diff --git a/tex/bounds-on-semistabilisers.tex b/tex/bounds-on-semistabilisers.tex
index b42370220ba3d9a3b9903db32cb379d2482ab5eb..15560e6d11adf980acbf3e57c7281f379230ec35 100644
--- a/tex/bounds-on-semistabilisers.tex
+++ b/tex/bounds-on-semistabilisers.tex
@@ -96,7 +96,7 @@ bound for the rank of $u$:
Which we can then immediately bound using Equation \ref{eqn-tilt-cat-cond}.
Alternatively, given that
$\chern_1^{\beta_{-}}(u)$, $\chern_1^{\beta_{-}}(v)\in\frac{1}{n}\ZZ$,
-we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that Equation to:
+we can tighten this bound on $\chern_1^{\beta_{-}}(u)$ given by that equation to:
\[
n\chern^{\beta_{-}}_1(u) \leq n\chern^{\beta_{-}}_1(v) - 1
\]
@@ -124,7 +124,7 @@ Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
tilt semistabilisers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
However, when computing all tilt semistabilisers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be 25. This will be a recurring example to
-illustrate the performance of later Theorems about rank bounds
+illustrate the performance of later theorems about rank bounds
\end{example}
\begin{sagesilent}
@@ -227,7 +227,7 @@ of travel.
\item $\chern^P_2(u) \geq 0$
\end{itemize}
\end{multicols}
-
+
\end{corollary}
\begin{proof}
@@ -422,7 +422,7 @@ see-saw principle.
&>
2 \frac{\chern^{\beta_0}_2(v)}{R}
\\
- \chern_2^{\beta_0}(v)
+ \chern_2^{\beta_0}(v)
- \frac{
\left(
q-\chern^{\beta_0}_1(v)
@@ -565,7 +565,7 @@ bounds do not share the same assymptote as the lower bound
Bounds on $d\coloneqq\chern_2(u)$ in terms of $r\coloneqq \chern_0(u)$ for a fixed
value $\chern_1^{\beta}(v)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
Where $\chern(v) = (3,2\ell,-2\ell^2)$ and $P$ chosen as the point on $\Theta_v$
- with $\beta(P)\coloneqq-2/3-1/99$ in the context of problem
+ with $\beta(P)\coloneqq-2/3-1/99$ in the context of problem
\ref{problem:problem-statement-1}.
}
\label{fig:problem1:d_bounds_xmpl_gnrc_q}
@@ -644,7 +644,7 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
\noindent
Therefore, $r$ is bounded above by the minimum of these two expressions which
can then be factored into the expression given in the Lemma.
-
+
\end{proof}
The above Lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
@@ -670,7 +670,7 @@ following Lemma \ref{lem:prob1:convenient_r_bound}.
$q_{\mathrm{max}}=\sage{problem1.maximising_q}$.
Substituting $q=q_{\mathrm{max}}$ into the bound in Lemma
\ref{lem:prob1:r_bound} gives the bound as stated in the current Lemma.
-
+
\end{proof}
\begin{remark}
@@ -973,7 +973,7 @@ proof of Theorem \ref{thm:rmax_with_uniform_eps}:
\frac{mn\aa}{\gcd(m,2n^2)}
\right)}
\end{equation*}
-
+
\end{lemmadfn}
\vspace{10pt}
@@ -1076,7 +1076,7 @@ from plots_and_expressions import main_theorem2
$\epsilon_{v,q} = \frac{k_{v,q}}{\lcm(m, 2n^2)}$ can be used instead of
$\epsilon_{v} = \frac{1}{\lcm(m, 2n^2)}$ as it satisfies the same required
property, as per Definition/Lemma \ref{lemdfn:epsilon_q}.
-
+
\end{proof}
Although the general form of this bound is quite complicated, it does simplify a
@@ -1218,7 +1218,7 @@ end}
However the reduction in the overall bound on $r$ is not as drastic, since all
possible values for $k_{v,q}$ in $\{1,2,\ldots,\sage{n}\}$ are iterated through
cyclically as we consider successive possible values for $q$.
-And for each $q$ where $k_{v,q}=1$, both Theorems give the same bound.
+And for each $q$ where $k_{v,q}=1$, both theorems give the same bound.
Calculating the maximums over all values of $q$ yields
$\sage{max(theorem2_bounds)}$ for Theorem \ref{thm:rmax_with_uniform_eps}, and
$\sage{max(theorem3_bounds)}$ for Theorem \ref{thm:rmax_with_eps1}.
diff --git a/tex/characteristic-curves.tex b/tex/characteristic-curves.tex
index a9d643196cbecf44d86ff72fd0d5493793188bd5..2da28b8fb3b6533fbffa38b3a4b166f1a7c8eb64 100644
--- a/tex/characteristic-curves.tex
+++ b/tex/characteristic-curves.tex
@@ -6,7 +6,7 @@ Considering the stability conditions with two parameters $\alpha, \beta$ on
Picard rank 1 surfaces.
We can draw 2 characteristic curves for any given Chern character $v$ with
$\Delta(v) \geq 0$ and positive rank.
-These are given by the Equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
+These are given by the equations $\chern_i^{\alpha,\beta}(v)=0$ for $i=1,2$.
\begin{definition}[Characteristic Curves $V_v$ and $\Theta_v$]
Given a Chern character $v$, with positive rank and $\Delta(v) \geq 0$, we
@@ -89,7 +89,7 @@ degenerate_characteristic_curves
\]
\noindent
In particular, this means $\beta_\pm(v)$ are the two roots of the quadratic
- Equation $\chern_2^{\beta}(v)=0$.
+ equation $\chern_2^{\beta}(v)=0$.
This definition will be extended to the rank 0 case in definition \ref{dfn:beta_-_rank0}.
\end{definition}
@@ -211,4 +211,3 @@ $\frac{\delta}{\delta\beta} \chern_2^{\alpha,\beta} = -\chern_1^{\alpha,\beta}$.
This fact, along with the hindsight knowledge that non-vertical walls are
circles with centers on the $\beta$-axis, gives an alternative view to see that
the circular walls must be nested and non-intersecting.
-
diff --git a/tex/computing-solutions.tex b/tex/computing-solutions.tex
index 9c59dc5160e2563952934266deb9603c2d1f8f92..023901104e848155da2f922c3cb5a476d1d243ed 100644
--- a/tex/computing-solutions.tex
+++ b/tex/computing-solutions.tex
@@ -84,7 +84,7 @@ alternative algorithm which will later be described in Section
Alongside this thesis, there is a library \cite{NaylorRust2023}
to compute the solutions to Problem \ref{problem:problem-statement-2},
-using the Theorems above.
+using the theorems above.
The source code is also shown in Appendix \ref{appendix:tilt-rs}, but is better
viewed digitally from source, or via the documentation \cite{naylorPseudo_tiltRust2024}
@@ -241,8 +241,8 @@ This could be due to a range of potential reasons:
For relatively small Chern characters (as those appearing in examples so far),
the difference in performance between the program \cite{NaylorRust2023} when
-patched with the results of the different Theorems above, do not show any
-significant difference in performance. The earlier, weaker Theorems occasionally
+patched with the results of the different theorems above, do not show any
+significant difference in performance. The earlier, weaker theorems occasionally
producing the results marginally faster.
Note that this program patched with Theorem \ref{thm:loose-bound-on-r} will be
diff --git a/tex/references.bib b/tex/references.bib
index 147e8b6ee62675745c53ca4d3c0247bdad359968..9ddc227f96f58ff9b0ccbe1c53ce42d9681de85f 100644
--- a/tex/references.bib
+++ b/tex/references.bib
@@ -3,7 +3,8 @@
author = {Maciocia, Antony},
date = {2014-03-31},
copyright = {info:eu-repo/semantics/openAccess},
- langid = {english}
+ langid = {english},
+ file = {/home/maths-lap-246/snap/zotero-snap/common/Zotero/storage/AUMM74LK/Maciocia - 2014 - Computing the walls associated to bridgeland stabi.pdf}
}
@incollection{alma994504533502466,
@@ -121,26 +122,6 @@
organization = {GitLab}
}
-@software{naylorLnayPseudo_tilt_py2023,
- title = {Lnay/Pseudo\_tilt\_py},
- author = {Naylor, Luke},
- date = {2023-05-24T12:42:56Z},
- origdate = {2023-05-24T12:42:03Z},
- url = {https://github.com/lnay/pseudo_tilt_py},
- urldate = {2024-05-11},
- annotation = {Programmers: \_:n121}
-}
-
-@online{naylorPseudo_tiltRust2024,
- title = {Pseudo\_tilt - {{Rust}}},
- author = {Naylor, Luke},
- date = {2024-07-18},
- url = {https://pseudowalls.gitlab.io/tilt.rs/pseudo_tilt/},
- urldate = {2024-07-18},
- abstract = {Documentation page for pseudo\_tilt Rust crate},
- file = {/home/luke/Zotero/storage/N4NYVCH4/pseudo_tilt.html}
-}
-
@manual{sagemath,
type = {manual},
title = {{{SageMath}}, the {{Sage Mathematics Software System}} ({{Version}} 9.6.0)},
diff --git a/tex/setting-and-problems.tex b/tex/setting-and-problems.tex
index f69c8007e10557d6025548d259ef79162322553c..d1bbeeccbcec3e2a2ec3f9a6e9a6261f4896dc78 100644
--- a/tex/setting-and-problems.tex
+++ b/tex/setting-and-problems.tex
@@ -11,7 +11,7 @@ affect the results.
% NOTE: SURFACE SPECIALIZATION
Given a Chern Character $v$, and a given stability
condition $\sigma_{\alpha,\beta}$,
- a \textit{pseudo-semistabilising} $u$ is a `potential' Chern character:
+ a \emph{pseudo-semistabilising} $u$ is a `potential' Chern character:
\[
u = \left(r, c\ell, \frac{e}{\lcm(m,2)} \ell^2\right)
\qquad
@@ -39,7 +39,7 @@ affect the results.
be considered but are left out for now as they do not have a great impact on
the finiteness of pseudo-walls.
In the case of a principally polarised abelian surface, the main example in
- this Thesis, the Euler characteristic condition is vacuous and the extension
+ this thesis, the Euler characteristic condition is vacuous and the extension
group condition eliminates possibities with lower rank, and often none at all
for small values of $\chern_0(v)$.
\end{remark}
@@ -52,11 +52,11 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
\begin{definition}[Pseudo-walls]
\label{dfn:pseudo-wall}
Let $u$ be a pseudo-semistabiliser of $v$, for some stability condition.
- Then the \textit{pseudo-wall} associated to $u$ is the set of all stablity
+ Then the \emph{pseudo-wall} associated to $u$ is the set of all stablity
conditions where $u$ is a pseudo-semistabiliser of $v$.
\end{definition}
-% TODO possibly reference forwards to Bertram's nested wall Theorem section to
+% TODO possibly reference forwards to Bertram's nested wall Theorem section to
% cover that being a pseudo-semistabiliser somewhere implies also on whole circle
\begin{lemma}[Sanity check for Pseudo-semistabilisers]
@@ -92,7 +92,7 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
This fact along with $c_0$, $c_2$ being an integers on surfaces, and
$m\coloneqq \ell^2$ implies that $\chern(H)$
(hence $\chern(E)$ too) is of the required form.
-
+
Since all the objects in the sequence are in $\firsttilt\beta$, we have
$\chern_1^{\beta} \geq 0$ for each of them. Due to additivity
diff --git a/tilt.rs b/tilt.rs
index 6689d9a00a4378a45842340f49533970ade268ef..76b62a20e9adf55bf0fed229c9c7b1972683b615 160000
--- a/tilt.rs
+++ b/tilt.rs
@@ -1 +1 @@
-Subproject commit 6689d9a00a4378a45842340f49533970ade268ef
+Subproject commit 76b62a20e9adf55bf0fed229c9c7b1972683b615