From 0c941a4335529c1e5073e9f5eb43fa92d4116297 Mon Sep 17 00:00:00 2001 From: youainti Date: Wed, 25 Aug 2021 16:45:24 -0700 Subject: [PATCH] Rearranged main layout, part way through updating sections, added sections. Updated current status in readme. --- CurrentWriting/Main.tex | 59 +++++++++++-------- CurrentWriting/readme.txt | 7 ++- CurrentWriting/sections/01_LawsOfMotion.tex | 2 +- .../sections/02_KesslerSyndrome.tex | 2 +- .../sections/04_ConstellationOperator.tex | 10 +++- CurrentWriting/sections/06_KesslerRegion.tex | 2 +- CurrentWriting/sections/09_Results.tex | 26 ++++++++ 7 files changed, 76 insertions(+), 32 deletions(-) create mode 100644 CurrentWriting/sections/09_Results.tex diff --git a/CurrentWriting/Main.tex b/CurrentWriting/Main.tex index a176600..e8b2b2f 100644 --- a/CurrentWriting/Main.tex +++ b/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{Model}\label{SEC:Models} + \subsection{Laws of motion}\label{SEC:Laws} + \subfile{sections/01_LawsOfMotion} %Roughly done 2021-07-15 -\section{Modeling the Environment}\label{SEC:Environment} -\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. -% I'll keep it here to add it back easily. - -\subsection{Kessler Syndrome}\label{SEC:Kessler} -\subfile{sections/02_KesslerSyndrome} %roughly done before 2021-07-14 -\subfile{sections/06_KesslerRegion} %roughly done before 2021-07-14 - + %\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 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 + \subsection{Constellation Operator's Program}\label{SEC:Operator} + \subfile{sections/04_ConstellationOperator} %Reasonably done. -\section{Model}\label{SEC:Models} -\subsection{Constellation Operator's Program}\label{SEC:Operator} -\subfile{sections/04_ConstellationOperator} %TODO - -\subsection{Social Planner's Program}\label{SEC:Planner} -\subfile{sections/05_SocialPlanner} %TODO + \subsection{Social Planner's Program}\label{SEC:Planner} + \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 diff --git a/CurrentWriting/readme.txt b/CurrentWriting/readme.txt index b99ef87..a55ee83 100644 --- a/CurrentWriting/readme.txt +++ b/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. + + diff --git a/CurrentWriting/sections/01_LawsOfMotion.tex b/CurrentWriting/sections/01_LawsOfMotion.tex index 728e96d..cd98075 100644 --- a/CurrentWriting/sections/01_LawsOfMotion.tex +++ b/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. diff --git a/CurrentWriting/sections/02_KesslerSyndrome.tex b/CurrentWriting/sections/02_KesslerSyndrome.tex index 6f0a768..df93436 100644 --- a/CurrentWriting/sections/02_KesslerSyndrome.tex +++ b/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. diff --git a/CurrentWriting/sections/04_ConstellationOperator.tex b/CurrentWriting/sections/04_ConstellationOperator.tex index b431816..e10afd0 100644 --- a/CurrentWriting/sections/04_ConstellationOperator.tex +++ b/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) diff --git a/CurrentWriting/sections/06_KesslerRegion.tex b/CurrentWriting/sections/06_KesslerRegion.tex index 880c9d8..4e9e5b7 100644 --- a/CurrentWriting/sections/06_KesslerRegion.tex +++ b/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} diff --git a/CurrentWriting/sections/09_Results.tex b/CurrentWriting/sections/09_Results.tex new file mode 100644 index 0000000..c1eb22e --- /dev/null +++ b/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}