%
\section{Introduction}%
%
\begin{frame}[t]%
\frametitle{Optimization Algorithms}%
\begin{itemize}%
\item Many questions in the real world are actually optimization problems\only<2->{, e.g.,
\begin{itemize}%
%
\item \only<-6>{Find the \emph{shortest} tour for a salesman to visit certain set of cities\only<-2>{ in China and return to Hefei!}}\only<7->{\alert<7>{Traveling Salesman Problem}\scitep{ABCC2006TTSPACS,LLKS1985TTSPAGTOCO,GP2004TTSPAIV,L2011SGEFTSPIMSS}}%
%
\item<3-> \only<-6>{I need to transport $n$ items from here to Feixi\only<-3>{ but they are too big to transport them all at once. How can I load them best into my car so that I have to travel back and forth the least times?}}\only<7->{\alert<7>{Bin Packing Problem}\scitep{KSH1995EHFTBPP}}%
%
\item<4-> \only<-6>{Which setting of $x_1$, $x_2$, $x_3$, and $x_4$ can make $(x_1\lor\lnot x_2 \lor x_3)\land(\lnot x_2\lor\lnot x_3 \lor x_4) \land (\lnot x_1\lor\lnot x_3 \lor \lnot x_4)$ become true\only<-4>{ (or, at least, as \emph{many} of its terms as possible)?}}\only<7->{\alert<7>{Maximum (3-)Satisfiability Problem}\scitep{HS2000SAORFROS,TH2004UAIAEEFSAFSAMS,S1978TCOSP,RMK2000EASP}}%
%
\item<5-> \only<-6>{I want to build a large factory with $n$ workshops.\only<-5>{ I know the flow of material between each two workshops and now need to choose the locations of the workshops such that the overall running cost incurred by material transportation is \emph{minimized}.}}\only<7->{\alert<7>{Quadratic Assignment Problem}\scitep{MF1999ACOMATSAACFTQAP,GTD1999ACOQAP}}%
\end{itemize}%
}%
%
\item<6-> Many optimization problems are \alert<7>{\NPHard}, meaning that finding the best possible solution will usually not be possible in feasible time.%
%
\item<8-> We use metaheuristic optimization algorithms to give us good approximate solutions within acceptable runtime.%
%
\item<9-> Examples of such algorithms are\only<-19>{ %
Evolutionary Algorithms\scitep{BFM1997EA,CWM2011VOEAFRWA,BFM2000EC1BAAO,BFM2000EC2BAAO,DLJD2000EC,EM1999EC,CDGDMPP1999NIIO,GT2002AIECTAA,WGOEB}\uncover<10->{, %
Ant Colony Optimization\scitep{DMC1996ACO,DS2004ACO,GM2002APBATDOP,ZBMD2004MBSFCOACS,WGOEB}\uncover<11->{, %
Evolution Strategies\scitep{R1965ES,R1973ES,R1994ES,S1965KYASDEFIDS,S1968EOEZDT1,S1975EUNO,WGOEB}\uncover<12->{, %
Differential Evolution\scitep{WGOEB}\uncover<13->{, %
Particle Swarm Optimization\scitep{WGOEB}\uncover<14->{, %
Estimation of Distribution Algorithms\scitep{PSL2005DE,F2006DE,MMVRCC2006DE,BZSM2006DE,LZ2000DE,MM2005DE,BVPK2006DE,S2010DEFCFOAABKPARS}\uncover<15->{, %
CMA-ES\scitep{HOG1995ESAD,HO1996AANMDIESTCMA,HO2001ESCMA,HMK2003RTTCOTDESWCMACE,HK2004ETCESOMTF,H2006TCESACR,AH2005ARCESWIPS,AH2005PEOAALSEA}\uncover<16->{, and %
Local Search methods\scitep{HS2005SLSFAA,AL1997LSICO,DBSD2001DOILSA}\uncover<17->{ such as %
Simulated Annealing\scitep{SSF2002FCAIFSA,LA1987SATAA,B1987GAASA,JCS2003HC,KGV1983SA,VC1985SA,DPSW1982MCTICO,DPSW1982MCTICO2,P1970AMCMFTASOCTOCOP,WGOEB}\uncover<18->{ or %
Tabu Search\scitep{G1989TSPI,G1990TSPII,GL1993TABU,DWH1989TSTATAAATNN,BT1994TABU}\uncover<19->{, %
as well as hybrids of local and global search, such as Memetic Algorithms\scitep{M1989MA,M2002MA,MC2003AGITMA,ES2003HWOTMA,HKS2005RAIMA,DM2004MA,RS1994FMA}%
}}}}}}}}}}}\only<20->{\dots\ many}%
%
\item<20-> \alert<20>{Which of them is best (for my problem)?}%
\item<21-> \alert<21>{How can I make a good algorithm better (for my problem)?}%
%
\end{itemize}%
%
\locate{2}{\includegraphics[width=0.6\paperwidth]{\sharedPath/graphics/optimization/tsp/tsp_example/tsp_example}}{0.2}{0.29}%
\locate{3}{\includegraphics[width=0.875\paperwidth]{\sharedPath/graphics/optimization/bin_packing/bin_packing_example/bin_packing_example}}{0.0625}{0.495}%
\locate{4}{\includegraphics[width=0.55\paperwidth]{\sharedPath/graphics/optimization/sat/sat_example/sat_example}}{0.225}{0.5}%
\locate{5}{\includegraphics[width=0.85\paperwidth]{\sharedPath/graphics/optimization/qap/qap_example/qap_example}}{0.075}{0.625}%
\locate{6-7}{\includegraphics[width=0.78\paperwidth]{\sharedPath/graphics/complexity/exponential_functions/exponential_functions}}{0.09}{0.55}%
%
\end{frame}%
%
\begin{frame}[t]%
\frametitle{Algorithm Analysis and Comparison}%
\begin{itemize}%
\item \alert{Which of the algorithms is best (for my problem)?}%
\item<2-> Traditional Approach {\`{a}} la \emph{\inQuotes{QuickSort is better than Bubble Sort because it needs \bigOOf{n \log n} while Bubble Sort needs \bigOOf{n^2} steps to sort $n$ elements in the average case.}}%
%
\item<3-> Complexity Analysis, Theoretical Bounds of Runtime and Solution Quality%
\item<4-> \alert<-9>{Usually not feasible}\uncover<5->{%
\begin{itemize}%
\item analysis extremely complicated\uncover<6->{ since%
\item<6-> algorithms are usually randomized\uncover<7->{ and%
\item<7-> have many parameters (e.g., crossover rate, population size)\uncover<8->{ and%
\item<8-> \inQuotes{sub-algorithms} (e.g., crossover operator, mutation operator, selection algorithm)%
\item<9-> optimization problems also differ in many aspects%
\item<10-> theoretical results only available for toy problems and extremely simplified algorithms.%
\item<11-> \alert<10>{Currently, not mature enough to be an easy-to-use tool for practitioners}%
}}}%
\end{itemize}%
}%
%
\item<12-> \alert{Experimental analysis and comparison only practical alternative.}%
%
\end{itemize}%
\end{frame}%
%
%
%
\begin{frame}%
\frametitle{Performance and Anytime Algorithms}%
%
\emph{\inQuotes{We use metaheuristic optimization algorithms to give us \alert<3->{good approximate solutions} within \alert<4->{acceptable runtime}.}}%
%
\uncover<2->{%
\begin{itemize}%
\item Algorithm performance has two dimensions\scitep{NAFR2010RPBBOB2ES,WCTLTCMY2014BOAAOSFFTTSP}:\uncover<3->{ solution quality\uncover<4->{ and required runtime}}%
\item<5-> Anytime Algorithms\scitep{BD1989STDPP2} are optimization methods which maintain an approximate solution at \emph{any time} during their run and iteratively improve this guess.%
\item<6-> All metaheuristics are Anytime Algorithms.%
\item<7-> Several exact methods like Branch-and-Bound\scitep{LMSK1963AAFTTSP,Z1993TBABACSOTATSP,Z1999TAADFBABACSOTATSP} are Anytime Algorithms.%
\item<8-> Consequence: Most optimization algorithms produce approximate solutions of different qualities at different points during their process.%
\item<9-> Experiments must capture solution quality and runtime data.%
\end{itemize}%
}%
%
\locate{3}{\includegraphics[width=0.55\paperwidth]{\sharedPath/graphics/optimization/performance/performance_dimensions/performance_dimensions_1}}{0.225}{0.542}%
\locate{4}{\includegraphics[width=0.55\paperwidth]{\sharedPath/graphics/optimization/performance/performance_dimensions/performance_dimensions_2}}{0.225}{0.542}%
\locate{5}{\includegraphics[width=0.55\paperwidth]{\sharedPath/graphics/optimization/performance/performance_dimensions/performance_dimensions}}{0.225}{0.542}%
%
\end{frame}%
%
\begin{frame}[t]%
\frametitle{Experimental Procedure}%
\begin{itemize}%
\item In optimization or Machine Learning, the following experimental procedure is often used\uncover<2->{%
\begin{enumerate}%
\item Select a \only<3->{\alert<3>{set of }}benchmark instance\only<3->{\alert<3>{s}}\only<3-7>{:%
\begin{itemize}%
\item multiple instances%
\item<4-> which cover some different problem features%
\item<5-> should be well-known to make results comparable%
\only<-6>{%
\item<6-> e.g., \tspLib\expandafter\scitep{\tspLibReferences} for the TSP has instances with different numbers of cities and geometries%
}%
\only<-7>{%
\item<7-> e.g., \bbob\expandafter\scitep{\bbobReferences} offers different benchmark functions for numerical optimization problems%
}%
\end{itemize}%
}%
\item<8-> Do experiment\only<12->{\alert<12-13>{s}}\only<9-13>{:%
\begin{itemize}%
\item conduct several independent runs of algorithm for each benchmark instance%
\item<10-> collect algorithm progress informatio, e.g., as \emph{\inQuotes{runtime bestObjectiveValue}} tuples%
\item<11-> one log file per run, each log file has several such tuples%
\item<12-> repeat for different algorithm parameter settings (e.g., different population sizes of an EA)%
\item<13-> repeat with other algorithms for comparison purposes%
\end{itemize}%
}%
\item<14-> Evaluate the gathered data\only<15->{:\alert<22>{%
\begin{itemize}%
\item draw diagrams of progress of solution quality over time%
\item<16-> draw diagrams of advanced statistical parameters such as ECDF\scitep{HAFR2012RPBBOBES,HS1998ELVAPAR,TH2004UAIAEEFSAFSAMS,WCTLTCMY2014BOAAOSFFTTSP}\only<17->{ and ERT\scitep{HAFR2012RPBBOBES,WCTLTCMY2014BOAAOSFFTTSP}} (over time)%
\item<18-> use statistical tests to compare results (at different points during the runs)%
\item<19-> analyze the impact of benchmark features and algorithm parameters on the above%
\end{itemize}%
}}%
%
\item<20-> Draw conclusions about algorithm performance and parameter settings%
\item<21-> But this is all \emph{very} cumbersome, involves much work and much data\dots
\end{enumerate}%
}%
\item<22-> The \optimizationBenchmarking\ Evaluator can automatize much of this work%
\end{itemize}%
%
\locateWithCaption{6}{%
\includegraphics[width=0.925\paperwidth]{\sharedPath/graphics/optimization/tsp/tspLib_features/tspLib_features_symmetric}%
}{%
The relative amounts of the instances of the 110 symmetric instances of \tspLib\ according to their features (the 10 asymmetric instances are not plotted).%
}{0.0375}{0.51}{0.925}%
%
\locateWithCaption{7}{%
\pgfuseimage{bbob_features}%
}{%
The relative amounts of \bbob\ benchmark functions according to their features.%
}{0.0375}{0.51}{0.925}%
%
\locateWithCaption{10}{%
\includegraphics[width=0.8\paperwidth]{\sharedPath/graphics/optimization/tsp/tspSuite_logfile_example/tspSuite_logfile_example}%
}{%
Example for data collected in a log file by \tspSuite\scitep{\tspSuiteReferences}.%
}{0.0375}{0.53}{0.925}%
%
\locateWithCaption{15-17}{
\strut%
\includegraphics[width=0.28\paperwidth]{\sharedPath/graphics/optimization/performance/progress_example/progress_example}%
\uncover<16->{%
\strut\hfill\strut%
\includegraphics[width=0.28\paperwidth]{\sharedPath/graphics/optimization/performance/ecdf_example/ecdf_example}%
\uncover<17->{%
\strut\hfill\strut%
\includegraphics[width=0.28\paperwidth]{\sharedPath/graphics/optimization/performance/ert_example/ert_example}%
}}\strut%
}{%
Examples for progress\only<16>{ and ERT}\only<17->{, ERT, and ECDF} diagrams for different algorithms (signified by different colors) over different sub-sets of the \tspLib\ data.%
}{0.0375}{0.525}{0.925}%
%
\end{frame}%
%
%


\section*{vEB - Peter Christensen}

\begin{enumerate}
	\item Define the goal: To get algorithm with $O(\log\log u)$ runtime for universe of size U
	\item A recursion tree that where the input shrinks $\sqrt{u}$ and constant work in the leafs would give that.
	\item Show an example of a bit vector tree, where the leafs are spint into cluster and the parent are the summart vector.
	\item Define the structure vEb with a $u$, summary, cluster, min, max.
	\item Go through an example of how the structure works, such as Figure~20.6.
	\item go through insert and predecessor algorithm and explain how it works.
\end{enumerate}


\documentclass[../psets.tex]{subfiles}

\pagestyle{main}
\renewcommand{\leftmark}{Problem Set \thesection}

\begin{document}

\section{Mechanical Waves}
\begin{enumerate}[label={\arabic*)}]
    \item \marginnote{8/9:}\textcite{bib:YoungFreedman}: Problem 15.9.\par
    Which of the following wave functions satisfies the wave equation?
    \begin{enumerate}
        \item $y(x,t)=A\cos(kx+\omega t)$.
        \item $y(x,t)=A\sin(kx+\omega t)$.
        \item $y(x,t)=A(\cos kx+\cos\omega t)$.
        \item For the wave of part (b), write the equations for the transverse velocity and transverse acceleration of a particle at point $x$.
    \end{enumerate}
    \item \textcite{bib:YoungFreedman}: Problem 15.26.\par
    A fellow student with a mathematical bent tells you that the wave function of a traveling wave on a thin rope is $y(x,t)=(\SI{2.30}{\milli\meter})\cos[(\SI{6.98}{\radian\per\meter})x+(\SI{742}{\radian\per\second})t]$. Being more practical, you measure the rope to have a length of $\SI{1.35}{\meter}$ and a mass of $\SI{0.00338}{\kilo\gram}$. You are then asked to determine:
    \begin{enumerate}
        \item Amplitude.
        \item Frequency.
        \item Wavelength.
        \item Wave speed.
        \item Direction the wave is traveling.
        \item Tension in the rope.
        \item Average power transmitted by the wave.
    \end{enumerate}
    \item \textcite{bib:YoungFreedman}: Problem 15.30.\par
    \textbf{Interference of Triangular Pulses.} Two triangular wave pulses are traveling toward each other on a stretched string as shown below. Each pulse is identical to the other and travels at $\SI{2.00}{\centi\meter\per\second}$. The leading edges of the pulses are $\SI{1.00}{\centi\meter}$ apart at $t=0$. Sketch the shape of the string at $t=\SI{0.250}{\second}$, $t=\SI{0.500}{\second}$, $t=\SI{0.750}{\second}$, $t=\SI{1.000}{\second}$, and $t=\SI{1.250}{\second}$.
    \begin{center}
        \begin{tikzpicture}[scale=1.2]
            \footnotesize
            \draw [orx,very thick] (0,0)
                -- (1,0)
                -- (2,1)
                -- (3,0)
                -- (4,0)
                -- (5,1)
                -- (6,0)
                -- (7,0)
            ;

            \draw [very thin,|-|] (0.85,0.05) -- node[left]{$\SI{1.00}{\centi\meter}$} ++(0,0.95);
            \draw [very thin,|-|] (1,-0.15) -- node[below]{$\SI{1.00}{\centi\meter}$} ++(1,0);
            \draw [very thin,|-|] (2,-0.15) -- node[below]{$\SI{1.00}{\centi\meter}$} ++(1,0);
            \draw [very thin,|-|] (3,0.3) -- node[above]{$\SI{1.00}{\centi\meter}$} ++(1,0);
            \draw [very thin,|-|] (4,-0.15) -- node[below]{$\SI{1.00}{\centi\meter}$} ++(1,0);
            \draw [very thin,|-|] (5,-0.15) -- node[below]{$\SI{1.00}{\centi\meter}$} ++(1,0);
            \draw [very thin,|-|] (6.15,0.05) -- node[right]{$\SI{1.00}{\centi\meter}$} ++(0,0.95);

            \draw [grx,ultra thick,-latex] (2,1.1) -- node[above=1mm,black]{$v=\SI{2.00}{\centi\meter\per\second}$} ++(1,0);
            \draw [grx,ultra thick,-latex] (5,1.1) -- node[above=1mm,black]{$v=\SI{2.00}{\centi\meter\per\second}$} ++(-1,0);
        \end{tikzpicture}
    \end{center}
    \item \textcite{bib:YoungFreedman}: Problem 15.32.\par
    \textbf{Interference of Rectangular Pulses.} The below figure shows two rectangular wave pulses on a stretched string traveling toward each other. Each pulse is traveling with a speed of $\SI{1.00}{\milli\meter\per\second}$ and has the height and width shown in the figure. If the leading edges of the pulses are $\SI{8.00}{\milli\meter}$ apart at $t=0$, sketch the shape of the string at $t=\SI{4.00}{\second}$, $t=\SI{6.00}{\second}$, and $t=\SI{10.0}{\second}$.
    \begin{center}
        \begin{tikzpicture}[scale=1.2]
            \footnotesize
            \draw [orx!60!black,line join=round,double=orx,double distance=1.2pt] (0,0)
                -- (1,0)
                -- (1,0.75)
                -- (2,0.75)
                -- (2,0)
                -- (4,0)
                -- (4,-1)
                -- (5,-1)
                -- (5,0)
                -- (6,0)
            ;

            \draw [very thin,|-|] (0.5,0.05) -- node[xshift=-0.05cm,fill=white,inner sep=2pt]{$\SI{3.00}{\milli\meter}$} ++(0,0.7);
            \draw [very thin,|-|] (1,0.9) -- node[above]{$\SI{4.00}{\milli\meter}$} ++(1,0);
            \draw [very thin,|-|] (2,-0.15) -- node[below]{$\SI{8.00}{\milli\meter}$} ++(1.95,0);
            \draw [very thin,|-|] (4,-1.15) -- node[below]{$\SI{4.00}{\milli\meter}$} ++(1,0);
            \draw [very thin,|-|] (5.5,-0.05) -- node[xshift=0.05cm,fill=white,inner sep=2pt]{$\SI{4.00}{\milli\meter}$} ++(0,-0.95);

            \draw [grx,ultra thick,-latex] (2.1,0.4) -- ++(0.5,0) node[right,black]{$v=\SI{1.00}{\milli\meter\per\second}$};
            \draw [grx,ultra thick,-latex] (3.9,-0.75) -- ++(-0.5,0) node[left,black]{$v=\SI{1.00}{\milli\meter\per\second}$};
        \end{tikzpicture}
    \end{center}
    \item \textcite{bib:YoungFreedman}: Problem 15.34.\par
    Adjacent antinodes of a standing wave on a string are $\SI{15.0}{\centi\meter}$ apart. A particle at an antinode oscillates in simple harmonic motion with amplitude $\SI{0.850}{\centi\meter}$ and period $\SI{0.0750}{\second}$. The string lies along the $+x$-axis and is fixed at $x=0$.
    \begin{enumerate}
        \item How far apart are the adjacent nodes?
        \item What are the wavelength, amplitude, and speed of the two traveling waves that form this pattern?
        \item Find the maximum and minimum transverse speeds of a point at an antinode.
        \item What is the shortest distance along the string between a node and an antinode?
    \end{enumerate}
    \item \textcite{bib:YoungFreedman}: Problem 15.64.\par
    A strong string of mass $\SI{3.00}{\gram}$ and length $\SI{2.20}{\meter}$ is tied to supports at each end and is vibrating in its fundamental mode. The maximum transverse speed of a point at the middle of the string is $\SI{9.00}{\meter\per\second}$. The tension in the string is $\SI{330}{\newton}$.
    \begin{enumerate}
        \item What is the amplitude of the standing wave at its antinode?
        \item What is the magnitude of the maximum transverse acceleration of a point at the antinode?
    \end{enumerate}
    \item A harmonic wave travels down a string in the $+x$ direction. At position $x=0$ and time $t=0$, the following is observed: the displacement of the string is $+\SI{1.0}{\centi\meter}$, the transverse velocity is $-\SI{2.0}{\centi\meter\per\second}$, and the transverse acceleration is $-\SI{4.0}{\centi\meter\per\square\second}$.
    \begin{enumerate}
        \item What is the frequency of the wave?
        \item What is the amplitude of the wave?
    \end{enumerate}
    \item A long, uniform rope of length $L$ hangs vertically. The only tension in the rope is that produced by its own weight.
    \begin{enumerate}
        \item Show that, as a function of the distance $y$ from the lower end of the rope, the speed of a transverse wave pulse on the rope is $\sqrt{gy}$.
        \item How much time does it take for a wave pulse to travel from one end of the rope to the other?
    \end{enumerate}
    \item Using continuity conditions on a string, we derived the relative amplitudes for transmitted and reflected waves at a boundary. Show that the average power of the \emph{transmitted} wave plus the average power of the \emph{reflected} wave is equal to the average power of the \emph{incident} wave. (Otherwise, energy would not be conserved.)
\end{enumerate}

\end{document}


\chapter{Seamless modeling of retarded vdW interactions}\label{chap:casimir}

{\sffamily\mathversion{sans} This chapter briefly discusses the extension of the MBD method to distances at which the finite speed of light cannot be neglected, resulting in the so-called retarded vdW (Casimir) interactions.
Previously, microscopic models of vdW interactions such as MBD were restricted to the non-retarded regime, whereas the macroscopic continuous models used for description of Casimir interactions could not be used at short distances and must have been parametrized from experimental data.
Here, we show that these two descriptions can be unified within a single framework, which then enables seamless calculation of vdW energies both at the non-retarded and retarded (Casimir) regimes.
This unification also extends the applicability of the new developments in Chapter~\ref{chap:polarizability}, because any improvements in a model of material response can be directly used in the study of Casimir physics.
The results discussed in this chapter have been published in \citep*{VenkataramPRL17}.
The Maxwell-equation scattering calculations were done by Prashanth Venkataram, the DFT and polarizability screening calculations by myself, and the unified theoretical framework is a result of joint work.
}\vspace{1em}

The ACFD formula and hence the MBD correlation energy in~\eqref{eq:mbd-rpa} originate from the nonrelativistic quantum mechanics, which assumes that the electromagnetic forces in the form of the Coulomb law acts instantly over any distance.
This limits the applicability of MBD to systems that are separated by less than hundreds of angstroms, at which point the time it takes for light to travel between the interacting objects becomes comparable to the frequency of the electronic oscillations that drive the vdW interactions.
(The speed of light, $c$, in atomic units is approximately 137, the inverse of the fine-structure constant.)
The well-known effect of this retardation of the electromagnetic force is the asymptotic $1/R^7$ attraction that replaces the nonrelativistic $1/R^6$ power law.

The extension of MBD to account for this retardation consists of two steps.
First, the instantaneous dipole operator (eq.~\ref{eq:dipole-op}) is replaced with its frequency-dependent retarded version, which is proportional to the Green's function, $\boldsymbol G_0$, of the electric field,
\begin{equation}
  \tilde{\mathbf T}(\mathbf R,u)=\frac{4\pi u^2}{c^2}\boldsymbol G_0=\big(\boldsymbol\nabla\otimes\boldsymbol\nabla'-\tfrac{u^2}{c^2}\mathbf I\big)\mathrm e^{-|\mathbf r-\mathbf r'|u/c}v(|\mathbf r-\mathbf r'|)\Big|_{\substack{\mathbf r=\mathbf R\\\mathbf r'=\mathbf 0}}
  \label{eq:green-maxwell}
\end{equation}
This substitution prevents one to perform the analytic integration over frequencies analytically (see eq.~\ref{eq:mbd-rpa}), but otherwise it is a straightforward modification of the MBD method.

The second step is necessary only because of the kind of systems that we want to study.
The prototypical systems studied in the context of Casimir interactions consist of small microscopic bodies such as molecules, and macroscopic objects with nontrivial shapes or surface gratings~\cite{RodriguezNP11,WoodsRMP16}.
The latter are typically large enough that microscopic description of individual atoms in them is unnecessary and, furthermore, such large atomic calculations would be unfeasible.
As an alternative, efficient approaches solve directly the continuous Maxwell equations either with scattering or finite-differencing methods~\cite{RodriguezPRA07,RahiPRD09}.
This raises the issue of connecting the continuous and microscopic descriptions.
It turns out that such a connection is naturally enabled by the form of the MBD expression for the interaction energy.
Consider the MBD interaction energy of two bodies, $A$ and $B$,
\begin{equation}
\begin{aligned}
E_\text{int}&=E_{AB}-E_{A}-E_{B} \\
&=\frac1{2\pi}\int_0^\infty\mathrm du\operatorname{Tr}\big(\ln(1+(\boldsymbol\alpha_A+\boldsymbol\alpha_B)\tilde{\mathbf T})-\ln(1+\boldsymbol\alpha_A\tilde{\mathbf T})-\ln(1+\boldsymbol\alpha_B\tilde{\mathbf T})\big) \\
&=\frac1{2\pi}\int_0^\infty\mathrm du\operatorname{Tr}\big(\ln((1+\boldsymbol\alpha_A\tilde{\mathbf T}+\boldsymbol\alpha_B\tilde{\mathbf T})(1+\boldsymbol\alpha_A\tilde{\mathbf T})^{-1}(1+\boldsymbol\alpha_B\tilde{\mathbf T})^{-1})\big) \\
&=\frac1{2\pi}\int_0^\infty\mathrm du\operatorname{Tr}\big(\ln((1+\boldsymbol\alpha_B\tilde{\mathbf T}(1+\boldsymbol\alpha_A\tilde{\mathbf T})^{-1})(1+\boldsymbol\alpha_B\tilde{\mathbf T})^{-1})\big) \\
&\equiv\frac1{2\pi}\int_0^\infty\mathrm du\operatorname{Tr}\big(\ln((1+\boldsymbol\alpha_B\tilde{\mathbf T}_A)(1+\boldsymbol\alpha_B\tilde{\mathbf T})^{-1})\big) \\
&=\frac1{2\pi}\int_0^\infty\mathrm du\operatorname{Tr}\big(\ln(1+\boldsymbol\alpha_B\tilde{\mathbf T}_A)-\ln(1+\boldsymbol\alpha_B\tilde{\mathbf T})\big) \\
&=E_B(\tilde{\mathbf T}_A)-E_B(\tilde{\mathbf T})
\end{aligned}
\label{eq:casimir-mbd}
\end{equation}
Here, we defined $\tilde{\mathbf T}_A=\tilde{\mathbf T}(1+\boldsymbol\alpha_A\tilde{\mathbf T})^{-1}$, which is the retarded dipole operator screened by the electromagnetic response of the body $A$, and the interaction energy between $A$ and $B$ was recast as the difference in the total energy of $B$ calculated with the bare and screened dipole operators.
The definition of $\tilde{\mathbf T}_A$ has the form of a Dyson-like equation analogous to that for the interacting nonlocal polarizability, which points to two equivalent points of view on the ground-state system of bodies of matter interacting via the electromagnetic force---one as fluctuating polarizations of the electronic density propagated (in the Green's function sense) by the electromagnetic field, the other as fluctuations in the electromagnetic field propagated by the electronic response of the matter.

\begin{figure}[t]
\centering
\includegraphics[width=10cm]{media/casimir.pdf}
\caption{\textbf{Retardation effects in vdW interactions.}
Interaction energies of a perpendicular (black) and parallel (red) carbyne wire, fullerene C$_{500}$, and a protein with a golden surface calculated with different models are plotted relative to the prediction of a pairwise approximation as a function of the vertical distance, $z$.
$\mathcal{E}$ is the full retarded MBD method (eq.~\ref{eq:casimir-mbd}), $\mathcal{E}_0$ is the nonrelativistic approximation ($c\rightarrow\infty$ in~\eqref{eq:green-maxwell}), and $\mathcal E_\text{CP}$ is the so-called ``Casimir--Polder approximation'' which approximates the whole microscopic object with a single point.
The inset shows the local power-law asymptote for the plate--fullerene system.
}\label{fig:casimir}
\end{figure}

Using the formulation in~\eqref{eq:casimir-mbd}, the MBD interaction energy of a macroscopic body (which can be also a collection of macroscopic bodies) and a set of microscopic objects can be calculated in the following way.
First, one obtains the Green's function of the electric field in the presence of the macroscopic body by an efficient continuous macroscopic method.
Second, the screened retarded dipole operator is calculated from the Green's function using~\eqref{eq:green-maxwell}.
Third, the vdW interaction energy is calculated using the regular MBD method with the screened and bare dipole operators according to~\eqref{eq:casimir-mbd}.
Figure~\ref{fig:casimir} illustrates the effects of the retardation on the interactions of several prototypical systems with a golden plate.
The full retarded MBD interaction energy calculated with~\ref{eq:casimir-mbd} transitions between the nonrelativistic approximation (the regular MBD), which becomes exact at short distances, and the relativistic Casimir--Polder approximation, which models the microscopic objects as point objects, and becomes exact at large separations.
In this regard, this unified framework represents a new seamless approach to multi-scale modeling that enables accurate description of intermolecular interactions at a range spanning several orders of magnitude.


\documentclass{article}
\usepackage{fullpage}
\usepackage{nopageno}
\usepackage{amsmath}
\allowdisplaybreaks

\newcommand{\abs}[1]{\left\lvert #1 \right\rvert}

\begin{document}
\title{Notes}
\date{December 4, 2013}
\maketitle
\section*{8.7}
application of laplace transform is what the section is about. we are only doing one application. L-R-C circuit
\begin{align*}
  Q(t)&=\text{charge}\\
  Q'(t)&=\frac{\mathrm{d}Q(t)}{\mathrm{d}t}=\text{current}\\
  L\frac{\mathrm{d}^2Q}{\mathrm{d}t^2}+R\frac{\mathrm{d}Q}{\mathrm{d}t}+\frac{Q}{C}&=E(t)\\
  Q(o)&=Q_0\\
  \frac{\mathrm{d}Q}{\mathrm{d}t}(0)&=I_0\\
  C&=\text{capacitor Faradey F}\\
  L&=\text{inductor Henry H}\\
  R&=\text{resistance Ohm \omega}\\
  E&=\text{voltage Volts V}
\end{align*}
\subsection*{exercise 1}
suppose that we consider a circuit ith a capacitor C a resistor R and a voltage supply $E(t)=\left\{\begin{aligned}100&,\quad0\leq t\leq1\\0&,\quad t\geq 1\end{aligned}\right.$. If  $L=0$ find $Q(t)$ and $I(t)$ if $Q(0)=0$, $C=\frac{1}{50}F$ and $R=50\omega$
\begin{align*}
  E(t)&=100-100U(t-1)\\
  50\frac{\mathrm{d}Q}{\mathrm{d}t}+50Q&=100-100U(t-1)\\
  Q'+Q=2-2U(t-1) \intertext{take laplace transform}\\
  \mathcal{L}\{Q'\}&=s\mathcal{L}\{Q\}-Q(0)\\
  s\mathcal{L}\{Q\}-Q(0)+\mathcal{L}\{Q\}&=\frac{2}{s}-2\frac{e^{-s}}{s}\\
  \mathcal{L}\{Q\}(s+1)&=2\left(\frac{1}{s}-\frac{e^-s}{s}\right)\\
  \mathcal{L}\{Q\}&=\frac{2}{s(s+1)}-\frac{2e^{-s}}{s(s+1)}\\
  \frac{2}{s(s+1)}&=\frac{A}{s}+\frac{B}{s+1}\\
  2&=A(s+1)+Bs\\
  A&=2,\quad A+B=0\\
  B&=-2\\
  \mathcal{L}\{Q\}&=(\frac{2}{s}-\frac{2}{s+1})-e^{-s}(\frac{2}{s}-\frac{2}{s+1})\\
  Q(t)&=2-2e^{-t}-(2-2e^{1-t})U(t-1)\intertext{case 1, $0<t<1$}\\
  \frac{\mathrm{d}U}{\mathrm{d}t}&=0\\
  \frac{\mathrm{d}Q}{\mathrm{d}t}&=2e^{-t}\intertext{case 2, $t>1$}\\
\end{align*}

\section*{6.3 linear system}
\begin{align*}
  2x+3y&=5\\
  3x+9y&=6\\
  Ax=b\\
  A&=\left(\begin{array}{cc}2&3\\3&9\end{array}\right),X=\left(\begin{array}{c}x\\y\end{array}\right),b=\left(\begin{array}{c}5\\6\end{array}\right)\\
  X&=A^{-1}b
\end{align*}
consider $y''+5y'+2y=\cos t$ can you write this in system?
\begin{align*}
  y'&=x\\
  y''&=x'\\
  y''&=-5y'-2y+\cos t\\
  &=-5x-2y+\cos t\\
  x'&=\\
  y'&=
\end{align*}
\end{document}


\chapter*{Introduction}

The vertical distribution of fish eggs in a water column is determined
by the buoyant forcing and turbulent mixing.  A model of the steady
state distribution has been developed by Sundby \shortcite{sund83}.
Westg{\aa}rd \shortcite{west89} developed two numerical models for the time
evolution of the concentration. These models are written in FORTRAN
and use the GPGS library for graphics.

The commercial software system Matlab and the free alternative Octave
provide interactive command-line environments for numerical computations
and visualisation.  VertEgg is a toolbox for scientific work on
vertical egg distributions in these environments. It contains tools for
analysing observed distributions and performing numerical simulations
including pre- and post-processing of the results. VertEgg is
\emph{not} a finished application, the tools are intended to be used
interactively or sewed together by the user to make the application
that solves her problem. This approach should give the competent user a
powerful and flexible environment for research on vertical distributions.

The author believes strongly in learning by problem solving and
examples. Therefore the main part of the manual is the example chapter.
The examples ranges from simple illustrations of the functions in the
toolbox, by verification and sensitivity studies to a ready-to-run
egg-distribution model complete with file I/O.
These examples are also available online as \emph{scripts}, programs in
the internal programming language of the packages. The recommended way
to make small applications is to take the closest example script and
try out modifications.

The first two chapters give an introduction to the theory of vertical
egg distributions and the numerical method used. The manual also has a
chapter containing the reference manual, a systematic description of the
tools provided in VertEgg.

% Ta noe om tilgjengelighet, copyright ??


% \chapter{Reproduction of GCN}

\section{Graph Convolution Layer and GCN model}

A neutral network based on graph convolution consists of layers of graph convolution and non-linear activation function. To reproduct the work of Kipf et al.\cite{DBLP:journals/corr/KipfW16}, a neutral network is modeled by the forward function in Equation \ref{forward-function-in-gcn}.

\begin{equation}
    Z = f(X, A) = \text{softmax} \left(\hat{A}\cdot \text{ReLU}\left(\hat{A}XW^{(0)}\right)W(1)\right)
    \label{forward-function-in-gcn}
\end{equation}

\section{Model Implement and Experiment}

The reproduced model is based on PyTorch package, which contains built-in neutral network model frameworks. The codes are attached in Appendix.

To regain the training result in the work of Kipf et al\cite{DBLP:journals/corr/KipfW16}, the model implements the same hyperparameters, as shown in Table \ref{hyperparam-gcn}.

\medskip

\begin{table}[H]
    \centering
    \small
    \begin{tabular}{cccccc}
        \hline
        random seed & hidden units & dropout rate & learning rate & weight decay for L2 loss & epochs \\
        24 & 16 & 0.5 & 0.01 & 5e-4 & 200 \\
        \hline
    \end{tabular}
    \caption{Hyperparameters in GCN Reproduction}
    \label{hyperparam-gcn}
\end{table}

\section{Comparison and Result}

The training is carried out on \textbf{Cora} dataset\cite{Sen_Namata_Bilgic_Getoor_Galligher_Eliassi-Rad_2008} in both reproducable random split measures. The result performance is calculated on the basis on 20 trainings, which

The original paper performance and comparison is shown in Table \ref{result-comparison}.

\begin{table}[H]
    \centering
    \small
    \begin{tabular}{lll}
        \hline
        model & Cora & Citeseer \\
        \hline
        ManiReg\cite{ManiReg} & 59.5 & 60.1 \\
        SemiEmb\cite{SemiEmb} & 59.0 & 59.6 \\
        LP\cite{LP} & 68.0 & 45.3 \\
        DeepWalk\cite{DeepWalk} & 67.2 & 43.2 \\
        ICA\cite{ICA} & 75.1 & 69.1 \\
        Planetoid*\cite{DBLP:journals/corr/YangCS16} & 75.7 & 64.7 \\
        GCN (paper) & 81.5 & 70.3 \\
        GCN (paper, rand splits) & 80.1$\pm$0.5 & 67.9$\pm$0.5 \\
        \hline
        GCN (reproduced) & 82.5 & 73.1 \\
        \hline
    \end{tabular}
    \caption{Comparison of GCN and Other Model Performance}
    \label{result-comparison}
\end{table}

The results of reproduction generally matches the original performance. The reproduction appears to have slightly higher accuracy rate, which can be caused by sample shuffling and difference in train-valid-test ratio. The reproduction is in all satisfactory.


\chapter{Object Definition}

Objects in {\rayshade} are composed of relatively simple {\em primitive}
objects.  These primitives may be used by themselves, or they
may be combined to form more complex objects known as {\em aggregates}.
A special family of aggregate objects,
{\em Constructive Solid Geometry} or CSG
objects, are the result of a boolean operations applied to
primitive, aggregate, or CSG objects.

This chapter describes objects from a strictly geometric point of
view.  Later chapters on surfaces, textures, and shading describe
how object appearances are defined.

An {\em instance} is an object that has optionally been transformed
and
textured.  They are the entities that are actually rendered by
{\rayshade}; when you specify that, for example, a textured
sphere is to be rendered, you are said to be instantiating
the textured sphere.
An instance
is specified as a primitive, aggregate, or CSG object that
is followed by optional transformation and texturing information.
Transformations and textures are described in Chapters 7 and 8 respectively.

\section{The World Object}

Writing a {\rayshade} input file is principally
a matter of defining a special aggregate object, the World object,
which is a list of the objects in the scene.  When writing a {\rayshade}
input file, all objects that are instantiated outside of object-definition
blocks are added to the World object; you need not (nor should you)
define the World object explicitly in the input file.

\section{Primitives}

Primitive objects are the building box with which other objects are
created.  Each primitive type has associated with it specialized
methods for
creation,
intersection with a ray,
bounding box calculation,
surface normal calculation,
ray enter/exit classification,
and for the computation 2D texture coordinates termed {\em u-v}
coordinates.
This latter method is often referred to as the {\em inverse mapping}
method.

While most of these methods should be of little concern to you, the
inverse mapping methods
will affect the way in which certain textures are applied to primitives.
Inverse mapping is a matter of computing normalized $u$ and $v$ coordinates
for a given point on the surface of the primitive.  For planar objects,
the $u$ and $v$ coordinates of a point are computed
by linear interpolation based upon the $u$ and $v$ coordinates assigned
to vertices or other known points on the primitive.  For non-planar
objects, $uv$ computation can be considerably more involved.

This section briefly describes each primitive and
the syntax that should be used to create an instance of the primitive.
It also describes the inverse mapping method, if any, for each type.

\begin{defprim}{blob}{{\em thresh st r} \evec{p} [{\em st r} \evec{p} \ldots]}
	Defines a blob with consisting of a threshold equal to {\em thresh},
	and a
	group of one or more metaballs.  Each metaball is defined by
	its position \evec{p}, radius {\em r}, and strength {\em st}.
\end{defprim}
The metaballs affect each other according to a superimposed
density distribution:
\[
F(x,y,z) = \sum_{i=0}^n b_{i}e^{-d_{i}} - T = 0
\]
There is no inverse mapping method for blobs.

\begin{defprim}{box}{\evec{corner1} \evec{corner2}}
	Creates an axis-aligned box
	which has \evec{corner1} and \evec{corner2} as
	opposite corners.
\end{defprim}
Transformations may be applied to the box if a non-axis-aligned instance
is required.  There is no inverse mapping method for boxes.

\begin{defprim}{sphere}{{\em radius} \evec{center}}
	Creates a sphere with the given {\em radius} and centered at the
	given position.
\end{defprim}
Note that ellipsoids may be created by applying the proper scaling
to a sphere.  Inverse mapping on the sphere is accomplished
by computing the longitude and latitude of the point on the sphere,
with the $u$ value corresponding to longitude and $v$ to latitude.
On an untransformed sphere, the $z$ axis defines the poles, and the
$x$ axis intersects the sphere at $u = 0$, $v = 0.5$.  There are
degeneracies at the poles: the south pole contains all points of
latitude 0., the north all points of latitude 1.

\begin{defprim}{torus}{{\em rmajor rminor} \evec{center} \evec{up}}
	Creates a torus centered at \evec{center} by rotating
	a circle with the given minor radius around the center
	point at a distance equal to the major radius.
\end{defprim}
In tori inverse mapping,
the $u$ value is computed using the angle of rotation about the
up vector, and the $v$ value is computing the angle of rotation
around the tube, with $v=0$ occuring on the innermost point of the tube.

\begin{defprim}{triangle}{\evec{p1} \evec{p2} \evec{p3}}
	Creates a triangle with the given vertices.
\end{defprim}

\begin{defprim}{triangle}{\evec{p1} \evec{n1} \evec{p2} \evec{n2}
	\evec{p3} \evec{n3}}
	Creates a Phong-shaded triangle with the given vertices and
	vertex normals.
\end{defprim}
For both Phong- and flat-shaded triangles, the $u$ axis is the
vector from \evec{p1} to \evec{p2}, and the $v$ axis the vector
from \evec{p1} to \evec{p3}.  There is a degeneracy at
\evec{p3}, which contains all points with $v = 1.0$.  This default
mapping may be modified using the {\tt triangleuv} primitive described
below.

\begin{defprim}{triangleuv}{\evec{p1} \evec{n1} \evec{uv1}
  \evec{p2} \evec{n2} \evec{uv2}
  \evec{p3} \evec{n3} \evec{uv3}}
	Creates a Phong-shaded triangle with the given vertices,
	vertex normals.  When performing texturing, the
	{\em uv} given for each vertex are used instead of the
	default values.
\end{defprim}
When computing $uv$ coordinates within the interior of the
triangle, linear interpolation of the coordinates associated with
each triangle vertex is used.

\begin{defprim}{poly}{\evec{p1} \evec{p2} \evec{p3} [\evec{p4} \ldots ]}
	Creates a polygon with the given vertices. The vertices
	should be given in counter-clockwise order as one is
	looking at the ``front'' side of the polygon.  The number of
	vertices in a polygon is limited only by available memory.
\end{defprim}
Inverse mapping for arbitrary polygons is problematical.
{\Rayshade}
punts and equates $u$ with the $x$ coordinate of the point of intersection,
and $v$ with the $y$ coordinate.

\begin{defprim}{heightfield}{{\em file}}
	Creates a height field defined by the altitude data stored
	in the named {\em file}.  The height field is based upon
	perturbations of the unit square in the $z=0$ plane, and is
	rendered as a surface tessellated by right isosceles triangles.
\end{defprim}
See Appendix C for a discussion of the format of a height field file.
Height field inverse mapping is straight-forward:  $u$ is the
$x$ coordinate of the point of intersection, $v$ the $y$ coordinate.

\begin{defprim}{plane}{\evec{point} \evec{normal}}
	Creates a plane that passes through the given point and
	has the specified normal.
\end{defprim}
Inverse mapping on the plane is identical to polygonal inverse mapping.

\begin{defprim}{cylinder}{{\em radius} \evec{bottom} \evec{top}}
	Creates a cylinder that extends from \evec{bottom} to \evec{top}
	and has the indicated {\em radius}.  Cylinders are rendered
	{\em without} endcaps.
\end{defprim}
The cylinder's axis defines the $v$ axis.  The $u$ axis wraps around the
cylinder, with $u=0$ dependent upon the orientation of the cylinder.

\begin{defprim}{cone}{$rad_{bottom}$ \evec{bottom} $rad_{top}$ \evec{top}}
	Creats a (truncated) cone that extends from \evec{bottom} to
	\evec{top}.  The cone will have a radius of $rad_{bottom}$ at
	\evec{bottom} and a radius of $rad_{top}$ at \evec{top}.
	Cones are rendered {\em without} endcaps.
\end{defprim}
Cone inverse mapping is analogous to cylinder mapping.

\begin{defprim}{disc}{{\em radius} \evec{pos} \evec{normal}}
	Creates a disc centered at the given position and with the
	indicated surface normal.
\end{defprim}
Discs are useful for placing
endcaps on cylinders and cones.
Inverse mapping for the disc is based on the computation of the
normalized polar coordinates of the point of intersection.  The
normalized radius
of the point of intersection is assigned to $u$, while the normalized angle
from a reference vector is assigned to $v$.

\section{Aggregate Objects}

An aggregate is a collection of primitives, aggregate, and CSG
objects.  An aggregate, once defined, may be instantiated at will,
which means that
copies that are optionally transformed and textured may be made.
If a scene calls for the presence of many geometrically identical
objects, only one such object need be defined; the one defined object
may then be instantiated many times.

An aggregate is one of several possible types.  These aggregate types
are differentiated by the type of ray/aggregate intersection algorithm
(often termed an {\em acceleration technique} or {\em efficiency scheme})
that is used.

Aggregates are defined by giving a keyword that defines the
type of the aggregate, followed by
a series of object instantiations and
surface definitions, and terminated using the {\tt end} keyword.
If a defined object contains no instantiations, a warning message
is printed.

The most basic type of aggregate, the {\em list}, performs
intersection testing in the simplest possible way:  Each object in the
list is tested for intersection with the ray in turn, and the closest
intersection is returned.

\begin{defkey}{list}{\ldots {\tt end}}
	Create a List object containing those objects instantiated between
	the {\tt list}/{\tt end} pair.
\end{defkey}

The {\em grid} aggregate
divides the region of space it occupies into a number of discrete
box-shaped
voxels.  Each of these voxels contains a list of the objects that
intersect the voxel.  This discretization makes it possible to
restrict the objects
tested for intersection to those that are likely to hit the ray,
and to test
the objects in nearly ``closest-first'' order.

\begin{defkey}{grid}{{\em xvox yvox zvox} \ldots {\tt end}}
	Create a Grid objects composed of {\em xvox} by {\em yvox} by
	{\em zvox} voxels containing those objects
	instantiated between the {\tt grid}/{\tt end} pair.
\end{defkey}
It is usually only worthwhile to ``engrid'' rather large,
complex collections of objects.  Grids also use a great deal more
memory than List objects.

\section {Constructive Solid Geometry}

Constructive Solid Geometry is
the process of building solid objects from other solids.
The three CSG
operators are Union, Intersection, and Difference.  Each operator
acts upon two objects and produces a single object result.
By combining multiple levels of CSG operators, complex
objects can be produced from simple primitives.

The union of two objects results in an
object that encloses the space occupied by the two given objects.
Intersection results in an object that encloses the space where the two
given objects overlap.  Difference is an order dependent operator; it
results in the
first given object minus the space where the second intersected
the first.

\subsection{CSG in {\Rayshade}}

CSG in {\rayshade} will generally operate properly when applied to
conjunction with
on boxes, spheres,
tori, and blobs.
These primitives are by nature consistent, as they all
enclose a portion of space (no hole from the ``inside'' to the
``outside''), have surface normals which point outward (they
are not ``inside-out''), and do not have any extraneous surfaces.

CSG objects may also be constructed from aggregate objects.
These aggregates contain
whatever is listed inside, and may therefore be inconsistent.
For example, an object which contains a single triangle will not
produce correct results in CSG models, because the triangle does not enclose
space.  However, a collection of four triangles which form a pyramid
does enclose space, and if the triangle normals
are oriented correctly,
the CSG operators should work correctly on the pyramid.

CSG objects are specified by surrounding the objects upon
which to operate, as well as any associated surface-binding commands,
by the operator verb on one side and the {\tt end}
keyword on the other:

\begin{defkey}{union}{$<${\em Object}$>$ $<${\em Object}$>$
[$<${\em Object}$>$ \ldots] {\tt end}}
	Specify a new object defined as the union of the
	given objects.
\end{defkey}

\begin{defkey}{difference}{$<${\em Object}$>$ $<${\em Object}$>$
[$<${\em Object}$>$ \ldots] {\tt end}}
	Specify a new object defined as the difference of the
	given objects.
\end{defkey}

\begin{defkey}{intersect}{$<${\em Object}$>$ $<${\em Object}$>$
[$<${\em Object}$>$ \ldots] {\tt end}}
	Specify a new object defined as the intersection of the
	given objects.
\end{defkey}

Note that the current implementation does not support more that two
objects in a CSG list (but it is planned for a future version).

% The following aren simple CSG objects using the four consistent
% primitives:
%
% union box ... difference ...

\subsection{Potential CSG Problems}

A consistent CSG model is one which is made
up of solid objects with no dangling surfaces.  In {\rayshade},
it is quite easy to construct inconsistent models, which will usually
appear incorrect in the final images.
In {\rayshade}, CSG is implemented by maintaining
the tree structure of the CSG operations.  This tree is traversed,
and the operators therein applied, on a per-ray basis.
It is therefore difficult to verify the consistency of
the model ``on the fly.''

One class of CSG problems occur when
surfaces of objects being operated upon
coincide.  For example, when subtracting a box from another box to make a
square cup, the result will be wrong if the tops of the two boxes
coincide.  To correct this, the inner box should be made
slightly taller than the outer box.
A related problem that must be
avoided occurs when two coincident surfaces are assigned
different surface properties.

It may seem that the union operator is unnecessary, since
listing two objects together in an aggregate results
in an image that appears to be the same.
While the result of such a short-cut
may appear the same on the exterior, the interior
of the resulting object will contain
extraneous surfaces.
The following examples show this quite clearly.

\begin{verbatim}
    difference
      box -2 0 -3  2 3 3
      union  /* change to list; note bad internal surfaces */
        sphere 2 1 0 0
        sphere 2 -1 0 0
      end
    end rotate 1 0 0 -40  rotate 0 0 1 50
\end{verbatim}

The visual evidence of an inconsistent CSG object varies depending
upon the operator being used.
When subtracting a consistent object from and
inconsistent one, the resulting object will appear to be
the union of the two objects, but the shading will be incorrect.
It will appear to be inside-out in places, while correct
in other places.  The inside-out sections indicate the areas
where the problems occur.
Such problems are often caused by
polygons with incorrectly specified
normals, or by surfaces that exactly coincide (which
appear as partial ``Swiss cheese'' objects).

The following example illustrates an attempt to subtract a sphere from
a pyramid defined using an incorrectly facing triangle.  Note
that the resulting image obviously points to which triangle is
reversed.

\begin{verbatim}
    name pyramid list
        triangle 1 0 0  0 1 0  0 0 1
        triangle 1 0 0  0 0 0  0 1 0
        triangle 0 1 0  0 0 0  0 0 1
        triangle 0 0 1  1 0 0  0 0 0  /* wrong order */
    end

    difference
        object pyramid scale 3 3 3 rotate 0 0 1 45
            rotate 1 0 0 -30 translate 0 -3.5 0
        sphere 2.4 0 0 0
    end
\end{verbatim}

By default, cylinders and cones do not have end caps, and thus
are not consistent primitives.  One must usually
add endcaps by listing the
cylinder or cone with (correctly-oriented) endcap discs in an aggregate.

\section {Named Objects}

A name may be associated with any primitive, aggregate, or CSG
object through the use of the {\tt name}
keyword:

\begin{defkey}{name}{{\em objname} $<${\em Instance\/}$>$}
	Associate {\em objname} with the given object.  The
	specified object is not actually instantiated; it
	is only stored under the given name.
\end{defkey}

An object thus named may then be instantiated (with possible
additional transforming and texturing) via the {\tt object} keyword:

\begin{defkey}{object}{{\em objname} [$<$Transformations$>$] [$<$Textures$>$]}
	Instantiate a copy of the object associated with {\em objname}.
	If given, the transformations and textures are composed
	with any already associated with
	the object being instantiated.
\end{defkey}


\documentclass[10pt,tgadventor, onlymath]{beamer}

\usepackage{graphicx,amsmath,amssymb,tikz,psfrag,neuralnetwork, stackengine,array, multirow, fontawesome}

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%% begin presentation

\title{\large \bfseries Power Allocation in Heterogeneous Networks for Base Stations with Multiple Antennas}

\author{Peter Hartig \\ \and Supervisor: Prof. Laura  Cottatellucci
}

\date{\today}

\begin{document}

\frame{
\thispagestyle{empty}
\titlepage
}

\section{System Description}
\begin{frame}
\frametitle{The Heterogeneous Network}
	\includegraphics[width=\textwidth]{het_net}
\end{frame}

\begin{frame}
\frametitle{The Heterogeneous Network Game}
\begin{columns}

\begin{column}{0.5\linewidth}
	\includegraphics[width=\textwidth]{het_net}

\end{column}
\begin{column}{0.5\linewidth}
\begin{table}
    \setlength{\extrarowheight}{2pt}
    \begin{tabular}{cc|c|c|}
      & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $2$}\\
      & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$A$}  & \multicolumn{1}{c}{$B$} \\\cline{3-4}
      \multirow{2}*{Player $1$}  & $A$ & $(8,8)$ & $(2,15)$ \\\cline{3-4}
      & $B$ & $(15,2)$ & $(3,3)$ \\\cline{3-4}
    \end{tabular}
  \end{table}
\end{column}
\end{columns}
\bigskip
\begin{itemize}
\item
	Players $=$ Femto Cell Base Stations (FCBS)
\item
	Player Strategy $=$ FCBS Transmission Scheme
\end{itemize}

\end{frame}

%\begin{frame}
%\frametitle{The Heterogeneous Network Game}
%\begin{table}
%    \setlength{\extrarowheight}{2pt}
%    \begin{tabular}{cc|c|c|}
%      & \multicolumn{1}{c}{} & \multicolumn{2}{c}{Player $2$}\\
%      & \multicolumn{1}{c}{} & \multicolumn{1}{c}{$A$}  & \multicolumn{1}{c}{$B$} \\\cline{3-4}
%      \multirow{2}*{Player $1$}  & $A$ & $(8,8)$ & $(2,15)$ \\\cline{3-4}
%      & $B$ & $(15,2)$ & $(3,3)$ \\\cline{3-4}
%    \end{tabular}
%  \end{table}
%\bigskip
%\begin{itemize}
%\item
%	Players $=$ Femto Cell Base Stations (FCBS)
%\item
%	Player Strategy $=$ FCBS Transmission Scheme
%\end{itemize}
%
%\end{frame}

\section{Project Goals}

\begin{frame}
\frametitle{Objectives}
\begin{enumerate}
\setlength\itemsep{2em}

\item Find a Nash Equilibrium between all players.
\begin{itemize}
\item Preferably a "social optimal" Nash Equilibrium.
\end{itemize}
\item Minimize resources required to reach Nash Equilibrium.
\end{enumerate}
%\pause
\begin{center}
%\begin{tikzpicture}{center}
%\node [block] (game) {Game with Many Players};
%\node [block, right = of game] (central) {Central Problem};
%\node [block, right = of central] (distributed) {Distributed Solution to Central Problem};
%
%\path[line] (game) -- (central);
%\path[line] (central) -- (distributed);
%
%\end{tikzpicture}
		\includegraphics[scale=.2]{het_net}

\end{center}

\end{frame}

\section{Key Tools}

\begin{frame}
\frametitle{Key Tools: The Normalized Nash Equilibrium}
A Nash equilibrium in which the dual variables (prices) corresponding to constraints are equal up to a constant.
\begin{equation}
\lambda_f = \frac{\lambda_{0}}{ r_f}, \; \forall \; [r_0 \cdots r_F] > 0
\end{equation}
Choosing $r_f =1$, all players must pay the same "price" to change their strategy.
\end{frame}

\begin{frame}
\frametitle{Key Tools: N-Person Concave Games}
Conditions
\begin{itemize}
\item All player utility functions must be concave with respect to their own strategy.
\item The set of strategies for the \emph{entire} game is convex.
\end{itemize}
\bigskip
Implications
\begin{itemize}
\item Immediately proves existence of a \emph{pure strategy} Nash Equilibrium.
\item Provides an additional sufficient condition for uniqueness of Nash Equilibrium (diagonally strict concavity).
\end{itemize}
\end{frame}

\begin{frame}
\frametitle{Key Tools: Potential Games}
\begin{enumerate}
\item
Potential Games
\begin{itemize}
\item "Central" optimization function $\Psi(\mathbf{s})$ with
\end{itemize}
\begin{equation}\label{potential_game_condition}
\frac{\partial \Psi(\mathbf{s})}{\partial \mathbf{s}_{f}}
 =
 \frac{\partial U_f(\mathbf{s})}{\partial \mathbf{s}_{f}}.
\end{equation}
\item
When all utility functions are concave, the optimum of the potential function corresponds to a Nash Equilibrium.
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Key Tools: Distributed Optimization}
Some optimization problems may be decomposed into "sub-problems".
\begin{equation}
f(x) = \sum_{i = 1}^{F} f_{i}(x_{i})
\end{equation}
Decomposing the corresponding Lagrangian gives
\begin{equation}
L(x,y) = \sum_{i = 1}^{F} L_i(x_i,y)
\end{equation}
allows for dual ascent with distributed updates.
\end{frame}
%

\section{System Model}
\begin{frame}
\frametitle{System Model: Femto Base Stations}
Each FCBS is a player in the game and is characterized by:
\\
\begin{itemize}
\setlength\itemsep{2em}

\item
	$T_{f}$ antennas to transmit to $K_{f}$ femtocell users ($T_{f} \geq K_{f}$).
\item
	The transmitted
	signal is $\mathbf{s}_{f
	}= \mathbf{U}_{f}\mathbf{x}_{f}$ with $E[\mathbf{x}_{f}\mathbf{x}_{f}^H] = \mathbf{I}$.
\item
	An average power constraint $E\{trace(\mathbf{U}_{f}^H\mathbf{U}_{f})\} \leq P^{Total}_{f} $.
\item
	A utility function $U_{f}(\boldsymbol{\gamma}_{f}) =
	\sum_{i=1}^{K_{f}}
    	 U_{f,i}(\gamma_{f,i}) $
    	with non-decreasing function $U_{f,i}(\cdot)$.
\end{itemize}
\end{frame}

%\begin{frame}
%\frametitle{System Model: FCBS Users}
%User $i$ of FBS $f$ has signal to interference plus noise ratio (SINR)
%	\begin{equation*}
%	\gamma_{f,i} = \frac{\|\mathbf{h}^H_{f,i}\mathbf{u}_{f,i}\|^2}
%	{\sigma^2_{\text{noise}}   +
%	\underbrace{
%	 \sum_{\tilde{f}=1, \tilde{f}\neq f}^{f} \sum_{u=1}^{K_{\tilde{f}}}
%	\|\mathbf{h}^H_{\tilde{f},u}\mathbf{u}_{\tilde{f},i}\|^2}_{\mathrm{inter-cell}}
%	 +
%	 \underbrace{
%	 \sum_{\tilde{k}=1, \tilde{k}\neq i}^{K_f}
%	 \|\mathbf{h}^H_{f,\tilde{k}}\mathbf{u}_{f,\tilde{k}}\|^2}_{\mathrm{intra-cell}}},
%	  \; i \in \{1 ... K_f\}
%	  \end{equation*}
%\end{frame}

\begin{frame}
\frametitle{System Model: Macro Users}
	Received interference constraint given by
	\begin{equation}
	E\{\sum^F_{f=1} \mathbf{\tilde{h}}_{m,f}^T  \mathbf{U}_{f}
	\mathbf{U}_{f}^{H} \mathbf{\tilde{h}}_{m,f}^*\} \leq I^{Threshold}
	_{m}
	\end{equation}
	\bigskip
	\centering
		\includegraphics[scale=.2]{het_net}
\end{frame}

\begin{frame}
\frametitle{System Model: General}
\begin{itemize}
\setlength\itemsep{2em}

\item
	No inter-femto cell interference
\item
	FCBS have knowledge of channel state information
	\begin{itemize}
	\item
	The downlink channel matrix $\mathbf{H}_f \in \mathbb{C}_{K_{f} \times T_{f}} $ to its $K_{f} $ users.

	\end{itemize}
\end{itemize}
	\bigskip
	\centering
		\includegraphics[scale=.2]{het_net}
\end{frame}

\section{General Setup}

\begin{frame}

\frametitle{General Problem Formulation}
\begin{enumerate}
\setlength\itemsep{2em}

\item  Player $f$ has trasnmitted signal $\mathbf{s}_{f
	}= \mathbf{U}_{f}\mathbf{x}_{f}$ with strategy $\mathbf{U}_f$.
\begin{itemize}
\item $\mathbf{U}_f$ has an implied power allocation
\end{itemize}
%\item Simplify by selecting $\mathbf{U}_f$ as a pseudo inverse of $\mathbf{H}_f$  (i.e $\mathbf{H}_f\mathbf{U}_f = \mathbf{I}$) such that
%	\begin{equation*}
%	\gamma_{f,i} = \frac{\|\mathbf{h}^H_{f,i}\mathbf{u}_{f,i}\|^2}
%	{\sigma^2_{\text{noise}}}
%	\end{equation*}
%	with
%	\begin{equation*}
%	U_{f}(\boldsymbol{\gamma}_{f}) =
%	\sum_{i=1}^{K_{f}}
%    	 U_{f,i}(\gamma_{f,i}) .
%	\end{equation*}
\pause
Does the problem admit the N-Person Concave game Framework?

\item  Set of strategies is convex. \faThumbsOUp
\pause
\item  $U_{f}(\boldsymbol{\gamma}_{f})$ is concave only when $U_{f,i}(\gamma_{f,i})$ is concave and non-increasing.
\faThumbsODown
\end{enumerate}
\end{frame}

\section{Convex Setup}

\begin{frame}
\frametitle{Convex Problem Formulation}
\begin{enumerate}
\setlength\itemsep{2em}
\item Pre-select $\mathbf{U}_f$ to be a psuedo inverse of $\mathbf{H}_f$  (i.e $\mathbf{H}_\mathrm{f}\mathbf{U}_\mathrm{f} = \mathbf{I}$).
\item
	Normalize the columns of $\mathbf{U}_{f}$ such that
	 $\|\mathbf{u}_{f,i}\|^2 =1 \;\forall i \in \{1 ... K_{f}\}$.
\item
	Player $f$ strategy is now the diagonal, power allocation
	matrix $\mathrm{diag}(\mathbf{p}_{f})$ with $p_{f,i} \geq 0, \forall i \in \{1 ... K_{f}\}$
such that the transmitted
	signal is
	$\mathbf{s}_{f	}= \mathbf{U}_{f}
	\mathrm{diag}(\mathbf{p}_{f})^{\frac{1}{2}}
	\mathbf{x}_{f}$.
\item
	Power constraint given by
	\begin{gather*}
	\sum_{i=1}^{K_{f}} p_{f,i}
	  \leq P^{Total}_{f}.
	  	\end{gather*}
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Problem Analysis}
Does the problem admit the N-Person Concave game Framework?
\\
\begin{enumerate}
\setlength\itemsep{2em}
\item  Set of strategies is convex. \faThumbsOUp
\item  $U_{f}(\boldsymbol{\gamma}_{f})$ is concave if
	$U_{f,i}(\cdot)$ is non-decreasing and concave (reasonable assumption). \faThumbsOUp
\item
	If $U_{f,i}(\cdot)$ is strictly concave, this satisfies conditions for the unique Normalized Nash Equilibrium (diagonally strict concavity) . \faThumbsOUp
\end{enumerate}

\end{frame}

\section{A Solution}
\begin{frame}
\frametitle{Player Optimization Problem}
Each player attempts to solve the optimization problem given by
	\begin{subequations}
	\begin{align}
	    \underset{\mathbf{p}_{f} }{\text{min}} \;
	    & - \sum_{i=1}^{K_f}
    	U_{f,i}(\gamma_{f,i}) \label{player_opt_c} \\
	    \text{subject to  }\\
	  &
	  \sum^F_{f=1} E\{ \mathbf{\tilde{h}}_{m,f}^T  \mathbf{s}_{f}
	\mathbf{s}_{f}^{H} \mathbf{\tilde{h}}_{m,f}^* \}
	\leq I^{Threshold}
	_{m} & m \in \{1 ...m\}
		\label{interference_const_c}\\
        &
        	\sum_{i=1}^{K_{f}} p_{f,i}
	   \leq P_{f}^{\text{Total}}  \label{power_const_c}\\
        & p_{f,i} \geq 0 &  i\in \{1 ...K_{f}\} \label{pos_power_const_c}
	\end{align}
	\end{subequations}

Note that $\mathbf{U}_{f}$ is pre-selected as a pseudo-inverse to  $\mathbf{H}_f$.
\end{frame}

\begin{frame}
\frametitle{Steps Outline}
\begin{enumerate}
\setlength\itemsep{2em}

\item
	Form Potential Function.
\item
	Setup distributed algorithm.
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{The Potential Function}
Things to note
\begin{enumerate}
\item
	Spacing assumption makes $U_{f}(\boldsymbol{\gamma}_{f})$ independent of all other player strategies.

\item
	The sum of concave functions over a convex set is also concave. Does this satisfy the potential function condition?
	\begin{equation*}\label{potential_game_condition}
\frac{\partial \Psi(\boldsymbol{\gamma})}{\partial \boldsymbol{\gamma}_{f}}
 =
 \frac{\partial U_f(\boldsymbol{\gamma})}{\partial \boldsymbol{\gamma}_{f}}
\end{equation*}
Yes!
\end{enumerate}

\end{frame}

\begin{frame}
\frametitle{Solving for a NNE}
\begin{enumerate}
\setlength\itemsep{2em}

\item
	The potential function of a game with strictly concave utility functions has optimum corresponding to NNE.
\item
	Satisfying the Diagonally Strict Concavity conditions means the NNE is unique.
\end{enumerate}
\bigskip
Just need to solve for the unique optimum of a convex problem.
\pause
\par
 Can the problem be distributed?
\end{frame}

\begin{frame}
\frametitle{Distributed Dual Ascent}
For $U_{f,i}(\gamma_{f,i}) = log(1+\gamma_{f,i})$ and potential function
\begin{gather*} \label{Potential_Function}
\Psi(\mathbf{p}) = \sum_{f = 1}^{F} U_{f}(\mathbf{p}_{f}).
\end{gather*}
We arrive at a central, convex optimization problem given by
		\begin{subequations}
	\label{optim}
	\begin{align}
	    \underset{\mathbf{p}}{\text{minimize  }}
	    & \; \Psi(\mathbf{p}) \label{potential_game} \\
	    \text{subject to  } \; &
	  \sum^F_{f=1} E\{\tilde{\mathbf{h}}_{m,f}^T  \mathbf{s}_{f}
	\mathbf{s}_{f}^{H} \tilde{\mathbf{h}}_{m,f}^* \}\leq I^{Threshold}
	_{m} & m \in \{1 ...m\}
		\label{interference_const}\\
        & E\{trace(\mathbf{s}_f\mathbf{s}_f^H)\}  \leq P_{f}^{\text{Total}}  \label{power_const}
        & \forall f \in \{1 ... f\}\\
        & p_{f,i} \geq 0 &  \forall i \in \{1 ...K_{f}\} \; \forall f \in \{1 ... F\}\label{pos_power_const}
	\end{align}
	\end{subequations}
\end{frame}

\section{Simulation Results}

\begin{frame}
\frametitle{Expected Results}
\begin{enumerate}
\setlength\itemsep{2em}

\item
	Power not limiting, at least one interference constraint should be active.
\item
	Power not limiting, increasing the number of antennas at the base stations should allow for higher utility.
\item
	The choice of the psuedoinverse, $\mathbf{U}_{f}$, should be further optimized.
\end{enumerate}
\end{frame}

\begin{frame}
\frametitle{Multiple Antennas with Constant Power}
\begin{figure}
	\includegraphics[width=\textwidth]{results/central_antenna}
	\caption{With $K_f = 5$ (users), increasing antennas allows players with constant power to increase utility.}
\end{figure}
\end{frame}

\begin{frame}
\frametitle{Increasing Power at FCBS}
\begin{figure}
	\includegraphics[width=\textwidth]{results/central_power}
	\caption{For $K_f = 5$ and $T_f = 15$, increasing FCBS power constraint is limited by interference.}
\end{figure}
\end{frame}

\begin{frame}
\frametitle{Selecting the Beamformer}
\begin{figure}
	\includegraphics[width=\textwidth]{results/central_beamformer}
\caption{With $K_f = 5$ (users) and $T_f = 10$ (antennas), choice of $\mathbf{U}_{f}$ may depend on the active system constraints.}
\end{figure}
\end{frame}

\section{Conclusion}
\begin{frame}
\frametitle{Continuing Work}
\begin{enumerate}
\item
	Implement distributed version.
\item
	Improve choice of beamformer using additional DOF.
\item
	Consider "fairness" with respect to FCBS utility.
\end{enumerate}
\end{frame}

\begin{frame}
  \centering \Large
  \emph{Thank You.}
  \\
	\bigskip
    \centering \Large
  \emph{Questions or Comments?}
\end{frame}

\end{document}


The water quality model in GIFMod has two major components including particle fate transport and constituent transport. The physical processes affecting each of these classes of water quality constituents are different but some processes are also shared between them. Constituents can be set to interact with particles through rate-limited sorption-desorption processes. In this chapter, first the governing equations for the transport of particles and constituents are presented and then the properties that can be assigned to "Particle" and "Constituent" group in GIFMod are described.
\section{Transport of Solid Particles}
\input{Particles.tex}

\section{Transport and Transformation of Dissolved and Solid-Associated Water Quality Constituents}
\input{Constituents.tex}

\subsection{External flux and aeration}\label{sssec:ExtFlux}
\input{External_Flux.tex}

\subsection{Build-up}\label{sssec:Buildup}
\input{Build-up.tex}


% !TeX root = ../../python-snippets.tex

\section{Search Algorithms}

This section contains implementations of certain search algorithms.

\subsection{Binary Search}

The binary search algortihm is a search algorithm, that finds the position of a target value within a sorted array.
It compares the target value to the middle element of the array.
If they are not equal, the half in which the target cannot lie is eliminated and the search continues on the remaining half, again taking the middle element to compare to the target value, and repeating this until the target value is found.

\lstinputlisting[caption=binary\_search.py]{../standard_lib/binary_search.py}


\chapter{外文资料原文}
\label{cha:engorg}

\title{The title of the English paper}

\textbf{Abstract:} As one of the most widely used techniques in operations
research, \emph{ mathematical programming} is defined as a means of maximizing a
quantity known as \emph{bjective function}, subject to a set of constraints
represented by equations and inequalities. Some known subtopics of mathematical
programming are linear programming, nonlinear programming, multiobjective
programming, goal programming, dynamic programming, and multilevel
programming$^{[1]}$.

It is impossible to cover in a single chapter every concept of mathematical
programming. This chapter introduces only the basic concepts and techniques of
mathematical programming such that readers gain an understanding of them
throughout the book$^{[2,3]}$.

\section{Single-Objective Programming}
Contents

\section*{References}
\begin{originbbl}
  \item M. Addlesee et al., ``Implementing a sentient computing system,'' IEEE Comput., vol. 34, no. 8, pp. 50–56, Aug. 2001.
  \item K. Al Nuaimi and H. Kamel, ``A survey of indoor positioning systems and algorithms,'' in Proc. IEEE IIT, 2011, pp. 185–190.
  \item P. Bahl, V. N. Padmanabhan, and A. Balachandran, ``Enhancements to the radar user location and tracking system,'' Microsoft Corp., Redmond, WA, USA, Tech. Rep., 2000.
\end{originbbl}


%!TEX root = ../thesis.tex
%*******************************************************************************
%*********************************** Fourth Chapter *****************************
%*******************************************************************************

\chapter{Lattice Configurations and the Gluon Propagator}\label{chapter:GluonPropagator}
\ifpdf
    \graphicspath{{Chapter4/Figs/Raster/}{Chapter4/Figs/PDF/}{Chapter4/Figs/}}
\else
    \graphicspath{{Chapter4/Figs/Vector/}{Chapter4/Figs/}}
\fi

Now that we have developed the required background understanding of lattice QCD and the topological objects of interest to this research, we can explain how our calculations are performed. This chapter will first describe how we calculate the Landau gauge gluon propagator on the lattice. This is the primary quantity of interest for the first part of the original research, and as such we will explicitly detail its calculation.  We will then motivate our choice of momentum variables, before proceeding to a description of the renormalisation scheme we employ. Finally, we will present the lattice parameters and data cuts utilised in this work.

\section{Lattice Definition of the Gluon Propagator}
In a gauge field theory the position-space propagator, $D_{\mu\nu}(x,y)$, of the gauge boson is the two-point correlation function. In the case of perturbative QCD this can be interpreted as the probability amplitude of a gluon being created at the space-time point $x$, propagating to $y$, and then being annihilated. The propagator therefore serves as a useful measure of the behaviour of gluons as a function of distance; or, correspondingly, as a function of momentum in the momentum-space representation. In this section we detail how the non-perturbative momentum-space Landau gauge gluon propagator is calculated on the lattice. We begin with the definition of the coordinate-space propagator as a two-point correlator~\cite{Zwanziger:1991gz,Cucchieri:1999sz,Langfeld:2001cz}.
\begin{equation}
D^{ab}_{\mu\nu}(x) = \langle A^a_\mu(x) \, A^b_\nu(0)\rangle.
\label{eq:coordGluonProp}
\end{equation}
The propagator in momentum space is simply related by the discrete Fourier transform,
\begin{equation}
D^{ab}_{\mu\nu}(p) = \sum_x e^{-ip\cdot x} \langle A^a_\mu(x) \, A^b_\nu(0) \rangle.
\end{equation}
Noting that the coordinate space propagator $D^{ab}_{\mu\nu}(x-y)$ only depends on the difference $x-y,$ such that
\begin{equation}
\langle A^a_\mu(x) \, A^b_\nu(0)\rangle = \langle A^a_\mu(x+y) \, A^b_\nu(y)\rangle\, ,
\end{equation}
we can make use of translational invariance to average over the four-dimensional volume to obtain the form for the momentum space propagator.
\begin{align}
D^{ab}_{\mu\nu}(p) &= \frac{1}{V}\sum_{x,y} e^{-ip\cdot x}\langle A^a_\mu(x+y) \, A^b_\nu(y) \rangle \nonumber \\
                &= \frac{1}{V}\sum_{x,y} \langle e^{-ip\cdot (x+y)} A^a_\mu(x+y) \, e^{+ip\cdot y}A^b_\nu(y) \rangle \nonumber \\
                &= \frac{1}{V}\langle A^a_\mu(p) \, A^b_\nu(-p) \rangle. \label{eq:gluPropxtop}
\end{align}

Hence we find that the momentum space gluon propagator on a finite lattice with four-dimensional volume $V$ is given by
%
\begin{equation}
D_{\mu\nu}^{ab}(p) \equiv \frac{1}{V}\left \langle A^a_\mu (p)\,A^b_\nu(-p)\right\rangle \, . \label{eq:gluonProp}
\end{equation}
%
In the continuum, the Landau-gauge momentum-space gluon propagator has the following form~\cite{Leinweber:1998im,Bonnet:2001uh}
%
\begin{equation}
D^{ab}_{\mu\nu}(p) = \left ( \delta_{\mu\nu} - \frac{p_\mu p_\nu}{p^2} \right )\,\delta^{ab}\,D(p^2) \, ,
\end{equation}
%
where $D(p^2)$ is the scalar gluon propagator.  Contracting Gell-Mann index $b$ with $a$ and
Lorentz index $\nu$ with $\mu$ one has
%
\begin{equation}
D^{aa}_{\mu\mu}(p) = (4-1)\,(n_c^2-1)\,D(p^2) \, ,
\end{equation}
%
such that the scalar function can be obtained from the gluon propagator via
%
\begin{equation}
D(p^2) = \frac{1}{3(n_c^2-1)}\,D^{aa}_{\mu\mu}(p) \, ,
\label{eq:scalarProp}
\end{equation}
%
where $n_c = 3$ is the number of colours.

As the lattice gauge links $U_\mu(x)$ naturally reside in the $3\times 3$ fundamental representation of $SU(3),$ we now wish to work in the matrix representation of $A_\mu(x)$, as introduced in Eq.~\eqref{eq:CovariantDerivative}. Using the orthogonality relation $\Tr(\lambda_a\lambda_b) = 2\delta_{ab}$ for the Gell-Mann matrices, it is straightforward to see that
%
\begin{equation}
2\Tr(A_\mu\,A_\mu) = A^a_\mu A^a_\mu\, ,
\end{equation}
%
which can be substituted into Eq.~\eqref{eq:scalarProp} to obtain the final expression for the lattice scalar gluon propagator,
%
\begin{equation}
D(p^2) = \frac{2}{3\,(n_c^2-1)\,V}\big\langle {\rm Tr}\, A_\mu(p)\,A_\mu(-p) \big\rangle \,. \label{eq:scalarProp2}
\end{equation}

To calculate Eq.~\eqref{eq:scalarProp2} on the lattice, we need to define $A_\mu(p)$. As defined in Eq.~\eqref{eq:GaugePotentialLat}, we make use of the midpoint definition of the coordinate-space gauge potential in terms of the lattice link variables such that
%
\begin{equation}
A_\mu\left(x+\frac{a}{2}\hat{\mu}\right) = \frac{i}{2ag}\left(U_\mu(x) - U_\mu^\dag(x)\right) - \frac{i}{6ag}\Tr\left(U_\mu(x) - U_\mu^\dag(x)\right)I + \mathcal{O}(a^2)\, .
\end{equation}
%
Once the link variables are fixed to Landau gauge following the procedure described in Sec.~\ref{sec:LandauGauge}, we can obtain the momentum-space gauge potential by performing a Fourier transform,
%
\begin{equation}
A_\mu(p) = \sum_x e^{-ip\cdot(x+\hat{\mu}/2)}\, A_\mu(x+\hat{\mu}/2)\, .
\end{equation}
%
We have now constructed a workable lattice definition to calculate the Landau gauge scalar gluon propagator within the lattice framework established in Chapter \ref{chapter:LatticeQCD}.

\section{Momentum Variables}\label{sec:MomentumVariables}
As discussed in Sec.~\ref{sec:Confinement}, it is understood that at high energies QCD is asymptotically free. With this understanding, we expect that at high momentum the Landau gauge gluon propagator will tend towards the Landau gauge photon propagator~\cite{ryder1996quantum}
%
\begin{equation}
D_\gamma(p^2) = \frac{1}{p^2}\, .
\end{equation}
%
However, lattice discretisation errors cause a deviation from this idealised behaviour that we would like to systematically account for. To do this, we follow the work of Refs.~\cite{Weisz:1982zw, Weisz:1983bn,Luscher:1985zq,Symanzik:1983dc,Symanzik:1983gh}. As we are considering the propagator at high momenta, it is sufficient to consider the behaviour of the photon propagator on the lattice. We therefore consider for this section only an Abelian theory. The commutator in the field strength tensor then vanishes, simplifying to
%
\begin{equation}
F_{\mu\nu} = \partial_\mu \,A_\nu - \partial_\nu\,A_\mu
\end{equation}
%
We consider this Abelian field to be on a lattice generated using the Wilson action (see Eq.~\eqref{eq:WilsonAction}). From Eq.~\eqref{eq:FieldStrengthPlaquette} we know that the Wilson action can be written as $\mathcal{S}_\text{W} = a^4\frac{1}{2}\sum_x F_{\mu\nu}F^{\mu\nu} + \mathcal{O}(a^4)$. As we are interested in the momentum-space propagator, we write the field strength tensor at the plaquette midpoint $\tilde{x}$ as
%
\begin{align}
F_{\mu\nu}(\tilde{x}) &= \frac{A_\nu\left(\tilde{x}+a\frac{\hat{\mu}}{2}\right) - A_\nu\left(\tilde{x}-a\frac{\hat{\mu}}{2} \right)}{a} - \frac{A_\mu\left(\tilde{x}+a\frac{\hat{\nu}}{2}\right) - A_\mu\left(\tilde{x}-a\frac{\hat{\nu}}{2} \right)}{a}\nonumber\\
&= \frac{1}{a} \sum_p e^{ip\cdot\tilde{x}} \left(\tilde{A}_\nu(p)\, e^{-iap\frac{\hat{\mu}}{2}} - \tilde{A}_\nu(p)\, e^{iap\frac{\hat{\mu}}{2}} - \tilde{A}_\mu(p)\, e^{-iap\frac{\hat{\nu}}{2}} + \tilde{A}_\mu(p)\, e^{iap\frac{\hat{\nu}}{2}}\right)\nonumber\\
&= -\frac{1}{a} \sum_p e^{ip\cdot\tilde{x}}\left(2i\sin\left(\frac{a p_\mu}{2}\right)\tilde{A}_\nu(p) - 2i\sin\left(\frac{a p_\nu}{2}\right)\tilde{A}_\mu(p)\right)\nonumber\\
&= - \sum_p e^{ip\cdot\tilde{x}}\tilde{f}_{\mu\nu}(p)\, ,
\end{align}
%
where
%
\begin{equation}
\tilde{f}_{\mu\nu}(p) = i\left(\hat{k}_\mu \tilde{A}_\nu(p) - \hat{k}_\nu \tilde{A}_\mu(p)\right)\, ,~\hat{k}_\mu = \frac{2}{a}\sin\left(\frac{ap_\mu}{2}\right)\, .
\end{equation}
%
The Wilson action can therefore be written as
%
\begin{align}
\mathcal{S}_\text{W} &= a^4\frac{1}{2}\sum_{\tilde{x}}\sum_{p,\,p^\prime}e^{i\tilde{x}(p+p^\prime)}\tilde{f}_{\mu\nu}(p) \, \tilde{f}^{\mu\nu}(p^\prime)\nonumber\\
&=a^4\frac{1}{2}\sum_{p,\,p^\prime} \delta(p+p^\prime)\tilde{f}_{\mu\nu}(p) \, \tilde{f}^{\mu\nu}(p^\prime) \nonumber\\
&= a^4\frac{1}{2}\sum_{p}\tilde{f}_{\mu\nu}(p) \, \tilde{f}^{\mu\nu}(-p) + \mathcal{O}(a^4)\, . \label{eq:WilsonMomentum}
\end{align}
%
We are now in a position to consider the propagator. Equivalent to the two-point correlator definition, the propagator is also the Green's function of the equations of motion, $M_{\mu\nu}$, satisfying
%
\begin{equation}
M_{\mu\nu}D^{\nu\lambda}(p) = \delta_\mu^\lambda\, .
\end{equation}
%
In the continuum, we can write the Abelian Lagrangian density in terms of momentum space variables as
%
\begin{align}
\mathcal{L} &= \frac{1}{2}\tilde{F}_{\mu\nu}\,\tilde{F}^{\mu\nu}\nonumber\\
&= \frac{1}{2}(p_\mu\,\tilde{A}_\nu - p_\nu\,\tilde{A}_\mu)\,(p^\mu\,\tilde{A}^\nu - p^\nu\,\tilde{A}^\mu)\nonumber\\
&= (p^2\delta_{\mu\nu} - p_\mu\,p_\nu)\tilde{A}^\mu\,\tilde{A}^\nu\, ,
\end{align}
%
and hence
%
\begin{equation}
M_{\mu\nu} = (p^2\delta_{\mu\nu} - p_\mu\,p_\nu)\, .
\label{eq:ContEquationsOfMotion}
\end{equation}
%
However, it is understood that in the continuum the equations of motion are not invertible unless an additional gauge fixing term is added, with a gauge fixing parameter $\alpha$. Hence, the equations of motion for the photon field in momentum space are given by Eq.~\eqref{eq:ContEquationsOfMotion} with an additional gauge fixing term~\cite{ryder1996quantum}
%
\begin{equation}
M_{\mu\nu} = p^2\delta_{\mu\nu} - \left(1-\frac{1}{\alpha}\right)p_\mu p_\nu\, .
\end{equation}
%
By inspection, we see that Eq.~\eqref{eq:WilsonMomentum} for the lattice Wilson action will have the same equations of motion, with the substitution $p_\mu\rightarrow \hat{k}_\mu$. In turn, this gives the propagator
%
\begin{equation}
D_{\mu\nu}(p) = \frac{1}{\hat{k}^2}\left[\delta_{\mu\nu} + (\alpha-1)\frac{\hat{k}_\mu \hat{k}_\nu}{\hat{k}^2}\right]\, .
\end{equation}
Landau gauge corresponds to setting $\alpha=0$, so we find that
%
\begin{equation}
D_{\mu\mu}(p) = \frac{3}{\hat{k}^2}\, ,
\end{equation}
%
and therefore by comparison with Eq.~\eqref{eq:scalarProp} we see that
%
\begin{equation}
D(p^2) = \frac{1}{\hat{k}^2}\, .
\end{equation}
%
This suggests that for the Wilson action we should make the substitution $p_\mu\rightarrow \hat{k}_\mu = \frac{2}{a}\sin\left(a p_\mu/2\right)$ so that at tree-level we observe the expected behaviour of the gluon propagator.\\

A similar analysis can be performed for the L\"uscher-Weisz action used in this work, taking into account the contributions from the rectangle terms. The L\"uscher-Weisz action written in the same form as Eq.~\eqref{eq:WilsonMomentum} is~\cite{Weisz:1982zw}
%
\begin{equation}
\mathcal{L}_\text{LW} = a^4\frac{1}{2}\sum_p \left(1+\frac{1}{12}a^2\hat{k}^2\right) \tilde{f}_{\mu\nu}(p) \, \tilde{f}^{\mu\nu}(-p) + \mathcal{O}(a^6)\, .
\end{equation}
%
The equations of motion then become
%
\begin{equation}
M_{\mu\nu} = \left(\hat{k}^2 + \frac{1}{12}a^2\hat{k}^4\right)\delta_{\mu\nu} - \left(1-\frac{1}{\alpha}\right)\left(\sqrt{\hat{k}_\mu^2 + \frac{1}{12}a^2\hat{k}_\mu^4}\right)\left(\sqrt{\hat{k}_\nu^2 + \frac{1}{12}a^2\hat{k}_\nu^4}\right)\, .
\end{equation}
Therefore the propagator is
%
\begin{equation}
D_{\mu\nu}(p) = \frac{1}{q^2}\left[\delta_{\mu\nu} + (\alpha-1)\frac{q_\mu q_\nu}{q^2}\right]\, ,
\end{equation}
with
\begin{equation}
q_\mu = \sqrt{\hat{k}_\mu^2 + \frac{1}{12}a^2\hat{k}_\mu^4} = \frac{2}{a}\sqrt { \sin ^ { 2 } \left( \frac { p _ { \mu } a } { 2 } \right) + \frac { 1 } { 3 } \sin ^ { 4 } \left( \frac { p _ { \mu } a } { 2 } \right) }\, .
\end{equation}
%
In Landau gauge, this tells us that the tree-level form for the scalar propagator is
%
\begin{equation}
D(p^2) = \frac{a^2}{4 \sin ^ { 2 } \left( \frac { p _ { \mu } a } { 2 } \right) + \frac { 1 } { 3 } \sin ^ { 4 } \left( \frac { p _ { \mu } a } { 2 } \right)} = \frac{1}{q^2}\, .
\label{eq:LWCorrection}
\end{equation}\\
Given Eq.~\eqref{eq:LWCorrection}, in this work we make the variable substitution $p_\mu\rightarrow q_\mu$ to ensure that at high momentum the gluon propagator tends towards tree level as required.

\section{Renormalisation}\label{sec:Renormalisation}
Before plotting the propagator, it is essential to discuss the issue of renormalisation. On the lattice, we calculate the bare dimensionless propagator $D_B(q^2,\Lambda)$, as the lattice introduces an explicit regularisation parameter in the form of the momentum cutoff, $\Lambda = \pi/a$. This is related to the renormalised propagator $D_R(q^2,\mu)$ through the relation
%
\begin{equation}
D_B(q^2,\Lambda) = Z_3(\mu, \Lambda)\, D_R(q^2,\mu)\, ,
\label{eq:LatticeRenormalisation}
\end{equation}
%
where $\mu$ is the renormalisation scale. To obtain the renormalisation constant $Z_3(\mu, \Lambda)$, and therefore the renormalised propagator, it is necessary to enforce a renormalisation scheme. Here we employ the momentum space subtraction (MOM) scheme~\cite{Bowman:2004jm,Leinweber:1998uu,Bonnet:2001uh}, which requires that for some sufficiently large $\mu$
%
\begin{equation}
D_R(q^2,\mu)\big|_{q^2=\mu^2}=\frac{1}{\mu^2}\, .
\end{equation}
%
This sets the value of the renormalisation constant to be
%
\begin{equation}
Z_3(\mu,\Lambda) = \, \mu^2 \, D_B(\mu^2,\Lambda)\, ,
\end{equation}
such that
%
\begin{equation}
D_R(q^2,\mu) = \frac{D_B(q^2,\Lambda)}{\mu^2 \, D_B(\mu^2,\Lambda)}\, .
\end{equation}
%
This renormalised propagator is what we plot in e.g. Fig.~\ref{fig:UntouchedPropagator}, and will be denoted as simply $D(q^2)$ hereafter. The value of $\mu$ is arbitrary, however to make contact with perturbation theory it is necessary that it is sufficiently large such that it is outside the infrared region where the gluon propagator exhibits substantial deviation from perturbative behaviour. Furthermore, $\mu$ must be away from the momentum cutoff, as the renormalisation constant is only independent of the cutoff in the limit that the cutoff tends towards infinity~\cite{Bonnet:2001uh,Boucaud:2006pc}.  Once the renormalisation scheme has been imposed, it is then possible to connect the lattice results with those obtained from perturbation theory by connecting the renormalisation constant from the MOM scheme with those obtained from the renormalisation schemes used in perturbation theory.\\

The crux of this argument is that it is the renormalised propagator, not the bare propagator, that carries the physical meaning.  More generally, it is the shape of the propagator that carries meaning, and we are free to impose a scaling constant without changing the physical significance of the result. To facilitate comparisons between vortex-modified ensembles, we make use of the original $Z_3(\mu, \Lambda)$ obtained for the untouched propagator unless specified otherwise. Maintaining this consistency is sufficient to comment on the qualitative shape of the propagator, which is the most significant point of interest in this research.\\

\section{Lattice Parameters and Data Cuts} \label{sec:LatticeParameters}
We calculate the gluon propagator on 100 configurations of a $20^3\times 40$ $SU(3)$ lattice with spacing $a=0.125\,\si{fm}$, as used in Refs.~\cite{Trewartha:2015nna,OMalley:2011aa}. The momentum variables chosen for both the Wilson and L\"uscher-Weisz action have been numerically verified to provide better tree level agreement in Refs.~\cite{Marenzoni:1994ap, Bonnet:2001uh}, and we present a comparison of the L\"uscher-Weisz and uncorrected variables in Fig.~\ref{fig:MomentumComparison}. To visualise this improvement, we plot $k^2\,D(k^2)$ (where $k_\mu=p_\mu,\,q_\mu$ is the momentum variable for the given case under consideration) such that the tree-level propagator appears as $k^2\,D(k^2)=1$, shown as the black dashed line. This choice of plotting $k^2\,D(k^2)$ against $ka$ has the benefit of aiding both the visualisation of the tree-level behaviour, and the onset of the non-perturbative infrared properties of the propagator. Via the method described in the previous section, we have renormalised the L\"uscher-Weisz corrected propagator such that $q^2\,D(q^2) = 1$ at $qa=6.0$, and applied this same renormalisation constant to the uncorrected propagator. While this choice of renormalisation point is near the cutoff, it provides a renormalised propagator that approaches tree-level from above. In subsequent sections we will select renormalisation points further away from the cutoff.\\

We can clearly see that at high momenta the corrected gluon propagator tends towards the expected tree-level behaviour, whereas the uncorrected propagator fans out considerably. This fanning is the result of asymmetry between the spatial and temporal components of the propagator, which is accounted for in the tree-level correction~\cite{Marenzoni:1994ap}. The results presented in Fig.~\ref{fig:MomentumComparison} clearly motivate the need for tree-level correction when calculating the gluon propagator on the lattice.\\
%
\begin{figure}[H]
\centering
\includegraphics[width=\linewidth]{./ScalarGluComp_q2_MomentumComparison.pdf}
\caption[The renormalised scalar gluon propagator is plotted with no tree-level momentum correction and the L\"uscher-Weisz correction.]{\label{fig:MomentumComparison} The scalar gluon propagator is plotted with no tree-level momentum correction (blue crosses) and the L\"uscher-Weisz correction (red dots) presented in Eq.~\eqref{eq:LWCorrection}. It is clear that the corrected momentum has improved tree-level behaviour, free from the fanning effect present in the uncorrected case.}
\end{figure}
%

When considering the gluon propagator we shall maintain the plotting convention introduced in Fig.~\ref{fig:MomentumComparison} of considering $q^2D(q^2)$ against $qa$ for the remainder of this work. To improve the momentum-corrected propagator presented in  Fig.~\ref{fig:MomentumComparison} we follow the procedure of Ref.~\cite{Bonnet:2001uh,Leinweber:1998im} and perform a momentum half-cut. The momentum half-cut corresponds to only considering lattice momenta in the range
%
\begin{equation}
p_\mu = \frac{2\pi n_\mu}{a N_\mu},~n_\mu\in \left(-\frac{N_\mu}{4},\,\frac{N_\mu}{4}\right]\, .
\end{equation}
%
This cut limits the positive range of the kinematically corrected $q_\mu$ to
%
\begin{equation}
q_\mu \in \left[0,\, \frac{2\sqrt{21}}{3a}\right]\approx\left[0,\frac{3.06}{a} \right]\, .
\end{equation}
%
Furthermore, a cylinder cut of radius $pa=2$ lattice units is performed, such that we only consider points within two lattice units of the diagonal. This cut is implemented by considering points satisfying
%
\begin{equation}
|pa|^2\, \sin(\theta_c) \leq 2\, ,
\end{equation}
%
where
%
\begin{equation}
\theta_c = \cos^{-1}\left(\frac{pa \cdot \hat{n}}{|pa|}\right)\, ,
\end{equation}
%
and $\hat{n} = \frac{1}{2}(1,\,1,\,1,\,1)$ is the unit vector along the diagonal. This is performed so that all directions are equally sampled, whilst omitting points where one direction dominates the signal. This reduces the impact of lattice cutoff artefacts. Finally, we can take advantage of the rotational symmetry of the scalar propagator to perform $Z(3)$ averaging over the Cartesian coordinates. This means that we average over all points with the same Cartesian radius; for example, we would average across the points $(n_x,n_y,n_z)=(2,1,1),\,(1,2,1)$ and $(1,1,2)$. These choices of cuts assist in producing a cleaner signal that accurately represents the behaviour of the continuum propagator. With the momentum half-cut, we now renormalise at $qa=3.0$. This choice of renormalisation point is both sufficiently large and away from the lattice momentum cutoff, as well as falling within the momentum half-cut range of $qa\in [0,\,3.06]$. This choice of renormalisation point will be used for the remainder of this work.\\

With these cuts implemented, the gluon propagator on the original untouched configurations appears as Fig.~\ref{fig:UntouchedPropagator}. We observe the expected tree-level behaviour at high momenta, with an infrared enhancement indicative of amplified low-momentum propagation. It should be noted that the difference in peak height observed between Fig.~\ref{fig:MomentumComparison} and Fig.~\ref{fig:UntouchedPropagator} is due to the different renormalisation constant. Due to the cuts we have made, we observe a much cleaner signal, particularly in the region $qa\geq 1.5$, in agreement with the results of Ref.~\cite{Bonnet:2001uh}. For the remainder of this work we will employ these data cuts and this choice of momentum variables when plotting the gluon propagator to ensure an accurate and clear signal.
%
\begin{figure}
\centering
\includegraphics[width=\linewidth]{./ScalarGluComp_q2_NoCoolU.pdf}
\caption[The untouched gluon propagator with all data cuts and correct momentum variables utilised.]{\label{fig:UntouchedPropagator} The untouched gluon propagator with all data cuts and correct momentum variables utilised. We observe a substantially cleaner signal when compared to the untouched propagator shown in Fig.~\ref{fig:MomentumComparison}.}
\end{figure}