{
"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 $\\originalDelta(v) \\geq 0$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"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 0x7f1bd72a2010>"
]
},
"execution_count": 2,
"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",
"execution_count": 3,
"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 0x7f1bccb53110>"
]
},
"execution_count": 3,
"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",
"execution_count": 4,
"id": "23d48b0b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle q\\)</html>"
],
"text/latex": [
"$\\displaystyle q$"
],
"text/plain": [
"q"
]
},
"execution_count": 4,
"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",
"execution_count": 5,
"id": "49be9bb7",
"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",
"execution_count": 6,
"id": "4cacebc7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle c = \\beta r + q\\)</html>"
],
"text/latex": [
"$\\displaystyle c = \\beta r + q$"
],
"text/plain": [
"c == beta*r + q"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q"
]
},
{
"cell_type": "markdown",
"id": "760b306b",
"metadata": {},
"source": [
"## $\\chern_2^{P}(u) > 0$"
]
},
{
"cell_type": "markdown",
"id": "1f866f7f",
"metadata": {},
"source": [
"For problem 2, this amounts to $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "b9bb6e1e",
"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_condition_with_q"
]
},
{
"cell_type": "markdown",
"id": "392671ee",
"metadata": {},
"source": [
"Separate out the terms of the corresponding lower bound on $d$:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "78a1301a",
"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",
"execution_count": 9,
"id": "5991acb3",
"metadata": {},
"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"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_with_q = ((0 <= Δ(u))\n",
" .subs(c_in_terms_of_q))\n",
"\n",
"bgmlv2_with_q"
]
},
{
"cell_type": "markdown",
"id": "723511fa",
"metadata": {},
"source": [
"Rearrange expression for $d$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "1f66d604",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}$"
],
"text/plain": [
"d <= 1/2*(beta*r + q)^2/r"
]
},
"execution_count": 10,
"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",
"execution_count": 11,
"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"
]
},
"execution_count": 11,
"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",
"execution_count": 12,
"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",
"execution_count": 13,
"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>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, \\frac{q^{2}}{2 \\, r}\\right)$"
],
"text/plain": [
"(1/2*beta^2*r, beta*q, 1/2*q^2/r)"
]
},
"execution_count": 13,
"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",
"execution_count": 14,
"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",
"execution_count": 15,
"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)"
]
},
"execution_count": 15,
"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",
"execution_count": 16,
"id": "8986af54",
"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)"
]
},
"execution_count": 16,
"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_ineq"
]
},
{
"cell_type": "markdown",
"id": "e7eb0057",
"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",
"execution_count": 17,
"id": "0acd0b7e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\psi, \\phi\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\psi, \\phi\\right)$"
],
"text/plain": [
"(psi, phi)"
]
},
"execution_count": 17,
"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": "0d880757",
"metadata": {},
"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": "f122a6d5",
"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>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\phi - \\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"1/2*beta^2*r + beta*q + phi - 1/2*(psi - q)^2/(R - r)"
]
},
"execution_count": 18,
"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": "9e813c96",
"metadata": {},
"source": [
"Verify that the expression above indeed is equal the upper bound on $d$ given by $\\originalDelta(v-u) \\geq 0$"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "10fb65aa",
"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": "a777246a",
"metadata": {},
"source": [
"# Specialize to problem 2"
]
},
{
"cell_type": "markdown",
"id": "9413516f",
"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": "dbc8e7a0",
"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": "95c39f2b",
"metadata": {},
"source": [
"View the specialized bounds:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "2e9a7cdc",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| 0\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| 0$"
],
"text/plain": [
"beta*q ' + ' 1/2*beta^2*r ' + ' 0"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"beta*q ' + ' 1/2*beta^2*r ' + ' 1/2*q^2/r"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"beta*q ' + ' 1/2*beta^2*r ' + ' -1/2*(psi - q)^2/(R - r)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"for bound_terms in [\n",
" positive_radius_lowerbound_terms,\n",
" bgmlv2_d_upperbound_terms,\n",
" bgmlv3_d_upperbound_terms\n",
"]:\n",
" pretty_print(bound_terms.problem2.const,\n",
" \" + \",bound_terms.problem2.linear,\n",
" \" + \",bound_terms.problem2.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "2997ec1a",
"metadata": {},
"source": [
"## Plots for all Bounds on $d$"
]
},
{
"cell_type": "code",
"execution_count": 22,
"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_condition.rhs()\n",
" .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": [
{
"ename": "NameError",
"evalue": "name 'bgmlv3_d_upperbound' is not defined",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mNameError\u001b[0m Traceback (most recent call last)",
"Cell \u001b[0;32mIn[23], line 1\u001b[0m\n\u001b[0;32m----> 1\u001b[0m bounds_on_d_qmin \u001b[38;5;241m=\u001b[39m \u001b[43mplot_d_bound\u001b[49m\u001b[43m(\u001b[49m\u001b[43mv_example\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mInteger\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m0\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mymin\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[38;5;241;43m-\u001b[39;49m\u001b[43mRealNumber\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43m0.5\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m)\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m 2\u001b[0m bounds_on_d_qmin\n",
"Cell \u001b[0;32mIn[22], line 21\u001b[0m, in \u001b[0;36mplot_d_bound\u001b[0;34m(v_example, q_example, ymax, ymin, xmax, aspect_ratio)\u001b[0m\n\u001b[1;32m 4\u001b[0m \u001b[38;5;28;01mdef\u001b[39;00m \u001b[38;5;21mplot_d_bound\u001b[39m(\n\u001b[1;32m 5\u001b[0m v_example,\n\u001b[1;32m 6\u001b[0m q_example,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 11\u001b[0m \n\u001b[1;32m 12\u001b[0m \u001b[38;5;66;03m# Equations to plot imminently representing the bounds on d:\u001b[39;00m\n\u001b[1;32m 13\u001b[0m eq2 \u001b[38;5;241m=\u001b[39m (bgmlv2_d_upperbound\n\u001b[1;32m 14\u001b[0m \u001b[38;5;241m.\u001b[39msubs(R \u001b[38;5;241m==\u001b[39m v_example\u001b[38;5;241m.\u001b[39mch[Integer(\u001b[38;5;241m0\u001b[39m)])\n\u001b[1;32m 15\u001b[0m \u001b[38;5;241m.\u001b[39msubs(C \u001b[38;5;241m==\u001b[39m v_example\u001b[38;5;241m.\u001b[39mch[Integer(\u001b[38;5;241m1\u001b[39m)])\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 18\u001b[0m \u001b[38;5;241m.\u001b[39msubs(q \u001b[38;5;241m==\u001b[39m q_example)\n\u001b[1;32m 19\u001b[0m )\n\u001b[0;32m---> 21\u001b[0m eq3 \u001b[38;5;241m=\u001b[39m (\u001b[43mbgmlv3_d_upperbound\u001b[49m\n\u001b[1;32m 22\u001b[0m \u001b[38;5;241m.\u001b[39msubs(R \u001b[38;5;241m==\u001b[39m v_example\u001b[38;5;241m.\u001b[39mch[Integer(\u001b[38;5;241m0\u001b[39m)])\n\u001b[1;32m 23\u001b[0m \u001b[38;5;241m.\u001b[39msubs(C \u001b[38;5;241m==\u001b[39m v_example\u001b[38;5;241m.\u001b[39mch[Integer(\u001b[38;5;241m1\u001b[39m)])\n\u001b[1;32m 24\u001b[0m \u001b[38;5;241m.\u001b[39msubs(D \u001b[38;5;241m==\u001b[39m v_example\u001b[38;5;241m.\u001b[39mch[Integer(\u001b[38;5;241m2\u001b[39m)])\n\u001b[1;32m 25\u001b[0m \u001b[38;5;241m.\u001b[39msubs(beta \u001b[38;5;241m=\u001b[39m beta_minus(v_example))\n\u001b[1;32m 26\u001b[0m \u001b[38;5;241m.\u001b[39msubs(q \u001b[38;5;241m==\u001b[39m q_example)\n\u001b[1;32m 27\u001b[0m )\n\u001b[1;32m 29\u001b[0m eq4 \u001b[38;5;241m=\u001b[39m (positive_radius_condition\u001b[38;5;241m.\u001b[39mrhs()\n\u001b[1;32m 30\u001b[0m \u001b[38;5;241m.\u001b[39msubs(q \u001b[38;5;241m==\u001b[39m q_example)\n\u001b[1;32m 31\u001b[0m \u001b[38;5;241m.\u001b[39msubs(beta \u001b[38;5;241m=\u001b[39m beta_minus(v_example))\n\u001b[1;32m 32\u001b[0m )\n\u001b[1;32m 34\u001b[0m example_bounds_on_d_plot \u001b[38;5;241m=\u001b[39m (\n\u001b[1;32m 35\u001b[0m plot(\n\u001b[1;32m 36\u001b[0m eq3,\n\u001b[0;32m (...)\u001b[0m\n\u001b[1;32m 55\u001b[0m )\n\u001b[1;32m 56\u001b[0m )\n",
"\u001b[0;31mNameError\u001b[0m: name 'bgmlv3_d_upperbound' is not defined"
]
}
],
"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": null,
"id": "47b30d7e",
"metadata": {},
"outputs": [],
"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": null,
"id": "25e4850b",
"metadata": {},
"outputs": [],
"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": "code",
"execution_count": null,
"id": "553bba31",
"metadata": {},
"outputs": [],
"source": [
"var(\"a_v b_q n\") # Define symbols introduce for values of beta and q\n",
"beta_value_expr = (beta == a_v/n)\n",
"q_value_expr = (q == b_q/n)\n",
"# RENDERED TO LATEX: positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()\n",
"# placeholder for the specific values of k (start with 1):\n",
"var(\"kappa\", domain=\"real\")\n",
"\n",
"assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_exp_term)\n",
"assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_exp_term_alt2)\n",
"\n",
"r_upper_bound1 = (\n",
" assymptote_gap_condition1\n",
" * r * 2*n^2 / kappa\n",
")\n",
"\n",
"assert r_upper_bound1.lhs() == r\n",
"\n",
"r_upper_bound2 = (\n",
" assymptote_gap_condition2\n",
" * (r-R) * 2*n^2 / kappa + R\n",
")\n",
"\n",
"assert r_upper_bound2.lhs() == r"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "990a2840",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound1.subs(kappa==1).rhs()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "5a3c7037",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound2.subs(kappa==1).rhs()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "82add957",
"metadata": {},
"outputs": [],
"source": [
"var(\"epsilon\")\n",
"var(\"chbv\") # symbol to represent \\chern_1^{\\beta}(v)\n",
"\n",
"# Tightness conditions:\n",
"\n",
"bounds_too_tight_condition1 = (\n",
" bgmlv2_d_upperbound_exp_term\n",
" < epsilon\n",
")\n",
"\n",
"bounds_too_tight_condition2 = (\n",
" bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)\n",
" < epsilon\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d33e3459",
"metadata": {},
"outputs": [],
"source": [
"bgmlv2_d_upperbound_exp_term"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "3728c192",
"metadata": {},
"outputs": [],
"source": [
"bgmlv3_d_upperbound_exp_term_alt2"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "90149eb2",
"metadata": {},
"outputs": [],
"source": [
"var(\"Delta nu\", domain=\"real\")\n",
"# Delta to represent bogomolov(v)\n",
"\n",
"q_sol = solve(\n",
" r_upper_bound1.subs(kappa==1).rhs()\n",
" == r_upper_bound2.subs(kappa==1).rhs()\n",
" , q\n",
")[0].rhs()\n",
"\n",
"r_upper_bound_all_q = (r_upper_bound1.rhs()\n",
" .expand()\n",
" .subs(q==q_sol)\n",
" .subs(kappa==1)\n",
" .subs(psi**2 == Delta/nu^2)\n",
" .subs(1/psi**2 == nu^2/Delta)\n",
")"
]
},
{
"cell_type": "markdown",
"id": "818e07fc",
"metadata": {},
"source": [
"$\\let\\originalDelta\\Delta$\n",
"$\\renewcommand\\Delta{\\originalDelta(v)}$\n",
"Redefine \\Delta in latex to be $\\Delta$"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8b0b0bf4",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound_all_q.expand()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "53bd2a9c",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound1.subs(kappa==1).rhs()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "cfa6e1af",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound2.subs(kappa==1).rhs()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "801a348a",
"metadata": {},
"outputs": [],
"source": [
"q_sol.expand()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "d4bf7486",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound_all_q.expand().subs([nu==1,Delta==psi^2])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "896d26dd",
"metadata": {},
"outputs": [],
"source": [
"c_in_terms_of_q.subs([q_value_expr,beta_value_expr])"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "51f22f7d",
"metadata": {},
"outputs": [],
"source": [
"rhs_numerator = (positive_radius_condition\n",
" .rhs()\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor()\n",
" .numerator()\n",
")"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "8148f5cd",
"metadata": {},
"outputs": [],
"source": [
"(positive_radius_condition\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor())"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "af5315c8",
"metadata": {},
"outputs": [],
"source": [
"var(\"delta\", domain=\"real\") # placeholder symbol to be replaced by k_{q,i}"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "e3e75309",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound1.rhs()"
]
},
{
"cell_type": "code",
"execution_count": null,
"id": "203b216b",
"metadata": {},
"outputs": [],
"source": [
"r_upper_bound2.rhs()"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.8",
"language": "sage",
"name": "sagemath"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.11.3"
}
},
"nbformat": 4,
"nbformat_minor": 5
}