakspace+\\gamma^{n-1}r(\\mathbf{s}_{t+n-1},\\mathbf{a}_{t+n-1})+\\gamma^n \\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi}[Q (\\mathbf{s}_{t+n},\\mathbf{a}_{t+n})|\\mathbf{s}_{t+n}]\\\\&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma r(\\mathbf{s}_{t+1},\\mathbf{a}_{t+1})+\\cdots\\\\&\\nobreakspace\\nobreakspace\\nobreakspace\\nobreakspace+\\gamma^{n-1}r(\\mathbf{s}_{t+n-1},\\mathbf{a}_{t+n-1})+\\gamma^n \\sum_\\mathbf{a}\\pi(\\mathbf{a}|\\mathbf{s}_{t+1})Q (\\mathbf{s}_{t+n},\\mathbf{a}_{t+n})\\\\\\end{split}\\end{equation} and the analogous expression when using the argmax is \\begin{equation}\\begin{split}\nG_{t:t+n}^{(\\text{argmax})}&=r(\\mathbf{s}_t,\\mathbf{a}_t)+\\gamma r(\\mathbf{s}_{t+1},\\mathbf{a}_{t+1})+\\cdots\\\\&\\nobreakspace\\nobreakspace\\nobreakspace\\nobreakspace+\\gamma^{n-1}r(\\mathbf{s}_{t+n-1},\\mathbf{a}_{t+n-1})+\\gamma^n \\argmax_{\\mathbf{a}'}Q (\\mathbf{s}_{t+n},\\mathbf{a}_{t+n})\\\\\\end{split}\\end{equation}Since the action- or $Q$-value is preferred for policies that depend on the evaluators, that is what we focus on in this section, while the state-value is preferred for the policy gradients presented in the next section.\n\nOne more common elaboration, featured in the TD($\\lambda$) algorithm (discussed later), is to work with a weighted sum of discounted sums of future rewards defined as:\n\\begin{equation}\\begin{split}\\\nG_{t:h}^{(\\lambda,\\text{action})} &=(1-\\lambda)\\sum_{n=1}^{h-t-1}\\lambda^{n-1}G_{t:t+n}^\\text{(action)}+\\lambda^{h-t-1}G_{t:h}^\\text{(action)} \\\\G_{t:h}^{(\\lambda,\\text{state})} &=(1-\\lambda)\\sum_{n=1}^{h-t-1}\\lambda^{n-1}G_{t:t+n}^\\text{(state)}+\\lambda^{h-t-1}G_{t:h}^\\text{(state)} \n\\end{split}\\end{equation} where $0\\leq t0\\\\\\min\\left(\\frac{\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}|\\mathbf{s})}{\\pi_{\\boldsymbol{\\theta}_k}(\\mathbf{a}|\\mathbf{s})},1-\\epsilon\\right)A_{\\pi_{\\boldsymbol{\\theta}_k}}(\\mathbf{s},\\mathbf{a}), &A_{\\pi_{\\boldsymbol{\\theta}_k}}(\\mathbf{s},\\mathbf{a})<0\\end{cases}\\end{equation}. In other words, it is off-policy Advantage Actor Critic (A2C) algorithm but clipped so that the target never exceeds the range of \\begin{equation}\\Psi_t^\\text{(PPO)}\\in\\left[(1-\\epsilon)\\left|A_{\\pi_{\\boldsymbol{\\theta}_k}}(\\mathbf{s},\\mathbf{a})\\right|,(1+\\epsilon)\\left|A_{\\pi_{\\boldsymbol{\\theta}_k}}(\\mathbf{s},\\mathbf{a})\\right|\\right] \\end{equation}\n\n\\section{Derivation of the policy gradient theorem and building intuition}\n\nAt first, the surprising introduction of logarithm makes one ask: why $\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\log\\pi_{\\boldsymbol{\\theta}}$ instead of $\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\pi_{\\boldsymbol{\\theta}}$? To cultivate intuition for the PGT, we first note that our goal is to relate the gradient of an expectation over trajectories to the expectation over trajectories of a gradient, in order to make it possible to compute. This comes up because the value we are attempting to maximize, the state-value function for a given policy at the start of all trajectories made by the policy, is the expectation of action-values over all actions. Meanwhile, the policy itself is a function of individual actions. We can relate this back to the expectation over trajectories formed by the policy by observing that since \\begin{equation}\\begin{split}\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}f(\\mathbf{a}_t,\\mathbf{s}_t)&=\\sum_{\\mathbf{a}\\sim\\mathcal{A}}\\pi(\\mathbf{a}|\\mathbf{s}_t)f(\\mathbf{a},\\mathbf{s}_t)\\\\\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}f(\\boldsymbol{\\tau})&=\\int_{\\boldsymbol{\\tau}}p(\\boldsymbol{\\tau})f(\\boldsymbol{\\tau})\\end{split}\\end{equation} and \\begin{equation}\\frac{\\partial f(x)}{\\partial x}=f(x)\\frac{\\partial \\log f(x)}{\\partial x}\\end{equation}and \\begin{equation}\\log p(\\boldsymbol{\\tau}|\\pi_{\\boldsymbol{\\theta}})=p(\\mathbf{s}_0)\\sum_{t=0}^H \\pi(\\mathbf{a}_t|\\mathbf{s}_t)p(\\mathbf{s}_{t+1}|\\mathbf{s}_t,\\mathbf{a}_t)\\end{equation} we have \\begin{equation}\\begin{split}\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}J(\\pi_{\\boldsymbol{\\theta}})&=\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\int_{\\boldsymbol{\\tau}}p(\\boldsymbol{\\tau},\\boldsymbol{\\theta})R(\\boldsymbol{\\tau})\\\\&=\\int_{\\boldsymbol{\\tau}}\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}p(\\boldsymbol{\\tau},\\boldsymbol{\\theta})R(\\boldsymbol{\\tau})\\\\&=\\int_{\\boldsymbol{\\tau}}p(\\boldsymbol{\\tau},\\boldsymbol{\\theta})\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\log p(\\boldsymbol{\\tau},\\boldsymbol{\\theta})R(\\boldsymbol{\\tau})\\\\&=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\log p(\\boldsymbol{\\tau},\\boldsymbol{\\theta})R(\\boldsymbol{\\tau})\\right]\\\\&=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\sum_{t=0}^{H}\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\log \\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}|\\mathbf{s})R(\\boldsymbol{\\tau})\\right]\\end{split}\\end{equation}where we can replace $R(\\boldsymbol{\\tau})$ with $\\Psi_t$. You'll also see, as in \\cite{sutton_barto_rl}, the policy gradient written as\\begin{equation}\\begin{split}\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}J(\\pi_{\\boldsymbol{\\theta}})&=\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}[V(\\mathbf{s}_0)]\\\\&\\propto\\sum_\\mathbf{s}p_{\\pi_{\\boldsymbol{\\theta}}}(\\mathbf{s})\\sum_\\mathbf{a}\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}|\\mathbf{s})Q_{\\pi_{\\boldsymbol{\\theta}}}(\\mathbf{s},\\mathbf{a})\\\\&=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\sum_\\mathbf{a}Q(\\mathbf{s}_t, \\mathbf{a})\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}|\\mathbf{s}_t)\\right]\\\\&=\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[Q(\\mathbf{s}_t, \\mathbf{a}_t)\\boldsymbol{\\nabla}_{\\boldsymbol{\\theta}}\\log\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}_t|\\mathbf{s}_t)\\right]\\end{split}\\end{equation}where we can replace $Q(\\mathbf{s}_t,\\mathbf{a}_t)$ with $\\Psi_t$.\n\nThe reader may have noticed that the PGT is written like the commonly-used cross-entropy loss function, and there is a good reason for that: they come from the same derivation! The cross entropy of normalized distribution $\\boldsymbol{\\pi}$ relative to normalized distribution $\\boldsymbol{\\pi}'$ is written as \\begin{equation}\\label{eq:cross-entropy}\nH(\\boldsymbol{\\pi},\\boldsymbol{\\pi}')=-\\sum_{\\mathbf{x}\\sim\\mathcal{X}}\\pi'(\\mathbf{x})\\log\\pi(\\mathbf{x})\n\\end{equation}This means that, in a standard supervised learning setting, the parameters fitting a loss function using cross-entropy to match a normalized predicted distribution to a target distribution can be written as \n\\begin{equation}\n\\boldsymbol{\\theta}=\\argmin_{\\boldsymbol{\\theta}}\\mathop{\\mathbb{E}}_{\\mathbf{x},\\pi_\\text{target}\\sim\\mathcal{D}}\\left[\\pi_\\text{target}(\\mathbf{x})\\log \\pi_{\\boldsymbol{\\theta}}(\\mathbf{x})\\right] \n\\end{equation} \n\nNote that with the cross-entropy loss, we are looking at the expectation over a dataset, and unlike the PGT, we are summing over all possible actions, or discrete entries in a probability distribution. However, the individual contribution of each action or discrete entry is treated the same in both distributions, and for certain model architectures, we may only have access to the outcomes of one action or discrete entry at a time. In this case, the formalisms line up exactly, except for the fact that in supervised learning, the target $\\pi'(\\mathbf{x})$ is a fixed quantity, while in the PGT, the target $\\Psi_t$ is dependent on the policy itself, we just pretend that it isn't when updating our model parameters, which means that our targets will change as soon as we update our parameters. For this reason, we never want to actually minimize our parameters over the loss function of a fixed dataset, so be aware if you see an abuse of notation writing the gradient update for policy evaluators as \\begin{equation}\\boldsymbol{\\phi}=\\argmin_{\\boldsymbol{\\phi}}\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\phi}}}\\left[\\left(U_t-Q(\\mathbf{s}_t,\\mathbf{a}_t)\\right)^2\\right]\\end{equation} and similarly for policy gradients \\begin{equation}\\boldsymbol{\\theta}=\\argmin_{\\boldsymbol{\\theta}}\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\log\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}_t,\\mathbf{s}_t)\\Psi_t\\right]\\end{equation}\nHowever, the regressions that we use to train our policy evaluators do use the above notation, and assume that as we add more data to our dataset, we will approximate to the final distributions we want to optimize over. But there are stronger reasons to avoid using this notation. As von Neumann once famously quipped, \"no one understands entropy very well\"\\cite{no_one_understands_entropy}, so building a greater intuition for cross-entropy will help our intuition for the PGT. \n\nThe form of cross-entropy loss can be grasped at a high level by first considering the likelihood of any normalized distribution prediction. For a binomial distribution of multiple coin flips with the individual probability of returning heads set to $p$, the likelihood of receiving $k$ heads over $n$ coin flips is \\begin{equation}\\mathcal{L}(n,k,p)=\\underbrace{\\frac{n!}{k!(n-k)!}}_{C(n,k)}p^k(1-p)^{n-k}\\end{equation}where the exponents come from the interchangeability of each equivalent coin flip, and the binomial coefficient $C(n,k)$ comes from the total number of such combinations. Taking the negative log of the likelihood to produce a loss function gives us \\begin{equation}L=-\\log(\\mathcal{L})=-\\log(C(n,k))-k\\log p -(n-k)\\log (1-p)\\end{equation}The single-trial case, coming from the Bernoulli distribution, takes $k\\rightarrow 1$, and the single-trial continuous case, which we need in order to compute derivatives, takes $k\\rightarrow \\infty$. This is why the continuous beta distribution takes a similar form.\n\nFor example, the targets $\\Psi_t$ \\textit{represent} a target distribution, but this distribution is not normalized nor is it the distribution we hope to ultimately approximate, since any movement to our model parameters will change the targets themselves. Rather, the targets only show us how to nudge our parameters to get to our desired policy, and this is related to why there are so many options available to us to define them. But the cross-entropy is a monotonically-increasing function that is maximized when the two distributions are identical, and which increases with the magnitude of their deviation. It can also be interpreted as the number of bits required to encode the targets using our current policy, but if the target distribution in unnormalized, the number of bits required to encode it goes to infinity, which means that minimizing the PGT loss over a static dataset will produce negative infinity loss.\n\nLastly, we observe that just like the PGT allows us to improve our model parameters using observations corresponding to the subset of actions taken during our data-collection trajectories, rather than over all possible actions, the standard supervised learning cross-entropy loss allows us to consider contributions from subsets of the possible actions rather than summing over all possible actions. However, this leaves open the challenge of properly normalizing the targets, which requires data for all possible actions to perform.\n\n\\section{Combining policy evaluation and policy function training in actor-critic algorithms}\nActor-critic algorithms like the $Q$-Actor Critic and the Advantage Actor Critic train the value function policy evaluator along with the policy itself. Among the A2C class of algorithms there are different estimators of the advantage function for the policy part of the training process (while the the state-value or policy evaluator part is handled according to Section \\ref{sec:value_training}):\n\\begin{description}\n\\item[MC advantage or Vanilla Policy Gradient (VPG):]\\hfill \\\\\n$\\Psi_t =G_t-V(\\mathbf{s}_t)$\n\\item[TD advantage]\\hfill \\\\\n$\\Psi_t =\\delta_t^\\text{(state)}\\coloneqq G_{t:t+1}^\\text{(state)}-V(\\mathbf{s}_t)= r_t+\\gamma V(\\mathbf{s}_{t+1})-V(\\mathbf{s}_t)$\n\\item[$n$-step advantage:]\\hfill \\\\\n$\\Psi_t =G_{t:t+n}^\\text{(state)}-V(\\mathbf{s}_t)$ for any $n$\n\\item[$Q$-advantage:]\\hfill \\\\\n$\\Psi_t =Q(\\mathbf{s}_t,\\mathbf{a}_t)-V(\\mathbf{s}_t)$ for any $n$\n\\item[Generalized Advantage Estimator (GAE) :]\\hfill \\\\ \n$\\Psi_t=\\sum_{i=0}^\\infty(\\lambda\\gamma)^i\\delta_{t+i}=G^{(\\lambda,\\text{state})}_t-V(\\mathbf{s}_{t})$\n\\end{description}\n\nWhen implementing policy gradient algorithms, in practice the training process waits until the end of each episode $t=H$ to the beginning at $t=0$ to propagate information, although for some choices we can update within an episode after a given number of time steps. (Another formulation called \\textit{eligibility traces} allows us to update immediately, but requires storing additional information and applying additional update rules, see Chapter 12 of \\cite{sutton_barto_rl}.) In this setting, the REINFORCE and TD algorithms use the returns from the episode that just played out, making them on-policy algorithms, while AC and A2C use the expected returns provided by an action-value or $Q$-value function, requiring us to optimize the $Q$ function in addition to the policy. In this case, the action that is \"played out\" is the $Q$-value-maximizing action, not necessarily the action that the policy would have used during the episode, making it an off-policy algorithm. We note the relationship between these algorithms and their constituent parts in Figure \\ref{fig:rl_venn_diagram}.\n\n\\begin{figure}\n\\label{fig:rl_venn_diagram}\n \\begin{center}\n \\includegraphics[width=0.75\\linewidth]{rl_venn_diagram}\n \\caption{Venn diagram relating the value-based algorithms like TD variants of $Q$-learning and SARSA, and policy-based algorithms like REINFORCE covered in this document. \\footnotemark}\n \\label{fig:basis_small_width}\n \\end{center}\n\\end{figure}\n\\footnotetext{Source: \\url{https://www.davidsilver.uk/wp-content/uploads/2020/03/pg.pdf}}\n\nActor-critic algorithms require us to a compute a state-value function, possibly in addition to an action-value $Q$-function, and certainly in addition to the policy function, which opens the question of how do we organize our algorithm to accomplish this goal? We can either learn the state-value function separately, or jointly in multi-task models with the action-value and/or policy functions. In fact, any splitting between the state-value, action-value, and policy functions is available to us, so that we may not have separate sets of parameters like $\\boldsymbol{\\theta}$ for the policy and $\\boldsymbol{\\phi}$ for the action- or state-value, but rather some parameters will be shared between them and other will not. \n\nNote that multi-task models are generally limited to neural networks and bespoke implementations. Moreover, they minimize a total loss \\begin{equation}L_\\text{total}=L_\\text{actor}+L_\\text{critic}+L_\\text{regularizer}\\end{equation} according to batch updates of Equation \\ref{eq:simple_gradient_update}\\cite{online_policy_gradient_review,openAI_part3}. For the critic, which predicts action- or state-values or both, the loss function depends on the difference between our predictions and the results returned from the environment, optionally bootstrapped off of prior predictions. For the normal distribution likelihood, we can use the squared error as in \\begin{equation}\\begin{split}L_\\text{critic}^{\\text{(state)}}&=\\frac{1}{2}\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\left(U_t-V(\\mathbf{s}_{t})\\right)^2\\right]\\\\L_\\text{critic}^{\\text{(action)}}&=\\frac{1}{2}\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\left(U_t-Q(\\mathbf{s}_{t},\\mathbf{a}_t)\\right)^2\\right]\\\\L_\\text{critic}^\\text{(both)}&=L_\\text{critic}^{\\text{(state)}}+L_\\text{critic}^{\\text{(action)}}\\end{split}\\end{equation} Some variations may use the absolute error or the Huber loss. \n\nFor the actor, i.e., the policy gradient, our loss function, according to Equations \\ref{eq:simple_gradient_update}, \\ref{eq:policy_gradient_theorem}, and \\ref{eq:policty_gradient_update}, is \\begin{equation}L_\\text{actor}=-\\mathop{\\mathbb{E}}_{\\boldsymbol{\\tau}\\sim\\pi_{\\boldsymbol{\\theta}}}\\left[\\sum_{t=0}^H\\log\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}_t|\\mathbf{s}_t)\\Psi_t\\right]\\end{equation} Lastly, we can also add additional losses to our final optimized quantity, such as regularization losses to reduce the model complexity, or a self-entropy term \\begin{equation}L_\\text{regularizer}^\\text{(entropy)}=\\beta H(\\pi, \\mathbf{s})=-\\beta \\sum_a\\pi(\\mathbf{a}|\\mathbf{s})\\log\\pi(\\mathbf{a}|\\mathbf{s})\\end{equation} weighted (multiplied) by a chosen positive or negative hyperparameter $\\beta$, to encourage more or less exploration as desired. L1 and L2 regularizer losses are always available, written as \\begin{equation}L_\\text{regularizer}^{\\text{(L2)}}=\\beta\\boldsymbol{\\theta}^\\top\\boldsymbol{\\theta}=\\frac{\\sigma^2}{\\sigma_p^2}\\boldsymbol{\\theta}^\\top\\boldsymbol{\\theta}\\end{equation} where $\\beta$ is the $L2$ regularization hyperparameter and $\\sigma_p$ is a noise parameter scale hyperparameter which is equivalent to the average noise parameter for the prior, where the model parameters are all set to zero, or $\\boldsymbol{\\theta}=\\mathbf{0}$. In addition, losses that allow us to simultaneously optimize either the homoskedastic the noise parameter $\\sigma$, or the heteroskedastic noise parameter $\\sigma(\\mathbf{a},\\mathbf{s})$, can be included, derived from the negative log of the normalization constant in a normal distribution, and which we can write as \\begin{equation}L_\\sigma=N_\\text{data}\\log\\sigma\\end{equation}such that $L_\\text{regularizer}^\\text{(L2)}$ and $L_\\sigma$ balance each other out against the data fit term $L_\\text{critic}$ and/or the policy gradient loss $L_\\text{actor}$.\n\n\\section{Training with custom loss functions}\n\nStochastic gradient descent has the benefit of only ever needing to store and compute as many components as there are parameters in the model, as opposed to least squares methods that require storage of the square of the number of model parameters and computation of the cube. But the parameter update rules for the action- and state-value functions can be performed using any regression technique against the target $U_t$, including memory-based techniques. One such method, K-nearest neighbors regression, requires prediction order complexity $\\mathcal{O}(k\\times N_\\text{data}\\times N_\\text{features})$ using brute force, or $\\mathcal{O}(k\\times \\log(N_\\text{data}))$ using a k-d tree or ball tree data structure. Another, Gaussian process regression (GPR), requires prediction order complexity $\\mathcal{O}(N_\\text{data}^3)$, which is much higher. Techniques for reducing the order complexity of the GPR revolve around selecting subsets of the training points, such as the Nystrom approximation\\cite{nystrom}. \n\nFor direct policy optimization, however, we are required to craft a custom loss function, which limits our techniques to methods with explicitly-defined loss functions. For example, almost all policy gradient implementations found online use neural networks, which come with infrastructure for defining a custom loss function and computing its gradient, but any algorithm that performs stochastic gradient descent over a custom loss function will also work. In special cases, the gradients can be analytically derived and programmed explicitly. \n\nFor example, if we use a linear model for action preferences $h_{\\boldsymbol{\\theta}}(\\mathbf{s},\\mathbf{a})=\\boldsymbol{\\theta}^\\top\\mathbf{x}(\\mathbf{s},\\mathbf{a})$, where $\\mathbf{x}(\\mathbf{s},\\mathbf{a})$ represents our observations or features covering both the state and the action, we can use the softmax function to define the policy as \\begin{equation}\\pi_{\\boldsymbol{\\theta}}(\\mathbf{a}|\\mathbf{s})=\\frac{\\exp[\\beta h_{\\boldsymbol{\\theta}}(\\mathbf{s},\\mathbf{a})]}{\\sum_{\\mathbf{a}'\\sim\\mathcal{A}}\\exp[\\beta h_{\\boldsymbol{\\theta}}(\\mathbf{s},\\mathbf{a}')]}=\\frac{\\exp[\\beta \\boldsymbol{\\theta}^\\top\\mathbf{x}(\\mathbf{s},\\mathbf{a})]}{\\sum_{\\mathbf{a}'\\sim\\mathcal{A}}\\exp[\\beta \\boldsymbol{\\theta}^\\top\\mathbf{x}(\\mathbf{s},\\mathbf{a}')]}=S(\\beta \\boldsymbol{\\theta}^\\top\\mathbf{x})\\end{equation}which becomes the greedy policy as $\\beta\\rightarrow \\infty$. We can use the analytical expression to programmatically derive the gradient with respect to the weights $\\boldsymbol{\\theta}$, but it will be unique to each problem as it depends on the number of actions in the action space $\\mathcal{A}$. For more complex model loss functions, such as those for neural networks, it becomes unwieldy to programmatically define the gradients without a robust automation process. In these cases, Tensorflow, PyTorch, or a stand-alone automatic differentiator like Python's \\textit{autograd}, or its more-efficient successor \\textit{JAX}, can be used. These last two libraries work with regular functions that use a special imported version of \\textit{NumPy} to help it keep track of the operations, although it can only handle certain operations and patterns that the user must be aware of. Unfortunately, problems that require any degree of simulation, or are undifferentiable, are not well suited to analytical automatic differentiation, and instead need numerical automatic differentiation methods, which are far slower and therefore rarely used. One such solution, if the reader should pursue it, is to use the Python \\textit{numdifftools} to perform the differentiation. Fortunately for policy gradients, interaction with the environment tends to be greater overhead than calculating derivatives, since only one gradient step is taken at a time, so that \\textit{numdifftools} is viable solution.\n\nWhile it would be possible to employ function optimization methods like SciPy 's minimize package to optimize either policy evaluators or the policy itself, or to use any other non-gradient based function optimizer, these will drive the loss to its optimum over a fixed dataset and this can produce poor results. This is because, unlike in supervised learning, we do not have a fixed distribution of data, but rather the data distribution changes as the parameters change. We can see that this is the case since our actor and critic losses and their gradients are defined as expectations over trajectories produced by the policies we are optimizing, rather than sums over all possibilities. In fact, it is because we measure our policies through the data distribution that they return to us that we can slowly learn optimal policies without any reinforcement learning tricks at all. \n\nThis is a major achievement of both semi-gradient and policy gradient methods, since it brings us back to earth by allowing us to sample these expectations using the most recent policy, and with importance sampling, we can account for the data distribution drift in off-policy training. However, we must keep in mind that the loss functions we compute at any step of optimization are limited by the expectations over trajectories we had to compute using a different policy than the one we end up with because it didn't exist yet. In other words, our goal is not to optimize the loss at any given stage of training, but to optimize the rewards of the system, which the losses help us achieve at each step. \n\nWe can gain greater intuition for why we use gradient descent in training the policy function by observing that the similarity between the Policy Gradient Theorem and cross-entropy loss used to train models that predict normalized distributions can be used to build a \"ladder of certainty\" when developing your algorithm. For example, a policy function can be trained using targets defined by the softmax function over the $Q$-values of another model, and targets for all possible actions don't need to be trained simultaneously in the policy function. This procedure becomes equivalent to learning a softmax policy over the $Q$-value and can form a bridge to training against other targets like the advantage function. However, when we learn a softmax policy over $Q$-values, we aim to strictly minimize our loss as in traditional supervised machine learning, which will fail dramatically when using non-normalized targets, or when the learning $\\alpha$ is too large, since the loss will overflow to negative infinity.\n\nAs a result, one rarely sees solutions that don't use gradient descent or require the tuning of a learning rate $\\alpha$ hyperparameter. For exact, online methods, like Bayesian linear regression, we account for the changing data distribution of the loss function because each step of training amounts to adding new rows of data, so that we are ultimately minimizing or maximizing the average of the expectations over the history of the parameters, similar to stochastic gradient descent. It is up to numerical experiments to fine-tune $\\alpha$, or the decay factor we use to drop out old rows of data, to see how the different methods compare. When using neural networks with gradient descent, it is also important to scale the inputs to the standard normal where the model has the most capacity, and to be aware that the model interpolates linearly through gaps of data, and extrapolates linearly outside the domain of the standard normal. This can be addressed by either stalling training until a data sample has been made, or one can apply batch normalization\\cite{batch_normalization} as a regularization technique.\n\n\n\n\\section{Engineering considerations and pitfalls}\n\nFrom an engineering perspective, note that the formalism for training the state- and value-functions, regardless of the target $U_t$, is equivalent to the formalism for \\textit{contextual multi-armed bandits} (CMABs), and both align between versions with discrete and continuous actions. While both policy evaluator learning and bandits may train against targets compiled during a loop at the end of each episode that rewinds the previous actions, states, and rewards, only $Q$-learning incorporates the $\\argmax_\\mathbf{a}$ over future steps before it is reflected in the policy being trained thanks to the Bellman equation. This allows information about successful policies to propagate back to earlier steps before they are reflected in the policy used to collect data, making training in situations where planning is relevant much faster. Since we incorporate the unique qualities of $Q$-learning entirely during the collection loop that passes targets to the function approximator, we can use a CMAB under the hood for both training and prediction in the collection loop. Direct policy optimization, meanwhile, enjoys a similar structure, except with a different loss function that doesn't strictly fit the regression structure.\n\nJust as properties of the update algorithm made true gradient methods using the state- or action-value functions unstable, requiring the development of semi-gradient methods, certain combinations of algorithm properties can produce other instabilities. One particular combination, known as the \\textit{deadly triad}, must be particularly avoided:\n\\begin{enumerate}\n\\item Bootstrapping the target $U_t$, i.e., using any target besides Monte Carlo such as TD\n\\item Off-policy training\n\\item Non-linear function approximation\n\\end{enumerate}\nTo learn more, see Section 11.3 in \\cite{sutton_barto_rl}.\n\nUsing function approximation for both the policy evaluation and the policy itself necessarily implies partial observability of our environment, and while losing observations can kill our ability to learn a successful policy, the methods described here will do as well as we possibly can given the handicap. However, we can go the other way and add observations, too, and an obvious choice is to add historical data from earlier the trajectory. This can be accommodated by either encoding them as truncated and padded concatenations to the original feature vector, or by using sequential models like RNNs and LSTMs to encode the historical data into a fluid representation. In addition, there are a host of feature extraction, kernel approximation, convolution, and dimensionality reduction methods that can be employed, in parallel, serial, or along any directed acyclic graph (DAG) that defines your feature pipeline, as shown in Figure \\ref{fig:ml_map}. But remember, as we add more expressiveness and/or features to our models, that is, as we add more parameters, we increase our model capacity but also dramatically reduce the speed at which it learns. A general rule of thumb is that by using $M$ times more features, it takes $M^2$ more samples to move out of exploration into exploitation.\n\\begin{figure}\n\\label{fig:ml_map}\n \\begin{center}\n \\includegraphics[width=0.75\\linewidth]{ml_map.png}\n \\caption{Building a feature extraction and transformation pipeline can be a challenge, and this flowchart is a terrific way to explore your options. The online version provides links to implementations in the Python package \\textit{scikit-learn} \\footnotemark}\n \\end{center}\n\\end{figure}\n\\footnotetext{Source: \\url{https://scikit-learn.org/stable/tutorial/machine_learning_map/index.html}}\n\nMoreover, policy and action-value functions generally only work when we incorporate interaction features between the state and action features, except for decision-tree based algorithms like random forests and gradient-boosted trees, whereas neural networks are designed to learn which interactions are most useful at the cost of extensive hyperparameter tuning. Since there are so many ways to add features, we can bring down the complexity of these computations and to help our agents learn faster by applying randomized projections onto any subset of our feature space, but this also introduces additional hyperparameters to tune. We recommend randomized projections over Principle Component Analysis (PCA) since random projections to not require an input data sample before we can train our function approximators, and because there is no particular reason to expect the signal to be insensitive to the low-variance features that are filtered out in PCA. Other options for dimensionality reduction include manifold learning, which also require data sampling, and for which there are many options, including VAE, t-SNE, and spectral embedding. See \\url{https://scikit-learn.org/stable/modules/manifold.html#spectral-embedding} for a list of well-implemented options.\n\nInterestingly, there is a parallel for randomized projections in the kernel space for problems that are better served that way. As covered in \\cite{GP_BLR}, we can project our inputs into basis functions that allow us to use a linear regression to approximate an equivalent Gaussian. This allows us to perform regressions in the feature space, for example, by using Bayesian linear regression, with order complexity $\\mathcal{O}(N_\\text{basis})$ for gradient methods and $\\mathcal{O}(N_\\text{basis}^3)$ for exact methods, as opposed to Gaussian process regression with order complexity $\\mathcal{O}(N_\\text{data}^3)$, all while approximating the kernel regression. Popular choices here include the Nystrom approximation, which amounts to using the top $N$ principle components of the kernel function evaluated between each data point by simply dropping data points at random, and Random Fourier Features.\n\nWhen computing the $\\argmax_\\mathbf{a}$ for $Q$-learning style algorithms, we may run into difficulties for problem spaces with continuous or large action spaces. However, we have several options at play. The first is to simply sample the action space randomly and rank the predicted $Q$-values for all of them. This is a popular approach in derivative-free Bayesian optimization. If the action space is continuous, another option is to apply gradient ascent, but this can be computationally expensive. And lastly, we can encourage exploration by using Thompson sampling when sampling for the $\\argmax$, as in \\cite{RLSVI}.\n\n\\section{Additional resources}\nAdditional resources can be found below:\n\\begin{itemize}\n\\item RLlib from Ray covers different algorithms and whether they can accept discrete or continuous actions, as well as additional support such as sequential models, multi-agent implementations, and multi-GPU training. See \\url{https://docs.ray.io/en/latest/rllib-algorithms.html}\n\\item Lilian Wang has created one of the most thorough overview of policy gradient based algorithms at \\url{https://lilianweng.github.io/lil-log/2018/04/08/policy-gradient-algorithms.html}\n\\item Spinning Up from OpenAI offers a terrific overview of reinforcement learning concepts as well as deep dives (and implementations) of some of the more advanced algorithms such as Proximal Policy Optimization (PPO), Deep Deterministic Policy Gradient (DDPG), and Soft Actor-Critic (SAC). However, some of their explanations, such as around the policy gradient, can be obscure and needlessly confusing. See \\url{https://spinningup.openai.com/en/latest/}\n\\item David Silver's lecture slides at University College London are a legendary resource for succinct coverage of the main concepts in reinforcement learning. See \\url{https://www.davidsilver.uk/teaching/}\n\\item Julien Vitay provides a sprawling overview of deep reinforcement learning at \\url{https://julien-vitay.net/deeprl/Introduction.html}\n\\end{itemize}\n\n\\printbibliography\n\\end{document}\n", "meta": {"hexsha": "dd60f98a31a0749ac2dc9e129c9dc0e13155f30a", "size": 72408, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "main.tex", "max_stars_repo_name": "KoyoteScience/real-world-reinforcement-learning", "max_stars_repo_head_hexsha": "120e071a27dd0ad6b1cf05bd18a8f6b4d8d82f32", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "main.tex", "max_issues_repo_name": "KoyoteScience/real-world-reinforcement-learning", "max_issues_repo_head_hexsha": "120e071a27dd0ad6b1cf05bd18a8f6b4d8d82f32", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "main.tex", "max_forks_repo_name": "KoyoteScience/real-world-reinforcement-learning", "max_forks_repo_head_hexsha": "120e071a27dd0ad6b1cf05bd18a8f6b4d8d82f32", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 182.3879093199, "max_line_length": 3591, "alphanum_fraction": 0.7604546459, "num_tokens": 19707, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.34226221457390893}} {"text": "\\section{An Extended Model}\n\nAfter asserting that a deep reinforcement learning agent can navigate the environment, the model is extended to address the issues with the simple model. The first issue that should be addressed is that women should not permanently withdraw from the labour force. Secondly, the log-function used for the sub-utilities caps the utility very hard, which implies that very small perturbations of $\\beta_L$ makes the agent substitute into one of the extremes. The agent end up working either 45 hours exclusively or working 0 hours exclusively, de facto making leisure and consumption perfect substitutes. The agent should ideally be more balanced, implying the functional form of the sub-utilities should have a steeper curvature. This leads to a new expression of the utility function:\n\n\\begin{equation}\n U_t = \\beta_L L_{t}^\\zeta + \\beta_Y Y_t^{\\zeta}\n\\end{equation}\n\nThe fertility process should also be extended. I now assume that each household can have a maximum of four children. The environment tracks how many children the family has, and the associated age. In each period the number of children in the household can potentially increment by one.\n\n\\begin{equation}\n K_{t+1} = K_t+ \\psi_t, \\qquad \\psi_t \\mid Q_t \\sim Bernoulli (p_\\psi(Q_t))\n\\end{equation}\n\nI let $B$ be a vector of size 4 equal to the maximum number of children. Furthermore, I let the vector $C$ denote the individual age of the children also of size 4. This implies that for index $i$ of vector $B$ corresponds to the $k$'th child, index $i$ of vector $C$ implies the same child's corresponding age.\n\n\\begin{equation}\n B_t[i] = \\begin{cases}\n 1 & \\text{if } i \\leq K_t \\\\\n 0 & \\text{else}\n \\end{cases}\n\\end{equation}\n\nThe age vector is defined as:\n\n\\begin{equation}\n C_{t+1} = C_t + B_{t+1}\n\\end{equation}\n\nThe formulation ensures the age of a child increments if and only if the child is born.\n\nFinally, the leisure component of the model needs to be adjusted to take into account the number of hours used per child. Let $J$ denote the number of hours used on children. Children below age 3 takes 10 hours per week, children in the age 3 up to age 15 takes 3.5 hours per week, and children above does not reduce the leisure time of women, using the numbers found by \\textcite{ekert-jaffe_time_2015}. \n\n\\begin{equation}\n J_t = \\sum_{i=1}^4 \\begin{cases}\n 0.0 & \\text{if } B_t[i] = 0 \\\\\n 10.0 & \\text{if } B_t[i] = 1 \\land C_t[i] < 3 \\\\\n 3.5 & \\text{if } B_t[i] = 1 \\land 3 \\leq C_t[i] < 16 \\\\\n 0 & \\text{if } B_t[i] = 1 \\land 16 \\leq C_t[i] \n \\end{cases}\n\\end{equation}\n\nAnother thing to address is, that empirically women tend to work less when they are young (in the age of 18 to 30), gradually increasing the number of working hours as shown in figure \\ref{fig:dqi_model1_average_path_sim_vs_empirical}. Looking to figure \\ref{fig:educ_empirical}, it could appear to be a consequence of women doing full time education, not having the time to work (more than a few hours a week). To address this problem I add an education variable to the model. In this setup the education does not add to higher salary instead it only goes in the model as additional time spent. Education is assumed to be exogenous, and over the life cycle the women will with some probability be done with education, not to return ever again to the education system. Using \\textbf{FOLK1A} from Statistics Denmark I find the total number of women in a given age group. The total number of women for a given age group doing full time education is found in the source \\textbf{UDDAKT10} also from Statistics Denmark. I fit this with a linear regression and let the number of women doing full time education decrease linearly from age 18 to 30. This can be seen in Figure \\ref{fig:prob_educ_full_time}. I model the education the following way:\n\n\\begin{figure}[ht]\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{figures/total_women_vs_education.png}\n \\caption{FTE vs. Total Number of Women}\n \\label{fig:educ_empirical}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n \\centering\n \\includegraphics[width=1\\linewidth]{figures/prop_women_doing_full_time_education.png}\n \\caption{Fraction of Women in FTE vs. Linear Fit}\n \\label{fig:prob_educ_full_time}\n\\end{subfigure}\n \\caption{Women and Full Time Education (FTE)}\n \\label{fig:educ_women}\n\\end{figure}\n\n\\begin{equation}\n E_{t+1} = E_{t} \\cdot \\iota_t, \\qquad \\iota_t \\mid Q_t \\sim Binomial(p_{\\iota}(Q_t))\n\\end{equation}\n\nSo in each period with some probability $E^{prob}_t \\equiv p_{\\iota}(Q_T)$ the women will end full time education. Both $E$ and $E^{prob}_t$ is part of the state space. Since women do full time education, I assume time used is 37 hours per week.\n\nIt is assumed for simplicity that human capital accumulation does not depreciate or accumulate when the agent is under full time education:\n\n\\begin{equation}\n G_{t+1} = \n \\begin{cases}\n G_t(1 - \\delta) + \\frac{H_t}{37}, & \\text{if } E_t > 0 \\\\\n G_t & \\text{else}\n \\end{cases}\n\\end{equation}\n\n\nFinally, note that the utility is calculated on a weekly basis instead of on an annual basis. This is again done, such that the sub utilities, do not squeeze the values too hard, as was the case in model 1, making estimating of the model very hard, due to small perturbations of $\\beta_L$ having too large effect. Compared to the original model $f^M(Q_t)$ now represents the total income per week of the husband rather than the annual total income.\n\n\\begin{equation}\n L_t = 24 \\cdot 7 - H_t - J_t - E_t \\cdot 37\n\\end{equation}\n\nIn Denmark education is coupled with a transfer from the state. This transfer is added as well, such that the household will receive a transfer of $tr = \\frac{6000 DKK \\times 12}{52} \\approx 1400 DKK$ each week from the government, while the women is studying. This ratio is approximately doubled if the women gets a child during the education \\parencite{noauthor_satser_nodate-1}. The total income of the household is equal to:\n\n\\begin{equation}\n Y_t = H_t \\cdot W_t + f^M(Q_t) + E_t (tr + \\mathbf{1} \\{ K_t > 0 \\} \\cdot tr) \n\\end{equation}\n\n\nSummarizing the model; using a lot of the framework from the simple model a couple of extensions is made. The state space is now of 14 dimensions containing $(Q)$ age, $(G)$ human capital, $(Z)$ idiosyncratic wage path, $(K)$ the number of children in at time $t$ in the household, a vector of size four $(B)$ corresponding to the number of children the household contain, a vector of size four $(C)$ containing the age of the individual children, a dummy $(E)$ that indicates if the women is under education, and lastly $(E^{prob})$ a number between 0 and 1 that indicates the probability of being enrolled in full time education next period. The action space is slightly tweaked to contain $\\{ 0, 15, 25, 37\\}$ representing the discreet choices of hours the woman can choose to work. This formulation is closer to that of \\textcite{francesconi_joint_2002}, where he only considers the choices: \\textit{no work}, \\textit{part-time}, \\textit{full-time}. In contrast with \\textcite{francesconi_joint_2002} this model distinguishes between two types of part-time work; 15 and 25 hours a week. The model specification is:\n\n\\begin{equation}\n \\textbf{State space: }\\statespace = \\R^2 \\times \\{0, 1, 2, 3, 4\\} \\times \\{18, 19, \\cdots , 60\\} \\times \\{0, 1\\}^4 \\times \\{0, 1, \\cdots, 18\\}^4 \\times \\{0,1\\} \\times [0, 1]\n\\end{equation}\n\n\\begin{equation}\n \\textbf{Action space: }\\actionspace = \\{0, 15, 25, 37\\}\n\\end{equation}\n\nAdditionally, the model contains the following parameters: $\\alpha=4.609, \\eta_G = 0.164, \\eta_{G^{2}}=0.015, \\delta=0.209, \\sigma_\\epsilon = 15.11, \\beta_L, \\beta_Y = 1, W_{min} = 120, tr = 1600, \\zeta=0.5$, using the values of the Mincer equation from the parameter calibration performed earlier! A recursive formulation of the model is presented below:\n\n\\begin{align}\n U_t(L_t, Y_t) &= \\beta_L L_t^{\\zeta} + \\beta_Y Y_t^{\\zeta}\\\\\n L_t(K_t, H_t, J_t, E_t) &= 24 \\cdot 7 - H_t - J_t - E_t \\cdot 37\\\\\n \\log \\tilde{W}_t (G_t) &= \\alpha + \\eta_G G_t + \\eta_{G^2} G_t^2 \\\\\n W_t(\\tilde{W}_t, Z_t) &= \\max(W_{min} , \\tilde{W}_t + Z_t) \\\\\n Y_t(Q_t,H_t, W_t, E_t, K_t) &= 46 \\cdot H_t \\cdot W_t + f^M(Q_t) + E_t (tr + \\mathbf{1} \\{ K_t > 0 \\} \\cdot tr)\\\\\n J_t (B_t, C_t) &= \\sum_{i=1}^4 \\begin{cases}\n 0.0 & \\text{if } B_t[i] = 0 \\\\\n 10.0 & \\text{if } B_t[i] = 1 \\land C_t[i] < 3 \\\\\n 3.5 & \\text{if } B_t[i] = 1 \\land 3 \\geq C_t[i] < 16 \\\\\n 0.0 & \\text{if } B_t[i] = 1 \\land 16 \\geq C_t[i] \n \\end{cases} \\\\\n B_t[i] (K_t) &= \\begin{cases}\n 1 & \\text{if } i \\leq K_t \\\\\n 0 & \\text{else}\n \\end{cases}\n\\end{align}\n\nLaw of motion:\n\n\\begin{align}\n Q_{t+1}(Q_t) &= Q_t \\\\\n K_{t+1}(K_t, Q_t) &= K_{t} + \\psi_t, \\qquad \\psi_t \\mid Q_t \\sim Bernoulli(p_\\psi(Q_t)) \\\\\n Z_{t+1}(Z_t) &= Z_t + \\epsilon_t, \\qquad \\epsilon_t \\sim \\ndist(0, \\sigma_\\epsilon)\\\\\n G_{t+1}(G_t, H_t, E_t) &= \n \\begin{cases}\n G_t(1 - \\delta) + \\frac{H_t}{37}, & \\text{if } E_t > 0 \\\\\n G_t & \\text{else}\n \\end{cases} \\\\\n C_{t+1}(C_t, B_{t+1}) &= C_{t} + B_{t+1} \\\\\n E_{t+1}(E_t, Q_t) &= E_t \\cdot \\iota_t, \\qquad \\iota_t \\mid Q_t \\sim Binomial(p_\\iota(Q_t))\n\\end{align}\n\n\n", "meta": {"hexsha": "c97efe43dcd92e4b0e51813374a8e6fd41bbc410", "size": 9371, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "chapters/model2.tex", "max_stars_repo_name": "JakartaLaw/speciale", "max_stars_repo_head_hexsha": "95d89c281b9d8f73065a823cba97a5bedcbf129d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "chapters/model2.tex", "max_issues_repo_name": "JakartaLaw/speciale", "max_issues_repo_head_hexsha": "95d89c281b9d8f73065a823cba97a5bedcbf129d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "chapters/model2.tex", "max_forks_repo_name": "JakartaLaw/speciale", "max_forks_repo_head_hexsha": "95d89c281b9d8f73065a823cba97a5bedcbf129d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 67.9057971014, "max_line_length": 1241, "alphanum_fraction": 0.702379682, "num_tokens": 2904, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5926665999540698, "lm_q2_score": 0.5774953651858118, "lm_q1q2_score": 0.34226221457390893}} {"text": "% ****** Start of file response.tex ****** %\n\\documentclass{article}\n\n%===============================================================================\n% Import packages\n%===============================================================================\n\n\\usepackage[top=0.8in,bottom=1in,left=1in,right=1in]{geometry}\n% Physics package\n\\usepackage{physics}\n% Paragraph skip package\n\\usepackage[parfill]{parskip}\n% AMS packages\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{mathtools}\n% Color package\n\\usepackage[dvipsnames]{xcolor}\n% Acronym package\n\\usepackage[acronym]{glossaries}\n% SI units package\n\\usepackage{siunitx}\n% Hypertext package\n\\usepackage[hidelinks]{hyperref}\n% Bold math\n\\usepackage{bm}\n% Other packages\n\\usepackage{graphicx}\n\\usepackage{textcomp}\n\\usepackage{float}\n\\usepackage{cite}\n\\usepackage{multicol}\n\\usepackage{multirow}\n% \\usepackage{kpfonts}\n\n% Bibliography style\n\\usepackage[numbers,sort&compress]{natbib}\n\\bibliographystyle{apsrev4-1}\n\n\\begin{document}\n\n% Define acronyms\n\\newacronym{tls}{TLS}{two-level system}\n\n%===============================================================================\n% Add date\n%===============================================================================\n\\today\\\\\n\nAshot Melikyan,\\\\\nAssociate Editor,\\\\\nPhysical Review B.\\\\\n\nDear Professor Melikyan,\n\nThank you very much for your effort in managing the review process of our manuscript. We are also thankful for the reviewer comments, and believe the second version of the manuscript we submit herewith has been significantly improved by the constructive criticism we received. Additionally, the subsequent sections of this document discuss the reviewer comments and our responses to them.\n\nPlease note that in the following sections, the statements in {\\color{RoyalBlue} \\textbf{blue}} are the comments of the reviewers. Our responses are shown in black letters, and the modifications we have done to the manuscript are given in {\\color{Red} red}.\n\n\\subsection*{General changes to the manuscript}\n\n\nWe have made some minor changes in language and presentation to improve the clarity and organization of the manuscript.\n\\begin{itemize}\n \\item Section I - seventh paragraph (page 2): \\\\\n {\\color{Red} In Sec. VII, we discuss the physical significance of our theoretical results and their possible use in future nanoelectronic devices.\n Finally, we summarize our findings and present our conclusions in Sec. VIII.}\n \\item Section VI - fourth paragraph (page 8): \\\\\n {\\color{Red} Considering the effects of the applied dressing field on the longitudinal conductivity of 2DEG, we can identify that the dressing field has sharpened the conductivity peaks.}\n \\item Section VIII - first paragraph (page 10): \\\\\n {\\color{Red} Finally, we derived analytical expressions for the diagonal components of the electric conductivity tensor concerning the 2DEG quantum Hall system operates at low temperatures.}\n\\end{itemize}\n\n\\subsection*{Response to the comments of Reviewer 1}\n\nWe would like to thank the reviewer for bringing the deficiencies of our manuscript to our attention and providing constructive feedback to improve the quality of our work. We have considered all of your suggestions seriously and revised our manuscript as described below.\n\n\\subsubsection*{Comment 1 -\n\\color{RoyalBlue} My concern is that the manuscript is heavily skewed towards a purely mathematical formulation of the problem. It has a minimal connection to realistic two-dimensional electron systems. The manuscript does not discuss how the results can be applied to understanding mechanisms of charge transport in nanoelectronic devices and can be used to optimize device performance. Without such discussion, the manuscript will have a minimal impact on the community working on developing nanoelectronics.}\n\nWe agree that our work has a theoretical bias, and the reader may benefit from some application perspective from our theoretical results to characterize or design nanoelectronic devices. Therefore, we have discussed the physical significance of our theoretical results and their possible use in optimizing nanoelectronic device performance. First, we added a detailed comparison between our theoretical results and experiment observations when no radiation is present. We also added a new section (Section VII) on this very aspect.\n\n\\begin{itemize}\n \\item Section VI - third paragraph (page 8):\\\\\n {\\color{Red}\n By comparing the theoretical \\cite{ando72,ando74_1,ando74_2,ando74_3,ando74_4,ando82,endo09} and experimental \\cite{endo09,wakabayashi78,ochiai190,mancoff96,arapov02,grbic04,caviglia10} studies on the magnetoresistance of 2DEG quantum Hall systems when no radiation is present against our results, we can identify that longitudinal conductivity oscillations in Fig. 4 are a repetition of the Shubnikov–de Haas(SdH) oscillations.\n As observed in the experimental work of Caviglia \\textit{et al.} \\cite{caviglia10}, the period of SdH oscillations depends only on the perpendicular component of the magnetic field to the plane of the 2DEG.\n Therefore, we can identify that many experiments on different types of 2DEGs at low-temperatures \\cite{endo09,ochiai190,mancoff96,arapov02,grbic04,caviglia10} have analyzed these SdH oscillations against the applied magnetic field's amplitude.\n The cyclotron frequency ($\\omega_0$) of the system depends on the magnetic field's perpendicular component to the plane of the 2DEG.\n The gate voltage modifies the Fermi level of the system.\n Since the Landau level energy is only dependent on the cyclotron frequency, this gate voltage variation under a constant magnetic field amplitude generates the same SdH oscillations.\n Interestingly, this oscillatory behavior against applied gate voltage has been observed experimentally by Wakabayashi \\textit{et al.} \\cite{wakabayashi78} in an inversion layer on a silicon surface at low temperature.\n By comparing these observations against our results in Fig. 4, we can identify that our oscillations also show the same characteristic behavior; i.e $\\hbar\\omega_0$ periodic SdH oscillations against the Fermi energy.\n Furthermore, when we increase the gate voltage, the Fermi level rises. As illustrated in Fig. 4, this will result in a higher conductivity peak value at the higher-order Landau levels. This same behavior was also observed in the experimental observations presented in Ref. \\cite{wakabayashi78}.\n In our work, we provide analytical results describing the controllability of these conductivity regions using a dressing field.\n }\n \\item Section VII (page 9):\\\\\n {\\color{Red}\n \\subsection*{VII. PHYSICAL SIGNIFICANCE OF THE OUTCOMES}\n\n With the realization of 2DEGs in Si-MOSFETs (Metal Oxide Semiconductor Field Effect Transistors) \\citep{fowler66}, Klitzing \\textit{et al.} \\cite{klitzing80} made the first transport measurements on such systems to reveal the quantum Hall effect. The empirical discovery of these unusual properties marked the beginning of a whole new realm in condensed matter physics that continues to produce phenomenal advancements in electronic systems. The quantum Hall effects in a 2DEG under a static magnetic field are described by plateaus quantized to integer values of the conductivity quantum ($\\flatfrac{e^2}{\\hbar}$) in the off-diagonal conductivity, with simultaneous peaks at inter-plateau transition for the diagonal conductivity \\cite{endo09}. This is due to the applied magnetic field and it changes the energy spectrum of 2DEG dramatically. The magnetic field causes the density of states in 2DEG to split up into a sequence of delta functions, separated by an energy $\\hbar\\omega_0$, with $\\omega_0$ the cyclotron frequency which depends on the applied magnetic field.\n However, experimental results demonstrate that these Landau levels are broadened and the main source of these broadening at low temperatures is the disorders in materials \\cite{ando85,dial07}. The broaden sequence of delta functions of the density of states implies the oscillating behavior in the experimental measurements of longitudinal conductivity which is known as SdH oscillations \\cite{endo09,wakabayashi78}.\n\n Our theoretical analysis on longitudinal conductivity behavior of dressed quantum Hall system was developed by considering low-temperature limit with Gaussian impurity broadening assumptions.\n As illustrated in Fig. 4, we can describe the experimentally observed SdH oscillation results in Refs.\\cite{endo09,wakabayashi78} through our model.\n Under the undressed condition, our results overlap with the conductivity measurement of quantum Hall systems \\cite{endo09}. Strikingly, we show in Fig. 5 that we can manipulate the broadening of these conductivity peaks using an external dressing field. At low temperatures, the principal cause of broadening of these conductivity peaks is impurity-induced scattering. Using an external dressing field, we can suppress the impurity-induced scattering and this results a reduction of both the scattering-induced broadening and the longitudinal conductivity peaks.\n\n Research on novel states of matter has driven the evolution of present-day nanoelectronic devices. In particular, controllable manipulation of material properties through a gate electric field has revolutionized the development of material science and technology \\cite{ahn03,deng18}.\n The charge carrier concentration of a system is an imperative parameter that defines its conductivity properties.\n As we can manipulate the charge carrier concentration using an electrostatic field-effect, we exert significant control over the conductivity externally.\n A 2DEG under static magnetic field with quantum Hall effects is an excellent example that illustrates how the gate electric field controls conductivity.\n We also observed that a considerable number of studies exist on charge transport in the quantum limit that employed different types of 2D field-effect transistors (FETs) subjected to magnetic fields \\cite{wakabayashi78,yang18,long20,li14}. Yang \\textit{et al.} \\cite{yang18} study shows quantized Hall plateaus and SdH oscillations for longitudinal conductivity against gate voltage in black phosphorus FET under static magnetic fields at low temperatures. Since the Fermi level of a system can be altered with the applied gate voltage, our Fig. 4 depicts this result.\n In particular, our analysis shows that we can manipulate the broadening of the conductivity regions using an external dressing field. Although Yang \\textit{et al.} \\cite{yang18}, achieved broadening in longitudinal conductivity peaks by changing the temperature in a low range, in this study we presented a general theory on manipulating longitudinal conductivity broadening using a high-intensity electromagnetic field.\n\n The realization of the underlying mechanism of 2D FETs in the quantum realm promises its potential in next-generation nanoelectronic applications. In a particular application that uses the switching mechanism of the above-discussed FETs with quantum Hall effects, we can achieve high and low output conductivities by changing the input gate voltage. As a result of manipulating the broadening of conductivity regions, we can limit the broadening of conductivity peaks around Landau levels using a dressing high-intensity electromagnetic field. This will enhance the sensitivity of FETs which provides the ability to observe narrow changes in gate voltage.\n Furthermore, adopting the mechanism presented in Ref. \\cite{hirakawa01}, we can manipulate conductivity peaks into very sensitive, narrowband high-frequency radiation detectors.\n We envision these advances in nanoelectronics from our theoretical model and knowledge generated from the insight it provides to the underlying dependencies.\n Furthermore, this theoretical model will aid in the development of simulation tools that will design the quantum effects in\n magneto-transport properties of 2D nanostructures.\n }\n\\end{itemize}\n\n\\subsubsection*{Comment 2 -\n\\color{RoyalBlue} Moreover, this current research direction has a significant overlap with previous experimental and theoretical studies of quantum Hall systems that started with the observation of zero-resistance states in high mobility systems [Zudov et al, Phys. Rev. B 64, 201311 (2001), Mani et al, Phys. Rev. Lett. 92, 146801 (2004)] and gave rise to theoretical models for the phenomenon [Durst et al, Phys. Rev. Lett. 91, 086803 (2003), Dmitriev et al., Phys. Rev. B 71, 115316 (2005), Dmitriev et al, Phys. Rev. B 80, 165327 (2009)]. The present manuscript needs to connect to various known phenomena discussed earlier in the literature on quantum Hall systems.}\n\nWe agree with the reviewer that it is useful to have a discussion comparing our theoretical model and previous work on the transport properties of quantum Hall systems.\nAmong the mentioned studies we can identify that experimental work reported in Refs. \\cite{zudov01,mani02,zudov03,mani04} are specifically aimed at the unusual oscillations of the magnetoresistance induced by the microwave (millimeter-wave) radiation in 2DEG quantum Hall systems.\nThese oscillations are known as \\textit{microwave-induced resistance oscillations} (MIROs).\nTo describe these behaviors Durst \\textit{et al.} \\cite{durst03} introduced a simple theoretical model assuming that the experimentally observed oscillations are a consequence of photoexcited disorder-scattered electrons.\nHowever, later Refs. \\cite{dmitriev03,dmitriev05,dmitriev09} proposed a novel model considering the changes were made on\nthe electron distribution function by the microwave field.\nThese more generalized models \\cite{dmitriev03,dmitriev05,dmitriev09} have successfully described the behavior of MIROs at experimentally relevant temperatures that were missed in the previous model \\cite{durst03}.\nFurthermore, we can recognize that the underlying mechanism of all these models \\cite{durst03,dmitriev03,dmitriev05,dmitriev09} is microwave photon absorption by an electron in the associated system. In our theoretical model, we take into account higher frequencies than microwaves as the dressing electromagnetic field.\nTherefore we can identify and quantify significant dissimilarities between our system and MIRO systems \\cite{zudov01,mani02,zudov03,mani04,durst03,dmitriev03,dmitriev05,dmitriev09}. They are:\n\\begin{itemize}\n \\item\n the experiments on magnetoresistance oscillations on 2DEG quantum Hall systems \\cite{zudov01,mani02,zudov03,mani04} were performed in the microwave frequency range ($30 -\\SI{150}{\\giga\\hertz}$). This leads to building the theoretical models presented in Refs. \\cite{durst03,dmitriev03} by assuming that these oscillations are caused by photoexcited electrons. Since these models consider on a relevant frequency range (microwave radiation), it will allow us to acknowledge the photon absorption by electrons. In contrast to that, our consideration is only focused on the systems with high-frequency range dressing fields which will not be associated with any photon absorption.\n \\item\n the applied microwave radiation power on the MIRO experiments \\cite{mani02,zudov03} varies around the $\\SI{1}{\\milli\\watt\\per\\square\\centi\\metre}$ range. However, in our analysis, we take the dressing field as a high-intensity electromagnetic field, where we cannot consider the dressing field as a perturbation in our analysis. This leads us to recognize the formation of Floquet states together with conventional Landau levels. As we mentioned in our results, we used dressing high-intensity electromagnetic fields of magnitude around the $\\SI{100}{\\watt\\per\\square\\centi\\metre}$ range in our numerical calculations.\n \\item\n these fascinating MIROs are only observed under the influence of weak magnetic fields ($B < \\SI{0.2}{\\tesla}$) in experiments performed in Refs. \\cite{zudov01,mani02,zudov03,mani04}. In this range of weak magnetic fields, we can only observe MIROs as Shubnikov–de Haas (SdH) oscillations reveal only with high-intensity magnetic fields. In comparison to our analysis, we are interested in the SdH oscillations and manipulation of their characteristics. Therefore in our analysis, we aim at the effects induced by higher intensity magnetic fields ($B \\sim \\SI{1}{\\tesla}$) than the MIRO experiments.\n \\item\n in our work, we analyzed the 2DEG quantum Hall system with a dressing high-intensity electromagnetic field which does not contribute to energy exchange between the high-frequency dressing field and electrons. Therefore we have assumed the applied electromagnetic radiation as a pure dressing field. There are two possible absorption mechanisms in a 2DEG quantum Hall system; namely electron transitions between distinct Landau levels and electron transitions between distinct states of the same broadened Landau level. To avoid these absorptions, we have tuned the dressing field into a high-frequency range in our system under the analysis. Furthermore, due to the dressing high-intensity electromagnetic field, the entire electronic states of the conventional 2DEG quantum Hall system will be restructured. We have addressed these modifications through the Floquet theory. However, the MIRO models \\cite{durst03,dmitriev03} are based on low-frequency cases where the system can absorb low-frequency photons from the field.\n\\end{itemize}\n\nBased on the above discussion, we can identify that the high-frequency and low-frequency illumination of 2DEG quantum Hall systems leads to two distinguishable\nmagneto-transport scenarios.\n\n\\begin{itemize}\n \\item Section I - fifth paragraph (page 1):\\\\\n {\\color{Red}\n Recently, experimental research on the effects of microwave illumination of 2DEG systems revealed microwave-induced resistance oscillations (MIROs) under weak magnetic fields \\cite{zudov01,mani02,zudov03,mani04}.\n This inspired investigations on the theoretical description of MIROs, and several semiclassical and quantum kinetic equation formalisms have been proposed to address the underlying mechanism of MIROs \\cite{durst03,dmitriev03,dmitriev05,dmitriev09}. These formalisms provide a proper explanation for the experimental observations of MIROs. However, these experimental and theoretical works have been linked to photon absorption from low-frequency (microwave) electromagnetic fields.\n In contrast to that, high-frequency external illumination on a 2DEG quantum Hall system needs to be studied as a pure dressing (nonabsorbable) field.\n The influence induced by a pure dressing field on\n magneto-transport properties of 2DEG quantum Hall system needs to be described by a non-absorption mechanism, and it has escaped the researchers’ attention before.\n Lately, Dini \\textit{et al.} \\cite{dini16} have investigated the one-directional conductivity behavior of dressed quantum Hall system subjected to a high-frequency field.\n }\n \\item Section VI - sixth paragraph (page 9):\\\\\n {\\color{Red}\n Lately, several experimental \\cite{zudov01,mani02,zudov03,mani04} and theoretical \\cite{durst03,dmitriev03,dmitriev05,dmitriev09} studies have uncovered various remarkable\n magneto-transport properties induced by microwave radiation on 2DEG quantum Hall systems. However, these experiments and theoretical models only analyzed the behavior of MIROs in 2DEG systems which are based on photon absorption from the applied low-frequency field.\n It is important to state that the difference between our SdH oscillations and MIROs \\cite{zudov01,mani02,zudov03,mani04} by considering the frequency range of the applied electromagnetic field. In our case, the applied dressing field is in the off-resonant range, and it will only change the broadening of Landau levels, but avoid any contribution towards the photon absorptions. This will clearly describe the crucial difference between low-frequency illumination and high-frequency illumination effects on 2DEG quantum Hall systems. Therefore, our theoretical analysis can help to fill the gap in knowledge of the high-frequency dressing field effects on 2DEG quantum Hall systems.\n }\n\\end{itemize}\n\n\\subsubsection*{Comment 3 -\n\\color{RoyalBlue} The manuscript will also provide more impact if it demonstrates how the new results can help to improve the future development of nanoelectronic devices. After these questions are addressed, the manuscript will be suitable for publication in Physical Review B. Otherwise, it will fit better to a more mathematically oriented journal.\n}\n\nThis is a valid comment. As we mentioned in comment 1, considering the importance of discussing the physical significance of our results, we included a new section (Section VII) to address this comment.\n\n\\begin{itemize}\n \\item Section VII (page 8):\\\\\n {\\color{Red}\n \\subsection*{VII. PHYSICAL SIGNIFICANCE OF THE OUTCOMES}\n\n With the realization of 2DEGs in Si-MOSFETs (Metal Oxide Semiconductor Field Effect Transistors) \\citep{fowler66}, Klitzing \\textit{et al.} \\cite{klitzing80} made the first transport measurements on such systems to reveal the quantum Hall effect. The empirical discovery of these unusual properties marked the beginning of a whole new realm in condensed matter physics that continues to produce phenomenal advancements in electronic systems. The quantum Hall effects in a 2DEG under a static magnetic field are described by plateaus quantized to integer values of the conductivity quantum ($\\flatfrac{e^2}{\\hbar}$) in the off-diagonal conductivity, with simultaneous peaks at inter-plateau transition for the diagonal conductivity \\cite{endo09}. This is due to the applied magnetic field and it changes the energy spectrum of 2DEG dramatically. The magnetic field causes the density of states in 2DEG to split up into a sequence of delta functions, separated by an energy $\\hbar\\omega_0$, with $\\omega_0$ the cyclotron frequency which depends on the applied magnetic field.\n However, experimental results demonstrate that these Landau levels are broadened and the main source of these broadening at low temperatures is the disorders in materials \\cite{ando85,dial07}. The broaden sequence of delta functions of the density of states implies the oscillating behavior in the experimental measurements of longitudinal conductivity which is known as SdH oscillations \\cite{endo09,wakabayashi78}.\n\n Our theoretical analysis on longitudinal conductivity behavior of dressed quantum Hall system was developed by considering low-temperature limit with Gaussian impurity broadening assumptions.\n As illustrated in Fig. 4, we can describe the experimentally observed SdH oscillation results in Refs.\\cite{endo09,wakabayashi78} through our model.\n Under the undressed condition, our results overlap with the conductivity measurement of quantum Hall systems \\cite{endo09}. Strikingly, we show in Fig. 5 that we can manipulate the broadening of these conductivity peaks using an external dressing field. At low temperatures, the principal cause of broadening of these conductivity peaks is impurity-induced scattering. Using an external dressing field, we can suppress the impurity-induced scattering and this results a reduction of both the scattering-induced broadening and the longitudinal conductivity peaks.\n\n Research on novel states of matter has driven the evolution of present-day nanoelectronic devices. In particular, controllable manipulation of material properties through a gate electric field has revolutionized the development of material science and technology \\cite{ahn03,deng18}.\n The charge carrier concentration of a system is an imperative parameter that defines its conductivity properties.\n As we can manipulate the charge carrier concentration using an electrostatic field-effect, we exert significant control over the conductivity externally.\n A 2DEG under static magnetic field with quantum Hall effects is an excellent example that illustrates how the gate electric field controls conductivity.\n We also observed that a considerable number of studies exist on charge transport in the quantum limit that employed different types of 2D field-effect transistors (FETs) subjected to magnetic fields \\cite{wakabayashi78,yang18,long20,li14}. Yang \\textit{et al.} \\cite{yang18} study shows quantized Hall plateaus and SdH oscillations for longitudinal conductivity against gate voltage in black phosphorus FET under static magnetic fields at low temperatures. Since the Fermi level of a system can be altered with the applied gate voltage, our Fig. 4 depicts this result.\n In particular, our analysis shows that we can manipulate the broadening of the conductivity regions using an external dressing field. Although Yang \\textit{et al.} \\cite{yang18}, achieved broadening in longitudinal conductivity peaks by changing the temperature in a low range, in this study we presented a general theory on manipulating longitudinal conductivity broadening using a high-intensity electromagnetic field.\n\n The realization of the underlying mechanism of 2D FETs in the quantum realm promises its potential in next-generation nanoelectronic applications. In a particular application that uses the switching mechanism of the above-discussed FETs with quantum Hall effects, we can achieve high and low output conductivities by changing the input gate voltage. As a result of manipulating the broadening of conductivity regions, we can limit the broadening of conductivity peaks around Landau levels using a dressing high-intensity electromagnetic field. This will enhance the sensitivity of FETs which provides the ability to observe narrow changes in gate voltage.\n Furthermore, adopting the mechanism presented in Ref. \\cite{hirakawa01}, we can manipulate conductivity peaks into very sensitive, narrowband high-frequency radiation detectors.\n We envision these advances in nanoelectronics from our theoretical model and knowledge generated from the insight it provides to the underlying dependencies.\n Furthermore, this theoretical model will aid in the development of simulation tools that will design the quantum effects in\n magneto-transport properties of 2D nanostructures.\n }\n\\end{itemize}\n\n\\subsubsection*{Comment 4 -\n\\color{RoyalBlue} The quantum Quantum Hall effect requires high mobility samples. In these samples, the structure of the disorder is usually complicated and combines both short-length potentials of impurities and long-length electrostatic inhomogeneities. The interplay of these components of disorder opens exciting questions about the transport properties of 2DEGs. What is the structure of disorder considered in the present manuscript and hidden in the notations for Vimp? What are the conditions for validity of eq. (15)?\n}\n\nWe thank the reviewer for raising this important question. In our previous manuscript, we have presented the detailed derivation of Eq.~[15] with a discussion on models of disorder under Appendix C. However, here we again elucidate our disorder model and approximations made to derive the Eq.~[15] for clarity and to respond to this question.\n\nWe modeled the effect caused by impurities in the considered system as a single perturbation potential. Analyzing the electric properties for a specific impurity distribution is a rather formidable task. We do not consider a specific impurity distribution here as it is unlikely to represent a measured impurity configuration in an experiment. Therefore, in this study, we consider the statistically averaged properties of 2DEG over impurity configurations.\nWe start our analysis by considering a static disorder that corresponds only to elastic electron scattering processes.\nWe adapt the Edwards model \\cite{akkermans10} to represent the randomly distributed impurities over the associated system and we approximate this into a Gaussian white noise.\n\nSince we are presenting the perturbation potential $V(\\vb{r})$ by a group of randomly localized impurities, we assume $N_{imp}$ number of identical single impurity potentials distributed randomly but in fixed positions $\\vb{r}_i$. Thus, we can describe the perturbation potential $V(\\vb{r})$ as the sum of uncorrelated single impurity potentials $\\upsilon(\\vb{r})$,\n\\begin{equation} \\label{eq:1}\n V(\\vb{r}) =\n \\sum_{i=1}^{N_{imp}}\n \\upsilon (\\vb{r}-\\vb{r}_i).\n\\end{equation}\nFurthermore, we approximate this perturbation potential as a Gaussian random potential where one can choose the zero of energy such that the potential is zero on average. This model is characterized by the following two equations \\cite{akkermans10}\n\\begin{subequations}\n\\begin{equation} \\label{eq:2}\n \\expval{\\upsilon(\\vb{r})}_{imp} =0,\n\\end{equation}\n\\begin{equation} \\label{eq:3}\n \\expval{\\upsilon(\\vb{r})\\upsilon(\\vb{r'})}_{imp} = \\Upsilon(\\vb{r}-\\vb{r'}),\n\\end{equation}\n\\end{subequations}\nwhere $\\expval{\\cdot}_{imp}$ represents the average over the impurity disorder and $\\Upsilon(\\vb{r}-\\vb{r'})$ is any decaying function that solely depends only on $\\vb{r}-\\vb{r'}$, and decays with a characteristic length $r_c$. Since this study considers the case where the wavelength of radiation or a scattering electron is much greater than $r_c$, it is reasonable to approximate its two-point correlation function by\n\\begin{equation} \\label{eq:4}\n \\expval{\\upsilon(\\vb{r})\\upsilon(\\vb{r'})}_{imp} = \\Upsilon_{imp}^2\\delta(\\vb{r}-\\vb{r'}),\n\\end{equation}\nwhere $\\Upsilon_{imp}^2$ is a positive constant. A random potential $V(\\vb{r})$ with this property is called white noise \\cite{akkermans10}. Then we can approximately write the total scattering potential as\n\\begin{equation} \\label{eq:5}\n V(\\vb{r}) =\n \\sum_{i=1}^{N_{imp}}\n \\Upsilon_{imp} \\delta(\\vb{r}-\\vb{r}_i).\n\\end{equation}\n\nUsing this expression, we can evaluate the Floquet-Fermi golden rule for a dressed quantum Hall system. Furthermore, considering the properties of a Gaussian white noise impurity distribution \\cite{wackerl20,akkermans10}, we assume $V_{imp}$ is a constant in the momentum space.\nHere $V_{imp} = \\expval{|V_{{k'}_x,k_x}|^2}_{imp}$ with $V_{{k'}_x,k_x} = \\mel**{k'_x}{\\upsilon(x) }{k_x}$, and $\\braket{x}{k_x} = e^{-ik_x x}$.\nAll the detailed steps for the derivation of the Floquet-Fermi golden rule for a dressed quantum Hall system are included in Appendix C.\nHowever, throughout the derivation, we use only the first-order contribution (Born approximation) of the impurity potential.\n\nSince previous studies on Floquet-Drude conductivity \\cite{wackerl20}, and magneto-transport properties in undressed \\cite{endo09} and dressed \\cite{dini16} quantum Hall systems have used this particular Gaussian white noise potential, we also selected this particular impurity model to describe our system. Such a choice enables us to compare our analytical results with these previous models. As you have mentioned in the comment, consideration of other impurity disorder models and their impact on the\nmagneto-transport properties of dressed quantum Hall system would be an intriguing future research possibility.\n\nSince we have only mentioned the validity conditions we used to derive the Eq.~[15] in Appendix C and not in the main text of the previous manuscript, we have added these validity conditions to the main text of the revised manuscript.\n\n\\begin{itemize}\n \\item Section IV - first paragraph (page 4):\\\\\n {\\color{Red}\n We have modeled the effect caused by impurities as a single short-range perturbation potential. Analyzing the electric properties for a specific impurity distribution is a rather formidable task. We do not consider a specific impurity distribution here as it is unlikely to represent a measured impurity configuration in an experiment.\n Therefore, in this study, we consider the statistically averaged properties of 2DEG over impurity configurations.\n Furthermore, we have assumed that a group of randomly distributed impurities forms our perturbation potential under the Edwards impurity model \\cite{akkermans10,wackerl20}.\n Essentially, all these assumptions mean that we can write the total scattering potential in the 2DEG as a sum of uncorrelated single impurity potentials $\\upsilon(\\vb{r})$. This enables us to approximate the impurity potential as a Gaussian white noise \\cite{akkermans10,wackerl20}.\n Here $\\eta_{imp}$ is the number of impurities in a unit area, $V_{imp} = \\expval{|V_{{k'}_x,k_x}|^2}_{imp}$ with $V_{{k'}_x,k_x} = \\mel**{k'_x}{\\upsilon(x) }{k_x}$, and $\\braket{x}{k_x} = e^{-ik_x x}$.\n Moreover, in this analysis, $\\expval{\\cdot}_{imp}$ represents the average over the impurity disorder. In this derivation, we only considered the first order (the Born approximation) contribution from the impurity potential.\n }\n\\end{itemize}\n\n\\subsubsection*{Comment 5 -\n\\color{RoyalBlue} If continuous illumination is applied to an electron system, the system will heat indefinitely. The distribution function is stabilized when the electron-electron and electron-phonon scattering mechanisms are included. What are the conditions of validity for eqs. (33-34)? The deviations of the distribution function from its equilibrium result in effects that can significantly overcome the equilibrium contributions, see Dmitriev et al, Phys. Rev. B 80, 165327 (2009). However, to capture these effects, a treatment within Floquet-Drude formalism is insufficient and a complete quantum kinetic equation has to be analyzed. Can the authors argue why their results still present some interest to the community even if they potentially disregard more significant contributions?\n}\n\nThis is a good, valid question. First, we discuss the validity of the Eq.~[33] and Eq.~[34]. Our analysis derived the longitudinal conductivity of a dressed quantum Hall system without any partial distribution function specification until Eq.~[31].\nIn contrast to the MIRO analysis \\cite{dmitriev03,dmitriev05,dmitriev09}, we consider the dressed quantum Hall system under the off-resonant condition, where the photon absorption does not happen. Therefore, we can select the Fermi-Dirac distribution as our particle distribution function for Eq.~[31]\n\\begin{equation} \\tag{5}\n f(\\varepsilon) = \\frac{1}{\\exp[(\\varepsilon - \\varepsilon_F)/k_B T]+1}.\n\\end{equation}\nHere, $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, and $\\varepsilon_F$ is the Fermi energy of the system.\nThen we consider a special scenario where the system operates at a low-temperature limit, i.e., the condition $k_BT \\ll \\varepsilon_F$.\nSince previous theoretical studies \\cite{wackerl20,dini16,endo09} on magneto-transport properties of 2DEG quantum Hall systems were derived under this assumption, we also adopted the same conditions as it enables us to compare results with those studies.\nIn addition, as mentioned in previous experimental studies on SdH oscillations \\cite{zudov03,mani02,arapov02} in 2DEG undressed quantum Hall systems with increasing temperature, we can identify the decaying behavior of SdH oscillations.\nTherefore, if we need to maintain a good oscillatory behavior, we need to operate in low temperatures.\nSince in our analysis, we study the manipulation of SdH oscillations using a dressing field, it is reasonable to simplify our general derivation given in Eq.~[31] for low-temperature limit. Under these assumptions, the derivative of the Fermi-Dirac distribution is sharply peaked around the Fermi energy, and can be represented by the delta function \\cite{endo09}\n\\begin{equation} \\tag{6}\n - \\pdv{f(\\varepsilon)}{\\varepsilon} \\approx \\delta(\\varepsilon - \\varepsilon_F).\n\\end{equation}\nApplying these adequate conditions, we can simplify our derivation into a single expression that can be compared with previous studies \\cite{wackerl20,dini16,endo09} on\nmagneto-transport in dressed quantum Hall systems. We have added clarification on the above assumptions in the revised manuscript.\n\\begin{itemize}\n \\item Section V - second paragraph (page 7):\\\\\n {\\color{Red}\n We consider a dressed quantum Hall system under off-resonant condition, where photon absorption does not happen. Therefore, we can select the Fermi-Dirac distribution as our partial distribution function ($f$) for our system\n \\begin{equation} \\tag{33}\n f(\\varepsilon) = \\frac{1}{\\exp[(\\varepsilon - \\varepsilon_F)/k_B T]+1},\n \\end{equation}\n where $k_B$ is the Boltzmann constant, $T$ is the absolute temperature, and $\\varepsilon_F$ is the Fermi energy of the system. At low-temperatures, i.e., $k_BT \\ll \\varepsilon_F$, the derivative of this distribution is sharply peaked around the Fermi energy, and can be approximated by a delta function \\cite{endo09}\n \\begin{equation} \\tag{34}\n - \\pdv{f(\\varepsilon)}{\\varepsilon} \\approx \\delta(\\varepsilon - \\varepsilon_F).\n \\end{equation}\n }\n\\end{itemize}\n\nYes, indeed, experiments on quantum hall systems subject to a strong dressing field show elevated temperatures. However, as demonstrated in previous studies by Dini \\textit{et al.} \\cite{dini16} and Kibis \\textit{et al.} \\cite{kibis15}, it is possible to maintain low system temperature by limiting the dressing field to narrow pulses.\n\nAlthough we have restricted our general derivation given in Eq.~[31] into a particular instance for comparison, there is no fundemental restriction in extending our theoretical model into different types of particle distribution functions. As presented in the previous work of Dmitriev \\textit{et al.} \\cite{dmitriev05}, a well-defined analytical discussion needs to address the changes in distribution functions in our derived general longitudinal conductivity formula, and this would be a noteworthy extension to our study.\n\n\\subsubsection*{Comment 6 -\n\\color{RoyalBlue} The authors presented the results for the conductivity tensor's xx- and yy-components. They seem to be identical, apart of a dimensional factor $(eB)^2$ in the denominator. The equality of these two components is expected for isotropic systems. Does the polarization of the electromagnetic field break the isotropy? Do the two components of the conductivity remain equal even for an arbitrary direction of a linearly polarized field? Is there a non-Hall contribution to the xy-component?\n}\n\nWe express our gratitude towards the reviewer for pointing out this inadvertent oversight done in the $yy$-component of the current operator calculation. This has led to an unexpected normalization factor in the yy-component of the conductivity expression. Since we have only used the normalized $xx$-component of the conductivity tensor for subsequent analysis, this oversight has no impact on the analysis reported in the manuscript. With this correction, we observe that both xx-component and $yy$-component show the same conductivity behaviors. We added these amendments to the current operator calculations done in Appendix D and the longitudinal conductivity calculation under Section V.\n\n\\begin{itemize}\n \\item Appendix D (page 16):\\\\\n {\\color{Red}\n \\subsection*{\\label{appendix_d}Appendix D: Current operators for a dressed quantum Hall system}\n\n In this section, we derive the current density operator for the $N$-th Landau level in a dressed quantum Hall system. We already found the exact solution for the time-dependent Schrödinger equation with the Hamiltonian give in Eq.~(1) and we identified them as the Floquet states in Eq.~(14). For the simplicity of notation, we can represent the Floquet modes derived in Eq.~(10) as quantum states using their corresponding quantum numbers as follows\n \\begin{equation} \\tag{D1}\n \\ket{\\phi_{n,m}} = \\ket{n,k_x}.\n \\end{equation}\n Using this complete set of quantum states \\cite{wackerl20,holthaus15,grifoni98}, we can represent the single particle current operator's matrix element as\n \\begin{equation} \\tag{D2}\n \\left(\\vb{j} \\right)_{nm,n'm'} = \\mel{n,k_x}{\\;\\hat{\\vb{j}}\\;}{n',k'_x}.\n \\end{equation}\n Next, we can identify the particle current operator for our system \\cite{mahan00,bruus04} as\n \\begin{equation} \\tag{D3}\n \\hat{\\vb{j}} = \\frac{1}{\\widetilde{m}} \\left\\{\\hat{\\vb{p}} - e\\left[\\vb{A}_s + \\vb{A}_d(t) \\right]\\right\\},\n \\end{equation}\n where $\\widetilde{m}$ is the mass of the associated particle.\n\n First, we consider the $x$-directional particle current operator component, and we can identify that as\n \\begin{equation} \\tag{D4}\n \\hat{j}_x = \\frac{1}{\\widetilde{m}} \\left(-i\\hbar\\pdv{x} + eBy \\right).\n \\end{equation}\n Next, we calculate the matrix elements of $x$-directional current operator against our Floquet mode basis\n \\begin{equation} \\tag{D5}\n \\left({j}_x \\right)_{nm,n'm'} =\n \\mel{n,k_x}{\\;\\frac{1}{\\widetilde{m}} \\left(-i\\hbar\\pdv{x} + eBy \\right)}{n',k'_x},\n \\end{equation}\n and we evaluate these using the Floquet modes derived in Eq.~(7) as follows\n \\begin{equation} \\tag{D6}\n \\begin{aligned}\n \\left({j}_x \\right)_{nm,n'm'} = &\n \\frac{1}{{\\widetilde{m}}}\n \\delta_{k_x,k'_x}\n \\int\n \\left(\\hbar k'_x + eBy\\right) \\\\\n & \\times\n \\chi_{n}\\bm{\\left(}y - y_0 - \\zeta(t)\\bm{\\right)}\n \\chi_{n'}\\bm{\\left(}y - y_0 - \\zeta(t)\\bm{\\right)}\n dy.\n \\end{aligned}\n \\end{equation}\n Let $[y - y_0 - \\zeta(t)] = \\bar{y}$, and we can obtain\n \\begin{equation} \\tag{D7}\n \\begin{aligned}\n \\left({j}_x \\right)_{nm,n'm'} =\n \\frac{1}{{\\widetilde{m}}}\n \\delta_{k_x,k'_x}\n \\int &\n \\left[ \\hbar k'_x + eB\\bar{y} -\\hbar k'_x + eB\\zeta(t)\\right]\n \\chi_{n}(\\bar{y})\n \\chi_{n'}(\\bar{y})\n d\\bar{y}.\n \\end{aligned}\n \\end{equation}\n Using the following integral identities of the Floquet modes that are made of Gauss-Hermite functions \\cite{vedenyapin11,szego59}\n \\begin{equation} \\tag{D8}\n \\int\n \\chi_{n}({y})\n \\chi_{n'}({y}) d{y} =\n \\delta_{n',n},\n \\end{equation}\n \\begin{equation} \\tag{D9}\n \\int\n y \\chi_{n}({y})\\chi_{n'}({y}) d{y} =\n \\frac{1}{\\kappa}\n \\left(\\sqrt{\\frac{n+1}{2}} \\delta_{n',n+1} + \\sqrt{\\frac{n}{2}}\n \\delta_{n',n-1} \\right),\n \\end{equation}\n we simplify the matrix elements of $x$-directional current operator to obtain\n \\begin{equation} \\tag{D10}\n \\begin{aligned}\n &\\left({j}_x \\right)_{nm,n'm'} =\n \\frac{eB}{{\\widetilde{m}\\kappa}}\n \\delta_{k_x,k'_x}\n \\left[\n \\left(\\sqrt{\\frac{n+1}{2}} \\delta_{n',n+1} + \\sqrt{\\frac{n}{2}}\n \\delta_{n',n-1}\\right)\n + \\zeta(t) \\delta_{n',n}\n \\right].\n \\end{aligned}\n \\end{equation}\n Due to high complexity of extract solution, in this study we only consider the constant contribution from the Fourier series components of the above expression. Therefore, we can identify the $0$-th component of the Fourier series as\n \\begin{equation} \\tag{D11}\n \\begin{aligned}\n \\left({j}_x \\right)_{nm,n'm'} =&\n \\frac{eB}{\\widetilde{m}\\kappa}\n \\delta_{k_x,k'_x}\n \\left(\\sqrt{\\frac{n+1}{2}} \\delta_{n',n+1} + \\sqrt{\\frac{n}{2}}\n \\delta_{n',n-1} \\right).\n \\end{aligned}\n \\end{equation}\n To calculate the electric current operator, we substitute the electron's charge and effective mass to the above derived equation. This leads to\n \\begin{equation} \\tag{D12}\n \\begin{aligned}\n \\Big({j}^x_{s=0}\\Big)_{nm,n'm'}^{electron} =&\n \\frac{e\\hbar}{{m_e}l_0}\n \\delta_{k_x,k'_x}\n \\left(\\sqrt{\\frac{n+1}{2}} \\delta_{n',n+1} + \\sqrt{\\frac{n}{2}}\n \\delta_{n',n-1} \\right).\n \\end{aligned}\n \\end{equation}\n where $l_0 = \\sqrt{\\flatfrac{\\hbar}{eB}}$ is the magnetic length.\n\n Moreover, we can identify the $y$-directional current operator component as\n \\begin{equation} \\tag{D13}\n \\hat{j}_y = \\frac{1}{\\widetilde{m}} \\left(-i\\hbar\\pdv{y} - \\frac{eE}{\\omega}\\cos(\\omega t) \\right).\n \\end{equation}\n Using this operator, we can represent the matrix elements of $y$-directional current operator in Floquet mode basis as\n \\begin{equation} \\tag{D14}\n \\left({j}_y \\right)_{nm,n'm'} =\n \\mel{n,k_x}{\\;\\frac{-1}{\\widetilde{m}} \\left(i\\hbar\\pdv{y} + \\frac{eE}{\\omega}\\cos(\\omega t) \\right)}{n',k'_x}.\n \\end{equation}\n After following the same steps done for the $x$-directional current operator, and recursion relation of the first derivative of Gauss-Hermite functions\n \\begin{equation} \\tag{D15}\n \\begin{aligned}\n \\pdv{\\chi_{n}({y})}{y} =\n \\kappa\\left[\n - \\sqrt{\\frac{n+1}{2}} \\chi_{n+1}({y})\n + \\sqrt{\\frac{n}{2}} \\chi_{n - 1}({y})\n \\right],\n \\end{aligned}\n \\end{equation}\n we can identify the $0$-th component of matrix elements for $y$-directional electric current operator as\n \\begin{equation} \\tag{D16}\n \\begin{aligned}\n \\Big({j}^y_{s=0}\\Big)_{nm,n'm'}^{electron} = &\n \\frac{ie\\hbar\\kappa}{m_e}\n \\delta_{k_x,k'_x}\n \\left(\n \\sqrt{\\frac{n}{2}} \\delta_{n',n-1}\n - \\sqrt{\\frac{n+1}{2}} \\delta_{n',n+1}\n \\right).\n \\end{aligned}\n \\end{equation}\n \\begin{equation} \\tag{D17}\n \\begin{aligned}\n \\Big({j}^y_{s=0}\\Big)_{nm,n'm'}^{electron} = &\n \\frac{ie \\hbar}{m_e l_0}\n \\delta_{k_x,k'_x}\n \\left(\n \\sqrt{\\frac{n}{2}} \\delta_{n',n-1}\n - \\sqrt{\\frac{n+1}{2}} \\delta_{n',n+1}\n \\right).\n \\end{aligned}\n \\end{equation}\n }\n \\item Section V - second paragraph (page 7):\\\\\n {\\color{Red}\n Moreover, let $\\Pi = \\varepsilon_F$ and the derived expression in Eq.~(31) leads to\n \\begin{equation} \\tag{35}\n \\begin{aligned}\n \\sigma^{xx} =\n \\frac{e^2l_0^2}{\\pi\\hbar A}\n \\sum_{n}\n \\frac{(n+1)}{\\gamma_{n}\\gamma_{n+1}}\n \\left[\n \\frac{1}\n {\n 1 + \\left(\\frac{X_F - n -1}{\\gamma_{n+1}}\\right)^2\n }\n \\right]\n \\left[\n \\frac{1}\n {\n 1 + \\left(\\frac{X_F - n}{\\gamma_{n}}\\right)^2\n }\n \\right],\n \\end{aligned}\n \\end{equation}\n where $X_F = \\left[\\flatfrac{\\varepsilon_F}{(\\hbar \\omega_0)} - \\flatfrac{1}{2}\\right]$,\n $\\gamma_n = \\flatfrac{\\widetilde{{\\Gamma}}(\\varepsilon_n)}{(\\hbar \\omega_0)}$, and $l_0 = \\sqrt{\\flatfrac{\\hbar}{eB}}$.\n Following the same steps as above derivation, we can derive the longitudinal conductivity in the $y$-direction by applying the electric current operator for $y$-direction derived in Appendix D\n \\begin{equation} \\tag{36}\n \\begin{aligned}\n {\\sigma}^{yy} =\n \\frac{e^2l_0^2}{\\pi\\hbar A}\n \\sum_{n}\n \\frac{(n+1)}{\\gamma_{n}\\gamma_{n+1}}\n \\left[\n \\frac{1}\n {\n 1 + \\left(\\frac{X_F - n -1}{\\gamma_{n+1}}\\right)^2\n }\n \\right]\n \\left[\n \\frac{1}\n {\n 1 + \\left(\\frac{X_F - n}{\\gamma_{n}}\\right)^2\n }\n \\right].\n \\end{aligned}\n \\end{equation}\n }\n\\end{itemize}\n\nSuppose we apply an arbitrary directional polarized dressing field. In that case, we could derive an expression for the longitudinal conductivity components by considering dressing field contribution for each direction ($x$ and $y$) with the same steps presented in our analysis. This would change the amplitude of the associated dressing field ($E$) in each direction. There will be a new component in parallel to the conductivity component of interest, which will modify the time-dependent Hamiltonian given in Eq.~[2]. Then, we need to solve the Schrödinger equation with the modified Hamiltonian to find the new Floquet modes for the new system. These will define the effect on the inverse scattering time matrix components. As mentioned in Ref. \\cite{wackerl20} and our work, applied strong dressing field tends to change the quantum state of the considered system and creates novel states called Floquet states. The properties of these states depend on the characteristics of the applied dressing field. Therefore, the polarization method also changes the behavior of the Floquet states. These polarization-dependent conductivity behaviors in 2DEG systems can be found in references \\cite{wackerl20,morina15} and in their work authors have illustrated the conductivity behavior with circular and linear polarized dressing fields. However, in a dressed quantum Hall system with a $y$-directional linear polarized field, we predict the same longitudinal conductivity behaviors in diagonal components of the conductivity tensor.\n\nFor analyzing our system, we mainly employ the diagonal components of the Floquet-Drude conductivity tensor introduced by Wackerl \\textit{et al.} \\cite{wackerl20}. As we mainly focus on SdH oscillations in longitudinal conductivity components, we limit our analysis to the dressing field influence on two diagonal components of the conductivity tensor. Therefore we are unable to predict the off-diagonal component behavior of the system.\n%\n% As mentioned in Ref.\\cite{wackerl20} and our work, applied strong dressing field tends to change the quantum state of the consideirng system and creates novel states called Floquet states. The properties of these states are depends on the characteristics of applied dressing field. Therefore, the polarization method also change the behavior of the Floquet states.\n% These polarization dependent conductivity behaviors in 2DEG systems can be found in references \\cite{wackerl20,morina15} and in their work authors have illustrated the conductivity behavior with circular and linear polarized dressing fields. In addition, they have mentioned the impact on the direction of the linearly polarization dressing field and this will break the isotropy behavior of the longitudinal conductivities in the consideirng system.\n%\n% We also followed the same formalism of Floquet states for 2DEG quantum Hall systems. In our work also we were able to identify that the applied dressing field's polarization direction can be effect the magnitude of the longitudinal conductivity component. A detailed derivation of these two longitudinal conductivity component's current operators are given in Appendix D. If we applied a arbitary directional polarized dressing field, we can derive a expression for the longitudinal conductivity components by considering dressing field contribution for each direction ($x$ and $y$) with same steps in this analysis. This would change the amplitude of the considering dressing field ($E$) in each direction. However, now there will be a new conponent in parallel to the considering conductivity component and we need to solve the Schrödinger equation with the modified Hamiltonian in Eq.~[2] for the new system with these changes.\n% However in our analysis, our main objective was to identify the manipulation capabilities of SdH oscillations using dressing field intensity. In addition, we only consider the longitudinal components in this analysis by extending the Floquet-Drude conductivity formula that derived only for the diagonal components in conductivity tensor. Therefore we are unable to predict the off-diagonal component behavior of the system through our analysis and we need a novel formalism to handle these type of behaviors of dressed 2DEG quantum Hall systems.\n\n\\subsection*{Response to the comments of Reviewer 2}\n\nWe want to thank the reviewer for his/her insightful comments on our work. To address your comments, we had to add new materials and reorganize our text. As a result, we have been able to considerably improve our discussion on the underlying assumptions and corresponding physical significance of our analysis. We hope our response and the corresponding changes we incorporated into the manuscript will be sufficient to clarify the issues raised.\n\n\n\\subsubsection*{Comment 1 -\n\\color{RoyalBlue} The paper is very low on comparison with experiments, for which there is a large amount of data available for the zero radiation case. I would like to see detailed analysis of how Figs 4 and 5 compare with available\nexperimental data. Without this addition, physical relevance of such\ndetailed calculations is questionable.\n}\n\nWe thank the reviewer for pointing out the importance of this fact. In the revised manuscript, we have included a detailed analysis of our results against the undressed 2DEG quantum Hall system behavior.\n\n\\begin{itemize}\n \\item Section VI - third paragraph (page 8):\\\\\n {\\color{Red}\n By comparing the theoretical \\cite{ando72,ando74_1,ando74_2,ando74_3,ando74_4,ando82,endo09} and experimental \\cite{endo09,wakabayashi78,ochiai190,mancoff96,arapov02,grbic04,caviglia10} studies on the magnetoresistance of 2DEG quantum Hall systems when no radiation is present against our results, we can identify that longitudinal conductivity oscillations in Fig. 4 are a repetition of the Shubnikov–de Haas(SdH) oscillations.\n As observed in the experimental work of Caviglia \\textit{et al.} \\cite{caviglia10}, the period of SdH oscillations depends only on the perpendicular component of the magnetic field to the plane of the 2DEG.\n Therefore, we can identify that many experiments on different types of 2DEGs at low-temperatures \\cite{endo09,ochiai190,mancoff96,arapov02,grbic04,caviglia10} have analyzed these SdH oscillations against the applied magnetic field's amplitude.\n The cyclotron frequency ($\\omega_0$) of the system depends on the magnetic field's perpendicular component to the plane of the 2DEG.\n The gate voltage modifies the Fermi level of the system.\n Since the Landau level energy is only dependent on the cyclotron frequency, this gate voltage variation under a constant magnetic field amplitude generates the same SdH oscillations.\n Interestingly, this oscillatory behavior against applied gate voltage has been observed experimentally by Wakabayashi \\textit{et al.} \\cite{wakabayashi78} in an inversion layer on a silicon surface at low temperature.\n By comparing these observations against our results in Fig. 4, we can identify that our oscillations also show the same characteristic behavior; i.e $\\hbar\\omega_0$ periodic SdH oscillations against the Fermi energy.\n Furthermore, when we increase the gate voltage, the Fermi level rises. As illustrated in Fig. 4, this will result in a higher conductivity peak value at the higher-order Landau levels. This same behavior was also observed in the experimental observations presented in Ref. \\cite{wakabayashi78}.\n In our work, we provide analytical results describing the controllability of these conductivity regions using a dressing field.\n }\n\\end{itemize}\n\n\\subsubsection*{Comment 2 -\n\\color{RoyalBlue} In the presence of disorder, Anderson localization and topologically protected edge modes are dominant considerations when calculating DC transport coefficients. How are these factors taken into account?\n}\n\nWe omitted a discussion on the Anderson localization and topologically protected edge modes as we neglected them in our work.\nWhile deriving Eq.~[31], we have assumed that the applied static magnetic field is placed in a lower intensity range where we can neglect several effects known to appear in a 2DEG quantum Hall system. Under this condition, we can neglect the localization in the tails of Landau level peaks, the formation of the edge states, the electron-electron interactions, and the spin splitting as well. As mentioned in the work of Endo \\textit{et al.} \\cite{endo09}, within the magnetic field range of interest, the undressed\nmagneto-transport properties of GaAs/AlGaAs 2DEG quantum Hall system provide an excellent agreement between the experimental observations and theory that was developed by neglecting the above-mentioned effects. Our work is very similar to theirs with an additional dressing field. Our results for the undressed system agree with the results in Ref. \\cite{endo09}, while we provide a better description to manipulate the width of the Landau level conductivity peaks. We have added this clarification in the manuscript.\n\n\\begin{itemize}\n \\item Section I - sixth paragraph (page 2):\\\\\n {\\color{Red}\n It is important to note that, we limit the applied magnetic field to a range where we can neglect effects such as Anderson localization, formation of the edge states, spin splitting, and the electron-electron interactions. These are not overly restrictive assumptions, and other have successfully used same assumptions in their work, e.g see Endo \\textit{et al.} \\cite{endo09}.\n }\n \\item Section VIII - first paragraph (page 10):\\\\\n {\\color{Red}\n We modeled the impurities in the material as a Gaussian random scattering potential, and restricted the applied magnetic field to a range where we can neglect effects of Anderson localization, formation of the edge states, spin splitting, and electron-electron interactions.\n }\n\\end{itemize}\n\n\n\\subsubsection*{Comment 3 -\n\\color{RoyalBlue} Can the authors reproduce the quantized Hall conductance of filled Landau levels in the presence of disorder? Please discuss.\n}\n\nFor analyzing our system, we mainly employ the diagonal components of the Floquet-Drude conductivity tensor introduced by Wackerl \\textit{et al.} \\cite{wackerl20}. As we mainly focus on SdH oscillations in longitudinal conductivity components, we limit our analysis to the dressing field influence on two diagonal components of the conductivity tensor.\nTherefore we cannot predict the off-diagonal component behavior of the system. However, we can use the already established formalism introduced by Endo \\textit{et al.} \\cite{endo09} to predict the behavior of off-diagonal components of the conductivity tensor. As mentioned in Ref. \\cite{endo09}, this theory on the relation between longitudinal and transverse conductivities highly agree with experiment observations\n\\begin{equation} \\tag{1}\n \\dv{\\widetilde{\\sigma}^{xy}(X_F,I)}{B} \\approx\n \\pi\\mu \\frac{\\hbar \\omega_0}{\\varepsilon_F} [\\widetilde{\\sigma}^{xx}(X_F,I)]^2.\n\\end{equation}\nBy examining the connection between longitudinal and transverse conductivities in the 2DEG quantum Hall system, we can expect the same Hall conductivity behavior in a dressed quantum Hall system. However, using the dressing field, we can squeeze the conductivity peaks on Landau levels, i.e., squeeze the region of longitudinal conductivity peaks placed near the Landau levels. This implies sudden Hall conductivity transitions between one plateau to another near a Landau level. As in Refs. \\cite{klitzing80,gusynin06}, we can also predict a staircase behavior of quantum Hall conductivity against the gate voltage. With an external dressing field, we will be able to get rapid transitions than the undressed system. However, to derive an analytical expression for the quantum Hall conductivity through Floquet-Drude conductivity needs to be reformulated from the beginning. It will be the subject of our future study.\n\n\\subsubsection*{Comment 4 -\n\\color{RoyalBlue} The way some previous works are cited is disappointing. Example from page 8: “Despite this behavior being identified in previous works, their results did not coincide with the more accurate description of conductivity components in undressed quantum Hall systems.” This is not informative. Where, specifically, did results from earlier works fall short of reality, which have been better addressed in this work?\n}\n\nWe made the following changes in language and presentation to improve the clarity of our manuscript.\n\n\\begin{itemize}\n \\item Section VIII (page 8):\\\\\n {\\color{Red}\n Despite this behavior being identified in the previous work of Dini \\textit{et al.} \\cite{dini16}, they cannot fully account for the experimental observations of longitudinal conductivity components in undressed quantum Hall systems presented in Ref. \\cite{endo09}. The authors of Ref. \\cite{dini16} have used the conventional expression of longitudinal conductivity from Refs. \\cite{ando74_1,ando82} and this theory yields a semi-elliptical broadening for the Landau levels. These predictions significantly deviate from the experimentally observed Landau level broadening \\cite{endo09}.\n However, our generalized analysis on the conductivity of dressed quantum Hall systems provides a well-suited description for empirically observed behaviors of undressed quantum Hall systems as well.\n }\n \\item Section III - first paragraph (page 3):\\\\\n {\\color{Red}\n In their study \\cite{wackerl20}, the authors have presented more accurate results than the former theoretical descriptions \\cite{morina15,pervishko15} for the conductivity of nanoscale systems in the presence of a dressing field.\n }\n \\item Section I - fifth paragraph (page 2):\\\\\n {\\color{Red}\n The authors of Ref. \\cite{dini16} used the conductivity models from Refs. \\cite{ando74_1,ando82}, and as mentioned in Endo \\textit{et al.} \\cite{endo09}, those models predict a semi-elliptical broadening against Fermi level for each Landau level and provide less agreement with the empirical results.\n }\n \\item Section IV - third paragraph (page 5):\\\\\n {\\color{Red}\n In the study presented in Ref. \\cite{endo09}, the authors have assumed that the effective mass of an electron in GaAs-based quantum well system is $m_e \\approx 0.07\\widetilde{m}_e$ where $\\widetilde{m}_e$ is the mass of the electron \\cite{endo09,winkler03,wackerl20}.\n }\n\\end{itemize}\n\n\\subsubsection*{Comment 5 -\n\\color{RoyalBlue} Adding some physical insight into the remarkable observation of radiation-induced narrowing of lineshapes (Figs 4, 5) will help elevate\nthis work.\n}\n\nWe agree with the reviewer that a discussion on our theoretical results and their physical significance in modern nanoelectronic devices is a vital requirement. We have added a new Section VII to overcome the above-mentioned deficiency of our manuscript. The total content of the section is given below,\n\n\\begin{itemize}\n \\item Section VII (page 8):\\\\\n {\\color{Red}\n \\subsection*{VII. PHYSICAL SIGNIFICANCE OF THE OUTCOMES}\n\n With the realization of 2DEGs in Si-MOSFETs (Metal Oxide Semiconductor Field Effect Transistors) \\citep{fowler66}, Klitzing \\textit{et al.} \\cite{klitzing80} made the first transport measurements on such systems to reveal the quantum Hall effect. The empirical discovery of these unusual properties marked the beginning of a whole new realm in condensed matter physics that continues to produce phenomenal advancements in electronic systems. The quantum Hall effects in a 2DEG under a static magnetic field are described by plateaus quantized to integer values of the conductivity quantum ($\\flatfrac{e^2}{\\hbar}$) in the off-diagonal conductivity, with simultaneous peaks at inter-plateau transition for the diagonal conductivity \\cite{endo09}. This is due to the applied magnetic field and it changes the energy spectrum of 2DEG dramatically. The magnetic field causes the density of states in 2DEG to split up into a sequence of delta functions, separated by an energy $\\hbar\\omega_0$, with $\\omega_0$ the cyclotron frequency which depends on the applied magnetic field.\n However, experimental results demonstrate that these Landau levels are broadened and the main source of these broadening at low temperatures is the disorders in materials \\cite{ando85,dial07}. The broaden sequence of delta functions of the density of states implies the oscillating behavior in the experimental measurements of longitudinal conductivity which is known as SdH oscillations \\cite{endo09,wakabayashi78}.\n\n Our theoretical analysis on longitudinal conductivity behavior of dressed quantum Hall system was developed by considering low-temperature limit with Gaussian impurity broadening assumptions.\n As illustrated in Fig. 4, we can describe the experimentally observed SdH oscillation results in Refs.\\cite{endo09,wakabayashi78} through our model.\n Under the undressed condition, our results overlap with the conductivity measurement of quantum Hall systems \\cite{endo09}. Strikingly, we show in Fig. 5 that we can manipulate the broadening of these conductivity peaks using an external dressing field. At low temperatures, the principal cause of broadening of these conductivity peaks is impurity-induced scattering. Using an external dressing field, we can suppress the impurity-induced scattering and this results a reduction of both the scattering-induced broadening and the longitudinal conductivity peaks.\n\n Research on novel states of matter has driven the evolution of present-day nanoelectronic devices. In particular, controllable manipulation of material properties through a gate electric field has revolutionized the development of material science and technology \\cite{ahn03,deng18}.\n The charge carrier concentration of a system is an imperative parameter that defines its conductivity properties.\n As we can manipulate the charge carrier concentration using an electrostatic field-effect, we exert significant control over the conductivity externally.\n A 2DEG under static magnetic field with quantum Hall effects is an excellent example that illustrates how the gate electric field controls conductivity.\n We also observed that a considerable number of studies exist on charge transport in the quantum limit that employed different types of 2D field-effect transistors (FETs) subjected to magnetic fields \\cite{wakabayashi78,yang18,long20,li14}. Yang \\textit{et al.} \\cite{yang18} study shows quantized Hall plateaus and SdH oscillations for longitudinal conductivity against gate voltage in black phosphorus FET under static magnetic fields at low temperatures. Since the Fermi level of a system can be altered with the applied gate voltage, our Fig. 4 depicts this result.\n In particular, our analysis shows that we can manipulate the broadening of the conductivity regions using an external dressing field. Although Yang \\textit{et al.} \\cite{yang18}, achieved broadening in longitudinal conductivity peaks by changing the temperature in a low range, in this study we presented a general theory on manipulating longitudinal conductivity broadening using a high-intensity electromagnetic field.\n\n The realization of the underlying mechanism of 2D FETs in the quantum realm promises its potential in next-generation nanoelectronic applications. In a particular application that uses the switching mechanism of the above-discussed FETs with quantum Hall effects, we can achieve high and low output conductivities by changing the input gate voltage. As a result of manipulating the broadening of conductivity regions, we can limit the broadening of conductivity peaks around Landau levels using a dressing high-intensity electromagnetic field. This will enhance the sensitivity of FETs which provides the ability to observe narrow changes in gate voltage.\n Furthermore, adopting the mechanism presented in Ref. \\cite{hirakawa01}, we can manipulate conductivity peaks into very sensitive, narrowband high-frequency radiation detectors.\n We envision these advances in nanoelectronics from our theoretical model and knowledge generated from the insight it provides to the underlying dependencies.\n Furthermore, this theoretical model will aid in the development of simulation tools that will design the quantum effects in\n magneto-transport properties of 2D nanostructures.\n }\n\\end{itemize}\n\n\\bibliography{response}\n\\medskip\nSincerely yours,\n\n\\def\\s#1#2#3{\\vbox{\\hsize=4.5cm\n\t\t\\kern2cm\n\t\t\\hrule\\kern1ex\n\t\t\\hbox to \\hsize{\\strut\\hfil #1 \\hfil}\n\t\t\\hbox to \\hsize{\\strut\\hfil #2 \\hfil}\n\t\t\\hbox to \\hsize{\\strut\\hfil #3 \\hfil}}}\n\n\\hbox to \\hsize{\\s{Malin Premaratne}{(Corresponding Author)}{\\href{malin.premaratne@monash.edu}{malin.premaratne@monash.edu}}}\n\n\n\\end{document}\n\n% ****** End of file dressed_quantum_hall.tex ****** %\n", "meta": {"hexsha": "d7d0ecdf487c9f7781629affdfaa0e0ff54c83ac", "size": 67130, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "response/response.tex", "max_stars_repo_name": "KosalaHerath/magnetic-2DEG-conductivity", "max_stars_repo_head_hexsha": "91c5df1b018579b4b9c91d84f2d60ee482a001de", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "response/response.tex", "max_issues_repo_name": "KosalaHerath/magnetic-2DEG-conductivity", "max_issues_repo_head_hexsha": "91c5df1b018579b4b9c91d84f2d60ee482a001de", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "response/response.tex", "max_forks_repo_name": "KosalaHerath/magnetic-2DEG-conductivity", "max_forks_repo_head_hexsha": "91c5df1b018579b4b9c91d84f2d60ee482a001de", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 103.9164086687, "max_line_length": 1531, "alphanum_fraction": 0.7795471473, "num_tokens": 15954, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.34226221457390893}} {"text": "\\chapter{Solving the De Novo Genome Assembly}\n\n\nIn this last chapter, we approach the genome assembly problem with the use of quantum annealing. Let us first understand what this problem is, and then translate it to a QUBO model and solve it using quantum annealing.\n\n\n\\section{The Genome Assembly Problem}\n\n\nThe genome of an organism is all its genetic material \\cite{Roth2019}. The deoxyribonucleic acid (DNA) is the carrier of that genetic information. It consists of two long chains twisted to form a double helix \\cite{Alberts2007}. Each of these chains is composed of a series of nucleotides or bases: adenine (A), guanine (G), cytosine (C), and thymine (T). Since these bases are matched in pairs in the DNA double helix, they are called base pairs (bp).\n\nA genome sequence is the complete list of nucleotides of every chromosome of an organism. With today's technology, automated sequence machines can read up to $10.000$ bp at a time \\cite{Reinert1015} while the human genome contains 3 Mbp, so we cannot simply read the whole genome. This is where genome assembly comes in.\n\nGenome assembly refers to the process of, given a large number of short DNA reads, stitch them together to form a large representation of the original chromosome where the reads came from. The two main techniques used to reconstruct these sequences are the ab initio reference-free alignment and the de novo reference-based assembly.\n\n\n\\subsection{Ab initio reference-based alignment}\n\n\nIn this method, the DNA reads are matched against a known trusted reference of the same organism. This is essentially a pattern matching problem, where we find the index of a given sub-string in a larger string. However, after the reconstruction is complete the result is compared to the reference in order to identify implications; therefore introducing bias based on the reference \\cite{Sarkar2020}.\n\nIn the naive approach, the short sub-string is compared to the reference starting at the first index. If the end of the sub-string is reached with a positive, a match is obtained. Otherwise, the sub-string is shifted a single position and we compare again. Heuristic methods that improve on this idea are based on shifting a greater number of spaces after a mismatch.\n\nDifferent number of strategies have been developed in this direction. For instance, the classic Boyer-Moore and Knuth-Pratt-Morris algorithms \\cite{Holmes1999}. However, these in particular are not adequate for genome assembly since these are exact string matching algorithms and DNA reads usually need approximate matches due to reads errors. Other algorithms worth mentioning are the Needleman-Wunsch algorithm \\cite{Needleman1970} and the Smith-Waterman algorithm \\cite{Smith1981}, for global and local alignment respectively. These are dynamic programming algorithms designed specifically with DNA reads in mind. \n\nState of the art algorithms trades off accuracy for speed and memory. Given enough computational power, the de novo reference-free method yields better results without introducing any reference bias.\n\n\n\\subsection{De novo reference-free assembly}\n\\label{sec:de-novo-genome-assembly}\n\nOn the other hand, the de novo reference-free method is, as its name suggests, reference free. Meaning, it is based only on the DNA reads. Thus, it has no reference bias but it is more computationally complex. It is usually used the first time a species DNA is read.\n\nIn this technique, multiple copies of the same DNA are made before slicing it. After chopping each copy at random places the data is redundant and the different reads overlap, making the assembly easier. There are multiple methods for de novo assembly based on different tools: Overlap-Layout Consensus (OLC) methods, de Bruijn graph (DBG) methods, string graphs, greedy and hybrid methods are some of the most famous examples (for a review see \\cite{Sohn2018}). Depending on the reading method and the number and length of the DNA reads, different methods excel from the rest. For instance, short-read technologies with a large number of reads favor DBG methods while high-quality long reads favor OCL methods. For our purposes, we will focus on the OCL method.\n\nIn the OLC graph used for the de novo whole-genome assembly, each node represents a different DNA read. Directed edges are associated a weight depending on how well these two reads are stitched together in a certain order. For example, the directed edge going from reading $r_1$ to read $r_2$ will be assigned a weight depending on how well $r_1r_2$ can be stiched.\n\nThe weights computation depends on the implementation. For the purpose of this thesis, we will use exact matches, no taking reading errors into consideration. Then, the weight assigned to an edge is the length of the overlap between both reads without any errors, with a change of sign. For example, given the reads $r_1 = AATT$ and $r_2 = TTCC$, the perfect stitching will produce $AATTCC$, so the overlap between both reads is $2$, giving a weight of $-2$. We may call this the \\emph{distance} between reads $r_1$ and $r_2$ (in that order). It does not fulfill the mathematical definition of distance since it is not even symmetric, but it will be useful for us anyway.\n\nA Hamiltonian path in our overlap graph will represent a series of reads in a certain order. By minimizing the total cost of our Hamiltonian path, we maximize the overlap between reads, resulting in the shortest possible final chain. This is exactly the same as solving the Travelling Salesman problem associated with our overlap graph.\n\nFigure \\ref{de-novo-process} shows the whole problem resolution \\cite{Boev2020}. Given the DNA reads (a) we compute the overlap graph (b) using a distance between the reads. We continue by viewing this problem as a traveling salesman and transforming it into a QUBO model (c). Then, either a simulated annealer (e) or the D-Wave quantum annealer (d) are used to obtain the Hamiltonian path/cycle of minimum cost (f). Finally, we traverse the cycle and build the resulting genome sequence.\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.4]{de-novo-process.png}\n\t\\centering\n\t\\caption{Resolution diagram for the genome assembly using quantum annealing \\cite{Boev2020}}\n\t\\label{de-novo-process}\n\\end{figure}\n\n\n\\newparagraph{Numerical example}\n\n\nLet us study a final numerical example based on \\cite{Sarkar2020} that shows the whole process. Suppose we are given the following reads:\n\n\\begin{itemize}\n\t\\item $r_0 = ATGGCGTGCA$\n\t\\item $r_1 = GCGTGCAATG$\n\t\\item $r_2 = TGCAATGGCG$\n\t\\item $r_3 = AATGGCGTGC$\n\\end{itemize}\n\nWe may compute the overlap between the different reads. This results in the overlap graph from figure \\ref{fig:overlap-graph}.\n\n\\begin{figure}[h]\n\t\\includegraphics[scale=1.1]{graphs/salesman-example.pdf}\n\t\\centering\n\t\\caption{OLC graph}\n\t\\label{fig:overlap-graph}\n\\end{figure}\n\nWhich is the same graph studied in the traveling salesman section, \\ref{sec:salesman-example}. From the study done in that section, we know that there are six types of cycles in the graph. They are displayedin table \\ref{tbl:salesman-cycles}. In figure \\ref{fig:overlap-cycles}, we see how these types of cycles represent different ordinations of our DNA reads, as well as their overlaps and the total length of the resulting chain.\n\n\\begin{figure}[h]\n\t\\makebox[\\textwidth][c]{\n\t\t\\includegraphics[scale=0.35]{overlap-cycles.png}\n\t}\n\t\\centering\n\t\\caption{Type of cycles and their corresponding overlap analysis \\cite{Sarkar2020}}\n\t\\label{fig:overlap-cycles}\n\\end{figure}\n\nAs we know, the type of Hamiltonian cycle that minimizes the cost is type A. This type translates into the shortest chain, with a total length of $19$. We can easily compute the resulting assembly by traversing the graph and weaving the reads together.\n\n\n\\section{Experimentation}\n\\label{sec:experimentation}\n\n\nIn this last section, we will explain the experiments reproduced using the D-Wave systems.\n\n\n\\subsection{How to reproduce the experiments}\n\n\nIn order to configure the samplers and submit jobs to D-Wave quantum annealers we used the \\emph{D-Wave Ocean Software}, a suite of tools provided by D-Wave to use their quantum systems \\cite{DWave-OceanDoc}. Using python and the provided packages we may connect to \\emph{Leap}, a 'quantum' cloud service that provides access the quantum computers \\cite{DWave-Leap}.\n\nLeap provides a minute of free QPU time with the (free) developer plan. We may extend this time by obtaining either commercial or research access. This can be done by contacting the Leap team.\n\nFor the pourpose of this thesis I used the developer plan. In total, I submitted $104$ jobs to Leap, out of the allowed $106$. I consumed $59.064$ QPU seconds, as can bee seen in Leap's breakdown in figure \\ref{fig:leap-breakdown}. The number of jobs and QPU time used in each system can be seen in table \\ref{tab:leap-breakdown}.\n\n\\begin{table}[H]\n\t\\centering\n\t\\begin{tabular}{lrr}\n\t\t\\textbf{System} & \\textbf{Number of jobs} & \\textbf{QPU time (s)} \\\\\n\t\t\\hline\n\t\tD-Wave 2000Q\t& 26\t& 11.015\t\\\\\n\t\tAdvantage\t\t& 77\t& 46.681\t\\\\\n\t\tTotal\t\t\t& 104\t& 59.064 \n\t\\end{tabular}\n\t\\caption{Leap usage summary breakdown between systems.}\n\t\\label{tab:leap-breakdown}\n\\end{table}\n\nApart from Leap, in order to reproduce these experiments you will need to install tthe following Python packages, available thorugh \\emph{pip}: \\textbf{\\emph{dimod}}, \\textbf{\\emph{neal}}, \\textbf{\\emph{minorminer}}, \\textbf{\\emph{dwave}} and \\textbf{\\emph{dwave\\_networkx}}. The code I used for these experiments can be found in \\cite{thesis-code}, which are inspired on \\cite{Sarkar2020}.\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.5]{experiments/leap-breakdown.png}\n\t\\centering\n\t\\caption{Leap monthly usage summary.}\n\t\\label{fig:leap-breakdown}\n\\end{figure}\n\n\n\\subsection{Experiment 1: Data preparation and exact solving}\n\n\nThe first experiment aims to solve the previously discussed example using an exact solver provided by the Ocean's \\emph{dimod} library \\cite{Dimod}. The steps taken to transform the given reads to the QUBO matrix that Ocean's classes and functions use will be shared between the simulated annealer and the real quantum solvers. Therefore, the data preparation applied will be shared between experiments. The pseudocode of the experiment can be seen in the following snippet.\n\n\\begin{algorithm}\n\t\\caption*{\\textbf{Experiment 1}}\n\t\n\tData preparation:\n\t\\begin{itemize}\n\t\t\\item Compute the distance between every two reads, creating an adjacency matrix.\n\t\t\\item Transform the TSP adjancency matrix into a QUBO $Q$ matrix, as explained in section \\ref{sec:tsp-qubo}.\n\t\t\\item Transform the QUBO matrix into an adjacency dictionary.\n\t\\end{itemize}\n\n\tSolving:\n\t\\begin{itemize}\n\t\t\\item Initiaize a sampler: \\textbf{\\emph{dimod.ExactSolver}}.\n\t\t\\item Solve the prepared QUBO model using the selected solver.\n\t\\end{itemize}\n\t\n\tPresent the results.\n\\end{algorithm}\n\nAs we already know, using quantum annealing does not guarantee that we will obtain the optimal solution for a given cost function. In order to overcome this problem we use multi-sampling: running the experiment multiple times and and look at the best obtained solutions. Ocean already implements different types of \\emph{samplers} to facilitate this task \\cite{DWave-OceanDoc-Samplers}. In this first experiment we use an \\emph{ExactSolver}, which simply check every possible solution. Although time-costly, this method will let us know if our data manipulation before solving the experiment is correct, and how the solutions landscape looks like.\n\nThe third step in data preparation is a formatting step. Ocean requires the QUBO and Ising models to be in an adjacency dictionary instead of a matrix. Let us look at an example of this transformation in order to better understand it. Consider a 2-reads, the associated TSP will have $2^2 = 4$ nodes: $n0t0$, $n0t1$, $n1t0$ and $n1t1$. Suppose the following (inconsistency) matrix is the associated $Q$ matrix:\n\n$$\nQ = \n\\left(\n\\begin{array}{cccc}\n\t1 & 2 & 3 & 4 \\\\\n\t5 & 6 & 7 & 8 \\\\\n\t9 & 10 & 11 & 12 \\\\\n\t13 & 14 & 15 & 16 \n\\end{array}\n\\right)\n$$\n\nThen, it will be transformed into the dictionary, with $11$ missing entries: \n\n\\begin{minted}[bgcolor=bg]{python}\nquboDict: {\n\t('n0t0', 'n0t0'): 1,\n\t('n0t0', 'n0t1'): 2,\n\t('n0t0', 'n1t0'): 3,\n\t('n0t0', 'n1t1'): 4,\n\t...\n\t('n1t1', 'n1t1'): 16,\n}\n\\end{minted}\n\nIn particular, this experiment is applied to the already studied example with the following four reads:\n\n\\begin{itemize}\n\t\\item $r_0 = ATGGCGTGCA$\n\t\\item $r_1 = GCGTGCAATG$\n\t\\item $r_2 = TGCAATGGCG$\n\t\\item $r_3 = AATGGCGTGC$\n\\end{itemize}\n\nThe pair-wise distances are computed, providing a direct measure of much two reads overlap. Then, these values are normalized for easier use. The resulting normalized TSP matrix is transformed into a QUBO matrix using $1.6$ as multi-location and repetition penalties, and $-1.6$ for self-bias, as done in \\cite{Sarkar2020}. Finally, we initialize an \\emph{ExactSolver} and use it to sample every possible solution.\n\nAfter the execution is completed, we find the lowest energy in the obtained solutions and print every solution with that energy:\n\n\\begin{minted}[bgcolor=bg]{python}\n{'n0t0': 0, 'n0t1': 0, 'n0t2': 1, 'n0t3': 0,\n 'n1t0': 0, 'n1t1': 0, 'n1t2': 0, 'n1t3': 1,\n 'n2t0': 1, 'n2t1': 0, 'n2t2': 0, 'n2t3': 0,\n 'n3t0': 0, 'n3t1': 1, 'n3t2': 0, 'n3t3': 0} --> -7.9811\n\n{'n0t0': 0, 'n0t1': 1, 'n0t2': 0, 'n0t3': 0,\n 'n1t0': 0, 'n1t1': 0, 'n1t2': 1, 'n1t3': 0, \n 'n2t0': 0, 'n2t1': 0, 'n2t2': 0, 'n2t3': 1, \n 'n3t0': 1, 'n3t1': 0, 'n3t2': 0, 'n3t3': 0} --> -7.9811\n\n{'n0t0': 1, 'n0t1': 0, 'n0t2': 0, 'n0t3': 0,\n 'n1t0': 0, 'n1t1': 1, 'n1t2': 0, 'n1t3': 0, \n 'n2t0': 0, 'n2t1': 0, 'n2t2': 1, 'n2t3': 0, \n 'n3t0': 0, 'n3t1': 0, 'n3t2': 0, 'n3t3': 1} --> -7.9811\n\n{'n0t0': 0, 'n0t1': 0, 'n0t2': 0, 'n0t3': 1,\n 'n1t0': 1, 'n1t1': 0, 'n1t2': 0, 'n1t3': 0,\n 'n2t0': 0, 'n2t1': 1, 'n2t2': 0, 'n2t3': 0,\n 'n3t0': 0, 'n3t1': 0, 'n3t2': 1, 'n3t3': 0} --> -7.9811\n\\end{minted}\n\nAs expected, these are the four minimums of the cost functions, representing the four ways of describing a type A loop in the used codification.\n\nFinally, let us plot the landscape of solutions for the given reads. We sort the solutions by increasing energy and simply plot their energy, as seen in figure \\ref{fig:exp1-landscape}.\n\n\\begin{figure}[H]\n\t\\makebox[\\textwidth][c]{\n\t\t\\includegraphics[scale=0.7]{experiments/experiment1.png}\n\t}\n\t\\centering\n\t\\caption{Landscape of solutions}\n\t\\label{fig:exp1-landscape}\n\\end{figure}\n\n\n\\subsection{Experiment 2: Simulated Quantum Annealing}\n\nFor our second experiment, we aim to solve our example using Simulated Annealing. Although \\emph{dimod} provides a simulated annealing sampler, we use for our final experiments the \\emph{SimulatedAnnealingSampler} from \\emph{Ocean}'s \\emph{Neal} package as it yields more distributed results and has far better computing time. The same experiment was reproduced $10$ times using both samplers in my machine. Dimod's sampler run in a mean time of $11.76$ seconds while \\emph{Neal}'s run in a mean time of $0.0610$ seconds.\n\nThe pseudo-code does not change much from the first experiment to the second one, we simply change the annealer:\n\n\\begin{algorithm}\n\t\\caption*{\\textbf{Experiment 2}}\n\t\n\tData preparation:\n\t\\begin{itemize}\n\t\t\\item Compute the distance between every two reads, creating an adjacency matrix.\n\t\t\\item Transform the TSP adjancency matrix into a QUBO $Q$ matrix, as explained in section \\ref{sec:tsp-qubo}.\n\t\t\\item Transform the QUBO matrix into an adjacency dictionary.\n\t\\end{itemize}\n\t\n\tSolving:\n\t\\begin{itemize}\n\t\t\\item Initiaize a sampler: \\textbf{\\emph{neal.SimulatedAnnealingSampler}}.\n\t\t\\item \\textbf{Sample from} the prepared QUBO model using the selected \\textbf{sampler}.\n\t\\end{itemize}\n\t\n\tPresent the obtained samples.\n\\end{algorithm}\n\nIt is worth mentioning that we do not \\emph{solve} the QUBO model in these experiments, we \\emph{sample} different results using simulated annealing.\n\nSince \\emph{Neal}'s sampler is quite efficient we were able to execute experiments with up to $10.000$ repetitions of the experiment, also called \\emph{sample reads} or simply \\emph{reads}. We developed an automated and scalable way to check whether a given cycle returned by the annealer was valid, and to recover its associated type. Using these tools we can easily check the number of occurrences each type of cycle appeared in the obtained sample reads. These results are presented, along with the associated energy to each result, in table \\ref{tab:exp2} and figure \\ref{fig:exp2-occ}. See figure \\ref{fig:overlap-cycles} to recall the cycle types and cost computing.\n\n\\begin{table}[H]\n\t\\centering\n\t\\begin{tabular}{lrr}\n\t\t\\textbf{Cycle type} & \\textbf{Occurences} & \\textbf{Energy} \\\\\n\t\t\\hline\n\t\tType A\t& 3722\t& -7.9811\t\\\\\n\t\tType C\t& 1474\t& -7.4541\t\\\\\n\t\tType D\t& 1469\t& -7.4541\t\\\\\n\t\tType F\t& 1458\t& -7.4541\t\\\\\n\t\tType E\t& 1431\t& -7.4541\t\\\\\n\t\tType B\t& 446\t& -6.927 \n\t\\end{tabular}\n\t\\caption{Results of experiment 2}\n\t\\label{tab:exp2}\n\\end{table}\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.8]{experiments/experiment2.png}\n\t\\centering\n\t\\caption{Ocurrences of each type in a $10.000$ reads experiment using Simulated Annealing}\n\t\\label{fig:exp2-occ}\n\\end{figure}\n\nWe can appreciate in figure \\ref{fig:exp2-occ} that the best type of cycle agglomerates most of the samples, $37.22\\%$. In the second place, the four types of cycles that have the exact same energy also have almost exactly the same number of samples. This matches our quantum annealing theory: the physical system has an equal probability of ending in eigenstates with equal eigenenergies.\n\nIt is worth mentioning that there was not a single sample that encoded an invalid cycle. This means that the penalties values used for the experiment ($1.6$ for multi-location and repetition, and $-1.6$ for self-bias) are working to prevent invalid cycles.\n\nNow that we are familiar with the type of results and codifications we may find, let us jump to the quantum realm by using the D-Wave's Quantum Computer Systems.\n\n\n\\subsection{Experiment 3: Quantum Annealing using D-Wave}\n\n\nFor our third experiment, we aim to solve our example using D-Wave's quantum annealers. For this purpose, we need to add some extra steps to our data processing. Section \\ref{sec:embeddings} explained how not every graph can be directly mapped to the existing Chimera / Pegasus topologies that D-Wave systems use, and how we can overcome this problem by embedding our problems graphs into these topologies. The extra steps in our data processing are related to these embeddings: we will need to embed our graph into the fixed topology that will be used. Then, after the anneal takes place, we will translate the solutions from the embedded graph. Ocean provides a set of utilities to deal with these embeddings \\cite{DWave-OceanDoc-Embedding}.\n\nAdditionally, we will need to connect to Leap to send our jobs to the quantum annealers using a \\emph{client}. First, a configuration file is created, which includes your API key and some extra configuration details:\n\n\\begin{minted}[bgcolor=bg]{bash}\n[ocete]\nsolver = {\"qpu\": true}\ntoken = DEV-\nendpoint = https://cloud.dwavesys.com/sapi\n\\end{minted}\n\nIn our code we initialize a \\emph{client} and a \\emph{solver} using the configuration file. These are objects that encapsulate the connection functionality and solving problems in the associated machine respectively. We also initialize a sampler as we did before, but this time a production sampler is used: a \\emph{DWaveSampler} that also needs our configuration file. The pseudo-code for this experiment can be found below. Each new step will be explained in detail below.\n\n\\begin{algorithm}\n\t\\caption*{\\textbf{Experiment 3}}\n\t\n\tInitialization:\n\t\\begin{itemize}\n\t\t\\item Initiaize a sampler: \\textbf{\\emph{system.samplers.DWaveSampler}}.\n\t\t\\item \\textbf{Initialize a client and obtain an associated solver}.\n\t\\end{itemize}\n\t\n\tData preparation:\n\t\\begin{itemize}\n\t\t\\item Compute the distance between every two reads, creating an adjacency matrix.\n\t\t\\item Transform the TSP adjacency matrix into a QUBO $Q$ matrix, as explained in section \\ref{sec:tsp-qubo}.\n\t\t\\item Transform the QUBO matrix into an adjacency dictionary.\n\t\t\\item \\textbf{Find an embedding from our graph to the selected machine's topology (either Chimera or Pegasus)}.\n\t\t\\item \\textbf{Use the previous embedding to create a new QUBO model equivalent for the embedded graph}.\n\t\\end{itemize}\n\t\n\tSolving:\n\t\\begin{itemize}\n\t\t\\item Use the solver to send a job to the client. This job samples from the prepared QUBO model multiple times.\n\t\\end{itemize}\n\t\n\tFormat the results:\n\t\\begin{itemize}\n\t\t\\item \\textbf{Use the computed embedding to translate our answers}.\n\t\\end{itemize}\n\\end{algorithm}\n\nThe first thing to be noticed is that the initialization now needs to be done before the data preparation. This is because, in order to find the embedding, we need to know which exact topology the quantum system will have. This information is provided through the \\emph{solver}, previously initialized. The initialization step is as simple as follows:\n\n\\begin{minted}[bgcolor=bg]{python}\nimport dwave\n\n# Create the solver (connecting to D-Wave) and the Sampler\nconfig_file='../dwave.conf'\nclient = cloud.Client.from_config(config_file, profile='ocete')\nsolver = client.get_solver()\ndwsampler = system.samplers.DWaveSampler(config_file=config_file)\n\\end{minted}\n\nSuppose $Q$ already holds our computed QUBO model. We can compute an embedding and obtained the new associated model as follows:\n\n\\begin{minted}[bgcolor=bg]{python}\nadjacency_dict = embedding.utils.edgelist_to_adjacency(solver.edges)\nembedding = minorminer.find_embedding(Q, solver.edges)\nQ_embedded = embed_qubo(Q, embedding, adjacency_dict)\n\\end{minted}\n\nBy using the same configuration file, the sampler already knows what machine it is associated to. We will sample from it, with the same syntax as in the second experiment:\n\n\\begin{minted}[bgcolor=bg]{python}\nresponse = dwsampler.sample_qubo(Q_embedded, num_reads=num_reads)\n\\end{minted}\n\nWhere $num\\_reads$ is a parameter that sets the number of samples to be read. However, these solutions are associated to the $Q\\_embedded$, not to our original $Q$ model. We need to translate the responses to understand them:\n\n\\begin{minted}[bgcolor=bg]{python}\nbqm = dimod.BinaryQuadraticModel.from_qubo(Q)\nunembedded_response = embedding.unembed_sampleset(response, embedding, bqm)\n\\end{minted}\n\nFor this experiment, the D-Wave 2000Q was used (see section \\ref{sec:d-wave-systems} to see its characteristics), since it was the default solver. After running the experiment with the same parameters as the simulated annealing experiment ($10.000$ reads and $(-1.6; 1.6; 1.6)$ for the QUBO parameters), we obtained that more than $90\\%$ of the samples were invalid, a huge difference with the astonishing $0\\%$ obtained in the second experiment.\n\nMy initial hypothesis was that the QUBO parameters were well adjusted for SA but not for QA, and thus allowed further tunning. After many attempts with different parameters values, this hypothesis was discarded. The only other set of parameters worth presenting is $(-1.5; 1.5; 1.5)$. These results are presented in table \\ref{tab:exp3}.\n\n\\begin{table}[H]\n\t\\centering\n\t\\begin{tabular}{lrrr}\n\t\t\\textbf{Cycle type} & \\textbf{Occurences (param=1.5)} & \\textbf{Occurences (param=1.6)} & \\textbf{Energy} \\\\\n\t\t\\hline\n\t\tType A\t& 131\t& 50\t& -7.9811\t\\\\\n\t\tType C\t& 124\t& 46\t& -7.4541\t\\\\\n\t\tType D\t& 40\t& 91\t& -7.4541\t\\\\\n\t\tType F\t& 55\t& 117\t& -7.4541\t\\\\\n\t\tType E\t& 112\t& 118\t& -7.4541\t\\\\\n\t\tType B\t& 99\t& 64\t& -7.4541\t\\\\ \n\t\tInvalid & 9221\t& 9194\t& $>$ -5.6433 \n\t\\end{tabular}\n\t\\caption{Results of experiment 3, $10.000$ reads using the quantum annealer.}\n\t\\label{tab:exp3}\n\\end{table}\n\nIn figures \\ref{fig:exp3-occ1} and \\ref{fig:exp3-occ2} we see a comparison of the distribution between the different type of cycles in both experiments. We cannot see a distribution as we did in \\ref{fig:exp2-occ}, potentially due to the low numbers of reads that actually represent a valid cycle. In fact, such distribution between the cycle types is not needed: a single sample with the lowest energy solution will completely solve our genome assembly problem.\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.75]{experiments/experiment3 (1.5).png}\n\t\\centering\n\t\\caption{Ocurrences of each cycle type in a $10.000$ reads experiment using Quantum Annealing, filtering out invalid cycles, with QUBO parameters $(-1.5; 1.5; 1.5)$}\n\t\\label{fig:exp3-occ1}\n\\end{figure}\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.75]{experiments/experiment3 (1.6).png}\n\t\\centering\n\t\\caption{Ocurrences of each cycle type in a $10.000$ reads experiment using Quantum Annealing, filtering out invalid cycles, with QUBO parameters $(-1.6; 1.6; 1.6)$}\n\t\\label{fig:exp3-occ2}\n\\end{figure}\n\nWith these results in mind, is it possible that we are sampling completely random solutions? In a 4-reads problem, we obtain a 16 binary variables QUBO model. That means there are up to $2^{16}$ possible solutions using our encoding, even more after we embed it in the Chimera topology. If we were to randomly sample from that solution space, we will never get almost $10\\%$ of samples from a subset of $24$ solutions that represent our $6$ valid cycles. So we do know that the quantum system is working, just not as good as expected.\n\nWe have not found a clear improvement precision-wise in our small example, although QA does solve the problem. What about time-wise? In figure \\ref{fig:exp3-time} we see the time breakdown provided by Leap after our last experiment. We can see a total QPU sampling time of $2.389$ seconds, while if we repeat the experiment with the same parameters using SA we obtain a mean of $2.525$ sampling seconds, over $10$ experiment repetitions. Keeping in mind that our QA time data comes from a single execution and the small difference between both time measures, we cannot conclude that either approach has any time advantage in such small problems.\n\nHowever, does QA actually scales better than SA with the problem size? In the next experiment, we will study the Pegasus topology to understand what is the maximum problem size that can be tackled using the Advantage system. Finally, the last couple of experiments will try to answer the scalability comparison inquiry.\n\n\\begin{figure}[h]\n\t\\includegraphics[scale=0.5]{experiments/exp3-time.png}\n\t\\centering\n\t\\caption{Time breakdown provided by Leap after the last experiment.}\n\t\\label{fig:exp3-time}\n\\end{figure}\n\n\n\\subsection{Displaying chimera embeddings}\n\n\nLet us plot some logical embeddings using the D-Wave 2000Q system, with relies on a Chimera graph. In figure \\ref{fig:exp2_4reads} we see the embedding of a 4-nodes complete graph into the Chimera topology. The grey nodes represent unused nodes while nodes with the same color represent a single logical qubit mapped into multiple physical qubits.\n\n\\begin{figure}[h]\n\t\\includegraphics[scale=0.6]{experiments/exp2_4reads.png}\n\t\\centering\n\t\\caption{Embedding of a 4-nodes complete graph into a Chimera topology.}\n\t\\label{fig:exp2_4reads}\n\\end{figure}\n\nThe last embedding had a \\emph{maximum chain length} of $2$. That is, the longest chain in a logical qubit is 2 physical qubits. This is a critical parameter to be minimized in embeddings: the longer the qubit chain is, the more probable it is for the qubits in a logical qubit to be desynchronized and end up in different states. Thus, the more difficult it is for the system to stay in the ground state.\n\nIn figure \\ref{fig:exp2_13reads} we see an embedding of a 13-nodes complete graph into a Chimera topology. In this case, the maximum chain length is $5$, obtained by both the dark blue, purple, red, dark green, lime, and cyan logical qubits. In the rest of the experiments, I looked at the maximum chain length obtained in the pegasus embeddings, obtained lengths up to $15$ qubits. Since I did not have the time to study this characteristic in-depth, it will be a future line of work.\n\n\\begin{figure}[h]\n\t\\includegraphics[scale=0.6]{experiments/exp2_13reads.png}\n\t\\centering\n\t\\caption{Embedding of a complete 13-nodes complete graph into a Chimera topology.}\n\t\\label{fig:exp2_13reads}\n\\end{figure}\n\nRecall that in every 16-nodes chimera cell, every node in the vertical axis is connected to every node in the horizontal axis, and to no other node from the vertical axis. For example, take the blue logical qubit, represented by 3 nodes at the very bottom in figure \\ref{fig:exp2_13reads}. In the first cell, it connects with lime, pink, dark green, and cyan. In the second cell, it connects with light green, purple, red, and dark blue. In the third cell, it connects to brown, light blue, pink, and yellow. That makes a total of 12 connections, as expected since the embedded started with a complete 13-nodes graph.\n\n\n\\FloatBarrier\n\\subsection{Experiment 4: D-Wave's Advantage limits}\n\n\nFor our fourth experiment, we aim to test the limits of the Advantage architecture for our problem. That is, given a number of DNA reads, we will compute the associated QUBO problem and try to embed the obtained graph into the Advantage architecture. We will test these embeddings against a Pegasus 16 ($P_{16}$) topology since that is the one supported by the Advantage system.\n\nIn fact, given a set of $n$ reads, this is simply finding if a $n^2$-nodes complete graph ($K_{n^2}$) can be embedded into a $P_{16}$. The reads themselves are not relevant for the embedding. However, I have developed an automated way of creating fake tests, originally for a general-purpose, but it can also be used here.\n\nThis automated test production works as follows: Given a number of reads, the size of each chain (default value $150$), and the required overlap between adjacent chains (default value $50$), it randomly creates the original DNA chain, and then chop it using the given parameters. This makes sure that adjacent reads have enough overlap so it is basically impossible for two random reads to overlap more than $50$ bps.\n\nUsing these random tests we created graphs of an increasing number of reads until we were unable to find an embedding from $K_{n^2}$ to $P_{16}$. The results of the experiment can be seen in table \\ref{tab:exp4}. We can see that a valid embedding was found to problem size up to 14 reads ($K_{196}$ associated graph). The embedding time, as well as the test generation time, are also displayed. I decided to display both since, as time grew larger and larger, I was not sure whether the test generation was adding too much of an overload.\n\n\\begin{table}[H]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\textbf{Number of reads} & \\textbf{Nodes in graph} & \\textbf{Test generation time} & \\textbf{Embedding time} & \\textbf{Total time} \\\\\n\t\t\t\\hline\n\t\t\t2 & 4 & 0.000157 & 0.081310 & 0.081708 \\\\\n\t\t\t3 & 9 & 0.000216 & 0.108525 & 0.109205 \\\\\n\t\t\t4 & 16 & 0.000269 & 0.194807 & 0.196045 \\\\\n\t\t\t5 & 25 & 0.000335 & 0.377673 & 0.379814 \\\\\n\t\t\t6 & 36 & 0.000392 & 1.147603 & 1.154559 \\\\\n\t\t\t7 & 49 & 0.000433 & 3.514801 & 3.523473 \\\\\n\t\t\t8 & 64 & 0.000497 & 11.876785 & 11.888235 \\\\\n\t\t\t9 & 81 & 0.000558 & 19.226392 & 19.244367 \\\\\n\t\t\t10 & 100 & 0.000610 & 43.457666 & 43.477268 \\\\\n\t\t\t11 & 121 & 0.000676 & 62.499896 & 62.528545 \\\\\n\t\t\t12 & 144 & 0.000724 & 89.277396 & 89.313831 \\\\\n\t\t\t13 & 169 & 0.000812 & 63.782066 & 63.827706 \\\\\n\t\t\t14 & 196 & 0.000840 & 152.262340 & 152.316223\n\t\t\\end{tabular}\n\t}\n\t\\caption{Results of experiment 4}\n\t\\label{tab:exp4}\n\\end{table}\n\nIt is worth noting that the algorithm used to find the embeddings does not guarantee to find a valid one if it exists. I tried to run the algorithm $50$ times with $15$ reads and could not find a valid embedding.\n\nThe reader might have already noticed that the embeddings are quite time-consuming. In fact, this is an NP problem itself and time will grow exponentially on input size. However, keep in mind that this embedding does not depend on the reads. Once a single embedding from a $K_{196}$ to a $P_{16}$ has been pre-computed it can be used -or further optimized- for different problems.\n\nNow that we know the maximum number of reads that the Advantage system may tackle, let us put it to the test with 'big' input problems in the next experiment.\n\n\n\\subsection{Experiment 5: Scalability comparison between Simulated Annealing and Quantum Annealing}\n\n\nIn our fifth experiment, we put the D-Wave's quantum annealers to the test with bigger input problems and study how they behave compared with the simulated annealing experiments. We will start with a $3$ reads and then progressively increase this value, creating tests of this size and trying to solve them using both the simulated annealer and the Advantage system.\n\nFor each fixed number of reads, we will sample $10.000$ times using the annealers, with the same set of parameters used for the previous experiments. The number of valid solutions (i.e. valid cycles) will be displayed, along with the number of cycles that reached the real solution, which we know in advance since we created the test.\n\nWe will also display the energy of the best sample obtained, along with the difference between the real solution energy and the best sample obtained, which I called \\emph{energy delta}. Finally, we will also display the sampling time, not including the time it took to create the test nor to prepare the data.\n\nThe results of this experiment using the simulated annealer can be seen in table \\ref{tab:exp5_1}. \n\n\\begin{table}[H]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\textbf{N. reads} & \\textbf{Valid cycles} & \\textbf{Times sol. reached} & \\textbf{Energy delta} & \\textbf{Solution energy} & \\textbf{Sampling time} \\\\\n\t\t\t\\hline\n\t\t\t3\t& 10000\t\t& 9564\t\t\t& 0\t\t\t\t& -6.214214\t\t& 1.881374 \\\\\n\t\t\t4\t& 10000\t\t& 6852\t\t\t& 0\t\t\t\t& -8.131935\t\t& 2.178147 \\\\\n\t\t\t5\t& 10000\t\t& 3427\t\t\t& 0\t\t\t\t& -9.999001\t\t& 3.487972 \\\\\n\t\t\t6\t& 10000\t\t& 1308\t\t\t& 0\t\t\t\t& -11.834192\t& 5.121054 \\\\\n\t\t\t7\t& 10000\t\t& 351\t\t\t& 0\t\t\t\t& -13.656181\t& 7.057235 \\\\\n\t\t\t8\t& 10000\t\t& 108\t\t\t& 0\t\t\t\t& -15.443562\t& 9.680610 \\\\\n\t\t\t9\t& 10000\t\t& 21\t\t\t& 0\t\t\t\t& -17.232667\t& 12.797996 \\\\\n\t\t\t10\t& 10000\t\t& 3\t\t\t\t& 0\t\t\t\t& -18.995278\t& 16.080461 \\\\\n\t\t\t11\t& 10000\t\t& 0\t\t\t\t& 0.625356\t\t& -20.764680\t& 20.118828 \\\\\n\t\t\t12\t& 10000\t\t& 0\t\t\t\t& 0.601820\t\t& -22.510011\t& 25.351562 \\\\\n\t\t\t13\t& 10000\t\t& 0\t\t\t\t& 0.570361\t\t& -24.256735\t& 30.612547 \\\\\n\t\t\t14\t& 10000\t\t& 0\t\t\t\t& 1.378872\t\t& -26.005003\t& 34.897780\n\t\t\\end{tabular}\n\t}\n\t\\caption{Results of experiment 5, $10.000$ reads using the simulated annealer.}\n\t\\label{tab:exp5_1}\n\\end{table}\n\nWe stopped at 15 reads since that is the maximum value that will be able to compare with using the quantum annealer. In the experiment results, we see how the number of valid cycles out of $10.000$, which turns out to be every single cycle up to 14 reads. However, the number of samples that reached the best solution go down from $95\\%$ to ground cero from $11$ reads onward. Although we do not obtain the real solution, how close are these samples to it? \n\nThe columns related to the energy of the solutions try to answer this question. We may see how the solution's energy goes down to around $-26$, which is totally normal: the more reads in our graph, the higher the number of reads overlap, and the lower is the resulting solution energy. As soon as we cannot find the solution, the energy delta starts to increase, going up to around $1.37$. This means the energy of the best sample is quite close to the actual value. Is this enough?\n\nWhen we tackle a classic TSP, a near-optimal solution is a really valid and useful solution. We do not necessarily need the actual minimum. However, although a valid cycle with low energy represents a valid way of sorting and sewing the genome reads, it does not provide the real genome the reads came originally from. It has some value since it could be further tuned to obtain the solution, but it does not solve our problem.\n\nLastly, the sampling time seems to grow linearly with the number of reads. However, using this simple simulated annealer for a real case with up to 3 Mbps would be simply impossible.\n\nFor the second part of this experiment, I run the same experiment but using the quantum annealer. Some extra steps come with it, like computing the embedding and connecting to Leap, but they have all been explained in previous experiments. We again sample $10.000$ times a maintain the same values for the annealing schedule self-bias, multi-location, and repetition parameters. The results of this experiment can be seen in table \\ref{tab:exp5_2}.\n\n\\begin{table}[H]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\n\t\t\\begin{tabular}{cccccc}\n\t\t\t\\textbf{N. reads} & \\textbf{Valid cycles} & \\textbf{Times sol. reached} & \\textbf{Energy delta} & \\textbf{Solution energy} & \\textbf{Sampling QPU time} \\\\\n\t\t\t\\hline\n\t\t\t3\t& 4574\t\t& 3295\t\t& 0\t\t\t\t& -6.214214\t\t& 1.035 \\\\\n\t\t\t4\t& 683\t\t& 156\t\t& 0\t\t\t\t& -8.131935\t\t& 0.962 \\\\\n\t\t\t5\t& 88\t\t& 6\t\t\t& 0\t\t\t\t& -9.999900\t\t& 1.272 \\\\\n\t\t\t6\t& 2\t\t\t& 0\t\t\t& -1.322536\t\t& -11.834014\t& 1.272 \\\\\n\t\t\t7\t& 1\t\t\t& 0\t\t\t& -1.647457\t\t& -13.671186\t& 1.206 \\\\\n\t\t\t8\t& 0\t\t\t& 0\t\t\t& -3.867139\t\t& -15.444996\t& 1.339 \\\\\n\t\t\t9\t& 0\t\t\t& 0\t\t\t& -36.592409\t& -17.227226\t& 1.570 \\\\\n\t\t\t10\t& 0\t\t\t& 0\t\t\t& -50.03568\t\t& -18.468571\t& 1.276\n\t\t\\end{tabular}\n\t}\n\t\\caption{Results of experiment 5, $10.000$ reads using the quantum annealer.}\n\t\\label{tab:exp5_2}\n\\end{table}\n\nThe first thing we may notice is that the experiment did not reach 14 reads. That is because I manually stopped it before it finished. The available QPU time was quite tight at the moment, and seeing the results of the experiment with a number of reads up to $10$ showed that we were not going to learn anything new by letting the experiment finished and wasting our remaining QPU time.\n\nBack to the results, we can appreciate the number of valid cycles rapidly goes down to $0$, and even quicker the number of reads that reached the solution. At 4 reads we see similar results to what we saw in experiment 3: around $700$ valid cycles and $150$ times the real solution (type A cycle in the third experiment) was reached. Although we have developed this experiment in different machines, D-Wave 2000Q and Advantage, keep in mind that the improvements from one generation to another are size-related: we may tackle bigger (and thus harder) problems by using the Advantage system. It does not mean that its computation is somehow better.\n\nNot only the responses are not valid, but they are also quite far from the solution. We can see that in how the delta grows hugely as we increase the number of reads. Leap provides a graph for each submitted experiment displaying the samples' distribution. In figures \\ref{fig:exp5_1} and \\ref{fig:exp5_2} we see this distribution for $4$ and $10$ reads respectively. \n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.5]{experiments/exp5_1.png}\n\t\\centering\n\t\\caption{$10.000$ samples' distribuition for $4$ reads using the quantum annealer.}\n\t\\label{fig:exp5_1}\n\\end{figure}\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.5]{experiments/exp5_2.png}\n\t\\centering\n\t\\caption{$10.000$ samples' distribuition for $10$ reads using the quantum annealer.}\n\t\\label{fig:exp5_2}\n\\end{figure}\n\nWe can see how in the $4$ reads experiment, most of the samples group around $0$ energy, and only a very few are close to the solution, which is actually reached in this execution. However, in the $10$ reads experiment, most of the samples are gathered around $100/150$ energy, these solutions are not only invalid but really far from being so.\n\nLastly, let us comment on the QPU sampling time. This value is provided directly by Leap. From our code, we cannot separate the connection overload and other computations from the actual QPU sampling time. We may appreciate how time does not grow with the input. This is because the time it takes for the annealer to converge is fixed by the annealing schedule.\n\nIn conclusion, these results are astonishingly awful. The simulated annealing sampler not only provides closer samples to the solution, but in every seen case it reaches valid samples. My current hypothesis is that either the annealing schedule is not optimal, or that the values of the parameters can be improved for our specific problem. In our last experiment, I will spend all our remaining QPU time checking these two hypotheses.\n\n\n\\subsection{Experiment 6: Advantage tunning}\n\n\nFor our last experiment, we aim to test our previous hypothesis and try to solve a $10$ reads problem using the Advantage system. In the first set of iterations, I tried to increase the multi-location and repetition penalties in order to increase the number of valid cycles sampled. For this experiment, a single test with $10$ reads was created. Its solution energy was $-18.994150$. The results of the experiment can be found in table \\ref{tab:exp6_1}, where the mentioned parameters are \\emph{(self-bias, multi-location, repetition)}.\n\n\\begin{table}[H]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\textbf{Parameters} & \\textbf{Valid cycles} & \\textbf{Times sol. reached} & \\textbf{Energy delta} & \\textbf{Sampling QPU time} \\\\\n\t\t\t\\hline\n\t\t\t(-1.6, 1.6, 1.6)\t& 0\t\t& 0\t\t\t& -76.344431\t&\t1.500 \\\\\n\t\t\t(-3, 3, 3)\t\t\t& 0\t\t& 0\t\t\t& -87.263801\t&\t1.624 \\\\\n\t\t\t(-5, 5, 5)\t\t\t& 0\t\t& 0\t\t\t& -83.657639\t&\t1.272 \\\\\n\t\t\t(-10, 10, 10)\t\t& 0\t\t& 0\t\t\t& -258.290739\t&\t1.248 \\\\\n\t\t\t(-20, 20, 20)\t\t& 0\t\t& 0\t\t\t& -357.917063\t&\t1.311 \\\\\n\t\t\t(-50, 50, 50)\t\t& 0\t\t& 0\t\t\t& -1017.605383 \t&\t1.441 \\\\\n\t\t\t(-100, 100, 100)\t& 0\t\t& 0\t\t\t& -1418.541757 \t&\t1.261\n\t\t\\end{tabular}\n\t}\n\t\\caption{Results of experiment 6, $10.000$ reads using the quantum annealer for different parameters configurations.}\n\t\\label{tab:exp6_1}\n\\end{table}\n\nAs we can see in the table, we did not find a single valid cycle in these iterations. The energy delta naturally increases as we increase the parameter values: every single penalty is greater penalized. However, the energy delta barely increases until our parameters get a value of $10$. This could show how the responses for \\emph{parameters} $= (-5, 5, 5)$ incurred in a lower number of penalties. In figure \\ref{fig:exp6_1} we find the samples distribution for \\emph{parameters} $= (-5, 5, 5)$. The overall energy is much higher than what we saw for \\emph{parameters} $= (-1.6, 1.6, 1.6)$ in figure \\ref{fig:exp5_2}, which occurs due to the higher penalty. This experiment concludes that either we have not been able to find the correct values for the parameters, or the values of the parameters are not the problem.\n\n\\begin{figure}[H]\n\t\\includegraphics[scale=0.5]{experiments/exp6_1.png}\n\t\\centering\n\t\\caption{$10.000$ samples distribuition for \\emph{parameters} $= (-5, 5, 5)$.}\n\t\\label{fig:exp6_1}\n\\end{figure}\n\nFinally, we try changing the annealing schedule. There are two possible parameters that we may modify in this sense: the \\emph{annealing time} -time let for the annealer to converge-, or the \\emph{annealing schedule} -which modifies the annealing functions as defined in figure \\ref{fig:dwave-annealing-functions}. Since my initial hypothesis was that the annealer did not have enough time to find good solutions, let us focussed on the former for our last experiment.\n\nThe anneal time is meassured in microseconds, with a default value of $20$. Although the initial idea was to start with that value and make it progressively higher, at this point I was able to submit only one more worthy experiment, with a anneal time of $40$ microseconds. This was due to a bug in the D-Wave $get\\_solver()$ function, which did not read the annealing time correctly and led to some invalid experiments at the last minute. We can see the obtained results in table \\ref{tab:exp6_2}.\n\n\\begin{table}[H]\n\t\\centering\n\t\\makebox[\\textwidth][c]{\n\t\t\\begin{tabular}{ccccc}\n\t\t\t\\textbf{Annealing time ($\\mu$s)} & \\textbf{Valid cycles} & \\textbf{Times sol. reached} & \\textbf{Energy delta} & \\textbf{Sampling QPU time} \\\\\n\t\t\t\\hline\n\t\t\t20\t& 0\t\t& 0\t\t\t& -76.344431\t& 1.500 \\\\\n\t\t\t40\t& 0\t\t& 0\t\t\t& -57.944162\t& 1.573\n\t\t\\end{tabular}\n\t}\n\t\\caption{Results of experiment 6, $10.000$ reads using different annealing times.}\n\t\\label{tab:exp6_2}\n\\end{table}\n\nAlthough the energy delta is reduced, the improvement is marginal: we still find no valid cycles and the responses have a high penalty. Probably with even more annealing time we will obtain further refined results.\n", "meta": {"hexsha": "ff9146d597d2405916bb972fbbd4d427981a4ba5", "size": 44568, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "thesis/chapters/chapter3_GenomeAssembly.tex", "max_stars_repo_name": "Ocete/TFG", "max_stars_repo_head_hexsha": "baf0853276aef8f6fd6c5cccbe69ede7ca7987c3", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2021-09-05T22:40:37.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-05T22:40:37.000Z", "max_issues_repo_path": "thesis/chapters/chapter3_GenomeAssembly.tex", "max_issues_repo_name": "Ocete/TFG", "max_issues_repo_head_hexsha": "baf0853276aef8f6fd6c5cccbe69ede7ca7987c3", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "thesis/chapters/chapter3_GenomeAssembly.tex", "max_forks_repo_name": "Ocete/TFG", "max_forks_repo_head_hexsha": "baf0853276aef8f6fd6c5cccbe69ede7ca7987c3", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-07-29T18:23:25.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-29T18:23:25.000Z", "avg_line_length": 70.7428571429, "max_line_length": 819, "alphanum_fraction": 0.7474645486, "num_tokens": 12611, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.34226221457390893}} {"text": "\\section{Preliminaries}\\label{sec:prelim}\n%\n{\\sc First Pass: assume fixed Godel T norm -- we can extend to other\n tnorms and to distribution semantics later. \n}\n\nThroughout this paper, we assume a general knowledge of logic\nprogramming terminology, including tabled resolution and the\nwell-founded semantics including both default and explicit negation as\nin \\cite{AlDP95}. Here, we include notation and definitions that will\nbe used for our proof of the correctness of Plow.\n\nAn {\\em annotated atom} is an atom $A$ associated with an annotation\n$n$ that is either a variable or $0 \\leq n \\leq 1$, denoted $A:n$.\nFrom an annotated atom {\\em A:n}, an {\\em objective literal} $O$ is\nformed as either $O = A:n$, termed a positive objective literal with\n$sign(O) = pos$; or aa $O = \\ourneg A:n$, termed a negative objective\nliteral with $sign(O) = \\ourneg$, and the \\ourneg{} symbol denotes\n{\\em explicit negation}. In either case the annotation, $n$, is\ndenoted as $ annotation(O)$, while the underlying atom, $A$, is\n$atom(A)$.\n%\nTwo objective literals $O_1$ and $O_2$ with the same underlying atom\nare {\\em homologs} if they have the same sign and {\\em conjugates}\notherwise. An objective literal $A:n$ is ground if both $A$ and $n$\nare ground.\n%\n% = A:n$ and $O_2 = \\neg A:m$ are termed {\\em conjugates}.\n%denoted as $O_1 = conjugate(O_2)$ or $O_2 = conjugate(O_1)$; \n%Otherwise if $atom(O_1) = atom(O_2)$ $O_1$ and $O_2$ are termed {\\em\n% homologs}.\n%\n%, denoted as $O_1 = homolog(O_2)$ or $O_2 = homolog(O_1)$.\nFrom an objective literal $O$ a default literal is formed as either a\npositive default literal $O$ or as a negative default literal $\\naf\nO$, where \\naf{} denotes {\\em default negation}. A default literal is\nsometimes simply called a {\\em literal}.\n\\footnote{\nWhen convenient, an objective literal $A:1$ ($\\neg A:1$) is denoted\nsimply as $A$ ($\\neg A$).}\n\nA rule has the form\n\\[r = O \\mif{} L_1,\\ldots,L_n\\]\nwhere $O$ is an objective literal and $L_0,\\ldots,L_n$ are default\nliterals.\n\nA rule $R$ is ground if all literals in $R$ are ground; a program\n$\\cP$ is ground if all rules in $\\cP$ are ground.\n\n\n%An objective literal $O$ is undefined in $\\cI$ if $O \\neg \\hat{\\in}\n\n%\\subsection{Three-Valued Models for Annotated Atoms} \n\nOur attention is restricted to three-valued (partial) interpretations\nand models such as those extending the well-founded model. Each such\ninterpretation $\\cI$ is represented as a pair of sets of ground\nobjective literals: ($\\cT$,$za\\cF$).\n\n%{\\sc TES: In the following, I dont yet feel confident about my $>$'s and my\n%$\\geq$'s. ... conflicting truth values are mapped to U, but in the\n%edge case, I'd like true to outweigh false (as with $r$ below).}\n\n\\begin{definition} \\label{def:satisfiable}\nLet $P$ be a ground objective literal $O$ with\n% $atom(O) = A$ and \n$annotation(O) = n$, and $\\cI$ an interpretation.\n\n$O$ is true in $\\cI$ if\n\\begin{itemize}\n\\item $\\cT$ contains an $O_T$ such that $O_T$ is a homolog of\n $O$ and $annotation(O_T) \\geq n$; and\n\\item $\\cT$ does not contain a conjugate $O_C$ of $O$ with\n $annotation(O_C) \\geq (1-n)$; and\n\\item $\\cF$ does not contain a homolog $O_F$ of $O$ with\n $annotation(O_F) \\geq (1-n)$.\n\\end{itemize}\n\n%---------------------------------------------------------------- \n$O$ is false in $\\cI$ if \n\\begin{itemize}\n\\item $\\cT$ does not contain a homolog $O_T$ of $O$ with\n $annotation(O_T) \\geq (1-n)$; and either\n\\begin{itemize} \n\\item $\\cF$ contains a homolog $O_F$ of $O$ with $annotation(O_F) \\geq\n n$; or \n\\item $\\cT$ contains a conjugate $O_C$ of $O$ with $annotation(O_C) \\geq n$.\n%Either $A:n_1 \\in \\cF$ for $n_1 \\geq m$ or $\\ourneg A:n_2 \\in \\cT$ for\n%$m_2 \\geq m$; and\n\\end{itemize}\n\\end{itemize}\n$O$ is $\\ourU$ in $\\cI$ if it is neither true nor false in $\\cI$.\n\nA positive literal $O$ is true (false) in $\\cI$ if $O$ is true (false)\nin $\\cI$; a negative literal $naf O$ is true in $\\cI$ if $O$ is false\nin $\\cI$ and is false in $\\cL$ if $O$ is true in $\\cI$. A literal is\n$\\ourU$ in $\\cI$ if it is neither true nore false in $\\cI$.\n\\end{definition}\n%---------------------------------------------------------------- \n\n%\\begin{definition} \\label{def:satisfiable}\n%A ground annotated atom $A:m$ is true in $\\cI$ if \n%\\begin{itemize}\n%\\item There exists an $A:n_1 \\in \\cT$ with $n_1\\geq m$, and \n%\\begin{itemize}\n%\\item $\\cT$ does not contain $\\ourneg A:n_2$ with $n_2 > (1-m)$ and\n%\\item $\\cF$ does not contain $A:n_3$ with $n_3 > (1-m)$.\n%\\end{itemize}\n%\\end{itemize}\n%A ground annotated atom $A:m$ is false in $\\cI$ if \n%\\begin{itemize}\n%\\item Either $A:n_1 \\in \\cF$ for $n_1 \\geq m$ or $\\ourneg A:n_2 \\in \\cT$\n% for $m_2 \\geq m$; and\n%\\item $\\cT$ does not contain $A:n_3$ for $n_3 \\geq (1-m)$.\n%\\end{itemize}\n%\\end{definition}\n\nNote that in the above definition the truth value \\ourU{} captures\nboth the traditional case where a literal is undefined, along with the\ncase where a literal is overdefined.\n\n\\begin{example} \n Consider the interpretation $\\cI_1$ where\n\\begin{itemize}\n\\item $\\cT = \\{p:0.7,q:0.5, r:0.6,\\ourneg r:0.5\\}$\n\\item $\\cF = \\{p:0.4, q:0.5\\}$.\n\\end{itemize}\n$p:0.6$ is true in $\\cI$ and $p:0.8$ false, but $p:0.7$ is \\ourU ;\n$q:0.5$ is \\ourU{} in $\\cI$. $r:0.4$ is true in $\\cI$ but $r:0.7$ is\nfalse; $r:n$ is \\ourU{} for $0.5 \\leq n \\leq 0.6$.\n\\end{example}\n\n%\\cT$ and $O \\neg \\hat{\\in} \\cF$.\n\n%----------------------------------------------------------------\n\n%---------------------------------------------------------------- \n\\subsubsection{Well-Founded Model}\n\nMotivation: consider the program\n\\begin{verbatim}\np:0.8:- not p:0.8.\np:0.5.\nneg p:0.3.\n\\end{verbatim}\nWe want a model with p:0.5 as true, p:0.7 as false and p:0.6 as u.\nHowever, if the rule {\\tt p:0.8:- not p:0.8.} were removed, p:0.6\nwould be false.\n\n\\comment{\n%For two interpretations, $\\cI$ and $\\cJ$, $\\cI\n%\\subseteq \\cJ$ iff $true(\\cI) \\subseteq true(\\cJ)$ and $false(\\cI)\n%%\\subseteq false(\\cJ)$. Alternatively, a three-value interpretation\n%can be represented as a set of literals.\n\nA program $P$ is {\\em safe} if each rule $r$ in $P$ is such that every\nvariable in $r$ occurs in a positive literal in the body of $r$. \n\nSymbols within a term may be represented through {\\em positions} which\nare members of the set $\\Pi$. A {\\em position\\/} in a term is either\nthe empty string $\\Lambda$ that reaches the root of the term, or the\nstring $\\pi.i$ that reaches the $i$th child of the term reached by\n$\\pi$, where $\\pi$ is a position and $i$ an integer. For a term $t$\nwe denote the symbol at position $\\pi$ in $t$ by $t_\\pi$. For\nexample, $p(a,f(X))_{2.1} = X$. We assume that a program $P$ is\ndefined over a language $\\cL$, containing a finite set $\\functions$ of\npredicate and function symbols, and a countable set of variables from\nthe set ${\\cal V} \\cup \\posvar$. Elements of the set $\\cal V$ are\nreferred to as {\\em program variables}. Elements of the set\n$\\posvar$, called {\\em position variables}, are of the form $X_\\pi$,\nwhere $\\pi$ is a position.\n%\nThese variables are used when it is convenient to mark certain\npositions of interest in a term. The Herbrand Universe of $\\cL$ is\ndenoted $\\cH_{\\cL}$, or as $\\cH_{P}$ if $\\cL$ consists of the\npredicate and function symbols in $P$; similarly the Herbrand Base is\ndenoted as $\\cB_{\\cL}$ or as $\\cB_P$. Throughout the paper variant\nterms are considered to be equal.\n}\n\nOne of the most important formulations of stratification is that of\n{\\em dynamic} stratification \\cite{Przy89d}, which shows that a\nprogram has a 2-valued well-founded model iff it is dynamically\nstratified, so that it is the weakest notion of stratification\nconsistent with the well-founded semantics. The original definition\nof dynamic stratification included neither explicit negation (neg) nor\nannotations; however the encapsulation of these features within the\ninterpretations of Definition~\\ref{def:satisfiable} allows the\ndefinitions in this section to be mostly unchanged from their original\nformulation.\n\n%\nAs presented in~\\cite{Przy89d}, dynamic stratification computes strata\nvia operators on interpretations of the form ($\\cT$,$\\cF$) where $\\cT$\nand $\\cF$ are subsets of $\\cH_P$. Given a set $\\cS$ of ground\nobjective literals, a ground objective literal $A:m \\hat{\\in} \\cS$ if\n$A:n \\in \\cS$ with $n \\geq m$.\n\n%-----------------------------------------------------------------------\nDynamic stratification is based on a series of reduction of a program\nin the usual manner.\n\n\\begin{definition}[Reduction of $\\cP$ modulo $\\cI$] \\label{def:reduction}\n% \nLet $\\cI$ be an interpretation and $\\cP$ a program, both over the same\nlangage $\\cL$. By the {\\em reduction of $P$ modulo $\\cI$} we mean a\nnew program $\\frac{P}{\\cI}$ obtained from $P$ by performing the\nfollowing operations:\n\\begin{enumerate}\n\\item Remove from $\\cP$ all rules that contain a literal that is false\n in $\\cI$.\n\\item Remove from all the remaining rules those literals that are true\n in $\\cI$\n\\end{enumerate}\n\\end{definition}\n\nEach stratum is then based on the intepretation of the previous\nstratum if it exists commbined with a reduction.\n\n%-----------------------------------------------------------------------\n\\begin{definition} \\label{def:dyn-ops}\nFor a normal program $P$, sets $\\cT$ and $\\cF$ of ground atoms and a\n3-valued interpretation $I = (\\cT,\\cF)$ (sometimes called a pre-interpretation):\n\\begin{description} \\item[$True^P_I(\\cT) =$]\n $\\{A|A$ is true in $I$; or there is a clause $B \\leftarrow\n L_1,...,L_n$ in $P$, a grounding substitution $\\theta$ such that $A\n = B\\theta$ and for every $1 \\leq i \\leq n$ either $L_i\\theta$ is\n true in $I$, or $L_i\\theta \\hat{\\in} \\cT$\\};\n \\item[$False^P_I(\\cF) =$] \n%$\\{A : val_I(A) \\neq {\\tt f}$ \n$\\{A|A$ is false in $I$; or for every clause $B \\leftarrow\n L_1,...,L_n$ in $P$ and grounding substitution $\\theta$ such that\n $A = B\\theta$ there is some $i$ $(1 \\leq i \\leq n)$ such that\n $L_i\\theta$ is false in $I$ or $L_i\\theta \\hat{\\in} \\cF\\}$.\n\\end{description}\n\\end{definition}\n%------------------------------------------------------------------------() \n%\n\\cite{Przy89d} shows that $True^P_I$ and $False^P_I$ are both\nmonotonic, and defines $\\kcaltrue^P_I$ as the least fixed point of $True^P_I(\\emptyset)$\nand $\\calfalse^P_I$ as the greatest fixed point of\n$False^P_I(\\cH_P)$.\n%\\footnote{Below, we will sometimes omit the program $P$ in\n% these operators when the context is clear.}.\n%, along with an\n%operator $\\cal I$ that assigns to every interpretation $I$ of $P$ a\n%new interpretation ${\\cal I}(I) = I \\cup \\langle \\cT_I ; \\cF_I \\rangle$.\n%\nIn words, the operator $\\kcaltrue^P_I$ extends the interpretation $I$ to add\nthe new atomic facts that can be derived from $P$ knowing $I$; $\\calfalse^P_I$\nadds the new negations of atomic facts that can be shown false in $P$\nby knowing $I$ (via the uncovering of unfounded sets). An iterated\nfixed point operator builds up dynamic strata by constructing\nsuccessive partial interpretations as follows.\n%---------------------------------------------------------------------- \n\\begin{definition}[Iterated Fixed Point and Dynamic Strata] \\label{def:wfm}\n\\label{def:IFP}\nFor a normal program $P$ let \n\n\\begin{center}\n$ \\begin{array}{rcl}\n WFM_0 & = & \\langle \\emptyset ; \\emptyset \\rangle; \\\\\n WFM_{\\alpha+1} & = & WFM_{\\alpha} \\cup\n \\langle \\kcaltrue^P_{WFM_\\alpha};\\calfalse^P_{WFM_\\alpha} \\rangle; \\\\\n WFM_\\alpha & = & \\bigcup_{\\beta < \\alpha} WFM_\\beta, \\mbox{ for limit ordinal }\\alpha.\n \\end{array}\n$\n\\end{center}\n\n\\noindent\n $WFM(P)$ denotes the fixed point interpretation $WFM_\\delta$,\n where $\\delta$ is the smallest (countable) ordinal such that both\n sets $\\kcaltrue^P_{WFM_\\delta}$ and $\\calfalse^P_{WFM_\\delta}$ are empty.\n%($\\delta$ exists, and is\n% a countable ordinal because both $\\kcaltrue_I$ and $\\calfalse^P_I$ are monotonically\n% increasing). \n% We refer to $\\delta$ as the {\\em depth} of program $P$. \nThe {\\em stratum} of atom $A$, is the least ordinal $\\beta$ such that\n $A \\in WFM_{\\beta}$.\n% (where $A$ may be either in the true or false\n% component of $WFM_{\\beta}$).\n\\end{definition}\n%------------------------------------------------------------------------() \n%\n\\cite{Przy89d} shows that %the iterated fixed point \n$WFM(P)$ is in fact the well-founded model and that any undefined\natoms of the well-founded model do not belong to any stratum --\ni.e. they are not added to $WFM_{\\delta}$ for any ordinal\n$\\delta$. Thus, a program is \\emph{dynamically stratified} if every\natom belongs to at least one stratum.\n\nThis section has considered normal logic programs extended both with\nexplicit negation. We note that if all atoms in a program $\\cP$ have\nthe annotation 1, Definition~\\ref{def:wfm} reduces to the definition\nof the Well0founded Semantics with Explicit Negation~\\cite{AlDP95},\nand if $\\cP$ also does not contain explicit negation, the definition\nreduces to the definition of Well-Founded Semantics in \\cite{Przy89d}.\n\n%------------------------------------------------------------------------\n% taking out fixed-order stuff until we put LPADs in (though maybe we\n% wont even need it then )\n\n%\\input{fixed-order-dynstrat}\n\n\n", "meta": {"hexsha": "a3002bce70d6bd0b0bea9329e1b516c11d3ab9a8", "size": 13315, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/paper/prelim.tex", "max_stars_repo_name": "theresasturn/plow", "max_stars_repo_head_hexsha": "0999214f33d71413ed9a029eb9dccbf4e6d35dbb", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2019-03-26T21:41:40.000Z", "max_stars_repo_stars_event_max_datetime": "2021-02-03T22:22:06.000Z", "max_issues_repo_path": "docs/paper/prelim.tex", "max_issues_repo_name": "theresasturn/plow", "max_issues_repo_head_hexsha": "0999214f33d71413ed9a029eb9dccbf4e6d35dbb", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/paper/prelim.tex", "max_forks_repo_name": "theresasturn/plow", "max_forks_repo_head_hexsha": "0999214f33d71413ed9a029eb9dccbf4e6d35dbb", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-01-23T15:26:21.000Z", "max_forks_repo_forks_event_max_datetime": "2020-01-23T15:26:21.000Z", "avg_line_length": 43.5130718954, "max_line_length": 101, "alphanum_fraction": 0.6513706346, "num_tokens": 4099, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5774953651858118, "lm_q2_score": 0.5926665999540698, "lm_q1q2_score": 0.34226221457390893}} {"text": "\\documentclass[12pt, a4paper]{article}\n\n\\include{preamble}\n\\usepackage{lipsum} % <---------- remove me\n\n\\title{My lovely paper}\n\\include{author_vincent}\n\\date{\\today}\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nShort introduction to subject of the paper \\ldots\n\\end{abstract}\n\n\\section{Introduction}\nThis is a reference \\citep{Vincent:2015dh}\n\n\\lipsum[1-2] % <---------- remove me\n\n\\begin{equation}\n\\label{eq1}\n y = mx+c\n\\end{equation}\n\n\\lipsum[2] % <---------- remove me\n\n\\begin{figure}[t!] \n\t\\centering\n\t\\includegraphics[]{bayes_net_general.pdf} \n\t\\caption{This is a caption}\n\t\\label{fig:first_figure}\n\\end{figure}\n\n\\lipsum[3-5] % <---------- remove me\n\n% REFERENCE SECTION\n\\clearpage\n\\bibliographystyle{apacite}\n\\bibliography{my_references}\n\n\\end{document}\n", "meta": {"hexsha": "508c93c90632a0f9edf2077bb792b4cd009dc5d8", "size": 765, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "latex templates/paper/paper.tex", "max_stars_repo_name": "drbenvincent/document_templates", "max_stars_repo_head_hexsha": "75b426d7a110026a48f2dc36a596258c64a66617", "max_stars_repo_licenses": ["Unlicense"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "latex templates/paper/paper.tex", "max_issues_repo_name": "drbenvincent/document_templates", "max_issues_repo_head_hexsha": "75b426d7a110026a48f2dc36a596258c64a66617", "max_issues_repo_licenses": ["Unlicense"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "latex templates/paper/paper.tex", "max_forks_repo_name": "drbenvincent/document_templates", "max_forks_repo_head_hexsha": "75b426d7a110026a48f2dc36a596258c64a66617", "max_forks_repo_licenses": ["Unlicense"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 17.0, "max_line_length": 49, "alphanum_fraction": 0.6954248366, "num_tokens": 245, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5039061705290805, "lm_q2_score": 0.6791787056691698, "lm_q1q2_score": 0.34224234067864884}} {"text": "\\chapter{Exponentials and Logarithms}\n\n\\section{Properties of the exponential function}\n\n\\section{Gradient of \\(e^{kx}\\)} % TODO: Typeset better\n\\section{Properties of the logarithm}\n\\section{Laws of logarithms}\n\\section{Equations involving exponentials}\n\\section{Reduction to linear form}\n\\section{Modelling using exponential functions}\n", "meta": {"hexsha": "fac99468ff3f597e226cc8012f7c584cd1a8d2ab", "size": 338, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "corpus/a-level-notes/math/content/pure/exp_log.tex", "max_stars_repo_name": "aDotInTheVoid/ltxmk", "max_stars_repo_head_hexsha": "ee461679e51e92a0e4b121f28ae5fe17d5e5319e", "max_stars_repo_licenses": ["Apache-2.0", "MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "corpus/a-level-notes/math/content/pure/exp_log.tex", "max_issues_repo_name": "aDotInTheVoid/ltxmk", "max_issues_repo_head_hexsha": "ee461679e51e92a0e4b121f28ae5fe17d5e5319e", "max_issues_repo_licenses": ["Apache-2.0", "MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "corpus/a-level-notes/math/content/pure/exp_log.tex", "max_forks_repo_name": "aDotInTheVoid/ltxmk", "max_forks_repo_head_hexsha": "ee461679e51e92a0e4b121f28ae5fe17d5e5319e", "max_forks_repo_licenses": ["Apache-2.0", "MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 30.7272727273, "max_line_length": 55, "alphanum_fraction": 0.798816568, "num_tokens": 84, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.679178699175393, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.34224233740639465}} {"text": "%!TEX root = main.tex\n\nThis chapter describes the technical details of the implementation of signature-based partition refinement.\nIt contains a condensed version of several sections from~\\cite{DBLP:conf/tacas/DijkP16}.\n\n\n\\section{Decision diagram algorithms in Sylvan}\n\n\nIn {symbolic model checking}, sets of states and transitions are represented by their characteristic function, rather than stored individually.\nWith states described by $N$ Boolean variables,\na set $S\\subseteq \\mathbb{B}^N$ can be represented by its characteristic function $f\\colon \\mathbb{B}^N\\rightarrow \\mathbb{B}$, where $S=\\{s\\mid f(s)\\}$.\n%\n%\n%\nBinary decision diagrams (BDDs) are a concise and canonical representation of Boolean functions.\n\nAn (ordered) BDD is a directed acyclic graph with leaves $0$ and $1$. Each internal node has a variable label $x_i$ and two outgoing edges labeled $0$ and $1$. Variables are encountered along each path according to a fixed variable ordering. Duplicate nodes and nodes with two identical outgoing edges are forbidden.\n%\n%\\begin{definition}\n%An (ordered) BDD is a directed acyclic graph with the following properties:\n%\\begin{compactenum}\n%\\item There is a single root node and two terminal nodes 0 and 1.\n%\\item Each non-terminal node $p$ has a variable label $x_i$ and two outgoing edges, labeled 0 and 1; we write $\\text{lvl}(p)=i$ and use $p[0]$ and $p[1]$ for the children.\n%\\item For each edge from node $p$ to non-terminal node $q$, $\\text{lvl}(p)<\\text{lvl}(q)$.\n%%\\item There are no {duplicate} nodes, i.e.,\\\\$\\forall p\\forall q\\cdot(\\text{lvl}(p)=\\text{lvl}(q)\\wedge p[0]=q[0]\\wedge p[1]=q[1])\\rightarrow p=q$.\n%\\item There are no duplicate nodes $\\langle x_i,p[0],p[1]\\rangle$ (i.e., maximum sharing)\n%\\item There are no redundant nodes (nodes with $p[0]=p[1]$)\n%\\end{compactenum}\n%\\end{definition}\n%\n%\n%%\nIt is well known that for a fixed variable ordering, every Boolean function is represented by a unique BDD.\n\n\nIn addition to BDDs with leaves $0$ and $1$, multi-terminal binary decision diagrams have been proposed with leaves other than $0$ and $1$, representing functions from the Boolean space $\\mathbb{B}^N$ onto any finite set. For example, MTBDDs can have leaves representing integers (encoding $\\mathbb{B}^N\\rightarrow\\mathbb{N}$), floating-point numbers (encoding $\\mathbb{B}^N\\rightarrow\\mathbb{R}$) and rational numbers (encoding $\\mathbb{B}^N\\rightarrow\\mathbb{Q}$).\nPartial functions are supported using a terminal leaf $\\bot$.% (in practice we use the Boolean leaf $0$ for non-Boolean codomains).\n\n\n\nSylvan~\\cite{DBLP:conf/tacas/DijkP15} implements parallelized operations on decision diagrams using parallel data structures and work-stealing.\nWork-stealing is a load balancing method for task-based parallelism.\nRecursive operations, such as most BDD operations, implicitly form a tree of tasks.\nIndependent subtasks are stored in queues and idle processors steal tasks from the queues of busy processors.\n\n\n\\begin{algorithm}[t!]\n\\Def{\\Apply{$x$, $y$, $\\textsf{F}$}}{\n \\lIf{$(x,y,\\textsf{F})\\in\\textsf{cache}$}{\n \\Return{$\\textsf{cache}[(x,y,\\textsf{F})]$}\\tcc*[f]{get from cache} \n }\n \\lIf{$x$ and $y$ are terminals}{\n \\Return{$\\textsf{F}(x,y)$}\\tcc*[f]{apply operator \\textsf{F}}\n }\n $v$ = topVar($x$,$y$) \\;\n low $\\leftarrow$ \\Apply{$x_{v=0}$, $y_{v=0}$, $\\textsf{F}$} \\tcc*[r]{execute in parallel}\n high $\\leftarrow$ \\Apply{$x_{v=1}$, $y_{v=1}$, $\\textsf{F}$} \\;\n result $\\leftarrow$ \\BDDnode{$v$, low, high} \\tcc*[r]{compute result}\n $\\textsf{cache}[(x,y,\\textsf{F})]$ $\\leftarrow$ result \\tcc*[r]{put in cache}\n \\Return{result}\n}\n\\caption{Generic algorithm that applies a binary operator \\textsf{F} to BDDs $x$ and $y$.}\n\\label{alg:generic}\n\\end{algorithm}\n\n\n\nAlgorithm~\\ref{alg:generic} describes the implementation of a generic binary operation $\\textsf{F}$.\n%\n%\n%\nBDD operations mainly consist of consulting an operation cache, performing some recursive step, and creating new BDD nodes using a unique table.\nThe operation cache is required to reduce the time complexity of BDD operations from exponential to polynomial in the size of the BDDs.\nSylvan uses a single shared unique table for all BDD nodes and a single shared operation cache for all operations.\nTo obtain high performance in a multi-core environment, the datastructures for the BDD nodes and the operation cache must be highly scalable.\nSylvan implements several non-blocking datastructures to enable good speedups.\n\n\n\nTo compute symbolic signature-based partition refinement, several basic operations must be supported by the BDD package (see also \\cite{DBLP:conf/atva/WimmerHHSB06}).\n%\n%\n%\nSylvan implements basic operations such as $\\wedge$ and \\texttt{if-then-else}, and existential quantification $\\exists$.\nNegation $\\neg$ is performed in constant time using complement edges.\nTo compute relational products of transition systems, there are operations \\texttt{relnext} (to compute successors) and \\texttt{relprev} (to compute predecessors and to concatenate relations), which combine the relational product with variable renaming.\nThe operation \\texttt{and\\_exists} computes the traditional relational product without variable renaming.\nSimilar operations are also implemented for MTBDDs.\n%\n%\n%\nSylvan is designed to support custom BDD algorithms.\nWe present two custom algorithms below.\n%We also use custom extensions of traditional BDD algorithms to avoid variable renaming.\n\n\n\n\\section{Encoding of signature refinement}\n\nWe implement symbolic signature refinement similar to~\\cite{DBLP:conf/atva/WimmerHHSB06}.\n%\n%\nUnlike~\\cite{DBLP:conf/atva/WimmerHHSB06}, we do not refine the partition with respect to a single block, but with respect to all blocks simultaneously.\n% like in~\\cite{DBLP:journals/entcs/BlomO03,DBLP:conf/qest/Derisavi07}.\n%\n%\n%\nWe use a binary encoding with variables $s$ for the current state, $s'$ for the next state, $a$ for the action labels and $b$ for the blocks.\nWe order BDD variables $a$ and $b$ after $s$ and $s'$, since this is required\nto efficiently replace signatures $(a,b)$ by new block numbers $b$ (see below).\nVariables $s$ and $s'$ are interleaved, which is common in the context of transition systems.\n\n\n\n\\begin{figure}[tp]\n\\begin{center}\n\\scalebox{0.9}{\n\\begin{tikzpicture}\n\\coordinate (a) at (250:2);\n\\coordinate (b) at (290:2);\n\\coordinate (c) at (250:3);\n\\coordinate (d) at (290:3);\n\\draw (0,0) -- (c);\n\\draw (0,0) -- (d);\n\\draw (a) -- (b);\n\\draw (c) -- (d);\n\\draw (0,-1.5) node {$s,s'$};\n\\draw (0,-2.35) node {$a$};\n\\draw (0,-4.5) node {BDD $\\mathcal{T}(s,s',a)$};\n\n\\coordinate (a0) at (3,0);\n\\coordinate (aa) at ($(a0) + (250:2)$);\n\\coordinate (ab) at ($(a0) + (290:2)$);\n\\coordinate (ac) at ($(a0) + (250:3)$);\n\\coordinate (ad) at ($(a0) + (290:3)$);\n\\coordinate (ae) at ($(a0) + (250:4)$);\n\\coordinate (af) at ($(a0) + (290:4)$);\n\\draw (a0) -- (ae);\n\\draw (a0) -- (af);\n\\draw (aa) -- (ab);\n%\\draw (ac) -- (ae);\n\\%draw (ad) -- (af);\n\\draw (ac) -- (ad);\n\\draw (ae) -- (af);\n\\draw (3,-1.5) node {$s$};\n\\draw (3,-2.35) node {$a$};\n\\draw (3,-3.25) node {$b$};\n\\draw (3,-4.5) node {BDD $\\sigma_T(s,a,b)$};\n\n\\coordinate (b0) at (6.5,0);\n\\coordinate (ba) at ($(b0) + (250:2)$);\n\\coordinate (bb) at ($(b0) + (290:2)$);\n\\coordinate (bc) at ($(b0) + (250:3)$);\n\\coordinate (bd) at ($(b0) + (290:3)$);\n\\coordinate (be) at ($(b0) + (250:4)$);\n\\coordinate (bf) at ($(b0) + (290:4)$);\n\\draw (b0) -- (ba);\n\\draw (b0) -- (bb);\n\\draw (ba) -- (bb);\n\\draw (bc) -- (be);\n\\draw (bd) -- (bf);\n\\draw (bc) -- (bd);\n\\draw (be) -- (bf);\n\\draw ($(b0) + (0,-1.5)$) node {$s$};\n\\draw ($(b0) + (0,-3.25)$) node {$b$};\n\\draw ($(b0) + (0,-4.5)$) node {MTBDD $\\sigma_R(s,b)$};\n\n\\coordinate (c0) at (-3,0);\n\\coordinate (ca) at ($(c0) + (250:2)$);\n\\coordinate (cb) at ($(c0) + (290:2)$);\n\\coordinate (cc) at ($(c0) + (250:3)$);\n\\coordinate (cd) at ($(c0) + (290:3)$);\n\\coordinate (ce) at ($(c0) + (250:4)$);\n\\coordinate (cf) at ($(c0) + (290:4)$);\n\\draw (c0) -- (ca);\n\\draw (c0) -- (cb);\n\\draw (ca) -- (cb);\n\\draw (cc) -- (ce);\n\\draw (cd) -- (cf);\n\\draw (cc) -- (cd);\n\\draw (ce) -- (cf);\n\\draw ($(c0) + (0,-1.5)$) node {$s$};\n\\draw ($(c0) + (0,-3.25)$) node {$b$};\n\\draw ($(c0) + (0,-4.5)$) node {BDD $\\mathcal{P}(s,b)$};\n\\end{tikzpicture}\n}\n\\end{center}\n\\caption{Schematic overview of the BDDs in signature refinement}\n\\label{fig:bdds}\n\\end{figure}\n\n\n\nTo perform symbolic bisimulation we represent a number of sets by their characteristic functions. See also Figure~\\ref{fig:bdds}.\n\\begin{itemize}\n\\item A set of states is represented by a BDD $\\mathcal{S}(s)$;\n% with $\\mathcal{S}(s)=1\\Leftrightarrow s\\in S$\n\\item Transitions are represented by a BDD $\\mathcal{T}(s,s',a)$;\n\\item Markovian transitions are represented by an MTBDD $\\mathcal{R}(s,s')$, with leaves containing rational numbers ($\\mathbb{Q}$);\n\\item Signatures $\\textbf{T}$ and $\\textbf{B}$ are represented by a BDD $\\sigma_T(s,a,b)$;\n\\item Signatures $\\textbf{R}^s$ and $\\textbf{R}^b$ are represented by an MTBDD $\\sigma_R(s,b)$.\n\\end{itemize}\n\n\n\n\nIn the literature,\n%~\\cite{DBLP:conf/cav/BoualiS92,DBLP:conf/tacas/Derisavi07,DBLP:conf/qest/Derisavi07,DBLP:conf/atva/WimmerHHSB06}, \nthree methods have been proposed to represent $\\pi$.\n%\n%\n%\n\\begin{enumerate}\n\\item As an equivalence relation, using a BDD $\\mathcal{E}(s,s')=1$ iff $s\\equiv_{\\pi} s'$.\n\\item As a partition, by assigning each block a unique number, encoded with variables $b$, using a BDD $\\mathcal{P}(s,b)=1$ iff $s\\in C_b$.\n\\item Using $k=\\ceil{\\log_2 n}$ BDDs $\\mathcal{P}_{0},\\dotsc,\\mathcal{P}_{k-1}$ such that $\\mathcal{P}_i(s)=1$ iff $s\\in C_b$ and the $i^\\text{th}$ bit of $b$ is 1. This requires significant time to restore blocks for the refinement procedure, but can require less memory.\n\\end{enumerate}\n\n%We use variables $t$ and $b$ instead of $s$ and $b$ in $\\mathcal{P}$ to avoid creating a custom BDD operation for computing $\\exists t\\colon \\mathcal{T}(s,t,a) \\wedge \\mathcal{P}(t,b)$.\n%Also, \\cite{DBLP:conf/atva/WimmerHHSB06} often requires renaming operations to get $\\mathcal{P}(t,b)$ from $\\mathcal{P}(s,b)$, which is unnecessary and expensive.\n\n%Furthermore, the encoding of block numbers is such that the the lowest significant bit comes first...\n\n%We represent the partition by the assignment to a block number.\n%An alternative method to represent the partition, used by \\cite{DBLP:conf/cav/BoualiS92} and \\cite{DBLP:journals/ijfcs/MummeC13}, stores the equivalence relation in a BDD $\\mathcal{R}(s,t)$.\nWe choose to use method 2, since in practice the BDD of $\\mathcal{P}(s,b)$ is smaller than the BDD of $\\mathcal{E}(s,s')$.\nUsing $\\mathcal{P}(s,b)$ also has the advantage of straight-forward signature computation.\n%In cases where only few blocks contain multiple states, $\\mathcal{P}(s,b)$ blows up to a large tree, assigning each state to a different block.\n%In such cases, the BDD for $\\mathcal{E}(s,t)$ would be much smaller, since less states are equivalent.\n%\n%\nThe logarithmic representation is incompatible with our approach, since we refine all blocks simultaneously. Their approach involves restoring individual blocks to the $\\mathcal{P}(s,b)$ representation, performing a refinement step, and compacting the result to the logarithmic representation. Restoring all blocks simply computes the full $\\mathcal{P}(s,b)$.\n\nWe represent Markovian transitions using rational numbers, since they offer better precision than floating-point numbers.\nThe manipulation of floating-point numbers typically introduces tiny rounding errors, resulting in different results of similar computations.\nThis significantly affects bisimulation reduction, often resulting in finer partitions than the maximal bisimulation~\\cite{DBLP:conf/mmb/WimmerB10}.\n\n\n\\section{The \\texttt{refine} algorithm}\n\\label{sec:refine}\n\\input{refine}\n\n\\section{Computing inert transitions}\n\\label{sec:inert}\n\\input{inert}\n", "meta": {"hexsha": "3ba78608c09d4a40518debb87f0d2e937e995091", "size": 11840, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/implementation.tex", "max_stars_repo_name": "alaarman/sigrefmc", "max_stars_repo_head_hexsha": "5afc0cc5d7f45260f71279289f2c1db67bfb04e1", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/implementation.tex", "max_issues_repo_name": "alaarman/sigrefmc", "max_issues_repo_head_hexsha": "5afc0cc5d7f45260f71279289f2c1db67bfb04e1", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-10-20T13:11:59.000Z", "max_issues_repo_issues_event_max_datetime": "2020-10-20T13:11:59.000Z", "max_forks_repo_path": "doc/implementation.tex", "max_forks_repo_name": "alaarman/sigrefmc", "max_forks_repo_head_hexsha": "5afc0cc5d7f45260f71279289f2c1db67bfb04e1", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2021-09-16T08:45:51.000Z", "max_forks_repo_forks_event_max_datetime": "2021-09-16T08:45:51.000Z", "avg_line_length": 47.9352226721, "max_line_length": 466, "alphanum_fraction": 0.7024493243, "num_tokens": 3580, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5039061705290806, "lm_q1q2_score": 0.3422423308618862}} {"text": "\\documentclass[12pt]{beamer}\n\n\\usepackage{tikz}\n\\usepackage{graphicx}\n\\usepackage[T1]{fontenc}\n\\usepackage{amsmath}\n\\usepackage{multicol}\n\\usepackage{booktabs, braket} % For formal tables\n\\usepackage{cleveref}\n\\usepackage{graphicx}\n\\usepackage{blkarray}\n\\usetikzlibrary{graphs, graphs.standard}\n\n\\DeclareMathOperator{\\E}{\\textrm{E}}\t\t % expected value\n\\DeclareMathOperator{\\pr}{\\mathrm{P}}\t\t % probability \n\\DeclareMathOperator{\\cov}{t_{cov}}\t % cover time\n\n\\DeclareMathOperator{\\X}{\\mathbb{X}}\t\t % expected value\n\\DeclareMathOperator{\\tr}{\\text{tr}}\t\t % trace\n\\DeclareMathOperator{\\Ap}{$A$ ^\\prime} \t\t % A'\n\\DeclareMathOperator{\\bp}{$b$ ^\\prime} \t\t % b'\n\\DeclareMathOperator{\\Xb}{\\mathcal{X}}\t\t % big X\n\n\\title{Semidefinite Programming and Quantum Algorithms}\n\\author{Michael Czekanski \\& R. Teal Witter}\n\\institute{Middlebury College}\n\\date{November 5, 2019}\n\n\\begin{document}\n\\graphicspath{{./../figures/}}\n\n\\frame{\\titlepage}\n\n\\begin{frame}{Overview}\n\\tableofcontents\n\\end{frame}\n\n\\section{Query Complexity}\n\\begin{frame}{Query Complexity}\n\\begin{center}\nThe number of times an algorithm queries the input.\n\\end{center}\n\\begin{table}[]\n\\begin{tabular}{cc}\n\\hline\n\\textbf{Input} & \\textbf{Output} \\\\ \\hline\n00 & 0 \\\\\n01 & 1 \\\\\n10 & 1 \\\\\n11 & 1 \\\\ \\hline\n\\end{tabular}\n\\end{table}\n\\end{frame}\n\n\\section{Semidefinite Programming}\n\n\\begin{frame}{Quantum Query Complexity as SDP: Reichardt \\cite{reichardt2009span}}\n\\begin{align} \\label{eq:reichardtObj} \n f_{\\text{bound}} = \\min_{\\X} M(\\X) = \\max_{y \\in D} \\sum_{j \\in [n]}\n \\bra{y,j}\\X\\ket{y,j} \n\\end{align}\nSubject to\n\\begin{align}\\label{Eq:reichardtSemi}\n \\X \\succcurlyeq 0 \n\\end{align}\n\\begin{align}\\label{Eq:reichardtOffDiag}\n \\forall (y,z) \\in F \\sum_{j \\in [n]: y_j \\ne z_j} \n \\bra{y,j} \\X \\ket{z, j} = 1.\n\\end{align}\n\\end{frame}\n\n\\begin{frame}{A Standard Form: Boyd\\cite{boyd2004convex}}\nMinimize\n\\begin{align}\\label{Eq:boyd_obj}\n M(\\X) = \\tr(C\\X) \n\\end{align}\nSubject to\n\\begin{align} \\label{Eq:boydSemi}\n \\X \\succcurlyeq 0 \n\\end{align}\n\n\\begin{align} \\label{Eq:boydTraceCon}\n \\tr(A_i \\X) = b_i \\text{$\\qquad$ for $i \\in \\{1,...,p\\}$}\n\\end{align}\n\\end{frame}\n\n\\section{Alternating Direction Methods}\n\\begin{frame}{We use ADM \\cite{adm}}\n\\centering\n\\includegraphics[scale=.15]{figures/adm_algorithm}\n\\bigskip\n\\begin{itemize}\n \\item Easy to implement\n \\item Exploits sparsity (yay zeroes!)\n \\item Exploits constraint orthogonality\n\\end{itemize}\n\\end{frame}\n\n\\section{Our Work}\n\\begin{frame}{Numerial Results: Our Algorithm is Good.}\n\\centering\n\\includegraphics[scale=.5]{figure_all_or_complexity.eps}\n\\end{frame}\n\n\\begin{frame}{Run Time Results: Our Algorithm is Bad.}\n\\centering\n\\includegraphics[scale=.5]{figure_all_or_time.eps}\n\\end{frame}\n\n\\begin{frame}{We Have Plans (For the Future)}\n\\begin{itemize}\n \\item Document\n \\item More optimization\n \\item Span program\n\\end{itemize} \n\\end{frame}\n\n\\begin{frame}{Thank you!}\n\\bibliographystyle{abbrv}\n\\bibliography{main}\n\\end{frame}\n\n\\end{document}", "meta": {"hexsha": "7bf1f3e834bd4cf933db7072182a37df636a99ce", "size": 3140, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/sneakpeek.tex", "max_stars_repo_name": "rtealwitter/QuantumQueryOptimizer", "max_stars_repo_head_hexsha": "64f68110ab088c271fad96976f2fa06af88d5a2e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "paper/sneakpeek.tex", "max_issues_repo_name": "rtealwitter/QuantumQueryOptimizer", "max_issues_repo_head_hexsha": "64f68110ab088c271fad96976f2fa06af88d5a2e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "paper/sneakpeek.tex", "max_forks_repo_name": "rtealwitter/QuantumQueryOptimizer", "max_forks_repo_head_hexsha": "64f68110ab088c271fad96976f2fa06af88d5a2e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 25.5284552846, "max_line_length": 82, "alphanum_fraction": 0.6732484076, "num_tokens": 1055, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6791786861878392, "lm_q2_score": 0.5039061705290805, "lm_q1q2_score": 0.3422423308618861}} {"text": "% Add a bit of space between paragraphs (copied from Karl Broman)\n\\setlength{\\parskip}{0.5\\baselineskip}\n\n% Add abstract (copied from Karl Broman)\n\\begin{quote}\n\\small\n\\articleABSTRACT\n\\end{quote}\n\n\\section{Introduction}\n\n\\subsection{Historical context}\n\n%Summarize the scientific question that your code was written to explore.\nThis paper aims to computationally reproduce my own statistical analysis from 16~years ago as part of the 10-year reproducibility challenge \\supercite{Hinsen2019}. The analysis examined seasonal patterns in coronary heart disease in multiple countries \\supercite{Barnett2004}. In most countries coronary heart disease deaths and emergency admissions to hospital have a strong seasonal pattern with a peak in winter and nadir in summer \\supercite{Stewart2017}. There is an interesting variability between-countries in the size and timing of the winter peak. A better understanding of the differences between countries could help our understanding of the underlying causes of the winter peak in disease.\n\nThe analysis examined monthly time series of coronary events (fatal and non-fatal) from 35~locations in 21~countries. The time series were 8 to 14 years in length and the earliest year of data was 1980. The analysis aimed to split the time series into a long-term trend, seasonal pattern(s) and remaining noise. There were two approaches, one that used a two-stage approach by first removing the trend, and another that estimated the trend and season together. The trend was estimated using the Kalman filter and the seasonal patterns were estimated using sinusoids. The estimates were made using Markov chain Monte Carlo. \n\n%Describe the computational context: Which hardware was used to run the code? Which software infrastructure? Which constraints existed on software development? Which technical choices (language, libraries, ...) were made? Were reproducibility and/or re-usability important criteria?\n\n\\subsection{Original source code}\n\nThe original code was written in SAS (version~8.00 for Windows) and chain convergence was checked using the ``coda'' package (version unknown) in R \\supercite{coda}. The code was based on Matlab code that I wrote during my PhD \\supercite{Barnett2003} at The University of Queensland. I converted the code to SAS because I did not have access to Matlab after finishing my PhD and I hoped that more people would be able to re-use my SAS code to apply to their own time series data. \n%It was not common practice at the time to share code and data in order to allow others to reproduce results. \nThe SAS code was first published in November 2004 with an update in August 2006, but I did not record what changes were made. \nThe paper states the code was published in September 2002 but this is likely a typo in the year given that the paper was not submitted until January 2004. \n\n%About the original source code: Was it published? Was it archived somewhere? Was there a license for it?\nThe original code was published in an online appendix to the paper that was separate from the journal in November 2004 at this address: \\url{http://www4.ktl.fi/publications/monica/chd_seasonal/appendix.htm}. However, the site has since moved to: \\url{https://www.thl.fi/publications/monica/chd_seasonal/appendix.htm} (accessed 25 January 2020). The web site was created by the WHO MONICA Project specifically for appendices to papers that used the MONICA data. MONICA stands for monitoring trends and determinants in cardiovascular disease. The MONICA Project was a large multi-country study that aimed to examine several aspects of coronary heart disease \\supercite{Tunstall2003}. It was a well-managed project that had staff who assisted with access to the data and helped add my code to the web. \n\n%Please provide only information you are certain about. It's OK to forget technical details after many years.\n\n\\section{Results}\n\n%Retrieval of the software\n%Was it easy to find a copy of your source code?\n%Was it easy to locate and setup the dependencies?\n%Provide a list of all dependencies (libraries but also tools such as compilers).\n\n\\subsection{Retrieval of the software}\n\nMost of my SAS code was easy to find, both on the web and on an old compact disc in a desk draw. I made this compact disc of my files when I moved jobs. I did not keep the data because I was more interested in other people using my code for their own data rather than replicating our published results. I believe that another reason for deleting the data was to save space, as the files were relatively large by the standards of the day (around 31,000~kilobytes for data in text format and 70,000~kilobytes for data in SAS format). Luckily the original data files were available in text files in fixed-width format on a compact disc attached to the MONICA monograph published in 2003 \\supercite{Tunstall2003}.\n\nMy SAS code contained the macros needed to run the two statistical analyses. There was also SAS code to simulate a seasonal time series as an example data set. I found SAS code to read the MONICA data on my compact disc, but not on the web. I could not find the SAS code that applied the two methods to each location.\n\n%Execution\n%Describe what you did in order to run the software today.\n%Did you succeed in running the software in a modern computational environment? Or did you have to search for old versions of tools and libraries?\n%Describe the computational environment of the reproduction: hardware, operating system, compiler versions etc.\n%Did you have to modify the software in order to make it run today?\n%Were the original instructions that came with the software sufficient, or did you have to modify or extend them?\n\n\\subsection{Replication execution}\n\nI used SAS version~9.4 for Windows (Microsoft Windows~10) to replicate the results; there was no need to use an earlier version of SAS. \n\nMy SAS files had relatively good instructions with a detailed header at the top of every file and comments throughout. However, some files had been adapted from the original analysis to secondary analyses, making the code a palimpsest with some commands commented out with notes such as, ``changed for weather analysis'' and ``sensitivity analysis of Ghent''. The correct data set was also ambiguous because alternative data sets had also been used (e.g., fatal vs non-fatal events). In hindsight I should have kept the exact data and syntax files needed to re-create the published results.\n%Mix of SAS files from different projects in the same folder.\n\nThe macros to run the analyses mainly needed only cosmetic edits, and I also put each macro in its own file rather than one overall file of macros. I also moved part of one macro, that estimated the standard deviation of the noise, into the macro that tested the periodogram for remaining seasonal structure. The largest change was having to re-write the files that applied the macros to the MONICA data. My SAS programming was rusty and I could not automate the process for each centre, so instead I created a new file for each analysis in each of three centres.\n\nI did not repeat my analysis in all 35 locations, but instead did the three locations in the first figure as this felt sufficient to test the code.\n\nThe code to estimate the trend without the seasonal pattern used a slightly different parameterisation with ``tau'' as a ratio rather than an absolute value. ``tau'' controls the amount of change over time in the trend, with smaller values creating more linear trends. I cannot be sure that this version was the same code as the original.\n\n\\subsection{Closeness of the replicated results to the original}\n\n%How close were the results you got to the originally published ones? Include the replicated data table and figure, as we do in other ReScience articles.\nThe original figure is shown in Figure~\\ref{fig:original} and my replication in Figure~\\ref{fig}. The figures show the estimates of the trend and season for three locations.\nThe results are similar except for the confidence intervals for the trend for the two-stage method, which are much narrower for the replication, but are centred on a similar mean.\nThis may be because I could not be certain that I had found the original code to perform the Kalman filter smoothing because of the difference in how ``tau'' was parameterised. \nI also needed some trial-and-error to select ``tau'' which controls the smoothness of the trend and is defined by the user rather than being estimated by an algorithm.\n%However, the estimates do look remarkably similar to my replicated estimates of the trend using the combined method. Hence another possibility is that the original figures were mislabeled as being from the two-stage method when in fact they came from the combined method.\n\n\\begin{figure}[!h]\n \\centerline{\\includegraphics[scale=0.9]{figures/original_figure.eps}}\n\n\\caption{Original results for three locations using the two-stage method (left and centre column) and combined method (right column). The left column shows the observed data, estimated trend and 95\\% confidence interval. The middle column shows the periodogram of the de-trended data and limit for the test of seasonal structure (dotted line). The right column shows the observed data, estimated trend, and estimated annual seasonal pattern.}\n\t\\label{fig:original}\n\\end{figure}\n\n\\begin{figure}[!h]\n\\centerline{Perth, Australia}\n\t\\centerline{\n \\includegraphics[scale=0.2]{figures/trend_twostage_perth.eps}\n \\includegraphics[scale=0.2]{figures/periodogram_twostage_perth.eps}\n \\includegraphics[scale=0.2]{figures/estimates_combined_perth.eps}\n }\n\n\\centerline{Belfast, UK}\n \\centerline{\n \\includegraphics[scale=0.2]{figures/trend_twostage_belfast.eps}\n \\includegraphics[scale=0.2]{figures/periodogram_twostage_belfast.eps}\n \\includegraphics[scale=0.2]{figures/estimates_combined_belfast.eps}\n }\n\n\\centerline{Warsaw, Poland}\n \\centerline{\n \\includegraphics[scale=0.2]{figures/trend_twostage_warsaw.eps}\n \\includegraphics[scale=0.2]{figures/periodogram_twostage_warsaw.eps}\n \\includegraphics[scale=0.2]{figures/estimates_combined_warsaw.eps}\n }\n\n\\caption{Replicated results for Figure~\\ref{fig:original}}\n\t\\label{fig}\n\\end{figure}\n\nThe estimates of the seasonal patterns for the three locations are replicated in Table~\\ref{tab:season} which also shows the original results. The results are similar for the two-stage method, with most differences being within $\\pm$~0.1. The biggest difference being for the estimate of the noise standard deviation in Belfast (9.0 versus 8.8).\n\n\\begin{table}[!h]\n \\centering\n \\begin{tabular}{rccc}\\hline\n & Frequency, & Amplitude, & Noise \\\\ \nLocation & months & rate per 100,000 & SD \\\\ \\hline\n\\multicolumn{4}{c}{Two-stage method} \\\\ \\hline\nPerth, Australia & 2.3 & 2.0 (2.0) & 4.3 (4.2)\\\\\n & 12 & 2.6 (2.6) \\\\\nBelfast, UK & 12 & 7.0 (7.0) & 8.7 (8.7) \\\\ \nWarsaw, Poland & 3.4 & 3.3 (3.3) & 7.5 (7.5)\\\\\n & 6 & 3.6 (3.6)\\\\ \n & 12 & 3.2 (3.2)\\\\\n\\hline\n\\multicolumn{4}{c}{Combined method} \\\\ \\hline\nPerth, Australia & 2.3 & 1.0 (1.0) & 3.9 (3.9)\\\\\n & 2.7 & ---- (0.8) \\\\\n & 3.8 & 1.3 (1.3)\\\\\n & 5.5 & 1.5 (1.4)\\\\\n & 12 & 2.7 (2.7)\\\\\nBelfast, UK & 2.8 & ---- (1.6) & 9.0 (8.8) \\\\\n & 12 & 7.1 (7.0) & \\\\ \nWarsaw, Poland & 3.4 & 3.4 (3.4) & 7.9 (7.8)\\\\\n & 6 & 3.7 (3.7)\\\\ \n & 12 & 3.3 (3.3)\\\\\n\\hline\n\\end{tabular}\n \\caption{Estimates of seasonal patterns using the two-stage and combined methods for three locations. The original results are in brackets. Each row in the table corresponds to a seasonal frequency at a location. SD = standard deviation.}\n \\label{tab:season} \n\\end{table}\n\nThere were differences in the frequencies for the combined method, as two seasonal frequencies were not included in the new results, one in Perth and one in Belfast. Although the amplitudes were relatively small (1.6 or less), hence these are smaller seasonal patterns that are marginally important.\n\nThe difference in seasonal frequencies could be because of a key ambiguity in the original paper regarding the decision to add additional seasonal components for the combined approach. After fitting a candidate model, the residuals were tested to look for additional seasonal structure. If structure existed, then the model was modified to add an additional seasonal component as necessary, followed by another test of the residuals. The code produced both the periodogram and the spectrum, which is a smoothed version of the periodogram. The periodogram tended to flag significant seasonal patterns when the spectrum did not, and I used the spectrum to make the decisions because some of the statistically significant patterns found by the periodogram were only just above the 0.05 threshold used to judge statistical significance. This part of the replication was therefore somewhat subjective.\n\nSome of the discrepancy in results in Table~\\ref{tab:season} could be because the estimates are made using Markov chain Monte Carlo which uses random steps to make estimates. Hence the estimates will depend on the original random number seed. However, I used 5,000 estimates with a burn-in of 500, so these differences should be minor and may explain some of the small differences in the first decimal place for the amplitude and noise estimates. This potential discrepancy would have been avoided if the original code had specified the random number seed.\n\n%If you had to make any modifications to the software or to the instructions that were supplied with it, try to describe which competence another researcher would need in order to do the same work. Would a general familiarity with your programming language and environment have been sufficient?\n\nOverall the replication took about 21~hours. This was mostly the fiddly task of working out which macros to run and re-creating the figures and tables. Given the amount of code that had to be re-created, I think that only someone familiar with SAS would have been able to replicate the results. My familiarity with the MONICA Project also helped me create the new code, hence I am not certain that an outsider could have replicated the results, perhaps only by trial-and-error.\n\n%Software\n%If the licenses applying to the original software allow, please make your old software available as a repository on GitHub or an equivalent site. In the best case scenario, in which the old code is publishable, here is the procedure:\n\n%Add your old software as a single commit (ideally the initial one) of a new source repository.\n%Edit the code as necessary to reproduce the results in today's computational environment, committing the changes in logical units in the process. Include the additional tools (scripts, makefiles, ...) that you wrote for re-running your software and the additional instructions.\n%Make sure your repository contains a license.\n%Finally, submit your repository to Software Heritage for archiving.\n\n\\section{Discussion}\n\n%In addition to the content covered below, we are interested in the insight resulting from the reproducibility experience in relation to the choice of technology, and in your opinion on the original source code (clarity, documentation, etc).\n\nIt was an interesting exercise to bring this old code back to life. It would have been easier and faster if my original code had been properly arranged and prepared solely to repeat the original analyses. However, I was not aware of guidelines for curating code back in 2004. My original code had many comments which were useful, but there were problems with the overall file structure and how the files related to one another. There were also ambiguities with the dates of the code, and the changes I made after posting the original files. This is now solved by sites that track and date-stamp all changes, such as \\textit{GitHub}.\n\nMy main reason for sharing the original code was so that others could adapt it to their own data. However, of the 21 papers that cited the original paper which I could access, none of them stated that they re-used my SAS code. \n%I was not focused on the nitty-gritty of reproducing the original results and my code was in bad shape to do this.\n\n% One instead cited the available R code to run the same analysis.\n% Not confident it was exactly the same, but broadly similar.\n\nOverall the replication was broadly successful in terms of re-creating the key figure and estimates for three locations. There was subjectivity in modelling decisions concerning the smoothness of the trend and number of seasonal components, which made perfect replication difficult.\n\n%The ``combined'' model in the SAS files are available in the ``season'' package in R \\supercite{season} using the ``nscosinor`` command.", "meta": {"hexsha": "6bc0268153a4a4def74ee0cc538970318295cd3f", "size": 16753, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "article/content.tex", "max_stars_repo_name": "khinsen/tenyears", "max_stars_repo_head_hexsha": "c6b4da5f298888c5f0f488ad0231713f1d11204b", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "article/content.tex", "max_issues_repo_name": "khinsen/tenyears", "max_issues_repo_head_hexsha": "c6b4da5f298888c5f0f488ad0231713f1d11204b", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "article/content.tex", "max_forks_repo_name": "khinsen/tenyears", "max_forks_repo_head_hexsha": "c6b4da5f298888c5f0f488ad0231713f1d11204b", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-11-25T16:47:52.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-25T16:47:52.000Z", "avg_line_length": 97.4011627907, "max_line_length": 896, "alphanum_fraction": 0.782427028, "num_tokens": 3848, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5312093733737563, "lm_q2_score": 0.6442251201477016, "lm_q1q2_score": 0.34221842238529343}} {"text": "\\chapter{Machine Learning and Random Forests}\n\\label{ch:ML}\n%\nThe analysis tasks described in \\autoref{ch:iact} make strong use of machine\nlearning tools to be performed in the most efficient way. There are a lot of\ndifferent machine learning techniques that find increasingly many and\nsuccessfull applications in modern physics. The following overview is based on\nthe works \\cite{bagging, goodfellow, mitchell}.\n\n\\section{Machine Learning}\n%\nMachine learning describes the field of applied statistics that uses computers\nto learn to solve certain problems. The \\textit{learning} is defined by Tom\nMitchell~\\cite{mitchell} as: \\enquote{A computer program is said to learn from\nexperience $E$ with respect to some class of tasks $T$ and performance measure\n$P$, if its performance at tasks in $T$, as measured by $P$, improves with\nexperience $E$.} There is a wide variety of different tasks this can be applied\nto and many performance measures. The most important preposition for well\nsuited problems is the available experience or in this case the amount of data\nto learn from.\n\nMachine learning aims at solving problems that profit from the computing power\nof modern technology but are not the kind of problem to be solved by a typical\nprogram written by a human \\cite{goodfellow}.\n\n\\subsection{The Experience}\n%\nTo make a generic machine learning algorithm work for a specific task it has to\nbe \\textit{trained} on data. Just like a human, it needs to be given\ninformation to base a decision on and to be shown how that decision based on\nthe information is supposed to look. The kind of data suitable for learning\ndepends on the kind of algorithm to be used and vice versa. Generally, machine\nlearning algorithms can be divided into \\textbf{supervised} and\n\\textbf{unsupervised} algorithms, both of which will be used in this work.\n\nData to derive experience from for supervised algorithms consists of data points\nwith a certain feature set and a specific label or true value. The data sets for\nthe \\textit{separation} of two classes for example contain a certain feature\nset for every data point and a label for the corresponding class each of these\npoints belongs to. Supervised learning therefore aims at learning to predict\na certain target value (or multiple) from a given feature set including this\nvalue. The used learning data in this analysis is provided by simulations, so\nthat the true values for the machine learning tasks are known. Unsupervised\nlearning, however, works on unlabelled data and is used in this work in form of\na clustering algorithm.\n\n\\subsection{The Task}\n%\nThe task a machine learning algorithm is supposed to do is the solution of a\nspecific problem in the best possible way. To reach this solution the process\nof learning is used. A machine learning algorithm uses data to adapt to a specific task upon the given data. The three desired tasks\nwithin the analysis of Cherenkov telescope images are the \\textit{separation} of\ngamma-rays from hadronic cosmic rays, as well as the \\textit{estimation} of the\nenergy and source position of cosmic gamma-rays. The kind of task already\ndetermines what machine learning techniques are best to be used or which ones\ndo not suit the problem and how working architectures have to look. A\nclassification like the separation of gamma-rays from hadronic cosmic rays\nrequires a different output and therefore network structure than a regression\ntask.\n\n\\subsection{The Performance Measure}\n%\nAs described above, machine learning is about improving on certain tasks.\nTherefore, it is essential to quantify the performance during, but also after the\nlearning process. This already implicates that performance measures are very\nspecific to the task. During the training of a machine learning algorithm a\nloss is being minimalized. This loss is dependant on the task at hand and improved during the training epochs until a saturation is reached. The performance of a classification task is naturally\nmeasured by the \\textbf{accuracy} of the model, because it simply describes how\nmany of the model's outputs are correct. When validating continuous outputs\nrather than discrete classifications an accuracy is not appropriate. The\nestimation of the energy, e.g., requires a continuous-valued metric.\n\nThe models are trained on specific, simulated data but only to be applied on\ndata sets they have \\textit{not} been trained on. Thus, the interesting metrics\nare those calculated on data sets complementary to the training data sets,\nbecause they resemble the real use case. To do so, the whole data set is divided\ninto a fraction determined for training and a test set on which the metrics can\nbe calculated. A frequently used method for this is the\n\\textbf{cross-validation}. The data set is divided into $n$ equaly sized, random\nsubsamples; $n-1$ samples are used for training whereas the single excluded\nsample is used for validation. This is done $n$ times for each one of the\nsubsamples and the metrics are calculated as the mean value of all the single\nvalidations.\n\n\\section{Random Forests}\n%\nOne of the most frequently used machine learning algorithms and the one used\nfor the tasks in this analysis is the so called random forest. This is a\nsupervised learning algorithm based on the so called \\textit{decision tree}.\n\n\\subsection{Decision Trees}\n%\nDecision trees classify data points based on consecutive binary decisions. The\ndecisions are based on the single features of the data point and result in a\npoint specific result that is being returned. The number of single binary\ndecisions (also called \\textit{node}) preceding the final classification is\ncalled \\textit{depth} of the tree. Decision trees are trained on labelled data\nand determine thresholds for every feature to classify the data point.\n%\n\\begin{figure}[H]\n \\centering\n \\begin{tikzpicture}[node distance = 3cm, auto]\n \\node (f1) [feature] {\\texttt{feature\\_1}};\n \\node (f2) [feature, below of=f1, xshift=-2.4cm] {\\texttt{feature\\_2}};\n \\node (f3) [feature, below of=f1, xshift=2.4cm] {\\texttt{feature\\_3}};\n\n \\node (c1) [class1, below of=f2, xshift=-1.2cm] {\\texttt{class 1}};\n \\node (c2) [class2, below of=f2, xshift=1.2cm] {\\texttt{class 2}};\n\n \\node (c3) [class1, below of=f3, xshift=-1.2cm] {\\texttt{class 1}};\n \\node (c4) [class2, below of=f3, xshift=1.2cm] {\\texttt{class 2}};\n\n \\draw [pil] (f1) -- node[anchor=east] {$\\mathbf{>15}$\\;} (f2);\n \\draw [pil] (f1) -- node[anchor=west] {\\;$\\mathbf{\\leq15}$} (f3);\n \\draw [pil] (f2) -- node[anchor=east] {$\\mathbf{<100}$\\;} (c1);\n \\draw [pil] (f2) -- node[anchor=west] {\\;$\\mathbf{\\geq100}$} (c2);\n \\draw [pil] (f3) -- node[anchor=east] {$\\symbf{<-\\sfrac{\\pi}{2}}$\\;} (c3);\n \\draw [pil] (f3) -- node[anchor=west] {\\;$\\symbf{\\geq-\\sfrac{\\pi}{2}}$} (c4);\n \\end{tikzpicture}\n \\caption{Example sketch of a decision tree. The tree decides whether an input data point is of class 1 or class 2. The available features are \\texttt{feature\\_1}, \\texttt{feature\\_2} and \\texttt{feature\\_3}. This decision tree has a depth of 2 and solves the task of a binary separation by performing binary decisions at each node (green boxes). The decision thresholds are written next to the respective connecting lines.}\n \\label{fig:tree}\n\\end{figure}\n%\nTo determine the best decision threshold for each node, a specific loss function\nto be minimalized is necessary. For a classification task the required\nloss would be the information gain. The thresholds are thus optimized to get the best gain in information at the respective node. This way the decision tree is build from the\ntop node downwards until a perfect classification is achieved or until a set\nmaximum depth is reached.\n\nDecision trees can also be used for regression tasks. The only difference when\nsolving such tasks is the metric to find the best thresholds at each node.\nWhile for classification tasks the information gain is the natural choice, regression\ntasks require a metric describing the error of the single result. Thus, the\nthresholds are determined by minimizing the variance of the target variable.\n\n\\subsection{Random forests}\n%\nA decision tree that is fitted to the data too extensively will reach\nvery high accuracies, but will suffer from a very bad generalization. At some\npoint the tree starts to adapt to the training data's specifics too much and\nwill not work on other data that does not have these specific characteristics.\nThis phenomenon is called \\textit{overfitting} and generally is represented by\na large gap between training and test errors \\cite{goodfellow}. A way to\nprevent this from happening is to constrain the complexity of the decision tree\nwhile using a large number of different trees. This way the trees are not\noverfitted but the complete model is still complex enough.\n\nRandom forests are generated by a process called\n\\textit{bagging}~\\cite{bagging}. Every tree within the forest is trained on a\nsubset of the data randomly sampled with replacement. This way, a large number\nof slightly different trees is generated. While every single decision tree is\nstill prone to overfitting to fluctuations in its respective data set, averaging over\nall of them is not, as long as the trees are not correlated. To prevent such\ncorrelations and further generalize the model, additionally every decision node\ninside a single tree is only given a random subset of all available features.\nRandom forests therefore have two additional parameters: the number of trees\n$n$ and the number $k$ of available features at each node.\n\nThe output of the model consisting of such a forest is then generated\nby counting trees with a specific decision. A confidence for a multi-class\nseparation, e.g., can be generated by counting the fraction of trees that\ndecided for the specific class. The output of a random forest therefore is generated by averaging over all decision trees. Thus, the output score for a classification lies between \\num{0} and \\num{1}, representing the average class decision of the trees. By setting a threshold on this confidence\nclassification, decisions can be performed. In case of regression tasks the\noutput of the forest is the mean of the regression results of the single trees,\nas well.\n\n\\section{Performance Measures}\n%\nAs mentioned earlier, the right performance measure depends on the task to be\nvalidated. Since there are several different tasks to be performed during this\nanalysis a number of performance measures is needed for the evaluation.\n\n\\textbf{Accuracy.} The accuracy of a model describes the proportion of input\nexamples for which the model computes the right output. This performance\nmeasure can be used for models with discrete outputs, such as classifications\nbecause it only takes perfect matches as the right output.\n\n\\textbf{Receiver Operating Characteristic.} For every classification task solved via the above mentioned forest output the\nso called \\textit{receiver operating characteristic} curve (ROC) can be\ndetermined. For a given class this represents the rate of correctly classified\nexamples dependant on the rate of falsely classified ones. The area under this\ncurve (AUC) can be used to quantify the performance of a classification model.\nWhen randomly deciding on an equally distributed data set of two classes, the\nAUC is expected to be about \\num{0.5}, representing the worst possible\nclassification performance. A perfect classification yields an AUC of \\num{1}.\n\n\\textbf{Confusion Matrix.} The confusion matrix shows the results of a machine\nlearning algorithm against the true values. It therefore shows how well the\nalgorithm performs by showing where mismatches occur and with what effect. For\nclassifications this matrix shows what classes are most frequently confused\nwith each other, while for a regression task it shows how big the spread of\nreconstructed values around the true value is.\n\n% \\textbf{Precision.} For a classification the precision describes the fraction\n% of events correctly classified as signal events over all events labelled as\n% signal events.\n% %\n% \\begin{equation}\n% \\text{precision} = \\frac{T_{\\text{p}}}{T_{\\text{p}} + F_{\\text{p}}}\n% \\end{equation}\n% %\n% \\textbf{Recall.} The recall of a classification model describes the fraction\n% of events correctly classified as signal events over all true signal events.\n% %\n% \\begin{equation}\n% \\text{recall} = \\frac{T_{\\text{p}}}{T_{\\text{p}} + F_{\\text{n}}}\n% \\end{equation}\n% %\n% \\textbf{$F_\\beta$ score.} The $F_\\beta$ score of a classification model describes the harmonic mean of the precision and recall.\n% %\n% \\begin{equation}\n% F_\\beta = (1 + \\beta^2)\\frac{\\text{precision} \\cdot \\text{recall}}{\\text{precision} + \\text{recall}}\n% \\end{equation}\n%\n\\textbf{$\\symbf{R^2}$ Score.} This metric quantifies the goodness of fit for a\nchosen model. For a regression task, the $\\symbf{R^2}$ score or sometimes\ncalled \\textit{coefficient of determination} measures how well the input data\npoints are approximated by the model. A regression perfectly describing the\ninput data is indicated by an $\\symbf{R^2}$ score of 1, whereas an\n$\\symbf{R^2}$ score closer to \\num{0} characterizes a decreasing quality of the\nmodel. Models with scores below zero can be interpreted as not suitable to the\ndata.\n", "meta": {"hexsha": "1782c1e481b4b45893e44e8f4b819676c8debe4f", "size": 13238, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "content/04_ML.tex", "max_stars_repo_name": "KevSed/Master-Thesis", "max_stars_repo_head_hexsha": "89175c8acef3c982a6c79ab641cd150f1cd91621", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "content/04_ML.tex", "max_issues_repo_name": "KevSed/Master-Thesis", "max_issues_repo_head_hexsha": "89175c8acef3c982a6c79ab641cd150f1cd91621", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "content/04_ML.tex", "max_forks_repo_name": "KevSed/Master-Thesis", "max_forks_repo_head_hexsha": "89175c8acef3c982a6c79ab641cd150f1cd91621", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 59.3632286996, "max_line_length": 425, "alphanum_fraction": 0.7813113763, "num_tokens": 3150, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.5312093733737563, "lm_q1q2_score": 0.34221842238529343}} {"text": "\\chapter{A Drift-Kinetic Model for Scrape-off Layer Plasma Dynamics}\n\\label{ch:dk}\n\nA physical theory describing the dynamics of magnetized plasma systems is considered to be closed and, more fundamentally, predictive, if it provides a constitutional relation for the sources of Maxwell's equations, namely the charge density $\\rho$ and current density $\\mathbf{J}$, in terms of the electromagnetic fields $(\\phi, \\mathbf{A})$.\n%\nKinetic theory achieves this goal by providing a distribution function $f_a$ for each species $a$ in the plasma, where $f_a$ is a measure of the number of particles of species $a$ near point $\\mathbf x$, having velocity $\\mathbf v$, at time $t$ per unit volume and is normalized such that $\\int f d\\mathbf x d\\mathbf v=N$ with $N$ the total number of particles in the system.\n%\nWhen $f_a$ is known, the charge density and current density can be obtained by taking velocity moments of $f_a$, namely with $\\rho=\\sum_a q_a \\int f_a d\\mathbf v$ and $\\mathbf{J}= \\sum_a q_a \\int \\mathbf v f_a d \\mathbf v$ where $q_a$ is the charge of the species $a$.\n\nThe equation for the evolution of $f_a$ is derived from the analysis of the trajectories of the particles in the plasma.\n%\nWhen the details of particular temporal or spatial scales can be neglected, the equation for the evolution of $f_a$ can be greatly simplified.\n%\nThis is the case of drift-kinetic theory, where the description of the charged particles inside the plasma is reduced to the behavior of its guiding-centers \\citep{Hazeltine2003}.\n%\nThis is particularly useful in the SOL, where fluctuations are characterized by frequencies lower than the ion gyrofrequency \\citep{Endler1995,Agostini2011,Carralero2014,Garcia2015}, and the turbulent eddies, which include coherent radial propagation of filamentary structures \\citep{DIppolito2002,DIppolito2011,Carreras2005,Serianni2007}, have a radial extension comparable to the time-averaged SOL pressure gradient length $L_p$ \\citep{Zweben2007}.\n\n{In recent years, there has been a significant development of first-principles simulations of the SOL dynamics} with both kinetic \\citep{Tskhakaya2012} and gyrokinetic \\citep{Xu2007, Shi2015, Chang2017,Shi2017} codes.\n%\nHowever, as kinetic simulations {of the SOL and edge regions} remain prohibitive as they still are computationally extremely expensive, the less demanding fluid simulations are the standard of reference.\n%\nThe fluid simulations are usually based on the drift-reduced Braginskii \\citep{Braginskii1965,Zeiler1997} or gyrofluid \\citep{Ribeiro2008a,Held2016} models to evolve plasma density, fluid velocity and temperature. Fluid models assume that the distribution function is close to a local Maxwellian, and that scale lengths along the magnetic field are longer than the mean free path.\n%\nHowever, kinetic simulations show that the plasma distribution function is far from Maxwellian in the SOL region \\citep{Tskhakaya2008,Lonnroth2006,Battaglia2014} and that collisionless effects in the SOL might become important \\citep{Batishchev1997}.\n%\nThis is expected to be particularly true in ITER and other future devices that will be operated in the high confinement mode (H-mode) regime \\citep{Martin2008}.\n%\nIn such cases, a transport barrier is formed that creates a steep pressure gradient at the plasma edge.\n%\nIf the pressure gradient exceeds a threshold value, ELMs are destabilized \\citep{Leonard2014}, expelling large amounts of heat and particles to the wall.\n%\nDescribing structures with such high temperatures (and therefore low collisionality) with respect to the background SOL plasma requires therefore a model that allows for the treatment of arbitrary collision frequencies.\n%\nA kinetic full-F description is therefore needed for a proper SOL description \\citep{Hazeltine1998}.\n\nLeveraging the development of previous models {\\citep{Hammett1993,Dorland1993,Beer1996,Sugama2001,Ji2010,Zocco2011,Schekochihin2016,Hatch2016,Parker2016,Hirvijoki2016,Mandell2018}}, we construct here a moment-hierarchy to evolve the SOL plasma dynamics.\n%\nOur model is valid in arbitrary magnetic field geometries and, making use of the full Coulomb collision operator, at arbitrary collision frequencies.\n%\nThe model is derived within a full-F framework, as the amplitude of the background and fluctuating components of the plasma parameters in the SOL have comparable amplitude. We work within the drift approximation \\citep{Hinton1976, Cary2009}, which assumes that plasma quantities have typical frequencies that are small compared to the ion gyrofrequency $\\Omega_i=e B /m_i$, and their perpendicular spatial scale is small compared to the ion sound Larmor radius $\\rho_s=c_s/\\Omega_i$, with $c_s^2 = T_e/m_i$, $T_e$ the electron temperature, $B$ the magnitude of the magnetic field, $e$ the electron charge, and $m_i$ the ion mass.\n\nIn this chapter, we use the methods of Lagrangian mechanics to derive the equations of motion of a charged particle in an electromagnetic field in the drift-kinetic approximation, that is, when the magnetic field is slowly varying with respect to the gyroradius, and when fluctuations occur on spatial scales larger than the ion gyroradius.\n%\nA detailed description of the drift-kinetic ordering is provided in \\cref{sec:ordering}.\n%\nIn \\cref{sec:solparticle}, we derive the drift-kinetic Lagrangian and state the equations governing the particle position and velocity in the drift-kinetic approximation, together with the equation for the evolution of the distribution function, the so-called drift-kinetic equation.\n%\nThe drift-kinetic equation, when coupled to Maxwell's equations, yields a system of equations describing the dynamics of the plasma system that is, in principle, closed.\n%\nHowever, the numerical solution of kinetic models such as the drift-kinetic one still remains computationally extremely demanding.\n%\nFor this reason, the drift-kinetic equation is converted into a moment-hierarchy equation for the evolution of the velocity moments of the distribution function $f_a$ using a suitable polynomial expansion of $f_a$ i.e., using a Hermite-Laguerre polynomial basis.\n%\nThe expansion of the distribution function in a Hermite-Laguerre basis is performed in \\cref{sec:momentexpansion}, while the moment-hierarchy equation is derived in \\cref{sec:momenthierarchy}.\n%\nA shifted-velocity formulation, which retains the presence of a finite flow velocity in the Hermite-Laguerre basis and better captures strong near-Maxwellian flows with fewer expansion coefficients, is used.\n%\nA particular novelty of the framework derived here is the inclusion of collisions by evaluating explicitly the velocity moments of the full Coulomb nonlinear collision operator (the prefix full is used here to state that both like-particle and unlike-particle collisions are included).\n%\nThis allows us to describe turbulent systems arbitrarily far from equilibrium using a model that is particularly efficient for numerical implementation.\n%\nIn \\cref{sec:poisson}, the system of equations is closed by deriving Poisson's equation in terms of coefficients of the Hermite-Laguerre expansion of the distribution function.\n%\nFinally, a a fluid model based on the truncation of the Hermite-Laguerre expansion in the high collisionality regime is presented, which allows the comparison to well-known fluid models used to describe the plasma dynamics in the SOL.\n%\nThe conclusions follow.\n%\nWe note that the results described in the present chapter have been published in \\citet{Jorge2017}.\n\n\\section{Ordering}\n\\label{sec:ordering}\n\nDenoting {$k_\\perp \\sim |\\nabla_\\perp \\log \\phi|\\sim |\\nabla_\\perp \\log n|\\sim |\\nabla_\\perp \\log T_{e}|$ and $\\omega \\sim |\\partial_t \\log \\phi| \\sim |\\partial_t \\log n| \\sim |\\partial_t \\log T_e|$}, with $\\phi$ the electrostatic potential, we introduce the drift-kinetic ordering parameter $\\epsilon$ such that\\footnote[1]\n%\n{\\label{note1}We note that while this ordering differs from the one presented in \\citet{Jorge2017}, the set of equations presented to describe SOL plasmas remains unchanged.\n%\nWe also point out that the ordering $\\omega \\sim k_\\parallel c_s$ may become marginal near separatrix where $k_\\parallel$ decreases to values below than $\\omega/c_s$.\n}\n\n\\begin{equation}\n \\epsilon \\sim k_\\perp \\rho_s \\sim \\frac{k_\\parallel}{k_\\perp} \\ll 1.\n \\label{eq:ordering}\n\\end{equation}\n\n\\noindent {On the other hand, we let $k_\\perp L_p \\sim 1$ since turbulent eddies {are observed to have} an extension comparable to the scale lengths of the {time}-averaged quantities.}\nThese assumptions are in agreement with experimental measurements of SOL plasmas \\citep{LaBombard2001,Zweben2004,Myra2013,Carralero2014}.\n%\nWe set turbulence to be correlated along the magnetic field lines by ordering $\\omega \\sim k_\\parallel c_s$ (see \\cref{note1}), such that\n%\n\\begin{equation}\n \\frac{\\omega}{\\Omega_i} \\sim \\epsilon^2,\n\\label{eq:ordering2}\n\\end{equation}\n%\nan ordering in agreement with previous drift-reduced fluid models for the SOL \\citep{Zeiler1997,Catto2004}.\n%\nWe also order the electron collision frequency $\\nu_{ei}$ as\n\n\\begin{equation}\n\t\\frac{\\nu_{ei}}{ \\Omega_i} \\sim \\epsilon_\\nu < \\epsilon,\n\t\\label{eq:orderingnu}\n\\end{equation}\n%\nIn addition, the ion collision frequency ${\\nu_i =} \\nu_{ii}$ is ordered as $\\nu_{ii} < \\epsilon^2 \\Omega_i$ that, noticing $\\nu_i \\sim \\sqrt{{m_e}/{m_i}}(T_e/T_i)^{3/2} \\nu_e$ {(with $\\nu_e = \\nu_{ei}$)}, yields\n\\begin{equation}\n \\left(\\frac{\\epsilon_\\nu}{\\epsilon^2}\\right)^{2/3}\\left(\\frac{m_e}{m_i}\\right)^{1/3}\\lesssim\\frac{T_i}{T_e}\\lesssim 1.\n \\label{eq:titebound}\n\\end{equation}\nThe ordering in \\cref{eq:titebound} can be used to justify applying our model in the cold ion limit, $T_i \\ll T_e$, {but allows for $T_i \\sim T_e$}.\nWe note that in the SOL the ratio $T_i/T_e$ is typically in the range $1 < T_i/T_e < 4$ \\citep{Kocan2011}. Furthermore, it is seen that the ion temperature in this range of values plays a negligible role in determining the SOL turbulent dynamics, usually due to a steeper electron temperature profile compared with the ion one, which is usually below the threshold limit of the ion temperature gradient instability \\citep{Mosetto2015}.\n\nThe ordering in Eqs. (\\ref{eq:ordering})-(\\ref{eq:titebound}) is justified in a wide variety of experimental conditions. For example, for a typical JET discharge \\citep{Erents2000,Liang2007,Xu2009} with the SOL parameters $B_T = 2.5$ T, $T_e \\sim T_i \\sim 20$ eV, $n_e \\simeq 10^{19}$ m$^{-3}$, and $k_\\perp \\sim 1$ cm$^{-1}$, we obtain $\\epsilon_\\nu \\sim 0.016$ and $\\epsilon \\sim 0.0182$. For a medium-size tokamak such as TCV \\citep{Rossel2012,Nespoli2017}, estimating $B_T = 1.5$ T, $T_e \\sim T_i \\sim 40$ eV, $n_e \\simeq 6 \\times 10^{18}$, and $k_\\perp \\sim 1$ cm$^{-1}$, we obtain $\\epsilon_\\nu \\sim 6.2 \\times 10^{-3}$ and $\\epsilon \\sim 0.043$. Finally, for small-size tokamaks such as ISTTOK \\citep{Silva2011a,Jorge2016}, with $B_T = 0.5$ T, $T_e \\sim T_i \\sim 20$ eV, $n_e \\simeq 0.8 \\times 10^{18}$, and $k_\\perp \\sim 1$ cm$^{-1}$, we obtain $\\epsilon_\\nu \\sim 0.0072$ and $\\epsilon \\sim 0.091$. Lower values of $\\epsilon_\\nu$, as in the presence of ELMs where temperatures can reach up to $100$ eV \\citep{Pitts2003}, are also included in the ordering considered here.\n%\n%Following typical SOL experimental measurements (see, e.g. \\citet{Zweben2007,Terry2009,Grulke2014}), we order $k_{\\parallel} \\sim 1/L_B \\sim 1/R$, with $L_B$ the background magnetic field spatial gradient scale and $R$ the tokamak major radius, and take $k_\\parallel \\rho_s \\sim \\epsilon^3$. This yields\n%\n%\\begin{equation}\n %\\frac{k_\\parallel}{k_\\perp} \\sim \\epsilon^2,\n %\\label{eq:kpar1}\n%\\end{equation}\n%\n%\\noindent a lower ratio than the ones used in most drift-kinetic and gyrokinetic deductions \\citep{Hahm1988a,Hazeltine2003,Abel2013}.\n%\nWe note that the orderings in \\cref{eq:orderingnu,eq:ordering,eq:ordering2} imply that\n\n\\begin{equation}\n k_\\parallel \\lambda_{mfp} \\sim \\sqrt{\\frac{m_i}{m_e}}\\frac{\\epsilon^2}{\\epsilon_\\nu},\n \\label{eq:kparlmfp}\n\\end{equation}\n%\nwhich includes both the collisional regime $k_{\\parallel} \\lambda_{mfp} \\ll 1$, when $\\epsilon_\\nu \\sim \\epsilon$, and the collisionless regime $(k_{\\parallel} \\lambda_{mfp})^{-1} \\ll 1$, when $\\epsilon_\\nu \\ll \\epsilon$.\n%\nFinally, the plasma parameter $\\beta = n T_e/(B^2/2\\mu_0)$ is ordered as $\\beta \\sim \\epsilon^3$, {implying that our equations describe plasma dynamics in an electrostatic regime}.\n%\nAlthough electromagnetic effects can lead to a non-negligible enhancement on heat and particle transport in the SOL \\citep{LaBombard2005}, we focus on devices with low-enough $\\beta$ such that the value of the MHD ballooning parameter $\\alpha_{\\text{MHD}} = \\beta R/L_p$ stays below the electromagnetic balloning instability threshold.\n%\nWe refer the reader to \\citet{Halpern2013a} for a detailed treatment of electromagnetic effects in the SOL within the drift-reduced fluid description and here we consider the electrostatic limit.\n\n\\section{SOL Guiding-Center Model}\n\\label{sec:solparticle}\n\n\\subsection{Single-Particle Motion}\n\nTo derive a convenient equation of motion in the presence of a strong magnetic field $\\mathbf B$, we start with the Hamiltonian of a charged particle of species $a$ \\citep{Jackson1999},\n%\n\\be\n H_a(\\mathbf q, \\mathbf p)=\\frac{[\\mathbf p - q_a \\mathbf A(\\mathbf q)]^2}{2m_a}+q_a \\phi(\\mathbf q),\n \\label{eq:hamiltonian}\n\\ee\n%\nand its associated Lagrangian,\n%\n\\begin{equation}\n L_a(\\mathbf x, \\mathbf v)=\\left[q_a \\mathbf A(\\mathbf x) + m_a \\mathbf v\\right]\\cdot \\dot{\\mathbf x}-\\left[\\frac{m_a v^2}{2}+q_a \\phi(\\mathbf x)\\right],\n \\label{eq:lagrangian}\n\\end{equation}\n%\nwhere $\\mathbf p = q_a \\mathbf A + m _a\\mathbf v$ is the canonical momentum conjugated to $\\mathbf q = \\mathbf x$, $\\mathbf v$ is the particle velocity, $\\mathbf A$ is the magnetic vector potential, $\\phi$ is the electrostatic potential, $m_a$ is the mass of the particle and $q_a$ its charge.\n\nWe now perform a coordinate transformation from the phase-space coordinates $\\mathbf z = (\\mathbf x, \\mathbf v)$ to the guiding-center coordinates $\\mathbf Z = (\\mathbf R, v_\\parallel, \\mu, \\theta)$ by writing the particle velocity as [see, e.g., \\citet{Littlejohn1983a}]\n%\n\\begin{align}\n \\mathbf v &= \\mathbf U+ v_\\perp' \\mathbf c,\n \\label{eq:GCcoordinates}\n\\end{align}\n%\nwith\n%\n\\begin{equation}\n \\mathbf U = \\mathbf v_E(\\mathbf R) + v_\\parallel \\mathbf b(\\mathbf R),\n\\end{equation}\n%\nand $\\mathbf v_E = \\mathbf E \\times \\mathbf B/B^2$ the $\\mathbf E \\times \\mathbf B$ velocity. The gyroangle $\\theta$, defined as\n%\n\\begin{equation}\n \\theta = \\tan^{-1} \\left[\\frac{\\left(\\mathbf v - \\mathbf U \\right)\\cdot \\mathbf e_2}{\\left(\\mathbf v - \\mathbf U \\right)\\cdot \\mathbf e_1} \\right]\n\\end{equation}\n%\nis introduced by defining the right-handed coordinate set $(\\mathbf e_1, \\mathbf e_2, \\mathbf b)$, such that $\\mathbf c = -\\mathbf a \\times \\mathbf b=\\mathbf d a(\\theta)/d\\theta$, with $\\mathbf a = \\cos \\theta \\mathbf e_1 + \\sin \\theta \\mathbf e_2$.\nThe decomposition in \\cref{eq:GCcoordinates} allows us to isolate the high-frequency gyromotion contained in the $v_\\perp' \\mathbf c$ term, from the dominant guiding-center velocity $\\mathbf U$.\nThe adiabatic invariant $\\mu$ is defined as\n%\n\\begin{equation}\n \\mu = \\frac{m_a v_\\perp^{'2}}{2B}\n\\label{eq:gcmu}\n\\end{equation}\n%\nwhereas the guiding-center position is\n%\n\\begin{equation}\n \\mathbf R = \\mathbf x - \\rho_a \\mathbf a,\n \\label{eq:GCx}\n\\end{equation}\n%\nwith $\\rho_a = \\sqrt{2 m_a \\mu/(q_a^2 B)}$ the Larmor radius. Incidentally, for the case of weakly varying magnetic fields, \\cref{eq:GCx} describes the circular motion of a particle around its guiding-center $\\mathbf R$ with radius $\\rho_a$, i.e., $(\\mathbf x - \\mathbf R)^2 = \\rho_a^2$.\n\nAs our goal is to develop a model that describes turbulent fluctuations occurring on a spatial scale longer than the sound Larmor radius $\\rho_s$, and a time scale larger than the gyromotion one, we keep terms in the Lagrangian up to $O(\\epsilon)$ and order $T_i \\sim T_e$, which implies\n%\n\\begin{equation}\n k_{\\perp} \\rho_i \\sim \\epsilon.\n \\label{eq:tiordering}\n\\end{equation}\n%\nWe therefore expand the electromagnetic fields around $\\mathbf R$, to first order in $\\epsilon$, i.e.,\n%\n\\begin{equation}\n \\phi(\\mathbf x) \\simeq \\phi(\\mathbf R) + \\rho_a \\mathbf a \\cdot \\nabla_{\\mathbf R} \\phi(\\mathbf R),\n \\label{eq:FLRexp}\n\\end{equation}\n%\nand similarly for $\\mathbf A$. In the following, if not specified, the electromagnetic fields and potentials are evaluated at the guiding-center position $\\mathbf R$, and we denote $\\nabla=\\nabla_{\\mathbf R}$.\nIn addition, to take advantage of the difference between the turbulent and gyromotion time scales, we use the gyroaveraged Lagrangian $\\lb L_a \\rb_{\\mathbf R}$ to evaluate the plasma particle motion, where the gyroaveraging operator $\\lb \\chi \\rb_{\\mathbf R}$ acting on a quantity $\\chi(\\theta)$ is defined as\n%\n\\begin{equation}\n \\lb \\chi \\rb_{\\mathbf R} = \\frac{1}{2\\pi}\\int_0^{2\\pi} \\chi (\\theta) d\\theta,\n\\label{gyaveroperator}\n\\end{equation}\n%\nwhich is performed at fixed guiding-center coordinates $\\mathbf R$, $v_\\parallel$ and $\\mu$.\n\nTo evaluate $\\lb L_a \\rb_{\\mathbf R}$ we note that, with the expansion for $\\phi$ and $\\mathbf A$, the Lagrangian in \\cref{eq:lagrangian} can be expressed as $L_a=L_{0a}+L_{1a}+\\tilde L_a$ where $L_{0a}$ is gyroangle independent,\n%\n\\begin{equation}\n \\begin{split}\n L_{0a} &= \\left(q_a \\mathbf A + m_a \\mathbf U\\right)\\cdot \\dot{\\mathbf R}-\\left(\\frac{m_a v_\\parallel^2}{2}+\\frac{m_a v_E^2}{2}+\\mu B+q_a \\phi\\right),\n \\end{split}\n \\label{eq:L0}\n\\end{equation}\n%\n$L_{1a}$ is proportional to $\\rho_a^2$ (and hence to $\\mu$) and is order $\\epsilon^0$\n%\n\\begin{equation}\n \\begin{split}\n &L_{1a} = \\rho_a^2 q_a \\dot \\theta \\left(\\mathbf a \\cdot \\nabla\\right) \\left(\\mathbf A \\cdot \\mathbf c\\right)+m_a \\rho_a^2 \\Omega \\dot \\theta+{\\rho_a \\dot \\rho_a}\\left[q_a \\left(\\mathbf a \\cdot \\nabla\\right)\\left(\\mathbf A \\cdot \\mathbf a\\right)\\right],\n \\end{split}\n \\label{eq:L2}\n\\end{equation}\n%\nand the $\\tilde L_{a}$ contribution contains the terms {linearly proportional to $\\cos \\theta$ or $\\sin \\theta$} \\citep{Cary2009} {which are not present in $\\lb L_a \\rb_{\\mathbf R}$, as $\\lb \\tilde L_a \\rb_{\\mathbf R} = 0$}.\n\nWe note that $\\lb L_{1a} \\rb_{\\mathbf R}$ can be simplified since $\\lb \\left(\\mathbf a \\cdot \\nabla\\right) \\mathbf A \\cdot \\mathbf c \\rb_{\\mathbf R} = -\\mathbf b \\cdot (\\nabla \\times \\mathbf A) /2$, and $\\lb \\left(\\mathbf a \\cdot \\nabla\\right) \\mathbf A \\cdot \\mathbf a \\rb_{\\mathbf R} = \\nabla_\\perp\\cdot\\mathbf{A}/2$. Subtracting the total derivative $-q_a d/dt(\\rho_a^2 \\nabla_\\perp \\mathbf A)/4$ from $\\lb L_a \\rb_{\\mathbf R}$, which does not alter the resulting equations of motion, we redefine the gyroaveraged Lagrangian as\n%\n\\begin{equation}\n \\begin{split}\n \\lb L_a \\rb_{\\mathbf R} &= \\left(q_a\\mathbf A + m_a{\\mathbf U}\\right) \\cdot \\dot{\\mathbf R} - \\left(\\frac{m_a v_\\parallel^2}{2}+\\frac{m_a v_E^2}{2} + q_a \\phi\\right)-\\mu B\\left(1- \\frac{\\dot \\theta}{\\Omega_a}\\right)\n -\\frac{\\rho_a^2}{4}\\frac{d }{dt}\\left[\\nabla_\\perp\\cdot\\left(q_a \\mathbf A\\right)\\right].\n \\end{split}\n \\label{eq:gyroLag}\n\\end{equation}\n%\nWe now order the terms appearing in $\\lb L_a \\rb_{\\mathbf R}$.\nAs imposed by the Bohm sheath conditions \\citep{Stangeby2000}, both electrons and ions stream along the field lines with parallel velocities comparable to the sound speed $c_s = \\sqrt{T_e/m_i}$ in the SOL.\nThe Bohm boundary conditions at the sheath also set the electrostatic potential $e \\phi \\sim \\Lambda T_e$ across the SOL, where $\\Lambda = \\ln \\sqrt{m_i/(m_e 2\\pi)}\\simeq 3$.\nTherefore, we keep the $m_a v_E^2/2$ term in the Lagrangian in \\cref{eq:gyroLag}, as to take into account the presence of the numerically large factor $\\Lambda^2$ in $v_E^2 \\sim \\epsilon^2 \\Lambda^2 c_s^2$.\n\nBy neglecting the higher-order terms in \\cref{eq:gyroLag}, i.e., $-(\\rho_a^2/4){d }\\left[\\nabla_\\perp\\cdot\\left(q_a \\mathbf A\\right)\\right]/{dt}$, the expression for the gyroaveraged Lagrangian describing SOL single-particle dynamics, up to $O(\\epsilon)$, can be written as\n%\n\\begin{equation}\n \\lb L_a \\rb_{\\mathbf R} = q_a \\mathbf A^* \\cdot \\dot{\\mathbf R} - q_a \\phi^* -\\frac{m_a v_\\parallel^2}{2}+ \\mu \\frac{m_a \\dot \\theta}{q_a}.\n \\label{eq:lagSOL}\n\\end{equation}\n%\nwhere\n%\n\\begin{equation}\n q_a\\phi^* = q_a\\phi+ m_a v_E^2/2+\\mu B\n\\label{eq:phis}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n q_a\\mathbf A^* = q_a \\mathbf A + m_a v_\\parallel \\mathbf b + m_a \\mathbf v_E.\n\\label{eq:As}\n\\end{equation}\n%\nThe Euler-Lagrange equations applied to the Lagrangian in \\cref{eq:lagSOL} for the coordinates $\\theta$, $v_\\parallel$, and $\\mu$, yield, respectively, $\\dot \\mu = 0$, $v_\\parallel = \\mathbf b \\cdot \\dot{\\mathbf R}$, and $\\dot \\theta = \\Omega_a$. For the ${\\mathbf R}$ coordinate, we obtain\n%\n\\begin{equation}\n m_a \\dot v_\\parallel \\mathbf b = q_a (\\mathbf E^* + \\dot{\\mathbf R}\\times \\mathbf B^*),\n \\label{eq:motLag}\n\\end{equation}\n%\nwhere the relation $[\\nabla \\mathbf A - (\\nabla \\mathbf A)^T] \\cdot \\dot{\\mathbf R} = \\dot{\\mathbf R}\\times (\\nabla \\times \\mathbf A)$ has been used, and we defined $\\mathbf E^* = -\\nabla \\phi^* - \\partial_t \\mathbf A^*$, and $\\mathbf B^* = \\nabla \\times \\mathbf A^*$, with the parallel component of $\\mathbf B^{*}$ given by\n%\n\\begin{equation}\n B_{\\parallel}^* = \\mathbf B^* \\cdot \\mathbf b = B + \\frac{m_a}{q_a}\\mathbf b \\cdot \\nabla \\times \\left(v_\\parallel \\mathbf b + \\mathbf v_E\\right).\n\\label{eq:defbpars} \n\\end{equation}\n%\nBy projecting \\cref{eq:motLag} along $\\mathbf B^*$, we derive $m \\dot v_\\parallel B_\\parallel^*= e \\mathbf E^* \\cdot \\mathbf B^*$, while crossing with $\\mathbf b$ yields the guiding-center velocity $\\dot{\\mathbf R} B_\\parallel^* = v_\\parallel \\mathbf B^* + \\mathbf E^* \\times \\mathbf B/B$. Using the expressions for the fields $\\mathbf E^{*}$ and $\\mathbf B^{*}$, we obtain \n%\n\\be\n \\dot{\\mathbf R} = \\mathbf U+\\frac{\\mathbf B}{\\Omega_a B_\\parallel^*}\\times\\left(\\frac{d \\mathbf U}{dt}+\\frac{\\mu\\nabla B}{m_a}\\right),\n \\label{eq:GC1}\n\\ee\n%\nand\n%\n\\be\n m_a \\dot v_\\parallel = q_a E_\\parallel - \\mu \\nabla_\\parallel B + m_a \\mathbf v_E \\cdot \\frac{d \\mathbf b}{dt}-m_a\\mathcal{A},\n \\label{eq:GC2}\n\\ee\n%\nIn \\cref{eq:GC1,eq:GC2}, {in addition to the time derivatives of the phase-space coordinates $\\dot{\\mathbf R}, \\dot v_{\\parallel}$, that only have an explicit time dependence, we define the total derivative $d/dt$ of a field $\\phi(\\mathbf R, t)$ that has an explicit time and $\\mathbf R$ dependence as\n%\n\\begin{equation}\n \\frac{d\\phi}{dt} = \\frac{\\partial \\phi}{\\partial t} + \\mathbf U \\cdot \\nabla \\phi.\n\\end{equation}\n%\nThe $\\mathcal{A}$ term represents the higher-order nonlinear terms in $\\dot v_\\parallel$ that ensure phase-space conservation properties \\citep{Cary2009}, and it is given by\n%\n\\begin{equation}\n \\mathcal{A}=\\frac{B}{B_\\parallel^*}\\left(\\left.\\frac{d \\mathbf U}{dt}\\right|_\\perp + \\mu \\nabla_\\perp B\\right)\\cdot \\frac{\\nabla \\times \\mathbf U}{\\Omega_a},\n\\end{equation}\n%\nwith $d_t\\mathbf U|_\\perp=-\\mathbf b \\times (\\mathbf b \\times d_t\\mathbf U)$.\n\nThe guiding-center equations of motion (\\ref{eq:GC1}) and (\\ref{eq:GC2}) satisfy the energy, $E_{gc} = q_a \\phi^*+m_a v_\\parallel^2/2$ \\citep{Cary2009}, and momentum, $\\mathbf P_{gc} = e \\mathbf A^*$ \\citep{Cary2009}, conservation laws, given by\n%\n\\begin{equation}\n \\frac{d E_{gc}}{dt} = q_a \\frac{\\partial \\phi^*}{\\partial t}- q_a \\frac{\\partial \\mathbf A^*}{\\partial t}\\cdot \\dot{\\mathbf R},\n \\label{eq:enconservation}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n \\frac{\\partial \\mathbf P_{gc}}{\\partial t} = -q_a \\nabla \\phi^* + q_a \\nabla \\mathbf A^* \\cdot \\dot{\\mathbf R}.\n \\label{eq:moconservation}\n\\end{equation}\n%\nIn addition, we note that using \\cref{eq:GC1,eq:GC2} and Maxwell's equations, a conservation equation for $B_{\\parallel}^*$ can be derived\n%\n\\begin{equation}\n \\frac{\\partial B_{\\parallel}^*}{\\partial t} + \\nabla \\cdot (\\dot{\\mathbf R} B_{\\parallel}^*) + \\frac{\\partial}{\\partial v_\\parallel}\\left(\\dot v_\\parallel B_{\\parallel}^* \\right)=0.\n \\label{eq:liouvilleGC}\n\\end{equation}\n%\nSince $B_{\\parallel}^*$ is the Jacobian of the guiding-center transformation, \\cref{eq:liouvilleGC} is in fact the phase-space volume conservation law for the guiding-center system of equations (also called Liouville's theorem), reflecting therefore their Hamiltonian nature.\n\n\\subsection{The Guiding-Center Boltzmann Equation}\n\\label{subsec:gcboltzmann}\n\nThe Boltzmann equation for the evolution of the distribution function $f_a(\\mathbf x, \\mathbf v)$ of the particles in $(\\mathbf x, \\mathbf v)$ coordinates is\n%\n\\be\n \\frac{\\partial f_a}{\\partial t}+\\dot{\\mathbf x}\\cdot \\nabla_{\\mathbf x} f_a + \\dot{\\mathbf v}\\cdot \\nabla_{\\mathbf v} f_a = C(f_a),\n \\label{eq:boltzmann}\n\\ee\n%\nwhere $C(f_a)=\\sum_b C(f_a,f_b) = \\sum_b C_{ab}$ is the collision operator.\n%\nBecause $f_a$ can significantly deviate from a Maxwellian distribution function in the SOL \\citep{Battaglia2014}, we consider the {bilinear} Coulomb operator $C_{ab}$ \\citep{Balescu1988}, to model collisions between particles of species $a$ and $b$\n%\n\\be\n \\begin{split}\n C_{ab}&=L_{ab} \\frac{\\partial}{\\partial v_i}\\left[\\frac{\\partial^2 G_b}{\\partial v_i \\partial v_j}\\frac{\\partial f_a}{\\partial v_j}-\\frac{m_a}{m_b}\\frac{\\partial H_b}{\\partial v_i}f_a\\right],\n \\end{split}\n \\label{eq:coulombop}\n\\ee\n%\nwith\n%\n\\be\n \\begin{split}\n H_b&=2\\int \\frac{f_b(\\mathbf v')}{|\\mathbf v - \\mathbf v'|}d\\mathbf v',\n \\end{split}\n\\label{eq:roshb}\n\\ee\n%\nand\n%\n\\be\n \\begin{split}\n G_b=\\int f_b(\\mathbf v')|\\mathbf v - \\mathbf v'|d\\mathbf v',\n \\end{split}\n\\ee\n%\nthe Rosenbluth potentials satisfying $\\nabla^2_v G_b = H_b$. In \\cref{eq:coulombop} we introduced $L_{ab}=q_a^2 q_b^2 \\lambda/(4 \\pi \\epsilon_0^2 m_a^2)=\\nu_{ab} v_{tha}^3/n_b$, where $\\lambda$ is the Coulomb logarithm, $\\nu_{ab}$ the collision frequency between species $a$ and $b$, and $v_{tha}^2=2 T_a/m_a$.\n\nTaking advantage of the small electron to ion mass ratio, the collision operator between unlike-species can be simplified [see, e.g. \\citet{Balescu1988,Helander2002}]. \nThe electron-ion collision operator, to first order in $m_e/m_i$, is given by the operator $C_{ei}(f_e)=C_{ei}^0+C_{ei}^1$, where $C_{ei}^0$ is the Lorentz pitch-angle scattering operator\n\\be\n \\begin{split}\n C_{ei}^0&=\\frac{n_i L_{ei}}{v_{the}^3}\\frac{\\partial}{\\partial \\mathbf c_e}\\cdot\\left[\\frac{1}{c_e}\\frac{\\partial f_e}{\\partial \\mathbf c_e}-\\frac{\\mathbf c_e}{c_e^3}\\left(\\mathbf c_e \\cdot \\frac{\\partial f_e}{\\partial \\mathbf c_e}\\right)\\right],\n \\end{split}\n \\label{eq:cei0}\n\\ee\n%\nand $C_{ei}^1$ the momentum-conserving term\n%\n\\be\n \\begin{split}\n C_{ei}^1&={\\frac{2 n_i L_{ei}}{v_{the}^4 c_e^3}f_{Me}{\\mathbf u_{i}} \\cdot \\mathbf c_e}.\n \\end{split}\n \\label{eq:cei1}\n\\ee\n%\nwith $\\mathbf c_a = (\\mathbf v - \\mathbf u_a)/v_{tha}$.\n%\nIon-electron collisions, to first order in $m_e/m_i$, are desribed using the operator\n%\n\\begin{equation}\n \\begin{split}\n C_{ie}&=\\frac{ \\mathbf R_{ei}}{m_i n_i v_{thi}}\\cdot \\frac{\\partial f_i}{\\partial \\mathbf c_i}%,\\\\\n %C_{ie}^1&=\n +\\nu_{ei}\\frac{n_e}{n_i}\\frac{m_e}{m_i}\\frac{\\partial}{\\partial \\mathbf c_i}\\cdot\\left(\\mathbf c_i f_i%,\\\\\n %C_{ie}^2&=\\nu_{ei}\\frac{m_e}{m_i}%\n +\\frac{T_e}{T_i}\n %\\frac{T_e}{m_i}\n \\frac{\\partial f_i}{\\partial \\mathbf c_i} \n \\right),\n \\end{split}\n \\label{eq:cie}\n\\end{equation}\n%\nwhere $\\mathbf R_{ei}=\\int m_e \\mathbf v C_{ei} d\\mathbf v$ is the electron-ion friction force.\n%\nWe take advantage of \\cref{eq:orderingnu} to order the electron collision frequency $\\nu_e$ and the ion collision frequency $\\nu_i$ as\n%\n\\begin{equation}\n \\frac{\\nu_i}{\\Omega_i} \\sim \\sqrt{\\frac{m_e}{m_i}}\\left(\\frac{T_e}{T_i}\\right)^{3/2}\\epsilon_{\\nu} < \\epsilon^2,\n \\label{eq:orderingnu2}\n\\end{equation}\n%\nwhere we used the relation $\\nu_i \\sim \\sqrt{{m_e}/{m_i}}(T_e/T_i)^{3/2} \\nu_e$. The orderings in \\cref{eq:orderingnu2,eq:FLRexp} yield the lower bound in \\cref{eq:titebound} for the ion to electron temperature ratio.\n\nWe now express the particle distribution function $f_a$ in terms of the guiding-center coordinates {by defining $F_a$, a function of guiding-center coordinates, as}\n%\n\\begin{equation}\n F_a(\\mathbf R, v_\\parallel, \\mu, \\theta) = f_a(\\mathbf x(\\mathbf R, v_\\parallel, \\mu, \\theta), \\mathbf v(\\mathbf R, v_\\parallel, \\mu, \\theta)).\n \\label{eq:fguidF}\n\\end{equation}\n%\nUsing the chain rule to rewrite \\cref{eq:boltzmann} in guiding-center coordinates, we obtain\n%\n\\be\n \\frac{\\partial F_a}{\\partial t}+\\dot{\\mathbf R}\\cdot \\nabla F_a + \\dot{v_\\parallel}\\frac{\\partial F_a}{\\partial v_\\parallel} + \\dot \\mu \\frac{\\partial F_a}{\\partial \\mu}+ \\dot \\theta \\frac{\\partial F_a}{\\partial \\theta} = C(F_a),\n \\label{eq:boltzmannSS}\n\\ee\n%\nwhere $\\dot{\\mathbf R}$ and $\\dot v_\\parallel$ are given by \\cref{eq:GC1} and \\cref{eq:GC2} respectively, $\\dot \\theta = \\Omega_a$, and $\\dot \\mu = 0$.\nEquation (\\ref{eq:boltzmannSS}) can be simplified by applying the gyroaveraging operator in \\cref{gyaveroperator}. This results in the drift-kinetic equation\n%\n\\begin{equation}\n \\frac{\\partial \\lb F_a \\rb_{\\mathbf R}}{\\partial t}+ \\dot{\\mathbf R} \\cdot \\nabla\\lb F_a \\rb_{\\mathbf R} + \\dot v_{\\parallel}\\frac{\\partial \\lb F_a \\rb_{\\mathbf R}}{\\partial v_\\parallel} = \\lb C(F_a)\\rb_{\\mathbf R}.\n \\label{eq:boltzmannGC1}\n\\end{equation}\n%\nWe now write \\cref{eq:boltzmannGC1} in a form useful to take gyrofluid moments of the form $\\int \\lb F_a \\rb_{\\mathbf R} B dv_\\parallel d\\mu d\\theta$ (see \\cref{sec:momenthierarchy}). Using the conservation law in \\cref{eq:liouvilleGC} for $B_{\\parallel}^{*}$, we can write the guiding-center Boltzmann equation in conservative form as\n%\n\\begin{equation}\n \\begin{split}\n &\\frac{\\partial (B_{\\parallel}^*\\lb F_a \\rb_{\\mathbf R})}{\\partial t}+ \\nabla \\cdot ( \\dot{\\mathbf R} B_{\\parallel}^*\\lb F_a \\rb_{\\mathbf R}) + \\frac{\\partial( \\dot v_{\\parallel a} B_{\\parallel}^*\\lb F_a \\rb_{\\mathbf R})}{\\partial v_\\parallel} = B_{\\parallel}^*\\lb C(F_a)\\rb_{\\mathbf R}. \n \\end{split}\n \\label{eq:boltzmannGC}\n\\end{equation}\n%\nMoreover, in order to relate the gyrofluid moments $\\int \\lb F_a \\rb_{\\mathbf R} B dv_\\parallel d\\mu d\\theta$ with the usual fluid moments $\\int f_a d^3 v$, we estimate the order of magnitude of the gyrophase dependent part of the distribution function $\\tilde F_a = F_a - \\lb F_a \\rb_{\\mathbf R}$ where $\\lb F_a \\rb_{\\mathbf R}$ obeys \\cref{eq:boltzmannGC1}. The equation for the evolution of $\\tilde F_a$ is obtained by subtracting \\cref{eq:boltzmannGC1} from the Boltzmann equation, \\cref{eq:boltzmannSS}, that is\n%\n\\begin{equation}\n \\frac{\\partial \\tilde F_a}{\\partial t}+\\dot{\\mathbf R}\\cdot \\nabla \\tilde F_a + \\dot{v_\\parallel}\\frac{\\partial \\tilde F_a}{\\partial v_\\parallel} + \\Omega_a \\frac{\\partial \\tilde F_a}{\\partial \\theta} = C(F_a)-\\lb C(F_a)\\rb_{\\mathbf R}.\n \\label{eq:boltztilde}\n\\end{equation}\n\nUsing the orderings in \\cref{eq:orderingnu,eq:orderingnu2}, as well as $\\partial_t \\sim \\dot{\\mathbf R} \\cdot \\nabla \\sim \\dot v_\\parallel \\partial_{v_\\parallel} \\sim \\epsilon \\Omega_i$ and $\n\\partial_\\theta \\sim 1$, the comparison of the leading-order term on the left-hand side of \\cref{eq:boltztilde} with the right-hand side of the same equation imply the following ordering for $\\tilde F_e$\n%\n\\begin{equation}\n \\frac{\\tilde F_e}{\\lb F_e \\rb_{\\mathbf R}} \\sim \\frac{m_e}{m_i}\\epsilon_\\nu< \\epsilon^2,\n \\label{eq:orderingftildee}\n\\end{equation}\n%\nand $\\tilde F_i$\n%\n\\begin{equation}\n \\frac{\\tilde F_i}{\\lb F_i \\rb_{\\mathbf R}} \\sim \\sqrt{\\frac{m_e}{m_i}}\\left(\\frac{T_e}{T_i}\\right)^{3/2}\\epsilon_\\nu < \\epsilon^2.\n \\label{eq:orderingftildei}\n\\end{equation}\n%\nTo evaluate the leading-order term of $\\tilde F_a$, we expand the collision operator $C(F_a) = C_0(\\lb F_a\\rb_{\\mathbf R}) + \\epsilon C_1(F_a) + ...$, such that\n%\n\\begin{equation}\n \\tilde F_a \\simeq \\frac{1}{\\Omega_a}\\int_0^\\theta\\left[C_0(\\lb F_a \\rb_{\\mathbf R} )-\\lb C_0( \\lb F_a \\rb_{\\mathbf R})\\rb_{\\mathbf R}\\right]d\\theta' + O( \\epsilon^3 \\lb F_a \\rb_{\\mathbf R}).\n \\label{eq:tildefapp}\n\\end{equation}\n%\nThe relation in \\cref{eq:tildefapp} can be further simplified by expanding the $\\theta$ dependence of $F_a$ in Fourier harmonics, \n%\n\\begin{equation}\n F_a=\\sum_m e^{i m \\theta} F_{m a},\n \\label{eq:fourftilde}\n\\end{equation}\n%\nso that for $m=0$ we have $\\lb F_a \\rb_{\\mathbf R} = F_{0a}$, and similarly for $C_0(\\lb F_a \\rb_{\\mathbf R})$\n%\n\\begin{equation}\n C_0(\\lb F_a \\rb_{\\mathbf R}) = \\sum_{m'} e^{i m' \\theta} C_{m' a}.\n\\end{equation}\n%\nWe can then write \\cref{eq:tildefapp} as\n%\n\\begin{equation}\n \\tilde F_{m a} = \\frac{C_{m a}}{i m \\Omega_a},\n \\label{eq:fmacmafourier}\n\\end{equation}\n%\nfor $m \\not=0$.\n\n\\section{Moment Expansion}\n\\label{sec:momentexpansion}\n\nWe now derive a polynomial expansion for the distribution function $\\lb F_a \\rb_{\\mathbf R}$ that simplifies the solution of \\cref{eq:boltzmannGC}, with the collision operators in Eqs. (\\ref{eq:coulombop}) - (\\ref{eq:cie}).\nThis section is organized as follows.\nIn \\cref{section:gcmoment} the Hermite-Laguerre basis is introduced, relating the corresponding expansion coefficients for $\\lb F_a \\rb_{\\mathbf R}$ with its usual gyrofluid moments.\nIn \\cref{section:jifluidexpansion}, we briefly review the fluid moment expansion of the Coulomb collision operator presented in \\citet{Ji2006, Ji2008}.\nIn \\cref{section:cabmomentexpansion}, leveraging the work in \\citet{Ji2006, Ji2008}, we expand $C_{ab}$ in terms of the product of the gyrofluid moments, for both like- and unlike-species collisions which, ultimately, allows us to solve \\cref{eq:boltzmannGC} in terms of gyrofluid moments.\n\n\\subsection{Guiding-Center Moment Expansion of \\texorpdfstring{$\\lb F_a \\rb_{\\mathbf R}$}{}}\n\\label{section:gcmoment}\n\nTo take advantage of the anisotropy introduced by a strong magnetic field, and efficiently treat the left-hand side of \\cref{eq:boltzmannGC} where the parallel and perpendicular directions appear decoupled, we express $\\lb F_a \\rb_{\\mathbf R}$ by using {a Hermite polynomial basis expansion for the parallel velocity coordinate \\citep{Grad1949,Armstrong1967a,Grant1967,Ng1999,Zocco2011,Loureiro2013a,Parker2015,Schekochihin2016,Tassi2016} and a Laguerre polynomial basis for the perpendicular velocity coordinate {\\citep{Zocco2015,Omotani2015,Mandell2018}}. More precisely, we use the following expansion\n%\n\\be\n \\begin{split}\n \\lb F_a \\rb_{\\mathbf R} &=\\sum_{p,j=0}^{\\infty} \\frac{N_a^{pj}}{\\sqrt{2^p p!}}F_{Ma} H_p(s_{\\parallel a})L_j(s_{\\perp a}^2),\n \\end{split}\n \\label{eq:gyrof}\n\\ee\n%\nwhere the {\\textit{physicists'}} Hermite polynomials $H_p$ of order $p$ are defined by the Rodrigues' formula \\citep{Abramowitz1972}\n%\n\\begin{equation}\n H_p(x)=(-1)^p e^{x^2}\\frac{d^p}{dx^p}e^{-x^2},\n\\end{equation}\n%\nand normalized via\n%\n\\begin{equation}\n \\int_{-\\infty}^{\\infty} dx H_p(x) H_{p'}(x) e^{-x^2} = 2^p p! \\sqrt{\\pi} \\delta_{p{p'}},\n\\end{equation}\n%\nand the Laguerre polynomials $L_j$ of order $j$ are defined by the Rodrigues' formula \\citep{Abramowitz1972} \n%\n\\begin{equation}\n L_j(x)=\\frac{e^x}{j!}\\frac{d^j}{dx^j}(e^{-x}x^j),\n\\end{equation}\n%\nwhich are orthonormal with respect to the weight $e^{-x}$\n%\n\\begin{equation}\n \\int_{0}^{\\infty} dx L_j(x) L_{j'}(x) e^{-x} = \\delta_{jj'}.\n\\end{equation}\n%\nBecause of the orthogonality of the Hermite-Laguerre basis, the coefficients $N_a^{pj}$ of the expansion in \\cref{eq:gyrof} are\n%\n\\be\n N_a^{pj}=\\frac{1}{N_{a}}\\int \\frac{H_p(s_{\\parallel a}) L_j(s_{\\perp a}^2) \\lb F_a \\rb_{\\mathbf R} }{\\sqrt{2^p p!}}\\frac{B}{m_a} d\\mu dv_\\parallel d\\theta,\n \\label{eq:gyromoments}\n\\ee\n%\nand correspond to the guiding-center moments of $\\lb F_a \\rb_{\\mathbf R}$.\n\nIn \\cref{eq:gyrof}, the shifted bi-Maxwellian is introduced\n%\n\\be\n\tF_{Ma}=N_a\\frac{e^{-{s_{\\parallel a}^2}-s_{\\perp a}^2}}{{\\pi}^{3/2}v_{th\\parallel a} v_{th\\perp a}^2},\n\t\\label{eq:bimax}\n\\ee\n%\nwhere $s_{\\parallel a}$ and $s_{\\perp a}$ are the normalized parallel and perpendicular shifted velocities respectively, defined by\n%\n\\begin{equation}\n s_{\\parallel a} = \\frac{v_\\parallel-u_{\\parallel a}}{v_{th\\parallel a}},~v_{th\\parallel a}^2=\\frac{2 T_{\\parallel a}}{m_a},\n \\label{eq:sparallela}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n s_{\\perp a}^2 = \\frac{v_\\perp^{'2}}{v_{th\\perp a}^{2}}=\\frac{\\mu B}{T_{\\perp a}},~v_{th\\perp a}^2=\\frac{2 T_{\\perp a}}{m_a},\n \\label{eq:sperpa}\n\\end{equation}\n%\n{which provide an efficient representation of the distribution function in both the weak ($u_{\\parallel a} \\ll v_{th a})$ and strong flow ($u_{\\parallel a} \\sim v_{th a}$) regimes by better capturing strong near-Maxwellian flows with fewer expansion coefficients \\citep{Hirvijoki2016}.}\n%\n\n\nThe guiding-center density $N_a$, appearing in \\cref{eq:bimax}, the guiding-center fluid velocity $u_{\\parallel a}$, in \\cref{eq:sparallela}, and the guiding-center parallel $T_{\\parallel a}=P_{\\parallel a}/N_a$ and perpendicular $T_{\\perp a}=P_{\\perp a}/N_a$ temperatures in \\cref{eq:sparallela,eq:sperpa} are defined as $N_a = ||1||_a$, $N_a u_{\\parallel a} = || v_{\\parallel}||_a$, $P_{\\parallel a} = m_a ||(v_\\parallel-u_{\\parallel a})^2||_a$, and $P_{\\perp a} = ||\\mu B ||_a$, where\n%\n\\begin{equation}\n ||\\chi||_a \\equiv \\int \\chi \\lb F_a \\rb_{\\mathbf R} \\frac{B}{m_a} d\\mu dv_\\parallel d\\theta.\n\\end{equation}\n%\nThe definition of $N_a$, $u_{\\parallel a}$, $P_{\\parallel a}$, and $P_{\\perp a}$ implies that $N_a^{00}=1,~N_a^{10}=0,~N_a^{20}=0,~N_a^{01}=0,$ respectively.\nLater, we will consider the parallel and perpendicular heat fluxes, defined as\n%\n\\begin{align}\n Q_{\\parallel a} &= m_a ||(v_\\parallel-u_{\\parallel a})^3||_a,~Q_{\\perp a} = ||(v_\\parallel-u_{\\parallel a}) \\mu B||_a,\n \\label{eq:fluidmoments1}\n\\end{align}\n%\nwhich are related to the coefficients $N_a^{30}$ and $N_a^{11}$ by %, N_a^{02}, N_a^{21},$ and $N_a^{40}$ by\n%\n\\be\n \\begin{split}\n N_a^{30}&=\\frac{Q_{\\parallel a}}{\\sqrt{3}P_{\\parallel a} v_{tha \\parallel}},\n ~N_a^{11}=-\\frac{\\sqrt{2} Q_{\\perp a}}{P_{\\perp a} v_{tha \\parallel}}.\n \\end{split}\n \\label{eq:kineticmoments1}\n\\ee\n\n\n\\subsection{Fluid Moment Expansion of the Collision Operator}\n\\label{section:jifluidexpansion}\n\nA polynomial expansion of the nonlinear Coulomb collision operator in \\cref{eq:coulombop} was carried out in \\citet{Ji2009}, while the treatment of finite fluid velocity and unlike-species collisions is described in \\citet{Ji2008}.\n%\nThis allowed expressing $C_{ab}$ as products of fluid moments of $f_a$ and $f_b$. \n%\nWe summarize here the main steps of \\citet{Ji2006, Ji2008}.\n%\nFor an alternative derivation of the fluid moment expansion in terms of multipole moments of the Coulomb operator, see \\cref{ch:op}.\n\nSimilarly to \\cref{eq:gyrof}, the particle distribution function $f_a$ is expanded as\n%\n\\be\n f_a = f_{aM} \\sum_{l,k=0}^{\\infty}\\frac{L_k^{l+1/2}(c_a^2) \\mathbf P^{l}(\\mathbf c_a) \\cdot {\\mathbf M_a}^{lk}}{\\sqrt{\\sigma_k^l}},\n \\label{eq:faji}\n\\ee\n\n\\noindent where \n%\n\\begin{equation}\n f_{aM}=\\frac{n_a}{\\pi^{3/2} v_{tha}^3} e^{-c_a^2}\n\\end{equation}\n%\nis a shifted Maxwell-Boltzmann distribution function, and $\\mathbf c_a$ the shifted velocity defined as $\\mathbf c_a=(\\mathbf v - \\mathbf U_a)/v_{tha}$, with $\\mathbf U_a=u_{\\parallel a}\\mathbf b+\\mathbf u_{\\perp a}$ the fluid velocity. The fluid variables $n_a, \\mathbf U_a,$ and $T_a$ are defined as the usual moments of the particle distribution function $f_a$, i.e. $n_a = \\int f_a d^3 \\mathbf v$, $n_a \\mathbf U_a = \\int f_a \\mathbf v d^3v$, $n_a T_a = \\int m f_a (\\mathbf v - \\mathbf u_a)^2 d^3v/3$.\n\nThe tensors $\\mathbf P_a^{lk}(\\mathbf c_a)=\\mathbf P^{l}(\\mathbf c_a)L_k^{l+1/2}(c_a^2)$ constitute an orthogonal basis, where $\\mathbf P^l(\\mathbf c_a)$ is the symmetric and traceless tensor\n%\n\\be\n \\begin{split}\n \\mathbf P^l(\\mathbf c_a) &= \\sum_{i=0}^{\\floor{l/2}}d_i^l S_i^l c_a^{2i}\\left\\{\\mathbf I^i \\hat c_a^{l-2i}\\right\\},\n \\end{split}\n\\ee\n%\nwith $\\mathbf I$ denoting the identity matrix, $\\{\\mathbf A^i\\}$ denoting the symmetrization of the tensor $\\mathbf A^i$, ${\\floor{l/2}}$ denoting the largest integer less than or equal to $l/2$, and the coefficients $d_i^l$ and $S_i^l$ defined by\n%\n\\begin{equation}\n d_i^l=\\frac{(-2)^i(2l-2i)!l!}{(2l)!(l-i)!},\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n S_i^l=\\frac{l!}{(l-2i)!2^i i!}.\n\\end{equation}\n%\nThe tensor $\\mathbf P^l(\\mathbf c_a)$ is can be also computed using the recursion relation\n%\n\\begin{equation}\n \\mathbf P^{l+1}(\\mathbf c)=\\mathbf c \\mathbf P^l(\\mathbf c)-\\frac{c^2}{2l+1}\\frac{\\partial \\mathbf P^{l}(\\mathbf c)}{\\partial \\mathbf c}\n\\end{equation}\n%\nand is normalized via\n%\n\\begin{equation}\n \\int d \\mathbf v \\mathbf P^{n}(\\mathbf v)\\mathbf P^l(\\mathbf v) \\cdot \\mathbf M^l g(v) = \\mathbf M^n \\delta_{n,l} \\sigma_n \\int d \\mathbf v v^{2n} g(v),\n\\label{eq:normplk}\n\\end{equation}\n%\nwith $\\sigma_l = l!/[2^l (l+1/2)!]$.\n%\nWe note that the tensor $\\mathbf A^i$ is formed by $i$ multiplications of the $\\mathbf A$ elements (e.g., if $\\mathbf A$ is a {rank-2 tensor}, $\\mathbf A^3 \\equiv \\mathbf A \\mathbf A \\mathbf A$, which in index notation can be written as $(\\mathbf A^3)_{ijlkmn} = A_{ij}A_{lk}A_{mn}$).\n\nIn the expansion in \\cref{eq:faji}, $L_k^{l+1/2}(x)$ are the associated Laguerre polynomials\n%\n\\be\n \\begin{split}\n L_{k}^{l+1/2}(x)&=\\sum_{m=0}^{k}L_{km}^{l} x^m,\n \\end{split}\n \\label{eq:asslaguerre}\n\\ee\n%\nnormalized via\n%\n\\begin{equation}\n \\int_0^\\infty e^{-x} x^{l+1/2} L_k^{l+1/2}(x) L_{k'}^{l+1/2}(x) dx = \\lambda_{k}^l\\delta_{k,k'}.\n\\label{eq:normlkl}\n\\end{equation}\n%\nwith $\\lambda_{k}^l={(l+k+1/2)!}/{k!}$ and $L_{km}^{l}=[{(-1)^m(l+k+1/2)!}]/[{(k-m)!(l+m+1/2)!m!}]$. The $\\sigma_k^l=\\sigma_l \\lambda_k^l$ term is a normalization factor from the orthogonality relations in \\cref{eq:normplk,eq:normlkl}.\n%\nFinally, the coefficients of the expansion in \\cref{eq:faji} $\\mathbf M_a^{lk}$ are\n%\n\\be\n \\mathbf M_a^{lk} = \\frac{1}{n_a}\\int d \\mathbf v f_a \\frac{L_k^{l+1/2}(c_a^2) \\mathbf P^{l}(\\mathbf c_a)}{\\sqrt{\\sigma_k^l}},\n \\label{eq:MlkCoulomb}\n\\ee\n%\nwhich correspond to the moments of $f_a$ due to the orthogonality relations in \\cref{eq:normplk,eq:normlkl}.\n\nBy using the expansion in \\cref{eq:faji} in the collision operator in \\cref{eq:coulombop}, a closed form for $C_{ab}$ in terms of products of $\\mathbf M_a^{lk}$ can be obtained. For like-species collisions it reads\n%\n\\be\n C_{aa}=\\sum_{l,k=0}^{\\infty}\\sum_{n,q=0}^{\\infty}\\sum_{m=0}^k\\sum_{r=0}^q\\frac{L_{km}^lL_{qr}^n}{\\sqrt{\\sigma_k^l \\sigma_q^n}}c\\left(f_a^{lkm},f_a^{nqr}\\right),\n \\label{eq:JiCab}\n\\ee\n%\nwith\n%\n\\be\n\\begin{split}\n c\\left(f_a^{lkm},f_a^{nqr}\\right)&=f_{aM}\\sum_{u=0}^{\\text{min}(2,l,n)}\\nu_{*aau}^{lm,nr}(c_a^2)\\sum_{i=0}^{\\text{min}(l,n)-u}d_i^{l-u,n-u}\\mathbf P^{l+n-2(i+u)}(\\hat{\\mathbf c_a})\\cdot ({\\mathbf M_a^{lk}\\cdot^{i+u}\\mathbf M_a^{nq}})_{TS},\n\\end{split}\n \\label{eq:ccjiheld}\n\\ee\n%\nwhere $\\hat{\\mathbf c_a} = \\mathbf c_a/c_a$, $\\cdot^n$ is the $n$-fold inner product (e.g., for the matrix $\\mathbf A = A_{ij}$, $(\\mathbf A \\cdot^1 \\mathbf A)_{ij} = \\sum_k A_{ki}A_{kj}$), and $({\\mathbf A})_{TS}$ the traceless symmetrization of $\\mathbf A$ (e.g., $({\\mathbf A})_{TS} = (A_{ij}+A_{ji})/2-\\delta_{ij}\\sum_k A_{kk}/3$).\n%\nWe refer the reader to \\citet{Ji2009} for the explicit form of the $\\nu_{*abu}^{lm,nr}$ coefficients.\n\n\\subsection{Guiding-Center Moment Expansion of the Collision Operator}\n\\label{section:cabmomentexpansion}\n\nIn order to apply the gyroaveraging operator to the like-species collision operator $C_{aa}$ in \\cref{eq:JiCab}, we expand the fluid moments as $\\mathbf M_{a}^{lk}=\\mathbf M_{a0}^{lk} + \\epsilon \\mathbf M_{a1}^{lk} + ...$, aiming at representing the collision operator up to $O(\\epsilon_\\nu \\epsilon)$.\nAn analytical expression for the leading-order $\\mathbf M_{a0}^{lk}$ in terms of guiding-center moments $N_{a}^{pj}$ can be obtained as follows.\nBy splitting $f_a = \\lb f_a \\rb_{\\mathbf R} + \\tilde f_a$ when evaluating the fluid moments $\\mathbf M_a^{lk}$ according to \\cref{eq:MlkCoulomb}, we obtain\n%\n\\begin{equation}\n \\mathbf M_a^{lk} = \\frac{1}{n_a}\\int d^3 x' d^3 v' \\delta(\\mathbf x' - \\mathbf x)\\frac{L_k^{l+1/2}(c_a^{'2}) \\mathbf P^{l}(\\mathbf c'_a)}{\\sqrt{\\sigma_k^l}}\\left(\\lb f_a \\rb_{\\mathbf R} + \\tilde f_a\\right).\n \\label{eq:malkexact1}\n\\end{equation}\n%\nwhere the Dirac delta function was introduced to convert the velocity integral into an $(\\mathbf x, \\mathbf v)$ integral that encompasses the full phase-space.\nSince the volume element in phase space can be written as $d^3 \\mathbf x d^3\\mathbf v = (B_\\parallel^*/m)d \\mathbf R dv_\\parallel d \\mu d\\theta$ \\citep{Cary2009}, and defining $\\mathbf x' = \\mathbf R + \\rho_a \\mathbf a$, we can write the fluid moments in \\cref{eq:malkexact1} as\n%\n\\begin{equation}\n\\begin{split}\n \\mathbf M_a^{lk} &= \\frac{1}{n_a}\\int d \\mathbf R d v_\\parallel d\\mu d\\theta\\frac{B_\\parallel^*}{m_a} \\delta(\\mathbf x-\\mathbf R - \\rho_a \\mathbf a)\\frac{L_k^{l+1/2}(c_a^{'2}) \\mathbf P^{l}(\\mathbf c'_a)}{\\sqrt{\\sigma_k^l}}\\left( \\lb F_a \\rb_{\\mathbf R} + \\tilde F_a\\right). \n\\end{split}\n\\label{eq:malkexact}\n\\end{equation}\n%\nwhere $\\lb f_a \\rb_{\\mathbf R}$ and $\\tilde f_a$ in \\cref{eq:malkexact1} are written in terms of guiding-center coordinates using \\cref{eq:fguidF}.\nNeglecting the higher-order $\\mathbf \\rho_a$ and $\\tilde F_a$ terms, the leading-order fluid moments $\\mathbf M_{a0}^{lk}$ are given by\n%\n\\begin{equation}\n \\mathbf M_{a0}^{lk} = \\frac{1}{n_a}\\int d v_\\parallel d\\mu d\\theta\\frac{B_\\parallel^*}{m_a} \\frac{L_k^{l+1/2}(c_a^{'2}) \\mathbf P^{l}(\\mathbf c'_a)}{\\sqrt{\\sigma_k^l}}\\lb F_a \\rb_{\\mathbf R} .\n \\label{eq:malk0exact}\n\\end{equation}\n%\nThe $\\theta$ integration can be performed by making use of the gyroaveraging formula of the $\\mathbf P^l$ tensor\n%\n\\be\n \\lb \\mathbf P^l(\\mathbf c_a) \\rb_{\\mathbf R} = c_a^{l} P_l\\left(\\xi_a\\right) \\mathbf P^l(\\mathbf b),\n \\label{eq:Pgyro}\n\\ee\n%\nwhere $\\xi_a=\\mathbf c_a \\cdot \\mathbf b/c_a$ is the pitch angle velocity coordinate, and $P_l$ is a Legendre polynomial defined by\n%\n\\begin{equation}\n P_l(x)=\\frac{1}{2^ll!}\\frac{d^l}{dx^l}\\left[(x^2-1)^l\\right],\n\\end{equation}\n%\nand normalized via\n%\n\\begin{equation}\n \\int_{-1}^1 P_l(x)P_{l'}(x)dx=\\frac{\\delta_{ll'}}{l+1/2},\n\\end{equation}\n%\nyielding\n%\n\\begin{equation}\n \\mathbf M_{a0}^{lk} = \\frac{\\mathbf P^l(\\mathbf b)}{n_a}\\int d v_\\parallel d\\mu d\\theta \\frac{B_\\parallel^*}{m_a} \\frac{L_k^{l+1/2}(c_a^{'2}) c_a^{l} P_l\\left(\\xi_a\\right) }{\\sqrt{\\sigma_k^l}}\\lb F_a \\rb_{\\mathbf R}.\n \\label{eq:malk0exact1}\n\\end{equation}\n%\nFor the derivation of \\cref{eq:Pgyro}, see \\cref{sec:gudingcentertransf}.\n%\nFinally, we use the basis transformation\n%\n\\be\n \\begin{split}\n c_a^l P_l(\\xi_a)L_k^{l+1/2}(c_a^2)=&\\sum_{p=0}^{l+2k}\\sum_{j=0}^{k+\\floor{l/2}}T_{alk}^{pj}\n H_p(s_{\\parallel a})L_j(s_{\\perp a}^2),\n \\end{split}\n \\label{eq:tlkpj}\n\\ee\n%\nwith the inverse\n%\n\\be\n \\begin{split}\n H_p(s_{\\parallel a})L_j(s_{\\perp a}^2) =& \\sum_{l=0}^{p+2j}\\sum_{k=0}^{j+\\floor{p/2}}\\left(T_a^{-1}\\right)_{pj}^{lk}\n c_a^l P_l(\\xi_a)L_k^{l+1/2}(c_a^2),\n \\end{split}\n \\label{eq:tminus1pjlk}\n\\ee\n%\nto obtain an expression for the integrand in \\cref{eq:malk0exact1} in terms of the Hermite-Laguerre basis.\nA numerical evaluation of $T_{alk}^{pj}$ and $\\left(T_a^{-1}\\right)_{pj}^{lk}$ was carried out in \\citet{Omotani2015}.\n%\nInstead, in Appendix \\ref{app:tlkpj}, we derive {the} analytic expressions {of both $T_{alk}^{pj}$ and $\\left(T_a^{-1}\\right)_{pj}^{lk}$}.\n%\nUsing the definition of guiding-center moments $N_a^{pj}$ in \\cref{eq:gyromoments}, the leading-order fluid moment $\\mathbf M_{a0}^{lk}$ is then given by\n%\n\\be\n n_a \\mathbf M_{a0}^{lk} = N_a \\mathbf P^l(\\mathbf b) \\mathcal{N}_a^{lk},\n \\label{eq:CoulDKmom}\n\\ee\n%\nwhere we define\n%\n\\begin{equation}\n \\mathcal{N}_a^{lk} = \\sum_{p=0}^{l+2k}\\sum_{j=0}^{k+\\floor{l/2}}T_{alk}^{pj}{N}_a^{pj}\\sqrt{\\frac{2^p p!}{\\sigma_k^l}}.\n\\label{eq:CoulDKmom1}\n\\end{equation}\n\nThe leading-order part $C_{aa0}$ of the collision operator $C_{aa}$ can be calculated by approximating $\\mathbf M_{a}^{lk}$ appearing in \\cref{eq:ccjiheld} with $\\mathbf M_{a0}^{lk}$.\nFor the ions, the largest contribution to $\\mathbf M_{i}^{lk}-\\mathbf M_{i0}^{lk}$ is of order $\\epsilon$ and it is given by the $\\rho_i$ appearing in \\cref{eq:malkexact} [the $\\tilde F_i$ correction is smaller since $\\tilde F_i < \\epsilon^2 \\lb F_i \\rb_{\\mathbf R}$, see \\cref{eq:orderingftildei}]. Therefore, by using the ordering in \\cref{eq:orderingnu2}, the largest correction to $C_{ii0}$ is $O(\\sqrt{m_e/m_i} \\epsilon \\epsilon_\\nu)$. The correction to $C_{ee0}$ is of the same order. It follows that we can approximate $C_{aa}$ appearing in \\cref{eq:ccjiheld} with $C_{aa0}$ to represent the collision operator up to $O(\\epsilon_{\\nu} \\epsilon)$.\n\nAs an aside, we note that the relationship between the guiding-center and fluid moments in \\cref{eq:CoulDKmom} provides, for the indices $(l,k)=(0,0)$,\n%\n\\begin{align}\n n_a &= N_a,\n\\end{align}\n%\nwhile, for $(l,k)=(0,1)$, yields\n%\n\\begin{align}\n T_a = \\frac{T_{\\parallel a}+2 T_{\\perp a}}{3}.\n\\end{align}\n%\nMoreover, the $(l,k)=(2,0)$ moment provides a relationship useful to express the viscosity tensor $\\mathbf \\Pi_a = \\int (\\mathbf c_a \\mathbf c_a - c_a^2 \\mathbf I) f_a d\\mathbf v$ as\n%\n\\begin{align}\n \\mathbf \\Pi_a = \\mathbf b \\mathbf b N (T_{\\parallel a} - T_{\\perp a}),\n\\label{eq:stressdk}\n\\end{align}\n%\nwhile for $(l,k)=(1,1)$ gives\n%\n\\begin{align}\n \\mathbf q_a &= \\left(\\frac{Q_{\\parallel a}}{2}+Q_{\\perp a}\\right)\\mathbf b,\n\\label{eq:heatfluxdk}\n\\end{align}\n%\nwith $\\mathbf q_a$ the heat flux density $\\mathbf q_a = m \\int \\mathbf c_a c_a^2 f_a d \\mathbf v/2$.\n\nIn order to express the Boltzmann equation, \\cref{eq:boltzmannGC}, in terms of the guiding-center moments $N_a^{pj}$, we evaluate the guiding-center moments of $\\lb C_{aa} \\rb_{\\mathbf R}$ which, up to $O(\\epsilon^2)$, are given by\n\\be\n \\begin{split}\n C_{aa}^{pj}=\\frac{1}{N_a} \\int \\lb C_{aa0} \\rb_{\\mathbf R} \\frac{H_p(s_{\\parallel a}) L_j(s_{\\perp a}^2)}{\\sqrt{2^p p!}} \\frac{B}{m_a} dv_{\\parallel} d\\mu d\\theta.\n \\label{eq:CoulIntGyro}\n \\end{split}\n\\ee\n%\nBy using the gyroaveraging property of $\\mathbf P^{l}(\\mathbf c_a)$ in \\cref{eq:Pgyro} in the like-species operator in \\cref{eq:JiCab,eq:ccjiheld} (with $\\mathbf M_{a}^{lk} = \\mathbf M_{a0}^{lk}$), and the relation between $\\mathbf M_{a0}^{lk}$ and $N_{a}^{pj}$ in \\cref{eq:CoulDKmom}, the gyroaveraged collision operator coefficients $\\lb c\\left(f_a^{lkm},f_a^{nqr}\\right) \\rb_{\\mathbf R}$ are given by\n%\n\\be\n \\begin{split}\n &\\lb c(f_a^{lkm},f_a^{nqr})\\rb_{\\mathbf R}=f_{aM}\\sum_{u=0}^{\\text{min}(2,l,n)}\\nu_{*aau}^{lm,nr}(c_a^2)\\sum_{i=0}^{\\text{min}(l,n)-u}d_i^{l-u,n-u} P_{l+n-2(i+u)}(\\xi) \\mathcal{N}_a^{lk}\\mathcal{N}_a^{nq}\\mathcal{P}^{l,n}_{i+u},\n \\end{split}\n \\label{eq:JiCabGyro}\n\\ee\n%\nwith $\\mathcal{P}^{l,n}_{i+u}=\\mathbf{P}^{l+n-2(i+u)}\\cdot ({\\mathbf P^{l}\\cdot^{i+u}{\\mathbf P}^{n}})_{TS}$.\n%\nUsing the basis transformation of \\cref{eq:tminus1pjlk} to express ${H_p(s_{\\parallel a}) L_j(s_{\\perp a}^2)}$ in \\cref{eq:CoulIntGyro} in terms of $c_a^l P_l(\\xi_a)L_k^{l+1/2}(c_a^2)$, and performing the resulting integral, we obtain\n%\n\\be\n \\begin{split}\n C_{aa}^{pj}=&\\sum_{l,k}\\sum_{n,q}\\sum_{u=0}^{\\text{min}(2,l,n)}\\sum_{i=0}^{\\text{min}(l,n)-u}\\sum_{e=0}^{p+2j}\\sum_{f=0}^{j+\\floor{p/2}}\\sum_{g=0}^f\\sum_{m=0}^k\\sum_{r=0}^q\\\\\n &\\frac{L_{km}^lL_{qr}^n L_{fg}^e d_i^{l-u,n-u}}{\\sqrt{\\sigma_k^l \\sigma_q^n} (e+1/2)4 \\pi}\\frac{C_{*aau}^{eg,lm,nr}}{\\sqrt{2^p p!}}\\delta_{e,l+n-2(i+u)}{\\left(T^{-1}\\right)}_{pj}^{ef} \\mathcal{N}_a^{lk}\\mathcal{N}_a^{nq}\\mathcal{P}^{l,n}_{i+u},\n \\end{split}\n \\label{eq:caapjexact}\n\\ee\n%\nwith $C_{*aabu}^{jw,lm,nr}=\\int d\\mathbf v c_a^{2w+j}f_{Ma}\\nu_{*aau}^{lm,nr}$ [for an efficient algorithmic representation of $C_{*aabu}^{jw,lm,nr}$ see \\citet{Ji2009}].\n\nWe now turn to the electron-ion collision operator, $C_{ei} = C_{ei}^0 + C_{ei}^1$, with $C_{ei}^0$ given by \\cref{eq:cei0} and $C_{ei}^1$ given by \\cref{eq:cei1}.\nAs the basis $ L_k^{l+1/2} \\mathbf P^l(\\mathbf c_a)$ is an eigenfunction of the Lorentz pitch-angle scattering operator $C_{ei}^0$ with eigenvalue $-l(l+1)$ \\citep{Ji2008}, we write $C_{ei}^0$ as\n%\n\\be\n C_{ei}^0=-\\sum_{l,k}\\frac{n_i L_{ei}}{v_{the}^3 c_e^3}\\frac{l(l+1)f_{eM}}{\\sqrt{\\sigma_k^l}}L_k^{l+1/2}(c_e^2) \\mathbf P^l(\\mathbf c_e) \\cdot {\\mathbf M_e}^{lk}.\n \\label{eq:cei0eig}\n\\ee\n%\nSimilarly to like-species collisions, we approximate $\\mathbf M_{e}^{lk} \\simeq \\mathbf M_{e0}^{lk}$ in \\cref{eq:cei0eig}, representing $C_{ei}^0$ accurately up to $O(\\epsilon_\\nu \\epsilon)$.\nUsing the basis transformation in \\cref{eq:tminus1pjlk} and the gyroaverage property of $\\mathbf P^l (\\mathbf c_a)$ in \\cref{eq:Pgyro}, we take guiding-center moments of $C_{ei}$ of the form (\\ref{eq:CoulIntGyro}), and obtain\n%\n\\be\n\\begin{split}\n C_{ei}^{pj} = -\\frac{\\nu_{ei}}{8 \\pi^{3/2}}&\n \\sum_{l=0}^{p+2j}\\sum_{f=0}^{j+\\floor{p/2}}\n \\frac{{\\left(T^{-1}_e\\right)}_{pj}^{lf}}{\\sqrt{2^p p!}}\\left[\\sum_{k=0}^\\infty A_{ei}^{lf,k} \\mathcal{N}_e^{lk} -\\delta_{l,1}\\frac{{u_{\\parallel i}}}{v_{the}}\\frac{{16}}{3 } \\frac{\\Gamma(f+3/2)}{f!\\sqrt{\\pi}}\\right],\n\\end{split}\n\\label{eq:ceipj}\n\\ee\n%\nwhere the $A_{ei}$ coefficients are given by\n%\n\\be\n \\begin{split}\n A_{ei,0}^{lf,k}=&\\frac{l(l+1)}{l+1/2}\\frac{(l!)^2 2^l}{(2l)!} \\sum_{m=0}^f \\sum_{n=0}^k \\frac{L_{fm}^l L_{kn}^l}{\\sqrt{\\sigma_k^l}} {(l+m+n-1)!},\n \\end{split}\n\\ee\n%\nwhere we used the identity $|\\mathbf P^l(\\mathbf b)|^2=2^l (l!)^2/(2l)!$ \\citep{Snider2018}.\n\nFinally, for the ion-electron collision operator, $C_{ie}$, we neglect $O(\\sqrt{m_e/m_i}\\epsilon_\\nu \\epsilon)$ corrections by approximating $F_i \\simeq \\lb F_i \\rb_{\\mathbf R}$, and use the transformation in \\cref{eq:GCcoordinates} to convert the $C_{ie}$ operator in \\cref{eq:cie} to guiding-center variables, yielding\n%\n\\begin{align}\n C_{ie} &= \\frac{\\mathbf R_{ei}}{m_i n_i v_{th i }}\\cdot\\left[\\mathbf c_\\perp \\frac{m_i v_{thi}^2}{B}\\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial \\mu}+\\mathbf b \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial c_{\\parallel i}}\\right]+\\nu_{ei}\\frac{m_e}{m_i}\\frac{n_e}{n_i}\\bigg[3 \\lb F_i \\rb_{\\mathbf R}\\nonumber\\\\\n &\\l. + c_{\\parallel i} \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial c_{\\parallel i}} +2\\mu \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial \\mu}+ \\frac{T_e}{2T_i}\\frac{\\partial^2 \\lb F_i \\rb_{\\mathbf R}}{\\partial c_{\\parallel i }^2}+\\frac{2 T_e}{B} \\frac{\\partial}{\\partial \\mu}\\left(\\mu \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial \\mu}\\right)\\right].\n \\label{eq:cie11}\n\\end{align}\n%\nBy evaluating $\\mathbf R_{ei}$ at the guiding-center position $\\mathbf R$ (neglecting higher order $\\epsilon$ effects), we write $\\mathbf R_{ei} \\cdot \\mathbf b = N_e m_e v_{th\\parallel e} C_{ei}^{10}/\\sqrt{2} + O(\\sqrt{m_e/m_i}\\epsilon_\\nu \\epsilon)$ and gyroaverage \\cref{eq:cie11}, yielding\n%\n\\begin{equation}\n \\begin{split}\n \\lb C_{ie} \\rb_{\\mathbf R}&=\\frac{C_{ei}^{10}}{\\sqrt{2}}\\frac{m_e}{m_i} \\frac{N_e}{n_i} \\frac{v_{th\\parallel e}}{v_{th\\parallel i}} \\frac{\\partial \\lb F_i\\rb_{\\mathbf R} }{\\partial s_{\\parallel}}+\\nu_{ei}\\frac{m_e}{m_i}\\frac{n_e}{n_i}\\bigg[3 \\lb F_i \\rb_{\\mathbf R}\\\\\n %\\lb C_{ie}^1 \\rb_{\\mathbf R}&=\n &\\l.+ s_{\\parallel i} \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial s_{\\parallel i}}+2\\mu \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial \\mu}+ \\frac{T_e}{2 T_{\\parallel i}}\\frac{\\partial^2 \\lb F_i \\rb_{\\mathbf R}}{\\partial s_{\\parallel i}^2} +\\frac{2 T_e }{B}\\frac{\\partial}{\\partial \\mu}\\left(\\mu \\frac{\\partial \\lb F_i \\rb_{\\mathbf R}}{\\partial \\mu}\\right) \\right],\n \\end{split}\n \\label{eq:gyrocie}\n\\end{equation}\n%\nwhere we used $c_{\\parallel i}^2 = s_{\\parallel i}^2 T_{\\parallel i}/T_{i}$.\nTaking guiding-center moments of the form (\\ref{eq:CoulIntGyro}) of $\\lb C_{ie} \\rb_{\\mathbf R}$ in \\cref{eq:gyrocie}, we obtain\n%\n\\begin{equation}\n C_{ie}^{pj}=\\nu_{ei}\\frac{m_e}{m_i}\\sum_{lk}B_{lk}^{pj}N_{i}^{lk},\n \\label{eq:ciepj}\n\\end{equation}\n%\nwith\n%\n\\begin{equation}\n\\begin{split}\n B_{lk}^{pj}&=2j\\delta_{lp}\\delta_{kj-1}\\left(1-\\frac{T_e}{T_{\\perp i}}\\right)-\\sqrt{ p}\\frac{v_{th\\parallel e}}{v_{th\\parallel i}}\\frac{C_{ei}^{10}}{\\nu_{ei}}\\delta_{lp-1}\\delta_{kj}\\\\\n &-(p+2j)\\delta_{lp}\\delta_{kj}+\\sqrt{ p (p-1)}\\delta_{l p-2}\\delta_{kj}\\left(\\frac{T_e}{T_{\\parallel i}}-1\\right).\n\\end{split}\n\\end{equation}\n\n\\section{Drift-Kinetic Moment-Hierarchy}\n\\label{sec:momenthierarchy}\n\nIn this section, we derive a set of equations that describe the evolution of the guiding-center moments $N_a^{pj}$, by integrating in guiding-center velocity space the conservative form of the Boltzmann equation, \\cref{eq:boltzmannGC}, with the weights $H_p(s_{\\parallel a})L_j(s_{\\perp a}^2)$.\nFirst, we highlight the dependence of $\\dot{\\mathbf R}$ and $\\dot v_{\\parallel}$ on $s_{\\parallel a}$ and $s_{\\perp a}^2$ by rewriting the equations of motion as\n%\n\\begin{equation}\n\\begin{split}\n \\dot{\\mathbf R} &= \\mathbf U_{0 a} + \\mathbf U_{p a}^* + s_{\\perp a}^2 \\mathbf U_{\\nabla B a}^* + s_{\\parallel a}^2 \\mathbf U_{k a}^* + s_{\\parallel a}(v_{th\\parallel a} \\mathbf b + \\mathbf U_{p a}^{*th}),\n\\end{split}\n\\label{eq:rdotGCform}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n\\begin{split}\n m_a \\dot{v}_\\parallel &= F_{\\parallel a}-s_{\\perp a}^2 F_{M a} +s_{\\parallel a} F_{p a}^{th}-m_a \\mathcal{A}.\n\\end{split}\n\\label{eq:vparGCform}\n\\end{equation}\n%\nIn \\cref{eq:rdotGCform,eq:vparGCform}, $\\mathbf U_{0 a} = \\mathbf v_E + u_{\\parallel a} \\mathbf b$ is the lowest-order guiding-center fluid velocity, $\\mathbf U_{\\nabla B a}^* = (T_{\\perp a}/m_a)(\\mathbf b \\times \\nabla B/\\Omega_a^{*} B)$ is the fluid grad-B drift, with $\\Omega_a^{*} = q_a B_{\\parallel}^* / m_a$, $\\mathbf U_{ka}^{*} = (2 T_{\\parallel a}/m_a)(\\mathbf b \\times \\mathbf k/\\Omega_a^*)$ is the fluid curvature drift with $\\mathbf k = \\mathbf b \\cdot \\nabla \\mathbf b$, $\\mathbf U_{pa}^* = ({\\mathbf b}/{\\Omega_a^*})\\times d_0 \\mathbf U_{0 a}/dt$ is the fluid polarization drift, $F_{\\parallel a} = q_a E_\\parallel+m_a\\mathbf v_E \\cdot d_0 \\mathbf b/{dt}$, $F_{M a} = {T_{\\perp a}}{}\\nabla_\\parallel \\ln B$ is the fluid mirror force, and both $\\mathbf U_{p a}^{*th}$ and $F_{p a}^{th}$ are related to gradients of the electromagnetic fields\n%\n\\begin{equation}\n\\begin{split}\n \\mathbf U_{p a}^{*th} &= v_{th\\parallel a}\\frac{\\mathbf b}{\\Omega_a^*}\\times \\left(\\mathbf b \\cdot \\nabla \\mathbf v_E+\\mathbf v_E \\cdot \\nabla \\mathbf b + 2 u_{\\parallel a} \\mathbf k\\right),\\\\\n F_{p a}^{th} &= m_a v_{th\\parallel a} \\mathbf E \\cdot \\left(\\frac{\\mathbf b \\times \\mathbf k}{B}\\right).\n\\end{split}\n\\end{equation}\n%\nThe fluid convective derivative operator is defined as \n%\n\\begin{equation}\n \\frac{d_{0 a}}{dt} = \\partial_t + \\mathbf U_{0 a} \\cdot \\nabla. \n \\label{eq:convdev0}\n\\end{equation}\n\nNext, to obtain an equation for the moment $N_a^{pj}$, we apply the guiding-center moment operator\n%\n\\begin{equation}\n\\begin{split}\n ||\\chi||_a^{*pj} &=\\frac{1}{N_a B} ||\\chi H_p(s_{\\parallel a}) L_j(s_{\\perp a}^2) B_{\\parallel}^*||\\\\\n &= \\frac{1}{N_a}\\int \\chi \\frac{B_\\parallel^*}{m_a} \\lb F_a \\rb_{\\mathbf R} \\frac{H_p(s_{\\parallel a}) L_j(s_{\\perp a}^2)}{\\sqrt{2^p p!}} dv_\\parallel d\\mu d\\theta,\n\\end{split}\n\\end{equation}\n%\nto Boltzmann's equation, \\cref{eq:boltzmannGC}. By defining $|| 1 ||_a^{*pj} = {N}_a^{*pj}$ such that\n%\n\\begin{equation}\n\\begin{split}\n {N}_a^{*pj} &= N_a^{pj}\\left(1+\\frac{\\mathbf b \\cdot \\nabla \\times \\mathbf v_E}{\\Omega_a}+u_{\\parallel a} \\frac{\\mathbf b \\cdot \\nabla \\times \\mathbf b}{\\Omega_a}\\right)\\\\\n &+ v_{th\\parallel a}\\frac{\\mathbf b \\cdot \\nabla \\times \\mathbf b}{\\sqrt{2}\\Omega_a}\\left(\\sqrt{p+1}N_a^{p+1~j}+\\sqrt{p}N_a^{p-1~j}\\right),\n\\end{split}\n\\label{eq:overlinenapj}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n\\begin{split}\n \\frac{d_a^{*pj}}{dt}={N}_a^{*pj}\\frac{\\partial}{\\partial t}+\\left|\\left|\\dot{\\mathbf R}\\right|\\right|_a^{*pj} \\cdot \\nabla,\n\\end{split}\n\\end{equation}\n%\nthe drift-kinetic moment-hierarchy conservation equation for species $a$ is\n%\n\\be\n \\begin{split}\n \\frac{\\partial {N}_a^{*pj}}{\\partial t} + \\nabla \\cdot \\left|\\left|{\\dot{\\mathbf R}}\\right|\\right|_a^{*pj}-\\frac{\\sqrt{2 p}}{v_{th\\parallel a}} \\left|\\left|\\dot v_\\parallel\\right|\\right|_a^{*p-1j} +\\mathcal{F}_a^{pj}= \\sum_b C_{ab}^{pj},\n \\end{split}\n \\label{eq:finalDKE}\n\\ee\n%\nwhere we define the fluid operator\n%\n\\be\n \\begin{split}\n \\mathcal{F}_a^{pj} &= \\frac{d_a^{*pj}}{dt}\\ln\\left(N_a T_{\\parallel a}^{p/2} T_{\\perp a}^jB^{-j}\\right)+\\frac{\\sqrt{2p}}{v_{th\\parallel a}}\\frac{d^{*p-1 j}u_{\\parallel a}}{dt}\\\\\n &+\\frac{\\sqrt{p(p-1)}}{2}\\frac{d_a^{*p-2 j}}{dt}\\ln T_{\\parallel a}-j\\frac{d_a^{*pj-1}}{dt}\\ln\\left(\\frac{ T_{\\perp a}}{B}\\right),\n \\end{split}\n \\label{eq:finalDKEF}\n\\ee\n%\nsince it is the key term that describes the evolution of the guiding-center fluid properties $N_a, u_{\\parallel a}, P_{\\perp a},$ and $P_{\\parallel a}$ (see \\cref{sec:fluidmodel}).\n%\nThe guiding-center moments of the particle's equations of motion are given by\n%\n\\be\n \\begin{split}\n \\left|\\left|{\\dot{\\mathbf R}}\\right|\\right|_a^{*pj}&= \\sum_{l,k}\\left(\\mathbf U_{0 a}\\delta_{pl}\\delta_{jk} + v_{th\\parallel a}\\mathbf b\\mathcal{V}_{lk}^{1pj}\\right){N}_a^{*lk}\\\\\n & +\\left(\\mathbf U_{pa}\\delta_{pl}\\delta_{jk} + \\mathbf U_{pa}^{th} \\mathcal{V}_{lk}^{1pj}+ \\mathbf U_{\\nabla B a}\\mathcal{M}_{lk}^{pj} + \\mathbf U_{k a}\\mathcal{V}_{lk}^{2pj}\\right) N_a^{lk},\n \\end{split}\n \\label{eq:finalDKE3}\n\\ee\n\\be\n \\begin{split}\n m_a\\left|\\left|{\\dot v_\\parallel}{}\\right|\\right|_a^{*pj}&=\\sum_{l,k}\\left[F_{\\parallel a} \\delta_{p,l}\\delta_{j,k} + F_{p a}^{th} \\mathcal{V}_{lk}^{1pj} +F_{M a}\\mathcal{M}_{lk}^{pj}\\right] {N}_a^{*lk}+m_a\\left|\\left| \\mathcal{A}\\right|\\right|_a^{*pj}.\n \\end{split}\n \\label{eq:finalDKE4}\n\\ee\n%\nwhere the phase-mixing operators read\n%\n\\begin{align}\n \\mathcal{V}_{lk}^{1pj}&=\\left(\\sqrt{\\frac{p+1}{2}}\\delta_{p+1, l}+\\sqrt{\\frac{p}{2}}\\delta_{p-1 ,l}\\right)\\delta_{k,j},\\label{eq:finalDKE5}\\\\\n \\mathcal{V}_{lk}^{2pj}&=\\left[\\delta_{p,l}\\left(p+\\frac{1}{2}\\right)+\\frac{\\sqrt{(p+2)(p+1)}}{2}{\\delta_{p+2 ,l}}{}+ \\frac{\\sqrt{p(p-1)}}{2}\\delta_{p-2,l} \\right]\\delta_{j,k},\\label{eq:finalDKE6}\\\\\n \\mathcal{M}_{lk}^{pj}&=(2j+1)\\delta_{p,l}\\delta_{j,k}-(j+1)\\delta_{p,l}\\delta_{j+1, k}-j\\delta_{p,l}\\delta_{j-1 ,k}.\\label{eq:finalDKE7}\n\\end{align}\n%\nThe expressions of $\\mathbf U_{p a}, U_{\\nabla B_a}, U_{p a}^{th}$, and $\\mathbf U_{k a}$ are derived from $\\mathbf U_{p a}^{*}, U_{\\nabla B a}^{*}, U_{p a}^{*th}$, and $\\mathbf U_{k a}^{*}$ by replacing $\\Omega_a^*$ with $\\Omega_a$.\n\nThe expression of $\\left|\\left| \\mathcal{A}\\right|\\right|^{*pj}$ \nIn \\cref{eq:finalDKE4} is given by\n%\n\\begin{equation}\n\\begin{split}\n || \\mathcal{A}||_a^{*pj} &= \\frac{1}{N_a \\Omega_a} \\sum_{l,k}\\left(A_{1 a}\\mathcal{V}_{lk}^{3pj}+A_{2 a} \\mathcal{V}_{lk}^{2pj}+A_{3 a} \\mathcal{V}_{lk}^{1pj}\\right.\\\\\n &\\left.+A_{4 a} \\mathcal{V}_{lk}^{1p'j'}\\mathcal{M}_{p'j'}^{pj}+A_{5 a} \\mathcal{M}_{lk}^{pj}+A_{6 a} \\delta_{pl}\\delta_{jk}\\right)N_a^{lk},\n\\end{split}\n\\label{eq:mathavv}\n\\end{equation}\n%\nwith the phase-mixing term\n%\n\\be\n \\begin{split}\n \\mathcal{V}_{lk}^{3pj}=&\\left[\\sqrt{(p+3)(p+2)(p+1)}\\delta_{p+3,l}+3\\sqrt{(p+1)^3}{\\delta_{p+1 ,l}}\\right.\\\\\n &\\left.+ 3\\sqrt{p^3}{\\delta_{p-1 ,l}}+\\sqrt{p(p-1)(p-2)}\\delta_{p-3,l} \\right]\\frac{\\delta_{j,k}}{\\sqrt{8}},\n \\end{split}\n \\label{eq:vv3pjlk}\n\\ee\n%\nand the coefficients $A_{ia}$\n%\n\\begin{align}\n A_{1 a}&= v_{th\\parallel a}^3 \\nabla_\\perp \\cdot \\nabla \\times \\mathbf b,\\\\\n A_{2 a} &= v_{th\\parallel a}^2 \\left[\\nabla_\\perp \\cdot (u_{\\parallel a} \\nabla \\times \\mathbf b + \\nabla \\times \\mathbf v_E) + \\nabla \\times \\mathbf b \\cdot \\mathbf A_a\\right],\\\\\n A_{3 a} &=v_{th\\parallel a} (u_{\\parallel a} \\nabla \\times \\mathbf b + \\nabla \\times \\mathbf v_E) \\cdot \\mathbf A_a + v_{th\\parallel a}^2 \\nabla \\times \\mathbf b \\cdot \\mathbf C,\\\\\n A_{4 a} &= v_{th\\parallel a} \\frac{T_\\perp}{m_a B}\\nabla_\\perp B \\cdot \\nabla \\times \\mathbf b,\\\\\n A_{5 a} &= \\frac{T_{\\perp a}}{m_a B}\\nabla_\\perp B \\cdot (u_{\\parallel a} \\nabla \\times \\mathbf b + \\nabla \\times \\mathbf v_E),\\\\\n A_{6 a} &= (v_{th\\parallel a} u_{\\parallel a} \\nabla \\times \\mathbf b + \\nabla \\times \\mathbf v_E) \\cdot \\mathbf C\n\\end{align}\n%\nwith\n%\n\\begin{align}\n \\mathbf A_a &= \\bm b \\times \\left[ \\frac{\\partial \\mathbf b}{\\partial t} + (\\mathbf b \\cdot \\nabla) \\mathbf v_E + (\\mathbf v_E \\cdot \\nabla) \\mathbf b +2 u_{\\parallel a} v_{th\\parallel a} \\mathbf k\\right]\\times \\bm b,\\\\\n \\mathbf C &= \\frac{\\bm b \\times}{v_{th\\parallel a}}\\left[\\frac{\\partial \\mathbf v_E}{\\partial t}+(\\mathbf v_E \\cdot \\nabla)\\mathbf v_E + u_{\\parallel a}^2 \\mathbf k\\right]\\times \\bm b.\n\\end{align}\n\n{Similar moment-hierarchy models (with uniform magnetic fields) have been numerically implemented, and successfully compared with their kinetic counterpart \\citep{Paskauskas2009,Loureiro2015,Schekochihin2016,Groselj2017}, and even shown to be more efficient than other velocity discretization techniques in the same region of validity \\citep{Camporeale2016}. Equation (\\ref{eq:finalDKE}) generalizes such models to spatially varying fields and full Coulomb collisions, while retaining phase-mixing operators that couple nearby Hermite and Laguerre moments and providing a close form for the projection of the Coulomb operator in velocity space. We also note that the use of shifted velocity polynomials in the Hermite-Laguerre basis, which gives rise to the fluid operator $\\mathcal{F}_a^{pj}$, allows us to have an efficient representation of the distribution function both in the weak ($u_{\\parallel a} \\ll v_{th a})$ and strong flow ($u_{\\parallel a} \\sim v_{th a}$) regimes. As we will see in \\cref{sec:fluidmodel}, the fluid operator $\\mathcal{F}_a^{pj}$ generates the lowest order fluid equations, as it is present even if all kinetic moments $N_a^{pj}$ (except $N_a^{00}$) are set to zero.}\n\n\\section{Drift-Kinetic Poisson's Equation}\n\\label{sec:poisson}\n\nWe use Poisson's equation to evaluate the electric field appearing in the moment-hierarchy equation, \\cref{eq:finalDKE}.\nIn $(\\mathbf x, \\mathbf v)$ coordinates, Poisson's equation reads\n%\n\\begin{equation}\n\\begin{split}\n \\epsilon_0 \\nabla \\cdot \\mathbf E &= \\sum_a q_a n_a=\\sum_a q_a \\int f_a d^3 v.\n\\end{split}\n\\label{eq:fmoment}\n\\end{equation}\n%\nFollowing the same steps used to derive \\cref{eq:malkexact} from \\cref{eq:MlkCoulomb}, we can write Poisson's equation, \\cref{eq:fmoment}, as\n%\n\\begin{equation}\n \\epsilon_0 \\nabla \\cdot \\mathbf E = \\sum_a q_a \\int d^3 \\mathbf R dv_\\parallel d\\mu d\\theta \\frac{B_\\parallel^*}{m} \\delta(\\mathbf R + \\rho_a \\mathbf a - \\mathbf x)F_a(\\mathbf R, v_\\parallel, \\mu, \\theta).\n \\label{eq:poissonexact1}\n\\end{equation}\n%\nEquation (\\ref{eq:poissonexact1}) shows that all particles that have a Larmor orbit crossing a given point $\\mathbf x$, give a contribution to the charge density at this location.\n\nPerforming the integral over $\\mathbf R$ and introducing the Fourier transform $F_a(\\mathbf x - \\rho_a \\mathbf a,v_\\parallel, \\mu, \\theta) = \\int d^3\\mathbf k F_a(\\mathbf k,v_\\parallel, \\mu, \\theta) e^{-i \\mathbf k \\cdot \\mathbf x} e^{i \\rho_a \\mathbf k \\cdot \\mathbf a}$, \\cref{eq:poissonexact1} can be rewritten as\n%\n\\begin{equation}\n \\epsilon_0 \\nabla \\cdot \\mathbf E = \\sum_a q_a \\int dv_\\parallel d\\mu d^3 \\mathbf k d\\theta \\frac{B_\\parallel^*}{m_a} F_a(\\mathbf k, v_\\parallel, \\mu, \\theta)e^{-i \\mathbf k \\cdot \\mathbf x} e^{i \\rho_a \\mathbf k \\cdot \\mathbf a}.\n \\label{eq:poissonexact2}\n\\end{equation}\n%\nTo perform the $\\mathbf k$ integration, we use the cylindrical coordinate system $(k_{\\perp}, \\alpha, k_\\parallel)$, expressing $\\mathbf k = k_\\perp \\cos \\theta \\mathbf e_1+k_\\parallel \\mathbf b$, such that $\\mathbf k \\cdot \\mathbf a = k_\\perp \\cos \\theta$. This coordinate system allows us to express $e^{i \\rho_a \\mathbf k \\cdot \\mathbf a}$ in \\cref{eq:poissonexact2} in terms of Bessel functions using the Jacobi-Anger expansion \\citep{Andrews1992}\n%\n\\begin{equation}\n e^{i k_\\perp \\rho_a \\cos \\theta} = J_0(k_\\perp \\rho_a) + 2 \\sum_{l=1}^{\\infty} J_l(k_\\perp \\rho_a) i^l \\cos (l \\theta) = \\sum_{l=-\\infty}^{\\infty}i^l J_l(k_\\perp \\rho_a)e^{i l \\theta},\n\\label{eq:jacobianger}\n\\end{equation}\n%\nwhere $J_l(k_\\perp \\rho_a)$ is the Bessel function of the first kind of order $l$. We can then write\n%\n\\begin{equation}\n\\begin{split}\n \\epsilon_0 \\nabla \\cdot \\mathbf E = \\sum_a &q_a \\int dv_\\parallel d\\mu d\\theta \\frac{B_\\parallel^*}{m_a} \\left( \\Gamma_0[F_a]+ 2\\sum_{l=1}^{\\infty} i^l \\Gamma_l[F_a \\cos(l\\theta) ]\\right). \n\\end{split}\n\\label{eq:poissonotexact3}\n\\end{equation}\n%\nwhere the Fourier-Bessel operator $\\Gamma_l[f]$ is defined as\n%\n\\begin{equation}\n \\Gamma_l[F_a(\\mathbf k, v_\\parallel, \\mu, \\theta)] \\equiv \\int d^3 \\mathbf k J_l(k_\\perp \\rho_a) F_a(\\mathbf k, v_\\parallel, \\mu, \\theta) e^{-i \\mathbf k \\cdot \\mathbf x}.\n\\label{eq:fourbesseloperator}\n\\end{equation}\n%\nIntroducing the Fourier decomposition of $\\tilde F_a$, \\cref{eq:fmacmafourier}, in \\cref{eq:poissonotexact3}, we obtain\n%\n\\begin{equation}\n\\begin{split}\n \\epsilon_0 \\nabla \\cdot \\mathbf E =& \\sum_a q_a \\int dv_\\parallel d\\mu \\frac{B_\\parallel^*}{m} \\left( \\Gamma_0[\\lb F_a \\rb_{\\mathbf R}] + 2\\pi\\sum_{l=1}^{\\infty} \\frac{i^{l-1}}{l\\Omega_a}\\Gamma_l[C_{l a}+C_{-l a}]\\right),\n\\end{split}\n\\label{eq:poissonpresqueexact}\n\\end{equation}\n%\nwhere the $\\theta$ integration was performed by using the identity $\\int_0^{2\\pi} e^{i\\theta(l-m)} d\\theta = 2\\pi \\delta(l-m)$.\nNotice that $\\int_0^{2\\pi}\\Gamma_0[F_a]d\\theta/2\\pi = \\Gamma_0(\\lb F_a \\rb_{\\mathbf R})$, and corresponds to the $J_0(k_\\perp \\rho_a)$ operator used in most gyrofluid closures \\citep{Hammett1992a,Dorland1993,Snyder2001,Madsen2013}, and in the gyrokinetic Poisson equation \\citep{Lee1983,Dubin1983a}.\n\nWe now order the terms appearing in \\cref{eq:poissonpresqueexact}. Using the Taylor series expansion of a Bessel function $J_l(x)$ of order $l$ \\citep{Abramowitz1972}, we find\n%\n\\begin{equation}\n \\Gamma_0[\\lb F_a \\rb_{\\mathbf R}] \\sim \\left[1 - \\frac{(k_{\\perp} \\rho_a)^2}{4} + O(\\epsilon^{4})\\right]\\lb F_a \\rb_{\\mathbf R},\n\\end{equation}\n%\nwhile using the orderings of $\\nu_e$ and $\\nu_i$ in \\cref{eq:orderingnu,eq:orderingnu2}\n%\n\\begin{equation}\n \\frac{\\Gamma_l[C_{la}]}{\\Omega_a} < \\epsilon_\\nu\\epsilon^{l+1} \\lb F_a \\rb_{\\mathbf R}.\n\\end{equation}\n%\nfor $l \\ge 1$.\n%\nConsistently with \\cref{section:cabmomentexpansion}, we neglect the $l\\ge1$ collisional terms, therefore representing Poisson's equation up to $O(\\epsilon_\\nu \\epsilon)$.\n%\nSuch terms are included in the gyrokinetic model in \\cref{ch:gk}.\n%\n%For the derivation of an higher-order Poisson equation, the treatment of finite $l\\ge1$ collisional effects are presented in Appendix \\ref{app:poisson}.\n%\nTaylor expanding $J_0(x) \\simeq 1-x^2/4$, Poisson's equation reads\n%\n\\begin{equation}\n\\begin{split}\n \\epsilon_0 \\nabla \\cdot \\mathbf E=&\\sum_a q_a\\left[N_{a}\\left(1+\\frac{\\mathbf b \\cdot \\nabla \\times \\mathbf b}{\\Omega_a} u_{\\parallel a}+\\frac{\\mathbf b \\cdot \\nabla \\times \\mathbf v_E}{\\Omega_a}\\right) +\\frac{1}{2m_a}\\nabla_\\perp^2 \\left(\\frac{P_{\\perp a}}{\\Omega_a^2}\\right)\\right].\n\\end{split}\n \\label{eq:poissonfin2}\n\\end{equation}\n\n\n\\section{Collisional Drift-Reduced Fluid Model}\n\\label{sec:fluidmodel}\n\nThe infinite set of equations that describe the evolution of the moments of the distribution function, \\cref{eq:finalDKE}, and Poisson's equation, \\cref{eq:poissonfin2}, constitute the drift-reduced model, which is valid for distribution functions arbitrarily far from equilibrium. For practical purposes, a closure scheme must be provided in order to reduce the model to a finite number of equations.\nIn this section, we derive a closure in the high collisionality regime. For this purpose, we first state in \\cref{sec:fluideqs} the evolution equations for the fluid moments (i.e. $n_a, u_{\\parallel a}, T_{\\parallel a}, T_{\\perp a}, Q_{\\parallel a}$ and $Q_{\\perp a}$), that correspond to the lowest-order indices of the moment-hierarchy equation. Then, in \\cref{sec:highcoll}, we apply a prescription for the higher-order parallel and perpendicular moment equations that allows a collisional closure for $Q_{\\parallel a}$ and $Q_{\\perp a}$ in terms of $n_a, u_{\\parallel a}, T_{\\parallel a}$ and $T_{\\perp a}$.\nThe nonlinear closure prescription used here, {sometimes called \\textit{semi-collisional closure} \\citep{Zocco2011}}, can be employed at arbitrary collisionalities by including a sufficiently high number of moments {[indeed, it was used in \\citet{Zocco2015,Loureiro2015} to consider low collisionality regimes]. It also allows us to retain the non-linear collision contributions inherent to a full-F description that may have the same size as its linear contributions, as pointed out in \\citet{Catto2004}.}\n\n\\subsection{Fluid Equations}\n\\label{sec:fluideqs}\n\nWe first look at the $(p,j)=(0,0)$ case of \\cref{eq:finalDKE}. Noting that $C_{ab}^{00}=0$, we obtain\n%\n\\begin{equation}\n \\frac{\\partial N_a^{*00}}{\\partial t}+\\nabla \\cdot \\left|\\left|\\dot{\\mathbf R}\\right|\\right|^{*00}_a + \\mathcal{F}_a^{00}=0.\n \\label{eq:cont1}\n\\end{equation}\n%\nEvaluating $\\left|\\left|\\dot{\\mathbf R}\\right|\\right|^{*pj}_a$ in \\cref{eq:finalDKE3} and $\\mathcal{F}_a^{pj}$ in \\cref{eq:finalDKEF}, for $(p,j)=(0,0)$, \\cref{eq:cont1} yields the continuity equation\n%\n\\begin{equation}\n \\frac{d_a^0 N_a}{dt} + \\frac{d_{0 a}}{dt}\\left(\\frac{N_a\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right) = -N_a \\nabla \\cdot \\mathbf u_{0 a} - \\frac{N_a\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\nabla \\cdot \\mathbf U_{0 a}.\n \\label{eq:continuity}\n\\end{equation}\n%\nThe upper convective derivative ${d_a^0}/{dt}$, defined by\n%\n\\begin{equation}\n \\frac{d_a^0}{dt}=\\frac{\\partial}{\\partial t} + \\mathbf u_{0a} \\cdot \\nabla,\n \\label{eq:convectop0}\n\\end{equation}\n%\nis related to the guiding-center fluid velocity $\\mathbf u_{0a}$\n%\n\\begin{equation}\n \\mathbf u_{0a} = \\mathbf U_{0a} + \\frac{T_{\\parallel a}+T_{\\perp a}}{m_a}\\frac{\\mathbf b \\times \\nabla B}{\\Omega_a B} +\\frac{\\mathbf b}{\\Omega_a}\\times \\frac{d_{0 a} \\mathbf U_{0 a}}{dt},\n \\label{eq:guidvel}\n\\end{equation}\n%\nand it differs from the lower-convective derivative ${d_{0a}}/{dt}$ in \\cref{eq:convdev0} by the addition of the last two terms in \\cref{eq:guidvel}.\nThe vorticity $\\nabla_\\perp^2 \\phi$ is related to the $\\mathbf E \\times \\mathbf B$ drift by\n%\n\\begin{equation}\n \\frac{\\mathbf b \\cdot \\nabla \\times \\mathbf v_E}{\\Omega_a} = \\frac{\\nabla_\\perp^2 \\phi}{B \\Omega_a} + O(\\epsilon^3),\n \\label{eq:vorticityapprox}\n\\end{equation}\n%\nand it appears in \\cref{eq:continuity} due to the difference between $N_a^{*00}$ and $N_a^{00}$ [see \\cref{eq:overlinenapj}].\nTo derive \\cref{eq:continuity}, we use the low-$\\beta$ limit expression for $\\mathbf b \\times \\mathbf k \\simeq (\\mathbf b \\times \\nabla B)/B$ and neglect $u_{\\parallel a}\\mathbf b \\cdot \\nabla \\times \\mathbf b/\\Omega_a$ as\n%\n\\begin{equation}\n \\frac{u_{\\parallel a} \\mathbf b \\cdot \\nabla \\times \\mathbf b}{\\Omega_a} \\sim \\frac{T_e}{T_i}\\beta \\sim \\epsilon^3,\n \\label{eq:bcurlbapprox}\n\\end{equation}\n%\ntherefore keeping up to $O(\\epsilon^2)$ terms [namely the $\\nabla_\\perp^2 \\phi$ term in \\cref{eq:vorticityapprox}].\n%\nWe note that, although the particle Lagrangian is kept up to $O(\\epsilon)$, the Euler-Lagrange equations set the particle equations of motion and Botlzmann equation to be second order accurate in $\\epsilon$.\n\nThe parallel momentum equation is obtained by setting $(p,j)=(1,0)$ in \\cref{eq:finalDKE}, yielding\n%\n\\begin{equation}\n \\begin{split}\n m_a\\frac{d_a^0 u_{\\parallel a}}{dt} &=\\frac{m_a v_{th\\parallel a}}{\\sqrt 2}\\sum_b C_{ab}^{10}-\\frac{m_a \\nabla_\\perp^2 \\phi}{\\Omega_a B}\\frac{d_0 u_{\\parallel a}}{dt} - \\frac{m_a}{\\sqrt 2N_a}\\nabla \\cdot\\left(\\mathbf u_a^1 N_a v_{th\\parallel a} \\right) \\\\\n &+m_a ||\\mathcal{A}||_a^{*00}+\\left(1+\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right)\\left(q_a E_\\parallel - T_{\\perp a} \\frac{\\nabla_\\parallel B}{B}+m_a \\mathbf v_E \\cdot \\frac{d_{0 a} \\mathbf b}{dt}\\right),\n \\end{split}\n \\label{eq:parallelvelgc}\n\\end{equation}\n%\nwith \n%\n\\begin{equation}\n \\begin{split}\n \\mathbf u_a^1 &= \\frac{\\mathbf U_{p a}^{th}}{\\sqrt{2}}+\\frac{\\sqrt 2}{m_a}\\frac{\\mathbf b \\times \\nabla B}{\\Omega_a B}\\frac{Q_{\\parallel a}+Q_{\\perp a}}{N_a v_{th\\parallel a}}+v_{th\\parallel a}\\frac{\\mathbf b}{2}\\left(1+\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right).\n \\end{split}\n\\end{equation}\n%\nThe expression for $C_{ab}^{10}$ is given in Appendix \\ref{app:cabmoments}, as well as all the $C_{ab}^{pj}$ coefficients relevant for the present fluid model.\nThe left-hand side of \\cref{eq:parallelvelgc} describes the convection of $u_{\\parallel a}$, while the first term in the right-hand side is related to pressure and heat flux gradients, the second term to resistivity (collisional effects), the third term consists of high-order terms kept to ensure phase-space conservation properties, and the last term is the parallel fluid acceleration, namely due to parallel electric fields, mirror force, and inertia.\n\nThe parallel and perpendicular temperature equations are obtained by setting $(p,j)=(2,0)$ and $(0,1)$ respectively in \\cref{eq:finalDKE}. This yields for the parallel temperature\n%\n\\begin{equation}\n \\begin{split}\n \\frac{N_a}{\\sqrt{2}}\\frac{d_a^0 T_{\\parallel a}}{dt} &=\n \\sqrt{2} Q_{\\perp a} \\frac{\\nabla_\\parallel B}{B}- \\frac{N_a \\nabla_\\perp^2 \\phi}{\\sqrt{2}\\Omega_a B}\\frac{d_{0a} T_{\\parallel a}}{dt}-2\\frac{N_a T_{\\parallel a}}{v_{th\\parallel a}}\\mathbf u_a^{1} \\cdot \\nabla u_{\\parallel a}\\\\\n &-\\nabla \\cdot (N_a T_{\\parallel a} \\mathbf u_a^{2\\parallel})+ N_a T_{\\parallel a}\\frac{\\mathbf E}{B}\\cdot \\frac{\\mathbf b \\times \\nabla B}{B}\\left(1+\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right)\\\\\n &+\\sum_b C_{ab}^{20}N T_{\\parallel a} +\\frac{2 N_a T_{\\parallel a}}{v_{th\\parallel a}}||\\mathcal{A}||_a^{*10},\n \\end{split}\n\\label{eq:paralleltempc}\n\\end{equation}\n%\nwhere\n%\n\\begin{equation}\n \\begin{split}\n \\mathbf u_a^{2\\parallel}&= \\frac{Q_\\parallel a}{2 N_a T_{\\parallel a}}\\frac{\\mathbf U_{pa}^{th}}{v_{th\\parallel a}}+\\frac{\\sqrt{2}T_{\\parallel a}}{m_a}\\frac{\\mathbf b \\times \\nabla B}{\\Omega_a B}+\\frac{\\mathbf b}{2}\\frac{Q_{\\parallel a} }{N_a T_{\\parallel a} }\\left(1+\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right),\n \\end{split}\n\\end{equation}\n%\nand for the perpendicular temperature\n%\n\\begin{equation}\n \\begin{split}\n &N_a\\frac{d_a^0 }{dt}\\left(\\frac{T_{\\perp a}}{B}\\right)+\\frac{N_a\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\frac{d_{0 a} }{dt}\\left(\\frac{T_{\\perp a}}{B}\\right) =\\nabla \\cdot \\left(\\frac{N_a T_{\\perp a}}{B} \\mathbf u_a^{2\\perp}\\right)-\\frac{ N_a T_{\\perp a}}{B}\\sum_b C_{ab}^{01},\n \\end{split}\n \\label{eq:perptempc}\n\\end{equation}\n%\nwith\n%\n\\begin{equation}\n \\begin{split}\n \\mathbf u_a^{2\\perp}&=-\\frac{Q_{\\perp a}}{N_a T_{\\perp a}}\\frac{\\mathbf U_{pa}^{th}}{v_{th\\parallel a}}-\\frac{T_{\\perp a}}{m_a}\\frac{\\mathbf b \\times \\nabla B}{\\Omega_a B}.\n \\end{split}\n\\end{equation}\n%\nThe equations for the evolution of the parallel $Q_{\\parallel a}$ and perpendicular $Q_{\\perp a}$ heat fluxes are obtained by setting $(p,j)=(3,0)$ and $(1,1)$ respectively in \\cref{eq:finalDKE}, yielding\n%\n\\begin{equation}\n \\begin{split}\n \\frac{d_a^0 Q_{\\parallel a}}{dt} &=- \\frac{d_{0 a}}{dt}\\left(Q_{\\parallel a}\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right)+N_a T_{\\parallel a} \\sqrt{3} v_{th\\parallel a} \\sum_b C_{ab}^{30} \\\\\n &-Q_{\\parallel a}\\nabla \\cdot \\mathbf u_a^0- \\frac{Q_{\\parallel a} \\nabla_\\perp^2 \\phi}{\\Omega_a B}\\nabla \\cdot \\mathbf U_{0 a} - 3 \\nabla \\cdot (\\mathbf u_{k a} Q_{\\parallel a})\\\\\n &-\\frac{3}{\\sqrt{2}}\\left(1+\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right)\\frac{\\mathbf E \\cdot \\mathbf b \\times \\nabla B}{B^2}Q_{\\parallel a}+3 \\sqrt{2}N_a T_{\\parallel a} ||\\mathcal{A}||_a^{*20}\\\\\n &-3 \\sqrt{2}N_a T_{\\parallel a} \\mathbf u_a^{2\\parallel} \\cdot \\nabla u_{\\parallel a} - 3 \\sqrt{2}N_a v_{th\\parallel a} \\mathbf u_a^1 \\cdot \\nabla T_{\\parallel a},\n \\end{split}\n \\label{eq:qpar}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n \\begin{split}\n \\frac{d_a^0}{dt}\\left(\\frac{Q_\\perp a}{B}\\right)&=- \\frac{d_{0 a}}{dt}\\left(\\frac{Q_{\\perp a}}{B}\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right)-\\frac{N_a v_{th\\parallel a}}{\\sqrt{2}}(\\mathbf u_a^1 \\cdot \\nabla) \\frac{T_{\\perp a}}{B}\\\\\n &+\\frac{N_a T_{\\perp a}}{B}(\\mathbf u_a^{2\\perp} \\cdot \\nabla) u_{\\parallel a}-\\left(\\frac{Q_{\\perp a}}{B}\\right)\\left(\\nabla \\cdot \\mathbf u_a^0 + \\frac{\\nabla_\\perp^2 \\phi}{\\Omega B}\\nabla \\cdot \\mathbf U_{0 a}\\right)\\\\\n &-(\\mathbf U_{k a} + 2 \\mathbf U_{\\nabla B})\\cdot \\nabla \\left(\\frac{Q_{\\perp a}}{B}\\right)-\\frac{\\sum_b C_{ab}^{11}}{\\sqrt{2}}\\frac{v_{th\\parallel a}N_a T_{\\perp a}}{B}\\\\\n &+\\left(\\frac{N_a T_{\\perp a}^2}{m_a}\\frac{\\nabla_\\parallel B}{B^2}+\\frac{Q_{\\perp a}}{B}\\mathbf E \\cdot \\frac{\\mathbf b \\times \\nabla B}{B^2}\\right)\\left(1+\\frac{\\nabla_\\perp^2 \\phi}{\\Omega_a B}\\right).\n \\end{split}\n \\label{eq:qperp}\n\\end{equation}\n%\n{In \\cref{eq:qpar,eq:qperp} we neglected the higher-order moments with respect to $N^{30}$ and $N^{11}$, an approximation that we will scrutinize in the next section.}\nEquations (\\ref{eq:continuity})-(\\ref{eq:qperp}) constitute a closed set of six coupled non-linear partial differential equations for both the fluid variables $n_a, u_{\\parallel a}, T_{\\parallel a}, T_{\\perp a}$, and the kinetic variables $Q_{\\parallel a}$ and $Q_{\\perp a}$.\n\nWith respect to previous $\\delta$F \\citep{Dorland1993,Brizard1994} and full-F gyrofluid models \\citep{Madsen2013}, our fluid model, Eqs. (\\ref{eq:continuity}-\\ref{eq:qperp}), while neglecting $k_{\\perp} \\rho_i \\sim 1$ effects, {includes the velocity contributions from} the $B_{\\parallel}^*$ denominator in the equations of motion, \\cref{eq:GC1,eq:GC2}, and includes the effects of full Coulomb collisions up to order $\\epsilon_\\nu \\epsilon$.\n{Also, due to the choice of basis functions with shifted velocity arguments $H_p(s_{\\parallel a})$ instead of $H_p(v_\\parallel/v_{tha})$, we obtain a set of equations that can efficiently describe both weak flow ($u_{\\parallel a} \\ll v_{th a})$ and strong flow ($u_{\\parallel a} \\sim v_{th a}$) regimes}.\n\n\\subsection{High Collisionality Regime}\n\\label{sec:highcoll}\n\nWe now consider the high collisionality regime, where the characteristic fluctuation frequency{, $\\omega$,} of the {fluid} variables, {satisfies}\n%\n\\begin{equation}\n\\begin{split}\n\t&\\omega \\sim v_{tha} |\\nabla_\\parallel \\ln N_a| \\sim v_{tha} |\\nabla_\\parallel \\ln T_{\\parallel a}|\\sim v_{tha} |\\nabla_\\parallel \\ln T_{\\perp a}| \\sim |\\nabla_\\parallel \\ u_{\\parallel a}| \\sim v_{th a}/ L_{\\parallel a},\n\\end{split}\n\\end{equation}\n%\nis much smaller than the collision frequency $\\nu_a \\simeq \\nu_{aa}$, that is\n%\n\\begin{equation}\n \\delta_a \\sim \\frac{\\omega}{\\nu_a} \\sim \\frac{\\lambda_{mfp a}}{L_{\\parallel a}} \\ll 1,\n\\label{eq:smallmfp}\n\\end{equation}\n%\nwhere the mean free path $\\lambda_{mfp a}$ in \\cref{eq:smallmfp} is defined as\n%\n\\begin{equation}\n \\lambda_{mfp a} = v_{th a}/\\nu_{aa}.\n\\end{equation}\n%\nEquation (\\ref{eq:smallmfp}) describes the so-called linear transport regime \\citep{Balescu1988}.\nIn this case, the distribution function can be expanded around a Maxwell-Boltzmann equilibrium, according to the Chapman-Enskog asymptotic closure scheme \\citep{Chapman1962} and, to first order in $\\delta_a$, we have\n%\n\\begin{equation}\n \\lb F_a \\rb_{\\mathbf R} \\simeq F_{Ma}\\left[1+\\delta_a f_{1 a}(\\mathbf R,v_{\\parallel},\\mu,t)\\right].\n\\label{eq:chapenskexpansion}\n\\end{equation}\n%\nAccording to \\cref{eq:chapenskexpansion}, all moments $N_a^{pj}$ in the Hermite-Laguerre expansion \\cref{eq:gyrof} with $(p,j)\\not=(0,0)$ are order $\\delta_a$. {Since $Q_{\\parallel a}$ and $Q_{\\perp a}$ are determined at first order in $\\delta_a$ only by the moments $(p,j)=(0,0),(3,0),(1,1)$, the truncation of Sec. (\\ref{sec:fluideqs}), i.e., neglecting $(p,j)\\not=(0,0),(3,0),(1,1)$ is justified. For a more detailed discussion {on this topic} see \\citet{Balescu1988}.\nMoreover, in the linear regime, a relationship between the hydrodynamical and kinetic variables can be obtained along the lines of the semi-collisional closure. This allows us to express $Q_{\\parallel a}$ and $Q_{\\perp a}$ as a function of $N_a, u_{\\parallel a}, T_{\\parallel a}$ and $T_{\\perp a}$, therefore reducing the number of equations.\nWe now derive this functional relationship.\n\nWe consider Eqs. (\\ref{eq:qpar})-(\\ref{eq:qperp}) in the linear regime, and neglect the polarization terms that are proportional to $\\nabla_\\perp^2 \\phi/(\\Omega_a B)$.\n{This yields $\\sqrt{{3}/{2}}{\\sum_b C_{ab}^{30}}/{v_{th\\parallel a}} \\simeq R_{\\parallel a}$ and ${\\sum_b C_{ab}^{11}}/(\\sqrt{{2}}{v_{th\\parallel a}}) \\simeq R_{\\perp a}$, with $R_{\\parallel a}$ and $R_{\\perp a}$ given by}\n%\n\\begin{equation}\n R_{\\parallel a} = \\frac{\\nabla_{\\parallel} T_{\\parallel a}}{T_{\\parallel a}} +u_{\\parallel a} \\frac{\\mathbf b \\times \\nabla B}{\\Omega_a B}\\cdot \\left(\\frac{\\nabla u_{\\parallel a}}{u_{\\parallel a}}+\\frac{\\nabla T_{\\parallel a}}{T_{\\parallel a}}\\right),\n \\label{eq:collimit1}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n R_{\\perp a} &= \\frac{T_{\\perp a}}{T_{\\parallel a}}\\frac{\\nabla_{\\parallel} B}{B} - \\frac{1}{2\\sqrt{2}}\\nabla_{\\parallel} \\ln \\frac{T_{\\perp a}}{B}-u_{\\parallel a} \\frac{\\mathbf b \\times \\nabla B}{\\Omega_a B}\\cdot \\left(\\frac{T_{\\perp a}}{T_{\\parallel a}}\\frac{\\nabla u_{\\parallel a}}{u_{\\parallel a}}+\\nabla \\ln \\frac{T_{\\perp a}}{B}\\right),\n\\end{split}\n\\label{eq:collimit2}\n\\end{equation}\n%\nsince $d_a^0/dt \\sim d_{0 a}/dt \\sim \\omega{}$ and $(d^0 Q_{\\parallel, \\perp }/dt)/Q_{\\parallel,\\perp a} \\sim \\delta_a^2 \\nu_a$.\nWe compute the guiding-center moments of the collision operator $C_{ab}^{30}$ and $C_{ab}^{11}$ by truncating the series for the like-species collision operator in \\cref{eq:caapjexact} at $(l,k,n,q)=(2,1,2,1)$.\nThe resulting $C_{ab}^{pj}$ coefficients are presented in Appendix \\ref{app:cabmoments}.\n\nWith the expression of $C_{ab}^{30}$ and $C_{ab}^{11}$, we can solve for $Q_{\\parallel a}$ and $Q_{\\perp a}$. In the regime $(T_{\\parallel a}-T_{\\perp a})/T_a \\sim \\delta$, at lowest order, we obtain for the electron species\n%\n\\begin{equation}\n \\frac{Q_{\\parallel e}}{N_e T_{ e} v_{th e}} =-0.362 \\frac{u_{\\parallel e}-u_{\\parallel i}}{v_{th e}}-10.6 \\lambda_{mfpe}\\frac{\\nabla_\\parallel T_e}{T_e},\n \\label{eq:qpare}\n\\end{equation}\n%\nand\n%\n\\begin{equation}\n \\frac{Q_{\\perp e}}{N_e T_{ e} v_{th e}} =-0.119 \\frac{u_{\\parallel e}-u_{\\parallel i}}{v_{th e}}-3.02 \\lambda_{mfpe}\\frac{\\nabla_\\parallel T_e}{T_e}.\n\\label{eq:qperpe}\n\\end{equation}\n%\nAnalogous expressions are obtained for the ion species.\n\nEquations (\\ref{eq:continuity}), (\\ref{eq:parallelvelgc}), (\\ref{eq:paralleltempc}), and (\\ref{eq:perptempc}), with $Q_{\\parallel a}$ and $Q_{\\perp a}$ given by \\cref{eq:qperpe,eq:qpare} are valid in the high collisionality regime, and can be compared with the drift-reduced Braginskii equations in \\citet{Zeiler1997}. We first rewrite the continuity equation, \\cref{eq:continuity}, in the form\n\\begin{equation}\n \\frac{\\partial N_e}{\\partial t} + \\nabla \\cdot \\left[N_e\\left(\\mathbf v_E + u_{\\parallel e} \\mathbf b+ \\frac{T_{\\parallel e}+T_{\\perp e}}{m_e}\\frac{\\mathbf b \\times \\nabla B}{\\Omega_e B} \\right)\\right]\n =0,\n\\end{equation}\n\\noindent where we expand the convective derivative $d^0{a}/dt$ using \\cref{eq:convectop0} and \\cref{eq:guidvel}, and neglect polarization terms proportional to the electron mass $m_e$. By noting that the diamagnetic drift $v_{de}$ can be written as \n\\begin{equation}\n \\mathbf v_{de} = \\frac{1}{e N_e}\\nabla \\times \\frac{p_e \\mathbf b}{B}-2\\frac{T_e}{m_e}\\frac{\\mathbf b \\times \\nabla B}{\\Omega_e B},\n\\end{equation}\n\\noindent and by considering the isotropic regime $T_{\\parallel e} \\sim T_{\\perp e} \\sim T_{e}$, we obtain\n\\begin{equation}\n \\frac{\\partial N_e}{\\partial t} + \\nabla \\cdot \\left[N_e\\left(\\mathbf v_E + u_{\\parallel e} \\mathbf b+ \\mathbf v_{de} \\right)\\right]\n =0,\n\\end{equation}\n\\noindent which corresponds to the continuity equation in the drift-reduced Braginskii model in \\citet{Zeiler1997}. In {that model}, the polarization equation is obtained by subtracting both electron and ion continuity equations, using Poisson's equation $n_e \\simeq n_i$ with $n_e$ and $n_i$ the electron and ion particle densities respectively, and neglecting {terms proportional to} the electron to ion mass ratio. Applying the same procedure to the present fluid model, we obtain\n%\n\\begin{equation}\n\\begin{split}\n 0&=\\nabla \\cdot \\left(\\frac{\\nabla_\\perp^2 \\phi N_i u_{\\parallel i} \\mathbf b}{\\Omega_i B}\\right)-\\nabla \\cdot \\left[\\frac{\\mathbf v_E}{2 m_i}\\nabla_\\perp^2\\left(\\frac{N_i T_{\\perp i}}{\\Omega_i^2}\\right)\\right]-\\frac{1}{2m_i}\\frac{\\partial}{\\partial t}\\nabla_\\perp^2 \\left(\\frac{N_i T_{\\perp i}}{\\Omega_i^2}\\right)\\\\\n &+\\nabla \\cdot \\left(\\frac{N_i}{\\Omega_i}\\mathbf b \\times \\frac{d_{0i}\\mathbf U_{0i}}{dt}\\right)+\\nabla \\cdot \\left[\\mathbf b\\left(N_i u_{\\parallel i} - N_e u_{\\parallel e}\\right)\\right]\\\\\n &+\\nabla \\cdot \\left[\\left({N_i T_{\\parallel i}+N_eT_{\\parallel e}+N_i T_{\\perp i}+N_eT_{\\perp e}}\\right)\\frac{\\mathbf b \\times \\nabla B}{e B^2}\\right].\n\\end{split}\n\\label{eq:gcpolarizationeq}\n\\end{equation}\n\nIn \\cref{eq:gcpolarizationeq}, the first three terms, which are not present in the drift-reduced Braginskii model, correspond to the difference between ion guiding-center density $N_i$ and particle density $n_i$, proportional to both $\\nabla_\\perp^2 \\phi$ and $\\nabla_\\perp^2 P_i$.\nThe parallel momentum and temperature equations, \\cref{eq:parallelvelgc} and \\cref{eq:paralleltempc}, with respect to \\citep{Zeiler1997}, contain the higher-order term $\\mathcal{A}{\\sim O(\\epsilon^2)}$ that ensures phase-space conservation, mirror force terms proportional to $(\\nabla_\\parallel B)/B$, and polarization terms proportional to $\\nabla_\\perp^2 \\phi/(\\Omega_a B)$ due to the difference between guiding-center and particle fluid quantities.\n{This set of fluid equations constitute an improvement over the drift-reduced Braginskii model. With respect to the original Braginskii equations \\citep{Braginskii1965}, they include the non-linear terms that arise when retaining full Coulomb collisions, and the effect of ion-electron collisions.}\n\n\\section{Conclusion}\n\nIn this chapter, a drift-kinetic model is developed, suitable to describe the plasma dynamics in the SOL region of tokamak devices at arbitrary collisionality. Taking advantage of the separation between the turbulent and gyromotion scales, a gyroaveraged Lagrangian and its corresponding equations of motion are obtained. This is the starting point to deduce a drift-kinetic Boltzmann equation with full Coulomb collisions for the gyroaveraged distribution function.\n\nThe gyroaveraged distribution function is then expanded into an Hermite-Laguerre basis, and the coefficients of the expansion are related to the lowest-order gyrofluid moments. The fluid moment expansion of the Coulomb operator described in \\citet{Ji2009} is reviewed, and its respective particle moments are written in terms of coefficients of the Hermite-Laguerre expansion, relating both expansions. This allows us to express analytically the moments of the collision operator in terms of guiding-center moments.\nA moment-hierarchy that describes the evolution of the guiding-center moments is derived, together with a Poisson's equation accurate up to $\\epsilon^2$. These are then used to derive a fluid model in the high collisionality limit.\n\nThe drift-kinetic model derived herein {will be considered in \\cref{ch:gk} as} a starting point for the development of a gyrokinetic Boltzmann equation suitable for the SOL region (e.g. \\citet{Qin2007, Hahm2009}). Indeed, using a similar approach, a gyrokinetic moment-hierarchy may be derived, allowing for the use of perpendicular wave numbers satisfying $k_\\perp \\rho_s \\sim 1$.", "meta": {"hexsha": "827f200afcd6a0375ac1cf9a6fdaa0cc5eecd09e", "size": 92424, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "IST Version/main/ch2_drift_kinetic.tex", "max_stars_repo_name": "rogeriojorge/Rogerio_PhD_Thesis", "max_stars_repo_head_hexsha": "955be22ad75b54d44a3c2d1499098824e7500d38", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "IST Version/main/ch2_drift_kinetic.tex", "max_issues_repo_name": "rogeriojorge/Rogerio_PhD_Thesis", "max_issues_repo_head_hexsha": "955be22ad75b54d44a3c2d1499098824e7500d38", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "IST Version/main/ch2_drift_kinetic.tex", "max_forks_repo_name": "rogeriojorge/Rogerio_PhD_Thesis", "max_forks_repo_head_hexsha": "955be22ad75b54d44a3c2d1499098824e7500d38", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 63.8287292818, "max_line_length": 1197, "alphanum_fraction": 0.6930018177, "num_tokens": 32895, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477016, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.3422184223852934}} {"text": "%!TeX root = brillouin_pulse_math\n\\documentclass[paper.tex]{subfiles}\n\\begin{document}\n\n\\section{Dispersion Timothy and the Brillouin Precursors}\n\nUzunoglu \\cite{Theoretical2020} mention a handy way to circumvent the lossy-flesh problem: the Brillouin and Sommerfeld precursors\\footnotemark, and pulse or wavepacket dispersion in tissues in general. The original text of these papers are in German; translations can be found in Léon Brillouin's 1960 book\\cite{Wave1960} on the topic.\n\n\\footnotetext{\"precursor\" is perhaps a misnomer when single pulses are used rather than trapezoidally-modulated tones - Oughston and Najafabadi both call these \\textit{Brillouin pulses}.\\\\ \\cite{Optimal2015} \\cite{Optimal2017} \\cite{Heat2010}.}\n\n\n\nThese are fascinating structures somewhat similar to solitons, which arise as a result of an interplay between two aspects (Albanese \n\\cite{Shortrisetime1989}): \n\n\\begin{itemize}\n\t\\item The highly dispersive nature of tissue (the variation in speed of light with frequency), which causes a peculiar distortion of short pulses (which necessarily have a broadband spectral content).\n\t\\item The variation in loss (the imaginary part of the complex permittivity and the penetration depth) versus frequency. \\wikinote{add image from that mathematica code}\n\\end{itemize}\n\n\nA third aspect leads to the formation of the prototypical Brillouin precursor specifically: any sharp \nbeginning or end of a tone, or other sharply defined structure in a waveform, includes harmonics beyond the tone's carrier wave up to the rise time\\wikinote{add image}.\n\nThe aspect that makes the precursors useful is that they sidestep Bragg's Law decay, with an amplitude that decays as $1/\\sqrt{z}$ rather than $e^{-z}$. Note that Bragg's law is still strictly obeyed in the frequency domain (excepting second-order \"bleaching\" (Lukofsky \\cite{Can}) or induced transparency-like (\\cite{Electromagnetically1997}) effects).\n\nBased on Yang's inactivation thresholds, for instance, at 9 GHz, assuming that a torso is about 10 penetration depths in radius and the hypothetical Brillouin train drives the virus equally effectively as a pure sine, implementing a precursor would decrease the field required at the skin to a perfectly practicable 1 kV/m, rather than an air-ionizing 6.6 MV/m ($\\frac{1}{\\sqrt{10}}=0.316$ versus $e^{-10}=4.5e^{-5}$). \n\nPrecursors have been used in practice, by Ong \\cite{Detection2003}\\footnote{Well, Ong don't really observe the precursor itself directly, it seems.} using simple square waves to detect respiration\\footnotemark, Ossberger \\cite{Noninvasive2004} for the same purpose; and various techniques for tumor detection.\\footnote{TODO: Also another really good paper that I can't find}\n\nWhile it is clearly possible to produce this abnormal penetration (any sharp edge sufficiently distant from suffices to provoke dispersive distortion), it was not obvious to us that a waveform could be constructed that would be an effective driver for the 10 GHz normal modes in question. \n\nFirst, if dispersive pulses are too close together, they interfere as they propagate and diminish the precursor thus formed \\cite{Dynamical2005}.\n\nAs an example, despite the waveform from nonlinear transmission lines\\cite{NLTL6275} producing strong wideband components, in cursory simulations they did not appear to satisfy the requirements for usable Brillouin pulses\\wikinote{dubious-discuss}\\wikinote{citation-needed}. Our working hypothesis is that this is because of destructive interference from the falling edge that immediately follows; but this is mere speculation.\\wikinote{dubious-discuss}\n\nSecond, and most critically, while the pulse amplitude decays slowly, the total energy contained in such a precursor decays very quickly. The biological ramifications of this are discussed in detail by Adair in \\cite{Biophysics2000}.\n\n%temporal Soliton-producing nonlinear transmission lines, and NLTL oscillators constructed therefrom, particularly well suited \n%\n%At a depth of 8 cm in Cole-Cole muscle, an 8 GHz resonance, attack pattern alpha appears to achieve 4 picometers of oscillation amplitude, rather than 0.01 picometers with $\\sin(t\\ \\omega_{resonance})$. The sharp edges are at approx. 250 GHz. \n\nDispersive pulse propagation is easily modelled by a Fourier transform, propagation, and an inverse fourier transform (see Franzen \\cite{Wideband1999}, see also \\cite{Comments1993} and \\cite{Shortrisetime1989})\\footnote{implemented in\\ghfile{/home/arthurdent/Projects/covidinator/electronics/propagation/propagation_numerical_optimize.py}} \\footnote{This isn't truly a transient process, of course - the fourier integral is still periodic, wrapping around the window; it's just more transient than a steady-state}.\n\nThe loss and refractive index can be obtained from a four term Cole-Cole model\\cite{gabriel1996compilation}\\footnote{implemented in\\ghfile{/electronics/simple_fdtd/fdtd_PCB_extensions/fdtd_PCB_extensions/tissue.py}}. We used parameters from the IT.IS Foundation tissue database\\cite{Tissue2018a}. IT.IS Foundation computational phantom 'duke V3.0' was used, \\cite{Duke2014}.\n\n%http://web.physics.ucsb.edu/~fratus/phys103/LN/DHM.pdf\n% section on Greens functions\n% and also \n% and\n% https://www.int.washington.edu/users/dbkaplan/228_01/green.pdf\t\n\n\\pagebreak\nA qualitative statement of the problem we were trying to solve (which may not a useful formulation, especially if Q turns out to be < 1 and the virus exhibits only relaxational and non-resonant behavior):\n\n\\begin{toolchain}\nAn arbitrary waveform propagates through a dispersive, lossy medium with a certain analytic complex \npermittivity. \n\nIt then drives a damped harmonic oscillator that is initially at rest.\\\\\n\nWhat waveform produces the peak transient amplitude on the oscillator?\n\\end{toolchain}\n\n\nEach problem has been solved separately. There are some nearly equivalent but not mathematically identical formulations for optimal control of quantum systems with dispersion.\\footnotemark However, in the classical domain, this appears to be unprecedented in the literature.\n\n\n\\footnotetext{While not strictly related, \\cite{Wavepacket1989} tune a light pulse to match the wavefunction of the desired chemical products in a photo-reaction, which is just pretty darned awesome}\n\nOptimal pulses for several situations in Debye and Lorentz media have been determined by Oughstun's asymptotic method, which appears to be highly satisfactory in terms of physical intution, and also avoids truncation issues that befall fourier-transform techniques. In general, these optimal pulses are simple gaussian monopulses or higher-order gaussian derivatives \\cite{Optimal2017} \\cite{Optimal2015}. \n\nMacke \\cite{Simple2012} have come up with a number of analytic dispersion representations for various signals, such as chirped gaussian pulses; however, as it turns out, it was important that we not constrain ourselves to parameterized shapes.\n\n\n\\subsection{Analytic techniques}\n\n\n\\begin{fquote}[Wait][ \\cite{Propagation1965}]\n\tIn such cases, it may be feasible to evaluate the integral by a purely numerical procedure. With the wide availability of the digital computer, this is certainly fashionable at the moment. Consequently, one might say that the problem has been solved and no further discussion is needed. \\\\\n\t\n\tHowever, it would be a pity if one accepted this answer since all physical insight into the nature of transient processes has been ignored. \n\\end{fquote}\n\n\n\n\\begin{autem}\n\tautem: \nReflection from tissue interfaces?\n\nThermal noise will kick the oscillator around; it won't necessarily start at rest. Will this affect the \n\\end{autem}\n\nOne might suggest try to find the optimal transient solution first, and then try to build an input waveform deconvolved through the transfer function of the tissue. This hardly seems liable to produce a truly optimal solution, however.\n\n\nAlso, this seems similar to a pulse propagating through the tissue's Cole medium and then dissipating the maximum power in a uniform Lorentz medium, which might be another route to attacking it.\n\nAnother route might be to set up two ODEs for the wave equation and the oscillator and solve them simultaneously.\n\n\\footnote{Progress was hindered by the number of conventions for Fourier normalization. It's all well and good for the signal processing specialists to make all sorts of excuses about brevity, but when crossing fields it's pretty annoying for such a fundamental concept to be so loosely defined. On the same topic, can we finally settle the i vs. j convention, and the choice of reduced units in molecular dynamics?}\n\n%\\printendnotes\n\n\\subsubsection{Attempt at analytical solution: Zhu et al 's dispersive waveform optimization propagation formulation}\n\nWe apologize in advance for the desecration of mathematics in this section.\n\nThe most promising analytical technique explored was a slight modification of a formulation for optimal-energy pulses, largely supplied by Prof. Costas Sarris in a yet-unpublished personal communication, based on Zhu, Hum, Costas formulation in \\cite{Microwave2012b}, previously discussed by Pozar \\cite{Waveform2003}. \n\nThis is based on lagrange multipliers and 'functional' or 'variational' calculus\\cite{Methods1989}. \n\nHere, rather than taking the derivative with respect to a small change in a variable, we take the derivative with respect to a small change in the function itself - a very useful technique \\footnote{that the author was not aware of!} which imposes little to no limitation on the form of the solution. For some intuition, the definition of the first variation and the Gateaux derivative may be of some use.\\footnote{The complex conjugate in Sarris' formulation, $\\nabla H() = H()^\\star$, seems to come from \\cite{Signal1969}, page 140 onwards; I cannot make heads or tails of his formulation. Pozar note that this complex conjugate arises from the concept of \"adjointness\".}\n\nA common setup for such a problem is to use the Euler-Lagrange equation; this is usually brought up in the case of the Brachistochrone problem, or the Principle of Least Action \\footnote{For background, see The Feynman Lectures, vol II chapter 19}. In these types of problems, it appears that the path of integration through space from endpoints a to b can be replaced by the whole frequency domain from -$\\infty$ to +$\\infty$. \n\n$$ e_x(z,t) = \\frac{1}{\\sqrt{2 \\pi}} \\int_{-\\infty}^{+\\infty}{F(\\omega) e^{- j (\\omega/c_0)n(\\omega)z}\\ e^{j\\omega t} d\\omega} $$\n\nwhere $\\int e^{j \\omega t} d\\omega$ is the inverse fourier transform. At this point, you could choose to perform a second integral over the time-domain Greens function solution to the harmonic oscillator\\cite{Complex2020} - or straightforwardly integrate over the frequency-domain Greens function.\n\n\\footnote{Many symbolic math tools have support for functional calculus. However, neither Maxima, SageMath, Mathematica's VariationalD, Matlab's Fundiff, nor Maple's functionalDerivative seem to yield any meaningful result for grad xi.}\\\\\n\n%$$W = \\frac{1}{\\eta_0} \\int_{-\\infty}^{+\\infty}{(n_r(\\omega))\\ |F(\\omega)|^2}\\ d\\omega$$\\\\\n%\n%Where $n(\\omega)$ is a complex refractive index per Debye relaxation equation, $n_r()$ is the real part of the same, everything else is a real constant. Set up a functional with a lagrange multiplier\\\\\n%\n%$$\\xi = -e_x(z,t) + \\lambda W$$\\\\\n%\n%Take the functional (\"variational\") gradient with respect to $F(\\omega)$.\n%\n%$$ \\nabla \\xi = -\\frac{1}{\\sqrt{2\\pi}} \\left(e^{- j (\\omega/c_0)n(\\omega)z}\\right)^\\star \\ e^{-j\\omega T} + \\lambda...$$\n%\n%At this point, you could choose to perform a second integral over the time-domain Greens function solution to the harmonic oscillator\\cite{Complex2020}, eq. 4 and 15, \n%\n%$$x(t) = \\int_{-\\infty}^{+\\infty}{G(t,t')\\ \\frac{q\\cdot e_x(z,t'))}{m}\\ } dt'$$\\\\\n%\n%(eq. 15, for the underdamped case)\n%\n%$$G(t,t') = UnitStep(t-t') \\frac{1}{\\sqrt{\\gamma^2-\\omega_0^2} }\\sinh\\left({\\sqrt{\\gamma^2-\\omega_0^2}}\\ (t-t')\\right)$$\n%\n%But a much more straightforward \n%\n%Where * is the complex conjugate.\\\\\n\n\n\n\n%\n%Then set\n%\n%$$x(t) = \\int_{-\\infty}^{+\\infty}{G(t,t')\\ \\frac{q\\cdot e_x(z,t'))}{m}\\ } dt'$$\\\\\n%\n%Anyhow, then modify the lagrange multiplier,\n%\n%$$\\xi = -x(t) + \\lambda W$$\\\\\n%\n%And now,\n%\n%$$\\nabla \\xi = 0$$\n\nThis lagrange multiplier technique has, however, a fundamental limitation (noted by Tufts \\cite{Optimum1964}): for reasons that the author does not yet comprehend, only the pulse energy can be used as a constraint, rather than the value of the peak field. This seems to turn out to be of some importance.\n\n\n\\footnote{Note the ni specifically in the complex conjugate!}\n\n\n\\subsection{Laziness prevails; a return to numerical optimization}\n\nNumerically solving optimal control problems using conjugate-gradient was noted by Lasdon \\cite{conjugate1967} and \nrevisited by Kek \\cite{Conjugate}. They reduce the computational workload by first computing the \ngradient function by hand.\n\nUsing an energy constraint as above appears to tend towards a square step function or a sharp impulse, perhaps as a result of Pontryagin's maximum principle\\cite{Optimum1964}.\n\n\\ghfile{electronics/propagation/propagation_numerical_optimize.py}\n\nHowever, a maximum absolute value constraint appears to produce fantastic results.\n\n\n\\begin{figure}[H]\n\t%\t\\makebox[\\textwidth][c]{\n\t\\includesvg[width=\\textwidth]{muscle_pulse_8cm_minimize_test1}\n\t\\caption{}\n\\end{figure}\n\n\nThe (noisy) output of the optimization for z=0.08 m in muscle. The low-frequency component is believed to be due to the FFT window size. Increasing the window size with the same sample rate caused CG convergence to stall. However, it's interesting that this appears to result in a larger amplitude. \n\n\n%\\begin{figure}[H]\n%\t%\t\\makebox[\\textwidth][c]{\n%\t\\includesvg[width=\\textwidth]{muscle_pulse_8cm_minimize_test2}\n%\t\\caption{}\n%\\end{figure}\n\n\n\\begin{figure}[H]\n\t%\t\\makebox[\\textwidth][c]{\n\t\\includesvg[width=\\textwidth]{propagated_waveforms_detail}\n\t\\caption{\\\\\n\t\tSome input (left) and output (right) waves (without considering oscillator coupling) transmitted through 6 cm of muscle.\\\\\n\t\tX axis is time; Y axis is amplitude.\\\\\n\t\t(a) Gaussian monopulse with 16 ps FWHM.\\\\\n\t\t(b) Optimized 10 GHz sawtooth wave. Note the slight high-frequency ripple.\\\\ This has a frequency of approximately 1.5 GHz.\\\\\n\t\t(c) Prototypical precursor 10 GHz sine burst.\\\\\n\t\t(d) Continuous 10 GHz sine tone.\n}\n\\end{figure}\n\n\\begin{figure}[H]\n\t%\t\\makebox[\\textwidth][c]{\n\t\\includesvg[width=\\textwidth]{propagated_waveforms}\n\t\\caption{The same four pulses plotted against depth (Y axis) and amplitude (Z axis)}\n\\end{figure}\n\nOf course, 10x attenuation at 1 MV/m still takes us beyond safe levels at the skin. However, the loss that makes phased-array near-field focusing systems impractical at baseband is no longer as significant of a concern. A dispersive phased array with more than 10 elements may improve penetration further.\n\n\n\n\n\n\n\n\n\\end{document}", "meta": {"hexsha": "b28b2f32a0e7c890e6a6e6ad9eae3d5e6ededa62", "size": 14951, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "documents/brillouin_pulse_math.tex", "max_stars_repo_name": "0xDBFB7/covidinator", "max_stars_repo_head_hexsha": "e9c103e5e62bc128169400998df5f5cd13bd8949", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "documents/brillouin_pulse_math.tex", "max_issues_repo_name": "0xDBFB7/covidinator", "max_issues_repo_head_hexsha": "e9c103e5e62bc128169400998df5f5cd13bd8949", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "documents/brillouin_pulse_math.tex", "max_forks_repo_name": "0xDBFB7/covidinator", "max_forks_repo_head_hexsha": "e9c103e5e62bc128169400998df5f5cd13bd8949", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 66.4488888889, "max_line_length": 673, "alphanum_fraction": 0.780549796, "num_tokens": 3709, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6442251201477015, "lm_q2_score": 0.5312093733737562, "lm_q1q2_score": 0.3422184223852933}} {"text": "\\documentclass[10pt,showpacs,preprintnumbers,footinbib,amsmath,amssymb,aps,prl,twocolumn,groupedaddress,superscriptaddress,showkeys]{revtex4-1}\n\\usepackage{graphicx}\n\\usepackage{dcolumn}\n\\usepackage{bm}\n\\usepackage[colorlinks=true,urlcolor=blue,citecolor=blue]{hyperref}\n\\usepackage{color}\n\\usepackage{listings}\n\n\n\\begin{document}\n\n\n\n\\title{Life on Devaron, a forest planet with an ancient Jedi Temple.}\n\n\\author{S.~O.~Mebody}\n\\affiliation{Department of Procrastination, University of Akiva}\n\\author{A.~N.~Ybody} \n\\affiliation{Department of Odd Travels, University of Anoat}\n\\author{N.~O.~Body}\n\\affiliation{Department of Even Travels, University of Ambria}\n\n\\begin{abstract}\nWe present the first calculations for rodlike microswimmers from outer space.\n\\end{abstract}\n\n\n\n\\maketitle\n\n\n\n\\section{Introduction}\nThe aim of this project is to get familiar with various vector and matrix operations,\nfrom dynamic memory allocation to the usage of programs in the library\npackage of the course. \nFor Fortran users memory handling and most matrix and vector operations\nare included in the ANSI standard of Fortran 90/95. Array handling in Python is also rather trivial. For C++ user however,\nthere are several possible options. Two are listed here.\n\n\\begin{itemize}\n \\item For this exercise we recommend that you make your own functions for dynamic memory allocation of a vector and a matrix. You don't need to write a class for this operations. Use then the library package lib.cpp with its header file lib.hpp for obtaining LU-decomposed matrices, solve linear equations etc.\n\n \\item A very good and often recommended library for C++ handling of arrays is the library Armadillo, to be found at \\url{arma.sourceforge.net}. We will discuss the usage of this library during the lab sessions and lectures. Armadillo has also an interface to Lapack functions for solving systems of linear equations.\n\\end{itemize}\n\n\\noindent\nYour program, whether it is written in C++, Python \nor Fortran2008, should include\ndynamic memory handling of matrices and vectors. \n\n\n\\section{Methods and algortihms}\n\n\nMany important differential equations in Science can be written as \nlinear second-order differential equations\n\n\\begin{equation*}\n\\frac{d^2y}{dx^2}+k^2(x)y = f(x),\n\\end{equation*}\nwhere $f$ is normally called the inhomogeneous term and $k^2$ is a real function.\n\nA classical equation from electromagnetism is Poisson's equation.\nThe electrostatic potential $\\Phi$ is generated by a localized charge\ndistribution $\\rho (\\mathbf{r})$. In three dimensions \nit reads\n\n\\begin{equation*}\n\\nabla^2 \\Phi = -4\\pi \\rho (\\mathbf{r}).\n\\end{equation*}\nWith a spherically symmetric $\\Phi$ and $\\rho (\\mathbf{r})$ the equations\nsimplifies to a one-dimensional equation in $r$, namely\n\n\\begin{equation*}\n\\frac{1}{r^2}\\frac{d}{dr}\\left(r^2\\frac{d\\Phi}{dr}\\right) = -4\\pi \\rho(r),\n\\end{equation*}\nwhich can be rewritten via a substitution $\\Phi(r)= \\phi(r)/r$ as\n\n\\begin{equation*}\n\\frac{d^2\\phi}{dr^2}= -4\\pi r\\rho(r).\n\\end{equation*}\nThe inhomogeneous term $f$ or source term is given by the charge distribution\n$\\rho$ multiplied by $r$ and the constant $-4\\pi$.\n\nWe will rewrite this equation by letting $\\phi\\rightarrow u$ and \n$r\\rightarrow x$. \nThe general one-dimensional Poisson equation reads then\n\n\\begin{equation*}\n-u''(x) = f(x).\n\\end{equation*}\n\n\nIn this project we will solve the one-dimensional Poisson equation\nwith Dirichlet boundary conditions by rewriting it as a set of linear equations.\n\n\nTo be more explicit we will solve the equation\n\n\\begin{equation*}\n-u''(x) = f(x), \\hspace{0.5cm} x\\in(0,1), \\hspace{0.5cm} u(0) = u(1) = 0.\n\\end{equation*}\nand we define the discretized approximation to $u$ as $v_i$ with \ngrid points $x_i=ih$ in the interval from $x_0=0$ to $x_{n+1}=1$.\nThe step length or spacing is defined as $h=1/(n+1)$. \nWe have then the boundary conditions $v_0 = v_{n+1} = 0$.\nWe approximate the second\nderivative of $u$ with\n\n\\begin{equation*}\n -\\frac{v_{i+1}+v_{i-1}-2v_i}{h^2} = f_i \\hspace{0.5cm} \\mathrm{for} \\hspace{0.1cm} i=1,\\dots, n,\n\\end{equation*}\nwhere $f_i=f(x_i)$.\nWe can rewrite this equation as a linear set of equations of the form\n\n\\begin{equation*}\n \\mathbf{A}\\mathbf{v} = \\tilde{\\mathbf{b}},\n\\end{equation*}\nwhere $\\mathbf{A}$ is an $n\\times n$ tridiagonal matrix which we rewrite as\n\n\\[\n \\mathbf{A} = \\begin{bmatrix}\n 2& -1& 0 &\\dots & \\dots &0 \\\\\n -1 & 2 & -1 &0 &\\dots &\\dots \\\\\n 0&-1 &2 & -1 & 0 & \\dots \\\\\n & \\dots & \\dots &\\dots &\\dots & \\dots \\\\\n 0&\\dots & &-1 &2& -1 \\\\\n 0&\\dots & & 0 &-1 & 2 \\\\\n \\end{bmatrix},\n\\]\nand $\\tilde{b}_i=h^2f_i$.\n\n\nIn our case we will assume that the source term is \n$f(x) = 100e^{-10x}$, and keep the same interval and boundary \nconditions. Then the above differential equation\nhas a closed-form solution given by $u(x) = 1-(1-e^{-10})x-e^{-10x}$ (convince yourself that this is correct by inserting the\nsolution in the Poisson equation). We will compare\nour numerical solution with this result in the next exercise. \n\n\nWe can rewrite our matrix $\\mathbf{A}$ in terms of one-dimensional vectors $a,b,c$ \nof length $1:n$. \nOur linear equation reads\n\n\\[\n \\mathbf{A} = \\begin{bmatrix}\n b_1& c_1 & 0 &\\dots & \\dots &\\dots \\\\\n a_1 & b_2 & c_2 &\\dots &\\dots &\\dots \\\\\n & a_2 & b_3 & c_3 & \\dots & \\dots \\\\\n & \\dots & \\dots &\\dots &\\dots & \\dots \\\\\n & & &a_{n-2} &b_{n-1}& c_{n-1} \\\\\n & & & &a_{n-1} & b_n \\\\\n \\end{bmatrix}\\begin{bmatrix}\n v_1\\\\\n v_2\\\\\n \\dots \\\\\n \\dots \\\\\n \\dots \\\\\n v_n\\\\\n \\end{bmatrix}\n =\\begin{bmatrix}\n \\tilde{b}_1\\\\\n \\tilde{b}_2\\\\\n \\dots \\\\\n \\dots \\\\\n \\dots \\\\\n \\tilde{b}_n\\\\\n \\end{bmatrix}.\n\\]\nWe can compute the relative error in the data set $i=1,\\dots, n$,by setting up\n\n\\[\n \\epsilon_i=log_{10}\\left(\\left|\\frac{v_i-u_i}\n {u_i}\\right|\\right),\n\\]\nas function of $log_{10}(h)$ for the function values $u_i$ and $v_i$.\nFor each step length extract the max value of the relative error. \nTry to increase $n$ to $n=10^7$. Make a table of the results and \ncomment your results. You can use either the algorithm from b) or c). \n\nTo compute the elapsed time in c++ you can use the following statements\n\\begin{lstlisting}\n...\n#include \"time.h\" // you have to include the time.h header\nint main()\n{\n // declarations of variables \n ...\n clock_t start, finish; // declare start and final time\n start = clock();\n // your code is here, do something and then get final time\n finish = clock();\n ( (finish - start)/CLOCKS_PER_SEC );\n...\n\\end{lstlisting}\n\n\\section{Our results}\n\nWe present our results in Fig.~\\ref{fig:figure1}.\n\n\\begin{figure} \\label{fig:figure1}\n%\\includegraphics[scale=0.4]{figure1.pdf}\n\\caption{Our results could not be better}\n\\end{figure}\n\n\\section{Conclusions and perspectives}\nWhat a wonderful world!\n\n\\begin{thebibliography}{99}\n\\bibitem{heisenberg} W. Heisenberg, Zeits. f. Physik {\\bf 77}, 1 (1932).\n\\bibitem{miller2006} G.~A.~Miller, A.~K.~Opper, and E.~J.~Stephenson, Annu.~Rev.~Nucl.~Sci.~{\\bf 56}, 253 (2006).\n\\bibitem{ekstrom2015} A.~Ekstr\\\"om, G.~R.~Jansen, K.~A.~Wendt, G.~Hagen, T.~Papenbrock, B.~D.~Carlsson, C.~Forssen, M.~Hjorth-Jensen, P.~Navratil, and W.~Nazarewicz, Phys.~Rev.~C {\\bf 91}, 051301(R) (2015).\n\\bibitem{brown1977} B. A. Brown, A. Arima and J. B. McGrory, Nucl. Phys. {\\bf A277}, 77 (1977) and references therein.\n\\end{thebibliography}\n\n\\end{document}\n\n\n\n", "meta": {"hexsha": "16c8faa99a26e6d52f8f9053c348c3cf49ac52f3", "size": 8086, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/Projects/2017/ReportExamplesLatexstyle/reportexample.tex", "max_stars_repo_name": "solisius/ComputationalPhysics", "max_stars_repo_head_hexsha": "94d32d177881695d443eea34af3410e886b8cb9a", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 220, "max_stars_repo_stars_event_min_datetime": "2016-08-25T09:18:33.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-31T14:09:16.000Z", "max_issues_repo_path": "doc/Projects/2017/ReportExamplesLatexstyle/reportexample.tex", "max_issues_repo_name": "cosmologist10/ComputationalPhysics", "max_issues_repo_head_hexsha": "c6642becb1036e2faaf4f1da78a31785b2033fe7", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2020-12-04T12:55:10.000Z", "max_issues_repo_issues_event_max_datetime": "2020-12-04T12:55:10.000Z", "max_forks_repo_path": "doc/Projects/2017/ReportExamplesLatexstyle/reportexample.tex", "max_forks_repo_name": "cosmologist10/ComputationalPhysics", "max_forks_repo_head_hexsha": "c6642becb1036e2faaf4f1da78a31785b2033fe7", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 136, "max_forks_repo_forks_event_min_datetime": "2016-08-25T09:04:56.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-12T09:54:21.000Z", "avg_line_length": 36.4234234234, "max_line_length": 319, "alphanum_fraction": 0.6377689834, "num_tokens": 2428, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5312093733737562, "lm_q2_score": 0.6442251064863697, "lm_q1q2_score": 0.3422184151282658}} {"text": "\\documentclass[11pt]{article}\n\n\\usepackage{myreport}\n\\usepackage[american]{babel}\n\\usepackage[backend=biber, sorting=none]{biblatex}\n\\addbibresource{report.bib}\n\n\\begin{document}\n\\title{\\vspace{-2.5em} Excitation and Detection Efficiency Under Light-Sheet\n Illumination \\vspace{-1.0em}} \\author{Talon Chandler}\n\\date{\\vspace{-1em}June 21, 2017\\\\ (Updated: June 26, 2017)\\vspace{-1em}}\n\\maketitle\n\\section{Introduction}\nIn these notes I will calculate the excitation and detection efficiency of a\nsingle fluorophore under polarized scanned light-sheet illumination. I will\nstart by finding the transverse and longitudinal fields at all positions in a\nGaussian beam using reasonable approximations. Next I will calculate the\nexcitation efficiency of a single fluorophore as the beam is scanned across the\nfluorophore. I will combine these results with the results from previous notes\nto write the complete forward model. Finally I will discuss the approximations\nwe may need to simplify reconstruction.\n\n\\section{Transverse and Longitudinal Fields In A Scanned Gaussian Beam}\nThe complex spatial electric field of a paraxial Gaussian beam propagating along the\n$\\mh{z}$ axis is \\cite{nov}\n\\begin{align}\n \\mb{E}(x,y,z) &= \\mb{A}\\frac{w_0}{w(z)}\\text{exp}\\left\\{{-\\frac{(x^2+y^2)}{w^2(z)}}+i\\left[kz - \\eta(z) + \\frac{k(x^2+y^2)}{2R(z)}\\right]\\right\\}\\label{eq:gauss}\n\\end{align}\nwhere\n\\begin{alignat}{2} \n \\mb{A} &= \\cos\\phi_{\\text{pol}}\\mh{x} + \\sin\\phi_{\\text{pol}}\\mh{y}\\qquad &&\\text{is the input Jones vector in 3D},\\\\\n w_0 &\\approx \\frac{2n}{k(\\text{NA})}\\qquad &&\\text{is the beam waist radius,}\\\\\n z_0 &= \\frac{kw_0^2}{2} &&\\text{is the Rayleigh range,}\\\\\n k &= \\frac{2\\pi n}{\\lambda}\\qquad &&\\text{is the wave number,}\\\\\n w(z) &= w_0\\sqrt{1+\\frac{z^2}{z_0^2}}\\qquad &&\\text{is the beam radius,}\\\\ \n R(z) &= z\\left(1+\\frac{z_0^2}{z^2}\\right)\\qquad &&\\text{is the wavefront radius,}\\\\\n \\eta(z) &= \\text{arctan}\\left(\\frac{z}{z_0}\\right)\\qquad &&\\text{is the phase correction.}\n\\end{alignat}\nEquation \\ref{eq:gauss} uses the paraxial approximation, so it can only be used\nfor beams with a waist that is significantly larger than the reduced wavelength\n($w_0 \\gg \\lambda/n$). Notice that under the paraxial approximation the beam is uniformly polarized in the transverse plane.\n\nWe would like to calculate the longitudinal component of a Gaussian beam when\nthe beam waist approaches the reduced wavelength ($w_0 > \\lambda/n$). One\napproach is to numerically evaluate the Richards-Wolf diffraction integral\n\\cite{richards, biobeam}. This approach is time consuming and too\naccurate for our needs---we only want to model weak longitudinal\nfields. Instead, I will follow Novotny et. al. \\cite{nov} and use a longitudinal\ncorrection to equation \\ref{eq:gauss}.\n\nAs written, equation \\ref{eq:gauss} doesn't satisfy Gauss' law\n($\\nabla \\cdot \\mb{E} = 0$), so we will add a longitudinal field to correct\nit. If the input beam is polarized along the $\\mh{x}$ axis\n($\\phi_{\\text{pol}}=0$), then we can rearrange Gauss' law to relate the\nlongitudinal and transverse fields with\n\\begin{align}\n E_z(x, y, z) = - \\int \\left[\\frac{\\partial}{\\partial x} E_x(x,y,z)\\right]dz. \n\\end{align}\nCarnicer et al. \\cite{carnicer} worked through this integral using the angular\nspectrum representation and found that\n\\begin{align}\n E_z(x,y,z) = -i\\frac{2x}{kw_0^2}E_x(x,y,z) = -i\\frac{x}{z_0}E_x(x,y,z)\\label{eq:longit}. \n\\end{align}\n\nEquation \\ref{eq:longit} means that:\n\\begin{itemize}\n\\item There is no longitudinal polarization in the $\\mh{y}-\\mh{z}$ plane\n because $E_z(0,y,z) = 0$.\n\\item There are longitudinal field lobes on both sides of the optical axis\n along the transverse polarization direction.\n\\item The factor of $i$ means that the longitudinal fields are $90^{\\circ}$ out\n of phase with the transverse fields which means that the total field is\n elliptically polarized off axis.\n\\item The longitudinal field strength is proportional to\n $\\lambda/w_0^2$---highly focused beams have the strongest longitudinal fields.\n\\end{itemize}\n\nIf the input is polarized along the $\\mh{x}$ axis, the corrected 3D Jones vector is\n\\begin{align}\n\\mb{A}(x,y,z) = \\mh{x}- i\\frac{x}{z_0}\\mh{z}.\n\\end{align}\n\nIf the input polarization is arbitrary then the corrected 3D Jones vector is\n\\begin{align}\n \\mb{A}(x,y,z) = \\cos\\phi_{\\text{pol}}\\mh{x} + \\sin\\phi_{\\text{pol}}\\mh{y} - i\\frac{x\\cos\\phi_{\\text{pol}} + y\\sin\\phi_{\\text{pol}}}{z_0}\\mh{z}.\n\\end{align}\n\nIf the beam is scanned along the $\\mh{y}$ axis with velocity $v$ then the time\ndependent 3D Jones vector is\n\\begin{align}\n \\mb{A}(x,y,z,t) = \\cos\\phi_{\\text{pol}}\\mh{x} + \\sin\\phi_{\\text{pol}}\\mh{y} - i\\frac{x\\cos\\phi_{\\text{pol}} + (y - vt)\\sin\\phi_{\\text{pol}}}{z_0}\\mh{z}\\label{eq:scanned_j}\n\\end{align}\nand the time-dependent electric field is\n\\begin{align}\n \\mb{E}(x,y,z,t) &= \\mb{A}(x,y,z,t)\\frac{w_0}{w(z)}\\text{exp}\\left\\{{-\\frac{(x^2+(y-vt)^2)}{w^2(z)}}+i\\left[kz - \\eta(z) + \\frac{k(x^2+(y-vt)^2)}{2R(z)}\\right]\\right\\}\\label{eq:scanned_e}.\n\\end{align}\nWe will use equations \\ref{eq:scanned_j} and \\ref{eq:scanned_e} to calculate\nthe excitation efficiency of a single fluorophore. Notice that as the Rayleigh\nrange $z_0$ increases the longitudinal electric field decreases, so we can ignore the\nlongitudinal component when the beam is weakly focused.\n\n\\section{Scanned Beam Excitation Efficiency}\nWe define the excitation efficiency of a fluorophore as the fraction of \nincident power that excites the fluorophore. If a fluorophore with absorption\ndipole moment $\\bs{\\mu}_{\\text{abs}}$ is placed in a time-independent complex\nelectric field $\\mb{E}$, then the excitation efficiency is given by\n\\begin{align}\n \\eta_{\\text{exc}} = \\frac{|\\bs{\\mu}_{\\text{abs}} \\cdot \\mb{E}(x,y,z)|^2}{|\\mb{E}(x,y,z) |^2}.\n\\end{align}\nIf the fluorophore is placed in the path of a scanned laser then the electric\nfield becomes time dependent. If the laser is scanned quickly we would need to\nconsider the coherence of the electric field, but we will only consider slow\nscanning here to simplify the calculation. Specifically, we require that\n$v \\ll w_0/\\tau_c$---the scan velocity must be much less than the beam width\ndivided by the coherence time. In this case the excitation efficiency is\n\\begin{align}\n \\eta_{\\text{exc}} = \\frac{\\intinf|\\bs{\\mu}_{\\text{abs}} \\cdot \\mb{E}(x,y,z,t)|^2dt}{\\intinf|\\mb{E}(x,y,z,t)|^2dt}\\label{eq:excitationx}\n\\end{align}\nWe plug equation \\ref{eq:scanned_e} into equation \\ref{eq:excitationx}, express\n$\\bs{\\mu}_{\\text{abs}}$ in spherical coordinates\n\\begin{align}\n \\bs{\\mu}_{\\text{abs}} = \\cos\\Phi\\sin\\Theta\\mh{x} + \\sin\\Phi\\sin\\Theta\\mh{y} + \\cos\\Theta\\mh{z},\n\\end{align}\nand evaluate the integrals (see Appendix for details) to write the\nexcitation efficiency as\n\\begin{align}\n \\eta_{\\text{exc}} = \\frac{\\sin^2\\Theta\\cos^2(\\Phi-\\phi_{\\text{pol}}) + \\cos^2\\Theta\\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z_0^2}}{1+\\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z_0^2}}.\\label{eq:strong}\n\\end{align}\nIf the beam is weakly focused then we can ignore the longitudinal excitation and\nthe excitation efficiency simplifies to\n\\begin{align}\n \\eta_{\\text{exc}} \\approx \\sin^2\\Theta\\cos^2(\\Phi-\\phi_{\\text{pol}}).\\label{eq:weak_approx}\n\\end{align}\n\n\\textbf{How good is the approximation in equation \\ref{eq:weak_approx}?}\n\nFor longitudinal fluorophores equation \\ref{eq:weak_approx} is a very bad\napproximation---it predicts that a longitudinal fluorophore will not be excited\nat all. We can't use a percentage error because the approximation completely\nignores the excitation of longitudinal dipoles.\n\nInstead, we can compare the size of signals from longitudinal and transverse\nfluorophores. If the signal from longitudinal fluorophores is less than the\nsignal from noise and background, then we can ignore the signal from\nlongitudinal fluorophores.\n\\begin{align}\n \\text{Excitation Ratio} &= \\frac{\\text{Max Longitudinal Excitation}}{\\text{Max Transverse Excitation}}\\\\\n &= \\frac{\\eta_{\\text{exc}}(\\Theta = 0)}{\\eta_{\\text{exc}}(\\Theta = \\pi/2, \\Phi = \\phi_{\\text{pol}})}\\\\\n &= \\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z_0^2}\\label{eq:ratio}\n\\end{align}\n\nAt first glance equation \\ref{eq:ratio} looks bleak---the excitation ratio grows\nwithout bound in the $x$ direction which means that the longitudinal excitation\nbecomes a larger fraction of the total excitation as we move away from the plane\nof the light-sheet. Fortunately, we only care about regions of the beam with a\nhigh intensity, so we look at the intensity-weighted excitation ratio instead.\n\\begin{align*}\n \\text{Intensity-Weighted Excitation Ratio} &= \\frac{w_0}{w(z)}e^{-\\frac{2x^2}{w^2(z)}}\\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z_0^2}\\label{eq:intratio} \\\\\n\\end{align*}\nWe can interpret the intensity-weighted excitation ratio as the fraction of the\nmaximum signal (created by a transverse fluorophore at the origin) that we\nignore by ignoring longitudinal excitation. Figure \\ref{fig:error} shows that\nthe intensity weighted excitation ratio is \\textless 2\\% for the imaging parameters used\nin Wu et. al. \\cite{wu2013}.\n\n\\fig{../figures/error.pdf}{1.0}{Intensity-weighted excitation ratio as a\n function of position using the parameters in Wu et. al. \\cite{wu2013}:\n $w_0 = 1.2\\ \\mu$m, $z_0 = 9\\ \\mu$m, $\\lambda = 488$ nm, FOV$\\approx$ 80$\\times$80\n $\\mu$m${}^2$. The maximum values are at the edge of the FOV and are \\textless 2\\%.}{error}\n\nAs a rough heuristic, if the fraction of the signal from noise and background is\ngreater than the intensity-weighted excitation ratio we can justifiably ignore\nthe longitudinal component.\n\nFigure 2 shows the maximum intensity-weighted excitation ratio for an\n80$\\times$80 $\\mu$m${}^2$ FOV as a function of $w_0$. The maximum\nintensity-weighted excitation ratio decreases as $w_0$ increases because wider\nbeams have smaller longitudinal components.\n\n\\fig{../figures/max-error.pdf}{0.5}{Maximum intensity-weighted excitation ration (see equation \\ref{eq:intratio}) for an 80$\\times$80 $\\mu$m${}^2$ FOV with $\\lambda = 488$ nm.}{max-error}\n\n\\section{Detection Efficiency}\nIf we detect fluorescence in wide-field mode with an orthogonal arm and no\npolarizer then the detection efficiency is\n\\begin{align}\n \\eta_{\\text{det}} = 2A + 2B\\sin^2\\Theta'\n\\end{align}\nwhere\n\\begin{align}\n A &= \\frac{1}{4} - \\frac{3}{8}\\cos\\alpha + \\frac{1}{8}\\cos^3\\alpha,\\\\\n B &= \\frac{3}{16}\\cos\\alpha - \\frac{3}{16}\\cos^3\\alpha, \n\\end{align}\n$\\alpha = \\text{arcsin(NA}/n)$ is the detection cone half angle, and $\\Theta'$\nis the angle between the dipole axis and the detection optical axis. See the\n2017-06-09 notes for the relationship between $\\Theta'$ and $\\Theta,\\Phi$. See\n\\cite{fourkas} and the 2017-04-25 notes for the derivation of the detection\nefficiencies and additional expression for detection arms that use a polarizer.\n\n\\section{Orientation Forward Model}\nThe detected intensity is proportional to the the product of the excitation and\ndetection efficiencies. Using a weakly focused excitation beam and an\nunpolarized detection arm gives us the following model\n\\begin{align}\n I_{\\phi_{\\text{pol}}} &= I_{\\text{tot}}\\sin^2\\Theta\\cos^2(\\Phi - \\phi_{\\text{pol}})(2A+2B\\sin^2\\Theta')\\label{eq:forward}\n\\end{align}\nwhere $I_{\\text{tot}}$ is the intensity we would collect if we had an excitation and detection efficiency of 1.\n\n\\section{Discussion}\nEquation \\ref{eq:strong} shows that for strongly focused beams the excitation\nefficiency is a function of position. This couples the orientation and location\nof the fluorophore and complicates our reconstruction. For now we'll use only\nweakly focused beams so that we can ignore the longitudinal component.\n\nTo ignore the longitudinal component, we require that the fraction of the signal\nfrom noise and background is greater than the intensity weighted excitation\nratio (\\textless 2\\% with current imaging parameters). We'll need to be careful about\nlongitudinal excitation if we want to use beams that are more strongly focused.\n\nUnder the weak-focusing approximation the orientation and position of\nfluorophores are decoupled. This will allow us to split the reconstruction into\ntwo steps (1) estimate the position of the fluorophores using unpolarized frames\n(or the sum of orthogonally polarized frames) with established reconstruction\ntechniques then (2) estimate the orientation or the fluorophores using polarized\nframes and equation \\ref{eq:forward}.\n\nNote that we are working in a different regime than Agrawal et. al. \\cite{agrawal}. They\nare considering imaging systems with better resolution than ours, so the\nposition and orientation are coupled and must be estimated together. At our\nresolution, the position and orientation are decoupled so we can estimate them\nseparately.\n\n\\section{References}\n\\setlength\\biblabelsep{0.025\\textwidth}\n\\printbibliography[heading=none]\n\\pagebreak\n\n\\section{Appendix}\nWe will evaluate the following integrals to find the excitation efficiency\n\\begin{align}\n \\eta_{\\text{exc}}(x,y,z) = \\frac{\\intinf|\\bs{\\mu}_{\\text{abs}} \\cdot \\mb{E}(x,y,z,t)|^2dt}{\\intinf|\\mb{E}(x,y,z,t)|^2dt}\\label{eq:excitation}\n\\end{align}\nwhere\n\\begin{align}\n \\bs{\\mu}_{\\text{abs}} &= \\cos\\Phi\\sin\\Theta\\mh{x} + \\sin\\Phi\\sin\\Theta\\mh{y} + \\cos\\Theta\\mh{z}\\\\\n \\mb{E}(x,y,z,t) &= \\mb{A}(x,y,z,t)\\frac{w_0}{w(z)}\\text{exp}\\left\\{{-\\frac{(x^2+(y-vt)^2)}{w^2(z)}}+i\\left[kz - \\eta(z) + \\frac{k(x^2+(y-vt)^2)}{2R(z)}\\right]\\right\\}\\\\\n \\mb{A}(x,y,z,t) &= \\cos\\phi_{\\text{pol}}\\mh{x} + \\sin\\phi_{\\text{pol}}\\mh{y} - i\\frac{x\\cos\\phi_{\\text{pol}} + (y - vt)\\sin\\phi_{\\text{pol}}}{z_0}\\mh{z}.\\label{eq:scanned_jones}\n\\end{align}\n\nWe'll need the following facts\n\\begin{align}\n \\intinf e^{-ax^2}dx &= \\sqrt{\\frac{\\pi}{a}}\\\\\n \\intinf xe^{-ax^2}dx &= 0 \\label{eq:odd}\\\\ \n \\intinf x^2e^{-ax^2}dx &= \\frac{1}{2a}\\sqrt{\\frac{\\pi}{a}}.\n\\end{align}\n\nThe numerator is\n\\begin{align}\n = &\\intinf|\\bs{\\mu}_{\\text{abs}} \\cdot \\mb{E}(x,y,z,t)|^2dt\\\\\n \\begin{split}\n = &\\intinf\\Bigg |\\left[\\cos\\Phi\\sin\\Theta\\cos\\phi_{\\text{pol}} + \\sin\\Phi\\sin\\Theta\\sin\\phi_{\\text{pol}} - i\\cos\\Theta\\frac{x\\cos\\phi_{\\text{pol}} + (y - vt)\\sin\\phi_{\\text{pol}}}{z_0}\\right]\\\\\n &\\ \\ \\ \\ \\ \\ \\ \\ \\frac{w_0}{w(z)}\\text{exp}\\left\\{{-\\frac{(x^2+(y-vt)^2)}{w^2(z)}}+i\\left[kz - \\eta(z) + \\frac{k(x^2+(y-vt)^2)}{2R(z)}\\right]\\right\\}\\Bigg |^2dt.\n \\end{split}\\\\\n \\intertext{After changing variables $y' = y - vt$ and moving constants outside the integral we get}\n = &\\frac{w_0^2}{w^2(z)}e^{-\\frac{2x^2}{w^2(z)}}\\intinf\\left|\\left[\\cos\\Phi\\sin\\Theta\\cos\\phi_{\\text{pol}} + \\sin\\Phi\\sin\\Theta\\sin\\phi_{\\text{pol}} - i\\cos\\Theta\\frac{x\\cos\\phi_{\\text{pol}} + y'\\sin\\phi_{\\text{pol}}}{z_0}\\right]\\right|^2e^{-\\frac{2y'^2}{w^2(z)}}dy'.\n \\intertext{After expanding the square brackets we get}\n = &\\frac{w_0^2}{w^2(z)}e^{-\\frac{2x^2}{w^2(z)}}\\intinf \\Bigg [(\\cos\\Phi\\sin\\Theta\\cos\\phi_{\\text{pol}} + \\sin\\Phi\\sin\\Theta\\sin\\phi_{\\text{pol}})^2 +\\cos^2\\Theta\\frac{(x\\cos\\phi_{\\text{pol}} +y'\\sin\\phi_{\\text{pol}})^2}{z_0^2} \\Bigg ]e^{-\\frac{2y'^2}{w^2(z)}}dy'\\\\\n = &\\frac{w_0^2}{w(z)}e^{-\\frac{2x^2}{w^2(z)}}\\sqrt{\\frac{\\pi}{2}}\\Bigg [\\sin^2\\Theta\\cos^2(\\Phi - \\phi_{\\text{pol}}) + \\cos^2\\Theta\\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z_0^2} \\Bigg ].\n\\end{align}\nThe denominator is\n\\begin{align}\n = &\\intinf|\\mb{E}(x,y,z,t)|^2dt\\\\\n = &\\frac{w_0^2}{w^2(z)}e^{-\\frac{2x^2}{w^2(z)}}\\intinf\\left|\\left[\\cos\\phi_{\\text{pol}}\\mh{x} + \\sin\\phi_{\\text{pol}}\\mh{y} - i\\frac{x\\cos\\phi_{\\text{pol}} + y'\\sin\\phi_{\\text{pol}}}{z_0}\\mh{z}\\right]\\right|^2e^{-\\frac{2y'^2}{w^2(z)}}dy'\\\\\n = &\\frac{w_0^2}{w^2(z)}e^{-\\frac{2x^2}{w^2(z)}}\\intinf\\left[1 + \\frac{(x\\cos\\phi_{\\text{pol}} + y'\\sin\\phi_{\\text{pol}})^2}{z^2_0}\\right]e^{-\\frac{2y'^2}{w^2(z)}}dy'\\\\\n = &\\frac{w_0^2}{w(z)}e^{-\\frac{2x^2}{w^2(z)}}\\sqrt{\\frac{\\pi}{2}}\\left[1 + \\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z^2_0}\\right].\n\\end{align}\nThe final excitation efficiency is\n\\begin{align*}\n\\eta_{\\text{exc}}(x,y,z) = \\frac{\\sin^2\\Theta\\cos^2(\\Phi - \\phi_{\\text{\\text{pol}}}) + \\cos^2\\Theta\\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z_0^2}}{1 + \\frac{x^2\\cos^2\\phi_{\\text{pol}} + \\frac{1}{4}w^2(z)\\sin^2\\phi_{\\text{pol}}}{z^2_0}}.\n\\end{align*}\n\n\\end{document}\n\n", "meta": {"hexsha": "9b36b4b5d7f7d5dde02fd6423c2cabd248df9110", "size": 16417, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "notes/2017-06-21-light-sheet-illumination/report/report.tex", "max_stars_repo_name": "talonchandler/dipsim", "max_stars_repo_head_hexsha": "04904871924276fd1662ca15b7224166d271c0d8", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "notes/2017-06-21-light-sheet-illumination/report/report.tex", "max_issues_repo_name": "talonchandler/dipsim", "max_issues_repo_head_hexsha": "04904871924276fd1662ca15b7224166d271c0d8", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notes/2017-06-21-light-sheet-illumination/report/report.tex", "max_forks_repo_name": "talonchandler/dipsim", "max_forks_repo_head_hexsha": "04904871924276fd1662ca15b7224166d271c0d8", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 58.0106007067, "max_line_length": 274, "alphanum_fraction": 0.7057318633, "num_tokens": 5630, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331462646255, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3421608175009801}} {"text": "\\chapter{Review of PDF determination}\n\\label{ch:pdfdet}\nUnderstanding the functional structure of parton distributions is a complex task that has been subject to a number of approaches over the years. As nonperturbative quantities describing the behaviour of QCD bound states, in principle they may be subject to analysis using Lattice QCD methods. While a great deal of effort and progress has been made in understanding PDFs through nonperturbative methods~\\cite{Dolgov:2000ca,Horsley:2004uq,Gockeler:2004wp,Schroers:2005rm}, results remain short of providing distributions for practical application at hadron colliders.\n\nThe majority of PDF analyses are therefore performed analogously to the determination of many other QCD parameters; via a fit to appropriate experimental data. The fundamental difficulty in PDF fits being that they are determinations of \\emph{functions} rather than single parameters and therefore one must attempt to find some optimum solution in an (in principle) infinite-dimensional functional parameter space. This is of course complicated by having only a finite set of experimental data points upon which to perform a fit. Moreover as the applications involving PDFs have become more precise, a detailed understanding of the uncertainties in the determination of PDFs has become vital. The problem of PDF fitting is therefore one of finding a reliable estimator for a probability distribution in a space of functions.\n\nThe complexity of the task, along with the inherent ambiguities in the QCD treatment of data, led to the emergence of several competing methodologies and determinations. Today there are a diverse array of fitting groups producing sets of parton distribution functions, the most important of which being the ABM~\\cite{Alekhin:2013nda,Alekhin:2012ig} (formerly ABKM~\\cite{Alekhin:2009ni}), CTEQ/CJ~\\cite{Gao:2013xoa,Lai:2010vv,Nadolsky:2008zw,Owens:2012bv}, JR/GJR~\\cite{JimenezDelgado:2008hf,Gluck:2007ck}, HERAPDF~\\cite{Aaron:2009aa,::2014uva}, MSTW~\\cite{Martin:2009iq,Martin:2012da} (formerly MRST~\\cite{Martin:1998sq,Martin:2001es,Martin:2002dr,Martin:2004dh}) and NNPDF~\\cite{Ball:2012cx,Ball:2011gg,Ball:2011uy,Ball:2011mu,Ball:2010de,Ball:2008by} groups. Typically PDF sets are provided for a variety of theory input parameters such as perturbative order, and value of the strong coupling. All modern PDF sets now include a quantitative assessment of their associated uncertainties. In this chapter we shall review the ingredients and methods utilised in a modern PDF determination, primarily focusing on the methodology of the three global PDF fits recommended for LHC phenomenology by the PDF4LHC working group~\\cite{Botje:2011sn}, namely the procedures of the CTEQ, MSTW and NNPDF collaborations.\n\nThese three groups produce PDF sets determined from a fit to a wide range of experimental data, including DIS, Drell-Yan and inclusive jet cross sections. The CTEQ and MSTW determinations follow a similar fitting procedure and method of uncertainty estimation, with the NNPDF group taking a rather different approach to both. We will now describe the basic fitting procedure of these groups, with an eye to detailing areas where the groups have different solutions.\n\n\\section{Experimental data on parton distributions before the LHC}\nThe most important ingredient in the determination of parton distributions is naturally the selection of the dataset from which to extract PDF constraints. The first step in performing a PDF fit is therefore to identify which datasets are most sensitive to input parton distributions, and offer precise and reliable data. As PDF determinations to date have relied only upon fixed-order perturbation theory, the dataset chosen should probe sufficiently inclusive observables which are therefore relatively insensitive to resummation effects. In general PDF fitting collaborations also require data to be taken at a sufficiently high scale that leading-twist factorisation remains reliable, although there are some exceptions which we shall discuss later in the section. Here we shall briefly discuss some of the most important processes in terms of PDF sensitivity, and review some of the most relevant experimental measurements. For this section we shall restrict ourselves to data available before the start of LHC operation in order to provide a background for the methodological developments made in the light of LHC data. \n\n\\subsubsection{Fixed-Target and collider DIS}\nDeep inelastic scattering data provides the backbone for much of a PDF analysis, and data is available from a wide array of sources. Precise electron-proton scattering data from HERA provides the cleanest probe of proton structure function data, while high-luminosity fixed-target experiments can provide important constraints, at the expense of potentially having to deal with additional data corrections due to nuclear and higher-twist effects. As DIS is one of the best understood scattering processes in QCD, precise theoretical predictions are available up to 3-loop order in the zero-mass scheme~\\cite{Vermaseren:2005qc,Moch:2008fj} and 2-loop order with full heavy quark masses intact~\\cite{Buza:1997mg,Buza:1996xr,Blumlein:2006mh,Bierenbaum:2007qe,Bierenbaum:2007dm,Bierenbaum:2009zt,Blumlein:2014fqa}.\n\nAt leading order, neutral current DIS measurements from a proton target directly probe the quark sea distributions $q_i+\\bar{q}_i$, with the relative power of each flavour contribution mediated via its coupling to $\\gamma,Z$. Charged current, and $Z$-mediated neutral current data can provide some constraint upon PDF flavour separation via the $F^3$ structure function. \n\nIn addition to proton structure function measurements, data obtained from scattering off deuterium targets can be important in constraining light quark flavour separation under the assumption of isospin symmetry. Data may be presented as direct measurements of $F_d$ structure functions or as the ratio $F_d/F_p$. A simultaneous fit to deuterium and proton data may therefore provide important constraints upon the $u-d$ and $u/d$ PDF combinations. Data determined via deuteron scattering are subject to nuclear corrections e.g. shadowing effects~\\cite{Badelek:1994qg} which may be estimated as part of the theoretical treatment or neglected; the corrections to be considered part of the theory uncertainty.\n\nAlongside the direct information on quark distributions, scaling violations present in structure function data provide constraints upon the gluon. While rather indirect, the wealth of DIS measurements available at a wide range of scales provides a great deal of information on the structure of the gluon distribution.\n\nDIS data may be presented either as experimental cross sections, or separated into structure functions. Fixed target structure function data on $F_2$ from muon scattering is available for both proton and deuteron targets from the BCDMS~\\cite{Benvenuti:1989rh,Benvenuti:1989fm}, NMC~\\cite{Arneodo:1996qe,Arneodo:1996kd} and Fermilab E665~\\cite{Adams:1996gu} experiments. Electron scattering $F_2$ data is also available from SLAC data on both proton and deuteron targets~\\cite{Whitlow:1991uw}. The longitudinal structure function $F_L$ is measured in fixed target experiments also by SLAC~\\cite{Whitlow:1990gk} , BCDMS~\\cite{Benvenuti:1989rh} and NMC~\\cite{Arneodo:1996qe}. \n\nIn addition to the large datasets available from fixed target experiments, HERA data provides a clean probe of DIS properties, although with HERA data the separation of cross-sections into structure functions is typically not performed. Neutral current cross-section data is provided by ZEUS~\\cite{Breitweg:1998dz,Chekanov:2001qu,Chekanov:2002ej,Chekanov:2003yv} and H1~\\cite{Adloff:2000qk,Adloff:2000qj,Adloff:2003uh}. Charged-current DIS data is also provided by the HERA collaborations \\cite{Chekanov:2003vw,Adloff:2003uh} along with information on the longitudinal structure function $F_L$~\\cite{Andreev:2013vha,Chekanov:2009na}. Information on charm hadroproduction in DIS is available via $F_2^{\\mathrm{charm}}$ measurements at HERA also~\\cite{Adloff:1996xq,Adloff:2001zj,Aktas:2005iw,Aktas:2004az,Breitweg:1999ad,Chekanov:2003rb,Chekanov:2007ch}. This data provides particular constraint upon the gluon PDF, and has been an important testing ground for heavy quark flavour schemes. The clean $ep$ environment means that data is unaffected by nuclear or deuteron corrections, although low energy datapoints may still suffer from substantial higher-twist corrections. These corrections are typically kept under control by kinematic cuts on the affected points, however some groups (notably the ABM/CJ groups) include the affected data and attempt to model the corrections.\n\nHERA measurements from the two collaborations have been examined as a combined analysis and dataset, so far resulting in two studies of direct interest to PDF determination; a combination of HERA-1 inclusive DIS data~\\cite{aaron:2009wt}, and of charm production cross-sections~\\cite{Abramowicz:1900rp}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{3-PDFdet/figs/d12-172f3.pdf}\n\\caption[Combined reduced charm cross-section data from HERA]{Reduced charm cross-section data from the HERA combined measurement. Data from the measurements contained in the combination analysis is shown for comparison. Figure from \\cite{Abramowicz:1900rp}.}\n\\label{fig:HERAF2c}\n\\end{figure}\n\nThe very large quantity of deep-inelastic scattering measurements performed at a variety of experimental facilities means that generally DIS data forms the backbone for PDF fits, providing a substantial proportion of the experimental data points used in a fit.\n\n\\subsubsection{Neutrino DIS}\nThere are a number of measurements available for the scattering of neutrino beams from heavy nuclear targets. For example the NuTeV~\\cite{Tzanov:2005kr} and CHORUS~\\cite{Onengut:2005kv} data on neutrino $F_2$ and $F_3$. Assuming an approximately isoscalar target, and neglecting CKM factors, the PDF dependence of the neutrino structure function data at leading order is given by~\\cite{Forte:2013wc}\n\\begin{eqnarray}\n\tF_2^\\nu(x) &=& x\\left( u^+(x) + d^+(x) + 2s(x) + 2\\bar{c}(x)\\right), \\\\\n\tF_2^{\\bar{\\nu}}(x) &=& x\\left( u^+(x) + d^+(x) + 2\\bar{s}(x) + 2c(x)\\right),\n\\end{eqnarray}\nand for the $F_3$ structure function,\n\\ba\n\tF_3^\\nu(x) &=& x\\left( u^-(x) + d^-(x) + 2s - 2\\bar{c}\\right), \\\\\n\tF_3^{\\bar{\\nu}}(x) &=& x\\left( u^-(x) + d^-(x) - 2\\bar{s}(x) + 2c(x)\\right).\n\\ea\nA simultaneous fit of these data points therefore provides a good handle upon the valence quark distributions $q-\\bar{q}$. These datasets are relatively precise; however they are subject to potentially large nuclear corrections which introduce an uncertainty that is poorly understood.\n\nNeutrino DIS becomes particularly valuable for PDF determination when considering the semi-inclusive DIS dimuon production process $\\nu N \\to \\mu\\mu X$ illustrated in Figure \\ref{fig:dimuon}. In this process the contribution from initial state strangeness is Cabbibo favoured, therefore providing a direct handle on the strange distribution whose contribution is ordinarily difficult to discern from total structure function measurements. Measurements of this process are therefore commonly used as a strangeness probe, and data has been provided by the NuTeV/CCFR collaborations~\\cite{Goncharov:2001qe}.\n%\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=0.5\\textwidth]{3-PDFdet/figs/dimuon.pdf}\n\\caption[Leading order diagram for dimuon production in neutrino DIS]{Leading order diagram for dimuon production in neutrino DIS.}\n\\label{fig:dimuon}\n\\end{figure}\n%\n\\subsubsection{Fixed-target and collider Drell-Yan}\nAfter DIS measurements, the production of electroweak vector bosons in hadronic collisions provides the next most important contribution to the constraint of parton densities, with precise predictions available at NNLO in QCD~\\cite{Anastasiou:2003ds,Catani:2009sm,Catani:2010en}. At leading order the neutral current Drell-Yan process is moderated by the PDF combination\n\t\t\\be q(x_1)\\bar{q}(x_2) + \\bar{q}(x_1)q(x_2),\\ee\nand provides a direct probe of various partonic combinations depending upon the experimental configuration. In the Drell-Yan process the relevant kinematic variables are the invariant mass of the lepton pair\n\\be M_{ll}^2 = (E_1 + E_2)^2 - (\\mathbf{p}_1 + \\mathbf{p}_2)^2,\\ee\nand the intermediate boson's rapidity, given in the detector frame by\n\\be y = \\frac{1}{2}\\log \\frac{E+ p_L}{E-p_L},\\ee\nwhere $E$ is the detector frame energy of the intermediate boson, and $p_L$ its longitudinal momentum. in terms of which the parton-$x$ is given by;\n\\be x_\\pm = M_{ll} e^{\\pm y} / \\sqrt{s}, \\ee\nwhere $s$ is the centre-of-mass energy squared of the reaction and the $\\pm$ denotes the parton direction with respect to the beam frame. High rapidity measurements therefore constrain PDFs at both high and low-$x$. \n\nAdditionally the charged-current process $qq^\\prime \\to l^\\pm \\nu_l$ provides information on quark flavour separation in the initial state hadrons. While the rapidity of the lepton pair resulting from $Z/\\gamma$ decay in neutral current Drell-Yan is experimentally straightforward to distinguish, the presence of a neutrino in the final state of $W$ production processes complicates the direct resolution of the $W$ rapidity. Therefore data is often presented in the pseudorapidity of the detected lepton,\n\\be \\eta = -\\log \\tan \\theta,\\ee\ndefined in terms of the angle $\\theta$ between the final state lepton and the beam axis. It can therefore be measured without knowledge of the particle mass and momentum. The pseudorapidity coincides with the standard rapidity in the case of massless particles where $E = |\\mathbf{\\bar{p}}|$.\n\nLepton asymmetries are another common form for experimental results in Drell-Yan, defined in terms of $W^{\\pm}\\to l^\\pm\\nu_l $ differential cross-sections $d\\sigma_{l^\\pm}/d\\eta_l$ as\n\\be \n A^l_W=\\frac{d\\sigma_{l^{+}}/d\\eta_{l}-d\\sigma_{l^{-}}/d\\eta_{l}}\n {d\\sigma_{l^{+}}/d\\eta_{l}+d\\sigma_{l^{-}}/d\\eta_{l}}, \n\\ee\nsuch measurements also benefit from the cancellation of shared systematic uncertainties. Measurements of lepton pair production from proton beams incident upon heavy nuclear targets, such as the E605\\cite{Moreno:1990sf} experiment determining dimuon production from a copper target are useful for the constraint of the light quark sea $q+\\bar{q}$. These measurements are typically very precise but suffer from poorly determined nuclear corrections. Several approaches have been performed to study the extent of these corrections~\\cite{deFlorian:2003qf,Hirai:2007sx,Kulagin:2007ju,Eskola:2009uj}, although the effects are typically small and may sometimes be discounted in comparison to experimental uncertainties~\\cite{Ball:2009mk}. Contributions from initial state heavy quarks and strangeness are typically suppressed in these measurements due to the relatively low scales.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{3-PDFdet/figs/ncdy.pdf}\n\\includegraphics[width=0.45\\textwidth]{3-PDFdet/figs/ccdy.pdf}\n\\caption[Leading order diagram for the Drell-Yan process]{Drell-Yan process at leading order, initiated by beam protons with PDF $f_b$ and target protons with PDF $f_t$. The neutral current process is shown on the left, and the charged current process on the right.}\n\\label{fig:ncdy}\n\\end{figure}\n\nFixed target experiments upon hydrogen or deuterium targets provide a relatively clean probe and the ratio of Drell-Yan cross sections in proton to deuteron targets can provide crucial information on the $u/d$ PDF combination. While relatively free of nuclear effects, deuteron data still suffers from poorly understood corrections, which have been the subject of extensive study\\cite{Martin:2012da, Badelek:1994qg,Accardi:2011fa,Brady:2011hb}.\nExperimental measurements from the Fermilab NuSea/E866 collaboration are commonly used, providing data from $pp$~\\cite{Webb:2003bj} and $pd/pp$~\\cite{Towell:2001nh} experiments. \n\nThe theoretically cleanest environment to examine the Drell-Yan process is at high scales at a collider. Several measurements are available from the Tevatron collaborations which provide information free of nuclear or deuteron corrections. As a $p\\bar{p}$ collider, neutral-current Drell-Yan at the Tevatron targets the quark valence contribution and asymmetry data provides information on the $u/d$ ratio. A measurement of the $Z$ rapidity distribution is available from D0~\\cite{Abazov:2007jy}, and several measurements are available for $W$ lepton asymmetries from both Tevatron collaborations \\cite{Acosta:2005ud,Abazov:2007pm,Abazov:2008qv,Abe:1998rv}.\n\nIn order to obtain a handle on the contribution of initial state strange quarks to the Drell-Yan process it is once again necessary to examine less inclusive processes. Of particular interest are measurements of $W$ production in association with a charm jet, analogous to the usefulness of dimuon measurements in neutrino DIS where the strange contribution is favoured in terms of CKM elements. Measurements of this process were initially made at the Tevatron by both CDF~\\cite{Aaltonen:2007dm} and D0~\\cite{Abazov:2008qz}. More precise determinations can be obtained by normalisation with respect to the total $W+$ jets rate~\\cite{Stirling:2012vh}. \n\n\\subsubsection{Jet production data}\nWhile DIS data provides constraints upon the gluon distribution via scaling violations and contribution to heavy quark and longitudinal structure functions, DIS and Drell-Yan data do not provide a substantial direct constraint upon gluon densities. The most constraining datasets for the gluon, particularly in the uncertain large-$x$ region, are those of jet production measurements. The large strong coupling of the gluon combined with a high gluon luminosity in the proton at high scales results in $gg$ initiated diagrams being the dominant sub channels for the production of inclusive jet and dijet events.\n\nCross-section calculations for inclusive jet and dijet data in hadron-hadron collisions are available at NLO in QCD~\\cite{Ellis:1992en,Giele:1994gf,Nagy:2001fj,Nagy:2003tz}, however a great deal of progress has been made in the determination of the NNLO corrections~\\cite{Currie:2013dwa,Glover:2001af,Glover:2001rd}, with the exact gluon-gluon sub channel calculation recently determined~\\cite{Currie:2013dza}. For the full calculation however, only approximate NNLO results are available via threshold resummation techniques~\\cite{deFlorian:2013qia,Kidonakis:2000gi,Kumar:2013hia}. Jet data may therefore only be included into an NNLO PDF fit through an approximate treatment if at all.\n\nJet data must be included via some clustering algorithm which takes a QCD final state and identifies suitable jet-like structures. Earlier measurements were performed with so-called cone algorithms, although these are potentially very sensitive to infrared and collinear effects. More recent experiments typically utilise sequential-combination algorithms such as the Cambridge-Aachen~\\cite{Dokshitzer:1997in,Wobisch:1998wt}, $k_T$~\\cite{Ellis:1993tq} or anti$-k_T$~\\cite{Cacciari:2008gp} algorithms, often used as implemented in the efficient {\\tt FastJet}~\\cite{Cacciari:2011ma} package.\n\nThe CDF collaboration has published precise measurements of inclusive jet~\\cite{Abulencia:2007ez,Aaltonen:2008eq} and dijet~\\cite{Aaltonen:2008dn} cross sections. Data is also available from the D0 experiment, once again for inclusive~\\cite{Abazov:2008ae} and dijet~\\cite{Abazov:2010fr} quantities.\nFigure \\ref{fig:CDFkTJet} shows the results of an inclusive jet measurement at CDF using the $k_T$ clustering algorithm.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{3-PDFdet/figs/CrossSectionAll.pdf}\n\\caption[Inclusive Jet data from the CDF experiment]{Inclusive jet data from CDF using the $k_T$ jet clustering algorithm, compared to predictions from the CTEQ6.1M PDF set. Figure from~\\cite{Abulencia:2007ez}. }\n\\label{fig:CDFkTJet}\n\\end{figure}\n\n\\subsubsection{Prompt photon measurements}\nComplementary to the data on jet production, measurements of prompt photon processes $pp/p\\bar{p} \\to \\gamma X$ can also provide an important handle on the gluon. The term \\emph{prompt} photon refers the production of a photon in the hard scatter rather than in subsequent emissions. Prompt photons in the final state can originate either from Compton scattering processes $gq \\to \\gamma q$ or annihilation events $q\\bar{q} \\to \\gamma g$, processes denoted \\emph{direct} photon production. Alternatively prompt photons may be produced via the fragmentation of final state hadrons into photons via so-called fragmentation functions\\cite{Bourhis:1997yu,Gluck:1992zx}. In $pp$ collisions the Compton scatter is typically the dominant process, particularly at higher scales where the fragmentation contribution is suppressed. For $p\\bar{p}$ events the annihilation contribution becomes more important due to the enhanced $q\\bar{q}$ PDF luminosity. Figure \\ref{fig:jrpromptphoton} demonstrates the relative fraction of these contributions to the cross-section for a range of photon transverse energy $E_T$.\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/subproc_ppgamma_jetphox_lhc_inc_stack.pdf}\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/subproc_ppgamma_jetphox_lhc_is_stack.pdf}\\\\\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/subproc_ppgamma_jetphox_tevatron_inc_stack.pdf}\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/subproc_ppgamma_jetphox_tevatron_is_stack.pdf}\n\\caption[Relative contribution of partonic subprocesses to $pp/p\\bar{p}\\to \\gamma X$]{Relative contribution of partonic subprocesses to $pp/p\\bar{p}\\to \\gamma X$. Figures on the left refer to the inclusive case, and on the right to the observable after isolation cuts on the final state photon. Figure from~\\cite{dEnterria:2012yj}. }\n\\label{fig:jrpromptphoton}\n\\end{figure}\n\nFor the purposes of PDF determination direct photon measurements which are free of the additional uncertainties introduced when performing calculations with photon fragmentation functions are the ideal measurement. While performing selection cuts to measure only the direct photon contribution is experimentally challenging, the relative contribution of fragmentation photons may be suppressed by making isolation cuts upon the final state photon. These cuts admit only photons with no hadronic material in close proximity. Smooth-cone cuts such as the Frixione isolation criterion~\\cite{Frixione:1998jh} in principle can remove entirely the fragmentation contribution. However these cuts remain challenging to implement experimentally, with experimental data usually obtained with simpler isolation cuts which aim to suppress rather than eliminate fragmentation photons.\n\nTheoretical predictions are available at NLO for the Compton process~\\cite{Owens:1987qy,Aurenche:1988vi} and commonly used as implemented in the {\\tt JETPHOX} program~\\cite{Catani:2002ny,Aurenche:2006vj,Belghobsi:2009hx}. While inclusive data is challenging to include in a PDF determination due to contamination by fragmentation photons, results are available from a wide range of isolated photon measurements. Isolated data is available from UA1/UA2 at the Sp$\\bar{\\mathrm{p}}$S~\\cite{Albajar:1988im,Alitti:1992hn,Ansari:1988te}, PHENIX at RHIC~\\cite{Adler:2006yt}, CDF~\\cite{Aaltonen:2009ty,Abazov:2005wc,Abe:1994rra,Acosta:2002ya,Acosta:2004bg} and D0~\\cite{Abachi:1996qz,Abazov:2001af,Abbott:1999kd}.\n\n\\subsubsection{Top quark pair production data}\nThe production of top-antitop pairs is potentially a process of great interest in the determination of PDFs, with calculations available up to NNLO for the total cross-section~\\cite{Czakon:2013goa,Baernreuther:2012ws,Czakon:2012zr,Czakon:2012pz}. The impact of the total top pair production cross-section upon PDFs is quite sensitive to the kinematics of the collider, with Tevatron data probing directly the quark content of the proton, while data from colliders with higher centre of mass energies being dominated by the gluon-gluon channel. Precise data from the Tevatron is available in the form of a combined D0-CDF analysis~\\cite{Aaltonen:2012ttbar}.\n\n\\subsubsection{Experimental cuts}\n A simple cut is typically performed on the hard scale $Q^2$ and for DIS the final state invariant mass $W^2$ to ensure the reliability of perturbative predictions. The MSTW2008 parton fit uses an initial scale for evolution of $Q_0^2=1$ $\\mathrm{ GeV}^2$, CT10 uses $Q_0^2=1.3$ $\\mathrm{ GeV}^2$ and NNPDF2.3 $Q_0^2=2$ $\\mathrm{ GeV}^2$. Most of the data included in global parton fits has a minimum of $Q^2\\sim2$ to $5$ GeV$^2$\\cite{DeRoeck:2011na}.\n\n\\section{Methodological elements}\n\\subsection{Parametrisation} \nGiven an experimental dataset, one must choose a convenient and effective parametrisation of the parton distribution functions such that their predictions may be compared to data. Nominally there are a total of 13 PDFs, six quarks, six antiquarks and a gluon. However as mentioned in the previous section, the heavy quarks $c$, $b$, $t$ are determined perturbatively. There are therefore typically seven free PDFs remaining to be fitted. The parton parametrisation basis is chosen for ease of fitting and perturbative evolution; a basis close to the DGLAP basis in Eqn.~\\ref{eq:DGLAP} is desirable for efficiency. However often a different basis is chosen to avoid fitting quantities that are poorly defined by the experimental dataset. \n\nFor example, MSTW2008\\cite{Martin:2009iq} uses the following basis for their determination:\n\\ba\n\tg,&& \\nonumber\\\\\n q_v &\\equiv & q - \\bar{q},\\nonumber \\\\\n\t\\Delta &\\equiv & \\bar{d} - \\bar{u},\\nonumber \\\\\n\tS &\\equiv & 2(\\bar{u}+\\bar{d})+s+\\bar{s}\\nonumber,\\\\\n\ts^\\pm &\\equiv & s \\pm \\bar{s},\n\t\\ea\t\nwhere g is the gluon PDF and the $q_v$ correspond to the $u$, $d$ quark valence PDFs. These fully parameterise the degrees of freedom to be determined. A functional form in $x$ is then chosen for each of the distributions (the value of $Q^2$ is kept fixed at the input scale for fitting). While all groups include the limiting-$x$ description of Eqn.~\\ref{eq:pdflimits}, the choice of parametrisation for the remainder function $r$ varies substantially between fitting groups. As an example, the valence quark PDF $q_v$ parametrisation in MSTW2008 is provided by the expression\n\\be xq_v(x,Q_0^2) = ax^{b}(1-x)^{c}(1+d\\sqrt{x}+e x),\\ee\nand the equivalent parametrisation in CT10\\cite{Lai:2010vv} is \n\\be xq_v(x,Q_0^2) = ax^b(1-x)^b \\exp{(cx + dx^2 + e\\sqrt{x})},\\ee\nwhere the $(a$,...,$e)$ are the parameters to be determined in the fit. In total the MSTW08 basis has 30 free parameters (taking into account sum rule constraints), the CT10 parametrisation is a little less flexible, having 26 free parameters. The problem is now reduced to finding the optimum parameters for the $7$ PDFs that minimise some measure of fit quality, the differing versions of which we shall discuss later in the chapter.\n\nThe NNPDF procedure is markedly different from that of the other PDF fitting groups and the first major difference lies in the choice of parametrisation. Unlike in the general procedure outlined above, neural networks are used to provide the functional $x$ dependence of the PDFs. Neural networks are a typical computational tool in machine learning environments, often used in regression applications where flexibility and a lack of bias with respect to a conventional fixed parametrisation are desired. A typical neural network in a fitting context will usually have considerably more functional freedom (and therefore parameters) than a normal parametric model, with the neural network compensating for its relative generality with respect to the problem by having much greater flexibility. \n\nThe use of neural networks as applied to the determination of the proton structure function $F_2^p$ was first suggested in Ref.~\\cite{Forte:2002fg} and subsequently developed in~\\cite{DelDebbio:2004qj}. The approach was later extended to the determination of quark distributions~\\cite{DelDebbio:2007ee} before becoming a global analysis of PDFs as of NNPDF2.0~\\cite{Ball:2010de} as part of the wider NNPDF methodology.\n\nIn the NNPDF approach the specific networks used in the parametrisation are multi-layer feed forward neural networks configured with 2-5-3-1 architecture. This architecture applied over seven PDFs results in a fit with a total of 259 free parameters, considerably more than in competing approaches. The architecture chosen in fact has considerable redundancy to minimise potential bias due to inflexibility or choice of architecture. The flexibility of the approach was demonstrated in Ref.~\\cite{Ball:2011eq} where the architecture was modified considerably, with no significant change in the fit results.\n\nDue to the redundant parametrisation provided by the neural networks, there is a great deal of freedom in the choice of the input parton distribution basis. In the more recent NNPDF analyses: sets NNPDF 2.1 and NNPDF 2.3, the basis is chosen for simplicity of evolution as:\n\\ba\n\\mathrm{gluon}\\quad \\quad&g,& \\nonumber \\\\\n\\mathrm{singlet}\\quad \\quad&\\Sigma &\\equiv \\sum_{i=1}^{n_f} (q_i + \\bar{q_i}),\\nonumber \\\\\n\\mathrm{valence}\\quad\\quad &V &\\equiv \\sum_{i=1}^{n_f} (q_i - \\bar{q_i}),\\nonumber \\\\\n\\mathrm{triplet}\\quad\\quad &T_3 &\\equiv (u + \\bar{u}) - (d+\\bar{d}),\\nonumber \\\\\n\\mathrm{sea}\\; \\mathrm{asymmetry}\\quad\\quad &\\Delta &\\equiv \\bar{d}-\\bar{u},\\nonumber \\\\\n\\text{strange sea/valence}\\quad\\quad &s\\pm &\\equiv s \\pm \\bar{s}.\n\\ea\n\\clearpage\nThe equivalent functional forms for the fitting in terms of the Neural Networks are;\n\\ba \\Sigma(x,Q_0^2)&=&x^{-\\alpha_\\Sigma}(1-x)^{\\beta_\\Sigma} \\mathrm{NN}_\\Sigma(x), \\nonumber \\\\\nV(x,Q_0^2)&=&A_V x^{-\\alpha_V} (1-x)^{\\beta_V} \\mathrm{NN}_V(x),\\nonumber \\\\\nT3(x,Q_0^2)&=&x^{-\\alpha_{T3}}(1-x)^{\\beta_{T3}} \\mathrm{NN}_{T3}(x),\\nonumber \\\\\n\\Delta(x,Q_0^2)&=&A_\\Delta x^{-\\alpha_\\Delta}(1-x)^{\\beta_\\Delta} \\mathrm{NN}_\\Delta(x),\\nonumber \\\\\ng(x,Q_0^2)&=&A_gx^{-\\alpha_g}(1-x)^{\\beta_g} \\mathrm{NN}_g(x)\\nonumber, \\\\\ns^+(x,Q_0^2)&=&x^{-\\alpha_{s^+}} (1-x)^{\\beta_{s^+}}\\mathrm{NN}_{s^+}(x),\\nonumber \\\\\ns^-(x,Q_0^2)&=&x^{-\\alpha_{s^-}}(1-x)^{\\beta_{s^-}} \\mathrm{NN}_{s^-}(x) - s_{\\mathrm{aux}}(x,Q_0^2), \\label{eq:NNPDF23param}\n\\ea\nwhere the NN denote the 2-5-3-1 neural network parametrisations and the $A$ are set by enforcing the appropriate sum rules. In the NNPDF approach the treatment of the limiting exponents $\\alpha$, $\\beta$ is rather different. These factors are introduced in order to speed up the convergence of the neural network fitting, with the intention of providing a rough preprocessing function as a backbone for the neural networks to deviate from, and ensuring that the functions have the correct behaviour under integration. These exponents are therefore randomised within an optimised range at the start of the fit and are not modified by the fitting procedure. The final results should therefore be reasonably independent of the preprocessing factor and of the coefficients involved. \n\\\\\nWhile determinations with fixed parametrisations typically design the strange valence functional form such that the strange valence sum rule is automatically satisfied, this cannot be done with a neural net parametrisation. In the determinations up to NNPDF2.3 the strange auxiliary term $s_{\\mathrm{aux}}(x,Q_0^2)$ in Eqn.~\\ref{eq:NNPDF23param} is therefore introduced to ensure the strange valence sum rule is followed, and has the form~\\cite{Ball:2009mk}:\n\\be s_{\\mathrm{aux}}(x,Q_0^2) = A_{s^-}(x^{r}(1-x))^s.\\ee\n\\subsection{Fit quality and minimisation}\nWith an experimental dataset selected and a choice made for the parametrisation of the PDFs, the optimal fit should be determined by varying fit parameters and attempting to minimise some measure of fit quality. Different groups make quite different choices not only in the minimisation method but also in the measure used to determine fit quality. The most general statement that can be made is that the global fit quality (generally denoted $\\chi^2$) is built from the quality of fit to individual datasets as\n\n\\be \\chi^2 = \\sum_k^{n} \\chi^2_k,\\ee\nfor a fit with $n$ data sets, each with a consistent normalisation. In the NNPDF approach the full covariance matrix of the data is used in determining the quality of fit, including all appropriate correlations within and between datasets. The $\\chi^2$ measure for a set of data with common correlations is then given by\n\\be \\chi^2_k=\\sum_{i,j=1}^{N_{\\mathrm{dat}}}\\frac{(D_{k,i}-T_{k,i})(D_{k,j}-T_{k,j})}{\\mathrm{Cov}[i,j]}.\\ee\nHere the $T$ are the theoretical predictions for the experimental data points $D$ calculated from the neural network parametrisation, and $\\mathrm{Cov}[i,j]$ is the covariance between data points $i$ and $j$. In practice there is a ensemble of neural networks each associated with a single Monte Carlo sample of the experimental data, for the purposes of error propagation. This point will be discussed in more detail later in the chapter. In NNPDF determinations the full experimental correlations should be available for a dataset to be included into the determination.\nOther groups take a different strategy, often with the suggestion that correlation effects are small to negligible with the exception of overall normalisations. Adopting the same practice as earlier MRST fits, the MSTW2008 fit uses an uncorrelated $\\chi^2$ measure over much of its dataset~\\cite{Martin:2009iq}, with the normalisation of the theory predictions set by a fitted parameter $\\mathcal{N}$\n\\be \\chi^2_k=\\sum_{i=1}^{N_{\\mathrm{dat}}} \\frac{(D_{k,i}-T_{k,i}/\\mathcal{N}_k)^2}{\\mathrm{Var}[i]} + \\left(\\frac{1-\\mathcal{N}_k}{\\sigma_k^{\\mathcal{N}}}\\right)^4, \\label{eq:MSTWchi2}\\ee \nwhere the final quartic penalty is intended to prevent the normalisation deviating too far from the experimental normalisation uncertainty $\\sigma_{\\mathcal{N}}$, and the variance $\\mathrm{Var}[i]$ is constructed by the sum in quadrature of the statistical and uncorrelated systematic errors. The CT series of fits utilise a $\\chi^2$ measure that includes systematic uncertainties in terms of explicit shifts~\\cite{Stump:2001gu,Pumplin:2002vw}. In this arrangement, the fit quality measure is given by\n\\begin{equation} \\label{eq:CTchi2}\n\\chi^{2}_k = \\sum_{i=1}^{N_{\\mathrm{dat}}} \\frac{1}{%\n\\mathrm{Var}[i]} \\left(D_{k,i}-T_{k,i}-\\sum_{n=1}^{N_{\\mathrm{corr}}}r_{n}\\sigma^{\\mathrm{corr}}_{k,n,i}\\right)^{2}\n+\\sum_{n=1}^{N_{\\mathrm{corr}}} r_{n}^{2},\n\\end{equation}\nwhere here the $\\sigma^{\\mathrm{corr}}$ are the $N_{\\mathrm{corr}}$ correlated systematic uncertainties. In this procedure the theory predictions $T$ are shifted parametrically by the variables $r$. The optimal shift values are found by minimising the $\\chi^2$ with respect to the $r$ analytically at each stage of the fit. This procedure was introduced to accommodate for overall shifts in the CT10 distributions. A similar method which was adopted in MSTW2008 for a limited number of datasets where correlations were deemed to be important, with the normalisations also determined in the fit as per the uncorrelated case.\n\n\\subsubsection{Normalisation uncertainty}\n\nA key point that must be addressed when constructing a measure of fit quality is the treatment of normalisation uncertainties, or multiplicative uncertainties in general. Even using the same definition of the fit quality measure, substantial deviations may be produced by defining the covariance matrix and therefore the breakdown into systematic errors, differently.\n\nThe full experimental uncertainty information is characterised by the sum of all uncorrelated errors for a datapoint $\\sigma^{\\mathrm{unc}}$; the set of $N_{\\mathrm{add}}$ correlated additive systematics $\\sigma^{\\mathrm{add}}$; and the set of $N_{\\mathrm{mul}}$ correlated multiplicative systematics $\\sigma^{\\mathrm{mul}}$.\nGiven this information one may naively define an \\emph{`experimental'} prescription~\\cite{Ball:2012wy} for constructing a covariance matrix as\n\\be\n\\label{eq:covmat}\n\\mathrm{Cov}[i,j]=\n\\delta_{ij}\\; \\sigma^{\\mathrm{unc}}_{i}\\sigma^{\\mathrm{unc}}_{j} + \n\\sum_{k=1}^{N_{\\mathrm{add}}}\\sigma^{\\mathrm{add}}_{i,k}\\sigma^{\\mathrm{add}}_{j,k}\n+ \\left( \\sum_{k=1}^{N_{\\mathrm{mul}}} \\sigma_{i,k}^{\\mathrm{mul}}\\sigma_{j,k}^{\\mathrm{mul}}\n\\right) D_{i} D_{j},\n\\ee\nwhere once again the $D$ represent the experimental data points. This method of constructing the covariance matrix is therefore unambiguously defined by the experimental results. While a perfectly valid definition for analysing the description of data after a PDF determination, it is unreliable for use directly within a fitting procedure. The use of the experimental definition has for some time been understood to result in a \\emph{d'Agostini bias}\\cite{DAgostini:1993uj}. That is, the theoretical values determined via a minimisation of a $\\chi^2$ function with the experimental covariance matrix are systematically shifted lower than the true value, an effect which only worsens as the number of data points subject to a common multiplicative error increases. The bias is generated by downward statistical fluctuations of data, if these low data points are used to generate the normalisation uncertainty, the result is a smaller uncertainty for the lower points, causing the fit to systematically undershoot the data. \n\nThe typical method employed to avoid the d'Agostini bias proceeds by including the normalisation as a fitted parameter and penalising large deviations as shown in Eqn. \\ref{eq:MSTWchi2}. This procedure largely corrects for the problem, although when applied to a dataset with several different normalisation uncertainties it still suffers from a bias. This effect was demonstrated by the NNPDF collaboration in Ref.~\\cite{Ball:2009qv}. The bias can be avoided by using the so-called $t_0$ prescription~\\cite{Ball:2009qv} for defining the covariance matrix. In this method the covariance matrix is constructed using the predictions from a previous fit rather than the experimental data values, to multiply with the multiplicative uncertainties.\n\\be\n\\label{eq:covmat_t0}\n\\mathrm{Cov}^{t_0}[i,j]=\n\\delta_{ij}\\; \\sigma^{\\mathrm{unc}}_{i}\\sigma^{\\mathrm{unc}}_{j} + \n\\sum_{k=1}^{N_{\\mathrm{add}}}\\sigma^{\\mathrm{add}}_{i,k}\\sigma^{\\mathrm{add}}_{j,k}\n+ \\left( \\sum_{k=1}^{N_{\\mathrm{mul}}} \\sigma_{i,k}^{\\mathrm{mul}}\\sigma_{j,k}^{\\mathrm{mul}}\n\\right) T_{i}\\, T_{j},\n\\ee\nwhere here the $T$ are theory predictions for the associated datapoint, generated by some prior (fixed) PDF set. The prior, or $t_0$ set should be determined self-consistently via an iterative procedure in which the $t_0$ set is obtained from the previous result for the full fit. As the theory predictions are not subject to the same fluctuations as the data, the fit is not subject to the aforementioned bias. This effect can be seen explicitly in a fit to artificial pseudodata, performed with the experimental and $t_0$ covariance matrix definitions in Figure \\ref{fig:expbias}.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.8\\textwidth]{3-PDFdet/figs/t0plot.pdf}\n\\caption[Demonstration of d'Agostini bias in a fit to pseudodata generated according to the kinematics of CDF inclusive jet data]{Demonstration of d'Agostini bias in a fit to pseudodata generated according to the kinematics of CDF inclusive jet data. Fit results are shown as a ratio to the `true' value used to generate the pseudodata. The fit performed with the experimental definition of the covariance matrix results in predictions shifted systematically downwards with respect to the underlying law. The predictions from the fit using a $t_0$ covariance matrix do not suffer from such a bias.}\n\\label{fig:expbias}\n\\end{figure}\n\n\\subsubsection{Minimisation}\nWith a figure of merit constructed, the PDF determination now becomes a problem of varying the free parameters in the PDF basis to minimise said measure. Even for those groups utilising a fixed parametrisation, performing a minimisation of the global $\\chi^2$ for a large, $n\\sim\\mathcal{O}(1000)$ dataset with a fairly large number of free parameters (approximately $50$ in the MSTW analysis once normalisation uncertainties are added as free parameters) is a challenging numerical task. For performing the minimisation, the MINUIT~\\cite{MINUIT} package is a common choice, although other function minimisation methods are applied such as the Levenberg-Marquardt~\\cite{levenberg,marquardt} method as used in the MSTW fits.\n\nIn the NNPDF case the minimisation is complicated by the very large number of parameters and highly nonlocal behaviour in the error function, making conventional methods of minimisation difficult. These difficulties are overcome in the NNPDF methodology by the use of \\emph{genetic algorithms}, which are particularly efficient at exploring large parameter spaces. The implementation of the genetic algorithm is discussed in detail in Refs.~\\cite{Ball:2010de,DelDebbio:2007ee}.\n\nIn addition to the basic difficulty of minimisation in a large parameter space, there is a further issue that arises when considering the fitting of a function with a great deal of redundant flexibility. Because of the flexibility of the parametrisation, it is possible that training the neural networks so that each reaches the global minimum in the error function actually results in the networks fitting to statistical noise. This effect is known as \\emph{overlearning} and is a problem often encountered in the training of large neural networks~\\cite{bishopnn,nnoverlearn}. In previous NNPDF determinations, the widely used \\emph{cross-validation} technique~\\cite{Ball:2010de,bishopnn} was employed in order to identify when overlearning occurred. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[scale=0.5]{3-PDFdet/figs/chi2ite-1004-NMC-pd.pdf}\n\\caption[Demonstration of overlearning in the cross-validation of a neural network fit]{A typical signal of overlearning in a neural network fit. E$_{\\mathrm{tr}}$ and E$_{\\mathrm{val}}$ represent the training and validation figures of merit respectively. As the number of genetic algorithm generations proceeds, eventually the network begins to fit statistical noise in the training set and the validation fit quality begins to decrease. Figure from~\\cite{Ball:2010de}.}\n\\label{fig:crossval}\n\\end{figure}\n\nIn this method the experimental data set is split into two separate sets. The first, a fitting set which is used for the minimisation of the error function, and a second validation set which is not used directly in the fitting procedure. For each iteration in the genetic algorithm minimisation the error function is computed between the neural network predictions and both data sets. In the early stages of the training both error functions should decrease. However in the latter stages of the training where statistical noise begins to become an important contribution, the goodness-of-fit calculated to the fitting data set may continue to decrease while the same value calculated to the validation set has stopped decreasing or even begun to increase. This is a clear signal of overlearning, where fitting to statistical noise in the fitting set means that the fit to the validation set is no longer improving. At this stage the training of the neural networks is stopped. A typical signal of overlearning in a cross-validated fit can be seen in Figure~\\ref{fig:crossval} which compares the fit quality for both the training and validated sets over a number of fit iterations.\n\n\\subsection{Error propagation}\n\\label{sec:errors}\nIn order to undertake precision QCD studies, some estimate of the uncertainty on PDFs is required for a meaningful interpretation of the measured observables. The need for PDF sets with quantified uncertainties has been long recognised, and all modern determinations provide sets with at least experimental uncertainty estimation. While performing a comprehensive quantification of the theoretical uncertainty in a PDF fit is challenging, many methods have been developed in order to propagate the uncertainty from the dataset to the fitted PDFs. Ideally, one would like to determine a representation of the probability distribution in the whole functional space. That is given a dataset ${d}$, we would like to find the probability of a certain PDF candidate $f$ such that our fitted PDF central value is given by\n\\be \\left\\left(x\\right) = \\int \\mathcal{D}f \\;f\\left(x\\right) \\mathcal{P}\\left(f \\middle| d \\right), \\ee\nand the uncertainty by\n\\be \\mathrm{Var}\\left[f\\right]\\left(x\\right) = \\int \\mathcal{D} f \\; \\left[ f\\left(x\\right) - \\left\\left(x\\right) \\right]^2 \\mathcal{P}\\left(f \\middle| d \\right). \\ee\nThe probability distribution for an observable $\\mathcal{O}$ is then simply $\\mathcal{O}\\left[f\\right]\\mathcal{P}\\left(f \\middle| d \\right)$, in terms of which an observable's central\nvalue and PDF uncertainty can be calculated by\n\\be \\left = \\int \\mathcal{D}f \\; \\mathcal{O}\\left[f\\right] \\mathcal{P}\\left(f \\middle| d \\right), \\ee\n\\be \\mathrm{Var}\\left[\\mathcal{O}\\right] = \\int \\mathcal{D}f \\; \\left(\\mathcal{O}\\left[f\\right] - \\left<\\mathcal{O}\\right> \\right)^2 \\;\\mathcal{P}\\left(f \\middle| d \\right). \\ee\nThe probability distribution $\\mathcal{P}\\left(f \\middle| d \\right)$ is however a difficult quantity to determine. In this section we shall examine a number of the methods used in the literature to provide an estimate of PDF uncertainties.\n\\subsubsection{The Hessian method}\nThe Hessian method is the most widely used method of uncertainty determination in PDFs. In essence, the method involves examining how the fit quality $\\chi^2$ varies when the $n$ fit parameters $a$ are perturbed about the values which minimise the $\\chi^2$, here denoted by $a^\\mathrm{min}$. A tolerance in the $\\chi^2$ variation is then chosen, and the error on an observable is determined geometrically from observables calculated with parameters perturbed by the selected tolerance. To examine this quantitatively, we first define the difference in $\\chi^2$ from the minimum value\n\\be \\Delta\\chi^2(a) \\equiv \\chi^2(a) - \\chi^2(a^\\mathrm{min}) = \\sum^n_{i,j=1}H_{ij}(a_i-a_i^\\mathrm{min})(a_j - a_j^\\mathrm{min}), \\ee\nwhere the $a_i$ represent the $i$th component of the parameter set $a$ (and likewise, for the minimised set $a^\\mathrm{min}$). Here we assume that the variation around the $\\chi^2$ minimum is approximately quadratic. The Hessian matrix $H$ has values determined by\n\\be H_{ij} = \\frac{1}{2} \\frac{ \\partial^2 \\chi^2(a) }{\\partial a_i\\partial a_j}\\bigg|_{\\mathrm{min}}, \\ee\nwhere the \\emph{min} subscript refers to the parameters obtained at the $\\chi^2$ minimum Early Hessian uncertainty estimates~\\cite{Adloff:2000qk,Alekhin:2002fv} were based upon the standard formula for linear error propagation\n\\be (\\Delta F)^2 = T^2\\sum^n_{i,j=1}\\frac{\\partial F}{\\partial a_i}C_{ij}\\frac{\\partial F}{\\partial a_j}, \\ee\nwhere $T^2=\\Delta\\chi^2$ is the tolerance in $\\chi^2$ variation and $C=H^{-1}$ is the inverse Hessian matrix. This procedure is however a little inconvenient due to the requirement of the partial derivatives of the observable with respect to the fit parameters. There are also numerical issues relating to this method which give rise to peculiar uncertainty estimates~\\cite{DeRoeck:2011na}. In order to overcome these issues the geometrical method outlined above was developed by the CTEQ collaboration~\\cite{Pumplin:2000vx,Pumplin:2001ct}.\n\nFor this method it is convenient to work in a rescaled orthogonal eigenbasis for the covariance matrix. The orthonormal eigenbasis is defined in the usual way\n\n\\be H v_i = \\lambda_i v_i, \\ee\nand the rescaled eigenbasis is defined as $e_{i}=1/\\sqrt{\\lambda_i} v_{i}$. The difference between a parameter set $a$ and $a^\\mathrm{min}$ can now be expanded as\n\\be a_i-a_i^\\mathrm{min} = \\sum^n_{k=1}e_{ik}z_k, \\ee\nwhere $e_{ik}$ is the $i$th component of the $k$th rescaled eigenvector, and the $z_k$ are the coefficients for the expansion of the parameter difference onto the rescaled eigenbasis. Therefore the expression for $\\Delta\\chi^2$ reduces to\n\\be\\Delta\\chi^2(a) = \\sum_{k=1}^n z_k^2 \\quad\\quad \\mathrm{or,}\\quad\\quad \\chi^2(a) = \\chi^2(a^\\mathrm{min}) + \\sum^n_{k=1}z_k^2.\\ee\nThis defines a hypersphere in the parameter space of radius $\\Delta\\chi^2$ centred around $a^\\mathrm{min}$, which corresponds to the variation in the parameters that is consistent with the tolerance $T=\\sqrt{\\Delta\\chi^2}$ in the quadratic approximation. It is now possible to construct an ensemble of $2n$ PDF sets corresponding to the fits on the boundaries of the volume. A PDF set $S^\\pm_k$ therefore has the parameter set\n\\be a_i(S_k^\\pm) = a_i^\\mathrm{min} \\pm te^{ik},\\ee\ni.e. each parameter is perturbed by $t$ in the direction of the $e_{k}$ eigenvector. In the quadratic approximation $t=T$, when the approximation breaks down $t$ can be determined by an iterative procedure to obtain the desired $\\Delta\\chi^2$. The error on an observable $F$ is then given simply by Pythagoras' theorem on the hypersphere\n\\be(\\Delta F)^2=\\frac{1}{2}\\sum_i^n(F(S^+_i)-F(S^-_i))^2.\\ee \n\nIn this procedure there is something of an ambiguity in the determination of the tolerance (and hence, the volume of the sphere in parameter space). Ideally the difference in $\\chi^2$ values should be exactly one for a confidence level of one-sigma\\footnote{It should be noted that this is only the case when, either the data errors are uncorrelated, or when the correlations are included in the definition of the global goodness-of-fit $\\chi^2$\\cite{Stump:2001gu}}. In the case of PDF fits, this tolerance often leads to uncertainties far lower than expected. In practice, the CTEQ group uses a value of $\\Delta \\chi^2\\sim 100$ and MSTW uses a value $\\sim50$. The more recent MSTW PDF sets have uncertainties calculated with a dynamically determined tolerance. More specialised fits such as ABM11 or the HERAPDF series, based upon relatively restrictive datasets may use the standard tolerance of $\\Delta\\chi^2=1$. Their use of a more restrictive dataset perhaps leading to fewer conflicts between experimental datasets that could require a more flexible tolerance.\n\nThe uncertainties produced via the Hessian procedure are difficult to analyse in a statistical sense due to the (occasional) inflation of the $\\Delta\\chi^2$ and the approximations made in the procedure. It is therefore difficult to find a representation in the Hessian approach of the full probability distribution $\\mathcal{P}\\left(f \\middle| d \\right)$. Furthermore the uncertainty in the choice of functional form, or estimation of parametrisation bias, is not typically take account of. The HERAPDF family of fits however do attempt to estimate this uncertainty by performing a series of fits with slightly modified parametrisations.\n\n\\subsubsection{Lagrange multiplier method}\nAnother method of error propagation that has been explored is the Lagrange multiplier method. The method has the advantage of not assuming that the $\\chi^2$ function is quadratic around the global minimum. We shall briefly discuss the method applied to the PDF error determination as suggested by Pumplin~\\cite{Pumplin:2000vx} and Stump~\\cite{Stump:2001gu}. A description of the process can also be found in~\\cite{DeRoeck:2011na,Martin:2009iq}.\n\nFirstly, a general global fit is performed to the data as described above. This yields a set of parameters ${a^\\mathrm{min}}$ which minimise the $\\chi^2$ measure. Using these parameters we calculate\nthe best fit prediction for the observable in question $F(a^\\mathrm{min})$. A new PDF fit can now be performed, where instead of minimising the $\\chi^2$ the following function is minimised\n\\be \\Psi= \\chi^2(a) + \\lambda (F(a)-F(a^\\mathrm{min}))\\ee\ni.e. we introduce the observable $F$ as a parameter in the fitting procedure and constrain the fit so that the minimal $\\Psi$ occurs when $F(a)=F(a^\\mathrm{min})$. The value $\\lambda$ in this function is the Lagrange multiplier. The fit above is performed for many values of $\\lambda$, each time leading to a parameter set that depends on that particular value of $\\lambda$, this parameter set will be denoted $a_\\lambda$. Using these parameters, we now calculate values for $\\chi^2(a_\\lambda )$ and $\\mathcal{O}(a_\\lambda )$.\n\nAt this stage we now have a set of values for $\\chi^2(a_\\lambda)$ and $\\mathcal{O}(a_\\lambda)$ over a large range of $\\lambda$ values. This allows a determination of the relationship between the goodness-of-fit and the prediction for $F$ via the parameter $\\lambda$. We obtain an approximate function $\\chi^2(F)$ over a range of observable values, with a minimum at $F=F(a^\\mathrm{min})$ i.e $\\lambda=0$ and $a_\\lambda=a^\\mathrm{min}$. Also we have a set of the $a$ parameters for every point on the curve which are optimised for the best fit to the observable $F$. This means that we have a set of fully optimised parameters for any arbitrary confidence level determined by the $\\Delta\\chi^2$ that we select as a tolerance. Uncertainties for the PDFs can therefore be given in a way that utilises the whole of the $a$ parameter space, rather than just perturbing around the global minimum as in the Hessian approach.\n\nOf course, the disadvantage of this method is that the PDF uncertainties must be calculated for each observable in a rather computationally intensive process. The errors are naturally optimised for the particular observable, but the process is inconvenient for a PDF end-user, and so it is not widely-used in error determination. In this sense the Lagrange multiplier approach suggests a method of estimating $\\mathcal{P}\\left(\\mathcal{O} \\middle| a \\right)$, or the probability density of an observable in the space of parameters. The Lagrange multiplier method also relies on the same somewhat arbitrary choice of tolerance in $\\chi^2$ as the Hessian method. The method has however been applied as a cross-check to the Hessian results~\\cite{Pumplin:2000vx,Martin:2002aw}.\n\n\\subsubsection{Monte Carlo method}\nAnother quite distinct method of PDF uncertainty determination is the Monte Carlo method, first suggested by Giele and Keller~\\cite{Giele:1998gw,Giele:2001mr} where a Monte Carlo procedure in the space of fit parameters was outlined. The NNPDF collaboration uses a similar method in all of its fits, although with the Monte Carlo performed in the space of experimental data. The method is designed to faithfully represent the uncertainties present in the initial data, and to propagate the errors in a way that does not assume anything of the nature of the error propagation. The Monte Carlo approach was also analysed and compared to the results of a Hessian fit by the MSTW group in~\\cite{Watt:2012tq}.\n\nIn the Monte Carlo procedure an ensemble of $N_{\\text{rep}}$ artificial data replicas is produced for every data point in the fit, generated according to the probability distribution of the initial data. Typically this distribution is multi-Gaussian with central values and variances provided by experimental results, but any probability distribution may be used if required. If we use $F_{p}^{(\\mathrm{art})(k)}$ to represent a single element $k$ of the pseudo-data sample (the \\emph{art} superscript designates the data as an artificial sample) of the observable $F$ at the kinematical point $\\left\\{ x_p,Q^2_p \\right\\} $. Then we can generate such a pseudo-data element as in~\\cite{Ball:2008by} by\n\\be F_{p}^{(\\mathrm{art})(k)}=S_{p,N}^{(k)}F_{p}^{(exp)}\\left( 1 + \\sum_{l=1}^{N_c}r_{p,l}^{(k)}\\sigma_{p,l} + r_{p}^{(k)}\\sigma_{p,s}\\right),\\ee\nwhere the $r$ are independent Gaussian random numbers centred upon the experimental central value. The $\\sigma_{p,s}$ term contains the uncorrelated systematic uncertainties and the statistical uncertainty added in quadrature. The $\\sigma_{p,l}$ are the correlated errors for the data provided by experiment. The normalisation of the probability distribution is fixed by the term $S_{p,N}$. Provided a large enough quantity of these artificial replicas ($N_{\\mathrm{rep}}$) is generated, this form of the generating distribution for the Monte Carlo ensemble reproduces all of the statistical qualities of the original experimental data. In Ref.~\\cite{Ball:2010de} it is demonstrated that $N_{\\mathrm{rep}}=1000$ is sufficient to reproduce the experimental central values and variances to an accuracy of better than one percent.\n\nNow that a good Monte Carlo sample of the experimental data is available, instead of performing just the one fit to the data, $N_{\\mathrm{rep}}$ independent fits are performed, one for each of the data replicas. At the end of the fitting procedure we obtain an ensemble of $N_{\\mathrm{rep}}$ equally probable PDFs which reliably describe the probability distribution of the PDFs based upon the original experimental uncertainties. The central values and uncertainties of an observable can be simply obtained by computing the average and the variance over the ensemble of PDFs.\n\\be \\left< F\\right> = \\frac{1}{N_{\\mathrm{rep}}}\\sum_i^{N_{\\mathrm{rep}}} F^{(k)}, \\label{eq:MCCV}\\ee\n \\be \\sigma^2[F] = \\frac{1}{N_{\\mathrm{rep}}-1} \\sum_{i=1}^{N_{\\mathrm{rep}}} (F^{(k)} - \\left< F\\right>)^2, \\label{eq:MCVAR}\\ee\n where $F^{(k)}$ denotes the observable $F$ computed using PDF replica $k$.\n\nThe Monte Carlo method therefore propagates the errors from the experimental data through to the PDFs in a natural way, without the need for a linear propagation of errors assumption, or the need for an inflated tolerance in the $\\chi^2$ distribution. Figure \\ref{fig:mcerror} demonstrates a Monte Carlo ensemble of PDF replicas for the gluon distribution.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/pdf_xg_log_rep.pdf}\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/pdf_xg_rep.pdf}\n\\caption[A Monte Carlo representation of the gluon PDF probability distribution]{A Monte Carlo representation of the gluon PDF probability distribution. Individual PDF replicas are shown as green lines, and the ensemble average, standard deviation and 68\\% confidence level are shown.}\n\\label{fig:mcerror}\n\\end{figure}\n\n%\n%In the NNPDF approach the method has another benefit. Considering again the method of cross-validation used to prevent overfitting in the neural network parametrisation. At first glance, it would seem that separating the data sets into a fitting and validation set means that not all of the data is directly used in a fit. However, as the selection of data points for each set is done at randomly on a replica, by replica basis, for a large enough $N_{\\mathrm{rep}}$ the whole data set is used for fitting across the ensemble. The use of an ensemble of replicas additionally helps with estimating the functional uncertainty present when determining a function from a finite set of data points. \n%\n%\\subsection{Sources of theoretical uncertainty}\n%\n%\n%Maybe have here a short summary of all the sources of theoretical uncertainty, and how they may be taken into account?\n%alphas, GM scheme, deuteron/nuclear corrections, fragmentation in prompt $\\gamma$, NP uncertainties e.g hadronisation/pileup.\n\n\\section{Status of PDF determination before the LHC}\nIn preparation for the application of parton distributions at the LHC, extensive studies were performed in order to benchmark and understand areas of agreement and discrepancy across fitting collaborations~\\cite{Watt:2011kp,Dittmar:2009ii}. While agreement had generally improved as the level of sophistication applied in parton fits increased, there were still notable regions where PDF fits from the widest datasets remained in disagreement at levels greater than their quoted uncertainties. Figure \\ref{fig:pdflumidiff} illustrates the situation for two important PDF luminosities before the LHC. These discrepancies extended not only to so far unmeasured quantities such as Higgs production cross sections, but also to PDF standard candle observables such as $W$ boson production (c.f. Figure \\ref{fig:standardcandleerror}).\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/ratiogglumi1_68cl.pdf}\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/ratioqqbarlumi1_68cl.pdf}\n\\caption[Luminosities for $gg$ and $q\\bar{q}$ PDF combinations at the $7$ TeV LHC]{Luminosities for $gg$ (left) and $q\\bar{q}$ (right) PDF combinations at the $7$ TeV LHC. Figure from~\\cite{Watt:2011kp}.}\n\\label{fig:pdflumidiff}\n\\end{figure}\n\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/ggH120GeVLHC7TeVnlo68cl.pdf}\n\\includegraphics[width=0.48\\textwidth]{3-PDFdet/figs/wpmLHC7TeVnlo68cl.pdf}\n\\caption[Predictions for example LHC processes based upon a number of PDF determinations]{Predictions for LHC processes based upon a number of PDF determinations. Left figure: cross section for Higgs production in gluon fusion. Right figure: cross section for the production of $W$ bosons. Figure from~\\cite{Watt:2011kp}.}\n\\label{fig:standardcandleerror}\n\\end{figure}\nThe Les Houches benchmark exercise~\\cite{Dittmar:2009ii} helped to elucidate the methodological source of many of these differences by testing fits from various methodologies to a standard dataset.\n\nMany of the observed discrepancies arise due to differences in the theoretical description of data, with the choice of flavour number scheme providing the largest differences. Dataset choice and methodological choices introducing significant differences also. These differences led to the conservative PDF4LHC recommendation for observables to be calculated as the central contour of the CTEQ-MSTW-NNPDF uncertainty envelope. Despite the differences, for the LHC Run-I the range of available sets allowed for experimental collaborations to effectively explore the differences in the resulting predictions.\n\nWhile providing accurate determinations for use at the LHC has been the primary concern in the years leading up to the LHC's first operation, there was substantial interest in the potential of the LHC to provide constraints upon PDFs and potentially provide discriminating power between sets. Data from the LHC provides the best opportunity for distinguishing the most effective approaches both theoretically and methodologically. Additionally LHC data provides particularly valuable input in the field of collider-only determinations, which aim to provide a cleaner description of data by avoiding the inclusion of nuclear-corrected and low energy data. The inclusion of a large LHC dataset into PDF fits is however a challenging problem, and one which has inspired a great deal of progress in the efficient calculation of collider observables. The remainder of this work will therefore deal with the both the technical inclusion of LHC data into parton distribution fits and the subsequent phenomenological results.\n\n", "meta": {"hexsha": "2631f34b7a0e527cbd543680ec7d0c05606d0545", "size": 62957, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "3-PDFdet/chapter3.tex", "max_stars_repo_name": "nhartland/thesis", "max_stars_repo_head_hexsha": "a58dc2b3d8b218b3893f4a8f0c1e6bcdcc545c55", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "3-PDFdet/chapter3.tex", "max_issues_repo_name": "nhartland/thesis", "max_issues_repo_head_hexsha": "a58dc2b3d8b218b3893f4a8f0c1e6bcdcc545c55", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "3-PDFdet/chapter3.tex", "max_forks_repo_name": "nhartland/thesis", "max_forks_repo_head_hexsha": "a58dc2b3d8b218b3893f4a8f0c1e6bcdcc545c55", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 178.3484419263, "max_line_length": 1377, "alphanum_fraction": 0.7912543482, "num_tokens": 15870, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3421608092704867}} {"text": "\\documentclass[11pt,oneside]{article}\t%use\"amsart\"insteadof\"article\"forAMSLaTeXformat\n\\usepackage{geometry}\t\t%Seegeometry.pdftolearnthelayoutoptions.Therearelots.\n\\geometry{letterpaper}\t\t%...ora4paperora5paperor...\n%\\geometry{landscape}\t\t%Activateforforrotatedpagegeometry\n%\\usepackage[parfill]{parskip}\t\t%Activatetobeginparagraphswithanemptylineratherthananindent\n\\usepackage{graphicx}\t\t\t\t%Usepdf,png,jpg,orepsßwithpdflatex;useepsinDVImode\n\t\t\t\t\t\t\t\t%TeXwillautomaticallyconverteps-->pdfinpdflatex\t\t\n\\usepackage{amssymb}\n\\usepackage[colorlinks]{hyperref}\n\n%----macros begin---------------------------------------------------------------\n\\usepackage{color}\n\\usepackage{amsthm}\n\n\\def\\conv{\\mbox{\\textrm{conv}\\,}}\n\\def\\aff{\\mbox{\\textrm{aff}\\,}}\n\\def\\E{\\mathbb{E}}\n\\def\\R{\\mathbb{R}}\n\\def\\Z{\\mathbb{Z}}\n\\def\\tex{\\TeX}\n\\def\\latex{\\LaTeX}\n\\def\\v#1{{\\bf #1}}\n\\def\\p#1{{\\bf #1}}\n\\def\\T#1{{\\bf #1}}\n\n\\def\\vet#1{{\\left(\\begin{array}{cccccccccccccccccccc}#1\\end{array}\\right)}}\n\\def\\mat#1{{\\left(\\begin{array}{cccccccccccccccccccc}#1\\end{array}\\right)}}\n\n\\def\\lin{\\mbox{\\rm lin}\\,}\n\\def\\aff{\\mbox{\\rm aff}\\,}\n\\def\\pos{\\mbox{\\rm pos}\\,}\n\\def\\cone{\\mbox{\\rm cone}\\,}\n\\def\\conv{\\mbox{\\rm conv}\\,}\n\\newcommand{\\homog}[0]{\\mbox{\\rm homog}\\,}\n\\newcommand{\\relint}[0]{\\mbox{\\rm relint}\\,}\n\n%----macros end-----------------------------------------------------------------\n\n\\title{Domain mapping with LAR\n\\footnote{This document is part of the \\emph{Linear Algebraic Representation with CoChains} (LAR-CC) framework~\\cite{cclar-proj:2013:00}. \\today}\n}\n\\author{Alberto Paoluzzi}\n%\\date{}\t\t\t\t\t\t\t%Activatetodisplayagivendateornodate\n\n\\begin{document}\n\\maketitle\n\\nonstopmode\n\n\\begin{abstract}\nIn this module a first implementation (no optimisations) is done of several \\texttt{LAR} operators, reproducing the behaviour of the plasm \\texttt{STRUCT} and \\texttt{MAP} primitives, but with better handling of the topology, including the stitching of decomposed (simplicial domains) about their possible sewing. A definition of specialised classes \\texttt{Model}, \\texttt{Mat} and \\texttt{Verts} is also contained in this module, together with the design and the implementation of the \\emph{traversal} algorithms for networks of structures.\n\\end{abstract}\n\n\\tableofcontents\n\n%===============================================================================\n\\section{Introduction}\n%===============================================================================\n\nThe \\texttt{mapper} module, introduced here, aims to provide the tools needed to apply both dimension-independent affine transformations and general simplicial maps to geometric objects and assemblies developed within the LAR scheme. \n\nFor this purpose, a simplicial decomposition of the $[0,1]^d$ hypercube ($d \\geq 1$) with any possible \\texttt{shape} is firstly given, followed by its scaled version with any according $\\texttt{size}\\in\\E^d$, being its position vector the mapped image of the point $\\mathbf{1}\\in\\E^d$. A general mapping mechanism is specified, to map any domain decomposition (either simplicial or not) with a given set of coordinate functions, providing a piecewise-linear approximation of any curved embedding of a $d$-dimensional domain in any $\\E^n$ space, with $n \\geq d$. \nA suitable function is also given to identify corresponding vertices when mapping a domain decomposition of the fundamental polygon (or polyhedron) of a closed manifold. \n\nThe geometric tools given in this chapter employ a normalised homogeneous representation of vertices of the represented shapes, where the added coordinate is the \\emph{last} of the ordered list of vertex coordinates. The homogeneous representation of vertices is used \\emph{implicitly}, by inserting the extra coordinate only when needed by the operation at hand, mainly for computing the product of the object's vertices times the matrix of an affine tensor. \n\nA set of primitive surface and solid shapes is also provided, via the mapping mechanism of a simplicial decomposition of a $d$-dimensional chart. A simplified version of the PLaSM specification of dimension-independent elementary affine transformation is given as well.\n\nThe second part of this module is dedicated to the development of a complete framework for the implementation of hierarchical assemblies of shapes and scene graphs, by using the simplest possible set of computing tools. In this case no hierarchical graphs or multigraph are employed, i.e.~no specialised data structures are produced. The ordered list model of hierarchical structures, inherited from PHIGS and PLaSM, is employed in this context. A recursive traversal is used to transform all the component parts of a hierarchical assembly into the reference frame of the first object of the assembly, i.e.~in world coordinates.\n\n\n%-------------------------------------------------------------------------------\n%===============================================================================\n\\section{Primitive objects}\n\\label{sec:generators}\n%===============================================================================\n\nA large number of primitive surfaces or solids is defined in this section, using the \\texttt{larMap} mechanism and the coordinate functions of a suitable chart.\n\n%-------------------------------------------------------------------------------\n\\subsection{1D primitives}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Circle}\n%-------------------------------------------------------------------------------\n@D Circle centered in the origin\n@{def larCircle(radius=1.,angle=2*PI,dim=1):\n\tdef larCircle0(shape=36):\n\t\tdomain = larIntervals([shape])([angle])\n\t\tV,CV = domain\n\t\tx = lambda p : radius*COS(p[0])\n\t\ty = lambda p : radius*SIN(p[0])\n\t\treturn larMap([x,y])(domain,dim)\n\treturn larCircle0\n@}\n%-------------------------------------------------------------------------------\n\\paragraph{Helix curve}\n%-------------------------------------------------------------------------------\n@D Helix curve about the $z$ axis\n@{def larHelix(radius=1.,pitch=1.,nturns=2,dim=1):\n\tdef larHelix0(shape=36*nturns):\n\t\tangle = nturns*2*PI\n\t\tdomain = larIntervals([shape])([angle])\n\t\tV,CV = domain\n\t\tx = lambda p : radius*COS(p[0])\n\t\ty = lambda p : radius*SIN(p[0])\n\t\tz = lambda p : (pitch/(2*PI)) * p[0]\n\t\treturn larMap([x,y,z])(domain,dim)\n\treturn larHelix0\n@}\n%-------------------------------------------------------------------------------\n%-------------------------------------------------------------------------------\n\\subsection{2D primitives}\n%-------------------------------------------------------------------------------\nSome useful 2D primitive objects either in $\\E^2$ or embedded in $\\E^3$ are defined here, including 2D disks and rings, as well as cylindrical, spherical and toroidal surfaces.\n\n\\paragraph{Disk surface}\n%-------------------------------------------------------------------------------\n@D Disk centered in the origin\n@{def larDisk(radius=1.,angle=2*PI):\n\tdef larDisk0(shape=[36,1]):\n\t\tdomain = larIntervals(shape)([angle,radius])\n\t\tV,CV = domain\n\t\tx = lambda p : p[1]*COS(p[0])\n\t\ty = lambda p : p[1]*SIN(p[0])\n\t\treturn larMap([x,y])(domain)\n\treturn larDisk0\n@}\n%-------------------------------------------------------------------------------\n\\paragraph{Helicoid surface}\n%-------------------------------------------------------------------------------\n@D Helicoid about the $z$ axis\n@{def larHelicoid(R=1.,r=0.5,pitch=1.,nturns=2,dim=1):\n\tdef larHelicoid0(shape=[36*nturns,2]):\n\t\tangle = nturns*2*PI\n\t\tdomain = larIntervals(shape,'simplex')([angle,R-r])\n\t\tV,CV = domain\n\t\tV = larTranslate([0,r,0])(V)\n\t\tdomain = V,CV\n\t\tx = lambda p : p[1]*COS(p[0])\n\t\ty = lambda p : p[1]*SIN(p[0])\n\t\tz = lambda p : (pitch/(2*PI)) * p[0]\n\t\treturn larMap([x,y,z])(domain,dim)\n\treturn larHelicoid0\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Ring surface}\n%-------------------------------------------------------------------------------\n@D Ring centered in the origin\n@{def larRing(r1,r2,angle=2*PI):\n\tdef larRing0(shape=[36,1]):\n\t\tV,CV = larIntervals(shape)([angle,r2-r1])\n\t\tV = larTranslate([0,r1])(V)\n\t\tdomain = V,CV\n\t\tx = lambda p : p[1] * COS(p[0])\n\t\ty = lambda p : p[1] * SIN(p[0])\n\t\treturn larMap([x,y])(domain)\n\treturn larRing0\n@}\n%-------------------------------------------------------------------------------\n\\paragraph{Cylinder surface}\n%-------------------------------------------------------------------------------\n@D Cylinder surface with $z$ axis\n@{from scipy.linalg import det\n\"\"\"\ndef makeOriented(model):\n\tV,CV = model\n\tout = []\n\tfor cell in CV: \n\t\tmat = scipy.array([V[v]+[1] for v in cell]+[[0,0,0,1]])\n\t\tif det(mat) < 0.0:\n\t\t\tout.append(cell)\n\t\telse:\n\t\t\tout.append([cell[1]]+[cell[0]]+cell[2:])\n\treturn V,out\n\"\"\"\ndef larCylinder(radius,height,angle=2*PI):\n\tdef larCylinder0(shape=[36,1]):\n\t\tdomain = larIntervals(shape)([angle,1])\n\t\tV,CV = domain\n\t\tx = lambda p : radius*COS(p[0])\n\t\ty = lambda p : radius*SIN(p[0])\n\t\tz = lambda p : height*p[1]\n\t\tmapping = [x,y,z]\n\t\tmodel = larMap(mapping)(domain)\n\t\t# model = makeOriented(model)\n\t\treturn model\n\treturn larCylinder0\n@}\n%-------------------------------------------------------------------------------\n\\paragraph{Spherical surface of given radius}\n%-------------------------------------------------------------------------------\n@D Spherical surface of given radius\n@{def larSphere(radius=1,angle1=PI,angle2=2*PI):\n\tdef larSphere0(shape=[18,36]):\n\t\tV,CV = larIntervals(shape,'simplex')([angle1,angle2])\n\t\tV = larTranslate([-angle1/2,-angle2/2])(V)\n\t\tdomain = V,CV\n\t\tx = lambda p : radius*COS(p[0])*COS(p[1])\n\t\ty = lambda p : radius*COS(p[0])*SIN(p[1])\n\t\tz = lambda p : radius*SIN(p[0])\n\t\treturn larMap([x,y,z])(domain)\n\treturn larSphere0\n@}\n%-------------------------------------------------------------------------------\n\\paragraph{Toroidal surface}\n%-------------------------------------------------------------------------------\n@D Toroidal surface of given radiuses\n@{def larToroidal(r,R,angle1=2*PI,angle2=2*PI):\n\tdef larToroidal0(shape=[24,36]):\n\t\tdomain = larIntervals(shape,'simplex')([angle1,angle2])\n\t\tV,CV = domain\n\t\tx = lambda p : (R + r*COS(p[0])) * COS(p[1])\n\t\ty = lambda p : (R + r*COS(p[0])) * SIN(p[1])\n\t\tz = lambda p : -r * SIN(p[0])\n\t\treturn larMap([x,y,z])(domain)\n\treturn larToroidal0\n@}\n%-------------------------------------------------------------------------------\n\\paragraph{Crown surface}\n%-------------------------------------------------------------------------------\n@D Half-toroidal surface of given radiuses\n@{def larCrown(r,R,angle=2*PI):\n\tdef larCrown0(shape=[24,36]):\n\t\tV,CV = larIntervals(shape,'simplex')([PI,angle])\n\t\tV = larTranslate([-PI/2,0])(V)\n\t\tdomain = V,CV\n\t\tx = lambda p : (R + r*COS(p[0])) * COS(p[1])\n\t\ty = lambda p : (R + r*COS(p[0])) * SIN(p[1])\n\t\tz = lambda p : -r * SIN(p[0])\n\t\treturn larMap([x,y,z])(domain)\n\treturn larCrown0\n@}\n%-------------------------------------------------------------------------------\n\n%-------------------------------------------------------------------------------\n\\subsection{3D primitives}\n%-------------------------------------------------------------------------------\n\n\n\\paragraph{Solid Box}\n%-------------------------------------------------------------------------------\n@D Solid box of given extreme vectors\n@{def larBox(minVect,maxVect):\n\tsize = VECTDIFF([maxVect,minVect])\n\tprint \"size =\",size\n\tbox = larApply(s(*size))(larCuboids([1]*len(size)))\n\tprint \"box =\",box\n\treturn larApply(t(*minVect))(box)\n@}\n%-------------------------------------------------------------------------------\n\n\n\\paragraph{Solid Ball}\n%-------------------------------------------------------------------------------\n@D Solid Sphere of given radius\n@{def larBall(radius=1,angle1=PI,angle2=2*PI):\n\tdef larBall0(shape=[18,36]):\n\t\tV,CV = checkModel(larSphere(radius,angle1,angle2)(shape))\n\t\treturn V,[range(len(V))]\n\treturn larBall0\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Solid cylinder}\n%-------------------------------------------------------------------------------\n@D Solid cylinder of given radius and height\n@{def larRod(radius,height,angle=2*PI):\n\tdef larRod0(shape=[36,1]):\n\t\tV,CV = checkModel(larCylinder(radius,height,angle)(shape))\n\t\treturn V,[range(len(V))]\n\treturn larRod0\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Hollow cylinder}\n%-------------------------------------------------------------------------------\n@D Hollow cylinder of given radiuses and height\n@{def larHollowCyl(r,R,height,angle=2*PI):\n\tdef larHollowCyl0(shape=[36,1,1]):\n\t\tV,CV = larIntervals(shape)([angle,R-r,height])\n\t\tV = larTranslate([0,r,0])(V)\n\t\tdomain = V,CV\n\t\tx = lambda p : p[1] * COS(p[0])\n\t\ty = lambda p : p[1] * SIN(p[0])\n\t\tz = lambda p : p[2] * height\n\t\treturn larMap([x,y,z])(domain)\n\treturn larHollowCyl0\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Hollow sphere}\n%-------------------------------------------------------------------------------\n@D Hollow sphere of given radiuses\n@{def larHollowSphere(r,R,angle1=PI,angle2=2*PI):\n\tdef larHollowSphere0(shape=[36,1,1]):\n\t\tV,CV = larIntervals(shape)([angle1,angle2,R-r])\n\t\tV = larTranslate([-angle1/2,-angle2/2,r])(V)\n\t\tdomain = V,CV\n\t\tx = lambda p : p[2]*COS(p[0])*COS(p[1])\n\t\ty = lambda p : p[2]*COS(p[0])*SIN(p[1])\n\t\tz = lambda p : p[2]*SIN(p[0])\n\t\treturn larMap([x,y,z])(domain)\n\treturn larHollowSphere0\n@}\n%-------------------------------------------------------------------------------\n\n\n\\paragraph{Solid torus}\n%-------------------------------------------------------------------------------\n@D Solid torus of given radiuses\n@{def larTorus(r,R,angle1=2*PI,angle2=2*PI):\n\tdef larTorus0(shape=[24,36,1]):\n\t\tdomain = larIntervals(shape)([angle1,angle2,r])\n\t\tV,CV = domain\n\t\tx = lambda p : (R + p[2]*COS(p[0])) * COS(p[1])\n\t\ty = lambda p : (R + p[2]*COS(p[0])) * SIN(p[1])\n\t\tz = lambda p : -p[2] * SIN(p[0])\n\t\treturn larMap([x,y,z])(domain)\n\treturn larTorus0\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Solid pizza}\n%-------------------------------------------------------------------------------\n@D Solid pizza of given radiuses\n@{def larPizza(r,R,angle=2*PI):\n\tassert angle <= PI\n\tdef larPizza0(shape=[24,36]):\n\t\tV,CV = checkModel(larCrown(r,R,angle)(shape))\n\t\tV += [[0,0,-r],[0,0,r]]\n\t\treturn V,[range(len(V))]\n\treturn larPizza0\n@}\n%-------------------------------------------------------------------------------\n\n%===============================================================================\n\\section{Computational framework}\n%===============================================================================\n\\subsection{Exporting the library}\n%-------------------------------------------------------------------------------\n@O larlib/larlib/mapper.py\n@{\"\"\" Mapping functions and primitive objects \"\"\"\nfrom larlib import *\n\n@< Basic tests of mapper module @>\n@< Circle centered in the origin @>\n@< Helix curve about the $z$ axis @>\n@< Disk centered in the origin @>\n@< Helicoid about the $z$ axis @>\n@< Ring centered in the origin @>\n@< Spherical surface of given radius @>\n@< Cylinder surface with $z$ axis @>\n@< Toroidal surface of given radiuses @>\n@< Half-toroidal surface of given radiuses @>\n@< Solid box of given extreme vectors @>\n@< Solid Sphere of given radius @>\n@< Solid helicoid about the $z$ axis @>\n@< Solid cylinder of given radius and height @>\n@< Solid torus of given radiuses @>\n@< Solid pizza of given radiuses @>\n@< Hollow cylinder of given radiuses and height @>\n@< Hollow sphere of given radiuses @>\n@< Symbolic utility to represent points as strings @>\n@< Remove the unused vertices from a LAR model pair @>\n@}\n%-------------------------------------------------------------------------------\n%===============================================================================\n\\subsection{Examples}\n%===============================================================================\n\n\\paragraph{3D rotation about a general axis}\nThe approach used by \\texttt{lar-cc} to specify a general 3D rotation is shown in the following example,\nby passing the rotation function \\texttt{r} the components \\texttt{a,b,c} of the unit vector \\texttt{axis} scaled by the rotation \\texttt{angle}. \n\n%-------------------------------------------------------------------------------\n@O test/py/mapper/test02.py\n@{\"\"\" General 3D rotation of a toroidal surface \"\"\"\nfrom larlib import *\n\nmodel = checkModel(larToroidal([0.2,1])())\nangle = PI/2; axis = UNITVECT([1,1,0])\na,b,c = SCALARVECTPROD([ angle, axis ])\nmodel = larApply(r(a,b,c))(model)\nVIEW(STRUCT(MKPOLS(model)))\n@}\n%-------------------------------------------------------------------------------\n\n\n\\paragraph{3D elementary rotation of a 2D circle}\nA simpler specification is needed when the 3D rotation is about a coordinate axis. In this case the rotation angle can be directly given as the unique non-zero parameter of the the rotation function \\texttt{r}. The rotation axis (in this case the $x$ one) is specified by the non-zero (angle) position.\n\n%-------------------------------------------------------------------------------\n@O test/py/mapper/test03.py\n@{\"\"\" Elementary 3D rotation of a 2D circle \"\"\"\nfrom larlib import *\n\nmodel = checkModel(larCircle(1)(),1)\nmodel = larEmbed(1)(model)\nmodel = larApply(r(PI/2,0,0))(model)\nVIEW(STRUCT(MKPOLS(model)))\n@}\n%-------------------------------------------------------------------------------\n\n\n\n\n%===============================================================================\n\\subsection{Tests about domain}\n%===============================================================================\n\n\\paragraph{Mapping domains}\nThe generations of mapping domains of different dimension (1D, 2D, 3D) is shown below.\n\t\n%-------------------------------------------------------------------------------\n@D Basic tests of mapper module\n@{\"\"\" Basic tests of mapper module \"\"\"\nfrom larlib import *\n\nif __name__==\"__main__\":\n\tV,EV = larDomain([5])\n\tVIEW(EXPLODE(1.5,1.5,1.5)(MKPOLS((V,EV))))\n\tV,EV = larIntervals([24])([2*PI])\n\tVIEW(EXPLODE(1.5,1.5,1.5)(MKPOLS((V,EV))))\n\t\t\n\tV,FV = larDomain([5,3])\n\tVIEW(EXPLODE(1.5,1.5,1.5)(MKPOLS((V,FV))))\n\tV,FV = larIntervals([36,3])([2*PI,1.])\n\tVIEW(EXPLODE(1.5,1.5,1.5)(MKPOLS((V,FV))))\n\t\t\n\tV,CV = larDomain([5,3,1])\n\tVIEW(EXPLODE(1.5,1.5,1.5)(MKPOLS((V,CV))))\n\tV,CV = larIntervals([36,2,3])([2*PI,1.,1.])\n\tVIEW(EXPLODE(1.5,1.5,1.5)(MKPOLS((V,CV))))\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Testing some primitive object generators}\nThe various model generators given in Section~\\ref{sec:generators} are tested here, including LAR 2D circle, disk, and ring, as well as the 3D cylinder, sphere, and toroidal surfaces, and the solid objects ball, rod, crown, pizza, and torus.\n\n%-------------------------------------------------------------------------------\n@O test/py/mapper/test01.py\n@{\"\"\" Testing some primitive object generators \"\"\"\nfrom larlib import *\n\nmodel = larCircle(1)()\nVIEW(EXPLODE(1.2,1.2,1.2)(MKPOLS(model)))\nmodel = larHelix(1,0.5,4)()\nVIEW(EXPLODE(1.2,1.2,1.2)(MKPOLS(model)))\nmodel = larDisk(1)([36,4])\nVIEW(EXPLODE(1.2,1.2,1.2)(MKPOLS(model)))\nmodel = larHelicoid(1,0.5,0.1,10)()\nVIEW(EXPLODE(1.2,1.2,1.2)(MKPOLS(model)))\nmodel = larRing(.9, 1.)([36,2])\nVIEW(EXPLODE(1.2,1.2,1.2)(MKPOLS(model)))\nmodel = larCylinder(.5,2.)([32,1])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larSphere(1,PI/6,PI/4)([6,12])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larBall(1)()\nVIEW(EXPLODE(1.2,1.2,1.2)(MKPOLS(model)))\nmodel = larSolidHelicoid(0.2,1,0.5,0.5,10)()\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larRod(.25,2.)([32,1])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larToroidal(0.5,2)()\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larCrown(0.125,1)([8,48])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larPizza(0.05,1,PI/3)([8,48])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larTorus(0.5,1)()\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larBox([-1,-1,-1],[1,1,1])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larHollowCyl(0.8,1,1,angle=PI/4)([12,2,2])\nVIEW(STRUCT(MKPOLS(model)))\nmodel = larHollowSphere(0.8,1,PI/6,PI/4)([6,12,2])\nVIEW(STRUCT(MKPOLS(model)))\n@}\n%-------------------------------------------------------------------------------\n\n\n\\subsection{Volumetric utilities}\n\n\n\\paragraph{Limits of a LAR Model}\n%-------------------------------------------------------------------------------\n@D Model limits\n@{def larLimits (model):\n\tif isinstance(model,tuple): \n\t\tV,CV = model\n\t\tverts = scipy.asarray(V)\n\telse: verts = model.verts\n\treturn scipy.amin(verts,axis=0).tolist(), scipy.amax(verts,axis=0).tolist()\n\t\nassert larLimits(larSphere()()) == ([-1.0, -1.0, -1.0], [1.0, 1.0, 1.0])\n@}\n%-------------------------------------------------------------------------------\n\n\\paragraph{Alignment}\n%-------------------------------------------------------------------------------\n@D Alignment primitive\n@{def larAlign (args):\n\tdef larAlign0 (args,pols):\n\t\tpol1, pol2 = pols\n\t\tbox1, box2 = (larLimits(pol1), larLimits(pol2))\n\t\tprint \"box1, box2 =\",(box1, box2)\n\t\t\n\treturn larAlign0\n@}\n%-------------------------------------------------------------------------------\n\n\n\\bibliographystyle{amsalpha}\n\\bibliography{mapper}\n\n\\end{document}", "meta": {"hexsha": "fa2aa9e8eb131924515e84e34f7db834437bdf76", "size": 21308, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/tex/mapper.tex", "max_stars_repo_name": "cvdlab/lar-cc", "max_stars_repo_head_hexsha": "7092965acf7c0c78a5fab4348cf2c2aa01c4b130", "max_stars_repo_licenses": ["MIT", "Unlicense"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2016-09-20T04:48:12.000Z", "max_stars_repo_stars_event_max_datetime": "2016-09-20T04:48:12.000Z", "max_issues_repo_path": "src/tex/mapper.tex", "max_issues_repo_name": "Ahdhn/lar-cc", "max_issues_repo_head_hexsha": "7092965acf7c0c78a5fab4348cf2c2aa01c4b130", "max_issues_repo_licenses": ["MIT", "Unlicense"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-02-20T21:57:07.000Z", "max_issues_repo_issues_event_max_datetime": "2018-02-21T07:18:11.000Z", "max_forks_repo_path": "src/tex/mapper.tex", "max_forks_repo_name": "Ahdhn/lar-cc", "max_forks_repo_head_hexsha": "7092965acf7c0c78a5fab4348cf2c2aa01c4b130", "max_forks_repo_licenses": ["MIT", "Unlicense"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2016-11-04T10:47:42.000Z", "max_forks_repo_forks_event_max_datetime": "2018-04-10T17:32:50.000Z", "avg_line_length": 41.1351351351, "max_line_length": 628, "alphanum_fraction": 0.5301295288, "num_tokens": 5516, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5964331319177487, "lm_q2_score": 0.5736784074525096, "lm_q1q2_score": 0.3421608092704867}} {"text": "\\lstset{\n captionpos=b,\n language=Haskell,\n basicstyle=\\scriptsize,\n numbers=left,\n numberstyle=\\tiny,\n columns=fullflexible,\n stepnumber=1,\n escapechar=\\#,\n keepspaces=true,\n belowskip=-10pt,\n literate={<}{{$\\langle$}}1 {>}{{$\\rangle$}}1,\n morekeywords={function,rr,int,float,bool,isnull,partition,as,downregion,upregion,reads,writes,rdwrs,reduces,read,write,reduce,using,unpack,pack,coloring,multicoloring,color,newcolor,atomic,simultaneous},\n deletekeywords={float,head,min,max}\n}\n\\begin{lstlisting}[float={t},label={lst:circuit_ex},caption={Circuit Simulation}]\n-- \ntype CircuitNode = \n-- < owned node, owned or ghost node, resistance, current>\ntype CircuitWire = \n\ntype node_list = < CircuitNode@rn, node_list@rl >\ntype wire_list= < CircuitWire@rw, wire_list@rl >\n\ntype CircuitPiece = rr[rpw,rpn,rg]\n < wire_list@rl, node_list@rl > \n where rpn #$\\le$# rn and rg #$\\le$# rn and rpw #$\\le$# rw and\n rn * rw and rl * rn and rl * rw\n\n-- Simulation initialization and invocation\nfunction simulate_circuit[rl,rw,rn] ( all_nodes : node_list@rl, \n all_wires : wire_list@rl, steps : int ), \n reads(rn,rw,rl), writes(rn,rw,rl) : bool = \n let pc : \n = color_circuit[rn,rw,rl](all_nodes,all_wires) in\n -- Disjoint partition for the owned nodes of each piece\n partition rn using pc.1 as rn0,rn1 in\n -- Aliased partition for ghost nodes of each piece\n partition rn using pc.2 as rg0,rg1 in\n -- Disjoint partition for the owned wires of each piece\n partition rw using pc.3 as rw0,rw1 in\n let lists0 : @rl,node_list@rl> = \n build_lists[rl,rw,rn,rw0,rn0,rg0](all_nodes,all_wires,pc.1,pc.2,pc.3,0) in\n let piece0 : CircuitPiece = \n pack lists0 as CircuitPiece[rw0,rn0,rg0] in\n let lists1 : @rl,node_list@rl> =\n build_lists[rl,rw,rn,rw1,rn1,rg1](all_nodes,all_wires,pc.1,pc.2,pc.3,1) in\n let piece1 : CircuitPiece = \n pack lists1 as CircuitPiece[rw1,rn1,rg1] in\n execute_time_steps[rl,rw,rn](piece0,piece1,steps)\n\n-- Time Step Loop\nfunction execute_time_steps[rl,rw,rn] ( p0 : CircuitPiece, \n p1 : CircuitPiece, steps : int ) , reads(rn,rw,rl), writes(rn,rw) : bool = \n if steps #$<$# 1 then true else\n unpack p0 as piece0 : CircuitPiece[rw0,rn0,rg0] in \n unpack p1 as piece1 : CircuitPiece[rw1,rn1,rg1] in\n let _ : bool = calc_new_currents[rl,rw0,rn0,rg0](piece0.1) in\n let _ : bool = calc_new_currents[rl,rw1,rn1,rg1](piece1.1) in\n let _ : bool = distribute_charge[rl,rw0,rn0,rg0](piece0.1) in\n let _ : bool = distribute_charge[rl,rw1,rn1,rg1](piece1.1) in\n let _ : bool = update_voltage[rl,rn0](piece0.2) in\n let _ : bool = update_voltage[rl,rn1](piece1.2) in\n execute_time_steps[rl,rw,rn](p0,p1,steps-1)\n\nfunction color_circuit[rn,rw,rl] ( all_nodes : node_list@rl, \n all_wires : wire_list@rl ), \n reads(rn,rw,rl) : = \n -- Invoke programmer chosen coloring algorithm (e.g. METIS)\n -- return owned, ghost, wire colorings\n\n-- Helper method\nfunction build_lists[rl,rw,rn,rpw,rpn,rg] ( nodes : node_list@rl, \n wires : wire_list@rl, oc : coloring(rn), gc : multicoloring(rn), \n wc : coloring(rw), c : int), reads(rn,rw,rl), writes(rl) \n : < wire_list@rl, node_list@rl > = \n -- Construct lists of node and wire pointers for the given colorings\n\\end{lstlisting}\n\n\\section{Circuit Example}\n\\label{sec:example}\n\nListing~\\ref{lst:circuit_ex} shows a circuit simulation written\nin the core Legion language (see\nSection~\\ref{sec:legioncore}). The input is a graph of circuit\nelements and connecting wires. To perform parallel\nsimulation the graph is partitioned into pieces, after which the\nsimulation is run for a number of time steps. Each time step has\nthree phases: calculation of each wire's current, distribution of charge to\nthe nodes connected to each wire, and updating of node voltages.\n\nThe entry function is {\\tt simulate\\_circuit} (line 15), which accepts a \nlinked list of nodes contained in region {\\tt rn} and a linked list of \nwires contained in region {\\tt rw}. The linked list elements themselves\nare in region {\\tt rl}.\nWhen {\\tt simulate\\_circuit} is invoked it must have both read and\nwrite privileges for the regions {\\tt rl}, {\\tt rw}, and {\\tt rn}\n(line 17); the type system statically verifies that\n{\\tt simulate\\_circuit} only accesses these regions and their subregions.\n\nPartitioning of the graph is done using {\\em colorings} built in the\n{\\tt color\\_circuit} function (line 18-19). A coloring maps region\nelements to colors (integers in Core Legion), with each color corresponding\n to a new subregion\nto be created. Note that it is impossible to pick a single static\npartitioning scheme that would do well for all graphs. Dynamic\npartitioning using colorings enables the programmer \nto make partitioning decisions based on input data.\n\nThe {\\tt color\\_circuit} function is responsible for specifying how the\ncircuit is to be partitioned. The body of this function is not shown as the\nideal algorithm for partitioning data will be application specific. The \nonly restriction imposed by the Legion programming model is that the result of\nthe algorithm be provided in the form of colorings. For our circuit example,\nthere are three colorings. The first maps each node to the graph piece that owns it.\nThe third maps each wire to a graph piece that\nowns one of its nodes. The second coloring captures, for each piece $p$,\nthe nodes in other pieces on the boundary of $p$, often referred to as\n{\\em ghost nodes}. \n%We describe how ghost node regions are used in\n%conjunction with the owned nodes shortly. \nNote that \nghost nodes may be on the boundary of multiple pieces and therefore be included in\nmultiple ghost regions. For conciseness, a bit of syntactic sugar is used here the\nform of a {\\em multicoloring}, which allows the mapping of an element to multiple \ncolors. Although multicolorings are not part of the Core Legion described in \nSection~\\ref{sec:legioncore}, they can be easily implemented using a separate \ncoloring for each color in the multicoloring.\n\nAfter creating the colorings, the application partitions the node and wire\nregions into {\\em subregions} according to the colorings (lines 21-25). For\nsimplicity, our example only partitions the circuit into two pieces.\nLine 21 uses coloring {\\tt pc.1} to partition {\\tt rn} into {\\tt rn0}, and {\\tt rn1}.\nSubregion {\\tt rn0} will contain all locations with a color of 0 in {\\tt pc.1},\nwhile {\\tt rn1} will contain all locations with a color of 1. This\npartitioning results in {\\em disjoint} subpartitions.\nConstraints are introduced into the static environment describing both the\ndisjointness of the subregions (i.e. {\\tt rn0 $*$ rn1}) and the inclusion of\neach subregion in the original region (e.g. {\\tt rn0 $\\leq$ rn}). Line\n23 is similar, but uses a multicoloring, so the subregions may be {\\em aliased},\nand only the inclusion constraints are introduced into the static environment.\n\nThe circuit simulation uses an {\\em allocate-then-partition} style of\ncomputation, where a large data structure is first allocated and then\npartitioned. Legion also supports a {\\em partition-then-allocate} style,\nwhere empty regions are first partitioned and later populated with\ndata. Both are useful; e.g., Sequoia \\cite{Fatahalian06} and DPJ\n\\cite{Bocchino09} support static forms of allocate-then-partition and\npartition-then-allocate, respectively. Legion supports both\napproaches dynamically.\n\nAfter partitioning the circuit into pieces, the application creates\ninstances of {\\tt CircuitPiece} (defined on lines 9-12) for each piece\n(lines 28-33). {\\tt CircuitPiece} is a {\\em region relationship}, a\nbounded existential type. Region relationships allow the programmer\nto {\\em pack} a group of regions and pointers into those regions together\nand remember properties\nabout them such as disjointness and subregion relationships (lines 29\nand 33). The type system verifies the properties hold statically when\npacking; when region relationships are {\\em unpacked} the same\nproperties are reintroduced into the static checking environment\n(lines 40-41). A key feature underlying the soundness of the Legion\ntype system is that privileges cannot be packed in a region\nrelationship---privileges belong to functions. When a function unpacks\na region relationship it must already hold privileges for the unpacked regions. \n%new regions it finds in the packed value. \nFor example, on line 40 the\nfunction {\\tt execute\\_time\\_steps} has read/write privileges for\nregion {\\tt rn0} because it has read/write privileges for {\\tt rn} and\nthe {\\tt CircuitPiece} region relationship constraints ensure $\\tt rn0\n\\leq rn$ (line 12).\n\nThe {\\tt execute\\_time\\_steps} function (lines 37-48) \nalso illustrates the importance of using different partitions to give\nmultiple views onto the same logical region. Both {\\tt\n calc\\_new\\_currents} and {\\tt distribute\\_charge} \nuse the owned and ghost regions of a piece, which are from different partitions. In\n{\\tt calc\\_new\\_currents} these regions only need read\nprivileges, allowing both instances of {\\tt calc\\_new\\_currents} to be\nrun in parallel. In {\\tt distribute\\_charge} the\nprivilege is for a reduction which can also be done in parallel\nbecause of the atomic and commutative nature of reductions. Finally,\nthe {\\tt update\\_voltage} function modifies only the owned regions, permitting\neach piece to be updated in parallel. No\nsingle partition of the nodes\ndescribes these data sharing patterns.\n\n%This function also demonstrates the benefit of being able to dynamically\n%(re-)discover disjointness. Safely running both instances of \n%{\\tt calc\\_new\\_currents} in parallel depends on knowing that {\\tt rw0} and\n%{\\tt rw1} in {\\tt execute\\_time\\_steps} are disjoint. Similarly, parallel\n%execution of the instances of {\\tt update\\_voltage} requires knowing that\n%{\\tt rn0} is disjoint from {\\tt rn1}. In both cases, this knowledge was\n%statically available in {\\tt simulate\\_circuit} and could have been captured\n%in a region relationship, but it would have resulted in much more complicated\n%code.\n%Instead, an inexpensive dynamic disjointness check is performed by the Legion\n%runtime, and the safety of parallel execution follows from the soundness of\n%the type system.\nThe {\\tt execute\\_time\\_steps} function also demonstrates the benefit of dynamically\nbeing able to discover disjointness. To run the two instances of \n{\\tt calc\\_new\\_currents} in parallel requires knowing that {\\tt rw0}\nand {\\tt rw1} are disjoint. There is a similar requirement for parallel\nexecution of the two {\\tt update\\_voltage} calls with {\\tt rn0} and {\\tt rn1}. In\nboth cases, this knowledge was statically available in {\\tt simulate\\_circuit}\nand could have been captured in a region relationship at the cost\nof much more complicated code. Instead, an inexpensive dynamic disjointness\ncheck by the Legion runtime will (re)discover the parallelism.\n\nListing~\\ref{lst:circuit_leaf} shows the leaf functions for each phase.\nEach function iterates over the list of wires or\nnodes for its piece. Each function specifies the region privileges it\nrequires (lines 2,14,23). \n%These privileges\n%are statically checked to match the operations performed inside of\n%each function (e.g. read and write). \nIn the case of {\\em reduce} the\nprivilege must also specify which reduction function is used\n(line 14).\n\nIn addition to privileges, functions can also specify {\\em coherence}\non regions. Coherence specifies what updates the function may\nobserve from other functions using aliased regions. \nIf not otherwise specified, coherence defaults to {\\em exclusive}, \nmeaning the function must appear to execute in program order relative to\nother function calls. \nLine 14 in Listing~\\ref{lst:circuit_leaf} shows\nan example of {\\tt atomic}, a relaxed coherence mode requiring\nthat operations to {\\tt rn} and {\\tt rg} appear atomic relative\nto other functions using regions which may alias. The most relaxed coherence\nmode is {\\tt simult}; simultaneous coherence allows concurrent access \nto the region by all functions that are using the region\nin a simultaneous mode. The interaction between {\\em tasks} (which are functions chosen to execute in parallel) using the same\nregion with different coherence\nmodes is formalized in Section~\\ref{sec:coherence}.\n\n% This is a description of how the listings should be formatted.\n% It can go anywhere before the listings.\n\\lstset{\n captionpos=b,\n language=Haskell,\n basicstyle=\\scriptsize,\n numbers=left,\n numberstyle=\\tiny,\n columns=fullflexible,\n stepnumber=1,\n escapechar=\\#,\n keepspaces=true,\n belowskip=-10pt,\n literate={<}{{$\\langle$}}1 {>}{{$\\rangle$}}1,\n morekeywords={function,rr,int,float,bool,isnull,partition,as,downregion,upregion,reads,writes,rdwrs,reduces,read,write,reduce,using,unpack,pack,coloring,multicoloring,color,newcolor,atomic,simultaneous},\n deletekeywords={float,head,min,max}\n}\n\n\\begin{lstlisting}[float={t},label={lst:circuit_leaf},caption={Circuit Leaf Functions}]\nfunction calc_new_currents[rl,rw,rn,rg] ( ptr_list : wire_list@rl ), \n reads(rl,rw,rn,rg), writes(rw) : bool =\n if isnull(ptr_list) then true else\n let wire_node : wire_list = read(ptr_list) in\n let wire : CircuitWire = read(wire_node.1) in\n let in_node : CircuitNode = read(wire.1) in\n let out_node : CircuitNode = read(wire.2) in\n let current : float = (in_node.1 - out_node.1) / wire.3 in \n let new_wire : CircuitWire = in\n let _ : CircuitWire@rw = write(wire_node.1, new_wire) in\n calc_new_currents[rl,rw,rn,rg](wire_node.2)\n\nfunction distribute_charge[rl,rw,rn,rg] ( ptr_list : wire_list@rl ), \n reads(rl,rw,rn), reduces(reduce_charge,rn,rg), atomic(rn,rg) : bool =\n if isnull(ptr_list) then true else\n let wire_node : wire_list = read(ptr_list) in\n let wire : CircuitWire = read(wire_node.1) in\n let _ : CircuitNode@rn = reduce(reduce_charge, wire.1, wire.4) in\n let _ : CircuitNode@(rn,rg) = reduce(reduce_charge, wire.2, wire.4) in\n distribute_charge[rl,rw,rn,rg](wire_node.2)\n\nfunction update_voltage[rl,rn] ( ptr_list : node_list@rl ), \n reads(rl,rn), writes(rn) : bool = \n if isnull(ptr_list) then true else\n let node_node : node_list = read(ptr_list) in\n let node : CircuitNode = read(node_node.1) in\n let voltage : float = (node.3/node.4) in\n let new_node : CircuitNode = in\n let _ : CircuitNode@rn = write(node_node.1, new_node) in\n update_voltage[rl,rn](node_node.2)\n\n-- Reduction function for distribute charge\nfunction reduce_charge ( node : CircuitNode, current : float ) : CircuitNode =\n let new_charge : float = node.3 + current in\n < node.1,new_charge,node.3,node.4>\n\\end{lstlisting}\n\n\n", "meta": {"hexsha": "181681ceed5cc475470d3d4cdf55c102fbef4f1e", "size": 15535, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/types-pldi2013/example.tex", "max_stars_repo_name": "lightsighter/LegionOrigins", "max_stars_repo_head_hexsha": "0180bb3a8ee6efd0d2efdb743f75d3fba86f18f7", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 2, "max_stars_repo_stars_event_min_datetime": "2021-11-10T06:29:39.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-14T20:56:13.000Z", "max_issues_repo_path": "doc/types-pldi2013/example.tex", "max_issues_repo_name": "lightsighter/LegionOrigins", "max_issues_repo_head_hexsha": "0180bb3a8ee6efd0d2efdb743f75d3fba86f18f7", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/types-pldi2013/example.tex", "max_forks_repo_name": "lightsighter/LegionOrigins", "max_forks_repo_head_hexsha": "0180bb3a8ee6efd0d2efdb743f75d3fba86f18f7", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 52.8401360544, "max_line_length": 205, "alphanum_fraction": 0.7401351786, "num_tokens": 4314, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.658417500561683, "lm_q2_score": 0.5195213219520929, "lm_q1q2_score": 0.3420619302881984}} {"text": "\\chapter*{Abstract}\n\nMaximising the economic effectiveness of a wind farm is essential in making wind a more economic source of energy. This effectiveness can be increased through the reduction of operation and maintenance costs, which can be achieved through continuously monitoring the condition of wind turbines. An alternative to expensive condition monitoring systems, which can be uneconomical especially for older wind turbines, is to implement classification algorithms on supervisory control and data acquisition (SCADA) signals, which are collected in most wind turbines. Several publications were reviewed, which were all found to use separate algorithms to predict specific faults in advance. In reality, wind turbines tend to have multiple faults which may happen simultaneously and have correlations with one another. This project focusses on developing a methodology to predict multiple wind turbine faults in advance simultaneously by implementing classification algorithms on SCADA signals for a wind farm with 25 turbines rated at 2,500 kW, spanning a period of 30 months. The data, which included measurements of wind speed, active power and pitch angle, was labelled using corresponding downtime data to detect normal behaviour, faults and varying timescales before a fault occurs. Three different classification algorithms, namely decision trees, random forests and k nearest neighbours were tested using imbalanced and balanced training data, initially to optimise a number of hyperparameters. The random forest classifier produced the best results. Upon conducting a more detailed analysis on the performance of specific faults, it was found that the classifier was unable to detect the varying timescales before a fault with accuracy comparable to that of normal or faulty behaviour. This could have been due to the SCADA data, which are used as features, being unsuitable for detecting the faults, and there is potential to improve this by balancing only these classes.\n\\\\[.5cm]\n\\noindent\\textbf{\\textit{Keywords:}} \\keywords\n", "meta": {"hexsha": "bfb8a902622f582374fd84e422ad3694c9022d16", "size": 2051, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/abstract.tex", "max_stars_repo_name": "nmstreethran/WindTurbineClassification", "max_stars_repo_head_hexsha": "b0ea6de909ccd5bb425cee291ca3c252c11df4eb", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 34, "max_stars_repo_stars_event_min_datetime": "2019-03-01T21:24:40.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-20T09:59:17.000Z", "max_issues_repo_path": "docs/abstract.tex", "max_issues_repo_name": "nmstreethran/WindTurbineClassification", "max_issues_repo_head_hexsha": "b0ea6de909ccd5bb425cee291ca3c252c11df4eb", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/abstract.tex", "max_forks_repo_name": "nmstreethran/WindTurbineClassification", "max_forks_repo_head_hexsha": "b0ea6de909ccd5bb425cee291ca3c252c11df4eb", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 8, "max_forks_repo_forks_event_min_datetime": "2019-03-01T21:24:46.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-26T15:04:14.000Z", "avg_line_length": 341.8333333333, "max_line_length": 1973, "alphanum_fraction": 0.8298391029, "num_tokens": 373, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6001883735630721, "lm_q2_score": 0.5698526514141571, "lm_q1q2_score": 0.3420189360228672}} {"text": "\\chapter{parelm}\n\n\\section{Introduction}\n\nAn LPE may have parameters that do not affect the behavior of that process in any way.\nThese parameters are said to be \\emph{inert}.\nRemoving inert parameters from an LPE reduces the size of state vectors and the state space of that LPE, which both benefits performance.\n\n\\section{Algorithm}\n\nThe algorithm consists of the following steps \\cite{groote2001computer}:\n\n\\begin{enumerate}\n\n\\item Make it so that all parameters of the LPE are \\labeledas{inert}.\n\n\\item Consider the guards of all summands of the LPE, and \\removelabelfrom{inert} from all LPE parameters that occur in the guard of a summand.\n\n\\item Let $X$ be the set of all LPE parameters that are \\labeledas{inert}, and consider all expressions $v_s(p)$ that define the value of an LPE parameter $p$ after application of a summand $s$.\nFor each such expression, if $p$ is \\emph{not} \\labeledas{inert}, \\removelabelfrom{inert} all LPE parameters that occur in $v_s(p)$.\n\n\\item Repeat the previous step until the fixpoint of $X$ is reached.\nThen remove all LPE parameters that are still \\labeledas{inert} from the LPE, substituting references to those parameters by their initial values.\n\n\\end{enumerate}\n\n\\clearpage\n\\section{Example}\n\nConsider the following LPE:\n\n\\begin{lstlisting}\n//Process definition:\nPROCDEF example[A :: Int, B](x, y, z :: Int)\n = A ? i [[x==0]] >-> example[A, B](i, y, z)\n + A ? i [[x==1]] >-> example[A, B](0, i, z)\n + B [[x==2]] >-> example[A, B](0, y, z)\n + B [[true]] >-> example[A, B](z, y, x)\n ;\n\n//Initialization:\nexample[A, B](0, 0, 0);\n\\end{lstlisting}\n\nFirst, the `inert' label is removed from $x$ because $x$ occurs in the guards of the first three summands.\n\nIn the fourth summand, $z$ is used in the expression of which the value is assigned to $x$.\nTherefore $z$ is also no longer \\labeledas{inert}.\n\n$y$ remains labeled with `inert': it does not occur in a guard, nor is it used in the assignment to a process parameter other than itself.\nRemoving $y$ gives\n\n\\begin{lstlisting}\n//Process definition:\nPROCDEF example[A :: Int, B](x, z :: Int)\n = A ? i [[x==0]] >-> example[A, B](i, z)\n + A ? i [[x==1]] >-> example[A, B](0, z)\n + B [[x==2]] >-> example[A, B](0, z)\n + B [[true]] >-> example[A, B](z, x)\n ;\n\n//Initialization:\nexample[A, B](0, 0);\n\\end{lstlisting}\n\n\\section{Benchmark results}\n\nThe following durations were measured with a benchmark for several models:\n\\begin{itemize}\n\\item The average duration of \\txs{} to make 500 steps in a model after it has been converted to LPE form;\n\\item The average duration of \\txs{} to make 500 steps in a model after it has been converted to LPE form and after the \\texttt{parelm} operation has been applied.\n\\end{itemize}\n\nWhen plotting the second series of measurements against the first (see Figure~\\ref{parelm-vs-lpe-only:fig}), it is easy to see that the impact is insignificant in most cases.\nThe only model for which a significant performance increase has been measured is the \\texttt{ControlLoop} model, which has indeed lost part of its state vector.\n\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=0.5\\linewidth]{charts/parelm-vs-lpe-only}\n\\caption{Benchmark results: parelm vs LPE transformation}\n\\label{parelm-vs-lpe-only:fig}\n\\end{center}\n\\end{figure}\n\n", "meta": {"hexsha": "6a3cc8f13509772064758cc549217a733963cd48", "size": 3273, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sys/lpeops/tex/lpeopsDoc/parelm.tex", "max_stars_repo_name": "ikbendedjurre/TorXakis", "max_stars_repo_head_hexsha": "a791ce9960e88df576733404fe4d60114c35e50a", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 44, "max_stars_repo_stars_event_min_datetime": "2017-06-09T08:17:13.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-08T02:17:01.000Z", "max_issues_repo_path": "sys/lpeops/tex/lpeopsDoc/parelm.tex", "max_issues_repo_name": "ikbendedjurre/TorXakis", "max_issues_repo_head_hexsha": "a791ce9960e88df576733404fe4d60114c35e50a", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 746, "max_issues_repo_issues_event_min_datetime": "2017-06-13T07:36:42.000Z", "max_issues_repo_issues_event_max_datetime": "2022-03-23T19:14:31.000Z", "max_forks_repo_path": "sys/lpeops/tex/lpeopsDoc/parElm.tex", "max_forks_repo_name": "ikbendedjurre/txs-develop", "max_forks_repo_head_hexsha": "bc11f4b93a15e220bf6941d395d5b4cd361bfe74", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 13, "max_forks_repo_forks_event_min_datetime": "2017-11-16T11:33:59.000Z", "max_forks_repo_forks_event_max_datetime": "2021-06-26T16:33:36.000Z", "avg_line_length": 38.5058823529, "max_line_length": 194, "alphanum_fraction": 0.719217843, "num_tokens": 946, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.34201892787234045}} {"text": "\\section{Building NWAs from other NWAs (namespace \\texttt{opennwa::construct})}\n\\label{Se:Building NWAs}\n\nThe library provides functions for performing automata-theoretic operations upon\nNWAs. The supported operations are union, intersection, concatenation,\nreversal, Kleene star, complement, and determinization.\n\nThe library supports two interfaces to each of these operations. In one, the\noperation allocates an NWA with \\texttt{new}, performs the construction, and\nreturns a \\texttt{NwaRefPtr} to the result. In the other, the operation takes\na reference to an NWA, clears it, and constructs the result in-place. The\nfirst form is usually more convenient to use, and creates a mini language for\nset expressions; the second form makes it possible to store the result of an\noperation in a subclass of \\texttt{Nwa}, or an existing \\texttt{Nwa} object.\n\nEach of these functions is in the namespace \\texttt{opennwa::construct}:\n\n\\begin{functionlist}\n\n \\functionDefEarly{NwaRefPtr}{unionNwa}{Nwa const \\& a, Nwa const \\& b}{}\n \\functionDef{void}{unionNwa}{Nwa \\& out, Nwa const \\& a, Nwa const \\& b}{}\n Computes the union of the NWAs \\texttt{a} and \\texttt{b},\n either returning the result or storing it in \\texttt{out}.\n See Section\n \\ref{Se:Union}. (This function is called \\texttt{unionNwa} instead of\n \\texttt{union} because the latter is a C++ keyword.) The two NWAs must not\n have any states in common, and the output will be nondeterministic even\n if both input NWAs are deterministic. The running time\\footnote{We count\n state lookups, additions, etc.\\ as constant time even though they are\n actually logarithmic, and occasionally linear.} is\n $O(|Q^a|+|Q^b|+|\\delta^a|+|\\delta^b|)$.\n\n \\functionDefFirstEarly{NwaRefPtr}{intersect}{Nwa const \\& a, Nwa const \\& b}{}\n \\functionDef{void}{intersect}{Nwa \\& out, Nwa const \\& a, Nwa const \\& b}{}\n Computes the intersection of the NWAs \\texttt{a} and \\texttt{b},\n either returning the result or storing it in \\texttt{out}.\n See Section \\ref{Se:Intersection}. If both input NWAs are deterministic,\n the output will be deterministic. The worst-case running time is\n $O(|Q^a| |Q^b| d_a d_b)$, where $d_a$ is the maximum out-degree\n of a state in \\texttt{a} and $d_b$ is the maximum out-degree of a state\n in \\texttt{b}.\n\n \\functionDefFirstEarly{NwaRefPtr}{concat}{Nwa const \\& left, Nwa const \\& right}{}\n \\functionDef{void}{concat}{Nwa \\& out, Nwa const \\& left, Nwa const \\& right}{}\n Computes the concatenation of the NWAs \\texttt{left} and\n \\texttt{right}, either returning the result or storing it in\n \\texttt{out}.\n See Section \\ref{Se:Concatenation}. The output automaton will be\n nondeterministic. The running time is\n $O(|Q^a|+|Q^b|+|\\delta^a|+|\\delta^b|+|Q^a||\\delta_r^b|)$.\n\n \\functionDefFirstEarly{NwaRefPtr}{star}{Nwa const \\& orig}{}\n \\functionDef{void}{star}{Nwa \\& out, Nwa const \\& orig}{}\n Computes the Kleene star of the NWA \\texttt{orig}, either\n returning the result or storing it in \\texttt{out}. See Section\n \\ref{Se:Star}. The output automaton will be nondeterministic. The\n running time is $O(|Q|+|\\delta|+|Q_0||Q_f||\\delta_c|)$.\n\n \\functionDefFirstEarly{NwaRefPtr}{reverse}{Nwa const \\& orig}{}\n \\functionDef{void}{reverse}{Nwa \\& out, Nwa const \\& orig}{}\n Computes the NWA that accepts the reverse of each nested word\n accepted by the NWA \\texttt{orig}, either returning the result or\n storing it in \\texttt{out}. See Section \\ref{Se:Reverse}. The running\n time is $O(|Q|+|\\delta|)$.\n\n \\functionDefFirstEarly{NwaRefPtr}{determinize}{Nwa const \\& orig}{}\n \\functionDef{NwaRefPtr}{determinize}{Nwa \\& out, Nwa const \\& orig}{}\n Computes a determinization of \\texttt{orig}, either returning the\n result or storing it in \\texttt{out}.\n See Section \\ref{Se:Determinize}. The worst-case running time for this\n algorithm is at least $O({|Q|^3} \\cdot2^{|Q|^2})$.\n\n\\clearpage\n \\functionDefFirstEarly{NwaRefPtr}{complement}{Nwa const \\& orig}{}\n \\functionDef{void}{complement}{Nwa \\& out, Nwa const \\& orig}{}\n Computes the complement of the determinization of \\texttt{orig},\n either returning the result or storing it in \\texttt{out}.\n See Section\n \\ref{Se:Complement}. %For complementing the set of final states to work\n %correctly, two things must hold: \\texttt{orig} must be deterministic and\n %the transition functions must be total. (That is, for a given state $q$\n %and symbol $\\sigma$, there must be exactly one internal transition $(q,\n %5\\sigma, q') \\in\\delta_i$, exactly one call transition $(q, \\sigma,\n %q')\\in\\delta_c$, and exactly one return transition $(q, p, \\sigma,\n %q')\\in\\delta_r$ for each state $p$.)\n \\texttt{complement()} determinizes \\texttt{orig}, so its worst-case\n running time is also at least $O({|Q|^3} \\cdot2^{|Q|^2})$.\n\n %\\vspace{0.5\\baselineskip}\n %If \\texttt{determinize} is \\texttt{true}, it will\n %automatically determinize the input NWA before complementation, which\n %also guarantees the input machine will be total. If you\n %know that \\texttt{orig} is already deterministic and total, you can set\n %\\texttt{determinize} to\n %\\texttt{false} to avoid this step. (Complementing a nondeterministic or\n %non-total NWA can easily lead to an incorrect result.) If\n %\\texttt{determinize} is \\texttt{true}, the worst-case running time is the\n %same as for the \\texttt{determinize} function (roughly\n %$O(2^{|Q|^2})$). If \\texttt{determinize} is \\texttt{false}, the running\n %time is $O(|Q|+|\\delta|)$.\n\n\\end{functionlist}\n\nAs mentioned above, these functions create a small language of set\nexpressions. For instance, to compute an automaton \\texttt{M} whose language\nis $\\texttt{A} \\cup (\\texttt{B} \\cap \\texttt{C})*$ (where \\texttt{A},\n\\texttt{B}, and \\texttt{C} are \\texttt{NwaRefPtr}s), one can write\n\\begin{center}\n \\texttt{NwaRefPtr M = unionNwa(*A, *star(*intersect(*B, *C)))}\n\\end{center}\n\n\n\\begin{figure}[tb]\n \\centering\n \\begin{minipage}{0.3\\textwidth}\n \\centering\n \\nwaimage[.6]{Figures/figure-1}\n \\caption{An example NWA.}\n \\label{Fi:Example1}\n \\end{minipage}\n \\hspace{0.025\\textwidth}\n \\begin{minipage}{0.3\\textwidth}\n \\centering\n \\nwaimage[0.49]{Figures/union-other}\n \\caption{A second example NWA.}\n \\label{Fi:Union1}\n \\end{minipage}\n \\hspace{0.025\\textwidth}\n \\begin{minipage}{0.3\\textwidth}\n \\centering\n \\nwaimage[0.8]{Figures/union-result}\n \\caption{The NWA resulting from the union of the NWA in Figure\n \\ref{Fi:Example1} and the NWA in Figure \\ref{Fi:Union1}. Note that there\n are two initial states.}\n \\label{Fi:Union2}\n \\end{minipage}\n\\end{figure}\n\n\n\n\n\\subsection{Union}\n\\label{Se:Union}\nThe union of two NWAs is constructed by taking the union of each of the\ncomponents of the NWAs. (In\nparticular, it does \\textsl{not} do a cross-product construction, and will\n\\textsl{always} produce a nondeterministic automaton as a result as long as\nboth machines have at least one initial state.)\n\nFormally, the union of NWAs $N = (Q_N, \\Sigma_N, Q_{0,N}, \\delta_N, F_N)$\nand $M = (Q_M, \\Sigma_M,$ $Q_{0,M}, \\delta_M, F_M)$ is\n$N \\cup M = (Q_N \\cup Q_M, \\Sigma_N \\cup \\Sigma_M, Q_{0,N} \\cup\n Q_{0,M}, \\delta_N \\cup \\delta_M, F_N \\cup F_M)$.\n\nAs an example, \\figref{Union2} illustrates the union of \\figref{Example1} and\n\\figref{Union1}.\n\n\nThe state sets of the NWAs must not overlap,\ni.e., $Q_1 \\cap Q_2 = \\emptyset$. \\textsl{Client code should not rely on\n this condition being checked} or any particular behavior occurring if it\ndoes not hold.\n\nClient information is copied directly from the original NWAs using\n\\texttt{ClientInfo::clone()}.\n\n\n\\subsection{Intersection}\n\\label{Se:Intersection}\n\nThe intersection of two NWAs is computed in the standard cross-product\nfashion, using a worklist algorithm to only compute those states that\nare reachable.\n\nThe algorithm traverses the original NWAs starting at\nthe initial states and incrementally adds transitions for each pair of\n``intersectable'' transitions that are encountered. By default, transitions\nare intersectable when the transitions are the same kind (internal, call,\nor return) and the symbols on the edge are identical or one is wild.\n\nFor example, \\figref{Intersect1} shows the intersection of \\figref{Example1} and\n\\figref{Intersect1}, and \\figref{Intersect4} shows the intersection of the\ntwo NWAs in \\figref{Intersect3}.\n\n \n\\begin{figure}[tp]\n \\centering\n \\begin{minipage}{0.48\\textwidth}\n \\centering\n \\nwaimage[0.45]{Figures/intersection-simple-other}\n \\caption{Simple NWA to intersect with the NWA in Figure \\ref{Fi:Example1}.}\n \\label{Fi:Intersect1}\n \\end{minipage}\\hspace{0.2cm}\n \\begin{minipage}{0.49\\textwidth}\n \\centering\n \\nwaimage{Figures/intersection-simple-result}\n \\caption{The NWA resulting from the intersection of the NWA in Figure\n \\ref{Fi:Example1} and the NWA in Figure \\ref{Fi:Intersect1}. Note that\n because each of the input NWAs only accepts a single word and those\n words are different, the language of the intersection is empty. The NWA\n built up by \\texttt{intersect()} got as far as it\n could. \\texttt{(key\\#2,3)} is the pair of states \\texttt{(start,start)}\n from the two input automata.}\n \\label{Fi:Intersect2}\n \\end{minipage}\n\\end{figure}\n\n\\begin{figure}[p]\n \\centering\n \\adjustbox{width=\\textwidth,height=\\textheight,keepaspectratio=true}{\n \\nwaimage{Figures/intersection-complex-first}\n \\nwaimage{Figures/intersection-complex-second}}\n \\caption{Two complex NWAs to intersect.}\n \\label{Fi:Intersect3}\n\\end{figure}\n\n\\begin{figure}[p]\n \\centering\n \\nwaimage[.45]{Figures/intersection-complex-result}\n \\caption{The NWA resulting from the intersection of the NWAs in Figure \\ref{Fi:Intersect3}.}\n \\label{Fi:Intersect4}\n\\end{figure}\n\n\\antistupidfloats\n\nIt is possible to customize what symbols are considered equivalent, or\notherwise impose constraints on what transitions can be intersected, by\noverriding the \\texttt{transitionIntersect} function in a subclass of\n\\texttt{Nwa}. In addition, it is possible to impose additional constraints on\nwhat states can be combined by overriding \\texttt{stateIntersect}. Both\nfunctions also produce the result of the intersection:\n\\texttt{transitionIntersect} produces the symbol that will be used on the\nresulting edge, and \\texttt{stateIntersect} produces the state that will be\nused as the target.\n\nThe default behavior of\n\\texttt{transitionIntersect} is that two transitions are intersectable if\nneither symbol is epsilon and either the symbols are the same or at least one of\nthe symbols is a wild. (Epsilon transitions are dealt with in\n\\texttt{intersect()} itself. If you override \\texttt{transitionIntersect}, it\nshould return \\texttt{false} if either input is epsilon.)\n\nThe default behavior of \\texttt{stateIntersect} is that any two\nstates can be combined, the resulting state is labeled with a\nkey that is uniquely generated from the pair of the keys of\nthe two states under consideration, and the client information associated\nwith the resulting state is \\texttt{null}.\n\n\nClient information is initially generated by the helper method \\texttt{stateIntersect},\nbut can be altered through the use of the helper methods\n\\texttt{intersectClientInfoInternal}, \\texttt{intersectClientInfoCall}, and\n\\texttt{intersect\\-Client\\-InfoReturn}, which are invoked by\n\\texttt{intersect()} as transitions of the three different kinds involving the\nassociated state are added. The default behavior of these three functions is\nto perform no changes to the \\texttt{ClientInfo}. These methods can be\noverridden in subclasses to specify alternative behaviors.\n\n\\goodbreak\nThe following operations are virtual functions in class \\texttt{Nwa} and are intended\nto be overridden to customize behavior:\n\n\\begin{functionlist}\n \\functionDefNoCloseParen{bool}{stateIntersect}{%\n \\parbox[t]{4in}\n {Nwa const \\& first, State state1,\\\\ \n Nwa const \\& second, State state2,\\\\\n State \\& resSt, ref\\_ptr \\& resCI)}}\n Determines whether the given states can be combined and,\n if so, creates the combined state. If the two states are incompatable,\n returns \\texttt{false}. If the two states are compatable, it computes the\n key of the combined state (storing it in \\texttt{resSt} and the client\n information (storing it in \\texttt{resCI}), then returns \\texttt{true}.\n\n \\functionDefFirstNoCloseParen{bool}{transitionIntersect}{%\n \\parbox[t]{4in}\n {Nwa const \\& first, Symbol sym1,\\\\\n Nwa const \\& second, Symbol sym2,\\\\ \n Symbol \\& resSym )}}\n Determines whether the given symbols are considered to be equivalent for\n the purposes of intersection. If so, it computes the symbol that should\n be associated with the combined transition (storing it in\n \\texttt{resSym}) and returns true. If not, returns false.\n\n \\functionDefFirstEarlyNoCloseParen{void}{intersectClientInfoInternal}{%\n \\parbox[t]{4in}\n {Nwa const \\& first, State src1, State tgt1,\\\\\n Nwa const \\& second, State src2, State tgt2,\\\\\n Symbol resSym, State resSt )}}\n \\nopagebreak\n \\functionDefEarlyNoCloseParen{void}{intersectClientInfoCall}{%\n \\parbox[t]{4in}\n {Nwa const \\& first ,State call1, State entry1,\\\\\n Nwa const \\& second, State call2, State entry2,\\\\\n Symbol resSym, State resSt )}} \\nopagebreak\n \\functionDefNoCloseParen{\\small void}{\\small intersectClientInfoReturn}{%\n \\parbox[t]{4in}\n {\\small Nwa const \\& first, State exit1, State call1,State ret1,\\\\\n Nwa const \\& second, State exit2,State call2, State ret2,\\\\\n Symbol resSym, State resSt )}}\n Called after a transition of the corresponding type is added to the\n automaton. It is intended to be used to\n alter the client information associated with \\texttt{resSt} given the\n endpoints of the new transition.\n\n\\end{functionlist}\n\n\n\n\n\\subsection{Concatenation}\n\\label{Se:Concatenation}\n\nThe concatenation of two NWAs is constructed by taking the union of all\nstates and transitions of the two automata, and adding\ninternal epsilon transitions from each final state of the first NWA to each\ninitial state of the second NWA. In the resulting NWA, the initial states\nare the initial states from the first NWA, and the final states are the final\nstates of the second NWA.\n\nHowever, the concatenation construction is a bit more complicated than just this,\nbecause we need to address the issue of an unbalanced-left\nword being concatenated with an unbalanced-right word. (Recall the\ndefinition of what happens when an NWA reads a pending return: it is\nallowed to take a return transition where the call predecessor is an initial\nstate of the automaton.) \n\nTo deal properly with the case where the first operand generates strings with\npending calls and the second operand generates strings with pending returns,\nthe\ntransitions from the second automaton are augmented in the\nfollowing manner. When computing the concatenation of \\texttt{left} and\n\\texttt{right}, for each transition $(q, p_0, a, q')\n\\in\\delta_r^{\\texttt{right}}$ where $p_0 \\in Q_0^\\texttt{right}$ (these are\ntransitions that \\texttt{right} can take when reading a pending return) and\neach $p \\in Q^\\texttt{left}$, we add $(q, p, a, q')$ to $\\delta_r$ of the\nresult. (It is actually only necessary to add such a transition for states\n$p$ that either appear in the call position of a call transition in\n\\texttt{left} or are in $Q_0^\\texttt{left}$.)\n\n\n % If the original NWAs are $(Q_1, \\Sigma_1,\n%{Q_0}_1, \\delta_1, {Q_f}_1)$ and $(Q_2, \\Sigma_2, {Q_0}_2, \\delta_2,\n%{Q_f}_2)$, then the resulting NWA is $(Q, \\Sigma, Q_0, \\delta, Q_f)$ where $Q\n%= Q_1 \\cup Q_2$, $\\Sigma = \\Sigma_1 \\cup \\Sigma_2$, $Q_0 = {Q_0}_1$, $\\delta\n%= \\delta_1 \\cup \\delta_2 \\cup \\delta_\\epsilon$ (where $\\delta_\\epsilon =\n%\\{(q,\\epsilon,q') | q \\in {Q_f}_1, q' \\in {Q_0}_2\\}$, and $Q_f = {Q_f}_2$ .\n%The NWA resulting from the concatenation of the NWA in Figure\n%\\ref{Fi:Example1} and the NWA shown in Figure \\ref{Fi:Concat1} is shown in\n%Figure \\ref{Fi:Concat2}.\n\n\\begin{figure}[p]\n \\centering\n \\begin{minipage}{0.5\\textwidth}\n \\begin{minipage}{\\textwidth}\n \\centering\n \\nwaimage[1]{Figures/concat-other}\n \\caption{Simple NWA to concatenate onto the NWA in Figure \\ref{Fi:Example1}.}\n \\label{Fi:Concat1}\n \\end{minipage}\n\n \\vspace{2\\baselineskip}\n \\begin{minipage}{0.8\\textwidth}\n \\centering\n \\nwaimage[0.55]{Figures/reverse-of-figure-1}\n \\caption{The NWA resulting from performing reverse on the NWA in Figure \\ref{Fi:Example1}.}\n \\label{Fi:Reverse1}\n \\end{minipage}\n \\end{minipage}\n \\begin{minipage}{0.49\\textwidth}\n \\centering\n \\nwaimage[0.75]{Figures/concat-result}\n \\caption{The NWA resulting from the concatenation of the NWA in Figure\n \\ref{Fi:Example1} with the NWA in Figure \\ref{Fi:Concat1}.}\n \\label{Fi:Concat2}\n \\end{minipage}\n\\end{figure}\n\n\\antistupidfloats\n\n\n\nThe state sets of the NWAs must not overlap,\ni.e., $Q_1 \\cap Q_2 = \\emptyset$. \\textsl{Client code should not rely on\n this condition being checked.}\n\nClient information is copied directly from the original NWAs using\n\\texttt{ClientInfo::clone()}.\n\n\\figref{Concat2} shows the result of concatenating \\figref{Example1} and\n\\figref{Concat1}.\n\n\\subsection{Kleene star}\n\\label{Se:Star}\n\nLike concatenation, Kleene-star is complicated in the case of NWAs because of\nthe ability to have unbalanced words in the automaton's\nlanguage. Relative to standard FAs, in the concatenation construction we only needed\nto add extra transitions; in this construction, we must add additional states\nas well.\n\nThe NWA resulting from performing Kleene-Star on the NWA shown in\n\\figref{Star1} is shown in \\figref{Star2}.\n\nThe construction presented in \\cite{JACM:AM2009} has a minor error. The error is\nanalogous to not adding a distinguished start state in the traditional\nThompson construction,\\footnote{The initial state of the automaton $A^*$\n must accept, because $\\epsilon$ is in $L^*$; and because of this property it is\n incorrect to merely add epsilon transitions from the old final states to\n the old initial states. If you do this and there is a cycle from the\n initial state back to itself (for example, a self-loop on the initial\n state), the word corresponding to that path would be accepted even though\n it should not be.} and in\nfact can be exhibited using the same example (it is not necessary to use NWA\ncalls or returns). Alur confirmed that our interpretation of the construction\nin \\cite{JACM:AM2009} is\ncorrect~\\cite{AlurNwaStarBroken}. Below, we present\na version that uses $\\epsilon$-transitions, and thus it looks a bit\ndifferent from the version in \\cite{JACM:AM2009}.\n\nWhen computing $R = A^*$ for some\nNWA $A$, the resulting NWA has two ``copies'' of $A$. These are denoted by\nprimed and unprimed version of states from $A$ in the definition below. \n\n\\begin{figure}[p]\n \\centering\n \\nwaimage[.4]{Figures/star-simple}\n \\caption{An NWA on which to perform Kleene-Star.}\n \\label{Fi:Star1}\n\\end{figure}\n\n\\begin{figure}[p]\n \\centering\n \\nwaimage[1.1]{Figures/star-simple-result}\n \\caption{The NWA resulting from performing Kleene-Star on the NWA in Figure \\ref{Fi:Star1}.}\n \\label{Fi:Star2}\n\\end{figure}\n\\antistupidfloats\n\n\nSuppose that $R$ is reading a word $w=w_1w_2\\cdots w_n$, where each $w_i \\in\nL(A)$. $R$ begins in a start state of $A'$. Henceforth it maintains the\nfollowing invariant on the state that $R$ is in with respect to the portion\nof $w$ read so far: if the next symbol $\\sigma$ is in a return position, then\nthat\nsymbol is a \\emph{pending} return in the current $w_i$ iff $R$ is in the $A'$\nportion, i.e., if the current state is primed. (Note that this return only\nneeds to be\npending in the current $w_i$. In the full string $w$, $\\sigma$ may match a\ncall in an earlier $w_j$, or it may be pending in the whole string.)\n\nInternal transitions thus keep $R$ in the same copy of $A$, and call\ntransitions always take $R$ to the unprimed copy of $A$ (because if it then\nreads a return, the return will match that call). Return transitions can\ntarget either copy of $A$: if the call predecessor is unprimed, then the\ntarget will be unprimed; if the call predecessor is primed, then the target\nwill be primed.\n\nThe description above describes $R$'s operation under ``normal''\nconditions. If $R$ is in a final state (either primed or unprimed) of the\nautomaton $A$, it is also allowed to guess that it should ``restart'' by taking\nan epsilon transition to a distinguished start and final state $q_0$. This\nguess is correct if it just read the\nlast character in $w_i$ (making the next character the first one in\n$w_{i+1}$). Note that $q_0$ only has transitions to\nthe $A'$ portion of $R$, maintaining the invariant.\n\nThe reason for the two copies of $A$ comes into play when $R$ reads a return\n$\\sigma$ while in the $A'$ portion. By the invariant, $\\sigma$ is pending in the\ncurrent $w_i$. In the original automaton $A$, the transitions that the\nmachine can use are return transitions $(q,r,\\sigma,p)$ where the call predecessor\n$r$ is in $Q_0$. We need to make sure that $R$ can take those same\ntransitions. There are two cases we need to consider:\n\\begin{enumerate}\n\\item For the cases where\n$\\sigma$ is pending in the whole string $w$, we need to have a version of the\nreturn transition with $q_0$ in the call-predecessor position, so we add $(q',\nq_0, \\sigma, p')$.\n\\item For the cases where $\\sigma$ is matched with a call in some\nearlier $w_j$, it does not matter what state the machine was in before that\ncall; thus we add $(q', s, \\sigma, p')$ for each state $s$ in $Q \\cup Q'$.\n\\end{enumerate}\n\\vspace{0.25\\baselineskip}\n\n``Normal'' operation corresponds to the transitions introduced by the\n\\textsc{Internal}, \\textsc{Call}, and \\textsc{Locally-Matched-Return}\ninference rules given below. The source states of both transitions added by\n\\textsc{Locally-Matched-Return} are unprimed, because if the current symbol\nis a return that matches a call in the same $w_i$, $R$ will be in\nthe $A$ portion by the invariant.\nThe \\textsc{Restart} rule allows $R$ to restart, and the\n\\textsc{Start} rule allows $R$ to get from $q_0$ to the $A'$ portion; these\ntransitions target the $A'$ portion because there have been no calls read in\nthe current $w_i$, and thus a return symbol would be pending.\nThe \\textsc{Globally-Pending-Return} rule addresses the situation where the\ncurrent symbol is a return that is pending in the whole string $w$. (This is\nthe first case in the previous paragraph.) The \\textsc{Locally-Pending-Return} rule\naddresses the situation where the current symbol is a return that is pending\nin the current $w_i$ but matches a call from a previous $w_j$.\n\n\nFormally, if the original NWA is $(Q, \\Sigma, Q_0, \\delta, Q_f)$,\nthen the result of performing Kleene-Star on that NWA is $(Q^*, \\Sigma,\nQ_0^*, \\delta^*, Q_f^*)$. The sets of states are defined by $Q^* = Q \\cup\nQ' \\cup \\{q_0\\}$ (with $Q' = \\{q'\\, |\\, q \\in Q\\}$ and $q_0 \\not \\in Q$),\nand $Q_0^* = Q_f^* = \\{q_0\\}$.\nThe transitions in $\\delta^*$ are defined by the following rules:\n\n\\begin{mathpar}\n{\\inferrule*[left=\\textsc{Internal}]\n {(q,\\sigma,p) \\in \\delta_i }\n { (q,\\sigma,p) \\in \\delta_i^* \\\\ (q',\\sigma,p') \\in \\delta_i^*}\n}\n\\and\n{\\inferrule*[right=\\textsc{Call}]\n { (q,\\sigma,p) \\in \\delta_c }\n { (q,\\sigma,p) \\in \\delta_c^* \\\\ (q',\\sigma,p) \\in \\delta_c^* }\n}\n\\and\n{\\inferrule*[left=\\textsc{Locally-Matched-Return}]\n { (q,r,\\sigma,p) \\in \\delta_r }\n {(q,r,\\sigma,p) \\in \\delta_r^* \\\\ (q,r',\\sigma,p') \\in \\delta_r^* }\n}\n\\\\\n\\and\n{\\inferrule*[left=\\textsc{Start}]\n { q \\in Q_0 }\n {(q_0, \\epsilon, q') \\in \\delta_i^*}\n}\n\\and\n{\\inferrule*[right=\\textsc{Retart}]\n { q \\in Q_f }\n {(q, \\epsilon, q_0) \\in \\delta_i^* \\\\ (q', \\epsilon, q_0) \\in \\delta_i^* }\n}\n\\and\n{\\inferrule*[left=\\textsc{Globally-Pending-Return}]\n { (q,r,\\sigma,p) \\in \\delta_r \\\\ r \\in Q_0 }\n {(q',q_0,\\sigma,p') \\in \\delta_r^* }\n}\n\\and\n{\\inferrule*[left=\\textsc{Locally-Pending-Return}]\n { (q,r,\\sigma,p) \\in \\delta_r \\\\ r \\in Q_0 \\\\ s \\in Q \\cup Q' }\n {(q',s,\\sigma,p') \\in \\delta_r^* }\n}\n\\end{mathpar}\n\n\nClient information is copied directly from the original NWA (using\n\\texttt{ClientInfo::clone()}) such that for each $q \\in Q$, $q$\nand $q'$ have (different copies of) the same client information.\n\n\\emph{Note:} The key for state $q'$ is generated from the key for state $q$\nusing the expression \\texttt{getKey(q, getKey(\"prime\"))}. The input automaton\nto the Kleene star function must not already contain both $q$ and $q'$ for\nany $q$.\n\n%Consider the slightly more complex example of computing the Kleene-Star of the NWA shown in Figure \\ref{Fi:Star3}. The resulting NWA is shown in Figure \\ref{Fi:Star4}.\n\n%\\begin{figure}[htbp]\n% \\centering\n% \\includegraphics[angle=270,width=10cm]{Figures/Figure13.pdf}\n% \\caption{Complex NWA on which to perform Kleene-Star.}\n% \\label{Fi:Star3}\n%\\end{figure}\n\n%\\begin{figure}[htbp]\n% \\centering\n% \\includegraphics[angle=270,width=12cm]{Figures/Figure14.pdf}\n% \\caption{The NWA resulting from performing Kleene-Star on the NWA in Figure \\ref{Fi:Star3}.}\n% \\label{Fi:Star4}\n%\\end{figure}\n\n\\subsection{Reverse}\n\\label{Se:Reverse}\n\nA nested word $n = (w, \\rightsquigarrow)$ is reversed by reversing the linear\nword $w$ and exchanging calls and returns. Formally,\n$n^{rev} = (w^{rev}, \\{(|w|+1-r, |w|+1-c) | (c,r)\n\\in\\rightsquigarrow\\})$. (Pending calls and returns are handled by defining\n$|w|+1-(+\\infty) = -\\infty$ and $|w|+1-(-\\infty) = +\\infty$.) Roughly\nspeaking, call transitions in $A$ correspond to return transitions in\n$A^{rev}$ and vice versa, and we reverse the direction of all transitions as\nin the standard FA construction. We describe the construction from the\nperspective of $A^{rev}$ --- that is, a ``call transition'' is a call\ntransition in $A^{rev}$, and a ``call'' is a call in the reversed string.\n\nPerhaps unsurprisingly, pending returns pose a problem because the\nrole of initial and final states are exchanged. Because of this complication,\nthe algorithm for reversing an NWA has a similar flavor to that of the\nKleene-star procedure. The automaton $A^{rev}$ has two ``copies'' of $A$\n(primed and unprimed), and maintains the same invariant as the Kleene-star\nconstruction: if\nthe next symbol $\\sigma$ is in a return position, then that symbol is a\n\\emph{pending} return iff $A^{rev}$ is in the $A'$ portion.\n(For those familiar with the construction in \\cite{JACM:AM2009}, ours is more\ncomplicated because the version in \\cite{JACM:AM2009} will not work as stated\nwith a weakly-hierarchical NWA.)\n\nIf the original NWA is $(Q, \\Sigma, Q_0, \\delta, Q_f)$, then the result of\nreversing that NWA is $(Q \\cup Q', \\Sigma, Q_f', \\delta^{rev}, Q_0)$ obtained using\nthe following rules:\n\n\\begin{mathpar}\n{\\inferrule*[left=\\textsc{Internal}]\n { (p,\\sigma,q) \\in \\delta_i }\n { (q,\\sigma,p) \\in \\delta^{rev}_i \\\\ (q',\\sigma,p') \\in \\delta^{rev}_i }\n} \n\\and\n{\\inferrule*[left=\\textsc{Call-Return}]\n { (q_c,\\sigma_c,q_e) \\in \\delta_c \\\\ (q_x,\\_\\!\\_\\,,\\sigma_r,q_r) \\in \\delta_r }\n { (q_r, \\sigma_r, q_x), (q_r', \\sigma_r, q_x) \\in \\delta^{rev}_c \\\\\n (q_e, q_r, \\sigma_c, q_c), (q_e, q_r', \\sigma_c, q_c') \\in \\delta^{rev}_c }\n}\n\\and\n{\\inferrule*[left=\\textsc{Pending-Return}]\n { (q_c,\\sigma,q_e) \\in \\delta_c \\\\ q_f \\in Q_f }\n { (q_e', q_f, \\sigma, q_c') \\in \\delta^{rev}_R }\n}\n\\end{mathpar}\n%The \\textsc{Pending-Return} rule is the most perplexing one. The name\n%\\textsc{Pending-Return} is written from the perspective of the reversed\n%automaton -- in other words, the reversed automaton will take a transition\n%added by this rule when it reads a pending return, and that pending return\n%would have been a pending call in the original automaton. However, $q_f \\in\n%Q_f$ is written from the perspective of the original automaton: because\n%$q_f$ is a final state in the original it is an initial state in the\n%reversed automaton, and this is why the transition can be taken on a pending\n%return.\n\n\nThe NWA resulting from performing reverse on the NWA shown in\nFigure \\ref{Fi:Example1} is shown in Figure \\ref{Fi:Reverse1}.\n \n\nClient\ninformation is copied directly from the original NWA using\n\\texttt{ClientInfo::clone()}.\n\n\\subsection{Determinize}\n\\label{Se:Determinize}\n\n\\begin{definition}\nAn NWA, $(Q,\\Sigma,Q_0,\\delta,Q_f)$, is \\textbf{deterministic} iff \n\n\\begin{enumerate} \n\n\\item $|Q_0| \\leq 1$, \n\n\\item For all $q \\in Q$, there is never a choice between reading $\\sigma$ and\n following a $\\sigma$ transition or following a wild (\\wild) transition:\n \\begin{itemize}\n \\item if $(q,\\wild,q') \\in \\delta_i$ then $|\\{q'|(q,\\sigma,q') \\in\n \\delta_i, {\\sigma\\neq@}\\}| = 0$; \\\\ otherwise, for all $\\sigma \\in \\Sigma - \\{\\wild\\}$,\n $|\\{q'|(q,\\sigma,q') \\in \\delta_i\\}| \\leq 1$,\n\n \\item if $(q,\\wild,q') \\in \\delta_c$ then $ |\\{q'|(q,\\sigma,q') \\in\n \\delta_c, {\\sigma\\neq@}\\}| = 0$;\\\\\n otherwise, for all $\\sigma \\in \\Sigma - \\{\\wild\\}$,\n $|\\{q'|(q,\\sigma,q') \\in \\delta_c\\}| \\leq 1$, and\n\n \\item \\mbox{for $q' \\in Q$, if $(q,q',\\wild,q'') \\in \\delta_r$ then\n $|\\{q''|(q,q',\\sigma,q'') \\in \\delta_r, {\\sigma\\neq@}\\}| = 0$;} \\\\\n otherwise, for all\n $\\sigma \\in \\Sigma - \\{\\wild\\}$, $|\\{q''|(q,q',\\sigma,q'') \\in \\delta_r\\}|\n \\leq 1$,\n \\end{itemize}\n\\item There are no $\\epsilon$ transitions:\n \\begin{itemize}\n \\item for all $(q,\\sigma,q') \\in \\delta_i, \\sigma \\neq \\epsilon$,\n \\item for all $(q,\\sigma,q') \\in \\delta_c, \\sigma \\neq \\epsilon$, and\n \\item for all $(q,q',\\sigma,q'') \\in \\delta_r, \\sigma \\neq \\epsilon$.\n \\end{itemize}\n\\end{enumerate}\nIf an NWA is not deterministic, then it is \\textbf{non-deterministic}.\n\\end{definition}\n\nDeterminizing an NWA operates like a\ngeneralization of the classical subset construction. Instead of the states\nin the determinized NWA being subsets of states in the original NWA, states of the\ndeterminized NWA are sets of state pairs (i.e., binary relations on states)\n\\cite{JACM:AM2009}. To support determinization, the library provides a\ntypedef of \\texttt{std::set$<$pair$<$State, State$>>$} as \n\\texttt{Nwa::BinaryRelation}. See also \\texttt{opennwa/RelationOps.hpp}.\n\nWe present the algorithm we use for determinization in\n\\appref{DeterminizeAlgorithm}.\n\n%There is very experimental support for using the BuDDY\n\n\nThe result of determinizing the automaton in \\figref{Det1} is shown in\n\\figref{Det2}.\n\n\\begin{figure}[p]\n \\centering\n \\nwaimage[0.45]{Figures/determinize}\n \\caption{Simple nondeterministic NWA.}\n \\label{Fi:Det1}\n\\end{figure}\n\n\n\\begin{figure}[p]\n \\centering\n \\nwaimage[.75]{Figures/determinize-result}\n \\caption{The NWA resulting from determinizing the NWA in Figure\n \\ref{Fi:Det1}. As mentioned in the text, states in the determinized NWA\n are relations on the states in the original NWA. The state\n $\\varnothing$ has been removed from this diagram.} \n \\label{Fi:Det2}\n\\end{figure}\n\n\nClient information is generated through the use of the helper method\n\\texttt{mergeClientInfo}, but can be altered through the use of the helper\nmethods \\texttt{mergeClientInfoInternal}, \\texttt{mergeClientInfoCall}, and\n\\texttt{mergeClientInfoReturn}, which are invoked by \\texttt{determinize} as\ntransitions of the three kinds involving the associated state are added. The\ndefault behavior of \\texttt{mergeClientInfo} is that the \\texttt{ClientInfo}\nassociated with the resulting state is \\texttt{null}. The default behavior\nof \\texttt{mergeClientInfoInternal}, \\texttt{mergeClientInfoCall}, and\n\\texttt{mergeClientInfoReturn} is to make no changes to the the\n\\texttt{ClientInfo}. These methods can be overridden to specify alternative\nbehaviors. As determinization is performed, \\texttt{mergeClientInfo} is\ncalled each time a new state is created. Then, as each transition is added,\n\\texttt{mergeClientInfoInternal}, \\texttt{mergeClientInfoCall}, or\n\\texttt{mergeClientInfoReturn} is called (depending on the type of transition\nbeing added) to update the \\texttt{ClientInfo} associated with the target\nstate of the transition being added.\n\nThe following functions can be overridden in a subclass of \\texttt{Nwa} to\ncustomize the behavior of determinization:\n\\begin{functionlist}\n \\functionDefNoCloseParen{void}{mergeClientInfo}{%\n \\parbox[t]{4in}{\n Nwa const \\& nondet, BinaryRelation const \\& binRel, \\\\\n St resSt, ref\\_ptr\\& resCI)}}\n Callback that gets called when a new state \\texttt{resSt} (representing the\n binary relation \\texttt{binRel}) is added to the determinized automaton.\n Intended to provide a hook for computing the client information that should\n be associated with the new state; the client information should be set in\n the output parameter \\texttt{resCI} (i.e., \\texttt{setClientInfo} should not\n be called directly). \\texttt{nondet} is the NWA being determinized.\n\n \\functionDefFirstEarlyNoCloseParen{void}{mergeClientInfoInternal}{%\n \\parbox[t]{4in}{\n Nwa const \\& nondet,\\\\\n BinaryRelation const \\& binRelSource,\\\\\n BinaryRelation const \\& binRelTarget,\\\\\n State sourceSt, Symbol resSym, State resSt,\\\\\n ref\\_ptr\\& resCI )}}\n \\functionDefEarlyNoCloseParen{void}{mergeClientInfoCall}{%\n \\parbox[t]{4in}{\n Nwa const \\& nondet,\\\\\n BinaryRelation const \\& binRelCall,\\\\\n BinaryRelation const \\& binRelEntry,\\\\\n State callSt, Symbol resSym, State resSt,\\\\\n ref\\_ptr\\& resCI )}}\n \\functionDefNoCloseParen{void}{mergeClientInfoReturn}{%\n \\parbox[t]{4in}{\n Nwa const \\& nondet,\\\\\n BinaryRelation const \\& binRelExit,\\\\\n BinaryRelation const \\& binRelCall,\\\\\n BinaryRelation const \\& binRelReturn,\\\\\n State exitSt, State callSt, Symbol resSym,\\\\\n State resSt, ref\\_ptr\\& resCI )}}\n Callbacks that get called when a new transition is added to the given\n automaton. The endpoints and their associated binary relations are\n given.\n Alters the client information associated with \\texttt{resSt} given\n information about the transition being added to the determinized\n automaton.\n \\end{functionlist}\n\n%Consider the slightly more complex determinization of the NWA shown in Figure \\ref{Fi:Det3}. The resulting NWA is shown in Figure \\ref{Fi:Det4}.\n\n%\\begin{figure}[htbp]\n% \\centering\n% \\includegraphics[angle=270,width=12cm]{Figures/Figure18.pdf}\n% \\caption{Complex nondeterministic NWA.}\n% \\label{Fi:Det3}\n%\\end{figure}\n\n%\\begin{figure}[htbp]\n% \\centering\n% \\includegraphics[angle=270,width=12cm]{Figures/Figure19.pdf}\n% \\caption{The NWA resulting from determinizing the NWA in Figure \\ref{Fi:Det3}.}\n% \\label{Fi:Det4}\n%\\end{figure}\n\n\\subsection{Complement}\n\\label{Se:Complement}\n\nComplementing an NWA is performed by determinizing the automaton and then\ncomplementing the set of final states.\n\\begin{comment}\n In our implementation, an extra flag\nto \\texttt{complement} controls whether the determinization step is to be\nperformed, so it can be bypassed if you have \\textsl{a priori} knowledge that\nthe input NWA must already be deterministic.\n\\end{comment}\nThe result of\ncomplementing the NWA shown in Figure \\ref{Fi:Det1} is\nshown in Figure \\ref{Fi:Comp1}.\n\nClient information is computed during determinization as described in the\nprevious section, then cloned during the complement step.\n\n\\begin{figure}[h]\n \\centering\n \\nwaimage[1]{Figures/complement-of-determinize}\n \\caption{The complement of the NWA in Figure \\ref{Fi:Det1} (the\n determinization of which is shown in Figure \\ref{Fi:Det2}). We omit\n transitions to the state $\\{\\}$; any action that does not appear in the\n diagram goes to the state $\\{\\}$.\n }\n \\label{Fi:Comp1}\n\\end{figure}\n\n\n\n", "meta": {"hexsha": "275b426e8937606f8cd92730759e8a42872a3a52", "size": 36227, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Doc/opennwa/03-building-from-others.tex", "max_stars_repo_name": "jusito/WALi-OpenNWA", "max_stars_repo_head_hexsha": "2bb4aca02c5a5d444fd038e8aa3eecd7d1ccbb99", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 15, "max_stars_repo_stars_event_min_datetime": "2015-03-07T17:25:57.000Z", "max_stars_repo_stars_event_max_datetime": "2022-02-04T20:17:00.000Z", "max_issues_repo_path": "src/wpds/Doc/opennwa/03-building-from-others.tex", "max_issues_repo_name": "ucd-plse/mpi-error-prop", "max_issues_repo_head_hexsha": "4367df88bcdc4d82c9a65b181d0e639d04962503", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2018-03-03T05:58:55.000Z", "max_issues_repo_issues_event_max_datetime": "2018-03-03T12:26:10.000Z", "max_forks_repo_path": "src/wpds/Doc/opennwa/03-building-from-others.tex", "max_forks_repo_name": "ucd-plse/mpi-error-prop", "max_forks_repo_head_hexsha": "4367df88bcdc4d82c9a65b181d0e639d04962503", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 15, "max_forks_repo_forks_event_min_datetime": "2015-09-25T17:44:35.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-18T18:25:38.000Z", "avg_line_length": 44.12545676, "max_line_length": 169, "alphanum_fraction": 0.7123140199, "num_tokens": 10726, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.34201892787234045}} {"text": "\\documentclass[11pt, a4paper, oneside]{article}\n\n\\usepackage{color}\n\\usepackage{tikz}\n\\usepackage{showframe}\n\\usepackage{blindtext}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{bm}\n\\usepackage{calc} \n\\usepackage{enumitem} \n\\usepackage{xcolor}\n\\usepackage{lipsum}\n\\usepackage{setspace}\n\\usepackage{glossaries}\n\\usepackage{latexsym}\n\n\\tikzset{\n vertex/.style = {\n circle,\n fill = black,\n outer sep = 2pt,\n inner sep = 1pt,\n }\n}\n\n\\SetLabelAlign{parright}{\\parbox[t]{\\labelwidth}{\\raggedleft#1}}\n\\setstretch{0.5}\n\n\\DeclareMathOperator*{\\argmin}{argmin}\n\n\\begin{document}\n\n\\title{}\n\\author{}\n\\date{}\n\n%Dataset\n%name: LS3D-W.\n%download: https://www.adrianbulat.com/face-alignment/\n\n\\part*{Studies dependency}.\n\n\\section*{Face alignment 2D}\n\nCao, X., Wei, Y., Wen, F., Sun, J.: Face alignment by explicit shape regression. In: CVPR. (2012)\n\nXiong, X., De la Torre, F.: Supervised descent method and its applications to face alignment. In: CVPR. (2013) 532–539\n\nJourabloo, A., Liu, X.: Pose-invariant 3d face alignment. In: CVPR. (2015) 3694–3702\n\nZhu, S., Li, C., Change Loy, C., Tang, X.: Face alignment by coarse-to-fine shape searching. In: CVPR. (2015) 4998–5006\n\nBulat, A., Tzimiropoulos, G.: Convolutional aggregation of local evidence for large pose face alignment. In: BMVC. (2016)\n\n\\section*{Face alignment 3D}\n\nBulat, A.,Tzimiropoulos, G.: Two-stage Convolutional Part Heatmap Regression for the 1st 3D Face Alignment in the Wild (3DFAW) Challenge. ???? (2016)\n\n\n\\part*{Bulat 2017 Face Alignment}\n\n\\noindent\nNormalized Mean Error:\n\n\\begin{equation}\n\\text{NME}\n= \\frac{1}{N} \\sum_{k=1}^N \\frac{\\| \\mathbf{x}_k - \\mathbf{y}_k \\|^2}{d}\n\\tag{}\n\\end{equation}\n\n\\begin{description}[labelwidth=\\widthof{\\bfseries 123456789012345},align=parright]\n\t\\item[Input variable: ]\t$d = \\sqrt{w_{ \\mathit{bbox}} * h_{\\mathit{bbox}} }$\n\t\\item[$\\mathit{bbox}:$] bounding box.\n\\end{description}\n\n\\bigskip\n\n\\part*{Bulat 2016 Two stage 3D pose}\n\n\\begin{equation}\nl_2 = \\frac{1}{N} \\sum_{n=1}^N (\\tilde{z}_n - z_n) _2\n\\tag{}\n\\end{equation}\n\n\\bigskip\n\n\\noindent\nGround Truth Error(GTE):\n\n\\begin{equation}\nE( \\mathbf{X}, \\mathbf{Y})\n= \\frac{1}{N} \\sum_{n=1}^N \\frac{\\| \\mathbf{X}_n - \\mathbf{Y}_n \\|_2}{d_i}\n\\tag{}\n\\end{equation}\n\n\\begin{description}[labelwidth=\\widthof{\\bfseries 123456789012345},align=parright]\n\t\\item[$\\mathbf{X}$:] Predicted set of points.\n\t\\item[$\\mathbf{Y}$:] Corresponding ground truth.\n\t\\item[$d_i$:] Interocular distance for the $i$th image.\n\\end{description}\n\n\\bigskip\n\n\\noindent\nCross View Ground Truth Consistency Error(CVGTCE):\n\\begin{equation}\nE_{vc}( \\mathbf{X}, \\mathbf{Y}, T )\n=\\frac{1}{N} \\sum_{n=1}^1 \\frac{\\| s\\mathbf{Rx}_n - \\mathbf{y}_n \\|_2}{d_i}\n\\tag{}\n\\end{equation}\n\n\\begin{equation}\nP = \\{ s, \\mathbf{R}, \\mathbf{t} \\}\n\\tag{}\n\\end{equation}\n\n\\begin{equation}\n\\{ s, \\mathbf{R}, \\mathbf{t} \\}\n= \\argmin_{s, \\mathbf{R}, \\mathbf{t}}\n\\sum_{n=1}^N \\| \\mathbf{y}_k - (s \\mathbf{Rx} + \\mathbf{•}mathbf{t}) \\|_2 ^2\n\\end{equation}\n\n\n\n\\end{document}", "meta": {"hexsha": "d85b73f2b3fd404d421a11767c1234b03418d16d", "size": 3002, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "book_reviews/face_alignment.tex", "max_stars_repo_name": "booiljung/booiljung.github.io", "max_stars_repo_head_hexsha": "1a861660b4bf3c049ed8f47fc8f5346ee8de7823", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "book_reviews/face_alignment.tex", "max_issues_repo_name": "booiljung/booiljung.github.io", "max_issues_repo_head_hexsha": "1a861660b4bf3c049ed8f47fc8f5346ee8de7823", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "book_reviews/face_alignment.tex", "max_forks_repo_name": "booiljung/booiljung.github.io", "max_forks_repo_head_hexsha": "1a861660b4bf3c049ed8f47fc8f5346ee8de7823", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 23.8253968254, "max_line_length": 149, "alphanum_fraction": 0.6805463025, "num_tokens": 1133, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5698526514141571, "lm_q2_score": 0.6001883592602049, "lm_q1q2_score": 0.3420189278723404}} {"text": "\\chapter[Agent Based Simulation]{Agent Based Simulation}\\label{chp:agent_based_simulation}\n\n\\chapterinitial{S}{ometimes} individual behaviours and interactions are well\nunderstood, and an understanding of how a whole population of such individuals\nmight behave needed. For example psychologists and economists may know a lot\nabout how individual spenders and vendors behave in response to given stimuli,\nbut an understanding of how these stimuli might effect the macro-economy is\nnecessary. Agent based\\index{agent based} simulation is a paradigm of thinking\nthat allows such emergent population level behaviour to be investigated from\nindividual rules and interactions.\n\n\\section{Problem}\\label{sec:agent_based_simulation_problem}\n\nConsider a city populated by two categories of household, for example a household\nmight be fans of Cardiff City FC or Swansea City AFC\\footnote{Swansea and\nCardiff are two cities in South Wales with rival football clubs.}.\nEach household has a preference for living close to households of the same kind,\nand will move around the city while their preferences are not satisfied.\nHow will these individual preferences affect the overall distribution of fans in\nthe city?\n\n\\section{Theory}\\label{sec:agent_based_simulation_theory}\n\nThe problem considered here is considered a `classic' one for the paradigm of\nagent based simulation, and is referred to as Schelling\\index{Schelling}'s\nsegregation model. It features in Thomas Schelling's book `Micromotives and\nMacrobehaviours'~\\cite{schelling2006micromotives}, whose title neatly summarises\nthe world view of agent based modelling: we know, understand, determine, or can\ncontrol individual micromotives; and from this we'd like to observe and\nunderstand macrobehaviours.\n\nIn general an agent based model consists of two components, agents, and an\nenvironment:\n\n\\begin{itemize}\n \\item agents are autonomous entities that will periodically choose to take one\n of a number of actions.\n These are chosen in order to maximise that agent's own given utility\n function;\n \\item an environment contains a number of agents and defines how their\n interactions affect each other.\n The agents may be homogeneous or heterogeneous, and the\n relationships may change over time, possibly due to the actions taken\n by the agents.\n\\end{itemize}\n\nIn general, an agent will first observe a subset of its environment, for\nexample it will consider some information about the agents it is currently\nclose to.\nThen it will update some information about itself based on these observations.\nThis could be recording relevant information from the observations, but could\nalso include some learning, maybe considering its own previous actions.\nIt will then decide on an action to take, and carry out this action. This\ndecision may be deterministic or random and/or based on its own attributes from\nsome learning process\\index{learning process}; with the ultimate aim of\nmaximising its own utility. In practice, a utility\\index{utility} can be\nrepresented by a function that maps the environment to some numeric value.\nThis process happens to all agents in the environment, possibly simultaneously.\nThis is summarised in Figure~\\ref{fig:abm_diagram}\n\n\\begin{figure}\n \\begin{center}\n \\includestandalone[width=\\textwidth]{./assets/abm-diagram}\n \\end{center}\n \\caption{Representation of an agent interacting with its environment.}\n \\label{fig:abm_diagram}\n\\end{figure}\n\nFor the football team supporters problem, each household is an agent.\nThe environment is the city.\nEach household's utility function is to satisfy their preference of living next\nto at least a given number of households supporting the same team as them.\nTheir choices of action are to move house or not to move house.\n\nAs a simplification the city will be modelled as a \\(50 \\times 50\\) grid.\nEach cell of the grid is a house that can either contain a household of Cardiff\nCity FC supporters, or contain a household of Swansea City AFC supporters.\nA house's neighbours are assumed to be the houses adjacent to it, horizontally,\nvertically, and diagonally.\nFor mathematical simplicity, it is also assumed that the grid is a torus,\nwhere houses in the top row are vertically adjacent to the bottom row, and\nhouses in the rightmost column are horizontally adjacent to the leftmost column.\n\nEvery household has a preference \\(p\\).\nThis corresponds to the minimum proportion of neighbours they are happy to live\nFigure~\\ref{fig:schelling_happyunhappy} shows a household of Cardiff City FC\nsupporters that are happy with their neighbours, and not happy with their\nneighbours, when \\(p=0.5\\). Households supporting Cardiff City FC are shaded\ngrey.\n\n\\begin{figure}\n \\begin{center}\n \\subfigure[A happy household, with 6 similar neighbours (\\(\\frac{6}{8} > p = 0.5\\))]{\\includestandalone[width=0.4\\textwidth]{./assets/schelling_happy}}\n \\subfigure[An unhappy household, with 2 similar neighbours (\\(\\frac{2}{8} < p = 0.5\\))]{\\includestandalone[width=0.4\\textwidth]{./assets/schelling_unhappy}}\n \\end{center}\n \\caption{Example of a household happy and unhappy with its neighbours, when\n \\(p=0.5\\). Households supporting Cardiff City FC are shaded grey, households\n supporting Swansea City AFC are white.}\n \\label{fig:schelling_happyunhappy}\n\\end{figure}\n\nThe original problem stated that households move around the city whenever they\nare unhappy with their neighbours.\nThis long process of selling, searching for, and buying houses can be simplified\nto randomly pairing two unhappy households and swapping their houses.\nIn fact, this can be simplified to consider the houses themselves as agents,\nwho swap households with each other.\n\nTherefore the model logic is:\n\n\\begin{enumerate}\n \\item initialise the model: fill each house in the grid with either a\n household of Cardiff City FC or Swansea City AFC supporters with\n probability \\(0.5\\) each;\n \\item at each discrete time step, for every house:\n \\begin{enumerate}\n \\item consider their household's neighbours (\\textit{observe});\n \\item determine if the household is happy (\\textit{update});\n \\item if unhappy (\\textit{decide}), swap household with another randomly\n chosen house with an unhappy household (\\textit{action}).\n \\end{enumerate}\n\\end{enumerate}\n\nAfter a number of time steps the overall structure of the city can be observed.\nThis agent based model only explicitly defines individual behaviours\nand interactions, therefore any population level behaviour that may have emerged\nwould occur without explicit definition.\n\n\\section{Solving with Python}\\label{sec:agent_based_simulation_solving-with-python}\n\nAgent based modelling lends itself well to a programming paradigm called\nobject-orientated programming\\index{object-orientated programming}.\nThis paradigm lets a number of \\textit{objects} from a set of instructions\ncalled a \\textit{class} to be built.\nThese objects can both store information (in Python these are called\n\\textit{attributes}\\index{attribute}), and do things (in Python these are called\n\\textit{methods}\\index{method}).\nObject-orientated programming allow for the creation of new classes which can be\nused to implement the individual behaviours of an agent based model.\n\nFor this problem two classes will be built: a\n\\mintinline{python}{House} and a \\mintinline{python}{City} for them to live in.\n\nThe following libraries will be used:\n\n\\begin{pyin}\nimport random\nimport itertools\nimport numpy as np\n\\end{pyin}\n\nNow to define the \\mintinline{python}{City}:\n\n\\begin{pyin}\nclass City:\n def __init__(self, size, threshold):\n \"\"\"Initialises the City object.\n\n Args:\n size: an integer number of rows and columns\n threshold: float between 0 and 1 representing the\n minimum acceptable proportion of similar neighbours\n \"\"\"\n self.size = size\n sides = range(size)\n self.coords = itertools.product(sides, sides)\n self.houses = {\n (x, y): House(x, y, threshold, self)\n for x, y in self.coords\n }\n\n def run(self, n_steps):\n \"\"\"Runs the simulation of a number of time steps.\n\n Args:\n n_steps: an integer number of steps\n \"\"\"\n for turn in range(n_steps):\n self.take_turn()\n\n def take_turn(self):\n \"\"\"Swaps all sad households.\"\"\"\n sad = [h for h in self.houses.values() if h.sad()]\n random.shuffle(sad)\n i = 0\n while i <= len(sad) / 2:\n sad[i].swap(sad[-i])\n i += 1\n\n def mean_satisfaction(self):\n \"\"\"Finds the average household satisfaction.\n\n Returns:\n The average city's household satisfaction\n \"\"\"\n return np.mean(\n [h.satisfaction() for h in self.houses.values()]\n )\n\\end{pyin}\n\nThis defines a class, a template or a set of instructions that can be used to\ncreate instances, called objects.\nFor the considered problem only one instance of the \\mintinline{python}{City}\nclass will be needed.\nHowever, it is useful to be able to produce more in order to run multiple trials\nwith different random seeds.\nThis class contains four methods: \\mintinline{python}{__init__},\n\\mintinline{python}{run}, \\mintinline{python}{take_turn} and\n\\mintinline{python}{mean_satisfaction}.\n\nThe \\mintinline{python}{__init__} method is run whenever the object is first\ncreated, and initialises the object.\nIn this case it sets a number of attributes.\n\n\\begin{itemize}\n \\item First the square grid's \\mintinline{python}{size} is defined, which\n is the number of rows and columns of houses it contains;\n \\item next the \\mintinline{python}{coords} are defined, a list of tuples\n representing all the possible coordinates of the grid, this uses the\n itertools library for efficient iteration;\n \\item finally \\mintinline{python}{houses} is defined, a dictionary with\n grid coordinates as keys, and instances of the\n \\mintinline{python}{House} class.\n\\end{itemize}\n\n\nThe \\mintinline{python}{run} method runs the simulation. For each\n\\mintinline{python}{n_steps} number of discrete time steps, the city runs the\nmethod \\mintinline{python}{take_turn}.\nIn this method, first a list of all the houses is created with households that\nare unhappy with their neighbours; these are put in a random order using the\nrandom library; and then working inwards from the boundary\nhouses with sad households are paired up and swap households.\n\nThe last method defined here is the \\mintinline{python}{mean_satisfaction}\nmethod, which is only used to observe any emergent behaviour\\index{emergent behaviour}.\nThis calculates the average satisfaction of all the houses in the grid, using\nthe numpy library for convenience.\n\nIn order to be able to create an instance of the above class, we need to define\na \\mintinline{python}{House} class:\n\n\\begin{pyin}\nclass House:\n def __init__(self, x, y, threshold, city):\n \"\"\"Initialises the House object.\n\n Args:\n x: the integer x-coordinate\n y: the integer y-coordinate\n threshold: a number between 0 and 1 representing\n the minimum acceptable proportion of similar\n neighbours\n city: an instance of the City class\n \"\"\"\n self.x = x\n self.y = y\n self.threshold = threshold\n self.kind = random.choice([\"Cardiff\", \"Swansea\"])\n self.city = city\n\n def satisfaction(self):\n \"\"\"Determines the household's satisfaction level.\n\n Returns:\n A proportion\n \"\"\"\n same = 0\n for x, y in itertools.product([-1, 0, 1], [-1, 0, 1]):\n ax = (self.x + x) % self.city.size\n ay = (self.y + y) % self.city.size\n same += self.city.houses[ax, ay].kind == self.kind\n return (same - 1) / 8\n\n def sad(self):\n \"\"\"Determines if the household is sad.\n\n Returns:\n a Boolean\n \"\"\"\n return self.satisfaction() < self.threshold\n\n def swap(self, house):\n \"\"\"Swaps two households.\n\n Args:\n house: the house object to swap household with\n \"\"\"\n self.kind, house.kind = house.kind, self.kind\n\\end{pyin}\n\nIt contains four methods: \\mintinline{python}{__init__},\n\\mintinline{python}{satisfaction}, \\mintinline{python}{sad} and\n\\mintinline{python}{swap}.\n\nThe \\mintinline{python}{__init__} methods sets a number of attributes at the\ntime the object is created: the house's \\mintinline{python}{x} and\n\\mintinline{python}{y} coordinates (its column and row numbers on the grid);\nits \\mintinline{python}{threshold} which corresponds to \\(p\\); its\n\\mintinline{python}{kind} which is randomly chosen between having a Cardiff City\nFC supporting household or a Swansea City AFC supporting household; and finally\nits \\mintinline{python}{city}, an instance of the \\mintinline{python}{City}\nclass, shared by all the houses.\n\nThe \\mintinline{python}{satisfaction} method loops though each of the house's\nneighbouring cells in the city grid, counts the number of neighbours that are\nof the same kind as itself, and returns this as a proportion.\nThen the \\mintinline{python}{sad} method returns a boolean indicating if the\nhousehold's satisfaction is below the minimum threshold.\n\nFinally the \\mintinline{python}{swap} method takes another house object, and\nswaps their household kinds.\n\nA function to create and run one of these simulations will now be written\nwith a given random seed\\index{random seed}, threshold, and number of steps.\nThis function returns the resulting mean happiness:\n\n\\begin{pyin}\ndef find_mean_happiness(seed, size, threshold, n_steps):\n \"\"\"Create and run an instance of the simulation.\n\n Args:\n seed: the random seed to use\n size: an integer number of rows and columns\n threshold: a number between 0 and 1 representing\n the minimum acceptable proportion of similar\n neighbours\n n_steps: an integer number of steps\n\n Returns:\n The average city's household satisfaction after\n n_steps\n \"\"\"\n random.seed(seed)\n C = City(size, threshold)\n C.run(n_steps)\n return C.mean_satisfaction()\n\\end{pyin}\n\nNow consider each household with a threshold of\n0.65, and compare the mean happiness after 0 steps and 100 steps.\nFirst 0 steps:\n\n\\begin{pyin}\ninitial_happiness = find_mean_happiness(\n seed=0, size=50, threshold=0.65, n_steps=0\n)\nprint(initial_happiness)\n\\end{pyin}\n\n\\begin{pyout}\n0.4998\n\\end{pyout}\n\nThis is well below the minimum threshold of \\(0.65\\), and so on average\nhouseholds are unhappy.\nAfter 100 steps:\n\n\\begin{pyin}\nfinal_happiness = find_mean_happiness(\n seed=0, size=50, threshold=0.65, n_steps=100\n)\nprint(final_happiness)\n\\end{pyin}\n\n\\begin{pyout}\n0.9078\n\\end{pyout}\n\nAfter 100 time steps the average satisfaction level is much higher.\nIn fact, it is much higher than each individual household's threshold.\nNow consider that this satisfaction level is really a level of how similar\neach households' neighbours are, it is actually a level of segregation.\nThis was the central premise of Schelling's original\nmodel~\\cite{schelling2006micromotives}\nthat overall emergent segregation levels are much higher than any individuals'\npersonal preference for segregation.\n\nMore analysis methods can be added, including plotting functions.\nFigure~\\ref{fig:schelling_python_plot} shows the grid at the beginning, after 20\ntime steps, and after 100 time steps, with households supporting Cardiff City FC\nin grey, and those supporting Swansea City AFC in white.\nIt visually shows the households segregating over time.\n\n\\begin{figure}\n\\begin{center}\n\\subfigure[At the beginning.]{\\includegraphics[width=0.32\\textwidth]{./assets/python_schelling_0}}\n\\subfigure[After 20 time steps.]{\\includegraphics[width=0.32\\textwidth]{./assets/python_schelling_20}}\n\\subfigure[After 100 time steps.]{\\includegraphics[width=0.32\\textwidth]{./assets/python_schelling_100}}\n\\end{center}\n\\caption{Plotted results from the Python code.}\n\\label{fig:schelling_python_plot}\n\\end{figure}\n\n\\section{Solving with R}\\label{sec:agent_based_simulation_solving-with-R}\n\nAgent based modelling lends itself well to a programming paradigm called\nobject-orientated programming\\index{object-orientated programming}.\nThis paradigm lets a number of \\textit{objects} from a set of instructions\ncalled a \\textit{class} to be built.\nThese objects can both store information (in the R package used here these are called\n\\textit{fields}\\index{field}), and do things (in the R package used here these are called\n\\textit{methods}\\index{method}).\nObject-orientated programming allow for the creation of new classes which can be\nused to implement the individual behaviours of an agent based model.\n\nThere are a number of ways of doing object-orientated programming in R.\nIn this chapter, a package called R6\\index{R6} will be used here.\n\nFor this problem two classes will be built: a\n\\mintinline{R}{House} and a \\mintinline{R}{City} for them to live in.\n\nNow to define the \\mintinline{R}{City}\\footnote{\nNote that no documentation is included in the definition\nof the class and the style is inconsistent with the other chapters in this book.\nThe authors do not know of any widely accepted convention for documenting classes\nin R, furthermore the text of this chapter serves as detailed documentation in this\ncontext.\n}\n\n\\begin{Rin}\nlibrary(R6)\nCity <- R6Class(\"City\", list(\n size = NA,\n houses = NA,\n initialize = function(size, threshold) {\n self$size <- size\n self$houses <- c()\n for (x in 1:size) {\n row <- c()\n for (y in 1:size) {\n row <- c(row, House$new(x, y, threshold, self))\n }\n self$houses <- rbind(self$houses, row)\n }\n },\n run = function(n_steps) {\n if (n_steps > 0) {\n for (turn in 1:n_steps) {\n self$take_turn()\n }\n }\n },\n take_turn = function() {\n sad <- c()\n for (house in self$houses) {\n if (house$sad()) {\n sad <- c(sad, house)\n }\n }\n sad <- sample(sad)\n num_sad <- length(sad)\n i <- 1\n while (i <= num_sad / 2) {\n sad[[i]]$swap(sad[[num_sad - i]])\n i <- i + 1\n }\n },\n mean_satisfaction = function() {\n mean(sapply(self$houses, function(x) x$satisfaction()))\n }\n) )\n\\end{Rin}\n\nThis defines an R6 class, a template or a set of instructions that can be used\nto create instances of it, called objects.\nFor this model only one instance of the \\mintinline{R}{City} class is needed,\nalthough it may be useful to be able to produce more in order to run multiple\ntrials with different random seeds.\nThis class contains four methods: \\mintinline{R}{initialize},\n\\mintinline{R}{run}, \\mintinline{R}{take_turn} and\n\\mintinline{R}{mean_satisfaction}.\n\nThe \\mintinline{R}{initialize} method is run at the time the object is first\ncreated.\nIt initialises the object by setting a number of its fields:\n\n\\begin{itemize}\n \\item First the square grid's \\mintinline{R}{size} is defined, which\n is the number of rows and columns of houses it contains;\n \\item then the \\mintinline{R}{houses} are defined by iteratively repeating\n the \\mintinline{R}{rbind} function to create a two-dimensional vector\n of instances of the, yet to be defined, \\mintinline{R}{House} class,\n representing the houses themselves.\n\\end{itemize}\n\nThe \\mintinline{R}{run} method runs the simulation. For each discrete time\nstep from 1 to \\mintinline{R}{n_steps}, the world runs the method\n\\mintinline{R}{take_turn}.\nIn this method, a list of all the houses with households that\nare unhappy with their neighbours is created;\nthese are put in a random order and then working inwards from the boundary,\nhouses with sad households are paired up and swap households.\n\nThe last method defined here is the \\mintinline{R}{mean_satisfaction}\nmethod, which is used to observe the emergent behaviour\\index{emergent behaviour}.\nThis calculates the average satisfaction of all the houses in the grid.\n\nIn order to be able to create an instance of the above class,\na \\mintinline{R}{House} class is needed:\n\n\\begin{Rin}\nHouse <- R6Class(\"House\", list(\n x = NA,\n y = NA,\n threshold = NA,\n city = NA,\n kind = NA,\n initialize = function(x = NA,\n y = NA,\n threshold = NA,\n city = NA) {\n self$x <- x\n self$y <- y\n self$threshold <- threshold\n self$city <- city\n self$kind <- sample(c(\"Cardiff\", \"Swansea\"), 1)\n },\n satisfaction = function() {\n same <- 0\n for (x in -1:1) {\n for (y in -1:1) {\n ax <- ( (self$x + x - 1) %% self$city$size) + 1\n ay <- ( (self$y + y - 1) %% self$city$size) + 1\n if (self$city$houses[[ax, ay]]$kind == self$kind) {\n same <- same + 1\n }\n }\n }\n (same - 1) / 8\n },\n sad = function() {\n self$satisfaction() < self$threshold\n },\n swap = function(house) {\n old <- self$kind\n self$kind <- house$kind\n house$kind <- old\n }\n) )\n\\end{Rin}\n\nIt contains four methods: \\mintinline{R}{initialize},\n\\mintinline{R}{satisfaction}, \\mintinline{R}{sad} and \\mintinline{R}{swap}.\n\nThe \\mintinline{R}{initialize} methods sets a number of the class' fields when\nthe object is created: the house's \\mintinline{R}{x} and \\mintinline{R}{y}\ncoordinates (its column and row numbers on the grid); its\n\\mintinline{R}{threshold} which corresponds to \\(p\\); its \\mintinline{R}{kind}\nwhich is randomly chosen between having a Cardiff City\nFC supporting household or a Swansea City AFC supporting household; and finally\nits \\mintinline{R}{city}, an instance of the \\mintinline{R}{City} class, shared\nby all the houses.\n\nThe \\mintinline{R}{satisfaction} method loops though each of the house's\nneighbouring cells in the city grid, counts the number of neighbours that are of\nthe same kind as itself, and returns this as a proportion.\nThe \\mintinline{R}{sad} method returns a boolean indicating of the\nhousehold's satisfaction is below its minimum threshold.\n\nFinally the \\mintinline{R}{swap} method takes another house object, and\nswaps their household kinds.\n\nA function to create and run one of these simulations will now be written\nwith a given random seed, threshold, and number of steps. This function return\nthe resulting mean happiness:\n\n\\begin{Rin}\n#' Create and run an instance of the simulation.\n#'\n#' @param seed: the random seed to use\n#' @param size: an integer number of rows and columns\n#' @param threshold: a number between 0 and 1 representing\n#' the minimum acceptable proportion of similar neighbours\n#' @param n_steps: an integer number of steps\n#'\n#' @return The average city's household satisfaction\n#' after n_steps\nfind_mean_happiness <- function(seed,\n size,\n threshold,\n n_steps){\n set.seed(seed)\n city <- City$new(size, threshold)\n city$run(n_steps)\n city$mean_satisfaction()\n}\n\\end{Rin}\n\nNow consider each household with a threshold of\n0.65, and compare the mean happiness after 0 steps and 100 steps.\nFirst 0 steps:\n\n\\begin{Rin}\ninitial_happiness <- find_mean_happiness(\n seed = 0,\n size = 50,\n threshold = 0.65,\n n_steps = 0\n)\nprint(initial_happiness)\n\\end{Rin}\n\n\\begin{Rout}\n[1] 0.4956\n\\end{Rout}\n\nThis is well below the minimum threshold of \\(0.65\\), and so on average\nhouseholds are unhappy here.\nLet's run the simulation for 100 generations and see how this changes:\n\n\\begin{Rin}\nfinal_happiness <- find_mean_happiness(\n seed = 0,\n size = 50,\n threshold = 0.65,\n n_steps = 100\n)\nprint(final_happiness)\n\\end{Rin}\n\n\\begin{Rout}\n[1] 0.9338\n\\end{Rout}\n\nAfter 100 time steps the average satisfaction has increased.\nIt is now actually much higher that each individual household's threshold.\nThis satisfaction level can be considered as a level of how similar each\nhouseholds' neighbours are, and so it is actually a level of segregation.\nThis was the central premise of Schelling's original\nmodel~\\cite{schelling2006micromotives}, that overall\nemergent segregation levels are much higher than any individuals' personal\npreference for segregation.\n\nMore analysis methods can be added, including plotting functions.\nFigure~\\ref{fig:schelling_R_plot} shows the grid at the beginning, after 20\ntime steps, and after 100 time steps, with households supporting Cardiff City FC\nin grey, and those supporting Swansea City AFC in white.\nIt shows the households segregating over time.\n\n\\begin{figure}\n\\begin{center}\n\\subfigure[At the beginning.]{\\includegraphics[width=0.32\\textwidth]{./assets/schelling_R_0}}\n\\subfigure[After 20 time steps.]{\\includegraphics[width=0.32\\textwidth]{./assets/schelling_R_20}}\n\\subfigure[After 100 time steps.]{\\includegraphics[width=0.32\\textwidth]{./assets/schelling_R_100}}\n\\end{center}\n\\caption{Plotted results from the R code.}\n\\label{fig:schelling_R_plot}\n\\end{figure}\n\n\\section{Wider context}\\label{sec:agent_based_simulation_wider_context}\n\nThe simulations described in this chapter come under the larger umbrella term of\nmulti agent systems, which discusses the theory of systems with multiple\nindependent agents interacting with one another. A good source on the topic\nis~\\cite{shoham2008multiagent}.\n\nThe model described here is called Schelling's\\index{Schelling} segregation\nmodel, and was first described in~\\cite{schelling2006micromotives}. Another model\nconsidered as classic in this domain is a model of a flock of birds presented\nin~\\cite{reynolds1987flocks}, otherwise referred to as Boids\\index{Boids}, where the\nbehaviours of flocks of birds are understood by capturing the individual\ninteractions between individual birds. Conway's Game of\nLife\\index{Conway's Game of Life}, described in~\\cite{gardener1970mathematical} is\nanother classic, which comes under the banner of cellular\nautomata\\index{cellular automata}. Here cells on a grid either become alive or\ndead depending on a certain simplistic set of rules. Emergent behaviours\nobserved due to these rules include self replicating as well as oscillating\nstructures. In the 1970s agent based tournaments were held by Robert\nAxelrod\\index{Axelrod}~\\cite{axelrod1990evolution},\nwhich was the first of a number of studies using agent based modelling and game\ntheory\\index{game theory} (see Chapter~\\ref{chp:game_theory}) to understand the\nemergence of cooperative behaviours.\n\nIn recent years, similar methodologies have been used in a variety of\napplications such as~\\cite{diaz2021agent} which models parents' choice of school,\nin~\\cite{romanowska2019agent} archaeological population migration and trade\ndynamics are modelled, and~\\cite{jing2020agent} offers a systematic\nliterature review for the use of agent based modelling of autonomous vehicles.\n\n", "meta": {"hexsha": "21060f6cd48f309fc19770befaecab42d7027cad", "size": 26828, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "src/chapters/07/main.tex", "max_stars_repo_name": "drvinceknight/amwoss", "max_stars_repo_head_hexsha": "8b0bf80f0a06dc5cf9bfeef4b9f9e174ccadf06d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-21T21:35:44.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-21T21:35:44.000Z", "max_issues_repo_path": "src/chapters/07/main.tex", "max_issues_repo_name": "drvinceknight/amwoss", "max_issues_repo_head_hexsha": "8b0bf80f0a06dc5cf9bfeef4b9f9e174ccadf06d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 71, "max_issues_repo_issues_event_min_datetime": "2019-11-18T11:00:25.000Z", "max_issues_repo_issues_event_max_datetime": "2021-10-21T22:49:40.000Z", "max_forks_repo_path": "src/chapters/07/main.tex", "max_forks_repo_name": "drvinceknight/amwoss", "max_forks_repo_head_hexsha": "8b0bf80f0a06dc5cf9bfeef4b9f9e174ccadf06d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2020-01-15T12:00:49.000Z", "max_forks_repo_forks_event_max_datetime": "2020-01-15T12:00:49.000Z", "avg_line_length": 39.3950073421, "max_line_length": 164, "alphanum_fraction": 0.7339346951, "num_tokens": 6501, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5698526368038304, "lm_q2_score": 0.600188359260205, "lm_q1q2_score": 0.34201891910339244}} {"text": "%===============================================================================\n% COVID-19 Reproduction Numbers in South Africa\n% Kevin Durant\n% May 2020\n%===============================================================================\n\n\\documentclass[12pt,a4paper]{article}\n\n% \\usepackage{booktabs} % better looking tables.\n\\usepackage{mathtools} % also loads amsmath.\n\\usepackage{microtype}\n\\usepackage{iftex}\n\\ifPDFTeX\n \\usepackage[T1]{fontenc}\n \\usepackage[utf8]{inputenc}\n \\usepackage{amssymb}\n\\else\n % Unicode-math should be loaded after other maths- or font-related packages.\n % It loads amsmath and fontspec if necessary, and enables a custom version of\n % the Latin Modern Math font by default.\n \\usepackage{unicode-math}\n% \\setmainfont{STIX Two Text}\n% \\setmathfont{STIX Two Math}\n\\fi\n\\usepackage{biblatex}\n\\usepackage{svg}\n\\usepackage{hyperref}\n\\addbibresource{cvza.bib}\n\n% General maths commands %======================================================\n\n\\newcommand\\ds{\\displaystyle} % large maths.\n\\newcommand\\ts{\\textstyle} % small maths.\n\\newcommand\\mb[1]{\\mathbb{#1}} % mathbb shorthand.\n\\newcommand\\mc[1]{\\mathcal{#1}} % mathcal shorthand.\n\\newcommand\\ub[1]{\\symbf{#1}} % unicode-math symbf shorthand.\n\\newcommand\\ff[1]{^{\\underline{#1}}} % falling factorial.\n\\newcommand\\rf[1]{^{\\overline{#1}}} % rising factorial.\n\\newcommand\\ul[1]{\\underline{#1}} % underline.\n\\newcommand\\ol[1]{\\overline{#1}} % overline.\n\\DeclareMathOperator\\Pb{P} % probability.\n\\DeclareMathOperator\\Ex{E} % expected value.\n\\DeclareMathOperator\\Va{V} % variance.\n\\DeclarePairedDelimiter\\lr{\\lparen}{\\rparen} % sized parentheses.\n\\DeclarePairedDelimiter\\lrb{\\lbrack}{\\rbrack} % sized brackets.\n\\DeclarePairedDelimiter\\abs{\\lvert}{\\rvert} % absolute value symbol.\n\\DeclarePairedDelimiter\\cl{\\lceil}{\\rceil} % ceiling symbol.\n\\DeclarePairedDelimiter\\fl{\\lfloor}{\\rfloor} % floor symbol.\n\\DeclareMathOperator\\B{B} % beta distribution.\n\\DeclareMathOperator\\BP{BP} % beta prime distribution.\n\\DeclareMathOperator\\NB{NB} % negative binomial distribution.\n\n% Title %=======================================================================\n\n\\title{Estimating COVID-19 Reproduction Numbers}\n\\author{Kevin Durant}\n\\date{June 2020}\n\n% Document %====================================================================\n\n\\begin{document}\n\n\\maketitle\n\n\\begin{abstract}\nThis is a short description of an analytic, \\emph{almost} fully Bayesian\nsolution to the problem of inferring effective reproduction numbers for an\nepidemic from reported infection counts. Under the assumptions of a negative\nbinomial likelihood function and a simple beta prime distributed predictive\nprior, the posterior distribution of the daily infection rate follows a beta\nprime distribution whose parameters can be expressed in closed form. These\nposteriors can in turn be used to estimate effective reproduction numbers.\n\\end{abstract}\n\n\\section{Introduction} %========================================================\n\nThe effective reproduction number of an epidemic, $R_t$, refers to the expected\nnumber of new infections caused by a single infected individual at a given stage\nof the epidemic's course. Along with other, complementary measures, these\nnumbers can help give an indication of the rate at which the epidemic is\nspreading.\n\nThe method for estimating effective reproduction numbers discussed here is a\nslightly different take on one described elsewhere by Kevin\nSystrom~\\cite{systrom2020}, which is itself based on work\nby~\\citeauthor{bettencourt2008}~\\cite{bettencourt2008}. Unlike those two\napproaches, the solution described here is almost entirely analytic---the only\nnumerical computation required being that of percentiles of a beta prime\ndistribution. All three methods are based on a key point made by the latter\nauthors mentioned above: that under the assumptions of a standard epidemic\nsusceptible-infected~(SIR) model, the effective reproduction number of a virus\nat time~$t$ can be estimated from the number of new cases recorded between\ntimes~$t-1$ and~$t$, and~$t$ and~$t+1$.\n\nMore specifically, let~$R_t$ be the effective reproduction number at time~$t$,\nand assume that this number remains constant over the interval~$(t-1, t]$ (most\nlikely a single day)\\footnote{Note that our notation differs slightly from that\nused by~\\citeauthor{bettencourt2008}---our~$R_t$ and~$\\lambda_t$ correspond to\ntheir~$R_{t-1}$ and~$\\Delta T(t)$ respectively.}. Likewise, let~$\\lambda_t$ be\nthe average number of new infections that occur during this interval---i.e., the\ncurrent rate of infection. One then has the following\napproximation~\\cite{bettencourt2008}:\n\\begin{equation*}\n \\lambda_t \\approx \\lambda_{t-1}\\exp(\\gamma(R_t - 1)),\n\\end{equation*}\nin which $\\gamma$ is the reciprocal of the infectious period of the virus.\nEquivalently,\n\\begin{equation}\\label{eqn:R_t}\n R_t \\approx \\frac{1}{\\gamma} \\log\\lr*{\\frac{\\lambda_t}{\\lambda_{t-1}}} + 1.\n\\end{equation}\n\nThe approach outlined here involves modelling the number of observed\ninfections~$k_t$ as a stochastic process, assuming that $k_t$ depends on an\nunderlying rate of infection~$\\lambda_t$ via a negative binomial distribution (a\nPoisson distribution can also be used). One can then infer the infection\nrates~$\\lambda_t$ analytically, and use them to estimate effective reproduction\nnumbers by applying equation~\\eqref{eqn:R_t}.\n\nSpecifically, one finds that the posterior rate of infection~$\\lambda_t$ can be\ndescribed using a beta prime distribution:\n\\begin{equation*}\n \\Pb\\lr*{\\frac{\\lambda_t}{r} \\Bigm\\vert k_1, \\dots, k_{t-1}}\n = \\BP\\lr*{\\frac{\\lambda_t}{r} \\Bigm\\vert \\alpha_t, \\beta_t},\n\\end{equation*}\nin which\n\\begin{align}\\label{eqn:recursion}\n\\begin{split} \n \\alpha_t &= \\frac{a_1}{c^{t-1}} + \\sum_{i=0}^{t-1} \\frac{k_{t-i}}{c^i}, \\\\\n \\beta_t &= \\frac{b_1}{c^{t-1}} + \\sum_{i=0}^{t-1} \\frac{r}{c^i}\n = \\frac{b_1}{c^{t-1}} + r\\frac{1 - \\frac{1}{c^t}}{1 - \\frac{1}{c}}.\n\\end{split}\n\\end{align}\nBoth $r$ and $c$ are model parameters that can easily be optimised, because\ntheir marginal likelihood~$\\Pb(k_1, \\dots, k_{t-1} \\mid r, c)$ is available in\nclosed form. The constants $a_1$ and~$b_1$ are parameters of the initial prior\non $\\lambda_1$.\n\nWe apply this model to the estimation of reproduction numbers in\nsection~\\ref{sec:application}, by making use of a simplified version of\nequation~\\eqref{eqn:R_t} in which $\\lambda_{t-1}$ is replaced with a point\nestimate~$\\lambda_{t-1}^*$. We use, for example, the median value of\n$\\Pb(\\lambda_{t-1} \\mid k_1, \\dots, k_{t-2})$ as such an estimate.\n\n\\section{The stochastic process} %==============================================\n\nLet~$\\ub{k} = k_1, \\dots, k_{t-1}$ be a sequence of observed infection counts,\nand~$\\ub{\\lambda}$ the corresponding sequence of unknown infection rates. The\nprimary assumption is that each~$k$ and~$\\lambda$ are related via a negative\nbinomial distribution:\n\\begin{equation}\\label{eqn:likelihood}\n \\Pb(k_t \\mid \\lambda_t, \\ub{k}) = \\Pb(k_t \\mid \\lambda_t)\n \\sim \\NB\\lr*{k_t \\Bigm\\vert r, \\frac{\\lambda_t}{r + \\lambda_t} = p_t},\n\\end{equation}\nwhere $r$ is an unknown dispersion parameter and $p_t$ is a reparameterisation\nof $\\lambda_t$ as a `success' probability. Parameterised in this way, the\nnegative binomial distribution converges to a Poisson distribution of\nrate~$\\lambda_t$ as~$r \\to \\infty$, and by adjusting $r$ we can control the\nlevel of variance inherent to the distribution (smaller values of $r$ result in\nhigher variance).\n\nInference of the rate sequence~$\\ub{\\lambda}$ is performed iteratively, by\nrepeated application of Bayes' rule:\n\\begin{equation*}\n \\Pb(\\lambda_t \\mid k_t, \\ub{k}) \\propto \\Pb(k_t \\mid \\lambda_t)\n \\Pb(\\lambda_t \\mid \\ub{k}).\n\\end{equation*}\nThe first term on the right-hand side---the likelihood function---is simply the\nnegative binomial distribution given above. The second term is a predictive\nprior on~$\\lambda_t$ given only the \\emph{past} infection counts~$\\ub{k}$.\nTechnically the inference will be done with respect to $\\ub{p} = p_1, \\dots,\np_{t-1}$, not $\\ub{\\lambda}$, but with the right choice of prior the translation\nbetween the two is seamless.\n\nThe conjugate prior for the negative binomial likelihood function (with known\ndispersion) is the beta distribution, so our second assumption is that the prior\ndistribution on $p_t$ is of this form. Note that the change-of-variable formula\nfor probability density functions implies that when $p_t$ follows a beta\ndistribution, $\\lambda_t/r$ is distributed according to a beta prime\ndistribution with identical parameters:\n\\begin{equation*}\n p_t \\sim \\B(p_t \\mid \\alpha, \\beta)\n \\Rightarrow \\frac{\\lambda_t}{r}\n \\sim \\BP\\lr*{\\frac{\\lambda_t}{r} \\Bigm\\vert \\alpha, \\beta}.\n\\end{equation*}\nAs mentioned above, this allows us to work with $p_t$ instead of $\\lambda_t$\nwhile deriving posteriors and marginal likelihoods, but still consider\n$\\lambda_t$ when computing $R_t$.\n\nThe third and final assumption we make involves the way in which the predictive\nprior $\\Pb(p_t \\mid \\ub{k})$ is derived from the previous posterior $\\Pb(p_{t-1}\n\\mid \\ub{k})$. In the case of a Gaussian stochastic process, one would derive\nthe prior by assuming additive Gaussian noise on the previous latent variable,\nresulting in a distribution that has the same mean as the previous posterior,\nbut higher variance. Doing so involves solving an integral of the form\n\\begin{equation*}\n \\Pb(p_t \\mid \\ub{k}) = \\int \\Pb(p_t \\mid p_{t-1})\n \\Pb(p_{t-1} \\mid \\ub{k})\\, dp_{t-1},\n\\end{equation*}\nwhich is tractable in the Gaussian case.\n\nAlthough the situation is not quite as straightforward here, we can achieve a\nsimilar outcome by simply assuming the relationship to the previous posterior\ndirectly: specifically, if\n\\begin{equation}\\label{eqn:posterior form}\n \\Pb(p_{t-1} \\mid \\ub{k}) \\sim \\B(p_{t-1} \\mid \\alpha_{t-1}, \\beta_{t-1}),\n\\end{equation}\nwe might assume a predictive prior of the form\n\\begin{equation}\\label{eqn:prior}\n \\Pb(p_t \\mid \\ub{k}) \\sim \\B(p_t \\mid \\alpha_{t-1}/c, \\beta_{t-1}/c)\n = \\B\\lr*{p_t \\mid a_t, b_t}.\n\\end{equation}\nThis is a straightforward prior that has the same mean as the predictive\nposterior on~$p_{t-1}$, but a variance that is larger \\emph{roughly} by a\nfactor~$c$---since the mean and variance of a beta distribution with parameters\n$\\alpha$ and $\\beta$ are given by\n\\begin{equation*}\n \\Ex[X] = \\frac{\\alpha}{\\alpha + \\beta}, \\quad\n \\Va[X] = \\frac{\\alpha\\beta}{(\\alpha + \\beta)^2 (\\alpha + \\beta + 1)},\n\\end{equation*}\nand in general $c$ will be small relative to $\\alpha + \\beta$.\n\nOne could just as easily make use of a predictive prior that introduces additive\nnoise, unlike the multiplicative noise described above. The main reason for\nchoosing multiplicative noise here is that changes in scale made to~$p_t$ (and\n$\\lambda_t$) result in additive changes to~$R_t$, so in this way one is\neffectively introducing additive noise into the overarching reproduction number\nprocess.\n\nThe remaining details of the stochastic process now follow from\nassumptions~\\eqref{eqn:likelihood} and~\\eqref{eqn:prior}. Firstly--and most\nimportantly---the posterior on~$p_t$ is given by\n\\begin{align*}\n \\Pb(p_t \\mid k_t, \\ub{k}) &\\propto \\Pb(k_t \\mid p_t) \\Pb(p_t \\mid \\ub{k}) \\\\\n &= \\NB(k_t \\mid r, p_t)\\, \\B(p_t \\mid a_t, b_t) \\\\\n &\\sim p_t^{k_t} (1 - p_t)^r \\cdot p_t^{a_t-1} (1 - p_t)^{b_t-1} \\\\\n &\\Rightarrow \\B(p_t \\mid a_t + k_t, b_t + r).\n\\end{align*}\nThat is (extending equation~\\eqref{eqn:posterior form}):\n\\begin{equation}\\label{eqn:posterior}\n \\Pb(p_t \\mid k_t, \\ub{k}) \\sim \\B(p_t \\mid a_t + k_t, b_t + r)\n = \\B(p_t \\mid \\alpha_t, \\beta_t).\n\\end{equation}\nCombining this with equation~\\eqref{eqn:prior} one can solve for $\\alpha_t$\nand~$\\beta_t$ recursively, leading to equation~\\eqref{eqn:recursion}. Note\nhowever that these solutions presume that the time series of data points is\ncomplete; in the presence of a missing datum $k_t$ one will need to specify\n$\\alpha_t$ and~$\\beta_t$ explicitly---for example simply by setting them to\n$a_t$ and~$b_t$ respectively.\n\nSecondly, we can derive the marginal likelihood of observation~$k_t$ given the\nprevious observations:\n\\begin{align}\\label{eqn:marginal}\n \\Pb(k_t \\mid \\ub{k}) &= \\int \\Pb(k_t, p_t \\mid \\ub{k})\\, dp_t \\nonumber \\\\\n &= \\int \\NB(k_t \\mid r, p_t)\\, \\B(p_t \\mid a_t, b_t)\\, dp_t \\nonumber \\\\\n &= \\binom{k_t + r - 1}{k_t} \\frac{1}{\\Beta(a_t, b_t)}\n \\int p_t^{a_t+k_t-1} (1 - p_t)^{b_t+r-1}\\, dp_t \\nonumber \\\\\n &= \\begin{dcases}\n \\frac{\\Beta(a_t, b_t + r)}{\\Beta(a_t, b_t)} &\\text{if } k_t = 0, \\\\\n \\frac{1}{k_t\\Beta(k_t, r)}\\frac{\\Beta(a_t + k_t, b_t + r)}{\\Beta(a_t, b_t)}\n &\\text{if } k_t > 0,\n \\end{dcases}\n\\end{align}\nwhere~$\\Beta(x, y)$ denotes the beta function. (The final expression can also be\nrephrased in terms of $\\alpha_t$ and $\\beta_t$ using\nequation~\\eqref{eqn:posterior}.) This allows us to compute the overall marginal\nlikelihood iteratively, since\n\\begin{equation*}\n \\Pb(\\ub{k}) = \\prod_{i=1}^{t-1} \\Pb(k_i \\mid k_1, \\dots, k_{i-1}).\n\\end{equation*}\nThe overall marginal likelihood will in turn allow us to compare the relative\nlikelihoods of values of~$r$ and~$c$, which are the model's two tunable\nparameters.\n\nEquations~\\eqref{eqn:prior} and~\\eqref{eqn:posterior} allow us to derive,\nvia~$p_t$, a posterior distribution for each~$\\lambda_t$. Ideally one would hope\nfor a distribution on~$\\lambda_t/\\lambda_{t-1}$, since this is the ratio on\nwhich~$R_t$ depends (equation~\\eqref{eqn:R_t}), however this would require one\nto either specify~$\\Pb(p_t \\mid p_{t-1})$---something we explicitly avoiding\ndoing above to keep things tractable---or treat $\\lambda_t$ and~$\\lambda_{t-1}$\nas independent for the purposes of deriving $R_t$.\n\nHere we adopt a simpler approach: replace $\\lambda_{t-1}$ in\nequation~\\eqref{eqn:R_t} with a point estimate~$\\lambda_{t-1}^*$, resulting in\nthe approximation\n\\begin{equation}\\label{eqn:R_t approximation}\n R_t \\approx \\frac{1}{\\gamma} \\log\\lr*{\\frac{\\lambda_t}{\\lambda_{t-1}}} + 1\n \\approx \\frac{1}{\\gamma}\n \\log\\lr*{\\frac{\\lambda_t}{r}\\frac{r}{\\lambda_{t-1}^*}} + 1,\n\\end{equation}\nto which our inferred posterior on $\\lambda_t/r$ can directly be applied. This\nis not unlike a simplification made by the other\nauthors~\\cite{bettencourt2008,systrom2020}, who set~$\\lambda_{t-1}^* = k_{t-1}$.\nIn what follows, we have set $\\lambda_{t-1}^*$ to the median of the posterior\ndistribution~$\\Pb(\\lambda_{t-1} \\mid k_1, \\dots k_{t-1})$ (which is simply $r$\ntimes the median of the posterior on $\\lambda_{t-1}/r$).\n\nBefore moving on, we note that~\\citeauthor{bettencourt2008} also describe a\nnegative binomial process in which $r$ is not constant, but rather $r_t =\nk_{t-1}$. We have briefly tested this approach, and in practice it performs\nsimilarly to the one we have described above; the main difference being that the\nprecision of the likelihood function (and thus posterior) varies with the\nobserved counts~$\\ub{k}$. The approach outlined in this section can still be\napplied, and the resulting analytic solution differs only slightly.\n\n\\subsection*{An application to South African data}\\label{sec:application} %=====\n\nIn this section we apply the steps described above to data stemming from the\nCOVID-19 epidemic in South Africa. The data set used here is obtained from South\nAfrica's National Institute for Communicable Diseases, via the University of\nPretoria~\\cite{dsfsi2020}. The data take the form of daily cumulative infection\ncounts per province, from which we derive new infection counts per day.\n\n\\begin{figure}[htb]\n \\centering\n \\includeinkscape{img/counts}\n \\caption{Daily new infection counts for the COVID-19 epidemic in South Africa,\n smoothed using a Gaussian window with a standard deviation of~$\\sigma =\n 3.5$. Shaded areas indicate different levels of the nationwide lockdown,\n with the first, grey area depicting the period between the announcement of\n the lockdown and its initiation.}\n \\label{fig:counts}\n\\end{figure}\n\nBefore inferring daily infection rates, we first need to decide whether or not\nthe raw counts should be smoothed, and if so, to what extent (see\nfigure~\\ref{fig:counts}). The obvious argument against doing so is that the\nvariance of $R_t$ depends indirectly on the variance of the $k_t$, so smoothed\ninfection counts may result in artificially precise posteriors.\n\nOn the other hand, one might argue that the underlying assumption of the model\nis that $k_t$ represents the \\emph{true} number of infections on a given day, of\nwhich smoothed counts are likely a more appropriate indication. A second, more\ntechnical argument for reducing the variance of the reported infection counts is\nthe fact that the equation $\\lambda_t \\approx \\lambda_{t-1}\\exp(\\gamma(R_t -\n1))$ contains an implicit assumption: that $R_t \\ge 0$, and thus\n$\\lambda_t/\\lambda_{t-1} \\ge \\exp(-\\gamma)$.\n\n\\begin{figure}[htb]\n \\centering\n \\includeinkscape{img/ratio}\n \\caption{Ratios of consecutive infection counts~$k_t/k_{t-1}$ for various\n levels of Gaussian smoothing. The red line indicates the bound implied by\n equation~\\eqref{eqn:R_t}.}\n \\label{fig:ratio}\n\\end{figure}\n\nIn figure~\\ref{fig:ratio} we plot the ratio~$k_t/k_{t-1}$ for various levels of\nGaussian averaging---roughly corresponding to smoothing windows of 0.5, 1, and~2\nweeks (if one uses four standard deviations as a guide). One sees that in this\ncase, windows with scales greater than $\\sigma = 7/3$ appear both to respect the\nimplicit bound mentioned above, as well as alleviate much of the periodicity\nvisible in the raw count sequence.\n\nFor the remainder of this example we will counts that have been averaged using a\nGaussian window with standard deviation~$\\sigma = 3.5$. This value will not\nnecessarily be suitable for data sets from other regions or sources, however.\n\nThe rest of the application is straightforward: we apply\nequation~\\eqref{eqn:marginal} to select values for $r$ and~$c$ that maximise the\nmarginal likelihood (for the countrywide South African data this yields $r =\n1592$ and~$c \\approx 3.56$), and then use equation~\\ref{eqn:posterior} to\ncompute the parameters of the posteriors on $\\lambda_1, \\lambda_2, \\dots$. These\nposteriors, shown in figure~\\ref{fig:posterior}, track the smoothed $k_t$\nrelatively closely, as one might expect.\n\nFinally, equation~\\eqref{eqn:R_t approximation} allows us to plot the estimated\nevolution of $R_t$ over time, simply by mapping percentiles of $\\lambda_t/r$ to\nthose of~$R-t$. This plot is shown in figure~\\ref{fig:countrywide} for the\nentire country, and figure~\\ref{fig:provincial} on a provincial level.\n\n\\begin{figure}[htbp]\n \\centering\n \\includeinkscape{img/posterior}\n \\caption{Daily posterior infection rates~$\\lambda_t$, plotted as a median and\n 5--95th percentile interval.}\n \\label{fig:posterior}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\centering\n \\includeinkscape{img/countrywide}\n \\caption{Estimated reproduction numbers~$R_t$ for the COVID-19 epidemic in\n South Africa, plotted as a median and 5--95th percentile interval.}\n \\label{fig:countrywide}\n\\end{figure}\n\n\\begin{figure}[htbp]\n \\centering\n \\includeinkscape{img/provincial}\n \\caption{Reproduction numbers for the COVID-19 epidemic in South Africa,\n estimated at a provincial level.}\n \\label{fig:provincial}\n\\end{figure}\n\nThe only comment we will make on these figures is that the effect of the hard\nlockdown (implemented on 28 March) on the estimated reproduction numbers is\nclearly visible, as is---to a lesser extent---the slow relaxation of lockdown\nrestrictions and gradual reopening of economic activity that follows.\n\n% Bibliography %================================================================\n\n\\printbibliography\n\n\\end{document}", "meta": {"hexsha": "638e11f72d07f6180e6ef0e101d79ece9d7c0454", "size": 19928, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/cvza.tex", "max_stars_repo_name": "kevdur/covid", "max_stars_repo_head_hexsha": "5893b911afb974b9b00dc6af871a7c7129fd7952", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/cvza.tex", "max_issues_repo_name": "kevdur/covid", "max_issues_repo_head_hexsha": "5893b911afb974b9b00dc6af871a7c7129fd7952", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2021-06-11T10:04:33.000Z", "max_issues_repo_issues_event_max_datetime": "2022-01-21T21:39:26.000Z", "max_forks_repo_path": "doc/cvza.tex", "max_forks_repo_name": "kevdur/covid", "max_forks_repo_head_hexsha": "5893b911afb974b9b00dc6af871a7c7129fd7952", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 49.3267326733, "max_line_length": 80, "alphanum_fraction": 0.7147731835, "num_tokens": 5704, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.629774621301746, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34188151857858695}} {"text": "\\documentclass[11pt, a4paper]{article}\n\n\\usepackage{tikz}\n\\usepackage{amsmath}\n\\usepackage{amsfonts}\n\\usepackage{mathtools}\n\\usepackage{graphicx}\n\\usepackage{multicol}\n\\usepackage{appendix}\n\\usepackage{pgfplots}\n\\usepackage{algorithm}\n\\usepackage{algpseudocode}\n\\usepackage[utf8]{inputenc}\n\\usepackage[english]{babel}\n\n\\pgfplotsset{compat=1.16}\n\n\\date{}\n\\title{Exoplanets detection using \\\\ Auto-Tuned Random Forest}\n\n\\author{Lorenzo Loconte \\\\ \\small Knowledge Engineering Project \\\\ \\small University of Bari Aldo Moro, Department of Informatics}\n\n\\begin{document}\n\n\\maketitle\n\\begin{abstract}\n The \\texttt{NASA Kepler} project consists of discovering exoplanets (i.e. planets outside our solar system).\n The data generated by the \\texttt{Kepler Space Telescope} is analyzed by humans and algorithms to discover new exoplanets.\n The main problem is that a lot of exoplanets are revealed false positives.\n This work consists of identifying exoplanets using random forest, a \\textit{Supervised Machine Learning} model.\n Furthermore, the fitted model is analyzed in order to determine which features are relevant.\n The hyperparameters are automatically optimized with techniques that come from the \\texttt{AutoML} research.\n In fact, the hyperparameters of the model are optimized and cross-validated with \\texttt{Hyperband}, a simple yet effective and scalable method for hyperparameters optimization.\n\\end{abstract}\n\n\\clearpage\n\n\\section{Introduction}\n \\paragraph{Kepler Object of Interest}\n Exoplanets are particular planets that are situated outside our solar system which orbit around one star (sometimes even multiple stars).\n The \\texttt{NASA Kepler} project aims to discover these objects by looking at some stars and observing their luminosity during time.\n Basically suppose we are observing a star.\n If a rapid decrease and consequently increase of its luminosity occurs it's most likely that an object passed the way between us and the star.\n Analyzing the variation of luminosity for a certain amount of time we can extrapolate features like the radius and the orbit eccentricity of the object.\n Sometimes these object are actually exoplanets but it's common they can be something else like asteroids or even another star.\n Sometimes they are just false positives due to measurements noise.\n\n The goal of this work is to build a model (using \\textit{Supervised Machine Learning}) that is capable to determinate if a hypothetical exoplanet can be confirmed or is just false positive.\n The dataset used is the cumulative list of \\texttt{Kepler Object of Interest} (\\texttt{KOI}) tables that come from the \\texttt{NASA Kepler} project of discovering exoplanets.\n The complete dataset can be found at \\texttt{https://exoplanetarchive.ipac.caltech.edu}.\n The dataset is composed by a list of hypothetical exoplanets.\n Each sample have a label indicating if the corresponding exoplanet is candidate, false positive or confirmed.\n The provided dataset consists of a lot of heterogeneous features.\n For the classification task the selected features can be found in Appendix \\ref{appendix:features}.\n \n The candidate exoplanets (i.e. the exoplanets which existence is uncertain) are discarded in order to reduce noise and improve the learning process.\n For simplicity all the samples having null values are discarded from the dataset.\n Furthermore, all the selected features are numeric and the preprocessing method applied to the dataset is the standard normalization.\n The resulting dataset contains \\texttt{5860} samples, of which only \\textasciitilde \\texttt{38\\%} are confirmed and the remaining are false positives.\n\n \\paragraph{Random Forests}\n As described before, the classification task consists to determinate if some numerical characteristics can be attributed to an existing exoplanet.\n The model used for the classification task is a random forest, a bagging ensemble of decision trees.\n The prediction of the random forest is computed as the mode of the predictions made by the decision trees.\n Note that each tree of the forest is trained on a subset of the features.\n Random forests correct the habit of decision trees of overfitting on their training set.\n In fact, decision trees tend to overfit their training sets because they have a low bias but an high variance.\n\n In \\textit{Supervised Machine Learning}, bias consists of erroneous assumptions about the data and variance is a measure of sensitivity to the noise in the training data.\n So, random forests generally obtain better performance.\n\n The law that regulates the bias and variance is called \\texttt{Bias-Variance tradeoff} \\cite{biasvariance}.\n This law states that models having low bias have an high variance and vice versa.\n In \\textit{Machine Learning} bias and variance corresponds to the models behaviors that we want to regulate in order to minimize the error.\n An high bias causes \\textit{underfitting} while an high variance causes \\textit{overfitting}.\n \\[\\mathrm{Error}=\\mathrm{Bias}^{2}+\\mathrm{Variance}+\\mathrm{Irreducible\\:Error}\\]\n\n \\paragraph{Hyperparameters}\n Often it's difficult (or nearly impossible) to select the correct hyperparameters that guarantee good performance on some specific datasets.\n \\texttt{AutoML} is a branch of \\textit{Machine Learning} that try to simplify the process of model selection and hyperparameters optimization.\n This work focuses on solving the hyperparameters search problem for random forests.\n The algorithm used is \\texttt{Hyperband} that, as it'll be shown in the next section, represents an evolution of random search designed especially for hyperparameters optimization.\n\n For the hyperparameters optimization algorithm chosen, the number of trees in the random forest is not considered an hyperparameter.\n Furthermore, the following hyperparameters are optimized:\n \\begin{itemize}\n \\item The split criterion (Gini Impurity or Information Gain, see Appendix \\ref{appendix:splitcriterions} for details)\n \\item The fraction of features to use for each tree\n \\item The fraction of samples used to train each tree\n \\item The maximum depth of each tree\n \\item The minimum number of samples required to split an internal node\n \\item The minimum number of samples to be at a leaf node\n \\end{itemize}\n\n\\clearpage\n\n\\section{Hyperparameters Optimization}\n In this section it will be shown a formalization of the hyperparameters optimization task as a search problem.\n In this work we refer to hyperparameters as the ones that cannot be trained (e.g. the ones described in the previous section).\n After that we briefly introduce the hyperparameters optimizer used in this work, \\texttt{Hyperband}, as described in \\cite{hyperband}.\n\n Given a training set $T$, a model $L$ and its hyperparameters search space $\\Omega$, the hyperparameters optimization task consists to find an hyperparameters configuration $\\omega \\in \\Omega$ such that the resulting model $L\\langle\\omega\\rangle$ maximizes a certain score $S$ using cross-validation on $T$.\n Note that most search algorithms don't converge to the global maximum.\n So, the goodness of the hyperparameters found is dependent of the amount of resources we have.\n\n There are a lot of different techniques for hyperparameters optimization, some of them are the following:\n \\begin{itemize}\n \\item Random Search\n \\item Grid Search\n \\item Hyperband\n \\item Bayesian Optimization\n \\item Hybrid Approaches (like \\texttt{BOHB} \\cite{bohb})\n \\end{itemize}\n\n Of the techniques cited above only \\texttt{Bayesian Optimization} guarantees the convergence to the global maximum in the hyperparameters search space (see \\cite{bayesianoptimization} for details).\n The simplest algorithm is \\texttt{Random Search} and it works well if the hyperparameters search space is not too big.\n \\texttt{Random Search} is expensive because each evaluation of the score function is done on models completely built from scratch.\n So, each evaluation of the score function on some randomly selected hyperparameters will use the same amount of computational resources, even if some hyperparameters of the search space obtain a very low score.\n The idea behind \\texttt{Hyperband} is that it's better to save up resources on these hyperparameters configurations that doesn't obtain a sufficiently good score by gradually building more complex models over time.\n \n Note that we refer to budget as the computational cost of cross-validating a certain model.\n For example, for neural networks the budget can be the number of training epochs, while for random forests it can be the number of trees.\n\n Algorithm \\ref{algorithm:hyperband} shows an implementation of \\texttt{Hyperband} as described now.\n Given a budget $R$, \\texttt{Hyperband} consists of random sampling some points (for example using the uniform distribution) in the hyperparameters space and cross-validating them using only a fraction of the budget $R$.\n\n After that we pick a fraction $1/\\eta$ of the best models.\n In this way we discard the models that didn't obtain a sufficiently good score saving up computational resources.\n\n After that, we strengthen the previous models using a larger portion of $R$ and iterate in this way until we use the complete budget $R$.\n The procedure of consequently eliminating hyperparameters samples that don't obtain a good score is called \\texttt{Successive Halving}.\n The fraction of the budget to use for each iteration grows geometrically in respect of $\\eta$.\n Furthermore, we execute the discussed algorithm multiple times using less and less initial random points and using an higher starting budget.\n So, initially \\textit{exploration} is preferred and \\textit{exploitation} is performed later.\n\n Even if \\texttt{Hyperband} is a simple algorithm, it is well scalable on multiple CPUs because we assume that every point in the hyperparameters space is independent from the others.\n\n In our case the model $L$ is a random forest.\n As said before, the budget $R$ is the maximum number of trees that the final random forest will have.\n The choice of the score $S$ is dependent from the task.\n For simplicity the $F_{1}$ score is used (i.e. we want to maximize both \\textit{Precision} and \\textit{Recall}).\n In this work the hyperparameters search space can be defined formally as:\n \\[\\Omega = \\overbrace{\\{gini, entropy\\}}^{\\text{Split criterion}} \\quad\\times \\overbrace{(0, 1]}^{\\text{Features fraction}} \\times \\overbrace{(0, 1]}^{\\text{Samples fraction}} \\times\\]\n \\[\\times \\overbrace{\\mathbb{N}^{*}}^{\\text{Max depth}} \\times \\overbrace{\\mathbb{N}^{*}}^{\\text{Min samples to split}} \\times \\overbrace{\\mathbb{N}^{*}}^{\\text{Min samples at leaf}}\\]\n For obvious reasons and for the hyperparameters optimization algorithm chosen, only a subset of this space can be explored.\n\n \\begin{algorithm}\n \\caption{Hyperband algorithm for hyperparameters optimization}\n \\label{algorithm:hyperband}\n \\begin{algorithmic}[1]\n \\Procedure{Hyperband}{$T,L,S,\\Omega,R,\\eta$}\n \\State $C \\gets \\emptyset$\n \\State $s_{max} \\gets \\lfloor \\log_{\\eta}{R} \\rfloor, \\quad B \\gets (s_{max} + 1)R$\n \\For{$s \\in \\{s_{max},s_{max}-1,...,0\\}$}\n \\State $n \\gets \\lceil \\frac{B}{R}\\frac{\\eta^{s}}{(s+1)} \\rceil, \\quad r \\gets R\\eta^{-s}$\n \\State $H \\gets getRandomHyperparameters(n,\\Omega)$\n \\For{$i \\in \\{0,...,s\\}$}\n \\State $n_{i} \\gets \\lfloor n\\eta^{-i} \\rfloor, \\quad r_{i} \\gets r\\eta^{i}$\n \\State $H \\gets top_{\\lfloor n_{i}/\\eta \\rfloor} \\arg sort_{\\omega} \\{S(L\\langle\\omega \\cup \\{r_{i}\\}\\rangle, T) \\mid \\omega \\in H\\}$\n \\EndFor\n \\State $C \\gets C \\cup H$\n \\EndFor\n \\State \\textbf{return} $\\arg max_{\\omega} \\{S(L\\langle\\omega \\cup \\{R\\}\\rangle, T) \\mid \\omega \\in C\\}$\n \\EndProcedure\n \\end{algorithmic}\n \\end{algorithm}\n\n\\clearpage\n\n\\section{Conclusion}\n In this section we analyze the results of the model chosen comparing it to other models.\n As you can see from Table \\ref{table:benchmark} the auto-tuned random forest used in this work obtained a way better \\textit{Recall}.\n\n Furthermore, as you can see from Figure \\ref{figure:confusion} the false positives count (the number of examples, which are predicted as exoplanets, but that are not) and the false negatives count (the number of actual exoplanets not being discovered), are pretty low.\n\n Models like neural networks and support vector machines are said \\textit{black-box} models.\n It means that predictions cannot be explained and the features importance cannot be evaluated easily.\n At contrary, random forests are said \\textit{white-box} models.\n So, predictions can be explained deeply and it's possible to evaluate the importance of each feature used.\n In \\textit{Artificial Intelligence} the explainability of a model is important because it guarantees transparency.\n Furthermore, in \\textit{Machine Learning}, explainable models can reveal hidden knowledge about the data that isn't known at prior.\n In the end we present Figure \\ref{figure:importances} that shows the importance of the five most important features.\n\n \\begin{table}\n \\centering\n \\begin{tabular}{|c c c c|}\n \\hline\n Model & Precision & Recall & $F_{1} \\downarrow$ \\\\\n \\hline\\hline\n Random Forest & 0.940 & 0.916 & 0.927 \\\\\n \\hline\n \\textbf{Random Forest w/HB} & \\textbf{0.934} & \\textbf{0.935} & \\textbf{0.935} \\\\\n \\hline\n \\end{tabular}\n \\caption{\\textit{Precision}, \\textit{Recall} and $F_{1}$ metrics of random forests with and without hyperparameters optimization (using the default ones in \\texttt{scikit-learn}). For \\texttt{Hyperband} $R=216$ and $\\eta=3$ are used.}\n \\label{table:benchmark}\n \\end{table}\n\n \\begin{figure}\n \\centering\n \\begin{tikzpicture}[scale=1.85]\n \\draw[thick] (0, 0) -- (2, 0);\n \\draw[thick] (0, 0) -- (0, 2);\n \\draw[thick] (2, 2) -- (2, 0);\n \\draw[thick] (2, 2) -- (0, 2);\n \\draw[thick] (0, 1) -- (2, 1);\n \\draw[thick] (1, 0) -- (1, 2);\n \\node[rotate=0] (h1) at (1, 2.5) {\\textbf{Actual}};\n \\node[rotate=0] (h11) at (0.5, 2.25) {Positive};\n \\node[rotate=0] (h12) at (1.5, 2.25) {Negative};\n \\node[rotate=90] (h2) at (-0.5, 1) {\\textbf{Predicted}};\n \\node[rotate=90] (h21) at (-0.25, 1.5) {Positive};\n \\node[rotate=90] (h22) at (-0.25, 0.5) {Negative};\n \\coordinate[label={\\textbf{412}}] (TP) at (0.5, 1.33);\n \\coordinate[label={ 28}] (FN) at (0.5, 0.33);\n \\coordinate[label={ 29}] (FP) at (1.5, 1.33);\n \\coordinate[label={\\textbf{703}}] (TN) at (1.5, 0.33);\n \\end{tikzpicture}\n \\caption{Confusion matrix over the test set.}\n \\label{figure:confusion}\n \\end{figure}\n\n In random forests the features importance are calculated by averaging the features importance in each decision tree.\n For a decision tree the features importance are calculated during the fit and are computed as the normalized total reduction of the criterion brought by each feature. \n\n The most important feature is PRF $\\Delta\\theta_{SQ}$ (\\texttt{Pixel Response Function}), that basically is the angular offset on the plane of the sky between the centroid of the star and the calculated centroid of the star during the passage of the object of interest.\n It's very important because a lot of false positives are due to the transit of another star in binary systems.\n So, if this value is big it's most likely that the object of interest is not an exoplanet.\n For some reason, the planetary radius represents another discriminant feature to determinate if an hypothetical exoplanet is real or a false positive.\n It is possible that objects that have a very small radius can be attributed to measurements noise.\n\n In conclusion, we can say that most of the false positive occurrences are attributed to binary star systems and measurements noise.\n\n \\begin{figure}\n \\centering\n \\begin{tikzpicture}\n \\begin{axis}[\n ybar,\n ylabel=Importance,\n xlabel=Feature \\#,\n symbolic x coords={21, 5, 3, 6, 1},\n bar width=16pt,\n ymin=0.0, ymax=0.35,\n ytick={0.0, 0.05, 0.10, 0.15, 0.20, 0.25, 0.30, 0.35},\n yticklabel style={/pgf/number format/fixed, /pgf/number format/fixed zerofill}\n ]\n \\addplot[style={fill=white}] coordinates {\n (21, 0.296)\n (5, 0.24)\n (3, 0.069)\n (6, 0.052)\n (1, 0.050)\n };\n \\end{axis}\n \\end{tikzpicture}\n \\caption{Five most important features importance.}\n \\label{figure:importances}\n \\end{figure}\n\n\\clearpage\n\\appendix\n\\appendixpage\n\n\\section{Exoplanet Selected Features}\n The documentation of the following exoplanet selected features can be found at \\texttt{https://exoplanetarchive.ipac.caltech.edu}.\n However most of them are auto-explicative.\n The features used in this work are regarding the observed star, the processed features of the object of interest and various spectrometric metrics.\n \\begin{multicols}{2}\n \\label{appendix:features}\n \\begin{enumerate}\n \\item Orbital Period [$days$]\n \\item Transit Duration [$hrs$]\n \\item Transit Depth [$ppm$]\n \\item Fitted Stellar Density [$g/cm^{3}$]\n \\item Planetary Radius [$Earth\\:radii$]\n \\item Inclination [$deg$]\n \\item Equilibrium Temperature [$K$]\n \\item Insolation Flux [$Earth\\:flux$]\n \\item Stellar Effective Temperature [$K$]\n \\item Stellar Surface Gravity [$\\log_{10}(cm/s^{2})$]\n \\item Stellar Metallicity [$dex$]\n \\item Stellar Radius [$Solar\\:radii$]\n \\item Stellar Mass [$Solar\\:mass$]\n \\item RA [$decimal\\:degrees$]\n \\item Dec [$decimal\\:degrees$]\n \\item Kepler-band [$mag$]\n \\item FW $\\Delta\\alpha$ [$sec$]\n \\item FW $\\Delta\\delta$ [$arcsec$]\n \\item PRF $\\Delta\\alpha_{SQ}$ [$arcsec$]\n \\item PRF $\\Delta\\delta_{SQ}$ [$arcsec$]\n \\item PRF $\\Delta\\theta_{SQ}$ [$arcsec$]\n \\end{enumerate}\n \\end{multicols}\n\n\\section{Decision tree split criterions}\n \\label{appendix:splitcriterions}\n \\subsection{Gini impurity}\n Citing \\texttt{wikipedia.org}, Gini impurity is a measure of how often a randomly chosen element from the set would be incorrectly labeled if it was randomly labeled according to the distribution of labels in the subset.\n To compute Gini impurity for a set of items with $J$ classes, suppose $i\\in\\{1,2,...,J\\}$, and let $p_{i}$ be the faction of items labeled with class $i$ in the set.\n \\[{I} _{G}(p)=\\sum _{i=1}^{J}p_{i}\\sum _{k\\neq i}p_{k}=\\sum _{i=1}^{J}p_{i}(1-p_{i})=1-\\sum _{i=1}^{J}{p_{i}}^{2}\\]\n \\subsection{Information gain}\n Information gain is based on the concept of entropy and information content from information theory.\n For each node of the tree, the information value represents the expected amount of information that would be needed to specify whether a new instance should be classified yes or no, given the example reached that node.\n Given $T$ the set of training examples and $X$ the selected feature.\n \\[IG(T, X)=\\mathrm{H}(T)-\\mathrm{H}(T|X)=\\mathrm{H}(T)-\\sum_{x}{\\Pr(X=x)\\mathrm{H}(T|X=x)}\\]\n\n\\bibliography{biblio.bib}\n\\bibliographystyle{unsrt}\n\n\\end{document}\n", "meta": {"hexsha": "398856957d5f78883302ee63a274964614c47357", "size": 19346, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/doc.tex", "max_stars_repo_name": "loreloc/exoplanet-detection", "max_stars_repo_head_hexsha": "d28e51581443e05b408f9bd6e2d1055eb75d9a0d", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2020-01-02T17:45:50.000Z", "max_stars_repo_stars_event_max_datetime": "2021-11-20T16:48:07.000Z", "max_issues_repo_path": "doc/doc.tex", "max_issues_repo_name": "loreloc/exoplanet-detection", "max_issues_repo_head_hexsha": "d28e51581443e05b408f9bd6e2d1055eb75d9a0d", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/doc.tex", "max_forks_repo_name": "loreloc/exoplanet-detection", "max_forks_repo_head_hexsha": "d28e51581443e05b408f9bd6e2d1055eb75d9a0d", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 62.2057877814, "max_line_length": 309, "alphanum_fraction": 0.7289879045, "num_tokens": 4995, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746213017459, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.3418815185785869}} {"text": "\\subsection{Carrier trapping and Shockley-Read-Hall recombination}\n\n\n\\begin{figure}\nTo describe charge becoming trapping into trap states and recombination associated with those states the model uses Shockley-Read-Hall (SRH) theory. A 0D depiction of this SRH recombination and trapping is shown in figure \\ref{fig:dos_structure}, the free electron and hole carrier distributions are labeled as n free and p free respectively. The trapped carrier populations are denoted with n trap and p trap , they are depicted with filled red and blue boxes. SRH theory describes the rates at which electrons and holes become captured and escape from the carrier traps. If one considers a single electron trap, the change in population of this trap can be described by four carrier capture and escape rates as depicted in figure \\ref{fig:dos_structure}. The rate rec describes the rate at which electrons become captured into the electron trap, $r_{ee}$ is the rate which electrons can escape from the trap back to the free electron population, $r_{hc}$ is the rate at which free holes get trapped and $r_{he}$ is the rate at which holes escape back to the free hole population. Recombination is described by holes becoming captured into electron space slice through our 1D traps. Analogous processes are also defined for the hole traps.\n\n\\centering\n\\includegraphics[width=40mm]{./images/dos_structure.jpg}\n\\caption{Trap filling in both energy and position space as the solar cell is taken from a negative bias\nCarrier trapping, de-trapping, and recombination}\n\\label{fig:dos_structure}\n\\end{figure}\n\n\n\\begin{table}\n\\begin{center}\n \\begin{tabular}{lll}\n \\hline\n Mechanism & Symbol & Description \\\\\n \\hline\nElectron capture rate & $r_{ec}$ & $n v_{th} \\sigma_{n} N_{t}(1-f)$ \\\\\nElectron escape rate & $r_{ee}$ & $e_{n} N_{t} f$ \\\\\nHole capture rate & $r_{hc}$ & $p v_{th} \\sigma_{p} N_{t} f$ \\\\\nHole escape rate & $r_{he}$ & $e_{p} N_{t} (1-f)$\\\\\n \\hline\n\\end{tabular}\n\\end{center}\n\\caption{Shockley-Read-Hall trap capture and emission rates, where $f$ is the fermi-Dirac occupation function and $N_{t}$ is the trap density of a single carrier trap.}\n\\label{tab:rates}\n\\end{table}\n\n\nFor each trap level the carrier balance \\ref{eq:srhrate} is solved, giving each trap level an independent quasi-Fermi level. Each point in position space can be allocated between 10 and 160 independent trap states. The rates of each process $r_{ec}$, $r_{ee}$, $r_{hc}$, and $r_{he}$ are give in table \\ref{tab:rates}.\n\n\\begin{equation}\n\\label{eq:srhrate}\n\\frac{\\delta n_t}{\\partial t}=r_{ec}-r_{ee}-r_{hc}+r_{he}\n\\end{equation}\n\nThe escape probabilities are given by:\n\n\\begin{equation}\n\\label{eq:taile}\ne_n=v_{th}\\sigma_{n} N_{c} exp \\left ( \\frac{E_t-E_c}{kT}\\right )\n\\end{equation}\n\nand\n\n\\begin{equation}\n\\label{eq:taile}\ne_p=v_{th}\\sigma_{p} N_{v} exp \\left ( \\frac{E_v-E_t}{kT}\\right )\n\\end{equation}\n\n where $\\sigma_{n,p}$ are the trap cross sections, $v_{th}$ is the thermal emission velocity of the carriers, and $N_{c,v}$ are the effective density of states for free electrons or holes. The distribution of trapped states (DoS) is defined between the mobility edges as\n\n\\begin{equation}\n\\label{eq:taile}\n\\rho^{e/h}(E)=N^{e/h}exp(E/E_{u}^{e/h})\n\\end{equation}\n\nwhere , $N_{e/h}$ is the density of trap states at the LUMO or HOMO band edge\nin states/eV and where $E_{U}^{e/h}$ is slope energy of the density of states. \n\nThe value of $N_{t}$ for any given trap level is calculated by averaging the DoS function over the energy ($\\Delta E$ ) which a trap occupies:\n\n\\begin{equation}\n\\label{eq:taile}\nN_{t}(E)=\\frac{\\int^{E+\\Delta E/2}_{E-\\Delta E/2} \\rho^{e}{E} dE}{\\Delta E}\n\\end{equation}\n\nThe occupation function is given by the equation,\n\\begin{equation}\nf(E_{t},F_{t})=\\frac{1}{e^{\\frac{E_{t}-F_{t}}{kT}}+1}\n\\end{equation}\nWhere, $E_{t}$ is the trap level, and $F_{t}$ is the Fermi-Level of the trap.\nThe carrier escape rates for electrons and holes are given by\n\n\n\n\\subsubsection{Free-to-free carrier recombination}\nA free-carrier-to-free-carrier recombination pathway is also included in the model. However, most organic solar cells have a great deal of trap states and an ideality factor greater than 1.0 suggesting that free to free recombination is not the dominant mechanism. It is also worth noting that since I included this recombination pathway in the model I have not found it useful to reproduce experimental results.\n\nFree-to-free recombination is described using equation \\ref{equ:freetofree}\n\n\\begin{equation}\nR_{free}=k_{r}(n_{f}p_{f}-n_{0}p_{0})\n\\label{equ:freetofree}\n\\end{equation}\n\n", "meta": {"hexsha": "92435275ef0fa685d5af5426cff14c73e5c5363d", "size": 4591, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "gpvdm_data/docs/man/electrical_srh.tex", "max_stars_repo_name": "roderickmackenzie/gpvdm", "max_stars_repo_head_hexsha": "914fd2ee93e7202339853acaec1d61d59b789987", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 12, "max_stars_repo_stars_event_min_datetime": "2016-09-13T08:58:13.000Z", "max_stars_repo_stars_event_max_datetime": "2022-01-17T07:04:52.000Z", "max_issues_repo_path": "gpvdm_data/docs/man/electrical_srh.tex", "max_issues_repo_name": "roderickmackenzie/gpvdm", "max_issues_repo_head_hexsha": "914fd2ee93e7202339853acaec1d61d59b789987", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2017-11-11T12:33:02.000Z", "max_issues_repo_issues_event_max_datetime": "2019-03-08T00:48:08.000Z", "max_forks_repo_path": "gpvdm_data/docs/man/electrical_srh.tex", "max_forks_repo_name": "roderickmackenzie/gpvdm", "max_forks_repo_head_hexsha": "914fd2ee93e7202339853acaec1d61d59b789987", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2019-01-03T06:17:12.000Z", "max_forks_repo_forks_event_max_datetime": "2022-01-01T15:59:00.000Z", "avg_line_length": 51.0111111111, "max_line_length": 1240, "alphanum_fraction": 0.7453713788, "num_tokens": 1316, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297746074044134, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34188151103423536}} {"text": "\\documentclass{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amsmath, amssymb}\n% \\usepackage{proof}\n\\usepackage{syntax}\n\\usepackage{listings}\n\\usepackage{xcolor}\n\\usepackage{hyperref}\n\n\n\\title{Pixy Semantics}\n\\author{Reed Mullanix, Finn Hackett}\n\\date{February 2018}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{Introduction}\nThe semantics of Pixy can be divided up into 3 portions: The term language, how that term language is evaluated, and the type system.\n\\section{Term Language}\nThe term language of Pixy is (roughly) as follows:\n\\begin{grammar}\n ::= \n \\alt \n \\alt \"nil\"\n \\alt \"?\" \n \\alt \"if\" \"then\" \"else\" \n \\alt \"fby\" \n \\alt \"where\" ( \"=\" )*\n \\alt \"fun\" \"=>\" \n \\alt \n\\end{grammar}\n\n{\\color{red} NOTE:} This is incomplete! We need to standardize on the term language.\n\n\\section{Evaluation}\nThe evaluation rules for Pixy are quite different from other languages.\nTo begin with, each expression can be seen as a taking a State and producing a value and a new State. This state is then fed back into the expression to produce a new State and value, and so on. However, some expressions pose some problems. For example, when evaluating \\lstinline{if ... then ... else} expression, we should only really evaluate one of the branches, but doing so may skip important stateful evaluation inside of the untaken branch. To reconcile this, we present a model of evaluation which we call \"Choked Evaluation\". Whenever we are presented with a branching construct, we still evaluate the branches, with the caveat that all variables \\textit{and} literals evaluate to \\lstinline{nil} on the branch that is not taken.\n\nAnother point we need to make is that evaluation is only valid on \\textbf{closed} expressions, or expressions that have no free variables.\n\n{\\color{red} NOTE:} Insert full evaluation semantics here.\n\n{\\color{red} NOTE:} We need to spec out when exactly evaluation terminates for a given step.\n\n\\section{Type Theory}\nTypically, type systems follow this general form:\n\\begin{itemize}\n \\item The user declares the construction and elimination rules for a type.\n \\item The user then uses theses construction rules to create programs.\n\\end{itemize}\nWe prefer to take a different approach, which has been strongly influenced \nby systems such as NuPRL. Generally speaking, our type system works as follows:\n\\begin{itemize}\n \\item The user writes a program.\n \\item The user then creates a proof that the program inhabits some type.\n\\end{itemize}\n\nThat of course raises the question: When does a program inhabit a type? To answer that, we must first answer what exactly a type is in Pixy. We define a type as having 2 components:\n\\begin{enumerate}\n \\item A collection of canonical inhabitants.\n \\item An equivalence relation over those inhabitants.\n\\end{enumerate}\n\nFor example, the canonical inhabitants of the type \\lstinline{Nat} are $0,1,2,3...$ and the equivalence relationship is just the equivalence relationship of natural numbers. When we say that $a \\in A$, what we are really saying is that $a = a$ under the equality relationship imposed by $A$. This point may seem slightly pedantic, but it has large implications. This can be extended to separate elements, so we could also propose that $a = b \\in A$, or that 2 terms $a$ and $b$ are equivalent under the equality relation of $A$. Note that the canonical inhabitants aren't the only members of a type. Any term that evaluates to a canonical inhabitant is also a member of the type. On top of that, if we have 2 terms $t, t'$ and they evaluate to $a, a'$ respectively, and $a = a' \\in A$, then $t, t' \\in A$ as well!\n\nContinuing in the spirit of NuPRL, what exactly is $a \\in A$? Well, if we use the logic of Propositions-as-Types, $a \\in A$ should really just be a type! We shall denote this type as $Eq\\ a\\ b\\ A$. We shall also include all of the standard portions of Martin-Löf Type Theory.\n\nTime is represented by using a the type $Next : \\star \\rightarrow \\star$. This type corresponds with the $\\circ$ operator in Linear Temporal logic. With this primitive type, we can begin to define the operators of Linear Temporal Logic using inductive and coinductive types.\n\\begin{align*}\n &\\circ A \\text{ corresponds to } Next(A) \\\\\n &\\square A \\text{ corresponds to } \\nu \\sigma . A \\times Next(\\sigma) \\\\\n &\\lozenge A \\text{ corresponds to } \\mu \\sigma . A + Next(\\sigma) \\\\\n &A \\triangleright B \\text{ corresponds to } \\mu \\sigma . A \\times Next(B + \\sigma) \\\\ \n\\end{align*}\n\nThere is an alternitave encoding by using temporally-indexed types, which makes quantification over time easier. This was not chosen, as the inductive/coinductive defintions make inductive/coinductive reasoning easier, which makes the kind of proofs we wish to do easier (For example, proving that 2 streams are in sync).\n\n% {\\color{red} NOTE:} This section is incomplete, as we have multiple ways of preceding. I have listed out the possible options.\n\n% \\begin{enumerate}\n% \\item Use a temporally indexed dependent type. This allows us to encode\n% certain properties such as \"$\\forall$ Times t, ...\" and \"$\\exists$ Time t, ...\" easily.\n% \\item Use a co-inductive stream type. This would allow us to more easily\n% prove relationships between 2 streams.\n% \\end{enumerate}\n\n\\section{Relating Programs to Types}\nNote again that we do not derive the types of programs from the bottom-up, as is the norm. Rather, we prove that programs inhabit types from the top-down, using a proof refinement system.\n\nTo begin, a proof is a tree of \\textbf{Judgments}, which consists of a number of \\textbf{hypotheses} of the form $x:A$ followed by a \\textbf{Goal}, which is of the form $term:T$. To proceed with the proof, we need to use refinement rules, which are ways of decomposing sub-goals. For example, say we had some term \\lstinline{fun x => x}, and we wanted to prove that this term is a member of $Bool \\rightarrow Bool$. An example proof would be as follows:\n\\begin{verbatim}\n H >> (fun x => x) in Bool -> Bool by intro-function.\n x:Bool, H >> x in Bool by hypothesis x.\n H >> Bool in U by bool-intro-universe.\n\\end{verbatim}\nNote that we use 3 rules here, \\lstinline{intro-function}, \\lstinline{hypothesis}, and \\lstinline{intro-universe}. These correspond to the standard type inference rules, but there is a catch: We cannot infer the types. This is because a term can inhabit many potential types. For example, we could also prove that \\lstinline{fun x => x} inhabits the type $\\Pi_{A:U}.A \\rightarrow A$:\n\\begin{verbatim}\n H >> (fun x => x) in (A:U) -> A -> A by intro-function-pi\n A:U, x:A, H >> x in A by hypothesis x.\n\\end{verbatim}\n\n{\\color{red} NOTE:} The above rule needs some thinking about. As such, I have decided to not include it in the rule section yet.\n\n{\\color{red} NOTE:} Write some examples that show how to use the rules to prove nil-safety.\n\n\\section{Rules}\n\\subsection{Bool}\n\\begin{verbatim}\n H >> true in Bool by intro-true.\n H >> false in Bool by intro-false\n H >> Bool in U1 by bool-intro-universe.\n\\end{verbatim}\n\\subsection{Nil}\n\\begin{verbatim}\n H >> nil in Nil by intro-nil.\n -- TODO: Write the choking type rules.\n -- Needs clarification\n\\end{verbatim}\n\\subsection{Functions}\n\\begin{verbatim}\n H >> (fun x => b) in (x:A) -> B by intro-function.\n x:A, H >> b in B.\n H >> A in Ui.\n H >> B(x) in Ui. -- We may have to be careful about universe levels?\n H >> (x:A) -> B in Ui by function-intro-universe.\n H >> A in Ui.\n H >> B in Ui.\n\\end{verbatim}\n\\subsection{Universes}\n\\begin{verbatim}\n H >> Ui in Uj by universe-cumulative.\n -- Note, i < j.\n\\end{verbatim}\n\n\\section{Bibliography}\n{\\color{red} NOTE:} This Section needs to be properly formatted, right now it is just a dumping ground for things that have influenced this work.\\\\\n\\url{http://www.nuprl.org/book/} \\\\\n\\url{http://www.nuprl.org/documents/Constable/naive.pdf} \\\\\n\\url{http://ect.bell-labs.com/who/ajeffrey/papers/plpv12.pdf} \\\\\n\n\\end{document}\n", "meta": {"hexsha": "ffacb6d48507b46fd11ad5bf67eae9e180f22be8", "size": 8136, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paper/semantics_v2.tex", "max_stars_repo_name": "fhackett/pixy-lang", "max_stars_repo_head_hexsha": "a697da339dfcc0c3d317a3a6448950f04a6c5e0d", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 19, "max_stars_repo_stars_event_min_datetime": "2017-10-14T06:35:56.000Z", "max_stars_repo_stars_event_max_datetime": "2020-08-19T14:26:19.000Z", "max_issues_repo_path": "paper/semantics_v2.tex", "max_issues_repo_name": "fhackett/pixy-lang", "max_issues_repo_head_hexsha": "a697da339dfcc0c3d317a3a6448950f04a6c5e0d", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-10-13T21:36:43.000Z", "max_issues_repo_issues_event_max_datetime": "2017-10-13T21:36:43.000Z", "max_forks_repo_path": "paper/semantics_v2.tex", "max_forks_repo_name": "fhackett/pixy-lang", "max_forks_repo_head_hexsha": "a697da339dfcc0c3d317a3a6448950f04a6c5e0d", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2017-10-13T21:34:21.000Z", "max_forks_repo_forks_event_max_datetime": "2019-02-19T17:07:36.000Z", "avg_line_length": 56.5, "max_line_length": 813, "alphanum_fraction": 0.7223451327, "num_tokens": 2141, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6297745935070806, "lm_q2_score": 0.5428632831725052, "lm_q1q2_score": 0.34188150348988366}} {"text": "%---------------------------------------------------------------------\n% Writeup of the improved search of Run II data for single top quark\n% production at DZero.\n% Started: Oct 2006\n% Authors: The Single Top Working Group\n%---------------------------------------------------------------------\n%\n\n\\appendix\n\\section*{Appendix 3 --- Normalization of Non-Top Backgrounds to Data}\n\\label{appendix-matrixmethod}\n\nWe use the matrix method~\\cite{mm_explained} to estimate how many\nevents in the preseleceted sample contain a fake lepton (originating\nfrom QCD-multijet production) and how many events have a real isolated\nlepton (like the ones originating from $W$+jets or $\\ttbar$). Two\ndata samples are defined: the \\emph{tight} sample, which is the signal\nsample after all selection cuts have been applied; and the\n\\emph{loose} sample, where the same selection has been applied but\nrequiring only loose lepton isolation, thus omitting the tight\nrequirements in the lepton isolation.\n\nThe tight data sample, with $N_{{\\rm tight}}$ events, is a subset of\nthe loose data sample with $N_{\\rm loose}$ events. The loose sample\ncontains $N_{\\rm loose}^{{\\rm real-}\\ell}$ events with a real lepton\n(signal-like events, mostly $W$+jets and $\\ttbar$) and $N_{\\rm\nloose}^{{\\rm fake-}\\ell}$ fake lepton events, which is the number of\nQCD events in the loose sample. We measure the efficiency for a real\nisolated lepton to pass the tight lepton isolation selection and call\nit $\\varepsilon_{{\\rm real-}\\ell}$; and the efficiency for a fake\nlepton to pass the tight lepton isolation criteria, $\\varepsilon_{{\\rm\nfake-}\\ell}$.\n\nWith these definitions, the matrix method is applied by using these\ntwo equations:\n\\begin{eqnarray*}\nN_{\\rm loose} & = & N_{\\rm loose}^{{\\rm fake-}\\ell}\n + N_{\\rm loose}^{{\\rm real-}\\ell} \\\\\nN_{{\\rm tight}} & = & N_{{\\rm tight}}^{{\\rm fake-}\\ell}\n + N_{{\\rm tight}}^{{\\rm real-}\\ell} \n = \\varepsilon_{{\\rm fake-}\\ell} \\;\n N_{\\rm loose}^{{\\rm fake-}\\ell}\n + \\varepsilon_{{\\rm real-}\\ell} \\;\n N_{\\rm loose}^{{\\rm real-}\\ell} \n\\end{eqnarray*}\n\\noindent to solve for $N_{\\rm loose}^{{\\rm fake-}\\ell}$ and $N_{\\rm\nloose}^{{\\rm real-}\\ell}$ so that the QCD and the $W$-like\ncontributions in the tight sample $N_{\\rm tight}^{{\\rm fake-}\\ell}$\nand $N_{\\rm tight}^{{\\rm real-}\\ell}$ can be determined.\n\nThe results of the matrix method, which we apply separately in each\njet multiplicity bin, are shown in Table~\\ref{mm-numbers}. The\npretagged orthogonal sample is scaled to $N_{\\rm tight}^{{\\rm\nfake-}\\ell}$ and the $W$+jets MC samples\n($Wb\\bar{b}$+$Wc\\bar{c}$+$Wjj$) are scaled to $N_{\\rm tight}^{{\\rm\nreal-}\\ell}$, after subtracting the expected $\\ttbar$ events in each\nbin of the tight sample.\n\n\\vspace{0.2in}\n\\begin{table}[!h!tbp]\n\\begin{center}\n\\begin{minipage}{6.5in}\n\\begin{ruledtabular}\n\\begin{tabular}{l||ccccc|ccccc}\n\\multicolumn{11}{c}{\\hspace{1in}\\underline{Normalization of $W$+Jets and Multijets to Data}}\\vspace{0.1in} \\\\\n& \\multicolumn{5}{c|}{Electron Channel} & \\multicolumn{5}{c}{Muon Channel} \\\\\n & 1 jet & 2 jets & 3 jets & 4 jets & 5+ jets\n & 1 jet & 2 jets & 3 jets & 4 jets & 5+ jets \\\\\n\\hline\n$N_{{\\rm loose}}$ & 38,935 & 15,213 & 7,118 & 2,191 & 654 & 18,714 & 7,092 & 3,054 & 878 & 221 \\\\\n$N_{{\\rm tight}}$ & 27,370 & 8,220 & 3,075 & 874 & 223 & 17,816 & 6,432 & 2,590 & 727 & 173 \\\\\n$N_{\\rm tight}^{{\\rm fake-}e}$ & 1,691 & 1,433 & 860 & 256 & 86 & 498 & 329 & 223 & 56 & 10 \\\\\n$N_{\\rm tight}^{{\\rm real-}e}$ & 25,679 & 6,787 & 2,215 & 618 & 137 & 17,319 & 6,105 & 2,369 & 669 & 162 \\\\\n$\\varepsilon_{{\\rm real-}e}$ & 0.87 & 0.87 & 0.87 & 0.87 & 0.87 & 0.99 & 0.99 & 0.99 & 0.96 & 0.88 \\\\\n$\\varepsilon_{{\\rm fake-}e}$ & 0.18 & 0.19 & 0.19 & 0.17 & 0.17 & 0.41 & 0.36 & 0.34 & 0.31 & 0.25\n\\end{tabular}\n\\end{ruledtabular}\n\\vspace{-0.1in}\n\\caption[mmnumbers]{Matrix method yields in the electron and muon channels:\nthe loose and tight selected events and the expected contribution from\nmultijet and $W$-like events.}\n\\label{mm-numbers}\n\\end{minipage}\n\\end{center}\n\\end{table}\n\n% from the old muon-only table:\n%$N_{\\rm tight}^{{\\rm real-}\\mu}$($W$+jets) & 17,313 & 6,064 & 2,276 & 563 & 116 \\\\\n%$N_{\\rm tight}^{{\\rm real-}\\mu}$($\\ttbar$) & 6 & 41 & 93 & 106 & 46 \\\\\n\n\n\\begin{figure}[!h!tbp]\n\\begin{center}\n\\includegraphics[width=0.32\\textwidth]{figures/electron/cc_EqOneJet_PreTag_WTransverseMass.eps\n\\includegraphics[width=0.32\\textwidth]{figures/electron/cc_EqTwoJet_PreTag_WTransverseMass.eps\n\\includegraphics[width=0.32\\textwidth]{figures/electron/cc_EqThreeJet_PreTag_WTransverseMass.eps\n\\includegraphics[width=0.32\\textwidth]{figures/muon/mu_EqOneJet_PreTag_WTransverseMass.eps \n\\includegraphics[width=0.32\\textwidth]{figures/muon/mu_EqTwoJet_PreTag_WTransverseMass.eps \n\\includegraphics[width=0.32\\textwidth]{figures/muon/mu_EqThreeJet_PreTag_WTransverseMass.eps\n\\end{center}\n\\vspace{-0.1in}\n\\caption[mu_2jet]{The leading jet $\\pt$, second leading jet $\\pt$, the muon\n$\\pt$, the $\\met$ and the $\\Delta R$(jet1,jet2) after selection but before tagging (first column), with 1\n$b$-tag (center column) and with 2 $b$-tags (right column).}\n\\label{fig:mu_2jet}\n\\end{figure}\n", "meta": {"hexsha": "248663a3bb7c9a1627891d31bd1945d0b733153e", "size": 5466, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "STnote/appendix_matrixmethod.tex", "max_stars_repo_name": "tgadf/thesis", "max_stars_repo_head_hexsha": "19d4a6bc7f7ac8660fce582322703d50e0d6bd31", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "STnote/appendix_matrixmethod.tex", "max_issues_repo_name": "tgadf/thesis", "max_issues_repo_head_hexsha": "19d4a6bc7f7ac8660fce582322703d50e0d6bd31", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "STnote/appendix_matrixmethod.tex", "max_forks_repo_name": "tgadf/thesis", "max_forks_repo_head_hexsha": "19d4a6bc7f7ac8660fce582322703d50e0d6bd31", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 51.0841121495, "max_line_length": 122, "alphanum_fraction": 0.6295279912, "num_tokens": 1920, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6039318479832804, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.34183661995377096}} {"text": "\\subsection{Controller Design using Dead-Time and PT1 element}\n\nNow that $T_u$, $T_g$, $K_{p,crit}$ and $\\tau_{crit}$ have been determined, we\nuse the formulas in tables \\ref{tab:ziegler_nichols}, \\ref{tab:CHR_rejection}\nand \\ref{tab:CHR_tracking} to create various P, PI and PID controllers. The\nresulting closed loop step functions are plotted in figures\n\\ref{fig:Tt_PT1_P}, \\ref{fig:Tt_PT1_PI} and \\ref{fig:Tt_PT1_PID}.\n\n\nThe P-controller obtained from the Ziegler-Nichols method is quite\nunbelievable. The motor requires about 30 seconds to reach its end speed, so\nwhy is it able to reach its target value after just three seconds? By\nsimulating what happens on the output of the controller (the voltage that gets\nfed into the motor), we see what's going on (see figure\n\\ref{fig:Tt_PT1_P_voltage}).\n\nThe voltage levels being used to control the motor grossly exceed the\ncapabilities of the driver and the maximum rating of the motor. One would have\nto decrease the $K_p$ parameter in this controller until the voltage reaches\nacceptable levels again in order to make this controller feasible. The issue\nthen, though, is that the target value doesn't get reached any more.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\imagewidth]{images/Tt_PT1_P}\n \\caption{Various P controllers}\n \\label{fig:Tt_PT1_P}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\imagewidth]{images/Tt_PT1_PI}\n \\caption{Various PI controllers}\n \\label{fig:Tt_PT1_PI}\n\\end{figure}\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\imagewidth]{images/Tt_PT1_PID}\n \\caption{Various PID controllers}\n \\label{fig:Tt_PT1_PID}\n\\end{figure}\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=\\imagewidth]{images/Tt_PT1_P_voltages}\n \\caption{Simulated voltage on the input of the motor}\n \\label{fig:Tt_PT1_P_voltage}\n\\end{figure}\n\n\\clearpage\n", "meta": {"hexsha": "6e4fb683ecd80dfea89895476965682ecef6fab0", "size": 2001, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "versuche/rtGL/labor2/sections/simulations/controllers_Tt_PT1.tex", "max_stars_repo_name": "TheComet93/laborjournal", "max_stars_repo_head_hexsha": "5b83c35ec2580a22106d755f466dc6371d7444ee", "max_stars_repo_licenses": ["WTFPL"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "versuche/rtGL/labor2/sections/simulations/controllers_Tt_PT1.tex", "max_issues_repo_name": "TheComet93/laborjournal", "max_issues_repo_head_hexsha": "5b83c35ec2580a22106d755f466dc6371d7444ee", "max_issues_repo_licenses": ["WTFPL"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "versuche/rtGL/labor2/sections/simulations/controllers_Tt_PT1.tex", "max_forks_repo_name": "TheComet93/laborjournal", "max_forks_repo_head_hexsha": "5b83c35ec2580a22106d755f466dc6371d7444ee", "max_forks_repo_licenses": ["WTFPL"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 38.4807692308, "max_line_length": 78, "alphanum_fraction": 0.7206396802, "num_tokens": 595, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3418366118838624}} {"text": "\\documentclass[12pt]{article}\n%\\usepackage{a4wide}\n\\usepackage[margin=2.5cm,a4paper]{geometry}\n%\\usepackage[latin1]{inputenc}\n\\usepackage{amsmath}\n\\usepackage{amsthm}\n\\usepackage{paralist}\n\\usepackage{bbm}\n\\usepackage{amssymb}\n\\usepackage{pdfpages}\n\\usepackage[T1]{fontenc}\n\n\\usepackage[pdftex,bookmarks,colorlinks,breaklinks,pdfpagelabels]{hyperref} \n\n\\usepackage{epic,eepic,latexsym,verbatim}\n%\\usepackage{youngtab}\n%\\usepackage{bbm}\n\\newcommand{\\ket}[1]{|#1\\rangle}\n\\newcommand{\\bra}[1]{\\langle #1|}\n\\newcommand{\\bracket}[2]{\\langle #1|#2\\rangle}\n\\newcommand{\\ketbra}[1]{|#1\\rangle\\langle #1|}\n\\newcommand{\\average}[1]{\\langle #1\\rangle}\n\\newcommand{\\minus}{\\!-\\!}\n\n\\newcommand{\\om}{\\omega}\n%\\newcommand{\\th}{\\theta}\n\\newcommand{\\rn}{\\mathbb{R}^n}\n\\newcommand{\\R}{\\mathbb{R}}\n\\setlength{\\parindent}{0pt}\n\\newtheorem{theorem}{Theorem}\n\\newcommand{\\question}[2]{{\\bf #1} {\\em #2}\\\\}\n\\newcommand{\\problem}[3]{{\\bf #1} {\\em #2}\\\\[5\\lineskip]\n{#3}}\n\n\\newcounter{dummy}\n\\setcounter{dummy}{0}\n\\newtheorem{p}{Exercise}[dummy]\n\\renewcommand\\thep{\\arabic{p}}\n\\setcounter{secnumdepth}{0}\n\\overfullrule=5pt\n\n\\author{Joseph M.\\ Renes}\n\\title{Solutions to Baez and Muniain's \\\\\\emph{Gauge Fields, Knots and Gravity}}\n\\begin{document}\n\\renewcommand{\\abstractname}{\\vspace{-\\baselineskip}}\n\n\n\\hypersetup{pageanchor=true}\n\n\\maketitle\n\n\\begin{abstract}\nHere are my solutions to some of the problems in \\emph{Gauge Fields, Knots and Gravity}. \nAt some point I also wrote up a few notes on the Lie derivative in the style of the book.\\\\\n\nversion 1.0 --- Importing original writeup of solutions and Lie derivative.\n\\end{abstract}\n\n\\pdfbookmark[1]{\\contentsname}{Contents}\n\\tableofcontents\n\n\n\n\\newpage\n\n\\section{Part I}\n\\input{src/I1}\n\n\\input{src/I2}\n\n\\input{src/I3}\n\n\\input{src/I4}\n\n\\input{src/I5}\n\n\\input{src/I6}\n\n\n\\section{Part II}\n\\setcounter{dummy}{1}\n\\setcounter{p}{0}\n\n\\input{src/II1}\n\n\\setcounter{p}{54}\n\n\\input{src/II2}\n\n\\input{src/II3}\n\n%\\section{The Lie derivative}\n\n%\\pdfbookmark[1]{The Lie derivative}{lie}\n\\includepdf[pages=-,addtotoc={1,section,1,The Lie derivative,lie}\n]{src/lie.pdf}\n\n\\includepdf[pages=-]{src/expmap.pdf}\n\n\n\n\n\n\\end{document}", "meta": {"hexsha": "e7b4c326fad2809b80bc9e7100cfa47475a075d7", "size": 2139, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "gfkg.tex", "max_stars_repo_name": "joerenes/Baez-Muniain-solutions", "max_stars_repo_head_hexsha": "e1e38de9acab877bc4200af59c7910d42de748ca", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 3, "max_stars_repo_stars_event_min_datetime": "2017-04-13T12:10:03.000Z", "max_stars_repo_stars_event_max_datetime": "2019-04-19T18:18:34.000Z", "max_issues_repo_path": "gfkg.tex", "max_issues_repo_name": "joerenes/Baez-Muniain-solutions", "max_issues_repo_head_hexsha": "e1e38de9acab877bc4200af59c7910d42de748ca", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 1, "max_issues_repo_issues_event_min_datetime": "2017-04-13T12:15:30.000Z", "max_issues_repo_issues_event_max_datetime": "2017-04-13T20:19:44.000Z", "max_forks_repo_path": "gfkg.tex", "max_forks_repo_name": "joerenes/Baez-Muniain-solutions", "max_forks_repo_head_hexsha": "e1e38de9acab877bc4200af59c7910d42de748ca", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 20.5673076923, "max_line_length": 91, "alphanum_fraction": 0.7223001403, "num_tokens": 744, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5660185351961015, "lm_q2_score": 0.6039318337259583, "lm_q1q2_score": 0.3418366118838624}} {"text": "\\chapter*{Introduction}\n\nIn math, people prefer to use ``$ \\log_e $'' or just ``$ \\log $'' rather than ``$ \\ln $'', which is the opposite to many scientific fields. Due to its simplicity (especially when typing), I will also stick to the usage of ``$ \\ln $'' which is identical to ``$ \\log_e $''. But to avoid any confusion, I will always specify the base 10 (``$ \\lg $'').\n\nAt many places in physics and astronomy, people use $ \\equiv $ as ``is defined as'' or any kind of ``trivial by the definition'' cases. But I will use $ := $ for the definition and use $ \\equiv $ to mean ``equality holds trivially by the definition''. For example, under 1-D linear homogeneous isotropic matter, linear electric polarization is written as $ P \\equiv \\epsilon_0 \\chi E $ or $ \\chi := P/\\epsilon_0 E $. The latter is the \\textit{definition} of electric susceptability, while the former is \\textit{trivially true by the definition}. \n\nFor distributions, $ \\sim $ means ``follows'', i.e., $ X \\sim \\mathcal{N}(\\mu, \\sigma^2) $ means ``the random variable $ X $ follows the Gaussian (normal) distribution with mean $ \\mu $ and standard deviation $ \\sigma $ (thus variance $ \\sigma^2 $).'' When this is an approximation, the symbol with dot ``$ \\simdot $'' is used, such as $ X \\simdot \\mathcal{N}(\\mu, \\sigma^2) $. In this case, it is interpreted as ``approximately follows.''\n\nIn this note, I used some math-like notations like theorem (Thm). However, the theorem here is not necessarily the same as that of mathematics. I tried to put some important assumptions and summary notes to theorem.\n\nThe references are \n\\begin{itemize}\n\\item ``Walpole'' is Walpole et al. 2013, ``Essentials of Probabilities and Statistics''.\n\\item ``Sivia'' is Sivia and Skilling 2006, ``Data Analysis A Bayesian Tutorial'', 2/e.\n\\end{itemize}", "meta": {"hexsha": "219f954ff92ca9b0ff472699c4aadbb7372526f6", "size": 1808, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Books/chaps/00_intro.tex", "max_stars_repo_name": "ysBach/SNU_AOclass", "max_stars_repo_head_hexsha": "e2e364b08c2e6e129c267db9cbd76cfd0ab77527", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2020-03-23T06:14:52.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-14T01:49:51.000Z", "max_issues_repo_path": "Books/chaps/00_intro.tex", "max_issues_repo_name": "ysBach/SNU_AOclass", "max_issues_repo_head_hexsha": "e2e364b08c2e6e129c267db9cbd76cfd0ab77527", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 9, "max_issues_repo_issues_event_min_datetime": "2020-05-04T17:21:49.000Z", "max_issues_repo_issues_event_max_datetime": "2021-05-24T11:41:55.000Z", "max_forks_repo_path": "Books/chaps/00_intro.tex", "max_forks_repo_name": "ysBach/SNU_AOclass", "max_forks_repo_head_hexsha": "e2e364b08c2e6e129c267db9cbd76cfd0ab77527", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2020-05-10T14:19:34.000Z", "max_forks_repo_forks_event_max_datetime": "2021-07-14T09:18:08.000Z", "avg_line_length": 120.5333333333, "max_line_length": 546, "alphanum_fraction": 0.7101769912, "num_tokens": 478, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.603931819468636, "lm_q2_score": 0.5660185351961015, "lm_q1q2_score": 0.34183660381395375}} {"text": "\\documentclass[10pt,journal,compsoc]{IEEEtran}\n\n\\usepackage[pdftex]{graphicx} \n\\usepackage{cite}\n\\usepackage{subcaption} \n\\hyphenation{op-tical net-works semi-conduc-tor}\n\\graphicspath{ {../images/} }\n\n\\begin{document}\n\n\\title{Robotics Software Engineer Nanodegree: Deep RL Arm Manipulation}\n\\author{Manuel Huertas L\\'opez}\n\n\\markboth{Deep RL Arm Manipulation project, Robotic Nanodegree, Udacity}%\n{}\n\n\\IEEEtitleabstractindextext{%\n\n\\begin{abstract}\n\nReinforcement learning is an area of ​​machine learning. It instructs an agent to take actions in an environment in order to maximize a cumulative reward. In this project a DQN algorithm will be used to guide a robotic arm to touch a cylindrical object. The idea behind the algorithm is to find an action value function; this function will give the cumulative expected reward for every state and action pair values. Once this function is obtained the action for every state will be the one that maximizes the reward. DQN is a variation of the algorithm where this function is replaced with a Deep Neural Network.\n\n\\end{abstract}\n\n\\begin{IEEEkeywords}\nRobot, Udacity, Deep Reinforcemente Learning.\n\\end{IEEEkeywords}}\n\n\\maketitle\n\\IEEEdisplaynontitleabstractindextext\n\\IEEEpeerreviewmaketitle\n\\section{Introduction}\n\nIn this project a Robotic Arm with three degree of freedom will be taught in order to touch a cylindrical shape in the scene. Rather than using the forward and backward kinematic to calculate the position of the arm with respect to the joint angles or to move the arm to a goal position sending demands to the joints, the robot will try some random movements and it will receive a positive or negative reward. A video camera will be recording constatinualsy the robot movement and this information is used to train a neural network, the network will try to reproduce the sequence of movement that maximise the cumulative reward. As the final goal the robot will find the combination of joint movement, without a model, to approximate and finally touch the object in the scene.\n\nEvery Episodic finalized when the robot: touch the ground, touch the cylinder or the maximum number of iteration is reached. The final reward will be positive just in case the robot touch the cylinder, and there are some intermediate positive reward bases on smooth movement and when the robot is closer to the cylinder. The problem can be defined as a Markov Decision Process \\cite{udacity} where:\n\\begin{itemize}\n\\item a finite set of states: the image from the video camera and all the pixel value combinations.\n\\item a finite set of actions: the movement every joint.\n\\item a finite set of rewards: the positive and negative reward depending on the robot behaviour.\n\\item the one-step dynamics of the environment: given the current state and the action, the expected new state and new reward. The neural network will obtain this during thetrainingg process.\n\\item a discount rate between [0,1]: the discount rate.\n\\end{itemize}\n\n\\label{sec:introduction}\n\n\\section{Background / Formulation}\n\nOur goal is to find a function that for every state, the image in this case, tell use which action to take, the movement of the joint. This problem is not easy since there are a big number of states, pixel combination of the image, and sequences of movements. In this project an algorithm based in DQN have been used. In DQN a Deep Neural Network will build a function where the input will be the pixel of the image and the output the expected cumulative reward for every possible action. Selecting the action with the highest cumulative reward will give us the desired action. In addition, a Long Short Term Memory network have been used to use the information of the sequence of multiple images, instead of a single frame.\n\n\\subsection{Deep Q Network}\n\nThe Q-Learning algoritm is a model-free algoritm. The values for each state-action pair are estimated based on observation of the environment \\cite{deepreinforcementlearning}. The Q-Learning algoritm estimate the state action funcition using the following equation:\n\n\\begin{align}\nQ(s_{t},a_{t})=(1-\\alpha) \\cdot Q(s_{t},a_{t})+\\alpha \\cdot (r_{t}+\\gamma max_a\\cdot Q(s_{t+1},a))\n\\end{align}\n\nThe meaning of this parameter are:\n\n\\begin{itemize}\n\\item s: state value at time t, from the observation.\n\\item a: action value a time t.\n\\item alpha: learning rate, the way in which new values replaces old ones.\n\\item r: reward at time t.\n\\item gamma: discount factor.\n\\end{itemize}\n\nIn addition, the greedy policy is used to select between a current best action, exploit, or to select a random action, explore. If during the learning process we always select the action with currently the best reward we are never going to explore new solutions whose can give a better discounted future reward. On the other one, we need to select the current best action in order to use the knowledge we are acquiring iteratively. One way to accomplish this is to explore more at the beginning of the learning process, and gradually reduce the exploration favouring the exploitation of the current knowledge. For doing this the following parameters are defined:\n\n\\begin{itemize}\n\\item EPS\\_START: starging gready value.\n\\item EPS\\_END: ending gready value.\n\\item EPS\\_DECAY: gready decay rate\n\\end{itemize}\n\nFinally, the state action function is replaced with a deep neural network. The input of the network is the image from the video camera, and the output the reward or every action. The loss function, for the gradient descent algorithm, is defined as:\n\n\\begin{align}\nLoss=1/2 \\cdot [r_{t} + max_{a_{t+1}}\\cdot Q(s_{t+a},a_{t+1};\\theta_{t-1}}) - Q(s,a;\\theta}]^2\n\\end{align}\n\nTwo extra techniques are included to make the neural network converge to the final solution. Experience replay, where a memory of experience is used to store previous values; mini-batches of these values are used as input of the neural network during learning. This technique helps to break the temporal dependencies of the learning process. Since the target of the loss function and the predictions have parameters in common, the process of learning can not converge to the correct solution, fixed Q targets is used to update the Q function after several iterations, in this way the small changes in the Q function approximation are not affecting the learning process.\n\nThe following extra parameters are defined.\n\n\\begin{itemize}\n\\item INPUT\\_WIDTH: Image from the camera, width\n\\item INPUT\\_HEIGHT: Image from the camera, height\n\\item OPTIMIZER: Optimizer for the gradient descent algorithm \n\\item LEARNING\\_RATE: Neural Network learning rate.\n\\item REPLAY\\_MEMORY: Size of the memory for the experience replay technique.\n\\item BATCH\\_SIZE: Batch size for the experience replay technique.\n\\end{itemize}\n\n\\subsection{Long Short Term Memory}\n\nLSTM is a special Recurrent Neural Network, capable of learning long-term dependencies. The Long short-term memory adds memory to the DQN to remember the sequence of movements of the joint to find and touch the object in the scene \\cite{lstmnetwork}.\n\n\\section{Robotic Arm}\n\nIn a reinforcement learning problem, a temporary series of interactions are produced until the episodic ends: S_{0},A_{0},R_{1},S_{1},A_{1},R_{2},...,R_{T},S_{T}\n\nIn this project, an episode ends when: the robot's arm touches the ground, a maximum of iterations is reached, or the robot touches the object in the scene; the states are the image, captured by the video camera, of the scene; the actions can be to move the joints at random using position or speed control; finally, the reward strategy is defined as a negative number when the robot finishes the episodic without touching the object, the positive number when it touches the object and an incremental or decremental reward when the arm approaches the target or not.\n\nDifferent reward strategies were tested, the final choose solution was: a big number positive number when the robot touches the object; a negative number when the robot touch the ground or the maximum iterations are reached, to this negative number a 1\\% of the distance to the goal is added, in order to decrease the negative reward depending on the closest to the goal; a smooth reward when the robot is searching the object in function of the smoothest of the movement and the distance to the goal.\n\n\\subsection{Rewards}\n\nThe positive reward when the object is touched is 1000000.0. The negative base reward is -1.0. When the episodice ends without success to the negative base reward is added 1\\% the distance to the goal. Finally when the Robot is searching a soomoth average goal delta is added to the distance to the goal as follow:\n\nAverageGoalDelta = ( AverageGoalDelta * 0.2 ) + (DistanceDelta * (1-0.2));\n\nReward Searching = AverageGoalDelta + 0.01*exp(-pow(DistanceGoal,2));\n\nThe same rewards were used for both objectives. \n\n\\subsection{Hiper Paramters}\n\nFinally the hiper-parameters selecting were:\n\n\\begin{itemize}\n\\item VELOCITY\\_CONTROL[false]: The position control was used.\n\\item NUMBER\\_ACTIONS[6]: The number of actions was calculated as number of DOF x 2.\n\\item ALLOW\\_RANDOM[true]: The learnning process allows random decisions to explore new solutions.\n\\item GAMMA[0.9]: discount factor, selected was 0.9.\n\\item EPS\\_START[0.9]: Starting greedy value.\n\\item EPS\\_END[0.01] Ending greedy value.\n\\item EPS\\_DECAY[200]: Greedy decay rate\n\\item INPUT\\_WIDTH[64]: Image from the camera width\n\\item INPUT\\_HEIGHT[64]: Image from the camera height\n\\item OPTIMIZER[RMSprop]: Optimizer for the gradient descent algorithm \n\\item LEARNING\\_RATE[0.01]: Neural Network learning rate.\n\\item REPLAY\\_MEMORY[10000]: Size of the memory for the experience replay technique.\n\\item BATCH\\_SIZE[64]: Batch size for the experience replay technique.\n\\item USE\\_LSTM[true]: Uses Long Short Memory.\n\\item LSTM\\_SIZE[512]: Sizes of the LSTM network.\n\\end{itemize}\n\nThe same Hiper Parameters were used for both objectives. \n\n\\section{Results}\n\nIn order to find the reward and hiper-parameters a trial and error strategy was used. At the beginning velocity control was used, but the desired accuracy of 80\\% of success touching the object with the gripper base was not achieve, for that reason the position control was selected. The reward strategy consists in give a high positive number when the robot touches the object, this big number was selected because the robot could accumulate small rewards during the searching process and it was important to stimulate the robot not to keep searching but to move toward the goal. During the searching process a small amount of the distance to goal was added to the reward to indicate the robot's arm that the action should move the arm in the object direction. Also, a factor depending on of the delta movement, was used to increase, when the movement was directing the robot to the goal, or decrease when the movement was in the other direction. This searching strategy was used to indicate the robot's arm to keep moving and to do it smoothly to the goal direction. Finally, when the robot ends the episodic without success, a negative reward is given. To this negative reward a factor of the distance to the goal is added to indicate the robot how close was to the solution, even having failed.\n\n\\begin{figure}[h]\n\\centering\n\\begin{subfigure}[b]{0.4\\textwidth}\n\\includegraphics[scale=0.25]{Learnning_Collision_ARM}\n\\caption{Robot's Arm touching the object}\n\\end{subfigure}\n\\begin{subfigure}[b]{0.4\\textwidth}\n\\includegraphics[scale=0.27]{Learnning_Collision_Gripper}\n\\caption{Robot's Arm touching the object with the base gripper}\n\\end{subfigure}\n\\caption{Robot's Arm}\n\\end{figure}\n\n\\section{Discussion}\n\nThe project consists in completing eight tasks:\n\n\\begin{enumerate}\n\\item Subscribe to camera and collision topics: This was easily achieved following the udacity instructions.\n\\item Create the DQN Agent: Udacity provided the class, it was just necessary to create an instance.\n\\item Velocity or position based control of arm joints: position control was selected because it was easier to define the rewards.\n\\item Reward for robot gripper hitting the ground: using a bounding box for the gripper it was detected when it was hitting the ground; a negative reward plus a 1\\% to the distance to the goal was used.\n\\item Issue an interim reward based on the distance to the object: an incremental, reward positive or negative, was used, depending on the robot movement to the goal or not.\n\\item Issue a reward based on collision between the arm and the object: a very high number was used to stimulate the robot to move to the goal.\n\\item Tuning the hyper parameters: a trail an error approach was used to find the final parameters.\n\\item Issue a reward based on collision between the arm’s gripper base and the object: the same reward and hyper parameters were used for the two objectives.\n\\end{itemize}\n\n\n\\section{Conclusion / Future work}\n\nIn order to improve the project a deeper understating about how the reward affects the learning process will be needed. A way to do this can be to visualize the Q-Function using a 2D plot where the x-axis can be the actions, y-axis the state, and the coloured value of the graph the reward. The rewards were selected using some logic about how the human will learn to solve the problem. Different strategies will be needed to be explored in order to find a solution that converges quicklier to the final solution.\n\n% Bibliography\n\\begin{thebibliography}{9}\n\\bibliographystyle{ieeetr}\n\n\\bibitem{udacity} \nRobotics Software Engineer Nanodegree program\n\\textit{https://eu.udacity.com}\n\n\\bibitem{deepreinforcementlearning} \n\\textit{Human-level control through deep reinforcement learning}. \n\n\\bibitem{lstmnetwork} \nUnderstanding LSTM Networks\n\\textit{http://colah.github.io/posts/2015-08-Understanding-LSTMs}\n\n\\end{thebibliography}\n\n\\end{document}", "meta": {"hexsha": "11ae583a6b55b6fbf5f56c9e63fc8841e1383eeb", "size": 13867, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/udacity-deeprl.tex", "max_stars_repo_name": "mahulo2009/RoboND-DeepRL-Project", "max_stars_repo_head_hexsha": "de9318327edd2c13dd87284d35ee26edeba0db4f", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/udacity-deeprl.tex", "max_issues_repo_name": "mahulo2009/RoboND-DeepRL-Project", "max_issues_repo_head_hexsha": "de9318327edd2c13dd87284d35ee26edeba0db4f", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/udacity-deeprl.tex", "max_forks_repo_name": "mahulo2009/RoboND-DeepRL-Project", "max_forks_repo_head_hexsha": "de9318327edd2c13dd87284d35ee26edeba0db4f", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 70.75, "max_line_length": 1299, "alphanum_fraction": 0.7913030937, "num_tokens": 3194, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.607663184043154, "lm_q2_score": 0.5621765008857981, "lm_q1q2_score": 0.34161396252250303}} {"text": "% Created 2022-05-03 Tue 11:38\n% Intended LaTeX compiler: pdflatex\n\\documentclass[t]{beamer}\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage{graphicx}\n\\usepackage{longtable}\n\\usepackage{wrapfig}\n\\usepackage{rotating}\n\\usepackage[normalem]{ulem}\n\\usepackage{amsmath}\n\\usepackage{amssymb}\n\\usepackage{capt-of}\n\\usepackage{hyperref}\n\\mode{\\usetheme{Amsterdam}}\n\\mode{\\usecolortheme{rose}}\n\\usepackage{fontspec}\n\\usepackage{polyglossia}\n\\setmainlanguage[babelshorthands=true]{german}\n\\usepackage{hyperref}\n\\usepackage{color}\n\\usepackage{xcolor}\n\\usepackage[misc]{ifsym}\n\\definecolor{darkblue}{rgb}{0,0,.5}\n\\definecolor{darkgreen}{rgb}{0,.5,0}\n\\definecolor{islamicgreen}{rgb}{0.0, 0.56, 0.0}\n\\definecolor{darkred}{rgb}{0.5,0,0}\n\\definecolor{mintedbg}{rgb}{0.95,0.95,0.95}\n\\definecolor{arsenic}{rgb}{0.23, 0.27, 0.29}\n\\definecolor{prussianblue}{rgb}{0.0, 0.19, 0.33}\n\\definecolor{coolblack}{rgb}{0.0, 0.18, 0.39}\n\\hypersetup{colorlinks=true, breaklinks=true, anchorcolor=blue,linkcolor=white, citecolor=islamicgreen, filecolor=darkred, urlcolor=darkblue}\n\\usepackage{booktabs}\n\\usepackage{pgf}\n\\usepackage{minted}\n\\RequirePackage{fancyvrb}\n\\DefineVerbatimEnvironment{verbatim}{Verbatim}{fontsize=\\scriptsize}\n\\usetheme{default}\n\\author{Göran Kirchner\\thanks{e\\_kirchnerg@doz.hwr-berlin.de}}\n\\date{2021-05-03}\n\\title{Funktionale Programmierung in F\\# (5)}\n\\subtitle{Parser Combinators}\n\\hypersetup{\n pdfauthor={Göran Kirchner},\n pdftitle={Funktionale Programmierung in F\\# (5)},\n pdfkeywords={},\n pdfsubject={},\n pdfcreator={Emacs 27.2 (Org mode 9.5.2)}, \n pdflang={English}}\n\\begin{document}\n\n\\maketitle\n\n\\section{Ziel }\n\\label{sec:org0ee434b}\n\\begin{frame}[label={sec:orgc9d6c29}]{Programm}\n\\begin{itemize}\n\\item Hausaufgaben\n\\item Test\n\\item Parser (Kombinatoren)\n\\end{itemize}\n\\end{frame}\n\n\\section{Hausaufgaben }\n\\label{sec:org62f30a3}\n\\begin{frame}[label={sec:orgfc210c3},fragile]{Accumulate}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet rec accumulateR func input acc = \n match input with\n | [] -> acc |> List.rev\n | head::tail -> accumulateR func tail (func head :: acc)\nlet accumulate func input = accumulateR func input []\nlet test1 = accumulate (fun x -> x * x) [1; 2; 3]\nlet test2 = accumulate (fun (x:string) -> x.ToUpper()) [\"hello\"; \"world\"]\ntest2\n\\end{minted}\n\n\\begin{verbatim}\nval accumulateR: func: ('a -> 'b) -> input: 'a list -> acc: 'b list -> 'b list\nval accumulate: func: ('a -> 'b) -> input: 'a list -> 'b list\nval test1: int list = [1; 4; 9]\nval test2: string list = [\"HELLO\"; \"WORLD\"]\nval it: string list = [\"HELLO\"; \"WORLD\"]\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgd3a40f9},fragile]{Space Age}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\ntype Planet = \n | Mercury\n | Venus\n | Earth\n | Mars\n | Jupiter\n | Saturn\n | Uranus\n | Neptune\nlet orbitalPeriodRelativeToEarthOn planet = \n match planet with\n | Mercury -> 0.2408467\n | Venus -> 0.61519726\n | Earth -> 1.0\n | Mars -> 1.8808158\n | Jupiter -> 11.862615\n | Saturn -> 29.447498\n | Uranus -> 84.016846\n | Neptune -> 164.79132\n\\end{minted}\n\n\\begin{verbatim}\ntype Planet =\n | Mercury\n | Venus\n | Earth\n | Mars\n | Jupiter\n | Saturn\n | Uranus\n | Neptune\nval orbitalPeriodRelativeToEarthOn: planet: Planet -> float\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:org155f1d4},fragile]{Space Age (II)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nopen System\n[]\nlet SecondsInOneEarthYear = 31557600.0\nlet secondsInAYearOn planet =\n SecondsInOneEarthYear * orbitalPeriodRelativeToEarthOn planet\nlet round (number : float) = Math.Round(number, 2)\nlet age (planet: Planet) (seconds: int64): float =\n float seconds / (secondsInAYearOn planet)\n |> round\nlet test1 = age Earth 1000000000L\n\\end{minted}\n\n\\begin{verbatim}\n[]\nval SecondsInOneEarthYear: float = 31557600\nval secondsInAYearOn: planet: Planet -> float\nval round: number: float -> float\nval age: planet: Planet -> seconds: int64 -> float\nval test1: float = 31.69\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:org48530c4},fragile]{Zusammenfassung}\n \\begin{itemize}\n\\item nutze \\href{https://exercism.io}{exercism.io}!\n\\item Formatierung (dotnet fantomas)\n\\item Vermeide \\texttt{mutable}!!\n\\item nur wichtiges verdient einen Namen\n\\item Vertraue der \\alert{Pipe} (\\texttt{>>}, \\texttt{|>}, \\ldots{})!!\n\\item If-Then-Else mit Boolean ist unnötig\n\\item Parametrisiere!\n\\item If-Then-Else vermeiden \\ldots{} besser \\texttt{match}!\n\\item Be lazy! (vermeide \\texttt{for}-loops)\n\\item \\href{https://fsharpforfunandprofit.com/troubleshooting-fsharp/}{Troubleshooting F\\#}\n\\item \\href{https://docs.microsoft.com/de-de/dotnet/fsharp/style-guide/}{F\\#-Styleguide}\n\\end{itemize}\n\\end{frame}\n\n\\section{Test }\n\\label{sec:org3061079}\n\\begin{frame}[label={sec:orgd6d78e6}]{Test}\n\\begin{itemize}\n\\item 90 Minuten\n\\item Ergebnis per \\href{mailto://e\\_kirchnerg@doz.hwr-berlin.de}{E-Mail} an \\href{mailto://e\\_kirchnerg@doz.hwr-berlin.de}{e\\_kirchnerg@doz.hwr-berlin.de}.\n\\end{itemize}\n\n\\(\\leadsto\\) \\href{../src/5/test.md}{Test}\n\\end{frame}\n\n\\section{Parser }\n\\label{sec:org0caa37e}\n\\begin{frame}[label={sec:orga6ffe1c},fragile]{Parser 1 (hard-coded character)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nopen System\nlet A_Parser str =\n if String.IsNullOrEmpty(str) then\n (false,\"\")\n else if str.[0] = 'A' then\n let remaining = str.[1..]\n (true,remaining)\n else\n (false,str)\nlet inputABC = \"ABCD\";;\nlet inputZBC = \"ZBCD\";;\nlet test11 = A_Parser inputABC\nlet test12 = A_Parser inputZBC\n\\end{minted}\n\n\\begin{verbatim}\nval test11: bool * string = (true, \"BCD\")\nval test12: bool * string = (false, \"ZBCD\")\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgc4ae52c},fragile]{Parser 2 (match a specified character)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet pchar (charToMatch,str) =\n if String.IsNullOrEmpty(str) then\n let msg = \"No more input\"\n (msg,\"\")\n else \n let first = str.[0] \n if first = charToMatch then\n let remaining = str.[1..]\n let msg = sprintf \"Found %c\" charToMatch\n (msg,remaining)\n else\n let msg = sprintf \"Expecting '%c'. Got '%c'\" charToMatch first\n (msg,str)\n\\end{minted}\n\n\\begin{verbatim}\nval pchar: charToMatch: char * str: string -> string * string\n\\end{verbatim}\n\\end{frame}\n\n\n\\begin{frame}[label={sec:org82f0e21},fragile]{Parser 2 (2)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet inputABC = \"ABCD\";;\nlet inputZBC = \"ZBCD\";;\nlet test21 = pchar('A',inputABC) \nlet test22 = pchar('A',inputZBC)\n\\end{minted}\n\n\\begin{verbatim}\nval test21: string * string = (\"Found A\", \"BCD\")\nval test22: string * string = (\"Expecting 'A'. Got 'Z'\", \"ZBCD\")\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgfd2993e},fragile]{Parser 3 (return a Result)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet pchar (charToMatch, s) =\n if String.IsNullOrEmpty(s) then\n Error \"No more input\"\n else\n let first = s.[0]\n if first = charToMatch then\n let remaining = s.[1..]\n Ok (charToMatch, remaining)\n else\n let msg = sprintf \"Expecting '%c'. Got '%c'\" charToMatch first\n Error msg\n\\end{minted}\n\n\\begin{verbatim}\nval pchar: charToMatch: char * s: string -> Result<(char * string),string>\n\\end{verbatim}\n\\end{frame}\n\n\n\n\\begin{frame}[label={sec:org4f16965},fragile]{Parser 3 (2)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet test31 = pchar('A',inputABC) \nlet test32 = pchar('A',inputZBC) \nlet test33 = pchar('Z',inputZBC)\n\\end{minted}\n\n\\begin{verbatim}\nval test31: Result<(char * string),string> = Ok ('A', \"BCD\")\nval test32: Result<(char * string),string> = Error \"Expecting 'A'. Got 'Z'\"\nval test33: Result<(char * string),string> = Ok ('Z', \"BCD\")\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgf25d48e},fragile]{Parser 4 (use currying)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet pchar charToMatch str = \n if String.IsNullOrEmpty(str) then\n Error \"No more input\"\n else\n let first = str.[0] \n if first = charToMatch then\n let remaining = str.[1..]\n Ok (charToMatch,remaining)\n else\n let msg = sprintf \"Expecting '%c'. Got '%c'\" charToMatch first\n Error msg\n\\end{minted}\n\n\\begin{verbatim}\nval pchar: charToMatch: char -> str: string -> Result<(char * string),string>\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:org5144ddb},fragile]{Parser 4 (2)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet parseA = pchar 'A'\nlet inputABC = \"ABC\"\nlet inputZBC = \"ZBC\"\nlet test41 = parseA inputABC\nlet test42 = parseA inputZBC\nlet parseZ = pchar 'Z' \nlet test43 = parseZ inputZBC\n\\end{minted}\n\n\\begin{verbatim}\nval parseA: (string -> Result<(char * string),string>)\nval inputABC: string = \"ABC\"\nval inputZBC: string = \"ZBC\"\nval test41: Result<(char * string),string> = Ok ('A', \"BC\")\nval test42: Result<(char * string),string> = Error \"Expecting 'A'. Got 'Z'\"\nval parseZ: (string -> Result<(char * string),string>)\nval test43: Result<(char * string),string> = Ok ('Z', \"BC\")\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:orga7e56d5},fragile]{Parser 5 (type to wrap the parser function)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\ntype Parser<'T> =\n | Parser of (string -> Result<'T , string>)\nlet pchar charToMatch = \n let innerFn str =\n if String.IsNullOrEmpty(str) then\n Error \"No more input\"\n else\n let first = str.[0] \n if first = charToMatch then\n let remaining = str.[1..]\n Ok (charToMatch, remaining)\n else\n let msg = sprintf \"Expecting '%c'. Got '%c'\" charToMatch first\n Error msg\n Parser innerFn\n\\end{minted}\n\n\\begin{verbatim}\ntype Parser<'T> = | Parser of (string -> Result<'T,string>)\nval pchar: charToMatch: char -> Parser\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:orgb7ce182},fragile]{Parser 5 (2)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet parseA = pchar 'A'\nlet inputABC = \"ABC\"\nparseA inputABC\n\\end{minted}\n\n\\begin{verbatim}\nparseA inputABC;;\n ^^^^^^\n\n...: error FS0003: This value is not a function and cannot be applied.\n\\end{verbatim}\n\\end{frame}\n\n\n\\begin{frame}[label={sec:org732898b},fragile]{Parser 5 (3)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet run parser input = \n let (Parser innerFn) = parser \n innerFn input\nlet parseA = pchar 'A' \nlet inputABC = \"ABC\"\nlet test1 = run parseA inputABC\nlet inputZBC = \"ZBC\"\nlet test2 = run parseA inputZBC\n\\end{minted}\n\n\\begin{verbatim}\nval run: parser: Parser<'a> -> input: string -> Result<'a,string>\nval parseA: Parser = Parser \nval inputABC: string = \"ABC\"\nval test1: Result<(char * string),string> = Ok ('A', \"BC\")\nval inputZBC: string = \"ZBC\"\nval test2: Result<(char * string),string> = Error \"Expecting 'A'. Got 'Z'\"\n\\end{verbatim}\n\\end{frame}\n\n\\section{Parser Kombinatoren }\n\\label{sec:org3348783}\n\\begin{frame}[label={sec:org7b54569}]{Understanding Parser Combinators}\n\\(\\leadsto\\) \\href{./5 Understanding parser combinators.pdf}{Understanding parser combinators} (Scott Wlashin)\n\\end{frame}\n\n\\begin{frame}[label={sec:orgb2e078e}]{FParsec Tutorial}\n\\begin{itemize}\n\\item \\href{http://www.quanttec.com/fparsec/tutorial.html\\#}{FParsec Tutorial}\n\\item \\href{http://www.quanttec.com/fparsec/users-guide/}{User’s Guide}\n\\item \\href{http://www.quanttec.com/fparsec/about/fparsec-vs-alternatives.html}{FParsec vs alternatives}\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[label={sec:org84ac197},fragile]{Using FParsec (1)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\n#r \"../src/5/02-fparsec/lib/FParsecCS.dll\";; \n#r \"../src/5/02-fparsec/lib/FParsec.dll\";;\nopen FParsec\nlet test p str =\n match run p str with\n | Success(result, _, _) -> printfn \"Success: %A\" result\n | Failure(errorMsg, _, _) -> printfn \"Failure: %s\" errorMsg;;\ntest pfloat \"1.25\"\ntest pfloat \"1.25E 2\"\n\\end{minted}\n\n\\begin{verbatim}\nSuccess: 1.25 \nFailure: Error in Ln: 1 Col: 6\n1.25E 2\n ^\nExpecting: decimal digit\n\nval it: unit = ()\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:org1359eae},fragile]{Using FParsec (2)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet str s = pstring s\nlet floatBetweenBrackets:Parser = str \"[\" >>. pfloat .>> str \"]\";;\n\ntest floatBetweenBrackets \"[1.0]\"\ntest floatBetweenBrackets \"[]\"\ntest floatBetweenBrackets \"[1.0]\"\n\\end{minted}\n\n\\begin{verbatim}\nSuccess: 1.0 \nFailure: Error in Ln: 1 Col: 2\n[]\n ^\nExpecting: floating-point number\n\nSuccess: 1.0 \nval it: unit = ()\n\\end{verbatim}\n\\end{frame}\n\n\\begin{frame}[label={sec:org9ef280b},fragile]{Using FParsec (3)}\n \\begin{minted}[bgcolor=mintedbg,frame=none,framesep=0pt,mathescape=true,fontsize=\\scriptsize,breaklines=true,linenos=false,numbersep=5pt,gobble=0]{fsharp}\nlet betweenStrings s1 s2 p = str s1 >>. p .>> str s2;;\nlet floatBetweenBrackets_:Parser = pfloat |> betweenStrings \"[\" \"]\";;\nlet floatBetweenDoubleBrackets_:Parser = pfloat |> betweenStrings \"[[\" \"]]\";;\ntest floatBetweenBrackets_ \"[1.0]\"\ntest floatBetweenDoubleBrackets_ \"[[1.0]]\"\nlet between_ pBegin pEnd p = pBegin >>. p .>> pEnd;;\nlet betweenStrings_ s1 s2 p = p |> between_ (str s1) (str s2);;\ntest (many floatBetweenBrackets) \"\"\ntest (many floatBetweenBrackets) \"[1.0]\"\ntest (many floatBetweenBrackets) \"[2][3][4]\"\ntest (many floatBetweenBrackets) \"[1][2.0E]\"\n\\end{minted}\n\n\\begin{verbatim}\nSuccess: [] \nSuccess: [1.0] \nSuccess: [2.0; 3.0; 4.0] \nFailure: Error in Ln: 1 Col: 9\n[1][2.0E]\n ^\nExpecting: decimal digit\n\nval it: unit = ()\n\\end{verbatim}\n\\end{frame}\n\n\\section{Ende }\n\\label{sec:orgd52857a}\n\\begin{frame}[label={sec:org00f27f4}]{Zusammenfassung (Kurs)}\n\\begin{itemize}\n\\item Wichtige Werkzeuge (git, dotnet, code)\n\\item Elementare Syntax\n\\item Funktionen, Pattern Matching, Discriminated Unions (DU)\n\\item Tuple, Record, List, Array, Seq\n\\item funktionale Operationen auf Listen (Tail-Rekursion)\n\\item funktionaler Umgang mit fehlenden Daten (Option)\n\\item funktionaler Umgang mit Fehlern (Result)\n\\item funktionales Design (statt Patterns: Funktionen \\& Verkettung)\n\\item funktionales Refactoring\n\\item funktionales Domain Modeling (DDD)\n\\item eigenschaftsbasiertes Testen (Property Based Testing) (cool!!)\n\\item funktionale Parser (Kombinatoren) (noch cooler!!)\n\\end{itemize}\n\\(\\leadsto\\) \\alert{\\alert{Was ist Funktionale Programmierung?}}\n\\end{frame}\n\n\\begin{frame}[label={sec:org36aba8b}]{Links}\n\\begin{itemize}\n\\item \\href{https://fsharp.org/}{fsharp.org}\n\\item \\href{https://docs.microsoft.com/de-de/dotnet/fsharp/}{docs.microsoft.com/../dotnet/fsharp}\n\\item \\href{https://sergeytihon.com/}{F\\# weekly}\n\\item \\href{https://fsharpforfunandprofit.com/}{fsharpforfunandprofit.com}\n\\item \\href{https://github.com/fsprojects/awesome-fsharp}{github.com/../awesome-fsharp}\n\\end{itemize}\n\\end{frame}\n\n\\begin{frame}[label={sec:org5528b6b}]{Ende}\n\\begin{itemize}\n\\item Wie geht es weiter?\n\\item \\href{https://exercism.io}{Exercism!}\n\\item Buchtipps\n\\begin{itemize}\n\\item \\href{https://pragprog.com/book/swdddf/domain-modeling-made-functional}{Domain Modeling Made Functional} (F\\#)\n\\item \\href{https://www.apress.com/gp/book/9781484239995}{Stylish F\\#} (F\\#)\n\\item \\href{https://www.cambridge.org/core/books/pearls-of-functional-algorithm-design/B0CF0AC5A205AF9491298684113B088F\\#}{Perls of Functional Algorithm Design} (Haskell)\n\\item \\href{https://www.cs.ox.ac.uk/publications/books/functional/}{Thinking Functional with Haskell} (Haskell)\n\\item \\href{http://www.paulgraham.com/onlisp.html}{On Lisp} (LISP)\n\\item \\href{http://www.iqool.de/FPMP.html}{Funktionale Programmierung und Metaprogrammierung} (LISP)\n\\item \\href{https://github.com/norvig/paip-lisp}{Paradigms of Artificial Intelligence Programming} (LISP)\n\\item \\href{https://adv-r.hadley.nz/}{Advanced R} (R)\n\\end{itemize}\n\\item Sprachen: \\href{https://www.r-project.org/}{R}, \\href{https://www.haskell.org/}{Haskell}, \\href{https://clojure.org/}{Clojure}, \\href{https://lisp-lang.org/}{Common Lisp}, \\href{https://elixir-lang.org/}{Elixir}, \\href{https://code.kx.com/q/}{q}\n\n\\item \\alert{\\alert{Have FUN!}}\n\\end{itemize}\n\\end{frame}\n\\end{document}", "meta": {"hexsha": "ad7d494a98a5d2d07f3a8e659f7048a7b0934957", "size": 18075, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "slides/slides-05.tex", "max_stars_repo_name": "kirchnergo/course.2022.hwr.fun", "max_stars_repo_head_hexsha": "47f14f58c50924fc794ce5e16aede83356d8ff02", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 4, "max_stars_repo_stars_event_min_datetime": "2022-03-04T12:08:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-04T13:43:19.000Z", "max_issues_repo_path": "slides/slides-05.tex", "max_issues_repo_name": "kirchnergo/course.2022.hwr.fun", "max_issues_repo_head_hexsha": "47f14f58c50924fc794ce5e16aede83356d8ff02", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "slides/slides-05.tex", "max_forks_repo_name": "kirchnergo/course.2022.hwr.fun", "max_forks_repo_head_hexsha": "47f14f58c50924fc794ce5e16aede83356d8ff02", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2022-03-04T12:58:54.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-04T13:22:44.000Z", "avg_line_length": 35.0290697674, "max_line_length": 251, "alphanum_fraction": 0.7032365145, "num_tokens": 5935, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5621765008857982, "lm_q2_score": 0.6076631698328916, "lm_q1q2_score": 0.34161395453382754}} {"text": "\\documentclass{article}\n\n\\usepackage{amssymb}\n\\usepackage{amsthm}\n\\usepackage[UKenglish]{babel}\n\\usepackage{enumitem}\n\\usepackage{fancyhdr}\n\\usepackage[margin=1in]{geometry}\n\\usepackage{graphicx}\n\\usepackage[utf8]{inputenc}\n\\usepackage{listings}\n\\usepackage{mathtools}\n\\usepackage{tikz-cd}\n\\usepackage{csquotes}\n\n\\newcommand{\\F}{\\mathbb{F}}\n\\newcommand{\\N}{\\mathbb{N}}\n\\newcommand{\\Z}{\\mathbb{Z}}\n\\newcommand{\\Q}{\\mathbb{Q}}\n\\newcommand{\\R}{\\mathbb{R}}\n\\newcommand{\\C}{\\mathbb{C}}\n\\newcommand{\\A}{\\mathbb{A}}\n\\renewcommand{\\P}{\\mathbb{P}}\n\n\\newcommand{\\val}[1]{\\left. #1 \\right\\rvert}\n\\newcommand{\\rb}[1]{\\left( #1 \\right)}\n\\renewcommand{\\sb}[1]{\\left[ #1 \\right]}\n\\newcommand{\\cb}[1]{\\left\\{ #1 \\right\\}}\n\\newcommand{\\ab}[1]{\\left\\langle #1 \\right\\rangle}\n\\newcommand{\\abs}[1]{\\left\\lvert #1 \\right\\rvert}\n\\newcommand{\\two}[2]{\\begin{pmatrix} #1 \\\\ #2 \\end{pmatrix}}\n\\newcommand{\\three}[3]{\\begin{pmatrix} #1 & #2 & #3 \\end{pmatrix}}\n\n\\theoremstyle{definition}\\newtheorem{definition}{Definition}[subsection]\n\\theoremstyle{definition}\\newtheorem{remark}[definition]{Remark}\n\\theoremstyle{definition}\\newtheorem*{example}{Example}\n\\theoremstyle{definition}\\newtheorem*{note}{Note}\n\\newtheorem{proposition}[definition]{Proposition}\n\\newtheorem{lemma}[definition]{Lemma}\n\\newtheorem{theorem}[definition]{Theorem}\n\\newtheorem{corollary}[definition]{Corollary}\n\n\\pagestyle{fancy}\n\\lhead{M3P8 Algebra III}\n\\rhead{Autumn 2018}\n\n\\title{M3P8 Algebra III}\n\\author{Lectured by Dr David Helm \\\\ Typeset by David Kurniadi Angdinata}\n\\date{Autumn 2018}\n\n\\setcounter{section}{-1}\n\n\\begin{document}\n\n\\maketitle\n\n\\vfill\n\n\\tableofcontents\n\n\\pagebreak\n\n\\marginpar{Lecture 1 \\\\ Friday \\\\ 05/10/18}\n\n\\section{Introduction}\n\nThis course is an introduction to ring theory. The topics covered will include ideals, factorisation, the theory of field extensions, finite fields, polynomial rings in several variables, and the theory of modules.\n\nIn addition to the lecture notes, the following will cover much of the material we will be studying.\n\n\\begin{enumerate}\n\\item M Artin, Algebra, 1991\n\\end{enumerate}\n\nRings are contexts in which it makes sense to add and multiply. For example, $ \\Z $, $ \\Q $, $ \\R $, $ \\C $, polynomials, functions $ \\cb{0, 1} \\to \\R $, and $ \\Z / n\\Z $ are rings. The goals of this course include\n\\begin{enumerate}\n\\item to unify arguments that apply in all of the above contexts, and\n\\item to study relationships between different rings.\n\\end{enumerate}\nThe applications of rings include\n\\begin{enumerate}\n\\item number theory, by studying extensions of $ \\Z $ in which particular Diophantine equations have solutions, for example $ n = x^2 + y^2 = \\rb{x + iy}\\rb{x - iy} $ to study solutions in $ \\Z\\cb{i} $ and pass to result about $ \\Z $,\n\\item algebraic geometry, by the study of zero sets of polynomials in several variables via rings of functions, and\n\\item topology, by cohomology classes of topological spaces.\n\\end{enumerate}\n\n\\section{Basic definitions and examples}\n\n\\subsection{Rings}\n\nRecall the definition of a commmutative ring.\n\n\\begin{definition}\nA \\textbf{commutative ring with identity} $ R $ is a set together with two binary operations $ +_R, \\cdot_R : R \\times R \\to R $, addition and multiplication, and two distinguished elements $ 0_R $ and $ 1_R $ such that the following holds.\n\\begin{enumerate}\n\\item The operation $ +_R $ makes $ R $ into an abelian group with identity $ 0_R $.\n\\begin{enumerate}\n\\item For all $ r \\in R $, $ 0_R +_R r = r +_R 0_R = 0_R $ (additive identity).\n\\item For all $ r, s, t \\in R $, $ \\rb{r +_R s} +_R t = r +_R \\rb{s +_R t} $ (associativity of $ +_R $).\n\\item For all $ r, s \\in R $, $ r +_R s = s +_R r $ (commutativity of $ +_R $).\n\\item For all $ r \\in R $, there exists $ -r \\in R $ such that $ r +_R \\rb{-r} = \\rb{-r} +_R r = 0_R $ (additive inverses).\n\\end{enumerate}\n\\item The operation $ \\cdot_R $ is associative and commutative with identity $ 1_R $.\n\\begin{enumerate}\n\\item For all $ r \\in R $, $ 1_R \\cdot_R r = r \\cdot_R 1_R = 1_R $ (multiplicative identity).\n\\item For all $ r, s, t \\in R $, $ \\rb{r \\cdot_R s} \\cdot_R t = r \\cdot_R \\rb{s \\cdot_R t} $ (associativity of $ \\cdot_R $).\n\\item For all $ r, s \\in R $, $ r \\cdot_R s = s \\cdot_R r $ (commutativity of $ \\cdot_R $).\n\\end{enumerate}\n\\item Multiplication distributes over addition.\n\\begin{enumerate}\n\\item For all $ r, s, t \\in R $, $ r \\cdot_R \\rb{s +_R t} = r \\cdot_R s +_R r \\cdot_R t $ and $ \\rb{s +_R t} \\cdot_R r = s \\cdot_R r +_R t \\cdot_R r $ (distributivity of $ \\cdot $ over $ + $).\n\\end{enumerate}\n\\end{enumerate}\n\\end{definition}\n\nThere is some redundancy here, of course. I have written things this way so that one obtains the definition of a noncommutative ring simply by removing the condition that multiplication is commutative. In this course, however, all rings will be commutative.\n\n\\begin{proposition}\nLet $ R $ be a ring. Then for all $ r \\in R $, $ r \\cdot_R 0_R = 0_R $.\n\\end{proposition}\n\n\\begin{proof}\n$ r \\cdot_R 0_R = r \\cdot_R \\rb{0_R +_R 0_R} = r \\cdot_R 0_R +_R r \\cdot_R 0_R $. Thus $ 0_R = -\\rb{r \\cdot_R 0_R} +_R \\rb{r \\cdot_R 0_R} = -\\rb{r \\cdot_R 0_R} +_R \\rb{r \\cdot_R 0_R +_R r \\cdot_R 0_R} = r \\cdot_R 0_R $.\n\\end{proof}\n\nSome people require $ 0_R \\ne 1_R $ in $ R $.\n\n\\begin{proposition}\nIf $ 0_R = 1_R $, then $ R = \\cb{0_R} $.\n\\end{proposition}\n\n\\begin{proof}\n$ 0_R = r \\cdot_R 0_R = r \\cdot_R 1_R = r $.\n\\end{proof}\n\nWhen it is clear from the context what ring we are working with, we will write $ 0_R $ and $ 1_R $ as $ 0 $ and $ 1 $, $ a +_R b $ as $ a + b $ and $ a \\cdot_R b $ as $ ab $.\n\n\\begin{definition}\nA ring $ R $ is a \\textbf{field} if $ R \\ne \\cb{0_R} $ and every nonzero element of $ R $ has a multiplicative inverse, that is for every $ r \\in R \\setminus \\cb{0_R} $ there exists $ r^{-1} \\in R $ such that $ rr^{-1} = r^{-1}r = 1_R $.\n\\end{definition}\n\nWe do not consider the zero ring $ \\cb{0_R} $ to be a field. We have seen many examples of rings at this point. The sets $ \\Z $, $ \\Q $, $ \\R $, $ \\C $ are all rings with their usual notion of addition and multiplication. All of them but $ \\Z $ are in fact fields. As another example, we have the ring $ \\Z / n\\Z $ of integers modulo $ n $. Let $ n \\in \\Z_{> 0} $, and recall that $ a $ and $ b $ are said to be congruent modulo $ n $ if $ a - b $ is divisible by $ n $. It is easy to check that this is an equivalence relation on $ \\Z $. Moreover, since any $ a \\in \\Z $ can uniquely be written as $ qn + r $ with $ q, r \\in \\Z $ and $ 0 \\le r < n $, the set $ \\cb{\\sb{0}_n, \\dots, \\sb{n - 1}_n} $ is a complete list of the equivalence classes under this relation, where $ \\sb{a}_n $ denotes the set of all integers congruent to $ a \\mod n $. We denote this $ n $-element set by $ \\Z / n\\Z $, and we can define addition and multiplication in $ \\Z / n\\Z $ by setting $ \\sb{a}_n + \\sb{b}_n = \\sb{a + b}_n $ and $ \\sb{a}_n\\sb{b}_n = \\sb{ab}_n $. This defines a ring structure on $ \\Z / n\\Z $ once one checks that it is well-defined. This is the first example of a general construction of the quotient of a ring by an ideal we will define later.\n\n\\marginpar{Lecture 2 \\\\ Monday \\\\ 08/10/18}\n\n\\subsection{Polynomial rings}\n\nA very important class of rings that we will study are the polynomial rings. Let $ R $ be any ring. Then we can form a new ring $ R\\sb{X} $, called the \\textbf{ring of polynomials in $ X $ with coefficients in $ R $}. Informally, a polynomial in $ R\\sb{X} $ is a finite sum of the form $ r_0 + \\dots + r_nX^n $ for some $ n \\in \\Z_{\\ge 0} $ and $ r_i \\in R $. If $ n > m $, we consider $ r_0 + \\dots + r_nX^n $ to represent the same polynomial of $ R\\sb{X} $ as $ s_0 + \\dots + s_mX^m $ if $ r_i = s_i $ for $ i \\le m $ and $ r_i = 0_R $ for $ i > m $. That is, you can pad out a polynomial with terms of the form $ 0_RX^i $ without changing it. From a formal standpoint, it is better to define a polynomial to be an infinite sum $ \\sum_{n = 0}^\\infty r_iX^i $ for $ r_i \\in R $ in which all but finitely many $ r_i $ are zero. This makes it easier to define addition and multiplication. The \\textbf{degree} of such an expression is the largest $ i $ such that $ r_i $ is nonzero. We add and multiply in $ R\\sb{X} $ just as we would any other polynomials, by\n$$ \\rb{\\sum_{i = 0}^\\infty r_iX^i} +_{R\\sb{X}} \\rb{\\sum_{i = 0}^\\infty s_iX^i} = \\sum_{i = 0}^\\infty \\rb{r_i +_R s_i} X^i, $$\n$$ \\rb{\\sum_{i = 0}^\\infty r_iX^i} \\cdot_{R\\sb{X}} \\rb{\\sum_{i = 0}^\\infty s_iX^i} = \\sum_{i = 0}^\\infty \\rb{\\sum_{j = 0}^i \\rb{r_j \\cdot_R s_{i - j}}} X^i. $$\nWhat about polynomial rings in more than one variable? Since the construction of polynomial rings takes an arbitrary ring as input, one can iterate it. Start with a ring $ R $, and consider first the ring $ R\\sb{X} $ and then the ring $ \\rb{R\\sb{X}}\\sb{Y} $. An polynomial of this has the form $ \\sum_{i = 0}^\\infty \\rb{\\sum_{j = 0}^\\infty r_{ij}X^j}Y^i $ for $ r_{ij} \\in R $. On the other hand, we can consider the ring $ \\rb{R\\sb{Y}}\\sb{X} $, whose polynomials have the form $ \\sum_{i = 0}^\\infty \\rb{\\sum_{j = 0}^\\infty s_{ij}Y^j}X^i $ for $ s_{ij} \\in R $. Alternatively, we could consider the ring $ R\\sb{X, Y} $ whose polynomials are formal expressions of the form $ \\sum_{i = 0}^\\infty \\rb{\\sum_{i = 0}^\\infty r_{ij}X^i}Y^j $ with only finitely many nonzero coefficients $ r_{ij} $ and define addition and multiplication in the usual way. It is not hard to see that all three approaches yield the same ring. There is a bijection between these expressions. We will therefore primarily use notation like $ R\\sb{X, Y} $ for polynomial rings in multiple variables, but we will occasionally need to know that this is the same as $ \\rb{R\\sb{X}}\\sb{Y} $ or $ \\rb{R\\sb{Y}}\\sb{X} $. The identification we have made here is an example of an isomorphism of rings, a notion we will make precise later.\n\n\\subsection{Subrings and extensions}\n\n\\begin{definition}\nLet $ R $ be a ring. A subset $ S $ of $ R $ is a \\textbf{subring} of $ R $ if\n\\begin{enumerate}\n\\item $ 0_R, 1_R, -1_R \\in S $.\n\\item $ S $ is closed under $ +_R $ and $ \\cdot_R $, so if $ r, s \\in S $, then so are $ r +_R s $ and $ r \\cdot_R s $.\n\\end{enumerate}\n\\end{definition}\n\nSubrings inherit the additive and multiplicative structures from the ring that contains them, and are thus themselves rings.\n\n\\begin{example}\n$ \\Z $ is a subring of $ \\R $, which is itself a subring of $ \\C $.\n\\end{example}\n\nIt is easy to see that the intersection of two subrings of $ R $, or even an arbitrary collection of subrings of $ R $, is also a subring of $ R $.\n\n\\begin{definition}\nLet $ S \\subseteq R $ be a subring of a ring $ R $, and let $ \\alpha $ be an element of $ R $. We can then form a subring $ S\\sb{\\alpha} $ of $ R $, called the \\textbf{subring of $ R $ generated by $ \\alpha $ over $ S $}, consisting of all elements of $ R $ that can be expressed as $ r_0 + \\dots + r_n\\alpha^n $ for some $ n \\in \\Z^* $, and $ r_i \\in S $.\n\\end{definition}\n\nThis operation is known as \\textbf{adjoining} the element $ \\alpha $ to the ring $ S $. An alternative way of defining the ring $ S\\sb{\\alpha} $ is to note that it is the smallest subring of $ R $ containing $ S $ and $ \\alpha $. In one direction, any such subring contains every expression of the form $ r_0 + \\dots + r_n\\alpha^n $, with $ r_i \\in S $, so any subring of $ R $ containing $ S $ and $ \\alpha $ contains $ S\\sb{\\alpha} $. One can thus construct $ S\\sb{\\alpha} $ as the intersection of every subring of $ R $ containing $ S $ and $ \\alpha $. Since the intersection of any collection of subrings of $ R $ is a subring of $ R $ it is clear that this intersection is equal to $ S\\sb{\\alpha} $ as defined above.\n\n\\begin{example}\nLet $ i $ denote a square root of $ -1 $ in $ \\C $. $ \\Z \\subseteq \\C $ and $ i $ form $ \\Z\\sb{i} $. Note $ -1 = i^2 = i^6 = i + i^3 + i^{10} $.\n\\end{example}\n\n\\begin{proposition}\nEvery element of $ \\Z\\sb{i} $ can be uniquely expressed as $ a + bi $ for $ a, b \\in \\Z $.\n\\end{proposition}\n\n\\begin{example}\nGiven $ \\sum_{n = 0}^\\infty a_ni^n $ with only finitely many $ a_n $ nonzero, set $ a = a_0 - a_2 + \\dots $ and $ b = a_1 - a_3 + \\dots $. Then $ \\sum_{n = 0}^\\infty a_ni^n = a + bi $. For uniqueness, if $ a + bi = c + di $ in $ \\C $ for $ a, b, c, d \\in \\Z $, then $ a = c $ or $ b = d $.\n\\end{example}\n\nIf $ \\alpha $ is more complicated then the elements of $ R\\sb{\\alpha} $ may well be harder to describe.\n\n\\begin{example}\nIf $ \\alpha $ is the real cube root of $ 2 $, then every element of $ \\Z\\sb{\\alpha} $ can be uniquely expressed as $ a + b\\alpha + c\\alpha^2 $ for $ a, b, c \\in \\Z $.\n\\end{example}\n\n\\begin{example}\nIn $ \\Z\\sb{\\pi} $, any element has a unique expression in the form $ \\sum_{n = 0}^\\infty a_n\\pi^n $ for all but finitely many $ a_n $ are zero. Suppose $ \\sum_{n = 0}^\\infty a_n\\pi^n = \\sum_{n = 0}^\\infty b_n\\pi^n $, then $ 0 = \\sum_{n = 0}^\\infty \\rb{a_n - b_n}\\pi^n $. Since $ \\pi $ is transcendental, this polynomial must be zero. Thus each $ a_n = b_n $.\n\\end{example}\n\n\\begin{example}\nThe elements of $ \\Z\\sb{\\tfrac{1}{2}} $ can be expressed uniquely as $ a / b $, where $ b $ is a power of $ 2 $ and $ a $ is odd unless $ b = 1 $.\n\\end{example}\n\n\\begin{example}\nLet $ \\alpha $ be a root of $ x^2 - \\tfrac{1}{2}x + 1 $. Then $ \\alpha^2 \\in \\Z\\sb{\\alpha} $ and $ \\alpha^2 = \\alpha / 2 - 1 $. Can show that every element of $ \\Z\\sb{\\alpha} $ can be expressed as $ a + b\\alpha $ for $ a, b \\in \\Z\\sb{\\tfrac{1}{2}} $, but not every $ a + b\\alpha $ arises $ a, b \\in \\Z\\sb{\\tfrac{1}{2}} $.\n\\end{example}\n\n\\marginpar{Lecture 3 \\\\ Wednesday \\\\ 10/10/18}\n\n\\subsection{Integral domains and rings of fractions}\n\n\\begin{definition}\nA \\textbf{zero divisor} in a ring $ R $ is a nonzero element $ r $ of $ R $ such that there exists a nonzero $ s \\in R $ with $ rs = 0 $. A ring $ R $ in which there are no zero divisors is called an \\textbf{integral domain}.\n\\end{definition}\n\n\\begin{example}\n$ \\Z $ is an integral domain and any subring of a field is an integral domain, but $ \\Z / 6\\Z $ is not an integral domain, as $ \\sb{2}\\sb{3} $ is zero modulo $ 6 $ even though neither $ \\sb{2} $ nor $ \\sb{3} $ is zero modulo $ 6 $.\n\\end{example}\n\nIf $ R $ is an integral domain, then we can form the field of fractions of $ R $ in analogy to the way we build $ \\Q $ from $ \\Z $.\n\n\\begin{definition}\nLet $ R $ be an integral domain. The \\textbf{field of fractions} $ K\\rb{R} $ is the set of equivalence classes of expressions of the form $ a / b $ for $ a, b \\in R $, $ b \\ne 0 $, where $ a / b \\sim a' / b' $ if and only if $ ab' = a'b $. We add and multiply elements of $ K\\rb{R} $ just as we do for fractions, by\n$$ \\dfrac{a}{b} + \\dfrac{a'}{b'} = \\dfrac{ab' + a'b}{bb'}, \\qquad \\dfrac{a}{b} \\cdot \\dfrac{a'}{b'} = \\dfrac{aa'}{bb'}, \\qquad 0_{K\\rb{R}} = \\dfrac{0_R}{1_R}, \\qquad 1_{K\\rb{R}} = \\dfrac{1_R}{1_R}. $$\nIf $ a \\ne 0 $ in $ R $, then $ b / a \\in K\\rb{R} $, so $ \\rb{a / b} \\cdot \\rb{b / a} = ab / ba \\sim 1 / 1 $.\n\\end{definition}\n\nThen $ K\\rb{R} $ is a field, and it contains $ R $ in a natural way as a subring if we identify $ r $ with $ r / 1_R \\in K\\rb{R} $. The field $ K\\rb{R} $ is in some sense the smallest field containing $ R $ as a subring. When we talk about homomorphisms and isomorphisms, we will be able to state this more precisely. More generally, a subset $ S $ of $ R $ is a \\textbf{multiplicative system} if $ 1 \\in S $ and $ 0 \\notin S $, and $ S $ is closed under multiplication, that is if $ a, b $ are in $ S $ then so is $ ab $. For any integral domain $ R $ and any multiplicative system $ S $, we can define $ S^{-1}R \\subseteq K\\rb{R} $ consisting of all fractions of the form $ a / b $ with $ b \\in S $. It is easy to see that his is closed under addition and multiplication, and defines a ring in between $ R $ and $ K\\rb{R} $.\n\n\\begin{example}\nIf $ R = \\Z $ and $ S $ is the set of powers of $ 2 $, then $ S^{-1}R = \\Z\\sb{\\tfrac{1}{2}} $. On the other hand, if $ S $ is the set of odd integers, then $ S^{-1}R $ is the set of all rational numbers of the form $ a / b $ with $ b $ odd.\n\\end{example}\n\nIn general $ S^{-1}R $ is the smallest subring of $ K\\rb{R} $ containing $ R $ in which every element of $ S $ has a multiplicative inverse, that is $ b^{-1} \\in S $ for all $ b \\in S $. The process of obtaining $ S^{-1}R $ from $ R $ is called \\textbf{localisation} and is an extremely powerful tool. One can even make sense of it when $ R $ is not an integral domain, but one has to be more careful. The equivalence relation on fractions is tricker, for example. We will not discuss this in this course but it will be quite useful in future courses.\n\n\\section{Homomorphisms, ideals, and quotients}\n\n\\subsection{Homomorphisms}\n\nLet $ R $ and $ S $ be rings. A ring homomorphism from $ R $ to $ S $ is, roughly, a way of interpreting elements of $ R $ as elements of $ S $, in a way that is compatible with the addition and multiplication laws on $ R $ and $ S $. More precisely is the following.\n\n\\begin{definition}\nA function $ f : R \\to S $ is a \\textbf{ring homomorphism} if\n\\begin{enumerate}\n\\item $ f\\rb{1_R} = 1_S $,\n\\item for all $ r, r' \\in R $, $ f\\rb{r +_R r'} = f\\rb{r} +_S f\\rb{r'} $, and\n\\item for all $ r, r' \\in R $, $ f\\rb{r \\cdot_R r'} = f\\rb{r} \\cdot_S f\\rb{r'} $.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{note}\nIf $ f $ is a homomorphism then $ f\\rb{0_R} = f\\rb{0_R + 0_R} = f\\rb{0_R} +_S f\\rb{0_R} $ gives $ f\\rb{0_R} = 0_S $. Thus we do not need to require this as an axiom. On the other hand we do need to require $ f\\rb{1_R} = 1_S $. For certain $ R, S $ one can construct examples of maps $ f : R \\to S $ that satisfy properties $ 2 $ and $ 3 $ of the definition without satisfying property $ 1 $.\n\\end{note}\n\n\\begin{example}\nIf $ R $ is a subring of $ S $, then the inclusion of $ R $ into $ S $, such as $ \\Z \\subset \\Q \\subset \\R \\subset \\C $, is a homomorphism. This is just a fancy way of saying that the addition and multiplication on $ R $ are induced from the corresponding operations on $ S $.\n\\end{example}\n\n\\begin{example}\nThe composition of two homomorphisms is a homomorphism, as is easily checked from the definitions.\n\\end{example}\n\n\\begin{example}\nThe map $ \\Z \\to \\Z / n\\Z $ that takes an integer $ m $ into its congruence class modulo $ n $ is also a homomorphism. In fact, this is a special case of the following construction.\n\\end{example}\n\n\\begin{proposition}\nLet $ R $ be any ring. Then there is a unique ring homomorphism $ f : \\Z \\to R $ such that\n$$ f\\rb{n} = \\begin{cases} 1_R + \\dots + 1_R & n > 0 \\\\ -\\rb{1_R + \\dots + 1_R} & n < 0 \\\\ 0_R & n = 0 \\end{cases}. $$\n\\end{proposition}\n\n\\begin{proof}\nLet $ f : \\Z \\to R $ be a homomorphism. Then, directly from the definition, we have $ f\\rb{0} = 0_R $ and $ f\\rb{1} = 1_R $. In particular for all $ n > 0 $, $ f\\rb{n} = f\\rb{1 + \\dots + 1} = 1_R +_R \\dots +_R 1_R $, where there are $ n $ copies of $ 1_R $ in the sum. Moreover, $ 0_R = f\\rb{n + \\rb{-n}} = f\\rb{n} + f\\rb{-n} $, so $ f\\rb{-n} = -\\rb{1_R +_R \\dots +_R 1_R} $. Thus $ f\\rb{n} $ is determined, for all $ n $, completely by the fact that $ f $ is a homomorphism. In the converse direction, it is not hard to check that the map defined above is in fact a homomorphism.\n\\end{proof}\n\nThus, for any ring $ R $, we can regard an integer as an element of $ R $ via this homomorphism.\n\n\\begin{definition}\nA bijective homomorphism $ f : R \\to S $ is called an \\textbf{isomorphism}. Write $ S \\cong R $ for $ S $ is isomorphic to $ R $.\n\\end{definition}\n\nIn this case one verifies easily that the inverse map $ f^{-1} : S \\to R $ is also a bijective homomorphism.\n\n\\subsection{Evaluation homomorphisms}\n\nLet $ R $ be a ring, and consider the ring $ R\\sb{X} $ of polynomials in $ X $ with coefficients in $ R $. If $ s $ is an element of $ R $, then we can define a homomorphism $ R\\sb{X} \\to R $ by \\textbf{evaluation at $ s $}. More precisely, given an element of $ R\\sb{X} $ of the form $ P\\rb{X} = r_0 + \\dots + r_nX^n $ for some $ n $ and $ r_i \\in R $. Then $ P\\rb{s} $ for $ s \\in R $ is defined to be $ P\\rb{s} = r_0 + \\dots + r_ns^n \\in R $. Consider the map $ \\phi_s : R\\sb{X} \\to R $ that sends $ \\phi_s\\rb{P} $ to $ P\\rb{s} $. In effect, it substitutes $ s $ for $ X $. It is easy to check that this is in fact a ring homomorphism. More generally, if $ R $ and $ S $ are rings and $ f : R \\to S $ is a homomorphism, and $ s $ is an element of $ S $, then we can define a map\n$$ \\phi_{s, f} : R\\sb{X} \\to S, $$\nby setting\n$$ \\phi_{s, f}\\rb{r_0 + \\dots + r_nX^n} = f\\rb{r_0} + \\dots + f\\rb{r_n}s^n. $$\nThat is, by appling $ f $ to the coefficients and substituting $ s $ for $ X $. Again, this is clearly a homomorphism. The evaluation homomorphisms $ \\phi_{s, f} $ are a fundamental property of polynomial rings. In some sense, they are the reason polynomial rings are worth studying. In fact, the ring $ R\\sb{X} $ is uniquely characterised by the fact that homomorphisms from $ R\\sb{X} $ to $ S $ are in bijection with pairs $ \\rb{s, f} $, where $ f : R \\to S $ is a homomorphism and $ s $ is an element of $ S $.\n\n\\subsection{Images, kernels, and ideals}\n\n\\begin{definition}\nLet $ f : R \\to S $ be a homomorphism. The \\textbf{image} of $ f $ is $ Im\\rb{f} = \\cb{f\\rb{r} \\mid r \\in R} \\subseteq S $. The \\textbf{kernel} of $ f $ is $ Ker\\rb{f} = \\cb{r \\in R \\mid f\\rb{r} = 0} \\subseteq R $.\n\\end{definition}\n\nThe image of a homomorphism $ f : R \\to S $ is easily seen to be a subring of $ S $.\n\n\\begin{example}\nIf $ R $ is a subring of $ S $, $ f : R \\to S $ is the inclusion and $ s $ lies in $ S $, then the image of the map $ \\phi_{s, f} : R\\sb{X} \\to S $ is precisely the subring $ R\\sb{s} $ of $ S $.\n\\end{example}\n\nBy contrast, the kernel of a homomorphism $ f $ is almost never a subring of $ R $. For instance, subrings contain the identity. However, it is an ideal of $ R $.\n\n\\marginpar{Lecture 4 \\\\ Friday \\\\ 12/10/18}\n\n\\begin{definition}\nA nonempty subset $ I $ of $ R $ is an \\textbf{ideal} of $ R $ if $ I $ is closed under addition, that is for all $ i, j \\in I $, $ i + j \\in I $, and for all $ i \\in I $, $ r \\in R $, $ ri \\in I $.\n\\end{definition}\n\nThen one can verify, directly from the definition, that the kernel of any homomorphism $ f : R \\to S $ is an ideal of $ R $.\n\n\\begin{note}\nAny ideal of $ R $ contains $ 0_R $, and conversely the subset $ \\cb{0_R} $ of $ R $ is an ideal, called the \\textbf{zero ideal}. A homomorphism $ f : R \\to S $ is injective if and only if its kernel is the zero ideal. Forward direction is easy. Conversely, if $ f\\rb{x} = f\\rb{y} $, $ f\\rb{x - y} = 0 $, so $ x - y \\in Ker\\rb{f} $. If $ Ker\\rb{f} = \\cb{0} $, $ x = y $.\n\\end{note}\n\nThe kernel of the homomorphism $ \\Z \\to R $ is either the zero ideal, or the ideal of multiples of $ n $ in $ \\Z $ for some positive $ n $. We say that $ R $ has characteristic zero or characteristic $ n $, respectively. If not zero, the \\textbf{characteristic} of $ R $ is the smallest $ n $ such that the sum of $ n $ copies of $ 1_R $ is equal to zero.\n\n\\subsection{Ideals: examples and basic operations}\n\nIf $ r $ is an element of $ R $, then any ideal containing $ R $ contains any multiple $ sr $ of $ R $, for any $ r $ in $ S $. Conversely, one checks easily that the set $ \\cb{sr \\mid s \\in R} $ is an ideal of $ R $. It is known as the \\textbf{ideal of $ R $ generated by $ r $}, and denoted $ \\ab{r} $. An ideal generated by one element in this way is called a \\textbf{principal ideal}.\n\n\\begin{note}\nThe ideal generated by $ 1_R $, or more generally by any element of $ R $ with a multiplicative inverse, is all of $ R $. This ideal is called the \\textbf{unit ideal} of $ R $.\n\\end{note}\n\n\\begin{proposition}\n$ R $ is a \\textbf{field} if and only if the only ideals of $ R $ are the zero ideal $ \\cb{0} $ and the unit ideal $ R $.\n\\end{proposition}\n\n\\begin{proof}\nIf $ R $ is a field, let $ I \\subseteq R $ be a nonzero ideal. There exists $ r \\in I \\ne 0 $. Then for all $ s \\in R $, $ \\rb{sr^{-1}}\\rb{r} \\in I $, so $ s \\in I $ for all $ s \\in R $. Conversely, if $ R $ has only zero ideal, unit ideal, let $ r \\in R \\ne 0 $, let $ I \\cb{sr \\mid s \\in R} $. This is an ideal not zero ideal, so it is all of $ R $. In particular, $ 1 \\in I $, so there exists $ s \\in R $ such that $ sr = 1 $.\n\\end{proof}\n\nMore generally is the following.\n\n\\begin{definition}\nIf $ S $ is a subset of elements of $ R $, then any ideal containing $ S $ consists of all elements of $ R $ the form $ r_0s_0 + \\dots + r_ns_n $ for some $ n \\in \\Z_{\\ge 0} $, $ r_i \\in R $, and $ s_i \\in S $. The intersection of all these ideals is an ideal of $ R $, known as the \\textbf{ideal of $ R $ generated by $ S $}, and denoted $ \\ab{S} $. It is also the smallest ideal of $ R $ containing $ S $.\n\\end{definition}\n\nIf $ S $ has one element, $ \\ab{S} $ is a principal ideal. We will show soon that any ideal of $ \\Z $ is a principal ideal, as is any ideal of the ring $ k\\sb{X} $ for any field $ k $. On the other hand, there are rings in which not every ideal is principal.\n\n\\begin{example}\nThe ideal $ \\ab{X, Y} $ of $ k\\sb{X, Y} $ is not a principal ideal.\n\\end{example}\n\nGiven ideals $ I $ and $ J $ there are several ways to create new ideals.\n\\begin{enumerate}\n\\item If $ I, J $ are ideals, then the intersection $ I \\cap J $ is an ideal. If $ I $ and $ J $ are given by generators, it might be hard to find generators for the intersection. Certainly it is not enough to intersect the generating sets.\n\\item The union of ideals is not usually an ideal. Taking $ R = \\Z $, $ \\ab{3} \\cup \\ab{5} $ contains $ 3, 5 $ but not $ 3 + 5 $.\n\\item If $ I, J $ are ideals, then the sum $ I + J $ is an ideal, which are all expressions of the form $ i + j $ for $ i \\in I $, $ j \\in J $. It is the smallest ideal containing both $ I $ and $ J $, and also the ideal generated by $ I \\cup J $.\n\\item If $ I, J $ are ideals, the product $ I \\cdot J $ or $ IJ $ is the ideal generated by elements of the form $ ij $ for $ i \\in I $, $ j \\in J $. This may be strictly larger than the set of such products.\n\\end{enumerate}\n\n\\begin{example}\nConsider the product of the ideals $ I = \\ab{X, Y} $ and $ J = \\ab{Z, W} $ in $ R = k\\sb{X, Y, Z, W} $ for $ k $ a field. The product $ IJ = \\ab{XZ, XW, YZ, YW} $ contains $ XZ + YW $, but the latter is not a product of an element in $ I $ with an element in $ J $.\n\\end{example}\n\n\\begin{note}\nLet $ I, J $ be general ideals. The product of $ I $ and $ J $ is always contained in the intersection of $ I $ and $ J $, but the two need not be equal, even in simple rings like $ \\Z $. $ \\ab{3} \\cdot \\ab{3} = \\ab{9} \\subseteq \\Z $ and $ \\ab{3} \\cap \\ab{3} = \\ab{3} $.\n\\end{note}\n\n\\subsection{Quotients}\n\nLet $ R $ be a ring and let $ I $ be an ideal of $ R $. If $ x, y $ are elements of $ R $, we say that $ x $ is \\textbf{congruent to $ y $ modulo $ I $} if $ x - y $ is in $ I $. This is an equivalence relation on $ R $. We denote the equivalence class of $ r $ by $ r + I $, or the alternative notations $ \\sb{r}_I $, $ \\bar{r} $. It is the set $ \\cb{r + s \\mid s \\in I} $. Let $ R / I $ denote the set of equivalence classes on $ R $ modulo $ I $. This set has the natural structure of a ring. The additive and multiplicative identities are $ 0_R + I $ and $ 1_R + I $, respectively, and addition and multiplication are defined by $ \\rb{r + I} + \\rb{s + I} = \\rb{r + s} + I $ and $ \\rb{r + I} \\cdot \\rb{s + I} = \\rb{rs + I} $ respectively. One has to check that these are well-defined, but this is not difficult. The ring $ R / I $ is called the \\textbf{quotient of $ R $ by the ideal $ I $}.\n\n\\begin{example}\nIf $ R = \\Z $ and $ I $ is the ideal generated by $ n $, then $ R / I $ is the ring $ \\Z / n\\Z $ that we have already seen.\n\\end{example}\n\n\\begin{note}\nThere is a \\textbf{reduction modulo $ I $} or \\textbf{natural quotient} homomorphism $ R \\to R / I $ defined by taking $ r $ to $ r + I $. This homomorphism is surjective with kernel $ I $.\n\\end{note}\n\nWe then have the following.\n\n\\begin{proposition}[Universal property of the quotient]\nLet $ I \\subseteq R $ be an ideal and let $ f : R \\to S $ be a homomorphism, and suppose that the kernel of $ f $ contains $ I $. Then there is a unique homomorphism $ \\bar{f} : R / I \\to S $ such that for all $ r \\in R $, $ \\bar{f}\\rb{r + I} = f\\rb{r} $.\n\\end{proposition}\n\n\\begin{proof}\n$ \\bar{f} $ is necessarily unique, as every element of $ R / I $ has the form $ r + I $ for some $ r $. It thus suffices to show that it is well-defined and gives a homomorphism. If $ r + I = r' + I $, then $ r - r' \\in I $, so $ f\\rb{r - r'} = 0 $ gives $ f\\rb{r} = f\\rb{r'} $. Thus $ \\bar{f} $ is well-defined. Checking that it is a homomorphism follows from $ f $ is a homomorphism.\n\\end{proof}\n\n\\begin{note}\nThe kernel of $ \\bar{f} $ in the above proposition is just the image of the kernel of $ f $ in $ R / I $. If the kernel of $ f $ is equal to $ I $, this image is the zero ideal and $ \\bar{f} $ is injective. In particular, any homomorphism of $ R $ to $ S $ can be thought of as an isomorphism of some quotient of $ R $ with a subring of $ S $.\n\\end{note}\n\n\\begin{example}\nLet $ R \\subseteq S $ be a subring, $ \\alpha \\in S $, and $ \\iota : R \\to S $ be the inclusion map. Recall that we have an evaluation at $ \\alpha $ by $ \\phi_{\\iota, \\alpha} : R\\sb{X} \\to S $. Image of this is $ R\\sb{\\alpha} $. Let $ I = Ker\\rb{\\phi_{\\iota, \\alpha}} $. Then $ \\phi_{\\iota, \\alpha} $ descends to a map $ \\phi_{\\iota, \\alpha} : R\\sb{\\alpha} / I \\to S $ that is injective with image $ R\\sb{\\alpha} $. So $ R\\sb{\\alpha} $ is isomorphic to a quotient of $ R\\sb{X} $.\n\\end{example}\n\n\\marginpar{Lecture 5 \\\\ Monday \\\\ 15/10/18}\n\n\\subsection{Prime and maximal ideals}\n\n\\begin{definition}\nAn ideal $ I $ of $ R $ is \\textbf{prime} if the quotient $ R / I $ is an integral domain. It is \\textbf{maximal} if $ R / I $ is a field.\n\\end{definition}\n\n\\begin{note}\nAs fields are integral domains, every maximal ideal is prime. The converse need not hold, of course. The zero ideal in $ \\Z $ is prime but not maximal.\n\\end{note}\n\n\\begin{proposition}\nAn ideal $ I $ is prime if and only if for every pair of elements $ s, r $ in $ R $ such that $ rs $ is in $ I $, either $ r $ is in $ I $ or $ s $ is in $ I $.\n\\end{proposition}\n\n\\begin{proof}\nThis is just a restatement of the definition. $ R / I $ integral domain if and only if for all whenever two elements $ r + I $ and $ s + I $ in $ R / I $ satisfy $ \\rb{r + I}\\rb{s + I} = 0 + I $ in $ R / I $, either $ r + I = 0 + I $ or $ s + I = 0 + I $ in $ R / I $. This is the same as saying $ rs $ lies in $ I $ if and only if either $ r $ or $ s $ lies in $ I $.\n\\end{proof}\n\n\\begin{proposition}\nAn ideal $ I $ is maximal if and only if the only ideals of $ R $ containing $ I $ are $ I $ and the unit ideal $ R $.\n\\end{proposition}\n\nThis justifies the name maximal for such ideals.\n\n\\begin{proof}\nFirst suppose that $ R / I $ is a field. Recall that $ R / I $ is a field if and only if only ideals of $ R / I $ are $ \\cb{0} $ and $ R / I $. Given an ideal $ J \\subseteq R / I $, let $ \\widetilde{J} $ be the preimage of $ J $ under $ R \\to R / I $. $ \\widetilde{J} $ is an ideal containing $ I $ and contained in $ R $. Then $ J $ is either the zero ideal of $ R / I $, in which case $ \\widetilde{J} $ is contained in, and thus equal to, $ I $, or $ J $ is all of $ R / I $, in which case $ \\widetilde{J} $ contains $ I $ and an element of $ 1_R + I $, so $ \\widetilde{J} $ contains $ 1_R $ and is thus the unit ideal of $ R $. Conversely, if the only ideals of $ R $ containing $ I $ are $ I $ and the unit ideal, then for any $ r $ in $ R \\setminus I $, the ideal of $ R $ generated by $ I $ and $ r $ contains $ 1_R $. We can thus write $ 1_R = rs + i $, where $ i \\in I $ and $ s \\in R $. This means that $ s + I $ and $ r + I $ are multiplicative inverses of each other in $ R / I $, so $ R / I $ is a field.\n\\end{proof}\n\n\\section{Factorisation}\n\nIn these notes $ R $ always denotes an integral domain.\n\n\\subsection{Divisibility, units, associates, and irreducibles}\n\n\\begin{definition}\nLet $ r, s $ be elements of $ R $. We say $ r $ \\textbf{divides} $ s $, written $ r \\mid s $, if there exists $ r' \\in R $ with $ rr' = s $, or, equivalently, $ s $ lies in the principal ideal $ \\ab{r} $ generated by $ r $. An element $ r $ that divides $ 1_R $ is called a \\textbf{unit} of $ R $, or, equivalently, $ \\ab{r} = R $.\n\\end{definition}\n\nThe set of units in $ R $ forms a group under multiplication denoted $ R^* $. For any element $ r \\in R $ and any unit $ u $ of $ R $, both $ u $ and $ ur $ divide $ r $.\n\n\\begin{definition}\nThe set of elements of $ R $ of the form $ ur $, with $ r \\in R^* $ are called \\textbf{associates} of $ R $, that is $ r, r' $ are associates if $ r = ur' $ for a unit $ u \\in R^* $.\n\\end{definition}\n\nThis implies $ r \\mid r' $, that is there exists $ u' $ with $ u'u = 1 $ and $ u'r = r' $.\n\n\\begin{note}\nThe principal ideals $ \\ab{r} $ and $ \\ab{r'} $ are equal if and only if $ r $ and $ r' $ are associates.\n\\end{note}\n\n\\begin{definition}\nA nonzero element $ r $ of $ R $ is called \\textbf{irreducible} if $ r $ is not a unit and the only elements of $ R $ that divide $ r $ are the units and the associates of $ r $.\n\\end{definition}\n\n\\subsection{Unique factorisation domains}\n\nAn interesting question is when elements of rings admit unique factorisations into irreducibles. To that end we define the following.\n\n\\begin{definition}\nA \\textbf{unique factorisation domain} (UFD) is a ring $ R $ in which\n\\begin{enumerate}\n\\item every nonunit, nonzero element $ r \\in R $ admits a factorisation as a finite product of irreducibles in $ R $, and\n\\item if $ r = p_1 \\dots p_n = q_1 \\dots q_m \\in R $ are two factorisations of $ r $ as products of irreducibles $ p_i, q_i $, then $ n = m $ and, up to permuting the $ q_i $, each $ q_i $ is an associate of $ p_i $.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{example}\nBoth conditions can fail.\n\\begin{enumerate}\n\\item There are certainly domains in which $ 1 $ can fail, although they are somewhat exotic. One example is to take the rational polynomial ring $ R = \\C\\sb{X^\\Q} $ with coefficients in $ \\C $, whose entries are finite formal sums $ \\sum_{i = 0}^N a_iX^{n_i} $ where the $ a_i $ are in $ \\C $ and the $ n_i $ are nonnegative rational numbers $ \\Q_{\\ge 0} $. Any element of $ R $ is a polynomial in $ X^{1 / n} $ for some $ n $. The element $ X $ of this ring is not a unit, and also not a finite product of irreducibles. In $ \\C\\sb{X^{1 / n}} $, $ X $ factors as $ \\rb{X^{1 / n}}^n $. $ X $ has no factorisation into irreducibles in $ R $. We will show later that a very mild finiteness condition on a domain $ R $, the condition that $ R $ is Noetherian, actually guarantees that $ 1 $ holds.\n\\item Even if $ 1 $ holds, $ 2 $ often fails. The classic example of this is $ R = \\Z\\sb{\\sqrt{-5}} $, in which $ 2, 3, 1 + \\sqrt{-5}, 1 - \\sqrt{-5} $ are all irreducibles, none are associates of each other, yet $ \\rb{2}\\rb{3} = \\rb{1 + \\sqrt{-5}}\\rb{1 - \\sqrt{-5}} $.\n\\end{enumerate}\n\\end{example}\n\nAnother way to interpret condition $ 2 $ is as follows.\n\n\\begin{definition}\nWe say an element $ r $ of $ R $ is \\textbf{prime} if the principal ideal $ \\ab{r} $ of $ R $ is a prime ideal. In other words, for any $ s, s' $ in $ R $, if $ r $ divides $ ss' $, then $ r \\mid s $ or $ r \\mid s' $.\n\\end{definition}\n\n\\begin{lemma}\nPrime elements are irreducible.\n\\end{lemma}\n\n\\begin{proof}\nIf $ r $ is prime and $ s $ divides $ r $, we can write $ r = ss' $. Then since $ r $ divides $ ss' $ we have that either $ r $ divides $ s $, in which case $ rs'' = s $, then $ ss's'' = s $ and $ s's'' = 1 $, so $ r $ is an associate of $ s $, or $ r $ divides $ s' $, in which case $ s' = rs'' $, then $ r = srs'' $ and $ ss'' = 1 $, so $ r $ is an associate of $ s' $ and $ s $ is a unit.\n\\end{proof}\n\nThe converse is not necessarily true, but we have the following observation as a criteria for $ R $ to be a UFD.\n\n\\begin{proposition}\nLet $ R $ be a domain in which condition $ 1 $ holds. Then condition $ 2 $ above holds for $ R $ if and only if every irreducible element of $ R $ is prime.\n\\end{proposition}\n\n\\begin{proof}\nFirst suppose condition $ 2 $ holds, and let $ r $ be an irreducible element of $ R $. If $ r $ divides $ ab $, we can write $ rs = ab $ for some $ s \\in R $. Expanding out $ s, a, b $ as products of irreducibles we see that $ r $ is an associate of some irreducible dividing $ a $ or $ b $, so $ r $ is prime. Conversely, if every irreducible element of $ R $ is prime, and we have $ p_1 \\dots p_n = q_1 \\dots q_m $ products of irreducibles, then, since $ p_1 $ is prime, it divides the product $ q_1 \\dots q_m $ and is thus an associate of some $ q_i $. We can thus cancel $ p_1 $ from the left and $ q_i $ from the right after introducing a unit on one side. This is possible because $ R $ is an integral domain. Repeating the process we find that, up to reordering the terms and multiplying by units, the two expressions coincide.\n\\end{proof}\n\n\\subsection{Principal ideal domains}\n\n\\begin{definition}\nAn integral domain $ R $ is a \\textbf{principal ideal domain} (PID) if every ideal of $ R $ is a principal ideal.\n\\end{definition}\n\n\\begin{theorem}\n\\label{thm:3.3.2}\nEvery PID is a UFD.\n\\end{theorem}\n\nWe first show $ 1 $. It is true for units trivially.\n\n\\begin{lemma}\nLet $ R $ be a PID. Then every nonzero nonunit $ r \\in R $ has a irreducible divisor.\n\\end{lemma}\n\n\\begin{proof}\nFix $ r = r_0 \\in R $. We first show $ r $ has an irreducible factor. If $ r_0 $ is irreducible we are done. Otherwise $ r_0 $ is not irreducible, we can choose an $ r_1 $, not a unit nor an associate of $ r_0 $, such that $ r_1 $ divides $ r_0 $, so $ r_0 = r_1s_1 $ with $ r_1, s_1 $ not units. If $ r_1 $ is not irreducible we choose $ r_2 $ similarly, and repeat. If this process ever terminates we have found an irreducible divisor of $ r $. Suffices to show this terminates. Suppose it does not terminate. We obtain an increasing tower of ideals\n$$ \\ab{r_0} \\subsetneq \\ab{r_1} \\subsetneq \\dots. $$\nLet $ I $ be the union of all these ideals generated by $ r_0, r_1, \\dots $. Then $ I $ is an ideal, so it is generated by some element $ s \\in I $. Thus $ s $ divides $ r_i $ for all $ i $. On the other hand, $ s $ lives in some $ \\ab{r_j} $, so $ r_j $ divides $ s $. Thus $ s $ is an associate of $ r_j $, and therefore an associate of $ r_i $ for all $ i > j $, that is $ I \\subseteq \\ab{r_j} $. This contradicts our construction because $ \\ab{r_{j + 1}} \\subseteq I $ and $ \\ab{r_{j + 1}} \\ne \\ab{r_j} $.\n\\end{proof}\n\nThus $ r $ has an irreducible divisor $ s_0 $.\n\n\\begin{lemma}\nLet $ R $ be a PID. Every nonzero nonunit $ r \\in R $ is a finite product of irreducibles.\n\\end{lemma}\n\n\\begin{proof}\nConsider $ rs_0^{-1} $. If this is a unit we are done. If not let $ s_1 $ be an irreducible divisor of $ rs_0^{-1} $. If $ r\\rb{s_0s_1}^{-1} $ is a unit we are done, otherwise repeat. We obtain a sequence of irreducibles $ s_0, s_1, \\dots $ such that $ s_0 \\dots s_i $ divides $ r $ for all $ i $, so $ r = r_0s_0 = r_0r_1s_1 = \\dots $ with $ r_0, r_1, \\dots $ irreducible. If this process ever terminates we are done. Suppose it does not. Then we have a strictly increasing tower of ideals\n$$ \\ab{r} \\subsetneq \\ab{s_0} \\subsetneq \\ab{s_1} \\subsetneq \\dots. $$\nThis cannot continue forever. Arguing as above we arrive at a contradiction.\n\\end{proof}\n\nNow we show $ 2 $.\n\n\\begin{proof}[Proof of Theorem \\ref{thm:3.3.2}]\nIt suffices to show that in a PID every irreducible is prime. Let $ r \\in R $ be irreducible, and suppose that $ r $ divides $ st $. Want $ r \\mid s $ or $ r \\mid t $. Let $ q $ be a generator of the ideal $ \\ab{r, s} $ of $ R $, so $ \\ab{r, s} = \\ab{q} $. Then $ q $ divides $ r $, so either $ q $ is a unit or $ q $ is an associate of $ r $. If $ q $ is an associate of $ r $, then since $ q $ divides $ s $, $ r $ divides $ s $. on the other hand, if $ q $ is a unit, then the ideal generated by $ r $ and $ s $ is the unit ideal and $ 1 \\in \\ab{r, s} $, so we can write $ 1 = xr + ys $ for $ x, y $ elements of $ R $. We then have $ t = xrt + yst $, and since $ r $ divides both $ yst $ and $ xrt $, $ r $ divides $ t $.\n\\end{proof}\n\n\\marginpar{Lecture 6 \\\\ Wednesday \\\\ 16/10/18}\n\n\\subsection{Euclidean domains}\n\nOne technique for proving that rings are PIDs is Euclid's algorithm. We formalise this in an abstract setting as follows.\n\n\\begin{definition}\nLet $ R $ be an integral domain.\n\\begin{enumerate}\n\\item A \\textbf{Euclidean norm} on $ R $ is a function $ N : R \\to \\Z_{\\ge 0} $ such that for all $ a, b \\in R $, with $ b \\ne 0 $, there exists $ q, r \\in R $ such that $ a = qb + r $, and either $ r = 0 $ or $ N\\rb{r} < N\\rb{b} $.\n\\item An integral domain $ R $ is called a \\textbf{Euclidean domain} if there is a Euclidean norm on $ R $.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{theorem}\nAny Euclidean domain is a PID.\n\\end{theorem}\n\n\\begin{proof}\nLet $ R $ be a Euclidean domain, $ N $ be a Euclidean norm on $ R $, and $ I \\subseteq R $ a nonzero ideal of $ R $. Let $ a \\in I $ be a nonzero element such that $ N\\rb{a} $ is minimal, that is if $ b \\in I $, $ b \\ne 0 $, then $ N\\rb{b} \\ge N\\rb{a} $. Claim that $ I = \\ab{a} $. Let $ b \\in I $. Then there exist $ q, r $ such that $ b = aq + r $, with either $ r = 0 $ or $ N\\rb{r} \\ge N\\rb{a} $. So $ r = 0 $ gives $ b = aq $. Thus $ I = \\ab{a} $.\n\\end{proof}\n\n\\begin{proof}\nLet $ R $ be a Euclidean domain, $ N $ be a Euclidean norm on $ R $, and $ I \\subseteq R $ be a nonzero ideal of $ R $. Let $ n $ be the smallest integer such that there exists a nonzero element $ a \\in I $ with $ N\\rb{a} = n $ minimal, that is if $ b \\in I $ and $ b \\ne 0 $, then $ N\\rb{b} < N\\rb{a} $. Claim that $ I = \\ab{a} $. Then for any $ b \\in I $, we can write $ b = qa + r $ with $ N\\rb{r} < N\\rb{a} $ unless $ r = 0 $. But since $ N\\rb{a} $ is the smallest possible norm in $ I $, we must have $ r = 0 $, so $ b = qa $. Thus $ I $ is generated by $ a $ and we are done.\n\\end{proof}\n\n\\subsection{Examples}\n\n\\begin{example}\n\\hfill\n\\begin{enumerate}\n\\item The classic example of a Euclidean domain is $ \\Z $, with $ N\\rb{x} = \\abs{x} $ for $ x \\in \\Z $.\n\\item The ring $ \\Z\\sb{i} $ is a Euclidean domain, with $ N\\rb{z} = z\\bar{z} = \\abs{z}^2 $, so $ N\\rb{x + yi} = \\abs{x + yi}^2 = x^2 + y^2 $. To see this, note that given $ a $ and $ b $ in $ \\Z\\sb{i} $ for $ b \\ne 0 $, set $ q' = a / b \\in \\Q\\sb{i} $. Write $ q' = x' + iy' $ with $ x', y' \\in \\Q $. Let $ x $ and $ y $ be the closest integers to $ x' $ and $ y' $, such that $ \\abs{x - x'}, \\abs{y - y'} \\le 1 / 2 $, and set $ q = x + iy $ in $ \\Z\\sb{i} $ and $ r = a - bq $. Then\n$$ N\\rb{r} = \\abs{r}^2 = \\abs{a - bq}^2 = \\abs{a - b\\rb{\\dfrac{a}{b} + \\rb{q - q'}}}^2 = \\abs{b\\rb{q - q'}}^2 = \\abs{b}^2\\abs{q - q'}^2 \\le \\dfrac{N\\rb{b}}{2}. $$\nSimilar arguments can be used to prove that $ \\Z\\sb{\\alpha} $ is a Euclidean domain for\n$$ \\alpha = \\sqrt{-2}, \\qquad \\alpha = \\dfrac{-1 + \\sqrt{-3}}{2}, \\qquad \\alpha = \\dfrac{-1 + \\sqrt{-7}}{2}. $$\nBeyond this one needs other tricks and for most $ \\alpha $ unique factorisation fails.\n\\item A critical example is the polynomial ring $ K\\sb{X} $ for $ K $ a field. Here we can take $ N\\rb{P\\rb{X}} $ to be the degree of $ P\\rb{X} $. Then, given polynomials $ P\\rb{X}, T\\rb{X} \\in K\\sb{X} $ and $ T\\rb{X} \\ne 0 $, we can use polynomial long division to write $ P\\rb{X} = Q\\rb{X}T\\rb{X} + R\\rb{X} $ for some $ Q\\rb{X} $ with the degree of $ R $ strictly less than that of $ T $, unless $ T $ is constant, in which case we can make $ R = 0 $. To prove this, fix $ T\\rb{X} $. If $ \\deg\\rb{T\\rb{X}} = 0 $, $ T\\rb{X} $ is constant, so $ T\\rb{X} = c \\ne 0 \\in K $. Take $ Q\\rb{X} = c^{-1}P\\rb{X} $, so $ R\\rb{X} = 0 $. Otherwise induct on $ \\deg\\rb{P\\rb{X}} $. If $ \\deg\\rb{P\\rb{X}} < \\deg\\rb{T\\rb{X}} $, set $ R\\rb{X} = P\\rb{X} $ and $ Q\\rb{X} = 0 $. Suppose the claim is true for polynomials of degree $ n $ and $ P\\rb{X} $ has degree $ n + 1 $, so\n$$ P\\rb{X} = \\sum_{i = 0}^{n + 1} a_iX^i, \\qquad T\\rb{X} = \\sum_{i = 0}^d b_iX^i, $$\nfor $ d < n + 1 $. Then $ S\\rb{X} = P\\rb{X} - \\rb{a_{n + 1} / b_d}X^{n + 1 - d}T\\rb{X} $ has degree $ n $. By inductive hypothesis there exist $ Q\\rb{X}, R\\rb{X} $ with $ \\deg\\rb{R\\rb{X}} < \\deg\\rb{T\\rb{X}} $ such that\n$$ S\\rb{X} = Q\\rb{X}T\\rb{X} + R\\rb{X} \\qquad \\implies \\qquad P\\rb{X} = \\rb{\\dfrac{a_{n + 1}}{b_d}X^{n + 1 - d} + Q\\rb{X}}T\\rb{X} + R\\rb{X}. $$\n\\end{enumerate}\n\\end{example}\n\nLater, will show if $ R $ UFD, then $ R\\sb{X} $ is also a UFD.\n\n\\section{The Chinese remainder theorem}\n\nIn elementary number theory, let $ m_1, m_2 \\in \\Z $ be relatively prime and $ a_1, a_2 \\in \\Z $. Then there exists $ a \\in \\Z $ such that\n$$ a \\equiv a_1 \\mod m_1, \\qquad a \\equiv a_2 \\mod m_2. $$\nMoreover, $ a $ is unique up to congruence modulo $ m_1m_2 $. Question is given ideals $ I_1, \\dots, I_r $ and $ a_1, \\dots, a_r \\in \\R $, when can we find a $ a \\in R $ with $ a \\in a_1 + I_1, \\dots a_r + I_r $?\n\n\\subsection{Products}\n\n\\begin{definition}\nLet $ R_1, \\dots, R_n $ be rings. The \\textbf{direct product} $ R \\times \\dots \\times R_n $ is a ring whose elements are $ n $-tuples $ \\rb{r_1, \\dots, r_n} $ with $ r_i \\in R_i $ for all $ i $. The addition and multiplication are given componentwise.\n$$ \\rb{r_1, \\dots, r_n} + \\rb{r_1', \\dots, r_n'} = \\rb{r_1 + r_1', \\dots, r_n + r_n'}, \\qquad \\rb{r_1, \\dots, r_n}\\rb{r_1', \\dots, r_n'} = \\rb{r_1r_1', \\dots, r_nr_n'}. $$\n\\end{definition}\n\n\\begin{note}\nThe product comes with natural homomorphisms for all $ i $, $ \\pi_i $, \\textbf{projection} onto the $ i $-th factor, defined by\n$$ \\pi_i\\rb{r_1, \\dots, r_n} = r_i : R_1 \\times \\dots \\times R_n \\to R_i, $$\nand the following universal property.\n\\end{note}\n\n\\begin{theorem}[Universal property of the product]\nLet $ S, R_1, \\dots, R_n $ be any rings. For any homomorphisms $ f_1 : S \\to R_1, \\dots, f_n : S \\to R_n $, there exists a unique homomorphism $ f : S \\to R_1 \\times \\dots \\times R_n $ such that $ \\pi_i \\circ f = f $ for all $ i $.\n\\end{theorem}\n\n\\begin{proof}\nGiven $ f_i $, the homomorphism $ f $ is defined by $ f\\rb{s} = \\rb{f_1\\rb{t}, \\dots, f_n\\rb{t}} $. Then $ \\rb{\\pi_i \\circ f}\\rb{s} = f_i\\rb{s} $. For uniqueness, if $ \\rb{\\pi \\circ g}\\rb{s} = f_i\\rb{s} $ for all $ i $, then $ g\\rb{s} = \\rb{f_1\\rb{s}, \\dots, f_n\\rb{s}} = f\\rb{s} $.\n\\end{proof}\n\nMore generally, if $ I $ is any index set, and for each $ i \\in I $ we have a ring $ R_i $, we can define the product $ \\prod_i R_i $. An element $ r $ of this product is a choice, for each $ i \\in I $, of an element of $ R_i $. We write such an element as $ \\rb{r_i}_{i \\in I} $. For each $ j \\in I $ we have a map $ \\pi_j : \\prod_i R_i \\to R_j $ given by $ \\pi_j\\rb{\\rb{r_i}_{i \\in I}} = r_j $. Such a product satisfies a very similar universal property. For any collection $ f_i : S \\to R_i $ of maps for each $ i \\in I $, we get a unique map $ f : S \\to \\prod_i R_i $ such that $ \\pi_j \\circ f = f_j $.\n\n\\subsection{The Chinese remainder theorem}\n\nLet $ R $ be a ring, and let $ I_1, \\dots, I_r $ be a finite collection of ideals of $ R $. We have the natural maps $ R \\to R / I_1, \\dots, R \\to R / I_r $, which are surjective with kernel $ I_j $. Consider the product map\n$$ R \\to \\dfrac{R}{I_1} \\times \\dots \\times \\dfrac{R}{I_r}. $$\nIt is easy to see that the kernel of this map is the set of $ r \\in R $ such that $ r $ maps to zero in $ R / I_j $ for all $ j $. That is, the kernel is the intersection $ I_1 \\cap \\dots \\cap I_r $. Call this ideal $ J $. We thus have an injective embedding\n$$ \\dfrac{R}{J} \\hookrightarrow \\dfrac{R}{I_1} \\times \\dots \\times \\dfrac{R}{I_r}. $$\nA natural question to ask is, what can we say about the image? In other words, given congruence classes modulo $ I_1, \\dots, I_r $, when is there a single element of $ R $ that lives in all those congruence classes simultaneously?\n\n\\begin{note}\nBecause the above map is injective, if one such element exists, then there is a unique congruence class modulo $ J $ that satisfies all of the required congruences.\n\\end{note}\n\nOf course, without further hypotheses we cannot expect this map to be surjective. Think about what happens when $ I_1 = I_2 $, for instance. Nonetheless, we have the following.\n\n\\begin{definition}\nWe will say $ I_1, \\dots, I_r $ are \\textbf{pairwise relatively prime} if for each $ i \\ne j $, the sum $ I_i + I_j $ is the unit ideal in $ R $.\n\\end{definition}\n\n(TODO Exercise: if $ R = \\Z $, then $ I_i = \\ab{n_i} $, and $ \\cb{I_i} $ is pairwise relatively prime if and only if for all $ i \\ne j $, $ n_i $ and $ n_j $ are relatively prime)\n\n\\begin{theorem}\nLet $ R $ be a ring and $ I_1, \\dots, I_r $ be pairwise relatively prime ideals. Then the natural map\n$$ \\dfrac{R}{J} \\hookrightarrow \\dfrac{R}{I_1} \\times \\dots \\times \\dfrac{R}{I_r} $$\nis an isomorphism.\n\\end{theorem}\n\n\\begin{proof}\nWe have to prove it is surjective. Fix any tuple $ \\rb{c_1, \\dots, c_r} $ of elements of $ R $. We need to find $ c \\in R $ such that $ c \\in c_i + I_i $ for all $ i $. It suffices to construct, for each $ i $, an element $ e_i $ of $ R $ such that $ e_i \\equiv 1 \\mod I_i $ and $ e_i \\equiv 0 \\mod I_j $ for $ i \\ne j $. Suppose we have such an element. Then the element $ c = c_1e_1 + \\dots + c_re_r $ is such that $ c \\equiv c_j \\mod I_j $ for all $ j $. Given $ i, j $ with $ i \\ne j $, we know that $ I_i + I_j $ is the unit ideal. That is, we can write $ a_{ij} + b_{ij} = 1 $ with $ a_{ij} \\in I_i $ and $ b_{ij} \\in I_j $. Then $ a_{ij} \\equiv 1 \\mod I_j $ and $ a_{ij} \\equiv 0 \\mod I_i $ as an element of $ R / I_1 \\times \\dots \\times R / I_r $, so $ a_{ij} $ has zero in the $ i $-th place and one in the $ j $-th place. Then for any $ j $ we can take $ e_j = \\prod_{i \\ne j} a_{ij} $ and $ e_j \\equiv 1 \\mod I_j $ and $ e_j \\equiv 0 \\mod I_i $ for all $ i \\ne j $, so $ e_j $ has one only in the $ j $-th place. So $ R \\to R / I_1 \\times \\dots \\times R / I_r $ is surjective. The result follows.\n\\end{proof}\n\n\\subsection{Examples}\n\nWhen $ R = \\Z $, then every ideal is principal, so we can write $ I_j = \\ab{n_j} $ for all $ j $. The condition that $ I_i + I_j $ is the unit ideal becomes the condition that $ n_i \\in \\Z $ are pairwise relatively prime. In this case the ideal $ J $ is generated by the product $ n $ of the $ n_i $. Specialising, we find the version of the Chinese remainder theorem from elementary number theory.\n\n\\begin{theorem}\nIf $ \\cb{n_j \\in \\Z} $ is a finite collection of pairwise relatively prime integers, and $ n $ is their product, then for any $ c_1, \\dots, c_r \\in \\Z $, there exists $ c \\in \\Z $ unique up to congruence modulo $ n $ such that $ c $ is congruent to $ c_i \\mod n_i $ for all $ i $.\n\\end{theorem}\n\nNow let $ K $ be a field and take $ R = K\\sb{X} $. If $ c_1, \\dots, c_r \\in K $ are distinct elements of $ K $, the ideals $ I_i = \\ab{X - c_i} \\subseteq R $ are such that $ I_i + I_j = \\ab{X - c_i} + \\ab{X - c_j} $ contains $ c_i - c_j \\in K^* $, so contains $ 1 $. That is, $ I_i + I_j $ is the unit ideal in $ R $ and the ideals $ I_i $ are pairwise relatively prime. Moreover, for each $ i $, $ I_i $ is the kernel of the evaluation map $ f_i : R \\to K $ by that takes $ P\\rb{X} $ to $ P\\rb{c_i} $. Let $ f : R \\to K \\times \\dots \\times K $ by $ P\\rb{X} \\mapsto \\rb{P\\rb{c_1}, \\dots, P\\rb{c_r}} $. Then the following diagram commutes.\n$$\n\\begin{tikzcd}[column sep=1in]\nR \\arrow{r}{f} \\arrow[two heads]{d} & K \\times \\dots \\times K \\\\\n\\dfrac{R}{J} \\arrow[swap]{r}{\\sim} & \\dfrac{R}{I_1} \\times \\dots \\times \\dfrac{R}{I_r} \\arrow[swap]{u}{\\sim}\n\\end{tikzcd}\n$$\nChinese remainder theorem gives that $ f $ is surjective. We thus have an isomorphism of $ R / I_i $ with $ K $ that takes $ P\\rb{X} $ to $ P\\rb{c_i} $ for all polynomials $ P $. We thus obtain the following.\n\n\\begin{theorem}\nFor any $ c_1, \\dots, c_n \\in K $, there is a polynomial $ P\\rb{X} $ in $ R $, unique up to congruence modulo $ \\rb{X - a_1} \\dots \\rb{X - a_n} $ such that $ P\\rb{a_i} = c_i $ for all $ i $.\n\\end{theorem}\n\n\\marginpar{Lecture 7 \\\\ Friday \\\\ 19/10/18}\n\n\\section{Fields and field extensions}\n\nNext we will use $ K\\sb{X} $ is a PID for $ K $ a field to study fields systematically.\n\n\\subsection{Prime fields}\n\nLet $ K $ be a field. We have a unique ring homomorphism $ \\iota : \\Z \\to K $ by $ n \\ge 0 \\mapsto n_K = 1_K + \\dots + 1_K $. Let $ I $ be the kernel. Then $ \\Z / I \\hookrightarrow K $ so $ \\Z / I $ is an integral domain, so $ I $ is a prime ideal. Thus $ I $ is either the zero ideal $ \\cb{0} $, if $ K $ has characteristic zero, or the ideal $ \\ab{p} $ for some prime $ p $ of $ \\Z $. In the former case $ I = \\cb{0} $, the injection $ \\Z \\hookrightarrow K $ extends to an inclusion $ \\Q \\hookrightarrow K $ sending $ a / b \\mapsto \\rb{\\iota a}\\rb{\\iota b^{-1}} = a_K / b_K $. In the latter case $ I = \\ab{p} $, we get an injection $ \\Z / p\\Z \\hookrightarrow K $, which we often denote $ \\F_p $ when we think of it as a field. Upshot is that every field $ K $ contains exactly one of $ \\Q $, $ \\F_p $, for $ p $ prime, in exactly one way depending on its characteristic. This field is called the \\textbf{prime field} of $ K $, and it is contained in $ K $ in a unique way.\n\n\\subsection{Field extensions}\n\nThe prime fields are in some sense the smallest possible fields. Once we know they exist, it makes sense to study fields by studying pairs $ K, L $ of fields such that $ K \\subseteq L $ of fields, trying to relate $ L $ to $ K $.\n\n\\begin{definition}\nA \\textbf{field extension} is a pair of fields $ K, L $ with $ K \\subseteq L $, and is often denoted $ L / K $.\n\\end{definition}\n\n\\begin{note}\nSuch an inclusion of fields $ L / K $ makes $ L $ into a $ K $-vector space, that is a vector space over $ K $.\n\\end{note}\n\n\\begin{definition}\nWe say that a field extension $ L / K $ is \\textbf{finite} if $ L $ is finite-dimensional as a $ K $-vector space. If this is the case, the \\textbf{degree} of such an extension is the dimension of $ L $ as a $ K $-vector space $ \\dim_KL $, and is denoted $ \\sb{L : K} $.\n\\end{definition}\n\n\\begin{proposition}\nLet $ K \\subseteq L \\subseteq M $ be fields. Then $ M / K $ is finite if and only if $ M / L $ and $ L / K $ are both finite. If this is the case then $ \\sb{M : K} = \\sb{M : L}\\sb{L : K} $.\n\\end{proposition}\n\n\\begin{proof}\nFirst suppose that $ M / K $ is finite. Then $ L $ is a $ K $-subspace of $ M $, so finite dimensional as a $ K $-vector space. Moreover, there exists a $ K $-basis $ m_1, \\dots, m_r $, and this basis spans $ M $ over $ K $ and thus also over $ L $. Thus $ M $ is finite-dimensional as an $ L $-vector space, so $ M / L $ is finite. Conversely, suppose $ L / K $, $ M / L $ are finite. Let $ e_1, \\dots, e_n $ be a $ K $-basis for $ L $, and let $ f_1, \\dots, f_n $ be an $ L $-basis for $ M $. Then claim that\n$$ e_1f_1, \\dots, e_1f_m, \\dots, e_nf_1, \\dots, e_nf_m $$\nis a $ K $-basis for $ M $. Every element $ x $ of $ M $ can be expressed uniquely as $ c_1f_1 + \\dots + c_mf_m $ with $ c_i \\in L $. Each $ c_i $ in turn can be expressed as $ d_{1, i}e_1 + \\dots + d_{n, i}e_n $ with $ d_{j, i} \\in K $. Thus we can express $ x $ as\n$$ d_{1, 1}e_1f_1 + \\dots + d_{n, 1}e_nf_1 + \\dots + d_{1, m}e_1f_m + \\dots + d_{n, m}e_nf_m. $$\nIn particular the set $ \\cb{e_if_j} $ for $ 1 \\le i \\le n $ and $ 1 \\le j \\le m $ spans $ M $ over $ K $. In this case the degree of $ L $ over $ K $ is $ n $ and the degree of $ M $ over $ L $ is $ m $, so it remains to show that $ \\cb{e_if_j} $ is linearly independent over $ K $. Suppose we have elements $ d_{i, j} $ of $ K $ such that $ \\sum_{i, j} d_{i, j}e_if_j = 0 $. Then, regrouping, we find that $ \\sum_j \\rb{\\sum_i d_{i, j}e_i}f_j = 0 $ is an $ L $-linear combination of the $ f_j $ that is zero. Since the $ f_j $ are linearly independent over $ L $ we must have $ \\sum_i d_{i, j}e_i = 0 $ for all $ j $. Since the $ e_i $ are linearly independent over $ K $ we must have $ d_{i, j} = 0 $ for all $ i, j $.\n\\end{proof}\n\n\\marginpar{Lecture 8 \\\\ Monday \\\\ 22/10/18}\n\n\\subsection{Extensions generated by one element}\n\nLet $ L / K $ be a field extension, and let $ \\alpha $ be an element of $ L $.\n\n\\begin{definition}\nWe let $ K\\rb{\\alpha} $ denote the subfield of $ L $ consisting of all elements of $ L $ that can be expressed in the form $ P\\rb{\\alpha} / Q\\rb{\\alpha} $, where $ P $ and $ Q $ are polynomials with coefficients in $ K $ and $ Q\\rb{\\alpha} $ is not zero. This is the smallest subfield of $ L $ containing $ K $ and $ \\alpha $.\n\\end{definition}\n\nRecall that if $ R, S $ are rings, $ f : R \\to S $ is a homomorphism, and $ \\alpha \\in S $, then have $ \\phi_{f, a} : R\\sb{X} \\to S $ by $ \\phi_{f, a}\\rb{\\sum_{i = 1}^n r_iX^i} = \\sum_{i = 1}^n f\\rb{r_i}\\alpha^i $. We have a natural map $ K\\sb{X} \\to K\\rb{\\alpha} \\subseteq L. $, inclusion on $ K $, that takes a polynomial $ P\\rb{X} $ to $ P\\rb{\\alpha} $. It is a ring homomorphism. Let $ I $ be the kernel of this homomorphism. We then get an injection of $ K\\sb{X} / I $ into the field $ K\\rb{\\alpha} $. Thus $ K\\sb{X} / I $ is an integral domain, so $ I $ is a prime ideal of $ K\\sb{X} $. Since $ K\\sb{X} $ is a PID, every nonzero prime ideal is maximal. (TODO Exercise) There are thus two cases. In the first $ I $ is the zero ideal that is not maximal. That is, there is no nonzero polynomial $ Q $ in $ K\\sb{X} $ such that $ Q\\rb{\\alpha} $ is zero in $ L $. We say that $ \\alpha $ is \\textbf{transcendental} over $ K $ in this case. In the second $ I $ is an ideal $ \\ab{Q} $ for $ Q \\in K\\sb{X} $ a nonzero irreducible polynomial that is a maximal ideal of $ K\\sb{X} $. In this case we say $ \\alpha $ is \\textbf{algebraic} over $ K $.\n\n\\begin{definition}\n$ K\\rb{X} $ is the \\textbf{field of rational functions} on $ X $,\n$$ K\\rb{X} = \\cb{\\dfrac{P\\rb{X}}{Q\\rb{X}} \\ \\Bigg| \\ P, Q \\in K\\sb{X}, \\ Q \\ne 0} / \\sim. $$\n\\end{definition}\n\nAssume first that $ \\alpha $ is transcendental over $ K $, that is $ I = \\cb{0} $. Recall $ I = \\cb{P\\rb{X} \\in K\\sb{X} \\mid P\\rb{\\alpha} = 0} $. So in this case there is no nonzero polynomial $ P \\in K\\sb{X} $ with $ P\\rb{\\alpha} = 0 $. In this case the map taking $ P\\rb{X} $ to $ P\\rb{\\alpha} $ is an injection of $ K\\sb{X} $ into $ K\\rb{\\alpha} \\subseteq L $. In particular every nonzero element of $ K\\sb{X} $ gets sent to a nonzero, hence invertible, element of $ L $. Thus the map from $ K\\sb{X} $ to $ L $ extends to an injective map from the field of fractions of $ K\\sb{X} $, which we denote $ K\\rb{X} $, to $ L $. This map takes $ P\\rb{X} / Q\\rb{X} $ to $ P\\rb{\\alpha} / Q\\rb{\\alpha} $. By definition of $ K\\rb{\\alpha} $, this map is surjective so the image of this map is $ K\\rb{\\alpha} $. In particular $ K\\rb{X} $ and $ K\\rb{\\alpha} $ are isomorphic. Thus the following diagram holds.\n$$\n\\begin{tikzcd}[column sep=1in]\n& L \\\\\nK\\rb{X} \\arrow{r}{\\sim}[swap]{g} & K\\rb{\\alpha} \\arrow[hookrightarrow, swap]{u}{\\subseteq} \\\\\nK\\sb{X} \\arrow[hookrightarrow]{u}{\\subseteq} \\arrow[hookrightarrow, swap]{ur}{f}\n\\end{tikzcd}\n\\qquad f : P\\rb{X} \\mapsto P\\rb{\\alpha} \\qquad g : \\dfrac{P\\rb{X}}{Q\\rb{X}} \\mapsto \\dfrac{P\\rb{\\alpha}}{Q\\rb{\\alpha}}\n$$\n\n\\begin{note}\nIn this case $ K\\rb{\\alpha} $ is infinite dimensional as a $ K $-vector space. It contains a subspace isomorphic to $ K\\sb{X} $, for instance.\n\\end{note}\n\nIf $ \\alpha $ is algebraic over $ K $, then $ I $ is a nonzero maximal ideal of the PID $ K\\sb{X} $, so it is generated by a single irreducible polynomial $ Q\\rb{X} $ in $ K\\sb{X} $. As a consequence, since the units in $ K\\sb{X} $ are just the constant polynomials, the polynomial $ Q\\rb{X} $ is well-defined up to a constant factor. It is called the \\textbf{minimal polynomial} of $ \\alpha $. By definition, it divides every polynomial $ P\\rb{X} $ such that $ P\\rb{\\alpha} = 0 $. Since $ \\ab{Q\\rb{X}} $ is maximal, the ring $ K\\sb{X} / \\ab{Q\\rb{X}} $ is a field. Recall that for any $ P \\in K\\sb{X} $, can write $ P\\rb{X} $ uniquely as $ A\\rb{X}Q\\rb{X} + R\\rb{X} $ with $ \\deg\\rb{R} < \\deg\\rb{Q} $. So $ 1, \\dots, X^{\\deg\\rb{Q} - 1} $ are a $ K $-basis of $ K\\sb{X} / \\ab{Q\\rb{X}} $. So its dimension as a $ K $-vector space is equal to the degree of $ Q\\rb{X} $. The map $ K\\sb{X} \\to K\\rb{\\alpha} \\subseteq L $ descends to an injection of $ K\\sb{X} / \\ab{Q\\rb{X}} $ into $ L $. Since its image is a subfield of $ K\\rb{\\alpha} $ containing $ K $ and $ \\alpha $, this map is an isomorphism of $ K\\rb{\\alpha} $ with $ K\\sb{X} / \\ab{Q\\rb{X}} $. Thus in this case the extension $ K\\rb{\\alpha} / K $ is a finite extension, of degree equal to the degree of $ Q\\rb{X} $. Thus the following diagram holds.\n$$\n\\begin{tikzcd}[column sep=1in]\nK\\sb{X} \\arrow[twoheadrightarrow]{dr}{f} & L \\\\\n\\dfrac{K\\sb{X}}{\\ab{Q\\rb{X}}} \\arrow[hookrightarrow]{u}{\\subseteq} \\arrow{r}{g}[swap]{\\sim} & K\\rb{\\alpha} \\arrow[hookrightarrow, swap]{u}{\\subseteq}\n\\end{tikzcd}\n\\qquad f : P\\rb{X} \\mapsto P\\rb{\\alpha} \\qquad g : \\sb{R\\rb{X}}_{\\ab{Q\\rb{X}}} \\to R\\rb{\\alpha}\n$$\n\nTo summarise, extend $ K $ by a single element by\n\\begin{enumerate}\n\\item building $ K\\sb{X} $, and\n\\item either passing to field of fractions $ K\\rb{X} $ to form a transcendental extension, or choosing an irreducible polynomial $ Q $ to form an algebraic extension $ K\\sb{X} / \\ab{Q\\rb{X}} $.\n\\end{enumerate}\nSlightly informally, instead of $ K\\sb{X} / \\ab{Q\\rb{X}} $, we sometimes write $ K\\rb{\\alpha} $, where $ \\alpha $ is a root of $ Q\\rb{X} $.\n\n\\begin{definition}\nAn extension $ L / K $ is \\textbf{algebraic} if every element of $ L $ is algebraic over $ K $.\n\\end{definition}\n\nAn observation is that if $ L / K $ is finite, then $ L / K $ is algebraic. Suppose not. Let $ \\alpha \\in L $ be transcendental over $ K $. $ K\\sb{X} $ is a polynomial ring in $ K\\rb{\\alpha} $ contained in $ L $, so $ L / K $ is not finite.\n\n\\begin{corollary}\nLet $ L / K $ be a field extension for $ \\alpha, \\beta \\in L $ algebraic over $ K $. Then $ \\alpha + \\beta $, $ \\alpha\\beta $ are algebraic over $ K $.\n\\end{corollary}\n\n\\begin{proof}\n$ \\sb{K\\rb{\\alpha} : K} = \\deg\\rb{\\alpha} $ and $ \\sb{K\\rb{\\alpha, \\beta} : K\\rb{\\alpha}} \\le \\deg\\rb{\\beta} $, so $ \\sb{K\\rb{\\alpha, \\beta} : K} \\le \\rb{\\deg\\rb{\\alpha}}\\rb{\\deg\\rb{\\beta}} $. Now $ K \\subseteq K\\rb{\\alpha + \\beta} \\subseteq K\\rb{\\alpha, \\beta} $, so $ \\deg\\rb{\\alpha + \\beta} $ over $ K $ is at most $ \\rb{\\deg\\rb{\\alpha}}\\rb{\\deg\\rb{\\beta}} $. Similarly for $ \\alpha\\beta $.\n\\end{proof}\n\n\\begin{corollary}\nIf $ L / K $, then the subset $ L^{alg} $ of elements of $ L $ algebraic over $ K $ is a field.\n\\end{corollary}\n\n\\begin{proof}\nIf $ a_0 + \\dots + a_n\\alpha^n = 0 $ then $ a_0\\rb{\\alpha^{-1}}^n + \\dots + a_n = 0 $.\n\\end{proof}\n\n\\begin{example}\n$ \\bar{\\Q} \\subseteq \\C $ is the subfield of elements of $ \\C $ that are algebraic over $ \\Q $.\n\\end{example}\n\n\\subsection{Example}\n\n\\begin{example}\nConsider the polynomial $ X^2 + X + 1 $ in $ \\F_2\\sb{X} $. It has no roots in $ \\F2 $, so it is irreducible, as a polynomial of degree 2 any nontrivial factor would be linear. The other polynomials of degree two are $ X^2 $, $ X^2 + X = X\\rb{X + 1} $, $ X^2 + 1 = \\rb{X + 1}^2 $, so $ X^2 + X + 1 $ is the unique irreducible polynomial of degree two. Let $ \\F_4 = \\F_2\\sb{X} / \\ab{X^2 + X + 1} $. Thus the quotient $ \\F_2\\sb{X} = \\ab{X^2 + X + 1} $ is a field extension of degree two of $ \\F_2 $, which is denoted $ \\F_4 $. Its four elements are $ 0, 1, X, X + 1 $, or more precisely, their classes modulo $ \\ab{X^2 + X + 1} $.\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$ \\cdot $ & $ 0 $ & $ 1 $ & $ X $ & $ X + 1 $ \\\\\n\\hline\n$ 0 $ & $ 0 $ & $ 0 $ & $ 0 $ & $ 0 $ \\\\\n\\hline\n$ 1 $ & $ 0 $ & $ 1 $ & $ X $ & $ X + 1 $ \\\\\n\\hline\n$ X $ & $ 0 $ & $ X $ & $ X + 1 $ & $ 1 $ \\\\\n\\hline\n$ X + 1 $ & $ 0 $ & $ X + 1 $ & $ 1 $ & $ X $ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nNote that $ X^2 = -X - 1 = X + 1 $, $ X^2 + X + 1 = 0 $, $ \\rb{X + 1}^2 = X $, and $ X^3 = X\\rb{X + 1} = 1 $ in $ \\F_4 $. In particular the multiplicative group of $ \\F_4 $ is cyclic of order three. This is not particularly surprising, as all groups of order three are cyclic. We will see later, though, that the multiplicative group of any finite field is cyclic.\n\\end{example}\n\n\\begin{proposition}\nLet $ K $ be a field with four elements. Then $ K \\cong \\F_4 $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ \\alpha \\in K $ with $ \\alpha \\ne 0 $ and $ \\alpha \\ne 1 $. Consider $ 1, \\alpha, \\alpha^2 $. Since $ K $ has dimension two over $ \\F_2 $, there is a linear dependence. So there exists a polynomial $ P $ in $ \\F_2\\sb{X} $ of degree at most two such that $ P\\rb{\\alpha} = 0 $. In fact $ P $ must be irreducible of degree two. If it is divisible by something of degree one, then a polynomial of degree one vanishes on $ \\alpha $, so $ \\alpha = 0 $ or $ \\alpha = 1 $. So $ \\alpha^2 + \\alpha + 1 = 0 $. The map $ \\F_2\\sb{X} \\to K $ sending $ X $ to $ \\alpha $ descends to $ \\F_2\\sb{X} / \\ab{X^2 + X + 1} \\to K $. So $ \\F_4 $ embeds in $ K $. Thus the following diagram holds and $ K \\cong \\F_4 $.\n$$\n\\begin{tikzcd}[column sep=1in]\n\\F_2\\sb{X} \\arrow[twoheadrightarrow]{dr}{P\\rb{X} \\mapsto P\\rb{\\alpha}} & \\\\\n\\dfrac{\\F_2\\sb{X}}{\\ab{X^2 + X + 1}} \\arrow[hookrightarrow]{u}{\\subseteq} \\arrow[swap]{r}{\\sim} & K\n\\end{tikzcd}\n$$\n\\end{proof}\n\n\\marginpar{Lecture 9 \\\\ Wednesday \\\\ 24/10/18}\n\n\\section{Finite fields}\n\n\\subsection{Finite fields}\n\nLet $ K $ be a finite field. That is, a field with only finitely many elements. Then $ K $ has characteristic $ p $ for some prime $ p $, and is in particular a finite dimensional $ \\F_p $ vector space. Thus its order is a power $ p^r $ of $ p $ for $ r > 0 \\in \\Z $. If we fix a particular prime power $ p^r $, then two questions naturally arise. Does there exist a field of order $ p^r $? If so, can we classify fields of order $ p^r $ up to isomorphism? We will see that in fact, up to isomorphism, there is a unique field $ \\F_{p^r} $ of order $ p^r $.\n\n\\subsection{The Frobenius automorphism}\n\nLet $ p $ be a prime. For any ring $ R $, the map $ x \\mapsto x^p $ on $ R $ certainly satisfies $ \\rb{xy}^p = x^py^p $ for all $ x, y \\in R $. On the other hand,\n$$ \\rb{x + y}^p = x^p + \\two{p}{1}xy^{p - 1} + \\dots + \\two{p}{p - 1}x^{p - 1}y + y^p. $$\nNow the binomial coefficients satisfy\n$$ p \\ \\Bigg| \\ \\two{p}{i} = \\dfrac{p!}{i!\\rb{p - i}!}, $$\nfor $ 1 \\le i \\le p - 1 $, so if $ R $ has characteristic $ p $, we have $ \\rb{x + y}^p = x^p + y^p $. So $ x \\mapsto x^p : R \\to R $ is a ring homomorphism from $ R $ to $ R $, called the \\textbf{Frobenius endomorphism} of $ R $. If $ R $ is a field of characteristic $ p $, then the Frobenius endomorphism is injective. If in addition $ R $ is finite, then any injective map from $ R $ to $ R $ is surjective. In particular the Frobenius endomorphism is a bijective and an isomorphism from $ R $ to $ R $ when $ R $ is a finite field of characteristic $ p $. In this case we call the map $ x \\mapsto x^p $ the Frobenius \\textbf{automorphism}. Composing the Frobenius endomorphism with itself, we find that for any $ r $, $ x \\mapsto x^{p^r} $ is also an endomorphism of any ring $ R $ of characteristic $ p $.\n\n\\begin{example}\nLet $ R = \\F_4 $. $ y \\to y^2 $ gives $ 0 \\mapsto 0 $, $ 1 \\mapsto 1 $, $ X \\mapsto X + 1 $, and $ X + 1 \\mapsto X $.\n\\end{example}\n\n\\begin{note}\nLet $ K $ be a field of $ p^r $ elements. Then $ \\alpha^{p^r} = \\alpha $ for all $ \\alpha \\in K $. If $ \\alpha = 0 $, clear. Otherwise $ \\alpha \\in K^* $, $ K^* $ is an abelian group of order $ p^r - 1 $. Lagrange's theorem gives $ \\alpha^{p^r - 1} = 1 $, so $ \\alpha^{p^r} = \\alpha $.\n\\end{note}\n\nWe have the following.\n\n\\begin{proposition}\nLet $ K $ be a field of characteristic $ p $, such that $ \\alpha^{p^r} = \\alpha $ for all $ \\alpha \\in K $. Let $ P\\rb{X} \\in K\\sb{X} $ be an irreducible factor of $ X^{p^r} - X $ over $ K\\sb{X} $. Then every element $ \\beta $ of $ K\\sb{X} / \\ab{P\\rb{X}} $ satisfies $ \\beta^{p^r} = \\beta $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ d = \\deg\\rb{P} $. Can write $ \\beta = c_0 + \\dots + c_{d - 1}X^{d - 1} $. Moreover, since $ P\\rb{X} = 0 $ in $ K\\sb{X} / \\ab{P\\rb{X}} $ and $ P\\rb{X} $ divides $ X^{p^r} - X $, we have $ X^{p^r} = X $ in $ K\\sb{X} / \\ab{P\\rb{X}} $. Thus\n$$ \\beta^{p^r} = c_0^{p^r} + \\dots + c_{d - 1}^{p^r}\\rb{X^{p^r}}^{d - 1} = c_0 + \\dots + c_{d - 1}\\rb{X^{p^r}}^{d - 1} = c_0 + \\dots + c_{d - 1}X^{d - 1} = \\beta. $$\n\\end{proof}\n\n\\begin{corollary}\nThere exists a field $ K $ of characteristic $ p $ such that\n\\begin{enumerate}\n\\item $ \\alpha^{p^r} = \\alpha $ for all $ \\alpha \\in K $, and\n\\item the polynomial $ X^{p^r} - X $ of $ K\\sb{X} $ factors into linear factors over $ K\\sb{X} $.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{proof}\nLet $ K_0 = \\F_p $. $ K_0 $ satisfies $ 1 $. We construct a tower of fields $ K_0 = \\F_p \\subsetneq K_1 \\subsetneq \\dots $ all satisfying $ 1 $ as follows. Suppose we have constructed $ K_i $ satisfying $ 1 $. If $ X^{p^r} - X $ factors into linear factors over $ K_i\\sb{X} $, we are done. Otherwise, choose a nonlinear irreducible factor $ P_i\\rb{X} $ of $ X^{p^r} - X $ in $ K_i\\sb{X} $ of degree at least two, and set $ K_{i + 1} = K_i\\sb{X} / \\ab{P_i\\rb{X}} $. Then $ K_{i + 1} $ is strictly larger than $ K_i $ and still satisfies $ 1 $. On the other hand, in any field $ K_i $ satisfying $ 1 $, every element is a root of $ X^{p^r} - X $, so $ \\#K_i \\le p^r $ for all $ i $. Since this polynomial can have at most $ p^r $ roots, this process must eventually terminate.\n\\end{proof}\n\nSince $ X^{p^r} - X $ has degree $ p^r $, we expect the field $ K $ constructed above to have $ p^r $ elements. So it suffices to show that over any field $ K $ of characteristic $ p $, $ X^{p^r} - X $ has no repeated roots. To prove this we need an additional tool.\n\n\\subsection{Derivatives}\n\n\\begin{definition}\nLet $ R $ be a ring, and let $ P\\rb{X} = r_0 + \\dots + r_dX^d $ be an element of $ R\\sb{X} $. The \\textbf{derivative} $ P'\\rb{X} $ of $ P\\rb{X} $ is the polynomial $ r_1 + \\dots + dr_dX^{d - 1} $.\n\\end{definition}\n\n\\begin{note}\nJust as for differentiation in calculus, we have a Leibniz rule. For $ P, Q \\in R\\sb{X} $, $ \\rb{PQ}'\\rb{X} = P\\rb{X}Q'\\rb{X} + P'\\rb{X}Q\\rb{X} $, by reducing to $ P, Q $ monomials.\n\\end{note}\n\nFrom this we deduce the following.\n\n\\begin{lemma}\nLet $ K $ be a field, and let $ P\\rb{X} $ be a polynomial in $ K\\sb{X} $ with a multiple root in $ K $. Then $ P\\rb{X} $ and $ P'\\rb{X} $ have a common factor of degree greater than zero.\n\\end{lemma}\n\n\\begin{proof}\nLet $ \\alpha \\in K $ be the multiple root. Then we can write $ P\\rb{X} = \\rb{X - \\alpha}^2Q\\rb{X} $. Applying the Leibniz rule we get $ P'\\rb{X} = 2\\rb{X - \\alpha}Q\\rb{X} + \\rb{X - \\alpha}^2Q'\\rb{X} $ and it is clear that $ X - \\alpha $ divides both $ P\\rb{X} $ and $ P'\\rb{X} $.\n\\end{proof}\n\n\\begin{corollary}\nLet $ K $ be a field of characteristic $ p $. Then $ X^{p^r} - X $ has no repeated roots in $ K $.\n\\end{corollary}\n\n\\begin{proof}\nLet $ P\\rb{X} = X^{p^r} - X $. Then $ P'\\rb{X} = -1 $, so $ P\\rb{X} $ and $ P'\\rb{X} $ have no common factor.\n\\end{proof}\n\n\\begin{corollary}\nThere exists a finite field of $ p^r $ elements.\n\\end{corollary}\n\n\\begin{proof}\nThe field $ K $ we constructed has $ p^r $ elements.\n\\end{proof}\n\n\\marginpar{Lecture 10 \\\\ Friday \\\\ 26/10/18}\n\n\\subsection{The multiplicative group}\n\nRather than show immediately that there is a unique finite field of $ p^r $ elements, we make a detour to study the multiplicative group of a finite field. This is not strictly necessary to prove uniqueness, but will simplify the proof, and is of interest in its own right. Let $ K $ denote a field of $ p^r $ elements. The goal of this section is to show that $ K^* $ is cyclic.\n\n\\begin{note}\nAs a multiplicative group, $ K^* $ is an abelian group of order $ p^r - 1 $, so by Lagrange's theorem, we have $ \\alpha^{p^r - 1} = 1 $ for all $ \\alpha \\in K^* $.\n\\end{note}\n\nRecall for an abelian group $ A $, operation written additively, that the order of an element $ a $ of $ A $ is the smallest $ d \\in \\Z_{> 0} $ such that $ da = 0 $.\n\\begin{enumerate}\n\\item The order of an element $ a $ of $ A $ divides the order of $ A $.\n\\item If $ d'a = 0 $ for some $ d' \\in \\Z $ then the order of $ a $ divides $ d' $.\n\\end{enumerate}\nThe order of an element $ a $ of $ K^* $ is the smallest $ d \\in \\Z_{> 0} $ such that $ a^d = 1 $. Since $ a^{p^r - 1} = 1 $, the order of $ a $ is a divisor of $ p^r - 1 $. On the other hand, if $ d $ is a divisor of $ p^r - 1 $, then any element of order dividing $ d $ is a root of the polynomial $ X^d - 1 $. Since $ K $ is a field, this polynomial has at most $ d $ roots, and we find that there are at most $ d $ elements of $ K^* $ of order dividing $ d $. Order of any element divides $ p^r - 1 $. Know $ X^{p^r - 1} - 1 $ has $ p^r - 1 $ distinct roots in $ K $. For $ d \\mid p^r - 1 $, $ X^d - 1 \\mid X^{p^r - 1} - 1 $, so $ X^d - 1 $ has exactly $ d $ roots in $ K $. That is, for all $ d \\mid p^r - 1 $, $ K^* $ has exactly $ d $ elements of order dividing $ d $. In fact, we have the following.\n\n\\begin{proposition}\n\\label{prop:6.4.1}\nLet $ A $ be a finite abelian group of order $ n $, and suppose that $ A $ has exactly $ d $ elements of order dividing $ d $, for all $ d $ dividing $ n $. Then $ A $ is cyclic.\n\\end{proposition}\n\nIn particular $ K^* $ is cyclic. The remainder of this section will be devoted to proving this proposition. As a corollary, we deduce that the multiplicative group of any finite field is cyclic. Consider the cyclic group $ \\Z / n\\Z $. The order of any element in this group is a divisor of $ n $.\n\n\\begin{definition}\nFor $ n \\in \\Z $, we let $ \\Phi\\rb{n} $ denote the number of elements in $ \\rb{\\Z / n\\Z, +} $ of exact order $ n $. This equals to the number of elements $ t \\in \\Z $ for $ 1 \\le t \\le n $ such that $ \\rb{t, n} = 1 $.\n\\end{definition}\n\n\\begin{note}\nSince $ \\sb{1} $ in $ \\Z / n\\Z $ has order $ n $, $ \\Phi\\rb{n} $ is nonzero for all $ n $.\n\\end{note}\n\n\\begin{lemma}\nFor any $ d $ dividing $ n $, the cyclic group $ \\Z / n\\Z $ contains a unique subgroup of order $ d $, and any element of $ \\Z / n\\Z $ of order dividing $ d $ is contained in this subgroup.\n\\end{lemma}\n\n\\begin{proof}\nThe cyclic subgroup $ C $ of $ \\Z / n\\Z $ generated by $ n / d $ is clearly a subgroup of order $ d $. This has $ d $ elements $ \\sb{0}, \\dots, \\rb{d - 1}\\sb{n / d} $. Conversely, if $ x $ is an element of a subgroup of $ \\Z / n\\Z $ of order $ d $, then the order of $ x $ divides $ d $, so $ dx $ is divisible by $ n $, and hence by unique factorisation $ x $ is divisible by $ n / d $. Thus $ x $ is in $ C $ and the claim follows.\n\\end{proof}\n\nAs a consequence, we deduce the following.\n\n\\begin{corollary}\nFor any $ d $ dividing $ n $, $ \\Phi\\rb{d} $ is the number of elements of $ \\Z / n\\Z $ of order $ d $.\n\\end{corollary}\n\n\\begin{corollary}\nFor any $ n \\in \\Z $, we have\n$$ \\sum_{d \\mid n} \\Phi\\rb{d} = n. $$\n\\end{corollary}\n\n\\begin{proof}\nSince every element of $ \\Z / n\\Z $ has order $ d $ for some $ d $ dividing $ n $, the sum over all possible $ d $ dividing $ n $ of the number of elements of order $ d $ is just the number of elements of $ \\Z / n\\Z $, which is $ n $.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:6.4.1}]\nLet $ A $ be as in the proposition. We must show that $ A $ contains an element of order $ n $. In fact, we will show, by induction on $ d $, that $ A $ contains exactly $ \\Phi\\rb{d} $ elements of order $ d $ for all $ d \\mid n $. In particular, $ A $ has $ \\Phi\\rb{n} > 0 $ elements of order $ n $, so it is cyclic. If $ d = 1 $, the only element of order one is the identity of $ A $. Since $ \\Phi\\rb{1} = 1 $ the base case holds. Assume the claim is true for all $ d' < d $. $ A $ has\n\\begin{enumerate}\n\\item $ d $ elements of order dividing $ d $, and\n\\item $ \\Phi\\rb{d} $ elements of order $ d' $ for $ d' \\mid d $ and $ d' < d $,\n\\end{enumerate}\nso the number of elements of exact order $ d $ is $ d - \\sum_{d' \\mid d, \\ d' < d} \\Phi\\rb{d'} $. By the corollary, this is precisely $ \\Phi\\rb{d} $.\n\\end{proof}\n\n\\subsection{Uniqueness}\n\nWe now turn to the question of showing that any two fields of $ p^r $ elements are isomorphic. Let $ K $ be such a field. The cyclicity of $ K^* $ immediately shows.\n\n\\begin{proposition}\nAny finite field $ K $ of characteristic $ p $ is generated over $ \\F_p $ by a single element $ \\alpha \\in K $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ \\alpha $ be an element of $ K $, that generates $ K^* $ as an abelian group. Then $ \\F_p\\rb{\\alpha} $ is contained in $ K $, but contains $ \\alpha^n $ for all $ n $ so contains $ K^* $, hence $ K = \\F_p\\rb{\\alpha} $.\n\\end{proof}\n\nAs a corollary, we deduce the following.\n\n\\begin{proposition}\nFor any prime $ p $ and any $ r \\in \\Z_{> 0} $, there exists an irreducible polynomial $ P\\rb{X} \\in \\F_p\\sb{X} $ of degree $ r $ in $ \\F_p\\sb{X} $.\n\\end{proposition}\n\n\\begin{proof}\nLet $ K $ be a finite field of $ p^r $ elements, $ \\alpha $ be an element of $ K $ that generates $ K $ over $ \\F_p $, and $ P $ the minimal polynomial of $ \\alpha $ over $ \\F_p $. We then have a surjective map $ \\F_p\\sb{X} \\to K $ taking $ X $ to $ \\alpha $. It is kernel is generated by irreducible $ P $ of degree $ \\deg\\rb{P} = \\sb{\\F_p\\rb{\\alpha} : \\F_p} = r $. Thus the following diagram holds.\n$$\n\\begin{tikzcd}[column sep=1in]\n\\F_p\\sb{X} \\arrow[twoheadrightarrow]{dr}{Q\\rb{X} \\mapsto Q\\rb{\\alpha}} & \\\\\n\\dfrac{\\F_p\\sb{X}}{\\ab{P\\rb{X}}} \\arrow[hookrightarrow]{u}{\\subseteq} \\arrow[swap]{r}{\\sim} & K = \\F_p\\rb{\\alpha}\n\\end{tikzcd}\n$$\n\\end{proof}\n\nWe thus have the following.\n\n\\begin{lemma}\nEvery irreducible polynomial $ P\\rb{X} $ of degree $ r $ in $ \\F_p\\sb{X} $ is a divisor of $ X^{p^r - 1} - 1 $.\n\\end{lemma}\n\n\\begin{proof}\nLet $ K = \\F_p\\rb{\\alpha} $ where $ \\alpha $ is a root of $ P $. $ \\#K = p^r $ so $ \\alpha^{p^r} - \\alpha $ is zero in $ K $. So $ P\\rb{X} \\mid X^{p^r} - X $.\n\\end{proof}\n\n\\begin{corollary}\nAny two finite fields $ K, K' $ of cardinality $ p^r $ are isomorphic.\n\\end{corollary}\n\n\\begin{proof}\nChoose $ \\alpha \\in K $ such that $ \\alpha $ generates $ K $ over $ \\F_p $. We can then write $ K = \\F_p\\rb{\\alpha} \\cong \\F_p\\sb{X} / \\ab{P\\rb{X}} $, where $ P\\rb{X} $ is the minimal polynomial of $ \\alpha $ over $ \\F_p $. In particular $ P\\rb{X} $ is irreducible of degree $ r $. Since $ P\\rb{X} $ divides $ X^{p^r - 1} - 1 $ in $ \\F_p\\sb{X} $, it also divides $ X^{p^r - 1} - 1 $ in $ K'\\sb{X} $. Since in $ K'\\sb{X} $, $ X^{p^r - 1} - 1 $ factors into linear factors, $ P\\rb{X} $ also factors into linear factors over $ K' $. In particular there exists a root $ \\alpha' \\in K' $ of $ P\\rb{X} $ in $ K'\\sb{X} $ such that $ P\\rb{\\alpha'} = 0 $. Then the map $ \\F_p\\sb{X} \\to K' $ that sends $ X $ to $ \\alpha' $ has kernel $ \\ab{P\\rb{X}} $ and induces a map\n$$\n\\begin{tikzcd}[column sep=1in]\nK \\arrow{r}{Q\\rb{\\alpha} \\mapsto Q\\rb{X}}[swap]{\\sim} & \\dfrac{\\F_p\\sb{X}}{\\ab{P\\rb{X}}} \\arrow[hookrightarrow]{r}{Q\\rb{X} \\mapsto Q\\rb{\\alpha'}} & K'\n\\end{tikzcd}\n$$\nSince this is map of fields from $ K $ to $ K' $ that takes $ \\alpha $ to $ \\alpha' $ it is injective. Since both fields $ K, K' $ have the same cardinality $ p^r $, it is also surjective and an isomorphism.\n\\end{proof}\n\nIf $ k = \\Q $, $ \\Q\\sb{X} / \\ab{X^2 - p} $ are pairwise nonisomorphic extensions of degree $ \\alpha $ for every prime $ p $.\n\n\\marginpar{Lecture 11 \\\\ Monday \\\\ 29/10/18}\n\nLecture 11 is a problem class.\n\n\\marginpar{Lecture 12 \\\\ Wednesday \\\\ 31/10/18}\n\n\\section{$ R $-modules}\n\n\\subsection{Definitions}\n\n\\begin{definition}\nAn $ R $-module $ M $ is a set, together with two operations $ + : M \\times M \\to M $ and $ \\cdot : R \\times M \\to M $, such that\n\\begin{enumerate}\n\\item $ \\rb{M, +} $ makes $ M $ into an abelian group with identity $ 0_M $,\n\\item $ r\\rb{m + m'} = rm + rm' $ for all $ r \\in R $, $ m, m' \\in M $,\n\\item $ \\rb{r + r'} = rm + r'm $ for all $ r, r' \\in R $, $ m \\in M $,\n\\item $ \\rb{rr'}m = r\\rb{r'm} $ for all $ r, r' \\in R $, $ m \\in M $, and\n\\item $ 1_R \\cdot m = m $ for all $ m \\in M $.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{note}\nFor an abelian group $ M $, let $ End\\rb{M} $ denote the set of homomorphisms $ M \\to M $ of abelian groups. $ End\\rb{M} $ is a noncommutative ring. $ 2 $ if and only if for all $ r \\in R $, $ \\cdot r : M \\to M $ lives in $ End\\rb{M} $. $ 3 $, $ 4 $, and $ 5 $ if and only if the map $ R \\to End\\rb{M} $ given by $ 2 $ is a homomorphism of rings.\n\\end{note}\n\n\\begin{example}\nThe usual addition and multiplication on $ R $ naturally makes $ R $ into an $ R $-module. More generally, any ideal $ I $ of $ R $ is an $ R $-module with the usual addition and multiplication.\n\\end{example}\n\n\\begin{example}\nIf $ f : R \\to S $, then $ f $ makes $ S $ into an $ R $-module, where the addition $ + $ is the usual addition in $ S $, and the multiplication law is defined by $ r \\cdot s = f\\rb{r} \\cdot_S s $ for all $ r \\in S $, $ s \\in S $. In particular any quotient $ R / I $ is an $ R $-module. More generally, if $ f : R \\to S $ is a homomorphism, and $ M $ is any $ S $-module, then $ M $ is also an $ R $-module via $ r \\cdot m = f\\rb{r} \\cdot m $. In particular, $ R \\to R / I $ lets us treat any $ R / I $-module $ M $ as an $ R $-module. Note that if $ M $ is an $ R / I $-module, then for all $ r \\in I $, $ m \\in M $, $ r \\cdot m = 0 $. We say that $ I $ \\textbf{annihilates} $ M $ in this situation. Conversely, if $ M $ is an $ R $-module and $ r \\cdot m = 0 $ for all $ r \\in I $, $ m \\in M $, then $ M $ naturally has the structure of an $ R / I $-module. Given $ r + I \\in R / I $, $ m \\in M $, we define $ \\rb{r + I} \\cdot m = rm $. If $ r + I = r' + I $, then $ r - r' \\in I $, so $ rm - r'm = \\rb{r - r'}m = 0 $ by assumption.\n\\end{example}\n\n\\begin{example}\nLet $ R = \\Z $, and let $ M $ be an abelian group. Then $ M $ has the unique natural structure of $ \\Z $-module, as follows. Property $ 3 $ from the module axioms shows that\n$$ n \\cdot m = \\begin{cases} m + \\dots + m & n > 0 \\\\ 0 & n = 0 \\\\ \\rb{-m} + \\dots + \\rb{-m} & n < 0 \\end{cases}. $$\nThus the multiplication law $ \\Z \\times M \\to M $ is forced on us, and one checks that it does satisfy properties $ 2 $ to $ 5 $ above. Informally, we say that abelian groups are $ \\Z $-modules.\n\\end{example}\n\n\\begin{example}\nIf $ R $ is a field, then $ R $-modules are just $ R $-vector spaces.\n\\end{example}\n\n\\begin{example}\nLet $ S $ be a set, and let $ M_S $ be the set of $ R $-valued functions $ f : S \\to R $. We add and multiply pointwise. For $ f, g \\in M_S $, we can define $ f + g $ as the function that takes $ s \\in S $ to $ f\\rb{s} + g\\rb{s} $, and $ rf $ as the function that takes $ s $ to $ r \\cdot f\\rb{s} $. $ M_S $ is clearly an $ R $-module. Also of interest is the $ R $-submodule $ F_S $ of $ M_S $ that consists of functions $ f : S \\to R $ such that $ f\\rb{s} = 0_R $ for all but finitely many $ s $. The $ R $-module $ F_S $ is called the \\textbf{free $ R $-module} on the set $ S $ and will be very important for us.\n\\end{example}\n\n\\subsection{Submodules, quotients, and direct sums}\n\n\\begin{definition}\nLet $ M $ be an $ R $-module. A subset $ N $ of $ M $ is an \\textbf{$ R $-submodule} of $ M $ if $ N $ is closed under addition and multiplication by elements of $ R $. That is, $ N $ is an additive subgroup of $ M $, and for all $ n \\in N $, $ r \\in R $, we have $ rN \\subseteq N $. In particular, the ideals of $ R $ are just the $ R $-submodules of $ R $.\n\\end{definition}\n\n\\begin{definition}\nIf $ S $ is any subset of $ M $, we define the \\textbf{$ R $-submodule of $ M $ generated by $ S $} to be the set of all elements of $ M $ of the form $ r_1s_1 + \\dots + r_ns_n $, where the $ r_i $ are elements of $ R $ and the $ s_i $ are elements of $ S $. It is the smallest $ R $-submodule of $ M $ containing $ S $.\n\\end{definition}\n\n\\begin{definition}\nAn $ R $-module $ M $ is a \\textbf{finitely generated} $ R $-module if $ M $ admits a finite subset $ S $ of $ M $ such that the $ R $-submodule of $ M $ generated by $ S $ is all of $ M $. We say $ S $ is a \\textbf{generating set} for $ M $.\n\\end{definition}\n\n\\begin{definition}\nLet $ M $ be an $ R $-module and $ N $ be an $ R $-submodule of $ M $. We say two elements $ m, m' $ of $ M $ are \\textbf{congruent modulo $ N $} if their difference $ m - m' $ lies in $ N $. This is easily seen to be an equivalence relation, and the equivalence classes are the cosets of the form $ m + N $, for $ m \\in M $. The set of equivalence classes is denoted $ M / N $. It has the natural structure of an $ R $-module, where $ \\rb{m + N} + \\rb{m' + N} = \\rb{m + m'} + N $ and $ r \\cdot \\rb{m + N} = rm + N $. This $ R $-module is called the \\textbf{quotient of $ M $ by $ N $}. If $ m + N = m' + N $, then $ m - m' \\in N $, so $ rm - rm' = r\\rb{m - m'} \\in N $. So well-defined. Have a natural map $ M \\to M / N $ taking $ m $ to $ m + N $.\n\\end{definition}\n\n\\begin{definition}\nGiven two $ R $-modules $ M_1 $ and $ M_2 $, the \\textbf{direct sum} $ M_1 \\oplus M_2 $ is the set of ordered pairs $ \\rb{m_1, m_2} $ with $ \\rb{m_1, m_2} + \\rb{m_1', m_2'} = \\rb{m_1 + m_1', m_2 + m_2'} $ and $ r\\rb{m_1, m_2} = \\rb{rm_1, rm_2} $ for $ m_1, m_1' \\in M_1 $, $ m_2, m_2' \\in M_2 $, and $ r \\in R $.\n\\end{definition}\n\n\\begin{example}\nLet $ M $ be an $ R $-module and $ I $ an ideal of $ R $. Then we can form the $ R $-submodule $ IM $ of $ M $ consisting of all elements of $ M $ of the form $ i_1m_1 + \\dots + i_rm_r $ where the $ i_j $ are in $ I $ and the $ m_j $ are in $ M $. This is an $ R $-submodule of $ M $, so we can form the quotient $ M / IM $. Then $ M / IM $ is certainly an $ R $-module, but it is also an $ R / I $-module. One can define multiplication $ R / I \\times M / IM \\to M / IM $ by $ \\rb{r + I}\\rb{m + IM} = rm + IM $. As always one has to check that this is well-defined, but this is straightforward. We need that if $ r - r' $ lies in $ I $, and $ m - m' $ lies in $ IM $, then $ rm - r'm' $ lies in $ IM $. But $ rm - r'm' = \\rb{r - r'}m + r'\\rb{m - m'} $ which is clearly in $ IM $.\n\\end{example}\n\n\\subsection{Module homomorphisms, kernels, and images}\n\n\\begin{definition}\nA map $ f : M \\to N $ of $ R $-modules is called a \\textbf{homomorphism of $ R $-modules} if\n\\begin{enumerate}\n\\item $ f $ is a homomorphism of the underlying abelian groups, and\n\\item $ f\\rb{rm} = rf\\rb{m} $ for all $ r \\in R $ and $ m \\in M $.\n\\end{enumerate}\n\\end{definition}\n\nWarning that a ring homomorphism $ R \\to R $ satisfies $ f\\rb{rr'} = f\\rb{r}f\\rb{r'} $, but an $ R $-module homomorphism $ R \\to R $ satisfies $ f\\rb{rr'} = rf\\rb{r'} $.\n\n\\begin{definition}\nThe kernel of $ f : M \\to N $ is the set $ \\cb{m \\in M \\mid f\\rb{m} = 0} $, an $ R $-submodule of $ M $. The image of $ f : M \\to N $ is the set $ \\cb{n \\in N \\mid \\exists m \\in M, \\ f\\rb{m} = n} $, an $ R $-submodule of $ N $.\n\\end{definition}\n\nIt is easy to see that the kernel and image of a homomorphism of $ R $-modules $ f : M \\to N $ are $ R $-submodules of $ M $ and $ N $, respectively.\n\n\\begin{note}\nIn particular there is a natural homomorphism from $ M $ to $ M / N $, taking $ m $ to $ m + N $. This homomorphism has the following universal property, exactly analogous to the universal property of the quotient construction for rings.\n\\end{note}\n\n\\begin{proposition}[Universal property of the quotient]\nLet $ N $ be an $ R $-submodule of $ M $, and let $ f : M \\to M' $ be an $ R $-module homomorphism whose kernel contains $ N $. Then there is unique homomorphism $ \\bar{f} : M / N \\to M' $ such that $ f\\rb{m + N} = f\\rb{m} $ for all $ m \\in M $. In particular the kernel of $ \\bar{f} $ is the image of $ Ker\\rb{f} $ in $ M / N $.\n\\end{proposition}\n\n\\begin{proof}\nThe proof is identical to that for quotient rings, and will be omitted.\n\\end{proof}\n\n\\subsection{Free modules}\n\n\\begin{definition}\nLet $ M $ be an $ R $-module. A subset $ S $ of $ M $ is a \\textbf{basis} for $ M $ if the following two conditions hold.\n\\begin{enumerate}\n\\item $ S $ \\textbf{spans $ M $ over $ R $}. For all $ m \\in M $, there exist $ s_1, \\dots, s_n \\in S $ finite and $ r_1, \\dots, r_n \\in R $ such that $ m = r_1s_1 + \\dots + r_ns_n $, that is the $ R $-submodule of $ M $ generated by $ S $ is all of $ M $.\n\\item $ S $ is \\textbf{$ R $-linearly independent}. For any collection $ s_1, \\dots, s_n $ of distinct elements of $ S $, and any $ r_1, \\dots, r_n \\in R $, $ r_1s_1 + \\dots + r_ns_n = 0 $ is nonzero in $ M $ unless all $ r_i $ are zero.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\nAn $ R $-module $ M $ that has a basis $ S $ is called a \\textbf{free} $ R $-module. The cardinality $ n $ of the basis $ S $ is called the \\textbf{rank} of the free $ R $-module $ M $ over $ R $.\n\\end{definition}\n\n\\marginpar{Lecture 13 \\\\ Friday \\\\ 02/11/18}\n\n\\begin{remark}\nIf $ R $ is a field, then the notion of a basis for an $ R $-module coincides with the usual notion for vector spaces. In this case, at least if one assumes the axiom of choice, every $ R $-module has a basis. When $ R $ is not a field only very special modules have bases. For instance any quotient $ R / I $ of $ R $, for $ I $ a nonzero ideal, has no basis.\n\\end{remark}\n\n\\begin{example}\nThe ring $ R $ is a free module of rank one over $ R $, with basis $ \\cb{1_R} $. More generally any unit $ u \\in R^* $ gives a basis of $ R $ as an $ R $-module.\n\\end{example}\n\nRecall that the free module $ F_S $ on a set $ S $ was defined to be the set of functions $ f : S \\to R $ such that $ f\\rb{s} = 0 $ for all but finitely many $ s \\in S $. For each $ s \\in S $, we have an element $ e_s $ of $ F_S $ defined by $ e_s\\rb{t} = 0 $ for all $ t \\in S $ with $ t \\ne s $, $ e_s\\rb{s} = 1 $. Claim that the $ e_s $ form a basis for $ F_S $. In particular, given $ f : R \\to S $ with $ f\\rb{s} = 0 $ for all but finitely many $ s $, let $ s_1, \\dots, s_n $ be the set of elements in $ S $ on which $ f\\rb{s_i} $ is nonzero. Set $ r_i = f\\rb{s_i} $. Claim that $ f = r_1e_{s_1} + \\dots + r_ne_{s_n} $. If $ f\\rb{s} = 0 $, then $ s \\notin \\cb{s_1, \\dots, s_n} $ so $ e_{s_i}\\rb{s} = 0 $ for all $ i $. For any $ i $, $ e_{s_i}\\rb{s_j} = 0 $ if $ i \\ne j $ and $ e_{s_i}\\rb{s_i} = 1 $, so $ \\rb{\\sum_{i = 1}^n r_ie_{s_i}}\\rb{s_j} = r_j = f\\rb{s_j} $. Then $ f $ can be written as $ r_1e_{s_1} + \\dots + r_ne_{s_n} $, so the $ e_s $ span $ F_S $. On the other hand, for all $ s_1, \\dots, s_n \\in S $ distinct with $ \\sum_{i = 1}^n r_ie_{s_i} = 0 $, $ \\sum_{i = 1}^n r_ie_{s_i} $ takes the value $ r_i $ by evaluating at $ s_i $ for all $ i $, and thus is only the zero function when all $ r_i $ are zero for all $ i $, so we do have $ R $-linear independence. Thus $ F_S $ is free, justifying its name.\n\n\\begin{proposition}\nLet $ F_1, F_2 $ be free modules with basis $ S_1, S_2 $. Then $ F_1 \\oplus F_2 $ is free with basis\n$$ \\cb{\\rb{s, 0} \\mid s \\in S_1} \\cup \\cb{\\rb{0, s'} \\mid s' \\in S_2}. $$ Moreover, if $ F_1 $ and $ F_2 $ are free of finite ranks $ n_1 $ and $ n_2 $ respectively, then $ F_1 \\oplus F_2 $ is free of rank $ n_1 + n_2 $.\n\\end{proposition}\n\n\\begin{proof}\nFor linear independence, let $ s_1, \\dots, s_m \\in S_1 $ and $ s_1', \\dots, s_l' \\in S_2 $ be distinct. Suppose we have $ r_1, \\dots, r_m, r_1', \\dots, r_l' \\in R $ such that $ r_1\\rb{s_1, 0} + \\dots + r_m\\rb{s_m, 0} + r_1'\\rb{0, s_1'} + \\dots + r_l'\\rb{0, s_l'} = 0 $. Then $ r_1s_1 + \\dots + r_ms_m = 0 $ in $ M_1 $ and $ r_1's_1' + \\dots + r_l's_l' = 0 $ in $ M_2 $ gives all $ r_i, r_i' $ are zero. For spanning set, let $ \\rb{m, m'} \\in M_1 \\oplus M_2 $. Write $ m = r_1s_1 + \\dots + r_ms_m $ for $ s_i \\in S_1 $ and $ m' = r_1's_1' + \\dots + r_l's_l' $ for $ s_i' \\in S_2 $, then $ \\rb{m, m'} = r_1\\rb{s_1, 0} + \\dots + r_m\\rb{s_m, 0} + r_1'\\rb{0, s_1'} + \\dots + r_l'\\rb{0, s_l'} $. Thus $ S_1 \\cup S_2 $ is a basis for $ F_1 \\oplus F_2 $, which immediately proves the claim.\n\\end{proof}\n\nFree modules have the following universal property.\n\n\\begin{proposition}[Universal property of free modules]\nLet $ F_S $ be a free $ R $-module on a set $ S $. Then for any $ R $-module $ M $, and any map of sets $ f : S \\to M $, there is a unique homomorphism of $ R $-modules $ \\phi_f : F_S \\to M $ such that $ \\phi_f\\rb{e_s} = f\\rb{s} $ for all $ s \\in S $.\n\\end{proposition}\n\n\\begin{proof}\nDefine $ \\phi_f $ by $ \\phi_f\\rb{g} = \\sum_{s \\in S, \\ g\\rb{s} \\ne 0} g\\rb{s}f\\rb{s} $. Note that this is a finite sum since all but finitely many $ s $ have $ g\\rb{s} = 0 $. Then it is clear that this is a homomorphism of $ R $-modules. On the other hand suppose $ \\phi $ is any other map $ F_S \\to N $ with $ \\phi\\rb{e_s} = f\\rb{s} $ for all $ s $. Then we can write $ g = \\sum_{s \\in S, \\ g\\rb{s} \\ne 0} g\\rb{s}e_s $, again a finite sum, so\n$$ \\phi\\rb{g} = \\sum_{s \\in S, \\ g\\rb{s} \\ne 0} g\\rb{s}\\phi\\rb{e_s} = \\sum_{s \\in S, \\ g\\rb{s} \\ne 0} g\\rb{s}f\\rb{s}, $$\nso uniqueness is clear.\n\\end{proof}\n\nThe image of $ \\phi_f $ is the submodule of $ N $ generated by the elements $ f\\rb{s} $, for $ s \\in S $.\n\n\\begin{corollary}\nLet $ M $ be a free $ R $-module with a basis $ T $ for $ M $. Let $ S $ be any set of the same cardinality as $ T $, and let $ g : T \\to S $ be any bijection. Then the map $ \\phi_f : F_S \\to M $ is an isomorphism. In particular, any two free $ R $-modules of the same rank are isomorphic.\n\\end{corollary}\n\n\\begin{proof}\nThe map $ \\phi_f : F_S \\to M $ is such that $ \\phi_f\\rb{e_s} = f\\rb{s} $. Since elements of $ T $ are linearly independent, this map is injective. Suppose $ \\phi_f\\rb{g} = 0 $. Can write $ g = \\sum r_ie_{s_i} $ for $ s_i $ distinct, then $ \\phi_f\\rb{g} = \\sum r_if\\rb{s_i} $. Since $ s_i $ are distinct, $ f\\rb{s_i} $ are distinct elements of $ T $, so $ \\sum r_if\\rb{s_i} = 0 $ gives all $ r_i $ are zero, so $ g = 0 $. Since elements of $ T $ span $ M $, this map is surjective. Given $ m \\in M $, write $ m = \\sum r_it_i $. For all $ i $, find $ s_i $, with $ f\\rb{s_i} = t_i $. Then $ \\phi_f\\rb{\\sum r_ie_{s_i}} = \\sum r_it_i = m $. Thus $ M $ is isomorphic to $ F_S $. Since $ M $ was arbitrary, any module of rank equal to the cardinality of $ S $ is isomorphic to $ F_S $ and the result follows.\n\\end{proof}\n\n\\begin{note}\nIt is also true, but harder to prove, that if $ M, N $ are free of different ranks, then $ M \\ncong N $.\n\\end{note}\n\n\\subsection{Generators and relations}\n\nNow let $ M $ be any $ R $-module, and let $ S = \\cb{m_1, \\dots, m_t} $ be a finite subset of $ M $ generating $ M $. Then we have a natural map $ F_S \\to M $ taking $ e_i $ to $ m_i $ for all $ m_i \\in S $, and this map is surjective. In particular, let $ K $ be the kernel of this map, then $ M \\cong F_S / K $. Elements of the kernel $ K $ are called \\textbf{relations} among $ S $.\n\n\\marginpar{Lecture 14 \\\\ Monday \\\\ 05/11/18}\n\nExplicitly, an element of $ K $ is a map $ f : S \\to R $ such that $ f\\rb{s} = 0 $ for all but finitely many $ s $, and $ \\sum_{s \\in S} f\\rb{s}s = 0 $. In other words, each element of $ K $ encodes a linear relation among the elements of $ S $. It is a measure of how far the elements of $ S $ are from being linearly independent. Let $ T = \\cb{k_1, \\dots, k_s} $ be a subset of $ K $ that generates $ K $. Then in the same way as above, we get a surjection $ F_T \\twoheadrightarrow K $ taking $ e_i $ to $ k_i $, with $ F_T $ a free module of rank $ s $. Composing with the inclusion of $ K $ in $ F_S $ gives us a map $ \\phi : F_T \\to F_S $ whose image is $ K $. The map $ \\phi $ determines $ M $ up to isomorphism with the quotient $ F_S / K $, and hence with $ F_S / \\phi\\rb{F_T} $. A description of a module as a quotient of a free module by the image of a map of free modules is called a \\textbf{presentation} of $ M $. If both modules have finite rank the presentation is called \\textbf{finite}. A module that has a finite presentation is called \\textbf{finitely presented}. Put another way, a presentation is a description of a module $ M $ in terms of\n\\begin{enumerate}\n\\item a generating set $ S $ for $ M $, and\n\\item a generating set $ T $ for the linear relations satisfied by $ S $.\n\\end{enumerate}\nWhen $ S $ and $ T $ are finite we can encode a presentation in a matrix, called the \\textbf{presentation matrix}. Write $ S = \\cb{e_1, \\dots, e_t} $ and $ T = \\cb{f_1, \\dots, f_s} $. Then $ \\phi $ is determined by $ \\phi\\rb{f_1}, \\dots, \\phi\\rb{f_s} $. For each $ i $ we can write $ \\phi\\rb{f_i} $ as a sum $ \\sum_{j = 1}^t r_{ij}e_{s_j} $, and let $ A $ be the $ s $ by $ t $ matrix whose $ i, j $ entry is $ r_{ij} $. Then $ A $ gives a map from $ R^t $ to $ R^s $, and the quotient of $ R^s $ by the submodule $ AR^t $ of $ R^s $ is isomorphic to $ M $.\n\n\\begin{example}\nLet $ R = \\Z $ and $ M = \\Z / n\\Z $ generated by $ \\sb{1}_n $. Map $ \\Z \\to M $ is the quotient map with kernel $ \\ab{n} $. So presentation matrix is just $ \\rb{n} $.\n\\end{example}\n\n\\begin{example}\nLet $ R = \\Z\\sb{\\sqrt{-5}} $ and $ I = \\ab{2, 1 + \\sqrt{-5}} $. $ R^2 \\twoheadrightarrow I $ by $ e_1 \\mapsto 2 $ and $ e_2 \\mapsto 1 + \\sqrt{-5} $. $ 2e_2 - \\rb{1 + \\sqrt{-5}}e_1 \\mapsto 0 $ in $ I $, and since $ \\rb{2}\\rb{3} = \\rb{1 + \\sqrt{-5}}\\rb{1 - \\sqrt{-5}} $, $ 3e_1 - \\rb{1 - \\sqrt{-5}}e_2 \\mapsto 0 $ in $ I $. Claim that the two relations $ \\rb{1 + \\sqrt{-5}}e_1 - 2e_2 $ and $ 3e_1 - \\rb{1 - \\sqrt{-5}}e_2 $ generate $ K $. Let $ ae_1 + be_2 $ be a relation, so $ a, b \\in R $ and $ 2a + \\rb{1 - \\sqrt{-5}}b = 0 $, that is $ a = \\rb{\\rb{1 + \\sqrt{-5}} / 2}b $. Question is for which $ b $ does $ \\rb{\\rb{1 + \\sqrt{-5}} / 2}b $ lie in $ R $. Claim that the set of such $ b $ is an ideal $ J $ of $ R $. $ 2 \\in J $ and $ 1 - \\sqrt{-5} \\in J $ so $ J $ contains $ \\ab{2, 1 - \\sqrt{-5}} $. $ 1 \\notin J $, since $ \\ab{2, 1 - \\sqrt{-5}} $ is maximal, $ J = \\ab{2, 1 - \\sqrt{-5}} $. So we have a map $ R^2 \\to R^2 $ with matrix\n$$ \\two{1 + \\sqrt{-5} & 3}{-2 & -1 + \\sqrt{-5}} $$\npresenting $ I $.\n\\end{example}\n\nGeneral idea is if we have a presentation matrix $ A : R^t \\to R^s $ for $ M $, with $ s $ rows and $ t $ columns, then $ BAC $ is also a presentation matrix for $ M $, where $ B $ is $ s \\times s $ and $ C $ is $ t \\times t $, and $ B $ and $ C $ are invertible matrices with inverse matrix entries in $ R $.\n\n\\section{Noetherian rings and modules}\n\n\\subsection{Definitions and basic properties}\n\n\\begin{definition}\nLet $ R $ be a ring and let $ M $ be an $ R $-module. We say $ M $ is \\textbf{Noetherian} if every increasing infinite chain\n$$ M_1 \\subseteq M_2 \\subseteq \\dots $$\nof $ R $-submodules $ M_i $ of $ M $ is \\textbf{eventually constant}. That is, for any such chain, there exists $ N $ such that we have $ M_i = M_N $ for all $ i \\ge N $. A ring $ R $ is Noetherian if $ R $ is Noetherian as an $ R $-module over itself. Since the $ R $-submodules of $ R $ are just the ideals of $ R $, a ring $ R $ is Noetherian if every increasing infinite chain\n$$ I_1 \\subseteq I_2 \\subseteq \\dots $$\nof ideals $ I_j $ of $ R $ is eventually constant.\n\\end{definition}\n\nThe following result about Noetherian $ R $-modules is fundamental.\n\n\\begin{proposition}\nAn $ R $-module $ M $ is Noetherian if and only if every $ R $-submodule $ N $ of $ M $ is finitely generated.\n\\end{proposition}\n\n\\begin{proof}\nSuppose first that $ M $ is Noetherian, and let $ N $ be an $ R $-submodule of $ M $. Choose an element $ n_0 $ of $ N $, and let $ N_0 $ be the $ R $-submodule of $ N $ generated by $ n_0 $. If $ N_0 $ is all of $ N $, then $ N $ is finitely generated. Otherwise, choose $ n_1 $ in $ N \\setminus N_0 $, and let $ N_1 $ be the $ R $-submodule of $ N $ generated by $ n_0 $ and $ n_1 $. If $ N $ is not finitely generated, we may continue this process indefinitely, choosing for each $ i $ an $ n_i $ in $ N \\setminus N_{i - 1} $, which is nonempty since $ N $ is not finitely generated, and letting $ N_i $ be generated by $ n_0, \\dots, n_i $. In this way we obtain a strictly increasing infinite chain\n$$ N_0 \\subsetneq N_1 \\subsetneq \\dots $$\nof submodules of $ M $, contradicting the fact that $ M $ is Noetherian. Conversely, suppose that every $ R $-submodule of $ M $ is finitely generated, and let\n$$ M_0 \\subseteq M_1 \\subseteq \\dots $$\nbe an increasing chain. We must show that this chain is eventually constant. Let $ N $ be the union of the submodules $ M_i $. Note that $ N $ is an $ R $-submodule of $ M $. Thus $ N $ is finitely generated, say by $ n_1, \\dots, n_s $. If $ n_1, n_2 \\in N $, then there exist $ i, j $ with $ n_1 \\in M_i $, $ n_2 \\in M_j $. If $ d \\ge i, j $, $ n_1, n_2 \\in M_d $, so $ n_1 + n_2 \\in M_d $ gives $ n_1 + n_2 \\in N $. Since $ N $ is the union of the $ M_j $, there exist $ i_1, \\dots, i_s $ such that $ n_j $ is in $ M_{i_j} $ for all $ j $. Let $ d $ be the largest of the $ i_j $. Then $ M_d $ contains $ n_1, \\dots, n_s $ so it contains $ N $. In particular for any $ d' \\ge d $ we have $ N \\subseteq M_d \\subseteq M_{d'} \\subseteq N $, so $ N = M_d = M_{d'} $ for all such $ d' $ and the chain is constant after $ M_d $.\n\\end{proof}\n\n\\marginpar{Lecture 15 \\\\ Wednesday \\\\ 07/11/18}\n\n\\begin{corollary}\nLet $ R $ be a PID. Then $ R $ is Noetherian.\n\\end{corollary}\n\n\\begin{proof}\nEvery ideal of $ R $ is principal, hence finitely generated.\n\\end{proof}\n\n\\begin{example}\nAny field is Noetherian.\n\\end{example}\n\n\\begin{example}\nThe ring $ \\C\\sb{X^{\\Q_{\\ge 0}}} $ is not Noetherian. The ideal consisting of all elements with no constant term is not finitely generated.\n\\end{example}\n\n\\subsection{Finitely generated modules over Noetherian rings}\n\nPlan is\n\\begin{enumerate}\n\\item show that Noetherianness has strong consequences, and\n\\item use these properties to show $ R $ is Noetherian gives $ R\\sb{X} $ is Noetherian and other consequences.\n\\end{enumerate}\nThe goal of this section is to prove the following theorem.\n\n\\begin{theorem}\n\\label{thm:8.2.1}\nAny finitely generated $ R $-module $ M $ over a Noetherian ring $ R $ is Noetherian.\n\\end{theorem}\n\nWe proceed in several steps.\n\n\\begin{proposition}\nLet $ M $ be a Noetherian $ R $-module. Then for any submodule $ N $ of $ M $,\n\\begin{enumerate}\n\\item $ N $ is Noetherian, and\n\\item $ M / N $ is Noetherian.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{proof}\n\\hfill\n\\begin{enumerate}\n\\item Since $ M $ is Noetherian, any submodule of $ M $ is finitely generated, and thus any submodule of $ N $ is finitely generated.\n\\item Given a submodule $ N' $ of $ M / N $, let $ \\widetilde{N'} $ be its preimage in $ N $ under the canonical quotient map $ f : M \\to M / N $. We have a surjection from $ \\widetilde{N'} $ to $ N' $ induced by $ f $. $ \\widetilde{N'} \\subseteq M $, so $ \\widetilde{N'} $ is finitely generated, say by $ n_1, \\dots, n_s $. Claim that $ f\\rb{n_1}, \\dots, f\\rb{n_s} $ generate $ N' $. Given $ n \\in N' $, there exists $ \\widetilde{n} \\in \\widetilde{N'} $ such that $ f\\rb{\\widetilde{n}} = n $. Write $ \\widetilde{n} = r_1n_1 + \\dots + r_sn_s $ for $ r_i \\in R $. Then $ n = f\\rb{\\widetilde{n}} = r_1f\\rb{n_1} + \\dots + r_sf\\rb{n_s} $.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{proposition}\nLet $ M $ be an $ R $-module, let $ N $ be a Noetherian submodule of $ M $, and suppose that $ M / N $ is Noetherian. Then $ M $ is Noetherian.\n\\end{proposition}\n\n\\begin{proof}\nLet $ M' $ be a submodule of $ M $. Then $ M' \\cap N $ is a submodule of $ M $, hence finitely generated. Let $ a_1, \\dots, a_s \\in M' \\cap N $ generate $ M' \\cap N $. Let $ \\bar{M'} $ denote the image of $ M' $ in $ M / N $. This is a submodule of $ M / N $ and thus finitely generated. Let $ \\bar{b_1}, \\dots, \\bar{b_t} \\in \\bar{M'} \\subseteq M / N $ generate $ \\bar{M'} $, and choose elements $ b_1, \\dots, b_t $ of $ M' $ mapping to $ \\bar{b_1}, \\dots, \\bar{b_t} $ in $ M / N $, respectively. We now show that $ a_1, \\dots, a_s, b_1, \\dots, b_t $ is a generating set for $ M' $, proving the claim. Given any $ m \\in M' $, let $ \\bar{m} $ be its image in $ M / N $ under $ f : M' \\to \\bar{M'} $. Then we can write $ \\bar{m} $ as a sum $ r_1\\bar{b_1} + \\dots + r_t\\bar{b_t} $ for some $ r_1, \\dots, r_t \\in R $. Let $ m' = m - r_1b_1 - \\dots - r_tb_t $. Then the image of $ m' $ in $ M / N $ is $ f\\rb{m'} = \\bar{m} - r_1\\bar{b_1} - \\dots - r_t\\bar{b_t} = 0 $. So $ m' $ lies in $ N $. $ m \\in M', r_1b_1 \\in M', \\dots, r_tb_t \\in M' $ so $ m' $ also lies in $ M' $. So it lies in $ M' \\cap N $. We can thus write $ m' $ as $ q_1a_1 + \\dots + q_sa_s $ for some $ q_1, \\dots, q_s \\in R $. We then have\n$$ m = q_1a_1 + \\dots + q_sa_s + r_1b_1 + \\dots + r_tb_t, $$\nproving the claim.\n\\end{proof}\n\n\\begin{corollary}\nLet $ M $ and $ N $ are Noetherian $ R $-modules, then so is $ M \\oplus N $.\n\\end{corollary}\n\n\\begin{proof}\nWe have a surjection $ M \\oplus N \\to M $ taking $ \\rb{m, n} $ to $ m $. Its kernel $ K $ is the set of pairs of $ M \\oplus N $ of the form $ \\rb{0, n} $, which is isomorphic to $ N $ by the map $ N \\to K $ taking $ n $ to $ \\rb{0, n} $, and hence Noetherian. The surjection $ M \\oplus N \\to M $ descends to an isomorphism $ \\rb{M \\oplus N} / K \\cong M $, so that $ \\rb{M \\oplus N} / K $ is Noetherian. Thus $ M \\oplus N $ is Noetherian.\n\\end{proof}\n\nNow assume $ R $ is Noetherian. Then $ R, R \\oplus R, \\dots $ are all Noetherian $ R $-modules, that is the following.\n\n\\begin{corollary}\nIf $ R $ is Noetherian, then any free $ R $-module of finite rank is Noetherian.\n\\end{corollary}\n\n\\begin{proof}\nA free $ R $-module of rank $ s $ is the direct sum of $ s $ copies of $ R $, each of which is Noetherian as an $ R $-module when $ R $ is Noetherian.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:8.2.1}]\nLet $ M $ be a finitely generated $ R $-module, and let $ m_1, \\dots, m_s $ be a set of generators for $ M $. Then if $ R^s $ is a free $ R $-module of rank $ s $, with generators $ e_1, \\dots, e_s $, we have a surjection of $ R^s $ onto $ M $ taking $ e_i $ to $ m_i $ for all $ i $. Let $ K $ be the kernel. Then $ M $ is isomorphic to $ R^s / K $, and $ R^s $ is a Noetherian $ R $-module, so $ M $ is Noetherian as well.\n\\end{proof}\n\n\\section{Polynomial rings in several variables}\n\n\\subsection{The Hilbert basis theorem}\n\nIn this section, we will use the ideas of the previous section to establish the following key result about polynomial rings, known as the Hilbert basis theorem.\n\n\\begin{theorem}[Hilbert basis theorem]\n\\label{thm:9.1.1}\nLet $ R $ be a Noetherian ring. Then $ R\\sb{X} $ is Noetherian.\n\\end{theorem}\n\n\\marginpar{Lecture 16 \\\\ Friday \\\\ 09/11/18}\n\nOver a field, if $ Q\\rb{X} = a_nX^n + \\dots $ and $ P\\rb{X} = b_mX^m + \\dots $ for $ m \\ge n $ and $ a_n, b_m \\ne 0 $, then\n$$ \\deg\\rb{P\\rb{X} - \\dfrac{b_m}{a_n}X^{m - n}Q\\rb{X}} < \\deg\\rb{P\\rb{X}}. $$\nOver a ring, this only goes so far. Over $ \\Z $, cannot use a multiple of $ 3X + 4 $ to reduce the degree of $ a_nX^n + \\dots $ unless $ 3 \\mid a_n $. Let $ P\\rb{X} = b_0 + \\dots + b_nX_n $, with $ b_n \\in R $ nonzero. We say that $ b_n $ is the \\textbf{leading coefficient} of $ P\\rb{X} $. In general, if I have $ Q_1\\rb{X}, \\dots, Q_r\\rb{X} $ with degrees $ d_1, \\dots, d_r $ and leading coefficients $ a_1, \\dots, a_r $ and $ P\\rb{X} $ of degree $ d \\ge d_1, \\dots, d_r $ then there exist $ n_1, \\dots, n_r \\in R $ such that\n$$ \\deg\\rb{P\\rb{X} - n_1X^{d - d_1}Q_1\\rb{X} - \\dots - n_rX^{d - d_r}Q_r\\rb{X}} < d, $$\nif and only if leading coefficient of $ P\\rb{X} $ is in the ideal generated by $ a_1, \\dots, a_r $. \n\n\\begin{lemma}\nLet $ R $ be Noetherian and $ I \\subseteq R\\sb{X} $ be an ideal. Let $ J \\subseteq R $ be the set of leading coefficients of polynomials in $ I $. That is, the set of $ a \\in R $ such that there exists a polynomial $ P\\rb{X} $ in $ I $ with leading coefficient $ a $. Then $ J $ is an ideal of $ R $.\n\\end{lemma}\n\n\\begin{proof}\nCertainly if $ a \\in J $ is the leading coefficient of $ P\\rb{X} \\in I $ such that $ P\\rb{X} = aX^n + \\dots $, then for any $ r \\in R $, $ ra $ is the leading coefficient of $ rP\\rb{X} = raX^n + \\dots $ so $ ra \\in J $, so $ J $ is closed under multiplication. On the other hand, if $ a, b \\in J $ are the leading coefficients of $ P\\rb{X} $ and $ Q\\rb{X} $ in $ I $, then let $ n, m $ be the degrees of $ P\\rb{X} = aX^n + \\dots $ and $ Q\\rb{X} = bX^m + \\dots $ respectively. Without loss of generality we may assume $ n \\ge m $. Then $ a + b $ is the leading coefficient of $ P\\rb{X} + X^{n - m}Q\\rb{X} = \\rb{a + b}X^{n + m} + \\dots $, and the latter polynomial is in $ I $ so $ a + b \\in J $. Thus $ J $ is closed under addition, and is therefore an ideal.\n\\end{proof}\n\nNow since $ R $ is Noetherian, $ J $ is finitely generated, say by $ a_1, \\dots, a_s \\in R $. By definition of $ J $, there are thus polynomials $ P_1, \\dots, P_s $ in $ I $, of degrees $ d_1, \\dots, d_n $, such that $ P_i = a_iX^{d_i} + \\dots $ has leading coefficient $ a_i $ for all $ i $. Let $ N $ be the largest of the $ d_i $.\n\n\\begin{lemma}\nGiven $ Q\\rb{X} \\in I $ of degree $ d \\ge N $. Then there exist $ R_1\\rb{X}, \\dots, R_s\\rb{X} \\in R\\sb{X} $ such that $ Q\\rb{X} - R_1\\rb{X}P_1\\rb{X} - \\dots - R_s\\rb{X}P_s\\rb{X} $ has degree less than $ N $.\n\\end{lemma}\n\n\\begin{proof}\nThe proof is by induction on $ d $ and the base case $ d < N $ is clear by setting $ R_i = 0 $ for all $ i $. Suppose the claim is true for polynomials of degree less than or equal to $ d - 1 $, with $ d \\ge N $. Let $ a \\in J $ be the leading coefficient of $ Q\\rb{X} = aX^d + \\dots $, so that $ Q\\rb{X} - aX^d $ has degree at most $ d - 1 $. Since $ a $ lies in $ J $ we can write $ a = r_1a_1 + \\dots + r_sa_s $. Then the leading term of the polynomial\n$$ r_1X^{d - d_1}P_1\\rb{X} + \\dots + r_sX^{d - d_s}P_s\\rb{X} $$\nis $ aX^d $, so the difference\n$$ Q\\rb{X} - r_1X^{d - d_1}P_1\\rb{X} - \\dots - r_sX^{d - d_s}P_s\\rb{X} $$\nhas degree at most $ d - 1 $ and lies in $ I $. By the inductive hypothesis this difference is an $ R\\sb{X} $-linear combination of the $ P_i\\rb{X} $,\n$$ R_1\\rb{X}P_1\\rb{X} + \\dots + R_s\\rb{X}P_s\\rb{X}. $$\nSo\n$$ Q\\rb{X} = \\rb{R_1\\rb{X} + r_1X^{d - d_1}}P_1\\rb{X} + \\dots + \\rb{R_s\\rb{X} + r_sX^{d - d_s}}P_s\\rb{X}. $$\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm:9.1.1}]\nThe following proof is due to Emmy Noether, and is a vast simplification of Hilbert's original proof. Let $ I $ be an ideal of $ R\\sb{X} $. We want to show that $ I $ is finitely generated. Let $ I_{\\le N} = I \\cap R\\sb{X}_{\\le N} $ be the subset of $ I $ consisting of all polynomials of degree at most $ N $. Then $ I_{\\le N} $ is an $ R $-submodule of the $ R $-module $ R\\sb{X}_{\\le N} $ of all polynomials of degree at most $ N $. The latter is free of rank $ N + 1 $ and generated by $ 1, \\dots, X^N $ as an $ R $-module, so it is finitely generated, hence Noetherian. In particular since $ R $ is Noetherian $ I_{\\le N} $ is also a finitely generated $ R $-module. Let $ T_1\\rb{X}, \\dots, T_k\\rb{X} $ generate $ I_{\\le N} $ as an $ R $-module. We will show that\n$$ P_1\\rb{X}, \\dots, P_s\\rb{X}, T_1\\rb{X}, \\dots, T_k\\rb{X} $$\ngenerate $ I $ as an $ R\\sb{X} $-module. More precisely, we will show that $ Q\\rb{X} $ is an $ R\\sb{X} $-linear combination of the $ P_i\\rb{X} $ and $ T_j\\rb{X} $. Given $ Q\\rb{X} \\in I $, there exist $ R_1\\rb{X}, \\dots, R_s\\rb{X} \\in R\\sb{X} $ such that\n$$ Q\\rb{X} = R_1\\rb{X}P_1\\rb{X} + \\dots + R_s\\rb{X}P_s\\rb{X} + T\\rb{X}, $$\nwith $ T\\rb{X} \\in I_{\\le N} $. There exist $ r_1, \\dots, r_k \\in R $ such that\n$$ T\\rb{X} = r_1T_1\\rb{X} + \\dots + r_kT_k\\rb{X}, $$\nso\n$$ Q\\rb{X} = R_1\\rb{X}P_1\\rb{X} + \\dots + R_s\\rb{X}P_s\\rb{X} + r_1T_1\\rb{X} + \\dots + r_kT_k\\rb{X}. $$\n\\end{proof}\n\nAs a corollary, we deduce the following.\n\n\\begin{corollary}\nLet $ R $ be any field or PID, or indeed any Noetherian ring. Then for any $ n $, the ring $ R\\sb{X_1, \\dots, X_n} $ is Noetherian.\n\\end{corollary}\n\nAn observation is that if $ R $ is Noetherian and $ I \\subseteq R $ is an ideal, then $ R / I $ is Noetherian. Let $ J $ be an ideal of $ R / I $ and $ \\widetilde{J} $ be preimage of $ J $ in $ R $. There exist $ \\widetilde{j_1}, \\dots, \\widetilde{j_n} $ generating $ \\widetilde{J} $ over $ R $. Let $ j_i = \\widetilde{j_i} + I \\in R / I $. These lie in $ J $ and generate $ J $ over $ R / I $. In particular, any quotient of polynomial ring over a field or PID is Noetherian. Indeed, since any quotient of a Noetherian ring is Noetherian, we can say more.\n\n\\begin{definition}\nLet $ R $ be a ring. An $ R $-algebra is a ring $ S $ together with a homomorphism $ f : R \\to S $. If $ S $ is an $ R $-algebra, we say that $ S $ is finitely generated as an $ R $-algebra over $ R $ if there exists a finite set of elements $ s_1, \\dots, s_n \\in S $ such that every element of $ S $ can be expressed as a polynomial in the $ s_i $ with coefficients in $ R $. Equivalently, $ S $ is generated over $ R $ by $ s_1, \\dots, s_n $ if the homomorphism $ R\\sb{X_1, \\dots, X_n} \\to S $ by $ f : R \\mapsto S $ and sending $ X_i $ to $ s_i $ is surjective.\n\\end{definition}\n\n\\begin{note}\nAny finitely generated $ R $-algebra $ S $ is isomorphic to a quotient $ R\\sb{X_1, \\dots, X_n} / I $ for some $ n $ and some ideal $ I $. Thus we can rephrase the Hilbert basis theorem as saying that if $ R $ is Noetherian, then any finitely generated $ R $-algebra is Noetherian.\n\\end{note}\n\n\\marginpar{Lecture 17 \\\\ Monday \\\\ 12/11/18}\n\nLecture 17 is a problem class.\n\n\\marginpar{Lecture 18 \\\\ Wednesday \\\\ 14/11/18}\n\n\\subsection{Polynomial rings over UFDs are UFDs}\n\nOur next goal is to study factorisation in polynomial rings of the form $ R\\sb{X} $. $ \\Z\\sb{X} $ is not a PID nor a UFD. Idea is relate factorisations in $ \\Z\\sb{X} $ to factorisations in $ \\Q\\sb{X} $. Warning that irreducibles in $ \\Q\\sb{X} $ does not give irreducibles in $ \\Z\\sb{X} $.\n\n\\begin{example}\n$ 3x + 15 $ irreducible in $ \\Q\\sb{X} $. In $ \\Z\\sb{X} $ $ 3x + 15 = 3\\rb{x + 15} $.\n\\end{example}\n\nCertainly if $ R $ is not a UFD then we cannot expect to have unique factorisation in $ R\\sb{X} $, since we do not even have it in $ R $. Assume $ R $ is a UFD. Then the ring $ R\\sb{X} $ might be quite complicated, but $ R\\sb{X} $ is contained in a much simpler ring where we do understand factorisation, the ring $ K\\sb{X} $, where $ K $ is the field of fractions of $ R $. Our goal will thus be to compare factorisations in $ K\\sb{X} $ with factorisations in $ R\\sb{X} $. Fundamental question is can we turn factorisations in $ K\\sb{X} $ of $ P\\rb{X} \\in R\\sb{X} $ into factorisations in $ R\\sb{X} $? The key to doing this is the following result, often called Gauss' lemma.\n\n\\begin{theorem}[Gauss' lemma]\n\\label{thm:9.2.1}\nLet $ R $ be a UFD and let $ K $ be its field of fractions. Let $ P\\rb{X} \\in R\\sb{X} $, and let $ Q\\rb{X} $ be a polynomial in $ K\\sb{X} $ that divides $ P\\rb{X} $ in $ K\\sb{X} $. Then there is an element $ \\alpha \\in K^* $ such that $ \\alpha Q\\sb{X} $ lies in $ R\\sb{X} $, and divides $ P\\rb{X} $ in $ R\\sb{X} $. In particular, if $ P\\rb{X} $ is reducible in $ K\\sb{X} $, then $ P\\rb{X} $ is also reducible in $ R\\sb{X} $.\n\\end{theorem}\n\n\\begin{proof}\nWrite $ P\\rb{X} = Q\\rb{X}T\\rb{X} \\in K\\sb{X} $, and choose nonzero elements $ e_1, e_2 \\in R $ such that $ e_1Q\\rb{X} $ and $ e_2T\\rb{X} $ have coefficients in $ R $, and so that the greatest common divisor of the coefficients of $ Q\\rb{X} $ is one, as is the greatest common divisor of the coefficients of $ T\\rb{X} $. Letting $ d = e_1e_2 $, we have $ dP\\rb{X} = Q'\\rb{X}T'\\rb{X} $ with $ Q'\\rb{X} = e_1Q\\rb{X} $ and $ T'\\rb{X} = e_2T\\rb{X} $. Suppose $ d $ is nota unit in $ R $. Then $ d $ is divisible by an irreducible element $ q $ of $ R $. Since $ R $ is a UFD, irreducibles are prime, so the ideal of $ R $ generated by $ q $ is a prime ideal. Thus $ R / \\ab{q} $ is an integral domain, so $ R / \\ab{q}\\sb{X} $ is as well. Moreover, if $ \\bar{Q'}\\rb{X} $ and $ \\bar{T'}\\rb{X} $ are the images of $ Q'\\rb{X} $ and $ T'\\rb{X} $ modulo $ \\ab{q} $ in $ R / \\ab{q}\\sb{X} $, $ dP\\rb{X} = Q'\\rb{X}T'\\rb{X} $ becomes $ 0 = \\bar{Q'}\\rb{X}\\bar{T'}\\rb{X} $ in $ R / \\ab{q}\\sb{X} $. Since $ R / \\ab{q}\\sb{X} $ is an integral domain we must have either $ \\bar{Q'}\\rb{X} = 0 $ or $ \\bar{T'}\\rb{X} = 0 $ in $ R / \\ab{q}\\sb{X} $. Without loss of generality assume $ \\bar{Q'}\\rb{X} = 0 $. Then all the coefficients of $ Q'\\rb{X} $ are divisible by $ q $. Thus $ d_1P\\rb{X} = Q_1\\rb{X}T_1\\rb{X} $ for $ Q_1\\rb{X}, T_1\\rb{X} \\in R\\sb{X} $ and $ Q_1\\rb{X} $ is a multiple of $ Q\\rb{X} $ in $ K\\sb{X} $. If $ d_1 $ is a unit, done. Otherwise write $ d_1 = d_2q_1 $ for $ q_1 $ irreducible. Same trick gives $ d_2P\\rb{X} = Q_2\\rb{X}T_2\\rb{X} $ for $ Q_2\\rb{X}, T_2\\rb{X} \\in R\\sb{X} $ and $ Q_2\\rb{X} $ is a multiple of $ Q_1\\rb{X} $ in $ K\\sb{X} $. Repeat, contradicting our construction of $ Q'\\rb{X} $. Thus $ \\alpha = d $ is a unit in $ P\\rb{X} $, and we have $ P\\rb{X} = e_1Q\\rb{X}d^{-1}e_2T\\rb{X} $, with $ e_1Q\\rb{X} $ and $ d^{-1}e_2T\\rb{X} $ in $ R\\sb{X} $.\n\\end{proof}\n\n\\begin{note}\nThe converse to the last claim of Theorem \\ref{thm:9.2.1} is not true. if $ P\\rb{X} $ is reducible in $ R\\sb{X} $, it might be irreducible in $ K\\sb{X} $.\n\\begin{example}\nThe polynomial $ 7x $ factors into irreducibles as $ 7 \\cdot x $ in $ \\Z\\sb{X} $, but since $ 7 $ is a unit in $ \\Q\\sb{X} $, $ 7x $ is irreducible in $ \\Q\\sb{X} $.\n\\end{example}\nThe following lemma shows that this kind of thing is all that can happen, however.\n\\end{note}\n\n\\begin{proposition}\nLet $ P\\rb{X} $ in $ R\\sb{X} $ be a polynomial and suppose that the greatest common divisor of all of its coefficients is one. Then $ P\\rb{X} $ is irreducible in $ K\\sb{X} $ if and only if it is also irreducible in $ R\\sb{X} $.\n\\end{proposition}\n\n\\begin{proof}\nSuppose $ P\\rb{X} $ is irreducible in $ R\\sb{X} $, and write $ P\\rb{X} = Q\\rb{X}T\\rb{X} $, with $ Q\\rb{X} $ and $ T\\rb{X} $ nonunits in $ R\\sb{X} $. If $ Q\\rb{X} $ or $ T\\rb{X} $ were constant with degree zero then it would divide every coefficient of $ P\\rb{X} $ and thus divide the GCD of those coefficients, making it a unit. Thus $ Q\\rb{X} $ and $ T\\rb{X} $ are nonconstant with positive degree and the factorisation $ P\\rb{X} = Q\\rb{X}T\\rb{X} $ is also a nontrivial factorisation in $ K\\sb{X} $, so $ P\\rb{X} $ is reducible in $ K\\sb{X} $. Conversely suppose $ P $ is reducible in $ K\\sb{X} $. Then there exist $ Q\\rb{X} \\in K\\sb{X} $ with $ 0 < \\deg\\rb{Q} < \\deg\\rb{P} $ such that $ Q\\rb{X} \\mid P\\rb{X} $ in $ K\\sb{X} $. Gauss' lemma shows that there exist $ \\alpha \\in K^* $ such that $ \\alpha Q\\rb{X} \\in R\\sb{X} $ and $ \\alpha Q\\rb{X} \\mid P\\rb{X} $ in $ R\\sb{X} $.\n\\end{proof}\n\nWe are now in a position to prove the following.\n\n\\begin{theorem}\nIf $ R $ is a UFD, then $ R\\sb{X} $ is a UFD.\n\\end{theorem}\n\n\\begin{proof}\nFor existence of factorisations, let $ P\\rb{X} $ be an element of $ R\\sb{X} $. We must show that $ P\\rb{X} $ factors into irreducibles. Let $ d $ be the greatest common divisor of the coefficients of $ P\\rb{X} $, and write $ P\\rb{X} = dQ\\rb{X} $ where the greatest common divisor of the coefficients of $ Q\\rb{X} $ is one. Since $ R $ is a UFD, $ d $ factors into irreducibles $ q_1, \\dots, q_s $ in $ R $, and these remain irreducible in $ R\\sb{X} $, so it suffices to show that $ Q\\rb{X} $ factors into irreducibles. Factor $ Q\\rb{X} $ into irreducibles in $ K\\sb{X} $, $ Q\\rb{X} = Q_1\\rb{X}, \\dots Q_r\\rb{X} $. By Gauss' lemma, there exist scalars $ \\alpha_1, \\dots, \\alpha_r \\in K^* $ such that $ \\alpha_1 \\dots \\alpha_r = 1 $ and $ \\alpha_iQ_i\\rb{X} \\in R\\sb{X} $. Let $ Q'_i \\rb{X} = \\alpha_iQ_i\\rb{X} $. GCD of coefficients of $ Q'_1\\rb{X}, \\dots, Q'_r\\rb{X} $ is one gives $ Q'_1\\rb{X}, \\dots, Q'_r\\rb{X} $ are irreducible in $ R\\sb{X} $ since they are irreducible in $ K\\sb{X} $. For uniqueness of factorisations, it remains to show that if $ P\\rb{X} \\in R\\sb{X} $ is irreducible in $ R\\sb{X} $ and divides $ A\\rb{X}B\\rb{X} $ in $ R\\sb{X} $ for $ A\\rb{X}, B\\rb{X} \\in R\\sb{X} $, then $ P\\rb{X} $ divides either $ A\\rb{X} $ or $ B\\rb{X} $ in $ R\\sb{X} $.\n\\begin{enumerate}\n\\item If $ P\\rb{X} $ is constant, then $ P\\rb{X} = c $ is irreducible in $ R $. In $ R / \\ab{c}\\sb{X} $ a domain, $ 0 = \\bar{A}\\rb{X}\\bar{B}\\rb{X} $ so $ \\bar{A}\\rb{X} = 0 $ or $ \\bar{B}\\rb{X} = 0 $ gives $ c \\mid A\\rb{X} $ or $ c \\mid B\\rb{X} $.\n\\item If $ P\\rb{X} $ is nonconstant, since $ P\\rb{X} $ is irreducible in $ R\\sb{X} $ it is irreducible in $ K\\sb{X} $ by Gauss' lemma, and hence divides either $ A\\rb{X} $ or $ B\\rb{X} $ in $ K\\sb{X} $. Suppose $ P\\rb{X} $ divides $ A\\rb{X} $ in $ K\\sb{X} $. Then $ A\\rb{X} = P\\rb{X}Q\\rb{X} $ in $ K\\sb{X} $. Then there is an element $ \\alpha = r / s \\in K^* $ for $ r, s \\in R $ and $ r \\ne 0 $ such that $ \\alpha P\\rb{X} $ lies in $ R\\sb{X} $ and divides $ A\\rb{X} $ in $ R\\sb{X} $, and $ A\\rb{X} = \\alpha P\\rb{X}\\alpha^{-1} Q\\rb{X} $ in $ R\\sb{X} $ by Gauss' lemma. On the other hand, since $ P\\rb{X} $ is irreducible in $ R\\sb{X} $ the GCD of its coefficients is one, so the only way $ \\alpha P\\rb{X} $ lies in $ R\\sb{X} $ is if $ s $ is a unit and $ \\alpha $ lies in $ R $. Thus $ \\alpha^{-1}Q\\rb{X} \\in R\\sb{X} $, $ \\alpha \\in R $, and $ P\\rb{X} \\in R\\sb{X} $, so $ P\\rb{X} $ also divides $ A\\rb{X} $.\n\\end{enumerate}\n\\end{proof}\n\n\\begin{corollary}\nIf $ K $ is a UFD, a field, or a PID, then $ K\\sb{X_1, \\dots, X_n} $ is a UFD for any $ n $.\n\\end{corollary}\n\nWarning that quotients of UFDs are only rarely UFDs themselves.\n\n\\begin{example}\n$ \\Z\\sb{X} $ is a UFD. $ \\Z\\sb{X} / \\ab{X^2 + 5} = \\Z\\sb{\\sqrt{-5}} $ is not a UFD.\n\\end{example}\n\n\\end{document}", "meta": {"hexsha": "206ed1daf86a6dca7960fe13eda4ca3c5d1da194", "size": 120852, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "M3P8 Algebra III/M3P8.tex", "max_stars_repo_name": "kckennylau/JMC3", "max_stars_repo_head_hexsha": "1ef0265a73457e0d379c71f9de2218f217e89a8a", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "M3P8 Algebra III/M3P8.tex", "max_issues_repo_name": "kckennylau/JMC3", "max_issues_repo_head_hexsha": "1ef0265a73457e0d379c71f9de2218f217e89a8a", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "M3P8 Algebra III/M3P8.tex", "max_forks_repo_name": "kckennylau/JMC3", "max_forks_repo_head_hexsha": "1ef0265a73457e0d379c71f9de2218f217e89a8a", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 83.7505197505, "max_line_length": 1855, "alphanum_fraction": 0.6261708536, "num_tokens": 43462, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165382362518, "lm_q2_score": 0.6477982315512489, "lm_q1q2_score": 0.34159472093717047}} {"text": "\\documentclass[dvisvgm,hypertex,aspectratio=169]{beamer}\n\\usefonttheme{serif}\n\n%\\usepackage[utf8]{inputenc}\n%\\usepackage[T1]{fontenc}\n\n%\\usepackage[draft]{animate}\n\\usepackage[final]{animate}\n\\usepackage{ifthen}\n\n\n%\\usepackage{pythontex} % <--\n\\usepackage{graphicx}\n\n\n\\usepackage{tikz}\n\\usepackage{pgfplots}\n\\usepackage{pgfplotstable}\n\\pgfplotsset{compat=1.16}\n\\usetikzlibrary{calc}\n\\usetikzlibrary{decorations.pathmorphing,patterns}\n\\usepackage{amsmath}\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \n% Define footer\n\\usepackage{ccicons}\n\n\\makeatletter\n\\setbeamertemplate{footline}\n{\n \\leavevmode%\n \\hbox{%\n %\\begin{beamercolorbox}[wd=.333333\\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%\n %\\usebeamerfont{title in head/foot}\\insertsubsection\n %\\end{beamercolorbox}%\n %\\begin{beamercolorbox}[wd=.333333\\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%\n % \\usebeamerfont{date in head/foot}\\insertshortdate{}\\hspace*{2em}\n % \\insertframenumber{} / \\inserttotalframenumber\\hspace*{2ex} \n %\\end{beamercolorbox}}%\n %\\vskip0pt%\n \\begin{beamercolorbox}[wd=.92\\paperwidth,ht=2.25ex,dp=1ex,right]{author in head/foot}%\n \\usebeamerfont{author in head/foot}\\insertauthor\n \\end{beamercolorbox}%\n \\begin{beamercolorbox}[wd=.08\\paperwidth,ht=2.25ex,dp=1ex,right]{date in head/foot}%\n \\ccbysa\n \\end{beamercolorbox}}%\n \\vskip0pt%\n}\n\\makeatother\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n\\author{\\href{mailto:kjartan@tec.mx}{kjartan@tec.mx}}\n\n\\begin{document}\n\n\\section{Create animation}\n\n \\def\\tend{30}\n \\def\\nframes{24}\n \\def\\nsamples{200}\n \\def\\wexpstart{-2}\n \\def\\wexpend{1}\n \\def\\ttau{1}\n \n\\begin{frame}[label=A]{First-order Bode plot}\n \\begin{center}\n \\begin{tikzpicture}[scale=0.7, transform shape,]\n \\tiny\n \\node[draw, minimum width=14mm, minimum height=12mm] (sys) {$G(s)=\\frac{1}{s\\tau + 1}$};\n \\node[coordinate, left of=sys, node distance=46mm] (input) {};\n \\node[coordinate, right of=sys, node distance=46mm] (output) {};\n \\draw[->] (input) -- node[ above] { $\\textcolor{blue!80!green}{u(t)=\\sin(\\omega_1 t)}$} (sys);\n \\draw[->] (sys) -- node[near end, above] { $\\textcolor{green!80!black}{y(t)=|G(i\\omega_1)|\\sin\\big(\\omega_1 t + \\arg G(i\\omega_1)\\big)}$} (output);\n \\end{tikzpicture}\n \\end{center}\n \\begin{center}\n \\begin{animateinline}[controls, loop, palindrome]{4}\n \\multiframe{\\nframes}{n=0+1}{\n \n \\pgfmathsetmacro{\\wexp}{\\wexpstart + (\\wexpend-\\wexpstart)/\\nframes*\\n}\n \\pgfmathsetmacro{\\ww}{pow(10, \\wexp)}\n \\pgfmathsetmacro{\\wstart}{pow(10, \\wexpstart)}\n \\pgfmathsetmacro{\\wend}{pow(10, \\wexpend)}\n \\pgfmathsetmacro{\\gain}{1/sqrt(pow(\\ttau*\\ww,2) + 1)}\n \\pgfmathsetmacro{\\phshift}{-atan2(\\ttau*\\ww,1)}\n\n \\begin{tikzpicture}[scale=0.5, transform shape]\n\n \\begin{loglogaxis} [\n width=7cm,\n height=5cm,\n ylabel=$|G|$,\n %xticklabels=\\empty,\n xtick={0.01, 0.1, 1, 10},\n xticklabels={$\\frac{0.01}{\\tau}$, $\\frac{0.1}{\\tau}$, $\\frac{1}{\\tau}$, $\\frac{10}{\\tau}$}, \n grid=both,\n minor y tick num=9,\n % extra y ticks={.5}, % how to convert to fixed point tick label ?\n % extra y tick style={log identify minor tick positions=true},\n every major grid/.style={red, opacity=0.5},\n ymin=0.01, ymax=2,\n xmin = 0.01, xmax=10,\n ]\n \\addplot+[thick, orange!80!black, no marks, domain=\\wstart:\\wend, samples=100]\n {1/sqrt(pow((\\ttau*\\x),2) + 1)};\n \\draw[black!90, ] (axis cs: \\ww, 0.01) -- (axis cs: \\ww, 10);\n \\node[magenta!80!black, circle, draw, inner sep=2pt, thick,] at (axis cs: \\ww, \\gain) {};\n \\end{loglogaxis}\n \\begin{semilogxaxis} [\n xlabel=$\\omega$,\n ylabel=$\\arg G$,\n xshift = 8cm, \n width=7cm,\n height=5cm,\n grid=both,\n ytick={0, -90},\n ymin=-90, ymax = 0,\n xtick={0.01, 0.1, 1, 10},\n xticklabels={$\\frac{0.01}{\\tau}$, $\\frac{0.1}{\\tau}$, $\\frac{1}{\\tau}$, $\\frac{10}{\\tau}$},\n xmin = 0.01, xmax = 10,\n minor y tick num=2,\n every major grid/.style={red, opacity=0.5},\n %legend entries={Bessel filter, Delay of one},\n %legend pos={south west},\n ]\n \\addplot+[thick, orange!80!black, no marks, domain=\\wstart:\\wend, samples=100]\n {-atan2(\\x*\\ttau,1)};\n \\draw[black!90,] (axis cs: \\ww, -90) -- (axis cs: \\ww, 10);\n \\node[magenta!80!black, circle, draw, inner sep=2pt, thick,] at (axis cs: \\ww, \\phshift) {};\n \\end{semilogxaxis}\n \\begin{axis} [\n width = 10cm,\n height= 5cm,\n yshift = -5cm,\n xshift = 5cm,\n xlabel = {$t$},\n ymin=-1.2,\n ymax = 1.2,\n ]\n \\addplot+[thick, blue!80!green, no marks, domain=0:\\tend, samples=\\nsamples, smooth] {sin(\\ww*180/3.14*x)};\n \\addplot+[thick, green!80!black, no marks, domain=0:\\tend, samples=\\nsamples, smooth] {\\gain*sin(\\ww*180/3.14*x + \\phshift)};\n \n \\end{axis}\n \\begin{scope}[yshift=-4cm, xshift=-4.5cm]\n \\pgfmathsetmacro{\\www}{min(\\ww, 8)};\n \\clip (-2, -1) rectangle (2, 10);\n \\draw[->] (-2, 0) -- (0.3, 0) node[below] {Re};\n \\draw[->] (0, -1) -- (0, 8) node[left] {Im};\n \\node[red] at (-1/\\ttau, 0) {\\large $\\times$};\n \\node[magenta!80!black, circle, draw, inner sep=2pt, thick,] at (0, \\www) {};\n \\draw[<->, black!60] (-1/\\ttau, 0) to (0, \\www);\n \\node at (0.9, \\www) {$\\frac{\\www}{\\tau}$};\n \\node[] at (-1/\\ttau, -0.5){$-\\frac{1}{\\tau}$};\n \\node at (-1, 6) {$s$-plane}; \n \\end{scope}\n \\begin{scope}[yshift=-3cm, xshift=0cm, ]\n \\pgfmathsetmacro{\\xx}{2*\\gain*cos(\\phshift)};\n \\pgfmathsetmacro{\\yy}{2*\\gain*sin(\\phshift)};\n \\draw[->] (-2, 0) -- (3, 0) node[below] {Re};\n \\draw[->] (0, -2) -- (0, 1) node[left] {Im};\n \\node[magenta!80!black, circle, draw, inner sep=2pt, thick,] at (\\xx, \\yy) {};\n \\node at (1.5, 1) {$G(i\\omega)$-plane};\n \\draw (2,0) -- (2, 0.2) node[above] {1};\n \\draw[orange!80!black, domain=0:100, smooth, samples at={0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.4, 0.64, 0.8, 1.0, 1.28, 2, 2.56, 3, 4, 5.12, 6, 8, 10.24, 20}, variable=\\t] plot ({2*(1/sqrt(1 + pow(\\ttau*\\t, 2)))*cos(-atan2(\\ttau*\\t, 1))}, {2*(1/sqrt(1 + pow(\\ttau*\\t, 2)))*sin(-atan2(\\ttau*\\t, 1))});\n \\end{scope}\n \\end{tikzpicture}\n }\n \\end{animateinline}\n \\end{center}\n\\end{frame}\n\n\\end{document}\n\n\n", "meta": {"hexsha": "f7b04d4bbe1f8c6b4b4abba8e4de6ebed9d1c82e", "size": 6986, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "classic-control/slides/first-order-bode/first-order-bode.tex", "max_stars_repo_name": "kjartan-at-tec/mr2025", "max_stars_repo_head_hexsha": "88c28aa76e84890c25d252167e5bbcd25318463e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "classic-control/slides/first-order-bode/first-order-bode.tex", "max_issues_repo_name": "kjartan-at-tec/mr2025", "max_issues_repo_head_hexsha": "88c28aa76e84890c25d252167e5bbcd25318463e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "classic-control/slides/first-order-bode/first-order-bode.tex", "max_forks_repo_name": "kjartan-at-tec/mr2025", "max_forks_repo_head_hexsha": "88c28aa76e84890c25d252167e5bbcd25318463e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 39.4689265537, "max_line_length": 316, "alphanum_fraction": 0.5302032637, "num_tokens": 2465, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5273165233795672, "lm_q2_score": 0.6477982179521103, "lm_q1q2_score": 0.341594704141986}} {"text": "\n\n\nThis chapter details the proposed framework and implementation. I start by giving an overview of the proposed framework, and then five crucial components in the framework are carefully explained: Deep Denoising Autoencoders on audio-visual modalities, Fisher Vector Encoding, Document Embedding on textual modality, Tree-based Feature Selection and Multi-Task Learning. \n\n\n\n\n\n\\section{Overview}\n\nFigure \\ref{fig:pipeline} illustrates the framework of my proposed method for automatic Bipolar Disorder BD recognition. For acoustic modality, I extract 39-dimensional Mel-Frequency Cepstrum Coefficients (MFCCs)\\footnotemark as the Low-Level Descriptors (LLDs) with OpenSMILE and for visual modality, I extract the 132-dimensional facial landmarks, 6-dimensional head pose, 6-dimensional eye gaze, and 35-dimensional Facial Action Units (FAUs) as the LLDs with OpenFace. To discover the correlation across audio-visual modalities and produce robust representations, I propose a Deep Denoising Autoencoder (DDAE) that learns a shared and joint representation on different number of modalities. More specifically, according to the number of modalities on which DDAEs are built, I define uni-DDAE, bi-DDAE, and multi-DDAE. Before feeding multimodal features into DDAE, features must be aligned on frame-level first to ensure they are extracted from the same time interval. I then compute the dynamic changes of the latent representations in DDAE as each representation is regarded as a specific movement of the subject. After computing the velocity ($1^{st}$-order derivative) and the acceleration ($2^{nd}$-order derivative) of the latent representations, the three features are concatenated as frame-level descriptors. Because the video clips vary in length, I encode the frame-level descriptors with a Fisher Vector (FV), a fixed-length descriptor on session-level, by fitting them into a Gaussian Mixture Model (GMM). To reduce redundancy and select the most discriminative feature set, a tree-based model is used for feature selection by evaluating the feature importance.\n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=13.5cm]{images/design/general_pipeline.png}\n \\caption{Pipeline of the proposed framework that includes different learning architectures on different modalities}\n \\label{fig:pipeline}\n\\end{figure}\n\nOn the other hand, I obtain the transcripts of video interviews and apply a document embedding model (doc2vec) to learn fixed-length representations of the transcripts on session-level. To boost the performance of the doc2vec model, the model is pre-trained on an additional Turkish corpus. \n\nWith both fixed-length FVs on audio-visual modalities and fixed-length document embeddings on textual modality, a Multi-Task Deep Neural Network (MT-DNN) is built to handle the overfitting due to the limited size of the BD corpus. The MT-DNN is learned with a weighted loss from the ternary classification task of Mania Level and the regression task of Young Mania Rating Scale (YMRS). Furthermore, to address the imbalance between three classes, I duplicate the training instances of the minority class and apply Unweighted Average Recall (UAR) as my metric for evaluation. Finally, with the best-performing MT-DNN, the final decision on each video interview is inferred: depression, hypo-mania, or mania.\n\n\n\\footnotetext{The 39-dimensional MFCC features are obtained by appending additional 13 delta and 13 acceleration coefficients to the conventional 13-dimensional MFCC features.}\n\n\n\n\n\n\n\n\n\n\\section{Deep Denoising Autoencoders}\n\\label{sec:DDAE}\n\nAutoencoders (AEs) are an unsupervised learning algorithm that learns a latent-space representation of given data with hidden layers constrained while reconstructing the data from the representation. The representations learnt from AEs generally have a lower dimensionality and have been proven effective as high-level features in the following classification tasks, which in many cases are competitive or even superior to the hand-engineered representations \\cite{ng2011}. Autoencoders are composed of two parts: a) encoder, which compresses the input into a latent representation with the function $h=f(x)$, and b) decoder, which reconstructs the input from the latent representation with the function $r=g(h)$, as shown in Figure \\ref{fig:unimodal_ae}. Therefore, autoencoders can be described by the function $g(f(x))=r$, and the reconstruction error $\\parallel x-r \\parallel ^ 2$ is minimized in the training processing. \n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[height=6.5cm]{images/design/autoencoder.png}\n \\caption{Schematic for Unimodal Deep Denoising Autoencoders}\n \\label{fig:unimodal_ae}\n\\end{figure}\n\n\nAutoencoders were first introduced as an approach to initialize the weights of neural networks \\cite{ballard1987}, and because of the \\textit{curse of dimensionality} \\cite{bellman1966}, the low-dimensional hidden layers enable AEs to be commonly employed in feature engineering and representation learning \\cite{charte2018}. Several variations of AEs have been developed with different constraints on the hidden layers, such as sparse AEs \\cite{ng2011}, denoising AEs \\cite{vincent2008}, and variational AEs \\cite{kingma2013}. \n\nDenoising AEs, proposed by \\cite{vincent2008}, stochastically corrupts part of the original input $x$ and reconstructs the original input $x$ with the corrupted version $\\tilde{x}$. The noise can be of different forms, such as additive isotropic Gaussian noise and masking noise \\cite{vincent2010}. The Gaussian noise $\\tilde{x}$ is obtained with $\\tilde{x}|x \\sim \\mathcal{x, \\sigma^2,I}$ and the masking noise is calculated by forcing a fraction $v$ of the elements of input $x$ to 0. Although the Gaussian noise is reported to be the natural choice for real-valued inputs \\cite{vincent2010}, I only consider the masking noise in the experiment as the Gaussian noise could introduce a small change in facial expressions, which could be interpreted as a misleading descriptor for the BD symptoms. The objective of denoising AEs accordingly becomes minimizing the reconstruction error $\\parallel \\tilde{x}-r \\parallel ^ 2$. The denoising AEs are reported to produce representations that are robust to small irrelevant changes in the input.\n\nWith the development of multimedia data processing, a more recent learning architecture is introduced for feature fusion: the multi-modal AEs \\cite{mangai2010}. Feature fusion aims to learn a shared representation cross modalities without redundant or irrelevant information, which is one of the biggest challenges in multi-modal data processing \\cite{mangai2010}. In the work of \\cite{hong2015}, authors aimed to perform multimodal fusion of 2D images and 3D human poses to obtain high-level representations. \nInspired by \\cite{vincent2008}, \\cite{mangai2010}, and \\cite{dibekliouglu2017}, I propose a series of 3-layer Deep Denoising Autoencoders (DDAEs) (i.e., DDAEs with 3 hidden layers) with different number of modalities to investigate the multimodal fusion in the BD detection task. \n\n\\subsection{Unimodal Deep Denoising Autoencoders}\n\nI first define the unimodal DDAEs on acoustic and visual features respectively as shown in Figure \\ref{fig:unimodal_ae}, where features of one modality are fed into DDAEs to learn a compact-size representation after adding masking noise. This architecture is considered as a baseline for the following architectures as it does not discover the correlation across modalities. \n\nIn many implementations of AEs, the binary cross-entropy (BCE) is set as the loss function, which measures the amount of information is preserved in the reconstruction compared to the original input \\cite{de2005}. BCE is calculated with Equation \\ref{eq:binarycrossentropy}, in which $x_k$ represents one node in the input layer and $r_k$ represents the corresponding node in the output layer. The input data must be therefore normalized to the range $[0,1]$ with the sigmoid activation function in the output layer. BCE could apply to binary-value image pixel intensities, such as MNIST dataset, because the range of original intensities lies in $[0, 255]$, but other features, like MFCC, rarely share the same range across the dataset and normalization could wrongly corrupt the correlation in features, thus hurting the extraction of emotion-related information. In my framework, instead, I define the loss function as the mean squared error (MSE) (Equation \\ref{eq:mse}) with the linear activation function in the output layer and no normalization. The DDAEs are trained on minimizing the distance between reconstructed input and original input, and the reconstruction could be easily interpreted and visualized in the evaluation.\n\n\\begin{equation}\n J(x,r) = - \\sum_{k=1}^d x_k \\log(r_k) + (1-x_k) \\log(1-r_k)\n \\label{eq:binarycrossentropy}\n\\end{equation}\n\n\\begin{equation}\n J(x,r) = \\frac{1}{N} \\sum_{k=1}^d (x_k - r_k)^2\n \\label{eq:mse}\n\\end{equation}\n\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[height=7.5cm]{images/design/bimodal_ae.png}\n \\caption{Schematic for Bimodal Deep Denoising Autoencoders (Different colours indicate different modalities)}\n \\label{fig:bimodal_ae}\n\\end{figure}\n\n\n\\subsection{Bimodal Deep Denoising Autoencoders}\n\nI continue to define the 3-layer bimodal DDAEs on acoustic and visual features altogether as displayed in Figure \\ref{fig:bimodal_ae}. The acoustic features are Low-Level Descriptors (LLDs), either MFCC or eGeMAPS features, and the visual features in this architecture are set as facial landmarks whose dynamics have been proven useful in depression detection \\cite{dibekliouglu2017}. As suggested in the work of \\cite{ngiam2011}, two separate encoders are merged into one shared hidden layer after one hidden layer, which outputs the first-order representations. From the shared hidden layer, two modalities are then reconstructed via their decoders with the sum of MSE on both reconstructed inputs as the loss function. \nAcoustic and visual modalities need to be aligned beforehand to ensure they are within the same window \\cite{ngiam2011}, and I concatenate 3 contiguous acoustic features as each input that has approximately the same duration as 1 visual feature. Because audio-visual modalities have different data formats and ranges, they are normalized and whitened separately.\n\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[height=7.5cm]{images/design/multimodal_ae.png}\n \\caption{Schematic for Multimodal Deep Denoising Autoencoders (Different colours indicate different modalities)}\n \\label{fig:multimodal_ae}\n\\end{figure}\n\n\\subsection{Multimodal Deep Denoising Autoencoders}\n\nGiven the importance of head pose, eye gaze, and action units in emotion recognition \\cite{adams2015, ekman2013}, the 3-layer multimodal DDAEs are defined on a total of five modalities, namely facial landmarks, MFCC / eGeMAPS, head pose, eye gaze, and action units (shown in Figure \\ref{fig:multimodal_ae}). Following the same design principle as the bimodal DDAEs, five modalities are merged into the shared representation layer with their ``mid-level\" representation output by their first hidden layer, and masking noise is added to modalities individually. The reconstruction loss of each modalities is assigned with equal weights in the joint reconstruction loss function. \n\n\\subsection{Implementation Details}\n\nThe list of investigated hyperparameters for all DDAEs is given in Table \\ref{tab:param_DDAE}, in which the hidden ratio is defined as the ratio between two consecutive hidden layers, and for instance, with hidden ratio 0.5, the dimensions of all hidden layers would be $\\{0.5d, 0.25d, 0.5d\\}$ where $d$ represents the input dimensionality. In addition, I evaluate the denoising effect in DDAEs by setting different masking noise levels.\n\n\n\\begin{table}[ht]\n \\centering\n \\caption{Hyperparameter settings to investigate for DDAEs}\n \\begin{tabular}{l|l}\n \\Xhline{2\\arrayrulewidth}\n Hyperparameter & Values \\\\\n \\hline\n hidden ratio & \\{0.4, 0.5\\} \\\\\n noise level & \\{0.1, 0.2, 0.4\\} \\\\\n batch size & 1024 \\\\\n learning rate & 0.01 \\\\\n epochs & 100 \\\\\n \\Xhline{2\\arrayrulewidth}\n \\end{tabular}\n \\label{tab:param_DDAE}\n\\end{table}\n\nThe dimensions of all input modalities are listed in Table \\ref{tab:dim_modality}. Because the shared representation is trained from only one acoustic features, either MFCC features or eGeMAPS features are input into bimodal DDAEs (Table \\ref{fig:bimodal_ae}) and multimodal DDAEs (Table \\ref{fig:multimodal_ae}). The sum of the input dimensions would thus be either 300 or 252, and the shared representation has four different dimensions with different hidden ratio settings (either 0.4 or 0.5), as shown in the last two row of Table \\ref{tab:dim_modality}. The dimension of facial landmarks is obtained by $x$ and $y$ coordinates of 68 landmarks, while to acoustic features, dimensions of MFCC and eGeMAPS are multiplied with 3 because of the alignment of audio-visual modalities.\n\nThe training of DDAEs can be considered as the unsupervised feature learning, in which more data benefit the encoding of DDAEs \\cite{ngiam2011}. All DDAEs are therefore trained with all available labelled (training set and development set) and unlabelled (test set) audio-visual data. \n\n\\begin{table}[ht]\n \\centering\n \\caption{Dimension of all five modalities. The two dimensions of representations in the last two rows are based on the different hidden ratios, defined in \\ref{tab:param_DDAE}}\n \\begin{tabular}{l|l}\n \\Xhline{2\\arrayrulewidth}\n Modality & Dimension \\\\\n \\hline\n Facial Landmarks & 136 ($68 \\times 2$) \\\\\n MFCC features & 117 ($39 \\times 3$) \\\\\n eGeMAPS features & 69 ($23 \\times 3$) \\\\\n Head Pose & 6 \\\\\n Eye Gaze & 6 \\\\\n Action Units & 35 \\\\\n \\hline\n Sum of input (MFCC) & 300 \\\\\n Sum of input (eGeMAPS) & 252 \\\\\n \\hline\n Representation (MFCC) & 48 (0.4) / 75 (0.5) \\\\\n Representation (eGeMAPS) & 40 (0.4) / 63 (0.5) \\\\\n \\Xhline{2\\arrayrulewidth}\n \\end{tabular}\n \\label{tab:dim_modality}\n\\end{table}\n\n\\subsection{Computing Dynamics of Latent Representations}\n\nAfter training, each DDAE (unimodal, bimodal, or multimodal) learns a presentation for the per-frame extracted features. These representations, however, only encode the static element of the input, such as locations of facial landmarks or loudness of audio signal, but without any temporary information. Following the ideas proposed in \\cite{dibekliouglu2017}, I extend these representations with the dynamics. Considering DDAE-based representations as a matrix $H \\in \\mathcal{R}^{n \\times d}$, in which $n$ denotes the number of frames and $d$ the final dimension of representations. Each column in $H_i(i \\in \\{1,2...d\\})$ corresponds to one node in the representation layer, or for example, to one static point in unimodal DDAEs. I then compute the first-order dynamics, velocity $V$, of $H$ by the $1^{st}$ derivative $V_i = \\frac{d H_i}{dt}$, measuring the velocity of the change between per-frame representations. I continue to calculate the second-order dynamics, acceleration $A$, of $H$ by the $2^{nd}$ derivative $A_i = \\frac{d^2 H_i}{d^2 t}$, measuring the acceleration of the change. To align $H$, $V$, and $A$, I discard the first two frames in each video sessions, and I concatenate $H$, $V$, and $A$ for the final representations on frame-level.\n\n\n\n\n\n\n\\section{Fisher Vector Encoding}\n\\label{sec:fisher}\n\nFeature aggregation is an approach via which low-level descriptors (LLDs) can be summarised to produce fixed-length high-level descriptors on variable-length audio-visual recordings, which encodes more global information. Many approaches within feature aggregation exist, such as the Bag-of-Words (BoW) \\cite{csurka2004} and the Fisher Vector (FV) \\cite{perronnin2010}. In my framework, to encode the frame-level representations learnt from DDAEs into a fixed-length vector on session-level, I investigate and implement the improved Fisher Vector (FV) \\cite{perronnin2010, sanchez2013}. The BoW representations are briefly explained first as they are one of the baseline features used in Chapter \\ref{ch:evaluation}.\n\n\\subsection{Bag-of-Words Representation}\n\nBag-of-Words originates from natural language processing and as a semi-supervised representation learning, it represents the distribution of LLDs based on a dictionary or codebook learned from them \\cite{csurka2004, peng2016}. Generally, BoW is composed of five steps: a) feature extraction, b) feature-preprocessing, c) codebook generation, d) feature encoding, and e) pooling and normalization. According to the modality of representations, BoW can be categorised into Bag-of-Audio-Words (BoAW) and Bag-of-Visual-Words (BoVW). The feature extraction is completed as described in Chapter \\ref{ch:background} and these extracted LLDs are usually high dimensional and strong correlated. Principal Component Analysis (PCA), a statistical procedure, is therefore used in BoW to preprocess the LLDs to low-dimensional and de-correlated features \\cite{peng2016}. For codebook generation, two approaches are often considered, (i) partitioning the feature space into regions, each represented by its centre, such as $k$-means \\cite{bishop2006}, and (ii) using a generative model to capture the probability distribution of features, such as Gaussian Mixture Model (GMM) \\cite{bishop2006}. With parameters indicating either cluster centres or parameters for GMM, the LLDs extracted from one video are encoded into a fixed-length vector via voting-based or reconstruction-based encoding method \\cite{peng2016}. In the last step, a pooling operation is used to obtain a global per-video representation and normalization enables the representations to be invariant to the variable number of LLDs extracted from different videos \\cite{peng2016}.\n\n\\subsection{Fisher Vector}\n\nFisher Vector (FV) extends the bag-of-words (BoW) representations to learn the distribution of LLDs with their mean and variance, and it is commonly used as a global image descriptors in image classification \\cite{perronnin2010, krapac2011}. More recently, FV has become popular for a variety of applications in social signal processing, such as depression estimation \\cite{jain2014, dhall2015} and emotion recognition \\cite{kaya2015}, because it combines advantages of both the generative and discriminative approaches \\cite{sanchez2013} in machine learning. In the workflow of FV encoding, a generative model, typically Gaussian Mixture Model (GMM), is firstly built on LLDs and the Fisher kernel is then computed from this generative model. FVs are quantified using first and second order statistics of the gradient of the sample log-likelihood with respect to GMMs' parameters \\cite{sanchez2013}.\n\nFormally, let $X = \\{x_t; t=1...T\\}$ be an element in the time-series representations with the number of frames, $T$. Let $\\Theta = \\{\\mu_k, \\Sigma_k, \\pi_k; k=1...K\\}$ be the parameter of a Gaussian Mixture Model (GMM) fitting the distribution of the representations where $\\mu_i$ and $\\Sigma_i$ are respectively the mixture mean vector and covariance matrix and priors of GMM. GMM estimates a probability distribution of multiple multivariate Gaussian distributions and it is trained while maximizing the likelihood $p(X|\\Theta)$:\n\n\\begin{equation}\n \\Theta^{*} = \\arg \\max_{\\Theta} p(X|\\Theta) = \\arg \\max_{\\Theta} \\prod_{i=1}^{N} p(x_i|\\Theta)\n\\end{equation}\n\nGMM associates each vector $x_t$ to a mode $k$ in the mixture with a strength given by the posterior probability:\n\n\\begin{equation}\n q_{tk} = \\frac{\\exp [-1/2 (x_t-\\mu_k)^{T} \\Sigma_k^{-1}(x_t-\\mu_k)]}{\\sum_{i=1}^{K} \\exp [-1/2 (x_t-\\mu_i)^{T} \\Sigma_k^{-1}(x_t-\\mu_i)]}\n\\end{equation}\n\nFor each mode $k$, the mean and covariance deviation vectors are defined as \n\n\\begin{align}\n u_{jk} &= \\frac{1}{N\\sqrt{\\pi_k}} \\sum_{t=1}^T q_{tk}\\frac{x_{jt}-\\mu_{jk}}{\\sigma_{jk}} \\\\\n v_{jk} &= \\frac{1}{N\\sqrt{2\\pi_k}} \\sum_{t=1}^T q_{tk}[(\\frac{x_{jt}-\\mu_{jk}}{\\sigma_{jk}})^2 - 1]\n\\end{align}\n\nwhere $j=1,2...D$ spans the vector dimensions. The FV of the time-series representations is the stacking of the vector $u_k$ and $v_k$ for each of the K modes in the Gaussian mixtures:\n\n\\begin{equation}\n \\Phi(X) = [...u_k...v_k...]^T\n\\end{equation}\n\n\\subsection{Improved Fisher Vectors}\n\nIn addition to the traditional FV, the improved FV was introduced by \\cite{perronnin2010} for better classification performance with the following ideas :\n\\begin{itemize}\n \\item Power normalization. As the number of Gaussians increases, FV becomes sparser and the distribution of features becomes more ``peaky\" around zero. The kernel is therefore replaced with the Laplacian kernel, which is more robust on sparse vectors, by applying the function $|z| \\sign z$ to each dimension of the vector $\\Phi(X)$. \n \\item $l_2$ normalization. Before using the representations in a linear model, the vector $\\Phi(X)$ is further by the $l_2$ normalization to discard the video-independent information, or descriptors which are likely to occur in any video. FV thus focus on video-specific features.\n\\end{itemize}\n\nI compute the Fisher Vectors using GMMs with empirically 16 or 32 Gaussian distributions \\cite{syed2018, dibekliouglu2017} to estimate the distribution on DDAE-based $d$-dimensional representations. The resulting feature vectors are $96 \\times d_{latent}$ dimensional ($16 \\times 2 \\times 3 \\times d_{latent}$) or $192 \\times d_{latent}$ dimensional ($32 \\times 2 \\times 3 \\times d_{latent}$), where $d_{latent}$ denotes the dimension of latent representation learnt from DDAEs.\n\n\n\n\n\n\\section{Tree-based Feature Selection}\n\nFeature selection is a key step to reduce redundancy and improve the accuracy of classifiers by selecting the most informative features \\cite{hira2015}. The reduced dimensional features should preserve as much information as the original high-dimensional features do. Considering the high dimension of the obtained Fisher Vectors with respect to the limited size of the dataset, feature selection is deemed necessary in my framework. Many approaches have been proposed to select the discriminative features, such as Min-Redundancy Max-Relevance (mRMR) algorithm \\cite{peng2005}, Analysis of Variance (ANOVA) \\cite{moran1918}, and correlation-based feature selection (CFS) \\cite{pudil1994}, I apply tree-based feature selection in my framework by evaluating and ranking the importance of each feature. More specifically, a Random Forest (RF) classifier is used to compute the information gain $(S,A)$ for a feature $A$ relative to a dataset $S$: \n\n\\begin{equation}\n Gain(S,A) = Entropy(S) - \\sum_{v \\in values(A)} \\frac{|S_v|}{|S|} Entropy(S_v)\n\\end{equation}\n\nwhere $values(A)$ is the set of all possible values for feature $A$, and $S_v$ is the subset of $S$ for which feature $A$ has value $v$. As RF is robust to redundant features and insensitive to the irrelevant information, the resulting feature importance leads to a reliable and discriminative subset of features. The split criterion is preset as entropy, and the number of trees as 800, determined by the baseline system describe in Chapter \\ref{ch:evaluation}. I select the top 100 features in the Fisher Vector ranked with the importance, and in other words, the dimension for features in audio-visual modality is 100.\n\n\n\n\\section{Document Embedding}\n\nSince strong correlations have been found between interview contents and depression symptoms \\cite{morales2016, pampouchidou2016}, analyzing emotion-related textual modality has been emerged as a new approach in depression detection. More specifically, the use of negatively-valenced words and pronouns has been stressed in \\cite{morales2016}. To incorporate textural modality into my framework, I first transcribe the recordings from BD corpus into plain texts using Google Speech-To-Text API \\footnote{\\url{https://cloud.google.com/speech-to-text/}}, and then make use of Paragraph Vector (PV) \\cite{mikolov2014} to embed to session-level transcripts into textual features of the same size.\n\nThe PV model, or doc2vec, is an unsupervised learning algorithm to learn the distributed representations of a variable-length piece of texts \\cite{mikolov2014}, and it is usually considered as an extension of word2vec, which aims to learn the word representations \\cite{mikolov2013}. \n\n\\subsection{word2vec}\n\nThe two architectures within word2vec, Continuous Bag-Of-Words (CBOW) and Skip-Gram (SG), are variants of neural networks with one hidden layer \\cite{mikolov2013}. While CBOW architecture takes the context words (multiple word vectors) as input and predicts the target word (one word vector), SG architecture infers the context words given the target word, as shown in Figure \\ref{fig:word2vec}. \n\nFormally, in word2vec-CBOW architecture, the objective is to maximize the log probability of the target word given context words with Equation \\ref{eq:cbow}.\n\n\\begin{equation}\n \\frac{1}{T} \\sum_{t=c}^{T-c} \\log p(w_t | w_{t-c} ... w_{t+c})\n \\label{eq:cbow}\n\\end{equation}\n\nwhere $c$ indicates the size of context windows (which equals to 2 in Figure \\ref{fig:word2vec}) and $T$ represents the length of the word sequence. On the other hand, in word2vec-SG architecture, the optimization function is based on the averaged log probability as Equation \\ref{eq:sg}, where $c$ and $T$ share the same notations as Equation \\ref{eq:cbow}.\n\n\\begin{equation}\n \\frac{1}{T} \\sum_{t=i}^{T} \\sum_{-c\\leq j \\leq c} \\log p(w_{t+j}|w_t)\n \\label{eq:sg}\n\\end{equation}\n\nA large $c$ means a broader context window and thus a higher probability to capture the semantics of the words, but it could also lead to more expensive computation \\cite{mikolov2013}. Furthermore, since the high dimensionality of input vectors can easily cause difficulties in the computation of condition probability $p(w_O|w_I)$ obtained by the softmax function, shown in Equation \\ref{eq:softmax}, where $w_O$ is denoted as the ``output vector\", $w_I$ as the ``input vector\", and $W$ as the weight matrix that is updated via backpropagation with loss function $E = -\\log p(w_O|w_I)$. Mikolov \\textit{et al.} \\cite{mikolov2013efficient} proposed three optimization techniques to improve the quality of output vectors and also the training speed: hierarchical softmax, negative sampling, and subsampling \\cite{mikolov2013, mikolov2013efficient}. \n\n\\begin{equation}\n p(w_O|w_I) = \\frac{\\exp({v^{'}_{w_O}}^{T} v_{w_I})}{\\sum_{w=1}^{W} \\exp({v^{'}_w}^{T} v_{w_{I}})}\n \\label{eq:softmax}\n\\end{equation}\n\nAs an efficient approach of computer softmax, the hierarchical softmax applies a binary tree to represent $W$ words in the output layer \\cite{mikolov2013}, and the evaluation of $W$ nodes in the output weight matrix is therefore reduced to $\\log_2 W$ nodes. An alternative to the hierarchical softmax is negative sampling: instead of updating the entire weight matrix, only a limited number of ``negative words\" are selected to update the weights \\cite{mikolov2013efficient}. The ideas underlying subsampling is straightforward: the most frequent words in a large corpus generally provides less semantic information, such as ``a\", ``the\" or ``some\" \\cite{mikolov2013efficient}. Subsampling of such words can decrease the number of training samples and also help the neural networks to assign higher weights to rare words, which reflects more emotion-related information.\n\n\\begin{figure}[ht]\n \\centering\n \\begin{minipage}{0.47\\textwidth}\n \\centering\n \\includegraphics[height=4.3cm]{images/design/word2vec_cbow.png} \\\\\n (a) word2vec-CBOW\n \\end{minipage}\n \\begin{minipage}{0.48\\textwidth}\n \\centering\n \\includegraphics[height=4.3cm]{images/design/word2vec_sg.png} \\\\\n (b) word2vec-SG\n \\end{minipage}\n \\caption{Schematic for two Word Vector (word2vec) architectures}\n \\label{fig:word2vec}\n\\end{figure}\n\n\\subsection{doc2vec}\n\nTo extend the embeddings to a higher level, the Paragraph Vector (doc2vec) is proposed to learn the representation for a variable-length of texts: sentence, paragraph and document. There are also two architectures within doc2vec, Paragraph Vector with Distributed Memory (PV-DM), corresponding to word2vec-CBOW, and Paragraph Vector with Distributed Bag-Of-Words (PV-DBOW), corresponding to word2vec-SG. As displayed in Figure \\ref{fig:doc2vec}, both architectures introduce an additional document token that could be considered as the topic of each document (or in my framework, the mania level of transcript). The input in PV-DM is the concatenation of document vectors and word vectors, both of which are learned in the training process, but only the document vector is used for the inferring process. On the contrary, the PV-DBOW ignores the context words and predicts the words randomly sampled from the inferred document. It is obvious that PV-DBOW requires fewer data storage as it only saves softmax weights and PV-DM also needs to save the word vectors \\cite{mikolov2014}. \n\n\\begin{figure}[ht]\n \\centering\n \\begin{minipage}{0.55\\textwidth}\n \\centering\n \\includegraphics[height=5.4cm]{images/design/doc2vec_dm-m.png} \\\\\n (a) doc2vec-DM\n \\end{minipage}\n \\begin{minipage}{0.43\\textwidth}\n \\centering\n \\includegraphics[height=5.4cm]{images/design/doc2vec_dbow.png} \\\\\n (b) doc2vec-DBOW\n \\end{minipage}\n \\caption{Schematic for two Paragraph Vector (doc2vec) architectures}\n \\label{fig:doc2vec}\n\\end{figure}\n\n\n\\subsection{Implementation Details}\nSome researchers have reported that PV-DBOW outperforms PV-DM in the sentiment analysis, contradictory results in the work of \\cite{mikolov2014}. In addition, Yang \\textit{et al.} only considers PV-DM architectures in their depression detection framework, and therefore, in my experiment, I investigate both PV-DM and PV-DBOW in the BD detection with different hyperparameter settings, as shown in Table \\ref{tab:param_doc2vec}.\n\n\n\\begin{table}[ht]\n \\centering\n \\caption{Hyperparameter settings to investigate for document embedding}\n \\begin{tabular}{l|l}\n \\Xhline{2\\arrayrulewidth}\n Hyperparameter & Values \\\\\n \\hline\n model & \\{PV-DM, PV-DBOW\\} \\\\\n vector size & \\{25, 50, 100\\} \\\\\n window size & \\{5, 10\\} \\\\\n negative words & \\{5, 10\\} \\\\\n hierarchical softmax & \\{0, 1\\} \\\\\n \\Xhline{2\\arrayrulewidth}\n \\end{tabular}\n \\label{tab:param_doc2vec}\n\\end{table}\n\nThe translation of transcripts to English prior to the document embedding has been considered as it might help to understand the transcripts and to evaluate the embeddings. Nonetheless, I apply the doc2vec model directly on the Turkish transcripts to ensure the completeness of semantic content, which could be compromised in the translation, and after learning process, the document embeddings are evaluated with translation for better understanding. \n\nDue to the limited size of the BD corpus, I pre-train the doc2vec models on an external, large-scale Turkish corpus,``trwiki\", to improve the performance of my models \\cite{lau2016}. The ``trwiki\" corpus is based on Wikimedia\\footnote{\\url{https://dumps.wikimedia.org/trwiki/}} dump service, which contains various kinds of texts in Turkish, such as articles, templates, media/file descriptions, and primary meta-pages. After training separate doc2vec models following the hyperparameter settings in Table \\ref{tab:param_doc2vec}, I first evaluate their performance with a simple Random Forest (RF) classifier, the same classifier used in the baseline, and select the best-performing doc2vec model to infer the document vector for each transcript. These document vectors are used in the final fusion stage to predict the mania level of each session. In addition, visualizing the similar groupings of transcripts via Embedding Projector\\footnote{\\url{http://projector.tensorflow.org}} and the similar words stored in PV-DM models are presented to qualitatively examine the embedding space inferred by the doc2vec models.\n\n\n\n\n\n\n\\section{Multi-Task Learning}\n\nIn the context of Deep Learning, the generalization of the model could be improved if representations being shared across related tasks \\cite{ruder2017}, and this approach is called Multi-Task Learning (MTL). There are two typical structures in MTL, hard parameter sharing and soft parameter sharing. Hard parameter sharing is applied by sharing the hidden layers between all tasks, with unchanged task-specific output layers (as shown in Figure \\ref{fig:mtl}). Soft parameter sharing, on the other hand, separate all tasks with their own model and parameters (as shown in Figure \\ref{fig:mtl}) with an objective to minimize the distance between parameters of different models.\n\n\\begin{figure}[ht]\n \\centering\n \\begin{minipage}[c]{0.42\\textwidth}\n \\centering\n \\includegraphics[height=4.2cm]{images/design/multitask_hard.png} \\\\\n (a) Hard parameter sharing\n \\end{minipage}\n \\begin{minipage}[c]{0.55\\textwidth}\n \\centering\n \\includegraphics[height=4.2cm]{images/design/multitask_soft.png} \\\\\n (b) Soft parameter sharing\n \\end{minipage}\n \\caption{Schematic for two architectures in Multi-Task Learning frameworks}\n \\label{fig:mtl}\n\\end{figure}\n\nHard parameter sharing architecture greatly reduces the risk of overfitting \\cite{baxter1997} and more specifically, the risk of overfitting the shared parameters is an order N smaller than the risk of overfitting the task-specific parameters, where N which represents the number of tasks. I therefore adjust my deep neural network (DNN) classifier to learn on both classification task (mania levels) and regression task (YMRS scores). The joint loss function is defined as the weighted sum of cross entropy loss for classification and the Euclidean loss for regression, as shown in Equation \\ref{eq:multi_loss}.\n\n\\begin{equation} \n\\begin{split}\n \\mathcal{L} & = \\mathcal{W}_c \\mathcal{L}_c + \\mathcal{W}_r \\mathcal{L}_r \\\\\n & = \\mathcal{W}_c (-\\sum_{c=1}^N y_c \\log(p_c)) + \\mathcal{W}_r (\\frac{1}{M} \\sum_{i=1}^M \\parallel y_r - p_r \\parallel ^ 2)\n\\end{split}\n\\label{eq:multi_loss}\n\\end{equation}\n\nwhere $\\mathcal{W}_c$ and $\\mathcal{W}_r$ represent the weight for two losses, $y_c$ and $y_r$ are values for two tasks while $p_c$ and $p_r$ are the predicted values, and $N$ is the number of classes and $M$ is the number of samples in the training set.\n", "meta": {"hexsha": "4c41d0ebe1e16bf1e2e66835fc7a7b8bcb42ca13", "size": 34618, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "paperwork/dissertation_final/mainSections/4-design.tex", "max_stars_repo_name": "ZihengZZH/bipolar-disorder", "max_stars_repo_head_hexsha": "deef966d65014175d6cb8f35320b2b33bfadfd13", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 17, "max_stars_repo_stars_event_min_datetime": "2019-08-07T11:21:01.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-01T00:57:09.000Z", "max_issues_repo_path": "paperwork/dissertation_final/mainSections/4-design.tex", "max_issues_repo_name": "ZihengZZH/bipolar-disorder", "max_issues_repo_head_hexsha": "deef966d65014175d6cb8f35320b2b33bfadfd13", "max_issues_repo_licenses": ["MIT"], "max_issues_count": 5, "max_issues_repo_issues_event_min_datetime": "2020-01-28T22:56:27.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-10T00:15:53.000Z", "max_forks_repo_path": "paperwork/dissertation_final/mainSections/4-design.tex", "max_forks_repo_name": "ZihengZZH/bipolar-disorder", "max_forks_repo_head_hexsha": "deef966d65014175d6cb8f35320b2b33bfadfd13", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 5, "max_forks_repo_forks_event_min_datetime": "2019-11-22T07:10:32.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-25T08:20:48.000Z", "avg_line_length": 98.3465909091, "max_line_length": 1675, "alphanum_fraction": 0.7715061529, "num_tokens": 8577, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6477982179521103, "lm_q2_score": 0.5273165233795671, "lm_q1q2_score": 0.34159470414198595}} {"text": "\\documentclass[10pt]{article}\n\\usepackage{a4wide}\n\\usepackage{listings}\n\\usepackage{listings}\n\\newcommand{\\be}{\\begin{equation}}\n\\newcommand{\\ee}{\\end{equation}}\n\\newcommand{\\OP}[1]{{\\bf\\widehat{#1}}}\n\n\\begin{document}\n\n\n\\section*{Project 3, Variational Monte Carlo studies of helium and beryllium}\n\nThe final aim of this project is to develop a variational Monte Carlo program which can be used to obtain ground state properties of atoms like He, Be, O, Ne, Si etc as well as diatomic molecules. \nfor important molecules \n\nThe final project builds on project 1 and 2 and adds the optimization procedure for the variational parameter $\\beta$ using for example Newton's methods, the steepest descent method or the conjugate gradient method. \nFurthermore, the single-particle wave functions used in the computation\nof the Slater determinant will be replaced by Gaussian type orbitals (GTO) using already optimized parameters. \n\nThus, in addition to these ingredients, the final project contains all the elements you have included inprojects 1 and 2. This means that you can include all results you have obtained in projects 1 and 2 in the wrap-up of the final report. It is the final report which counts for the final mark.\n\nYou will thus see that this report contains much of the same text as included in projects 1 and 2. \n\n{\\bf The deadline for project 3 is June 15, at noon}. See below for delivery format.\n\n\n\n\\section*{Part 1: Variational Monte Carlo calculations of the helium atom}\n\nWe will start with the simplest possible system beyond hydrogen, namely the helium atom.\nWe label $r_1$ the distance from electron 1 to the nucleus and similarly \n$r_2$ the distance between electron 2 and the nucleus.\nThe contribution to the potential energy from the interactions between the \nelectrons and the nucleus is\n\\be\n -\\frac{2}{r_1}-\\frac{2}{r_2},\n\\ee \nand if we add the electron-electron repulsion with\n$r_{12}=|{\\bf r}_1-{\\bf r}_2|$, the total potential energy \n$V(r_1, r_2)$ is\n\\be\n V(r_1, r_2)=-\\frac{2}{r_1}-\\frac{2}{r_2}+\n \\frac{1}{r_{12}},\n\\ee\nyielding the total Hamiltonian\n\\be\n \\OP{H}=-\\frac{\\nabla_1^2}{2}-\\frac{\\nabla_2^2}{2}\n -\\frac{2}{r_1}-\\frac{2}{r_2}+\n \\frac{1}{r_{12}},\n\\ee\nand Schr\\\"odinger's equation reads\n\\be\n \\OP{H}\\psi=E\\psi.\n\\ee\nAll equations are in so-called atomic units. The distances\n$r_i$ and $r_{12}$ are dimensionless. To have energies in electronvolt\nyou need to multiply all results with \n$2\\times E_0$,\nwhere $E_0=13.6$ eV.\nThe experimental binding energy for helium in atomic units a.u. is $E_{\\mathrm{He}}=-2.9037$ a.u..\n\n\n\\begin{enumerate}\n\n\\item[1a)] We want to perform a Variational Monte Carlo calculation of the ground state of the helium atom.\nIn our first attempt we will use a brute force Metropolis sampling with a trial wave function which has the following form\n\\begin{equation}\n \\psi_{T}({\\bf r_1},{\\bf r_2}, {\\bf r_{12}}) = \n \\exp{\\left(-\\alpha(r_1+r_2)\\right)}\n \\exp{\\left(\\frac{r_{12}}{2(1+\\beta r_{12})}\\right)}, \n\\label{eq:trial}\n\\end{equation}\nwith $\\alpha$ and $\\beta$ as variational parameters.\n\nYour task is to perform a Variational Monte Carlo calculation\nusing the Metropolis algorithm to compute the integral\n\\begin{equation}\n \\langle E \\rangle =\n \\frac{\\int d{\\bf r_1}d{\\bf r_2}\\psi^{\\ast}_T({\\bf r_1},{\\bf r_2}, {\\bf r_{12}})\\OP{H}({\\bf r_1},{\\bf r_2}, {\\bf r_{12}})\\psi_T({\\bf r_1},{\\bf r_2}, {\\bf r_{12}})}\n {\\int d{\\bf r_1}d{\\bf r_2}\\psi^{\\ast}_T({\\bf r_1},{\\bf r_2}, {\\bf r_{12}})\\psi_T({\\bf r_1},{\\bf r_2}, {\\bf r_{12}})}.\n\\end{equation}\nYou should parallelize your program. Find the energy minimum and compute also the mean distance\n$r_{12}$ between the two electrons for the optimal set of the variational parameters.\nA code for doing a VMC calculation for the helium atom can be \nfound on the webpage of the course, see under programs.\n\nYour Monte Carlo moves are determined by\n\\begin{equation}\n {\\bf R}' = {\\bf R} +\\delta \\times r,\n\\end{equation}\nwhere $r$ is a random number from the uniform distribution and $\\delta$\na chosen step length.\nIn solving this exercise you need to devise an algorithm which finds\nan optimal value of $\\delta$ for the variational parameters $\\alpha$ and $\\beta$,\nresulting in roughly $50\\%$ accepted moves. \n\nGive a physical interpretation of the best value of $\\alpha$.\nMake a plot of the variance as a function of the number of Monte Carlo\ncycles.\n\\item[1b)]\nFind closed form expressions for the local energy (see below) for the above \ntrial wave function and explain shortly how this \ntrial function satisfies \nthe cusp condition when $r_1\\rightarrow 0$ or\n$r_2\\rightarrow 0$ or $r_{12}\\rightarrow 0$. Show that\nclosed-form expression for the trial wave function is\n\\[ \nE_{L2} = E_{L1}+\\frac{1}{2(1+\\beta r_{12})^2}\\left\\{\\frac{\\alpha(r_1+r_2)}{r_{12}}(1-\\frac{\\mathbf{r}_1\\mathbf{r}_2}{r_1r_2})-\\frac{1}{2(1+\\beta r_{12})^2}-\\frac{2}{r_{12}}+\\frac{2\\beta}{1+\\beta r_{12}}\\right\\},\n\\]\nwhere\n\\[ \nE_{L1} = \\left(\\alpha-Z\\right)\\left(\\frac{1}{r_1}+\\frac{1}{r_2}\\right)+\\frac{1}{r_{12}}-\\alpha^2.\n\\]\n\nCompare the results of with and without the closed-form expressions (in terms of CPU time).\n\\item[1c)] Introduce now importance sampling and study the dependence of the results as a function of the time step $\\delta t$. \nCompare the results with those obtained under 1a) and comment eventual differences.\nIn performing the Monte Carlo analysis you should use blocking as a technique to make the statistical analysis of the numerical data.\nThe code has to run in parallel. \n\\item[1d)] With the optimal parameters for the ground state wave function, compute the onebody density. Discuss your results and compare the results with those obtained with a pure hydrogenic wave functions. Run a Monte Carlo calculations without the Jastrow factor as well\nand compute the same quantities. How important are the correlations induced by the Jastrow factor?\n\n\\item[1e)] Repeat step 1c) by varying the energy using the \nconjugate gradient method or similar methods to obtain the best possible parameter\n$\\beta$. Replace now the hydrogen-like single-particle wave functions with the 3-21G basis defined \nat the EMSL website \\url{https://bse.pnl.gov/bse/portal}. \n\n\\end{enumerate}\n\n\n\\section*{Part 2: Variational Monte Carlo calculations of the Beryllium and Neon atoms}\nThe previous exercise has prepared you for extending your calculational machinery to other systems.\nHere we will focus on the neon and beryllium atoms.\nIt is convenient to make modules or classes of trial wave functions, both many-body wave functions\nand single-particle wave functions and the quantum numbers involved, such as spin, orbital momentum and principal\nquantum numbers.\n\nThe new item you need to pay attention to is the calculation of the Slater Determinant. This is an additional complication\nto your VMC calculations.\nIf we stick to hydrogen-like wave functions,\nthe trial wave function for Beryllium can be written as \n\\begin{equation}\n \\psi_{T}({\\bf r_1},{\\bf r_2}, {\\bf r_3}, {\\bf r_4}) = \n Det\\left(\\phi_{1}({\\bf r_1}),\\phi_{2}({\\bf r_2}),\n \\phi_{3}({\\bf r_3}),\\phi_{4}({\\bf r_4})\\right)\n \\prod_{i \n\nclass MyMultiplyClass{ \npublic: \n double multiply(double x, double y) { \n return x * y; \n } \n}; \nTEST(MyMath) { \n MyMultiplyClass my; CHECK_EQUAL(56, my.multiply(7,8)); \n} \nint main() \n{ \nreturn UnitTest::RunAllTests(); \n}\n\\end{verbatim}\nFor Fortran users, the link at \\url{http://sourceforge.net/projects/fortranxunit/} contains a similar software for unit testing.\n\\end{document}\n\n\n", "meta": {"hexsha": "85715be0b22ad3fc150e8436ee0d7596be707302", "size": 20908, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/Projects/2015/project3_2015.tex", "max_stars_repo_name": "GabrielSCabrera/ComputationalPhysics2", "max_stars_repo_head_hexsha": "a840b97b651085090f99bf6a11abab57100c2e85", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 87, "max_stars_repo_stars_event_min_datetime": "2015-01-21T08:29:56.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-28T07:11:53.000Z", "max_issues_repo_path": "doc/Projects/2015/project3_2015.tex", "max_issues_repo_name": "GabrielSCabrera/ComputationalPhysics2", "max_issues_repo_head_hexsha": "a840b97b651085090f99bf6a11abab57100c2e85", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 3, "max_issues_repo_issues_event_min_datetime": "2020-01-18T10:43:38.000Z", "max_issues_repo_issues_event_max_datetime": "2020-02-08T13:15:42.000Z", "max_forks_repo_path": "doc/Projects/2015/project3_2015.tex", "max_forks_repo_name": "GabrielSCabrera/ComputationalPhysics2", "max_forks_repo_head_hexsha": "a840b97b651085090f99bf6a11abab57100c2e85", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 54, "max_forks_repo_forks_event_min_datetime": "2015-02-09T10:02:00.000Z", "max_forks_repo_forks_event_max_datetime": "2022-03-07T10:44:14.000Z", "avg_line_length": 47.3031674208, "max_line_length": 538, "alphanum_fraction": 0.751147886, "num_tokens": 5646, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.5273165085228825, "lm_q2_score": 0.647798211152541, "lm_q1q2_score": 0.3415946909323269}} {"text": "\\chapter{Constraining the equation of state with astrophysics}\n%\\chapter{Astrophysics around neutrons stars}\n\\chapterimage[width=15cm]{wordcloud/chap3b.png}\n\n\\subsection{Accretion disks}\n\nGravitational potential energy release\n\\be\n\\Delta E_{\\mathrm{acc}} = \\frac{G M m}{R} \\sim 10^{20} \\left( \\frac{10\\km}{R} \\right) \\left( \\frac{M}{\\Msun} \\right) \\unitspace\\erg\\unitspace\\g^{-1}\n\\ee\n\nRoche Lobe \\cite{PRP02} \\cite{LL15}\n\nLMXB \\cite{TH06}\n\nHard and soft state \\cite{HvdK89}\nAlternates between these two states \\cite{MDF14} \\cite{DGK07}\n\n\n\n\\subsection{Between the disk and the star: boundary layers}\n\n\\be\n\\Omega(R) \\approx \\Omega_{\\mathrm{K}}(R) = \\left( \\frac{G M}{R^3} \\right)^{1/2}\n\\ee\n\nLayer of thickness $b$ equals $\\Omega(R + b) \\approx \\Omega_{\\mathrm{K}}(R + b)$ that must slow down to $\\Omega_{*}$.\n\nEnergy difference\n\\be\n\\dot{E} = \\frac{1}{2} \\Mdot R^2 (\\Omega_{\\mathrm{K}}^2 - \\Omega_{*}^2) = \n\\frac{1}{2} \\Mdot \\frac{GM}{R} \\left[ 1 - \\left(\\frac{\\Omega_*}{\\Omega_{\\mathrm{K}}} \\right)^2 \\right] \n\\ee\n\nViscous torque $G_{\\mathrm{T}} = \\Mdot R^2 (\\Omega_{\\mathrm{K}} - \\Omega_*)$\nHence,\n\\be\n\\dot{E} = \\frac{1}{2} \\frac{G M \\Mdot}{R} \\left( 1 - \\frac{\\Omega_*}{\\Omega_{\\mathrm{K}}} \\right)^2\n\\ee\n\n\n\\section{X-ray bursts}\n\\subsection{Unstable thermonuclear burning on top of neutron stars}\n\n\\subsection{Constraining the size of bursting source}\n\n\n\n\n%--------------------------------------------------\n\\section{Accretion}\n%infalling matter leads to X-rays \\cite{Lewin93}\n\n\\subsection{Source of energy}\nIn the heart of this whole problem is an astrophysical process called accretion.\n\nGravitational potential energy release\n\\be\n\\Delta E_{\\mathrm{acc}} = \\frac{G M m}{R} \\sim 10^{20} \\left( \\frac{10\\km}{R} \\right) \\left( \\frac{M}{\\Msun} \\right) \\unitspace\\erg\\unitspace\\g^{-1}\n\\ee\n\nEddington luminosity\n\\be\nL_{\\mathrm{Edd}} = \\frac{ 4 \\pi G M \\mproton c }{\\sigma_{\\mathrm{T}} } \\approx \\Ten{1.3}{38} \\left( \\frac{M}{\\Msun} \\right) \\ergs\n\\ee\n\nAccretion luminosity\n\\be\nL_{\\mathrm{acc}} = \\frac{G M \\Mdot}{R} \n\\ee\n\n\\subsection{Binary systems}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=7.5cm]{figs/astro/roche.pdf}\n\\includegraphics[width=7.5cm]{figs/astro/roche.pdf}\n\\caption{\\label{fig:roche}\nRoche potential for binary systems.\n}\n\\end{figure}\n\n\\subsubsection{Roche lobes and mass transfer}\nRoche Lobe \\cite{PRP02} \\cite{LL15}\n\nA flow of gas between two stars can be described by the Euler equation.\nIt gives the time evolution of the velocity $\\vec{v}$ of the gas that has a pressure of $P$ and density $\\rho$.\nIn a reference frame rotating together with the binary system with angular velocity $\\omega$ the Euler equation takes the form \n\\be\n\\frac{ \\partial \\vec{v} }{\\partial t} + (\\vec{v} \\cdot \\nabla)\\vec{v} = -\\nabla \\Phi_{\\mathrm{R}} - 2 \\vec{ \\omega } \\times \\vec{v} - \\frac{1}{\\rho} \\nabla P,\n\\ee\nwhere the angular velocity of the binary is then\n\\be\n\\vec{ \\omega } = \\left( \\frac{ G M }{a^3} \\right)^{1/2} \\vec{e},\n\\ee\nas given with the unit vector $\\vec{e}$, normal to the orbital plane.\nHere $M$ is the total mass of the system, i.e. $M = M_1 + M_2$, where $M_1$ and $M_2$ are the individual masses of the two stars in the system, respectively, and $a$ is their orbital separation.\n\nThe effects originating from the gravitation and from the centrifugal forces are encapsulated in the so-called Roche potential, given as a function of radial vector $\\vec{r}$ as\n\\be\n\\Phi_{\\mathrm{R}}(\\vec{r}) = -\\frac{G M_1}{|\\vec{r} - \\vec{r_1}|} -\\frac{G M_2}{|\\vec{r} - \\vec{r_2}|} - \\frac{1}{2} ( \\vec{ \\omega } \\times \\vec{v} )^2,\n\\ee\nwhere the location of the stars is given with $\\vec{r_1}$ and $\\vec{r_2}$.\n\nBy studying the shape of the potential, we see that in between the stars, in the so-called $L_1$ point there exists a location where the individual gravitational pull from the stars is balanced.\nThis leads to a kinda of a nozzle in the system from which the material can leak from the less massive star to the more massive object.\nSuch a leaking, or a Roche lobe overflow, will then occur if the companion star's radius exceeds the size of its Roche lobe.\nTypically such a thing can happen when the star evolves and expands at the end of its life cycle. \n\nLMXB \\cite{TH06}\n\n\n\\subsection{Accretion disks}\nHard and soft state \\cite{HvdK89}\nAlternates between these two states \\cite{MDF14} \\cite{DGK07}\n\n\n\n\\section{Accretion to a neutron star}\n\n\\subsection{Boundary layers}\n\n\\be\n\\Omega(R) \\approx \\Omega_{\\mathrm{K}}(R) = \\left( \\frac{G M}{R^3} \\right)^{1/2}\n\\ee\n\nLayer of thickness $b$ equals $\\Omega(R + b) \\approx \\Omega_{\\mathrm{K}}(R + b)$ that must slow down to $\\Omega_{*}$.\n\nEnergy difference\n\\be\n\\dot{E} = \\frac{1}{2} \\Mdot R^2 (\\Omega_{\\mathrm{K}}^2 - \\Omega_{*}^2) = \n\\frac{1}{2} \\Mdot \\frac{GM}{R} \\left[ 1 - \\left(\\frac{\\Omega_*}{\\Omega_{\\mathrm{K}}} \\right)^2 \\right] \n\\ee\n\nViscous torque $G_{\\mathrm{T}} = \\Mdot R^2 (\\Omega_{\\mathrm{K}} - \\Omega_*)$\nHence,\n\\be\n\\dot{E} = \\frac{1}{2} \\frac{G M \\Mdot}{R} \\left( 1 - \\frac{\\Omega_*}{\\Omega_{\\mathrm{K}}} \\right)^2\n\\ee\n\n\n\\subsection{X-ray bursts}\nThermonuclear runaway.\n\n\\subsection{Constraining the size of the star}\nCooling tail method.\n\n\n\n\n\n% --------------------------------------------------\n\\newpage\n\\section{Appendix A: Real astrophysics}\n\n%\\begin{itemize}\n% \\item Accretion\n% \\item Disks \n% \\item Soft and hard states\n% \\item Boundary layers\n% \\item Ignition?\n% \\item X-ray bursts\n% \\item Cooling tail method\n%\\end{itemize}\n%\n%Possible layout:\n%\n%v1\n%\\begin{enumerate}\n% \\item Accretion\n% \\begin{itemize}\n% \\item Disks\n% \\item Soft and hard states\n% \\item Boundary layers\n% \\end{itemize}\n% \\item X-ray bursts\n% \\begin{itemize}\n% \\item Unstable thermonuclear burning\n% \\item Cooling tail method\n% \\end{itemize}\n%\\end{enumerate}\n%\n%v2\n%\\begin{enumerate}\n% \\item Accretion disks\n% \\begin{itemize}\n% \\item Accretion\n% \\item ?Roche lobe overflow\n% \\item Soft and hard state\n% \\item Boundary layers\n% \\end{itemize}\n% \\item X-ray bursts\n% \\begin{itemize}\n% \\item Unstable thermonuclear burning\n% \\item Cooling tail method\n% \\end{itemize}\n%\\end{enumerate}\n\n\n\\sect{Energetics}\nLet us try and estimate the energetics of different phenomena of what we can observe from neutron stars.\\mnote{Energy output}\nFew possible stable sources of energy exists: thermal, gravitational, rotational, and magnetic.\nIn addition, unstable fusion processes can also power some observable phenomena.\n\n\n\n", "meta": {"hexsha": "57267b04c4febce146dca075466df0b73123ff8a", "size": 6501, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "notes/roche_lobe_notes.tex", "max_stars_repo_name": "natj/thesis", "max_stars_repo_head_hexsha": "342586960e1d956ed6edf6aae4bfd0410747f011", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2019-04-16T20:30:36.000Z", "max_stars_repo_stars_event_max_datetime": "2019-04-16T20:30:36.000Z", "max_issues_repo_path": "notes/roche_lobe_notes.tex", "max_issues_repo_name": "natj/thesis", "max_issues_repo_head_hexsha": "342586960e1d956ed6edf6aae4bfd0410747f011", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "notes/roche_lobe_notes.tex", "max_forks_repo_name": "natj/thesis", "max_forks_repo_head_hexsha": "342586960e1d956ed6edf6aae4bfd0410747f011", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 31.4057971014, "max_line_length": 194, "alphanum_fraction": 0.6742039686, "num_tokens": 2162, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6992544335934766, "lm_q2_score": 0.48828339529583464, "lm_q1q2_score": 0.3414343290106885}} {"text": "\\section{Interpretation of QM}\n\\subsection{Supporting definitions}\n\nTo provide our interpretation of quantum mechanics we need to develop\na number of supporting definitions. As the reader familiar with\nprocess algebraic systems can readily verify, these definitions make\n\\emph{essential} use of the reflective operations and as such identify\nthis calculus as uniquely suited to this particular task.\n\nAmong these operations we find a notion of \\emph{multiplication} of\nnames that interacts well with a notion of \\emph{tensor product} of\nprocesses. Even more intriguingly, we find a notion of a \\emph{dual}\nto a process in the form of maps from processes to names. While\nnotions of composite names have been investigated in the process\nalgebraic literature, it is the fact that names reflect process\nstructure that enables the collection of duals to enjoy an algebraic\nstructure dual to the collection of processes (i.e. there are\noperations available to duals that reflect the operations on\nprocesses). Moreover, it is this structure that enables an effective\ndefinition of inner product.\n\n\\subsubsection{Multiplication}\n\\begin{mathpar}\n \\quotep{Q} \\cdot \\quotep{R} := \\quotep{Q|R}\n \\and \\\\\n \\quotep{Q} \\cdot P := P\\{ \\quotep{Q|R} / \\quotep{R} : \\quotep{R} \\in \\freenames{P} \\}\n\\end{mathpar}\n\n\\paragraph{Discussion}\nThe first equation needs little explanation; the second says that each\nfree name of the process is replaced with the multiplication of that\nname by the scalar. Multiplication of a scalar (name) by a state\n(process) results in a process all the names of which have been `moved\nover' by parallel composition with the process the scalar\nquotes. There is a subtlety that the bound names have to be\nmanipulated so that multiplied names aren't accidentally\ncaptured. There are many ways to achieve this.\n\n\\begin{remark}\\label{rem:multiplication_identities}\n The reader is invited to verify that for all $x,y,z \\in \\QProc$ and $P \\in \\Proc$\n \\begin{mathpar}\n x \\cdot \\quotep{0} \\equiv x \n \\and\n x \\cdot y \\equiv y \\cdot x\n \\and\n x \\cdot (y \\cdot z) \\equiv (x \\cdot y) \\cdot z\n \\and \\\\\n \\quotep{0} \\cdot P \\equiv P\n \\and \\\\\n x \\cdot (y \\cdot P) \\equiv (x \\cdot y) \\cdot P\n \\and \\\\\n x \\cdot (P|Q) \\equiv (x \\cdot P) | (x \\cdot Q)\n \\and \\\\ \n \\end{mathpar}\n\\end{remark}\n\n\\subsubsection{Tensor product}\n\nWe define a tensor product on processes by structural induction.\n\n\\paragraph{Tensor of sums} First note that all summations, including\n$\\pzero$ and sequence, can be written $\\Sigma_{i} x_{i}.A_{i} +\n\\Sigma_{j} x_{j}.C_{j}$, where we have grouped input-guarded processes\ntogether and output-guarded processes together.\n\nThus, we can define the tensor product of two summations, $N_{1}\\otimes N_{2}$, where\n\n\\begin{mathpar}\n N_{1} := \\Sigma_{i} x_{i}.A_{i} + \\Sigma_{j} x_{j}.C_{j}\n \\and\n N_{2} := \\Sigma_{i'} y_{i'}.B_{i'} + \\Sigma_{j'} y_{j'}.D_{j'} \n\\end{mathpar}\n\nas follows.\n\n\\begin{mathpar}\n \\Sigma_{i} x_{i}.A_{i} + \\Sigma_{j} x_{j}.C_{j} \\otimes \\Sigma_{i'}\n y_{i'}.B_{i'} + \\Sigma_{j'} y_{j'}.D_{j'} \n \\and \\\\\n := \\; \\Sigma_{i} \\Sigma_{i'} \\quotep{\\stackrel{\\vee}{x_{i}}| \\stackrel{\\vee}{y_{i'}}}.(A_{i}\\otimes B_{i'}) \\; | \\; \\Sigma_{i'} \\Sigma_{i} \\quotep{\\stackrel{\\vee}{y_{i'}}|\\stackrel{\\vee}{x_{i}}}.(B_{i'}\\otimes A_{i})\n \\and\n \\;\\; | \\;\\; \\Sigma_{j} \\Sigma_{j'} \\quotep{\\stackrel{\\vee}{x_{j}}|\\stackrel{\\vee}{y_{j'}}}.(A_{j}\\otimes B_{j'}) \\; | \\; \\Sigma_{j'} \\Sigma_{j} \\quotep{\\stackrel{\\vee}{y_{j'}}|\\stackrel{\\vee}{x_{j}}}.(B_{j'}\\otimes A_{j})\n\\end{mathpar}\n\n\\begin{remark}\n Do we need to $x^{L}$ and $y^{R}$ for this construction as well?\n\\end{remark}\n\n\\paragraph{Tensor of parallel compositions} Next, we distribute tensor\nover par.\n\n\\begin{mathpar}\n P_{1}|P_{2} \\otimes Q_{1}|Q_{2} := (P_{1} \\otimes Q_{1}) | (P_{1}\n \\otimes Q_{2}) | (P_{2} \\otimes Q_{1}) | (P_{2} \\otimes Q_{2})\n\\end{mathpar}\n\n\\paragraph{Tensor with dropped names} We treat tensor of a\nprocess with a dropped name as parallel composition.\n\n\\begin{mathpar}\n P \\otimes \\dropn{x} := P | \\dropn{x}\n\\end{mathpar}\n\n\\paragraph{Tensor of agents}\n\nFinally, we need to define tensor on agents. Note that the definition\nof tensor on summations only tensors inputs with inputs and outputs\nwith outputs. Thus, we only have to define the operation on\n``homogeneous'' pairings.\n\n\\begin{mathpar}\n (\\vec{x})P \\otimes (\\vec{y})Q\n \\and \\\\\n := (x_{0}^{L}|y_{0}^{R},\\ldots,x_{0}^{L}|y_{n}^{R},\\ldots,x_{m}^{L}|y_{0}^{R},\\ldots,x_{m}^{L}|y_{n}^R)(P\\{ \\vec{x}^{L}/\\vec{x}\\} \\otimes Q \\{ \\vec{y}^{R}/\\vec{y}\\})\n \\and \\\\\n \\clift{\\vec{P}} \\otimes \\clift{\\vec{Q}}\n \\and \\\\\n := \\clift{P_{0}\\otimes Q_{0},\\ldots,P_{0}\\otimes Q_{n},\\ldots,P_{m}\\otimes Q_{0},\\ldots,P_{m}\\otimes Q_{n}}\n\\end{mathpar}\n\n\\begin{remark}\n Observe that arities of tensored abstractions matches arities of\n tensored concretions if the original arities matched. Note also that\n the length of the arities corresponds to the increase in dimension\n we see in ordinary vector space tensor product.\n\\end{remark}\n\n\\begin{remark}\n Operationally, this definition distributes the tensor down to\n components ``linked'' by summation. Tensor over summation is\n intriguing in that it mixes names. Moreover, as a consequence of the\n way it mixes names we have the identities for all $x \\in \\QProc$ and\n $P,Q \\in \\Proc$\n\n \\begin{mathpar}\n (x \\cdot P) \\otimes Q \\equiv x \\cdot (P \\otimes Q) \\equiv P \\otimes (x \\cdot Q)\n \\and \\\\\n P \\otimes \\pzero \\equiv P\n \\end{mathpar}\n\n that the reader is invited to verify.\n\\end{remark}\n\n\\subsubsection{Annihilation}\n\\begin{mathpar}\n P^{\\perp} := \\{ Q : \\forall R. P|Q \\red^{*} R \\Rightarrow R \\red^{*} \\pzero \\}\n \\and \\\\\n \\annihilate{P} := \\Sigma_{Q \\in P^{\\perp}} \\quotep{Q}?(y).(\\dropn{y}|Q) | \\Sigma_{Q \\in P^{\\perp}} \\quotep{Q}\\clift{\\Box}\n\\end{mathpar}\n\n\\paragraph{Discussion} The reader will note that $P^{\\perp}$ is a\n\\emph{set} of processes, while $\\annihilate{P}$ is a\n\\emph{context}. We call the set $P^{\\perp}$ the \\emph{annihilators} of\n$P$. The parallel composition of a process in the annihilators of $P$\nwith $P$ will result in a process, the state space of which has all\npaths eventually leading to $\\pzero$. Execution may endure loops; but\nunder reasonable conditions of fairness (naturally guaranteed under\nmost notions of bisimulation) such a composite process cannot get\nstuck in such a loop and will, eventually pop out and terminate.\n\nThe context $\\annihilate{P}$ is ready and willing to ``take the\n$P$ out of'' the process to which it is applied. It will effectively\ntransmit the code of the process to which it is applied to one of the\nannihilators and run the process against it.\n\n\\begin{remark}\n Note that ${\\annihilate{P}}^*$ is the abstraction corresponding to\n context $\\annihilate{P}$. We will set $\\dualize{P} := {\\annihilate{P}}^*$.\n\\end{remark}\n\n\\subsubsection{Evaluation}\nWe fix $M$ a domain of fully abstract interpretation with an equality\ncoincident with bisimulation. We take $\\meaningof{\\cdot} : \\Proc \\to\nM$ to be the map interpreting processes and $\\nmeaningof{\\cdot} : \\M\n\\to Proc$ to be the map running the other way. Then we define\n\n\\begin{mathpar}\n \\int P := \\nmeaningof{\\meaningof{P}}\n\\end{mathpar}\n\n\\paragraph{Discussion}\nThere are many fully abstract interpretations of Milner's\n$\\pi$-calculus. Any of them can be used as a basis for interpreting\nthe reflective calculus here. Equipped with such a domain it is\nlargely a matter of grinding through to check that the Yoneda\nconstruction for the normalization-by-evaluation program can be\nextended to this setting.\n\n\\begin{remark}\n The reader is invited to verify that $\\int (\\annihilate{P}[P]) = 0$,\n and equivalently $(\\nu\\; x)\\int \\dualize{P}\\langle x \\rangle |\n x\\clift{P} = 0$.\n\\end{remark}\n\n\\subsection{Quantum mechanics}\n\n\\subsubsection{What is the quantum mechanical notion of continuation?}\\label{sec:quantum_continuation}\n\nImagine the following experimental set-up. Alice, our intrepid quantum\ninvestigator, prepares a state by performing some operation on some\ninitial state. Then she performs some measurement to obtain an\nobservation. Using the information of the observation, she selects a\nnew initial state, operation and measurement and repeats the steps\nabove. She iterates this procedure until she obtains some desired\nobservation. What is the expression of this procedure in the language\nof quantum mechanics?\n\n\\begin{figure}[htp]\\label{fig:iterated_experiment}\n% \\fbox{\n \\begin{lstlisting}[mathescape]\n $\\mathcal{E} ::=$\n let S = $U \\state{L}$ in (* prepare state*)\n let m = $\\innerprod{M}{S}^2$ in (* take measurement *)\n match m with (* use m to decide next experiment *)\n v$_0$ -> $\\mathcal{E}$\n | $\\ldots$\n | v$_N$ -> $\\mathcal{E}$\n | v$_{Exit}$ -> m (* return observation *)\n \\end{lstlisting}\n% }\n \\caption{Iterated experiment schematic}\n\\end{figure}\n\nFigure \\ref{fig:iterated_experiment} gives a schematic description of\nsuch an iterated experimental procedure. The question is how do we\nwrite down this iterated procedure without stepping outside the\nlanguage of quantum mechanics? Note that accounts of famous composite\nquantum experiments, like the Stern-Gerlach experiment leave the\nlanguage of quantum mechanics to describe the iterated experiment.\n\nWe ask this question for many reasons, but one of them is to help set\nup the exegesis of our interpretation. In our framework\n\\emph{everything} is a computation, both the quantum operations and\nprocesses (classical or quantum) that invoke those operations. There\nis no need to step out of the conceptual (and more pragmatically, the\ncomputational) framework to describe these kinds of experiments. More\nto the point, the framework we are proposing is -- like the hybrid\nfunctional language employed in the schema -- \\emph{compositional}:\nexperiments, computations are built out of experiments and\ncomputations. This is of enormous pragmatic value if we are to build\nand reason about systems of significant scale.\n\n\\begin{remark}\n It is also worth noting in this connection that this schema is the\n core of a wide range of recursive functions. Further, this\n connection to calculations of fixpoints makes it a close neighbor of\n search techniques like natural selection and the scientific\n method. This is a theme to which we will return, for quantum\n information seems very \\emph{unlife-like} in it's uncloneable,\n undeleteable nature.\n\\end{remark}\n\nReturning the matter of the computational interpretation, our\ninterpretation will take the form of a map, written\n$\\meaningof{-}(-)$, from expressions in Dirac notation to expressions\nin our target reflective calculus. The map takes an \\emph{ancillary}\nargument, a channel along which to communicate results to subsequent\ncomputations. This is how we communicate, for example, the results of\ntaking a measurement to a subsequent step in an experiment.\n\n\\subsubsection{Interpretation}\n\nTable \\ref{tbl:core_qm_op_defns} gives the core operational\ncorrespondences. It is meant as an intuitive guide.\n\n\\begin{table}[htp]\\label{tbl:core_qm_op_defns}\n \\center{\n \\fbox{\n \\begin{tabular}{c|c}\n quantum mechanics & process calculus \\\\\n \\hline\n scalar & $x := \\quotep{P}$ \\\\\n state vector & $\\state{P} := P$ \\\\\n dual & $\\state{P}^{*} := \\event{\\annihilate{P}} := \\quotep{\\annihilate{P}}[-]$ \\\\\n matrix & $ \\Sigma_{\\alpha} \\state{P_{\\alpha}}x_{\\alpha}\\event{Q_{\\alpha}}$ \\\\\n vector addition & $\\state{P} + \\state{Q} := \\state{P | Q}$ \\\\\n tensor product & $\\state{P} \\otimes \\state{Q} := \\state{P \\otimes Q}$ \\\\\n inner product & $\\innerprod{P}{Q} := \\quotep{\\int \\annihilate{P}[Q]}$ \\\\\n \\end{tabular}\n }\n }\n \\caption{QM - operational definitions}\n\\end{table}\n\n\\paragraph{Discussion}\nThe process algebraic view of a state is called, ironically, a\nprocess, and that is what we map vectors to in our interpretation. It\nhas long been noted in the process algebraic community that names play\na role somewhat similar to scalars in a vector space. What is unique\nabout the reflective calculus, and makes it suitable for an\ninterpretation of this form is that with the structure of names\nreflecting the structure of processes we can both make this similarity\nin a precision instrument; and find a notion of \\emph{dual} to a\nstate. \n\nIf we posit names as scalars, then in perfect analogy with vector\nspaces a dual is a map from processes to names. We actually have two\ncandidates for this interpretation: nominal contexts, $\\quotep{M}$,\nand their corresponding abstraction, $\\quotep{M}^{*}$. The goal of\nsupporting a notion of continuation selects the latter of the two for\nour interpretation.\n\nTaking these as the basis of the interpretation together with the\nalgebraic identities required by the Dirac notation more or less fixes\nthe definitions of the rest of the operations. Among the interesting\nparticularities, the definition of inner product finds near perfect\nmirroring of the Feyman interpretation.\n\n\\begin{mathpar}\n \\inferrule* [lab=states] {} {\\meaningof{\\state{P}}(c) = c?(l,r).r\\clift{P}}\n \\and\n \\inferrule* [lab=events] {} {\\meaningof{\\event{P}}(c) = (x)c?(l,r).l\\clift{\\dualize{P}\\langle x \\rangle} } \n \\and\n \\inferrule* [lab=vector addition] {} {\\meaningof{\\state{P} + \\state{Q}}(c) = \\meaningof{\\state{P | Q}}(c)}\n \\and\n \\inferrule* [lab=tensor product] {} {\\meaningof{\\state{P} \\otimes \\state{Q}}(c) = \\meaningof{\\state{P\\otimes Q}}(c)}\n \\and\n \\inferrule* [lab=inner product] {} {\\meaningof{\\innerprod{P}{Q}}(c) = (\\nu\\; x)c\\clift{\\int \\dualize{P}\\langle x \\rangle | x\\clift{Q}}}\n \\and\n \\inferrule* [lab=matrix] {} {\\meaningof{\\fprmatrix{P}{x}{Q}}(c) = \\\\\\\\ (u)(\\nu \\; lr) c!(l,r).(l?(e). (\\nu\\; y)\\meaningof{\\innerprod{\\dropn{e}}{Q}}(x) | x?(z).x?(a).c\\clift{(\\dualize{P}\\sigma(z,a))\\langle u \\rangle}|y!(x) \\\\\\\\\n + r?(e). (\\nu\\; x)\\meaningof{\\innerprod{P}{\\dropn{e}}}(x) | x?(z).x?(a).c\\clift{Q\\sigma(z,a)}|x!(x_{\\alpha}))} \n \\and\n \\inferrule* [lab=matrix application] {} {\\meaningof{(\\fprmatrix{P}{x}{Q})(\\state{S})}(c) = (\\nu\\; c'u)\\meaningof{\\fprmatrix{P}{x}{Q}}(c')\\langle u \\rangle | \\meaningof{\\state{S}}(c') | c'?(a).c!(a)\n \\\\\\\\\n \\meaningof{(\\fprmatrix{P}{x}{Q})(\\event{S})}(c) = (\\nu\\; c'u)\\meaningof{\\fprmatrix{P}{x}{Q}}(c')\\langle u \\rangle | \\meaningof{\\event{S}}(c') | c'?(a).c!(a)}\n\\end{mathpar}\n\nwhere\n\n\\begin{mathpar}\n P\\sigma(z,a) := P\\{ z\\cdot a\\cdot r/r : r \\in \\freenames{P} \\}\n\\end{mathpar}\n\n\\begin{remark}\n The reader is invited to verify that\n \\begin{mathpar}\n \\meaningof{\\innerprod{P}{Q}}(c)\n \\and \\\\\n \\wbbisim \n \\and \\\\\n %\\and\n (\\nu\\; x)(\\nu c'lr)\\meaningof{\\event{P}}(c')\\langle x \\rangle \\;|\\; \\meaningof{\\state{Q}}(c')\n %\\and\n |\\; c'!(l,r).l?(p).r?(q).c\\clift{\\int \\dropn{p} | x!(q)}\n \\end{mathpar}\n This provides a (more) compositional definition of inner product. It\n also illustrates an important point of the computational\n interpretation. We have a notion of equivalence providing a crucial\n proof method: bisimulation.\n\\end{remark}\n\n\\begin{remark}\n Assuming $\\int (\\annihilate{P}[P]) = 0$, the reader is\n invited to verify that $(\\fprmatrix{P}{x}{P})(\\state{P}) = x \\cdot \\state{P}$.\n\\end{remark}\n\n% \\begin{remark}\n% The reader is invited to verify that $\\innerprod{P}{Q}$ could\n% equally well have been written $\\quotep{\\int \\stackrel{\\vee}{x}}$\n% where $x = \\event{\\annihilate{P}}(Q)$.\n\n% One of the motivations for this remark is that there is another way\n% to factor these operations. We could package up evaluation in the dual:\n\n% \\begin{mathpar}\n% \\state{P}^{*} := \\event{\\int \\annihilate{P}} := \\quotep{\\int \\annihilate{P}}[-]\n% \\end{mathpar}\n\n% and then have inner product defined by\n \n% \\begin{mathpar}\n% \\innerprod{M}{Q} := \\event{M}(Q)\n% \\end{mathpar}\n\n% where we use $M$ to label the dual to emphasize that it is a context.\n\n% Hopefully, experience with the calculations will provide guidance on\n% the best factoring.\n% \\end{remark}\n\n\\begin{remark}\\label{rem:abstract_scalars}\n Assuming $\\int (\\annihilate{P}[P]) = 0$, the reader is\n invited to verify that $\\forall P,Q. (\\prmatrix{0}{Q})(\\state{0}) =\n \\state{0}$ and dually $(\\prmatrix{P}{0})(\\event{0}) = \\event{0}$.\n\\end{remark}\n\n\\subsubsection{Interpreting continuations}\n\nAs promised, we can combine these interpretations with standard\nsemantics for conditionals and continuations to provide an\ninterpretation of the iterated experiment.\n\n\\begin{lstlisting}[mathescape] \n $\\ldb$ let S = $U \\state{L}$ in \n let m = $\\innerprod{M}{S}^2$ in \n match m with \n v$_0$ -> $\\mathcal{E}$\n | $\\ldots$\n | v$_N$ -> $\\mathcal{E}$\n | v$_{Exit}$ -> m $\\rdb(c)$\n $=$\n $(\\nu\\; c')\\meaningof{\\innerprod{M}{U\\state{L}}}(c')$\n $| c'?(m).m!(m) | \\Sigma_{i=0}^{N}v_{i}?(m).\\meaningof{\\mathcal{E}} + v_{Exit}?(m).c!(m)$\n\\end{lstlisting}\n\n\\paragraph{A quick tally}\nAlready the interpretation is beginning to show signs of\npromise. First of all, it is no more notationally cumbersome than the\nnotation used in QM calculations. Beyond syntax, we have a new proof\nprinciple in hand and the ability to reason about more complex\nexperimental situations than is directly calculable in ordinary\nquantum mechanics.\n\n\\subsubsection{Adjointness}\n\nWe need to give a definition of $(\\cdot)^{\\dagger}$ for matrices. The\nobvious candidate definition is\n\\begin{mathpar}\n\\meaningof{(\\fprmatrix{P}{x}{Q})^{\\dagger}}(c)\n= (\\nu\\; u)\\meaningof{\\fprmatrix{\\dualize{Q}\\langle u \\rangle}{\\overline{x}}{\\dualize{P}\\langle u \\rangle}}(c) \n\\end{mathpar}\n\n% But, $(Q_{\\alpha}^{\\underline{\\perp}})^{*}$ requires a name along\n% which to communicate the process to achieve the context application.\n\n\\begin{remark}\n i'm a little worried that i don't (yet) have proper support for\n complex conjugacy. But, the observation above may give us a\n clue. According to Abramsky, it must be the case that the scalars\n are iso to the homset of the identity for the tensor -- which the\n observation above (\\ref{rem:abstract_scalars}) characterizes. For\n now, we will simply bookmark the notion with $\\overline{x}$.\n\\end{remark}\n\n\\subsubsection{Basis for a basis}\nIf processes label states and ``addition'' of states (a.k.a. vector\naddition) is interpreted as parallel composition, what corresponds to\nnotions of linear independence and basis? Here, we recall that Yoshida\nhas developed a set of \\emph{combinators} for an asynchronous verison\nof Milner's $\\pi$-calculus \\cite{DBLP:conf/concur/Yoshida98}. These\nare a finite set of processes such any process can be expressed as\nparallel composition of these combinators together with liberal uses\nof the new operator and replication. We can simply give a translation\nof these into the present calculus and have reasonable expectation\nthat the property carries over. That is, that the resultant set allows\nto express all processes via parallel composition. Note, however, that\nthere is no new operator or replication in this calculus. As a result,\nwe expect that the corresponding set is actually infinite. That is, we\nexpect that the space is actually infinite dimensional.\n\n\\begin{remark}\n The reader familiar with the lambda calculus may reasonably object:\n certainly, the collection $S$, $K$ and $I$ is a finite set of\n combinators \\cite{Barendregt84}. Shouldn't we expect to see a finite\n set of combinators for an effectively equivalent system? i am very\n sympathetic to this critique and feel it warrants full attention. On\n the other hand, i also have in mind the following analogy. The\n natural numbers, as a monoid under addition, has exactly $1$\n generator, while the natural numbers, as a monoid under\n multiplication, has countably many generators (the primes). We\n observe that the application of the lambda calculus is much less\n resource sensitive than the parallel composition of the\n $\\pi$-calculus. Could it be the case that we have an analogy of the\n form\n \n \\begin{mathpar}\n m + n : MN :: m*n : M|N\n \\end{mathpar}\n\n giving a similar blow up in the set of ``primes''? This is such a\n wonderful thought that, even if it's not true, i think it's worth\n writing down.\n\\end{remark}\n", "meta": {"hexsha": "a6b58eb12e8ca99cc437be5e22c588e30944b0af", "size": 20141, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "qm2pi/qm2pi.qmops.tex", "max_stars_repo_name": "leithaus/pi4u", "max_stars_repo_head_hexsha": "c87163938857589153eb5225d0e4ac17597fd189", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 13, "max_stars_repo_stars_event_min_datetime": "2015-10-12T20:35:01.000Z", "max_stars_repo_stars_event_max_datetime": "2020-06-16T00:37:17.000Z", "max_issues_repo_path": "qm2pi/qm2pi.qmops.tex", "max_issues_repo_name": "leithaus/pi4u", "max_issues_repo_head_hexsha": "c87163938857589153eb5225d0e4ac17597fd189", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": 6, "max_issues_repo_issues_event_min_datetime": "2018-07-06T19:01:06.000Z", "max_issues_repo_issues_event_max_datetime": "2019-08-19T22:39:58.000Z", "max_forks_repo_path": "qm2pi/qm2pi.qmops.tex", "max_forks_repo_name": "leithaus/pi4u", "max_forks_repo_head_hexsha": "c87163938857589153eb5225d0e4ac17597fd189", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 7, "max_forks_repo_forks_event_min_datetime": "2016-10-03T06:03:03.000Z", "max_forks_repo_forks_event_max_datetime": "2020-06-16T00:37:25.000Z", "avg_line_length": 42.4915611814, "max_line_length": 228, "alphanum_fraction": 0.7062707909, "num_tokens": 6053, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO\n\n", "lm_q1_score": 0.6992544210587585, "lm_q2_score": 0.4882833952958347, "lm_q1q2_score": 0.3414343228901938}} {"text": "\\subsubsection{\\stid{3.05} Enabling Time Integrators for Exascale Through SUNDIALS} \r\n\r\n\\paragraph{Overview} \r\n\r\nThis project is enhancing the SUNDIALS library of numerical software packages for integrating differential systems in time using state-of-the-art adaptive time step technologies for use on exascale systems. Through software infrastructure developments, this project is enabling the efficient and robust SUNDIALS time integrator packages to easily interoperate with external linear and nonlinear solver packages developed for exascale computing. In addition, this project is providing a many-vector capability so that SUNDIALS time integrators can more easily operate on data divided over heterogeneous architectures. Lastly, this project is supporting the deployment and use of SUNDIALS packages within ECP applications, mainly through incorporation into the discretization-based Co-Design Centers, AMReX and CEED.\r\n\r\nEfficient time integrators are essential for ECP because they are at the core of every time-dependent simulation application. However, many applications do not use state-of-the-art methods, and if they do, they often do not yet use them fully on their systems. For example, at the start of the ECP the astrophysics code, Nyx, used an adaptive integration package for solving individual reactions. However, by applying a time integration package to a larger reaction system, the code is able to vectorize more of the calculations and get an accurate solution much faster. By allowing for solvers tuned to exascale systems and vectors that are heterogeneous, SUNDIALS will be more applicable for use in multiphysics systems running on exascale platforms.\r\n\r\n\r\n\r\n\\paragraph{Key Challenges}\r\n\r\nCurrent implementations of efficient time integrators face challenges on many fronts. First, integrators typically have treated the full physical problem with a single step size or have relied on low order operator splitting methods to couple physical processes at different time scales. While research is moving forward within the time integration community on methods for multirate systems, the software infrastructure needs to be in place to accommodate these schemes once they are developed. Second, typical integrators operate on problem data in the form of vectors. These operations suffer from low arithmetic intensity, and their efficiency is often memory bandwidth limited. Lastly, implicit integrators, which are required in many exascale systems, require efficient linear and nonlinear solvers to be highly effective. In addition, by applying integrator-dependent controls on these solvers, their efficiency can be significantly increased. Applying these controls, however, often requires that information about the integrator and its progress be passed to the solver, and software must be designed to effectively pass that information while ensuring adequate encapsulation to provide ease of maintenance and software extension.\r\n\r\n\\paragraph{Solution Strategy}\r\n\r\nThis project includes a number of implementation activities that will prepare the SUNDIALS suite of time integrators for exascale systems. The main activity is a redesign of all linear solver interfaces and encapsulation of the nonlinear solvers within the time integrators in SUNDIALS. The new linear solver interfaces will make it much easier to interface external solver packages while maintaining the efficiency of SUNDIALS integrators. Encapsulating the nonlinear solvers, will reduce redundant code and allow the time integrators to better leverage common code thus lowering the code maintenance burden with SUNDIALS. In addition, the integrators will be able to take advantage of outside nonlinear solvers. \r\n\r\nThis project is also introducing a set of optional fused vector kernels into SUNDIALS. The goal of these kernels is to execute multiple vector operations at once thereby reducing the number of kernel launches in GPU environments and also reducing the number of communications required for reduction operations. These new kernels will be added to all supplied SUNDIALS vectors and will be invoked through optional interfaces.\r\n\r\nLastly, this project is developing a many-vector capability for SUNDIALS. Due to the tight data encapsulation within SUNDIALS, users are able to supply any vector they would like underneath the integrators. This project will supply the infrastructure needed to make it easy to place a vector of vectors underneath the integrators. This vector of vectors is essential for later implementation of time integrators that will advance various parts of the system with different time step sizes. This many-vector capability will also ease the use of different programming environments as differing vectors can be instantiated on different parts of a hybrid machine. \r\n\r\n\\paragraph{Recent Progress}\r\n\r\nIn September of 2017, SUNDIALS 3.0.0 including new linear solver and matrix interfaces, was released \\cite{SUNDIALSweb}. Figure \\ref{fig:sunorg1} shows the new organization of SUNDIALS where separate linear solver interfaces are provided for direct and iterative linear solver methods. These interfaces are shared across all SUNDIALS integrators. Individual integrators have the freedom to supply specific information from the integrator that controls the linear solver. In addition, a single interface to each external solver, such as SuperLU\\_MT or KLU, is shared across the suite, as opposed to the prior situation where each integrator included its own interface to each solver. In addition, a SUNMATRIX class was developed that can be instantiated with a dense, banded, or sparse matrix. Again, this class is shared by all integrators. \r\n\r\n\\begin{figure}[htb]\r\n\t\\centering\r\n\t\\includegraphics[width=6in]{projects/2.3.3-MathLibs/2.3.3.05-SUNDIALS/sunorg1.pdf}\r\n\t\\caption{\\label{fig:sunorg1}New structure of SUNDIALS showing options for the new SUNLINEARSOLVER and SUNMATRIX classes.}\r\n\\end{figure}\r\n\r\n\\paragraph{Next Steps}\r\n\r\nDuring the remainder of FY18, this project team will:\r\n\\begin{enumerate}\r\n\\item Complete a release of SUNDIALS with the new fused vector routines implemented within each supplied vector and optionally used from the integrator packages.\r\n\\item Complete a release of SUNDIALS with all nonlinear solvers encapsulated and with implementations of the new nonlinear solver interfaces for both the Newton and fixed point solvers used in the integrator packages.\r\n\\end{enumerate}\r\n", "meta": {"hexsha": "fb03fd88bd78e363cc018646ada09cc488ef7ed3", "size": 6472, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "projects/2.3.3-MathLibs/2.3.3.05-SUNDIALS/2.3.3.05-SUNDIALS.tex", "max_stars_repo_name": "tgamblin/ECP-ST-CAR-PUBLIC", "max_stars_repo_head_hexsha": "74d6fb18bae7ff1c32b78dd8cd7ae29e91218c33", "max_stars_repo_licenses": ["BSD-2-Clause"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "projects/2.3.3-MathLibs/2.3.3.05-SUNDIALS/2.3.3.05-SUNDIALS.tex", "max_issues_repo_name": "tgamblin/ECP-ST-CAR-PUBLIC", "max_issues_repo_head_hexsha": "74d6fb18bae7ff1c32b78dd8cd7ae29e91218c33", "max_issues_repo_licenses": ["BSD-2-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "projects/2.3.3-MathLibs/2.3.3.05-SUNDIALS/2.3.3.05-SUNDIALS.tex", "max_forks_repo_name": "tgamblin/ECP-ST-CAR-PUBLIC", "max_forks_repo_head_hexsha": "74d6fb18bae7ff1c32b78dd8cd7ae29e91218c33", "max_forks_repo_licenses": ["BSD-2-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 161.8, "max_line_length": 1247, "alphanum_fraction": 0.8204573548, "num_tokens": 1301, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723317123102955, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.34141802024232093}} {"text": "\\def\\Arccosh{{\\rm Arccosh}}\n\\advance\\sectioncount by -1\n\\section{Introduction}\n\nThis paper documents a computer-assisted procedure\nfor rigorously analyzing small hyperbolic $3$-manifolds.\nBriefly, we will define a compact six-dimensional space ${\\cal P}$\nthat parameterizes pairs of elements of ${\\rm Isom}({\\bf H}^3)$,\nand then construct a regular binary space partition (BSP) tree\nwhich subdivides ${\\cal P}$ into subregions ${\\cal P_i}$\nand whose leaves are --- with a few exceptions --- labeled with killerwords.\nThese killerwords will encode miniature proofs that ${\\cal P_i}$\ncannot contain any points which correspond to\nparticular choices of pairs of generators of\nany torsion-free discrete group of ${\\rm Isom}({\\bf H}^3)$.\nThis tree of mini-proofs will then be used\nto exhaustively isolate all possible manifolds\nwhich can have properties related to the dimensions of ${\\cal P}$.\n\nThe first two applications of the procedure\nwere used to prove a proposition from [GMT]:\n\\proclaim{Proposition [GMT]}\nLet $M$ be an orientable hyperbolic $3$-manifold, and let $\\delta$ be\na shortest geodesic. Then, either ${\\it tuberadius}(\\delta) > \\ln(3)/2$,\nor $M$ lies within one of seven tightly constrained\nexceptional shortest-geodesic-geometry regions.\n\\endproclaim\nand a related proposition; both of which were used in the proof of\nthe main technical result of [GMT]:\n\\proclaim{Theorem [GMT]}\nEvery closed hyperbolic $3$-manifold\nhas a non-coalescible insulator family,\nindeed one coming from a shortest geodesic.\nAs a consequence,\nhomotopy hyperbolic $3$-manifolds are hyperbolic.\n\\endproclaim\nMore recently, the proposition has been sharpened:\n\\proclaim{Theorem}\nLet $M$ be an orientable hyperbolic $3$-manifold,\nand let $\\delta$ be a shortest geodesic.\nThen, either ${\\it tuberadius}(\\delta) > \\ln(3)/2$, or\n$M$ is isometric to one of seven specific manifolds.\n\\endproclaim\n\\demo{Proof}\nLet $X_0, \\cdots, X_6$ denote the exceptional regions of Theorem FIXME(2.XX),\nwith $N_i$ the corresponding conjectural manifold.\n[GMT] showed that $N_0$=Vol3 is the unique manifold in the region\nand [JR] showed that Vol3 covered no 3-manifold.\n[JR] also proved that $N_5$ and $N_6$ are isometric.\n[CLLMR] and [L] identify a manifold in each region\nand [CLLMR] show that these manifolds are the unique ones in its region.\n[CLLMR] show that $N_1$, $N_5$ and $N_6$ cover no manifold.\nIn [GT] the proof is completed by showing that $N_3$ covers no manifold\nand each of $N_2$ and $N_4$ 2-fold cover manifolds,\nhowever the quotients are all non exceptional,\ni.e. each shortest geodesic has a $ln(3)/2$ tube.\n\\enddemo\n\nOther applications of the procedure:\n\nTheorem [G] (Smale conjecture of hyperbolic 3-manifolds)\nIf N is a closed hyperbolic 3-manifold,\nthen the natural inclusion ${\\rm Isom}(N)\\to {\\rm Diff}(N)$ is a homotopy equivalence.\n\nThe proof made essential use of the fact that a shortest geodesic\nof a closed hyperbolic 3-manifold satisfies the insulator condition [GMT].\n\nTheorem [GMM], [Mi] The Week's manifold\nis the unique closed orientable hyperbolic 3-manifold of minimal volume.\n\nThis result relies on \n\nLemma [ACS] Suppose that M is a closed, orientable hyperbolic 3-manifold,\nand that C is a shortest geodesic in M.\nSet $N = \\rm{drill}_C(M)$. If $\\rm{tuberad}(C) \\ge \\ln(3)/2$ then\n$\\rm{vol} N < 3:0177 \\rm{vol}M$.\n\nThis is based on a result of [ADST],\nthat makes essential use of Perelman's [Pe] monotonicty result.\na key element of his proof of geometrization and the $\\ln(3)/2$\ntheorem of [GMT].\nLemma [ACS] is used to prove other results such as \n\nTheorem [ACS] Suppose that M is a closed, orientable, hyperbolic\n3-manifold such that ${\\rm vol}(M) \\le 1.22$.\nThen $H_1(M:Zp)$ has dimension at most 3 for every prime p.\n\n(The statement is sharpened slightly to take into account [GT])\n\nTheorem [LM] Let M be a compact orientable 3-manifold with boundary a torus,\nand with interior admitting a complete finite-volume hyperbolic structure.\nThen the number of exceptional slopes on M is at most 10.\n\nThis result, known as the Gordon conjecture,\nrelies on rigorous computer assistance using the AffApprox technology.\n\nFinally, the procedure is currently in use\nby Gabai, Haraway, Meyerhoff, Thurston, and Yarmola\nto rigorously analyze small orientable cusped hyperbolic $3$-manifolds,\nand to rigorously analyze small non-orientable closed hyperbolic $3$-manifolds.\n\n\nThis paper is organized as follows.\n\nIn Section 1 we introduce [GMT; Proposition 2.8] as Theorem FIXME(2.19),\nand sketch its proof.\n\nIn Section 2 we provide a formal statement and proof of Theorem FIXME(2.19).\n\nIn Section 3,\nthe method for describing the decomposition\nof the parameter space ${\\cal W}$ into sub-regions is given,\nand the conditions used to eliminate sub-regions are discussed.\nNear the end of this section, the first part of a detailed example is given.\n\nEliminating \na sub-region requires that a certain function is shown to be bounded \nappropriately over the entire sub-region. This is carried out by using a \nfirst-order Taylor approximation of the function together with a \nremainder \nbound.\nOur computer version of such a Taylor approximation with remainder bound \nis called an {\\it AffApprox} and in Section 4,\nthe relevant theory is developed.\nAt this point, the detailed example of Section 3 can be completed.\n\nIn Sections 5 and 6, round-off error analysis appropriate to our \nset-up is introduced. Specifically, in Section 6, round-off error is \nincorporated \ninto the {\\it AffApprox} formulas introduced in Section 4. The proofs here \nrequire an analysis of round-off error for complex numbers, which is carried \nout in Section 5.\n\nIn Section 7 we give some hints about the search for a atree,\nin the hope that they will help others endeavoring to apply these methods.\n\nFinally, in Section 8, we present an updated version of\nthe code used to check that the proof is valid,\nwith self-contained copies of the proofs required for the reader\nto check its own validity.\n\n{\\it Prior publication}\nSections 2 through 6 originally appeared in [GMT],\nand are reproduced here with only minor revision.\nSection 1 is an abridged version of the parts of the rest of [GMT].\nSection 7 borrows from material which originally appeared in remarks in [GMT].\n\n{\\it Acknowledgements}.\nFIXME\n", "meta": {"hexsha": "56b277acb017484f685d8eeaae088d7dd91bdc7b", "size": 6270, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "thesis/TeX_and_Figures_Files/Chapter_0.tex", "max_stars_repo_name": "njt99/findingkillerwords", "max_stars_repo_head_hexsha": "71271dca14a9986d631608929544bcd6d68813f0", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "thesis/TeX_and_Figures_Files/Chapter_0.tex", "max_issues_repo_name": "njt99/findingkillerwords", "max_issues_repo_head_hexsha": "71271dca14a9986d631608929544bcd6d68813f0", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "thesis/TeX_and_Figures_Files/Chapter_0.tex", "max_forks_repo_name": "njt99/findingkillerwords", "max_forks_repo_head_hexsha": "71271dca14a9986d631608929544bcd6d68813f0", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 41.8, "max_line_length": 86, "alphanum_fraction": 0.7685805423, "num_tokens": 1686, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.672331699179286, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.3414180135742385}} {"text": "% \\documentclass[a4paper, 10px]{ctexart}\n\\documentclass{ctexart}\n\\usepackage[left=1in, right=1in, top=1.2in, bottom=1.2in]{geometry}\n\\usepackage{ctex}\n\\usepackage[utf8]{inputenc}\n\\usepackage{boxproof}\n\\usepackage{fontspec}\n\\usepackage{a4wide}\n% \\setmainfont[Scale = 1]{SF Display}\n% \\setCJKmainfont{Songti SC}\n\\usepackage{fancyhdr}\n\n% Automata\n\\usepackage{tikz}\n\\usetikzlibrary{automata, positioning, arrows}\n\n\\pagestyle{fancy}\n\\fancypagestyle{plain}{\n \\fancyhead[L]{East China Normal University}\n \\fancyhead[R]{}\n \\fancyfoot[C]{\\thepage}\n}\n\n\\tikzset{\n->, % makes the edges directed\n>=stealth, % makes the arrow heads bold\nnode distance=3cm, % specifies the minimum distance between two nodes. Change if necessary.\nevery state/.style={thick, fill=gray!10}, % sets the properties for each ’state’ node\ninitial text=$start$, % sets the text that appears on the start arrow\n}\n\n\\def\\premise{\\mathrm{premise}}\n\\def\\assumption{\\mathrm{assumption}}\n\\def\\MT{\\mathrm{MT\\ }}\n\\def\\LEM{\\mathrm{LEM}}\n\\def\\intro{\\mathrm{i\\ }}\n\\def\\elim{\\mathrm{e\\ }}\n\\def\\introa{\\mathrm{i_1\\ }}\n\\def\\elima{\\mathrm{e_1\\ }}\n\\def\\introb{\\mathrm{i_2\\ }}\n\\def\\elimb{\\mathrm{e_2\\ }}\n\n\\title{Computation Theory Assignment 4}\n\\author{10185101210 陈俊潼}\n\\date{September 2020}\n\n\\begin{document}\n\n\\maketitle\n\n\\section{Automata}\n\n\\subsection{This is an automata.}\n\nTo draw an automata, you need to first import \\texttt{tikz} package.\n\nFigure \\ref{aut1} is an example:\n\n\n\\begin{figure}[ht]\n \\centering\n \\begin{tikzpicture}[scale=2]\n \\node[state, initial] (0) {$q_0$};\n \\node[state, above right of=0] (1) {$q_1$};\n \\node[state, right of=1] (2) {$q_2$};\n \\node[state, above right of=2, accepting] (3) {$q_3$};\n \\node[state, below right of=2, accepting] (4) {$q_4$};\n \\node[state, below right of=0] (5) {$q_5$};\n \\node[state, right of=5] (6) {$q_6$};\n \\node[state, right of=6] (7) {$q_7$};\n \\node[state, right of=7, accepting] (8) {$q_8$};\n \\draw (0) edge[above] node{$a$} (1)\n (1) edge[above] node{$b$} (2)\n (2) edge[above] node{$c$} (3)\n (2) edge[above] node{$d$} (4)\n (0) edge[above] node{$c$} (5)\n (5) edge[above] node{$c$} (6)\n (6) edge[above] node{$c$} (7)\n (7) edge[above] node{$c$} (8);\n \\end{tikzpicture}\n \\caption{An automata example}\n \\label{aut1}\n\\end{figure}\n\\end{document}\n", "meta": {"hexsha": "5672a700775dedd251ac502e4f96fb16184a1022", "size": 2438, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "Course/ComputationTheory/Assignment1/Assignment1.tex", "max_stars_repo_name": "AixMoon/LearnigRepo", "max_stars_repo_head_hexsha": "ee98fb352735e2b4f97304847b6c0311bc30195e", "max_stars_repo_licenses": ["MIT"], "max_stars_count": 11, "max_stars_repo_stars_event_min_datetime": "2020-05-02T20:06:07.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-24T10:01:29.000Z", "max_issues_repo_path": "Course/ComputationTheory/Assignment1/Assignment1.tex", "max_issues_repo_name": "AixMoon/LearnigRepo", "max_issues_repo_head_hexsha": "ee98fb352735e2b4f97304847b6c0311bc30195e", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "Course/ComputationTheory/Assignment1/Assignment1.tex", "max_forks_repo_name": "AixMoon/LearnigRepo", "max_forks_repo_head_hexsha": "ee98fb352735e2b4f97304847b6c0311bc30195e", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2020-06-04T04:29:28.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-15T08:15:01.000Z", "avg_line_length": 28.6823529412, "max_line_length": 91, "alphanum_fraction": 0.6255127153, "num_tokens": 853, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5078118642792044, "lm_q2_score": 0.6723316926137812, "lm_q1q2_score": 0.34141801024019724}} {"text": "\\documentclass{article}\n\\usepackage{a4wide}\n\\usepackage{amsmath}\n\\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue]{hyperref}\n\\usepackage[sort&compress,comma,authoryear]{natbib}\n\\newcommand{\\Prob}[0]{\\mbox{Prob}}\n\\newcommand{\\ul}[1]{\\underline{#1}}\n\\newcommand{\\ol}[1]{\\overline{#1}}\n\\usepackage{bm}\n\\usepackage{latexsym}\n\n\n\\title{DEB model description: 'hex'}\n%\\author{S.A.L.M. Kooijman and K. Lika and S. Augustine and N. Marn and others?}\n\n\n\\begin{document}\n\n\\maketitle\n\nThis document specifies the standard DEB model 'hex'.\nThe {\\sc deb} model for holometabolic insects (and some other hexapods). Its characteristics are\n\\begin{description}\n \\item[$\\circ$] morphological life stages: egg, larva, (pupa), imago; functional stages: embryo, adult, (pupa), imago\n\t\n \\item[$\\circ$] the embryo still behaves like the std-model\n\t\n \\item[$\\circ$] the larval stage accelerates (V1-morph) and behaves as adult, i.e. no maturation, allocation to reproduction and $E_H^b = E_H^p$.\n\t\n \\item[$\\circ$] pupation occurs when reproduction buffer density hits a threshold, $E_R/ L^3 = [E_R^j]$\n\t\n \\item[$\\circ$] pupa behaves like an isomorphic embryo of the std-model, emergence occurs at $E_H = E_H^e$\n\t Larval structure rapidly transforms to pupal reserve just after start of pupation, and sets $E_H = 0$ at $j$.\n\t\t\n \\item[$\\circ$] the reproduction buffer remains unchanged during the pupal stage\n\t\n \\item[$\\circ$] the imago does not grow or allocate to reproduction. \n\t Imago's reserve mobilisation matched somatic plus maturity maintenance $\\dot{p}_C = \\dot{p}_M + \\dot{p}_J$.\n\\end{description}\nHemi-metabolic insects skip the pupal stage, don't convert larval structure to reserve. \nImago structure equals larval structure when reproduction buffer density hits a threshold.\nThe model is discussed in the comments for Section 7.8.\n\nFor $\\dot{k}_E = \\dot{v}/ L_b$, reserve mobilisation prior to pupation (i.e. during acceleration) is $\\dot{p}_C = E (\\dot{k}_E - \\dot{r})$ \n with $\\dot{r} = \\frac{\\kappa [E] \\dot{k}_E - [\\dot{p}_M]} {\\kappa [E] + [E_G]} = g \\dot{k}_M \\frac{e/ l_b - 1} {e + g}$.\nThe larva allocates to reproduction as $\\dot{p}_R = (1- \\kappa) \\dot{p}_C - \\dot{p}_J$, with $\\dot{p}_J = \\dot{k}_J E_H^p$.\n$[E_R]$ has a maximum at $[E_R^m] = [E_R^{\\mbox{\\tiny ref}}] f \\frac{1 - l_b} {f - l_b}$ with$[E_R^{\\mbox{\\tiny ref}}] = (1 - \\kappa) [E_m] \\frac{g + l_b} {1 - l_b}$, so pupation occurs when $[E_R] = s_j [E_R^{\\mbox{\\tiny ref}}]$, with $s_j = [E_R^j]/ [E_R^{\\mbox{\\tiny ref}}]$.\n\nReserve mobilisation of the imago is $\\dot{p}_C = \\dot{p}_M^e + \\dot{p}_J^e$, where $\\dot{p}_M^e = [\\dot{p}_M] L_e^3$ and $\\dot{p}_J^e = \\dot{k}_J E_H^e$.\n\n\n\\section{Background}\n\n\\input{model_intro}\n\n\\input{std_description}\n\n\\bibliographystyle{apalike}\n\\bibliography{debmodels}\n\n\\end{document}", "meta": {"hexsha": "942c7303379e3be8cbed8e5f5f0bd1dbde36b58f", "size": 2798, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "docs/models/hex.tex", "max_stars_repo_name": "add-my-pet/AmPtool-new", "max_stars_repo_head_hexsha": "f60cdd10c65c0190ca41ba6fbd4d5018c5aea771", "max_stars_repo_licenses": ["0BSD"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "docs/models/hex.tex", "max_issues_repo_name": "add-my-pet/AmPtool-new", "max_issues_repo_head_hexsha": "f60cdd10c65c0190ca41ba6fbd4d5018c5aea771", "max_issues_repo_licenses": ["0BSD"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "docs/models/hex.tex", "max_forks_repo_name": "add-my-pet/AmPtool-new", "max_forks_repo_head_hexsha": "f60cdd10c65c0190ca41ba6fbd4d5018c5aea771", "max_forks_repo_licenses": ["0BSD"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 45.868852459, "max_line_length": 278, "alphanum_fraction": 0.6940671909, "num_tokens": 961, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6723316860482763, "lm_q2_score": 0.5078118642792044, "lm_q1q2_score": 0.341418006906156}} {"text": "\\documentclass[10pt]{beamer} % aspect ratio 4:3, 128 mm by 96 mm\n%\\documentclass[10pt,aspectratio=169]{beamer} % aspect ratio 16:9\n\\graphicspath{{../journal_papers/Elsevier/figs/}}\n%\\includeonlyframes{frame1,frame2,frame3}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Packages\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\usepackage{appendixnumberbeamer}\n\\usepackage{booktabs}\n\\usepackage[scale=2]{ccicons}\n\\usepackage{pgfplots}\n\\usepackage{xspace}\n\\usepackage{amsmath}\n\\usepackage{totcount}\n\\usepackage{tikz}\n%\\usepackage{comment}\n%\\usetikzlibrary{external} % speedup compilation\n%\\tikzexternalize % activate!\n%\\usetikzlibrary{shapes,arrows} \n\n%\\usepackage{bibentry}\n\n\\usepackage{caption}%\n\\captionsetup[figure]{labelformat=empty}%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Metropolis theme custom modification file\n\\input{metropolis_mods.tex}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Custom commands\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% matrix command \n\\newcommand{\\matr}[1]{\\mathbf{#1}} % bold upright (Elsevier, Springer)\n%\\newcommand{\\matr}[1]{#1} % pure math version\n%\\newcommand{\\matr}[1]{\\bm{#1}} % ISO complying version\n% vector command \n\\newcommand{\\vect}[1]{\\mathbf{#1}} % bold upright (Elsevier, Springer)\n\\newcommand{\\ud}{\\mathrm{d}}\n\\renewcommand{\\vec}[1]{\\mathbf{#1}}\n\\newcommand{\\veca}[2]{\\mathbf{#1}{#2}}\n\\newcommand{\\bm}[1]{\\mathbf{#1}}\n\\newcommand{\\bs}[1]{\\boldsymbol{#1}}\n% derivative upright command\n\\DeclareRobustCommand*{\\drv}{\\mathop{}\\!\\mathrm{d}}\n% \n\\newcommand{\\themename}{\\textbf{\\textsc{metropolis}}\\xspace}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Title page options\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% \\date{\\today}\n\\date{15.08.2019}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% option 1\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\title{Parallel spectral element method for guided wave based structural health monitoring}\n%\\subtitle{}\n%\\author{\\textbf{Paweł Kudela}\\\\Jochen Moll \\\\Piotr Fiborek}\n% logo align to Institute \n%\\institute{Institute of Fluid Flow Machinery\\\\Polish Academy of Sciences \\\\ \\vspace{-1.5cm}\\flushright \\includegraphics[width=4cm]{../images/logo/logo_eng_40mm.eps}}\n%\\institute{Institute of Fluid Flow Machinery\\\\Polish Academy of Sciences \\\\ \\vspace{-1.5cm}}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% option 2 - authors in one line\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\t\\title{Parallel spectral element method for guided wave based structural health monitoring}\n%\t\\subtitle{Lamb-opt}\n\t\\author{\\textbf{Paweł Kudela}\\textsuperscript{1}, Jochen Moll\\textsuperscript{2}, Piotr Fiborek\\textsuperscript{1} }\n%\t% logo align to Institute \n%\t\\institute{\\textsuperscript{1}Xi'an Jiaotong University \\\\ \\textsuperscript{2}Institute of Fluid Flow Machinery\\\\ \\hspace*{1pt} Polish Academy of Sciences \\\\ \\vspace{-1.5cm}\\flushright \\includegraphics[width=4cm]{../images/logo/logo_eng_40mm.eps}}\n\t\\institute{ \\textsuperscript{1}Institute of Fluid Flow Machinery\\\\ \\hspace*{1pt} Polish Academy of Sciences \\\\ \\\\ \\textsuperscript{2}J.W. Goethe University\\\\ \\hspace*{1pt} Department of Physics \\vspace{-1.5cm}}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% option 3 - multilogo vertical\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\title{Elastic constants identification of composite laminates by using Lamb wave dispersion curves and optimization methods}\n%\\subtitle{Lamb-opt}\n%\t\\author{\\textbf{Paweł Kudela}\\inst{1}, Maciej Radzieński\\inst{1}, Wiesław Ostachowicz\\inst{1}, Zhibo Yang\\inst{2} }\n%\t% logo under Institute \n%\t\\institute%\n%\t{ \n%\t\t\\inst{1}%\n%\t\tInstitute of Fluid Flow Machinery\\\\ \\hspace*{1pt} Polish Academy of Sciences \\\\ \\includegraphics[height=0.85cm]{../images/logo/logo_eng_40mm.eps} \\\\\n%\t\t\\and\n%\t\t\\inst{2}%\n%\t Xi'an Jiaotong University \\\\ \\includegraphics[height=0.85cm]{../images/logo/logo_box.eps}\n% }\n% end od option 3\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%% option 4 - 3 Institutes and logos horizontal centered\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\title{Elastic constants identification of composite laminates by using Lamb wave dispersion curves and optimization methods}\n%\\subtitle{Lamb-opt }\n%\\author{\\textbf{Paweł Kudela}\\textsuperscript{1}, Maciej Radzieński\\textsuperscript{1}, Marco Miniaci\\textsuperscript{2}, Zhibo Yang\\textsuperscript{3} }\n%\n%\\institute{ \n%\\begin{columns}[T,onlytextwidth]\n%\t\\column{0.39\\textwidth}\n%\t\\begin{center}\n%\t\t\\textsuperscript{1}Institute of Fluid Flow Machinery\\\\ \\hspace*{3pt}Polish Academy of Sciences\n%\t\\end{center}\n%\t\\column{0.3\\textwidth}\n%\t\\begin{center}\n%\t\t\\textsuperscript{2}Zurich University\n%\t\\end{center}\n%\t\\column{0.3\\textwidth}\n%\t\\begin{center}\n%\t\t\\textsuperscript{3}Xi'an Jiaotong University\n%\t\\end{center}\n%\\end{columns}\n%\\vspace{6pt}\n%% logos \n%\\begin{columns}[b,onlytextwidth]\n%\t\\column{0.39\\textwidth}\n%\t\t\\centering \n%\t\t\\includegraphics[width=\\textwidth,height=0.85cm,keepaspectratio]{../images/logo/logo_eng_40mm.eps}\n%\t\\column{0.3\\textwidth}\n%\t\t\\centering \n%\t\t\\includegraphics[width=\\textwidth,height=0.85cm,keepaspectratio]{../images/logo/logo_box.eps}\n%\t\\column{0.3\\textwidth}\n%\t\t\\centering \n%\t\t\\includegraphics[width=\\textwidth,height=0.85cm,keepaspectratio]{../images/logo/logo_box2.eps}\n%\\end{columns}\n%}\n%\\makeatletter\n%\\setbeamertemplate{title page}{\n%\t\\begin{minipage}[b][\\paperheight]{\\textwidth}\n%\t\t\\centering % <-- Center here\n%\t\t\\ifx\\inserttitlegraphic\\@empty\\else\\usebeamertemplate*{title graphic}\\fi\n%\t\t\\vfill%\n%\t\t\\ifx\\inserttitle\\@empty\\else\\usebeamertemplate*{title}\\fi\n%\t\t\\ifx\\insertsubtitle\\@empty\\else\\usebeamertemplate*{subtitle}\\fi\n%\t\t\\usebeamertemplate*{title separator}\n%\t\t\\ifx\\beamer@shortauthor\\@empty\\else\\usebeamertemplate*{author}\\fi\n%\t\t\\ifx\\insertdate\\@empty\\else\\usebeamertemplate*{date}\\fi\n%\t\t\\ifx\\insertinstitute\\@empty\\else\\usebeamertemplate*{institute}\\fi\n%\t\t\\vfill\n%\t\t\\vspace*{1mm}\n%\t\\end{minipage}\n%}\n%\n%\\setbeamertemplate{title}{\n%\t% \\raggedright% % <-- Comment here\n%\t\\linespread{1.0}%\n%\t\\inserttitle%\n%\t\\par%\n%\t\\vspace*{0.5em}\n%}\n%\\setbeamertemplate{subtitle}{\n%\t% \\raggedright% % <-- Comment here\n%\t\\insertsubtitle%\n%\t\\par%\n%\t\\vspace*{0.5em}\n%}\n%\\makeatother\n% end of option 4\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% option 5 - 2 Institutes and logos horizontal centered\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\title{Elastic constants identification of composite laminates by using Lamb wave dispersion curves and optimization methods}\n%\\subtitle{Lamb-opt }\n%\\author{\\textbf{Paweł Kudela}\\textsuperscript{1}, Maciej Radzieński\\textsuperscript{1}, Marco Miniaci\\textsuperscript{2}}\n%\n%\\institute{ \n%\t\\begin{columns}[T,onlytextwidth]\n%\t\t\\column{0.5\\textwidth}\n%\t\t\t\\centering\n%\t\t\t\\textsuperscript{1}Institute of Fluid Flow Machinery\\\\ \\hspace*{3pt}Polish Academy of Sciences\n%\t\t\\column{0.5\\textwidth}\n%\t\t\t\\centering\n%\t\t\t\\textsuperscript{2}Zurich University\n%\t\\end{columns}\n%\t\\vspace{6pt}\n%\t% logos \n%\t\\begin{columns}[b,onlytextwidth]\n%\t\t\\column{0.5\\textwidth}\n%\t\t\\centering \n%\t\t\\includegraphics[width=\\textwidth,height=0.85cm,keepaspectratio]{../images/logo/logo_eng_40mm.eps}\n%\t\t\\column{0.5\\textwidth}\n%\t\t\\centering \n%\t\t\\includegraphics[width=\\textwidth,height=0.85cm,keepaspectratio]{../images/logo/logo_box.eps}\n%\t\\end{columns}\n%}\n%\\makeatletter\n%\\setbeamertemplate{title page}{\n%\t\\begin{minipage}[b][\\paperheight]{\\textwidth}\n%\t\t\\centering % <-- Center here\n%\t\t\\ifx\\inserttitlegraphic\\@empty\\else\\usebeamertemplate*{title graphic}\\fi\n%\t\t\\vfill%\n%\t\t\\ifx\\inserttitle\\@empty\\else\\usebeamertemplate*{title}\\fi\n%\t\t\\ifx\\insertsubtitle\\@empty\\else\\usebeamertemplate*{subtitle}\\fi\n%\t\t\\usebeamertemplate*{title separator}\n%\t\t\\ifx\\beamer@shortauthor\\@empty\\else\\usebeamertemplate*{author}\\fi\n%\t\t\\ifx\\insertdate\\@empty\\else\\usebeamertemplate*{date}\\fi\n%\t\t\\ifx\\insertinstitute\\@empty\\else\\usebeamertemplate*{institute}\\fi\n%\t\t\\vfill\n%\t\t\\vspace*{1mm}\n%\t\\end{minipage}\n%}\n%\n%\\setbeamertemplate{title}{\n%\t% \\raggedright% % <-- Comment here\n%\t\\linespread{1.0}%\n%\t\\inserttitle%\n%\t\\par%\n%\t\\vspace*{0.5em}\n%}\n%\\setbeamertemplate{subtitle}{\n%\t% \\raggedright% % <-- Comment here\n%\t\\insertsubtitle%\n%\t\\par%\n%\t\\vspace*{0.5em}\n%}\n%\\makeatother\n% end of option 5\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% End of title page options\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% logo option - alternative manual insertion by modification of coordinates in \\put()\n%\\titlegraphic{%\n%\t%\\vspace{\\logoadheight}\n%\t\\begin{picture}(0,0)\n%\t\\put(305,-185){\\makebox(0,0)[rb]{\\includegraphics[width=4cm]{../images/logo/logo_eng_40mm.eps}}}\n%\t\\end{picture}}\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%\\tikzexternalize % activate!\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{document}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\maketitle\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% SLIDES\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Table of contents}\n \\setbeamertemplate{section in toc}[sections numbered]\n \\tableofcontents[hideallsubsections]\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Introduction}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[fragile,label=frame1]{Funding}\n\t\n\tNAWA – Polish National Agency for Academic Exchange\n\t\n\tMobility project to carry out research at Johann Wolfgang Goethe University, Frankfurt am Main, Germany, 2019 (3 months)\n\t\n\t\\emph{Towards model assisted structural health monitoring}\n\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[fragile,label=frame2]{Context}\n\t\n\tNational Science Centre, Poland\n\t\n\t2019--2021 (3 years)\n\t\n \t\\emph{Feasibility studies of artificial intelligence-driven diagnostics}\n \n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[label=frame3]{Small project within big project}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\only<1>{\n\t\t\\includegraphics[width=\\textwidth]{beamer_figs/Plan-scheme-nn-small-eng.png}\t\n\t\t\\label{fig:plan_scheme}\n\t\t}\n\t\t\\only<2>{\n\t\t\\includegraphics[width=\\textwidth]{beamer_figs/Plan-scheme-nn-small-eng2.png}\t\n\t\t\\label{fig:plan_scheme2}\n\t\t}\n\t\\end{figure}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Motivation}\n\t\\biblioref{C.A.C Leckey, K.R. Wheeler,V.N. Hafiychuk, H. Hafiychuk, D.A. Timucin}{2018}{Simulation of guided-wave ultrasound propagation in composite laminates: Benchmark comparisons of numerical codes and experiment}{Ultrasonics}\n\t\\begin{figure}\n\t\t\\includegraphics[width=\\textwidth]{beamer_figs/Leckey-2018-table.png}\n\t\t\\caption{Performance metrics of commercial software for wave propagation modelling}\n\t\\label{fig:Leckey_table}\n\t\\end{figure}\n\t\\begin{alertblock}{Potential Problems}\n\t\tShortest run time \\textbf{19.5 h} is for COMSOL but requires huge amount of~memory space\\\\\n\t\t19.5 h $\\times$ 1000 simulations $\\approx$ \\textbf{812 days}\n\t\\end{alertblock}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Numerical modelling}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Flat shell spectral element (theoretical background)}\n\t\\def\\myindenta{0.0\\textwidth} % define myindenta variable for correcting caption placement\n\t\\only<1->{\n\t\\begin{columns}[T]\t\n\t\t\\column{0.5\\textwidth}\n\t\t\\begin{figure}\n\t\t\t\\includegraphics[width=\\textwidth]{shell.png}\n\t\t\t\\caption{\\hspace{\\myindenta}36--node spectral shell element.}\n\t\t\t\\label{fig:spectral_shell_element}\n\t\t\\end{figure}\n\t\t\\column{0.5\\textwidth}\n\t\tThe displacement field is based on Mindlin--Reissner theory:\n\t\t\\begin{equation*}\n\t\t\\begin{split}\n\t\t& u(x,y,z)=u_0(x,y) - \\varphi_x(x,y) \\cdot z\\\\\n\t\t& v(x,y,z)=v_0(x,y) - \\varphi_y(x,y) \\cdot z\\\\\n\t\t& w(x,y,z)=w_0(x,y) \\label{eq:delam_platedispl}\n\t\t\\end{split}\n\t\t\\end{equation*}\n\t\\end{columns}\n\t}\n\t\\only<2>{\n\t\t\\begin{alertblock}{Important properties}\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item Non--uniform distribution of nodes which coincide with Gauss--Lobatto--Legendre integration points\n\t\t\t\t\\item Diagonality of mass matrix\n\t\t\t\t\\item Fast convergence (spectral, exponential)\n\t\t\t\\end{itemize}\n\t\t\\end{alertblock}\n\t}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Approximation}\n\t\\begin{equation*}\n\t\\left[\\begin{array}{l} u_0^e(\\xi, \\eta) \\\\ \\varphi_x^e(\\xi, \\eta)\\\\ v_0^e(\\xi, \\eta) \\\\ \\varphi_y^e(\\xi, \\eta)\\\\ w_0^e(\\xi, \\eta)\\\\ \\end{array}\\right] = \\bm{N}^e \\vec{\\hat{u}}^e = \\sum \\limits_{j=1}^{6} \\sum \\limits_{i=1}^{6} N^e_i(\\xi) N^e_i(\\eta)\\, \\bm{I} \\left[ \\begin{array}{l} {\\hat{u}_0}^e(\\xi_i,\\eta_j)\\\\\\hat{\\varphi}_x^e(\\xi_i,\\eta_j)\\\\{\\hat{v}_0}^e(\\xi_i,\\eta_j) \\\\\\hat{\\varphi}_y^e(\\xi_i,\\eta_j) \\\\ \\hat{w}_0^e(\\xi_i,\\eta_j)\\end{array} \\right]\\label{eq:delam_plateaproxim}\n\t\\end{equation*} \n\twhere $\\bm{N}^e$ are shape functions, $\\vec{\\hat{u}}^e$ are nodal degrees of freedom in the element, $\\bm{I}$ is the unit matrix of the size 5x5.\t\n\t\\begin{equation*}\n\t\\left[\\begin{array}{l} x_0^e(\\xi, \\eta) \\\\ y_0^e(\\xi, \\eta) \\end{array}\\right] = \\sum \\limits_{j=1}^{6} \\sum \\limits_{i=1}^{6} N^e_i(\\xi) N^e_j(\\eta)\\, \\left[ \\begin{array}{l} x^e(\\xi_i,\\eta_j)\\\\y^e(\\xi_i,\\eta_j)\\end{array} \\right]\\label{eq:delam_plategeom}\n\t\\end{equation*} \n\t\\begin{alertblock}{Shape functions are orthogonal Lagrange polynomials}\n\t\\end{alertblock}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Equation of motion}\n\t\\only<1->{\n\t\\begin{equation*}\n\t\\bm{M} \\vec{\\ddot{U}} + \\bm{C} \\vec{\\dot{U}} + \\bm{K} \\vec{U} = \\vec{F} \\label{eq:motion}\n\t\\end{equation*} \n\t}\n\t\\vspace{0.4cm}\n\t\\only<2->{\n\t\\begin{equation*}\n\t\\ddot{\\vec{U}}\\simeq \\frac{1}{\\Delta t^2} \\left(\\vec{u}_{t+\\Delta t} - 2\\,\\vec{u}_t + \\vec{u}_{t-\\Delta t}\\right) \\label{eq:central_scheme}\n\t\\end{equation*}\n\t\\begin{equation*}\n\t\\dot{\\vec{U}}\\simeq \\frac{\\vec{u}_{t+\\Delta t} -\\vec{u}_{t-\\Delta t}}{2 \\Delta t}\n\t\\label{eq:first_derivative_scheme}\n\t\\end{equation*}\n\t}\n\t\\vspace{0.4cm}\n\t\\only<3>{\n\t\\begin{equation*}\n\t\\begin{split}\n\t\\underbrace{\\left(\\frac{1}{\\Delta t^2} \\,\\bm{M} + \\frac{1}{2 \\Delta t} \\bm{C}\\right)}_{\\vec{M}_0} \\vec{u}_{t+\\Delta t} &= \\vec{F}_t - \\underbrace{\\left(\\bm{K} \\vec{u}_t\\right)}_{\\vec{F}^i} + \\underbrace{\\left(\\frac{2}{\\Delta t^2} \\,\\bm{M} \\right)}_{\\vec{M}_1}\\vec{u}_t \\\\\n\t&+ \\underbrace{\\left(- \\frac{1}{\\Delta t^2} \\,\\bm{M} + \\frac{1}{2 \\Delta t} \\bm{C}\\right)}_{\\vec{M}_2} \\vec{u}_{t-\\Delta t}.\n\t\\label{eq:explicit_integration}\n\t\\end{split}\n\t\\end{equation*}\n}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Vectorization of the code}\n\t\\vspace{-0.4cm}\n\t\\begin{equation*}\n\t\t\\underbrace{\\left(\\bm{K} \\vec{u}_t\\right)}_{\\vec{F}^i} \n\t\\end{equation*}\n\t\\vspace{0.4cm}\n\t\\begin{equation*}\n\t\\begin{split}\n\t\\vec{F}_u^i&=\\bm{N},_{\\xi}^T \\left(\\bs{\\sigma}_{xx}\\,.*(\\vec{J}^{-1})_{11}\\,.*\\vec{W}\\right)+\\bm{N},_{\\eta}^T \\left(\\bs{\\sigma}_{xx}\\,.*(\\vec{J}^{-1})_{21}\\,.*\\vec{W}\\right)\\\\\n\t&+\\bm{N},_{\\xi}^T \\left(\\bs{\\sigma}_{xy}\\,.*(\\vec{J}^{-1})_{12}\\,.*\\vec{W}\\right)+\\bm{N},_{\\eta}^T \\left(\\bs{\\sigma}_{xy}\\,.*(\\vec{J}^{-1})_{22}\\,.*\\vec{W}\\right), \n\t\\label{eq:internal_forces_u}\n\t\\end{split}\n\t\\end{equation*}\n\t\\vspace{0.4cm}\n\t\\begin{equation*}\n\t\\begin{split}\n\t\\bs{\\sigma}_{xx}&=\\left((\\bm{N},_{\\xi}\\vec{U}_x).*(\\vec{J}^{-1})_{11}+(\\bm{N},_{\\eta}\\vec{U}_x).*(\\vec{J}^{-1})_{21}\\right).*\\vec{A}_{11}\\\\\n\t&+\\left((\\bm{N},_{\\xi}\\bs{\\Phi}_x).*(\\vec{J}^{-1})_{11}+(\\bm{N},_{\\eta}\\bs{\\Phi}_x).*(\\vec{J}^{-1})_{21}\\right).*\\vec{B}_{11}\\\\\n\t&+\\left((\\bm{N},_{\\xi}\\vec{U}_y).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\vec{U}_y).*(\\vec{J}^{-1})_{22}\\right).*\\vec{A}_{12}\\\\\n\t&+\\left((\\bm{N},_{\\xi}\\bs{\\Phi}_y).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\bs{\\Phi}_y).*(\\vec{J}^{-1})_{22}\\right).*\\vec{B}_{12}\\\\\n\t&+\\left((\\bm{N},_{\\xi}\\vec{U}_x).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\vec{U}_x).*(\\vec{J}^{-1})_{22}\\right).*\\vec{A}_{16}\\ldots,\n\t\\end{split}\n\t\\end{equation*}\n\t\n\twhere $.*$ denotes element-wise operation known as Hadamarad product (the same symbol for element-wise operation is used in Matlab).\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Mesh colouring}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\t\\includegraphics[width=0.8\\textwidth]{beamer_figs/mesh_colouring_scheme4.jpg}\t\n\t\t\t\\label{fig:mesh_colouring}\n\t\\end{figure}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Delamination modelling}\n\t\\begin{figure} [h!]\n\t\t\\centering\n\t\t\\includegraphics[width=0.9\\textwidth]{delam_modelling_shell.png}\t\n\t\t\\caption{The concept of delamination modelling: cross-section through the composite laminate and delamination showing three regions: (I) undelaminated, (II) above delamination interface, and (III) below delamination interface and corresponding elements.}\n\t\t\\label{fig:delam_modelling_shell}\n\t\\end{figure}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Computation speedup analysis}\n\t\\begin{table}\n\t\t\\renewcommand{\\arraystretch}{1.3}\n\t\t\\centering \\small\n\t\tComputation run times depending on the number of degrees of freedom (NDOF) for the wave propagation duration 400~$\\mu$s; GPU: Nvidia Tesla K20X; CPU: Intel Xeon X5660 2.8~GHz.\n\t\t\n\t\t\\begin{tabular}{ccccccc} \n\t\t\t%\\hline\n\t\t\t\\toprule\t\n\t\t\tplate size [cm] & 30$\\times$30 & 40$\\times$40 & 50$\\times$50 & 70$\\times$70 & 90$\\times$90 & 100$\\times$100 \\\\\n\t\t\tNDOF $\\cdot 10^6$ & 1.02 & 1.46 & 1.98 & 3.09 & 5.23 & 6.36 \\\\\n\t\t\t\\midrule\n\t\t\tCPU time [h]& 4.03& 5.71 & 8.42 & 13.13 & 22.37 & 27.33\\\\\n\t\t\t\\midrule\n\t\t\tGPU time [h]& 1.00& 1.19 & 1.26 & 1.50 & 2.00 & 2.28\\\\\n\t\t\t%\\hline \n\t\t\t\\bottomrule \n\t\t\\end{tabular} \n\t\t\\label{tab:run_time}\n\t\\end{table}\n\t\\begin{alertblock}{Run time}\n\t\tThe longest run time on GPU was \\textbf{2.28 h}\\\\\n\t\tThe typical run time for simulations performed next was about 3 hours ($t = 1300\\, \\mu$s, NDOF $= 2.23\\cdot 10^6$) \\\\\n\t\tThe GPU memory usage was about 925 MB\\\\\n\t\t3 h $\\times$ 1000 simulations $\\approx$ \\textbf{125 days}\n\t\\end{alertblock}\n\\end{frame}\n\\begin{frame}{Computation speedup analysis}\n\t\\begin{figure} [h!]\n\t\t\\centering\n\t\t\\includegraphics{speedup.png}\t\n\t\t\\caption{Computation speedup; NDOF - number of degrees of freedom.}\n\t\t\\label{fig:speedup}\n\t\\end{figure}\n\t\\begin{equation*}\n\tspeedup = \\frac{CPU_{time}}{GPU_{time}}\n\t\\label{eq:speedup}\n\t\\end{equation*}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Experimental validation}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Open Guided Waves}\n\t\\begin{block}{\\url{http://www.open-guided-waves.de}}\n\t\tOpen Guided Waves is an extendable online platform where high--quality and well documented datasets for guided wave--based inspections are provided.\n\t\\end{block}\n\n\tCFRP plates [45/0/-45/90/-45/0/45/90]\\textsubscript{S}\n\t\n\t500 mm $\\times$ 500 mm, thickness of 2 mm. \n\t\n\tPiezoelectric transducers diameter 10 mm, thickness 0.2 mm.\n\t\n\t\\begin{table}\n\t\t\\renewcommand{\\arraystretch}{1.3}\n\t\t\\centering \\small\n\t\tMaterial properties of unidirectional Hexply M21/34\\%/UD134/T700/300; Units: GPa.\n\t\t\n\t\t\\begin{tabular}{cccccc} \n\t\t\t%\\hline\n\t\t\t\\toprule\n\t\t\t$Q_{11}$ & $Q_{12}$ & $Q_{22}$ & $Q_{44}$ & $Q_{55}$ & $Q_{66}$\\\\\n\t\t\t% \\cmidrule(lr){1-3} \\cmidrule(lr){4-6} \\cmidrule(lr){7-7}\n\t\t\t%\\hline\n\t\t\t\\midrule\n\t\t\t130& 6.1& 11.2 & 3.0 & 4.2 & 4.2\\\\\n\t\t\t%\\hline \n\t\t\t\\bottomrule \n\t\t\\end{tabular} \n\t\t\\label{tab:mat_prop}\n\t\\end{table}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Open Guided Waves}\n\n\t\\begin{table}\n\t\t\\renewcommand{\\arraystretch}{1.3}\n\t\t\\centering \\small\n\t\tCoordinates of analysed defects; Units: [m].\n\t\t\n\t\t\\begin{tabular}{cccc} \n\t\t\t%\\hline\n\t\t\t\\toprule\n\t\t\t\\multicolumn{2}{c}{\\textbf{Data set I} }\t& \\multicolumn{2}{c}{\\textbf{Data set II} } \\\\\n\t\t\t\\multicolumn{2}{c}{\\textbf{(wavefield plate)} }\t& \\multicolumn{2}{c}{\\textbf{(SHM plate)} } \\\\\n\t\t\t%\\midrule\n\t\t\t\\cmidrule(lr){1-2} \\cmidrule(lr){3-4}\n\t\t\tx & y & x & y \\\\\n\t\t\t% \\cmidrule(lr){1-3} \\cmidrule(lr){4-6} \\cmidrule(lr){7-7}\n\t\t\t%\\hline\n\t\t\t0.191 & 0.145 & 0.250 & 0.427 \\\\ \n\t\t\t%\\hline \n\t\t\t\\bottomrule \n\t\t\\end{tabular} \n\t\t\\label{tab:defect_coordinates}\n\t\\end{table}\t\t\n\t\\begin{block}{Wavefield plate}\n\t\tWavefield measured by scanning laser Doppler vibrometer\n\t\\end{block}\n\t\\begin{block}{SHM plate}\n\t\tSignals excited and registered by an array of piezoelectric transducers\n\t\\end{block}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[b]{Meshing - GMSH quadrilateral elements}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{delam_Jochen_signals_D5_a_5mm_b_5mm_angle_0added_mass.png}\t\t\n\t\t\\caption{red -- piezoelectric transducers; green -- defect location}\n\t\t\\label{fig:quad_mesh}\n\t\\end{figure}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Meshing - spectral element mesh close up}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\includegraphics[width=0.7\\textwidth]{delam_Jochen_signals_D5_a_5mm_b_5mm_angle_0added_mass_spec_zoom.png}\t\n\t\t\\label{fig:spec_mesh_zoom}\n\t\\end{figure}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{Wavefield plate: comparison of wavefields}\n%\\vspace{6pt}\n\\def\\myindenta{0.001\\textwidth} % define myindenta variable for correcting caption placement\n\\begin{columns}[T]\n\t\\column{0.5\\textwidth}\n\t\t\\begin{figure}\n\t\t\t\\only<1>{\n\t\t\t\\includegraphics[width=\\textwidth]{exp_frame64.png}\n\t\t\t\\caption{\\hspace{\\myindenta}Experimental wavefield: $t=112.5\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<2>{\n\t\t\t\\includegraphics[width=\\textwidth]{exp_frame100.png}\n\t\t\t\\caption{\\hspace{\\myindenta}Experimental wavefield: $t=169\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<3>{\n\t\t\t\t\\includegraphics[width=\\textwidth]{exp_frame64.png}\n\t\t\t\t\\caption{\\hspace{\\myindenta}Experimental wavefield: $t=112.5\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<4>{\n\t\t\t\t\\includegraphics[width=\\textwidth]{exp_frame100.png}\n\t\t\t\t\\caption{\\hspace{\\myindenta}Experimental wavefield: $t=169\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<5>{\n\t\t\t\t\\includegraphics[width=\\textwidth]{cut_Vz_1_frame128_top.png}\n\t\t\t\t\\caption{\\hspace{\\myindenta}Simulation (delamination): $t=200\\, \\mu$s}\n\t\t\t}\n\t\t\\end{figure}\n\t\\column{0.5\\textwidth}\n\t\t\\begin{figure}\n\t\t\t\\only<1>{\n\t\t\t\\includegraphics[width=\\textwidth]{Vz_1_frame72_bottom.png}\n\t\t\t\\caption{\\hspace{\\myindenta}Simulation (\\textbf{delamination}): $t=112.5\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<2>{\n\t\t\t\\includegraphics[width=\\textwidth]{Vz_1_frame108_bottom.png}\n\t\t\t\\caption{\\hspace{\\myindenta}Simulation (\\textbf{delamination}): $t=169\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<3>{\n\t\t\t\t\\includegraphics[width=\\textwidth]{added_mass_Vz_1_frame72_bottom.png}\n\t\t\t\t\\caption{\\hspace{\\myindenta}Simulation (\\textbf{added mass}): $t=112.5\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<4>{\n\t\t\t\t\\includegraphics[width=\\textwidth]{added_mass_Vz_1_frame108_bottom.png}\n\t\t\t\t\\caption{\\hspace{\\myindenta}Simulation (\\textbf{added mass}): $t=169\\, \\mu$s}\n\t\t\t}\n\t\t\t\\only<5>{\n\t\t\t\t\\includegraphics[width=\\textwidth]{added_mass_cut_Vz_1_frame128_top.png}\n\t\t\t\t\\caption{\\hspace{\\myindenta}Simulation (added mass): $t=200\\, \\mu$s}\n\t\t\t}\n\t\t\t\\label{fig:wavefield}\n\t\t\\end{figure}\n\\end{columns}\n\t\\only<6>{\n\t\\begin{alertblock}{Remarks}\n\t\t\\begin{itemize}\n\t\t\t\\item Simulated wavefield patterns matches the experimental data very well.\n\t\t\t\\item The interaction of wave front with added mass is slightly different than with delamination.\n\t\t\t\\item The phenomenon of wave entrapment is visible for the case of delamination whereas it is not present in case of added mass.\n\t\t\t\\item A reversible damage in the form of added mass coupled to the structure by tacky tape is a good way for collecting benchmark data set. Such benchmark data set is a good starting point for development of algorithms based on full wavefield data for visualization of defect location even in the form of delamination.\n\t\t\\end{itemize}\n\t\\end{alertblock}\n\t}\n\\end{frame}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}[t]{SHM plate: comparison of signals}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\only<1>{\n\t\t\\includegraphics[width=0.9\\textwidth]{path_1_7.png}\t\n\t\t\\label{fig:path1_7}\n\t\t}\n\t\t\\only<2>{\n\t\t\t\\includegraphics[width=0.9\\textwidth]{path_2_4.png}\t\n\t\t\t\\label{fig:path2_4}\n\t\t}\n\t\t\\only<3>{\n\t\t\t\\includegraphics[width=0.9\\textwidth]{path_4_8.png}\t\n\t\t\t\\label{fig:path4_8}\n\t\t}\n\t\t\\only<4>{\n\t\t\t\\includegraphics[width=0.9\\textwidth]{path_2_4_diff.png}\t\n\t\t\t\\label{fig:path2_4_diff}\n\t\t}\n\t\\end{figure}\n\t\\only<5>{\n\t\t\\begin{alertblock}{Remarks}\n\t\t\t\\begin{itemize}\n\t\t\t\t\\item Differences between experimental and numerical signals are evident and are related to amplitude rather than wave velocity.\n\t\t\t\t\\item The amplitude difference can be attributed to the damping coefficient selected in numerical simulation.\n\t\t\t\t\\item The delamination affect signals more than added mass.\n\t\t\t\\end{itemize}\n\t\t\\end{alertblock}\n\t}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Conclusions}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Summary}\n\t\\begin{itemize}\n\t\t\\item The developed code is highly parallel and enables GPU computation. It allows up to \\textbf{12$\\times$} computation speedup in comparison to CPU. It also \\textbf{does not require assembly of stiffness matrix}. Therefore, larger guided wave propagation problems can be solved than in case of conventional finite element approach.\n\t\t\\item The simulated wave propagation in composite laminates \\textbf{matches well the experimental data} from Open Guided Waves platform in terms of full wavefield patterns.\n\t\t\\item There are differences in wave packets' amplitudes -- proper \\textbf{wave attenuation modelling is still a challenge}.\n\t\t\\item The current implementation is very well suited for any parametric studies or \\textbf{model--assisted} approaches for structural health monitoring but is not feasible for \\textbf{model--based} approach in which numerical signals are directly compared to experimental one.\n\t\t\n\t\\end{itemize}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Future works}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Random excitation}\n\t\\begin{figure}\n\t\t\\centering\t\n\t\t\\includegraphics[width=0.9\\textwidth]{beamer_figs/random_exc_sig.png}\t\n\t\t\\label{fig:random_exc_sig} \n\t\\end{figure}\n \\begin{alertblock}{Advantage}\n Wide frequency range increases probability that the guided wave will interact with damage.\n \\end{alertblock}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Random excitation: full wavefield}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\only<1>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/Vz_5_frame16_top.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_16}\n\t\t}\n\t\t\\only<2>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/Vz_5_frame32_top.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_32}\n\t\t}\n\t\t\\only<3>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/Vz_5_frame48_top.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_48}\n\t\t}\n\t\t\\only<4>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/Vz_5_frame64_top.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_64}\n\t\t}\n\t\t\\only<5>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/Vz_5_frame80_top.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_80}\n\t\t}\n\t\\end{figure}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Random excitation: RMS}\n\t\\begin{figure}\n\t\t\\centering\n\t\t\\only<1>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/RMS5.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_RMS5}\n\t\t}\n\t\t\\only<2>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/RMS14.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_RMS14}\n\t\t}\n\t\t\\only<3>{\n\t\t\t\\includegraphics[width=0.75\\textwidth]{beamer_figs/RMS22.png}\t\n\t\t\t\\label{fig:random_exc_wavefield_RMS22}\n\t\t}\n\t\\end{figure}\t\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n{\\setbeamercolor{palette primary}{fg=black, bg=white}\n\\begin{frame}[standout]\n Thank you for your attention!\\\\ \\vspace{12pt}\n Questions?\\\\ \\vspace{12pt}\n \\url{pk@imp.gda.pl}\n\\end{frame}\n}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% END OF SLIDES\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\appendix\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Model--based vs model--assisted SHM}\n\t In the \\textbf{model--based} approach, numerical signals must match very well experimental signals. It is because numerical signals are used in an inverse method in which certain objective function include terms such as:\n\t\\begin{equation*}\n\tg(t) = \\sum_{i=1}^{N} \\left(s(t)_i^{exp} - s(t)_i^{num}\\right)^2,\n\t\\label{eq:obj_fun}\n\t\\end{equation*}\n\twhere $s(t)^{exp}$ and $s(t)^{num}$ are experimental and numerical signals (or damage indexes), respectively.\n\t\n\tIn the \\textbf{model--assisted} approach, on the other hand, the modelled phenomenon must reflect the real behaviour, e.g. the amplitude of transmitted wave packet decreases when the size of the damage increases. But in this case, the perfect match between numerical and experimental signals is not necessary. The aim of the presented method is model--assisted structural health monitoring.\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Backup slides}\n \\begin{equation*}\n \\begin{split}\n \\bs{\\sigma}_{xx}&=\\left((\\bm{N},_{\\xi}\\vec{U}_x).*(\\vec{J}^{-1})_{11}+(\\bm{N},_{\\eta}\\vec{U}_x).*(\\vec{J}^{-1})_{21}\\right).*\\vec{A}_{11}\\\\\n &+\\left((\\bm{N},_{\\xi}\\bs{\\Phi}_x).*(\\vec{J}^{-1})_{11}+(\\bm{N},_{\\eta}\\bs{\\Phi}_x).*(\\vec{J}^{-1})_{21}\\right).*\\vec{B}_{11}\\\\\n &+\\left((\\bm{N},_{\\xi}\\vec{U}_y).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\vec{U}_y).*(\\vec{J}^{-1})_{22}\\right).*\\vec{A}_{12}\\\\\n &+\\left((\\bm{N},_{\\xi}\\bs{\\Phi}_y).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\bs{\\Phi}_y).*(\\vec{J}^{-1})_{22}\\right).*\\vec{B}_{12}\\\\\n &+\\left((\\bm{N},_{\\xi}\\vec{U}_x).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\vec{U}_x).*(\\vec{J}^{-1})_{22}\\right).*\\vec{A}_{16}\\\\\n &+\\left((\\bm{N},_{\\xi}\\bs{\\Phi}_x).*(\\vec{J}^{-1})_{12}+(\\bm{N},_{\\eta}\\bs{\\Phi}_x).*(\\vec{J}^{-1})_{22}\\right).*\\vec{B}_{16}\\\\\n &+\\left((\\bm{N},_{\\xi}\\vec{U}_y).*(\\vec{J}^{-1})_{11}+(\\bm{N},_{\\eta}\\vec{U}_y).*(\\vec{J}^{-1})_{21}\\right).*\\vec{A}_{16}\\\\\n &+\\left((\\bm{N},_{\\xi}\\bs{\\Phi}_y).*(\\vec{J}^{-1})_{11}+(\\bm{N},_{\\eta}\\bs{\\Phi}_y).*(\\vec{J}^{-1})_{21}\\right).*\\vec{B}_{16},\n \\end{split}\n \\end{equation*}\n where $.*$ denotes element-wise operation known as Hadamarad product (the same symbol for element-wise operation is used in Matlab). It should be noted that $\\vec{A}$, $\\vec{B}$, $\\vec{D}$ and $(\\vec{J}^{-1})$ with appropriate indexes are arranged in the form of vectors respective to GLL nodes of consecutive disjoint spectral elements.\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Backup slides}\n\t\\begin{equation*}\n\t\\vec{U}_x = \\left[\n\t\\begin{array}{c} \n\t\\hat{\\vec{u}}_0^{e=1} \\\\[2pt]\n\t\\hat{\\vec{u}}_0^{e=2} \\\\[2pt]\n\t\\vdots\\\\[2pt]\n\t\\hat{\\vec{u}}_0^{e=n}\\\\[2pt]\n\t\\end{array}\\right],\n\t\\quad\n\t\\vec{U}_y = \\left[\n\t\\begin{array}{c} \n\t\\hat{\\vec{v}}_0^{e=1} \\\\[2pt]\n\t\\hat{\\vec{v}}_0^{e=2} \\\\[2pt]\n\t\\vdots\\\\[2pt]\n\t\\hat{\\vec{v}}_0^{e=n}\\\\[2pt]\n\t\\end{array}\\right],\n\t\\quad\n\t\\vec{U}_z = \\left[\n\t\\begin{array}{c} \n\t\\hat{\\vec{w}}_0^{e=1} \\\\[2pt]\n\t\\hat{\\vec{w}}_0^{e=2} \\\\[2pt]\n\t\\vdots\\\\[2pt]\n\t\\hat{\\vec{w}}_0^{e=n}\\\\[2pt]\n\t\\end{array}\\right],\n\t\\end{equation*}\n\t\\begin{equation*}\n\t\\bs{\\Phi}_x = \\left[\n\t\\begin{array}{c} \n\t\\hat{\\bs{\\varphi}}_x^{e=1} \\\\[2pt]\n\t\\hat{\\bs{\\varphi}}_x^{e=2} \\\\[2pt]\n\t\\vdots\\\\[2pt]\n\t\\hat{\\bs{\\varphi}}_x^{e=n}\\\\[2pt]\n\t\\end{array}\\right],\n\t\\quad\n\t\\bs{\\Phi}_y = \\left[\n\t\\begin{array}{c} \n\t\\hat{\\bs{\\varphi}}_y^{e=1} \\\\[2pt]\n\t\\hat{\\bs{\\varphi}}_y^{e=2} \\\\[2pt]\n\t\\vdots\\\\[2pt]\n\t\\hat{\\bs{\\varphi}}_y^{e=n}\\\\[2pt]\n\t\\end{array}\\right],\n\t\\end{equation*}\n\t$n$ is the total number of elements.\\\\\t\n\\end{frame}\t\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Backup slides}\n\tSparse matrices of shape function derivatives:\n\t\\begin{equation*}\n\t\\bm{N},_{\\xi} = \\left[\n\t\\begin{array}{cccc} \n\t\\bm{N},_{\\xi}^{e=1} & 0 & \\ldots & 0\\\\[2pt]\n\t0& \\bm{N},_{\\xi}^{e=2} & \\ldots& 0\\\\[2pt]\n\t\\vdots&\\vdots&\\ddots&0\\\\[2pt]\n\t0& 0 &0&\\bm{N},_{\\xi}^{e=n}\\\\[2pt]\n\t\\end{array}\\right].\n\t\\end{equation*}\n\\end{frame}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\begin{frame}{Backup slides}\n\\begin{equation*}\n\\begin{split}\n\\vec{F}_u^i&=\\bm{N},_{\\xi}^T \\left(\\bs{\\sigma}_{xx}\\,.*(\\vec{J}^{-1})_{11}\\,.*\\vec{W}\\right)+\\bm{N},_{\\eta}^T \\left(\\bs{\\sigma}_{xx}\\,.*(\\vec{J}^{-1})_{21}\\,.*\\vec{W}\\right)\\\\\n&+\\bm{N},_{\\xi}^T \\left(\\bs{\\sigma}_{xy}\\,.*(\\vec{J}^{-1})_{12}\\,.*\\vec{W}\\right)+\\bm{N},_{\\eta}^T \\left(\\bs{\\sigma}_{xy}\\,.*(\\vec{J}^{-1})_{22}\\,.*\\vec{W}\\right), \\\\ \n\\vec{F}_v^i&=\\ldots ,\\\\\n\\vec{F}_w^i&=\\ldots ,\\\\\n\\vec{M}_x^i&=\\bm{N},_{\\xi}^T \\left(\\bs{\\sigma}_{xxz}\\,.*(\\vec{J}^{-1})_{11}\\,.*\\vec{W}\\right)+\\bm{N},_{\\eta}^T \\left(\\bs{\\sigma}_{xxz}\\,.*(\\vec{J}^{-1})_{21}\\,.*\\vec{W}\\right)\\\\\n&+\\bm{N},_{\\xi}^T \\left(\\bs{\\sigma}_{xyz}\\,.*(\\vec{J}^{-1})_{12}\\,.*\\vec{W}\\right)+\\bm{N},_{\\eta}^T \\left(\\bs{\\sigma}_{xyz}\\,.*(\\vec{J}^{-1})_{22}\\,.*\\vec{W}\\right)\\\\\n&+\\bs{\\sigma}_{xz}\\,.*\\vec{W},\\\\\n\\vec{M}_y^i&=\\ldots ,\n\\label{eq:internal_forces}\n\\end{split}\n\\end{equation*}\nwhere $^T$ is the matrix transpose and $\\vec{W}$ is the vector resulting from multiplication of integration weights and Jacobian determinant at appropriate GLL points:\n\\begin{equation*}\n\\vec{W} = \\vec{w}_{\\xi}\\,.*\\vec{w}_{\\eta}\\,.*\\left(\\det\\vec{J}\\right).\n\\end{equation*}\n\\end{frame}\n\\end{document}", "meta": {"hexsha": "74543121c1305af70fbb12404d68d6d1ea228ad2", "size": 33120, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "reports/beamer_presentations/seminar_Goethe_University.tex", "max_stars_repo_name": "pawelkudela/ma-shm", "max_stars_repo_head_hexsha": "b0403ba4e98e7d8176cbce00a6102b7deef82629", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1, "max_stars_repo_stars_event_min_datetime": "2022-03-03T05:39:06.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-03T05:39:06.000Z", "max_issues_repo_path": "reports/beamer_presentations/seminar_Goethe_University.tex", "max_issues_repo_name": "pawelkudela/ma-shm", "max_issues_repo_head_hexsha": "b0403ba4e98e7d8176cbce00a6102b7deef82629", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "reports/beamer_presentations/seminar_Goethe_University.tex", "max_forks_repo_name": "pawelkudela/ma-shm", "max_forks_repo_head_hexsha": "b0403ba4e98e7d8176cbce00a6102b7deef82629", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 40.8384710234, "max_line_length": 483, "alphanum_fraction": 0.622192029, "num_tokens": 11492, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5389832354982647, "lm_q2_score": 0.6334102636778401, "lm_q1q2_score": 0.3413975133148912}} {"text": "% !TeX root = ../main.tex\n\\section{Basic Conditional Narrowing}\nFrom the same motivations as in the unconditional narrowing case we pursuit the objective of define basic conditional narrowing derivations.\n\n\\begin{definition}\n Let $T$ be a goal clause. A \\textit{position constraint} for $T$ is a mapping $B$ that assigns to every goal $e \\in T$ a subset of $\\basicPos{e}$. The position constraint that assigns to every $e \\in T$ the set $\\basicPos{e}$ will be denoted by $\\overline{T}$.\n\\end{definition}\n\n\\begin{definition}\n \\begin{enumerate}\n \\item A narrowing derivation\n $$T \\narrow_{[e_1, p_1, l_1 \\contr r_1 \\Leftarrow c_1, \\sigma_1]} \\narrow \\cdots \\narrow_{[e_{n-1}, p_{n_1}, l_{n-1} \\contr r_{n-1} \\Leftarrow c_{n-1}, \\sigma_{n-1}]} T_n$$\n is basic if $p_i \\in B_i(e_i)$ for $1 \\leq i \\leq n - 1$ where the position constraints $B_1, \\dots, B_n$ are inductively defined by $B_1 = \\overline{T}_1$ and\n \\begin{displaymath}\n B_{i+1}(e) =\n \\begin{cases}\n B_i(e') & \\text{ if } e' \\in T_i \\setminus \\{e_i\\} \\\\\n \\mathcal{B}(B_i(e_i), p_i, r_i) & \\text{ if } e' = e_i[r_i]_{p_i} \\\\\n \\basicPos{e'} & \\text{ if } e' \\in \\overline{c}_i\n \\end{cases}\n \\end{displaymath}\n \\item A rewrite sequence\n $$T \\rewriteOnGoals_{[e_1, p_1, l_1 \\contr r_1 \\Leftarrow c_1, \\sigma_1]} \\rewriteOnGoals \\cdots \\rewriteOnGoals_{[e_{n-1}, p_{n_1}, l_{n-1} \\contr r_{n-1} \\Leftarrow c_{n-1}, \\sigma_{n-1}]} T_n$$\n is \\textit{based} on a position constraint $B_1$ for $T_1$ if $p_i \\in B_i$ for $1 \\leq i \\leq n - 1$ with $B_2, \\dots, B_n$ defined by\n \\begin{displaymath}\n B_{i+1}(e) =\n \\begin{cases}\n B_i(e) & \\text{ if } e' \\in T_i \\setminus \\{e_i\\} \\\\\n \\mathcal{B}(B_i(e_i), p_i, r_i) & \\text{ if } e = e_i[r_i\\sigma_i]_{p_i} \\\\\n \\basicPos{e'} & \\text{ if } e=e'\\sigma \\text{ with } e' \\in \\overline{c}_i\n \\end{cases}\n \\end{displaymath}\n for all $1 \\leq i < n$ and $e \\in T_{i+1}$.\n \\end{enumerate}\n\\end{definition}\n\nHolldobler \\cite{Holldobler:1989:FEL:76924}, showed that basic conditional narrowing is complete for complete 1-CTRSs. However, the following example shows that this result is indeed not valid.\n\n\\begin{example}\n Consider the 1-CTRS\n \\begin{displaymath}\n \\trs =\n \\begin{cases}\n f(x) & \\contr a \\\\\n d & \\contr b \\\\\n d & \\contr c\\\\\n b & \\contr c \\Leftarrow f(d) = a\n \\end{cases}\n \\end{displaymath}\n\\end{example}\n\nThe authors in \\cite{Middeldorp1994} have shown that $\\trs$ is complete. Note that basic conditional narrowing is not able to solve the goal $eq(f(d),a)$ as can be seen form the figure below.\n\n\\begin{figure}[!ht]\n \\begin{displaymath}\n \\xymatrix{\n & & eq(f(d),a) \\ar@{~>}[dll] \\ar@{~>}[drr] \\ar@{~>}[d] & & \\\\\n eq(\\underline{d},b),eq(\\underline{d},c) \\ar@{~>}[d] & & eq(f(b),a) \\ar@{~>}[dl] \\ar@{~>}[dr] & & eq(f(c),a) \\ar@{~>}[d] \\\\\n eq(\\underline{d},c), eq(f(d),a) & eq(\\underline{b},c) & & eq(f(c),a), eq(f(d),a) & eq(\\underline{c},b) \\ar@{~>}[d]\\\\\n & & & & eq(f(d),a)\n }\n \\end{displaymath}\n \\caption{Derivation tree for the objective $eq(f(d),a)$}\n \\label{figure:counterexample-basic-conditional-narrowing}\n\\end{figure}\nNote that this derivation tree is basic-normalised, i.e., we explore all basic reductions. However, the following non-basic narrowing derivation shows that the goal can be solved:\n\\begin{align*}\n eq(f(d),a) &\\narrow eq(a,a), eq(\\underline{d},b), eq(\\underline{d},c) \\\\\n & \\narrow eq(a,a), eq(b,b), eq(\\underline{d},c) \\\\\n & \\narrow eq(a,a), eq(b,b), eq(c,c) \\\\\n & \\narrow^* \\top\n\\end{align*}\nNote further that basic conditional narrowing is unable to solve the normalised goal $eq(f(x),a)$. In \\cite{Middeldorp1994}, Holldobler himself has found the problem in his previous proof. The mistake was due to the incorrect assumption that the strong normalization of $\\rewriteOnGoals_\\trs$ is implied by the strong normalization of $\\trs$. We now show, following his own steps, that completeness of basic conditional narrowing can be ensured by strengthening strong normalization. We begin with a definition.\n\n\\begin{definition}\n A 1-CTRS $\\trs$ is decreasing if there exists a well-founded extension $>$ of the rewrite relation $\\contr_\\trs$ with the following properties:\n \\begin{enumerate}\n \\item $>$ has the subterm property, i.e. $t > \\restr{t}{p}$ for all positions $p \\in \\pos{t} \\setminus \\{\\Lambda\\}$\n \\item if $l \\contr r \\Leftarrow c \\in \\trs$ and $\\sigma$ is a substitution then $l\\sigma > s \\sigma$ for all $s = t \\in c$.\n \\end{enumerate}\n\\end{definition}\n\nEvery decreasing 1-CTRS is strongly normlizing and moreover -- when the number of rewrite rules is finite -- its rewrite relation is decidable.\n\n\\begin{example}\n The CTRS of the above example is not decreasing: as $f(d) \\contr a \\Leftarrow d = b, d = c$ is an instance of the first rewrite rule we must have $f(d) > b$, but the rule $b \\contr c \\Leftarrow f(d) = a$ requires $b > f(d)$.\n\\end{example}\n\nWith the decreasing hypothesys one can show:\n\\begin{lemma}\n If $\\trs$ is a decreasing 1-CTRS then $\\rewriteOnGoals_\\trs$ is strongly normalizing.\n\\end{lemma}\n\nIs now standard to give statements as in the basic non-conditional case. That is the content of the next lemma.\n\\begin{lemma}\n Let $\\trs$ be a 1-CTRS, $T$ a goal clause, and $\\sigma$ a normalized substitution. Every innermost $\\rewriteOnGoals_\\trs$-sequence starting from $T \\sigma$ is based on $\\overline{T}$.\n\\end{lemma}\n\nFinally, we have our desired completeness result for conditional basic narrowing.\n\\begin{theorem}\n Basic conditional narrowing is complete for decreasing and confluent 1-CTRS.\n\\end{theorem}\n", "meta": {"hexsha": "c132b716c69f64b537ba7e0bffa30b013b767956", "size": 5935, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "sections/basic_conditional_narrowing.tex", "max_stars_repo_name": "deividrvale/report-narrowing", "max_stars_repo_head_hexsha": "1e3ce34a1afb5268b4307fcc9af9374d2e121a27", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "sections/basic_conditional_narrowing.tex", "max_issues_repo_name": "deividrvale/report-narrowing", "max_issues_repo_head_hexsha": "1e3ce34a1afb5268b4307fcc9af9374d2e121a27", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "sections/basic_conditional_narrowing.tex", "max_forks_repo_name": "deividrvale/report-narrowing", "max_forks_repo_head_hexsha": "1e3ce34a1afb5268b4307fcc9af9374d2e121a27", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 57.6213592233, "max_line_length": 511, "alphanum_fraction": 0.6323504634, "num_tokens": 1881, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5583269943353745, "lm_q2_score": 0.6113819732941511, "lm_q1q2_score": 0.3413510595401536}} {"text": "\\section{Background} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\r\n\\label{sec:bg}\r\n\r\nPattern matching in the context of a programming language was first introduced \r\nin a string manipulation language SNOBOL\\cite{SNOBOL64}. Its fourth \r\nreincarnation SNOBOL4 had patterns as first-class data types providing \r\noperations of concatenation and alternation on them\\cite{SNOBOL71}. The first \r\nreference to a pattern-matching construct that resembles the one found in \r\nstatically typed functional languages today is usually attributed to Burstall \r\nand his work on structural induction\\cite{Burstall69provingproperties}.\r\n\r\nIn the context of object-oriented programming, pattern matching has been first \r\nexplored in Pizza programming language\\cite{Odersky97pizzainto}. These efforts \r\nhave been continued in Scala\\cite{Scala2nd} and together with notable work of \r\nBurak Emir on \\emph{Object-Oriented Pattern Matching}\\cite{EmirThesis} have \r\nresulted in incorporation of pattern matching into the language.\r\n\r\n%The first tree based pattern matching methods were found in Fred McBride's \r\n%extension of LISP in 1970.\r\n\r\n%ML and Haskell further popularized pattern matching ...\r\n\r\nPattern matching has been closely related to \\emph{algebraic data types} and \r\n\\emph{equational reasoning} since the early days of functional programming.\r\nIn languages like ML and Haskel an \\emph{Algebraic Data Type} is a data type \r\neach of whose values is picked from a disjoint sum of (possibly recursive) data \r\ntypes, called \\emph{variants}. Each of the variants is marked with a unique \r\nsymbolic constant called \\emph{constructor}, while the set of all constructors \r\nof a given type is called \\emph{signature}. Constructors provide a convenient \r\nway of creating a value of its variant type as well as a way of discriminating \r\nits variant type from the algebraic data type through pattern matching.\r\n\r\nAlgebraic data type \\codeocaml{expr} from Section~\\ref{sec:intro} consists of 5 \r\nvariants, marked with constructors \\codeocaml{Value}, \\codeocaml{Plus}, \r\n\\codeocaml{Minus}, \\codeocaml{Times} and \\codeocaml{Divide} respectively. \r\nConstructor \\codeocaml{Value} expects a value of type \\codeocaml{int} during \r\nconstruction, as well as any pattern that admits values of type \\codeocaml{int} \r\nduring decomposition through pattern matching. Similarly, the other four \r\nconstructors expect a value of a cartesian product of two \\codeocaml{expr} \r\ntypes during construction, as well as any pattern that would admit a value of \r\nsuch type during decomposition.\r\n\r\nAlgebraic data types can be parameterized and recursive, as demonstrated by the \r\nfollowing Haskell code that defines a binary tree parameterized on type \r\n\\codehaskell{k} of keys and type \\codehaskell{d} of data stored in the nodes:\r\n\r\n\\begin{lstlisting}[language=Haskell]\r\ndata Tree k d = Node k d (Tree k d) (Tree k d) | Leaf\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nNaturally, they can be decomposed in a generic algorithm like the function \r\n\\code{find} below, defined through case analysis on the tree's structure:\r\n\r\n\\begin{lstlisting}[language=Haskell]\r\nfind :: (Ord k) => k -> Tree k d -> Maybe d\r\nfind i Leaf = Nothing\r\nfind i (Node key item left right) = \r\n if i == key \r\n then Just item \r\n else \r\n if i [<] key \r\n then find i left \r\n else find i right\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nThe set of values described by such an algebraic data type is defined \r\ninductively as the least set closed under constructor functions of its variants.\r\nAlgebraic data types draw their name from the practice of using case distinction \r\nin mathematical function definitions and proofs that involve \\emph{algebraic \r\nterms}.\r\n\r\nOne of the main differences of algebraic data types from classes in \r\nobject-oriented languages is that an algebraic data type definition is \r\n\\emph{closed} because it fixes the structure of its instances once and for all. \r\nOnce we have listed all the variants a given algebraic data type may have we \r\ncannot extend it with new variants without modifying its definition. This is not \r\nthe case in object-oriented languages, where classes are \\emph{open} to \r\nextension through subclassing. Notable exceptions to this restriction in \r\nfunctional community are \\emph{polymorphic variants} in OCaml\\cite{garrigue-98} \r\nand \\emph{open data types} in Haskell\\cite{LohHinze2006}, which allow addition \r\nof new variants later. These extensions, however, are simpler than object-oriented \r\nextensions as neither polymorphic variants nor open data types form subtyping \r\nrelation between themselves: open data types do not introduce any subtyping \r\nrelation, while the subtyping relation on polymorphic variants is a \r\n\\emph{semantic subtyping} similar to that of XDuce\\cite{HosoyaPierce2000}, which \r\nis based on the subset relation between values of the type. In either case they \r\nmaintain the important property that each value of the underlying algebraic data \r\ntype belongs to exactly one disjoint subset tagged with a constructor. The \r\n\\emph{nominative subtyping} of object-oriented languages does not usually have \r\nthis disjointness making classes effectively have multiple types. In particular, \r\nthe case of disjoint constructors can be seen as a degenerated case of a flat \r\nclass hierarchy among the multitude of possible class hierarchies.\r\n\r\nClosedness of algebraic data types is particularly useful for reasoning about \r\nprograms by case analysis and allows the compiler to perform an automatic \r\n\\emph{incompleteness} check -- test of whether a given \\emph{match statement} \r\ncovers all possible cases. Similar reasoning about programs involving extensible \r\ndata types is more involved as we are dealing with potentially open set of \r\nvariants. \\emph{Completeness} check in such scenario reduces to checking presence \r\nof a case that handles the static type of the subject. Absence of such a case,\r\nhowever, does not necessarily imply incompleteness, only potential incompleteness, \r\nas the answer will depend on the actual set of variants available at run-time.\r\n\r\nA related notion of \\emph{redundancy} checking arises from the \r\ntradition of using \\emph{first-fit} strategy in pattern matching. It warns the \r\nuser of any \\emph{case clause} inside a match statement that will \r\nnever be entered because of a preceding one being more general. Object-oriented \r\nlanguages, especially C++, typically prefer \\emph{best-fit} strategy (e.g. for \r\noverload resolution and class template specialization) because it is not prone \r\nto errors where semantics of a statement might change depending on the ordering \r\nof preceding definitions. The notable exception in C++ semantics that prefers \r\nthe \\emph{first-fit} strategy is ordering of \\code{catch} handlers of a \r\n\\code{try}-block. Similarly to functional languages the C++ compiler will perform \r\n\\emph{redundancy} checking on catch handlers and issue a warning that lists the \r\nredundant cases. We use this property of the C++ type system to perform redundancy \r\nchecking of our match statements in \\textsection\\ref{sec:redun}.\r\n\r\nThe patterns that work with algebraic data types we have seen so far are \r\ngenerally called \\emph{tree patterns} or \\emph{constructor patterns}. Their \r\nanalog in object-oriented languages is often referred to as \\emph{type pattern} \r\nsince it may involve type testing and type casting. Special cases of these patterns \r\nare \\emph{list patterns} and \\emph{tuple patterns}. The former lets one split a \r\nlist into a sequence of elements in its beginning and a tail with the help of \r\nlist constructor \\codehaskell{:} and an empty list constructor \\codehaskell{[]} \r\ne.g. \\codehaskell{[x:y:rest]}. The latter does the same with tuples using tuple\r\nconstructor \\codehaskell{(,,...,)} e.g. \\codehaskell{([x:xs],'b',(1,2.0),\"hi\",True)}.\r\n\r\nPattern matching is not used solely with algebraic data types and can equally \r\nwell be applied to built-in types. The following Haskell code defines factorial \r\nfunction in the form of equations:\r\n\r\n\\begin{lstlisting}[language=Haskell]\r\nfactorial 0 = 1\r\nfactorial n = n * factorial (n-1)\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nHere 0 in the left hand side of the first \\emph{equation} is an example of a \r\n\\emph{value pattern} (also known as \\emph{constant pattern}) that will only \r\nmatch when the actual argument passed to the function factorial is 0. The \r\n\\emph{variable pattern} \\codehaskell{n} (also referred to as \\emph{identifier \r\npattern}) in the left hand side of the second equation will match any value, \r\n\\emph{binding} variable \\codehaskell{n} to that value in the right hand side of \r\nequation. Similarly to variable pattern, the \\emph{wildcard pattern} \\codehaskell{_} \r\nwill match any value, neither binding it to a variable nor even obtaining it. \r\nValue patterns, variable patterns and wildcard patterns are \r\ngenerally called \\emph{primitive patterns}. Patterns like variable and wildcard \r\npatterns that never fail to match are called \\emph{irrefutable}, in contrast to \r\n\\emph{refutable} patterns like value patterns, which may fail to match.\r\n\r\nIn Haskell 98\\cite{Haskell98Book} the above definition of factorial could also \r\nbe written as:\r\n\r\n\\begin{lstlisting}[language=Haskell]\r\nfactorial 0 = 1\r\nfactorial (n+1) = (n+1) * factorial n\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nThe \\codehaskell{(n+1)} pattern in the left hand side of equation is an example of \r\n\\emph{n+k pattern}. According to its informal semantics ``Matching an $n+k$ \r\npattern (where $n$ is a variable and $k$ is a positive integer literal) against \r\na value $v$ succeeds if $v \\ge k$, resulting in the binding of $n$ to $v-k$, and \r\nfails otherwise''\\cite{haskell98}. n+k patterns were introduced into Haskell to \r\nlet users express inductive functions on natural numbers in much the same way as \r\nfunctions defined through case analysis on algebraic data types. Besides \r\nsuccinct notation, such language feature could facilitate automatic proof of \r\ntermination of such functions by compiler. Peano numbers, used as an analogy to \r\nalgebraic data type representation of natural numbers, is not always the best \r\nabstraction for representing other mathematical operations however. This, \r\ntogether with numerous ways of defining semantics of generalized n+k patterns \r\nwere some of the reasons why the feature was never generalized in Haskell to \r\nother kinds of expressions, even though there were plenty of known applications. \r\nMoreover, numerous debates over semantics and usefulness of the feature \r\nresulted in n+k patterns being removed from the language altogether in Haskell \r\n2010 standard\\cite{haskell2010}. Generalization of n+k patterns, called \r\n\\emph{application patterns} has been studied by Nikolaas N. Oosterhof in his \r\nMaster's thesis\\cite{OosterhofThesis}. Application patterns essentially treat \r\nn+k patterns as equations, while matching against them attempts to solve or \r\nvalidate the equation.\r\n\r\nWhile n+k patterns were something very few languages had, another common feature of \r\nmany programming languages with pattern matching are guards. A \\emph{guard} \r\nis a predicate attached to a pattern that may make use of the variables bound in \r\nit. The result of its evaluation will determine whether the case clause and the \r\nbody associated with it will be \\emph{accepted} or \\emph{rejected}. The \r\nfollowing OCaml code for $exp$ language from Section~\\ref{sec:intro} defines the \r\nrules for factorizing expressions $e_1e_2+e_1e_3$ into $e_1(e_2+e_3)$ and \r\n$e_1e_2+e_3e_2$ into $(e_1+e_3)e_2$ with the help of guards spelled out after \r\nkeyword \\codeocaml{when}:\r\n\r\n\\begin{lstlisting}[language=Caml,keepspaces,columns=flexible]\r\nlet factorize e =\r\n match e with\r\n Plus(Times(e1,e2), Times(e3,e4)) when e1 = e3 \r\n -> Times(e1, Plus(e2,e4))\r\n | Plus(Times(e1,e2), Times(e3,e4)) when e2 = e4 \r\n -> Times(Plus(e1,e3), e4)\r\n | e -> e\r\n ;;\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nOne may wonder why we could not simply write the above case clause as \r\n\\codeocaml{Plus(Times(e,e2), Times(e,e4))} to avoid the guard? Patterns that \r\npermit use of the same variable in them multiple times are called \r\n\\emph{equivalence patterns}, while the requirement of absence of such patterns \r\nin a language is called \\emph{linearity}. Neither OCaml nor Haskell support such \r\npatterns, while Miranda\\cite{Miranda85} as well as Tom's pattern matching \r\nextension to C, Java and Eiffel\\cite{Moreau:2003} supports \\emph{non-linear \r\npatterns}.\r\n\r\nThe example above illustrates yet another common pattern-matching facility -- \r\n\\emph{nesting of patterns}. In general, a constructor pattern composed of a \r\nlinear vector of (distinct) variables is called a \\emph{simple pattern}. The \r\nsame pattern composed not only of variables is called \\emph{nested pattern}.\r\nUsing nested patterns, with a simple expression in the case clause we could\r\ndefine a predicate that tests the top-level expression to be tagged with a\r\n\\codeocaml{Plus} constructor, while both of its arguments to be marked with \r\n\\codeocaml{Times} constructor, binding their arguments (or potentially pattern \r\nmatching further) respectively. Note that the visitor design pattern does not \r\nprovide this level of flexibility and each of the nested tests might have \r\nrequired a new visitor to be written. Nesting of patterns like the one above is \r\ntypically where users resort to \\emph{type tests} and \\emph{type casts} that in \r\ncase of C++ can be combined into a single call to \\code{dynamic_cast}.\r\n\r\nRelated to nested patterns are \\emph{as-patterns} that help one take a value \r\napart while still maintaining its integrity. The following rule could have been \r\na part of a hypothetical rewriting system in OCaml similar to the one above. Its \r\nintention is to rewrite expressions of the form $\\frac{e_1/e_2}{e_3/e_4}$ into \r\n$\\frac{e_1}{e_2}\\frac{e_4}{e_3} \\wedge e_2\\neq0 \\wedge e_3\\neq0 \\wedge e_4\\neq0$.\r\n\r\n\\begin{lstlisting}[language=Caml]\r\n | Divide(Divide(_,e2) as x, Divide(e3,e4))\r\n -> Times(x, Divide(e4, e3))\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nWe introduced a name ``x'' as a synonym of the result of matching the \r\nentire sub-expression \\codeocaml{Divide(_,e2)} in order to refer it without \r\nrecomposing in the right-hand side of the case clause. We omitted the \r\nconjunction of relevant non-zero checks for brevity, one can see that we will \r\nneed access to \\codeocaml{e2} in it however.\r\n\r\nDecomposing algebraic data types through pattern matching has an important \r\ndrawback that was originally spotted by Wadler\\cite{Wadler87}: they expose \r\nconcrete representation of an abstract data type, which conflicts with the \r\nprinciple of \\emph{data abstraction}. To overcome the problem he proposed the \r\nnotion of \\emph{views} that represent conversions between different \r\nrepresentations that are implicitly applied during pattern matching. As an \r\nexample, imagine polar and cartesian representations of complex numbers. A user \r\nmight choose polar representation as a concrete representation for the abstract \r\ndata type \\codeocaml{complex}, treating cartesian representation as view or vice \r\nversa:\\footnote{We use the syntax from Wadler's original paper for this example}\r\n\r\n\\begin{lstlisting}[language=Haskell,columns=flexible]\r\ncomplex ::= Pole real real\r\nview complex ::= Cart real real\r\n in (Pole r t) = Cart (r * cos t) (r * sin t)\r\n out (Cart x y) = Pole (sqrt(x^2 + y^2)) (atan2 x y)\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nThe operations then might be implemented in whatever representation is the most \r\nsuitable, while the compiler will implicitly convert representation if needed:\r\n\r\n\\begin{lstlisting}[language=Haskell,columns=flexible]\r\n add (Cart x1 y1) (Cart x2 y2) = Cart (x1 + x2) (y1 + y2)\r\n mult (Pole r1 t1) (Pole r2 t2) = Pole (r1 * r2) (t1 + t2)\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nThe idea of views were later adopted in various forms in several languages: \r\nHaskell\\cite{views96}, Standard ML\\cite{views98}, Scala (in the form of \r\n\\emph{extractors}\\cite{EmirThesis}) and F$\\sharp$ (under the name of \r\n\\emph{active patterns}\\cite{Syme07}). We demonstrate our support of views in \r\n\\textsection\\ref{sec:view}.\r\n\r\n%Views in functional programming languages [92, 71] are conversions from one data type to\r\n%another that are implicitly applied in pattern matching. They play a role similar to extractors\r\n%in Scala, in that they permit to abstract from the concrete data-type of the matched objects.\r\n%However, unlike extractors, views are anonymous and are tied to a particular target data\r\n%type.\r\n\r\nLogic programming languages like Prolog take pattern matching to even greater \r\nlevel. The main difference between pattern matching in logic languages and \r\nfunctional languages is that functional pattern matching is a ``one-way'' \r\nmatching where patterns are matched against values, possibly binding some \r\nvariables in the pattern along the way. Pattern matching in logic programming is \r\n``two-way'' matching based on \\emph{unification} where patterns can be matched \r\nagainst other patterns, possibly binding some variables in both patterns and \r\npotentially leaving some variables \\emph{unbound} or partially bound -- i.e. \r\nbound to patterns. A hypothetical example of such functionality can be matching \r\na pattern \\codeocaml{Plus(x,Times(x,1))} against another pattern \r\n\\codeocaml{Plus(Divide(y,2),z)}, which will result in binding \\codeocaml{x} to a \r\n\\codeocaml{Divide(y,2)} and \\codeocaml{z} to \\codeocaml{Times(Divide(y,2),1)} \r\nwith \\codeocaml{y} left unbound, leaving both \\codeocaml{x} and \\codeocaml{z} \r\neffectively a pattern.\r\n\r\n\\subsection{Algebraic Data Types in C++}\r\n\\label{sec:adt}\r\n\r\nC++ does not have a direct support of algebraic data types, but they can usually \r\nbe emulated in a number of ways. A pattern-matching solution that strives to be \r\ngeneral will have to account for different encodings and be applicable to all of \r\nthem.\r\n\r\nConsider an ML data type of the form:\r\n\r\n\\begin{lstlisting}[language=ML,keepspaces,columns=flexible,escapechar=@]\r\ndatatype DT = @$C_1$@ of {@$L_{11}:T_{11},...,L_{1m}:T_{1m}$@} \r\n | ...\r\n | @$C_k$@ of {@$L_{k1}:T_{k1},...,L_{kn}:T_{kn}$@}\r\n\\end{lstlisting}\r\n\r\n\\noindent There are at least 3 different ways to represent it in C++. Following \r\nEmir, we will refer to them as \\emph{encodings}~\\cite{EmirThesis}:\r\n\r\n\\begin{itemize}\r\n\\setlength{\\itemsep}{0pt}\r\n\\setlength{\\parskip}{0pt}\r\n\\item Polymorphic Base Class (or \\emph{polymorphic encoding} for short)\r\n\\item Tagged Class (or \\emph{tagged encoding} for short)\r\n\\item Discriminated Union (or \\emph{union encoding} for short)\r\n\\end{itemize}\r\n\r\n\\noindent\r\nIn polymorphic and tagged encoding, base class \\code{DT} represents algebraic \r\ndata type, while derived classes represent variants. The only difference between \r\nthe two is that in polymorphic encoding base class has virtual functions, while \r\nin tagged encoding it has a dedicated member of integral type that uniquely \r\nidentifies the variant -- derived class. \r\n\r\nThe first two encodings are inherently \\emph{open} because the classes can be \r\narbitrarily extended through subclassing. The last encoding is inherently \r\n\\emph{closed} because we cannot add more members to the union without modifying \r\nits definition.\r\n\r\n%In order to be able to provide a common syntax for these representations, we \r\n%need to understand better similarities and differences between them. Before we \r\n%look into them let's fix some terminology.\r\n\r\nWhen we deal with pattern matching, the static type of the original expression \r\nwe are matching may not necessarily be the same as the type of expression we \r\nmatch it with. We call the original expression a \\emph{subject} and its static \r\ntype -- \\emph{subject type}. We call the type we are trying to match subject \r\nagainst -- a \\emph{target type}.\r\n\r\nIn the simplest case, detecting that the target type is a given type or a type \r\nderived from it, is everything we want to know. We refer to such a use-case as \r\n\\emph{type testing}. In the next simplest case, besides testing we might want to \r\nget a pointer or a reference to the target type of subject as casting it to such \r\na type may involve a non-trivial computation only a compiler can safely \r\ngenerate. We refer to such a use-case as \\emph{type identification}. Type \r\nidentification of a given subject against multiple target types is typically \r\nreferred to as \\emph{type switching}.\r\n\r\nOnce we uncovered the target type, we may want to be able to decompose it \r\n\\emph{structurally} (when the target type is a \\emph{structured} data type like \r\narray, tuple or class) or \\emph{algebraically} (when the target type is a scalar \r\ndata type like \\code{int} or \\code{double}). Structural decomposition in our \r\nlibrary can be performed with the help of \\emph{tree patterns}, while algebraic \r\ndecomposition can be done with the help of \\emph{generalized n+k patterns}.\r\n\r\n\\subsubsection{Polymorphic Base Class}\r\n\\label{sec:pbc}\r\n\r\nIn this encoding user declares a polymorphic base class \\code{DT} that will \r\nbe extended by classes representing all the variants. Base class might declare \r\nseveral virtual functions that will be overridden by derived classes, for example \r\n\\code{accept} used in a Visitor Design Pattern.\r\n\r\n\\begin{lstlisting}[keepspaces,columns=flexible]\r\nclass DT { virtual @$\\sim$@DT{} };\r\nclass @$C_1$@ : public DT {@$T_{11} L_{11}; ... T_{1m} L_{1m};$@} \r\n...\r\nclass @$C_k$@ : public DT {@$T_{k1} L_{k1}; ... T_{kn} L_{kn};$@} \r\n\\end{lstlisting}\r\n\r\nThe uncover the actual variant of such an algebraic data type, the user might \r\nuse \\code{dynamic_cast} to query one of the $k$ expected run-time types (an \r\napproach used by Rose\\cite{SQ03}) or she might employ a visitor design pattern \r\ndevised for this algebraic data type (an approach used by Pivot\\cite{Pivot09} \r\nand Phoenix\\cite{Phoenix}). The most attractive feature of this approach is that \r\nit is truly open as we can extend classes arbitrarily at will (leaving the \r\northogonal issues of visitors aside).\r\n\r\n\\subsubsection{Tagged Class}\r\n\\label{sec:tc}\r\n\r\nThis encoding is similar to the \\emph{Polymorphic Base Class} in that we use \r\nderived classes to encode the variants. The main difference is that the user \r\ndesignates a member in the base class, whose value will uniquely \r\ndetermine the most derived class a given object is an instance of. Constructors \r\nof each variant $C_i$ are responsible for properly initializing the dedicated \r\nmember with a unique value $c_i$ associated with that variant. Clang\\cite{Clang} \r\namong others uses this approach.\r\n\r\n\\begin{lstlisting}[keepspaces,columns=flexible]\r\nclass DT { enum kinds {@$c_1, ..., c_k$@} m_kind; };\r\nclass @$C_1$@ : public DT {@$T_{11} L_{11}; ... T_{1m} L_{1m};$@} \r\n...\r\nclass @$C_k$@ : public DT {@$T_{k1} L_{k1}; ... T_{kn} L_{kn};$@} \r\n\\end{lstlisting}\r\n\r\nIn such scenario the user might use a simple switch statement to uncover the \r\ntype of the variant combined with a \\code{static_cast} to properly cast the \r\npointer or reference to an object. People might prefer this encoding to the one \r\nabove for performance reasons as it is possible to avoid virtual dispatch with \r\nit altogether. Note, however, that once we allow for extensions and not limit \r\nourselves with encoding algebraic data types only it also has a significant \r\ndrawback in comparison to the previous approach: we can easily check that given \r\nobject belongs to the most derived class, but we cannot say much about whether \r\nit belongs to one of its base classes. A visitor design pattern can be \r\nimplemented to take care of this problem, but control inversion that comes along \r\nwith it will certainly diminish the convenience of having just a switch \r\nstatement. Besides, forwarding overhead might lose some of the performance \r\nbenefits gained originally by putting a dedicated member into the base class.\r\n\r\n\\subsubsection{Discriminated Union}\r\n\\label{sec:du}\r\n\r\nThis encoding is popular in projects that are either implemented in C or \r\noriginated from C before coming to C++. It involves a type that contains a union \r\nof its possible variants, discriminated with a dedicated value stored as a part \r\nof the structure. The approach is used by EDG front-end\\cite{EDG} and many others.\r\n\r\n\\begin{lstlisting}[keepspaces,columns=flexible]\r\nstruct DT\r\n{\r\n enum kinds {@$c_1, ..., c_k$@} m_kind;\r\n union {\r\n struct @$C_1$@ {@$T_{11} L_{11}; ... T_{1m} L_{1m};$@} @$C_1$@;\r\n ...\r\n struct @$C_k$@ {@$T_{k1} L_{k1}; ... T_{kn} L_{kn};$@} @$C_k$@; \r\n };\r\n};\r\n\\end{lstlisting}\r\n\r\nAs before, the user can use a switch statement to identify the variant $c_i$ and \r\nthen access its members via $C_i$ union member. This approach is truly closed, as \r\nwe cannot add new variants to the underlying union without modifying class \r\ndefinition. \r\n\r\nNote also that in this case both subject type and target types are the same and \r\nwe use an integral constant to distinguish which member(s) of the underlying union \r\nis active now. In the other two cases the type of a subject is a base class of \r\nthe target type and we use either run-time type information or the integral \r\nconstant associated by the user with the target type to uncover the target type. \r\n\r\n\\subsection{Expression Templates}\r\n\r\nInterestingly enough C++ has a pure functional sublanguage in it that has a \r\nstriking similarity to ML and Haskell. The sublanguage in question is template \r\nfacilities of C++ that has been shown to be Turing \r\ncomplete\\cite{veldhuizen:templates_turing_complete}. \r\n\r\nHaskell definition of \\code{factorial} we saw earlier can be rewritten in \r\ntemplate sublanguage of C++ as following:\r\n\r\n\\begin{lstlisting}\r\ntemplate \r\n struct factorial { enum { result = N*factorial::result }; };\r\ntemplate <>\r\n struct factorial<0> { enum { result = 1 }; };\r\n\\end{lstlisting}\r\n\r\n\\noindent\r\nOne can easily see similarity with equational definitions in Haskell, with the \r\nexception that more specific cases (specialization for 0) have to follow the \r\ngeneral definition in C++. The main difference between Haskell definition and \r\nits C++ counterpart is that the former describes computations on \\emph{run-time \r\nvalues}, while the latter can only work with \\emph{compile-time values}.\r\n\r\nTurns out we can even express our $exp$ language using this functional \r\nsublanguage:\r\n\r\n\\begin{lstlisting}\r\ntemplate \r\nstruct value {\r\n value(const T& t) : m_value(t) {}\r\n T m_value;\r\n};\r\n\r\ntemplate \r\nstruct variable {\r\n variable() : m_value() {}\r\n T m_value;\r\n};\r\n\r\ntemplate \r\nstruct plus {\r\n plus(const E1& e1, const E2& e2) : m_e1(e1), m_e2(e2) {}\r\n const E1 m_e1; const E2 m_e2;\r\n};\r\n\r\n// ... definitions of other expressions\r\n\\end{lstlisting}\r\n\r\n\\noindent The idea is that expressions can be composed out of subexpressions, \r\nwhose shape (type) is passed as arguments to above templates. Explicit \r\ndescription of such expressions is very tedious however and is thus never \r\nexpressed directly, but as a result of corresponding operations: \r\n\r\n\\begin{lstlisting}[keepspaces,columns=flexible]\r\ntemplate \r\n value val(const T& t) { return value(t); }\r\ntemplate \r\n plus operator+(const E1& e1, const E2& e2)\r\n { return plus(e1,e2); }\r\n\\end{lstlisting}\r\n\r\n\\noindent With this, one can now capture various expressions as following:\r\n\r\n\\begin{lstlisting}\r\nvariable v;\r\nauto x = v + val(3);\r\n\\end{lstlisting}\r\n\r\n\\noindent The type of variable \\code{x} -- \\code{plus,value>}\r\n -- captures the structure of the expression, while the values inside of it \r\nrepresent various subexpressions the expression was created with. Such an \r\nexpression can be arbitrarily, but finitely nested. Note that value 3 is not \r\nadded to the value of variable \\code{v} here, but the expression \\code{v+3} is \r\nrecorded, while the meaning to such expression can be given differently in \r\ndifferent contexts. A general observation is that only the shape of the \r\nexpression becomes fixed at compile time, while the values of variables involved \r\nin it can be changed arbitrarily at run time, allowing for \\emph{lazy \r\nevaluation} of the expression. Polymorphic function \\code{eval} below implements \r\njust that:\r\n\r\n\\begin{lstlisting}[keepspaces,columns=flexible]\r\ntemplate \r\n T eval(const value& e) { return e.m_value; }\r\ntemplate \r\n T eval(const variable& e) { return e.m_value; }\r\ntemplate \r\n auto eval(const plus& e) \r\n -> decltype(eval(e.m_e1) + eval(e.m_e2))\r\n { return eval(e.m_e1) + eval(e.m_e2); }\r\n\\end{lstlisting}\r\n\r\n\\noindent One can now modify value of the variable \\code{v} and re-evaluate \r\nexpression as following:\r\n\r\n\\begin{lstlisting}\r\nv = 7; // assumes overloading of assignment\r\nint r = eval(x); // returns 10\r\n\\end{lstlisting}\r\n\r\n\\noindent The above technique for lazy evaluation of expressions was \r\nindependently invented by Todd Veldhuizen and David Vandevoorde and is generally \r\nknown in the C++ community by the name \\emph{Expression Templates} that Todd \r\ncoined\\cite{Veldhuizen95expressiontemplates, vandevoorde2003c++}. \r\n\r\nNote again how implementation of \\code{eval} resembles equations in Haskell that \r\ndecompose an algebraic data type. The similarities are so striking that there \r\nwere attempts to use Haskell as a pseudo code language for template \r\nmetaprogramming in C++\\cite{Milewski11}. A key observation in this analogy is \r\nthat partial and explicit template specialization of C++ class templates are \r\nsimilar to defining equations for Haskell functions. Variables introduced via \r\ntemplate clause of each equation serve as \\emph{variable patterns}, while the \r\nnames of actual templates describing arguments serve as \\emph{variant \r\nconstructors}. An important difference between the two is that Haskell's \r\nequations use \\emph{first-fit} strategy making order of equations important, \r\nwhile C++ uses \\emph{best-fit} strategy, thus making the order irrelevant.\r\n\r\nPatterns expressed this way can be arbitrarily nested as long as they can be \r\nexpressed in terms of the types involved and not the values they store. Using \r\nthe above example, for instance, it is very easy to specialize \\code{eval} for \r\nan expression of form $c_1*x+c_2$ where $c_i$ are some (not known) constant \r\nvalues and $x$ is any variable. Specializing for a concrete instance of that \r\nexpression $2*x+3$ will be much harder, because in the representation we chose \r\nvalues 2 and 3 become run-time values and thus cannot participate in \r\ncompile-time computations anymore. In this case we could have devised a template \r\nthat allocates a dedicated type for each constant making such value part of the \r\ntype:\r\n\r\n\\begin{lstlisting}[keepspaces,columns=flexible]\r\ntemplate struct constant {};\r\n\r\ntemplate \r\n T eval(const constant& e) { return t; }\r\ntemplate \r\n auto eval(const times,E>& e) \r\n -> decltype(eval(e.m_e2)) \r\n { return (decltype(eval(e.m_e2)))(0); }\r\ntemplate \r\n auto eval(const times>& e) \r\n -> decltype(eval(e.m_e1)) \r\n { return (decltype(eval(e.m_e1)))(0); }\r\n\\end{lstlisting}\r\n\r\n\\noindent Here the first equation for \\code{eval} describes the necessary general \r\ncase for handling expressions of type \\code{constant}, while the other two \r\nare redundant cases that can be seen as an optimization detecting expressions of \r\nthe form $e*0$ and $0*e$ for any arbitrary expression $e$ and returning 0 \r\nwithout actually computing $e$.\r\n\r\nUnfortunately, a similar pattern to detect expressions of the form $x-x$ for any \r\nvariable $x$ cannot be expressed because expression templates are blind to \r\nobject identity and can only see their types. This means that expression \r\ntemplates of the form $x-y$ are indisthinguishable at compile time from \r\nexpressions of the form $x-x$ because their types are identical.\r\n\r\nNevertheless, with all the limitations, expression templates provide an \r\nextremely powerful abstraction mechanism, which we use to express a \r\npattern-language for our SELL. Coincidentally, we employ the compile-time \r\npattern-matching facility already supported by C++ as a meta-language to \r\nimplement its run-time counterpart.\r\n", "meta": {"hexsha": "56c366eb2a120e4011f89cc2b5a346c82c68d9f7", "size": 32449, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "media/papers/TR/sec-2-background.tex", "max_stars_repo_name": "akrzemi1/Mach7", "max_stars_repo_head_hexsha": "eef288eb9fe59712ff153dd70791365391b7b118", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 1310, "max_stars_repo_stars_event_min_datetime": "2015-01-04T03:44:04.000Z", "max_stars_repo_stars_event_max_datetime": "2022-03-18T04:44:01.000Z", "max_issues_repo_path": "media/papers/TR/sec-2-background.tex", "max_issues_repo_name": "akrzemi1/Mach7", "max_issues_repo_head_hexsha": "eef288eb9fe59712ff153dd70791365391b7b118", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 62, "max_issues_repo_issues_event_min_datetime": "2015-01-12T07:59:17.000Z", "max_issues_repo_issues_event_max_datetime": "2021-11-14T22:02:14.000Z", "max_forks_repo_path": "media/papers/TR/sec-2-background.tex", "max_forks_repo_name": "akrzemi1/Mach7", "max_forks_repo_head_hexsha": "eef288eb9fe59712ff153dd70791365391b7b118", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": 108, "max_forks_repo_forks_event_min_datetime": "2015-02-13T17:39:07.000Z", "max_forks_repo_forks_event_max_datetime": "2021-11-18T11:06:59.000Z", "avg_line_length": 52.9347471452, "max_line_length": 97, "alphanum_fraction": 0.7529661931, "num_tokens": 7885, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6113819591324418, "lm_q2_score": 0.5583269943353745, "lm_q1q2_score": 0.34135105163328894}} {"text": "\\input{templates/lec_template.tex}\n\n\\begin{document}\n\n\\title[Recursion]{Week 10: Recursion}\n\\subtitle{Basic Programming in Python}\n\n\\date{\\today}\n\n\\begin{frame}[plain]\n \\titlepage\n\\end{frame}\n\n\\begin{frame}\n \\tableofcontents\n\\end{frame}\n\n\\section{Fibonacci}\n\n\\begin{frame}\n \\sectionpage\n\\end{frame}\n\n\n\\begin{frame}{The Fibonacci Number}\n\n\\textbf{Definition:}\n\n\n$F_0 = 0, F_1 = 1$\n\n$F_n = F_{n-1} + F_{n-2}$\n\n\\vspace{1em}\n\\textbf{Example:}\n\n\n$F_2 = F_1 + F_0 = 0 + 1 = 1$\n\n$F_3 = F_2 + F_1 = 1 + 1 = 2$\n\n$F_{100} = ?$\n\n\\note{\n \\begin{center}\n \\includegraphics[height=0.8\\textheight]{10_Recursion/rabbits.png}\n \\end{center}\n\n}\n\\end{frame}\n\n\\end{document}\n", "meta": {"hexsha": "7f0bffedfcb22d1dca2f7b86988ce2d98e66b488", "size": 676, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "2021/10_Recursion/10_Recursion_Lec.tex", "max_stars_repo_name": "lfrommelt/monty", "max_stars_repo_head_hexsha": "e8cabf0e4ac01ab3d97eecee5e699139076d6544", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "2021/10_Recursion/10_Recursion_Lec.tex", "max_issues_repo_name": "lfrommelt/monty", "max_issues_repo_head_hexsha": "e8cabf0e4ac01ab3d97eecee5e699139076d6544", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "2021/10_Recursion/10_Recursion_Lec.tex", "max_forks_repo_name": "lfrommelt/monty", "max_forks_repo_head_hexsha": "e8cabf0e4ac01ab3d97eecee5e699139076d6544", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 12.7547169811, "max_line_length": 73, "alphanum_fraction": 0.6420118343, "num_tokens": 259, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230157, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.34130061746091017}} {"text": "%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n% Short Sectioned Assignment\n% LaTeX Template\n% Version 1.0 (5/5/12)\n%\n% This template has been downloaded from:\n% http://www.LaTeXTemplates.com\n%\n% Original author:\n% Frits Wenneker (http://www.howtotex.com)\n%\n% License:\n% CC BY-NC-SA 3.0 (http://creativecommons.org/licenses/by-nc-sa/3.0/)\n%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n%----------------------------------------------------------------------------------------\n%\tPACKAGES AND OTHER DOCUMENT CONFIGURATIONS\n%----------------------------------------------------------------------------------------\n\\documentclass[a4paper]{article}\n\n\\usepackage[hmarginratio=1:1,top=32mm,margin=20mm]{geometry} % Document margins\n\\usepackage{lmodern}\n\\usepackage[T1]{fontenc}\t% Use 8-bit encoding that has 256 glyphs\n\\usepackage[english]{babel}\t % English language/hyphenation\n%\\usepackage[utf8x]{inputenc}\n\\usepackage{amsmath,amsfonts,amsthm,bm} % AMS - Math, Fonts, Theorems, BoldMath\n\\usepackage{graphicx} % For putting pictures\n\\usepackage{booktabs} % Horizontal rules in tables\n\\usepackage{hyperref} % For hyperlinks in the PDF\n\n\\usepackage{fancyhdr} % Headers and footers\n\\pagestyle{fancy} % All pages have headers and footers\n\\fancyhead{} % Blank out the default header\n\\fancyfoot{} % Blank out the default footer\n\\fancyhead[C]{Thesis} % Custom header text\n\\fancyfoot[L]{MathMods $\\bullet$ 2015 Apr}\n\\fancyfoot[R]{Page \\huge{\\thepage}} % Custom footer text\n\n\\numberwithin{equation}{section} % Number equations within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)\n\\numberwithin{figure}{section} % Number figures within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)\n\\numberwithin{table}{section} % Number tables within sections (i.e. 1.1, 1.2, 2.1, 2.2 instead of 1, 2, 3, 4)\n\n\\setlength\\parindent{0pt} % Removes all indentation from paragraphs - comment this line for an assignment with lots of text\n\n\\newtheorem{thm}{Theorem}\n\\newtheorem{corr}{Corollary}\n\\newtheorem{lem}{Lemma}\n\\newtheorem{prop}{Proposition}\n\\newtheorem*{rem}{Remark}\n\\newtheorem{defn}{Definition}\n\n%----------------------------------------------------------------------------------------\n%\tTITLE SECTION\n%----------------------------------------------------------------------------------------\n\n\\newcommand{\\horrule}[1]{\\rule{\\linewidth}{#1}} % Create horizontal rule command with 1 argument of height\n\n\\title{\t\n\\normalfont \\normalsize \n\\textsc{Thesis} \\\\ [25pt]\n\\horrule{0.5pt} \\\\[0.4cm] % Thin top horizontal rule\n\\huge The Singular Points method for Asian American options for local volatility models % The assignment title\n\\horrule{2pt} \\\\[0.5cm] % Thick bottom horizontal rule\n}\n\n\\author{Sudip Sinha}\n\n\\date{\\normalsize\\today}\n\n\\begin{document}\n\n\\maketitle\t% Print the title\n\n%\\begin{abstract}\n%Your abstract.\n%\\end{abstract}\n\n%----------------------------------------------------------------------------------------\n%\tPROBLEM 1\n%----------------------------------------------------------------------------------------\n\n\n\\section{Introduction}\nThe Singular Points method for Asian American options for local volatility models.\n\n\n\\section{Notations}\n$[n] = \\{0, 1, 2, \\dots, n\\}$\n\n\n\\section{Basic formulae}\n\n\\begin{subequations} \\label{eq:arithmeticmean}\nArithmetic average\n\\begin{align}\nA_{n} &= \\frac{\\sum_{i=0}^n S_i}{n+1} \\\\\n\\implies (n+1) A_{n} &= \\sum_{i=0}^n S_i\n\\end{align}\n\\end{subequations}\n\n\n\\section{Results}\n\n\\begin{defn}[Path]\nA path is a sequence $(j_i)_{i \\in [n]}$ such that $j_{i+1} \\in \\{ j_i,j_i+1 \\}$.\n\\end{defn}\n\n\n\\begin{lem}\nLet there be two paths $\\alpha$ and $\\beta$, such that $S_{i,j_i^\\alpha} >= S_{i,j_i^\\beta} \\; \\forall i$. Then $A^\\alpha >= A^\\beta$.\n\\end{lem}\n\n\\begin{proof}\nClearly if $S_{i,j_i^\\alpha} = S_{i,j_i^\\beta} \\; \\forall i$, then $A^\\alpha = A^\\beta$.\n\nWe only need to show the case of inequality.\nLet $S_{i,j_i^\\alpha} = S_{i,j_i^\\beta} \\; \\forall i \\in [n] \\setminus \\{l\\}$. That is, $S_{l,j_l^\\alpha} > S_{l,j_l^\\beta}$.\n\nNow, from equation \\ref{eq:arithmeticmean}, we have:\n\\begin{align*}\n(n+1) A_{n,j}^\\alpha &= \\sum_{i=0}^{l-1} S_{i,j_i} + S_{l,j_l^\\alpha} + \\sum_{i=l+1}^{n} S_{i,j_i} \\\\\n(n+1) A_{n,j}^\\beta &= \\sum_{i=0}^{l-1} S_{i,j_i} + S_{l,j_l^\\beta} + \\sum_{i=l+1}^{n} S_{i,j_i} \\\\\n\\implies (n+1) \\left(A_{n,j}^\\alpha - A_{n,j}^\\beta\\right) &= S_{l,j_l^\\alpha} - S_{l,j_l^\\beta} \\\\\n\t&= S_{l-1,j_{l-1}} u_l - S_{l-1,j_{l-1}} d_l \\\\\n\t&= S_{l-1,j_{l-1}} (u_l - d_l) > 0 \\\\\n\\implies A_{n,j}^\\alpha > A_{n,j}^\\beta\n\\end{align*}\n\n\\end{proof}\n\n\n\\begin{rem}\nThe path $\\alpha$ signifies the path above and $\\beta$ signifies the path below. Thus, the path above always has a higher arithmetic mean.\n\\end{rem}\n\n\n\\begin{corr}\nAt each node $N(n,j)$, the average values vary between a minimum average $A_{n,j}^{\\mathrm{min}}$ (corresponding to the path with $(n-j)$ down movements followed by $j$ up movements) and a maximum average $A_{n,j}^{\\mathrm{max}}$ (corresponding to the path with $j$ up movements followed by $(n-j)$ down movements).\n\\end{corr}\n\n\\begin{proof}\nThe path 'min' is the bottom-most one and 'max' is the topmost one.\n\\end{proof}\n\n\n\\begin{lem}[Lemma 3]\nThe price function at maturity $v_{n,j}$ is convex and piecewise linear.\n\\end{lem}\n\\begin{proof}\nBy construction. See the paper.\n\\end{proof}\n\n\n\\begin{lem}[Lemma 4]\nThe price function $v_{i,j}$ is concave and non-linear.\n\\end{lem}\n\\begin{proof}\n\\ref{eq:arithmeticmean}\n\n\\end{proof}\n\n\\section{Conclusion}\nThe singular points method may not be used to price Geometric Asian options.\n\n\\end{document}\n\n%%% Local Variables:\n%%% mode: latex\n%%% TeX-master: t\n%%% End:\n", "meta": {"hexsha": "b920f9b311c6bc607b6616aa30cf198806e93884", "size": 5568, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "MathMods/Thesis/docs/misc/tr_asian_singularpoints.tex", "max_stars_repo_name": "homdx/edu", "max_stars_repo_head_hexsha": "a32c9f1777f80a54c3d4a3fc8389748fe27739c0", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "MathMods/Thesis/docs/misc/tr_asian_singularpoints.tex", "max_issues_repo_name": "homdx/edu", "max_issues_repo_head_hexsha": "a32c9f1777f80a54c3d4a3fc8389748fe27739c0", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "MathMods/Thesis/docs/misc/tr_asian_singularpoints.tex", "max_forks_repo_name": "homdx/edu", "max_forks_repo_head_hexsha": "a32c9f1777f80a54c3d4a3fc8389748fe27739c0", "max_forks_repo_licenses": ["MIT"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-09-15T21:30:43.000Z", "max_forks_repo_forks_event_max_datetime": "2018-09-15T21:30:43.000Z", "avg_line_length": 32.3720930233, "max_line_length": 315, "alphanum_fraction": 0.6154813218, "num_tokens": 1752, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5156199157230156, "lm_q2_score": 0.6619228758499942, "lm_q1q2_score": 0.3413006174609101}} {"text": "\n\n\n\n\n\n\\chapter{Conclusions}\n\\label{chap:conclusions}\nWe have taken, over the course of this work, a tour through a rather wide expanse of material, so by way of conclusion we present a short summary of the main paradigms we have explored and the principal results we have uncovered.\n\n\n\n\\subsection*{Part I: Ionization}\n\n\\subsubsection[Analytical R-matrix Theory]{Analytical $R$-Matrix theory}\nIn chapter~\\ref{chap:R-matrix} we lay down the groundwork for the ionization parts of this thesis, reviewing in detail the construction of the Analytical $R$-Matrix theory of photoionization. We showed how one can split space into an inner and an outer region, using the inner region as a source term for the Schrödinger equation on the outer region, and how to employ the ionic and Coulomb-corrected continuum wavefunctions to produce a simple trajectory-based description of the photoionization, both on the single-active-electron direct channels as well as for the multi-electron correlation-driven mechanism. In addition, we presented in section~\\ref{sec:molecular-shape-factors} simple analytical formulas for a suitable model of a molecular orbital with nontrivial geometry.\n\n\n\n\n\\subsubsection{Multi-channel geometrical effects}\nIn chapter~\\ref{chap:multi-channel} we built on the multi-electron Analytical $R$-Matrix expressions for the direct and correlation-driven yields for photoelectrons that leave behind an excited ion, probing the ionization of a suitable model of an aligned carbon dioxide molecule, with a nontrivial combination of orbital geometries. There we showed how to implement a suitable modification of the saddle-point approximation for the correlation-driven geometrical factors, giving clean and physically transparent expressions. \n\nFinally, we showed how the nontrivial angular distributions obtained in this geometry admit a simple interpretation as double-slit diffraction fringes that originate in a pair of `slits' caused by the correlation-driven interaction with the ion inside the tunnelling~barrier.\n\n\n\\subsubsection{Analytical continuation of electrostatic potentials}\nIn chapter~\\ref{chap:complex-space-potentials} we examined in detail one of the crucial ingredients of the multi-electron ARM yield for correlation-driven ionization, the correlation interaction electrostatic potentials\n\\begin{equation}\n\\Vnm{\\vbr}=\\matrixel**{n}{\\sum_{j=1}^{N-1} \\frac{1-\\delta_{nm}}{\\| \\vbr - \\hat{\\vbr}_j\\|} }{m}\n,\n\\backtag{e4-correlation-interaction-potential-initial}\n\\end{equation}\nand their analytical continuations into complex positions $\\vbr$. We examined the naive analytical continuation for an elementary gaussian orbital via direct numerical integration, and we found it wanting, as it does not obey the Cauchy-Riemann equations.\n\nWe then focussed on the behaviour of simpler models -- exponential and gaussian type orbitals --, for which we can find exact expressions for the potentials, and therefore examine their analytical continuations directly. Here we find that, for points that are `real enough', in the sense that\n\\begin{equation}\n\\Re(\\vbr^2)>0,\n\\backtag{e4-re-r2-less-than-0}\n\\end{equation}\nthe different models agree surprisingly well, but that immediately upon leaving that region the potential for the gaussian-type orbital catastrophically diverges.\n\nThis means, then, that as long as we keep our evaluations of $\\Vnm{\\rl(t)}$ to trajectories that obey \\eqref{e4-re-r2-less-than-0} throughout, we can be rather confident that the analytical continuation is accurate, even if using gaussian-based quantum chemical calculations. Moreover, as we showed in chapter~\\ref{chap:quantum-orbits}, it is in fact possible to choose trajectories that adhere to this constraint.\n\nOn the other hand, the other region -- the points that are so imaginary that $\\Re(\\vbr^2)<0$~-- is much more challenging. As we argued, it is difficult to obtain any information, even of a qualitative type, about the behaviour of the potential in this region, even though there are formal existence theorems that guarantee us the existence of analytical continuations of our interaction potential.\n\n\n\n\\subsubsection{Quantum orbits in complex time and complex space}\nIn chapter~\\ref{chap:quantum-orbits} we explored the origin and meaning of the imaginary part of the ARM trajectory, which is of the form\n\\begin{equation}\n\\rl(t) = \\int_{\\ts}^{t} \\left[\\vbp+\\vba(\\tau) \\right] \\: \\d\\tau\n,\n\\backtag{e5-laser-driven-trajectory}\n\\end{equation}\nand is generally complex-valued for real times $t$. This imaginary part emerges directly from the Schrödinger equation, and more particularly from the boundary matching of our eikonal Volkov states with the WKB asymptotic expressions for the states of the system in the inner ARM region, and it is a crucial ingredient in allowing ARM do describe the Coulomb enhancement of ionization.\n\nHowever, this imaginary part of the position also combines with the branch cut of the square root in the Coulomb potential,\n\\begin{equation}\nU(\\rl(t)) = -\\frac{1}{\\sqrt{\\rl(t)^2}},\n\\backtag{e5-coulomb-potential-at-the-trajectory}\n\\end{equation}\nto imprint a series of branch cuts on the complex time plane, which we need to integrate over to obtain the ARM yield. Moreover, these branch cuts can and do intersect the standard contour along the real axis, which needs to be accordingly modified, ideally in an algorithmic way which can be performed by a computer to calculate photoelectron spectra.\n\nThis modification is possible by hinging on the concept of a time of closest approach, that is, times $\\tca$ that obey equations of the type\n\\begin{equation}\n\\rl(\\tca)\\cdot\\vbv(\\tca)=0,\n\\backtag{e5-tca-equation}\n\\end{equation}\nand which are always present as saddle points in the middle of any gate formed by two branch cuts. These have a fascinatingly rich geometry, with the complex-valued quantum versions forming multiple sheets of a surface encasing the corresponding classical version, and they show interesting topological transitions -- with strong effects on the requirements for the ARM integration path -- at soft recollisions, where the laser-driven trajectory has a turning point close to the ion.\n\nMoreover, we were able to successfully and algorithmically choose the appropriate $\\tca$s to use as waypoints by requiring that they be `out of the tunnelling barrier', having positive real part of the kinetic energy $\\vbv(t)^2$, after which it is possible to programmatically choose appropriate integration paths that avoid complex, and fast-changing, configurations of branch cuts. In addition, this navigation algorithm is also automatically able to steer us clear of the problematic regions of chapter~\\ref{chap:complex-space-potentials}.\n\n\n\n\n\n\\subsubsection{Low-Energy Structures and Near-Zero Energy Structures}\nIn chapter~\\ref{chap:LES-NZES}, after a review of the available experimental evidence on the Low-Energy and Near-Zero Energy Structures of mid-IR photoionization, we brought our branch-cut navigation algorithm from chapter~\\ref{chap:quantum-orbits} to bear on experiment, by using it to analyse the LES regions, which emerge naturally from one of the unavoidable (but resolvable) difficulties of the algorithm: the soft recollisions, where multiple branch cuts come into close proximity, interact, and undergo topological transitions.\n\nFurther, these LES peaks, which are well understood to be associated with these soft recollisions, are joined within the ARM formalism by a dynamically equivalent set at much lower energy, which approaches the ion on a forwards turning point, after an integral number of periods, instead of the more usual backwards turning points after a half-integral number of laser cycles.\n\nWe study the peaks caused by this new series of trajectories, showing that it produces a peak at energies consistent with those observed for the NZES, and that it produces qualitatively similar transverse photoelectron to those observed in high-resolution experiments. Moreover, the identification of this series as a contributor to the NZES opens clear ways to test this mechanism experimentally: since the new series only needs to advance by the tunnel exit over a laser period, instead of by the laser oscillation quiver radius $\\zquiv=F/\\omega^2$, its momentum and energy scale as\n\\begin{equation}\n\\pzsr \n\\approx \\frac{\\zexit}{\\Delta t}\n= \\frac{I_p/F}{(n+1)\\pi/\\omega}\n\\propto \\frac{I_p\\omega}{F}\n%\n\\quad\\text{and}\\quad\n%\n\\frac12\\left(\\pzsr\\right)^2 \n\\sim \\frac{I_p^2}{U_p}\n\\sim I_p\\gamma^2,\n\\backtagtwo{e6-pzsr-odd-n-summary}{e6-odd-n-energy-scaling}\n\\end{equation}\nrespectively. Thus, the new series scales inversely with the ponderomotive potential $U_p$ (as opposed to the known series, which scales as $\\frac12\\left(\\pzsr\\right)^2 \\propto U_p$), which means that it should be possible to probe the role of the new series by using experiments in harder targets with higher ionization potentials.\n\n\n\n\n\\section*{Part II: High-order harmonic generation}\n\n\\subsubsection{Conservation of spin angular momentum in bicircular HHG}\nAfter reviewing the standard theory of high-order harmonic generation in chapter~\\ref{chap:HHG-intro}, we turned in chapter~\\ref{chap:spin-HHG} to the generation of high-order harmonics by bicircular fields -- counter-rotating circularly polarized fields, one at the fundamental at $\\SI{800}{nm}$ and one detuned from its second harmonic, at $\\SI{410}{nm}$. We reviewed the experimental evidence that establishes clear selection rules coming from the conservation of spin angular momentum, together with the breakdown of these selection rules when the polarization of one of the drivers is degraded from circular through linear.\n\nWe then provided a suitable photon-picture model that is able to explain the observed harmonic emission while retaining a parametric picture of harmonic generation which conserves spin angular momentum on a per-channel basis~-- and, indeed, while maintaining consistency with known results when applied to the lowest-order channel, which reduces to perturbative four-wave mixing. \n\nOur model, based essentially on arguments lowest-order perturbation theory by separating an elliptical driver into circular components and treating them separately, is surprisingly effective at predicting the dependence of the harmonic emission on the ellipticity of the driver. Moreover, it correctly matches the results of a numerical experiment, which in principle can also be experimentally realized, where the two circular components of an elliptical field are taken as separate and detuned independently\n\n\n\n\\subsubsection{Nondipole effects in HHG through noncollinear bicircular beams}\nFinally, in chapter~\\ref{chap:nondipole-HHG}, we turned to the generation of harmonics in fields that are particularly strong, or at a very long wavelength, or both, looking to extend the harmonic cutoff past the barrier posed by the magnetic field of the driving laser as the velocity of the continuum electron increases.\n\nHere we proposed a simple and flexible scheme for addressing the continuum electron's motion along the direction, by combining two counter-rotating circularly polarized fields of the same frequency in a non-collinear configuration to produce a field with a forwards ellipticity that acts in the same direction as the magnetic Lorentz force.\n\nWe then extended the existing beyond-dipole Strong-Field Approximation HHG formalism to deal with arbitrary beam configurations, showing that to do so one needs to use slightly more complex nondipole Volkov states, and related this to a nontrivial average force acting on charged particles across the focus.\n\nFinally, with this extended beyond-dipole SFA in hand, we showed that the field configuration can indeed help recover the harmonic emission from its exponential quenching at the hands of the magnetic Lorentz force, and, moreover, that it can be used at much lower intensities and wavelengths to produce even harmonics that can be used to demonstrate the presence of the effect, for the first time in HHG, using currently available laser sources.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n ", "meta": {"hexsha": "6f146af50bb28daca4b881d62c8184c47ec5cd87", "size": 12064, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "10-Conclusions/Conclusions.tex", "max_stars_repo_name": "episanty/PhD-Thesis", "max_stars_repo_head_hexsha": "3347dfb59c11db5572a4139ee3b784ad56260e76", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": 5, "max_stars_repo_stars_event_min_datetime": "2016-11-16T19:28:08.000Z", "max_stars_repo_stars_event_max_datetime": "2021-12-19T12:32:56.000Z", "max_issues_repo_path": "10-Conclusions/Conclusions.tex", "max_issues_repo_name": "episanty/PhD-Thesis", "max_issues_repo_head_hexsha": "3347dfb59c11db5572a4139ee3b784ad56260e76", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 21, "max_issues_repo_issues_event_min_datetime": "2017-02-22T19:26:54.000Z", "max_issues_repo_issues_event_max_datetime": "2021-08-31T16:38:58.000Z", "max_forks_repo_path": "10-Conclusions/Conclusions.tex", "max_forks_repo_name": "episanty/PhD-Thesis", "max_forks_repo_head_hexsha": "3347dfb59c11db5572a4139ee3b784ad56260e76", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 3, "max_forks_repo_forks_event_min_datetime": "2016-12-17T06:06:55.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-26T11:08:08.000Z", "avg_line_length": 84.9577464789, "max_line_length": 780, "alphanum_fraction": 0.8032990716, "num_tokens": 2654, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. NO", "lm_q1_score": 0.7341195269001831, "lm_q2_score": 0.4649015713733885, "lm_q1q2_score": 0.34129332163178366}} {"text": "\\chapter{Astronomy}\n\n\\section{Reading sky map to find celestial objects}\n\n% https://en.wikipedia.org/wiki/Celestial_sphere\n\nCelestial sphere.\nThe sky looks as if it were projected to a spherical screen.\nIf a star is far enough, it will look as if it were fixed in the sky.\n\nYou find a star in the sky.\nYou write a letter to your friend.\nHow do you write where that star is?\nHow do you explain to him which direction he should look at?\n\nYou use the \\emph{equatorial coordinate system}.\n\n\\emph{Right ascension}, \\emph{declination}, and \\emph{epoch}.\n\nExample: Alpha Centauri A.\nRight ascension 14 h 39 m 35.06311 s.\nDeclination \\(-60\\deg\\) 50' 15.0992\".\nEpoch J2000.\n\n% https://en.wikipedia.org/wiki/Epoch_(astronomy)#Julian_years_and_J2000\nJ2000 is the Gregorian date 2000-01-01 12:00 TT (terrestrial time).\n\nStar chart, star map, sky map\n\n% https://en.wikipedia.org/wiki/Celestial_coordinate_system\n\nCelestial coordinate system\n\n% https://en.wikipedia.org/wiki/Star_chart\n\n% https://en.wikipedia.org/wiki/Celestial_coordinate_system\n\nEquatorial coordinate system\n\n% https://en.wikipedia.org/wiki/Alpha_Centauri\n\n\\section{Distance}\n\n% https://en.wikipedia.org/wiki/Parsec\n\n1 au (astronomical unit) is roughly the distance between the Sun and the Earth.\nIt is about 150 million km.\n\nParsec is a unit of length.\n\\( 648000/\\pi \\).\n1 pc is about 3.26 ly.\n\nA \\emph{light year} is the distance traveled by light in one year.\n\\emph{Light year} (ly) is a unit of \\emph{distance}, not time.\n1 au is about 6 light minutes.\n\n\\section{Objects}\n\nA \\emph{planet} is?\n\n% https://en.wikipedia.org/wiki/Stellar_evolution\nA \\emph{star} is a luminous sphere of plasma held together by its own gravity.\n(\"Star\", Wikipedia)\nEvery star begins from collapsing clouds of gas and dust.\nA \\emph{protostar} is ...\nA \\emph{main-sequence star} is ...\n\nA \\emph{solar system} is?\n\nA \\emph{galaxy} is?\n\nA \\emph{nebula} is?\n\nA \\emph{constellation} is?\n\nA \\emph{satellite} is?\n\nA \\emph{moon} is?\n\nA \\emph{comet} is?\n\nAn \\emph{asteroid} is?\n\nA \\emph{supernova} is?\n\nA \\emph{brown dwarf} is?\n\nA \\emph{white dwarf} is?\n\nA \\emph{black hole} is?\n\n\\section{Cosmology}\n\n\\section{Cosmogony}\n", "meta": {"hexsha": "b5fde2997c8b5ddc54c185c9be63fe0a54b93b87", "size": 2148, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "research/physics/astronomy.tex", "max_stars_repo_name": "edom/work", "max_stars_repo_head_hexsha": "df55868caa436efc631e145a43e833220b8da1d0", "max_stars_repo_licenses": ["Apache-2.0", "CC0-1.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "research/physics/astronomy.tex", "max_issues_repo_name": "edom/work", "max_issues_repo_head_hexsha": "df55868caa436efc631e145a43e833220b8da1d0", "max_issues_repo_licenses": ["Apache-2.0", "CC0-1.0"], "max_issues_count": 4, "max_issues_repo_issues_event_min_datetime": "2020-12-02T18:37:37.000Z", "max_issues_repo_issues_event_max_datetime": "2022-02-16T00:55:32.000Z", "max_forks_repo_path": "research/physics/astronomy.tex", "max_forks_repo_name": "edom/work", "max_forks_repo_head_hexsha": "df55868caa436efc631e145a43e833220b8da1d0", "max_forks_repo_licenses": ["Apache-2.0", "CC0-1.0"], "max_forks_count": 1, "max_forks_repo_forks_event_min_datetime": "2018-10-02T15:20:22.000Z", "max_forks_repo_forks_event_max_datetime": "2018-10-02T15:20:22.000Z", "avg_line_length": 22.6105263158, "max_line_length": 79, "alphanum_fraction": 0.7397579143, "num_tokens": 643, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878696277513, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.341048057064018}} {"text": "\\chapter{Actuator Disc Method}\n\n\\section{Coordinate Systems}\n\n\\subsection{Rotor Axis System}\n\nOrigin of the Rotor Axis System is coincident with the rotor hub center, the x-axis is positive in forward direction, the y-axis is positive right and z-axis is positive downward direction and it is coincident with the rotor shaft axis.\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=110mm]{eps/coordinate_system_RAS.eps}\n \\caption{Rotor Axis System}\n\\end{figure}\n\n\\subsection{Rotor-Wind Axis System}\n\nRotor-Wind Axis System is very much like Rotor Axis System, the only difference is that it is rotated about z-axis in such a manner that x-axis points directly into relative wind, so there is no lateral airspeed component.\n\n\\subsection{Control Axis System}\n\nFor most purposes, using the Rotor Axis System causes unnecessary complications. It is convenient to use no cyclic feathering axes system. \\cite{GessowMyers1985} Origin of the Control Axis System is coincident with the origin of the Rotor Axis System, but it is rotated by angles of the swashplate roll and pitch so there is no cyclic feathering in this coordinate system.\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=120mm]{eps/rotor_planes.eps}\n \\caption{Rotor reference planes}\n\\end{figure}\n\n\\subsection{Disc Axis System}\n\nOrigin of the Disc Axis System is coincident with the origin of the Rotor Axis System, but it is rotated by angles of the rotor cone roll and pitch in such a manner that z-axis is perpendicular to the tip path plane so there is no cyclic flapping in this coordinate system.\n\n\\subsection{Control-Wind Axis System}\n\nControl-Wind Axis System is very much like Control Axis System, the only difference is that it is rotated about z-axis in such a manner that x-axis points directly into relative wind, so there is no lateral airspeed component.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Assumptions}\n\nFollowing assumptions are made for the purpose of modeling helicopter rotor aerodynamics:\n\\begin{itemize}\n \\item[---] forces and moments generated by the rotor are considered to be quasi-steady,\n \\item[---] rotor lift force is a linear function of blade incidence angle and drag force is a quadratic function of lift, \\cite{Padfield2007}\n \\item[---] rotor blades have 3 degrees of freedom movement,\n \\item[---] inflow is uniformly distributed over rotor disc, \\cite{Padfield2007}\n \\item[---] reversed flow effects are ignored,\n \\item[---] airflow is considered to be quasi-steady and incompressible,\n \\item[---] thrust is considered to be parallel to the z-axis of the Control Axis System and magnitude of the thrust is considered to be magnitude of the resulting rotor force. \\cite{GessowMyers1985}\n\\end{itemize}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Momentum Theory}\n\n\\subsection{Momentum Theory for Axial Flight}\n\nMass flow through the rotor disc, momentum change and change in kinetic energy are given by the following formulas. \\cite{Padfield2007}\n\\begin{align}\n \\label{eq-aero-mass-flow}\n \\dot m\n &=\n \\rho A_1 V_C = \\rho A_R \\left( V_C + V_i \\right)\n =\n \\rho A_{i \\infty} \\left( V_C + V_{i \\infty} \\right) \\\\\n \\label{eq-aero-thrust-1}\n T\n &=\n \\dot m \\left( V_C + V_{i \\infty} \\right)\n -\n \\dot m V_C = \\dot m V_{ i \\infty } \\\\\n T \\left( V_C + V_{ i \\infty } \\right)\n &=\n \\frac{1}{2} \\dot m \\left( V_C + V_{ i \\infty } \\right)^2\n -\n \\frac{1}{2} \\dot m V_C^2\n =\n \\frac{1}{2} \\dot m \\left( 2V_C V_{ i \\infty } + V_{ i \\infty }^2 \\right)\n\\end{align}\n\nWhere:\n\\begin{description}[align=right,labelwidth=3cm]\n \\item [$A_1 = \\pi R_1^2$] [m\\textsuperscript{2}] control volume section area\n \\item [$A_R = \\pi R^2$] [m\\textsuperscript{2}] rotor disc area\n \\item [$A_{\\infty} = \\pi R_{\\infty}^2$] [m\\textsuperscript{2}] far wake slipstream section area\n \\item [$\\dot m$] [kg/s] mass flow\n \\item [$V_C$] [m/s] climb velocity\n \\item [$V_i$] [m/s] induced velocity\n \\item [$V_{i \\infty}$] [m/s] far wake induced velocity\n \\item [$T$] [N] rotor thrust\n\\end{description}\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=51mm]{eps/momentum_theory_axial.eps}\n \\caption{Flow trough a rotor in axial flight}\n\\end{figure}\n\nFrom these relationships it can be deduced that induced velocity in the far wake is twice the rotor inflow. \\cite{Padfield2007}\n\\begin{equation}\n \\label{eq-aero-indeced-vel}\n V_{i \\infty} = 2 V_i\n\\end{equation}\n\nSubstituting equations (\\ref{eq-aero-mass-flow}) and (\\ref{eq-aero-indeced-vel}) into (\\ref{eq-aero-thrust-1}) rotor thrust is given as follows:\n\\begin{equation}\n \\label{eq-aero-thrust-climb}\n T = 2 \\rho A_R \\left( V_C + V_i \\right) V_i\n\\end{equation}\n\nIn hover flight, this equation can be expressed as:\n\\begin{equation}\n \\label{eq-aero-thrust-hover}\n T = 2 \\rho A_R V_{ih}^2\n\\end{equation}\n\nTransforming equations (\\ref{eq-aero-thrust-climb}) and (\\ref{eq-aero-thrust-hover}) gives:\n\\begin{align}\n V_i &= \\frac{T}{ 2 \\rho A_R \\left( V_C + V_i \\right) } \\\\\n V_{ih} &= \\sqrt{ \\frac{T}{ 2 \\rho A_R } }\n\\end{align}\n\nWriting velocities in normalized form:\n\\begin{align}\n \\label{eq-aero-lambda-i}\n \\lambda_i &= \\frac{ V_i } { \\Omega_R R } \\\\\n \\label{eq-aero-lambda-ih}\n \\lambda_{ih} &= \\frac{ V_{ih} } { \\Omega_R R } \\\\\n \\mu_C &= \\frac{ V_C } { \\Omega_R R }\n\\end{align}\n\nRotor thrust coefficient is:\n\\begin{equation}\n \\label{eq-aero-thrust-coef}\n C_T = \\frac{ T }{ \\rho A_R \\Omega_R^2 R^2 }\n\\end{equation}\n\nThen equations (\\ref{eq-aero-lambda-i}) and (\\ref{eq-aero-lambda-ih}) can be expressed as:\n\\begin{align}\n \\lambda_i &= \\frac{ C_T }{ 2 \\left( \\mu_C + \\lambda_i \\right) } \\\\\n \\lambda_{ih} &= \\sqrt{ \\frac{ C_T }{ 2 } }\n\\end{align}\n\nCombining these equations gives:\n\\begin{equation}\n \\lambda_{ih}^2 = \\lambda_i \\left( \\mu_C + \\lambda_i \\right)\n\\end{equation}\n\nThis equation can be transformed into following form:\n\\begin{equation}\n \\label{eq-aero-lambda-i-2}\n \\lambda_i\n =\n - \\frac{\\mu_C}{2}\n +\n \\sqrt{ \\left( \\frac{\\mu_C}{2} \\right)^2 + \\lambda_{ih}^2 }\n\\end{equation}\n\nFor descent velocity $V_D = - V_C$ formula (\\ref{eq-aero-lambda-i-2}) can be written as:\n\\begin{equation}\n \\lambda_i\n =\n \\frac{\\mu_D}{2}\n -\n \\sqrt{ \\left( \\frac{\\mu_D}{2} \\right)^2 - \\lambda_{ih}^2 }\n\\end{equation}\n\nThis relationship is valid only in windmill brake state where the wake is fully established and the flow is upwards. \\cite{Padfield2007} It can be assumed that such a condition occurs when descent velocity is two times greater than induced velocity in hover. \\cite{Stepniewski1984}\n\nYoung’s approximation can be used to determine induced velocity outside range of momentum theory application. \\cite{Padfield2007}\n\\begin{align}\n \\lambda_i\n =\n \\lambda_{ih} \\left( 1 + \\frac{\\mu_D}{\\lambda_{ih}} \\right)\n &\\mathrm{~for~} 0 \\leq \\mu_D \\leq -1.5 \\lambda_{ih} \\\\\n \\lambda_i\n =\n \\lambda_{ih} \\left( 7 - 3 \\frac{\\mu_D}{\\lambda_{ih}} \\right)\n &\\mathrm{~for~} -1.5 \\lambda_{ih} < \\mu_D \\leq -2 \\lambda_{ih}\n\\end{align}\n\n\\subsection{Momentum Theory in Forward Flight}\n\nIn forward flight induced velocity in the far wake is twice the flow at the rotor. \\cite{Padfield2007} Expression for thrust is given as follows:\n\\begin{equation}\n T = \\dot m 2 V_i = \\left( \\rho A_R V' \\right) 2 V_i\n\\end{equation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=120mm]{eps/momentum_theory_forward.eps}\n \\caption{Flow trough a rotor in forward flight}\n\\end{figure}\n\nTransforming this equation for induced velocity gives:\n\\begin{equation}\n \\label{eq-aero-lambda-i-3}\n V_i = \\frac{T}{ 2 \\rho A_R V' }\n\\end{equation}\n\nWhere $V'$ is the resultant velocity at the rotor.\n\\begin{equation}\n \\label{eq-aero-vel-rotor}\n V' = \\sqrt{\n V^2 \\cos^2 \\alpha_R\n + \\left( V \\sin \\alpha_R - V_i \\right)^2\n }\n\\end{equation}\n\nWriting velocities in normalized form:\n\\begin{align}\n \\label{eq-aero-lambda-i-4}\n \\lambda_i &= \\frac{V_i} { \\Omega_R R } \\\\\n \\label{eq-aero-mu-x}\n \\mu_X &= \\frac{u_{rw}} { \\Omega_R R } \\\\\n \\label{eq-aero-mu-z}\n \\mu_Z &= \\frac{w_{rw}} { \\Omega_R R }\n\\end{align}\n\nWhere:\n\\begin{align}\n u_{rw} &= V \\cos \\alpha_R \\\\\n w_{rw} &= V \\sin \\alpha_R\n\\end{align}\n\nSubstituting equations (\\ref{eq-aero-lambda-i-4}), (\\ref{eq-aero-mu-x}), (\\ref{eq-aero-mu-z}) and (\\ref{eq-aero-vel-rotor}) into (\\ref{eq-aero-lambda-i-3}) formula for the normalized induced velocity can be expressed as follows.\n\\begin{equation}\n \\label{eq-aero-lambda-i-5}\n \\lambda_i\n =\n \\frac{C_T}{ 2 \\sqrt{ \\mu_X^2 + \\left( \\mu_Z - \\lambda_i \\right)^2 } }\n\\end{equation}\n\nIn high speed flight summing helicopter translational velocity and velocity due to rotor shaft rotation causes strong non-uniformities of rotor induced velocity. Glauert’s model is used to describe this phenomena. \\cite{Padfield2007, Bramwell2001}\n\\begin{equation}\n \\lambda_i \\left( r, \\Psi \\right)\n =\n \\lambda_{i0} + \\frac{r}{R} \\lambda_{1c} \\cos \\Psi\n\\end{equation}\n\nWhere:\n\\begin{align}\n \\lambda_{1c} &= \\lambda_{i0} \\tan \\left( \\frac{\\chi}{2} \\right)\n \\mathrm{~for~} \\chi < \\frac{\\pi}{2} \\\\\n \\lambda_{1c} &= \\lambda_{i0} \\cot \\left( \\frac{\\chi}{2} \\right)\n \\mathrm{~for~} \\chi > \\frac{\\pi}{2}\n\\end{align}\n\nThe wake angle is given by the following formula:\n\\begin{equation}\n \\chi = \\arctan \\left( \\frac{\\mu}{ \\lambda_{i0} - \\mu_Z } \\right)\n\\end{equation}\n\nWhere $\\lambda_{i0}$ is given by formula (\\ref{eq-aero-lambda-i-5}).\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Forces Acting on the Blade Segment}\n\nDetermining forces and moments acting on segment of the blade is made, assuming that blade is composed of aerodynamically independent, narrow strips of elements. \\cite{Stepniewski1984} High aspect ratio of the blade justifies usage of two-dimensional flow, while lift loss at the blade tip and root can be accounted by using tip-loss factor. \\cite{Padfield2007, Stepniewski1984, Bramwell2001}\n\nControl-Wind Axis System is used to determine forces and moments generated by the rotor, such computed forces and moments are the transformed to the Body Axis System.\n\n\\begin{figure}[h!]\n \\centering\n \\includegraphics[width=120mm]{eps/blade_element_theory_01.eps}\n \\caption{Rotor blade element}\n\\end{figure}\n\nLift and drag acting on a blade section is given by the following expressions:\n\\begin{align}\n \\label{eq-aero-blade-section-lift}\n dL &= \\frac{1}{2} \\rho U^2 \\left( r, \\Psi \\right) C_L c_b dr \\\\\n \\label{eq-aero-blade-section-drag}\n dD &= \\frac{1}{2} \\rho U^2 \\left( r, \\Psi \\right) C_D c_b dr\n\\end{align}\n\nLift and drag coefficients are given as follows: \\cite{Padfield2007}\n\\begin{align}\n \\label{eq-aero-blade-section-lift-coef}\n C_L &= a \\alpha_{BE} \\left( r, \\Psi \\right) \\\\\n C_D &= \\delta_0 + \\delta_2 C_T^2\n\\end{align}\n\nBlade section angle of attack is given by the following formula.\n\\begin{equation}\n \\alpha_{BE} \\left( r , \\Psi \\right) = \\theta + \\phi \\left( r , \\Psi \\right) \n\\end{equation}\n\nWhere:\n\\begin{equation}\n \\phi \\left( r , \\Psi \\right)\n =\n \\arctan \\frac{ U_P \\left( r, \\Psi \\right) }{ U_T \\left( r, \\Psi \\right) }\n\\end{equation}\n\nLinear velocity of the rotor hub and angular velocity expressed Rotor Axis System are given by the following formulas: \\cite{Stepniewski1984, Bramwell2001}\n\\begin{align}\n {\\vec V}_{RH,r}\n &=\n {\\boldsymbol T} \\left( \\varepsilon \\right)\n \\left(\n {\\vec V}_{O,b} + {\\vec \\omega}_b \\times {\\vec r}_{RH,b}\n \\right) \\\\\n {\\vec \\omega}_r\n &=\n {\\boldsymbol T} \\left( \\varepsilon \\right)\n {\\vec \\omega}_b\n\\end{align}\n\nRotattion matrix ${\\boldsymbol T} \\left( \\varepsilon \\right)$ is given as:\n\\begin{equation}\n {\\boldsymbol T} \\left( \\varepsilon \\right)\n =\n \\left[\n \\begin{matrix}\n \\cos \\varepsilon & 0 & \\sin \\varepsilon \\\\\n 0 & 1 & 0 \\\\\n -\\sin \\varepsilon & 0 & \\cos \\varepsilon \\\\\n \\end{matrix}\n \\right]\n\\end{equation}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=100mm]{eps/blade_element_theory_02.eps}\n \\caption{Velocity components at the blade element}\n\\end{figure}\n\nFollowing formulas can be used to transform this values to Control Axis System:\n\\begin{align}\n {\\vec V}_{RH,c} \n &=\n {\\boldsymbol T} \\left( \\theta_{1c}, \\theta_{1s} \\right)\n {\\vec V}_{RH,r} \\\\\n {\\vec \\omega}_c\n &=\n {\\boldsymbol T} \\left( \\theta_{1c}, \\theta_{1s} \\right)\n {\\vec \\omega}_r\n\\end{align}\n\nWhere rotation matrices are:\n\n--- for counterclockwise direction of rotor:\n\\begin{equation}\n {\\boldsymbol T} \\left( \\theta_{1c}, \\theta_{1s} \\right)\n =\n \\left[\n \\begin{matrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos \\theta_{1c} & -\\sin \\theta_{1c} \\\\\n 0 & \\sin \\theta_{1c} & \\cos \\theta_{1c} \\\\\n \\end{matrix}\n \\right]\n \\left[\n \\begin{matrix}\n \\cos \\theta_{1s} & 0 & -\\sin \\theta_{1s} \\\\\n 0 & 1 & 0 \\\\\n \\sin \\theta_{1s} & 0 & \\cos \\theta_{1s} \\\\\n \\end{matrix}\n \\right]\n\\end{equation}\n\n--- for clockwise direction of rotor:\n\\begin{equation}\n {\\boldsymbol T} \\left( \\theta_{1c}, \\theta_{1s} \\right)\n =\n \\left[\n \\begin{matrix}\n 1 & 0 & 0 \\\\\n 0 & \\cos \\theta_{1c} & \\sin \\theta_{1c} \\\\\n 0 & -\\sin \\theta_{1c} & \\cos \\theta_{1c} \\\\\n \\end{matrix}\n \\right]\n \\left[\n \\begin{matrix}\n \\cos \\theta_{1s} & 0 & -\\sin \\theta_{1s} \\\\\n 0 & 1 & 0 \\\\\n \\sin \\theta_{1s} & 0 & \\cos \\theta_{1s} \\\\\n \\end{matrix}\n \\right]\n\\end{equation}\n\nFollowing formulas can be used to transform linear and angular velocity vector to Control-Wind Axis System:\n\\begin{align}\n {\\vec V}_{RH,cw}\n &=\n {\\boldsymbol T} \\left( \\beta \\right) {\\vec V}_{RH,c} \\\\\n {\\vec \\omega}_{cw}\n &=\n {\\boldsymbol T} \\left( \\beta \\right) {\\vec \\omega}_c\n\\end{align}\n\nRotation matrix ${\\boldsymbol T} \\left( \\beta \\right)$ is given by formula (\\ref{eq-aero-matrix-beta}).\n\nAssuming that flapping angle is positive upwards and writing velocity components as:\n\\begin{align}\n {\\vec V}_{RH,cw} &= \\left[ u_{cw}, 0, w_{cw} \\right]^T \\\\\n {\\vec \\omega}_{cw} &= \\left[ p_{cw}, q_{cw}, 0 \\right]^T\n\\end{align}\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=120mm]{eps/blade_element_velocity.eps}\n \\caption{Air velocities at the blade element}\n\\end{figure}\n\nAir velocity at the blade segment is:\n\n--- for counterclockwise direction of rotor:\n\\begin{gather}\n U_T = \\Omega_R r \\cos \\beta + u_{cw} \\sin \\Psi \\\\\n \\begin{multlined}[0.88\\textwidth]\n U_P =\n w_{cw} \\cos \\beta - V_i \\cos \\beta\n - \\dot \\beta r - u_{cw} \\sin \\beta \\cos \\Psi \\\\\n + p_{cw} r \\sin \\Psi + q_{cw} r \\cos \\Psi\n \\end{multlined}\n\\end{gather}\n\n--- for clockwise direction of rotor:\n\\begin{gather}\n U_T = \\Omega_R r \\cos \\beta + u_{cw} \\sin \\Psi \\\\\n \\begin{multlined}[0.88\\textwidth]\n U_P =\n w_{cw} \\cos \\beta - V_i \\cos \\beta\n - \\dot \\beta r - u_{cw} \\sin \\beta \\cos \\Psi \\\\\n - p_{cw} r \\sin \\Psi + q_{cw} r \\cos \\Psi\n \\end{multlined}\n\\end{gather}\n\nAssuming that for small angles:\n\\begin{align}\n \\label{eq-aero-sin-beta-approx}\n \\sin \\beta \\approx \\beta \\\\\n \\label{eq-aero-cos-beta-approx}\n \\cos \\beta \\approx 1\n\\end{align}\n\nThis expressions can be simplified to:\n\n--- for counterclockwise direction of rotor:\n\\begin{align}\n U_T &= \\Omega_R r + u_{cw} \\sin \\Psi \\\\\n U_P &=\n w_{cw} - V_i - \\dot \\beta r - u_{cw} \\beta \\cos \\Psi\n + p_{cw} r \\sin \\Psi + q_{cw} r \\cos \\Psi\n\\end{align}\n\n--- for clockwise direction of rotor:\n \\begin{align}\n U_T &= \\Omega_R r + u_{cw} \\sin \\Psi \\\\\n U_P &=\n w_{cw} - V_i - \\dot \\beta r - u_{cw} \\beta \\cos \\Psi\n - p_{cw} r \\sin \\Psi + q_{cw} r \\cos \\Psi\n \\end{align}\n\nUsing normalized velocities:\n\n\\begin{align}\n \\mu &= \\frac{u_{cw}}{\\Omega_R R} \\\\\n \\lambda &= \\frac{w_{cw} - V_i}{\\Omega_R R}\n\\end{align}\n\nExpressions for air velocity at the blade segment can be written in the following form:\n\n--- for counterclockwise direction of rotor:\n\\begin{align}\n U_T &= \\Omega_R r + \\mu \\Omega_R R \\sin \\Psi \\\\\n U_P &=\n \\lambda \\Omega_R R - \\dot \\beta r - \\mu \\Omega_R R \\beta \\cos \\Psi\n + p_{cw} r \\sin \\Psi + q_{cw} r \\cos \\Psi\n\\end{align}\n\n--- for clockwise direction of rotor:\n\\begin{align}\n U_T &= \\Omega_R r + \\mu \\Omega_R R \\sin \\Psi \\\\\n U_P &=\n \\lambda \\Omega_R R - \\dot \\beta r - \\mu \\Omega_R R \\beta \\cos \\Psi\n - p_{cw} r \\sin \\Psi + q_{cw} r \\cos \\Psi\n\\end{align}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Rotor Thrust}\n\nAssuming that for small angles:\n\\begin{align}\n \\label{eq-aero-approx-phi}\n \\phi = \\arctan &\\frac{U_P}{U_T} \\approx \\frac{U_P}{U_T} \\\\\n \\label{eq-aero-approx-u}\n U &\\approx U_T \\\\\n \\label{eq-aero-approx-dt}\n dT &\\approx dL\n\\end{align}\n\nExpression for the blade segment angle of attack is given as follows:\n\\begin{equation}\n \\alpha_{BE} = \\theta + \\frac{U_P}{U_T}\n\\end{equation}\n\nThen expression (\\ref{eq-aero-blade-section-lift-coef}) can be written as. \\cite{GessowMyers1985}\n\\begin{equation}\n \\label{eq-aero-rotor-list-coef}\n C_L = a \\left( \\theta + \\frac{U_P}{U_T} \\right)\n\\end{equation}\n\nSubstituting expression (\\ref{eq-aero-blade-section-lift}) and taking into account simplifications (\\ref{eq-aero-approx-phi}), (\\ref{eq-aero-approx-u}) and (\\ref{eq-aero-approx-dt}) then rotor thrust is given by the following formula.\n\\begin{equation}\n dT\n \\approx\n \\frac{1}{2} \\rho a c_b U_T^2 \\left( \\theta + \\frac{U_P}{U_T} \\right) dr\n\\end{equation}\n\nTransforming this relationship gives:\n\\begin{equation}\n \\label{eq-aero-blade-section-thrust}\n dT\n \\approx\n \\frac{1}{2} \\rho a c_b\n \\left( \\theta U_T^2 + U_P U_T \\right) dr\n\\end{equation}\n\nTotal thrust generated by the rotor of $n_b$ blades can be determined by integrating differential equation (\\ref{eq-aero-blade-section-thrust}) first with respect to the azimuth then along the blade span. \\cite{GessowMyers1985} Total thrust is given as follows:\n\\begin{equation}\n \\label{eq-aero-thrust-2}\n T = \\frac{ n_b }{ 2 \\pi } \\int_{0}^{2 \\pi} \\int_{0}^{BR}\n \\frac{dT}{dr} dr d\\Psi\n\\end{equation}\n\nWhere $B$ is a tip loss factor.\n\nSubstituting (\\ref{eq-aero-blade-section-thrust}) into (\\ref{eq-aero-thrust-2}) gives:\n\\begin{equation}\n T = \\frac{1}{2} \\rho a c_b n_b\n \\left(\n \\theta \\frac{1}{2 \\pi}\n \\int_{0}^{2\\pi} \\int_{0}^{BR} \\frac{U_T^2}{dr} dr d\\Psi\n +\n \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\int_{0}^{BR} U_P U_T dr d\\Psi\n \\right)\n\\end{equation}\n\nNeglecting helicopter angular velocity expressions for $U_T^2$ and $U_P U_T$ can be written as:\n\\begin{gather}\n \\label{eq-aero-u-t-sq}\n U_T^2 =\n r^2 \\Omega_R^2 + 2 \\Omega_R^2 R r \\mu \\sin \\Psi\n + \\mu^2 R^2 \\Omega_R^2 \\sin^2 \\Psi \\\\\n \\label{eq-aero-u-p-u-t}\n \\begin{multlined}[0.88\\textwidth]\n U_P U_T =\n \\lambda \\Omega_R^2 R r - \\dot \\beta \\Omega_R r^2\n - \\beta \\mu \\Omega_R^2 R r \\cos \\Psi \\\\\n + \\lambda \\mu \\Omega_R^2 R^2 \\sin \\Psi\n - \\dot \\beta \\mu \\Omega_R R r \\sin \\Psi\n - \\beta \\mu^2 \\Omega_R^2 R^2 \\sin \\Psi \\cos \\Psi\n \\end{multlined}\n\\end{gather}\n\nExpression for the blade flapping angle can be written as follows: \\cite{GessowMyers1985, Padfield2007, NASA-TT-F-494}\n\\begin{equation}\n \\label{eq-aero-blade-flapping-angle}\n \\beta \\left( \\Psi \\right) = \n \\beta_0 + \\beta_{1c} \\cos \\Psi + \\beta_{1s} \\sin \\Psi\n\\end{equation}\n\nAssuming constant rotor revolution speed, blade flapping angle derivatives with respect to time can be written as derivatives with respect to the azimuth. \\cite{GessowMyers1985}\n\\begin{gather}\n \\dot \\beta = \\frac{d\\beta}{dt}\n = \\frac{d\\beta}{d\\Psi} \\frac{d\\Psi}{dt} =\n \\bar \\beta \\Omega_R =\n \\Omega_R\n \\left( \\beta_{1s} \\cos \\Psi - \\beta_{1c} \\sin \\Psi \\right) \\\\\n \\ddot \\beta = \\frac{d^2\\beta}{dt^2}\n = \\frac{d^2\\beta}{d\\Psi^2} \\left( \\frac{d\\Psi}{dt} \\right)^2 = \n \\bar{\\bar \\beta} \\Omega_R^2\n =\n -\\Omega_R^2\n \\left( \\beta_{1c} \\cos \\Psi + \\beta_{1s} \\sin \\Psi \\right)\n\\end{gather}\n\nThen expressions for $U_T^2$ and $U_P U_T$ can be written as:\n\\begin{gather}\n U_T^2 = r^2 \\Omega_R^2 + 2 \\Omega_R^2 R r \\mu \\sin \\Psi\n + \\mu^2 R^2 \\Omega_R^2 \\sin^2 \\Psi \\\\\n \\begin{multlined}[0.88\\textwidth]\n U_P U_T = \\lambda \\Omega_R^2 R r\n - \\beta_{1s} \\Omega_R^2 r^2 \\cos \\Psi\n + \\beta_{1c} \\Omega_R^2 r^2 \\sin \\Psi \\\\\n - \\beta \\mu \\Omega_R^2 R r \\cos \\Psi\n + \\lambda \\mu \\Omega_R^2 R^2 \\sin \\Psi\n - \\beta_{1s} \\mu \\Omega_R^2 R r \\sin \\Psi \\cos \\Psi \\\\\n + \\beta_{1c} \\mu \\Omega_R^2 R r \\sin^2 \\Psi\n - \\beta \\mu^2 \\Omega_R^2 R^2 \\sin \\Psi \\cos \\Psi\n \\end{multlined}\n\\end{gather}\n\nKnowing that: \\cite{GessowMyers1985}\n\\begin{gather}\n \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\sin \\Psi d\\Psi =\n \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\cos \\Psi d\\Psi = 0 \\\\\n \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\sin^2 \\Psi d\\Psi =\n \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\cos^2 \\Psi d\\Psi = \\frac{1}{2} \\\\\n \\frac{1}{2\\pi} \\int_{0}^{2\\pi} \\sin \\Psi \\cos \\Psi d\\Psi = 0\n\\end{gather}\n\nRotor thrust can be written as:\n\\begin{equation}\n T = \\frac{1}{2} \\rho a c_b n_b \\Omega_R^2 R^3 B\n \\left[\n \\frac{\\lambda B}{2} \n +\n \\frac{\\theta}{3} \\left( B^2 + \\frac{3}{2} \\mu^2 \\right)\n +\n \\frac{\\beta_{1c} B}{4} \\mu\n \\right]\n\\end{equation}\n\nUsing expression (\\ref{eq-aero-thrust-coef}) rotor thrust coefficient is given by the following formula:\n\\begin{equation}\n C_T = \\frac{1}{2} a s B\n \\left[\n \\frac{\\lambda B}{2}\n +\n \\frac{\\theta}{3} \\left( B^2 + \\frac{3}{2} \\mu^2 \\right)\n +\n \\frac{\\beta_{1c} B}{4} \\mu\n \\right]\n\\end{equation}\n\nWhere $s$ is rotor solidity.\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Rotor H Force}\n\n%TODO\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Rotor Y Force}\n\n%TODO\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Rotor Torque}\n\nThe torque on a blade element is given by the following formula. \\cite{GessowMyers1985, Bramwell2001}\n\\begin{equation}\n \\label{eq-aero-blade-section-torque}\n dQ = r \\left( dD \\cos \\phi - dL \\sin \\phi \\right) dr\n\\end{equation}\n\nAssuming that for small angles:\n\\begin{align}\n \\sin \\phi &\\approx \\phi \\\\\n \\cos \\phi &\\approx 1\n\\end{align}\n\nAnd assuming that drag coefficient is constant along blade span, expression (\\ref{eq-aero-blade-section-torque}) can be written as: \\cite{GessowMyers1985, Bramwell2001}\n\\begin{equation}\n dQ =\n \\frac{1}{2} \\rho U_T^2 C_D c_b r dr -\n \\frac{1}{2} \\rho U_T^2 C_L c_b r \\phi dr\n\\end{equation}\n\nTorque due to the profile drag can be expressed as: \\cite{Bramwell2001}\n\\begin{equation}\n Q_p = \\frac{n_b}{2\\pi}\n \\int_{0}^{R} \\int_{0}^{2\\pi} \\frac{1}{2} \\rho U_T^2 C_D c_b r d \\Psi dr\n\\end{equation}\n\nSubstituting (\\ref{eq-aero-u-t-sq}) and integrating this equation first with respect to the azimuth then along the blade span gives:\n\\begin{equation}\n Q_p = \\frac{1}{2} \\rho n_b c_b \\Omega_R^2 R^4 C_D\n \\left( \\frac{1}{4} + \\frac{1}{4} \\mu^2 \\right)\n\\end{equation}\n\nInduced torque is given by the following formula: \\cite{Bramwell2001}\n\\begin{equation}\n Q_i = \\frac{n_b}{2\\pi} \\int_{0}^{R} \\int_{0}^{2\\pi}\n \\frac{1}{2} \\rho U_T^2 C_L c_b r \\phi d \\Psi dr\n\\end{equation}\n\nSubstituting (\\ref{eq-aero-approx-phi}) and (\\ref{eq-aero-rotor-list-coef}) gives:\n\\begin{equation}\n \\label{eq-aero-q-i}\n Q_i = \\frac{n_b}{2\\pi} \\frac{1}{2} \\rho a c_b\n \\int_{0}^{R} \\int_{0}^{2\\pi}\n \\left( \\theta U_P U_T r + U_P^2 r \\right) d \\Psi dr\n\\end{equation}\n\nNeglecting helicopter angular velocity expressions for $U_P^2$ can be written as:\n\\begin{multline}\n \\label{eq-aero-u-p-sq}\n U_P^2 = \\dot \\beta^2 r^2\n + 2 \\beta \\dot \\beta \\mu \\Omega_R R r \\cos \\Psi\n - 2 \\dot \\beta \\lambda \\Omega_R R r \\\\\n + \\beta^2 \\mu^2 \\Omega_R^2 R^2 \\cos^2 \\Psi\n - 2 \\beta \\lambda \\mu \\Omega_R^2 R^2 \\cos \\Psi\n + \\lambda^2 \\Omega_R^2 R^2\n\\end{multline}\n\nSubstituting (\\ref{eq-aero-u-p-u-t}) and (\\ref{eq-aero-u-p-sq}) into (\\ref{eq-aero-q-i}) and integrating equation (\\ref{eq-aero-q-i}) first with respect to the azimuth then along the blade span gives:\n\\begin{multline}\nQ_i = \\frac{1}{2} \\rho a c_b n_b \\Omega_R^2 R^4\n\\left[\n \\frac{1}{3} \\lambda \\theta\n + \\frac{1}{2} \\lambda^2\n - \\frac{1}{8} \\left( \\beta_{1c}^2 + \\beta_{1s}^2 \\right)\n \\right.\n \\\\\n \\left.\n + \\frac{1}{2} \\mu^2\n \\left(\n \\frac{\\beta_0^2}{2}\n - \\frac{3}{8} \\beta_{1c}^2\n - \\frac{1}{8} \\beta_{1s}^2\n \\right)\n - \\frac{1}{2} \\mu \\lambda \\beta_{1c}\n + \\frac{1}{3} \\mu \\beta_0 \\beta_{1s}\n\\right]\n\\end{multline}\n\nThe total rotor torque is given as follows: \\cite{Bramwell2001, NASA-TT-F-494}\n\\begin{gather}\n Q = Q_p - Q_i \\\\\n \\begin{multlined}[0.88\\textwidth]\n Q = \\frac{1}{2} \\rho a c_b n_b \\Omega_R^2 R^4\n \\left[\n \\frac{C_D}{4a} \\left( 1 + \\mu^2 \\right)\n - \\frac{1}{3} \\lambda \\theta\n - \\frac{1}{2} \\lambda^2\n + \\frac{1}{8} \\left( \\beta_{1c}^2 + \\beta_{1s}^2 \\right)\n \\right.\n \\\\\n \\left.\n - \\frac{1}{2} \\mu^2\n \\left(\n \\frac{\\beta_0^2}{2}\n - \\frac{3}{8} \\beta_{1c}^2\n - \\frac{1}{8} \\beta_{1s}^2\n \\right)\n + \\frac{1}{2} \\mu \\lambda \\beta_{1c}\n - \\frac{1}{3} \\mu \\beta_0 \\beta_{1s}\n \\right]\n \\end{multlined}\n\\end{gather}\n\nRotor torque coefficient can be written as:\n\\begin{multline}\n C_Q = \\frac{1}{2} a s\n \\left[\n \\frac{C_D}{4a} \\left( 1 + \\mu^2 \\right)\n - \\frac{1}{3} \\lambda \\theta\n - \\frac{1}{2} \\lambda^2\n + \\frac{1}{8} \\left( \\beta_{1c}^2 + \\beta_{1s}^2 \\right)\n \\right.\n \\\\\n \\left.\n - \\frac{1}{2} \\mu^2\n \\left(\n \\frac{\\beta_0^2}{2}\n - \\frac{3}{8} \\beta_{1c}^2\n - \\frac{1}{8} \\beta_{1s}^2\n \\right)\n + \\frac{1}{2} \\mu \\lambda \\beta_{1c}\n - \\frac{1}{3} \\mu \\beta_0 \\beta_{1s}\n \\right]\n\\end{multline}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\n\\section{Flapping Coefficients}\n\nExpressions for blades flapping coefficients can be derived from the equation of moments equilibrium about flapping hinge using method described in \\cite{NASA-TT-F-494}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=100mm]{eps/blade_element_moments.eps}\n \\caption{Forces acting on the blade element}\n\\end{figure}\n\n\\begin{minipage}{\\textwidth}\n Moments equilibrium about flapping hinge can be written as follows: \\cite{GessowMyers1985}\n \\begin{equation}\n \\label{eq-aero-blade-moments-equilibrium}\n M_I + M_{CF} + M_C + M_T + M_W = 0\n \\end{equation}\n\n Where:\n \\begin{description}[align=right,labelwidth=1.5cm]\n \\item [$M_I$] [N$\\cdot$m] moment due to inertia forces of flapping\n \\item [$M_{CF}$] [N$\\cdot$m] moment due to centrifugal force\n \\item [$M_C$] [N$\\cdot$m] moment due to Coriolis force\n \\item [$M_T$] [N$\\cdot$m] moment due to thrust\n \\item [$M_W$] [N$\\cdot$m] moment due to weight\n \\end{description}\n\\end{minipage}\n\n\\subsection{Moments of Inertia Forces}\n\nAssuming that rotor revolution speed is constant, helicopter angular velocities are constant and rotor blades are able to move about flapping hinge axis while neglecting helicopter yaw motion and blades pitching and lagging motion then centrifugal force can be written as:\n\\begin{equation}\n dF_{CF} = m_b \\Omega_R^2 r dr\n\\end{equation}\n\nComponent of the Coriolis force laying on the flapping plane is given:\n\n--- for counterclockwise direction of rotor:\n\\begin{equation}\n dF_C =\n 2 m_b p_{cw} \\Omega_R r \\cos \\Psi dr\n - 2 m_b q_{cw} \\Omega_R r \\sin \\Psi dr\n\\end{equation}\n\n--- for clockwise direction of rotor:\n\\begin{equation}\n dF_C =\n - 2 m_b p_{cw} \\Omega_R r \\cos \\Psi dr\n - 2 m_b q_{cw} \\Omega_R r \\sin \\Psi dr\n\\end{equation}\n\nMoment of inertia forces of flapping is given by the following formula:\n\\begin{equation}\n M_I = - \\ddot \\beta \\int_{0}^{R} m_b r^2 dr\n\\end{equation}\n\nAssuming that rotor blades are homogeneous rods blade first moment of mass and moment of inertia can be written as: \\cite{NASA-TT-F-494}\n\\begin{align}\n \\label{eq-aero-j-b}\n I_b &\\approx \\int_{0}^{R} m_b r^2 dr \\\\\n \\label{eq-aero-s-b}\n S_b &\\approx \\int_{0}^{R} m_b r dr\n\\end{align}\n\nHence:\n\\begin{equation}\n M_I = - \\ddot \\beta I_b\n\\end{equation}\n\nTaking into account (\\ref{eq-aero-sin-beta-approx}) and (\\ref{eq-aero-cos-beta-approx}) moment of centrifugal forces is:\n\\begin{equation}\n \\label{eq-aero-m-cf}\n M_{CF} =\n - \\int_{0}^{R} \\beta m_b \\Omega_R^2 r^2 dr =\n - \\Omega_R^2 \\beta \\int_{0}^{R} m_b r^2 dr\n\\end{equation}\n\nSubstituting (\\ref{eq-aero-j-b}) into (\\ref{eq-aero-m-cf}) gives:\n\\begin{equation}\n M_{CF} = - \\Omega_R^2 \\beta I_b\n\\end{equation}\n\nMoment of Coriolis forces can be writes as:\n\n--- for counterclockwise direction of rotor:\n\\begin{multline}\n M_C =\n 2 \\int_{0}^{R} m_b p_{cw} \\Omega_R r^2 \\cos \\Psi dr\n - 2 \\int_{0}^{R} m_b q_{cw} \\Omega_R r^2 \\sin \\Psi dr\n = \\\\ =\n 2 p_{cw} \\Omega_R I_b \\cos \\Psi - 2 q_{cw} \\Omega_R I_b \\sin \\Psi\n\\end{multline}\n\n--- for clockwise direction of rotor:\n\\begin{multline}\n M_C =\n - 2 \\int_{0}^{R} m_b p_{cw} \\Omega_R r^2 \\cos \\Psi dr\n - 2 \\int_{0}^{R} m_b q_{cw} \\Omega_R r^2 \\sin \\Psi dr\n = \\\\ =\n -2 p_{cw} \\Omega_R I_b \\cos \\Psi - 2 q_{cw} \\Omega_R I_b \\sin \\Psi\n\\end{multline}\n\nUsing approximation (\\ref{eq-aero-s-b}) moment due to weight can expressed as:\n\\begin{equation}\n M_W = -g \\int_{0}^{R} m_b r dr = -g S_b\n\\end{equation}\n\n\\subsection{Moment of Thrust}\n\nUsing equation (\\ref{eq-aero-blade-section-thrust}) expression for moment of thrust about flapping hinge can be written as follows:\n\\begin{equation}\n M_T =\n \\int_{0}^{BR} dT r = \n \\frac{1}{2} \\rho a c_b\n \\int_{0}^{BR} \\left( \\theta U_T^2 + U_P U_T \\right) dr\n\\end{equation}\n\n\\subsection{Equilibrium of Moments about Flapping Hinge}\n\nSubstituting into (\\ref{eq-aero-blade-moments-equilibrium}) expressions for moments of thrust, weight, inertia, centrifugal and Coriolis forces moments equilibrium equation is given as:\n\n--- for counterclockwise direction of rotor:\n\\begin{gather}\n \\begin{multlined}[0.88\\textwidth]\n \\int_{0}^{BR} dT r\n - \\ddot \\beta I_b\n - \\beta \\Omega_R^2 I_b\n + 2 p_{cw} \\Omega_R I_b \\cos \\Psi \\\\\n - 2 q_{cw} \\Omega_R I_b \\sin \\Psi\n - g S_b\n = 0\n \\end{multlined}\n \\\\\n \\begin{multlined}[0.88\\textwidth]\n \\label{eq-aero-equilibrium-blade-moments-ccw-1}\n - I_b \\ddot \\beta\n - I_b \\beta \\Omega_R^2\n + \\int_{0}^{BR} dT r\n = 2 I_b q_{cw} \\Omega_R \\sin \\Psi \\\\\n - 2 I_b p_{cw} \\Omega_R \\cos \\Psi\n + g S_b\n \\end{multlined}\n\\end{gather}\n \n--- for clockwise direction of rotor:\n\\begin{gather}\n \\begin{multlined}[0.88\\textwidth]\n \\int_{0}^{BR} dT r\n - \\ddot \\beta I_b\n - \\beta \\Omega_R^2 I_b\n - 2 p_{cw} \\Omega_R I_b \\cos \\Psi \\\\\n - 2 q_{cw} \\Omega_R I_b \\sin \\Psi\n - g S_b\n = 0\n \\end{multlined}\n \\\\\n \\begin{multlined}[0.88\\textwidth]\n \\label{eq-aero-equilibrium-blade-moments-cw-1}\n - I_b \\ddot \\beta\n - I_b \\beta \\Omega_R^2\n + \\int_{0}^{BR} dT r\n = 2 I_b q_{cw} \\Omega_R \\sin \\Psi \\\\\n + 2 I_b p_{cw} \\Omega_R \\cos \\Psi\n + g S_b\n \\end{multlined}\n\\end{gather}\n\nDividing both sides of equations (\\ref{eq-aero-equilibrium-blade-moments-ccw-1}) and (\\ref{eq-aero-equilibrium-blade-moments-cw-1}) by $I_b \\Omega_R^2$ gives:\n\\begin{gather}\n \\label{eq-aero-equilibrium-blade-moments-ccw-2}\n \\bar{\\bar \\beta} + \\beta\n = \\frac{1}{ I_b \\Omega_R^2 } \\int_{0}^{BR} dT r\n - 2 \\frac{q_{cw}}{\\Omega_R} \\sin \\Psi\n + 2 \\frac{p_{cw}}{\\Omega_R} \\cos \\Psi\n - \\frac{g S_b}{I_b \\Omega_R^2} \\\\\n \\label{eq-aero-equilibrium-blade-moments-cw-2}\n \\bar{\\bar \\beta} + \\beta\n = \\frac{1}{ I_b \\Omega_R^2 } \\int_{0}^{BR} dT r\n - 2 \\frac{q_{cw}}{\\Omega_R} \\sin \\Psi\n - 2 \\frac{p_{cw}}{\\Omega_R} \\cos \\Psi\n - \\frac{g S_b}{I_b \\Omega_R^2}\n\\end{gather}\n\nSubstituting expressions (\\ref{eq-aero-u-t-sq}) and (\\ref{eq-aero-u-p-u-t}) into (\\ref{eq-aero-equilibrium-blade-moments-ccw-2}) and (\\ref{eq-aero-equilibrium-blade-moments-cw-2}) gives:\n\\begin{multline}\n \\label{eq-aero-blade-moment-thrust-ccw}\n M_T =\n \\int_{0}^{BR} dT r =\n \\frac{1}{2} \\rho a c_b\n \\int_{0}^{BR} \\left( \\theta U_T^2 + U_P U_T \\right) r dr = \\\\\n =\n \\frac{1}{2} \\rho a c_b R^4 \\Omega_R^2\n \\left(\n \\frac{B^4}{4} \\theta\n + \\frac{2}{3} B^3 \\theta \\mu \\sin \\Psi\n + \\frac{B^2}{2} \\theta \\mu^2 \\sin^2 \\Psi\n \\right.\n \\\\\n + \\frac{B^3}{3} \\lambda\n - \\frac{B^4}{4} \\bar \\beta\n - \\frac{B^3}{3} \\beta \\mu \\cos \\Psi\n + \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R} \\sin \\Psi \\\\\n + \\frac{B^2}{2} \\lambda \\mu \\sin \\Psi\n - \\frac{B^2}{2} \\beta \\mu^2 \\sin \\Psi \\cos \\Psi\n - \\frac{B^3}{3} \\bar \\beta \\mu \\sin \\Psi\n \\\\\n \\left.\n + \\frac{B^3}{3} \\frac{p_{cw}}{\\Omega_R} \\mu \\sin^2 \\Psi\n + \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R} \\cos \\Psi\n + \\frac{B^3}{3} \\frac{q_{cw}}{\\Omega_R} \\mu \\sin \\Psi \\cos \\Psi\n \\right)\n\\end{multline}\n\\begin{multline}\n \\label{eq-aero-blade-moment-thrust-cw}\n M_T =\n \\int_{0}^{BR} dT r =\n \\frac{1}{2} \\rho a c_b\n \\int_{0}^{BR} \\left( \\theta U_T^2 + U_P U_T \\right) r dr = \\\\\n =\n \\frac{1}{2} \\rho a c_b R^4 \\Omega_R^2\n \\left(\n \\frac{B^4}{4} \\theta\n + \\frac{2}{3} B^3 \\theta \\mu \\sin \\Psi\n + \\frac{B^2}{2} \\theta \\mu^2 \\sin^2 \\Psi\n \\right.\n \\\\\n + \\frac{B^3}{3} \\lambda\n - \\frac{B^4}{4} \\bar \\beta\n - \\frac{B^3}{3} \\beta \\mu \\cos \\Psi\n - \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R} \\sin \\Psi \\\\\n + \\frac{B^2}{2} \\lambda \\mu \\sin \\Psi\n - \\frac{B^2}{2} \\beta \\mu^2 \\sin \\Psi \\cos \\Psi\n - \\frac{B^3}{3} \\bar \\beta \\mu \\sin \\Psi\n \\\\\n \\left.\n - \\frac{B^3}{3} \\frac{p_{cw}}{\\Omega_R} \\mu \\sin^2 \\Psi\n + \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R} \\cos \\Psi\n + \\frac{B^3}{3} \\frac{q_{cw}}{\\Omega_R} \\mu \\sin \\Psi \\cos \\Psi\n \\right)\n\\end{multline}\n\nKnowing that:\n\\begin{align}\n \\label{eq-aero-trigonometric-4}\n \\sin \\Psi \\cos \\Psi &= \\frac{ \\sin \\left( 2 \\Psi \\right) }{2} \\\\\n \\label{eq-aero-trigonometric-5}\n \\sin^2 \\Psi &= \\frac{ 1 - \\cos \\left( 2 \\Psi \\right) }{2} \\\\\n \\label{eq-aero-trigonometric-6}\n \\cos^2 \\Psi &= \\frac{ 1 + \\cos \\left( 2 \\Psi \\right) }{2} \\\\\n \\label{eq-aero-trigonometric-7}\n \\cos \\Psi \\sin \\left( 2 \\Psi \\right) &=\n \\frac{ \\sin \\Psi + \\sin \\left( 3 \\Psi \\right) }{2} \\\\\n \\label{eq-aero-trigonometric-8}\n \\sin \\Psi \\sin \\left( 2 \\Psi \\right) &=\n \\frac{ \\cos \\Psi - \\cos \\left( 3 \\Psi \\right) }{2}\n\\end{align}\n\nSubstituting (\\ref{eq-aero-blade-moment-thrust-ccw}) into (\\ref{eq-aero-equilibrium-blade-moments-ccw-2}) and (\\ref{eq-aero-blade-moment-thrust-cw}) into (\\ref{eq-aero-equilibrium-blade-moments-cw-2}) gives:\n\\begin{multline}\n \\label{eq-aero-flapping-coefs-ccw-1}\n \\bar{\\bar \\beta} + \\bar \\beta \\frac{\\gamma}{2}\n \\left( \\frac{B^4}{4} + \\frac{B^3}{3} \\mu \\sin \\Psi \\right)\n + \\beta \\left(\n 1 + \\frac{B^3}{6} \\gamma \\mu \\cos \\Psi\n + \\frac{B^2}{8} \\gamma \\mu^2 \\sin \\left( 2 \\Psi \\right)\n \\right)\n = \\\\ =\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n + \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R} \\sin \\Psi\n + \\frac{B^2}{2} \\lambda \\mu \\sin \\Psi\n + \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu \\cos \\left( 2 \\Psi \\right)\n \\right.\n \\\\\n + \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R} \\cos \\Psi\n + \\frac{B^3}{6} \\frac{q_{cw}}{\\Omega_R} \\mu \\sin \\left( 2 \\Psi \\right)\n + \\frac{B^4}{4} \\theta\n \\\\\n \\left.\n + \\frac{2}{3} B^3 \\theta \\mu \\sin \\Psi\n + \\frac{B^2}{4} \\theta \\mu^2\n - \\frac{B^2}{4} \\theta \\mu^2 \\cos \\left( 2 \\Psi \\right)\n \\right)\n \\\\\n - 2 \\frac{q_{cw}}{\\Omega_R} \\sin \\Psi\n + 2 \\frac{p_{cw}}{\\Omega_R} \\cos \\Psi\n - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n\\end{multline}\n\\begin{multline}\n \\label{eq-aero-flapping-coefs-cw-1}\n \\bar{\\bar \\beta} + \\bar \\beta \\frac{\\gamma}{2}\n \\left( \\frac{B^4}{4} + \\frac{B^3}{3} \\mu \\sin \\Psi \\right)\n + \\beta \\left(\n 1 + \\frac{B^3}{6} \\gamma \\mu \\cos \\Psi\n + \\frac{B^2}{8} \\gamma \\mu^2 \\sin \\left( 2 \\Psi \\right)\n \\right)\n = \\\\ =\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n - \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R} \\sin \\Psi\n + \\frac{B^2}{2} \\lambda \\mu \\sin \\Psi\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu \\cos \\left( 2 \\Psi \\right)\n \\right.\n \\\\\n + \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R} \\cos \\Psi\n + \\frac{B^3}{6} \\frac{q_{cw}}{\\Omega_R} \\mu \\sin \\left( 2 \\Psi \\right)\n + \\frac{B^4}{4} \\theta\n \\\\\n \\left.\n + \\frac{2}{3} B^3 \\theta \\mu \\sin \\Psi\n + \\frac{B^2}{4} \\theta \\mu^2\n - \\frac{B^2}{4} \\theta \\mu^2 \\cos \\left( 2 \\Psi \\right)\n \\right)\n \\\\\n - 2 \\frac{q_{cw}}{\\Omega_R} \\sin \\Psi\n - 2 \\frac{p_{cw}}{\\Omega_R} \\cos \\Psi\n - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n\\end{multline}\n\nWhere $\\gamma$ is a blade Lock number.\n\nTransforming equations (\\ref{eq-aero-flapping-coefs-ccw-1}) and (\\ref{eq-aero-flapping-coefs-cw-1}) gives:\n\\begin{multline}\n \\label{eq-aero-flapping-coefs-ccw-2}\n \\bar{\\bar \\beta} + \\bar \\beta \\frac{\\gamma}{2}\n \\left( \\frac{B^4}{4} + \\frac{B^3}{3} \\mu \\sin \\Psi \\right)\n + \\beta \\left(\n 1 + \\frac{B^3}{6} \\gamma \\mu \\cos \\Psi\n + \\frac{B^2}{8} \\gamma \\mu^2 \\sin \\left( 2 \\Psi \\right)\n \\right)\n = \\\\ =\n \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n + \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right)\n - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\right]\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left( \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R} \\right)\n + 2 \\frac{p_{cw}}{\\Omega_R}\n \\right] \\cos \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R}\n + \\frac{B^2}{2} \\lambda \\mu\n + \\frac{2}{3} B^3 \\theta \\mu\n \\right)\n - 2 \\frac{q_{cw}}{\\Omega_R}\n \\right] \\sin \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n - \\frac{B^2}{4} \\theta \\mu^2\n \\right)\n \\right] \\cos \\left( 2 \\Psi \\right)\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left( \\frac{B^3}{6} \\frac{q_{cw}}{\\Omega_R} \\mu \\right)\n \\right] \\sin \\left( 2 \\Psi \\right)\n\\end{multline}\n\n\\vfill\n\n\\begin{multline}\n \\label{eq-aero-flapping-coefs-cw-2}\n \\bar{\\bar \\beta} + \\bar \\beta \\frac{\\gamma}{2}\n \\left( \\frac{B^4}{4} + \\frac{B^3}{3} \\mu \\sin \\Psi \\right)\n + \\beta \\left(\n 1 + \\frac{B^3}{6} \\gamma \\mu \\cos \\Psi\n + \\frac{B^2}{8} \\gamma \\mu^2 \\sin \\left( 2 \\Psi \\right)\n \\right)\n = \\\\ =\n \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right)\n - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\right]\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left( \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R} \\right)\n - 2 \\frac{p_{cw}}{\\Omega_R}\n \\right] \\cos \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n - \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R}\n + \\frac{B^2}{2} \\lambda \\mu\n + \\frac{2}{3} B^3 \\theta \\mu\n \\right)\n - 2 \\frac{q_{cw}}{\\Omega_R}\n \\right] \\sin \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n - \\frac{B^2}{4} \\theta \\mu^2\n \\right)\n \\right] \\cos \\left( 2 \\Psi \\right)\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left( \\frac{B^3}{6} \\frac{q_{cw}}{\\Omega_R} \\mu \\right)\n \\right] \\sin \\left( 2 \\Psi \\right)\n\\end{multline}\n\n\\vfill\n\nDifferentiating equation (\\ref{eq-aero-blade-flapping-angle}) gives:\n\\begin{align}\n \\label{eq-aero-beta-prim}\n \\bar \\beta &=\n \\left( \\beta_{1s} \\cos \\Psi - \\beta_{1c} \\sin \\Psi \\right) \\\\\n \\label{eq-aero-beta-bis}\n \\bar{\\bar \\beta} &= -\n \\left( \\beta_{1c} \\cos \\Psi + \\beta_{1s} \\sin \\Psi \\right)\n\\end{align}\n\nSubstituting expressions (\\ref{eq-aero-beta-prim}) and (\\ref{eq-aero-beta-bis}) into equations (\\ref{eq-aero-flapping-coefs-ccw-2}) and (\\ref{eq-aero-flapping-coefs-cw-2}) as well as using trigonometric identities (\\ref{eq-aero-trigonometric-4}), (\\ref{eq-aero-trigonometric-5}), (\\ref{eq-aero-trigonometric-6}), (\\ref{eq-aero-trigonometric-7}) and (\\ref{eq-aero-trigonometric-8}) gives:\n\\begin{gather}\n \\label{eq-aero-flapping-coefs-ccw-3}\n \\begin{multlined}[0.88\\textwidth]\n \\beta_0\n + \n \\beta_{1s} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} + B^2 \\right) \\cos \\Psi\n +\n \\beta_{1c} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} - B^2 \\right) \\sin \\Psi\n \\\\\n +\n \\beta_{1c} \\gamma \\frac{B^3}{6} \\mu \\cos \\left( 2 \\Psi \\right)\n +\n \\left(\n \\beta_{1s} \\gamma \\frac{B^3}{6} \\mu\n + \\beta_0 \\gamma \\frac{B^2}{8} \\mu^2\n \\right) \\sin \\left( 2 \\Psi \\right)\n \\\\\n +\n \\beta_{1c} \\gamma \\frac{B^2}{16} \\mu^2 \\sin \\left( 3 \\Psi \\right)\n -\n \\beta_{1s} \\gamma \\frac{B^2}{16} \\mu^2 \\cos \\left( 3 \\Psi \\right)\n = \\\\ =\n \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n + \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right) - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\right]\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R}\n - \\frac{B^3}{3} \\beta_0 \\mu\n \\right) + 2 \\frac{p_{cw}}{\\Omega_R}\n \\right] \\cos \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R}\n + \\frac{B^2}{2} \\lambda \\mu\n + \\frac{2}{3} B^3 \\theta \\mu\n \\right) - 2 \\frac{q_{cw}}{\\Omega_R}\n \\right] \\sin \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n - \\frac{B^2}{4} \\theta \\mu^2\n \\right)\n \\right] \\cos \\left( 2 \\Psi \\right)\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left( \\frac{B^3}{6} \\frac{q_{cw}}{\\Omega_R} \\mu \\right)\n \\right] \\sin \\left( 2 \\Psi \\right)\n \\end{multlined}\n \\\\\n \\label{eq-aero-flapping-coefs-cw-3}\n \\begin{multlined}[0.88\\textwidth]\n \\beta_0\n + \n \\beta_{1s} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} + B^2 \\right) \\cos \\Psi\n +\n \\beta_{1c} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} - B^2 \\right) \\sin \\Psi\n \\\\\n +\n \\beta_{1c} \\gamma \\frac{B^3}{6} \\mu \\cos \\left( 2 \\Psi \\right)\n +\n \\left(\n \\beta_{1s} \\gamma \\frac{B^3}{6} \\mu\n + \\beta_0 \\gamma \\frac{B^2}{8} \\mu^2\n \\right) \\sin \\left( 2 \\Psi \\right)\n \\\\\n -\n \\beta_{1s} \\gamma \\frac{B^2}{16} \\mu^2 \\cos \\left( 3 \\Psi \\right)\n +\n \\beta_{1c} \\gamma \\frac{B^2}{16} \\mu^2 \\sin \\left( 3 \\Psi \\right)\n = \\\\ =\n \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right) - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\right]\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R}\n - \\frac{B^3}{3} \\beta_0 \\mu\n \\right) - 2 \\frac{p_{cw}}{\\Omega_R}\n \\right] \\cos \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n - \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R}\n + \\frac{B^2}{2} \\lambda \\mu\n + \\frac{2}{3} B^3 \\theta \\mu\n \\right) - 2 \\frac{q_{cw}}{\\Omega_R}\n \\right] \\sin \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n - \\frac{B^2}{4} \\theta \\mu^2\n \\right)\n \\right] \\cos \\left( 2 \\Psi \\right)\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left( \\frac{B^3}{6} \\frac{q_{cw}}{\\Omega_R} \\mu \\right)\n \\right] \\sin \\left( 2 \\Psi \\right)\n \\end{multlined}\n\\end{gather}\n\nNeglecting all blade flapping harmonics above the first \\cite{GessowMyers1985} equations (\\ref{eq-aero-flapping-coefs-ccw-3}) and (\\ref{eq-aero-flapping-coefs-cw-3}) can be simplified as follows:\n\n\\begin{multline}\n \\label{eq-aero-flapping-coefs-ccw-4}\n \\beta_0\n +\n \\beta_{1s} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} + B^2 \\right) \\cos \\Psi\n +\n \\beta_{1c} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} - B^2 \\right) \\sin \\Psi\n = \\\\ =\n \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n + \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right) - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\right]\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R}\n - \\frac{B^3}{3} \\beta_0 \\mu\n \\right) + 2 \\frac{p_{cw}}{\\Omega_R}\n \\right] \\cos \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R}\n + \\frac{B^2}{2} \\lambda \\mu\n + \\frac{2}{3} B^3 \\theta \\mu\n \\right) - 2 \\frac{q_{cw}}{\\Omega_R}\n \\right] \\sin \\Psi\n\\end{multline}\n\n\\begin{multline}\n \\label{eq-aero-flapping-coefs-cw-4}\n \\beta_0\n +\n \\beta_{1s} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} + B^2 \\right) \\cos \\Psi\n +\n \\beta_{1c} \\gamma \\frac{B^2}{8}\n \\left( \\frac{\\mu^2}{2} - B^2 \\right) \\sin \\Psi\n = \\\\ =\n \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda\n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right) - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\right]\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^4}{4} \\frac{q_{cw}}{\\Omega_R}\n - \\frac{B^3}{3} \\beta_0 \\mu\n \\right) - 2 \\frac{p_{cw}}{\\Omega_R}\n \\right] \\cos \\Psi\n \\\\\n + \\left[\n \\frac{\\gamma}{2}\n \\left(\n - \\frac{B^4}{4} \\frac{p_{cw}}{\\Omega_R}\n + \\frac{B^2}{2} \\lambda \\mu\n + \\frac{2}{3} B^3 \\theta \\mu\n \\right) - 2 \\frac{q_{cw}}{\\Omega_R}\n \\right] \\sin \\Psi\n\\end{multline}\n\n\\clearpage\n\n\\subsection{Final Form}\n\nEquations (\\ref{eq-aero-flapping-coefs-ccw-4}) and (\\ref{eq-aero-flapping-coefs-cw-4}) can be transformed to get blade flapping coefficients:\n\n--- for counterclockwise direction of rotor:\n\\begin{align}\n \\beta_0 &=\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda \n + \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right) - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\\\\n \\beta_{1c} &=\n 2 \\mu \\left( \\lambda + \\frac{4}{3} B \\theta \\right)\n \\cfrac{1}{\\cfrac{\\mu^2}{2} - B^2}\n +\n \\left(\n B^4 \\frac{p_{cw}}{\\Omega_R}\n - 16 \\frac{q_{cw}}{\\gamma \\Omega_R}\n \\right)\n \\cfrac{1}{B^2\\left(\\cfrac{\\mu^2}{2}-B^2\\right)}\n \\\\\n \\beta_{1s} &=\n - \\frac{4}{3} \\beta_0 \\mu\n \\cfrac{B}{\\cfrac{\\mu^2}{2} + B^2}\n +\n \\left(\n B^4 \\frac{q_{cw}}{\\Omega_R} \n + 16 \\frac{p_{cw}}{\\gamma \\Omega_R}\n \\right)\n \\cfrac{1}{B^2 \\left(\\cfrac{\\mu^2}{2}+B^2\\right)}\n\\end{align}\n\n--- for clockwise direction of rotor:\n\\begin{align}\n \\beta_0 &=\n \\frac{\\gamma}{2}\n \\left(\n \\frac{B^3}{3} \\lambda \n - \\frac{B^3}{6} \\frac{p_{cw}}{\\Omega_R} \\mu\n + \\frac{B^4}{4} \\theta\n + \\frac{B^2}{4} \\theta \\mu^2\n \\right) - \\frac{ g S_b }{ I_b \\Omega_R^2 }\n \\\\\n \\beta_{1c} &=\n 2 \\mu \\left( \\lambda + \\frac{4}{3} B \\theta \\right)\n \\cfrac{1}{\\cfrac{\\mu^2}{2} - B^2}\n -\n \\left(\n B^4 \\frac{p_{cw}}{\\Omega_R}\n + 16 \\frac{q_{cw}}{\\gamma \\Omega_R}\n \\right)\n \\cfrac{1}{B^2\\left(\\cfrac{\\mu^2}{2}-B^2\\right)}\n \\\\\n \\beta_{1s} &=\n - \\frac{4}{3} \\beta_0 \\mu\n \\cfrac{B}{\\cfrac{\\mu^2}{2} + B^2}\n +\n \\left(\n B^4 \\frac{q_{cw}}{\\Omega_R} \n - 16 \\frac{p_{cw}}{\\gamma \\Omega_R}\n \\right)\n \\cfrac{1}{B^2 \\left(\\cfrac{\\mu^2}{2}+B^2\\right)}\n\\end{align}\n", "meta": {"hexsha": "f05f1b36a611d4e7e28ccd62ae1a55ad512022f1", "size": 46896, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "tex/rotor_2.tex", "max_stars_repo_name": "marek-cel/mscsim-docs", "max_stars_repo_head_hexsha": "9984f33c84787c4420f11f2834bb35e040e1f36f", "max_stars_repo_licenses": ["CC0-1.0"], "max_stars_count": 7, "max_stars_repo_stars_event_min_datetime": "2019-12-01T02:27:28.000Z", "max_stars_repo_stars_event_max_datetime": "2021-09-09T07:02:20.000Z", "max_issues_repo_path": "tex/rotor_2.tex", "max_issues_repo_name": "marek-cel/mscsim-docs", "max_issues_repo_head_hexsha": "9984f33c84787c4420f11f2834bb35e040e1f36f", "max_issues_repo_licenses": ["CC0-1.0"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "tex/rotor_2.tex", "max_forks_repo_name": "marek-cel/mscsim-docs", "max_forks_repo_head_hexsha": "9984f33c84787c4420f11f2834bb35e040e1f36f", "max_forks_repo_licenses": ["CC0-1.0"], "max_forks_count": 2, "max_forks_repo_forks_event_min_datetime": "2019-12-01T10:56:23.000Z", "max_forks_repo_forks_event_max_datetime": "2019-12-01T19:41:05.000Z", "avg_line_length": 31.4105827194, "max_line_length": 392, "alphanum_fraction": 0.5997952917, "num_tokens": 18074, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3410480492395055}} {"text": "\\graphicspath{ {imgs/} }\n\\documentclass[main.tex]{subfiles}\n\\externaldocument{02data.tex}\n\\externaldocument{04methods.tex}\n\\begin{document}\n\\chapter{Model}\\label{chap:model}\nWith the building blocks defined in the previous chapter it is possible to construct a complete model for the task of lung nodule detection. This section describes in detail the used model and the learning process that was used to train it.\n\n\\section{Network Architecture}\nThe model is inspired by the network presented by Huang~\\cite{huang2017lung}. It has three 3D convolutional layers and three dense layers. The full architecture can be seen in Figure~\\ref{fig:net_struct} and is in the following sections explained from the input to output.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.5]{net_struct.png}\n\\end{center}\n\\caption{Architecture of the neural network. Each of the convolutional layers is composed of a 3D convolution layer with the respective filter size followed by a batch normalization and a pooling layer. The pool size is $(2,2,2)$ with a stride of $(1,1,1)$. The structure of the neural network resembles the one described by Huang~\\cite{huang2017lung}.}\n\\label{fig:net_struct}\n\\end{figure}\n\n\\subsection{Input} \nThe input to the network are the patches that have been cut and stored from the complete lung scan as described in Chapter~\\ref{chap:data} with a shape of $(50x50x5)$. The patches are randomly augmented by flipping them in x and y plane (examples can be seen in \\ref{fig:input}). The augmentation is applied to make the learned classification more robust against distortions in the input and aiding in generalization. This makes sense in the specific scenario since the nodules are growing in different shapes and locations in the lung and flipping them is not producing an impossible input to the network. There is also an additional parameter that allows for scaling the input in the $x,y$ plane.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.25]{patches-health.png}\n\\includegraphics[scale=0.25]{patches-nodule.png}\n\\end{center}\n\\caption{Input data for the cases of healthy and nodule patches. The image is taken from Tensorboard and shows in the case of nodules the random permutation of the input data.}\n\\label{fig:input}\n\\end{figure}\n\n\\subsection{Hidden Layers}\nThe convolutional part has three convolutional layers with $40$, $20$ and $20$ kernels each. The kernel size is $(3,3,3)$. This is in accordance with Huang et al.'s~\\cite{huang2017lung} implementation. Each of them is followed by a batch normalization layer. Batch normalization is in TensorFlow implemented as described by Ioffe and Szegedy~\\cite{ioffe2015batch}. The input of it is fed into a max-pooling layer with a pool size of $(2,2,2)$. The output of the last convolutional layer is flattened and fed to the dense layers. Two dense layers with $64$ neurons each and a ReLU activation function are used in this model. Their output is finally combined in two neurons, forming a 2D output of the network. \n\n\n\\subsection{Output}\nThe output of the network is the activation of the final two neurons. The class of the input is determined by the neuron with the higher activation in an one-hot design. Given the index of the maximum, is $0$ encoding a healthy $1$ encoding a nodule patch.\n\n\n\\section{Training}\nDuring training, the network is operated with batches of the input data. Regularization methods used for this network include batch normalization and dropout. Batch normalization is already described in Section~\\ref{ss:convlayer}. Dropout is another regularization method which during training drops random neurons of the network - training effectively several models at once, as discovered by Srivastava et al.~\\cite{srivastava2014dropout}, which should increase the overall performance of the network. During training, no improved performance could be observed when applying dropout throughout the whole network. It was rather harmful if applied to the convolutional layers. Thus, the final model uses dropout only in the fully connected layers. The loss of the network is then computed by the softmax cross-entropy between the labels in their one-hot form and the output of the two neurons at the end of the network. The Adam Optimizer is used on this loss for adapting the network parameters. For each epoch, the training is done on the complete training set with randomly permuted batches of size $10$. The training time is set to 3 full days.\n \n\n\n\\end{document}\n", "meta": {"hexsha": "7399559281746d6b1f9100c5ef9028e1aee97cff", "size": 4522, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "doc/chapters/05model.tex", "max_stars_repo_name": "AndreaSuckro/acts_thesis", "max_stars_repo_head_hexsha": "24ed8d2d85d246d30b15e3f98a9f0b0f7da77f65", "max_stars_repo_licenses": ["MIT"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "doc/chapters/05model.tex", "max_issues_repo_name": "AndreaSuckro/acts_thesis", "max_issues_repo_head_hexsha": "24ed8d2d85d246d30b15e3f98a9f0b0f7da77f65", "max_issues_repo_licenses": ["MIT"], "max_issues_count": null, "max_issues_repo_issues_event_min_datetime": null, "max_issues_repo_issues_event_max_datetime": null, "max_forks_repo_path": "doc/chapters/05model.tex", "max_forks_repo_name": "AndreaSuckro/acts_thesis", "max_forks_repo_head_hexsha": "24ed8d2d85d246d30b15e3f98a9f0b0f7da77f65", "max_forks_repo_licenses": ["MIT"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 98.3043478261, "max_line_length": 1148, "alphanum_fraction": 0.7872622733, "num_tokens": 1021, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.6150878555160665, "lm_q2_score": 0.5544704649604273, "lm_q1q2_score": 0.3410480492395055}} {"text": "\\documentclass{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage{amssymb,amsfonts,amsmath}\n\\usepackage{graphicx}\n\\usepackage{caption}\n\\usepackage{subfig}\n\\usepackage{color}\n\\usepackage[affil-it]{authblk}\n\\usepackage{multirow}\n\\usepackage{fullpage}\n\\usepackage{booktabs}\n\\usepackage{pdfsync}\n\n%\\usepackage[nofiglist, notablist, nomarkers]{endfloat}\n\\input{utils.tex}\n%\\title{Probabilistic learning of confounding factors in genetical\n%genomics studies}\n\\title{Derivations and mathematical details of GPmix}\n\\author[1]{Christoph Lippert, Oliver Stegle}\n\n\\affil[1]{Department Empirical Inference,\nMax Planck Institutes T\\\"ubingen, Germany}\n\\date{}\n\n\\captionsetup[subfloat]{listofformat=parens}\n\\newcommand{\\OLI}[1]{{\\color{blue}\\fbox{OLI} #1}}\n\\makeatletter\n\\newcommand{\\rmnum}[1]{\\romannumeral #1}\n\\newcommand{\\Rmnum}[1]{\\expandafter\\@slowromancap\\romannumeral #1@}\n\\makeatother\n\n\\newcommand{\\fix}{\\marginpar{FIX}}\n\\newcommand{\\new}{\\marginpar{NEW}}\n\n%\\nipsfinalcopy % Uncomment for camera-ready version\n%\\newcommand{\\B}[1]{\\bm{#1}} Christoph:removed this to make things consistent\n\\newcommand{\\B}[1]{{\\bf{#1}}}\n\\newcommand{\\Exp}{\\mathbb{E}}\n\\newcommand{\\norma}[1]{\\mathcal{N}\\left(#1\\right)}\n\\newcommand{\\eref}[1]{(\\ref{#1})}\n\\renewcommand{\\R}{\\mathbb{R}}\n\\newcommand\\norm[1]{\\left\\Vert {#1} \\right\\Vert}\n\\newcommand\\rank{\\mathrm{rank}}\n\\newcommand\\tr{\\mathrm{Tr}}\n\\newcommand\\Normal[3]{\\normal{#1}{{#2}\\;;\\;{#3}}}\n\\newcommand\\ve[1]{\\text{vec}\\left(#1\\right)}\n\\newcommand\\Real{\\mathbb{R}}\n\\newcommand{\\Ykron}{\\B{U}_R^\\T\\B{Y}\\B{U}_C}\n\\newcommand{\\XWAkron}[1]{\\B{U}_R^\\T\\B{X}_{#1}\\B{W}_{#1}\\B{A}_{#1}\\B{U}_C}\n\\newcommand{\\XWAkronT}[1]{\\B{U}_C^\\T\\B{A}_{#1}^\\T\\B{W}_{#1}^{\\T}\\B{X}_{#1}^{\\T}\\B{U}_R}\n\\begin{document}\n\\maketitle\n\n\n\\section{Some definitions}\n\\begin{itemize}\n\\item $\\B{A}=\\B{U}\\B{S}\\B{U}^{\\T}$ is the eigenvalue decomposition of\n the symmetric $D$-by-$D$ matrix $\\B{A}$, where $\\B{U}$ is an\n $D$-by-$D$ orthonormal matrix, holding the $D$ eigenvectors of\n $\\B{A}$ and $\\B{S}$ is an $D$-by-$D$ diagonal matrix, holding the\n corresponding eigenvalues of $\\B{A}$ as diagonal entries. \n\\item $\\B{A}\\odot\\B{B}$ is the pointwise or Hadamard product of $\\B{A}$ and $\\B{B}$.\n\\item $\\B{A}\\otimes\\B{B}$ is the Kronecker product of $\\B{A}$ and $\\B{B}$.\n\\item $\\B{Y}\\in\\mathbb{R}^{N\\times G}$ is the matrix holding all samples, having $N$ rows and $G$ columns.\n\\end{itemize}\n\n\n\\section{Kronecker testing strategies and models}\n$\\B{A}_j\\in\\Real^{C\\times M_j}$ is a matrix, that replicates the\n$j$-th fixed effects matrix $\\B{X}_j\\in\\Real^{R\\times D_j}$. $\\B{A}_j^\\T$\ntypically would be a binary matrix, but could in principle be\nanything. Using different versions of $\\B{A}_j^\\T$ corresponds to chosing\na testing strategy. For example, when $\\B{A}_j^\\T$ is the $C\\times C$\nIdentity matrix, then one would fit an independent weight to every\ncolumn of $\\B{Y}$, when $\\B{A}^\\T_j$ is a row-vector of ones, then one\nwould fit a single joint weight to all columns of $\\B{Y}$.\n\\begin{equation}\n\\Normal{\\ve{\\B{Y}}}{\\sum_j\\B{A}^{\\T}_j\\otimes\\B{X}_j\\ve{\\B{W}_j}}{\\B{C}\\otimes\\B{R}\n+ \\sigma^2\\B{I}}\n\\end{equation}\nAs long as $D_j\\leq R$, $M_j\\leq C$, the rank of $\\B{X}_j$ is $D_j$\nand the rank of $\\B{A}_j$ is $M_j$, and the rank of\n$\\left[\\B{X}_1...\\B{X}_J\\right]$ is $\\sum_{j=1}^JM_j$ the number of\ndegrees of freedom of a single $\\B{W}_j$ is $D_j \\cdot M_j$\n(sufficient condition).\n\\section{Efficient computation of tensor GP models}\n\\begin{equation}\n\\log\\Normal{\\ve{\\B{Y}}}{\\sum_{j=1}^J\\B{A}^{\\T}_j\\otimes\\B{X}_j\\ve{\\B{W}_j}}{\\B{C}\\otimes\\B{R}\n+ \\sigma^2\\B{I}}\n\\end{equation}\nApply the vec-trick to the mean term:\n\\begin{equation}\n\\log\\Normal{\\ve{\\B{Y}}}{\\ve{\\sum_{j=1}^J\\B{X}_j\\B{W}_j\\B{A}_j}}{\\B{C}\\otimes\\B{R}\n+ \\sigma^2\\B{I}}\n\\end{equation}\n\\begin{equation}\n-\\frac{C\\cdot R}{2}\\log(2\\pi)-\\frac{1}{2}\\log|\\B{C}\\otimes\\B{R} +\n\\sigma^2\\B{I}|\n-\\frac{1}{2}\\ve{\\B{Y} -\\sum_{j=1}^J\\B{X}_j\\B{W}_j\\B{A}_j}^\\T\\left(\\B{C}\\otimes\\B{R} +\n\\sigma^2\\B{I}\\right)^{-1}\\ve{\\B{Y} -\\sum_{j=1}^J\\B{X}_j\\B{W}_j\\B{A}_j}\n\\end{equation}\n\\subsection{Derivative of the squared form wrt. $\\B{W}$}\n\\begin{equation}\n\\frac{\\partial}{\\partial [\\B{W}_k]_{ab}}\\left(-\\frac{1}{2}\\ve{\\B{Y} -\\sum_{j=1}^J\\B{X}_j\\B{W}_j\\B{A}_j}^\\T\\left(\\B{C}\\otimes\\B{R} +\n\\sigma^2\\B{I}\\right)^{-1}\\ve{\\B{Y} -\\sum_{j=1}^J\\B{X}_j\\B{W}_j\\B{A}_j}\\right)\n\\end{equation}\nWe define matrix $\\B{D}\\in\\Real^{R\\times C}$, such that $\\ve{\\B{D}} = \\diag\\left(\\B{S}_C\\otimes\\B{S}_R +\n\\sigma^2\\B{I}\\right)^{-1}$ and rotate the data:\n\\begin{equation}\n\\frac{\\partial}{\\partial [\\B{W}_k]_{ab}}\n-\\frac{1}{2}\\tr\\left(\\left(\\Ykron\n -\\sum_{j=1}^J\\XWAkron{j}\\right)^\\T\\left(\\left(\\Ykron\n -\\sum_{j=1}^J\\XWAkron{j}\\right)\\odot \\B{D}\\right)\\right)\n\\end{equation}\n\\begin{eqnarray}\\nonumber\n\\frac{\\partial}{\\partial [\\B{W}_k]_{ab}}\n-\\frac{1}{2}\\tr\\left(\\Ykron^\\T (\\Ykron \\odot \\B{D})\\right)\n +\\tr\\left( \\sum_{j=1}^J\\XWAkronT{j}(\\Ykron \\odot \\B{D})\\right) \\\\\n-\\tr\\left( \\sum_{j=1}^J\\sum_{i=j+1}^J\\XWAkronT{j}(\\XWAkron{i} \\odot\n \\B{D}) \\right) \n-\\frac{1}{2}\\tr\\left(\\sum_{j=1}^J\\XWAkronT{j}(\\XWAkron{j} \\odot \\B{D}) \\right)\n\\end{eqnarray}\nLeaving out all terms without $\\B{W}_k$:\n\\begin{eqnarray}\\nonumber\n\\frac{\\partial}{\\partial [\\B{W}_k]_{ab}}\n \\tr\\left( \\XWAkronT{k}(\\Ykron \\odot \\B{D})\\right) \\\\\n-\\tr\\left( \\sum_{j\\neq k}\\XWAkronT{k}(\\XWAkron{j} \\odot\n \\B{D}) \\right) \n-\\frac{1}{2}\\tr\\left(\\XWAkronT{k}(\\XWAkron{k} \\odot \\B{D}) \\right)\n\\end{eqnarray}\n\n\\begin{eqnarray}\\nonumber\n \\tr\\left( {\\B{U}_C^\\T[\\B{A}_{k}]_{b:}^{\\T}[\\B{X}_{k}]_{:a}^{\\T}\\B{U}_R} (\\Ykron \\odot \\B{D})\\right) \\\\\n-\\tr\\left( \\sum_{j\\neq k}{\\B{U}_C^\\T[\\B{A}_{k}]_{b:}^{\\T}[\\B{X}_{k}]_{:a}^{\\T}\\B{U}_R} (\\XWAkron{j} \\odot\n \\B{D}) \\right) \n-\\tr\\left({\\B{U}_C^\\T[\\B{A}_{k}]_{b:}^{\\T}[\\B{X}_{k}]_{:a}^{\\T}\\B{U}_R} (\\XWAkron{k} \\odot \\B{D}) \\right)\n\\end{eqnarray}\nMoving around the terms in the trace and combining the last two traces\ninto one sum:\n\\begin{eqnarray}\\nonumber\n \\tr\\left( [\\B{X}_{k}]_{:a}^{\\T}\\B{U}_R (\\Ykron \\odot \\B{D}) \\B{U}_C^\\T[\\B{A}_{k}]_{b:}^{\\T}\\right) \\\\\n-\\tr\\left( \\sum_{j=1}^J{[\\B{X}_{k}]_{:a}^{\\T}\\B{U}_R} (\\XWAkron{j} \\odot\n \\B{D}) \\B{U}_C^\\T[\\B{A}_{k}]_{b:}^{\\T}\\right) \n\\end{eqnarray}\nObserving that this is a scalar, we can eliminate the trace and re-arrange:\n\\begin{eqnarray}\n [\\B{X}_{k}]_{:a}^{\\T}\\B{U}_R ((\\Ykron-\\sum_{j=1}^J \\XWAkron{j}) \\odot \\B{D}) \\B{U}_C^\\T[\\B{A}_{k}]_{b:}^{\\T} \\\\\n\\end{eqnarray}\nStacking together these terms for all $a$ and $b$, the gradient becomes\n\\begin{eqnarray}\n \\B{X}_{k}^{\\T}\\B{U}_R ((\\Ykron-\\sum_{j=1}^J \\XWAkron{j}) \\odot \\B{D}) \\B{U}_C^\\T\\B{A}_{k}^{\\T} \\\\\n\\end{eqnarray}\n\n\\newpage\n\\end{document}\n", "meta": {"hexsha": "a14342e2a29d0ede4534e69646e124e1554e0483", "size": 6571, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "svca_limix/derivations/derivations.tex", "max_stars_repo_name": "DenisSch/svca", "max_stars_repo_head_hexsha": "bd029c120ca8310f43311253e4d7ce19bc08350c", "max_stars_repo_licenses": ["Apache-2.0"], "max_stars_count": 65, "max_stars_repo_stars_event_min_datetime": "2015-01-20T20:46:26.000Z", "max_stars_repo_stars_event_max_datetime": "2021-06-27T14:40:35.000Z", "max_issues_repo_path": "svca_limix/derivations/derivations.tex", "max_issues_repo_name": "DenisSch/svca", "max_issues_repo_head_hexsha": "bd029c120ca8310f43311253e4d7ce19bc08350c", "max_issues_repo_licenses": ["Apache-2.0"], "max_issues_count": 29, "max_issues_repo_issues_event_min_datetime": "2015-02-01T22:35:17.000Z", "max_issues_repo_issues_event_max_datetime": "2017-08-07T08:18:23.000Z", "max_forks_repo_path": "svca_limix/derivations/derivations.tex", "max_forks_repo_name": "DenisSch/svca", "max_forks_repo_head_hexsha": "bd029c120ca8310f43311253e4d7ce19bc08350c", "max_forks_repo_licenses": ["Apache-2.0"], "max_forks_count": 35, "max_forks_repo_forks_event_min_datetime": "2015-02-01T17:26:50.000Z", "max_forks_repo_forks_event_max_datetime": "2019-09-13T07:06:16.000Z", "avg_line_length": 42.1217948718, "max_line_length": 131, "alphanum_fraction": 0.6399330391, "num_tokens": 2950, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704649604273, "lm_q2_score": 0.6150878555160665, "lm_q1q2_score": 0.3410480492395055}} {"text": "\\documentclass[american,ignorenonframetext,notheorems,aspectratio=1610]{beamer}\n\\usetheme[compress]{Madrid}\n\\usecolortheme{albatross}\n\\usepackage{tikz,tikzscale}\n\\usetikzlibrary{snakes}\n\\input{mixed/fig/tikzsettings}\n\\def\\restrict{r}\n\\def\\prolongate{p}\n\n\\usepackage{../mathsim}\n\\usepackage{times}\n\\usepackage{xr}\n\\externaldocument{main}\n\\usepackage{mfirstuc}\n\\usepackage{mathtools} \n\\mathtoolsset{showonlyrefs}\n\n\\def\\footnote#1{}\n\\def\\putindex#1{#1}\n\\title{Finite Elements}\n\\author{Guido Kanschat}\n\\date{\\today}\n\\begin{document}\n\\frame{\\maketitle}\n\\frame{\\frametitle{Overview}\\tableofcontents[hideallsubsections]}\n\\section{Elliptic PDE and Their Weak Formulation}\n\\frame{\\sectoc}\n\\subsection{Elliptic boundary value problems}\n\n\\frame {\\input{blocks/Notation-coordinates.tex}}\n\\frame {\\input{blocks/Notation-partial-derivative.tex}}\n\\frame {\\input{blocks/Notation-elim-coord.tex}\n \\input{blocks/Definition-lin-pde-2order.tex}}\n\\frame {\\input{blocks/Definition-poisson-eqn.tex}}\n\\frame {\\input{blocks/Definition-domain.tex}}\n\\frame {\\input{blocks/Definition-boundary-conditions.tex}}\n\\frame {\\input{blocks/Definition-dirichlet-problem-differential.tex}}\n\\frame {\\input{blocks/Theorem-Dirichlet-principle.tex}}\n\\frame {\\input{blocks/Theorem-Dirichlet-variational-principle.tex}\n \\input{blocks/Corollary-Dirichlet-uniqueness.tex}}\n\\frame {\\input{blocks/Lemma-reduction-to-zero-bc.tex}}\n\\frame {\\input{blocks/Notation-l2.tex}}\n\\frame {\\input{blocks/Lemma-Friedrichs-continuous.tex}\n \\input{blocks/Problem-Friedrichs.tex}}\n\\frame {\\input{blocks/Lemma-h1-norm.tex}}\n\\frame {\\input{blocks/Lemma-Dirichlet-energy-boundedness.tex}\n \\input{blocks/Lemma-minimizing-sequence.tex}}\n\\frame {\\input{blocks/Definition-h10.tex}\n \\input{blocks/Lemma-Friedrichs-h1.tex}}\n\\frame {\\input{blocks/Definition-weak-formulation.tex}\n \\input{blocks/Theorem-weak-unique-solution-1.tex}}\n\n\\frame {\\input{blocks/Lemma-neumann-weak.tex}}\n\\frame {\\input{blocks/Definition-natural-bc.tex}}\n\\frame {\\input{blocks/Lemma-mixed-bc-weak.tex}}\n\n\\subsection{Hilbert Spaces and Bilinear Forms}\n\\frame{\\tableofcontents[currentsection,subsectionstyle=show/shaded/hide]}\n\n\\frame {\\input{blocks/Definition-inner-product.tex}}\n\\frame {\\input{blocks/Theorem-bcs-inequality.tex}\n \\input{blocks/Lemma-inner-product-norm.tex}}\n\\frame {\\input{blocks/Definition-complete.tex}}\n\\frame {\\input{blocks/Definition-Banach-hilbert.tex}}\n\\frame {\\input{blocks/Definition-orthogonal.tex}}\n\\frame {\\input{blocks/Definition-orthogonal-complement.tex}}\n\\frame {\\input{blocks/Lemma-orthogonal-closed.tex}}\n\\frame {\\input{blocks/Theorem-orthogonal-complement.tex}}\n% \\frame {\\input{blocks/Corollary-ortho-density.tex}}\n\\frame {\\input{blocks/Definition-ortho-projection.tex}}\n\\frame {\\input{blocks/Definition-dual-space.tex}}\n\\frame {\\input{blocks/Theorem-Riesz-representation.tex}}\n\\frame {\\input{blocks/Definition-bilinear-form.tex}}\n\\frame {\\input{blocks/Lemma-pde-bilinear.tex}}\n\\frame {\\input{blocks/Lemma-lax-milgram.tex}}\n\\frame {\\input{blocks/Lemma-weak-well-posed.tex}\n \\input{blocks/Definition-elliptic.tex}}\n\n\\subsection{Fast facts on Sobolev spaces}\n\\frame{\\tableofcontents[currentsection,subsectionstyle=show/shaded/hide]}\n\n\\frame {\\input{blocks/Notation-multi-index.tex}}\n\\frame {\\input{blocks/Definition-distributional-derivative.tex}}\n\\frame {\\input{blocks/Definition-Wkp.tex}\n \\input{blocks/Corollary-wkp-embedding.tex}}\n\\frame {\\input{blocks/Definition-hkp.tex}\n \\input{blocks/Theorem-meyers-serrin.tex}}\n\n\\frame {\\input{blocks/Definition-boundary-smoothness.tex}}\n\\frame {\\input{blocks/Definition-continuous-embedding.tex}}\n\\frame {\\input{blocks/Theorem-sobolev-embedding.tex}}\n\\frame {\\input{blocks/Lemma-trace-continuous.tex}\n \\input{blocks/Theorem-trace.tex}}\n\\frame {\\input{blocks/Definition-hoelder-spaces.tex}}\n\\frame {\\input{blocks/Theorem-hoelder-embedding.tex}\n \\input{blocks/Corollary-sobolev-continuous.tex}}\n\n\\subsection{Properties of solutions}\n\\frame{\\tableofcontents[currentsection,subsectionstyle=show/shaded/hide]}\n\n\\frame {\\input{blocks/Definition-wkp-loc.tex}\n \\input{blocks/Theorem-gt-8-8.tex}\n \\input{blocks/Theorem-gt-8-10.tex}\n \\input{blocks/Corollary-gt-8-10.tex}}\n\\frame {\\input{blocks/Theorem-gt-8-13.tex}\n \\input{blocks/Corollary-h2-solution-bvp.tex}\n \\input{blocks/Remark-classical-smooth.tex}\n \\input{blocks/Remark-classical-convex.tex}}\n\\frame {\\input{blocks/Theorem-kondratev.tex}}\n\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\section{Conforming Finite Element Methods}\n\\frame{\\sectoc}\n\\subsection{Meshes, shape functions, and degrees of freedom}\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%\n\\frame {\\input{blocks/Definition-facets.tex}\n \\input{blocks/Definition-mesh.tex}}\n\\frame {\\input{blocks/Definition-finite-element.tex}\n\\input{blocks/Notation-dofs.tex}}\n\\frame {\\input{blocks/Definition-node-topology.tex}\n \\input{blocks/Definition-fe-space.tex}}\n\\frame {\\input{blocks/Notation-global-local.tex}}\n\\frame {\\input{blocks/Definition-local-global.tex}}\n\\frame {\\input{blocks/Lemma-fe-support.tex}}\n\\frame {\\input{blocks/Lemma-mesh-continuity.tex}\n \\input{blocks/Lemma-nodal-continuity.tex}}\n\n\\frame {\\input{blocks/Definition-barycentric-coordinates.tex}}\n\\frame {\\input{blocks/Lemma-barycentric-affine.tex}\n \\input{blocks/Corollary-barycentric-interpolation.tex}}\n\n\\begin{frame}\n \\frametitle{The $P_1$ element in barycentric coordinates}\n \\begin{columns}\n \\begin{column}{.5\\textwidth}\n \\begin{center}\n \\includegraphics[width=.6\\textwidth]{mixed/fig/p1-p.tikz}\n \\end{center}\n \\end{column}\n \\begin{column}{.5\\textwidth}\n \\begin{gather*}\n \\phi_i = \\lambda_i,\n \\quad i=0,1,2\n \\end{gather*}\n \\end{column}\n \\end{columns}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{The $P_2$ element in barycentric coordinates}\n \\begin{columns}\n \\begin{column}{.5\\textwidth}\n \\begin{center}\n \\includegraphics[width=.6\\textwidth]{mixed/fig/p2-p.tikz}\n \\end{center}\n \\end{column}\n \\begin{column}{.5\\textwidth}\n \\begin{xalignat*}2\n \\phi_{ii} &= 2\\lambda_i^2 - \\lambda_i,\n &i&=0,1,2\\\\\n \\phi_{ij} &= 4\\lambda_i\\lambda_j\n &j&\\neq i\n \\end{xalignat*}\n \\end{column}\n \\end{columns}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{The $P_3$ element in barycentric coordinates}\n \\begin{columns}\n \\begin{column}{.5\\textwidth}\n \\begin{center}\n \\includegraphics[width=.6\\textwidth]{mixed/fig/p3-p.tikz}\n \\end{center}\n \\end{column}\n \\begin{column}{.5\\textwidth}\n \\begin{xalignat*}2\n \\phi_{iii} &= \\tfrac12 \\lambda_i(3\\lambda_i-1)(3\\lambda_i-2)\n &i&=0,1,2\\\\\n \\phi_{ij} &= \\tfrac92\\lambda_i\\lambda_j(3\\lambda_j-1)\n &j&\\neq i\\\\\n \\phi_0 &= 27\\lambda_0\\lambda_1\\lambda_2\n \\end{xalignat*}\n \\end{column}\n \\end{columns}\n\\end{frame}\n\n\\frame {\\input{blocks/Definition-galerkin-approximation.tex}}\n\\frame {\\input{blocks/Corollary-galerkin-equations.tex}}\n\\frame {\\input{blocks/Lemma-discrete-lax-milgram.tex}\n \\input{blocks/Lemma-cea.tex}}\n\\frame {\\input{blocks/Lemma-fe-matrix.tex}}\n\\frame {\\input{blocks/Algorithm-matrix-assembling.tex}}\n\n\\frame {\\input{blocks/Definition-tensor-product-polynomials.tex}}\n\\frame {\\input{blocks/Lemma-tensor-product-node-functionals.tex}}\n\n\\begin{frame}\n \\frametitle{Example: $Q_2$ shape functions}\n \\begin{center}\n \\includegraphics[height=.27\\textheight]{graph/shape0}\n \\includegraphics[height=.27\\textheight]{graph/shape1}\n \\includegraphics[height=.27\\textheight]{graph/shape2}\n\n \\includegraphics[height=.27\\textheight]{graph/shape3}\n \\includegraphics[height=.27\\textheight]{graph/shape4}\n \\includegraphics[height=.27\\textheight]{graph/shape5}\n\n \\includegraphics[height=.27\\textheight]{graph/shape6}\n \\includegraphics[height=.27\\textheight]{graph/shape7}\n \\includegraphics[height=.27\\textheight]{graph/shape8}\n \\end{center}\n\\end{frame}\n\n\\frame {\\input{blocks/Lemma-tensor-product-trace.tex}}\n\n\\begin{frame}\n \\frametitle{Example: $Q_2$ continuous basis (selection)}\n \\begin{center}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-02}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-03}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-15}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-16}\n\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-07}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-13}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-18}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-22}\n\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-17}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-23}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-20}\n \\includegraphics[height=.27\\textheight]{graph/cgbasis1-24}\n \\end{center}\n\\end{frame}\n\n\\begin{frame}\n \\frametitle{Example: $Q_2$ discontinuous basis (selection)}\n \\begin{center}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-08}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-15}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-20}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-27}\n\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-07}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-19}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-23}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-30}\n\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-26}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-33}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-24}\n \\includegraphics[height=.27\\textheight]{graph/dgbasis1-22}\n \\end{center}\n\\end{frame}\n\n\\subsubsection{Mapped finite elements}\n\n\\frame {\\input{blocks/Definition-mapped-mesh.tex}}\n\\frame {\\input{blocks/Example-mapping-linear.tex}}\n\\frame {\\input{blocks/Example-mapping-bilinear.tex}}\n\\frame {\\input{blocks/Definition-mapped-fe.tex}}\n\\frame {\\input{blocks/Lemma-mapped-norms-affine.tex}}\n\\frame {\\input{blocks/Lemma-shape-regular-transformation.tex}}\n\\frame {\\input{blocks/Assumption-mapping-decomposition.tex}}\n\\frame {\\input{blocks/Lemma-scaling-1.tex}}\n\n\\subsection{A priori error analysis}\n\\frame{\\subtoc}\n\n\\frame {\\input{blocks/Lemma-poincare.tex}}\n\\frame {\\input{blocks/Lemma-bramble-hilbert.tex}}\n%\\frame {\\input{blocks/Lemma-b-h-projector.tex}}\n\\frame {\\input{blocks/Definition-mesh-family.tex}}\n\\frame {\\input{blocks/Definition-nodal-interpolation.tex}\n \\input{blocks/Lemma-nodal-interpolation.tex}}\n\\frame {\\input{blocks/Definition-broken-sobolev-norm.tex}}\n\\frame {\\input{blocks/Theorem-fe-interpolation.tex}\n \\input{blocks/Corollary-fe-approximation.tex}}\n\\frame {\\input{blocks/Theorem-fe-convergence.tex}}\n\n\\subsubsection{Estimates of stronger norms}\n\n\\frame {\\input{blocks/Definition-stronger-norm.tex}}\n\\frame {\\input{blocks/Lemma-inverse-estimate.tex}}\n\\frame {\\input{blocks/Theorem-h2-error.tex}}\n\n\\subsubsection{Estimates of weaker norms}\n\n\\frame {\\input{blocks/Definition-dual-problem.tex}}\n\\frame {\\input{blocks/Lemma-poisson-dual.tex}}\n\\frame {\\input{blocks/Assumption-elliptic-regularity.tex}}\n\\frame {\\input{blocks/Theorem-fe-l2.tex}\n \\input {blocks/Corollary-fe-functional.tex}}\n\n\\subsubsection{Green's function and maximum norm estimates}\n\n\\frame {\\input {blocks/Definition-greens-function.tex}\n \\input {blocks/Theorem-greens-function-rd.tex}}\n\\frame {\\input {blocks/Lemma-greens-function-domain.tex}}\n\\frame {\\input {blocks/Theorem-greens-function-representation.tex}}\n\\frame {\\input {blocks/Theorem-linfty-error.tex}}\n\n\\subsection{A posteriori error analysis}\n\\frame{\\subtoc}\n\n\\frame {\\input {blocks/Definition-a-posteriori.tex}}\n\\frame {\\input {blocks/Definition-locally-quasi-uniform.tex}}\n\\frame {\\input {blocks/Definition-fem-neighborhood.tex}}\n\\frame {\\input {blocks/Theorem-clement.tex}}\n\\frame {\\input {blocks/Theorem-scott-zhang.tex}}\n\\frame {\\input {blocks/Theorem-schoeberl-interpolation.tex}}\n\\frame {\\input {blocks/Definition-residual.tex}}\n\\frame {\\input {blocks/Lemma-residual-hm1.tex}}\n\\frame {\\input {blocks/Definition-mvl-jmp.tex}}\n\\frame {\\input {blocks/Definition-residual-strong.tex}}\n\\frame {\\input {blocks/Lemma-residual-upper-bound.tex}}\n\\frame {\\input {blocks/Definition-osc.tex}}\n\\frame {\\input {blocks/Definition-residual-estimator.tex}}\n\\frame {\\input {blocks/Theorem-residual-estimate.tex}}\n\n\\section{Variational Crimes}\n\\frame{\\sectoc}\n\\subsection{Numerical quadrature}\n\\frame {\\input {blocks/Lemma-strang-1.tex}}\n\\frame {\\input {blocks/Definition-tensor-product-quadrature.tex}\n \\input {blocks/Lemma-tensor-product-gauss.tex}}\n\\frame {\\input {blocks/Assumption-bilinear-form-quadrature.tex}}\n\\frame {\\input {blocks/Definition-bilinear-quadrature.tex}}\n\\frame {\\input {blocks/Lemma-quadrature-stability.tex}}\n\\frame {\\input {blocks/Theorem-quadrature-error-bilinear.tex}}\n\\frame {\\input {blocks/Lemma-product-sobolev-norm.tex}}\n\\frame {\\input {blocks/Theorem-quadrature-error-rhs.tex}}\n\n\\section{Solving the Discrete Problem}\n\\frame{\\sectoc}\n\\subsection{Richardson's method}\n\\frame {\\input {blocks/Definition-spd-condition-number.tex}}\n\\frame {\\input {blocks/Definition-matrix-condition-number.tex}}\n\\frame {\\input {blocks/Lemma-richardson-error-step.tex}}\n\\frame {\\input {blocks/Theorem-richardson-convergence1.tex}}\n\\frame {\\input {blocks/Definition-richardson-method.tex}}\n\\frame {\\input {blocks/Theorem-richardson-convergence2.tex}}\n\\frame {\\input {blocks/Definition-riesz-isomorphism.tex}}\n\\frame {\\input {blocks/Definition-p-richardson-method.tex}}\n\\subsection{The conjugate gradient method}\n\\frame{\\subtoc}\n\\frame {\\input {blocks/Definition-steepest-descent.tex}}\n\\frame {\\input {blocks/Lemma-steepest-descent.tex}}\n\\frame {\\input {blocks/Definition-cg-method.tex}}\n\\frame {\\input {blocks/Definition-pcg-method.tex}}\n\\frame {\\input {blocks/Definition-krylov-space.tex}}\n\\frame {\\input {blocks/Lemma-cg-orthogonality.tex}}\n\\frame {\\input {blocks/Lemma-cg-minimization.tex}}\n\\frame {\\input {blocks/Theorem-cg-convergence.tex}}\n\n\\subsection{Condition numbers of finite element matrices}\n\\frame{\\subtoc}\n\\frame {\\input {blocks/Notation-up-to-a-constant.tex}}\n\\frame {\\input {blocks/Definition-mass-stiffness.tex}}\n\\frame {\\input {blocks/Lemma-condition-number-mass-matrix.tex}}\n\\frame {\\input {blocks/Corollary-refined-condition-number.tex}}\n\\frame {\\input {blocks/Theorem-condition-number-stiffness-matrix.tex}}\n\n\\subsection{Multigrid methods}\n\\frame {\\input {blocks/Algorithm-mgm.tex}}\n\n\\section{Bibliography}\n\\frame{\\bibliographystyle{alpha}\n\\bibliography{all}}\n\\end{document}\n\n%%% Local Variables:\n%%% mode: latex\n%%% TeX-master: t\n%%% End:\n", "meta": {"hexsha": "4b122e5433bd526970d6db6e6577321ed59ad1b3", "size": 14780, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "fem/slides.tex", "max_stars_repo_name": "arimiftari/notes", "max_stars_repo_head_hexsha": "737b95ed6a4163bd1d395c0379410513dcb03ef1", "max_stars_repo_licenses": ["CC-BY-4.0"], "max_stars_count": null, "max_stars_repo_stars_event_min_datetime": null, "max_stars_repo_stars_event_max_datetime": null, "max_issues_repo_path": "fem/slides.tex", "max_issues_repo_name": "arimiftari/notes", "max_issues_repo_head_hexsha": "737b95ed6a4163bd1d395c0379410513dcb03ef1", "max_issues_repo_licenses": ["CC-BY-4.0"], "max_issues_count": 2, "max_issues_repo_issues_event_min_datetime": "2018-05-24T07:31:37.000Z", "max_issues_repo_issues_event_max_datetime": "2018-08-31T12:58:14.000Z", "max_forks_repo_path": "fem/slides.tex", "max_forks_repo_name": "arimiftari/notes", "max_forks_repo_head_hexsha": "737b95ed6a4163bd1d395c0379410513dcb03ef1", "max_forks_repo_licenses": ["CC-BY-4.0"], "max_forks_count": 6, "max_forks_repo_forks_event_min_datetime": "2018-05-15T19:28:53.000Z", "max_forks_repo_forks_event_max_datetime": "2020-11-05T19:07:29.000Z", "avg_line_length": 38.9973614776, "max_line_length": 79, "alphanum_fraction": 0.7352503383, "num_tokens": 4312, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES", "lm_q1_score": 0.5544704502361149, "lm_q2_score": 0.6150878555160666, "lm_q1q2_score": 0.3410480401827598}} {"text": "\n\\section{Language}~\\label{sec:language}\n\nCopilot is embedded into the functional programming language Haskell\n\\cite{PeytonJones02}, and a working knowledge of Haskell is necessary to use\nCopilot effectively. Copilot is a pure declarative language; i.e., expressions\nare free of side-effects and are referentially transparent. A program written\nin Copilot, which from now on will be referred to as a \\emph{specification}, has\na cyclic behavior, where each cycle consists of a fixed series of steps:\n\n\\begin{itemize}\n\\item Sample external variables and arrays.\n\\item Update internal variables.\n\\item Fire external triggers. (In case the specification is violated.)\n\\item Update observers (for debugging purpose).\n\\end{itemize}\n\n\\noindent We refer to a single cycle as an \\emph{iteration} or a \\emph{step}.\n\nAll transformation of data in Copilot is propagated through streams.\nA stream is an infinite, ordered sequence of values which must conform to the same type.\nE.g., we have the stream of Fibonacci numbers:\n\n\\begin{center}\n$s_{fib} = \\{0, 1, 1, 2, 3, 5, 8, 13, 21, \\dots \\}$\n\\end{center}\n\n\\noindent We denote the $n$th value of the stream $s$ as $s(n)$, and the first\nvalue in a sequence $s$ as $s(0)$. For example, for $s_{fib}$ we have that $s_{fib}(0) = 0$,\n$s_{fib}(1) = 1$, $s_{fib}(2) = 1$, and so forth.\n\nConstants as well as arithmetic, boolean, and relational operators are\nlifted to work pointwise on streams:\n\n\\noindent\n%\\begin{minipage}{0.3\\textwidth}\n\\begin{lstlisting}[language = Copilot, frame = single]\nx :: Stream Int32\nx = 5 + 5\n\ny :: Stream Int32\ny = x * x\n\nz :: Stream Bool\nz = x == 10 && y < 200\n\\end{lstlisting}\n%\\end{minipage}\n\n\n\\noindent Here the streams {\\tt x}, {\\tt y}, and {\\tt z} are simply\n\\emph{constant streams}:\n\n\\begin{center}\n$\\mathtt{x} \\leadsto \\{10, 10, 10, \\dots \\}$,\n$\\mathtt{y} \\leadsto \\{100, 100, 100, \\dots \\}$,\n$\\mathtt{z} \\leadsto \\{\\mbox{T},\\; \\mbox{T},\\; \\mbox{T},\\; \\dots \\}$\n\\end{center}\n\nTwo types of \\emph{temporal} operators are provided, one for delaying streams and one for\nlooking into the future of streams:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\n(++) :: [a] -> Stream a -> Stream a\ndrop :: Int -> Stream a -> Stream a\n\\end{lstlisting}\n%\nHere {\\tt xs ++ s} prepends the list {\\tt xs} at the front of the stream {\\tt s}.\nFor example the stream {\\tt w} defined as follows, given our previous definition\nof {\\tt x}:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nw = [5,6,7] ++ x\n\\end{lstlisting}\n%\nevaluates to the sequence\n$\\mathtt{w} \\leadsto \\{5, 6, 7, 10, 10, 10, \\dots\\}$.\nThe expression {\\tt drop k s} skips the first {\\tt k} values of the stream {\\tt\n s}, returning the remainder of the stream.\nFor example we can skip the first two values of {\\tt w}:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nu = drop 2 w\n\\end{lstlisting}\n%\nwhich yields the sequence\n$\\mathtt{u} \\leadsto \\{7, 10, 10, 10, \\dots\\}$.\n\n\\subsection{Streams as Lazy-Lists} \\label{sec:stream}\n\nA key design choice in Copilot is that streams should mimic \\emph{lazy lists}.\nIn Haskell, the lazy-list of natural numbers can be programmed like this:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nnats_ll :: [Int32]\nnats_ll = [0] ++ zipWith (+) (repeat 1) nats_ll\n\\end{lstlisting}\n%\nAs both constants and arithmetic operators are lifted to work pointwise on\nstreams in Copilot, there is no need for {\\tt zipWith} and {\\tt repeat} when\nspecifying the stream of natural numbers:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nnats :: Stream Int32\nnats = [0] ++ (1 + nats)\n\\end{lstlisting}\n%\nIn the same manner, the lazy-list of Fibonacci numbers can be specified in Haskell as follows:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nfib_ll :: [Int32]\nfib_ll = [1, 1] ++ zipWith (+) fib_ll (drop 1 fib_ll)\n\\end{lstlisting}\n%\nIn Copilot we simply throw away {\\tt zipWith}:\n\\begin{lstlisting}[language = Copilot, frame = single]\nfib :: Stream Int32\nfib = [1, 1] ++ (fib + drop 1 fib)\n\\end{lstlisting}\n\nCopilot specifications must be \\emph{causal}, informally meaning that\nstream values cannot depend on future values. For example, the following stream\ndefinition is allowed:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nf :: Stream Word64\nf = [0,1,2] ++ f\n\ng :: Stream Word64\ng = drop 2 f\n\\end{lstlisting}\n%\n\nBut if instead {\\tt g} is defined as {\\tt g = drop 4 f}, then the definition is\ndisallowed. While an analogous stream is definable in a lazy language, we bar\nit in Copilot, since it requires future values of {\\tt f} to be\ngenerated before producing values for {\\tt g}. This is not possible since\nCopilot programs may take inputs in real-time from the environment (see\nSection~\\ref{subsec:interacting}).\n\n\n\\subsection{Structs}\nStructs require some special attentation in Copilot, as we cannot magically\nimport the definition of the struct in Copilot. In this section we discuss the\nsteps that need to be taken by following the code of \\texttt{Struct.hs} in the\n\\texttt{Examples} directory of the Copilot distribution, or the repository\n\\footnote{\\url{https://github.com/Copilot-Language/Copilot/blob/master/Examples/Struct.hs}}.\n\nLet's assume that we have defined a 2d-vector type in our C code:\n\\begin{lstlisting}\nstruct vec {\n\tfloat x;\n\tfloat y;\n};\n\\end{lstlisting}\nFor us to be able to use this vector inside Copilot, we need to follow a number\nof steps:\n\\begin{enumerate}\n \\item Enable \\texttt{DataKinds} compiler extension.\n \\item Define a datatype to mimic the C definition.\n \\item Write an instance of the \\texttt{Struct} class, containing a definition\n for the struct name and function to translate the fields to a heterogeneous\n list.\n \\item Write an instance of the \\texttt{Typed} class.\n\\end{enumerate}\n\n\\subsubsection*{Enabling compiler extensions}\nFirst and foremost, we need to enable the \\texttt{DataKinds} extension to GHC,\nby putting:\n\\begin{lstlisting}[language=Copilot]\n{-# LANGUAGE DataKinds #-}\n\\end{lstlisting}\nat the top of our specification file. This allows us to define \\emph{kinds},\nwhich are the types of types. Our datatype needs to carry the names of the\nfields in C as well. Using the \\texttt{DataKinds} extension we are able to\nwrite the names of the fields as part of our types.\n\n\n\\subsubsection*{Defining the datatype}\nA suitable representation of structs in Haskell is provided by the\n\\emph{record-syntax}, this allows us to use named fields as part of the\ndatatype. For Copilot this is not enough though: we still need to define the\nnames of the fields in our C code. Therefore we introduce new \\texttt{Field}\ndatatype, which takes two arguments: the name of field, and it's type. Now we\ncan mimic our vector struct in Copilot as follows:\n\\begin{lstlisting}[language=Copilot]\ndata Vec = Vec\n { x :: Field \"x\" Float\n , y :: Field \"y\" Float\n }\n\\end{lstlisting}\nHere we created two fields, $x$ and $y$, each with their corresponding C names\nand types. Note that the name inside Haskell and the C names do not necessarily\nneed to match, nor is it always possible to have them match. For type-safety,\ninside Copilot we will typically only use the Haskell level names (i.e. the\nunquoted ones). The C names are only used by Copilot internally.\n\n\n\\subsubsection*{Instance of \\texttt{Struct}}\nOur next task is to inform Copilot about our new type, therefore we need to\nwrite and instance of the \\texttt{Struct}-class. This class has the purpose of\ndefining the datatype as a struct, it provides the code generator of Copilot the\nname of struct in C, and provides a function to translate the struct to a list\nof values:\n\\begin{lstlisting}[language=Copilot]\ninstance Struct Vec where\n -- typename :: Vec -> String\n typename _ = \"vec\" -- Name of the type in C\n\n -- Function to translate Vec to list of Value's, order should match struct.\n -- tovalues :: Vec -> [Value Vec]\n toValues v = [ Value Float (x v)\n , Value Float (y v)\n ]\n\\end{lstlisting}\nBoth definitions should be pretty self-explanatory. Note however that\n\\texttt{Value a} is a wrapper around the \\texttt{Field} datatype to hide the\nactual type of \\texttt{Field}. It takes the type of the field, and the field\nitself as its arguments. The elements in the list should be in the same order\nas the fields in the struct.\n\nBoth \\texttt{typename} and \\texttt{toValues} have to be defined by the user,\nbut neither should ever be used by the user. Both functions are only used by\nthe code generator of Copilot.\n\n\n\\subsubsection*{Instance of \\texttt{Typed}}\nIn Copilot, streams can only of types that are instances of the \\texttt{Typed}\nclass. To be able to create streams of vectors, \\texttt{Vec} needs to be an\ninstance of \\texttt{Typed} as well. The class only provides a \\texttt{typeOf}\nfunction, returning the type:\n\\begin{lstlisting}[language=Copilot]\ninstance Typed Vec where\n typeOf = Struct (Vec (Field 0) (Field 0))\n\\end{lstlisting}\nFor \\texttt{Vec} this means we need to return something of the \\texttt{Vec}\ntype wrapped in the \\texttt{Struct} constructor. In this case it does not\nmatter what the values of the fields are, we just need to return something of\nthe correct type.\n\n\n\\subsubsection*{Simple operations}\nBuilding streams of structs works like building any other stream, but we need\nto wrap the values of a struct using the \\texttt{Field} constructor. The reason\nfor this is quite straightforward: the fields of our struct are defined in\nterms of \\texttt{Field}:\n\\begin{lstlisting}[language=Copilot]\nv :: Stream Vec\nv = [ Vec (Field 0) (Field 1) ] ++ v\n\\end{lstlisting}\n\nWe can also turn a field of a struct into its own stream using the\n\\texttt{(\\#)}-operator:\n\\begin{lstlisting}[language=Copilot]\nvx :: Stream Float\nvx = v # x\n\\end{lstlisting}\nNote the we use the Haskell level accessor \\texttt{x} to retrieve the field\nfrom the stream of vectors.\n\n\n\\subsubsection*{Example code}\n\\begin{example}\n\\label{exm:struct}\nNow that we defined all there is, we can make streams of structs. The following\ncode has been taken from the \\texttt{Struct.hs} example, and shows the basic\nusage of structs.\n\n\\begin{lstlisting}[language=Copilot]\n{-# LANGUAGE DataKinds #-}\n\nmodule Struct where\n\nimport Language.Copilot\nimport Copilot.Compile.C99\n\nimport Prelude hiding ((>), (<), div, (++))\n\n\ndata Vec = Vec\n { x :: Field \"x\" Float\n , y :: Field \"y\" Float\n }\n\ninstance Struct Vec where\n typename _ = \"vec\" -- Name of the type in C\n\n -- Function to translate Vec to list of Value's, order should match struct.\n toValues v = [ Value Float (x v)\n , Value Float (y v)\n ]\n\n-- We need to provide an instance to Typed with a bogus Vec\ninstance Typed Vec where\n typeOf = Struct (Vec (Field 0) (Field 0))\n\n\nvecs :: Stream Vec\nvecs = [ Vec (Field 1) (Field 2)\n , Vec (Field 12) (Field 8)\n ] ++ vecs\n\n\nspec = do\n -- Trigger that always executes, splits the vec into seperate args.\n trigger \"split\" true [arg $ vecs # x, arg $ vecs # y]\n\\end{lstlisting}\n\\end{example}\n\n\\subsection{Functions on Streams} \\label{sec:FnOnStreams}\n\nGiven that constants and operators work pointwise on streams, we can use Haskell\nas a macro-language for defining functions on streams. The idea of using\nHaskell as a macro language is powerful since Haskell is a\ngeneral-purpose higher-order functional language.\n\n\\begin{example}\nWe define the function {\\tt even}, which given a stream of\nintegers returns a boolean stream which is true whenever the input stream\ncontains an even number, as follows:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\neven :: Stream Int32 -> Stream Bool\neven x = x `mod` 2 == 0\n\\end{lstlisting}\n%\nApplying {\\tt even} on {\\tt nats} (defined above) yields the sequence\n$\\{T, F, T, F, T, F, \\dots\\}$.\n\\end{example}\n\nIf a function is required to return multiple results, we simply use plain\nHaskell tuples:\n\n\\begin{example}\nWe define complex multiplication as follows:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nmul_comp\n :: (Stream Double, Stream Double)\n -> (Stream Double, Stream Double)\n -> (Stream Double, Stream Double)\n(a, b) `mul_comp` (c, d) = (a * c - b * d, a * d + b * c)\n\\end{lstlisting}\n%\nHere {\\tt a} and {\\tt b} represent the real and imaginary part of the left\noperand, and {\\tt c} and {\\tt d} represent the real and imaginary part\nof the right operand.\n\\end{example}\n\n\\subsection{Stateful Functions} \\label{sec:stateful}\n\nIn addition to pure functions, such as {\\tt even} and {\\tt mul\\_comp},\nCopilot also facilitates \\emph{stateful} functions. A \\emph{stateful} function\nis a function which has an internal state, e.g. as a latch (as in electronic\ncircuits) or a low/high-pass filter (as in a DSP).\n\n\\begin{figure*}\n\\begin{minipage}{0.25\\linewidth}\n\\begin{tabular}{c|c||c}\n$\\mathtt{x}_i$: & $\\mathtt{y}_{i-1}$: & $\\mathtt{y}_i$:\\\\\n\\hline\n$F$ & $F$ & $F$ \\\\\n\\hline\n$F$ & $T$ & $T$ \\\\\n\\hline\n$T$ & $F$ & $T$ \\\\\n\\hline\n$T$ & $T$ & $F$\n\\end{tabular}\n\\end{minipage}\n\\begin{minipage}{0.35\\linewidth}\n\\begin{lstlisting}[frame=none]\nlatch :: Stream Bool -> Stream Bool\nlatch x = y\n where\n y = if x then not z else z\n z = [False] ++ y\n\\end{lstlisting}\n\\end{minipage}\n\\hspace{1cm}\n\\begin{minipage}{0.3\\linewidth}\n\\begin{tabular}{c|c|c|c|c|c}\n & 0 & 1 & 2 & 3 & 4\\\\\n\\hline\nx & $F$ & $T$ & $T$ & $F$ & $F$ \\\\\n\\hline\ny & $F$ & $T$ & $F$ & $F$ & $F$ \\\\\n\\end{tabular}\n\\end{minipage}\n\\caption{A latch [Example 3]. The specification function is provided at the left and the\nimplementation in copilot is provided in the middle. The right shows an example of\nthe latch, where x is $\\{F, T, T, F, F, \\dots \\}$ and the initial value of y (used with $x_0$ to find\n$y_0$ since there is no $y_{-1}$) is False.}\n\\label{fig:jk_latch}\n\\end{figure*}\n\n\\begin{example}\nWe consider a simple latch, as described in \\cite{Farhat2004}, with a single\ninput and a boolean state. A latch is a way of simulating memory in circuits by feeding\nback output gates as inputs. Whenever the input is true the internal state is reversed.\nThe operational behavior and the implementation of the latch is shown in Figure\n\\ref{fig:jk_latch}.\\footnote\n{In order\nto use conditionals (i.e., if-then-else) in Copilot specifications,\nas in Figures~\\ref{fig:jk_latch} and~\\ref{fig:counter}, the GHC\nlanguage extension {\\tt RebindableSyntax} must be set on.}\n\\end{example}\n\n\\begin{figure*}\n\\begin{minipage}{0.4\\linewidth}\n\\begin{tabular}{c|c||c}\n$\\mathtt{inc}_i$: & $\\mathtt{reset}_i$: & $\\mathtt{cnt}_i$: \\\\\n\\hline\n$F$ & $F$ & $\\mathtt{cnt}_{i-1}$ \\\\\n\\hline\n* & $T$ & $0$ \\\\\n\\hline\n$T$ & $F$ & $\\mathtt{cnt}_{i-1} + 1$ \\\\\n\\hline\n\\end{tabular}\n\\end{minipage}\n\\hspace{1cm}\n\\begin{minipage}{0.6\\linewidth}\n\\begin{lstlisting}[language = Copilot, frame = none]\ncounter :: Stream Bool -> Stream Bool\n -> Stream Int32\ncounter inc reset = cnt\n where\n cnt = if reset then 0\n else if inc then z + 1\n else z\n z = [0] ++ cnt\n\\end{lstlisting}\n\\end{minipage}\n\\caption{A resettable counter. The specification is provided at the left and the\nimplementation is provided at the right.\n}\n\\label{fig:counter}\n\\end{figure*}\n\n\\begin{example}\nWe consider a resettable counter with two inputs, {\\tt inc} and {\\tt reset}.\nThe input {\\tt inc} increments the counter and the input {\\tt reset} resets the\ncounter. The internal state of the counter, {\\tt cnt}, represents the value of the\ncounter and is initially set to zero. At each cycle, $i$, the value of\n$\\mathtt{cnt}_i$ is determined as shown in the left table in Figure\n\\ref{fig:counter}.\n\\end{example}\n\n%\\begin{figure}\n%\\begin{code}\n%fir2pole :: Double -> Double -> Double -> Double\n% -> Double -> Sig Double -> Sig Double\n%fir2pole a1 a2 b0 b1 b2 x0 = y0\n% where\n% y0 = - (constant a1)*y1 - (constant a2)*y2\n% + (constant b0)*x0 + (constant b1)*x1 + (constant b2)*x2\n% x2 = [0, 0] ++ x0 ; x1 = drop 1 x2\n% y2 = [0, 0] ++ y0 ; y1 = drop 1 y2\n%\\end{code}\n%\\caption{A $2$-pole IIR filter.}\n%\\label{fig:2_pole_iir_filter}\n%\\end{figure}\n\n\\subsection{Types} \\label{sec:types}\n\nCopilot is a typed language, where types are enforced by the Haskell type system\nto ensure generated C programs are well-typed. Copilot is \\emph{strongly typed}\n(i.e., type-incorrect function application is not possible) and \\emph{statically\n typed} (i.e., type-checking is done at compile-time). The base types are\nBooleans, unsigned and signed words of width 8, 16, 32, and 64, floats, and\ndoubles. All elements of a stream must belong to the same base\ntype. These types have instances for the class {\\tt Typed a}, used to constrain\nCopilot programs.\n\nWe provide a {\\tt cast} operator\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\ncast :: (Typed a, Typed b) => Stream a -> Stream b\n\\end{lstlisting}\n%\nthat casts from one type to another. The cast operator is only defined for\ncasts that do not lose information, so an unsigned word type {\\tt a} can only be\ncast to another unsigned type at least as large as {\\tt a} or to a signed word\ntype strictly larger than {\\tt a}. Signed types cannot be cast to unsigned\ntypes but can be cast to signed types at least as large.\n\nThere also exists an {\\tt unsafeCast} operator which allows casting from any\ntype to any other (except from floating point numbers to integer types):\n\n\\begin{lstlisting}[language = Copilot, frame = single]\nunsafeCast :: (Typed a, Typed b) => Stream a -> Stream b\n\\end{lstlisting}\n\n\\subsection{Interacting With the Target Program}\n\\label{subsec:interacting}\n\nAll interaction with the outside world is done by sampling \\emph{external\n symbols} and by evoking \\emph{triggers}. External symbols are symbols that\nare defined outside Copilot and which reflect the visible state of the target\nprogram that we are monitoring. They include variables and arrays.\nAnalogously, triggers are functions that are defined outside Copilot and which\nare evoked when Copilot needs to report that the target program has violated a\nspecification constraint.\n\n\\paragraph{External Variables.}\n\n\nAs discussed in Section~\\ref{sampling}, \\emph{sampling} is an approach\nfor monitoring the state of an executing system based on sampling\nstate-variables, while assuming synchrony between the monitor and the\nobserved software. Copilot targets hard real-time embedded C programs\nso the state variables that are observed by the monitors are variables\nof C programs. Copilot monitors run either in the same thread or a\nseparate thread as the system under observation and the only variables\nthat can be observed are those that are made available through shared\nmemory. This means local variables cannot be observed. Currently,\nCopilot supports basic C datatypes, arrays and structs. Combinations of each of\nthose work as well: nested arrays, arrays of structs, structs containg arrays\netc. All of these variables containing actual data; pointers to data are not\nsupported by design.\n\n\nCopilot has both an interpreter and a compiler.The compiler must be\nused to monitor an executing program. The Copilot reification process\ngenerates a C monitor from a Copilot specification. The variables that\nare observed in the C code must be declared as \\emph{external}\nvariables in the monitor. The external variables have the same name as\nthe variables being monitored in the C code are treated as shared\nmemory. The interpreter is intended for exploring ideas and algorithms\nand is not intended to monitor executing C\nprograms. It may seem external variables would have no meaning if the\nmonitor was run in the interpreter, but Copilot gives the user the\nability to specify default stream values for an external variable that\nget used when the monitor interpreted.\n\n A Copilot specification is \\emph{open} if defined with external symbols in the\nsense that values must be provided externally at runtime. To simplify writing\nCopilot specifications that can be interpreted and tested, constructs for\nexternal symbols take an optional environment for interpretation.\n\nExternal variables are similar to global variables in other languages. They\nare defined by using the {\\tt extern} construct:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nextern :: Typed a => String -> Maybe [a] -> Stream a\n\\end{lstlisting}\n%\n\\noindent\nIt takes the name of an external variable, a possible Haskell list to serve as\nthe environment for the interpreter, and generates a stream by sampling the\nvariable at each clock cycle. For example,\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nsumExterns :: Stream Word64\nsumExterns = let ex1 = extern \"e1\" (Just [0..])\n ex2 = extern \"e2\" Nothing\n in ex1 + ex2\n\\end{lstlisting}\n%\nis a stream that takes two external variables {\\tt e1} and {\\tt e2} and adds\nthem. The first external variable contains the infinite list {\\tt [0,1,2,...]}\nof values for use when interpreting a Copilot specification containing the\nstream. The other variable contains no environment ({\\tt sumExterns} must have\nan environment for both of its variables to be interpreted).\n\nSometimes, type inference cannot infer the type of an external variable. For\nexample, in the stream definition\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nextEven :: Stream Bool\nextEven = e0 `mod` 2 == 0\n where e0 = externW8 \"x\" Nothing\n\\end{lstlisting}\n%\n\\noindent\nthe type of {\\tt extern \"x\"} is ambiguous, since it cannot be inferred from a\nBoolean stream and we have not given an explicit type signature. For\nconvenience, typed {\\tt extern} functions are provided, e.g., {\\tt externW8} or\n{\\tt externI64} denoting an external unsigned 8-bit word or signed 64-bit word,\nrespectively.\n% Please see the grammar in Appendix~\\ref{sec:BNF} for the list of\n% all sampling functions.\n\nIn general it is best practice to define external symbols with\ntop-level definitions, e.g.,\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\ne0 :: Stream Word8\ne0 = extern \"e0\" (Just [2,4..])\n\\end{lstlisting}\n\n\\noindent\nso that the symbol name and its environment can be shared between streams.\n\nJust like regular variables, arrays can be sampled as well. Copilot threats\narrays in the same way as it does for scalars. \n\\begin{example}\n\\label{exmp:pitot}\nLets take the example where we\nhave the readouts of four pitot tubes, giving us the measured airspeed:\n\\begin{code}[frame=single]\n/* Array containing readouts of 4 pitot tubes. */\ndouble airspeeds[4] = ... ;\n\\end{code}\nIn our Copilot specification, we need to provide the type of our array, because\nCopilot need to know the length of the array we refer to. Apart from that,\nreferring to an external array is like referring to any other variable:\n\\begin{lstlisting}[language=Copilot, frame=single]\nairspeeds :: Stream (Array 4 Double)\nairspeeds = extern \"airspeeds\" Nothing\n\\end{lstlisting}\n\\end{example}\n\n\n\\paragraph{Triggers.}\nTriggers, the only mechanism for Copilot streams to effect the outside world,\nare defined by using the {\\tt trigger construct}:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\ntrigger :: String -> Stream Bool -> [TriggerArg] -> Spec\n\\end{lstlisting}\n%\nThe first parameter is the name of the external function, the second parameter is the\nguard which determines when the trigger should be evoked, and the third parameter\nis a list of arguments which is passed to the trigger when evoked.\nTriggers can be combined into a specification by using the \\emph{do}-notation:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\nspec :: Spec\nspec = do\n trigger \"f\" (even nats) [arg fib, arg (nats * nats)]\n trigger \"g\" (fib > 10) []\n let x = externW64 \"x\" (Just [1..])\n trigger \"h\" (x < 10) [arg x]\n\\end{lstlisting}\n%\nThe order in which the triggers are defined is irrelevant. To interpret this spec we run:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\ninterpret 10 spec\n\\end{lstlisting}\n%\nwhich will yield the following output:\n%\n\\begin{code}\nf: g:\t h:\n(1,0) () (1)\n-- () (2)\n(2,4) () (3)\n-- () (4)\n(5,16) () (5)\n-- () (6)\n(13,36) --\t(7)\n-- -- (8)\n(34,64) --\t(9)\n-- -- --\n\\end{code}\n%\n\n\\begin{example}\n\\label{exm:engine}\nWe consider an engine controller with the following property: If the temperature\nrises more than 2.3 degrees within 0.2 seconds, then the fuel injector should\nnot be running. Assuming that the global sample rate is 0.1 seconds, we can\ndefine a monitor that surveys the above property:\n%\n\\begin{lstlisting}[language = Copilot, frame = single]\npropTempRiseShutOff :: Spec\npropTempRiseShutOff =\n trigger \"over_temp_rise\"\n (overTempRise && running) []\n\n where\n max = 500 -- maximum engine temperature\n\n temps :: Stream Float\n temps = [max, max, max] ++ temp\n\n temp = extern \"temp\" Nothing\n\n overTempRise :: Stream Bool\n overTempRise = drop 2 temps > (2.3 + temps)\n\n running :: Stream Bool\n running = extern \"running\" Nothing\n\\end{lstlisting}\n%\n\nHere, we assume that the external variable {\\tt temp} denotes the temperature of\nthe engine and the external variable {\\tt running} indicates whether the fuel\ninjector is running. The external function {\\tt over\\_temp\\_rise} is called\nwithout any arguments if the temperature rises more than 2.3 degrees within 0.2\nseconds and the engine is not shut off. Notice there is a latency of one tick between when the property is violated and when the guard becomes true.\n\\end{example}\n", "meta": {"hexsha": "f571b1cf9b537db8354b4cfea39dbe672a9208c6", "size": 25297, "ext": "tex", "lang": "TeX", "max_stars_repo_path": "TutorialAndDevGuide/Tutorial/Language.tex", "max_stars_repo_name": "Copilot-Language/copilot-discussion", "max_stars_repo_head_hexsha": "caccad918b23dae991095344a845827ddccd6047", "max_stars_repo_licenses": ["BSD-3-Clause"], "max_stars_count": 6, "max_stars_repo_stars_event_min_datetime": "2015-06-10T00:44:21.000Z", "max_stars_repo_stars_event_max_datetime": "2021-05-17T13:20:09.000Z", "max_issues_repo_path": "TutorialAndDevGuide/Tutorial/Language.tex", "max_issues_repo_name": "Copilot-Language/copilot-discussion", "max_issues_repo_head_hexsha": "caccad918b23dae991095344a845827ddccd6047", "max_issues_repo_licenses": ["BSD-3-Clause"], "max_issues_count": 30, "max_issues_repo_issues_event_min_datetime": "2019-04-01T20:24:19.000Z", "max_issues_repo_issues_event_max_datetime": "2021-09-07T22:34:17.000Z", "max_forks_repo_path": "TutorialAndDevGuide/Tutorial/Language.tex", "max_forks_repo_name": "Copilot-Language/copilot-discussion", "max_forks_repo_head_hexsha": "caccad918b23dae991095344a845827ddccd6047", "max_forks_repo_licenses": ["BSD-3-Clause"], "max_forks_count": null, "max_forks_repo_forks_event_min_datetime": null, "max_forks_repo_forks_event_max_datetime": null, "avg_line_length": 36.3985611511, "max_line_length": 149, "alphanum_fraction": 0.7264497767, "num_tokens": 7048, "lm_name": "Qwen/Qwen-72B", "lm_label": "1. YES\n2. YES\n\n", "lm_q1_score": 0.6513548782017746, "lm_q2_score": 0.523420348936324, "lm_q1q2_score": 0.34093239762974975}} {"text": "\\documentclass{article}\n\\usepackage[utf8]{inputenc}\n\\usepackage[T1]{fontenc}\n\\usepackage{lmodern,textcomp}\n\\usepackage[frenchb]{babel}\n\\usepackage{amsmath,amsfonts}\n\\usepackage{graphicx}\n\\title{Machine Learning - Lesson n°11}\n\\author{Chloé-Agathe Azencott}\n\\date{December 2016}\n\\DeclareMathOperator*{\\argmax}{arg\\,max}\n\\begin{document}\n\n\\maketitle\n\nScribes : Rémi Lalanne, Adam Hotait and Devang Thakkar\n\n\\section{Introduction}\nIn Machine Learning, it appears necessary to reduce the dimensionality of data set. There are various reasons to do so :\n\n\\begin{itemize}\n \\item Computational complexity : the goal is to reduce the computational cost in time and space and therefore to avoid data overfitting.\n \n \\item Interpretability, so that models are easier to interpret by human users.\n \n \\item To simplify the models and make them more robust since there is less variance.\n \n \\item Data visualization : the goal is to make this visualization easier for a human, for instance doctors who want to visualize plenty of data of their patients.\n \n \\item To curb the cost of data acquisition, that can be very high in some cases.\n \n \\item To keep only relevant and non-redundant attributes, since irrelevant and redundant attributes can prevent the algorithm from being efficient.\n \n\\end{itemize}\n\n\\bigskip\n\n\\textbf{Feature selection and feature extraction} are two ways of accomplishing dimensionality reduction.\n\nOn the one hand, the idea of feature selection is to select $m