From 69e352c66a97f6e1d7a3d57ef6e1d596d146684c Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 11 Jun 2024 14:44:50 +0100
Subject: [PATCH] Capitalise Figure
---
tex/content.tex | 18 +++++++++---------
1 file changed, 9 insertions(+), 9 deletions(-)
diff --git a/tex/content.tex b/tex/content.tex
index f16d26e..eed348e 100644
--- a/tex/content.tex
+++ b/tex/content.tex
@@ -1168,7 +1168,7 @@ bounds_on_d_qmax
Recalling that $q \coloneqq \chern^{\beta}_1(E) \in [0, \chern^{\beta}_1(F)]$,
it is worth noting that the extreme values of $q$ in this range lead to the
-tightest bounds on $d$, as illustrated in figure
+tightest bounds on $d$, as illustrated in Figure
(\ref{fig:d_bounds_xmpl_extrm_q}).
In fact, in each case, one of the weak upper bounds coincides with one of the
weak lower bounds, (implying no possible destabilizers $E$ with
@@ -1182,7 +1182,7 @@ In the other case, $q=\chern^{\beta}_1(F)$, it is the right hand sides of
The more generic case, when $0 < q\coloneqq\chern_1^{\beta}(E) < \chern_1^{\beta}(F)$
-for the bounds on $d$ in terms of $r$ is illustrated in figure
+for the bounds on $d$ in terms of $r$ is illustrated in Figure
(\ref{fig:d_bounds_xmpl_gnrc_q}).
The question of whether there are pseudo-destabilizers of arbitrarily large
rank, in the context of the graph, comes down to whether there are points
@@ -1266,7 +1266,7 @@ This means that the lower bound for $d$ will be large than either of the two
upper bounds for sufficiently large $r$, and hence those values of $r$ would yield no
solution to problem \ref{problem:problem-statement-1}.
-A generic example of this is plotted in figure
+A generic example of this is plotted in Figure
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}.
\begin{figure}
@@ -1286,13 +1286,13 @@ A generic example of this is plotted in figure
\ref{problem:problem-statement-1}}
As discussed at the end of subsection \ref{subsubsect:all-bounds-on-d-prob1}
-(and illustrated in figure \ref{fig:problem1:d_bounds_xmpl_gnrc_q}),
+(and illustrated in Figure \ref{fig:problem1:d_bounds_xmpl_gnrc_q}),
there are no solutions $u$ to problem \ref{problem:problem-statement-1}
with large $r=\chern_0(u)$, since the lower bound on $d=\chern_2(u)$ is larger
than the upper bounds.
Therefore, we can calculate upper bounds on $r$ by calculating for which values,
the lower bound on $d$ is equal to one of the upper bounds on $d$
-(i.e. finding certain intersection points of the graph in figure
+(i.e. finding certain intersection points of the graph in Figure
\ref{fig:problem1:d_bounds_xmpl_gnrc_q}).
\begin{lemma}[Problem \ref{problem:problem-statement-1} upper Bound on $r$]
@@ -1907,7 +1907,7 @@ $\beta_{-}(v)$ is rational.
Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
Iterate over $q = \frac{b}{n} \in [0,\chern_1^{\beta_{-}}(v)]\cap\frac{1}{n}\ZZ$.
-The code used to generate the corresponding values for $b$ is shown in figure
+The code used to generate the corresponding values for $b$ is shown in Figure
\ref{fig:code:consideredb}.
\lstinputlisting[
@@ -1918,7 +1918,7 @@ The code used to generate the corresponding values for $b$ is shown in figure
We can therefore reduce the problem of finding solutions to the problem to
finding the solutions $u$ with each fixed possible $\chern_1^\beta(u)$
(i.e. choice of $b$).
-The code representing this appears in figure
+The code representing this appears in Figure
\ref{fig:code:reducingtoeachb}
For any $u = (r,c\ell,\frac{e}{2}\ell^2)$, satisfying
@@ -2053,7 +2053,7 @@ indicators to the size of the bounds on the pseudo-semistabiliser ranks.
\label{fig:benchmark}
\end{figure}
-As shown in figure \ref{fig:benchmark}, there can be a significant improvement
+As shown in Figure \ref{fig:benchmark}, there can be a significant improvement
by using Theorems \ref{thm:rmax_with_uniform_eps} \ref{thm:rmax_with_eps1}
which specialise to different values of $\chern_1^{\beta_{-}(v)}(u)$
of solutions $u$ of problem \ref{problem:problem-statement-2}.
@@ -2074,7 +2074,7 @@ average looser times the mean average of $1/{k_{v,q}}$:
This certainly tends to 0 for large $n$. But in the current example, with
$n=15$, this gives us approximately 0.2 for the ratio of the average tighter bound
versus the average looser.
-However, the actual ratio in the benchmark shown in figure \ref{fig:benchmark}
+However, the actual ratio in the benchmark shown in Figure \ref{fig:benchmark}
between the two instances of the program patched with the two corresponding
bounds is around 0.6 instead.
Not as good as the improvement on the bound, however still not an insignificant
--
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