From 5b04798231121045b39c696ab7144c6b9580f9a9 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 2 Jul 2024 15:35:31 +0100
Subject: [PATCH] Make numerical conditions for rpoblem solutions slightly
stricter
---
tex/setting-and-problems.tex | 7 ++++---
1 file changed, 4 insertions(+), 3 deletions(-)
diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index 073d467..ffa24c7 100644
--- a/tex/setting-and-problems.tex
+++ b/tex/setting-and-problems.tex
@@ -156,12 +156,13 @@ are equivalent to the following more numerical conditions:
\item $u$ has positive rank
\item $\beta(P)<\mu(u)<\mu(v)$, i.e. $V_u$ is strictly between $P$ and $V_v$.
\label{lem:ps-wall-num-test:num-cond-slope}
- \item $\chern_1^{\beta(P)}(u) \leq \chern_1^{\beta(P)}(v)$
+ \item $\chern_1^{\beta(P)}(u) < \chern_1^{\beta(P)}(v)$
\item $\Delta(v-u) \geq 0$
\item $\chern_2^{P}(u)>0$
\end{enumerate}
\end{lemma}
+\noindent\textbf{\color{red} TODO ADJUST TO SLIGHTLY STRONGER CONDITION 3}
\begin{proof}[Proof for $\chern_0(v)>0$ case.]
Let $u,v$ be Chern characters with
$\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
@@ -430,7 +431,7 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
\item $\Delta(v-u) \geq 0$
\label{item:bgmlvv-u:lem:num_test_prob1}
\item $\mu(u)<\mu(v)$
- \item $0\leq\chern_1^{\beta(P)}(u)\leq\chern_1^{\beta(P)}(v)$
+ \item $0<\chern_1^{\beta(P)}(u)<\chern_1^{\beta(P)}(v)$
\label{item:chern1bound:lem:num_test_prob1}
\item $\chern_2^{P}(u)>0$
\label{item:radiuscond:lem:num_test_prob1}
@@ -472,7 +473,7 @@ problem using Lemma \ref{lem:pseudo_wall_numerical_tests}.
\label{item:bgmlvv-u:lem:num_test_prob2}
\item $\mu(u)<\mu(v)$
\label{item:mubound:lem:num_test_prob2}
- \item $0\leq\chern_1^{\beta_{-}}(u)\leq\chern_1^{\beta_{-}}(v)$
+ \item $0<\chern_1^{\beta_{-}}(u)<\chern_1^{\beta_{-}}(v)$
\label{item:chern1bound:lem:num_test_prob2}
\item $\chern_2^{\beta_{-}}(u)>0$
\label{item:radiuscond:lem:num_test_prob2}
--
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