\section{Code Listings}

Parts of this code have been released on \texttt{CRAN} under the package name \texttt{nnR}, and can be found in \cite{nnR-package}, with the corresponding repository being found at \cite{Rafi_nnR_2024}: 

\lstinputlisting[language = R, style = rstyle, label = nn_creator, caption = {R code for neural network generation}]{"/Users/shakilrafi/R-simulations/nn_creator.R"}

\lstinputlisting[language = R, style = rstyle, label = aux_fun, caption = {R code for auxilliary functions}]{"/Users/shakilrafi/R-simulations/aux_fun.R"}

\lstinputlisting[language = R, style = rstyle, label = activations, caption = {R code for activation functions ReLU and Sigmoid}]{"/Users/shakilrafi/R-simulations/activations.R"}

\lstinputlisting[language = R, style = rstyle, label = instantiation, caption = {R code for intanitation}]{"/Users/shakilrafi/R-simulations/instantiation.R"}

\lstinputlisting[language = R, style = rstyle, label = stk, caption = {R code for parallelizing two neural networks}]{"/Users/shakilrafi/R-simulations/stacking.R"}


\lstinputlisting[language = R, style = rstyle, label = affn, caption = {R code for affine neural networks}]{"/Users/shakilrafi/R-simulations/Aff.R"}


\lstinputlisting[language = R, style = rstyle, label = comp, caption = {R code for composition of two neural networks}]{"/Users/shakilrafi/R-simulations/comp.R"}


\lstinputlisting[language = R, style = rstyle, label = scalar_mult, caption = {R code for scalar multiplication}]{"/Users/shakilrafi/R-simulations/scalar_mult.R"}


\lstinputlisting[language = R, style = rstyle, label = nn_sum, caption = {R code for sum of two neural networks}]{"/Users/shakilrafi/R-simulations/nn_sum.R"}


\lstinputlisting[language = R, style = rstyle, label = i, caption = {R code for $\mathfrak{i}$}]{"/Users/shakilrafi/R-simulations/i.R"}
\lstinputlisting[language = R, style = rstyle, label = Id, caption = {R code for $\mathsf{Id}$ neural networks}]{"/Users/shakilrafi/R-simulations/Id.R"}
\lstinputlisting[language = R, style = rstyle, label = Tun, caption = {R code for $\tun$}]{"/Users/shakilrafi/R-simulations/Tun.R"}


\lstinputlisting[language = R, style = rstyle, label = Phi_k, caption = {R code for $\Phi_k$}]{"/Users/shakilrafi/R-simulations/Phi_k.R"}
\includegraphics{"/Users/shakilrafi/R-simulations/Phi_k_properties/Phi_k_diff.png"}


\lstinputlisting[language = R, style = rstyle, label = Phi_k_properties, caption = {R code for simulations involving $\Phi_k$}]{"/Users/shakilrafi/R-simulations/Phi_k_properties.R"}


\lstinputlisting[language = R, style = rstyle, label = Phi, caption = {R code for $\Phi$}]{"/Users/shakilrafi/R-simulations/Phi.R"}


\includegraphics{"/Users/shakilrafi/R-simulations/Phi_properties/Phi_diff_contour.png"}


\lstinputlisting[language = R, style = rstyle, label = Phi_properties, caption = {R code for simulations involving $\Phi$}]{"/Users/shakilrafi/R-simulations/Phi_properties.R"}


\lstinputlisting[language = R, style = rstyle, label = Sqr, caption = {R code for $\sqr$}]{"/Users/shakilrafi/R-simulations/Sqr.R"}

\lstinputlisting[language = R, style = rstyle, label = Sqr_properties, caption = {R code simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Sqr_properties.R"}

\lstinputlisting[language = R, style = rstyle, label = Pwr, caption = {R code for $\pwr^{q,\ve}$ networks}]{"/Users/shakilrafi/R-simulations/Pwr.R"}

\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\pwr_3^{q,\ve}$}]{"/Users/shakilrafi/R-simulations/Pwr_3_properties.R"}

\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\nrm^d_1$}]{"/Users/shakilrafi/R-simulations/Nrm.R"}

\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\mxm_d$}]{"/Users/shakilrafi/R-simulations/Mxm.R"}

\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\tay$}]{"/Users/shakilrafi/R-simulations/Tay.R"}

\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\etr$}]{"/Users/shakilrafi/R-simulations/Etr.R"}


\newpage

\begin{center}
	\textbf{Vita}
\end{center}

The author was born in November 1\textsuperscript{st}, 1992 in the city of Dhaka in the heart of Bangladesh. He grew up in the large city with a childhood that included setting things on fire, and very occasionally focusing on mathematics. He failed to achieve his childhood goal of becoming an astronomer however when he entered college at Troy University in 2011 and realized it would involve cold nights outside, and so chose mathematics instead. He has continued his pursuits in mathematics and is now a graduate student at the University of Arkansas trying to graduate.