From e4cf3a424a9439f41962f4527886348acab76612 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 10 May 2023 00:31:46 +0100
Subject: [PATCH] Correct bgmlv2 'lowerbounds' to 'upperbounds'
---
main.tex | 48 ++++++++++++++++++++++++++++++------------------
1 file changed, 30 insertions(+), 18 deletions(-)
diff --git a/main.tex b/main.tex
index 893d31f..0da3ee1 100644
--- a/main.tex
+++ b/main.tex
@@ -360,45 +360,45 @@ bgmlv2_d_ineq = (
).expand()
# Keep hold of lower bound for d
-bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
+bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()
\end{sagesilent}
\begin{equation}
- \label{eqn-bgmlv2_d_lowerbound}
+ \label{eqn-bgmlv2_d_upperbound}
\sage{bgmlv2_d_ineq}
\end{equation}
\begin{sagesilent}
# Seperate out the terms of the lower bound for d
-bgmlv2_d_lowerbound_without_hyp = (
- bgmlv2_d_lowerbound
+bgmlv2_d_upperbound_without_hyp = (
+ bgmlv2_d_upperbound
.subs(1/r == 0)
)
-bgmlv2_d_lowerbound_const_term = (
- bgmlv2_d_lowerbound_without_hyp
+bgmlv2_d_upperbound_const_term = (
+ bgmlv2_d_upperbound_without_hyp
.subs(r==0)
)
-bgmlv2_d_lowerbound_linear_term = (
- bgmlv2_d_lowerbound_without_hyp
- - bgmlv2_d_lowerbound_const_term
+bgmlv2_d_upperbound_linear_term = (
+ bgmlv2_d_upperbound_without_hyp
+ - bgmlv2_d_upperbound_const_term
).expand()
-bgmlv2_d_lowerbound_exp_term = (
- bgmlv2_d_lowerbound
- - bgmlv2_d_lowerbound_without_hyp
+bgmlv2_d_upperbound_exp_term = (
+ bgmlv2_d_upperbound
+ - bgmlv2_d_upperbound_without_hyp
).expand()
\end{sagesilent}
-Viewing equation \ref{eqn-bgmlv2_d_lowerbound} as a lower bound for $d$ in term
+Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
of $r$ again, there's a constant term
-$\sage{bgmlv2_d_lowerbound_const_term}$,
+$\sage{bgmlv2_d_upperbound_const_term}$,
a linear term
-$\sage{bgmlv2_d_lowerbound_linear_term}$,
+$\sage{bgmlv2_d_upperbound_linear_term}$,
and a hyperbolic term
-$\sage{bgmlv2_d_lowerbound_exp_term}$.
+$\sage{bgmlv2_d_upperbound_exp_term}$.
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}.
@@ -554,10 +554,22 @@ 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).
\begin{equation*}
- \sage{ bgmlv3_d_upperbound_const_term }
+ 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*}
- \sage{bgmlv3_d_upperbound_const_term_alt1.subs(chbv == 0)}
+ 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{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*}
--
GitLab