Changes to computation section

CurrentWriting/sections/07_ComputationalApproach.tex

@@ -15,6 +15,10 @@ $0 = f^2(x(\theta),\theta)$,
 allowing one to find $x(\dot)$ as the solution to a minimization problem.
 
 By choosing a neural network as the functional approximation, we are able to 
+use the fact that a NN with a single hidden layer can be used to approximate
+functions arbitrarily well 
+under certain conditions\footnote{FIND REFERENCE, SEE MALIAR}.
+We can also
 take advantage of the significant computational and practical improvements
 currently revolutionizing Machine Learning.
 In particular, we can now use common frameworks, such as python, PyTorch,
@@ -22,7 +26,7 @@ and various online accerators (Google Colab)
 which have been optimized for relatively high performance and 
 straightforward development.
 
-\subsubsection{Computational Plan}
+\subsection{Computational Plan}
 I have decided to use python and the PyTorch Neural Network library for this project.
 
 The most difficult step is creating the euler equations. 
@@ -120,4 +124,17 @@ I am currently working on a plan to guarantee existence of solutions.
 Some of what I want to do is check numerically crucial values as well as
 examine the necessary Inada conditions.
 
+
+\subsection{Computational Results}
+Cases to consider
+\begin{itemize}
+    \item Reproduce Rao Rondina model
+    \item Reproduce Adilov, perfect competition, cornot-like market.
+    \item Add military operators to Adilov's model. 
+        This will involve some sort of complementarities.
+    \item Competitive market where the number of satellites improves quality, i.e. allows 
+        for pricing differences (Orbital Internet and comms).
+    \item Interacting orbital shells, using a vector of debris.
+\end{itemize}
+
 \end{document}