Newer
Older
{
"cells": [
{
"cell_type": "markdown",
"id": "8ebd0216",
"metadata": {},
"source": [
"# Utilities"
]
},
{
"cell_type": "markdown",
"id": "b22eb2a9",
"metadata": {},
"source": [
"Define \\chern command in latex $\\newcommand{\\chern}{\\operatorname{ch}}$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"id": "b87a49bc",
"metadata": {},
"outputs": [],
"source": [
"# Requires extra package:\n",
"#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
"%display latex\n",
"\n",
"from pseudowalls import *\n",
"\n",
"Δ = lambda v: v.Q_tilt()\n",
"alpha = stability.Tilt().alpha\n",
"beta = stability.Tilt().beta\n",
"\n",
"def beta_minus(v):\n",
" solutions = solve(\n",
" stability.Tilt(alpha=0).degree(v)==0,\n",
" beta)\n",
" return min(map(lambda s: s.rhs(), solutions))\n",
"\n",
"class Object(object):\n",
" pass"
]
},
{
"cell_type": "markdown",
"id": "6d374a2a",
"metadata": {},
"source": [
"Fix a Chern character $v$ with positive rank and $\\Delta(v) \\geq 0$"
]
},
{
"cell_type": "code",
"id": "ab162897",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
],
"text/plain": [
"<pseudowalls.chern_character.Chern_Char object at 0x7fddcf532150>"
"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 0x7fddc57d52d0>"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u = Chern_Char(*var(\"r c d\", domain=\"real\"))\n",
"u"
]
},
{
"cell_type": "markdown",
"id": "cb9c11e7",
"metadata": {},
"source": [
"# Bounds on $\\operatorname{ch}_2(u)=d$"
]
},
{
"cell_type": "code",
"id": "23d48b0b",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle q\\)</html>"
],
"text/latex": [
"$\\displaystyle q$"
],
"text/plain": [
"q"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"var(\"q\", domain=\"real\") # Symbol for q=\\chern_1^{\\beta}(u)"
]
},
{
"cell_type": "markdown",
"id": "377c2843",
"metadata": {},
"source": [
"Express $c$ in terms of $q:=\\chern_1^{\\beta}(u)$"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [],
"source": [
"c_in_terms_of_q = solve(q == u.twist(beta).ch[1], c)[0]\n",
"assert c_in_terms_of_q.lhs() == c, \"Meant to be an expression for c\""
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle c = \\beta r + q\\)</html>"
"$\\displaystyle c = \\beta r + q$"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"c_in_terms_of_q"
]
},
{
"cell_type": "markdown",
"id": "5f6f18b1",
"metadata": {},
"source": [
"## $\\chern_2^{P}(u) > 0$"
]
},
{
"cell_type": "markdown",
"id": "8e635cb8",
"metadata": {},
"source": [
"For problem 2, this amounts to $\\chern_2^{\\beta}(u) > 0$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"id": "1a169293",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d > \\frac{1}{2} \\, \\beta^{2} r + \\beta q\\)</html>"
],
"text/latex": [
"$\\displaystyle d > \\frac{1}{2} \\, \\beta^{2} r + \\beta q$"
],
"text/plain": [
"d > 1/2*beta^2*r + beta*q"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"positive_radius_condition_with_q = (\n",
" (\n",
" (0 > - u.twist(beta).ch[2])\n",
" + d # rearrange for d\n",
" )\n",
" .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q\n",
" .expand()\n",
")\n",
"positive_radius_d_lowerbound = positive_radius_condition_with_q.rhs()\n",
"\n",
"positive_radius_condition_with_q"
]
},
{
"cell_type": "markdown",
"id": "b08c74f1",
"metadata": {},
"source": [
"Separate out the terms of the corresponding lower bound on $d$:"
]
},
{
"cell_type": "code",
"execution_count": 8,
"id": "8fe70d6b",
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, 0\\right)\\)</html>"
],
"text/latex": [
"$\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, 0\\right)$"
],
"text/plain": [
"(1/2*beta^2*r, beta*q, 0)"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"positive_radius_lowerbound_terms = Object()\n",
"\n",
"positive_radius_lowerbound_terms.const = positive_radius_condition_with_q.rhs().subs(r==0)\n",
"\n",
"positive_radius_lowerbound_terms.linear = (\n",
" positive_radius_condition_with_q.rhs()\n",
" - positive_radius_lowerbound_terms.const\n",
")\n",
"\n",
"positive_radius_lowerbound_terms.hyperbolic = 0\n",
"\n",
"(positive_radius_lowerbound_terms.linear,\n",
" positive_radius_lowerbound_terms.const,\n",
" positive_radius_lowerbound_terms.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "900f332b",
"metadata": {},
"source": [
"## $\\Delta(u) \\geq 0$"
]
},
{
"cell_type": "markdown",
"id": "62529298",
"metadata": {},
"source": [
"Express this inequality in terms of $q$"
]
},
{
"cell_type": "code",
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\leq {\\left(\\beta r + q\\right)}^{2} - 2 \\, d r\\)</html>"
],
"text/latex": [
"$\\displaystyle 0 \\leq {\\left(\\beta r + q\\right)}^{2} - 2 \\, d r$"
],
"text/plain": [
"0 <= (beta*r + q)^2 - 2*d*r"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"bgmlv2_with_q = ((0 <= Δ(u))\n",
" .subs(c_in_terms_of_q))\n",
"bgmlv2_with_q"
]
},
{
"cell_type": "markdown",
"id": "723511fa",
"metadata": {},
"source": [
"Rearrange expression for $d$"
]
},
{
"cell_type": "code",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}\\)</html>"
"$\\displaystyle d \\leq \\frac{{\\left(\\beta r + q\\right)}^{2}}{2 \\, r}$"
"d <= 1/2*(beta*r + q)^2/r"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_ineq = (bgmlv2_with_q\n",
" + 2*d*r # move d to rhs\n",
") / (2*r) # scale-out d coefficient (r>0)\n",
"\n",
"assert bgmlv2_d_ineq.lhs() == d, \"Should be ineq for d\"\n",
"\n",
"bgmlv2_d_ineq"
]
},
{
"cell_type": "markdown",
"id": "425bcb7c",
"metadata": {},
"source": [
"Keep hold of the upper bound for $d$:"
]
},
{
"cell_type": "code",
"id": "6ae4f2e7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}$"
],
"text/plain": [
"1/2*beta^2*r + beta*q + 1/2*q^2/r"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs().expand()\n",
"bgmlv2_d_upperbound"
]
},
{
"cell_type": "markdown",
"id": "988aaf5b",
"metadata": {},
"source": [
"Separate out the terms of this lower bound for d"
]
},
{
"cell_type": "code",
"id": "653a3340",
"metadata": {},
"outputs": [],
"source": [
"bgmlv2_d_upperbound_terms = Object()\n",
"\n",
"bgmlv2_d_upperbound_without_hyp = (\n",
" bgmlv2_d_upperbound\n",
" .subs(1/r == 0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_terms.const = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" .subs(r==0)\n",
")\n",
"\n",
"bgmlv2_d_upperbound_terms.linear = (\n",
" bgmlv2_d_upperbound_without_hyp\n",
" - bgmlv2_d_upperbound_terms.const\n",
").expand()\n",
"\n",
"bgmlv2_d_upperbound_terms.hyperbolic = (\n",
" bgmlv2_d_upperbound\n",
" - bgmlv2_d_upperbound_without_hyp\n",
").expand()"
]
},
{
"cell_type": "code",
"id": "326bb656",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, \\frac{q^{2}}{2 \\, r}\\right)\\)</html>"
"$\\displaystyle \\left(\\frac{1}{2} \\, \\beta^{2} r, \\beta q, \\frac{q^{2}}{2 \\, r}\\right)$"
"(1/2*beta^2*r, beta*q, 1/2*q^2/r)"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(bgmlv2_d_upperbound_terms.linear,\n",
" bgmlv2_d_upperbound_terms.const,\n",
" bgmlv2_d_upperbound_terms.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "5cc08c62",
"metadata": {},
"source": [
"Sanity check:"
]
},
{
"cell_type": "code",
"id": "7ff937ff",
"metadata": {},
"outputs": [],
"source": [
"assert ( bgmlv2_d_upperbound\n",
"- bgmlv2_d_upperbound_terms.const\n",
"- bgmlv2_d_upperbound_terms.linear\n",
"- bgmlv2_d_upperbound_terms.hyperbolic) == 0, \"Error in terms separation\""
]
},
{
"cell_type": "markdown",
"id": "024e8c41",
"metadata": {},
"source": [
"## $\\Delta(v-u) \\geq 0$"
]
},
{
"cell_type": "markdown",
"id": "a2647f43",
"metadata": {},
"source": [
"Express this inequality in terms of $q$"
]
},
{
"cell_type": "code",
"id": "87544e6e",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle 0 \\leq {\\left(\\beta r - C + q\\right)}^{2} - 2 \\, {\\left(D - d\\right)} {\\left(R - r\\right)}\\)</html>"
],
"text/latex": [
"$\\displaystyle 0 \\leq {\\left(\\beta r - C + q\\right)}^{2} - 2 \\, {\\left(D - d\\right)} {\\left(R - r\\right)}$"
],
"text/plain": [
"0 <= (beta*r - C + q)^2 - 2*(D - d)*(R - r)"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_with_q = ((0 <= Δ(v-u))\n",
" .subs(c_in_terms_of_q)\n",
")\n",
"\n",
"bgmlv3_with_q"
]
},
{
"cell_type": "markdown",
"id": "d36504bb",
"metadata": {},
"source": [
"Rearrange in terms of $d$ assuming $r>R$"
]
},
{
"cell_type": "code",
"metadata": {
"scrolled": false
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d \\leq D - \\frac{{\\left(\\beta r - C + q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
],
"text/latex": [
"$\\displaystyle d \\leq D - \\frac{{\\left(\\beta r - C + q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
],
"text/plain": [
"d <= D - 1/2*(beta*r - C + q)^2/(R - r)"
]
},
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_ineq = (\n",
" (\n",
" bgmlv3_with_q\n",
" + 2*(D-d)*(R-r) # move d term to lhs\n",
" )/2/(r-R) # assume r>R\n",
") + D\n",
"\n",
"assert bgmlv3_d_ineq.lhs() == d, \"Should be bound for d\"\n",
"assert not bgmlv3_d_ineq.rhs().has(d), \"Should be bound for d\"\n",
"\n",
"bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs()\n",
"bgmlv3_d_ineq"
]
},
{
"cell_type": "markdown",
"id": "7bdaf020",
"metadata": {},
"source": [
"$\\renewcommand{\\psi}{\\chern_1^{\\beta}(v)}$\n",
"$\\renewcommand{\\phi}{\\chern_2^{\\beta}(v)}$\n",
"Redefine psi and phi in latex to be $\\psi$ and $\\phi$"
]
},
{
"cell_type": "code",
"id": "e64456bd",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\left(\\psi, \\phi\\right)\\)</html>"
"$\\displaystyle \\left(\\psi, \\phi\\right)$"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"ch1bv, ch2bv = var(\"psi phi\", domain=\"real\") # symbol to represent ch_1^\\beta(v) and\n",
"# ch_2^\\beta(v)\n",
"ch1bv, ch2bv"
"cell_type": "markdown",
"id": "31c2cee4",
"source": [
"Define expression for the different terms of this bound of $d$ in terms of $\\phi$ and $\\psi$"
]
},
{
"cell_type": "code",
"execution_count": 18,
"id": "d962282f",
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\phi - \\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
"$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\phi - \\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
"1/2*beta^2*r + beta*q + phi - 1/2*(psi - q)^2/(R - r)"
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"bgmlv3_d_upperbound_terms = Object()\n",
"\n",
"bgmlv3_d_upperbound_terms.linear = bgmlv2_d_upperbound_terms.linear\n",
"bgmlv3_d_upperbound_terms.const = ch2bv + bgmlv2_d_upperbound_terms.const\n",
"bgmlv3_d_upperbound_terms.hyperbolic = (ch1bv - q)^2/2/(r-R)\n",
"\n",
"(bgmlv3_d_upperbound_terms.linear\n",
" + bgmlv3_d_upperbound_terms.const\n",
" + bgmlv3_d_upperbound_terms.hyperbolic)"
]
},
{
"cell_type": "markdown",
"id": "10d89cb2",
"metadata": {},
"source": [
"Verify that the expression above indeed is equal the upper bound on $d$ given by $\\Delta(v-u) \\geq 0$"
]
},
{
"cell_type": "code",
"id": "3598bfdf",
"metadata": {},
"outputs": [],
"source": [
"assert (\n",
" (bgmlv3_d_upperbound_terms.linear\n",
" + bgmlv3_d_upperbound_terms.const\n",
" + bgmlv3_d_upperbound_terms.hyperbolic\n",
" - bgmlv3_d_ineq.rhs())\n",
" .subs(ch2bv == v.twist(beta).ch[2])\n",
" .subs(ch1bv == v.twist(beta).ch[1])\n",
") == 0, \"Sanity check\""
]
},
{
"cell_type": "markdown",
"id": "97c575cd",
"metadata": {},
"source": [
"# Specialize to problem 2"
]
},
{
"cell_type": "markdown",
"id": "d35f737d",
"metadata": {},
"source": [
"Add extra attributes to the bound objects above with a specialization to the case $\\chern_2^{\\beta}(v)=0$"
]
},
{
"cell_type": "code",
"execution_count": 20,
"id": "2b45d576",
"metadata": {},
"outputs": [],
"source": [
"for bound_terms in [\n",
" positive_radius_lowerbound_terms,\n",
" bgmlv2_d_upperbound_terms,\n",
" bgmlv3_d_upperbound_terms\n",
"]:\n",
" bound_terms.problem2 = Object()\n",
" bound_terms.problem2.const = bound_terms.const.subs(ch2bv == 0)\n",
" bound_terms.problem2.linear = bound_terms.linear.subs(ch2bv == 0)\n",
" bound_terms.problem2.hyperbolic = bound_terms.hyperbolic.subs(ch2bv == 0)"
]
},
{
"cell_type": "markdown",
"id": "cbb5ee0f",
"metadata": {},
"source": [
"View the specialized bounds:"
]
},
{
"cell_type": "code",
"execution_count": 21,
"id": "cd81504c",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\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": 22,
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
"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",
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
" .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": [
{
"image/png": "iVBORw0KGgoAAAANSUhEUgAAAnQAAAHWCAYAAAD+VRS3AAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjcuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/bCgiHAAAACXBIWXMAAA9hAAAPYQGoP6dpAABt+klEQVR4nO3dd3yN9///8cfJjkwJsWKPENSMXVuVFjVaoy1Vq3ZpVXVR3dqiCC2qaLU6jKrRUiOUIvaMrWLGzJZ1rt8ffs33k1oJSa6T5Hm/3XK7Oee6zvV+nsh45b0ui2EYBiIiIiKSY9mZHUBEREREHo5pBZ1hGLRr145y5crh7e3NJ598YlYUERERkRzNtILOYrHw22+/MWjQICIjI2nUqJFZUURERERyNNOHXP/++2/c3d2pW7eu2VFEREREciRTCzqr1cratWtp0qQJDg4OZkYRERERybFMLei2b9/O9evXadGihZkxRERERHK0bOsWi46O5t133+Xo0aMULlyY4sWL8++OKS1btsyuGCIiIiK5jiU79qG7du0aLVq0oHLlynz77bdYLBbmzJnDyJEjcXZ25uLFi1gslqyOISIiIpIrZcuQ69NPP01kZCRfffVVauHWrl07IiMjadGihYo5ERERkYeQ5QXdwoULWbduHYMGDcLNzS31+d27dwNo/pyIiIjIQ8rygu7LL78E4KmnnkrzfEhICKD5cyIiIiIPK0vn0CUnJ+Pi4kLRokU5c+ZMmmONGjXi4sWLHD9+PKuaFxEREckTsrSH7urVq6SkpFCjRo00z8fHxxMaGpo63Lpo0SIiIyOzMoqIiIhIrpWlBZ2fnx9ubm74+vqmef6HH34gMTExdbh1yZIleHl5ZWUUERERkVwrSws6i8VC37592bVrV+qec2vWrOGXX34BwN/fn+PHj1OiRImsjCEiIiKSq2X5PnRxcXG89NJLXLt2jSJFihAQEMDIkSN599132bJlCwULFmTq1Km39eKJiIiISPpky8bCIiIiIpJ1TL2Xq4iIiIg8PBV0IiIiIjmcCjoRERGRHE4FnYiIiEgOp4JOREREJIdTQSciIiKSw6mgExEREcnhVNCJiIiI5HAq6ERERERyOBV0IiIiIjmcCjoRERGRHE4FnYiIiEgOp4JOREREJIdTQSciIiKSw2V7QWcYBlFRURiGkd1Ni4iIiORK2V7QRUdH4+XlRXR0dHY3LTlQdIK+TkRERO5HQ65i0zycPcyOICIiYvNU0IlN23VhF2WnlOX0jdNmRxEREbFZKujEphX1KMrJ6ycJPRdqdhQRERGblaGCbty4cVgsljQfhQsXzqpsIhR2L0xxz+JsO7fN7CgiIiI2yyGjL6hcuTJ//vln6mN7e/tMDSTyXw2KN2BL+BazY4iIiNisDBd0Dg4O6pWTbNWweEOWhC3hZvJNXBxczI4jIiJiczI8h+7YsWMULVqU0qVL061bN06ePHnP8xMSEoiKikrzIZIRz1R+htB+oTjZO5kdRURExCZZjAzs8Ltq1Sri4uKoUKECly5d4v333ycsLIyDBw/i6+t7x9eMGzeOd99997bnIyMj8fT0fPDkIiIiIgJksKD7r9jYWMqWLctrr73GyJEj73hOQkICCQkJqY+joqIoXry4CjrJkFk7Z3Hqxik+bPGh2VFERERszkNtW+Lm5kbVqlU5duzYXc9xdnbG09MzzYdIRoVHhTN712zdMk5EROQOHqqgS0hI4PDhwxQpUiSz8ojcUcPiDbkcd5lj1+7+x4OIiEhelaGC7tVXXyUkJIRTp06xbds2unTpQlRUFL169cqqfCIA1POvhwULm89sNjuKiIiIzclQQXf27Fm6d+9OQEAAnTp1wsnJia1bt1KyZMmsyicCgJeLF1X8qrA5XAWdiIjIfz3UoogHERUVhZeXlxZFSIYtP7ocH1cfGhRvYHYUERHJQ67FX+Na/DXK+ZQzO8pd6V6ukmM8WeFJFXMiIpJtohKiGB8yntJflGbIyiFmx7mnDN8pQsQshmEweetkahSpQdNSTc2OIyIiuVRcUhzTtk/jk82fEJsYy8DaA3m90etmx7onFXSSY1gsFubsmUPdy3VV0ImISKYzDAOLxUJCcgKfbfmMrpW78uajb1LMs5jZ0e5LBZ3kKM1KNWP50eVmxxARkVwkMSWRObvnMHX7VDa+sBHffL6cGn4KNyc3s6Olm+bQSY7SvHRzTt04xekbp82OIiIiOVyyNZm5e+YSMC2AQSsGUb1wdRJSbt3dKicVc6CCTnKYJiWbYMHC+lPrzY4iIiI5XP/f+tP7197UKlKL/QP3s6DTAop6FDU71gPRtiWS44wPGU/LMi214lVERDLEMAyWhC2hYL6CPFryUfZf2k+SNYmaRWqaHe2hqaATERGRXM0wDFYeW8nb699m98XdvFz3ZSY9PsnsWJkq24Zcg4ODCQwMJCgoKLualFwqLimO7/Z9x7moc2ZHERERG3fi2gkazGnAkz88ibuTOxt6bch1xRyoh05yoKiEKHw+8WHGEzPoV6uf2XFERMQGnb5xmlLepYhJjOGZn5/h5Xov06pMKywWi9nRsoQWRUiO4+nsSe2itVl3ep3ZUURExMZsPrOZlvNbUim4EhdjLuLu5M7KZ1fyWNnHcm0xByroJIdqVqoZ60+tJ5s7mEVExEZtPbuV1t+1ptE3jbgUe4kFnRbg5+Zndqxso4JOcqQWZVpwKfYS+yP2mx1FRERswEd/fcTZqLP81OUn9r60l06VOmFnyTtlju4UITnSoyUepVOlTiRbk82OIiIiJth1YRdjN4ylV7VedAnswjcdvsHL2Qt7O3uzo5lCBZ3kSM4Ozix6ZpHZMUREJJvtubiHcRvG8euRXynvU558jvkA8HH1MTmZufJOX6TkOgnJCfx+/HeiE6LNjiIiItlgxdEV1PiqBvsj9jPvqXkcGnyItuXbmh3LJqigkxzrfPR52ixow5qTa8yOIiIiWeTQ5UN8vetr4Nb86XlPzSNscBg9q/XEwU4Djf9SQSc5Vun8pQnwDWDVsVVmRxERkUwWdiWMHot6UGV6FT7e/DEJyQm4OLjQs1pPHO0dzY5nc1TQSY7WplwbVh1fpe1LRERyCathpdfSXlSeXplNZzYx/YnpHBx0EGcHZ7Oj2TQVdJKjtSnfhnPR5zgQccDsKCIi8hBOXj9JsjUZO4sdvq6+THl8CseHHuel2i/hZO9kdjybp4JOcrTGJRvTuGRjohKizI4iIiIP4NT1U/T5tQ8VplZg4YGFAExsPZHBdQarVy4DNJtQcjQXBxdCXggxO4aIiGTQmcgzfLDxA+bsmYOPqw8TWk2gU6VOZsfKsVTQSY5nGAa7L+6mTP4yeLt4mx1HRETuwTAMLBYLa0+uZXHYYj5s/iGDggbh5uRmdrQczWJk02zy4OBggoODSUlJ4ejRo0RGRuLp6ZkdTUsuFxEbQeHPCjOnwxxeqP6C2XFEROQO/rnxDx9u+pAkaxJzOswh2ZrMzeSbuDu5mx0tV8i2OXSDBw/m0KFDhIaGZleTkkf4ufnRoHgDloYtNTuKiIj8x6nrp+i3rB/lppZjSdgSKhesDICDnYOKuUykIVfJFZ6q+BTvrH+HuKS41NvAiIiIua7HX6fy9Mp4OHvwUYuPGFh7oIZWs4hWuUqu0CGgA/HJ8fxx/A+zo4iI5GnHrx1nxO8jSEhOIL9rfpZ0XcKp4ad4tcGrKuaykAo6yRXK+5anTbk2xCXFmR1FRCRPOnb1GL2W9qLitIosPLiQo1ePAtC6XGuNnGSDbFsU8a+oqCi8vLy0KEJERCSXGLt+LO9vep/C7oUZ3XA0/Wr2w9XR1exYeYrm0EmuEh4ZToqRQinvUmZHERHJ1Q5fPoyjvSPlfMoRVCyILx7/gr41++Li4GJ2tDxJQ66SaxiGQbN5zfhw04dmRxERybUOXT5E90XdqTy9Mp/89QkAT1Z4kiF1hqiYM5EKOsk1LBYLXQK7sPjwYpJSksyOIyKSq5yJPEPXX7pSZXoVtoRvYfoT05nWdprZseT/U0EnuUrXyl25Gn+VdafWmR1FRCRXiLwZCdzaN27vxb18+eSXHBt6jJdqv6R7rdoQFXSSq1QvXJ3yPuX58eCPZkcREcnR9lzcQ+efOhMwLYC4pDiKehTl8ODD9K/VHyd7J7PjyX+ooJNcxWKx0KdGH1wdtLpKRORBbD+3nfY/tKfGVzXYc3EPH7b4EEc7R+DWz9i8yDDgp5/g1xm/mx3lrrTKVXKd0Y1Gmx1BRCRHMgyDwSsHE50Qzfyn5tO9ancc7PJ2qXBsxwHeGOfDLyuKMqpfMdpbDSx2tlfYqodOcqWYxBg2/rPR7BgiIjbNMAzWnFhDk7lN+Dv8bywWC792+5WDgw7yfLXn83wxl3gzEffQ1rQp/TmrVsGEmVVtspgDFXSSS80InUHr71oTlRBldhQREZtjGAbLjy6n/tf1eey7x9LcZaeoR1Hs7exNTGcua4qVTfPnczE8EicXJ6JrreK5Tz/i8cfNTnZv2VbQBQcHExgYSFBQUHY1KXlY96rdSUhOYPHhxWZHERGxOV9s+4J2P7TDwc6B35/9ne19t1O/eH2zY5kuNBTaP3aRGsmDOLT6NwAq1HkEJxfbXwSiW39JrtVifgtSrClseGGD2VFEREyVbE1m4YGFxCfF069WP67EXeFAxAGalGySZxc6/K8r5y6zbd5kOo0dS8VAJ2Z+cZ66TYuaHStDNOQquVbv6r0J+SeEk9dPmh1FRMQUiSmJfL3raypOq8jzS55n05lNABTIV4CmpZqqmPv/rp2/SB3fWcyfepCdO8lxxRxolavkYp0qdeL7/d+nboopIpKXXIq5RNCsIMKjwulcqTM/P/0zNYrUMDuWzdi1ZhvXQ2fR+NVZVAiqys3AM3R1y7m3LlNBJ7lWPsd8rHx2pdkxRESyTWxiLEvClvBs1Wfxc/Ojb82+dK7Umcp+lc2OZjPOnoVRo+DivjimvHiQq+evUrhUAVxycDEHmkMnuZxhGGw4vQEfVx+qFa5mdhwRkSwReTOS6aHTmbh1Ijdu3mD/wP1ULFDR7Fg2Z+OcmZzbs4mXf/yWTz6Bns8b2NnnjmFn9dBJrjdk1RCq+lVlYZeFZkcREcl0EzZP4INNH5CQnECfGn14reFrlPQuaXYsm5IQn4yzqwOObvkpUqogR4+k4OVtD+SOYg5U0EkuZ7FY6FujL6P/HM2lmEsUci9kdiQRkYd2+sZpPJw88M3ni73FngG1BjCi3giKeBQxO5pNOXEC/lnYkyTDndZvTad+16eBp82OlSW0ylVyvV7Ve2FvZ883e74xO4qIyEM5GHGQnkt6Um5KOaZsmwLAKw1eYUKrCSrm/kdcVBzvvnOTypVheWgLPMo3NztSltMcOskTev/amw2nN3B86PE8vQO6iORMByMO8ua6N/n1yK/4e/rzav1X6VuzL25ObmZHszmJNxM5P6sy3256jsSAsbz+OrjlgU+ThlwlTxhaZyhl85clMSURVztXs+OIiNyXYRhci7+Gbz5fohOjOXzlMHPaz+HZR57Fyd7271yQ3U7sPkqximVxcXXiXP6x9Bxbn5J5aHGveuhERERsiNWwsjRsKR/99RGGYRDaLxSLxYLVsGJn0Uyp/4qMhEkfnOWNKmXYED+bxwb0NDuSKfSVIXlG5M1IXv/zdY5dPWZ2FBGR2ySlJDF3z1wqT69M55864+HkwYctPkw9rmIuLWuKlTXzVlGhgsFn0/359dpimjzf1exYptGQq+QZzg7OfL37a+KS4pjSZorZcUREAEixpmBvZ0+yNZkxa8dQt1hdvunwDfX865kdzabtXb2BVo5t6ddpOwPfCqJYsSfNjmQqDblKnvLO+neY+PdEzo48i7eLt9lxRCQPux5/neDQYL7c8SXb+m6jmGcxIm9G4uXiZXY0m3X1/BV2L19By/69MKwGR7fvJaBedbNj2YRs678NDg4mMDCQoKCg7GpS5DYDaw8kMSWRObvnmB1FRPKoC9EXeG3Na5ScXJL3N75P+4D2qavvVczdWXIyBAfD+H5LqGUZwbmTV7HYWVTM/Q/10Eme03NJT0L+CeHEsBM42GnWgYhkrx6LerD86HIGBQ3i5XovU9i9sNmRbNrePzex/Lv9vD1/EH37pPDhuGsUKFbQ7Fg2RwWd5DnHrx3nevx1goqpt1hEst62s9uYsGUCXSp1oXvV7pyNOou7k7umfaTThqnv4nVzLclN1hNUR/uI3o0KOsmz/v3St1hyz738RMQ2WA0rK4+tZMLmCWw6s4kKvhWY0HICHSp2MDuazUtMhM2zP+dmsjtthg0gKSEJewd77Oy1yvde9NmRPOl6/HVqzqzJ8qPLzY4iIrnQzwd/pt0P7Ui2JrOk6xIODz6sYi4dfl9lULUqHAw9jWPyWQAcnR1VzKWDeugkz2o0pxEpRgpbXtyiXjoReShRCVHM3DmTiNgIJrSaQEJyAjvO76BhiYZmR8sREuIS2BP8DJMWd+eSSzemTjGoUlU/lzNCJa/kWW83fputZ7fy+/HfzY4iIjnU+ejzjF4zmuKTivPG2jeISojCMAycHZxVzKVDfGwihgHO+Zy5aVecwcPdWbcOFXMPQD10kmcZhsGj3zzKzeSbqbfWERFJrxs3b1BsYjEc7Bx4qdZLDK83nKIeRc2OlSMYBqxadIaqlxpzsuA3NHmmmdmRcjz10EmeZbFYeL/5+5yPPs8/kf+YHUdEbJxhGIScDqH3r71JTEnE28WbhZ0XEj4inE9afaJiLp2O7LtMq1bw5DP+bLvQlZKBJcyOlCuoh07yvITkBJwdnM2OISI2KikliV8O/cLnf3/Ozgs7CSwYyLJuyyjrU9bsaDnO7t/XU/FiWzrPCmXwG1V44gmzE+UeD9VD99FHH2GxWHj55ZczKY5I9nN2cCY8MpxtZ7eZHUVEbFCHhR3osbgHPq4+/P7s7xwYeEDFXAZYU6wc3HIQgMpNG7I96VOWrCmvYi6TPXBBFxoaysyZM3nkkUcyM4+IKYauGkrPpT1JtiabHUVETHb6xmlG/jGSPRf3ADCm0Rj2vrSX1c+vpnW51ppvmwG7dsGUIVMoGVaX86ev4eTiRJN+Q3DOp1GRzPZABV1MTAzPPvsss2bNIn/+/JmdSSTbvd34bY5ePcp3+74zO4qImGT7ue10/aUrZaeUZe6euRy7egyAR0s+yiOF1HmREVfPX2HCqM3Urg0/hvbmeLHlFC3lY3asXO2BCrrBgwfzxBNP0LJly8zOI2KKWkVr0alSJ95Z/w5xSXFmxxGRbDZ121Tqzq7Lrgu7mNpmKuEjwnm68tNmx8qxDn7/Bh2LvsjkyVY2bfWieuumZkfK9TJ8Z/KFCxeya9cuQkND03V+QkICCQkJqY+joqIy2qRItvik5SdUnl6ZCZsnMK7pOLPjiEgWik2M5Zs93+Dq4Eqfmn3oWKkjxb2K065CO+ztdL/QB7Hvz7+IjbVSv0NjKnV7H4Bh/tpMI7tkqKALDw9n+PDhrF69GhcXl3S95qOPPuLdd999oHAi2amcTzlmtZtF3WJ1zY4iIlkkPDKc4NBgZu6cSVRCFIODBgPg7+mPv6e/yelypgsX4LXXoFfxcTh5FIQOjSno72d2rDwnQ9uWLF26lI4dO2Jv/39/vaSkpGCxWLCzsyMhISHNMbhzD13x4sW1bYnYtGRrMg52Ge7AFhEbdiDiANW/rI67kzt9avRhWN1hlPQuaXasHCspIYnNcybzTnAzwiJqM/Gjq/R4Ib/uu2qSDBV00dHR/PNP2g1Ye/fuTcWKFRk9ejRVqlS57zW0D53YuvDIcJrPb86c9nN4tOSjZscRkQeUmJLITwd/4u/wvwl+IhjDMPh237d0qtQJdyd3s+PleMmJyRwLrsPB+N60HDQUb2+zE+VtGeqC8PDwuK1oc3Nzw9fXN13FnEhOUMyzGPld8jP89+GE9gvVfBqRHCYiNoIvd3zJjB0zuBhzkcfKPkZ8Ujyujq70rNbT7Hg52j9HIzi7ZCjeTT+ict0ylB24lUouTmbHEnTrL5Hb2FnsmNJmCrsv7mbGjhlmxxGRDEhKSaLqjKp8svkTngp4ioODDvLHc3/g6uhqdrQcLS7WYOxYqBnkhnNKOPFXzwLgpGLOZujWXyJ38dLyl/h+//ccGnxIk6VFbFSKNYVfj/zKVzu/4ofOP+Dj6sP6U+upVrgaPq7a9ywzHNoUinXrAB77eDUvDCjAG2MM3D20ubKt0axvkbv4uOXH7L20l/PR51XQidiYGzdv8PWur5m6fSr/RP5Dw+INiYiNwMfVh2alm5kdL1eIi04gn4czBUuX5NjfFdi88SalKwOomLNF6qETuQfDMHSbHxEb8u/3ZKcfO7H86HK6VenG8LrDqVW0ltnRco2oKFg5bR513N7Hq/tefP3ymR1J0kFz6ETuwWKxcD76PN1+6cbl2MtmxxHJkxJTEll4YCGPfvMoy44sA271oJ8ZcYb5HeermMskhtXKz99GEBAAH8xswBn7F3Bz06KwnEIFnch92Fvs+fPkn7y04iWyuUNbJE87F3WOd9a/Q8nJJem+qDuOdo6p8+Iq+FagsHthkxPmLpsmDaFceGuaNLayclN5mg55Exc3Z7NjSTqpoBO5j0LuhfjyyS9ZfHgxC/YvMDuOSK5mGEbq/ZQXHV7EpK2T6FypMwcHHWRdr3XaGzKTXbtwlZP7zwBQsH5fqDmRhT/aUby4ycEkwzSHTiSdnlv8HMuOLGPXgF2U8ylndhyRXCU6IZpv931LcGgwTUo2YfoT04lNjCXFSMHTWb8rMltKCsyaZRB0rR6xhj+N31xkdiR5SOqhE0mn6U9Mp6R3SXZf2G12FJFcIzwynCErh1BsYjGGrRpGxQIV6ValGwBuTm4q5rLAvrWbadP0AgMHWlh+aTqVnp9udiTJBOqhE8kA3eNV5OElpiRy+sZpKvhWIOxKGM3mNaNvjb4MqD1AWwRlsfiYeGIWlGTpwYE88uy71K1rdiLJLCroRDIoxZrCq6tfpZ5/PbpW6Wp2HJEc4/i148zcOZO5e+bi7eLNkSFHsFgs+kMpiyXeTGTz3K+o3O4F/Ip5cHLvEUpVKY+dvQbpcpNs+98MDg4mMDCQoKCg7GpSJEvYWey4FHuJfr/149jVY2bHEbF5EbERtJjfgvJTyzN712yerfosS7stTd3jUcVc1lmzBlo3uUQtxzcJC1kLQJlqASrmciH10Ik8gOiEaGrNrIWroyt/9f4LD2cPsyOJ2JTj146z7tQ6+tfqj9Ww0n1Rd9pVaEeXwC64OLiYHS/XO3f0DLt/mEjH9z6lQUNHpk+6RuWauhVabqaCTuQBHYg4QMM5DWlUohG/dvtVvQyS5yWmJPJr2K/M3DWTP0/+iY+rDyeGncDbxdvsaHnOkW17cdvRnr35f6dt90rohje5n/pcRR5QFb8q/Pz0zxRxL4LVsJodR8RUKdYUKgVX4plfniE+KZ55T83j7IizKuayiWHAX4vWsv6D50lKNAioW43C/U7wRA8Vc3mFuhREHsJjZR/jsbKPAXAl7goF8hUwOZFI9ohPimdJ2BIW7F/Aws4L8XD24P1m71O1UFWq+FUxO16eciTMYNhwCynnDD7qdYWoq1H4FvHCwUm/4vMS/W+LZIL9l/bTYE4Dvuv4HR0qdjA7jkiW2Xl+J1/v/prv939PZEIkjUs25mLMRTycPehetbvZ8fKcDdMncOHwXo4fX8CkSS2p3a6leuTyKBV0Ipmgsl9lWpdtTY/FPdjQawNBxbSaW3KPK3FXyO+SH3s7e0b/OZrDVw4zOGgwL1R/gfK+5c2Ol+cYVoO4mATcPF1w8ilFkYBEDk4wcHFVJZeXaVGESCaJT4qn+fzmHLlyhPW91lOtcDWzI4k8sBRrCqtPrGbOnjn8GvYry3ss57Gyj3Ex5iIF8hXQIiCT7NltELeqHVHW0jz+1lSz44gN0aIIkUzi6ujKqmdXUSZ/GXou7amFEpJjTds+jZKTS9L2+7YcuXKET1t9Sq0itQAo7F5YxZwJrl+8xvAhN6lV28Kve7rhV/MpsyOJjdF3pUgm8nbxZs3za7gWfw07i/5ekpwhIjaChQcW8ni5x6ngWwELFtoHtOfFGi9Sq0it1A2AxRxxUXEkLKmC7+UhfPbZGwwZ8hyOjmanElujgk4kk+V3zU9+1/xEJUTxwtIXeL/5+wQWDDQ7lkgaN5Nv8tuR35i/bz6rjq3CzmKHt4s3FXwrMLjOYLPjCXBg4y7K16lOPs987PX5gpcmPIpfSbNTia1SF4JIFklMSeT4teM0/qYx289tNzuOCIZhkJSSBMCYP8fwzC/PcDn2MlPaTOHCKxfoWa2nyQkF4OJFeG3gMQLP1GbT94sAqN/1afxKFjY5mdgyLYoQyULX4q/R7od27Lm4h0XPLOLxco+bHUnyoLArYfyw/we+3fctI+uPZEidIZy+cZqE5AQCCgSYHU/+v6SEJH6fvYxnx3TGyQnmfrKWNj2bYu9ob3Y0yQFU0IlksbikOLr90o1Vx1cRNjiMsj5lzY4kecSvYb8ydsNY9l7ai6ezJ10qdWFg0EBqF61tdjS5g9ClK6gZ056P9u5j0JjK+OjWq5IBKuhEskGyNZnfj//OkxWexDAMrIYVezv91S2Z60L0BX4+9DNV/arSrHQzfjvyG9/t/45ulbvRpnwbXBxczI4o/3Hu6D8cWPMHrQf3x7Aa/HPwOKWqam8/ybhsK+iCg4MJDg4mJSWFo0ePqqCTPGva9mksDVvKwi4LdasweWjX4q+x6NAiFh5cyIbTG7C32DO+2Xheb/S62dHkHm7ehE8/hat/T+e1Jz/EocMh/Irpd6I8OPXQiWSz9afW88wvz+Dm6MbiroupWaSm2ZEkh7lx8wY3k29S2L0wwduDGfb7MJqXbk63yt3oVKkT+V3zmx1R7mH7kuWs+vkkH/wyjJEjUnjz9Tg88nuYHUtyOBV0IiY4E3mGzj915kDEAb584kt6Ve9ldiSxcVfirrA0bCmLDi9i7cm1DKg1gKltpxKVEEVcUhyF3bUCMqfYMOUdnOL24fvUEgIqao8/yRwq6ERMcjP5JoNXDOZ05GnWPL9GGxHLbQzDwGKxMGf3HPr/1h8Dg0dLPEqXwC50rNiRYp7FzI4o6RAdZRD69Zsk2JegzbCXSElKwc7eDoudijnJPNpYWMQkLg4ufN3ha+KS4rCz2LHh9Abik+JpU76N2dHERMeuHmPZkWUsPbKUtuXaMubRMTQq0YjpT0znqYpP4efmZ3ZESSfDavDDQhg1ysKYx+IIrB0HoG1IJEuooBMxWT7HfAB8v/97Zu2aRY+qPZjcejIF3QqanEyy09KwpYxZO4awK2G4OLjQskxLqhaqCkAF3wpU8K1gckLJiNjIWI7PasOSn4dSv/7TtBs3mZK6y4NkIRV0Ijbiqye/4tESj/LyHy9T6XglJraeyPOPPK/7aOZCMYkxrDmxhmVHl9GqTCt6VO1Bfpf8NCzekE9afkLLMi1TC33JWaKux+HhnQ83Lzeu29Vm9Fg/arc1O5XkBZpDJ2JjImIjGPHHCLaEbyFscBjODs5mR5JM8vvx3/li2xesP7WehJQEAgsGMrrhaN1yKxdISYFfvjlGC2sDDvksonGXxmZHkjxGs7BFbIyfmx8LOi0gtF8ozg7OnLh2gi4/dSHsSpjZ0SQDohKiWBq2lJeWv8Qfx/8A4Hr8dZKtyXzY4kOODT3GwUEHVczlArv+OkvdutC9f1k2XRxCQG1tDCzZT0OuIjbq302Hz0efZ8f5HVSZXoW+NfvyVuO38Pf0Nzmd3M2PB35k+o7pbAnfQrI1mfI+5WlSsgkA3at2p3vV7iYnlMy0fclyqkd3orjXfqb8FUCDBmPNjiR5lHroRGzcoyUfJWxIGBNaTeDnQz9T+ovSzN0z1+xYeZ5hGBy9epQvd3xJ11+68nf43wBcjb+Kt4s3Ux6fwolhJzg69KiKuFwmKSGJ3et2AlCtdSu2W2bzy+/laNDA5GCSp2kOnUgOEp0Qzexds3mywpOU9y3Pb0d+w9XRlRalW2jxRBb790elxWJh3IZxzNo1i/PR57G32BNULIj3mr1HyzItTU4pWW3dOtj3w4f0rvsJSU+cpUAR3eFBbIMKOpEcrNOPnVgStoTAgoH0r9mf7lW7a5+yTJKUksTeS3vZfGYzm8Nvfax5fg2BBQOZtn0aZyLP0KxUMxqVaISHs36p53bnjp5h5qSTjP+yKa2b3WDyB6eoWL+G2bFEUqmgE8nBDMNg/en1fLnjS5aELcEwDLb3207NIjVT7zIg6XM9/jp7L+2laammAFQKrkTYlTCc7J0IKhpEg+INGBQ0iFLepUzNKeb469OeeBkH2Ft0J88+a0HfWmJrVNCJ5BL/3uuzd/Xe2NvZ0/6H9rg6utK+QntalmlJIfdCZke0KXFJcUwPnc7OCzvZcX4Hx68dB+DyqMsUyFeA3478hm8+X2oVqaWtY/Igw4DtS5eTggcNOjbhyrnLOLk44+mr31tim7KtoAsODiY4OJiUlBSOHj2qgk4kCxmGwcS/J/L9ge/ZdWEXANULV2dBpwUEFgw0OV32MQyD8Khw9l/az75L+9gfsR8HOwfmd5xPsjWZIp8XIcA3gNpFa1O7aG3qFKtDeZ/y6tnM444dg+HDDV6r2QyreyWavz7D7Egi96UeOpFc7lLMJdacXMPqE6uZ1nYans6evPjri4RHhVO3WF3qFqtLnWJ1cnQPXrI1mdM3TnPkyhGOXj1KQIEA2pZvy+oTq2n9XWsAPJ09qepX9dbdGFp9AoDVsGJn0WJ/ueVm7E22znqX0dOfJiKpJlM+j+LJpzyw2KnAF9ungk4kD/pyx5esPLaSbee2EREbAcBPXX7i6cpP83f43+yP2E9gwUAqFqiIr6uv6T1WhmEQERvBP5H/8M+NfzgTeYZOlTpROn9pPtj4Ae+GvEuSNQkAVwdXXq73Mh+2+JAbN2/w15m/qOpXlRJeJUx/H2LbkhKSOBbcgCMM5fGBPXF1NTuRSPqpoBPJwwzD4EzkGbaf20794vXx9/Tn/Y3vM3bDWKyGFYB8jvkYEjSET1p9wsWYi0zbPo3C7oXJ75IfbxdvfPP5Us+/HgBX465isVhwtHPE0d4RRztH7O3sgVu9aAnJCSRZk4hJjCEqIYp8jvko5V2KGzdvsDRsKVEJUVyJu8KlmEvEJMWwoNMCAB6Z8Qj7I/an5nZzdOPnp3+mTfk2bPpnE/su7SOgQAAVfCvg7+mvXjdJt8M7wole1w/3Zl8SGFSKlKQU7B3tzY4lkmEq6ETkNgnJCRy7doywK2GER4ZTsUBF2pRvw+4Lu+mwsAMXYy6m9ogVyFeAy6MuA1BuSjlOXD+R5lore6ykTfk2jF0/lvEbx6c59vwjzzO/43yOXztO+anlcXFwoUC+Avi5+VHYvTDLui3D3s6eJYeXYGexo4RXCUp6lyS/S371tslDuX7NythxdsydHcMfb3bAvdEEqjapZXYskQemgk5EMswwDOKT47lx8waxibGU971178o/T/5JVEIUSSlJJKYkkmRN4rGyj+Hv6c++S/s4GHEQBzsH3J3c8XT2pJhnMUp5l8JqWEmxpuBo72jyO5O8YM/qEPIdeIlWE/5i6Ku+DBsGTk5mpxJ5OCroREQkT7hxJQbvAu5cPnOJwz+9RcDTH1KoZEGzY4lkChV0IiKSq0VEwJrgqTT0nYxXjwPkL6DVDpL7aOawiIjkSkkJScyeeo4KFeCLHx/jjNureHppbFVyJwezA4iIiGSF7VN688jNo3Trto333w+gQIEAsyOJZBkVdCIikmucPx5OXEwK5aqXwrfRKxiGwZcNtCJacj8VdCIikuMlJMDnnxt0cGhLRHI1ylX/jor1a5gdSyTbaA6diIjkaNuXrKBJ3YuMHWvh9+jvqDVgutmRRLKdCjoREcmxoq5GUe5KT15s9i379sEr71XD01c7KEjeo21LREQkR4mNjGXrvMk88vRwChZx5+yRfyhWvgQWO82Vk7wr23rogoODCQwMJCgoKLuaFBGRXMQw4McfoeWjl6np+hnH//4bAP+AkirmJM9TD52IiNi8k3vCOLT4czp+OJ0n2zky6dMYSpVzNzuWiM3QKlcREbF5ifHxVPLdzNplZ2nctjSgYk7kf6mHTkREbI7VCuvnL8H+3EIavbYQB0cL1hQrdvZayydyJ/rOEBERmxK6PYX69eGTiW44OjkQGx0PoGJO5B405CoiIjZjwxdjuHD8NAkJP/B28GM0fPQxsyOJ5Agq6ERExFTJicnExiTi5ZMPp0I1KOpWnJ2TDOwdtHJVJL00h05EREwTssHAc3sjLiYF0ebNyWbHEcmxNCFBRESy3YUTZ3muRwJNm1lYcmAIpZo+b3YkkRxNQ64iIpKtIq9Ekm99FcqmvMPcuSN5/vnu2Kl7QeShZOhbaMaMGTzyyCN4enri6elJ/fr1WbVqVVZlExGRXGTHyo0kJljxKuBFWP55jAzuQ69eqJgTyQQZ+jby9/fn448/ZseOHezYsYPmzZvToUMHDh48mFX5REQkhztxAoY8t5/aN5qwZdGtToC6nTvgVcDL5GQiucdDL4rw8fHh008/pU+fPuk6X4siRETyhtjIWFbNXMyzbz1PoUIw97OtNOtSV/ddFckCD9zRnZKSwsKFC4mNjaV+/fqZmUlERHKB/WtW86RfPz5+6wRhYdD8mXoq5kSySIZ76Pbv30/9+vW5efMm7u7ufP/997Rt2/au5yckJJCQkJD6OCoqiuLFi6uHTkQkFzq+8yDHN6/j8WFDMawGF0+do0hZf7NjieR6Ge6hCwgIYM+ePWzdupWBAwfSq1cvDh06dNfzP/roI7y8vFI/ihcv/lCBRUTE9kRGwogRMPWtdVQgmOtX4rHYWVTMiWSTh55D17JlS8qWLctXX311x+PqoRMRyd02f/cdfyy7zMSVI3jn7WSGD03BOZ+z2bFE8pSH3ofOMIw0Bdt/OTs74+ysb2wRkdzGMMBigaSrYTSvdZ4BEw2K+TugLU5Fsl+GvuveeOMN2rRpQ/HixYmOjmbhwoVs2LCB33//PavyiYiIjbkcYeXQvEHEOlWn7fCXaDL0PS12EDFZhubQXbp0ieeff56AgABatGjBtm3b+P3332nVqlVW5RMRERuRnJjCtGkGFQLs2B/mhpuHE4CKOREb8NBz6DJK+9CJiOQ8kZdvcHFBE974fiw+1Trx4YdQsKDZqUTkX5roICIid3Xl/A18i3jjVdCb3bThvYmlCGxkdioR+S/10ImIyG0SEuC74P08492Avfl/p1HHhmZHEpF70C2RRUQkjY0rjlG1Krw0ujLrL79D1UaVzY4kIvehIVcREUm15YcfqZ/0HLUrHmXx4tJUqTLK7Egikg7qoRMRyePiouLYvnIzADXbd2BXvp9ZsLQUVaqYHExE0k0FnYhIHmUY8Msv8PXIDykT/hSR1+JxcXOhbpentBWJSA6jgk5EJA86vusQo55fy9NPw+ZrrxDb8G+8fFzNjiUiD0irXEVE8qAtn3bFPvESV2tsoG1bs9OIyMPSoggRkTzAmmKwZcF8cC9Fo05NqPjcdNy83HHOZ3YyEckM2TbkGhwcTGBgIEFBQdnVpIiIADt3QqNHwTg+m/iTfwDgU8QX53zOJicTkcyiIVcRkVwq+no0u74exfBpA0nxqEbwlHgaN9M8OZHcSIsiRERymX//THfJ50J+y34+GHOS3btRMSeSi6mHTkQkF9mx/hiOO17ApcUPBNQsgWE1tAWJSB6gHjoRkVzg/NlknnsOmj9RiMibvpAcC6BiTiSP0CpXEZEcLvTXlRQ4PYRtm3YyeVp+Gr2wDDv9uS6Sp6igExHJoSLOXsPP34cS1Wpw+GQnQnfY413Q7FQiYgbNoRMRyWFOnYItM9+nQeG5+PY8iGd+bT8iktepU15EJIeIi4pj0nunqVQJZv3RhQt+H+Dh5WR2LBGxAeqhExHJIbZ92oGk2GusStrEG2+Am5vZiUTEVmgOnYiIDTuxO4wUw5kKNUvj3Xg8Lm5ufFDF7FQiYms05CoiYoOiomDUq1as69sR/sdHAATUrUbJKuVMTiYitkgFnYiIDbGmWPnr229pUCuC6TPsWJeyjIbDppgdS0RsnAo6EREbEnn5BoE3X2Z4xyWEhcGAUZVwcXMxO5aI2LhsK+iCg4MJDAwkKCgou5oUEckRrl24xtrPX+fKxVjyF/YhpfUh+k0YQPHiZicTkZxCq1xFREySkgIzZ8KsSSdZNaIB4aUWU7tNA7NjiUgOpCFXERETHN6yi9/HPsfQIclUb1QGnvpHxZyIPDBtWyIiko0Mq4HFzgKGQUmf42zfGEHNhkUB3e1BRB6chlxFRLJBYiKsnTUP9xtLaTB6MfYOlv8r7kREHpKGXEVEstiaPxJ55BEI/rogVhd/EuITAVTMiUim0ZCriEgW2vjZACJOxVCo0AI+nNqWRx5pa3YkEcmFVNCJiGSy+Jh4bsZbyV/QDcfiLSldLIUN08CiDjkRySKaQycikkkMA5YuSSHwn2qcjH+cNm98ZnYkEckjNIdORCQTnNxzhCfbJtKpsz2/HBlHxSf7mx1JRPIQDbmKiDykK+cuU2RPdWp7f8ZLywbz5JNdNLwqItlKPXQiIg/AsBps/mk5SYkGBYoV5KDvUsbM7kO7dporJyLZTwWdiEgG7dkD/buE0jC5HduXhwBQu11rXNxczA0mInmWCjoRkXS6duEq34//ilq1YMuROmzz3UfDTk3NjiUiooJORCS9wjb8yRPFXuOrSWfZswfqtq5qdiQRESAbty0JDg4mODiYlJQUjh49qm1LRCRH2LduK+f2bqbNiFcwrAbXLl7Ft2gBs2OJiKShfehERO7g4kUYPRryR0ymb8sfKNVvE+6eTmbHEhG5Iw25ioj8R8isqUwd/BkrVkCVTkOoNGyLijkRsWnah05E5P8zrAYWOwtG3CWa1UvglVng46MfkyJi+zTkKiJ53j+nkzn/U3euu7Si7bD+qYWdiEhOoSFXEcmz4mMSGD/eoFKgA7uOlcG7kC+AijkRyXE0liAiedLlsxHEL63H3hWTGDasAz3f/AQPD7NTiYg8GBV0IpKnnD8ZQdEyfhQoWpAQoyefzqxEmWpmpxIReTiaQycieUJ0NMybuI1+pRqzy3cj9Z+sa3YkEZFMozl0IpKrGVaDFQv2UrEijP6kNmtvTKZ600fMjiUikqlU0IlIrvbX/Dm0SgmibbPzHDpsT9vhA3F1dzU7lohIplJBJyK5zvWL1/h76XoAanfpwUG/P5j1XVFKljQ5mIhIFtGiCBHJNVJS4OuvISX0fdpV/4XY6BO4ebhS4/FmZkcTEclS6qETkVxh//q/GfbMagYMgP3GWzg9uR03D0ezY4mIZAutchWRXGHrp524GZ+CU8tfadDA7DQiItkr24Zcg4ODCQ4OJiUlJbuaFJFcLCkxhc1zpmBfsCaPdm5CpRfm4O7tgb065UQkD8q2IdfBgwdz6NAhQkNDs6tJEcml1q6F6tUt2F9YzM1z2wDwKuiNvaO9yclERMyhRREikmNcv3iNw/MH8vK0t8hfoioeT63j0RrqkhMRUUEnIjbPmmJgZ2/BzdsdV7vLTP7oIs17VMViUTEnIgJaFCEiNswwIGTpPgqfeh675iuoUN3f7EgiIjZJ25aIiE06FnaTtm2h/bOluBBXGWenZLMjiYjYLA25iojN2bLwJ4pffoVzp/cz/3tvmnb4HovF7FQiIrZLPXQiYhMMq0H40QsAlK7TkBOW/mwLdeapp1AxJyJyH5pDJyKm27cPDn73GnWKLqVI30Pkc9fggYhIRqiHTkRMcyPiOu++eoIaNeC7v3tzo/w0FXMiIg9APXQiYppdn7XkRqSF3d5rGDoUnJzMTiQikjNlqIfuo48+IigoCA8PD/z8/Hjqqac4cuRIVmUTkVzoQMg2juw6BYBH08lU7jOfV15RMSci8jAyVNCFhIQwePBgtm7dypo1a0hOTuaxxx4jNjY2q/KJSC5x6RL0fTEJj73PcGbNVADK165CoVJFTE4mIpLzPdSQ6+XLl/Hz8yMkJITGjRun6zUachXJW5ISktg890t6v9udqIQCTP3wBF1fLKX7roqIZKKHmn0cGRkJgI+Pz13PSUhIICEhIfVxVFTUwzQpIjlM5JUbVLW8y9t9fekwvAe+vmXNjiQikus8cA+dYRh06NCB69evs2nTprueN27cON59993bnlcPnUjude7YOY4s/pBqvT/F1y8fkZdv4FXQ2+xYIiK51gNvWzJkyBD27dvHDz/8cM/zxowZQ2RkZOpHeHj4gzYpIjbu5k344ANo3Sqe8m4rOHf41qIpFXMiIlnrgYZchw4dyrJly9i4cSP+/ve+WbazszPOzs4PFE5Eco69azZyYcMXjP/0J4YNL4fXc8cp7q095UREskOGftoahsHQoUNZsmQJGzZsoHTp0lmVS0RyiJRkA3sHCw7OLhTwjmH/zhtUqOqLbhUtIpJ9MjSHbtCgQXz//ff8+uuvBAQEpD7v5eWFq6truq6hVa4iuUNMDGz46gu8b/5J/dHLsHfQDVdFRMySoTl0M2bMIDIykqZNm1KkSJHUjx9//DGr8omIjTGsBj//EEfFijBrYXmSPGqRkpxidiwRkTwtw0OuIpK3bf38GeJPuhIUNJ+JE9tSunRbsyOJiOR5muQiIvcVefk6yVZHfAu5Q8luVK7kQc8nzU4lIiL/eqg7RTwIzaETyTmsVpj3TSIt4itwMKY7bV7/yOxIIiJyBw+8D52I5G4HNu6kYYMkXuzrxC8nv6BG16FmRxIRkbtQQScit7lw8hwBp+vxWPlv2bQJRk7sQOHSRc2OJSIid6GCTkQASE5MZsO8hSQlGhQpU4yDRdbxzpxeNGpkdjIREbkfFXQiwoYN0KfD3zxq/yx71oUCUL3Vo9g72psbTERE0kUFnUgeduHEWeaN+YJmzeBY5KMcKnuMoMfrmB1LREQySAWdSB52cmsIbUp+xMJ5l/nrL6hav4zZkURE5AFk27YlwcHBBAcHk5KSwtGjR7VtiYhJti39g2vHd9Hm1TEYVoOYyBg88nuYHUtERB6C9qETySOOH4eXX4YA43O6NVlD1aErcHHVHDkRkdxAd4oQyQM2TH2XP/7Mx/79o+j9+cvU7jQSi53F7FgiIpJJVNCJ5FKG1cBqNbB3sAMjmRbNk3n7B8iXT71yIiK5jYZcRXKhg/sSSfijDRddutN2aF+z44iISBbTKleRXOTGlRhefhmq1XRi64kGFCpTyuxIIiKSDTTkKpJLnD8ejuPa2pzd9jUffvgkfV9+Dycns1OJiEh2UA+dSA53cv8/ABQp48/BlBFMXVCD115DxZyISB6igk4kh7p8GT59dT0l95Zh+x+7sdhZaDrodYqUKWZ2NBERyWYq6ERymOTEZH6asZUKFeDjOY1ZHz+Pms2qmh1LRERMpIJOJIfZPDeYdq7NeKH7VcKO2NOy33M4OGk6rIhIXqZtS0RygAsnznJi10EaPd2auKg4/tl/mEoNa5kdS0REbIT+rBexYQkJMGkS+Jz4iBZV1pJw8yD5PPOpmBMRkTQ05Cpio0J/Xcmgjr/z1lvwj/f7FHh2G84uusuDiIjcTj10IjbKevRLutbNz8hPH6dy5fxmxxERERuWbQVdcHAwwcHBpKSkZFeTIjlKbHQioXPG41D8MRp1akzlft/j5umGRf3oIiJyH9n2q2Lw4MEcOnSI0NDQ7GpSJEcwDPj5Zwis4oDd1b9IvHIYAHdvdyx2FpPTiYhITqAhVxETRfxzkdM/vsDYKROpXisQ/17rKFNWXXIiIpIxKuhETJCUmIKjkz1eBfNjsdjx9YxI6rcDrVMSEZEHoX3oRLKR1Qorv91GlchnSW4aQrlHdJsuERF5eOoOEMkmu7ZH07AhPDcogBOxLXD30BYkIiKSOTTkKpINNs39mjIx70DSYX5d6U2TJl+ZHUlERHIRFXQiWSQ5MZkzR85RpmpJyj3amqPr4ti0JR8OTmYnExGR3EYFnUgW2LQJLi4bQvWiISRWOEiRsv4UKTvU7FgiIpJLaQ6dSCa6cPIcr/Q/SuPGsOjAcJJqz8fJWd9mIiKStdRDJ5KJrizpQktfP6rM+ZVevSphp1pORESygQo6kYe0Y9nvuBerSMVapcjXfA4N/IvgVdDsVCIikpeo/0DkAZ08CV063qTwmb6c2TAXgLI1KuFV0NvUXCIikveooBPJoLioONZ+8T6Ngq6ybYcLuwtupdWIsWbHEhGRPEwFnUgGxVyP4hGXKXwyagthYdCuqz8WO4vZsUREJA/Ltlt/BQcHExwcTEpKCkePHtWtvyRHObb7OOf+eI9qfb8kfwFX4qLiyOeZz+xYIiIigO7lKnJPkZHw7rvw+09HWfpKJ4xGiwgICjA7loiISBpa5SpyF6FLV3AjdAazZv7KG29WoOTA/Ti7aGhVRERsjwo6kf9ITEjBydkeFy8f3LzdObw/Fv/SnoCKORERsU0achX5/65cgS0zx+NrbKP+68uxs1cBJyIiOYNWuUqel5KUwqzp0VSoAN+uqE2yX2uy+e8cERGRh6IhV8nTDKvB3i9a43CsOJ06fcOHH7bFz8/sVCIiIhmjgk7ypIunzuPg6kmBwu4klRpCnTpF6d3Y7FQiIiIPRkOukqckJsKkT+Ox/6M6ofM/A6Bul6eo3LiOyclEREQenAo6yTO2L99AzepJjBrjyi/n59Og98tmRxIREckUGnKVPOHMoZPUutGCZ+ovoOOP3aha9XGzI4mIiGQa9dBJrhUXHc/ar2aTnGRQIrAMR0pt4+1ZXala1exkIiIimUsFneQ6hgGLF0Pfp7bQyGUwB7ccACCwUW0sdtpbTkREch8VdJKrnNwTxtevfELnzhDp0oLzQaeo1kRdciIikrupoJNc5fz+7bQqPYtVyyJZsQJKBxY1O5KIiEiWy7ZbfwUHBxMcHExKSgpHjx7Vrb8kUxgGbFrwEzHnj9D2tbexplhJvJmIi5uL2dFERESyTbb10A0ePJhDhw4RGhqaXU1KLrd7NzRqBL8t/AdPwkhOMrCzt1MxJyIieY62LZEcx7AabJz8MivXFyMy8jXavvcqjZprsYOIiORdmkMnOUZKUgpJiVYsdhYMB29aP+nJ7t3QTMWciIjkcdk2h+5fUVFReHl5aQ6dZMjfm+Lw3tGIU45DaTukt9lxREREbIp66MSmXTxzjZ49oUHjfGw88RQlqlYxO5KIiIjN0Rw6sVn/HDhO/m01STj1E7NmPc6LL76Dnf4EERERuY1+PYrNObT1MAAlAsuyO+U9vlpUl759UTEnIiJyF/oVKTbj9Gn4cPByKh6vzO4NB7HYWWjSfzjefvnNjiYiImLTVNCJ6eJj4pk7YQOVKsGXyx7nL7slVG8caHYsERGRHCPDBd3GjRtp164dRYsWxWKxsHTp0iyIJXnJtvlf0NGnA6NHRnHosAONe3TAYqetSERERNIrwwVdbGws1apVY9q0aVmRR/KIk3vCCPl+OQBBzw7lWp1Qxn3gibu7ycFERERyoAyvcm3Tpg1t2rTJiiySB0RHw/jxEBA5kXoBu0lOegI3LzdKP1LB7GgiIiI5lubQSbYwrAabFyxgYIeVBAfD9ZKfUu6lTTg4amhVRETkYWX5PnQJCQkkJCSkPo6KisrqJsUGWewsOJ3/nu5Ny/Ph3LaUKOFldiQREZFcI8sLuo8++oh33303q5sRG3Ttcjz7572KfdnuNOrYiEcGL8Y5n7PZsURERHKdLB9yHTNmDJGRkakf4eHhWd2kmCwl2eCrryAg0Blr5FGsMbf+z1XMiYiIZI0s76FzdnbG2Vm/yPOKs0f+4dpv3Zg08RuefLIilQatpnARzZMTERHJShku6GJiYjh+/Hjq41OnTrFnzx58fHwoUaJEpoaTnCM+NhFXNycK+Bci3CjGj98nUK0pgIo5ERGRrGYxDMPIyAs2bNhAs2bNbnu+V69ezJ07976vj4qKwsvLi8jISDw9PTPStNigpCRY+tU6Gtr1JL5xKGWrFDE7koiISJ6T4R66pk2bksEaUHKpkNVXGPhyAS6F1+DHd7pTq6CL2ZFERETypCyfQye504avJlEh+VP8Cx3lhx/yU63ap2ZHEhERybNU0Em63Yy9yZmws1SoVY7yzTpzKtSPP9a6YdH21CIiIqZSQSf3ZRiwbBmwpQ/l/Q6SUm03xSqUoFiFZ82OJiIiIujWX3IfJ/cepV/XMJ56Cn479TYuLX/E3kErV0VERGyJeujkrgyrQdK6Z2hfOpB2S7+nffuKWFTLiYiI2BwVdJKGYTX4e+H3+FZ8lICaJXBu+TOtSvvj6m52MhEREbkbDblKqr174fGWsZS6Pop//loEQKmq5XF1dzU5mYiIiNyLCjoh8vIN/vx0FM0bXefMBXeOld/LY8NGmB1LRERE0kkFnXAzNp7Kbj8w48Pd7N0LTR4raHYkERERyYAM3/rrQQUHBxMcHExKSgpHjx7Vrb9Mti9kH5Fb3qNq/2/x9nUh8WYiTi5OZscSERGRB5BtPXSDBw/m0KFDhIaGZleTcgeXLhr07g3PdHfC2zGca+cvAaiYExERycG0yjUP2fLDQm4enMNvv/3OBx9UJLDP39pTTkREJBdQQZcHxEUnkM/DGfeC/iTnL8eRQzfx9csHqJgTERHJDbJtDt2/oqKi8PLy0hy6bHDmDOybO4L89seoP2Y5dloCIyIikivpV3wudDP2Jp9+EEnFirBwQyso2Q0L2Vq3i4iISDZSD10uY1gNDk2uy9awahzxnsXbb4OHh9mpREREJCtpDl0ucWrfMTz8ilKgsBuxpd+hWfNy9KludioRERHJDhpyzeFiYmDcm1H4bK9N6IKpANTp+CRlqlc0OZmIiIhkFxV0OZRhNdj4wzIqBybzyURPll5bStMBw82OJSIiIiZQQZdDndhzmEYpTzGg3SoOH4ZerzXD1d3V7FgiIiJiAhV0OciNiOusmTaFlGSDcjUDORl4gDeC21GqlNnJRERExEwq6HIAqxVmz4YBnbdRL9+bHN11HIByNQNNTiYiIiK2QAWdjTu0KZSZw96jXz9wLv04sS3/oVKd8mbHEhERERuibUts3NWTB2hadjFbNo6k/qNugI/ZkURERMTGZNvGwsHBwQQHB5OSksLRo0e1sfBdJCXBxjlfkXDjAm1Hj8OaYsWwGtg72psdTURERGxUtg25Dh48mEOHDhEaGppdTeY469cZ1KgBa1bcwNX+GtYUAzt7OxVzIiIick8acrUBhtVgy+fPsTKkOl5eo3hm3Ghq1jQ7lYiIiOQUWhRhopuxCSQlGljsLCS5BNC+W3H++gsVcyIiIpIh2TaH7l9RUVF4eXnl+Tl0v/8WSaXTtThoP5a2g543O46IiIjkYOqhy2YnD17giSegTXsv1pzqR0D92mZHEhERkRxOc+iy0dHQ/ZQ8VBuvmytZtKgFHTuOxmIxO5WIiIjkdOqhy2KG1WD3up0AlK9Vhe3WYGYvqU+nTqiYExERkUyhgi4L7d8P7/X7iUfO1+HQ9hNY7Cw82rsv+TzzmR1NREREchENuWaByMs3+H7qVoZ++DgB5TvRpuOfBNUpa3YsEZEHlpiYSHJystkxRGyOg4MDTk5OZsdQQZcVdi+cTJei07n50T8MHu6Kk1MzsyOJiDywxMREDh48iNVqNTuKiM2xs7OjcuXKphd12rYkkxzcuJ2Ify7Q7PkOxNyIIfpaJEXKFDM7lojIQ4uLi+Pw4cOUKlUKV1dXs+OI2Iz4+HhOnz5N0aJFcXZ2xtnZGTc3N1OyqIfuIUVEwBtvQEP7GVQpE441pT3u3u64e7ubHU1EJFO5urqSL5/mAIv81/Lly4mNjcXT05OuXbuaUtRpUcQDsqZYCZk1lcEdV7J4MSRXm0LNkX9gZ6+lqyIiInmJm5sbTk5OREVFkZCQYEoGFXQPyGKxkO/G7/R8IpSjR6HfIA/sHe3NjiUiIiLZ7N/hVjNl25BrcHAwwcHBpKSkZFeTme7c6ShO/zQAo8IwGj1VnxrDf8XBSaPWIiIiYq5s66EbPHgwhw4dIjQ0NLuazDQJNw0++ggqVnEjLjIa+5RrACrmRERExCaoIrmPk3uPkLyuM/ODF9N/QAXqvLYcLy+zU4mIiIj8HxV0dxF1PQ7P/PkoUrYEO1fXZNkyC+Vrmp1KRERE5HZaFPEfsbHw7ce/cfPH0pw6fAlXd1cajZpP+ZrlzY4mIiIickcq6P4/w2qw/KezVKoEr33akP0JQyhUzMPsWCIiYrKmTZvy8ssvmx3jrrIz3+HDh7OlHck4FXT/X0jwu9SIqEvd2vFsDvWhxfC3yeepDTRFREQAJk+ebPrWHJcvX+azzz675znnz5/PpjS2JU8XdJFXbnDw7zAAyrV6gfMlZvPzYlfKlDE5mIiIZKvExESzI5imadOmLF68+J7nrF69moIFC1LG5F+QBQsWpHHjxgQHB9/1nA0bNtC/f3/Wr1+f7uuGhoby8ssvM2/ePPr378+pU6cyI262ypMFndUK33wDWz5/gaRNvTEM8K9YiqD2bcyOJiKSq5UqVYrJkyenea569eqMGzcOuFVcDBkyhCFDhuDt7Y2vry9vvfUW/3vb8fScYxgGEyZMoEyZMri6ulKtWjV++eWX264xcuRIChQoQKtWre6ZOzk5+a7tJSQkMGzYMPz8/HBxcaFRo0a3bdF1v/f9b6Zhw4bx2muv4ePjQ+HChdMcB4iNjaVnz564u7tTpEgRPv/883vmvp+lS5dib2/P66+/TlJS0l3PmzJlCt26dXuotjJLnTp12L59O1euXLnj8R49ejB9+nTCw8Pp27cvq1atuuf1EhISePrppxkzZgy9evWid+/e9O7dOyuiZ6k8V9Ad+msH3Z8I48UX4c+rEyjU5RcsuluXiEiGXYi+wM3kmwCcun6KXRd2pfm4EH3hga47b948HBwc2LZtG1OmTGHSpEnMnj07Q+e89dZbfPPNN8yYMYODBw8yYsQInnvuOUJCQm67xubNm/nqq68eONNrr73GokWLmDdvHrt27aJcuXK0bt2aa9euPdB7d3NzY9u2bUyYMIHx48ezZs2a1OOjRo1i/fr1LFmyhNWrV7NhwwZ27tyZenzu3LlY0vlLLTk5mZ07d/LTTz8RERFx18/Bvn378Pf3x97edu6G1L59e+bOnXvX4w4ODvTs2ZOZM2cSFRXFiy++yJIlS9IU/f/auHEjnp6eFCpUCLhVMG7dupVLly5lVfysYWSzyMhIAzAiIyOzu2kjJTnFOP5FRePnUf2NkJBsb15EJEeKjY01duzYYcTGxqZ5fuz6scaxq8cMwzCMZxc9azCONB9j14+97VolS5Y0Jk2alOa5atWqGWPH3jq3SZMmRqVKlQyr1Zp6fPTo0UalSpVSH9/vnJiYGMPFxcXYsmVLmnb69OljdO/ePfUa1atXT9f7v1d7MTExhqOjo7FgwYLUY4mJiUbRokWNCRMmpPt9/9tOo0aN0pwTFBRkjB492jAMw4iOjjacnJyMhQsXph6/evWq4erqagwfPtwwDMNYvHixERAQkK73NXv2bOPYsVv/fx9//LFRsGDBO/5unjhxojF79uzUx8nJyca0adOM5557zti+fbthGIbRo0cPY968eelq916Sk5ONqVOnGr169Uq9dnh4uNGkSZM0512+fNlo1qxZuq9rtVqNpUuXGs8++6zx559/pjk2c+bM265foEABY+3atfe97r/fG7/88osxe/ZsY9KkScbVq1fTnSsz5foeuuTEZEJmBXNkdzh29nY4P7aKp94PpnFjs5OJiORsA2oNwN/TH4D3mr3Hzv4703wMqDXgga5br169NL1M9evX59ixY2luHXmvcw4dOsTNmzdp1aoV7u7uqR/z58/nxIkTqa+pXbv2Q2c6fvw4SUlJNGzYMPWYo6MjderUeaAVoY888kiax0WKFCEiIgKAEydOkJiYSP369VOP+/j4EBAQkPq4Y8eOhIWF3bedmJgYLl++TLly5QAYPnw4+fLl45NPPrnt3LNnz+Ln55f6+Ndff6Vr167ExcVx+vRpANq1a0dUVFT63+hdLFu2jG7duhEfH586j23NmjX4+/unOa9AgQJp/i/v58qVK+zatQtfX19Kly592zEXF5c0z7m4uHDjxo0HexMmydUbC2/cCK+/EscvL3zAoe0uBNTog3/FUmbHEhHJFYp4FEn9d+n8pSlN6XucfYudnd1tw173mrv1IKxWKwArVqygWLFiaY797ypNNze3TGvzv8OchmGkeS6979vR0fG26/77fv77+ocxa9Ys+vXrl/rYxcWFDz74gP79+zNo0KA0n7eYmBhcXV1THzdv3hyr1cqmTZv47rvvAChbtuwdF0x88cUX91xg0KRJEzp27Jj6uEWLFgCsW7eOOXPmALcWOdxpjmN6hpbPnz/P1KlTSUpKYujQoZQsWfK2c7y8vG773MbExFCgQIH7Xt+W5MqCLuKfi+z7fhxd3v+YilW9uVg7jJZ1Pc2OJSKS5xUsWJALF/5vbl1UVNRtv/C3bt162+Py5cunmcN1r3MCAwNxdnbmzJkzNGnSJFNy3629cuXK4eTkxF9//UWPHj2AW4Xajh070uwNl573fT/lypXD0dGRrVu3UqJECQCuX7/O0aNHM/Q+L168iIuLC76+vmme79GjB5MmTeKdd97h66+/Tn2+QIECXL9+PfWxt7c33377LU2bNk0t9Pbv388LL7xwW1vDhw/PyFvE09OThQsX0rhx49SCOyQkhA8//JAbN27g7e2deq6Dw91LmNOnTxMcHIyTkxPDhg2jSJEidz23YsWKzJo1K/VxQkIC0dHRdyz+bFmuLOhSUlIo7/4nC6aH0eb5etjZqZgTEbEFzZs3Z+7cubRr1478+fPz9ttv3zbZPjw8nJEjRzJgwAB27drF1KlTb1vNea9zPDw8ePXVVxkxYgRWq5VGjRoRFRXFli1bcHd3p1evXhnOfbf23NzcGDhwIKNGjcLHx4cSJUowYcIE4uLi6NOnT4be9/24u7vTp08fRo0aha+vL4UKFeLNN9/Ezu7/Zk8tWbKEMWPG3HPYdcaMGfTv3/+Oq0Rff/11unXrxogRI6hSpQoAlSpV4syZM2nOu3TpUpqi0t3dPU2OhxEeHp46FHzgwAEcHR3x8/Nj3rx59O3bF7hVNHt43Hnz/88++4y4uDhef/3124rWO2ncuDERERGcPXsWf39/QkJCCAoKUkF3N8HBwQQHB6eZA5GZ/v7tbzjwPpVfWkyRMsVIeekIJR1tZ0WOiIjAmDFjOHnyJE8++SReXl689957t/VU9ezZk/j4eOrUqYO9vT1Dhw6lf//+GTrnvffew8/Pj48++oiTJ0/i7e1NzZo1eeONNx4o973a+/jjj7FarTz//PNER0dTu3Zt/vjjD/Lnz5+h950en376KTExMbRv3x4PDw9eeeUVIiMjU49HRkZy5MiRu77+6NGjfPDBB4wfP/6e7bzxxhssW7YMgMcff5x+/frxyiuvpB7v1q0bI0eO5NtvvyU5OfmBiuS76dy5M6NHj+ann34CoFatWkybNi1ND2BoaCjNmze/4+tfffXVDLXn4ODAvHnz+OCDD6hXrx4hISEsWLDggfObxWJk5qB8OkRFReHl5UVkZCSeng/fc3byhJURI+04tvMQ84aNoGjnuRQre/euVRERyZi4uDgOHz5MpUqVyJcva++g07RpU6pXr37bnm0ZPUcyV//+/Rk/fjyFCxc2Owpwa1ua9u3bU6dOHVNz/Pu9cfr0aW7cuEF0dDQ9e/bEx8cn27Pk6CHXjXNmYpz6gT171vLZpEBqd/lDe8qJiIhksnfeeYdp06bx/vvvmx2FqKgoLl++fNdibtKkSfdcAdusWTM6d+6cVfFMk+MKOsNqEHktDu8CbngVr8j16IYcOpCMm4eT2dFERERyJX9/fzp27MjKlStp27ataTkMw2DSpEn3LCxHjBiRjYlsR44q6A4dgvCfepPP8TqN3viVaq0aQyttKCciklts2LAhU86RzFerVi2zI3D16lUGDhxIwYIFzY6SZfbu3ctnn33GqVOneP3112nYsCHvvvsuCQkJXLx4kbFjx1K9evXbXpcjNhaOvBLFm6OuU60a/LStKy5VBmhoVUREJI8pUKBAmk2Oc6MvvviCb775hscff5wXX3yRfv36MWrUKJ555hlWrFiRZkuZ/2XzBZ01xcqV7+tS4sZbvPceTF/ahqAO5nX3ioiIiGSFEydOULRoURwcHDh//jzXrl3jjTfeoFixYly9ehV3d3fatWt3x9fa7JDroc27KFS+Ir5++bheahLtn6hCkbJmpxIRERHJGpcuXUpdsPHXX3/RqFEjatasCUCXLl3o0qXLXV9rcz10V67AyMFXKXW0EaE/3Nq5uXb7xylS1v8+rxQREcl8TZs2TXPXB1uTnfke5P60kn4NGjSgRo0aXLlyhQMHDtC0adN0v9ZmCrqUpBTWzFpAxYAU5izwZcXNtbQcONjsWCIiIgJMnjw5zb1w7+by5ct89tln9zzn/PnzmRUrV1q/fj2GYeTMgu7Yjn00d+3J6y+GcPQoPD2wPg5ONjsiLCIiuUhiYqLZEUzTtGlTFi9efM9zVq9eTcGCBSlTpsx9r1ewYEEaN25McHDwXc/ZsGED/fv3Z/369enOGRoayssvv8y8efPo37//A91pI6dYv349zs7O1KtXL92vMbWgu3jqPH9M+oSUZIOK9WtwofYJXv20Obl8AYuISJ5VqlSp2+7wUL16dcaNGwfcKi6GDBnCkCFD8Pb2xtfXl7feeov/valRes4xDIMJEyZQpkwZXF1dqVatGr/88stt1xg5ciQFChSgVatW98ydnJx81/YSEhIYNmwYfn5+uLi40KhRI0JDQzP0vv/NNGzYMF577TV8fHwoXLhwmuMAsbGx9OzZE3d3d4oUKXLbPW4zaunSpdjb2/P666+TlJR01/OmTJlCt27d0n3dOnXqsH379jveLxagR48eTJ8+nfDwcPr27cuqVavueb2EhASefvppxowZQ69evejduze9e/dOd56cZv369dSrVw8XF5d0v8a0gm7yZBj67C5q5vuc04fPAuBfsZRZcURExEbMmzcPBwcHtm3bxpQpU5g0aRKzZ8/O0DlvvfUW33zzDTNmzODgwYOMGDGC5557jpCQkNuusXnzZr766qsHzvTaa6+xaNEi5s2bx65duyhXrhytW7fm2rVrD/Te3dzc2LZtGxMmTGD8+PGsWbMm9fioUaNYv349S5YsYfXq1WzYsIGdO3emHp87dy6WdO7rlZyczM6dO/npp5+IiIi46+dg3759+Pv7Y2+fsfujt2/fnrlz5971uIODAz179mTmzJlERUXx4osvsmTJEu50R9KNGzfi6elJoUKFgFsF49atW7l06VKGMuUEFy9eJCwsLEPDrZAJq1wNwyA6OvquxxMSEkhISEh9vGPVWgDGjYtiwIDGxD+2h4K+7kRFRT1sFBERyQI3b9684/MXLkD+/ODiAqdOwfXraY8XKXLrI6OKFy/OpEmTsFgsBAQEsH//fiZNmkS/fv3SdU5sbCwTJ05k3bp11K9fH4AyZcrw119/8dVXX9GkSRMAypUrx4QJEx4qU48ePZgxYwZz586lTZs2AMyaNYs1a9bw9ddfM2rUqAy990ceeYSxY8cCUL58eaZNm8batWtp1aoVMTExfP3118yfPz+1R3HevHn4+//fokEvLy8CAgLS1da8efPo1asXvr6+jBkzhvHjx9OzZ8/b7rO+du1agoKCUh+npKQwY8YMduzYweDBgwkKCuLs2bM899xzaTZ9btKkCcHBwbz66qv3zGFnZ0fXrl155plnWLZsGc8//zy9e/emRYsWqeecPn06zf1R7e3t8fDw4ODBg6lFnpmSkpJITEwkMTGR6OhoHBz+r7zy8PBId5ENt1a6FipUiGeeeSZDGR66oIuOjsbLyyvDrzOM4nz5JXz55cMmEBGRrBQQEMCCBQtue/6rr+C556BcOXj7bfjvKWPHwn9GDNOlXr16aX4B1q9fn88//5yUlJTUXqJ7nXPo0CFu3rx52zBqYmIiNWrUSH1cu3bth850/PhxkpKSaNiwYeoxR0dH6tSp80ArQh955JE0j4sUKUJERARwa4+yxMTE1CIVwMfHJ00B17FjRzp27HjfdmJiYrh8+TLlypUDYPjw4cyYMYNPPvmEDz74IM25Z8+eTdNbtGzZMrp168amTZs4deoUQUFBrFmzJk1hCbc2Ab7XPVX/68qVK+zatQtfX19Kly5927H/Dj+6uLhw48aNdF8/K33zzTeEh4fj7OzMO++8k6YjKzIy8rYi+V6qVavGxYsXM5zhoQs6Dw8PIiMj73r8vz10586eo179ehw6dIhixYo9bPPpEhQUdNt8BrWXM9qLioqiePHihIeHZ+gb4mHk5s9nbm9PXy9Z097GjRsJDw+/7diAAbd66ADeew9Gjkx7/E69c3Z2drcNqd1r7taDsFqtAKxYseK23zP/u0rTzc0t09r8bw+MYRhpnkvv+3Z0dLztuv++nzsNRT6oWbNmpenxdHFx4YMPPqB///4MGjQozectJiYGV1fX1Mf/9pytW7eOOXPmALcWOdxpHmJ6eqbOnz/P1KlTSUpKYujQoZQsWfK2c7y8vG57/zExMRQoUOC+188OvXv35vr160RHR9O1a1fy//uNwa06KTs8dEFnsVge6Aenh4dHtv3Atbe3z7a21F7W8PT01NeL2ks3fb1kbnvu7u53PPa/BVvp0rc+7qdgwYJcuHAh9XFUVNRtqxW3bt162+Py5cunmcN1r3MCAwNxdnbmzJkzqcOrD+tu7ZUrVw4nJyf++usvevToAdwq1Hbs2JFmb7j0vO/7KVeuHI6OjmzdupUSJUoAcP36dY4ePZqh93nx4kVcXFzw9fVN83yPHj2YNGkS77zzTprbSxUoUIDr/zOe7unpycKFC2ncuHFqURwSEsKHH37IjRs38Pb2Tj33f4ce/+v06dMEBwfj5OTEsGHDKHKP8fmKFSsya9as1McJCQlER0ffsfgzg6OjI05OTjg5OWVrffO/8sS+IIMHZ+9+dmovZ8vtn8/c3l52y+2fz8xur3nz5sydO5d27dqRP39+3n777dsm24eHhzNy5EgGDBjArl27mDp16m2rOe91joeHB6+++iojRozAarXSqFEjoqKi2LJlC+7u7vTq1SvDue/WnpubGwMHDmTUqFH4+PhQokQJJkyYQFxcHH369MnQ+74fd3d3+vTpw6hRo/D19aVQoUK8+eab2Nn93/rGJUuWMGbMGMLCwu56nRkzZtC/f/87rkB9/fXX6datGyNGjKBKlSoAVKpUiTNnztz2+fh3uPbAgQM4Ojri5+fHvHnz6Nu3L3CrsL1b79Rnn31GXFwcr7/++m2F5Z00btyYiIgIzp49i7+/PyEhIQQFBdlMQWcTjGwWHh5uAEZ4eHh2Ny05UGRkpAEYkZGRZkeRHEBfL1kjNjbW2LFjhxEbG/vQ14qMjDSeeeYZw9PT0yhevLgxd+5co1q1asbYsWMNwzCMJk2aGIMGDTJeeuklw9PT08ifP7/x+uuvG1arNfUa6TnHarUaX3zxhREQEGA4OjoaBQsWNFq3bm2EhISkXmP48OHpyny/9uLj442hQ4caBQoUMJydnY2GDRsa27dvz9D7vlumDh06GL169Up9HB0dbTz33HNGvnz5jEKFChkTJkxI87pvvvnGuNev9iNHjhj29vYGcM+Pdu3apb7m8uXLxlNPPZXmOidOnDC6dOli/Pjjj8aPP/5odO3a1Zg4caJx7dq11HM2b95sjBw5Mj2f4nRZs2aN8dJLLxlz5841evfubZw4cSLTrv2g/v3e+OWXX4zZs2cbkyZNMq5evWpKFothZOKgfDpcvnwZPz8/IiIiKFiwYHY2LTlQQkICH330EWPGjEnXDuWSt+nrJWvExcVx+PBhKlWqRL58+bK0raZNm1K9evXb9mzL6DmSufr378/48eMpXLhwul/z1ltv0b59e+rUqZOFycz17/fG6dOnuXHjBtHR0fTs2TPNitzsku1Drv/+kNUPW0kPZ2fn2zbWFLkbfb2IZI133nmHadOm8f7776fr/KioKC5fvnzXYm7SpEn3XAHbrFmz1JvUS/rkiTl0IiIi8uD8/f3p2LEjK1eupG3btvc81zAMJk2adM/ib8SIEZkdMc9TQSciIjbjfzemfZhzJPPVqlUrXeddvXqVgQMHalpVNlNBJyIiIpnGVvaGy2tMu5eriIiIiGQOFXRic8aNG4fFYknzkZGVVZL7bdy4kXbt2lG0aFEsFgtLly5Nc9wwDMaNG0fRokVxdXWladOmHDx40JywIpLrLV++nKtXr5qaQQWd2KTKlStz4cKF1I/9+/ebHUlsSGxsLNWqVWPatGl3PD5hwgQmTpzItGnTCA0NpXDhwrRq1Yro6OhsTiqSe+zcuZMtW7awd+9es6PYHFdXV3755ZdMv41dRmgOndgkBwcH9crJXbVp04Y2bdrc8ZhhGEyePJk333yTTp06ATBv3jwKFSrE999/z4ABA7IzqkiusGTJEh5//HHs7e3p378/c+fONTuSTWnWrBlbt27l5MmTpmVQD53YpGPHjlG0aFFKly5Nt27dTP0mkZzl1KlTXLx4kcceeyz1OWdnZ5o0acKWLVtMTCZwa1Pg/73Hqa2y9ZzZmW///v3UrFmT1atX8/zzz9O/f/90v/bw4cNZmMx22NnZ4e/vT0REhHkZsquh4OBgAgMDCQoKyq4mJYeqW7cu8+fP548//mDWrFlcvHiRBg0amD4/QXKGixcvAlCoUKE0zxcqVCj1mIj8n7i4OGJjY+96/Pz58/j7+5OcnMylS5dwcEjf4N7kyZPTfROBy5cv89lnn6XrXFvl6urKzZs3TWs/2wq6wYMHc+jQIUJDQ7OrScmh2rRpQ+fOnalatSotW7ZkxYoVwK1hM5H0slgsaR4bhnHbcyKJiYlmRzDVihUrqFmzJp9++uldzzEMA3t7ezp37kz79u25cePGfa+7evVqChYsSJkyZdKVo2DBgjRu3Jjg4OD0Rk+X0NBQXn75ZebNm0f//v05depUpl7flmjIVWyem5sbVatW5dixY2ZHkRzg37mX/+2Ni4iIuK3XTsyVkJDAsGHD8PPzw8XFhUaNGqX5o/+3337D29sbq9UKwJ49e7BYLIwaNSr1nAEDBtC9e3fgVuExYcIEypQpg6urK9WqVeOXX35J02bTpk0ZMmQII0eOpECBArRq1equ+ZKTkxkyZAje3t74+vry1ltv8b+3P79f/lKlSt12v9nq1aunuT1d06ZNGTZsGK+99ho+Pj4ULlz4ttvXxcbG0rNnT9zd3SlSpAiff/75vT+xGfDEE0/QtWvXux6PiIggLCwMgPj4eMLCwmjWrBlnz57l999/T/3Ys2dPmtdNmTKFbt26ZShLnTp12L59O1euXMnw+7iThIQEnn76acaMGUOvXr3o3bs3vXv3zpRr30l8fDwuLi5Zdv37UUEnNi8hIYHDhw9TpEgRs6NIDlC6dGkKFy7MmjVrUp9LTEwkJCSEBg0amJhM/uu1115j0aJFzJs3j127dlGuXDlat27NtWvXAGjcuDHR0dHs3r0bgJCQEAoUKEBISEjqNTZs2ECTJk2AWzeD/+abb5gxYwYHDx5kxIgRPPfcc2nOh1u9/Q4ODmzevJmvvvrqrvn+PW/btm1MmTKFSZMmMXv27HTnT6958+bh5ubGtm3bmDBhAuPHj0/z9Ttq1CjWr1/PkiVLWL16NRs2bGDnzp1prjF37twH7oG+1+t27txJ69at2bx5M2vXruWLL77A0dERf39/Hn/88dSP6tWrp75m3759+Pv7Y29vn+Es7du3v+uCiy+++ILx48ene+rExo0b8fT0TP1Drk6dOmzdupVLly5lONf9WK1Wzp49i5+fX6ZfO92MbBYZGWkARmRkZHY3LTnEK6+8YmzYsME4efKksXXrVuPJJ580PDw8jNOnT5sdTWxEdHS0sXv3bmP37t0GYEycONHYvXu38c8//xiGYRgff/yx4eXlZSxevNjYv3+/0b17d6NIkSJGVFSUyclzptjYWGPHjh1GbGxs2gNx5w0jOf7Wv2POGMaNg/937Pq+W8f/o0mTJsbw4cONmJgYw9HR0ViwYEHqscTERKNo0aLGhAkTUp+rWbOm8dlnnxmGYRhPPfWU8cEHHxhOTk5GVFSUceHCBQMwDh8+bMTExBguLi7Gli1b0rTXp08fo3v37mnar169+n3fc5MmTYxKlSoZVqs19bnRo0cblSpVuvV205G/ZMmSxqRJk9Jct1q1asbYsWPTtNOoUaM05wQFBRmjR482DOPW17qTk5OxcOHC1ONXr141XF1djeHDh6c+t3jxYiMgIOCe72nz5s3G0KFDjZkzZxqjR482Nm3aZBiGYYwdO9Z49dVXjblz5xpz5841nnrqKSMmJsYwDMNYtWrVPa95JxMnTjRmz56d5rnk5GRj6tSpRq9evYzt27cbhmEY4eHhRpMmTdKcd/nyZaNZs2Z3vfbFixeNcePGGaNGjTLCwsLumWPmzJm3Xb9AgQLG2rVr0/9m7uPf742BAwcaQUFBxoQJE4yrV69m2vUzQj10YnPOnj1L9+7dCQgIoFOnTjg5ObF161ZKlixpdjSxETt27KBGjRrUqFEDgJEjR1KjRg3eeecd4FbPycsvv8ygQYOoXbs2586dY/Xq1Xh4eJgZO/c59hXEnb3178Ofwub/GWJb3+bW8bs4ceIESUlJNGzYMPU5R0dH6tSpk2ZlZNOmTdmwYQOGYbBp0yY6dOhAlSpV+Ouvv1i/fj2FChWiYsWKHDp0iJs3b9KqVSvc3d1TP+bPn8+JEyfStF27du10vb169eql6b2qX78+x44dIyUlJd350+ORRx5J87hIkSKpqyVPnDhBYmIi9evXTz3u4+NDQEBAmtd07NgxdWj0Ti5cuEDnzp1544036NevHy4uLml6+bZt20avXr3o1asXKSkp/PHHH8Ct7Tgy6k49VcuWLaNbt27Ex8enzmNbs2YN/v7+ac4rUKDAbf9f/6tQoUKMHTuWt99+m5UrVzJ06FD+/vvvO5575cqV24ZAXVxc0jUHMKPi4uLo3Lkzjo6OmX7t9NI+dGJzFi5caHYEsXFNmzZNM5fpvywWC+PGjbttLpJksvIDwCn/rX9XGgXlX/q/Y81WgfPd7+n57//f/RavNG3alK+//pq9e/diZ2dHYGAgTZo0ISQkhOvXr6cOt/47z27FihUUK1YszTX/u9LSzc0tY+/zAfPb2dnd9nV6p41n/1sEWCyW1Pdzr6/zjFi0aBElSpRInWP63++NWrVqpf7bx8eHqKgo4PbPXXrExMTg6uqa5rkWLVoAsG7dOubMmQPcGi6/0xzG9Awde3h4MGLECBITE1mwYAFz5syhV69eNGrUKPUcLy+v2z5/MTExWXKv2Xbt2nHjxg1TNy9XD52IiDwY1yJg//97QNyKg1fg/x3zrnrr+F2UK1cOJycn/vrrr9TnkpKS2LFjB5UqVUp97t95dJMnT6ZJkyZYLBaaNGnChg0b0syfCwwMxNnZmTNnzlCuXLk0H8WLF3+gt7d169bbHpcvXx57e/t05S9YsCAXLlxIPR4VFZXhVZblypXD0dExTZbr169z9OjRDF3HarXeszh8kPlud1OgQAGuX7+e5jlPT09WrlxJ48aNUwvqkJAQWrZseVuPWXq3RYFbn9Nz587h7u5+W69gxYoV0+wLl5CQQHR0dK4d7VEPnYiIZDs3NzcGDhzIqFGj8PHxoUSJEkyYMIG4uDj69OmTep6XlxfVq1fnu+++44svvgBuFXlPP/00SUlJNG3aFLjVY/Pqq68yYsQIrFYrjRo1Iioqii1btuDu7k6vXr0ynDE8PJyRI0cyYMAAdu3axdSpU1NXmKYnf/PmzZk7dy7t2rUjf/78vP322xkunNzd3enTpw+jRo3C19eXQoUK8eabb2Jnl7Y/ZsmSJYwZM+auw64dO3bkvffe48yZM5QoUQK4NRpyv5WoERERrF69Gjs7O3bu3Mknn3xy34KrUqVKnDlz5rbnw8PDKVeuHAAHDhzA0dERPz8/5s2bR9++fYFbRXF6pkacOnWKmTNnYmdnx0svvXTHor1x48ZERERw9uxZ/P39CQkJISgoSAWdiIhIZvr444+xWq08//zzREdHU7t2bf744w/y58+f5rxmzZqxa9eu1OItf/78BAYGcv78+TS9ee+99x5+fn589NFHnDx5Em9vb2rWrMkbb7zxQPl69uxJfHw8derUwd7enqFDh6a5S8L98o8ZM4aTJ0/y5JNP4uXlxXvvvfdA+6B9+umnxMTE0L59ezw8PHjllVeIjIxMc05kZCRHjhy56zWKFy/OokWLeOutt2jUqBFWq5W2bdvy+++/s3z5cpKSkli+fDnR0dFs3ryZs2fPUrFiRU6fPs2NGzcYMmQIW7du5c8//+Txxx+/Z97HH3+cfv368corr6R5vnPnzowePZqffvoJuDXMO23aNF544YXUc0JDQ2nevPldr717927mzZtH4cKFee211277WvlfDg4OzJs3jw8++IB69eoREhLCggUL7pk9J7MYmTVAn05RUVF4eXkRGRmJp6dndjYtIiIPIC4ujsOHD1OpUiXy5ctndhwxyTPPPMOECRMoVarUfc/t378/48ePz/A9ud966y3at29PnTp1bjv2xRdf4OLiQq9evUzd7+1//fu98W/hGx0dTc+ePfHx8cn2LJpDJyIiIve0YsUKnn766XQVcwDvvPMO06ZNy1AbUVFRXL58+Y7FHMDw4cMZMGCAzRRztkYFnYiIiNxVaGgohQoV4umnn+bQoUPpeo2/vz8dO3Zk5cqV6TrfMAwmTZrE+++//zBR8zTNoRMREZE7+vvvvxkwYAB+fn4kJiYyY8aMdL/2f7dCuZ+rV68ycOBAChYs+CAxBRV0IiIichf169dn3759Wd5OVuwNl9dk25BrcHAwgYGBBAUFZVeTIiIiInlCthV0gwcP5tChQ4SGhmZXkyIiIiJ5ghZFiIiIiORwKuhEREREcjgVdCIiIiI5nAo6ERERkRxOBZ2IiIhIDqeCTkRERCSH08bCIiKSLvHx8WZHELEptvQ9oYJORETuycHBATs7O06fPm12FBGbYxgGKSkpZsdQQSciIvfm5ORE5cqVSU5OJjIykuXLl+Pm5oazs7PZ0URMl5KSQnJystkxVNCJiMj9OTk54eTkxM2bN4mNjSUpKUkFncj/SEhIMLV9FXQiIpJuzs7OeHp6EhUVRWJiotlxRGyKp6enaX/oWAzDMLKzwaioKLy8vIiMjMTT0zM7mxYRkUwQGxtrem+EiC1ydnbGzc3NlLbVQyciIhni5uZm2i8tEbmzbNuHLjg4mMDAQIKCgrKrSREREZE8QUOuIiIiIjlcthd0hmEQHR2Nh4cHFoslO5sWERERyZWyvaATERERkcyle7mKiIiI5HAq6ERERERyOBV0IiIiIjmcCjoRERGRHE4FnYiIiEgOp4JOREREJIdTQSciIiKSw/0/sn0hcDr87VoAAAAASUVORK5CYII=\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 $r=\\operatorname{ch}_0(u)$"
]