From 91448bd796d359e3bacd2286bf1226daac829a3f Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue, 13 Aug 2024 15:04:25 +0100
Subject: [PATCH] Capitalisations
---
tex/bounds-on-semistabilisers.tex | 2 +-
tex/characteristic-curves.tex | 6 +++---
tex/computing-solutions.tex | 4 ++--
3 files changed, 6 insertions(+), 6 deletions(-)
diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index 06fd0f6..3957171 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -645,7 +645,7 @@ idea will be pursued in Subsection \ref{subsec:bounds-on-semistab-rank-prob-1}.
\ref{problem:problem-statement-1}}
\label{subsec:bounds-on-semistab-rank-prob-1}
-As discussed at the end of subsection \ref{subsubsect:all-bounds-on-d-prob1}
+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}),
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
diff --git a/tex/characteristic-curves.tex b/tex/characteristic-curves.tex
index 2da28b8..e1511bc 100644
--- a/tex/characteristic-curves.tex
+++ b/tex/characteristic-curves.tex
@@ -23,7 +23,7 @@ define two characteristic curves on the $(\alpha, \beta)$-plane:
These characteristic curves for a Chern character $v$ with $\Delta(v)\geq0$ are
not affected by flipping the sign of $v$ so it's only necessary to consider
non-negative rank.
-As discussed in subsection \ref{subsect:relevance-of-V_v}, making this choice
+As discussed in Subsection \ref{subsect:relevance-of-V_v}, making this choice
has Gieseker stable coherent sheaves appearing in the heart of the stability
condition $\firsttilt{\beta}$ as we move `left' (decreasing $\beta$).
@@ -91,7 +91,7 @@ degenerate_characteristic_curves
In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
equation $\chern_2^{\beta}(v)=0$.
- This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
+ This definition will be extended to the rank 0 case in Definition \ref{dfn:beta_-_rank0}.
\end{definition}
@@ -196,7 +196,7 @@ the pseudo-wall where $u$ pseudo-semistabilizes $v$. This is done by finding the
intersection of $\Theta_v$ and $\Theta_u$, the point $(\beta,\alpha)$ where
$\nu_{\alpha,\beta}(u)=\nu_{\alpha,\beta}(v)=0$, and a pseudo-wall point on
$\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0)$
-(as per subsection \ref{subsect:bertrams-nested-walls}).
+(as per Subsection \ref{subsect:bertrams-nested-walls}).
\subsection{Bertram's Nested Wall Theorem}
diff --git a/tex/computing-solutions.tex b/tex/computing-solutions.tex
index 0efaf39..f28d9a3 100644
--- a/tex/computing-solutions.tex
+++ b/tex/computing-solutions.tex
@@ -21,7 +21,7 @@ Recalling Consequence 2 of Lemma \ref{lem:pseudo_wall_numerical_tests}, we can
iterate through the possible values of $\mu(u)=\frac{c}{r}$ taking a decreasing
sequence of all fractions between $\mu(v)$ and $\beta_{-}$, whos denominators
are no large than $r_{\mathrm{max}}$ (giving a finite sequence). This can be done with
-Farey sequences \cite[chapter 6]{alma994504533502466}, for which there exist
+Farey sequences \cite[Chapter 6]{alma994504533502466}, for which there exist
formulae to generate.
These $\mu(u)$ values determine pairs $r,c$ up to multiples, we can then take
@@ -248,7 +248,7 @@ producing the results marginally faster.
Note that this program patched with Theorem \ref{thm:loose-bound-on-r} will be
using the same bound as was used in the previously existing program
\cite{SchmidtGithub2020}. However the difference of performance can be of
-several orders of magnitude as illustrated in the table in section
+several orders of magnitude as illustrated in the table in Section
\ref{table:bench-schmidt-vs-nay}.
This will be attributed to the difference in programming language and algorithm,
the latter having already been discussed in that same section.
--
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