MCE_NM_Julia_automatic_differentiation.ipynb

{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<img\n",
    "src=\"https://www.imt-atlantique.fr/sites/default/files/Images/Ecole/charte-graphique/IMT_Atlantique_logo_RVB_Baseline_400x272.jpg\"\n",
    "WIDTH=200 HEIGHT=200>\n",
    "\n",
    "<CENTER>\n",
    "</br>\n",
    "<p><font size=\"5\"> TAF MCE </span></p>\n",
    "<p><font size=\"4\">  UE Numerical Methods </font></p>\n",
    "<p></p>\n",
    "<p><font size=\"5\">  Automatic differentiation </font></p>\n",
    "</p></br>\n",
    "</p>\n",
    "</CENTER>\n",
    "\n",
    "----------------------------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <span style=\"color:#00B8DE\"> Table of contents</span> \n",
    "<a name=\"table-of-contents\"></a>\n",
    "    \n",
    "1. [Finite Difference](#Finite-difference)\n",
    "2. [Improved Finite Difference](#Improved-finite-difference)\n",
    "3. [Dual Numbers](#Dual-numbers)\n",
    "4. [Forward Automatic Differentiation](#Forward-automatic-differentiation)\n",
    "5. [Backward Automatic Differentiation](#Backward-automatic-differentiation)\n",
    "6. [Exercise: Optimization Problem](#Optimization-problem)\n",
    "7. [Appendix](#Appendix)\n",
    "8. [References](#References)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <span style=\"color:#00B8DE\"> Differentiation:</span>\n",
    "## <span style=\"color:#00B8DE\"> From finite difference to automatic differentiation</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:#00B8DE\"> Finite difference </span> \n",
    "<a name=\"Finite-difference\"></a> \n",
    "[Table of contents](#table-of-contents)\n",
    "\n",
    "To compute derivatives or partial differentials of a function $f$, one can use finite difference approximation schemes such as $\\dfrac{f(x+h)-f(x)}{h}$ or $\\dfrac{f(x+h)-f(x-h)}{2h}$, with $h$ small. However, such a scheme lakes precision for high $h$ and suffers from division robustness issue as $h$ decreases. As an example, consider the code below where the function \n",
    "\n",
    "$$\n",
    "f(x)    = x(1-x)(1-2x)^2(1-8x+8x^2)^2e^{1+e^x}\n",
    "$$ \n",
    "\n",
    "is used. Although more precise, symmetric finite difference shows the same behavior as forward finite difference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "\u001b[32m\u001b[1m   Updating\u001b[22m\u001b[39m registry at `~/.julia/registries/General`\n",
      "######################################################################### 100,0%\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m ZeroMQ_jll ────── v4.3.2+5\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m OpenSpecFun_jll ─ v0.5.3+4\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m Zlib_jll ──────── v1.2.11+18\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Project.toml`\n",
      " \u001b[90m [276daf66] \u001b[39m\u001b[92m+ SpecialFunctions v0.10.3\u001b[39m\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Manifest.toml`\n",
      " \u001b[90m [efe28fd5] \u001b[39m\u001b[93m↑ OpenSpecFun_jll v0.5.3+3 ⇒ v0.5.3+4\u001b[39m\n",
      " \u001b[90m [8f1865be] \u001b[39m\u001b[93m↑ ZeroMQ_jll v4.3.2+4 ⇒ v4.3.2+5\u001b[39m\n",
      " \u001b[90m [83775a58] \u001b[39m\u001b[93m↑ Zlib_jll v1.2.11+17 ⇒ v1.2.11+18\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m SymPy ─ v1.0.31\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Project.toml`\n",
      " \u001b[90m [24249f21] \u001b[39m\u001b[92m+ SymPy v1.0.31\u001b[39m\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Manifest.toml`\n",
      " \u001b[90m [24249f21] \u001b[39m\u001b[92m+ SymPy v1.0.31\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m Calculus ──── v0.5.1\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m DualNumbers ─ v0.6.2\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Project.toml`\n",
      " \u001b[90m [fa6b7ba4] \u001b[39m\u001b[92m+ DualNumbers v0.6.2\u001b[39m\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Manifest.toml`\n",
      " \u001b[90m [49dc2e85] \u001b[39m\u001b[92m+ Calculus v0.5.1\u001b[39m\n",
      " \u001b[90m [fa6b7ba4] \u001b[39m\u001b[92m+ DualNumbers v0.6.2\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m DiffResults ────────── v1.0.2\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m DiffRules ──────────── v1.0.1\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m CommonSubexpressions ─ v0.3.0\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m ForwardDiff ────────── v0.10.12\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Project.toml`\n",
      " \u001b[90m [f6369f11] \u001b[39m\u001b[92m+ ForwardDiff v0.10.12\u001b[39m\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Manifest.toml`\n",
      " \u001b[90m [bbf7d656] \u001b[39m\u001b[92m+ CommonSubexpressions v0.3.0\u001b[39m\n",
      " \u001b[90m [163ba53b] \u001b[39m\u001b[92m+ DiffResults v1.0.2\u001b[39m\n",
      " \u001b[90m [b552c78f] \u001b[39m\u001b[92m+ DiffRules v1.0.1\u001b[39m\n",
      " \u001b[90m [f6369f11] \u001b[39m\u001b[92m+ ForwardDiff v0.10.12\u001b[39m\n",
      "\u001b[32m\u001b[1m  Resolving\u001b[22m\u001b[39m package versions...\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m FunctionWrappers ─ v1.1.1\n",
      "\u001b[32m\u001b[1m  Installed\u001b[22m\u001b[39m ReverseDiff ────── v1.4.3\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Project.toml`\n",
      " \u001b[90m [37e2e3b7] \u001b[39m\u001b[92m+ ReverseDiff v1.4.3\u001b[39m\n",
      "\u001b[32m\u001b[1mUpdating\u001b[22m\u001b[39m `~/.julia/environments/v1.5/Manifest.toml`\n",
      " \u001b[90m [069b7b12] \u001b[39m\u001b[92m+ FunctionWrappers v1.1.1\u001b[39m\n",
      " \u001b[90m [37e2e3b7] \u001b[39m\u001b[92m+ ReverseDiff v1.4.3\u001b[39m\n"
     ]
    }
   ],
   "source": [
    "import Pkg\n",
    "Pkg.add(\"SpecialFunctions\")\n",
    "Pkg.add(\"SymPy\")\n",
    "Pkg.add(\"DualNumbers\") # https://github.com/JuliaDiff/DualNumbers.jl\n",
    "Pkg.add(\"ForwardDiff\") # https://github.com/JuliaDiff/ForwardDiff.jl\n",
    "Pkg.add(\"ReverseDiff\") # https://github.com/JuliaDiff/DualNumbers.jl"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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"
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "pyplot()\n",
    "\n",
    "f(x)     = x*(1-x)*(1-2x)^2*(1-8x+8x^2)^2*exp(1+exp(x)) \n",
    "df(x)    = f(x)*(1/x-1/(1-x)-4/(1-2x)+2*(16x-8)/(1-8x+8x^2)+exp(x))\n",
    "δf(x,δ)  = (f(x+δ)-f(x))/δ\n",
    "δf2(x,δ) = (f(x+δ)-f(x-δ))/(2δ)\n",
    "h        = [1/10^k for k in 16:-.1:3]\n",
    "x        = 0.3\n",
    "ϵ1       = [abs(δf(x,δ)-df(x)) for δ in h]\n",
    "ϵ2       = [abs(δf2(x,δ)-df(x)) for δ in h]\n",
    "\n",
    "plot(h,ϵ1,xscale=:log10, yscale=:log10,label=\"ϵ1 (forward)\", \n",
    "               xlabel=\"δ\",ylabel=\"|δf(x0,δ)-f'(x0)|\")\n",
    "plot!(h,ϵ2,xscale=:log10, yscale=:log10,label=\"ϵ2 (symmetric)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On another hand, the expressions of derivatives generated by symbolic computation software tend to become bulky as the complexity of functions, the number of variables or the order of derivatives increase:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*}64 x \\left(1 - 2 x\\right)^{2} \\left(1 - x\\right) \\left(32 x - 16\\right) \\left(8 x^{2} - 8 x + 1\\right) e^{e^{x} + 1} + 64 x \\left(1 - 2 x\\right)^{2} \\left(1 - x\\right) \\left(8 x^{2} - 8 x + 1\\right)^{2} e^{x} e^{e^{x} + 1} - 64 x \\left(1 - 2 x\\right)^{2} \\left(8 x^{2} - 8 x + 1\\right)^{2} e^{e^{x} + 1} + 64 x \\left(1 - x\\right) \\left(8 x - 4\\right) \\left(8 x^{2} - 8 x + 1\\right)^{2} e^{e^{x} + 1} + 64 \\left(1 - 2 x\\right)^{2} \\left(1 - x\\right) \\left(8 x^{2} - 8 x + 1\\right)^{2} e^{e^{x} + 1}\\end{equation*}"
      ],
      "text/plain": [
       "                                                       x                      \n",
       "              2                     ⎛   2          ⎞  ℯ  + 1                 2\n",
       "64⋅x⋅(1 - 2⋅x) ⋅(1 - x)⋅(32⋅x - 16)⋅⎝8⋅x  - 8⋅x + 1⎠⋅ℯ       + 64⋅x⋅(1 - 2⋅x) \n",
       "\n",
       "                         2      x                                       2   x \n",
       "         ⎛   2          ⎞   x  ℯ  + 1                 2 ⎛   2          ⎞   ℯ  \n",
       "⋅(1 - x)⋅⎝8⋅x  - 8⋅x + 1⎠ ⋅ℯ ⋅ℯ       - 64⋅x⋅(1 - 2⋅x) ⋅⎝8⋅x  - 8⋅x + 1⎠ ⋅ℯ   \n",
       "\n",
       "                                             2   x                            \n",
       "+ 1                          ⎛   2          ⎞   ℯ  + 1               2        \n",
       "    + 64⋅x⋅(1 - x)⋅(8⋅x - 4)⋅⎝8⋅x  - 8⋅x + 1⎠ ⋅ℯ       + 64⋅(1 - 2⋅x) ⋅(1 - x)\n",
       "\n",
       "                 2   x    \n",
       " ⎛   2          ⎞   ℯ  + 1\n",
       "⋅⎝8⋅x  - 8⋅x + 1⎠ ⋅ℯ      "
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using SymPy\n",
    "\n",
    "@vars x\n",
    "df_x = diff(64x*(1-x)*(1-2x)^2*(1-8x+8x^2)^2*exp(1+exp(x)))\n",
    "df_x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "\\begin{equation*}- 64 \\left(2 x - 1\\right) \\left(8 x^{2} - 8 x + 1\\right) \\left(16 x^{5} e^{x} - 40 x^{4} e^{x} + 128 x^{4} + 34 x^{3} e^{x} - 256 x^{3} - 11 x^{2} e^{x} + 160 x^{2} + x e^{x} - 32 x + 1\\right) e^{e^{x} + 1}\\end{equation*}"
      ],
      "text/plain": [
       "                                                                              \n",
       "              ⎛   2          ⎞ ⎛    5  x       4  x        4       3  x       \n",
       "-64⋅(2⋅x - 1)⋅⎝8⋅x  - 8⋅x + 1⎠⋅⎝16⋅x ⋅ℯ  - 40⋅x ⋅ℯ  + 128⋅x  + 34⋅x ⋅ℯ  - 256⋅\n",
       "\n",
       "                                            x    \n",
       " 3       2  x        2      x           ⎞  ℯ  + 1\n",
       "x  - 11⋅x ⋅ℯ  + 160⋅x  + x⋅ℯ  - 32⋅x + 1⎠⋅ℯ      "
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "simplify(factor(df_x))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:#00B8DE\"> Improved finite difference\n",
    "<a name=\"Improved-finite-difference\"></a> \n",
    "[Table of contents](#table-of-contents)\n",
    "\n",
    "Going back to finite difference method, assume that $f$ admits an analytical extension in the neigbourhood of the real axis and consider the finite difference $Im\\left(\\dfrac{f(x+ih)-f(x)}{h}\\right)$ or $Im\\left(\\dfrac{f(x+ih)-f(x-ih)}{2h}\\right)$, where $Im(.)$ stands for the imaginary part. The following results show a tremendous improvement compared to the finite difference with real valued increment. Do you have any explanation for it? Hint: compare the analytical development of the finite difference in both cases."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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bV3AZ8RFmZSqYurz7mJWpgF3bviTsWJE9KSiAMSONmKUziF83H1v/PriO/hStreNjHUPnWg4Lz8Yk/Wew+7+5WSns6KZDp4H2W/TcTpGQJYQQouAYVZXFF43UXqnng6NGhtfWcOVpHRMbarHUFe4YaQlYeWBWtgqKuSUZoeeIW/0daWf/xPn5d7Hw8DR1abmy6/gMGms74td/D4A++g6RX79B2umDOA99G8deI1C02ic6hm2r7mSGXSTjxqVct6lgo7Cjq450AwRs1hOZKiFLCCFE/ttx04jvWj3P7zHQ1E0hZICOr1tocbE0zc1nErDyQNFqMa9ci4QdK0g+sAmngeOxqtvM1GU9kMbCCsceL5J6cj8JW38hYtYrGNNTKfPaV1j7tM2XY1jWaYrW0TXHwe7/VtVeIairjqh06PS7nrh0CVlCCCHyx9kYlW5b9QRsMWChVTjQU8uqAB01HUx3Vz9IwMoz8yqeqClJ2Hd+DpvmnU1dTp5YNW6HedU6JGxdglnl2ri/MRvz8tXyrX1Fq8WmeRdSgnc9dAb52o4K27voCE2CrlsNxGdIyBJCCPH47qSojNpnoMEaPefjVH5rr+VgTy0t3YtGtDHtvALFiG3bvujKVcG68VOmLiXPFEXBecgk0i8ex7pZRxTNk10SzIlNi84kbPuFlKM7HjqDfQMXhT+6aAnYbKD2Sj0fN9YyvLaCTmPaTxlCCCGKj+RMlVmnjcw4acRcC7OaaXi5jgYLbdF6LykaMa8Y0No5YuPb3qQTiT4OnUtZbFp0KZBwBaB1cMGqXguSDmzK0+SmTcpoON1fR4cKCqP2G2i4Ws+WMGOeJ0YVQghROhmMKj+eN1JzZdYM7C/XyRrA/lp9bZELVyABS+QDm1bd0N8JI+Pq2TxtX9lWYUk7HUd76yhjpdDtDwMdfzdwIlpClhBCiHupqsrWG0a81+h5aZ8B/3IK5wfomNlMi5NF0QtWf5OAJZ6YRU1vdK7lHzhlQ04al1HY1U3Lho5abiSp+KzRM2yPnpvJErSEEELAiWiVjr8b6LLVgJOFwuFeWpY9paOqfdENVn+TgCWemKLRYNOqG6kn92NIjHu0fRWFHlWyLhvObaVhU5hKzRV6PjhqIFEGwgshRKkUnqQybI8enzV6wpJU1nXQsqe7lqZuxSe2FJ9KRZFm3bQDKArJh7c91v5mGoUxdbRcflrHq/U0zDiVdZ19wTkjeqMELSGEKA0SM1Qm/2Wg1ko9m8JU5rbScKa/jl4emmI3BloClsgXWht7rL3bkHxoC6rR+NjtOJgrfNZUy8WB/wyE916j5/cbMhBeCCFKKr1R5bsQAzVW6pl12shr9TVcflrHmDpazIrpneYSsES+sWnVDUP0HdIvBD9xW38PhP+rtxZXS4WuW7MGwp+UgfBCCFFiqKrKhutG6q/WM/aAkc4VFS4O1PFpEy0O5sUzWP1NApbIN+YeXpiVr/rIg90fxLeMhl3dtKz//0D4Rmv0DJeB8EIIUewdjTTSbrOBXtsMlLdWONZHx6K2OirZFu9g9TcJWCLfKIqCTavupJ09gj42Ml/b7fn/gfBzWmrYGKZSa6WeD48ZiEhV5dKhEEIUI6GJKoN36mmyzkBkqsrmTlqCumpp5FoygtXfZCZ3ka+sG7cjfsMPJB/6HYeuQ/O1bTONwti6WobU1PD5CSPTTxr5ONiIrRl42EJVO4Wqdgoedvf+u7h3MwshREkQl67y6Qkjs88acTKHBa21DKtVclfzkIAl8pXG0hrrxk+R/OdW7Ds9i6LN/1Ps74Hw4+pq+DNCJTRR5VoiXEtU2X7TSGgipBr+2d7JAqragYet8v/gBR7/CmDWupL5wy2EEEVBhkHlu3NZH4jTDPB2Qw1vNtBga1ayf/dKwBL5zqZVN5IPbib1zJ9YN/QrsONUsFHoV/X+H1BVVYlIzQpc1xIhNEnN/vf660auJ0Hmv250HJh2iBdjNmFeqSaeDbwoU6sOWnvnAqtbCCFKA1VVWX1N5e2/DFxLhOG1FD5qrKW8TckOVn8rVQErMDCQwMBAUlNTTV1KiWZeoRrmHl4kH9hUoAErN4qi4G4N7tYKzd3vf95gVLmVkhXA7ty6S+Pl3xBl7oISsgf9qdXcBlLsy2JXzQu7GnWyBu+Xq4qiLZj1HIUQoqQ5dNfIhD+NHIpQ6VJJYV0HLfWcS0ew+lupClhjx45l7NixBAcH07hxY1OXU6LZtOpG7C9fkBkRjplbRVOXcw+tRqGSLVS0NhK56ksMtjY0mfglMRobtpyOIORUCJY3z+F74Tz1T+5FqxrA3BKLKp6Ye3hiXrUOFlW80NjYmfqlCCFEkXI5XuWdvwysuqbS0Bm2d9USUKF03k9XqgKWKDzW3m2IXzuf5INbcOw90tTl5Chx5yoyrp6hzNjpaKxtcQWGNnGHJu7cTmnLqqtGvriURsr1SzRPO0/nhPN47t+K+fblAOjcKmFe1QsLDy/Mq9ZB51YJRVM6f5EIIUq36DSVT44b+TbEiJsV/OyvZUgNBW0JHcCeFxKwRIFQzMyxbtaR5MN/4ND1eRRzC1OXdI+MG5dI2LIYu6cGYFGjwX3Pl7NWeKWellfq2RCW1JCVV+vzwRWVY5FGPI13eNHqAu0N5yl34xwpR4JANaJY2WJRrS4OvUdiVqaCCV6VEEIUrjS9ytwQI1OPGzGqMMVHw2v1NXLzEBKwRAGyadmVpF2rSTmxD5umAaYuJ5sxI42YJdMxK18V+y7PPXT7yrYKbzbQ8maDrO7vFVcrsOhKOSbGtsXBHQb5pDLE8jJ1k8+T/lcQEV++isvQt7H08i2EVyOEEIXPqKqsuJJ1OTA8GUZ5afjQR4OblQSrv8n1DFFgzMpUwKJWI5LzcWb3/BC/7nsMsZE4P/cWis7skfat4aDwXiMtp/ubcaafjvF1NeyMtqL1ufrUihjAF62+QlfFi6gFH5C48zeZBFUIUeLsuW2k2ToDz+4y4O2icKa/jsBWWglX/yEBSxQo21bdyLh+nozwK6YuBYDUM3+SfHBz1mU890pP1FZdZ4WPfbVcGKgjuI+O4bU0fBdqzbuek7FrP4D4DT8Ss2Q6xoy0fKpeCCFM53ycSq9tetpuMqAosKe7lnUddXg6SrDKiQQsUaAs6zVH4+BC8sHNpi4FQ0IMscu+wrJuM2xads23dhVFoZGrwvRmWr5tpWXhJQ0baj+P89B3SDt9iMjZE9DHRuTb8YQQojBFpKqMPWCg3io9J6NVfm2n5c9eWtqUkwjxIPLdEQVK0eqwad6ZlKM7MaYlm6wOVVWJ+fVLUBScBr2OohTMJ67na2l4vqbCy/sNhFVrQ5lXv8SYnEjErFdIv3KmQI4phBAFIUWv8ulxAzVW6Pn1spHPm2o4P0DHMzU0aArod2hJIgFLFDibFp1RMzNIObrLZDUk799I+vmjOD37Blo7xwI9VmArLZVtYeAOPYay1XCbMBude2UiA98i6YDpe/KEEOJBDEaVRReN1FqpZ0qwkRdra7j8tI43G2ixlLsD80wClihwOscyWNZtRvKBTSYZ9J155zpxG37Axq8HVnWaFvjxbMwUVrbXcTkeXj1kQGvrSJkxn2HTqhtxv80hdsU3qPrMAq9DCCEeVdBNI43X6nlhj4GWbgrnBuj4qoUWF0sJVo9KApYoFLatupF5O5SMayGFelxVn0HM4unonMvi2POlQjtuPWeFOS21fH9eZdllI4pWh1O/MTgNeo3kI9uJDHwLQ2JsodUjhBAPciZGpcvvejpsMWBjpnCwp5aVATqq20uwelwSsEShsKjtg9alXKEPdo/fvIjMu2FZUzIU8mSnw2srDK6hMHK/gYtxWT13Ns07U2bcDPTRt4mY9QoZYRcLtSYhhPi3W8kqL+3V03CNnssJKqsCtOzvoaWFu8SDJyXfQVEoFI0G25ZdSTm+D0NSfKEcM+3iCZJ2r8Gh2/OYV6xeKMf8N0VR+K6VlvLWWeOx0vRZIcuiah3c35iDxt6FiDlvknx0R6HXJoQo3ZIyVaYcM1BzpZ51oSpfNddwtr+OflU1BXYTUGkjAUsUGutmHQBIObK9wI9lTE4k9pcvsKjRENu2/Qr8eLmxM88aj3U+Hl7/05j9uNbRFbdXZmLt3YbYpTOJW/89qsFgsjqFEKWD3qjy/XkjNVfo+fykkXF1sgawj6+nxVwrwSo/ScAShUZr64iVtx9JBzejGo0P3+ExqapK7G+zMWam4TR4gskXYG7oojC7hZZ554ysvPLP61bMzHF6dgIOfUaTtGctUQvex5icaMJKhRAllaqqbAkz4r1Gz8h9BtpXULgwQMf0ZlocLSRYFQQJWKJQ2bbqhiHqNkm7VxfYnXQpfwWRemIfTgPHo3MsUyDHeFQjPBUGVVd4aZ+By/H/3EmpKAp2/r1xHT2NzBuXuPvVeDJvh5quUCFEiXM8SqXDFgPd/jBQxlLhr95alrbTUcVOglVBkoAlCpV51bpYN+1A/IYfuTN1GIl71mFMz7+lZPRRt4lb/S3WTQKw9m6Tb+0+KUVRmO+nxd0Knt6pJ91w73QVlrUa4fbGbDRmFkR89Rqppw6YqFIhRElxM0Xh+d16Gq/VczNFZUNHLTu7afEtI2/9hUG+y6JQKYqC87MTcH9rHhY1GhK/fgF3Ph5KwrZfMaY82eUx1WAgZukMNLYOOPZ7OZ8qzj/2/x+PdSYG3vzz/kukOtdylHntKyy9fIle+AkpJ/aaoEohRHGXkKHy7l8Gmv5hw9ZwlW9baTjdT0ePKjKAvTBJwBImYVbOA+chEyn73kKsGrUhYdsybn/0PHEbfsSQEPNYbSZuX07G9Qs4D5mExtImnyvOH41cFb5qoWFuiJHV1+4PWRoLK5xfeA/LOk1J2LwI1SgD34UQeZNpVAk8m7W0zdenjYytlcHlgTpG19Gi00iwKmylKmAFBgZSp04d+vUz3V1l4l46l7I49R9HuQ8WYevXneQDm7n98fPE/jYXffSdPLeTHnqOhG2/YNfxGSyq1inAip/cy14a+ldVGL7HwNWE+2e2VxQF+85D0EfeJPXEPhNUKIQoTlRVZV2okXqr9Lxy0Ej3ygqXntbxbt0M7MwlWJmKoppi7RITCw4OpnHjxhw7dgwfHx9Tl1NsxMTE4OzsXKDHMKYkkbR/I0l71mFMTcTapy127QdiVs4j933SUrj7xVi0Ng6UGT8LRast0BrzQ3yGis8aPc4WCvt7arHI4fboyHmTMcRF4j7pO5PfCVnSFca5LURBOBJh5M3DRvbdUelYQWFGMy0NXbJ+n8h5bVryW1sUKRprW+w7PkPZDxbh0Gsk6ZdPcXf6aKJ+/JiM6xdy3Cdu7TyMCbE4D5lULMIVgIO5wor2Wk7GqLx1JOcpK+w7PYv+znXSTh8s5OqEEEXdtQSVZ3bqabbeQFyGytbOWv7oqssOV8L0JGCJIkljYYmdf2/KTv4Jp2feQH/nOhFfvUrkt2+TdvF49qLRqSf3k3J4G459X0ZXpryJq340vmU0fNFMwzdnjKwLvT9kWVStg0VNbxK2LzPJItlCiKInNl3lzT8NeP6mZ+9tlR/baDneR0enSvJ2XtToTF2AEA+i6MywadYR6ybtST11kMSg5UR9+w5mlWtj69eD+HXzsWrQCutmHU1d6mN5pa6G3bdVhu0x4O2i4PGfeWnsOj5DVOBbpIUcwapuMxNVKYQwtXSDyrchRj45biTDAJMbaXijvgYbM+mxKqok8opiQdFosfZujduEubiOmopiZk7sr1+AmTmOT79abG89VhSFhW20OJrDoJ0GMv4zP5ZFjQaYV61Dwh+/Si+WEKWQqqqsvGLE6zc9bx42MrBq1tI27/toJVwVcdKDJYoVRVGw9PLF0suXjOsXUKxs0NrYm7qsJ+JokTUey2+jgXf/MvJF83/GkSmKgn3HZ4maP5n0C8FYejY2YaVCiMK0/07WAPbDESrdKyts6qSljpOEquJCerBEsWVepTZmbhVNXUa+aOqmYXpTDbNOG9l4/d7xWBaejTGrXIuEbdKLJURpcDFOpe92Pa03Gsg0quzspmVjJ52Eq2JGApYQRcRr9TT0rKLw/B4DYUn3rldo3/FZMq6eJf3yKRNWKIQoSJGpKq8cMFB3lZ5jUSpL2mr5q7eOduXlrbo4kv81IYoIRVH4qY0WOzMYtMNAXLqa3WNlWbcZZhWqk7htmYmrFELkt1S9yucnsmZgX3zJyFRfDecH6BhSU4OmmI4vFTIGS4gixdlSYflTWtpsNOC0WI+ZBlwtoYwldHYZyCunPuOz9afJqFjn/48rWX9bZf3taglmsiSGEMWCUVX55bLKe38ZuJ0CL9fR8IGPBldL+RkuCSRgCVHEtHDXcLKfwukYlag0lcg0iEqD6w4tCL9SGa8TyxiT9BFRaZCZwxylDuZZgcz1/+GrgbNC/6oavF0otndbClHS7LxpZOIRA8FR0NdD4fOmWmo6yM9nSSIBS4giqI6TksOAVi0pTs9Qccl0QltfxaxSLRIyITIVotJUotL4fxhTs/+OSIXvzhn59ISRGvbQv6qGAdU0NJKwJYRJhMSqTDpsYPMNlWZuCvt7aGhVVkbrlEQSsIQoRqwatUG3dSkJ25bh+tIUHMyzeqxqPOCTb6ZRZcdNlVXXjCw4b+Tzk0aq2cGAahoGVNXg4yphS4iCdidF5cNjRn64YMTDFla219K/qiI/eyWYBCwhihFFo8WuwyBif51Fxs2rmFeo9tB9zDQKnSspdK6k4Ts/lV23VH67auSH80am/z9sZfVsKTR2zfkXvqqqqOmpaCytC+JlCVFiJWeqzDptZMZJI+Za+KKZhjF1NDku8C5KFumXFKKYsW7cDq1LWRK3/frI+5ppFDpW1PB9Gx23h+jY1kVL+woKP14w0mSdgeor9Ew6bOCvSGP2HYyq0UjM0hnc/uBZkg9vk7m4hMgDg1Hlx/NGaq7UM+24kdFeGq48reP1+loJV6VEiQtYoaGheHl5MXr0aKZMmWLqcoTId4pWh13A06SeOkDmneuP3Y6ZRqFDRQ0LWuu4M0TH9q5aOlTQ8NNFI03XGai2XM/EwwYurPyR1ODdmFerR+yyL4lZ9BnGlKR8fEVClCx/3DDSaK2el/YZ8C+ncH6Aji+aa3GykGBVlBzbuYuzhw4VWPtFImCNHz8eDw8PFEXhzJkz9zx36dIlWrZsSa1atWjatCkhISEPbc/BwYG0tDSqVq1aUCULYVI2TQLQOriSsH15vrSn0ygEVNAwv7WW24N1BHXV0qmihrT967H9czVfV36J75t9hNPQd0g7f4y7M8eQfvVsvhxbiJLiZLRKxy16Om814GiucLiXlmVP6ahqL8GqqDn/11/E/G5NyB93CuwYRSJg9e/fn/3791OlSpX7nhs1ahQjR47k4sWLTJo0iRdffBGAu3fvEhAQcM+fSZMmUaVKFf78809++uknNmzYQGhoaCG/GiEKnqIzw679AFKD95AZeTNf29ZpFNpX0PCl/SHeCv+eGN++xDfuxXtHjXyj+OE+6Vu0jq5EzplIwtZfUI2GfD2+EMVNeJLKsD16Gq3Rcz1JZV0HLXu6a2nqViTeYsV/3Lh4nqur0kgxS6T9qGYFdpwiMci9TZs2OT4eERFBcHAw27ZtA6Bfv36MGzeO0NBQPDw8CAoKyrVNRVFwd3cnMTGxQGoWwtRsmncmYftyErcvx/nZCfnadvrVM8QsmY6VdxvqP/sS32s0lLcx8N5RI7UcytBv3EwStv1Kwh+/kHbxOM7PTULn5JavNQhR1CVmqEw/aeTL00ZszWBOSw0jvTQy2W8RFhd7m78WXccSaxoPq4xzufIFdqwiEbByc+PGDcqXL49Ol1WmoihUrlyZsLAwPDw8ctxn165dLF68GABHR0fq16+fa/vjxo3DwcGBvn370q9fv3yvv6SJjY01dQniP7TNu5Ky/VeMzbqhyaeAY4y8SdpPU9BUqAldhhMbFwfAeA84HWHJ0N0qjm1S8GnWDYuy1clYG8id6aMx7z4CXZ2C+zRYkOTcFo9Cb4TF18yYcc6cxEyFl2tmML52BvZmkBhn6ur+Ief1vdLSkpjyx5toy5dnRL0+WLu4EhMTk2/tOzs73/N1kQ5YcP/8PA+7g6ldu3a0a9cuT23PnTsXHx+fx66tNPrvCSRMyxjQnzsHN6I9+gdOT7/6xO0Z4qOJWD4TnVMZ3EZ9jMba9p7nf+mg0m6zgef+tOFIbx2VGrXEWLs+sSu+IXXVN5g174xDn9FoLCyfuJbCJue2eBhVVdkYpvLWEQMX4uC5mgpTfbVUsjU3dWm5kvM6S0ZaOr/+/BOhrhnMaNMF7/r+BX7MIn2BuFKlSoSHh6PX64Gsk/vGjRtUrlzZxJUJUTRozC2xbduP5CPb0cdGPlFbxrQUohZ8AEYjriM/uS9cAVjpFNZ10GKuhZ7b9CRlqmis7XB+4T2cBr1GSvAuIma9Qkb4lSeqRYii5mikkXabDfTaZqCCtcKxPjoWtdVRyVYuBxZ1RqOR+Ys+p+7NdrzrPBTv+p0K5bhFOmC5ubnRqFEjli5dCsDq1avx8PDI9fKgEKWRrV93NBZWJO787bHbUA16on+aij76Nq6jPkHnVCbXbctaK2zsqONyAgzeZcBgVFEUBZvmnXGbMBfFzJyIr14jcfdamTNLFHuhiSqDd+ppss5AVJrKls5atnfV0shVglVxMW/tO6yyDSax3RHadu9baMctEgFr7NixVKxYkfDwcAICAqhRo0b2c/Pnz2f+/PnUqlWLzz//nB9//NGElQpR9GgsrbH1703yod8xxEc/8v6qqhK7/GvSL5/G5cUPMSv/8OlNGrgoLH9Ky6YwlXf++mfFaTP3Sri9/hW2ft2JXzef6AUfYEiUcSCi+IlLz1oz0PM3PTtvqXzfWsuJvjq6VNLI8jbFyMYFy4i7XJnnbbzp2X10oR5bUUvhR8zg4GAaN27MsWPHZAzWI4iJiZHr+UWUMSWJ2x8PxaZ5Zxx7j3ykfeM3/5x1J+Jzb2HdOG/jF//29WkDr/9p5IfWWl70vPfzWtq5o8T88gUoCs7PTsDSy/eR2i5Mcm6Lv2UYVL47Z+TjYCNpBpjUQMOEBhpszYpfqCrt53XQsjWY/1Wdu2XP0nfC02i12kI9fpHowRJCPBmNtS22bXqRfHAzhqS4PO+XdGAziduX49DzxUcOVwCv1tMwylPD6P0Gdt8y3vOcpZcv7m99h1mF6kTNn0zcugWo+oxHPoYQhUFVVVZfM1JnlZ43/jTS10Ph8tM6PmysLZbhqrQ7tGUruqNVueN0kT6vDyj0cAUSsIQoMWz9+4CiIWn32jxtn3rmEHGrArFt3RPbdv0f65iKojCnlQb/cgp9gwxcir+3Q1xr54TryI9x6D2SpH0biPj6dTLv3nisYwlRUA7dNeK30UD/IAO1HBRO9tXxfRsd5awlWBVHpw8eIHmnC1E2N+j5Rld0ZmYmqUMClhAlhNbGHptW3UnatxFj8oMn2E0PPUfMos+xqt8Chz6jnmhMiZlG4bcALW6W0P0PPTFp94YsRaPBrm1f3F7/GjUjnYhZr5B5J+yxjydEfrkcrzIgSE/LDQZS9Crbu2rZ0llHPWcJVsXV1TOnublOIdEihg7jW2BpY2OyWiRgCVGC2LXrB0YDiXvX5bpNZuRNor+fglnF6jgPmYSiefKucycLhc2ddUSnQf8gAxmG+4d2mlesgduEOWhsHYjf8MMTH1OIxxWdpvLaIQN1Vun5M0LlZ38tx/roCKggb4nF2d2wUEKWRqHXZNBiZG0cXHO/G7owyNkkRAmitXPEpmVXkvaux5iWfN/zhsQ4ouZNRmNjj+tLH6GYW+TbsavbK6zpoGX/XZUxBww5TtGgsbDCoftw0kKOkHbxeL4dW4i8SNOrzDxpoPoKPQsvGPmosYaLA3U8X0uDRu4MLNaSk2PZ8eNRdEYL6gx2pmwVD1OXJAFLiJLG7qn+qJnpJO3beM/jxvQ0or7/ADUzDddRn6Cxscv3Y7cpp2FBay0/XlD58rQxx22sGrXBvIon8et/kIWiRaEwqiq/Xjbi+Zued/4yMqSGhstP63jHW4uVToJVcZeRnsI7q8ayoeImyvc0UK1+A1OXBEjAEqLE0Tq4YNO8E0m712BMTwVANRiIWTQN/d0buI6cis6lbIEd/4VaGt5uqGHiYSMbrt8fshRFwaH3SDJvXiHlrx0FVocQAHtuG2m2zsDgXQYauSic7a9jbistblYSrEoCfWYmS+cs4opRz2uNetLAr7WpS8omAUuIEsjuqQEYU5NJPrAZVVWJ+20OaeeDcRk2GfOK1Qv8+NOaaOjtofDsTgMnou+/VGhRtQ5W3q2J37IIY3pagdcjSp/zcSq9tulpu8mAosDe7lrWdtRR21GCVUky/9ePqRDRiNesB9Dct/Bmac8LCVhClEA6Z3esmwaQuGs1CZt/JvnPrTgNeg1Lz8aFcnyNorCkrZbajtDjDz23U+4PWQ7dh2NMSiBp1+pCqUmUDndTVF7eb6DeKj2nolWWPaXlz15aWpeTt7uSZtH6j1ipO0Ws3346PD3A1OXcR5fTg1WrVn2i27avXr362PsKIfKHfcDTpBzZTmLQCuy7Po9N0w6FenwbM4UNHXU0Xaen1zYDu7trsf7XeBedazls2/Qkcedv2LTojNbBpVDrEyVLil7lq9NGPj9pRKfA9KYaxtXVYKGVHquSaOuilWSG1uFpr1QG9XzT1OXkKMeANWXKlEIuQwiR33Su5bHvOBjVqMeuwyCT1FDBRmFjJx2tN+p5freBFe2199ytZd9hECmHtxH/+2KcB71ukhpF8WYwqiy5rDL5qIGIVBhXR8PkRhqcLSVYlVT71m7C6mQt7F3P06/fNFOXk6scA9bzzz9f2HUIIQqAfefBpi4BH1eFX9pp6bvdwIfHjHzi+8+8WxprO+w6DSZ+3Xxs2/TCvHw1E1Yqiptt4UYmHjZwKgaerqbwaRMt1ewlWJVkx3buIvNAOaLtr9H7jT4mWQInr+SitBCiwPX20PB5Uw1TjxtZeuneOwttW3VD51o+a9qG0rf2vHgMp2NUOv+up9PvBuzNFP7spWV5e52EqxLu4rFjRP9uRZzlXbq81hZzy/ybx68g5NiD9V+qqnLu3Dnu3LlDamoqLi4u1KpVq1Sv0i2EeDQTG2g4H6fy/B4D008aqOukUNdJoY6TlgZPDcdqxSeknTuKVZ0mpi5VFFG3klXeP2rg50sq1e1gTYCW3h7KE40ZFsXDzcuXuLQyGYMukzZjGmLr6GTqkh4q14BlMBjYtGkTixYtYufOnSQmJt7z6VJRFLy8vBgwYAAvvPACVapUKZSChRDFk6IozPPT0txN5Xi0SkisStBNI9HpgNqEVTZ1Kbt0AYHtGuDpossOYNXts9Y7FKVXYobKzFNGZp02YqWFb1poGOWlkfOilIi9c5ujC69jjiX1n6+Aa/kKpi4pT3IMWMuWLeP999/nzp07dOnShSlTpuDt7Y2rqyuWlpbExsZy7do1/vrrL3777TemTZvG888/z0cffUT58uUL+zXkWWBgIIGBgaSmppq6FCFKJXOtwkivf94UVVUlIhVC4lRuXhxBlXWvUfvyNmbf7pwVvAAzDdR2gDr/D1x1nRTqOCrUcJDgVdLpjSoLL6h8cMxAXAa8Xk/D294aHMzl/720SEtLYvWPv1Mlsy6VBkBlT09Tl5RniprDoAdPT08mTZrEoEGDsLa2fmgjp06d4uuvv6ZWrVq8/fbbBVJofgoODqZx48YcO3YMHx8fU5dTbMTExMhlYVGgYpbOIO18MO7v/UCUakNInMrZWJWzsRASm/Xvv4OXuSZrYPPkRlpqPeHkkXJuFy2qqrLlhsrEwwbOxcGQGgpTfbVUsZNg9SiK+3mt12fw7q8jOGNM4eMqo/BtF2Dqkh5Jjj1Y586de6Rr2g0aNGDhwoUyQFUI8UTsuw0j5eR+knauwr3bC7hbK7T7V6e4qqpEpsHZWJUjESpzQoz8ckXP4Or5E7SE6QVHZQWrnbdU2pbLmrC2cRn5fy1tDAYDK77+hWuOWiZ69cS3VfEKV5DLXYSPO2BQBhoKIZ6EzqkMdm37krh7DfrYyPueVxQFNyuFduU1vOWt5fJAHbNbaNh5W8VrlZ6hu/RcjJMPesVRWJLK0F16Gq/VcytFZWNHLTu7SbgqrRau/gjLmEoM03WiXavnTF3OY3ngXYQJCQns2LGDc+fOERMTg6IoODk54eXlRfv27bG3ty+sOoUQpYRd+4EkH9pKwuafcR4y8YHbWuoUxtbV8mJtDT9eMPLZSenRKm7iM1Q+P2HkqzNGHMxhnp+GF2tr0Mn4ulLrty1fsDTzJM/7KvTr+5Gpy3lsOQYsVVX58MMPmTVrFqmpqVhbW+Pk5JS1aGxcHCkpKVhZWfHmm28yZcoU6bkSQuQbjaU19l2eI+63Odj698a8Us2H7iNBq/jJNKosOGdkSrCR5EyY1EDDxAYa7GQAe6m2c/laNKe86eV1lxd6f2jqcp5IjpcIP/nkE2bNmsXkyZMJDQ0lKSmJGzduEB4eTlJSEqGhobz//vvMmjWLqVOnFnbNQogSzqZ5Z3RlKxO3bsEjje38O2jJpcOiS1VV1oUaqbdKzysHjfSorHDpaR0f+2olXJVyh7duQ/NXFQyWsYwf9CkaTfGeCz3H6r///ntmzpzJO++8Q+XKle97vnLlyrz99tvMmDGDBQsWFHiRQojSRdFqceg5gowrp0k78+cj7y9Bq2j6K9KI/yYDfbYbqGKrcLyvjoX+OirYSLAq7c4eOkRikCNRNuH0mNAFnZmZqUt6YjkGrOjoaLy8vB66s6enJ9HR0flelBBCWHr5YlHbh/gNP6Aa9I/XhgStIiEsSWXwTj1N1xmITVfZ2lnLtq46GrpIsBJw7cxpbqw1kmQeR8ArzbCysTV1SfkixzFYPj4+zJ49Gz8/P8xySZF6vZ45c+aUuHmkwsLCiIqKMnUZRVJ8fDwODg6mLkM8IVdX1xx7posaRVFw6PUSETPHknxgM7Ztej12Ww8bo+Waj3WLfyRkqHz2/wHsjubwfWstw2opaGUAu/i/iBthnFkahaLR0GxkDRzLuJu6pHyTY8D68ssv6dChAzVr1qRPnz54eXnh6OiIoijExsZy7tw51q1bR2xsLNu3by/smgtMWFgYXl5epKSkmLoUIQqMtbU1586dKxYhy7x8NaybdSThj1+w9m2PxvrJPtnmFrT6V7LksxaqLBacT/RGlR/OG/ngmJGkzKx1KCfJAHbxHwmxMRyYF4K10YHqg60o51HN1CXlqxwDVtOmTTlx4gQzZsxg7dq1fPPNN/c8X7lyZbp27crEiROpWrVqoRRaGKKiokhJSWHp0qV5ukQqRHFz7tw5hgwZQlRUVLEIWAAOXYaSGryHhO3LcOw1Il/a/HfQ+uGCkanBWmqv1DPCU8PkRhrKy5igx6KqKr//fwb2kDgYWlNhmq+Wirby/RT3ysxMY8mPi/FM98O1Zwo1GrYwdUn5Ltd5sKpWrcp3330HQEpKCnFxcQA4OTlhZWVVKMWZipeXV4m79ClEcaV1cMHuqQEkbF+OrV8PdC5l861tS53CuLpaepeJ59fbDkw/aeSni0bG1dXwVkMNrpYSDPLqVLTKm4cNbL+p4l9O4ajMwC5yYTDombpiHPvdIvm4UVUaPsHl/6LsofdApqSkYG1tTfny5Slfvvx94SosLKzAihNCCADbdv3Q2NoTv3FhgbRvrYNJDbVcHaRjUkMN884ZqbZcz5RjBhIyZDD8g9xJURmxV0+jtXpCE1XWddCyS2ZgFw+w+utl3EpwYkKlrrRqXzLDFeQhYDVo0IBDhw7l+NzChQtp0KBBvhclhBD/prGwxKHr86Se2Ev6tZACO46DucJHjbVcfVrHSE8Nn5/MClpfnDKQqpeg9W8pepVPgg3UWKFnTajKV801nOmvo5eHRiafFrlavH4qcUnm9FB86NpulKnLKVAPDVienp60adOG9957D70+61bpyMhIevfuzciRIxkxIn/GRAghxINYNwnArEJ14h9x8tHHUcZK4YvmWdM79K+q4Z0jRmqs0PNdiIEMQ+kOWkZVZfFFI7VW6vnkuJHRXhouP61jfD0t5loJViJ3m7fN58ekv0jz+oueLz9j6nIK3EMD1qZNmwgMDGTOnDk0a9aMefPmUbduXU6ePMnOnTuZOXNmYdQphCjlFI0Gh14jyLh+ntQT+wrlmBVtFea11nJ+oI6nyiuMPWDE8zc9iy8aMRhLX9DafctIk3V6nt9joIWbwrkBOr5orsXJQoKVeLD96zejDWpAj1RPxvWfYepyCkWe5qEfOXIke/fuJSQkhLFjx1K5cmVOnz5NmzZtCro+IYTIZlnLG8u6zYjftBBVn1Fox61ur7CknY5T/XQ0dFZ4fo+BBqv1rLlmLPDetKLgcrxK72162m02oFMU9vXQ8luAjuoyrYXIg+O7d5Oxz51EqzuMHzKl2C+Bk1d5epX79u2jb9++ODo6MnjwYI4fP87LL79MfHx8QdcnhBD3cOj5EobYCJL2bij0Y9dzVljbUcfhXloq2Cj0CzLQZJ2BP26U3KD121UjjdbqOR6t8ms7LYd6afErWzreIMWTuxh8jMjNlsRZRtLl9baYW5bsWQj+7aE/JZMmTaJdu3Y0atSIM2fOsHjxYn7//Xd2795NvXr1StREo0KIos/MvRI2LbuSsG0ZhuQEk9TQ1E3Dtq46dnXTYqGFzlsNtN1k4OBdo0nqKQiZRpUJfxoYuMNA98oKZ/vreKaGBo0MYBd5dOvKZS6tSCZNl4zfy/WwdXQydUmF6qEBa8GCBSxcuJDVq1fj4uICQMeOHTl9+jR+fn507ty5wIvML4GBgdSpU4d+/fqZupQC079/f8qXL4+iKCQlJT1w2xMnTtCtW7fsr+fPn4+npyfe3t5Fbo3JKVOm8OabbwKwceNGRo0q2XefiAez7zwEVCNJe9aatI625TXs76FlUyct8RkqrTYYGLJLz63k4t2bdSdFpf1mA7PPGPmmhYZf22mxNZNgJfIuNuIuR368hgYNDZ8vh1uFSqYuqdA9NGCdPn2aoUOH3ve4o6Mjy5YtY9myZQVSWEEYO3YsISEhrF692tSlFJjRo0dz4sSJPG377rvv8tZbb2V//fXXX7NkyRJOnDiRHabzwmAwPGqZD/T33aq56dGjB3/99RdXrlzJ1+OK4kNr64h1kwCSD21F1WeatBZFUehWWUNwXx0L22jZFq5S+7esqR2K4x2H++8Y8Vmr51K8yq7uWsbX08q0C+KRpCQmsHPOEWwyHfDoa0YVzzqmLskkHhqwKlV6cOocOHBgvhVTXNxOUQmOyv8/t1Py9ss4ODiYDh064Ovri6+vL0FBQdnPBQQE4Obm9tA2wsLCOHv2bPaNCv379+fKlSs899xz9O/fH4AlS5ZQv359GjRoQLdu3bh58yYAP//8M507d2bo0KH4+vqyf/9+ateuDWQtleHq6sp7770HwI4dO2jfvj0Av/76K82aNaNRo0Z4e3uzZcuW7Ho8PDyYNm0a7dq14/nnnyc+Pp7+/ftTp04dOnXqxOXLl++pf+DAgSxcWDCTToriwdavO8bEWFJP7jd1KQBoFIVhtTVcGKhjWC0Nbx0x0nCNnu3hxeOyoaqqfHPGQLtNBmrYKwT31clYK/HI9PoM5i36GqfU8jh0iMerWTNTl2QyOS6VM3fuXEaMGIGFhUWeGzp58iSRkZEEBATkW3FF1fxzRj4Kzv9fmh/6aJjSWPvAbWJjYxk+fDgbNmx4orXk9uzZQ/PmzbO/XrVqFR4eHqxatYp69epx5swZJk6cyLFjx6hQoQLTpk1j5MiRbN68GYD9+/dz/PhxatasCUBqaiphYWFER0dTq1Ytdu7cCUBQUFD2OdGpUyeeeeYZFEUhNDSUli1bcv36dczMzICs0Ldz504URWHChAnY29sTEhJCVFQUPj4+94T5li1bMmnSJKZNm/bY3wNRvJmVrYJFzYYk7d+IdeN2pi4nm5OFwuyWWl6qreGVgwY6/m6gX1Ujs5ppqWJXNHuCkjJVXtprYMVVlTfqa/i8qQYzTdGsVRRdRqORL357na2ON/DuXIOmAc+auiSTyjFgLV68mKlTp/Lss88yYMAAfH19s98E/+3WrVv8/vvvLFu2jMOHD/Pzzz8XdL1FwigvDT2r5P8nu3LWD9/m0KFDhIaG0rNnz+zHpk+fTqdOnR7pWOHh4ZQtm/uabrt27aJ79+5UqFABgDFjxjB16tTsO6X8/PyywxVA+/btCQoKIjo6miFDhrBgwQLi4+MJCgri22+/BeDatWsMHjyY8PBwdDodUVFRXL9+nRo1agAwbNiw7EsRu3btYs6cOQC4urrSt2/fe+orW7Ys4eHhj/SaRclj49eDmJ+mknHzKuYVqpm6nHs0cFHY3V3L8itZa/R5/abnHW8NExtosNQVnfByIU6l73Y915NgxVNaBlaXXivxeNbNWUaSviqv1KvOU6U8XEEuAevIkSOsWbOGb775hq+//hpzc3Nq1apFmTJlsLCwIC4ujmvXrhEREYGLiwtDhw7ll19+wd3dvbDrN4ly1kqewlBBUFWVatWqERwc/ETtWFtbk5qa+sDj/HvcxX/HYNja2t7zdUBAAJs3byYmJobZs2dz6dIl1qxZw7Vr12jcuDEAgwYN4osvvqB3794AODs7k5aWlmObD7vlPS0trcQvOi4ezqpeC7QOriTv34j506+atJb0K2dICzmCQ4/h2Y8pisIzNRS6V1aYetzIx8FGfr5o5OsWWnoUwIe0R7XmmpEX9hioYAN/9dbh5VR0gp8oXlb9MYuQDB0+5uXo22mwqcspEnL9Ce/bty979uzhypUrzJkzh5YtW2Jra4vBYKBKlSqMGjWKrVu3cvPmTb744otSE65MrVWrVty6dYuVK1dmP5aZ+eiDfBs0aMD58+dzfb59+/Zs2bKFO3fuADBv3jzat2+f62DXgIAAduzYwfXr16lVqxYBAQF89NFH+Pv7Z08qFxsbi4eHBwBLly4lNjb2gcf/6aefAIiJiWHt2nvvFjt37hwNGzbM8+sVJZOi1WLTsgspx3ZiTHnwXbMFSVVV4tZ+R+KOlaRfOXPf83bmCtObaTndP2tyzp7bDHTbmjWQ3BT0RpVJhw30CzLQqaLCkV4SrsTj2xn0C3Mj92BR8RK9Xyn5S+Dk1UM/QlWtWpURI0Ywb9481q9fzx9//MGyZcuYMmUKHTp0yPHSoSg4jo6ObN26lXnz5lG/fn0aNWp0zyD3nj17UrFiRQBq165N27Ztc2zHz8+PsLAwYmJicny+bt26fPbZZ3Ts2JEGDRqwb98+5s+fn2td7u7uuLu706JFCwD8/f25devWPWPyvvnmG/r06YOfnx8nT5584Biy999/n9jYWOrUqcPgwYPp0KHDPc9v3bq1RE+3IfLOpkUXVL2e5COmm5Mv7dxRMsOvoLFxIGF77ndWezoq/NFFy5oALWdjVeqt0vPeXwaSMwsvaN1NUemwxcCXp43MaqZhZXstduYSrsTjOfLHdtham+7xjZgw4MtSM0t7XihqSZ1++AGCg4Np3Lgxx44dw8fH56GPl1QzZsxAURQmTpxo6lIeSVRUFO3bt+evv/7C3Nzc1OUUKyX1HI9e9BmZ4Zdxf+d7lMf4BR8TE4Ozs/NjHz9i9gQwGLBt04uYJdNxmzAH80o1H7hPil5l+kkj008aKWMJXzbX0r+qUqBTIhy6a6R/kAG9Civba/EvJ2+GJdmTntcPE/Lnn9xcrSHB8i5d326HlY3tw3cqRXL86fr444/v+SNKpldffRUbGxtTl/HIrly5wnfffSfhSmSz9euBPvIm6ZdOFPqx06+cJuPqWew6DMKqURt0ruVJ3L78oftZ6xQ+aqwlpL8OH1eFgTsMBGwxEBKb/595VVVl7lkD/psMeNgpHO+jk3AlnkjouRCurzGQbBbHU680lXCVgxwHue/atSv734qi8MEHHxRaQaLwWFhYMGbMGFOX8cialeJ5VUTOzKvVxaycB0n7NmJZu3B75hK2L8esfFUs6zZDURTsAgYSu/xrMu9cx6xslYfuX81eYX1HHVvCjLx6yEDD1XpGeWnwcVVwtQQXC3C1zPq3ozloH3H6hORMlVH7DfxyWWV8XQ0zm2kw18olQfH4Im7e4PTiO2g0Wpq+WA0nNxmDnZMcA9b69euxtbWVa6lCiGJBURRs/HoQtyoQfWwEOqeHT7abHzJuXCL9/DGch76dfWnP2rc9CVuXkhi0Eucheb/83rWyhvYVFL48beTL00YCQ+7fRgGcLcDF8p/Q9XcA+3cQ+/v5NAM8t0vP1UT4tZ2WZ2rI73TxZBJiYzjw3RmsDY5Ue9aS8tVrmLqkIivHgOXk5MSff/5JkyZNeOqpp/j222/x9PQs7NqEECLPrH2fIn7jjyQf2IxD92GFcszE7cvRuZbHyrt19mOKzgzbdv2JX78A+y7PoXPJfb65/7LQKrzjreUdby0ZBpWYdIhOg6h0NevvNIhOV4lKg6i0rMdC4iA6zUhUGsRl3N9mTXs43EtHPWfptRJPJj0tlT++3otLWkWcuidT07uFqUsq0nIMWJaWltlzJO3evZuEBNOsWC+EEHmlsbDCpmkHkv/cin3nwSi6gh2jl3nnOqmnDuA06DUUzb0rMNi06EzitmUk7vgNp4GvPFb75lqFstZQ1hqy+q4eTm/8VyhLU4nPgDblFOzlLkHxhIxGI3OWfEyDlN6Yt75Lo7bdTF1SkZdjwGrQoAETJ06kS5cuAPzwww9s3bo1xwYUReH9998vuAqFECKPbFp1J2nvelJO7MPGt32BHisxaCVaR1escziOxtwS27Z9SNj6C/adnkXrkPfF05+ETqPgZgVuVpDXUCZEXsxdNZGN1pfxCqiKX6eRpi6nWMgxYH377beMHTuWb775BkVR+OWXX3IdjyUBq+i4desWw4YNIzQ0FAsLCzw9PZk3b16ut+meOHGC9957L3t9waLuxIkTXLx48YELjG/YsIF9+/Yxc+bMXLdJS0ujVatW7Ny5EwcHh4IoVZiImXslLGo1InnfxgINWProO6QE78Kh10gUXc5zAdr69SBxx0oSd6/BsdeIAqtFiIK2/rtf0Ub58JK3I90kXOVZjqmpUaNGHDx4kNjYWFRVZdeuXSQmJub4Ry4fFh1arZb333+fCxcucOrUKapUqcLbb7+d6/bvvvsub731ViFW+GROnDhxzwz2/6XX6+nZs+cDwxVkXQIfPHgwX331VX6XKIoAW7/uZFw/T8aNSwV2jMQdv6GxtsOmRedct9FY2WDbuifJBzZjSJbfk6J42rLrO/4wnKGMtcpzPaQz5VE89JaSXbt2UadOncKoReRRcHAwHTp0wNfXF19f3+yZ3N3d3fHz88verlmzZly9ejXHNsLCwjh79ixt2rQBIDIyko4dO1K/fn0aNGjAsGFZg4Tr16/PoUOHsvebP38+Tz/9NABt27Zl4sSJtGnThkqVKjFz5kyWL19Oy5YtqVKlCsuX/zMXkKIofPbZZzRt2pRq1aoRFBTEO++8Q6NGjahbty5nz57N3nbJkiU0a9YMHx8f/P39OXPmDBEREXzwwQcEBQXh7e3N6NGjs9udNWsWbdu25Z133uHnn3+mf//+2W399NNPeHt707BhQ3x9fQkNDQXgmWee4Ycffnjs/wNRdFnWbY7WsQxJBzYVSPuG+GiSD2/D1r8PGnPLB25r698bVJWkvesLpBYhCtL+Hev44sbvlHGKpP/rsnjzo8rxEuG/+fv7Z/9bVVVefPFFpkyZ8sBlTko6Q3w0hoScl5h5Elp754eO1YiNjWX48OFs2LDhgf8HBoOBwMDA7IWV/2vPnj00b948++ulS5fi4eHBtm3bALKX0Bk/fjyBgYHZS+AEBgYSGBiYvV9YWBi7d+/mzp07VK9enQkTJnDw4EGOHDlC7969GTRoUPa29vb2HDlyhN9++41evXqxcuVKPvvsM2bMmMG0adP49ddfOXDgAMuXL2fv3r1YWFiwb98+Bg8ezMmTJ/n444/ZtGkTq1atuue1pKens3v3bgB+/vnn7Md3797NtGnT2LdvH+XKlSMlJSX7uXLlymFubs758+flDtkSJmt9wq4kbl+OsedLaKzt8rX9xN1rUMzMsfXr8dBttbaO2LToQtLe9di164fG0kSrxAvxiE7s3Uv67xXpXc6fl8ePlWmbHsNDA9a/GY1GFi1axLhx40p1wEo6uIXEP37J93btOg3GoctzD9zm0KFDhIaG0rNnz+zHpk+fTqdOnbK/VlWVMWPG4OjoyCuv5HwHU3h4OGXL/nP7ePPmzfnqq6+YMGEC/v7+2e0NGTKEDz/8kIiICM6dO4eiKLRu/c8t6QMGDECj0VC+fHlcXV2zA13jxo25ffs2aWlpWFpmfcr/u+fLx8cHjUZDt27dsrdds2YNkDUH28mTJ++ZTDQyMpKMjBzuP/+/4cOH5/j45s2bGTp0KOXKlQPA2vreN7eyZcsSHh4uAasEsmnRmYQ/fiH58Dbs2uXfmpWG5ASSD2zG1r83Gqu8rYJg+1Q/kg5sIvnAZuzaD8i3WoQoKJdPnuDuJjNSLCN5YfgwzMwe3FMrcvZIAQuy3ryLq797X/6eguJx2bbsilW95g/f8BFp7R++ZpSqqlSrVo3g4OBctxk/fjw3btxg3bp1uX7qsLa2vuf70KJFC06cOEFQUBCrV69m8uTJHD9+HCsrK55//nl++OEHjh8/zrhx4+5p5+/wBFljwP7+WqvNum1dr9fft61Wq8XCwuKe/f7eTlVVhg8f/khLNNnaPt4SDWlpaVhZWT3WvqJo09o5YeXdmqQDm7D17/NY6xPmJOtSn5p16S+PdI5lsGkSQOLuNdi27olibvHwnYQwkdtXr3B+WTyq1kCrUXWxdyq4tQxLukcOWMXZ2LFjGTt2bPaCt49L6+BSaLdd/1erVq24desWK1euzL6bLjMzEzOzrDuZxo8fz+XLl1m3bt0D1+pr0KABq1evzv762rVrVKhQgYEDB9K5c2fc3NxISkrCwcGBsWPH0qJFC9LT01m0aFGBvr4ePXowdOhQRowYQaVKlTAajQQHB+Pr64u9vT3x8fGP1Nbw4cMZOXIkZcuWzb5EaG1tjcFg4OrVq9SrV6+gXoowMVu/HkQe20X6hWAsvXyfuD1jWgpJe9dj06IrWlvHR9rXrv1Akg9vI/nItjxdWhTCFOIi73L4hytYGG3wGuaKW6XSe6UqPzzSxzqtVstPP/1E1apVC6oe8RCOjo5s3bqVefPmUb9+fRo1apQ9yP3AgQPMmTOH0NBQmjVrhre3N3369MmxHT8/P8LCwrLHWu3evZvGjRvj7e1Nq1atmDlzZvYUBhUrVsTb25vnnnvuvsts+a1NmzZ8+umn9OrVi4YNG1KvXj1WrFgBQPv27UlOTqZhw4bZg9wf1tbkyZPp2LEjDRs2xN/fn8jISAD2799Ps2bNZJqGEszcwwuzCtVJ2r8xX9pLPrAZNSMN23Z9H3lfXZnyWDVqQ+KO31AN+ofvIEQhS01OImjOEWwzHKncR8HDq66pSyr2FLU4X/N7TH/3YB07dgwfH5+HPl5SzZgxA0VRmDjxweulJSUl4enpyb59+0pMuB40aBAvvfQSAQEBpi6lUJW2czzp0O/ErZxN2ck/PXDJmpiYmFzniwNQM9K5/ckLWNVrjtPTrz5WLZm3rnF3xss4PTsBm6YdHqsNIR7Fw87rv+kzM1nz2Xrc4qth3SGKpp07FkJ1Jd9De7CMRiPHjh3j22+/5YMPPmDixIl8/vnnrF27Nrs3QBRPr776KjY2Dx6oO2/ePDw9PRkzZkyJCVdpaWm0bdu21IWr0sjapx2KhTVJB55sMt3kI9swJsVj99TjD1I3K18Vy3rNSQxagWo0PFE9QuQXo9HInGXvYZ7qgtr0uoSrfJTrGKzLly8TGBjIL7/8QlRUFBqNBkdHRywtLYmNjSU1NTX7jrIRI0bwzDPPyG2cxYyFhQVjxox54DajR4/O0+W44sTS0rLEvSaRM42FJTbNOpJy+A8cujyHYvbo6xOqBj2JO37DyscfXZnyT1SPXcAgIr9+jdRTB7H+1wLRQpjKwnVTWKe9wLjW7rTrOsHU5ZQoOSaikSNHUq9ePY4fP86HH35IcHAw6enpREVFER4eTnJyMnfu3GH9+vXUrVuXCRMmUKdOHQ4ePFjY9QshxAPZ+HXHmJxAyvG9j7V/yrFdGGIjsG+f+xJNeWXh4YlFLW8Sg5YX6zuyRcmw5afl2AW3ZIhZAwZIuMp3OQYsVVU5c+YMu3fvZuzYsXh7e2ffdv83Nzc3unfvTmBgIDdv3uTdd9/l2rVrhVK0EELklVmZCljU9nmswe6q0UBi0Aos6zXHrHz+XCK3CxhEZvgV0s4dzZf2hHgcuw4uZVHmDgwOdxjeb4qpyymRcgxY33//PTVq1MhzI1qtlqFDhzJ48OB8K0wIIfKLbeseZIZdICPs4iPtl3rqIPqIcOwCBj184zyyqNkQ8yqeJG5fJr1YwiSO7t7O9Etr8bDMYODrz97XgSLyxyMNmlJVlcTERPmlIIQoVizrNEXr5PZIvViqqpIYtByLWt5YeOTfbP+KomDXYRAZ10LIuHom39r9L0NSHLErviFu9bekHNuFPvqO/O4WnDtyhNjN9vS72ZmPBgai0z36uESRNzkGrMzMzHvWbQsJCaFr167Y2Njg6OiIlZUVAQEBHDhwoNAKFQ+XnJxMs2bNaNiwIQ0bNqRz587Zixvn5MSJE9nL1UDWQs6enp54e3sTHR1dCBXn3ZQpU3jzzTdNXcY9/ruw9KNSVZXWrVvLpfVCoGiy1idMOb4HQ3JCnvZJO3eUzPAr2HV4Jt/rsazbDLPyVUnYtizf2wZIDz1HxBfjSD11gLRzR4lZMp07n7zA7fefIeqHKSQErSDt0kmM6U+2qoUoXsIunCd0dQYpZon0f7EHlpaPtwqGyJsc7yJ84YUXqFmzJlOmTCEyMpKnnnoKOzs73nrrLcqWLcutW7dYsWIFTz31FL///jtPPfVUYdctcmBlZUVQUBB2dlmL23799de88cYb2ev8/de7777L22+/nf31119/zZIlS2jSpMkjHddgMORrF7Ner0enK/mLDCiKwuuvv85HH310zyLVomDYtOhMwtZfSDn8x0OnW1BVlcTtyzCv4olFjQb5XouiKNgFPE3M4s/JCLuIeeVa+dKuqqok799I3LoFmFeqicsL76F1dMWQFEfG9QtkhJ4j4/p5ErcvR01PBUWDWfmqmFfxxNzDE/MqnujcKqIoSr7UI4qOqNs3OfHzTXSY4TusMk7uuc8LJ/JHjj1Yu3btomXLlgDMmTMHDw8Pzp49y4cffsioUaP46KOPCAkJoUOHDrz77ruFWrDImiyyQ4cO+Pr64uvrmz2Tu0ajyQ5XqqqSkJCQ69QZYWFhnD17ljZt2gDQv39/rly5wnPPPZfdK7NkyRLq169PgwYN6NatGzdv3gSyem46d+7M0KFD8fX1Zf/+/dSuXTv7uK6urrz33nsA7Nixg/bt2wPw66+/0qxZMxo1aoS3tzdbtmzJrsfDw4Np06bRrl07nn/+eeLj4+nfvz916tShU6dOXL58Odfvx6FDh2jdujUNGzakQYMGrF+/HoCjR4/SokULGjRoQNOmTbN7XENDQ3F1dWXy5Mk0atQIT09Pjh49ysiRI7O3vXXrVvZr7dChA/369cPb2xt/f3/CwsJyrGPJkiU0a9YMHx8f/P39OXPmDEajkc6dOzNr1iwArly5QqVKlbh06RKQtZzPli1bSExMzPX1ifyhtXXEulFrkg5sfug8VBlXz5BxLQS7js8UWNiw8m6NzrU8CUHL86U9Y3oasUtnELf6W2xbdaPMuBloHV2BrNduVbcZDt1eoMyYzyn/2SrcJ32H48BXMKtYnfSrp4n9dRZ3PxvBrfcGEDX/fRK2/kLa+WMYU5PzpT5hOslxcewNPIml3oYaA22oUDN/Ar14sBy7CWJiYrIX0N20aRPvvPPOfevaaTQaxo4dS79++bdSfXERnRpDdGpsvrfrYuWEi9WDZ92NjY1l+PDhbNiwgcqVc14nKiAggNOnT1OmTBm2bduW4zZ79uyhefN/FqxetWoVHh4erFq1inr16nHmzBkmTpzIsWPHqFChAtOmTWPkyJFs3pw1YeP+/fs5fvw4NWvWBCA1NZWwsDCio6OpVasWO3fuBCAoKCh7Qs9OnTrxzDNZb1ihoaG0bNmS69evZ6+jGBYWxs6dO1EUhQkTJmBvb09ISAhRUVH4+Phkr734bzExMfTp04c1a9bQsmVLjEYjcXFxZGRk0LdvX77//ns6derE/v376d+/f3ZQi46OpkWLFkydOpWZM2cSEBDA7t27WbBgAWPGjGHu3Ll8+umn2a/1xIkT1K5dmxkzZjB69Oh7wiFkLVO0fPly9u7di4WFBfv27WPw4MGcPHmSpUuX0rRpUxo3bsyECROYOXNm9vfNzMyMevXqceDAATp37vzA/3vx5Gz8epBydCdp545iVbdZrtslbFuWNTFonaYFVoui0WIXMJDY5V+Teec6ZmWrPHZbmRHhRC/8BEPMXZyHvo21T9uHHtusfNWsOyNbdAHAmJJERtgFMkLPZ/Vy7V2LujUJFAWdeyXs2vbDpnmnx65RmEZGWjpbvt6FS1plHLsmUKtx84fvJPJFjgHLycmJmzdvEh4ezunTp7N7J/7r5s2b2UGsNNlw6Q9+Pp0/nzr/7YX6gxjW4MHjPQ4dOkRoaCg9e/bMfmz69Ol06vTPL76goCCMRiPTpk1j6tSpfPvtt/e1Ex4eTtmyuXcR79q1i+7du1OhQgUAxowZw9SpU7MHyfr5+WWHBMhaJzAoKIjo6GiGDBnCggULiI+PJygoKPv4165dY/DgwYSHh6PT6YiKiuL69evZd6wOGzYsu7dg165dzJkzBwBXV1f69s15/bdDhw5Rp06d7B5XjUaDs7Mzp0+fxtzcPPv74ufnh5ubG6dOnaJcuXLY2tpmjz/z8fHJXm8RoHHjxmzfvj37GH5+ftk/AyNHjuTDDz+8b7Dw+vXrOXnyJM2a/fOmHRkZSUZGBq6urixZsoR27doxbNgwBg269460smXLEh4enuv/hcg/5lU8MatYg+T9m3INWBlhF0m/EIzz0HcK/FKZtW97ErYuJTFoBc5DJj1WG6kn9xPz65doHZxxe/1rzMp5PFY7GmtbLD0bY+nZGMjqjdZHhJNx/Typpw4Su+IbdGUqYFFdFkgvLgwGA+u+XId7Qm00rW/g85QsNF6YcgxYbdq0YcKECTg5OeHq6sro0aP59ttvs9+AALZs2cI777xTKnuwetbsRKuK+f/J1sXK6aHbqKpKtWrVCA4OfuB2Go2GESNGULNmzRwDlrW1NampuQ9wVVX1njeX/77R/DdYBwQEsHnzZmJiYpg9ezaXLl1izZo1XLt2jcaNs35hDxo0iC+++ILevXsD4OzsTFpaWo5tPundTv+t/7+vw8LCIvsxrVaLpaXlPV/r9Y+2IK+qqgwfPpyPP/44x+ePHz+Oi4sL4eHh99WWlpaGlZXVIx1PPB5FUbD160Hsiq/RR91C53r/zOwJQcvRlamAlbdfwdejM8O2XX/i1y/AvvNz6FzL5Xlf1WAgftNCknatxqqhH07PvIHGMv8WY1cUBTP3Spi5V8K68VNEfvs20Ys/w/3NQLR2jvl2HFFwft0+A8f0VqQ1vEinPk8+Ua54NDkO0Pn6669p0qQJLi4u/P777wwcOJAmTZpQpkwZateujYODAz169MDT05MZM2YUds0m52LlTC3n6vn+52GXBwFatWrFrVu3WLlyZfZjmZmZANy9e5eYmJjsx5cvX06DBjkP0G3QoAHnz5/P9Tjt27dny5Yt3LlzB8hak7B9+/a5fqIPCAhgx44dXL9+nVq1ahEQEMBHH32Ev79/9jiw2NhYPDw8AFi6dCmxsblfZm3fvj0//fQTkHUZcO3atTlu17JlS86dO5e9ioDRaCQmJgZPT0/S09OzL1UePHiQiIgI6tevn+sxc3PgwAEuXsyaP+mHH37gqaeeuu/70KNHDxYvXsyNGzey6zh6NGsiyeDgYL744guCg4NRVfW+n5lz587RsGHDR65LPB4rH38UK5sc1yfMvHOdtFMHsWs/EEVTOHMD2bTojMbajsSdv+V5H0NCDJHfvk3SnrU49B6J8wvv5Wu4+i9Fq8Vl6Ntg0BOzdAaq0VhgxxL5Y/mmWaxRL0Kjk3R6XsKVKeTYg1WuXDlWr16d/bW3tzcdO3Zk5cqV3Lp1C2dnZ/z9/encubPcbVLIHB0d2bp1K2+88QaffPIJOp2OTz/9lC5duhAeHs6IESPQ6/Woqkr16tVZunRpju34+fkRFhaW62rrdevW5bPPPqNjx6yFPytVqsSCBQtyrcvd3R13d3d8fX0B8Pf359atW7z11lvZ23zzzTf06dOHChUq0KJFi1zHkAG8//77DB8+nDp16lClShU6dOiQ43ZOTk6sXbuWCRMmkJiYiKIofPLJJ/Ts2ZPVq1czfvx4kpOTsbS05LfffsPGxuaRFyn39/dnypQphISE4ODgwOLFi+/bpk2bNnz66af06tULg8FAZmYm3bp1o1atWgwaNIiFCxdSrlw5Fi9eTJMmTfDz86NVq1bZ02jUqyeXXQqLxtwSm2adSDm8DYcuQ1HM/+nNTAxaidaxDNa+hXdntMbcElv/PiRs/QX7ToPROrg8cPv0K2eIXjQNgDJjZxTaJTutgwvOQyYRNX8yiTtWYt8h/yZfFfkr6Ld1OPzVgr5eEQx99gNTl1NqKWopnHkuODiYxo0bc+zYMXx8fB76eEk1Y8YMFEVh4sSJpi6lyPr555/ZtGkTq1atKpD23377bWrWrMmLL75YIO3/V2k7x3Ojj7zFnWnDcXrmDWyadSQmJgZ7Yzp3Pn0Rx96jsG3Tq1DrMaYmc/ujodg074Rj75E5bqOqKkl71hK/4QfMq9bF5fl30No/vNc7v8VvWUTi9hWUGft5gUxhIZ7Mn6d/Z+pfi+kX1YZuL/TFzc3d1CWVWo80k7soWV599VVsbGxMXUapVr58eYYNG2bqMkodXZnyWHj6krR/Y/Z4v8Sdq9BY22FtgjvlNFY22LbuQfLBzTlOhGpMSyHm52nEr1uArX8fyoz53CThCsC+0xDMq9UlevHnGBLjTFKDyNnpo8HMOvwL1TEy+JXn0OnMTF1SqfZEAUuj0VC+fHlmzZp1z8zvRVVgYCB16tQplQPzc2JhYcGYMWNMXUaR9sILLxRY7xXA+PHjc52rTBQs29Y9yLxxiYzrFzAmxpJ8eBu2/n3QmFs+fOeCqMe/NwBJe9bd83jmnetEfDmetPPBuAybjGOvESgmXDsuezyW0fj/8VgPnlNMFI6rly5zc2UavW70YFqvLzG3Kn13+Bc1T/SbfejQofj5+TF79myqVHn8OVwKy9ixYwkJCblnfJkQonSy9PJF6+xO8v6N6A9tQTEzx9bPdLexa20dsWnRlaR9GzCmZU3umRK8m4gvXwWtDrcJs7FqWPB3NuaF1sEF5+cmkX7xOIlBK0xdTql39+5dQn68iV6TTvtBdbB1qWDqkgS5DHLPq38v7/H33VNCCFEcKBottq26E79lMWg02LXtg8bKtJfMbdv1JWn/RpL2rseYFE/S3vVYNW6H08BX0ViYpmctN5a1fbDr8AwJvy/FvGpdLGvKnbCmkJCYxKE5p7A0WlOjdyblvB79TmlRMPLt2kSlSpXyqykhhCgU1s06wv9vhLb172PaYgCdYxlsmnYgYctikg5sxrHfGJyHTCpy4epv9p0HY1G9HjFLPseQmP+rW4gHS0/PYPuXe7BNc6Rc67vUaNnW1CWJf3lgD1ZCQgI7duzg3LlzxMTEoCgKTk5OeHl50b59e+zt7QurTiGEyHdaWwfsOw0mVW9Aa+tg6nIAsOswCENCDHYdBmHh4WXqch5I0Whxfu4t7n4xjpglM3AdPbXQ5g8r7QwGAxu+2kqZhEroGpygYQ+5WaaoyTFgqarKhx9+yKxZs0hNTcXa2honJydUVSUuLo6UlBSsrKx48803mTJlisyFJYQotuw7DEL/rwl6TU3n7I7riI9MXUae/T0eK+q7d0ncthz7zoNNXVKJZzQaWRe4GfeIKqRW3Yf/0LGmLknkIMdLhJ988gmzZs1i8uTJhIaGkpSUxI0bNwgPDycpKYnQ0FDef/99Zs2axdSpUwu7ZpEHw4cPR1EUkpKSct3mxIkT2evxFQcnTpy4Zwb7nGzYsKFYzev1sP+jh5kzZw6fffZZPlYkxKOzrNUIu47PkvDHUtIunjB1OSXeL0d+5mryHWLd99Dx5VHSyVFE5Riwvv/+e2bOnMk777yT42zblStX5u2332bGjBkPnN1bmMbGjRvz9AP37rvv3jPTelH3sICl1+vp2bMnM2fOLMSqTGvUqFF8//33JCTcP3eSEIXJvtOzWNRoSMyS6RgSik6PYEmz5dhmfriyHgv7XfR4ZRiKzHVVZOUYsKKjo/Hyevi1f09PT6Kjo/O9KPFgwcHBdOjQAV9fX3x9fQkKCsp+Ljo6mo8++ogvv/zygW2EhYVx9uxZ2rRpA0BkZCQdO3akfv36NGjQIHvyy/r163Po0KHs/ebPn8/TTz8NQNu2bZk4cSJt2rShUqVKzJw5k+XLl9OyZUuqVKnC8uXLs/dTFIXPPvuMpk2bUq1aNYKCgnjnnXdo1KgRdevW5ezZs9nbLlmyhGbNmuHj44O/vz9nzpwhIiKCDz74gKCgILy9vRk9enR2u7NmzaJt27a88847/Pzzz/Tv3z+7rZ9++glvb28aNmyIr69v9tI0/xYfH89LL71E/fr1adiwIcOHDwcgKSmJ4cOHU69ePerVq8dHH/1z2eZRX/uUKVNo1aoVtWrVYtmyZTn+n1y6dIlu3brRpEkTGjZsmL1I9y+//IKvry/p6emoqkqPHj2YPn06AObm5nTs2JEVK+RWeWFaWeOxJgEQs6Rg58fKvB1K4s5VpW6i0wN/Hsby17L0uV2Zl5/+Eo21nalLEg+i5qBVq1Zq79691YyMjJyeVlVVVTMzM9XevXurrVq1ynWbourYsWMqoB47dixPj/9XenyGmngj5YF/MhIzVVVV1YzETDXxRkr2vsl303LdJz0+9+/332JiYtSGDRuq169fz/H5p59+Wt24caOqqqoKqImJiTlut3jxYnXgwIHZX3/55ZfqiBEjsr+Ojo5WVVVVFyxYoA4ePDj78fr166t79+5VVVVV/f391YEDB6oGg0G9efOmamlpqb733nuqqqrq4cOH1XLlymXvB6hz585VVVVVV65cqVpbW6ubNm1SVVVVp0+frj7zzDOqqqrq/v371a5du6ppaWmqqqrq3r171QYNGqiqqqo//fST2q9fv3teB6BOmzYt++t/b7Nr1y61evXq6q1bt1RVVdXk5GQ1OTn5vu/FCy+8oI4bN041GAyqqqpqRESEqqqqOmnSJHXw4MGqwWBQk5KSVG9vb3XlypWP9dqnTJmiqqqqXrlyRXVxcVHDwsKyn0tMTFT1er3q6+urnjt3LrvW+vXrZ5+LI0eOVMeOHavOmDFD7dKli2o0GrPbX7Rokfr000/f97pyktdzvLT5+3wXTy714gn1xmtd1Lgti/O97Yxb19Son6aqN17tpN54tZN6872n1ZSzh/P9OEXR6btn1Y6L+6pzp32qJl89m6d95Lw2rRwHuX/55Zd06NCBmjVr0qdPH7y8vHB0dERRFGJjYzl37hzr1q0jNjaW7du3F1YWLDJuH4zhxraIB25TY2AFyjZ3JvpMApdX3sTvy6y5SS4tu0Hi9dQc96nU0Y0qnR+8btShQ4cIDQ2lZ8+e2Y9Nnz6dTp068dtvv2Fubk737t0f+hrCw8MpW7Zs9tfNmzfnq6++YsKECfj7+9OpU9ZyIUOGDOHDDz8kIiKCc+fOoSgKrVu3zt5vwIAB2TP6u7q60rt3bwAaN27M7du3SUtLw9Iy6xbzv3u+fHx80Gg02eO/GjduzJo1awBYv349J0+epFmzZtnHiIyMJCMjI9fX8neP039t3ryZoUOHUq5cOQCsra1z3G7Tpk0cO3Yse0b1MmXKABAUFMQ333yDRqPBxsaGoUOHEhQUxIABAx75tb/00ksAVKtWDT8/P/bt28ezzz6bXcOFCxc4e/Ysgwb9s4BuYmIiISEh+Pj4MHv2bJo2bcqGDRsIDg6+5xJw2bJlCQ8Pz/X7I0RhsqzZEPtOg0n4YykW1ethWavRE7eZeTuUhD9+IfXEPrTO7jgNeg2L2o2JWzmb6AUfYOPXHYeeL5lsFv6CdulyKLN3LKCKmsHgbq2wrlrH1CWJPMgxYDVt2pQTJ04wY8YM1q5dyzfffHPP85UrV6Zr165MnDiRqlWrFkqhRUm5ls641HvwFBUWjlnXxV3q2WNb0Sr78ZrPVMKYYcxxH3P7h8/7qqoq1apVIzg4+L7ndu3axc6dO/Hw8Mh+rG7dumzatIn69e+dfM7a2prU1H+CXosWLThx4gRBQUGsXr2ayZMnc/z4caysrHj++ef54YcfOH78OOPGjbunnb8DBIBWq83+Wvv/pTz0ev1922q1WiwsLO7Z7+/tVFVl+PDhfPzxxw/9XvzN1rZgloRQVfW+sWz//vpRXvt//bddVVVxdXXlxIkTOW4fERFBbGwsRqORuLg4XF1ds59LS0vDysoqx/2EMAW7joNIv3qGmCUzcJ8Y+NjrJuYUrKybBKBos35Xuoz8mOQDm4hb/z3pF0/i/NwkzCvVzM+XYnI3b9/l0g/hdNJ0pEVAIo4N25i6JJFHuU40WrVqVb777rvsuwjDw8MJDw8nOTmZ0NBQvv3221IZrgDM7c2wrWj1wD9mtlm/AMxsdfcELGs3i1z3Mbd/+GDFVq1acevWrXsGe2dmZgLw7bffEh4eTmhoaPZYo7Nnz94XrgAaNGjA+fPns7++du0atra2DBw4kDlz5nDx4sXsu9vGjh3Ld999x549exg8uGBvwe7RoweLFy/OXhnAaDRy9OhRAOzt7YmPj3/ktu7cuQNASkpKjmtm/j0w3mjMCr6RkZEAdOjQge+//x5VVUlOTmbp0qUEBAQ81utauHAhAKGhoezfvx8/v3uXPKlduzbW1tYsXrw4+7HLly8TExODXq/n6aef5pNPPuGLL75gwIABpKenZ2937tw5GjaUWbRF0ZE9HkuBmCXTH3k8VubtUKJ/nsbd6aPJCLuI06DXKPvej9g075wdriDrg4qtXw/cJ8xFMbcg4qvXSAhaUWLWR4yNi+dYYAioRurXu075p542dUniETx0JveUlBSsra0pX7485cuXv++TclhYWIEVJ+7n6OjI1q1bmTdvHvXr16dRo0b3DHLPKz8/P8LCwoj5//w/u3fvpnHjxnh7e9OqVStmzpyJg0PWxIsVK1bE29ub5557LtfLbPmlTZs2fPrpp/Tq1YuGDRtSr1697AHc7du3Jzk5mYYNG2YPcn9YW5MnT6Zjx440bNgQf3//7PD0b1999RUpKSnUq1cPb29v3n33XQDef/99FEWhfv36NGvWjJ49e94zgP5RWFhY0KpVKzp27MicOXPuW/lAp9OxceNGVq5cSYMGDahbty4vvfQSqampvP3229SuXZvnn3+eQYMG0axZM1577bXsfbdu3SoLmIsiR2vnhPNzb5N++TQJf/yap33yGqz+y6xsZdxe+wq7dv1I2PwzkXPfQh9zN79eikmkpqay+5sjWKVbU7bKAWoOGmnqksQjUlRVVR+0QY0aNViyZAktWrS477mFCxfyxhtvEBcXV1D1FYjg4GAaN27MsWPH8PHxeejjJdWMGTNQFOWh80YlJSXh6enJvn37Sm2v5ZNQFIXExMQCuZQZEhLC6NGj2bt3b562L23neF7FxMTg7Px4l7HEgyVs+5WE35fgOnoalrVzPuf+eynQvuMz91wKfBTpV04Ts3QmxtQknPqPw6pxu2I3T1Rmpp71s3bgHOWKuft6Wo5/97GWS5Lz2rQe2oPl6elJmzZteO+997LHlERGRtK7d29GjhzJiBEjCrxIUTBeffVVbGwevLjtvHnz8PT0ZMyYMRKuiqAbN24wb948U5chRK7sAp7GolajrPmx4u+d1udxe6wexKJ6fdwnfotV3WbELJ1BzOLPMaYk5sdLKRRGo5EN3wVRJsIdveNaWox+vciuRSke7KE9WAALFizgzTffpGbNmowYMYIPPvgAGxsbFi1alD2PUmGLjY3lzTffZNu2bdnjddLT0xk1ahT29vYYDAYCAwNz3Fd6sERpJed4zuSTfsEyJMZxd+YYdGUqUGbM5+gjbuRbj9WDpBzbRexvc9FYWuE0eCKWNYv+WMV1P2/H9VRZou3X0HX0MMzK3j/Zd17JeW1aD+3BAhg5ciR79+4lJCSEsWPHUrlyZU6fPv3I4Wr8+PF4eHigKApnzpy557lLly7RsmVLatWqRdOmTQkJCXlgW05OTvz444/Url07+7E1a9bg7+/P7NmzcXJyumeCTCGEEKahtXPEZejbZFw9S8SsV/K1x+pBrBu3w/2t79C5lifq27eJ2/ADqj73KV9Mbe2FzeyNP8Rtu9/pNLj3E4UrYXp5Clj79u2jb9++ODo6MnjwYI4fP87LL7/8SHd0AfTv35/9+/dTpUqV+54bNWoUI0eO5OLFi0yaNIkXX3wRgLt37xIQEHDPn0mTJuXY/vXr17OnKKhWrRrXr19/pPqEEEIUDIsaDXDo+SKqQV/gwerfdE5uuI75DIcew0nas46IL18j83ZogR7zcew+dZBvji7ATX+YLt0aFYveNvFgDz2zJ02axJdffkmvXr1YsGABLi4uDBkyhBdffJF69eqxcOFCOnTokKeD5dbjFRERQXBwMNu2bQOgX79+jBs3jtDQUDw8PPJ8l1ylSpWyQ1VoaCidO3fO035CCCEKnl27fti1K/w7XhWNFrunBmBRy4eYJdO5O+sVHHq8iG3rniiaPPUzFKij507BIiv6utZhaKO62DbN23uqKNoeGrAWLFjAwoULGTp0aPZjHTt25PTp07z88st07twZg+HJ5hy5ceMG5cuXR6fLKkdRFCpXrkxYWNg9k2b+1+jRozl//jyjR49m4sSJ9OvXj9GjR3Pq1ClSU1Np2bLlA487btw4HBwc6Nu3L/369XvkHjkhiqv4+PjsKTpE1phOUQpYO2E2/GPYsYz4tfNIPHUQi56jUOycTFbStYQw3j8xDX/3GjzrYouhSZd8+9mU87pw/Xe820MD1unTp++bswey5mNatmwZffr0yZfCcprZ+mFyunvq559/zvMx586de89A37/nfRKipHNwcJDBr/8h349S5NnXSWvUmphls0hf8A7OQ9/OdQqJghQWdotVW37H3TyTZ20NVHrhrXy/ZCrntek8tG80p3D1bwMHDnziIipVqkR4ePg9y6XcuHGDypVlgJ8QQoj8Z+nli/ukeZhVqknUgg9ICd5TqMePjIrm1LwrtAxrzjsxZak47MMCH48mCleOAWvu3Ln3LMWRFydPnnysGcUB3NzcaNSoEUuXLgVg9erVeHh4PPDyoBBCCPEktLYOuI74COtGbYhZ8jlJ+zYUynETk5I5MPckOr2W8pYrqD7iEzRWD56TUBQ/OcblxYsXM3XqVJ599lkGDBiAr68vZmb3r5N369Ytfv/9d5YtW8bhw4cfenlu7NixrF+/njt37hAQEICtrS2XL18GYP78+bzwwgt8+umn2Nvbs2jRoid/dY/p3LlzJju2EAVJzm0h7qVodTg9+yYaW0fiVn+LITEW+y5DC2z29/T0DLbNPohjkiMWlj9S/6U30TmVKZBjCdPKdaLRNWvW8M0337Bv3z7Mzc2pVasWZcqUwcLCgri4OK5du0ZERAQuLi4MHTqUiRMn4u7uXtj1P5bcJlsMCwvDy8srxwWBhSgprK2tOXfunFyC/xeZkFGoqkrSzlXEb/wRmxZdcBwwDkWjzddjGAwG1s7egWt4GTItFuI3ZChWdZrk6zH+Tc5r08r1gm/fvn3p27cvV69eZceOHRw7dozbt2+TlpZGlSpV6NixI61ataJt27Y59m4VR5UrV+bcuXNERUWZupQiKT4+Xm4EKAFcXV0lXAnxH4qiYNd+ABpbB2JXfI0hOR6X595GMTPPl/aNRiPrf9yJ2w13Eqx/pW3XLgUaroTp5diDtWHDBvz9/Uvsm6ksF/J45NOQKKnk3Bb/lnrmT6IXfYp55dq4vjQlX8ZHbVyxE6fDZbhrs46AphVw6DE8Hyp9MDmvTSvHQe59+vThwoULAGi1Wo4cOVKoRQkhhBCmYlWvOWVe/ozMW9eInDPxvkWqH9XWqztZGbeGcPs/aFtDh323F/KnUFGk5RiwHBwciIyMBLKuSxfUYD8hhBCiKLKoVhe38V9gSI4nYvYE9JG3HqudQ8HBzDwYSPXMW7RzvoXzsxOKxOzxouDlOAbL39+f5557jgYNGgDw8ssvY29vn2MDiqKwY8eOgqtQCCGEMAGzch64vfolUfPeI+KbN3Ad9QnmlWrmef8z4edJWqZnoHMbuhtP4vralHwb0yWKvhxj9MKFCxk1ahROTk4oioKVlRU2NjY5/rG2ti7smoUQQohCoXN2p8z4WWid3Ymc+xZpF0/kab+whHDeOTSV3R4r6JB8GPdRU9Ha5NxRIUqmHHuwnJyc+OyzzwDQaDTMmjWLpk2bFmphQgghRFGgtXWgzNjPif5pKlHz38f5uUlYe7fOdfvbdyJYueJ3HJz1PHfjEhVGT0fnWr4QKxZFwUMvBBuNRglXQgghSjWNhRWuL03BqmErYhZ9StL+TTluFxcfz1+B56hz24sJp/RUeWYSFh5ehVytKArytPBRamoqc+bMISgoiOjoaJydnfH392fs2LE4OZluFXIhhBCisCg6M5yHTCLe1pG4VXOzZn3vPCT7RrDUtDR2zj6KXZotdppvqdHtWawa+pm4amEqOfZgffbZZ3zxxRcA6PV62rVrx+TJk8nMzKRWrVpkZmby8ccf06BBA65evVqoBQshhBCmomg0OPQZhX33YST+8Qtxv81FNRrQ6/VsnrMXp3gnzHQ/49m0Gbb+fUxdrjChHHuwfvzxR2bNmgXAggULCAsL48yZM9SqVSt7m9DQUDp37sybb77JmjVrCqdaIYQQwsQURcE+4Gm0tg7ErpiNPimOvSlNKXPbnVSbX/GtUhaHPqNkiqNSLseAdevWLdzc3ABYvXo1H3744T3hCsDDw4OPP/6YESNGFHyVQgghRBFj07wzGlsHNv52mPKJ5Yl2XE9z2zScn/s439cxFMVPjpcIraysiI+PJzk5mUOHDtGqVascd9bpdBiNxgItUAghhCiq9lzLoHxie+7Yb6cpl3Ad8REaC0tTlyWKgBwDVsOGDZk+fTovv/wyGRkZzJ49m4yMjHu2iY6OZurUqbRp06ZQChVCCCGKkr1hh/gueiHh3tfpNX4Ibm98g9ZObvwSWXK8RDh16lR69erF4cOHWbRoERs2bKBy5cq0bt0aZ2dnbt26xa5du7C2tubXX38t7JqFEEIIkzp2/DRfnP6WxjUaMLBVVzSKLH8j7qWoqqrm9mRGRgbm5uaoqsqCBQtYtmwZt27dyp6m4bXXXqNcuXKFWe8TCQwMJDAwkNTUVEJDQzl27Bg+Pj6mLqvYkJXZRUkl57Z4FJejrxEyK4x4t1iefqUb5lozU5eUIzmvTeuBAaukCg4OpnHjxhKwHpH8sIqSSs5tkVe3k+4ydttbeKhVmNJxIvZ2tqYuKVdyXpuW9GkKIYQQeRAVHcvG73Zhp9oyuevrRTpcCdPLcQzW3r17n6hRGfguhBCiJElKTmH/nONUSfWg40A/nK0cTV2SKOJyDFht27Z97AYVRcFgMDz2/kIIIURRkpGZyR+zD+KY5EjZYQ54VKlo6pJEMZBjwJK5rYQQQggwGAxsCNyFa5Qb1gMV6tataeqSRDEhY7CEEEKIXGxYuBO3MHfolEbTZg1NXY4oRiRgCSGEEDnYvGIXZc6VJalVHG07Njd1OaKYkYAlhBBC/EfQrgM4HHYlqt5duvbzN3U5ohiSgCWEEEL8y5Fbwcy4+Q1XfC7S84WnTF2OKKZyHOQuhBBClEanT19g7tGf8KnagMFteqPRSD+EeDwSsIQQQgjgRsJNTqy5QHezbvR8rh06jdbUJYliTAKWEEKIUi86NYY3d07BtoENM9tOwVJnYeqSRDEnfZ9CCCFKtfiERLZ+fQC7FDs+6/gezk6Opi5JlAASsIQQQpRaqWlp7Jh9BLcYdyb5voKbTRlTlyRKCLlEKIQQolTS6/VsmbsX51hXHAdbUqtWVVOXJEoQ6cESQghR6hiNRtYv2EmZW+6Y9TTSyKeOqUsSJYwELCGEEKXOpl924X65HGltE/Hz9zV1OaIEkoAlhBCiVPlj/T6cj7sR4xNBx55+pi5HlFASsIQQQpQa+44ewWqPPXdr3qb7s+1MXY4owUpVwAoMDKROnTr069fP1KUIIYQoZCfvnuXjizM40+QkvUY8JbO0iwJVqs6usWPHEhISwurVq01dihBCiEJ0/vwVflq7knpuXrw08Bl0OrmJXhQsOcOEEEKUaHeSIti+8QBtk/wJGNYMc62ZqUsSpYAELCGEECVWbGo8E3dNIbOmntltPsPWwsbUJYlSQgKWEEKIEikpJYWgL//Esawzbz09Fjc7F1OXJEqRUjUGSwghROmQkZnJH7MP4BznwqjmQ6loV87UJYlSRnqwhBBClChGg5GNgbtwiXTDqj/Uq1fL1CWJUkh6sIQQQpQo63/aiXtYOdROqTRr4W3qckQpJQFLCCFEibF5xW7KhLiT0DKadh1bmLocUYpJwBJCCFEi7Nx6EIfDLkTVvUPX/m1NXY4o5SRgCSGEKPaOXDyOst2Sux636TmsvanLEUIClhBCiOLtXPQlPjj+OX81/5OeL8sSOKJokLsIhRBCFFtXr4axeuNWqtauwviAFzGTJXBEESFnohBCiGIpOjWGX7auxTvSmzHDnsNSZ2HqkoTIJv2oQgghip3EtCQm7fqYk2VP4vumF872jqYuSYh7SA+WEEKIYiU1LY1ts/6kvH1FJj83kHKObqYuSYj7SA+WEEKIYsNgMLA5cC/OMc70bdmVqo6VTV2SEDmSgCWEEKJYMBqNrP9+B2433dH1MNCocR1TlyREriRgCSGEKBY2L9+D28VypLSJp3XbJqYuR4gHkoAlhBCiyNux5SBOR12JanCXzr3bmLocIR6qVA1yDwwMJDAwkNTUVFOXIoQQIo8OHzyBboc1d6veps/QAFOXI0SelKoerLFjxxISEsLq1atNXYoQQog8uBBxmej1yUSViaTn6HYyS7soNuRMFUIIUSTdSrzDW/s+Zk+TnXQa3wozMzNTlyREnknAEkIIUeRERcWwdkEQ9hp73uk+Hltra1OXJMQjKVVjsIQQQhR9qfo0Zm/7kaZRzek+sB2Olg6mLkmIRyY9WEIIIYqMzMxMpuydwRHtUWq+UY4qlSqYuiQhHov0YAkhhCgSjEYjG+btolxiZfoP60ntMjVMXZIQj00ClhBCiCJhw+KduF8rh2MHW5qU8zZ1OUI8EblEKIQQwuS2rt2L6yl3YptE0b5LS1OXI8QTk4AlhBDCpPbt+gvrfQ5E1LpNt6f9TV2OEPlCLhEKIYQwmePBIeg3aYmucJc+I9rLRKKixJCAJYQQwiQuXb5G1LIkEp0S6Ta2DVqt1tQlCZFv5KOCEEKIQheZEs2RX8+SapnKU6/4YmVpaeqShMhX0oMlhBCiUCVlJPPWro/J9NQzvc2HODrIRKKi5JGAJYQQotCkpaez+rs/SHFN4/Me71Hewc3UJQlRIOQSoRBCiEJhVI18vWsBDhFOTKozHg+HyqYuSYgCIwFLCCFEgTMajcw98iN/xOzEeZQlPj51TV2SEAVKApYQQogCt3HpLlw3VeB1n1G0qdLC1OUIUeAkYAkhhChQW1bvweWEG3aelvT07GzqcoQoFBKwhBBCFJidvx/E/oAzkV536PZMW1OXI0ShKVV3EQYGBhIYGEhqaqqpSxFCiBLv4N5gtNutuetxmz4vBsgs7aJUKVVn+9ixYwkJCWH16tWmLkUIIUq0Y0fPkL4eIstG0PPlpyRciVKnVPVgCSGEKHhnQy4RuyKVBOd4uo1vg5mZvNWI0kc+UgghhMg34Qm32L/+KEk2SXR4tbksgSNKLflYIYQQIl/cTY5kws4PsaxjyZdtPsbO1tbUJQlhMtKDJYQQ4olFx8Sy94vjuCW4MbPDh7g4OZm6JCFMSgKWEEKIJ5KSmcKUAzNI1Cbyut8o3KxdTV2SECYnAUsIIcRjS0tPZ9ofX3M55Rp+Y7ypVk3WFxQCZAyWEEKIx5SZqWfz7D00i2lF/1d6UMu5uqlLEqLIkIAlhBDikRmNRjbM24nrbTe0vfQ0Kl/f1CUJUaTIJUIhhBCPbP3CHbhfK0dmhxT8/H1NXY4QRY4ELCGEEI9k4687KRNSloSW0QR0aWnqcoQokiRgCSGEyLM/1u/D6WgZohvepWv/tqYuR4giSwKWEEKIPNkddBirPfbcrXmbHs89ZepyhCjSJGAJIYR4qL9uH+fg8aNEVLpL75HtZfFmIR5CfkKEEEI80Jmb55m85zPifaPp9Uo7tFqtqUsSosgr1gErNjaWF198kUqVKmU/FhoaipeXF6NHj2bKlCmmK04IIUqAW3fvcmNODP761nzU+i3MdGamLkmIYqHQA9b48ePx8PBAURTOnDlzz3OXLl2iZcuW1KpVi6ZNmxISEvLAtpycnPjxxx+pXbv2PY87ODiQlpZG1apV871+IYQoLTIMmUw9NouzZc8ysvNgLHUWpi5JiGKj0ANW//792b9/P1WqVLnvuVGjRjFy5EguXrzIpEmTePHFFwG4e/cuAQEB9/yZNGlSju1XqVKFP//8k59++okNGzYQGhpakC9HCCFKrAW7F3Mx/grdnmuLq6uzqcsRolgp9Jnc27Rpk+PjERERBAcHs23bNgD69evHuHHjCA0NxcPDg6CgoDy1ryhK9t/u7u4kJibmT+FCCFGKbNuwn3p7fKg+qCpeLjVNXY4QxU6RWSrnxo0blC9fHp0uqyRFUahcuTJhYWF4eHjkut/o0aM5f/48o0ePZuLEiYSFhbF48WIAHB0dqV8/9+Ubxo0bh4ODA3379qVfv375+npKotjYWFOXIESBkHP7XhcuXMN8jw0RVe4QUL0ZMTExpi5JPAY5rwuXs/O9vbxFJmDBP71Pf1NV9aH7zJs3756vq1evTrt27fJ0vLlz5+Lj45P3AsV9J5AQJYWc21miomKI25BGul06PUc9haWFjLsqzuS8Np0icxdhpUqVCA8PR6/XA1nh6saNG1SuXNnElQkhROmQmaln7/xgNAYtTUbVlXAlxBMoMgHLzc2NRo0asXTpUgBWr16Nh4fHAy8PCiGEyD+bFu3CJdoVlwE2VCjnbupyhCjWCj1gjR07looVKxIeHk5AQAA1atTIfm7+/9q777Aor7Tx41+GXqU3CyAWsGBBFBvYu7GgcV2zMcYkutE1bl7XtN8mvkn2XZNr37SNG7OuLdGY2IOxoUGISomKHUVAQZTe6wxl5vcHb0xUVJRhZoD7c13zh8M857nn8czh5jxnzv3ll3z55Zd069aN1atXs379el2HJ4QQbVLU0XhcEt0pDy5mQNCD164KIRpH52uw1qxZw5o1axr8Wffu3YmNjdVxREII0bal5qRTHQH5HbKYOWuMvsMRolUwmFuEQgjtu12WRXrJLX2HIQxYeXUFb8evJqLvYSYsHiY1BoXQEoP6FqEQQntyK/J4+fBrmJSYMrdgDt1mdyTAx1/fYQkDUldXx1ff7qbUrIwPpr6NjZWVvkMSotWQBEuIVqhKqSTqkzN08fBldNfhlGaqWH7iLXqn+DPXNYyBvfvKTIVgZ/R+epzvRd+ne9DB1kPf4QjRqsgIKxoluzyX/znzKTdLW9/tpsgTMbx59G+cz72s71C0Qq1Wc2DdT7gWurFg4FwmhYzi6f83nlUj/oJRhTGqzQr2rfqJqIi4O9uiiLYn7vYZvsjaQMasVIYMCtR3OEK0OpJgGaiy6nIqair1HcYdP60/g8lFS147+j45+Xn6Dkdr4lMT0Ow1wyLRhmVH3mTdl9+SmJis77Ca5Ej4SdxueFAzspJevboBoDBSENJxMB89tQqrOQpqLWoxOWTNoXdiORx+HKVKpZPYqpRKUlLTUavVOjlfY+TlF1BWVq7vMHQqLf0WUd/+zGCPIOYFz9R3OEK0SnKL0MBoNBoij8Vy60QuMYEn+GLShyiM9JsHX867yjVVKn28/PFN6kzcucuM+ssAbG1t9BpXU10rTOWd0x8QMmwob0x+meE3BlC+uY7C/yjZ1SGC/jP88fHpqO8wH8v5c1cwP25LTpcswqaMu+/nCoWCoEEBBA2CixeTyDlchV2UG1ExCdT2VRI6JQhbG+39v5ZXVnL5/DWykvJRZxjhWOyIicaEK9bHMQtSEDJ2AJaWllo73+NIKkhhx5Vw/A8EYI45Zsss8HB31UssulRRWcWF9Sn0ru3F0AF99D6+CNFaGWkaU4+mlUlISCAwMJAzZ84YVKmcYmUJH/38BdevZDC5bBIbXDexPPglJncbq7eYNBoNfzryJhU1FXwY/DYFeSXcWl9IqUMJk/8rBHNzM73F1hQ3MzLZuf0Ayf5JfDTuPSxNLID6nayPHYylLsYYq2or8jvnEhzWp0VsupiTm8/Zj5Kpsqpk0mvDG/1/k3o9nXP7k3BKc6VGUU321JtMHTgWe4t2jx1DubKCC6eTuKpO4nTVWVwvejImczSVJpWUuZRg6WOGnYsNOfFFuGS7ojRRkjkjjal9x+Jk2fwlPerUdZy8cor83eVs8/gOhTM8bTYTm2gHqk1VBC/vhYuzU7PH8TBqtZrU6zfp1NFT658vtVrN7k+P4njbCc8X7PHz89Vq+8KwFBYWSqkcPZIEy0ASrBPRp0k+fpO93ffwysBFjPIaxr92fkXHBG+Gv9YX+3Z2eonrePQpfjx1gklzR9DF0gtHR0dOxV+gYnsd+R3zmLlsTItbLJ1fUETcRxcx0igYtKIHzg0MQFVKJZHfx2F6xhKzOjOK/PIJCRuAk6ODHiJ+tOqaGn5Y/RNW5Vb0+bPvE83E3M7KISbiDOvNN6NGzXzNPEJDgunQ3v2BxxQVl3D5fDK3b+fwo0MkKYU3ePPsG8R3iqe2fxX9rALwt+2Gr6/Xff3kRtot4o+fY5PZ19TU1fJ8+bP0G9GTHt26POBsT660rJyo2Fi+qdpBbmk+L2YuxHOUE0MDgzBRGHP+wmWytpZSba5i2J/74uhgr/UYGqOmppbwtZG43fDgtNtpqkJKCfUcSqB7gFaSrQM7o7CLcaJmQjkjxw3WQsTCkEmCpV+SYOk5wSopLePIpljc0zzIcctm+Ev9cXWo/ws6MyuXq/+bQXG3Qma+pPtZrOqaGiJWxaGyURL2xti7PqzHImIxPWRDfq8cpj/fcjYmrKisIuIfMViVW9P95Q54e3d46OtLy8o5tjsem4v1szmZs9IIC5yElalhfZ1999oInJJdsZtnRr/+PZrUVomqlO8vHMZ9RycOdjiEa5A9v/Objo+DF/n5hSSeTyE/uRjjTDMcyh1QoKDYophzk07Tx70nvSz88fJs3+jEu6y6nEMXj2G905HdHfdg1dWM2Z2eYrDvAExMmraK4WZGJqcPXqLdNUcUGgVxU44zPWAS/k5d73pdYWEhBYUl3Ph3LpVWlYx8dQDt7GybdO7HpVSp2P9ZNM5ZrlQMLibfPY8jxcdwS/VgXOY4zk79mZDOgwl0C8DM5PGTrTOnL1G+rZaC7nl6GU+E7kmCpV+SYPXvz63b2cTvvsDw3wXi6qK72wM/x5+nYG85ZjXm1I1QMmbSkPt+KYVviaTdWUfav+xAF18vncUGcHBXNNYn7XF9yQY/P9/7Pqz7vztGu3hnyoYVMXFmiE5jexI1NbWEfxyJQ64TLvNt6N27e6OPzc8v5MiRk2xgM1YmVrxo+Rxjxg41iGK4ETei2LP/ME91nMDEaaFaa7e8spKDaUf5LmkvIy+NwrvKm3bK+kSzxKKEKvdK2nW2pnsfHzq0d2/yTGZNTS0nM+PZcTWcoJjBuNQ4YdS/juETBjzWujC1Ws3Zs4nciMzENcsNlYmKCr9SBk4KwPMBM3u/9O3ExGRubyyi3LaMMa8GY2tj3aT31Fhl5RUc+TQWh0JHTJ9SMyx0AFB/i/5a6g0unLtKuPl+bpVk8ufE5dS4qfDo70zgwF6NmtnKzM7l4sfXqbAtZ+prIzA1leW3bYEkWPrV5hMs/x49OLw6FudSZ7I7ZTFr+f0Lg7WtvLKSiM0ncE32INcph6D5PenYoeE9aKqUSiLfPY3StooZr43W2e24ktIy4t9PpKxjCWF/qr8m935Y1Wo1e/9zFOerbjBVRcjIgTqJ7Umo1Wr2fHEU51RXTGeoGTL8yWYucyvy2H58HwERgYT3+p5RoUMY5zMSE4WxliNunKsZqfzpxOuM8h7G68HLMDIy0vo5aupqiDoUT+nNShx8bOge0Jn2ng++bagNFy9cJenoTVxuu1KjqKG0azEDJvakU0fPBx6jUlVzMvI0pXEqnMucKbIqwmwADBsXhLXVwxfS/7ZvX7iQRN5XZZQ4FDNpxTAszC20+t7uVVBYxPF/nsW2zA77OeYEPqAOoEajITUvnUsHUzBONsOh0gGlsZKSDkW493MicFCvBhN+pUrFwdUnsaiyYMCK7npfYyZ0RxIs/WrzCdaNmHycbrhQ2C0P1yQPLJ+HwF7NV+j03NlEbu0swFppTdXgUsbPGI6x8cN/OR8/dgqjfRYYTa9maIhu9qvZu/4Idlcc8VvR8c5f/Q19WOvq6tjzyY9UlFXR749dCXBr2u2p5hK+9Uccz7hSObKEcVOHNbm96+npbE7/jqiMk8zL+j3d+/kwfOQAna5HKyovJuZvl7jR/jovLp6DuYn+Z9O07XZWDj8fuIDN1XaY15mT55lD51Ed6NPX7861LlaWsC8lgnNxV5iROIMcl2w8hjsxaHCfR362fnFv3044c5ndxw+gCqjgb6FvYWZs2izvLys7l1NrrmChssDzWYc722o8ilqtJjkljaux11Ekm+JQ6fhrshXsSNCAPndi/mf0epyPetBzlg8BfWQn/7ZEEiz9arMJ1rx58/nrq+/R6YovlSNKGD15MO9u/5g02zTWT/oEE4V2p9Cr62rYdGEbxYeVdFN1ofcffPHt3Lhbfmq1mj2rIzGvMGfM2wOb/bbUrdvZpHyURUlAAdPm/7q+6kEf1iqlkjej/kZy2XU+H7sab3vD2trgyKETWEa0o7BfLk/9YbRW276alczlTWm45bmTb5ePx3gHBg3uq9VzNKROXcfKY++iSDZl2ZSFD5wBbS3KKys5cfg0tafBvsqBSz4X6TjRhZKTlVTcVLG9y3bGeo9guvNkunX1eez2G+rbZ7LP8/qx95ismMDLM+djZqrdJOt2WRYbv95BYEYgXV/o8MRLAO5NtmId4zjf4Tyj24XiqnZlXeEmVgxcwtSuzT87LwyLJFj61WYTrMot9YNlTucswpbWDzzJhdd56eB/8XLn55k9eKrWzpd4JZldMfuJsjrOAv+5zOk5/bHXQFxLvkHWFyWUBhUxde5IrcXWkJ2fRGCbZceQv/a+aw3Kwz6sZapy3t7zIYOvDiXkj/1xd3Np1hgbKz4zgb9FfMwczWzmzp/SbDNMp09dJONAHi4lLuS4ZTHgdz3x8mrfLOcC+DpiJxvytvKP0asIdO/TbOcxNHV1dcTHnCciP5JjlccZWBHEcKvBDA8LxMHS/onbfVDfjr1yBtV/FCT1v8zCuXO1dis4OfMGK2P/G2sTSz4cugpPZ+1sA6LRaLhelE70rRiKf6qk821f0sek8MroF7XSvmhZJMHSrzaVYK1Zs4Y1a9ZQVVXF1zPCiXeJZ/nKBXfdRti2LRy3M+3xe60Dni5NG/Rq1XVsS9xF9v4SfKt86L2sM10dOz9xe7vWRqDIMGH4m31xtm6eD83ly8kUrVdSEVrM+GnD7/rZoz6stzOzObohjlM94/lgyl+x1vM37S4nXuPthNV07dCZ90PeaPZ1Umq1muPHTlF5tBbLaitKAgoZ9/QQrW+kGf3jzxjvtyRv/G1mjJ+g1bZbkrzKAtqZ22nl9t3D+vZPZ+NZlbia0T4hvDH4lSZvzHk+OZHcdeXE9DjJst8twMHCvkntPczNjEw8PVyb/G1M0TJJgqVfLWsDoyZasmQJiYmJ7Nq1C4DkDtfuW6MxafJIfuj+A2uvbmrSuVKvp/PhV/9iw4Vt2I+2ZNIbw5uUXAGM/n0wWwK+Zt2Fr5vUzoNoNBr+c+srTnT/iVGTHn+PnPae7gxb0pfrNWm8f+RjVKrqZoiycW6XZnF1201m3p7BO8NW6GQRukKhIHT0IIb/tR/FPQtod8GJyPfOcCI1Hm39HZOccoO6g8Zkt89i2ti2fcvHxcqp2dZG/VZIv0G8NfTPZCUUsPuLI00q8/NzZgIrT68iqctVls98oVmTK4BOHT0luRJCT9pUgnWvLoX372Lczs6WiWNGEJ0eQ3xKwmO3WVdXx4EdUdxcU4D/9Z78c8zfeXHAM1pZN2VvZ8ez/Z4m7koCF69cbXJ794rJOE1C/gVCpwx64q9x+zp4897QNxh4YjD7vojSS825YmUpK6Pe5XDvw4xfOPTOLu26YmNlxfTnx9BpiSMZPum8Ffc/vHbsPW7cbFqh7JLSMpI23qLcspzxi4a2uA1eW7LR3iFM95mIe4one/999In6ddTReDZ+v4N+HgG8+Pwcne+zJYTQrTY9QjtXNrxOaIx3CM+UziVvYwVVSmWj27uZkcn3fz+GXawThV3zGff6YHq5+mkrXACmdBnH/Iw/cGlvKmqN9pKXmppa8v5TzkzVdII9m/ZNxcD2ATiMsML9pgfhG3/UUoSNU1FZxc61h6itquWdia/i7Ky/6fHOnTux6IXf837IG5ilWJL+aR6bTnyHsvbxCyvX1dVxdG0cFtUW9H6+s872ZxK/GjN5KOXDi3C95sH3G398rCTrcPhxFAcsGFUTyvshr7fKb3wKIe7WphOsS+0vNPi8kZERI8YFY6O05ciOk49sR61WExF+guRPM7GosMR8joawxeMeuffOkzBWGOMzx50N7TcRcSNKa+0eTD3KWbtzjAkeppW9lEaMC6ZkUAHOl905uCtaCxE+Wk1NLYc+P07nW778tecK2tvq/5t1RkZGDO8YzBvPLCV3+G22Zuzg2fAlRJ+If6xf0Pu/icIl2xXLqYoWV4C6NZkwI4SS4HxcLruzb8uxRh2z75tIrKPsyeuaw8wlY7X+DWUhhGFq0wmWqfrBOyB37eJNcY98bM85kJ5++4Gvy87JY8+HP2IV1Y7iToUMfSOAoEEBzRHuHf38ejHUJ4hv4vZQVl7e5PYqayrZcPkbzIcY0bNH10cf0EiT54wg1y8b65P2/BT5s9babYharSb835E4Z7tiMc2IXr0av0u7LlhbWfL09ClsmvJPgmoDMd5txZ7VkaReT3/ksTHHE7A/60xR33yd7YMmHmzy0yMpDMzF6Zwr+7ZFPvB1arWaPeuP4HDahYI+OcxYNKbR+3IJIVq+Np1gdc19+KZ+Y+YOQWmm5PS2xAZnGyJvHOfE5+ewKbKFaSrClo3T2bqKRb2e45lzz3D0u9gmt3X425MMuDGQF/rM00Jkd5u2cBS57XOo229CQsJlrbf/ix++qa/ZVj2y/Il3adeF9rYe/OXpP2I0vRqLcgsy1hSxZ/0RysorGnx9Rkkm2YeLyHPLYcq8EboNVjzQU/NGkx+Qg8MpFw40MENbW1vLnn8dxeWyOyWD6/eTkzVzQrQtbfoTb1398HUsNlZWWI8zwS3XnZjoXxe8FxQWsfrIZ/x3zD+4MTCFwJXd79QO0xUPB1eUPctxSHTmRtqTL57OzM7F/rwTvRz9cLXW/t5VxsbGTFkaSpFDIQXbKkhOSdP6OY78cBLHBFcK++UxVgu7tOvC0JBARr09gJK+BdgnOnHi/fNEHbn7tmFVrZK3T6zmQN/9jF4cLLMfBuapZ0eR55+NZYwtP5w9eud5lUrF3k8jcbnuhnJ0KZNnj9BfkEIIvWnTCVa5ZdkjXzMsdAA5rtmURqgor6wk5tZpfvz0Z1ziPfnr0P9ixZTFODs56CDa+42ePYQqsyoSvkt84jZit5+jRlHDqFmDtBjZ3SzMzRn1pyCqLKrYt+9H8isLtNZ27MmzmEVak9M5q8XN8FhaWDDtD6PxXe5OpWMFJget2Pv3SJJTbqBWq9m+aT9lhRW8NWY59u3s9B2uuIdCoWDawtEkTbzEPxI/J+JGFFW1Sv5335c4ZjlhNLWaMZOH6jtMIYSetOnVlmrFoxcZKxQKAuf6k/5ZPlvXfs83bt8yru9onh/8Ozz+r0afvlhbWWI5yhjbgx7EnTxL8NB+j3X8lSspuKS5UTG0GLtmvrVp364dQcv9+Sp6M8lHkpjn8DRG7TUYmYMmzwjKHrGw3k2NkTVoioyg1AgjLzUV2UqM91hR5FbAU4tGtdhbMJ06etJppScxJxIwO2jO/u3R5PfJYkhyCEvHvYSPfSd9hygeQKFQ8NLoZyiNK+Hytusc6xDLWfNzjH95BIGdm3ctphDCsLXpBMvS8cGL3H/L26sDF/tfwyTVlL88tYTJvmO18k07bQgZHcTe+GOYHjSlemDNY9VLS9x7A3NLc8Y9NaQZI/yVh5MrH456h7/v+SdEm/OZ/+dkW2UzLf0pBuYPfOixX/l+TZJ9EiOyQgnODWZ1nw8wVhsz2yeMZxfMfOJ9uwzJkGH9UQapKL6cy4Fr+/Ga1YkXg7S/Lk5ol8JIwatBf2RvfCTV+dV8PP99/J2092URIUTL1KZK5fwiISGBi9/fxCPUjnGjRjX6OJWqGnPzxiVlunTlSgp56yooCy5i8tMjGnVM3Mmz1O4ygakqho1s3PoxbZVdqFIpKS0px9hKgZGxEXVKNZqah3dDhYUChakRapUadY0GE5v69Uj25nYY62CXdl2rrqvRyS7lop42+rZaraamptYgxwjRNkmpHP1q+X/2P6EN3TbyltWfHusYQx04/f27kNglgnanHMgbVYCLs9NDX19bW0vuoVJqHWqZHtq8haMbYmlugaXrb3ZXf5ztwrS/tZhBkuSq5VEoFAY7RgghdK9lLlrRAtM6U2hFc3ehc4IotChkx6kfHvnayCNxOFY40mVa+xa7bkkIIYQwZG32t+uqc+9gVNF6bi05OzlgPK+G7YV7uFaY+sDXVdUq+XfFBs4NPUNAgHbL+AghhBCiXptNsFqjad0mEmDak6itpx5YhmXHuXCKa0oIGz9Rx9EJIYQQbYckWK2IicKYZ7yexv22B5GJ99dQzM7Jx/M7HxZY/AEPGzc9RCiEEEK0DZJgtTIDB/Xh1IRY1l7bSFWt8q6fbU3dwfGOx5k8UvcL24UQQoi2RBKsVujlAQtQlBizb9+Pd567XpTOD2kR9B7XFXv7dnqMTgghhGj9WmyCFR0dzeTJk/H19aVbt24sXboUlUrFrFmzcHR0pGfPnqxatUrfYeqFh40bv2cOric8uZmRCUDCv68yOX8i07pO0HN0QgghROun0wRr2bJl2NnZYWRkhJGREZcuXbrzs+TkZPr374+lpSUWFhb07t2bxMQH19gLDQ1l//79pKam4ufnx/Xr19m6dSuDBg2iW7duGBsbU1dXp4u3ZZDGzBlClWkVp769xM/x53HP8SSoZx9Mm7C/0q5du7QYoRCGQ/q2aI2kX+uXThOsWbNmsWHDBjp06ICHh8ddP1u0aBFVVVV88cUXbNmyBZVKxcKFC8nJySEkJARHR8c7Dy8vL1auXAnAgQMHMDMzo3fv3uTm5hIYGEhcXByvvPIKhw8fJi0tTZdv0WDYWFlhPsIItywPCvdUkmefy/CRQU1qc/fu3VqKTgjDIn1btEbSr/VLpzu5h4SEALBixYq7Zpdyc3M5c+YMAM888wzGxsYsXbqUoqIiqqqq+Omnnxpsb926dYSHhxMYGMiqVavYunUrN2/eBCA9PZ0OHTpQVlZ233FVVVXkGhdRnJFBQkKCtt+mwbBzNiPNLgv3Ug+qB5dw7ty5JrVXUlLSqq+XaLukb4vWSPq17vn5+WFlZVX/D40eeHl5aTw8PDQXL17UaDQazenTpzU+Pj4af3//O68JCgrS+Pn5aaKjoxtsY+fOnRonJyeNtbW1xt/fX/Pss89qbt68qRk/frymR48emu7du2uWL1/e4LFbtmzRUL+PuzzkIQ95yEMe8pCHVh5nzpy5k2sYTC3CX9Zl/UKj0aB5SB3qsLAwwsLC7nv+0KFDjzzX+PHj2bJlC97e3lhaGm5xu7CwMLmH/giGdo0MLR4wvJgMLR5DZIjXyNBiMrR4wDBjMjSGdo20HY+f368VUgwiwerYsSN5eXlAfSFiY2NjMjIyqKuro1OnTlo/n7OzM/PmzdN6u9pmaWlJ//799R2GQTO0a2Ro8YDhxWRo8RgiQ7xGhhaTocUDhhmToTG0a9Sc8RhEguXq6kr//v3Jzs5my5Yt2NjYYGdnh6OjI97e3voOT2+WLFmi7xAMnqFdI0OLBwwvJkOLxxAZ4jUytJgMLR4wzJgMjaFdo+aMx0jzsPtwWrZkyRI2b95MRUUFAMbGxnh7e5OSkkJSUhJz5swhKSkJjUaDr68v27dvp2fPnroKTwghhBBCK3SaYAkhhBBCtAUtdid3IYQQQghDZRBrsAxFdHQ0X331FUqlEgcHBz7//HNUKhWLFi3Czs6Ouro61qxZo+8wDU5RURErVqwgIiKCjIwMANLS0pg4cSKhoaG4u7u32bJFouVqqF/LeCBaExmnm1eLn8FatmwZ3t7e95XegfryO0OGDKFbt24MHDjwoaV3oL78zvr16+9sWFpWVsbu3bsJDQ3ls88+w8HBgdjY2OZ8Ozqjzevm4ODA+vXr6d69+13Pt2vXDqVSiY+Pj9bjF6Ihzd2vW+t4IFoObfZxkHG6ObX4BGvWrFmcOHECLy+v+362aNEiXnrpJa5du8bKlStZuHAhADk5OYwZM+auxy+ld6C+/I6/vz+2trakp6ff+SZj586dSU9P18n7am7Ncd1+y8vLi7i4ODZu3Eh4eHibLVkkdKu5+3VrHQ9Ey6HNPi7jdPNq8bcIfym/c6/c3FwSEhKIiIgA6jcTW7p0KWlpaXh7e3P06NEGj1u3bh23b9/mgw8+AOr36PplEE1LS2PChAnN8C50T9vX7V6/bBprZGSEm5tbgyWLhNC25u7XrXU8EC1Hc/RxGaebR4ufwXqQjIwMPD09MTGpzyGNjIzo1KnTnVqFDdm1axfvvvsu2dnZLF68mLy8PMLCwoiKiuLVV18lLy+PIUOG6Oot6MWTXDeAxYsXc/XqVRYvXkxqairHjh1jwYIFLFiwAHNzc3r37q2L8IVokLb6dVsbD0TL8SR9XMbp5tXiZ7Ae5reld4CHlt6BB5ff2bRpkzbDMniPe90A1q5de9e/fX19GTlypFbjEqIptNGvoe2NB6LleNw+PnLkSBmnm1GrncHq2LEjt27dora2FqjvaBkZGc1Seqc1kesmWiPp16K1kz5ueFptguXq6kq/fv3YsmULUH/7z9vbu02X3mkMuW6iNZJ+LVo76eOGp8Xv5L5kyRK+//57srOzcXZ2xsbGhpSUFACSkpJ47rnnKCgowM7Ojs2bN0vpnf8j1020RtKvRWsnfbzlaPEJlhBCCCGEoWm1twiFEEIIIfRFEiwhhBBCCC2TBEsIIYQQQsskwRJCCCGE0DJJsIQQQgghtEwSLCGEEEIILZMESwghhBBCyyTBEkKIBmzYsIGAgABsbGywtbWlb9++hIeH6zssIUQL0aqLPQshxJO4ePEiCxcuZNGiRXzyySeYmpqSlpZGRUWFvkMTQrQQkmAJIcQ9CgsLAejZsyehoaEYGxszfPhwPUclhGhJ5BahEELcIyQkhEWLFrFs2TJcXFyYPXs2+/bt03dYQogWRBIsIYS4R3Z2NjExMaxdu5b9+/fTvXt3wsLCWLJkib5DE0K0EFLsWQgh7jF37lzKy8vvmrVav349L7zwArGxsQQHB+sxOiFESyAzWEII8RtKpZLdu3czbty4u56fP38+JiYmHD16VE+RCSFaEkmwhBDiNwoKCqiursbR0fGu501MTHBycqKgoEBPkQkhWhJJsIQQ4jfs7e1RKBRkZmbe9XxNTQ35+fn3JV5CCNEQSbCEEOI3rK2tCQwMZPv27Xc9/91331FXV0dISIieIhNCtCSyyF0IIe5x6NAhJk+ezKxZs3juuee4fv06r7/+OkOGDOHw4cP6Dk8I0QJIgiWEEA3Yt28f77zzDpcvX8be3p7Zs2ezevVqbGxs9B2aEKIFkARLCCGEEELLZA2WEEIIIYSWSYIlhBBCCKFlkmAJIYQQQmiZJFhCCCGEEFomCZYQQgghhJZJgiWEEEIIoWX/H2oEQczTsQ3UAAAAAElFTkSuQmCC"
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "δfi(x,δ)  = imag((f(x+δ*im)-f(x))/δ)\n",
    "δfi2(x,δ) = imag((f(x+δ*im)-f(x-δ*im))/(2δ))\n",
    "\n",
    "h       = [1/10^k for k in 20:-.5:3]\n",
    "x       = 0.3\n",
    "ϵ1      = [abs(δf(x,δ)-df(x))+1.e-25 for δ in h]\n",
    "ϵ2      = [abs(δf2(x,δ)-df(x))+1.e-25 for δ in h]\n",
    "ϵ3      = [abs(δfi(x,δ)-df(x))+1.e-25 for δ in h]\n",
    "ϵ4      = [abs(δfi2(x,δ)-df(x))+1.e-25 for δ in h]\n",
    "\n",
    "plot(h,ϵ1,xscale=:log10,yscale=:log10, label=\"ϵ1 (forward)\",\n",
    "               xlabel=\"δ\",ylabel=\"|δf(x0,δ)-f'(x0)|\")\n",
    "plot!(h,ϵ2,xscale=:log10, yscale=:log10,label=\"ϵ2 (symmetric)\")\n",
    "plot!(h,ϵ3,xscale=:log10,yscale=:log10,label=\"ϵ3 (forward complex)\")\n",
    "plot!(h,ϵ4,xscale=:log10,yscale=:log10,linestyle=:auto,label=\"ϵ4 (symmetric complex)\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:#00B8DE\"> Dual Numbers </span>\n",
    "<a name=\"Dual-numbers\"></a> \n",
    "[Table of contents](#table-of-contents)\n",
    "\n",
    "\n",
    "Before going to automatic differentiation, let us go beyond the finite difference with complex increment by searching an alternative kind of increment of the variable for which any higher order approximation term will cancel. To this end, we introduce dual numbers [1]. \n",
    "\n",
    "First, let us recall that complex numbers $a+jb$ can be introduced in a matrix form $aI+bJ=\\begin{bmatrix} a&-b\\\\ b&a \\end{bmatrix}$. The relationship $J^2=-I$ catches the definition $j=\\sqrt{-1}$. \n",
    "\n",
    "In the same way, dual numbers $a+\\varepsilon b$ can be seen has representions of $2\\times 2$ matrices $aI+bE=\\begin{bmatrix} a&b\\\\ 0&a \\end{bmatrix}$. Note that $E=\\begin{bmatrix} 0&1\\\\ 0&0 \\end{bmatrix}$ and $E^2$ is the $2\\times 2$ null matrix. in other words, $\\varepsilon^2=0$.\n",
    "\n",
    "Dual numbers are of interest for differentiation. Indeed, il we consider a polynomial $p(x)=\\sum_{k=0:n}p_kx^k$, since $(a+\\varepsilon b)^k = a^k+k a^{k-1}\\varepsilon b$ (because $\\varepsilon^m=0$ for $m>1$), we get\n",
    "\n",
    "$$\n",
    "p(a+\\varepsilon b) = p(a)+\\varepsilon b p'(a).\n",
    "$$\n",
    "\n",
    "Now, analytic functions write $f(x)=\\sum_{k=0:\\infty}f^{(k)}(x_0)\\dfrac{(x-x_0)^k}{k!}$ and the previous expression extends to \n",
    "\n",
    "$$\n",
    "f(a+\\varepsilon b) =\\sum_{k=0:\\infty}f^{(k)}(a)\\dfrac{(\\varepsilon b)^k}{k!}= f(a)+\\varepsilon b f'(a)\n",
    "$$\n",
    "\n",
    "With a view to implement chain rules for derivation based on dual numbers let us consider usual binary operators for dual numbers. Clearly, $(a+\\varepsilon b) + (c+\\varepsilon d) = (a+c)+\\varepsilon (b+d)$ and $(a+\\varepsilon b) \\times (c+\\varepsilon d) = (ac)+\\varepsilon (ad+bc)$. For division, assume first that $c\\neq 0$:\n",
    "\n",
    "$$\n",
    "\\dfrac{a+b\\varepsilon}{c+d\\varepsilon}= \\dfrac{(a+b\\varepsilon)(c-d\\varepsilon)}{(c+d\\varepsilon)(c-d\\varepsilon)}\n",
    "= \\dfrac{ac-ad\\varepsilon+bc\\varepsilon}{c^2+cd\\varepsilon-cd\\varepsilon}\n",
    "= \\dfrac{a}{c} + \\dfrac{(bc - ad)}{c^2}\\varepsilon\n",
    "$$\n",
    "\n",
    "Now, if $c=0$, since ${a+b\\varepsilon = (x+y\\varepsilon) d\\varepsilon} = xd\\varepsilon$, there is no solution if $a\\neq 0$, while if $a= 0$, $\\dfrac{a+b\\varepsilon}{c+d\\varepsilon}$ can be represented by any dual number in the form $\\dfrac{b}{d} + {y\\varepsilon}$.\n",
    "\n",
    "In addition, note that $f(g(a+\\varepsilon b))= f(g(a)+\\varepsilon g'(b))= f(g(a)) + \\varepsilon f'(g(a))g'(b)$.\n",
    "\n",
    "These results, together with knowledge of the expression of derivatives of usual functions can be used to efficiently implement the derivation chain rule to compute the derivative of complex functions. Here is an example using library **DualNumbers** [3]."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "f(0.3 + 1.0ɛ) = 0.16288752190025318 + 0.43432091320486865ɛ \n",
      "\n",
      "error with dual numbers: 2.220446049250313e-16 \n",
      "\n",
      "error on finite difference: -1.2470025012589758e-11"
     ]
    }
   ],
   "source": [
    "using DualNumbers\n",
    "\n",
    "f(x)  = x*(1-x)*(1-2x)^2*(1-8x+8x^2)^2*exp(1+exp(x)) \n",
    "df(x) = f(x)*(1/x-1/(1-x)-4/(1-2x)+2*(16x-8)/(1-8x+8x^2)+exp(x))\n",
    "\n",
    "x = .3\n",
    "y = x+ε\n",
    "δ = 1.e-6 # optimal step size for this function and symmetric difference\n",
    "print(\n",
    "\"f($y) = $(f(y)) \\n\n",
    "error with dual numbers: $(df(x)-dualpart(f(y))) \\n\n",
    "error on finite difference: $( (f(x+δ)-f(x-δ))/(2δ)-dualpart(f(y)) )\" )"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Automatic differentiation is intended to provide automated propagation of the chain rules for complex differentiation tasks. \n",
    "Several modes can be considered to this end. \n",
    "\n",
    "For differentiation tasks **DualNumbers** library [3], that is not maintained currently, has been superseded by **ForwardDiff** library [4] that implements the forward automatic differentiation. Forward differentiation implements the principle of dual numbers calculation: the function evaluations and derivatives are both propagated in the chain rule in a way similar to what is done with dual numbers. \n",
    "\n",
    "## <span style=\"color:#00B8DE\"> Forward Automatic Differentiation</span>\n",
    "<a name=\"Forward-automatic-differentiation\"></a> \n",
    "[Table of contents](#table-of-contents)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "df(x) - ForwardDiff.derivative(f,x) = 2.220446049250313e-16"
     ]
    }
   ],
   "source": [
    "using ForwardDiff\n",
    "\n",
    "f(x)  = x*(1-x)*(1-2x)^2*(1-8x+8x^2)^2*exp(1+exp(x)) \n",
    "df(x) = f(x)*(1/x-1/(1-x)-4/(1-2x)+2*(16x-8)/(1-8x+8x^2)+exp(x))\n",
    "\n",
    "x = .3\n",
    "print(\"df(x) - ForwardDiff.derivative(f,x) = $( df(x)  - ForwardDiff.derivative(f,x) )\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us consider the example presented in [2]:  $f(x_1,x_2)=x_1\\sqrt{\\log\\dfrac{x_1}{\\sin x_2^2}}$. We want to compute the partial derivative \n",
    "\n",
    "$$\n",
    "\\dfrac{\\partial f}{\\partial x_2}=-x_1x_2\\cot x_2^2\\;\\left(\\log\\dfrac{x_1}{\\sin x_2^2}\\right)^{-1/2}\n",
    "$$\n",
    "\n",
    "We introduce notations for input variables $v_{-1}=x_1$ and $v_{0}=x_0$,intermediate variables $v_1$ to $v_5$ that represent intermediary results provided by elementary operations in the function  definition, and output variable $v_6=f(x_1,x_2)$. Often, as here, indices are set so that the first index of intermediate variables is 1. The flow from $v_{-1:0}=x_{1:2}$ to $v_6=f(x_1,x_2)=y$ is summerized in the following graph.\n",
    "\n",
    "<center><img src=\"operations_graph.png\" alt=\"drawing\" width=\"600\"/></center>\n",
    "<center> Graph of operations for function $f(x_1,x_2)$ [1] </center>\n",
    "\n",
    "These operations can also be represented by the successive operations in the left column in the following table. \n",
    "\n",
    "<center><img src=\"forward_chain_rule.png\" alt=\"drawing\" width=\"500\"/></center>\n",
    "<center> Forward chain rule [1] </center>\n",
    "\n",
    "The right part of this table represent elementary derivations associated to the elementary operations on the left side. Here setting $\\dot{v}_{-1}=0$ and $\\dot{v}_0=1$ comes from the fact that we compute partial derivatives w.r.t. $x_2$: $\\dot{v}_{-1}=\\dfrac{\\partial x_1}{\\partial x_2}=0$ and $\\dot{v}_{0}=\\dfrac{\\partial x_2}{\\partial x_2}=1$. \n",
    "\n",
    "In practice, we first propagate calculations through the first column to get the $v_i$s and then through the second one, replacing at each step the right hand side by its numerical value. Finaly, we get \n",
    "\n",
    "$$\n",
    "\\begin{array}{ll}\n",
    "v_{-1:6}&=\n",
    "[x_1\\;\\; x_2\\;\\; \\sin x_2^2\\;\\; \\dfrac{x_1}{\\sin x_2^2}\\;\\; \\log \\dfrac{x_1}{\\sin x_2^2}\\;\\; \\sqrt{\\log \\dfrac{x_1}{\\sin x_2^2}} \\; \\;x_1\\sqrt{\\log \\dfrac{x_1}{\\sin x_2^2}}]\\\\\n",
    "\\\\\n",
    "& = [v_{-1}\\;\\; v_0\\;\\; \\sin v_0^2\\;\\; \\dfrac{v_{-1}}{v_1}\\;\\; \\log v_2\\;\\; \\sqrt{v_3}\\;\\; v_{-1}v_4]\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "where each term $v_i$ is computed recursively from the previous ones $v_{-1:i-1}$. Clearly, the interest is that we don't propagate formulas but only associated numerical values $v_{-1:i-1}$, as classicaly done when we compute the value of a function at a given point.\n",
    "\n",
    "Then the same principle is used to get the partial derivatives, $\\dot{v}_{i}$ being calculated as a function of $v_{-1:i-1}$ and $\\dot{v}_{-1:i-1}$:\n",
    "\n",
    "$$\n",
    "\\begin{array}{ll}\n",
    "\\dot{v}_{-1:6}&=\n",
    "[0\\;\\; 1\\;\\; 2x_2\\cos x_2^2\\;\\; \\dfrac{-2x_1x_2\\cos x_2^2}{\\sin x_2^2}\\;\\; \\ldots]\\\\\n",
    "\\\\\n",
    "& = [0\\;\\; 1\\;\\; 2v_0\\dot{v}_0 \\;\\; (\\cos v_1) \\dot{v}_1 \\;\\; \\dfrac{\\dot{v}_{-1}v_2-v_{-1}\\dot{v}_2}{v_2^2} \\;\\;\\ldots]\n",
    "\\end{array}\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Complete gradient calculation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "ForwardDiff.gradient(f,x)-[∇f_1(x),∇f_2(x)]= [0.0, -4.440892098500626e-16]"
     ]
    }
   ],
   "source": [
    "f(x)   = x[1]*√log(x[1]/sin(x[2]^2))\n",
    "∇f_1(x) = √log(x[1]/sin(x[2]^2)) + 1/(2*√log(x[1]/sin(x[2]^2)))\n",
    "∇f_2(x) = -x[1]*x[2]*cot(x[2]^2)*1/√log(x[1]/sin(x[2]^2))\n",
    "\n",
    "x      = [1,1]\n",
    "print(\"ForwardDiff.gradient(f,x)-[∇f_1(x),∇f_2(x)]= $(ForwardDiff.gradient(f,x)-[∇f_1(x),∇f_2(x)])\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### General chain rule"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In a more formal way, the algorithm writes as follows. For $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, relationships are in the form $v_i=g_i(v_{s_i})$ with $s_i\\subset -n+1:i-1$ and for $i=1:K$ and $y=f(x)=v_K$ . \n",
    "$s_i$ is the set of indices of variables that generated $v_i$. Then the forward mode for computing the directional derivative in direction **d** can be implemented as in the following algorithmic form:\n",
    "\n",
    "1. $v_{-n+1:0}= x_{1:n}$; $\\;\\dot{v}_{-n+1:0}=$**d**\n",
    "\n",
    "2. for $i=1:K$\n",
    "\n",
    "$\\quad\\quad$ $v_i=g_i(v_{s_i})$\n",
    "\n",
    "$\\quad\\quad$ $\\dot{v}_i = \\sum_{j\\in s_i} \\dot{v}_j \\dfrac{\\partial g_i(v_{s_i})}{\\partial v_j}$\n",
    "\n",
    "3. $\\dfrac{dy}{dt}(x+t$**d$)=$d** $^T\\nabla f(x)=\\dot{v}_K$\n",
    "    \n",
    "In particular for **d${=}$e**$_i$, the i-th vector of the canonical basis, we get $\\dfrac{dy}{dt}(x+t$**e**$_i)=\\dfrac{\\partial f(x)}{\\partial x_i}$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:#00B8DE\"> Backward Automatic Differentiation</span>\n",
    "<a name=\"Backward-automatic-differentiation\"></a> \n",
    "[Table of contents](#table-of-contents)\n",
    "\n",
    "Note that in the forward automatic differentiation process of the example described above, if we want to compute the partial derivative w.r.t. $x_1$, we must repeat the computations in the second column with $\\dot{v}_{-1}=\\dfrac{\\partial x_1}{\\partial x_1}=1$ and $\\dot{v}_{0}=\\dfrac{\\partial x_2}{\\partial x_1}=0$. If we consider functions $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, the total amount of calculations to get the gradient of $f$ increases linearly with $n$. However, being able to compute gradients at lower cost would be of high interest for optimization purpose.\n",
    "\n",
    "The backward mode adresses this kind of issue. It involves variables $\\overline{v}_i=\\dfrac{\\partial f}{\\partial v_i}$ and is closely related to backpropagation techniques developped for learning neural networks. $\\overline{v}_i$ represents the sensibility of the output to variations of intermediate variables. \n",
    "\n",
    "To make the reverse mode clearer, consider first a simple case where $f:\\mathbb{R}\\rightarrow\\mathbb{R}$ and $f$ is a composition of one to one transforms: $f(x)=g_n\\circ g_{n-1}\\circ\\ldots \\circ g_1(x)$, that is $v_0=x$ and $v_i=g_i(v_{i-1})$ for $i=1:n$. The graph of $f$ is made of a simple line and the derivation chain rule yields\n",
    "\n",
    "$$\n",
    "\\begin{array}{ll}\n",
    "\\overline{v}_{i-1}\n",
    "&=\\dfrac{\\partial y}{\\partial v_{i-1}}=\\dfrac{\\partial}{\\partial v_{i-1}}g_n\\circ g_{n-1}\\circ ...\\circ g_{i}(v_{i-1}) \\\\\n",
    "\\\\\n",
    "&= \\dfrac{\\partial }{\\partial v_i} g_n\\circ g_{n-1}\\circ ...\\circ g_{i+1}(v_{i})\n",
    "\\times \\dfrac{\\partial g_{i}(v_{i-1})}{\\partial v_{i-1}}\\\\\n",
    "\\\\\n",
    "&= \\dfrac{\\partial y}{\\partial v_i}\\times\\dfrac{\\partial v_i}{\\partial v_{i-1}} = \\overline{v}_{i}\\dfrac{\\partial v_i}{\\partial v_{i-1}}\n",
    "\\end{array}\n",
    "$$\n",
    "\n",
    "More generally, if $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}$, relationships are in the form $v_i=g_i(v_{s_i})$ with $s_i\\subset -n+1:i-1$. \n",
    "$s_i$ is the set of indices of variables that generated $v_i$ while $\\{j;i\\in s_j\\}$ represents the set of next indices in the graph when going from $v_i$ to $y$.\n",
    "\n",
    "$$\n",
    "\\overline{v}_i =\\dfrac{\\partial y}{\\partial v_i}=\\sum_{\\{j;i\\in s_j\\}}\\dfrac{\\partial y}{\\partial v_j}\\dfrac{\\partial g_j(v_{s_j})}{\\partial v_i}= \\sum_{\\{j;i\\in s_j\\}} \\overline{v}_j\\dfrac{\\partial v_j}{\\partial v_i}\n",
    "$$\n",
    "\n",
    "The graph of the example that we considered in forward differentiation section is quite complex due in particular to multiple occurences of the variable $x_1$ in the definition and the presence of binary operators.\n",
    "Initially, variables $\\overline{v}_i$ are set to 0. Then, the sensitivities are propagated backward until obtaining the entries of $\\nabla f$, here $\\overline{v}_{-1}= \\dfrac{\\partial f}{\\partial x_1}$ and $\\overline{v}_{0}=\\dfrac{\\partial f}{\\partial x_2}$. Unlike with the forward mode, we see here that the second column is swept only once to obtain the gradient at a given point. \n",
    "\n",
    "Here, the computation of $f(x_1,x_2)$ is done from top to bottom for the left hand side, like for the forward mode, while operations on the right hand side are performed from the bottom to the top. \n",
    "\n",
    "In particular, we see that at the 4th line of the second column of the table for reverse mode, $\\overline{v}_{-1}$ is added to itself. This comes from the fact that in the graph of operations $y$ is influenced by ${v}_{-1}$ through the direct path but also through the path passing by $v_3$. \n",
    "<!-- To explain further the relationships in the right column, note that \n",
    "$$\n",
    "\\overline{v}_i =\\dfrac{\\partial y}{\\partial v_i}=\\sum_{j\\in ch(i)}\\dfrac{\\partial y}{\\partial v_j}\\dfrac{\\partial v_j}{\\partial v_i}= \\sum_{j\\in ch (i)}\\overline{v}_j\\dfrac{\\partial v_j}{\\partial v_i}\n",
    "$$\n",
    "where $ch (i)$ is the set of children $\\{v_j\\}$ of $v_i$ in the graph oriented from entries ${x_k}$ to output $y$. In particular, for $i=-1$, $ch(i)=\\{3,6\\}$ and two terms contribute to the caculation of $\\overline{v}_{-1}$ -->\n",
    "\n",
    "<center><img src=\"operations_graph.png\" alt=\"drawing\" width=\"600\"/></center>\n",
    "<center> Graph of operations for function $f(x_1,x_2)$ [1] </center>\n",
    "\n",
    "\n",
    "<center><img src=\"backward_chain_rule.png\" alt=\"drawing\" width=\"400\"/></center>\n",
    "<center> Backward chain rule [1] </center>\n",
    "\n",
    "\n",
    "The reverse mode can be implemented as in the following algorithmic form where $y=v_K$:\n",
    "\n",
    "1. $v_{-n+1:0}=x_{1:n}$\n",
    "\n",
    "2. for j= 1:K \n",
    "\n",
    "$\\quad\\quad$ $v_i=g_i(v_{s_i})$\n",
    "\n",
    "3. $\\overline{v}_K=1$, $\\overline{v}_{-n+1:K-1}=0$\n",
    "\n",
    "4. for $j=K-1:-1:1$\n",
    "\n",
    "$\\quad\\quad$ for $i\\in s_j$ \n",
    "\n",
    "$\\quad\\quad\\quad$ $\\dfrac{\\partial v_j}{\\partial v_i}=\\dfrac{\\partial g_j(v_{s_j})}{\\partial v_i}$\n",
    "\n",
    "$\\quad\\quad\\quad$ $\\overline{v}_i = \\overline{v}_i + \\overline{v}_j\\dfrac{\\partial v_j}{\\partial v_i}$\n",
    "    \n",
    "You can find other examples illustrating the reverse mode in [5], [6] or [7] (p.13) for instance.\n",
    "\n",
    "For functions $f:\\mathbb{R}^n\\rightarrow\\mathbb{R}^m$ where the input dimension is higher than the output dimension ($n>m$), it is generally more efficient to use the reverse mode while the forward mode is recommended when $m>n$. In fact, forward and reverse modes computational complexity are about $n×ops(f)$ and $m×ops(f)$ operations respectively, where $ops(f)$ is the number of operations required to compute $f$. It is also possible to combine both modes.\n",
    "\n",
    "Concerning libraries **ForwardDiff.jl** [4] and **ReverseDiff.jl** [9-10], it is added in [9] that ReverseDiff generally performs better for gradients when $n>100$ or when $n$ is larger than the number of operations performed to compute $f$. ReverseDiff is also recommanded for functions involving array-based operations.\n",
    "\n",
    "Let us compute the gradients "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using ForwardDiff, ReverseDiff\n",
    "\n",
    "f(x)    = x[1]*√log(x[1]/sin(x[2]^2))\n",
    "∇f_1(x) = √log(x[1]/sin(x[2]^2)) + 1/(2*√log(x[1]/sin(x[2]^2)))\n",
    "∇f_2(x) = -x[1]*x[2]*cot(x[2]^2)*1/√log(x[1]/sin(x[2]^2))\n",
    "\n",
    "x      = [1.,1.]\n",
    "@time println(\"ForwardDiff.gradient error: $(ForwardDiff.gradient(f,x)-[∇f_1(x),∇f_2(x)])\")\n",
    "@time println(\"ReverseDiff.gradient error: $(ReverseDiff.gradient(f,x)-[∇f_1(x),∇f_2(x)])\") # note : woud not work with x=[1,1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:#00B8DE\"> Exercise: an Optimization Problem</span>\n",
    "<a name=\"Optimization-problem\"></a> \n",
    "[Table of contents](#table-of-contents)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Consider the function $f({\\bf x})=\\dfrac{\\parallel {\\bf x}\\parallel^4}{1+\\parallel{\\bf x}\\parallel^2}$\n",
    "\n",
    "> - Plot the function\n",
    "> - Implement gradient and Newton algorithm with backtracking using forward differentiation to compute the gradient an the Hessian\n",
    "> - Plot the current error at each loop for both algorithms (choose for instance ${\\bf x}\\in\\mathbb{R}^{10}$)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "..."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "------"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# <span style=\"color:#00B8DE\"> Appendix</span>\n",
    "<a name=\"Appendix\"></a> \n",
    "[Table of contents](#table-of-contents)\n",
    "\n",
    "\n",
    "## <span style=\"color:#00B8DE\">A - Simple dual number algebra implementation</span>\n",
    "\n",
    "see also https://www.youtube.com/watch?v=vAp6nUMrKYg"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Dual structure and associated operators\n",
    "\n",
    "struct Dual_ <: Number # class of dual numbers \n",
    "    d::Tuple{Float64,Float64}\n",
    "end\n",
    "\n",
    "import Base: +, -, *, /, convert, promote_rule, show\n",
    "\n",
    "+(x::Dual_,y::Dual_) = Dual_(x.d .+ y.d)\n",
    "-(x::Dual_,y::Dual_) = Dual_(x.d .- y.d)\n",
    "-(x::Dual_)          = Dual_((-x.d[1],-x.d[2]))\n",
    "*(x::Dual_,y::Dual_) = Dual_((x.d[1]*y.d[1], x.d[2]*y.d[1]+x.d[1]*y.d[2]))\n",
    "/(x::Dual_,y::Dual_) = Dual_((x.d[1]/y.d[1], (x.d[2]*y.d[1]-x.d[1]*y.d[2])/y.d[1]^2))\n",
    "\n",
    "ϵ                   = Dual_((0,1))\n",
    "realPart(x::Dual_)  = x.d[1]\n",
    "dualPart(x::Dual_)  = x.d[2]\n",
    "conj(x::Dual_)      = Dual_((x.d[1],-x.d[2]))\n",
    "\n",
    "convert(::Type{Dual_},x::Real) = Dual_((x,zero(x)))\n",
    "promote_rule(::Type{Dual_},::Type{<:Number}) = Dual_\n",
    "show(io::IO,x::Dual_)= x.d[2]>=0 ? print(io,x.d[1],\"+ϵ\",x.d[2]) : print(io,x.d[1],\"-ϵ\",abs(x.d[2]))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x=Dual_((1,2))\n",
    "y=Dual_((3,4))\n",
    "x+y, x/y, Dual_((1.,0))==1., Dual_((1,-2)), Dual_((0,0))^2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# let us redefine a few clasic functions from basic operators\n",
    "\n",
    "#cos_(x) = sum([(-x^2)^k/factorial(2k) for k=0:10])     # Poor definition !\n",
    "#sin_(x) = sum([x*(-x^2)^k/factorial(2k+1) for k=0:9])\n",
    "\n",
    "exp_(x;n=20) = 1 + sum( accumulate((u,v)->x*u/v, 1:n, init=1) )\n",
    "cos_(x;n=20) = 1 + sum( accumulate((u,v)-> -x^2*u/(2v*(2v-1)), 1:n, init=1) )\n",
    "sin_(x;n=20) = x + sum( accumulate((u,v)-> -x^2*u/(2v*(2v+1)), 1:n, init=x) )\n",
    "\n",
    "print(exp_(1)-exp(1),'\\n',cos_(π/4)-√2/2,'\\n', sin_(π/4)-√2/2)\n",
    "\n",
    "function root_(x,k::Int;n=10)\n",
    "    t=1\n",
    "    for _=1:n\n",
    "        t += (x/t^(k-1)-t)/k\n",
    "    end\n",
    "    t\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "root_(1+ϵ,5)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "pyplot()\n",
    "\n",
    "x = range(0,stop=10,length=100)\n",
    "y = [sin_(rem(u,2π)+ϵ) for u in x]\n",
    "\n",
    "p1 = plot(x,realPart.(y),label=\"sin_\")\n",
    "p2 = plot(x,dualPart.(y),label=\"sin_'\")\n",
    "plot(p1,p2)#,layout=(1,1))\n",
    "x = range(0,stop=10,length=20)\n",
    "scatter!(x,[sin.(x),cos.(x)],label=\"\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## <span style=\"color:#00B8DE\">B - An example for a large problem </span>\n",
    "\n",
    "\n",
    "Let us consider a somewhat complex example from https://github.com/JuliaDiff/ReverseDiff.jl/blob/master/examples/gradient.jl that enables optimized operations, that can be useful/necessary for big problems. See the doc of  ReverseDiff for more information"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "WARNING: redefinition of constant f_tape. This may fail, cause incorrect answers, or produce other errors.\n",
      "WARNING: redefinition of constant compiled_f_tape. This may fail, cause incorrect answers, or produce other errors.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "([95.15001684876657 92.6767134811275 … 94.4146061509646 98.68007823884541; 94.8795615103044 92.40625814266534 … 94.14415081250243 98.40962290038324; … ; 95.10267440673721 92.62937103909815 … 94.36726370893524 98.63273579681605; 97.76499416734758 95.2916907997085 … 97.02958346954561 101.29505555742642], [94.94333567849267 100.17912776670433 … 95.3015478682614 100.88947817254314; 95.26415672315574 100.49994881136739 … 95.62236891292446 101.2102992172062; … ; 91.43230251314628 96.66809460135795 … 91.79051470291502 97.37844500719675; 96.0656637054263 101.30145579363796 … 96.42387589519504 102.01180619947677])"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Code from https://github.com/JuliaDiff/ReverseDiff.jl/blob/master/examples/gradient.jl\n",
    "################\n",
    "using ReverseDiff: GradientTape, GradientConfig, gradient, gradient!, compile, DiffResults\n",
    "\n",
    "#########\n",
    "# setup #\n",
    "#########\n",
    "\n",
    "# some objective function to work with\n",
    "f(a, b) = sum(a' * b + a * b')\n",
    "\n",
    "# pre-record a GradientTape for `f` using inputs of shape 100x100 with Float64 elements\n",
    "const f_tape = GradientTape(f, (rand(100, 100), rand(100, 100)))\n",
    "\n",
    "# compile `f_tape` into a more optimized representation\n",
    "const compiled_f_tape = compile(f_tape)\n",
    "\n",
    "# some inputs and work buffers to play around with\n",
    "a, b = rand(100, 100), rand(100, 100)\n",
    "inputs = (a, b)\n",
    "results = (similar(a), similar(b))\n",
    "all_results = map(DiffResults.GradientResult, results)\n",
    "cfg = GradientConfig(inputs)\n",
    "\n",
    "####################\n",
    "# taking gradients #\n",
    "####################\n",
    "\n",
    "# with pre-recorded/compiled tapes (generated in the setup above) #\n",
    "#-----------------------------------------------------------------#\n",
    "\n",
    "# this should be the fastest method, and non-allocating\n",
    "gradient!(results, compiled_f_tape, inputs)\n",
    "\n",
    "# the same as the above, but in addition to calculating the gradients, the value `f(a, b)`\n",
    "# is loaded into the the provided `DiffResult` instances (see DiffResults.jl documentation).\n",
    "gradient!(all_results, compiled_f_tape, inputs)\n",
    "\n",
    "# this should be the second fastest method, and also non-allocating\n",
    "gradient!(results, f_tape, inputs)\n",
    "\n",
    "# you can also make your own function if you want to abstract away the tape\n",
    "∇f!(results, inputs) = gradient!(results, compiled_f_tape, inputs)\n",
    "\n",
    "# with a pre-allocated GradientConfig #\n",
    "#-------------------------------------#\n",
    "# these methods are more flexible than a pre-recorded tape, but can be\n",
    "# wasteful since the tape will be re-recorded for every call.\n",
    "\n",
    "gradient!(results, f, inputs, cfg)\n",
    "\n",
    "gradient(f, inputs, cfg)\n",
    "\n",
    "# without a pre-allocated GradientConfig #\n",
    "#----------------------------------------#\n",
    "# convenient, but pretty wasteful since it has to allocate the GradientConfig itself\n",
    "\n",
    "gradient!(results, f, inputs)\n",
    "\n",
    "gradient(f, inputs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "-----------\n",
    "# <span style=\"color:#00B8DE\"> References</span>\n",
    "<a name=\"References\"></a> \n",
    "[Table of contents](#table-of-contents)\n",
    "\n",
    ">- [1] https://en.wikipedia.org/wiki/Dual_number\n",
    ">- [2] https://www.sanyamkapoor.com/machine-learning/autograd-magic/\n",
    ">- [3] https://github.com/JuliaDiff/DualNumbers.jl\n",
    ">- [4] http://www.juliadiff.org/ForwardDiff.jl/stable/index.html\n",
    ">- [5] https://en.wikipedia.org/wiki/Automatic_differentiation#Reverse_accumulation\n",
    ">- [6] http://www.columbia.edu/~ahd2125/post/2015/12/5/ \n",
    ">- [7] https://arxiv.org/pdf/1502.05767.pdf#page31\n",
    ">- [8] https://www.youtube.com/watch?v=xtfNug-htcs\n",
    ">- [9] https://github.com/JuliaDiff/ReverseDiff.jl\n",
    ">- [10] http://www.juliadiff.org/ReverseDiff.jl/api/\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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