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{
"cells": [
{
"cell_type": "markdown",
"id": "8ebd0216",
"metadata": {},
"source": [
"# Utilities"
]
},
{
"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 0x7f8bce292c50>"
"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 0x7f8bce293040>"
"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": "code",
"id": "e64456bd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left({\\operatorname{ch}_1^{\\beta}(v)}, {\\operatorname{ch}_2^{\\beta}(v)}\\right)\\)</html>"
"$\\displaystyle \\left({\\operatorname{ch}_1^{\\beta}(v)}, {\\operatorname{ch}_2^{\\beta}(v)}\\right)$"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# symbols to represent ch_i^\\beta(v) and\n",
"var(\"ch1bv\", domain=\"real\", latex_name=r\"\\operatorname{ch}_1^{\\beta}(v)\")\n",
"var(\"ch2bv\", domain=\"real\", latex_name=r\"\\operatorname{ch}_2^{\\beta}(v)\")\n",
"\n",
"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 - \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}} + {\\operatorname{ch}_2^{\\beta}(v)}\\)</html>"
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q - \\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}} + {\\operatorname{ch}_2^{\\beta}(v)}$"
"1/2*beta^2*r + beta*q - 1/2*(ch1bv - q)^2/(R - r) + ch2bv"
"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({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
"$\\displaystyle 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + \\frac{q^{2}}{2 \\, r} \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + -\\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
"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({\\operatorname{ch}_1^{\\beta}(v)} - 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 $\\frac{k_{v,q}}{2n^2}$ higher than the lowerbound given by $\\operatorname{ch}_2^{\\beta}(u) > 0$"
{
"cell_type": "code",
Luke Naylor
committed
"execution_count": 26,
"id": "1377923b",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{{k_{v,q}}}{{\\operatorname{lcm}(m,2n^2)}} < \\frac{q^{2}}{2 \\, r}, \\frac{{k_{v,q}}}{{\\operatorname{lcm}(m,2n^2)}} < -\\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{{k_{v,q}}}{{\\operatorname{lcm}(m,2n^2)}} < \\frac{q^{2}}{2 \\, r}, \\frac{{k_{v,q}}}{{\\operatorname{lcm}(m,2n^2)}} < -\\frac{{\\left({\\operatorname{ch}_1^{\\beta}(v)} - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)$"
],
"text/plain": [
"(k/lcm_m_2n2 < 1/2*q^2/r, k/lcm_m_2n2 < -1/2*(ch1bv - q)^2/(R - r))"
Luke Naylor
committed
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
"var(\"a_v b_q n\") # Define symbols introduce for values of beta and q\n",
"var(\"lcm_m_2n2\", latex_name = r\"\\operatorname{lcm}(m,2n^2)\") # more semantic variable name\n",
Luke Naylor
committed
"\n",
"beta_value_expr = (beta == a_v/n)\n",
"q_value_expr = (q == b_q/n)\n",
"# placeholder for the specific values of k (start with 1):\n",
"var(\"k\", domain=\"real\", latex_name=\"k_{v,q}\")\n",
"assymptote_gap_condition1 = (k/lcm_m_2n2 < bgmlv2_d_upperbound_terms.hyperbolic)\n",