Refactor recurring example into notebook

main.tex

@@ -749,28 +749,19 @@ bound for the rank of $E$:
 
 \end{proof}
 
-\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
-\label{exmpl:recurring-first}
 \begin{sagesilent}
-recurring = Object()
-recurring.chern = Chern_Char(3, 2, -2)
-recurring.b = beta_minus(recurring.chern)
-recurring.twisted = recurring.chern.twist(recurring.b)
+from plots_and_expressions import recurring
 \end{sagesilent}
+
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\label{exmpl:recurring-first}
 Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
 that $m=2$, $\beta_-=\sage{recurring.b}$,
 giving $n=\sage{recurring.b.denominator()}$ and
 $\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
 
-\begin{sagesilent}
-n = recurring.b.denominator()
-m = 2
-loose_bound = (
-  m*n^2*recurring.twisted.ch[1]^2
-) / gcd(m, 2*n^2)
-\end{sagesilent}
 Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
-tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
+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
@@ -1671,22 +1662,12 @@ $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
 that $m=2$, $\beta=\sage{recurring.b}$,
 giving $n=\sage{recurring.b.denominator()}$.
 
-\begin{sagesilent}
-recurring.n = recurring.b.denominator()
-recurring.bgmlv = recurring.chern.Q_tilt()
-corrolary_bound = (
-  r_upper_bound_all_q.expand()
-  .subs(Delta==recurring.bgmlv)
-  .subs(nu==1) ## \ell^2=1 on P^2
-  .subs(R==recurring.chern.ch[0])
-  .subs(n==recurring.n)
-)
-\end{sagesilent}
 Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
 the ranks of tilt semistabilizers for $v$ are bounded above by
-$\sage{corrolary_bound} \approx  \sage{float(corrolary_bound)}$,
+$\sage{recurring.corrolary_bound} \approx  \sage{float(recurring.corrolary_bound)}$,
 which is much closer to real maximum 25 than the original bound 144.
 \end{example}
+
 \begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
 \label{exmpl:extravagant-second}
 Just like in example \ref{exmpl:extravagant-first}, take
@@ -1887,40 +1868,7 @@ The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabi
 in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
 
 \begin{sagesilent}
-import numpy as np
-
-def bound_comparisons(example):
-    n = example.b.denominator()
-    a_v = example.b.numerator()
-
-    def theorem_bound(v_twisted, q_val, k):
-      return int(min(
-        n^2*q_val^2/k
-      ,
-        v_twisted.ch[0]
-        + n^2*(v_twisted.ch[1] - q_val)^2/k
-      ))
-
-    def k(n, a_v, b_q):
-      n = int(n)
-      a_v = int(a_v)
-      b_q = int(b_q)
-      k = -a_v*b_q % n
-      return k if k > 0 else k + n
-
-    b_qs = list(range(example.twisted.ch[1]*n+1))
-    qs = list(map(lambda x: x/n,b_qs))
-    ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))
-    theorem2_bounds = [
-        theorem_bound(example.twisted, q_val, 1)
-        for q_val in qs
-    ]
-    theorem3_bounds = [
-        theorem_bound(example.twisted, q_val, k)
-        for q_val, k in zip(qs,ks)
-    ]
-    return qs, theorem2_bounds, theorem3_bounds
-
+from plots_and_expressions import bound_comparisons
 qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
 \end{sagesilent}