From d7354a803d8e0f9f0ac8ef4673a597a2326f4f1b Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 29 May 2023 16:34:11 +0100
Subject: [PATCH] Update formula for bgmlv3 to nicer alt form 1
---
main.tex | 9 +++++++--
1 file changed, 7 insertions(+), 2 deletions(-)
diff --git a/main.tex b/main.tex
index 0847d0d..e282d78 100644
--- a/main.tex
+++ b/main.tex
@@ -661,6 +661,9 @@ vertical wall (TODO as discussed in ref).
\let\originalbeta\beta
\renewcommand\beta{{\originalbeta_{-}}}
+\bgroup
+% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
+\def\psi{\chern_1^{\beta}(F)}
\begin{align}
d &\geq&
\sage{bgmlv1_d_lowerbound_linear_term}
@@ -678,11 +681,13 @@ vertical wall (TODO as discussed in ref).
\\
d &\leq&
\sage{bgmlv3_d_upperbound_linear_term}
- &+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
- +& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)},
+ &+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)}
+ % ^ ch_2^\beta(F)=0 for beta_{-}
+ +& \sage{bgmlv3_d_upperbound_exp_term_alt2},
&\qquad\text{when\:} r > R
\label{eqn:bgmlv3_d_bound_betamin}
\end{align}
+\egroup
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
--
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