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...@@ -13,7 +13,7 @@ a different algorithm will be presented making use of theorems from Section ...@@ -13,7 +13,7 @@ a different algorithm will be presented making use of theorems from Section
with the goal of cutting down the run time. with the goal of cutting down the run time.
\subsubsection{Finding possible \texorpdfstring{$r$}{r} and \subsubsection{Finding possible \texorpdfstring{$r$}{r} and
\texorpdfstring{$c$}{c}} \texorpdfstring{$c$}{c}}
To do this, first calculate the upper bound $r_{\mathrm{max}}$ on the ranks of tilt To do this, first calculate the upper bound $r_{\mathrm{max}}$ on the ranks of tilt
semistabilisers, as given by Theorem \ref{thm:loose-bound-on-r}. semistabilisers, as given by Theorem \ref{thm:loose-bound-on-r}.
...@@ -36,7 +36,7 @@ the Bogomolov inequalities and Consequence 3 of Lemma ...@@ -36,7 +36,7 @@ the Bogomolov inequalities and Consequence 3 of Lemma
($\chern_2^{\beta_{-}}(u)>0$). ($\chern_2^{\beta_{-}}(u)>0$).
\subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r} \subsubsection{Finding \texorpdfstring{$d$}{d} for fixed \texorpdfstring{$r$}{r}
and \texorpdfstring{$c$}{c}} and \texorpdfstring{$c$}{c}}
$\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the $\Delta(u) \geq 0$ induces an upper bound $\frac{c^2}{2r}$ on $d$, and the
$\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$. $\chern_2^{\beta_{-}}(u)>0$ condition induces a lower bound on $d$.
...@@ -51,8 +51,8 @@ end up not yielding any solutions for the problem. ...@@ -51,8 +51,8 @@ end up not yielding any solutions for the problem.
In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to In fact, solutions $u$ to our problem with high rank must have $\mu(u)$ close to
$\beta_{-}(v)$: $\beta_{-}(v)$:
\begin{align*} \begin{align*}
0 &\leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\ 0 & \leq \chern_1^{\beta_{-}}(u) \leq \chern_1^{\beta_{-}}(u) \\
0 &\leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r} 0 & \leq \mu(u) - \beta_{-} \leq \frac{\chern_1^{\beta_{-}}(v)}{r}
\end{align*} \end{align*}
In particular, it is the $\chern_1^{\beta_{-}}(v-u) \geq 0$ condition which In particular, it is the $\chern_1^{\beta_{-}}(v-u) \geq 0$ condition which
fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$. fails for $r$,$c$ pairs with large $r$ and $\frac{c}{r}$ too far from $\beta_{-}$.
...@@ -66,17 +66,17 @@ alternative algorithm which will later be described in Section ...@@ -66,17 +66,17 @@ alternative algorithm which will later be described in Section
\ref{sect:prob2-algorithm}. \ref{sect:prob2-algorithm}.
\begin{center} \begin{center}
\label{table:bench-schmidt-vs-nay} \label{table:bench-schmidt-vs-nay}
\begin{tabular}{ |r|l|l| } \begin{tabular}{ |r|l|l| }
\hline \hline
Choice of $v$ on $\mathbb{P}^2$ Choice of $v$ on $\mathbb{P}^2$
& $(3, 2\ell, -2)$ & $(3, 2\ell, -2)$
& $(3, 2\ell, -\frac{15}{2})$ \\ & $(3, 2\ell, -\frac{15}{2})$ \\
\hline \hline
\cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\ \cite[\texttt{tilt.walls_left}]{SchmidtGithub2020} exec time & \sim 20s & >1hr \\
\cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\ \cite{NaylorRust2023} exec time & \sim 50ms & \sim 50ms \\
\hline \hline
\end{tabular} \end{tabular}
\end{center} \end{center}
\section{Computing Solutions to Problem \ref{problem:problem-statement-2}} \section{Computing Solutions to Problem \ref{problem:problem-statement-2}}
...@@ -94,7 +94,7 @@ The algorithm yields solutions ...@@ -94,7 +94,7 @@ The algorithm yields solutions
$u=(r,c\ell,d\ell^2)$ to the problem as follows. $u=(r,c\ell,d\ell^2)$ to the problem as follows.
\subsubsection{Iterating Over Possible \subsubsection{Iterating Over Possible
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}} \texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}}
Given a Chern character $v$, the domain of the problem are first verified: that Given a Chern character $v$, the domain of the problem are first verified: that
$v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that $v$ has positive rank, that it satisfies $\Delta(v) \geq 0$, and that
...@@ -102,6 +102,7 @@ $\beta_{-}(v)$ is rational. ...@@ -102,6 +102,7 @@ $\beta_{-}(v)$ is rational.
Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms. Take $\beta_{-}(v)=\frac{a_v}{n}$ in simplest terms.
Iterate over $q = \frac{b_q}{n} \in (0,\chern_1^{\beta_{-}}(v))\cap\frac{1}{n}\ZZ$. Iterate over $q = \frac{b_q}{n} \in (0,\chern_1^{\beta_{-}}(v))\cap\frac{1}{n}\ZZ$.
The code used to generate the corresponding values for $b_q$ is shown in Listing The code used to generate the corresponding values for $b_q$ is shown in Listing
% texlab: ignore
\ref{fig:code:consideredb}. \ref{fig:code:consideredb}.
\lstinputlisting[ \lstinputlisting[
...@@ -118,6 +119,7 @@ We can therefore reduce the problem of finding solutions to the more specialised ...@@ -118,6 +119,7 @@ We can therefore reduce the problem of finding solutions to the more specialised
problem of finding the solutions $u$ with each fixed possible $q=\chern_1^\beta(u)$ problem of finding the solutions $u$ with each fixed possible $q=\chern_1^\beta(u)$
(i.e. choice of $b$). (i.e. choice of $b$).
The code representing this appears in Listing The code representing this appears in Listing
% texlab: ignore
\ref{fig:code:reducingtoeachb}. \ref{fig:code:reducingtoeachb}.
Line 16 refers to creating an objects representing the context the specialised Line 16 refers to creating an objects representing the context the specialised
problem for the fixed $q$ value, with the next line `solving' the specialised problem for the fixed $q$ value, with the next line `solving' the specialised
...@@ -133,9 +135,9 @@ and collect up the results. ...@@ -133,9 +135,9 @@ and collect up the results.
]{../tilt.rs/src/tilt_stability/find_all.git-untrack.rs.tex.git-untrack} ]{../tilt.rs/src/tilt_stability/find_all.git-untrack.rs.tex.git-untrack}
\subsubsection{Iterating Over Possible \subsubsection{Iterating Over Possible
\texorpdfstring{$r=\chern_0(u)$}{r} \texorpdfstring{$r=\chern_0(u)$}{r}
for Fixed for Fixed
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q} \texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
} }
Let $q=\frac{b_q}{n}$, for which we are now solving the more specialised problem of finding Let $q=\frac{b_q}{n}$, for which we are now solving the more specialised problem of finding
...@@ -159,6 +161,7 @@ Fixing $r$ and $q$ also determines $c\coloneqq\chern_1(u)$, and so we can genera ...@@ -159,6 +161,7 @@ Fixing $r$ and $q$ also determines $c\coloneqq\chern_1(u)$, and so we can genera
the corresponding values of $c$, as we generate the $r$ values. the corresponding values of $c$, as we generate the $r$ values.
It now remains to solve the problem for each of the combinations of fixed values It now remains to solve the problem for each of the combinations of fixed values
for $q$ and $r$ (and consequently $c$) considered. for $q$ and $r$ (and consequently $c$) considered.
% texlab: ignore
This is shown in Listing \ref{fig:code:reducingtoeachr}. This is shown in Listing \ref{fig:code:reducingtoeachr}.
\lstinputlisting[ \lstinputlisting[
...@@ -170,11 +173,11 @@ This is shown in Listing \ref{fig:code:reducingtoeachr}. ...@@ -170,11 +173,11 @@ This is shown in Listing \ref{fig:code:reducingtoeachr}.
\subsubsection{Iterating Over Possible \subsubsection{Iterating Over Possible
\texorpdfstring{$d=\chern_2(u)/\ell^2$}{d} \texorpdfstring{$d=\chern_2(u)/\ell^2$}{d}
for Fixed for Fixed
\texorpdfstring{$r=\chern_0(u)$}{r} \texorpdfstring{$r=\chern_0(u)$}{r}
and and
\texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q} \texorpdfstring{$q=\chern^{\beta_{-}}(u)$}{q}
} }
At this point we are considering a specialisation of the problem At this point we are considering a specialisation of the problem
...@@ -198,8 +201,10 @@ equivalent to bounds on $d$ given by the equations ...@@ -198,8 +201,10 @@ equivalent to bounds on $d$ given by the equations
in Subsubsection \ref{subsubsect:all-bounds-on-d-prob2} in Subsubsection \ref{subsubsect:all-bounds-on-d-prob2}
It therefore remains to just pick values It therefore remains to just pick values
$d\in\frac{1}{\lcm(m,2n^2)}\ZZ$ within the bounds. $d\in\frac{1}{\lcm(m,2n^2)}\ZZ$ within the bounds.
% texlab: ignore
Listing \ref{fig:code:solveforfixedr} is the code for solving this Listing \ref{fig:code:solveforfixedr} is the code for solving this
specialisation of the problem, where the possible $d$ values are computed in specialisation of the problem, where the possible $d$ values are computed in
% texlab: ignore
Listing \ref{fig:code:possible_chern2}. Listing \ref{fig:code:possible_chern2}.
The explicit code for the bounds can be found in Appendix The explicit code for the bounds can be found in Appendix
\ref{appendix:subsubsec:fixed-r}. \ref{appendix:subsubsec:fixed-r}.
...@@ -232,11 +237,11 @@ decrease in computational time to find the solutions to the problem. ...@@ -232,11 +237,11 @@ decrease in computational time to find the solutions to the problem.
This could be due to a range of potential reasons: This could be due to a range of potential reasons:
\begin{itemize} \begin{itemize}
\item Unexpected optimisations from the compiler for a certain form of the \item Unexpected optimisations from the compiler for a certain form of the
program. program.
\item Increased complexity to computing the formulae for the tighter bounds. \item Increased complexity to computing the formulae for the tighter bounds.
\item Modern CPU architecture such as branch predictors \item Modern CPU architecture such as branch predictors
\cite{BranchPredictor2024} may offset the overhead of considering ranks that \cite{BranchPredictor2024} may offset the overhead of considering ranks that
turn out to be too large to have any solutions. turn out to be too large to have any solutions.
\end{itemize} \end{itemize}
For relatively small Chern characters (as those appearing in examples so far), For relatively small Chern characters (as those appearing in examples so far),
......