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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 $\\newcommand{\\chern}{\\operatorname{ch}}$"
]
},
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{
"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",
"mu = stability.Mumford().slope\n",
"ts = stability.Tilt\n",
"\n",
"var(\"beta\", domain=\"real\")\n",
"\n",
"def beta_minus(v):\n",
" beta = stability.Tilt().beta\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": "cb9c11e7",
"metadata": {},
"source": [
"# Bounds on $\\operatorname{ch}_2(u)=d$"
]
},
{
"cell_type": "code",
"execution_count": 2,
"id": "4cacebc7",
"metadata": {},
"outputs": [],
"source": [
"v = Chern_Char(*var(\"R C D\", domain=\"real\"))\n",
"u = Chern_Char(*var(\"r c d\", domain=\"real\"))\n",
"\n",
"alpha = ts().alpha\n",
"beta = ts().beta\n",
"\n",
"\n",
"c_lower_bound = -(\n",
" ts(beta=beta).rank(u)\n",
" /alpha\n",
").expand() + c\n",
"\n",
"var(\"q\", domain=\"real\")\n",
"c_in_terms_of_q = c_lower_bound + q"
]
},
{
"cell_type": "code",
"execution_count": 3,
"id": "f53cee90",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta r + q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta r + q$"
],
"text/plain": [
"beta*r + q"
]
},
"execution_count": 3,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q"
]
},
{
"cell_type": "markdown",
"id": "900f332b",
"metadata": {},
"source": [
"## $\\Delta(u) \\geq 0$"
]
},
{
"cell_type": "code",
"execution_count": 4,
"id": "6ae4f2e7",
"metadata": {},
"outputs": [],
"source": [
"# First Bogomolov-Gieseker form expression that must be non-negative:\n",
"bgmlv2 = Δ(u)\n",
"bgmlv2_with_q = (\n",
" bgmlv2\n",
" .expand()\n",
" .subs(c == c_in_terms_of_q)\n",
")\n",
"# RENDERED TO LATEX: 0 <= bgmlv2_with_q\n",
"bgmlv2_d_ineq = (\n",
" (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0\n",
" + d # Rearrange for d\n",
").expand()\n",
"\n",
"# Keep hold of lower bound for d\n",
"bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()"
]
},
{
"cell_type": "code",
"execution_count": 5,
"id": "c0c12e43",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle d \\leq \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"d <= 1/2*beta^2*r + beta*q + 1/2*q^2/r"
]
},
"execution_count": 5,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_ineq"
]
},
{
"cell_type": "code",
"execution_count": 6,
"id": "653a3340",
"metadata": {},
"outputs": [],
"source": [
"# Seperate out the terms of the lower bound for d\n",
"\n",
"bgmlv2_d_upperbound_without_hyp = (\n",
" bgmlv2_d_upperbound\n",
" .subs(1/r == 0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_const_term = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" .subs(r==0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_linear_term = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" - bgmlv2_d_upperbound_const_term\n",
").expand()\n",
"\n",
"bgmlv2_d_upperbound_exp_term = (\n",
" bgmlv2_d_upperbound\n",
" - bgmlv2_d_upperbound_without_hyp\n",
").expand()"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "326bb656",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q$"
],
"text/plain": [
"beta*q"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_const_term"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "fcd9fa2a",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 9,
"id": "ade0f7b2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*q^2/r"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_exp_term"
]
},
{
"cell_type": "markdown",
"id": "024e8c41",
"metadata": {},
"source": [
"## $\\Delta(v-u) \\geq 0$"
]
},
{
"cell_type": "code",
"execution_count": 10,
"id": "b7799156",
"metadata": {},
"outputs": [],
"source": [
"# Third Bogomolov-Gieseker form expression that must be non-negative:\n",
"bgmlv3 = Δ(v-u)\n",
"bgmlv3_with_q = (\n",
" bgmlv3\n",
" .expand()\n",
" .subs(c == c_in_terms_of_q)\n",
")\n",
"var(\"r_alt\",domain=\"real\") # r_alt = r - R temporary substitution\n",
"\n",
"bgmlv3_with_q_reparam = (\n",
" bgmlv3_with_q\n",
" .subs(r == r_alt + R)\n",
" /r_alt # This operation assumes r_alt > 0\n",
").expand()\n",
"\n",
"bgmlv3_d_ineq = (\n",
" ((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d\n",
" .subs(r_alt == r - R) # Resubstitute r back in\n",
" .expand()\n",
")\n",
"\n",
"# Check that this equation represents a bound for d\n",
"assert bgmlv3_d_ineq.lhs() == d\n",
"\n",
"bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d\n",
"\n",
"# Seperate out the terms of the lower bound for d\n",
"\n",
"bgmlv3_d_upperbound_without_hyp = (\n",
" bgmlv3_d_upperbound\n",
" .subs(1/(R-r) == 0)\n",
")\n",
"\n",
"bgmlv3_d_upperbound_const_term = (\n",
" bgmlv3_d_upperbound_without_hyp\n",
" .subs(r==0)\n",
")\n",
"\n",
"bgmlv3_d_upperbound_linear_term = (\n",
" bgmlv3_d_upperbound_without_hyp\n",
" - bgmlv3_d_upperbound_const_term\n",
").expand()\n",
"\n",
"bgmlv3_d_upperbound_exp_term = (\n",
" bgmlv3_d_upperbound\n",
" - bgmlv3_d_upperbound_without_hyp\n",
").expand()\n",
"\n",
"# Verify the simplified forms of the terms that will be mentioned in text\n",
"\n",
"var(\"psi phi\", domain=\"real\") # symbol to represent ch_1^\\beta(v) and\n",
"# ch_2^\\beta(v)\n",
"\n",
"assert bgmlv3_d_upperbound_const_term == ( \n",
" (\n",
" # keep hold of this alternative expression:\n",
" bgmlv3_d_upperbound_const_term_alt := (\n",
" phi\n",
" + beta*q\n",
" )\n",
" )\n",
" .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
" .expand()\n",
")\n",
"\n",
"assert bgmlv3_d_upperbound_exp_term == (\n",
" (\n",
" # Keep hold of this alternative expression:\n",
" bgmlv3_d_upperbound_exp_term_alt :=\n",
" (\n",
" R*phi\n",
" + (C - q)^2/2\n",
" + R*beta*q\n",
" - D*R\n",
" )/(r-R)\n",
" )\n",
" .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
" .expand()\n",
")\n",
"\n",
"assert bgmlv3_d_upperbound_exp_term == (\n",
" (\n",
" # Keep hold of this alternative expression:\n",
" bgmlv3_d_upperbound_exp_term_alt2 :=\n",
" (\n",
" (psi - q)^2/2/(r-R)\n",
" )\n",
" )\n",
" .subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\\beta(v)\n",
" .expand()\n",
")"
]
},
{
"cell_type": "markdown",
"id": "9809bd85",
"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 $\\phi$"
]
},
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{
"cell_type": "code",
"execution_count": 11,
"id": "38bbbcd2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "6701d8a7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q + \\phi\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q + \\phi$"
],
"text/plain": [
"beta*q + phi"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_const_term_alt"
]
},
{
"cell_type": "code",
"execution_count": 13,
"id": "7a82d747",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"-1/2*(psi - q)^2/(R - r)"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_exp_term_alt2"
]
},
{
"cell_type": "code",
"execution_count": 14,
"id": "927e6929",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 14,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "ff25eea1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q$"
],
"text/plain": [
"beta*q"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_const_term"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "e8359c0f",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*q^2/r"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_exp_term"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "6823c5b8",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
],
"text/plain": [
"1/2*beta^2*r"
]
},
"execution_count": 17,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_linear_term"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "160871f2",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle \\beta q$"
],
"text/plain": [
"beta*q"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)"
]
},
{
"cell_type": "code",
"execution_count": 19,
"id": "1da23b29",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"-1/2*(psi - q)^2/(R - r)"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_exp_term_alt2"
]
},
{
"cell_type": "markdown",
"id": "2997ec1a",
"metadata": {},
"source": [
"## Plots for all Bounds on $d$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "d7235dc3",
"metadata": {},
"outputs": [],
"source": [
"positive_radius_condition = (\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",
"\n",
"\n",
"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": 21,
"id": "f5b1d9bf",
"metadata": {},
"outputs": [
{
"data": {
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\n",
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"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 21,
"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": 22,
"id": "47b30d7e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 22,
"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": 23,
"id": "25e4850b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
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"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 23,
"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": "code",
"execution_count": 24,
"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": 25,
"id": "990a2840",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle n^{2} q^{2}\\)</html>"
],
"text/latex": [
"$\\displaystyle n^{2} q^{2}$"
],
"text/plain": [
"n^2*q^2"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r_upper_bound1.subs(kappa==1).rhs()"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "5a3c7037",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R\\)</html>"
],
"text/latex": [
"$\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R$"
],
"text/plain": [
"n^2*(psi - q)^2 + R"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"r_upper_bound2.subs(kappa==1).rhs()"
]
},
{
"cell_type": "code",
"execution_count": 27,
"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": 28,
"id": "d33e3459",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*q^2/r"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound_exp_term"
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "3728c192",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"-1/2*(psi - q)^2/(R - r)"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_exp_term_alt2"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "90149eb2",
"metadata": {},
"outputs": [],
"source": [
"var(\"Delta nu\", domain=\"real\")\n",
"# Delta to represent bogomolov(v)\n",