{
"cells": [
{
"cell_type": "markdown",
"id": "63724666",
"metadata": {},
"source": [
"# Utilities"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "abe149f0",
"metadata": {},
"outputs": [],
"source": [
"# Requires extra package:\n",
"#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
"%display latex\n",
"\n",
"from pseudowalls import *\n",
"\n",
"Δ = lambda v: v.Q_tilt()\n",
"mu = stability.Mumford().slope\n",
"ts = stability.Tilt\n",
"\n",
"var(\"beta\", domain=\"real\")\n",
"\n",
"def beta_minus(v):\n",
" beta = stability.Tilt().beta\n",
" solutions = solve(\n",
" stability.Tilt(alpha=0).degree(v)==0,\n",
" beta)\n",
" return min(map(lambda s: s.rhs(), solutions))\n",
"\n",
"class Object(object):\n",
" pass"
]
},
{
"cell_type": "markdown",
"id": "0b574b63",
"metadata": {},
"source": [
"# Define Cherns in Examples"
]
},
{
"cell_type": "markdown",
"id": "f1c3a606",
"metadata": {},
"source": [
"Define two recurring examples used to illustrate performance of the different theorems to find semistabilizer bounds:"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "30d3738a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\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}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = 3 \\\\ \\mathrm{ch}_{1} = 2 \\ell^{1} \\\\ \\mathrm{ch}_{2} = -2 \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f25db3a9990>"
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurring = Object()\n",
"recurring.chern = Chern_Char(3, 2, -2)\n",
"recurring.chern"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "fe394909",
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\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}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = 29 \\\\ \\mathrm{ch}_{1} = 13 \\ell^{1} \\\\ \\mathrm{ch}_{2} = -\\frac{3}{2} \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f25d1953e20>"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"extravagant = Object()\n",
"extravagant.chern = Chern_Char(29, 13, -3/2)\n",
"extravagant.chern"
]
},
{
"cell_type": "markdown",
"id": "09692ba5",
"metadata": {},
"source": [
"# Calculate Preliminary Quantities"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "7576ebe4",
"metadata": {},
"outputs": [],
"source": [
"m = 1 # value of $\\ell^2$\n",
"\n",
"# add attributes for ...\n",
"for example in [recurring, extravagant]:\n",
" example.betaminus = beta_minus(example.chern)\n",
" example.twisted = example.chern.twist(example.betaminus)\n",
" example.n = example.betaminus.denominator()\n",
" example.bgmlv = example.chern.Q_tilt()\n",
"\n",
"# Actual maximal rank of Pseudo-Semistabilizers\n",
"# (needs to be calculated elsewhere)\n",
"recurring.actual_rmax = 25\n",
"extravagant.actual_rmax = 49313"
]
},
{
"cell_type": "markdown",
"id": "500abba3",
"metadata": {},
"source": [
"# First Loose Bound"
]
},
{
"cell_type": "markdown",
"id": "f3e61899",
"metadata": {},
"source": [
"Formula for loose bound:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "77fff07c",
"metadata": {},
"outputs": [],
"source": [
"def loose_bound(example):\n",
" n = example.n\n",
" twisted = example.twisted\n",
" return ( m*n^2*twisted.ch[1]^2\n",
" ) / gcd(m, 2*n^2)\n",
"\n",
"for example in [recurring, extravagant]:\n",
" example.loose_bound = loose_bound(example)"
]
},
{
"cell_type": "markdown",
"id": "218fa169",
"metadata": {},
"source": [
"Loose bounds for the two examples:"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "9edbb954",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 144\\)</html>"
],
"text/latex": [
"$\\displaystyle 144$"
],
"text/plain": [
"144"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"recurring.loose_bound"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "21b52c4d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 215296\\)</html>"
],
"text/latex": [
"$\\displaystyle 215296$"
],
"text/plain": [
"215296"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"extravagant.loose_bound"
]
},
{
"cell_type": "markdown",
"id": "22bc4b5f",
"metadata": {},
"source": [
"# Stronger Convenient Bound"
]
},
{
"cell_type": "markdown",
"id": "ee140573",
"metadata": {},
"source": [
"Use expression for bound used in main document"
]
},
{
"cell_type": "markdown",
"id": "8de156d1",
"metadata": {},
"source": [
"$\\renewcommand\\nu\\ell$\n",
"Redefine \\nu to $\\nu$ in latex\n",
"$\\let\\originalDelta\\Delta$\n",
"$\\renewcommand\\Delta{\\originalDelta(v)}$\n",
"Redefine \\Delta in latex to be $\\Delta$"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "712b7324",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
],
"text/plain": [
"1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"from plots_and_expressions import main_theorem1, Delta, nu, R, n\n",
"# Delta: symbol for Δ(v)\n",
"# n: symbol for denominator for β_(v)\n",
"# R : symbol for chern_0(v)\n",
"# nu : ...\n",
"main_theorem1.corollary_r_bound"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "43aa5f56",
"metadata": {},
"outputs": [],
"source": [
"def corrolary_bound(example):\n",
" return (\n",
" main_theorem1.corollary_r_bound\n",
" .subs(Delta==example.bgmlv)\n",
" .subs(nu==1)\n",
" .subs(R==example.chern.ch[0])\n",
" .subs(n==example.n)\n",
" )\n",
"\n",
"\n",
"for example in [recurring, extravagant]:\n",
" example.corrolary_bound = corrolary_bound(example)"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "5baed51c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 53838.5009765625\\)</html>"
],
"text/latex": [
"$\\displaystyle 53838.5009765625$"
],
"text/plain": [
"53838.5009765625"
]
},
"execution_count": 10,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"float(extravagant.corrolary_bound)"
]
},
{
"cell_type": "code",
"execution_count": 11,
"id": "fcc15f60",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 37.515625\\)</html>"
],
"text/latex": [
"$\\displaystyle 37.515625$"
],
"text/plain": [
"37.515625"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"float(recurring.corrolary_bound)"
]
},
{
"cell_type": "markdown",
"id": "e678a488",
"metadata": {},
"source": [
"# Stronger Complicated Bounds"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "ef8f09ff",
"metadata": {},
"outputs": [],
"source": [
"import numpy as np\n",
"\n",
"def bound_comparisons(example):\n",
" n = example.n\n",
" a_v = example.betaminus.numerator()\n",
"\n",
" def theorem_bound(v_twisted, q_val, k):\n",
" return int(min(\n",
" n^2*q_val^2/k,\n",
" v_twisted.ch[0]\n",
" + n^2*(v_twisted.ch[1] - q_val)^2/k\n",
" ))\n",
"\n",
" def k(n, a_v, b_q):\n",
" n = int(n)\n",
" a_v = int(a_v)\n",
" b_q = int(b_q)\n",
" k = -a_v*b_q % n\n",
" return k if k > 0 else k + n\n",
"\n",
" b_qs = list(range(example.twisted.ch[1]*n+1))\n",
" qs = list(map(lambda x: x/n,b_qs))\n",
" ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))\n",
" theorem2_bounds = [\n",
" theorem_bound(example.twisted, q_val, 1)\n",
" for q_val in qs\n",
" ]\n",
" theorem3_bounds = [\n",
" theorem_bound(example.twisted, q_val, k)\n",
" for q_val, k in zip(qs,ks)\n",
" ]\n",
" return qs, theorem2_bounds, theorem3_bounds"
]
},
{
"cell_type": "markdown",
"id": "b1c51cd8",
"metadata": {},
"source": [
"Array content:\n",
"- First row: $q$-values\n",
"- Second row: Theorem 1 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\\beta}=q$\n",
"- Second row: Theorem 2 bounds on $ch_0(u)$ for u solutions to prob with $ch_1^{\\beta}=q$"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "9cd102ac",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\begin{array}{l}\n",
"\\verb|[[0|\\verb| |\\verb|1/3|\\verb| |\\verb|2/3|\\verb| |\\verb|1|\\verb| |\\verb|4/3|\\verb| |\\verb|5/3|\\verb| |\\verb|2|\\verb| |\\verb|7/3|\\verb| |\\verb|8/3|\\verb| |\\verb|3|\\verb| |\\verb|10/3|\\verb| |\\verb|11/3|\\verb| |\\verb|4]|\\\\\n",
"\\verb| |\\verb|[0|\\verb| |\\verb|1|\\verb| |\\verb|4|\\verb| |\\verb|9|\\verb| |\\verb|16|\\verb| |\\verb|25|\\verb| |\\verb|36|\\verb| |\\verb|28|\\verb| |\\verb|19|\\verb| |\\verb|12|\\verb| |\\verb|7|\\verb| |\\verb|4|\\verb| |\\verb|3]|\\\\\n",
"\\verb| |\\verb|[0|\\verb| |\\verb|0|\\verb| |\\verb|4|\\verb| |\\verb|3|\\verb| |\\verb|8|\\verb| |\\verb|25|\\verb| |\\verb|12|\\verb| |\\verb|15|\\verb| |\\verb|19|\\verb| |\\verb|6|\\verb| |\\verb|5|\\verb| |\\verb|4|\\verb| |\\verb|3]]|\n",
"\\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\begin{array}{l}\n",
"\\verb|[[0|\\verb| |\\verb|1/3|\\verb| |\\verb|2/3|\\verb| |\\verb|1|\\verb| |\\verb|4/3|\\verb| |\\verb|5/3|\\verb| |\\verb|2|\\verb| |\\verb|7/3|\\verb| |\\verb|8/3|\\verb| |\\verb|3|\\verb| |\\verb|10/3|\\verb| |\\verb|11/3|\\verb| |\\verb|4]|\\\\\n",
"\\verb| |\\verb|[0|\\verb| |\\verb|1|\\verb| |\\verb|4|\\verb| |\\verb|9|\\verb| |\\verb|16|\\verb| |\\verb|25|\\verb| |\\verb|36|\\verb| |\\verb|28|\\verb| |\\verb|19|\\verb| |\\verb|12|\\verb| |\\verb|7|\\verb| |\\verb|4|\\verb| |\\verb|3]|\\\\\n",
"\\verb| |\\verb|[0|\\verb| |\\verb|0|\\verb| |\\verb|4|\\verb| |\\verb|3|\\verb| |\\verb|8|\\verb| |\\verb|25|\\verb| |\\verb|12|\\verb| |\\verb|15|\\verb| |\\verb|19|\\verb| |\\verb|6|\\verb| |\\verb|5|\\verb| |\\verb|4|\\verb| |\\verb|3]]|\n",
"\\end{array}$"
],
"text/plain": [
"array([[0, 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4],\n",
" [0, 1, 4, 9, 16, 25, 36, 28, 19, 12, 7, 4, 3],\n",
" [0, 0, 4, 3, 8, 25, 12, 15, 19, 6, 5, 4, 3]], dtype=object)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"np.array(bound_comparisons(recurring))"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.7",
"language": "sage",
"name": "sagemath"
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"language_info": {
"codemirror_mode": {
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"file_extension": ".py",
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