From 8bf8cabcdf62a60d7f5fc92469af1d8fac58abd8 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 10 May 2023 00:34:13 +0100
Subject: [PATCH] Add simplified bounds for d at beta_{-}
---
main.tex | 35 +++++++++++++++++------------------
1 file changed, 17 insertions(+), 18 deletions(-)
diff --git a/main.tex b/main.tex
index 0da3ee1..5120dc1 100644
--- a/main.tex
+++ b/main.tex
@@ -550,27 +550,26 @@ $(r,c,d)$ that satisfy all inequalities to give a pseudowall.
\subsubsection{All circular pseudowalls left of vertical wall}
-Suppose we take $\beta = \beta_{-}$ in the previous subsections, to find all
-circular walls to the left of the vertical wall (TODO as discussed in ref).
+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).
-\begin{equation*}
- d \geq
+\begin{align}
+ d &\geq&
\sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
- + \sage{bgmlv1_d_lowerbound_linear_term}
- + \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
-\end{equation*}
-\begin{equation*}
- d \geq
- \sage{bgmlv2_d_lowerbound_const_term}
- + \sage{bgmlv2_d_lowerbound_linear_term}
- + \sage{bgmlv2_d_lowerbound_exp_term}
-\end{equation*}
-\begin{equation*}
- d \leq
+ &+ \sage{bgmlv1_d_lowerbound_linear_term}
+ &+ \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
+\\
+ d &\leq&
+ \sage{bgmlv2_d_upperbound_const_term}
+ &+ \sage{bgmlv2_d_upperbound_linear_term}
+ &+ \sage{bgmlv2_d_upperbound_exp_term}
+\\
+ d &\leq&
\sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
- + \sage{bgmlv3_d_upperbound_linear_term}
- + \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
-\end{equation*}
+ &+ \sage{bgmlv3_d_upperbound_linear_term}
+ &+ \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
+\end{align}
\section{Conclusion}
--
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