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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 0x7f0ea1caf450>"
"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 0x7f0e97d8bfd0>"
"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": [
Luke Naylor
committed
"$\\renewcommand\\Omega{\\operatorname{lcm}(m, 2n^2)}$\n",
"Redifine \\Omega in latex to be $\\Omega$"
]
},
{
"cell_type": "code",
Luke Naylor
committed
"execution_count": 26,
"id": "1377923b",
"outputs": [
{
"data": {
"text/html": [