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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 $\\originalDelta(v) \\geq 0$"
]
},
{
"cell_type": "code",
"id": "ab162897",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f9c79700bd0>"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"v = Chern_Char(*var(\"R C D\", domain=\"real\"))\n",
"v"
]
},
{
"cell_type": "markdown",
"id": "06d8357b",
"metadata": {},
"source": [
"Let $u$ be a semistabilizer fitting problem 1 or 2 (destabilizing $v$ going down $\\Theta_v^{-}$)"
]
},
{
"cell_type": "code",
"id": "0d33d7e1",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7f9c6efabdd0>"
"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",
"metadata": {},
"source": [
"## $\\chern_2^{P}(u) > 0$"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For problem 2, this amounts to $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "code",
"execution_count": 24,
"id": "31cd035a",
"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": 24,
"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",
"metadata": {},
"source": [
"Separate out the terms of the corresponding lower bound on $d$:"
]
},
{
"cell_type": "code",
"execution_count": 8,
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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",
"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",
"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",
"source": [
"Define expression for the different terms of this bound of $d$ in terms of $\\phi$ and $\\psi$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"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",
"metadata": {},
"source": [
"Verify that the expression above indeed is equal the upper bound on $d$ given by $\\originalDelta(v-u) \\geq 0$"
]
},
{
"cell_type": "code",
"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",
"metadata": {},
"source": [
"# Specialize to problem 2"
]
},
{
"cell_type": "markdown",
"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,
"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",
"metadata": {},
"source": [
"View the specialized bounds:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| 0\\)</html>"
"$\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| 0$"
"beta*q ' + ' 1/2*beta^2*r ' + ' 0"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| \\frac{q^{2}}{2 \\, r}\\)</html>"
"$\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| \\frac{q^{2}}{2 \\, r}$"
"beta*q ' + ' 1/2*beta^2*r ' + ' 1/2*q^2/r"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
"$\\displaystyle \\beta q \\verb| |\\verb|+| \\frac{1}{2} \\, \\beta^{2} r \\verb| |\\verb|+| -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
"beta*q ' + ' 1/2*beta^2*r ' + ' -1/2*(psi - q)^2/(R - r)"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"for bound_terms in [\n",
" positive_radius_lowerbound_terms,\n",
" bgmlv2_d_upperbound_terms,\n",
" bgmlv3_d_upperbound_terms\n",
"]:\n",
" pretty_print(bound_terms.problem2.const,\n",
" \" + \",bound_terms.problem2.linear,\n",
" \" + \",bound_terms.problem2.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "2997ec1a",
"metadata": {},
"source": [
"## Plots for all Bounds on $d$"
]
},
{
"cell_type": "code",
"execution_count": 25,
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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": 26,
"id": "f5b1d9bf",
"metadata": {},
"outputs": [
{
"data": {
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JLmcYButOrcPLxYua/jXNjiMikiki4iOYsnUKU7ZOISYphv2v7qecTzmzY2U767/6krO7N/DaT9/yySfQ/XkDO/vcMeysHjrJ1SwWC0NWDaFU/lIs6qwVryKS+3y04SPGbhyL1Wald63evFH/DQI8A8yOla0kxqfg7OqAo1sB/Ev6cuSwFa/89kDuKOZABZ3kAS9Wf5Ehq4ZwMeYihdwLmR1HROS+nY48jZujGz75fHBxcGFAnQG89vBr+Ln5mR0tWzl+HP6e151kw50W735Ovc4dgY5mx8oUWuUqud6zDz2LvcWeb/Z8Y3YUEZH7cuTqEV785UXKTCnDlK1TABhSbwgfNftIxdy/xEXF8cH7CVSuDMtCm+FR9lGzI2U6zaGTPKHHkh6s/3s9xwYcy/O3shGRnOfI1SOMWDeC+Qfm4+fmxxv13qBP7T64O7mbHS3bSUpI4tzMyny74TmSyo/grbfAzc3sVJlPQ66SJwx+eDBlvcuSZE3C1c7V7DgiIndlGAbXE65TwLUAEfERbDmzhWmtptGzRk9cHHL2iszMcHzXEYpWKIOLqxNnC4yg+4h6lMhDi3vVQyciIpKN2Awby44s4+ONH5NkTSK0VygWiwWrzaoRhluIjISJY87wdpXSrIufxeN9upsdyRSaQyd5RnRiNMP/GM7+S/vNjiIicpNkazLf7PmGqtOr0m5eO+zt7PmgyQepx1XMpWWz2lg9dwXlyhmM/zyAXyIW0fj5zmbHMo2GXCXPcHFw4Zu933A1/ipftvnS7DgiIgCk2FJwsHMgxZbCsNXDCCoaxIwnZ9CweEOzo2Vre1ato7lja3o9vY2+7wZRtOiTZkcylXroJM9wtHekb+2+fLf3OyLiI8yOIyJ5XER8BKNCRlFiUgnORp3F1dGVQ/0P8WvXX1XM3cbVc1f448u5AFRv0ZTDpXfx4fQgihY1OVg2kGUFXXBwMJUqVSIoKCirmhS5Se9avbEaVmbvnG12FBHJo85EnWHIyiEUn1icsRvH8nSFp1OHU/O75Dc3XDaVkgLBwTCq12JqWQZz9sRVLHYWyj9c3exo2YYWRUie0/OXnqw5uYbjA4/jYKdZByKStbos6MLK4yvpH9SfAXUHaP+4u9jzxwaWfbeP9755lZdfsvLRyAgKFvU1O1a2o4JO8pxjEce4Fn+NoKLqLRaRzBd6NpSPN33MMxWfoWvVrpyNOouHsweezvodmB7rpn6AV8KfpDReS1AdLQy5HRV0kmcZhoGBgZ1FU0lF5MEyDIOVx1fy6eZPWXNyDWW9y/Jp809pV6Gd2dGyvaQk2DTrMxJS3Gk1sA/JicnYO9hjZ6+f1Xeiz47kSVGJUdT8siaLDi4yO4qI5EI/h/1Mq+9bEZ0Yzc8df+Zgv4Mq5tLh9xUGVavCgdBTOKacAcDR2VHFXDqoh07yrMe+eYyI+Ah29N6BxWIxO46I5GCRCZHM2DGDK3FXGNd8HIkpiWw9u5VHij+iny/pkBiXyO7gTkxc1JWLLl2YOsWgSlV93jJCJa/kWW8/8ja7Luzi92O/mx1FRHKo8Mhw3lj1BsUmFuO9te8RlRiFYRg4OzjTqEQjFXN3ER+bhGGAcz5nEuyK0W+QO2vWoGLuHqiHTvIswzCoN7seDnYObOi5QT94RSRDrsVfo8iEIrg4uNC3dl8G1BmAv4e/2bFyBMOAFQtPU/ViI074fk3jTk3NjpTjqYdO8iyLxcK7jd7l+LXjXIi5YHYcEcnmDMPgjxN/8OyiZ0myJlHAtQALOy3k9Gun+ajZRyrm0unw3ss0bw5Pdgpg6/nOlKhU3OxIuYJ66CRPMwyDhJQEXB1dzY4iItlUsjWZ+QfmM/6v8ey+sJsahWuwqPMiSuYvaXa0HGfX72upcKE1HWaG0u/tKjzxhNmJco/76qEbO3YsFouF11577QHFEclaFosFV0dXwiPD2X1ht9lxRCQbajevHc8tfo5CboX44/k/2NF7h4q5DLBZbRzYfACAyk0asC35UxavLqti7gG75x660NBQOnXqhKenJ02bNmXSpEnpep166CQ7avFdC85Fn2N3n92pt+ARkbwpPDKcadum0bVqV6oXrs6Gvzfg5eLFQ4UeMjtajrNzJ6yfOYmXg94l6tHTFCnpbXakXOueeuhiYmJ49tlnmTlzJgUKFHjQmUSy3IdNP2T/pf18vftrs6OIiEm2nd1G14VdKTW5FDN2zODQlUMAPFLiERVzGXT13BXGDd1E7drwU2hPjhVdpmIuk91TQdevXz+eeOIJHnvssQedR8QUQUWD6Fa1G++tfY+YpBiz44hIFpuydQp1Z9Ul9GwoE1tMJHxwOF2qdDE7Vo514Ie3aV/kRSZNsrFhixfVWzQxO1Kul+E7k8+bN4+dO3cSGhqarvMTExNJTExMfRwVFZXRJkWyxEePfkT5aeUZt2kco5qOMjuOiGSiyIRIZu+ajYeTB71q9eLpik9TwqsET5Z7UtMu7tHePzYSG2ujXrtGVOzyIQADA7SZRlbJUEEXHh7OoEGDWLVqFS4uLul6zdixY/nggw/uKZxIViqRvwSz2s6idpHaZkcRkUxy4toJpmydwle7viIhJYFBdQcBEOAZQIBngMnpcqbz52HYMOhRbCROHr7QrhG+AX5mx8pzMrQoYsmSJbRv3x57+//99WK1WrFYLNjZ2ZGYmJjmGNy6h65YsWJaFCHZms2wYWfRX5YiucmeC3uo+WVNCrgU4JXar/Bq0KsU8ShidqwcKzkxmU1fTeL94KYculSbCWOv0u2FArrvqkkyVNBFR0fz999/p3muZ8+eVKhQgTfffJMqVarc9Rpa5SrZ3clrJ2n5fUvmtJtDvWL1zI4jIvcoyZrEzwd+5q8zfzGt9TQMw2De/nm0q9COfI75zI6X46UkpXA0uA4H4nvy2KsDyJ/f7ER5W4aGXD08PG4q2tzc3PDx8UlXMSeSExT3Ko6nsyd9lvVhR+8dONo7mh1JRDLgatxVvtzxJdNCp3Eu+hyPl3mc+OR4XB1d6Vq1q9nxcrS/j1zizOIB5G8ylsp1S1Om7xYqujiZHUvQrb9EbmJvZ8+MJ2dw4PIBJm+dbHYcEcmAFFsKVadX5YOQD2gd2Jr9ffez8rmVuhvMfYqLNRgxAmoGueFsDSf+6hkAnFTMZRu69ZfIbbz2+2vM3DmTsFfDKJG/hNlxROQWUmwp/HLoF77c+SU/dvgRb1dvVh1fRY3CNfB18zU7Xq4QtiEU25Y+PP7xKl7oU5C3hxu4e1jMjiX/keFtS0TyitFNR7P93HbOx5xXQSeSzVyJu8KsnbP4PPRzwqPCaVCsARdjLuLt6s3jZR43O16uEBedSD4PZ3xLleDoX+XYtD6BUpUBVMxlR+qhExGRHMMwDCwWC0/Ne4rfj/1Ot6rdGFBnADX8a5gdLdeIioLfps2ljtuHeHXdg4+fFpDkBJpDJ3IX56PP0+nnTpyPPm92FJE8KdmazE/7f6LhVw1ZengpAOMfH8+ZIWf4qt1XKuYeEMNm4+dvL1G+PIz5sj6n7V/AzU2bLOcUKuhE7sLR3pH1f6+n16+9yOIObZE87VLsJT5c/yElJ5eky8IuONg5UMD1xv3DA70DKZivoMkJc5cNE/sTGN6Cxo1s/LahLE36v4OLm7PZsSSdVNCJ3EXBfAWZ1XYWy48uZ9bOWWbHEcnVDMMgPjkegHn75/HRho94ouwT7HllD+teWEejEo1MTpi7RJy/yol9pwHwrfcy1JzAvJ/sKFbM5GCSYZpDJ5JOvZb24sf9P7LnlT2U8S5jdhyRXCUqMYrv9n7H9O3TeaT4I3z+xOfEJsWSaE3E29Xb7Hi5jtUKM2caBEU8TKwRQKN3FpodSe6TeuhE0mlCiwmUyF+Cned3mh1FJNf4+/rfvLLsFYpOKMrAFQMp612WjpU6AuDm5KZiLhPs/XMTrZqcp29fC8sufk7F5z83O5I8AOqhE8mAFFsKDnba7UfkfiSkJHDq+ikqFKzAoSuHeOybx3i55sv0qtmLop5FzY6Xq8XHxBPzfQmWHOjLQ89+QN26ZieSB0UFnUgGpdhSGLpqKA2LN6RDpQ5mxxHJMY5HHOeL7V/w9e6v8cnnw6F+h7BYLFhtVuzttJoysyQlJLFpzgwqt3kBv6IenNhzmJJVymJnr0G63CTL/jeDg4OpVKkSQUFBWdWkSKawt9hzNvosLy59kaNXj5odRyTbuxx7mRbftSBwaiCzd82mR7UeLOu6DIvlxga1KuYyz+rV0KLxRWo5vsOhkD8BKF2tvIq5XEg9dCL3ICoxiqCZQTjbO/PXS3/h5uRmdiSRbOXU9VOsObmGF2u8iNVmpdOCTrQt15ZOlTvpvqpZ4OyR0+z6cQLtR39K/QaOfD4xgso1NR8xN1NBJ3KP9l/az8OzHuax0o+xsNNC9TJInpeYksgvh39h1s5Z/HHiDzydPTn12inyu+Q3O1qec3jrHty2t2VPgd9p3bUiFt2tK9dTn6vIPariV4WfnvkJH1cfrIbV7DgiprLarFQIrkDnBZ2JS47jq3ZfcXbIWRVzWcQwYOPCP1k75nmSkwzK161G4V7HeaKbirm8Qsv1RO7DE+We4IlyTwAQER+hLRYkz4hNiuXnsJ/5ft/3LOq0CA9nDz5u9jEPFXqIir4VzY6Xpxw+ZDBwkAXrWYOxPa4QdTUKH38vHJz0Kz4vUQ+dyAOw58IeSk4qyW9HfzM7ikim2nFuB68sewX/z/zp+UtPLFi4HHcZgM5VOquYy2LrPh/HzuDnOHYMBn70GLXfWIGPv5fZscQEKt9FHoAqflVoWqopnX7uxLoX1lG7SG2zI4k8MBHxEXg5e2FvZ8+bf7zJoSuHeO3h1+hZvSelCpQyO16eY9gM4mIScfN0wcm7JP7lkzgwzsDFVWOreZkWRYg8ILFJsTz27WMcuXqEkBdCqOJXxexIIvcs2ZrMyuMrmbtnLksPL2VZ12U0L9OcCzEX8M3nq0VAJtm9yyBuRRuibKVo+e5Us+NINqIhV5EHxM3Jjd+6/UZxr+J0X9ydLP5bSeSBmfDXBAImBtDmxzYcuXqEsc3GUr1wdQAKuxdWMWeCaxciGNQ/gVq1Lfyyuwt+NZ8yO5JkMxpyFXmACrgWYOVzK4lOjE7dNFUku7sUe4nv937PE+WeoJxPOVwdXOlWpRs9qvdILeTEPHFRcSQuroLP5f6MH/82/fs/h6Oj2akku1FBJ/KA+bn54efmR3RiNF0XdmVkk5GaUyfZTmJKIr8e+ZW5e+ay4ugK7O3sKZivIOV8ytE3qK/Z8QTYv34nZetUJ59nPvZ4T+aVcY/gV8LsVJJdachVJJNYDSsR8RE8OvdR1v+93uw4IhiGQZI1CYDhfw6n488duRhzkSmtpnD+9fM8X+15kxMKwIULMKzvUSqdrs2GHxYCUK9zR/xKFDY5mWRnWhQhkolikmJoN68df4X/xaLOi2gZ2NLsSJIH7bu4jx/2/cC8A/N4vd7r9K/Tn5PXThKfEk8l30pmx5P/l5yYzO+zlvLs8A44OcGcT/6kVfcm2DtqzqLcnQo6kUyWkJJAp587sfL4Sg73P0zJ/CXNjiR5xOKDi3l/3fvsv7SfAi4F6FipI71r9aZWkVpmR5NbCF2ynJoxbRm7Zy+vDq+Mt/YplwxQQSeSBZKtySw/upynKjyFYRgYGNhZNONBHqyLMRf5OexnKvtWpmmppvx6+Fd+3P8j3ap24/Eyj+Nk72R2RPmPs0f+Zv/qlbTo1xvDZvD3gWOUrFrW7FiSA2VZQRccHExwcDBWq5UjR46ooJM8a+rWqfxx8g++a/8dHs4eZseRHO56wnWWHFrCj/t/5M8Tf2KxWBjVZBTDHxludjS5g4QE+PRTuPrX5wx78iMc2oXhV1S/E+XeqYdOJIstP7Kcrgu7UtSzKD93/FkbEEuGRcRHkJiSiL+HP9O2TWPgioE0KtGIblW70aFiB3zy+ZgdUe5g2+JlrPj5BGMWDGTIYCvvvBWHRwH9cSf3RwWdiAkOXzlMx587ciziGNNaT6Nn9Z7at07u6ErcFZYcWsKCsAX8efJPXqn1ClNbTyUyIZLY5FiKeBQxO6Kk07op7+MUtxefpxZTvoK+7+XBUEEnYpK45DgGrRjE8WvH+aP7H5pTJzcxDAOLxcLXu76m16+9MDBoVKIRHSt1pH2F9vh7+JsdUdIhOsogdPY7JNoXp9XAV7AmW7Gzt8Nip2JOHhxtLCxiknyO+ZjZdiYJKQnYWexYd2od0YnRtCnfxuxoYqLDVw6z5NASfjn8C0+We5K3H3mbBsUbMK31NNpXaE8h90JmR5R0MmwGP86DoUMtDH88jkq14wC0DYlkChV0IiZzcXAB4Kf9P/HFji947qHnmNxyMt6u2rMgL1lyaAnD/xzOoSuHyOeYjxZlWqTedqucTznK+ZQzN6BkSGxkLMdmtmLxzwOoV68jbUZOooTu8iCZSAWdSDbx+ROfU69YPQb9PojVx1czscVEulTporl1uVB0YjR/nPiD347+RrPSzehSpQsFXApQP6A+4x4bx2OlH8PV0dXsmHIPoq7F4ZE/H25eblyzq82bI/yo3drsVJIXaA6dSDZzLvocA1cMZPu57Rzsd1C/2HORZUeWMeGvCWw8vZFkWzLlfMrxZoM3ebHGi2ZHk/tktcKCr4/SzFafMO+FNHqmkdmRJI/RLGyRbKaIRxEWdFrAjt47cHV05XjEcbou7MrxiONmR5MMiEmKYenhpbyy7BVWHV+V+pyroysTW0zk+MDjHO5/WMVcLrBz4xnq1oWuvcuw4UJ/ytfWxsCS9TTkKpJN/bOXWHhUOBv+3kDF4Ir0rtWbofWHUiK/JuNkVwvDFvLFji9Y//d6kqxJBHoH0qRkEwC6VOlClypdzA0oD9S2xcuoHv00xbz2MWVjeerXH2F2JMmj1EMnks01KdmEowOOMqrpKObtn0fg1EDm7J5jdqw8zzAMjlw9wvTQ6Twz/xn+Cv8LgPMx53Gwc+DT5p9ypP8Rjg44qiIul0lOTGbXmh0AVGvRnG2WWSz4PZD69U0OJnma5tCJ5CCxSbHM3DmTJ8o+QVmfsiw7sgxne2ealW6mfewy2T8/Ki0WC6NDRvPlzi85E3UGBzsH6haty6imo3i01KMmp5TMtmYN7P3xI3rW/YTkJ85Q0F93eJDsQQWdSA7W/qf2LDm0hEDvQPrU6kPP6j1126cHJMWWwu4Lu9nw9wY2hm9k4+mNrO2xlkq+lZi0ZRLhkeE0K92MR4o/onvy5gFnj5zmy4knGPVFE1o0vc6kMSepUK+G2bFEUqmgE8nBDMNgU/gmpm+fzoKwBRiGwbZe26heuHrqXQYkfa7EXWHfxX00LdUUgMqfVybschjO9s7UDajLI8UfoXet3hT3Km5yUjHDxk+742XsZ0+RHTz7rAV9a0l2o4JOJJe4HHuZ+Qfm80rtV7C3s+fJH57E3s6epys8TdvybSngWsDsiNlKXHIcU7dOZfv57YSeDeXvyL8BuDrsKt6u3vx6+Fd88vlQy78Wzg7OJqeVrGYYsG3JMqx4UL99Y66cvYyTizOePvq9JdlTlhV0wcHBBAcHY7VaOXLkiAo6kUw2ectk5ofNZ3P4ZhzsHHg44GFmt52dp+44YBgG4VHh7Lu4j/2X9rP74m7sLfZ89/R3pNhSCJgQQIWCFQgqEkTtIrWpXaQ2pQuUVs9mHnf0KAwaZDCsZlNs7hV59K3pZkcSuSv10Inkcueiz7H08FL+OPEHs9vOxsvFixeWvMDZ6LPUD6hPrSK1qF2kNkU8ipgd9Z4lW5P5O/JvjkUc48jVI5T1Lkursq1YfXw1j3/3OAAeTh5ULVSVxiUa81GzjwA0LC1pJMQmsGXmB7z5eUcuJddkymdRPPmUBxY7fY1I9qeCTiQPmrljJkuPLGXb2W1cir0EwA9P/0DXql3ZHL6ZPRf2EOgdSKB3IMW8iuFgZ+6WlSm2FC7FXuJc9DnORp3l5PWTtK/QnhL5SzA6ZDQfhHyA1bAC4GzvzOCHBzP2sbFcT7jO5vDNVPGrQjHPYire5I6SE5M5GlyfwwygZd/uuOomLZKDqKATycMMw+BM1Bl2nN9B3aJ18ffwZ8z6MYwMGUmKLQUARztHBtYdyPjHx3Mu+hyTtkzCN58vBfMVpGC+gvi6+fJwwMMARMRH4GjniIuDCxaLBQsW7Cx2WCwWkq3JxCXHkZCSkPrh5uRGgGcA1xOus+TQEq7FX+NawjWuxF0hMjGS75/+HvjfAoV/uDi4sKjTIlqVbcWm05vYf2l/agEa4BmAvZ191n8yJUc6uD2c6DW9cG/6BZWCSmJNtmLvqK8fyXlU0InITVJsKZyOPM2xiGMcizhGmQJlaBHYgt0XdvPM/GdSCy6AgvkKcnnoZQACpwRy/FraW5T9/uzvtAhswftr32f0+tFpjj3/0PN80/4bjl49Srlp5XBzdKOAawF8XH0o6lmUpV2WYm9nz6+Hf8VisVDUoyj+Hv74uflp3z25L9cibIwYacecWTGsfKcd7g3HUbVxLbNjidwzFXQick+SrElExEcQlRiVutBi1fFVRMRHkJCSgGEYGBi0KNOCop5F2XdxH4euHMLFwSX1w9/Dn0DvQGyGjRRbCk72Tia/K8kLdq8KId/+V2g+biMD3vBh4EBw0pee5HAq6EREJE+4fiWG/AXduXz6Igfnv0v5jh9RqISv2bFEHggVdCIikqtdugSrg6fSwGcSXt32U6CgVjtI7qNJKCIikislJyYza+pZypWDyT89zmm3N/D00tiq5E7m7kUgIiKSSbZN6clDCUfo0mUrH35YnoIFy5sdSSTTqKATEZFc49yxcOJirARWL4lPw9cxDIMv6mv/Qcn9VNCJiEiOl5gIn31m0M6hNZdSqhFY/Tsq1KthdiyRLKM5dCIikqNtW7ycxnUvMGKEhd+jv6NWn8/NjiSS5VTQiYhIjhV1NYrAK915sem37N0Lr4+uhqePdlCQvEfbloiISI4SGxnLlrmTeKjjIHz93Tlz+G+Kli2OxU5z5STvyrIeuuDgYCpVqkRQUFBWNSkiIrmIYcBPP8Fjj1ymput4jv31FwAB5UuomJM8Tz10IiKS7Z3YfYiwRZ/R/qPPebKNIxM/jaFkoLvZsUSyDa1yFRGRbC8pPp6KPpv4c+kZGrUuBaiYE/k39dCJiEi2Y7PB2m8WY392Hg2HzcPB0YLNasPOXmv5RG5F3xkiIpKthG6zUq8efDLBDUcnB2Kj4wFUzIncgYZcRUQk21g3eTjnj50iMfFH3gt+nAaPPG52JJEcQQWdiIiYKiUphdiYJLy88+FUqAZF3IqxY6KBvYNWroqkl+bQiYiIaULWGXhua8iF5CBavTPJ7DgiOZYmJIiISJY7f/wMz3VLpElTC4v396dkk+fNjiSSo2nIVUREslTklUjyra1CGev7zJkzhOef74qduhdE7kuGvoWmT5/OQw89hKenJ56entSrV48VK1ZkVjYREclFtv+2nqREG14FvThUYC5Dgl+iRw9UzIk8ABn6NgoICODjjz9m+/btbN++nUcffZR27dpx4MCBzMonIiI53PHj0P+5fdS+3pjNC290AtTt0A6vgl4mJxPJPe57UYS3tzeffvopL730UrrO16IIEZG8ITYylhVfLuLZd5+nUCGYM34LTZ+pq/uuimSCe+7otlqtzJs3j9jYWOrVq/cgM4mISC6wb/UqnvTrxcfvHufQIXi008Mq5kQySYZ76Pbt20e9evVISEjA3d2dH374gdatW9/2/MTERBITE1MfR0VFUaxYMfXQiYjkQsd2HODYpjW0HDgAw2Zw4eRZ/MsEmB1LJNfLcA9d+fLl2b17N1u2bKFv37706NGDsLCw254/duxYvLy8Uj+KFSt2X4FFRCT7iYyEwYNh6rtrKEcw167EY7GzqJgTySL3PYfuscceo0yZMsyYMeOWx9VDJyKSu2367jtWLr3MhN8G8/57KQwaYMU5n7PZsUTylPveh84wjDQF2385Ozvj7KxvbBGR3MYwwGKB5KuHeLTWOfpMMCga4IC2OBXJehn6rnv77bdp1aoVxYoVIzo6mnnz5rFu3Tp+//33zMonIiLZzOVLNsLmvkqsU3VaD3qFxgNGa7GDiMkyNIfu4sWLPP/885QvX55mzZqxdetWfv/9d5o3b55Z+UREJJtISbIybZpBufJ27DvkhpuHE4CKOZFs4L7n0GWU9qETEcl5Ii9f58L3jXn7hxF4V3uajz4CX1+zU4nIPzTRQUREbuvKuev4+OfHyzc/u2jF6AklqdTQ7FQi8l/qoRMRkZskJsJ3wfvolL8+ewr8TsP2DcyOJCJ3oFsii4hIGuuXH6VqVXjlzcqsvfw+VRtWNjuSiNyFhlxFRCTV5h9/ol7yc9SucIRFi0pRpcpQsyOJSDqoh05EJI+Li4pj22+bAKjZth078/3M90tKUqWKycFEJN1U0ImI5FGGAQsWwOwhH1E6/CkiI+JxcXOh7jNPaSsSkRxGBZ2ISB50bGcYQ5//k44dYVPE68Q2+Asvb1ezY4nIPdIqVxGRPGjzp52xT7rI1RrraN3a7DQicr+0KEJEJA+wWQ02f/8NuJek4dONqfDc57h5ueOcz+xkIvIgZNmQa3BwMJUqVSIoKCirmhQREWDHDmj4CBjHZhF/YiUA3v4+OOdzNjmZiDwoGnIVEcmloq9Fs3P2UAZN64vVoxrBU+Jp1FTz5ERyIy2KEBHJZf75M90lnwsFLPsYM/wEu3ahYk4kF1MPnYhILrJ97VEct7+AS7MfKV+zOIbN0BYkInmAeuhERHKBc2dSeO45ePSJQkQm+EBKLICKOZE8QqtcRURyuNBffqPgqf5s3bCDSdMK0PCFpdjpz3WRPEUFnYhIDnXpTAR+Ad4Ur1aDgyeeJnS7Pfl9zU4lImbQHDoRkRzm5EnY/OWH1C88B5/uB/AsoO1HRPI6dcqLiOQQcVFxTBx9iooVYebKZzjvNwYPLyezY4lINqAeOhGRHGLrp+1Ijo1gRfIG3n4b3NzMTiQi2YXm0ImIZGPHdx3CajhTrmYp8jcahYubG2OqmJ1KRLIbDbmKiGRDUVEw9A0btrVtCF85FoDydatRokqgyclEJDtSQSciko3YrDY2fvst9Wtd4vPpdqyxLqXBwClmxxKRbE4FnYhINhJ5+TqVEl5jUPvFHDoEfYZWxMXNxexYIpLNZVlBFxwcTKVKlQgKCsqqJkVEcoSI8xH8+dlbXLkQS4HC3lhbhNFrXB+KFTM7mYjkFFrlKiJiEqsVvvwSZk48wYrB9QkvuYjareqbHUtEciANuYqImODg5p38PuI5BvRPoXrD0vDU3yrmROSeadsSEZEsZNgMLHYWMAxKeB9j2/pL1GxQBNDdHkTk3mnIVUQkCyQlwZ8z5+J+fQn131yEvYPlf8WdiMh90pCriEgmW70yiYceguDZvthcAkiMTwJQMSciD4yGXEVEMtH68X24dDKGQoW+56OprXnoodZmRxKRXEgFnYjIAxYfE09CvI0Cvm44FnuMUkWtrJsGFnXIiUgm0Rw6EZEHxDBgyWIrlf6uxon4lrR6e7zZkUQkj9AcOhGRB+DE7sM82TqJpzvYs+DwSCo82dvsSCKSh2jIVUTkPl05exn/3dWpnX88ryztx5NPPqPhVRHJUuqhExG5B4bNYNP8ZSQnGRQs6ssBnyUMn/USbdporpyIZD0VdCIiGbR7N/R+JpQGKW3YtiwEgNptWuDi5mJuMBHJs1TQiYikU8T5q/wwaga1asHmw3XY6rOXBk83MTuWiIgKOhGR9Dq07g+eKDqMGRPPsHs31G1R1exIIiJAFm5bEhwcTHBwMFarlSNHjmjbEhHJEfau2cLZPZtoNfh1DJtBxIWr+BQpaHYsEZE0tA+diMgtXLgAb74JBS5N4uXHfqRkrw24ezqZHUtE5JY05Coi8h8hM6cytd94li+HKk/3p+LAzSrmRCRb0z50IiL/z7AZWOwsGHEXafpwIq/PBG9v/ZgUkexPQ64ikuf9fSqFc/O7cs2lOa0H9k4t7EREcgoNuYpInhUfk8ioUQYVKzmw82hp8hfyAVAxJyI5jsYSRCRPunzmEvFLHmbP8okMHNiO7u98goeH2alERO6NCjoRyVPOnbhEkdJ+FCziS4jRnU+/rEjpamanEhG5P5pDJyJ5QnQ0zJ2wlV4lG7HTZz31nqxrdiQRkQdGc+hEJFczbAbLv99DhQrw5ie1+fP6JKo3ecjsWCIiD5QKOhHJ1TZ+8xXNrUG0bnqOsIP2tB7UF1d3V7NjiYg8UCroRCTXuXYhgr+WrAWg9jPdOOC3kpnfFaFECZODiYhkEi2KEJFcw2qF2bPBGvohbaovIDb6OG4ertRo2dTsaCIimUo9dCKSK+xb+xcDO62iTx/YZ7yL05PbcPNwNDuWiEiW0CpXEckVtnz6NAnxVpwe+4X69c1OIyKStbJsyDU4OJjg4GCsVmtWNSkiuVhykpVNX03B3rcmj3RoTMUXvsI9vwf26pQTkTwoy4Zc+/XrR1hYGKGhoVnVpIjkUn/+CdWrW7A/v4iEs1sB8PLNj72jvcnJRETMoUURIpJjXLsQwcFv+vLatHcpULwqHk+t4ZEa6pITEVFBJyLZns1qYGdvwS2/O652l5k09gKPdquKxaJiTkQEtChCRLIxw4CQJXspfPJ57B5dTrnqAWZHEhHJlrRtiYhkS0cPJdC6NbR9tiTn4yrj7JRidiQRkWxLQ64iku1snjefYpdf5+ypfXzzQ36atPsBi8XsVCIi2Zd66EQkWzBsBuFHzgNQqk4Djlt6szXUmaeeQsWciMhdaA6diJhu71448N0w6hRZgv/LYeRz1+CBiEhGqIdORExz/dI1PnjjODVqwHd/9eR62Wkq5kRE7oF66ETENDvHP8b1SAu78q9mwABwcjI7kYhIzpShHrqxY8cSFBSEh4cHfn5+PPXUUxw+fDizsolILrQ/ZCuHd54EwKPJJCq/9A2vv65iTkTkfmSooAsJCaFfv35s2bKF1atXk5KSwuOPP05sbGxm5RORXOLiRXj5xWQ89nTi9OqpAJStXYVCJf1NTiYikvPd15Dr5cuX8fPzIyQkhEaNGqXrNRpyFclbkhOT2TTnC3p+0JWoxIJM/eg4nV8sqfuuiog8QPc1+zgyMhIAb2/v256TmJhIYmJi6uOoqKj7aVJEcpjIK9epavmA9172od2gbvj4lDE7kohIrnPPPXSGYdCuXTuuXbvGhg0bbnveyJEj+eCDD256Xj10IrnX2aNnObzoI6r1/BQfv3xEXr6Ol29+s2OJiORa97xtSf/+/dm7dy8//vjjHc8bPnw4kZGRqR/h4eH32qSIZHMJCTBmDLRoHk9Zt+WcPXhj0ZSKORGRzHVPQ64DBgxg6dKlrF+/noCAO98s29nZGWdn53sKJyI5x57V6zm/bjKjPp3PwEGBeD13jGL5taeciEhWyNBPW8MwGDBgAIsXL2bdunWUKlUqs3KJSA5hTTGwd7Dg4OxCwfwx7NtxnXJVfdCtokVEsk6G5tC9+uqr/PDDD/zyyy+UL18+9XkvLy9cXV3TdQ2tchXJHWJiYN2MyeRP+IN6by7F3kE3XBURMUuG5tBNnz6dyMhImjRpgr+/f+rHTz/9lFn5RCSbMWwGP/8YR4UKMHNeWZI9amFNsZodS0QkT8vwkKuI5G1bPutE/AlXgoK+YcKE1pQq1drsSCIieZ4muYjIXUVevkaKzRGfQu5QoguVK3rQ/UmzU4mIyD/u604R90Jz6ERyDpsN5n6dRLP4chyI6Uqrt8aaHUlERG7hnvehE5Hcbf/6HTSon8yLLzux4MRkanQeYHYkERG5DRV0InKT8yfOUv7Uwzxe9ls2bIAhE9pRuFQRs2OJiMhtqKATEQBSklJYN3ceyUkG/qWLcsB/De9/1YOGDc1OJiIid6OCTkRYtw5eavcXj9g/y+41oQBUb/4I9o725gYTEZF0UUEnkoedP36GucMn07QpHI18hLAyRwlqWcfsWCIikkEq6ETysBNbQmhVYizz5l5m40aoWq+02ZFEROQeZNm2JcHBwQQHB2O1Wjly5Ii2LRExydYlK4k4tpNWbwzHsBnERMbgUcDD7FgiInIftA+dSB5x7Bi89hqUNz6jS+PVVB2wHBdXzZETEckNdKcIkTxg3dQPWPlHPvbtG0rPz16j9tNDsNhZzI4lIiIPiAo6kVzKsBnYbAb2DnZgpNDs0RTe+xHy5VOvnIhIbqMhV5Fc6MDeJBJXtuKCS1daD3jZ7DgiIpLJtMpVJBe5fiWG116DajWd2HK8PoVKlzQ7koiIZAENuYrkEueOheP4Z23ObJ3NRx89ycuvjcbJyexUIiKSFdRDJ5LDndj3NwD+pQM4YB3M1O9rMGwYKuZERPIQFXQiOdTly/DpG2spsac021buwmJnocmrb+FfuqjZ0UREJIupoBPJYVKSUpg/fQvlysHHXzVibfxcajatanYsERExkQo6kRxm05xg2rg25YWuVzl02J7Hej2Hg5Omw4qI5GXatkQkBzh//AzHdx6gYccWxEXF8fe+g1RsUMvsWCIikk3oz3qRbCwxESZOBO/jY2lW5U8SEw6QzzOfijkREUlDQ64i2VToL7/xavvfefdd+Dv/hxR8divOLrrLg4iI3Ew9dCLZlO3IF3SuW4Ahn7akcuUCZscREZFsLMsKuuDgYIKDg7FarVnVpEiOEhudROhXo3Ao9jgNn25E5V4/4ObphkX96CIichdZ9quiX79+hIWFERoamlVNiuQIhgE//wyVqjhgd3UjSVcOAuCe3x2LncXkdCIikhNoyFXERJf+vsCpn15gxJQJVK9ViYAeayhdRl1yIiKSMSroREyQnGTF0ckeL98CWCx2zJ4eSb02oHVKIiJyL7QPnUgWstngt2+3UiXyWVKahBD4kG7TJSIi90/dASJZZOe2aBo0gOdeLc/x2Ga4e2gLEhEReTA05CqSBTbMmU3pmPch+SC//Jafxo1nmB1JRERyERV0IpkkJSmF04fPUrpqCQIfacGRNXFs2JwPByezk4mISG6jgk4kE2zYABeW9qd6kRCSyh3Av0wA/mUGmB1LRERyKc2hE3mAzp84y+u9j9CoESzcP4jk2t/g5KxvMxERyVzqoRN5gK4sfobHfPyo8tUv9OhRETvVciIikgVU0Incp+1Lf8e9aAUq1CpJvke/on6AP16+ZqcSEZG8RP0HIvfoxAl4pn0ChU+/zOl1cwAoU6MiXr75Tc0lIiJ5jwo6kQyKi4rjz8kf0jDoKlu3u7DLdwvNB48wO5aIiORhKuhEMijmWhQPuUzhk6GbOXQI2nQOwGJnMTuWiIjkYVl266/g4GCCg4OxWq0cOXJEt/6SHOXormOcXTmaai9/QYGCrsRFxZHPM5/ZsURERADdy1XkjiIj4YMP4Pf5R1jy+tMYDRdSPqi82bFERETS0CpXkdsIXbKc66HTmfnlL7z9TjlK9N2Hs4uGVkVEJPtRQSfyH0mJVpyc7XHx8sYtvzsH98USUMoTUDEnIiLZk4ZcRf7flSuw+ctR+BhbqffWMuzsVcCJiEjOoFWukudZk63M/DyacuXg2+W1SfFrQRb/nSMiInJfNOQqeZphM9gzuQUOR4vx9NNf89FHrfHzMzuViIhIxqigkzzpwslzOLh6UrCwO8kl+1OnThF6NjI7lYiIyL3RkKvkKUlJMPHTeOxXVif0m/EA1H3mKSo3qmNyMhERkXungk7yjG3L1lGzejJDh7uy4Nw31O/5mtmRREREHggNuUqecDrsBLWuN6NTve9p/1MXqlZtaXYkERGRB0Y9dJJrxUXH8+eMWaQkGxSvVJrDJbfy3szOVK1qdjIREZEHSwWd5DqGAYsWwctPbaahSz8ObN4PQKWGtbHYaW85ERHJfVTQSa5yYvchZr/+CR06QKRLM84FnaRaY3XJiYhI7qaCTnKVc/u20bzUTFYsjWT5cihVqYjZkURERDJdlt36Kzg4mODgYKxWK0eOHNGtv+SBMAzY8P18Ys4dpvWw97BZbSQlJOHi5mJ2NBERkSyTZT10/fr1IywsjNDQ0KxqUnK5XbugYUP4dd7feHKIlGQDO3s7FXMiIpLnaNsSyXEMm8H6Sa/x29qiREYOo/XoN2j4qBY7iIhI3qU5dJJjWJOtJCfZsNhZMBzy0+JJT3btgqYq5kREJI/Lsjl0/4iKisLLy0tz6CRD/toQR/7tDTnpOIDW/XuaHUdERCRbUQ+dZGsXTkfQvTvUb5SP9cefonjVKmZHEhERyXY0h06yrb/3H6PA1poknpzPzJktefHF97HTnyAiIiI30a9HyXbCthwEoHilMuyyjmbGwrq8/DIq5kRERG5DvyIl2zh1Cj7qt4wKxyqza90BLHYWGvceRH6/AmZHExERydZU0Inp4mPimTNuHRUrwhdLW7LRbjHVG1UyO5aIiEiOkeGCbv369bRp04YiRYpgsVhYsmRJJsSSvGTrN5Np792ON4dEEXbQgUbd2mGx01YkIiIi6ZXhgi42NpZq1aoxbdq0zMgjecSJ3YcI+WEZAEHPDiCiTigjx3ji7m5yMBERkRwow6tcW7VqRatWrTIji+QB0dEwahSUj5zAw+V3kZL8BG5ebpR6qJzZ0URERHIszaGTLGHYDDZ9/z192/1GcDBcK/Epga9swMFRQ6siIiL3K9P3oUtMTCQxMTH1cVRUVGY3KdmQxc6C07kf6NqkLB/NaU3x4l5mRxIREck1Mr2gGzt2LB988EFmNyPZUMTlePbNfQP7Ml1p2L4hD/VbhHM+Z7NjiYiI5DqZPuQ6fPhwIiMjUz/Cw8Mzu0kxmTXFYMYMKF/JGVvkEWwxN/7PVcyJiIhkjkzvoXN2dsbZWb/I84ozh/8m4tcuTJzwNU8+WYGKr66isL/myYmIiGSmDBd0MTExHDt2LPXxyZMn2b17N97e3hQvXvyBhpOcIz42CVc3JwoGFCLcKMpPPyRSrQmAijkREZHMZjEMw8jIC9atW0fTpk1ver5Hjx7MmTPnrq+PiorCy8uLyMhIPD09M9K0ZEPJybBkxhoa2HUnvlEoZar4mx1JREQkz8lwD12TJk3IYA0ouVTIqiv0fa0gF8Nr8NP7Xanl62J2JBERkTwp0+fQSe60bsZEyqV8SkChI/z4YwGqVfvU7EgiIiJ5lgo6SbeE2AROHzpDuVqBlG3agZOhfqz80w2LtqcWERExlQo6uSvDgKVLgc0vUdbvANZquyharjhFyz1rdjQRERFBt/6Suzix5wi9Oh/iqafg15Pv4fLYT9g7aOWqiIhIdqIeOrktw2aQvKYTbUtVos2SH2jbtgIW1XIiIiLZjgo6ScOwGfw17wd8KjxC+ZrFcX7sZ5qXCsDV3exkIiIicjsacpVUe/ZAy8diKXltKH9vXAhAyaplcXV3NTmZiIiI3IkKOiHy8nX++HQojza8xunz7hwtu4fHBw42O5aIiIikkwo6ISE2nspuPzL9o13s2QONH/c1O5KIiIhkQIZv/XWvgoODCQ4Oxmq1cuTIEd36y2R7Q/YSuXk0VXt/S34fF5ISknBycTI7loiIiNyDLOuh69evH2FhYYSGhmZVk3ILFy8Y9OwJnbo6kd8xnIhzFwFUzImIiORgWuWah2z+cR4JB77i119/Z8yYClR66S/tKSciIpILqKDLA+KiE8nn4Yy7bwApBQI5HJaAj18+QMWciIhIbpBlc+j+ERUVhZeXl+bQZYHTp2HvnMEUsD9KveHLsNMSGBERkVxJv+JzoYTYBD4dE0mFCjBvXXMo0QULWVq3i4iISBZSD10uY9gMwibVZcuhahzOP5P33gMPD7NTiYiISGbSHLpc4uTeo3j4FaFgYTdiS71P00cDeam62alEREQkK2jINYeLiYGR70Thva02od9PBaBO+ycpXb2CyclEREQkq6igy6EMm8H6H5dSuVIKn0zwZEnEEpr0GWR2LBERETGBCroc6vjugzS0PkWfNis4eBB6DGuKq7ur2bFERETEBCrocpDrl66xetoUrCkGgTUrcaLSft4ObkPJkmYnExERETOpoMsBbDaYNQv6dNjKw/ne4cjOYwAE1qxkcjIRERHJDlTQZXNhG0L5cuBoevUC51ItiX3sbyrWKWt2LBEREclGtG1JNnf1xH6alFnE5vVDqPeIG+BtdiQRERHJZrJsY+Hg4GCCg4OxWq0cOXJEGwvfRnIyrP9qBonXz9P6zZHYrDYMm4G9o73Z0URERCSbyrIh1379+hEWFkZoaGhWNZnjrF1jUKMGrF5+HVf7CGxWAzt7OxVzIiIickcacs0GDJvB5s+e47eQ6nh5DaXTyDepWdPsVCIiIpJTaFGEiRJiE0lOMrDYWUh2KU/bLsXYuBEVcyIiIpIhWTaH7h9RUVF4eXnl+Tl0v/8aScVTtThgP4LWrz5vdhwRERHJwdRDl8VOHDjPE09Aq7ZerD7Zi/L1apsdSURERHI4zaHLQkdC91EirDZeCb+xcGEz2rd/E4vF7FQiIiKS06mHLpMZNoNda3YAULZWFbbZgpm1uB5PP42KOREREXkgVNBlon37YHSv+Tx0rg5h245jsbPwSM+XyeeZz+xoIiIikotoyDUTRF6+zg9TtzDgo5aUL/s0rdr/QVCdMmbHEhG5Z0lJSaSkpJgdQyTbcXBwwMnJyewYKugyw655k3imyOckjP2bfoNccXJqanYkEZF7lpSUxIEDB7DZbGZHEcl27OzsqFy5sulFnbYteUAOrN/Gpb/P0/T5dsRcjyE6IhL/0kXNjiUict/i4uI4ePAgJUuWxNXV1ew4ItlGfHw8p06dokiRIjg7O+Ps7Iybm5spWdRDd58uXYK334YG9tOpUjocm7Ut7vndcc/vbnY0EZEHytXVlXz5NAdY5L+WLVtGbGwsnp6edO7c2ZSiTosi7pHNaiNk5lT6tf+NRYsgpdoUag5ZiZ29lq6KiIjkJW5ubjg5OREVFUViYqIpGVTQ3SOLxUK+67/T/YlQjhyBXq96YO9ob3YsERERyWL/DLeaKcuGXIODgwkODsZqtWZVkw/c2VNRnJrfB6PcQBo+VY8ag37BwUmj1iIiImKuLOuh69evH2FhYYSGhmZVkw9MYoLB2LFQoYobcZHR2FsjAFTMiYiISLagiuQuTuw5TMqaDnwTvIjefcpRZ9gyvLzMTiUiIiLyPyrobiPqWhyeBfLhX6Y4O1bVZOlSC2Vrmp1KRERE5GZaFPEfsbHw7ce/kvBTKU4evIiruysNh35D2ZplzY4mIiIicksq6P6fYTNYNv8MFSvCsE8bsC+xP4WKepgdS0RETNakSRNee+01s2PcVlbmO3jwYJa0Ixmngu7/hQR/QI1LdalbO55Nod40G/Qe+Ty1gaaIiAjApEmTTN+a4/Lly4wfP/6O55w7dy6L0mQvebqgi7xynQN/HQIgsPkLnCs+i58XuVK6tMnBREQkSyUlJZkdwTRNmjRh0aJFdzxn1apV+Pr6UtrkX5C+vr40atSI4ODg256zbt06evfuzdq1a9N93dDQUF577TXmzp1L7969OXny5IOIm6XyZEFns8HXX8Pmz14geUNPDAMCKpQkqG0rs6OJiORqJUuWZNKkSWmeq169OiNHjgRuFBf9+/enf//+5M+fHx8fH959913+fdvx9JxjGAbjxo2jdOnSuLq6Uq1aNRYsWHDTNYYMGULBggVp3rz5HXOnpKTctr3ExEQGDhyIn58fLi4uNGzY8KYtuu72vv/JNHDgQIYNG4a3tzeFCxdOcxwgNjaW7t274+7ujr+/P5999tkdc9/NkiVLsLe356233iI5Ofm2502ZMoUuXbrcV1sPSp06ddi2bRtXrly55fFu3brx+eefEx4ezssvv8yKFSvueL3ExEQ6duzI8OHD6dGjBz179qRnz56ZET1T5bmCLmzjdro+cYgXX4Q/ro6j0DMLsOhuXSIiGXY++jwJKQkAnLx2kp3nd6b5OB99/p6uO3fuXBwcHNi6dStTpkxh4sSJzJo1K0PnvPvuu3z99ddMnz6dAwcOMHjwYJ577jlCQkJuusamTZuYMWPGPWcaNmwYCxcuZO7cuezcuZPAwEBatGhBRETEPb13Nzc3tm7dyrhx4xg1ahSrV69OPT506FDWrl3L4sWLWbVqFevWrWPHjh2px+fMmYMlnb/UUlJS2LFjB/Pnz+fSpUu3/Rzs3buXgIAA7O2zz92Q2rZty5w5c2573MHBge7du/Pll18SFRXFiy++yOLFi9MU/f9Yv349np6eFCpUCLhRMG7ZsoWLFy9mVvzMYWSxyMhIAzAiIyOzumnDmmI1jk2uYPw8tLcREpLlzYuI5EixsbHG9u3bjdjY2DTPj1g7wjh69ahhGIbx7MJnDUaS5mPE2hE3XatEiRLGxIkT0zxXrVo1Y8SIG+c2btzYqFixomGz2VKPv/nmm0bFihVTH9/tnJiYGMPFxcXYvHlzmnZeeuklo2vXrqnXqF69erre/53ai4mJMRwdHY3vv/8+9VhSUpJRpEgRY9y4cel+3/+007BhwzTnBAUFGW+++aZhGIYRHR1tODk5GfPmzUs9fvXqVcPV1dUYNGiQYRiGsWjRIqN8+fLpel+zZs0yjh698f/38ccfG76+vrf83TxhwgRj1qxZqY9TUlKMadOmGc8995yxbds2wzAMo1u3bsbcuXPT1e6dpKSkGFOnTjV69OiReu3w8HCjcePGac67fPmy0bRp03Rf12azGUuWLDGeffZZ448//khz7Msvv7zp+gULFjT+/PPPu173n++NBQsWGLNmzTImTpxoXL16Nd25HqRc30OXkpRCyMxgDu8Kx87eDufHV/DUh8E0amR2MhGRnK1PrT4EeAYAMLrpaHb03pHmo0+tPvd03YcffjhNL1O9evU4evRomltH3umcsLAwEhISaN68Oe7u7qkf33zzDcePH099Te3ate8707Fjx0hOTqZBgwapxxwdHalTp849rQh96KGH0jz29/fn0qVLABw/fpykpCTq1auXetzb25vy5cunPm7fvj2HDh26azsxMTFcvnyZwMBAAAYNGkS+fPn45JNPbjr3zJkz+Pn5pT7+5Zdf6Ny5M3FxcZw6dQqANm3aEBUVlf43ehtLly6lS5cuxMfHp85jW716NQEBAWnOK1iwYJr/y7u5cuUKO3fuxMfHh1KlSt10zMXFJc1zLi4uXL9+/d7ehEly9cbC69fDW6/HseCFMYRtc6F8jZcIqFDS7FgiIrmCv4d/6r9LFShFKUrd4ewb7Ozsbhr2utPcrXths9kAWL58OUWLFk1z7N+rNN3c3B5Ym/8d5jQMI81z6X3fjo6ON133n/fz39ffj5kzZ9KrV6/Uxy4uLowZM4bevXvz6quvpvm8xcTE4Orqmvr40UcfxWazsWHDBr777jsAypQpc8sFE5MnT77jAoPGjRvTvn371MfNmjUDYM2aNXz11VfAjUUOt5rjmJ6h5XPnzjF16lSSk5MZMGAAJUqUuOkcLy+vmz63MTExFCxY8K7Xz05yZUF36e8L7P1hJM98+DEVqubnQu1DPFbX0+xYIiJ5nq+vL+fP/29uXVRU1E2/8Lds2XLT47Jly6aZw3WncypVqoSzszOnT5+mcePGDyT37doLDAzEycmJjRs30q1bN+BGobZ9+/Y0e8Ol533fTWBgII6OjmzZsoXixYsDcO3aNY4cOZKh93nhwgVcXFzw8fFJ83y3bt2YOHEi77//PrNnz059vmDBgly7di31cf78+fn2229p0qRJaqG3b98+XnjhhZvaGjRoUEbeIp6ensybN49GjRqlFtwhISF89NFHXL9+nfz586ee6+Bw+xLm1KlTBAcH4+TkxMCBA/H397/tuRUqVGDmzJmpjxMTE4mOjr5l8Zed5cqCzmq1Utb9D77//BCtnn8YOzsVcyIi2cGjjz7KnDlzaNOmDQUKFOC99967abJ9eHg4Q4YMoU+fPuzcuZOpU6fetJrzTud4eHjwxhtvMHjwYGw2Gw0bNiQqKorNmzfj7u5Ojx49Mpz7du25ubnRt29fhg4dire3N8WLF2fcuHHExcXx0ksvZeh93427uzsvvfQSQ4cOxcfHh0KFCvHOO+9gZ/e/2VOLFy9m+PDhdxx2nT59Or17977lKtG33nqLLl26MHjwYKpUqQJAxYoVOX36dJrzLl68mKaodHd3T5PjfoSHh6cOBe/fvx9HR0f8/PyYO3cuL7/8MnCjaPbwuPXm/+PHjycuLo633nrrpqL1Vho1asSlS5c4c+YMAQEBhISEEBQUpILudoKDgwkODk4zB+JB+uvXv2D/h1R+ZRH+pYtifeUwJRyzz4ocERGB4cOHc+LECZ588km8vLwYPXr0TT1V3bt3Jz4+njp16mBvb8+AAQPo3bt3hs4ZPXo0fn5+jB07lhMnTpA/f35q1qzJ22+/fU+579Texx9/jM1m4/nnnyc6OpratWuzcuVKChQokKH3nR6ffvopMTExtG3bFg8PD15//XUiIyNTj0dGRnL48OHbvv7IkSOMGTOGUaNG3bGdt99+m6VLlwLQsmVLevXqxeuvv556vEuXLgwZMoRvv/2WlJSUeyqSb6dDhw68+eabzJ8/H4BatWoxbdq0ND2AoaGhPProo7d8/RtvvJGh9hwcHJg7dy5jxozh4YcfJiQkhO+///6e85vFYjzIQfl0iIqKwsvLi8jISDw977/n7MRxG4OH2HF0RxhzBw6mSIc5FC1z+65VERHJmLi4OA4ePEjFihXJly9z76DTpEkTqlevftOebRk9Rx6s3r17M2rUKAoXLmx2FODGtjRt27alTp06pub453vj1KlTXL9+nejoaLp37463t3eWZ8nRQ67rv/oS4+SP7N79J+MnVqL2Myu1p5yIiMgD9v777zNt2jQ+/PBDs6MQFRXF5cuXb1vMTZw48Y4rYJs2bUqHDh0yK55pclxBZ9gMIiPiyF/QDa9iFbgW3YCw/Sm4eTiZHU1ERCRXCggIoH379vz222+0bt3atByGYTBx4sQ7FpaDBw/OwkTZR44q6MLCIHx+T/I5XqPh279QrXkjaK4N5UREcot169Y9kHPkwatVq5bZEbh69Sp9+/bF19fX7CiZZs+ePYwfP56TJ0/y1ltv0aBBAz744AMSExO5cOECI0aMoHr16je9LkdsLBx5JYp3hl6jWjWYv7UzLlX6aGhVREQkjylYsGCaTY5zo8mTJ/P111/TsmVLXnzxRXr16sXQoUPp1KkTy5cvT7OlzL9l+4LOZrVx5Ye6FL/+LqNHw+dLWhHUzrzuXhEREZHMcPz4cYoUKYKDgwPnzp0jIiKCt99+m6JFi3L16lXc3d1p06bNLV+bbYdcwzbtpFDZCvj45eNayYm0faIK/mXMTiUiIiKSOS5evJi6YGPjxo00bNiQmjVrAvDMM8/wzDPP3Pa12a6H7soVGNLvKiWPNCT0xxs7N9du2xL/MgF3eaWIiMiD16RJkzR3fchusjLfvdyfVtKvfv361KhRgytXrrB//36aNGmS7tdmm4LOmmxl9czvqVDeylff+7A84U8e69vP7FgiIiICTJo0Kc29cG/n8uXLjB8//o7nnDt37kHFypXWrl2LYRg5s6A7un0vj7p2560XQzhyBDr2rYeDU7YdERYRkVwkKSnJ7AimadKkCYsWLbrjOatWrcLX15fSpUvf9Xq+vr40atSI4ODg256zbt06evfuzdq1a9OdMzQ0lNdee425c+fSu3fve7rTRk6xdu1anJ2defjhh9P9GlMLugsnz7Fy4idYUwwq1KvB+drHeePTR8nlC1hERPKskiVL3nSHh+rVqzNy5EjgRnHRv39/+vfvT/78+fHx8eHdd9/l3zc1Ss85hmEwbtw4SpcujaurK9WqVWPBggU3XWPIkCEULFiQ5s2b3zF3SkrKbdtLTExk4MCB+Pn54eLiQsOGDQkNDc3Q+/4n08CBAxk2bBje3t4ULlw4zXGA2NhYunfvjru7O/7+/jfd4zajlixZgr29PW+99RbJycm3PW/KlCl06dIl3detU6cO27Ztu+X9YgG6devG559/Tnh4OC+//DIrVqy44/USExPp2LEjw4cPp0ePHvTs2ZOePXumO09Os3btWh5++GFcXFzS/RrTCrpJk2DAszupme8zTh08A0BAhZJmxRERkWxi7ty5ODg4sHXrVqZMmcLEiROZNWtWhs559913+frrr5k+fToHDhxg8ODBPPfcc4SEhNx0jU2bNjFjxox7zjRs2DAWLlzI3Llz2blzJ4GBgbRo0YKIiIh7eu9ubm5s3bqVcePGMWrUKFavXp16fOjQoaxdu5bFixezatUq1q1bx44dO1KPz5kzB0s69/VKSUlhx44dzJ8/n0uXLt32c7B3714CAgKwt8/Y/dHbtm3LnDlzbnvcwcGB7t278+WXXxIVFcWLL77I4sWLudUdSdevX4+npyeFChUCbhSMW7Zs4eLFixnKlBNcuHCBQ4cOZWi4FR7AKlfDMIiOjr7t8cTERBITE1Mfb1/xJwAjR0bRp08j4h/fja+PO1FRUfcbRUREMkFCQsItnz9/HgoUABcXOHkSrl1Le9zf/8ZHRhUrVoyJEydisVgoX748+/btY+LEifTq1Std58TGxjJhwgTWrFlDvXr1AChdujQbN25kxowZNG7cGIDAwEDGjRt3X5m6devG9OnTmTNnDq1atQJg5syZrF69mtmzZzN06NAMvfeHHnqIESNGAFC2bFmmTZvGn3/+SfPmzYmJiWH27Nl88803qT2Kc+fOJSDgf4sGvby8KF++fLramjt3Lj169MDHx4fhw4czatQounfvftN91v/880+CgoJSH1utVqZPn8727dvp168fQUFBnDlzhueeey7Nps+NGzcmODiYN95444457Ozs6Ny5M506dWLp0qU8//zz9OzZk2bNmqWec+rUqTT3R7W3t8fDw4MDBw6kFnlmSk5OJikpiaSkJKKjo3Fw+F955eHhke4iG26sdC1UqBCdOnXKUIb7Luiio6Px8vLK8OsMoxhffAFffHG/CUREJDOVL1+e77///qbnZ8yA556DwEB47z347ykjRsB/RgzT5eGHH07zC7BevXp89tlnWK3W1F6iO50TFhZGQkLCTcOoSUlJ1KhRI/Vx7dq17zvTsWPHSE5OpkGDBqnHHB0dqVOnzj2tCH3ooYfSPPb39+fSpUvAjT3KkpKSUotUAG9v7zQFXPv27Wnfvv1d24mJieHy5csEBgYCMGjQIKZPn84nn3zCmDFj0px75syZNL1FS5cupUuXLmzYsIGTJ08SFBTE6tWr0xSWcGMT4DvdU/W/rly5ws6dO/Hx8aFUqVI3Hfvv8KOLiwvXr19P9/Uz09dff014eDjOzs68//77aTqyIiMjbyqS76RatWpcuHAhwxnuu6Dz8PAgMjLytsf/20N39sxZHq73MGFhYRQtWvR+m0+XoKCgm+YzqL2c0V5UVBTFihUjPDw8Q98Q9yM3fz5ze3v6esmc9tavX094ePhNx/r0udFDBzB6NAwZkvb4rXrn7OzsbhpSu9PcrXths9kAWL58+U2/Z/69StPNze2BtfnfHhjDMNI8l9737ejoeNN1/3k/txqKvFczZ85M0+Pp4uLCmDFj6N27N6+++mqaz1tMTAyurq6pj//pOVuzZg1fffUVcGORw63mIaanZ+rcuXNMnTqV5ORkBgwYQIkSJW46x8vL66b3HxMTQ8GCBe96/azQs2dPrl27RnR0NJ07d6bAP98Y3KiTssJ9F3QWi+WefnB6eHhk2Q9ce3v7LGtL7WUOT09Pfb2ovXTT18uDbc/d3f2Wx/5dsJUqdePjbnx9fTl//nzq46ioqJtWK27ZsuWmx2XLlk0zh+tO51SqVAlnZ2dOnz6dOrx6v27XXmBgIE5OTmzcuJFu3boBNwq17du3p9kbLj3v+24CAwNxdHRky5YtFC9eHIBr165x5MiRDL3PCxcu4OLigo+PT5rnu3XrxsSJE3n//ffT3F6qYMGCXPvXeLqnpyfz5s2jUaNGqUVxSEgIH330EdevXyd//vyp5/576PG/Tp06RXBwME5OTgwcOBD/O4zPV6hQgZkzZ6Y+TkxMJDo6+pbFnxkcHR1xcnLCyckpS+ubf8sT+4L065e1+9mpvZwtt38+c3t7WS23fz4fdHuPPvooc+bMoU2bNhQoUID33nvvpsn24eHhDBkyhD59+rBz506mTp1602rOO53j4eHBG2+8weDBg7HZbDRs2JCoqCg2b96Mu7s7PXr0yHDu27Xn5uZG3759GTp0KN7e3hQvXpxx48YRFxfHSy+9lKH3fTfu7u689NJLDB06FB8fHwoVKsQ777yDnd3/1jcuXryY4cOHc+jQodteZ/r06fTu3fuWK1DfeustunTpwuDBg6lSpQoAFStW5PTp0zd9Pv4Zrt2/fz+Ojo74+fkxd+5cXn75ZeBGYXu73qnx48cTFxfHW2+9dVNheSuNGjXi0qVLnDlzhoCAAEJCQggKCso2BV22YGSx8PBwAzDCw8OzumnJgSIjIw3AiIyMNDuK5AD6eskcsbGxxvbt243Y2Nj7vlZkZKTRqVMnw9PT0yhWrJgxZ84co1q1asaIESMMwzCMxo0bG6+++qrxyiuvGJ6enkaBAgWMt956y7DZbKnXSM85NpvNmDx5slG+fHnD0dHR8PX1NVq0aGGEhISkXmPQoEHpyny39uLj440BAwYYBQsWNJydnY0GDRoY27Zty9D7vl2mdu3aGT169Eh9HB0dbTz33HNGvnz5jEKFChnjxo1L87qvv/7auNOv9sOHDxv29vYGcMePNm3apL7m8uXLxlNPPZXmOsePHzeeeeYZ46effjJ++ukno3PnzsaECROMiIiI1HM2bdpkDBkyJD2f4nRZvXq18corrxhz5swxevbsaRw/fvyBXfte/fO9sWDBAmPWrFnGxIkTjatXr5qSxWIYD3BQPh0uX76Mn58fly5dwtfXNyublhwoMTGRsWPHMnz48HTtUC55m75eMkdcXBwHDx6kYsWK5MuXL1PbatKkCdWrV79pz7aMniMPVu/evRk1ahSFCxdO92veffdd2rZtS506dTIxmbn++d44deoU169fJzo6mu7du6dZkZtVsnzI9Z8fsvphK+nh7Ox808aaIrejrxeRzPH+++8zbdo0Pvzww3SdHxUVxeXLl29bzE2cOPGOK2CbNm2aepN6SZ88MYdORERE7l1AQADt27fnt99+o3Xr1nc81zAMJk6ceMfib/DgwQ86Yp6ngk5ERLKNf29Mez/nyINXq1atdJ139epV+vbtq2lVWUwFnYiIiDww2WVvuLzGtHu5ioiIiMiDoYJOsp2RI0disVjSfGRkZZXkfuvXr6dNmzYUKVIEi8XCkiVL0hw3DIORI0dSpEgRXF1dadKkCQcOHDAnrIjkesuWLePq1aumZlBBJ9lS5cqVOX/+fOrHvn37zI4k2UhsbCzVqlVj2rRptzw+btw4JkyYwLRp0wgNDaVw4cI0b96c6OjoLE4qknvs2LGDzZs3s2fPHrOjZDuurq4sWLDggd/GLiM0h06yJQcHB/XKyW21atWKVq1a3fKYYRhMmjSJd955h6effhqAuXPnUqhQIX744Qf69OmTlVFFcoXFixfTsmVL7O3t6d27N3PmzDE7UrbStGlTtmzZwokTJ0zLoB46yZaOHj1KkSJFKFWqFF26dDH1m0RylpMnT3LhwgUef/zx1OecnZ1p3LgxmzdvNjGZwI1Ngf99j9PsKrvnzMp8+/bto2bNmqxatYrnn3+e3r17p/u1Bw8ezMRk2YednR0BAQFcunTJvAxZ1VBwcDCVKlUiKCgoq5qUHKpu3bp88803rFy5kpkzZ3LhwgXq169v+vwEyRkuXLgAQKFChdI8X6hQodRjIvI/cXFxxMbG3vb4uXPnCAgIICUlhYsXL+LgkL7BvUmTJqX7JgKXL19m/Pjx6To3u3J1dSUhIcG09rOsoOvXrx9hYWGEhoZmVZOSQ7Vq1YoOHTpQtWpVHnvsMZYvXw7cGDYTSS+LxZLmsWEYNz0nkpSUZHYEUy1fvpyaNWvy6aef3vYcwzCwt7enQ4cOtG3bluvXr9/1uqtWrcLX15fSpUunK4evry+NGjUiODg4vdHTJTQ0lNdee425c+fSu3dvTp48+UCvn51oyFWyPTc3N6pWrcrRo0fNjiI5wD9zL//bG3fp0qWbeu3EXImJiQwcOBA/Pz9cXFxo2LBhmj/6f/31V/Lnz4/NZgNg9+7dWCwWhg4dmnpOnz596Nq1K3Cj8Bg3bhylS5fG1dWVatWqsWDBgjRtNmnShP79+zNkyBAKFixI8+bNb5svJSWF/v37kz9/fnx8fHj33Xf59+3P75a/ZMmSN91vtnr16mluT9ekSRMGDhzIsGHD8Pb2pnDhwjfdvi42Npbu3bvj7u6Ov78/n3322Z0/sRnwxBNP0Llz59sev3TpEocOHQIgPj6eQ4cO0bRpU86cOcPvv/+e+rF79+40r5syZQpdunTJUJY6deqwbds2rly5kuH3cSuJiYl07NiR4cOH06NHD3r27EnPnj0fyLVvJT4+HhcXl0y7/t2ooJNsLzExkYMHD+Lv7292FMkBSpUqReHChVm9enXqc0lJSYSEhFC/fn0Tk8l/DRs2jIULFzJ37lx27txJYGAgLVq0ICIiAoBGjRoRHR3Nrl27AAgJCaFgwYKEhISkXmPdunU0btwYuHEz+K+//prp06dz4MABBg8ezHPPPZfmfLjR2+/g4MCmTZuYMWPGbfP9c97WrVuZMmUKEydOZNasWenOn15z587Fzc2NrVu3Mm7cOEaNGpXm63fo0KGsXbuWxYsXs2rVKtatW8eOHTvSXGPOnDn33AN9p9ft2LGDFi1asGnTJv78808mT56Mo6MjAQEBtGzZMvWjevXqqa/Zu3cvAQEB2NvbZzhL27Ztb7vgYvLkyYwaNSrdUyfWr1+Pp6dn6h9yderUYcuWLVy8eDHDue7GZrNx5swZ/Pz8Hvi1083IYpGRkQZgREZGZnXTkkO8/vrrxrp164wTJ04YW7ZsMZ588knDw8PDOHXqlNnRJJuIjo42du3aZezatcsAjAkTJhi7du0y/v77b8MwDOPjjz82vLy8jEWLFhn79u0zunbtavj7+xtRUVEmJ8+ZYmNjje3btxuxsbFpD8SdM4yU+Bv/jjltGNcP/O/Ytb03jv9H48aNjUGDBhkxMTGGo6Oj8f3336ceS0pKMooUKWKMGzcu9bmaNWsa48ePNwzDMJ566iljzJgxhpOTkxEVFWWcP3/eAIyDBw8aMTExhouLi7F58+Y07b300ktG165d07RfvXr1u77nxo0bGxUrVjRsNlvqc2+++aZRsWLFG283HflLlChhTJw4Mc11q1WrZowYMSJNOw0bNkxzTlBQkPHmm28ahnHja93JycmYN29e6vGrV68arq6uxqBBg1KfW7RokVG+fPk7vqdNmzYZAwYMML788kvjzTffNDZs2GAYhmGMGDHCeOONN4w5c+YYc+bMMZ566ikjJibGMAzDWLFixR2veSsTJkwwZs2alea5lJQUY+rUqUaPHj2Mbdu2GYZhGOHh4Ubjxo3TnHf58mWjadOmt732hQsXjJEjRxpDhw41Dh06dMccX3755U3XL1iwoPHnn3+m/83cxT/fG3379jWCgoKMcePGGVevXn1g188I9dBJtnPmzBm6du1K+fLlefrpp3FycmLLli2UKFHC7GiSTWzfvp0aNWpQo0YNAIYMGUKNGjV4//33gRs9J6+99hqvvvoqtWvX5uzZs6xatQoPDw8zY+c+R2dA3Jkb/z74KWz61xDb2lY3jt/G8ePHSU5OpkGDBqnPOTo6UqdOnTQrI5s0acK6deswDIMNGzbQrl07qlSpwsaNG1m7di2FChWiQoUKhIWFkZCQQPPmzXF3d0/9+Oabbzh+/HiatmvXrp2ut/fwww+n6b2qV68eR48exWq1pjt/ejz00ENpHvv7+6euljx+/DhJSUnUq1cv9bi3tzfly5dP85r27dunDo3eyvnz5+nQoQNvv/02vXr1wsXFJU0v39atW+nRowc9evTAarWycuVK4MZ2HBl1q56qpUuX0qVLF+Lj41Pnsa1evZqAgIA05xUsWPCm/69/K1SoECNGjOC9997jt99+Y8CAAfz111+3PPfKlSs3DYG6uLikaw5gRsXFxdGhQwccHR0f+LXTS/vQSbYzb948syNINtekSZM0c5n+y2KxMHLkyJvmIskDVrYPOBW48e+KQ6HsK/871nQFON/+np7//P/dbfFKkyZNmD17Nnv27MHOzo5KlSrRuHFjQkJCuHbtWupw6z/z7JYvX07RokXTXPO/Ky3d3Nwy9j7vMb+dnd1NX6e32nj2v0WAxWJJfT93+jrPiIULF1K8ePHUOab//d6oVatW6r+9vb2JiooCbv7cpUdMTAyurq5pnmvWrBkAa9as4auvvgJuDJffag5jeoaOPTw8GDx4MElJSXz//fd89dVX9OjRg4YNG6ae4+XlddPnLyYmJlPuNdumTRuuX79u6ubl6qETEZF74+oP9v/fA+JWDLwq/e9Y/qo3jt9GYGAgTk5ObNy4MfW55ORktm/fTsWKFVOf+2ce3aRJk2jcuDEWi4XGjRuzbt26NPPnKlWqhLOzM6dPnyYwMDDNR7Fixe7p7W3ZsuWmx2XLlsXe3j5d+X19fTl//nzq8aioqAyvsgwMDMTR0TFNlmvXrnHkyJEMXcdms92xOLyX+W63U7BgQa5du5bmOU9PT3777TcaNWqUWlCHhITw2GOP3dRjlt5tUeDG5/Ts2bO4u7vf1CtYoUKFNPvCJSYmEh0dnWtHe9RDJyIiWc7NzY2+ffsydOhQvL29KV68OOPGjSMuLo6XXnop9TwvLy+qV6/Od999x+TJk4EbRV7Hjh1JTk6mSZMmwI0emzfeeIPBgwdjs9lo2LAhUVFRbN68GXd3d3r06JHhjOHh4QwZMoQ+ffqwc+dOpk6dmrrCND35H330UebMmUObNm0oUKAA7733XoYLJ3d3d1566SWGDh2Kj48PhQoV4p133sHOLm1/zOLFixk+fPhth13bt2/P6NGjOX36NMWLFwdujIbcbSXqpUuXWLVqFXZ2duzYsYNPPvnkrgVXxYoVOX369E3Ph4eHExgYCMD+/ftxdHTEz8+PuXPn8vLLLwM3iuL0TI04efIkX375JXZ2drzyyiu3LNobNWrEpUuXOHPmDAEBAYSEhBAUFKSCTkRE5EH6+OOPsdlsPP/880RHR1O7dm1WrlxJgQIF0pzXtGlTdu7cmVq8FShQgEqVKnHu3Lk0vXmjR4/Gz8+PsWPHcuLECfLnz0/NmjV5++237ylf9+7diY+Pp06dOtjb2zNgwIA0d0m4W/7hw4dz4sQJnnzySby8vBg9evQ97YP26aefEhMTQ9u2bfHw8OD1118nMjIyzTmRkZEcPnz4ttcoVqwYCxcu5N1336Vhw4bYbDZat27N77//zrJly0hOTmbZsmVER0ezadMmzpw5Q4UKFTh16hTXr1+nf//+bNmyhT/++IOWLVveMW/Lli3p1asXr7/+eprnO3TowJtvvsn8+fOBG8O806ZN44UXXkg9JzQ0lEcfffS21961axdz586lcOHCDBs27KavlX9zcHBg7ty5jBkzhocffpiQkBC+//77O2bPySzGgxqgT6eoqCi8vLyIjIzE09MzK5sWEZF7EBcXx8GDB6lYsSL58uUzO46YpFOnTowbN46SJUve9dzevXszatSoDN+T+91336Vt27bUqVPnpmOTJ0/GxcWFHj16mLrf27/9873xT+EbHR1N9+7d8fb2zvIsmkMnIiIid7R8+XI6duyYrmIO4P3332fatGkZaiMqKorLly/fspgDGDRoEH369Mk2xVx2o4JOREREbis0NJRChQrRsWNHwsLC0vWagIAA2rdvz2+//Zau8w3DYOLEiXz44Yf3EzVP0xw6ERERuaW//vqLPn364OfnR1JSEtOnT0/3a/+9FcrdXL16lb59++Lr63svMQUVdCIiInIb9erVY+/evZneTmbsDZfXZNmQa3BwMJUqVSIoKCirmhQRERHJE7KsoOvXrx9hYWGEhoZmVZMiIiIieYIWRYiIiIjkcCroRERERHI4FXQiIiIiOZwKOhEREZEcTgWdiIiISA6ngk5EREQkh9PGwiIiki7x8fFmRxDJVrLT94QKOhERuSMHBwfs7Ow4deqU2VFEsh3DMLBarWbHUEEnIiJ35uTkROXKlUlJSSEyMpJly5bh5uaGs7Oz2dFETGe1WklJSTE7hgo6ERG5OycnJ5ycnEhISCA2Npbk5GQVdCL/kpiYaGr7KuhERCTdnJ2d8fT0JCoqiqSkJLPjiGQrnp6epv2hYzEMw8jKBqOiovDy8iIyMhJPT8+sbFpERB6A2NhY03sjRLIjZ2dn3NzcTGlbPXQiIpIhbm5upv3SEpFby7J96IKDg6lUqRJBQUFZ1aSIiIhInqAhVxEREZEcLssLOsMwiI6OxsPDA4vFkpVNi4iIiORKWV7QiYiIiMiDpXu5ioiIiORwKuhEREREcjgVdCIiIiI5nAo6ERERkRxOBZ2IiIhIDqeCTkRERCSHU0EnIiIiksP9H213GfziZDtjAAAAAElFTkSuQmCC\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 26,
"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": 27,
"id": "47b30d7e",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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LL+H7780/774b/vjDnPCXF6Skp/DBpg/48Y/ZLC18hrKhTTjU/xDuro5dMqmFzkIPPmg2TYuIiDi6lBT45BOoVAm6dIErVyAuzjyWF4o5wzBYueFNNn1WkOHLBlGy6D3E3r8dwlo5fDEHmhQhIiIit2EY5lChDRvgscfgtdfMXR7yjLREOi/uxZa9c5ldqggF7/2KShFNrE6VLSroLJaQAElJEBxsdRIREZG/Xb0KM2fCo49C8eLmDNZSpfJGa9w1+87vI/TgFApd3szKCm/j4uJGs9LOOQBQXa4Wq1oVxo+3OoWIiIgpMRHefRfKlIGBA82JDgAPPZR3irlzF3fxn++6cNcHdzH95DGoOIDmpZs7bTEHmhRhuQYNICbG6hQiIiLw7bfmvqqXL5vbcg0fDmXLWp0q51xJvcLkde/S9sgIaqS6Mfa+sfS7px+4O/+AdhV0FmvcGL76CuLjwd/f6jQiIpLfXLhgLnJfvTqULGkuQTJsmPl1nmFLh6OfcdmvGhN/m4R3lSd5ttGbBBUsb3WyHKMxdBbbuxciI+HHH6F1a6vTiIhIfnH6NLzzDkyfDnfdZU54yIuWHVrGR7+NY773NlxqjCcu4kkCvPJeTaExdBarWBHCw+HQIauTiIhIfhAbC337mhMcZsyAAQPMCQ95zd4jS1g5N4z2c+/nbGoqxxv/DOV65MliDtRC5xBSU8297kREROzl3DlzUfuUFKhXz1x+pE8fKFjQ6mQ5zLAx5pexzPzlNZZGeHKi6iSa1+iLi7PvQXYbDjmGLioqiqioKNLT062Okis8PCA93Vznx90h/0VERMRZ7dgBb78NCxfCvn1my9yWLc6/x+r/ikuOI+XADIJPfE3DCm9TqFUUZWo+T0V3T6uj5Qq10DmAkyfNqeBffKFxdCIikjM2boQxY2DxYnOCw7Bh0K1b3tuhKDX1Cp9vmsIrv06mY7FyvF+lGVR5Hdzy2Bu9DbUHOYCwMPD1heXLVdCJiMidMwzz5uoKU6eaLXJz5kCnTnlvaI9hGCzauwj/Dc8QkZbIgxW6MbTZaAgItzqaJTQpwgG4uEDLlrBsmdVJRETEGRmGuVpCo0ZmAQdmQbdrl7meXF4r5ji/ntiLf9Dt225851aZiCZfMfvh2YTn02IOVNA5jFatYOdOs/tVREQkK2w2+OYbqF0b2rQx75cubR4rWBDc3CyNl+MOXzpMz0Vdsa1pT8HjX7DjhR1MeWYjFcs/YXU0y6nL1UHcd5/ZUrdunbmoo4iIyO0sWmTOVm3WDFasMP/Ma5MdAC5ePsTvKzvy1J5tePoUYWfbqVQr256SrnmsYv0XNCnCgZw6ZY6nExERuZHkZPj0UzhwACZMgLQ0c8bqPfdYncx+vtr5Fa//0IOYogmsCnuWR5pOpYBHAatjORx1uTqQa8Wc45bYIiJihStXYMoUc1/VXr3g2DFzuSt397xZzNkMG5f+XAgrmlPCvwgt73oGt/bH6NTyExVzN6EuVwdy4IA5BmL+fHNPPRERkaQkqFDB3Krr6afN5UciI61OZT9rDv7AoJVvUNY1mXkVKlC/yF3UL9nU6lgOTwWdAylRAv76y1y+RAWdiEj+deYMfPghDB0KPj4wbhw0bPj3hIe8aPe53exe3paIpMO4u9ahb8sPcCl5r9WxnIbG0DmYBx4wm9G1hImISP5z9ChMnAizZpndqcuWQf36Vqeys6unSUlNoMSMRtxbwI1B1TpQr8G7uLhqVFh2qIXOwTzwAAwZAvHx4O9vdRoREcktb70FI0eay428+ir07QuFClmdyn4SUhKYvO5dhsV/iWfhWiztvJTI4Ei83L2sjuaU1ELnYA4fNge9LlwIjzxidRoREbGnDRvMX94rVzYXBt6/H55/3tw9KK9KS0tiXUxP+uxYyv7EWH58YAzNq3YHzzxcveYChyzooqKiiIqKIj09nf379+ergg5g+3aoWtXcukVERPIWw4Cff4axY2HVKnjxRYiKsjpV7vjt+G8MWvIsS/z2Md/rHu5vNY9SBUtZHStPcMiSoU+fPuzevZtNmzZZHcUS1aurmBMRyYt27IA6dczdgeLizFUN3n/f6lT2l3RxB6ztiJ+rCwV8i3Os8c/0emK9irkcpLLBAZ06ZQ6C/e03q5OIiMi/lZJiFnIAoaEQHGxOdti0ydzlIa9tz/VPxy4f4ZmFz9D+6ycw4vZS1a8gK7qsoGbJ+6yOludoUoQDCg2Fgwdh8eJ8MLtJRCSPSkyEmTPhnXfMteROnICiRWHpUquT2d/lpMv8vOwpSp3/iZUXQxjRbDS2Gs/h5qayw17UQueA3NzMBYa/+87qJCIikl1paeaM1VKl4OWXoUkTWLkSPD2tTpYL0lMwrpyk6ZymTNm/ioTgxuzrs4uetXqqmLMz/e06qLZtzf36jhzJ2wtJiojkFefPm92pbm6wYgU88QQMHpw/PsMNw2D+7vk8ePJDCri5Mq3NNMoWKkuYvzYozy1qoXNQrVqBh4da6UREHN3hw9C7N4SHw9q14OJiFnRRUfmjmNuxczqPzK7FE/OfYKFbJNz9Ho1KNFIxl8vUQuegAgIgJgZq1rQ6iYiI3Mj27TB+PHz1FRQuDCNGmEtOQf5YqeDo5aO8snQA76Uspq1rUQZ0WUHz0s2tjpVvqaBzYJoQISLiWAwDUlPN8XDffGOuRjBlCjz3HBQoYHW63GFLOofr7vGkFevIlrO72Fh/Es/VGoCrq0oKKznkwsLX5MedIv7JMODJJ83u1+eftzqNiEj+lZ5uFnATJkDr1jB6NFy5YhZ27vmkjrmaepX31r/Hqj1f8FPIJVwafI4ttDGuLvmgOdIJ6F/Bgbm4mHu6fv651UlERPKnq1fhww+hYkVzkkNAADT/b69igQL5o5izGTZ+WvsK2+YW5K3Vb1C5xH0ktN4BRZqqmHMg+eBb0bl16GC2zp0+ba5fJCIi9mcY5i/Vv/9ubs31+OMwbx7Urm11slyWGs8TC59l96EFfFIqnJ09vqF00TpWp5IbUJerg7t4EYoUMcdovPii1WlERPK2Eydg8mTYts3cb9XFBY4dgxIlrE6Wu3ac2UGJQ1MoGL+LleXH4ONRgPoRGtjtyNRW6uCCguC++2DBAquTiIjkXbt3w7PPQpkyMGsW1KtnTn6A/FXMnTqzmcELn6DGhzWYee4SVB5K89LNVcw5AYdsoYuKiiIqKor09HT279+fr1voAPbtM6fEBwdbnUREJO9JTTXXkPPwgEGDoEcP8Pe3OlXuik+OZ/zacTx9fCw70jw5V20SvWr1wsPNw+pokkUOWdBdoy5XERHJaTYbLFkC770Hc+dCWBjs2AGVKuWT7bn+yZYKh2ZyIrAu1Wa3ZGS1h+na6E0C/fNRs2QeoS5XJzF79t8zq0REJPtSUszP0ipV4OGHzfsXL5rHqlXLX8WcYRh8u/db2n9+H8b21whP2MXxgcfp/8BsFXNOSgWdkwgJgVWrzIG6IiKSfX36mAsAly9vbtG1di3cdZfVqXLfjv1fsfrzUDp9/QgJLl6cbBoDZbrg6+lrdTT5F9Tl6iRSU6F4cejcGd591+o0IiKO7+RJmDoV7rkHHnnEHI+cng6VK1udzCK2NF5f/SZzf3uL7yJ8uFDzfe6t0h0XFxerk0kO0Dp0TsLDA55+2hzvMWFC/ljMUkTkTvzxB7zzDnzxBfj4QESE+XjFitbmssrFqxex7Z9O8KmFNK8wjtKFSlOpWhfc3PSDJC9Rl6sT6dIFzp6FX3+1OomIiGP64QdzPNyKFTB2rLmGXH5dwzMpOZbpa96g7PtleXvXTxD+CM1K3stzNZ9TMZcHqcvViRgGHDgAFSpYnURExDGkpsLXX8ORI/Daa5CUBAsXmjs7eOTTFTdsho15O+dRbHN3ktOS+LZYb0Y0GUERvyJWRxM7UkHnhAzDHAeiblcRya/i4uDjj82lR06cgLZtYdEicM3v/U5nYrjoVpBSH9/LC6Vq8EKDYZQp2cbqVJILVBI4mZQUqFULevfOv90IIpK/xcdDqVKQkGCOLR40CKpWtTqVtfac28OENaP4hJUElX+BPX32UDyguNWxJBfZ9XeZNWvW0LZtW4oVK4aLiwuLFi2y5+XyBU9Pc8r99OlmS52ISH6wbRv062d2sfr7m7NXjxwx15XLz8XcmQu7WDG/Go0+qkLMiQ3sqTkTqo5QMZcP2bWgS0xMpHr16kybNs2el8l3XnwRdu4011ASEcmrDAN++glatoSaNWHxYjh82Dz29NPmUk752afbP6XhzDpUurKT6Q1fZE+fPdxVpi245Pd+5/zJrl2uDzzwAA888IA9L5EvNW9uttJ98AHce6/VaURE7OPJJ+H//g/uvhu+/NKc6JDfxw6n29K5fORrCh+eTrnyo3ik5osUaPgKT/iGWB1NLOZQ/zWSk5NJTk7OuB8XF2dhGsfl6mqOoZs2zRxTl5+2qxGRvOvSJZgxA9q0MbtRe/QweySaNIH8vvatYbPx875vGLT6Tap5ufJ5xbtoEFaNBqWaWh1NHIRDFXRjx47lzTfftDqGU+jd2xxPkt9/WxUR57d/P0yZAnPmQFoaBAebBV3LllYncwy/n/qdIyvaEZZ8gsI+jXmp5SRcitexOpY4GIfqaB8+fDixsbEZt+PHj1sdyWF5e5vF3NGj5gegiIgz+uQTqFTJ7FodMsRcCLh7d6tTOYjEYyTF7qPV5634vwRXXKv8h1VdVlFHxZzcgEO173h5eeHl5WV1DKdx6JA5lm7RImjXzuo0IiK3l5IC8+aZQ0c6d4YWLcz15J5+2vxFVSA2KZYJa8fz5pX/wzu0ESu7rCQyJBJ3V4f6kS0OxqFa6CR7ypaFunXh/fetTiIicmvnz8Nbb0HJktC1K6xebT5esqTZIqdiDlJSEli97HHumVaa9zZO4bcS/aHW+1QtUlXFnNyWXb9DEhISOHjwYMb9I0eOsG3bNoKCgihRooQ9L51vDBpkzgTbutWcCSYi4mj27Pn786lrVxgwACIjrc3kaGKOxjB4ybMsDTzC4NJNadNqLsX8i1kdS5yIXVvoNm/eTM2aNalZsyYAgwYNombNmrzxxhv2vGy+8uijUKYMTJhgdRIREZNhwPLl8Oqr5v1KlWDSJDh+HD78UMXcP105txHWPEqguwehQZGcaf4rzz+2SsWcZJv2cs0DZs6EjRvho480tV9ErJOUBHPnmvur7twJNWrAL7+An5/VyRzPgXN7GL7qda5c+oPvSxXEpcEX4F/W6ljixFTQiYjIv2azQcWK5mSthx4yh4No/bjrnb9ynpU/PU7ZSzE8FhvOqOZj6FytM67a3UH+JY2yzCOSkuDTT+GRRyA01Oo0IpIfbNtm7is9bhwUKgTjx5vrx5Uvb3UyB5R2FSP5Io3ntKRI8jFGVLyfPU9/hY9XoNXJJI9QC10ecfEilCgBAwfC6NFWpxGRvCotDb791pxdv2YNhIfD/Plwzz1WJ3NMNsPG5zs+p8OZmfh4+PFbudcpF1SOEG3VJTlMbbx5RFCQuU3OtGkQG2t1GhHJq7p3N/dUNQxzMeAjR1TM3cyW3yfQ5uOqdF3UlSXedaDWFOpH1FcxJ3ahFro85ORJc226oUNBO6iJSE7YsQOmToXHHoPWrc1uVptNyyTdyoELBxiytA8fpS/nWyOCKs3n0SCigdWxJI9TC10eUqwY9OkD775rLuIpInIn0tNh4UJo1gyqV4cffoDLl81jNWqomLuZ9CunYHN/jJTL7Lt8nK1VP6DHM0dVzEmucMhJEVFRUURFRZGenm51FKczbBjUqWN2wYqI3ImPP4bevaFBA3ObrkcfBQ8Pq1M5rvikOCb+Non1+xfwU5EEKpTsyK4Xd2nmquQqdbnmYTabuV+iiMit7NpldqsGBpozVWNj4cABqF3b6mSOLc2Wxk+/vETw4Rm0POlKn3sG8vq9wyngpZ9XkvscsoVO/r2ePcHT05wkISLyv9LTYckSc7bqypUQFgaDB5vHAgNVzN2SYUDqZR6e35ljf/7A9DJl2NVrERHBVa1OJvmYWujyqPHj4fXXYf9+KFXK6jQi4iiutdzv2QOVK0P9+tC/v9mt6ulpdTrHt+mvTVQ4MoXAK4dZXX4Mgd4FqRlW0+pYIiro8qrERHPGa8uW8NlnVqcREav9/jtERcGWLebN1RX27jX3WZXbO/7XGib8Nolpu75j8t0deKnWcxB2v7bCEIehEVZ5lK8vjBoFn38OGzZYnUZErJCWZn4G1K9vzkz96Sdz+ZGUFPO4irnbu3T1EoOXDiJpeRPqX/6ZmW1n0u/BL6FYaxVz4lA0hi4P694dPvnEXDdKC3+K5B9nz5pbALq4wBtvQLly5jIkDz0E7vrUz5r0ZDgwnYRCjZmz41PK3/0inRuMwLeA9lYUx6Qu1zwuPR3c3KxOISL2ZrOZLXAffAA//gj79pnDLuLjwd/f6nTOwzAMvtr1FZ9vmsJ3/gdwqRPF1WLt8PHwsTqayC2pyzWPc3ODq1fNxYYTEqxOIyL2EBUF5ctDmzZw4gR8+CEULWoeUzGXdVt2z2bNZ0E8981TuPmEcva+36DkkyrmxCmo8T0fOHMGXnvN3Bps0iSr04hITti0CSIjwc/PnNzQoAHMnWsOr9DQrmyypTL45+Es3PwO30T4seqJL7in0lNWpxLJFnW55hPjxplF3ZYt5lY+IuJ8rl6Fr74yW+Q2b4ZZs+DZZ81l0VTEZd+ZhDO4H/iAwme+Z2WFsZy9coEn7npCOzyIU9J3bT7x8svmjLZevcxxdSLiXGbOhPBws4ALDobvvoMuXcxjKuay58rV80xZNZRyU8sxYf8aKPkUzUs1pWOVjirmxGmpyzWf8PCAjz6CRo3MAdMPPWR1IhG5lZQUc2Zq6dJQty4UKQLPPQcvvGBOdpDsS7elE709morbelM+LZWedw/klcb/AR9tfi3OT12u+cyWLeZ6VPqNXsQxHTwIM2bAnDlw7pw5VGL0aKtTOTnDgFM/cc6jCGVnNqF/2Xvo2fA1ShRvYnUykRzjkC10UVFRREVFka6+wRxXq5b559Kl0KKF1qQScSRffAFPPw2FCkHXrtCjh7k9l9y5bae3MWnNm3zqtpaQSgM50O8ARfyKWB1LJMephS4fOnDAHE/35pvmb/8iYo2DB82xcYGBMHy4uSDwsmXmbg4+WinjX/nrzBb2xDzN4/v2ERZUiUVt36diRAt1T0iepdGf+VD58jBsmFnQ/f671WlE8peUFPi//zP3WS5f3hzbem0rrtBQ6NxZxdy/Yhh8vOVjmsxuSIWkA8xqOoQ/ev9BxRItVcxJnqYWunwqJcUcaJ2WZi5/4O1tdSKRvC0x0dxjeds2qFkTGjY0Z50//rgKuJyQmp5K3OEvKHxkBmvLv8Wyo6sYUn8Q/t4FrY4mkitU0OVjf/wBderAyJFmi52I5KzERPj6a3NP5YQEs0XcxcUc9lC+vNXp8gbDZmPJrs94efUY6vv7EV2pJtz9LngGWh1NJFdpSHw+VrUq/PyzWdSJSM65dAmGDoV588xCrkUL6Nv37wWAVczljN+O/8bZVe0JTj1D2aD7GdxiIhSpanUsEUuohU4As+WgQAGoWNHqJCLO6dw5WLsW2reH1FRzzcfWrc2FgEuVsjpdHhN/kKvpaYTPaMhDhYLoX6s7te5WN4PkbyroBMMwlzNJSoLffjNn3InI7aWnw08/mVtwLV5sLuB96hTo48o+zl85z/hfxjIu6Rvcwlqxp/RLVChcATdXN6ujiVhOs1wFFxf48kvzB9GTT5oTJUTk1lJTzRbtBx+E/fth4kT4808Vc/ZwNekSq354kLrTSvPR1o/ZUGYY3P0ekSGRKuZE/ksFnQDmD6b5880xdQMHWp1GxPHExZmTG9q0gatXzda4YcNg40bYvh0GDDD3WJWctfzQcup9WIWqF37gPxUacaj/IRpU7QXumhos8k8q6CTDfffBBx/AV1/B6dNWpxFxDMuWQadO5l6qPXqYLdjnzpnHnn/enFSk5c1yXsKZtbC6LUGePpQrVo9LrbbQ/ZEfCfENsTqaiEPSGDq5zsWLEKS9qiUf27vXbLV2cYEGDeDyZXMrrqefhvBwq9PlbbtPb2PIyv/gEn+I78qE4tLgM/AtaXUsEYenFjq5TlAQxMaaC55qJwnJL86fh6lToXZtiIyEX381H1+yBHbtgldeUTFnT6fiT/H1/Hpc/aEmB87vpdu9b0GLGBVzIlmkgk5uyNUVjh0zl13Yu9fqNCL21bcvhIXBoEEQEQHffGPupALmLzjqUrWj1ARsice4d/a9zDyxh4Ti7dnZ+w8er/w4LvqLF8kyh+xyjYqKIioqivT0dPbv368uV4ucPw/NmsGFCxATo8VQJW+w2WDNGvjiC3NSQ5kyMHOmOdHhqac0sSG3pNnS+GTrJ3Q7Pxsvn1A2lP0PFYMrUlBbdYncEYcs6K7RGDrrnTkDTZtCfLy5VVihQlYnErkz27bB55+buzf89Ze52O+sWeYvLZJ7DJuNjZtGMmzLl6w+d5CF9w3nkZq91LUq8i9p6y+5pSJFYOVK+PZbFXPifA4cgKJFwd/fHB/33XfmWoudOkG9eupKzW27z+3m5e97Ee26lqf8y/JO+y3cHXa31bFE8gSNoZPbCguDF14wv541y9xNQsRRnToFU6aYY+AqVID/+z/z8YkT4eRJs7CrX1/FXG5KSzwGG1/ALS2Rk0lx7KwVTY+n96uYE8lB6nKVLLPZzLXq1q83u60eftjqRCKZ/ec/MG4cuLubCwA/9RQ89JC5T7HkvktXLvL22rFsO7yEZWEpuDSahxFUW5MdROxALXSSZa6u8OOP0LYtPPqouQixiFXOn4ePP4ZWrcz9VMEc7zljhrkw9sKF8MQTKuaskJyWzOKfn+PgV6F8vPkD7o18iuQHdkDhOirmROxEY+gkW7y9zda5l1+GPn3MlpCePa1OJfnJt9/CtGmwahUYBjRpAp6e5rGWLa3Nlu8ZBqRc5MGvnuT8yZW8X7YS+3p/S5FCmiIvYm8q6CTbXF1h8mSoUgUeecTqNJLXnTtntrY1bgyVKpkTHWw2s6h79FEIDbU6oQD88ucv1Dj6Pv4pp3j93rcI8XufyiGVrY4lkm9oDJ38aydOmJMmPvgASpSwOo3kBSdPwuLFMH++2RIH8OGH5l6qhqEJDY7k8J8/MnbdZGbuX05Unad5sVZ3KKK1YERym8bQyb926RLs3AnVq5s/gEWyyzBg925zXBzApEnm7g0uLjB9ujkmrkcP85iKOcdwLvEcfZa8gMvqNjRO/I25j87lhQc+VTEnYhG10EmOuHTJHEs3fz507w7vvQd+flanEkeWnm7OmF60yBwXd+CAudxI//5mAefpaW67JY7FSE3EZf9UjoW0ou6nbRhXuzMd67+Gt1dBq6OJ5Gsq6CTHGAbMng1DhsDq1VC1qtWJxNFcvWqOwfTygl69zBmpoaHmEjgPP2wui+PtbXVKuZF0Wzqfbv+Ur7dG8UPgn7jcM4OUYg/i6eZpdTQRQQWd2EFCgtk6l5Rkdp0NGGCu1C/509Gj5nI3P/4IK1bAJ59Ax46wfTskJsI994Cbm9Up5VZ+2z6N1G2vcv+f8Txc+UmmthxHSGApq2OJyD84ZEEXFRVFVFQU6enp7N+/XwWdk/r1V3ONsEKF4P33oX17jX/KD5KTzQLt2pI2H39s3m/YEB54wFzst6S27XQO6cn0WTqIZds/4KsSgbjU+5Sa5dpZnUpEbsAhC7pr1ELn/P78E/r1M/fQfPBBs7ArU8bqVJLTjhz5uxVu5Upzhup995ktcpcvQ4sWEBhodUrJqhNxJ/A5MJXC51awssI44lMSaVexnRYFFnFgmuUqdlWypPnDfdEi2LED/vjDfNxxf42QrLh69e9/w8cfN4v0AQMgPh7eeMPcQxXMou6xx1TMOYu4hBNMWNaPClMrMPnw71DmOZqXasrDlR5WMSfi4NRCJ7nm6lVzwLuLi1kElC0Lr7yimYzOIDUVNm82W9xWroR162DTJnPiy6JF5kK/LVqA/ps6p9T0VGZsmUHtXS9xNs3GhrLDGdpwKAFe+gcVcRYq6CTX2WwwYoS524SbG7z4orlURViY1cnkGpsN9u83d2YAs3DbudMs2Jo0MVveOnaEIkWszSn/kmHAiUWc8SpJ+U+aMqRiE55v+AZhRepYnUxEskkFnVjmzBmYOBE++shcxuKvv8w/JffZbLBxI/zyi3lbu9ZcW/DUKShaFJYsgZAQqFXLnOwgzm/9ifW898tovvDYgOtdr3KhZFcKFyhsdSyRfG/79u1MmjSJI0eOMGzYMBo2bMibb75JcnIyp0+fZsSIEdSoUeO656mgE8tdvgxbt0Lz5hAbC23bQqdOZgtQwYJWp8ub4uPNRX2PHjV3YEhLM2cj22xQvz7ce6+5d2rDhn9vfC95w9GTazkU04VHDhyhTGg1Fjw0lXLhja2OJSL/9dxzzzFjxgzGjRvH+++/T+PGjZkyZQr79+/n/vvvp1evXkydOvW65+l3bbFcwYJmMQdw8aI5gL5PHxg40BxQ3707NNNuQv/ayZPmhIUNG8xttmw2KF4cnnvObHXbvNmc3ODhYXVSsQvDYOrGaUxeMYifww3mthzBg/e8jpurFgEUcRSHDh2iWLFiuLu7c/LkSS5evMirr75K8eLF+e233/Dz86Nt27Y3fK5a6MQhnTwJn30Gs2ZB6dKwdCmkpJjLYrRoAb6+Vid0TGlpsG8fbNtm3jZtMmcaR0ebLaFNm0LduuZivvXrQ2Sk1gbM65LSkkg4GE3wn7P5pcLb/HJiPS/d058CntqbT8TRrFu3Dh8fH2rWrEm1atUICgpi9erVWXquCjpxaIZhFiKFCsGaNeaAfC8vs0WvbVto0yb/LlIbG2suBbN9O9SoAY0aweefwzPPmMdLljTHvLVu/ffG9pJ/2GxpfLNtJoPXjKNFoRBmVr4HaowFD23bIuLozp8/T2hoKG+88QYjR47M0nPU5SoOzcXFLObAHNO1f785QP+778yZsbNnm4P5bTaYORPq1TNbnfJKt6FhwLlzZqtb9ermLNN33oEPPoDDh81zPDxg9GizoGvVytxHt3p1jT/Mz1YdWUXimscpknaRu4s+wtCW46FwBatjiUgWrVq1CsMwaNq0aZafoxY6cVqXL8OJE1ClilncVKxodjl6eJhFXdWqZsHn4WGeGxjomN2Lyclw/DicPQsNGpiP9e9vFqr79pnZAX7+2Vwu5MsvzfFuNWqYhVulSpq4IP8Vu5uEdIOIjxvxeEgx+tXpQ7WqL1qdSkSy6cUXX2TWrFlcvnwZb2/vLD1HBZ3kGVeuwJYtf+9IcfKkuUsFmDsXnDpljscrVcr8s39/c3HjAwfg/HkIDjaX5sipwu/yZXMplrNn/76VLg0PPWQWoJ06mVujnT5tnu/tbb4HFxdzooLNZhap127ly6twkxs7FX+Kt9eM5r2UJbiFt+NAmQGUCyqn3R1EnFRkZCRFihTJ8vg5sKDL1TAM4uPjb3gsOTmZ5OTkjPvXzouLi8uVbOL8qlc3b9dc+9Z56y2ztevYMXOpjh9/hEcfNQu4yZNh+vS/n+PmBr17w5gxsGePOSbNy8u8ubiYEzKuFYpPPmnuY5qUBAkJkJhojmNr2RLGj4dx4/5+XU9P6NzZ7Do2DLOYbNYMwsMhIsK8xcWZ13jvvevfW1KSeRO5JuHKWbbEPMcLezZx1dWH9q3fona5DhRx87rp56yI5C5/f/9s/XJ1+vRp9u7dy5NPPpmt6+R6C921VjcRERGRvC67vYzbt2/n/vvvZ+XKlVSuXDnLz8v1gi47LXSnTp2ibt267N69m+LFi+dKvjp16rBp06Y8d63cvl5uXisuLo6IiAiOHz+eK13z+ndzvuvl9vcI2Oe9GYZB3KlVBB6MYmuZwUzfOos3Gg3h0fufzpP/brl9LX2WOOe1cvN6d/I9kt0WujuV612uLi4u2f6P4u/vn2sfwm5ubnnyWrl9vdx+bwABAQG5ck39uznv9XLrewRy/r1tPb6OwStfwz/5FIvKFqdpqSo0rfyVXa51O3n5exL0WeJs17Liern5WZJVWrbkf/Tp0ydPXiu3r5fb7y036d/Nea+Xm3Lqvf15+U82LHuYMgnbOZsUycstJkH5Nplm7uTlfzd9jzjn9fLye3NUDj3L9cSJExlNm+Hh4VbHEQel2dByO075PZJymfTky5T9pCmVXeMZflcb6jeZibu7l9XJ8iyn/D6RXOXI3yMO3ULn5eWV6U+RG/Hy8mLEiBH6PpGbcqbvkZT0FKZvmk7vi9F4+pVi/hPzqRRcCT9t1WV3zvR9ItZw5O8Rh26hc+RKWEQkJxk2G7/99jIv//4NGy+eYHGr13mwei/wCbM6mog4AYduoRMRyQ+2n97OwCXd+dprC92CKjPziR3cFXqX1bFExIm4Wh1ARCS/Sok7AOufxdOWTILhyr668+nVaZeKORHJNnW5iojksrMJZ3gzZhT7j//MsuLg0ugrjILVtVWXiNwxdbmKiOSSK6lX+GlldyJOfs28M34Mu/c1Uu/ph6e7NyrlROTfUEEnImJvtnRIuUCrLx4l+dwG3ilXjf2PfUvhgBJWJxORPEJj6MQpjRw5EhcXl0y3okWLWh1LLLZmzRratm1LsWLFcHFxYdGiRZmOG4bByJEjKVasGD4+PjRt2pRdu3bZNdPyQ8tJiGkPazsw9r6xzOuxl8aP/a5iziK3+x7p1q3bdZ8t9erVsyasWGLs2LHUqVMHf39/QkNDeeSRR9i3b1+mc6z4LLkdFXTitO666y5OnTqVcfvjjz+sjiQWS0xMpHr16kybNu2GxydMmMC7777LtGnT2LRpE0WLFqVly5Y33V/639h/cD5dPm9Cq89b8WVKMNQYz70l76VsUNkcv5Zk3e2+RwBat26d6bPlhx9+yMWEYrWYmBj69OnD+vXrWb58OWlpabRq1YrExMSMc3LzsyTLDAc0bdo0IzIy0qhQoYIBGLGxsVZHEgczYsQIo3r16lbHEAcGGAsXLsy4b7PZjKJFixrjxo3LeCwpKckIDAw0Pvzwwxy77qn4U0b3hV2NP+dgzJ5Z0Phm9zeGzWbLsdeXnPO/3yOGYRhdu3Y1Hn74YUvyiGM6e/asARgxMTGGYeTeZ0l2OWQLXZ8+fdi9ezebNm2yOoo4sAMHDlCsWDFKly5Nx44dOXz4sNWRxIEdOXKE06dP06pVq4zHvLy8aNKkCevWrfvXr2+kxMGOkSQl/sXSwz+ztuxrPN3lL9pHttfsVSezevVqQkNDqVChAj169ODs2bNWRxILxcbGAhAUFATY/7PkTmlShDile+65h08//ZQKFSpw5swZ3nrrLRo0aMCuXbsoXLiw1fHEAZ0+fRqAIkWKZHq8SJEi/Pnnn3f8uqnpqczYMoPF22eyNOgkpQrX4ciAI3i4efyrvGKNBx54gA4dOlCyZEmOHDnC66+/TvPmzdmyZYtDbvck9mUYBoMGDaJRo0ZUqVIFsN9nyb+lgk6c0gMPPJDxddWqValfvz5ly5YlOjqaQYMGWZhMHN3/tpYZhnFHLWiGYbBmy3hcd41m8J9XeLJaVy61WEKQX3FUyjmvJ598MuPrKlWqULt2bUqWLMn333/Po48+amEysULfvn3ZsWMHa9euve5YTn2W5BSH7HIVyS5fX1+qVq3KgQMHrI4iDuraLOhrv11fc/bs2et+076ttKt0X9ydnsuG4+Xuw6auPzHnkTkE+RXPqbjiIMLCwihZsqQ+W/Khfv36sXjxYlatWkV4eHjG4zn6WZKDVNBJnpCcnMyePXsIC9NG5nJjpUuXpmjRoixfvjzjsZSUFGJiYmjQoEGWXuPwpcNc3PgS/Hwvnat2YsoTP1Kn0zmqlGx12+eKc7pw4QLHjx/XZ0s+YhgGffv25ZtvvmHlypWULl060/Gc+CyxB3W5ilMaPHgwbdu2pUSJEpw9e5a33nqLuLg4unbtanU0sVBCQgIHDx7MuH/kyBG2bdtGUFAQJUqU4KWXXuLtt9+mfPnylC9fnrfffpsCBQrQqVOnW77upcsHmbpuHG9t+ZT/VL6PETX60Lx0c3DR78TO5lbfI0FBQYwcOZLHHnuMsLAwjh49yquvvkpwcDDt27e3MLXkpj59+vDFF1/w7bff4u/vn9ESFxgYiI+PDy4uLnf8WWJXls2vzYLY2FgtWyI39OSTTxphYWGGh4eHUaxYMePRRx81du3aZXUssdiqVasM4Lpb165dDcMwlxsYMWKEUbRoUcPLy8to3Lix8ccff9z09a6mXjUmrp1gbJntZnwzw814K+YtIzElMZfejdjDrb5Hrly5YrRq1coICQkxPDw8jBIlShhdu3Y1jh07ZnVsyUU3+v4AjNmzZ2eck93PktzgYhiGketVZBbFxcURGBhIbGwsAQEBVscRkfzCsMGf8zjtW4mKnzTj1chWPNdoBCGFq1idTETkhtTlKiLyD6uPrmbar2P42msrRauN4s+X/qSgd0GrY4mI3JIKOhER4MCxZRxf+xwPH/yLysXqcqz5j5QKq0tBq4OJiGSBRvSKSP5m2Jj460Raft6a8LSz/N8D41nffT2lwupanUxEJMvUQici+VJiSiJXDnxCyPHPqVtxLO7N3qFk7d5U8PC2OpqISLY5ZEEXFRVFVFQU6enpVkcRkTwmPS2ZeVujGPLLJB4OjWB6lSY0iahPk9L3WR1NROSOaZariOQLhmHw48Efcf31KbzS4pgZ1IkxzcdQqmApq6OJiPxrDtlCJyKSoy5uJR4POi3oxNPFytKn0UvMrdTF6lQiIjlGBZ2I5FnHYo8xNmY009K+J6DUU/ze63dKFSxl6QbaIiL2oIJORPKcy/HH2LKiI513b8HwLESPtpO5u3wHSrvqI09E8iYtWyIieco3e76h/kc1qBL/G+PvfpwD/Q5wd8WnQMWciORhKuhExOkZhsGlY0tgZUvCfYNpVOEx0tseokvrufh7+VsdT0TE7vQrq4g4tfVHVzJoxWsUtcWyoHwZ6oZUoG67j62OJSKSq1TQiYhTOnDhANuWtaPU1b0kp9Wgb6upuJRubnUsERFLqKATEeeSdI7UlDiaRTejjmcag6v2YtO903DVGDkRycf0CSgiTuFq6lXeXz+FgbFz8SxYme+e+o5KwZXw8fCxOpqIiOVU0ImIQ7PZ0lj3y4u8tO17tsedpeb9o2hVrQc1vYOtjiYi4jBU0ImIw9p8cjMDv3uWhQV20qPI3dz3TAzlgspZHUtExOFo2RIRcTjJl3fDr0/jbaTh4lmIIw2+p1fHLSrmRERuwiFb6KKiooiKiiI9Pd3qKCKSi/6KPc4bq0dy7OQvLCvhSRVff9Y8u8bqWCIiDs/FMAzD6hA3ExcXR2BgILGxsQQEBFgdR0TsJD45np9+7kzps9/R5lwhXmsykhdq9cLD3dPqaCIiTsEhW+hEJJ+wpWIknaf53Ha4X97B2PL3sP+JhQT6FrU6mYiIU1FBJyK5zjAMvtv/HfediMKXVCbfP5kSgSUoEVjC6mgiIk5JkyJEJFft3hPNk9H1eHjew8w3SsPd79KoRCMVcyIi/4Ja6EQkV5yIO8GwZYMYf+X/aG0E89zTP3J/2fvBxcXqaCIiTk8FnYjYlZF8CZc9E0gJe5xf/9rEb/eMoWvtwbhpwoOISI5RQScidpGUlkTUxiiW7pzDsuALlCl6Hwf7HcTN1c3qaCIieY7G0IlIjrIZNn5e/wZbPy/E6z8PpVyxRsS13g5FW6iYExGxE7XQiUjOSU2g8+KebN33JXNKFWV79/mUL97Q6lQiInmeFhYWkX9tz7k9FD30PoUub2ZVhXG4urrTpFQTq2OJiOQb6nIVkTt27vwf/GdxZ6pOr8qHp/6CSoNoVrq5ijkRkVymLlcRybbElETeXfcOjxx9k5qpbkxoOYE+dfqAu5fV0URE8iWHLOiioqKIiooiPT3d6igi8k+2dDgSTax/Dd7dMBm/qp14tuGbFAwsY3UyEZF8TWPoROS2DMNg6cGlzFg/nm98/sCl5gTiw5/A38vf6mgiIoKDttCJiOPYfehbTv/Wiw6Hz1CrRGNONF5BRGgNVMqJiDgOTYoQkRuzpTM6ZjQPznuEorbLLG43jdVdVxMRWsPqZCIi8j/UQicimcQmxZJ6YAbBf31F44rjCbl/OhVqdqeym4fV0URE5CbUQiciAKSkJPDJr29Rbmo5Ru1YBGEP0KREQ16o/QLuKuZERByaWuhE8jnDMFiwZwEFN3ajZFoiD1fszrBmo8C/mNXRREQki1TQieRn59YR6+pH98Xd6R5Rld6NhzCzzKNWpxIRkWxSQSeSDx28eJBxMW8yw/iZgmWeZWfvnUQERlgdS0RE7pAKOpF85MLlA2xb0ZGOe7fjXSCMnW2jqFb2ESJcNJxWRMSZ6VNcJJ+Yt3Me9WbcTWTi70y551n2991PtXKPgoo5ERGnZ9dP8jFjxtCgQQMKFChAwYIF7XkpEbkBm2Hj4tEF8HMzSvmH0bpKNzzaH6dTi4/x8fCxOp6IiOQQu3a5pqSk0KFDB+rXr88nn3xiz0uJyP9YfeA7Bq0cQTm3VL6qWIl6RSpTr+RUq2OJiIgd2LWge/PNNwGYM2dOls5PTk4mOTk5435cXJw9YonkaTvP7mTv8naUSD6Cj3s9BrScikuJhlbHEhERO3KowTNjx44lMDAw4xYRoVl3Ill25SQpcQdo8WkLvopNIb3SINY++ysNVcyJiOR5DjXLdfjw4QwaNCjjflxcnIo6kduIT47n3XXv8GrCV3gWrsVPnX8iMiQSTzdPq6OJiEguyXYL3ciRI3FxcbnlbfPmzXcUxsvLi4CAgEw3EbmxtLQk1vzcmfpRZRj76zjWFu8BtadRvWh1FXMiIvlMtlvo+vbtS8eOHW95TqlSpe40j4hkwbrj63j5u2dZ4r+f/hENaN3qS0oElrA6loiIWCTbBV1wcDDBwcH2yCIit3H1wu/47BmLf+lB+PuX5K8mM+hZoonVsURExGJ2HUN37NgxLl68yLFjx0hPT2fbtm0AlCtXDj8/P3teWiRPOXrpEK+ufJ0L5zaztKQ/Vf0LseyZZVbHEhERB2HXgu6NN94gOjo6437NmjUBWLVqFU2bNrXnpUXyhEtXL/Hz8icpc+FnfrlYhBHNRmOr3g03N4eazyQiIhZzMQzDsDrEzcTFxREYGEhsbKwmSEj+kp6MkXSeGp+1ISBhP6Mr1qdOi/n4+gRZnUxERByQfs0XcSCGYfD1rq9pe+ojCri5M/3B6ZQpVIaifkWtjiYiIg7MoRYWFsnPtu+YSrtPatBxQUe+da8Ktd6jQUQDFXMiInJbaqETsdiRS0cYurQ/76cu4RH3MAZ3XU2TUpq5KiIiWaeCTsQitqSzuO4aR3rxTmw/v49NDSbzbM2+uLrqv6WIiGSPfnKI5LIrqVd497d3WbXnC34OjaNc+MPs7bsXVxeNgBARkTujgk4kl6Tb0lm+7hUCD05l3HEbPev2I7HJ6/h5F9JgVhER+VccsqCLiooiKiqK9PR0q6OI/HuGAalxPL6wG/sOL+Lj0iXY2fMbShWpZXUyERHJI7QOnYgdbT+9nZKH3qNgwm5WVxiHj0cB7gm/x+pYIiKSx6inR8QOTp3ZxMvfPE7Nj2oy83wc3PUqTUs1VTEnIiJ24ZBdriLOKi45jnG/jOWZE+Opm+ZFVJsonr/7eXDzsDqaiIjkYSroRHKCLRUOziC+YH0+2jqD4tW706XhSPz9iludTERE8gEVdCL/gmEYLNy7kNkb3mGx316K13qP4wOPU8CjgNXRREQkH1FBJ3KHtu/7kksb+/HMkQvcW+Z+TjX7hWKFK6NSTkREcpsKOpHssqXxn1Uj+GL92ywuUYCfHptNo7u6WZ1KRETyMRV0Ill0/sp5jP3TCTm1iBYVx1OhcAUqV+uMm6ub1dFERCSfU0EnchtJyZeZtX4Sr/42jedKVOPd6o/TrGRjcPO0OpqIiAiggk7kpmyGjS/++ILwLT0om55M52ovMqzJG+AbanU0ERGRTFTQidzI6ZVcdi/Mi9+/SO/S9/BCg+FMK3G/1alERERuSAWdyD/sOruLCWveZLZLDEEVXmRf332E+YdZHUtEROSWHLKgi4qKIioqivT0dKujSD5x5vxO/ljVkSf27SYooAx7286icqk2hLm4WB1NRETktlwMwzCsDnEzcXFxBAYGEhsbS0BAgNVxJC8yDOZsj2b0Ty+yOiyZ9SX68vC9E/HUhAcREXEirlYHELFCmi2NC4fmws+NqVCwFI/f3Q//x8/QoekUFXMiIuJ0HLLLVcReDJuNn/Z+zaBVo6jh48HcStVoUKw6DUo1tTqaiIjIHVNBJ/nG5pObObbiEYql/EVRv6a83HISLsVqWR1LRETkX1NBJ3lfwlGS0pN5YO4DtAzw46WaI1hR+w1cXDXiQERE8gYVdJJnXbp6iYlrxzPqyny8izRmddfVVAyuiLurvu1FRCRv0U82yXNSUhJYt+oZev2xmr9SUnngwXHcW7kbd3n4WR1NRETELlTQSZ6y+uhqXv6uGz8V/JMhZZrzUKu5FPUranUsERERu9IgIskTEs9ugDXtKejuSfHgapxvsYHnH12hYk5ERPIFtdCJU9t3dievrHyN5Mt7+aF0IWoEFmHxU4utjiUiIpKrVNCJUzqbeJZVPz1Guctr2R4XwVvNx2JUfQoXFzU6i4hI/qOCTpxL2hWM5Is0mdOSosnHGVGpDXs6z8Pb09/qZCIiIpZRQSdOId2Wzmc7PqPjmZl4ewYw++HZlA8qT+ECha2OJiIiYjmHLOiioqKIiooiPT3d6ijiALZsGcvwzZ+y/PRe/JoN5vGaL1DPv6zVsURERByGi2EYhtUhbiYuLo7AwEBiY2MJCAiwOo7ksv0X9jP4xxf52LaCxZSgWvOvuSf8HqtjiYiIOByNIBeHk37lJGzqg0tKHIfjTrOt+kc83/mIijkREZGbcMguV8mf4pJimbBuIr/tX8DPYVcpX6ozf/T+AxcXF6ujiYiIODQVdGK51PRUlv0ygJAjH/P+STf61XuZq/cOo4CnPyrlREREbk8FnVjHMCDlEg/P78zxYz/yQdly7HlhEcUL32V1MhEREaeiSRFiifUn1hN55H0CrxwhpsJYCvoUonrR6lbHEhERcUqaFCG56s8TK3nx6wep/0l9Zl9Oh6ojaVK6qYo5ERGRf0FdrpIrLl69yFurR9H7zBQaphWgTrtZdKneBVzdrI4mIiLi9FTQiX2lJ8H+KBKDmvL5zi+IrNmXpxuOoIBPsNXJRERE8gwVdGIXNsPGvJ3z+HzTFL4POExEnQiODTyGt7u31dFERETyHI2hkxy3addM1n4WxPMLn8bLtxjnWqyHkk+omBMREbETtdBJzklPYdDPw/h2y2QWRPgT0/Er6lR4wupUIiIieZ4KOvnXTsWfwuPABwSf+YG2FcdTL7we1St30A4PIiIiucQhu1yjoqKoXLkyderUsTqK3ELCldO8u2IQ5aeWZ9KBX6H0MzQr1YQn7npCxZyIiEgu0sLCkm1ptjRm/T6Lu3b0IzYtlVUlX+bVe1+lkE8hq6OJiIjkS+pylawzDDj5I5c8wxi8bDADyjelV4PXaVOskdXJRERE8jUVdJIlW09tZdKaN/ncfR0hlV7mUP9DhPiGWB1LREREUEEnt3Hi9Cb2xXTi0f0HKR4UyYG2X1Ix4j5CNEZORETEYTjkpAhxAIbBR5s/ovHsRpRNPsKcZsPY0XsHFUu0ABVzIiIiDkUFnWSSkp7C+f2zYHlDqgRXpEv94QQ/eYH2jcbi7qoGXREREUekn9ACgGGzsfiPOQyKGUOjgECiI2vRsFhNGpZqanU0ERERuQ0VdMKvx37l/OpHCUk9S6XgBxjaYiKE3mV1LBFxICkpKaSlpVkdQ8ThuLu74+npaXUMFXT5Wtx+rtpstJvXjraFghlQZyjf13jZ6lQi4mBSUlLYtWsXNpvN6igiDsfV1ZW77rrL8qLObgXd0aNHGT16NCtXruT06dMUK1aMzp0785///MfyN53fnUs8x9g1bzEx+Vt8irVm3XPrKF+4PK4uGlIpItdLS0vDZrNRqlQpfHx8rI4j4jCuXr3K0aNHSUtLs7y2sVtBt3fvXmw2Gx999BHlypVj586d9OjRg8TERCZNmmSvy8otXEm6wPoVT9N951ouGe50eGgi9SOfoaKbt9XRRMQJ+Pj4UKBAAatjiDic2NhYkpKS8PLywtfX15IMubr118SJE5k+fTqHDx++4fHk5GSSk5Mz7sfFxREREaGtv3LAskPLGPxdN1YUPsV3gW14uMWnFC5Q2OpYIuIErly5wp49e4iMjFRBJ/IP1/5vbN26lcTERAICAnjyySctKepytY8tNjaWoKCgmx4fO3YsgYGBGbeIiIhcTJdHXdwKqx8ixMuPSsUbEX//dp5r972KORERkRzi6+uLp6cncXFxmRqmclOuFXSHDh1i6tSpvPDCCzc9Z/jw4cTGxmbcjh8/nlvx8p70/35DeQRAWiI1CxXn6w5fUya0mrW5RERE8hgvLy+8vLwszZDtgm7kyJG4uLjc8rZ58+ZMzzl58iStW7emQ4cOPP/88zd9bS8vLwICAjLd5A7smQTL6oMtHfzLQYtV4FvS6lQiIiJiJ9meFNG3b186dux4y3NKlSqV8fXJkydp1qwZ9evXZ8aMGdkOKFmUGg8pl8E3AkLuBVdvINeGR4qIiIiFsl3QBQcHExwcnKVz//rrL5o1a0atWrWYPXs2rq5aFsNuVj8IngWhyWIIvse8iYiISL5gt2VLTp48SdOmTSlRogSTJk3i3LlzGceKFi1qr8vmH4YBfy2GoFpQIBzufge89fcqIiKSH9mtoFu2bBkHDx7k4MGDhIeHZzqWiyul5F1pibChB1QeCpGDoXAdqxOJiIiIRezWB9qtWzcMw7jhTe7Qlb/MIi41Djz84IHfoZK26hIRsaemTZvy0ksvWR3jpnIz3549e3LlOpJ9GtTmDDKKYAPOrIa4/ebdAsXBxcWqVCIiko+89957li/Nce7cudvuNnXy5MlcSuNYVNA5uuML/7sESao5Vq7tPihc2+pUIiJ5SkpKitURLNO0aVO++eabW56zbNkyQkJCKFOmTC6lurGQkBAaN25MVFTUTc9ZvXo1PXv2ZNWqVVl+3U2bNvHSSy8RHR1Nz549OXLkSE7EzVUq6ByRYYOk8+bXfmWhUE1ITzLvu+ifTEScV6lSpXjvvfcyPVajRg1GjhwJmMVF37596du3LwULFqRw4cK89tprmYbrZOUcwzCYMGECZcqUwcfHh+rVqzN//vzrXmPQoEEEBwfTsmXLW+ZOS0u76fWSk5Pp378/oaGheHt706hRIzZt2pSt930tU//+/Rk6dChBQUEULVo003GAxMREunTpgp+fH2FhYbzzzju3zH07ixYtws3NjWHDhpGamnrT895///3bLlmWW+rWrcvGjRs5f/78DY936tSJDz74gOPHj/P888/z448/3vL1kpOT6dChA8OHD6dr1648++yzPPvss/aIbleqDhzRhu6w5mGzq7VQNag7HTz8rU4lIpLJqfhTJKWZv2weuXSErae2Zrqdij91R68bHR2Nu7s7GzZs4P3332fy5MnMnDkzW+e89tprzJ49m+nTp7Nr1y4GDhxI586diYmJue41fv31Vz766KM7zjR06FAWLFhAdHQ0W7dupVy5ctx///1cvHjxjt67r68vGzZsYMKECYwaNYrly5dnHB8yZAirVq1i4cKFLFu2jNWrV7Nly5aM43PmzMEli0Nx0tLS2LJlC19//TVnz5696d/Bjh07CA8Px83NLdvvx17atWvHnDlzbnrc3d2dLl26MGPGDOLi4njuuedYuHDhDcfxr1mzhoCAAIoUKQKYBeP69es5c+aMveLbh+HAYmNjDcCIjY21Oor9Xd5tGAlHza/PrjOMMzHW5hER+a/ExERj8+bNRmJiYqbHR6waYRy4cMAwDMN4esHTBiPJdBuxasR1r1WyZElj8uTJmR6rXr26MWKEeW6TJk2MyMhIw2azZRx/5ZVXjMjIyIz7tzsnISHB8Pb2NtatW5fpOt27dzeeeuqpjNeoUaNGlt7/ra6XkJBgeHh4GHPnzs04lpKSYhQrVsyYMGFClt/3tes0atQo0zl16tQxXnnlFcMwDCM+Pt7w9PQ05s2bl3H8woULho+PjzFgwADDMAzjm2++MSpWrJil9zVz5kzjwAHz32/cuHFGSEjIDX/evvvuu8bMmTMz7qelpRnTpk0zOnfubGzcuNEwDMPo1KmTER0dnaXr3kpaWpoxdepUo2vXrhmvffz4caNJkyaZzjt37pzRrFmzLL+uzWYzFi1aZDz99NPGzz//nOnYjBkzrnv94OBgY8WKFbd93Wv/N+bPn2/MnDnTmDx5snHhwoUs58pJaqFzBLZ0WN0Gdo8374fUh9DG1mYSEbmNXrV6ER5gLks1utlotvTckunWq1avO3rdevXqZWplql+/PgcOHCA9PT1L5+zevZukpCRatmyJn59fxu3TTz/l0KFDGc+pXTvr45Fvdr2DBw+SmppKw4YNM455eHhQt27dO5oRWq1a5v22w8LCOHv2LGDuiZ6SkkL9+vUzjgcFBVGxYsWM++3bt2fv3r23vU5CQgLnzp2jXLlyAAwYMIACBQowfvz46849ceIEoaGhGfe//fZbnnzySa5cucLRo0cBaNu2LXFxcVl/ozexePFiOnbsyNWrVzPGsS1fvvy65c+Cg4Mz/Vvezvnz59m6dSuFCxemdOnS1x3z9vbO9Ji3tzeXL1++szdhEbutQ/dvREVFERUVlek/b56Tlgj7pkDZnuAdDE2WmPuuiog4iTD/sIyvSxcqTWlK3+Jsk6ur63XdXrcau3UnbDYbAN9//z3FixfPdOyfszR9fX1z7Jr/281pGEamx7L6vj08PK573Wvv53+f/298/PHH9OjRI+O+t7c3Y8aMoWfPnrz44ouZ/t4SEhLw8fHJuN+8eXNsNhu//PILn3/+OQBly5a94YSJKVOm3HKCQZMmTWjfvn3G/fvuuw+AlStXMmvWLMCc5HCjMY5Z6Vo+efIkU6dOJTU1lX79+lGy5PX7mgcGBl73d5uQkJDlXbEchUMWdH369KFPnz7ExcURGBhodRz7SLsCe9+DwLsg/GEoeJfViURE7C4kJIRTp/4eWxcXF3fdD/z169dfd798+fKZxnDd6pzKlSvj5eXFsWPHaNKkSY7kvtn1ypUrh6enJ2vXrqVTp06AWaht3rw509pwWXnft1OuXDk8PDxYv349JUqUAODSpUvs378/W+/z9OnTeHt7U7hw4UyPd+rUicmTJ/PGG2/wySefZDweHBzMpUuXMu4XLFiQzz77jKZNm2YUen/88QfdunW77loDBgzIzlskICCAefPm0bhx44yCOyYmhrfffpvLly9TsGDBjHPd3W9ewhw9epSoqCg8PT3p378/YWFhNz23UqVKfPzxxxn3k5OTiY+Pv2Hx58jU5ZqbLmyCVQ9A2lXwDoGHj5rFnIhIPtG8eXM+++wzfvnlF3bu3EnXrl2vG2x//PhxBg0axL59+/jyyy+ZOnXqdYXBrc7x9/dn8ODBDBw4kOjoaA4dOsTvv/9OVFQU0dHRd5T7Ztfz9fWld+/eDBkyhKVLl7J792569OjBlStX6N69e7be9+34+fnRvXt3hgwZwooVK9i5cyfdunXLtE/6woULqVSp0i1fZ/r06bRr147z589nul24cIFhw4YRHR3Nzp07M86PjIzk2LFjmV7jzJkzmYpKPz+/HNuv/fjx4xldwTt37sTDw4PQ0NBMs5RTU1Px97/xZMFJkybx6aefMmzYMMaMGXPLYg6gcePGnD17lhMnTgBmAVmnTh2nK+gcsoUuz0lPAjdv8CwERhqkXAD3cHAvYHUyEZFcNXz4cA4fPsxDDz1EYGAgo0ePvq6lqkuXLly9epW6devi5uZGv3796NmzZ7bOGT16NKGhoYwdO5bDhw9TsGBB7r77bl599dU7yn2r640bNw6bzcYzzzxDfHw8tWvX5qeffqJQoULZet9ZMXHiRBISEmjXrh3+/v68/PLLxMbGZhyPjY1l3759N33+/v37GTNmDKNGjbrldV599VUWL14MQOvWrenRowcvv/z3zkQdO3Zk0KBBfPbZZ6SlpdG1a9dsv5ebeeyxx3jllVf4+uuvAahVqxbTpk3L1AK4adMmmjdvfsPnDx48OFvXc3d3Jzo6mjFjxlCvXj1iYmKYO3fuHee3iouRk53yOexal2tsbCwBAQFWx7kzu8bBsf+D1pu0hpyIOKUrV66wZ88eIiMjKVDAvr+INm3alBo1aly3Zlt2z5Gc1bNnT0aNGkXRokWtjgKYy9K0a9eOunXrWprj2v+No0ePcvnyZeLj4+nSpQtBQUG5nkUtdPaQEmvut+obAUWagWdBc0057dIlIiJO6I033mDatGm89dZbVkchLi6Oc+fO3bSYmzx58i1nwDZr1ozHHnvMXvEso4LOHlY/AN5FoPFCCL7HvImIiDip8PBw2rdvzw8//ECbNm0sy2EYBpMnT75lYTlw4MBcTOQ4VNDlBMOA499AcD0oUBxqTQGf4rd/noiIZLJ69eocOUdyXq1atayOwIULF+jduzchISFWR3E4GtSVE9ISYNMLcMwcwEnhOlCgmLWZRERE8pjg4OBMixzL31TQ3anE47D+OXOsnIc/tNkBlfJnM6+IiIhYSwVddl2bFOziAud+hfj/Drz0ufU6NyIiIiL2ooIuO479H/x0D9hSoUA4PLQHgmpanUpEROyoadOmmXZ9cDS5me9O9qeV3KGC7nYMGySdM7/2r2BOfEhPNu9rXTkREckn3nvvvUx74d7MuXPnmDRp0i3POXnyZE7Fkv9SRXI7v3WFXx41u1oLVYfa74OHn9WpREQkB6WkpFgdwTJNmzblm2++ueU5y5YtIyQkhDJlytz29UJCQmjcuDFRUVE3PWf16tX07NmTVatWZTnnpk2beOmll4iOjqZnz553tNNGXuaQBV1UVBSVK1emTp061gS4/Ack/ml+Xf5FqDHeHDMnIiL/SqlSpa7b4aFGjRqMHDkSMIuLvn370rdvXwoWLEjhwoV57bXX+OemRlk5xzAMJkyYQJkyZfDx8aF69eqZ9gK99hqDBg0iODiYli1b3jJ3WlraTa+XnJxM//79CQ0Nxdvbm0aNGrFp06Zsve9rmfr378/QoUMJCgqiaNGimY4DJCYm0qVLF/z8/AgLC+Odd965Ze7bWbRoEW5ubgwbNozU1NSbnvf+++/TsWPHLL9u3bp12bhxI+fPn7/h8U6dOvHBBx9w/Phxnn/+eX788cdbvl5ycjIdOnRg+PDhdO3alWeffZZnn302y3nyA4cs6Pr06cPu3buv+w+RK2xpENMW9vz3P0lIfQhpkPs5REQc3KlTkJRkfn3kCGzdmvl26tSdvW50dDTu7u5s2LCB999/n8mTJzNz5sxsnfPaa68xe/Zspk+fzq5duxg4cCCdO3cmJibmutf49ddf+eijj+4409ChQ1mwYAHR0dFs3bqVcuXKcf/993Px4sU7eu++vr5s2LCBCRMmMGrUKJYvX55xfMiQIaxatYqFCxeybNkyVq9ezZYtWzKOz5kzB5csNkCkpaWxZcsWvv76a86ePXvTv4MdO3YQHh6Om5tbtt5Lu3btmDNnzk2Pu7u706VLF2bMmEFcXBzPPfccCxcu5EY7kq5Zs4aAgACKFCkCmAXj+vXrOXPmTLYy5WmGA4uNjTUAIzY21r4XSok3jB1vGkbSefP+5d2GkZZs32uKiDiJxMREY/PmzUZiYmKmx0eMMIwDB8yvn37aMMyxKX/fRoy4/rVKlixpTJ48OdNj1atXN0b89+QmTZoYkZGRhs1myzj+yiuvGJGRkRn3b3dOQkKC4e3tbaxbty7Tdbp372489dRTGa9Ro0aNLL3/W10vISHB8PDwMObOnZtxLCUlxShWrJgxYcKELL/va9dp1KhRpnPq1KljvPLKK4ZhGEZ8fLzh6elpzJs3L+P4hQsXDB8fH2PAgAGGYRjGN998Y1SsWDFL72vmzJnGgf/+A44bN84ICQm54c/bd99915g5c2bG/bS0NGPq1KlG165djY0bNxqGYRjHjx83mjRpkul5586dM5o1a5alLIZhGDabzVi0aJHx9NNPGz///HOmYzNmzLju9YODg40VK1Zk+fXt4dr/jfnz5xszZ840Jk+ebFy4cMGSLA7ZQpfr0q/CgSg4t868HxgJbp7WZhIRcXC9ekF4uPn16NGwZUvmW69ed/a69erVy9TKVL9+fQ4cOEB6enqWztm9ezdJSUm0bNkSPz+/jNunn36aaY/P2rVr/+tMBw8eJDU1lYYNG2Yc8/DwoG7dunc0I7RatWqZ7oeFhXH27FkADh06REpKCvXr1884HhQURMWKFTPut2/fnr179972OgkJCZw7d45y5coBMGDAAAoUKMD48eOvO/fEiROZFvNdvHgxHTt25OrVqxnj2JYvX074tW+G/woODr7lnqr/6/z582zdupXChQtTunTp6455e3tneszb25vLly9n+fXzuvy79df5DbDjDWjyLXiHQLuj4O5jdSoREacR9o/lN0uXNm+34+rqel2X2q3Gbt0Jm80GwPfff0/x4pm3YfznLE1fX98cu+b/dnMahpHpsay+bw8Pj+te99r7+d/n/xsff/wxPXr0yLjv7e3NmDFj6NmzJy+++GKmv7eEhAR8fP7++XjfffcBsHLlSmbNmgWYkxxuNA4xK92/J0+eZOrUqaSmptKvXz9Klix53TmBgYHXvf+EhASCg4Nv+/r5Rf5roUu7av7pWcic6JB8wbyvYk5ExO5CQkI49Y/BdXFxcdfNVly/fv1198uXL59pDNetzqlcuTJeXl4cO3aMcuXKZbpFRETcUe6bXa9cuXJ4enqydu3ajGOpqals3ryZyMjIbL3v2ylXrhweHh6Zsly6dIn9+/dn63VOnz6Nt7c3hQsXzvR4p06diIyM5I033sj0eHBwMJcuXcq4HxAQwA8//EDjxo0ziuKYmBhatGhxXYuZu/vN242OHj3KkCFDiIqKon///kyaNOmGxRxApUqVMloqwZwkER8ff9Pz86P8VdDtfAuWNzLXlguoAM2WQoHit3+eiIjkiObNm/PZZ5/xyy+/sHPnTrp27XrdYPvjx48zaNAg9u3bx5dffsnUqVMZMGBAls/x9/dn8ODBDBw4kOjoaA4dOsTvv/9OVFQU0dHRd5T7Ztfz9fWld+/eDBkyhKVLl7J792569OjBlStX6N69e7be9+34+fnRvXt3hgwZwooVK9i5cyfdunXD1fXvH+ULFy6kUqVKt3yd6dOn065dO86fP5/pduHCBYYNG0Z0dDQ7d+7MOD8yMpJjx45d9/dxrbt2586deHh4EBoammkmcWpqKv7+/jfMMGnSJD799FOGDRvGmDFjCAu79W5LjRs35uzZs5w4cQIwC8g6deqooPuHvN/lmnwR0hLAtwQUbWlu0WUYoFVIRERy3fDhwzl8+DAPPfQQgYGBjB49+rqWqi5dunD16lXq1q2Lm5sb/fr1o2fPntk6Z/To0YSGhjJ27FgOHz5MwYIFufvuu3n11VfvKPetrjdu3DhsNhvPPPMM8fHx1K5dm59++olChQpl631nxcSJE0lISKBdu3b4+/vz8ssvExsbm3E8NjaWffv23fT5+/fvZ8yYMYwaNeqW13n11VdZvHgxAK1bt6ZHjx68/PLLGccfe+wxXnnlFb7++msAatWqxbRp0+jWrVvGOZs2baJ58+Y3fP3Bgwff9r3+k7u7O9HR0YwZM4Z69eoRExPD3Llzs/UaeZ2LkZOd8jksLi6OwMBAYmNjCQgIuLMX+ekec5uuexfkbDgRkXziypUr7Nmzh8jISAoUKGDXazVt2pQaNWpct2Zbds+RnNWzZ09GjRpF0aJFs/yc1157jXbt2lG3bl07JrPWtf8bR48e5fLly8THx9OlSxeCgoJyPUvea6EzbPDn1xDaGAoUg9pRUODOxkyIiIgIvPHGG0ybNo233norS+fHxcVx7ty5mxZzkydPvuUM2GbNmvHYY4/dUdb8Ku8VdGkJsKUfVHkDKvaDwlmfli4iIiLXCw8Pp3379vzwww+0adPmlucahsHkyZNvWfwNHDgwpyPme3mjoEv801yCpPY08AiANjvBp4jVqUREJJtWr16dI+dIzqtVq1aWzrtw4QK9e/cmJCTEzonkn5x7lqth++8XrnBhEyQcNu+qmBMREbFEcHBwpoWIJXc4ZEEXFRVF5cqVqVOnzs1POjrPnPBgSwXfCHhwFxSqnnshRURERByEQxZ0ffr0Yffu3WzatCnzAVs6XP3vRryBlcyJD7YU834WNyMWERERyWucawzduqch6RS0iIFCNcybiIiI2N2WLVtITk7G19eX6tXVI+ZoHLKF7jqJ/12huuIAqDHR2iwiIiL5zMKFC6lcuTK1a9dm8uTJVseRG3Dsgs6WZv65/wPzz5D6EJx3FygUEcnrmjZtyksvvWR1jNty9Jy5me+PP/7g7rvvZtmyZTzzzDPX7dpxK3v27LFjMvknxy7oXP/bI1x9tLU5RERE8qgrV66QmJh40+MnT54kPDyctLQ0zpw5g7t71kZrvffee3h5eWXp3HPnzjFp0qQsnSs35tgF3TWuHlYnEBGRPCglJcXqCJb6/vvvufvuu5k48ebDmQzDwM3Njccee4x27dpx+fLl277usmXLCAkJoUyZMlnKERISQuPGjYmKispq9CzZtGkTL730EtHR0fTs2fOO9s91Fs5R0ImISJ6TnJxM//79CQ0Nxdvbm0aNGmVa3eC7776jYMGC2GzmmqPbtm3DxcWFIUOGZJzTq1cvnnrqKcAsPCZMmECZMmXw8fGhevXqzJ8/P9M1mzZtSt++fRk0aBDBwcG0bNnypvnS0tLo27cvBQsWpHDhwrz22mv8c/vz2+UvVarUdfvN1qhRg5EjR2bK079/f4YOHUpQUBBFixbNdBwgMTGRLl264OfnR1hYGO+8886t/2Kz4cEHH+TJJ5+86fGzZ8+yd+9eAK5evcrevXtp1qwZJ06cYOnSpRm3bdu2ZXre+++/T8eOHbOVpW7dumzcuJHz589n+33cSHJyMh06dGD48OF07dqVZ599lmeffTZHXtsRqaATERFLDB06lAULFhAdHc3WrVspV64c999/PxcvXgSgcePGxMfH8/vvvwMQExNDcHAwMTExGa+xevVqmjRpApibwc+ePZvp06eza9cuBg4cSOfOnTOdDxAdHY27uzu//vorH3300U3zXTtvw4YNvP/++0yePJmZM2dmOX9WRUdH4+vry4YNG5gwYQKjRo1i+fLlGceHDBnCqlWrWLhwIcuWLWP16tVs2bIl02vMmTMHlztcvutWz9uyZQv3338/v/76KytWrGDKlCl4eHgQHh5O69atM241atTIeM6OHTsIDw/Hzc0t21natWvHnDlzbnhsypQpjBo1itOnT2fptdasWUNAQABFipibDdStW5f169dz5syZbOdyBiroRETkzlw9BelJ5teJxyF299/HLv9hHr+JxMREpk+fzsSJE3nggQeoXLkyH3/8MT4+PnzyyScABAYGUqNGjYytvlavXs3AgQPZvn078fHxnD59mv3799O0aVMSExN59913mTVrFvfffz9lypShW7dudO7c+bqirVy5ckyYMIGKFStSqVKlm2aMiIhg8uTJVKxYkaeffpp+/fplzPDMSv6sqlatGiNGjKB8+fJ06dKF2rVrs2LFCgASEhL45JNPmDRpEi1btqRq1apER0eTnp6e6TUCAwOpWLHiLa+zbt06+vfvz8cff8ywYcNYu3ZtxrHExESio6OJjo6mffv2GWPqXFxciIyMpGHDhjz00EP4+Pjc9v2sWLHiuo0B0tPTmTZtGt26dctoxTxx4gRNmzbNdF6TJk344Ycfbvi6AwYMoFevXnz00UcMHTqUffv23TLH0aNHCQoKyrjv5uaGv78/u3btuu17cEYq6ERE5M4c+AiunDC/3jMRfv1HF9uqB8zjN3Ho0CFSU1Np2LBhxmMeHh7UrVs308zIpk2bsnr1agzD4JdffuHhhx+mSpUqrF27llWrVlGkSBEqVarE7t27SUpKomXLlvj5+WXcPv30Uw4dOpTp2rVr187S26tXr16m1qv69etz4MAB0tPTs5w/K6pVq5bpflhYGGfPngXMv6eUlBTq16+fcTwoKOi64q19+/YZXaM3curUKR577DFeffVVevTogbe3d6ZWvg0bNtC1a1e6du1Keno6P/30EwDNmjXL1nsBs1D7362/Fi9eTMeOHbl69WrGOLbly5cTHh6e6bzg4ODr/r3+qUiRIowYMYLXX3+dH374gX79+vHbb7/d8Nzz58/j7e2d6TFvb+8sjQF0Rs61sLCIiDiO8r3As5D5deQQKP/C38ea/QhewTd96rWxaP/b3WcYRqbHmjZtyieffML27dtxdXWlcuXKNGnShJiYGC5dupTR3XptnN33339P8eLFM73m/8609PX1zd77vMP8rq6umcbcAaSmpl73Wh4emSf+ubi4ZLyf/33+nVqwYAElSpSgaNGiANeN06tVq1bG10FBQcTFxQHX/91lRUJCwnUteffddx8AK1euZNasWYDZ4nqjMYxZ6Tr29/dn4MCBpKSkMHfuXGbNmkXXrl1p1KhRxjmBgYHX/f0lJCQQHHzz70tnphY6ERG5Mz5h4PbfFhDfCAis/PexglXN4zdRrlw5PD09M3X7paamsnnzZiIjIzMeuzaO7r333qNJkya4uLjQpEkTVq9enWn8XOXKlfHy8uLYsWOUK1cu0y0iIuKO3t769euvu1++fHnc3NyylD8kJIRTp/7udo6Li8v2LMty5crh4eGRKculS5fYv39/tl7HZrPdsji8k/FuNxMcHMylS5cyPRYQEMAPP/xA48aNMwrqmJgYWrRocV2LWVaXRQHz7/Svv/7Cz8/vulbBSpUqZbR0gjlJIj4+npIlS2bzHTkHtdCJiEiu8/X1pXfv3gwZMoSgoCBKlCjBhAkTuHLlCt27d88479o4us8//5wpU6YAZpHXoUMHUlNTM8Zg+fv7M3jwYAYOHIjNZqNRo0bExcWxbt06/Pz86Nq1a7YzHj9+nEGDBtGrVy+2bt3K1KlTM2aYZiV/8+bNmTNnDm3btqVQoUK8/vrr2S6c/Pz86N69O0OGDKFw4cIUKVKE//znP7i6Zm6PWbhwIcOHD79pt2v79u0ZPXo0x44do0SJEgDMmzfvtjNRz549y7Jly3B1dWXLli2MHz/+tgVXZGQkx44du+7x48ePU65cOQB27tyJh4cHoaGhREdH8/zzzwNmUezv73/L1wc4cuQIM2bMwNXVlRdeeOGGRXvjxo05e/YsJ06cIDw8nJiYGOrUqaOCLjdFRUURFRV13aBPERHJO8aNG4fNZuOZZ54hPj6e2rVr89NPP1GoUKFM5zVr1oytW7dmFG+FChWicuXKnDx5MlNr3ujRowkNDWXs2LEcPnyYggULcvfdd/Pqq6/eUb4uXbpw9epV6tati5ubG/369cu0S8Lt8g8fPpzDhw/z0EMPERgYyOjRo+9oHbSJEyeSkJBAu3bt8Pf35+WXXyY2NjbTObGxsbecJBAREcGCBQt47bXXaNSoETabjTZt2rB06VKWLFlCamoqS5YsIT4+nl9//ZUTJ05QqVIljh49yuXLl+nbty/r16/n559/pnXr1rfM27p1a3r06MHLL7+c6fHHHnuMV155ha+//howu3mvTZS4ZtOmTTRv3vymr/37778THR1N0aJFGTp06HXfK//k7u5OdHQ0Y8aMoV69esTExDB37txbZndmLkZOddDbQVxcHIGBgcTGxhIQEGB1HBGRfOnKlSvs2bOHyMhIChQoYHUcscgTTzzBhAkTKFWq1G3P7dmzJ6NGjcoYs5dVr732Gu3ataNu3eu3+ZwyZQre3t507dr1uskOVrn2f+Na4RsfH0+XLl0yza7NLRpDJyIiIrf0/fff06FDhywVcwBvvPEG06ZNy9Y14uLiOHfu3A2LOfh72RJHKeYcjQo6ERERualNmzZRpEgROnTowO7du2//BCA8PJz27dvfdE25/2UYBpMnT+att976N1HzNYccQyciIiLW++233+jVqxehoaGkpKQwffr0LD/3n0uh3M6FCxfo3bs3ISEhdxJTUEEnIiIiN1G/fn127Nhh9+vk1bXhcpO6XEVEREScnAo6ERERESengk5ERETEyamgExEREXFydi3o2rVrR4kSJfD29iYsLIxnnnmGkydP2vOSIiIiIvmOXQu6Zs2a8fXXX7Nv3z4WLFjAoUOHePzxx+15SREREZF8x67LlgwcODDj65IlSzJs2DAeeeQRUlNT8fDwsOelRURERPKNXFuH7uLFi8ydO5cGDRrctJhLTk4mOTk5435cXFxuxRMRERFxWnYv6F555RWmTZvGlStXqFevHkuWLLnpuWPHjuXNN9+0dyQREbkDV69etTqCiENxpP8TLoZhGNl5wsiRI29bdG3atInatWsDcP78eS5evMiff/7Jm2++SWBgIEuWLMHFxeW6592ohS4iIoLY2FgCAgKyE1NERHJISkoKu3btwmazWR1FxOEYhsHhw4eJj48nPj6eLl26EBQUlOs5sl3QnT9/nvPnz9/ynFKlSuHt7X3d4ydOnCAiIoJ169ZRv379214rLi6OwMBAFXQiIhZLSUkhLS2N2NhYlixZgq+vL15eXlbHErFceno6aWlpXL161dKCLttdrsHBwXe859q12vGfrXAiIuL4PD098fT0JCkpicTERFJTU1XQifyD1bWN3cbQbdy4kY0bN9KoUSMKFSrE4cOHeeONNyhbtmyWWudERMTxeHl5ERAQQFxcHCkpKVbHEXEoAQEBlv2iY7eCzsfHh2+++YYRI0aQmJhIWFgYrVu3Zt68efqtTkTESfn6+vLkk09a3hoh4oi8vLzw9fW15NrZHkOXmzSGTkREROT2tJeriIiIiJNTQSciIiLi5By6y9UwDOLj4/H397/hunUiIiIi4uAFnYiIiIjcnrpcRURERJycCjoRERERJ6eCTkRERMTJqaATERERcXIq6EREREScnAo6ERERESengk5ERETEyf0/P2MAacDvi+4AAAAASUVORK5CYII=\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 27,
"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": 28,
"id": "25e4850b",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"Graphics object consisting of 3 graphics primitives"
]
},
"execution_count": 28,
"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 $r=\\operatorname{ch}_0(u)$"
]