diff --git a/examples.ipynb b/examples.ipynb
index f14242b98a437f4c8faaa3096d3934355d58d4d9..ec5739e12506f13b4a17b6bcc43900ecd574d466 100644
--- a/examples.ipynb
+++ b/examples.ipynb
@@ -69,7 +69,7 @@
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = 3 \\\\ \\mathrm{ch}_{1} = 2 \\ell^{1} \\\\ \\mathrm{ch}_{2} = -2 \\ell^{2} \\end{array}$"
],
"text/plain": [
- "<pseudowalls.chern_character.Chern_Char object at 0x7ff85e32f1f0>"
+ "<pseudowalls.chern_character.Chern_Char object at 0x7f1718d278e0>"
]
},
"execution_count": 2,
@@ -100,7 +100,7 @@
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = 29 \\\\ \\mathrm{ch}_{1} = 13 \\ell^{1} \\\\ \\mathrm{ch}_{2} = -\\frac{3}{2} \\ell^{2} \\end{array}$"
],
"text/plain": [
- "<pseudowalls.chern_character.Chern_Char object at 0x7ff8548d70d0>"
+ "<pseudowalls.chern_character.Chern_Char object at 0x7f1718d263b0>"
]
},
"execution_count": 3,
@@ -263,13 +263,13 @@
{
"data": {
"text/html": [
- "<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta \\Omega}{8 \\, m} + \\frac{R^{2} m}{2 \\, \\Delta \\Omega}\\)</html>"
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{{\\Delta(v)} {\\operatorname{lcm}(m,2n^2)}}{8 \\, m} + \\frac{R^{2} m}{2 \\, {\\Delta(v)} {\\operatorname{lcm}(m,2n^2)}}\\)</html>"
],
"text/latex": [
- "$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta \\Omega}{8 \\, m} + \\frac{R^{2} m}{2 \\, \\Delta \\Omega}$"
+ "$\\displaystyle \\frac{1}{2} \\, R + \\frac{{\\Delta(v)} {\\operatorname{lcm}(m,2n^2)}}{8 \\, m} + \\frac{R^{2} m}{2 \\, {\\Delta(v)} {\\operatorname{lcm}(m,2n^2)}}$"
],
"text/plain": [
- "1/2*R + 1/8*Delta*Omega/m + 1/2*R^2*m/(Delta*Omega)"
+ "1/2*R + 1/8*bgmlv_v*lcm_m_2n2/m + 1/2*R^2*m/(bgmlv_v*lcm_m_2n2)"
]
},
"execution_count": 8,
@@ -278,7 +278,7 @@
}
],
"source": [
- "from plots_and_expressions import main_theorem1, Delta, m, R, n, lcm_m_2n2\n",
+ "from plots_and_expressions import main_theorem1, bgmlv_v, m, R, n, lcm_m_2n2\n",
"# Delta: symbol for Δ(v)\n",
"# n: symbol for denominator for β_(v)\n",
"# R : symbol for chern_0(v)\n",
@@ -296,7 +296,7 @@
"def corrolary_bound(example):\n",
" return (\n",
" main_theorem1.corollary_r_bound\n",
- " .subs(Delta==example.bgmlv)\n",
+ " .subs(bgmlv_v==example.bgmlv)\n",
" .subs(m==example.m)\n",
" .subs(R==example.chern.ch[0])\n",
" .subs(n==example.n)\n",
diff --git a/main.tex b/main.tex
index 2c14f5599f73535ed7e5dedcbded9a6b1a3494b4..facd95285d121c0c9efb85d7f0623a08439bf6e1 100644
--- a/main.tex
+++ b/main.tex
@@ -511,8 +511,6 @@ Let $P$ be a point on $\Theta_v^-$.
\noindent
The following conditions:
-\bgroup
-\renewcommand{\labelenumi}{\alph{enumi}.}
\begin{enumerate}
\item $u$ is a pseudo-semistabilizer of $v$ at some point on $\Theta_v^-$ above
$P$
@@ -520,7 +518,6 @@ The following conditions:
$\nu_{\alpha,\beta}(u)<\nu_{\alpha,\beta}(v)$ outside the pseudo-wall, and
$\nu_{\alpha,\beta}(u)>\nu_{\alpha,\beta}(v)$ inside.
\end{enumerate}
-\egroup
\noindent
are equivalent to the following more numerical conditions:
@@ -1121,9 +1118,6 @@ $d$ yields:
from plots_and_expressions import bgmlv3_d_upperbound_terms
\end{sagesilent}
-\bgroup
-\def\psi{\chern_1^{\beta}(v)}
-\def\phi{\chern_2^{\beta}(v)}
\begin{equation*}
\label{eqn-bgmlv3_d_upperbound}
d \leq
@@ -1133,7 +1127,6 @@ from plots_and_expressions import bgmlv3_d_upperbound_terms
\qquad
\text{where }r>R
\end{equation*}
-\egroup
\noindent
@@ -1175,9 +1168,6 @@ These give bounds with the same assymptotes when we take $r\to\infty$
\let\originalbeta\beta
\renewcommand\beta{{\originalbeta_{-}}}
-\bgroup
-% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
-\def\psi{\chern_1^{\beta}(F)}
\begin{align}
d &>&
\frac{1}{2}\beta^2 r
@@ -1201,7 +1191,6 @@ These give bounds with the same assymptotes when we take $r\to\infty$
&\qquad\text{when\:} r > R
\label{eqn:bgmlv3_d_bound_betamin}
\end{align}
-\egroup
\begin{sagesilent}
@@ -1340,9 +1329,6 @@ from plots_and_expressions import main_theorem1
with $\chern_1^\beta = q$
are bounded above by the following expression.
- \bgroup
- \def\psi{\chern_1^{\beta}(F)}
- \renewcommand\Omega{{\lcm(m,2n^2)}}
\begin{align*}
\min
\left(
@@ -1350,7 +1336,6 @@ from plots_and_expressions import main_theorem1
\sage{main_theorem1.r_upper_bound2}
\right)
\end{align*}
- \egroup
Taking the maximum of this expression over
$q \in [0, \chern_1^{\beta}(F)]\cap \frac{1}{n}\ZZ$
@@ -1376,32 +1361,20 @@ considering equations
\ref{eqn:bgmlv3_d_bound_betamin},
\ref{eqn:radiuscond_d_bound_betamin}.
-\bgroup
-
-\let\originalepsilon\epsilon
-\renewcommand\epsilon{{\originalepsilon_{v}}}
-
\begin{sagesilent}
from plots_and_expressions import \
-assymptote_gap_condition1, assymptote_gap_condition2, kappa
+assymptote_gap_condition1, assymptote_gap_condition2, k
\end{sagesilent}
-\bgroup
-\def\psi{\chern_1^{\beta}(F)}
-\renewcommand\Omega{{\lcm(m,2n^2)}}
\begin{align}
- &\sage{assymptote_gap_condition1.subs(kappa==1)} \\
- &\sage{assymptote_gap_condition2.subs(kappa==1)}
+ &\sage{assymptote_gap_condition1.subs(k==1)} \\
+ &\sage{assymptote_gap_condition2.subs(k==1)}
\end{align}
-\egroup
\noindent
This is equivalent to:
-\bgroup
-\renewcommand\Omega{{\lcm(m,2n^2)}}
-\def\psi{\chern_1^{\beta}(F)}
\begin{equation}
\label{eqn:thm-bound-for-r-impossible-cond-for-r}
r \leq
@@ -1414,15 +1387,12 @@ This is equivalent to:
}
\right)
\end{equation}
-\egroup
-
-\egroup % end scope where epsilon redefined
\end{proof}
\begin{sagesilent}
-from plots_and_expressions import q_sol, Delta, psi
+from plots_and_expressions import q_sol, bgmlv_v, psi
\end{sagesilent}
\begin{corollary}[Bound on $r$ \#2]
@@ -1433,21 +1403,12 @@ from plots_and_expressions import q_sol, Delta, psi
which are solutions to problem \ref{problem:problem-statement-2},
are bounded above by the following expression.
- \bgroup
- \let\originalDelta\Delta
- \renewcommand\Delta{{\originalDelta(v)}}
- \renewcommand\Omega{{\lcm(m,2n^2)}}
\begin{equation*}
\sage{main_theorem1.corollary_r_bound}
\end{equation*}
- \egroup
\end{corollary}
\begin{proof}
-\bgroup
-\renewcommand\Omega{{\lcm(m,2n^2)}}
-\def\psi{\chern_1^{\beta}(F)}
-\let\originalDelta\Delta
The ranks of the pseudo-semistabilizers for $v$ are bounded above by the
maximum over $q\in [0, \chern_1^{\beta}(F)]$ of the expression in theorem
\ref{thm:rmax_with_uniform_eps}.
@@ -1474,7 +1435,6 @@ And evaluating $f_1$ at this $q$-value gives:
\end{equation*}
Finally, noting that $\originalDelta(v)=\psi^2\ell^2$, we get the bound as
stated in the corollary.
-\egroup
\end{proof}
@@ -1666,10 +1626,6 @@ from plots_and_expressions import main_theorem2
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
are bounded above by the following expression:
- \bgroup
- \def\kappa{k_{v,q}}
- \def\psi{\chern_1^{\beta}(F)}
- \renewcommand\Omega{{\lcm(m,2n^2)}}
\begin{align*}
\min
\left(
@@ -1677,7 +1633,6 @@ from plots_and_expressions import main_theorem2
\sage{main_theorem2.r_upper_bound2}
\right)
\end{align*}
- \egroup
Where $k_{v,q}$ is defined as in definition/lemma \ref{lemdfn:epsilon_q},
and $R = \chern_0(v)$
@@ -1702,9 +1657,6 @@ from plots_and_expressions import main_theorem2_corollary
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
are bounded above by the following expression:
- \bgroup
- \def\kappa{k_{v,q}}
- \def\psi{\chern_1^{\beta}(F)}
\begin{align*}
\min
\left(
@@ -1712,7 +1664,6 @@ from plots_and_expressions import main_theorem2_corollary
\sage{main_theorem2_corollary.r_upper_bound2}
\right)
\end{align*}
- \egroup
Where $R = \chern_0(v)$ and $k_{v,q}$ is the least
$k\in\ZZ_{>0}$ satisfying
\begin{equation*}