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