From 4ae777b711bd93e196636c8fb0bd692889d43e89 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 10 May 2023 12:31:48 +0100
Subject: [PATCH] Redefine beta to beta_{-} in 6.4.1
---
main.tex | 10 ++++++++--
1 file changed, 8 insertions(+), 2 deletions(-)
diff --git a/main.tex b/main.tex
index 259a22b..c3f8668 100644
--- a/main.tex
+++ b/main.tex
@@ -554,6 +554,12 @@ 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
vertical wall (TODO as discussed in ref).
+% redefine \beta (especially coming from rendered SageMath expressions)
+% to be \beta_{-} for the rest of this subsubsection
+\bgroup
+\let\originalbeta\beta
+\renewcommand\beta{{\originalbeta_{-}}}
+
\begin{align}
d &\geq&
\sage{bgmlv1_d_lowerbound_linear_term}
@@ -573,7 +579,7 @@ vertical wall (TODO as discussed in ref).
Furthermore, we get an extra bound for $d$ resulting from the condition that the
radius of the circular wall must be positive. As discussed in (TODO ref), this
-is equivalent to $\chern^{\beta_{-}}_2(E) > 0$, which yields:
+is equivalent to $\chern^{\beta}_2(E) > 0$, which yields:
\begin{sagesilent}
positive_radius_condition = (
@@ -589,7 +595,7 @@ is equivalent to $\chern^{\beta_{-}}_2(E) > 0$, which yields:
\begin{equation*}
\sage{positive_radius_condition}
\end{equation*}
-
+\egroup
\section{Conclusion}
--
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