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main.tex

@@ -47,6 +47,8 @@ Practical Methods for Finding Pseudowalls}
 
 \maketitle
 
+\tableofcontents
+
 \section{Introduction}
 \label{sec:intro}
 
@@ -209,7 +211,33 @@ for the rank of $E$:
 
 \section{B.Schmidt's Method}
 
-\section{Limitations}
+Goals:
+\begin{itemize}
+	\item intro
+	\item link repo
+\end{itemize}
+
+\subsection{Strategy}
+
+Goals:
+\begin{itemize}
+	\item link repo
+	\item Calc max destab rank
+	\item Decrease mu(E) starting from mu(F) taking on all poss frac vals
+	\item iterate over something else
+	\item Stop when conditions fail
+	\item method works same way for both rational beta_{-} but also for walls
+		larger than certain amount
+\end{itemize}
+
+\subsection{Limitations}
+
+Goals:
+\begin{itemize}
+	\item large rank forces mu to beta_{-}, so many vals of mu(E) checked
+		needlessly
+	\item noticeably slow (show benchmark)
+\end{itemize}
 
 \section{Refinement}
 \label{sec:refinement}
@@ -267,7 +295,9 @@ For the next subsections, we consider $q$ to be fixed with one of these values,
 and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
 
 
-\subsection{
+\subsection{Numerical Inequalities}
+
+\subsubsection{
 	\texorpdfstring{
 		$\Delta(E) + \Delta(G) \leq \Delta(F)$
 	}{
@@ -390,7 +420,7 @@ In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
 $\chern^{\beta}_2(F) = 0$,
 so some of these expressions simplify.
 
-\subsection{
+\subsubsection{
 	\texorpdfstring{
 		$\Delta(E) \geq 0$
 	}{
@@ -480,7 +510,7 @@ Notice that for $\beta = \beta_{-}$ (or $\beta_{+}$), that is when
 $\chern^{\beta}_2(F)=0$, the constant and linear terms match up with the ones
 for the bound found for $d$ in subsection \ref{subsect-d-bound-bgmlv1}.
 
-\subsection{
+\subsubsection{
 	\texorpdfstring{
 		$\Delta(G) \geq 0$
 	}{
@@ -642,14 +672,11 @@ $\chern^{\beta}_2(F) = 0$,
 so some of these expressions simplify, and in particular, the constant and
 linear terms match those of the other bounds in the previous subsections.
 
-\subsection{Bounds on \texorpdfstring{$r$}{r}}
-
-Now, the inequalities from the last three subsections will be used to find, for
-each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
-no possible solutions for $d$. At that point, there are no Chern characters
-$(r,c,d)$ that satisfy all inequalities to give a pseudowall.
+\subsubsection{All Bounds on $d$ together}
+%% RECAP ON INEQUALITIES TOGETHER
 
-\subsubsection{All circular pseudowalls left of vertical wall}
+%%%% RATIONAL BETA MINUS
+\minorheading{Special Case: Rational $\beta_{-}$}
 
 Suppose we take $\beta = \beta_{-}$ (so that $\chern^{\beta}_2(F)=0$)
 in the previous subsections, to find all circular walls to the left of the
@@ -877,8 +904,16 @@ Some of the details around the associated numerics are explored next.
 \label{fig:d_bounds_xmpl_gnrc_q}
 \end{figure}
 
+\subsection{Bounds on Semistabilizer Rank \texorpdfstring{$r$}{r}}
+
+Now, the inequalities from the above (TODO REF) will be used to find, for
+each given $q=\chern^{\beta}_1(E)$, how large $r$ needs to be in order to leave
+no possible solutions for $d$. At that point, there are no Chern characters
+$(r,c,d)$ that satisfy all inequalities to give a pseudowall.
+
+
+\subsubsection{All Semistabilizers Left of Vertical Wall for Rational Beta min}
 
-\minorheading{Rational $\beta\not=0$}
 
 The strategy here is similar to what was shown in (sect \ref{sec:twisted-chern}),
 % ref to Schmidt?
@@ -1135,7 +1170,7 @@ proof of theorem \ref{thm:rmax_with_uniform_eps}:
 
 	\begin{equation*}
 		d - \frac{(\aa r + 2\bb)\aa}{2n^2}
-		\geq \epsilon_{q,2} \geq \epsilon_{q,1}  0
+		\geq \epsilon_{q,2} \geq \epsilon_{q,1} > 0
 \end{equation*}
 
 Where $\epsilon_{q,1}$ and $\epsilon_{q,2}$ are defined as follows:
@@ -1252,7 +1287,26 @@ eps_k_i_subs = nu == (2*m*n^2)/delta
 
 \egroup % end scope where beta redefined to beta_{-}
 
-\section{Conclusion}
+\subsubsection{All Semistabilizers Giving Sufficiently Large Circular Walls Left
+of Vertical Wall}
+
+
+Goals:
+\begin{itemize}
+	\item refresher on strategy
+	\item point out no need for rational beta
+	\item calculate intersection of bounds?
+\end{itemize}
+
+\subsection{Irrational $\beta_{-}$}
+
+Goals:
+\begin{itemize}
+	\item Point out if only looking for sufficiently large wall, look at above
+		subsubsection
+	\item Relate to Pell's equation through coordinate change?
+	\item Relate to numerical condition described by Yanagida/Yoshioka
+\end{itemize}
 
 \newpage
 \section{Appendix - SageMath code}