
\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 simulations involving $\sqr$}]{"/Users/shakilrafi/R-simulations/Pwr.R"}

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

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

\lstinputlisting[language = R, style = rstyle, label = Pwr_3_properties, caption = {R code simulations involving $\sqr$}]{"/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"}





