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Luke Naylor's avatar
Refactor sagesilent content into separate notebook
Luke Naylor committed
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{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "8ebd0216",
   "metadata": {},
   "source": [
    "# Utilities"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "b87a49bc",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Requires extra package:\n",
    "#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
    "%display latex\n",
    "\n",
    "from pseudowalls import *\n",
    "\n",
    "Δ = lambda v: v.Q_tilt()\n",
    "mu = stability.Mumford().slope\n",
    "ts = stability.Tilt\n",
    "\n",
    "var(\"beta\", domain=\"real\")\n",
    "\n",
    "def beta_minus(v):\n",
    "    beta = stability.Tilt().beta\n",
    "    solutions = solve(\n",
    "        stability.Tilt(alpha=0).degree(v)==0,\n",
    "        beta)\n",
    "    return min(map(lambda s: s.rhs(), solutions))\n",
    "\n",
    "class Object(object):\n",
    "  pass"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cb9c11e7",
   "metadata": {},
   "source": [
    "# Bounds on $\\operatorname{ch}_2(u)=d$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "4cacebc7",
   "metadata": {},
   "outputs": [],
   "source": [
    "v = Chern_Char(*var(\"R C D\", domain=\"real\"))\n",
    "u = Chern_Char(*var(\"r c d\", domain=\"real\"))\n",
    "\n",
    "alpha = ts().alpha\n",
    "beta = ts().beta\n",
    "\n",
    "\n",
    "c_lower_bound = -(\n",
    "    ts(beta=beta).rank(u)\n",
    "    /alpha\n",
    ").expand() + c\n",
    "\n",
    "var(\"q\", domain=\"real\")\n",
    "c_in_terms_of_q = c_lower_bound + q"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "f53cee90",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\beta r + q\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\beta r + q$"
      ],
      "text/plain": [
       "beta*r + q"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "c_in_terms_of_q"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "900f332b",
   "metadata": {},
   "source": [
    "## $\\Delta(u) \\geq 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "6ae4f2e7",
   "metadata": {},
   "outputs": [],
   "source": [
    "# First Bogomolov-Gieseker form expression that must be non-negative:\n",
    "bgmlv2 = Δ(u)\n",
    "bgmlv2_with_q = (\n",
    "    bgmlv2\n",
    "    .expand()\n",
    "    .subs(c == c_in_terms_of_q)\n",
    ")\n",
    "# RENDERED TO LATEX: 0 <= bgmlv2_with_q\n",
    "bgmlv2_d_ineq = (\n",
    "    (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0\n",
    "    + d # Rearrange for d\n",
    ").expand()\n",
    "\n",
    "# Keep hold of lower bound for d\n",
    "bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "c0c12e43",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle d \\leq \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle d \\leq \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}$"
      ],
      "text/plain": [
       "d <= 1/2*beta^2*r + beta*q + 1/2*q^2/r"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_ineq"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "653a3340",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Seperate out the terms of the lower bound for d\n",
    "\n",
    "bgmlv2_d_upperbound_without_hyp = (\n",
    "    bgmlv2_d_upperbound\n",
    "    .subs(1/r == 0)\n",
    ")\n",
    "\n",
    "bgmlv2_d_upperbound_const_term = (\n",
    "    bgmlv2_d_upperbound_without_hyp\n",
    "    .subs(r==0)\n",
    ")\n",
    "\n",
    "bgmlv2_d_upperbound_linear_term = (\n",
    "    bgmlv2_d_upperbound_without_hyp\n",
    "    - bgmlv2_d_upperbound_const_term\n",
    ").expand()\n",
    "\n",
    "bgmlv2_d_upperbound_exp_term = (\n",
    "    bgmlv2_d_upperbound\n",
    "    - bgmlv2_d_upperbound_without_hyp\n",
    ").expand()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "326bb656",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\beta q\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\beta q$"
      ],
      "text/plain": [
       "beta*q"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_const_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "fcd9fa2a",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
      ],
      "text/plain": [
       "1/2*beta^2*r"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_linear_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "ade0f7b2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
      ],
      "text/plain": [
       "1/2*q^2/r"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_exp_term"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "024e8c41",
   "metadata": {},
   "source": [
    "## $\\Delta(v-u) \\geq 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "b7799156",
   "metadata": {},
   "outputs": [],
   "source": [
    "# Third Bogomolov-Gieseker form expression that must be non-negative:\n",
    "bgmlv3 = Δ(v-u)\n",
    "bgmlv3_with_q = (\n",
    "    bgmlv3\n",
    "    .expand()\n",
    "    .subs(c == c_in_terms_of_q)\n",
    ")\n",
    "var(\"r_alt\",domain=\"real\") # r_alt = r - R temporary substitution\n",
    "\n",
    "bgmlv3_with_q_reparam = (\n",
    "    bgmlv3_with_q\n",
    "    .subs(r == r_alt + R)\n",
    "    /r_alt # This operation assumes r_alt > 0\n",
    ").expand()\n",
    "\n",
    "bgmlv3_d_ineq = (\n",
    "    ((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d\n",
    "    .subs(r_alt == r - R) # Resubstitute r back in\n",
    "    .expand()\n",
    ")\n",
    "\n",
    "# Check that this equation represents a bound for d\n",
    "assert bgmlv3_d_ineq.lhs() == d\n",
    "\n",
    "bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d\n",
    "\n",
    "# Seperate out the terms of the lower bound for d\n",
    "\n",
    "bgmlv3_d_upperbound_without_hyp = (\n",
    "    bgmlv3_d_upperbound\n",
    "    .subs(1/(R-r) == 0)\n",
    ")\n",
    "\n",
    "bgmlv3_d_upperbound_const_term = (\n",
    "    bgmlv3_d_upperbound_without_hyp\n",
    "    .subs(r==0)\n",
    ")\n",
    "\n",
    "bgmlv3_d_upperbound_linear_term = (\n",
    "    bgmlv3_d_upperbound_without_hyp\n",
    "    - bgmlv3_d_upperbound_const_term\n",
    ").expand()\n",
    "\n",
    "bgmlv3_d_upperbound_exp_term = (\n",
    "    bgmlv3_d_upperbound\n",
    "    - bgmlv3_d_upperbound_without_hyp\n",
    ").expand()\n",
    "\n",
    "# Verify the simplified forms of the terms that will be mentioned in text\n",
    "\n",
    "var(\"chb1v chb2v\",domain=\"real\") # symbol to represent ch_1^\\beta(v)\n",
    "var(\"psi phi\", domain=\"real\") # symbol to represent ch_1^\\beta(v) and\n",
    "# ch_2^\\beta(v)\n",
    "\n",
    "assert bgmlv3_d_upperbound_const_term == ( \n",
    "    (\n",
    "        # keep hold of this alternative expression:\n",
    "        bgmlv3_d_upperbound_const_term_alt := (\n",
    "            phi\n",
    "            + beta*q\n",
    "        )\n",
    "    )\n",
    "    .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
    "    .expand()\n",
    ")\n",
    "\n",
    "assert bgmlv3_d_upperbound_exp_term == (\n",
    "    (\n",
    "        # Keep hold of this alternative expression:\n",
    "        bgmlv3_d_upperbound_exp_term_alt :=\n",
    "        (\n",
    "            R*phi\n",
    "            + (C - q)^2/2\n",
    "            + R*beta*q\n",
    "            - D*R\n",
    "        )/(r-R)\n",
    "    )\n",
    "    .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
    "    .expand()\n",
    ")\n",
    "\n",
    "assert bgmlv3_d_upperbound_exp_term == (\n",
    "    (\n",
    "        # Keep hold of this alternative expression:\n",
    "        bgmlv3_d_upperbound_exp_term_alt2 :=\n",
    "        (\n",
    "            (psi - q)^2/2/(r-R)\n",
    "        )\n",
    "    )\n",
    "    .subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\\beta(v)\n",
    "    .expand()\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "38bbbcd2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
      ],
      "text/plain": [
       "1/2*beta^2*r"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_linear_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "6701d8a7",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\beta q + \\phi\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\beta q + \\phi$"
      ],
      "text/plain": [
       "beta*q + phi"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_const_term_alt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "7a82d747",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
      ],
      "text/plain": [
       "-1/2*(psi - q)^2/(R - r)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_exp_term_alt2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "927e6929",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
      ],
      "text/plain": [
       "1/2*beta^2*r"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_linear_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "ff25eea1",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\beta q\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\beta q$"
      ],
      "text/plain": [
       "beta*q"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_const_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "id": "e8359c0f",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
      ],
      "text/plain": [
       "1/2*q^2/r"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_exp_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "id": "6823c5b8",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
      ],
      "text/plain": [
       "1/2*beta^2*r"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_linear_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "160871f2",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\beta q\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\beta q$"
      ],
      "text/plain": [
       "beta*q"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "1da23b29",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
      ],
      "text/plain": [
       "-1/2*(psi - q)^2/(R - r)"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_exp_term_alt2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2997ec1a",
   "metadata": {},
   "source": [
    "## Plots for all Bounds on $d$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "d7235dc3",
   "metadata": {},
   "outputs": [],
   "source": [
    "positive_radius_condition = (\n",
    "    (\n",
    "        (0 > - u.twist(beta).ch[2])\n",
    "        + d # rearrange for d\n",
    "    )\n",
    "    .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q\n",
    "    .expand()\n",
    ")\n",
    "\n",
    "\n",
    "v_example = Chern_Char(3,2,-2)\n",
    "q_example = 7/3\n",
    "\n",
    "def plot_d_bound(\n",
    "    v_example,\n",
    "    q_example,\n",
    "    ymax=5,\n",
    "    ymin=-2,\n",
    "    xmax=20,\n",
    "    aspect_ratio=None):\n",
    "\n",
    "    # Equations to plot imminently representing the bounds on d:\n",
    "    eq2 = (bgmlv2_d_upperbound\n",
    "        .subs(R == v_example.ch[0])\n",
    "        .subs(C == v_example.ch[1])\n",
    "        .subs(D == v_example.ch[2])\n",
    "        .subs(beta = beta_minus(v_example))\n",
    "        .subs(q == q_example)\n",
    "    )\n",
    "\n",
    "    eq3 = (bgmlv3_d_upperbound\n",
    "        .subs(R == v_example.ch[0])\n",
    "        .subs(C == v_example.ch[1])\n",
    "        .subs(D == v_example.ch[2])\n",
    "        .subs(beta = beta_minus(v_example))\n",
    "        .subs(q == q_example)\n",
    "    )\n",
    "\n",
    "    eq4 = (positive_radius_condition.rhs()\n",
    "        .subs(q == q_example)\n",
    "        .subs(beta = beta_minus(v_example))\n",
    "    )\n",
    "\n",
    "    example_bounds_on_d_plot = (\n",
    "        plot(\n",
    "            eq3,\n",
    "            (r,v_example.ch[0],xmax),\n",
    "            color='green',\n",
    "            linestyle = \"dashed\",\n",
    "            legend_label=r\"upper bound: $\\Delta(v-u) \\geq 0$\",\n",
    "        )\n",
    "        + plot(\n",
    "            eq2,\n",
    "            (r,0,xmax),\n",
    "            color='blue',\n",
    "            linestyle = \"dashed\",\n",
    "            legend_label=r\"upper bound: $\\Delta(u) \\geq 0$\"\n",
    "        )\n",
    "        + plot(\n",
    "            eq4,\n",
    "            (r,0,xmax),\n",
    "            color='orange',\n",
    "            linestyle = \"dotted\",\n",
    "            legend_label=r\"lower bound: $\\mathrm{ch}_2^{\\beta_{-}}(u)>0$\"\n",
    "        )\n",
    "    )\n",
    "    example_bounds_on_d_plot.ymin(ymin)\n",
    "    example_bounds_on_d_plot.ymax(ymax)\n",
    "    example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])\n",
    "    if aspect_ratio:\n",
    "        example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)\n",
    "    return example_bounds_on_d_plot"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "683ac3f7",
   "metadata": {},
   "source": [
    "### Bounds on $d$ with Minimal $q=\\operatorname{ch}^{\\beta}_1(u)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "f5b1d9bf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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YdmEZVlxcgczcTNwaeQs17WuKHUvjnNywBk+vncKEbb9iwQJg8CABevq6MezMHjrSeWMOjoGHgwd29tkpdhQiolL3/cnv8ePpHyFAwIjGI/BV869QyaqS2LE0iiIrD8amBjA0t4VzVUdE3VXC2kYfgG4UcwB76KgckF6UYvzf4/F4wmP+kCMinfA49TEsjCxgb2aPJeeWIDkrGeObjoejuaPY0TRKdDTweOtg5AoW6DD9Z7HjlCmuciWdN6j+IJgYmGDtlbViRyEiei93E+9iyN4hcF/hjuUXlgMAJjWfhO/bfM9i7l8y5ZmYPSMbdeoA+8PbwrJmG7EjlTkWdKTzrIytMOiDQVhzeQ1ylblixyEiKraopCj03dEXnlJPHIk+goXtFmJyy8lix9JIOdk5iA+pD707CxAUBMzdEoAWfT8VO1aZ4xw6KhfG+IxBJatKyFHmwFDfUOw4RETvJAgCUrNTYWtqi+SsZIQ/Dceqj1fh8wafw9jAWOx4Gif6ahQq1a4BE1MjPLWdicEzm6NKHbFTqQ/n0BEREWkQlaDC/qj9WHBmAbLzsnFp+CVIJBIoVUro6+mLHU/jyGRA8LxYTKtbHaFZ6/DRiMFiRxIFh1yp3JAr5Jh2bBruJN4ROwoRUSG5ylxsvr4Z9VbVQ7et3aAn0cNs/9n5x1nMFaRSqnA05BBq1RKw+GdX7E3eBb9BfcWOJRoOuVK5YaxvjHVX1iE9Jx3LOy0XOw4REYCX91k10DOAUlBiyj9T4O3ijTWfrEHLyi3FjqbRrh8JRXvDzhje8yJGTvdGpUqfiB1JVBxypXLl22PfYvnF5YiZGAMbExux4xBROZaUmYSVF1di9eXVCB8ejkpWlZCancqfTW+R9CwRV/cfQLvAAAgqAVEXr8OjWQOxY2kEtQ25SqVSeHl5wdvbW11NEhUyxmdM/s2qiYjE8ET2BBP+noDKSytjwZkF6O3VGwZ6LwfMWMy9Xl4eIJUCc4bvRmPJRDx9kASJnoTF3L+wh47KnS/2foEj0UfwaMKj/B+iRETq0m9HPxyJPoKxPmMxtulYOJg5iB1Jo13/5xT2/3YT320ehWFDlfhhVjIcKnHPvf9iQUflTnRyNJKykuBTyUfsKERUDpyLOYcfz/yIfnX6oX+9/niW9gxWxlawMLIQO5pWCF0xG9bZx5DndwLePlwY8ibsnqByp4ZdDdRADbz6W0Yi0Z17+RGRZhAEAYfuH8KCMwtw8vFJ1HaonV/AuVi6iJxOs+XkAGfW/YTsPAt0GjcCLQOnQd/gO+jpc2OOt+Fnh8ql1OxUNFrTCHvv7hU7ChHpoD8j/8THv38MRZ4Cu/vuRsSoCHTx6CJ2LI339yEB9eoBEeGPYJgXCwAwNDZkMVcEHHKlcqt1SGvIsmW4HHiZvXRE9F7kCjnWXVmHF+kvsKD9AijyFLjw9AI+rPwhf74UgSJTgWvSPgje1R8vTPphxXIBdevx81YcLHmp3Pqu1Xe4GncVB+8dFDsKEWmpp/KnmHx0MtyC3fDNP98gOSsZgiDA2MAYraq0YjH3DlkZORAEwNjMGNl6bhg93gLHj4PFXAmwh47KLUEQ8OHGD5GrysX5oef5g5eIiiUlKwWVllSCkb4RvmzyJcb6jEUlq0pix9IKggAc2vkE9V60wgPHjfDr01rsSFqPPXRUbkkkEnzX6js8kT1BrDxW7DhEpOEEQcDR6KP4bNdnyFHmwNbUFn/2/hMxE2PwY7sfWcwV0d0bCWjfHvikjysuPO+LKl6VxY6kE9hDR+WaIAhQKBUwMTAROwoRaahcZS62RWzD4rOLcf3FdTSo2AC7+uxCNdtqYkfTOlf/PoHacZ3Ra204Rk+ri48/FjuR7nivHrr58+dDIpFgwoQJpRSHSL0kEglMDEwQK4/FxacXxY5DRBqoyx9dMGj3IDhbOuOfQf/gSuAVFnPFoFKqEHE2AgBQx78lLuYuwu6jNVnMlbISF3Th4eFYs2YNPvjgg9LMQySK0QdHY+CugchV5oodhYhEFiuPRdCRIFx9fhUAML3VdNz48gYOfXYIbau35XzbYrhyBVg+Zjmq3GmKZ4+SYWRiBL/hY2BsZix2NJ1TooIuPT0dn332GdauXQtbW9vSzkSkdnP85+B+8n2su7JO7ChEJJLLzy5j0O5BqLasGtZeWYuopCgAgG9lX9SrUE/kdNol6VkiFgadQZMmwLbwIbhfaT9cqtqJHUunlaigGz16ND7++GO0a9eutPMQiaJ+xfoYVH8QZoXNQpoiTew4RKRmKy+uRJO1TXDmyRksar8IMRNj0LduX7Fjaa2I36ehh8sXWLpUhVPnrdGgg7/YkXResW/9tXXrVly5cgXh4eFFOl+hUEChUOQ/lsvlxW2SSC3mtp6LbbdeTnye3Xq22HGIqAzJFXKsv7IelsaWGNZoGLp5dIOLpQu6eXSDvh7vF1oSN/45jYwMFZp3awXPft8DAMa5cjMNdSlWQRcTE4Px48fjyJEjMDEp2qrA+fPnY/Zs/nIkzVfZujLWdV2Hxs6NxY5CRGXkQcoDLL+wHBuubkBWXhbGNx0PAHCzdoObtZvI6bTT8+fA5MlAgNssGFk6At1awdHVSexY5U6xti3Zs2cPevToAX39//31olQqIZFIoKenB4VCUeAY8PoeOjc3N25bQhpNqVLyr3QiHXM97joarm4IW1NbfNn4S4z2GQ0XSxexY2mtXEUuzmxYihnS1rgT3wRL5idhwOe2vO+qSIpV0KWlpeHx48cFnhsyZAhq166NKVOmoG7duu+8BvehI033OPUx2v/aHpt7bEYz12ZixyGiEspR5mDbrW04F3sOP3/8MwRBwB+3/kD32t1hZmgmdjytl5eTh3tSH0RkDUG7UWNhYyN2ovKtWEOulpaWhYo2c3Nz2NvbF6mYI9IGrlausDS2xOiDo3Fx2EX21BFpmcTMRPxy6RdIw6WIS49DhxodkJWbBVNDUwyoN0DseFrtcVQ8YnePhY3/fNRpWh01Rp6Hp4mR2LEIvPUXUSH6evpY2Wklrjy/gl8u/SJ2HCIqhjxVHuqtqod5p+aha62uiBgVgb8H/g1TQ1Oxo2m1zAwBM2cCjbzNYayMQVbSy9slGrGY0xi89RfRG4z4awT+uPUHIkdHwtXKVew4RPQaeao87I/ajzWX1+C3nr/BztQO/zz4Bw0qNoCDmYPY8XRC5KlwqM6PwEc/HsHnIxwwbaoAC0turqxpir1tCVF5saD9AtyIv4Gn8qcs6Ig0TGJmItZfWY+fL/2MJ7InaOHWAnHpcbAztUO76twjtTRkpilgZmkMx2pVcO9cLZw5mY1qdQCAxZwmYg8d0VsIgsDb/BBpkFffkz229cChe4fQv15/jPUZi0bOjcSOpjPkcuDgyhD4mH8P6/7XYe/EBSTagHPoiN5CIpHgedpz9N3RFy/SX4gdh6hcylHmYOutrWi5oSX23d0HAFjUfhFiJ8ViY7eNLOZKiaBS4c9f4+HhAcxb0wJP9D+HuTkXhWkLFnRE76An0cOJhyfwxb4voOYObaJyLS49DrNDZ6Pq0qrov7M/jPSNYGf68n6g7nbunCNXyk4Fj4F7TAf4tVLh4Kma8B/zLUzMjcWORUXEgo7oHSpYVMCGbhtw8N5B/Bz+s9hxiHSaIAjIzssGAGyP2I6FZxeiq0dX3Bx5EycCTuDDKh+KnFC3JD9PwoObTwAAjs2HAY2WYOs2Pbjxphlah3PoiIpo9IHR2HBtAy4HXoaXo5fYcYh0SpoiDb/f/B2rLq1CC7cW+Pnjn5GRk4EcZQ5sTW3FjqdzlEpg7VoB3snNkCG4otW3O8WORO+JPXRERbToo0XwdPDEjRc3xI5CpDMepz7GqAOjUGlJJYw6OApVbKqgl2cvAIC5kTmLuTJw49gZdPJ/jpEjJdj/4md4DuLIgy5gDx1RMfAer0TvLzsvGw9THsLT0RO3E26j7ea2GNZoGIY3Gg43a471laWs9Cykb6mCPREj8cFns9G0qdiJqLSwoCMqpjxVHoKOBMG/qj+61e4mdhwirXE/+T5WX1qNjdc2ws7UDnfH3IVEIuEfSmUsJzsHZzatRp0un8OpkiUeXL+LqnVrQk+fg3S6RG3/m1KpFF5eXvD29lZXk0RlQl+ijyfyJxi0exDuJN4ROw6RxkvISMBHv36EmitqYv3V9QioH4D9A/bn7/HIYq7sHD0KdPB7gcaG3+JO2DEAQPX6HizmdBB76IhKIE2RhmbrmyFPlYeLwy7C2sRa7EhEGuVR6iOceHgCQxoOgVKlRN8dfdGlVhf0qdOH91VVg6dRT3D1jyXoMXcRWrQ0xM/ByajTyE7sWFSGWNARldC9pHvwXuuND6t8iL399kJPwr94qXzLUeZg7529WHd1HY5GH4WlsSUejX/EhQ0iuHvhOswvdcV127/Rub8neMMb3cffQEQlVNO+Jv7o9QcczRyRp8oTOw6RqJQqJWqvrI0+O/ogPScd67uux9NJT1nMqYkgAKd3HsOJeYOQmyPAo2l9VBwejY8HsJgrLwzEDkCkzTrV7IRONTsBAJKzkvN3sSfSdRk5GdgRuQO/3fwNO/vshJWxFX5o+wPqOdVDHac6YscrV+7eETBuvATKpwLmByRCniSHvbM1DIz4K748YQ8dUSm4FncN1ZZVw8F7B8WOQlSmLj+7jJH7R8JliQs+3/s5JJAgISMBANCvbj8Wc2oW+vNCXJEOxP37wLgf2qHJ14dg78w5veURy3eiUlDXqS78q/qj95+9ERoQCu9KXM1NuiM5KxnWxtbQ19PHlH+m4HbibYzzGYcvGn6BarbVxI5X7ggqAZnpCphbmcDIriqcPXIQsVCAiSnHVsszLoogKiWZuZlot7kdopKiEPp5KOo61RU7ElGJ5SpzcTj6MEKuh2Df3X3Y338/2tdoj+dpz+Fo7ggDPfYHiOHaVQGZh7pArqqGjtNXiB2HNAiHXIlKiZmhGfYP2A83azcM3j0Yav5biajULDm3BK7BrujyRxdEJUVhftv5aFCxAQDA2dKZxZwIUuKSMX5MNho3kWDvtX5watRd7EikYfhdSVSK7EztcHTQUaRmp+Zvmkqk6eIz4rHlxhZ8XOtj1LKvBVMDUwyoOwABDQLyCzkST6Y8E4rddWGfMAaLF0/DmDEDYWgodirSNCzoiEqZg5kDHMwckJ6TjoG7BmKW/yz+UiSNo8hT4K+ovxByPQSH7h2Cvp4+HMwcUMu+FkZ6jxQ7HgG4dfIKavo0gJmVGa7bLcOXCz+EUxWxU5Gm4pArURnJzstGrDwWfpv8cPLxSbHjEEEQBOQqcwEAU49NRe8/eyM+Ix7LOy3H86+eY1D9QSInJACIiwMmj7wHrydNcOr3nQCA5n17w6lKRZGTkSbjogiiMpSmSEP3bd1xNuYstn26DV09uoodicqhiPgI/H7zd/x+63d81fwrjPEZg0epj5CVmwVPR0+x49H/y1Xk4u91+/DZ1F4wMgI2LTiGToP9oW/Ie93Su7GgIypjijwFPtv1Gfbc2YPI0ZGoZV9L7EhUTuy9sxczQmfgxosbsDWxRW+v3ghsHIjGLo3FjkavEb7nABqld8X86zcwamod2HGfcioGFnREaqBUKXEk+gg61ewEQRAgQOC9X6nUJWYm4s+IP+Hl6AW/qn7YH7UfW25uwYC6A9DBvQOM9I3Ejkj/8TTqMW4dPYwOowMhqAQ8jriPqvVqih2LtJDaCjqpVAqpVAqlUomoqCgWdFRurby4EgfvHcTvvX6HjYmN2HFIyyVnJWPvnb34M/JPHH1wFIIgYE7rOZj24TSxo9FbZGcDixYBSed+xuRPfoBBt0g4VeLvRCo59tARqdnh+4fRb2c/OJo5Yk+/PfBy9BI7EmmZpMwk5Chz4GzpjJUXV2LcoXH4sMqH6FunL3p79YajuaPYEektLu7ej0N/PsC8HeMwaaIS336TCUtbS7FjkZZjQUckgvvJ99F9a3c8Sn2ElZ1XIqB+APeto7dKzEzE7tu7seP2Dhx7cAyjvEdheaflkGXLkJmbCWdLZ7EjUhGFLp8Bo8wbsO++Gx61+X1PpYMFHZFI0nPSMe7QODxKfYR/Bv/DOXVUiCAIkEgk2Hh1I4b/NRwCBPhV8UNvr97o6dkTFSwqiB2RiiBNLiB8/bdQ6FdGp3FfQpmrhJ6+HiR6LOao9HBjYSKRWBhZYEO3DcjOy4aeRA8nH59Edl42PqrxkdjRSER3Eu9gz5092Ht3L7rU6oJpH05Dy8otsbLzSvSo3YNFnBYRVAL+2AoEBUkw9aNMeDXJBABuQ0JlggUdkchMDEwAAJuubcLGaxsxvNFwLGi3ALamtiInI3Xac2cPpvwzBVFJUTAzNENH945oWLEhAKCWfS1ud6NlMmQZuL+2E3b/ORbNm/dGl1lLUYV3eaAyxIKOSEOs77oeTSs1RdDRIOy5swcL2y/E4PqDORSrgzJyMnDs4THsu7sPbau1Rf96/WFrYotWlVvhp49+QttqbWFqaCp2TCoBeUomLG3MYG5tjhS9Jpgy0wlNOoudisoDzqEj0jDP0p4h6GgQzsacReSoSP5i1yGH7h1C8PlghD0OQ44yB7Xsa2Gq71R83uBzsaPRe1IqgR0b76GtqgUi7Xai1aetxI5E5Qz/9CfSMC6WLtjScwuujrgKU0NT3E++j95/9kZEfITY0agYMnMzcSDqAEYfGI2j0UcBADKFDHoSPSxstxB3x9zF3TF3WczpgCunY9G0KdA/sAZOxY2BRxNuDEzqxyFXIg31atPhZ2nPcOnZJdRbVQ8D6g3ALP9ZcLdzFzccvdGfEX9izZU1OPX4FBRKBarZVEPLyi0BAP3q9kO/uv1ETkil6eLu/WiQ1hNu1jex/LQHWrSYKXYkKqfYQ0ek4VpVaYW7Y+5C2lmKE49OoPbK2th0bZPYsco9QRBwO+E2Vl5ciR7beuBszFkAwIuMFzDUM8T8tvNxZ/QdRI+LxoB6A0ROS6UpV5GLq8cvAwDqd2iPi5J12PG3O1q0EDkYlWucQ0ekRbJys7Dq0ip08+iGGnY1cCDqAMyNzOFXxY8bE5exVz8qJRIJZoXOwprLa/A8/TkM9QzR3K05ZvnNQutqrUVOSWXt+HHgxh8/YEjTBcj9OBYOzrzDA2kGFnREWqzLH12wP2o/POw9ENg4EIPrD4aDmYPYsXRCnioP1+Ou4/ST0zgdcxqnn5zGscHH4OXohRUXViBGHoO21drCt7IvzI3MxY5LZexp1BOsCX6AOb/4o0PrVCyd9xC1mzcUOxZRPhZ0RFpMEASEPQ7DmstrsPP2TgiCgPDh4ahfsX7+XQaoaBIzE3Hzxc38XrY6P9dBZEIkjPSN4FPJB75uvviyyZeoYsPNxMqj04sGw1q4hesul/HZZxLwW4s0DQs6Ih2RmJmI7RHbEdg4EAZ6Bui+tTtMDU3RzaMbOrp3zF9kQS9l5mZi5cWVCH8WjkvPLuFR6iMAQGJQIuzN7LHv7j7Ym9qjsUvj/M2fqfwQBODinv1QwhItevgh8WkCjEyMYWXP31ukmdRW0EmlUkilUiiVSkRFRbGgIypDgiBg0dlF2HprK67GXYWBngGauzbH+q7rUdO+/GypoBJUeJz6GLfib+Fm/E3cjL8JQz1DbO6xGXmqPFRaUgm1HWrD28UbTVyawNvFG9Vtq7Nns5y7dw8YP17A5EatobLwRJtvVokdieid2ENHpONiZDHYH7UfRx8cxcZuG2FtYo3P93yOp2lP0dKtJZpWaoomLk3gaO4odtQSy1Xm4rHsMaKTo3En8Q5q2ddCp5qdcCT6CDr81gEAYG1sjXoV6qGFawssaL8AADgsTQVkZ2Tj/NrZmPJzb8TnNsLyn+T4pLslJHr8GiHNx4KOqBxac3kN9t7di/Ox55GclQwA2NprK/rW7YsLsRdwJ/EOatrXRDWbaqhoUVH0okepUiIhMwHP0p4hRhaD6JRo9PTsiao2VTE3bC5mh82GUlACeHlv3InNJuKHtj9Ali3DudhzqOtUF5UsK4n+Pkiz5SpycU/aAncxFh1HDoYpb9JCWoQFHVE5JggColOicS3uGlq4tYCLpQtmhc7C7LDZ+eeYGJhgnM84LGi/AM/SnmHZ+WVwNHeEg5kDHMwc4GjmiKauTQEACRkJMNAzgLGBMYz1jfMLKD2JHrLzspGRk4E8VR7yVHnIysuCsb4x3KzdkJKVgp23d0KWLUNqdiqSspIgU8iwpecWAEDdn+siIuF/d8owMzTDjt470KlmJ5x5cga34m/B3c4dNexqwM3KDfp6+mr8LJI2u30pBmnHh8Oi9S/w8q4KZa4S+ob8+iHtw4KOiApJz0nHw5SHeJj6EA9SHsDD3gOdanbCtbhr+HT7p0jMTIRMIQMAOJg5ICEoAQDgvtwd0SnRBa51eOBhfFTjI8w4MQNzT84tcGzQB4Owucdm3E++j5orasLK2Ao2JjawM7WDi6UL9vXbB309ffx19y8AL2+L5mLpohG9hqTdUpJVmDlLD5vWpePwt91g4bsQ9fwaix2LqMRY0BFRieQoc5CUmQS5Qg4PBw8AwPGHx5GanYrsvGxk5WZBIpHgoxofwdXKFZEJkbibeBcGegb5vXhVrKughl0NqAQVBEFgzxqpxbUjYTC79SXaLzyNsV/bY9w4wMhI7FRE74cFHRERlQupiemwcbBAwpMXuL19Ojx6/4AKVbR3MRDRv7GgIyIinRYfDxyVrkBL+6WwHnALtg5c7UC6R0/sAERERGUhV5GLdSueolYtYNm2j/DE/GtYWXNslXSTgdgBiIiIysLF5UPwQXYU+vW7gO+/94DD/8/1JNJFLOiIiEhnPLsfg8x0JdwbVIW971cQBAG/tOCKaNJ9LOiIiEjrKRTATz8J6GbQGfF59eHe4DfUbt5Q7FhEasM5dEREpNUu7j4Av6ZxmDlTgr/TfkPjET+LHYlI7VjQERGR1pInyeGeOBhftP4VN24AX82tDyt77qBA5Q+3LSEiIq2SIcvA+ZCl+KD3eDg6WyD27mNUqlkZEj3OlaPyS209dFKpFF5eXvD29lZXk0REpEMEAdi2DWj3YQIamS7G/XPnAACuHlVYzFG5xx46IiLSeA+u3UHkrp/Q44ef8UkXQwQvSkdVdwuxYxFpDK5yJSIijZeTlQVP+zM4ti8WrTpXA8Bijujf2ENHREQaR6UCTmzeDf2nW+E7eSsMDCVQKVXQ0+daPqLX4XcGERFplPCLSjRvDixYYg5DIwNkpGUBAIs5orfgkCsREWmM0GVT8fz+IygUf+A76Udo+eFHYkci0gos6IiISFR5OXnISM+BtZ0ZjCo0hIu5Gy4HC9A34MpVoqLiHDoiIhJNWKgAq4u+iMv1Rqdvl4odh0hrcUICERGp3fPoWAwcoIB/awl23xqDqv6DxI5EpNU45EpERGolS5TB7ERd1FDOwKZNkzBoUH/osXuB6L0U61to1apV+OCDD2BlZQUrKys0b94chw4dKqtsRESkQy4dPIkchQrWDta4YxuCSdKhCAgAizmiUlCsbyNXV1f8+OOPuHTpEi5duoQ2bdqgW7duiIiIKKt8RESk5aKjgTEDb6JJqh/O7nzZCdC0VzdYO1iLnIxId7z3ogg7OzssWrQIQ4cOLdL5XBRBRFQ+ZMgycGjNLnw2fRAqVAA2LT6P1p825X1XicpAiTu6lUoltm7dioyMDDRv3rw0MxERkQ64efQIPnEajh+nR+POHaBNn2Ys5ojKSLF76G7evInmzZsjOzsbFhYW+P3339G5c+c3nq9QKKBQKPIfy+VyuLm5sYeOiEgH3b8cgftnjqPjuLEQVALiHj6Fcw1XsWMR6bxi99B5eHjg2rVrOH/+PEaOHImAgABERka+8fz58+fD2to6/8PNze29AhMRkeaRyYCJE4EV04+jFqRIScyCRE/CYo5ITd57Dl27du1Qo0YNrF69+rXH2UNHRKTbzvz2Gw7vS8CSgxMx47s8jB+rhLGZsdixiMqV996HThCEAgXbfxkbG8PYmN/YRES6RhAAiQTITbqDNo2fYcQSAZVcDcAtTonUr1jfddOmTUOnTp3g5uaGtLQ0bN26FaGhofj777/LKh8REWmYhHgVIkNGIcOoATqP/xJ+Y+dysQORyIo1h+7FixcYNGgQPDw80LZtW1y4cAF///032rdvX1b5iIhIQ+TlKLFypYBaHnq4eccc5pZGAMBijkgDvPccuuLiPnRERNpHlpCKuC1+mPb7TNjV74kffgAcHcVORUSvcKIDERG9UeKzVNg728Da0QZX0Qlzl1SFl6/YqYjov9hDR0REhSgUwG/Sm+hj0wLXbf+Gb4+WYkciorfgLZGJiKiAkwfuoV494MspdXAiYQbq+dYROxIRvQOHXImIKN/ZP7ahee5ANKkdhV27qqFu3SCxIxFREbCHjoionMuUZ+LiwTMAgEZdu+GK2Z/Ysqcq6tYVORgRFRkLOiKickoQgB07gPWTfkD1mO6QJWfBxNwETT/tzq1IiLQMCzoionLo/pVIBA06ht69gTPJXyGj5TlY25mKHYuISoirXImIyqGzi/pCP+cFkhqGonNnsdMQ0fvioggionJApRRwdstmwKIqfHv6ofbAn2FubQFjM7GTEVFpUNuQq1QqhZeXF7y9vdXVJBERAbh8GfD9EBDur0PWg8MAADtnexibGYucjIhKC4dciYh0VFpKGq6sD8L4lSOhtKwP6fIstGrNeXJEuoiLIoiIdMyrP9NNzExgK7mJeVMf4OpVsJgj0mHsoSMi0iGXTtyD4aXPYdL2D3g0qgxBJXALEqJygD10REQ64FlsHgYOBNp8XAGybHsgLwMAWMwRlRNc5UpEpOXC9x6Ew6MxuHDqMpautIXv5/ugxz/XicoVFnRERFoqPjYZTq52qFy/IW4/6InwS/qwcRQ7FRGJgXPoiIi0zMOHwNk136NFxU2wHxwBK1tuP0JU3rFTnohIS2TKMxE89xE8PYG1hz/Fc6d5sLQ2EjsWEWkA9tAREWmJC4u6ITcjGYdyT2HaNMDcXOxERKQpOIeOiEiDRV+9A6VgjFqNqsGm1RyYmJtjXl2xUxGRpuGQKxGRBpLLgaCvVVCd6IKYw/MBAB5N66NKXXeRkxGRJmJBR0SkQVRKFU7/+itaNI7Hz6v0cFy5Dy3HLRc7FhFpOBZ0REQaRJaQCq/sCRjfYzfu3AFGBHnCxNxE7FhEpOHUVtBJpVJ4eXnB29tbXU0SEWmF5OfJOPbTN0iMy4BtRTsoO0Ri+MIRcHMTOxkRaQuuciUiEolSCaxZA6wNfoBDE1sgpuouNOnUQuxYRKSFOORKRCSC22ev4O+ZAzF2TB4a+FYHuj9mMUdEJcZtS4iI1EhQCZDoSQBBQBW7+7h4Mh6NWroA4N0eiKjkOORKRKQGOTnAsbUhsEjdgxZTdkHfQPK/4o6I6D1xyJWIqIwdPZyDDz4ApOsdoTJxhSIrBwBYzBFRqeGQKxFRGTq5eATiH6ajQoUt+GFFZ3zwQWexIxGRDmJBR0RUyrLSs5CdpYKtozkM3dqhWiUlQlcCEnbIEVEZ4Rw6IqJSIgjAnt1KeD2ujwdZHdFp2mKxIxFROcE5dEREpeDBtbv4pHMOevbSx467s1D7k0CxIxFROcIhVyKi95T4NAHO1xqgic1ifLlvND755FMOrxKRWrGHjoioBASVgDPb9yM3R4BDJUdE2O/B1HVD0aUL58oRkfqxoCMiKqZr14DAT8PRMq8LLu4PAwA06dIBJuYm4gYjonKLBR0RURElP0/C73NWo3Fj4OxdH1ywv4GWPf3FjkVExIKOiKio7oT+g48rTcbq4FhcuwY07VBP7EhERADUuG2JVCqFVCqFUqlEVFQUty0hIq1w4/h5PL1+Bp0mfgVBJSA5Lgn2Lg5ixyIiKoD70BERvUZcHDBlCmAbvxTD2v2BqsNPwcLKSOxYRESvxSFXIqL/CFu7AitGL8aBA0DdnmPgOe4sizki0mjch46I6P8JKgESPQmEzBdo3UyBr9YCdnb8MUlEmo9DrkRU7j1+lIdn2/sjxaQ9Oo8LzC/siIi0BYdciajcykpXYM4cAZ5eBrhyrzpsKtgDAIs5ItI6HEsgonIpITYeWXua4fqBYIwb1w2Dv10AS0uxUxERlQwLOiIqV549iIdLdSc4uDgiTBiMRWs8Ub2+2KmIiN4P59ARUbmQlgaELLmA4VVb4Yr9STT/pKnYkYiISg3n0BGRThNUAg5suY7atYEpC5rgWOpSNPD/QOxYRESligUdEem005s3oL3SG51bP0PkbX10Hj8SphamYsciIipVLOiISOekxCXj3J4TAIAmnw5AhNNhrP3NBVWqiByMiKiMcFEEEekMpRJYvx5Qhn+PLg12ICMtGuaWpmjYsbXY0YiIyhR76IhIJ9w8cQ7j+hzBiBHATWE6jD65CHNLQ7FjERGpBVe5EpFOOL+oJ7KzlDBqtxctWoidhohIvdQ25CqVSiGVSqFUKtXVJBHpsNwcJc5sWA59x0b4sJcfPD/fAAsbS+izU46IyiG1DbmOHj0akZGRCA8PV1eTRKSjjh0DGjSQQP/5LmQ/vQAAsHa0gb6hvsjJiIjEwUURRKQ1UuKScXvzSExYOR22levBsvtxfNiQXXJERCzoiEjjqZQC9PQlMLexgKleApbOj0ObAfUgkbCYIyICuCiCiDSYIABhe26g4sNB0GtzALUauIodiYhII3HbEiLSSPfuZKNzZ6DrZ1XxPLMOjI3yxI5ERKSxOORKRBrn7NbtcEv4Ck8f3cTm323g3+13SCRipyIi0lzsoSMijSCoBMREPQcAVPNpiWhJIC6EG6N7d7CYIyJ6B86hIyLR3bgBRPw2GT4ue+A8LBJmFhw8ICIqDvbQEZFoUuNTMPvraDRsCPx2bghSa65kMUdEVALsoSMi0VxZ3A6pMgmu2hzF2LGAkZHYiYiItFOxeujmz58Pb29vWFpawsnJCd27d8fdu3fLKhsR6aBbYRdw98pDAICl/1LUGboZX33FYo6I6H0Uq6ALCwvD6NGjcf78eRw9ehR5eXn46KOPkJGRUVb5iEhHvHgBDPsiF5bX++DJ0RUAgJpN6qJCVWeRkxERab/3GnJNSEiAk5MTwsLC0KpVqyK9hkOuROVLriIXZzb9giGz+0OucMCKH6LR94uqvO8qEVEpeq/ZxzKZDABgZ2f3xnMUCgUUCkX+Y7lc/j5NEpGWkSWmop5kNr4bZo9u4wfA3r6G2JGIiHROiXvoBEFAt27dkJKSglOnTr3xvFmzZmH27NmFnmcPHZHuenrvKe7u+gH1hyyCvZMZZAmpsHa0ETsWEZHOKvG2JWPGjMGNGzfwxx9/vPW8qVOnQiaT5X/ExMSUtEki0nDZ2cC8eUCH9lmoaX4AT2+/XDTFYo6IqGyVaMh17Nix2LdvH06ePAlX17ffLNvY2BjGxsYlCkdE2uP60ZN4HroMcxZtx7jx7rAeeB9uNtxTjohIHYr101YQBIwdOxa7d+9GaGgoqlWrVla5iEhLKPME6BtIYGBsAgebdNy8nIpa9ezBW0UTEalPsebQjRo1Cr///jv27t0LDw+P/Oetra1hampapGtwlSuRbkhPB0JXL4NN9j9oPmUf9A14w1UiIrEUaw7dqlWrIJPJ4O/vD2dn5/yPbdu2lVU+ItIwgkrAn39konZtYO3Wmsi1bAxlnlLsWERE5Vqxh1yJqHw7/1MfZD0whbf3ZixZ0hnVqnUWOxIRUbnHSS5E9E6yhBTkqQxhX8ECqNIPdTwtMfgTsVMREdEr73WniJLgHDoi7aFSASEbc9A2qxYi0vuj0zfzxY5ERESvUeJ96IhIt906eRktW+Tii2FG2PFgGRr2HSt2JCIiegMWdERUyPMHT+HxqBk+qvkrTp0CJi3phorVXMSORUREb8CCjogAAHk5eQgN2YrcHAHO1Sshwvk4ZmwIgK+v2MmIiOhdWNAREUJDgaHdzuFD/c9w7Xg4AKBB+w+hb6gvbjAiIioSFnRE5djz6FiETF2G1q2Be7IPEVnjHrw7+ogdi4iIiokFHVE59uB8GDpVmY+tIQk4fRqo17y62JGIiKgE1LZtiVQqhVQqhVKpRFRUFLctIRLJhT2HkXz/Cjp9PRWCSkC6LB2WtpZixyIiovfAfeiIyon794EJEwAP4Sf08zuKemMPwMSUc+SIiHQB7xRBVA6ErpiNw/+Y4ebNIAz5aQKa9JwEiZ5E7FhERFRKWNAR6ShBJUClEqBvoAcIeWjbJg/f/QGYmbFXjohI13DIlUgHRdzIgeJwJ8SZ9EfnscPEjkNERGWMq1yJdEhqYjomTADqNzLC+egWqFC9qtiRiIhIDTjkSqQjnt2PgeGxJoi9sB4//PAJhk2YCyMjsVMREZE6sIeOSMs9uPkYAOBc3RURyolYsaUhJk8GizkionKEBR2RlkpIABZ9fQJVrlfHxcNXIdGTwH/UN3CuXknsaEREpGYs6Ii0TF5OHravOo9atYAfN7TCiawQNGpdT+xYREQkIhZ0RFrmzCYpupi2xuf9k3Dnrj7aDR8IAyNOhyUiKs+4bQmRFngeHYvoKxHw7d0BmfJMPL55G54tG4sdi4iINAT/rCfSYAoFEBwM2EXPR9u6x6DIjoCZlRmLOSIiKoBDrkQaKnzvQYzq8TemTwce23wPh88uwNiEd3kgIqLC2ENHpKFUUb+gb1NbTFrUEXXq2Iodh4iINJjaCjqpVAqpVAqlUqmuJom0SkZaDsI3zIGB20fw7dkKdYb/DnMrc0jYj05ERO+gtl8Vo0ePRmRkJMLDw9XVJJFWEATgzz8Br7oG0Es6jZzE2wAACxsLSPQkIqcjIiJtwCFXIhHFP47Do22fY+byJWjQ2AuuAcdRvQa75IiIqHhY0BGJIDdHCUMjfVg72kIi0cP6VTI07wJwnRIREZUE96EjUiOVCjj46wXUlX2GPP8wuH/A23QREdH7Y3cAkZpcuZiGli2BgaM8EJ3RFhaW3IKEiIhKB4dcidTg1Kb1qJ4+A8i9jb0HbeDnt1rsSEREpENY0BGVkbycPDy5+xTV61WB+4cdEHU8E6fOmsHASOxkRESka1jQEZWBU6eAuH1j0MAlDDm1IuBcwxXONcaKHYuIiHQU59ARlaLnD57iq8AotGoF7Lw1HrlNNsPImN9mRERUtthDR1SKEnd/inb2Tqi7YS8CAjyhx1qOiIjUgAUd0Xu6tO9vWFSqjdqNq8KszQa0cHWGtaPYqYiIqDxh/wFRCT14AHzaIxsVnwzDk9BNAIAaDT1h7Wgjai4iIip/WNARFVOmPBPHln0PX+8kXLhkgquO59F+4kyxYxERUTnGgo6omNJT5PjAZDkWBJ3FnTtAl76ukOhJxI5FRETlmNpu/SWVSiGVSqFUKhEVFcVbf5FWuXf1Pp4enov6w36BrYMpMuWZMLMyEzsWERERAN7LleitZDJg9mzg7+1R2PNVTwi+O+Hh7SF2LCIiogK4ypXoDcL3HEBq+CqsXbMX076thSojb8LYhEOrRESkeVjQEf1HjkIJI2N9mFjbwdzGArdvZsC1mhUAFnNERKSZOORK9P8SE4Gza+bAXriA5t/sh54+CzgiItIOXOVK5Z4yV4m1P6ehVi3g1wNNkOfUAWr+O4eIiOi9cMiVyjVBJeD6sg4wuOeGnj034ocfOsPJSexURERExcOCjsqluIfPYGBqBYeKFsitOgY+Pi4Y0krsVERERCXDIVcqV3JygOBFWdA/3ADhmxcDAJp+2h11WvmInIyIiKjkWNBRuXFxfygaNchF0FRT7Hi2GS2GTBA7EhERUangkCuVC08iH6Bxalv0ab4FPbb1Q716HcWOREREVGrYQ0c6KzMtC8dWr0NeroDKXtVxt+oFfLe2L+rVEzsZERFR6WJBRzpHEIBdu4Bh3c/C12Q0Is7eAgB4+TaBRI97yxERke5hQUc65cG1O1j/1QL06gXITNrimfdD1PdjlxwREek2FnSkU57dvIj21dbi0D4ZDhwAqnm5iB2JiIiozKnt1l9SqRRSqRRKpRJRUVG89ReVCkEATm3ZjvRnd9F58ndQKVXIyc6BibmJ2NGIiIjURm09dKNHj0ZkZCTCw8PV1STpuKtXAV9f4K+tj2GFO8jLFaCnr8dijoiIyh1uW0JaR1AJOLl0Ag6eqASZbDI6z/0avm242IGIiMovzqEjraHMVSI3RwWJngSCgQ06fGKFq1eB1izmiIionFPbHLpX5HI5rK2tOYeOiuXcqUzYXPLFQ8Ox6DxmiNhxiIiINAp76EijxT1JxuDBQItWZjgZ3R2V69UVOxIREZHG4Rw60liPb92H7YVGUDzcjrVrO+KLL2ZAj3+CEBERFcJfj6RxIs/fBgBU9qqBq8q5WL2zKYYNA4s5IiKiN+CvSNIYjx4BP4zej9r36+BqaAQkehL4BY6HjZOt2NGIiIg0Ggs6El1WehY2LQyFpyfwy76OOK23Gw1aeYkdi4iISGsUu6A7efIkunTpAhcXF0gkEuzZs6cMYlF5cmHzMvSw64Ypk+SIvG2AVgO6QaLHrUiIiIiKqtgFXUZGBurXr4+VK1eWRR4qJx5cu4Ow3/cDALw/G4tkn3DMmmcFCwuRgxEREWmhYq9y7dSpEzp16lQWWagcSEsD5swBPGRL0MzjKvJyP4a5tTmqfVBL7GhERERai3PoSC0ElYAzW7ZgZLeDkEqBlCqL4P7lKRgYcmiViIjofZX5PnQKhQIKhSL/sVwuL+smSQNJ9CQwevY7+vvXxA+bOqNyZWuxIxEREemMMi/o5s+fj9mzZ5d1M6SBkhOycDPka+jX6A/fHr74YPQuGJsZix2LiIhI55T5kOvUqVMhk8nyP2JiYsq6SRKZMk/A6tWAh5cxVLIoqNJf/p+zmCMiIiobZd5DZ2xsDGNj/iIvL2LvPkbyX/0QvGQjPvmkNjxHHUFFZ86TIyIiKkvFLujS09Nx//79/McPHz7EtWvXYGdnh8qVK5dqONIeWRk5MDU3goNrBcQIlbDtdwXq+wMAizkiIqKyJhEEQSjOC0JDQ9G6detCzwcEBGDTpk3vfL1cLoe1tTVkMhmsrKyK0zRpoNxcYM/q42ipNxhZrcJRo66z2JGIiIjKnWL30Pn7+6OYNSDpqLAjiRg5wQEvYhpi24z+aOxoInYkIiKicqnM59CRbgpdHYxaeYvgWiEKf/xhi/r1F4kdiYiIqNxiQUdFlp2RjSd3YlGrsTtqtu6Fh+FOOHzMHBJuT01ERCQqFnT0ToIA7NsH4OxQ1HSKgLL+VVSqVRmVan0mdjQiIiICb/1F7/DgehSG972D7t2Bvx5+B5N226BvwJWrREREmoQ9dPRGgkpA7vE+6FrNC132/I6uXWtDwlqOiIhI47CgowIElYBzW3+Hfe0P4dGoMozb/Yn21VxhaiF2MiIiInoTDrlSvuvXgY7tMlA1JQiPT+8EAFStVxOmFqYiJyMiIqK3YUFHkCWk4p9FQWjjm4Inzy1wr+Z1fDRuotixiIiIqIhY0BGyM7JQx/wPrPrhKq5fB/w+chQ7EhERERVDsW/9VVJSqRRSqRRKpRJRUVG89ZfIboTdgOzsXNQL/BU29ibIyc6BkYmR2LGIiIioBNTWQzd69GhERkYiPDxcXU3Sa7yIEzBkCNCnvxFsDGOQ/OwFALCYIyIi0mJc5VqOnP1jK7IjNuCvv/7GvHm14TX0HPeUIyIi0gEs6MqBzDQFzCyNYeHoijxbd9yNzIa9kxkAFnNERES6QG1z6F6Ry+WwtrbmHDo1ePIEuLFpImz176H51P3Q4xIYIiIincRf8TooOyMbi+bJULs2sDW0PVClHyRQa91OREREasQeOh0jqARELm2K83fq467NWnz3HWBpKXYqIiIiKkucQ6cjHt64B0snFzhUNEdGtRlo3cYdQxuInYqIiIjUgUOuWi49HZj1rRx2F5sgfMsKAIBPj09QvUFtkZMRERGRurCg01KCSsDJP/ahjlceFiyxwp7kPfAfMV7sWERERCQCFnRaKvrabfgqu2NEl0O4fRsImNwaphamYsciIiIiEbCg0yKp8Sk4unI5lHkC3Bt54YHXLUyTdkHVqmInIyIiIjGxoNMCKhWwbh0wotcFNDP7FlFX7gMA3Bt5iZyMiIiINAELOg0XeSoca8bNxfDhgHG1jsho9xiePjXFjkVEREQahNuWaLikB7fgX2MXzp6chOYfmgOwEzsSERERaRi1bSwslUohlUqhVCoRFRXFjYXfIDcXOLlhNRSpz9F5yiyolCoIKgH6hvpiRyMiIiINpbYh19GjRyMyMhLh4eHqalLrnDguoGFD4OiBVJjqJ0OlFKCnr8dijoiIiN6KQ64aQFAJOPvTQBwMawBr6yD0mTUFjRqJnYqIiIi0BRdFiCg7Q4HcHAESPQlyTTzQtZ8bTp8GizkiIiIqFrXNoXtFLpfD2tq63M+h+/svGTwfNUaE/kx0HjVI7DhERESkxdhDp2YPIp7j44+BTl2tcfThcHg0byJ2JCIiItJynEOnRlHhN1Elsgmssw9i58626NFjCiQSsVMRERGRtmMPXRkTVAKuHr8MAKjZuC4uqqRYt7s5evYEizkiIiIqFSzoytDNm8Dc4dvxwTMfRF6MhkRPgg+HDIOZlZnY0YiIiEiHcMi1DMgSUvH7ivMY+0NHeNTsiU49/oG3Tw2xYxERlVhOTg7y8vLEjkGkcQwMDGBkZCR2DBZ0ZeHq1qX41OVnZM9/jNHjTWFk1FrsSEREJZaTk4OIiAioVCqxoxBpHD09PdSpU0f0oo7blpSSiJMXEf/4OVoP6ob01HSkJcvgXL2S2LGIiN5bZmYmbt++japVq8LU1FTsOEQaIysrC48ePYKLiwuMjY1hbGwMc3NzUbKwh+49xccD06YBLfVXoW71GKiUXWFhYwELGwuxoxERlSpTU1OYmXEOMNF/7d+/HxkZGbCyskLfvn1FKeq4KKKEVEoVwtauwOgeB7FrF5BXfzkaTToMPX0uXSUiIipPzM3NYWRkBLlcDoVCIUoGFnQlJJFIYJb6NwZ/HI6oKGD4KEvoG+qLHYuIiIjU7NVwq5jUNuQqlUohlUqhVCrV1WSpe/pIjkfbR0CoNQ6+3Zuj4fi9MDDiqDURERGJS209dKNHj0ZkZCTCw8PV1WSpUWQLmD8fqF3XHJmyNOgrkwGAxRwRERFpBFYk7/Dg+l3kHe+FzdJdCBxRCz6T98PaWuxURERERP/Dgu4N5CmZsLI1g3ONyrh8pBH27ZOgZiOxUxEREREVxkUR/5GRAfz641/I3lYND2+/gKmFKXyDNqNmo5piRyMiIiJ6LRZ0/09QCdi/PRaensDkRS1xUzEGFSpZih2LiIhE5u/vjwkTJogd443Ume/27dtqaYeKjwXd/wuTzkbD+KZo2iQLZ8Lt0Hb8dzCz4gaaREREALB06VLRt+ZISEjA4sWL33rOs2fP1JRGs5Trgk6WmIqIc3cAAO7tP8ezyuvw5y5TVK8ucjAiIlKrnJwcsSOIxt/fH7t27XrrOUeOHIGjoyOqi/wL0tHREa1atYJUKn3jOaGhoQgMDMSJEyeKfN3w8HBMmDABISEhCAwMxMOHD0sjrlqVy4JOpQI2bgTO/vQ5ck8NgSAArrWrwrtrJ7GjERHptKpVq2Lp0qUFnmvQoAFmzZoF4GVxMWbMGIwZMwY2Njawt7fH9OnT8e/bjhflHEEQsHDhQlSvXh2mpqaoX78+duzYUegakyZNgoODA9q3b//W3Hl5eW9sT6FQYNy4cXBycoKJiQl8fX0LbdH1rvf9KtO4ceMwefJk2NnZoWLFigWOA0BGRgYGDx4MCwsLODs746effnpr7nfZs2cP9PX18c033yA3N/eN5y1fvhz9+vV7r7ZKi4+PDy5evIjExMTXHh8wYAB+/vlnxMTEYNiwYTh06NBbr6dQKNC7d29MnToVAQEBGDJkCIYMGVIW0ctUuSvoIk9fQv+P7+CLL4B/khaiwqc7IOHduoiIiu152nNk52UDAB6mPMSV51cKfDxPe16i64aEhMDAwAAXLlzA8uXLERwcjHXr1hXrnOnTp2Pjxo1YtWoVIiIiMHHiRAwcOBBhYWGFrnHmzBmsXr26xJkmT56MnTt3IiQkBFeuXIG7uzs6dOiA5OTkEr13c3NzXLhwAQsXLsScOXNw9OjR/ONBQUE4ceIEdu/ejSNHjiA0NBSXL1/OP75p0yZIivhLLS8vD5cvX8b27dsRHx//xs/BjRs34OrqCn19zbkbUteuXbFp06Y3HjcwMMDgwYOxZs0ayOVyfPHFF9i9e3eBov+VkydPwsrKChUqVADwsmA8f/48Xrx4UVbxy4agZjKZTAAgyGQydTctKPOUwv1ltYU/gwKFsDC1N09EpJUyMjKES5cuCRkZGQWen3lipnAv6Z4gCILw2c7PBMxCgY+ZJ2YWulaVKlWE4ODgAs/Vr19fmDnz5bl+fn6Cp6enoFKp8o9PmTJF8PT0zH/8rnPS09MFExMT4ezZswXaGTp0qNC/f//8azRo0KBI7/9t7aWnpwuGhobCli1b8o/l5OQILi4uwsKFC4v8vl+14+vrW+Acb29vYcqUKYIgCEJaWppgZGQkbN26Nf94UlKSYGpqKowfP14QBEHYtWuX4OHhUaT3tW7dOuHevZf/fz/++KPg6Oj42t/NS5YsEdatW5f/OC8vT1i5cqUwcOBA4eLFi4IgCMKAAQOEkJCQIrX7Nnl5ecKKFSuEgICA/GvHxMQIfn5+Bc5LSEgQWrduXeTrqlQqYc+ePcJnn30m/PPPPwWOrVmzptD1HRwchGPHjr3zuq++N3bs2CGsW7dOCA4OFpKSkoqcqzTpfA9dXk4ewtZKcfdqDPT09WD80SF0/16KVq3ETkZEpN1GNB4BVytXAMDc1nNxOfBygY8RjUeU6LrNmjUr0MvUvHlz3Lt3r8CtI992TmRkJLKzs9G+fXtYWFjkf2zevBnR0dH5r2nSpMl7Z7p//z5yc3PRsmXL/GOGhobw8fEp0YrQDz74oMBjZ2dnxMfHAwCio6ORk5OD5s2b5x+3s7ODh4dH/uMePXrgzp0772wnPT0dCQkJcHd3BwCMHz8eZmZmWLBgQaFzY2Nj4eTklP9479696Nu3LzIzM/Ho0SMAQJcuXSCXy4v+Rt9g37596NevH7KysvLnsR09ehSurq4FznNwcCjwf/kuiYmJuHLlCuzt7VGtWrVCx0xMTAo8Z2JigtTU1JK9CZHo9MbCJ08C33yViR2fz0PkRRN4NBwK19pVxY5FRKQTnC2d8/9dzbYaqqHaW85+SU9Pr9Cw19vmbpWESqUCABw4cACVKlUqcOzfqzTNzc1Lrc3/DnMKglDguaK+b0NDw0LXffV+/vv697F27VoMHz48/7GJiQnmzZuHwMBAjBo1qsDnLT09HaampvmP27RpA5VKhVOnTuG3334DANSoUeO1CyaWLVv21gUGfn5+6NGjR/7jtm3bAgCOHz+ODRs2AHi5yOF1cxyLMrT87NkzrFixArm5uRg7diyqVKlS6Bxra+tCn9v09HQ4ODi88/qaRCcLuvjHcbjx+yx8+v2PqF3PBnFN7qBdUyuxYxERlXuOjo54/vx/c+vkcnmhX/jnz58v9LhmzZoF5nC97RwvLy8YGxvjyZMn8PPzK5Xcb2rP3d0dRkZGOH36NAYMGADgZaF26dKlAnvDFeV9v4u7uzsMDQ1x/vx5VK5cGQCQkpKCqKioYr3PuLg4mJiYwN7evsDzAwYMQHBwMGbMmIH169fnP+/g4ICUlJT8xzY2Nvj111/h7++fX+jdvHkTn3/+eaG2xo8fX5y3CCsrK2zduhWtWrXKL7jDwsLwww8/IDU1FTY2NvnnGhi8uYR59OgRpFIpjIyMMG7cODg7O7/x3Nq1a2Pt2rX5jxUKBdLS0l5b/GkynSzolEolalr8gy0/30GnQc2gp8dijohIE7Rp0wabNm1Cly5dYGtri++++67QZPuYmBhMmjQJI0aMwJUrV7BixYpCqznfdo6lpSW+/vprTJw4ESqVCr6+vpDL5Th79iwsLCwQEBBQ7Nxvas/c3BwjR45EUFAQ7OzsULlyZSxcuBCZmZkYOnRosd73u1hYWGDo0KEICgqCvb09KlSogG+//RZ6ev+bPbV7925MnTr1rcOuq1atQmBg4GtXiX7zzTfo168fJk6ciLp16wIAPD098eTJkwLnvXjxokBRaWFhUSDH+4iJickfCr516xYMDQ3h5OSEkJAQDBs2DMDLotnS8vWb/y9evBiZmZn45ptvChWtr9OqVSvEx8cjNjYWrq6uCAsLg7e3Nwu6N5FKpZBKpQXmQJSmc3+dA259jzpf7oJz9UpQfnkXVQw1Z0UOEREBU6dOxYMHD/DJJ5/A2toac+fOLdRTNXjwYGRlZcHHxwf6+voYO3YsAgMDi3XO3Llz4eTkhPnz5+PBgwewsbFBo0aNMG3atBLlflt7P/74I1QqFQYNGoS0tDQ0adIEhw8fhq2tbbHed1EsWrQI6enp6Nq1KywtLfHVV19BJpPlH5fJZLh79+4bXx8VFYV58+Zhzpw5b21n2rRp2LdvHwCgY8eOGD58OL766qv84/369cOkSZPw66+/Ii8vr0RF8pv06tULU6ZMwfbt2wEAjRs3xsqVKwv0AIaHh6NNmzavff3XX39drPYMDAwQEhKCefPmoVmzZggLC8OWLVtKnF8sEqE0B+WLQC6Xw9raGjKZDFZW799z9iBahYmT9HDvciRCxk2ES69NqFTjzV2rRERUPJmZmbh9+zY8PT1hZla2d9Dx9/dHgwYNCu3ZVtxzqHQFBgZizpw5qFixothRALzclqZr167w8fERNcer741Hjx4hNTUVaWlpGDx4MOzs7NSeRauHXE9uWAPh4R+4du0YFgd7ocmnh7mnHBERUSmbMWMGVq5cie+//17sKJDL5UhISHhjMRccHPzWFbCtW7dGr169yiqeaLSuoBNUAmTJmbBxMIe1W22kpLVE5K08mFsaiR2NiIhIJ7m6uqJHjx44ePAgOnfuLFoOQRAQHBz81sJy4sSJakykObSqoIuMBGK2D4GZYQp8p+1F/fatgPbcUI6ISFeEhoaWyjlU+ho3bix2BCQlJWHkyJFwdHQUO0qZuX79OhYvXoyHDx/im2++QcuWLTF79mwoFArExcVh5syZaNCgQaHXacXGwrJEOb4NSkH9+sD2C31hUncEh1aJiIjKGQcHhwKbHOuiZcuWYePGjejYsSO++OILDB8+HEFBQejTpw8OHDhQYEuZf9P4gk6lVCHx96aonDodc+cCP+/pBO9u4nX3EhEREZWF6OhouLi4wMDAAM+ePUNycjKmTZuGSpUqISkpCRYWFujSpctrX6uxQ66RZ66gQs3asHcyQ0rVYHT9uC6ca4idioiIiKhsvHjxIn/BxunTp+Hr64tGjRoBAD799FN8+umnb3ytxvXQJSYCk0YnoWqUL8L/eLlzc5OuHeFcw/UdryQiIip9/v7+Be76oGnUma8k96elomvRogUaNmyIxMRE3Lp1C/7+/kV+rcYUdMpcJY6u3YLaHkps2GKPA9nH0G7kaLFjEREREYClS5cWuBfumyQkJGDx4sVvPefZs2elFUsnnThxAoIgaGdBd+/SDbQxHYxvvghDVBTQe2RzGBhp7IgwERHpkJycHLEjiMbf3x+7du166zlHjhyBo6Mjqlev/s7rOTo6olWrVpBKpW88JzQ0FIGBgThx4kSRc4aHh2PChAkICQlBYGBgie60oS1OnDgBY2NjNGvWrMivEbWgi3v4DIeDF0CZJ6B284Z43iQaXy9qAx1fwEJEVG5VrVq10B0eGjRogFmzZgF4WVyMGTMGY8aMgY2NDezt7TF9+nT8+6ZGRTlHEAQsXLgQ1atXh6mpKerXr48dO3YUusakSZPg4OCA9u3bvzV3Xl7eG9tTKBQYN24cnJycYGJiAl9fX4SHhxfrfb/KNG7cOEyePBl2dnaoWLFigeMAkJGRgcGDB8PCwgLOzs6F7nFbXHv27IG+vj6++eYb5ObmvvG85cuXo1+/fkW+ro+PDy5evPja+8UCwIABA/Dzzz8jJiYGw4YNw6FDh956PYVCgd69e2Pq1KkICAjAkCFDMGTIkCLn0TYnTpxAs2bNYGJiUuTXiFbQLV0KjP3sChqZ/YRHt2MBAK61q4oVh4iINERISAgMDAxw4cIFLF++HMHBwVi3bl2xzpk+fTo2btyIVatWISIiAhMnTsTAgQMRFhZW6BpnzpzB6tWrS5xp8uTJ2LlzJ0JCQnDlyhW4u7ujQ4cOSE5OLtF7Nzc3x4ULF7Bw4ULMmTMHR48ezT8eFBSEEydOYPfu3Thy5AhCQ0Nx+fLl/OObNm2CpIj7euXl5eHy5cvYvn074uPj3/g5uHHjBlxdXaGvX7z7o3ft2hWbNm1643EDAwMMHjwYa9asgVwuxxdffIHdu3fjdXckPXnyJKysrFChQgUALwvG8+fP48WLF8XKpA3i4uJw586dYg23AqWwylUQBKSlpb3xuEKhgEKhyH986dAxAMCsWXKMGNEKWR9dg6O9BeRy+ftGISKiMpCdnf3a558/B2xtARMT4OFDICWl4HFn55cfxeXm5obg4GBIJBJ4eHjg5s2bCA4OxvDhw4t0TkZGBpYsWYLjx4+jefPmAIDq1avj9OnTWL16Nfz8/AAA7u7uWLhw4XtlGjBgAFatWoVNmzahU6dOAIC1a9fi6NGjWL9+PYKCgor13j/44APMnDkTAFCzZk2sXLkSx44dQ/v27ZGeno7169dj8+bN+T2KISEhcHX936JBa2treHh4FKmtkJAQBAQEwN7eHlOnTsWcOXMwePDgQvdZP3bsGLy9vfMfK5VKrFq1CpcuXcLo0aPh7e2N2NhYDBw4sMCmz35+fpBKpfj666/fmkNPTw99+/ZFnz59sG/fPgwaNAhDhgxB27Zt88959OhRgfuj6uvrw9LSEhEREflFnphyc3ORk5ODnJwcpKWlwcDgf+WVpaVlkYts4OVK1woVKqBPnz7FyvDeBV1aWhqsra2L/TpBcMMvvwC//PK+CYiIqCx5eHhgy5YthZ5fvRoYOBBwdwe++w747ykzZwL/GTEskmbNmhX4Bdi8eXP89NNPUCqV+b1EbzsnMjIS2dnZhYZRc3Jy0LBhw/zHTZo0ee9M9+/fR25uLlq2bJl/zNDQED4+PiVaEfrBBx8UeOzs7Iz4+HgAL/coy8nJyS9SAcDOzq5AAdejRw/06NHjne2kp6cjISEB7u7uAIDx48dj1apVWLBgAebNm1fg3NjY2AK9Rfv27UO/fv1w6tQpPHz4EN7e3jh69GiBwhJ4uQnw2+6p+l+JiYm4cuUK7O3tUa1atULH/jv8aGJigtTU1CJfvyxt3LgRMTExMDY2xowZMwp0ZMlkskJF8tvUr18fcXFxxc7w3gWdpaUlZDLZG4//t4fuaexTNGveDJGRkahUqdL7Nl8k3t7eheYzsD3taE8ul8PNzQ0xMTHF+oZ4H7r8+dT19vj1UjbtnTx5EjExMYWOjRjxsocOAObOBSZNKnj8db1zenp6hYbU3jZ3qyRUKhUA4MCBA4V+z/x7laa5uXmptfnfHhhBEAo8V9T3bWhoWOi6r97P64YiS2rt2rUFejxNTEwwb948BAYGYtSoUQU+b+np6TA1Nc1//Krn7Pjx49iwYQOAl4scXjcPsSg9U8+ePcOKFSuQm5uLsWPHokqVKoXOsba2LvT+09PT4eDg8M7rq8OQIUOQkpKCtLQ09O3bF7avvjHwsk5Sh/cu6CQSSYl+cFpaWqrtB66+vr7a2mJ7ZcPKyopfL2yvyPj1UrrtWVhYvPbYvwu2atVefryLo6Mjnj9/nv9YLpcXWq14/vz5Qo9r1qxZYA7X287x8vKCsbExnjx5kj+8+r7e1J67uzuMjIxw+vRpDBgwAMDLQu3SpUsF9oYryvt+F3d3dxgaGuL8+fOoXLkyACAlJQVRUVHFep9xcXEwMTGBvb19gecHDBiA4OBgzJgxo8DtpRwcHJDyr/F0KysrbN26Fa1atcovisPCwvDDDz8gNTUVNjY2+ef+e+jxvx49egSpVAojIyOMGzcOzm8Zn69duzbWrl2b/1ihUCAtLe21xZ8YDA0NYWRkBCMjI7XWN/9WLvYFGT1avfvZsT3tpuufT11vT910/fNZ2u21adMGmzZtQpcuXWBra4vvvvuu0GT7mJgYTJo0CSNGjMCVK1ewYsWKQqs533aOpaUlvv76a0ycOBEqlQq+vr6Qy+U4e/YsLCwsEBAQUOzcb2rP3NwcI0eORFBQEOzs7FC5cmUsXLgQmZmZGDp0aLHe97tYWFhg6NChCAoKgr29PSpUqIBvv/0Wenr/W9+4e/duTJ06FXfu3HnjdVatWoXAwMDXrkD95ptv0K9fP0ycOBF169YFAHh6euLJkyeFPh+vhmtv3boFQ0NDODk5ISQkBMOGDQPwsrB9U+/U4sWLkZmZiW+++aZQYfk6rVq1Qnx8PGJjY+Hq6oqwsDB4e3trTEGnEQQ1i4mJEQAIMTEx6m6atJBMJhMACDKZTOwopAX49VI2MjIyhEuXLgkZGRnvfS2ZTCb06dNHsLKyEtzc3IRNmzYJ9evXF2bOnCkIgiD4+fkJo0aNEr788kvByspKsLW1Fb755htBpVLlX6Mo56hUKmHZsmWCh4eHYGhoKDg6OgodOnQQwsLC8q8xfvz4ImV+V3tZWVnC2LFjBQcHB8HY2Fho2bKlcPHixWK97zdl6tatmxAQEJD/OC0tTRg4cKBgZmYmVKhQQVi4cGGB123cuFF426/2u3fvCvr6+gKAt3506dIl/zUJCQlC9+7dC1wnOjpa+PTTT4Vt27YJ27ZtE/r27SssWbJESE5Ozj/nzJkzwqRJk4ryKS6So0ePCl9++aWwadMmYciQIUJ0dHSpXbukXn1v7NixQ1i3bp0QHBwsJCUliZJFIgilOChfBAkJCXByckJ8fDwcHR3V2TRpIYVCgfnz52Pq1KlF2qGcyjd+vZSNzMxM3L59G56enjAzMyvTtvz9/dGgQYNCe7YV9xwqXYGBgZgzZw4qVqxY5NdMnz4dXbt2hY+PTxkmE9er741Hjx4hNTUVaWlpGDx4cIEVueqi9iHXVz9k+cOWisLY2LjQxppEb8KvF6KyMWPGDKxcuRLff/99kc6Xy+VISEh4YzEXHBz81hWwrVu3zr9JPRVNuZhDR0RERCXn6uqKHj164ODBg+jcufNbzxUEAcHBwW8t/iZOnFjaEcs9FnRERKQx/r0x7fucQ6WvcePGRTovKSkJI0eO5LQqNWNBR0RERKVGU/aGK29Eu5crEREREZUOFnSkcWbNmgWJRFLgozgrq0j3nTx5El26dIGLiwskEgn27NlT4LggCJg1axZcXFxgamoKf39/REREiBOWiHTe/v37kZSUJGoGFnSkkerUqYPnz5/nf9y8eVPsSKRBMjIyUL9+faxcufK1xxcuXIglS5Zg5cqVCA8PR8WKFdG+fXukpaWpOSmR7rh8+TLOnj2L69evix1F45iammLHjh2lfhu74uAcOtJIBgYG7JWjN+rUqRM6der02mOCIGDp0qX49ttv0bNnTwBASEgIKlSogN9//x0jRoxQZ1QinbB792507NgR+vr6CAwMxKZNm8SOpFFat26N8+fP48GDB6JlYA8daaR79+7BxcUF1apVQ79+/UT9JiHt8vDhQ8TFxeGjjz7Kf87Y2Bh+fn44e/asiMkIeLkp8L/vcaqpND2nOvPdvHkTjRo1wpEjRzBo0CAEBgYW+bW3b98uw2SaQ09PD66uroiPjxcvg7oakkql8PLygre3t7qaJC3VtGlTbN68GYcPH8batWsRFxeHFi1aiD4/gbRDXFwcAKBChQoFnq9QoUL+MSL6n8zMTGRkZLzx+LNnz+Dq6oq8vDy8ePECBgZFG9xbunRpkW8ikJCQgMWLFxfpXE1lamqK7Oxs0dpXW0E3evRoREZGIjw8XF1Nkpbq1KkTevXqhXr16qFdu3Y4cOAAgJfDZkRFJZFICjwWBKHQc0Q5OTliRxDVgQMH0KhRIyxatOiN5wiCAH19ffTq1Qtdu3ZFamrqO6975MgRODo6onr16kXK4ejoiFatWkEqlRY1epGEh4djwoQJCAkJQWBgIB4+fFiq19ckHHIljWdubo569erh3r17YkchLfBq7uV/e+Pi4+ML9dqRuBQKBcaNGwcnJyeYmJjA19e3wB/9f/31F2xsbKBSqQAA165dg0QiQVBQUP45I0aMQP/+/QG8LDwWLlyI6tWrw9TUFPXr18eOHTsKtOnv748xY8Zg0qRJcHBwQPv27d+YLy8vD2PGjIGNjQ3s7e0xffp0/Pv25+/KX7Vq1UL3m23QoEGB29P5+/tj3LhxmDx5Muzs7FCxYsVCt6/LyMjA4MGDYWFhAWdnZ/z0009v/8QWw8cff4y+ffu+8Xh8fDzu3LkDAMjKysKdO3fQunVrxMbG4u+//87/uHbtWoHXLV++HP369StWFh8fH1y8eBGJiYnFfh+vo1Ao0Lt3b0ydOhUBAQEYMmQIhgwZUirXfp2srCyYmJiU2fXfhQUdaTyFQoHbt2/D2dlZ7CikBapVq4aKFSvi6NGj+c/l5OQgLCwMLVq0EDEZ/dfkyZOxc+dOhISE4MqVK3B3d0eHDh2QnJwMAGjVqhXS0tJw9epVAEBYWBgcHBwQFhaWf43Q0FD4+fkBeHkz+I0bN2LVqlWIiIjAxIkTMXDgwALnAy97+w0MDHDmzBmsXr36jflenXfhwgUsX74cwcHBWLduXZHzF1VISAjMzc1x4cIFLFy4EHPmzCnw9RsUFIQTJ05g9+7dOHLkCEJDQ3H58uUC19i0aVOJe6Df9rrLly+jQ4cOOHPmDI4dO4Zly5bB0NAQrq6u6NixY/5HgwYN8l9z48YNuLq6Ql9fv9hZunbt+sYFF8uWLcOcOXOKPHXi5MmTsLKyyv9DzsfHB+fPn8eLFy+KnetdVCoVYmNj4eTkVOrXLjJBzWQymQBAkMlk6m6atMRXX30lhIaGCg8ePBDOnz8vfPLJJ4KlpaXw6NEjsaORhkhLSxOuXr0qXL16VQAgLFmyRLh69arw+PFjQRAE4ccffxSsra2FXbt2CTdv3hT69+8vODs7C3K5XOTk2ikjI0O4dOmSkJGRUfBA5jNByMt6+e/0J4KQGvG/Yyk3Xh7/Dz8/P2H8+PFCenq6YGhoKGzZsiX/WE5OjuDi4iIsXLgw/7lGjRoJixcvFgRBELp37y7MmzdPMDIyEuRyufD8+XMBgHD79m0hPT1dMDExEc6ePVugvaFDhwr9+/cv0H6DBg3e+Z79/PwET09PQaVS5T83ZcoUwdPT8+XbLUL+KlWqCMHBwQWuW79+fWHmzJkF2vH19S1wjre3tzBlyhRBEF5+rRsZGQlbt27NP56UlCSYmpoK48ePz39u165dgoeHx1vf05kzZ4SxY8cKa9asEaZMmSKcOnVKEARBmDlzpvD1118LmzZtEjZt2iR0795dSE9PFwRBEA4dOvTWa77OkiVLhHXr1hV4Li8vT1ixYoUQEBAgXLx4URAEQYiJiRH8/PwKnJeQkCC0bt36jdeOi4sTZs2aJQQFBQl37tx5a441a9YUur6Dg4Nw7Nixor+Zd3j1vTFy5EjB29tbWLhwoZCUlFRq1y8O9tCRxomNjUX//v3h4eGBnj17wsjICOfPn0eVKlXEjkYa4tKlS2jYsCEaNmwIAJg0aRIaNmyIGTNmAHjZczJhwgSMGjUKTZo0wdOnT3HkyBFYWlqKGVv33FsNZMa+/PftRcCZfw2xnej08vgbREdHIzc3Fy1btsx/ztDQED4+PgVWRvr7+yM0NBSCIODUqVPo1q0b6tati9OnT+PEiROoUKECateujcjISGRnZ6N9+/awsLDI/9i8eTOio6MLtN2kSZMivb1mzZoV6L1q3rw57t27B6VSWeT8RfHBBx8UeOzs7Jy/WjI6Oho5OTlo3rx5/nE7Ozt4eHgUeE2PHj3yh0Zf5/nz5+jVqxemTZuG4cOHw8TEpEAv34ULFxAQEICAgAAolUocPnwYwMvtOIrrdT1V+/btQ79+/ZCVlZU/j+3o0aNwdXUtcJ6Dg0Oh/69/q1ChAmbOnInvvvsOBw8exNixY3Hu3LnXnpuYmFhoCNTExKRIcwCLKzMzE7169YKhoWGpX7uouA8daZytW7eKHYE0nL+/f4G5TP8lkUgwa9asQnORqJTVHAEY2b78t2cQUPPL/x1rfQgwfvM9PV/9/71r8Yq/vz/Wr1+P69evQ09PD15eXvDz80NYWBhSUlLyh1tfzbM7cOAAKlWqVOCa/11paW5uXrz3WcL8enp6hb5OX7fx7H+LAIlEkv9+3vZ1Xhw7d+5E5cqV8+eY/vd7o3Hjxvn/trOzg1wuB1D4c1cU6enpMDU1LfBc27ZtAQDHjx/Hhg0bALwcLn/dHMaiDB1bWlpi4sSJyMnJwZYtW7BhwwYEBATA19c3/xxra+tCn7/09PQyuddsly5dkJqaKurm5eyhIyKikjF1BvT/vwfE3A2w9vrfMZt6L4+/gbu7O4yMjHD69On853Jzc3Hp0iV4enrmP/dqHt3SpUvh5+cHiUQCPz8/hIaGFpg/5+XlBWNjYzx58gTu7u4FPtzc3Er09s6fP1/occ2aNaGvr1+k/I6Ojnj+/Hn+cblcXuxVlu7u7jA0NCyQJSUlBVFRUcW6jkqlemtxWJL5bm/i4OCAlJSUAs9ZWVnh4MGDaNWqVX5BHRYWhnbt2hXqMSvqtijAy8/p06dPYWFhUahXsHbt2gX2hVMoFEhLS9PZ0R720BERkdqZm5tj5MiRCAoKgp2dHSpXroyFCxciMzMTQ4cOzT/P2toaDRo0wG+//YZly5YBeFnk9e7dG7m5ufD39wfwssfm66+/xsSJE6FSqeDr6wu5XI6zZ8/CwsICAQEBxc4YExODSZMmYcSIEbhy5QpWrFiRv8K0KPnbtGmDTZs2oUuXLrC1tcV3331X7MLJwsICQ4cORVBQEOzt7VGhQgV8++230NMr2B+ze/duTJ069Y3Drj169MDcuXPx5MkTVK5cGcDL0ZB3rUSNj4/HkSNHoKenh8uXL2PBggXvLLg8PT3x5MmTQs/HxMTA3d0dAHDr1i0YGhrCyckJISEhGDZsGICXRXFRpkY8fPgQa9asgZ6eHr788svXFu2tWrVCfHw8YmNj4erqirCwMHh7e7OgIyIiKk0//vgjVCoVBg0ahLS0NDRp0gSHDx+Gra1tgfNat26NK1eu5Bdvtra28PLywrNnzwr05s2dOxdOTk6YP38+Hjx4ABsbGzRq1AjTpk0rUb7BgwcjKysLPj4+0NfXx9ixYwvcJeFd+adOnYoHDx7gk08+gbW1NebOnVuifdAWLVqE9PR0dO3aFZaWlvjqq68gk8kKnCOTyXD37t03XsPNzQ07d+7E9OnT4evrC5VKhc6dO+Pvv//G/v37kZubi/379yMtLQ1nzpxBbGwsateujUePHiE1NRVjxozB+fPn8c8//6Bjx45vzduxY0cMHz4cX331VYHne/XqhSlTpmD79u0AXg7zrly5Ep9//nn+OeHh4WjTps0br3316lWEhISgYsWKmDx5cqGvlX8zMDBASEgI5s2bh2bNmiEsLAxbtmx5a3ZtJhFKa4C+iORyOaytrSGTyWBlZaXOpomIqAQyMzNx+/ZteHp6wszMTOw4JJI+ffpg4cKFqFq16jvPDQwMxJw5c4p9T+7p06eja9eu8PHxKXRs2bJlMDExQUBAgKj7vf3bq++NV4VvWloaBg8eDDs7O7Vn4Rw6IiIieqsDBw6gd+/eRSrmAGDGjBlYuXJlsdqQy+VISEh4bTEHAOPHj8eIESM0ppjTNCzoiIiI6I3Cw8NRoUIF9O7dG5GRkUV6jaurK3r06IGDBw8W6XxBEBAcHIzvv//+faKWa5xDR0RERK917tw5jBgxAk5OTsjJycGqVauK/Np/b4XyLklJSRg5ciQcHR1LEpPAgo6IiIjeoHnz5rhx40aZt1MWe8OVN2obcpVKpfDy8oK3t7e6miQiIiIqF9RW0I0ePRqRkZEIDw9XV5NERERE5QIXRRARERFpORZ0RERERFqOBR0RERGRlmNBR0RERKTlWNARERERaTkWdERERERajhsLExFRkWRlZYkdgUijaNL3BAs6IiJ6KwMDA+jp6eHRo0diRyHSOIIgQKlUih2DBR0REb2dkZER6tSpg7y8PMhkMuzfvx/m5uYwNjYWOxqR6JRKJfLy8sSOwYKOiIjezcjICEZGRsjOzkZGRgZyc3NZ0BH9i0KhELV9FnRERFRkxsbGsLKyglwuR05OjthxiDSKlZWVaH/oSARBENTZoFwuh7W1NWQyGaysrNTZNBERlYKMjAzReyOINJGxsTHMzc1FaZs9dEREVCzm5uai/dIiotdT2z50UqkUXl5e8Pb2VleTREREROUCh1yJiIiItJzaCzpBEJCWlgZLS0tIJBJ1Nk1ERESkk9Re0BERERFR6eK9XImIiIi0HAs6IiIiIi3Hgo6IiIhIy7GgIyIiItJyLOiIiIiItBwLOiIiIiItx4KOiIiISMv9H7u3EBBQR7IfAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "Graphics object consisting of 3 graphics primitives"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bounds_on_d_qmin = plot_d_bound(v_example, 0, ymin=-0.5)\n",
    "bounds_on_d_qmin"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "24dd62c1",
   "metadata": {},
   "source": [
    "### Bounds on $d$ with Maximal $q=\\operatorname{ch}^{\\beta}_1(u)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "id": "47b30d7e",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 3 graphics primitives"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bounds_on_d_qmax = plot_d_bound(v_example, 4, ymin=-3, ymax=3)\n",
    "bounds_on_d_qmax"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "133ccbe7",
   "metadata": {},
   "source": [
    "### Bounds on $d$ with Mid-way $q=\\operatorname{ch}^{\\beta}_1(u)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "id": "25e4850b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "Graphics object consisting of 3 graphics primitives"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "typical_bounds_on_d = plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)\n",
    "typical_bounds_on_d"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1c6f5622",
   "metadata": {},
   "source": [
    "# Bounds on Semistabilizer Rank $r=\\operatorname{ch}_0(u)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "553bba31",
   "metadata": {},
   "outputs": [],
   "source": [
    "var(\"a_v b_q n\") # Define symbols introduce for values of beta and q\n",
    "beta_value_expr = (beta == a_v/n)\n",
    "q_value_expr = (q == b_q/n)\n",
    "# RENDERED TO LATEX: positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()\n",
    "# placeholder for the specific values of k (start with 1):\n",
    "var(\"kappa\", domain=\"real\")\n",
    "\n",
    "assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_exp_term)\n",
    "assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_exp_term_alt2)\n",
    "\n",
    "r_upper_bound1 = (\n",
    "    assymptote_gap_condition1\n",
    "    * r * 2*n^2 / kappa\n",
    ")\n",
    "\n",
    "assert r_upper_bound1.lhs() == r\n",
    "\n",
    "r_upper_bound2 = (\n",
    "    assymptote_gap_condition2\n",
    "    * (r-R) * 2*n^2 / kappa + R\n",
    ")\n",
    "\n",
    "assert r_upper_bound2.lhs() == r"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "990a2840",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle n^{2} q^{2}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle n^{2} q^{2}$"
      ],
      "text/plain": [
       "n^2*q^2"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_upper_bound1.subs(kappa==1).rhs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "5a3c7037",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R$"
      ],
      "text/plain": [
       "n^2*(psi - q)^2 + R"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "r_upper_bound2.subs(kappa==1).rhs()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "82add957",
   "metadata": {},
   "outputs": [],
   "source": [
    "var(\"epsilon\")\n",
    "var(\"chbv\") # symbol to represent \\chern_1^{\\beta}(v)\n",
    "\n",
    "# Tightness conditions:\n",
    "\n",
    "bounds_too_tight_condition1 = (\n",
    "    bgmlv2_d_upperbound_exp_term\n",
    "    < epsilon\n",
    ")\n",
    "\n",
    "bounds_too_tight_condition2 = (\n",
    "    bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)\n",
    "    < epsilon\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "d33e3459",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
      ],
      "text/plain": [
       "1/2*q^2/r"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv2_d_upperbound_exp_term"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "id": "3728c192",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
      ],
      "text/latex": [
       "$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
      ],
      "text/plain": [
       "-1/2*(psi - q)^2/(R - r)"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bgmlv3_d_upperbound_exp_term_alt2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "id": "90149eb2",
   "metadata": {},
   "outputs": [],
   "source": [
    "var(\"Delta nu\", domain=\"real\")\n",
    "\n",
    "q_sol = solve(\n",
    "    r_upper_bound1.subs(kappa==1).rhs()\n",
    "    == r_upper_bound2.subs(kappa==1).rhs()\n",
    "    , q\n",
    ")[0].rhs()\n",
    "\n",
    "r_upper_bound_all_q = (r_upper_bound1.rhs()\n",
    "    .expand()\n",
    "    .subs(q==q_sol)\n",
    "    .subs(kappa==1)\n",
    "    .subs(psi**2 == Delta/nu^2)\n",
    "    .subs(1/psi**2 == nu^2/Delta)\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
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