{
"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",
"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 0x7f6298a16c20>"
]
},
"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 0x7f628ef97b20>"
]
},
"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": "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",
"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_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",
"execution_count": 17,
"id": "e64456bd",
"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": "31c2cee4",
"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": "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>"
],
"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": "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",
"execution_count": 19,
"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>"
],
"text/latex": [
"$\\displaystyle 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + 0 \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + \\frac{q^{2}}{2 \\, r} \\\\ \\beta q + \\frac{1}{2} \\, \\beta^{2} r + -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"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)}}"
]
},
"execution_count": 21,
"metadata": {},
"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,
"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",
" .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": [
{
"data": {
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g/K9wvF29aVCqgdlxRERyxZWkK0zfNp0pW6cQmxLL/oH7CSoeZHYsu7Nhzkz+3r2RF7//mg8/hD5PGzg42ucyJNmlHjop8F789UUqFK3A0p5LzY4iIpLj3t/4PhP+mEC6LZ1+Dfrx8v0vU86nnNmx7EpKUjqu7k44exSjVAV/jhy24lPUESgYxRyooJMCzmKx8EKjFxj8y2DOxJ4h0DvQ7EgiIvfs1LVTeLl44VfEDzcnN4Y2HsqL971IgEeA2dHsyvHj8NeCPqQZnrR78zOaPtED6GF2rFyhu1ylwOtduzfuTu58sfMLs6OIiNyTyOhIQpeGUmVqFaZunQrAyKYjeb/N+yrm/kdibCLvvJ1MzZqwPKINXlUfMDtSrlNBJwWet6s3T9Z+klk7Z5FuSzc7johIth2MPki377tR87OarD2xlk8e+oRXmr1idiy7lJqcysV5dXE49CGjRsG780O5/4nHzI6V6zTkKoXC4MaDKetTllRrKk4O+rIXEftnGAbXkq9RzL0YMSkx7L+4n9mdZvN03adxcbT/nQvy2vFdRyhTvTJu7i78XWwMfcY0pXxNs1PlHd3lKiIiYkcMw+CXo7/w/h/vk5yezPZ+27FYLNgMW6HeY/VWYmJg0vgzvF6rEuuTZvPQgD5mRzKFvjKk0IhNieW1317j6OWjZkcREbmB1WZlwf4F1Pu8Ho989wgA41qNyziuYi4zm9XGmnkrqVbN4OPPAvnpyo+0fPoJs2OZRmNPUmi4OLrw5e4viU2J5bOHPzM7jogIcL2Qc3RwJNWayou/vkjdknWZ1mEa/yn3HyyWgrOsRk7bs3o9bZ070q/bNga+GUKZMo+YHclUKvel0HBzcmNIyBC+3P0l0QnRZscRkULuatJV3tvwHhWmVODv2L9xd3bnwKADrHpqFS3Kt1AxdxOXz17it5nzAKjXrjWHK+3ivRkhlCljcjA7kGcFXVhYGMHBwYSEhORVkyI3GBgyEAsWZmyfYXYUESmkzsSe4aVVL1F2Ulne2/Aej1R9JGM41a+In8np7FN6OoSFwbh+S2hoGcHfJy5jcbAQdF89s6PZDd0UIYXO4BWD+SHyB6JGROHq5Gp2HBEpZHou6smq46sYHDKYoY2HUsKzhNmR7Nqe3zay/Jt9vPXVIJ5/zsr7Y69QvIy/2bHsjgo6KXROx5zmQvwFQsqot1hEct+m05v4cNOH9KzVk961e3M27ixeLl54uXqZHS1fWD/tHXyS15Lech0hjR3NjmO3dFOEFDrlfMpRzqcc//wto3kqIpLTbIaNFUdW8OGmD9kUtYkaxWvg5XK9gCvtVdrkdPYtNRU2zf6E5HRPOgwbQLP+r+Po9BYOjpr2fzv67EihdC35Gg1mNuDnIz+bHUVECqCFBxbSeUFnDAx+6vkT+wftp1NQJ7Nj2b1fVxrUrg0HIk7hnH4GAGdXZxVzWaAhVym0ms9pjtWwsvnZzeqlE5F7EpcSx6yds7gQf4EP235ISnoK289up1m5ZmZHyxdSElPYHfY4k37sxQW3nkybalCrtn4uZ4dKXim03mrxFlvObGHV8VVmRxGRfOps3FlG/zaacpPL8epvr3I56TKGYeDq5KpiLguSElIxDHAt4kqyQ1kGD/fk999RMXcX1EMnhZZhGDSb0wyrYWXLc1vUSyci2XIt+RplPi2Do8WRfg36Mfy+4ZTzKWd2rHzBMGDl4tPUvtCCE/5f0vLx1mZHyvfUQyeFlsVi4Z1W73A65jRRsVFmxxERO2cYBmuOr+HJH58k1ZpKUbeiLOi+gKgRUXzS7hMVc1l0eG80bdvCI48HsvXcE5QP1uctJ6iHTgo1wzBIsabg5uRmdhQRsVOp1lQW7F/AJ39+wt4Le6lfsj6LH19MxWIVzY6W7+z6dR3Vz3ek+6wIBr9ei4cfNjtRwXFPPXQTJkzAYrHw4osv5lAckbxlsVhwc3IjKiaK3ed3mx1HROxQp+86Ebo0lEDvQH7v8zs7+u9QMZcNNquNA5sPAFCzVTO2pX3EkjVVVczlsLsu6CIiIpg5cyZ16tTJyTwipnj+5+cJXRqK1WY1O4qImOzUtVO8+OuLGX/kvd3ibQ4MOsCK3itoXbG15ttmw86dMHXIVMofasLZU1dwcXOhZb8huBbRLj057a4Kuvj4eJ588klmzZpFsWLFcjqTSJ4b12ocey/s5eu9X5sdRURMsu3vbTyx6AkqT63M13u/5viV4wA0K9eMYP9gk9PlL5fPXmLiqE00agTfR/TlWJnllK7ga3asAu2uCrrBgwfz8MMP8+CDD+Z0HhFTNAlswuM1H+eN398gMS3R7DgiksembZ1Gk9lN2HluJ9M7TCdqRBTdg7ubHSvfOvDt63Qt/SyTJ9vYuMWHeu1amR2pwMv21l8LFixg586dREREZOn8lJQUUlJSMh7HxsZmt0mRPDGhzQSqT6/OpD8n8UaLN8yOIyK5KD41nq/2fIW7kzt96/ela42uBHoH0jmoM44O2i/0buz97Q8SEmw07dKCGj3fA2BYoBbTyCvZ+kxHRUUxfPhwvvnmG9zcsnZX4IQJE/Dx8cn4KFu27F0FFcltlYpV4vNHPqdbjW5mRxGRXPLXtb8YtXoUZSeVZejKoew8txOAQO9AutboqmLuLpw7B08/DRd/H0ta5AwA/AMD8A8MMDlZ4ZKtZUuWLl1K165dcXT8vy94q9WKxWLBwcGBlJSUTMfg5j10ZcuW1bIlYtesNqt+sIsUMPsu7KPe5/XwdvWmX4N+DA4ZTPmi5c2OlW+lpaSxac5k3g5rzaGLjfh0wmV6P1NM+66aJFsFXVxcHH/99Vem5/r27Uv16tV59dVXqVWr1h2voXXoxN6dunaKh75+iG+6fUPjMo3NjiMidynVmsoPB35gc9Rmwh4OwzAMvtn7Dd1qdMPDxcPsePleemo6R8MacyCpLw8OGkrRomYnKtyyNYfOy8vrhqLNw8MDPz+/LBVzIvlBWe+yeLh4MGjFILY+v1U9dSL5THRCNDN3zCQsIoxz8ed4qPJDJKUl4e7sztN1nzY7Xr7215GLnFkylKKtJlCzSSUqD9xCDTcXs2MJ2vpL5AaODo7MeHjG9bvdtk03O46IZEO6LZ06/63Dexvfo1O1TuwfuJ9VT63C3dnd7Gj5WmKCwZgx0CDEA1drFEmXzwDgomLObmjrL5FbGPrLUL7c/SUHBh3QPBsRO2W1WVl+ZDkzd87k665f4+vuy9oTa6lbsi7FixQ3O16BELkxAtuWATz0wWqeGVCc10cbeHppcWV7k+1lS0QKi/FtxrPr/C7OxZ9TQSdiZy4nXuaLXV/wWcRn/BXzF/cF3se5uHP4uvvSplIbs+MVCIlxKRTxcsW/YnmO/lmNTRuSqVgTQMWcPVIPnYiI5BuGYWCxWOj6fVdWHl1Jr9q9GBwymEalG5kdrcCIjYVfps+jscd7+PTag19AEbMjSRZoDp3IHZyJPcOTPz5JdEK02VFECqVUayrf7fuOZnOa8fORnwH4qO1HnBl5hi+7fKliLocYNhs/fH2RoCAYP/N+Tjs+g4eHbgrLL1TQidyBk4MTq46tov/y/uRxh7ZIoXY27ixj1o2h3KRy9P6xN25Obvi6X98PtIpvFc2Ry2EbJw2hSlQ7Wraw8cvGqrQa8gZuHq5mx5IsUkEncgclPUsys9NMlh5ayrw988yOI1KgGYZBcnoyAIsiF/HJn5/QrUY3Dgw6wNo+a2lerrnJCQuWK+cuc2LfaQD8mz4PDT5lwfcOaFOn/Edz6ESyqO9PfVkcuZg9L+yhYrGKZscRKVDiU+P5bt93fLb9M+4PvJ+wh8NISE0g3ZaOj5uP2fEKHKsVZs0yCLlyHwlGIC3eWGx2JLlH6qETyaIp7adQ2qs0u87vMjuKSIFxOuY0Q34ZQplPyzBg+QACvQMz9lP2cPFQMZcL9q7dRIdW5xg40MLyC59R4+nPzI4kOUA9dCLZkGpNxcVRC2mK3Ivk9GROXj1JDf8aHLp0iAfmPcBz9Z/j+QbPa4mgXJYUn0T8/PIsPTCQOk++Q5MmZieSnKKCTiSb0m3pjFo9ihblW9C1Rlez44jkG0cvH2Xmjpl8uftLfN19OTzkMBaLBavNqi32clFqciqb5n5OzU7PEFDGixN7DlOhVlUcHDVIV5Dk2f9mWFgYwcHBhISE5FWTIrnC0eLI6djT9P2pLyeunjA7jojdu5hwkbZft6Xa9GrM2T2H0LqhLO+9HIvl+gK1KuZyz5o10K7lBRo6v8Gh8LUAVKobpGKuAFIPnchdiEmOoeHMhni4eLDp2U14uniaHUnErkTFRLH25FqeqfcMVpuVnot70rlaZx4Lfkz7quaBv4+cZtd3n9L13Y+4v5kzn026Qs0GvmbHklykgk7kLu2/uJ+mXzSlbaW2LHp8EQ4W/cUrhVuaNY3lR5Yze9dsfj32K54unpwafopi7sXMjlboHN66B4/tndlT7Fc69qqBRbt1FXj6DSRyl2oF1GJ+t/kUcytGui3d7DgiprLarNQIq0G3hd24nHiZ/z78X86MOKNiLo8YBvyxeC3rxj9NWqpBUJO6lOx3nId7q5grLJzMDiCSn3UO6kznoM7A9c3C/Yr4mZxIJG8kpiWyKHIR8/fNZ1GPRXi5evFu63epFVCL2iVqmx2vUDl8yGDYcAvWvw0mhF4i9nIsfqV8cHLRr/jCRD10Ijlgz/k9VJhSgSUHl5gdRSRX7Ty3k0ErBlHqk1KELg3FarNyMeEiAL1q91Ixl8fWfzaRnWFPcewYDHv/QRq9vBK/Ulq7rzBS+S6SA2oF1KJj1Y70XNyTX5/8ldYVW5sdSSTHXEm6go+rD44Ojryy5hUOXjrIsMbD6Fu/L5WKVTI7XqFj2AwS41Pw8HbDxbcCpYJSOTDRwM1dY6uFmW6KEMkhqdZUOn3Xic1Rm1kXuo5GpRuZHUnkrqXb0ll1bBVz98xl2eFlLO+1nLaV23I+/jzFixTHyUH9AWbYvcsgcWUnYm0Vaf/mNLPjiB3RkKtIDnFxdGHx44upFVCLZ396FpthMzuSyF2ZvGUygZ8G8sh3j3D40mEmtJlAvZL1ACjpWVLFnAmunr/C8CHJNGxk4afdPQlo8KjZkcTO6LtSJAd5uniyovcKriVf0zImkm9cSrzEt/u+pX2V9lTzq4aroytP1HyCZ+o9Q72S9TIWABZzJMYmkrKkFn7RQ/j449cZMuQpnJ3NTiX2RgWdSA7zdffF192X+NR4ei3uxdiWY2lYuqHZsUQySbWmsvLoSubumcvyI8uxYMHH1YdqftUYGDLQ7HgC7N+wk6qN61HEuwh7fKfwwsT/EKCtbuUW1IUgkkvSrGlEJ0TTel5rNvy1wew4ItgMG6nWVABG/zaaR79/lNMxp/n0oU85+9JZQuuFmpxQAM6fh1cGHiX4dCM2frsYgKZP9CCgfEmTk4k9000RIrkoPjWeLgu6sDlqMz8+/iMdqnYwO5IUQnsv7OXbfd/y3f7vGHX/KIY0HsKJqydISE3QMiN2JC0ljV9nL+PJ0d1xcYG5H66lQ59WODprr1u5MxV0IrksOT2Znot68svRXzg69Cjli2rMRPLGkoNLeHv92+y/uB9fd18eD36c/g37U79UfbOjyU1ELF1Bg/jOTNizl0Gja+KrrVclG1TQieSBf/a47FqjK4ZhYDNsODror27JWZcSL/HDgR8I9g+mZYWWLD+ynG/3fUvv2r15qPJDuDi6mB1R/uXvI3+xf80q2g3uj2Ez+OvAMSrUrmp2LMmH8qygCwsLIywsDKvVypEjR1TQSaE1bes0lh1ZxvePfY+vu/4El3tzOfEySw8t5YfIH1h7ci2GYfBOq3d4o8UbZkeT20hOho8+gst/fsYrj7yPU5dIAsrod6LcPfXQieSxdSfX0eOHHvi4+fBTz5+oFVDL7EiSz1xKvESqNZXSXqWZvm06w1YOo0X5FvQI7kGPmj0I8AgwO6LcxrYly1n5wwnGLxrGyBFW3ngtEa9iXmbHknxOBZ2ICU5ePcmj3z/K8SvHmd5xOqF1Q7XWl9xWdEI0Sw4t4YfIH1h3ch0DGw1kWsdpxCTHkJSeRElP3QGZX6yf+jYuiXvxe3QJQdX1fS85QwWdiEniU+MZtnIYp66d4rc+v2khYrmBYRhYLBbm7JpDv5/7AdC6Qmt6BPfg0eqPUsKzhMkJJSviYg0ivniDFMdydBj2AtY0Kw6ODlgcVMxJztHCwiIm8XTxZE6XOSSnJ+NgcSD8VDgp1hQeqvyQ2dHEJIZhcPDSQZYdXsayw8t4pNojvP6f1/lPuf8w4+EZdK3eFX8Pf7NjShYZNoPvFsCoURZGP5RIcKNEAC1DIrlCBZ2Iydyc3ACYt2ceX+7+ktC6oXza7lPdMFHI/HjwR1797VWOXTlGEecitKvcjrol6gJQ1a8qVf1052N+khCTwLFZHVjyw1CaNu1Bp7GTKa8ViyQXqaATsRNfdP6C5uWa89Lql1h5bCWfPPQJvWv31lBsARSXEseq46tYdngZ7Sq348k6T+Ln7scDFR5gcrvJtKnUJqPQl/wl9moiXkWL4OHjwVWHRrw6JoBGHc1OJYWB5tCJ2JlzcecY9uswIv6O4NCQQ/rFXoCsOraKKVunsPbkWlKtqdQOqM0rzV7hqTpPmR1N7pHVCou+PEob2/1E+i6mxWMtzI4khYz+9BexM6W8SvFDjx/YOWAnbk5uHL9ynF6Le3Hi6gmzo0k2xKXE8dOhnxi0YhCrj68G4ErSFVKsKUx8cCLHhx1n78C9KuYKgJ1/nKFJE+jVvzIbzw8hqJGGxyXvachVxE79M4fudMxpNvy1gRphNXiu/nO82uxVbR9mxxYeWMiM7TP44/QfpNvSqVysMs3LNQegV+1e9Krdy+SEkpO2LVlOvbhulPXZx9Q/grj//jFmR5JCSj10InaudcXWHBlyhDEtx7DwwEKqTKvC3N1zzY5V6BmGQWR0JGHbwnhs4WP8GfUncH29OE8XTya3m8zRoUc5NuwYvWv3Njmt5KS0lDR2/b4DgLrt2rLNMptFv1bh/vtNDiaFmubQieQj8anxfL79czoHdaaqX1V+OvQTTg5OtK/SXnvD5rJ/flRaLBbe+v0tZu2cxYWECzg5ONG4TGPGtRpHm0ptTE4pue3332Hvd+/Tt8mHpD18huKltMOD2AcVdCL52BOLnmDhgYVULFqRFxq9wLP1n6V4keJmxyoQ0m3p7Luwj81Rm9l8ZjPhp8JZ/fRqgv2DCdsWRlRsFK0rtKZZuWZ4uniaHVdy2d9HTjNz0gnG/bcV7VpfY/L4k1RvWt/sWCIZVNCJ5HPb/t5GWEQY3+//HpthY1u/bdQrWS9jlwHJmqtJV9l9fjetK7YGoOZnNYmMjsTZwZmGpRvSvGxzBjceTIWiFcwNKqb446M++Bj72VN6B08+aUHfWmJvVNCJFBCXEi+x8MBC+jfsj5ODEw9/+zAOFgceD36cTkGdKOpW1OyIdiUxLZEZETPYcW4H289u5+iVowBEj4qmeJHiLDu8DF93XxqVbqSlYwohw4BtS5djxYv7u7bk0t/RuLi54u2n31tin/KsoAsLCyMsLAyr1cqRI0dU0InksunbpvPd/u/YHLUZR4sj9wXex5wuc6jmV83saHnGMAxOx5xm38V97Luwj30X9+Hk4MRXXb8i3ZZOqU9KEeQXRMNSDWlYuiFNA5tSxbeKejYLuaNHYfhwg1catMbmWYMHXpthdiSRO1IPnUgBFxUTxS9Hf2H1idXM6TwHHzcfQpeGcuraKZoGNuW+wPtoGtg0X2/0nmpN5eTVkxy7coxjV44RVDyI9lXas/r4atp90w4Ab1dvagXUonnZ5nzY9kMAbIZNO3FIhuSEZLbMeodXP+vBxbQGTP0klkce9cLioAJf7J8KOpFC6IudX/Dr8V/5M+pP/o77G4AfevzAY8GPseXMFg5cPEDNgJpU8a2Cn7uf6T1WhmFwKfESZ2LPEBUbxYmrJ+havSvli5ZnXPg43gl/B5thA67vjTus8TA+bPshMckxbIraRK2AWpT1Lmv6+xD7lpaSxtGw+znMUNoP7IO7u9mJRLJOBZ1IIXcm9gx/Rv1J07JNCfQOZFz4OMauH4vB9R8NHs4eDGk8hA8e/ICzcWeZunUqAR4B+Bfxx9/DH/8i/jQs3RCAiwkXcXZwxt3ZHRdHl4weMAeLA6nWVFKtqaTb0olPjSc+NZ4izkUo51OOmOQYlh9ZTnRiNNEJ0VxMuEhCWgLfdv8WgLr/rcveC3szMrs7ubP48cV0qNqBzVGb2X9xP1V9r29gX9qrtHrdJMsObo8i7vd+eLb+L8EhFbCmWXF01hJAkv+ooBORGySlJXHo0iFOXD3BqWunqF68Og9Xe5jd53fTfWF3LiZcJD41HoDiRYoTPSoagMpTK9+wRdmvT/5KuyrtGLNuDOM2jMt07Ok6T/NV1684evko1aZXw93J/Xqx6OFPgEcAP/f6GQeLA0sOLgGgrE9ZAr0DKeFRQr1tck+uXrExZqwDc2fHs+qNLng2n0jtlg3NjiVy11TQichdSUpLIjoxmoTUBGr41wBg7Ym1XE2+SlJaEmm2NBwsDrSt1JYy3mXYd2EfkdGRODo44uniiaeLJ6W9SlOpWCVsho3k9GSKOBcx+V1JYbB7dThF9r9A24l/MPRlP4YNAxcXs1OJ3BsVdCIiUihcuxRP0eKeRJ++wMGFbxLU431KlPc3O5ZIjlBBJyIiBdrFi7AmbBrN/Cbj03s/xYrrbgcpeDRzWERECqS0lDRmT/ubatVgyvcPcdrjZbx9NLYqBZOT2QFERERyw7apfamTfISePbfy3ntBFC8eZHYkkVyjgk5ERAqMs8eiSIy3UqVeBfyav4RhGPz3ft0RLQWfCjoREcn3UlLgk08Mujh15GJ6XarU+4bqTeubHUskz2gOnYiI5GvblqygZZPzjBlj4de4b2g44DOzI4nkORV0IiKSb8VejqXKpT482/pr9u6Fl96ti7efVlCQwkfLloiISL6SEJPAlnmTqdNjOP6lPDlz+C/KVC2HxUFz5aTwyrMeurCwMIKDgwkJCcmrJkVEpAAxDPj+e3jwP9E0cP+YY3/+CUBgUHkVc1LoqYdORETs3ondh4j88RO6vv8Zj3RyZtJH8VSo4ml2LBG7obtcRUTE7qUmJVHDbxNrl52hRceKgIo5kf+lHjoREbE7Nhus+2oJjn8voPkrC3BytmCz2nBw1L18Ijej7wwREbErEdusNG0KH37qgbOLEwlxSQAq5kRuQ0OuIiJiN9ZPGc25Y6dISfmOt8Ieotl/HjI7kki+oIJORERMlZ6aTkJ8Kj6+RXApUZ/SHmXZMcnA0Ul3ropklebQiYiIacLXG3hva875tBA6vDHZ7Dgi+ZYmJIiISJ47d/wMT/VOoVVrC0v2D6FCq6fNjiSSr2nIVURE8lTMpRiKrKtFZevbzJ07kqef7oWDuhdE7km2voVmzJhBnTp18Pb2xtvbm6ZNm7Jy5crcyiYiIgXI9l82kJpiw6e4D4eKzWNk2HOEhqJiTiQHZOvbKDAwkA8++IDt27ezfft2HnjgAbp06cKBAwdyK5+IiORzx4/DkKf20ehaSzYvvt4J0KR7F3yK+5icTKTguOebInx9ffnoo4947rnnsnS+booQESkcEmISWDnzR55882lKlIC5H2+h9WNNtO+qSC64645uq9XKggULSEhIoGnTpjmZSURECoB9a1bzSEA/PnjzOIcOwQOP36diTiSXZLuHbt++fTRt2pTk5GQ8PT359ttv6dix4y3PT0lJISUlJeNxbGwsZcuWVQ+diEgBdGzHAY5t+p32w4Zi2AzOn/ybUpUDzY4lUuBlu4cuKCiI3bt3s2XLFgYOHEhoaCiRkZG3PH/ChAn4+PhkfJQtW/aeAouIiP2JiYERI2Dam79TjTCuXkrC4mBRMSeSR+55Dt2DDz5I5cqV+fzzz296XD10IiIF26ZvvmHVsmg+/WUEb7+VzvChVlyLuJodS6RQued16AzDyFSw/ZurqyuurvrGFhEpaAwDLBZIu3yIBxqeZcCnBmUCndASpyJ5L1vfda+//jodOnSgbNmyxMXFsWDBAtavX8+vv/6aW/lERMTORF+0ETlvEAku9eg4/AVaDn1XNzuImCxbc+guXLjA008/TVBQEG3atGHr1q38+uuvtG3bNrfyiYiInUhPtTJ9ukG1IAf2HfLAw8sFQMWciB245zl02aV16ERE8p+Y6Gucn9+S178dg2/dbrz/Pvj7m51KRP6hiQ4iInJLl85ew69UUXz8i7KLDrz7aQWCm5udSkT+TT10IiJyg5QU+CZsH48XvZ89xX6leddmZkcSkdvQlsgiIpLJhhVHqV0bXni1Juui36Z285pmRxKRO9CQq4iIZNj83fc0TXuKRtWP8OOPFalVa5TZkUQkC9RDJyJSyCXGJrLtl00ANOjchZ1FfmD+0grUqmVyMBHJMhV0IiKFlGHAokXwxcj3qRT1KDFXknDzcKPJY49qKRKRfEYFnYhIIXRsZySjnl5Ljx6w6cpLJDT7Ex9fd7Njichd0l2uIiKF0OaPnsAx9QKX66+nY0ez04jIvdJNESIihYDNarB5/lfgWYHm3VpS/anP8PDxxLWI2clEJCfk2ZBrWFgYwcHBhISE5FWTIiIC7NgBzf8DxrHZJJ1YBYBvKT9ci7ianExEcoqGXEVECqi4q3Hs/GIUw6cPxOpVl7CpSbRorXlyIgWRbooQESlg/vkz3a2IG8Us+xg/+gS7dqFiTqQAUw+diEgBsn3dUZy3P4Nbm+8IalAOw2ZoCRKRQkA9dCIiBcDZM+k89RQ88HAJYpL9ID0BQMWcSCGhu1xFRPK5iJ9+ofipIWzduIPJ04vR/JllOOjPdZFCRQWdiEg+dfHMFQICfSlXtz4HT3QjYrsjRf3NTiUiZtAcOhGRfObkSdg88z3uLzkXvz4H8C6m5UdECjt1youI5BOJsYlMevcUNWrArFWPcS5gPF4+LmbHEhE7oB46EZF8YutHXUhLuMLKtI28/jp4eJidSETshebQiYjYseO7DmE1XKnWoCJFW4zDzcOD8bXMTiUi9kZDriIidig2Fka9bMO2rhNRqyYAENSkLuVrVTE5mYjYIxV0IiJ2xGa18cfXX3N/w4t8NsOB363LaDZsqtmxRMTOqaATEbEjMdHXCE5+keFdl3DoEAwYVQM3DzezY4mIncuzgi4sLIzg4GBCQkLyqkkRkXzhyrkrrP3kNS6dT6BYSV+s7SLpN3EAZcuanUxE8gvd5SoiYhKrFWbOhFmTTrByxP1EVfiRRh3uNzuWiORDGnIVETHBwc07+XXMUwwdkk695pXg0b9UzInIXdOyJSIieciwGVgcLGAYlPc9xrYNF2nQrDSg3R5E5O5pyFVEJA+kpsLaWfPwvLaU+1/9EUcny/8VdyIi90hDriIiuWzNqlTq1IGwL/yxuQWSkpQKoGJORHKMhlxFRHLRho8HcPFkPCVKzOf9aR2pU6ej2ZFEpABSQSciksOS4pNITrJRzN8D57IPUrGMlfXTwaIOORHJJZpDJyKSQwwDli6xEvxXXU4ktafD6x+bHUlECgnNoRMRyQEndh/mkY6pdOvuyKLDY6n+SH+zI4lIIaIhVxGRe3Tp72hK7a5Ho6If88KywTzyyGMaXhWRPKUeOhGRu2DYDDYtXE5aqkHxMv4c8FvK6NnP0amT5sqJSN5TQScikk27d0P/xyJolt6JbcvDAWjUqR1uHm7mBhORQksFnYhIFl05d5lvx31Ow4aw+XBjtvrtpVm3VmbHEhFRQSciklWH1v/Gw2Ve4fNJZ9i9G5q0q212JBERIA+XLQkLCyMsLAyr1cqRI0e0bImI5At7f9/C33s20WHESxg2gyvnL+NXurjZsUREMtE6dCIiN3H+PLz6KhS7OJnnH/yOCv024untYnYsEZGb0pCriMi/hM+axrTBH7NiBdTqNoQawzarmBMRu6Z16ERE/j/DZmBxsGAkXqD1fSm8NAt8ffVjUkTsn4ZcRaTQ++tUOmcX9uKqW1s6DuufUdiJiOQXGnIVkUIrKT6FceMMagQ7sfNoJYqW8ANQMSci+Y7GEkSkUIo+c5GkpfexZ8Ukhg3rQp83PsTLy+xUIiJ3RwWdiBQqZ09cpHSlAIqX9ifc6MNHM2tQqa7ZqURE7o3m0IlIoRAXB/M+3Uq/Ci3Y6beBpo80MTuSiEiO0Rw6ESnQDJvBivl7qF4dXv2wEWuvTaZeqzpmxxIRyVEq6ESkQPvjqzm0tYbQsfVZIg860nH4QNw93c2OJSKSo1TQiUiBc/X8Ff5cug6ARo/15kDAKmZ9U5ry5U0OJiKSS3RThIgUGFYrfPEFWCPeo1O9RSTEHcfDy5367VubHU1EJFeph05ECoR96/5k2OOrGTAA9hlv4vLINjy8nM2OJSKSJ3SXq4gUCFs+6kZykhWXB3/i/vvNTiMikrfybMg1LCyMsLAwrFZrXjUpIgVYWqqVTXOm4ujfgP90b0mNZ+bgWdQLR3XKiUghlGdDroMHDyYyMpKIiIi8alJECqi1a6FePQuO534k+e+tAPj4F8XR2dHkZCIi5tBNESKSb1w9f4WDXw3kxelvUqxcbbwe/Z3/1FeXnIiICjoRsXs2q4GDowWPop64O0QzecJ5HuhdG4tFxZyICOimCBGxY4YB4Uv3UvLk0zg8sIJq9QLNjiQiYpe0bImI2KWjh5Lp2BE6P1mBc4k1cXVJNzuSiIjd0pCriNidzQsWUjb6Jf4+tY+vvi1Kqy7fYrGYnUpExH6ph05E7IJhM4g6cg6Aio2bcdzSn60Rrjz6KCrmRETuQHPoRMR0e/fCgW9eoXHppZR6PpIinho8EBHJDvXQiYhprl28yjsvH6d+ffjmz75cqzpdxZyIyF1QD52ImGbnxw9yLcbCrqJrGDoUXFzMTiQikj9lq4duwoQJhISE4OXlRUBAAI8++iiHDx/OrWwiUgDtD9/K4Z0nAfBqNZmaz33FSy+pmBMRuRfZKujCw8MZPHgwW7ZsYc2aNaSnp/PQQw+RkJCQW/lEpIC4cAGefzYNrz2Pc3rNNACqNqpFiQqlTE4mIpL/3dOQa3R0NAEBAYSHh9OiRYssvUZDriKFS1pKGpvm/pe+7/QiNqU4094/zhPPVtC+qyIiOeieZh/HxMQA4Ovre8tzUlJSSElJyXgcGxt7L02KSD4Tc+katS3v8NbzfnQZ3hs/v8pmRxIRKXDuuofOMAy6dOnC1atX2bhx4y3PGzt2LO+8884Nz6uHTqTg+vvo3xz+8X3q9v0Iv4AixERfw8e/qNmxREQKrLtetmTIkCHs3buX77777rbnjR49mpiYmIyPqKiou21SROxccjKMHw/t2iZR1WMFfx+8ftOUijkRkdx1V0OuQ4cOZdmyZWzYsIHAwNtvlu3q6oqrq+tdhROR/GPPmg2cWz+FcR8tZNjwKvg8dYyyRbWmnIhIXsjWT1vDMBg6dChLlixh/fr1VKxYMbdyiUg+YU03cHSy4OTqRvGi8ezbcY1qtf3QVtEiInknW3PoBg0axLfffstPP/1EUFBQxvM+Pj64u7tn6Rq6y1WkYIiPh/WfT6Fo8m80fXUZjk7acFVExCzZmkM3Y8YMYmJiaNWqFaVKlcr4+P7773Mrn4jYGcNm8MN3iVSvDrMWVCXNqyHWdKvZsURECrVsD7mKSOG25ZPHSTrhTkjIV3z6aUcqVuxodiQRkUJPk1xE5I5ioq+SbnPGr4QnlO9JzRpe9HnE7FQiIvKPe9op4m5oDp1I/mGzwbwvU2mTVI0D8b3o8NoEsyOJiMhN3PU6dCJSsO3fsINm96fx7PMuLDoxhfpPDDU7koiI3IIKOhG5wbkTfxN06j4eqvo1GzfCyE+7ULJiabNjiYjILaigExEA0lPTWT9vAWmpBqUqleFAqd95e04ozZubnUxERO5EBZ2IsH49PNflT/7j+CS7f48AoF7b/+Do7GhuMBERyRIVdCKF2LnjZ5g3egqtW8PRmP8QWfkoIe0bmx1LRESySQWdSCF2Yks4HcpPYMG8aP74A2o3rWR2JBERuQt5tmxJWFgYYWFhWK1Wjhw5omVLREyydekqrhzbSYeXR2PYDOJj4vEq5mV2LBERuQdah06kkDh2DF58EYKMT+jZcg21h67AzV1z5ERECgLtFCFSCKyf9g6rfivCvn2j6PvJizTqNhKLg8XsWCIikkNU0IkUUIbNwGYzcHRyACOdNg+k89Z3UKSIeuVERAoaDbmKFEAH9qaSsqoD59160XHo82bHERGRXKa7XEUKkGuX4nnxRajbwIUtx++nRKUKZkcSEZE8oCFXkQLi7LEonNc24szWL3j//Ud4/sV3cXExO5WIiOQF9dCJ5HMn9v0FQKlKgRywjmDa/Pq88goq5kREChEVdCL5VHQ0fPTyOsrvqcS2VbuwOFhoNeg1SlUqY3Y0ERHJYyroRPKZ9NR0Fs7YQrVq8MGcFqxLmkeD1rXNjiUiIiZSQSeSz2yaG0Yn99Y80+syhw478mC/p3By0XRYEZHCTMuWiOQD546f4fjOAzTv0Y7E2ET+2neQGs0amh1LRETshP6sF7FjKSkwaRL4Hp9Am1prSUk+QBHvIirmREQkEw25itipiJ9+YVDXX3nzTfir6HsUf3Irrm7a5UFERG6kHjoRO2U78l+eaFKMkR+1p2bNYmbHERERO5ZnBV1YWBhhYWFYrda8alIkX0mISyVizjicyj5E824tqNnvWzy8PbCoH11ERO4gz35VDB48mMjISCIiIvKqSZF8wTDghx8guJYTDpf/IPXSQQA8i3picbCYnE5ERPIDDbmKmOjiX+c59f0zjJn6KfUaBhMY+juVKqtLTkREskcFnYgJ0lKtOLs44uNfDIvFgS9mxNC0E+g+JRERuRtah04kD9ls8MvXW6kV8yTprcKpUkfbdImIyL1Td4BIHtm5LY5mzeCpQUEcT2iDp5eWIBERkZyhIVeRPLBx7hdUin8b0g7y0y9Fadnyc7MjiYhIAaKCTiSXpKemc/rw31SqXZ4q/2nHkd8T2bi5CE4uZicTEZGCRgWdSC7YuBHOLxtCvdLhpFY7QKnKgZSqPNTsWCIiUkBpDp1IDjp34m9e6n+EFi1g8f7hpDX6ChdXfZuJiEjuUg+dSA66tOQxHvQLoNacnwgNrYGDajkREckDKuhE7tH2Zb/iWaY61RtWoMgDc7g/sBQ+/manEhGRwkT9ByJ36cQJeKxrMiVPP8/p9XMBqFy/Bj7+RU3NJSIihY8KOpFsSoxNZO2U92gecpmt293Y5b+FtiPGmB1LREQKMRV0ItkUfzWWOm5T+XDUZg4dgk5PBGJxsJgdS0RECrE82/orLCyMsLAwrFYrR44c0dZfkq8c3XWMv1e9S93n/0ux4u4kxiZSxLuI2bFEREQA7eUqclsxMfDOO/DrwiMsfakbRvPFBIUEmR1LREQkE93lKnILEUtXcC1iBrNm/sTrb1Sj/MB9uLppaFVEROyPCjqRf0lNseLi6oibjy8eRT05uC+BwIregIo5ERGxTxpyFfn/Ll2CzTPH4Wdspelry3FwVAEnIiL5g+5ylULPmmZl1mdxVKsGX69oRHpAO/L47xwREZF7oiFXKdQMm8GeKe1wOlqWbt2+5P33OxIQYHYqERGR7FFBJ4XS+ZNncXL3pnhJT9IqDKFx49L0bWF2KhERkbujIVcpVFJTYdJHSTiuqkfEVx8D0OSxR6nZorHJyURERO6eCjopNLYtX0+DemmMGu3OorNfcX/fF82OJCIikiM05CqFwunIEzS81obHm86n6/c9qV27vdmRREREcox66KTASoxLYu3ns0lPMygXXInDFbby1qwnqF3b7GQiIiI5SwWdFDiGAT/+CM8/upnmboM5sHk/AMHNG2Fx0NpyIiJS8KigkwLlxO5DfPHSh3TvDjFubTgbcpK6LdUlJyIiBZsKOilQzu7bRtuKs1i5LIYVK6BicGmzI4mIiOS6PNv6KywsjLCwMKxWK0eOHNHWX5IjDAM2zl9I/NnDdHzlLWxWG6nJqbh5uJkdTUREJM/kWQ/d4MGDiYyMJCIiIq+alAJu1y5o3hx+XvAX3hwiPc3AwdFBxZyIiBQ6WrZE8h3DZrBh8ov8sq4MMTGv0PHdl2n+gG52EBGRwktz6CTfsKZZSUu1YXGwYDgVpd0j3uzaBa1VzImISCGXZ3Po/hEbG4uPj4/m0Em2/LkxkaLbm3PSeSgdh/Q1O46IiIhdUQ+d2LXzp6/Qpw/c36IIG44/SrnatcyOJCIiYnc0h07s1l/7j1FsawNSTi5k1qz2PPvs2zjoTxAREZEb6Nej2J3ILQcBKBdcmV3Wd/l8cROefx4VcyIiIregX5FiN06dgvcHL6f6sZrsWn8Ai4OFlv2HUzSgmNnRRERE7JoKOjFdUnwScyeup0YN+O+y9vzhsIR6LYLNjiUiIpJvZLug27BhA506daJ06dJYLBaWLl2aC7GkMNn61RS6+nbh1ZGxRB50okXvLlgctBSJiIhIVmW7oEtISKBu3bpMnz49N/JIIXFi9yHCv10OQMiTQ7nSOIKx473x9DQ5mIiISD6U7btcO3ToQIcOHXIjixQCcXEwbhwExXzKfUG7SE97GA8fDyrWqWZ2NBERkXxLc+gkTxg2g03z5zOwyy+EhcHV8h9R5YWNODlraFVERORe5fo6dCkpKaSkpGQ8jo2Nze0mxQ5ZHCy4nP2WXq2q8v7cjpQr52N2JBERkQIj1wu6CRMm8M477+R2M2KHrkQnsW/eyzhW7kXzrs2pM/hHXIu4mh1LRESkwMn1IdfRo0cTExOT8REVFZXbTYrJrOkGn38OQcGu2GKOYIu//n+uYk5ERCR35HoPnaurK66u+kVeWJw5/BdXfu7JpE+/5JFHqlNj0GpKltI8ORERkdyU7YIuPj6eY8eOZTw+efIku3fvxtfXl3LlyuVoOMk/khJScfdwoXhgCaKMMnz/bQp1WwGomBMREcltFsMwjOy8YP369bRu3fqG50NDQ5k7d+4dXx8bG4uPjw8xMTF4e3tnp2mxQ2lpsPTz32nm0IekFhFUrlXK7EgiIiKFTrZ76Fq1akU2a0ApoMJXX2Lgi8W5EFWf79/uRUN/N7MjiYiIFEq5PodOCqb1n0+iWvpHBJY4wnffFaNu3Y/MjiQiIlJoqaCTLEtOSOb0oTNUa1iFqq27czIigFVrPbBoeWoRERFTqaCTOzIMWLYM2PwcVQMOYK27izLVylGm2pNmRxMRERG09ZfcwYk9R+j3xCEefRR+PvkWbg9+j6OT7lwVERGxJ+qhk1sybAZpvz9O54rBdFr6LZ07V8eiWk5ERMTuqKCTTAybwZ8LvsWv+n8IalAO1wd/oG3FQNw9zU4mIiIit6IhV8mwZw+0fzCBCldH8dcfiwGoULsq7p7uJicTERGR21FBJ8REX+O3j0bxQPOrnD7nydGqe3ho2AizY4mIiEgWqaATkhOSqOnxHTPe38WePdDyIX+zI4mIiEg2ZHvrr7sVFhZGWFgYVquVI0eOaOsvk+0N30vM5nep3f9rivq5kZqcioubi9mxRERE5C7kWQ/d4MGDiYyMJCIiIq+alJu4cN6gb194vJcLRZ2juHL2AoCKORERkXxMd7kWIpu/W0DygTn8/POvjB9fneDn/tSaciIiIgWACrpCIDEuhSJernj6B5JerAqHI5PxCygCqJgTEREpCPJsDt0/YmNj8fHx0Ry6PHD6NOydO4JijkdpOno5DroFRkREpEDSr/gCKDkhmY/Gx1C9OixY3xbK98RCntbtIiIikofUQ1fAGDaDyMlN2HKoLoeLzuKtt8DLy+xUIiIikps0h66AOLn3KF4BpSle0oOEim/T+oEqPFfP7FQiIiKSFzTkms/Fx8PYN2Lx3daIiPnTAGjc9REq1atucjIRERHJKyro8inDZrDhu2XUDE7nw0+9WXplKa0GDDc7loiIiJhABV0+dXz3QZpbH2VAp5UcPAihr7TG3dPd7FgiIiJiAhV0+ci1i1dZM30q1nSDKg2CORG8n9fDOlGhgtnJRERExEwq6PIBmw1mz4YB3bdyX5E3OLLzGABVGgSbnExERETsgQo6Oxe5MYKZw96lXz9wrdiehAf/okbjqmbHEhERETuiZUvs3OUT+2lV+Uc2bxhJ0/94AL5mRxIRERE7k2cLC4eFhREWFobVauXIkSNaWPgW0tJgw5zPSbl2jo6vjsVmtWHYDBydHc2OJiIiInYqz4ZcBw8eTGRkJBEREXnVZL6z7neD+vVhzYpruDtewWY1cHB0UDEnIiIit6UhVztg2Aw2f/IUv4TXw8dnFI+PfZUGDcxOJSIiIvmFboowUXJCCmmpBhYHC2luQXTuWZY//kDFnIiIiGRLns2h+0dsbCw+Pj6Ffg7drz/HUONUQw44jqHjoKfNjiMiIiL5mHro8tiJA+d4+GHo0NmHNSf7EdS0kdmRREREJJ/THLo8dCRiH+UjG+GT/AuLF7eha9dXsVjMTiUiIiL5nXrocplhM9j1+w4AqjasxTZbGLOXNKVbN1TMiYiISI5QQZeL9u2Dd/stpM7ZxkRuO47FwcJ/+j5PEe8iZkcTERGRAkRDrrkgJvoa307bwtD32xNUtRsduv5GSOPKZscSEblrqamppKenmx1DxO44OTnh4uJidgwVdLlh14LJPFb6M5In/MXg4e64uLQ2O5KIyF1LTU3lwIED2Gw2s6OI2B0HBwdq1qxpelGnZUtyyIEN27j41zlaP92F+GvxxF2JoVSlMmbHEhG5Z4mJiRw8eJAKFSrg7u5udhwRu5GUlMSpU6coXbo0rq6uuLq64uHhYUoW9dDdo4sX4fXXoZnjDGpVisJm7YxnUU88i3qaHU1EJEe5u7tTpIjmAIv82/Lly0lISMDb25snnnjClKJON0XcJZvVRvisaQzu+gs//gjpdafSYOQqHBx166qIiEhh4uHhgYuLC7GxsaSkpJiSQQXdXbJYLBS59it9Ho7gyBHoN8gLR2dHs2OJiIhIHvtnuNVMeTbkGhYWRlhYGFarNa+azHF/n4rl1MIBGNWG0fzRptQf/hNOLhq1FhEREXPlWQ/d4MGDiYyMJCIiIq+azDEpyQYTJkD1Wh4kxsThaL0CoGJORERE7IIqkjs4secw6b9356uwH+k/oBqNX1mOj4/ZqURERET+jwq6W4i9moh3sSKUqlyOHasbsGyZhaoNzE4lIiIiciPdFPEvCQnw9Qc/k/x9RU4evIC7pzvNR31F1QZVzY4mIiIiclMq6P4/w2awfOEZatSAVz5qxr6UIZQo42V2LBERMVmrVq148cUXzY5xS3mZ7+DBg3nSjmSfCrr/LzzsHepfbEKTRklsivClzfC3KOKtBTRFREQAJk+ebPrSHNHR0Xz88ce3Pefs2bN5lMa+FOqCLubSNQ78eQiAKm2f4Wy52fzwozuVKpkcTERE8lRqaqrZEUzTqlUrfvzxx9ues3r1avz9/alk8i9If39/WrRoQVhY2C3PWb9+Pf3792fdunVZvm5ERAQvvvgi8+bNo3///pw8eTIn4uapQlnQ2Wzw5Zew+ZNnSNvYF8OAwOoVCOncwexoIiIFWoUKFZg8eXKm5+rVq8fYsWOB68XFkCFDGDJkCEWLFsXPz48333yT/912PCvnGIbBxIkTqVSpEu7u7tStW5dFixbdcI2RI0dSvHhx2rZte9vc6enpt2wvJSWFYcOGERAQgJubG82bN79hia47ve9/Mg0bNoxXXnkFX19fSpYsmek4QEJCAn369MHT05NSpUrxySef3Db3nSxduhRHR0dee+010tLSbnne1KlT6dmz5z21lVMaN27Mtm3buHTp0k2P9+7dm88++4yoqCief/55Vq5cedvrpaSk0KNHD0aPHk1oaCh9+/alb9++uRE9VxW6gi7yj+30evgQzz4Lv12eSInHFmHRbl0iItl2Lu4cyenJAJy8epKd53Zm+jgXd+6urjtv3jycnJzYunUrU6dOZdKkScyePTtb57z55pt8+eWXzJgxgwMHDjBixAieeuopwsPDb7jGpk2b+Pzzz+860yuvvMLixYuZN28eO3fupEqVKrRr144rV67c1Xv38PBg69atTJw4kXHjxrFmzZqM46NGjWLdunUsWbKE1atXs379enbs2JFxfO7cuViy+EstPT2dHTt2sHDhQi5evHjLz8HevXsJDAzE0dF+dkPq3Lkzc+fOveVxJycn+vTpw8yZM4mNjeXZZ59lyZIlmYr+f2zYsAFvb29KlCgBXC8Yt2zZwoULF3Irfu4w8lhMTIwBGDExMXndtGFNtxrHplQ3fhjV3wgPz/PmRUTypYSEBGP79u1GQkJCpufHrBtjHL181DAMw3hy8ZMGY8n0MWbdmBuuVb58eWPSpEmZnqtbt64xZsz1c1u2bGnUqFHDsNlsGcdfffVVo0aNGhmP73ROfHy84ebmZmzevDlTO88995zRq1evjGvUq1cvS+//du3Fx8cbzs7Oxvz58zOOpaamGqVLlzYmTpyY5ff9TzvNmzfPdE5ISIjx6quvGoZhGHFxcYaLi4uxYMGCjOOXL1823N3djeHDhxuGYRg//vijERQUlKX3NXv2bOPo0ev/fx988IHh7+9/09/Nn376qTF79uyMx+np6cb06dONp556yti2bZthGIbRu3dvY968eVlq93bS09ONadOmGaGhoRnXjoqKMlq2bJnpvOjoaKN169ZZvq7NZjOWLl1qPPnkk8Zvv/2W6djMmTNvuH7x4sWNtWvX3vG6/3xvLFq0yJg9e7YxadIk4/Lly1nOlZMKfA9demo64bPCOLwrCgdHB1wfWsmj74XRooXZyURE8rcBDQcQ6B0IwLut32VH/x2ZPgY0HHBX173vvvsy9TI1bdqUo0ePZto68nbnREZGkpycTNu2bfH09Mz4+Oqrrzh+/HjGaxo1anTPmY4dO0ZaWhrNmjXLOObs7Ezjxo3v6o7QOnXqZHpcqlQpLl68CMDx48dJTU2ladOmGcd9fX0JCgrKeNy1a1cOHTp0x3bi4+OJjo6mSpUqAAwfPpwiRYrw4Ycf3nDumTNnCAgIyHj8008/8cQTT5CYmMipU6cA6NSpE7GxsVl/o7ewbNkyevbsSVJSUsY8tjVr1hAYGJjpvOLFi2f6v7yTS5cusXPnTvz8/KhYseINx9zc3DI95+bmxrVr1+7uTZikQC8svGEDvPZSIoueGU/kNjeC6j9HYPUKZscSESkQSnmVyvh3xWIVqUjF25x9nYODww3DXrebu3U3bDYbACtWrKBMmTKZjv3vXZoeHh451ua/hzkNw8j0XFbft7Oz8w3X/ef9/Pv192LWrFn069cv47Gbmxvjx4+nf//+DBo0KNPnLT4+Hnd394zHDzzwADabjY0bN/LNN98AULly5ZveMDFlypTb3mDQsmVLunbtmvG4TZs2APz+++/MmTMHuH6Tw83mOGZlaPns2bNMmzaNtLQ0hg4dSvny5W84x8fH54bPbXx8PMWLF7/j9e1JgSzoLv51nr3fjuWx9z6geu2inG90iAebeJsdS0Sk0PP39+fcuf+bWxcbG3vDL/wtW7bc8Lhq1aqZ5nDd7pzg4GBcXV05ffo0LVu2zJHct2qvSpUquLi48Mcff9C7d2/geqG2ffv2TGvDZeV930mVKlVwdnZmy5YtlCtXDoCrV69y5MiRbL3P8+fP4+bmhp+fX6bne/fuzaRJk3j77bf54osvMp4vXrw4V69ezXhctGhRvv76a1q1apVR6O3bt49nnnnmhraGDx+enbeIt7c3CxYsoEWLFhkFd3h4OO+//z7Xrl2jaNGiGec6Od26hDl16hRhYWG4uLgwbNgwSpUqdctzq1evzqxZszIep6SkEBcXd9Piz54VyILOarVS1fM35n92iA5P34eDg4o5ERF78MADDzB37lw6depEsWLFeOutt26YbB8VFcXIkSMZMGAAO3fuZNq0aTfczXm7c7y8vHj55ZcZMWIENpuN5s2bExsby+bNm/H09CQ0NDTbuW/VnoeHBwMHDmTUqFH4+vpSrlw5Jk6cSGJiIs8991y23vedeHp68txzzzFq1Cj8/PwoUaIEb7zxBg4O/zd7asmSJYwePfq2w64zZsygf//+N71L9LXXXqNnz56MGDGCWrVqAVCjRg1Onz6d6bwLFy5kKio9PT0z5bgXUVFRGUPB+/fvx9nZmYCAAObNm8fzzz8PXC+avbxuvvj/xx9/TGJiIq+99toNRevNtGjRgosXL3LmzBkCAwMJDw8nJCREBd2thIWFERYWlmkORE768+c/Yf971HzhR0pVKoP1hcOUd7afO3JERARGjx7NiRMneOSRR/Dx8eHdd9+9oaeqT58+JCUl0bhxYxwdHRk6dCj9+/fP1jnvvvsuAQEBTJgwgRMnTlC0aFEaNGjA66+/fle5b9feBx98gM1m4+mnnyYuLo5GjRqxatUqihUrlq33nRUfffQR8fHxdO7cGS8vL1566SViYmIyjsfExHD48OFbvv7IkSOMHz+ecePG3bad119/nWXLlgHQvn17+vXrx0svvZRxvGfPnowcOZKvv/6a9PT0uyqSb6V79+68+uqrLFy4EICGDRsyffr0TD2AERERPPDAAzd9/csvv5yt9pycnJg3bx7jx4/nvvvuIzw8nPnz5991frNYjJwclM+C2NhYfHx8iImJwdv73nvOThy3MWKkA0d3RDJv2AhKd59Lmcq37loVEZHsSUxM5ODBg9SoUYMiRXJ3B51WrVpRr169G9Zsy+45krP69+/PuHHjKFmypNlRgOvL0nTu3JnGjRubmuOf741Tp05x7do14uLi6NOnD76+vnmeJV8PuW6YMxPj5Hfs3r2WjycF0+ixVVpTTkREJIe9/fbbTJ8+nffee8/sKMTGxhIdHX3LYm7SpEm3vQO2devWdO/ePbfimSbfFXSGzSDmSiJFi3vgU7Y6V+OaEbk/HQ8vF7OjiYiIFEiBgYF07dqVX375hY4dO5qWwzAMJk2adNvCcsSIEXmYyH7kq4IuMhKiFvaliPNVmr/+E3XbtoC2WlBORKSgWL9+fY6cIzmvYcOGZkfg8uXLDBw4EH9/f7Oj5Jo9e/bw8ccfc/LkSV577TWaNWvGO++8Q0pKCufPn2fMmDHUq1fvhtfli4WFYy7F8saoq9StCwu3PoFbrQEaWhURESlkihcvnmmR44JoypQpfPnll7Rv355nn32Wfv36MWrUKB5//HFWrFiRaUmZ/2X3BZ3NauPSt00od+1N3n0XPlvagZAu5nX3ioiIiOSG48ePU7p0aZycnDh79ixXrlzh9ddfp0yZMly+fBlPT086dep009fa7ZBr5KadlKhaHb+AIlytMInOD9eiVGWzU4mIiIjkjgsXLmTcsPHHH3/QvHlzGjRoAMBjjz3GY489dsvX2l0P3aVLMHLwZSocaU7Ed9dXbm7UuT2lKgfe4ZUiIiI5r1WrVpl2fbA3eZnvbvanlay7//77qV+/PpcuXWL//v20atUqy6+1m4LOmmZlzaz5VA+yMme+HyuS1/LgwMFmxxIRERFg8uTJmfbCvZXo6Gg+/vjj255z9uzZnIpVIK1btw7DMPJnQXd0+14ecO/Da8+Gc+QI9BjYFCcXux0RFhGRAiQ1NdXsCKZp1aoVP/74423PWb16Nf7+/lSqVOmO1/P396dFixaEhYXd8pz169fTv39/1q1bl+WcERERvPjii8ybN4/+/fvf1U4b+cW6detwdXXlvvvuy/JrTC3ozp88y6pJH2JNN6jetD7nGh3n5Y8eoIDfwCIiUmhVqFDhhh0e6tWrx9ixY4HrxcWQIUMYMmQIRYsWxc/PjzfffJP/3dQoK+cYhsHEiROpVKkS7u7u1K1bl0WLFt1wjZEjR1K8eHHatm1729zp6em3bC8lJYVhw4YREBCAm5sbzZs3JyIiIlvv+59Mw4YN45VXXsHX15eSJUtmOg6QkJBAnz598PT0pFSpUjfscZtdS5cuxdHRkddee420tLRbnjd16lR69uyZ5es2btyYbdu23XS/WIDevXvz2WefERUVxfPPP8/KlStve72UlBR69OjB6NGjCQ0NpW/fvvTt2zfLefKbdevWcd999+Hm5pbl15hW0E2eDEOf3EmDIp9w6uAZAAKrVzArjoiI2Il58+bh5OTE1q1bmTp1KpMmTWL27NnZOufNN9/kyy+/ZMaMGRw4cIARI0bw1FNPER4efsM1Nm3axOeff37XmV555RUWL17MvHnz2LlzJ1WqVKFdu3ZcuXLlrt67h4cHW7duZeLEiYwbN441a9ZkHB81ahTr1q1jyZIlrF69mvXr17Njx46M43PnzsWSxXW90tPT2bFjBwsXLuTixYu3/Bzs3buXwMBAHB2ztz96586dmTt37i2POzk50adPH2bOnElsbCzPPvssS5Ys4WY7km7YsAFvb29KlCgBXC8Yt2zZwoULF7KVKT84f/48hw4dytZwK+TAXa6GYRAXF3fL4ykpKaSkpGQ83r5yLQBjx8YyYEALkh7ajb+fJ7GxsfcaRUREckFycvJNnz93DooVAzc3OHkSrl7NfLxUqesf2VW2bFkmTZqExWIhKCiIffv2MWnSJPr165elcxISEvj000/5/fffadq0KQCVKlXijz/+4PPPP6dly5YAVKlShYkTJ95Tpt69ezNjxgzmzp1Lhw4dAJg1axZr1qzhiy++YNSoUdl673Xq1GHMmDEAVK1alenTp7N27Vratm1LfHw8X3zxBV999VVGj+K8efMIDPy/mwZ9fHwICgrKUlvz5s0jNDQUPz8/Ro8ezbhx4+jTp88N+6yvXbuWkJCQjMdWq5UZM2awfft2Bg8eTEhICGfOnOGpp57KtOhzy5YtCQsL4+WXX75tDgcHB5544gkef/xxli1bxtNPP03fvn1p06ZNxjmnTp3KtD+qo6MjXl5eHDhwIKPIM1NaWhqpqamkpqYSFxeHk9P/lVdeXl5ZLrLh+p2uJUqU4PHHH89Whnsu6OLi4vDx8cn26wyjLP/9L/z3v/eaQEREclNQUBDz58+/4fnPP4ennoIqVeCtt+Dfp4wZA/8aMcyS++67L9MvwKZNm/LJJ59gtVozeolud05kZCTJyck3DKOmpqZSv379jMeNGjW650zHjh0jLS2NZs2aZRxzdnamcePGd3VHaJ06dTI9LlWqFBcvXgSur1GWmpqaUaQC+Pr6ZirgunbtSteuXe/YTnx8PNHR0VSpUgWA4cOHM2PGDD788EPGjx+f6dwzZ85k6i1atmwZPXv2ZOPGjZw8eZKQkBDWrFmTqbCE64sA325P1X+7dOkSO3fuxM/Pj4oVK95w7N/Dj25ubly7di3L189NX375JVFRUbi6uvL2229n6siKiYm5oUi+nbp163L+/PlsZ7jngs7Ly4uYmJhbHv93D93fZ/7mvqb3ERkZSZkyZe61+SwJCQm5YT6D2ssf7cXGxlK2bFmioqKy9Q1xLwry57Ogt6evl9xpb8OGDURFRd1wbMCA6z10AO++CyNHZj5+s945BweHG4bUbjd3627YbDYAVqxYccPvmf+9S9PDwyPH2vx3D4xhGJmey+r7dnZ2vuG6/7yfmw1F3q1Zs2Zl6vF0c3Nj/Pjx9O/fn0GDBmX6vMXHx+Pu7p7x+J+es99//505c+YA129yuNk8xKz0TJ09e5Zp06aRlpbG0KFDKV++/A3n+Pj43PD+4+PjKV68+B2vnxf69u3L1atXiYuL44knnqDYP98YXK+T8sI9F3QWi+WufnB6eXnl2Q9cR0fHPGtL7eUOb29vfb2ovSzT10vOtufp6XnTY/9bsFWseP3jTvz9/Tl37lzG49jY2BvuVtyyZcsNj6tWrZppDtftzgkODsbV1ZXTp09nDK/eq1u1V6VKFVxcXPjjjz/o3bs3cL1Q2759e6a14bLyvu+kSpUqODs7s2XLFsqVKwfA1atXOXLkSLbe5/nz53Fzc8PPzy/T871792bSpEm8/fbbmbaXKl68OFf/Zzzd29ubBQsW0KJFi4yiODw8nPfff59r165RtGjRjHP/d+jx306dOkVYWBguLi4MGzaMUrcZn69evTqzZs3KeJySkkJcXNxNiz8zODs74+LigouLS57WN/+rUKwLMnhw3q5np/byt4L++Szo7eW1gv75zOn2HnjgAebOnUunTp0oVqwYb7311g2T7aOiohg5ciQDBgxg586dTJs27Ya7OW93jpeXFy+//DIjRozAZrPRvHlzYmNj2bx5M56enoSGhmY7963a8/DwYODAgYwaNQpfX1/KlSvHxIkTSUxM5LnnnsvW+74TT09PnnvuOUaNGoWfnx8lSpTgjTfewMHh/+5vXLJkCaNHj+bQoUO3vM6MGTPo37//Te9Afe211+jZsycjRoygVq1aANSoUYPTp0/f8Pn4Z7h2//79ODs7ExAQwLx583j++eeB64XtrXqnPv74YxITE3nttdduKCxvpkWLFly8eJEzZ84QGBhIeHg4ISEhdlPQ2QUjj0VFRRmAERUVlddNSz4UExNjAEZMTIzZUSQf0NdL7khISDC2b99uJCQk3PO1YmJijMcff9zw9vY2ypYta8ydO9eoW7euMWbMGMMwDKNly5bGoEGDjBdeeMHw9vY2ihUrZrz22muGzWbLuEZWzrHZbMaUKVOMoKAgw9nZ2fD39zfatWtnhIeHZ1xj+PDhWcp8p/aSkpKMoUOHGsWLFzdcXV2NZs2aGdu2bcvW+75Vpi5duhihoaEZj+Pi4oynnnrKKFKkiFGiRAlj4sSJmV735ZdfGrf71X748GHD0dHRAG770alTp4zXREdHG48++mim6xw/ftx47LHHjO+//974/vvvjSeeeML49NNPjStXrmScs2nTJmPkyJFZ+RRnyZo1a4wXXnjBmDt3rtG3b1/j+PHjOXbtu/XP98aiRYuM2bNnG5MmTTIuX75sShaLYeTgoHwWREdHExAQwMWLF/H398/LpiUfSklJYcKECYwePTpLK5RL4aavl9yRmJjIwYMHqVGjBkWKFMnVtlq1akW9evVuWLMtu+dIzurfvz/jxo2jZMmSWX7Nm2++SefOnWncuHEuJjPXP98bp06d4tq1a8TFxdGnT59Md+TmlTwfcv3nh6x+2EpWuLq63rCwpsit6OtFJHe8/fbbTJ8+nffeey9L58fGxhIdHX3LYm7SpEm3vQO2devWGZvUS9YUijl0IiIicvcCAwPp2rUrv/zyCx07drztuYZhMGnSpNsWfyNGjMjpiIWeCjoREbEb/7sw7b2cIzmvYcOGWTrv8uXLDBw4UNOq8pgKOhEREckx9rI2XGFj2l6uIiIiIpIzVNCJ3Rk7diwWiyXTR3burJKCb8OGDXTq1InSpUtjsVhYunRppuOGYTB27FhKly6Nu7s7rVq14sCBA+aEFZECb/ny5Vy+fNnUDCroxC7VrFmTc+fOZXzs27fP7EhiRxISEqhbty7Tp0+/6fGJEyfy6aefMn36dCIiIihZsiRt27YlLi4uj5OKFBw7duxg8+bN7Nmzx+wodsfd3Z1Fixbl+DZ22aE5dGKXnJyc1Csnt9ShQwc6dOhw02OGYTB58mTeeOMNunXrBsC8efMoUaIE3377LQMGDMjLqCIFwpIlS2jfvj2Ojo7079+fuXPnmh3JrrRu3ZotW7Zw4sQJ0zKoh07s0tGjRyldujQVK1akZ8+epn6TSP5y8uRJzp8/z0MPPZTxnKurKy1btmTz5s0mJhO4vijw/+5xaq/sPWde5tu3bx8NGjRg9erVPP300/Tv3z/Lrz148GAuJrMfDg4OBAYGcvHiRfMy5FVDYWFhBAcHExISkldNSj7VpEkTvvrqK1atWsWsWbM4f/48999/v+nzEyR/OH/+PAAlSpTI9HyJEiUyjonI/0lMTCQhIeGWx8+ePUtgYCDp6elcuHABJ6esDe5Nnjw5y5sIREdH8/HHH2fpXHvl7u5OcnKyae3nWUE3ePBgIiMjiYiIyKsmJZ/q0KED3bt3p3bt2jz44IOsWLECuD5sJpJVFosl02PDMG54TiQ1NdXsCKZasWIFDRo04KOPPrrlOYZh4OjoSPfu3encuTPXrl2743VXr16Nv78/lSpVylIOf39/WrRoQVhYWFajZ0lERAQvvvgi8+bNo3///pw8eTJHr29PNOQqds/Dw4PatWtz9OhRs6NIPvDP3Mt/98ZdvHjxhl47MVdKSgrDhg0jICAANzc3mjdvnumP/p9//pmiRYtis9kA2L17NxaLhVGjRmWcM2DAAHr16gVcLzwmTpxIpUqVcHd3p27duixatChTm61atWLIkCGMHDmS4sWL07Zt21vmS09PZ8iQIRQtWhQ/Pz/efPNN/nf78zvlr1Chwg37zdarVy/T9nStWrVi2LBhvPLKK/j6+lKyZMkbtq9LSEigT58+eHp6UqpUKT755JPbf2Kz4eGHH+aJJ5645fGLFy9y6NAhAJKSkjh06BCtW7fmzJkz/Prrrxkfu3fvzvS6qVOn0rNnz2xlady4Mdu2bePSpUvZfh83k5KSQo8ePRg9ejShoaH07duXvn375si1byYpKQk3N7dcu/6dqKATu5eSksLBgwcpVaqU2VEkH6hYsSIlS5ZkzZo1Gc+lpqYSHh7O/fffb2Iy+bdXXnmFxYsXM2/ePHbu3EmVKlVo164dV65cAaBFixbExcWxa9cuAMLDwylevDjh4eEZ11i/fj0tW7YErm8G/+WXXzJjxgwOHDjAiBEjeOqppzKdD9d7+52cnNi0aROff/75LfP9c97WrVuZOnUqkyZNYvbs2VnOn1Xz5s3Dw8ODrVu3MnHiRMaNG5fp63fUqFGsW7eOJUuWsHr1atavX8+OHTsyXWPu3Ll33QN9u9ft2LGDdu3asWnTJtauXcuUKVNwdnYmMDCQ9u3bZ3zUq1cv4zV79+4lMDAQR0fHbGfp3LnzLW+4mDJlCuPGjcvy1IkNGzbg7e2d8Ydc48aN2bJlCxcuXMh2rjux2WycOXOGgICAHL92lhl5LCYmxgCMmJiYvG5a8omXXnrJWL9+vXHixAljy5YtxiOPPGJ4eXkZp06dMjua2Im4uDhj165dxq5duwzA+PTTT41du3YZf/31l2EYhvHBBx8YPj4+xo8//mjs27fP6NWrl1GqVCkjNjbW5OT5U0JCgrF9+3YjISEh84HEs4aRnnT93/GnDePagf87dnXv9eP/0rJlS2P48OFGfHy84ezsbMyfPz/jWGpqqlG6dGlj4sSJGc81aNDA+Pjjjw3DMIxHH33UGD9+vOHi4mLExsYa586dMwDj4MGDRnx8vOHm5mZs3rw5U3vPPfec0atXr0zt16tX747vuWXLlkaNGjUMm82W8dyrr75q1KhR4/rbzUL+8uXLG5MmTcp03bp16xpjxozJ1E7z5s0znRMSEmK8+uqrhmFc/1p3cXExFixYkHH88uXLhru7uzF8+PCM53788UcjKCjotu9p06ZNxtChQ42ZM2car776qrFx40bDMAxjzJgxxssvv2zMnTvXmDt3rvHoo48a8fHxhmEYxsqVK297zZv59NNPjdmzZ2d6Lj093Zg2bZoRGhpqbNu2zTAMw4iKijJatmyZ6bzo6GijdevWt7z2+fPnjbFjxxqjRo0yDh06dNscM2fOvOH6xYsXN9auXZv1N3MH/3xvDBw40AgJCTEmTpxoXL58Oceunx3qoRO7c+bMGXr16kVQUBDdunXDxcWFLVu2UL58ebOjiZ3Yvn079evXp379+gCMHDmS+vXr8/bbbwPXe05efPFFBg0aRKNGjfj7779ZvXo1Xl5eZsYueI5+Dolnrv/74Eew6X+G2NZ1uH78Fo4fP05aWhrNmjXLeM7Z2ZnGjRtnujOyVatWrF+/HsMw2LhxI126dKFWrVr88ccfrFu3jhIlSlC9enUiIyNJTk6mbdu2eHp6Znx89dVXHD9+PFPbjRo1ytLbu++++zL1XjVt2pSjR49itVqznD8r6tSpk+lxqVKlMu6WPH78OKmpqTRt2jTjuK+vL0FBQZle07Vr14yh0Zs5d+4c3bt35/XXX6dfv364ubll6uXbunUroaGhhIaGYrVaWbVqFXB9OY7sullP1bJly+jZsydJSUkZ89jWrFlDYGBgpvOKFy9+w//X/ypRogRjxozhrbfe4pdffmHo0KH8+eefNz330qVLNwyBurm5ZWkOYHYlJibSvXt3nJ2dc/zaWaV16MTuLFiwwOwIYudatWqVaS7Tv1ksFsaOHXvDXCTJYVUHgEux6/+uMQqqvvB/x1qvBNdb7+n5z//fnW5eadWqFV988QV79uzBwcGB4OBgWrZsSXh4OFevXs0Ybv1nnt2KFSsoU6ZMpmv++05LDw+P7L3Pu8zv4OBww9fpzRae/XcRYLFYMt7P7b7Os2Px4sWUK1cuY47pv783GjZsmPFvX19fYmNjgRs/d1kRHx+Pu7t7pufatGkDwO+//86cOXOA68PlN5vDmJWhYy8vL0aMGEFqairz589nzpw5hIaG0rx584xzfHx8bvj8xcfH58pes506deLatWumLl6uHjoREbk77qXA8f/3gHiUBZ/g/ztWtPb147dQpUoVXFxc+OOPPzKeS0tLY/v27dSoUSPjuX/m0U2ePJmWLVtisVho2bIl69evzzR/Ljg4GFdXV06fPk2VKlUyfZQtW/au3t6WLVtueFy1alUcHR2zlN/f359z585lHI+Njc32XZZVqlTB2dk5U5arV69y5MiRbF3HZrPdtji8m/lut1K8eHGuXr2a6Tlvb29++eUXWrRokVFQh4eH8+CDD97QY5bVZVHg+uf077//xtPT84ZewerVq2daFy4lJYW4uLgCO9qjHjoREclzHh4eDBw4kFGjRuHr60u5cuWYOHEiiYmJPPfccxnn+fj4UK9ePb755humTJkCXC/yevToQVpaGq1atQKu99i8/PLLjBgxApvNRvPmzYmNjWXz5s14enoSGhqa7YxRUVGMHDmSAQMGsHPnTqZNm5Zxh2lW8j/wwAPMnTuXTp06UaxYMd56661sF06enp4899xzjBo1Cj8/P0qUKMEbb7yBg0Pm/pglS5YwevToWw67du3alXfffZfTp09Trlw54PpoyJ3uRL148SKrV6/GwcGBHTt28OGHH96x4KpRowanT5++4fmoqCiqVKkCwP79+3F2diYgIIB58+bx/PPPA9eL4qxMjTh58iQzZ87EwcGBF1544aZFe4sWLbh48SJnzpwhMDCQ8PBwQkJCVNCJiIjkpA8++ACbzcbTTz9NXFwcjRo1YtWqVRQrVizTea1bt2bnzp0ZxVuxYsUIDg7m7NmzmXrz3n33XQICApgwYQInTpygaNGiNGjQgNdff/2u8vXp04ekpCQaN26Mo6MjQ4cOzbRLwp3yjx49mhMnTvDII4/g4+PDu+++e1froH300UfEx8fTuXNnvLy8eOmll4iJicl0TkxMDIcPH77lNcqWLcvixYt58803ad68OTabjY4dO/Lrr7+yfPly0tLSWL58OXFxcWzatIkzZ85QvXp1Tp06xbVr1xgyZAhbtmzht99+o3379rfN2759e/r168dLL72U6fnu3bvz6quvsnDhQuD6MO/06dN55plnMs6JiIjggQceuOW1d+3axbx58yhZsiSvvPLKDV8r/8vJyYl58+Yxfvx47rvvPsLDw5k/f/5ts+dnFiOnBuizKDY2Fh8fH2JiYvD29s7LpkVE5C4kJiZy8OBBatSoQZEiRcyOIyZ5/PHHmThxIhUqVLjjuf3792fcuHHZ3pP7zTffpHPnzjRu3PiGY1OmTMHNzY3Q0FBT13v7X/98b/xT+MbFxdGnTx98fX3zPIvm0ImIiMhtrVixgh49emSpmAN4++23mT59erbaiI2NJTo6+qbFHMDw4cMZMGCA3RRz9kYFnYiIiNxSREQEJUqUoEePHkRGRmbpNYGBgXTt2pVffvklS+cbhsGkSZN477337iVqoaY5dCIiInJTf/75JwMGDCAgIIDU1FRmzJiR5df+71Iod3L58mUGDhyIv7//3cQUVNCJiIjILTRt2pS9e/fmeju5sTZcYZNnQ65hYWEEBwcTEhKSV02KiIiIFAp5VtANHjyYyMhIIiIi8qpJERERkUJBN0WIiIiI5HMq6ERERETyORV0IiIiIvmcCjoRERGRfE4FnYiIiEg+p4JOREREJJ/TwsIiIpIlSUlJZkcQsSv29D2hgk5ERG7LyckJBwcHTp06ZXYUEbtjGAZWq9XsGCroRETk9lxcXKhZsybp6enExMSwfPlyPDw8cHV1NTuaiOmsVivp6elmx1BBJyIid+bi4oKLiwvJyckkJCSQlpamgk7kf6SkpJjavgo6ERHJMldXV7y9vYmNjSU1NdXsOCJ2xdvb27Q/dCyGYRh52WBsbCw+Pj7ExMTg7e2dl02LiEgOSEhIML03QsQeubq64uHhYUrb6qETEZFs8fDwMO2XlojcXJ6tQxcWFkZwcDAhISF51aSIiIhIoaAhVxEREZF8Ls8LOsMwiIuLw8vLC4vFkpdNi4iIiBRIeV7QiYiIiEjO0l6uIiIiIvmcCjoRERGRfE4FnYiIiEg+p4JOREREJJ9TQSciIiKSz6mgExEREcnnVNCJiIiI5HP/D0GW6ddnhPNaAAAAAElFTkSuQmCC\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 23,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bounds_on_d_qmin = plot_d_bound(v_example, 0, ymin=-0.5)\n",
"bounds_on_d_qmin"
]
},
{
"cell_type": "markdown",
"id": "24dd62c1",
"metadata": {},
"source": [
"### Bounds on $d$ with Maximal $q=\\operatorname{ch}^{\\beta}_1(u)$"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "47b30d7e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 24,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bounds_on_d_qmax = plot_d_bound(v_example, 4, ymin=-3, ymax=3)\n",
"bounds_on_d_qmax"
]
},
{
"cell_type": "markdown",
"id": "133ccbe7",
"metadata": {},
"source": [
"### Bounds on $d$ with Mid-way $q=\\operatorname{ch}^{\\beta}_1(u)$"
]
},
{
"cell_type": "code",
"execution_count": 25,
"id": "25e4850b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 25,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"typical_bounds_on_d = plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)\n",
"typical_bounds_on_d"
]
},
{
"cell_type": "markdown",
"id": "1c6f5622",
"metadata": {},
"source": [
"# Bounds on Semistabilizer Rank $r=\\operatorname{ch}_0(u)$"
]
},
{
"cell_type": "markdown",
"id": "fdf7befd",
"metadata": {},
"source": [
"Express the two conditions corresponding to the upper bounds on $d$ from $\\Delta(u) \\geq 0$ or $\\Delta(v-u) \\geq 0$ being more than $\\frac{\\kappa}{2n^2}$ higher than the lowerbound given by $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "code",
"execution_count": 26,
"id": "1377923b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{\\kappa}{2 \\, n^{2}} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{2 \\, n^{2}} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{\\kappa}{2 \\, n^{2}} < \\frac{q^{2}}{2 \\, r}, \\frac{\\kappa}{2 \\, n^{2}} < -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\right)$"
],
"text/plain": [
"(1/2*kappa/n^2 < 1/2*q^2/r, 1/2*kappa/n^2 < -1/2*(psi - q)^2/(R - r))"
]
},
"execution_count": 26,
"metadata": {},
"output_type": "execute_result"
}
],
"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",
"\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_terms.hyperbolic)\n",
"assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_terms.hyperbolic)\n",
"\n",
"assymptote_gap_condition1, assymptote_gap_condition2"
]
},
{
"cell_type": "markdown",
"id": "ce8bc94f",
"metadata": {},
"source": [
"Rearrange these two conditions into bounds for $r$:"
]
},
{
"cell_type": "code",
"execution_count": 27,
"id": "553bba31",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(r < \\frac{n^{2} q^{2}}{\\kappa}, r < \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(r < \\frac{n^{2} q^{2}}{\\kappa}, r < \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R\\right)$"
],
"text/plain": [
"(r < n^2*q^2/kappa, r < n^2*(psi - q)^2/kappa + R)"
]
},
"execution_count": 27,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"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\n",
"\n",
"(r_upper_bound1, r_upper_bound2)"
]
},
{
"cell_type": "markdown",
"id": "7f4476b5",
"metadata": {},
"source": [
"### Main Theorem 1"
]
},
{
"cell_type": "markdown",
"id": "f6f4b131",
"metadata": {},
"source": [
"The first main theorem is about these two upper bounds on $r$ needing to be satisfied for $\\kappa = 1$ (weakest form)"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "602840cc",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle r < \\min\\left( n^{2} q^{2} , n^{2} {\\left(\\psi - q\\right)}^{2} + R \\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle r < \\min\\left( n^{2} q^{2} , n^{2} {\\left(\\psi - q\\right)}^{2} + R \\right)$"
],
"text/plain": [
"r < \\min\\left( n^{2} q^{2} , n^{2} {\\left(\\psi - q\\right)}^{2} + R \\right)"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"main_theorem1 = Object()\n",
"main_theorem1.r_upper_bound1 = r_upper_bound1.subs(kappa == 1).rhs()\n",
"main_theorem1.r_upper_bound2 = r_upper_bound2.subs(kappa == 1).rhs()\n",
"\n",
"r\"r < \\min\\left(\" + latex(main_theorem1.r_upper_bound1) + \",\" + latex(main_theorem1.r_upper_bound2) + r\"\\right)\""
]
},
{
"cell_type": "markdown",
"id": "8bf4b71c",
"metadata": {},
"source": [
"### Main Theorem 1 Corollary"
]
},
{
"cell_type": "markdown",
"id": "ddd87bcf",
"metadata": {},
"source": [
"$\\renewcommand\\nu\\ell$\n",
"Redefine \\nu to $\\nu$ in latex\n",
"$\\let\\originalDelta\\Delta$\n",
"$\\renewcommand\\Delta{\\originalDelta(v)}$\n",
"Redefine \\Delta in latex to be $\\Delta$"
]
},
{
"cell_type": "code",
"execution_count": 29,
"id": "90149eb2",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
],
"text/plain": [
"1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2)"
]
},
"execution_count": 29,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"Delta nu\", domain=\"real\")\n",
"# Delta to represent bogomolov(v)\n",
"# nu to represent \\ell\n",
"\n",
"q_sol = solve(\n",
" main_theorem1.r_upper_bound1\n",
" == main_theorem1.r_upper_bound2\n",
" , q\n",
")[0].rhs()\n",
"\n",
"main_theorem1.corollary_r_bound = (main_theorem1.r_upper_bound1\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",
").expand()\n",
"\n",
"main_theorem1.corollary_r_bound"
]
},
{
"cell_type": "markdown",
"id": "6c1e0d68",
"metadata": {},
"source": [
"# Unsorted Extras"
]
},
{
"cell_type": "code",
"execution_count": 30,
"id": "896d26dd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle c = \\frac{a_{v} r}{n} + \\frac{b_{q}}{n}\\)</html>"
],
"text/latex": [
"$\\displaystyle c = \\frac{a_{v} r}{n} + \\frac{b_{q}}{n}$"
],
"text/plain": [
"c == a_v*r/n + b_q/n"
]
},
"execution_count": 30,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q.subs([q_value_expr,beta_value_expr])"
]
},
{
"cell_type": "code",
"execution_count": 31,
"id": "51f22f7d",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle {\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}\\)</html>"
],
"text/latex": [
"$\\displaystyle {\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}$"
],
"text/plain": [
"(a_v*r + 2*b_q)*a_v"
]
},
"execution_count": 31,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"rhs_numerator = (positive_radius_condition_with_q\n",
" .rhs()\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor()\n",
" .numerator()\n",
")\n",
"rhs_numerator"
]
},
{
"cell_type": "code",
"execution_count": 32,
"id": "8148f5cd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d > \\frac{{\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}}{2 \\, n^{2}}\\)</html>"
],
"text/latex": [
"$\\displaystyle d > \\frac{{\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}}{2 \\, n^{2}}$"
],
"text/plain": [
"d > 1/2*(a_v*r + 2*b_q)*a_v/n^2"
]
},
"execution_count": 32,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(positive_radius_condition_with_q\n",
" .subs([q_value_expr,beta_value_expr])\n",
" .factor())"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.7",
"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": "3.10.8"
}
},
"nbformat": 4,
"nbformat_minor": 5
}