Capitalise Theorem/Proposition/Lemma

tex/content.tex

@@ -148,7 +148,7 @@ $d \in \frac{1}{\lcm(m,2)}\ZZ$.
 	conditions where $u$ is a pseudo-semistabilizer 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-semistabilizer somewhere implies also on whole circle
 
 \begin{lemma}[Sanity check for Pseudo-semistabilizers]
@@ -413,7 +413,7 @@ $\Theta_v$, and hence the apex of the circular pseudo-wall with centre $(\beta,0
 \subsection{Bertram's Nested Wall Theorem}
 \label{subsect:bertrams-nested-walls}
 
-Although Bertram's nested wall theorem can be proved more directly, it's also
+Although Bertram's nested wall Theorem can be proved more directly, it's also
 important for the content of this document to understand the connection with
 these characteristic curves.
 Emanuele Macri noticed in (TODO ref) that any circular wall of $v$ reaches a critical
@@ -432,9 +432,9 @@ for solutions to the problems
 tackled in this article (to be introduced later).
 In particular, problem (\ref{problem:problem-statement-1}) will be translated to
 a list of numerical inequalities on it's solutions $u$.
-% ref to appropriate lemma when it's written
+% ref to appropriate Lemma when it's written
 
-The next lemma is a key to making this translation and revolves around the
+The next Lemma is a key to making this translation and revolves around the
 geometry and configuration of the characteristic curves involved in a
 semistabilizing sequence.
 
@@ -469,7 +469,7 @@ Let $u,v$ be Chern characters with
 $\Delta(u),\Delta(v) \geq 0$, and $v$ has positive rank.
 
 
-For the forwards implication, assume that the suppositions of the lemma are
+For the forwards implication, assume that the suppositions of the Lemma are
 satisfied. Let $Q$ be the point on $\Theta_v^-$ (above $P$) where $u$ is a
 pseudo-semistabilizer of $v$.
 Firstly, consequence 3 is part of the definition for $u$ being a
@@ -629,7 +629,7 @@ Note also that the condition on $\beta_-(v)$ always holds for $v$ rank 0.
 
 The problems introduced in this section are phrased in the context of stability
 conditions. However, these can be reduced down completely to purely numerical
-problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
+problem with the help of Lemma \ref{lem:pseudo_wall_numerical_tests}.
 
 \begin{lemma}[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
 	\label{lem:num_test_prob1}
@@ -666,7 +666,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 	Lemma \ref{lem:pseudo_wall_numerical_tests} gives that the remaining
 	conditions for $u$ being a solution to problem
 	\ref{problem:problem-statement-1} are precisely equivalent to the
-	remaining conditions in this lemma.
+	remaining conditions in this Lemma.
 	% TODO maybe make this more explicit
 	% (the conditions are not exactly the same)
 
@@ -699,7 +699,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 \end{corollary}
 
 \begin{proof}
-	This is a specialization of the previous lemma, using $P=(\beta_{-},0)$.
+	This is a specialization of the previous Lemma, using $P=(\beta_{-},0)$.
 
 \end{proof}
 
@@ -709,7 +709,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
 \subsection{Bound on \texorpdfstring{$\chern_0(u)$}{ch0(u)} for Semistabilizers}
 \label{subsect:loose-bound-on-r}
 
-The proof for the following theorem \ref{thm:loose-bound-on-r} was hinted at in
+The proof for the following Theorem \ref{thm:loose-bound-on-r} was hinted at in
 \cite{SchmidtBenjamin2020Bsot}, but the value appears explicitly in
 \cite{SchmidtGithub2020}. The latter reference is a SageMath \cite{sagemath}
 library for computing certain quantities related to Bridgeland stabilities on
@@ -779,11 +779,11 @@ that $m=1$, $\beta_-=\sage{recurring.betaminus}$,
 giving $n=\sage{recurring.n}$ and
 $\chern_1^{\sage{recurring.betaminus}}(F) = \sage{recurring.twisted.ch[1]}$.
 
-Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
+Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
 tilt semistabilizers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
 However, when computing all tilt semistabilizers 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}
@@ -797,7 +797,7 @@ that $m=1$, $\beta_-=\sage{extravagant.betaminus}$,
 giving $n=\sage{extravagant.n}$ and
 $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
 
-Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
+Using the above Theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
 tilt semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
 However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
 rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
@@ -813,15 +813,15 @@ made in the presentation to concentrate on the case we are interested in:
 problem \ref{problem:problem-statement-2}, finding all pseudo-walls when $\beta_{-}\in\QQ$.
 % FUTURE add reference to section explaining new alg
 In section [ref], a different
-algorithm will be presented making use of the later theorems in this article,
+algorithm will be presented making use of the later Theorems in this article,
 with the goal of cutting down the run time.
 
 \subsubsection{Finding possible \texorpdfstring{$r$}{r} and
 \texorpdfstring{$c$}{c}}
 To do this, first calculate the upper bound $r_{max}$ on the ranks of tilt
-semistabilizers, as given by theorem \ref{thm:loose-bound-on-r}.
+semistabilizers, as given by Theorem \ref{thm:loose-bound-on-r}.
 
-Recalling consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, we can
+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 $\beta_{-}$, who's denominators
 are no large than $r_{max}$ (giving a finite sequence). This can be done with
@@ -833,9 +833,9 @@ all multiples which satisy $0<r\leq r_{max}$.
 
 We now have a finite sequence of pairs $r,c$ for which there might be a solution
 $(r,c\ell,d\ell^2)$ to our problem. In particular, any $(r,c\ell,d\ell^2)$
-satisfies consequence 2 of lemma \ref{lem:pseudo_wall_numerical_tests}, and the
+satisfies consequence 2 of Lemma \ref{lem:pseudo_wall_numerical_tests}, and the
 positive rank condition. What remains is to find the $d$ values which satisfy
-the Bogomolov inequalities and consequence 3 of lemma
+the Bogomolov inequalities and consequence 3 of Lemma
 \ref{lem:pseudo_wall_numerical_tests}
 ($\chern_2^{\beta_{-}}(u)>0$).
 
@@ -948,7 +948,7 @@ 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}
+from Lemma \ref{lem:num_test_prob1}
 (or corollary \ref{cor:num_test_prob2}).
 
 In the case where we are tackling problem \ref{problem:problem-statement-2}
@@ -992,7 +992,7 @@ Expressing this as a bound on $d$, then yields:
 }
 This condition refers to condition
 \ref{item:bgmlvu:lem:num_test_prob1}
-from lemma \ref{lem:num_test_prob1}
+from Lemma \ref{lem:num_test_prob1}
 (or corollary \ref{cor:num_test_prob2}).
 
 
@@ -1013,7 +1013,7 @@ from plots_and_expressions import 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 corollary
+in Lemma \ref{lem:num_test_prob1} or corollary
 \ref{cor:num_test_prob2} that $r>0$):
 
 
@@ -1051,7 +1051,7 @@ 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}
+from Lemma \ref{lem:num_test_prob1}
 (or corollary \ref{cor:num_test_prob2}).
 
 Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
@@ -1257,7 +1257,7 @@ and $\chern_2^B(v)$ are all strictly positive:
 \begin{itemize}
 	\item $R > 0$ from the setting of problem
 	\ref{problem:problem-statement-1}
-	\item $r > 0$ from lemma \ref{lem:num_test_prob1}
+	\item $r > 0$ from Lemma \ref{lem:num_test_prob1}
 	\item $\chern_2^B(v)>0$ because $B < \originalbeta_{-}$ due to the choice of $P$ being
 	a point on $\Theta_v^{-}$
 \end{itemize}
@@ -1322,14 +1322,14 @@ the lower bound on $d$ is equal to one of the upper bounds on $d$
 	\end{equation}
 
 	Therefore, $r$ is bounded above by the minimum of these two expressions which
-	can then be factored into the expression given in the lemma.
+	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 $q$.
+The above Lemma \ref{lem:prob1:r_bound} gives an upper bound on $r$ in terms of $q$.
 But given that $0 \leq q \leq \chern_1^{B}(v)$, we can take the maximum of this
 bound, over $q$ in this range, to get a simpler (but weaker) bound in the
-following lemma \ref{lem:prob1:convenient_r_bound}.
+following Lemma \ref{lem:prob1:convenient_r_bound}.
 
 \begin{lemma}
 \label{lem:prob1:convenient_r_bound}
@@ -1341,14 +1341,14 @@ following lemma \ref{lem:prob1:convenient_r_bound}.
 \end{lemma}
 
 \begin{proof}
-	The first term of the minimum in lemma \ref{lem:prob1:r_bound}
+	The first term of the minimum in Lemma \ref{lem:prob1:r_bound}
 	increases linearly in $q$, and the second
 	decreases linearly. So the maximum is achieved with the value of
 	$q=q_{\mathrm{max}}$ where they are equal.
 	Solving for the two terms in the minimum to be equal yields:
 	$q_{\mathrm{max}}=\sage{problem1.maximising_q}$.
-	Substituting $q=q_{\mathrm{max}}$ into the bound in lemma
-	\ref{lem:prob1:r_bound} gives the bound as stated in the current lemma.
+	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}
 
@@ -1558,7 +1558,7 @@ original bound 215296.
 
 These bound can be refined a bit more by considering restrictions from the
 possible values that $r$ take.
-Furthermore, the proof of theorem \ref{thm:rmax_with_uniform_eps} uses the fact
+Furthermore, the proof of Theorem \ref{thm:rmax_with_uniform_eps} uses the fact
 that, given an element of $\frac{1}{2n^2}\ZZ$, the closest non-equal element of
 $\frac{1}{2}\ZZ$ is at least $\frac{1}{2n^2}$ away. However this a
 conservative estimate, and a larger gap can sometimes be guaranteed if we know
@@ -1566,7 +1566,7 @@ this value of $\frac{1}{2n^2}\ZZ$ explicitly.
 
 The expressions that will follow will be a bit more complicated and have more
 parts which depend on the values of $q$ and $\beta$, even their numerators
-$\aa,\bb$ specifically. The upcoming theorem (TODO ref) is less useful as a
+$\aa,\bb$ specifically. The upcoming Theorem (TODO ref) is less useful as a
 `clean' formula for a bound on the ranks of the pseudo-semistabilizers, but has a
 purpose in the context of writing a computer program to find
 pseudo-semistabilizers. Such a program would iterate through possible values of
@@ -1593,7 +1593,7 @@ $n$, and so invertible mod $n$).
 Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
 
 Next, we seek to find a larger $\epsilon$ to use in place of $\epsilon_F$ in the
-proof of theorem \ref{thm:rmax_with_uniform_eps}:
+proof of Theorem \ref{thm:rmax_with_uniform_eps}:
 
 \begin{lemmadfn}[
 	Finding a better alternative to $\epsilon_v$:
@@ -1723,7 +1723,7 @@ from plots_and_expressions import main_theorem2
 			\sage{main_theorem2.r_upper_bound2}
 		\right)
 	\end{align*}
-	Where $k_{v,q}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
+	Where $k_{v,q}$ is defined as in definition/Lemma \ref{lemdfn:epsilon_q},
 	and $R = \chern_0(v)$
 
 	Furthermore, if $\aa \not= 0$ then
@@ -1767,7 +1767,7 @@ from plots_and_expressions import main_theorem2_corollary
 \end{corollary}
 
 \begin{proof}
-This is a specialisation of theorem \ref{thm:rmax_with_eps1}, where we can
+This is a specialisation of Theorem \ref{thm:rmax_with_eps1}, where we can
 drastically simplify the $\lcm$ and $\gcd$ terms by noting that $m$ divides both
 $2$ and $2n^2$, and that $a_v$ is coprime to $n$.
 \end{proof}
@@ -1813,10 +1813,10 @@ end}
 \vspace{1em}
 
 \noindent
-It's worth noting that the bounds given by theorem \ref{thm:rmax_with_eps1}
+It's worth noting that the bounds given by Theorem \ref{thm:rmax_with_eps1}
 reach, but do not exceed the actual maximum rank 25 of the
 pseudo-semistabilizers of $v$ in this case.
-As a reminder, the original loose bound from theorem \ref{thm:loose-bound-on-r}
+As a reminder, the original loose bound from Theorem \ref{thm:loose-bound-on-r}
 was 144.
 
 \end{example}
@@ -1829,8 +1829,8 @@ take $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so that
 $\beta=\sage{extravagant.betaminus}$, giving $n=\sage{n:=extravagant.n}$
 and $\chern_1^{\sage{extravagant.betaminus}}(F) = \sage{extravagant.twisted.ch[1]}$.
 This example was chosen because the $n$ value is moderatly large, giving more
-possible values for $k_{v,q}$, in dfn/lemma \ref{lemdfn:epsilon_q}. This allows
-for a larger possible difference between the bounds given by theorems
+possible values for $k_{v,q}$, in dfn/Lemma \ref{lemdfn:epsilon_q}. This allows
+for a larger possible difference between the bounds given by Theorems
 \ref{thm:rmax_with_uniform_eps} and \ref{thm:rmax_with_eps1}, with the bound
 from the second being up to $\sage{n}$ times smaller, for any given $q$ value.
 The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabilizer $u$ of $v$
@@ -1873,10 +1873,10 @@ 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}.
+$\sage{max(theorem2_bounds)}$ for Theorem \ref{thm:rmax_with_uniform_eps}, and
+$\sage{max(theorem3_bounds)}$ for Theorem \ref{thm:rmax_with_eps1}.
 \end{example}
 
 \egroup % end scope where beta redefined to beta_{-}
@@ -1885,7 +1885,7 @@ $\sage{max(theorem3_bounds)}$ for theorem \ref{thm:rmax_with_eps1}.
 \label{sect:prob2-algorithm}
 
 Alongside this article, there is a library \cite{NaylorRust2023} to compute
-the solutions to problem \ref{problem:problem-statement-2}, using the theorems
+the solutions to problem \ref{problem:problem-statement-2}, using the Theorems
 above.
 
 The way it works, is by yielding solutions to the problem
@@ -1957,7 +1957,7 @@ So condition \ref{item:mubound:lem:num_test_prob2} in corollary
 Note that the right hand-side is greater than, or equal, to 0, so such $r$ also
 satisfies \ref{item:rankpos:lem:num_test_prob2}.
 
-Then theorem \ref{thm:rmax_with_eps1} gives an upper on possible $r$ values
+Then Theorem \ref{thm:rmax_with_eps1} gives an upper on possible $r$ values
 for which it is possible to satisfy conditions
 \ref{item:bgmlvu:lem:num_test_prob2},
 \ref{item:bgmlvv-u:lem:num_test_prob2}, and
@@ -2000,7 +2000,7 @@ to the problem for this choice of $v$.
 
 The bounds of the ranks of solutions to problem
 \ref{problem:problem-statement-2}
-given by theorems
+given by Theorems
 \ref{thm:loose-bound-on-r}
 \ref{thm:rmax_with_uniform_eps}
 \ref{thm:rmax_with_eps1}, have been shown in passing to be tighter than the
@@ -2019,11 +2019,11 @@ 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
+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
@@ -2043,7 +2043,7 @@ indicators to the size of the bounds on the pseudo-semistabiliser ranks.
 	\includegraphics[width=\linewidth]{../figures/benchmark.png}
 	\caption{
 		Comparing the performance of program \cite{NaylorRust2023}
-		with different patches corresponding to the results of theorems
+		with different patches corresponding to the results of Theorems
 		\ref{thm:loose-bound-on-r}
 		\ref{thm:rmax_with_uniform_eps}
 		\ref{thm:rmax_with_eps1}
@@ -2054,16 +2054,16 @@ indicators to the size of the bounds on the pseudo-semistabiliser ranks.
 \end{figure}
 
 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}
+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}.
 the program to eliminate.
 
-As for the difference between theorems \ref{thm:rmax_with_uniform_eps}
+As for the difference between Theorems \ref{thm:rmax_with_uniform_eps}
 and \ref{thm:rmax_with_eps1}, the biggest indicator is the `$n$'-value, that is,
 the denominator of $\beta_{-}(v)$. For this example, it is 15.
-The bound from theorem \ref{thm:rmax_with_eps1} is roughly $1/{k_{v,q}}$ times
-that of theorem \ref{thm:rmax_with_uniform_eps}.
+The bound from Theorem \ref{thm:rmax_with_eps1} is roughly $1/{k_{v,q}}$ times
+that of Theorem \ref{thm:rmax_with_uniform_eps}.
 Note that $k_{v,q}$ iterates through all its possible values
 $\{1, 2, \ldots, n\}$ cyclically.
 So we could expect the average tighter bound to be approximately that of the