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{
"cells": [
{
"cell_type": "markdown",
"id": "8ebd0216",
"metadata": {},
"source": [
"# Utilities"
]
},
{
"cell_type": "markdown",
"id": "b22eb2a9",
"metadata": {},
"source": [
"Define \\chern command in latex $\\newcommand{\\chern}{\\operatorname{ch}}$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "b87a49bc",
"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",
"alpha = stability.Tilt().alpha\n",
"beta = stability.Tilt().beta\n",
"\n",
"def beta_minus(v):\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": "6d374a2a",
"metadata": {},
"source": [
"Fix a Chern character $v$ with positive rank and $\\Delta(v) \\geq 0$"
]
},
{
"cell_type": "code",
"id": "ab162897",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f9075cd3e20>"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = Chern_Char(*var(\"R C D\", domain=\"real\"))\n",
"v"
]
},
{
"cell_type": "markdown",
"id": "06d8357b",
"metadata": {},
"source": [
"Let $u$ be a semistabilizer fitting problem 1 or 2 (destabilizing $v$ going down $\\Theta_v^{-}$)"
]
},
{
"cell_type": "code",
"id": "0d33d7e1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f906c28bac0>"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u = Chern_Char(*var(\"r c d\", domain=\"real\"))\n",
"u"
]
},
{
"cell_type": "markdown",
"id": "cb9c11e7",
"metadata": {},
"source": [
"# Bounds on $\\operatorname{ch}_2(u)=d$"
]
},
{
"cell_type": "code",
"id": "23d48b0b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle q\\)</html>"
],
"text/latex": [
"$\\displaystyle q$"
],
"text/plain": [
"q"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"q\", domain=\"real\") # Symbol for q=\\chern_1^{\\beta}(u)"
]
},
{
"cell_type": "markdown",
"id": "377c2843",
"metadata": {},
"source": [
"Express $c$ in terms of $q:=\\chern_1^{\\beta}(u)$"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [],
"source": [
"c_in_terms_of_q = solve(q == u.twist(beta).ch[1], c)[0]\n",
"assert c_in_terms_of_q.lhs() == c, \"Meant to be an expression for c\""
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle c = \\beta r + q\\)</html>"
"$\\displaystyle c = \\beta r + q$"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q"
]
},
{
"cell_type": "markdown",
"id": "5f6f18b1",
"metadata": {},
"source": [
"## $\\chern_2^{P}(u) > 0$"
]
},
{
"cell_type": "markdown",
"id": "8e635cb8",
"metadata": {},
"source": [
"For problem 2, this amounts to $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "1a169293",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d > \\frac{1}{2} \\, \\beta^{2} r + \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle d > \\frac{1}{2} \\, \\beta^{2} r + \\beta q$"
],
"text/plain": [
"d > 1/2*beta^2*r + beta*q"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"positive_radius_condition_with_q = (\n",
" (\n",
" (0 > - u.twist(beta).ch[2])\n",
" + d # rearrange for d\n",
" )\n",
" .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q\n",
" .expand()\n",
")\n",
"positive_radius_d_lowerbound = positive_radius_condition_with_q.rhs()\n",
"\n",
"positive_radius_condition_with_q"
]
},
{
"cell_type": "markdown",
"id": "b08c74f1",
"metadata": {},
"source": [
"Separate out the terms of the corresponding lower bound on $d$:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "8fe70d6b",
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"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, 0\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, 0\\right)$"
],
"text/plain": [
"(1/2*beta^2*r, beta*q, 0)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"positive_radius_lowerbound_terms = Object()\n",
"\n",
"positive_radius_lowerbound_terms.const = positive_radius_condition_with_q.rhs().subs(r==0)\n",
"\n",
"positive_radius_lowerbound_terms.linear = (\n",
" positive_radius_condition_with_q.rhs()\n",
" - positive_radius_lowerbound_terms.const\n",
")\n",
"\n",
"positive_radius_lowerbound_terms.hyperbolic = 0\n",
"\n",
"(positive_radius_lowerbound_terms.linear,\n",
" positive_radius_lowerbound_terms.const,\n",
" positive_radius_lowerbound_terms.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "900f332b",
"metadata": {},
"source": [
"## $\\Delta(u) \\geq 0$"
]
},
{
"cell_type": "markdown",
"id": "62529298",
"metadata": {},
"source": [
"Express this inequality in terms of $q$"
]
},
{
"cell_type": "code",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\leq {\\left(\\beta r + q\\right)}^{2} - 2 \\, d r\\)</html>"
],
"text/latex": [
"$\\displaystyle 0 \\leq {\\left(\\beta r + q\\right)}^{2} - 2 \\, d r$"
],
"text/plain": [
"0 <= (beta*r + q)^2 - 2*d*r"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"bgmlv2_with_q = ((0 <= Δ(u))\n",
" .subs(c_in_terms_of_q))\n",
"bgmlv2_with_q"
]
},
{
"cell_type": "markdown",
"id": "723511fa",
"metadata": {},
"source": [
"Rearrange expression for $d$"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}\\)</html>"
"$\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}$"
"d <= 1/2*(beta*r + q)^2/r"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_ineq = (bgmlv2_with_q\n",
" + 2*d*r # move d to rhs\n",
") / (2*r) # scale-out d coefficient (r>0)\n",
"\n",
"assert bgmlv2_d_ineq.lhs() == d, \"Should be ineq for d\"\n",
"\n",
"bgmlv2_d_ineq"
]
},
{
"cell_type": "markdown",
"id": "425bcb7c",
"metadata": {},
"source": [
"Keep hold of the upper bound for $d$:"
]
},
{
"cell_type": "code",
"id": "6ae4f2e7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*beta^2*r + beta*q + 1/2*q^2/r"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs().expand()\n",
"bgmlv2_d_upperbound"
]
},
{
"cell_type": "markdown",
"id": "988aaf5b",
"metadata": {},
"source": [
"Separate out the terms of this lower bound for d"
]
},
{
"cell_type": "code",
"id": "653a3340",
"metadata": {},
"outputs": [],
"source": [
"bgmlv2_d_upperbound_terms = Object()\n",
"\n",
"bgmlv2_d_upperbound_without_hyp = (\n",
" bgmlv2_d_upperbound\n",
" .subs(1/r == 0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_terms.const = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" .subs(r==0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_terms.linear = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" - bgmlv2_d_upperbound_terms.const\n",
").expand()\n",
"\n",
"bgmlv2_d_upperbound_terms.hyperbolic = (\n",
" bgmlv2_d_upperbound\n",
" - bgmlv2_d_upperbound_without_hyp\n",
").expand()"
]
},
{
"cell_type": "code",
"id": "326bb656",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, \\frac{q^{2}}{2 \\, r}\\right)\\)</html>"
"$\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, \\frac{q^{2}}{2 \\, r}\\right)$"
"(1/2*beta^2*r, beta*q, 1/2*q^2/r)"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(bgmlv2_d_upperbound_terms.linear,\n",
" bgmlv2_d_upperbound_terms.const,\n",
" bgmlv2_d_upperbound_terms.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "5cc08c62",
"metadata": {},
"source": [
"Sanity check:"
]
},
{
"cell_type": "code",
"id": "7ff937ff",
"metadata": {},
"outputs": [],
"source": [
"assert ( bgmlv2_d_upperbound\n",
"- bgmlv2_d_upperbound_terms.const\n",
"- bgmlv2_d_upperbound_terms.linear\n",
"- bgmlv2_d_upperbound_terms.hyperbolic) == 0, \"Error in terms separation\""
]
},
{
"cell_type": "markdown",
"id": "024e8c41",
"metadata": {},
"source": [
"## $\\Delta(v-u) \\geq 0$"
]
},
{
"cell_type": "markdown",
"id": "a2647f43",
"metadata": {},
"source": [
"Express this inequality in terms of $q$"
]
},
{
"cell_type": "code",
"id": "87544e6e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\leq {\\left(\\beta r - C + q\\right)}^{2} - 2 \\, {\\left(D - d\\right)} {\\left(R - r\\right)}\\)</html>"
],
"text/latex": [
"$\\displaystyle 0 \\leq {\\left(\\beta r - C + q\\right)}^{2} - 2 \\, {\\left(D - d\\right)} {\\left(R - r\\right)}$"
],
"text/plain": [
"0 <= (beta*r - C + q)^2 - 2*(D - d)*(R - r)"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_with_q = ((0 <= Δ(v-u))\n",
" .subs(c_in_terms_of_q)\n",
")\n",
"\n",
"bgmlv3_with_q"
]
},
{
"cell_type": "markdown",
"id": "d36504bb",
"metadata": {},
"source": [
"Rearrange in terms of $d$ assuming $r>R$"
]
},
{
"cell_type": "code",
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq D - \\frac{{\\left(\\beta r - C + q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle d \\leq D - \\frac{{\\left(\\beta r - C + q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"d <= D - 1/2*(beta*r - C + q)^2/(R - r)"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_ineq = (\n",
" (\n",
" bgmlv3_with_q\n",
" + 2*(D-d)*(R-r) # move d term to lhs\n",
" )/2/(r-R) # assume r>R\n",
") + D\n",
"\n",
"assert bgmlv3_d_ineq.lhs() == d, \"Should be bound for d\"\n",
"assert not bgmlv3_d_ineq.rhs().has(d), \"Should be bound for d\"\n",
"\n",
"bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs()\n",
"bgmlv3_d_ineq"
]
},
{
"cell_type": "markdown",
"id": "7bdaf020",
"metadata": {},
"source": [
"$\\renewcommand{\\psi}{\\chern_1^{\\beta}(v)}$\n",
"$\\renewcommand{\\phi}{\\chern_2^{\\beta}(v)}$\n",
"Redefine psi and phi in latex to be $\\psi$ and $\\phi$"
]
},
{
"cell_type": "code",
"id": "e64456bd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\psi, \\phi\\right)\\)</html>"
"$\\displaystyle \\left(\\psi, \\phi\\right)$"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ch1bv, ch2bv = var(\"psi phi\", domain=\"real\") # symbol to represent ch_1^\\beta(v) and\n",
"# ch_2^\\beta(v)\n",
"ch1bv, ch2bv"
"cell_type": "markdown",
"id": "31c2cee4",
"source": [
"Define expression for the different terms of this bound of $d$ in terms of $\\phi$ and $\\psi$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "d962282f",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\phi - \\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\phi - \\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
"1/2*beta^2*r + beta*q + phi - 1/2*(psi - q)^2/(R - r)"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_terms = Object()\n",
"\n",
"bgmlv3_d_upperbound_terms.linear = bgmlv2_d_upperbound_terms.linear\n",
"bgmlv3_d_upperbound_terms.const = ch2bv + bgmlv2_d_upperbound_terms.const\n",
"bgmlv3_d_upperbound_terms.hyperbolic = (ch1bv - q)^2/2/(r-R)\n",
"\n",
"(bgmlv3_d_upperbound_terms.linear\n",
" + bgmlv3_d_upperbound_terms.const\n",
" + bgmlv3_d_upperbound_terms.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "10d89cb2",
"metadata": {},
"source": [
"Verify that the expression above indeed is equal the upper bound on $d$ given by $\\Delta(v-u) \\geq 0$"
]
},
{
"cell_type": "code",
"id": "3598bfdf",
"metadata": {},
"outputs": [],
"source": [
"assert (\n",
" (bgmlv3_d_upperbound_terms.linear\n",
" + bgmlv3_d_upperbound_terms.const\n",
" + bgmlv3_d_upperbound_terms.hyperbolic\n",
" - bgmlv3_d_ineq.rhs())\n",
" .subs(ch2bv == v.twist(beta).ch[2])\n",
" .subs(ch1bv == v.twist(beta).ch[1])\n",
") == 0, \"Sanity check\""
]
},
{
"cell_type": "markdown",
"id": "97c575cd",
"metadata": {},
"source": [
"# Specialize to problem 2"
]
},
{
"cell_type": "markdown",
"id": "d35f737d",
"metadata": {},
"source": [
"Add extra attributes to the bound objects above with a specialization to the case $\\chern_2^{\\beta}(v)=0$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "2b45d576",
"metadata": {},
"outputs": [],
"source": [
"for bound_terms in [\n",
" positive_radius_lowerbound_terms,\n",
" bgmlv2_d_upperbound_terms,\n",
" bgmlv3_d_upperbound_terms\n",
"]:\n",
" bound_terms.problem2 = Object()\n",
" bound_terms.problem2.const = bound_terms.const.subs(ch2bv == 0)\n",
" bound_terms.problem2.linear = bound_terms.linear.subs(ch2bv == 0)\n",
" bound_terms.problem2.hyperbolic = bound_terms.hyperbolic.subs(ch2bv == 0)"
]
},
{
"cell_type": "markdown",
"id": "cbb5ee0f",
"metadata": {},
"source": [
"View the specialized bounds:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "cd81504c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + \\frac{q^{2}}{2 \\, r} \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
"$\\displaystyle 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + \\frac{q^{2}}{2 \\, r} \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
"0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + \\frac{q^{2}}{2 \\, r} \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}"
"output_type": "execute_result"
}
],
"source": [
"sum(\n",
" r\" \\\\ \" + (latex(bound_terms.problem2.const)\n",
" + \" + \" + latex(bound_terms.problem2.linear)\n",
" + \" + \" + latex(bound_terms.problem2.hyperbolic))\n",
" for bound_terms in [\n",
" positive_radius_lowerbound_terms,\n",
" bgmlv2_d_upperbound_terms,\n",
" bgmlv3_d_upperbound_terms\n",
" ]\n",
")"
]
},
{
"cell_type": "markdown",
"id": "2997ec1a",
"metadata": {},
"source": [
"## Plots for all Bounds on $d$"
]
},
{
"cell_type": "code",
"execution_count": 22,
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"id": "d7235dc3",
"metadata": {},
"outputs": [],
"source": [
"v_example = Chern_Char(3,2,-2)\n",
"q_example = 7/3\n",
"\n",
"def plot_d_bound(\n",
" v_example,\n",
" q_example,\n",
" ymax=5,\n",
" ymin=-2,\n",
" xmax=20,\n",
" aspect_ratio=None):\n",
"\n",
" # Equations to plot imminently representing the bounds on d:\n",
" eq2 = (bgmlv2_d_upperbound\n",
" .subs(R == v_example.ch[0])\n",
" .subs(C == v_example.ch[1])\n",
" .subs(D == v_example.ch[2])\n",
" .subs(beta = beta_minus(v_example))\n",
" .subs(q == q_example)\n",
" )\n",
"\n",
" eq3 = (bgmlv3_d_upperbound\n",
" .subs(R == v_example.ch[0])\n",
" .subs(C == v_example.ch[1])\n",
" .subs(D == v_example.ch[2])\n",
" .subs(beta = beta_minus(v_example))\n",
" .subs(q == q_example)\n",
" )\n",
"\n",
" eq4 = (positive_radius_d_lowerbound\n",
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" .subs(q == q_example)\n",
" .subs(beta = beta_minus(v_example))\n",
" )\n",
"\n",
" example_bounds_on_d_plot = (\n",
" plot(\n",
" eq3,\n",
" (r,v_example.ch[0],xmax),\n",
" color='green',\n",
" linestyle = \"dashed\",\n",
" legend_label=r\"upper bound: $\\Delta(v-u) \\geq 0$\",\n",
" )\n",
" + plot(\n",
" eq2,\n",
" (r,0,xmax),\n",
" color='blue',\n",
" linestyle = \"dashed\",\n",
" legend_label=r\"upper bound: $\\Delta(u) \\geq 0$\"\n",
" )\n",
" + plot(\n",
" eq4,\n",
" (r,0,xmax),\n",
" color='orange',\n",
" linestyle = \"dotted\",\n",
" legend_label=r\"lower bound: $\\mathrm{ch}_2^{\\beta_{-}}(u)>0$\"\n",
" )\n",
" )\n",
" example_bounds_on_d_plot.ymin(ymin)\n",
" example_bounds_on_d_plot.ymax(ymax)\n",
" example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])\n",
" if aspect_ratio:\n",
" example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)\n",
" return example_bounds_on_d_plot"
]
},
{
"cell_type": "markdown",
"id": "683ac3f7",
"metadata": {},
"source": [
"### Bounds on $d$ with Minimal $q=\\operatorname{ch}^{\\beta}_1(u)$"
]
},
{
"cell_type": "code",
"execution_count": 23,
"id": "f5b1d9bf",
"metadata": {},
"outputs": [
{
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\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bounds_on_d_qmin = plot_d_bound(v_example, 0, ymin=-0.5)\n",
"bounds_on_d_qmin"
]
},
{
"cell_type": "markdown",
"id": "24dd62c1",
"metadata": {},
"source": [
"### Bounds on $d$ with Maximal $q=\\operatorname{ch}^{\\beta}_1(u)$"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "47b30d7e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bounds_on_d_qmax = plot_d_bound(v_example, 4, ymin=-3, ymax=3)\n",
"bounds_on_d_qmax"
]
},
{
"cell_type": "markdown",
"id": "133ccbe7",
"metadata": {},
"source": [
"### Bounds on $d$ with Mid-way $q=\\operatorname{ch}^{\\beta}_1(u)$"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "25e4850b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"typical_bounds_on_d = plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)\n",
"typical_bounds_on_d"
]
},
{
"cell_type": "markdown",
"id": "1c6f5622",
"metadata": {},
"source": [
"# Bounds on Semistabilizer Rank $r=\\operatorname{ch}_0(u)$"
]
},
{
"cell_type": "markdown",
"id": "fdf7befd",
"metadata": {},
"source": [
"Express the two conditions corresponding to the upper bounds on $d$ from $\\Delta(u) \\geq 0$ or $\\Delta(v-u) \\geq 0$ being more than $\\frac{\\kappa}{2n^2}$ higher than the lowerbound given by $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "markdown",
"id": "6e5e2671",
"metadata": {},
"source": [
"$\\renewcommand\\Omega{\\operatorname{lcm}(2n^2,m)}$\n",
"Redifine \\Omega in latex to be $\\Omega$"
]
},
{
"cell_type": "code",
"id": "1377923b",
"outputs": [
{
"data": {
"text/html": [