From e3839a820af327728117bbc7f5219227575178cf Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 10 May 2023 18:23:19 +0100
Subject: [PATCH] Add plots with example of bounds on d for large r
---
main.tex | 110 +++++++++++++++++++++++++++++++++++++++++++++++++++----
1 file changed, 103 insertions(+), 7 deletions(-)
diff --git a/main.tex b/main.tex
index df4844f..a2e6ffe 100644
--- a/main.tex
+++ b/main.tex
@@ -585,19 +585,115 @@ radius of the circular wall must be positive. As discussed in (TODO ref), this
is equivalent to $\chern^{\beta}_2(E) > 0$, which yields:
\begin{sagesilent}
- positive_radius_condition = (
- (
- (0 > - u.twist(beta).ch[2])
- + d # rearrange for d
- )
- .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
- .expand()
+positive_radius_condition = (
+ (
+ (0 > - u.twist(beta).ch[2])
+ + d # rearrange for d
)
+ .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
+ .expand()
+)
\end{sagesilent}
\begin{equation*}
\sage{positive_radius_condition}
\end{equation*}
+\begin{sagesilent}
+def beta_min(chern):
+ ts = stability.Tilt()
+ return min(
+ map(
+ lambda soln: soln.rhs(),
+ solve(
+ (ts.degree(chern))
+ .expand()
+ .subs(ts.alpha == 0),
+ beta
+ )
+ )
+ )
+
+v_example = Chern_Char(3,2,-2)
+q_example = 7/3
+
+def plot_d_bound(v_example, q_example, ymax=5, ymin=-2, xmax=20):
+
+ # Equations to plot imminently representing the bounds on d:
+ eq1 = (
+ bgmlv1_d_lowerbound
+ .subs(R == v_example.ch[0])
+ .subs(C == v_example.ch[1])
+ .subs(D == v_example.ch[2])
+ .subs(beta = beta_min(v_example))
+ .subs(q == q_example)
+ )
+
+ eq2 = (
+ bgmlv2_d_upperbound
+ .subs(R == v_example.ch[0])
+ .subs(C == v_example.ch[1])
+ .subs(D == v_example.ch[2])
+ .subs(beta = beta_min(v_example))
+ .subs(q == q_example)
+ )
+
+ eq3 = (
+ bgmlv3_d_upperbound
+ .subs(R == v_example.ch[0])
+ .subs(C == v_example.ch[1])
+ .subs(D == v_example.ch[2])
+ .subs(beta = beta_min(v_example))
+ .subs(q == q_example)
+ )
+
+ eq4 = (
+ positive_radius_condition.rhs()
+ .subs(q == q_example)
+ .subs(beta = beta_min(v_example))
+ )
+
+ example_bounds_on_d_plot = (
+ plot(
+ eq3,
+ (r,v_example.ch[0],xmax),
+ color='green',
+ linestyle = "dashed",
+ legend_label=r"upper bound: $\Delta(G) \geq 0$",
+ title=r"$q :=\mathrm{ch}_1^{\beta_{-}}(E)=" + latex(q_example) + r"$"
+ )
+ + plot(
+ eq2,
+ (r,0,xmax),
+ color='blue',
+ linestyle = "dashed",
+ legend_label=r"upper bound: $\Delta(E) \geq 0$"
+ )
+ + plot(
+ eq4,
+ (r,0,xmax),
+ color='orange',
+ linestyle = "dotted",
+ legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(E)>0$"
+ )
+ + plot(
+ eq1,
+ (r,v_example.ch[0]/2,xmax),
+ color='red',
+ linestyle = "dotted",
+ legend_label=r"lower bound: $\Delta(E) + \Delta(G) \leq \Delta(F)$"
+ )
+ )
+ example_bounds_on_d_plot.ymin(ymin)
+ example_bounds_on_d_plot.ymax(ymax)
+ example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])
+ return example_bounds_on_d_plot
+
+\end{sagesilent}
+
+\sageplot{plot_d_bound(v_example, 0)}
+\sageplot{plot_d_bound(v_example, 2)}
+\sageplot{plot_d_bound(v_example, 4)}
+
\egroup
\section{Conclusion}
--
GitLab