Rearranged main layout, part way through updating sections, added sections. Updated current status in readme.

CurrentWriting/Main.tex

@@ -23,44 +23,50 @@
 
 %Describe sections
 The paper is organized as follows.
-\cref{SEC:Environment}, describes the laws of motion 
+Section \ref{SEC:Model}, describes the laws of motion 
 governing satellites and debris(\ref{SEC:Laws})%, 
 %places limits on various measures of risk (\ref{SEC:Survival}),
 and reviews various definitions of kessler syndrome in 
 the context of dynamic modeling, their advantages, and their disadvanteges (\ref{SEC:Kessler}).
-In \cref{SEC:Models} the dynamic problem faced by constellation operators and 
-social planners is first described.
-Finally \cref{SEC:Computation} describes the computational approach and results.
+It then describes the dynamic problem faced by constellation operators 
+(\ref{SEC:Operator}) and social planners (\ref{SEC:Planner}).
+Section \ref{SEC:Computation} describes the computational approach and the
+results are reported in \cref{SEC:Results}.
+Section \ref{SEC:Conclusion} concludes with a discussion of future changes and 
 
-
-\section{Modeling the Environment}\label{SEC:Environment}
+%\section{Modeling the Environment}\label{SEC:Environment}
+\section{Model}\label{SEC:Models}
     \subsection{Laws of motion}\label{SEC:Laws}
     \subfile{sections/01_LawsOfMotion} %Roughly done 2021-07-15
 
     %\subsection{Marginal survival rates}\label{SEC:Survival}
     %\subfile{sections/03_SurvivalAnalysis} %roughly done 2021-07-14
     % Thoughts on removal: This doesn't add much to the actual questions.
-% It is interesting, but as a paper is just too much.
+    % It is interesting, but in a paper is just too much.
     % I'll keep it here to add it back easily.
 
     \subsection{Kessler Syndrome}\label{SEC:Kessler}
+    % Kessler syndrome follows laws of motion because it is the main
+    % threat of orbital pollution and needs to be included.
+    % Also, there is not really a better place to place it.
     \subfile{sections/02_KesslerSyndrome}  %roughly done before 2021-07-14
     \subfile{sections/06_KesslerRegion}  %roughly done before 2021-07-14
 
-
-
-
-\section{Model}\label{SEC:Models}
     \subsection{Constellation Operator's Program}\label{SEC:Operator}
-\subfile{sections/04_ConstellationOperator} %TODO
+    \subfile{sections/04_ConstellationOperator} %Reasonably done.
 
     \subsection{Social Planner's Program}\label{SEC:Planner}
-\subfile{sections/05_SocialPlanner} %TODO
+    \subfile{sections/05_SocialPlanner} %Reasonably done?
 
 \section{Computation}\label{SEC:Computation}
-\subfile{sections/07_ComputationalApproach} %TODO
+\subfile{sections/07_ComputationalApproach} %needs some clarifications.
+
+\section{Results}\label{SEC:Results}
+\subfile{sections/09_Results} %TODO
 
 \section{Conclusion}\label{SEC:Conclusion}
+\subfile{sections/08_Conclusion} %TODO
+
 
 \newpage
 \section{References}
@@ -85,6 +91,9 @@ in a single place first.
 %With my current computational idea; each constellation provides the same risk to each other constellation
 % That can be easily adjusted in the computational models.
 
+\newpage
+%Just for simplicity, remove later
+\tableofcontents
 \end{document}
 
 %%% Notes to keep track of

CurrentWriting/readme.txt

@@ -3,6 +3,11 @@
 # TODO
  - Add section to computation that includes specifications of loss and cost functions. (Pg 9)
  - Explain utility better in the constellation operators program (pg 8) especially why I refer to it as utility. Maybe rename to "benefit function"?
+    Currently working on this. Decided to rename to benefit function.
  - In training section (4.0.1.1) clarify loss function. There is the satellite loss function and
-    the estimation loss function. Need to think of names.
+    the Approximation loss function.
  - Add a conclusion
+ - Reorganize the sections.
+    Currently working on this, mostly done.
+
+

CurrentWriting/sections/01_LawsOfMotion.tex

@@ -79,7 +79,7 @@ for the home constellation of the additional satellite.
 Note that it is reasonable to assume that the loss of satellites to collisions should be
 increasing in the level of debris: $\parder{l^i}{D_t}{} >0$.
 
-\subsection{Debris}
+\subsubsection{Debris}
 Debris is generated by various processes, including:
 \begin{itemize}
     \item Naturally occuring debris is captured from interplanetary space.

CurrentWriting/sections/02_KesslerSyndrome.tex

@@ -42,7 +42,7 @@ constitutes as kessler syndrome.
 
 
 
-\subsection{Two approaches to kessler syndrome}
+\subsubsection{Two approaches to kessler syndrome}
 I propose to analyze kessler syndrome in two slightly more restricted 
 fashions than \cite{RaoRondina2020}, for which I term the regions
  the $\epsilon$-kessler region and the proto-kesslerian region.

CurrentWriting/sections/04_ConstellationOperator.tex

@@ -5,8 +5,12 @@
 With the laws of motion introduced in sections \cref{SEC:Laws}, we can now describe
 the optimization problem facing each constellation operator.
 
-Each operator recieve utility in each period per 
-their per period utility $u^i(\vec s_t,D_t)$, which depends 
+Each operator recieve benefits\footnote{
+These benefits may take many forms, including profits for firms 
+and utility for militaries.
+}
+from their constellation
+according to $u^i(\vec s_t,D_t)$, which depends 
 on the current sizes of constellations and the level of debris.
 In addition, the operator pays for the launch of $x^i_t$ satellites
 according to the cost function $F(x)$.
@@ -27,7 +31,7 @@ Thus the $M$-period (possibly infinite), problem is:
 % - Identical debris production from destruction.
 % 
 
-\subsection{Infinite Period (Bellman) Equation}
+%\subsection{Infinite Period (Bellman) Equation} % Not sure how much help a new header is.
 The inifinite period version of the problem above can be rewritten in the bellman form as
 \begin{align}
     V^i(\vec s_t, \vec x^{\sim i}_t, D_t) = \max_{x^i_t} u^i(\vec s_t, D_t) -F(x) 

CurrentWriting/sections/06_KesslerRegion.tex

@@ -2,7 +2,7 @@
 \graphicspath{{\subfix{Assets/img/}}}
 
 \begin{document}
-\subsection{Defining the Proto-Kessler Region}
+\subsubsection{Defining the Proto-Kessler Region}
 With the definitions of kessler syndrome and the law of debris given above, we can now 
 explicitly describe the proto-kessler region.
 \begin{align}

CurrentWriting/sections/09_Results.tex

@@ -0,0 +1,26 @@
+\documentclass[../Main.tex]{subfiles}
+\graphicspath{{\subfix{Assets/img/}}}
+
+\begin{document}
+So far, I have not been able to actually analyze any models, 
+but the following are cases of interest.
+\begin{itemize}
+    \item Reproduce Rao-Rondina single satellite model.
+    \item Reproduce Adilov, perfect competition, cornot-like market.
+    \item Add military operators to Adilov's model. 
+        This will involve some sort of competitive complementarity 
+        with diminishing marginal returns.
+    \item Competitive market where the number of satellites improves quality, i.e. allows 
+        for pricing differences (Orbital Internet, e.g. Starlink).
+    \item Interacting orbital shells, using a vector representation of heterogeneous risk
+        imposed by constellations and debris.
+\end{itemize}
+
+Among these, policies that would be interesting to analyse include:
+\begin{itemize}
+    \item Launch and Operation Taxes
+    \item Deorbit contingent bonds.
+\end{itemize}
+
+
+\end{document}