\paragraph{}
In this example, a plane strain bracket with a downward uniform distributed load on the top is considered (see Fig.~\ref{adp_fig:ex_bracket_geo_bc}).
The material properties are: Young’s modulus $E = \SI{2e5}{\newton \per \square \meter}$ and Poisson’s ratio $\nu = 0.3$.
    \begin{figure}
        \centering
        \begin{subfigure}[b]{1\linewidth}
            \centering
            \scalebox{1.2}{
                \includegraphics{quadtree/ex_images/ex_bracket_geo.eps}
            }
        \end{subfigure}
    \end{figure}
    \begin{figure}\ContinuedFloat
        \begin{subfigure}[b]{1\linewidth}
            \centering
            \scalebox{1.3}{
                \includegraphics{quadtree/ex_images/ex_bracket_load.eps}
            }
        \end{subfigure}
        \caption{ Plane strain bracket: geometry and boundary conditions}
        \label{adp_fig:ex_bracket_geo_bc}
    \end{figure}

\paragraph{}
A total strain energy of $\SI{282.927}{\joule}$ is determined by ANSYS with the mesh shown in Fig.~\ref{qdt_fig:ex_bracket_ansys_mesh}

% /* cSpell:disable */
% -1 - 282.927
% 382*2  - 274.5638 (32-4/8-3)
% 950*2  - 280.1255 (64-4/8-3)
% 2110*2 - 281.0799 (128-4/15-4)
% 4414*2 - 282.0363 (256-4/15-4)

% mlp
% 530 - 279.8076
% 1554 - 282.0745
% 4208 - 282.5681
% err = abs([279.8076 ,282.0745,282.5681]-282.927)/282.927; dof = [530,1554,4208]; polyfit(log(dof),log(err),1)

% uniform
% 530 - 279.8076
% 1810 - 282.3186
% 6464 - 282.674
% err_un = abs([279.8076 ,282.3186 ,282.674]-282.927)/282.927; dof_un = [530,1810,6464]; polyfit(log(dof_un),log(err_un),1)

% ansys-2nd
% 406*2 -  282.412
% 1144*2 - 282.895
% 4270*2 - 282.923

% 16462*2 -  282.926
% err_a2 = abs([282.412,282.895,282.923]-282.927)/282.927; dof_a2 = [406,1144,4270]*2; polyfit(log(dof_a2),log(err_a2),1)

% ansys-1st
% 185*2 - 275.635
% 650*2 -  280.006
% 1227*2 -  281.502
% 2405*2 - 282.136
% err_a1 = abs([275.635,280.006  ,281.502 ,282.136]-282.927)/282.927; dof_a1 = [185,650,1227,2405]*2; polyfit(log(dof_a1),log(err_a1),1)

% disp > 0.01
% 530 - 279.8076
% 1348 - 281.4530
% 2768 - 282
% 5438 - 282.2
% err_d = abs([279.8076, 281.4530  ,282 ,282.2]-282.927)/282.927; dof_d = [530,1348,2768,5438]; polyfit(log(dof_d),log(err_d),1)

% str > 0.1
% 530 - 279.8076
% 1514 - 281.8970
% 4336 - 282.1295

% err_s = abs([279.8076, 281.8970  ,282.1295 ]-282.927)/282.927; dof_s = [530,1514,4336]; polyfit(log(dof_s),log(err_s),1)

% /* cSpell:enable */
\paragraph{}
% result analysis
The rate of convergence in terms of the total strain energy is shown in Fig.~\ref{adap_fig:ex_bracket_conv} and corresponding mesh development are presented in Fig.~\ref{adap_fig:ex_bracket_mesh_sbfem} (SBFEM) and Fig.~\ref{adap_fig:ex_bracket_mesh_ansys} ( Ansys 4-node quadrilateral element ).
Higher convergence rate compared to the result determined in ANSYS is observed.

\begin{figure}[h!]
    \centering
    \scalebox{0.5}{
        \includegraphics{adaptivity/ex_images/ex_adap_bracket_conv.eps}
    }
    \caption{Plane strain bracket: Convergence study}
    \label{adap_fig:ex_bracket_conv}
\end{figure}

\begin{figure}[h!]
    \centering
    \begin{subfigure}[b]{0.4\linewidth}
        \centering
        \scalebox{0.6}{
            \includegraphics{adaptivity/ex_images/ex_adap_bracket_530.png}
        }
        \caption{Initial mesh, 530 DOFs}
    \end{subfigure}
    \begin{subfigure}[b]{0.4\linewidth}
        \centering
        \scalebox{0.6}{
            \includegraphics{adaptivity/ex_images/ex_adap_bracket_1554.png}
        }
        \caption{1st refinement, 1554 DOFs}
    \end{subfigure} \\
    \begin{subfigure}[b]{1\linewidth}
        \centering
        \scalebox{0.8}{
            \includegraphics{adaptivity/ex_images/ex_adap_bracket_4208.png}
        }
        \caption{2nd refinement, 4208 DOFs}
    \end{subfigure}
    \caption{Plane strain bracket: Mesh development (SBFEM)}
    \label{adap_fig:ex_bracket_mesh_sbfem}
\end{figure}

\begin{figure}[h!]
    \centering
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \scalebox{0.4}{
            \includegraphics{adaptivity/ex_images/ex_adp_bracket_ansys_185.png}
        }
        \caption{Initial mesh, 138 DOFs}
    \end{subfigure}
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \scalebox{0.4}{
            \includegraphics{adaptivity/ex_images/ex_adp_bracket_ansys_650.png}
        }
        \caption{1st refinement, 482 DOFs}
    \end{subfigure}
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \scalebox{0.4}{
            \includegraphics{adaptivity/ex_images/ex_adp_bracket_ansys_1227.png}
        }
        \caption{2nd refinement, 1794 DOFs}
    \end{subfigure}
    \begin{subfigure}[b]{0.48\linewidth}
        \centering
        \scalebox{0.4}{
            \includegraphics{adaptivity/ex_images/ex_adp_bracket_ansys_2405.png}
        }
        \caption{3rd refinement, 6914 DOFs}
    \end{subfigure}
    \caption{Plane strain bracket: Mesh development (Ansys)}
    \label{adap_fig:ex_bracket_mesh_ansys}
\end{figure}


\section{Graph-based Machine Learning for Program Analysis}
\label{sec:graph-based-machine-learning}

We formulate our system in a Message Passing Neural Network (MPNN)
framework~\citep{Gilmer2017}. Our design mimics the \emph{transfer functions}
and \emph{meet operators} of classical iterative data flow
analysis~\citep{Kam1977,Cooper2003}, replacing the rule-based implementations
with learnable analogues (message and update functions). This single unified
model can be specialized through training to solve a diverse set of problems
without human intervention or algorithm design.

The \programl model is an adaptation of GGNN~\citep{Li2015a} that takes as input
an attributed directed multigraph as presented in
Section~\ref{sec:graph-representation}. It consists of three logical phases:
input encoding, message propagation and update, and result readout.

\paragraph{(I) Input Encoding} Starting from the augmented graph representation
$G = (V, E)$, we capture the semantics of the program graph vertices by mapping
every instruction, constant, and variable vertex $v \in V$ to a vector
representation $h_v^0 \in \mathbb{R}^{d}$ by lookup in a fixed-size learnable
embedding table. The mapping from vertex to embedding vector $f: v \mapsto
h_v^0$ must be defined for each IR.

For LLVM-IR, we construct an embedding key from each vertex using the name of
the instruction, e.g., \texttt{store}, and the data type for variables and
constants, e.g., \texttt{i32*} (a pointer to a 32-bit integer). In this manner,
we derive the set of unique embedding keys using the graph vertices of a
training set of LLVM-IRs described in Section~\ref{subsec:dataset}. This defines
the embedding table used for training and deployment. An \emph{unknown element}
embedding is used during deployment to map embedding keys  which were not
observed in the training data. Since composite types make the size of the
vocabulary unbounded in principle, our data-driven approach trades a certain
amount of semantic resolution against good coverage of the vocabulary by the
available datasets (cf. Table 1). The embedding vectors are trained jointly with
the rest of the model.

\paragraph{(II) Message Propagation} Each iteration step is divided into a
message propagation  followed by vertex state update. Receiving messages
$M(h_w^{t-1}, e_{wv})$ are a function of neighboring states and the respective
edge. Messages are mean-aggregated over the neighborhood after transformation
with a custom position-augmented transfer function that scales $h_w$ elementwise
with a position-gating vector $p(e_{wv})$:%
\begin{equation*}
	M(h^{t-1}_w,e_{wv}) = W_{\mathrm{type}(e_{wv})} \Big(h_w^{t-1} \odot p(e_{wv})\Big) + b_{\mathrm{type}(e_{wv})}
\end{equation*}
The position-gating $p(e_{wv}) = 2 \sigma (W_p \operatorname{emb}(e_{wv}) +
b_p)$ is implemented as a sigmoid-activated linear layer mapping from a constant
sinusoidal position embedding ~\citep{Vaswani2017,Gehring2017}. It enables the
network to distinguish non-commutative operations such as division, and the
branch type in diverging control-flow. In order to allow for reverse-propagation
of information, which is necessary for backward compiler analyses, we add
backward edges for each edge in the graph as separate edge-types. In all our
experiments, we employ Gated Recurrent Units (GRU)~\citep{Cho2014} as our update
function.

Step (II) is iterated $T$ times to extract vertex representations that are
contextualized with respect to the graph structure.

\paragraph{(III) Result Readout} Data flow analyses compute value sets composed
of instructions or variables. We support per-instruction and per-variable
classification tasks using a \emph{readout head} on top of the iterated feature
extraction, mapping, for each vertex, the extracted vertex features $h_v^T$ to
probabilities $R_v(h_v^T, h_v^0)$:
\begin{equation*}
	R_{v}(h_v^T, h_v^0) = \sigma\left(f(h_v^T, h_v^0)\right) \cdot g(h_v^T) \\
\end{equation*}
where $f(\cdot)$ and $g(\cdot)$ are linear layers and $\sigma(\cdot)$ is the
sigmoid activation function. Allowing the readout head to access the initial
node state $h_v^0$ in its gating function $\sigma(f(\cdot))$ acts as a skip
connection from the input embedding to the readout.


%%========================================
%% Section 4.3: The Direct Scattering Map
%%========================================

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

\begin{document}
\section{The Direct Scattering Map}\label{sec4:DM}

We now have the tools we need to prove that the direct scattering map is well-defined
as a map from $B_X(0, c_0)$ to $L^\infty(\mathbb R)$ and Lipschitz continuous
as a map from $B_X(0, c_0)$ to $L^\infty\big(\mathbb R \sm (-k,k)\mathbb)$
(for any $k > 0$). As a warm-up exercise, we first verify in
Proposition \ref{prop4:SD} the validity of the scattering equations first
presented (without proof) in \cite{Kodama1982}, which are key to the
construction of the inverse scattering map for the ILW. Following Proposition
\ref{prop4:SD}, we turn our attention to proving that the direct scattering map
is well-defined (Lemma \ref{lma4:rBnd} and Theorem \ref{thm4:Dwelldefined}), and
wrap up this dissertation with results on the Lipschitz continuity of the direct
scattering map (Theorem \ref{prop4:exist} and Corollary \ref{cor4:Lip}).

\begin{prop}[Scattering Equations]\label{prop4:SD}
	Suppose that $u$ satisfies the hypotheses of Proposition \ref{prop4:exist}.
	Let
	\begin{subequations}
		\label{eq4:SD}
		\begin{align}
			% \label{eq4:SDa}
			a(\lambda)
				&:= 1 + i \alpha(\lambda) \,
					\int_{\mathbb R} u(x) \, M_1^+(x; \lambda, u) \, \mathrm{d}x \\
			% \label{eq4:SDb}
			b(\lambda)
				&= i \beta(\lambda) \,
					\int_{\mathbb R} e^{-ix\lambda} \, u(x) \, M_1^+(x; \lambda, u) \, \mathrm{d}x
		\end{align}
	\end{subequations}
	\vspace{-\baselineskip}
	\begin{subequations}
		\label{eq4:SDbrev}
		\begin{align}
			% \label{eq4:SDa}
			\breve{a}(\lambda)
				&:= 1 + \alpha(\lambda)
					\int_{\mathbb R} u(x) \, N_1(x; \lambda, u) \, \mathrm{d}x \\
			% \label{eq4:SDb}
			\breve{b}(\lambda)
				&= i \beta(\lambda)
					\int_{\mathbb R} e^{-ix\lambda} \, u(x) \, N_1(x; \lambda, u) \, \mathrm{d}x
		\end{align}
	\end{subequations}
	For $\lambda \in \mathbb R \sm \{0\}$,
	\begin{align}
		M_1 (x; \lambda)
			&= a(\lambda) \, N_1(x; \lambda) + b(\lambda) \, N_e(x; \lambda)
			\label{eq3:M1N} \\
		N_1 (x; \lambda)
			&= \breve{a}(\lambda) \, M_1(x; \lambda)
				+ \breve{b}(\lambda) \, M_e(x; \lambda)
			\label{eq3:N1M}
	\end{align}
\end{prop}
\begin{proof}
	An immediate consequence of formulas \eqref{eq1:GFrep} is the jump
	relation\footnote{This jump relation can also be proven by taking a contour
	around the real axis. For details, please see
	\hyperref[app:JumpRelation]{Appendix 3: Jump Relation}.}
	\begin{align}\label{eq4:GFjump}
		G_L^+ - G_R^+ = i \alpha(\lambda) + i \beta(\lambda) e^{i \lambda x}.
	\end{align}
	Under the jump relation \eqref{eq4:GFjump}, the integral equation
	\eqref{eq4:JostIEleft} for $M_1^+$ becomes
	\begin{align}\label{eq4:altM1}
		M_1^+(x; \lambda)
			&= 1 + \int_{\mathbb R} G_L^+(x-x'; \lambda) u(x') M_1^+(x')\,dx' \\
			&= 1
				+ \int_{\mathbb R}
					\big(
						i \alpha(\lambda)
						+ i \beta(\lambda) \, e^{i \lambda (x-x')}
					\big)
					u(x') M_1^+(x')
				\, \mathrm{d}x \nonumber \\
			&= \left(
					1
					+ i \alpha(\lambda)
						\int_{\mathbb R} u(x')M_1^+(x'; \lambda)\,dx
				\right)
				\nonumber \\
			&\qquad+ i \beta(\lambda) \, e^{i\lambda x}
					\int_{\mathbb R} e^{-i\lambda x'} (x')M_1^+(x'; \lambda)\,dx
				\nonumber \\
			&\qquad+ \int_{\mathbb R} G_R^+(x-x'; \lambda) u(x') M_1^+(x')\,dx'
				\nonumber \\
			&= a(\lambda) + b(\lambda) \,e^{i\lambda x}
				+ G_R^+(\dotarg; \lambda) * \big[uM_1^+(\dotarg; \lambda)\big].
				\nonumber
	\end{align}
	Recalling from Proposition \ref{prop2:Tasymp} that
	$\lim_{x\to\infty} G_R^+*uM_1^+(x) = 0$, we see that $M_1^+$ satisfies the
	asymptotic condition
	\begin{align} \label{eq4:asympM}
		\lim_{x\to+\infty}
			\big|M_1^+(x; \lambda) - a(\lambda) - b(\lambda)e^{i\lambda x}\big|
			= 0.
	\end{align}

	The simple computation
	\begin{align*}
		G_R^+ * \big[ u(a\,N_1 + b\, N_e)\big]
			&= a \, G_R^+ *(u\, N_1) + b \, G_R^+ *(u\, N_e) \\
			&= a(N_1 - 1) + b(N_e - e^{i\lambda x}) \\
			&= a\, N_1 + b \, N_e - a - b\, e^{i\lambda x}
	\end{align*}
	shows that $a\, N_1 + b \, N_e$ is a solution to \eqref{eq4:altM1}. Further,
	since $a\, N_1 + b \, N_e$ also satisfies \eqref{eq4:asympM} as
	$\lim_{x\to+\infty} |N_1-1| = \lim_{x\to+\infty} |N_e - e^{ix\lambda}| = 0$,
	equation \eqref{eq3:M1N} follows from the uniqueness of Jost solutions.
	Equation \eqref{eq3:N1M} is verified analogously.
\end{proof}

While perhaps not immediately apparent, the significance of the following lemma,
Lemma \ref{lma4:rBnd}, is that it allows us to conclude
that the reflection coefficient $r(\lambda) = b(\lambda)/a(\lambda)$ is bounded
in $\lambda.$ That is,
$r \in L_\lambda^\infty(\mathbb R)$, which is the final piece we need to prove
that the ILW direct scattering map \label{sym:DM}
$\mathscr D: B_X(0, c_0) \ni u \mapsto r \in L_\lambda^\infty(\mathbb R)$ is
well-defined.\label{sym4:dsm}

%%==============
%% Allen's Lemma
%%==============
\begin{lma}\label{lma4:rBnd}
	For $\lambda \ne 0$, the functions $a$ and $b$ defined in Proposition \ref{prop4:SD} satisfy
	the equation
	\begin{align}\label{eq4:rbnd}
		\big| a(\lambda) \big|^2
			= 1 + \frac{2\zeta(-\lambda) - 1}{1 - 2 \zeta(\lambda)}
				\big| b(\lambda) \big|^2.
	\end{align}
\end{lma}
The following proof is taken from the unpublished notes of Professor Allen Wu.
% {\color{red} I thank Prof. Allen Wu from the University of Oklahoma for the following clever
% proof of Lemma \ref{lma4:rBnd}.}
\begin{proof}
	In this proof, we use the identity
	\begin{align*}
		\inn{G_L^+(\dotarg; \lambda)*f, \, g}
			&= \inn{f, \, G_L^+(\dotarg; \lambda)*g} \\
			&\quad+ i\alpha(\lambda) \inn{f,1}\inn{1,g}
				+ i\beta(\lambda) \inn{f, \, e^{i(\dotarg)\lambda}}
					\inn{e^{i(\dotarg)\lambda}, \, g}
	\end{align*}
	which follows from Proposition \ref{prop1:DiaConj} identity (ii) and
	the jump relation \eqref{eq4:GFjump} as
	\begin{align*}
		\ol{G_L^+(x; \lambda)}
			= G_R^+(-x; \lambda)
			= G_L^+(-x; \lambda) - i \alpha(\lambda) - i \beta(\lambda) \, e^{-i\lambda x}.
	\end{align*}

	Using $M_1^+ = 1 + G_L^+ (\dotarg; \lambda) * u M_1$ we compute
	\begin{align*}
		\inn{M_1^+, \, uM_1^+}
			&= \inn{1 + G_L^+ (\dotarg; \lambda) * u M_1^+, \, uM_1^+} \\
			&= \inn{1, \, uM_1^+} + \inn{uM_1^+, \, G_L^+ (\dotarg; \lambda) * u M_1^+} \\
			&\quad+ i \alpha(\lambda) |\inn{uM_1^+,\,1}|^2
				+ i \beta(\lambda) |\inn{uM_1^+,\,e^{i(\dotarg)\lambda}}|^2 \\
			&=  \inn{1, \, uM_1^+} + \inn{uM_1^+, \, M_1^+ - 1} \\
			&\quad+ i \alpha(\lambda) |\inn{uM_1^+,\,1}|^2
				+ i \beta(\lambda) |\inn{uM_1^+,\,e^{i(\dotarg)\lambda}}|^2
	\end{align*}
	Since $u$ is real, we have $\inn{M_1^+, \, u M_1^+} = \inn{uM_1^+, \, M_1^+}$ and
	\begin{align*}
		0 =
			\ol{\inn{uM_1^+, \, 1}} - \inn{uM_1^+, \, 1}
			+ i \alpha(\lambda) \left|\inn{uM_1^+, \, 1} \right|^2
			+ i \beta(\lambda) \left|\inn{uM_1^+, \, e^{i(\dotarg)\lambda}} \right|^2.
	\end{align*}
	Identity \ref{eq4:rbnd} then follows, as
	$\inn{uM_1^+, \,1} = \frac{1}{i} \frac{a-1}{\alpha(\lambda)}$,
	$\inn{uM_1^+, \, e^{i(\dotarg)\lambda}} = \frac{1}{i} \frac{b}{\beta(\lambda)}$,
	$\alpha(\lambda) = \frac{1}{1-2\zeta(\lambda)}$, and $\beta(\lambda) = \frac{1}{1-2\zeta(-\lambda)}$.
\end{proof}

While Lemma \ref{lma4:rBnd} holds only for $\lambda \ne 0$, we can nonetheless use
Lemma \ref{lma4:rBnd} to show that the $r$ remains bounded near $\lambda = 0$ and is
therefore at least essentially bounded and hence in $L_\lambda^\infty$.

\begin{thm}\label{thm4:Dwelldefined}
	Let $r(\lambda) := b(\lambda) / a(\lambda)$. The direct scattering map
	$\mathscr D$ given by
	\begin{align*}
		\begin{array}{rcl}
			\mathscr D: B_X(0, c_0) &\to& L_\lambda^\infty(\mathbb R)  \\
			            u           &\mapsto& r
		\end{array}
	\end{align*}
	is well-defined.
\end{thm}
\begin{proof}
	Since $M_1^+$ and $N_1^+$ exist and are unique for each $u \in B_X(0, c_0)$, the
	map $u \mapsto r$ is well-defined as a function. Moreover, it is easy to check
	that $\frac{1-2\zeta(\lambda)}{2\zeta(-\lambda)-1}$ is both positive and uniformly
	bounded in $\lambda$ for all real $\lambda \ne 0$, which means that \eqref{eq4:rbnd} and Lemma
	\ref{lma4:rBnd} implies both that $|a(\lambda)| \geq 1$ for all
	$\lambda \ne 0$ and, as a consequence
	\[
		\left| \frac{b(\lambda)}{a(\lambda)} \right|^2
			=
				\frac{1-2\zeta(\lambda)}{2\zeta(-\lambda)-1}
				\left[
					1 - \left|\frac{1}{a(\lambda)}\right|^2
				\right].
	\]
	Since $|a(\lambda)| \to \infty$ as $\lambda \to 0$ and a simple computation shows that
	\[
		\lim_{\lambda \to 0} \frac{1-2\zeta(\lambda)}{2\zeta(-\lambda)-1}
			= 1,
	\]
	$r = b/a$ is at least essentially bounded near $\lambda = 0$. Moreover,
	since
	\begin{align}
		\label{eq4:rNegLamLimit}
		\lim_{\lambda\to-\infty}\frac{1-2\zeta(\lambda)}{2\zeta(-\lambda)-1}
			= 0,
	\end{align}
	we need only prove $r(\lambda)$ stays bounded for large \textit{positive}
	$\lambda$. Indeed, a straight forward computation shows
	\begin{align}\label{eq4:reslimits}
		\lim_{\lambda\to+\infty} |\beta(\lambda)| = 1,
		\qquad \text{and} \qquad
		\lim_{\lambda\to+\infty} \alpha(\lambda) = 0.
	\end{align}
	Recalling that
	\begin{align*}
		a(\lambda)
				&:= 1 + i \alpha(\lambda) \,
					\int_{\mathbb R} u(x) \, M_1^+(x; \lambda, u) \, \mathrm{d}x \\
			b(\lambda)
				&= i \beta(\lambda) \,
					\int_{\mathbb R}
						e^{-ix\lambda} \, u(x) \, M_1^+(x; \lambda, u)
					\, \mathrm{d}x,
	\end{align*}
	we see that
	\[
		\left|\frac{b(\lambda)}{a(\lambda)}\right|
			\lesssim \nm{u\,M_1^+(\dotarg; \lambda)}_{L^1}
			\leq
				\nm{M_1^+(\dotarg; \lambda)}_{\inn{\dotarg}L^\infty}
				\nm{u}_{L^{1,1}}
	\]
	for $\lambda \gg 1$. Given $M_1^+ = \left(1 - T_{L, \lambda, u} \right)^{-1} 1$
	and the operator $T_{L, \lambda, u}$ is bounded uniformly in
	$\lambda$\textemdash{}in fact,
	$\nm{T_{L, \lambda, u}}_{\inn{\dotarg}L^\infty\toitself}
	<\frac{1}{2}$\textemdash{}we conclude by Neumann series that
	$\nm{M_1^+(\dotarg; \lambda)}_{\inn{\dotarg}L^\infty}$ is also uniformly
	bounded in $\lambda$. The result therefore follows.
\end{proof}

While, we do not yet have a proof that the ILW direct
scattering map is Lipschitz as a map from $B_X(0, c_0)$ into
$L_\lambda^\infty$, we are able to prove that it is Lipschitz in
more restrictive regimes.

%%===================
%% Lipschitz Theorems
%%===================
\begin{thm}\label{thm4:DlipR}
	For $c_0>0$ from Proposition \ref{prop4:exist} and for all fixed
	$k > 0$, the ILW direct scattering map
	\[
		\mathscr D: B_X(0, c_0) \ni u \mapsto r \in
			L_\lambda^\infty\big((-\infty,-k]\cup[k,\infty)\big)
	\]
	is Lipschitz continuous with Lipschitz constant depending on $k$.
\end{thm}
\begin{proof}
	Let $u_1, u_2 \in B_X(0, c_0)$ be arbitrary and respectively denote by
	$r_1 = b_1 / a_1$, $r_2 = b_2 / a_2$ the corresponding ILW direct scattering
	map $\mathscr D$ outputs. Since $|a_1(\lambda)| \geq 1$ for all $\lambda \in \mathbb R$
	by Lemma \ref{lma4:rBnd}, we find
	\begin{align}\label{eq4:DlipPnT}
		\left| \frac{b_1}{a_1} - \frac{b_2}{a_2} \right|
			&\leq \left|\frac{b_1}{a_1} - \frac{b_2}{a_1}\right|
				+ \left|\frac{b_2}{a_1} - \frac{b_2}{a_2}\right| \\
			&= \frac{1}{|a_1|} |b_1 - b_2|
				+ \frac{1}{|a_1|}
					\left|\frac{b_2}{a_2}\right|
					\left| a_1 - a_2 \right|.
					\nonumber
	\end{align}
	Proposition \ref{prop4:uMcont} implies the map $u \mapsto uM_1^+$ is Lipschitz as a map
	into $L^1_x$. As such,
	\begin{align}\label{eq4:DlipB}
		\frac{1}{|a_1|} |b_1 - b_2|
			&= \frac{|\beta(\lambda)|}{|a_1(\lambda)|}
				\left|
					\int_{\mathbb R}
						e^{ix\lambda}
						\big(
							u_1(x) \, M_1^+( x; \lambda, u_1)
								- u_2(x) \, M_1^+( x; \lambda, u_2)
						\big)
					\, \mathrm{d}x
				\right| \\
			&\leq
				\frac{|\beta(\lambda)|}{|a_1(\lambda)|}
				\nm{
					u_1\, M_1^+(\dotarg; \lambda, u_1)
					- u_2\, M_1^+(\dotarg; \lambda, u_2)
				}_{L^1}
				\nonumber \\
			&\lesssim
				\frac{|\beta(\lambda)|}{|a_1(\lambda)|}
				\nm{u_1 - u_2}_X,
				\nonumber
	\end{align}
	where the implied constant is uniform in $\lambda$. Similarly,
	\begin{align}\label{eq4:DlipA}
		|a_1 - a_2|
			& \leq
				|\alpha(\lambda)|
				\nm{
					u_1 \, M_1^+(\dotarg; \lambda, u_1)
					- u_2 \, M_1^+(\dotarg; \lambda, u_2)
				}_{L^1}
				\\
			&\lesssim |\alpha(\lambda)| \nm{u_1 - u_2}_X,
				\nonumber
	\end{align}
	where the implied constant is again uniform in $\lambda$.
	Now, the proof of Theorem \ref{thm4:Dwelldefined} implies
	that the term
	\[
		\frac{1}{|a_1|} \left| \frac{b_2}{a_2} \right| |a_1 - a_2|
	\]
	is bounded for $|\lambda| > 0$. Through direct computation,
	it is straightforward to show
	\begin{align*}
		\lim_{\lambda \to -\infty} |\alpha(\lambda)|
			= \lim_{\lambda \to +\infty} |\beta(\lambda)|
			= 1,
			\\
		\lim_{\lambda \to +\infty} |\alpha(\lambda)|
			= \lim_{\lambda \to -\infty} |\beta(\lambda)|
			= 0,
	\end{align*}
	which implies $|\alpha(\lambda)|$ and $|\beta(\lambda)|$ are bounded for
	$|\lambda| \gg 1$. Further, since $\alpha$ and $\beta$ have exactly one
	singularity, namely $\lambda = 0$, we conclude by estimates
	\eqref{eq4:DlipPnT} through \eqref{eq4:DlipA} that
	\begin{align}\label{eq4:DlipFinal}
		\nm{r_1 - r_2}_{L_\lambda^\infty}
			\lesssim_k \|u_1 - u_2\|_X
	\end{align}
	as $|a_1| \geq 1$ for $\lambda \ne 0$, where the implied constant depends
	on $k$ but is otherwise independent of $\lambda$.
\end{proof}

\begin{rmk}\label{rmk4:Lip}
	The difficulty in extending Theorem \ref{thm4:DlipR} to all values of real
	$\lambda$ is due to the possibility of the implied constant in
	\eqref{eq4:DlipFinal} ``blowing-up'' as $k\to 0$\textemdash{}especially when
	either $\int_{\mathbb R} u_1 \, M_1^+(x; \lambda = 0, u_1) \, \mathrm{d}x$ or
	$\int_{\mathbb R} u_2 \, M_1^+(x; \lambda = 0, u_2) \, \mathrm{d}x$ are zero. To see
	why this is so, note that
	\[
		\frac{\alpha(\lambda)}{a(\lambda)}
			= \frac{\alpha(\lambda)}
				{
					1
						+ \alpha(\lambda)
							\int_{\mathbb R}
								u \, M_1^+(x; \lambda)
							\, \mathrm{d}x
				}
			=\frac{1}
				{
					\frac{1}{\alpha(\lambda)}
						+ \int_{\mathbb R} u \, M_1^+(x; \lambda) \, \mathrm{d}x,
				}
	\]
	and
	\[
		\frac{\beta(\lambda)}{a(\lambda)}
			= \frac{\beta(\lambda)}
				{
					1
						+ \alpha(\lambda)
							\int_{\mathbb R}
								u \, M_1^+(x; \lambda)
							\, \mathrm{d}x
				}
			=\frac{1}
				{
					\frac{1}{\beta(\lambda)}
						+ \frac{\alpha(\lambda)}{\beta(\lambda)}
							\int_{\mathbb R}
								u \, M_1^+(x; \lambda)
							\, \mathrm{d}x.
				}
	\]
	which means
	\[
		\frac{\alpha(\lambda)}{a(\lambda)},~\frac{\beta(\lambda)}{a(\lambda)}
			\sim \mathcal O
				\left(
					\frac{1}
						{
							\int_{\mathbb R}
								u \, M_1^+(x; \lambda)
							\, \mathrm{d}x
						}
				\right)
	\]
	for $|\lambda| \ll 1$, as
	$\lim_{\lambda\to0} \alpha(\lambda) / \beta(\lambda) = 1$.
	Thus, if either $\int_{\mathbb R} u_1 \, M_1^+(x; \lambda = 0, u_1) \, \mathrm{d}x$ or
	$\int_{\mathbb R} u_2 \, M_1^+(x; \lambda = 0, u_2) \, \mathrm{d}x$ are zero, then the
	approach in the proof of Theorem \ref{thm4:DlipR} fails miserably for
	$|\lambda|$ that is not controlled below.
\end{rmk}

In light of Remark \ref{rmk4:Lip}, we obtain the following easy ``extension''
of Theorem \ref{thm4:DlipR}, which emphasizes the challenge in actually extending
Theorem \ref{thm4:DlipR} to all real values of $\lambda$.

\begin{cor}\label{cor4:Lip}
	For every $u \in B_X(0, c_0)$ with the property that
	\[
		\int_{\mathbb R} u\,M_1^+(x; \lambda = 0, u) \, \mathrm{d}x \ne 0,
	\]
	there is a neighborhood $\mathcal N(u)$ in $B_X(0, c_0)$ about $u$
	for which the map $\mathscr D : \mathcal N(u) \mapsto
	L_\lambda^\infty(\mathbb R)$ is Lipschitz continuous.
\end{cor}
\begin{proof}
	Fix $\varepsilon > 0$ so that
	$\left| \int_{\mathbb R} u\,M_1^+(x; \lambda = 0, u) \, \mathrm{d}x \right| > 2\varepsilon$
	Using the Lipschitz continuity of the map
	$B_X(0, c_0) \ni w(x) \to  w(x)\, M_1^+(x; \lambda, w)
	\in L_x^1(\mathbb R)$, we may choose $\mathcal N_\varepsilon(u)$ so that
	every $w$ in $\mathcal N(u)$ satisfies
	\[
		\left| \int_{\mathbb R} w(x)\, M_1^+(x; \lambda=0, w)\,dx \right|
			\geq \varepsilon.
	\]
	Then, Corollary \ref{cor4:Lip} follows from the proof of Theorem
	\ref{thm4:DlipR}, Remark \ref{rmk4:Lip}, and the Dominated Convergence
	Theorem.
\end{proof}

The following lemma, Lemma \ref{lma4:Mtexpansion}, was developed as part of an, as
yet, unsuccessful bid to
extend Theorem \ref{thm4:DlipR} to all real $r$. We include this lemma here in the
hopes that it may eventually be useful in completing the proof that the ILW
direct scattering map is Lipschitz continuous.

\begin{lma}\label{lma4:Mtexpansion}
	For $u \in X\cap \inn{x}^{-5} L_x^\infty(\mathbb R)$, the Jost solution
	boundary value
	$M_1^+$ has the following $\inn{x}^4L_x^\infty(\mathbb R)$
	linear approximation in $\lambda$ centered at $\lambda = 0$
	\begin{align}
		M_1^+(x; \lambda, u)
			= M^{(0)}(x; u) + \lambda \, M^{(1)}(x; u) + o(\lambda),
	\end{align}
	where $M^{(0)}(x; u) = M_1^+(x; \lambda = 0, u)$ and
	$M^{(1)}$ is $(\partial M_1^+ / \partial \lambda)(x; 0)$
\end{lma}
\begin{proof}
	It suffices to prove that $M_1^+$ has an $\inn{x}^4L^\infty(\mathbb R)$
	derivative in $\lambda$ at $\lambda = 0$. To do so, we
	define $M_h$ as the difference quotient
	\[
		M_h(\lambda; x) :=
			\frac{M_1^+(x; \lambda +h) - M_1^+(x; \lambda)}{h}.
	\]
	In order to simplify notation, throughout the rest of this proof,
	we denote $G_L^+$ by $G$, $M_1^+$ by $M$, and suppress $x$ dependency.
	That is, $G(\lambda):= G_L^+(x; \lambda)$ and $M(\lambda):=M_1^+(x; \lambda)$.
	Please note that while not explicitly indicated by the notation in this
	proof, all convolutions are with respect to the variable $x$.

	By linearity of convolution operators
	\begin{align*}
		M_h(\lambda; x)
			&=
				\frac{
					G(\lambda+h)* \big[u\,M(\lambda+h)\big]
					-G(\lambda) * \big[u\,M(\lambda)\big]
				}
				{h} \\[.1\baselineskip]
			&=
				\frac{
					\big[
						G(\lambda+h) - G(\lambda)
					\big]
						* \big[u\,M(\lambda+h)\big]
			% &\qquad
					- G(\lambda)
						*
							\big\{
								u
								\big[
									M(\lambda+h)
									-M(\lambda)
								\big]
							\big\}
				}
				{h} \\
			&=
				\left(
					\frac{G(\lambda+h)-G(\lambda)}{h}
				\right)
					* \big[u\,M(\lambda+h)\big]
				- G(\lambda)
					* \big[ u \, M_h(\lambda) \big]
	\end{align*}
	Define $G_h$ to be the difference quotient
	$G_h = \frac{1}{h}\big[G(\lambda+h) - G(\lambda)\big]$ and let
	$T_\lambda$ denote the operator given by $T_\lambda f = G(\lambda)*(u\,f)$.
	Since, as we see in the proof of Proposition \ref{prop4:exist}, $I + T_\lambda$
	is invertible, the following formula for $M_h$ follows from the above
	computation:
	\begin{align}\label{eq3:Mh}
		M_h(\lambda) := (I + T_\lambda)^{-1} \big(G_h * u\, M(\lambda+h)\big).
	\end{align}
	Given the continuity of $(I + T_\lambda)^{-1}$, equation \eqref{eq3:Mh}
	implies that $M_1^+$ is differentiable in $\lambda$ ($\lambda \in \mathbb R$)
	if and only if the limit
	\[
		\lim_{h\to0} G_h * u\, M(\lambda+h)
	\]
	holds pointwise for each $x \in \mathbb R$.

	Since a natural candidate for the limit of $G_h * u\, M(\lambda+h)$
	as $h\to 0$ is $\left(\frac{\partial}{\partial \lambda}G\right) * (uM)$,
	note that
	\begin{align}
		&G_h*\big(uM(\lambda+h)\big)
				- \left(\frac{\partial}{\partial \lambda}G\right)
					* \big(uM(\lambda)\big) \\
			&\qquad=
				\left(G_h - \frac{\partial}{\partial \lambda}G\right)
						* \big(uM(\lambda +h)\big)
					+ \left(\frac{\partial}{\partial \lambda}G\right)
						* u \big(M(\lambda+h) - M(\lambda)\big).
				\nonumber
	\end{align}

	By Technical Lemma \ref{tlma2:1},
	\begin{align*}
		\left\|
			\left(G_h - \frac{\partial}{\partial \lambda}G\right)
			* \big( uM(\lambda+h)\big)
		\right\|_{\inn{x}^4L_x^\infty}
			&\leq
				\left\|
					G_h - \frac{\partial}{\partial \lambda}G
				\right\|_{\inn{x}^4 L_x^1}
				\left\|
					\inn{\dotarg}^4
					u\,M(\lambda+h)
				\right\|_{L_x^\infty}.
	\end{align*}
	Hence
	\begin{align} \label{eq3:??}
		\lim_{h\to\infty}
				\left\|
					\left(G_h - \frac{\partial}{\partial \lambda}G\right)
					* \big( uM(\lambda+h)\big)
				\right\|_{\inn{x}^4 L_x^\infty}
			= 0,
	\end{align}
	as
	\begin{align*}
		\left\|
			\inn{\dotarg}^4
			u\,M(\lambda+h)
		\right\|_{L_x^\infty}
		=
			\esssup_{x\in\mathbb R} \big(\inn{x}^{-1} M\big) \big(\inn{x}^5 u\big)
		\leq \|M\|_{\inn{x}L_x^\infty} \|u\|_{\inn{x}^{-5}L_x^\infty}.
	\end{align*}

	Similarly,
	\begin{align*}
		&
			\left\|
				\left(\frac{\partial}{\partial \lambda} G\right)
				*
				u\big(M(\lambda+h) - M(\lambda)\big)
			\right\|_{\inn{x}^4 L_x^\infty} \\
		&\qquad\leq
			\left\|
				\frac{\partial}{\partial \lambda} G
			\right\|_{\inn{x}^4 L^1}
			\left\|
				\inn{\dotarg}^4 u
				\big[
					M(\lambda + h) - M(\lambda)
				\big]
			\right\|_{L_x^\infty}
			\nonumber \\
		&\qquad\leq
			\left\|
				\frac{\partial}{\partial \lambda} G
			\right\|_{\inn{x}^4 L^1}
			\|u\|_{\inn{x}^{-5} L_x^\infty}
			\|M(\lambda+h) - M(\lambda)\|_{\inn{x}L_x^\infty}
			\nonumber
	\end{align*}
	Hence, by Lemma \ref{lma4:Mlamcont}, we also find
	\begin{align}
		\lim_{h\to\infty}
			\left\|
				\left(\frac{\partial}{\partial \lambda} G\right)
				*
				u\big(M(\lambda+h) - M(\lambda)\big)
			\right\|_{\inn{x}^4 L_x^\infty}
			=0,
	\end{align}
	from which the result follows.
\end{proof}

\end{document}


\begin{landscape}
\chapter{Inference of Guarded Recursion: Proofs}
\label{app:infer-guard-recurs}
\label{cha:example-inferences}
\begin{figure}[h]
  \begin{prooftree}
\AxiomC{(1)}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash Z : Nat \infer Z : Nat$}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$\begin{matrix} IE \vdash (::)\,Nat\,Z \ : Stream\,Nat \to Stream\,Nat \infer \\
  (_g::)\,Nat\,Z \  : \later ^\kappa_1  \onk{Stream}\,Nat \to
  \onk{Stream}\,Nat\end{matrix}$}
\AxiomC{}
\RightLabel{}
\UnaryInfC{$IE \vdash zeros \phieq\ _gzeros$}
\AxiomC{}
\RightLabel{Var}
\UnaryInfC{$IE \vdash\ _gzeros : \forallk{Stream}\,Nat$}
\RightLabel{$\infer _{Rec\causal}$}
\BinaryInfC{$\begin{matrix} IE \vdash zeros : Stream\,Nat \infer \\ \onk{Next}\,
  _gzeros[\kappa] : \later ^\kappa_1  \onk{Stream}\,Nat\end{matrix}$}
\RightLabel{$\infer _{app}$}
\BinaryInfC{ $IE \vdash (::)\,Nat\,Z\,zeros : Stream\,Nat\infer
 \,(_g::)\,Nat\,Z\,(\onk{Next}\,
  _gzeros[\kappa]) : Stream^\kappa \,Nat$}
  \end{prooftree}

\begin{prooftree}
\AxiomC{}
\RightLabel{}
  \UnaryInfC{$IE \vdash (::) \phieq\ (_g::) $}
\AxiomC{}
\RightLabel{Var}
  \UnaryInfC{$IE \vdash (_g::) : (a : Type) \to a \to
    \later^\kappa_1\onk{Stream}\,a \to \onk{Stream}\,a$}
\RightLabel{$\infer _{\phi}$}
\BinaryInfC{$\begin{matrix}IE \vdash (::) : (a : Type) \to a \to Stream\,a \to
    Stream\,a \infer \\
    (_g::) : (a : Type) \to a \to \later^\kappa_1\onk{Stream}\,a \to \onk{Stream}\,a
\end{matrix}$}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash Nat : Type \infer Nat : Type$}
\LeftLabel{(1)}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$
IE \vdash (::)\,Nat : Nat \to Stream\,Nat \to Stream\,Nat \infer (_g::)\,Nat \  : Nat \to \later ^\kappa_1  \onk{Stream}\,Nat \to
  \onk{Stream}\,Nat$}
\end{prooftree}

$IE = zeros;\, \causal ;\,\phi, (::,\ _g::), (zeros,\, _gzeros);\, \open;\,
\Gamma,Z : Nat,\ zeros : Stream Nat,\,_gzeros : \forallk{Stream} Nat$

  \caption{Inference of zeros.}
  \label{fig:zero_inference}
\end{figure}
\newpage

\begin{figure}[h]
\begin{prooftree}
\AxiomC{(2)}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash Z : Nat \infer Z : Nat$}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$\begin{matrix} IE \vdash (::)\,Nat\,Z \ : Stream\,Nat \to Stream\,Nat \infer \\
  (_g::)\,Nat\,Z \  : \later ^\kappa_1  \onk{Stream}\,Nat \to
  \onk{Stream}\,Nat\end{matrix}$}
\AxiomC{(3)}
\RightLabel{$\infer_{app}$}
\BinaryInfC{$\begin{matrix}IE \vdash (::)\,Nat\,Z\,(map\ Nat\ Nat\ S\ nats) :
    Stream\,Nat \infer
\\
  (_g::)\,Nat\,Z\,(\onk{Next} ((\, _gmap[\kappa])\ Nat \ Nat\,S) \tensor^\kappa_1
  \,(\onk{Next}(\, _gnats[\kappa]))) : \onk{Stream}\ Nat\end{matrix}$}
\end{prooftree}

\begin{prooftree}
\AxiomC{}
\RightLabel{}
  \UnaryInfC{$IE \vdash (::) \phieq\ (_g::) $}
\AxiomC{}
\RightLabel{Var}
  \UnaryInfC{$IE \vdash (_g::) : (a : Type) \to a \to
    \later^\kappa_1\onk{Stream}\,a \to \onk{Stream}\,a$}
\RightLabel{$\infer _{\phi}$}
\BinaryInfC{$\begin{matrix}IE \vdash (::) : (a : Type) \to a \to Stream\,a \to
    Stream\,a \infer \\
    (_g::) : (a : Type) \to a \to \later^\kappa_1\onk{Stream}\,a \to \onk{Stream}\,a
\end{matrix}$}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash Nat : Type \infer Nat : Type$}
\LeftLabel{(2)}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$
IE \vdash (::)\,Nat : Nat \to Stream\,Nat \to Stream\,Nat \infer (_g::)\,Nat \  : Nat \to \later ^\kappa_1  \onk{Stream}\,Nat \to
  \onk{Stream}\,Nat$}
\end{prooftree}

\begin{prooftree}
\AxiomC{(4)}
\AxiomC{}
\RightLabel{}
\UnaryInfC{$IE \vdash \phi(nats) =\ _gnats$}
\AxiomC{}
\RightLabel{Var}
\UnaryInfC{$IE \vdash\ _gnats : \forallk{Stream}\,Nat$}
\RightLabel{$\infer _{Rec\causal}$}
\BinaryInfC{$IE \vdash nats : Stream\ Nat \infer\ (\onk{Next}(\, _gnats[\kappa])) : \onk{Stream}\ Nat$}
\LeftLabel{(3)}
\RightLabel{$\infer_{\tensor^\kappa_n}$}
\BinaryInfC{$\begin{matrix} IE \vdash map\,Nat\,Nat\,S\,nats : Stream\,Nat\infer
\\
    \,\onk{Next} ((\, _gmap[\kappa])\ Nat \ Nat\,S) \tensor^\kappa_1 \,(\onk{Next}(\, _gnats[\kappa])) :
    \later^\kappa_1 Stream^\kappa \,Nat\end{matrix}$}
\end{prooftree}

\begin{prooftree}
\AxiomC{(5)}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash Nat : Type \infer Nat : Type$}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$\begin{matrix} IE \vdash
map\ Nat\ Nat : (Nat \to Nat) \to Stream\ Nat \to Stream\ Nat \infer
\\
(\, _gmap[\kappa])\ Nat\ Nat : (Nat \to Nat) \to \onk{Stream}\ Nat \to \onk{Stream}\ Nat
\end{matrix}$}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash S : Nat \to Nat \infer S : Nat \to Nat$}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$\begin{matrix}IE \vdash\ map\ Nat\ Nat\ S : Stream\ Nat \to Stream\
    Nat \infer
\\
  (\, _gmap[\kappa])\ Nat\ Nat\ S) : \onk{Stream}\ Nat
  \to \onk{Stream}\ Nat \end{matrix}$}
\RightLabel{$\infer _{Next^\kappa}$}
\LeftLabel{(4)}
\UnaryInfC{$\begin{matrix}IE \vdash\ map\ Nat\ Nat\ S : Stream\ Nat \to Stream\
    Nat \infer
\\
  \onk{Next} ((\, _gmap[\kappa])\ Nat\ Nat\ S) : \later^\kappa_1 (\onk{Stream}\ Nat \to \onk{Stream}\ Nat) \end{matrix}$}
\end{prooftree}

$IE = nats;\, \noncausal ;\,\phi, (::, _g::), (nats, _gnats), (map,
_gmap);\, \sqcup;\, \Gamma,Z : Nat,\ _gnats :
\forallk{Stream} Nat, \ _gmap : \forall \kappa.((a : Type) \to (b : Type) \to
        (a \to b) \to \onk{Stream}\ a \to \onk{Stream}\ b)$
  \caption{Inference of nats.}
  \label{fig:nats_inference}
\end{figure}

\begin{figure}[h]
\begin{prooftree}
\AxiomC{(6)}
\AxiomC{}
\RightLabel{$\infer _{Refl}$}
\UnaryInfC{$IE \vdash Nat : Type \infer Nat : Type$}
\LeftLabel{(5)}
\RightLabel{$\infer _{app}$}
\BinaryInfC{$\begin{matrix}IE \vdash map\ Nat : (b : Type) \to (Nat \to b) \to Stream\ Nat \to
  Stream\ Nat \infer
\\
(\, _gmap[\kappa]) Nat : (b : Type) \to (Nat \to b) \to \onk{Stream}\ Nat \to
2222  \onk{Stream}\ Nat
\end{matrix}$}
\end{prooftree}

\begin{prooftree}
  \AxiomC{}
\RightLabel{}
\UnaryInfC{$IE \vdash \phi(map) =\ _gmap$}
\AxiomC{}
\RightLabel{Var}
\UnaryInfC{$IE \vdash\ _gmap : \forall \kappa . ((a : Type) \to (b : Type) \to (a \to b) \to \onk{Stream}\ a \to
\onk{Stream}\ b)$}
\RightLabel{$\infer _\phi$}
\BinaryInfC{$\begin{matrix}IE \vdash map : (a : Type) \to (b : Type) \to (a \to b) \to Stream\
  a \to Stream\ b \infer
\\
_gmap : \forall \kappa . ((a : Type) \to (b : Type) \to (a \to b) \to \onk{Stream}\ a \to
\onk{Stream}\ b)
\end{matrix}$}
\RightLabel{$\infer _{[\kappa]}$}
\LeftLabel{(6)}
\UnaryInfC{$\begin{matrix}IE \vdash map : (a : Type) \to (b : Type) \to (a \to b) \to Stream\
  a \to Stream\ b \infer
\\
\ _gmap[\kappa] : (a : Type) \to (b : Type) \to (a \to b) \to \onk{Stream}\ a \to
\onk{Stream} b
\end{matrix}$}

\end{prooftree}
  \caption{Inference of nats continued.}
  \label{fig:nats_inference_2}
\end{figure}
\end{landscape}

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\documentclass[modern]{aastex631}

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\usepackage{blindtext}

% Commands
\newcommand{\dd}{\mathrm{d}}
\newcommand{\diff}[2]{\frac{\dd #1}{\dd #2}}
\newcommand{\mean}[1]{\left\langle #1 \right\rangle}
\newcommand{\chieff}{\chi_{\mathrm{eff}}}

% Begin!
\begin{document}

% Title
\title{Gar\c{c}on: Spins Please (Or Some Other Pun-y Title)}

% Institutional Addresses
\newcommand{\cca}{Center for Computational Astrophysics, Flatiron Institute, New York NY 10010, USA}
\newcommand{\sbu}{Department of Physics and Astronomy, Stony Brook University, Stony Brook NY 11794, USA}

% Author list
\author[0000-0003-1540-8562]{Will M. Farr}
\email{will.farr@stonybrook.edu}
\affiliation{\sbu}
\affiliation{\cca}

\author{Maya Fishbach}

\author{Thomas Callister}

\author{Vicky Kalogera}

\author{Katie Breivik}

\author{Daniel Holz}

\author{Simona Miller}

\author{Sharan Banagiri}

\author{and friends}

% Abstract with filler text
\begin{abstract}
We show various models for the distribution of black hole spin vectors in
GWTC-3 and discuss what they might imply about the evolution of these systems.
\end{abstract}

% Main body with filler text
\section{Introduction}

Blah.

\section{Building Intuition}

\begin{figure}
    \includegraphics[width=\columnwidth]{static/SpinFigure.png}
    \caption{\label{fig:SpinFigure} A figure showing the coordinate system we
    use for spins.}
\end{figure}

[This argument was originally suggested by Maya, and fleshed out by Vicky and
Will at Aspen.]  Note that
\begin{equation}
    \chieff = \frac{s_{1,z} + q s_{2,z}}{1+q}
\end{equation}
where $s_{i,z}$ is the $z$-component (the out-of-orbital-plane component) of the
spin of the $i$ object and $q = m_2 / m_1 < 1$.  If we assume that the
distribution of the $\vec{s}_i$ in the population do not depend on $q$, then by
linearity of expectation values, we have
\begin{equation}
    \mean{\chieff} = \frac{\mean{s_{1,z}} + q \mean{s_{2,z}}}{1+q}
\end{equation}
and
\begin{equation}
    \diff{\mean{\chieff}}{q} = \frac{\mean{s_{2,z}} - \mean{s_{1,z}}}{\left( 1 + q \right)^2}.
\end{equation}
\citet{Callister2021} showed that the GWTC-2 \citep{Abbott2021GWTC-2} population
of binary black hole mergers have
\begin{equation}
    \mean{\chieff}\left( q = 0.8 \right) \simeq 0.04
\end{equation}
and
\begin{equation}
    \diff{\mean{\chieff}}{q} \simeq -0.46
\end{equation}
(we have chosen to evaluate the \citet{Callister2021} mean $\chieff$ at the
best-constrained mass ratio of $q \simeq 0.8$).  The GWTC-3 catalog exhibits the
same correlations, with reduced uncertainty \citep{Abbott2021GWTC-3}.  These
properties of the population imply
\begin{eqnarray}
    \mean{s_{1,z}} & \simeq & 0.70 \\
    \mean{s_{2,z}} & \simeq & -0.79.
\end{eqnarray}
Even accounting for the uncertainties in \citet{Callister2021}'s population,
that the mean $\chieff$ is close to zero at nearly-equal masses imples that
$\mean{s_{1,z}} \simeq - \mean{s_{2,z}}$.  That $\chieff$ decreases with
increasing $q$ implies that $\mean{s_{2,z}} < \mean{s_{1,z}}$, so that
$\mean{s_{2,z}}$ must be negative while $\mean{s_{1,z}}$ is positive
\emph{assuming that the spin distribution is independent of the system mass
ratio $q$}.

\bibliography{bib}

\end{document}


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\title{Introduction to Homology}
\author{Colton Grainger (MATH 6220 Topology 2)}
\date{2019-01-14}
\thanks{\emph{Git Repo:} \url{https://github.com/coltongrainger/fy19top2}}

\begin{document}

\maketitle

\subsection{Problems due 2019-01-30}

\subsubsection{}

Let \(X = \bigcup_{i\in I} X_i\) where the \(X_i\) are the (disjoint)
path components of \(X\). Prove that
\[H_*(X) \cong \bigoplus_{i\in I} H_*(X_i).\]

\subsubsection{}

Let \(X\) and \(Y\) be path connected spaces.

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Prove that any map \(f \colon X \to Y\) induces an isomorphism
  \(f_* \colon H_0(X) \xrightarrow{\cong} H_0(Y)\).
\item
  Prove that any map \(f \colon X \to X\) induces the identity on
  \(H_0(X)\).
\end{enumerate}

\subsubsection{}

Use the Hurewicz theorem to solve the following problems:

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Compute \(H_1(K)\) for the Klein bottle \(K\).
\item
  Compute \(H_1\) of \(X=\prod_{j \in J} X_j\) for topological spaces
  \(X_j\) in terms of \(H_1(X_j)\).
\item
  Let \(X_i\) with base points \(x_i \in X_i\). Suppose that there are
  open sets \(x_i\in U_i \subseteq X_i\) such that \(x_i\) is a
  deformation retract of \(U_i\). Show that
  \[{H}_1 \left( \bigvee_{i\in I}  X_i \right)  \cong \bigoplus_{i\in I} {H}_1(X_i).\]
\end{enumerate}

\subsubsection{}

Let \(f \colon X \to Y\) be a covering space of path connected spaces
with \(f(x_0) = y_0\). By the fundamental theorem of covering spaces,
\(f_{\#} \colon \pi_1(X,x_0) \to \pi_1(Y, y_0)\) is a monomorphism. Is
\(f_* \colon H_*(X) \to H_*(Y)\) also a monomorphism. Prove that it is,
or give a counterexample.

\end{document}


\documentclass[%
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\begin{document}

\title{Fractional Calculus}% Force line breaks with \

\author{Luke A. Siemens}
\email{luke.siemens@lsiemens.com}
\noaffiliation

\date{\today}

\maketitle

\section{Derive fractional calculus}
I would like to derive fractional calculus, it seems like it should exist as a natural extesion to the field of calculus.

\subsection{consepts}
Riemann–Liouville Integral, Riemann–Liouville Derivative, Caputo derivative, gamma function, analytic continuation, Ramanujan's master theorem

\section{Black Board 05-21-2020}

A transcription of the contents of my black board, as of 05/21/2020

Define the functions,

\[A(x, \alpha) = e^x\]
\[B(x, \alpha) = \left(x + \alpha \right)e^x\]
\[C(x, \alpha) = \left(x^2 + 2\alpha x + \alpha (\alpha - 1) \right)e^x\]

notice that,

\[\frac{d}{dx} A(x, \alpha) = e^x = A(x, \alpha + 1)\]
\[\frac{d}{dx} B(x, \alpha) = \left(x + \alpha + 1 \right)e^x = B(x, \alpha + 1)\]
\[\frac{d}{dx} C(x, \alpha) = \left(x^2 + 2(\alpha + 1) x + (\alpha + 1)\alpha \right)e^x = C(x, \alpha + 1)\]

and,

\[\int_{-\infty}^xA(t, \alpha)dt = e^x = A(x, \alpha - 1)\]
\[\int_{-\infty}^xB(t, \alpha)dt = \left(x + \alpha - 1 \right)e^x = B(x, \alpha - 1)\]
\[\int_{-\infty}^xC(t, \alpha)dt = \left(x^2 + 2(\alpha - 1) x + (\alpha - 1)(\alpha - 2)\right)e^x = B(x, \alpha - 1)\]

defining the operator $J^\alpha$ based on the Riemann-Liouvill integrals as,

\begin{equation}
J^\alpha f(x) = \frac{1}{\Gamma(\alpha)}\int_{-\infty}^x (x - t)^{\alpha - 1}f(t)dt
\label{operator_fractional_integral}
\end{equation}

I think that the folowing is true for $\beta \in \mathbb{R}$

\[J^\beta A(x, \alpha) = A(x, \alpha - \beta)\]
\[J^\beta B(x, \alpha) = B(x, \alpha - \beta)\]
\[J^\beta C(x, \alpha) = C(x, \alpha - \beta)\]

I propose that the folowing is true

\[J^\beta \left(P_n^\alpha(x)\right) = P_n^{\alpha-\beta}(x)\]

\[P_n^\alpha(x) = \left( \sum_{k=0}^n \binom{n}{k}\frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 + k - n)}x^k \right)e^x\]

where $\beta \in \mathbb{R}$

Assuming this worked, then

\[J^\beta \left(a F(x) + b G(x)\right) = a J^\beta F(x) + b J^\beta G(x)\]

\[J^\beta \left(J^\gamma \left( P_n^\alpha(x) \right)\right) = P_n^{\alpha - \gamma - \beta}(x)= P_n^{\alpha - \beta - \gamma}(x) = J^\gamma \left(J^\beta \left( P_n^\alpha(x) \right)\right)\]

\[J^{1} \left(P_n^\alpha(x)\right) = \int_{-\infty}^x P_n^\alpha(t)dt = P_n^{\alpha - 1}(x)\]
\[J^{-1} \left(P_n^\alpha(x)\right) = \frac{d}{dx} P_n^\alpha(x) = P_n^{\alpha + 1}(x)\]

so on the vector space formed by the set of functions $P_n^\alpha(x)$ the operator $J^\beta$ has all of the properties nessisary for a well defined fractional calculus. I expect that $J^\beta$ is not valid for $\beta \le 0$ but that there are equivelent definitions that are defined in that range (for example cauchy's differentiation formula generalized for fractional derivatives).

Using this fractional calculus is defined for a predefined set of exponential polynomials, but linear combinations of them can be used to construct arbitrary polynomial expoentials. Then this fractional calculus can be extended even further by taking aproximating some arbitrary function $F(x)$ and then computing the $n$th order taylor expansion of the function $F(x)e^{-x}$ denote its taylor expansion as $\mathfrak{T}_n\left(F(x)e^{-x}\right)$ and then using the function $\mathfrak{T}_n\left(F(x)e^{-x}\right)e^x \approx F(x)$ to approximate fractional calculus on $F(x)$ for sufficiently large $n$.

\section{Black Board 05-22-2020}

A transcription of the contents of my black board, as of 05/22/2020

\[P_n^\alpha(x) = \left( \sum_{k=0}^n \binom{n}{k}\frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 + k - n)}x^k \right)e^x = \frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 -n)}{}_1F_1(\alpha + 1, \alpha + 1 -n, x)\]

using this equation for $P_n^\alpha(x)$ makes it simple to prove the folowing,

\[\frac{d}{dx}P_n^\alpha(x) = \frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 -n)}\frac{d}{dx}{}_1F_1(\alpha + 1, \alpha + 1 -n, x) = \frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 -n)}\frac{\alpha + 1}{\alpha + 1 - n}{}_1F_1(\alpha + 2, \alpha + 2 -n, x) \]\[= \frac{\Gamma(\alpha + 2)}{\Gamma(\alpha + 2 -n)}{}_1F_1(\alpha + 2, \alpha + 2 -n, x) = P_n^{\alpha + 1}(x)\]

and also to prove the folowing,

\[\int_{-\infty}^x P_n^\alpha(t)dt = \frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 -n)} \int_{-\infty}^x  {}_1F_1(\alpha + 1, \alpha + 1 -n, t)dt = \frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 -n)}\frac{\alpha - n}{\alpha}{}_1F_1(\alpha, \alpha -n, x) \]\[= \frac{\Gamma(\alpha)}{\Gamma(\alpha - n)}{}_1F_1(\alpha, \alpha - n, x) = P_n^{\alpha - 1}(x)\]

finaly the folowing integral may be useful,

\[\int_{-\infty}^x \frac{\left(x - t\right)^{\alpha - 1}t^ne^t}{\Gamma(\alpha)}dt = \sum_{k=0}^n \binom{n}{k}\frac{\Gamma(\alpha + k)}{\Gamma(\alpha)}(-1)^kx^{n - k}e^x\]

\section{polynomials}
note that the fractional derivative of $x^k$ is often expressed as,

\[\frac{d^\alpha}{dx^\alpha}x^k = \frac{\Gamma(k + 1)}{\Gamma(k + 1 - \alpha)}x^{k - \alpha}\]

these fractional monomials can be found using $J^\alpha$ as the solution to the fractional integral of an impulse,

\[J^\alpha\delta(x) = \int_{-\infty}^x \frac{\left(x - t\right)^{\alpha - 1}}{\Gamma(\alpha)}\delta(t)dt = \frac{x^{\alpha - 1}}{\Gamma(\alpha)}\]

so the singularities produced at the origin is a side effect of the impulse.

These polynomials may be useful for fractional time derivatives (only past events have an effect), can the equations be reformulated for frational space derivates (using a combination of left and right derivatives). For example fractionaly integrate $H(x)H(1-x)$ a square bump function.

\section{Black Board 05-27-2020}

A transcription of the contents of my black board, as of 05/27/2020

Looking at the first four exponential polynomials $P_n^\alpha(x)$ I came up with the folowing recursive formula,

\[
P_{0}^{\alpha}(x) = e^x
\]
\begin{equation}
P_n^\alpha(x) = xP_{n-1}^\alpha(x) + \alpha P_{n-1}^{\alpha - 1}(x)
\label{polynomial_exponential_recursive}
\end{equation}

now fractionaly integrate the base case,

\[J^\beta P_0^\alpha(x) = \frac{1}{\Gamma(\beta)}\int_{-\infty}^x \left(x - t\right)^{\beta - 1}e^{t}dt = e^x\]

note that $e^x = P_0^{\alpha - \beta}(x)$, so the identity $J^\beta P_0^\alpha(x) = P_0^{\alpha - \beta}(x)$ is true, though also trivial. Given this identity let us suppose $J^\beta P_{n-1}^\alpha(x) = P_{n - 1}^{\alpha - \beta}(x)$, then

\[J^\beta P_n^\alpha(x) = J^\beta \left(xP_{n - 1}^\alpha(x) + \alpha P_{n - 1}^{\alpha - 1}(x)\right)\]
using linearity of $J^\beta$ and our asumption about $P_{n - 1}^\alpha$,
\[J^\beta P_n^\alpha(x) = J^\beta \left(xP_{n - 1}^\alpha(x)\right) + \alpha P_{n - 1}^{\alpha - \beta - 1}(x)\]
now using the generalized Leibniz rule given by \cite{Leibniz} equation (15.11),

\[
J^\beta \left(xP_{n - 1}^\alpha(x)\right) = \sum_{k=0}^{\infty} \binom{-\beta}{k}\left(J^{\beta + k}P_{n - 1}^\alpha(x)\right)\left(\frac{d^k}{dx^k}x\right) = \binom{-\beta}{0}xJ^\beta P_{n - 1}^\alpha(x) + \binom{-\beta}{1}J^{\beta + 1}P_{n - 1}^\alpha(x)
\]

all other terms are zero, simplifying

\[
J^\beta \left(xP_{n - 1}^\alpha(x)\right) = xP_{n - 1}^{\alpha - \beta}(x) - \beta P_{n - 1}^{\alpha - \beta - 1}(x)
\]

Using this the equation becomes,

\[
J^\beta P_n^\alpha(x) = xP_{n - 1}^{\alpha - \beta}(x) - \beta P_{n - 1}^{\alpha - \beta - 1}(x) + \alpha P_{n - 1}^{\alpha - \beta - 1}(x)
\]

simplifying,

\[
J^\beta P_n^\alpha(x) = xP_{n - 1}^{\alpha - \beta}(x) + \left(\alpha - \beta\right) P_{n - 1}^{\alpha - \beta - 1}(x) = P_n^{\alpha - \beta}(x)
\]

by induction then the folowing identity is true for all of the polynomial exponentials $P_n^\alpha(x)$

\begin{equation}
J^\beta P_n^\alpha(x) = P_n^{\alpha - \beta}(x)
\label{polynomial_exponential_integral}
\end{equation}

So the set $P_n^\alpha(x)$ with operator $J^\beta$ acts as an abelean group. Fractional calculus does not produce any contridictions when acting exclusivly on polynomial exponentials.

\section{Example problem using exponential polynomials}
Find the fractional derivative of the equation $e^{\lambda x}$, I will use this since the solution is an elementry function $\lambda^\alpha e^{\lambda x}$.

\[e^{\lambda x} = e^{(\lambda - 1)x}e^x = \sum_{k=0}^\infty \frac{(\lambda - 1)^k x^k e^x}{k!} = \sum_{k=0}^\infty \frac{(\lambda - 1)^k}{k!}P_k^0(x)\]

note that $P_n^0(x) = x^n e^x$. Taking the fractional derivative of this function,

\[J^{-\alpha} e^{\lambda x} = J^{-\alpha} \left(\sum_{n=0}^\infty \frac{(\lambda - 1)^n}{n!}P_n^0(x)\right) = \sum_{n=0}^\infty \frac{(\lambda - 1)^n}{n!}P_n^\alpha(x) = \sum_{n=0}^\infty \frac{(\lambda - 1)^n}{n!} \left( \sum_{k=0}^n \binom{n}{k}\frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 + k - n)}x^k \right)e^x \]

bringing out the inner summation,

\[J^{-\alpha} e^{\lambda x} = e^x \sum_{n=0}^\infty \sum_{k=0}^n \frac{(\lambda - 1)^n}{n!} \binom{n}{k}\frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 + k - n)}x^k \]

Simplify, and raise the limit of the inner summation since $\binom{n}{n + k} = 0$ if $k \in \mathbb{Z}^+$ so $\binom{n}{k}=0$ if $k > n$,

\[J^{-\alpha} e^{\lambda x} = e^x \sum_{n=0}^\infty \sum_{k=0}^\infty \frac{(\lambda - 1)^n}{k!(n - k)!}\frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 + k - n)}x^k \]

reindex the equation using the substitution $m = n - k$, note that any term with $m < 0$ evaluates to zero,

\[J^{-\alpha} e^{\lambda x} = e^x \sum_{m=0}^\infty \sum_{k=0}^\infty \frac{(\lambda - 1)^{m + k}}{k!m!}\frac{\Gamma(\alpha + 1)}{\Gamma(\alpha + 1 - m)}x^k \]

rearanging the equation,

\[J^{-\alpha} e^{\lambda x} = \left(\sum_{m=0}^\infty \frac{\Gamma(\alpha + 1)}{m!\Gamma(\alpha + 1 - m)}(\lambda - 1)^m\right) \sum_{k=0}^\infty \frac{(\lambda - 1)^k}{k!}x^k e^x = \left(\sum_{m=0}^\infty \binom{\alpha}{m}(\lambda - 1)^m 1^{\alpha - m}\right) e^{\lambda x}\]

this series only converges for $\left|1 - \lambda\right| < 1$, taking the analytic continuation results in the solution

 \[J^{-\alpha} e^{\lambda x} = \lambda^\alpha e^{\lambda x}\]

\section{Black Board 06-02-2020}

A transcription of the contents of my black board, as of 06/02/2020

I would like to define a function space that behaves nicely under fractional calculus. Towards that end we have observed that exponential functions and polynomials multipliyed by exponential functions behave nicely under fractional calculus. Let us define two sets, first a set of positive monotonic increasing functions $\mathbb{M}$, and a set of functions that are in some sence bounded by exponential functions $\mathbb{B}$,

\[\mathbb{M} := \left\lbrace a \in C(\mathbb{R}) \middle| (\forall x, y \in \mathbb{R}) x > y, a(x) > a(y) \geq 0 \right\rbrace\]

\[\mathbb{B}(a, b) := \left\lbrace f \in C^\infty(\mathbb{R}) \middle| (\forall x, x_0 \in \mathbb{R}) x \leq x_0, |f(x)| \leq a(x_0)e^{bx} \right\rbrace\]

Using these two sets we can construct a set of "nice" functions,

\begin{equation}
\mathbb{S} := \left\lbrace f \in C^\infty(\mathbb{R}) \middle| (\forall n \in \mathbb{Z}^+)(\exists a_n \in \mathbb{M}, b_n \in \mathbb{R}) b_n > 0, \frac{d^n}{dx^n}f(x) \in \mathbb{B}(a_n, b_n) \right\rbrace
\label{exponentialy_bounded}
\end{equation}

Given $\alpha \in \mathbb{R}$ and $f, g \in \mathbb{S}$ with $a_{fn}, a_{gn} \in \mathbb{M}$, $ b_{fn}, b_{gn} \in \mathbb{R}$ such that $b_{fn} \geq b_{gn} > 0$ and $\frac{d^n}{dx^n} f(x) \in \mathbb{B}(a_{fn}, b_{fn}), \frac{d^n}{dx^n} g(x) \in \mathbb{B}(a_{gn}, b_{gn})$ then,

\[(\forall x, x_0 \in \mathbb{R}) x \leq x_0, \left|\frac{d^n}{dx^n} (\alpha f(x))\right| \leq |\alpha| \left|\frac{d^n}{dx^n}f(x)\right| \leq (|\alpha| a_{fn}(x_0))e^{b_{fn}x}\]

and

\[(\forall x, x_0 \in \mathbb{R}) x \leq x_0, \left|\frac{d^n}{dx^n}(f(x) + g(x))\right| \leq \left|\frac{d^n}{dx^n} f(x)\right| + \left|\frac{d^n}{dx^n} g(x)\right| \leq \]
\[\leq a_{fn}(x)e^{b_{fn}x} + a_{gn}(x)e^{b_{gn}x} \leq \left(a_{fn}(x_0)e^{(b_{fn} - b_{gn})x_0} + a_{gn}(x_0)\right)e^{b_{gn}x}\]

and finally

\[(\forall x, x_0 \in \mathbb{R}) x \leq x_0, \left|\int_{-\infty}^x f(t)dt\right| \leq \int_{-\infty}^x \left|f(t)\right| dt \leq \int_{-\infty}^x a_{fn}(x_0)e^{b_{fn}t}dt = \frac{a_{fn}(x_0)}{b_{fn}}e^{b_{fn}x}\]

thus, if $f, g \in \mathbb{S}, \alpha \in \mathbb{R}$ then $\alpha f(x) \in \mathbb{S}$ and $f(x) + g(x) \in \mathbb{S}$. So the set of functions $\mathbb{S}$ is a vector space and is a sub-space of $C^\infty(\mathbb{R})$. Also by definition $\frac{d}{dx}f(x) \in \mathbb{S}$ and we found that $\int_{-\infty}^x f(t)dt$ nesisaraly exists for all x and that $\int_{-\infty}^x f(t)dt \in \mathbb{S}$. Note equivelent results can be produced when only requireing $b_{fn}>0, b_{gn}>0$ not $b_{fn}\geq b_{gn} > 0$

\section{Black Board 06-07-2020}

A transcription of the contents of my black board, as of 06/07/2020

Ramanujan's Master Theorem (RMT) can be stated as,

\[g(u) = \sum_{k=0}^\infty \frac{\phi(k)(-u)^k}{k!}\]

Given appropriate conditions on $\phi(k)$ the summ converges and the folowing result holds,

\[\int_0^{\infty} u^{s-1}g(u)du = \Gamma(s)\phi(-s)\]

When applicable this theorem acts to interpolate the sequence $\phi(k), k \in \mathbb{Z}^+$, finding an analytic function $\phi(-s)$ reproducing the sequence when $s = -k, k \in \mathbb{Z}^+$. Let us define $\phi(k)$ interms of a function $f(x) \in C^{\omega}(\mathbb{C})$,

\[\phi(k) = \left. \frac{d^k}{dx^k}f(x)\right|_{x = x_0}\]

then in this case $g(u)$ is,

\[g(u) = \sum_{k=0}^\infty \frac{(-u)^k}{k!} \left. \frac{d^k}{dx^k}f(x)\right|_{x=x_0}\]

recognize that $g(u)$ is the taylor expansion of $f(x_0 - u)$ in terms of $u$. Now using $f(x_0 - u)$ in the integral,

\[\int_0^{\int} u^{s-1}f(x_0 - u)du = \Gamma(s)\phi(-s)\]

using the substitution $t = x_0 - u$, the integral becomes,

\[\int_{x_0}^{-\infty} -(x_0 - t)^{s-1}f(t)dt = \Gamma(s)\phi(-s)\]

and finally rearanging,

\[\phi(-s) = \frac{1}{\Gamma(s)} \int_{-\infty}^{x_0} (x_0 - t)^{s-1} f(t)dt = \left. J^s f(x)\right|_{x = x_0}\]

So using RMT to interpolate betwean the derivatives of $f(x)$ at the point $x_0$ yields a definition for a fractional integral operator which is analytic in respect to $s$ and defined on a strip $s \in \mathbb{C}, a < \mathfrak{R}(s) < b$.

Note that the function $\phi(-s)$ produced from RMT is not unique.

\[\psi(s) \in C^{\omega}(\mathbb{C}), \psi(k) = 0, k \in \mathbb{Z}^+\]
\[\phi'(k) = \psi(k) + \phi(k) = \phi(k), k \in \mathbb{Z}^+\]

So applying RMT to $\phi'(k)$ will yield $\phi(-s)$ and not $\phi(-s) + \psi(-s)$. This demenstrates that while $J^\alpha$ has many of the properties required for fractional calculus it is not unique. Lets denote an arbitrary compatible fractional calculus operator as $I^\alpha$, then

\[R^\alpha f(x) = I^\alpha f(x) - J^\alpha f(x)\]

where $J^\alpha$ is operator found using RMT the RL integral. If we can prove that $R^\alpha = 0$ when some aditional constraint is applied to the definition of fractional calculus, then that constraint would force the operator $I^\alpha$ to be uniquely defined. Apply the generalized Leibniz rule given by \cite{Leibniz} equation (15.11) to the operator $I^\alpha$,

\[I^\alpha f(x)g(x) = \sum_{k=0}^\infty \binom{-\alpha}{k}\left( I^{\alpha + k}f(x) \right)\left( \frac{d^k}{dx^k} g(x)\right) = \sum_{k=0}^\infty \binom{-\alpha}{k}\left( \left(J^{\alpha + k} + R^{\alpha + k}\right)f(x) \right)\left( \frac{d^k}{dx^k} g(x)\right)\]

then subtracting $J^\alpha f(x)g(x)$ from both sides,

\begin{equation}
R^\alpha f(x)g(x) = \sum_{k=0}^\infty \binom{-\alpha}{k}\left( R^{\alpha + k}f(x) \right)\left( \frac{d^k}{dx^k} g(x)\right)
\label{RemainderOperatorProductRule}
\end{equation}

Now that we are setup, look at the set of ODEs $\frac{d^n}{dx^n}f(x) = f(x)$. All solutions to these equations are summes of one or more exponentials, and inpaticular

\[\exists f(x) \forall n \in \mathbb{Z}, \frac{d^n}{dx^n}f(x) = f(x)\]

where interpreting negetive intagers as repeated integrals of the form $\int_{-\infty}^x f(t)dt$. This statement is true and the only nontrivial $f(x)$ that statisfies it is $f(x) = e^x$ (ignoring the scaling constant). Any ambiguity in the statment can be removed by phrasing it as,

\begin{equation}
\exists f(x) \forall n \in \mathbb{Z}, \frac{d^n}{dx^n}f(x) = f(x), f(0) = 1
\label{ConstaintOnCalculus}
\end{equation}

Let us generalize this statment to fractional calculus,

\begin{equation}
\exists f(x) \forall \alpha \in \mathbb{C}, \frac{d^\alpha}{dx^\alpha}f(x) = f(x), f(0) = 1
\label{ConstaintOnFractionalCalculus}
\end{equation}

From statement \eqref{ConstaintOnFractionalCalculus} we can conclude that if $f(x)$ exists it must be $f(x) = e^x$, since it nesesitates that $\frac{d}{dx}f(x) = f(x), f(0) = 1$. We will now re	quire that statment \eqref{ConstaintOnFractionalCalculus} applies to fractional calculus. So $I^\alpha e^x = J^\alpha e^x + R^\alpha e^x = e^x$, but since $J^\alpha e^x = e^x$ then $R^\alpha e^x = 0$. Now apply this result to \eqref{RemainderOperatorProductRule} with $f(x) = e^x$ and $g(x) = e^{-x}h(x)$ with $h(x)$ in the domain of $I^\alpha$

\[R^\alpha h(x) = \sum_{k=0}^\infty \binom{-\alpha}{k}\left( R^{\alpha + k}e^x \right)\left( \frac{d^k}{dx^k} h(x)e^{-x}\right) = \sum_{k=0}^\infty 0 \binom{-\alpha}{k}\left( \frac{d^k}{dx^k} h(x)e^{-x}\right) = 0\]

So given \eqref{ConstaintOnFractionalCalculus} is true, then $R^\alpha = 0$, and the fractional calculus operator $J^\alpha$ derived from RMT is the only operator satisfying all of our constraints.

\section{Black Board 06-15-2020}

A transcription of the contents of my black board, as of 06/15/2020

Given $\alpha, \beta \in \mathbb{R}, \beta > 0$ and $f, g \in \mathbb{S}$ with $a_{fn}, a_{gn} \in \mathbb{M}$, $ b_{fn}, b_{gn} \in \mathbb{R}$ such that $b_{fn} > 0$, $b_{gn} > 0$ and $\frac{d^n}{dx^n} f(x) \in \mathbb{B}(a_{fn}, b_{fn}), \frac{d^n}{dx^n} g(x) \in \mathbb{B}(a_{gn}, b_{gn})$ then let $h(x) = f(\beta x + \alpha)$,

\[\left| \frac{d^n}{dx^n}h(x) \right| = \beta^n \left| \frac{d^n}{dx^n}f(\beta x + \alpha) \right| \leq \beta^n a_{fn}(\beta x_0 + \alpha)e^{b_{fn}(\beta x + \alpha)} = \beta^n e^{b_{fn}\alpha} a_{fn}(\beta x_0 + \alpha)e^{b_{fn}\beta x}\]

or let $h(x) = f(x)g(x)$, note if some function $\left| \frac{d^n}{dx^n}f(x) \right| \leq a_{fn}(x_0)e^{b_{fn}x}$ then by integrating $\left| \frac{d^{n-k}}{dx^{n-k}}f(x) \right| \leq b_{fn}^{-k}a_{fn}(x_0)e^{b_{fn}x}$,

\[\left| \frac{d^n}{dx^n}h(x) \right| \leq \sum_{k=0}^n \binom{n}{k}\left| \frac{d^{n-k}}{dx^{n-k}}f(x) \right|\left| \frac{d^{n-(n-k)}}{dx^{n-(n-k)}}g(x) \right| \leq \sum_{k = 0}^n \binom{n}{k}b_{fn}^{-k}b_{gn}^{-(n-k)}a_{fn}(x_0)a_{gn}(x_0)e^{(b_{fn} + b_{gn})x} \]

Identifying the closed form of the summation,

\[\left| \frac{d^n}{dx^n}h(x) \right| \leq \left(\frac{1}{b_{fn}} + \frac{1}{b_{gn}} \right)^n a_{fn}(x_0)a_{gn}(x_0) e^{(b_{fn} + b_{gn})x}\]

So if $f(x) \in \mathbb{S}$ then $f(\beta x + \alpha) \in \mathbb{S}$, and if $f(x), g(x) \in \mathbb{S}$ then $f(x)\cdot g(x) \in \mathbb{S}$

Note some sets of functions found to be in $\mathbb{S}$,

\[\left| x^ne^x \right| \leq n^n e^{2x - n} + 2^n n^n e^{x/2 - n} \leq n^n e^{-n}\left(e^{3x_0/2} + 2^n\right)e^{x/2}\]

Using this to construct arbitrary exponential polynomials and their derivates it is clear that $e^x\sum_{k=0}^n c_k x^k \in \mathbb{S}$ for $n<\infty$

\[a = \frac{1 + \sqrt{1 + 8n}}{4}\]

\[\left|x^n e^{-x^2}\right| \leq a^n e^{-a^2+a}e^x\]

Using this to construct arbitrary gaussian polynomials and their derivates it is clear that $e^{-x^2}\sum_{k=0}^n c_k x^k \in \mathbb{S}$ for $n<\infty$

\section{Black Board 10-22-2020}

A transcription of the contents of my black board, as of 10/22/2020

The Grunwald-Letnikov derivative can be defined as

\[\mathbb{D}^{\alpha}f(x) = \lim_{h \to 0^+} \frac{1}{h^{\alpha}} \sum_{n=0}^{\infty} (-1)^n \binom{\alpha}{n} f(x-nh)\]

I would like to show that $\mathbb{D}^{\alpha} f(x)$ is equivelent to $J^{\alpha}f(x)$ when $f(x) \in \mathbb{S}$,

\[J^{\alpha}f(x) = \frac{1}{\Gamma(\alpha + n)} \int_{-\infty}^x (x-t)^{\alpha + n -1} \left( \frac{d^n}{dt^n} f(t) \right) dt\]

where $\mathfrak{R}(\alpha + n) \geq 1$. One major issue with the Grunwald-Letnikov derivative when compairing it to Riemann-Louiville type integrals is the ambiguity about the bounds of integration. That is that $\lim_{h \to 0^+} x-nh$ is indeterminate. So we will redefine the derivative as follows,

\[\mathbb{D}_{x_0}^{\alpha}f(x) = \lim_{h \to 0^+} \frac{1}{h^{\alpha}} \sum_{n=0}^{\left\lfloor \frac{x-x_0}{h} \right\rfloor} (-1)^n \binom{\alpha}{n} f(x-nh)\]

Using this derivative the ambiguity is cleared up, $\lim_{h \to 0^+} x-nh$ ranges from $x$ to $x_0$ depending on whitch term of the summ is beaing checked.

Let $\mathfrak{R}(\alpha) \geq 1$ and check $\mathbb{D}^{-\alpha}f(x) = \mathbb{D}_{-\infty}^{-\alpha}f(x)$ for $f(x) \in \mathbb{S}$.

Expanding the binomial coeficint and simplifying,

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_0 \to -\infty} \lim_{h \to 0^+} h^{\alpha} \sum_{n=0}^{\left\lfloor \frac{x-x_0}{h} \right\rfloor} (-1)^n \frac{\Gamma(1-\alpha)}{\Gamma(n + 1)\Gamma(1 - \alpha - n)} f(x-nh) \]

Using the property $\Gamma(\epsilon - N) = (-1)^{N-1}\frac{\Gamma(-\epsilon)\Gamma(\epsilon + 1)}{\Gamma(N + 1 - \epsilon)}$, derived from Euler's reflection formula, with $\epsilon = -\alpha$ and $N = n - 1$. Using the substitution and simplifying, $\Gamma(1 - n - \alpha) = (-1)^n\frac{\Gamma(\alpha)\Gamma(1-\alpha)}{\Gamma(n + \alpha)}$, using this in the derivative produces,

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_0 \to -\infty} \lim_{h \to 0^+} h^{\alpha} \sum_{n=0}^{\left\lfloor \frac{x-x_0}{h} \right\rfloor} \frac{\Gamma(n+\alpha)}{\Gamma(\alpha)\Gamma(n + 1)} f(x-nh) \]

From Gautschi's inequality we get the limit, $\lim_{n \to \infty} \frac{\Gamma(n + \alpha)}{\Gamma(n + 1)n^{\alpha - 1}} = 1$. This could be used to simplify the terms in the sum but it does not apply for all terms. To fix this we will split the sum into two parts, depending on whether $(x - nh) - x_1$ is positive of negative. Then under the limit that $h$ goes to zero and if $(x - nh) - x_1 < 0$ we can say that the terms $\frac{\Gamma(n + \alpha)}{\Gamma(n+1)}$ go to $n^{\alpha - 1}$.

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_1 \to x^-} \lim_{x_0 \to -\infty} \lim_{h \to 0^+} \left( h^{\alpha} \sum_{n=\left\lfloor \frac{x - x_1}{h} \right\rfloor}^{\left\lfloor \frac{x-x_0}{h} \right\rfloor} \frac{\Gamma(n+\alpha)}{\Gamma(\alpha)\Gamma(n + 1)} f(x-nh) +  h^{\alpha} \sum_{n=0}^{\left\lfloor \frac{x-x_1}{h} \right\rfloor} \frac{\Gamma(n+\alpha)}{\Gamma(\alpha)\Gamma(n + 1)} f(x-nh)\right)\]

Let us call the second sum $E$,

\[E = \lim_{x_1 \to x^-} \lim_{x_0 \to -\infty} \lim_{h \to 0^+} h^{\alpha} \sum_{n=0}^{\left\lfloor \frac{x-x_1}{h} \right\rfloor} \frac{\Gamma(n+\alpha)}{\Gamma(\alpha)\Gamma(n + 1)} f(x-nh) \]

Setting the term $E$ asside for the moment, using the asymtotic limit described in the previuos paragraph and simplify.

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_1 \to x^-} \lim_{x_0 \to -\infty} \lim_{h \to 0^+} h^{\alpha} \sum_{n=\left\lfloor \frac{x - x_1}{h} \right\rfloor}^{\left\lfloor \frac{x-x_0}{h} \right\rfloor} \frac{n^{\alpha - 1}}{\Gamma(\alpha)} f(x-nh) +  E\]

Rearanging,

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_1 \to x^-} \lim_{x_0 \to -\infty} \lim_{h \to 0^+} \frac{1}{\Gamma(\alpha)} \sum_{n=\left\lfloor \frac{x - x_1}{h} \right\rfloor}^{\left\lfloor \frac{x-x_0}{h} \right\rfloor} (nh)^{\alpha - 1} f(x-nh)h +  E\]

Interpreting this as a Riemann integral, where the variable of integration is $nh=u$, then

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_1 \to x^-} \lim_{x_0 \to -\infty} \frac{1}{\Gamma(\alpha)} \int_{x-x_1}^{x-x_0} u^{\alpha - 1} f(x-u)du +  E\]

Using the varible substitution $t = x - u$, then

\[\mathbb{D}^{-\alpha}f(x) = \lim_{x_1 \to x^-} \lim_{x_0 \to -\infty} \frac{1}{\Gamma(\alpha)} \int_{x_0}^{x_1} (x - t)^{\alpha - 1} f(t)dt +  E\]

Finaly appling the last two limits,

\[\mathbb{D}^{-\alpha}f(x) = \frac{1}{\Gamma(\alpha)} \int_{-\infty}^x (x - t)^{\alpha - 1} f(t)dt +  E\]

\subsection{The Error term $E$}
Returning to the definition of $E$. We will attempt to evaluate $E$ using Gautschi's inequality, but first the inequality needs to be genralized. The inequality can be expresssed as,
\[N^{\epsilon - 1} > \frac{\Gamma(N + \epsilon)}{\Gamma(N + 1)}\]
where, $N, \epsilon \in \mathbb{R}$, $N > 0$ and $0 < \epsilon < 1$. Using the variable substitution $N=n + \left\lfloor \alpha \right\rfloor$ and $\epsilon = \alpha - \left\lfloor \alpha \right\rfloor$, then we get the inequality,
\[(n + \left\lfloor \alpha \right\rfloor)^{\alpha - \left\lfloor \alpha \right\rfloor - 1} > \frac{\Gamma(n + \alpha)}{\Gamma(n + \left\lfloor \alpha \right\rfloor + 1)} \]
Pochhammer symbol is related to the gamma function by the equation, $x^{(n)} = \frac{\Gamma(x + n)}{\Gamma(x)}$. Also if $x$ and $n$ are positive integers note that $x^{(n)} = x(x+1)(x+2)\ldots(x+n-1) < (x+n-1)^n$. So using these properties,
\[(n + \left\lfloor \alpha \right\rfloor)^{\alpha - \left\lfloor \alpha \right\rfloor - 1} > \frac{\Gamma(n + \alpha)}{(n + 1)^{(\left\lfloor \alpha \right\rfloor)}\Gamma(n + 1)} \]
Multipling both sides by $(n + 1)^{(\left\lfloor \alpha \right\rfloor)}$ gives,
\[(n + 1)^{(\left\lfloor \alpha \right\rfloor)}(n + \left\lfloor \alpha \right\rfloor)^{\alpha - \left\lfloor \alpha \right\rfloor - 1} > \frac{\Gamma(n + \alpha)}{\Gamma(n + 1)} \]
And then note that $(n + 1)^{(\left\lfloor \alpha \right\rfloor)} < (n+\left\lfloor \alpha \right\rfloor)^{\left\lfloor \alpha \right\rfloor}$,
\[(n + \left\lfloor \alpha \right\rfloor)^{\alpha - 1} > (n + 1)^{(\left\lfloor \alpha \right\rfloor)}(n + \left\lfloor \alpha \right\rfloor)^{\alpha - \left\lfloor \alpha \right\rfloor - 1} > \frac{\Gamma(n + \alpha)}{\Gamma(n + 1)} \]
So the new inequality is $(n + \left\lfloor \alpha \right\rfloor)^{\alpha - 1} > \frac{\Gamma(n + \alpha)}{\Gamma(n + 1)}$, where $n \in \mathbb{Z}^+$, $\alpha \in \mathbb{R}$ and $\alpha > 1$. Using this to evaluate $E$, note that there are no terms of $x_0$ and simplifying,
\[\left|E\right| = \lim_{x_1 \to x^-} \lim_{h \to 0^+} h^{\alpha} \sum_{n=0}^{\left\lfloor \frac{x-x_1}{h} \right\rfloor} \frac{\Gamma(n+\alpha)}{\Gamma(\alpha)\Gamma(n + 1)} \left| f(x-nh) \right| < \lim_{x_1 \to x^-} \lim_{h \to 0^+} h^{\alpha} \sum_{n=0}^{\left\lfloor \frac{x-x_1}{h} \right\rfloor} \frac{(n + \left\lfloor \alpha \right\rfloor)^{\alpha - 1}}{\Gamma(\alpha)} \left| f(x-nh) \right|\]
Simplifying
\[\left|E\right| < \lim_{x_1 \to x^-} \lim_{h \to 0^+} \frac{1}{\Gamma(\alpha)} \sum_{n=0}^{\left\lfloor \frac{x-x_1}{h} \right\rfloor} (nh + \left\lfloor \alpha \right\rfloor h)^{\alpha - 1} \left| f(x-nh) \right|h \]
Interpreting this as a Riemann integral, with a variable of integration $nh=u$ and noting that $\left\lfloor \alpha \right\rfloor h$ fanishes under the limit. Then we get,
\[\left|E\right| < \lim_{x_1 \to x^-} \frac{1}{\Gamma(\alpha)} \int_{0}^{x-x_1} u^{\alpha - 1} \left| f(x-u) \right|du \]
Using the variable substitution $t = x-u$,
\[\left|E\right| < \lim_{x_1 \to x^-} \frac{1}{\Gamma(\alpha)} \int_{x_1}^{x} (x-t)^{\alpha - 1} \left| f(t) \right|dt \]
And finaly noting that since $f(x) \in \mathbb{S}$ then it is a continuous function. So then,
\[\left|E\right| < \lim_{x_1 \to x^-} \frac{1}{\Gamma(\alpha)} \int_{x_1}^{x} (x-t)^{\alpha - 1} \left| f(t) \right|dt = 0\]
If $f(x) \in \mathbb{S}$, $\alpha \in \mathbb{R}$ and $\alpha > 1$ then $E = 0$ in that case.

\subsection{Conclusion}
Assuming that $E=0$ in general, then
\[\mathbb{D}^{-\alpha}f(x) = \frac{1}{\Gamma(\alpha)} \int_{-\infty}^x (x - t)^{\alpha - 1} f(t)dt\]
given that $f(x) \in \mathbb{S}$ and $\mathfrak{R}(\alpha) \geq 1$. Then relating it to the fractional integral $J^{\alpha}$,
\begin{equation}
\mathbb{D}^{-(\alpha+n)}\frac{d^n}{dx^n}f(x) = \frac{1}{\Gamma(\alpha + n)} \int_{-\infty}^x (x-t)^{\alpha + n - 1} \left( \frac{d^n}{dt^n} f(t) \right) dt = J^{\alpha}f(x)
\label{Grunwald-LetnikovCalculus}
\end{equation}

where $f(x) \in \mathbb{S}$ and $\mathfrak{R}(\alpha + n) \geq 1$. So the R-L integral/derivative, Caputo derivative and G-L derivative are all equivelent when taking the apropriate limits such that they are compatible with $J^\alpha$ and acting on functions in $\mathbb{S}$.

\begin{thebibliography}{1}

\bibitem{Leibniz}
  S.G. Samko, A. A. Kilbas and O. L. Marichev
  \textit{Fractional Integrals and Deriatives},
  Gordon and Breach Science Pubblishers
  1987.

\end{thebibliography}

\end{document}


\subsection{Nested logit}

With IID all goods are equal substitutes. If prices rise customers will switch to others in proportion to market size.

In practice if the price of a cheap car rises, there will be more substitution to other cheap cars than expensive cars.

Nested logit can address this.

\subsection{Estimating the nested logit model}

As with the multinomial logit model, we have prices, market shares and product characteristics.

\subsection{Own-price elasticity of demand}

\subsection{Cross-price elasticty of demand}


\thispagestyle{empty}
\section*{Extended abstract}

{\Large{\textbf{Secure layer-agnostic agreement protocol for cryptographically
agile communication}}} \\

Securing data and communication channels implies the use of cryptography. To
achieve primary security requirements (confidentiality, integrity and
availability) different types of cryptographic algorithms are used.
Cryptographic algorithms can be divided into three groups: symmetric encryption,
asymmetric encryption and cryptographic hashes. These algorithms are often used
in different combinations to achieve better performance and various security
requirements.
There is a vast amount of cryptographic algorithms currently available. They are
all constantly being improved but at the same time new vulnerabilities are being
discovered and exposed. An algorithm with vulnerabilities, from the
cryptanalysis point of view, can still be safe for usage, but the identified
vulnerability points to a weakness in its definition. Such a
weakness can eventually be used for various attacks on systems that are secured
by that algorithm, so vulnerable algorithms should be replaced as soon as
possible.

However, replacing cryptographic algorithms that are tightly integrated into
communication protocols would eventually require changing the protocols
themselves. This usually means that the running software or firmware needs to be
updated. Changing protocols may prove to be an issue when
dealing with typically large and complex communication systems. This is the
reason why cryptographic algorithms should never be tightly integrated into a
communication protocol and the solution is to use the concept of cryptographic
agility. Cryptographic agility is characterized by the ability to change
cryptographic algorithms and keys while using the same protocols and deployed
systems. On the other hand, using agility makes it possible to use the same
cryptographic algorithms or protocols over a wide array of communication
protocols thus enabling easier replacement of potentially insecure cryptographic
algorithms.

Cryptographic agility is a known principle that is already integrated into most
widely deployed secure channel protocols, such as IPsec, SSL, TLS and SSH.
However, each of the implemented agility mechanisms is strictly tied to that
solution and cannot be easily extracted as a standalone mechanism
applicable for other purposes. The possible solution that would enable this
functionality is an adaptable cryptographically agile protocol that is proposed
in this thesis. This solution can be used on various network layers,
architectures and devices. Furthermore, the aim is the agreement of
cryptographic prerequisites (algorithms and keys) in order to enable various
security requirements in an interconnected environment. Using the proposed
cryptographically agile solution enables protection of protocols that don’t have
any security mechanisms deployed and enable development of new secure protocols
on top of the agreed primitives.

The doctoral thesis describes a novel, lightweight and adaptable solution which
is
easily applicable in existing networked environments and currently emerging
areas, such as Internet of Things (IoT).
Security isn’t always integrated in current IoT solutions and is typically
planned at a later stage as an upgrade. The communication between IoT devices
needs to be secure which requires an agreement protocol to enable communicating
parties to agree on a cryptographic algorithm and keys used to protect the
exchanged messages. The IoT ecosystem needs a flexible and adaptable agreement
mechanisms because IoT devices have limited computing and bandwidth resources.
Secure communication is possible only after communicating devices agree on a set
of cryptographic algorithms and keys. Since IoT services are driven by the
underlying data sources, it is critical to validate a sensor as a credible data
source and to protect the exchanged data.

\subsection*{Chapter 2 - Cryptography}

The second chapter describes basic cryptographic primitives, cryptographic
hashes, symmetric and asymmetric cryptosystems and focuses on their application
in current communication systems. It specifically describes the primitives used
for achieving an agile and adaptive agreement on cryptographic keys and
algorithms. The needed primitives include a HMAC (Hashed Message Authentication
Code) algorithm based on cryptographic hashes with a secret key, digital
signatures for authentication that combine cryptographic hashes with asymmetric
encryption algorithms and the Diffie-Hellman key exchange that uses asymmetric
cryptography to generate secret data to conduct a secure exchange. These three
mechanisms are needed to conduct a SIGMA (sign-and-mac) exchange between two
communicating parties. The SIGMA exchange represents an authenticated key
exchange algorithm that secures the exchange from man in the middle attacks by
protecting the data used for digital signature verification with HMAC and
vice versa. The resulting secret data needs to be further processed by
using a key derivation function (KDF) based on hash functions (Hashed KDF). This
is to ensure that keys generated from the same secret data are computationally
independent. Computational independence represents that the discovery of one key
won't reveal any data on other keys calculated from the same secret data.

\subsection*{Chapter 3 - Cryptographycally agile communication}

The third chapter describes modern cryptographically agile protocols. Most of
these protocols are secure channel protocols that have many components and thus
tend to be fairly complex (e.g. SSL/TLS, IPsec, SSH). That complexity makes it
hard, or even impossible in certain cases, to create a formal model that can be
thoroughly verified to achieve provable security. The lack of security proofs is
probably one of the causes of the recent increase of attacks on SSL/TLS.
Verification of certain solutions can also reveal previously unknown
vulnerabilities. Protocol complexity also increases the possibility
of implementation problems which can cause significant security problems even if
the protocol design is provably secure. The chapter focuses on describing
established cryptographically agile solutions and compares them to modern
solutions like tcpcrypt, MinimaLT and QUIC that employ a less complex approach
to secure channels. All these solutions are compared to ACAP, the lightweight
layer-agnostic agreement protocol described in the thesis. The main advantages
of ACAP are a layer-agnostic architecture that enables its application on all
communication layers and a simple architecture that enables formal verification
and reduces the possibility of implementation problems. ACAP represents an easy
to integrate and adaptable building block that distributes secure communication
prerequisites i.e. cryptographic keys and algorithms.

\subsection*{Chapter 4 - Secure cryptographic agreement protocol}

The ACAP protocol is described in the fourth chapter. It consists of four
messages: \initi{}, \initr{}, \listi{} and \listr{}. After the message
exchange,
both communicating parties have access to the following data: secret key
generated from the Diffie-Hellman shared secret, public keys of both sides for
authentication and a list of cryptographic algorithms supported by both sides.
The list of supported cryptographic protocols is divided into groups and agreed
upon for each group. The first algorithm that is on the client list and is
present on the server list is chosen as the agreed algorithm. ACAP employs the
following security mechanisms: computational independence of secret keys by
using HKDF, denial of service attack mitigation by separating the most expensive
operations in an asynchronous background process, trust establishment that
enables both manual and automatic trust verification through PKI. Additionally,
the SIGMA design prevents the man in the middle and replay attack on the
protocol. At the end of the chapter a comprehensive comparison of the SSH
transport layer protocol negotiation and ACAP is given.

\subsection*{Chapter 5 - Formal model specification}

The formal security model and formal model verification is described in chapter
five. The formal security model is described in the security protocol
description language (SPDL) used by the Scyther security protocol verifier.
Models in SPDL are defined by using protocols and roles. Each protocol can
contain two or more roles that define multiple send and receive events. A
comprehensive explanation of the model specification process is given on the
Needham-Schroeder-Loewe protocol is give. After that the ACAP protocol model is
described. The model is specified by using two protocols, a main protocol with
the initiator and responder roles and a helper protocol that enabled the
specification of the Diffie-Hellman key exchange. After the message exchange
specification the following security requirements (Scyther claims) were defined,
immutability of the DH exponentials, secrecy of the shared secret used to derive
session keys and the inability of the attacker to interfere with the message
exchange. All three security requirements were successfully verified by using an
unbounded amount of protocol runs.

\no{
The main design motivation was to achieve provable security by extracting and
integrating
key exchange with algorithm agreement into a standalone solution. As a result,
the designed lightweight solution is formally verified and achieves
distribution of keys and algorithms in a provably secure manner. The conducted
formal verification proves resilience to (pre)replay attacks and
man-in-the-middle attacks.
}

\subsection*{Chapter 6 - Secure communication in the Internet of things}

Chapter six describes how the designed solution can be introduced in the
Internet of Things environment. It focuses on the device capabilities and
introduces three device tiers, sensors, middle and cloud tier. All secure
communication scenarios are conducted between devices from all tiers. The main
proposed secure communication scenarios include authenticated device management,
reliable data collection and sensitive data protection that are built on a
platform independent architecture. After scenario descriptions a secure
communication model for the IoT environment is given. The model consists of
devices, public keys, communication layers, hardware capabilities, cryptographic
algorithms and a set of secure communication operations. Secure communication
operations are performed to enable all previously described communication
scenarios. Furthermore, this chapter describes an adaptive cryptographic
algorithm agreement procedure that takes into account the required level of
security, cryptographic algorithm complexity, device battery power, bandwidth
and processing power to choose a cryptographic algorithm that is best suited in
the current environment.

\subsection*{Chapter 7 - ACAP protocol implementation}

A prototype implementation in Python and measurement results are described in
chapter seven.
The proposed prototype architecture is composed from five key components:
calculation and verification for cryptographic operations, message handling,
layer independent communication, cryptographic algorithm agreement and
communication with external applications. The prototype implementation supports
communication on the following layers: Ethernet, IPv4, IPv6, TCP, UDP and SCTP
with the same message formats that are shown in the appendix. It also supports
standard and elliptic curve asymmetric algorithms for Diffie-Hellman exchanges
and digital signatures.The prototype was tested in an emulated network
environment in the IMUNES tool, which simplified network setup and enabled
reliable measurements through environment automation. The measurements include
the effect of public key size on protocol message size, agreement duration in
regards to link delay and available bandwidth. They have shown that the solution
is bandwidth efficient as the entire negotiation is performed in under two
seconds on a 10 Kbps link. The use of elliptic curve cryptography is encouraged,
especially for devices with constrained resources. Furthermore, this chapter
provides an overview of cryptographic operation complexity for client and
server which shows the deployed denial of service mitigation mechanisms.

Finally, chapter eight concludes the thesis by giving a comprehensive overview
of the designed layer-agnostic protocol that enables cryptographic agility for
various devices and platforms.

The scope of this doctoral thesis covers the following scientific contributions:

1. Secure layer-agnostic agreement protocol for cryptographically agile
communication that is application and operating system independent.

2. Methods for integrating the protocol with applications that use different
communication models, especially P2P networks and client-server environments.

3. Evaluation of protocol communication and agreement efficiency in a simulated
and real network environment.

\vspace{1cm}
\textbf{Keywords}: cryptographic agility, secure communication, layer-agnostic
design, authenticated key exchange, algorithm agreement, formal model
verification

\no{
Secure communication is an important aspect of today’s interconnected
environments and it can be achieved by the use of cryptographic algorithms and
protocols. However, many existing cryptographic mechanisms are tightly
integrated into communication protocols. Issues emerge when security
vulnerabilities are discovered in cryptographic mechanisms because their
replacement would eventually require replacing deployed protocols. The concept
of cryptographic agility is the solution to these issues because it allows
dynamic switching of cryptographic algorithms and keys prior to and during the
communication. Most of today’s secure protocols implement cryptographic agility,
but cryptographic agility mechanisms cannot be used in a
standalone manner. In order to deal with the aforementioned limitations, this
thesis describes a lightweight cryptographically agile agreement protocol. The
design of the protocol is formally verified in an automated security
verification tool. The agreement of cryptographic keys and algorithms is done in
a computationally efficient way and has a low communication bandwidth
requirement. Protocol integration and usage is demonstrated in an Internet of
things environment with special emphasis on the agreement procedure which adapts
to current device capabilities. Additionally, a prototype implementation which
can communicate on different network layers is presented. The implementation is
tested and evaluated in an emulated network environment. The tests demonstrate
agreement efficiency on different communication layers when using different
types of cryptography.
}


\documentclass{article}
\usepackage{amsmath,amsfonts,amssymb}
\usepackage[mathscr]{euscript}
\usepackage[top=1in, left=1.5in, right=1.5in, bottom=1in]{geometry}

\title{Practicum in Artificial Intelligence: Optical Character Recognition via Neural Networks}
\date{October 15, 2012}
\author{Bryan Cuccioli (blc72@cornell.edu), Renato Amez (ra374@cornell.edu)}

\begin{document}

\maketitle

\emph{
We propose the design, implementation, training, and evaluation of a feed-forward neural network adapted to the task of optical character recognition (OCR). We define a precise metric for measuring the success of such a network on real-world data. We experimentally measure the efficacy of the neural network under variation of certain parameters of its construction and seek to maximize a precise quantity expressing the success of the neural network versus the difficulty in constructing and training it.
}

\section{Description}

We propose to implement a system for optical character recognition using a feed-forward neural network with backpropagation learning. We will prepare a set of training data consisting of individual letters and words along with the desired output for each input item. Characters included in the training data will be both handwritten and rendered by computer in both serif and sans-serif font faces. Images of words will first be segmented into individual characters; images of handwritten characters will be represented in the system as two-dimensional arrays of pixels normalized to a common size. Inference rules will be applied in the case of whole words to make educated guesses about difficult-to-recognize characters.

For data set $\Psi$, and character $x\in\Psi$, we define
\begin{equation}\delta(x)=\begin{cases}1 & \mathrm{character\ correctly\ recognized}\\
0 & \mathrm{otherwise}\end{cases}.\end{equation}
We then define the {\bf success of the neural network} $\mathscr{S}$ on data set $\Psi$ as
\begin{equation}\mathscr{S}=\sqrt{\sum_{a\in\Psi} \delta(a)}.\end{equation}
This definition is chosen so as to more heavily weight improvements to the neural network  once the network is already performing well, under the guise that e.g. 70\% recognition is not particularly better than 40\% recognition.

Training of the network will be complete once $\mathscr{S}$ falls above a pre-determined threshold $\alpha$. For the purposes of experimentation, $\alpha$ will be varied and the success of the neural network in interpreting the real data will be measured as a function of $\alpha$.

\section{Evaluation}

Evaluation of the neural network will be performed using data distinct from the training data. Samples rendered by computer and printed out as well as samples handwritten by multiple third parties will be used. We will evaluate the system varying parameters such as
\begin{enumerate}
\item the training error threshold, $\alpha$;
\item the number of layers in the neural network, $\ell$;
\item the size $\sigma$ and variety of the training data.
\end{enumerate}
For experimentation purposes, the success of the neural network in performing OCR will be measured as a function of each of these parameters separately. We will experimentally determine the values of these parameters that maximize the \emph{training efficiency}
\begin{equation}\mathscr{E}=\frac{\mathscr{S}^2}{\alpha\cdot\ell\cdot\sigma}.\end{equation}

\section{Timeline}

We propose the following timeline for the implementation and evaluation of our system:

\begin{itemize}
\item \emph{October 15}: Submission of project proposal;
\item \emph{October 26}: Design of a low-level outline of the architecture of our neural network, data storage, and test harness;
\item \emph{November 15}: Completion of implementing a basic neural network, system for storing training data and real data, and test harness for evaluating the efficiency of our neural network;
\item \emph{November 26}: Completion of gathering data based on the performance of the neural network on separate data under variation of the above parameters.
\item \emph{November 29}: Completion of final presentation and final project report.
\end{itemize}

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