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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 0x7fb73d92ee60>"
"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 0x7fb73d92f1f0>"
"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": "code",
"execution_count": 26,
"id": "1377923b",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{\\kappa}{2 \\, n^{2}} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{2 \\, n^{2}} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{\\kappa}{2 \\, n^{2}} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{2 \\, n^{2}} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)$"
],
"text/plain": [
"(1/2*kappa/n^2 < 1/2*q^2/r, 1/2*kappa/n^2 < -1/2*(psi - q)^2/(R - r))"
]
},