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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": [
Luke Naylor
committed
"<pseudowalls.chern_character.Chern_Char object at 0x7f5ef5b6f190>"
"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": [
Luke Naylor
committed
"<pseudowalls.chern_character.Chern_Char object at 0x7f5ee792a3d0>"
"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": [
{
Luke Naylor
committed
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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": {
Luke Naylor
committed
"image/png": 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fYdYsqFPHXCC4qmPWOVmWnJbMZxs+4/cdM1hY8AzlQptyqN8hPN0du2RSC52FHntM6+6IiIhzSE6G6dOhUiXo3Bni480JD5A7ijnDMFi2bhgbvsnPkEUDKFX4XmJabYMiDzp8MQeaFCEiIiK3YRjm2O/16+GJJ+Cdd6BmTatTZaPUBDot6MmmvbOYUboQ+e/7D5VKNLU6VZaooLNYfDxcuaI16URExLFcuQJffglPPQVFi5ozWEuVyh2tcdfsO7+P0IMTKHB5I8sqfIibmwfNyzS3OtYdUZerxWrVMqd2i4iIOIKEBPj4YyhTBvr3h2XLzMcffTT3FHPnLu7i7Z87c89n9zDl5DGo+CotyrRw2mIONCnCcg0bQnS01SlERETgp5/gpZfg8mVznNyQIRAWZnWq7JOYksj41Z/Q5shQaqR4MOr+UfS9ty94+lgd7a6poLNYkybmfnaXL0P+/FanERERV3PhgrkuarVqULKkuQTJ4MFQurTVybKRLQ2OfsNlv2qMXTMOnyrP8ELjYQTlL291smyjMXQWO3TI/O3n55/N5mwREZGccPq02bU6ZQpUqQJr11qdyD4WHVrEF2tG84PPVtxqfERsiWcI8M59NYXG0FmsbFnzt6DDh61OIiIiruDyZejTx/y/Z+pUePVVs1Eht9l75BeWzSpCu1mtOJuSwvEmSyCse64s5kAtdA4hNRU81fktIiJ2dO6cuaJCUhLce685e7VPn1w43MewMfLPUUz78x0WlsjDiarjaFGjD27OvgfZbThkGREREUFERARpaWlWR8kRnp7mXniGAR4eVqcREZHcZPt2+PBD+PFH2LfPXHpkyxbn32P1f8UmxZJ8YCrBJ+bQqMKHFHgwgrI1X6Kip2us4K8WOgfw99/m+IXZs6FVK6vTiIhIbrB+vbks1oIFZhH35pvwwgvg4/wTOjNISUnk2w0TeHPVeDoUDWNileZQ5V3wyGVv9DYcsoXO1RQtCr6+sHixCjoREblz15po3Nxg4kTYuxdmzoSOHcHLy9Jo2c4wDH7c+yP+656nRGoCj1ToyhvNR0BAcaujWUItdA6ia1fYvNlsGhcREckKw4CFC80WuZdeMv9PuXwZ/P1z6VCe82u57JaXUl/eR9diFelVfyAVyz9tdSpLaZarg2jVCnbsgFOnrE4iIiLOwmaDefOgTh14+GFzkl2pUuax/PlzXzF3+NJhevzYBduKduQ//h3bX97OhOfXu3wxB+pydRgPPGA2ka9aZc48EhERuZ1586B9e2jeHJYsgRYtct9kB4CLlw+xZVkHnt2zlTy+hdjZZhLVyrWjlHsuq1jvgrpcHcipU1CkiNUpRETEUSUnm7sLHTgAH30EKSmwaRPUr291Mvv5z87/8O5v3YkuHM/yIi/weLNJ5PXKa3Ush6MuVwdyrZhz3BJbRESskJhoTnIoVw569IC//oK0NHOiQ24s5myGjUt/zYelLSjpX4iW9zyPR7tjdGw5XcXcTajL1YEcPAitW8P//R/UrGl1GhERcQRXr0LFimYvTseOMGQIVK5sdSr7WXHwNwYse49y7knMrlCBBoXuoUGpZlbHcngq6BxIyZLm3np//KGCTkTElZ05A59/bq4d5+MDo0ZBw4bmdpG51e5zu9m9uA0lrh7G070ufVp+hlup+6yO5TQ0hs7BtGkDCQmwbJnVSUREJKcdOQJjx8JXX5ndqYsX584u1QyunCY5JZ6SUxtzX14PBlRrT/2Gn+DmrlFhWaEWOgfz4IPw+usQHw9+flanERGRnDJ8uHkrUADeeQd69za/zq3ik+MZv/oTBsd9T56CtVnYaSGVgyvj7eltdTSnpBY6B7N/vzlW4pdf4JFHrE4jIiL2tHYtBARAeDj8+iscOgTdukG+fFYns5/U1Kusju5B7+0L2Z8Qw+8PjaRF1W6QJxdXrznAIQu6iIgIIiIiSEtLY//+/S5V0AFs3QrVqoFam0VEch/DMLtSR42CqCjo0wcmTbI6Vc5Yc3wNA355gV/89vGD9720enA2pfOXtjpWruCQJUPv3r3ZvXs3GzZssDqKJWrUUDEnIpIbbd8OdeuauwPFx8PcuTBhgtWp7O/qxe2wsgN+7m7kzVeMY02W0PPptSrmspHKBgd07Jg5m2nLFquTiIjI3UpO/mef7tBQCA6GRYtg/Xp44onc/Qv8sctHeH7+87Sb8zRG7F6q+uVnaeel1Cx1v9XRch1NinBARYrAnj0wf76WLxERcVYJCTBtGnz8sbmW3IkTULgwLFxodTL7u3z1MksWPUvp83+w7GIIQ5uPwFbjRTw8VHbYSy7+vcB5eXlB27bmHn0iIuJcUlJgxAgoVcpctaBZM1i+HPLksTpZDkhLxkg8SbOZzZiwfznxwU3Y13sXPWr3UDFnZw45KeIaV5zles1PP8Hjj8O+fVChgtVpRETkds6dg5AQc9JD8+ZQpQoMHAilS1udzP4Mw+CH3T/wyMnPyevhzsqwoZQrUI4i/tqgPKeohc5BPfgg5M0LP/5odRIREbmVQ4egZ08oXhxWrQI3N3Nx+MmTXaOY275zCo/PqM3TPzzNfI/KUOtTGpdsrGIuh6n900H5+kJ0tPkbnoiIOJ5t22D0aJgzx5zoMGzYP5/ZuXmiwzVHLx/lzYWv8mnyAtq4F+bVzktpUaaF1bFclgo6B1anjtUJRETk3wwDUlPNsc4//ABr1phryL3wgvmLuCuwXT2H++6PSC3agU1nd7G+wTherP0q7u4qKaykMXQOzDDg6afN7tfu3a1OIyLiutLSzJUHxoyB1q3NLboSE82JDp4uUsdcSbnCp2s/Zfme7/gj5BJuDb/FFtoEdzcXaI50AvpbcGBubhAXB99/b3USERHXdPUqfPEFVKoE7duDv785axXMcc6uUMzZDBt/rHyTrbPy80HUe4SXvJ/41tuhUDMVcw7EBX4UndsTT8Arr8D58+YYDRERsT/DMH+p3rTJ/Ax+8kn47jtzlweXkhLH0/NfYPehuUwvXZyd3edRprCrfROcg7pcHdyZM1C0KEyZAj16WJ1GRCR3O3ECxo83JzwsXmwWdceOQcmSVifLWdvPbKfkoQnkj9vFsvIj8fXKS4MSDayOJbegtlIHV6gQPPAAzJ5tdRIRkdxr925zYkPZsvDVV3DvveYCweBaxdypMxsZOP9panxeg2nnLkH4G7Qo00LFnBNwyBa6iIgIIiIiSEtLY//+/S7dQgfm4sJBQeaClSIikr1SUsw15PLkgf79zUlo/v5Wp8pZcUlxfLRyNM8dH8X21DycqzaOnrV74uXhZXU0ySSHLOiuUZeriIhkN5sNfv4ZPv3UnHRWuDBs325OfHCJ7bn+zZYCh6ZxIrAe1Wa05P1qj9Gl8TAC/V2oWTKXUJerk/jqK7PrVURE7kxyMsyYYS7++/jj5npyFy6Yx6pVc61izjAMftr7E+2+vR9j2zsUj9/F8f7H6ffQDBVzTkoFnZMoWBCWLoVdu6xOIiLinHr1ghdfNPfHXrUK/vwT7rnH6lQ5b/v+/xD1bSgd5zxOvJs3J5tFQ9nO5MuTz+pochfU5eokkpLMboFevWDkSKvTiIg4vlOnYOJEc4LD44+b45ENw+xadUm2VN6NGsasNR/wcwlfLtScyH1VuuHm5mZ1MskGWofOSXh7m4tafvstjBjhGvsEiojciZ074eOPYdYs8PExJzwAVKxobS6rXLxyEdv+KQSfmk+LCqMpU6AMlap1xsNDJUBuorLAibzwgrke0rp1VicREXFMv/4KVavCkiUwahQcPw69e1udyhpXk2KYsuI9yk0sx4e7/oDij9O81H28WPNFFXO5kLpcnYhhmF0GLttdICLyP1JS4D//gb/+grffNrfqmjfP7NHwctEVN2yGjdk7Z1N0YzeSUq/yU9FXGNp0KIX8ClkdTexIBZ0TMgxzo2hX2ENQRORGYmLgyy9hwgRzd4e2bWH+fA1H4Uw0Fz3yU/rL+3i5dA1ebjiYsqUetjqV5ACVBE4mJQVq14aXXzYnSIiIuJrYWChdGhIS4LnnYMAAs5vVle05t4cxK4YznWUElX+ZPb33UCygmNWxJAfZ9XeZFStW0KZNG4oWLYqbmxs//vijPS/nEry8oEwZ8zdTx21bFRHJXlu2QN++5i+1AQEwaRIcPWquK+fKxdyZC7tY+kM1Gn9RhegT69hTcxpUHapizgXZtaBLSEigevXqTJ482Z6XcTk9esDWrbB+vdVJRETsxzDg99/h/vuhVi345Rc4csQ81qkTFC1qbT6rfb3taxpNq0ulxJ1MadSLPb33cE/ZNuDm6v3OrsmuXa4PPfQQDz30kD0v4ZJat4ZSpeDzz831lUREcqP27WHuXKhTx5z48MQTGjucZkvj8pE5FDw8hbDyw3m8Zi/yNnqTp/Nps29X51D/NJKSkkhKSkq/Hxsba2Eax+XhAT17wsyZ5tY1rv4BJyK5w8WL5i+qbdua23P17An9+sF994Grr31r2Gws2TePAVHDqObtzrcV76FhkWo0LN3M6mjiIByqFBg1ahTDhg2zOoZTeO01GDRIxZyIOL9Dh8zZqtOnmzP4ixY1C7qWLa1O5hi2nNrCkaVtKZJ0goK+TXit5TjcitW1OpY4GIfqaB8yZAgxMTHpt+PHj1sdyWH5+prF3JEj5gegiIgzmj4dypeH776DgQPNxdO7drU6lYNIOMbVmH08+O2D/F+8O+5V3mZ55+XUVTEnN+BQ7Tve3t54e3tbHcNpHD5sfhDOmQNPPml1GhGR20tONsfDububS47cf7/Zzfr88+YvqgIxV2MYs/IjhiX+Hz6hjVnWeRmVQyrj6e5Q/2WLg3GoFjrJmrJloUkTGDtWS5iIiGM7fx5GjjTXj+vcGZYvNx8vXdqcua9iDpKT44la9BT3Ti7Dp+snsKZkP6g9kaqFqqqYk9uy609IfHw8Bw8eTL9/5MgRtm7dSlBQECVLlrTnpV3GwIHw6KOwejU0amR1GhGR6+3ebS6IDmYx9+qrEB5ubSZHE300moG/vMDCwCMMLNOMhx+cRVF/F1+XRbLErlt/RUVF0bx58+se79KlCzNnzrzt87X11+3ZbObg4YoVzW1vRESsZhiwaBGsWGG2yhkGRERAhw4QHGx1OseSeG49efeMZmvZgby9YiRjm75NeLGGVscSJ6S9XHOBr7+GP/+EL77QPoYiYp0rV+Dbb+HTT81WuVq1IDoa/PysTuZ4Dpzbw5Dl75J4aQe/ls6PW8PvwL+c1bHEiamgExGRu5aWBhUqmDPvH3sM+vfX+nE3cj7xPMv+eIpyl6J5MqY4w1uMpFO1Trhrdwe5SxplmUtcvWq21D3+OISGWp1GRFzBli0weTJ8/DHkzw/jxkG1alBODU3XS72CkXSRJjNbUijpGEMrtmLPc//B1zvQ6mSSS6igyyUSEszfiP/+G7Q2s4jYS1qauafq+PFmd2qpUnDwoLk9V7t2VqdzPDbDxrfbv6X9mWn4evkxve10woLCCNFWXZLN1MabSxQsCC+9BJMmgXZMExF76dbN7AlISYH/+79/ijm53qYtY3j4y6p0+bELv/jUhdoTaFCigYo5sQuNoctFTp4016Z79114+22r04hIbrBrl/mLYrt20KoVbN5s7iFdr57VyRzXgQsHGLSwN1+kLeYnowRVWsymYQnNXBX7UgtdLlK0KHTvbo5nUSudiNyptDT48UdzF4cqVWDBArh40TxWq5aKuZtJSzwFG/thJF9m3+XjbK76Gd2fP6piTnKEQ46hi4iIICIigjRtUpplgwebCwxrmQARuVNffgmvvAINGsD338MTT0CePFanclxxV2MZu2Yca/fP5Y9C8VQo1YFdvXZp5qrkKHW55mI2m9alE5Hb27nT7FYtUABGj4aYGDhwQGPjbifVlsoff75G8OGptDzpTu97+/PufUPI663/ryTnOWQLndy9bt0gIMCciSYi8r9SU+Hnn2HiRIiKModsvP66eSwwUMXcLRkGpFzmsR86ceyv35hStiy7ev5IieCqVicTF6YWulzqgw9gxAjYv99cVkBEBP5pud+9G+65xxyi0bev2a3q5WV1Ose34e8NVDgygcDEw0SVH0mgT35qFqlpdSwRFXS5VXy8ubhny5bmVjwi4tq2bze7VTduhE2bzKJu3z5zH2i5veN/r2DMmnFM3vUz42u157XaL0KRVtoKQxyGRljlUn5+ZivdrFmwbp3VaUTECqmp5qSG++6D6tXh99/hqafMNeRAxVxmXLpyiYELB3B1cVMaXF7CtDbT6PvI91C0tYo5cSgq6HKxF180lxfYts3qJCKSk86fN/90c4MhQ8yu1DlzzH1W334bvL2tzecU0pJg76fExx5h5vavWVa8F491PEq3Wt3wcPewOp3IddTlmsulpYGHPntEcj2bDZYsgYgI+O03szu1bFlzxmqgtgvNNMMw+M+u//Dthgn87H8At7oRXCnaFl8vX6ujidySWuhyOQ8PSEqCsWO12LBIbvXZZ2b3aatWcPQoTJ4MoaHmMRVzmbdp9wxWfBPEi/OexcM3lLP3r4FSz6iYE6egZUtcwNmzMGwYHD9uLlEgIs5v0yaziPPzM7fnqlcPZs6Ehg01tCvLbCkMXDKE+Rs/Zl4JP5Y//R33VnrW6lQiWaIuVxcxfry5xtSaNXDvvVanEZE7ceWKORbus89g/Xr46it44QVzWTQVcVl3Jv4Mngc+o+CZX1lWYRRnEy/w9D1Pa4cHcUoq6FxEairUr2/Obtu4UetNiTibL780t/a7eBFat4ZeveDhhzVG9k4kXjnPl2vH8M7aKfQqW4ePqj0KFfuBuz4YxXmpy9VFeHrC1KlQt6656Xb79lYnEpFbSUoy/62WLWv+uy1UyJy53rMnhIVZnc45pdnSiNwWScWtr1A+NYUetfrzZpO3wTfI6mgid00tdC5m505zdXh1z4g4pgMHzF++Zs40lx95910YPtzqVE7OMODUH5zzKkS5aU3pV+5eejR6h5LFmlqdTCTbOGQLXUREBBEREaSlpVkdJdepUsX8c8kSaNHCXC1eRBzDrFnQqRMEBUGXLtC9O1SubHUq57b19FbGrRjG1x4rCanUnwN9D1DIr5DVsUSynVroXNCuXWZhN348vPaa1WlEXNf+/ebYuKAgcwHgs2dh8WJ48knw8bE6nXP7+8wm9kQ/x1P79lEkqBI/tplIxRIPqHtCci21z7ige+4xC7nBg83iTkRyTlISzJ5ttpBXrGjOVL3WGREaCs89p2LurhgGX276kqYzGlHh6gG+ajaIHa/soGLJlirmJFdTC52LunIF6tQxZ8itWwe+WjdTxK4SEyFvXti8GWrXhiZNzAkOTzyhAi47pKSlEHv4OwoemcrK8h+w6OhyBjUYgL9PfqujieQIFXQubOdOczHSt96Cd96xOo1I7pOYCHPnmq1wly+bxZybGxw6BOXKWZ0udzBsNn7Z9Q2vR42kgb8fkZVqQq1PII+2yBDX4pCTIiRnVKliTo6oXdvqJCK5y6VL5pi47783t9xr3hwGDvxnAWAVc9ljzfE1nF3ejuCUM5QLasXAB8ZCoapWxxKxhFroBICtW81u14oVrU4i4pzOn4eVK+Hxx80FvBs2NBcA7tpVBVy2izvIlbRUik9txKMFguhXuxu1aw22OpWIpVTQCYZhttIlJZlbg+lbLZI5aWmwaJHZpfrTT+YC3qdP69+QvZxPPM9Hf45i9NV5eBR5kD1lXqNCwQp4uGu7DBHNchXc3OC77+DECXOGnZb/E7m9lBSzRfvhh2HvXhgzBv76S8WcPVy5eonlvz1Cvcll+GLzl6wrOxhqfUrlkMoq5kT+SwWdAFCpkrmUwm+/mSvTi0hGsbFmS9wjj8DVq+Z+yIMHw/r1sH27uRRQSIjVKXOfxYcWU//zKlS98BtvV2jMoX6HaFi1J3hqar7Iv6mgk3QPPWS2Mkyfbg7qFhH44w/o2NHcS/WllyA52VwAGMz7detqeTN7iD+zEqLaEJTHl7Ci9bn04Ca6Pf47IflUNYvciMbQSQaGAWfOQOHCVicRsc6/9zxu2BBiYsytuDp2hOLFrU6Xu+0+vZVBy97GLe4QP5cNxa3hN5CvlNWxRByeWugkAzc3s5iLi4O2bc1JEiKu4PRp+OQTqFEDqlb952f/11/NAu+NN1TM2dOpuFPM+aE+V36ryYHze+l63wfwQLSKOZFMUkEnN+TpaXa7Pvoo7NljdRoR++rRwyzWhgyB8uVhwQKzKxWgQAF1qdpVSjy2hGPcN+M+pp3YQ3yxdux8ZQdPhT+Fm77xIpnmkF2uERERREREkJaWxv79+9XlapFLl8ztiS5dguXLzf/oRJydzQarVsHXX8Pbb0Pp0ua40dRUePpps4AT+0u1pTJ983S6np+Bt28o68q9TcXgiuTXVl0id8QhC7prNIbOeqdOmZuIX74Mu3ZBUJDViUTuzI4dMGuWOZv7r7/MQm7GDGjWzOpkrsWw2Vi/4X0Gb/qeqHMHmX//EB6v2VNdqyJ3SVt/yS0VKQLR0TB/voo5cT4HDphjQv39zfFxCxZA+/bm5IbGjcFdg05y1O5zu3n9155Euq/kWf9yfNxuE7WK1LI6lkiuoI8zua3QUOjZ0/x65kzYssXSOCK3dOIEfPwx1KkDFSrA3Lnm42PHmhMfPv/cHEqgYi7npCYcg/Uv45GawMmrseysHUn35/armBPJRupylUxLTTVbNfbtMxcgbtDA6kQiGb35plm45clj7uDw7LPmxB5frUFriUuJF/lw5Si2Hv6FRUWScWs8GyOojiY7iNiBfkeVTPP0NBdZrVrVHFf3f/9ndSJxZZcuQWSkWbgtXmw+1rKluZvDmTMwb57ZvapiLuclpSaxYMmLHPxPKF9u/Iz7Kj9L0kPboWBdFXMidqIxdJIlgYHmZuQvvmjOCPz+e+jQwepU4kp+/hk++wyWLDH3HW7U6J/u0wcesDabyzMMSL7II/95hvMnlzGxXCX2vfIThQpoiryIvamgkyzz8TFnC9arBw8+aHUaye3OnoUffzTHvVWqZK6LePUqfPoptGsHRYtanVAA/vzrT2ocnYh/8ineve8DQvwmEh4SbnUsEZehMXRy106cgM6dzVaTSpWsTiO5walTZpfpDz/AihXmY59/Dt27m41A6rVzHIf/+p1Rq8czbf9iIuo+R6/a3aBQc6tjibgcjaGTu5aQYP4HXLeuucaXyJ3YuxcuXDC//ugjeO018PaGL74wZ6d2724eUzHnGM4lnKP3Ly/jFvUwTRLWMOuJWbz80Ncq5kQsohY6yRbx8ebSJt99B716wbhxGowut2azwdq18NNP5m3fPpgwAfr1Mwu4PHm09qEjMlIScNs/iWMhD1Lv64cZXacTHRq8g493fqujibg0FXSSbQwDpk41NzFfuxYqV7Y6kTiaK1fMCQze3uYvAFOnQkgItGkDjz1mTmrIm9fqlHIjabY0vt72NXM2R/Bb4F+43TuV5KKPkMcjj9XRRAQVdGIHMTHmbNirV80FXl97DfLlszqVWOXYMfj9d3PtwiVLzGVFnnkGtm0zW3br1wcPD6tTyq2s2TaZlK1v0eqvOB4Lf4ZJLUcTElja6lgi8i8OWdBFREQQERFBWloa+/fvV0HnpFauNNcFK1oUJk0y1wuT3C8lxWyF8/CAHj3gyy/Nrxs2/Gex31LattM5pCXRe+EAFm37jP+UDMSt/tfUDGtrdSoRuQGHLOiuUQud8ztwAF5+GZYtg9atYfx4zYTNjf7+GxYuNFvhFi82x8Q1b25+ffmyWdjnz291SsmsE7En8D0wiYLnlrKswmjikhNoW7GtFgUWcWCa5Sp2Vb682c02f75Z3O3bZz7uuL9GSGbEx//zd/jcc1C8uNkad/q0uf1W2bLmsZYtzd0aVMw5h9j4E4xZ1JcKkyow/vAWKPsiLUo347FKj6mYE3FwaqGTHJOcDF5e5rIT7dtDWBgMGGAOihfHlpwM69bB0qXmbe1a2LIFqlQxt4BLSzMXmdasVOeUkpbC1E1TqbPrNc6m2lhXbghvNHqDAG997oo4CxV0kuPS0uDtt2HyZPN+r17w+utQqJC1ueQfNpvZmnptpvI998Du3WZLW/Pm5mzU9u1VjDs9w4ATP3LGuxTlpzdjUMWmvNToPYoUqmt1MhHJIhV0Ypnz580xdZMmmctY/P23ufaY5Ly0NLMFbuVK+PNPWLUKLl0yu1ALFTLHxoWGQs2ampGaW6w9sZZP/xzBd17rcL/nLS6U6kLBvAWtjiXi8rZt28a4ceM4cuQIgwcPplGjRgwbNoykpCROnz7N0KFDqVGjxnXPU0Enlrt4ETZuNLvsYmLgySfhxRfNfTq1OLF9XL4Ma9aYS4r07AmpqVCggNlg06ABNG5s7p3aqJGK7Nzm6MmVHIruzOMHjlA2tBpzH51EWPEmVscSkf968cUXmTp1KqNHj2bixIk0adKECRMmsH//flq1akXPnj2ZNGnSdc/ztCCrSAZBQWYxB3DunFlUPPccBATA009D165mYSF35+RJeP99syVuxw7z+1yqlLmllqcnbNpkTmbw1KdC7mQYTFo/mfFLB7CkuMGslkN55N538XBXk6uIozh06BBFixbF09OTkydPcvHiRd566y2KFSvGmjVr8PPzo02bNjd8rlroxCEdPAjffANff22O4/rtN3Ng/vLl5hgutRrdmM0G+/fDhg1mq+eGDVCunPm9vHQJmjY199xt3Bjuu888psmLudvV1KvEH4wk+K8Z/FnhQ/48sZbX7u1H3jx+VkcTkf+xevVqfH19qVmzJtWqVSMoKIioqKhMPVcFnTg0m83skg0ONteyu/9+cxeKli3NgfkPPGAWJa7o0iWzpW3nTqhRw1y495tvoHNn83j58mbx1rKl2coprsVmS2Xe1mkMXDGaBwqEMC38XqgxCrz8rY4mIrdx/vx5QkNDee+993j//fcz9Rx1rohDc3c3izkwW+a2boUff4RFi6B3b6hVC9avNwu/zz6D2rXN4iY3jb27eNFcwy88HPz94cMPISLC7EIFs4t0xAizoHvwQXMx39q1zTFx4pqWH1lOwoqnKJR6kVqFH+eNlh9BwQpWxxKRTFq+fDmGYdCsWbNMP0ctdOK0YmPNoqZSJTh6FCpWNLtlPTzM9dHq1IEpU8y172JizDF5jti9eOUKHD9uzvpt2NB8rHdvc0zbgQNmQQfm+m8tWsCcOeY+qFWqQNWqUKGCuqDlv2J2E59mUOLLxjwVUpS+dXtTrWovq1OJSBb16tWLr776isuXL+Pj45Op56igk1wjOdnsfrw2fuz4cXM7KjC7H0+ehNKloUwZ889XXzUfP3TInPUZEmLefHzurvAzDPP5ly//U6idP29O+ChTBh56yCzUnnnGnGV64YL5PF9fSEgwn/vSS+bM0/Ll/7lVqpS7Wh4l+5yKO8WHK0bwafIveBRvy4GyrxIWFKbdHUScVOXKlSlUqFCmx8+BBV2uhmEQFxd3w2NJSUkkJSWl3792XmxsbI5kE+cXFmbenn3WvH/tR2fECDhyBP76y2zNW7rUPKdQIRg3Dj7/POPrvPwyfPSRuZhux45mkefjY7b2+fvDvHnmec88Y05CSEw0bwkJZgvaAw+Yzx89+p/X9PCATp3MGbuenmbX8COPmNtmFStm3mJjzYLuk0+uf28pKeZN5Jr4xLNsin6Rl/ds4Iq7L+1af0CdsPYU8vC+6eesiOQsf3//LP1ydfr0afbu3cszzzyTpevkeAvdtVY3ERERkdwuq72M27Zto1WrVixbtozw8PBMPy/HC7qstNCdOnWKevXqsXv3booVK5Yj+erWrcuGDRty3bVy+no5ea3Y2FhKlCjB8ePHc6RrXn9vzne9nP4ZAfu8N8MwiD21nMCDEWwuO5Apm7/ivcaDeKLVc7ny7y2nr6XPEue8Vk5e705+RrLaQnencrzL1c3NLcv/UPz9/XPsQ9jDwyNXXiunr5fT7w0gICAgR66pvzfnvV5O/YxA9r+3zcdXM3DZO/gnneLHcsVoVroKzcL/Y5dr3U5u/pkEfZY427WsuF5OfpZklpYt+R+9e/fOldfK6evl9HvLSfp7c97r5aTsem9/Xf6LdYseo2z8Ns5erczrD4yD8g9nmLmTm//e9DPinNfLze/NUTn0LNcTJ06kN20WL17c6jjioDQbWm7HKX9Gki+TlnSZctObEe4ex5B7HqZB02l4enpbnSzXcsqfE8lRjvwz4tAtdN7e3hn+FLkRb29vhg4dqp8TuSln+hlJTktmyoYpvHIxkjx+pfnh6R+oFFwJP23VZXfO9HMi1nDknxGHbqFz5EpYRCQ7GTYba9a8zutb5rH+4gkWPPguj1TvCb5FrI4mIk7AoVvoRERcwbbT2+j/SzfmeG+ia1A4057ezj2h91gdS0SciLvVAUREXFVy7AFY+wJ5bEnEG+7sq/cDPTvuUjEnIlmmLlcRkRx2Nv4Mw6KHs//4EhYVA7fG/8HIX11bdYnIHVOXq4hIDklMSeSPZd0ocXIOs8/4Mfi+d0i5ty95PH1QKScid0MFnYiIvdnSIPkCD373BEnn1vFxWDX2P/kTBQNKWp1MRHIJjaETp/T+++/j5uaW4Va4cGGrY4nFVqxYQZs2bShatChubm78+OOPGY4bhsH7779P0aJF8fX1pVmzZuzatcuumRYfWkx8dDtY2Z5R949idve9NHlyi4o5i9zuZ6Rr167XfbbUr1/fmrBiiVGjRlG3bl38/f0JDQ3l8ccfZ9++fRnOseKz5HZU0InTuueeezh16lT6bceOHVZHEoslJCRQvXp1Jk+efMPjY8aM4ZNPPmHy5Mls2LCBwoUL07Jly5vuL3039h/8gc7fNuXBbx/k++RgqPER95W6j3JB5bL9WpJ5t/sZAWjdunWGz5bffvstBxOK1aKjo+nduzdr165l8eLFpKam8uCDD5KQkJB+Tk5+lmSa4YAmT55sVK5c2ahQoYIBGDExMVZHEgczdOhQo3r16lbHEAcGGPPnz0+/b7PZjMKFCxujR49Of+zq1atGYGCg8fnnn2fbdU/FnTK6ze9i/DUTY8a0/Ma83fMMm82Wba8v2ed/f0YMwzC6dOliPPbYY5bkEcd09uxZAzCio6MNw8i5z5KscsgWut69e7N79242bNhgdRRxYAcOHKBo0aKUKVOGDh06cPjwYasjiQM7cuQIp0+f5sEHH0x/zNvbm6ZNm7J69eq7fn0jORa2v8/VhL9ZeHgJK8u9w3Od/6Zd5XaavepkoqKiCA0NpUKFCnTv3p2zZ89aHUksFBMTA0BQUBBg/8+SO6VJEeKU7r33Xr7++msqVKjAmTNn+OCDD2jYsCG7du2iYMGCVscTB3T69GkAChUqlOHxQoUK8ddff93x66akpTB101QWbJvGwqCTlC5YlyOvHsHLw+uu8oo1HnroIdq3b0+pUqU4cuQI7777Li1atGDTpk0Oud2T2JdhGAwYMIDGjRtTpUoVwH6fJXdLBZ04pYceeij966pVq9KgQQPKlStHZGQkAwYMsDCZOLr/bS0zDOOOWtAMw2DFpo9w3zWCgX8l8ky1Llx64BeC/IqhUs55PfPMM+lfV6lShTp16lCqVCl+/fVXnnjiCQuTiRX69OnD9u3bWbly5XXHsuuzJLs4ZJerSFbly5ePqlWrcuDAAaujiIO6Ngv62m/X15w9e/a637RvK/UK3RZ0o8eiIXh7+rKhyx/MfHwmQX7FsiuuOIgiRYpQqlQpfba4oL59+7JgwQKWL19O8eLF0x/P1s+SbKSCTnKFpKQk9uzZQ5Ei2shcbqxMmTIULlyYxYsXpz+WnJxMdHQ0DRs2zNRrHL50mIvrX4Ml99GpakcmPP07dTueo0qpB2/7XHFOFy5c4Pjx4/pscSGGYdCnTx/mzZvHsmXLKFOmTIbj2fFZYg/qchWnNHDgQNq0aUPJkiU5e/YsH3zwAbGxsXTp0sXqaGKh+Ph4Dh48mH7/yJEjbN26laCgIEqWLMlrr73Ghx9+SPny5SlfvjwffvghefPmpWPHjrd83UuXDzJp9Wg+2PQ1b4ffz9AavWlRpgW46XdiZ3Orn5GgoCDef/99nnzySYoUKcLRo0d56623CA4Opl27dhamlpzUu3dvvvvuO3766Sf8/f3TW+ICAwPx9fXFzc3tjj9L7Mqy+bWZEBMTo2VL5IaeeeYZo0iRIoaXl5dRtGhR44knnjB27dpldSyx2PLlyw3guluXLl0MwzCXGxg6dKhRuHBhw9vb22jSpImxY8eOm77elZQrxtiVY4xNMzyMeVM9jA+iPzASkhNy6N2IPdzqZyQxMdF48MEHjZCQEMPLy8soWbKk0aVLF+PYsWNWx5YcdKOfD8CYMWNG+jlZ/SzJCW6GYRg5XkVmUmxsLIGBgcTExBAQEGB1HBFxFYYN/prN6XyVqDi9OW9VfpAXGw8lpGAVq5OJiNyQulxFRP4l6mgUk1eNZI73ZgpXG85fr/1Ffp/8VscSEbklFXQiIsCBY4s4vvJFHjv4N+FF63Gsxe+ULlKP/FYHExHJBI3oFRHXZtgYu2osLb9tTfHUs/zfQx+xtttaShepZ3UyEZFMUwudiLikhOQEEg9MJ+T4t9SrOArP5h9Tqs4rVPDysTqaiEiWOWRBFxERQUREBGlpaVZHEZFcJi01idmbIxj05zgeCy3BlCpNaVqiAU3L3G91NBGRO6ZZriLiEgzD4PeDv+O+6lm8U2OZFtSRkS1GUjp/aaujiYjcNYdsoRMRyVYXNxOHFx3nduS5ouXo3fg1ZlXqbHUqEZFso4JORHKtYzHHGBU9gsmpvxJQ+lm29NxC6fylLd1AW0TEHlTQiUiucznuGJuWdqDT7k0YeQrQvc14apVvTxl3feSJSO6kZUtEJFeZt2ceDb6oQZW4NXxU6ykO9D1ArYrPgoo5EcnFVNCJiNMzDINLx36BZS0pni+YxhWeJK3NITq3noW/t7/V8URE7E6/soqIU1t7dBkDlr5DYVsMc8uXpV5IBeq1/dLqWCIiOUoFnYg4pQMXDrB1UVtKX9lLUmoN+jw4CbcyLayOJSJiCRV0IuJcrp4jJTmW5pHNqZsnlYFVe7Lhvsm4a4yciLgwfQKKiFO4knKFiWsn0D9mFnnyh/Pzsz9TKbgSvl6+VkcTEbGcCjoRcWg2Wyqr/+zFa1t/ZVvsWWq2Gs6D1bpT0yfY6mgiIg5DBZ2IOKyNJzfS/+cXmJ93J90L1eL+56MJCwqzOpaIiMPRsiUi4nCSLu+GVc/hY6TilqcARxr+Ss8Om1TMiYjchEO20EVERBAREUFaWprVUUQkB/0dc5z3ot7n2Mk/WVQyD1Xy+bPihRVWxxIRcXhuhmEYVoe4mdjYWAIDA4mJiSEgIMDqOCJiJ3FJcfyxpBNlzv7Mw+cK8E7T93m5dk+8PPNYHU1ExCk4ZAudiLgIWwrG1fO0mNUWz8vbGVX+XvY/PZ/AfIWtTiYi4lRU0IlIjjMMg5/3/8z9JyLIRwrjW42nZGBJSgaWtDqaiIhT0qQIEclRu/dE8kxkfR6b/Rg/GGWg1ic0LtlYxZyIyF1QC52I5IgTsScYvGgAHyX+H62NYF587ndalWsFbm5WRxMRcXoq6ETEroykS7jtGUNykadY9fcG1tw7ki51BuKhCQ8iItlGBZ2I2MXV1KtErI9g4c6ZLAq+QNnC93Ow70E83D2sjiYikutoDJ2IZCubYWPJ2vfY/G0B3l3yBmFFGxPbehsUfkDFnIiInaiFTkSyT0o8nRb0YPO+75lZujDbuv1A+WKNrE4lIpLraWFhEblre87tofChiRS4vJHlFUbj7u5J09JNrY4lIuIy1OUqInfs3PkdvL2gE1WnVOXzU39DpQE0L9NCxZyISA5Tl6uIZFlCcgKfrP6Yx48Oo2aKB2NajqF33d7g6W11NBERl+SQBV1ERAQRERGkpaVZHUVE/s2WBkciifGvwSfrxuNXtSMvNBpG/sCyVicTEXFpGkMnIrdlGAYLDy5k6tqPmOe7A7eaY4gr/jT+3v5WRxMRERy0hU5EHMfuQz9xek1P2h8+Q+2STTjRZCklQmugUk5ExHFoUoSI3JgtjRHRI3hk9uMUtl1mQdvJRHWJokRoDauTiYjI/1ALnYhkEHM1hpQDUwn++z80qfgRIa2mUKFmN8I9vKyOJiIiN6EWOhEBIDk5numrPiBsUhjDt/8IRR6iaclGvFznZTxVzImIODS10Im4OMMwmLtnLvnXd6VUagKPVezG4ObDwb+o1dFERCSTVNCJuLJzq4lx96Pbgm50K1GVV5oMYlrZJ6xOJSIiWaSCTsQFHbx4kNHRw5hqLCF/2RfY+cpOSgSWsDqWiIjcIRV0Ii7kwuUDbF3agQ57t+GTtwg720RQrdzjlHDTcFoREWemT3ERFzF752zqT61F5YQtTLj3Bfb32U+1sCdAxZyIiNOz6yf5yJEjadiwIXnz5iV//vz2vJSI3IDNsHHx6FxY0pzS/kVoXaUrXu2O0/GBL/H18rU6noiIZBO7drkmJyfTvn17GjRowPTp0+15KRH5H1EHfmbAsqGEeaTwn4qVqF8onPqlJlkdS0RE7MCuBd2wYcMAmDlzZqbOT0pKIikpKf1+bGysPWKJ5Go7z+5k7+K2lEw6gq9nfV5tOQm3ko2sjiUiInbkUINnRo0aRWBgYPqtRAnNuhPJtMSTJMce4IGvH+A/McmkVRrAyhdW0UjFnIhIrudQs1yHDBnCgAED0u/HxsaqqBO5jbikOD5Z/TFvxf+HPAVr80enP6gcUpk8HnmsjiYiIjkkyy1077//Pm5ubre8bdy48Y7CeHt7ExAQkOEmIjeWmnqVFUs60SCiLKNWjWZlse5QZzLVC1dXMSci4mKy3ELXp08fOnTocMtzSpcufad5RCQTVh9fzes/v8Av/vvpV6IhrR/8npKBJa2OJSIiFslyQRccHExwcLA9sojIbVy5sAXfPaPwLzMAf/9S/N10Kj1KNrU6loiIWMyuY+iOHTvGxYsXOXbsGGlpaWzduhWAsLAw/Pz87HlpkVzl6KVDvLXsXS6c28jCUv5U9S/AoucXWR1LREQchF0Luvfee4/IyMj0+zVr1gRg+fLlNGvWzJ6XFskVLl25xJLFz1D2whL+vFiIoc1HYKveFQ8Ph5rPJCIiFnMzDMOwOsTNxMbGEhgYSExMjCZIiGtJS8K4ep4a3zxMQPx+RlRsQN0HfiCfb5DVyURExAHp13wRB2IYBnN2zaHNqS/I6+HJlEemULZAWQr7FbY6moiIODCHWlhYxJVt2z6JttNr0GFuB37yrAq1P6VhiYYq5kRE5LbUQidisSOXjvDGwn5MTPmFxz2LMLBLFE1La+aqiIhkngo6EYvYrp7Ffddo0op1ZNv5fWxoOJ4XavbB3V3/LEVEJGv0P4dIDktMSeSTNZ+wfM93LAmNJaz4Y+ztsxd3N42AEBGRO6OCTiSHpNnSWLz6TQIPTmL0cRs96vUloem7+PkU0GBWERG5Kw5Z0EVERBAREUFaWprVUUTunmFASixPze/KvsM/8mWZkuzsMY/ShWpbnUxERHIJrUMnYkfbTm+j1KFPyR+/m6gKo/H1ysu9xe+1OpaIiOQy6ukRsYNTZzbw+rynqPlFTaadj4V73qJZ6WYq5kRExC4csstVxFnFJsUy+s9RPH/iI+qlehPxcAQv1XoJPLysjiYiIrmYCjqR7GBLgYNTicvfgC82T6VY9W50bvQ+/n7FrE4mIiIuQAWdyF0wDIP5e+czY93HLPDbS7Han3K8/3HyeuW1OpqIiLgQFXQid2jbvu+5tL4vzx+5wH1lW3Gq+Z8ULRiOSjkREclpKuhEssqWytvLh/Ld2g9ZUDIvfzw5g8b3dLU6lYiIuDAVdCKZdD7xPMb+KYSc+pEHKn5EhYIVCK/WCQ93D6ujiYiIi1NBJ3IbV5Mu89Xacby1ZjIvlqzGJ9WfonmpJuCRx+poIiIigAo6kZuyGTa+2/EdxTd1p1xaEp2q9WJw0/cgX6jV0URERDJQQSdyI6eXcdmzIL1+7cUrZe7l5YZDmFyyldWpREREbkgFnci/7Dq7izErhjHDLZqgCr3Y12cfRfyLWB1LRETklhyyoIuIiCAiIoK0tDSro4iLOHN+JzuWd+DpfbsJCijL3jZfEV76YYq4uVkdTURE5LbcDMMwrA5xM7GxsQQGBhITE0NAQIDVcSQ3MgxmbotkxB+9iCqSxNqSfXjsvrHk0YQHERFxIu5WBxCxQqotlQuHZsGSJlTIX5qnavXF/6kztG82QcWciIg4HYfschWxF8Nm44+9cxiwfDg1fL2YVakaDYtWp2HpZlZHExERuWMq6MRlbDy5kWNLH6do8t8U9mvG6y3H4Va0ttWxRERE7poKOsn94o9yNS2Jh2Y9RMsAP16rOZSldd7DzV0jDkREJHdQQSe51qUrlxi78iOGJ/6AT6EmRHWJomJwRTzd9WMvIiK5i/5nk1wnOTme1cufp+eOKP5OTuGhR0ZzX3hX7vHyszqaiIiIXaigk1wl6mgUr//clT/y/8Wgsi149MFZFPYrbHUsERERu9IgIskVEs6ugxXtyO+Zh2LB1Tj/wDpeemKpijkREXEJaqETp7bv7E7eXPYOSZf38luZAtQILMSCZxdYHUtERCRHqaATp3Q24SzL/3iSsMsr2RZbgg9ajMKo+ixubmp0FhER16OCTpxLaiJG0kWazmxJ4aTjDK30MHs6zcYnj7/VyURERCyjgk6cQpotjW+2f0OHM9PwyRPAjMdmUD6oPAXzFrQ6moiIiOUcsqCLiIggIiKCtLQ0q6OIA9i0aRRDNn7N4tN78Ws+kKdqvkx9/3JWxxIREXEYboZhGFaHuJnY2FgCAwOJiYkhICDA6jiSw/Zf2M/A33vxpW0pCyhJtRZzuLf4vVbHEhERcTgaQS4OJy3xJGzojVtyLIdjT7O1+he81OmIijkREZGbcMguV3FNsVdjGLN6LGv2z2VJkSuUL92JHa/swM3NzepoIiIiDk0FnVguJS2FRX++SsiRL5l40oO+9V/nyn2DyZvHH5VyIiIit6eCTqxjGJB8icd+6MTxY7/zWbkw9rz8I8UK3mN1MhEREaeiSRFiibUn1lL5yEQCE48QXWEU+X0LUL1wdatjiYiIOCVNipAc9deJZfSa8wgNpjdgxuU0qPo+Tcs0UzEnIiJyF9TlKjni4pWLfBA1nFfOTKBRal7qtv2KztU7g7uH1dFEREScngo6sa+0q7A/goSgZny78zsq1+zDc42Gktc32OpkIiIiuYYKOrELm2Fj9s7ZfLthAr8GHKZE3RIc638MH08fq6OJiIjkOhpDJ9luw65prPwmiJfmP4d3vqKce2AtlHpaxZyIiIidqIVOsk9aMgOWDOanTeOZW8Kf6A7/oW6Fp61OJSIikuupoJO7diruFF4HPiP4zG+0qfgR9YvXp3p4e+3wICIikkMcsss1IiKC8PBw6tata3UUuYX4xNN8snQA5SeVZ9yBVVDmeZqXbsrT9zytYk5ERCQHaWFhybJUWypfbfmKe7b3JSY1heWlXuet+96igG8Bq6OJiIi4JHW5SuYZBpz8nUt5ijBw0UBeLd+Mng3f5eGija1OJiIi4tJU0EmmbD61mXErhvGt52pCKr3OoX6HCMkXYnUsERERQQWd3MaJ0xvYF92RJ/YfpFhQZQ60+Z6KJe4nRGPkREREHIZDTooQB2AYfLHxC5rMaEy5pCPMbD6Y7a9sp2LJB0DFnIiIiENRQScZJKclc37/V7C4EVWCK9K5wRCCn7lAu8aj8HRXg66IiIgj0v/QAoBhs7Fgx0wGRI+kcUAgkZVr06hoTRqVbmZ1NBEREbkNFXTCqmOrOB/1BCEpZ6kU/BBvPDAWQu+xOpaIOJDk5GRSU1OtjiHicDw9PcmTJ4/VMVTQubTY/Vyx2Wg7uy1tCgTzat03+LXG61anEhEHk5yczK5du7DZbFZHEXE47u7u3HPPPZYXdXYr6I4ePcqIESNYtmwZp0+fpmjRonTq1Im3337b8jft6s4lnGPUig8Ym/QTvkVbs/rF1ZQvWB53Nw2pFJHrpaamYrPZKF26NL6+vlbHEXEYV65c4ejRo6Smplpe29itoNu7dy82m40vvviCsLAwdu7cSffu3UlISGDcuHH2uqzcQuLVC6xd+hzddq7kkuFJ+0fH0qDy81T08LE6mog4AV9fX/LmzWt1DBGHExMTw9WrV/H29iZfvnyWZMjRrb/Gjh3LlClTOHz48A2PJyUlkZSUlH4/NjaWEiVKaOuvbLDo0CIG/tyVpQVP8XPgwzz2wNcUzFvQ6lgi4gQSExPZs2cPlStXVkEn8i/X/m1s3ryZhIQEAgICeOaZZywp6nK0jy0mJoagoKCbHh81ahSBgYHptxIlSuRgulzq4maIepQQbz8qFWtMXKttvNj2VxVzIiIi2SRfvnzkyZOH2NjYDA1TOSnHCrpDhw4xadIkXn755ZueM2TIEGJiYtJvx48fz6l4uU/af3+gvAIgNYGaBYoxp/0cyoZWszaXiIhILuPt7Y23t7elGbJc0L3//vu4ubnd8rZx48YMzzl58iStW7emffv2vPTSSzd9bW9vbwICAjLc5A7sGQeLGoAtDfzD4IHlkK+U1alERETETrI8KaJPnz506NDhlueULl06/euTJ0/SvHlzGjRowNSpU7McUDIpJQ6SL0O+EhByH7j7ADk2PFJEREQslOWCLjg4mODg4Eyd+/fff9O8eXNq167NjBkzcHfXshh2E/UI5MkPTRdA8L3mTURERFyC3ZYtOXnyJM2aNaNkyZKMGzeOc+fOpR8rXLiwvS7rOgwD/l4AQbUhb3Go9TH46PsqIiLiiuxW0C1atIiDBw9y8OBBihcvnuFYDq6UknulJsC67hD+BlQeCAXrWp1IRERELGK3PtCuXbtiGMYNb3KHEv82i7iUWPDyg4e2QCVt1SUiYk/NmjXjtddeszrGTeVkvj179uTIdSTrNKjNGaQXwQaciYLY/ebdvMXAzc2qVCIi4kI+/fRTy5fmOHfu3G13mzp58mQOpXEsKugc3fH5/12CJMUcK9dmHxSsY3UqEZFcJTk52eoIlmnWrBnz5s275TmLFi0iJCSEsmXL5lCqGwsJCaFJkyZERETc9JyoqCh69OjB8uXLM/26GzZs4LXXXiMyMpIePXpw5MiR7Iibo1TQOSLDBlfPm1/7lYMCNSHtqnnfTX9lIuK8SpcuzaeffprhsRo1avD+++8DZnHRp08f+vTpQ/78+SlYsCDvvPNOhuE6mTnHMAzGjBlD2bJl8fX1pXr16vzwww/XvcaAAQMIDg6mZcuWt8ydmpp60+slJSXRr18/QkND8fHxoXHjxmzYsCFL7/tapn79+vHGG28QFBRE4cKFMxwHSEhIoHPnzvj5+VGkSBE+/vjjW+a+nR9//BEPDw8GDx5MSkrKTc+bOHHibZcsyyn16tVj/fr1nD9//obHO3bsyGeffcbx48d56aWX+P3332/5eklJSbRv354hQ4bQpUsXXnjhBV544QV7RLcrVQeOaF03WPGY2dVaoBrUmwJe/lanEhHJ4FTcKa6mmr9sHrl0hM2nNme4nYo7dUevGxkZiaenJ+vWrWPixImMHz+eadOmZemcd955hxkzZjBlyhR27dpF//796dSpE9HR0de9xqpVq/jiiy/uONMbb7zB3LlziYyMZPPmzYSFhdGqVSsuXrx4R+89X758rFu3jjFjxjB8+HAWL16cfnzQoEEsX76c+fPns2jRIqKioti0aVP68ZkzZ+KWyaE4qampbNq0iTlz5nD27Nmbfg+2b99O8eLF8fDwyPL7sZe2bdsyc+bMmx739PSkc+fOTJ06ldjYWF588UXmz59/w3H8K1asICAggEKFCgFmwbh27VrOnDljr/j2YTiwmJgYAzBiYmKsjmJ/l3cbRvxR8+uzqw3jTLS1eURE/ishIcHYuHGjkZCQkOHxocuHGgcuHDAMwzCem/ucwftkuA1dPvS61ypVqpQxfvz4DI9Vr17dGDrUPLdp06ZG5cqVDZvNln78zTffNCpXrpx+/3bnxMfHGz4+Psbq1aszXKdbt27Gs88+m/4aNWrUyNT7v9X14uPjDS8vL2PWrFnpx5KTk42iRYsaY8aMyfT7vnadxo0bZzinbt26xptvvmkYhmHExcUZefLkMWbPnp1+/MKFC4avr6/x6quvGoZhGPPmzTMqVqyYqfc1bdo048AB8+9v9OjRRkhIyA3/v/3kk0+MadOmpd9PTU01Jk+ebHTq1MlYv369YRiG0bFjRyMyMjJT172V1NRUY9KkSUaXLl3SX/v48eNG06ZNM5x37tw5o3nz5pl+XZvNZvz444/Gc889ZyxZsiTDsalTp173+sHBwcbSpUtv+7rX/m388MMPxrRp04zx48cbFy5cyHSu7KQWOkdgS4Ooh2H3R+b9kAYQ2sTaTCIit9Gzdk+KB5jLUo1oPoJNPTZluPWs3fOOXrd+/foZWpkaNGjAgQMHSEtLy9Q5u3fv5urVq7Rs2RI/P7/029dff82hQ4fSn1OnTubHI9/segcPHiQlJYVGjRqlH/Py8qJevXp3NCO0WrWM+20XKVKEs2fPAuae6MnJyTRo0CD9eFBQEBUrVky/365dO/bu3Xvb68THx3Pu3DnCwsIAePXVV8mbNy8fffTRdeeeOHGC0NDQ9Ps//fQTzzzzDImJiRw9ehSANm3aEBsbm/k3ehMLFiygQ4cOXLlyJX0c2+LFi69b/iw4ODjD3+XtnD9/ns2bN1OwYEHKlClz3TEfH58Mj/n4+HD58uU7exMWsds6dHcjIiKCiIiIDP94c53UBNg3Acr1AJ9gaPqLue+qiIiTKOJfJP3rMgXKUIYytzjb5O7ufl23163Gbt0Jm80GwK+//kqxYsUyHPv3LM18+fJl2zX/t5vTMIwMj2X2fXt5eV33utfez/8+/258+eWXdO/ePf2+j48PI0eOpEePHvTq1SvD9y0+Ph5fX9/0+y1atMBms/Hnn3/y7bffAlCuXLkbTpiYMGHCLScYNG3alHbt2qXfv//++wFYtmwZX331FWBOcrjRGMfMdC2fPHmSSZMmkZKSQt++fSlV6vp9zQMDA6/73sbHx2d6VyxH4ZAFXe/evenduzexsbEEBgZaHcc+UhNh76cQeA8Ufwzy32N1IhERuwsJCeHUqX/G1sXGxl73H/7atWuvu1++fPkMY7hudU54eDje3t4cO3aMpk2bZkvum10vLCyMPHnysHLlSjp27AiYhdrGjRszrA2Xmfd9O2FhYXh5ebF27VpKliwJwKVLl9i/f3+W3ufp06fx8fGhYMGCGR7v2LEj48eP57333mP69OnpjwcHB3Pp0qX0+/nz5+ebb76hWbNm6YXejh076Nq163XXevXVV7PyFgkICGD27Nk0adIkveCOjo7mww8/5PLly+TPnz/9XE/Pm5cwR48eJSIigjx58tCvXz+KFCly03MrVarEl19+mX4/KSmJuLi4GxZ/jkxdrjnpwgZY/hCkXgGfEHjsqFnMiYi4iBYtWvDNN9/w559/snPnTrp06XLdYPvjx48zYMAA9u3bx/fff8+kSZOuKwxudY6/vz8DBw6kf//+REZGcujQIbZs2UJERASRkZF3lPtm18uXLx+vvPIKgwYNYuHChezevZvu3buTmJhIt27dsvS+b8fPz49u3boxaNAgli5dys6dO+natWuGfdLnz59PpUqVbvk6U6ZMoW3btpw/fz7D7cKFCwwePJjIyEh27tyZfn7lypU5duxYhtc4c+ZMhqLSz88v2/ZrP378eHpX8M6dO/Hy8iI0NDTDLOWUlBT8/W88WXDcuHF8/fXXDB48mJEjR96ymANo0qQJZ8+e5cSJE4BZQNatW9fpCjqHbKHLddKugocP5CkARiokXwDP4uCZ1+pkIiI5asiQIRw+fJhHH32UwMBARowYcV1LVefOnbly5Qr16tXDw8ODvn370qNHjyydM2LECEJDQxk1ahSHDx8mf/781KpVi7feeuuOct/qeqNHj8Zms/H8888TFxdHnTp1+OOPPyhQoECW3ndmjB07lvj4eNq2bYu/vz+vv/46MTEx6cdjYmLYt2/fTZ+/f/9+Ro4cyfDhw295nbfeeosFCxYA0Lp1a7p3787rr/+zM1GHDh0YMGAA33zzDampqXTp0iXL7+VmnnzySd58803mzJkDQO3atZk8eXKGFsANGzbQokWLGz5/4MCBWbqep6cnkZGRjBw5kvr16xMdHc2sWbPuOL9V3Izs7JTPZte6XGNiYggICLA6zp3ZNRqO/R+03qA15ETEKSUmJrJnzx4qV65M3rz2/UW0WbNm1KhR47o127J6jmSvHj16MHz4cAoXLmx1FMBclqZt27bUq1fP0hzX/m0cPXqUy5cvExcXR+fOnQkKCsrxLGqhs4fkGHO/1XwloFBzyJPfXFNOu3SJiIgTeu+995g8eTIffPCB1VGIjY3l3LlzNy3mxo8ff8sZsM2bN+fJJ5+0VzzLqKCzh6iHwKcQNJkPwfeaNxERESdVvHhx2rVrx2+//cbDDz9sWQ7DMBg/fvwtC8v+/fvnYCLHoYIuOxgGHJ8HwfUhbzGoPQF8i93+eSIikkFUVFS2nCPZr3bt2lZH4MKFC7zyyiuEhIRYHcXhaFBXdkiNhw0vwzFzACcF60LeotZmEhERyWWCg4MzLHIs/1BBd6cSjsPaF82xcl7+8PB2qOSazbwiIiJiLRV0WXVtUrCbG5xbBXH/HXjpe+t1bkRERETsRQVdVhz7P/jjXrClQN7i8OgeCKppdSoREbGjZs2aZdj1wdHkZL472Z9WcoYKutsxbHD1nPm1fwVz4kNaknlf68qJiIiL+PTTTzPshXsz586dY9y4cbc85+TJk9kVS/5LFcntrOkCfz5hdrUWqA51JoKXn9WpREQkGyUnJ1sdwTLNmjVj3rx5tzxn0aJFhISEULZs2du+XkhICE2aNCEiIuKm50RFRdGjRw+WL1+e6ZwbNmzgtddeIzIykh49etzRThu5mUMWdBEREYSHh1O3bl1rAlzeAQl/mV+X7wU1PjLHzImIyF0pXbr0dTs81KhRg/fffx8wi4s+ffrQp08f8ufPT8GCBXnnnXf496ZGmTnHMAzGjBlD2bJl8fX1pXr16hn2Ar32GgMGDCA4OJiWLVveMndqaupNr5eUlES/fv0IDQ3Fx8eHxo0bs2HDhiy972uZ+vXrxxtvvEFQUBCFCxfOcBwgISGBzp074+fnR5EiRfj4449vmft2fvzxRzw8PBg8eDApKSk3PW/ixIl06NAh069br1491q9fz/nz5294vGPHjnz22WccP36cl156id9///2Wr5eUlET79u0ZMmQIXbp04YUXXuCFF17IdB5X4JAFXe/evdm9e/d1/yByhC0VotvAnv/+IwlpACENcz6HiIiDO3UKrl41vz5yBDZvzng7derOXjcyMhJPT0/WrVvHxIkTGT9+PNOmTcvSOe+88w4zZsxgypQp7Nq1i/79+9OpUyeio6Ove41Vq1bxxRdf3HGmN954g7lz5xIZGcnmzZsJCwujVatWXLx48Y7ee758+Vi3bh1jxoxh+PDhLF68OP34oEGDWL58OfPnz2fRokVERUWxadOm9OMzZ87ELZMNEKmpqWzatIk5c+Zw9uzZm34Ptm/fTvHixfHw8MjSe2nbti0zZ8686XFPT086d+7M1KlTiY2N5cUXX2T+/PncaEfSFStWEBAQQKFChQCzYFy7di1nzpzJUqZczXBgMTExBmDExMTY90LJcYaxfZhhXD1v3r+82zBSk+x7TRERJ5GQkGBs3LjRSEhIyPD40KGGceCA+fVzzxmGOTbln9vQode/VqlSpYzx48dneKx69erG0P+e3LRpU6Ny5cqGzWZLP/7mm28alStXTr9/u3Pi4+MNHx8fY/Xq1Rmu061bN+PZZ59Nf40aNWpk6v3f6nrx8fGGl5eXMWvWrPRjycnJRtGiRY0xY8Zk+n1fu07jxo0znFO3bl3jzTffNAzDMOLi4ow8efIYs2fPTj9+4cIFw9fX13j11VcNwzCMefPmGRUrVszU+5o2bZpx4L9/gaNHjzZCQkJu+P/tJ598YkybNi39fmpqqjFp0iSjS5cuxvr16w3DMIzjx48bTZs2zfC8c+fOGc2bN89UFsMwDJvNZvz444/Gc889ZyxZsiTDsalTp173+sHBwcbSpUsz/fr2cO3fxg8//GBMmzbNGD9+vHHhwgVLsjhkC12OS7sCByLg3GrzfmBl8MhjbSYREQfXsycUL25+PWIEbNqU8daz5529bv369TO0MjVo0IADBw6QlpaWqXN2797N1atXadmyJX5+fum3r7/+OsMen3Xq1LnrTAcPHiQlJYVGjRqlH/Py8qJevXp3NCO0WrVqGe4XKVKEs2fPAnDo0CGSk5Np0KBB+vGgoCAqVqyYfr9du3bs3bv3tteJj4/n3LlzhIWFAfDqq6+SN29ePvroo+vOPXHiRIbFfBcsWECHDh24cuVK+ji2xYsXU/zaD8N/BQcH33JP1f91/vx5Nm/eTMGCBSlTpsx1x3x8fDI85uPjw+XLlzP9+rmd6279dX4dbH8Pmv4EPiHQ9ih4+lqdSkTEaRT51/KbZcqYt9txd3e/rkvtVmO37oTNZgPg119/pVixjNsw/nuWZr58+bLtmv/bzWkYRobHMvu+vby8rnvda+/nf59/N7788ku6d++eft/Hx4eRI0fSo0cPevXqleH7Fh8fj6/vP/8/3n///QAsW7aMr776CjAnOdxoHGJmun9PnjzJpEmTSElJoW/fvpQqVeq6cwIDA697//Hx8QQHB9/29V2F67XQpV4x/8xTwJzokHTBvK9iTkTE7kJCQjj1r8F1sbGx181WXLt27XX3y5cvn2EM163OCQ8Px9vbm2PHjhEWFpbhVqJEiTvKfbPrhYWFkSdPHlauXJl+LCUlhY0bN1K5cuUsve/bCQsLw8vLK0OWS5cusX///iy9zunTp/Hx8aFgwYIZHu/YsSOVK1fmvffey/B4cHAwly5dSr8fEBDAb7/9RpMmTdKL4ujoaB544IHrWsw8PW/ebnT06FEGDRpEREQE/fr1Y9y4cTcs5gAqVaqU3lIJ5iSJuLi4m57vilyroNv5ASxubK4tF1ABmi+EvMVu/zwREckWLVq04JtvvuHPP/9k586ddOnS5brB9sePH2fAgAHs27eP77//nkmTJvHqq69m+hx/f38GDhxI//79iYyM5NChQ2zZsoWIiAgiIyPvKPfNrpcvXz5eeeUVBg0axMKFC9m9ezfdu3cnMTGRbt26Zel9346fnx/dunVj0KBBLF26lJ07d9K1a1fc3f/5r3z+/PlUqlTplq8zZcoU2rZty/nz5zPcLly4wODBg4mMjGTnzp3p51euXJljx45d9/241l27c+dOvLy8CA0NzTCTOCUlBX9//xtmGDduHF9//TWDBw9m5MiRFCly692WmjRpwtmzZzlx4gRgFpB169ZVQfcvub/LNekipMZDvpJQuKW5RZdhgFYhERHJcUOGDOHw4cM8+uijBAYGMmLEiOtaqjp37syVK1eoV68eHh4e9O3blx49emTpnBEjRhAaGsqoUaM4fPgw+fPnp1atWrz11lt3lPtW1xs9ejQ2m43nn3+euLg46tSpwx9//EGBAgWy9L4zY+zYscTHx9O2bVv8/f15/fXXiYmJST8eExPDvn37bvr8/fv3M3LkSIYPH37L67z11lssWLAAgNatW9O9e3def/319ONPPvkkb775JnPmzAGgdu3aTJ48ma5du6afs2HDBlq0aHHD1x84cOBt3+u/eXp6EhkZyciRI6lfvz7R0dHMmjUrS6+R27kZ2dkpn81iY2MJDAwkJiaGgICAO3uRP+41t+m6b272hhMRcRGJiYns2bOHypUrkzdvXrteq1mzZtSoUeO6Nduyeo5krx49ejB8+HAKFy6c6ee88847tG3blnr16tkxmbWu/ds4evQoly9fJi4ujs6dOxMUFJTjWXJfC51hg7/mQGgTyFsU6kRA3jsbMyEiIiLw3nvvMXnyZD744INMnR8bG8u5c+duWsyNHz/+ljNgmzdvzpNPPnlHWV1V7ivoUuNhU1+o8h5U7AsFMz8tXURERK5XvHhx2rVrx2+//cbDDz98y3MNw2D8+PG3LP769++f3RFdXu4o6BL+MpcgqTMZvALg4Z3gW8jqVCIikkVRUVHZco5kv9q1a2fqvAsXLvDKK68QEhJi50Tyb849y9Ww/fcLd7iwAeIPm3dVzImIiFgiODg4w0LEkjMcsqCLiIggPDycunXr3vyko7PNCQ+2FMhXAh7ZBQWq51xIEREREQfhkAVd79692b17Nxs2bMh4wJYGV/67EW9gJXPigy3ZvJ/JzYhFREREchvnGkO3+jm4egoeiIYCNcybiIiI2N2mTZtISkoiX758VK+uHjFH45AtdNdJ+O8K1RVfhRpjrc0iIiLiYubPn094eDh16tRh/PjxVseRG3Dsgs6Wav65/zPzz5AGEJx7FygUEcntmjVrxmuvvWZ1jNty9Jw5mW/Hjh3UqlWLRYsW8fzzz1+3a8et7Nmzx47J5N8cu6Bz/2+PcPUR1uYQERHJpRITE0lISLjp8ZMnT1K8eHFSU1M5c+YMnp6ZG6316aef4u3tnalzz507x7hx4zJ1rtyYYxd017h7WZ1ARERyoeTkZKsjWOrXX3+lVq1ajB178+FMhmHg4eHBk08+Sdu2bbl8+fJtX3fRokWEhIRQtmzZTOUICQmhSZMmREREZDZ6pmzYsIHXXnuNyMhIevTocUf75zoL5yjoREQk10lKSqJfv36Ehobi4+ND48aNM6xu8PPPP5M/f35sNnPN0a1bt+Lm5sagQYPSz+nZsyfPPvssYBYeY8aMoWzZsvj6+lK9enV++OGHDNds1qwZffr0YcCAAQQHB9OyZcub5ktNTaVPnz7kz5+fggUL8s477/Dv7c9vl7906dLX7Tdbo0YN3n///Qx5+vXrxxtvvEFQUBCFCxfOcBwgISGBzp074+fnR5EiRfj4449v/Y3NgkceeYRnnnnmpsfPnj3L3r17Abhy5Qp79+6lefPmnDhxgoULF6bftm7dmuF5EydOpEOHDlnKUq9ePdavX8/58+ez/D5uJCkpifbt2zNkyBC6dOnCCy+8wAsvvJAtr+2IVNCJiIgl3njjDebOnUtkZCSbN28mLCyMVq1acfHiRQCaNGlCXFwcW7ZsASA6Oprg4GCio6PTXyMqKoqmTZsC5mbwM2bMYMqUKezatYv+/fvTqVOnDOcDREZG4unpyapVq/jiiy9umu/aeevWrWPixImMHz+eadOmZTp/ZkVGRpIvXz7WrVvHmDFjGD58OIsXL04/PmjQIJYvX878+fNZtGgRUVFRbNq0KcNrzJw5E7c7XL7rVs/btGkTrVq1YtWqVSxdupQJEybg5eVF8eLFad26dfqtRo0a6c/Zvn07xYsXx8PDI8tZ2rZty8yZM294bMKECQwfPpzTp09n6rVWrFhBQEAAhQqZmw3Uq1ePtWvXcubMmSzncgYq6ERE5M5cOQVpV82vE45DzO5/jl3eYR6/iYSEBKZMmcLYsWN56KGHCA8P58svv8TX15fp06cDEBgYSI0aNdK3+oqKiqJ///5s27aNuLg4Tp8+zf79+2nWrBkJCQl88sknfPXVV7Rq1YqyZcvStWtXOnXqdF3RFhYWxpgxY6hYsSKVKlW6acYSJUowfvx4KlasyHPPPUffvn3TZ3hmJn9mVatWjaFDh1K+fHk6d+5MnTp1WLp0KQDx8fFMnz6dcePG0bJlS6pWrUpkZCRpaWkZXiMwMJCKFSve8jqrV6+mX79+fPnllwwePJiVK1emH0tISCAyMpLIyEjatWuXPqbOzc2NypUr06hRIx599FF8fX1v+36WLl163cYAaWlpTJ48ma5du6a3Yp44cYJmzZplOK9p06b89ttvN3zdV199lZ49e/LFF1/wxhtvsG/fvlvmOHr0KEFBQen3PTw88Pf3Z9euXbd9D85IBZ2IiNyZA19A4gnz6z1jYdW/utiWP2Qev4lDhw6RkpJCo0aN0h/z8vKiXr16GWZGNmvWjKioKAzD4M8//+Sxxx6jSpUqrFy5kuXLl1OoUCEqVarE7t27uXr1Ki1btsTPzy/99vXXX3Po0KEM165Tp06m3l79+vUztF41aNCAAwcOkJaWlun8mVGtWrUM94sUKcLZs2cB8/uUnJxMgwYN0o8HBQVdV7y1a9cuvWv0Rk6dOsWTTz7JW2+9Rffu3fHx8cnQyrdu3Tq6dOlCly5dSEtL448//gCgefPmWXovYBZq/7v114IFC+jQoQNXrlxJH8e2ePFiihcvnuG84ODg6/6+/q1QoUIMHTqUd999l99++42+ffuyZs2aG557/vx5fHx8Mjzm4+OTqTGAzsi5FhYWERHHUb4n5Clgfl15EJR/+Z9jzX8H7+CbPvXaWLT/7e4zDCPDY82aNWP69Ols27YNd3d3wsPDadq0KdHR0Vy6dCm9u/XaOLtff/2VYsWKZXjN/51pmS9fvqy9zzvM7+7unmHMHUBKSsp1r+XllXHin5ubW/r7+d/n36m5c+dSsmRJChcuDHDdOL3atWunfx0UFERsbCxw/fcuM+Lj469rybv//vsBWLZsGV999RVgtrjeaAxjZrqO/f396d+/P8nJycyaNYuvvvqKLl260Lhx4/RzAgMDr/v+xcfHExx8859LZ6YWOhERuTO+RcDjvy0g+UpAYPg/x/JXNY/fRFhYGHny5MnQ7ZeSksLGjRupXLly+mPXxtF9+umnNG3aFDc3N5o2bUpUVFSG8XPh4eF4e3tz7NgxwsLCMtxKlChxR29v7dq1190vX748Hh4emcofEhLCqVP/dDvHxsZmeZZlWFgYXl5eGbJcunSJ/fv3Z+l1bDbbLYvDOxnvdjPBwcFcunQpw2MBAQH89ttvNGnSJL2gjo6O5oEHHriuxSyzy6KA+T39+++/8fPzu65VsFKlSuktnWBOkoiLi6NUqVJZfEfOQS10IiKS4/Lly8crr7zCoEGDCAoKomTJkowZM4bExES6deuWft61cXTffvstEyZMAMwir3379qSkpKSPwfL392fgwIH0798fm81G48aNiY2NZfXq1fj5+dGlS5csZzx+/DgDBgygZ8+ebN68mUmTJqXPMM1M/hYtWjBz5kzatGlDgQIFePfdd7NcOPn5+dGtWzcGDRpEwYIFKVSoEG+//Tbu7hnbY+bPn8+QIUNu2u3arl07RowYwbFjxyhZsiQAs2fPvu1M1LNnz7Jo0SLc3d3ZtGkTH3300W0LrsqVK3Ps2LHrHj9+/DhhYWEA7Ny5Ey8vL0JDQ4mMjOSll14CzKLY39//lq8PcOTIEaZOnYq7uzsvv/zyDYv2Jk2acPbsWU6cOEHx4sWJjo6mbt26KuhyUkREBBEREdcN+hQRkdxj9OjR2Gw2nn/+eeLi4qhTpw5//PEHBQoUyHBe8+bN2bx5c3rxVqBAAcLDwzl58mSG1rwRI0YQGhrKqFGjOHz4MPnz56dWrVq89dZbd5Svc+fOXLlyhXr16uHh4UHfvn0z7JJwu/xDhgzh8OHDPProowQGBjJixIg7Wgdt7NixxMfH07ZtW/z9/Xn99deJiYnJcE5MTMwtJwmUKFGCuXPn8s4779C4cWNsNhsPP/wwCxcu5JdffiElJYVffvmFuLg4Vq1axYkTJ6hUqRJHjx7l8uXL9OnTh7Vr17JkyRJat259y7ytW7eme/fuvP766xkef/LJJ3nzzTeZM2cOYHbzXpsocc2GDRto0aLFTV97y5YtREZGUrhwYd54443rflb+zdPTk8jISEaOHEn9+vWJjo5m1qxZt8zuzNyM7Oqgt4PY2FgCAwOJiYkhICDA6jgiIi4pMTGRPXv2ULlyZfLmzWt1HLHI008/zZgxYyhduvRtz+3RowfDhw9PH7OXWe+88w5t27alXr3rt/mcMGECPj4+dOnS5brJDla59m/jWuEbFxdH586dM8yuzSkaQyciIiK39Ouvv9K+fftMFXMA7733HpMnT87SNWJjYzl37twNizn4Z9kSRynmHI0KOhEREbmpDRs2UKhQIdq3b8/u3btv/wSgePHitGvX7qZryv0vwzAYP348H3zwwd1EdWkOOYZORERErLdmzRp69uxJaGgoycnJTJkyJdPP/fdSKLdz4cIFXnnlFUJCQu4kpqCCTkRERG6iQYMGbN++3e7Xya1rw+UkdbmKiIiIODkVdCIiIiJOTgWdiIiIiJNTQSciIiLi5Oxa0LVt25aSJUvi4+NDkSJFeP755zl58qQ9LykiIiLicuxa0DVv3pw5c+awb98+5s6dy6FDh3jqqafseUkRERERl2PXZUv69++f/nWpUqUYPHgwjz/+OCkpKXh5ednz0iIiIiIuI8fWobt48SKzZs2iYcOGNy3mkpKSSEpKSr8fGxubU/FEREREnJbdC7o333yTyZMnk5iYSP369fnll19ueu6oUaMYNmyYvSOJiMgduHLlitURRByKI/2bcDMMw8jKE95///3bFl0bNmygTp06AJw/f56LFy/y119/MWzYMAIDA/nll19wc3O77nk3aqErUaIEMTExBAQEZCWmiIhkk+TkZHbt2oXNZrM6iojDMQyDw4cPExcXR1xcHJ07dyYoKCjHc2S5oDt//jznz5+/5TmlS5fGx8fnusdPnDhBiRIlWL16NQ0aNLjttWJjYwkMDFRBJyJiseTkZFJTU4mJieGXX34hX758eHt7Wx1LxHJpaWmkpqZy5coVSwu6LHe5BgcH3/Gea9dqx3+3womIiOPLkycPefLk4erVqyQkJJCSkqKCTuRfrK5t7DaGbv369axfv57GjRtToEABDh8+zHvvvUe5cuUy1TonIiKOx9vbm4CAAGJjY0lOTrY6johDCQgIsOwXHbsVdL6+vsybN4+hQ4eSkJBAkSJFaN26NbNnz9ZvdSIiTipfvnw888wzlrdGiDgib29v8uXLZ8m1szyGLidpDJ2IiIjI7WkvVxEREREnp4JORERExMk5dJerYRjExcXh7+9/w3XrRERERMTBCzoRERERuT11uYqIiIg4ORV0IiIiIk5OBZ2IiIiIk1NBJyIiIuLkVNCJiIiIODkVdCIiIiJOTgWdiIiIiJP7f062L3bVABeKAAAAAElFTkSuQmCC\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": {
Luke Naylor
committed
"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": [