From 9a91722ea07b587d183dfe06ef3ebf38184ac1f9 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Mon, 24 Jul 2023 17:02:19 +0100
Subject: [PATCH] Correct corrolary to corollary
---
main.tex | 28 ++++++++++++++--------------
1 file changed, 14 insertions(+), 14 deletions(-)
diff --git a/main.tex b/main.tex
index 52e2334..14e8b49 100644
--- a/main.tex
+++ b/main.tex
@@ -33,7 +33,7 @@ sorting=ynt
\newcommand{\minorheading}[1]{{\noindent\normalfont\normalsize\bfseries #1}}
\newtheorem{theorem}{Theorem}[section]
-\newtheorem{corrolary}{Corrolary}[section]
+\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{fact}{Fact}[section]
\newtheorem{example}{Example}[section]
@@ -730,7 +730,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\end{proof}
-\begin{corrolary}[Numerical Tests for All `left' Pseudo-walls]
+\begin{corollary}[Numerical Tests for All `left' Pseudo-walls]
\label{cor:num_test_prob2}
Given a Chern character $v$ with positive rank and $\Delta(v) \geq 0$,
such that $\beta_{-}\coloneqq\beta_{-}(v) \in \QQ$.
@@ -750,7 +750,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\item $\chern_2^{\beta_{-}}(u)>0$
\label{item:radiuscond:lem:num_test_prob2}
\end{enumerate}
-\end{corrolary}
+\end{corollary}
\begin{proof}
This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.
@@ -996,7 +996,7 @@ u = Chern_Char(*var("r c d", domain="real"))
Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
lemma \ref{lem:num_test_prob1}
-(or corrolary \ref{cor:num_test_prob2})
+(or corollary \ref{cor:num_test_prob2})
that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
and so we can write:
@@ -1030,7 +1030,7 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
This section studies the numerical conditions that $u$ must satisfy as per
lemma \ref{lem:num_test_prob1}
-(or corrolary \ref{cor:num_test_prob2})
+(or corollary \ref{cor:num_test_prob2})
\subsubsection{Size of pseudo-wall: $\chern_2^P(u)>0$ }
\label{subsect-d-bound-radiuscond}
@@ -1038,7 +1038,7 @@ lemma \ref{lem:num_test_prob1}
This condition refers to condition
\ref{item:radiuscond:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
-(or corrolary \ref{cor:num_test_prob2}).
+(or corollary \ref{cor:num_test_prob2}).
In the case where we are tackling problem \ref{problem:problem-statement-2}
(with $\beta = \beta_{-}$), this condition, when expressed as a bound on $d$,
@@ -1061,7 +1061,7 @@ amounts to:
This condition refers to condition
\ref{item:bgmlvu:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
-(or corrolary \ref{cor:num_test_prob2}).
+(or corollary \ref{cor:num_test_prob2}).
\begin{sagesilent}
@@ -1089,7 +1089,7 @@ bgmlv2_with_q = (
\noindent
This can be rearranged to express a bound on $d$ as follows
(recall from condition \ref{item:rankpos:lem:num_test_prob1}
-in lemma \ref{lem:num_test_prob1} or corrolary
+in lemma \ref{lem:num_test_prob1} or corollary
\ref{cor:num_test_prob2} that $r>0$):
\begin{sagesilent}
@@ -1156,7 +1156,7 @@ for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
This condition refers to condition
\ref{item:bgmlvv-u:lem:num_test_prob1}
from lemma \ref{lem:num_test_prob1}
-(or corrolary \ref{cor:num_test_prob2}).
+(or corollary \ref{cor:num_test_prob2}).
Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
$d$ yields:
@@ -1297,7 +1297,7 @@ $d=\chern^{\beta_{-}}_2(u)$ induced by conditions
\ref{item:bgmlvu:lem:num_test_prob2},
\ref{item:bgmlvv-u:lem:num_test_prob2}, and
\ref{item:radiuscond:lem:num_test_prob1}
-from corrolary \ref{cor:num_test_prob2} have the same constant and linear
+from corollary \ref{cor:num_test_prob2} have the same constant and linear
terms in $r$, but different hyperbolic terms.
These give bounds with the same assymptotes when we take $r\to\infty$
(for any fixed $q=\chern_1^{\beta_{-}}(u)$).
@@ -1685,7 +1685,7 @@ r_upper_bound_all_q = (
)
\end{sagesilent}
-\begin{corrolary}[Bound on $r$ \#2]
+\begin{corollary}[Bound on $r$ \#2]
\label{cor:direct_rmax_with_uniform_eps}
Let $v$ be a fixed Chern character and
$R\coloneqq\chern_0(v) \leq n^2\Delta(v)$.
@@ -1700,7 +1700,7 @@ r_upper_bound_all_q = (
\sage{r_upper_bound_all_q.expand()}
\end{equation*}
\egroup
-\end{corrolary}
+\end{corollary}
\begin{proof}
\bgroup
@@ -1749,7 +1749,7 @@ corrolary_bound = (
.subs(n==recurring.n)
)
\end{sagesilent}
-Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
which is much closer to real maximum 25 than the original bound 144.
@@ -1772,7 +1772,7 @@ corrolary_bound = (
.subs(n==extravagant.n)
)
\end{sagesilent}
-Using the above corrolary \ref{cor:direct_rmax_with_uniform_eps}, we get that
+Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
--
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